0$ .","As $\\sigma _{*}^{k_{n,i}}\\left(c_{*,i}^{\\prime \\prime }\\right)=c_{*}$ , it follows that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{0}\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .","Similarly, as $\\sigma _{*}^{k_{n,0}}\\left(c_{*,i}^{\\prime }\\right)=g_{i}^{-1}\\left(c_{*}\\right)$ , we have that $d\\left(g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{0}\\right)\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .","Together, we are done.","Now we have to take care of the other half of $\\star $ .","This is done similarly, using right existence and the second part of Claim REF .","The following proposition explains why we needed to take a conjugate of $\\sigma $ .","The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.","In Section REF , we mentioned that it is a Ramsey structure.","Note that the underlying order is dense (by Proposition REF ).","Proposition 6.9 Let $M=\\left(V,<,R\\right)$ be the countable ordered random graph.","Then there is no automorphism $\\sigma \\in G=\\operatorname{Aut}\\left(M\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\in X$ .","First we find $a\\ne b$ in $M$ such that $\\sigma ^{n}\\left(a\\right)\\ne \\sigma ^{m}\\left(b\\right)$ for all $m,n\\in \\mathbb {Z}$ .","To do that, take any $a\\in M$ .","Then $\\left\\lbrace \\sigma ^{n}\\left(a\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is discrete (in the order sense: it is either a $\\mathbb {Z}$ -chain or just $a$ ).","Since $\\left(V,<\\right)$ is dense, there is some $b\\ne \\sigma ^{n}\\left(a\\right)$ for all $n\\in \\mathbb {Z}$ .","It follows that $b$ is as required.","Let $X=S_{x}\\left(M\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).","Let $p\\in X$ be any completion of the partial type $\\left\\lbrace R\\left(x,\\sigma ^{n}\\left(a\\right)\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace \\cup \\left\\lbrace \\lnot R\\left(x,\\sigma ^{m}\\left(b\\right)\\right)\\,|\\,m\\in \\mathbb {Z}\\right\\rbrace $ .","Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\in \\operatorname{cl}\\left\\lbrace \\sigma ^{n}\\left(p\\right)\\,|\\,n<\\omega \\right\\rbrace $ which is a fixed point of $G$ .","In other words, $p_{0}$ is an invariant type over $M$ .","However $R\\left(x,a\\right)\\wedge \\lnot R\\left(x,b\\right)\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).","The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.","If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\left(X,x_{0}\\right)$ .","Thus, there would be a conjugate $\\sigma _{*}$ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow.","But then if $\\left(Y,y_{0}\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\pi :X\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .","Thus, $\\pi $ maps $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ to $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(y\\right)\\,|\\,n<\\omega \\right\\rbrace $ , and the latter contains a $G$ -subflow.","Thus we get that $\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .","Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\left\\lbrace <,\\wedge \\right\\rbrace $ , and let $M\\models T$ be countable.","Then $\\operatorname{Aut}\\left(M\\right)$ has no shifty automorphism.","In particular, $M$ has no CIR.","Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .","Suppose that $\\sigma $ was shifty.","Let $\\bar{m}=\\left\\langle m\\,|\\,m\\in M\\right\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\bar{x}=\\left\\langle x_{m}\\,|\\,m\\in M\\right\\rangle $ .","Let $X=S_{\\bar{m}}\\left(M\\right)$ be the space of $\\bar{x}$ -complete types $p$ over $M$ such that $p\\upharpoonright \\emptyset =\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .","Then $X$ is a compact metric space.","Let $x_{*}=\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .","By Theorem REF , there is some conjugate $\\tau $ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\tau ^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{+}\\subseteq X$ and similarly, $\\operatorname{cl}\\left\\lbrace \\tau ^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{-}$ .","By Proposition REF , $\\tau $ fixes a branch or a point.","Suppose that $\\tau \\left(m\\right)=m$ for some $m\\in M$ .","Then for every $p\\in Y^{+}$ , $p\\models x_{m}=m$ .","However $G=\\operatorname{Aut}\\left(M\\right)$ acts transitively on $M$ , so we have a contradiction.","Now suppose that $\\tau $ fixes a branch $B\\subseteq M$ , but does not fix any point.","Suppose that $\\tau \\left(m\\right)>m$ for some $m\\in B$ .","Then $\\tau ^{n}\\left(m\\right)>m$ for all $n<\\omega $ , so for any $p\\in Y^{+}$ , $p\\models x_{m}>m$ .","There is some $m^{\\prime }\\in M$ such that $m^{\\prime }>m$ and $m^{\\prime }\\notin B$ .","Since $m<\\tau ^{n}\\left(m\\right)\\in B$ for all $n<\\omega $ , it follows that $p\\models x_{m}\\wedge m^{\\prime }=m$ for all $p\\in Y^{+}$ .","Let $\\tau ^{\\prime }\\in G$ fix $m$ and map $m^{\\prime }$ to $B$ .","Then $\\tau ^{\\prime }\\left(p\\right)\\models \\left(x_{m}\\wedge \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\right)=mm$ , so we can apply the same argument to $Y^{-}$ .","For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .","In addition, letting $H=\\operatorname{Aut}\\left(N\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).","In addition, if $B\\subseteq N$ is a branch, $m\\in B$ , there is always some $m^{\\prime }>m$ , $m^{\\prime }\\notin B$ and for any $n^{\\prime }>m$ in $B$ , $m^{\\prime }m\\equiv n^{\\prime }m$ .","Hence, we can apply Proposition REF and the same proof will work.","Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\omega $ -categorical structures.","We state here a few general conjectures and questions.","If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.","The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.","Conjecture 7.1 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion which admits a CIR.","Suppose that $M$ is a structure and $\\mathfrak {C}$ a monster model for $\\operatorname{Th}\\left(M\\right)$ .","The group of Lascar strong automorphisms of $M$, denoted by $\\operatorname{Aut}f\\left(M\\right)$ is the group of automorphisms of $M$ generated by the set $\\left\\lbrace \\sigma \\upharpoonright M\\,|\\,\\exists N\\prec \\mathfrak {C},\\left|N\\right|=\\left|T\\right|,\\sigma \\upharpoonright N=\\operatorname{id}\\right\\rbrace $ .","If $\\sigma $ is Lascar strong, then $\\sigma \\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{id}$ so $\\operatorname{Aut}f\\left(M\\right)$ is contained in $G^{0}$ .","However, there are examples (even $\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\operatorname{Aut}f\\left(M\\right)$ , see [16], [31].","The Lascar group of $M$ is the quotient $\\operatorname{Aut}\\left(M\\right)/\\operatorname{Aut}f\\left(M\\right)$ .","For more on the Lascar group, see [39].","In the $\\omega $ -categorical case, the quotient $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is also called the compact Lascar group.","If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$.","In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.","Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.","Forinstance, by Lemma \\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\operatorname{Aut}\\left(M\\right)$ has no compact quotients then it has finite topological rank.","However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.","Thus, potentially, the Lascar group — as a quotient of $\\operatorname{Aut}\\left(M\\right)$ — can be an obstruction to having finite topological rank.","During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.","Conjecture REF (together with Theorem REF ) implies that it could not.","Conjecture 7.2 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion with trivial Lascar group.","By the above, this second conjecture is implied by Conjecture REF .","Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.","It would be interesting to investigate other consequences of having a CIR.","For instance a CIR might have something to say about normal subgroups.","The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ for some CIR $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\aleph _{0}}$ is open.","Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.","We refer to [26] for a survey on this.","As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.","In fact the compact quotients are hidden in the stabilizer of a finite set.","It seems that one can avoid this counterexample by restricting to dense subgroups.","This leads us to the following questions.","Question 7.3 Let $M$ be $\\omega $ -categorical such that $G=\\operatorname{Aut}\\left(M\\right)$ has no compact quotient.","Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open (and hence is equal to $G$ )?","Note that the assumption of having no compact quotient is necessary.","Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\operatorname{Aut}\\left(M\\right)$ has a dense subgroup of index 2, see [23].","Question 7.4 Let $M$ be $\\omega $ -categorical and $N$ an $\\omega $ -categorical expansion of $M$ .","Set $G=\\operatorname{Aut}\\left(M\\right)$ and $H=\\operatorname{Aut}\\left(N\\right)\\le G$ .","Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.","Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open?","Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.","We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.","Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].","We would also like to thank Daoud Siniora for his comments.","Finally, we would like to thank the anonymous referee for his comments.","Definition 6.1 Suppose that $M$ is a countable structure.","Call an automorphism $\\sigma \\in G$ shifty if there is some invariant binary relation on finite sets in $M$ , $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ (the base will always be $$) such that:\\begin{itemize}\\item (Monotonicity) If A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{itemize}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ and $ A'A$, $ B'B$then $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B'$.\\item (Right existence) For every finite tuple $ a$ there is some $ a'a$such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$ (by this we mean that sets enumerated by $ a$,$ a'$ are independent).\\item (Right shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$, then there exists some $ n<$such that $ b'An(b)$.$ Lemma 6.2 If $\\sigma $ is shifty then it also satisfies: (Left existence) For every finite tuple $a$ there is some $a^{\\prime }\\equiv a$ such that $a^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.\\item (Left shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ b'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$, then there exists some $ n<$such that $ b'A-n(b)$.$ Suppose that $\\sigma $ is shifty, as witnessed by $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.","Given $ a$,there is some $ a'a$ such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.","Applying an automorphismtaking $ a'$ to $ a$ we get some $ a”a$ such that $ a”$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$,which shows left existence.$ As for left shiftiness, suppose that $A$ is finite and enumerated by $a$ , $b,b^{\\prime }$ are finite tuples such that $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ and $ bb'$.Then applying an automorphism, we get some $ a'$ such that $ ab'a'b$,so $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.","Hence for some $ n<$, $ a'bn(a)$.From $ a'bn(a)b$ we get that $ ab'a'b-n(a')-n(b)a-n(b)$,i.e., $ b'A-n(b)$.$ Proposition 6.3 The automorphism $\\sigma $ is a shifty automorphism on $M$ iff for any type $p\\in S\\left(\\emptyset \\right)$ (with finitely many variables), letting $Y_{a}=\\bigcap \\left\\lbrace \\bigcup \\left\\lbrace \\operatorname{tp}\\left(a,\\sigma ^{n}\\left(a^{\\prime }\\right)\\right)\\,|\\,n<\\omega \\right\\rbrace \\,|\\,a^{\\prime }\\equiv a\\right\\rbrace $ for any $a\\models p$ , the intersection $Y_{p}=\\bigcap \\left\\lbrace Y_{a}\\,|\\,a\\models p\\right\\rbrace $ is nonempty.","Suppose that $\\sigma $ is shifty, and fix some type $p\\in S\\left(\\emptyset \\right)$ .","Let $a\\models p$ .","By existence, there is some $a^{\\prime }\\equiv a$ with $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.","Let $ q=tp(a,a')$ and fix some $ bp$.Let $ Aut(M)$ map $ a$ to $ b$ and let $ b'=(a')$.We have that $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$ and hence by right shiftiness, $ q=tp(b,b')Yb$.Since $ b$ was arbitrary, $ qYp$.$ Suppose that the right hand side holds.","Given a finite tuple $a$ and $a^{\\prime }\\equiv a$ , write $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ iff $ tp(a,a')Yp$where $ p=tp(a/)$.","For general finite sets $ A,B$,write $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ iff there is some $ C$ containing $ A$ and $ C'$containing $ B$ such that $ C$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*C'$.","Obviously, $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is invariantand monotone.","Right existence follows from the assumption that $ Yp$for all $ pS()$.","Right shiftiness: supposethat $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ and $ a”a'$.","Then $ tp(a,a')Yp$and in particular it belongs to $ Ya$.","By definition of $ Ya$,$ tp(a,a'){ tp(a,n(a”)) | n<} $,so for some $ n<$, $ aa'ain(a”)$.$ Proposition 6.4 If $M$ is an ultrahomogeneous structure and $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR on finite subsets of $ M$ which respects substructures,then there exists a shifty automorphism $$ on $ M$, as witnessedby $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.$ Monotonicity and right existence are parts of the properties of a CIR, so we only have to prove right shiftiness.","Suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$and $ b'b$.","By the proof of Theorem \\ref {thm:existence of repulsive automorphism-ultrahomogeneous},the repulsive automorphism $$ constructed there satisfies thatfor some $ n<$, $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $n(b')$.","By stationarity,$ bA(b')$.$ Recall the definitions of flow and subflow from Section REF .","Theorem 6.5 Let $M$ be a countable homogeneous structure and $G=\\operatorname{Aut}\\left(M\\right)$ .","Suppose that $\\sigma \\in G$ is a shifty automorphism and that $\\left(X,d\\right)$ is a compact metric $G$ -flow.","Then for every $x_{*}\\in X$ there is some conjugate $\\sigma _{*}\\in G$ of $\\sigma $ such that: (*) Both $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ and $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contain a subflow of $X$ .","Remark 6.6 Note that Theorem REF implies that both $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ and $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ contain a subflow of $X$ : if e.g., $Y_{0}$ is a flow contained in the left space, then $GY_{0}=Y_{0}$ , so $\\sigma _{*}^{-k}\\left(Y_{0}\\right)\\subseteq Y_{0}\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ , hence $Y_{0}\\subseteq \\sigma _{*}^{k}\\left(\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace \\right)=\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace $ .","Before the proof we note the following useful lemma.","Lemma 6.7 Suppose that $G$ is a topological group acting continuously on a compact metric space $\\left(X,d\\right)$ .","Then for every $0<\\varepsilon $ there is some open neighborhood $U$ of $\\operatorname{id}\\in G$ such that for every $g,h\\in G$ if $gh^{-1}\\in U$ then for all $x\\in X$ we have that $d\\left(gx,hx\\right)<\\varepsilon $ .","It is enough to show that there is some open neighborhood $U$ of $\\operatorname{id}$ such that if $g\\in U$ then for all $x\\in X$ , $d\\left(gx,x\\right)<\\varepsilon $ (since then if $gh^{-1}\\in U$ then $d\\left(gh^{-1}\\left(hx\\right),hx\\right)<\\varepsilon $ ).","For every $x\\in X$ , there is some neighborhood $V_{x}$ of $x$ in $X$ and some neighborhood $U_{x}$ of $\\operatorname{id}$ in $G$ such that for all $g\\in U_{x}$ , $x^{\\prime }\\in V_{x}$ , $d\\left(gx^{\\prime },x^{\\prime }\\right)<\\varepsilon $ .","By compactness, a finite union of $V_{x}$ 's covers $X$ .","Let $U$ be the intersection of the corresponding $U_{x}$ 's.","Suppose that $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ witnesses that $$ is shifty.","Let $ G0$be a countable dense subset of $ G$, enumerated as $ gi | i<$,such that $ g0=id$.$ We construct an automorphism $\\tau :M\\rightarrow M$ by back and forth such that eventually $\\sigma _{*}=\\tau ^{-1}\\sigma \\tau $ and such that at each finite stage, $\\tau $ will be an elementary map.","For the construction it is actually better to think of the domain and range of $\\tau $ as two different structures, so we have $M=M_{*}$ and suppose that $\\sigma :M\\rightarrow M$ , $\\sigma _{*}:M_{*}\\rightarrow M_{*}$ and $\\tau :M_{*}\\rightarrow M$ .","The subscript $*$ will denote tuples from $M_{*}$ throughout.","Suppose that we have constructed a partial elementary map $f:A_{*}\\rightarrow A$ (that will be part of $\\tau $ eventually) with $A_{*}\\subseteq M_{*},A\\subseteq M$ finite, enumerated by $a_{*},a$ .","Here is the main tool in the construction.","Claim 6.8 Suppose that $b_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a*$ and$ b*'ba$.","Let $ b*=f-1(b)$.","Thenthere is $ k<$ and an extension $ f'$ of $ f$ such that anyautomorphism $ '$ extending $ f'$ will satisfy that for $ *'='-1'$,$ *'k(b'*)=b*$.$ Similarly, if $a_{*}\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b*'$ then there is some $ k<$ andan extension $ f'$ of $ f$ such that any automorphism $ '$ extending$ f'$ will satisfy that for $ *'='-1'$,$ *'k(b*)=b'*$.$ First, find some tuple $b^{\\prime }$ in $M$ such that $b^{\\prime }a\\equiv b_{*}^{\\prime }a_{*}$ .","In particular, $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.","By left shiftiness, there is some $ k<$such that $ -k(b)ab'ab*'a*$.Extend $ f$ to $ f'$ which sends $ b*'$ to $ -k(b)$.Then, for any $ '$ extending $ f'$, $ '-1k'(b*')='-1(b)=b*$.$ The second statement is proved similarly, using right shiftiness.","We will make sure that for each $n<\\omega $ , the following condition holds.","$\\star $ There are $k_{n,0},\\ldots ,k_{n,n-1}<\\omega $ such that for all $i0$ .","As $\\sigma _{*}^{k_{n,i}}\\left(c_{*,i}^{\\prime \\prime }\\right)=c_{*}$ , it follows that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{0}\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .","Similarly, as $\\sigma _{*}^{k_{n,0}}\\left(c_{*,i}^{\\prime }\\right)=g_{i}^{-1}\\left(c_{*}\\right)$ , we have that $d\\left(g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{0}\\right)\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .","Together, we are done.","Now we have to take care of the other half of $\\star $ .","This is done similarly, using right existence and the second part of Claim REF .","The following proposition explains why we needed to take a conjugate of $\\sigma $ .","The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.","In Section REF , we mentioned that it is a Ramsey structure.","Note that the underlying order is dense (by Proposition REF ).","Proposition 6.9 Let $M=\\left(V,<,R\\right)$ be the countable ordered random graph.","Then there is no automorphism $\\sigma \\in G=\\operatorname{Aut}\\left(M\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\in X$ .","First we find $a\\ne b$ in $M$ such that $\\sigma ^{n}\\left(a\\right)\\ne \\sigma ^{m}\\left(b\\right)$ for all $m,n\\in \\mathbb {Z}$ .","To do that, take any $a\\in M$ .","Then $\\left\\lbrace \\sigma ^{n}\\left(a\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is discrete (in the order sense: it is either a $\\mathbb {Z}$ -chain or just $a$ ).","Since $\\left(V,<\\right)$ is dense, there is some $b\\ne \\sigma ^{n}\\left(a\\right)$ for all $n\\in \\mathbb {Z}$ .","It follows that $b$ is as required.","Let $X=S_{x}\\left(M\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).","Let $p\\in X$ be any completion of the partial type $\\left\\lbrace R\\left(x,\\sigma ^{n}\\left(a\\right)\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace \\cup \\left\\lbrace \\lnot R\\left(x,\\sigma ^{m}\\left(b\\right)\\right)\\,|\\,m\\in \\mathbb {Z}\\right\\rbrace $ .","Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\in \\operatorname{cl}\\left\\lbrace \\sigma ^{n}\\left(p\\right)\\,|\\,n<\\omega \\right\\rbrace $ which is a fixed point of $G$ .","In other words, $p_{0}$ is an invariant type over $M$ .","However $R\\left(x,a\\right)\\wedge \\lnot R\\left(x,b\\right)\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).","The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.","If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\left(X,x_{0}\\right)$ .","Thus, there would be a conjugate $\\sigma _{*}$ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow.","But then if $\\left(Y,y_{0}\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\pi :X\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .","Thus, $\\pi $ maps $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ to $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(y\\right)\\,|\\,n<\\omega \\right\\rbrace $ , and the latter contains a $G$ -subflow.","Thus we get that $\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .","Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\left\\lbrace <,\\wedge \\right\\rbrace $ , and let $M\\models T$ be countable.","Then $\\operatorname{Aut}\\left(M\\right)$ has no shifty automorphism.","In particular, $M$ has no CIR.","Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .","Suppose that $\\sigma $ was shifty.","Let $\\bar{m}=\\left\\langle m\\,|\\,m\\in M\\right\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\bar{x}=\\left\\langle x_{m}\\,|\\,m\\in M\\right\\rangle $ .","Let $X=S_{\\bar{m}}\\left(M\\right)$ be the space of $\\bar{x}$ -complete types $p$ over $M$ such that $p\\upharpoonright \\emptyset =\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .","Then $X$ is a compact metric space.","Let $x_{*}=\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .","By Theorem REF , there is some conjugate $\\tau $ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\tau ^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{+}\\subseteq X$ and similarly, $\\operatorname{cl}\\left\\lbrace \\tau ^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{-}$ .","By Proposition REF , $\\tau $ fixes a branch or a point.","Suppose that $\\tau \\left(m\\right)=m$ for some $m\\in M$ .","Then for every $p\\in Y^{+}$ , $p\\models x_{m}=m$ .","However $G=\\operatorname{Aut}\\left(M\\right)$ acts transitively on $M$ , so we have a contradiction.","Now suppose that $\\tau $ fixes a branch $B\\subseteq M$ , but does not fix any point.","Suppose that $\\tau \\left(m\\right)>m$ for some $m\\in B$ .","Then $\\tau ^{n}\\left(m\\right)>m$ for all $n<\\omega $ , so for any $p\\in Y^{+}$ , $p\\models x_{m}>m$ .","There is some $m^{\\prime }\\in M$ such that $m^{\\prime }>m$ and $m^{\\prime }\\notin B$ .","Since $m<\\tau ^{n}\\left(m\\right)\\in B$ for all $n<\\omega $ , it follows that $p\\models x_{m}\\wedge m^{\\prime }=m$ for all $p\\in Y^{+}$ .","Let $\\tau ^{\\prime }\\in G$ fix $m$ and map $m^{\\prime }$ to $B$ .","Then $\\tau ^{\\prime }\\left(p\\right)\\models \\left(x_{m}\\wedge \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\right)=mm$ , so we can apply the same argument to $Y^{-}$ .","For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .","In addition, letting $H=\\operatorname{Aut}\\left(N\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).","In addition, if $B\\subseteq N$ is a branch, $m\\in B$ , there is always some $m^{\\prime }>m$ , $m^{\\prime }\\notin B$ and for any $n^{\\prime }>m$ in $B$ , $m^{\\prime }m\\equiv n^{\\prime }m$ .","Hence, we can apply Proposition REF and the same proof will work.","Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\omega $ -categorical structures.","We state here a few general conjectures and questions.","If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.","The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.","Conjecture 7.1 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion which admits a CIR.","Suppose that $M$ is a structure and $\\mathfrak {C}$ a monster model for $\\operatorname{Th}\\left(M\\right)$ .","The group of Lascar strong automorphisms of $M$, denoted by $\\operatorname{Aut}f\\left(M\\right)$ is the group of automorphisms of $M$ generated by the set $\\left\\lbrace \\sigma \\upharpoonright M\\,|\\,\\exists N\\prec \\mathfrak {C},\\left|N\\right|=\\left|T\\right|,\\sigma \\upharpoonright N=\\operatorname{id}\\right\\rbrace $ .","If $\\sigma $ is Lascar strong, then $\\sigma \\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{id}$ so $\\operatorname{Aut}f\\left(M\\right)$ is contained in $G^{0}$ .","However, there are examples (even $\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\operatorname{Aut}f\\left(M\\right)$ , see [16], [31].","The Lascar group of $M$ is the quotient $\\operatorname{Aut}\\left(M\\right)/\\operatorname{Aut}f\\left(M\\right)$ .","For more on the Lascar group, see [39].","In the $\\omega $ -categorical case, the quotient $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is also called the compact Lascar group.","If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$.","In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.","Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.","Forinstance, by Lemma \\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\operatorname{Aut}\\left(M\\right)$ has no compact quotients then it has finite topological rank.","However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.","Thus, potentially, the Lascar group — as a quotient of $\\operatorname{Aut}\\left(M\\right)$ — can be an obstruction to having finite topological rank.","During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.","Conjecture REF (together with Theorem REF ) implies that it could not.","Conjecture 7.2 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion with trivial Lascar group.","By the above, this second conjecture is implied by Conjecture REF .","Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.","It would be interesting to investigate other consequences of having a CIR.","For instance a CIR might have something to say about normal subgroups.","The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ for some CIR $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\aleph _{0}}$ is open.","Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.","We refer to [26] for a survey on this.","As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.","In fact the compact quotients are hidden in the stabilizer of a finite set.","It seems that one can avoid this counterexample by restricting to dense subgroups.","This leads us to the following questions.","Question 7.3 Let $M$ be $\\omega $ -categorical such that $G=\\operatorname{Aut}\\left(M\\right)$ has no compact quotient.","Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open (and hence is equal to $G$ )?","Note that the assumption of having no compact quotient is necessary.","Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\operatorname{Aut}\\left(M\\right)$ has a dense subgroup of index 2, see [23].","Question 7.4 Let $M$ be $\\omega $ -categorical and $N$ an $\\omega $ -categorical expansion of $M$ .","Set $G=\\operatorname{Aut}\\left(M\\right)$ and $H=\\operatorname{Aut}\\left(N\\right)\\le G$ .","Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.","Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open?","Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.","We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.","Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].","We would also like to thank Daoud Siniora for his comments.","Finally, we would like to thank the anonymous referee for his comments."],["Further questions","The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\omega $ -categorical structures.","We state here a few general conjectures and questions.","If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.","The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.","Conjecture 7.1 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion which admits a CIR.","Suppose that $M$ is a structure and $\\mathfrak {C}$ a monster model for $\\operatorname{Th}\\left(M\\right)$ .","The group of Lascar strong automorphisms of $M$, denoted by $\\operatorname{Aut}f\\left(M\\right)$ is the group of automorphisms of $M$ generated by the set $\\left\\lbrace \\sigma \\upharpoonright M\\,|\\,\\exists N\\prec \\mathfrak {C},\\left|N\\right|=\\left|T\\right|,\\sigma \\upharpoonright N=\\operatorname{id}\\right\\rbrace $ .","If $\\sigma $ is Lascar strong, then $\\sigma \\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{id}$ so $\\operatorname{Aut}f\\left(M\\right)$ is contained in $G^{0}$ .","However, there are examples (even $\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\operatorname{Aut}f\\left(M\\right)$ , see [16], [31].","The Lascar group of $M$ is the quotient $\\operatorname{Aut}\\left(M\\right)/\\operatorname{Aut}f\\left(M\\right)$ .","For more on the Lascar group, see [39].","In the $\\omega $ -categorical case, the quotient $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is also called the compact Lascar group.","If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$.","In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.","Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.","Forinstance, by Lemma \\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\operatorname{Aut}\\left(M\\right)$ has no compact quotients then it has finite topological rank.","However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.","Thus, potentially, the Lascar group — as a quotient of $\\operatorname{Aut}\\left(M\\right)$ — can be an obstruction to having finite topological rank.","During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.","Conjecture REF (together with Theorem REF ) implies that it could not.","Conjecture 7.2 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion with trivial Lascar group.","By the above, this second conjecture is implied by Conjecture REF .","Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.","It would be interesting to investigate other consequences of having a CIR.","For instance a CIR might have something to say about normal subgroups.","The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ for some CIR $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\aleph _{0}}$ is open.","Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.","We refer to [26] for a survey on this.","As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.","In fact the compact quotients are hidden in the stabilizer of a finite set.","It seems that one can avoid this counterexample by restricting to dense subgroups.","This leads us to the following questions.","Question 7.3 Let $M$ be $\\omega $ -categorical such that $G=\\operatorname{Aut}\\left(M\\right)$ has no compact quotient.","Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open (and hence is equal to $G$ )?","Note that the assumption of having no compact quotient is necessary.","Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\operatorname{Aut}\\left(M\\right)$ has a dense subgroup of index 2, see [23].","Question 7.4 Let $M$ be $\\omega $ -categorical and $N$ an $\\omega $ -categorical expansion of $M$ .","Set $G=\\operatorname{Aut}\\left(M\\right)$ and $H=\\operatorname{Aut}\\left(N\\right)\\le G$ .","Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.","Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open?"],["Acknowlegements","Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.","We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.","Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].","We would also like to thank Daoud Siniora for his comments.","Finally, we would like to thank the anonymous referee for his comments."]],"string":"[\n [\n \"Automorphism groups of finite topological rank\"\n ],\n [\n \"Abstract We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class.\",\n \"We then show how finite topological rank of the automorphism group of an $\\\\omega$-categorical structure can go down to reducts.\",\n \"Together, those results prove that a large number of $\\\\omega$-categorical structures that appear in the literature have an automorphism group of finite topological rank.\",\n \"In fact, we are not aware of any $\\\\omega$-categorical structure to which they do not apply (assuming the automorphism group has no compact quotients).\",\n \"We end with a few questions and conjectures.\"\n ],\n [\n \"Introduction\",\n \"Many automorphism groups of Fraïssé structures are known to admit a 2-generated dense subgroup.\",\n \"This is the case for example for dense linear orders and the random graph [25], [7].\",\n \"It is however not true that all automorphism groups of say $\\\\omega $ -categorical structures have even a finitely generated dense subgroup.\",\n \"For instance a construction of Cherlin and Hrushovski yields an $\\\\omega $ -categorical structure whose automorphism groups admits $(\\\\mathbb {Z}/2\\\\mathbb {Z})^{\\\\omega }$ as a quotient, which implies that it cannot have a finitely generated dense subgroup (see Remark REF ).\",\n \"However, it seems that the existence of such a large compact quotient is the only known obstruction.\",\n \"We speculate that this might indeed be the case and ask: Let $G$ is the automorphism group of an $\\\\omega $ -categorical structure; assume $G$ has no compact quotient, then does it have a finitely generated dense subgroup?\",\n \"This paper is our attempt at answering this question.\",\n \"We fall short of providing a definitive answer, but we succeed in finding sufficient conditions for such a $G$ to admit a finitely generated dense subgroup which seem to apply to all known examples.\",\n \"It is even plausible that those conditions are actually satisfied by all $\\\\omega $ -categorical structures with trivial $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ , see Conjecture REF .\",\n \"We now describe our main results.\",\n \"We first define a notion of a canonical independence relation, or CIR.\",\n \"It is a ternary independence relation $\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ which satisfies in particular stationarity over $$and transitivity on both sides.\",\n \"Importantly, we do not assume symmetry.We show that if an ultrahomogeneous structure $ M$ admits such a CIR,then it has a 2-generated dense subgroup, and even a cyclically denseconjugacy class (that is, for some $ f,gG$, the set $ { f-ngfn | nZ} $is dense).\",\n \"Many, but not all, classical $$-categorical structureshave a CIR.\",\n \"Examples that do not include the dense circular order(Corollary \\\\ref {cor:Circular order has no CIR but degree <=00003D3})and the dense infinitely-branching tree (Corollary \\\\ref {cor:Dense trees don^{\\\\prime }t have a groovy ind relation}).However, an expansion of any of those structures obtained by naminga point does have a CIR (Example \\\\ref {exa:trees with points}, Remark\\\\ref {rem:expanding a circular order by a point has a CIR}).\",\n \"Thisleads to our second main theorem which completely solves a relativeversion of our initial question: we show that if $ Aut(M)$ has nocompact quotients and $ N$ is an $$-categorical expansion of$ M$, then if $ Aut(N)$ has a finitely generated densesubgroup, then so does $ Aut(M)$.\",\n \"More precisely, we showthat adding two elements from $ Aut(M)$ to $ Aut(N)$yields a dense subgroup of $ Aut(M)$: see Theorem \\\\ref {thm:If no CQ then finitely generated}.$ In Section , we give a dynamical consequence of having a CIR.\",\n \"Our main motivation is to relate it to the Ramsey property.\",\n \"We observe in Proposition REF that Ramsey structures admit a weaker form of independence relation.\",\n \"We know by [20] that the Ramsey property is equivalent to extreme amenability of the automorphism group.\",\n \"It is then natural to look for a dynamical interpretation of having a CIR, one goal being to understand to what extent the Ramsey property is not sufficient to imply it.\",\n \"We give a necessary condition for having a CIR which sheds some light on this notion and can be used to prove negative results.\",\n \"The theorems presented in this paper lead to a number of open questions.\",\n \"In particular, it would be interesting to understand the obstructions to having a CIR and to prove more results about the automorphism groups of structures with a CIR.\",\n \"It is also our hope that some ideas introduced here could be used to develop a general theory of $\\\\omega $ -categorical structures.\",\n \"One strategy we have in mind is to show that $\\\\omega $ -categorical structures admit nice expansions and then to prove relative statements which pull down properties from an expansion to the structure itself.\",\n \"See Section for some precise conjectures.\",\n \"We end this introduction by mentioning some previous work done on this question: The existence of a cyclically dense conjugacy class was shown for the random graph by Macpherson [25], for the Urysohn space by Solecki [36], for dense linear orders by Darji and Mitchell [7] and recently for generic posets by Glab, Gordinowicz and Strobin [11].\",\n \"Kechris and Rosendal [21] study the property of having a dense conjugacy class for Polish groups.\",\n \"They show that a number of Polish groups admit cyclically dense conjugacy classes (see Theorem 2.10).\",\n \"Those include the group of homeomorphisms of the Cantor space and the automorphism group of a standard Borel space.\",\n \"Those groups do not fit in our context, although it should be possible to generalize our results so as to include them.\",\n \"In fact, the proof of Theorem 2.10 is very much in the same spirit as the proofs in this paper.\",\n \"Kwiatkowska and Malicki [19] give sufficient conditions for an automorphism group $G$ to have a cyclically dense conjugacy class, which gives new examples such as structures with the free amalgamation property and tournaments.\",\n \"Their conditions do not seem to formally imply ours, but all the examples that they give (and in particular structures with free amalgamation) are covered by our theorems.\",\n \"They also show that under the same hypothesis $L_{0}(G)$ has a cyclically dense conjugacy class.\",\n \"We did not study this.\"\n ],\n [\n \"$\\\\omega $ -categoricity,\\nUltrahomogeneous structures, Fraïssé limits and model companions\",\n \"Here we recall the basic facts we need for this paper.\",\n \"Let $L$ be some countable first order language (vocabulary).\",\n \"An $L$ -theory is called $\\\\omega $-categorical if it has a unique infinite countable model up to isomorphism.\",\n \"A countable model $M$ is $\\\\omega $ -categorical if its complete theory $Th\\\\left(M\\\\right)$ is.\",\n \"By a theorem of Engeler, Ryll-Nardzewski and Svenonius, see e.g., [15], this is equivalent to saying that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ is oligomorphic: for every $n<\\\\omega $ , there are only finitely many orbits of the action of $G$ on $M^{n}$ .\",\n \"It is also equivalent to the property that every set $X\\\\subseteq M^{n}$ which is invariant under $G$ is $\\\\emptyset $ -definable in $M$ .\",\n \"Also, if $M$ is $\\\\omega $ -categorical and $A$ is a finite subset of $M$ then $M_{A}$ is also $\\\\omega $ -categorical, where $M_{A}$ is the expansion of $M$ for the language $L_{A}$ which adds a constant for every element in $A$ .\",\n \"A countable $L$ -structure $M$ is called ultrahomogeneous if whenever $f:A\\\\rightarrow B$ is an isomorphism between two finitely generated substructures $A,B$ of $M$ , there is $\\\\sigma \\\\in \\\\operatorname{Aut}\\\\left(M\\\\right)$ extending $f$ .\",\n \"The age of an $L$ -structure $M$ , $\\\\operatorname{Age}\\\\left(M\\\\right)$ , is the class of all finitely generated substructures which can be embedded into $M$ .\",\n \"Recall that a class of finitely generated $L$ -structures $K$ closed under isomorphisms has the hereditary property (HP) if whenever $A\\\\in K$ and $B\\\\subseteq A$ ($B$ is a substructure of $A$ ), $B\\\\in K$ .\",\n \"The class $K$ has the joint embedding property (JEP) if whenever $A,B\\\\in K$ there is some $C$ such that both $A,B$ embed into $C$ .\",\n \"It has the amalgamation property (AP) if whenever $A,B,C\\\\in K$ and $f_{B}:A\\\\rightarrow B$ , $f_{C}:A\\\\rightarrow C$ are embeddings, then there is some $D\\\\in K$ and embeddings $g_{B}:B\\\\rightarrow D$ , $g_{C}:C\\\\rightarrow D$ such that $g_{B}\\\\circ f_{B}=g_{C}\\\\circ f_{C}$ .\",\n \"We say that $K$ is uniformly locally finite if for some function $f:\\\\omega \\\\rightarrow \\\\omega $ , for every $A\\\\in K$ and $X$ a subset of $A$ of size $n$ , the structure generated by $X$ has size $\\\\le f\\\\left(\\\\left|X\\\\right|\\\\right)$ .\",\n \"In the following, “essentially countable” means that $K$ contains at most countably many isomorphism types of structures.\",\n \"We also recall the notions of model companions and model completions.\",\n \"A theory $T$ is called model complete if whenever $M\\\\subseteq N$ are models of $T$ , $M\\\\prec N$ .\",\n \"Suppose that $T_{\\\\forall }$ is a universal theory.\",\n \"A theory $T^{\\\\prime }$ is the model companion of $T_{\\\\forall }$ if $T^{\\\\prime }$ is model complete and $T^{\\\\prime }_{\\\\forall }=T_{\\\\forall }$ (they have the same universal consequences).\",\n \"In other words, every model of $T_{\\\\forall }$ can be embedded in a model of $T^{\\\\prime }$ .\",\n \"The theory $T^{\\\\prime }$ is a model completion of $T_{\\\\forall }$ if in addition it has elimination of quantifiers.\",\n \"Models companions are unique, if they exist.\",\n \"For more, see [37] and [15].\",\n \"Fact 2.1 Suppose that that $K$ is an essentially countable class of finite $L$ -structures which has HP, JEP and AP.\",\n \"[15] The class $K$ has a Fraïssé limit: a unique countable ultrahomogeneous model $M$ with the same age.\",\n \"[15] If $K$ is uniformly locally finite in a finite language (or without assuming that $L$ is finite but instead that for each $n<\\\\omega $ there are finitely many isomorphism types of structures generated by $n$ elements), then $M$ is $\\\\omega $ -categorical and has quantifier elimination.\",\n \"(See remark below.)\",\n \"If $T$ is a countable universal $L$ -theory, $L$ is finite, $K$ is the class of finitely generated models of $T$ and $K$ is uniformly locally finite (or without assuming that $L$ is finite but instead that $K$ has at most finitely many isomorphism types of structures generated by $n$ elements for all $n<\\\\omega $ ), then the theory $Th\\\\left(M\\\\right)$ is $\\\\omega $ -categorical and is the model completion of $T$ .\",\n \"(See remark below.)\",\n \"[15]The converse to (1) also holds: if $M$ is an ultrahomogeneous then the age of $M$ satisfies HP, JEP and AP.\",\n \"Remark 2.2 The assumptions in parenthesis in (2) is not stated in [15], but a straightforward modification of that proof gives this.\",\n \"We could not find an explicit reference for (3) (which is well-known), so here is a short argument.\",\n \"Since $Th\\\\left(M\\\\right)$ eliminates quantifiers by (2), it is enough to show that $Th\\\\left(M\\\\right)_{\\\\forall }=T_{\\\\forall }=T$ (since $T$ is universal).\",\n \"If $\\\\psi $ is universal and $T\\\\models \\\\psi $ , then since $\\\\operatorname{Age}\\\\left(M\\\\right)\\\\subseteq K$ , $M\\\\models \\\\psi $ .\",\n \"On the other hand, if $M\\\\models \\\\psi $ where $\\\\psi $ is universal, and $T\\\\lnot \\\\models \\\\psi $ , then there is a model $A^{\\\\prime }\\\\models T$ such that $A^{\\\\prime }\\\\models \\\\lnot \\\\psi $ , and since $\\\\lnot \\\\psi $ is existential, the same is true for some finitely generated model $A\\\\subseteq A^{\\\\prime }$ , so $A\\\\in K$ and since $A\\\\in \\\\operatorname{Age}\\\\left(M\\\\right)$ , we get a contradiction.\",\n \"Classes $K$ as in Fact REF are called Fraïssé classes or amalgamation classes.\",\n \"Some examples of Fraïssé limits include DLO (dense linear order), i.e., $Th\\\\left(\\\\mathbb {Q},<\\\\right)$ (here we identify the Fraïssé limit and its theory), the random graph, the random poset (partially ordered set), the random tournament, and more.\",\n \"One example that we will be interested in is that of dense trees.\",\n \"Example 2.3 Let $L_{dt}=\\\\left\\\\lbrace <,\\\\wedge \\\\right\\\\rbrace $ , and let $T_{dt,\\\\forall }$ be the universal theory of trees with a meet function $\\\\wedge $ .\",\n \"Then $T_{dt,\\\\forall }$ has an $\\\\omega $ -categorical model completion by Fact REF (note that the tree generated by a finite set $B$ is just $B\\\\cup \\\\left\\\\lbrace x\\\\wedge y\\\\,|\\\\,x,y\\\\in B\\\\right\\\\rbrace $ ).\",\n \"We denote the model companion by $T_{dt}$ and call the unique countable model the dense tree.\",\n \"See also [35].\",\n \"Recall that a structure $M$ is homogeneous if whenever $a,b$ are finite tuples of the same length, and $a\\\\equiv b$ (which means $\\\\operatorname{tp}\\\\left(a/\\\\emptyset \\\\right)=\\\\operatorname{tp}\\\\left(b/\\\\emptyset \\\\right)$ , i.e., the tuples $a,b$ have the same type), then there is an automorphism taking $a$ to $b$ .\",\n \"Note that ultrahomogeneous structures are homogeneous and the same is true for $\\\\omega $ -categorical ones.\",\n \"When $M$ is homogeneous, an elementary map $f:A\\\\rightarrow B$ for $A,B\\\\subseteq M$ is just a restriction of an automorphism of $M$ .\",\n \"Finally, we use $\\\\mathfrak {C}$ to represent a monster model of the appropriate theory.\",\n \"This is a big saturated (so also homogeneous) model that contains all the models and sets we will need.\",\n \"This is standard in model theory.\",\n \"For more, see [37].\"\n ],\n [\n \"A mix of two Fraïsé limits\",\n \"Suppose that $K_{1},K_{2}$ are two amalgamation classes of finite structures in the languages $L_{1},L_{2}$ respectively.\",\n \"Assume the following properties: The symmetric difference $L_{1}\\\\mathrel {\\\\triangle }L_{2}$ is relational.\",\n \"The class $K$ of finite $L_{1}\\\\cup L_{2}$ -structures $A$ such that $A\\\\upharpoonright L_{1}\\\\in K_{1}$ and $A\\\\upharpoonright L_{2}\\\\in K_{2}$ is an amalgamation class.\",\n \"Let $M$ be its Fraïsé limit.\",\n \"For every $A\\\\in K_{1}$ , there is some expansion $A^{\\\\prime }$ of $A$ to an $L_{1}\\\\cup L_{2}$ -structure such that $A^{\\\\prime }\\\\upharpoonright L_{2}\\\\in K_{2}$ , and similarly that for every $B\\\\in K_{2}$ there is some expansion $B^{\\\\prime }$ to $L_{1}\\\\cup L_{2}$ whose restriction to $L_{1}$ is in $K_{1}$ .\",\n \"If $A\\\\in K$ and $A\\\\upharpoonright L_{1}\\\\subseteq B\\\\in K_{1}$ then there is an expansion $B^{\\\\prime }$ of $B$ to $L_{1}\\\\cup L_{2}$ such that $A\\\\subseteq B^{\\\\prime }$ and $B^{\\\\prime }\\\\upharpoonright L_{2}\\\\in K_{2}$ , and similarly for $L_{2}$ .\",\n \"Under all these conditions we have the following.\",\n \"Proposition 2.4 The structure $M\\\\upharpoonright L_{1}=M_{1}$ is the Fraïsé limit of $K_{1}$ and $M\\\\upharpoonright L_{2}=M_{2}$ is the Fraïsé limit of $K_{2}$ .\",\n \"Start with $M_{1}$ (for $M_{2}$ , the proof is the same).\",\n \"It is enough to show that $M_{1}$ is ultrahomogeneous and that $\\\\operatorname{Age}\\\\left(M_{1}\\\\right)=K_{1}$ .\",\n \"The second statement follows from (2), (3) above.\",\n \"For the first, by [15] it is enough to show that if $A\\\\subseteq B$ are from $K_{1}$ and $f:A\\\\rightarrow M_{1}$ then there is some $g:B\\\\rightarrow M_{1}$ extending $f$ .\",\n \"Using $f$ we can expand $A$ to an $L_{1}\\\\cup L_{2}$ -structure $A^{\\\\prime }$ in such a way that $f$ is an embedding to $M$ (this uses the fact that $L_{2}\\\\backslash L_{1}$ is relational).\",\n \"By (4) we can expand $B$ to an $L_{1}\\\\cup L_{2}$ -structure $B^{\\\\prime }$ such that $A^{\\\\prime }\\\\subseteq B^{\\\\prime }$ and $B^{\\\\prime }\\\\in K$ .\",\n \"Since $M$ is the Fraïssé limit of $K$ , it follows by [15] again that $f$ can be extended to $g:B^{\\\\prime }\\\\rightarrow M$ , and in particular, $g\\\\upharpoonright L_{1}$ is the embedding we seek.\"\n ],\n [\n \"The maximal compact quotient of\\nthe automorphism group of a countable $\\\\omega $ -categorical structure\",\n \"Assume in this section that $M$ is $\\\\omega $ -categorical and countable, and let $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ , considered as a topological group in the product topology.\",\n \"The contents of this section are folklore but we give the details for the sake of readability.\",\n \"Recall that for a structure $M$ and $A\\\\subseteq M$ , $\\\\operatorname{acl}\\\\left(A\\\\right)$ is the set of all algebraic elements over $A$ (elements satisfying an algebraic formula over $A$ : one with finitely many solutions).\",\n \"Similarly, $\\\\operatorname{dcl}\\\\left(A\\\\right)$ is the set of all elements definable over $A$ .\",\n \"In the context of $\\\\omega $ -categorical structures, $\\\\operatorname{acl}\\\\left(A\\\\right)$ and $\\\\operatorname{dcl}\\\\left(A\\\\right)$ are defined in terms of the size of the orbit of the action of $G$ fixing $A$ being finite or a singleton respectively.\",\n \"In the next proposition, we describe the maximal compact quotient of $\\\\operatorname{Aut}\\\\left(M\\\\right)$ in model theoretic terms.\",\n \"This uses the notion of $M^{\\\\operatorname{eq}}$ : the expansion of $M$ obtained by adding a new sort for every $\\\\emptyset $ -definable quotient of some $\\\\emptyset $ -definable set.\",\n \"See [37] for more.\",\n \"For $A\\\\subseteq M$ , $\\\\operatorname{Aut}\\\\left(M/A\\\\right)$ is the group of automorphisms of $M$ fixing $A$ .\",\n \"This is a closed subgroup of $G$ which is normal when $A$ is invariant under $\\\\operatorname{Aut}\\\\left(M\\\\right)$ , thus the quotient $G/\\\\operatorname{Aut}\\\\left(M/A\\\\right)$ is a Hausdorff topological group (with the quotient topology).\",\n \"We identify $\\\\operatorname{Aut}\\\\left(M\\\\right)$ and $\\\\operatorname{Aut}\\\\left(M^{\\\\operatorname{eq}}\\\\right)$ so that we can put $A=\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ .\",\n \"In this case, it is also compact as the next proposition says.\",\n \"Proposition 2.5 The group $G/\\\\operatorname{Aut}\\\\left(M/\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ is a compact Hausdorff (in fact — profinite) group.\",\n \"Let $H$ be the group of all elementary maps from $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ to $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ , also denoted by $\\\\operatorname{Aut}\\\\left(\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ .\",\n \"The group $H$ is naturally profinite as an inverse system of the family $\\\\left\\\\lbrace H_{X}\\\\,|\\\\,X\\\\subseteq M^{\\\\operatorname{eq}},\\\\text{ a finite }\\\\emptyset \\\\text{-definable set}\\\\right\\\\rbrace ,$ where $H_{X}$ is the group of elementary permutations of $X$ .\",\n \"Let $\\\\operatorname{res}:G\\\\rightarrow H$ be the restriction map $\\\\operatorname{res}\\\\left(\\\\sigma \\\\right)=\\\\sigma \\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ .\",\n \"We will show that $\\\\operatorname{res}$ is onto.\",\n \"Using back-and-forth, it is enough to show that given any complete type $p\\\\left(x\\\\right)$ for $x$ in the home sort over $A\\\\cup \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ where $A\\\\subseteq M$ is finite, $p$ can be realized in $M$ .\",\n \"We work in the monster model $\\\\mathfrak {C}$ .\",\n \"Let $E=\\\\mathord {\\\\equiv _{A\\\\cup \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)}}$ be the equivalence relation of having the same type over $A\\\\cup \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ .\",\n \"Then $E$ is $A$ -invariant and has boundedly many classes in $\\\\mathfrak {C}$ .\",\n \"By $\\\\omega $ -categoricity, as $A$ is finite $E$ is definable over $A$ , and by compactness, $E$ has finitely many classes.\",\n \"But then for every $E$ -class there must be a representative in $M$ .\",\n \"Since a realization of $p$ must have an $E$ -equivalent element in $M$ , $p$ is realized in $M$ .\",\n \"Note that by compactness we get that every such $p$ is isolated by its restriction to $A\\\\cup X$ where $X$ is some finite $\\\\emptyset $ -definable set in $M^{\\\\operatorname{eq}}$ .\",\n \"The kernel of $\\\\operatorname{res}$ is precisely $G^{0}=\\\\operatorname{Aut}\\\\left(M/\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ , so $\\\\operatorname{res}$ induces an isomorphism of groups $G/G^{0}\\\\rightarrow H$ .\",\n \"The group $G/G^{0}$ is also a topological group when equipped with the quotient topology.\",\n \"This map is easily seen to be continuous.\",\n \"To see that it is open, it is enough to show that the image of an open neighborhood $V$ of $\\\\operatorname{id}\\\\cdot G^{0}$ in $G/G^{0}$ contains an open neighborhood of $\\\\operatorname{id}$ in $H$ .\",\n \"The preimage of $V$ in $G$ is some open set $U\\\\subseteq G$ containing $\\\\operatorname{id}$ .\",\n \"Suppose $\\\\operatorname{id}\\\\in U_{b}=\\\\left\\\\lbrace \\\\sigma \\\\in G\\\\,|\\\\,\\\\sigma \\\\left(b\\\\right)=b\\\\right\\\\rbrace \\\\subseteq U$ is some basic open set.\",\n \"As we noted above, there is some finite $\\\\emptyset $ -definable set $X\\\\subseteq M^{\\\\operatorname{eq}}$ such that $\\\\operatorname{tp}\\\\left(b/\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ is isolated by $\\\\operatorname{tp}\\\\left(b/X\\\\right)$ .\",\n \"Then if $\\\\tau \\\\in G$ is such that $\\\\tau \\\\upharpoonright X=\\\\operatorname{id}_{X}$ , then there is some $\\\\sigma \\\\in \\\\operatorname{Aut}\\\\left(M/\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ such that $\\\\sigma \\\\tau \\\\left(b\\\\right)=b$ , so $\\\\sigma \\\\tau \\\\in U_{b}$ , but then $\\\\tau \\\\in U$ (because $U$ is a union of cosets of $\\\\operatorname{Aut}\\\\left(M/\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ ).\",\n \"Hence, the image of $V$ contains the open set $\\\\left\\\\lbrace \\\\tau \\\\in H\\\\,|\\\\,\\\\tau \\\\upharpoonright X=\\\\operatorname{id}_{X}\\\\right\\\\rbrace $ .\",\n \"Together these two groups are isomorphic as topological groups, so are profinite.\",\n \"Definition 2.6 We let $G^{0}=\\\\operatorname{Aut}\\\\left(M/\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ .\",\n \"Proposition 2.7 If $H\\\\trianglelefteq G$ is normal and closed and $G/H$ is compact then $G^{0}\\\\le H$ .\",\n \"Let $E_{n}$ be the equivalence relation on $M^{n}$ of having the same $H$ -orbit.\",\n \"Then $E_{n}$ refines $\\\\equiv $ (having the same type over $\\\\emptyset $ ) and is definable in $M$ .\",\n \"Indeed, suppose that $\\\\sigma \\\\in G$ .\",\n \"Then for every $a\\\\in M^{n}$ , $\\\\sigma O_{H}\\\\left(a\\\\right)=O_{H}\\\\left(\\\\sigma \\\\left(a\\\\right)\\\\right)$ (because $H$ is normal), where $O_{H}\\\\left(a\\\\right)$ denotes the orbit of $a$ under $H$ .\",\n \"Hence $E_{n}$ is invariant under $G$ so $\\\\emptyset $ -definable.\",\n \"In addition, $G$ acts transitively on the $H$ -orbits within each $\\\\equiv $ -class by $\\\\sigma \\\\cdot O_{H}\\\\left(a\\\\right)=O_{H}\\\\left(\\\\sigma \\\\left(a\\\\right)\\\\right)$ .\",\n \"The stabilizer of $O_{H}\\\\left(a\\\\right)$ is $\\\\left\\\\lbrace \\\\sigma \\\\in G\\\\,|\\\\,\\\\sigma \\\\left(O_{H}\\\\left(a\\\\right)\\\\right)=O_{H}\\\\left(a\\\\right)\\\\right\\\\rbrace $ so open (if $\\\\sigma $ is there, then to make sure that $\\\\sigma ^{\\\\prime }$ is there, it is enough that $\\\\sigma ^{\\\\prime }\\\\left(a\\\\right)=\\\\sigma \\\\left(a\\\\right)$ ).\",\n \"Note that this action factors through $H$ (i.e., the action $\\\\sigma H\\\\cdot O_{H}\\\\left(a\\\\right)=O_{H}\\\\left(\\\\sigma \\\\left(a\\\\right)\\\\right)$ is well-defined).\",\n \"Hence, the stabilizer is open in $G/H$ and hence has finite index in $G/H$ , so the number of orbits of $H$ under this action in every $\\\\equiv $ -class is finite.\",\n \"By $\\\\omega $ -categoricity, the number of $\\\\equiv $ -classes (of $n$ -tuples) is finite, so the number of orbits of $H$ is finite.\",\n \"In summary, $E_{n}$ is definable and has finitely many classes.\",\n \"Hence these classes belong to $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ .\",\n \"Given $\\\\sigma \\\\in G^{0}$ , $\\\\sigma $ fixes the orbits of $H$ under its action on $M^{n}$ .\",\n \"As $H$ is closed, this means that $\\\\sigma \\\\in H$ .\",\n \"Corollary 2.8 The group $G/G^{0}$ is the maximal compact Hausdorff quotient of $G$ .\",\n \"In light of Corollary REF , $G$ has no compact quotients (by which we mean that there is no nontrivial compact Hausdorff group which is an image of $G$ under a continuous group homomorphism) iff $G^{0}=G$ .\",\n \"If $N$ is a normal closed subgroup of $G$ , then we would like to say that $\\\\left(G,N\\\\right)$ has no compact quotients iff $G/N$ has no compact quotients.\",\n \"Let us generalize this to any closed subgroup.\",\n \"Definition 2.9 Suppose that $H\\\\le G$ is closed.\",\n \"We will say that the pair $\\\\left(G,H\\\\right)$ has no compact quotients if for all $g\\\\in G$ , there is some $h\\\\in H$ such that $g\\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=h\\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ .\",\n \"Proposition 2.10 Suppose that $H\\\\le G$ is closed.\",\n \"Then $\\\\left(G,H\\\\right)$ has no compact quotients iff for every closed and normal $N\\\\trianglelefteq G$ such that $G/N$ is compact, $NH=G$ .\",\n \"First note that $\\\\left(G,H\\\\right)$ has no compact quotient iff $\\\\left\\\\lbrace gH\\\\,|\\\\,g\\\\in G\\\\right\\\\rbrace =\\\\left\\\\lbrace gH\\\\,|\\\\,g\\\\in G^{0}\\\\right\\\\rbrace $ .\",\n \"This happens iff $G=G^{0}H$ .\",\n \"Hence the direction from right to left follows by taking $N=G^{0}$ .\",\n \"The direction from left to right is immediate by Proposition REF .\",\n \"Corollary 2.11 If $H\\\\le G$ is closed and normal then $\\\\left(G,H\\\\right)$ has no compact quotients iff $G/H$ has no compact quotients as a topological group (i.e., there is no nontrivial compact Hausdorff quotient).\",\n \"Left to right: suppose that $G/H$ has a compact quotient.\",\n \"Then there is a normal closed subgroup $N\\\\trianglelefteq G$ such that $H\\\\le N$ and $G/N$ is compact.\",\n \"By Proposition REF , $N$ contains $G^{0}$ .\",\n \"Thus, $G^{0}H\\\\le N$ and hence $G=N$ (by Proposition REF ).\",\n \"Right to left: for a finite $\\\\emptyset $ -definable subset $X\\\\subseteq \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ , let $G_{X}^{0}=\\\\operatorname{Aut}\\\\left(M/X\\\\right)\\\\le G$ (where we identify $G$ with $\\\\operatorname{Aut}\\\\left(M^{\\\\operatorname{eq}}\\\\right)$ ).\",\n \"Note that as $X$ is definable, $G_{X}^{0}$ is normal, hence so is the product with $H$ .\",\n \"Since $\\\\left[G:G_{X}^{0}\\\\right]$ is finite, $HG_{X}^{0}$ is closed as a finite union of translates of $G_{X}^{0}$ so that $G/HG_{X}^{0}$ is a compact (even finite) Hausdorff quotient of $G/H$ , so it must be trivial and hence that $HG_{X}^{0}=G$ .\",\n \"Take $g\\\\in G$ , we need to find $h\\\\in H$ such that $g\\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=h\\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ .\",\n \"By what we just said, we have that (*) for every finite definable $X\\\\subseteq \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ there is some $h_{X}\\\\in H$ such that $g\\\\upharpoonright X=h_{X}\\\\upharpoonright X$ .\",\n \"But since every finite $X\\\\subseteq \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ is contained in a finite $\\\\emptyset $ -definable set $X^{\\\\prime }\\\\subseteq \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ ($X^{\\\\prime }$ is just the union of all conjugates of $X$ ), (*) is true for all finite subset $X\\\\subseteq \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ .\",\n \"For every finite tuple $a$ from $M$ , $O_{H}\\\\left(a\\\\right)$ (the orbit of $a$ under $H$ ) is $a$ -definable (because $H$ is normal, $g\\\\cdot O_{H}\\\\left(a\\\\right)=O_{H}\\\\left(g\\\\left(a\\\\right)\\\\right)$ for any $g\\\\in G$ , so that $O_{H}\\\\left(a\\\\right)$ is $a$ -invariant thus $a$ -definable by $\\\\omega $ -categoricity).\",\n \"In $M^{\\\\operatorname{eq}}$ , every definable set $X$ has a code $X\\\\in M^{\\\\operatorname{eq}}$ (such that the automorphisms fixing $X$ setwise in $M$ are precisely the automorphisms fixing $X$ ).\",\n \"(This notation is a bit misleading since there could be many possible codes for $X$ .)\",\n \"Let $D\\\\subseteq M^{\\\\operatorname{eq}}$ be the collection of all possible codes $O_{H}\\\\left(a\\\\right)$ for all finite tuples $a$ from $M$ .\",\n \"Then $D$ is invariant under $G$ since $g\\\\left(O_{H}\\\\left(a\\\\right)\\\\right)=O_{H}\\\\left(g\\\\left(a\\\\right)\\\\right)$ for all $g\\\\in G$ (i.e., $g\\\\left(O_{H}\\\\left(a\\\\right)\\\\right)$ is a code for $O_{H}\\\\left(g\\\\left(a\\\\right)\\\\right)$ ).\",\n \"Let $\\\\bar{c}$ be a tuple enumerating $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ .\",\n \"Since every $h\\\\in H$ fixes $D$ pointwise, (*) gives us that $\\\\bar{c}\\\\equiv _{D}g\\\\left(\\\\bar{c}\\\\right)$ (because to check this equation it is enough to consider finite subtuples).\",\n \"Thus, the map $f$ taking $\\\\bar{c}$ to $g\\\\left(\\\\bar{c}\\\\right)$ fixing $D$ is an elementary map.\",\n \"By a back-and-forth argument almost identical to the one given in the proof of Proposition REF , there is some automorphism $h\\\\in G$ extending $f$ (the point is that the relation $\\\\equiv _{\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\cup D\\\\cup A}$ is bounded and $A$ -invariant for any finite set $A$ , hence definable and hence has finitely many classes, all of them realized in $M$ ).\",\n \"Since $H$ is closed, and $h$ fixes all $H$ -orbits setwise (as it fixes $D$ ), $h\\\\in H$ .\",\n \"Finally, $h\\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=g\\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ as requested.\",\n \"Example 2.12 If $M^{\\\\prime }$ is an expansion of $M$ and $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=\\\\operatorname{dcl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ in $M$ then $\\\\left(G,\\\\operatorname{Aut}\\\\left(M^{\\\\prime }\\\\right)\\\\right)$ has no compact quotients.\",\n \"This is because in that case, $G^{0}=G$ .\"\n ],\n [\n \"Expansions and reducts of $\\\\omega $ -categorical\\nstructures\",\n \"A group $H$ acts oligomorphically on a set $X$ if for all $n<\\\\omega $ , the number of orbits of $X^{n}$ under the action of $H$ is finite for every $n<\\\\omega $ .\",\n \"If $M$ is countable and $\\\\omega $ -categorical, and $H\\\\le G$ is closed, then $H=\\\\operatorname{Aut}\\\\left(M^{\\\\prime }\\\\right)$ for some expansion of $M$ .\",\n \"In addition, if $H$ acts oligomorphically on $M$ , then $M^{\\\\prime }$ is $\\\\omega $ -categorical by Ryll-Nardzewski.\",\n \"On the other hand, if $G\\\\le H$ where $H$ is a closed subgroup of the group of permutations of $M$ , then $H=\\\\operatorname{Aut}\\\\left(M^{\\\\prime }\\\\right)$ for some ($\\\\omega $ -categorical) reduct $M^{\\\\prime }$ of $M$ .\",\n \"Two such reducts $M^{\\\\prime }$ , $M^{\\\\prime \\\\prime }$ are the same up to bi-definability if they have the same definable sets, which is equivalent to $\\\\operatorname{Aut}\\\\left(M^{\\\\prime }\\\\right)=\\\\operatorname{Aut}\\\\left(M^{\\\\prime \\\\prime }\\\\right)$ .\",\n \"Proposition 2.13 Let $M$ be $\\\\omega $ -categorical and $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ .\",\n \"Then $G^{0}\\\\le G$ acts oligomorphically on $M$ and $G^{0}=\\\\operatorname{Aut}\\\\left(M^{\\\\prime }\\\\right)$ for an $\\\\omega $ -categorical expansion $M^{\\\\prime }$ of $M$ with no compact quotients.\",\n \"Let $M^{\\\\prime }_{0}$ be the expansion of $M^{\\\\operatorname{eq}}$ obtained by naming (i.e., adding constants for) every element in $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ .\",\n \"Then $M_{0}^{\\\\prime }$ is still $\\\\omega $ -categorical (as a many sorted structure) since for any given sort (or finite collection of sorts) $S$ , the equivalence relation $\\\\equiv _{\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)}$ on $S$ -tuples, of having the same type over $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ , is bounded, definable and hence finite, as in the arguments above.\",\n \"Let $M^{\\\\prime }$ be the reduct to the home sort (so it is also $\\\\omega $ -categorical).\",\n \"By definition, $G^{0}=\\\\operatorname{Aut}\\\\left(M^{\\\\prime }\\\\right)$ .\",\n \"Moreover, letting $H=\\\\operatorname{Aut}\\\\left(M^{\\\\prime }\\\\right)$ , we have that $H^{0}=H$ .\",\n \"This is because $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ .\"\n ],\n [\n \"Ramsey Classes\",\n \"Let us start with the definition.\",\n \"Definition 2.14 For two $L$ -structures $A,B$ , we let ${B \\\\atopwithdelims ()A}$ be the set substructures of $B$ isomorphic to $A$ .\",\n \"Suppose that $K$ is a class of finite $L$ -structures.\",\n \"We say that $K$ is a Ramsey class if for every $A,B\\\\in K$ and $k<\\\\omega $ there is some $C\\\\in K$ such that $C\\\\rightarrow \\\\left(B\\\\right)_{k}^{A}$ : for every function $f:{C \\\\atopwithdelims ()A}\\\\rightarrow k$ there is some $B^{\\\\prime }\\\\in {C \\\\atopwithdelims ()B}$ such that $f$ is constant on ${B^{\\\\prime } \\\\atopwithdelims ()A}$ .\",\n \"Say that an ultrahomogeneous $L$ -structure $M$ with a quantifier-free definable linear order is a Ramsey structure if $\\\\operatorname{Age}\\\\left(M\\\\right)$ is a Ramsey class.\",\n \"Ramsey classes are extremely important classes of finite structure.\",\n \"There are many examples of Ramsey classes, in particular the class of finite linear orders (this is just Ramsey's theorem) and furthermore, by a theorem of Nešetřil and Rödl [30], proved independently by Harrington and Abramson [1], the class of all finite linearly ordered graphs, or more generally the class of all finite linearly ordered structures in a fixed finite relational language is Ramsey.\",\n \"In fact, [13] generalizes this to allow function symbols as well.\",\n \"There is another definition of Ramsey structures that colors embeddings instead of copies, see [3].\",\n \"This is equivalent to our definition since we asked for a quantifier-free definable linear order (so that finite substructures are rigid).\",\n \"If we drop the requirement that there is a definable linear order then these definitions do not agree in general.\",\n \"In fact, using the alternative definition there must be a definable order in the $\\\\omega $ -categorical case, so these are equivalent in this case.\",\n \"Fact 2.15 [3] If $M$ is an $\\\\omega $ -categorical ultrahomogeneous Ramsey structure according to [3], then there is a definable linear order on $M$ .\",\n \"What about dense trees?\",\n \"Adding a generic linear order to a dense tree will not result in a Ramsey structure.\",\n \"By this we mean the model completion of the theory $T_{dt,<,\\\\forall }$ in the language $\\\\left\\\\lbrace <,\\\\wedge ,<^{\\\\prime }\\\\right\\\\rbrace $ which says that the $\\\\left\\\\lbrace <,\\\\wedge \\\\right\\\\rbrace $ -part is a meet tree, and $<^{\\\\prime }$ is a linear order.\",\n \"The class of finite structures of $T_{dt,<,\\\\forall }$ easily has HP, JEP and HP, thus this model completion exists (see Fact REF ).\",\n \"Call its Fraïssé limit the generically linearly ordered tree.\",\n \"It turns out that this structure is not Ramsey, see Claim REF .\",\n \"In any case, we can add a linear order to the tree structure and make it Ramsey.\",\n \"Example 2.16 ([33], and see there for more references) Let $L=\\\\left\\\\lbrace <,<_{\\\\operatorname{lex}},\\\\wedge \\\\right\\\\rbrace $ and let $M$ be the $L$ -structure whose universe is the tree $\\\\omega ^{<\\\\omega }$ with the natural interpretations of $<$ as the tree order, $\\\\wedge $ as the meet function ($s\\\\wedge t=s\\\\upharpoonright \\\\operatorname{len}\\\\left(s\\\\wedge t\\\\right)$ where $\\\\operatorname{len}\\\\left(s\\\\wedge t\\\\right)=\\\\max \\\\left\\\\lbrace k\\\\,|\\\\,s\\\\upharpoonright k=t\\\\upharpoonright k\\\\right\\\\rbrace $ ), $<_{\\\\operatorname{lex}}$ as the lexicographical order ($s<_{\\\\operatorname{lex}}t$ iff $sP$ .\",\n \"As $T_{dt}^{B}$ is model-complete, $M$ is existentially closed (see Fact REF ) so there is some $b\\\\in P$ (from $M)$ such that $f\\\\left(a\\\\right)b>f\\\\left(a\\\\right)$ which is a contradiction to the definition of $f$ .\",\n \"For three sets $A,B,C$ , let $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ iff $ ACBCC$and for all $ aAC$ with $ f(a)C$and $ bBC$ with $ f(b)C$such that $ f(a)Cf(b)$ (which is thesame as $ f(a)f(C)f(b)$),$ f(a)p withno c\\\\in \\\\left\\\\langle C\\\\right\\\\rangle such that a\\\\wedge c>p, andall b\\\\in \\\\left\\\\langle BC\\\\right\\\\rangle with b>p and no c\\\\in \\\\left\\\\langle C\\\\right\\\\rangle such that b\\\\wedge c>p, a\\\\wedge b=p.\\\\end{enumerate}Then $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ is canonical.\",\n \"The only nontrivial axioms to check arestationarity over $$, extension and transitivity.$ It is stationary over $\\\\emptyset $ by elimination of quantifiers, since $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $B$ iff $ A'={ aA | apap$ for all $ aAC$with $ apCdp$.\",\n \"Then find $ d'$ such that $ d'BCd$with $ d'pd”$ (and $ BCd'AC=C$).Now assume that $ d>p$.\",\n \"If there is some $ bBC$with $ bd>p$, any $ d'BCd$ such that $ BCd'AC=C$will work.\",\n \"Otherwise find some $ d'BCd$ such that $ d'a=p$for all $ aAC$ with $ a>p$.$ Transitivity: suppose that $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $DCB$ and $ D$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ and wehave to show that $ AD$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$.\",\n \"If we are in case (1) of the definition,($ aADC$, $ apC$,etc.)\",\n \"then we proceed exactly as in Example \\\\ref {exa:trees with predicate}.Otherwise, suppose that $ aADC$, $ bBC$are as in case (2).\",\n \"If there is some $ dCD$with $ ad>p$, then for no $ cC$is it the case that $ cd>p$ (otherwise $ ac>p$).\",\n \"Thus$ db=p$ because $ D$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ hence $ ab=p$ as required.If there is no such $ d$ then $ ab=p$ because $ A$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $DCB$.$ Trees also satisfy the following interesting phenomenon.\",\n \"Proposition 4.17 If $M$ is a dense tree as in Example REF (i.e., the model companion of the theory of trees in $\\\\left\\\\lbrace <,\\\\wedge \\\\right\\\\rbrace $ ) then for every $\\\\sigma \\\\in \\\\operatorname{Aut}\\\\left(M\\\\right)$ which does not have any fixed points, there is a branch $B\\\\subseteq M$ such that $\\\\sigma \\\\left(B\\\\right)=B$ .\",\n \"Let $B$ be a maximal linearly ordered set such that $\\\\sigma \\\\left(B\\\\right)=B$ (which exists by Zorn's lemma).\",\n \"We will show that $B$ is a branch.\",\n \"Note that if $x\\\\in B$ and $y0$ .\",\n \"As $\\\\sigma _{*}^{k_{n,i}}\\\\left(c_{*,i}^{\\\\prime \\\\prime }\\\\right)=c_{*}$ , it follows that $d\\\\left(\\\\sigma _{*}^{k_{n,i}}\\\\left(x_{0}\\\\right),z_{j_{i}}\\\\right)<\\\\varepsilon /2$ .\",\n \"Similarly, as $\\\\sigma _{*}^{k_{n,0}}\\\\left(c_{*,i}^{\\\\prime }\\\\right)=g_{i}^{-1}\\\\left(c_{*}\\\\right)$ , we have that $d\\\\left(g_{i}\\\\left(\\\\sigma _{*}^{k_{n,0}}\\\\left(x_{0}\\\\right)\\\\right),z_{j_{i}}\\\\right)<\\\\varepsilon /2$ .\",\n \"Together, we are done.\",\n \"Now we have to take care of the other half of $\\\\star $ .\",\n \"This is done similarly, using right existence and the second part of Claim REF .\",\n \"The following proposition explains why we needed to take a conjugate of $\\\\sigma $ .\",\n \"The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.\",\n \"In Section REF , we mentioned that it is a Ramsey structure.\",\n \"Note that the underlying order is dense (by Proposition REF ).\",\n \"Proposition 6.9 Let $M=\\\\left(V,<,R\\\\right)$ be the countable ordered random graph.\",\n \"Then there is no automorphism $\\\\sigma \\\\in G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\\\in X$ .\",\n \"First we find $a\\\\ne b$ in $M$ such that $\\\\sigma ^{n}\\\\left(a\\\\right)\\\\ne \\\\sigma ^{m}\\\\left(b\\\\right)$ for all $m,n\\\\in \\\\mathbb {Z}$ .\",\n \"To do that, take any $a\\\\in M$ .\",\n \"Then $\\\\left\\\\lbrace \\\\sigma ^{n}\\\\left(a\\\\right)\\\\,|\\\\,n\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace $ is discrete (in the order sense: it is either a $\\\\mathbb {Z}$ -chain or just $a$ ).\",\n \"Since $\\\\left(V,<\\\\right)$ is dense, there is some $b\\\\ne \\\\sigma ^{n}\\\\left(a\\\\right)$ for all $n\\\\in \\\\mathbb {Z}$ .\",\n \"It follows that $b$ is as required.\",\n \"Let $X=S_{x}\\\\left(M\\\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).\",\n \"Let $p\\\\in X$ be any completion of the partial type $\\\\left\\\\lbrace R\\\\left(x,\\\\sigma ^{n}\\\\left(a\\\\right)\\\\right)\\\\,|\\\\,n\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace \\\\lnot R\\\\left(x,\\\\sigma ^{m}\\\\left(b\\\\right)\\\\right)\\\\,|\\\\,m\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace $ .\",\n \"Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\\\in \\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma ^{n}\\\\left(p\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ which is a fixed point of $G$ .\",\n \"In other words, $p_{0}$ is an invariant type over $M$ .\",\n \"However $R\\\\left(x,a\\\\right)\\\\wedge \\\\lnot R\\\\left(x,b\\\\right)\\\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).\",\n \"The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.\",\n \"If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\\\left(X,x_{0}\\\\right)$ .\",\n \"Thus, there would be a conjugate $\\\\sigma _{*}$ of $\\\\sigma $ such that $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow.\",\n \"But then if $\\\\left(Y,y_{0}\\\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\\\pi :X\\\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .\",\n \"Thus, $\\\\pi $ maps $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ to $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(y\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ , and the latter contains a $G$ -subflow.\",\n \"Thus we get that $\\\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .\",\n \"Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\\\left\\\\lbrace <,\\\\wedge \\\\right\\\\rbrace $ , and let $M\\\\models T$ be countable.\",\n \"Then $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no shifty automorphism.\",\n \"In particular, $M$ has no CIR.\",\n \"Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .\",\n \"Suppose that $\\\\sigma $ was shifty.\",\n \"Let $\\\\bar{m}=\\\\left\\\\langle m\\\\,|\\\\,m\\\\in M\\\\right\\\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\\\bar{x}=\\\\left\\\\langle x_{m}\\\\,|\\\\,m\\\\in M\\\\right\\\\rangle $ .\",\n \"Let $X=S_{\\\\bar{m}}\\\\left(M\\\\right)$ be the space of $\\\\bar{x}$ -complete types $p$ over $M$ such that $p\\\\upharpoonright \\\\emptyset =\\\\operatorname{tp}\\\\left(\\\\bar{m}/M\\\\right)$ .\",\n \"Then $X$ is a compact metric space.\",\n \"Let $x_{*}=\\\\operatorname{tp}\\\\left(\\\\bar{m}/M\\\\right)$ .\",\n \"By Theorem REF , there is some conjugate $\\\\tau $ of $\\\\sigma $ such that $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\tau ^{n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow $Y^{+}\\\\subseteq X$ and similarly, $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\tau ^{-n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow $Y^{-}$ .\",\n \"By Proposition REF , $\\\\tau $ fixes a branch or a point.\",\n \"Suppose that $\\\\tau \\\\left(m\\\\right)=m$ for some $m\\\\in M$ .\",\n \"Then for every $p\\\\in Y^{+}$ , $p\\\\models x_{m}=m$ .\",\n \"However $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ acts transitively on $M$ , so we have a contradiction.\",\n \"Now suppose that $\\\\tau $ fixes a branch $B\\\\subseteq M$ , but does not fix any point.\",\n \"Suppose that $\\\\tau \\\\left(m\\\\right)>m$ for some $m\\\\in B$ .\",\n \"Then $\\\\tau ^{n}\\\\left(m\\\\right)>m$ for all $n<\\\\omega $ , so for any $p\\\\in Y^{+}$ , $p\\\\models x_{m}>m$ .\",\n \"There is some $m^{\\\\prime }\\\\in M$ such that $m^{\\\\prime }>m$ and $m^{\\\\prime }\\\\notin B$ .\",\n \"Since $m<\\\\tau ^{n}\\\\left(m\\\\right)\\\\in B$ for all $n<\\\\omega $ , it follows that $p\\\\models x_{m}\\\\wedge m^{\\\\prime }=m$ for all $p\\\\in Y^{+}$ .\",\n \"Let $\\\\tau ^{\\\\prime }\\\\in G$ fix $m$ and map $m^{\\\\prime }$ to $B$ .\",\n \"Then $\\\\tau ^{\\\\prime }\\\\left(p\\\\right)\\\\models \\\\left(x_{m}\\\\wedge \\\\tau ^{\\\\prime }\\\\left(m^{\\\\prime }\\\\right)\\\\right)=mm$ , so we can apply the same argument to $Y^{-}$ .\",\n \"For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .\",\n \"In addition, letting $H=\\\\operatorname{Aut}\\\\left(N\\\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).\",\n \"In addition, if $B\\\\subseteq N$ is a branch, $m\\\\in B$ , there is always some $m^{\\\\prime }>m$ , $m^{\\\\prime }\\\\notin B$ and for any $n^{\\\\prime }>m$ in $B$ , $m^{\\\\prime }m\\\\equiv n^{\\\\prime }m$ .\",\n \"Hence, we can apply Proposition REF and the same proof will work.\",\n \"Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\\\omega $ -categorical structures.\",\n \"We state here a few general conjectures and questions.\",\n \"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.\",\n \"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.\",\n \"Conjecture 7.1 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion which admits a CIR.\",\n \"Suppose that $M$ is a structure and $\\\\mathfrak {C}$ a monster model for $\\\\operatorname{Th}\\\\left(M\\\\right)$ .\",\n \"The group of Lascar strong automorphisms of $M$, denoted by $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is the group of automorphisms of $M$ generated by the set $\\\\left\\\\lbrace \\\\sigma \\\\upharpoonright M\\\\,|\\\\,\\\\exists N\\\\prec \\\\mathfrak {C},\\\\left|N\\\\right|=\\\\left|T\\\\right|,\\\\sigma \\\\upharpoonright N=\\\\operatorname{id}\\\\right\\\\rbrace $ .\",\n \"If $\\\\sigma $ is Lascar strong, then $\\\\sigma \\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=\\\\operatorname{id}$ so $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is contained in $G^{0}$ .\",\n \"However, there are examples (even $\\\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ , see [16], [31].\",\n \"The Lascar group of $M$ is the quotient $\\\\operatorname{Aut}\\\\left(M\\\\right)/\\\\operatorname{Aut}f\\\\left(M\\\\right)$ .\",\n \"For more on the Lascar group, see [39].\",\n \"In the $\\\\omega $ -categorical case, the quotient $\\\\operatorname{Aut}\\\\left(\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ is also called the compact Lascar group.\",\n \"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $nsM$.\",\n \"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.\",\n \"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.\",\n \"Forinstance, by Lemma \\\\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotients then it has finite topological rank.\",\n \"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.\",\n \"Thus, potentially, the Lascar group — as a quotient of $\\\\operatorname{Aut}\\\\left(M\\\\right)$ — can be an obstruction to having finite topological rank.\",\n \"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.\",\n \"Conjecture REF (together with Theorem REF ) implies that it could not.\",\n \"Conjecture 7.2 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion with trivial Lascar group.\",\n \"By the above, this second conjecture is implied by Conjecture REF .\",\n \"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.\",\n \"It would be interesting to investigate other consequences of having a CIR.\",\n \"For instance a CIR might have something to say about normal subgroups.\",\n \"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $A$ for some CIR $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\\\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\\\aleph _{0}}$ is open.\",\n \"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.\",\n \"We refer to [26] for a survey on this.\",\n \"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.\",\n \"In fact the compact quotients are hidden in the stabilizer of a finite set.\",\n \"It seems that one can avoid this counterexample by restricting to dense subgroups.\",\n \"This leads us to the following questions.\",\n \"Question 7.3 Let $M$ be $\\\\omega $ -categorical such that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotient.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open (and hence is equal to $G$ )?\",\n \"Note that the assumption of having no compact quotient is necessary.\",\n \"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has a dense subgroup of index 2, see [23].\",\n \"Question 7.4 Let $M$ be $\\\\omega $ -categorical and $N$ an $\\\\omega $ -categorical expansion of $M$ .\",\n \"Set $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ and $H=\\\\operatorname{Aut}\\\\left(N\\\\right)\\\\le G$ .\",\n \"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open?\",\n \"Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.\",\n \"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.\",\n \"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].\",\n \"We would also like to thank Daoud Siniora for his comments.\",\n \"Finally, we would like to thank the anonymous referee for his comments.\"\n ],\n [\n \"Examples of theories with a canonical independence relation\",\n \"There are many examples of countable ultrahomogeneous structures with a CIR.\",\n \"Here we will give some of them.\",\n \"We will define the relation $\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$, but sometimes leave most of the details of checking thatit satisfies the axioms to the reader.\",\n \"All the CIRs we define aredefined on finitely generated substructures.\\\\begin{example}The most trivial ultrahomogeneous structureis of course the structure with universe \\\\omega and no relationsbut equality.\",\n \"Its automorphism group is S_{\\\\infty }.\",\n \"For finitesets A,B,C define A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}\\\\end{example}{\\\\textstyle \\\\textstyle x}\\\\hspace{0.0pt}\\\\hbox{t}o 0pt{\\\\hss \\\\textstyle \\\\mid \\\\hss } \\\\hss $$\\\\hss $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ by $ ABC$.\",\n \"Thisis a CIR.$ Example 4.1 If $T$ is stable and $\\\\emptyset $ is a base (i.e., $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=\\\\operatorname{dcl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ so that every type over $\\\\emptyset $ has a unique non-forking extension), then $\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $f$ (i.e., non-forking independence) is canonical.$ Example 4.2 Let $\\\\left(\\\\mathcal {B},<,\\\\wedge ,\\\\vee ,0,1,\\\\square ^{c}\\\\right)$ be the atomless Boolean algebra.\",\n \"For finite sets $A,B,C$ , define $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ iff$ C=ACBC$and for every atom $ aAC$ and everyatom $ bBC$, if there is an atom $ cC$such that $ a,bc$ then $ ab0$.\",\n \"Let us show that $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$is a (symmetric) CIR.$ Stationarity over $\\\\emptyset $ : $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $B$ says that the atoms in$ AB$ are in bijection with (atoms of $ A$)$$ (atoms of $ B$).\",\n \"Thus, if $ A'$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $B'$ and $ f:AA'$,$ g:BB'$are isomorphisms, then $ h:ABA'B'$taking an atom $ ab$ to $ f(a)g(b)$is an isomorphism.\",\n \"This easily implies stationarity.$ Transitivity: suppose that $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CDB$ and $ D$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$, and wewant to show that $ AD$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$.\",\n \"Suppose that $ aACD$,$ bBC$ are atoms, and $ cC$is an atom such that $ a,bc$.\",\n \"Let $ dDC$be an atom such that $ adc$.\",\n \"Then $ db0$.\",\n \"Let$ b'db$ be an atom of $ BCD$,so that $ a,b'd$.\",\n \"Hence $ ab'0$ and thus $ ab0$.Transitivity to the right follows by symmetry.$ Extension: suppose that $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ and $ d$ is given.\",\n \"We want tofind $ d'BCd$ such that $ A$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CBd$.\",\n \"We may assume that$ dBC$.\",\n \"An atom in $ BCd$has the form $ db$ or $ dcb$ for some atom $ b$ of$ BC$.\",\n \"The type $ tp(d/BC)$is determined by knowing which of these terms $ db,dcb$is nonzero (for $ bBC$ an atom).\",\n \"As$ B$ is atomless and $ A$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$, we can find some $ d'$ suchthat for every $ a,b,c$ atoms in $ AC,BC,C$respectively such that $ a,bc$, if $ db0$ then $ ad'b0$(else $ d'b=0$) and if $ dcb0$ then $ a(d')cb0$(else $ (d')cb=0$).\",\n \"In addition, we ask thatif $ db,dcb0$ then both $ ad'b$and $ a(d')cb$ are not in $ ABC$.It now follows that $ ACBCd'=C$:suppose that $ eACBCd'$.Then as $ eBCd'$, it can be writtenas $ b0(b1d')(b2(d')c)$where $ b0,b1,b2BC$ are pairwisedisjoint and for every atom $ b'b1b2$ from $ BC$,$ d'b',(d')cb'0$.\",\n \"If both $ b1,b2=0$,then $ eBCAC$so $ eC$.\",\n \"If $ b10$, let $ b1'b1$be an atom of $ BC$.\",\n \"So $ eb1'ABC$(because $ eAC$) and has the form $ b1'd'$.Let $ aAC$ and $ cC$be atoms such that $ a,b1'c$.\",\n \"Then $ eb1'aABC$and has the form $ ab1'd'$ which is not in $ ABC$by construction, contradiction.\",\n \"Similarly $ b2=0$ and we are done.$ Existence and monotonicity are clear.\",\n \"Example 4.3 Let $\\\\left(M,R\\\\right)$ be the random tournament (a tournament is a complete directed graph such that for all $x,y$ , it cannot be that both $R\\\\left(x,y\\\\right)$ and $R\\\\left(y,x\\\\right)$ , and the random tournament is the Fraïssé limit of the class of finite tournaments).\",\n \"Given finite sets $A,B,C$ , write $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ iff $ AB=C$ andif $ aAC$, $ bBC$ then $ R(a,b)$.This easily satisfies all the requirements.$ Remark 4.4 The following definition is from [19].\",\n \"Let $K$ be a class of finite $L$ -structures where $L$ is a relational language.\",\n \"Then $K$ has the strong$^{+}$ amalgamation property if for all $A,B,C\\\\in K$ with $A\\\\subseteq B,C$ there is $D\\\\in K$ such that $D=C^{\\\\prime }\\\\cup B^{\\\\prime }$ with $C^{\\\\prime }\\\\cong _{A}C$ , $B^{\\\\prime }\\\\cong _{A}B$ and for every $n$ -ary relation $R$ and every $x_{1},\\\\ldots ,x_{n}$ and $y_{1},\\\\ldots ,y_{n}$ from $D\\\\backslash A$ which intersect both $C^{\\\\prime }$ and $B^{\\\\prime }$ , if [$x_{i}\\\\in B^{\\\\prime }$ iff $y_{i}\\\\in B^{\\\\prime }$ for all $1\\\\le i\\\\le n$ ], then $R^{D}\\\\left(x_{1},\\\\ldots ,x_{n}\\\\right)$ iff $R^{D}\\\\left(y_{1},\\\\ldots ,y_{n}\\\\right)$ (where $R^{D}$ is the interpretation of $R$ in $D$ ).\",\n \"For example, the random tournament satisfies this property.\",\n \"In [19] it is proved that if $M$ is ultrahomogeneous and $\\\\operatorname{Age}\\\\left(M\\\\right)$ has the strong$^{+}$ amalgamation property, then $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has a cyclically dense conjugacy class.\",\n \"They prove it using a condition they denote by $(\\\\Delta _{n})$ , see there, Theorem 5.12.\",\n \"We do not know if this condition implies the existence of a CIR.\"\n ],\n [\n \"Free amalgamation classes\",\n \"In [38], [27], [4] there is an axiomatic framework for defining an abstract ternary relation close to our CIR.\",\n \"More precisely, in [27] and [38], the notion of a stationary independence relation (SIR) is introduced (in [27] for finitely generated structures and in [38] for sets in general).\",\n \"A similar notion is defined in [4], with more axioms.\",\n \"In any case, all these notions imply ours, except perhaps that stationarity over $\\\\emptyset $ becomes stationarity over $\\\\operatorname{acl}\\\\left(\\\\emptyset \\\\right)$ , so that this becomes a CIR in the expansion $\\\\left(M,\\\\operatorname{acl}\\\\left(\\\\emptyset \\\\right)\\\\right)$ (note that our extension follows from full stationarity and their version of existence).\",\n \"Thus, we can apply our results to the examples studied there.\",\n \"In particular, we get the following examples.\",\n \"Example 4.5 The rational Urysohn space $\\\\mathbb {Q}\\\\mathbb {U}$ is the Fraïssé limit of the class of finite metric spaces with rational distances.\",\n \"Pick a point $q\\\\in \\\\mathbb {Q}\\\\mathbb {U}$ , and consider the structure $\\\\left(\\\\mathbb {Q}\\\\mathbb {U},q\\\\right)$ where we add a constant for $q$ .\",\n \"In [37], it is proved that the relation $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ which holds for finite $ A,B,C$ ifffor every $ aAC$, $ bBC$, $ d(a,b)={ d(a,c)+d(c,b) | cC} $is a CIR in $ (QU,q)$.$ In all examples given by Conant [4] which we list now, $\\\\operatorname{acl}\\\\left(\\\\emptyset \\\\right)=\\\\emptyset $ , so we actually get a CIR in the structure (i.e., no need to take an expansion) by [4].\",\n \"Example 4.6 Fraïssé limits with free amalgamation: suppose that $L$ is a relational language and $K$ is an essentially countable (see above Fact REF ) class of finite $L$ -structures, such that if $A,B,C\\\\in K$ and $A\\\\subseteq C,B$ , then the free amalgam of $A,B,C$ is in $K$ (i.e., a structure $D=C^{\\\\prime }\\\\cup B^{\\\\prime }$ with $C^{\\\\prime }\\\\cong _{A}C$ , $B^{\\\\prime }\\\\cong _{A}B$ , $B^{\\\\prime }\\\\cap C^{\\\\prime }\\\\subseteq A$ and for every tuple $a\\\\in D$ in the length of some relation $R\\\\in L$ , if $R\\\\left(a\\\\right)$ then $a\\\\in C^{\\\\prime }$ or $a\\\\in B^{\\\\prime }$ ) (here we also include the case $A=\\\\emptyset $ ).\",\n \"Let $M$ be the Fraïssé limit of $K$ , and define $B\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $AC$ iff $ ABC$ is the free amalgam of$ A,AB,AC$.\",\n \"If the language is finite or more generally in the contextof Fact \\\\ref {fact:Fraisse limits} (2), it is easy to see that inthis case $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ is a CIR (this is also proved, for finite languages,in \\\\cite [Proposition 3.4]{MR3663421}).$ This class of examples contain e.g., the random graph, the universal $K_{n}$ -free graph (the Henson graph), and their hypergraph analogs.\",\n \"Example 4.7 Let $L=\\\\left\\\\lbrace P_{n}\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ and let $K$ be the class of finite $L$ -structures in which $P_{n}\\\\left(x_{0},\\\\ldots ,x_{n-1}\\\\right)$ implies that $x_{i}\\\\ne x_{j}$ for $i\\\\ne j$ .\",\n \"Then $K$ is essentially countable and is a free amalgamation class, and moreover for each $n<\\\\omega $ there are finitely many isomorphism types of structures of size $n$ (so we can use Fact REF ).\",\n \"Let $M$ be the limit.\",\n \"Now recall the example $N$ described in Remark REF , with infinitely many independent equivalence relations with two classes.\",\n \"Then $M$ is $N$ expanded by naming the classes.\",\n \"In other words, $\\\\operatorname{Aut}\\\\left(M\\\\right)=\\\\operatorname{Aut}\\\\left(N\\\\right)^{0}$ .\",\n \"Example 4.8 In [5] the authors describe a generic $K_{n}+K_{3}$ -free graph, where $K_{n}+K_{3}$ is the free amalgam of the complete graph on $n$ vertices and a triangle over a single vertex.\",\n \"This structure is $\\\\aleph _{0}$ -categorical, with $\\\\operatorname{acl}\\\\left(\\\\emptyset \\\\right)=\\\\emptyset $ .\",\n \"By [4] there is a CIR on this graph.\",\n \"Example 4.9 $\\\\omega $ -categorical Hrushovski constructions.\",\n \"Let $L$ be a finite relational language, and let $f:\\\\mathbb {R}_{\\\\ge 0}\\\\rightarrow \\\\mathbb {R}_{\\\\ge 0}$ be a “control function”.\",\n \"According to [10], there is an $\\\\omega $ -categorical generic Hrushovski construction $M_{f}$ for a “free amalgamation class” $K_{f}$ if $f$ satisfies certain conditions.\",\n \"It follows from [4] that given extra conditions on the algebraic closure, $M_{f}$ admits a CIR.\",\n \"See more details in [4], [10].\"\n ],\n [\n \"Ultrahomogenous partial orders\",\n \"In [11] the authors prove that the automorphism group of every ultrahomogeneous poset (partially order set) is topologically 2-generated.\",\n \"They also characterize when they have a cyclically dense conjugacy class.\",\n \"We can find such a conjugacy class by finding a CIR whenever possible.\",\n \"We should remark that they prove more on the automorphism groups of those structures.\",\n \"By [32] there are four types of ultrahomogeneous posets.\",\n \"Fact 4.10 [32] Suppose that $\\\\left(H,<\\\\right)$ is an ultrahomogeneous poset.\",\n \"Then $H$ is isomorphic to one of the following: The random poset: the Fraïssé limit of the class of finite partial orders.\",\n \"The orders $\\\\mathcal {A}_{n}$ for $1\\\\le n\\\\le \\\\omega $ : $\\\\left(n,<\\\\right)$ where $<$ is trivial i.e., empty.\",\n \"The orders $\\\\mathcal {B}_{n}$ for $1\\\\le n\\\\le \\\\omega $ : $\\\\left(n\\\\times \\\\mathbb {Q},<\\\\right)$ where $\\\\left(k,q\\\\right)<\\\\left(m,p\\\\right)$ iff $k=m$ and $pP$ .\",\n \"As $T_{dt}^{B}$ is model-complete, $M$ is existentially closed (see Fact REF ) so there is some $b\\\\in P$ (from $M)$ such that $f\\\\left(a\\\\right)b>f\\\\left(a\\\\right)$ which is a contradiction to the definition of $f$ .\",\n \"For three sets $A,B,C$ , let $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ iff $ ACBCC$and for all $ aAC$ with $ f(a)C$and $ bBC$ with $ f(b)C$such that $ f(a)Cf(b)$ (which is thesame as $ f(a)f(C)f(b)$),$ f(a)p withno c\\\\in \\\\left\\\\langle C\\\\right\\\\rangle such that a\\\\wedge c>p, andall b\\\\in \\\\left\\\\langle BC\\\\right\\\\rangle with b>p and no c\\\\in \\\\left\\\\langle C\\\\right\\\\rangle such that b\\\\wedge c>p, a\\\\wedge b=p.\\\\end{enumerate}Then $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ is canonical.\",\n \"The only nontrivial axioms to check arestationarity over $$, extension and transitivity.$ It is stationary over $\\\\emptyset $ by elimination of quantifiers, since $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $B$ iff $ A'={ aA | apap$ for all $ aAC$with $ apCdp$.\",\n \"Then find $ d'$ such that $ d'BCd$with $ d'pd”$ (and $ BCd'AC=C$).Now assume that $ d>p$.\",\n \"If there is some $ bBC$with $ bd>p$, any $ d'BCd$ such that $ BCd'AC=C$will work.\",\n \"Otherwise find some $ d'BCd$ such that $ d'a=p$for all $ aAC$ with $ a>p$.$ Transitivity: suppose that $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $DCB$ and $ D$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ and wehave to show that $ AD$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$.\",\n \"If we are in case (1) of the definition,($ aADC$, $ apC$,etc.)\",\n \"then we proceed exactly as in Example \\\\ref {exa:trees with predicate}.Otherwise, suppose that $ aADC$, $ bBC$are as in case (2).\",\n \"If there is some $ dCD$with $ ad>p$, then for no $ cC$is it the case that $ cd>p$ (otherwise $ ac>p$).\",\n \"Thus$ db=p$ because $ D$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $CB$ hence $ ab=p$ as required.If there is no such $ d$ then $ ab=p$ because $ A$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $DCB$.$ Trees also satisfy the following interesting phenomenon.\",\n \"Proposition 4.17 If $M$ is a dense tree as in Example REF (i.e., the model companion of the theory of trees in $\\\\left\\\\lbrace <,\\\\wedge \\\\right\\\\rbrace $ ) then for every $\\\\sigma \\\\in \\\\operatorname{Aut}\\\\left(M\\\\right)$ which does not have any fixed points, there is a branch $B\\\\subseteq M$ such that $\\\\sigma \\\\left(B\\\\right)=B$ .\",\n \"Let $B$ be a maximal linearly ordered set such that $\\\\sigma \\\\left(B\\\\right)=B$ (which exists by Zorn's lemma).\",\n \"We will show that $B$ is a branch.\",\n \"Note that if $x\\\\in B$ and $y0$ .\",\n \"As $\\\\sigma _{*}^{k_{n,i}}\\\\left(c_{*,i}^{\\\\prime \\\\prime }\\\\right)=c_{*}$ , it follows that $d\\\\left(\\\\sigma _{*}^{k_{n,i}}\\\\left(x_{0}\\\\right),z_{j_{i}}\\\\right)<\\\\varepsilon /2$ .\",\n \"Similarly, as $\\\\sigma _{*}^{k_{n,0}}\\\\left(c_{*,i}^{\\\\prime }\\\\right)=g_{i}^{-1}\\\\left(c_{*}\\\\right)$ , we have that $d\\\\left(g_{i}\\\\left(\\\\sigma _{*}^{k_{n,0}}\\\\left(x_{0}\\\\right)\\\\right),z_{j_{i}}\\\\right)<\\\\varepsilon /2$ .\",\n \"Together, we are done.\",\n \"Now we have to take care of the other half of $\\\\star $ .\",\n \"This is done similarly, using right existence and the second part of Claim REF .\",\n \"The following proposition explains why we needed to take a conjugate of $\\\\sigma $ .\",\n \"The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.\",\n \"In Section REF , we mentioned that it is a Ramsey structure.\",\n \"Note that the underlying order is dense (by Proposition REF ).\",\n \"Proposition 6.9 Let $M=\\\\left(V,<,R\\\\right)$ be the countable ordered random graph.\",\n \"Then there is no automorphism $\\\\sigma \\\\in G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\\\in X$ .\",\n \"First we find $a\\\\ne b$ in $M$ such that $\\\\sigma ^{n}\\\\left(a\\\\right)\\\\ne \\\\sigma ^{m}\\\\left(b\\\\right)$ for all $m,n\\\\in \\\\mathbb {Z}$ .\",\n \"To do that, take any $a\\\\in M$ .\",\n \"Then $\\\\left\\\\lbrace \\\\sigma ^{n}\\\\left(a\\\\right)\\\\,|\\\\,n\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace $ is discrete (in the order sense: it is either a $\\\\mathbb {Z}$ -chain or just $a$ ).\",\n \"Since $\\\\left(V,<\\\\right)$ is dense, there is some $b\\\\ne \\\\sigma ^{n}\\\\left(a\\\\right)$ for all $n\\\\in \\\\mathbb {Z}$ .\",\n \"It follows that $b$ is as required.\",\n \"Let $X=S_{x}\\\\left(M\\\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).\",\n \"Let $p\\\\in X$ be any completion of the partial type $\\\\left\\\\lbrace R\\\\left(x,\\\\sigma ^{n}\\\\left(a\\\\right)\\\\right)\\\\,|\\\\,n\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace \\\\lnot R\\\\left(x,\\\\sigma ^{m}\\\\left(b\\\\right)\\\\right)\\\\,|\\\\,m\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace $ .\",\n \"Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\\\in \\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma ^{n}\\\\left(p\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ which is a fixed point of $G$ .\",\n \"In other words, $p_{0}$ is an invariant type over $M$ .\",\n \"However $R\\\\left(x,a\\\\right)\\\\wedge \\\\lnot R\\\\left(x,b\\\\right)\\\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).\",\n \"The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.\",\n \"If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\\\left(X,x_{0}\\\\right)$ .\",\n \"Thus, there would be a conjugate $\\\\sigma _{*}$ of $\\\\sigma $ such that $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow.\",\n \"But then if $\\\\left(Y,y_{0}\\\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\\\pi :X\\\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .\",\n \"Thus, $\\\\pi $ maps $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ to $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(y\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ , and the latter contains a $G$ -subflow.\",\n \"Thus we get that $\\\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .\",\n \"Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\\\left\\\\lbrace <,\\\\wedge \\\\right\\\\rbrace $ , and let $M\\\\models T$ be countable.\",\n \"Then $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no shifty automorphism.\",\n \"In particular, $M$ has no CIR.\",\n \"Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .\",\n \"Suppose that $\\\\sigma $ was shifty.\",\n \"Let $\\\\bar{m}=\\\\left\\\\langle m\\\\,|\\\\,m\\\\in M\\\\right\\\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\\\bar{x}=\\\\left\\\\langle x_{m}\\\\,|\\\\,m\\\\in M\\\\right\\\\rangle $ .\",\n \"Let $X=S_{\\\\bar{m}}\\\\left(M\\\\right)$ be the space of $\\\\bar{x}$ -complete types $p$ over $M$ such that $p\\\\upharpoonright \\\\emptyset =\\\\operatorname{tp}\\\\left(\\\\bar{m}/M\\\\right)$ .\",\n \"Then $X$ is a compact metric space.\",\n \"Let $x_{*}=\\\\operatorname{tp}\\\\left(\\\\bar{m}/M\\\\right)$ .\",\n \"By Theorem REF , there is some conjugate $\\\\tau $ of $\\\\sigma $ such that $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\tau ^{n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow $Y^{+}\\\\subseteq X$ and similarly, $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\tau ^{-n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow $Y^{-}$ .\",\n \"By Proposition REF , $\\\\tau $ fixes a branch or a point.\",\n \"Suppose that $\\\\tau \\\\left(m\\\\right)=m$ for some $m\\\\in M$ .\",\n \"Then for every $p\\\\in Y^{+}$ , $p\\\\models x_{m}=m$ .\",\n \"However $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ acts transitively on $M$ , so we have a contradiction.\",\n \"Now suppose that $\\\\tau $ fixes a branch $B\\\\subseteq M$ , but does not fix any point.\",\n \"Suppose that $\\\\tau \\\\left(m\\\\right)>m$ for some $m\\\\in B$ .\",\n \"Then $\\\\tau ^{n}\\\\left(m\\\\right)>m$ for all $n<\\\\omega $ , so for any $p\\\\in Y^{+}$ , $p\\\\models x_{m}>m$ .\",\n \"There is some $m^{\\\\prime }\\\\in M$ such that $m^{\\\\prime }>m$ and $m^{\\\\prime }\\\\notin B$ .\",\n \"Since $m<\\\\tau ^{n}\\\\left(m\\\\right)\\\\in B$ for all $n<\\\\omega $ , it follows that $p\\\\models x_{m}\\\\wedge m^{\\\\prime }=m$ for all $p\\\\in Y^{+}$ .\",\n \"Let $\\\\tau ^{\\\\prime }\\\\in G$ fix $m$ and map $m^{\\\\prime }$ to $B$ .\",\n \"Then $\\\\tau ^{\\\\prime }\\\\left(p\\\\right)\\\\models \\\\left(x_{m}\\\\wedge \\\\tau ^{\\\\prime }\\\\left(m^{\\\\prime }\\\\right)\\\\right)=mm$ , so we can apply the same argument to $Y^{-}$ .\",\n \"For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .\",\n \"In addition, letting $H=\\\\operatorname{Aut}\\\\left(N\\\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).\",\n \"In addition, if $B\\\\subseteq N$ is a branch, $m\\\\in B$ , there is always some $m^{\\\\prime }>m$ , $m^{\\\\prime }\\\\notin B$ and for any $n^{\\\\prime }>m$ in $B$ , $m^{\\\\prime }m\\\\equiv n^{\\\\prime }m$ .\",\n \"Hence, we can apply Proposition REF and the same proof will work.\",\n \"Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\\\omega $ -categorical structures.\",\n \"We state here a few general conjectures and questions.\",\n \"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.\",\n \"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.\",\n \"Conjecture 7.1 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion which admits a CIR.\",\n \"Suppose that $M$ is a structure and $\\\\mathfrak {C}$ a monster model for $\\\\operatorname{Th}\\\\left(M\\\\right)$ .\",\n \"The group of Lascar strong automorphisms of $M$, denoted by $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is the group of automorphisms of $M$ generated by the set $\\\\left\\\\lbrace \\\\sigma \\\\upharpoonright M\\\\,|\\\\,\\\\exists N\\\\prec \\\\mathfrak {C},\\\\left|N\\\\right|=\\\\left|T\\\\right|,\\\\sigma \\\\upharpoonright N=\\\\operatorname{id}\\\\right\\\\rbrace $ .\",\n \"If $\\\\sigma $ is Lascar strong, then $\\\\sigma \\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=\\\\operatorname{id}$ so $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is contained in $G^{0}$ .\",\n \"However, there are examples (even $\\\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ , see [16], [31].\",\n \"The Lascar group of $M$ is the quotient $\\\\operatorname{Aut}\\\\left(M\\\\right)/\\\\operatorname{Aut}f\\\\left(M\\\\right)$ .\",\n \"For more on the Lascar group, see [39].\",\n \"In the $\\\\omega $ -categorical case, the quotient $\\\\operatorname{Aut}\\\\left(\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ is also called the compact Lascar group.\",\n \"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $nsM$.\",\n \"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.\",\n \"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.\",\n \"Forinstance, by Lemma \\\\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotients then it has finite topological rank.\",\n \"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.\",\n \"Thus, potentially, the Lascar group — as a quotient of $\\\\operatorname{Aut}\\\\left(M\\\\right)$ — can be an obstruction to having finite topological rank.\",\n \"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.\",\n \"Conjecture REF (together with Theorem REF ) implies that it could not.\",\n \"Conjecture 7.2 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion with trivial Lascar group.\",\n \"By the above, this second conjecture is implied by Conjecture REF .\",\n \"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.\",\n \"It would be interesting to investigate other consequences of having a CIR.\",\n \"For instance a CIR might have something to say about normal subgroups.\",\n \"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $A$ for some CIR $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\\\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\\\aleph _{0}}$ is open.\",\n \"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.\",\n \"We refer to [26] for a survey on this.\",\n \"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.\",\n \"In fact the compact quotients are hidden in the stabilizer of a finite set.\",\n \"It seems that one can avoid this counterexample by restricting to dense subgroups.\",\n \"This leads us to the following questions.\",\n \"Question 7.3 Let $M$ be $\\\\omega $ -categorical such that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotient.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open (and hence is equal to $G$ )?\",\n \"Note that the assumption of having no compact quotient is necessary.\",\n \"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has a dense subgroup of index 2, see [23].\",\n \"Question 7.4 Let $M$ be $\\\\omega $ -categorical and $N$ an $\\\\omega $ -categorical expansion of $M$ .\",\n \"Set $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ and $H=\\\\operatorname{Aut}\\\\left(N\\\\right)\\\\le G$ .\",\n \"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open?\",\n \"Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.\",\n \"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.\",\n \"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].\",\n \"We would also like to thank Daoud Siniora for his comments.\",\n \"Finally, we would like to thank the anonymous referee for his comments.\"\n ],\n [\n \"Having finite topological rank \",\n \"In this section we will find some criteria that ensure that $G$ has finite topological rank.\"\n ],\n [\n \"$\\\\omega $ -categorical stable theories \",\n \"Proposition 5.1 If $T$ is stable $\\\\omega $ -categorical, $M\\\\models T$ is countable and $\\\\operatorname{Aut}\\\\left(\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ is finite, then $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has finite topological rank.\",\n \"Without loss of generality, $M=M^{\\\\operatorname{eq}}$ (if $S\\\\subseteq \\\\operatorname{Aut}\\\\left(M^{\\\\operatorname{eq}}\\\\right)$ generates a dense subgroup, then $S\\\\upharpoonright M=\\\\left\\\\lbrace f\\\\upharpoonright M\\\\,|\\\\,f\\\\in S\\\\right\\\\rbrace $ generates a dense subgroup of $\\\\operatorname{Aut}\\\\left(M\\\\right)$ ).\",\n \"Let $N=M_{\\\\operatorname{acl}\\\\left(\\\\emptyset \\\\right)}$ (i.e., name the elements in $\\\\operatorname{acl}\\\\left(\\\\emptyset \\\\right)$ ).\",\n \"Then $N$ is $\\\\omega $ -categorical by Propsition REF .\",\n \"Then in $N$ , $\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=\\\\operatorname{dcl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)$ , so by Example REF , there is a canonical independence relation in $N$ , so $G^{0}=\\\\operatorname{Aut}\\\\left(N\\\\right)$ is topologically 2-generated by Corollary REF , say by $\\\\left\\\\lbrace f_{1},f_{2}\\\\right\\\\rbrace $ .\",\n \"Now, $\\\\operatorname{Aut}\\\\left(M\\\\right)/G^{0}$ is finite by assumption, so let $S\\\\subseteq \\\\operatorname{Aut}\\\\left(M\\\\right)$ be a finite set of representatives.\",\n \"Then $S\\\\cup \\\\left\\\\lbrace f_{1},f_{2}\\\\right\\\\rbrace $ generates a dense subgroup $\\\\operatorname{Aut}\\\\left(M\\\\right)$ : given two finite tuples $\\\\bar{a},\\\\bar{b}$ from $M$ such that $\\\\bar{a}\\\\equiv \\\\bar{b}$ , there is an automorphism $\\\\sigma \\\\in \\\\operatorname{Aut}\\\\left(M\\\\right)$ such that $\\\\sigma \\\\left(\\\\bar{a}\\\\right)=\\\\bar{b}$ .\",\n \"Also, there is some $f\\\\in S$ such that $f^{-1}\\\\sigma \\\\in \\\\operatorname{Aut}\\\\left(N\\\\right)$ .\",\n \"Hence for some $g$ in the group generated by $\\\\left\\\\lbrace f_{1},f_{2}\\\\right\\\\rbrace $ , $g\\\\left(\\\\bar{a}\\\\right)=f^{-1}\\\\sigma \\\\left(\\\\bar{a}\\\\right)=f^{-1}\\\\left(\\\\bar{b}\\\\right)$ , so $fg\\\\left(\\\\bar{a}\\\\right)=\\\\bar{b}$ .\",\n \"The following fact implies immediately the next result.\",\n \"Fact 5.2 [9] If $T$ is $\\\\omega $ -categorical and $\\\\omega $ -stable and $M\\\\models T$ is countable, then $\\\\operatorname{Aut}\\\\left(\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ is finite.\",\n \"Corollary 5.3 If $T$ is $\\\\omega $ -stable and $\\\\omega $ -categorical and $M\\\\models T$ is countable, then $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has finite topological rank.\"\n ],\n [\n \"Reducing finite topological rank to expansions\",\n \"Suppose that $M$ is countable let $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ .\",\n \"We now want to explore the idea that perhaps by expanding $M$ (i.e., moving to a subgroup), we can show that the topological rank of $G$ is small by showing that the rank of the automorphism group of the expansion is.\",\n \"Suppose that $H\\\\le G$ .\",\n \"If $\\\\left(G,H\\\\right)$ has a compact quotient (see Definition REF ), then we cannot hope to deduce anything.\",\n \"For example, by Proposition REF we have that $G^{0}$ acts oligomorphically on $M$ and it can be that $G^{0}$ has a cyclically dense conjugacy class (so topological rank 2) while $G/G^{0}=\\\\left(\\\\mathbb {Z}/2\\\\mathbb {Z}\\\\right)^{\\\\omega }$ (so $G$ is not topologically finitely generated) — this happens in the example described in in Remark REF , see Example REF .\",\n \"Indeed, we will see that $\\\\left(G,H\\\\right)$ having a compact quotient is the only obstruction.\"\n ],\n [\n \"$\\\\omega $ -categorical structures with finitely many reducts\",\n \"Theorem 5.4 Suppose that $H\\\\le G$ is closed and that $\\\\left(G,H\\\\right)$ has no compact quotients.\",\n \"If there are only finitely many closed groups between $G$ and $H$ then there is some $g\\\\in G$ such that $H\\\\cup \\\\left\\\\lbrace g\\\\right\\\\rbrace $ topologically generate $G$ .\",\n \"Remark 5.5 The condition of having finitely many closed groups in the theorem holds when for instance $M$ is a reduct of an $\\\\omega $ -categorical structure $M^{\\\\prime }$ where $H=\\\\operatorname{Aut}\\\\left(M^{\\\\prime }\\\\right)$ , and $M^{\\\\prime }$ has only finitely many reducts up to bi-definability.\",\n \"Let $\\\\left\\\\lbrace H_{i}\\\\,|\\\\,i0$ .\",\n \"As $\\\\sigma _{*}^{k_{n,i}}\\\\left(c_{*,i}^{\\\\prime \\\\prime }\\\\right)=c_{*}$ , it follows that $d\\\\left(\\\\sigma _{*}^{k_{n,i}}\\\\left(x_{0}\\\\right),z_{j_{i}}\\\\right)<\\\\varepsilon /2$ .\",\n \"Similarly, as $\\\\sigma _{*}^{k_{n,0}}\\\\left(c_{*,i}^{\\\\prime }\\\\right)=g_{i}^{-1}\\\\left(c_{*}\\\\right)$ , we have that $d\\\\left(g_{i}\\\\left(\\\\sigma _{*}^{k_{n,0}}\\\\left(x_{0}\\\\right)\\\\right),z_{j_{i}}\\\\right)<\\\\varepsilon /2$ .\",\n \"Together, we are done.\",\n \"Now we have to take care of the other half of $\\\\star $ .\",\n \"This is done similarly, using right existence and the second part of Claim REF .\",\n \"The following proposition explains why we needed to take a conjugate of $\\\\sigma $ .\",\n \"The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.\",\n \"In Section REF , we mentioned that it is a Ramsey structure.\",\n \"Note that the underlying order is dense (by Proposition REF ).\",\n \"Proposition 6.9 Let $M=\\\\left(V,<,R\\\\right)$ be the countable ordered random graph.\",\n \"Then there is no automorphism $\\\\sigma \\\\in G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\\\in X$ .\",\n \"First we find $a\\\\ne b$ in $M$ such that $\\\\sigma ^{n}\\\\left(a\\\\right)\\\\ne \\\\sigma ^{m}\\\\left(b\\\\right)$ for all $m,n\\\\in \\\\mathbb {Z}$ .\",\n \"To do that, take any $a\\\\in M$ .\",\n \"Then $\\\\left\\\\lbrace \\\\sigma ^{n}\\\\left(a\\\\right)\\\\,|\\\\,n\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace $ is discrete (in the order sense: it is either a $\\\\mathbb {Z}$ -chain or just $a$ ).\",\n \"Since $\\\\left(V,<\\\\right)$ is dense, there is some $b\\\\ne \\\\sigma ^{n}\\\\left(a\\\\right)$ for all $n\\\\in \\\\mathbb {Z}$ .\",\n \"It follows that $b$ is as required.\",\n \"Let $X=S_{x}\\\\left(M\\\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).\",\n \"Let $p\\\\in X$ be any completion of the partial type $\\\\left\\\\lbrace R\\\\left(x,\\\\sigma ^{n}\\\\left(a\\\\right)\\\\right)\\\\,|\\\\,n\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace \\\\lnot R\\\\left(x,\\\\sigma ^{m}\\\\left(b\\\\right)\\\\right)\\\\,|\\\\,m\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace $ .\",\n \"Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\\\in \\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma ^{n}\\\\left(p\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ which is a fixed point of $G$ .\",\n \"In other words, $p_{0}$ is an invariant type over $M$ .\",\n \"However $R\\\\left(x,a\\\\right)\\\\wedge \\\\lnot R\\\\left(x,b\\\\right)\\\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).\",\n \"The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.\",\n \"If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\\\left(X,x_{0}\\\\right)$ .\",\n \"Thus, there would be a conjugate $\\\\sigma _{*}$ of $\\\\sigma $ such that $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow.\",\n \"But then if $\\\\left(Y,y_{0}\\\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\\\pi :X\\\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .\",\n \"Thus, $\\\\pi $ maps $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ to $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(y\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ , and the latter contains a $G$ -subflow.\",\n \"Thus we get that $\\\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .\",\n \"Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\\\left\\\\lbrace <,\\\\wedge \\\\right\\\\rbrace $ , and let $M\\\\models T$ be countable.\",\n \"Then $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no shifty automorphism.\",\n \"In particular, $M$ has no CIR.\",\n \"Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .\",\n \"Suppose that $\\\\sigma $ was shifty.\",\n \"Let $\\\\bar{m}=\\\\left\\\\langle m\\\\,|\\\\,m\\\\in M\\\\right\\\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\\\bar{x}=\\\\left\\\\langle x_{m}\\\\,|\\\\,m\\\\in M\\\\right\\\\rangle $ .\",\n \"Let $X=S_{\\\\bar{m}}\\\\left(M\\\\right)$ be the space of $\\\\bar{x}$ -complete types $p$ over $M$ such that $p\\\\upharpoonright \\\\emptyset =\\\\operatorname{tp}\\\\left(\\\\bar{m}/M\\\\right)$ .\",\n \"Then $X$ is a compact metric space.\",\n \"Let $x_{*}=\\\\operatorname{tp}\\\\left(\\\\bar{m}/M\\\\right)$ .\",\n \"By Theorem REF , there is some conjugate $\\\\tau $ of $\\\\sigma $ such that $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\tau ^{n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow $Y^{+}\\\\subseteq X$ and similarly, $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\tau ^{-n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow $Y^{-}$ .\",\n \"By Proposition REF , $\\\\tau $ fixes a branch or a point.\",\n \"Suppose that $\\\\tau \\\\left(m\\\\right)=m$ for some $m\\\\in M$ .\",\n \"Then for every $p\\\\in Y^{+}$ , $p\\\\models x_{m}=m$ .\",\n \"However $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ acts transitively on $M$ , so we have a contradiction.\",\n \"Now suppose that $\\\\tau $ fixes a branch $B\\\\subseteq M$ , but does not fix any point.\",\n \"Suppose that $\\\\tau \\\\left(m\\\\right)>m$ for some $m\\\\in B$ .\",\n \"Then $\\\\tau ^{n}\\\\left(m\\\\right)>m$ for all $n<\\\\omega $ , so for any $p\\\\in Y^{+}$ , $p\\\\models x_{m}>m$ .\",\n \"There is some $m^{\\\\prime }\\\\in M$ such that $m^{\\\\prime }>m$ and $m^{\\\\prime }\\\\notin B$ .\",\n \"Since $m<\\\\tau ^{n}\\\\left(m\\\\right)\\\\in B$ for all $n<\\\\omega $ , it follows that $p\\\\models x_{m}\\\\wedge m^{\\\\prime }=m$ for all $p\\\\in Y^{+}$ .\",\n \"Let $\\\\tau ^{\\\\prime }\\\\in G$ fix $m$ and map $m^{\\\\prime }$ to $B$ .\",\n \"Then $\\\\tau ^{\\\\prime }\\\\left(p\\\\right)\\\\models \\\\left(x_{m}\\\\wedge \\\\tau ^{\\\\prime }\\\\left(m^{\\\\prime }\\\\right)\\\\right)=mm$ , so we can apply the same argument to $Y^{-}$ .\",\n \"For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .\",\n \"In addition, letting $H=\\\\operatorname{Aut}\\\\left(N\\\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).\",\n \"In addition, if $B\\\\subseteq N$ is a branch, $m\\\\in B$ , there is always some $m^{\\\\prime }>m$ , $m^{\\\\prime }\\\\notin B$ and for any $n^{\\\\prime }>m$ in $B$ , $m^{\\\\prime }m\\\\equiv n^{\\\\prime }m$ .\",\n \"Hence, we can apply Proposition REF and the same proof will work.\",\n \"Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\\\omega $ -categorical structures.\",\n \"We state here a few general conjectures and questions.\",\n \"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.\",\n \"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.\",\n \"Conjecture 7.1 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion which admits a CIR.\",\n \"Suppose that $M$ is a structure and $\\\\mathfrak {C}$ a monster model for $\\\\operatorname{Th}\\\\left(M\\\\right)$ .\",\n \"The group of Lascar strong automorphisms of $M$, denoted by $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is the group of automorphisms of $M$ generated by the set $\\\\left\\\\lbrace \\\\sigma \\\\upharpoonright M\\\\,|\\\\,\\\\exists N\\\\prec \\\\mathfrak {C},\\\\left|N\\\\right|=\\\\left|T\\\\right|,\\\\sigma \\\\upharpoonright N=\\\\operatorname{id}\\\\right\\\\rbrace $ .\",\n \"If $\\\\sigma $ is Lascar strong, then $\\\\sigma \\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=\\\\operatorname{id}$ so $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is contained in $G^{0}$ .\",\n \"However, there are examples (even $\\\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ , see [16], [31].\",\n \"The Lascar group of $M$ is the quotient $\\\\operatorname{Aut}\\\\left(M\\\\right)/\\\\operatorname{Aut}f\\\\left(M\\\\right)$ .\",\n \"For more on the Lascar group, see [39].\",\n \"In the $\\\\omega $ -categorical case, the quotient $\\\\operatorname{Aut}\\\\left(\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ is also called the compact Lascar group.\",\n \"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $nsM$.\",\n \"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.\",\n \"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.\",\n \"Forinstance, by Lemma \\\\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotients then it has finite topological rank.\",\n \"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.\",\n \"Thus, potentially, the Lascar group — as a quotient of $\\\\operatorname{Aut}\\\\left(M\\\\right)$ — can be an obstruction to having finite topological rank.\",\n \"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.\",\n \"Conjecture REF (together with Theorem REF ) implies that it could not.\",\n \"Conjecture 7.2 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion with trivial Lascar group.\",\n \"By the above, this second conjecture is implied by Conjecture REF .\",\n \"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.\",\n \"It would be interesting to investigate other consequences of having a CIR.\",\n \"For instance a CIR might have something to say about normal subgroups.\",\n \"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $A$ for some CIR $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\\\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\\\aleph _{0}}$ is open.\",\n \"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.\",\n \"We refer to [26] for a survey on this.\",\n \"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.\",\n \"In fact the compact quotients are hidden in the stabilizer of a finite set.\",\n \"It seems that one can avoid this counterexample by restricting to dense subgroups.\",\n \"This leads us to the following questions.\",\n \"Question 7.3 Let $M$ be $\\\\omega $ -categorical such that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotient.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open (and hence is equal to $G$ )?\",\n \"Note that the assumption of having no compact quotient is necessary.\",\n \"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has a dense subgroup of index 2, see [23].\",\n \"Question 7.4 Let $M$ be $\\\\omega $ -categorical and $N$ an $\\\\omega $ -categorical expansion of $M$ .\",\n \"Set $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ and $H=\\\\operatorname{Aut}\\\\left(N\\\\right)\\\\le G$ .\",\n \"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open?\",\n \"Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.\",\n \"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.\",\n \"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].\",\n \"We would also like to thank Daoud Siniora for his comments.\",\n \"Finally, we would like to thank the anonymous referee for his comments.\",\n \"Definition 6.1 Suppose that $M$ is a countable structure.\",\n \"Call an automorphism $\\\\sigma \\\\in G$ shifty if there is some invariant binary relation on finite sets in $M$ , $\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ (the base will always be $$) such that:\\\\begin{itemize}\\\\item (Monotonicity) If A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}\\\\end{itemize}{\\\\textstyle \\\\textstyle x}\\\\hspace{0.0pt}\\\\hbox{t}o 0pt{\\\\hss \\\\textstyle \\\\mid \\\\hss } \\\\hss $$\\\\hss $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $B$ and $ A'A$, $ B'B$then $ A'$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $B'$.\\\\item (Right existence) For every finite tuple $ a$ there is some $ a'a$such that $ a$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $a'$ (by this we mean that sets enumerated by $ a$,$ a'$ are independent).\\\\item (Right shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ A$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $b'$, then there exists some $ n<$such that $ b'An(b)$.$ Lemma 6.2 If $\\\\sigma $ is shifty then it also satisfies: (Left existence) For every finite tuple $a$ there is some $a^{\\\\prime }\\\\equiv a$ such that $a^{\\\\prime }\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $a$.\\\\item (Left shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ b'$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $A$, then there exists some $ n<$such that $ b'A-n(b)$.$ Suppose that $\\\\sigma $ is shifty, as witnessed by $\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$.\",\n \"Given $ a$,there is some $ a'a$ such that $ a$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $a'$.\",\n \"Applying an automorphismtaking $ a'$ to $ a$ we get some $ a”a$ such that $ a”$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $a$,which shows left existence.$ As for left shiftiness, suppose that $A$ is finite and enumerated by $a$ , $b,b^{\\\\prime }$ are finite tuples such that $b^{\\\\prime }\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $A$ and $ bb'$.Then applying an automorphism, we get some $ a'$ such that $ ab'a'b$,so $ b$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $a'$.\",\n \"Hence for some $ n<$, $ a'bn(a)$.From $ a'bn(a)b$ we get that $ ab'a'b-n(a')-n(b)a-n(b)$,i.e., $ b'A-n(b)$.$ Proposition 6.3 The automorphism $\\\\sigma $ is a shifty automorphism on $M$ iff for any type $p\\\\in S\\\\left(\\\\emptyset \\\\right)$ (with finitely many variables), letting $Y_{a}=\\\\bigcap \\\\left\\\\lbrace \\\\bigcup \\\\left\\\\lbrace \\\\operatorname{tp}\\\\left(a,\\\\sigma ^{n}\\\\left(a^{\\\\prime }\\\\right)\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace \\\\,|\\\\,a^{\\\\prime }\\\\equiv a\\\\right\\\\rbrace $ for any $a\\\\models p$ , the intersection $Y_{p}=\\\\bigcap \\\\left\\\\lbrace Y_{a}\\\\,|\\\\,a\\\\models p\\\\right\\\\rbrace $ is nonempty.\",\n \"Suppose that $\\\\sigma $ is shifty, and fix some type $p\\\\in S\\\\left(\\\\emptyset \\\\right)$ .\",\n \"Let $a\\\\models p$ .\",\n \"By existence, there is some $a^{\\\\prime }\\\\equiv a$ with $a\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $a'$.\",\n \"Let $ q=tp(a,a')$ and fix some $ bp$.Let $ Aut(M)$ map $ a$ to $ b$ and let $ b'=(a')$.We have that $ b$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $b'$ and hence by right shiftiness, $ q=tp(b,b')Yb$.Since $ b$ was arbitrary, $ qYp$.$ Suppose that the right hand side holds.\",\n \"Given a finite tuple $a$ and $a^{\\\\prime }\\\\equiv a$ , write $a\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $*a'$ iff $ tp(a,a')Yp$where $ p=tp(a/)$.\",\n \"For general finite sets $ A,B$,write $ A$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $B$ iff there is some $ C$ containing $ A$ and $ C'$containing $ B$ such that $ C$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $*C'$.\",\n \"Obviously, $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ is invariantand monotone.\",\n \"Right existence follows from the assumption that $ Yp$for all $ pS()$.\",\n \"Right shiftiness: supposethat $ a$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $*a'$ and $ a”a'$.\",\n \"Then $ tp(a,a')Yp$and in particular it belongs to $ Ya$.\",\n \"By definition of $ Ya$,$ tp(a,a'){ tp(a,n(a”)) | n<} $,so for some $ n<$, $ aa'ain(a”)$.$ Proposition 6.4 If $M$ is an ultrahomogeneous structure and $\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ is a CIR on finite subsets of $ M$ which respects substructures,then there exists a shifty automorphism $$ on $ M$, as witnessedby $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$.$ Monotonicity and right existence are parts of the properties of a CIR, so we only have to prove right shiftiness.\",\n \"Suppose that $A\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $b$and $ b'b$.\",\n \"By the proof of Theorem \\\\ref {thm:existence of repulsive automorphism-ultrahomogeneous},the repulsive automorphism $$ constructed there satisfies thatfor some $ n<$, $ A$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $n(b')$.\",\n \"By stationarity,$ bA(b')$.$ Recall the definitions of flow and subflow from Section REF .\",\n \"Theorem 6.5 Let $M$ be a countable homogeneous structure and $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ .\",\n \"Suppose that $\\\\sigma \\\\in G$ is a shifty automorphism and that $\\\\left(X,d\\\\right)$ is a compact metric $G$ -flow.\",\n \"Then for every $x_{*}\\\\in X$ there is some conjugate $\\\\sigma _{*}\\\\in G$ of $\\\\sigma $ such that: (*) Both $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ and $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{-n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contain a subflow of $X$ .\",\n \"Remark 6.6 Note that Theorem REF implies that both $\\\\bigcap \\\\left\\\\lbrace \\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,k\\\\le n<\\\\omega \\\\right\\\\rbrace \\\\,|\\\\,k<\\\\omega \\\\right\\\\rbrace $ and $\\\\bigcap \\\\left\\\\lbrace \\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{-n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,k\\\\le n<\\\\omega \\\\right\\\\rbrace \\\\,|\\\\,k<\\\\omega \\\\right\\\\rbrace $ contain a subflow of $X$ : if e.g., $Y_{0}$ is a flow contained in the left space, then $GY_{0}=Y_{0}$ , so $\\\\sigma _{*}^{-k}\\\\left(Y_{0}\\\\right)\\\\subseteq Y_{0}\\\\subseteq \\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ , hence $Y_{0}\\\\subseteq \\\\sigma _{*}^{k}\\\\left(\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace \\\\right)=\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,k\\\\le n<\\\\omega \\\\right\\\\rbrace $ .\",\n \"Before the proof we note the following useful lemma.\",\n \"Lemma 6.7 Suppose that $G$ is a topological group acting continuously on a compact metric space $\\\\left(X,d\\\\right)$ .\",\n \"Then for every $0<\\\\varepsilon $ there is some open neighborhood $U$ of $\\\\operatorname{id}\\\\in G$ such that for every $g,h\\\\in G$ if $gh^{-1}\\\\in U$ then for all $x\\\\in X$ we have that $d\\\\left(gx,hx\\\\right)<\\\\varepsilon $ .\",\n \"It is enough to show that there is some open neighborhood $U$ of $\\\\operatorname{id}$ such that if $g\\\\in U$ then for all $x\\\\in X$ , $d\\\\left(gx,x\\\\right)<\\\\varepsilon $ (since then if $gh^{-1}\\\\in U$ then $d\\\\left(gh^{-1}\\\\left(hx\\\\right),hx\\\\right)<\\\\varepsilon $ ).\",\n \"For every $x\\\\in X$ , there is some neighborhood $V_{x}$ of $x$ in $X$ and some neighborhood $U_{x}$ of $\\\\operatorname{id}$ in $G$ such that for all $g\\\\in U_{x}$ , $x^{\\\\prime }\\\\in V_{x}$ , $d\\\\left(gx^{\\\\prime },x^{\\\\prime }\\\\right)<\\\\varepsilon $ .\",\n \"By compactness, a finite union of $V_{x}$ 's covers $X$ .\",\n \"Let $U$ be the intersection of the corresponding $U_{x}$ 's.\",\n \"Suppose that $\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ witnesses that $$ is shifty.\",\n \"Let $ G0$be a countable dense subset of $ G$, enumerated as $ gi | i<$,such that $ g0=id$.$ We construct an automorphism $\\\\tau :M\\\\rightarrow M$ by back and forth such that eventually $\\\\sigma _{*}=\\\\tau ^{-1}\\\\sigma \\\\tau $ and such that at each finite stage, $\\\\tau $ will be an elementary map.\",\n \"For the construction it is actually better to think of the domain and range of $\\\\tau $ as two different structures, so we have $M=M_{*}$ and suppose that $\\\\sigma :M\\\\rightarrow M$ , $\\\\sigma _{*}:M_{*}\\\\rightarrow M_{*}$ and $\\\\tau :M_{*}\\\\rightarrow M$ .\",\n \"The subscript $*$ will denote tuples from $M_{*}$ throughout.\",\n \"Suppose that we have constructed a partial elementary map $f:A_{*}\\\\rightarrow A$ (that will be part of $\\\\tau $ eventually) with $A_{*}\\\\subseteq M_{*},A\\\\subseteq M$ finite, enumerated by $a_{*},a$ .\",\n \"Here is the main tool in the construction.\",\n \"Claim 6.8 Suppose that $b_{*}^{\\\\prime }\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $a*$ and$ b*'ba$.\",\n \"Let $ b*=f-1(b)$.\",\n \"Thenthere is $ k<$ and an extension $ f'$ of $ f$ such that anyautomorphism $ '$ extending $ f'$ will satisfy that for $ *'='-1'$,$ *'k(b'*)=b*$.$ Similarly, if $a_{*}\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $b*'$ then there is some $ k<$ andan extension $ f'$ of $ f$ such that any automorphism $ '$ extending$ f'$ will satisfy that for $ *'='-1'$,$ *'k(b*)=b'*$.$ First, find some tuple $b^{\\\\prime }$ in $M$ such that $b^{\\\\prime }a\\\\equiv b_{*}^{\\\\prime }a_{*}$ .\",\n \"In particular, $b^{\\\\prime }\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $a$.\",\n \"By left shiftiness, there is some $ k<$such that $ -k(b)ab'ab*'a*$.Extend $ f$ to $ f'$ which sends $ b*'$ to $ -k(b)$.Then, for any $ '$ extending $ f'$, $ '-1k'(b*')='-1(b)=b*$.$ The second statement is proved similarly, using right shiftiness.\",\n \"We will make sure that for each $n<\\\\omega $ , the following condition holds.\",\n \"$\\\\star $ There are $k_{n,0},\\\\ldots ,k_{n,n-1}<\\\\omega $ such that for all $i0$ .\",\n \"As $\\\\sigma _{*}^{k_{n,i}}\\\\left(c_{*,i}^{\\\\prime \\\\prime }\\\\right)=c_{*}$ , it follows that $d\\\\left(\\\\sigma _{*}^{k_{n,i}}\\\\left(x_{0}\\\\right),z_{j_{i}}\\\\right)<\\\\varepsilon /2$ .\",\n \"Similarly, as $\\\\sigma _{*}^{k_{n,0}}\\\\left(c_{*,i}^{\\\\prime }\\\\right)=g_{i}^{-1}\\\\left(c_{*}\\\\right)$ , we have that $d\\\\left(g_{i}\\\\left(\\\\sigma _{*}^{k_{n,0}}\\\\left(x_{0}\\\\right)\\\\right),z_{j_{i}}\\\\right)<\\\\varepsilon /2$ .\",\n \"Together, we are done.\",\n \"Now we have to take care of the other half of $\\\\star $ .\",\n \"This is done similarly, using right existence and the second part of Claim REF .\",\n \"The following proposition explains why we needed to take a conjugate of $\\\\sigma $ .\",\n \"The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.\",\n \"In Section REF , we mentioned that it is a Ramsey structure.\",\n \"Note that the underlying order is dense (by Proposition REF ).\",\n \"Proposition 6.9 Let $M=\\\\left(V,<,R\\\\right)$ be the countable ordered random graph.\",\n \"Then there is no automorphism $\\\\sigma \\\\in G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\\\in X$ .\",\n \"First we find $a\\\\ne b$ in $M$ such that $\\\\sigma ^{n}\\\\left(a\\\\right)\\\\ne \\\\sigma ^{m}\\\\left(b\\\\right)$ for all $m,n\\\\in \\\\mathbb {Z}$ .\",\n \"To do that, take any $a\\\\in M$ .\",\n \"Then $\\\\left\\\\lbrace \\\\sigma ^{n}\\\\left(a\\\\right)\\\\,|\\\\,n\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace $ is discrete (in the order sense: it is either a $\\\\mathbb {Z}$ -chain or just $a$ ).\",\n \"Since $\\\\left(V,<\\\\right)$ is dense, there is some $b\\\\ne \\\\sigma ^{n}\\\\left(a\\\\right)$ for all $n\\\\in \\\\mathbb {Z}$ .\",\n \"It follows that $b$ is as required.\",\n \"Let $X=S_{x}\\\\left(M\\\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).\",\n \"Let $p\\\\in X$ be any completion of the partial type $\\\\left\\\\lbrace R\\\\left(x,\\\\sigma ^{n}\\\\left(a\\\\right)\\\\right)\\\\,|\\\\,n\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace \\\\lnot R\\\\left(x,\\\\sigma ^{m}\\\\left(b\\\\right)\\\\right)\\\\,|\\\\,m\\\\in \\\\mathbb {Z}\\\\right\\\\rbrace $ .\",\n \"Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\\\in \\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma ^{n}\\\\left(p\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ which is a fixed point of $G$ .\",\n \"In other words, $p_{0}$ is an invariant type over $M$ .\",\n \"However $R\\\\left(x,a\\\\right)\\\\wedge \\\\lnot R\\\\left(x,b\\\\right)\\\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).\",\n \"The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.\",\n \"If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\\\left(X,x_{0}\\\\right)$ .\",\n \"Thus, there would be a conjugate $\\\\sigma _{*}$ of $\\\\sigma $ such that $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow.\",\n \"But then if $\\\\left(Y,y_{0}\\\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\\\pi :X\\\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .\",\n \"Thus, $\\\\pi $ maps $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(x_{0}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ to $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\sigma _{*}^{n}\\\\left(y\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ , and the latter contains a $G$ -subflow.\",\n \"Thus we get that $\\\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .\",\n \"Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\\\left\\\\lbrace <,\\\\wedge \\\\right\\\\rbrace $ , and let $M\\\\models T$ be countable.\",\n \"Then $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no shifty automorphism.\",\n \"In particular, $M$ has no CIR.\",\n \"Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .\",\n \"Suppose that $\\\\sigma $ was shifty.\",\n \"Let $\\\\bar{m}=\\\\left\\\\langle m\\\\,|\\\\,m\\\\in M\\\\right\\\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\\\bar{x}=\\\\left\\\\langle x_{m}\\\\,|\\\\,m\\\\in M\\\\right\\\\rangle $ .\",\n \"Let $X=S_{\\\\bar{m}}\\\\left(M\\\\right)$ be the space of $\\\\bar{x}$ -complete types $p$ over $M$ such that $p\\\\upharpoonright \\\\emptyset =\\\\operatorname{tp}\\\\left(\\\\bar{m}/M\\\\right)$ .\",\n \"Then $X$ is a compact metric space.\",\n \"Let $x_{*}=\\\\operatorname{tp}\\\\left(\\\\bar{m}/M\\\\right)$ .\",\n \"By Theorem REF , there is some conjugate $\\\\tau $ of $\\\\sigma $ such that $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\tau ^{n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow $Y^{+}\\\\subseteq X$ and similarly, $\\\\operatorname{cl}\\\\left\\\\lbrace \\\\tau ^{-n}\\\\left(x_{*}\\\\right)\\\\,|\\\\,n<\\\\omega \\\\right\\\\rbrace $ contains a subflow $Y^{-}$ .\",\n \"By Proposition REF , $\\\\tau $ fixes a branch or a point.\",\n \"Suppose that $\\\\tau \\\\left(m\\\\right)=m$ for some $m\\\\in M$ .\",\n \"Then for every $p\\\\in Y^{+}$ , $p\\\\models x_{m}=m$ .\",\n \"However $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ acts transitively on $M$ , so we have a contradiction.\",\n \"Now suppose that $\\\\tau $ fixes a branch $B\\\\subseteq M$ , but does not fix any point.\",\n \"Suppose that $\\\\tau \\\\left(m\\\\right)>m$ for some $m\\\\in B$ .\",\n \"Then $\\\\tau ^{n}\\\\left(m\\\\right)>m$ for all $n<\\\\omega $ , so for any $p\\\\in Y^{+}$ , $p\\\\models x_{m}>m$ .\",\n \"There is some $m^{\\\\prime }\\\\in M$ such that $m^{\\\\prime }>m$ and $m^{\\\\prime }\\\\notin B$ .\",\n \"Since $m<\\\\tau ^{n}\\\\left(m\\\\right)\\\\in B$ for all $n<\\\\omega $ , it follows that $p\\\\models x_{m}\\\\wedge m^{\\\\prime }=m$ for all $p\\\\in Y^{+}$ .\",\n \"Let $\\\\tau ^{\\\\prime }\\\\in G$ fix $m$ and map $m^{\\\\prime }$ to $B$ .\",\n \"Then $\\\\tau ^{\\\\prime }\\\\left(p\\\\right)\\\\models \\\\left(x_{m}\\\\wedge \\\\tau ^{\\\\prime }\\\\left(m^{\\\\prime }\\\\right)\\\\right)=mm$ , so we can apply the same argument to $Y^{-}$ .\",\n \"For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .\",\n \"In addition, letting $H=\\\\operatorname{Aut}\\\\left(N\\\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).\",\n \"In addition, if $B\\\\subseteq N$ is a branch, $m\\\\in B$ , there is always some $m^{\\\\prime }>m$ , $m^{\\\\prime }\\\\notin B$ and for any $n^{\\\\prime }>m$ in $B$ , $m^{\\\\prime }m\\\\equiv n^{\\\\prime }m$ .\",\n \"Hence, we can apply Proposition REF and the same proof will work.\",\n \"Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\\\omega $ -categorical structures.\",\n \"We state here a few general conjectures and questions.\",\n \"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.\",\n \"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.\",\n \"Conjecture 7.1 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion which admits a CIR.\",\n \"Suppose that $M$ is a structure and $\\\\mathfrak {C}$ a monster model for $\\\\operatorname{Th}\\\\left(M\\\\right)$ .\",\n \"The group of Lascar strong automorphisms of $M$, denoted by $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is the group of automorphisms of $M$ generated by the set $\\\\left\\\\lbrace \\\\sigma \\\\upharpoonright M\\\\,|\\\\,\\\\exists N\\\\prec \\\\mathfrak {C},\\\\left|N\\\\right|=\\\\left|T\\\\right|,\\\\sigma \\\\upharpoonright N=\\\\operatorname{id}\\\\right\\\\rbrace $ .\",\n \"If $\\\\sigma $ is Lascar strong, then $\\\\sigma \\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=\\\\operatorname{id}$ so $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is contained in $G^{0}$ .\",\n \"However, there are examples (even $\\\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ , see [16], [31].\",\n \"The Lascar group of $M$ is the quotient $\\\\operatorname{Aut}\\\\left(M\\\\right)/\\\\operatorname{Aut}f\\\\left(M\\\\right)$ .\",\n \"For more on the Lascar group, see [39].\",\n \"In the $\\\\omega $ -categorical case, the quotient $\\\\operatorname{Aut}\\\\left(\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ is also called the compact Lascar group.\",\n \"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $nsM$.\",\n \"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.\",\n \"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.\",\n \"Forinstance, by Lemma \\\\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotients then it has finite topological rank.\",\n \"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.\",\n \"Thus, potentially, the Lascar group — as a quotient of $\\\\operatorname{Aut}\\\\left(M\\\\right)$ — can be an obstruction to having finite topological rank.\",\n \"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.\",\n \"Conjecture REF (together with Theorem REF ) implies that it could not.\",\n \"Conjecture 7.2 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion with trivial Lascar group.\",\n \"By the above, this second conjecture is implied by Conjecture REF .\",\n \"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.\",\n \"It would be interesting to investigate other consequences of having a CIR.\",\n \"For instance a CIR might have something to say about normal subgroups.\",\n \"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $A$ for some CIR $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\\\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\\\aleph _{0}}$ is open.\",\n \"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.\",\n \"We refer to [26] for a survey on this.\",\n \"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.\",\n \"In fact the compact quotients are hidden in the stabilizer of a finite set.\",\n \"It seems that one can avoid this counterexample by restricting to dense subgroups.\",\n \"This leads us to the following questions.\",\n \"Question 7.3 Let $M$ be $\\\\omega $ -categorical such that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotient.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open (and hence is equal to $G$ )?\",\n \"Note that the assumption of having no compact quotient is necessary.\",\n \"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has a dense subgroup of index 2, see [23].\",\n \"Question 7.4 Let $M$ be $\\\\omega $ -categorical and $N$ an $\\\\omega $ -categorical expansion of $M$ .\",\n \"Set $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ and $H=\\\\operatorname{Aut}\\\\left(N\\\\right)\\\\le G$ .\",\n \"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open?\",\n \"Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.\",\n \"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.\",\n \"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].\",\n \"We would also like to thank Daoud Siniora for his comments.\",\n \"Finally, we would like to thank the anonymous referee for his comments.\"\n ],\n [\n \"Further questions\",\n \"The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\\\omega $ -categorical structures.\",\n \"We state here a few general conjectures and questions.\",\n \"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.\",\n \"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.\",\n \"Conjecture 7.1 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion which admits a CIR.\",\n \"Suppose that $M$ is a structure and $\\\\mathfrak {C}$ a monster model for $\\\\operatorname{Th}\\\\left(M\\\\right)$ .\",\n \"The group of Lascar strong automorphisms of $M$, denoted by $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is the group of automorphisms of $M$ generated by the set $\\\\left\\\\lbrace \\\\sigma \\\\upharpoonright M\\\\,|\\\\,\\\\exists N\\\\prec \\\\mathfrak {C},\\\\left|N\\\\right|=\\\\left|T\\\\right|,\\\\sigma \\\\upharpoonright N=\\\\operatorname{id}\\\\right\\\\rbrace $ .\",\n \"If $\\\\sigma $ is Lascar strong, then $\\\\sigma \\\\upharpoonright \\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)=\\\\operatorname{id}$ so $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ is contained in $G^{0}$ .\",\n \"However, there are examples (even $\\\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\\\operatorname{Aut}f\\\\left(M\\\\right)$ , see [16], [31].\",\n \"The Lascar group of $M$ is the quotient $\\\\operatorname{Aut}\\\\left(M\\\\right)/\\\\operatorname{Aut}f\\\\left(M\\\\right)$ .\",\n \"For more on the Lascar group, see [39].\",\n \"In the $\\\\omega $ -categorical case, the quotient $\\\\operatorname{Aut}\\\\left(\\\\operatorname{acl}^{\\\\operatorname{eq}}\\\\left(\\\\emptyset \\\\right)\\\\right)$ is also called the compact Lascar group.\",\n \"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $nsM$.\",\n \"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.\",\n \"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.\",\n \"Forinstance, by Lemma \\\\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotients then it has finite topological rank.\",\n \"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.\",\n \"Thus, potentially, the Lascar group — as a quotient of $\\\\operatorname{Aut}\\\\left(M\\\\right)$ — can be an obstruction to having finite topological rank.\",\n \"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.\",\n \"Conjecture REF (together with Theorem REF ) implies that it could not.\",\n \"Conjecture 7.2 Any $\\\\omega $ -categorical structure has an $\\\\omega $ -categorical expansion with trivial Lascar group.\",\n \"By the above, this second conjecture is implied by Conjecture REF .\",\n \"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.\",\n \"It would be interesting to investigate other consequences of having a CIR.\",\n \"For instance a CIR might have something to say about normal subgroups.\",\n \"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\\\mathop {\\\\mathchoice{\\\\displaystyle \\\\displaystyle x}{\\\\hspace{0.0pt}}{}{0}\\\\hbox{t}o 0pt{\\\\hss \\\\displaystyle \\\\mid \\\\hss } \\\\hss \\\\displaystyle \\\\smile \\\\hss }\\\\hspace{0.0pt}$$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $A$ for some CIR $$\\\\displaystyle \\\\mid $ $\\\\displaystyle \\\\smile $$\\\\textstyle \\\\mid $ $\\\\textstyle \\\\smile $$\\\\scriptstyle \\\\mid $ $\\\\scriptstyle \\\\smile $$\\\\scriptscriptstyle \\\\mid $ $\\\\scriptscriptstyle \\\\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\\\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\\\aleph _{0}}$ is open.\",\n \"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.\",\n \"We refer to [26] for a survey on this.\",\n \"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.\",\n \"In fact the compact quotients are hidden in the stabilizer of a finite set.\",\n \"It seems that one can avoid this counterexample by restricting to dense subgroups.\",\n \"This leads us to the following questions.\",\n \"Question 7.3 Let $M$ be $\\\\omega $ -categorical such that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has no compact quotient.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open (and hence is equal to $G$ )?\",\n \"Note that the assumption of having no compact quotient is necessary.\",\n \"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ has a dense subgroup of index 2, see [23].\",\n \"Question 7.4 Let $M$ be $\\\\omega $ -categorical and $N$ an $\\\\omega $ -categorical expansion of $M$ .\",\n \"Set $G=\\\\operatorname{Aut}\\\\left(M\\\\right)$ and $H=\\\\operatorname{Aut}\\\\left(N\\\\right)\\\\le G$ .\",\n \"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.\",\n \"Is it true that any dense subgroup of $G$ of index less than $2^{\\\\aleph _{0}}$ is open?\"\n ],\n [\n \"Acknowlegements\",\n \"Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.\",\n \"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.\",\n \"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].\",\n \"We would also like to thank Daoud Siniora for his comments.\",\n \"Finally, we would like to thank the anonymous referee for his comments.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01918"}}},{"rowIdx":889,"cells":{"text":{"kind":"list like","value":[["The ALMA Early Science View of FUor/EXor objects. IV. Misaligned\n Outflows in the Complex Star-forming Environment of V1647 Ori and McNeil's\n Nebula"],["Abstract We present Atacama Large Millimeter/sub-millimeter Array (ALMA) observations of the star-forming environment surrounding V1647 Ori, an outbursting FUor/EXor pre-MS star.","Dust continuum and the (J = 2 - 1) $^{12}$CO, $^{13}$CO, C$^{18}$O molecular emission lines were observed to characterize the V1647 Ori circumstellar disc and any large scale molecular features present.","We detect continuum emission from the circumstellar disc and determine a radius r = 40 au, inclination i = 17$^{\\circ}$$^{+6}_{-9}$ and total disc mass of M$_{\\mathrm{disk}}$ of ~0.1 M$_{\\odot}$.","We do not identify any disc structures associated with nearby companions, massive planets or fragmentation.","The molecular cloud environment surrounding V1647 Ori is both structured and complex.","We confirm the presence of an excavated cavity north of V1647 Ori and have identified dense material at the base of the optical reflection nebula (McNeil's Nebula) that is actively shaping its surrounding environment.","Two distinct outflows have been detected with dynamical ages of ~11,700 and 17,200 years.","These outflows are misaligned suggesting disc precession over ~5500 years as a result of anisotropic accretion events is responsible.","The collimated outflows exhibit velocities of ~2 km s$^{-1}$, similar in velocity to that of other FUor objects presented in this series but significantly slower than previous observations and model predictions.","The V1647 Ori system is seemingly connected by an \"arm\" of material to a large unresolved structure located ~20$\"$ to the west.","The complex environment surrounding V1647 Ori suggests it is in the early stages of star formation which may relate to its classification as both an FUor and EXor type object."],["Introduction"," Stars form as a result of gravitational contraction of rotating molecular clouds.","As the gravitational collapse proceeds, a system emerges composed of an accreting star, disc and envelope.","Observations of this stage of pre-main sequence (pre-MS) stellar evolution display large scale molecular outflows that act to transport energy back into the surrounding environment and are likely the main dispersing mechanism of molecular cloud material, responsible for the transition from a deeply embedded Class 0 protostar to the more evolved Class I/II pre-MS stellar systems.","An empirical relationship has been identified between the opening angle of these outflows and the pre-MS stage of stellar evolution .","The earliest stages of evolution (Class 0) exhibit outflows that are highly collimated and have opening angles of 20-50$^{\\circ }$ whereas the more evolved Class I and II stages are associated with less collimated opening angles of 80-120$^{\\circ }$ and 100-160$^{\\circ }$ , respectively .","The origin of these outflows are debated due to the au spatial scales required to resolve them however several theories relate the outflows to intense mass accretion events from the disc and/or envelope onto the central pre-MS star .","Collimated and wide-angle outflows appear to represent two distinct structures in the star-forming environment of their host stars.","In particular, wide-angle outflows tend to be more massive and slower (velocities ranging from $\\sim $ 10-30 km s$^{-1}$ ) compared to highly collimated outflows which can reach velocities of $\\sim $ 100-1000 km s$^{-1}$ .","Their distinct velocities and masses arise from the mechanisms responsible for their production; wide angle outflows are likely produced from the ambient molecular material getting swept up when interacting with a jet bow shock whereas highly collimated jets likely arise from the interaction between a strongly magnetized central star and the inner edge of its accretion disc during intense mass accretion events .","Recent observations indicate that extended disc winds may also drive protostellar outflows .","Mass accretion events where material gets loaded from the disc onto the protostar are essential to star formation as these events are the primary means by which stars gain their mass.","The standard model of star formation indicates that a molecular cloud gravitationally contracts and forms a star-disc system in about $\\sim $ 5 $\\times $ 10$^{5}$ years .","The mass accretion rate necessary to produce a star-disc system after this timescale is inconsistent with the age and the implied accretion rates derived from the observed accretion luminosity of a large sample of stars in Taurus .","One solution to this \"luminosity problem\" is short-lived major accretion events of significant mass.","In the past several decades, a phenomenon has been observed in a small subset of pre-MS stars that is consistent with brief and intense mass accretion events identified as \"outbursts\" , , .","These outbursting pre-MS stars have been named FUor/EXor objects based on their namesakes FU Ori and EX Lup.","This phenomenon is identifiable when pre-MS stars suddenly increase in brightness by several magnitudes at optical/near-IR wavelengths as a result of intense mass accretion outbursts where accretion rates can reach as high as 10$^{-4}$ – 10$^{-5}$ M$_{\\odot }$ yr$^{-1}$ .","FU Ori type (FUor) and EX Lup type (EXor) are generally distinguished by the frequency and length of their protostellar outbursts where FUor type objects tend to have longer (years to decades) timescale and EXors are characterized by more frequent and shorter outbursts (months-years).","While it is evident these outbursts are caused by large mass-accretion events, it is not clear what underlying mechanism is responsible.","Several theories have been proposed which include binary companions and/or massive planets generating disc instabilities , , disc fragmentation , and a combination of magneto rotational instability (MRI) and gravitational instabilities .","A detailed summary of these mechanisms and their relevant references can be found in .","Regardless of the underlying mechanism, it is clear that sudden intense outbursts can affect the circumstellar disc of protostars and thus, may impact planet formation , .","Recently, an analysis of high resolution ALMA observations of the outbursting FUor type object V883 Ori revealed the first detection of a water snow line in a circumstellar disc .","Such a discovery was feasible because the water snow line, which typically resides < 5 au from a $\\sim $ 1 M$_{\\odot }$ central star, was temporarily extended to $\\sim $ 40 au in response to the increase in luminosity of the FUor object during outburst.","V1647 Ori, a protostar in the Lynds 1630 (L1630) region of the Orion Molecular Cloud , was first noted in 2003 when a large ($\\sim $ 1$^{\\prime }$ ) reflection nebula (McNeil's Nebula) appeared coincident with a sudden increase in optical stellar brightness , , .","After its identification in 2003, two more outbursts were identified: a 1966 outburst identified using archival photometric plates and a 2008 outburst which increased the flux values of V1647 Ori to similar values as in 2003.","During the 2008 outburst, carried out a 4.5 year photometric study of V1647 Ori and report that at the end of their campaign in 2012, V1647 Ori was still in outburst after almost half a decade.","The frequency and length of these outbursts, combined with several spectral features characteristic of both FUors and EXors emphasize the ambiguity between the FUor or EXor classification of V1647 Ori .","Spectral parameters of V1647 Ori were determined from data taken during quiescence and presented in .","They identify V1647 Ori as a young (< 0.5 Myr) pre-MS star of spectral type M0 $\\pm $ 2 with an approximate bolometric luminosity, mass and radius of $\\sim $ 5.2 L$_{\\odot }$ , 0.8 M$_{\\odot }$ , and 5 R$_{\\odot }$ , respectively.","During its 2003 outburst, its bolometric luminosity was measured to be L$_\\mathrm {bol}$ = 44 L$_{\\odot }$ .","Given the transient nature of V1647 Ori, several multiwavelength campaigns were setup to observe the star in its outbursting state.","The 2003 outburst was characterized by a $\\sim $ 3 magnitude increase in the near-IR , a factor of 25 increase in 12 $\\mu $ m flux and a factor of 50 increase in X-ray flux .","report no apparent changes in the (350 $\\mu $ m-1.3mm) submm dust continuum brightness during outburst compared with its quiescent state.","Plasma temperatures derived from the X-ray imaging spectrum presented in were too high to be produced via accretion alone suggesting extreme changes in magnetic field configuration (i.e., magnetic reconnection events) were be present.","Spectroscopic measurements during the 2003 outburst indicated accretion rates of a few 10$^{-6}$ to 10$^{-5}$ M$_{\\odot }$ year $^{-1}$ and H$\\alpha $ P Cygni profiles indicate wind velocities up to $\\sim $ 600 km s$^{-1}$ .","Mid-IR interferometric observations probing the circumstellar disc indicate a moderately flaring disc with no signatures of close companions at radii < 100 au [1].","All three outbursts re-illuminated the reflection nebula, where dust grains preferentially scatter blue light, just north of V1647 Ori and several studies have identified small variations in the local environment as probed by this extended nebular emission , .","The time delay between brightness variations of the star and of clumps in the nebulosity were used to estimate a stellar inclination angle of $\\sim $ 61$^{\\circ }$ .","An X-ray periodicity study of V1647 Ori found a similar stellar inclination angle of i $\\sim $ 68$^{\\circ }$ based on modeling the rotation of accretion hot spots on the stellar surface .","These inclinations are in disagreement with studies that suggest a more face-on inclination of $<$ 30$^{\\circ }$ based on the absence of a 9.7 $\\mu $ m amorphous silicate feature and a low column density of CO in absorption from the disc .","Here, we present ALMA 1.3 mm observations of the gas and dust surrounding V1647 Ori in order to characterize its circumstellar disc and local star-forming environment.","V1647 Ori is one of 8 FUor/EXor type objects observed as part of an ALMA Cycle 2 project (2013.1.00710.S ; PI L. Cieza) to investigate star-forming environments at this critical stage of pre-MS stellar evolution.","All other FUor/EXor objects observed as part of this program, including V883 Ori, HBC 494, V2775 Ori, NY Ori, V1143 Ori, V1118 Ori, ASASSN-13db, are presented in papers I, II, III, IV (this paper) and V , , , .","These recently published results indicate that the star-forming environments of these objects are structured and complex.","In particular, features such as rings/shells and both wide-angle and collimated outflows have all been identified.","Outflow velocities measured for V883 Ori, HBC 494, V2775 Ori, and V1647 Ori (Section 3.6) as part of this series of papers indicate that significantly slower velocities (v$\\sim $ 1 - 4 km $s^{-1}$ ) are present than those typically observed and associated with protostellar outflows at these wavelengths .","The resolving power and sensitivity of the observations presented in this series also provides strict constraints on circumstellar disc characteristics such as disc size, orientation, and mass."],["Observations","V1647 Ori was observed with ALMA band-6 at three epochs (December 12th, 2014, April 5th, 2015 and August 30th, 2015).","The first two observations were performed with an ALMA configuration of 45 antennas (12 meter diameter) and baselines ranging from 14.6 to 348.5 meters which achieved an angular resolution of $\\sim $ 1.0 $^{\\prime \\prime }$ .","The final epoch had a configuration of 35 antennas and longer baselines of 42-1574 meters resulting in a higher angular resolution of 0.2 $^{\\prime \\prime }$ .","The precipitable water vapor (PWV) levels were 0.7, 1.3 and 1 mm for the December 2014, April 2015, and August 2015 observations, respectively.","Each ALMA observation of V1647 Ori include two broad (2 GHz) continuum spectral windows centered at $\\sim $ 232 GHz and $\\sim $ 218 GHz and three narrow (59 MHz) windows centered near the rest frequencies of $^{12}$ CO (J = 2-1; 230.5380 GHz), $^{13}$ CO (J = 2-1; 220.3987 GHz) and C180 (J = 2-1; 219.5603 GHz).","Flux calibration was performed with Ganymede and J0423-013 while bandpass calibration was performed with quasars J0538-4405 and J0607-0834.","The time dependence variations of the complex gains were calibrated by alternating V1647 Ori with nearby phase calibrators J0541-0541, J0532-0307 and/or J0529-0519.","The ALMA pipeline-calibrated observations were further reduced with the Common Astronomical Software Application .","The two epochs of $\\sim $ 1.0$^{\\prime \\prime }$ resolution data were continuum subtracted in the visibility domain, concatenated, and cleaned to form a single dataset for the continuum and each spectral line.","The same procedure was performed for the single epoch of $\\sim $ 0.2$^{\\prime \\prime }$ resolution data.","While features of the V1647 Ori disc/outflows are clearly present in the $\\sim $ 1$^{\\prime \\prime }$ resolution datasets, none of these features are identifiable in the high resolution line imaging from the 0.2$^{\\prime \\prime }$ resolution dataset.","This is consistent with most of the flux from the system being fairly uniform on large scales.","Therefore we only present the detection of continuum emission from the 0.2$^{\\prime \\prime }$ dataset and the line and continuum emission from the combined two $\\sim $ 1$^{\\prime \\prime }$ datasets.","The beam size, position angle, and 3$\\sigma $ sensitivities for each spectral line channel map and the continuum are indicated in Table REF .","Integrated intensity images (moment 0) were creating using the CASA routine $immoments$ .","The moment 0 images were created using only pixels with a signal detection of 3$\\sigma $ or higher.","To better describe the morphology and dynamics of the V1647 Ori star-forming environment, we created moment 0 images for each spectral line with three distinct velocity ranges: $5.0-9.49$ km s$^{-1}$ , $9.5-10.5$ km s$^{-1}$ and $10.51-13.0$ km s$^{-1}$ .","These velocity bands will henceforth be referred to in this paper as blueshifted, systemic, and redshifted, respectively.","The images presented in this work are not primary-beam corrected.","However, primary-beam corrected channel maps were used when estimating physical parameters of the outflow from the line fluxes (Section 3.6).","Optical Gemini-GMOS imaging observations of V1647 Ori were performed on September 22, 2008 with an exposure time of 60 seconds in R band.","Optical imaging data was retrieved from the Gemini Science Archive and presented here to supplement the ALMA observations.","Optical analysis of these and similar data are presented in other work and the discussion in this paper is limited only to correlations between large-scale optical and mm emission morphology.","Table: V1647 Ori ALMA Continuum and Integrated Intensity Image Parameters"],["Continuum Emission","The integrated intensity image of the resolved 0.2$^{\\prime \\prime }$ V1647 Ori continuum is displayed in Figure REF .","Two-dimensional gaussian fitting of the $\\sim $ 225 GHz continuum emission performed in CASA indicate a flux, inclination, position angle and centroid of 82.62 $\\pm $ 8.26 mJy, ${\\it i} = 17^{\\circ }$ $^{+6}_{-9}$ , PA$ = 109^{\\circ } \\pm $ 19, and RA= 05:46:13.139, Dec= -00:06:04.90, respectively.","The inclination was calculated using the aspect ratio of an assumed circular disc.","All position angles reported in this work are east of north.","A semi major axis FWHM angular size of 189.0 $\\pm $ 3.6 milliarcseconds and a distance of 414 $\\pm {7}$ pc results in a disc dust radius of $\\sim $ 40 au.","Evidence that the continuum emission from the circumstellar disc is spatially resolved is identified by its amplitude as a function of UV distance and asymmetry in the continuum emission (Figure REF ) indicates a potentially non-gaussian distribution near the outer edges of the image.","A more in-depth analysis of continuum emission from V1647 Ori as well as the other FUor/EXor objects in this series will be presented in .","The mass of the V1647 Ori circumstellar disc dust can be estimated from the continuum emission (assuming continuum emission is thermal and optically thin) following where: $\\hspace{86.72377pt}M_{\\rm {disc}}= \\frac{F_\\nu d^2}{\\kappa _\\nu B_\\nu (T)},$ with $B_\\nu (T)$ representing the Planck function for dust at a temperature $T$ and $\\kappa _\\nu $ as the dust opacity, which is a power-law function of submm frequency, i.e., $\\kappa _\\nu =0.1(\\nu / 1000$ $\\mathrm {GHz})^\\beta $ cm$^{2}$ g$^{-1}$ .","The power law opacity index $\\beta $ is related to the size distribution and composition of the disc dust particles.","While $\\beta $ can range from $\\beta $ =0 for certain grain mineralogies or size distributions to $\\beta $ =2 for interstellar dust grains in the ISM , it is more likely to be in the range of 0.5-1.5 for circumstellar discs .","Without more sophisticated continuum modeling to estimate the disc dust temperature of V1647 Ori, a dust temperature and range of $T=20 \\pm 10$ K is chosen based on the median temperature of discs in Taurus-Auriga and the relatively small temperature range indicated in modeling of other circumstellar discs .","Assuming values of $\\beta $ =1, d= 414 pc, $\\kappa _\\nu $ = 0.023 cm$^{2}$ g$^{-1}$ , and a dust temperature of $T=20$ K, we estimate the disc dust mass of V1647 Ori to be $M_\\mathrm {dust}$ = 427 M$_\\mathrm {E}$ .","This corresponds to a total disc mass $M_\\mathrm {disc}$ = 0.13 $M_{\\odot }$ ($\\sim $ 136 M$_\\mathrm {Jup}$ ) assuming a gas to dust mass ratio of 100.","When converting from flux to disc mass using Equation 1, the largest sources of uncertainty are the ranges in $\\beta $ and dust temperature.","While the uncertainty in the continuum flux is dominated by the 10% absolute flux uncertainty of ALMA Band 6, the potential range in $\\beta $ ($\\beta $ = 1 $\\pm {0.5}$ ) and dust temperature (T = 20 $\\pm {10}$ ) introduce much higher uncertainties in dust mass that can result in masses in the extreme ranges of M$_{\\rm {dust}}$ = [110 - 2688] M$_\\mathrm {E}$ or $M_{\\rm {disc}}$ = [0.03 - 0.80] $M_{\\odot }$ assuming a gas to dust ratio of 100.","A summary of V1647 Ori disc parameters are displayed in Table REF .","Table: V1647 Ori Disc Parameters Derived From Continuum EmissionFigure: Resolved continuum emission of the V1467 Ori circumstellar disc with contours overlaid of values 1.75, 3, 7, 20, 30, 45 and 60 ×\\times 3σ3\\sigma .","A two-dimensional gaussian fit indicates a disc with inclination of 17 ∘ ^{\\circ } -9 +6 ^{+6}_{-9} and PA = 109±19\\pm 19 ∘ ^{\\circ }.","The beam size is displayed in the bottom left and north is up and east is left."],["Gas Emission Lines","Spectral flux density as a function of radial velocity for the $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O lines are displayed in 0.25 km s$^{-1}$ bins in Figure REF .","Spectral extraction was performed using two different apertures both centered at the position of V1647 Ori (J2000 RA= 05:46:13.135, Dec= -00:06:04.82).","An aperture diameter of 2.0 $^{\\prime \\prime }$ was chosen to extract spectral information at the precise location of the continuum emission (Fig REF top) and an aperture diameter of 20$^{\\prime \\prime }$ was chosen to encompass the nebulosity surrounding V1647 Ori (Fig REF bottom).","In both the small and large apertures, a double peaked $^{12}$ CO and single peaked $^{13}$ CO emission feature is observed.","A clear detection of C$^{18}$ O material is seen in the large 20$^{\\prime \\prime }$ aperture indicating an abundance of dense material traveling at a radial velocity equidistant between the two $^{12}$ CO peaks.","Due to the embedded nature of this object, the rest velocity of the system (i.e., systemic velocity) is not clear.","In the absence of a literature value of the systemic velocity of V1647 Ori, we estimate a systemic velocity of $\\sim $ 10.0 km s$^{-1}$ assuming the star forms in the densest part of the molecular cloud (i.e.","at the same velocity as the bulk of the dense C$^{18}$ O emission; Figure REF ).","This systemic velocity is also consistent with being equidistant between the peaks of the blue and red shifted $^{12}$ CO emission lines.","The blueshifted, systemic, and redshifted velocity ranges are indicated at the top of Figure REF .","Individual channel maps for each spectral line where emission associated with the V1647 Ori environment is observed is shown in Appendix Figures 1-3.","The integrated intensity images of $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O emission over their entire velocity range are displayed in Figure REF with white circles corresponding to the extraction apertures associated with the line profile displayed in Figure REF .","Several features are readily identifiable and will be discussed in detail in the following sections.","An illustration of the three-dimensional system geometry is displayed in Figure REF .","Figure: Line profiles from apertures centered on the dust continuum emission of V1647 Ori with diameters of 2 '' ^{\\prime \\prime } (top) and 20 '' ^{\\prime \\prime } (bottom) and velocity bins of 0.25 km s -1 ^{-1}.","A systemic velocity of 10.0 km s -1 ^{-1} is indicated with a vertical dashed line.","The blue, green and red lines above the plot represent the velocity ranges described in Section 2 for the blueshifted, systemic, and redshifted images presented throughout the rest of this paper.","Individual line channel maps are included in the Appendix.Figure: Integrated intensity (moment 0) images of 12 ^{12}CO, 13 ^{13}CO and C 18 ^{18}O integrated over each lines entire velocity range.","V1647 Ori is at the center of each panel (offset [0,0] corresponds to RA= 05:46:13.135, Dec= -00:06:04.82).","The small solid and large dashed white lines indicate the spectral extraction regions from Figure (aperture diameters of 2 '' ^{\\prime \\prime } and 20 '' ^{\\prime \\prime }, respectively) with north is up and east is left.","The beam is displayed in the lower left.Figure: A three-dimensional illustration of V1647 Ori and its surrounding misaligned outflows (labels D and E discussed in Sections 3.3 and 3.4).","The nearly face-on circumstellar disc is shown in green.","Features in the illustration are not to scale."],["Emission from $^{12}$ CO","The $^{12}$ CO emission line is typically optically thick in molecular cloud environments and is generally not a good probe of column density.","Instead, $^{12}$ CO emission traces molecular material of a particular temperature and is useful for probing excavated cavities as well as wide-angle outflows , .","Figure REF displays the blue, systemic and red shifted $^{12}$ CO emission summed over the velocity ranges indicated in the top of Figure REF and in Section 2.","The base of McNeil's Nebula, the familiar reflection nebula first identified from optical observations during its 2003 outburst is clearly seen in the blue shifted $^{12}$ CO emission line (label A).","This northern cavity has a well-defined wall and an opening angle to the north of $\\sim $ 100$^{\\circ }$ which extends $\\sim $ 10$^{\\prime \\prime }$ ($\\sim $ 4140 au).","A bright extended region spatially coincident with the star-disc system is identifiable at radii of $r \\lesssim 3^{\\prime \\prime }$ (1245 au) from V1647 Ori.","A distinct clump of material is seen $\\sim $ 2$^{\\prime \\prime }$ south-west from the central emission (label B) appearing to be cut off from the main nebula.","Meanwhile, the redshifted morphology of the $^{12}$ CO emission line is distinct from the blueshifted emission.","Emission coincident with the position of the circumstellar disc of V1647 Ori is detected in the arc-like shape bending from the north-west to the south-west.","Such a well defined structure is seemingly connected with an 'arm' to an atypically large column of $^{12}$ CO emission located 20$^{\\prime \\prime }$ (8300 au) west of V1647 Ori near the edge of the ALMA field of view (label C).","The northern bowl-like nebula may be faintly detected behind the arc-like feature in the red-shifted $^{12}$ CO emission.","More prominently displayed is an apparent southern feature extending in a straight column south of the V1647 Ori disc with an angular length of $\\sim $ 20$^{\\prime \\prime }$ ($\\sim $ 8300 au; label D ).","Parts of both the large southern and western columns are identified in the blueshifted $^{12}$ CO emission (Figure REF ; label D).","The systemic $^{12}$ CO emission of V1647 Ori appears to be resolved out or undetected and instead displays features that may originate from foreground cloud contamination.","Figure: Side by side comparison of blue, systemic, and red shifted moment 0 images of 12 ^{12}CO.","The central location of the V1647 Ori disc continuum (Fig ) is shown with a black cross and the beam is displayed in the lower left.","Figures are labeled according to the text in Section 3.","All 12 ^{12}CO channels associated with emission can be seen in Figure 1 of the Appendix."],["Emission from $^{13}$ CO","Unlike $^{12}$ CO, typical densities of $^{13}$ CO in molecular cloud environments are low enough such that the emission line is optically thin and is a better indicator of column density.","As such, the blue and redshifted emission features displayed in Figure REF likely trace material outflowing away from the central star whereas the systemic emission features likely represent material located near the central star-disc system, out of which the star has been forming.","Similar to the case of the $^{12}$ CO blueshifted emission, a clump of material is identifiable in the blueshifted $^{13}$ CO emission at position angle east of north of $\\sim $ 220$^{\\circ }$ seemingly disconnected or 'pinched' from the central region (label B).","The prominent blueshifted outflow feature (Figure REF left) identifiable at position angle $\\sim $ 330$^{\\circ }$ (label E) has an opening angle of $\\sim $ 50$^{\\circ }$ extending to distances of $\\sim $ 11$^{\\prime \\prime }$ ($\\sim $ 4500 au).","The peak intensity in the blueshifted image is located $\\sim $ 1$^{\\prime \\prime }$ north-west of V1647 Ori, in the direction of the outflow.","A second collimated outflow is detected in the redshifted line emission pointing almost directly south and extending a distance of 12.8$^{\\prime \\prime }$ ($\\sim $ 5300 au; label D).","Similar to the case of the blueshifted emission, the peak intensity of the redshifted image is located $\\sim $ 1$^{\\prime \\prime }$ south-east of V1647 Ori, in the direction of the southern outflow.","The systemic $^{13}$ CO emission feature (Figure REF center) displays an apparent \"hole\" coincident with the location of the collimated $^{13}$ CO blueshifted outflow (label F).","Figure: Side by side comparison of blue, systemic, and red shifted moment 0 images of 13 ^{13}CO.","The central location of the V1647 Ori disc continuum (Fig ) is shown with a black cross.","Figures are labeled according to the text in Section 3.","All 13 ^{13}CO channels associated with emission can be seen in Figure 1 of the Appendix."],["Emission from C$^{18}$ O","C$^{18}$ O emission, typically a tracer of higher column density gas in molecular cloud environments, is displayed in Figure REF .","Distinct features are seen in the blue, systemic and red shifted velocity channels.","The blue shifted C$^{18}$ O features trace the collimated outflowing material (label E) previously identified in the $^{13}$ CO integrated intensity images (Figure REF ).","This feature has roughly the same opening angle and angular size as its blueshifted $^{13}$ CO counterpart.","Similar to the case of the blueshifted $^{13}$ CO emission, the peak intensity location in the blueshifted C$^{18}$ O is located $\\sim $ 1$^{\\prime \\prime }$ northwest of V1647 Ori in the direction of the blueshifted outflow.","The same clump located at position angle $\\sim $ 220 from the $^{12}$ CO and $^{13}$ CO blueshifted integrated intensity images, apparently disconnected ('pinched') from the main emission features of V1647 Ori, is also observed to the south west in blueshifted C$^{18}$ O emission (label B).","While the redshifted C$^{18}$ O emission does not display the same collimated feature as is evident in the $^{13}$ CO redshifted emission, it does however display two bright peaks, one of which is coincident with the location of the peak intensity of the $^{13}$ CO redshifted emission south east of V1647 Ori (i.e., in the direction of the southern collimated outflow).","The second bright peak in the C$^{18}$ O redshifted image is very close to the location of the V1647 Ori disc and may be a detection of circumstellar disc gas.","Figure: Side by side comparison of blue, systemic, and red shifted moment 0 images of C 18 ^{18}O.","The central location of the V1647 Ori disc continuum (Fig ) is shown with a black cross.","Figures are labeled according to the text in Section 3.","All C 18 ^{18}O channels associated with emission can be seen in Figure 1 of the Appendix.A strong detection of C$^{18}$ O at systemic velocities likely represents the dense material from which the V1647 Ori protostar formed and may impact the structures identifiable in the other spectral lines.","To emphasize this role, the systemic C$^{18}$ O emission contours are overlaid on blueshifted $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O integrated intensity images in Figure REF .","The systemic C$^{18}$ O emission rests as the base of the bowl identified in blueshifted $^{12}$ CO (Figure REF left) and also is spatially coincident with an apparent discontinuity between the blueshifted $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O emission at position angle $\\sim $ 220$^{\\circ }$ (label B; i.e., these emission features appear to 'bend' around the material probed by systemic C$^{18}$ O emission).","Moreover, the large ($\\sim $ 10$^{\\prime \\prime }$ ) bright extended emission feature in the south east part of the systemic C$^{18}$ O emission appears to be in line with the axis of the $^{13}$ CO and C$^{18}$ O collimated outflow (label G in Figure REF ).","The redshifted C$^{18}$ O material may also play a role in shaping of the nebula as displayed in Figure REF .","Redshifted C$^{18}$ O emission material may be shaping the long extended arm observed in redshifted $^{12}$ CO that is apparently connecting the V1647 Ori system to an extended feature at a projected distance of $\\sim $ 8300 au.","Figure: C 18 ^{18}O systemic emission (white contours) is shown overlaid on blueshifted 12 ^{12}CO, 13 ^{13}CO and C 18 ^{18}O integrated intensity images.","The contour levels for the systemic C 18 ^{18}O emission are 4.0, 5.5, 6.0, 6.5, and 6.75 ×\\times 3σ\\sigma .","The beam is displayed in the lower left.Figure: Redshifted C 18 ^{18}O contours (0.6, 0.7, 0.85, 1.0, 1.1, and 1.2 ×\\times 3σ\\sigma ) overlaid on the redshifted 12 ^{12}CO integrated intensity image of V1647 Ori."],["Dynamics of the V1647 Ori Environment","Figure REF highlights the various outflows/features identified in the previous sections (labels D and E from Figures REF , REF , and REF ).","These features can be summarized in the following way: a $\\sim $ 50$^{\\circ }$ collimated outflow at position angle of $\\sim $ 330$^{\\circ }$ is detected in blueshifted $^{13}$ CO and C$^{18}$ O (Figure REF left), a $\\sim $ 100$^{\\circ }$ excavated cavity pointing directly north of V1647 Ori is detected in both blueshifted $^{12}$ CO and systemic C$^{18}$ O and misaligned with the aforementioned collimated outflow (Figure REF center), and a $\\sim $ 30$^{\\circ }$ collimated outflow detected in redshifted $^{12}$ CO and $^{13}$ CO pointing directly south of V1647 Ori (Figure REF right).","The dynamical age ($\\tau _{d}$ ) of these outflows can be estimated using their physical extent measured from the location of the V1647 Ori disc ($r_\\mathrm {outflow}$ ) and their maximum velocities relative to the velocity of the system (i.e., $\\tau _{d}$ = $\\frac{r_\\mathrm {outflow}}{v_\\mathrm {system}-v_\\mathrm {max}}$ ).","However, these dynamical ages are relatively uncertain given the low inclination of this system (i = 17$^{\\circ }$ ).","Since each of the two collimated outflows were detected with multiple emission lines, we can estimate two independent dynamical ages for each outflow.","These dynamical ages and the values used to determine them are displayed in Table REF and indicate the outflow identified in blueshifted $^{13}$ CO and C$^{18}$ O has a dynamical age of $\\sim $ 11,700 years whereas the outflow identified in redshifted $^{12}$ CO and $^{13}$ CO has a dynamical age of $\\sim $ 17,200 years.","Given an uncertainty of $\\pm $ 7 pc for a distance of 414 towards the Orion Nebula , these dynamical age estimates may be different by $\\sim $ 200 and 300 years, respectively.","However, we caution that the dynamical age of the redshifted $^{12}$ CO feature is highly uncertain given that it extends to radii larger than the ALMA primary beam.","We do not estimate the dynamical age of the $^{12}$ CO bowl-like feature pointed directly north because this feature is more likely a cavity rather than an outflow.","However, it should be noted that the northern bowl-like emission has a similar blueshifted $^{12}$ CO maximum velocity ($v_\\mathrm {max}$ = 3) as the collimated redshifted $^{12}$ CO feature pointed directly south ($v_\\mathrm {max}$ = 2.5).","Figure: Contours overplotted on blueshifted 13 ^{13}CO, C 18 ^{18}O and redshifted 12 ^{12}CO integrated intensity images of V1647 Ori to highlight outflow/cavity morphology.","Left: Blueshifted C 18 ^{18}O contours with levels of 1.0, 1.75, 4.0, 6.0, and 8.0 ×\\times 3σ\\sigma .","Middle: Blueshifted 12 ^{12}CO contours with levels 4.0, 10.0, 12.0, 25.0 and 35.0 ×\\times 3σ\\sigma .","Right: Redshifted 13 ^{13}CO contours with levels 1.0, 1.5, 2.0, 2.5, and 3.2 ×\\times 3σ\\sigma .Table: Dynamical Ages of the Outflows in V1647 OriPhysical characteristics (e.g., mass, momentum, and kinetic energy) of the molecular gas surrounding the V1647 Ori protostar can be estimated using primary beam corrected $^{12}$ CO and $^{13}$ CO line emission.","We follow the procedure outlined in and adopt a $^{12}$ CO to H$_{2}$ abundance ratio of 10$^{-4}$ and a $^{12}$ CO to $^{13}$ CO relative abundance of 62 .","This analysis assumes a constant excitation temperature of T$_\\mathrm {ex}$ = 50 K along the line of sight and a beam filling factor of 1.","We then calculate the outflow mass M =$\\sum _{v}$ M($x,y,v$ ), momentum P = $\\sum _{v}$ M($x,y,v$ ) $\\times $ $v$ , and kinetic energy KE = $\\sum _{v}$ M($x,y,v$ ) $\\times $ $\\frac{1}{2}$$v^{2}$ for each velocity channel within a radius of 10$^{\\prime \\prime }$ of V1646 Ori (e.g., see Figure REF ).","This calculation was performed on primary beam-corrected channel maps.","Parameters for the blue and redshifted regions of $^{12}$ CO and $^{13}$ CO are displayed in Table REF and correspond to the velocity ranges indicated at the top of Figure REF and displayed in Figures REF and REF .","Optical depth corrections, typically performed using the ratio of line fluxes between the optically thick $^{12}$ CO and optically thin $^{13}$ CO , are unable to be applied to this dataset because there are few channels where both $^{13}$ CO and $^{12}$ CO are detected.","Therefore, the $^{12}$ CO parameters estimated in this work (Table REF ) are lower limits, and not necessarily representative of the entire mass of the outflow in $^{12}$ CO. Table: Physical Properties of the V1647 Ori Outflows"],["Optical Reflection Nebula (McNeil's Nebula)","The $\\sim $ 1$^{\\prime }$ diameter optical r-band reflection nebula to the north of V1647 Ori is displayed in Figure REF .","These data were taken during the 2008 outburst and display a generally diffuse nebulosity of dust with a clump of material to the north west near RA and Dec offset [-4, 10].","Overlays of the $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O contours on the optical data indicate the millimeter gas emission is well aligned with the base of the reflection nebula.","Moreover, the collimated outflow observed in $^{13}$ CO and C$^{18}$ O appear to be traveling in the direction of the bright dust clump observable at optical wavelengths.","Figure: Optical Gemini GMOS r band observation of V1647 Ori and its reflection nebula taken during the 2008 outburst.","Overlaid are integrated intensity image contours of 12 ^{12}CO, 13 ^{13}CO, and C 18 ^{18}O as indicated in the lower left of each panel.","North is up and east is left"],["The Circumstellar Disc of V1647 and Potential Mechanism for Accretion Outbursts","The resolved continuum image of the V1647 Ori circumstellar disc (Figure REF ) does not display clear evidence of spirals or clumps associated with disc fragmentation or perturbations from a potential binary companion.","Therefore, it is unlikely there are any binary or massive substellar companions $\\gtrsim $ 40 au from the central star capable of producing the intense outbursts observed in V1647 Ori.","However, the potential non-gaussian radial brightness profile identifiable at large radii in Figure REF suggests some amount of asymmetry and warrants a more detailed investigation than that performed for the continuum data and presented here.","While the circumstellar disc is clearly detected in the dust continuum, it is less evident in the molecular line data.","Molecular emission spatially coincident with the location of the disc is identifiable in the redshifted $^{12}$ CO integrated intensity image (Figure REF ) and is potentially detected but unresolved from extended cloud/outflow emission at blueshifted $^{12}$ CO and $^{13}$ CO and redshifted $^{13}$ CO and C$^{18}$ O.","Such emission in these cases may not be from the disc but instead could be from a small region surrounding the disc.","Moreover, gaseous emission from the disc was not detected in any of the emission lines at the systemic velocities (e.g., see Figure REF top and Appendix).","Given the inconsistency of its detection, we were unable to search for signatures of Keplerian rotation.","The non-detection of the disc at systemic velocities is likely due to the deeply embedded nature of this system which is consistent with its high extinction at optical and X-ray wavelengths.","The radius of the disc derived from the continuum emission (Table REF ) is similar to that of the namesake of the FUor class as well as HBC 494 and V2775 Ori , .","However, compared with V883 Ori, the V1647 Ori disc is a factor of $\\sim $ 2 smaller and its star/disc bolometric luminosity is a factor of 10 fainter .","The V1647 Ori disc mass derived from the continuum is the least massive compared with the other FUor objects presented in this series.","This suggests objects that have been classified strictly as FUors may be more massive than EXor objects where V1647 Ori is in the middle of this classification as it displays outbursts indicative of both classifications."],["Excavated Cavities and Outflows","The molecular emission line data indicate that the star-forming environment surrounding V1647 Ori is both structured and complex.","Several features indicating the three-dimensional structure of this system are readily identifiable and show that the protostar and disc system are still deeply embedded in its parent molecular cloud.","The orientation of the system can be further constrained when considering the direction of the blueshifted outflow (label E) coincident with northern reflection nebula (Figure REF ).","The dust cavity appears in the north at optical wavelengths because it is preferentially forward scattering emission from its illuminating source (i.e., V1647 Ori during outburst).","This indicates that the cavity is located between V1647 Ori and our line of sight which is consistent with the cavity detection in blueshifted $^{12}$ CO emission (label A).","The co-alignment of this excavated northern cavity with the outflow at PA$\\sim $ 330$^{\\circ }$ is further substantiated by the location of the bright clumps visible during outburst in the optical reflection nebula since they are aligned with direction of the collimated outflow (Figure REF ).","This orientation also explains the absence of a reflection nebula at optical wavelengths to the south of V1647 Ori even though a southern cavity is present (Figure REF ; i.e., the southern cavity is pointed away from us and any reflected light will be backscattered and too faint to detect in shallow observations).","Given the appearance of its complex environment, the strongest constraints on the orientation of the V1647 Ori system is measured by the gaussian fit to the continuum data which indicates the disc is almost face-on (i = 17$^{\\circ }$ ) with a position angle of 109$^{\\circ }$ .","This disc inclination is in good agreement with previous estimates of i $<$ 30$^{\\circ }$ , but is not consistent with the stellar inclination modeled from periodic X-ray emission generated by rotating accretion hot spots .","The measured position angle of the disc (109$^{\\circ }$ $\\pm $ 19) appears to align more with the outflow at PA$\\sim $ 330$^{\\circ }$ (label E) than the southern collimated outflow at PA$\\sim $ 180$^{\\circ }$ (label D) assuming that outflows travel perpendicular to the plane of the disc and the disc is inclined such that the northern extended emission is blueshifted towards our line of sight.","Since the disc orientation is a strong constraint on the system as a whole, our interpretation of the large scale (r $\\gtrsim $ 1-3$^{\\prime \\prime }$ ) emission features (e.g., collimated outflows) assumes that these features were formed from material traveling roughly perpendicular to the circumstellar disc of V1647 Ori at the time of their respective outflow triggering events.","Such an assumption is reasonable given our current understanding of launching mechanism of material during protostellar evolution .","The blueshifted $^{12}$ CO emission whose bowl-like shape conforms well to the base of the optical reflection nebula (Figure REF ) is seemingly misaligned with the aforementioned blueshifted $^{13}$ CO and C$^{18}$ O collimated feature (Figure REF , center).","Moreover, the non-detection in $^{13}$ CO of this $^{12}$ CO blueshifted bowl-like shape suggests that there are no active wide-angle outflows at this location and that the $^{12}$ CO is probing the temperature of the walls of the excavated cavity.","The misalignment between the collimated southern outflow (Figures REF , REF , label D) and the collimated northern outflow at PA$\\sim $ 330$^{\\circ }$ (Figures REF , REF , label E) may be the result of a re-orientation or precession of the mass-launching area (i.e., the orientation of the circumstellar disc may have changed from one outburst to another resulting in two outflows orientated with different position angles).","The difference in position angle and the dynamical age between these collimated outflows (Table REF ) suggest each outflow was likely the result of a specific mass-accretion event with the southern outflow occurring $\\sim $ 5000 years before the north-west outflow.","Given that a radical change in disc position angle is unlikely, these collimated outflows were probably launched from opposite sides of the circumstellar disc during their respective mass accretion events.","While some asymmetric jets, like those detected from Herbig-Haro objects, appear to be shaped by their local star-forming environment , others systems exhibit clear evidence that the jet-launching region (i.e., the region perpendicular to the plane of the disc) has precessed over time resulting in outflows with asymmetric and misaligned features , , , .","While the amount of precession over the lifetime of the outflow varies from case to case, it appears that jet precession of more than a few degrees may be rare .","In cases where the degree of disc precession is high (like in the case of V1647 Ori), several mechanisms have been proposed capable of significantly re-orienting the outflow/jet: (a) radiative-induced disc warping where a jet, presumably oriented perpendicular to a disc, precesses in response to a disc warp which is generated in cases of high-incident radiation ; (b) tidal interaction from a companion in a non-coplanar orbit truncating and/or distorting the disc , ; or (c) anisotropic accretion events where an impact/merging of material onto the disc during anisotropic accretion can change the orientation of the disc angular momentum vector resulting in a net torque in the rotation of the disc , .","Estimating the critical radius at which the disc becomes unstable to warping following equation 5 in , the assumption that $\\eta $ =1 in , and the physical characteristics of V1647 Ori during outburst (M = 0.8 $M_{\\odot }$ , $\\overset{.","}{M}$$_\\mathrm {acc}$ = 4 $\\times $ $10^{-6}$ $M_{\\odot }$ yr$^{-1}$ , $L = 44 L_{\\odot }$ ), $R_\\mathrm {crit}$ = 2 $\\times $ 10$^{6}$ au.","Given that this radius is orders of magnitude larger than the radius of the V1647 Ori disc, it is unlikely a disc warp is the origin of its misaligned outflows.","However, it should be noted that the stellar parameters during the outburst event that produced these outflows ($\\sim $ 14,000 years ago; Table REF ) are not necessarily the same as those measured during the 2003 outburst.","As noted in Section 4.1, the non-detection of any companions to V1647 Ori at radii > 40 au likely rules out tidal interactions via a companion causing disc precession unless such a companion exists at smaller radii and has not disrupted the V1647 Ori disc in such a way as to be evident from the continuum observations.","Therefore, given the intrinsic connection between accretion and FUor/EXor outbursts, it appears that significant mass-loading (e.g., from a large clump of envelope material or a companion) during anisotropic accretion may be the most likely scenario to have caused the disc to precess between each outflow-generating outburst in V1647 Ori.","If this anisotropic accretion scenario is true in the case of V1647 Ori, one might expect more FUor/EXor type objects to display multiple misaligned outflows.","However, there appears to be little evidence for this, at least in the case of the other FUors studied as part of this series , , .","Given the complex geometry of the V1647 Ori star-forming environment, it may be that other FUor/EXor objects underwent smoother mass-accretion outbursts which did not result in disc precession or simply had fewer large-scale mass accretion events.","Z CMa, a binary FUor object that also exhibits properties of an EXor, displays two slightly misaligned outflows where a wiggling shape of one of the outflows may be generated via precession from an unseen additional companion , .","Given that Z CMa and V1647 Ori represent a rare class of objects that exhibit properties of both FUor and EXor objects, we speculate that disc precession may be related to this intermediate classification.","While the disc precession argument seems the most plausible, other scenarios could be invoked to describe the complicated geometry of these two collimated emission features.","The two outflows could originate from separate binary members ( e.g., each with their own outflow) if V1647 Ori has an undetected companion.","In this case, outflow misalignment may be a signature that the potential binary formation of V1647 Ori resulted from turbulent fragmentation .","However, this scenario generally considers binaries with separations much larger than what would likely be the case for V1647 Ori ($\\lesssim $ 40 au).","Another scenario alternative to changes in disc orientation is the case that the outflows instead are located at different position angles as a result of a dense material within the molecular cloud shaping the outflow.","V1647 Ori clearly resides in a complex and embedded environment and the outflow structure could simply be following a path of least resistance due to some unobserved molecular cloud component whose mass would be difficult to determine without knowledge of the pre-outflow geometry.","This scenario would also affect the dynamical age calculations if the outflowing material does not travel in a straight line.","Another explanation alternative to disc precession could be that the collimated features observed are actually inflows and not outflows.","This interpretation would require our understanding of the system geometry to be reversed (i.e., the blueshifted outflow situated between V1647 Ori and our line of sight would instead be behind V1647 Ori and traveling towards the system presumably in a large accretion stream).","However, this scenario is unlikely to be true given the spatial correlation between the blueshifted $^{13}$ CO and C$^{18}$ O emission with the northern cavity of the optical reflection nebula which independently indicates this northern region is between V1647 Ori and our line of sight and thus, the emission feature is more likely an outflow."],["The Shaping Potential of Dense Molecular Gas Surrounding V1647 Ori","Given the structure of the red and blueshifted molecular line emitting material compared to that of the systemic C$^{18}$ O emission (Figure REF ), it is evident the dense material in which the protostar is embedded influences the overall structure of the system.","While it is not clear whether this dense material can result in two misaligned outflows (Section 4.2), it is still evident that the blueshifted $^{12}$ CO bowl-like shape is surrounded by the dense C$^{18}$ O material of similar shape (Figure REF left).","This is in good agreement with the location of the wall of the excavated cavity as seen in $^{12}$ CO, suggesting the shaping mechanism that produced this cavity compressed the low-density ambient molecular material until it collided with the dense material as probed by C$^{18}$ O.","Moreover, the western arm of the systemic C$^{18}$ O material appears to have cut off or 'pinched' the material associated with blueshifted $^{12}$ CO and $^{13}$ CO material $\\sim $ 3$^{\\prime \\prime }$ southwest of the V1647 Ori disc (label B).","The outflowing blueshifted material appears to travel around the dense material at systemic velocities as it may be more energetically favorable.","If this interpretation is true then the location of the dense systemic C$^{18}$ O material must be between our line of sight and the V1647 Ori disc (i.e., in the foreground but still part of the V1647 Ori star-forming environment).","This interpretation suggests that the local dense gas plays a large role in the shaping of cavities and molecular outflows during this stage of pre-MS stellar evolution.","A similar shaping scenario may be taking place where the redshifted C$^{18}$ O emission appears along the border of the 'arm' seemingly connecting the V1647 Ori system to the large unresolved feature $\\sim $ 20$^{\\prime \\prime }$ (8300 au) to the west (Figure REF ).","This large $\\sim $ 40$^{\\prime \\prime }$ (17,000 au) column of material is also detected in a few channels of $^{13}$ CO emission traveling at the same velocity (Figure 2 in the appendix).","The source of this structure is unclear and it may just be leftover cloud material that either has yet to form a star or does not meet the physical requirements to do so.","However, we caution that this feature may be an artifact from emission outside the primary beam.","The closest pre-MS star (in projection) to this structure (2MASS J05461162-0006279) is located $\\sim $ 7$^{\\prime \\prime }$ (3150 au) from this extended structure's south-western edge.","It is identified as a Class II object in the L1630 star-forming region based on its infrared-excess and detection in X-rays and has a moderate absorption of N$_{H}$ = 5 $\\times $ 10$^{21}$ cm$^{-2}$ ."],["Outflow Dynamics and a Comparison with Other FUors","The complex structures identified in the V1647 Ori at spatial scales of $\\sim $ 1.0$^{\\prime \\prime }$ can be compared with the other similarly structured FUor/EXor type objects presented in this series , , .","Of the eight FUor/EXor objects observed, only V2775 Ori, V883 Ori, HBC 494 and V1647 Ori display significant amounts of extended $^{12}$ CO, $^{13}$ CO and/or C$^{18}$ O emission.","Given the high A$_V$ , N$_H$ , complex C$^{18}$ O environment, and ambiguous detection of its gaseous circumstellar disc, V1647 Ori appears to be in the earliest stage of stellar evolution compared with the other FUor objects presented.","In particular, the structures displayed in these ALMA observations of V1647 Ori could represent the progenitor stages of those of V883 Ori and V2775 Ori given their similar inclination (and thus similar viewing angle).","Similar to the case of V2775 Ori , V1647 Ori also exhibits a small outflow opening angle ($\\sim $ 30$^{\\circ }$ ) in $^{13}$ CO, C$^{18}$ O, further indicating its primordial nature if opening angle is a correlation to age.","In contrast, HBC 494 and V883 Ori , have much larger opening angles of $\\sim $ 150$^{\\circ }$ .","Almost counterintuitively, the outflows of V1647 Ori have the oldest dynamical ages (11,700 and 17,200 years) when compared to that of V2775 Ori , HBC 494 and V883 Ori .","These dynamical ages are very small given the timescale of pre-MS stellar evolution with respect to disc dissipation (e.g., $\\sim $ 1-3 Myr).","Therefore the older dynamical timescale of V1647 Ori's outflows do not necessarily mean it is inconsistent with the interpretation here that this system is less evolved than its other FUor counterparts.","It could simply mean that whatever mechanism initiates these outbursts may have started earlier in V1647 Ori.","Given the complex environment surrounding V1647 Ori, it may indicate that inhomogeneous patches of envelope material could have prompted more frequent or earlier accretion outbursts.","The outflow mass surrounding V1647 Ori (as indicated in Section 3.6) appears similar to those obtained for other Class 0-II pre-MS stars with outflows as well as the small sample of FUor type objects whose outflows have been measured .","However, when comparing the outflow velocities associated with these objects, it is clear that the outflows surrounding V1647 Ori are quite slow ($\\sim $ 1-2 km s$^{-1}$ ).","Given the similar sensitivity and resolution of the other FUors detected as part of this series, we compare the results of V1647 Ori directly with those of V2775 Ori, HBC 494, and V883 Ori.","Of these sources, V1647 Ori has the least massive circumstellar disc and is the only source considered both an FUor and EXor type object based on its unusual outburst frequency.","V1647 Ori is the most similar in disc mass and inclination to V2775 Ori , however their local environments are quite distinct.","While V2775 Ori displays an impressive ring-like structure in its blue and red shifted emission , there is little evidence of complex structure surrounding the large-scale environment of the protostar indicating that this system may be more evolved than V1647 Ori.","Moreover, V2775 Ori exhibits a redshifted $^{12}$ CO outflow velocity of 4.4 km s$^{-1}$ , a factor of $\\sim $ 2 faster than those of V1647 Ori, potentially indicative of a emptier environment surrounding the star where the outflow doesn't slow down from interactions with surrounding material.","A comparison of outflow mass and energy indicates the outflow of V1647 Ori as probed by $^{12}$ CO and $^{13}$ CO is more massive than those in V2775 Ori by a factor of 3 and 15, respectively.","A comparison of V1647 Ori with that of HBC 494 and V883 Ori, both of which have more massive and larger circumstellar discs, indicates the outflow mass of V1647 Ori is significantly smaller , .","Both these systems exhibit slow outflow speeds similar to or smaller than V1647 Ori possibly indicating they too still are interacting with or have been influenced by surrounding material."],["Summary and Conclusions","We have presented ALMA observations of the V1647 Ori continuum at a resolution of 0.2$^{\\prime \\prime }$ as well as molecular line emission of $^{12}$ CO,$^{13}$ CO and C$^{18}$ O at a resolution of $\\sim $ 1$^{\\prime \\prime }$ .","No emission line features were detected in the 0.2$^{\\prime \\prime }$ dataset indicating that structure surrounding V1647 Ori are fairly uniform on large scales.","These observations are presented as part of a series and we compare these results of V1647 Ori to that of other FUor type objects with extended emission (e.g., V2775 Ori, HBC 494, and V883 Ori).","We have identified a complex yet structured system surrounding the V1647 Ori protostar and circumstellar disc and are able to draw several conclusions: $\\bullet $ The circumstellar disc of V1647 Ori is resolved and has a nearly face-on inclination (i = 17$^{\\circ }$ ), a radius of 40 au and a total disc mass of $\\sim $ 0.1 M$_{\\odot }$ .","The continuum resolved disc does not display any disc structures (e.g., spirals and/or fragmentation) associated with nearby interacting companions (r $\\gtrsim $ 40 au) and therefore such a mechanism, under these constraints, is unlikely to be responsible for the various observed accretion outbursts.","The mass of the disc is less than the other FUor type objects presented in this series may be related to its classification as both an FUor and EXor type object.","$\\bullet $ Blueshifted $^{12}$ CO molecular line emission spatially coincident with the V1647 Ori optical reflection nebula (McNeil's Nebula) and a non-detection of this feature in $^{13}$ CO and C$^{18}$ O emission confirm that this structure is a cavity excavated during a previous accretion outburst.","A collimated $^{13}$ CO and C$^{18}$ O emission feature aligns with clumps in the nebula identified during the recent multiwavelength outburst.","$\\bullet $ Dense molecular material as probed via C$^{18}$ O emission traveling at a systemic velocity of $\\sim $ 10 km s$^{-1}$ is spatially coincident with the base of the northern excavated cavity and likely shaped the outflow material identified in blueshifted $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O emission.","$\\bullet $ Two distinct collimated ($\\sim $ 30-50$^{\\circ }$ ) outflows with position angles of 330$^{\\circ }$ and 180$^{\\circ }$ have been detected with dynamical ages of $\\sim $ 11,700 and $\\sim $ 17,100 years, respectively.","These outflows are likely the result of two distinct mass accretion events where outflows were launched from opposite sides of a disc that has changed orientation over the last $\\sim $ 5000 years from previous anisotropic accretion events.","$\\bullet $ Compared to the other FUor objects included in our series, V1647 Ori displays the most complex large-scale structures likely the result of its comparatively early stage of stellar evolution.","A comparison of outflow parameters supports this notion and also indicates that the outflow velocities of these FUor and FUor/EXor type objects ($\\sim $ 0.5 - 4 km s$^{-1}$ ) are relatively small compared to those observed in typical pre-MS stars and predicted by simulations ($v$$\\sim $ 10-30 km s$^{-1}$ ).","The complex star-forming environment surrounding V1647 Ori may result in more frequent yet still extreme outburst events which justify its classification as both an FUor and EXor type object.","$\\bullet $ A large ($\\sim $ 17,000 au) unresolved structure is identified $\\sim $ 8300 au from V1647 Ori.","This structure is not associated with any obvious source but is seemly connected to V1647 Ori by an 'arm' of molecular material as probed with redshifted $^{12}$ CO emission."],["Acknowledgements","This research was supported by CONICYT-FONDECYT awards (3150550, 1171246) and support from the Millennium Science Initiative (Chilean Ministry of Economy; grant Nucleus RC 130007).","This paper makes use of the following ALMA data: ADS/JAO.ALMA #2013.1.00710.S .","ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.","The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.","The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.. D. P. acknowledges funding by the National Aeronautics and Space Administration through Chandra Award Number GO6-17013A.","K.M acknowledges funding by the Joint Committee of ESO/Government of Chile, and funding by the Science and Technology Foundation of Portugal (FCT), grant No.","IF/00194/2015.","H.C. acknowledges support from the Spanish Ministerio de Economía y Competitividad under grant AYA 2014-55840-P. J.J.T acknowledges support from the University of Oklahoma, the Homer L. Dodge endowed chair, and grant 639.041.439 from the Netherlands Organisation for Scientific Research (NWO).","The authors would like to thank the referee whose comments and suggestions increased the quality of this work."]],"string":"[\n [\n \"The ALMA Early Science View of FUor/EXor objects. IV. Misaligned\\n Outflows in the Complex Star-forming Environment of V1647 Ori and McNeil's\\n Nebula\"\n ],\n [\n \"Abstract We present Atacama Large Millimeter/sub-millimeter Array (ALMA) observations of the star-forming environment surrounding V1647 Ori, an outbursting FUor/EXor pre-MS star.\",\n \"Dust continuum and the (J = 2 - 1) $^{12}$CO, $^{13}$CO, C$^{18}$O molecular emission lines were observed to characterize the V1647 Ori circumstellar disc and any large scale molecular features present.\",\n \"We detect continuum emission from the circumstellar disc and determine a radius r = 40 au, inclination i = 17$^{\\\\circ}$$^{+6}_{-9}$ and total disc mass of M$_{\\\\mathrm{disk}}$ of ~0.1 M$_{\\\\odot}$.\",\n \"We do not identify any disc structures associated with nearby companions, massive planets or fragmentation.\",\n \"The molecular cloud environment surrounding V1647 Ori is both structured and complex.\",\n \"We confirm the presence of an excavated cavity north of V1647 Ori and have identified dense material at the base of the optical reflection nebula (McNeil's Nebula) that is actively shaping its surrounding environment.\",\n \"Two distinct outflows have been detected with dynamical ages of ~11,700 and 17,200 years.\",\n \"These outflows are misaligned suggesting disc precession over ~5500 years as a result of anisotropic accretion events is responsible.\",\n \"The collimated outflows exhibit velocities of ~2 km s$^{-1}$, similar in velocity to that of other FUor objects presented in this series but significantly slower than previous observations and model predictions.\",\n \"The V1647 Ori system is seemingly connected by an \\\"arm\\\" of material to a large unresolved structure located ~20$\\\"$ to the west.\",\n \"The complex environment surrounding V1647 Ori suggests it is in the early stages of star formation which may relate to its classification as both an FUor and EXor type object.\"\n ],\n [\n \"Introduction\",\n \" Stars form as a result of gravitational contraction of rotating molecular clouds.\",\n \"As the gravitational collapse proceeds, a system emerges composed of an accreting star, disc and envelope.\",\n \"Observations of this stage of pre-main sequence (pre-MS) stellar evolution display large scale molecular outflows that act to transport energy back into the surrounding environment and are likely the main dispersing mechanism of molecular cloud material, responsible for the transition from a deeply embedded Class 0 protostar to the more evolved Class I/II pre-MS stellar systems.\",\n \"An empirical relationship has been identified between the opening angle of these outflows and the pre-MS stage of stellar evolution .\",\n \"The earliest stages of evolution (Class 0) exhibit outflows that are highly collimated and have opening angles of 20-50$^{\\\\circ }$ whereas the more evolved Class I and II stages are associated with less collimated opening angles of 80-120$^{\\\\circ }$ and 100-160$^{\\\\circ }$ , respectively .\",\n \"The origin of these outflows are debated due to the au spatial scales required to resolve them however several theories relate the outflows to intense mass accretion events from the disc and/or envelope onto the central pre-MS star .\",\n \"Collimated and wide-angle outflows appear to represent two distinct structures in the star-forming environment of their host stars.\",\n \"In particular, wide-angle outflows tend to be more massive and slower (velocities ranging from $\\\\sim $ 10-30 km s$^{-1}$ ) compared to highly collimated outflows which can reach velocities of $\\\\sim $ 100-1000 km s$^{-1}$ .\",\n \"Their distinct velocities and masses arise from the mechanisms responsible for their production; wide angle outflows are likely produced from the ambient molecular material getting swept up when interacting with a jet bow shock whereas highly collimated jets likely arise from the interaction between a strongly magnetized central star and the inner edge of its accretion disc during intense mass accretion events .\",\n \"Recent observations indicate that extended disc winds may also drive protostellar outflows .\",\n \"Mass accretion events where material gets loaded from the disc onto the protostar are essential to star formation as these events are the primary means by which stars gain their mass.\",\n \"The standard model of star formation indicates that a molecular cloud gravitationally contracts and forms a star-disc system in about $\\\\sim $ 5 $\\\\times $ 10$^{5}$ years .\",\n \"The mass accretion rate necessary to produce a star-disc system after this timescale is inconsistent with the age and the implied accretion rates derived from the observed accretion luminosity of a large sample of stars in Taurus .\",\n \"One solution to this \\\"luminosity problem\\\" is short-lived major accretion events of significant mass.\",\n \"In the past several decades, a phenomenon has been observed in a small subset of pre-MS stars that is consistent with brief and intense mass accretion events identified as \\\"outbursts\\\" , , .\",\n \"These outbursting pre-MS stars have been named FUor/EXor objects based on their namesakes FU Ori and EX Lup.\",\n \"This phenomenon is identifiable when pre-MS stars suddenly increase in brightness by several magnitudes at optical/near-IR wavelengths as a result of intense mass accretion outbursts where accretion rates can reach as high as 10$^{-4}$ – 10$^{-5}$ M$_{\\\\odot }$ yr$^{-1}$ .\",\n \"FU Ori type (FUor) and EX Lup type (EXor) are generally distinguished by the frequency and length of their protostellar outbursts where FUor type objects tend to have longer (years to decades) timescale and EXors are characterized by more frequent and shorter outbursts (months-years).\",\n \"While it is evident these outbursts are caused by large mass-accretion events, it is not clear what underlying mechanism is responsible.\",\n \"Several theories have been proposed which include binary companions and/or massive planets generating disc instabilities , , disc fragmentation , and a combination of magneto rotational instability (MRI) and gravitational instabilities .\",\n \"A detailed summary of these mechanisms and their relevant references can be found in .\",\n \"Regardless of the underlying mechanism, it is clear that sudden intense outbursts can affect the circumstellar disc of protostars and thus, may impact planet formation , .\",\n \"Recently, an analysis of high resolution ALMA observations of the outbursting FUor type object V883 Ori revealed the first detection of a water snow line in a circumstellar disc .\",\n \"Such a discovery was feasible because the water snow line, which typically resides < 5 au from a $\\\\sim $ 1 M$_{\\\\odot }$ central star, was temporarily extended to $\\\\sim $ 40 au in response to the increase in luminosity of the FUor object during outburst.\",\n \"V1647 Ori, a protostar in the Lynds 1630 (L1630) region of the Orion Molecular Cloud , was first noted in 2003 when a large ($\\\\sim $ 1$^{\\\\prime }$ ) reflection nebula (McNeil's Nebula) appeared coincident with a sudden increase in optical stellar brightness , , .\",\n \"After its identification in 2003, two more outbursts were identified: a 1966 outburst identified using archival photometric plates and a 2008 outburst which increased the flux values of V1647 Ori to similar values as in 2003.\",\n \"During the 2008 outburst, carried out a 4.5 year photometric study of V1647 Ori and report that at the end of their campaign in 2012, V1647 Ori was still in outburst after almost half a decade.\",\n \"The frequency and length of these outbursts, combined with several spectral features characteristic of both FUors and EXors emphasize the ambiguity between the FUor or EXor classification of V1647 Ori .\",\n \"Spectral parameters of V1647 Ori were determined from data taken during quiescence and presented in .\",\n \"They identify V1647 Ori as a young (< 0.5 Myr) pre-MS star of spectral type M0 $\\\\pm $ 2 with an approximate bolometric luminosity, mass and radius of $\\\\sim $ 5.2 L$_{\\\\odot }$ , 0.8 M$_{\\\\odot }$ , and 5 R$_{\\\\odot }$ , respectively.\",\n \"During its 2003 outburst, its bolometric luminosity was measured to be L$_\\\\mathrm {bol}$ = 44 L$_{\\\\odot }$ .\",\n \"Given the transient nature of V1647 Ori, several multiwavelength campaigns were setup to observe the star in its outbursting state.\",\n \"The 2003 outburst was characterized by a $\\\\sim $ 3 magnitude increase in the near-IR , a factor of 25 increase in 12 $\\\\mu $ m flux and a factor of 50 increase in X-ray flux .\",\n \"report no apparent changes in the (350 $\\\\mu $ m-1.3mm) submm dust continuum brightness during outburst compared with its quiescent state.\",\n \"Plasma temperatures derived from the X-ray imaging spectrum presented in were too high to be produced via accretion alone suggesting extreme changes in magnetic field configuration (i.e., magnetic reconnection events) were be present.\",\n \"Spectroscopic measurements during the 2003 outburst indicated accretion rates of a few 10$^{-6}$ to 10$^{-5}$ M$_{\\\\odot }$ year $^{-1}$ and H$\\\\alpha $ P Cygni profiles indicate wind velocities up to $\\\\sim $ 600 km s$^{-1}$ .\",\n \"Mid-IR interferometric observations probing the circumstellar disc indicate a moderately flaring disc with no signatures of close companions at radii < 100 au [1].\",\n \"All three outbursts re-illuminated the reflection nebula, where dust grains preferentially scatter blue light, just north of V1647 Ori and several studies have identified small variations in the local environment as probed by this extended nebular emission , .\",\n \"The time delay between brightness variations of the star and of clumps in the nebulosity were used to estimate a stellar inclination angle of $\\\\sim $ 61$^{\\\\circ }$ .\",\n \"An X-ray periodicity study of V1647 Ori found a similar stellar inclination angle of i $\\\\sim $ 68$^{\\\\circ }$ based on modeling the rotation of accretion hot spots on the stellar surface .\",\n \"These inclinations are in disagreement with studies that suggest a more face-on inclination of $<$ 30$^{\\\\circ }$ based on the absence of a 9.7 $\\\\mu $ m amorphous silicate feature and a low column density of CO in absorption from the disc .\",\n \"Here, we present ALMA 1.3 mm observations of the gas and dust surrounding V1647 Ori in order to characterize its circumstellar disc and local star-forming environment.\",\n \"V1647 Ori is one of 8 FUor/EXor type objects observed as part of an ALMA Cycle 2 project (2013.1.00710.S ; PI L. Cieza) to investigate star-forming environments at this critical stage of pre-MS stellar evolution.\",\n \"All other FUor/EXor objects observed as part of this program, including V883 Ori, HBC 494, V2775 Ori, NY Ori, V1143 Ori, V1118 Ori, ASASSN-13db, are presented in papers I, II, III, IV (this paper) and V , , , .\",\n \"These recently published results indicate that the star-forming environments of these objects are structured and complex.\",\n \"In particular, features such as rings/shells and both wide-angle and collimated outflows have all been identified.\",\n \"Outflow velocities measured for V883 Ori, HBC 494, V2775 Ori, and V1647 Ori (Section 3.6) as part of this series of papers indicate that significantly slower velocities (v$\\\\sim $ 1 - 4 km $s^{-1}$ ) are present than those typically observed and associated with protostellar outflows at these wavelengths .\",\n \"The resolving power and sensitivity of the observations presented in this series also provides strict constraints on circumstellar disc characteristics such as disc size, orientation, and mass.\"\n ],\n [\n \"Observations\",\n \"V1647 Ori was observed with ALMA band-6 at three epochs (December 12th, 2014, April 5th, 2015 and August 30th, 2015).\",\n \"The first two observations were performed with an ALMA configuration of 45 antennas (12 meter diameter) and baselines ranging from 14.6 to 348.5 meters which achieved an angular resolution of $\\\\sim $ 1.0 $^{\\\\prime \\\\prime }$ .\",\n \"The final epoch had a configuration of 35 antennas and longer baselines of 42-1574 meters resulting in a higher angular resolution of 0.2 $^{\\\\prime \\\\prime }$ .\",\n \"The precipitable water vapor (PWV) levels were 0.7, 1.3 and 1 mm for the December 2014, April 2015, and August 2015 observations, respectively.\",\n \"Each ALMA observation of V1647 Ori include two broad (2 GHz) continuum spectral windows centered at $\\\\sim $ 232 GHz and $\\\\sim $ 218 GHz and three narrow (59 MHz) windows centered near the rest frequencies of $^{12}$ CO (J = 2-1; 230.5380 GHz), $^{13}$ CO (J = 2-1; 220.3987 GHz) and C180 (J = 2-1; 219.5603 GHz).\",\n \"Flux calibration was performed with Ganymede and J0423-013 while bandpass calibration was performed with quasars J0538-4405 and J0607-0834.\",\n \"The time dependence variations of the complex gains were calibrated by alternating V1647 Ori with nearby phase calibrators J0541-0541, J0532-0307 and/or J0529-0519.\",\n \"The ALMA pipeline-calibrated observations were further reduced with the Common Astronomical Software Application .\",\n \"The two epochs of $\\\\sim $ 1.0$^{\\\\prime \\\\prime }$ resolution data were continuum subtracted in the visibility domain, concatenated, and cleaned to form a single dataset for the continuum and each spectral line.\",\n \"The same procedure was performed for the single epoch of $\\\\sim $ 0.2$^{\\\\prime \\\\prime }$ resolution data.\",\n \"While features of the V1647 Ori disc/outflows are clearly present in the $\\\\sim $ 1$^{\\\\prime \\\\prime }$ resolution datasets, none of these features are identifiable in the high resolution line imaging from the 0.2$^{\\\\prime \\\\prime }$ resolution dataset.\",\n \"This is consistent with most of the flux from the system being fairly uniform on large scales.\",\n \"Therefore we only present the detection of continuum emission from the 0.2$^{\\\\prime \\\\prime }$ dataset and the line and continuum emission from the combined two $\\\\sim $ 1$^{\\\\prime \\\\prime }$ datasets.\",\n \"The beam size, position angle, and 3$\\\\sigma $ sensitivities for each spectral line channel map and the continuum are indicated in Table REF .\",\n \"Integrated intensity images (moment 0) were creating using the CASA routine $immoments$ .\",\n \"The moment 0 images were created using only pixels with a signal detection of 3$\\\\sigma $ or higher.\",\n \"To better describe the morphology and dynamics of the V1647 Ori star-forming environment, we created moment 0 images for each spectral line with three distinct velocity ranges: $5.0-9.49$ km s$^{-1}$ , $9.5-10.5$ km s$^{-1}$ and $10.51-13.0$ km s$^{-1}$ .\",\n \"These velocity bands will henceforth be referred to in this paper as blueshifted, systemic, and redshifted, respectively.\",\n \"The images presented in this work are not primary-beam corrected.\",\n \"However, primary-beam corrected channel maps were used when estimating physical parameters of the outflow from the line fluxes (Section 3.6).\",\n \"Optical Gemini-GMOS imaging observations of V1647 Ori were performed on September 22, 2008 with an exposure time of 60 seconds in R band.\",\n \"Optical imaging data was retrieved from the Gemini Science Archive and presented here to supplement the ALMA observations.\",\n \"Optical analysis of these and similar data are presented in other work and the discussion in this paper is limited only to correlations between large-scale optical and mm emission morphology.\",\n \"Table: V1647 Ori ALMA Continuum and Integrated Intensity Image Parameters\"\n ],\n [\n \"Continuum Emission\",\n \"The integrated intensity image of the resolved 0.2$^{\\\\prime \\\\prime }$ V1647 Ori continuum is displayed in Figure REF .\",\n \"Two-dimensional gaussian fitting of the $\\\\sim $ 225 GHz continuum emission performed in CASA indicate a flux, inclination, position angle and centroid of 82.62 $\\\\pm $ 8.26 mJy, ${\\\\it i} = 17^{\\\\circ }$ $^{+6}_{-9}$ , PA$ = 109^{\\\\circ } \\\\pm $ 19, and RA= 05:46:13.139, Dec= -00:06:04.90, respectively.\",\n \"The inclination was calculated using the aspect ratio of an assumed circular disc.\",\n \"All position angles reported in this work are east of north.\",\n \"A semi major axis FWHM angular size of 189.0 $\\\\pm $ 3.6 milliarcseconds and a distance of 414 $\\\\pm {7}$ pc results in a disc dust radius of $\\\\sim $ 40 au.\",\n \"Evidence that the continuum emission from the circumstellar disc is spatially resolved is identified by its amplitude as a function of UV distance and asymmetry in the continuum emission (Figure REF ) indicates a potentially non-gaussian distribution near the outer edges of the image.\",\n \"A more in-depth analysis of continuum emission from V1647 Ori as well as the other FUor/EXor objects in this series will be presented in .\",\n \"The mass of the V1647 Ori circumstellar disc dust can be estimated from the continuum emission (assuming continuum emission is thermal and optically thin) following where: $\\\\hspace{86.72377pt}M_{\\\\rm {disc}}= \\\\frac{F_\\\\nu d^2}{\\\\kappa _\\\\nu B_\\\\nu (T)},$ with $B_\\\\nu (T)$ representing the Planck function for dust at a temperature $T$ and $\\\\kappa _\\\\nu $ as the dust opacity, which is a power-law function of submm frequency, i.e., $\\\\kappa _\\\\nu =0.1(\\\\nu / 1000$ $\\\\mathrm {GHz})^\\\\beta $ cm$^{2}$ g$^{-1}$ .\",\n \"The power law opacity index $\\\\beta $ is related to the size distribution and composition of the disc dust particles.\",\n \"While $\\\\beta $ can range from $\\\\beta $ =0 for certain grain mineralogies or size distributions to $\\\\beta $ =2 for interstellar dust grains in the ISM , it is more likely to be in the range of 0.5-1.5 for circumstellar discs .\",\n \"Without more sophisticated continuum modeling to estimate the disc dust temperature of V1647 Ori, a dust temperature and range of $T=20 \\\\pm 10$ K is chosen based on the median temperature of discs in Taurus-Auriga and the relatively small temperature range indicated in modeling of other circumstellar discs .\",\n \"Assuming values of $\\\\beta $ =1, d= 414 pc, $\\\\kappa _\\\\nu $ = 0.023 cm$^{2}$ g$^{-1}$ , and a dust temperature of $T=20$ K, we estimate the disc dust mass of V1647 Ori to be $M_\\\\mathrm {dust}$ = 427 M$_\\\\mathrm {E}$ .\",\n \"This corresponds to a total disc mass $M_\\\\mathrm {disc}$ = 0.13 $M_{\\\\odot }$ ($\\\\sim $ 136 M$_\\\\mathrm {Jup}$ ) assuming a gas to dust mass ratio of 100.\",\n \"When converting from flux to disc mass using Equation 1, the largest sources of uncertainty are the ranges in $\\\\beta $ and dust temperature.\",\n \"While the uncertainty in the continuum flux is dominated by the 10% absolute flux uncertainty of ALMA Band 6, the potential range in $\\\\beta $ ($\\\\beta $ = 1 $\\\\pm {0.5}$ ) and dust temperature (T = 20 $\\\\pm {10}$ ) introduce much higher uncertainties in dust mass that can result in masses in the extreme ranges of M$_{\\\\rm {dust}}$ = [110 - 2688] M$_\\\\mathrm {E}$ or $M_{\\\\rm {disc}}$ = [0.03 - 0.80] $M_{\\\\odot }$ assuming a gas to dust ratio of 100.\",\n \"A summary of V1647 Ori disc parameters are displayed in Table REF .\",\n \"Table: V1647 Ori Disc Parameters Derived From Continuum EmissionFigure: Resolved continuum emission of the V1467 Ori circumstellar disc with contours overlaid of values 1.75, 3, 7, 20, 30, 45 and 60 ×\\\\times 3σ3\\\\sigma .\",\n \"A two-dimensional gaussian fit indicates a disc with inclination of 17 ∘ ^{\\\\circ } -9 +6 ^{+6}_{-9} and PA = 109±19\\\\pm 19 ∘ ^{\\\\circ }.\",\n \"The beam size is displayed in the bottom left and north is up and east is left.\"\n ],\n [\n \"Gas Emission Lines\",\n \"Spectral flux density as a function of radial velocity for the $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O lines are displayed in 0.25 km s$^{-1}$ bins in Figure REF .\",\n \"Spectral extraction was performed using two different apertures both centered at the position of V1647 Ori (J2000 RA= 05:46:13.135, Dec= -00:06:04.82).\",\n \"An aperture diameter of 2.0 $^{\\\\prime \\\\prime }$ was chosen to extract spectral information at the precise location of the continuum emission (Fig REF top) and an aperture diameter of 20$^{\\\\prime \\\\prime }$ was chosen to encompass the nebulosity surrounding V1647 Ori (Fig REF bottom).\",\n \"In both the small and large apertures, a double peaked $^{12}$ CO and single peaked $^{13}$ CO emission feature is observed.\",\n \"A clear detection of C$^{18}$ O material is seen in the large 20$^{\\\\prime \\\\prime }$ aperture indicating an abundance of dense material traveling at a radial velocity equidistant between the two $^{12}$ CO peaks.\",\n \"Due to the embedded nature of this object, the rest velocity of the system (i.e., systemic velocity) is not clear.\",\n \"In the absence of a literature value of the systemic velocity of V1647 Ori, we estimate a systemic velocity of $\\\\sim $ 10.0 km s$^{-1}$ assuming the star forms in the densest part of the molecular cloud (i.e.\",\n \"at the same velocity as the bulk of the dense C$^{18}$ O emission; Figure REF ).\",\n \"This systemic velocity is also consistent with being equidistant between the peaks of the blue and red shifted $^{12}$ CO emission lines.\",\n \"The blueshifted, systemic, and redshifted velocity ranges are indicated at the top of Figure REF .\",\n \"Individual channel maps for each spectral line where emission associated with the V1647 Ori environment is observed is shown in Appendix Figures 1-3.\",\n \"The integrated intensity images of $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O emission over their entire velocity range are displayed in Figure REF with white circles corresponding to the extraction apertures associated with the line profile displayed in Figure REF .\",\n \"Several features are readily identifiable and will be discussed in detail in the following sections.\",\n \"An illustration of the three-dimensional system geometry is displayed in Figure REF .\",\n \"Figure: Line profiles from apertures centered on the dust continuum emission of V1647 Ori with diameters of 2 '' ^{\\\\prime \\\\prime } (top) and 20 '' ^{\\\\prime \\\\prime } (bottom) and velocity bins of 0.25 km s -1 ^{-1}.\",\n \"A systemic velocity of 10.0 km s -1 ^{-1} is indicated with a vertical dashed line.\",\n \"The blue, green and red lines above the plot represent the velocity ranges described in Section 2 for the blueshifted, systemic, and redshifted images presented throughout the rest of this paper.\",\n \"Individual line channel maps are included in the Appendix.Figure: Integrated intensity (moment 0) images of 12 ^{12}CO, 13 ^{13}CO and C 18 ^{18}O integrated over each lines entire velocity range.\",\n \"V1647 Ori is at the center of each panel (offset [0,0] corresponds to RA= 05:46:13.135, Dec= -00:06:04.82).\",\n \"The small solid and large dashed white lines indicate the spectral extraction regions from Figure (aperture diameters of 2 '' ^{\\\\prime \\\\prime } and 20 '' ^{\\\\prime \\\\prime }, respectively) with north is up and east is left.\",\n \"The beam is displayed in the lower left.Figure: A three-dimensional illustration of V1647 Ori and its surrounding misaligned outflows (labels D and E discussed in Sections 3.3 and 3.4).\",\n \"The nearly face-on circumstellar disc is shown in green.\",\n \"Features in the illustration are not to scale.\"\n ],\n [\n \"Emission from $^{12}$ CO\",\n \"The $^{12}$ CO emission line is typically optically thick in molecular cloud environments and is generally not a good probe of column density.\",\n \"Instead, $^{12}$ CO emission traces molecular material of a particular temperature and is useful for probing excavated cavities as well as wide-angle outflows , .\",\n \"Figure REF displays the blue, systemic and red shifted $^{12}$ CO emission summed over the velocity ranges indicated in the top of Figure REF and in Section 2.\",\n \"The base of McNeil's Nebula, the familiar reflection nebula first identified from optical observations during its 2003 outburst is clearly seen in the blue shifted $^{12}$ CO emission line (label A).\",\n \"This northern cavity has a well-defined wall and an opening angle to the north of $\\\\sim $ 100$^{\\\\circ }$ which extends $\\\\sim $ 10$^{\\\\prime \\\\prime }$ ($\\\\sim $ 4140 au).\",\n \"A bright extended region spatially coincident with the star-disc system is identifiable at radii of $r \\\\lesssim 3^{\\\\prime \\\\prime }$ (1245 au) from V1647 Ori.\",\n \"A distinct clump of material is seen $\\\\sim $ 2$^{\\\\prime \\\\prime }$ south-west from the central emission (label B) appearing to be cut off from the main nebula.\",\n \"Meanwhile, the redshifted morphology of the $^{12}$ CO emission line is distinct from the blueshifted emission.\",\n \"Emission coincident with the position of the circumstellar disc of V1647 Ori is detected in the arc-like shape bending from the north-west to the south-west.\",\n \"Such a well defined structure is seemingly connected with an 'arm' to an atypically large column of $^{12}$ CO emission located 20$^{\\\\prime \\\\prime }$ (8300 au) west of V1647 Ori near the edge of the ALMA field of view (label C).\",\n \"The northern bowl-like nebula may be faintly detected behind the arc-like feature in the red-shifted $^{12}$ CO emission.\",\n \"More prominently displayed is an apparent southern feature extending in a straight column south of the V1647 Ori disc with an angular length of $\\\\sim $ 20$^{\\\\prime \\\\prime }$ ($\\\\sim $ 8300 au; label D ).\",\n \"Parts of both the large southern and western columns are identified in the blueshifted $^{12}$ CO emission (Figure REF ; label D).\",\n \"The systemic $^{12}$ CO emission of V1647 Ori appears to be resolved out or undetected and instead displays features that may originate from foreground cloud contamination.\",\n \"Figure: Side by side comparison of blue, systemic, and red shifted moment 0 images of 12 ^{12}CO.\",\n \"The central location of the V1647 Ori disc continuum (Fig ) is shown with a black cross and the beam is displayed in the lower left.\",\n \"Figures are labeled according to the text in Section 3.\",\n \"All 12 ^{12}CO channels associated with emission can be seen in Figure 1 of the Appendix.\"\n ],\n [\n \"Emission from $^{13}$ CO\",\n \"Unlike $^{12}$ CO, typical densities of $^{13}$ CO in molecular cloud environments are low enough such that the emission line is optically thin and is a better indicator of column density.\",\n \"As such, the blue and redshifted emission features displayed in Figure REF likely trace material outflowing away from the central star whereas the systemic emission features likely represent material located near the central star-disc system, out of which the star has been forming.\",\n \"Similar to the case of the $^{12}$ CO blueshifted emission, a clump of material is identifiable in the blueshifted $^{13}$ CO emission at position angle east of north of $\\\\sim $ 220$^{\\\\circ }$ seemingly disconnected or 'pinched' from the central region (label B).\",\n \"The prominent blueshifted outflow feature (Figure REF left) identifiable at position angle $\\\\sim $ 330$^{\\\\circ }$ (label E) has an opening angle of $\\\\sim $ 50$^{\\\\circ }$ extending to distances of $\\\\sim $ 11$^{\\\\prime \\\\prime }$ ($\\\\sim $ 4500 au).\",\n \"The peak intensity in the blueshifted image is located $\\\\sim $ 1$^{\\\\prime \\\\prime }$ north-west of V1647 Ori, in the direction of the outflow.\",\n \"A second collimated outflow is detected in the redshifted line emission pointing almost directly south and extending a distance of 12.8$^{\\\\prime \\\\prime }$ ($\\\\sim $ 5300 au; label D).\",\n \"Similar to the case of the blueshifted emission, the peak intensity of the redshifted image is located $\\\\sim $ 1$^{\\\\prime \\\\prime }$ south-east of V1647 Ori, in the direction of the southern outflow.\",\n \"The systemic $^{13}$ CO emission feature (Figure REF center) displays an apparent \\\"hole\\\" coincident with the location of the collimated $^{13}$ CO blueshifted outflow (label F).\",\n \"Figure: Side by side comparison of blue, systemic, and red shifted moment 0 images of 13 ^{13}CO.\",\n \"The central location of the V1647 Ori disc continuum (Fig ) is shown with a black cross.\",\n \"Figures are labeled according to the text in Section 3.\",\n \"All 13 ^{13}CO channels associated with emission can be seen in Figure 1 of the Appendix.\"\n ],\n [\n \"Emission from C$^{18}$ O\",\n \"C$^{18}$ O emission, typically a tracer of higher column density gas in molecular cloud environments, is displayed in Figure REF .\",\n \"Distinct features are seen in the blue, systemic and red shifted velocity channels.\",\n \"The blue shifted C$^{18}$ O features trace the collimated outflowing material (label E) previously identified in the $^{13}$ CO integrated intensity images (Figure REF ).\",\n \"This feature has roughly the same opening angle and angular size as its blueshifted $^{13}$ CO counterpart.\",\n \"Similar to the case of the blueshifted $^{13}$ CO emission, the peak intensity location in the blueshifted C$^{18}$ O is located $\\\\sim $ 1$^{\\\\prime \\\\prime }$ northwest of V1647 Ori in the direction of the blueshifted outflow.\",\n \"The same clump located at position angle $\\\\sim $ 220 from the $^{12}$ CO and $^{13}$ CO blueshifted integrated intensity images, apparently disconnected ('pinched') from the main emission features of V1647 Ori, is also observed to the south west in blueshifted C$^{18}$ O emission (label B).\",\n \"While the redshifted C$^{18}$ O emission does not display the same collimated feature as is evident in the $^{13}$ CO redshifted emission, it does however display two bright peaks, one of which is coincident with the location of the peak intensity of the $^{13}$ CO redshifted emission south east of V1647 Ori (i.e., in the direction of the southern collimated outflow).\",\n \"The second bright peak in the C$^{18}$ O redshifted image is very close to the location of the V1647 Ori disc and may be a detection of circumstellar disc gas.\",\n \"Figure: Side by side comparison of blue, systemic, and red shifted moment 0 images of C 18 ^{18}O.\",\n \"The central location of the V1647 Ori disc continuum (Fig ) is shown with a black cross.\",\n \"Figures are labeled according to the text in Section 3.\",\n \"All C 18 ^{18}O channels associated with emission can be seen in Figure 1 of the Appendix.A strong detection of C$^{18}$ O at systemic velocities likely represents the dense material from which the V1647 Ori protostar formed and may impact the structures identifiable in the other spectral lines.\",\n \"To emphasize this role, the systemic C$^{18}$ O emission contours are overlaid on blueshifted $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O integrated intensity images in Figure REF .\",\n \"The systemic C$^{18}$ O emission rests as the base of the bowl identified in blueshifted $^{12}$ CO (Figure REF left) and also is spatially coincident with an apparent discontinuity between the blueshifted $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O emission at position angle $\\\\sim $ 220$^{\\\\circ }$ (label B; i.e., these emission features appear to 'bend' around the material probed by systemic C$^{18}$ O emission).\",\n \"Moreover, the large ($\\\\sim $ 10$^{\\\\prime \\\\prime }$ ) bright extended emission feature in the south east part of the systemic C$^{18}$ O emission appears to be in line with the axis of the $^{13}$ CO and C$^{18}$ O collimated outflow (label G in Figure REF ).\",\n \"The redshifted C$^{18}$ O material may also play a role in shaping of the nebula as displayed in Figure REF .\",\n \"Redshifted C$^{18}$ O emission material may be shaping the long extended arm observed in redshifted $^{12}$ CO that is apparently connecting the V1647 Ori system to an extended feature at a projected distance of $\\\\sim $ 8300 au.\",\n \"Figure: C 18 ^{18}O systemic emission (white contours) is shown overlaid on blueshifted 12 ^{12}CO, 13 ^{13}CO and C 18 ^{18}O integrated intensity images.\",\n \"The contour levels for the systemic C 18 ^{18}O emission are 4.0, 5.5, 6.0, 6.5, and 6.75 ×\\\\times 3σ\\\\sigma .\",\n \"The beam is displayed in the lower left.Figure: Redshifted C 18 ^{18}O contours (0.6, 0.7, 0.85, 1.0, 1.1, and 1.2 ×\\\\times 3σ\\\\sigma ) overlaid on the redshifted 12 ^{12}CO integrated intensity image of V1647 Ori.\"\n ],\n [\n \"Dynamics of the V1647 Ori Environment\",\n \"Figure REF highlights the various outflows/features identified in the previous sections (labels D and E from Figures REF , REF , and REF ).\",\n \"These features can be summarized in the following way: a $\\\\sim $ 50$^{\\\\circ }$ collimated outflow at position angle of $\\\\sim $ 330$^{\\\\circ }$ is detected in blueshifted $^{13}$ CO and C$^{18}$ O (Figure REF left), a $\\\\sim $ 100$^{\\\\circ }$ excavated cavity pointing directly north of V1647 Ori is detected in both blueshifted $^{12}$ CO and systemic C$^{18}$ O and misaligned with the aforementioned collimated outflow (Figure REF center), and a $\\\\sim $ 30$^{\\\\circ }$ collimated outflow detected in redshifted $^{12}$ CO and $^{13}$ CO pointing directly south of V1647 Ori (Figure REF right).\",\n \"The dynamical age ($\\\\tau _{d}$ ) of these outflows can be estimated using their physical extent measured from the location of the V1647 Ori disc ($r_\\\\mathrm {outflow}$ ) and their maximum velocities relative to the velocity of the system (i.e., $\\\\tau _{d}$ = $\\\\frac{r_\\\\mathrm {outflow}}{v_\\\\mathrm {system}-v_\\\\mathrm {max}}$ ).\",\n \"However, these dynamical ages are relatively uncertain given the low inclination of this system (i = 17$^{\\\\circ }$ ).\",\n \"Since each of the two collimated outflows were detected with multiple emission lines, we can estimate two independent dynamical ages for each outflow.\",\n \"These dynamical ages and the values used to determine them are displayed in Table REF and indicate the outflow identified in blueshifted $^{13}$ CO and C$^{18}$ O has a dynamical age of $\\\\sim $ 11,700 years whereas the outflow identified in redshifted $^{12}$ CO and $^{13}$ CO has a dynamical age of $\\\\sim $ 17,200 years.\",\n \"Given an uncertainty of $\\\\pm $ 7 pc for a distance of 414 towards the Orion Nebula , these dynamical age estimates may be different by $\\\\sim $ 200 and 300 years, respectively.\",\n \"However, we caution that the dynamical age of the redshifted $^{12}$ CO feature is highly uncertain given that it extends to radii larger than the ALMA primary beam.\",\n \"We do not estimate the dynamical age of the $^{12}$ CO bowl-like feature pointed directly north because this feature is more likely a cavity rather than an outflow.\",\n \"However, it should be noted that the northern bowl-like emission has a similar blueshifted $^{12}$ CO maximum velocity ($v_\\\\mathrm {max}$ = 3) as the collimated redshifted $^{12}$ CO feature pointed directly south ($v_\\\\mathrm {max}$ = 2.5).\",\n \"Figure: Contours overplotted on blueshifted 13 ^{13}CO, C 18 ^{18}O and redshifted 12 ^{12}CO integrated intensity images of V1647 Ori to highlight outflow/cavity morphology.\",\n \"Left: Blueshifted C 18 ^{18}O contours with levels of 1.0, 1.75, 4.0, 6.0, and 8.0 ×\\\\times 3σ\\\\sigma .\",\n \"Middle: Blueshifted 12 ^{12}CO contours with levels 4.0, 10.0, 12.0, 25.0 and 35.0 ×\\\\times 3σ\\\\sigma .\",\n \"Right: Redshifted 13 ^{13}CO contours with levels 1.0, 1.5, 2.0, 2.5, and 3.2 ×\\\\times 3σ\\\\sigma .Table: Dynamical Ages of the Outflows in V1647 OriPhysical characteristics (e.g., mass, momentum, and kinetic energy) of the molecular gas surrounding the V1647 Ori protostar can be estimated using primary beam corrected $^{12}$ CO and $^{13}$ CO line emission.\",\n \"We follow the procedure outlined in and adopt a $^{12}$ CO to H$_{2}$ abundance ratio of 10$^{-4}$ and a $^{12}$ CO to $^{13}$ CO relative abundance of 62 .\",\n \"This analysis assumes a constant excitation temperature of T$_\\\\mathrm {ex}$ = 50 K along the line of sight and a beam filling factor of 1.\",\n \"We then calculate the outflow mass M =$\\\\sum _{v}$ M($x,y,v$ ), momentum P = $\\\\sum _{v}$ M($x,y,v$ ) $\\\\times $ $v$ , and kinetic energy KE = $\\\\sum _{v}$ M($x,y,v$ ) $\\\\times $ $\\\\frac{1}{2}$$v^{2}$ for each velocity channel within a radius of 10$^{\\\\prime \\\\prime }$ of V1646 Ori (e.g., see Figure REF ).\",\n \"This calculation was performed on primary beam-corrected channel maps.\",\n \"Parameters for the blue and redshifted regions of $^{12}$ CO and $^{13}$ CO are displayed in Table REF and correspond to the velocity ranges indicated at the top of Figure REF and displayed in Figures REF and REF .\",\n \"Optical depth corrections, typically performed using the ratio of line fluxes between the optically thick $^{12}$ CO and optically thin $^{13}$ CO , are unable to be applied to this dataset because there are few channels where both $^{13}$ CO and $^{12}$ CO are detected.\",\n \"Therefore, the $^{12}$ CO parameters estimated in this work (Table REF ) are lower limits, and not necessarily representative of the entire mass of the outflow in $^{12}$ CO. Table: Physical Properties of the V1647 Ori Outflows\"\n ],\n [\n \"Optical Reflection Nebula (McNeil's Nebula)\",\n \"The $\\\\sim $ 1$^{\\\\prime }$ diameter optical r-band reflection nebula to the north of V1647 Ori is displayed in Figure REF .\",\n \"These data were taken during the 2008 outburst and display a generally diffuse nebulosity of dust with a clump of material to the north west near RA and Dec offset [-4, 10].\",\n \"Overlays of the $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O contours on the optical data indicate the millimeter gas emission is well aligned with the base of the reflection nebula.\",\n \"Moreover, the collimated outflow observed in $^{13}$ CO and C$^{18}$ O appear to be traveling in the direction of the bright dust clump observable at optical wavelengths.\",\n \"Figure: Optical Gemini GMOS r band observation of V1647 Ori and its reflection nebula taken during the 2008 outburst.\",\n \"Overlaid are integrated intensity image contours of 12 ^{12}CO, 13 ^{13}CO, and C 18 ^{18}O as indicated in the lower left of each panel.\",\n \"North is up and east is left\"\n ],\n [\n \"The Circumstellar Disc of V1647 and Potential Mechanism for Accretion Outbursts\",\n \"The resolved continuum image of the V1647 Ori circumstellar disc (Figure REF ) does not display clear evidence of spirals or clumps associated with disc fragmentation or perturbations from a potential binary companion.\",\n \"Therefore, it is unlikely there are any binary or massive substellar companions $\\\\gtrsim $ 40 au from the central star capable of producing the intense outbursts observed in V1647 Ori.\",\n \"However, the potential non-gaussian radial brightness profile identifiable at large radii in Figure REF suggests some amount of asymmetry and warrants a more detailed investigation than that performed for the continuum data and presented here.\",\n \"While the circumstellar disc is clearly detected in the dust continuum, it is less evident in the molecular line data.\",\n \"Molecular emission spatially coincident with the location of the disc is identifiable in the redshifted $^{12}$ CO integrated intensity image (Figure REF ) and is potentially detected but unresolved from extended cloud/outflow emission at blueshifted $^{12}$ CO and $^{13}$ CO and redshifted $^{13}$ CO and C$^{18}$ O.\",\n \"Such emission in these cases may not be from the disc but instead could be from a small region surrounding the disc.\",\n \"Moreover, gaseous emission from the disc was not detected in any of the emission lines at the systemic velocities (e.g., see Figure REF top and Appendix).\",\n \"Given the inconsistency of its detection, we were unable to search for signatures of Keplerian rotation.\",\n \"The non-detection of the disc at systemic velocities is likely due to the deeply embedded nature of this system which is consistent with its high extinction at optical and X-ray wavelengths.\",\n \"The radius of the disc derived from the continuum emission (Table REF ) is similar to that of the namesake of the FUor class as well as HBC 494 and V2775 Ori , .\",\n \"However, compared with V883 Ori, the V1647 Ori disc is a factor of $\\\\sim $ 2 smaller and its star/disc bolometric luminosity is a factor of 10 fainter .\",\n \"The V1647 Ori disc mass derived from the continuum is the least massive compared with the other FUor objects presented in this series.\",\n \"This suggests objects that have been classified strictly as FUors may be more massive than EXor objects where V1647 Ori is in the middle of this classification as it displays outbursts indicative of both classifications.\"\n ],\n [\n \"Excavated Cavities and Outflows\",\n \"The molecular emission line data indicate that the star-forming environment surrounding V1647 Ori is both structured and complex.\",\n \"Several features indicating the three-dimensional structure of this system are readily identifiable and show that the protostar and disc system are still deeply embedded in its parent molecular cloud.\",\n \"The orientation of the system can be further constrained when considering the direction of the blueshifted outflow (label E) coincident with northern reflection nebula (Figure REF ).\",\n \"The dust cavity appears in the north at optical wavelengths because it is preferentially forward scattering emission from its illuminating source (i.e., V1647 Ori during outburst).\",\n \"This indicates that the cavity is located between V1647 Ori and our line of sight which is consistent with the cavity detection in blueshifted $^{12}$ CO emission (label A).\",\n \"The co-alignment of this excavated northern cavity with the outflow at PA$\\\\sim $ 330$^{\\\\circ }$ is further substantiated by the location of the bright clumps visible during outburst in the optical reflection nebula since they are aligned with direction of the collimated outflow (Figure REF ).\",\n \"This orientation also explains the absence of a reflection nebula at optical wavelengths to the south of V1647 Ori even though a southern cavity is present (Figure REF ; i.e., the southern cavity is pointed away from us and any reflected light will be backscattered and too faint to detect in shallow observations).\",\n \"Given the appearance of its complex environment, the strongest constraints on the orientation of the V1647 Ori system is measured by the gaussian fit to the continuum data which indicates the disc is almost face-on (i = 17$^{\\\\circ }$ ) with a position angle of 109$^{\\\\circ }$ .\",\n \"This disc inclination is in good agreement with previous estimates of i $<$ 30$^{\\\\circ }$ , but is not consistent with the stellar inclination modeled from periodic X-ray emission generated by rotating accretion hot spots .\",\n \"The measured position angle of the disc (109$^{\\\\circ }$ $\\\\pm $ 19) appears to align more with the outflow at PA$\\\\sim $ 330$^{\\\\circ }$ (label E) than the southern collimated outflow at PA$\\\\sim $ 180$^{\\\\circ }$ (label D) assuming that outflows travel perpendicular to the plane of the disc and the disc is inclined such that the northern extended emission is blueshifted towards our line of sight.\",\n \"Since the disc orientation is a strong constraint on the system as a whole, our interpretation of the large scale (r $\\\\gtrsim $ 1-3$^{\\\\prime \\\\prime }$ ) emission features (e.g., collimated outflows) assumes that these features were formed from material traveling roughly perpendicular to the circumstellar disc of V1647 Ori at the time of their respective outflow triggering events.\",\n \"Such an assumption is reasonable given our current understanding of launching mechanism of material during protostellar evolution .\",\n \"The blueshifted $^{12}$ CO emission whose bowl-like shape conforms well to the base of the optical reflection nebula (Figure REF ) is seemingly misaligned with the aforementioned blueshifted $^{13}$ CO and C$^{18}$ O collimated feature (Figure REF , center).\",\n \"Moreover, the non-detection in $^{13}$ CO of this $^{12}$ CO blueshifted bowl-like shape suggests that there are no active wide-angle outflows at this location and that the $^{12}$ CO is probing the temperature of the walls of the excavated cavity.\",\n \"The misalignment between the collimated southern outflow (Figures REF , REF , label D) and the collimated northern outflow at PA$\\\\sim $ 330$^{\\\\circ }$ (Figures REF , REF , label E) may be the result of a re-orientation or precession of the mass-launching area (i.e., the orientation of the circumstellar disc may have changed from one outburst to another resulting in two outflows orientated with different position angles).\",\n \"The difference in position angle and the dynamical age between these collimated outflows (Table REF ) suggest each outflow was likely the result of a specific mass-accretion event with the southern outflow occurring $\\\\sim $ 5000 years before the north-west outflow.\",\n \"Given that a radical change in disc position angle is unlikely, these collimated outflows were probably launched from opposite sides of the circumstellar disc during their respective mass accretion events.\",\n \"While some asymmetric jets, like those detected from Herbig-Haro objects, appear to be shaped by their local star-forming environment , others systems exhibit clear evidence that the jet-launching region (i.e., the region perpendicular to the plane of the disc) has precessed over time resulting in outflows with asymmetric and misaligned features , , , .\",\n \"While the amount of precession over the lifetime of the outflow varies from case to case, it appears that jet precession of more than a few degrees may be rare .\",\n \"In cases where the degree of disc precession is high (like in the case of V1647 Ori), several mechanisms have been proposed capable of significantly re-orienting the outflow/jet: (a) radiative-induced disc warping where a jet, presumably oriented perpendicular to a disc, precesses in response to a disc warp which is generated in cases of high-incident radiation ; (b) tidal interaction from a companion in a non-coplanar orbit truncating and/or distorting the disc , ; or (c) anisotropic accretion events where an impact/merging of material onto the disc during anisotropic accretion can change the orientation of the disc angular momentum vector resulting in a net torque in the rotation of the disc , .\",\n \"Estimating the critical radius at which the disc becomes unstable to warping following equation 5 in , the assumption that $\\\\eta $ =1 in , and the physical characteristics of V1647 Ori during outburst (M = 0.8 $M_{\\\\odot }$ , $\\\\overset{.\",\n \"}{M}$$_\\\\mathrm {acc}$ = 4 $\\\\times $ $10^{-6}$ $M_{\\\\odot }$ yr$^{-1}$ , $L = 44 L_{\\\\odot }$ ), $R_\\\\mathrm {crit}$ = 2 $\\\\times $ 10$^{6}$ au.\",\n \"Given that this radius is orders of magnitude larger than the radius of the V1647 Ori disc, it is unlikely a disc warp is the origin of its misaligned outflows.\",\n \"However, it should be noted that the stellar parameters during the outburst event that produced these outflows ($\\\\sim $ 14,000 years ago; Table REF ) are not necessarily the same as those measured during the 2003 outburst.\",\n \"As noted in Section 4.1, the non-detection of any companions to V1647 Ori at radii > 40 au likely rules out tidal interactions via a companion causing disc precession unless such a companion exists at smaller radii and has not disrupted the V1647 Ori disc in such a way as to be evident from the continuum observations.\",\n \"Therefore, given the intrinsic connection between accretion and FUor/EXor outbursts, it appears that significant mass-loading (e.g., from a large clump of envelope material or a companion) during anisotropic accretion may be the most likely scenario to have caused the disc to precess between each outflow-generating outburst in V1647 Ori.\",\n \"If this anisotropic accretion scenario is true in the case of V1647 Ori, one might expect more FUor/EXor type objects to display multiple misaligned outflows.\",\n \"However, there appears to be little evidence for this, at least in the case of the other FUors studied as part of this series , , .\",\n \"Given the complex geometry of the V1647 Ori star-forming environment, it may be that other FUor/EXor objects underwent smoother mass-accretion outbursts which did not result in disc precession or simply had fewer large-scale mass accretion events.\",\n \"Z CMa, a binary FUor object that also exhibits properties of an EXor, displays two slightly misaligned outflows where a wiggling shape of one of the outflows may be generated via precession from an unseen additional companion , .\",\n \"Given that Z CMa and V1647 Ori represent a rare class of objects that exhibit properties of both FUor and EXor objects, we speculate that disc precession may be related to this intermediate classification.\",\n \"While the disc precession argument seems the most plausible, other scenarios could be invoked to describe the complicated geometry of these two collimated emission features.\",\n \"The two outflows could originate from separate binary members ( e.g., each with their own outflow) if V1647 Ori has an undetected companion.\",\n \"In this case, outflow misalignment may be a signature that the potential binary formation of V1647 Ori resulted from turbulent fragmentation .\",\n \"However, this scenario generally considers binaries with separations much larger than what would likely be the case for V1647 Ori ($\\\\lesssim $ 40 au).\",\n \"Another scenario alternative to changes in disc orientation is the case that the outflows instead are located at different position angles as a result of a dense material within the molecular cloud shaping the outflow.\",\n \"V1647 Ori clearly resides in a complex and embedded environment and the outflow structure could simply be following a path of least resistance due to some unobserved molecular cloud component whose mass would be difficult to determine without knowledge of the pre-outflow geometry.\",\n \"This scenario would also affect the dynamical age calculations if the outflowing material does not travel in a straight line.\",\n \"Another explanation alternative to disc precession could be that the collimated features observed are actually inflows and not outflows.\",\n \"This interpretation would require our understanding of the system geometry to be reversed (i.e., the blueshifted outflow situated between V1647 Ori and our line of sight would instead be behind V1647 Ori and traveling towards the system presumably in a large accretion stream).\",\n \"However, this scenario is unlikely to be true given the spatial correlation between the blueshifted $^{13}$ CO and C$^{18}$ O emission with the northern cavity of the optical reflection nebula which independently indicates this northern region is between V1647 Ori and our line of sight and thus, the emission feature is more likely an outflow.\"\n ],\n [\n \"The Shaping Potential of Dense Molecular Gas Surrounding V1647 Ori\",\n \"Given the structure of the red and blueshifted molecular line emitting material compared to that of the systemic C$^{18}$ O emission (Figure REF ), it is evident the dense material in which the protostar is embedded influences the overall structure of the system.\",\n \"While it is not clear whether this dense material can result in two misaligned outflows (Section 4.2), it is still evident that the blueshifted $^{12}$ CO bowl-like shape is surrounded by the dense C$^{18}$ O material of similar shape (Figure REF left).\",\n \"This is in good agreement with the location of the wall of the excavated cavity as seen in $^{12}$ CO, suggesting the shaping mechanism that produced this cavity compressed the low-density ambient molecular material until it collided with the dense material as probed by C$^{18}$ O.\",\n \"Moreover, the western arm of the systemic C$^{18}$ O material appears to have cut off or 'pinched' the material associated with blueshifted $^{12}$ CO and $^{13}$ CO material $\\\\sim $ 3$^{\\\\prime \\\\prime }$ southwest of the V1647 Ori disc (label B).\",\n \"The outflowing blueshifted material appears to travel around the dense material at systemic velocities as it may be more energetically favorable.\",\n \"If this interpretation is true then the location of the dense systemic C$^{18}$ O material must be between our line of sight and the V1647 Ori disc (i.e., in the foreground but still part of the V1647 Ori star-forming environment).\",\n \"This interpretation suggests that the local dense gas plays a large role in the shaping of cavities and molecular outflows during this stage of pre-MS stellar evolution.\",\n \"A similar shaping scenario may be taking place where the redshifted C$^{18}$ O emission appears along the border of the 'arm' seemingly connecting the V1647 Ori system to the large unresolved feature $\\\\sim $ 20$^{\\\\prime \\\\prime }$ (8300 au) to the west (Figure REF ).\",\n \"This large $\\\\sim $ 40$^{\\\\prime \\\\prime }$ (17,000 au) column of material is also detected in a few channels of $^{13}$ CO emission traveling at the same velocity (Figure 2 in the appendix).\",\n \"The source of this structure is unclear and it may just be leftover cloud material that either has yet to form a star or does not meet the physical requirements to do so.\",\n \"However, we caution that this feature may be an artifact from emission outside the primary beam.\",\n \"The closest pre-MS star (in projection) to this structure (2MASS J05461162-0006279) is located $\\\\sim $ 7$^{\\\\prime \\\\prime }$ (3150 au) from this extended structure's south-western edge.\",\n \"It is identified as a Class II object in the L1630 star-forming region based on its infrared-excess and detection in X-rays and has a moderate absorption of N$_{H}$ = 5 $\\\\times $ 10$^{21}$ cm$^{-2}$ .\"\n ],\n [\n \"Outflow Dynamics and a Comparison with Other FUors\",\n \"The complex structures identified in the V1647 Ori at spatial scales of $\\\\sim $ 1.0$^{\\\\prime \\\\prime }$ can be compared with the other similarly structured FUor/EXor type objects presented in this series , , .\",\n \"Of the eight FUor/EXor objects observed, only V2775 Ori, V883 Ori, HBC 494 and V1647 Ori display significant amounts of extended $^{12}$ CO, $^{13}$ CO and/or C$^{18}$ O emission.\",\n \"Given the high A$_V$ , N$_H$ , complex C$^{18}$ O environment, and ambiguous detection of its gaseous circumstellar disc, V1647 Ori appears to be in the earliest stage of stellar evolution compared with the other FUor objects presented.\",\n \"In particular, the structures displayed in these ALMA observations of V1647 Ori could represent the progenitor stages of those of V883 Ori and V2775 Ori given their similar inclination (and thus similar viewing angle).\",\n \"Similar to the case of V2775 Ori , V1647 Ori also exhibits a small outflow opening angle ($\\\\sim $ 30$^{\\\\circ }$ ) in $^{13}$ CO, C$^{18}$ O, further indicating its primordial nature if opening angle is a correlation to age.\",\n \"In contrast, HBC 494 and V883 Ori , have much larger opening angles of $\\\\sim $ 150$^{\\\\circ }$ .\",\n \"Almost counterintuitively, the outflows of V1647 Ori have the oldest dynamical ages (11,700 and 17,200 years) when compared to that of V2775 Ori , HBC 494 and V883 Ori .\",\n \"These dynamical ages are very small given the timescale of pre-MS stellar evolution with respect to disc dissipation (e.g., $\\\\sim $ 1-3 Myr).\",\n \"Therefore the older dynamical timescale of V1647 Ori's outflows do not necessarily mean it is inconsistent with the interpretation here that this system is less evolved than its other FUor counterparts.\",\n \"It could simply mean that whatever mechanism initiates these outbursts may have started earlier in V1647 Ori.\",\n \"Given the complex environment surrounding V1647 Ori, it may indicate that inhomogeneous patches of envelope material could have prompted more frequent or earlier accretion outbursts.\",\n \"The outflow mass surrounding V1647 Ori (as indicated in Section 3.6) appears similar to those obtained for other Class 0-II pre-MS stars with outflows as well as the small sample of FUor type objects whose outflows have been measured .\",\n \"However, when comparing the outflow velocities associated with these objects, it is clear that the outflows surrounding V1647 Ori are quite slow ($\\\\sim $ 1-2 km s$^{-1}$ ).\",\n \"Given the similar sensitivity and resolution of the other FUors detected as part of this series, we compare the results of V1647 Ori directly with those of V2775 Ori, HBC 494, and V883 Ori.\",\n \"Of these sources, V1647 Ori has the least massive circumstellar disc and is the only source considered both an FUor and EXor type object based on its unusual outburst frequency.\",\n \"V1647 Ori is the most similar in disc mass and inclination to V2775 Ori , however their local environments are quite distinct.\",\n \"While V2775 Ori displays an impressive ring-like structure in its blue and red shifted emission , there is little evidence of complex structure surrounding the large-scale environment of the protostar indicating that this system may be more evolved than V1647 Ori.\",\n \"Moreover, V2775 Ori exhibits a redshifted $^{12}$ CO outflow velocity of 4.4 km s$^{-1}$ , a factor of $\\\\sim $ 2 faster than those of V1647 Ori, potentially indicative of a emptier environment surrounding the star where the outflow doesn't slow down from interactions with surrounding material.\",\n \"A comparison of outflow mass and energy indicates the outflow of V1647 Ori as probed by $^{12}$ CO and $^{13}$ CO is more massive than those in V2775 Ori by a factor of 3 and 15, respectively.\",\n \"A comparison of V1647 Ori with that of HBC 494 and V883 Ori, both of which have more massive and larger circumstellar discs, indicates the outflow mass of V1647 Ori is significantly smaller , .\",\n \"Both these systems exhibit slow outflow speeds similar to or smaller than V1647 Ori possibly indicating they too still are interacting with or have been influenced by surrounding material.\"\n ],\n [\n \"Summary and Conclusions\",\n \"We have presented ALMA observations of the V1647 Ori continuum at a resolution of 0.2$^{\\\\prime \\\\prime }$ as well as molecular line emission of $^{12}$ CO,$^{13}$ CO and C$^{18}$ O at a resolution of $\\\\sim $ 1$^{\\\\prime \\\\prime }$ .\",\n \"No emission line features were detected in the 0.2$^{\\\\prime \\\\prime }$ dataset indicating that structure surrounding V1647 Ori are fairly uniform on large scales.\",\n \"These observations are presented as part of a series and we compare these results of V1647 Ori to that of other FUor type objects with extended emission (e.g., V2775 Ori, HBC 494, and V883 Ori).\",\n \"We have identified a complex yet structured system surrounding the V1647 Ori protostar and circumstellar disc and are able to draw several conclusions: $\\\\bullet $ The circumstellar disc of V1647 Ori is resolved and has a nearly face-on inclination (i = 17$^{\\\\circ }$ ), a radius of 40 au and a total disc mass of $\\\\sim $ 0.1 M$_{\\\\odot }$ .\",\n \"The continuum resolved disc does not display any disc structures (e.g., spirals and/or fragmentation) associated with nearby interacting companions (r $\\\\gtrsim $ 40 au) and therefore such a mechanism, under these constraints, is unlikely to be responsible for the various observed accretion outbursts.\",\n \"The mass of the disc is less than the other FUor type objects presented in this series may be related to its classification as both an FUor and EXor type object.\",\n \"$\\\\bullet $ Blueshifted $^{12}$ CO molecular line emission spatially coincident with the V1647 Ori optical reflection nebula (McNeil's Nebula) and a non-detection of this feature in $^{13}$ CO and C$^{18}$ O emission confirm that this structure is a cavity excavated during a previous accretion outburst.\",\n \"A collimated $^{13}$ CO and C$^{18}$ O emission feature aligns with clumps in the nebula identified during the recent multiwavelength outburst.\",\n \"$\\\\bullet $ Dense molecular material as probed via C$^{18}$ O emission traveling at a systemic velocity of $\\\\sim $ 10 km s$^{-1}$ is spatially coincident with the base of the northern excavated cavity and likely shaped the outflow material identified in blueshifted $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O emission.\",\n \"$\\\\bullet $ Two distinct collimated ($\\\\sim $ 30-50$^{\\\\circ }$ ) outflows with position angles of 330$^{\\\\circ }$ and 180$^{\\\\circ }$ have been detected with dynamical ages of $\\\\sim $ 11,700 and $\\\\sim $ 17,100 years, respectively.\",\n \"These outflows are likely the result of two distinct mass accretion events where outflows were launched from opposite sides of a disc that has changed orientation over the last $\\\\sim $ 5000 years from previous anisotropic accretion events.\",\n \"$\\\\bullet $ Compared to the other FUor objects included in our series, V1647 Ori displays the most complex large-scale structures likely the result of its comparatively early stage of stellar evolution.\",\n \"A comparison of outflow parameters supports this notion and also indicates that the outflow velocities of these FUor and FUor/EXor type objects ($\\\\sim $ 0.5 - 4 km s$^{-1}$ ) are relatively small compared to those observed in typical pre-MS stars and predicted by simulations ($v$$\\\\sim $ 10-30 km s$^{-1}$ ).\",\n \"The complex star-forming environment surrounding V1647 Ori may result in more frequent yet still extreme outburst events which justify its classification as both an FUor and EXor type object.\",\n \"$\\\\bullet $ A large ($\\\\sim $ 17,000 au) unresolved structure is identified $\\\\sim $ 8300 au from V1647 Ori.\",\n \"This structure is not associated with any obvious source but is seemly connected to V1647 Ori by an 'arm' of molecular material as probed with redshifted $^{12}$ CO emission.\"\n ],\n [\n \"Acknowledgements\",\n \"This research was supported by CONICYT-FONDECYT awards (3150550, 1171246) and support from the Millennium Science Initiative (Chilean Ministry of Economy; grant Nucleus RC 130007).\",\n \"This paper makes use of the following ALMA data: ADS/JAO.ALMA #2013.1.00710.S .\",\n \"ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.\",\n \"The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.\",\n \"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.. D. P. acknowledges funding by the National Aeronautics and Space Administration through Chandra Award Number GO6-17013A.\",\n \"K.M acknowledges funding by the Joint Committee of ESO/Government of Chile, and funding by the Science and Technology Foundation of Portugal (FCT), grant No.\",\n \"IF/00194/2015.\",\n \"H.C. acknowledges support from the Spanish Ministerio de Economía y Competitividad under grant AYA 2014-55840-P. J.J.T acknowledges support from the University of Oklahoma, the Homer L. Dodge endowed chair, and grant 639.041.439 from the Netherlands Organisation for Scientific Research (NWO).\",\n \"The authors would like to thank the referee whose comments and suggestions increased the quality of this work.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01924"}}},{"rowIdx":890,"cells":{"text":{"kind":"list like","value":[["Linear gyrokinetic investigation of the geodesic acoustic modes in\n realistic tokamak configurations"],["Abstract Geodesic acoustic modes (GAMs) are studied by means of the gyrokinetic global particle-in-cell code ORB5.","Linear electromagnetic simulations in the low electron beta limit have been performed, in order to separate acoustic and Alfv\\'enic time scales and obtain more accurate measurements.","The dependence of the frequency and damping rate on several parameters such as the safety factor, the GAM radial wavenumber and the plasma elongation is studied.","All simulations have been performed with kinetic electrons with realistic electron/ion mass ratio.","Interpolating formulae for the GAM frequency and damping rate, based on the results of the gyrokinetic simulations, have been derived.","Using these expressions, the influence of the temperature gradient on the damping rate is also investigated.","Finally, the results are applied to the study of a real discharge of the ASDEX Upgrade tokamak."],["Introduction","The ion heat transport in the plasma core is governed by turbulence formed by a class of microinstabilities such as toroidal ion temperature gradient (ITG) driven modes [1].","ITG turbulence is known to self-organize to form macroscopic structures [2].","These structures take the form of a macroscopic radial electric field which depends only on the radial coordinate.","$E \\times B$ poloidal flows associated with this electric field are referred to as zonal flows (ZFs) [3], [4], [5], [6].","The action of the toroidal magnetic field curvature on the ZF gives rise to oscillations of the radial electric field.","These oscillations of the ZFs are called geodesic acoustic modes (GAMs) [7], [8].","The modes are observed predominantly in the edge region of the tokamak plasmas with characteristic frequency of the order of the sound frequency $\\sim c_s/R$ , where $c_s = \\sqrt{T_e/m_i}$ is the sound speed, $R$ is the major radius.","One of the main linear damping mechanisms for the stationary ZF are collisional processes and for the GAM it is a collisionless wave-particle interaction, namely the Landau damping, and collisional damping at the very edge of the plasma, where equilibrium temperatures drastically decrease [9].","A recent comparison of collisionless and collisional damping of GAMs, using existing analytical theories, for experimentally relevant plasmas was done in Ref.","[10].","The importance of the ZF is that they can regulate the drift-wave (DW) turbulence [11].","But it is still a question how the GAMs influence the ZF efficiency of the DW suppression [8], [12], [13].","On the other hand, the development of zonal structures can play a key role in the transition from the low to the high confinement regime (L-H transition) [14].","In Ref.","[15] interaction of the mean and oscillatory poloidal flows with the turbulence were experimentally observed.","The turbulence suppression by the ZFs was observed in experiments described in Ref.","[16].","On the other hand, in Ref.","[17] the role of the mean flow in the dynamic evolution towards the H-mode is emphasized.","In Ref.","[12] two predators - one prey system, including ZF, GAM and turbulence, was developed to study transitions between states with different combinations of the ZF and GAM.","In this paper, we investigate the GAM frequency and collisionless damping rate, carrying out linear collisionless simulations with kinetic electrons.","The electromagnetic global gyrokinetic particle-in-cell code ORB5 is used [18], [19].","As it has been reported previously [20], [21], models, numerical or analytical, derived with adiabatic electrons, result in considerably smaller GAM damping rate in comparison to simulations performed with kinetic electrons.","By adiabatic electron models, we mean here models treating the $m\\ne 0$ component of the electrons as adiabatic, and setting the zonal component of the electron density perturbation to zero.","In simulations considered in this paper, electrondocumentclasss are treated drift-kinetically, and a realistic ion-electron mass ratio is used.","Moreover, to study the influence of the plasma elongation on the GAM dynamics, magnetic equilibria with realistic plasma shapes are considered.","To summarize the results obtained in different plasma regimes, interpolating formulae for the GAM frequency and damping rate, based on the gyrokinetic simulations with ORB5, are derived.","Due to the so-called phase mixing effect, the GAM damping rate is increased in the presence of a temperature gradient or the safety factor profile [8], [22], [23].","This effect arises when the damping rate of the wave depends on its wavenumber.","In the case of the GAM the damping rate increases with the GAM radial wavenumber (more precisely, with the radial wavenumber of the radial electric field).","Since the GAM frequency depends on the temperature and safety factor, the GAM oscillates with different frequencies at different radial points in presence of the temperature gradient or magnetic shear.","Distorting the GAM radial structure and creating higher radial wavenumbers, this process can strongly increase the GAM damping rate [8], [24].","A section of our paper is dedicated to the extension of previous works [21], [23], which were done treating the electrons as adiabatic, and in circular flux surfaces, to the inclusion of kinetic electrons and realistic tokamak configurations.","Finally, the last section of this paper is dedicated to the investigation of a realistic discharge of ASDEX Upgrade, described in Ref. [25].","In Appendix we have shown a comparison between ORB5 and GENE for the case of non-flat temperature profile."],["Model","The gyrokinetic simulations presented in this work have been performed with the code ORB5 [18], [19].","ORB5 is a nonlinear gyrokinetic multi-species global particle-in-cell (PIC) code, which solves the Vlasov-Maxwell system in the electrostatic or electromagnetic limit, and has a capability of handling true MHD equilibrium for an axisymmetric toroidal plasma.","The particle-in-cell method consists of coupling a particle-based algorithm for the Vlasov equation with a grid-based method for the computation of the self-consistent electromagnetic fields.","Several physical models are available in ORB5, all of them derived from a systematic Hamiltonian theory [19], [26] to provide exact energy and momentum conservation.","In this work, only one ion species (deuterium) has been considered while the electrons are assumed to be drift-kinetic.","This corresponds to the following gyrokinetic total Lagrangian: $L =&& \\sum _{\\rm {sp}}\\int \\mathrm {d}V \\mathrm {d}W \\left(\\left(\\frac{q}{c}{A}+p_\\parallel {b}\\right)\\cdot \\dot{{R}} + \\frac{m c}{q}\\mu \\dot{\\theta } - H_0-H_1\\right)f \\\\&&+ \\int \\mathrm {d}V \\mathrm {d}W H_2 f_{M,ions}-\\int \\mathrm {d}V \\frac{B_\\perp ^2}{8\\pi }.\\nonumber $ The velocity variables are the magnetic moment $\\mu \\equiv (mv_\\perp ^2)/(2B)$ , the canonical parallel momentum $p_\\parallel $ and the gyroangle $\\theta $ .","The equilibrium magnetic field is ${B}=\\nabla \\times {A}$ , $m$ and $q$ are the mass and charge of the particle species $sp$ and $c$ is the speed of light.","The volume element of the velocity space is $\\mathrm {d}W\\equiv (2\\pi )/m^2 B^*_\\parallel \\mathrm {d}p_\\parallel \\mathrm {d}\\mu $ with $B^*_\\parallel ={B^*}\\cdot {{b}}$ , ${b}={B}/B$ and ${B^*}={B}+(c/q)p_\\parallel \\nabla \\times {b}$ ; $\\mathrm {d}V$ denotes the volume element in physical space.","Here $f$ is the distribution function for the species $sp$ , while $f_{M,ions}$ is the equilibrium time independent distribution function of the ions.","In this system, only long wavelength electrostatic perturbation and magnetic perturbations perpendicular to the equilibrium magnetic field are considered.","Note that no second order term in the fields is retained for the electrons, this is equivalent to neglect the electron polarization density in the Polarization equation (drift-kinetic approximation, see Ref.","[19] for details).","The first two terms in the total Lagrangian define the charged particles Lagrangian [27].","The GK Hamiltonian in general depends on the electrostatic potentials $\\Phi $ and on the parallel component of the fluctuation magnetic potential $A_\\parallel $ .","The third term in the total Lagrangian is the electromagnetic field Lagrangian, in which the electric field component has been neglected (quasi-neutrality approximation, see [19] for details).","In this work we used the following Hamiltonian: $H&=&H_0+H_1+H_2\\\\H_0&=&\\frac{p_\\parallel ^2}{2m}+\\mu B\\nonumber \\\\H_1&=& e(J_0\\Phi - \\frac{p_\\parallel }{mc} J_0A_\\parallel )\\nonumber \\\\H_2 &=& - \\frac{q^2}{2mc^2}(J_0A_\\parallel )^2+\\frac{mc^2}{2B^2}|\\nabla _\\perp \\Phi |^2\\nonumber $ the gyroaveraging (Hermitian) operator $J_0$ , applied to an arbitrary function $\\psi $ in configuration space, is defined by $(J_0\\psi ) ( \\mathbf {R},\\mu ) = \\frac{1}{2\\pi }\\int _0^{2\\pi } \\psi ( \\mathbf {R}+{\\rho }(\\alpha )) \\,d \\alpha ,$ where ${\\rho }$ is the vector going from the guiding center position to the particle position.","In this work we have assumed $J_0=1$ for the electrons (drift-kinetic approximation).","The gyrokinetic equations for the particle distribution function and the GK field equations can be derived from the GK Lagrangian using variational principles.","In summary, the GK model used in the following is: $\\bullet $ gyrokinetic full-f Vlasov equation for the ions $&&\\frac{\\partial {f_i}}{\\partial {t}}+\\dot{{R}_i}\\cdot \\nabla f_i+\\dot{p_{\\parallel ,i}}\\frac{\\partial {f_i}}{\\partial {p_{\\parallel ,i}}}=0,\\\\&&\\dot{{R}_i}=\\left(\\frac{p_{\\parallel ,i}}{m_i}-\\frac{Z_ie}{m_ic}J_0A_\\parallel \\right)\\frac{{B_i^*}}{B^*_{\\parallel ,i}}+\\frac{c}{Z_ieB^*_{\\parallel ,i}}{b}\\times \\left[\\mu _i\\nabla B + Z_ie \\nabla J_0 \\Psi _i \\right],\\\\&&\\dot{p_{\\parallel ,i}}=-\\frac{{B_i^*}}{B^*_{\\parallel ,i}}\\cdot \\left[\\mu _i \\nabla B + Z_ie\\nabla J_0\\Psi _i \\right],$ $\\bullet $ drift-kinetic full-f Vlasov equation for the electrons: $&&\\frac{\\partial {f_e}}{\\partial {t}}+\\dot{{R}_e}\\cdot \\nabla f_e+\\dot{p_{\\parallel ,e}}\\frac{\\partial {f_e}}{\\partial {p_{\\parallel ,e}}}=0,\\\\&&\\dot{{R}_e}=\\left(\\frac{p_{\\parallel ,e}}{m_e}+\\frac{e}{m_ec}A_\\parallel \\right)\\frac{{B_e^*}}{B^*_{\\parallel ,e}}-\\frac{c}{eB^*_{\\parallel ,e}}{b}\\times \\left[\\mu _e\\nabla B - e \\nabla \\Psi _e \\right],\\\\&&\\dot{p_{\\parallel ,e}}=-\\frac{{B_e^*}}{B^*_{\\parallel ,e}}\\cdot \\left[\\mu _e \\nabla B -e\\nabla \\Psi _e \\right],$ having introduced the generalized potential $\\Psi \\equiv \\Phi - \\frac{p_{\\parallel ,sp} }{m{_{sp}}c} A_\\parallel .$ $\\bullet $ Linear polarization equation in the long wave-length limit (and drift-kinetic electrons): $&&\\int \\mathrm {d}W_i Z_ieJ_0 f_i-\\int \\mathrm {d}W_e e f_e=-\\nabla \\cdot \\left(\\frac{n_0 m_ic^2}{B^2} \\nabla _\\perp \\Phi \\right)$ $\\bullet $ Linear Ampère's law: $\\int \\mathrm {d}W_i \\frac{4\\pi Z_ie }{m_i c} p_{\\parallel ,i} J_0 f_i &-& \\int \\mathrm {d}W_e \\frac{4\\pi e }{m_e c} p_{\\parallel ,e} f_e = \\\\&&\\frac{1}{d_e^2}A_\\parallel +\\frac{1}{d_i^2}A_\\parallel -\\nabla _\\perp ^2 A_\\parallel -\\nabla \\cdot \\frac{\\beta _{i}}{4}\\nabla _\\perp A_\\parallel $ where $n_0$ is the density associated with the equilibrium Maxwellian $f_M$ and $\\beta _{sp}=(4\\pi n_{sp} T_{sp})/B^2$ .","The skin depth is defined by $d^{-2}_{sp}=\\beta _{sp}/\\rho ^2_{sp}$ , where $\\rho ^2_{sp} = T_{sp}m_{sp}c^2/q^2_{sp}B^2$ , and it appears on the right-hand-side of the Ampére's law because of the choice of the velocity space variables $(p_\\parallel ,\\mu )$ instead of the usual $(v_\\parallel ,\\mu )$ .","The indexes $i$ and $e$ indicate ions and electrons respectively and $q_i=Z_ie$ is the ion charge while $q_e=-e$ is the electron charge.","Despite all the approximations made, this model is highly physically relevant and it can be used to describe not only the GAM and ZF dynamics, but also a large class of micro-instabilities excited by the density and temperature gradients, like ion temperature gradient (ITG) driven modes, trapped electron modes (TEM) or kinetic ballooning modes (KBM).","It also contained the reduced MHD model as a subset (see, among other, Ref.","[28]).","According to the PIC method the particle distribution function is discretized with macroparticles, known as markers.","The motion of the markers is calculated using the equations of motions of the gyrokinetic model while the electromagnetic fields are evolved on a spatial grid using the two field equations.","The charge and current density, that are necessary to solve the field equations, are calculated by projecting the marker weights on a spatial grid.","After that, the fields are calculated using a finite elements method.","The code is based on a straight-field-line coordinate system $(s, \\chi , \\phi )$ .","Here, radial coordinate is $s = \\sqrt{\\psi /\\psi _{edge}}$ (where $\\psi $ is the poloidal flux), $\\chi = \\displaystyle {\\frac{1}{q(s)}} \\int _0^\\Theta \\displaystyle {\\frac{{B}\\cdot \\nabla \\phi }{{B}\\cdot \\nabla \\Theta _1}} d\\Theta _1$ is the straight-field-line coordinate (where $\\Theta $ and $q$ are the poloidal angle and the safety factor respectively) and $\\phi $ is a toroidal angle.","Two different kinds of magnetic equilibria are implemented: analytical equilibria with circular concentric magnetic surfaces and ideal MHD realistic equilibria.","For the latter case, the ORB5 code is coupled with the CHEASE code [29], which solves the Grad-Shafranov equation with a fixed plasma boundary."],["Equilibrium and simulations parameters","Linear electromangetic gyrokinetic collisionless simulations with drift-kinetic electrons and a realistic electron - ion (deuterium) mass ratio $m_e / m_i = 2.5\\cdot 10^{-4}$ have been performed.","Electrostatic simulations with kinetic electrons are, in principle, faster than electromagnetic simulations, due to the smaller number of equations to be solved.","Nevertheless, a high frequency oscillation, called the $\\omega _H$ -mode [30], is observed to be often numerically unstable.","To decrease the level of the high-frequency oscillations, electromagnetic simulations in the small-$\\beta _e$ ($\\beta _e = 10^{-5}$ ) limit have been performed instead of the electrostatic ones.","MHD equilibria of the circular and elongated plasma have been calculated with an external code CHEASE [29].","Simulations have been carried out with a flat density profile, which have been shown to not impact the GAM frequency and damping rate in linear simulations (see Appendix ).","To focus on the Landau damping in the absence of the phase mixing effect, flat temperature profile has been considered in simulations used for the results of this section.","Since the safety factor profiles have been taken from the CHEASE, there is a magnetic shear, that also causes the phase mixing, but its influence on the GAM damping rate is much smaller in comparison to the temperature gradient effect.","Plasma parameters have been taken close to the ASDEX Upgrade parameters near the plasma edge [25]: the major radius $R = 1.65$ m, the minor radius $a = 0.5$ m (inverse aspect ratio is $\\epsilon = 0.303$ ), the magnetic field on the axis $B = 2$ T. Since in the CHEASE code the plasma elongation is defined at the edge and changes gradually to the plasma center, the GAM frequency and damping rate have been measured at the same radial position $s_0 = 0.90$ to perform more accurate scan on the elongation.","The temperature has been taken to be $T_i = T_e = 70$ eV.","It means that $c_s = \\sqrt{q_e T_e / 2 m_p} = 5.8 \\cdot 10^{3}$ m/s, $\\rho _s = c_s / \\omega _{ci} = 6.1\\cdot 10^{-4}$ m and $\\rho ^* = \\rho _s / a = 1.2\\cdot 10^{-3}$ .","Here, $\\omega _{ci} = q_e B/2m_p$ ($\\omega _{ci}/2\\pi = 15.2$ MHz) is the ion gyro-frequency, $q_e$ is an electron charge, and $m_p$ is a proton mass.","To weaken the constraint on the space step and to reduce the effect of the charge accumulation at the edge of the numerical work box, we have simulated only a ring from $s_1 = 0.85$ to $s_2 = 0.95$ in a poloidal cross section with the Dirichlet condition for the potential $\\phi $ on inner boundaries ($\\phi (s_1) = \\phi (s_2) = 0$ ).","A typical simulation has the following parameters.","Number of nodes in radial direction is taken to be $n_s = 256$ , in toroidal directions $n_{\\phi } = 4$ and along the straight-field-line coordinate $\\chi $ the number of nodes is $n_{\\chi } = 64$ .","Time step is $dt \\cdot \\omega _{ci} = 2$ .","The GAM damping rate and frequency have been calculated (see Appendix ) for different GAM radial wavenumbers $k = k_r\\rho _i \\in [0.054, 0.377]$ (where $\\rho _i = \\sqrt{2}v_{Ti}/\\omega _{ci} = 8.57 \\cdot 10^{-4}$ m is the ion Larmor radius of the deuterium and $v_{Ti} = \\sqrt{q_eT_i/2m_p}$ is the thermal speed), the safety factor $q \\in [3.5, 5.0]$ at $s_0 = 0.90$ and the plasma elongation $e \\in [1.0, 1.6]$ at the edge.","This is the regime where GAMs are typically observed in tokamak plasmas (see, for example, Ref.","[25]).","To simulate the GAM dynamics, the ORB5 simulations have been initialized by introducing an axisimmetric density perturbation designed to produce an initial electric potential field of the form $\\sim \\sin (ks)$ , where $s \\in [s_1, s_2]$ (as in the so-called Rosenbluth-Hinton test [4]).","All toroidal modes $n \\ne 0$ and poloidal modes $|m| > 10$ have been filtered out.","To study the GAM dynamics, the frequency and damping rate of the poloidally averaged radial electric field have been calculated."],["Results of gyrokinetic simulations","In Fig.","REF , a comparison of the GAM frequencies and damping rate obtained from numerical simulations with two analytical theories of Qiu 2009 [32] and Gao 2010 [31] is shown.","A good agreement between numerical results and analytical predictions of the GAM frequency has been found.","Nevertheless, the GAM damping rate, obtained from the theories, derived using adiabatic electrons, is smaller in comparison to numerical simulations with kinetic electrons, and the divergence increases for smaller values of the GAM radial wavenumber.","Moreover, since the frequency stops increasing in the domain of higher wavenumbers (subplot $a$ of Fig.","REF ), a divergence between numerical results and analytical theories is observed.","The same effect was observed in Ref.","[33], where the GAMs were studied using drift reduced Braginskii equations.","The Gao 2010 theory describes the GAM dependence on the plasma elongation and it is in a good agreement with numerical results for the frequency.","Although the Gao 2010 theory provides considerably smaller damping rate, it seems to give similar trend of the damping coefficient with the plasma elongation, i.e., the damping rate is weakened by the elongation.","The Gao 2010 theory was derived in the large orbit drift width limit, where the dominant damping mechanism is the resonance $\\omega \\sim \\omega _d$ (here, $\\omega _d = {k_r}\\cdot {v_d}$ is a magnetic drift frequency, ${k_r}$ is a wave vector of the zonal potential in the radial direction and ${v_d}$ is a magnetic drift velocity) [34].","As explained in Ref.","[31], the GAM frequency decreases with the elongation less rapidly than the drift frequency.","To satisfy the resonance $\\omega \\sim \\omega _d$ particles have to have higher drift velocities, which involves fewer particles in the wave-particle interaction and, as a result, the GAM damping rate decreases.","Figure: Comparison between numerically simulated values (dots, traingles and squares) of the GAM frequency and values obtained using the interpolating expression provided in Eq.","() (solid lines).","Dotted lines indicate 95% confidence bounds of the fitting."],["Interpolating formulae","To provide a scaling of the GAM frequency and damping rate, corresponding interpolating expressions have been fitted to the results of the gyrokinetic simulations described in Sec.","REF .","For consistency, the regime has been chosen for the GAM wavenumbers $k = k_r \\rho _i$ in the range $[0.054, 0.377]$ , safety factor $q \\in [3.5, 5.0]$ and plasma elongation $e \\in [1.0, 1.6]$ .","Figure: Comparison between numerically simulated values (dots, triangles or squares) of the GAM damping rate and values obtained by using the interpolating expression provided in Eq.","() (solid lines).","Dotted lines indicate 95% confidence bounds of the fitting.To derive an interpolating expression for the frequency several assumptions have been used.","The experimentally obtained dependence [25] on the plasma elongation $1 / (1 + e)$ has been slightly modified to $1 / (1 + g_6 e)$ , where $g_6$ is an adjustable coefficient.","The dependence on the safety factor has been taken in the form $\\exp (-g_5 q^2)$ .","In fact, the $q$ -dependence in a form of $\\sqrt{1 + g_5/q^2}$ , that is given in Ref.","[21], gives the same results.","To describe how the frequency changes with the radial wavenumber, a polynomial has been taken.","Moreover, to take into account the frequency saturation for higher wavenumbers[33] we have introduced a function of the form $1 / (1 + g_4 k)$ .","Here, $k = k_r \\rho _{i}$ , $v_{Ti} = \\sqrt{q_eT_i/2m_p}$ .","The resulting frequency interpolating formula is the following one: $f_{\\omega } \\left[\\displaystyle {\\frac{\\sqrt{2} v_{Ti}}{R}} \\right] = \\displaystyle {\\frac{g_1 + g_2 k^2 + g_3 k^4}{1 + g_4 k}} \\displaystyle {\\frac{\\exp \\left( -g_5 q^2 \\right)}{1 + g_6 e}}.$ Among different tested functions, this form gives the best approximation to numerically simulated values of the GAM frequency, it has one of the smallest 95% confidential bounds and is not overfitted.","The corresponding coefficients $g$ with their 95% confidential bounds (lower $g_{lc}$ and upper $g_{uc}$ bounds) are $g = &&[3.7733,\\ 6.3505,\\ -1.9741e1,\\ 1.3557e-1,\\ 1.4620e-3,\\ 1.1684],\\\\*g_{lc} = &&[3.6745,\\ 3.3168,\\ -2.8800e1,\\ -6.0078e-2,\\ 1.1373e-3,\\ 1.1234],\\\\*g_{uc} = &&[3.8720,\\ 9.3843,\\ -1.0682e1,\\ 3.3121e-1,\\ 1.7866e-3,\\ 1.2135].$ Results for the Eq.","(REF ) are depicted in Fig.","REF .","For the damping rate we have derived the following expression (here, the damping rate is normalized to $\\sqrt{2} v_{Ti}/R$ ): $f_{\\gamma } \\left[\\displaystyle {\\frac{\\sqrt{2} v_{Ti}}{R}} \\right] = \\displaystyle {\\frac{\\left(h_1 + h_2 k^2\\right)\\exp \\left[-h_3 q^2\\right]}{1 + h_4 e^2}} + \\displaystyle {\\frac{\\left(h_5 + h_6 k^2\\right)\\exp \\left[-h_7 q^2 \\right]}{1+h_8 e^4}}.$ with interpolating coefficients $h =&& [-1.2494e-2,\\ -8.9688e-1,\\ 4.5498e-2,\\ -1.9884e-1,\\\\*&&-1.1248e-2, -2.5481,\\ -5.3340e-3,\\ 7.7748e-1],\\\\h_{lc} =&& [-2.3115e-2,\\ -1.6490,\\ 2.5215e-2,\\ -3.3573e-1,\\\\*&&-2.5523e-2, -3.1909,\\ -1.9665e-2,\\ 5.1924e-2],\\\\h_{uc} =&& [-1.8723e-3,\\ -1.4471e-1,\\ 6.5781e-2,\\ -6.1955e-2,\\\\*&&3.0272e-3, -1.9053,\\ 8.9973e-3,\\ 1.5030].$ Comparison between the results from the gyrokinetic simulations and the interpolation expression for the GAM damping rate is shown in Fig.","REF for some specific values of parameters taken as examples."],["Phase mixing","To investigate the influence of the phase mixing on the GAM dynamics the same parameters as described in chapter REF has been used, but the temperature gradient (the same for both the electrons and ions to have $\\tau _e = T_e/T_i = 1$ , $T_i(s_0) = 70$ eV) at a radial position $s_0 = 0.90$ has been introduced.","The radial point $s_0 = 0.90$ has been chosen here to be in agreement with the section REF .","Initial radial wavenumber of the radial electric field is $k = 0.108$ .","The safety factor is $q(s_0) = 4.0$ .","We consider a temperature profile of the following form, similarly to Ref.","[23]: $\\displaystyle {\\frac{T_e(s)}{T_e(s_0)}} = \\exp \\left[-\\Delta \\cdot k_T \\cdot \\tanh \\left(\\displaystyle {\\frac{s - s_0}{\\Delta }}\\right)\\right],$ where $\\Delta = 0.04$ , $k_T = - \\left.","d[\\ln (T)]/ds \\right|_{s = s_0}$ .","The temperature profiles and the corresponding temperature gradient profiles for different $k_T$ in a radial interval $s = [0.85, 0.95]$ , are shown in Fig.","REF .","Dependence of the GAM half-decay time $t_{1/2}$ on the temperature gradient has been investigated in the domain $k_T \\in [1, 15]$ .","Figure: Temperature and temperature gradient radial profiles for different k T k_T: k T =[1,5,15]k_T = [1, 5, 15].A scan of gyrokinetic simulations with the temperature gradient $k_T$ has been performed, and the results are depicted in Fig.","REF .","In presence of a temperature gradient, the GAM is observed to oscillate with different frequencies at different radial points, that leads to the distortion of the initial GAM radial structure.","Producing higher radial wavenumbers, this distortion amplifies the GAM damping.","This combined effect, already investigated for a more simplified configuration in Ref.","[22], [23], has been observed even more pronounced in the simulations described here.","In fact, here the phase mixing effect is investigated using gyrokinetic simulations with kinetic electrons that significantly influences the GAM damping, and, as a consequence, the GAM half-decay time.","For example, using the Sugama-Watanabe model[35], which is derived with adiabatic electrons, for the Landau damping and combining with phase mixing, we have obtained $t_{1/2}[R/ \\sqrt{2} v_{Ti}] = 118$ for the $k_T = 1$ and $t_{1/2}[R/ \\sqrt{2} v_{Ti}] = 23.4$ for $k_T = 10$ , that predicts much longer half-decay time of the GAM in comparison to the calculations based on the simulations with the kinetic electrons (compare with a Fig.","REF ).","In order to verify the results of gyrokinetic simulations, we have used a theoretical simplified model of the phase mixing, proposed in Ref.","[8], [22], [23], where the linear growth in time of the radial wavenumber is considered.","In the phase mixing simulations a space point $s_0$ is considered with a certain temperature $T(s_0)$ and temperature gradient $k_T(s_0)$ .","Initial radial electric field has the following radial structure: $E(s) = E_0 \\cos (k_0 s)$ with an initial amplitude $E_0$ and initial normalized radial wavenumber $k_0$ .","The electric field is assumed to evolve in time at a point $s_0$ according to a simple rule $E(s_0, t) = E_{a}(s_0, t) \\cos (\\omega (s_0) t),$ where $E_{a}(s_0, t)$ is an amplitude of the electric field, that changes in time due to the damping, $E_{a}(0) = E_0$ .","The general form of the GAM frequency is $\\omega (s, t) = \\sqrt{\\frac{2 T_e(s)}{2m_p}} \\omega ^*(k, q, e),$ where $\\omega ^*(k, q, e)$ describes frequency dependence on the radial wavenumber, the safety factor and the elongation.","The safety factor profile is taken to be flat, and plasma with a circular cross-section is considered: $e = 1.00$ , $r \\approx a s$ .","The damping rate is defined as $\\gamma (s_0, t) = \\displaystyle {\\frac{1}{E(s_0,t)}}\\displaystyle {\\frac{ d E(s_0,t)}{d t}}.$ At the beginning of every time interval $[t_1, t_1 + \\Delta t]$ , new values of the damping rate $\\gamma (s_0, t_1)$ and frequency $\\omega (s_0, t_1)$ are found with the scaling formulae given in Eq.","(REF ) and (REF ), using a current value of the wavenumber $k(s_0, t_1)$ .","A new value of the electric field can be found, assuming that the damping rate is constant at the lapse of time $[t_1, t_1 + \\Delta t]$ : $E(s_0, t_1 + \\Delta t) = E(s_0, t_1)\\cdot (1 + \\gamma (s_0, t_1) \\Delta t).$ After that, a new value of the wavenumber $k(s_0, t_1+\\Delta t)$ is calculated using the radial derivative of the frequency $\\left.","\\frac{\\partial {\\omega (s, t_1)}}{\\partial {s}} \\right|_{s = s_0} = - \\displaystyle {\\frac{1}{2}} \\omega (s_0, t_1) k_T.$ With that, the wavenumber is assumed to change linearly in time as $k(s_0, t_1 + \\Delta t) = k(s_0, t_1) - \\sqrt{2} \\rho ^* \\left.","\\frac{\\partial {\\omega (s, t_1)}}{\\partial {s}} \\right|_{s = s_0} \\Delta t,$ where $\\rho ^* = \\rho _s / a$ , $\\rho _s = c_s/ \\omega _{ci}$ .","Another option, it is to estimate the time evolution of the radial wavenumber directly from numerical calculations in ORB5.","Substituting new value of the normalized wavenumber $k(s_0, t_1+\\Delta t)$ into Eq.","(REF ), we can find the damping rate $\\gamma (s_0, t_1+\\Delta t)$ at the next time point.","The results obtained with this reduced theoretical model are also shown in Fig.","REF .","As it can be seen in Fig.","REF , the qualitative dependence of the half-decay time on the temperature gradient finds a good match of gyrokinetic simulations of ORB5 and analytical theory.","The difference is due to the global dynamics of the ORB5 simulations, which is compared here with a theory where the phase mixing follows a local estimation given in Ref. [8].","Figure: Dependence of the GAM half-decay time on the temperature gradient obtained from the simulations in ORB5 (green squares), from the theory using a linear estimation Eq.","() (blue dots) and estimation from ORB5 (red triangles) of the radial wavenumber."],["Comparison with experimental data","The dispersion relations obtained in Sec.","REF as an interpolation of gyrokinetic simulations and given in Eqs.","(REF ), (REF ) can be used to compare numerical estimations of the GAM behaviour to measurements of the GAM frequency, performed on ASDEX Upgrade tokamak [25] using Doppler reflectometry.","More precisely, we consider the discharge AUG#20787 with the plasma elongation at the edge $e = 1.09$ (and we assume that it is constant at the considered radial region $\\rho = r/a = [0.8, 1.0]$ ).","The GAM radial wavenumber is considered to be constant and is estimated to be $k_ra = 40\\pi $ from the experimental radial profile of the GAM amplitude (see Fig.","5f in Ref.","[25]).","Experimental safety factor and ion temperature profiles have been taken to estimate the GAM frequency and damping rate using the scaling formulae (REF ), (REF ) at different radial points $\\rho $ .","In Fig.","REF the GAM frequency profiles with corresponding theoretical prediction are depicted.","A good general agreement is found in the central region of interest, where the GAM intensity, measured in the experiments, is peaked.","On the other hand, the linear dispersion relation REF can not explain neither the staircase nature (the plateaus) of the frequencies nor the GAM peak splitting that is observed experimentally at the radius positions $\\rho = 0.922$ or $\\rho = 0.932$ (although the presence of GAM eigenmodes has been suggested by simplified analytical models [36], [37], whose detailed analysis is out of the scope of this paper).","For this reason, we can conjecture that the coherent phenomena at the basis of the formation of GAM extended eigenmode or frequency splitting must have a nonlinear origin.","For reference we have given here estimation of the GAM collisional damping rate using formulae, derived by Gao in Ref.","[9].","Introducing normalized ion collision rate $\\hat{\\nu }_i = \\nu _i qR / v_{ti}$ , the collisional damping rate is calculated as[9] $\\displaystyle {\\frac{\\gamma ^{col}}{v_{Ti}/qR}} = - \\displaystyle {\\frac{3\\hat{\\nu }_i}{14 + 8\\tau _i}},$ if $\\hat{\\nu }_i \\ll 1$ , and as: $\\displaystyle {\\frac{\\gamma ^{col}}{v_{Ti}/qR}} = - \\displaystyle {\\frac{3}{8}} \\hat{\\nu }_i \\left(\\displaystyle {\\frac{7}{4}} + \\tau _i + \\displaystyle {\\frac{\\hat{\\nu }_i^2}{q^2}} \\right)^{-1},$ if $\\hat{\\nu }_i \\ge 1$ .","To find the ion collisional rate we have used classical expressions: $\\nu _i &=& 4.8\\cdot 10^{-8} Z^4 \\mu ^{-1/2} n_i[cm^{-3}] T_i[eV]^{-3/2}\\ln \\Lambda ,\\\\\\ln \\Lambda &=& 23 - \\ln \\left[ \\sqrt{2n_i[cm^{-3}]}\\displaystyle {\\frac{Z^3}{T_i[eV]^{3/2}}} \\right],$ where $\\mu \\equiv m_i / m_p = 2$ , $Z = 1$ .","Figure: Comparison of the experimental GAM frequencies to the numerical values, obtained with the formula given in Eq.","().","Numerical damping rate is depicted on the right plot.","The grey dotted line is an estimation of the collisional damping rate of the GAM found using expressions given by Gao in Ref.",".","The red dotted lines are the 95% confidential bounds of the approximated damping rate.According to the Fig.","REF , the collisional damping is found to be negligible in the radial domain where GAMs are experimentally measured, except in a very narrow region close to the separatrix, where it can be of the same order of magnitude as the Landau damping."],["Conclusions","In tokamak plasmas, the drift-wave turbulence gives rise to the zonal flows that in their turn shear and distort convective and turbulent cells leading to the saturation of turbulence and, consequently, to a reduction of the radial heat transport.","Action of the magnetic curvature results in the oscillatory zonal flows, so-called geodesic acoustic modes.","The peculiarity of the GAM oscillations resides in the different shearing efficiency that the ZF have in relation to their oscillatory behavior.","The nonlinear interactions between the GAM and the DW turbulence is defined in a high degree by the GAM damping rate.","Lack of the experimental data of this characteristic of the GAM makes the results from linear gyrokinetic simulations particularly important for analytical and numerical investigation of the nonlinear GAM-DW systems.","In this work, linear gyrokinetic simulations have been performed with kinetic electrons to study the GAM dynamics.","Numerical results have been compared to analytical theories, derived with adiabatic electrons.","It has been shown that analytical theories, derived with adiabatic electrons, result in smaller values of the damping rate and for higher wavenumbers diverge from numerical calculations of the frequency.","That is why, investigating the GAM dependence on the plasma safety factor, elongation and radial wavenumber, we have found approximating analytic expressions for the frequency and damping rate to predict the GAM behaviour in different plasma regimes.","The derived expressions can be used to estimate the GAM linear characteristics used in analytical models of the nonlinear interactions between the GAM and the DW, such as different reduced models [38], [8].","Using these formulae, the phase mixing effect on the damping rate has also been calculated.","Based on the gyrokinetic simulations with kinetic electrons, the results have shown smaller half-decay times of the GAM in comparison with the Ref.","[22], [23].","The GAM is one of the special features of the I-mode and can be observed in the L-mode [15], [39], [40].","Comparison of the characteristic drive time of the GAM $t_{RD} \\sim 1/\\gamma _{RD}$ , which is given by the nonlinear coupling with the ion-temperature-mode (ITG)[8], [38], with the GAM half-decay time $t_{1/2}$ confirms the results of the Ref.","[22], [23].","Indeed, we estimate the GAM drive time to be $t_{RD} < t_s$ (where $t_s \\sim 2^{-1/2} R/v_{Ti}$ ) in the L-mode, $t_{RD} \\sim t_s$ in the I-mode and $t_{RD} \\sim 10t_s$ in the H-mode, according to Ref.","[23].","In this case, it can be seen from the Fig.","REF that the GAM half-decay time, which is defined by both the Landau damping and the phase mixing effect, is much higher than the drive-time in the L and I modes, $t_{1/2} > t^{L,I}_{RD}$ , for all considered values of $k_T$ .","This means that the energy transfer rate from the ITG turbulence to the GAM exceeds the Landau-phase mixing damping rate of the GAM.","As a result, the GAM can be observed in the L and I modes, but not in the H-mode, where $t_{1/2} < t^H_{RD}$ already for $k_T > 3$ .","This could be the explanation for the result that the GAM are not observed in the high-confinement mode, as proposed in Ref.","[22], [23].","The approximating expressions have showed quite good agreement with experimental data, but estimations of the GAM frequency, obtained from linear gyrokinetic simulations, does not explain the staircase radial profile of the frequency and the GAM peak splitting.","The frequency expression describes only the continuum or dispersive mode in contrast to eigenmode.","The latter is characterized by the GAM mode frequencies which are predicted to remain constant over a large radial extent, but a significant radial overlap in the frequency radial profile can be observed, that lead to the GAM frequency peak splitting[40]."],["Acknowledgments","Part of this work was done while two of the authors, I. Novikau and A. Biancalani, were visiting LPP-Palaiseau (France), whose team is kindly acknowledged for the hospitality.","The authors acknowledge discussions with F. Jenko, L. Villard, X. Garbet and T. Görler."],["Numerical convergence tests","To calculate the GAM damping rate and frequency, poloidally averaged radial electric field has been fitted, using the Levenberg–Marquardt algorithm [41], to a function of the form $\\exp (\\gamma t) \\cos (\\omega t)$ , where $\\gamma $ , $\\omega $ are sought-for damping rate and frequency.","Before the fitting it's necessary to filter the radial electric field to get ride of the high-frequency Alfvén oscillations.","In Fig.","REF the fitting is depicted for the case: $e = 1.30$ , $q = 4.0$ , $k = 0.108$ .","It is worth to mention that the choice of a time interval, where the fitting is performed, can influence the result damping rate, and it is not so crucial for the GAM frequency calculation.","This ambiguity in the choice of the time interval can be explained by the fact that at the beginning of the simulations there are some transient processes that must be excluded from the damping rate measurements.","Moreover, with the time, the global effects start to play a significant role, distorting initial radial structure of the radial electric field, that makes the damping rate to be variable in time.","The GAM dynamics (frequency and damping rate) doesn't depend on the plasma density in linear calculations (see Fig.","REF and REF ).","But the frequency of the Alfvén waves decreases with the increase of the density, and for high values of the plasma density it becomes difficult to separate acoustic and Alfvénic time scales (see Fig.","REF ).","Figure: Fourier spectrum of the radial electric field for different values of the electron beta: β e =5e-5\\beta _e = 5e-5 (blue line), β e =1e-3\\beta _e = 1e-3 (red line).","Only the Alfvén frequency changes with the electron beta.","The GAM frequency remains the same.","Here, the frequency is normalized to ω ci \\omega _{ci}.The transition from the simulations with the adiabatic electrons to the ones with the kinetic electrons applies additional restrictions on several numerical parameters such as the time step and the number of markers (see Fig.","REF ).","Figure: Convergence tests on the number of points in the radial space grid n s n_s (e=1.60e = 1.60, q=4.0q = 4.0, k r ρ i =0.108k_r \\rho _i = 0.108) (Fig.","), on the normalized time step dt norm dt_{norm} for different values of the plasma elongation e=1.00,1.60e = 1.00, 1.60 (q=3.5q = 3.5, k r ρ i =0.108k_r \\rho _i = 0.108) (Fig. )","and on the number of markers N markers N_{markers} (e=1.60e = 1.60, q=4.0q = 4.0, k r ρ i =0.108k_r \\rho _i = 0.108) (Fig.",").In projects with adiabatic electrons the normalized time step $dt_{norm} = dt\\cdot \\omega _{ci}$ can be of the order of 20, but in case of the kinetic electrons it has to be significantly reduced till 2 because of the high parallel velocity of the passing electrons.","Also electrostatic simulations of the kinetic electrons reveal high-frequency oscillations[30].","These oscillations can lead to numerical instabilities in case of low number of markers.","To reduce their level we have passed to electromagnetic simulations with small values of electron beta that gave us an opportunity to keep the number of markers on the level of $10^7$ .","The radial space step (or number of the points in the radial space grid) is determined by, among other parameters, the GAM wavenumber.","To investigate the GAM dynamics with higher values of the radial wavenumber, we simulated a narrow poloidal ring near the edge instead of the full plasma cross-section to reduce the number of radial space points."],["Comparison with the code GENE","A complete cross-code verification between the gyrokinetic ORB5 and GENE [42], [43] codes has already been done in Ref.","[21] on the linear collisionless dynamics of the GAMs with adiabatic and kinetic electrons in the specific case of flat temperature profiles.","For completeness, in this paper a comparison between these different codes is shown including the additional phase mixing physical effect, which is driven by non-flat temperature profiles.","The motivation behind this study is that although the linear physical models between ORB5 and GENE are equivalent [44], the numerical schemes are different.","GENE is an Eulerian code, where the distribution function is not discretized with markers, but it is discretized on a 5D fixed grid in phase-space $({R}, v_{\\parallel }, \\mu )$ , where ${R}$ is the gyrocenter position, $v_{\\parallel }$ is the parallel velocity, and $\\mu $ is the magnetic momentum.","The simulation plasma parameters have been taken as in Sec.","for both GENE and ORB5.","A sinusoidal perturbation in the potential field is initialised, as defined in Sec.","REF and is let evolved in time.","In GENE, the radial box size is $60 \\rho _s$ .","We have used 128 grid points in radial direction in order to have at least two points per ion Larmor radius.","Along the field line 68 points have been used.","In velocity space, 68 points and 128 equidistant symmetric grid points have been used for resolving respectively the $\\mu $ and the $v_{\\parallel }$ space.","The velocity space domain has been fixed to 3 and 9 times the thermal velocity, respectively in the $v_{\\parallel }$ and $\\mu $ space.","In order to avoid any recurrence problem, an hyperdiffusivity scheme has been used in the $v_{\\parallel }$ direction.","In Fig.","REF peaks of the flux-surface averaged radial electric field, measured at the radial position $s_0 = 0.90$ for the ion-electron temperature gradient $k_T = 10$ , are shown for both GENE and ORB5.","Half-decay time, calculated in ORB5, is $t_{1/2}^{orb}[R/\\sqrt{2} v_{Ti}] = 4.9$ and for the case of GENE it is $t_{1/2}^{GENE}[R/\\sqrt{2} v_{Ti}] = 4.8$ ."]],"string":"[\n [\n \"Linear gyrokinetic investigation of the geodesic acoustic modes in\\n realistic tokamak configurations\"\n ],\n [\n \"Abstract Geodesic acoustic modes (GAMs) are studied by means of the gyrokinetic global particle-in-cell code ORB5.\",\n \"Linear electromagnetic simulations in the low electron beta limit have been performed, in order to separate acoustic and Alfv\\\\'enic time scales and obtain more accurate measurements.\",\n \"The dependence of the frequency and damping rate on several parameters such as the safety factor, the GAM radial wavenumber and the plasma elongation is studied.\",\n \"All simulations have been performed with kinetic electrons with realistic electron/ion mass ratio.\",\n \"Interpolating formulae for the GAM frequency and damping rate, based on the results of the gyrokinetic simulations, have been derived.\",\n \"Using these expressions, the influence of the temperature gradient on the damping rate is also investigated.\",\n \"Finally, the results are applied to the study of a real discharge of the ASDEX Upgrade tokamak.\"\n ],\n [\n \"Introduction\",\n \"The ion heat transport in the plasma core is governed by turbulence formed by a class of microinstabilities such as toroidal ion temperature gradient (ITG) driven modes [1].\",\n \"ITG turbulence is known to self-organize to form macroscopic structures [2].\",\n \"These structures take the form of a macroscopic radial electric field which depends only on the radial coordinate.\",\n \"$E \\\\times B$ poloidal flows associated with this electric field are referred to as zonal flows (ZFs) [3], [4], [5], [6].\",\n \"The action of the toroidal magnetic field curvature on the ZF gives rise to oscillations of the radial electric field.\",\n \"These oscillations of the ZFs are called geodesic acoustic modes (GAMs) [7], [8].\",\n \"The modes are observed predominantly in the edge region of the tokamak plasmas with characteristic frequency of the order of the sound frequency $\\\\sim c_s/R$ , where $c_s = \\\\sqrt{T_e/m_i}$ is the sound speed, $R$ is the major radius.\",\n \"One of the main linear damping mechanisms for the stationary ZF are collisional processes and for the GAM it is a collisionless wave-particle interaction, namely the Landau damping, and collisional damping at the very edge of the plasma, where equilibrium temperatures drastically decrease [9].\",\n \"A recent comparison of collisionless and collisional damping of GAMs, using existing analytical theories, for experimentally relevant plasmas was done in Ref.\",\n \"[10].\",\n \"The importance of the ZF is that they can regulate the drift-wave (DW) turbulence [11].\",\n \"But it is still a question how the GAMs influence the ZF efficiency of the DW suppression [8], [12], [13].\",\n \"On the other hand, the development of zonal structures can play a key role in the transition from the low to the high confinement regime (L-H transition) [14].\",\n \"In Ref.\",\n \"[15] interaction of the mean and oscillatory poloidal flows with the turbulence were experimentally observed.\",\n \"The turbulence suppression by the ZFs was observed in experiments described in Ref.\",\n \"[16].\",\n \"On the other hand, in Ref.\",\n \"[17] the role of the mean flow in the dynamic evolution towards the H-mode is emphasized.\",\n \"In Ref.\",\n \"[12] two predators - one prey system, including ZF, GAM and turbulence, was developed to study transitions between states with different combinations of the ZF and GAM.\",\n \"In this paper, we investigate the GAM frequency and collisionless damping rate, carrying out linear collisionless simulations with kinetic electrons.\",\n \"The electromagnetic global gyrokinetic particle-in-cell code ORB5 is used [18], [19].\",\n \"As it has been reported previously [20], [21], models, numerical or analytical, derived with adiabatic electrons, result in considerably smaller GAM damping rate in comparison to simulations performed with kinetic electrons.\",\n \"By adiabatic electron models, we mean here models treating the $m\\\\ne 0$ component of the electrons as adiabatic, and setting the zonal component of the electron density perturbation to zero.\",\n \"In simulations considered in this paper, electrondocumentclasss are treated drift-kinetically, and a realistic ion-electron mass ratio is used.\",\n \"Moreover, to study the influence of the plasma elongation on the GAM dynamics, magnetic equilibria with realistic plasma shapes are considered.\",\n \"To summarize the results obtained in different plasma regimes, interpolating formulae for the GAM frequency and damping rate, based on the gyrokinetic simulations with ORB5, are derived.\",\n \"Due to the so-called phase mixing effect, the GAM damping rate is increased in the presence of a temperature gradient or the safety factor profile [8], [22], [23].\",\n \"This effect arises when the damping rate of the wave depends on its wavenumber.\",\n \"In the case of the GAM the damping rate increases with the GAM radial wavenumber (more precisely, with the radial wavenumber of the radial electric field).\",\n \"Since the GAM frequency depends on the temperature and safety factor, the GAM oscillates with different frequencies at different radial points in presence of the temperature gradient or magnetic shear.\",\n \"Distorting the GAM radial structure and creating higher radial wavenumbers, this process can strongly increase the GAM damping rate [8], [24].\",\n \"A section of our paper is dedicated to the extension of previous works [21], [23], which were done treating the electrons as adiabatic, and in circular flux surfaces, to the inclusion of kinetic electrons and realistic tokamak configurations.\",\n \"Finally, the last section of this paper is dedicated to the investigation of a realistic discharge of ASDEX Upgrade, described in Ref. [25].\",\n \"In Appendix we have shown a comparison between ORB5 and GENE for the case of non-flat temperature profile.\"\n ],\n [\n \"Model\",\n \"The gyrokinetic simulations presented in this work have been performed with the code ORB5 [18], [19].\",\n \"ORB5 is a nonlinear gyrokinetic multi-species global particle-in-cell (PIC) code, which solves the Vlasov-Maxwell system in the electrostatic or electromagnetic limit, and has a capability of handling true MHD equilibrium for an axisymmetric toroidal plasma.\",\n \"The particle-in-cell method consists of coupling a particle-based algorithm for the Vlasov equation with a grid-based method for the computation of the self-consistent electromagnetic fields.\",\n \"Several physical models are available in ORB5, all of them derived from a systematic Hamiltonian theory [19], [26] to provide exact energy and momentum conservation.\",\n \"In this work, only one ion species (deuterium) has been considered while the electrons are assumed to be drift-kinetic.\",\n \"This corresponds to the following gyrokinetic total Lagrangian: $L =&& \\\\sum _{\\\\rm {sp}}\\\\int \\\\mathrm {d}V \\\\mathrm {d}W \\\\left(\\\\left(\\\\frac{q}{c}{A}+p_\\\\parallel {b}\\\\right)\\\\cdot \\\\dot{{R}} + \\\\frac{m c}{q}\\\\mu \\\\dot{\\\\theta } - H_0-H_1\\\\right)f \\\\\\\\&&+ \\\\int \\\\mathrm {d}V \\\\mathrm {d}W H_2 f_{M,ions}-\\\\int \\\\mathrm {d}V \\\\frac{B_\\\\perp ^2}{8\\\\pi }.\\\\nonumber $ The velocity variables are the magnetic moment $\\\\mu \\\\equiv (mv_\\\\perp ^2)/(2B)$ , the canonical parallel momentum $p_\\\\parallel $ and the gyroangle $\\\\theta $ .\",\n \"The equilibrium magnetic field is ${B}=\\\\nabla \\\\times {A}$ , $m$ and $q$ are the mass and charge of the particle species $sp$ and $c$ is the speed of light.\",\n \"The volume element of the velocity space is $\\\\mathrm {d}W\\\\equiv (2\\\\pi )/m^2 B^*_\\\\parallel \\\\mathrm {d}p_\\\\parallel \\\\mathrm {d}\\\\mu $ with $B^*_\\\\parallel ={B^*}\\\\cdot {{b}}$ , ${b}={B}/B$ and ${B^*}={B}+(c/q)p_\\\\parallel \\\\nabla \\\\times {b}$ ; $\\\\mathrm {d}V$ denotes the volume element in physical space.\",\n \"Here $f$ is the distribution function for the species $sp$ , while $f_{M,ions}$ is the equilibrium time independent distribution function of the ions.\",\n \"In this system, only long wavelength electrostatic perturbation and magnetic perturbations perpendicular to the equilibrium magnetic field are considered.\",\n \"Note that no second order term in the fields is retained for the electrons, this is equivalent to neglect the electron polarization density in the Polarization equation (drift-kinetic approximation, see Ref.\",\n \"[19] for details).\",\n \"The first two terms in the total Lagrangian define the charged particles Lagrangian [27].\",\n \"The GK Hamiltonian in general depends on the electrostatic potentials $\\\\Phi $ and on the parallel component of the fluctuation magnetic potential $A_\\\\parallel $ .\",\n \"The third term in the total Lagrangian is the electromagnetic field Lagrangian, in which the electric field component has been neglected (quasi-neutrality approximation, see [19] for details).\",\n \"In this work we used the following Hamiltonian: $H&=&H_0+H_1+H_2\\\\\\\\H_0&=&\\\\frac{p_\\\\parallel ^2}{2m}+\\\\mu B\\\\nonumber \\\\\\\\H_1&=& e(J_0\\\\Phi - \\\\frac{p_\\\\parallel }{mc} J_0A_\\\\parallel )\\\\nonumber \\\\\\\\H_2 &=& - \\\\frac{q^2}{2mc^2}(J_0A_\\\\parallel )^2+\\\\frac{mc^2}{2B^2}|\\\\nabla _\\\\perp \\\\Phi |^2\\\\nonumber $ the gyroaveraging (Hermitian) operator $J_0$ , applied to an arbitrary function $\\\\psi $ in configuration space, is defined by $(J_0\\\\psi ) ( \\\\mathbf {R},\\\\mu ) = \\\\frac{1}{2\\\\pi }\\\\int _0^{2\\\\pi } \\\\psi ( \\\\mathbf {R}+{\\\\rho }(\\\\alpha )) \\\\,d \\\\alpha ,$ where ${\\\\rho }$ is the vector going from the guiding center position to the particle position.\",\n \"In this work we have assumed $J_0=1$ for the electrons (drift-kinetic approximation).\",\n \"The gyrokinetic equations for the particle distribution function and the GK field equations can be derived from the GK Lagrangian using variational principles.\",\n \"In summary, the GK model used in the following is: $\\\\bullet $ gyrokinetic full-f Vlasov equation for the ions $&&\\\\frac{\\\\partial {f_i}}{\\\\partial {t}}+\\\\dot{{R}_i}\\\\cdot \\\\nabla f_i+\\\\dot{p_{\\\\parallel ,i}}\\\\frac{\\\\partial {f_i}}{\\\\partial {p_{\\\\parallel ,i}}}=0,\\\\\\\\&&\\\\dot{{R}_i}=\\\\left(\\\\frac{p_{\\\\parallel ,i}}{m_i}-\\\\frac{Z_ie}{m_ic}J_0A_\\\\parallel \\\\right)\\\\frac{{B_i^*}}{B^*_{\\\\parallel ,i}}+\\\\frac{c}{Z_ieB^*_{\\\\parallel ,i}}{b}\\\\times \\\\left[\\\\mu _i\\\\nabla B + Z_ie \\\\nabla J_0 \\\\Psi _i \\\\right],\\\\\\\\&&\\\\dot{p_{\\\\parallel ,i}}=-\\\\frac{{B_i^*}}{B^*_{\\\\parallel ,i}}\\\\cdot \\\\left[\\\\mu _i \\\\nabla B + Z_ie\\\\nabla J_0\\\\Psi _i \\\\right],$ $\\\\bullet $ drift-kinetic full-f Vlasov equation for the electrons: $&&\\\\frac{\\\\partial {f_e}}{\\\\partial {t}}+\\\\dot{{R}_e}\\\\cdot \\\\nabla f_e+\\\\dot{p_{\\\\parallel ,e}}\\\\frac{\\\\partial {f_e}}{\\\\partial {p_{\\\\parallel ,e}}}=0,\\\\\\\\&&\\\\dot{{R}_e}=\\\\left(\\\\frac{p_{\\\\parallel ,e}}{m_e}+\\\\frac{e}{m_ec}A_\\\\parallel \\\\right)\\\\frac{{B_e^*}}{B^*_{\\\\parallel ,e}}-\\\\frac{c}{eB^*_{\\\\parallel ,e}}{b}\\\\times \\\\left[\\\\mu _e\\\\nabla B - e \\\\nabla \\\\Psi _e \\\\right],\\\\\\\\&&\\\\dot{p_{\\\\parallel ,e}}=-\\\\frac{{B_e^*}}{B^*_{\\\\parallel ,e}}\\\\cdot \\\\left[\\\\mu _e \\\\nabla B -e\\\\nabla \\\\Psi _e \\\\right],$ having introduced the generalized potential $\\\\Psi \\\\equiv \\\\Phi - \\\\frac{p_{\\\\parallel ,sp} }{m{_{sp}}c} A_\\\\parallel .$ $\\\\bullet $ Linear polarization equation in the long wave-length limit (and drift-kinetic electrons): $&&\\\\int \\\\mathrm {d}W_i Z_ieJ_0 f_i-\\\\int \\\\mathrm {d}W_e e f_e=-\\\\nabla \\\\cdot \\\\left(\\\\frac{n_0 m_ic^2}{B^2} \\\\nabla _\\\\perp \\\\Phi \\\\right)$ $\\\\bullet $ Linear Ampère's law: $\\\\int \\\\mathrm {d}W_i \\\\frac{4\\\\pi Z_ie }{m_i c} p_{\\\\parallel ,i} J_0 f_i &-& \\\\int \\\\mathrm {d}W_e \\\\frac{4\\\\pi e }{m_e c} p_{\\\\parallel ,e} f_e = \\\\\\\\&&\\\\frac{1}{d_e^2}A_\\\\parallel +\\\\frac{1}{d_i^2}A_\\\\parallel -\\\\nabla _\\\\perp ^2 A_\\\\parallel -\\\\nabla \\\\cdot \\\\frac{\\\\beta _{i}}{4}\\\\nabla _\\\\perp A_\\\\parallel $ where $n_0$ is the density associated with the equilibrium Maxwellian $f_M$ and $\\\\beta _{sp}=(4\\\\pi n_{sp} T_{sp})/B^2$ .\",\n \"The skin depth is defined by $d^{-2}_{sp}=\\\\beta _{sp}/\\\\rho ^2_{sp}$ , where $\\\\rho ^2_{sp} = T_{sp}m_{sp}c^2/q^2_{sp}B^2$ , and it appears on the right-hand-side of the Ampére's law because of the choice of the velocity space variables $(p_\\\\parallel ,\\\\mu )$ instead of the usual $(v_\\\\parallel ,\\\\mu )$ .\",\n \"The indexes $i$ and $e$ indicate ions and electrons respectively and $q_i=Z_ie$ is the ion charge while $q_e=-e$ is the electron charge.\",\n \"Despite all the approximations made, this model is highly physically relevant and it can be used to describe not only the GAM and ZF dynamics, but also a large class of micro-instabilities excited by the density and temperature gradients, like ion temperature gradient (ITG) driven modes, trapped electron modes (TEM) or kinetic ballooning modes (KBM).\",\n \"It also contained the reduced MHD model as a subset (see, among other, Ref.\",\n \"[28]).\",\n \"According to the PIC method the particle distribution function is discretized with macroparticles, known as markers.\",\n \"The motion of the markers is calculated using the equations of motions of the gyrokinetic model while the electromagnetic fields are evolved on a spatial grid using the two field equations.\",\n \"The charge and current density, that are necessary to solve the field equations, are calculated by projecting the marker weights on a spatial grid.\",\n \"After that, the fields are calculated using a finite elements method.\",\n \"The code is based on a straight-field-line coordinate system $(s, \\\\chi , \\\\phi )$ .\",\n \"Here, radial coordinate is $s = \\\\sqrt{\\\\psi /\\\\psi _{edge}}$ (where $\\\\psi $ is the poloidal flux), $\\\\chi = \\\\displaystyle {\\\\frac{1}{q(s)}} \\\\int _0^\\\\Theta \\\\displaystyle {\\\\frac{{B}\\\\cdot \\\\nabla \\\\phi }{{B}\\\\cdot \\\\nabla \\\\Theta _1}} d\\\\Theta _1$ is the straight-field-line coordinate (where $\\\\Theta $ and $q$ are the poloidal angle and the safety factor respectively) and $\\\\phi $ is a toroidal angle.\",\n \"Two different kinds of magnetic equilibria are implemented: analytical equilibria with circular concentric magnetic surfaces and ideal MHD realistic equilibria.\",\n \"For the latter case, the ORB5 code is coupled with the CHEASE code [29], which solves the Grad-Shafranov equation with a fixed plasma boundary.\"\n ],\n [\n \"Equilibrium and simulations parameters\",\n \"Linear electromangetic gyrokinetic collisionless simulations with drift-kinetic electrons and a realistic electron - ion (deuterium) mass ratio $m_e / m_i = 2.5\\\\cdot 10^{-4}$ have been performed.\",\n \"Electrostatic simulations with kinetic electrons are, in principle, faster than electromagnetic simulations, due to the smaller number of equations to be solved.\",\n \"Nevertheless, a high frequency oscillation, called the $\\\\omega _H$ -mode [30], is observed to be often numerically unstable.\",\n \"To decrease the level of the high-frequency oscillations, electromagnetic simulations in the small-$\\\\beta _e$ ($\\\\beta _e = 10^{-5}$ ) limit have been performed instead of the electrostatic ones.\",\n \"MHD equilibria of the circular and elongated plasma have been calculated with an external code CHEASE [29].\",\n \"Simulations have been carried out with a flat density profile, which have been shown to not impact the GAM frequency and damping rate in linear simulations (see Appendix ).\",\n \"To focus on the Landau damping in the absence of the phase mixing effect, flat temperature profile has been considered in simulations used for the results of this section.\",\n \"Since the safety factor profiles have been taken from the CHEASE, there is a magnetic shear, that also causes the phase mixing, but its influence on the GAM damping rate is much smaller in comparison to the temperature gradient effect.\",\n \"Plasma parameters have been taken close to the ASDEX Upgrade parameters near the plasma edge [25]: the major radius $R = 1.65$ m, the minor radius $a = 0.5$ m (inverse aspect ratio is $\\\\epsilon = 0.303$ ), the magnetic field on the axis $B = 2$ T. Since in the CHEASE code the plasma elongation is defined at the edge and changes gradually to the plasma center, the GAM frequency and damping rate have been measured at the same radial position $s_0 = 0.90$ to perform more accurate scan on the elongation.\",\n \"The temperature has been taken to be $T_i = T_e = 70$ eV.\",\n \"It means that $c_s = \\\\sqrt{q_e T_e / 2 m_p} = 5.8 \\\\cdot 10^{3}$ m/s, $\\\\rho _s = c_s / \\\\omega _{ci} = 6.1\\\\cdot 10^{-4}$ m and $\\\\rho ^* = \\\\rho _s / a = 1.2\\\\cdot 10^{-3}$ .\",\n \"Here, $\\\\omega _{ci} = q_e B/2m_p$ ($\\\\omega _{ci}/2\\\\pi = 15.2$ MHz) is the ion gyro-frequency, $q_e$ is an electron charge, and $m_p$ is a proton mass.\",\n \"To weaken the constraint on the space step and to reduce the effect of the charge accumulation at the edge of the numerical work box, we have simulated only a ring from $s_1 = 0.85$ to $s_2 = 0.95$ in a poloidal cross section with the Dirichlet condition for the potential $\\\\phi $ on inner boundaries ($\\\\phi (s_1) = \\\\phi (s_2) = 0$ ).\",\n \"A typical simulation has the following parameters.\",\n \"Number of nodes in radial direction is taken to be $n_s = 256$ , in toroidal directions $n_{\\\\phi } = 4$ and along the straight-field-line coordinate $\\\\chi $ the number of nodes is $n_{\\\\chi } = 64$ .\",\n \"Time step is $dt \\\\cdot \\\\omega _{ci} = 2$ .\",\n \"The GAM damping rate and frequency have been calculated (see Appendix ) for different GAM radial wavenumbers $k = k_r\\\\rho _i \\\\in [0.054, 0.377]$ (where $\\\\rho _i = \\\\sqrt{2}v_{Ti}/\\\\omega _{ci} = 8.57 \\\\cdot 10^{-4}$ m is the ion Larmor radius of the deuterium and $v_{Ti} = \\\\sqrt{q_eT_i/2m_p}$ is the thermal speed), the safety factor $q \\\\in [3.5, 5.0]$ at $s_0 = 0.90$ and the plasma elongation $e \\\\in [1.0, 1.6]$ at the edge.\",\n \"This is the regime where GAMs are typically observed in tokamak plasmas (see, for example, Ref.\",\n \"[25]).\",\n \"To simulate the GAM dynamics, the ORB5 simulations have been initialized by introducing an axisimmetric density perturbation designed to produce an initial electric potential field of the form $\\\\sim \\\\sin (ks)$ , where $s \\\\in [s_1, s_2]$ (as in the so-called Rosenbluth-Hinton test [4]).\",\n \"All toroidal modes $n \\\\ne 0$ and poloidal modes $|m| > 10$ have been filtered out.\",\n \"To study the GAM dynamics, the frequency and damping rate of the poloidally averaged radial electric field have been calculated.\"\n ],\n [\n \"Results of gyrokinetic simulations\",\n \"In Fig.\",\n \"REF , a comparison of the GAM frequencies and damping rate obtained from numerical simulations with two analytical theories of Qiu 2009 [32] and Gao 2010 [31] is shown.\",\n \"A good agreement between numerical results and analytical predictions of the GAM frequency has been found.\",\n \"Nevertheless, the GAM damping rate, obtained from the theories, derived using adiabatic electrons, is smaller in comparison to numerical simulations with kinetic electrons, and the divergence increases for smaller values of the GAM radial wavenumber.\",\n \"Moreover, since the frequency stops increasing in the domain of higher wavenumbers (subplot $a$ of Fig.\",\n \"REF ), a divergence between numerical results and analytical theories is observed.\",\n \"The same effect was observed in Ref.\",\n \"[33], where the GAMs were studied using drift reduced Braginskii equations.\",\n \"The Gao 2010 theory describes the GAM dependence on the plasma elongation and it is in a good agreement with numerical results for the frequency.\",\n \"Although the Gao 2010 theory provides considerably smaller damping rate, it seems to give similar trend of the damping coefficient with the plasma elongation, i.e., the damping rate is weakened by the elongation.\",\n \"The Gao 2010 theory was derived in the large orbit drift width limit, where the dominant damping mechanism is the resonance $\\\\omega \\\\sim \\\\omega _d$ (here, $\\\\omega _d = {k_r}\\\\cdot {v_d}$ is a magnetic drift frequency, ${k_r}$ is a wave vector of the zonal potential in the radial direction and ${v_d}$ is a magnetic drift velocity) [34].\",\n \"As explained in Ref.\",\n \"[31], the GAM frequency decreases with the elongation less rapidly than the drift frequency.\",\n \"To satisfy the resonance $\\\\omega \\\\sim \\\\omega _d$ particles have to have higher drift velocities, which involves fewer particles in the wave-particle interaction and, as a result, the GAM damping rate decreases.\",\n \"Figure: Comparison between numerically simulated values (dots, traingles and squares) of the GAM frequency and values obtained using the interpolating expression provided in Eq.\",\n \"() (solid lines).\",\n \"Dotted lines indicate 95% confidence bounds of the fitting.\"\n ],\n [\n \"Interpolating formulae\",\n \"To provide a scaling of the GAM frequency and damping rate, corresponding interpolating expressions have been fitted to the results of the gyrokinetic simulations described in Sec.\",\n \"REF .\",\n \"For consistency, the regime has been chosen for the GAM wavenumbers $k = k_r \\\\rho _i$ in the range $[0.054, 0.377]$ , safety factor $q \\\\in [3.5, 5.0]$ and plasma elongation $e \\\\in [1.0, 1.6]$ .\",\n \"Figure: Comparison between numerically simulated values (dots, triangles or squares) of the GAM damping rate and values obtained by using the interpolating expression provided in Eq.\",\n \"() (solid lines).\",\n \"Dotted lines indicate 95% confidence bounds of the fitting.To derive an interpolating expression for the frequency several assumptions have been used.\",\n \"The experimentally obtained dependence [25] on the plasma elongation $1 / (1 + e)$ has been slightly modified to $1 / (1 + g_6 e)$ , where $g_6$ is an adjustable coefficient.\",\n \"The dependence on the safety factor has been taken in the form $\\\\exp (-g_5 q^2)$ .\",\n \"In fact, the $q$ -dependence in a form of $\\\\sqrt{1 + g_5/q^2}$ , that is given in Ref.\",\n \"[21], gives the same results.\",\n \"To describe how the frequency changes with the radial wavenumber, a polynomial has been taken.\",\n \"Moreover, to take into account the frequency saturation for higher wavenumbers[33] we have introduced a function of the form $1 / (1 + g_4 k)$ .\",\n \"Here, $k = k_r \\\\rho _{i}$ , $v_{Ti} = \\\\sqrt{q_eT_i/2m_p}$ .\",\n \"The resulting frequency interpolating formula is the following one: $f_{\\\\omega } \\\\left[\\\\displaystyle {\\\\frac{\\\\sqrt{2} v_{Ti}}{R}} \\\\right] = \\\\displaystyle {\\\\frac{g_1 + g_2 k^2 + g_3 k^4}{1 + g_4 k}} \\\\displaystyle {\\\\frac{\\\\exp \\\\left( -g_5 q^2 \\\\right)}{1 + g_6 e}}.$ Among different tested functions, this form gives the best approximation to numerically simulated values of the GAM frequency, it has one of the smallest 95% confidential bounds and is not overfitted.\",\n \"The corresponding coefficients $g$ with their 95% confidential bounds (lower $g_{lc}$ and upper $g_{uc}$ bounds) are $g = &&[3.7733,\\\\ 6.3505,\\\\ -1.9741e1,\\\\ 1.3557e-1,\\\\ 1.4620e-3,\\\\ 1.1684],\\\\\\\\*g_{lc} = &&[3.6745,\\\\ 3.3168,\\\\ -2.8800e1,\\\\ -6.0078e-2,\\\\ 1.1373e-3,\\\\ 1.1234],\\\\\\\\*g_{uc} = &&[3.8720,\\\\ 9.3843,\\\\ -1.0682e1,\\\\ 3.3121e-1,\\\\ 1.7866e-3,\\\\ 1.2135].$ Results for the Eq.\",\n \"(REF ) are depicted in Fig.\",\n \"REF .\",\n \"For the damping rate we have derived the following expression (here, the damping rate is normalized to $\\\\sqrt{2} v_{Ti}/R$ ): $f_{\\\\gamma } \\\\left[\\\\displaystyle {\\\\frac{\\\\sqrt{2} v_{Ti}}{R}} \\\\right] = \\\\displaystyle {\\\\frac{\\\\left(h_1 + h_2 k^2\\\\right)\\\\exp \\\\left[-h_3 q^2\\\\right]}{1 + h_4 e^2}} + \\\\displaystyle {\\\\frac{\\\\left(h_5 + h_6 k^2\\\\right)\\\\exp \\\\left[-h_7 q^2 \\\\right]}{1+h_8 e^4}}.$ with interpolating coefficients $h =&& [-1.2494e-2,\\\\ -8.9688e-1,\\\\ 4.5498e-2,\\\\ -1.9884e-1,\\\\\\\\*&&-1.1248e-2, -2.5481,\\\\ -5.3340e-3,\\\\ 7.7748e-1],\\\\\\\\h_{lc} =&& [-2.3115e-2,\\\\ -1.6490,\\\\ 2.5215e-2,\\\\ -3.3573e-1,\\\\\\\\*&&-2.5523e-2, -3.1909,\\\\ -1.9665e-2,\\\\ 5.1924e-2],\\\\\\\\h_{uc} =&& [-1.8723e-3,\\\\ -1.4471e-1,\\\\ 6.5781e-2,\\\\ -6.1955e-2,\\\\\\\\*&&3.0272e-3, -1.9053,\\\\ 8.9973e-3,\\\\ 1.5030].$ Comparison between the results from the gyrokinetic simulations and the interpolation expression for the GAM damping rate is shown in Fig.\",\n \"REF for some specific values of parameters taken as examples.\"\n ],\n [\n \"Phase mixing\",\n \"To investigate the influence of the phase mixing on the GAM dynamics the same parameters as described in chapter REF has been used, but the temperature gradient (the same for both the electrons and ions to have $\\\\tau _e = T_e/T_i = 1$ , $T_i(s_0) = 70$ eV) at a radial position $s_0 = 0.90$ has been introduced.\",\n \"The radial point $s_0 = 0.90$ has been chosen here to be in agreement with the section REF .\",\n \"Initial radial wavenumber of the radial electric field is $k = 0.108$ .\",\n \"The safety factor is $q(s_0) = 4.0$ .\",\n \"We consider a temperature profile of the following form, similarly to Ref.\",\n \"[23]: $\\\\displaystyle {\\\\frac{T_e(s)}{T_e(s_0)}} = \\\\exp \\\\left[-\\\\Delta \\\\cdot k_T \\\\cdot \\\\tanh \\\\left(\\\\displaystyle {\\\\frac{s - s_0}{\\\\Delta }}\\\\right)\\\\right],$ where $\\\\Delta = 0.04$ , $k_T = - \\\\left.\",\n \"d[\\\\ln (T)]/ds \\\\right|_{s = s_0}$ .\",\n \"The temperature profiles and the corresponding temperature gradient profiles for different $k_T$ in a radial interval $s = [0.85, 0.95]$ , are shown in Fig.\",\n \"REF .\",\n \"Dependence of the GAM half-decay time $t_{1/2}$ on the temperature gradient has been investigated in the domain $k_T \\\\in [1, 15]$ .\",\n \"Figure: Temperature and temperature gradient radial profiles for different k T k_T: k T =[1,5,15]k_T = [1, 5, 15].A scan of gyrokinetic simulations with the temperature gradient $k_T$ has been performed, and the results are depicted in Fig.\",\n \"REF .\",\n \"In presence of a temperature gradient, the GAM is observed to oscillate with different frequencies at different radial points, that leads to the distortion of the initial GAM radial structure.\",\n \"Producing higher radial wavenumbers, this distortion amplifies the GAM damping.\",\n \"This combined effect, already investigated for a more simplified configuration in Ref.\",\n \"[22], [23], has been observed even more pronounced in the simulations described here.\",\n \"In fact, here the phase mixing effect is investigated using gyrokinetic simulations with kinetic electrons that significantly influences the GAM damping, and, as a consequence, the GAM half-decay time.\",\n \"For example, using the Sugama-Watanabe model[35], which is derived with adiabatic electrons, for the Landau damping and combining with phase mixing, we have obtained $t_{1/2}[R/ \\\\sqrt{2} v_{Ti}] = 118$ for the $k_T = 1$ and $t_{1/2}[R/ \\\\sqrt{2} v_{Ti}] = 23.4$ for $k_T = 10$ , that predicts much longer half-decay time of the GAM in comparison to the calculations based on the simulations with the kinetic electrons (compare with a Fig.\",\n \"REF ).\",\n \"In order to verify the results of gyrokinetic simulations, we have used a theoretical simplified model of the phase mixing, proposed in Ref.\",\n \"[8], [22], [23], where the linear growth in time of the radial wavenumber is considered.\",\n \"In the phase mixing simulations a space point $s_0$ is considered with a certain temperature $T(s_0)$ and temperature gradient $k_T(s_0)$ .\",\n \"Initial radial electric field has the following radial structure: $E(s) = E_0 \\\\cos (k_0 s)$ with an initial amplitude $E_0$ and initial normalized radial wavenumber $k_0$ .\",\n \"The electric field is assumed to evolve in time at a point $s_0$ according to a simple rule $E(s_0, t) = E_{a}(s_0, t) \\\\cos (\\\\omega (s_0) t),$ where $E_{a}(s_0, t)$ is an amplitude of the electric field, that changes in time due to the damping, $E_{a}(0) = E_0$ .\",\n \"The general form of the GAM frequency is $\\\\omega (s, t) = \\\\sqrt{\\\\frac{2 T_e(s)}{2m_p}} \\\\omega ^*(k, q, e),$ where $\\\\omega ^*(k, q, e)$ describes frequency dependence on the radial wavenumber, the safety factor and the elongation.\",\n \"The safety factor profile is taken to be flat, and plasma with a circular cross-section is considered: $e = 1.00$ , $r \\\\approx a s$ .\",\n \"The damping rate is defined as $\\\\gamma (s_0, t) = \\\\displaystyle {\\\\frac{1}{E(s_0,t)}}\\\\displaystyle {\\\\frac{ d E(s_0,t)}{d t}}.$ At the beginning of every time interval $[t_1, t_1 + \\\\Delta t]$ , new values of the damping rate $\\\\gamma (s_0, t_1)$ and frequency $\\\\omega (s_0, t_1)$ are found with the scaling formulae given in Eq.\",\n \"(REF ) and (REF ), using a current value of the wavenumber $k(s_0, t_1)$ .\",\n \"A new value of the electric field can be found, assuming that the damping rate is constant at the lapse of time $[t_1, t_1 + \\\\Delta t]$ : $E(s_0, t_1 + \\\\Delta t) = E(s_0, t_1)\\\\cdot (1 + \\\\gamma (s_0, t_1) \\\\Delta t).$ After that, a new value of the wavenumber $k(s_0, t_1+\\\\Delta t)$ is calculated using the radial derivative of the frequency $\\\\left.\",\n \"\\\\frac{\\\\partial {\\\\omega (s, t_1)}}{\\\\partial {s}} \\\\right|_{s = s_0} = - \\\\displaystyle {\\\\frac{1}{2}} \\\\omega (s_0, t_1) k_T.$ With that, the wavenumber is assumed to change linearly in time as $k(s_0, t_1 + \\\\Delta t) = k(s_0, t_1) - \\\\sqrt{2} \\\\rho ^* \\\\left.\",\n \"\\\\frac{\\\\partial {\\\\omega (s, t_1)}}{\\\\partial {s}} \\\\right|_{s = s_0} \\\\Delta t,$ where $\\\\rho ^* = \\\\rho _s / a$ , $\\\\rho _s = c_s/ \\\\omega _{ci}$ .\",\n \"Another option, it is to estimate the time evolution of the radial wavenumber directly from numerical calculations in ORB5.\",\n \"Substituting new value of the normalized wavenumber $k(s_0, t_1+\\\\Delta t)$ into Eq.\",\n \"(REF ), we can find the damping rate $\\\\gamma (s_0, t_1+\\\\Delta t)$ at the next time point.\",\n \"The results obtained with this reduced theoretical model are also shown in Fig.\",\n \"REF .\",\n \"As it can be seen in Fig.\",\n \"REF , the qualitative dependence of the half-decay time on the temperature gradient finds a good match of gyrokinetic simulations of ORB5 and analytical theory.\",\n \"The difference is due to the global dynamics of the ORB5 simulations, which is compared here with a theory where the phase mixing follows a local estimation given in Ref. [8].\",\n \"Figure: Dependence of the GAM half-decay time on the temperature gradient obtained from the simulations in ORB5 (green squares), from the theory using a linear estimation Eq.\",\n \"() (blue dots) and estimation from ORB5 (red triangles) of the radial wavenumber.\"\n ],\n [\n \"Comparison with experimental data\",\n \"The dispersion relations obtained in Sec.\",\n \"REF as an interpolation of gyrokinetic simulations and given in Eqs.\",\n \"(REF ), (REF ) can be used to compare numerical estimations of the GAM behaviour to measurements of the GAM frequency, performed on ASDEX Upgrade tokamak [25] using Doppler reflectometry.\",\n \"More precisely, we consider the discharge AUG#20787 with the plasma elongation at the edge $e = 1.09$ (and we assume that it is constant at the considered radial region $\\\\rho = r/a = [0.8, 1.0]$ ).\",\n \"The GAM radial wavenumber is considered to be constant and is estimated to be $k_ra = 40\\\\pi $ from the experimental radial profile of the GAM amplitude (see Fig.\",\n \"5f in Ref.\",\n \"[25]).\",\n \"Experimental safety factor and ion temperature profiles have been taken to estimate the GAM frequency and damping rate using the scaling formulae (REF ), (REF ) at different radial points $\\\\rho $ .\",\n \"In Fig.\",\n \"REF the GAM frequency profiles with corresponding theoretical prediction are depicted.\",\n \"A good general agreement is found in the central region of interest, where the GAM intensity, measured in the experiments, is peaked.\",\n \"On the other hand, the linear dispersion relation REF can not explain neither the staircase nature (the plateaus) of the frequencies nor the GAM peak splitting that is observed experimentally at the radius positions $\\\\rho = 0.922$ or $\\\\rho = 0.932$ (although the presence of GAM eigenmodes has been suggested by simplified analytical models [36], [37], whose detailed analysis is out of the scope of this paper).\",\n \"For this reason, we can conjecture that the coherent phenomena at the basis of the formation of GAM extended eigenmode or frequency splitting must have a nonlinear origin.\",\n \"For reference we have given here estimation of the GAM collisional damping rate using formulae, derived by Gao in Ref.\",\n \"[9].\",\n \"Introducing normalized ion collision rate $\\\\hat{\\\\nu }_i = \\\\nu _i qR / v_{ti}$ , the collisional damping rate is calculated as[9] $\\\\displaystyle {\\\\frac{\\\\gamma ^{col}}{v_{Ti}/qR}} = - \\\\displaystyle {\\\\frac{3\\\\hat{\\\\nu }_i}{14 + 8\\\\tau _i}},$ if $\\\\hat{\\\\nu }_i \\\\ll 1$ , and as: $\\\\displaystyle {\\\\frac{\\\\gamma ^{col}}{v_{Ti}/qR}} = - \\\\displaystyle {\\\\frac{3}{8}} \\\\hat{\\\\nu }_i \\\\left(\\\\displaystyle {\\\\frac{7}{4}} + \\\\tau _i + \\\\displaystyle {\\\\frac{\\\\hat{\\\\nu }_i^2}{q^2}} \\\\right)^{-1},$ if $\\\\hat{\\\\nu }_i \\\\ge 1$ .\",\n \"To find the ion collisional rate we have used classical expressions: $\\\\nu _i &=& 4.8\\\\cdot 10^{-8} Z^4 \\\\mu ^{-1/2} n_i[cm^{-3}] T_i[eV]^{-3/2}\\\\ln \\\\Lambda ,\\\\\\\\\\\\ln \\\\Lambda &=& 23 - \\\\ln \\\\left[ \\\\sqrt{2n_i[cm^{-3}]}\\\\displaystyle {\\\\frac{Z^3}{T_i[eV]^{3/2}}} \\\\right],$ where $\\\\mu \\\\equiv m_i / m_p = 2$ , $Z = 1$ .\",\n \"Figure: Comparison of the experimental GAM frequencies to the numerical values, obtained with the formula given in Eq.\",\n \"().\",\n \"Numerical damping rate is depicted on the right plot.\",\n \"The grey dotted line is an estimation of the collisional damping rate of the GAM found using expressions given by Gao in Ref.\",\n \".\",\n \"The red dotted lines are the 95% confidential bounds of the approximated damping rate.According to the Fig.\",\n \"REF , the collisional damping is found to be negligible in the radial domain where GAMs are experimentally measured, except in a very narrow region close to the separatrix, where it can be of the same order of magnitude as the Landau damping.\"\n ],\n [\n \"Conclusions\",\n \"In tokamak plasmas, the drift-wave turbulence gives rise to the zonal flows that in their turn shear and distort convective and turbulent cells leading to the saturation of turbulence and, consequently, to a reduction of the radial heat transport.\",\n \"Action of the magnetic curvature results in the oscillatory zonal flows, so-called geodesic acoustic modes.\",\n \"The peculiarity of the GAM oscillations resides in the different shearing efficiency that the ZF have in relation to their oscillatory behavior.\",\n \"The nonlinear interactions between the GAM and the DW turbulence is defined in a high degree by the GAM damping rate.\",\n \"Lack of the experimental data of this characteristic of the GAM makes the results from linear gyrokinetic simulations particularly important for analytical and numerical investigation of the nonlinear GAM-DW systems.\",\n \"In this work, linear gyrokinetic simulations have been performed with kinetic electrons to study the GAM dynamics.\",\n \"Numerical results have been compared to analytical theories, derived with adiabatic electrons.\",\n \"It has been shown that analytical theories, derived with adiabatic electrons, result in smaller values of the damping rate and for higher wavenumbers diverge from numerical calculations of the frequency.\",\n \"That is why, investigating the GAM dependence on the plasma safety factor, elongation and radial wavenumber, we have found approximating analytic expressions for the frequency and damping rate to predict the GAM behaviour in different plasma regimes.\",\n \"The derived expressions can be used to estimate the GAM linear characteristics used in analytical models of the nonlinear interactions between the GAM and the DW, such as different reduced models [38], [8].\",\n \"Using these formulae, the phase mixing effect on the damping rate has also been calculated.\",\n \"Based on the gyrokinetic simulations with kinetic electrons, the results have shown smaller half-decay times of the GAM in comparison with the Ref.\",\n \"[22], [23].\",\n \"The GAM is one of the special features of the I-mode and can be observed in the L-mode [15], [39], [40].\",\n \"Comparison of the characteristic drive time of the GAM $t_{RD} \\\\sim 1/\\\\gamma _{RD}$ , which is given by the nonlinear coupling with the ion-temperature-mode (ITG)[8], [38], with the GAM half-decay time $t_{1/2}$ confirms the results of the Ref.\",\n \"[22], [23].\",\n \"Indeed, we estimate the GAM drive time to be $t_{RD} < t_s$ (where $t_s \\\\sim 2^{-1/2} R/v_{Ti}$ ) in the L-mode, $t_{RD} \\\\sim t_s$ in the I-mode and $t_{RD} \\\\sim 10t_s$ in the H-mode, according to Ref.\",\n \"[23].\",\n \"In this case, it can be seen from the Fig.\",\n \"REF that the GAM half-decay time, which is defined by both the Landau damping and the phase mixing effect, is much higher than the drive-time in the L and I modes, $t_{1/2} > t^{L,I}_{RD}$ , for all considered values of $k_T$ .\",\n \"This means that the energy transfer rate from the ITG turbulence to the GAM exceeds the Landau-phase mixing damping rate of the GAM.\",\n \"As a result, the GAM can be observed in the L and I modes, but not in the H-mode, where $t_{1/2} < t^H_{RD}$ already for $k_T > 3$ .\",\n \"This could be the explanation for the result that the GAM are not observed in the high-confinement mode, as proposed in Ref.\",\n \"[22], [23].\",\n \"The approximating expressions have showed quite good agreement with experimental data, but estimations of the GAM frequency, obtained from linear gyrokinetic simulations, does not explain the staircase radial profile of the frequency and the GAM peak splitting.\",\n \"The frequency expression describes only the continuum or dispersive mode in contrast to eigenmode.\",\n \"The latter is characterized by the GAM mode frequencies which are predicted to remain constant over a large radial extent, but a significant radial overlap in the frequency radial profile can be observed, that lead to the GAM frequency peak splitting[40].\"\n ],\n [\n \"Acknowledgments\",\n \"Part of this work was done while two of the authors, I. Novikau and A. Biancalani, were visiting LPP-Palaiseau (France), whose team is kindly acknowledged for the hospitality.\",\n \"The authors acknowledge discussions with F. Jenko, L. Villard, X. Garbet and T. Görler.\"\n ],\n [\n \"Numerical convergence tests\",\n \"To calculate the GAM damping rate and frequency, poloidally averaged radial electric field has been fitted, using the Levenberg–Marquardt algorithm [41], to a function of the form $\\\\exp (\\\\gamma t) \\\\cos (\\\\omega t)$ , where $\\\\gamma $ , $\\\\omega $ are sought-for damping rate and frequency.\",\n \"Before the fitting it's necessary to filter the radial electric field to get ride of the high-frequency Alfvén oscillations.\",\n \"In Fig.\",\n \"REF the fitting is depicted for the case: $e = 1.30$ , $q = 4.0$ , $k = 0.108$ .\",\n \"It is worth to mention that the choice of a time interval, where the fitting is performed, can influence the result damping rate, and it is not so crucial for the GAM frequency calculation.\",\n \"This ambiguity in the choice of the time interval can be explained by the fact that at the beginning of the simulations there are some transient processes that must be excluded from the damping rate measurements.\",\n \"Moreover, with the time, the global effects start to play a significant role, distorting initial radial structure of the radial electric field, that makes the damping rate to be variable in time.\",\n \"The GAM dynamics (frequency and damping rate) doesn't depend on the plasma density in linear calculations (see Fig.\",\n \"REF and REF ).\",\n \"But the frequency of the Alfvén waves decreases with the increase of the density, and for high values of the plasma density it becomes difficult to separate acoustic and Alfvénic time scales (see Fig.\",\n \"REF ).\",\n \"Figure: Fourier spectrum of the radial electric field for different values of the electron beta: β e =5e-5\\\\beta _e = 5e-5 (blue line), β e =1e-3\\\\beta _e = 1e-3 (red line).\",\n \"Only the Alfvén frequency changes with the electron beta.\",\n \"The GAM frequency remains the same.\",\n \"Here, the frequency is normalized to ω ci \\\\omega _{ci}.The transition from the simulations with the adiabatic electrons to the ones with the kinetic electrons applies additional restrictions on several numerical parameters such as the time step and the number of markers (see Fig.\",\n \"REF ).\",\n \"Figure: Convergence tests on the number of points in the radial space grid n s n_s (e=1.60e = 1.60, q=4.0q = 4.0, k r ρ i =0.108k_r \\\\rho _i = 0.108) (Fig.\",\n \"), on the normalized time step dt norm dt_{norm} for different values of the plasma elongation e=1.00,1.60e = 1.00, 1.60 (q=3.5q = 3.5, k r ρ i =0.108k_r \\\\rho _i = 0.108) (Fig. )\",\n \"and on the number of markers N markers N_{markers} (e=1.60e = 1.60, q=4.0q = 4.0, k r ρ i =0.108k_r \\\\rho _i = 0.108) (Fig.\",\n \").In projects with adiabatic electrons the normalized time step $dt_{norm} = dt\\\\cdot \\\\omega _{ci}$ can be of the order of 20, but in case of the kinetic electrons it has to be significantly reduced till 2 because of the high parallel velocity of the passing electrons.\",\n \"Also electrostatic simulations of the kinetic electrons reveal high-frequency oscillations[30].\",\n \"These oscillations can lead to numerical instabilities in case of low number of markers.\",\n \"To reduce their level we have passed to electromagnetic simulations with small values of electron beta that gave us an opportunity to keep the number of markers on the level of $10^7$ .\",\n \"The radial space step (or number of the points in the radial space grid) is determined by, among other parameters, the GAM wavenumber.\",\n \"To investigate the GAM dynamics with higher values of the radial wavenumber, we simulated a narrow poloidal ring near the edge instead of the full plasma cross-section to reduce the number of radial space points.\"\n ],\n [\n \"Comparison with the code GENE\",\n \"A complete cross-code verification between the gyrokinetic ORB5 and GENE [42], [43] codes has already been done in Ref.\",\n \"[21] on the linear collisionless dynamics of the GAMs with adiabatic and kinetic electrons in the specific case of flat temperature profiles.\",\n \"For completeness, in this paper a comparison between these different codes is shown including the additional phase mixing physical effect, which is driven by non-flat temperature profiles.\",\n \"The motivation behind this study is that although the linear physical models between ORB5 and GENE are equivalent [44], the numerical schemes are different.\",\n \"GENE is an Eulerian code, where the distribution function is not discretized with markers, but it is discretized on a 5D fixed grid in phase-space $({R}, v_{\\\\parallel }, \\\\mu )$ , where ${R}$ is the gyrocenter position, $v_{\\\\parallel }$ is the parallel velocity, and $\\\\mu $ is the magnetic momentum.\",\n \"The simulation plasma parameters have been taken as in Sec.\",\n \"for both GENE and ORB5.\",\n \"A sinusoidal perturbation in the potential field is initialised, as defined in Sec.\",\n \"REF and is let evolved in time.\",\n \"In GENE, the radial box size is $60 \\\\rho _s$ .\",\n \"We have used 128 grid points in radial direction in order to have at least two points per ion Larmor radius.\",\n \"Along the field line 68 points have been used.\",\n \"In velocity space, 68 points and 128 equidistant symmetric grid points have been used for resolving respectively the $\\\\mu $ and the $v_{\\\\parallel }$ space.\",\n \"The velocity space domain has been fixed to 3 and 9 times the thermal velocity, respectively in the $v_{\\\\parallel }$ and $\\\\mu $ space.\",\n \"In order to avoid any recurrence problem, an hyperdiffusivity scheme has been used in the $v_{\\\\parallel }$ direction.\",\n \"In Fig.\",\n \"REF peaks of the flux-surface averaged radial electric field, measured at the radial position $s_0 = 0.90$ for the ion-electron temperature gradient $k_T = 10$ , are shown for both GENE and ORB5.\",\n \"Half-decay time, calculated in ORB5, is $t_{1/2}^{orb}[R/\\\\sqrt{2} v_{Ti}] = 4.9$ and for the case of GENE it is $t_{1/2}^{GENE}[R/\\\\sqrt{2} v_{Ti}] = 4.8$ .\"\n ]\n]"},"id":{"kind":"string","value":"1709.01818"}}},{"rowIdx":891,"cells":{"text":{"kind":"list like","value":[["Interfacing MHD Single Fluid and Kinetic Exospheric Solar Wind Models\n and Comparing Their Energetics"],["Abstract An exospheric kinetic solar wind model is interfaced with an observation-driven single fluid magnetohydrodynamic (MHD) model.","Initially, a photospheric magnetogram serves as observational input in the fluid approach to extrapolate the heliospheric magnetic field.","Then semi-empirical coronal models are used for estimating the plasma characteristics up to a heliocentric distance of 0.1AU.","From there on a full MHD model which computes the three-dimensional time-dependent evolution of the solar wind macroscopic variables up to the orbit of the Earth is used.","After interfacing the density and velocity at the inner MHD boundary, we compare with the results of a kinetic exospheric solar wind model based on the assumption of Maxwell and Kappa velocity distribution functions for protons and electrons respectively, as well as with \\textit{in situ} observations at 1AU.","This provides insight on more physically detailed processes, such as coronal heating and solar wind acceleration, that naturally arise by inclusion of suprathermal electrons in the model.","We are interested in the profile of the solar wind speed and density at 1AU, in characterizing the slow and fast source regions of the wind and in comparing MHD with exospheric models in similar conditions.","We calculate the energetics of both models from low to high heliocentric distances."],["Introduction","Solar wind heating and acceleration mechanisms are still subjects of active research.","In computational models, physical quantities estimated or observationally inferred close to the Sun serve as boundary or initial conditions that will examine the solar wind evolution and its underlying physics as it propagates through interplanetary space.","The solar wind plasma can be studied macroscopically through the magnetohydrodynamic (MHD) approach or microscopically when using the kinetic approach.","In the following Subsections REF and REF , we briefly review the main aspects of both approaches used in this paper, which are here for the first time interfaced in a global model.","[32] presented an empirical relation between the expansion of magnetic flux tubes and the solar wind speed, showing that they evolve inversely proportional to each other.","This assumption was tested using more than two decades of observations, and can give predictions of the solar wind speed at Earth.","The model involves synoptic magnetograms of photospheric field, which allow a Potential Field Source Surface (PFSS, with the source surface typically at 2.5$\\mathrm {R}_\\odot $ ) extrapolation which quantifies the expansion factors.","It was found that at the Earth's orbit, greater expansion corresponded to magnetic field lines near the centre of coronal holes, which diverge more slowly than the ones coming from the hole boundaries.","This is consistent with the fact that lower densities are found in the fast wind regions.","[1] improved the Wang-Sheeley model by using daily updated magnetogram data from the Wilcox Solar Observatory (WSO) and relating the magnetic flux tube expansion factor with the solar wind speed at the source surface, while including effects of stream interactions from the source surface to the Earth.","A statistical study which covered three years and compared the Wang-Sheeley model predictions with data from the WindNASA spacecraft at the L$_1$ Lagrangian point of the Earth designed for long-term solar wind measurements and its effects on the terrestrial magnetosphere (https://wind.nasa.gov).","satellite was presented.","The interplanetary magnetic field (IMF) polarity was properly predicted 75% of the time, while solar wind speeds were within 10-15% of actual values, when a 6-month period with data gaps was removed.","In the computational work of [14], solar wind variations were examined in the corotating frame with a three-dimensional MHD model with a CME (Coronal Mass Ejection) injection scheme in the streamer belt.","Such MHD models take into account magnetic field variations due to the CME interaction with the solar wind during the CME's evolution.","The CME movement depends on the background solar wind density and velocity and the vector properties of the solar wind magnetic field and velocity are affected by the passing disturbance.","ENLIL [16], [17], [15] is a heliospheric MHD model that provides a three-dimensional description of the time-dependent solar wind evolution.","It can use the Wang-Sheeley-Arge (WSA) [32], [1] semi-empirical model as its boundary condition.","The WSA model was used by [13] to study the solar wind at low heliocentric distances including an empirical method to link the magnetic field information with the velocity at 21.5$\\mathrm {R}_\\odot $ .","The new method was cross-validated using the 3D MHD code ENLIL and by comparing the results with observations at 1AU and at further distances as provided by Ulysses.","The estimation of the solar wind speed at 21.5$\\mathrm {R}_\\odot $ was indeed better than previous models and it captured both fast and slow solar wind.","Similar to ENLIL, we use a fully 3D MHD code EUHFORIA (European heliospheric forecasting information asset) that from 0.1AU onwards models the evolution of the plasma environment in the inner heliosphere.","The code details are discussed in [26] and in this paper we adopt it to get the macroscopic description of the solar wind."],["Kinetic Exospheric Models","Exospheric kinetic models are simplified collisionless, stationary models, that are meant to explain the acceleration of the solar wind in a self-consistent way and they were first established by [7] and [10].","The model was one-dimensional and time-independent and provided the state of the solar wind plasma along a magnetic field line.","The acceleration of the solar wind was due to the induced electric field even without suprathermal electrons (Maxwellian distribution), but when accounting for the presence of suprathermal electrons, the terminal speed at 1AU increased.","The original exospheric model [7], [10] assumed Maxwellian velocity distribution functions (VDFs) for protons and electrons and supersonic winds of 300 $\\mathrm {km\\ s^{-1}}$ could be reached at 1AU with temperatures of the order of 1 MK for both species at the exobase (the distance beyond which collisions become negligible).","Nevertheless, it remained difficult for the model to achieve higher bulk velocities, such as the ones observed in the fast solar wind, without increasing the temperature to unrealistic high values (10MK) at the exobase, or by adding other sources of solar wind acceleration.","After the induced electric field is calculated, the solar wind acceleration spontaneously follows giving the solution of the solar wind from sub- to supersonic, without any extra energy terms assumed.","A Lorentzian (Kappa) velocity distribution function is used instead of the classic Maxwellian in search for better agreement with observations in [20].","Indeed, suprathermal electrons are generally observed in the velocity distribution functions measured in situ in the solar wind.","[20] have shown that the presence of such suprathermal electrons accelerates the wind to higher bulk velocities, so that no other source of energy needs to be considered to reach the values observed in the high speed solar wind.","[11] applied the kinetic model developed by [20] using Kappa VDFs for both electron and proton populations that escape from the Sun to describe the solar wind.","Since the first exospheric model, the semi-analytic kinetic model has been able to describe not only the fast but also the slow solar wind together with their sources, in the cold coronal hole and hot equatorial regions respectively, without unrealistic assumptions of too high temperatures and extra heating in the corona, as required by non-turbulence driven fluid models.","While previous exospheric models placed the exobase at a distance of about 5-10$\\mathrm {R}_\\odot $ , from where on the proton total potential energy was a monotonic function of the heliocentric distance, [8] calculated the exobase to be positioned at about 1.1-5$\\mathrm {R}_\\odot $ .","This deeper location of the exobase, lowered under the radial location of the maximum of the total potential energy of the protons, gives the solar wind the observed acceleration to high velocities.","A low exobase leads indeed to higher bulk velocities at 1AU in the case of suprathermal electrons.","Collisionless (exospheric) theoretical models and collisional simulations were compared in [35].","Including suprathermal tails in the velocity distribution function of the electrons and employing a self-consistently computed heat flux, the models were able to reproduce fast solar wind speeds.","Results of collisional kinetic simulations with non-Maxwellian velocity distribution functions and collisionless exospheric models are in good agreement.","Taking into account that the exospheric and collisional models provide comparable results, in this paper we will go a step further and try to interface exospheric models with MHD ones.","On the way to developing predictive tools and 3D solar wind models, a 2D observationally driven kinetic exospheric solar wind model was developed by [21], presenting solar wind variations on the ecliptic plane and how they compare to observations from close to the Sun up to 1AU.","For the ecliptic variational study OMNIMulti-source data set for the near Earth solar wind of combined and normalized observational data from ACE (Advanced Composition Explorer), Wind, IMP 8 (Interplanetary Monitoring Platform) and GOES (Geostationary Operational Environmental Satellite) satellite missions.","observations were used for the time period 26 September to 23 October 2008.","The $\\kappa $ parameter was chosen as 2.35 and 3.82 for fast and slow wind respectively, to match bulk speed observations close to the orbit of the Earth.","We will present a three dimensional generalization of this exospheric model, i.e.","we find the solar wind characteristics along a collection of magnetic field lines each passing through a point on the spherical shell at the exobase level in latitude and longitude $(\\theta ,\\phi )$ .","The basic principles, boundary conditions, physical assumptions and computational methods used by the MHD and the exospheric kinetic models are described in Section 2.","The specific criteria and the observational data that are chosen for this work are explained and presented in Section 3.","The interfacing method as well as explicit results of both approaches and their energetics are discussed in Section 4, while in Section 5 we compare the two approaches and we close by discussing the main conclusions of the study in Section 6."],["MHD Modeling: EUHFORIA","The inner heliosphere model EUHFORIA [26] is used for our MHD approach.","EUHFORIA is a three-dimensional observationally driven model providing an accurate description of the large-scale time-dependent solar wind including transient events such as CMEs.","As such, it allows to inject CMEs at the inner radial boundary at 0.1AU as a time-dependent boundary condition.","Apart from CMEs, the variables at the inner radial boundary are constructed in order to capture the large-scale variations in the solar wind for the particular time period under study.","This is accomplished using a model for the coronal magnetic field and employing empirical relations between the coronal magnetic topology and the state of the solar wind.","The magnetic field model consists of a potential field source surface (PFSS) model in the low corona coupled with a current sheet model higher in the corona.","The PFSS model requires a magnetogram to be provided as input.","To finally compute the super-sonic state of the solar wind at 0.1AU, empirical relations inspired by the success of the WSA (Wang-Sheeley-Arge) model [32], [1] are used.","The MHD model is able to provide density and speed profiles at the Earth's orbit, it allows for slow and fast solar wind source region tracing, and can serve as the MHD counterpart in a comparison project together with kinetic exospheric models that correspond to similar initial and boundary conditions at 0.1AU.","This will be our first goal in this paper, and the way the two approaches are coupled will be described next.","EUHFORIA uses a finite volume discretization scheme to solve the hyperbolic conservative MHD equations.","The equations solved are those of ideal MHD with gravity included as a source term in the equations of momentum and energy: $\\frac{\\partial \\rho }{\\partial t}+\\nabla \\cdot (\\rho {v})=0{,} \\\\\\frac{\\partial (\\rho {v})}{\\partial t}+\\nabla \\cdot \\left[ \\rho {v}{v}+\\left( p+\\frac{B^2}{2\\mu _0}\\right)\\mathcal {I}-\\frac{1}{\\mu _0}{B}{B}\\right]=\\rho {g}{,} \\\\\\frac{\\partial \\mathcal {E}}{\\partial t}+\\nabla \\cdot \\left[\\left(\\mathcal {E}+p-\\frac{B^2}{2\\mu _0}\\right){v}+\\frac{1}{\\mu _0}{B}\\times \\left( {v}\\times {B}\\right)\\right]=\\rho {v}\\cdot {g}{,} \\\\\\frac{\\partial {B}}{\\partial t}-\\nabla \\times \\left( {v}\\times {B}\\right)=0{,} \\\\\\nabla \\cdot {B}=0{,} \\\\\\mathcal {E}= \\frac{p}{\\gamma -1}+\\frac{\\rho v^2}{2}+\\frac{B^2}{2\\mu _0} {,} \\qquad \\gamma =1.5$ where $\\rho $ is the mass density, ${v}$ the velocity vector, ${g}$ the gravitational acceleration, ${B}$ the magnetic field vector, $p$ the thermal pressure, $\\gamma $ the polytropic index, $\\mathcal {E}$ the total energy density, $\\mu _0$ the magnetic permeability and $\\mathcal {I}$ the unit tensor.","Note that we are working in the inertial frame, so no Coriolis nor centrifugal forces need to be added in Equation .","The polytropic index is chosen to be slightly smaller than the expected $\\gamma =5/3$ value for a monatomic gas.","This causes a finite energy to be injected into the system in the form of heat [27].","The use of either a non-adiabatic polytropic index or explicit source terms in the momentum and energy equation to drive the solar wind and heat the corona have been used in several works.","In EUHFORIA, the reduced polytropic index is used in order to slightly accelerate the solar wind further out in the heliosphere, from the speed values at the boundary at 0.1AU.","The single fluid MHD description still leaves freedom to vary $\\gamma $ , which allows to account for expected deviations from the mono-atomic ideal gas value of 5/3.","For a discussion of more self-consistent models that attempt to capture and explain the physical mechanisms resulting in the observed coronal heating and acceleration, we refer to [2].","The employed numerical grid is uniform in spherical coordinates, with the number of cells in $r$ , $\\theta $ , $\\phi $ chosen to be 800, 60, 180, respectively.","The outer boundary is set at 2AU.","Further details of the numerical solution scheme are described in [26]."],["Boundary Conditions","The essential input to EUHFORIA is a synoptic magnetogram.","We select a magnetogram from the GONG (Global Oscillation Network Group) standard synoptic data product, which are available with one hour cadence from GONG.","The chosen magnetogram corresponds closely to Carrington Rotation (CR) 2059.","During this Carrington rotation, an equatorial coronal hole was visible near the central meridian to about $60^\\circ $ degrees west.","The solar wind plasma state at 0.1AU in EUHFORIA is determined using a semi-empirical approach similar to the Wang-Sheeley-Arge model.","The method consists of constructing a model of the coronal magnetic field consisting of a PFSS extrapolation in the low corona while the \"Schatten\" Current Sheet is used from $2.5\\mathrm {R}_\\odot $ to 0.1AU.","The solar wind speed is then given through an empirical relation which is a function of the magnetic flux tube expansion factor.","The formula used in this work is given by $V(f_s)=240.0+675.0(1+f_s)^{-0.22} \\mathrm {km\\ s^{-1}} {,}$ where $f_s$ is given by $f_s=\\left(\\frac{\\mathrm {R}_\\odot }{r}\\right)^2\\frac{B_r(\\mathrm {R}_\\odot ,\\theta _0,\\phi _0)}{B_r(r,\\theta ,\\phi )}$ and it quantifies the expansion factor of the flux tube from the photospheric footpoint $(\\mathrm {R}_\\odot ,\\theta _0,\\phi _0)$ of the specific field line to its position further outwards $(r,\\theta ,\\phi )$ at a heliocentric distance $r$ [33].","As explained in [33], the expansion factor takes values greater or equal to unity for flux divergence more rapid than or equal to $r^2$ , respectively.","Simple scaling laws that are functions of $V$ are used in order to determine the plasma density and temperature.","For further details of the empirical model, see [26].","For the kinetic component of our analysis, we are using an exospheric model, which is a way to simulate low density plasmas, where the importance of collisions is limited.","The solar atmosphere is considered to have a collision-dominated barosphere at low altitude [8] and a collisionless exosphere, which is the region modeled kinetically.","These regions are separated by a surface called the exobase $r_0$ , beyond which collisions become negligible.","This exobase level is defined as the altitude where the particle mean free path $l_f$ and the local density scale height $H$ become equal, i.e.","where the dimensionless Knudsen number $K_n=l_\\mathrm {f}/H$ is equal to unity.","The kinetic modelA 1D version of the kinetic exospheric model developed by the group in IASB-BIRA and collaborators can be found in CCMC (http://ccmc.gsfc.nasa.gov/models/exo.php) and it can run online for user-defined setups.","[10], [20], [11], [8] gives different temperatures for electrons and protons as indeed observed [9] and can include different characteristics of any other ion species.","Sources of the fast solar wind are considered to be coronal holes and in these regions the electron VDFs are assumed to correspond to a Lorentzian function with a small $\\kappa $ -value and thus have a large suprathermal tail [11].","The low speed solar wind usually comes from equatorial regions, with larger $\\kappa $ -values.","When $\\kappa \\rightarrow \\infty $ the VDF tends to a Maxwellian.","In this work, we consider two particle species, namely electrons and protons and therefore their respective exobase levels need to be defined.","The proton exobase is located where the Coulomb mean free path for the protons $l_{\\mathrm {f},p}$ according to [29] as estimated for coronal values by [11] is equal to the local density scale height $H$ , where $l_{\\mathrm {f},p}\\approx 7.2\\times 10^7\\frac{T^2_p}{n_e}{,} { \\ \\ \\ \\ }H=\\left(-\\frac{\\mathrm {d}\\ln n_e}{\\mathrm {d}r} \\right)^{-1} {,}$ where all the quantities are in SI.","The proton mean free path is shown in Figure REF as a function of the proton temperature and the electron number density.","For the electrons, a similar electron exobase height can be estimated from the Coulomb mean free path in a plasma consisting only of electrons and protons: $l_{\\mathrm {f},e}=0.416 \\left(\\frac{T_e}{T_p}\\right)^2l_{\\mathrm {f},p} {,}$ as in [11] for a hydrogen plasma, assuming that the electrons have the mean thermal velocity $(8k_\\mathrm {B}T_e/m_e\\pi )^{1/2}$ , with $l_{\\mathrm {f},e},l_{\\mathrm {f},p}$ in meters and $T_e,T_p$ in kelvin.","A crude estimate of the scale height can be obtained assuming hydrostatic equilibrium in a stratified atmosphere with isothermal plane parallel layers.","This is not the case for the solar wind, since expansion is taking place and rather hydrodynamic conditions apply, but it gives a good approximation of the order of magnitude of the scale height.","Figure: The Coulomb mean free path l f,p l_{\\mathrm {f},p} in solar radii as a function of proton temperature and electron density.","This figure quantifies the variations in Equation .When we adopt the same proton and electron temperature in the above formulae, the electron mean free path becomes smaller than the proton one $l_{\\mathrm {f},e}0.01$ , due to the velocity dependence of the mean free path of the particles.","This shows that it is not especially important that the exobase is chosen to correspond exactly to the level where $K_n=1$ , but it will appear where the density gradient is very sharp and thus where the plasma becomes collisionless to a good approximation."],["Velocity Distribution Functions","When collisions are ignored as in the exospheric theory developed by [10], the Fokker-Planck equation reduces to the Vlasov equation for the evolution of the velocity distribution function: $\\frac{\\partial f}{\\partial t}+{v}\\cdot \\frac{\\partial f}{\\partial {r}}+{a}\\cdot \\frac{\\partial f}{\\partial {v}}=0 {.","}$ Our kinetic exospheric model works by constructing stationary solution to the Vlasov equation, starting from an exact stationary solution for protons and electrons prescribed at the exobase.","Kinetic models based on this equation were developed and are discussed in [11] for radial magnetic field lines and in [23] taking into account the spiral interplanetary magnetic field topology.","It was shown in [11] that the specific moments in the solar wind, namely densities and temperatures, as well as the electrostatic potential characteristics from the corona to the interplanetary space are already well described, agreeing with observations at 1AU, when the collision term is neglected, since the collisions would rather modify the temperature anisotropies.","Apart from the Maxwellians the generalised Lorentzian or Kappa function is also a solution of the Vlasov equation and can be used as a boundary condition to study the effect of suprathermal particles on the kinetic moments.","Observations suggest that the velocity distribution functions of the electrons have strong suprathermal tails.","We therefore assume a Lorentzian VDF for the electrons and a Maxwellian VDF for the protons at the exobase: $& f^p_{\\mathrm {Maxwell}}(r_0,v)=n_p(r_0)\\left(\\frac{m_p}{2\\pi k_\\mathrm {B}T_p(r_0)} \\right)^{3/2}\\exp \\left( -\\frac{m_pv_p^2}{2k_\\mathrm {B}T_p(r_0)}\\right) {,} \\\\& f^e_{\\mathrm {kappa}}(r_0,v)=\\frac{n_e(r_0)}{2\\pi \\kappa ^{3/2}}\\left(\\frac{m_e}{2k_\\mathrm {B}T_e(r_0)} \\right)^{3/2}A(\\kappa )\\left(1+\\frac{m_ev_e^2}{2k_\\mathrm {B}T_e(r_0)\\kappa } \\right)^{-(\\kappa +1)} {,} \\qquad $ where $A(\\kappa )=\\frac{\\Gamma (\\kappa +1)}{\\Gamma (\\kappa -1/2)\\Gamma (3/2)} {.","}$ We note in passing that the moments of the Lorentzian VDF are not well defined for every $\\kappa $ value, but rather every $i$ th moment is defined for $\\kappa >(i+1)/2$ [20].","Suprathermal protons have almost no influence on the solar wind velocity, so for them a Maxwellian VDF can suffice [11].","Liouville's theorem [3] implies that any function that depends on the constants of motion of a collection of particles satisfies the Vlasov equation.","The relevant constants of motion in this study are the total energy and the magnetic moment.","Knowing the velocity distribution functions for our particle species at the exobase, the velocity distribution as a function of the radial distance can be deduced from energy conservation [20].","Thereby, the electron and proton VDFs can be computed as a function of radius and speed along a purely radial magnetic field line.","The analytic expressions for the kinetic moments of the exospheric models were derived for a Maxwellian VDF by [10] and for a Lorentzian VDF by [20].","As explained above the exospheric model used in this paper includes only radial velocities along open radial magnetic field lines.","Like in previous exospheric models, it is assumed that there are no particles coming from the interplanetary space to the Sun.","The anisotropy of the distribution leads to the solar wind flux.","The density, temperature and $\\kappa $ -index are determined at the exobase by either the MHD model or constrained via observations.","The model provides then the velocity distribution function at any other distance as well as the rest of the kinetic moments.","We use the code and the analytical expressions for the Maxwellian and the $\\kappa $ distributions by [20].","Provided the number density, the electron and proton temperatures and the $\\kappa $ index for the electron VDF at the exobase, the quasi-neutrality and zero-current conditions are solved iteratively at a fixed radial distance $r_m$ using a Newton-Raphson scheme.","The value of $r_m$ is iteratively modified using a dichotomy method until the electric field is found to be continuous within a predefined tolerance [8].","On the other hand, EUHFORIA solves the 3D MHD equations taking self-consistently stream interactions into account thereby providing a $v_\\phi $ for any point, whereas in the kinetic model we impose that this velocity is constant on each spherical shell.","The kinetic model thus proceeds without accounting for stream interactions instead keeping the same topology of fast and slow solar wind sources at every radial distance as at the exobase, reducing the computational time.","As was argued in [23], the effects due to rotation as compared to the purely radial case change only the estimated proton and electron temperatures and their anisotropies.","More specifically, the spiral structure predicts higher electron temperatures $T_e$ and lower proton temperatures $T_p$ than the radial case, but the number density, the electric potential and the bulk speed remain almost unaffected up to $300\\mathrm {R}_\\odot $ .","It's worth noting here that stream interactions are once again not taken into account.","Therefore, in this study we use the radial magnetic field topology and simply rotate by $v_\\phi $ each spherical shell to account for solar rotation.","The advantage of the kinetic model used in this paper is the direct quantification of species-specific temperature profiles, densities, speeds, energy fluxes etc.","once the electric potential is calculated.","Even if $T_e=T_p$ is chosen at the exobase, the kinetic model self-consistently calculates the species-specific heating with distance and the two temperatures depart from each other.","The $T_e$ is indeed observed to be different than $T_p$ at 1AU for slow and fast wind cases, e.g.","as reported in [9]."],["Observational Input: Cases and Selection Criteria","Several missions have observed, or continue to observe the Sun, as well as measure the physical parameters that characterize the solar wind, at different heliocentric distances as well as heliographic latitudes.","They provide high resolution data, not only in the ecliptic plane, but also at higher latitudes, requiring simulations that explain and predict the behavior of the solar wind not only in the ecliptic plane, but rather in three dimensions.","In this study we will be using OMNI data, and data from the Ulysses spacecraft due to its large latitudinal coverage.","We based our synoptic magnetogram and solar state selection on the following criteria, that allow us to perform a crosscheck on their prediction ability with available spacecraft observations: i) quiet Sun periods (since the exospheric models are particularly tailored to quiet Sun conditions); ii) the presence of equatorial coronal holes, such that significant differences between high and slow speed wind may be expected at the orbit of the Earth; iii) position of Ulysses for combination of simulations and different spacecraft observations from different angles/telescopes; and iv) very few CME events according to the available catalogue CACTUSMore information can be found at http://sidc.oma.be/cactus/..","In this study we focus on the year 2007, as it was a mostly quiet Sun year coinciding with the third orbit of Ulysses.","For the global 3D comparison we focus on the period July-August 2007, when Ulysses crossed the ecliptic plane.","We have confirmed the relative paucity of CME events in the selected time period using the CACTUS CME list."],["Constraining the Kinetic Model using EUHFORIA","First, we constrain the input to the kinetic model based on data-driven results at the inner radial boundary of EUHFORIA at $21.5 \\mathrm {R}_\\odot $ and examine both models' efficiency at capturing the solar wind bulk quantities by comparing with observations at 1AU and at 1.4AU (the Ulysses orbit).","We compare the results of the MHD and kinetic models after interfacing their velocities and densities at the inner MHD boundary at $21.5\\mathrm {R}_\\odot $ .","In Figure REF we show a slice at the equator (left panels) and a slice in latitude (right panels) that corresponds to the mass density scaled with the inverse square of the heliocenteric distance, and the radial speed also indicating the positions of Mercury, Venus, Earth, Mars and the STEREO (Solar Terrestrial Relations Observatory) A spacecraft.","The grey part of the colorscale used in the Figure corresponds to high velocities and densities that occur especially during CME events.","The velocity ranges ocurring at this time are roughly from 350 to 650 $\\mathrm {km\\ s^{-1}}$ with a clear separation between streams of different speeds, forming a clear Parker spiral-like configuration.","There is a configuration of several distinct high and slow speed streams.","We can see the different streams with high density associated to low speed and vice versa, while the highest speed captured is around $650\\mathrm {km\\ s^{-1}}$ and corresponds to a density similar to the one measured at 1AU.","The highest density contrast with respect to the one measured at 1AU is about 10.","The latitudinal panels suggest that there is compression and rarefaction as the plasma flows outwards with the corresponding speed and density.","We thus conclude that the plasma is not following exactly the ideal Parker spiral moving on perfect cones as it expands, but following a rather more complicated motion, as the simulation is observation-driven and the photospheric magnetogram shows a complex topology, as discussed previously.","Figure: EUHFORIA longitudinal and latitudinal variations of the three velocity and the three magnetic field components at 0.1AU (left panels) and at 1AU (right panels) for a standard magnetogram centered at 10 August 2007.In Figure REF , we show the components of the velocity and magnetic field in the spherical ($r,\\theta ,\\phi $ ) basis at two different distances, 0.1AU and 1AU, as functions of heliospheric longitude and latitude, as calculated by EUHFORIA.","We observe a clear and narrow undulating current sheet showing clear discrimination between low- and high-latitude solar wind.","This undulating sheet is characterized by lower velocity than other regions at 0.1AU.","It separates the outward (southern hemisphere) and inward (northern hemisphere) magnetic field topologies.","As the solar wind propagates outwards, the interactions between streams of different speeds become increasingly important with the sharp features at 0.1AU turn into more diffused, smoothed and extended ones at larger distances.","At the Earth's orbit the current sheet and thus the slow speed region is thicker covering about $10^\\circ $ in latitude.","The speed difference at 1AU with respect to the inner boundary is about $50\\mathrm {km\\ s^{-1}}$ .","The $\\theta $ , $\\phi $ -components of the velocity increase in magnitude from zero to about $\\approx 15 \\mathrm {km\\ s^{-1}}$ at the Earth's orbit, roughly following the pattern that the slow speeds appear at small latitudes contrary to the high solar wind speeds.","The radial component of the magnetic field decreases by about 2 orders of magnitude from the inner boundary to the orbit of the Earth showing a broader current sheet ($\\approx 5^\\circ $ ), where the magnetic field is close to zero.","The $\\theta $ -component of the magnetic field increases from zero to about $0.36$ nT in magnitude up to 1AU.","On the other hand, the $B_\\phi $ decreases by almost a factor of 8 up to 1AU and is at both distances opposite in polarity to the radial magnetic field, thus having positive polarity at the North Pole and negative at the South.","In all the depicted quantities the rotation of the plasma shows as the entire structure in the 2D maps moves to the left.","[htbp] Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONEUHFORIA longitudinal and latitudinal variations of the radial velocity (first row), the number density (second row) and the temperature (third row) at 0.1AU (left panels), at 1AU (middle panels) and at 1.4AU (right panels).","They can be contrasted with their kinetic counterparts in Figure REF .","In Figure REF , we show the velocity (first row), the density (second row) and the temperature (third row) variation at three different distances, 0.1AU, 1AU, and at the orbit of Ulysses at 1.4 AU, i.e.","$\\approx 300\\mathrm {R}_\\odot $ , corresponding to left, middle and right columns, respectively.","Most of the acceleration has already taken place at 1AU and only a small increase is happening above that distance due to the heating implemented by the reduced polytropic index ($\\gamma =1.5$ ) as the wind flows away from the Sun.","The density decreases by 2 orders of magnitude from the inner boundary of the MHD simulation to the Earth's orbit and only by 50% from there onwards up to 1.4AU.","The current sheet, seen as the high density structure that appears in the middle of the Figures in the second row seems to have expanded from $2^\\circ $ at the inner boundary to about $10^\\circ $ from there on, while it diffuses.","The temperature decreases by a factor 8 from the inner boundary up to 1AU and only by 30% up to Ulysses' orbit.","There is a temperature reversal captured, in the sense that while at 0.1AU the equatorial region appears cold and the poles hot, the opposite is happening from 1AU onwards, with the temperature shows a peak at a longitude of $-100^\\circ $ .","The current sheet region gets very diffused outwards and from the Earth's orbit outwards appears discontinuous in this temperature view of the expanding plasma."],["Interfacing","In order to interface the two models at 0.1AU, we run the kinetic model up to $21.5\\mathrm {R}_\\odot $ with $r_0=r_s=2.5\\mathrm {R}_\\odot $ , $T_e=1\\mathrm {MK}$ , $T_p=1\\mathrm {MK}$ , using 600 $\\kappa $ -indexes in the range [2,8] with step 0.01 and with $n_e=n_p=3\\times 10^{10}\\mathrm {m^{-3}}$ [8].","With the results we create a matrix with solar wind speeds at $21.5\\mathrm {R}_\\odot $ and by comparison with the EUHFORIA results for $v_r$ , we estimate the appropriate $\\kappa $ for every speed at the internal boundary of the MHD run.","We obtain a 2D map of $N_\\theta \\times N_\\phi $ values of $\\kappa (\\theta ,\\phi )$ each corresponding to a field line.","From the matrix, we also compare the number density given by EUHFORIA at the same distance (0.1AU) with the number density given by the kinetic model and get a scaling factor that we assume to be valid throughout all the considered radial distances [8].","Thereby, we can estimate the appropriate initial density at the exobase that would give us the same density with EUHFORIA at 0.1AU.","For the temperatures, the relation is more complicated, but [8] have demonstrated that the temperature does not affect the kinetic moments as much as the $\\kappa $ -indexes and the exobase altitude even for extreme changes (1 – 2MK) and thus for convenience we take $T_e=T_p=1$ MK for all the latitudes and longitudes at $r_0=2.5\\mathrm {R}_\\odot $ and focus instead on the $\\kappa $ and the density parameters.","Note that EUHFORIA is not solving the MHD equations below 0.1AU, while the kinetic model provides results at any distance above the exobase and especially in the crucial region close to the Sun where the solar wind is being accelerated."],["Kinetic Model","Similar to Figure REF , in Figure REF , we show the velocity (first row), the density (second row), the electron temperature (third row) and the proton temperature (fourth row) at three different distances, 0.1AU, 1AU, and at the orbit of Ulysses at 1.4AU, i.e.","$\\approx 300\\mathrm {R}_\\odot $ , corresponding to the left, middle and right columns, respectively.","Unlike the MHD results discussed previously, in the kinetic approach the sharp structures that appear at the interfacing boundary (0.1AU) remain unchanged as the plasma moves outwards, as we don't account for stream interactions.","In other words, neighboring streams with different speeds will not interact as they propagate and they will keep the same topological features up to large radial distances.","The $\\kappa $ indexes corresponding to the kinetic velocities of this simulation lie in the range $(2,4)$ .","The bulk speed accelerates more than in the MHD case, reaching terminal speeds about 100 $\\mathrm {km\\ s^{-1}}$ higher.","Thus in the kinetic approach, where the acceleration is self-consistent and due to the induced electric field, the acceleration is more efficient than in the MHD approach, where semi-empirical schemes are used to accelerate the wind.","Similarly to the MHD case, the terminal speed is reached at 1AU and from there on the acceleration is very slow up to 1.4AU.","The density decreases faster by 20% at 1AU, to reach the orbit of Ulysses 30% smaller than the MHD case.","The temperatures of the electron and proton populations are depicted in the last two rows.","We have started at the exobase by setting both temperatures equal to 1MK independent of longitude and latitude.","The electron temperature drops by a factor 5 at the orbit of the Earth and it doesn't change much up to 1.4AU, while the proton temperature decreases by a factor 2 and remains about the same up to the orbit of Ulysses.","The temperature of the MHD plasma is always in between the electron and proton temperatures, being one order of magnitude smaller than the electron and one order of magnitude larger than the proton temperature.","The MHD temperature is not the average between the two particle species temperatures, since the two species are not in thermodynamic equilibrium.","[htbp] Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONKinetic results in the form of color maps in heliographic longitude and latitude from top to bottom rows: of the bulk speed, the number density, the electron and proton temperatures of the solar wind respectively at 0.1AU (left panels), at 1AU (middle panels) and at Ulysses' orbit 1.4AU (right panels).","These can be contrasted directly with the MHD counterparts in Figure REF ."],["Models ","In Figure REF , the speed, number density and temperature of the solar wind are shown for i) the MHD model, ii) the kinetic model and iii) near-Earth observations using the OMNI dataset.","From the speed plot we conclude that both models reproduce the number of peaks, their position with respect to each other and have a similar width.","The most prominent difference between the models and the observations appears at the double peak of August 13th.","EUHFORIA reproduces peaks of similar amplitude as observed, but showing a higher global minimum at about 400 $\\mathrm {km\\ s^{-1}}$ .","The heights of the peaks and their relative ratio to one another are not reproduced exactly by any model.","The kinetic model systematically overestimates the speeds varying from 400 to 800 $\\mathrm {km\\ s^{-1}}$ .","This can be explained by the fact that the acceleration in that model is more efficient than the MHD case and continues at a significant rate at distances larger than 0.1AU to about 50$\\mathrm {R}_\\odot $ .","This indicates that the final velocity obtained with the kinetic model can be improved by adapting the boundary conditions, for instance by lowering the empirical solar wind speed close to the Sun.","The second panel shows the number density variations for the models and the observations.","We observe that both models and the OMNI observations of the number density agree roughly in order of magnitude and in number of peaks, with the peaks not coinciding perfectly.","The peak amplitudes are about 2.5 times larger in the observations than in both models, that are in agreement with one another.","For the Alfvénic Mach number (third panel), we conclude that the Alfvénic Mach number of the MHD simulation is about a factor 3 higher than the average measured value most of the time, with three lows that reach lower than the observed values.","The average observed proton temperature agrees with the EUHFORIA plasma temperature in order of magnitude, varying from the kinetic proton temperature $T_p$ at its minimum to the kinetic electron temperature $T_e$ at its maximum, while it is located between the two species' temperatures, about an order of magnitude larger than the kinetic proton temperature and an order of magnitude smaller than the kinetic electron temperature, at all times.","The variation profile of the observations doesn't match with any model.","Finally, the plasma $\\beta $ shown in the final panel for the MHD simulation lies most of the time at the lower limit of the observed one, with three peaks that reach values of about a factor 3 higher than the observed highest values, showing the opposite trend when compared to the Alfvénic Mach number shown in the third panel.","Thus we conclude that, in the MHD simulation the magnetic field is higher with respect to the plasma pressure and the number density than the one indicated by observations at 1AU.","The same case corresponding to CR 2059 was analyzed in a comparative study published in [4] for different observational inputs and MHD modeling schemes.","None of their models seemed to accurately represent observation.","The case directly comparable to ours was the one using GONG magnetograms and the WSA empirical model and ENLIL for the MHD modeling (dark red curve of lower panel in Figure 3).","Before 27 July their maxima are not synchronous nor do they reach the same values, but after that the model captures the times of the peaks, but underestimates their value on the panel in the figure showing the speed.","Figure: Time variations as measured by Ulysses (black) and calculated by the kinetic (blue) and the MHD (red) model, for i) top left panel: the position of Ulysses in heliographic coordinates, ii) top right panel: the speed, iii) bottom left panel: the number density and iv) bottom right panel: the Ulysses small (black) and large (green) proton, the kinetic proton (magenta), the kinetic electron (blue) and MHD (red) temperature.For a further comparison of the models with respect to observations, we overplot in Figure REF observations from Ulysses at 1.4AU and at latitudes between $-10^\\circ $ and $5^\\circ $ together with the results of the kinetic and MHD models.","In the first panel (top left), we show the trajectory of Ulysses during the period of interest.","At the same 3D heliographic spherical coordinates we extract the quantity of interest from the MHD and the kinetic model to accommodate further comparisons.","More specifically, for each $(r,\\theta ,\\phi )$ position of Ulysses at each hour of a specific date, we choose the closest available point for the specific resolution of each model.","In the second panel, we show the observed speed by Ulysses with black, together with the kinetic prediction in blue and the MHD results in red.","All three exhibit different time profiles for each case.","The kinetic model systematically overestimates the speeds reaching maximum values of about 800 $\\mathrm {km\\ s}^{-1}$ , whereas the MHD model ranges from 400 to 650 $\\mathrm {km\\ s}^{-1}$ , while at the same period the Ulysses measurements lie between 300 and 650 $\\mathrm {km\\ s}^{-1}$ .","The peaks for both models are not well synchronized with the measured temporal velocity profiles.","For the density the two models don't reproduce the observed number of peaks and they are not synchronized either.","The number densities of the kinetic and MHD models agree with each other varying from 1 – 5 $\\mathrm {cm}^{-3}$ whereas the observed number density profile is much more variable with a number of peaks and reaching densities of 20 $\\mathrm {cm}^{-3}$ .","The average observed proton temperatureAs shown in Figure REF , there are two different proton temperatures estimated, that in general bracket the real temperature at 1AU.","As $T$ -large we denote the integral in the 3D velocity space of the distribution over all measured angles and energy bandwidths.","The $T$ -small is calculated by the sum over all angles for a determined energy, then summing the moments of the estimated spectrum of the plasma and by taking the radial component of the temperature tensor (http://www.cosmos.esa.int/web/ulysses/swoops-ions-user-notes).","agrees with the plasma temperature predicted by the MHD model and lies in between the kinetic proton (magenta) and kinetic electron (blue) temperatures of the kinetic model, being one order of magnitude smaller than the electron and one order of magnitude larger than the proton temperatures.","No model reproduces in high accuracy the temporal variations of the observed temperature profile.","The models show some correspondence with the Ulysses observations and the correct orders of magnitude are roughly reproduced, but certainly there is need for improvement.","To reach better agreement with observations, we can modify the parameters in the semi-empirical model of EUHFORIA to better adjust and reproduce the measurements in each case individually."],["Heat Flux","[27] presented an analysis of the different ways that energy source considerations can be used in MHD solar wind models.","A generalized formulation of the energy Equation with an extra energy source term at the right hand side $S$ was examined for different cases of $S$ .","The relevant models studied therein include i) a model with a polytropic index with spatial dependance $\\Gamma (r)$ and ii) a model with a polytropic index fixed at $\\gamma =5/3$ constant in the entire coronal volume.","In particular, the authors showed that a steady-state solar wind solution accelerated by a given non-adiabatic polytropic wind can equivalently be re-written using an energy source term, given by: $S=\\nabla \\cdot \\Bigg [{v}P\\bigg ( \\frac{1}{\\Gamma -1}-\\frac{1}{\\gamma -1}\\bigg ) \\Bigg ] {,}$ with ${v}$ , $P$ and $\\Gamma $ being obtained by the model, as explained in [27], with the non-adiabatic index $\\Gamma (r)=1.5$ for our case.","Figure: Heat flux along magnetic field lines in the equatorial plane as given by EUHFORIA.In this section, we will discuss the heat flux of the MHD and the kinetic models.","According to the formulation presented above we have that the analytic expression of the energy flux responsible for accelerating the wind in the MHD model is given by ${v}P\\bigg (\\frac{1}{\\Gamma -1}-\\frac{1}{\\gamma -1}\\bigg )$ .","In Figure REF , we present the heat flux of the MHD model along selected magnetic field lines as a function of distance from 0.1AU to 2AU in the equatorial plane.","Figure: Electron (top panel) and proton (bottom panel) heat fluxes for CR2059 with red, orange, light green, green and blue curves corresponding to κ\\kappa values of 3, 4, 5, 6 and 7, respectively.In Figure REF , we illustrate the radial profiles of the electron (top panel) and proton (bottom panel) heat fluxes, as calculated by the kinetic model for an exobase at 2.5$\\mathrm {R}_\\odot $ , 1MK electron and proton temperatures at the exobase $T_e=T_p=1\\mathrm {MK}$ and the same densities as our default case for $\\kappa $ -indexes 3, 4, 5, 6 and 7 corresponding to red, orange, light green, green and blue respectively.","We observe that the more suprathermal particles are present, i.e.","lower $\\kappa $ , the higher the heat flux is.","The electron heat flux is higher by an order of magnitude than the proton heat flux close to the Sun and it decreases faster than the proton heat flux with the heliocentric distance to reach similar values at 1AU.","We conclude that the differences between the proton heat flux curves corresponding to different $\\kappa $ -indexes are smaller than the respective electron flux curves.","In general, the heat fluxes are overestimated by exospheric models, since the corresponding VDFs have the highest possible anisotropy due to lack of collisions.","As shown in [35], exospheric models and kinetic simulations that include collisions are in good agreement, making the exospheric model a convenient tool for the study of weakly collisional plasmas.","Both the exospheric model and the collisional simulations found heat fluxes several times larger than the classical value, suggesting that the classical formulation is not appropriate for weakly collisional plasmas, since it was based on the assumption of a collision-dominated medium.","If one really needs to prescribe a realistic heat flux, then collisions need to be taken into account and further improvements in the kinetic model need to be considered.","The inclusion of interactions, through e.g.","Alfvén or whistler waves, will decrease the anisotropies and it will improve the higher moments, i.e.","temperatures and heat fluxes for the considered species [22], [24], [31].","Using such sophisticated schemes to improve the heat flux agreement with observations is outside of the scope of this paper, due to i) the consequent increased computational expense, ii) the fact that the temperatures are not the most important geo-effective parameters, making these improvements rather impractical for future operational space weather applications.","Thus, the heat flux profiles for the electron and protons, that are quantified by the kinetic model (Figure REF ) can serve as upper limits and are more physics-based than the MHD heating prescriptions that are based on the empirical determination of the polytropic index value.","The heat flux profiles along the magnetic field lines of the MHD model are in agreement in order of magnitude with the profiles of the protons of the kinetic model closer to the Sun, but they drop faster further out reaching electron heat flux values at the orbit of the Earth.","Our results for the heat flux for the MHD model depicted in Figure REF are roughly in agreement with the literature [5], [30].","In accordance to the kinetic model results, the electron heat flux is higher than the proton heat flux according to [5] and [30], respectively, with our results roughly being closer to the electron energetics profile, but at 1AU approaching the heat flux values expected for the protons.","In this study, we are interested in making a first comparison between single fluid and kinetic models, that have the potential to be used in space weather applications.","The two models are very different in nature, making use of very different formulations for the plasma physics.","EUHFORIA is a single fluid code that uses the MHD equations to describe the plasma.","Whereas the kinetic model used in this study begins by making an observationally-inspired assumption for the VDFs of the electron and proton species, that are considered to constitute the solar wind plasma, and based on that all the physically interesting quantities are calculated as moments over the velocity space.","EUHFORIA is observationally-driven and provides three-dimensional information that accounts for stream interactions and complicated magnetic topologies.","The kinetic model is a semi-analytic model solving for the plasma characteristics along a magnetic field line and accounting for heating and acceleration in a self consistent way.","Note that the electric field (used in the kinetic model) is hidden in the pressure term .","Parker explained that the momentum equation of the electrons is given by $\\frac{\\mathrm {d}p_e}{\\mathrm {d}r}+neE_\\mathrm {LS}=0 {,}$ which is the hydrodynamic condition, with $E_\\mathrm {LS}$ the Lemaire-Scherer electric field [10].","In [18], Parker showed that the electric field of Lemaire-Scherer is in the fluid approach given by $E_\\mathrm {LS}=-\\frac{1}{ne}\\frac{\\mathrm {d}p_e}{\\mathrm {d}r}=\\frac{m_p}{e}\\left( v\\frac{\\mathrm {d}v}{\\mathrm {d}r}+\\frac{GM_\\odot }{r^2}\\right) {,}$ which finally, taking into account that the Pannekoek-Rosseland electric field is $E_\\mathrm {PR}=\\frac{m_pg}{e}$ , is written as $E_\\mathrm {LS}=E_\\mathrm {PR}+\\frac{m_pv}{e} \\frac{\\mathrm {d}v}{\\mathrm {d}r} {.","}$ The Lemaire-Scherer electric field (used in exospheric models) is several times larger than the Pannekoek-Rosseland one, corresponding to hydrodynamic equilibrium that is used to describe the solar wind expansion.","It is able to lift and accelerate to supersonic speeds the initially slow and heavy protons through trapping the fast and light electrons, while keeping the quasi-neutrality and almost zero-current condition.","Any difference in the bulk speeds of the two populations would lead to the generation of a current, which due to Ampère's law has to remain small.","From Figure REF , we deduce that at large radial distances from the Sun, the MHD current sheet gets expanded and becomes thicker, evolving from about $2^\\circ $ at the inner boundary to about $10^\\circ $ at 1AU, while the fine structures that appear in the 0.1AU longitudinal and latitudinal map get diffused and smoothed.","On the contrary, in Figure REF we see that the fine structures and the current sheet size don't change under the kinetic approach, as there are no stream interactions taken into account.","Both models give speeds of the same order of magnitude at every altitude and have most of their acceleration taking place already before 1AU.","The kinetic model shows a more efficient acceleration, reaching terminal speeds of about 100 $\\mathrm {km\\ s^{-1}}$ larger than the MHD one up to 1AU.","The acceleration in the MHD approach is taking place due to the reduced polytropic index, while in the kinetic approach the acceleration is related to the induced electric field that assures quasi-neutrality and equal outward electron and proton fluxes.","Regarding the number density, both models give densities of the same order of magnitude, with the kinetic model though showing a sharper profile that decreases faster outwards.","The number density of the kinetic model is 20% and 30% lower at 1AU and 1.4 AU respectively than the MHD number density.","Moreover, for the temperature in the MHD approach, as is shown in the bottom row of Figure REF , there is a reversal of the hot-cold regions from the inner boundary up to large distances.","A faster cooling takes place at higher latitudes, making the initially colder equatorial region appear hotter at large radial distances with respect to the rest of the latitudes.","Furthermore, the temperature ranges fall by a factor 4 from 0.1AU up to 1AU, and continue decreasing by 30% up to the orbit of Ulysses.","For the kinetic approach, the proton temperature only falls by factor two up to the Earth's orbit and seems constant up to 1.4 AU, while the electron temperature drops by a factor 5 up to 1AU and doesn't seem to vary much from there on.","Contrary to the MHD single fluid results, in the kinetic case there is no temperature profile change and the cooling seems to take place uniformly at all latitudes.","There is a structure close to the equator at a heliographic longitude of $-70^\\circ $ that appears like a spike in contact with the current sheet in both Figures REF and REF .","In Figure REF though, in the middle and right panels, corresponding to large distances, that spike appears to change inclination from pointing to the left of the figure at the panels corresponding to 0.1AU to pointing to the right from 1AU outwards, unlike Figure REF , where the spike's inclination remains unchanged.","This difference is then likely due to stream interactions that are captured in the 3D MHD model, while these are excluded in the essentially 1D, radial field kinetic approach."],["Conclusions","After the parallelization of the kinetic model and its use in a quasi-3D approach we linked it to more robust 3D MHD models and compared both to observations at 1AU, using similar boundary conditions at $21.5\\mathrm {R}_\\odot $ .","When the two models were compared, starting from the same boundary conditions, the kinetic model gives systematically higher speeds than the MHD model at large radial distances.","This is due to the fact that the acceleration of the solar wind continues at a higher rate in the kinetic model after $21.5\\mathrm {R}_\\odot $ .","The acceleration mechanism in the kinetic model is due to the induced electric field that assures quasi-neutrality and prevents charge separation and also \"bounds\" the two species to move with the same bulk speed.","There is no explicit heating term in the MHD equations used by EUHFORIA.","The MHD model accelerates further the solar wind due to the reduced polytropic index $\\gamma $ , but at a very slow rate accounting for an acceleration of about 50 $\\mathrm {km\\ s}^{-1}$ from 0.1AU to 1AU.","The exospheric models overestimate the heat flux, especially for the electrons that have a thermal speed comparable to their bulk speed, but such a heat flux can be improved by inclusion of interactions through waves, e.g.","Alfvén or whistler waves [22], [24], [31].","The heat flux calculated by the kinetic exospheric model can be used as an upper limit for more physically-driven heat flux prescriptions in a global MHD model.","The heat flux of the kinetic model is in qualitative agreement with other studies [5], [30] and the heat flux profile of the MHD models close to the Sun resembles the proton profile, only to decrease faster outwards resembling the electron profile at 1AU.","In the exospheric model the fast electrons are slowed down and the protons are accelerated by the Lemaire-Scherer electric field that eventually leads to the observed supersonic solar wind.","As shown by [18] the acceleration of the solar wind in collisionless plasmas to supersonic values lies in the hydrodynamic equation in combination to the mass ratio between electrons and protons.","According to Parker, the exospheric model describes a very efficient heat transport mechanism with an electron temperature that decreases very slowly at large distances and through the induced electric field it elevates the protons and causes the transonic solar wind.","There is some agreement in the high- and slow-speed streams in the velocity profiles for both OMNI and Ulysses observations.","The high speed positions of the models are better synchronized for OMNI rather than Ulysses observations, as we showed earlier.","The number densities of both models were approximately in the same order of magnitude with the observed ones at 1AU and at the orbit of Ulysses.","The MHD temperature is one order of magnitude smaller than the electron temperature $T_e$ and one order of magnitude larger than the proton temperature $T_p$ of the kinetic model.","The average observed proton temperatures agreed with the temperature predicted by the MHD model in order of magnitude and thus were also lying in the range between the electron and proton kinetic temperatures.","In this study we assumed that the $\\kappa $ -value is independent of the radial distance, but observations such as [12] indicate that the $\\kappa $ can actually change as the distance from the Sun increases.","However, the fitting model used in [12] was not with a Kappa distribution for the full range, but a sum of a Maxwellian for the core and a Kappa function for the halo, so that the parameter $\\kappa $ does not represent the same quantity as in the model used in the current study.","The density ratio between the core and the halo remains constant with the distance, as analyzed by [25].","A more realistic exobase profile with temperatures having a latitudinal and even longitudinal variation can be taken into account and will be the next step towards a fully 3D kinetic numerical code of the solar corona and the solar wind.","The kinetic model is a semi-analytic model ignoring stream interactions and thus conserves the slow and fast wind distributions for every radial distance.","Accounting for stream interactions as the solar wind propagates outwards will be one of the main points of interest for a more realistic 3D model.","Shocks associated to sharp velocity gradients can be included in an empirical way or by using more sophisticated kinetic models including collisions and wave-particle interactions [19].","Here we have ignored the spiral shape of the magnetic field in the calculation of the moments in the kinetic model, adopting purely radial magnetic fields, because as argued in [23] this aspect won't affect the main average quantities apart from the temperature anisotropies.","But in the future we are planing on including the spiral magnetic field effects in a similar study.","[23] quantified the effect that the spiral magnetic field topology has on the particle temperatures and their anisotropies.","More specifically, in the radial case the electron temperature is slightly underestimated, whereas the opposite is true for the proton species.","The temperature anisotropies are overestimated by the kinetic model in comparison to observations [9].","Another important aspect that can be improved and would provide a deeper comparison between operational MHD codes and kinetic exospheric ones, would be to change the formulation of the kinetic model, so that boundary conditions for the speed, the density and temperature directly from MHD models can be used at each grid point including the magnetic field information.","These aspects will allow us to reach more fundamental conclusions about the two different models and will upgrade the kinetic exospheric model into a computational equivalent to the robust 3D MHD code.","When the 3D magnetic field topology from the source surface, through the Schatten current sheet region, and throughout the region covered by the MHD model, is used directly within the kinetic description, we can use its predicted heat fluxes and higher order moment information to turn the model into a self-consistent hybrid kinetic-MHD modeling tool.","SPM acknowledges financial support by the FWO and NASA Living with a Star grant number NNX16AC11G.","This research was supported by projects GOA/2015 – 014 (KU Leuven, 2014 – 2018), and the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7/08 CHARM)."],["Conflict of Interest","The authors declare that they have no conflict of interest."]],"string":"[\n [\n \"Interfacing MHD Single Fluid and Kinetic Exospheric Solar Wind Models\\n and Comparing Their Energetics\"\n ],\n [\n \"Abstract An exospheric kinetic solar wind model is interfaced with an observation-driven single fluid magnetohydrodynamic (MHD) model.\",\n \"Initially, a photospheric magnetogram serves as observational input in the fluid approach to extrapolate the heliospheric magnetic field.\",\n \"Then semi-empirical coronal models are used for estimating the plasma characteristics up to a heliocentric distance of 0.1AU.\",\n \"From there on a full MHD model which computes the three-dimensional time-dependent evolution of the solar wind macroscopic variables up to the orbit of the Earth is used.\",\n \"After interfacing the density and velocity at the inner MHD boundary, we compare with the results of a kinetic exospheric solar wind model based on the assumption of Maxwell and Kappa velocity distribution functions for protons and electrons respectively, as well as with \\\\textit{in situ} observations at 1AU.\",\n \"This provides insight on more physically detailed processes, such as coronal heating and solar wind acceleration, that naturally arise by inclusion of suprathermal electrons in the model.\",\n \"We are interested in the profile of the solar wind speed and density at 1AU, in characterizing the slow and fast source regions of the wind and in comparing MHD with exospheric models in similar conditions.\",\n \"We calculate the energetics of both models from low to high heliocentric distances.\"\n ],\n [\n \"Introduction\",\n \"Solar wind heating and acceleration mechanisms are still subjects of active research.\",\n \"In computational models, physical quantities estimated or observationally inferred close to the Sun serve as boundary or initial conditions that will examine the solar wind evolution and its underlying physics as it propagates through interplanetary space.\",\n \"The solar wind plasma can be studied macroscopically through the magnetohydrodynamic (MHD) approach or microscopically when using the kinetic approach.\",\n \"In the following Subsections REF and REF , we briefly review the main aspects of both approaches used in this paper, which are here for the first time interfaced in a global model.\",\n \"[32] presented an empirical relation between the expansion of magnetic flux tubes and the solar wind speed, showing that they evolve inversely proportional to each other.\",\n \"This assumption was tested using more than two decades of observations, and can give predictions of the solar wind speed at Earth.\",\n \"The model involves synoptic magnetograms of photospheric field, which allow a Potential Field Source Surface (PFSS, with the source surface typically at 2.5$\\\\mathrm {R}_\\\\odot $ ) extrapolation which quantifies the expansion factors.\",\n \"It was found that at the Earth's orbit, greater expansion corresponded to magnetic field lines near the centre of coronal holes, which diverge more slowly than the ones coming from the hole boundaries.\",\n \"This is consistent with the fact that lower densities are found in the fast wind regions.\",\n \"[1] improved the Wang-Sheeley model by using daily updated magnetogram data from the Wilcox Solar Observatory (WSO) and relating the magnetic flux tube expansion factor with the solar wind speed at the source surface, while including effects of stream interactions from the source surface to the Earth.\",\n \"A statistical study which covered three years and compared the Wang-Sheeley model predictions with data from the WindNASA spacecraft at the L$_1$ Lagrangian point of the Earth designed for long-term solar wind measurements and its effects on the terrestrial magnetosphere (https://wind.nasa.gov).\",\n \"satellite was presented.\",\n \"The interplanetary magnetic field (IMF) polarity was properly predicted 75% of the time, while solar wind speeds were within 10-15% of actual values, when a 6-month period with data gaps was removed.\",\n \"In the computational work of [14], solar wind variations were examined in the corotating frame with a three-dimensional MHD model with a CME (Coronal Mass Ejection) injection scheme in the streamer belt.\",\n \"Such MHD models take into account magnetic field variations due to the CME interaction with the solar wind during the CME's evolution.\",\n \"The CME movement depends on the background solar wind density and velocity and the vector properties of the solar wind magnetic field and velocity are affected by the passing disturbance.\",\n \"ENLIL [16], [17], [15] is a heliospheric MHD model that provides a three-dimensional description of the time-dependent solar wind evolution.\",\n \"It can use the Wang-Sheeley-Arge (WSA) [32], [1] semi-empirical model as its boundary condition.\",\n \"The WSA model was used by [13] to study the solar wind at low heliocentric distances including an empirical method to link the magnetic field information with the velocity at 21.5$\\\\mathrm {R}_\\\\odot $ .\",\n \"The new method was cross-validated using the 3D MHD code ENLIL and by comparing the results with observations at 1AU and at further distances as provided by Ulysses.\",\n \"The estimation of the solar wind speed at 21.5$\\\\mathrm {R}_\\\\odot $ was indeed better than previous models and it captured both fast and slow solar wind.\",\n \"Similar to ENLIL, we use a fully 3D MHD code EUHFORIA (European heliospheric forecasting information asset) that from 0.1AU onwards models the evolution of the plasma environment in the inner heliosphere.\",\n \"The code details are discussed in [26] and in this paper we adopt it to get the macroscopic description of the solar wind.\"\n ],\n [\n \"Kinetic Exospheric Models\",\n \"Exospheric kinetic models are simplified collisionless, stationary models, that are meant to explain the acceleration of the solar wind in a self-consistent way and they were first established by [7] and [10].\",\n \"The model was one-dimensional and time-independent and provided the state of the solar wind plasma along a magnetic field line.\",\n \"The acceleration of the solar wind was due to the induced electric field even without suprathermal electrons (Maxwellian distribution), but when accounting for the presence of suprathermal electrons, the terminal speed at 1AU increased.\",\n \"The original exospheric model [7], [10] assumed Maxwellian velocity distribution functions (VDFs) for protons and electrons and supersonic winds of 300 $\\\\mathrm {km\\\\ s^{-1}}$ could be reached at 1AU with temperatures of the order of 1 MK for both species at the exobase (the distance beyond which collisions become negligible).\",\n \"Nevertheless, it remained difficult for the model to achieve higher bulk velocities, such as the ones observed in the fast solar wind, without increasing the temperature to unrealistic high values (10MK) at the exobase, or by adding other sources of solar wind acceleration.\",\n \"After the induced electric field is calculated, the solar wind acceleration spontaneously follows giving the solution of the solar wind from sub- to supersonic, without any extra energy terms assumed.\",\n \"A Lorentzian (Kappa) velocity distribution function is used instead of the classic Maxwellian in search for better agreement with observations in [20].\",\n \"Indeed, suprathermal electrons are generally observed in the velocity distribution functions measured in situ in the solar wind.\",\n \"[20] have shown that the presence of such suprathermal electrons accelerates the wind to higher bulk velocities, so that no other source of energy needs to be considered to reach the values observed in the high speed solar wind.\",\n \"[11] applied the kinetic model developed by [20] using Kappa VDFs for both electron and proton populations that escape from the Sun to describe the solar wind.\",\n \"Since the first exospheric model, the semi-analytic kinetic model has been able to describe not only the fast but also the slow solar wind together with their sources, in the cold coronal hole and hot equatorial regions respectively, without unrealistic assumptions of too high temperatures and extra heating in the corona, as required by non-turbulence driven fluid models.\",\n \"While previous exospheric models placed the exobase at a distance of about 5-10$\\\\mathrm {R}_\\\\odot $ , from where on the proton total potential energy was a monotonic function of the heliocentric distance, [8] calculated the exobase to be positioned at about 1.1-5$\\\\mathrm {R}_\\\\odot $ .\",\n \"This deeper location of the exobase, lowered under the radial location of the maximum of the total potential energy of the protons, gives the solar wind the observed acceleration to high velocities.\",\n \"A low exobase leads indeed to higher bulk velocities at 1AU in the case of suprathermal electrons.\",\n \"Collisionless (exospheric) theoretical models and collisional simulations were compared in [35].\",\n \"Including suprathermal tails in the velocity distribution function of the electrons and employing a self-consistently computed heat flux, the models were able to reproduce fast solar wind speeds.\",\n \"Results of collisional kinetic simulations with non-Maxwellian velocity distribution functions and collisionless exospheric models are in good agreement.\",\n \"Taking into account that the exospheric and collisional models provide comparable results, in this paper we will go a step further and try to interface exospheric models with MHD ones.\",\n \"On the way to developing predictive tools and 3D solar wind models, a 2D observationally driven kinetic exospheric solar wind model was developed by [21], presenting solar wind variations on the ecliptic plane and how they compare to observations from close to the Sun up to 1AU.\",\n \"For the ecliptic variational study OMNIMulti-source data set for the near Earth solar wind of combined and normalized observational data from ACE (Advanced Composition Explorer), Wind, IMP 8 (Interplanetary Monitoring Platform) and GOES (Geostationary Operational Environmental Satellite) satellite missions.\",\n \"observations were used for the time period 26 September to 23 October 2008.\",\n \"The $\\\\kappa $ parameter was chosen as 2.35 and 3.82 for fast and slow wind respectively, to match bulk speed observations close to the orbit of the Earth.\",\n \"We will present a three dimensional generalization of this exospheric model, i.e.\",\n \"we find the solar wind characteristics along a collection of magnetic field lines each passing through a point on the spherical shell at the exobase level in latitude and longitude $(\\\\theta ,\\\\phi )$ .\",\n \"The basic principles, boundary conditions, physical assumptions and computational methods used by the MHD and the exospheric kinetic models are described in Section 2.\",\n \"The specific criteria and the observational data that are chosen for this work are explained and presented in Section 3.\",\n \"The interfacing method as well as explicit results of both approaches and their energetics are discussed in Section 4, while in Section 5 we compare the two approaches and we close by discussing the main conclusions of the study in Section 6.\"\n ],\n [\n \"MHD Modeling: EUHFORIA\",\n \"The inner heliosphere model EUHFORIA [26] is used for our MHD approach.\",\n \"EUHFORIA is a three-dimensional observationally driven model providing an accurate description of the large-scale time-dependent solar wind including transient events such as CMEs.\",\n \"As such, it allows to inject CMEs at the inner radial boundary at 0.1AU as a time-dependent boundary condition.\",\n \"Apart from CMEs, the variables at the inner radial boundary are constructed in order to capture the large-scale variations in the solar wind for the particular time period under study.\",\n \"This is accomplished using a model for the coronal magnetic field and employing empirical relations between the coronal magnetic topology and the state of the solar wind.\",\n \"The magnetic field model consists of a potential field source surface (PFSS) model in the low corona coupled with a current sheet model higher in the corona.\",\n \"The PFSS model requires a magnetogram to be provided as input.\",\n \"To finally compute the super-sonic state of the solar wind at 0.1AU, empirical relations inspired by the success of the WSA (Wang-Sheeley-Arge) model [32], [1] are used.\",\n \"The MHD model is able to provide density and speed profiles at the Earth's orbit, it allows for slow and fast solar wind source region tracing, and can serve as the MHD counterpart in a comparison project together with kinetic exospheric models that correspond to similar initial and boundary conditions at 0.1AU.\",\n \"This will be our first goal in this paper, and the way the two approaches are coupled will be described next.\",\n \"EUHFORIA uses a finite volume discretization scheme to solve the hyperbolic conservative MHD equations.\",\n \"The equations solved are those of ideal MHD with gravity included as a source term in the equations of momentum and energy: $\\\\frac{\\\\partial \\\\rho }{\\\\partial t}+\\\\nabla \\\\cdot (\\\\rho {v})=0{,} \\\\\\\\\\\\frac{\\\\partial (\\\\rho {v})}{\\\\partial t}+\\\\nabla \\\\cdot \\\\left[ \\\\rho {v}{v}+\\\\left( p+\\\\frac{B^2}{2\\\\mu _0}\\\\right)\\\\mathcal {I}-\\\\frac{1}{\\\\mu _0}{B}{B}\\\\right]=\\\\rho {g}{,} \\\\\\\\\\\\frac{\\\\partial \\\\mathcal {E}}{\\\\partial t}+\\\\nabla \\\\cdot \\\\left[\\\\left(\\\\mathcal {E}+p-\\\\frac{B^2}{2\\\\mu _0}\\\\right){v}+\\\\frac{1}{\\\\mu _0}{B}\\\\times \\\\left( {v}\\\\times {B}\\\\right)\\\\right]=\\\\rho {v}\\\\cdot {g}{,} \\\\\\\\\\\\frac{\\\\partial {B}}{\\\\partial t}-\\\\nabla \\\\times \\\\left( {v}\\\\times {B}\\\\right)=0{,} \\\\\\\\\\\\nabla \\\\cdot {B}=0{,} \\\\\\\\\\\\mathcal {E}= \\\\frac{p}{\\\\gamma -1}+\\\\frac{\\\\rho v^2}{2}+\\\\frac{B^2}{2\\\\mu _0} {,} \\\\qquad \\\\gamma =1.5$ where $\\\\rho $ is the mass density, ${v}$ the velocity vector, ${g}$ the gravitational acceleration, ${B}$ the magnetic field vector, $p$ the thermal pressure, $\\\\gamma $ the polytropic index, $\\\\mathcal {E}$ the total energy density, $\\\\mu _0$ the magnetic permeability and $\\\\mathcal {I}$ the unit tensor.\",\n \"Note that we are working in the inertial frame, so no Coriolis nor centrifugal forces need to be added in Equation .\",\n \"The polytropic index is chosen to be slightly smaller than the expected $\\\\gamma =5/3$ value for a monatomic gas.\",\n \"This causes a finite energy to be injected into the system in the form of heat [27].\",\n \"The use of either a non-adiabatic polytropic index or explicit source terms in the momentum and energy equation to drive the solar wind and heat the corona have been used in several works.\",\n \"In EUHFORIA, the reduced polytropic index is used in order to slightly accelerate the solar wind further out in the heliosphere, from the speed values at the boundary at 0.1AU.\",\n \"The single fluid MHD description still leaves freedom to vary $\\\\gamma $ , which allows to account for expected deviations from the mono-atomic ideal gas value of 5/3.\",\n \"For a discussion of more self-consistent models that attempt to capture and explain the physical mechanisms resulting in the observed coronal heating and acceleration, we refer to [2].\",\n \"The employed numerical grid is uniform in spherical coordinates, with the number of cells in $r$ , $\\\\theta $ , $\\\\phi $ chosen to be 800, 60, 180, respectively.\",\n \"The outer boundary is set at 2AU.\",\n \"Further details of the numerical solution scheme are described in [26].\"\n ],\n [\n \"Boundary Conditions\",\n \"The essential input to EUHFORIA is a synoptic magnetogram.\",\n \"We select a magnetogram from the GONG (Global Oscillation Network Group) standard synoptic data product, which are available with one hour cadence from GONG.\",\n \"The chosen magnetogram corresponds closely to Carrington Rotation (CR) 2059.\",\n \"During this Carrington rotation, an equatorial coronal hole was visible near the central meridian to about $60^\\\\circ $ degrees west.\",\n \"The solar wind plasma state at 0.1AU in EUHFORIA is determined using a semi-empirical approach similar to the Wang-Sheeley-Arge model.\",\n \"The method consists of constructing a model of the coronal magnetic field consisting of a PFSS extrapolation in the low corona while the \\\"Schatten\\\" Current Sheet is used from $2.5\\\\mathrm {R}_\\\\odot $ to 0.1AU.\",\n \"The solar wind speed is then given through an empirical relation which is a function of the magnetic flux tube expansion factor.\",\n \"The formula used in this work is given by $V(f_s)=240.0+675.0(1+f_s)^{-0.22} \\\\mathrm {km\\\\ s^{-1}} {,}$ where $f_s$ is given by $f_s=\\\\left(\\\\frac{\\\\mathrm {R}_\\\\odot }{r}\\\\right)^2\\\\frac{B_r(\\\\mathrm {R}_\\\\odot ,\\\\theta _0,\\\\phi _0)}{B_r(r,\\\\theta ,\\\\phi )}$ and it quantifies the expansion factor of the flux tube from the photospheric footpoint $(\\\\mathrm {R}_\\\\odot ,\\\\theta _0,\\\\phi _0)$ of the specific field line to its position further outwards $(r,\\\\theta ,\\\\phi )$ at a heliocentric distance $r$ [33].\",\n \"As explained in [33], the expansion factor takes values greater or equal to unity for flux divergence more rapid than or equal to $r^2$ , respectively.\",\n \"Simple scaling laws that are functions of $V$ are used in order to determine the plasma density and temperature.\",\n \"For further details of the empirical model, see [26].\",\n \"For the kinetic component of our analysis, we are using an exospheric model, which is a way to simulate low density plasmas, where the importance of collisions is limited.\",\n \"The solar atmosphere is considered to have a collision-dominated barosphere at low altitude [8] and a collisionless exosphere, which is the region modeled kinetically.\",\n \"These regions are separated by a surface called the exobase $r_0$ , beyond which collisions become negligible.\",\n \"This exobase level is defined as the altitude where the particle mean free path $l_f$ and the local density scale height $H$ become equal, i.e.\",\n \"where the dimensionless Knudsen number $K_n=l_\\\\mathrm {f}/H$ is equal to unity.\",\n \"The kinetic modelA 1D version of the kinetic exospheric model developed by the group in IASB-BIRA and collaborators can be found in CCMC (http://ccmc.gsfc.nasa.gov/models/exo.php) and it can run online for user-defined setups.\",\n \"[10], [20], [11], [8] gives different temperatures for electrons and protons as indeed observed [9] and can include different characteristics of any other ion species.\",\n \"Sources of the fast solar wind are considered to be coronal holes and in these regions the electron VDFs are assumed to correspond to a Lorentzian function with a small $\\\\kappa $ -value and thus have a large suprathermal tail [11].\",\n \"The low speed solar wind usually comes from equatorial regions, with larger $\\\\kappa $ -values.\",\n \"When $\\\\kappa \\\\rightarrow \\\\infty $ the VDF tends to a Maxwellian.\",\n \"In this work, we consider two particle species, namely electrons and protons and therefore their respective exobase levels need to be defined.\",\n \"The proton exobase is located where the Coulomb mean free path for the protons $l_{\\\\mathrm {f},p}$ according to [29] as estimated for coronal values by [11] is equal to the local density scale height $H$ , where $l_{\\\\mathrm {f},p}\\\\approx 7.2\\\\times 10^7\\\\frac{T^2_p}{n_e}{,} { \\\\ \\\\ \\\\ \\\\ }H=\\\\left(-\\\\frac{\\\\mathrm {d}\\\\ln n_e}{\\\\mathrm {d}r} \\\\right)^{-1} {,}$ where all the quantities are in SI.\",\n \"The proton mean free path is shown in Figure REF as a function of the proton temperature and the electron number density.\",\n \"For the electrons, a similar electron exobase height can be estimated from the Coulomb mean free path in a plasma consisting only of electrons and protons: $l_{\\\\mathrm {f},e}=0.416 \\\\left(\\\\frac{T_e}{T_p}\\\\right)^2l_{\\\\mathrm {f},p} {,}$ as in [11] for a hydrogen plasma, assuming that the electrons have the mean thermal velocity $(8k_\\\\mathrm {B}T_e/m_e\\\\pi )^{1/2}$ , with $l_{\\\\mathrm {f},e},l_{\\\\mathrm {f},p}$ in meters and $T_e,T_p$ in kelvin.\",\n \"A crude estimate of the scale height can be obtained assuming hydrostatic equilibrium in a stratified atmosphere with isothermal plane parallel layers.\",\n \"This is not the case for the solar wind, since expansion is taking place and rather hydrodynamic conditions apply, but it gives a good approximation of the order of magnitude of the scale height.\",\n \"Figure: The Coulomb mean free path l f,p l_{\\\\mathrm {f},p} in solar radii as a function of proton temperature and electron density.\",\n \"This figure quantifies the variations in Equation .When we adopt the same proton and electron temperature in the above formulae, the electron mean free path becomes smaller than the proton one $l_{\\\\mathrm {f},e}0.01$ , due to the velocity dependence of the mean free path of the particles.\",\n \"This shows that it is not especially important that the exobase is chosen to correspond exactly to the level where $K_n=1$ , but it will appear where the density gradient is very sharp and thus where the plasma becomes collisionless to a good approximation.\"\n ],\n [\n \"Velocity Distribution Functions\",\n \"When collisions are ignored as in the exospheric theory developed by [10], the Fokker-Planck equation reduces to the Vlasov equation for the evolution of the velocity distribution function: $\\\\frac{\\\\partial f}{\\\\partial t}+{v}\\\\cdot \\\\frac{\\\\partial f}{\\\\partial {r}}+{a}\\\\cdot \\\\frac{\\\\partial f}{\\\\partial {v}}=0 {.\",\n \"}$ Our kinetic exospheric model works by constructing stationary solution to the Vlasov equation, starting from an exact stationary solution for protons and electrons prescribed at the exobase.\",\n \"Kinetic models based on this equation were developed and are discussed in [11] for radial magnetic field lines and in [23] taking into account the spiral interplanetary magnetic field topology.\",\n \"It was shown in [11] that the specific moments in the solar wind, namely densities and temperatures, as well as the electrostatic potential characteristics from the corona to the interplanetary space are already well described, agreeing with observations at 1AU, when the collision term is neglected, since the collisions would rather modify the temperature anisotropies.\",\n \"Apart from the Maxwellians the generalised Lorentzian or Kappa function is also a solution of the Vlasov equation and can be used as a boundary condition to study the effect of suprathermal particles on the kinetic moments.\",\n \"Observations suggest that the velocity distribution functions of the electrons have strong suprathermal tails.\",\n \"We therefore assume a Lorentzian VDF for the electrons and a Maxwellian VDF for the protons at the exobase: $& f^p_{\\\\mathrm {Maxwell}}(r_0,v)=n_p(r_0)\\\\left(\\\\frac{m_p}{2\\\\pi k_\\\\mathrm {B}T_p(r_0)} \\\\right)^{3/2}\\\\exp \\\\left( -\\\\frac{m_pv_p^2}{2k_\\\\mathrm {B}T_p(r_0)}\\\\right) {,} \\\\\\\\& f^e_{\\\\mathrm {kappa}}(r_0,v)=\\\\frac{n_e(r_0)}{2\\\\pi \\\\kappa ^{3/2}}\\\\left(\\\\frac{m_e}{2k_\\\\mathrm {B}T_e(r_0)} \\\\right)^{3/2}A(\\\\kappa )\\\\left(1+\\\\frac{m_ev_e^2}{2k_\\\\mathrm {B}T_e(r_0)\\\\kappa } \\\\right)^{-(\\\\kappa +1)} {,} \\\\qquad $ where $A(\\\\kappa )=\\\\frac{\\\\Gamma (\\\\kappa +1)}{\\\\Gamma (\\\\kappa -1/2)\\\\Gamma (3/2)} {.\",\n \"}$ We note in passing that the moments of the Lorentzian VDF are not well defined for every $\\\\kappa $ value, but rather every $i$ th moment is defined for $\\\\kappa >(i+1)/2$ [20].\",\n \"Suprathermal protons have almost no influence on the solar wind velocity, so for them a Maxwellian VDF can suffice [11].\",\n \"Liouville's theorem [3] implies that any function that depends on the constants of motion of a collection of particles satisfies the Vlasov equation.\",\n \"The relevant constants of motion in this study are the total energy and the magnetic moment.\",\n \"Knowing the velocity distribution functions for our particle species at the exobase, the velocity distribution as a function of the radial distance can be deduced from energy conservation [20].\",\n \"Thereby, the electron and proton VDFs can be computed as a function of radius and speed along a purely radial magnetic field line.\",\n \"The analytic expressions for the kinetic moments of the exospheric models were derived for a Maxwellian VDF by [10] and for a Lorentzian VDF by [20].\",\n \"As explained above the exospheric model used in this paper includes only radial velocities along open radial magnetic field lines.\",\n \"Like in previous exospheric models, it is assumed that there are no particles coming from the interplanetary space to the Sun.\",\n \"The anisotropy of the distribution leads to the solar wind flux.\",\n \"The density, temperature and $\\\\kappa $ -index are determined at the exobase by either the MHD model or constrained via observations.\",\n \"The model provides then the velocity distribution function at any other distance as well as the rest of the kinetic moments.\",\n \"We use the code and the analytical expressions for the Maxwellian and the $\\\\kappa $ distributions by [20].\",\n \"Provided the number density, the electron and proton temperatures and the $\\\\kappa $ index for the electron VDF at the exobase, the quasi-neutrality and zero-current conditions are solved iteratively at a fixed radial distance $r_m$ using a Newton-Raphson scheme.\",\n \"The value of $r_m$ is iteratively modified using a dichotomy method until the electric field is found to be continuous within a predefined tolerance [8].\",\n \"On the other hand, EUHFORIA solves the 3D MHD equations taking self-consistently stream interactions into account thereby providing a $v_\\\\phi $ for any point, whereas in the kinetic model we impose that this velocity is constant on each spherical shell.\",\n \"The kinetic model thus proceeds without accounting for stream interactions instead keeping the same topology of fast and slow solar wind sources at every radial distance as at the exobase, reducing the computational time.\",\n \"As was argued in [23], the effects due to rotation as compared to the purely radial case change only the estimated proton and electron temperatures and their anisotropies.\",\n \"More specifically, the spiral structure predicts higher electron temperatures $T_e$ and lower proton temperatures $T_p$ than the radial case, but the number density, the electric potential and the bulk speed remain almost unaffected up to $300\\\\mathrm {R}_\\\\odot $ .\",\n \"It's worth noting here that stream interactions are once again not taken into account.\",\n \"Therefore, in this study we use the radial magnetic field topology and simply rotate by $v_\\\\phi $ each spherical shell to account for solar rotation.\",\n \"The advantage of the kinetic model used in this paper is the direct quantification of species-specific temperature profiles, densities, speeds, energy fluxes etc.\",\n \"once the electric potential is calculated.\",\n \"Even if $T_e=T_p$ is chosen at the exobase, the kinetic model self-consistently calculates the species-specific heating with distance and the two temperatures depart from each other.\",\n \"The $T_e$ is indeed observed to be different than $T_p$ at 1AU for slow and fast wind cases, e.g.\",\n \"as reported in [9].\"\n ],\n [\n \"Observational Input: Cases and Selection Criteria\",\n \"Several missions have observed, or continue to observe the Sun, as well as measure the physical parameters that characterize the solar wind, at different heliocentric distances as well as heliographic latitudes.\",\n \"They provide high resolution data, not only in the ecliptic plane, but also at higher latitudes, requiring simulations that explain and predict the behavior of the solar wind not only in the ecliptic plane, but rather in three dimensions.\",\n \"In this study we will be using OMNI data, and data from the Ulysses spacecraft due to its large latitudinal coverage.\",\n \"We based our synoptic magnetogram and solar state selection on the following criteria, that allow us to perform a crosscheck on their prediction ability with available spacecraft observations: i) quiet Sun periods (since the exospheric models are particularly tailored to quiet Sun conditions); ii) the presence of equatorial coronal holes, such that significant differences between high and slow speed wind may be expected at the orbit of the Earth; iii) position of Ulysses for combination of simulations and different spacecraft observations from different angles/telescopes; and iv) very few CME events according to the available catalogue CACTUSMore information can be found at http://sidc.oma.be/cactus/..\",\n \"In this study we focus on the year 2007, as it was a mostly quiet Sun year coinciding with the third orbit of Ulysses.\",\n \"For the global 3D comparison we focus on the period July-August 2007, when Ulysses crossed the ecliptic plane.\",\n \"We have confirmed the relative paucity of CME events in the selected time period using the CACTUS CME list.\"\n ],\n [\n \"Constraining the Kinetic Model using EUHFORIA\",\n \"First, we constrain the input to the kinetic model based on data-driven results at the inner radial boundary of EUHFORIA at $21.5 \\\\mathrm {R}_\\\\odot $ and examine both models' efficiency at capturing the solar wind bulk quantities by comparing with observations at 1AU and at 1.4AU (the Ulysses orbit).\",\n \"We compare the results of the MHD and kinetic models after interfacing their velocities and densities at the inner MHD boundary at $21.5\\\\mathrm {R}_\\\\odot $ .\",\n \"In Figure REF we show a slice at the equator (left panels) and a slice in latitude (right panels) that corresponds to the mass density scaled with the inverse square of the heliocenteric distance, and the radial speed also indicating the positions of Mercury, Venus, Earth, Mars and the STEREO (Solar Terrestrial Relations Observatory) A spacecraft.\",\n \"The grey part of the colorscale used in the Figure corresponds to high velocities and densities that occur especially during CME events.\",\n \"The velocity ranges ocurring at this time are roughly from 350 to 650 $\\\\mathrm {km\\\\ s^{-1}}$ with a clear separation between streams of different speeds, forming a clear Parker spiral-like configuration.\",\n \"There is a configuration of several distinct high and slow speed streams.\",\n \"We can see the different streams with high density associated to low speed and vice versa, while the highest speed captured is around $650\\\\mathrm {km\\\\ s^{-1}}$ and corresponds to a density similar to the one measured at 1AU.\",\n \"The highest density contrast with respect to the one measured at 1AU is about 10.\",\n \"The latitudinal panels suggest that there is compression and rarefaction as the plasma flows outwards with the corresponding speed and density.\",\n \"We thus conclude that the plasma is not following exactly the ideal Parker spiral moving on perfect cones as it expands, but following a rather more complicated motion, as the simulation is observation-driven and the photospheric magnetogram shows a complex topology, as discussed previously.\",\n \"Figure: EUHFORIA longitudinal and latitudinal variations of the three velocity and the three magnetic field components at 0.1AU (left panels) and at 1AU (right panels) for a standard magnetogram centered at 10 August 2007.In Figure REF , we show the components of the velocity and magnetic field in the spherical ($r,\\\\theta ,\\\\phi $ ) basis at two different distances, 0.1AU and 1AU, as functions of heliospheric longitude and latitude, as calculated by EUHFORIA.\",\n \"We observe a clear and narrow undulating current sheet showing clear discrimination between low- and high-latitude solar wind.\",\n \"This undulating sheet is characterized by lower velocity than other regions at 0.1AU.\",\n \"It separates the outward (southern hemisphere) and inward (northern hemisphere) magnetic field topologies.\",\n \"As the solar wind propagates outwards, the interactions between streams of different speeds become increasingly important with the sharp features at 0.1AU turn into more diffused, smoothed and extended ones at larger distances.\",\n \"At the Earth's orbit the current sheet and thus the slow speed region is thicker covering about $10^\\\\circ $ in latitude.\",\n \"The speed difference at 1AU with respect to the inner boundary is about $50\\\\mathrm {km\\\\ s^{-1}}$ .\",\n \"The $\\\\theta $ , $\\\\phi $ -components of the velocity increase in magnitude from zero to about $\\\\approx 15 \\\\mathrm {km\\\\ s^{-1}}$ at the Earth's orbit, roughly following the pattern that the slow speeds appear at small latitudes contrary to the high solar wind speeds.\",\n \"The radial component of the magnetic field decreases by about 2 orders of magnitude from the inner boundary to the orbit of the Earth showing a broader current sheet ($\\\\approx 5^\\\\circ $ ), where the magnetic field is close to zero.\",\n \"The $\\\\theta $ -component of the magnetic field increases from zero to about $0.36$ nT in magnitude up to 1AU.\",\n \"On the other hand, the $B_\\\\phi $ decreases by almost a factor of 8 up to 1AU and is at both distances opposite in polarity to the radial magnetic field, thus having positive polarity at the North Pole and negative at the South.\",\n \"In all the depicted quantities the rotation of the plasma shows as the entire structure in the 2D maps moves to the left.\",\n \"[htbp] Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONEUHFORIA longitudinal and latitudinal variations of the radial velocity (first row), the number density (second row) and the temperature (third row) at 0.1AU (left panels), at 1AU (middle panels) and at 1.4AU (right panels).\",\n \"They can be contrasted with their kinetic counterparts in Figure REF .\",\n \"In Figure REF , we show the velocity (first row), the density (second row) and the temperature (third row) variation at three different distances, 0.1AU, 1AU, and at the orbit of Ulysses at 1.4 AU, i.e.\",\n \"$\\\\approx 300\\\\mathrm {R}_\\\\odot $ , corresponding to left, middle and right columns, respectively.\",\n \"Most of the acceleration has already taken place at 1AU and only a small increase is happening above that distance due to the heating implemented by the reduced polytropic index ($\\\\gamma =1.5$ ) as the wind flows away from the Sun.\",\n \"The density decreases by 2 orders of magnitude from the inner boundary of the MHD simulation to the Earth's orbit and only by 50% from there onwards up to 1.4AU.\",\n \"The current sheet, seen as the high density structure that appears in the middle of the Figures in the second row seems to have expanded from $2^\\\\circ $ at the inner boundary to about $10^\\\\circ $ from there on, while it diffuses.\",\n \"The temperature decreases by a factor 8 from the inner boundary up to 1AU and only by 30% up to Ulysses' orbit.\",\n \"There is a temperature reversal captured, in the sense that while at 0.1AU the equatorial region appears cold and the poles hot, the opposite is happening from 1AU onwards, with the temperature shows a peak at a longitude of $-100^\\\\circ $ .\",\n \"The current sheet region gets very diffused outwards and from the Earth's orbit outwards appears discontinuous in this temperature view of the expanding plasma.\"\n ],\n [\n \"Interfacing\",\n \"In order to interface the two models at 0.1AU, we run the kinetic model up to $21.5\\\\mathrm {R}_\\\\odot $ with $r_0=r_s=2.5\\\\mathrm {R}_\\\\odot $ , $T_e=1\\\\mathrm {MK}$ , $T_p=1\\\\mathrm {MK}$ , using 600 $\\\\kappa $ -indexes in the range [2,8] with step 0.01 and with $n_e=n_p=3\\\\times 10^{10}\\\\mathrm {m^{-3}}$ [8].\",\n \"With the results we create a matrix with solar wind speeds at $21.5\\\\mathrm {R}_\\\\odot $ and by comparison with the EUHFORIA results for $v_r$ , we estimate the appropriate $\\\\kappa $ for every speed at the internal boundary of the MHD run.\",\n \"We obtain a 2D map of $N_\\\\theta \\\\times N_\\\\phi $ values of $\\\\kappa (\\\\theta ,\\\\phi )$ each corresponding to a field line.\",\n \"From the matrix, we also compare the number density given by EUHFORIA at the same distance (0.1AU) with the number density given by the kinetic model and get a scaling factor that we assume to be valid throughout all the considered radial distances [8].\",\n \"Thereby, we can estimate the appropriate initial density at the exobase that would give us the same density with EUHFORIA at 0.1AU.\",\n \"For the temperatures, the relation is more complicated, but [8] have demonstrated that the temperature does not affect the kinetic moments as much as the $\\\\kappa $ -indexes and the exobase altitude even for extreme changes (1 – 2MK) and thus for convenience we take $T_e=T_p=1$ MK for all the latitudes and longitudes at $r_0=2.5\\\\mathrm {R}_\\\\odot $ and focus instead on the $\\\\kappa $ and the density parameters.\",\n \"Note that EUHFORIA is not solving the MHD equations below 0.1AU, while the kinetic model provides results at any distance above the exobase and especially in the crucial region close to the Sun where the solar wind is being accelerated.\"\n ],\n [\n \"Kinetic Model\",\n \"Similar to Figure REF , in Figure REF , we show the velocity (first row), the density (second row), the electron temperature (third row) and the proton temperature (fourth row) at three different distances, 0.1AU, 1AU, and at the orbit of Ulysses at 1.4AU, i.e.\",\n \"$\\\\approx 300\\\\mathrm {R}_\\\\odot $ , corresponding to the left, middle and right columns, respectively.\",\n \"Unlike the MHD results discussed previously, in the kinetic approach the sharp structures that appear at the interfacing boundary (0.1AU) remain unchanged as the plasma moves outwards, as we don't account for stream interactions.\",\n \"In other words, neighboring streams with different speeds will not interact as they propagate and they will keep the same topological features up to large radial distances.\",\n \"The $\\\\kappa $ indexes corresponding to the kinetic velocities of this simulation lie in the range $(2,4)$ .\",\n \"The bulk speed accelerates more than in the MHD case, reaching terminal speeds about 100 $\\\\mathrm {km\\\\ s^{-1}}$ higher.\",\n \"Thus in the kinetic approach, where the acceleration is self-consistent and due to the induced electric field, the acceleration is more efficient than in the MHD approach, where semi-empirical schemes are used to accelerate the wind.\",\n \"Similarly to the MHD case, the terminal speed is reached at 1AU and from there on the acceleration is very slow up to 1.4AU.\",\n \"The density decreases faster by 20% at 1AU, to reach the orbit of Ulysses 30% smaller than the MHD case.\",\n \"The temperatures of the electron and proton populations are depicted in the last two rows.\",\n \"We have started at the exobase by setting both temperatures equal to 1MK independent of longitude and latitude.\",\n \"The electron temperature drops by a factor 5 at the orbit of the Earth and it doesn't change much up to 1.4AU, while the proton temperature decreases by a factor 2 and remains about the same up to the orbit of Ulysses.\",\n \"The temperature of the MHD plasma is always in between the electron and proton temperatures, being one order of magnitude smaller than the electron and one order of magnitude larger than the proton temperature.\",\n \"The MHD temperature is not the average between the two particle species temperatures, since the two species are not in thermodynamic equilibrium.\",\n \"[htbp] Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONKinetic results in the form of color maps in heliographic longitude and latitude from top to bottom rows: of the bulk speed, the number density, the electron and proton temperatures of the solar wind respectively at 0.1AU (left panels), at 1AU (middle panels) and at Ulysses' orbit 1.4AU (right panels).\",\n \"These can be contrasted directly with the MHD counterparts in Figure REF .\"\n ],\n [\n \"Models \",\n \"In Figure REF , the speed, number density and temperature of the solar wind are shown for i) the MHD model, ii) the kinetic model and iii) near-Earth observations using the OMNI dataset.\",\n \"From the speed plot we conclude that both models reproduce the number of peaks, their position with respect to each other and have a similar width.\",\n \"The most prominent difference between the models and the observations appears at the double peak of August 13th.\",\n \"EUHFORIA reproduces peaks of similar amplitude as observed, but showing a higher global minimum at about 400 $\\\\mathrm {km\\\\ s^{-1}}$ .\",\n \"The heights of the peaks and their relative ratio to one another are not reproduced exactly by any model.\",\n \"The kinetic model systematically overestimates the speeds varying from 400 to 800 $\\\\mathrm {km\\\\ s^{-1}}$ .\",\n \"This can be explained by the fact that the acceleration in that model is more efficient than the MHD case and continues at a significant rate at distances larger than 0.1AU to about 50$\\\\mathrm {R}_\\\\odot $ .\",\n \"This indicates that the final velocity obtained with the kinetic model can be improved by adapting the boundary conditions, for instance by lowering the empirical solar wind speed close to the Sun.\",\n \"The second panel shows the number density variations for the models and the observations.\",\n \"We observe that both models and the OMNI observations of the number density agree roughly in order of magnitude and in number of peaks, with the peaks not coinciding perfectly.\",\n \"The peak amplitudes are about 2.5 times larger in the observations than in both models, that are in agreement with one another.\",\n \"For the Alfvénic Mach number (third panel), we conclude that the Alfvénic Mach number of the MHD simulation is about a factor 3 higher than the average measured value most of the time, with three lows that reach lower than the observed values.\",\n \"The average observed proton temperature agrees with the EUHFORIA plasma temperature in order of magnitude, varying from the kinetic proton temperature $T_p$ at its minimum to the kinetic electron temperature $T_e$ at its maximum, while it is located between the two species' temperatures, about an order of magnitude larger than the kinetic proton temperature and an order of magnitude smaller than the kinetic electron temperature, at all times.\",\n \"The variation profile of the observations doesn't match with any model.\",\n \"Finally, the plasma $\\\\beta $ shown in the final panel for the MHD simulation lies most of the time at the lower limit of the observed one, with three peaks that reach values of about a factor 3 higher than the observed highest values, showing the opposite trend when compared to the Alfvénic Mach number shown in the third panel.\",\n \"Thus we conclude that, in the MHD simulation the magnetic field is higher with respect to the plasma pressure and the number density than the one indicated by observations at 1AU.\",\n \"The same case corresponding to CR 2059 was analyzed in a comparative study published in [4] for different observational inputs and MHD modeling schemes.\",\n \"None of their models seemed to accurately represent observation.\",\n \"The case directly comparable to ours was the one using GONG magnetograms and the WSA empirical model and ENLIL for the MHD modeling (dark red curve of lower panel in Figure 3).\",\n \"Before 27 July their maxima are not synchronous nor do they reach the same values, but after that the model captures the times of the peaks, but underestimates their value on the panel in the figure showing the speed.\",\n \"Figure: Time variations as measured by Ulysses (black) and calculated by the kinetic (blue) and the MHD (red) model, for i) top left panel: the position of Ulysses in heliographic coordinates, ii) top right panel: the speed, iii) bottom left panel: the number density and iv) bottom right panel: the Ulysses small (black) and large (green) proton, the kinetic proton (magenta), the kinetic electron (blue) and MHD (red) temperature.For a further comparison of the models with respect to observations, we overplot in Figure REF observations from Ulysses at 1.4AU and at latitudes between $-10^\\\\circ $ and $5^\\\\circ $ together with the results of the kinetic and MHD models.\",\n \"In the first panel (top left), we show the trajectory of Ulysses during the period of interest.\",\n \"At the same 3D heliographic spherical coordinates we extract the quantity of interest from the MHD and the kinetic model to accommodate further comparisons.\",\n \"More specifically, for each $(r,\\\\theta ,\\\\phi )$ position of Ulysses at each hour of a specific date, we choose the closest available point for the specific resolution of each model.\",\n \"In the second panel, we show the observed speed by Ulysses with black, together with the kinetic prediction in blue and the MHD results in red.\",\n \"All three exhibit different time profiles for each case.\",\n \"The kinetic model systematically overestimates the speeds reaching maximum values of about 800 $\\\\mathrm {km\\\\ s}^{-1}$ , whereas the MHD model ranges from 400 to 650 $\\\\mathrm {km\\\\ s}^{-1}$ , while at the same period the Ulysses measurements lie between 300 and 650 $\\\\mathrm {km\\\\ s}^{-1}$ .\",\n \"The peaks for both models are not well synchronized with the measured temporal velocity profiles.\",\n \"For the density the two models don't reproduce the observed number of peaks and they are not synchronized either.\",\n \"The number densities of the kinetic and MHD models agree with each other varying from 1 – 5 $\\\\mathrm {cm}^{-3}$ whereas the observed number density profile is much more variable with a number of peaks and reaching densities of 20 $\\\\mathrm {cm}^{-3}$ .\",\n \"The average observed proton temperatureAs shown in Figure REF , there are two different proton temperatures estimated, that in general bracket the real temperature at 1AU.\",\n \"As $T$ -large we denote the integral in the 3D velocity space of the distribution over all measured angles and energy bandwidths.\",\n \"The $T$ -small is calculated by the sum over all angles for a determined energy, then summing the moments of the estimated spectrum of the plasma and by taking the radial component of the temperature tensor (http://www.cosmos.esa.int/web/ulysses/swoops-ions-user-notes).\",\n \"agrees with the plasma temperature predicted by the MHD model and lies in between the kinetic proton (magenta) and kinetic electron (blue) temperatures of the kinetic model, being one order of magnitude smaller than the electron and one order of magnitude larger than the proton temperatures.\",\n \"No model reproduces in high accuracy the temporal variations of the observed temperature profile.\",\n \"The models show some correspondence with the Ulysses observations and the correct orders of magnitude are roughly reproduced, but certainly there is need for improvement.\",\n \"To reach better agreement with observations, we can modify the parameters in the semi-empirical model of EUHFORIA to better adjust and reproduce the measurements in each case individually.\"\n ],\n [\n \"Heat Flux\",\n \"[27] presented an analysis of the different ways that energy source considerations can be used in MHD solar wind models.\",\n \"A generalized formulation of the energy Equation with an extra energy source term at the right hand side $S$ was examined for different cases of $S$ .\",\n \"The relevant models studied therein include i) a model with a polytropic index with spatial dependance $\\\\Gamma (r)$ and ii) a model with a polytropic index fixed at $\\\\gamma =5/3$ constant in the entire coronal volume.\",\n \"In particular, the authors showed that a steady-state solar wind solution accelerated by a given non-adiabatic polytropic wind can equivalently be re-written using an energy source term, given by: $S=\\\\nabla \\\\cdot \\\\Bigg [{v}P\\\\bigg ( \\\\frac{1}{\\\\Gamma -1}-\\\\frac{1}{\\\\gamma -1}\\\\bigg ) \\\\Bigg ] {,}$ with ${v}$ , $P$ and $\\\\Gamma $ being obtained by the model, as explained in [27], with the non-adiabatic index $\\\\Gamma (r)=1.5$ for our case.\",\n \"Figure: Heat flux along magnetic field lines in the equatorial plane as given by EUHFORIA.In this section, we will discuss the heat flux of the MHD and the kinetic models.\",\n \"According to the formulation presented above we have that the analytic expression of the energy flux responsible for accelerating the wind in the MHD model is given by ${v}P\\\\bigg (\\\\frac{1}{\\\\Gamma -1}-\\\\frac{1}{\\\\gamma -1}\\\\bigg )$ .\",\n \"In Figure REF , we present the heat flux of the MHD model along selected magnetic field lines as a function of distance from 0.1AU to 2AU in the equatorial plane.\",\n \"Figure: Electron (top panel) and proton (bottom panel) heat fluxes for CR2059 with red, orange, light green, green and blue curves corresponding to κ\\\\kappa values of 3, 4, 5, 6 and 7, respectively.In Figure REF , we illustrate the radial profiles of the electron (top panel) and proton (bottom panel) heat fluxes, as calculated by the kinetic model for an exobase at 2.5$\\\\mathrm {R}_\\\\odot $ , 1MK electron and proton temperatures at the exobase $T_e=T_p=1\\\\mathrm {MK}$ and the same densities as our default case for $\\\\kappa $ -indexes 3, 4, 5, 6 and 7 corresponding to red, orange, light green, green and blue respectively.\",\n \"We observe that the more suprathermal particles are present, i.e.\",\n \"lower $\\\\kappa $ , the higher the heat flux is.\",\n \"The electron heat flux is higher by an order of magnitude than the proton heat flux close to the Sun and it decreases faster than the proton heat flux with the heliocentric distance to reach similar values at 1AU.\",\n \"We conclude that the differences between the proton heat flux curves corresponding to different $\\\\kappa $ -indexes are smaller than the respective electron flux curves.\",\n \"In general, the heat fluxes are overestimated by exospheric models, since the corresponding VDFs have the highest possible anisotropy due to lack of collisions.\",\n \"As shown in [35], exospheric models and kinetic simulations that include collisions are in good agreement, making the exospheric model a convenient tool for the study of weakly collisional plasmas.\",\n \"Both the exospheric model and the collisional simulations found heat fluxes several times larger than the classical value, suggesting that the classical formulation is not appropriate for weakly collisional plasmas, since it was based on the assumption of a collision-dominated medium.\",\n \"If one really needs to prescribe a realistic heat flux, then collisions need to be taken into account and further improvements in the kinetic model need to be considered.\",\n \"The inclusion of interactions, through e.g.\",\n \"Alfvén or whistler waves, will decrease the anisotropies and it will improve the higher moments, i.e.\",\n \"temperatures and heat fluxes for the considered species [22], [24], [31].\",\n \"Using such sophisticated schemes to improve the heat flux agreement with observations is outside of the scope of this paper, due to i) the consequent increased computational expense, ii) the fact that the temperatures are not the most important geo-effective parameters, making these improvements rather impractical for future operational space weather applications.\",\n \"Thus, the heat flux profiles for the electron and protons, that are quantified by the kinetic model (Figure REF ) can serve as upper limits and are more physics-based than the MHD heating prescriptions that are based on the empirical determination of the polytropic index value.\",\n \"The heat flux profiles along the magnetic field lines of the MHD model are in agreement in order of magnitude with the profiles of the protons of the kinetic model closer to the Sun, but they drop faster further out reaching electron heat flux values at the orbit of the Earth.\",\n \"Our results for the heat flux for the MHD model depicted in Figure REF are roughly in agreement with the literature [5], [30].\",\n \"In accordance to the kinetic model results, the electron heat flux is higher than the proton heat flux according to [5] and [30], respectively, with our results roughly being closer to the electron energetics profile, but at 1AU approaching the heat flux values expected for the protons.\",\n \"In this study, we are interested in making a first comparison between single fluid and kinetic models, that have the potential to be used in space weather applications.\",\n \"The two models are very different in nature, making use of very different formulations for the plasma physics.\",\n \"EUHFORIA is a single fluid code that uses the MHD equations to describe the plasma.\",\n \"Whereas the kinetic model used in this study begins by making an observationally-inspired assumption for the VDFs of the electron and proton species, that are considered to constitute the solar wind plasma, and based on that all the physically interesting quantities are calculated as moments over the velocity space.\",\n \"EUHFORIA is observationally-driven and provides three-dimensional information that accounts for stream interactions and complicated magnetic topologies.\",\n \"The kinetic model is a semi-analytic model solving for the plasma characteristics along a magnetic field line and accounting for heating and acceleration in a self consistent way.\",\n \"Note that the electric field (used in the kinetic model) is hidden in the pressure term .\",\n \"Parker explained that the momentum equation of the electrons is given by $\\\\frac{\\\\mathrm {d}p_e}{\\\\mathrm {d}r}+neE_\\\\mathrm {LS}=0 {,}$ which is the hydrodynamic condition, with $E_\\\\mathrm {LS}$ the Lemaire-Scherer electric field [10].\",\n \"In [18], Parker showed that the electric field of Lemaire-Scherer is in the fluid approach given by $E_\\\\mathrm {LS}=-\\\\frac{1}{ne}\\\\frac{\\\\mathrm {d}p_e}{\\\\mathrm {d}r}=\\\\frac{m_p}{e}\\\\left( v\\\\frac{\\\\mathrm {d}v}{\\\\mathrm {d}r}+\\\\frac{GM_\\\\odot }{r^2}\\\\right) {,}$ which finally, taking into account that the Pannekoek-Rosseland electric field is $E_\\\\mathrm {PR}=\\\\frac{m_pg}{e}$ , is written as $E_\\\\mathrm {LS}=E_\\\\mathrm {PR}+\\\\frac{m_pv}{e} \\\\frac{\\\\mathrm {d}v}{\\\\mathrm {d}r} {.\",\n \"}$ The Lemaire-Scherer electric field (used in exospheric models) is several times larger than the Pannekoek-Rosseland one, corresponding to hydrodynamic equilibrium that is used to describe the solar wind expansion.\",\n \"It is able to lift and accelerate to supersonic speeds the initially slow and heavy protons through trapping the fast and light electrons, while keeping the quasi-neutrality and almost zero-current condition.\",\n \"Any difference in the bulk speeds of the two populations would lead to the generation of a current, which due to Ampère's law has to remain small.\",\n \"From Figure REF , we deduce that at large radial distances from the Sun, the MHD current sheet gets expanded and becomes thicker, evolving from about $2^\\\\circ $ at the inner boundary to about $10^\\\\circ $ at 1AU, while the fine structures that appear in the 0.1AU longitudinal and latitudinal map get diffused and smoothed.\",\n \"On the contrary, in Figure REF we see that the fine structures and the current sheet size don't change under the kinetic approach, as there are no stream interactions taken into account.\",\n \"Both models give speeds of the same order of magnitude at every altitude and have most of their acceleration taking place already before 1AU.\",\n \"The kinetic model shows a more efficient acceleration, reaching terminal speeds of about 100 $\\\\mathrm {km\\\\ s^{-1}}$ larger than the MHD one up to 1AU.\",\n \"The acceleration in the MHD approach is taking place due to the reduced polytropic index, while in the kinetic approach the acceleration is related to the induced electric field that assures quasi-neutrality and equal outward electron and proton fluxes.\",\n \"Regarding the number density, both models give densities of the same order of magnitude, with the kinetic model though showing a sharper profile that decreases faster outwards.\",\n \"The number density of the kinetic model is 20% and 30% lower at 1AU and 1.4 AU respectively than the MHD number density.\",\n \"Moreover, for the temperature in the MHD approach, as is shown in the bottom row of Figure REF , there is a reversal of the hot-cold regions from the inner boundary up to large distances.\",\n \"A faster cooling takes place at higher latitudes, making the initially colder equatorial region appear hotter at large radial distances with respect to the rest of the latitudes.\",\n \"Furthermore, the temperature ranges fall by a factor 4 from 0.1AU up to 1AU, and continue decreasing by 30% up to the orbit of Ulysses.\",\n \"For the kinetic approach, the proton temperature only falls by factor two up to the Earth's orbit and seems constant up to 1.4 AU, while the electron temperature drops by a factor 5 up to 1AU and doesn't seem to vary much from there on.\",\n \"Contrary to the MHD single fluid results, in the kinetic case there is no temperature profile change and the cooling seems to take place uniformly at all latitudes.\",\n \"There is a structure close to the equator at a heliographic longitude of $-70^\\\\circ $ that appears like a spike in contact with the current sheet in both Figures REF and REF .\",\n \"In Figure REF though, in the middle and right panels, corresponding to large distances, that spike appears to change inclination from pointing to the left of the figure at the panels corresponding to 0.1AU to pointing to the right from 1AU outwards, unlike Figure REF , where the spike's inclination remains unchanged.\",\n \"This difference is then likely due to stream interactions that are captured in the 3D MHD model, while these are excluded in the essentially 1D, radial field kinetic approach.\"\n ],\n [\n \"Conclusions\",\n \"After the parallelization of the kinetic model and its use in a quasi-3D approach we linked it to more robust 3D MHD models and compared both to observations at 1AU, using similar boundary conditions at $21.5\\\\mathrm {R}_\\\\odot $ .\",\n \"When the two models were compared, starting from the same boundary conditions, the kinetic model gives systematically higher speeds than the MHD model at large radial distances.\",\n \"This is due to the fact that the acceleration of the solar wind continues at a higher rate in the kinetic model after $21.5\\\\mathrm {R}_\\\\odot $ .\",\n \"The acceleration mechanism in the kinetic model is due to the induced electric field that assures quasi-neutrality and prevents charge separation and also \\\"bounds\\\" the two species to move with the same bulk speed.\",\n \"There is no explicit heating term in the MHD equations used by EUHFORIA.\",\n \"The MHD model accelerates further the solar wind due to the reduced polytropic index $\\\\gamma $ , but at a very slow rate accounting for an acceleration of about 50 $\\\\mathrm {km\\\\ s}^{-1}$ from 0.1AU to 1AU.\",\n \"The exospheric models overestimate the heat flux, especially for the electrons that have a thermal speed comparable to their bulk speed, but such a heat flux can be improved by inclusion of interactions through waves, e.g.\",\n \"Alfvén or whistler waves [22], [24], [31].\",\n \"The heat flux calculated by the kinetic exospheric model can be used as an upper limit for more physically-driven heat flux prescriptions in a global MHD model.\",\n \"The heat flux of the kinetic model is in qualitative agreement with other studies [5], [30] and the heat flux profile of the MHD models close to the Sun resembles the proton profile, only to decrease faster outwards resembling the electron profile at 1AU.\",\n \"In the exospheric model the fast electrons are slowed down and the protons are accelerated by the Lemaire-Scherer electric field that eventually leads to the observed supersonic solar wind.\",\n \"As shown by [18] the acceleration of the solar wind in collisionless plasmas to supersonic values lies in the hydrodynamic equation in combination to the mass ratio between electrons and protons.\",\n \"According to Parker, the exospheric model describes a very efficient heat transport mechanism with an electron temperature that decreases very slowly at large distances and through the induced electric field it elevates the protons and causes the transonic solar wind.\",\n \"There is some agreement in the high- and slow-speed streams in the velocity profiles for both OMNI and Ulysses observations.\",\n \"The high speed positions of the models are better synchronized for OMNI rather than Ulysses observations, as we showed earlier.\",\n \"The number densities of both models were approximately in the same order of magnitude with the observed ones at 1AU and at the orbit of Ulysses.\",\n \"The MHD temperature is one order of magnitude smaller than the electron temperature $T_e$ and one order of magnitude larger than the proton temperature $T_p$ of the kinetic model.\",\n \"The average observed proton temperatures agreed with the temperature predicted by the MHD model in order of magnitude and thus were also lying in the range between the electron and proton kinetic temperatures.\",\n \"In this study we assumed that the $\\\\kappa $ -value is independent of the radial distance, but observations such as [12] indicate that the $\\\\kappa $ can actually change as the distance from the Sun increases.\",\n \"However, the fitting model used in [12] was not with a Kappa distribution for the full range, but a sum of a Maxwellian for the core and a Kappa function for the halo, so that the parameter $\\\\kappa $ does not represent the same quantity as in the model used in the current study.\",\n \"The density ratio between the core and the halo remains constant with the distance, as analyzed by [25].\",\n \"A more realistic exobase profile with temperatures having a latitudinal and even longitudinal variation can be taken into account and will be the next step towards a fully 3D kinetic numerical code of the solar corona and the solar wind.\",\n \"The kinetic model is a semi-analytic model ignoring stream interactions and thus conserves the slow and fast wind distributions for every radial distance.\",\n \"Accounting for stream interactions as the solar wind propagates outwards will be one of the main points of interest for a more realistic 3D model.\",\n \"Shocks associated to sharp velocity gradients can be included in an empirical way or by using more sophisticated kinetic models including collisions and wave-particle interactions [19].\",\n \"Here we have ignored the spiral shape of the magnetic field in the calculation of the moments in the kinetic model, adopting purely radial magnetic fields, because as argued in [23] this aspect won't affect the main average quantities apart from the temperature anisotropies.\",\n \"But in the future we are planing on including the spiral magnetic field effects in a similar study.\",\n \"[23] quantified the effect that the spiral magnetic field topology has on the particle temperatures and their anisotropies.\",\n \"More specifically, in the radial case the electron temperature is slightly underestimated, whereas the opposite is true for the proton species.\",\n \"The temperature anisotropies are overestimated by the kinetic model in comparison to observations [9].\",\n \"Another important aspect that can be improved and would provide a deeper comparison between operational MHD codes and kinetic exospheric ones, would be to change the formulation of the kinetic model, so that boundary conditions for the speed, the density and temperature directly from MHD models can be used at each grid point including the magnetic field information.\",\n \"These aspects will allow us to reach more fundamental conclusions about the two different models and will upgrade the kinetic exospheric model into a computational equivalent to the robust 3D MHD code.\",\n \"When the 3D magnetic field topology from the source surface, through the Schatten current sheet region, and throughout the region covered by the MHD model, is used directly within the kinetic description, we can use its predicted heat fluxes and higher order moment information to turn the model into a self-consistent hybrid kinetic-MHD modeling tool.\",\n \"SPM acknowledges financial support by the FWO and NASA Living with a Star grant number NNX16AC11G.\",\n \"This research was supported by projects GOA/2015 – 014 (KU Leuven, 2014 – 2018), and the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7/08 CHARM).\"\n ],\n [\n \"Conflict of Interest\",\n \"The authors declare that they have no conflict of interest.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01605"}}},{"rowIdx":892,"cells":{"text":{"kind":"list like","value":[["Did we learn from LLC Side Channel Attacks? A Cache Leakage Detection\n Tool for Crypto Libraries"],["Abstract This work presents a new tool to verify the correctness of cryptographic implementations with respect to cache attacks.","Our methodology discovers vulnerabilities that are hard to find with other techniques, observed as exploitable leakage.","The methodology works by identifying secret dependent memory and introducing forced evictions inside potentially vulnerable code to obtain cache traces that are analyzed using Mutual Information.","If dependence is observed, the cryptographic implementation is classified as to leak information.","We demonstrate the viability of our technique in the design of the three main cryptographic primitives, i.e., AES, RSA and ECC, in eight popular up to date cryptographic libraries, including OpenSSL, Libgcrypt, Intel IPP and NSS.","Our results show that cryptographic code designers are far away from incorporating the appropriate countermeasures to avoid cache leakages, as we found that 50% of the default implementations analyzed leaked information that lead to key extraction.","We responsibly notified the designers of all the leakages found and suggested patches to solve these vulnerabilities."],["Introduction","In the last decade we have witnessed the cloud revolution enabled by sandboxing mechanisms such as virtualization.","These technologies allow cloud service providers to place data and processes of various customers on the same hardware without jeopardizing their security.","IaaS clouds for instance rent expensive hardware resource to multiple customers by offering guest OS instances sandboxed inside virtualized machines (VMs).","PaaS cloud services go one step further and allow users to share the application space while sandboxed, e.g.","using containers, at the OS level.","Similar sandboxing techniques are used to isolate semi-trusted apps running on mobile devices.","Even browsers use sandboxing to execute untrusted code without the risk of harming the local host.","Resource sharing improves utilization and thereby helps reducing IT costs and saves power.","However, resource sharing also enables information leakages at the hardware level which sandboxing techniques cannot prevent and that can be exploited by malicious code.","One of the most prominent leakage sources, i.e.","the Last Level Cache (LLC), has been heavily targeted.","From the adversaries' point of view, the LLC leakage channel has a number of advantages, including high resolution information and cross core information leakage.LLC attacks have demonstrated to recover a wide range of information, ranging from private cryptographic keys to user's private shopping behavior .","Furthermore, LLC attacks have been successfully applied in a wide range of scenarios, e.g., malicious VMs in IaaS or PaaS public clouds, malicious Javascript execution in web browsers or even as malicious smartphone apps .","In short, cache attacks have demonstrated to pose a severe threat applicable in a wide range of use scenarios.","These attacks can be stopped at three different layers; at the hardware , the OS/hypervisor or at the application layer .","While the first two involve overheads that hardware/OS designers might not be willing to pay, preventing leakage in security critical applications can be achieved by aware code designers through proper implementation techniques.","Indeed, cache attacks usually exploit human coding mistakes that lead to private information leakage.","Although these leakages might sometimes be difficult to detect, they need to be carefully verified, especially when designing cryptographic code.","Cryptographic primitives make the core of the security infrastructure used to protect our digital communications.","If the used cryptographic code is poorly designed, any solution built on top is also susceptible to fail.","In summary, cache attacks diverge from traditional attacks in the following ways: Stealthy: Leakage attacks are extremely hard to detect.","The effects of an attack may only be felt through performance degradation for a short duration while leakage traces are being collected.","After the attack is performed no digital footprint is left.","Hence standard detection techniques such as traffic, access and privilege monitoring are completely blind to leakage attacks.","For instance, any application submitted to the app store is checked for access violations to devices, however, any attack code exploiting hardware leakage would only monitor legitimate memory access time variations.","Hard to Prevent: It is rather difficult to design leakage proof code, especially if good performance is also an objective.","Leakages are quite subtle and may not be detected after deployment for a long time.","Even if the code is not leaky on one platform, with gradual optimizations implemented at the microarchitectural level, new leakage channels may emerge with newer releases of a platform.","Difficult to Test: Verification of leakage resistant code is painfully slow and typically requires inspection by experts well versed in cache attacks.","In this paper, we introduce a tool that helps cryptographic code designers to analyze and expose any leakages in cryptographic implementations.","We demonstrate the viability of the technique by analyzing cache leakages in a number of popular cryptographic libraries with respect to three main cryptographic primitives we rely on every day to securely communicate over the internet: AES, RSA and ECC.","In order to achieve this goal, we first find secret dependent instructions/data with dynamic taint analysis, introduce cache evictions inside code routines, record cache traces and then verify the existing dependence with respect to the secret.","Our results show that, despite the efforts of cryptography library designers in reaction to the multitude of recently published LLC cache attacks several popular libraries are still vulnerable.","Our Contributions.","This paper presents a proactive tool to analyze leakage behavior of security critical code.","Unlike other approaches, our tool can be used to find real exploitable leakage in the design of cryptographic implementations.","The tool identifies secret dependent data, obtains cache traces obtained from the code execution on the microarchitecture and uses the generic Mutual Information Analysis (MIA) as a metric to determine the existence of cache leakage.","The detection technique is agnostic to the implementation, i.e.","the testing code can be run across all target platforms without having to redesign it, yet pinpoints parts of code that cause found leakages.","We perform the first big scale analysis of popular cryptographic libraries on three of the most commonly used primitives: AES, RSA and ECC.","We demonstrate that several cache leakages are still present in up-to-date cryptographic libraries (i.e.","50% still leak information) and need to be fixed."],["Preliminaries","The proposed tool employs techniques from modern cache attacks to monitor cache activity and measures information leakage using Mutual Information."],["LLC Attacks","LLC cache attacks are one of the most dangerous side channel attacks since they do not rely on physical proximity.","Furthermore, these attacks do not require root privileges and can be executed in user mode.","There are two main classes of LLC attacks: Flush and Reload Attack The Flush and Reload attack was first introduced in Gullasch et al.","but acquired its name in Yarom and Falkner , who were able to extract RSA cryptographic keys across VMs situated in different cores.","This attack was later shown to be successful also breaking symmetric key algorithms , , TLS sessions or the isolation in PaaS clouds and smartphone devices , .","In particular, the attack assumes the following; The attacker and the victim are executing processes in the same CPU but in different cores.","Attacker and victim share read-only memory pages.","The Flush and Reload attack can only succeed when attacking memory blocks shared with the potential victim.","These usually imply static code and global variables, but never dynamically allocated variables.","Prime and Probe Attack The Prime and Probe attack was first introduced for the L1 cache by Osvik et al.","and was later utilized by Ristenpart et al.","and Zhang et al.","to recover, in IaaS clouds, keystrokes and El Gamal decryption keys respectively.","Recently the attack was expanded to attack the LLC by Liu et al.","and Irazoqui et al.",", and has been successfully applied in commercial IaaS clouds , as Javascript executions and as smartphone applications .","The attack presents advantages and disadvantages over Flush and Reload: Prime and Probe does not require any memory sharing between victim and attacker.","Prime and Probe can increase the attack vector to dynamically allocated variables.","Prime and Probe implies some reverse engineering to know which sets in the cache the attacker is using, while Flush and Reload does not.","Prime and Probe is noisier than Flush and Reload.","Taking these two attacks into account, we explore the cryptographic libraries looking for leakages exploitable by either attack.","Thus, we examine both statically and dynamically allocated memory in our analysis."],["Mutual Information Analysis","The metric that we use to quantify leakage is Mutual Information (MI), that is given by the following equation: $I(X;Y) = H(X) - H(X|Y) = H(X) + H(Y) - H(X, Y)$ Where $H(X)$ is the entropy of random variable $X$ and $H(X|Y)$ is the entropy of the random variable $X$ given the knowledge of the random variable $Y$ .","In a way $I(X; Y)$ gives us how related variables $X$ and $Y$ are.","Note that if $X$ and $Y$ are independent, $I(X;Y)=0$ since $H(X|Y)=H(X)$ .","In contrast, if $X$ and $Y$ are fully dependent we obtain maximum MI, i.e., $I(X;Y)=H(X)$ , since $H(X|Y)=0$ ($Y$ fully determines $X$ ).","MI has been used in prior work for side channel attacks and leakage quantification.","Gierlichs et al.","utilize MI as a side channel distinguisher to mount differential side channel attacks.","Standaert et al.","utilized MI as a leakage quantifier to evaluate side channel attack security.","Prouff et al.","further expanded on the limitations and strengths of MI as a side channel attack metric.","Recently Zhang and Lee used MI to measure the security level of several cache architectures which they modeled as finite state machines."],["Methodology","Our goal is to determine whether specific memory addresses being used inside a cryptographic routine reveal information based on their cache accesses."],["Main Idea","Our approach involves the monitorization of the cache usage for all the memory addresses considered susceptible of carrying information.","In order to achieve this goal we utilize common known techniques from cache attacks, like the usage of cache flushing or cycle counter instructions.","We illustrate our approach on a toy example shown in Figure REF .","The Hello code snippet simply returns different messages depending on whether the caller is a female or a male.","We assume that the designer of this simple code does not want a potential attacker to know the gender of the caller.","Assume that the code designers would like to know whether they did a good job, i.e., whether the cache traces do not reveal the gender of the user.","Here, it is easy to observe that there is a $gender$ dependent branch that could reveal whether the user is a male or female.","We call $B1$ and $B2$ the two possible outputs of the branch.","Figure: NO_CAPTION"]],"string":"[\n [\n \"Did we learn from LLC Side Channel Attacks? A Cache Leakage Detection\\n Tool for Crypto Libraries\"\n ],\n [\n \"Abstract This work presents a new tool to verify the correctness of cryptographic implementations with respect to cache attacks.\",\n \"Our methodology discovers vulnerabilities that are hard to find with other techniques, observed as exploitable leakage.\",\n \"The methodology works by identifying secret dependent memory and introducing forced evictions inside potentially vulnerable code to obtain cache traces that are analyzed using Mutual Information.\",\n \"If dependence is observed, the cryptographic implementation is classified as to leak information.\",\n \"We demonstrate the viability of our technique in the design of the three main cryptographic primitives, i.e., AES, RSA and ECC, in eight popular up to date cryptographic libraries, including OpenSSL, Libgcrypt, Intel IPP and NSS.\",\n \"Our results show that cryptographic code designers are far away from incorporating the appropriate countermeasures to avoid cache leakages, as we found that 50% of the default implementations analyzed leaked information that lead to key extraction.\",\n \"We responsibly notified the designers of all the leakages found and suggested patches to solve these vulnerabilities.\"\n ],\n [\n \"Introduction\",\n \"In the last decade we have witnessed the cloud revolution enabled by sandboxing mechanisms such as virtualization.\",\n \"These technologies allow cloud service providers to place data and processes of various customers on the same hardware without jeopardizing their security.\",\n \"IaaS clouds for instance rent expensive hardware resource to multiple customers by offering guest OS instances sandboxed inside virtualized machines (VMs).\",\n \"PaaS cloud services go one step further and allow users to share the application space while sandboxed, e.g.\",\n \"using containers, at the OS level.\",\n \"Similar sandboxing techniques are used to isolate semi-trusted apps running on mobile devices.\",\n \"Even browsers use sandboxing to execute untrusted code without the risk of harming the local host.\",\n \"Resource sharing improves utilization and thereby helps reducing IT costs and saves power.\",\n \"However, resource sharing also enables information leakages at the hardware level which sandboxing techniques cannot prevent and that can be exploited by malicious code.\",\n \"One of the most prominent leakage sources, i.e.\",\n \"the Last Level Cache (LLC), has been heavily targeted.\",\n \"From the adversaries' point of view, the LLC leakage channel has a number of advantages, including high resolution information and cross core information leakage.LLC attacks have demonstrated to recover a wide range of information, ranging from private cryptographic keys to user's private shopping behavior .\",\n \"Furthermore, LLC attacks have been successfully applied in a wide range of scenarios, e.g., malicious VMs in IaaS or PaaS public clouds, malicious Javascript execution in web browsers or even as malicious smartphone apps .\",\n \"In short, cache attacks have demonstrated to pose a severe threat applicable in a wide range of use scenarios.\",\n \"These attacks can be stopped at three different layers; at the hardware , the OS/hypervisor or at the application layer .\",\n \"While the first two involve overheads that hardware/OS designers might not be willing to pay, preventing leakage in security critical applications can be achieved by aware code designers through proper implementation techniques.\",\n \"Indeed, cache attacks usually exploit human coding mistakes that lead to private information leakage.\",\n \"Although these leakages might sometimes be difficult to detect, they need to be carefully verified, especially when designing cryptographic code.\",\n \"Cryptographic primitives make the core of the security infrastructure used to protect our digital communications.\",\n \"If the used cryptographic code is poorly designed, any solution built on top is also susceptible to fail.\",\n \"In summary, cache attacks diverge from traditional attacks in the following ways: Stealthy: Leakage attacks are extremely hard to detect.\",\n \"The effects of an attack may only be felt through performance degradation for a short duration while leakage traces are being collected.\",\n \"After the attack is performed no digital footprint is left.\",\n \"Hence standard detection techniques such as traffic, access and privilege monitoring are completely blind to leakage attacks.\",\n \"For instance, any application submitted to the app store is checked for access violations to devices, however, any attack code exploiting hardware leakage would only monitor legitimate memory access time variations.\",\n \"Hard to Prevent: It is rather difficult to design leakage proof code, especially if good performance is also an objective.\",\n \"Leakages are quite subtle and may not be detected after deployment for a long time.\",\n \"Even if the code is not leaky on one platform, with gradual optimizations implemented at the microarchitectural level, new leakage channels may emerge with newer releases of a platform.\",\n \"Difficult to Test: Verification of leakage resistant code is painfully slow and typically requires inspection by experts well versed in cache attacks.\",\n \"In this paper, we introduce a tool that helps cryptographic code designers to analyze and expose any leakages in cryptographic implementations.\",\n \"We demonstrate the viability of the technique by analyzing cache leakages in a number of popular cryptographic libraries with respect to three main cryptographic primitives we rely on every day to securely communicate over the internet: AES, RSA and ECC.\",\n \"In order to achieve this goal, we first find secret dependent instructions/data with dynamic taint analysis, introduce cache evictions inside code routines, record cache traces and then verify the existing dependence with respect to the secret.\",\n \"Our results show that, despite the efforts of cryptography library designers in reaction to the multitude of recently published LLC cache attacks several popular libraries are still vulnerable.\",\n \"Our Contributions.\",\n \"This paper presents a proactive tool to analyze leakage behavior of security critical code.\",\n \"Unlike other approaches, our tool can be used to find real exploitable leakage in the design of cryptographic implementations.\",\n \"The tool identifies secret dependent data, obtains cache traces obtained from the code execution on the microarchitecture and uses the generic Mutual Information Analysis (MIA) as a metric to determine the existence of cache leakage.\",\n \"The detection technique is agnostic to the implementation, i.e.\",\n \"the testing code can be run across all target platforms without having to redesign it, yet pinpoints parts of code that cause found leakages.\",\n \"We perform the first big scale analysis of popular cryptographic libraries on three of the most commonly used primitives: AES, RSA and ECC.\",\n \"We demonstrate that several cache leakages are still present in up-to-date cryptographic libraries (i.e.\",\n \"50% still leak information) and need to be fixed.\"\n ],\n [\n \"Preliminaries\",\n \"The proposed tool employs techniques from modern cache attacks to monitor cache activity and measures information leakage using Mutual Information.\"\n ],\n [\n \"LLC Attacks\",\n \"LLC cache attacks are one of the most dangerous side channel attacks since they do not rely on physical proximity.\",\n \"Furthermore, these attacks do not require root privileges and can be executed in user mode.\",\n \"There are two main classes of LLC attacks: Flush and Reload Attack The Flush and Reload attack was first introduced in Gullasch et al.\",\n \"but acquired its name in Yarom and Falkner , who were able to extract RSA cryptographic keys across VMs situated in different cores.\",\n \"This attack was later shown to be successful also breaking symmetric key algorithms , , TLS sessions or the isolation in PaaS clouds and smartphone devices , .\",\n \"In particular, the attack assumes the following; The attacker and the victim are executing processes in the same CPU but in different cores.\",\n \"Attacker and victim share read-only memory pages.\",\n \"The Flush and Reload attack can only succeed when attacking memory blocks shared with the potential victim.\",\n \"These usually imply static code and global variables, but never dynamically allocated variables.\",\n \"Prime and Probe Attack The Prime and Probe attack was first introduced for the L1 cache by Osvik et al.\",\n \"and was later utilized by Ristenpart et al.\",\n \"and Zhang et al.\",\n \"to recover, in IaaS clouds, keystrokes and El Gamal decryption keys respectively.\",\n \"Recently the attack was expanded to attack the LLC by Liu et al.\",\n \"and Irazoqui et al.\",\n \", and has been successfully applied in commercial IaaS clouds , as Javascript executions and as smartphone applications .\",\n \"The attack presents advantages and disadvantages over Flush and Reload: Prime and Probe does not require any memory sharing between victim and attacker.\",\n \"Prime and Probe can increase the attack vector to dynamically allocated variables.\",\n \"Prime and Probe implies some reverse engineering to know which sets in the cache the attacker is using, while Flush and Reload does not.\",\n \"Prime and Probe is noisier than Flush and Reload.\",\n \"Taking these two attacks into account, we explore the cryptographic libraries looking for leakages exploitable by either attack.\",\n \"Thus, we examine both statically and dynamically allocated memory in our analysis.\"\n ],\n [\n \"Mutual Information Analysis\",\n \"The metric that we use to quantify leakage is Mutual Information (MI), that is given by the following equation: $I(X;Y) = H(X) - H(X|Y) = H(X) + H(Y) - H(X, Y)$ Where $H(X)$ is the entropy of random variable $X$ and $H(X|Y)$ is the entropy of the random variable $X$ given the knowledge of the random variable $Y$ .\",\n \"In a way $I(X; Y)$ gives us how related variables $X$ and $Y$ are.\",\n \"Note that if $X$ and $Y$ are independent, $I(X;Y)=0$ since $H(X|Y)=H(X)$ .\",\n \"In contrast, if $X$ and $Y$ are fully dependent we obtain maximum MI, i.e., $I(X;Y)=H(X)$ , since $H(X|Y)=0$ ($Y$ fully determines $X$ ).\",\n \"MI has been used in prior work for side channel attacks and leakage quantification.\",\n \"Gierlichs et al.\",\n \"utilize MI as a side channel distinguisher to mount differential side channel attacks.\",\n \"Standaert et al.\",\n \"utilized MI as a leakage quantifier to evaluate side channel attack security.\",\n \"Prouff et al.\",\n \"further expanded on the limitations and strengths of MI as a side channel attack metric.\",\n \"Recently Zhang and Lee used MI to measure the security level of several cache architectures which they modeled as finite state machines.\"\n ],\n [\n \"Methodology\",\n \"Our goal is to determine whether specific memory addresses being used inside a cryptographic routine reveal information based on their cache accesses.\"\n ],\n [\n \"Main Idea\",\n \"Our approach involves the monitorization of the cache usage for all the memory addresses considered susceptible of carrying information.\",\n \"In order to achieve this goal we utilize common known techniques from cache attacks, like the usage of cache flushing or cycle counter instructions.\",\n \"We illustrate our approach on a toy example shown in Figure REF .\",\n \"The Hello code snippet simply returns different messages depending on whether the caller is a female or a male.\",\n \"We assume that the designer of this simple code does not want a potential attacker to know the gender of the caller.\",\n \"Assume that the code designers would like to know whether they did a good job, i.e., whether the cache traces do not reveal the gender of the user.\",\n \"Here, it is easy to observe that there is a $gender$ dependent branch that could reveal whether the user is a male or female.\",\n \"We call $B1$ and $B2$ the two possible outputs of the branch.\",\n \"Figure: NO_CAPTION\"\n ]\n]"},"id":{"kind":"string","value":"1709.01552"}}},{"rowIdx":893,"cells":{"text":{"kind":"list like","value":[["An active-learning algorithm that combines sparse polynomial chaos\n expansions and bootstrap for structural reliability analysis"],["Abstract Polynomial chaos expansions (PCE) have seen widespread use in the context of uncertainty quantification.","However, their application to structural reliability problems has been hindered by the limited performance of PCE in the tails of the model response and due to the lack of local metamodel error estimates.","We propose a new method to provide local metamodel error estimates based on bootstrap resampling and sparse PCE.","An initial experimental design is iteratively updated based on the current estimation of the limit-state surface in an active learning algorithm.","The greedy algorithm uses the bootstrap-based local error estimates for the polynomial chaos predictor to identify the best candidate set of points to enrich the experimental design.","We demonstrate the effectiveness of this approach on a well-known analytical benchmark representing a series system, on a truss structure and on a complex realistic frame structure problem."],["Introduction","Structural reliability analysis aims at computing the probability of failure of a system with respect to some performance criterion in the presence of uncertainty in its structural and operating parameters.","Such uncertainty can be modelled by a random vector $\\mathbf {X}\\in {\\mathbb {R}}^M$ with prescribed joint probability density function $f_{\\mathbf {X}}$ .","The limit-state function $g$ is defined over the support of $\\mathbf {X}$ such that $\\left\\lbrace \\mathbf {x}: g(\\mathbf {x})\\le 0\\right\\rbrace $ defines the failure domain, while $\\left\\lbrace \\mathbf {x}: g(\\mathbf {x})>0\\right\\rbrace $ defines the safe domain.","The limit state surface implicitly defined by $g(\\mathbf {x}) = 0$ lies at the boundary between the two domains.","The probability of failure of such a system can be defined as [30], [25]: $P_F = \\int _{\\left\\lbrace \\mathbf {x}: g(\\mathbf {x})\\le 0\\right\\rbrace } f_{\\mathbf {X}}(\\mathbf {x}) d\\mathbf {x}.$ A straightforward approach to compute the integral in Eq.","(REF ) is to use of Monte Carlo Simulation (MCS).","However, standard MCS approaches can often not be used in the presence of complex and computationally expensive engineering models, because of the large number of samples they require to estimate small probabilities (typically in the order of $\\sim 10^{k+2}$ for $P_F \\approx 10^{-k}$ ) with acceptable accuracy.","Well-known methods based on local approximation of the limit-state function close to the failure domain (such as FORM [20] and SORM [35]) can be more efficient, yet they are usually based on linearisation and tend to fail in real-case scenarios with highly non-linear structural models.","In contrast, methods based on surrogate modelling have gradually gained momentum in the last few years.","Due to the nature of the problem of estimating low probabilities, most recent methods combine active-learning-based greedy algorithms with Gaussian process surrogate models (Kriging).","Among the first works to propose this approach, the earliest applications in this context were the efficient global reliability analysis method (EGRA) by [2], [3], and the active-learning reliability (AK-MCS) method based on Kriging by [14].","More recently, Kriging has been employed to devise quasi-optimal importance densities in [12], [10].","Amongst other variations, polynomial-chaos-based Kriging has also been used as an alternative metamodelling technique [37] to overcome some of the limitations of pure Kriging-based methods.","Additional works on the topic of Kriging and structural reliability can be found, including extensions of the original AK-MCS algorithm to more advanced sampling techniques [15], [1], system reliability [18] and for the exploration of multiple-failure regions [7].","Polynomial chaos expansions (PCE) [19] are a well-established tool in the context of uncertainty quantification, with applications in uncertainty propagation [43], sensitivity analysis [23] and, to a lesser degree, structural reliability [38].","While often considered as an efficient surrogate modelling technique due to their global convergence behaviour, PCEs have been employed only seldom in reliability analysis (see, e.g.","[32]) due to their lack of accuracy in the tails of the model response distribution, which are essential in this field.","In addition, most active-learning approaches with surrogates require some form of local error estimate to adaptively enrich a small set of model evaluations close to the limit state surface.","Kriging-based methods can rely on the Kriging variance for this task, but PCEs do not provide a natural equivalent.","In this paper, we leverage on the properties of regression-based sparse-PCE [6] to derive a local error estimator based on bootstrap resampling.","We then use this estimator to construct an active-learning strategy that adaptively approximates the limit-state function with PCE by minimizing a misclassification probability-based learning function at every iteration.","The method is then showcased on a standard benchmark functions representing a series system and on a realistic structural frame engineering example."],["Polynomial Chaos Expansions","Consider a finite variance model $Y = {\\mathcal {M}}(\\mathbf {X})$ representing the response of some quantity of interest (QoI) $Y$ to the random input parameters $\\mathbf {X}\\in {\\mathbb {R}}^M$ , modelled by a joint probability distribution function (PDF) $f_{\\mathbf {X}}$ .","Also consider the functional inner product defined by: $\\left \\equiv \\int _{\\mathbf {x}\\in \\Omega _{\\mathbf {X}}}g(\\mathbf {x})h(\\mathbf {x})f_{\\mathbf {X}}(\\mathbf {x})d\\mathbf {x}= {\\mathbb {E}}\\left[ g(\\mathbf {X}) h(\\mathbf {X}) \\right]$ where $\\Omega _{\\mathbf {X}}$ represents the input domain.","Under the assumption of independence of the input variables, that is $f_{\\mathbf {X}}(\\mathbf {x}) =\\prod \\limits _{i=1}^M f_{X_i}(x_i)$ , one can represent ${\\mathcal {M}}(\\mathbf {X})$ as the following generalised polynomial chaos expansion (see, e.g.","[19], [43]): $Y = {\\mathcal {M}}(\\mathbf {X}) = \\sum \\limits _{\\mathbf {\\alpha }\\in {\\mathbb {N}}^M}y_{\\mathbf {\\alpha }}\\Psi _{\\mathbf {\\alpha }}(\\mathbf {X}),$ where the $y_{\\mathbf {\\alpha }}$ are real coefficients and $\\mathbf {\\alpha }$ is a multi-index that identifies the degree of the multivariate polynomial $\\Psi _{\\mathbf {\\alpha }}$ in each of the input variables $X_i$ : $\\Psi _{\\mathbf {\\alpha }}= \\prod \\limits _{i = 1}^M \\phi ^{(i)}_{\\alpha _i}(X_i).$ Here $\\phi ^{(i)}_{\\alpha _i}$ is a polynomial of degree $\\alpha _i$ that belongs to the family of orthogonal polynomials w.r.t.","the marginal PDF $f_{X_i}$ .","For more details on the construction of such polynomials for both standard and arbitrary distributions, the reader is referred to [43].","In the presence of a complex dependence structure between the input variables, it is always possible to construct isoprobabilistic transforms (e.g.","Rosenblatt or Nataf transforms, see e.g.","[24]) to decorrelate the input variables prior to the expansion, even in the case of complex dependence modelled by vine copulas [40].","For the sake of notational simplicity and without loss of generality, we will hereafter assume independent input variables.","In practical applications, the series expansion in Eq.","(REF ) is traditionally truncated based on the maximal degree $p$ of the expansion, thus yielding a set of basis elements identified by the multi-indices ${\\mathbf {\\alpha }\\in {\\mathcal {A}}: \\sum \\limits _{i = 1}^M \\alpha _i\\le p}$ , with $\\text{card}({\\mathcal {A}}) \\equiv P = \\binom{M+p}{p}$ , or using more advanced truncation schemes that favour sparsity, e.g.","hyperbolic truncation [4].","The corresponding expansion coefficients $\\mathbf {y_{\\mathbf {\\alpha }}}$ can then be calculated efficiently via least-square analysis based on an existing sample of the input random vector ${\\mathcal {X}}= \\left\\lbrace \\mathbf {x}^{(1)},\\cdots ,\\mathbf {x}^{(N)}\\right\\rbrace $ , known as the experimental design (ED), and the corresponding model responses ${\\mathcal {Y}}=\\left\\lbrace y^{(1)},\\cdots ,y^{(N)}\\right\\rbrace $ as follows: $\\mathbf {y_{\\mathbf {\\alpha }}} = \\text{argmin}\\frac{1}{N} \\sum \\limits _{i = 1}^N\\left[y^{(i)} - \\sum \\limits _{\\mathbf {\\alpha }\\in {\\mathcal {A}}} y_{\\mathbf {\\alpha }}\\Psi _{\\mathbf {\\alpha }}(\\mathbf {x}^{(i)})\\right]^2.$ When the number of unknown coefficients $P$ is high (e.g.","for high-dimensional inputs or high-degree expansions), regression strategies that favour sparsity are needed to avoid over-fitting in the presence of a limited-size experimental design and to make the analysis at all feasible with a reasonable sample size $N$ .","Amongst them, least angle regression (LARS, [17]), based on a regularized version of Eq.","(REF ), has proven to be very effective in tackling realistic engineering problems even in relatively high dimensions (i.e.","$M \\sim 100$ ).","In this paper, we adopt the full degree-adaptive, sparse PCE based on hybrid-LARS introduced in [6]), as implemented in the UQLab Matlab software ([28], [29])."],["Bootstrap in least-square regression","Adopting a least-square regression strategy to calculate the coefficients in Eq.","(REF ) allows one to use the bootstrap resampling method [16] to obtain information on the variability in the estimated coefficients due to the finite size of the experimental design.","Suppose that a set of estimators $\\mathbf {\\theta }$ is a function of a finite-size sample ${\\mathcal {X}}= \\left\\lbrace \\mathbf {x}^{(1)},\\cdots ,\\mathbf {x}^{(N)}\\right\\rbrace $ drawn from the random vector $\\mathbf {X}$ .","Then the bootstrap method consists in drawing $B$ new sample sets $\\left\\lbrace {\\mathcal {X}}^{(1)},\\cdots ,{\\mathcal {X}}^{(B)}\\right\\rbrace $ from the original ${\\mathcal {X}}$ by resampling with substitution.","This is achieved by randomly assembling $B-$ times $N$ realizations $\\mathbf {x}^{(i)}\\in {\\mathcal {X}}$ , possibly including repeatedly the same realization multiple times within each sample.","The set of estimated quantities can then be re-calculated from each of the $B$ samples, thus yielding a set of estimators $\\mathbf {\\Theta } =\\left\\lbrace \\mathbf {\\theta }^{(1)},\\cdots ,\\theta ^{(B)}\\right\\rbrace $ .","This set of estimators can then be used to directly assess the variability of $\\mathbf {\\theta }$ due to the finite size of the experimental design ${\\mathcal {X}}$ , at no additional costs, e.g.","by calculating statistics, or directly using each realization separately.","Application of the bootstrap method combined with PCE to provide confidence bounds in the estimated $P_F$ in structural reliability applications can be found in e.g.","[32], [33]."],["Bootstrap-PCE","We propose to use the bootstrap technique to provide local error estimates to the PCE predictions.","The rationale is the following: the PCE coefficients $\\mathbf {y_{\\mathbf {\\alpha }}}$ in Eq.","(REF ) are estimated from the experimental design ${\\mathcal {X}}$ , therefore they can be resampled through bootstrap.","This can be achieved by first generating a set of bootstrap-resampled experimental designs $\\left\\lbrace {\\mathcal {X}}^{(b)},{\\mathcal {Y}}^{(b)}, b = 1,\\cdots ,B\\right\\rbrace $ .","For each of the generated designs, one can calculate a corresponding set of coefficients $\\mathbf {y_{\\mathbf {\\alpha }}}^{(b)}$ , effectively resulting in a set of $B$ different PCEs.","Correspondingly, the response of each PCE can be evaluated at a point $\\mathbf {x}$ as follows: $Y_{PC}^{(b)}(\\mathbf {x}) = \\sum \\limits _{\\mathbf {\\alpha }\\in {\\mathcal {A}}}y^{(b)}_{\\mathbf {\\alpha }}\\,\\Psi _{\\mathbf {\\alpha }}(\\mathbf {x}),$ thus yielding a full response sample at each point $\\left\\lbrace Y_{PC}^{(b)}(\\mathbf {x}), b =1,\\cdots ,B\\right\\rbrace $ .","Therefore, empirical quantiles can be employed to provide local error bounds on the PCE prediction at each point, as well as to any derived quantity (e.g.","$P_F$ or sensitivity indices, see e.g.","[33], [13]).","This bootstrap-resampling strategy in Eq.","(REF ) yields in fact a family of $B$ surrogate models that can be interpreted as trajectories.","Figure showcases how such trajectories can be directly employed to assess confidence bounds on point-wise predictions on a simple 1D test function given by: $f(x) = x\\sin (x),\\qquad x\\in [0, 2\\pi ],$ where the single random variable is assumed to be uniformly distributed within the bounds $X\\sim {\\mathcal {U}}(0,2\\pi ) $ , and where $B = 100$ bootstrap samples have been used.","Figure: Bootstrap-resampled trajectories (B=100B = 100) of the simple 1Danalytical function in Eq.","().The black line represents the true model, sampled at the 8 experimental designpoints 𝐱 (i) \\mathbf {x}^{(i)} (green dots).The PCE surrogate is represented by the blue line, while the bootstraptrajectories are given by the gray lines.","The corresponding 95% empiricalinter quantile-range is given by the dashed blue lines.This process of bootstrap-based trajectory resampling to provide better estimates of point-wise confidence bounds has been recently explored in the Gaussian process modelling literature, see e.g., [8], [41].","We refer to this approach as to bootstrap-PCE, or bPCE in short."],["Fast bPCE","Because the training of a PCE model with sparse least-square analysis may be time consuming, especially in high dimension and/or when an already large experimental design is available (i.e.","$N\\sim 10^3$ ), and because in this particular application we do not need very accurate estimates on the bounds of the derived quantities, we adopt a fast bPCE approach.","In this approach, the sparse polynomial basis identified by the LARS algorithm during calibration is calculated only once from the available full experimental design ${\\mathcal {X}}$ , and bootstrapping is applied only to the final hybrid step, which consists in a classic ordinary least-square regression on the sparse basis [6].","In the presence of a very expensive model, however (i.e.","requiring several hours for a single model run), we recommended to adopt full bootstrapping, including the estimation of the sparse PCE basis for each of the $B$ bootstrapped experimental designs ${\\mathcal {X}}^{(1,\\cdots ,B)}$ ."],["Active bPCE-based reliability analysis","In this section we present an adaptation of the Adaptive PC-Kriging MCS algorithm in [37] (based in turn on the original AK-MCS algorithm by [14]), that makes use of the bPCE just introduced.","Consistently with [14], [37], in the following we will refer to this algorithm as active bootstrap-polynomial-chaos Monte-Carlo simulation (A-bPCE).","We follow the original idea of adaptively building a surrogate of the limit-state function starting from a small initial experimental design and subsequently refining it to optimize the surrogate performance for structural reliability.","The ultimate goal of the adaptation is to retrieve an estimate of $P_F$ that is comparable to that of a direct Monte Carlo simulation (MCS) using a large sample set with a much smaller experimental design.","The algorithm is summarized as follows: Initialization: Generate an initial experimental design (e.g.","through Latin hypercube sampling or uniform sampling of a ball [9]) and calculate the corresponding bPCE surrogate (see Section REF ).","Generate a large reference MCS sample ${\\mathcal {X}}_{MCS} =\\left\\lbrace \\mathbf {x}_{MCS}^{(i)},~i = 1,\\cdots ,N_{MCS}\\right\\rbrace $ of size $N_{MCS}$ (e.g.","$N_{MCS} = 10^6\\gg N$ ).","A discussion on the choice of a suitable MCS sample is given in Section ).","Calculate a set of MCS estimators of the probability of failure: $\\left\\lbrace \\widehat{P}_F^{(b)},~b=1,\\cdots ,B\\right\\rbrace $ with the current bPCE surrogate.","Evaluate one or more suitable convergence criteria on $\\widehat{P}_F^{(1,\\cdots ,B)}$ (see Section REF ).","If they are met, go to Step REF (terminate the algorithm).","Otherwise continue to the next step.","Evaluate a suitable learning function on the MCS sample ${\\mathcal {X}}_{MCS}$ (see Section REF ).","Choose one or more additional $\\mathbf {x}_{MCS}^{(i)}\\in {\\mathcal {X}}_{MCS}$ and add them to the ED (see Section REF ).","Update the bPCE surrogate on the new ED and return to Step REF Algorithm termination: return the $\\widehat{P}_F$ resulting from the PCE on the current ED, as well as the error bounds derived e.g.","from the extremes or the empirical quantiles of the current $\\widehat{P}_F^{(1,\\cdots ,B)}$ set.","A detailed description of each step of the algorithm is given in the following sections."],["Initial experimental design","The initial experimental design is usually generated by space-filling sampling techniques of the random vector $\\mathbf {X}$ , such as Latin hypercube sampling (LHS) or pseudo-random sequences (e.g.","Sobol' sequence).","Alternative sampling techniques, such as the uniform sampling of a ball, have also proven effective in the context of structural reliability when low probabilities of failure are expected [9].","Note that this initial set of model evaluations does not need to be a subset of the reference sample ${\\mathcal {X}}_{MCS}$ used later to evaluate the $P_F$ estimates during the iterations of the algorithm."],["Inner MCS-based estimate of $P_F$","While the estimation of the $P_F$ via MCS is trivial, as it simply entails counting the number of samples that belong to the failure domain, some discussion about the number of samples $N_{MCS}$ in this step is needed.","Throughout this paper, we opted to choose a single MCS sample ${\\mathcal {X}}_{MCS}$ large enough to ensure a relatively small CoV for the $P_F$ estimate at every iteration.","This is by no means a requirement of this algorithm, but it simplifies significantly the notation (because ${\\mathcal {X}}_{MCS}$ becomes independent on the current iteration) and in some cases (as noted in both [14] and [37]) it can result in stabler convergence, due to the lowered MCS noise in the estimation of $P_F$ during each iteration.","This technique is known as common random numbers in the context of repeated reliability analysis .e.g.","in reliability-based design optimization [39].","It is entirely possible to redraw the ${\\mathcal {X}}_{MCS}$ during every iteration, possibly each time with a different number of samples $N_{MCS}$ .","The choice of $N_{MCS} = 10^6$ ensures that the CoV estimated probabilities of failure in the order of $P_F\\ge 10^{-3} $ is always smaller than $5\\%$ , which we found suitable in our application examples.","The choice of a single MCS sample drawn during the algorithm initialization also allows us to use the application examples to focus more on the convergence of the active learning part of A-bPCE, which is the focus of this paper.","In more general applications, the order of magnitude of $P_F$ may be unknown.","In this case, it is recommended instead to set a target desired CoV for the estimation of $P_F$ at each iteration (as proposed in the original AK-MCS algorithm in [14]), and gradually add samples to ${\\mathcal {X}}_{MCS}$ until it is reached."],["Convergence criteria","The proposed convergence criterion of choice is directly inspired by [37], [32] and it depends on the stability of the $P_F$ estimate at the current iteration.","Let us define: $\\begin{split}\\widehat{P}_F^+ &= \\max \\limits _{b = 1,\\cdots ,B}\\left(\\widehat{P}_F^{(b)}\\right)\\\\\\widehat{P}_F^- &= \\min \\limits _{b = 1,\\cdots ,B}\\left(\\widehat{P}_F^{(b)}\\right).\\end{split}$ Convergence is achieved when the following condition is satisfied for at least two consecutive iterations of the algorithm: $\\frac{\\widehat{P}_F^+ - \\widehat{P}_F^-}{\\widehat{P}_F} \\le \\epsilon _{\\widehat{P}_F},$ with $0.05\\le \\epsilon _{\\widehat{P}_F}\\le 0.15$ in typical usage scenarios."],["Learning function","A learning function is a function that allows one to rank a set of candidate points based on some utility criterion that depends on the desired application.","In this case, we adopt the same heuristic approach proposed in [37], by focusing on the probability of misclassification of the bPCE model on the candidate set given by ${\\mathcal {X}}_{MCS}$ .","Due to the availability of the bootstrap response samples ${\\mathcal {Y}}_{MCS}^{(1,\\cdots ,B)}$ , it is straightforward to define a measure of the misclassification probability $U_{FBR}$ (where the subscript FBR stands for failed bootstrap replicates) at each point $\\mathbf {x}_{MCS}^{(i)}\\in {\\mathcal {X}}_{MCS}$ as follows: $U_{FBR}(\\mathbf {x}_{MCS}^{(i)}) =\\left|\\frac{B_S(\\mathbf {x}_{MCS}^{(i)})-B_F(\\mathbf {x}_{MCS}^{(i)})}{B}\\right|$ where $B_S(\\mathbf {x}_{MCS}^{(i)})$ and $B_F(\\mathbf {x}_{MCS}^{(i)})$ are the number of safe (resp.","failed) bPCE replicate predictions at point $\\mathbf {x}_{MCS}^{(i)}$ (with $B_S(\\mathbf {x}_{MCS}^{(i)}) + B_F(\\mathbf {x}_{MCS}^{(i)}) = B$ ).","When all the $B$ replicates consistently classify $\\mathbf {x}_{MCS}^{(i)}$ in the safe or in the failure domain, $U_{FBR} = 1$ (minimum misclassification probability).","In contrast, $U_{FBR} = 0$ corresponds to the case when the replicates are equally distributed between the two domains.","In the latter case, 50% of the $B$ bootstrap PCEs predict that $\\mathbf {x}_{MCS}^{(i)}$ is in the safe domain, while the other 50% predicts that $\\mathbf {x}_{MCS}^{(i)}$ belongs to the failure domain.","Therefore, maximum epistemic uncertainty on the classification of a point $\\mathbf {x}^{(i)}_{MCS}$ is attained when $U_{FBR}$ is minimum."],["Enrichment of the experimental design","The aim of the iterative algorithm described in Section REF is to obtain a surrogate model that minimizes the misclassification probability.","As a consequence, the learning function in Eq.","(REF ) can be directly used to obtain a single-point enrichment criterion.","The next best candidate point for the ED $\\mathbf {x}^*\\in {\\mathcal {X}}_{MCS}$ is given by: $\\mathbf {x}^* = \\operatornamewithlimits{argmin}\\limits _{\\mathbf {x}^{(i)} \\in {\\mathcal {X}}_{MCS}}\\left(U_{FBR}(\\mathbf {x}^{(i)})\\right).$ Due to the global character of regression-based PCE, it can be beneficial to add multiple points in each iteration to sample several interesting regions of the parameter space simultaneously.","The criterion in Eq.","(REF ) can be extended to include $K$ distinct points simultaneously by following the approach in [37].","A limit state margin region is first defined as the set of points such that $U_{FBR}<1$ (i.e.","those point with non-zero misclassification probability at the current iteration).","Subsequently, $k$ -means clustering techniques (see, e.g., [44]) can be used at each iteration to identify $K$ disjoint regions $\\left\\lbrace {\\mathcal {X}}^{(1,\\cdots ,K)}_{MCS}\\right\\rbrace $ in the limit-state margin.","Then, Eq.","(REF ) can be directly applied to each of the subregions to obtain $K$ different enrichment points: $\\mathbf {x}^{*}_k = \\operatornamewithlimits{argmin}\\limits _{\\mathbf {x}^{(i)} \\in {\\mathcal {X}}_{MCS}^{(k)}}\\left(U^{(k)}_{FBR}(\\mathbf {x}_k^{(i)})\\right), \\quad k =1,\\cdots ,K$ where $\\mathbf {x}^{*}_k\\in {\\mathcal {X}}^{(k)}_{MCS}$ is the $k-$ th enrichment sample and $U^{(k)}_{FBR}$ is the learning function evaluated on the $k-$ th region of the parameter space.","Note that this approach is also convenient when parallel computing facilities are available and in the presence of computationally expensive objective functions, as the evaluation of the $K$ enrichment points can be carried out simultaneously."],["Results on benchmark applications","All the algorithm development and the final calculations presented in this section were performed with the polynomial chaos expansions and reliability analysis modules of the UQLab software for uncertainty quantification [28], [29], [27]."],["Series system","A common benchmark for reliability analysis functions is given by the four-branch function, originally proposed in [42], that represents a series system comprising four components with different failure criteria.","Although it is a simple analytical function, it shows multiple failure regions and a composite limit-state surface.","Its two-dimensional limit state function reads: $g(\\mathbf {x}) = \\min \\left\\lbrace \\begin{matrix}3+0.1(x_1+x_2)^2 - \\frac{x_1 + x_2}{\\sqrt{2}}\\\\3+0.1(x_1+x_2)^2 + \\frac{x_1 + x_2}{\\sqrt{2}}\\\\(x_1-x_2) + \\frac{6}{\\sqrt{2}}\\\\(x_2-x_1) + \\frac{6}{\\sqrt{2}}\\\\\\end{matrix} \\right\\rbrace $ where the two random input variables $X_1\\sim {\\mathcal {N}}(0,1)$ and $X_2\\sim {\\mathcal {N}}(0,1)$ are modelled as independent standard normals.","Failure occurs when $g(\\mathbf {x})\\le 0$ .","Due to the multi-failure shape of the limit-state surface (represented as a solid black line in Figure REF ), classic methods like FORM/SORM and importance sampling tend to fail with this benchmark problem.","The reference failure probability of $P_F = 4.460\\cdot 10^{-3}$ is obtained through an extremely large MCS ($N_{MCS} = 10^8$ ).","Figure: Four branch function: limit state surface (black line) andexperimental design before (black circles) and after (red crosses)enrichmentThe initial experimental design for the A-bPCE algorithm was obtained with a space-filling LHS sample consisting of $N_{ini} = 20$ points drawn from the input distributions (black dots in Figure REF ).","Three points at a time were added to the experimental design during the enrichment phase of the algorithm.","The number of replications for the A-bPCE algorithm is set to $B = 100$ .","After extensive testing, the algorithm was found to be very weakly dependent on the number of bootstrap replications, provided a minimum of $B \\ge 20$ was provided.","Indeed, the boostrap samples are used to identify areas of relatively large prediction variability, but an accurate estimate of such variability is never really needed by the algorithm.","Degree adaptive sparse PCE (with maximum degree in the range $p\\in [2, 10]$ ) based on LARS [6] was used to calibrate the PCE metamodel at each iteration.","For validation and comparison purposes, a similar analysis was performed on the same initial ED with the AK-MCS module of UQLab, with an anisotropic Matérn 5/2 ellipsoidal multivariate Kriging correlation function [36], [27], [22].","The convergence criterion in Eq.","(REF ) was set to $\\epsilon _{\\widehat{P}_F} = 0.05$ for both the AK-MCS and A-bPCE algorithms.","Convergence was achieved after 49 iterations, resulting in a total cost (including the initial experimental design) of $N_{tot} = 185$ model evaluations.","The experimental design points added during the iterations are marked by red crosses on panel (b) of Figure REF .","As expected, the adaptive algorithm tends to enrich the experimental design close to the limit state surface as it is adaptively learned during the iterations.","A graphical representation of the convergence of the algorithm is shown in Figure REF , where the estimated $\\widehat{P}_F$ is plotted against the total number of model evaluations $N_{tot}$ .","The shaded area represents the $95\\%$ confidence bounds based on the empirical quantiles as estimated from the bootstrap sample.","Figure: Convergence curves of the four-branchlimit-state function.","The reference P F P_F and β\\beta are given by dotted lines.The final results of the analysis are summarized in Table , where the generalised reliability index $\\beta = -\\Phi ^{-1}(P_F)$ is also given for reference.","For comparison, the reference MCS probability as well as an estimate from AK-MCS are also given.","The latter converged to a comparably accurate estimate of $P_F$ , at the cost of a slightly higher number of model evaluations.","Note that for engineering purposes, the algorithm could have been stopped earlier, i.e.","when a $5\\%$ accuracy on the generalized reliability index is attained.","In this case, the algorithm would have converged to a comparable result ($\\widehat{\\beta } = 2.56$ ) with only 50 runs of the model.","The final sparse PCE model after enrichment contained a total of $P = 12$ basis elements of degree up to $p = 5$ .","Table: Comparison of different reliability analysis methods for the fourbranch function"],["Two-dimensional truss structure","To test the algorithm on a more realistic engineering benchmark, consider the two-dimensional truss structure sketched in Figure REF .","This structure has been previously analysed in several works, see e.g.","[6], [5], [37].","The truss comprises 23 bars and 13 nodes, with deterministic geometry yet with uncertain material properties and random loads.","The components of the input random vector $\\mathbf {X}=\\left[A_{1,2},E_{1,2},P_{1,...,6}\\right]^{\\textsf {T}}$ include the cross-section and the Young's modulus $\\left\\lbrace A_1, E_1\\right\\rbrace $ of the horizontal bars, the cross-section and the Young's modulus $\\left\\lbrace A_2, E_2\\right\\rbrace $ of the diagonal bars and the six random loads $\\left\\lbrace P_1,\\cdots ,P_6\\right\\rbrace $ .","They are considered mutually independent and their distributions are given in Table REF .","An in-house developed Matlab-based finite-element solver is used to calculate the displacement at midspan $u(\\mathbf {X})$ , counted positively downwards.","Figure: Two-dimensional truss structure with uncertain parameters.Probabilitydistributions of the geometrical parameters A i ,E i \\left\\lbrace A_i,E_i\\right\\rbrace and of theloadsP 1 ,⋯,P 6 \\left\\lbrace P_1,\\cdots ,P_6\\right\\rbrace are given in Table Table: Two-dimensional truss structure: definition of the probabilisticmodel of the input variables This structure operates in the nominal range as long as the midspan displacement is smaller than a critical threshold $\\tau = 12$ cm, which can be cast as the following limit-state function: $g(\\mathbf {x}) = \\tau - u(\\mathbf {x})$ where $g(\\mathbf {x}) \\le 0$ if the system is in a failure state.","Because the FEM computational model is relatively cheap to evaluate, we could run a direct MCS-analysis with $N = 10^6$ samples to provide the reference $P_F= 1.52\\cdot 10^{-3}$ for validation purposes.","Additionally, standard FORM and SORM analyses were run to estimate the non-linearity of the limit-state surface.","FORM underestimated the failure probability of a factor of almost 2 and a cost of $N_{FORM} = 160$ model runs, while SORM achieved a good accuracy at a cost of $N_{SORM} = 372$ model runs, which suggests that the underlying problem is non-linear.","Neither of the two methods, however, provides confidence interval on their estimates.","The A-bPCE algorithm was initialized with an experimental design consisting in a uniform sampling of a ball (for details, see e.g.","[9]) of size $N_{ini} = 30$ , while the sparse adaptive PCE was given a polynomial degree range $1 \\le p \\le 10$ , hyperbolic truncation with q-norm $q = 0.75$ [6] and maximum allowed interaction $r = 2$ [29].","The internal MCS sample size was $N_{MCS} = 10^6$ and the algorithm was set to add $K = 3$ new samples per iteration.","The stopping criterion in Eq.","(REF ) was set to $\\epsilon _{\\widehat{P}_F} = 0.10$ .","For comparison purposes, we also ran a standard AK-MCS analysis with the same initial experimental design and convergence criterion.","The covariance family of choice for the underlying Kriging model was chosen as Gaussian.","Table: Comparison of the estimation of P F P_F with several algorithms forthe truss structure example.Table REF presents a comparison of the estimated $\\widehat{P}_F$ with the aforementioned analyses.","Both AK-MCS and A-bPCE estimates of $P_F$ include the reference value within the confidence bounds set by the convergence criterion.","However, for this particular example and choice of convergence criterion, A-bPCE achieved convergence significantly faster than AK-MCS, with a total cost of 129 model evaluations, as compared to the 300 required by AK-MCS, resulting in a final PCE of degree $p=3$ with $P =43$ basis elements.","Overall, A-bPCE provides a stable estimate of the failure probability and confidence intervals at a cost that is lower than FORM for this example."],["Top-floor displacement of a structural frame","Figure REF shows a well known, high dimensional benchmark in structural reliability applications [26], [4].","It consists on a three-span, five story frame structure that is subject to horizontal loads.","Both the loads and the properties of the elements of the frame (see Table REF ) are uncertain.","Of interest is the top-floor horizontal displacement at the top right corner $u$ .","Figure: 20-dimensional structural frame inSection .The distributions of the input variables are reported inTable Table: Frame structure: properties of the elements shown inFigure The uncertainties on the applied loads $P_1,...,P_3$ , the Young's moduli $E_1$ and $E_2$ , the moments of inertia $I_6,...,I_{13}$ and the cross sections $A_{14},...,A_{21}$ are modelled by a 21-dimensional joint random vector $\\mathbf {Z} = \\left\\lbrace P_1,...,A_{21}\\right\\rbrace $ with marginal distributions given in Table REF .","Table: Frame structure: definition of the probabilisticmodel of the input variables Additionally, a Gaussian copula [24] is used to model dependence between the variables.","The elements of the Gaussian copula correlation matrix $\\mathbf {R}$ are given as: $R_{E_1,E_2} = 0.9$ – the two Young's moduli are highly correlated; $R_{A_i,I_i} = 0.95$ – each element's cross-sectional area is highly correlated to the corresponding moment of inertia; ${R_{A_i,I_j} = R_{I_i,I_j} = R_{A_i,A_j} = 0.13}$ – the correlation between the properties of different elements is much lower; All the remaining elements of $\\mathbf {R}$ are set to 0.","A critical displacement of $\\tau = 5$ cm is identified as the maximum admissible threshold for the displacement $u$ , hence resulting in the limit-state function: $g(\\mathbf {z}) = \\tau - u(\\mathbf {z})$ where $u(\\mathbf {z})$ is the displacement on the top right corner calculated with an in-house FEM code.","Due to the associated computational costs, the maximum available budget for the calculation of a reference solution is in this case limited to $N_{REF} = 4\\cdot 10^4$ .","Therefore, the reference solution is calculated with standard importance sampling (IS) [30] instead of direct MCS.","In addition to Importance sampling, we also ran FORM and SORM.","Due to the non-linearity of the problem, FORM significantly underestimated $P_F$ , while SORM provided an accurate estimate.","However, due to the high dimensionality of the input space, the associated cost in terms of model evaluation was relatively high, with $N_{SORM} = 1146$ model runs, since all the gradients of the limit-state function are computed using finite-differences.","The A-bPCE algorithm was initialized with an experimental design consisting of an LHS sampling of the input random vector of size $N_{ini} = 40$ .","Sparse PCE was carried out with a q-norm truncation with $q = 0.75$ and maximum allowed interaction $r = 2$ .","Note that the initialization is essentially the same as for the truss structure in the previous application.","The internal MCS sample size was $N_{MCS} = 10^6$ , with single point enrichment per iteration.","The stopping criterion in Eq.","(REF ) was in this case set to $\\epsilon _{\\widehat{P}_F} = 0.15 $ .","For comparison purposes, an AK-MCS analysis was also run on the same initial design, with similar settings and a Gaussian covariance family.","A comparison of the results is gathered in Table REF .","Due to the different estimation method between the reference probability (importance sampling) and the active learning-based methods (which rely on an inner MCS), no direct comparison of the results is possible as in the previous cases.","Indeed, even fixing the same random seeds would result in different estimates due to the different methodologies.","Therefore, confidence bounds are given for all the three methods: 95% confidence bounds for IS [30], and $P_F^{\\pm }$ for both AK-MCS and A-bPCE.","The three methods give comparable results, albeit with significant differences in the convergence behaviour.","In particular, both AK- and A-bPCE resulted in a slight underestimation of the probability of failure w.r.t.","the reference solution by IS, which in turn is slightly overestimated with respect to the reference result quoted in the literature [4].","However, AK-MCS did not converge in the allotted maximum number of model evaluations, and its confidence bounds remained remarkably large with respect to A-bPCE.","A-bPCE converged instead at a total cost of approximately 200 model evaluations to the target $\\epsilon _{\\widehat{P}_F}$ , with a final sparse PCE of degree 2, counting 30 non-zero coefficients.","For both active-learning-based methods, the reference solution lies within the given confidence bounds.","Moreover, the confidence bounds on the reliability index $\\widehat{\\beta }$ show that the results are stable to within $2\\%$ of the calculated values.","Finally, it is interesting to mention that for this example the costs of FORM and A-bPCE were comparable, but the latter provides a much less biased estimate, and includes confidence bounds.","Table: Comparison of the estimation of P F P_F with several algorithms forthe top-floor displacement of a structural frame example"],["Conclusions and outlook","A novel approach to solving reliability problems with polynomial chaos expansions has been proposed.","The combination of the bootstrap method and sparse regression enabled us to introduce local error estimation in the standard PCE predictor.","In turn, this allows one to construct active learning algorithms similar to AK-MCS to greedily enrich a relatively small initial experimental design so as to efficiently estimate the probability of failure of complex systems.","This approach has shown comparable performance w.r.t.","to the well established AK-MCS method on both a simple analytical benchmark function and in two high-dimensional engineering applications of increasing complexity.","Extensions of this approach can be envisioned in two main directions: the simulation-based reliability analysis method can be extended beyond simple MCS (e.g.","by using importance sampling [12], line sampling [34] or subset simulation [11]) to achieve better $\\widehat{P}_F$ estimates at each iteration, especially for very low probabilities of failure; remote parallel computing facilities may be used during the enrichment phase of the algorithms with expensive computational models when adding more than one point at a time; the use of bootstrap to enable local error estimation in an active learning context can be used also with different regression-based surrogate modelling techniques, including e.g.","low rank tensor approximations [21].","Additionally, the bPCE approach itself introduced in this work can be used also outside of a pure reliability analysis context, as it provides an effective local error estimate for PCE.","It has been used, e.g.","in the context of reliability-based design optimization in [31].","Indeed the lack of this feature (as opposed to Kriging) has somewhat hindered its usage in more advanced active-learning applications."]],"string":"[\n [\n \"An active-learning algorithm that combines sparse polynomial chaos\\n expansions and bootstrap for structural reliability analysis\"\n ],\n [\n \"Abstract Polynomial chaos expansions (PCE) have seen widespread use in the context of uncertainty quantification.\",\n \"However, their application to structural reliability problems has been hindered by the limited performance of PCE in the tails of the model response and due to the lack of local metamodel error estimates.\",\n \"We propose a new method to provide local metamodel error estimates based on bootstrap resampling and sparse PCE.\",\n \"An initial experimental design is iteratively updated based on the current estimation of the limit-state surface in an active learning algorithm.\",\n \"The greedy algorithm uses the bootstrap-based local error estimates for the polynomial chaos predictor to identify the best candidate set of points to enrich the experimental design.\",\n \"We demonstrate the effectiveness of this approach on a well-known analytical benchmark representing a series system, on a truss structure and on a complex realistic frame structure problem.\"\n ],\n [\n \"Introduction\",\n \"Structural reliability analysis aims at computing the probability of failure of a system with respect to some performance criterion in the presence of uncertainty in its structural and operating parameters.\",\n \"Such uncertainty can be modelled by a random vector $\\\\mathbf {X}\\\\in {\\\\mathbb {R}}^M$ with prescribed joint probability density function $f_{\\\\mathbf {X}}$ .\",\n \"The limit-state function $g$ is defined over the support of $\\\\mathbf {X}$ such that $\\\\left\\\\lbrace \\\\mathbf {x}: g(\\\\mathbf {x})\\\\le 0\\\\right\\\\rbrace $ defines the failure domain, while $\\\\left\\\\lbrace \\\\mathbf {x}: g(\\\\mathbf {x})>0\\\\right\\\\rbrace $ defines the safe domain.\",\n \"The limit state surface implicitly defined by $g(\\\\mathbf {x}) = 0$ lies at the boundary between the two domains.\",\n \"The probability of failure of such a system can be defined as [30], [25]: $P_F = \\\\int _{\\\\left\\\\lbrace \\\\mathbf {x}: g(\\\\mathbf {x})\\\\le 0\\\\right\\\\rbrace } f_{\\\\mathbf {X}}(\\\\mathbf {x}) d\\\\mathbf {x}.$ A straightforward approach to compute the integral in Eq.\",\n \"(REF ) is to use of Monte Carlo Simulation (MCS).\",\n \"However, standard MCS approaches can often not be used in the presence of complex and computationally expensive engineering models, because of the large number of samples they require to estimate small probabilities (typically in the order of $\\\\sim 10^{k+2}$ for $P_F \\\\approx 10^{-k}$ ) with acceptable accuracy.\",\n \"Well-known methods based on local approximation of the limit-state function close to the failure domain (such as FORM [20] and SORM [35]) can be more efficient, yet they are usually based on linearisation and tend to fail in real-case scenarios with highly non-linear structural models.\",\n \"In contrast, methods based on surrogate modelling have gradually gained momentum in the last few years.\",\n \"Due to the nature of the problem of estimating low probabilities, most recent methods combine active-learning-based greedy algorithms with Gaussian process surrogate models (Kriging).\",\n \"Among the first works to propose this approach, the earliest applications in this context were the efficient global reliability analysis method (EGRA) by [2], [3], and the active-learning reliability (AK-MCS) method based on Kriging by [14].\",\n \"More recently, Kriging has been employed to devise quasi-optimal importance densities in [12], [10].\",\n \"Amongst other variations, polynomial-chaos-based Kriging has also been used as an alternative metamodelling technique [37] to overcome some of the limitations of pure Kriging-based methods.\",\n \"Additional works on the topic of Kriging and structural reliability can be found, including extensions of the original AK-MCS algorithm to more advanced sampling techniques [15], [1], system reliability [18] and for the exploration of multiple-failure regions [7].\",\n \"Polynomial chaos expansions (PCE) [19] are a well-established tool in the context of uncertainty quantification, with applications in uncertainty propagation [43], sensitivity analysis [23] and, to a lesser degree, structural reliability [38].\",\n \"While often considered as an efficient surrogate modelling technique due to their global convergence behaviour, PCEs have been employed only seldom in reliability analysis (see, e.g.\",\n \"[32]) due to their lack of accuracy in the tails of the model response distribution, which are essential in this field.\",\n \"In addition, most active-learning approaches with surrogates require some form of local error estimate to adaptively enrich a small set of model evaluations close to the limit state surface.\",\n \"Kriging-based methods can rely on the Kriging variance for this task, but PCEs do not provide a natural equivalent.\",\n \"In this paper, we leverage on the properties of regression-based sparse-PCE [6] to derive a local error estimator based on bootstrap resampling.\",\n \"We then use this estimator to construct an active-learning strategy that adaptively approximates the limit-state function with PCE by minimizing a misclassification probability-based learning function at every iteration.\",\n \"The method is then showcased on a standard benchmark functions representing a series system and on a realistic structural frame engineering example.\"\n ],\n [\n \"Polynomial Chaos Expansions\",\n \"Consider a finite variance model $Y = {\\\\mathcal {M}}(\\\\mathbf {X})$ representing the response of some quantity of interest (QoI) $Y$ to the random input parameters $\\\\mathbf {X}\\\\in {\\\\mathbb {R}}^M$ , modelled by a joint probability distribution function (PDF) $f_{\\\\mathbf {X}}$ .\",\n \"Also consider the functional inner product defined by: $\\\\left \\\\equiv \\\\int _{\\\\mathbf {x}\\\\in \\\\Omega _{\\\\mathbf {X}}}g(\\\\mathbf {x})h(\\\\mathbf {x})f_{\\\\mathbf {X}}(\\\\mathbf {x})d\\\\mathbf {x}= {\\\\mathbb {E}}\\\\left[ g(\\\\mathbf {X}) h(\\\\mathbf {X}) \\\\right]$ where $\\\\Omega _{\\\\mathbf {X}}$ represents the input domain.\",\n \"Under the assumption of independence of the input variables, that is $f_{\\\\mathbf {X}}(\\\\mathbf {x}) =\\\\prod \\\\limits _{i=1}^M f_{X_i}(x_i)$ , one can represent ${\\\\mathcal {M}}(\\\\mathbf {X})$ as the following generalised polynomial chaos expansion (see, e.g.\",\n \"[19], [43]): $Y = {\\\\mathcal {M}}(\\\\mathbf {X}) = \\\\sum \\\\limits _{\\\\mathbf {\\\\alpha }\\\\in {\\\\mathbb {N}}^M}y_{\\\\mathbf {\\\\alpha }}\\\\Psi _{\\\\mathbf {\\\\alpha }}(\\\\mathbf {X}),$ where the $y_{\\\\mathbf {\\\\alpha }}$ are real coefficients and $\\\\mathbf {\\\\alpha }$ is a multi-index that identifies the degree of the multivariate polynomial $\\\\Psi _{\\\\mathbf {\\\\alpha }}$ in each of the input variables $X_i$ : $\\\\Psi _{\\\\mathbf {\\\\alpha }}= \\\\prod \\\\limits _{i = 1}^M \\\\phi ^{(i)}_{\\\\alpha _i}(X_i).$ Here $\\\\phi ^{(i)}_{\\\\alpha _i}$ is a polynomial of degree $\\\\alpha _i$ that belongs to the family of orthogonal polynomials w.r.t.\",\n \"the marginal PDF $f_{X_i}$ .\",\n \"For more details on the construction of such polynomials for both standard and arbitrary distributions, the reader is referred to [43].\",\n \"In the presence of a complex dependence structure between the input variables, it is always possible to construct isoprobabilistic transforms (e.g.\",\n \"Rosenblatt or Nataf transforms, see e.g.\",\n \"[24]) to decorrelate the input variables prior to the expansion, even in the case of complex dependence modelled by vine copulas [40].\",\n \"For the sake of notational simplicity and without loss of generality, we will hereafter assume independent input variables.\",\n \"In practical applications, the series expansion in Eq.\",\n \"(REF ) is traditionally truncated based on the maximal degree $p$ of the expansion, thus yielding a set of basis elements identified by the multi-indices ${\\\\mathbf {\\\\alpha }\\\\in {\\\\mathcal {A}}: \\\\sum \\\\limits _{i = 1}^M \\\\alpha _i\\\\le p}$ , with $\\\\text{card}({\\\\mathcal {A}}) \\\\equiv P = \\\\binom{M+p}{p}$ , or using more advanced truncation schemes that favour sparsity, e.g.\",\n \"hyperbolic truncation [4].\",\n \"The corresponding expansion coefficients $\\\\mathbf {y_{\\\\mathbf {\\\\alpha }}}$ can then be calculated efficiently via least-square analysis based on an existing sample of the input random vector ${\\\\mathcal {X}}= \\\\left\\\\lbrace \\\\mathbf {x}^{(1)},\\\\cdots ,\\\\mathbf {x}^{(N)}\\\\right\\\\rbrace $ , known as the experimental design (ED), and the corresponding model responses ${\\\\mathcal {Y}}=\\\\left\\\\lbrace y^{(1)},\\\\cdots ,y^{(N)}\\\\right\\\\rbrace $ as follows: $\\\\mathbf {y_{\\\\mathbf {\\\\alpha }}} = \\\\text{argmin}\\\\frac{1}{N} \\\\sum \\\\limits _{i = 1}^N\\\\left[y^{(i)} - \\\\sum \\\\limits _{\\\\mathbf {\\\\alpha }\\\\in {\\\\mathcal {A}}} y_{\\\\mathbf {\\\\alpha }}\\\\Psi _{\\\\mathbf {\\\\alpha }}(\\\\mathbf {x}^{(i)})\\\\right]^2.$ When the number of unknown coefficients $P$ is high (e.g.\",\n \"for high-dimensional inputs or high-degree expansions), regression strategies that favour sparsity are needed to avoid over-fitting in the presence of a limited-size experimental design and to make the analysis at all feasible with a reasonable sample size $N$ .\",\n \"Amongst them, least angle regression (LARS, [17]), based on a regularized version of Eq.\",\n \"(REF ), has proven to be very effective in tackling realistic engineering problems even in relatively high dimensions (i.e.\",\n \"$M \\\\sim 100$ ).\",\n \"In this paper, we adopt the full degree-adaptive, sparse PCE based on hybrid-LARS introduced in [6]), as implemented in the UQLab Matlab software ([28], [29]).\"\n ],\n [\n \"Bootstrap in least-square regression\",\n \"Adopting a least-square regression strategy to calculate the coefficients in Eq.\",\n \"(REF ) allows one to use the bootstrap resampling method [16] to obtain information on the variability in the estimated coefficients due to the finite size of the experimental design.\",\n \"Suppose that a set of estimators $\\\\mathbf {\\\\theta }$ is a function of a finite-size sample ${\\\\mathcal {X}}= \\\\left\\\\lbrace \\\\mathbf {x}^{(1)},\\\\cdots ,\\\\mathbf {x}^{(N)}\\\\right\\\\rbrace $ drawn from the random vector $\\\\mathbf {X}$ .\",\n \"Then the bootstrap method consists in drawing $B$ new sample sets $\\\\left\\\\lbrace {\\\\mathcal {X}}^{(1)},\\\\cdots ,{\\\\mathcal {X}}^{(B)}\\\\right\\\\rbrace $ from the original ${\\\\mathcal {X}}$ by resampling with substitution.\",\n \"This is achieved by randomly assembling $B-$ times $N$ realizations $\\\\mathbf {x}^{(i)}\\\\in {\\\\mathcal {X}}$ , possibly including repeatedly the same realization multiple times within each sample.\",\n \"The set of estimated quantities can then be re-calculated from each of the $B$ samples, thus yielding a set of estimators $\\\\mathbf {\\\\Theta } =\\\\left\\\\lbrace \\\\mathbf {\\\\theta }^{(1)},\\\\cdots ,\\\\theta ^{(B)}\\\\right\\\\rbrace $ .\",\n \"This set of estimators can then be used to directly assess the variability of $\\\\mathbf {\\\\theta }$ due to the finite size of the experimental design ${\\\\mathcal {X}}$ , at no additional costs, e.g.\",\n \"by calculating statistics, or directly using each realization separately.\",\n \"Application of the bootstrap method combined with PCE to provide confidence bounds in the estimated $P_F$ in structural reliability applications can be found in e.g.\",\n \"[32], [33].\"\n ],\n [\n \"Bootstrap-PCE\",\n \"We propose to use the bootstrap technique to provide local error estimates to the PCE predictions.\",\n \"The rationale is the following: the PCE coefficients $\\\\mathbf {y_{\\\\mathbf {\\\\alpha }}}$ in Eq.\",\n \"(REF ) are estimated from the experimental design ${\\\\mathcal {X}}$ , therefore they can be resampled through bootstrap.\",\n \"This can be achieved by first generating a set of bootstrap-resampled experimental designs $\\\\left\\\\lbrace {\\\\mathcal {X}}^{(b)},{\\\\mathcal {Y}}^{(b)}, b = 1,\\\\cdots ,B\\\\right\\\\rbrace $ .\",\n \"For each of the generated designs, one can calculate a corresponding set of coefficients $\\\\mathbf {y_{\\\\mathbf {\\\\alpha }}}^{(b)}$ , effectively resulting in a set of $B$ different PCEs.\",\n \"Correspondingly, the response of each PCE can be evaluated at a point $\\\\mathbf {x}$ as follows: $Y_{PC}^{(b)}(\\\\mathbf {x}) = \\\\sum \\\\limits _{\\\\mathbf {\\\\alpha }\\\\in {\\\\mathcal {A}}}y^{(b)}_{\\\\mathbf {\\\\alpha }}\\\\,\\\\Psi _{\\\\mathbf {\\\\alpha }}(\\\\mathbf {x}),$ thus yielding a full response sample at each point $\\\\left\\\\lbrace Y_{PC}^{(b)}(\\\\mathbf {x}), b =1,\\\\cdots ,B\\\\right\\\\rbrace $ .\",\n \"Therefore, empirical quantiles can be employed to provide local error bounds on the PCE prediction at each point, as well as to any derived quantity (e.g.\",\n \"$P_F$ or sensitivity indices, see e.g.\",\n \"[33], [13]).\",\n \"This bootstrap-resampling strategy in Eq.\",\n \"(REF ) yields in fact a family of $B$ surrogate models that can be interpreted as trajectories.\",\n \"Figure showcases how such trajectories can be directly employed to assess confidence bounds on point-wise predictions on a simple 1D test function given by: $f(x) = x\\\\sin (x),\\\\qquad x\\\\in [0, 2\\\\pi ],$ where the single random variable is assumed to be uniformly distributed within the bounds $X\\\\sim {\\\\mathcal {U}}(0,2\\\\pi ) $ , and where $B = 100$ bootstrap samples have been used.\",\n \"Figure: Bootstrap-resampled trajectories (B=100B = 100) of the simple 1Danalytical function in Eq.\",\n \"().The black line represents the true model, sampled at the 8 experimental designpoints 𝐱 (i) \\\\mathbf {x}^{(i)} (green dots).The PCE surrogate is represented by the blue line, while the bootstraptrajectories are given by the gray lines.\",\n \"The corresponding 95% empiricalinter quantile-range is given by the dashed blue lines.This process of bootstrap-based trajectory resampling to provide better estimates of point-wise confidence bounds has been recently explored in the Gaussian process modelling literature, see e.g., [8], [41].\",\n \"We refer to this approach as to bootstrap-PCE, or bPCE in short.\"\n ],\n [\n \"Fast bPCE\",\n \"Because the training of a PCE model with sparse least-square analysis may be time consuming, especially in high dimension and/or when an already large experimental design is available (i.e.\",\n \"$N\\\\sim 10^3$ ), and because in this particular application we do not need very accurate estimates on the bounds of the derived quantities, we adopt a fast bPCE approach.\",\n \"In this approach, the sparse polynomial basis identified by the LARS algorithm during calibration is calculated only once from the available full experimental design ${\\\\mathcal {X}}$ , and bootstrapping is applied only to the final hybrid step, which consists in a classic ordinary least-square regression on the sparse basis [6].\",\n \"In the presence of a very expensive model, however (i.e.\",\n \"requiring several hours for a single model run), we recommended to adopt full bootstrapping, including the estimation of the sparse PCE basis for each of the $B$ bootstrapped experimental designs ${\\\\mathcal {X}}^{(1,\\\\cdots ,B)}$ .\"\n ],\n [\n \"Active bPCE-based reliability analysis\",\n \"In this section we present an adaptation of the Adaptive PC-Kriging MCS algorithm in [37] (based in turn on the original AK-MCS algorithm by [14]), that makes use of the bPCE just introduced.\",\n \"Consistently with [14], [37], in the following we will refer to this algorithm as active bootstrap-polynomial-chaos Monte-Carlo simulation (A-bPCE).\",\n \"We follow the original idea of adaptively building a surrogate of the limit-state function starting from a small initial experimental design and subsequently refining it to optimize the surrogate performance for structural reliability.\",\n \"The ultimate goal of the adaptation is to retrieve an estimate of $P_F$ that is comparable to that of a direct Monte Carlo simulation (MCS) using a large sample set with a much smaller experimental design.\",\n \"The algorithm is summarized as follows: Initialization: Generate an initial experimental design (e.g.\",\n \"through Latin hypercube sampling or uniform sampling of a ball [9]) and calculate the corresponding bPCE surrogate (see Section REF ).\",\n \"Generate a large reference MCS sample ${\\\\mathcal {X}}_{MCS} =\\\\left\\\\lbrace \\\\mathbf {x}_{MCS}^{(i)},~i = 1,\\\\cdots ,N_{MCS}\\\\right\\\\rbrace $ of size $N_{MCS}$ (e.g.\",\n \"$N_{MCS} = 10^6\\\\gg N$ ).\",\n \"A discussion on the choice of a suitable MCS sample is given in Section ).\",\n \"Calculate a set of MCS estimators of the probability of failure: $\\\\left\\\\lbrace \\\\widehat{P}_F^{(b)},~b=1,\\\\cdots ,B\\\\right\\\\rbrace $ with the current bPCE surrogate.\",\n \"Evaluate one or more suitable convergence criteria on $\\\\widehat{P}_F^{(1,\\\\cdots ,B)}$ (see Section REF ).\",\n \"If they are met, go to Step REF (terminate the algorithm).\",\n \"Otherwise continue to the next step.\",\n \"Evaluate a suitable learning function on the MCS sample ${\\\\mathcal {X}}_{MCS}$ (see Section REF ).\",\n \"Choose one or more additional $\\\\mathbf {x}_{MCS}^{(i)}\\\\in {\\\\mathcal {X}}_{MCS}$ and add them to the ED (see Section REF ).\",\n \"Update the bPCE surrogate on the new ED and return to Step REF Algorithm termination: return the $\\\\widehat{P}_F$ resulting from the PCE on the current ED, as well as the error bounds derived e.g.\",\n \"from the extremes or the empirical quantiles of the current $\\\\widehat{P}_F^{(1,\\\\cdots ,B)}$ set.\",\n \"A detailed description of each step of the algorithm is given in the following sections.\"\n ],\n [\n \"Initial experimental design\",\n \"The initial experimental design is usually generated by space-filling sampling techniques of the random vector $\\\\mathbf {X}$ , such as Latin hypercube sampling (LHS) or pseudo-random sequences (e.g.\",\n \"Sobol' sequence).\",\n \"Alternative sampling techniques, such as the uniform sampling of a ball, have also proven effective in the context of structural reliability when low probabilities of failure are expected [9].\",\n \"Note that this initial set of model evaluations does not need to be a subset of the reference sample ${\\\\mathcal {X}}_{MCS}$ used later to evaluate the $P_F$ estimates during the iterations of the algorithm.\"\n ],\n [\n \"Inner MCS-based estimate of $P_F$\",\n \"While the estimation of the $P_F$ via MCS is trivial, as it simply entails counting the number of samples that belong to the failure domain, some discussion about the number of samples $N_{MCS}$ in this step is needed.\",\n \"Throughout this paper, we opted to choose a single MCS sample ${\\\\mathcal {X}}_{MCS}$ large enough to ensure a relatively small CoV for the $P_F$ estimate at every iteration.\",\n \"This is by no means a requirement of this algorithm, but it simplifies significantly the notation (because ${\\\\mathcal {X}}_{MCS}$ becomes independent on the current iteration) and in some cases (as noted in both [14] and [37]) it can result in stabler convergence, due to the lowered MCS noise in the estimation of $P_F$ during each iteration.\",\n \"This technique is known as common random numbers in the context of repeated reliability analysis .e.g.\",\n \"in reliability-based design optimization [39].\",\n \"It is entirely possible to redraw the ${\\\\mathcal {X}}_{MCS}$ during every iteration, possibly each time with a different number of samples $N_{MCS}$ .\",\n \"The choice of $N_{MCS} = 10^6$ ensures that the CoV estimated probabilities of failure in the order of $P_F\\\\ge 10^{-3} $ is always smaller than $5\\\\%$ , which we found suitable in our application examples.\",\n \"The choice of a single MCS sample drawn during the algorithm initialization also allows us to use the application examples to focus more on the convergence of the active learning part of A-bPCE, which is the focus of this paper.\",\n \"In more general applications, the order of magnitude of $P_F$ may be unknown.\",\n \"In this case, it is recommended instead to set a target desired CoV for the estimation of $P_F$ at each iteration (as proposed in the original AK-MCS algorithm in [14]), and gradually add samples to ${\\\\mathcal {X}}_{MCS}$ until it is reached.\"\n ],\n [\n \"Convergence criteria\",\n \"The proposed convergence criterion of choice is directly inspired by [37], [32] and it depends on the stability of the $P_F$ estimate at the current iteration.\",\n \"Let us define: $\\\\begin{split}\\\\widehat{P}_F^+ &= \\\\max \\\\limits _{b = 1,\\\\cdots ,B}\\\\left(\\\\widehat{P}_F^{(b)}\\\\right)\\\\\\\\\\\\widehat{P}_F^- &= \\\\min \\\\limits _{b = 1,\\\\cdots ,B}\\\\left(\\\\widehat{P}_F^{(b)}\\\\right).\\\\end{split}$ Convergence is achieved when the following condition is satisfied for at least two consecutive iterations of the algorithm: $\\\\frac{\\\\widehat{P}_F^+ - \\\\widehat{P}_F^-}{\\\\widehat{P}_F} \\\\le \\\\epsilon _{\\\\widehat{P}_F},$ with $0.05\\\\le \\\\epsilon _{\\\\widehat{P}_F}\\\\le 0.15$ in typical usage scenarios.\"\n ],\n [\n \"Learning function\",\n \"A learning function is a function that allows one to rank a set of candidate points based on some utility criterion that depends on the desired application.\",\n \"In this case, we adopt the same heuristic approach proposed in [37], by focusing on the probability of misclassification of the bPCE model on the candidate set given by ${\\\\mathcal {X}}_{MCS}$ .\",\n \"Due to the availability of the bootstrap response samples ${\\\\mathcal {Y}}_{MCS}^{(1,\\\\cdots ,B)}$ , it is straightforward to define a measure of the misclassification probability $U_{FBR}$ (where the subscript FBR stands for failed bootstrap replicates) at each point $\\\\mathbf {x}_{MCS}^{(i)}\\\\in {\\\\mathcal {X}}_{MCS}$ as follows: $U_{FBR}(\\\\mathbf {x}_{MCS}^{(i)}) =\\\\left|\\\\frac{B_S(\\\\mathbf {x}_{MCS}^{(i)})-B_F(\\\\mathbf {x}_{MCS}^{(i)})}{B}\\\\right|$ where $B_S(\\\\mathbf {x}_{MCS}^{(i)})$ and $B_F(\\\\mathbf {x}_{MCS}^{(i)})$ are the number of safe (resp.\",\n \"failed) bPCE replicate predictions at point $\\\\mathbf {x}_{MCS}^{(i)}$ (with $B_S(\\\\mathbf {x}_{MCS}^{(i)}) + B_F(\\\\mathbf {x}_{MCS}^{(i)}) = B$ ).\",\n \"When all the $B$ replicates consistently classify $\\\\mathbf {x}_{MCS}^{(i)}$ in the safe or in the failure domain, $U_{FBR} = 1$ (minimum misclassification probability).\",\n \"In contrast, $U_{FBR} = 0$ corresponds to the case when the replicates are equally distributed between the two domains.\",\n \"In the latter case, 50% of the $B$ bootstrap PCEs predict that $\\\\mathbf {x}_{MCS}^{(i)}$ is in the safe domain, while the other 50% predicts that $\\\\mathbf {x}_{MCS}^{(i)}$ belongs to the failure domain.\",\n \"Therefore, maximum epistemic uncertainty on the classification of a point $\\\\mathbf {x}^{(i)}_{MCS}$ is attained when $U_{FBR}$ is minimum.\"\n ],\n [\n \"Enrichment of the experimental design\",\n \"The aim of the iterative algorithm described in Section REF is to obtain a surrogate model that minimizes the misclassification probability.\",\n \"As a consequence, the learning function in Eq.\",\n \"(REF ) can be directly used to obtain a single-point enrichment criterion.\",\n \"The next best candidate point for the ED $\\\\mathbf {x}^*\\\\in {\\\\mathcal {X}}_{MCS}$ is given by: $\\\\mathbf {x}^* = \\\\operatornamewithlimits{argmin}\\\\limits _{\\\\mathbf {x}^{(i)} \\\\in {\\\\mathcal {X}}_{MCS}}\\\\left(U_{FBR}(\\\\mathbf {x}^{(i)})\\\\right).$ Due to the global character of regression-based PCE, it can be beneficial to add multiple points in each iteration to sample several interesting regions of the parameter space simultaneously.\",\n \"The criterion in Eq.\",\n \"(REF ) can be extended to include $K$ distinct points simultaneously by following the approach in [37].\",\n \"A limit state margin region is first defined as the set of points such that $U_{FBR}<1$ (i.e.\",\n \"those point with non-zero misclassification probability at the current iteration).\",\n \"Subsequently, $k$ -means clustering techniques (see, e.g., [44]) can be used at each iteration to identify $K$ disjoint regions $\\\\left\\\\lbrace {\\\\mathcal {X}}^{(1,\\\\cdots ,K)}_{MCS}\\\\right\\\\rbrace $ in the limit-state margin.\",\n \"Then, Eq.\",\n \"(REF ) can be directly applied to each of the subregions to obtain $K$ different enrichment points: $\\\\mathbf {x}^{*}_k = \\\\operatornamewithlimits{argmin}\\\\limits _{\\\\mathbf {x}^{(i)} \\\\in {\\\\mathcal {X}}_{MCS}^{(k)}}\\\\left(U^{(k)}_{FBR}(\\\\mathbf {x}_k^{(i)})\\\\right), \\\\quad k =1,\\\\cdots ,K$ where $\\\\mathbf {x}^{*}_k\\\\in {\\\\mathcal {X}}^{(k)}_{MCS}$ is the $k-$ th enrichment sample and $U^{(k)}_{FBR}$ is the learning function evaluated on the $k-$ th region of the parameter space.\",\n \"Note that this approach is also convenient when parallel computing facilities are available and in the presence of computationally expensive objective functions, as the evaluation of the $K$ enrichment points can be carried out simultaneously.\"\n ],\n [\n \"Results on benchmark applications\",\n \"All the algorithm development and the final calculations presented in this section were performed with the polynomial chaos expansions and reliability analysis modules of the UQLab software for uncertainty quantification [28], [29], [27].\"\n ],\n [\n \"Series system\",\n \"A common benchmark for reliability analysis functions is given by the four-branch function, originally proposed in [42], that represents a series system comprising four components with different failure criteria.\",\n \"Although it is a simple analytical function, it shows multiple failure regions and a composite limit-state surface.\",\n \"Its two-dimensional limit state function reads: $g(\\\\mathbf {x}) = \\\\min \\\\left\\\\lbrace \\\\begin{matrix}3+0.1(x_1+x_2)^2 - \\\\frac{x_1 + x_2}{\\\\sqrt{2}}\\\\\\\\3+0.1(x_1+x_2)^2 + \\\\frac{x_1 + x_2}{\\\\sqrt{2}}\\\\\\\\(x_1-x_2) + \\\\frac{6}{\\\\sqrt{2}}\\\\\\\\(x_2-x_1) + \\\\frac{6}{\\\\sqrt{2}}\\\\\\\\\\\\end{matrix} \\\\right\\\\rbrace $ where the two random input variables $X_1\\\\sim {\\\\mathcal {N}}(0,1)$ and $X_2\\\\sim {\\\\mathcal {N}}(0,1)$ are modelled as independent standard normals.\",\n \"Failure occurs when $g(\\\\mathbf {x})\\\\le 0$ .\",\n \"Due to the multi-failure shape of the limit-state surface (represented as a solid black line in Figure REF ), classic methods like FORM/SORM and importance sampling tend to fail with this benchmark problem.\",\n \"The reference failure probability of $P_F = 4.460\\\\cdot 10^{-3}$ is obtained through an extremely large MCS ($N_{MCS} = 10^8$ ).\",\n \"Figure: Four branch function: limit state surface (black line) andexperimental design before (black circles) and after (red crosses)enrichmentThe initial experimental design for the A-bPCE algorithm was obtained with a space-filling LHS sample consisting of $N_{ini} = 20$ points drawn from the input distributions (black dots in Figure REF ).\",\n \"Three points at a time were added to the experimental design during the enrichment phase of the algorithm.\",\n \"The number of replications for the A-bPCE algorithm is set to $B = 100$ .\",\n \"After extensive testing, the algorithm was found to be very weakly dependent on the number of bootstrap replications, provided a minimum of $B \\\\ge 20$ was provided.\",\n \"Indeed, the boostrap samples are used to identify areas of relatively large prediction variability, but an accurate estimate of such variability is never really needed by the algorithm.\",\n \"Degree adaptive sparse PCE (with maximum degree in the range $p\\\\in [2, 10]$ ) based on LARS [6] was used to calibrate the PCE metamodel at each iteration.\",\n \"For validation and comparison purposes, a similar analysis was performed on the same initial ED with the AK-MCS module of UQLab, with an anisotropic Matérn 5/2 ellipsoidal multivariate Kriging correlation function [36], [27], [22].\",\n \"The convergence criterion in Eq.\",\n \"(REF ) was set to $\\\\epsilon _{\\\\widehat{P}_F} = 0.05$ for both the AK-MCS and A-bPCE algorithms.\",\n \"Convergence was achieved after 49 iterations, resulting in a total cost (including the initial experimental design) of $N_{tot} = 185$ model evaluations.\",\n \"The experimental design points added during the iterations are marked by red crosses on panel (b) of Figure REF .\",\n \"As expected, the adaptive algorithm tends to enrich the experimental design close to the limit state surface as it is adaptively learned during the iterations.\",\n \"A graphical representation of the convergence of the algorithm is shown in Figure REF , where the estimated $\\\\widehat{P}_F$ is plotted against the total number of model evaluations $N_{tot}$ .\",\n \"The shaded area represents the $95\\\\%$ confidence bounds based on the empirical quantiles as estimated from the bootstrap sample.\",\n \"Figure: Convergence curves of the four-branchlimit-state function.\",\n \"The reference P F P_F and β\\\\beta are given by dotted lines.The final results of the analysis are summarized in Table , where the generalised reliability index $\\\\beta = -\\\\Phi ^{-1}(P_F)$ is also given for reference.\",\n \"For comparison, the reference MCS probability as well as an estimate from AK-MCS are also given.\",\n \"The latter converged to a comparably accurate estimate of $P_F$ , at the cost of a slightly higher number of model evaluations.\",\n \"Note that for engineering purposes, the algorithm could have been stopped earlier, i.e.\",\n \"when a $5\\\\%$ accuracy on the generalized reliability index is attained.\",\n \"In this case, the algorithm would have converged to a comparable result ($\\\\widehat{\\\\beta } = 2.56$ ) with only 50 runs of the model.\",\n \"The final sparse PCE model after enrichment contained a total of $P = 12$ basis elements of degree up to $p = 5$ .\",\n \"Table: Comparison of different reliability analysis methods for the fourbranch function\"\n ],\n [\n \"Two-dimensional truss structure\",\n \"To test the algorithm on a more realistic engineering benchmark, consider the two-dimensional truss structure sketched in Figure REF .\",\n \"This structure has been previously analysed in several works, see e.g.\",\n \"[6], [5], [37].\",\n \"The truss comprises 23 bars and 13 nodes, with deterministic geometry yet with uncertain material properties and random loads.\",\n \"The components of the input random vector $\\\\mathbf {X}=\\\\left[A_{1,2},E_{1,2},P_{1,...,6}\\\\right]^{\\\\textsf {T}}$ include the cross-section and the Young's modulus $\\\\left\\\\lbrace A_1, E_1\\\\right\\\\rbrace $ of the horizontal bars, the cross-section and the Young's modulus $\\\\left\\\\lbrace A_2, E_2\\\\right\\\\rbrace $ of the diagonal bars and the six random loads $\\\\left\\\\lbrace P_1,\\\\cdots ,P_6\\\\right\\\\rbrace $ .\",\n \"They are considered mutually independent and their distributions are given in Table REF .\",\n \"An in-house developed Matlab-based finite-element solver is used to calculate the displacement at midspan $u(\\\\mathbf {X})$ , counted positively downwards.\",\n \"Figure: Two-dimensional truss structure with uncertain parameters.Probabilitydistributions of the geometrical parameters A i ,E i \\\\left\\\\lbrace A_i,E_i\\\\right\\\\rbrace and of theloadsP 1 ,⋯,P 6 \\\\left\\\\lbrace P_1,\\\\cdots ,P_6\\\\right\\\\rbrace are given in Table Table: Two-dimensional truss structure: definition of the probabilisticmodel of the input variables This structure operates in the nominal range as long as the midspan displacement is smaller than a critical threshold $\\\\tau = 12$ cm, which can be cast as the following limit-state function: $g(\\\\mathbf {x}) = \\\\tau - u(\\\\mathbf {x})$ where $g(\\\\mathbf {x}) \\\\le 0$ if the system is in a failure state.\",\n \"Because the FEM computational model is relatively cheap to evaluate, we could run a direct MCS-analysis with $N = 10^6$ samples to provide the reference $P_F= 1.52\\\\cdot 10^{-3}$ for validation purposes.\",\n \"Additionally, standard FORM and SORM analyses were run to estimate the non-linearity of the limit-state surface.\",\n \"FORM underestimated the failure probability of a factor of almost 2 and a cost of $N_{FORM} = 160$ model runs, while SORM achieved a good accuracy at a cost of $N_{SORM} = 372$ model runs, which suggests that the underlying problem is non-linear.\",\n \"Neither of the two methods, however, provides confidence interval on their estimates.\",\n \"The A-bPCE algorithm was initialized with an experimental design consisting in a uniform sampling of a ball (for details, see e.g.\",\n \"[9]) of size $N_{ini} = 30$ , while the sparse adaptive PCE was given a polynomial degree range $1 \\\\le p \\\\le 10$ , hyperbolic truncation with q-norm $q = 0.75$ [6] and maximum allowed interaction $r = 2$ [29].\",\n \"The internal MCS sample size was $N_{MCS} = 10^6$ and the algorithm was set to add $K = 3$ new samples per iteration.\",\n \"The stopping criterion in Eq.\",\n \"(REF ) was set to $\\\\epsilon _{\\\\widehat{P}_F} = 0.10$ .\",\n \"For comparison purposes, we also ran a standard AK-MCS analysis with the same initial experimental design and convergence criterion.\",\n \"The covariance family of choice for the underlying Kriging model was chosen as Gaussian.\",\n \"Table: Comparison of the estimation of P F P_F with several algorithms forthe truss structure example.Table REF presents a comparison of the estimated $\\\\widehat{P}_F$ with the aforementioned analyses.\",\n \"Both AK-MCS and A-bPCE estimates of $P_F$ include the reference value within the confidence bounds set by the convergence criterion.\",\n \"However, for this particular example and choice of convergence criterion, A-bPCE achieved convergence significantly faster than AK-MCS, with a total cost of 129 model evaluations, as compared to the 300 required by AK-MCS, resulting in a final PCE of degree $p=3$ with $P =43$ basis elements.\",\n \"Overall, A-bPCE provides a stable estimate of the failure probability and confidence intervals at a cost that is lower than FORM for this example.\"\n ],\n [\n \"Top-floor displacement of a structural frame\",\n \"Figure REF shows a well known, high dimensional benchmark in structural reliability applications [26], [4].\",\n \"It consists on a three-span, five story frame structure that is subject to horizontal loads.\",\n \"Both the loads and the properties of the elements of the frame (see Table REF ) are uncertain.\",\n \"Of interest is the top-floor horizontal displacement at the top right corner $u$ .\",\n \"Figure: 20-dimensional structural frame inSection .The distributions of the input variables are reported inTable Table: Frame structure: properties of the elements shown inFigure The uncertainties on the applied loads $P_1,...,P_3$ , the Young's moduli $E_1$ and $E_2$ , the moments of inertia $I_6,...,I_{13}$ and the cross sections $A_{14},...,A_{21}$ are modelled by a 21-dimensional joint random vector $\\\\mathbf {Z} = \\\\left\\\\lbrace P_1,...,A_{21}\\\\right\\\\rbrace $ with marginal distributions given in Table REF .\",\n \"Table: Frame structure: definition of the probabilisticmodel of the input variables Additionally, a Gaussian copula [24] is used to model dependence between the variables.\",\n \"The elements of the Gaussian copula correlation matrix $\\\\mathbf {R}$ are given as: $R_{E_1,E_2} = 0.9$ – the two Young's moduli are highly correlated; $R_{A_i,I_i} = 0.95$ – each element's cross-sectional area is highly correlated to the corresponding moment of inertia; ${R_{A_i,I_j} = R_{I_i,I_j} = R_{A_i,A_j} = 0.13}$ – the correlation between the properties of different elements is much lower; All the remaining elements of $\\\\mathbf {R}$ are set to 0.\",\n \"A critical displacement of $\\\\tau = 5$ cm is identified as the maximum admissible threshold for the displacement $u$ , hence resulting in the limit-state function: $g(\\\\mathbf {z}) = \\\\tau - u(\\\\mathbf {z})$ where $u(\\\\mathbf {z})$ is the displacement on the top right corner calculated with an in-house FEM code.\",\n \"Due to the associated computational costs, the maximum available budget for the calculation of a reference solution is in this case limited to $N_{REF} = 4\\\\cdot 10^4$ .\",\n \"Therefore, the reference solution is calculated with standard importance sampling (IS) [30] instead of direct MCS.\",\n \"In addition to Importance sampling, we also ran FORM and SORM.\",\n \"Due to the non-linearity of the problem, FORM significantly underestimated $P_F$ , while SORM provided an accurate estimate.\",\n \"However, due to the high dimensionality of the input space, the associated cost in terms of model evaluation was relatively high, with $N_{SORM} = 1146$ model runs, since all the gradients of the limit-state function are computed using finite-differences.\",\n \"The A-bPCE algorithm was initialized with an experimental design consisting of an LHS sampling of the input random vector of size $N_{ini} = 40$ .\",\n \"Sparse PCE was carried out with a q-norm truncation with $q = 0.75$ and maximum allowed interaction $r = 2$ .\",\n \"Note that the initialization is essentially the same as for the truss structure in the previous application.\",\n \"The internal MCS sample size was $N_{MCS} = 10^6$ , with single point enrichment per iteration.\",\n \"The stopping criterion in Eq.\",\n \"(REF ) was in this case set to $\\\\epsilon _{\\\\widehat{P}_F} = 0.15 $ .\",\n \"For comparison purposes, an AK-MCS analysis was also run on the same initial design, with similar settings and a Gaussian covariance family.\",\n \"A comparison of the results is gathered in Table REF .\",\n \"Due to the different estimation method between the reference probability (importance sampling) and the active learning-based methods (which rely on an inner MCS), no direct comparison of the results is possible as in the previous cases.\",\n \"Indeed, even fixing the same random seeds would result in different estimates due to the different methodologies.\",\n \"Therefore, confidence bounds are given for all the three methods: 95% confidence bounds for IS [30], and $P_F^{\\\\pm }$ for both AK-MCS and A-bPCE.\",\n \"The three methods give comparable results, albeit with significant differences in the convergence behaviour.\",\n \"In particular, both AK- and A-bPCE resulted in a slight underestimation of the probability of failure w.r.t.\",\n \"the reference solution by IS, which in turn is slightly overestimated with respect to the reference result quoted in the literature [4].\",\n \"However, AK-MCS did not converge in the allotted maximum number of model evaluations, and its confidence bounds remained remarkably large with respect to A-bPCE.\",\n \"A-bPCE converged instead at a total cost of approximately 200 model evaluations to the target $\\\\epsilon _{\\\\widehat{P}_F}$ , with a final sparse PCE of degree 2, counting 30 non-zero coefficients.\",\n \"For both active-learning-based methods, the reference solution lies within the given confidence bounds.\",\n \"Moreover, the confidence bounds on the reliability index $\\\\widehat{\\\\beta }$ show that the results are stable to within $2\\\\%$ of the calculated values.\",\n \"Finally, it is interesting to mention that for this example the costs of FORM and A-bPCE were comparable, but the latter provides a much less biased estimate, and includes confidence bounds.\",\n \"Table: Comparison of the estimation of P F P_F with several algorithms forthe top-floor displacement of a structural frame example\"\n ],\n [\n \"Conclusions and outlook\",\n \"A novel approach to solving reliability problems with polynomial chaos expansions has been proposed.\",\n \"The combination of the bootstrap method and sparse regression enabled us to introduce local error estimation in the standard PCE predictor.\",\n \"In turn, this allows one to construct active learning algorithms similar to AK-MCS to greedily enrich a relatively small initial experimental design so as to efficiently estimate the probability of failure of complex systems.\",\n \"This approach has shown comparable performance w.r.t.\",\n \"to the well established AK-MCS method on both a simple analytical benchmark function and in two high-dimensional engineering applications of increasing complexity.\",\n \"Extensions of this approach can be envisioned in two main directions: the simulation-based reliability analysis method can be extended beyond simple MCS (e.g.\",\n \"by using importance sampling [12], line sampling [34] or subset simulation [11]) to achieve better $\\\\widehat{P}_F$ estimates at each iteration, especially for very low probabilities of failure; remote parallel computing facilities may be used during the enrichment phase of the algorithms with expensive computational models when adding more than one point at a time; the use of bootstrap to enable local error estimation in an active learning context can be used also with different regression-based surrogate modelling techniques, including e.g.\",\n \"low rank tensor approximations [21].\",\n \"Additionally, the bPCE approach itself introduced in this work can be used also outside of a pure reliability analysis context, as it provides an effective local error estimate for PCE.\",\n \"It has been used, e.g.\",\n \"in the context of reliability-based design optimization in [31].\",\n \"Indeed the lack of this feature (as opposed to Kriging) has somewhat hindered its usage in more advanced active-learning applications.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01589"}}},{"rowIdx":894,"cells":{"text":{"kind":"list like","value":[["Deep and Confident Prediction for Time Series at Uber"],["Abstract Reliable uncertainty estimation for time series prediction is critical in many fields, including physics, biology, and manufacturing.","At Uber, probabilistic time series forecasting is used for robust prediction of number of trips during special events, driver incentive allocation, as well as real-time anomaly detection across millions of metrics.","Classical time series models are often used in conjunction with a probabilistic formulation for uncertainty estimation.","However, such models are hard to tune, scale, and add exogenous variables to.","Motivated by the recent resurgence of Long Short Term Memory networks, we propose a novel end-to-end Bayesian deep model that provides time series prediction along with uncertainty estimation.","We provide detailed experiments of the proposed solution on completed trips data, and successfully apply it to large-scale time series anomaly detection at Uber."],["Introduction","Accurate time series forecasting and reliable estimation of the prediction uncertainty are critical for anomaly detection, optimal resource allocation, budget planning, and other related tasks.","This problem is challenging, especially during high variance segments (e.g., holidays, sporting events), because extreme event prediction depends on numerous external factors that can include weather, city population growth, or marketing changes (e.g., driver incentives) [1] that all contribute to the uncertainty of the forecast.","These exogenous variables, however, are difficult to incorporate in many classical time series models, such as those found in the standard $R$ forecast[2] package.","In addition, these models usually require manual tuning to set model and uncertainty parameters.","Relatively recently, time series modeling based on the Long Short Term Memory (LSTM) model [3] has gained popularity due to its end-to-end modeling, ease of incorporating exogenous variables, and automatic feature extraction abilities [4].","By providing a large amount of data across numerous dimensions, it has been shown that an LSTM network can model complex nonlinear feature interactions [5], which is critical for modeling complex extreme events.","A recent paper [6] has shown that a neural network forecasting model is able to outperform classical time series methods in cases with long, interdependent time series.","However, the problem of estimating the uncertainty in time-series predictions using neural networks remains an open question.","The prediction uncertainty is important for assessing how much to trust the forecast produced by the model, and has profound impact in anomaly detection.","The previous model proposed in [6] had no information regarding the uncertainty.","Specifically, this resulted in a large false anomaly rates during holidays where the model prediction has large variance.","In this paper, we propose a novel end-to-end model architecture for time series prediction, and quantify the prediction uncertainty using Bayesian Neural Network, which is further used for large-scale anomaly detection.","Recently, Bayesian neural networks (BNNs) have garnered increasing attention as a principled framework to provide uncertainty estimation for deep models.","Under this framework, the prediction uncertainty can be decomposed into three types: model uncertainty, inherent noise, and model misspecification.","Model uncertainty, also referred to as epistemic uncertainty, captures our ignorance of the model parameters, and can be reduced as more samples being collected.","Inherent noise, on the other hand, captures the uncertainty in the data generation process and is irreducible.","These two sources have been previously recognized with successful application in computer visions [7].","The third uncertainty from model misspecification, however, has been long-overlooked.","This captures the scenario where the testing samples come from a different population than the training set, which is often the case in time series anomaly detection.","Similar ideas have gained attention in deep learning under the concept of adversarial examples in computer vision [8], but its implication in prediction uncertainty remains unexplored.","Here, we propose a principled solution to incorporate this uncertainty using an encoder-decoder framework.","To the best of our knowledge, this is the first time that misspecification uncertainty has been successfully applied to prediction and anomaly detection in a principled way.","In summary, this paper makes the following contributions: Provides a generic and scalable uncertainty estimation implementation for deep prediction models.","Quantifies the prediction uncertainty from three sources: (i) model uncertainty, (ii) inherent noise, and (iii) model misspecification.","The third uncertainty has been previously overlooked, and we propose a potential solution with an encoder-decoder.","Motivates a real-world anomaly detection use-case at Uber that uses Bayesian Neural Networks with uncertainty estimation to improve performance at scale.","The rest of this paper is organized as follows: Section gives an overview of previous work on time series prediction for both classical and deep learning models, as well as the various approaches for uncertainty estimation in neural networks.","The approach of Monte Carlo dropout (MC dropout) is used in this paper due to its simplicity, strong generalization ability, and scalability.","In Section , we present our uncertainty estimation algorithm that accounts for the three different sources of uncertainty.","Section provides detailed experiments to evaluate the model performance on Uber trip data, and lays out a successful application to large-scale anomaly detection for millions of metrics at Uber.","Finally, Section concludes the paper.","Classical time series models, such as those found in the standard $R$ forecast[2] package are popular methods to provide an univariate base-level forecast.","These models usually require manual tuning to set seasonality and other parameters.","Furthermore, while there are time series models that can incorporate exogenous variables [9], they suffer from the curse of dimensionality and require frequent retraining.","To more effectively deal with exogenous variables, a combination of univariate modeling and a machine learning model to handle residuals was introduced in [10].","The resulting two-stage model, however, is hard to tune, requires manual feature extraction and frequent retraining, which is prohibitive to millions of time series.","Relatively recently, time series modeling based on LSTM [3] technique gained popularity due to its end-to-end modeling, ease of incorporating exogenous variables, and automatic feature extraction abilities [4].","By providing a large amount of data across numerous dimensions, it has been shown that an LSTM approach can model complex extreme events by allowing nonlinear feature interactions [5], [6].","While uncertainty estimation for classical forecasting models has been widely studied [11], this is not the case for neural networks.","Approaches such as a modified loss function or using a collection of heterogenous networks [12] were proposed, however they require changes to the underlying model architecture.","A more detailed review is given in the next section.","In this work, we use a simple and scalable approach for deep model uncertainty estimation that builds on [13].","This framework provides a generic error estimator that runs in production at Uber-scale to mitigate against bad decisions (e.g., false anomaly alerts) resulting from poor forecasts due to high prediction variance."],["Bayesian Neural Networks","Bayesian Neural Networks (BNNs) introduce uncertainty to deep learning models from a Bayesian perspective.","By giving a prior to the network parameters $W$ , the network aims to find the posterior distribution of $W$ , instead of a point estimation.","This procedure is usually referred to as posterior inference in traditional Bayesian models.","Unfortunately, due to the complicated non-linearity and non-conjugacy in deep models, exact posterior inference is rarely available; in addition, most traditional algorithms for approximate Bayesian inference cannot scale to the large number of parameters in most neural networks.","Recently, several approximate inference methods are proposed for Bayesian Neural Networks.","Most approaches are based on variational inference that optimizes the variational lower bound, including stochastic search [14], variational Bayes [15], probabilistic backpropagation [16], Bayes by BackProp [17] and its extension [18].","Several algorithms further extend the approximation framework to $\\alpha $ -divergence optimization, including [19], [20].","We refer the readers to [21] for a more detailed and complete review of these methods.","All of the aforementioned algorithms require different training methods for the neural network.","Specifically, the loss function must be adjusted to different optimization problems, and the training algorithm has to be modified in a usually non-trivial sense.","In practice, however, an out-of-the-box solution is often preferred, without changing the neural network architecture and can be directly applied to the previously trained model.","In addition, most existing inference algorithms introduce extra model parameters, sometimes even double, which is difficult to scale given the large amount of parameters used in practice.","This paper is inspired by the Monte Carlo dropout (MC dropout) framework proposed in [13] and [22], which requires no change of the existing model architecture and provides uncertainty estimation almost for free.","Specifically, stochastic dropouts are applied after each hidden layer, and the model output can be approximately viewed as a random sample generated from the posterior predictive distribution [21].","As a result, the model uncertainty can be estimated by the sample variance of the model predictions in a few repetitions.","Details of this algorithm will be reviewed in the next section.","The MC dropout framework is particularly appealing to practitioners because it is generic, easy to implement, and directly applicable to any existing neural networks.","However, the exploration of its application to real-world problems remains extremely limited.","This paper takes an important step forward by successfully adapting this framework to conduct time series prediction and anomaly detection at large scale."],["Method","Given a trained neural network $f^{\\hat{W}}(\\cdot )$ where $\\hat{W}$ represents the fitted parameters, as well as a new sample $x^*$ , our goal is to evaluate the uncertainty of the model prediction, $\\hat{y}^* = f^{\\hat{W}}(x^*)$ .","Specifically, we would like to quantify the prediction standard error, $\\eta $ , so that an approximate $\\alpha $ -level prediction interval can be constructed by $[\\hat{y}^* - z_{\\alpha /2} \\eta , ~ \\hat{y}^* + z_{\\alpha /2} \\eta ]$ where $z_{\\alpha /2}$ is the upper $\\alpha /2$ quantile of a standard Normal.","This prediction interval is critical for various tasks.","For example, in anomaly detection, anomaly alerts will be fired when the observed value falls outside the constructed 95% interval.","As a result, underestimating $\\eta $ will lead to high false positive rates.","In the rest of this section, we will present our uncertainty estimation algorithm in Section REF , which accounts for three different sources of prediction uncertainties.","This framework can be generalized to any neural network architectures.","Then, in Section REF , we will present our neural network design for predicting time series at Uber."],["Prediction Uncertainty","We denote a neural network as function $f^W(\\cdot )$ , where $f$ captures the network architecture, and $W$ is the collection of model parameters.","In a Bayesian neural network, a prior is introduced for the weight parameters, and the model aims to fit the optimal posterior distribution.","For example, a Gaussian prior is commonly assumed: $W \\sim N(0, I)$ We further specify the data generating distribution $p(y \\,|\\, f^W(x))$ .","In regression, we often assume $ y\\,|\\, W \\sim N(f^W(x), \\sigma ^2) $ with some noise level $\\sigma $ .","In classification, the softmax likelihood is often used.","For time series prediction, we will focus on the regression setting in this paper.","Given a set of $N$ observations $X=\\lbrace x_1, ..., x_N\\rbrace $ and $Y=\\lbrace y_1, ..., y_N\\rbrace $ , Bayesian inference aims at finding the posterior distribution over model parameters $p(W \\,|\\, X, Y)$ .","With a new data point $x^*$ , the prediction distribution is obtained by marginalizing out the posterior distribution: $ p(y^* \\,|\\, x^*) = \\int _W p(y^* \\,|\\, f^W(x^*)) p(W \\,|\\, X, Y)\\, dW $ In particular, the variance of the prediction distribution quantifies the prediction uncertainty, which can be further decomposed using law of total variance: $\\begin{split}\\textrm {Var}(y^* \\,|\\, x^*) & =\\textrm {Var}\\left[ \\mathbb {E}(y^* \\,|\\, W, x^*) \\right] +\\mathbb {E}\\left[\\textrm {Var}(y^* \\,|\\, W, x^*) \\right] \\\\& = \\textrm {Var}(f^W(x^*)) + \\sigma ^2\\end{split}$ Immediately, we see that the variance is decomposed into two terms: (i) $\\textrm {Var}(f^W(x^*))$ , which reflects our ignorance over model parameter $W$ , referred to as the model uncertainty; and (ii) $\\sigma ^2$ which is the noise level during data generating process, referred to as the inherent noise.","An underlying assumption for (REF ) is that $y^*$ is generated by the same procedure.","However, this is not always the case in practice.","In anomaly detection, in particular, it is expected that certain time series will have unusual patterns, which can be very different from the trained model.","Therefore, we propose that a complete measurement of prediction uncertainty should be a combination from three sources: (i) model uncertainty, (ii) model misspecification, and (iii) inherent noise level.","The following sections provide details on how we handle these three terms.","The key to estimating model uncertainty is the posterior distribution $p(W\\,|\\, X, Y)$ , also referred to as Bayesian inference.","This is particularly challenging in neural networks because the non-conjugacy due to nonlinearities.","There have been various research efforts on approximate inference in deep learning (see Section REF for a review).","Here, we follow the idea in [13] and [22] to approximate model uncertainty using Monte Carlo dropout (MC dropout).","The algorithm proceeds as follows: given a new input $x^*$ , we compute the neural network output with stochastic dropouts at each layer.","That is, randomly dropout each hidden unit with certain probability $p$ .","This stochastic feedforward is repeated $B$ times, and we obtain $\\lbrace \\hat{y}^*_{(1)}, ..., \\hat{y}^*_{(B)}\\rbrace $ .","Then the model uncertainty can be approximated by the sample variance: $\\widehat{\\textrm {Var}}(f^W(x^*)) = \\frac{1}{B} \\sum _{b=1}^B \\left(\\hat{y}^*_{(b)} - \\overline{\\hat{y}}^* \\right)^2$ where $\\overline{\\hat{y}}^* = \\frac{1}{B} \\sum _{b=1}^B \\hat{y}^*_{(b)}$ [13].","There has been recent work done on choosing the optimal dropout probability $p$ adaptively by treating it as part of the model parameter, but this approach requires modifying the training phase [12].","In practice, we find that the uncertainty estimation is usually robust within a reasonable range of $p$ ."],["Model misspecification","Next, we address the problem of capturing potential model misspecification.","In particular, we would like to capture the uncertainty when predicting unseen samples with very different patterns from the training data set.","We propose to account for this source of uncertainty by introducing an encoder-decoder to the model framework.","The idea is to train an encoder that extracts the representative features from a time series, in the sense that a decoder can reconstruct the time series from the encoded space.","At test time, the quality of encoding of each sample will provide insight on how close it is to the training set.","Another way to think of this approach is that we first fit a latent embedding space for all training time series using an encoder-decoder framework.","Then, we measure the distance between test cases and training samples in the embedded space.","The next question is how to incorporate this uncertainty in the variance calculation.","Here, we take a principled approach by connecting the encoder, $g(\\cdot )$ , with a prediction network, $h(\\cdot )$ , and treat them as one large network $f = h(g(\\cdot ))$ during inference.","REF illustrates such an inference network, and Algorithm REF presents the MC dropout algorithm.","Specifically, given an input time series $x = \\lbrace x_1, ..., x_T\\rbrace $ , the encoder $g(\\cdot )$ constructs the learned embedding $e = g(x)$ , which is further concatenated with external features, and the final vector is fed to the final prediction network $h$ .","During this feedforward pass, MC dropout is applied to all layers in both the encoder $g$ and the prediction network $h$ .","As a result, the random dropout in the encoder perturbs the input intelligently in the embedding space, which accounts for potential model misspecification and gets further propagated through the prediction network.","Here, variational dropout for recurrent neural networks [22] is applied to the LSTM layers in the encoder, and regular dropout [13] is applied to the prediction network.","[H] [1] data $x^*$ , encoder $g(\\cdot )$ , prediction network $h(\\cdot )$ , dropout probability $p$ , number of iterations $B$ prediction $\\hat{y}^*_{mc}$ , uncertainty $\\eta _1$ $b = 1$ to $B$ $e^*_{(b)} \\leftarrow $ VariationalDropout$(g(x^*), p)$ $z^*_{(b)} \\leftarrow \\textrm {Concatenate}(e^*_{(b)}, \\textrm {extFeatures})$ $\\hat{y}^*_{(b)} \\leftarrow $ Dropout $(h(z^*_{(b)}), p)$ // prediction $\\hat{y}^*_{mc} \\leftarrow \\frac{1}{B} \\sum _{b=1}^B \\hat{y}^*_{(b)}$ // model uncertainty and misspecification $\\eta _1^2 \\leftarrow \\frac{1}{B} \\sum _{b=1}^B (\\hat{y}^*_{(b)} - \\hat{y}^* )^2$ $\\hat{y}^*_{mc}, \\, \\eta _1$ MCdropout"],["Inherent noise","Finally, we estimate the inherent noise level $\\sigma ^2$ .","In the original MC dropout algorithm [13], this parameter is implicitly determined by a prior over the smoothness of $W$ .","As a result, the model could end up with drastically different estimations of the uncertainty level depending on this pre-specified smoothness (see [21], chapter 4).","This dependency is undesirable in anomaly detection, because we want the uncertainty estimation to also have robust frequentist coverage, but it is rarely the case that we would know the correct noise level a priori.","Here, we propose a simple and adaptive approach that estimates the noise level via the residual sum of squares, evaluated on an independent held-out validation set.","Specifically, let $f^{\\hat{W}}(\\cdot )$ be the fitted model on training data, and $X^{\\prime }=\\lbrace x^{\\prime }_1, ..., x^{\\prime }_V\\rbrace , Y^{\\prime }=\\lbrace y^{\\prime }_1, ..., y^{\\prime }_V\\rbrace $ be an independent validation set, then we estimate $\\sigma ^2$ via $\\hat{\\sigma }^2 = \\frac{1}{V} \\sum _{v=1}^V \\left( y^{\\prime }_v - f^{\\hat{W}}(x^{\\prime }_v) \\right)^2 \\,.$ Note that $(X^{\\prime }, Y^{\\prime })$ are independent from $f^{\\hat{W}}(\\cdot )$ , and if we further assume that $f^{\\hat{W}}(x^{\\prime }_v)$ is an unbiased estimation of the true model, we have $\\begin{split}\\mathbb {E}(\\hat{\\sigma }^2) &=\\sigma ^2 + \\frac{1}{V} \\sum _{v=1}^V \\mathbb {E} \\left[ f^{\\hat{W}}(x^{\\prime }_v) - f^{W}(x^{\\prime }_v) \\right]^2 \\\\&= \\sigma ^2 + \\textrm {Var}_{\\rm TRN}(f^{\\hat{W}}(x^{\\prime }_v))\\end{split}$ where $\\textrm {Var}_{\\rm TRN}$ is w.r.t the training data, which decreases as the training sample size increases, and $\\rightarrow 0$ as the training sample size $N \\rightarrow \\infty $ .","Therefore, $\\hat{\\sigma }^2$ provides an asymptotically unbiased estimation on the inherent noise level.","In the finite sample scenario, it always overestimates the noise level and tends to be more conservative.","The final inference algorithm combines inherent noise estimation with MC dropout, and is presented in Algorithm REF .","[H] [1] data $x^*$ , encoder $g(\\cdot )$ , prediction network $h(\\cdot )$ , dropout probability $p$ , number of iterations $B$ prediction $\\hat{y}^*$ , predictive uncertainty $\\eta $ // prediction, model uncertainty and misspecification $\\hat{y}^*, \\, \\eta _1 \\leftarrow $ MCdropout $(x^*, g, h, p, B)$ // Inherent noise $x^{\\prime }_v$ in validation set $\\lbrace x^{\\prime }_1, ..., x^{\\prime }_V\\rbrace $ $\\hat{y^{\\prime }}_v \\leftarrow h(g(x^{\\prime }_v))$ $\\eta _2^2 \\leftarrow \\frac{1}{V} \\sum _{v=1}^V \\left( \\hat{y^{\\prime }}_v - y^{\\prime }_v \\right)^2$ // total prediction uncertainty $\\eta \\leftarrow \\sqrt{\\eta _1^2 + \\eta _2^2}$ $\\hat{y}^*, \\, \\eta $ Inference"],["Model Design","The complete architecture of the neural network is shown in REF .","The network contains two major components: (i) an encoder-decoder framework that captures the inherent pattern in the time series, which is learned during pre-training step, and (ii) a prediction network that takes input from both the learned embedding from encoder-decoder, as well as any potential external features to guide the prediction.","We discuss the two components in more details below.","Figure: Neural network architecture, with a pre-training phase using a LSTM encoder-decoder, followed by a prediction network, with input being the learned embedding concatenated with external features."],["Encoder-decoder","Prior to fitting the prediction model, we first conduct a pre-training step to fit an encoder that can extract useful and representative embeddings from a time series.","The goals are to ensure that (i) the learned embedding provides useful features for prediction and (ii) unusual input can be captured in the embedded space, which will get further propagated to the prediction network in the next step.","Here, we use an encoder-decoder framework with two-layer LSTM cells.","Specifically, given a univariate time series $\\lbrace x_t\\rbrace _t$ , the encoder reads in the first $T$ timestamps $\\lbrace x_1, ..., x_T\\rbrace $ , and constructs a fixed-dimensional embedding state.","After then, from this embedding state, the decoder constructs the following $F$ timestamps $\\lbrace x_{T+1}, ..., x_{T+F}\\rbrace $ with guidance from $\\lbrace x_{T-F+1}, ..., x_T\\rbrace $ ( REF , bottom panel).","The intuition is that in order to construct the next few timestamps, the embedding state must extract representative and meaningful features from the input time series.","This design is inspired from the success of video representation learning using a similar architecture [23]."],["Prediction network","After the encoder-decoder is pre-trained, it is treated as an intelligent feature-extraction blackbox.","Specifically, the last LSTM cell states of the encoder are extracted as learned embedding.","Then, a prediction network is trained to forecast the next one or more timestamps using the learned embedding as features.","In the scenario where external features are available, these can be concatenated to the embedding vector and passed together to the final prediction network.","Here, we use a multi-layer perceptron as the prediction network.","We will show in Section REF that the learned embedding from the encoder successfully captures interesting patterns from the input time series.","In addition, including external features significantly improves the prediction accuracy during holidays and special events (see Section )"],["Inference","After the full model is trained, the inference stage involves only the encoder and the prediction network ( REF , left panel).","The complete inference algorithm is presented in Algorithm REF , where the prediction uncertainty, $\\eta $ , contains two terms: (i) the model and misspecification uncertainty, estimated by applying MC dropout to both the encoder and the prediction network, as presented in Algorithm REF ; and (ii) the inherent noise level, estimated by the residuals on a held-out validation set.","Finally, an approximate $\\alpha $ -level prediction interval is constructed by $[\\hat{y}^* - z_{\\alpha /2} \\eta , ~ \\hat{y}^* + z_{\\alpha /2} \\eta ]$ , where $z_{\\alpha /2}$ is the upper $\\alpha /2$ quantile of a standard Normal.","Two hyper-parameters need to be specified in Algorithm REF : the dropout probability, $p$ , and the number of iterations, $B$ .","As for the dropout probability, we find in our experiments that the uncertainty estimation is relatively stable across a range of $p$ , and we choose the one that achieves the best performance on the validation set.","As for the number of iterations, the standard error of the estimated prediction uncertainty is proportional to $1/\\sqrt{B}$ .","We measure the standard error across different repetitions, and find that a few hundreds of iterations are usually suffice to achieve a stable estimation."],["Evaluation","This section contains two sets of results.","We first evaluate the model performance on a moderately sized data set of daily trips processed by the Uber platform.","We will evaluate the prediction accuracy and the quality of uncertain estimation during both holidays and non-holidays.","We will also present how the encoder recognizes the day of the week pattern in the embedding space.","Next, we will illustrate the application of this model to real-time large-scale anomaly detection for millions of metrics at Uber."],["Experimental settings","In this section, we illustrate the model performance using the daily completed trips over four years across eight representative large cities in U.S. and Canada, including Atlanta, Boston, Chicago, Los Angeles, New York City, San Francisco, Toronto, and Washington D.C. We use three years of data as the training set, the following four months as the validation set, and the final eight months as the testing set.","The encoder-decoder is constructed with two-layer LSTM cells, with 128 and 32 hidden states, respectively.","The prediction network has three fully connected layers with tanh activation, with 128, 64, and 16 hidden units, respectively.","Samples are constructed using a sliding window with step size one, where each sliding window contains the previous 28 days as input, and aims to forecast the upcoming day.","The raw data are log-transformed to alleviate exponential effects.","Next, within each sliding window, the first day is subtracted from all values, so that trends are removed and the neural network is trained for the incremental value.","At test time, it is straightforward to revert these transformations to obtain predictions at the original scale."],["Prediction performance","We compare the prediction accuracy among four different models: Last-Day: A naive model that uses the last day's completed trips as the prediction for the next day.","QRF: Based on the naive last-day prediction, a quantile random forest (QRF) is further trained to estimate the holiday lifts, i.e., the ratio to adjust the forecast during holidays.","The final prediction is calculated from the last-day forecast multiplied by the estimated ratio.","LSTM: A vanilla LSTM model with similar size as our model.","Specifically, a two-layer sacked LSTM is constructed, with 128 and 32 hidden states, respectively, followed by a fully connected layer for the final output.","This neural network also takes 28 days as input, and predicts the next day.","Our Model: Our model that combines an encoder-decoder and a prediction network, as described in REF .","Table REF reports the Symmetric Mean Absolute Percentage Error (SMAPE) of the four models, evaluated on the testing set.","We see that using a QRF to adjust for holiday lifts is only slightly better than the naive prediction.","On the other hand, a vanilla LSTM neural network provides an average of 26% improvement across the eight cities.","As we further incorporate the encoder-decoder framework and introduce external features for holidays to the prediction network ( REF ), our proposed model achieves another 36% improvement in prediction accuracy.","Note that when using LSTM and our model, only one generic model is trained, where the neural network is not tuned for any city-specific patterns; nevertheless, we still observe significant improvement on SMAPE across all cities when compared to traditional approaches.","Table: SMAPE of Four Different Prediction Models, Evaluated on the Test Data.Finally, REF visualizes the true values and our predictions during the testing period in San Francisco as an example.","We observe that accurate predictions are achieved not only in regular days, but also during holiday seasons.","Figure: Daily completed trips in San Francisco during eight months of the testing set.","True values are shown with the orange solid line, and predictions are shown with the blue dashed line, where the 95% prediction band is shown as the grey area.","Exact values are anonymized."],["Uncertainty estimation","Next, we evaluate the quality of our uncertainty estimation by calibrating the empirical coverage of the prediction intervals.","Here, the dropout probability is set to be 5% at each layer, and Table REF reports the empirical coverage of the 95% predictive intervals under three different scenarios: PredNet: Use only model uncertainty estimated from MC dropout in the prediction network, with no dropout layers in the encoder.","Enc+Pred: Use MC dropout in both the encoder and the prediction network, but without the inherent noise level.","This is the term $\\eta _1$ in Algorithm REF .","Enc+Pred+Noise: Use the full prediction uncertainty $\\eta $ as presented in Algorithm REF , including $\\eta _1$ as in 2), as well as the inherent noise level $\\eta _2$ .","Table: Empirical Coverage of 95% Predictive Intervals, Evaluated on the Test Data.By comparing PredNet with Enc+Pred, it is clear that introducing MC dropout to the encoder network is critical, which significantly improves the empirical coverage from 78% to 90% by capturing potential model misspecification.","In addition, by further accounting for the inherent noise level, the empirical coverage of the final uncertainty estimation, Enc+Pred+Noise, nicely centers around 95% as desired.","One important use-case of the uncertainty estimation is to provide insight for unusual patterns in the time series.","REF shows the estimated predictive uncertainty on six U.S. holidays in the testing data.","We see that New Year's Eve has significantly higher uncertainty than all other holidays.","This pattern is consistent with our previous experience, where New Year's Eve is usually the most difficult day to predict.","Figure: Estimated prediction standard deviations on six U.S. holidays during testing period for eight cities.","Exact values are anonymized."],["Embedding features","As illustrated previously, the encoder is critical for both improving prediction accuracy, as well as for estimating prediction uncertainty.","One natural follow-up question is whether we can interpret the embedding features extracted by the encoder.","This can also provide valuable insights for model selection and anomaly detection.","Here, we visualize our training data, each being a 28-day time series segment, in the embedding space.","We use the last LSTM cell in the encoder, and project its cell states to 2D for visualization using PCA ( REF ).","The strongest pattern we observe is day of the week, where weekdays and weekends form different clusters, with Fridays usually sitting in between.","We do not observe city-level clusters, which is probably due to the fact all cities in this data set are large cities in North America, where riders and drivers tend to have similar behaviors.","Figure: Training set of time series, visualized in the embedding space.","Each point represents a 28-day segment, colored by the day of the week of the last day.","We evaluate the cell states of the two LSTM layers, where the first layer with dimension 128 is plotted on the left, and second layer with dimension 32 is plotted on the right.","PCA is used to project into 2D space for visualization."],["Application to Anomaly Detection\nat Uber\n","At Uber, we track millions of metrics each day to monitor the status of various services across the company.","One important application of uncertainty estimation is to provide real-time anomaly detection and deploy alerts for potential outages and unusual behaviors.","A natural approach is to trigger an alarm when the observed value falls outside of the 95% predictive interval.","There are two main challenges we need to address in this application: Scalability: In order to provide real-time anomaly detection at the current scale, each predictive interval must be calculated within a few milliseconds during inference stage.","Performance: With highly imbalanced data, we aim to reduce the false positive rate as much as possible to avoid unnecessary on-call duties, while making sure the false negative rate is properly controlled so that real outages will be captured."],["Scalability","Our model inference is implemented in Go.","Our implementation involves efficient matrix manipulation operations, as well as stochastic dropout by randomly setting hidden units to zero with pre-specified probability.","A few hundred stochastic passes are executed to calculate the prediction uncertainty, which is updated every few minutes for each metric.","We find that the uncertainty estimation step adds only a small amount of computation overhead and can be conducted within ten milliseconds per metric."],["Performance","Here, we illustrate the precision and recall of this framework on an example data set containing 100 metrics with manual annotation available, where 17 of them are true anomalies.","Note that the neural network was previously trained on a separate and much larger data set.","By adding MC dropout layers in the neural network, the estimated predictive intervals achieved 100% recall rate and a 80.95% precision rate.","REF visualizes the neural network predictive intervals on four representative metrics, where alerts are correctly fired for two of them.","When applying this framework to all metrics, we observe a 4% improvement in precision compared to the previous ad-hoc solution, which is substantial at Uber's scale.","Figure: Four example metrics during a 12-hour span, and anomaly detection is performed for the following 30 minutes.","All metrics are evaluated by minutes.","The neural network constructs predictive intervals for the following 30 minutes, visualized by the shaded area in each plot.","(a) A normal metric with large fluctuation, where the observation falls within the predictive interval.","(b) A normal metric with small fluctuation, and an unusual inflation has just ended.","The predictive interval still captures the observation.","(c) An anomalous metric with a single spike that falls outside the predictive interval.","(d) An anomalous metric with two consecutive spikes, also captured by our model."],["Conclusion","We have presented an end-to-end neural network architecture for uncertainty estimation used at Uber.","Using the MC dropout technique and model misspecification distribution, we showed a simple way to provide uncertainty estimation for a neural network forecast at scale while providing a 95% uncertainty coverage.","A critical feature about our framework is its applicability to any neural network without modifying the underlying architecture.","We have used the proposed uncertainty estimate to measure special event (e.g., holiday) uncertainty and to improve anomaly detection accuracy.","For special event uncertainty estimation, we found New Year's Eve to be the most uncertain time.","Using the uncertainty information, we adjusted the confidence bands of an internal anomaly detection model to improve precision during high uncertainty events, resulting in a 4% accuracy improvement, which is large given the number of metrics we track at Uber.","Our future work will be focused on utilizing the uncertainty information for neural network debugging during high error periods."]],"string":"[\n [\n \"Deep and Confident Prediction for Time Series at Uber\"\n ],\n [\n \"Abstract Reliable uncertainty estimation for time series prediction is critical in many fields, including physics, biology, and manufacturing.\",\n \"At Uber, probabilistic time series forecasting is used for robust prediction of number of trips during special events, driver incentive allocation, as well as real-time anomaly detection across millions of metrics.\",\n \"Classical time series models are often used in conjunction with a probabilistic formulation for uncertainty estimation.\",\n \"However, such models are hard to tune, scale, and add exogenous variables to.\",\n \"Motivated by the recent resurgence of Long Short Term Memory networks, we propose a novel end-to-end Bayesian deep model that provides time series prediction along with uncertainty estimation.\",\n \"We provide detailed experiments of the proposed solution on completed trips data, and successfully apply it to large-scale time series anomaly detection at Uber.\"\n ],\n [\n \"Introduction\",\n \"Accurate time series forecasting and reliable estimation of the prediction uncertainty are critical for anomaly detection, optimal resource allocation, budget planning, and other related tasks.\",\n \"This problem is challenging, especially during high variance segments (e.g., holidays, sporting events), because extreme event prediction depends on numerous external factors that can include weather, city population growth, or marketing changes (e.g., driver incentives) [1] that all contribute to the uncertainty of the forecast.\",\n \"These exogenous variables, however, are difficult to incorporate in many classical time series models, such as those found in the standard $R$ forecast[2] package.\",\n \"In addition, these models usually require manual tuning to set model and uncertainty parameters.\",\n \"Relatively recently, time series modeling based on the Long Short Term Memory (LSTM) model [3] has gained popularity due to its end-to-end modeling, ease of incorporating exogenous variables, and automatic feature extraction abilities [4].\",\n \"By providing a large amount of data across numerous dimensions, it has been shown that an LSTM network can model complex nonlinear feature interactions [5], which is critical for modeling complex extreme events.\",\n \"A recent paper [6] has shown that a neural network forecasting model is able to outperform classical time series methods in cases with long, interdependent time series.\",\n \"However, the problem of estimating the uncertainty in time-series predictions using neural networks remains an open question.\",\n \"The prediction uncertainty is important for assessing how much to trust the forecast produced by the model, and has profound impact in anomaly detection.\",\n \"The previous model proposed in [6] had no information regarding the uncertainty.\",\n \"Specifically, this resulted in a large false anomaly rates during holidays where the model prediction has large variance.\",\n \"In this paper, we propose a novel end-to-end model architecture for time series prediction, and quantify the prediction uncertainty using Bayesian Neural Network, which is further used for large-scale anomaly detection.\",\n \"Recently, Bayesian neural networks (BNNs) have garnered increasing attention as a principled framework to provide uncertainty estimation for deep models.\",\n \"Under this framework, the prediction uncertainty can be decomposed into three types: model uncertainty, inherent noise, and model misspecification.\",\n \"Model uncertainty, also referred to as epistemic uncertainty, captures our ignorance of the model parameters, and can be reduced as more samples being collected.\",\n \"Inherent noise, on the other hand, captures the uncertainty in the data generation process and is irreducible.\",\n \"These two sources have been previously recognized with successful application in computer visions [7].\",\n \"The third uncertainty from model misspecification, however, has been long-overlooked.\",\n \"This captures the scenario where the testing samples come from a different population than the training set, which is often the case in time series anomaly detection.\",\n \"Similar ideas have gained attention in deep learning under the concept of adversarial examples in computer vision [8], but its implication in prediction uncertainty remains unexplored.\",\n \"Here, we propose a principled solution to incorporate this uncertainty using an encoder-decoder framework.\",\n \"To the best of our knowledge, this is the first time that misspecification uncertainty has been successfully applied to prediction and anomaly detection in a principled way.\",\n \"In summary, this paper makes the following contributions: Provides a generic and scalable uncertainty estimation implementation for deep prediction models.\",\n \"Quantifies the prediction uncertainty from three sources: (i) model uncertainty, (ii) inherent noise, and (iii) model misspecification.\",\n \"The third uncertainty has been previously overlooked, and we propose a potential solution with an encoder-decoder.\",\n \"Motivates a real-world anomaly detection use-case at Uber that uses Bayesian Neural Networks with uncertainty estimation to improve performance at scale.\",\n \"The rest of this paper is organized as follows: Section gives an overview of previous work on time series prediction for both classical and deep learning models, as well as the various approaches for uncertainty estimation in neural networks.\",\n \"The approach of Monte Carlo dropout (MC dropout) is used in this paper due to its simplicity, strong generalization ability, and scalability.\",\n \"In Section , we present our uncertainty estimation algorithm that accounts for the three different sources of uncertainty.\",\n \"Section provides detailed experiments to evaluate the model performance on Uber trip data, and lays out a successful application to large-scale anomaly detection for millions of metrics at Uber.\",\n \"Finally, Section concludes the paper.\",\n \"Classical time series models, such as those found in the standard $R$ forecast[2] package are popular methods to provide an univariate base-level forecast.\",\n \"These models usually require manual tuning to set seasonality and other parameters.\",\n \"Furthermore, while there are time series models that can incorporate exogenous variables [9], they suffer from the curse of dimensionality and require frequent retraining.\",\n \"To more effectively deal with exogenous variables, a combination of univariate modeling and a machine learning model to handle residuals was introduced in [10].\",\n \"The resulting two-stage model, however, is hard to tune, requires manual feature extraction and frequent retraining, which is prohibitive to millions of time series.\",\n \"Relatively recently, time series modeling based on LSTM [3] technique gained popularity due to its end-to-end modeling, ease of incorporating exogenous variables, and automatic feature extraction abilities [4].\",\n \"By providing a large amount of data across numerous dimensions, it has been shown that an LSTM approach can model complex extreme events by allowing nonlinear feature interactions [5], [6].\",\n \"While uncertainty estimation for classical forecasting models has been widely studied [11], this is not the case for neural networks.\",\n \"Approaches such as a modified loss function or using a collection of heterogenous networks [12] were proposed, however they require changes to the underlying model architecture.\",\n \"A more detailed review is given in the next section.\",\n \"In this work, we use a simple and scalable approach for deep model uncertainty estimation that builds on [13].\",\n \"This framework provides a generic error estimator that runs in production at Uber-scale to mitigate against bad decisions (e.g., false anomaly alerts) resulting from poor forecasts due to high prediction variance.\"\n ],\n [\n \"Bayesian Neural Networks\",\n \"Bayesian Neural Networks (BNNs) introduce uncertainty to deep learning models from a Bayesian perspective.\",\n \"By giving a prior to the network parameters $W$ , the network aims to find the posterior distribution of $W$ , instead of a point estimation.\",\n \"This procedure is usually referred to as posterior inference in traditional Bayesian models.\",\n \"Unfortunately, due to the complicated non-linearity and non-conjugacy in deep models, exact posterior inference is rarely available; in addition, most traditional algorithms for approximate Bayesian inference cannot scale to the large number of parameters in most neural networks.\",\n \"Recently, several approximate inference methods are proposed for Bayesian Neural Networks.\",\n \"Most approaches are based on variational inference that optimizes the variational lower bound, including stochastic search [14], variational Bayes [15], probabilistic backpropagation [16], Bayes by BackProp [17] and its extension [18].\",\n \"Several algorithms further extend the approximation framework to $\\\\alpha $ -divergence optimization, including [19], [20].\",\n \"We refer the readers to [21] for a more detailed and complete review of these methods.\",\n \"All of the aforementioned algorithms require different training methods for the neural network.\",\n \"Specifically, the loss function must be adjusted to different optimization problems, and the training algorithm has to be modified in a usually non-trivial sense.\",\n \"In practice, however, an out-of-the-box solution is often preferred, without changing the neural network architecture and can be directly applied to the previously trained model.\",\n \"In addition, most existing inference algorithms introduce extra model parameters, sometimes even double, which is difficult to scale given the large amount of parameters used in practice.\",\n \"This paper is inspired by the Monte Carlo dropout (MC dropout) framework proposed in [13] and [22], which requires no change of the existing model architecture and provides uncertainty estimation almost for free.\",\n \"Specifically, stochastic dropouts are applied after each hidden layer, and the model output can be approximately viewed as a random sample generated from the posterior predictive distribution [21].\",\n \"As a result, the model uncertainty can be estimated by the sample variance of the model predictions in a few repetitions.\",\n \"Details of this algorithm will be reviewed in the next section.\",\n \"The MC dropout framework is particularly appealing to practitioners because it is generic, easy to implement, and directly applicable to any existing neural networks.\",\n \"However, the exploration of its application to real-world problems remains extremely limited.\",\n \"This paper takes an important step forward by successfully adapting this framework to conduct time series prediction and anomaly detection at large scale.\"\n ],\n [\n \"Method\",\n \"Given a trained neural network $f^{\\\\hat{W}}(\\\\cdot )$ where $\\\\hat{W}$ represents the fitted parameters, as well as a new sample $x^*$ , our goal is to evaluate the uncertainty of the model prediction, $\\\\hat{y}^* = f^{\\\\hat{W}}(x^*)$ .\",\n \"Specifically, we would like to quantify the prediction standard error, $\\\\eta $ , so that an approximate $\\\\alpha $ -level prediction interval can be constructed by $[\\\\hat{y}^* - z_{\\\\alpha /2} \\\\eta , ~ \\\\hat{y}^* + z_{\\\\alpha /2} \\\\eta ]$ where $z_{\\\\alpha /2}$ is the upper $\\\\alpha /2$ quantile of a standard Normal.\",\n \"This prediction interval is critical for various tasks.\",\n \"For example, in anomaly detection, anomaly alerts will be fired when the observed value falls outside the constructed 95% interval.\",\n \"As a result, underestimating $\\\\eta $ will lead to high false positive rates.\",\n \"In the rest of this section, we will present our uncertainty estimation algorithm in Section REF , which accounts for three different sources of prediction uncertainties.\",\n \"This framework can be generalized to any neural network architectures.\",\n \"Then, in Section REF , we will present our neural network design for predicting time series at Uber.\"\n ],\n [\n \"Prediction Uncertainty\",\n \"We denote a neural network as function $f^W(\\\\cdot )$ , where $f$ captures the network architecture, and $W$ is the collection of model parameters.\",\n \"In a Bayesian neural network, a prior is introduced for the weight parameters, and the model aims to fit the optimal posterior distribution.\",\n \"For example, a Gaussian prior is commonly assumed: $W \\\\sim N(0, I)$ We further specify the data generating distribution $p(y \\\\,|\\\\, f^W(x))$ .\",\n \"In regression, we often assume $ y\\\\,|\\\\, W \\\\sim N(f^W(x), \\\\sigma ^2) $ with some noise level $\\\\sigma $ .\",\n \"In classification, the softmax likelihood is often used.\",\n \"For time series prediction, we will focus on the regression setting in this paper.\",\n \"Given a set of $N$ observations $X=\\\\lbrace x_1, ..., x_N\\\\rbrace $ and $Y=\\\\lbrace y_1, ..., y_N\\\\rbrace $ , Bayesian inference aims at finding the posterior distribution over model parameters $p(W \\\\,|\\\\, X, Y)$ .\",\n \"With a new data point $x^*$ , the prediction distribution is obtained by marginalizing out the posterior distribution: $ p(y^* \\\\,|\\\\, x^*) = \\\\int _W p(y^* \\\\,|\\\\, f^W(x^*)) p(W \\\\,|\\\\, X, Y)\\\\, dW $ In particular, the variance of the prediction distribution quantifies the prediction uncertainty, which can be further decomposed using law of total variance: $\\\\begin{split}\\\\textrm {Var}(y^* \\\\,|\\\\, x^*) & =\\\\textrm {Var}\\\\left[ \\\\mathbb {E}(y^* \\\\,|\\\\, W, x^*) \\\\right] +\\\\mathbb {E}\\\\left[\\\\textrm {Var}(y^* \\\\,|\\\\, W, x^*) \\\\right] \\\\\\\\& = \\\\textrm {Var}(f^W(x^*)) + \\\\sigma ^2\\\\end{split}$ Immediately, we see that the variance is decomposed into two terms: (i) $\\\\textrm {Var}(f^W(x^*))$ , which reflects our ignorance over model parameter $W$ , referred to as the model uncertainty; and (ii) $\\\\sigma ^2$ which is the noise level during data generating process, referred to as the inherent noise.\",\n \"An underlying assumption for (REF ) is that $y^*$ is generated by the same procedure.\",\n \"However, this is not always the case in practice.\",\n \"In anomaly detection, in particular, it is expected that certain time series will have unusual patterns, which can be very different from the trained model.\",\n \"Therefore, we propose that a complete measurement of prediction uncertainty should be a combination from three sources: (i) model uncertainty, (ii) model misspecification, and (iii) inherent noise level.\",\n \"The following sections provide details on how we handle these three terms.\",\n \"The key to estimating model uncertainty is the posterior distribution $p(W\\\\,|\\\\, X, Y)$ , also referred to as Bayesian inference.\",\n \"This is particularly challenging in neural networks because the non-conjugacy due to nonlinearities.\",\n \"There have been various research efforts on approximate inference in deep learning (see Section REF for a review).\",\n \"Here, we follow the idea in [13] and [22] to approximate model uncertainty using Monte Carlo dropout (MC dropout).\",\n \"The algorithm proceeds as follows: given a new input $x^*$ , we compute the neural network output with stochastic dropouts at each layer.\",\n \"That is, randomly dropout each hidden unit with certain probability $p$ .\",\n \"This stochastic feedforward is repeated $B$ times, and we obtain $\\\\lbrace \\\\hat{y}^*_{(1)}, ..., \\\\hat{y}^*_{(B)}\\\\rbrace $ .\",\n \"Then the model uncertainty can be approximated by the sample variance: $\\\\widehat{\\\\textrm {Var}}(f^W(x^*)) = \\\\frac{1}{B} \\\\sum _{b=1}^B \\\\left(\\\\hat{y}^*_{(b)} - \\\\overline{\\\\hat{y}}^* \\\\right)^2$ where $\\\\overline{\\\\hat{y}}^* = \\\\frac{1}{B} \\\\sum _{b=1}^B \\\\hat{y}^*_{(b)}$ [13].\",\n \"There has been recent work done on choosing the optimal dropout probability $p$ adaptively by treating it as part of the model parameter, but this approach requires modifying the training phase [12].\",\n \"In practice, we find that the uncertainty estimation is usually robust within a reasonable range of $p$ .\"\n ],\n [\n \"Model misspecification\",\n \"Next, we address the problem of capturing potential model misspecification.\",\n \"In particular, we would like to capture the uncertainty when predicting unseen samples with very different patterns from the training data set.\",\n \"We propose to account for this source of uncertainty by introducing an encoder-decoder to the model framework.\",\n \"The idea is to train an encoder that extracts the representative features from a time series, in the sense that a decoder can reconstruct the time series from the encoded space.\",\n \"At test time, the quality of encoding of each sample will provide insight on how close it is to the training set.\",\n \"Another way to think of this approach is that we first fit a latent embedding space for all training time series using an encoder-decoder framework.\",\n \"Then, we measure the distance between test cases and training samples in the embedded space.\",\n \"The next question is how to incorporate this uncertainty in the variance calculation.\",\n \"Here, we take a principled approach by connecting the encoder, $g(\\\\cdot )$ , with a prediction network, $h(\\\\cdot )$ , and treat them as one large network $f = h(g(\\\\cdot ))$ during inference.\",\n \"REF illustrates such an inference network, and Algorithm REF presents the MC dropout algorithm.\",\n \"Specifically, given an input time series $x = \\\\lbrace x_1, ..., x_T\\\\rbrace $ , the encoder $g(\\\\cdot )$ constructs the learned embedding $e = g(x)$ , which is further concatenated with external features, and the final vector is fed to the final prediction network $h$ .\",\n \"During this feedforward pass, MC dropout is applied to all layers in both the encoder $g$ and the prediction network $h$ .\",\n \"As a result, the random dropout in the encoder perturbs the input intelligently in the embedding space, which accounts for potential model misspecification and gets further propagated through the prediction network.\",\n \"Here, variational dropout for recurrent neural networks [22] is applied to the LSTM layers in the encoder, and regular dropout [13] is applied to the prediction network.\",\n \"[H] [1] data $x^*$ , encoder $g(\\\\cdot )$ , prediction network $h(\\\\cdot )$ , dropout probability $p$ , number of iterations $B$ prediction $\\\\hat{y}^*_{mc}$ , uncertainty $\\\\eta _1$ $b = 1$ to $B$ $e^*_{(b)} \\\\leftarrow $ VariationalDropout$(g(x^*), p)$ $z^*_{(b)} \\\\leftarrow \\\\textrm {Concatenate}(e^*_{(b)}, \\\\textrm {extFeatures})$ $\\\\hat{y}^*_{(b)} \\\\leftarrow $ Dropout $(h(z^*_{(b)}), p)$ // prediction $\\\\hat{y}^*_{mc} \\\\leftarrow \\\\frac{1}{B} \\\\sum _{b=1}^B \\\\hat{y}^*_{(b)}$ // model uncertainty and misspecification $\\\\eta _1^2 \\\\leftarrow \\\\frac{1}{B} \\\\sum _{b=1}^B (\\\\hat{y}^*_{(b)} - \\\\hat{y}^* )^2$ $\\\\hat{y}^*_{mc}, \\\\, \\\\eta _1$ MCdropout\"\n ],\n [\n \"Inherent noise\",\n \"Finally, we estimate the inherent noise level $\\\\sigma ^2$ .\",\n \"In the original MC dropout algorithm [13], this parameter is implicitly determined by a prior over the smoothness of $W$ .\",\n \"As a result, the model could end up with drastically different estimations of the uncertainty level depending on this pre-specified smoothness (see [21], chapter 4).\",\n \"This dependency is undesirable in anomaly detection, because we want the uncertainty estimation to also have robust frequentist coverage, but it is rarely the case that we would know the correct noise level a priori.\",\n \"Here, we propose a simple and adaptive approach that estimates the noise level via the residual sum of squares, evaluated on an independent held-out validation set.\",\n \"Specifically, let $f^{\\\\hat{W}}(\\\\cdot )$ be the fitted model on training data, and $X^{\\\\prime }=\\\\lbrace x^{\\\\prime }_1, ..., x^{\\\\prime }_V\\\\rbrace , Y^{\\\\prime }=\\\\lbrace y^{\\\\prime }_1, ..., y^{\\\\prime }_V\\\\rbrace $ be an independent validation set, then we estimate $\\\\sigma ^2$ via $\\\\hat{\\\\sigma }^2 = \\\\frac{1}{V} \\\\sum _{v=1}^V \\\\left( y^{\\\\prime }_v - f^{\\\\hat{W}}(x^{\\\\prime }_v) \\\\right)^2 \\\\,.$ Note that $(X^{\\\\prime }, Y^{\\\\prime })$ are independent from $f^{\\\\hat{W}}(\\\\cdot )$ , and if we further assume that $f^{\\\\hat{W}}(x^{\\\\prime }_v)$ is an unbiased estimation of the true model, we have $\\\\begin{split}\\\\mathbb {E}(\\\\hat{\\\\sigma }^2) &=\\\\sigma ^2 + \\\\frac{1}{V} \\\\sum _{v=1}^V \\\\mathbb {E} \\\\left[ f^{\\\\hat{W}}(x^{\\\\prime }_v) - f^{W}(x^{\\\\prime }_v) \\\\right]^2 \\\\\\\\&= \\\\sigma ^2 + \\\\textrm {Var}_{\\\\rm TRN}(f^{\\\\hat{W}}(x^{\\\\prime }_v))\\\\end{split}$ where $\\\\textrm {Var}_{\\\\rm TRN}$ is w.r.t the training data, which decreases as the training sample size increases, and $\\\\rightarrow 0$ as the training sample size $N \\\\rightarrow \\\\infty $ .\",\n \"Therefore, $\\\\hat{\\\\sigma }^2$ provides an asymptotically unbiased estimation on the inherent noise level.\",\n \"In the finite sample scenario, it always overestimates the noise level and tends to be more conservative.\",\n \"The final inference algorithm combines inherent noise estimation with MC dropout, and is presented in Algorithm REF .\",\n \"[H] [1] data $x^*$ , encoder $g(\\\\cdot )$ , prediction network $h(\\\\cdot )$ , dropout probability $p$ , number of iterations $B$ prediction $\\\\hat{y}^*$ , predictive uncertainty $\\\\eta $ // prediction, model uncertainty and misspecification $\\\\hat{y}^*, \\\\, \\\\eta _1 \\\\leftarrow $ MCdropout $(x^*, g, h, p, B)$ // Inherent noise $x^{\\\\prime }_v$ in validation set $\\\\lbrace x^{\\\\prime }_1, ..., x^{\\\\prime }_V\\\\rbrace $ $\\\\hat{y^{\\\\prime }}_v \\\\leftarrow h(g(x^{\\\\prime }_v))$ $\\\\eta _2^2 \\\\leftarrow \\\\frac{1}{V} \\\\sum _{v=1}^V \\\\left( \\\\hat{y^{\\\\prime }}_v - y^{\\\\prime }_v \\\\right)^2$ // total prediction uncertainty $\\\\eta \\\\leftarrow \\\\sqrt{\\\\eta _1^2 + \\\\eta _2^2}$ $\\\\hat{y}^*, \\\\, \\\\eta $ Inference\"\n ],\n [\n \"Model Design\",\n \"The complete architecture of the neural network is shown in REF .\",\n \"The network contains two major components: (i) an encoder-decoder framework that captures the inherent pattern in the time series, which is learned during pre-training step, and (ii) a prediction network that takes input from both the learned embedding from encoder-decoder, as well as any potential external features to guide the prediction.\",\n \"We discuss the two components in more details below.\",\n \"Figure: Neural network architecture, with a pre-training phase using a LSTM encoder-decoder, followed by a prediction network, with input being the learned embedding concatenated with external features.\"\n ],\n [\n \"Encoder-decoder\",\n \"Prior to fitting the prediction model, we first conduct a pre-training step to fit an encoder that can extract useful and representative embeddings from a time series.\",\n \"The goals are to ensure that (i) the learned embedding provides useful features for prediction and (ii) unusual input can be captured in the embedded space, which will get further propagated to the prediction network in the next step.\",\n \"Here, we use an encoder-decoder framework with two-layer LSTM cells.\",\n \"Specifically, given a univariate time series $\\\\lbrace x_t\\\\rbrace _t$ , the encoder reads in the first $T$ timestamps $\\\\lbrace x_1, ..., x_T\\\\rbrace $ , and constructs a fixed-dimensional embedding state.\",\n \"After then, from this embedding state, the decoder constructs the following $F$ timestamps $\\\\lbrace x_{T+1}, ..., x_{T+F}\\\\rbrace $ with guidance from $\\\\lbrace x_{T-F+1}, ..., x_T\\\\rbrace $ ( REF , bottom panel).\",\n \"The intuition is that in order to construct the next few timestamps, the embedding state must extract representative and meaningful features from the input time series.\",\n \"This design is inspired from the success of video representation learning using a similar architecture [23].\"\n ],\n [\n \"Prediction network\",\n \"After the encoder-decoder is pre-trained, it is treated as an intelligent feature-extraction blackbox.\",\n \"Specifically, the last LSTM cell states of the encoder are extracted as learned embedding.\",\n \"Then, a prediction network is trained to forecast the next one or more timestamps using the learned embedding as features.\",\n \"In the scenario where external features are available, these can be concatenated to the embedding vector and passed together to the final prediction network.\",\n \"Here, we use a multi-layer perceptron as the prediction network.\",\n \"We will show in Section REF that the learned embedding from the encoder successfully captures interesting patterns from the input time series.\",\n \"In addition, including external features significantly improves the prediction accuracy during holidays and special events (see Section )\"\n ],\n [\n \"Inference\",\n \"After the full model is trained, the inference stage involves only the encoder and the prediction network ( REF , left panel).\",\n \"The complete inference algorithm is presented in Algorithm REF , where the prediction uncertainty, $\\\\eta $ , contains two terms: (i) the model and misspecification uncertainty, estimated by applying MC dropout to both the encoder and the prediction network, as presented in Algorithm REF ; and (ii) the inherent noise level, estimated by the residuals on a held-out validation set.\",\n \"Finally, an approximate $\\\\alpha $ -level prediction interval is constructed by $[\\\\hat{y}^* - z_{\\\\alpha /2} \\\\eta , ~ \\\\hat{y}^* + z_{\\\\alpha /2} \\\\eta ]$ , where $z_{\\\\alpha /2}$ is the upper $\\\\alpha /2$ quantile of a standard Normal.\",\n \"Two hyper-parameters need to be specified in Algorithm REF : the dropout probability, $p$ , and the number of iterations, $B$ .\",\n \"As for the dropout probability, we find in our experiments that the uncertainty estimation is relatively stable across a range of $p$ , and we choose the one that achieves the best performance on the validation set.\",\n \"As for the number of iterations, the standard error of the estimated prediction uncertainty is proportional to $1/\\\\sqrt{B}$ .\",\n \"We measure the standard error across different repetitions, and find that a few hundreds of iterations are usually suffice to achieve a stable estimation.\"\n ],\n [\n \"Evaluation\",\n \"This section contains two sets of results.\",\n \"We first evaluate the model performance on a moderately sized data set of daily trips processed by the Uber platform.\",\n \"We will evaluate the prediction accuracy and the quality of uncertain estimation during both holidays and non-holidays.\",\n \"We will also present how the encoder recognizes the day of the week pattern in the embedding space.\",\n \"Next, we will illustrate the application of this model to real-time large-scale anomaly detection for millions of metrics at Uber.\"\n ],\n [\n \"Experimental settings\",\n \"In this section, we illustrate the model performance using the daily completed trips over four years across eight representative large cities in U.S. and Canada, including Atlanta, Boston, Chicago, Los Angeles, New York City, San Francisco, Toronto, and Washington D.C. We use three years of data as the training set, the following four months as the validation set, and the final eight months as the testing set.\",\n \"The encoder-decoder is constructed with two-layer LSTM cells, with 128 and 32 hidden states, respectively.\",\n \"The prediction network has three fully connected layers with tanh activation, with 128, 64, and 16 hidden units, respectively.\",\n \"Samples are constructed using a sliding window with step size one, where each sliding window contains the previous 28 days as input, and aims to forecast the upcoming day.\",\n \"The raw data are log-transformed to alleviate exponential effects.\",\n \"Next, within each sliding window, the first day is subtracted from all values, so that trends are removed and the neural network is trained for the incremental value.\",\n \"At test time, it is straightforward to revert these transformations to obtain predictions at the original scale.\"\n ],\n [\n \"Prediction performance\",\n \"We compare the prediction accuracy among four different models: Last-Day: A naive model that uses the last day's completed trips as the prediction for the next day.\",\n \"QRF: Based on the naive last-day prediction, a quantile random forest (QRF) is further trained to estimate the holiday lifts, i.e., the ratio to adjust the forecast during holidays.\",\n \"The final prediction is calculated from the last-day forecast multiplied by the estimated ratio.\",\n \"LSTM: A vanilla LSTM model with similar size as our model.\",\n \"Specifically, a two-layer sacked LSTM is constructed, with 128 and 32 hidden states, respectively, followed by a fully connected layer for the final output.\",\n \"This neural network also takes 28 days as input, and predicts the next day.\",\n \"Our Model: Our model that combines an encoder-decoder and a prediction network, as described in REF .\",\n \"Table REF reports the Symmetric Mean Absolute Percentage Error (SMAPE) of the four models, evaluated on the testing set.\",\n \"We see that using a QRF to adjust for holiday lifts is only slightly better than the naive prediction.\",\n \"On the other hand, a vanilla LSTM neural network provides an average of 26% improvement across the eight cities.\",\n \"As we further incorporate the encoder-decoder framework and introduce external features for holidays to the prediction network ( REF ), our proposed model achieves another 36% improvement in prediction accuracy.\",\n \"Note that when using LSTM and our model, only one generic model is trained, where the neural network is not tuned for any city-specific patterns; nevertheless, we still observe significant improvement on SMAPE across all cities when compared to traditional approaches.\",\n \"Table: SMAPE of Four Different Prediction Models, Evaluated on the Test Data.Finally, REF visualizes the true values and our predictions during the testing period in San Francisco as an example.\",\n \"We observe that accurate predictions are achieved not only in regular days, but also during holiday seasons.\",\n \"Figure: Daily completed trips in San Francisco during eight months of the testing set.\",\n \"True values are shown with the orange solid line, and predictions are shown with the blue dashed line, where the 95% prediction band is shown as the grey area.\",\n \"Exact values are anonymized.\"\n ],\n [\n \"Uncertainty estimation\",\n \"Next, we evaluate the quality of our uncertainty estimation by calibrating the empirical coverage of the prediction intervals.\",\n \"Here, the dropout probability is set to be 5% at each layer, and Table REF reports the empirical coverage of the 95% predictive intervals under three different scenarios: PredNet: Use only model uncertainty estimated from MC dropout in the prediction network, with no dropout layers in the encoder.\",\n \"Enc+Pred: Use MC dropout in both the encoder and the prediction network, but without the inherent noise level.\",\n \"This is the term $\\\\eta _1$ in Algorithm REF .\",\n \"Enc+Pred+Noise: Use the full prediction uncertainty $\\\\eta $ as presented in Algorithm REF , including $\\\\eta _1$ as in 2), as well as the inherent noise level $\\\\eta _2$ .\",\n \"Table: Empirical Coverage of 95% Predictive Intervals, Evaluated on the Test Data.By comparing PredNet with Enc+Pred, it is clear that introducing MC dropout to the encoder network is critical, which significantly improves the empirical coverage from 78% to 90% by capturing potential model misspecification.\",\n \"In addition, by further accounting for the inherent noise level, the empirical coverage of the final uncertainty estimation, Enc+Pred+Noise, nicely centers around 95% as desired.\",\n \"One important use-case of the uncertainty estimation is to provide insight for unusual patterns in the time series.\",\n \"REF shows the estimated predictive uncertainty on six U.S. holidays in the testing data.\",\n \"We see that New Year's Eve has significantly higher uncertainty than all other holidays.\",\n \"This pattern is consistent with our previous experience, where New Year's Eve is usually the most difficult day to predict.\",\n \"Figure: Estimated prediction standard deviations on six U.S. holidays during testing period for eight cities.\",\n \"Exact values are anonymized.\"\n ],\n [\n \"Embedding features\",\n \"As illustrated previously, the encoder is critical for both improving prediction accuracy, as well as for estimating prediction uncertainty.\",\n \"One natural follow-up question is whether we can interpret the embedding features extracted by the encoder.\",\n \"This can also provide valuable insights for model selection and anomaly detection.\",\n \"Here, we visualize our training data, each being a 28-day time series segment, in the embedding space.\",\n \"We use the last LSTM cell in the encoder, and project its cell states to 2D for visualization using PCA ( REF ).\",\n \"The strongest pattern we observe is day of the week, where weekdays and weekends form different clusters, with Fridays usually sitting in between.\",\n \"We do not observe city-level clusters, which is probably due to the fact all cities in this data set are large cities in North America, where riders and drivers tend to have similar behaviors.\",\n \"Figure: Training set of time series, visualized in the embedding space.\",\n \"Each point represents a 28-day segment, colored by the day of the week of the last day.\",\n \"We evaluate the cell states of the two LSTM layers, where the first layer with dimension 128 is plotted on the left, and second layer with dimension 32 is plotted on the right.\",\n \"PCA is used to project into 2D space for visualization.\"\n ],\n [\n \"Application to Anomaly Detection\\nat Uber\\n\",\n \"At Uber, we track millions of metrics each day to monitor the status of various services across the company.\",\n \"One important application of uncertainty estimation is to provide real-time anomaly detection and deploy alerts for potential outages and unusual behaviors.\",\n \"A natural approach is to trigger an alarm when the observed value falls outside of the 95% predictive interval.\",\n \"There are two main challenges we need to address in this application: Scalability: In order to provide real-time anomaly detection at the current scale, each predictive interval must be calculated within a few milliseconds during inference stage.\",\n \"Performance: With highly imbalanced data, we aim to reduce the false positive rate as much as possible to avoid unnecessary on-call duties, while making sure the false negative rate is properly controlled so that real outages will be captured.\"\n ],\n [\n \"Scalability\",\n \"Our model inference is implemented in Go.\",\n \"Our implementation involves efficient matrix manipulation operations, as well as stochastic dropout by randomly setting hidden units to zero with pre-specified probability.\",\n \"A few hundred stochastic passes are executed to calculate the prediction uncertainty, which is updated every few minutes for each metric.\",\n \"We find that the uncertainty estimation step adds only a small amount of computation overhead and can be conducted within ten milliseconds per metric.\"\n ],\n [\n \"Performance\",\n \"Here, we illustrate the precision and recall of this framework on an example data set containing 100 metrics with manual annotation available, where 17 of them are true anomalies.\",\n \"Note that the neural network was previously trained on a separate and much larger data set.\",\n \"By adding MC dropout layers in the neural network, the estimated predictive intervals achieved 100% recall rate and a 80.95% precision rate.\",\n \"REF visualizes the neural network predictive intervals on four representative metrics, where alerts are correctly fired for two of them.\",\n \"When applying this framework to all metrics, we observe a 4% improvement in precision compared to the previous ad-hoc solution, which is substantial at Uber's scale.\",\n \"Figure: Four example metrics during a 12-hour span, and anomaly detection is performed for the following 30 minutes.\",\n \"All metrics are evaluated by minutes.\",\n \"The neural network constructs predictive intervals for the following 30 minutes, visualized by the shaded area in each plot.\",\n \"(a) A normal metric with large fluctuation, where the observation falls within the predictive interval.\",\n \"(b) A normal metric with small fluctuation, and an unusual inflation has just ended.\",\n \"The predictive interval still captures the observation.\",\n \"(c) An anomalous metric with a single spike that falls outside the predictive interval.\",\n \"(d) An anomalous metric with two consecutive spikes, also captured by our model.\"\n ],\n [\n \"Conclusion\",\n \"We have presented an end-to-end neural network architecture for uncertainty estimation used at Uber.\",\n \"Using the MC dropout technique and model misspecification distribution, we showed a simple way to provide uncertainty estimation for a neural network forecast at scale while providing a 95% uncertainty coverage.\",\n \"A critical feature about our framework is its applicability to any neural network without modifying the underlying architecture.\",\n \"We have used the proposed uncertainty estimate to measure special event (e.g., holiday) uncertainty and to improve anomaly detection accuracy.\",\n \"For special event uncertainty estimation, we found New Year's Eve to be the most uncertain time.\",\n \"Using the uncertainty information, we adjusted the confidence bands of an internal anomaly detection model to improve precision during high uncertainty events, resulting in a 4% accuracy improvement, which is large given the number of metrics we track at Uber.\",\n \"Our future work will be focused on utilizing the uncertainty information for neural network debugging during high error periods.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01907"}}},{"rowIdx":895,"cells":{"text":{"kind":"list like","value":[["Half-vicinity model and a phase diagram for quantum oscillations in\n confined and degenerate Fermi gases"],["Abstract We propose an analytical model for the accurate calculation of size and density dependent quantum oscillations in thermodynamic and transport properties of confined and degenerate non-interacting Fermi gases.","We provide a universal, material independent, recipe that explicitly separates oscillatory quantum regime from stationary classical regime.","Our model quite accurately estimates quantum oscillations depending on confinement and degeneracy.","We construct a phase diagram representing stationary and oscillatory regimes on degeneracy-confinement space.","Analytical expressions of phase transition interfaces are derived for different dimensions.","The critical point on the phase diagram, which separates entirely stationary and entirely oscillatory regions, is determined and their aspect ratio dependencies are examined.","Quantum oscillations as well as their periods are analytically expressed for one-dimensional case.","Accuracy of our model is verified through quantum oscillations in electronic specific heat capacity.","We also compare the predictions of our half-vicinity model, based on bounded sums, with those of infinite sums, for the oscillatory violation of entropy-heat capacity equivalence in degenerate limit to show the accuracy of our model.","Furthermore, similarities between functional behaviors of total occupancy variance and conventional density of states functions at Fermi level are discussed."],["Introduction","A great deal of attention has been given to the physics of low-dimensional nanostructures associated with the intensive developments of nanoscience and nanotechnology in recent years.","When the mean de Broglie wavelength of particles inside a domain becomes comparable to the size of the domain, quantum size effects start to reveal themselves and their presence has to be taken into account in the calculation of the physical properties of such nanoscale systems [1], [2], [3], [4].","In Fermi gases, instead of fully occupied states, only partially occupied ones around Fermi level contribute to some thermodynamic quantities like entropy and specific heat and transport properties like electrical and thermal conductivities.","Definitional expressions of such quantities contain the variance of distribution function instead of the distribution function itself.","Hence, variance function plays a crucial role in the behavior of these quantities.","In degenerate and strongly confined ideal Fermi gases, the quantities containing variance function exhibit oscillatory behaviors.","The basic reason of this behavior is the oscillatory behavior of total occupancy variance (TOV) function due to changes in size and density.","It has been shown experimentally in literature that, size and density dependent oscillations occur in some certain thermodynamic and transport properties of charge carriers in semiconductors/metals or Fermi gases confined at nano domains in general.","First experimental studies of size dependent oscillations due to quantum size effects has been reported in late 60's [5], [6], [7], [8].","Thermodynamic properties like entropy and specific heat capacity at constant volume, thermoelectric properties such as Seebeck coefficient and electronic transport properties such as charge carrier mobility, electrical and thermal conductivities are some examples of these oscillatory quantities [4], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35].","Examination of size dependent oscillations also has crucial importance in superconductors and topological insulators [33], [34], [36], [37], [38], [39].","Size and density dependent oscillations in aforementioned quantities are attributed to the quantization of energy spectrum and fluctuations in density of states around Fermi level at nanoscale [15], [16], [17], [18], [19], [20], [40].","For Fermi systems with linear energy dispersion relation (e.g.","a Fermi system under an applied magnetic field or harmonic trap potential), quantum oscillations and its implications are extensively studied especially for specific materials [41], [15], [16], [17], [18], [19], [20], [42], [43].","For the systems obeying quadratic energy-momentum dispersion relation, on the other hand, there is a need for a mathematical model predicting quantum oscillations and explaining their nature in a comprehensive and material/case independent way.","Moreover, a phase diagram separating oscillatory and stationary regimes on confinement-degeneracy space has never been proposed for any dispersion relation.","Constructing a mathematical model and a phase diagram may help to deeply understand and efficiently control the oscillations to design novel nanoscale devices [20], [27].","In this article, we propose a theoretical model, half-vicinity model (HVM), that accurately predicts the oscillations appearing in confined Fermi gases.","A rectangular confinement domain is chosen, because it is one of the most suitable geometries for common manufacturing methods of low dimensional nanostructures.","In the following section, by examining the nature of TOV of quantum states, we introduce the model by defining a half-vicinity shell around Fermi level.","$d$ -dimensional discrete and continuum expressions of TOV function are given and thickness of half-vicinity shell, which plays an important role in HVM, is obtained for various dimensions.","For 1D case, we obtain analytical expressions for TOV oscillations and their periods.","In Sec.","III, a phase diagram of quantum oscillations is established and phase transition interfaces between stationary (classical, continuous) and oscillatory (quantum, discrete) regimes are analytically given for 1D, 2D and 3D cases.","Critical confinement and degeneracy values, which separates entirely stationary and oscillatory regions, are also examined by considering their aspect ratio dependencies.","In Sec.","IV, results of exact (definitional expressions based on infinite sums) and HVMs are compared for size and density dependent oscillations in the electronic specific heat capacity of strongly degenerate and confined ideal Fermi gases.","Furthermore, broken equivalence of entropy-heat capacity in quantum degenerate limit of ideal Fermi gases is also well predicted by HVM.","Finally, we discuss (Sec.","V) the relationship between TOV and conventional density of states functions at Fermi level."],["Half-vicinity model for total occupancy variance","Oscillations in thermodynamic and transport quantities originate from the nature of occupancy variance function.","For this reason, in order to establish a model to understand the nature of oscillations, we need to examine TOV in detail."],["Total occupancy variance revisited","Occupancies of quantum states for Fermions are described by Fermi-Dirac distribution function, $f=g_s/\\left[\\exp (-\\Lambda +\\tilde{\\varepsilon })+1\\right]$ where $g_s$ is spin degree of freedom, $\\Lambda =\\mu /(k_BT)$ is dimensionless chemical potential (or degeneracy parameter) indicating the strength of degeneracy, in which $\\mu $ is chemical potential, $k_B$ is Boltzmann's constant, $T$ is temperature.","$\\tilde{\\varepsilon }$ represents dimensionless energy eigenvalues (normalized to thermal energy $k_B T$ ).","For quadratic dispersion relation of massive particles confined in a $d$ -dimensional rectangular domain, $\\tilde{\\varepsilon }=\\varepsilon /(k_BT)=(\\alpha _1 i_1)^2+\\cdots +(\\alpha _d i_d)^2$ where $i_n$ is quantum state variable for a particular direction $n=\\lbrace 1,2,\\cdots ,d\\rbrace $ .","Here $\\alpha _n$ is confinement parameter indicating the strength of quantum confinement in a particular direction $n$ and defined as $\\alpha _n=h/\\left(\\sqrt{8m k_B T}L_n\\right)$ where $h$ is Planck's constant, $m$ is mass of the particle and $L_n$ is domain size in direction $n$ .","Derivative of Fermi-Dirac distribution function with respect to $\\Lambda $ , a bell-shaped function peaked at Fermi level, is called occupancy variance function.","Occupancy variance of an energy state $\\tilde{\\varepsilon }=\\tilde{\\varepsilon }(i_1,\\ldots ,i_d)$ is $\\sigma ^2(i_1,\\ldots ,i_d)=\\frac{\\partial f}{\\partial \\Lambda }=-\\frac{\\partial f}{\\partial \\tilde{\\varepsilon }}=\\frac{g_s}{4}\\operatorname{sech}^2\\left(\\frac{\\tilde{\\varepsilon }-\\Lambda }{2}\\right).$ In literature, it's sometimes called also as thermal broadening function, since it represents thermal broadening of energy states around Fermi level [44].","Due to the fact that Fermi-Dirac distribution is a Bernoulli-type distribution (basically a sigmoid function), occupancy variance can also be represented by a Bernoulli-type variance form $\\sigma ^2=g_s\\tilde{f}(1-\\tilde{f})$ , where $\\tilde{f}=f/g_s$ is the spin normalized Fermi-Dirac distribution function.","Spin factor $g_s$ is taken simply two in all numerical calculations in this article.","In order to understand the behavior of oscillatory quantities and how occupancy variance behaves under the accumulation operators such as summation or integration, we need to investigate the nature of TOV.","TOV of Fermi particles inside a $d$ -dimensional box is written in its exact form based on infinite sums as $\\Sigma _D^2=\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }\\sigma ^2(i_1,\\ldots ,i_d),$ where subscript $D$ stands for discrete calculations.","When the summations in Eq.","(2) are calculated by using the first two terms of Poisson summation formula (PSF) [45], [30], the following expression can be obtained after some mathematical operations, $\\begin{split}\\Sigma _{W}^2=& g_s\\sum _{n=0}^d\\frac{(-1)^{d-n+1}\\hspace{2.22214pt}\\pi ^{n/2}}{2^d \\hspace{2.22214pt}d^{1-\\Theta (n)\\Theta (d-n)}}Li_{\\frac{n-2}{2}}\\left[-\\exp (\\Lambda )\\right] \\\\& \\times \\sum _{k=1}^{d}\\prod _{m=k}^{k+n-1}{\\alpha _{\\text{mod}(m,d)+1}^{-1}},\\end{split}$ where $Li$ is polylogarithm function [46] and $\\Theta $ is left-continuous Heaviside step function.","Since the expressions calculated by using the first two terms of PSF are equivalent to those obtained from Weyl conjecture [45], [3], [47], we denote this expression by subscript $W$ .","In strongly degenerate conditions ($\\Lambda >>1$ ), by using the asymptotic forms of polylogarithm functions [48], Eq.","(3) can be approximated by $\\begin{split}\\Sigma _{W\\hspace{-2.77771pt}A}^2=& g_s\\sum _{n=0}^d\\frac{(-1)^{d-n}\\hspace{2.22214pt}\\pi ^{n/2}}{2^d \\hspace{2.22214pt}d^{1-\\Theta (n)\\Theta (d-n)}}\\frac{\\Lambda ^{(n-2)/2}}{\\Gamma (n/2)} \\\\& \\times \\sum _{k=1}^{d}\\prod _{m=k}^{k+n-1}{\\alpha _{\\text{mod}(m,d)+1}^{-1}},\\end{split}$ where $\\Gamma $ is gamma function.","When confinement parameters are much smaller than unity, namely in macroscale, summations may be converted directly into integrals by using continuum approximation and continuous TOV is obtained $\\Sigma _C^2=-\\frac{g_s\\pi ^{d/2}}{2^d \\alpha _1 \\cdots \\alpha _d}Li_{\\frac{d-2}{2}}\\left[-\\exp (\\Lambda )\\right],$ where $C$ subscript denotes the continuous expressions obtained under continuum approximation.","In strongly degenerate case ($\\Lambda >>1$ ), Eq.","(5) can be approximated by using asymptotic expansions of polylogarithm functions $\\Sigma _{C\\hspace{-2.77771pt}A}^2=\\frac{g_s\\pi ^{d/2}}{2^d \\alpha _1 \\cdots \\alpha _d}\\frac{\\Lambda ^{(d-2)/2}}{\\Gamma (d/2)}.$ As it should be, Eqs.","(5) and (6) are actually the first (bulk) terms of Eqs.","(3) and (4) respectively.","Expressions of thermodynamic and transport properties exhibiting oscillatory behavior contain Eq.","(1) multiplied by some relevant functions of the quantity to be calculated.","Then, it's proper to construct our HVM on TOV, in order to predict oscillatory behaviors in thermodynamic and transport properties."],["Basic concept of half-vicinity model","Contributions to oscillatory physical quantities substantially come from partially occupied states around Fermi level.","Under strongly confined and degenerate conditions, sharpness of the occupancy variance function around Fermi level increases and only few states contribute to TOV.","In that case, it becomes possible to make a quantitative treatment of these states by constructing a shell which contains them inside.","Therefore, for the accurate calculation of oscillatory physical quantities, instead of summing over all possible quantum states from unity to infinity, considering only those several states around Fermi level (inside the half-vicinity shell) is enough to represent the oscillations.","Half-vicinity shell in state space is defined as the interval containing the states in the half-vicinities ($\\pm 1/2$ ) of Fermi level.","It can be extended to any dimension, by considering the $\\pm 1/2$ intervals of Fermi level in each direction at quantum state space.","Magnitude of the contribution of each state to TOV is determined by its proximity to the Fermi level.","Contribution of a state is maximum when it lies exactly on the Fermi level, and exponentially decays as the state is away from it.","The number of half-vicinity states and their proximities to Fermi level change with degeneracy and/or confinement.","When confinement becomes stronger, spacing between energy levels increases and causes a decrement in number of half-vicinity states.","Because of this decrement, addition (removal) of a state to (from) half-vicinity shell leads to a strong change in TOV.","This causes strong fluctuations both in number of half-vicinity states and their proximities to the Fermi level.","In other words, half-vicinity states reorganize their distribution and proximities to the Fermi level when for instance sizes of the domain or density of the gas change.","Each state gives distinct contribution to TOV.","For macro domains with highly populated Fermi level, effect of this reorganization becomes statistically insignificant.","Conversely, in confined and degenerate Fermi gases these reorganization of half-vicinity states manifest themselves as quantum oscillations in several thermodynamic and transport quantities due to changes in size and density.","For strongly confined and degenerate Fermi gases, TOV, Eq.","(2), can approximately be calculated by restricting the summations over infinite number of quantum states to only the states inside the constructed half-vicinity shell.","TOV based on HVM is then given by $\\Sigma _{HV}^2=\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }\\sigma ^2w_{HV}.$ Here, half-vicinity window function is define as $w_{HV}=\\Theta \\left(\\Lambda -\\tilde{\\varepsilon }_{-1/2}\\right)\\Theta \\left(\\tilde{\\varepsilon }_{1/2}-\\Lambda \\right),$ where $\\Theta $ is Heaviside step function and $\\tilde{\\varepsilon }_{\\pm 1/2}=\\alpha _1^2\\left(i_1\\pm 1/2\\right)^2+\\ldots +\\alpha _d^2(i_d\\pm 1/2)^2$ give upper and lower energy bounds of half-vicinity window.","Window function $w_{HV}$ filters the states which do not significantly contribute to the oscillatory quantities and serves as a half-vicinity states pass filter, that restricts the upper and lower limits of summations.","By definition, in state space, half-vicinity shell thickness is equal to unity in any particular direction, which is the minimum possible thickness in state space for each direction.","On the other hand, in energy space its thickness varies with the strength of confinement and degeneracy.","Half-vicinity window can equivalently be written as $w_{HV}=\\Theta (\\tilde{\\varepsilon }_H-\\tilde{\\varepsilon })\\Theta (\\tilde{\\varepsilon }-\\tilde{\\varepsilon }_L),$ where $\\tilde{\\varepsilon }_H=\\Lambda +\\delta _{\\tilde{\\varepsilon }_H}$ , $\\tilde{\\varepsilon }_L=\\Lambda -\\delta _{\\tilde{\\varepsilon }_L}$ , $\\delta _{\\tilde{\\varepsilon }_H}=\\delta _{\\tilde{\\varepsilon }}/2+\\tilde{\\varepsilon }_0/4$ , $\\delta _{\\tilde{\\varepsilon }_L}=\\delta _{\\tilde{\\varepsilon }}/2-\\tilde{\\varepsilon }_0/4$ .","Here, the thickness of half-vicinity shell, $\\delta _{\\tilde{\\varepsilon }}$ , in its discrete form and the ground state energy in normalized energy space, $\\tilde{\\varepsilon }_0$ , are given respectively as follows =2(12 i1+...+d2 id) 0=12+...+d2 Instead of discrete definition of $\\delta _{\\tilde{\\varepsilon }}$ , it's also possible to define it based on continuous expressions which simplifies mathematical operations and will be given in the following subsection.","Half-vicinity shell thickness is a crucial parameter since contributions to oscillatory properties mainly come from the states within this half-vicinity shell.","Figure: (Color online) (a) Spin normalized 1D Fermi distribution and occupancy variance functions in energy space for Λ=25\\Lambda =25 and α=0.5\\alpha =0.5.","Thickness of half-vicinity shell in energy space (δ ε ˜ =5\\delta _{\\tilde{\\varepsilon }}=5) denoted by green line.","(b) Spin normalized occupancy variance function in 1D state space for weakly confined (α 1 =0.25\\alpha _1=0.25) and confined (α 1 =1\\alpha _1=1) domains for a constant chemical potential (Λ=10\\Lambda =10).","Shaded region denotes the thickness of half-vicinity shell which is equal to unity in state space.","Red and green points represent the half-vicinity states (Fermi points in 1D) and off-half-vicinity states respectively.In Fig.","1(a), spin normalized 1D Fermi distribution and occupancy variance functions in energy space are shown by dashed-blue and solid-red curves respectively.","Thickness of half-vicinity shell is represented by the solid horizontal green line.","In Fig.","1(b), occupancy variance is plotted in 1D state space for two different confinement values for a constant chemical potential ($\\Lambda =10$ ).","As is seen from Fig.","1(b), when confinement increases (denoted by the leftward arrow), occupancy variance peak becomes sharper.","Red states are the closest states to the respective Fermi levels, so their contributions are the largest.","Light blue shaded region represents HV window.","These closest states to Fermi level in 1D state space are called Fermi points.","They constitute a Fermi line in 2D and Fermi surface in 3D cases.","Green points represent the states outside of half-vicinity shell, which we call off-half-vicinity (OHV) states.","Note that in strongly confined case contributions of OHV states are negligible.","Thus, HVM represents the true behavior of occupancy variance function accurately in strongly degenerate cases, where oscillations are strong.","On the contrary, for weakly confined or unconfined cases, where oscillations weaken or disappear, HVM representation of variance function becomes inaccurate because of non-negligible contributions of OHV states (Fig.","1(b) for $\\alpha =0.25$ ).","However, in this case, continuum expressions of variance function already represent the true behavior properly."],["Thickness of half-vicinity shell in energy space and renormalized HVM","Thickness of half-vicinity shell in energy space plays an important role in HVM.","To describe the thickness in a general way, we invoke number of states (NOS) and density of states (DOS) concepts.","NOS inside the half-vicinity shell is found by $NOS_{HV}=\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }w_{HV}$ , which is an exact equation but does not lead to analytical expressions, which we seek here.","Therefore, we use continuum approximation to convert summations into integrals and find the thickness of half-vicinity shell in energy space analytically as $\\delta _{\\tilde{\\varepsilon }}=CNOS_{HV}/CDOS(\\Lambda )$ , where $CNOS_{HV}=\\int _{0}^{\\infty }\\cdots \\int _0^{\\infty }w_{HV}di_1\\ldots di_d$ .","By calculating the integrals of CNOS and using the known expressions of CDOS [47], $\\delta _{\\tilde{\\varepsilon }}$ can be obtained for various dimensions in degenerate limit as given in Table I.","Note that when $\\sqrt{\\Lambda }/\\alpha _n$ value of a system becomes less than unity for a particular direction $n$ , momentum modes in that direction becomes empty except the ground state and the system can be considered as if it has one dimension less.","Table: CNOS, CDOS and thickness of half-vicinity shell in energy space for various dimensions.$\\delta _{\\tilde{\\varepsilon }}$ can be generalized into $d$ -dimensions as $\\delta _{\\tilde{\\varepsilon }}(d)=2\\sqrt{\\frac{\\Lambda }{\\pi }}\\frac{\\Gamma (d/2)}{\\Gamma [(d+1)/2]}\\sum _{n=1}^{d}\\alpha _n.$ In continuum case, $(\\alpha _1,\\ldots ,\\alpha _d\\rightarrow 0)$ , HVM gives the following expression for TOV, HVC2 =02 wHVdi1...did =02 wHVCDOS()d =LH2 CDOS()d. It should be noted that $\\tilde{\\varepsilon }_0\\rightarrow 0$ in continuous case and $\\tilde{\\varepsilon }_H$ and $\\tilde{\\varepsilon }_L$ become $\\Lambda +\\delta _{\\tilde{\\varepsilon }}/2$ and $\\Lambda -\\delta _{\\tilde{\\varepsilon }}/2$ respectively.","It's possible to obtain an analytical solution for the integral in Eq.","(12c) for any dimension by considering strongly degenerate conditions $(\\Lambda >>1)$ , $\\Sigma _{HVC}^2\\cong \\tanh \\left(\\frac{\\delta _{\\tilde{\\varepsilon }}}{4}\\right)\\Sigma _{CA}^2.$ Here, $\\tanh $ factor acts as an amplitude renormalization factor.","Although this factor is found by a continuous approach, the similar renormalization factor is expected also for the discrete case, because amplitude renormalization process is independent of the type of accumulation operators which is verified by the numerical results in this article.","Thus, we may do the following approximation $\\begin{aligned}\\Sigma _{HV}^2 &=\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }\\sigma ^2w_{HV} \\\\&\\cong \\tanh \\left(\\frac{\\delta _{\\tilde{\\varepsilon }}}{4}\\right)\\left[\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }\\sigma ^2\\right]=\\tanh \\left(\\frac{\\delta _{\\tilde{\\varepsilon }}}{4}\\right)\\Sigma _{D}^2.\\end{aligned}$ Then, we can express discrete TOV, $\\Sigma _{D}^2$ , in terms of $\\Sigma _{HV}^2$ as follows, $\\Sigma _{D}^2\\approx \\frac{\\Sigma _{HV}^2}{\\tanh \\left(\\delta _{\\tilde{\\varepsilon }}/4\\right)}=\\Sigma _{RHV}^2,$ where $RHV$ subscript denotes renormalized half-vicinity.","Factor of $1/\\tanh $ renormalizes the amplitude of variance function found by HVM (which only considers HV states) in order to represent also the contributions of OHV states.","In other words, $\\Sigma _{RHV}^2$ is the renormalized TOV based on HVM and it includes contributions of both HV and OHV states.","Accuracy of this expression depends on the thickness of half-vicinity shell inside the factor of $\\tanh $ , which is a function of confinement and degeneracy.","The remarkable accuracy of Eq.","(15) in comparison with the exact equation (Eq.","(2)) is shown in Fig.","2.","Although Weyl representation (Eq.","(3)) does not predict the oscillations but only their trend, HVM quite perfectly matches with the exact results especially for high degeneracy and confinement conditions where oscillations are significant.","Consequently, by considering Eqs.","(2), (7) and (15), HVM proposes the following transformation: $\\sigma ^2\\xrightarrow{}\\sigma ^2\\frac{w_{HV}}{\\tanh \\left(\\delta _{\\tilde{\\varepsilon }}/4\\right)}$ for the calculation of any thermodynamic and transport quantity containing $\\sigma ^2$ term.","By this transformation, infinite sums are converted into finite sums over the minimum number of states near to Fermi level.","As is clear from the comparisons in Fig.","2, HVM can be used as a reliable tool to predict and represent quantum oscillations.","Figure: (Color online) Comparison of TOV results based on exact model, HVM and Weyl expressions.","(a) 1D (b) 2D (c) 3D occupancy variance functions changing with degeneracy and (d) 1D (e) 2D (f) 3D occupancy variance functions changing with isometric confinement in all available directions.","Red, blue and green solid curves represent Eq.","(2) in 1D, 2D and 3D respectively.","Eq.","(15) (HVM estimation) is represented by dashed black curves.","Dot-dashed purple curves are Weyl representations of TOV functions from Eq.","(3)."],["Analytical expression for TOV in 1D case","In highly degenerate and strongly confined 1D case, summation of TOV in Eq.","(2) vanishes and HVM represents only the contribution of Fermi point inside $i_F\\pm 1/2$ interval in state space, where $i_F=\\sqrt{\\Lambda }/\\alpha _1$ .","Therefore, for this special case, there is no need to do summation over half-vicinity shell states (since there is only one) and a special value of that state variable can be found analytically as $i_{*}=\\left[i_F\\right]=i_F-\\frac{1}{\\pi }\\arctan {\\left[\\tan \\left(\\pi i_F\\right)\\right]},$ where $\\left[\\cdots \\right]$ denotes the round function which can be represented by elementary functions on the right hand side.","Then, renormalized TOV in 1D case is $\\Sigma _{RHV}^2=\\frac{\\sigma ^2(i_{*})}{\\tanh \\left(\\alpha _1\\sqrt{\\Lambda }/2\\right)}.$ A comparison for variations of exact and HVM's TOV functions with confinement is given in Fig.","3.","Solid red, solid blue and dashed black curves are results of Eqs.","(2), (5) and (18) respectively.","Accuracy of HVM in representing the prominent oscillations is quite good, even though it only considers contribution of Fermi points.","Since Eqs.","(3) and (5) are equal to each other for 1D case, there is no need to examine $\\Sigma _W^2$ additionally.","It's seen from the red curve in Fig.","3 that transitions between stationary and oscillatory regimes are smooth and $\\ln 3/\\sqrt{\\Lambda }$ line perfectly separates these regimes, which will be examined in detail in the following section.","Figure: (Color online) A comparison of exact and HVM results for the variation of TOV with confinement when Λ=30\\Lambda =30.","Green vertical line, α 1 =ln3/Λ\\alpha _1=\\ln 3/\\sqrt{\\Lambda }, separates the oscillatory regime from the stationary one.","Δα pp \\Delta \\alpha _{pp} denotes the period of TOV varying with confinement.Extremum points of oscillations in Fig.","3 correspond to the values where the total number of particles is integer.","We can determine the periodicity of TOV changing with $\\Lambda $ and $\\alpha $ , considering the fact that maxima and minima of oscillations correspond to odd and even integer particle numbers respectively, for $g_s=2$ .","Special to the 1D case, expression found by considering first two terms of PSF exactly represent number of particles for a given chemical potential in degenerate case [30].","We find the values of $\\Lambda $ and $\\alpha $ correspond to integer numbers of particles as $\\Lambda =[\\alpha _1(\\tilde{N}+1/2)]^2$ and $\\alpha _1=\\sqrt{\\Lambda }/(\\tilde{N}+1/2)$ respectively.","Periods of oscillations in $\\alpha $ and $\\Lambda $ spaces can then easily be found respectively as pp=4(2N+1)(2N+3)=212N+3, pp=212(N+1)=8N+1(2N+1)2, where $\\tilde{N}=N/g_s$ is spin normalized particle number.","Considering the fact that $\\Sigma _{RHV}^2$ mimics the oscillations in an almost perfect way, an intriguing question comes to the mind: Is it possible to determine analytically where quantum oscillations start as the confinement and/or degeneracy changes?"],["A phase diagram for quantum oscillations: Transition from classical to quantum behavior","HVM takes the discreteness of quantum states into account and allows us to accurately calculate the thermodynamic and transport properties exhibiting size and density dependent quantum oscillations without using infinite summations.","In addition to that, HVM can accurately estimate where the transition from classical behavior to quantum behavior starts, in the framework of quantum oscillations.","The states contributing to TOV consist of HV states and OHV states (i.e., $\\Sigma _{D}^2=\\Sigma _{HV}^2+\\Sigma _{OHV}^2$ ).","HV states are responsible from the oscillatory part of TOV, whereas OHV states represent the stationary part.","They constitute two competing parts of TOV.","Domination of contributions of HV states over OHV ones leads to oscillations and vice versa.","Hence, by considering $\\Sigma _{HV}^2=\\Sigma _{OHV}^2$ balance, we can compare contributions of HV states and OHV states to define a universal (material independent) recipe for the separation of stationary regime (SR) from oscillatory regime (OR).","According to the recipe, when $\\Sigma _{HV}^2<\\Sigma _{OHV}^2$ , oscillations disappear (SR) and on the opposite condition $\\Sigma _{HV}^2>\\Sigma _{OHV}^2$ , oscillations reveal (OR).","Since $\\Sigma _{OHV}^2=\\Sigma _{D}^2-\\Sigma _{HV}^2$ , SR-OR transition can be quantified as the balance of the contributions of HV and OHV states ($\\Sigma _{HV}=\\Sigma _{OHV}$ ) by $\\Sigma _{D}^2=2\\Sigma _{HV}^2$ .","Since the transition is not sharp but smooth, we can safely use analytical expressions of TOV, instead of their exact expressions, to find an analytical expression for SR-OR separation.","From the balance condition between contributions of HV and OHV states as well as Eq.","(13), we can determine the following analytical condition for the transition between SR and OR, $\\Sigma _{CA}^2=2\\Sigma _{HVC}^2 \\;\\Rightarrow \\; 1=2\\tanh \\left(\\delta _{\\tilde{\\varepsilon }}/4\\right) \\;\\Rightarrow \\; \\delta _{\\tilde{\\varepsilon }}=2\\ln 3.$ It's seen from red curve in Fig.","3, balance condition quantified in Eq.","(20) clearly gives the SR-OR separation.","By decreasing confinement or degeneracy, the number of states around Fermi level increases, contributions of OHV states become also appreciable and their contributions make oscillation amplitude smaller.","The more number of states around Fermi level, the smaller oscillation amplitude.","When contributions of OHV states exceeds that of HV states, oscillations disappear.","To complete the construction of the phase diagram, it is necessary to check also the number of particles, ($N$ ), inside the system, since we are dealing with extremely confined systems having relatively low number of particles.","For statistical representations there has to be sufficiently large number of particles inside the system.","Nevertheless, the physically meaningful region in a phase diagram can be stated by the condition $N\\ge 1$ .","Full list of recipes and conditions to establish the phase diagram is given in the Table 2.","Table: Recipes and conditions for the construction of the phase diagram of quantum oscillations.According to the recipes and conditions given in Table 2, phase diagrams of quantum oscillations in degeneracy-confinement space for various dimensions can be determined.","For isometric 3D domains, the phase diagram is constructed and given in Fig.","4.","Solid-black curves represent interfaces defined by the conditions in Table II.","$\\delta _{\\tilde{\\varepsilon }}=2\\ln 3$ condition, representing the balance of HV and OHV states, separates OR and SR.","When both confinement and degeneracy sufficiently increase, system enters into the quantum regime where certain thermodynamic and transport quantities oscillate with varying size or density.","Figure: (Color online) A phase diagram on degeneracy-confinement space for quantum oscillations.","δ ε ˜ =2ln3\\delta _{\\tilde{\\varepsilon }}=2\\ln 3 condition defines the boundary between stationary (classical) and oscillatory (quantum) regimes.","Blue dot represents the critical point in the phase diagram for isometric 3D domains.Intersection of SR-OR interface curve with $N=1$ curve denotes the critical point which is represented by blue dot in Fig.","4.","Critical value of the confinement parameter at this point is $\\alpha _{*}^{3D}=0.78$ and the corresponding degeneracy value is $\\Lambda _{*}^{3D}=0.88$ .","These values are universal for isometric rectangular confinement domains.","For confinement values below $\\alpha _{*}$ , existence of oscillations can be controlled and they can even be suppressed by decreasing degeneracy (through density or temperature).","Similarly, for degeneracy values higher than $\\Lambda _*$ , existence of oscillations can be controlled by changing confinement.","This region defined by $\\lbrace \\Lambda >\\Lambda _{*},\\alpha <\\alpha _*\\rbrace $ is the crossover region restricted by dashed lines in the phase diagram.","On the contrary, for higher confinement values than $\\alpha _{*}$ , quantum oscillations cannot be suppressed and the region is called entirely oscillatory region.","Below the critical degeneracy values, system does not exhibit oscillatory behaviors regardless of the values of confinements.","Hence, this critical point is used to define different regions (crossover region, entirely OR and entirely SR).","Regimes on phase diagram and corresponding conditions are summarized in Table III.","Table: Conditions and regions of the phase diagram.Although the phase diagram in Fig.","4 represents only isometric 3D domain, the form of the phase diagrams for 1D and isometric 2D domains are also very similar to the 3D one.","Only SR-OR interface slightly shifts upward while $N=1$ curve slightly rotates in clockwise direction around the critical point as the dimension decreases.","For 1D and isometric 2D domains, universal critical values can also be found as $\\alpha _{*}^{1D}=1.07$ , $\\alpha _{*}^{2D}=0.87$ and $\\Lambda _{*}^{1D}=1.05$ , $\\Lambda _{*}^{2D}=0.98$ .","It's also possible to investigate the anisometric domains by defining aspect ratios.","For 2D domain, $r_{12}=L_1/L_2=\\alpha _2/\\alpha _1$ denotes the aspect ratio.","For 3D, additionally we define $r_{13}=L_1/L_3=\\alpha _3/\\alpha _1$ .","Here, the most confined direction is chosen as direction 1.","In Fig.","5, we show how the critical confinement values are changing with respect to aspect ratios of 2D and 3D domains.","Figure: (Color online) Aspect ratio dependencies of critical confinement values for 2D and 3D cases.Although the values in Fig.","5 are calculated in an exact way by using Tables I and II, $\\alpha _{1*}^{2D}$ can be approximated by $\\alpha _{1*}^{2D}\\approx 0.87-0.40\\ln r_{12}$ with less than $1.5\\%$ mean absolute percentage error (MAPE) for $0.1\\le r_{12} \\le 1$ interval.","Similarly, $\\alpha _{1*}^{3D}$ is approximated by $\\alpha _{1*}^{3D}\\approx 0.78-0.27\\ln r_{12}-0.27\\ln r_{13}$ with less than $2.2\\%$ MAPE for $0.1\\le \\lbrace r_{12},r_{13}\\rbrace \\le 1$ intervals.","Critical degeneracy values for anisometric cases can easily be found for 3D, 2D and 1D cases respectively, *3D=[231*3D(1+r12+r13)]2, *2D=[321*2D(1+r12)]2, *1D=(31*1D)2.","It's important to note that thickness of half-vicinity shell indicates the quantumness of the system.","When volume $V\\rightarrow \\infty $ or temperature $T\\rightarrow \\infty $ or density $n\\rightarrow 0$ , thickness $\\delta _{\\tilde{\\varepsilon }}\\rightarrow 0 <2\\ln 3$ and system behaves classically.","Conversely, for sets of $(V,T,n)$ making $\\delta _{\\tilde{\\varepsilon }}>2\\ln 3$ , the system shows quantum oscillatory behaviors.","As a matter of fact, $\\delta _{\\tilde{\\varepsilon }}$ is a precise indicator of whether system is in quantum regime or not in terms of oscillations.","In order to see the full picture of quantum oscillations in 1D case, variation of TOV, Eq.","(2), with confinement and degeneracy is given in Fig.","6.","It is seen now even more clearly that $\\delta _{\\tilde{\\varepsilon }}=2\\ln 3$ plane perfectly separates oscillatory region from the stationary one regardless of the values of $\\alpha $ and $\\Lambda $ .","Increment in degeneracy and/or confinement make $\\delta _{\\tilde{\\varepsilon }}$ larger and cause strong oscillations.","Figure: (Color online) Variation of discrete TOV (Σ D 2 \\Sigma ^2_D) with confinement (α\\alpha ) and degeneracy (Λ\\Lambda ).","Gray 2ln32\\ln 3 plane separates the oscillatory (quantum) and stationary (classical) regimes.HVM is constructed to predict oscillations appearing in strongly confined and highly degenerate conditions.","Accuracy of its predictions increases with increasing confinement and degeneracy, while accuracy becomes weaker for the regions near to SR-OR interface in phase diagram, Fig.","4.","To be precise, HVM predicts oscillations almost perfectly, as long as confinement and degeneracy are larger than their critical values ($\\alpha _*$ and $\\Lambda _*$ )."],["Size and density dependent oscillations in electronic heat capacity","In previous sections we investigated the oscillatory behavior of TOV function and constructed a theoretical model (HVM) for its calculation.","In this subsection, we examine quantum oscillations of heat capacity of an electron gas confined in a nanowire by considering both exact and HVM.","Electronic heat capacity from the derivative of internal energy with respect to temperature is written in its exact (based on infinite sums) form as $C_V=g_s k_B\\left[\\sum _{i_n=1}^{\\infty }\\tilde{\\varepsilon }^2\\sigma ^2-\\frac{\\left(\\sum _{i_n=1}^{\\infty }\\tilde{\\varepsilon }\\sigma ^2\\right)^2}{\\sum _{i_n=1}^{\\infty }\\sigma ^2}\\right].$ Heat capacity expression, Eq.","(22), contains zeroth, first and second order energy moments of occupancy variance.","Each summation in Eq.","(22) has distinct oscillatory behavior, which leads to the characteristic oscillations in heat capacity.","In order to reveal quantum oscillations and make electronic contribution of heat capacity appreciable, we focus on very low temperature ($T=5$ K), high electron density $(\\sim 10^{25} \\text{m}^{-3})$ and nanoscale confinements in which quantum effects are observable.","To make our discussions quantity-independent, we examine electronic heat capacity per particle $(c_V=C_V/N)$ .","In the calculations, chemical potential is obtained numerically from particle number equation as functions of temperature, density and confinement.","In Fig.","7, accuracy of HVM is examined by considering size and density dependent oscillations of normalized electronic specific heat.","$c_V^0$ represents the specific heat expression under continuum approximation, where all summations in Eq.","(22) replaced by integrals.","In all cases in Fig.","7, an electron gas confined in a quantum wire with 200 nm long $(L_3)$ is considered at 5K temperature.","Red, blue and green curves in Fig.","7(a), 7(b) and 7(c) respectively represent the results of the exact model, while the black dashed curves are the results of HVM.","Although confinement domain is a nanowire, it cannot be considered as a pure 1D structure since there are still a few excited states in transverse directions.","Thus, calculations are done over triple sums.","Figure: (Color online) Accuracy of HVM on predicting (a) size dependent and (b), (c) density dependent oscillations in normalized heat capacity for a nanowire with 200 nm long at 5K temperature.","Solid curves are the results obtained by using exact expressions (based on infinite sums), while dashed black curves are obtained by using HVM.In Fig.","7(a), sizes of the domain in transverse directions ($L_1$ and $L_2$ ) are changed simultaneously from 10 nm to 40 nm for $n=10^{25}\\text{m}^{-3}$ where $\\Lambda $ is always larger than $\\Lambda _{*}$ .","Confinement in transverse directions spans the values larger and smaller than $\\alpha _*$ .","HVM predicts oscillatory size dependence of electronic heat capacity pretty well between 10 nm and 25 nm, where confinement is strong, oscillations are large and unavoidable ($\\alpha \\ge \\alpha _*$ ).","For 25 nm and 40 nm interval, however, confinement becomes weaker ($\\alpha \\le \\alpha _*$ ) and an appreciable difference appears in between exact and HVM results while oscillations are also weak.","This is an expected result, since the system approaches to SR-OR interface in phase diagram.","In Fig.","7(b), quantum oscillations in heat capacity varying with electron density are shown for a square wire having 20 nm size in transverse directions.","Note that density variation for a constant confinement corresponds to a vertical movement on the phase diagram.","HVM perfectly matches with the exact model almost everywhere in Fig.","7(b) because of strong confinement, $\\alpha _{1}=\\alpha _{2}=1.48>\\alpha _{1*}^{3D}=1.27$ .","System shows oscillatory behavior regardless of the value of degeneracy (or electron density).","In other words, oscillations are unavoidable for the system discussed in Fig.","7(b) as long as the confinement is sustained.","By increasing the domain size from 20 nm to 25 nm in confined directions (reducing the confinement), we can put the system into the crossover regime where $\\alpha _{1}=\\alpha _2=1.18<\\alpha _{1*}^{3D}=1.23$ as it's shown in Fig.","7(c).","For this case, HVM does not match well with the exact model for lower electron densities, where the system is getting closer to SR-OR interface.","Oscillations decay by decreasing confinement and degeneracy but the decay rate of oscillations with degeneracy depends on how much the confinement smaller than the critical confinement value.","Since the confinement is not much smaller than $\\alpha _*$ , there is no considerable decay in oscillations in Fig.","7(c).","All these behaviors confirm the predictions of the phase diagram presented in Fig.","4 as well as aspect ratio dependencies of critical confinement parameters given in Fig.","5.","In contrast to size and density, variation of temperature does not lead to oscillatory behaviors in thermodynamic and transport quantities.","The reason can clearly be seen from Eqs.","(17) and (18) where oscillations are originated from the variation of round of $i_F$ .","Since temperature dependence of chemical potential disappears in strongly degenerate case, $[i_F]=[\\sqrt{\\Lambda }/\\alpha ]$ becomes independent of temperature.","Hence, in strongly degenerate case, temperature variation does not affect the proximities of states to the Fermi level in state space, but just changes the occupation probabilities of the states.","In other words, thermal broadening happens only in energy space, not in state space.","Since the proximities do not variate, oscillations do not appear.","This explanation can be directly extended to higher dimensional cases also, due to the orthogonality of state space dimensions."],["Oscillatory violation of entropy-heat capacity equivalence in degenerate limit","Continuum expressions of entropy and heat capacity of an ideal Fermi gas are equal to each other in degenerate limit [3].","When we consider the discrete nature of quantum states, however, entropy-heat capacity equivalence is also broken and quantum behaviors lead to an oscillatory non-equivalence of entropy and heat capacity of an ideal degenerate Fermi gas at nanoscale.","The well-known form of entropy is written as $S=-g_s k_B\\left[\\sum _{i_n=1}^{\\infty }{\\tilde{f}\\ln (\\tilde{f})+(1-\\tilde{f})\\ln (1-\\tilde{f})}\\right].$ In degenerate limit, both Eq.","(22) and (23) give the same expression (for instance, $g_s\\pi ^2Nk_B/(2\\Lambda )$ for 3D case) under continuum approximation.","However, as it is seen from Fig.","8, this equivalence is broken.","Heat capacity/entropy ratio deviates from unity and variates with domain size and density in an oscillatory fashion.","A nanowire in the previous subsection is reconsidered with the same specifications here for the examination of heat capacity-entropy ratio.","Both electronic heat capacity and entropy oscillate in confined and degenerate conditions.","Although their scales are almost the same, due to the phase and magnitude differences of both function's characteristic oscillations, their ratio also oscillates around unity.","Increasing domain size or temperature decreases the amplitude of oscillations.","Similar to heat capacity results, the success of HVM is quite good for higher electron density and confinement conditions.","Figure: (Color online) Oscillatory violation of entropy-heat capacity equivalence in degenerate limit depending on (a) size and (b), (c) density for a nanowire with 200 nm long at 5K temperature.","Solid curves are the results obtained by using exact expressions, while dashed black curves are obtained by using HVM.Very similar to the behavior of TOV, fully occupied or completely unoccupied states do not contribute to the entropy of the gas and contributions come only from the states around Fermi level, which are identified as HV states in HVM.","Therefore, it is possible to use HVM also for the quantities which does not directly contain occupancy variance, as long as HV states are dominant over the quantity to be calculated like in entropy."],["Resemblance of total occupancy variance and density of states functions","There is a direct relationship between DOS at Fermi level and TOV.","It's well-known that, in strongly degenerate case, spin normalized distribution function nearly turns into a Heaviside step function $f(\\tilde{\\varepsilon })\\approx g_s\\Theta (\\Lambda -\\tilde{\\varepsilon })$ .","Since the number of states below Fermi level $\\Lambda $ is approximately equal to the number of Fermions per spin state, $N=\\sum f(\\tilde{\\varepsilon })\\approx g_s\\sum \\Theta (\\Lambda -\\tilde{\\varepsilon })=NOS(\\Lambda )$ , derivative of $NOS(\\Lambda )$ with respect to $\\Lambda =\\tilde{\\varepsilon }_F$ gives $DOS(\\Lambda )$ and then TOV is approximately equal to $DOS(\\Lambda )$ .","$\\Sigma _{D}^2=-\\sum _{i_n}{\\frac{\\partial f}{\\partial \\tilde{\\varepsilon }}}\\approx g_s\\sum _{i_n}{\\delta (\\Lambda -\\tilde{\\varepsilon })}=DOS(\\Lambda ).$ In Fig.","9, resemblance of TOV and DOS functions is shown for two different subbands with constant confinement parameters for various dimensions.","Note that the results of lower dimensions are obtained by strongly confining a 3D domain in different directions.","Even the lower dimensional domains are represented by triple sums where sums in strongly confined directions represent subbands.","Hence, it's possible to directly obtain Eq.","(6) by using continuous DOS functions at Fermi level $\\tilde{\\varepsilon }_F=\\Lambda $ given in Table I.","While HVM predicts oscillations quite good, DOS functions represents only the trend of the oscillations (Fig.","9) like Weyl representations of TOV functions in Fig.","2 and continuous TOV function in Fig.","3.","Figure: (Color online) Similarity of TOV functions (solid curves) and continuous density of states functions at Fermi level (dashed curves) for various dimensions.Although the structure of two functions are very similar, due to thermal broadening at Fermi level there are small differences around the beginning of each subband especially for low dimensional cases.","The origin of this resemblance is basically coming from the fact that occupancy variance function has a peakwise nature at Fermi level and vanishing behavior elsewhere, which is directly related with density of states at Fermi level."],["Conclusion","In this article, investigation of size and density dependent quantum oscillations is done by proposing the HVM for the calculation of oscillatory quantities.","Proposition of our model not only allows to calculate oscillatory thermodynamic or transport quantities in a more efficient way, but also offers a phase diagram which predict quantum oscillations clearly.","Moreover, HVM introduces analytical conditions separating oscillatory and stationary regimes on the phase diagram.","Occupancy variance function appearing in many oscillatory quantities is examined in detail and the model is constructed on the half-vicinity of states around Fermi level.","Two main factors determine the value of TOV: the number of states inside the half-vicinity shell and their proximities to the Fermi level.","Confinement and degeneracy directly affect the distribution of the states around Fermi level besides the thickness of half-vicinity shell and so the number of states inside the shell.","The results show that HVM can be used as a reliable tool to predict and calculate quantum oscillations of a degenerate and confined ideal Fermi gas.","The proposed phase diagram and analytical conditions for quantum oscillations evidently describe entirely stationary/oscillatory and crossover regions on degeneracy-confinement space.","Consequently, our model and the phase diagram not only clarify the theoretical understanding of the quantum oscillations but also may help to determine the proper material parameters for experimental studies."]],"string":"[\n [\n \"Half-vicinity model and a phase diagram for quantum oscillations in\\n confined and degenerate Fermi gases\"\n ],\n [\n \"Abstract We propose an analytical model for the accurate calculation of size and density dependent quantum oscillations in thermodynamic and transport properties of confined and degenerate non-interacting Fermi gases.\",\n \"We provide a universal, material independent, recipe that explicitly separates oscillatory quantum regime from stationary classical regime.\",\n \"Our model quite accurately estimates quantum oscillations depending on confinement and degeneracy.\",\n \"We construct a phase diagram representing stationary and oscillatory regimes on degeneracy-confinement space.\",\n \"Analytical expressions of phase transition interfaces are derived for different dimensions.\",\n \"The critical point on the phase diagram, which separates entirely stationary and entirely oscillatory regions, is determined and their aspect ratio dependencies are examined.\",\n \"Quantum oscillations as well as their periods are analytically expressed for one-dimensional case.\",\n \"Accuracy of our model is verified through quantum oscillations in electronic specific heat capacity.\",\n \"We also compare the predictions of our half-vicinity model, based on bounded sums, with those of infinite sums, for the oscillatory violation of entropy-heat capacity equivalence in degenerate limit to show the accuracy of our model.\",\n \"Furthermore, similarities between functional behaviors of total occupancy variance and conventional density of states functions at Fermi level are discussed.\"\n ],\n [\n \"Introduction\",\n \"A great deal of attention has been given to the physics of low-dimensional nanostructures associated with the intensive developments of nanoscience and nanotechnology in recent years.\",\n \"When the mean de Broglie wavelength of particles inside a domain becomes comparable to the size of the domain, quantum size effects start to reveal themselves and their presence has to be taken into account in the calculation of the physical properties of such nanoscale systems [1], [2], [3], [4].\",\n \"In Fermi gases, instead of fully occupied states, only partially occupied ones around Fermi level contribute to some thermodynamic quantities like entropy and specific heat and transport properties like electrical and thermal conductivities.\",\n \"Definitional expressions of such quantities contain the variance of distribution function instead of the distribution function itself.\",\n \"Hence, variance function plays a crucial role in the behavior of these quantities.\",\n \"In degenerate and strongly confined ideal Fermi gases, the quantities containing variance function exhibit oscillatory behaviors.\",\n \"The basic reason of this behavior is the oscillatory behavior of total occupancy variance (TOV) function due to changes in size and density.\",\n \"It has been shown experimentally in literature that, size and density dependent oscillations occur in some certain thermodynamic and transport properties of charge carriers in semiconductors/metals or Fermi gases confined at nano domains in general.\",\n \"First experimental studies of size dependent oscillations due to quantum size effects has been reported in late 60's [5], [6], [7], [8].\",\n \"Thermodynamic properties like entropy and specific heat capacity at constant volume, thermoelectric properties such as Seebeck coefficient and electronic transport properties such as charge carrier mobility, electrical and thermal conductivities are some examples of these oscillatory quantities [4], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35].\",\n \"Examination of size dependent oscillations also has crucial importance in superconductors and topological insulators [33], [34], [36], [37], [38], [39].\",\n \"Size and density dependent oscillations in aforementioned quantities are attributed to the quantization of energy spectrum and fluctuations in density of states around Fermi level at nanoscale [15], [16], [17], [18], [19], [20], [40].\",\n \"For Fermi systems with linear energy dispersion relation (e.g.\",\n \"a Fermi system under an applied magnetic field or harmonic trap potential), quantum oscillations and its implications are extensively studied especially for specific materials [41], [15], [16], [17], [18], [19], [20], [42], [43].\",\n \"For the systems obeying quadratic energy-momentum dispersion relation, on the other hand, there is a need for a mathematical model predicting quantum oscillations and explaining their nature in a comprehensive and material/case independent way.\",\n \"Moreover, a phase diagram separating oscillatory and stationary regimes on confinement-degeneracy space has never been proposed for any dispersion relation.\",\n \"Constructing a mathematical model and a phase diagram may help to deeply understand and efficiently control the oscillations to design novel nanoscale devices [20], [27].\",\n \"In this article, we propose a theoretical model, half-vicinity model (HVM), that accurately predicts the oscillations appearing in confined Fermi gases.\",\n \"A rectangular confinement domain is chosen, because it is one of the most suitable geometries for common manufacturing methods of low dimensional nanostructures.\",\n \"In the following section, by examining the nature of TOV of quantum states, we introduce the model by defining a half-vicinity shell around Fermi level.\",\n \"$d$ -dimensional discrete and continuum expressions of TOV function are given and thickness of half-vicinity shell, which plays an important role in HVM, is obtained for various dimensions.\",\n \"For 1D case, we obtain analytical expressions for TOV oscillations and their periods.\",\n \"In Sec.\",\n \"III, a phase diagram of quantum oscillations is established and phase transition interfaces between stationary (classical, continuous) and oscillatory (quantum, discrete) regimes are analytically given for 1D, 2D and 3D cases.\",\n \"Critical confinement and degeneracy values, which separates entirely stationary and oscillatory regions, are also examined by considering their aspect ratio dependencies.\",\n \"In Sec.\",\n \"IV, results of exact (definitional expressions based on infinite sums) and HVMs are compared for size and density dependent oscillations in the electronic specific heat capacity of strongly degenerate and confined ideal Fermi gases.\",\n \"Furthermore, broken equivalence of entropy-heat capacity in quantum degenerate limit of ideal Fermi gases is also well predicted by HVM.\",\n \"Finally, we discuss (Sec.\",\n \"V) the relationship between TOV and conventional density of states functions at Fermi level.\"\n ],\n [\n \"Half-vicinity model for total occupancy variance\",\n \"Oscillations in thermodynamic and transport quantities originate from the nature of occupancy variance function.\",\n \"For this reason, in order to establish a model to understand the nature of oscillations, we need to examine TOV in detail.\"\n ],\n [\n \"Total occupancy variance revisited\",\n \"Occupancies of quantum states for Fermions are described by Fermi-Dirac distribution function, $f=g_s/\\\\left[\\\\exp (-\\\\Lambda +\\\\tilde{\\\\varepsilon })+1\\\\right]$ where $g_s$ is spin degree of freedom, $\\\\Lambda =\\\\mu /(k_BT)$ is dimensionless chemical potential (or degeneracy parameter) indicating the strength of degeneracy, in which $\\\\mu $ is chemical potential, $k_B$ is Boltzmann's constant, $T$ is temperature.\",\n \"$\\\\tilde{\\\\varepsilon }$ represents dimensionless energy eigenvalues (normalized to thermal energy $k_B T$ ).\",\n \"For quadratic dispersion relation of massive particles confined in a $d$ -dimensional rectangular domain, $\\\\tilde{\\\\varepsilon }=\\\\varepsilon /(k_BT)=(\\\\alpha _1 i_1)^2+\\\\cdots +(\\\\alpha _d i_d)^2$ where $i_n$ is quantum state variable for a particular direction $n=\\\\lbrace 1,2,\\\\cdots ,d\\\\rbrace $ .\",\n \"Here $\\\\alpha _n$ is confinement parameter indicating the strength of quantum confinement in a particular direction $n$ and defined as $\\\\alpha _n=h/\\\\left(\\\\sqrt{8m k_B T}L_n\\\\right)$ where $h$ is Planck's constant, $m$ is mass of the particle and $L_n$ is domain size in direction $n$ .\",\n \"Derivative of Fermi-Dirac distribution function with respect to $\\\\Lambda $ , a bell-shaped function peaked at Fermi level, is called occupancy variance function.\",\n \"Occupancy variance of an energy state $\\\\tilde{\\\\varepsilon }=\\\\tilde{\\\\varepsilon }(i_1,\\\\ldots ,i_d)$ is $\\\\sigma ^2(i_1,\\\\ldots ,i_d)=\\\\frac{\\\\partial f}{\\\\partial \\\\Lambda }=-\\\\frac{\\\\partial f}{\\\\partial \\\\tilde{\\\\varepsilon }}=\\\\frac{g_s}{4}\\\\operatorname{sech}^2\\\\left(\\\\frac{\\\\tilde{\\\\varepsilon }-\\\\Lambda }{2}\\\\right).$ In literature, it's sometimes called also as thermal broadening function, since it represents thermal broadening of energy states around Fermi level [44].\",\n \"Due to the fact that Fermi-Dirac distribution is a Bernoulli-type distribution (basically a sigmoid function), occupancy variance can also be represented by a Bernoulli-type variance form $\\\\sigma ^2=g_s\\\\tilde{f}(1-\\\\tilde{f})$ , where $\\\\tilde{f}=f/g_s$ is the spin normalized Fermi-Dirac distribution function.\",\n \"Spin factor $g_s$ is taken simply two in all numerical calculations in this article.\",\n \"In order to understand the behavior of oscillatory quantities and how occupancy variance behaves under the accumulation operators such as summation or integration, we need to investigate the nature of TOV.\",\n \"TOV of Fermi particles inside a $d$ -dimensional box is written in its exact form based on infinite sums as $\\\\Sigma _D^2=\\\\sum _{i_1=1}^{\\\\infty }\\\\cdots \\\\sum _{i_d=1}^{\\\\infty }\\\\sigma ^2(i_1,\\\\ldots ,i_d),$ where subscript $D$ stands for discrete calculations.\",\n \"When the summations in Eq.\",\n \"(2) are calculated by using the first two terms of Poisson summation formula (PSF) [45], [30], the following expression can be obtained after some mathematical operations, $\\\\begin{split}\\\\Sigma _{W}^2=& g_s\\\\sum _{n=0}^d\\\\frac{(-1)^{d-n+1}\\\\hspace{2.22214pt}\\\\pi ^{n/2}}{2^d \\\\hspace{2.22214pt}d^{1-\\\\Theta (n)\\\\Theta (d-n)}}Li_{\\\\frac{n-2}{2}}\\\\left[-\\\\exp (\\\\Lambda )\\\\right] \\\\\\\\& \\\\times \\\\sum _{k=1}^{d}\\\\prod _{m=k}^{k+n-1}{\\\\alpha _{\\\\text{mod}(m,d)+1}^{-1}},\\\\end{split}$ where $Li$ is polylogarithm function [46] and $\\\\Theta $ is left-continuous Heaviside step function.\",\n \"Since the expressions calculated by using the first two terms of PSF are equivalent to those obtained from Weyl conjecture [45], [3], [47], we denote this expression by subscript $W$ .\",\n \"In strongly degenerate conditions ($\\\\Lambda >>1$ ), by using the asymptotic forms of polylogarithm functions [48], Eq.\",\n \"(3) can be approximated by $\\\\begin{split}\\\\Sigma _{W\\\\hspace{-2.77771pt}A}^2=& g_s\\\\sum _{n=0}^d\\\\frac{(-1)^{d-n}\\\\hspace{2.22214pt}\\\\pi ^{n/2}}{2^d \\\\hspace{2.22214pt}d^{1-\\\\Theta (n)\\\\Theta (d-n)}}\\\\frac{\\\\Lambda ^{(n-2)/2}}{\\\\Gamma (n/2)} \\\\\\\\& \\\\times \\\\sum _{k=1}^{d}\\\\prod _{m=k}^{k+n-1}{\\\\alpha _{\\\\text{mod}(m,d)+1}^{-1}},\\\\end{split}$ where $\\\\Gamma $ is gamma function.\",\n \"When confinement parameters are much smaller than unity, namely in macroscale, summations may be converted directly into integrals by using continuum approximation and continuous TOV is obtained $\\\\Sigma _C^2=-\\\\frac{g_s\\\\pi ^{d/2}}{2^d \\\\alpha _1 \\\\cdots \\\\alpha _d}Li_{\\\\frac{d-2}{2}}\\\\left[-\\\\exp (\\\\Lambda )\\\\right],$ where $C$ subscript denotes the continuous expressions obtained under continuum approximation.\",\n \"In strongly degenerate case ($\\\\Lambda >>1$ ), Eq.\",\n \"(5) can be approximated by using asymptotic expansions of polylogarithm functions $\\\\Sigma _{C\\\\hspace{-2.77771pt}A}^2=\\\\frac{g_s\\\\pi ^{d/2}}{2^d \\\\alpha _1 \\\\cdots \\\\alpha _d}\\\\frac{\\\\Lambda ^{(d-2)/2}}{\\\\Gamma (d/2)}.$ As it should be, Eqs.\",\n \"(5) and (6) are actually the first (bulk) terms of Eqs.\",\n \"(3) and (4) respectively.\",\n \"Expressions of thermodynamic and transport properties exhibiting oscillatory behavior contain Eq.\",\n \"(1) multiplied by some relevant functions of the quantity to be calculated.\",\n \"Then, it's proper to construct our HVM on TOV, in order to predict oscillatory behaviors in thermodynamic and transport properties.\"\n ],\n [\n \"Basic concept of half-vicinity model\",\n \"Contributions to oscillatory physical quantities substantially come from partially occupied states around Fermi level.\",\n \"Under strongly confined and degenerate conditions, sharpness of the occupancy variance function around Fermi level increases and only few states contribute to TOV.\",\n \"In that case, it becomes possible to make a quantitative treatment of these states by constructing a shell which contains them inside.\",\n \"Therefore, for the accurate calculation of oscillatory physical quantities, instead of summing over all possible quantum states from unity to infinity, considering only those several states around Fermi level (inside the half-vicinity shell) is enough to represent the oscillations.\",\n \"Half-vicinity shell in state space is defined as the interval containing the states in the half-vicinities ($\\\\pm 1/2$ ) of Fermi level.\",\n \"It can be extended to any dimension, by considering the $\\\\pm 1/2$ intervals of Fermi level in each direction at quantum state space.\",\n \"Magnitude of the contribution of each state to TOV is determined by its proximity to the Fermi level.\",\n \"Contribution of a state is maximum when it lies exactly on the Fermi level, and exponentially decays as the state is away from it.\",\n \"The number of half-vicinity states and their proximities to Fermi level change with degeneracy and/or confinement.\",\n \"When confinement becomes stronger, spacing between energy levels increases and causes a decrement in number of half-vicinity states.\",\n \"Because of this decrement, addition (removal) of a state to (from) half-vicinity shell leads to a strong change in TOV.\",\n \"This causes strong fluctuations both in number of half-vicinity states and their proximities to the Fermi level.\",\n \"In other words, half-vicinity states reorganize their distribution and proximities to the Fermi level when for instance sizes of the domain or density of the gas change.\",\n \"Each state gives distinct contribution to TOV.\",\n \"For macro domains with highly populated Fermi level, effect of this reorganization becomes statistically insignificant.\",\n \"Conversely, in confined and degenerate Fermi gases these reorganization of half-vicinity states manifest themselves as quantum oscillations in several thermodynamic and transport quantities due to changes in size and density.\",\n \"For strongly confined and degenerate Fermi gases, TOV, Eq.\",\n \"(2), can approximately be calculated by restricting the summations over infinite number of quantum states to only the states inside the constructed half-vicinity shell.\",\n \"TOV based on HVM is then given by $\\\\Sigma _{HV}^2=\\\\sum _{i_1=1}^{\\\\infty }\\\\cdots \\\\sum _{i_d=1}^{\\\\infty }\\\\sigma ^2w_{HV}.$ Here, half-vicinity window function is define as $w_{HV}=\\\\Theta \\\\left(\\\\Lambda -\\\\tilde{\\\\varepsilon }_{-1/2}\\\\right)\\\\Theta \\\\left(\\\\tilde{\\\\varepsilon }_{1/2}-\\\\Lambda \\\\right),$ where $\\\\Theta $ is Heaviside step function and $\\\\tilde{\\\\varepsilon }_{\\\\pm 1/2}=\\\\alpha _1^2\\\\left(i_1\\\\pm 1/2\\\\right)^2+\\\\ldots +\\\\alpha _d^2(i_d\\\\pm 1/2)^2$ give upper and lower energy bounds of half-vicinity window.\",\n \"Window function $w_{HV}$ filters the states which do not significantly contribute to the oscillatory quantities and serves as a half-vicinity states pass filter, that restricts the upper and lower limits of summations.\",\n \"By definition, in state space, half-vicinity shell thickness is equal to unity in any particular direction, which is the minimum possible thickness in state space for each direction.\",\n \"On the other hand, in energy space its thickness varies with the strength of confinement and degeneracy.\",\n \"Half-vicinity window can equivalently be written as $w_{HV}=\\\\Theta (\\\\tilde{\\\\varepsilon }_H-\\\\tilde{\\\\varepsilon })\\\\Theta (\\\\tilde{\\\\varepsilon }-\\\\tilde{\\\\varepsilon }_L),$ where $\\\\tilde{\\\\varepsilon }_H=\\\\Lambda +\\\\delta _{\\\\tilde{\\\\varepsilon }_H}$ , $\\\\tilde{\\\\varepsilon }_L=\\\\Lambda -\\\\delta _{\\\\tilde{\\\\varepsilon }_L}$ , $\\\\delta _{\\\\tilde{\\\\varepsilon }_H}=\\\\delta _{\\\\tilde{\\\\varepsilon }}/2+\\\\tilde{\\\\varepsilon }_0/4$ , $\\\\delta _{\\\\tilde{\\\\varepsilon }_L}=\\\\delta _{\\\\tilde{\\\\varepsilon }}/2-\\\\tilde{\\\\varepsilon }_0/4$ .\",\n \"Here, the thickness of half-vicinity shell, $\\\\delta _{\\\\tilde{\\\\varepsilon }}$ , in its discrete form and the ground state energy in normalized energy space, $\\\\tilde{\\\\varepsilon }_0$ , are given respectively as follows =2(12 i1+...+d2 id) 0=12+...+d2 Instead of discrete definition of $\\\\delta _{\\\\tilde{\\\\varepsilon }}$ , it's also possible to define it based on continuous expressions which simplifies mathematical operations and will be given in the following subsection.\",\n \"Half-vicinity shell thickness is a crucial parameter since contributions to oscillatory properties mainly come from the states within this half-vicinity shell.\",\n \"Figure: (Color online) (a) Spin normalized 1D Fermi distribution and occupancy variance functions in energy space for Λ=25\\\\Lambda =25 and α=0.5\\\\alpha =0.5.\",\n \"Thickness of half-vicinity shell in energy space (δ ε ˜ =5\\\\delta _{\\\\tilde{\\\\varepsilon }}=5) denoted by green line.\",\n \"(b) Spin normalized occupancy variance function in 1D state space for weakly confined (α 1 =0.25\\\\alpha _1=0.25) and confined (α 1 =1\\\\alpha _1=1) domains for a constant chemical potential (Λ=10\\\\Lambda =10).\",\n \"Shaded region denotes the thickness of half-vicinity shell which is equal to unity in state space.\",\n \"Red and green points represent the half-vicinity states (Fermi points in 1D) and off-half-vicinity states respectively.In Fig.\",\n \"1(a), spin normalized 1D Fermi distribution and occupancy variance functions in energy space are shown by dashed-blue and solid-red curves respectively.\",\n \"Thickness of half-vicinity shell is represented by the solid horizontal green line.\",\n \"In Fig.\",\n \"1(b), occupancy variance is plotted in 1D state space for two different confinement values for a constant chemical potential ($\\\\Lambda =10$ ).\",\n \"As is seen from Fig.\",\n \"1(b), when confinement increases (denoted by the leftward arrow), occupancy variance peak becomes sharper.\",\n \"Red states are the closest states to the respective Fermi levels, so their contributions are the largest.\",\n \"Light blue shaded region represents HV window.\",\n \"These closest states to Fermi level in 1D state space are called Fermi points.\",\n \"They constitute a Fermi line in 2D and Fermi surface in 3D cases.\",\n \"Green points represent the states outside of half-vicinity shell, which we call off-half-vicinity (OHV) states.\",\n \"Note that in strongly confined case contributions of OHV states are negligible.\",\n \"Thus, HVM represents the true behavior of occupancy variance function accurately in strongly degenerate cases, where oscillations are strong.\",\n \"On the contrary, for weakly confined or unconfined cases, where oscillations weaken or disappear, HVM representation of variance function becomes inaccurate because of non-negligible contributions of OHV states (Fig.\",\n \"1(b) for $\\\\alpha =0.25$ ).\",\n \"However, in this case, continuum expressions of variance function already represent the true behavior properly.\"\n ],\n [\n \"Thickness of half-vicinity shell in energy space and renormalized HVM\",\n \"Thickness of half-vicinity shell in energy space plays an important role in HVM.\",\n \"To describe the thickness in a general way, we invoke number of states (NOS) and density of states (DOS) concepts.\",\n \"NOS inside the half-vicinity shell is found by $NOS_{HV}=\\\\sum _{i_1=1}^{\\\\infty }\\\\cdots \\\\sum _{i_d=1}^{\\\\infty }w_{HV}$ , which is an exact equation but does not lead to analytical expressions, which we seek here.\",\n \"Therefore, we use continuum approximation to convert summations into integrals and find the thickness of half-vicinity shell in energy space analytically as $\\\\delta _{\\\\tilde{\\\\varepsilon }}=CNOS_{HV}/CDOS(\\\\Lambda )$ , where $CNOS_{HV}=\\\\int _{0}^{\\\\infty }\\\\cdots \\\\int _0^{\\\\infty }w_{HV}di_1\\\\ldots di_d$ .\",\n \"By calculating the integrals of CNOS and using the known expressions of CDOS [47], $\\\\delta _{\\\\tilde{\\\\varepsilon }}$ can be obtained for various dimensions in degenerate limit as given in Table I.\",\n \"Note that when $\\\\sqrt{\\\\Lambda }/\\\\alpha _n$ value of a system becomes less than unity for a particular direction $n$ , momentum modes in that direction becomes empty except the ground state and the system can be considered as if it has one dimension less.\",\n \"Table: CNOS, CDOS and thickness of half-vicinity shell in energy space for various dimensions.$\\\\delta _{\\\\tilde{\\\\varepsilon }}$ can be generalized into $d$ -dimensions as $\\\\delta _{\\\\tilde{\\\\varepsilon }}(d)=2\\\\sqrt{\\\\frac{\\\\Lambda }{\\\\pi }}\\\\frac{\\\\Gamma (d/2)}{\\\\Gamma [(d+1)/2]}\\\\sum _{n=1}^{d}\\\\alpha _n.$ In continuum case, $(\\\\alpha _1,\\\\ldots ,\\\\alpha _d\\\\rightarrow 0)$ , HVM gives the following expression for TOV, HVC2 =02 wHVdi1...did =02 wHVCDOS()d =LH2 CDOS()d. It should be noted that $\\\\tilde{\\\\varepsilon }_0\\\\rightarrow 0$ in continuous case and $\\\\tilde{\\\\varepsilon }_H$ and $\\\\tilde{\\\\varepsilon }_L$ become $\\\\Lambda +\\\\delta _{\\\\tilde{\\\\varepsilon }}/2$ and $\\\\Lambda -\\\\delta _{\\\\tilde{\\\\varepsilon }}/2$ respectively.\",\n \"It's possible to obtain an analytical solution for the integral in Eq.\",\n \"(12c) for any dimension by considering strongly degenerate conditions $(\\\\Lambda >>1)$ , $\\\\Sigma _{HVC}^2\\\\cong \\\\tanh \\\\left(\\\\frac{\\\\delta _{\\\\tilde{\\\\varepsilon }}}{4}\\\\right)\\\\Sigma _{CA}^2.$ Here, $\\\\tanh $ factor acts as an amplitude renormalization factor.\",\n \"Although this factor is found by a continuous approach, the similar renormalization factor is expected also for the discrete case, because amplitude renormalization process is independent of the type of accumulation operators which is verified by the numerical results in this article.\",\n \"Thus, we may do the following approximation $\\\\begin{aligned}\\\\Sigma _{HV}^2 &=\\\\sum _{i_1=1}^{\\\\infty }\\\\cdots \\\\sum _{i_d=1}^{\\\\infty }\\\\sigma ^2w_{HV} \\\\\\\\&\\\\cong \\\\tanh \\\\left(\\\\frac{\\\\delta _{\\\\tilde{\\\\varepsilon }}}{4}\\\\right)\\\\left[\\\\sum _{i_1=1}^{\\\\infty }\\\\cdots \\\\sum _{i_d=1}^{\\\\infty }\\\\sigma ^2\\\\right]=\\\\tanh \\\\left(\\\\frac{\\\\delta _{\\\\tilde{\\\\varepsilon }}}{4}\\\\right)\\\\Sigma _{D}^2.\\\\end{aligned}$ Then, we can express discrete TOV, $\\\\Sigma _{D}^2$ , in terms of $\\\\Sigma _{HV}^2$ as follows, $\\\\Sigma _{D}^2\\\\approx \\\\frac{\\\\Sigma _{HV}^2}{\\\\tanh \\\\left(\\\\delta _{\\\\tilde{\\\\varepsilon }}/4\\\\right)}=\\\\Sigma _{RHV}^2,$ where $RHV$ subscript denotes renormalized half-vicinity.\",\n \"Factor of $1/\\\\tanh $ renormalizes the amplitude of variance function found by HVM (which only considers HV states) in order to represent also the contributions of OHV states.\",\n \"In other words, $\\\\Sigma _{RHV}^2$ is the renormalized TOV based on HVM and it includes contributions of both HV and OHV states.\",\n \"Accuracy of this expression depends on the thickness of half-vicinity shell inside the factor of $\\\\tanh $ , which is a function of confinement and degeneracy.\",\n \"The remarkable accuracy of Eq.\",\n \"(15) in comparison with the exact equation (Eq.\",\n \"(2)) is shown in Fig.\",\n \"2.\",\n \"Although Weyl representation (Eq.\",\n \"(3)) does not predict the oscillations but only their trend, HVM quite perfectly matches with the exact results especially for high degeneracy and confinement conditions where oscillations are significant.\",\n \"Consequently, by considering Eqs.\",\n \"(2), (7) and (15), HVM proposes the following transformation: $\\\\sigma ^2\\\\xrightarrow{}\\\\sigma ^2\\\\frac{w_{HV}}{\\\\tanh \\\\left(\\\\delta _{\\\\tilde{\\\\varepsilon }}/4\\\\right)}$ for the calculation of any thermodynamic and transport quantity containing $\\\\sigma ^2$ term.\",\n \"By this transformation, infinite sums are converted into finite sums over the minimum number of states near to Fermi level.\",\n \"As is clear from the comparisons in Fig.\",\n \"2, HVM can be used as a reliable tool to predict and represent quantum oscillations.\",\n \"Figure: (Color online) Comparison of TOV results based on exact model, HVM and Weyl expressions.\",\n \"(a) 1D (b) 2D (c) 3D occupancy variance functions changing with degeneracy and (d) 1D (e) 2D (f) 3D occupancy variance functions changing with isometric confinement in all available directions.\",\n \"Red, blue and green solid curves represent Eq.\",\n \"(2) in 1D, 2D and 3D respectively.\",\n \"Eq.\",\n \"(15) (HVM estimation) is represented by dashed black curves.\",\n \"Dot-dashed purple curves are Weyl representations of TOV functions from Eq.\",\n \"(3).\"\n ],\n [\n \"Analytical expression for TOV in 1D case\",\n \"In highly degenerate and strongly confined 1D case, summation of TOV in Eq.\",\n \"(2) vanishes and HVM represents only the contribution of Fermi point inside $i_F\\\\pm 1/2$ interval in state space, where $i_F=\\\\sqrt{\\\\Lambda }/\\\\alpha _1$ .\",\n \"Therefore, for this special case, there is no need to do summation over half-vicinity shell states (since there is only one) and a special value of that state variable can be found analytically as $i_{*}=\\\\left[i_F\\\\right]=i_F-\\\\frac{1}{\\\\pi }\\\\arctan {\\\\left[\\\\tan \\\\left(\\\\pi i_F\\\\right)\\\\right]},$ where $\\\\left[\\\\cdots \\\\right]$ denotes the round function which can be represented by elementary functions on the right hand side.\",\n \"Then, renormalized TOV in 1D case is $\\\\Sigma _{RHV}^2=\\\\frac{\\\\sigma ^2(i_{*})}{\\\\tanh \\\\left(\\\\alpha _1\\\\sqrt{\\\\Lambda }/2\\\\right)}.$ A comparison for variations of exact and HVM's TOV functions with confinement is given in Fig.\",\n \"3.\",\n \"Solid red, solid blue and dashed black curves are results of Eqs.\",\n \"(2), (5) and (18) respectively.\",\n \"Accuracy of HVM in representing the prominent oscillations is quite good, even though it only considers contribution of Fermi points.\",\n \"Since Eqs.\",\n \"(3) and (5) are equal to each other for 1D case, there is no need to examine $\\\\Sigma _W^2$ additionally.\",\n \"It's seen from the red curve in Fig.\",\n \"3 that transitions between stationary and oscillatory regimes are smooth and $\\\\ln 3/\\\\sqrt{\\\\Lambda }$ line perfectly separates these regimes, which will be examined in detail in the following section.\",\n \"Figure: (Color online) A comparison of exact and HVM results for the variation of TOV with confinement when Λ=30\\\\Lambda =30.\",\n \"Green vertical line, α 1 =ln3/Λ\\\\alpha _1=\\\\ln 3/\\\\sqrt{\\\\Lambda }, separates the oscillatory regime from the stationary one.\",\n \"Δα pp \\\\Delta \\\\alpha _{pp} denotes the period of TOV varying with confinement.Extremum points of oscillations in Fig.\",\n \"3 correspond to the values where the total number of particles is integer.\",\n \"We can determine the periodicity of TOV changing with $\\\\Lambda $ and $\\\\alpha $ , considering the fact that maxima and minima of oscillations correspond to odd and even integer particle numbers respectively, for $g_s=2$ .\",\n \"Special to the 1D case, expression found by considering first two terms of PSF exactly represent number of particles for a given chemical potential in degenerate case [30].\",\n \"We find the values of $\\\\Lambda $ and $\\\\alpha $ correspond to integer numbers of particles as $\\\\Lambda =[\\\\alpha _1(\\\\tilde{N}+1/2)]^2$ and $\\\\alpha _1=\\\\sqrt{\\\\Lambda }/(\\\\tilde{N}+1/2)$ respectively.\",\n \"Periods of oscillations in $\\\\alpha $ and $\\\\Lambda $ spaces can then easily be found respectively as pp=4(2N+1)(2N+3)=212N+3, pp=212(N+1)=8N+1(2N+1)2, where $\\\\tilde{N}=N/g_s$ is spin normalized particle number.\",\n \"Considering the fact that $\\\\Sigma _{RHV}^2$ mimics the oscillations in an almost perfect way, an intriguing question comes to the mind: Is it possible to determine analytically where quantum oscillations start as the confinement and/or degeneracy changes?\"\n ],\n [\n \"A phase diagram for quantum oscillations: Transition from classical to quantum behavior\",\n \"HVM takes the discreteness of quantum states into account and allows us to accurately calculate the thermodynamic and transport properties exhibiting size and density dependent quantum oscillations without using infinite summations.\",\n \"In addition to that, HVM can accurately estimate where the transition from classical behavior to quantum behavior starts, in the framework of quantum oscillations.\",\n \"The states contributing to TOV consist of HV states and OHV states (i.e., $\\\\Sigma _{D}^2=\\\\Sigma _{HV}^2+\\\\Sigma _{OHV}^2$ ).\",\n \"HV states are responsible from the oscillatory part of TOV, whereas OHV states represent the stationary part.\",\n \"They constitute two competing parts of TOV.\",\n \"Domination of contributions of HV states over OHV ones leads to oscillations and vice versa.\",\n \"Hence, by considering $\\\\Sigma _{HV}^2=\\\\Sigma _{OHV}^2$ balance, we can compare contributions of HV states and OHV states to define a universal (material independent) recipe for the separation of stationary regime (SR) from oscillatory regime (OR).\",\n \"According to the recipe, when $\\\\Sigma _{HV}^2<\\\\Sigma _{OHV}^2$ , oscillations disappear (SR) and on the opposite condition $\\\\Sigma _{HV}^2>\\\\Sigma _{OHV}^2$ , oscillations reveal (OR).\",\n \"Since $\\\\Sigma _{OHV}^2=\\\\Sigma _{D}^2-\\\\Sigma _{HV}^2$ , SR-OR transition can be quantified as the balance of the contributions of HV and OHV states ($\\\\Sigma _{HV}=\\\\Sigma _{OHV}$ ) by $\\\\Sigma _{D}^2=2\\\\Sigma _{HV}^2$ .\",\n \"Since the transition is not sharp but smooth, we can safely use analytical expressions of TOV, instead of their exact expressions, to find an analytical expression for SR-OR separation.\",\n \"From the balance condition between contributions of HV and OHV states as well as Eq.\",\n \"(13), we can determine the following analytical condition for the transition between SR and OR, $\\\\Sigma _{CA}^2=2\\\\Sigma _{HVC}^2 \\\\;\\\\Rightarrow \\\\; 1=2\\\\tanh \\\\left(\\\\delta _{\\\\tilde{\\\\varepsilon }}/4\\\\right) \\\\;\\\\Rightarrow \\\\; \\\\delta _{\\\\tilde{\\\\varepsilon }}=2\\\\ln 3.$ It's seen from red curve in Fig.\",\n \"3, balance condition quantified in Eq.\",\n \"(20) clearly gives the SR-OR separation.\",\n \"By decreasing confinement or degeneracy, the number of states around Fermi level increases, contributions of OHV states become also appreciable and their contributions make oscillation amplitude smaller.\",\n \"The more number of states around Fermi level, the smaller oscillation amplitude.\",\n \"When contributions of OHV states exceeds that of HV states, oscillations disappear.\",\n \"To complete the construction of the phase diagram, it is necessary to check also the number of particles, ($N$ ), inside the system, since we are dealing with extremely confined systems having relatively low number of particles.\",\n \"For statistical representations there has to be sufficiently large number of particles inside the system.\",\n \"Nevertheless, the physically meaningful region in a phase diagram can be stated by the condition $N\\\\ge 1$ .\",\n \"Full list of recipes and conditions to establish the phase diagram is given in the Table 2.\",\n \"Table: Recipes and conditions for the construction of the phase diagram of quantum oscillations.According to the recipes and conditions given in Table 2, phase diagrams of quantum oscillations in degeneracy-confinement space for various dimensions can be determined.\",\n \"For isometric 3D domains, the phase diagram is constructed and given in Fig.\",\n \"4.\",\n \"Solid-black curves represent interfaces defined by the conditions in Table II.\",\n \"$\\\\delta _{\\\\tilde{\\\\varepsilon }}=2\\\\ln 3$ condition, representing the balance of HV and OHV states, separates OR and SR.\",\n \"When both confinement and degeneracy sufficiently increase, system enters into the quantum regime where certain thermodynamic and transport quantities oscillate with varying size or density.\",\n \"Figure: (Color online) A phase diagram on degeneracy-confinement space for quantum oscillations.\",\n \"δ ε ˜ =2ln3\\\\delta _{\\\\tilde{\\\\varepsilon }}=2\\\\ln 3 condition defines the boundary between stationary (classical) and oscillatory (quantum) regimes.\",\n \"Blue dot represents the critical point in the phase diagram for isometric 3D domains.Intersection of SR-OR interface curve with $N=1$ curve denotes the critical point which is represented by blue dot in Fig.\",\n \"4.\",\n \"Critical value of the confinement parameter at this point is $\\\\alpha _{*}^{3D}=0.78$ and the corresponding degeneracy value is $\\\\Lambda _{*}^{3D}=0.88$ .\",\n \"These values are universal for isometric rectangular confinement domains.\",\n \"For confinement values below $\\\\alpha _{*}$ , existence of oscillations can be controlled and they can even be suppressed by decreasing degeneracy (through density or temperature).\",\n \"Similarly, for degeneracy values higher than $\\\\Lambda _*$ , existence of oscillations can be controlled by changing confinement.\",\n \"This region defined by $\\\\lbrace \\\\Lambda >\\\\Lambda _{*},\\\\alpha <\\\\alpha _*\\\\rbrace $ is the crossover region restricted by dashed lines in the phase diagram.\",\n \"On the contrary, for higher confinement values than $\\\\alpha _{*}$ , quantum oscillations cannot be suppressed and the region is called entirely oscillatory region.\",\n \"Below the critical degeneracy values, system does not exhibit oscillatory behaviors regardless of the values of confinements.\",\n \"Hence, this critical point is used to define different regions (crossover region, entirely OR and entirely SR).\",\n \"Regimes on phase diagram and corresponding conditions are summarized in Table III.\",\n \"Table: Conditions and regions of the phase diagram.Although the phase diagram in Fig.\",\n \"4 represents only isometric 3D domain, the form of the phase diagrams for 1D and isometric 2D domains are also very similar to the 3D one.\",\n \"Only SR-OR interface slightly shifts upward while $N=1$ curve slightly rotates in clockwise direction around the critical point as the dimension decreases.\",\n \"For 1D and isometric 2D domains, universal critical values can also be found as $\\\\alpha _{*}^{1D}=1.07$ , $\\\\alpha _{*}^{2D}=0.87$ and $\\\\Lambda _{*}^{1D}=1.05$ , $\\\\Lambda _{*}^{2D}=0.98$ .\",\n \"It's also possible to investigate the anisometric domains by defining aspect ratios.\",\n \"For 2D domain, $r_{12}=L_1/L_2=\\\\alpha _2/\\\\alpha _1$ denotes the aspect ratio.\",\n \"For 3D, additionally we define $r_{13}=L_1/L_3=\\\\alpha _3/\\\\alpha _1$ .\",\n \"Here, the most confined direction is chosen as direction 1.\",\n \"In Fig.\",\n \"5, we show how the critical confinement values are changing with respect to aspect ratios of 2D and 3D domains.\",\n \"Figure: (Color online) Aspect ratio dependencies of critical confinement values for 2D and 3D cases.Although the values in Fig.\",\n \"5 are calculated in an exact way by using Tables I and II, $\\\\alpha _{1*}^{2D}$ can be approximated by $\\\\alpha _{1*}^{2D}\\\\approx 0.87-0.40\\\\ln r_{12}$ with less than $1.5\\\\%$ mean absolute percentage error (MAPE) for $0.1\\\\le r_{12} \\\\le 1$ interval.\",\n \"Similarly, $\\\\alpha _{1*}^{3D}$ is approximated by $\\\\alpha _{1*}^{3D}\\\\approx 0.78-0.27\\\\ln r_{12}-0.27\\\\ln r_{13}$ with less than $2.2\\\\%$ MAPE for $0.1\\\\le \\\\lbrace r_{12},r_{13}\\\\rbrace \\\\le 1$ intervals.\",\n \"Critical degeneracy values for anisometric cases can easily be found for 3D, 2D and 1D cases respectively, *3D=[231*3D(1+r12+r13)]2, *2D=[321*2D(1+r12)]2, *1D=(31*1D)2.\",\n \"It's important to note that thickness of half-vicinity shell indicates the quantumness of the system.\",\n \"When volume $V\\\\rightarrow \\\\infty $ or temperature $T\\\\rightarrow \\\\infty $ or density $n\\\\rightarrow 0$ , thickness $\\\\delta _{\\\\tilde{\\\\varepsilon }}\\\\rightarrow 0 <2\\\\ln 3$ and system behaves classically.\",\n \"Conversely, for sets of $(V,T,n)$ making $\\\\delta _{\\\\tilde{\\\\varepsilon }}>2\\\\ln 3$ , the system shows quantum oscillatory behaviors.\",\n \"As a matter of fact, $\\\\delta _{\\\\tilde{\\\\varepsilon }}$ is a precise indicator of whether system is in quantum regime or not in terms of oscillations.\",\n \"In order to see the full picture of quantum oscillations in 1D case, variation of TOV, Eq.\",\n \"(2), with confinement and degeneracy is given in Fig.\",\n \"6.\",\n \"It is seen now even more clearly that $\\\\delta _{\\\\tilde{\\\\varepsilon }}=2\\\\ln 3$ plane perfectly separates oscillatory region from the stationary one regardless of the values of $\\\\alpha $ and $\\\\Lambda $ .\",\n \"Increment in degeneracy and/or confinement make $\\\\delta _{\\\\tilde{\\\\varepsilon }}$ larger and cause strong oscillations.\",\n \"Figure: (Color online) Variation of discrete TOV (Σ D 2 \\\\Sigma ^2_D) with confinement (α\\\\alpha ) and degeneracy (Λ\\\\Lambda ).\",\n \"Gray 2ln32\\\\ln 3 plane separates the oscillatory (quantum) and stationary (classical) regimes.HVM is constructed to predict oscillations appearing in strongly confined and highly degenerate conditions.\",\n \"Accuracy of its predictions increases with increasing confinement and degeneracy, while accuracy becomes weaker for the regions near to SR-OR interface in phase diagram, Fig.\",\n \"4.\",\n \"To be precise, HVM predicts oscillations almost perfectly, as long as confinement and degeneracy are larger than their critical values ($\\\\alpha _*$ and $\\\\Lambda _*$ ).\"\n ],\n [\n \"Size and density dependent oscillations in electronic heat capacity\",\n \"In previous sections we investigated the oscillatory behavior of TOV function and constructed a theoretical model (HVM) for its calculation.\",\n \"In this subsection, we examine quantum oscillations of heat capacity of an electron gas confined in a nanowire by considering both exact and HVM.\",\n \"Electronic heat capacity from the derivative of internal energy with respect to temperature is written in its exact (based on infinite sums) form as $C_V=g_s k_B\\\\left[\\\\sum _{i_n=1}^{\\\\infty }\\\\tilde{\\\\varepsilon }^2\\\\sigma ^2-\\\\frac{\\\\left(\\\\sum _{i_n=1}^{\\\\infty }\\\\tilde{\\\\varepsilon }\\\\sigma ^2\\\\right)^2}{\\\\sum _{i_n=1}^{\\\\infty }\\\\sigma ^2}\\\\right].$ Heat capacity expression, Eq.\",\n \"(22), contains zeroth, first and second order energy moments of occupancy variance.\",\n \"Each summation in Eq.\",\n \"(22) has distinct oscillatory behavior, which leads to the characteristic oscillations in heat capacity.\",\n \"In order to reveal quantum oscillations and make electronic contribution of heat capacity appreciable, we focus on very low temperature ($T=5$ K), high electron density $(\\\\sim 10^{25} \\\\text{m}^{-3})$ and nanoscale confinements in which quantum effects are observable.\",\n \"To make our discussions quantity-independent, we examine electronic heat capacity per particle $(c_V=C_V/N)$ .\",\n \"In the calculations, chemical potential is obtained numerically from particle number equation as functions of temperature, density and confinement.\",\n \"In Fig.\",\n \"7, accuracy of HVM is examined by considering size and density dependent oscillations of normalized electronic specific heat.\",\n \"$c_V^0$ represents the specific heat expression under continuum approximation, where all summations in Eq.\",\n \"(22) replaced by integrals.\",\n \"In all cases in Fig.\",\n \"7, an electron gas confined in a quantum wire with 200 nm long $(L_3)$ is considered at 5K temperature.\",\n \"Red, blue and green curves in Fig.\",\n \"7(a), 7(b) and 7(c) respectively represent the results of the exact model, while the black dashed curves are the results of HVM.\",\n \"Although confinement domain is a nanowire, it cannot be considered as a pure 1D structure since there are still a few excited states in transverse directions.\",\n \"Thus, calculations are done over triple sums.\",\n \"Figure: (Color online) Accuracy of HVM on predicting (a) size dependent and (b), (c) density dependent oscillations in normalized heat capacity for a nanowire with 200 nm long at 5K temperature.\",\n \"Solid curves are the results obtained by using exact expressions (based on infinite sums), while dashed black curves are obtained by using HVM.In Fig.\",\n \"7(a), sizes of the domain in transverse directions ($L_1$ and $L_2$ ) are changed simultaneously from 10 nm to 40 nm for $n=10^{25}\\\\text{m}^{-3}$ where $\\\\Lambda $ is always larger than $\\\\Lambda _{*}$ .\",\n \"Confinement in transverse directions spans the values larger and smaller than $\\\\alpha _*$ .\",\n \"HVM predicts oscillatory size dependence of electronic heat capacity pretty well between 10 nm and 25 nm, where confinement is strong, oscillations are large and unavoidable ($\\\\alpha \\\\ge \\\\alpha _*$ ).\",\n \"For 25 nm and 40 nm interval, however, confinement becomes weaker ($\\\\alpha \\\\le \\\\alpha _*$ ) and an appreciable difference appears in between exact and HVM results while oscillations are also weak.\",\n \"This is an expected result, since the system approaches to SR-OR interface in phase diagram.\",\n \"In Fig.\",\n \"7(b), quantum oscillations in heat capacity varying with electron density are shown for a square wire having 20 nm size in transverse directions.\",\n \"Note that density variation for a constant confinement corresponds to a vertical movement on the phase diagram.\",\n \"HVM perfectly matches with the exact model almost everywhere in Fig.\",\n \"7(b) because of strong confinement, $\\\\alpha _{1}=\\\\alpha _{2}=1.48>\\\\alpha _{1*}^{3D}=1.27$ .\",\n \"System shows oscillatory behavior regardless of the value of degeneracy (or electron density).\",\n \"In other words, oscillations are unavoidable for the system discussed in Fig.\",\n \"7(b) as long as the confinement is sustained.\",\n \"By increasing the domain size from 20 nm to 25 nm in confined directions (reducing the confinement), we can put the system into the crossover regime where $\\\\alpha _{1}=\\\\alpha _2=1.18<\\\\alpha _{1*}^{3D}=1.23$ as it's shown in Fig.\",\n \"7(c).\",\n \"For this case, HVM does not match well with the exact model for lower electron densities, where the system is getting closer to SR-OR interface.\",\n \"Oscillations decay by decreasing confinement and degeneracy but the decay rate of oscillations with degeneracy depends on how much the confinement smaller than the critical confinement value.\",\n \"Since the confinement is not much smaller than $\\\\alpha _*$ , there is no considerable decay in oscillations in Fig.\",\n \"7(c).\",\n \"All these behaviors confirm the predictions of the phase diagram presented in Fig.\",\n \"4 as well as aspect ratio dependencies of critical confinement parameters given in Fig.\",\n \"5.\",\n \"In contrast to size and density, variation of temperature does not lead to oscillatory behaviors in thermodynamic and transport quantities.\",\n \"The reason can clearly be seen from Eqs.\",\n \"(17) and (18) where oscillations are originated from the variation of round of $i_F$ .\",\n \"Since temperature dependence of chemical potential disappears in strongly degenerate case, $[i_F]=[\\\\sqrt{\\\\Lambda }/\\\\alpha ]$ becomes independent of temperature.\",\n \"Hence, in strongly degenerate case, temperature variation does not affect the proximities of states to the Fermi level in state space, but just changes the occupation probabilities of the states.\",\n \"In other words, thermal broadening happens only in energy space, not in state space.\",\n \"Since the proximities do not variate, oscillations do not appear.\",\n \"This explanation can be directly extended to higher dimensional cases also, due to the orthogonality of state space dimensions.\"\n ],\n [\n \"Oscillatory violation of entropy-heat capacity equivalence in degenerate limit\",\n \"Continuum expressions of entropy and heat capacity of an ideal Fermi gas are equal to each other in degenerate limit [3].\",\n \"When we consider the discrete nature of quantum states, however, entropy-heat capacity equivalence is also broken and quantum behaviors lead to an oscillatory non-equivalence of entropy and heat capacity of an ideal degenerate Fermi gas at nanoscale.\",\n \"The well-known form of entropy is written as $S=-g_s k_B\\\\left[\\\\sum _{i_n=1}^{\\\\infty }{\\\\tilde{f}\\\\ln (\\\\tilde{f})+(1-\\\\tilde{f})\\\\ln (1-\\\\tilde{f})}\\\\right].$ In degenerate limit, both Eq.\",\n \"(22) and (23) give the same expression (for instance, $g_s\\\\pi ^2Nk_B/(2\\\\Lambda )$ for 3D case) under continuum approximation.\",\n \"However, as it is seen from Fig.\",\n \"8, this equivalence is broken.\",\n \"Heat capacity/entropy ratio deviates from unity and variates with domain size and density in an oscillatory fashion.\",\n \"A nanowire in the previous subsection is reconsidered with the same specifications here for the examination of heat capacity-entropy ratio.\",\n \"Both electronic heat capacity and entropy oscillate in confined and degenerate conditions.\",\n \"Although their scales are almost the same, due to the phase and magnitude differences of both function's characteristic oscillations, their ratio also oscillates around unity.\",\n \"Increasing domain size or temperature decreases the amplitude of oscillations.\",\n \"Similar to heat capacity results, the success of HVM is quite good for higher electron density and confinement conditions.\",\n \"Figure: (Color online) Oscillatory violation of entropy-heat capacity equivalence in degenerate limit depending on (a) size and (b), (c) density for a nanowire with 200 nm long at 5K temperature.\",\n \"Solid curves are the results obtained by using exact expressions, while dashed black curves are obtained by using HVM.Very similar to the behavior of TOV, fully occupied or completely unoccupied states do not contribute to the entropy of the gas and contributions come only from the states around Fermi level, which are identified as HV states in HVM.\",\n \"Therefore, it is possible to use HVM also for the quantities which does not directly contain occupancy variance, as long as HV states are dominant over the quantity to be calculated like in entropy.\"\n ],\n [\n \"Resemblance of total occupancy variance and density of states functions\",\n \"There is a direct relationship between DOS at Fermi level and TOV.\",\n \"It's well-known that, in strongly degenerate case, spin normalized distribution function nearly turns into a Heaviside step function $f(\\\\tilde{\\\\varepsilon })\\\\approx g_s\\\\Theta (\\\\Lambda -\\\\tilde{\\\\varepsilon })$ .\",\n \"Since the number of states below Fermi level $\\\\Lambda $ is approximately equal to the number of Fermions per spin state, $N=\\\\sum f(\\\\tilde{\\\\varepsilon })\\\\approx g_s\\\\sum \\\\Theta (\\\\Lambda -\\\\tilde{\\\\varepsilon })=NOS(\\\\Lambda )$ , derivative of $NOS(\\\\Lambda )$ with respect to $\\\\Lambda =\\\\tilde{\\\\varepsilon }_F$ gives $DOS(\\\\Lambda )$ and then TOV is approximately equal to $DOS(\\\\Lambda )$ .\",\n \"$\\\\Sigma _{D}^2=-\\\\sum _{i_n}{\\\\frac{\\\\partial f}{\\\\partial \\\\tilde{\\\\varepsilon }}}\\\\approx g_s\\\\sum _{i_n}{\\\\delta (\\\\Lambda -\\\\tilde{\\\\varepsilon })}=DOS(\\\\Lambda ).$ In Fig.\",\n \"9, resemblance of TOV and DOS functions is shown for two different subbands with constant confinement parameters for various dimensions.\",\n \"Note that the results of lower dimensions are obtained by strongly confining a 3D domain in different directions.\",\n \"Even the lower dimensional domains are represented by triple sums where sums in strongly confined directions represent subbands.\",\n \"Hence, it's possible to directly obtain Eq.\",\n \"(6) by using continuous DOS functions at Fermi level $\\\\tilde{\\\\varepsilon }_F=\\\\Lambda $ given in Table I.\",\n \"While HVM predicts oscillations quite good, DOS functions represents only the trend of the oscillations (Fig.\",\n \"9) like Weyl representations of TOV functions in Fig.\",\n \"2 and continuous TOV function in Fig.\",\n \"3.\",\n \"Figure: (Color online) Similarity of TOV functions (solid curves) and continuous density of states functions at Fermi level (dashed curves) for various dimensions.Although the structure of two functions are very similar, due to thermal broadening at Fermi level there are small differences around the beginning of each subband especially for low dimensional cases.\",\n \"The origin of this resemblance is basically coming from the fact that occupancy variance function has a peakwise nature at Fermi level and vanishing behavior elsewhere, which is directly related with density of states at Fermi level.\"\n ],\n [\n \"Conclusion\",\n \"In this article, investigation of size and density dependent quantum oscillations is done by proposing the HVM for the calculation of oscillatory quantities.\",\n \"Proposition of our model not only allows to calculate oscillatory thermodynamic or transport quantities in a more efficient way, but also offers a phase diagram which predict quantum oscillations clearly.\",\n \"Moreover, HVM introduces analytical conditions separating oscillatory and stationary regimes on the phase diagram.\",\n \"Occupancy variance function appearing in many oscillatory quantities is examined in detail and the model is constructed on the half-vicinity of states around Fermi level.\",\n \"Two main factors determine the value of TOV: the number of states inside the half-vicinity shell and their proximities to the Fermi level.\",\n \"Confinement and degeneracy directly affect the distribution of the states around Fermi level besides the thickness of half-vicinity shell and so the number of states inside the shell.\",\n \"The results show that HVM can be used as a reliable tool to predict and calculate quantum oscillations of a degenerate and confined ideal Fermi gas.\",\n \"The proposed phase diagram and analytical conditions for quantum oscillations evidently describe entirely stationary/oscillatory and crossover regions on degeneracy-confinement space.\",\n \"Consequently, our model and the phase diagram not only clarify the theoretical understanding of the quantum oscillations but also may help to determine the proper material parameters for experimental studies.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01816"}}},{"rowIdx":896,"cells":{"text":{"kind":"list like","value":[["Blind image deblurring using class-adapted image priors"],["Abstract Blind image deblurring (BID) is an ill-posed inverse problem, usually addressed by imposing prior knowledge on the (unknown) image and on the blurring filter.","Most of the work on BID has focused on natural images, using image priors based on statistical properties of generic natural images.","However, in many applications, it is known that the image being recovered belongs to some specific class (e.g., text, face, fingerprints), and exploiting this knowledge allows obtaining more accurate priors.","In this work, we propose a method where a Gaussian mixture model (GMM) is used to learn a class-adapted prior, by training on a dataset of clean images of that class.","Experiments show the competitiveness of the proposed method in terms of restoration quality when dealing with images containing text, faces, or fingerprints.","Additionally, experiments show that the proposed method is able to handle text images at high noise levels, outperforming state-of-the-art methods specifically designed for BID of text images."],["Introduction","Blind image deblurring (BID) is an inverse problem where the observed image is modeled as the convolution of an underlying (sharp) image and an unknown blurring filter, often followed by additive noise.","The goal of BID is usually to estimate both the underlying image and the blurring filter.","The problem is obviously severely ill-posed.","In addition, since the convolution operator itself is typically ill-conditioned, the inverse problem is highly sensitive to the presence of noise.","In recent years, researchers have investigated a variety approaches to single image BID, mostly considering generic natural images [1], [2], [3], [4], [5], [6].","To deal with the ill-posed nature of the BID problem, most methods use prior information on both the image and the blurring filter.","The most common choice for the image prior exploits the statistics of natural images [1], [3], [4], [7], [8], [5], [9] and is usually based on implicit or explicit restoration of salient edges.","Although that approach gives good results for natural images, the prior itself is not designed for images that belong to specific classes (e.g., text, face, medical structures, fingerprints) appearing in many important applications, like document analysis, surveillance, and forensics.","Methods that use priors that capture the properties of images belonging to specific classes are more likely to provide better results, when dealing with those images, e.g., text [10], [11], [12] or face images [13], [14], [15].","Furthermore, images that belong to different specific classes may have different characteristics that are hard to capture with a unique prior.","For example, face images do not contain much texture and text images have specific structure due to the contents of interest being mainly in two tones (commonly, black and white).","Here, we proposed a method that uses patch-based image priors learned from a set of clean images of the specific class of interest.","The method is based on the so-called plug-and-play approach, recently proposed in [16].","In contrast with [16], we do not use a fixed denoiser, but a denoiser based on a Gaussian mixture model (GMM) that is learned from patches of clean images belonging to a specific class.","A similar idea was recently proposed for non-blind image deblurring and compressive imaging [17].","Here, in addition to the GMM-based image prior, we also adopt a weak prior on the blurring filter.","Considering the blur, earlier methods typically impose hard constrains for the (arguably) most relevant case of a generic motion blur by encouraging sparsity of the blur filter estimate [4], [18], [1], [8], [3], [19], [7].","In this paper, we use a weaker prior on the blur (limited support), thus being able to recover a wide variety of filters then those methods."],["Observation model","Consider the linear observation model $\\textbf {y} = \\textbf {H} \\textbf {x} + \\textbf {n}$ , where $\\textbf {y} \\in \\mathbb {R}^{n}$ , $\\textbf {x} \\in \\mathbb {R}^{m}$ denote the vectorized (lexicographically ordered) observed data and the (unknown) original image, respectively, and $\\textbf {n}$ is noise, assumed to be Gaussian, with zero mean and known variance $\\sigma ^2$ .","For computational convenience, $\\textbf {H} \\in \\mathbb {R}^{n \\times m}$ is the matrix that represents the convolution with the blurring filter $\\textbf {h}$ with periodic boundary conditions, thus with $n = m$ .","As explained above, to deal with the blind image deblurring problem, prior information (a regularizer) is imposed on both the underlying image and the blurring filter.","The image $\\textbf {x}$ and the blurring operator $\\textbf {H}$ (equivalently, the filter $\\textbf {h}$ ) are estimated by minimizing the cost function $O_\\lambda (\\textbf {x},\\textbf {h}) = \\frac{1}{2} ||\\textbf {y} - \\textbf {H x}||_2^2 + \\lambda \\phi (\\textbf {x}) + \\Psi _\\textit {S}(\\textbf {h}).$ We assume a weak prior on the blurring filter, $\\Psi _{\\textit {S}}$ , the indicator function of the set $\\textit {S}$ (set of filters with positive entries on a given support).","The rationale behind using a weak prior is that it covers a wider variety of blurring filters.","$\\Psi _{\\textit {S}}(\\textbf {u}) ={\\left\\lbrace \\begin{array}{ll}0 & \\quad \\text{if } \\textbf {u} \\in \\textit {S} \\\\\\infty & \\quad \\text{if } \\textbf {u} \\notin \\textit {S}.\\\\\\end{array}\\right.","}$ The function $\\phi $ represents the prior on the image used to promote characteristics that the underlying sharp image is assumed to have, while parameter $\\lambda $ controls the trade-off between data-fidelity term and the regularizer.","As shown recently [20], [6], good results can be obtained by alternating estimation of the image and the blur kernel (Algorithm 1).","Both steps are performed by using the alternating direction method of multipliers (ADMM) [6].","[H] Blind Image Deblurring Algorithm [1] Blurred image $\\textbf {y}$ Estimated sharp image $\\hat{\\textbf {x}}$ and the blur kernel $\\hat{\\textbf {h}}$ Initialization: Initial estimate $ \\hat{\\textbf {x}} = \\textbf {y}$ , $\\hat{\\textbf {h}}$ set to the identity filter, $\\lambda > 0$ stopping criterion is not satisfied $\\hat{\\textbf {x}} \\leftarrow \\underset{\\textbf {x}}{\\text{argmin}} \\hspace{5.69054pt} O_\\lambda (\\textbf {x}, \\hat{\\textbf {h}})$ estimating $\\textbf {x}$ with $\\textbf {h}$ fixed $\\hat{\\textbf {h}} \\leftarrow \\underset{\\textbf {h}}{\\text{argmin}} \\hspace{5.69054pt} O_\\lambda (\\hat{\\textbf {x}},\\textbf {h})$ estimating $\\textbf {h}$ with $\\textbf {x}$ fixed In contrast with [6], we do not decrease the regularization parameter $\\lambda $ at every iteration.","We found that a fixed parameter yields better results arguably due to the more expressive prior herein used, when compared with the total-variation regularizer in [6]."],["ADMM for image inverse problems","As discussed above, we use the ADMM optimization algorithm to perform estimation of both the image and the blurring filter, and therefore, in this section, we will briefly explain the ADMM for image inverse problems.","Consider an unconstrained optimization problem in which the objective function is the sum of two functions $\\underset{\\textbf {z}}{\\text{min}} \\hspace{5.69054pt} f_1 ( \\textbf {z}) + f_2 ( \\textbf {z}).$ By using a variable splitting procedure, we introduce a new variable $\\textbf {v}$ as the argument of the function $f_2$ , under the constrain that $\\textbf {z}=\\textbf {v}$ .","This leads to rewriting the unconstrained problem from above as a constrained one: $\\underset{\\textbf {z}, \\textbf {v}}{\\text{min}} \\hspace{5.69054pt} f_1(\\textbf {z}) + f_2 (\\textbf {v})\\hspace{11.38109pt} \\text{subject to} \\hspace{11.38109pt} \\textbf {z} = \\textbf {v}.$ The rationale behind variable splitting methods, such as the method of multipliers or augmented Lagrangian method (ALM), is that it may be easier to solve the constrained problem (4) instead of the unconstrained one (3).","The main idea behind the ALM is to minimize alternatingly the so-called augmented Lagrangian function $\\hat{\\textbf {z}}, \\hat{\\textbf {v}} \\leftarrow \\underset{\\textbf {z}, \\textbf {v}}{\\text{min}} \\hspace{5.69054pt} f_1(\\textbf {z}) + f_2 (\\textbf {v})+ \\textbf {d}^T(\\textbf {z}-\\textbf {v}) + \\frac{\\mu }{2}||\\textbf {z} - \\textbf {v}||_2^2,$ and updating the vector of Lagrange multipliers $\\textbf {d}$ (Algorithm 2).","In Equation (5), $\\mu \\ge 0$ is called the penalty parameter.","If we recall that, by definition, the proximity operator (PO) of some convex function $g$ , computed at the point $\\textbf {u}$ is defined as $\\text{prox}_g(\\textbf {u}) = \\underset{\\textbf {x}}{\\text{argmin}} \\hspace{5.69054pt}\\frac{1}{2}||\\textbf {x} - \\textbf {u}||_2^2 + g(\\textbf {x}),$ it is clear that in Algorithm 2, lines 3 and 4 are the PO of $f_1$ and $f_2$ , computed at $\\textbf {v}^k + \\textbf {d}^k$ and $\\textbf {z}^{k+1} - \\textbf {d}^k$ , respectively.","Formulation (6) can be considered as the solution to a denoising problem, with $\\textbf {u}$ as the noisy observation and $g$ the regularizer.","ADMM [1] Initialization: Set $k = 0$ , $\\mu > 0$ , initialize $\\textbf {v}_0$ and $\\textbf {d}_0 $ stopping criterion is not satisfied $ \\textbf {z}^{k+1} \\leftarrow \\underset{\\textbf {z}}{\\text{min}} \\hspace{5.69054pt} f_1(\\textbf {z}) + \\frac{\\mu }{2}||\\textbf {z} - \\textbf {v}^k - \\textbf {d}^k||_2^2$ $ \\textbf {v}^{k+1} \\leftarrow \\underset{\\textbf {v}}{\\text{min}} \\hspace{5.69054pt} f_2(\\textbf {v}) + \\frac{\\mu }{2}||\\textbf {z}^{k+1} - \\textbf {v} - \\textbf {d}^k||_2^2$ $\\textbf {d}^{k+1} \\leftarrow \\textbf {d}^k - ( \\textbf {z}^{k+1} - \\textbf {v}^{k+1} )$ $k \\leftarrow k + 1$"],["GMM-based Denoiser","For the image estimate update (line 3 of Algorithm 1), instead of using the PO of a convex regularizer (line 4 of the Algorithm 2), we implemented a state-of-the-art denoiser considering the fact that the PO itself is a denoising function.","This approach, also known as plug-and-play, was recently exploited in [16], but instead of using a fixed denoiser, such as BM3D [21] or K-SVD [22], we consider a class-adapted GMM-based denoiser [23].","In [23], the authors show that clean image patches are well modeled by a GMM estimated from a collection of clean images using the expectation-maximization (EM) algorithm.","Furthermore, for a GMM-based prior for the clean patches, the corresponding minimum mean squared error (MMSE) estimate can be obtained in closed form [24].","We use these facts to obtain a GMM-based prior learned from the set of clean images that belong to the specific class.","The rationale behind this approach is that with the class-adapted image prior, we may achieve better performance than with a fixed, generic denoiser, when we process images that do belong to the same specific class."],["Proposed method","The proposed method uses ADMM for solving each of the inner minimization problems in Algorithm 1 (lines 3 and 4), with $O_\\lambda (\\textbf {x},\\textbf {h})$ as defined in (1) with the PO of $\\phi $ replaced by the MMSE estimator using a class-adapted GMM prior."],["Image Estimate","The image estimation problem (line 3 of Algorithm 1) can be formulated as: $\\hat{\\textbf {x}} = \\underset{\\textbf {x}}{\\text{argmin}} \\hspace{5.69054pt} \\frac{1}{2} ||\\textbf {y} - \\textbf {H x}||_2^2 + \\lambda \\phi (\\textbf {x}).$ This problem can be written in the form (3) by setting $f_1(\\textbf {x}) = \\frac{1}{2} ||\\textbf {y} - \\textbf {H x}||_2^2$ and $f_2(\\textbf {x}) = \\lambda \\phi (\\textbf {x})$ .","Applying ADMM to problem (7), yields the so-called SALSA algorithm [25].","Line 3 of Algorithm 2 becomes a quadratic optimization problem, which has a linear solution: $\\textbf {x}^{k+1} = (\\textbf {H}^T\\textbf {H}+\\mu \\textbf {I})^{-1} (\\textbf {H}^T \\textbf {y} + \\mu (\\textbf {v}^k + \\textbf {d}^k)).$ As shown in some previous work [25], [26], the matrix inversion in (8) can be efficiently computed in the discrete Fourier transform (DFT) domain (using the FFT) in the case of cyclic deblurring, which we consider in this paper.","Extension to other boundary conditions can be obtained via the technique proposed in [6]."],["Blur Estimate","The blur estimation problem (line 4 of the Algorithm 1) can be formulated as: $\\hat{\\textbf {h}} = \\underset{\\textbf {h}}{\\text{argmin}} \\hspace{5.69054pt} \\frac{1}{2} ||\\textbf {y} - \\textbf {Xh}||_2^2 + \\Psi _\\textit {S}(\\textbf {h}),$ where $\\textbf {h} \\in \\mathbb {R}^{n}$ is the vector containing the lexicographically ordered blurring filter elements and $\\textbf {X} \\in \\mathbb {R}^{n \\times n}$ is the square matrix representing the convolution of the image $\\textbf {x}$ and the filter $\\textbf {h}$ .","Considering formulation (3) we have $f_1(\\textbf {h}) = \\frac{1}{2} ||\\textbf {y} - \\textbf {X h}||_2^2$ and $f_2(\\textbf {h}) = \\Psi _\\textit {S}(\\textbf {h})$ .","The resulting instance of line 3 of Algorithm 2 has the same form as (8) and, as previously explained, the matrix inversion can be efficiently computed in the DFT domain, using the FFT.","Since the proximity operator of the indicator of a convex set is the orthogonal projection on that set [27], line 4 of the Algorithm 2 becomes $\\text{prox}_{\\Psi _{\\textit {S}}}(\\textbf {u}) = P_{\\textit {S}}(\\textbf {u}),$ which simply sets to zero all negative elements and any elements outside of the given support."],["Experiments","In all the experiments, we use the following setting for the two ADMM algorithms described in Section 5: the image estimate is computed with 20 iterations of the algorithm in Subsection 5.1, initialized with the image estimate from the previous iteration, $\\textbf {d}_0 = 0$ , and $\\mu $ hand-tuned for the best visual results or best ISNR (improvement in SNR [6]) in terms of synthetic data.","The blur estimate is computed with two iteration of the algorithm explained in Subsection 5.2, initialized with the blur estimate from the previous iteration, $\\textbf {d}_0 = 0$ , and $\\mu = 0.01$ .","Furthermore, the experiments were performed on three sets of images: (a) a dataset containing 10 text images that is available from the author of [28] (one for testing and nine for training the mixture), (b) a dataset containing 100 face images from the same author as the text dataset, and (c) a dataset containing 128 fingerprints from the publicly available the UPEK fingerprint database.","The GMM-based prior is obtained by using patches of size $6 \\times 6$ pixels and a 20-component mixture."],["Results","For the experiments with text images, we created five test images using the same clean image of text and $15 \\times 15$ synthetic kernels that represent Gaussian, linear motion, out-of-focus, uniform, and nonlinear motion blur, respectively, and noise level corresponding to BSNR = 30 dB (Table 1).","For the face images, we created four $11 \\times 11$ synthetic blur kernels that represent Gaussian, linear motion, out-of-focus, and uniform blurs, respectively, and for the fifth experiment, we used blur kernel number 5 from [2], with noise level corresponding to BSNR = 40 dB (Table 2).","Experiments on the image containg fingerprints are performed with the $15 \\times 15$ linear motion blur kernel and noise level corresponding to BSNR = 40 dB (Fig.","2).","Results of all experiments are compared with two state-of-the-art BID algorithms constructed for natural images ([6] and [5]), and additionally with the algorithm explained in Subsection 5.1 with the BM3D denoiser plugged into it (PlugBM3D), instead of the GMM-based denoiser (PlugGMM).","Note that the generic algorithm [6] is designed for a wide variety of blur filters, while [5], like the majority of blind deblurring algorithms, is designed mostly for motion blur.","Figure: Text image blurred with nonlinear motion blur number 2 from and high noise level (BSNR = 20 dB): (a) Original image and ground truth kernel; (b) Blurred image; (c) Results of , ISNR = -2.72; (d) PlugBM3D, ISNR = 9.97; (e) PlugGMM, ISNR = 11.16.Figure: Fingerprint image blurred with 9×99 \\times 9 linear motion blur and noise level (BSNR = 40 dB): (a) Original image and ground truth kernel; (b) Blurred image; (c) Results of , ISNR = 0.36; (d) Results of , ISNR = -0.64; (e) PlugBM3D, ISNR = 0.56; (f) PlugGMM, ISNR = 1.19.Table: Results in terms of ISNR of the generic methods and , our method using the BM3D denoiser, and our method with the class-adapted GMM prior, tested for text images (BSNR = 30 dB).Table: Results in terms of ISNR of the generic methods and , our method using the BM3D denoiser, and our method with the class-adapted GMM prior, tested for face images (BSNR = 40 dB).Moreover, we tested our method on text images blurred with the blurring filter number 2 from [2], followed by a higher noise level (BSNR = 20 dB) (Fig.","1).","Results are compared, as previously explained, with the BM3D denoiser plugged into the ADMM loop and the method from Pan et al.","[11], which was designed for BID of text images.","As the BM3D denoiser is based on exploiting non-local patch similarities, which is highly present in the images we tested, visual results of using PlugBM3D are very good, but in terms of ISNR, PlugGMM clearly outperforms it."],["Conclusion","In this paper, we have proposed a class-adapted blind image deblurring method, built upon the so-called plug-and-play approach.","The method uses Gaussian mixture model (GMM) based denoisers, adapted to specific image classes, plugged into the ADMM optimization algorithm, and a weak prior (positivity and limited support) on the blurring filter.","Experiments show that the proposed method yields state-of-the-art results, when applied to images that belong to a specific class (e.g., text, face, and fingerprints), outperforming several generic techniques for blind image deblurring [5], [6].","In addition, experiments show that the proposed method can be used for a variety of blurring filters and is able to handle strong noise in the case of images known to contain text, outperforming the state-of-the-art method for BID of text images [11].","The proposed method suffers from some potential limitations, such as setting of the regularization parameter and stopping criteria for the inner ADMM algorithms, as well as for the outer iterations, that we aim to improve in future work."]],"string":"[\n [\n \"Blind image deblurring using class-adapted image priors\"\n ],\n [\n \"Abstract Blind image deblurring (BID) is an ill-posed inverse problem, usually addressed by imposing prior knowledge on the (unknown) image and on the blurring filter.\",\n \"Most of the work on BID has focused on natural images, using image priors based on statistical properties of generic natural images.\",\n \"However, in many applications, it is known that the image being recovered belongs to some specific class (e.g., text, face, fingerprints), and exploiting this knowledge allows obtaining more accurate priors.\",\n \"In this work, we propose a method where a Gaussian mixture model (GMM) is used to learn a class-adapted prior, by training on a dataset of clean images of that class.\",\n \"Experiments show the competitiveness of the proposed method in terms of restoration quality when dealing with images containing text, faces, or fingerprints.\",\n \"Additionally, experiments show that the proposed method is able to handle text images at high noise levels, outperforming state-of-the-art methods specifically designed for BID of text images.\"\n ],\n [\n \"Introduction\",\n \"Blind image deblurring (BID) is an inverse problem where the observed image is modeled as the convolution of an underlying (sharp) image and an unknown blurring filter, often followed by additive noise.\",\n \"The goal of BID is usually to estimate both the underlying image and the blurring filter.\",\n \"The problem is obviously severely ill-posed.\",\n \"In addition, since the convolution operator itself is typically ill-conditioned, the inverse problem is highly sensitive to the presence of noise.\",\n \"In recent years, researchers have investigated a variety approaches to single image BID, mostly considering generic natural images [1], [2], [3], [4], [5], [6].\",\n \"To deal with the ill-posed nature of the BID problem, most methods use prior information on both the image and the blurring filter.\",\n \"The most common choice for the image prior exploits the statistics of natural images [1], [3], [4], [7], [8], [5], [9] and is usually based on implicit or explicit restoration of salient edges.\",\n \"Although that approach gives good results for natural images, the prior itself is not designed for images that belong to specific classes (e.g., text, face, medical structures, fingerprints) appearing in many important applications, like document analysis, surveillance, and forensics.\",\n \"Methods that use priors that capture the properties of images belonging to specific classes are more likely to provide better results, when dealing with those images, e.g., text [10], [11], [12] or face images [13], [14], [15].\",\n \"Furthermore, images that belong to different specific classes may have different characteristics that are hard to capture with a unique prior.\",\n \"For example, face images do not contain much texture and text images have specific structure due to the contents of interest being mainly in two tones (commonly, black and white).\",\n \"Here, we proposed a method that uses patch-based image priors learned from a set of clean images of the specific class of interest.\",\n \"The method is based on the so-called plug-and-play approach, recently proposed in [16].\",\n \"In contrast with [16], we do not use a fixed denoiser, but a denoiser based on a Gaussian mixture model (GMM) that is learned from patches of clean images belonging to a specific class.\",\n \"A similar idea was recently proposed for non-blind image deblurring and compressive imaging [17].\",\n \"Here, in addition to the GMM-based image prior, we also adopt a weak prior on the blurring filter.\",\n \"Considering the blur, earlier methods typically impose hard constrains for the (arguably) most relevant case of a generic motion blur by encouraging sparsity of the blur filter estimate [4], [18], [1], [8], [3], [19], [7].\",\n \"In this paper, we use a weaker prior on the blur (limited support), thus being able to recover a wide variety of filters then those methods.\"\n ],\n [\n \"Observation model\",\n \"Consider the linear observation model $\\\\textbf {y} = \\\\textbf {H} \\\\textbf {x} + \\\\textbf {n}$ , where $\\\\textbf {y} \\\\in \\\\mathbb {R}^{n}$ , $\\\\textbf {x} \\\\in \\\\mathbb {R}^{m}$ denote the vectorized (lexicographically ordered) observed data and the (unknown) original image, respectively, and $\\\\textbf {n}$ is noise, assumed to be Gaussian, with zero mean and known variance $\\\\sigma ^2$ .\",\n \"For computational convenience, $\\\\textbf {H} \\\\in \\\\mathbb {R}^{n \\\\times m}$ is the matrix that represents the convolution with the blurring filter $\\\\textbf {h}$ with periodic boundary conditions, thus with $n = m$ .\",\n \"As explained above, to deal with the blind image deblurring problem, prior information (a regularizer) is imposed on both the underlying image and the blurring filter.\",\n \"The image $\\\\textbf {x}$ and the blurring operator $\\\\textbf {H}$ (equivalently, the filter $\\\\textbf {h}$ ) are estimated by minimizing the cost function $O_\\\\lambda (\\\\textbf {x},\\\\textbf {h}) = \\\\frac{1}{2} ||\\\\textbf {y} - \\\\textbf {H x}||_2^2 + \\\\lambda \\\\phi (\\\\textbf {x}) + \\\\Psi _\\\\textit {S}(\\\\textbf {h}).$ We assume a weak prior on the blurring filter, $\\\\Psi _{\\\\textit {S}}$ , the indicator function of the set $\\\\textit {S}$ (set of filters with positive entries on a given support).\",\n \"The rationale behind using a weak prior is that it covers a wider variety of blurring filters.\",\n \"$\\\\Psi _{\\\\textit {S}}(\\\\textbf {u}) ={\\\\left\\\\lbrace \\\\begin{array}{ll}0 & \\\\quad \\\\text{if } \\\\textbf {u} \\\\in \\\\textit {S} \\\\\\\\\\\\infty & \\\\quad \\\\text{if } \\\\textbf {u} \\\\notin \\\\textit {S}.\\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ The function $\\\\phi $ represents the prior on the image used to promote characteristics that the underlying sharp image is assumed to have, while parameter $\\\\lambda $ controls the trade-off between data-fidelity term and the regularizer.\",\n \"As shown recently [20], [6], good results can be obtained by alternating estimation of the image and the blur kernel (Algorithm 1).\",\n \"Both steps are performed by using the alternating direction method of multipliers (ADMM) [6].\",\n \"[H] Blind Image Deblurring Algorithm [1] Blurred image $\\\\textbf {y}$ Estimated sharp image $\\\\hat{\\\\textbf {x}}$ and the blur kernel $\\\\hat{\\\\textbf {h}}$ Initialization: Initial estimate $ \\\\hat{\\\\textbf {x}} = \\\\textbf {y}$ , $\\\\hat{\\\\textbf {h}}$ set to the identity filter, $\\\\lambda > 0$ stopping criterion is not satisfied $\\\\hat{\\\\textbf {x}} \\\\leftarrow \\\\underset{\\\\textbf {x}}{\\\\text{argmin}} \\\\hspace{5.69054pt} O_\\\\lambda (\\\\textbf {x}, \\\\hat{\\\\textbf {h}})$ estimating $\\\\textbf {x}$ with $\\\\textbf {h}$ fixed $\\\\hat{\\\\textbf {h}} \\\\leftarrow \\\\underset{\\\\textbf {h}}{\\\\text{argmin}} \\\\hspace{5.69054pt} O_\\\\lambda (\\\\hat{\\\\textbf {x}},\\\\textbf {h})$ estimating $\\\\textbf {h}$ with $\\\\textbf {x}$ fixed In contrast with [6], we do not decrease the regularization parameter $\\\\lambda $ at every iteration.\",\n \"We found that a fixed parameter yields better results arguably due to the more expressive prior herein used, when compared with the total-variation regularizer in [6].\"\n ],\n [\n \"ADMM for image inverse problems\",\n \"As discussed above, we use the ADMM optimization algorithm to perform estimation of both the image and the blurring filter, and therefore, in this section, we will briefly explain the ADMM for image inverse problems.\",\n \"Consider an unconstrained optimization problem in which the objective function is the sum of two functions $\\\\underset{\\\\textbf {z}}{\\\\text{min}} \\\\hspace{5.69054pt} f_1 ( \\\\textbf {z}) + f_2 ( \\\\textbf {z}).$ By using a variable splitting procedure, we introduce a new variable $\\\\textbf {v}$ as the argument of the function $f_2$ , under the constrain that $\\\\textbf {z}=\\\\textbf {v}$ .\",\n \"This leads to rewriting the unconstrained problem from above as a constrained one: $\\\\underset{\\\\textbf {z}, \\\\textbf {v}}{\\\\text{min}} \\\\hspace{5.69054pt} f_1(\\\\textbf {z}) + f_2 (\\\\textbf {v})\\\\hspace{11.38109pt} \\\\text{subject to} \\\\hspace{11.38109pt} \\\\textbf {z} = \\\\textbf {v}.$ The rationale behind variable splitting methods, such as the method of multipliers or augmented Lagrangian method (ALM), is that it may be easier to solve the constrained problem (4) instead of the unconstrained one (3).\",\n \"The main idea behind the ALM is to minimize alternatingly the so-called augmented Lagrangian function $\\\\hat{\\\\textbf {z}}, \\\\hat{\\\\textbf {v}} \\\\leftarrow \\\\underset{\\\\textbf {z}, \\\\textbf {v}}{\\\\text{min}} \\\\hspace{5.69054pt} f_1(\\\\textbf {z}) + f_2 (\\\\textbf {v})+ \\\\textbf {d}^T(\\\\textbf {z}-\\\\textbf {v}) + \\\\frac{\\\\mu }{2}||\\\\textbf {z} - \\\\textbf {v}||_2^2,$ and updating the vector of Lagrange multipliers $\\\\textbf {d}$ (Algorithm 2).\",\n \"In Equation (5), $\\\\mu \\\\ge 0$ is called the penalty parameter.\",\n \"If we recall that, by definition, the proximity operator (PO) of some convex function $g$ , computed at the point $\\\\textbf {u}$ is defined as $\\\\text{prox}_g(\\\\textbf {u}) = \\\\underset{\\\\textbf {x}}{\\\\text{argmin}} \\\\hspace{5.69054pt}\\\\frac{1}{2}||\\\\textbf {x} - \\\\textbf {u}||_2^2 + g(\\\\textbf {x}),$ it is clear that in Algorithm 2, lines 3 and 4 are the PO of $f_1$ and $f_2$ , computed at $\\\\textbf {v}^k + \\\\textbf {d}^k$ and $\\\\textbf {z}^{k+1} - \\\\textbf {d}^k$ , respectively.\",\n \"Formulation (6) can be considered as the solution to a denoising problem, with $\\\\textbf {u}$ as the noisy observation and $g$ the regularizer.\",\n \"ADMM [1] Initialization: Set $k = 0$ , $\\\\mu > 0$ , initialize $\\\\textbf {v}_0$ and $\\\\textbf {d}_0 $ stopping criterion is not satisfied $ \\\\textbf {z}^{k+1} \\\\leftarrow \\\\underset{\\\\textbf {z}}{\\\\text{min}} \\\\hspace{5.69054pt} f_1(\\\\textbf {z}) + \\\\frac{\\\\mu }{2}||\\\\textbf {z} - \\\\textbf {v}^k - \\\\textbf {d}^k||_2^2$ $ \\\\textbf {v}^{k+1} \\\\leftarrow \\\\underset{\\\\textbf {v}}{\\\\text{min}} \\\\hspace{5.69054pt} f_2(\\\\textbf {v}) + \\\\frac{\\\\mu }{2}||\\\\textbf {z}^{k+1} - \\\\textbf {v} - \\\\textbf {d}^k||_2^2$ $\\\\textbf {d}^{k+1} \\\\leftarrow \\\\textbf {d}^k - ( \\\\textbf {z}^{k+1} - \\\\textbf {v}^{k+1} )$ $k \\\\leftarrow k + 1$\"\n ],\n [\n \"GMM-based Denoiser\",\n \"For the image estimate update (line 3 of Algorithm 1), instead of using the PO of a convex regularizer (line 4 of the Algorithm 2), we implemented a state-of-the-art denoiser considering the fact that the PO itself is a denoising function.\",\n \"This approach, also known as plug-and-play, was recently exploited in [16], but instead of using a fixed denoiser, such as BM3D [21] or K-SVD [22], we consider a class-adapted GMM-based denoiser [23].\",\n \"In [23], the authors show that clean image patches are well modeled by a GMM estimated from a collection of clean images using the expectation-maximization (EM) algorithm.\",\n \"Furthermore, for a GMM-based prior for the clean patches, the corresponding minimum mean squared error (MMSE) estimate can be obtained in closed form [24].\",\n \"We use these facts to obtain a GMM-based prior learned from the set of clean images that belong to the specific class.\",\n \"The rationale behind this approach is that with the class-adapted image prior, we may achieve better performance than with a fixed, generic denoiser, when we process images that do belong to the same specific class.\"\n ],\n [\n \"Proposed method\",\n \"The proposed method uses ADMM for solving each of the inner minimization problems in Algorithm 1 (lines 3 and 4), with $O_\\\\lambda (\\\\textbf {x},\\\\textbf {h})$ as defined in (1) with the PO of $\\\\phi $ replaced by the MMSE estimator using a class-adapted GMM prior.\"\n ],\n [\n \"Image Estimate\",\n \"The image estimation problem (line 3 of Algorithm 1) can be formulated as: $\\\\hat{\\\\textbf {x}} = \\\\underset{\\\\textbf {x}}{\\\\text{argmin}} \\\\hspace{5.69054pt} \\\\frac{1}{2} ||\\\\textbf {y} - \\\\textbf {H x}||_2^2 + \\\\lambda \\\\phi (\\\\textbf {x}).$ This problem can be written in the form (3) by setting $f_1(\\\\textbf {x}) = \\\\frac{1}{2} ||\\\\textbf {y} - \\\\textbf {H x}||_2^2$ and $f_2(\\\\textbf {x}) = \\\\lambda \\\\phi (\\\\textbf {x})$ .\",\n \"Applying ADMM to problem (7), yields the so-called SALSA algorithm [25].\",\n \"Line 3 of Algorithm 2 becomes a quadratic optimization problem, which has a linear solution: $\\\\textbf {x}^{k+1} = (\\\\textbf {H}^T\\\\textbf {H}+\\\\mu \\\\textbf {I})^{-1} (\\\\textbf {H}^T \\\\textbf {y} + \\\\mu (\\\\textbf {v}^k + \\\\textbf {d}^k)).$ As shown in some previous work [25], [26], the matrix inversion in (8) can be efficiently computed in the discrete Fourier transform (DFT) domain (using the FFT) in the case of cyclic deblurring, which we consider in this paper.\",\n \"Extension to other boundary conditions can be obtained via the technique proposed in [6].\"\n ],\n [\n \"Blur Estimate\",\n \"The blur estimation problem (line 4 of the Algorithm 1) can be formulated as: $\\\\hat{\\\\textbf {h}} = \\\\underset{\\\\textbf {h}}{\\\\text{argmin}} \\\\hspace{5.69054pt} \\\\frac{1}{2} ||\\\\textbf {y} - \\\\textbf {Xh}||_2^2 + \\\\Psi _\\\\textit {S}(\\\\textbf {h}),$ where $\\\\textbf {h} \\\\in \\\\mathbb {R}^{n}$ is the vector containing the lexicographically ordered blurring filter elements and $\\\\textbf {X} \\\\in \\\\mathbb {R}^{n \\\\times n}$ is the square matrix representing the convolution of the image $\\\\textbf {x}$ and the filter $\\\\textbf {h}$ .\",\n \"Considering formulation (3) we have $f_1(\\\\textbf {h}) = \\\\frac{1}{2} ||\\\\textbf {y} - \\\\textbf {X h}||_2^2$ and $f_2(\\\\textbf {h}) = \\\\Psi _\\\\textit {S}(\\\\textbf {h})$ .\",\n \"The resulting instance of line 3 of Algorithm 2 has the same form as (8) and, as previously explained, the matrix inversion can be efficiently computed in the DFT domain, using the FFT.\",\n \"Since the proximity operator of the indicator of a convex set is the orthogonal projection on that set [27], line 4 of the Algorithm 2 becomes $\\\\text{prox}_{\\\\Psi _{\\\\textit {S}}}(\\\\textbf {u}) = P_{\\\\textit {S}}(\\\\textbf {u}),$ which simply sets to zero all negative elements and any elements outside of the given support.\"\n ],\n [\n \"Experiments\",\n \"In all the experiments, we use the following setting for the two ADMM algorithms described in Section 5: the image estimate is computed with 20 iterations of the algorithm in Subsection 5.1, initialized with the image estimate from the previous iteration, $\\\\textbf {d}_0 = 0$ , and $\\\\mu $ hand-tuned for the best visual results or best ISNR (improvement in SNR [6]) in terms of synthetic data.\",\n \"The blur estimate is computed with two iteration of the algorithm explained in Subsection 5.2, initialized with the blur estimate from the previous iteration, $\\\\textbf {d}_0 = 0$ , and $\\\\mu = 0.01$ .\",\n \"Furthermore, the experiments were performed on three sets of images: (a) a dataset containing 10 text images that is available from the author of [28] (one for testing and nine for training the mixture), (b) a dataset containing 100 face images from the same author as the text dataset, and (c) a dataset containing 128 fingerprints from the publicly available the UPEK fingerprint database.\",\n \"The GMM-based prior is obtained by using patches of size $6 \\\\times 6$ pixels and a 20-component mixture.\"\n ],\n [\n \"Results\",\n \"For the experiments with text images, we created five test images using the same clean image of text and $15 \\\\times 15$ synthetic kernels that represent Gaussian, linear motion, out-of-focus, uniform, and nonlinear motion blur, respectively, and noise level corresponding to BSNR = 30 dB (Table 1).\",\n \"For the face images, we created four $11 \\\\times 11$ synthetic blur kernels that represent Gaussian, linear motion, out-of-focus, and uniform blurs, respectively, and for the fifth experiment, we used blur kernel number 5 from [2], with noise level corresponding to BSNR = 40 dB (Table 2).\",\n \"Experiments on the image containg fingerprints are performed with the $15 \\\\times 15$ linear motion blur kernel and noise level corresponding to BSNR = 40 dB (Fig.\",\n \"2).\",\n \"Results of all experiments are compared with two state-of-the-art BID algorithms constructed for natural images ([6] and [5]), and additionally with the algorithm explained in Subsection 5.1 with the BM3D denoiser plugged into it (PlugBM3D), instead of the GMM-based denoiser (PlugGMM).\",\n \"Note that the generic algorithm [6] is designed for a wide variety of blur filters, while [5], like the majority of blind deblurring algorithms, is designed mostly for motion blur.\",\n \"Figure: Text image blurred with nonlinear motion blur number 2 from and high noise level (BSNR = 20 dB): (a) Original image and ground truth kernel; (b) Blurred image; (c) Results of , ISNR = -2.72; (d) PlugBM3D, ISNR = 9.97; (e) PlugGMM, ISNR = 11.16.Figure: Fingerprint image blurred with 9×99 \\\\times 9 linear motion blur and noise level (BSNR = 40 dB): (a) Original image and ground truth kernel; (b) Blurred image; (c) Results of , ISNR = 0.36; (d) Results of , ISNR = -0.64; (e) PlugBM3D, ISNR = 0.56; (f) PlugGMM, ISNR = 1.19.Table: Results in terms of ISNR of the generic methods and , our method using the BM3D denoiser, and our method with the class-adapted GMM prior, tested for text images (BSNR = 30 dB).Table: Results in terms of ISNR of the generic methods and , our method using the BM3D denoiser, and our method with the class-adapted GMM prior, tested for face images (BSNR = 40 dB).Moreover, we tested our method on text images blurred with the blurring filter number 2 from [2], followed by a higher noise level (BSNR = 20 dB) (Fig.\",\n \"1).\",\n \"Results are compared, as previously explained, with the BM3D denoiser plugged into the ADMM loop and the method from Pan et al.\",\n \"[11], which was designed for BID of text images.\",\n \"As the BM3D denoiser is based on exploiting non-local patch similarities, which is highly present in the images we tested, visual results of using PlugBM3D are very good, but in terms of ISNR, PlugGMM clearly outperforms it.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we have proposed a class-adapted blind image deblurring method, built upon the so-called plug-and-play approach.\",\n \"The method uses Gaussian mixture model (GMM) based denoisers, adapted to specific image classes, plugged into the ADMM optimization algorithm, and a weak prior (positivity and limited support) on the blurring filter.\",\n \"Experiments show that the proposed method yields state-of-the-art results, when applied to images that belong to a specific class (e.g., text, face, and fingerprints), outperforming several generic techniques for blind image deblurring [5], [6].\",\n \"In addition, experiments show that the proposed method can be used for a variety of blurring filters and is able to handle strong noise in the case of images known to contain text, outperforming the state-of-the-art method for BID of text images [11].\",\n \"The proposed method suffers from some potential limitations, such as setting of the regularization parameter and stopping criteria for the inner ADMM algorithms, as well as for the outer iterations, that we aim to improve in future work.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01710"}}},{"rowIdx":897,"cells":{"text":{"kind":"list like","value":[["Measurement of the shape of the $\\Lambda_b^0\\to\\Lambda_c^+ \\mu^-\n \\overline{\\nu}$ differential decay rate"],["Abstract A measurement of the shape of the differential decay rate and the associated Isgur-Wise function for the decay $\\Lambda_b^0\\to\\Lambda_c^+\\mu^-\\overline{\\nu}$ is reported, using data corresponding to $3 fb^{-1}$ collected with the LHCb detector in proton-proton collisions.","The $\\Lambda_c^+\\mu^-\\overline{\\nu}$(+ anything) final states are reconstructed through the detection of a muon and a $\\Lambda_c^+$ baryon decaying into $pK^-\\pi^+$, and the decays $\\Lambda_b^0\\to\\Lambda_c^+\\pi^+\\pi^-\\mu^-\\overline{\\nu}$ are used to determine contributions from $\\Lambda_b^0\\to \\Lambda_c^{\\star+}\\mu ^-\\bar{\\nu}$ decays.","The measured dependence of the differential decay rate upon the squared four-momentum transfer between the heavy baryons, $q^2$, is compared with expectations from heavy-quark effective theory and from unquenched lattice QCD predictions."],["0.5em"]],"string":"[\n [\n \"Measurement of the shape of the $\\\\Lambda_b^0\\\\to\\\\Lambda_c^+ \\\\mu^-\\n \\\\overline{\\\\nu}$ differential decay rate\"\n ],\n [\n \"Abstract A measurement of the shape of the differential decay rate and the associated Isgur-Wise function for the decay $\\\\Lambda_b^0\\\\to\\\\Lambda_c^+\\\\mu^-\\\\overline{\\\\nu}$ is reported, using data corresponding to $3 fb^{-1}$ collected with the LHCb detector in proton-proton collisions.\",\n \"The $\\\\Lambda_c^+\\\\mu^-\\\\overline{\\\\nu}$(+ anything) final states are reconstructed through the detection of a muon and a $\\\\Lambda_c^+$ baryon decaying into $pK^-\\\\pi^+$, and the decays $\\\\Lambda_b^0\\\\to\\\\Lambda_c^+\\\\pi^+\\\\pi^-\\\\mu^-\\\\overline{\\\\nu}$ are used to determine contributions from $\\\\Lambda_b^0\\\\to \\\\Lambda_c^{\\\\star+}\\\\mu ^-\\\\bar{\\\\nu}$ decays.\",\n \"The measured dependence of the differential decay rate upon the squared four-momentum transfer between the heavy baryons, $q^2$, is compared with expectations from heavy-quark effective theory and from unquenched lattice QCD predictions.\"\n ],\n [\n \"0.5em\"\n ]\n]"},"id":{"kind":"string","value":"1709.01920"}}},{"rowIdx":898,"cells":{"text":{"kind":"list like","value":[["Logarithmic Correction to BMSFT Entanglement Entropy"],["Abstract Using Rindler method we derive the logarithmic correction to the entanglement entropy of a two dimensional BMS-invariant field theory (BMSFT).","In particular, we present a general formula for extraction of the logarithmic corrections to both the thermal and the entanglement entropies.","We also present a CFT formula related to the logarithmic correction of the BTZ inner horizon entropy which results in our formula after taking appropriate limit."],["Introduction","One of the lessons of the AdS/CFT correspondence [1] is that the asymptotic symmetry of the asymptotically AdS spacetimes in d+1 dimensions is the same as conformal symmetry in one dimension lower.","One can use this idea to generalize the gauge/gravity duality beyond the AdS/CFT correspondence.","Accordingly , in any non-AdS/non-CFT correspondence, the symmetry of the dual field theory should be the same as the asymptotic symmetry of the gravity solutions.","The asymptotic symmetry of asymptotically flat spacetimes were known long before Maldacena's conjecture.","Such symmetries are known as BMS, which were first found at null infinity of four-dimensional asymptotically flat spacetimes [2], and later were generalized to the three-dimensional case [3].","Recently, Barnich and his collaborates [4] have shown that imposing just locally well-definiteness condition is enough to enhance both the translation and the rotation symmetry of the Poincare group at null infinity to an infinite-dimensional symmetry group.","In this work BMS group refers to this infinite-dimensional group which consists of super-rotation and super-translation generators.","In one dimension lower than gravity theory, the BMS algebra is given by an ultra-relativistic contraction of the conformal algebra.","Thus BMS symmetry can be the symmetry of a lower dimensional field theory.","It is proposed that the holographic dual of d+1-dimensional asymptotically flat spacetimes are ultra-relativistic d-dimensional BMS-invariant field theories (BMSFTs) [5],[6].","In flat-space holography, BMSFT entanglement entropy and its holographic dual were first studied in [7] and the entanglement entropy of a two-dimensional BMSFT was computed by a method similar to the CFT [8].","A holographic interpretation of the BMSFT entanglement entropy in [7] (similar to the Ryu-Takayanagi proposal in the context of AdS/CFT correspondence [9]) is given using Chern-Simons formulation of three-dimensional flat-space gravity [10].","Other aspects of BMSFT entanglement entropy has been studied in [11],[12].","A remarkable progress in this subject has been achieved recently in [13] where Rindler method [14] is used to derive not only the BMSFT entanglement entropy formula but also the holographic description in terms of some curves length in the bulk theory.","The central idea of the Rindler method is to find local unitary transformations which map entangled states to the thermal states in the field theory.","Then one can use the thermal entropy formula and find the entanglement entropy.","Calculation of thermal entropies in BMSFTs is performed using Cardy-like formula first introduced in [15].","Using Rindler method, the entanglement entropy is given by Cardy-like formula.","Similar to the Cardy-formula in CFT , the saddle point approximation is used to derive this formula.","Thus, it is possible to improve approximation and find possible corrections of this formula.","In [16], employing a method first used for the Cardy formula in [17]We note however that the Cardy (high temperature) limit is not always reliable for extracting the black hole entropy [18].","On the other hand, in two dimensional CFTs the logarithmic corrections are only universal in the Cardy regime.","From the dual three dimensional gravity perspective, the universality of the correction is usually because heat kernels do not contain logarithmic terms [19].","Thus the logarithmic correction to the BMSFT thermal and entanglement entropies are universal in the limit that BMSFT Cardy-like formula is reliable., logarithmic correction to the Cardy-like formula has been derived.","In this paper, we use the logarithmic correction to Cardy-like formula along with Rindler method to find the logarithmic correction of BMSFT entanglement entropy.","Similar to the leading term, this correction depends on central charges of BMS algebra, the interval of the sub-system and the cut-off.","However, the interesting point is that we can rewrite the corrections in a universal form: $S=S_0-3\\log \\left(C_M^{\\frac{1}{3}}\\dfrac{\\partial S_0}{\\partial C_L}\\right)$ where $S_0$ is the leading term and $C_L$ and $C_M$ are the central charges of BMS algebra.","$S$ can be both of the thermal entropy (which is given by the Cardy-like formula) and the entanglement entropy.","This formula works for all BMSFTs on plane or cylinder at zero or finite temperature.","The entanglement entropy of the boundary theory can be used to reconstruct the bulk dynamics [20], [21].","In this view, the logarithmic correction of the entanglement entropy might be helpful to find quantum correction to the Einstein field equations.","This paper is the first step on this road.","The idea is to use the first law of the entanglement entropy as the analogue of the first law of (black hole) thermodynamics [20], [21], [22].","The better understanding of BMSFT modular hamiltonian might allow us to achieve not only the classical bulk dynamics but also the quantum correction to the Einstein equation without the cosmological constant.","One approach to study Flat/BMSFT is to take limit from the AdS/CFT calculations.","According to the proposal of [6], the flat space limit in the gravity side corresponds to taking an ultra-relativistic limit from the CFT calculations.","Hence one can find all BMSFT formulas by taking limit from a CFT counterpart.","It was shown in [23] and [24] that the Cardy-like formula of BMSFT is given by taking limit from a formula in the CFT which is related to the inner horizon entropy of the gravity solution.","If we assume the same relation for the logarithmic correction, the logarithmic term in (REF ) is related to a logarithmic correction in the entropy of the inner horizon.","We propose a suitable logarithmic correction in the inner horizon formula which corresponds to the logarithmic term of (REF ) after taking limit.","Using logarithmically corrected inner and outer horizon entropy formulas we can calculate their multiplication.","The observation is that the previously known result, the multiplication being mass independent for the Einstein gravity [25] (see also [26]) , is violated in the presence of the logarithmic terms.","The structure of this paper is as follows: In section two we review the results of [13] which uses the Rindler method to derive the BMSFT entanglement entropy and its holographic description.","In section three we review the derivation of BMSFT Cardy-like formula and its logarithmic correction.","In section four we put together the results of section two and three to derive the logarithmic correction of BMSFT entanglement entropy.","Section five is devoted to the derivation of the BMSFT entanglememnt entropy formula by taking flat-space limit."],["Entanglement Entropy of BMSFT using Rindler method","One of the advantages of the Rindler method is the convenience it provides for the calculation of entanglement entropy in the context of gauge/gravity duality.","In this view, the thermal entropy of the boundary theory is mapped to the horizon entropy of the black hole (object) in the gravity side.","Thus, one can use it to prove Ryu-Takayanagi (RT) formula for the holographic entanglement of CFTs.","Moreover, since many thermal properties of the gravitational systems are known for the non-AdS cases, we expect that the Rindler method gives a lot of insights to find the analogue of RT formula for the dualities beyond the AdS/CFT correspondence.","In Rindler method, the asymptotic symmetries play an essential role.","We are interested in BMSFTs which are proposed to be the holographic dual of asymptotically flat spacetimes.","In our case, a Rindler transformation is of the form of BMS transformation and final coordinates are invariant under specific thermal identifications.","These transformations act like unitary operators $U_{\\mathcal {R}}$ on the fields and map the reduced density matrix in the subregion $\\mathcal {A}$ (which we want to calculate the entanglement entropy for) to a thermal density matrix in the interval $\\mathcal {B}$ , $\\rho _{\\mathcal {A}} = U_{\\mathcal {R}}\\, \\rho _{\\mathcal {B}}\\, U_{\\mathcal {R}}^{-1}.$ Since the unitary transformations do not change the entropy, the thermal entropy of the subregion $\\mathcal {B}$ is the same as entanglement entropy of the subregion $\\mathcal {A}$ .","A two-dimensional BMSFT has the following symmetry [27]: $\\nonumber \\tilde{u} &= \\partial _{\\phi }{f(\\phi )} u +g(\\phi ),\\\\\\tilde{\\phi } &= f(\\phi ),$ where $f(\\phi )$ and $g(\\phi )$ are arbitrary functions of the original coordinate $(u,\\phi )$ .","The BMS algebra is given by using infinitesimal BMS transformation as $\\nonumber [L_n,L_m]&=(n-m)L_{n+m}+{C_L\\over 12}n(n^2-1)\\delta _{n+m,0},\\\\\\nonumber [L_n,M_m]&=(n-m)M_{n+m}+{C_M\\over 12}n(n^2-1)\\delta _{n+m,0},\\\\{[}M_n,M_m]&=0.$ where for a BMSFT on a plane with coordinates $(u,\\phi )$ the generators $L_n$ and $M_n$ are given by $\\nonumber L_{n} &= -u (n+1) \\phi ^n \\partial _{u} - \\phi ^{n+1} \\partial _{\\phi }\\\\M_{n} &= \\phi ^{n+1}\\partial _{u}.$ The global part of this algebra is identified with $n=0,\\pm 1$ .","A Rindler transformation is of the form $\\tilde{x}=T(x)$ , but it should be invariant under some imaginary identification (thermal identification) of the new coordinate $\\tilde{x}^i\\sim \\tilde{x}^i+i\\tilde{\\beta }^i$ .","Moreover, vectors $\\partial _{\\tilde{x}^i}$ annihilate the vacuum and hence should be written as the linear combination of the global part of the BMS algebra: $\\partial _{\\tilde{x}^i} = \\sum _{n=-1}^{1}{( b_{n} L_{n} +d_{n} M_{n} )}.$ Using (REF ) and (REF ) we conclude that $\\partial _{\\phi }{f(\\phi )} = \\frac{1}{Y}, \\qquad \\partial _{\\phi }{g(\\phi )} = -\\frac{T}{Y^2},$ where $Y &= - b_{-1} -b_{0} \\phi -b_{1} \\phi ^2,\\\\T &= d_{-1} + d_{0} \\phi + d_{1} \\phi ^2.$ Solutions to (REF ) determine vector $ \\partial _{\\tilde{x}^i}$ .","It is assumed that entangled region is given by $\\mathcal {A} = \\lbrace (\\frac{-l_{\\phi }}{2} , \\frac{-l_{u}}{2}) \\cup (\\frac{l_{\\phi }}{2} , \\frac{l_{u}}{2})\\rbrace $ .","We can use the following constraints to find vector $ \\partial _{\\tilde{x}^i}$ and the geometric (modular) flow $ k_t = -\\tilde{\\beta _{\\phi }} \\partial _{\\tilde{\\phi }} + \\tilde{\\beta _{u}} \\partial _{\\tilde{u}} $ : The finite interval on $\\mathcal {A}$ should be mapped to the infinite interval on $\\mathcal {B}$ , $( \\frac{-l_{\\phi }}{2} , \\frac{-l_u}{2} ) &\\rightarrow ( -\\infty , -\\infty ),\\\\( \\frac{l_{\\phi }}{2} , \\frac{l_u}{2} ) &\\rightarrow ( \\infty , \\infty ).$ The origin of the entangled interval is mapped to the origin of the thermal interval $(0, 0) \\rightarrow (0,0)$ The thermal interval (tilde coordinate) obeys a thermal identification of the following form $( \\tilde{\\phi } , \\tilde{u} ) \\sim ( \\tilde{\\phi } + i \\tilde{\\beta }_{\\phi } ,\\tilde{u} - i \\tilde{\\beta }_{u} ) .$ The modular flow $k_t$ vanishes on the boundary of the entangled region $k_t(\\partial {\\mathcal {A}} ) = 0 \\Rightarrow {\\left\\lbrace \\begin{array}{ll}k_t( \\frac{-l_\\phi }{2} ,\\frac{-l_u}{2} ) = 0,\\\\k_t( \\frac{l_\\phi }{2} ,\\frac{l_u}{2} ) =0\\end{array}\\right.","}$ Using these conditions, [13] completely determines the Rindler transformation and modular flow of a BMSFT on the plane as below: $\\tilde{\\phi } &= \\frac{ \\tilde{ \\beta }_{\\phi } }{ \\pi } \\tanh ^{-1}{ \\frac{2 \\phi }{ l_{\\phi }} } ,\\\\\\tilde{u} + \\frac{ \\tilde{\\beta }_{u} }{ \\tilde{\\beta }_{\\phi } } \\tilde{\\phi } &= \\frac{2 \\tilde{\\beta }_{\\phi } ( u l_{\\phi } - l_{u} \\phi )}{\\pi (l_{\\phi }^2 - 4 \\phi ^2)},$ $k_t = -\\tilde{\\beta }_{\\phi } \\partial _{\\phi } + \\tilde{\\beta }_{u} \\partial _{u} =\\frac{- \\pi }{2 l_{\\phi }} \\left( \\left(l_{\\phi }^2 - 4 \\phi ^2\\right)\\partial _{\\phi } + \\left(l_u l_{\\phi } +4 \\frac{lu}{l_{\\phi }} \\phi ^2- 8 u \\phi \\right)\\partial _{u} \\right)$ For a BMSFT with identification of coordinates as $(\\tilde{u} , \\tilde{\\phi } ) \\sim ( \\tilde{u} + i \\bar{a} , \\tilde{\\phi } - i a) \\sim ( \\tilde{u} + 2 \\pi \\bar{b} , \\tilde{\\phi } - 2 \\pi b),$ The degeneracy of states is given by a Cardy-like formula [15], [13], [28] (see next section).","$S_{\\bar{b}|b}(\\bar{a}|a) = \\frac{- \\pi ^2}{3} \\left(C_{L} \\frac{b}{a} +C_{M} \\frac{(\\bar{a} b - a \\bar{b} )}{a^2}\\right).$ Using (REF ), the entanglement entropy of a BMSFT on the plane for the interval $\\mathcal {A} = \\lbrace (\\frac{-l_{\\phi }}{2} + \\epsilon _{\\phi } , \\frac{-l_{u}}{2} + \\epsilon _{u} ) \\cup (\\frac{l_{\\phi }}{2} -\\epsilon _{\\phi } , \\frac{-l_{u}}{2} - \\epsilon _{\\phi } )\\rbrace $ becomes [13] $S_{EE} = \\frac{C_{L}}{6} \\log { \\frac{l_{\\phi }}{\\epsilon _{\\phi }} } + \\frac{C_{M} }{6} \\left( \\frac{l_{u}}{l_{\\phi } }- \\frac{\\epsilon _{u}}{\\epsilon _{\\phi }}\\right),$ where $\\epsilon _u$ and $\\epsilon _\\phi $ are ultraviolet cut-offs in $u$ and $\\phi $ coordinates.","Similarly the entanglement entropy for finite temperature BMSFT on the cylinder has been calculated in [13]."],["Logarithmic Correction to Cardy-like Formula of BMSFT","The entropy of CFT thermal states is calculated using Cardy formula.","Using the saddle point approximation, it is possible to find a similar formula for the degeneracy of thermal states in a BMSFT [15], [13], [28].","Due to the Rindler method, any correction to Cardy-like formula has influence on the entanglement entropy formula.","In this section, we review the logarithmic correction to Cardy-like formula [16] and then use it to find the logarithmic correction to the entanglement entropy.","Method of [16] is based on [17] that introduces the first order logarithmic correction to the Cardy formula (For a calculation of all order corrections see [29]) We start from the modular invariant partition function of BMSFT on a torus defined by $Z_{0}( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ) = \\operatorname{Tr}e^{-\\hat{\\beta }_{u} (M_{0} - \\frac{C_{M}}{24}) + \\hat{\\beta }_{\\phi } (L_{0} -\\frac{C_{L}}{24}) }= e^{\\hat{\\beta }_{u} \\frac{C_{M}}{24} - \\hat{\\beta }_{\\phi } \\frac{C_{L}}{24} } Z( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ),$ where $ Z( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } )$ is $Z( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ) =\\operatorname{Tr}e^{-\\hat{\\beta }_{u} M_{0} + \\hat{\\beta }_{\\phi } L_{0}}= \\sum _{h_M,h_L} e^{( -\\hat{\\beta }_{u} h_{M} + \\hat{\\beta }_{\\phi } h_{L})} d(h_{M},h_{L}),$ and identification of torus are $(\\hat{u},\\hat{\\phi })\\sim (\\hat{u}+i\\hat{\\beta }_u,\\hat{\\phi }-i\\hat{\\beta }_\\phi )\\sim (\\hat{u},\\hat{\\phi }-2\\pi ).$ Here, $h_L$ and $h_M$ are respectively the eigenvalues of $L_0$ and $M_0$ .","It is shown that the BMS modular invariant partition function satisfies [13] $Z_{0}( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ) =Z_{0}\\left(-4 \\pi ^2 \\frac{\\hat{\\beta }_{u}}{\\hat{\\beta }_{\\phi }^2} | \\frac{4 \\pi ^2}{\\hat{\\beta }_{\\phi }}\\right).$ Plugging equation (REF ) into (REF ) gives, $Z( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ) = e^{ \\hat{\\beta }_{u} \\frac{C_M}{24} - \\hat{\\beta }_{\\phi } \\frac{C_L}{24} - \\frac{ \\pi ^2 \\hat{\\beta }_{u} C_{M}}{6 \\hat{\\beta }_{\\phi }^2} - \\frac{ \\pi ^2 C_{L}}{6 \\hat{\\beta }_{\\phi }} } Z\\left(-4 \\pi ^2 \\frac{\\hat{\\beta }_{u}}{\\hat{\\beta }_{\\phi }^2} | \\frac{4 \\pi ^2}{\\hat{\\beta }_{\\phi }}\\right).$ By using the inverse Laplace transformation in last term of (REF ) we find $d(h_{L} , h_{M} ) = \\int d\\hat{\\beta }_{u} d\\hat{\\beta }_{\\phi } e^{ \\hat{\\beta }_{u} \\frac{C_M}{24} - \\hat{\\beta }_{\\phi } \\frac{C_L}{24} - \\frac{ \\pi ^2 \\hat{\\beta }_{u} C_{M}}{6 \\hat{\\beta }_{\\phi }^2} - \\frac{ \\pi ^2 C_{L}}{6 \\hat{\\beta }_{\\phi }} + \\hat{\\beta }_{u} h_M - \\hat{\\beta }_{\\phi } h_L } Z\\left(-4 \\pi ^2 \\frac{\\hat{\\beta }_{u}}{\\hat{\\beta }_{\\phi }^2} | \\frac{4 \\pi ^2}{\\hat{\\beta }_{\\phi }} \\right).$ In order to simplify (REF ), we use two approximations.","First, we consider large charges which yields $d(h_{L} , h_{M} ) = \\int d\\hat{\\beta }_{u} d\\hat{\\beta }_{\\phi } e^{ - \\frac{ \\pi ^2 \\hat{\\beta }_{u} C_{M}}{6 \\hat{\\beta }_{\\phi }^2} - \\frac{ \\pi ^2 C_{L}}{6 \\hat{\\beta }_{\\phi }} + \\hat{\\beta }_{u} h_M - \\hat{\\beta }_{\\phi } h_L } Z\\left(-4 \\pi ^2 \\frac{\\hat{\\beta }_{u}}{\\hat{\\beta }_{\\phi }^2} | \\frac{4 \\pi ^2}{\\hat{\\beta }_{\\phi }}\\right).$ Then, we approximate (REF ) around the saddle point given by $(\\hat{\\beta }_{\\phi }^{s})^2 = \\frac{\\pi ^2 C_{M}}{6 h_{M}},\\qquad \\hat{\\beta }_{u}^{s} ={\\hat{\\beta }_{\\phi }^{s}\\over 2h_M\\,C_M} (C_{M} h_{L} - C_{L} h_{M} ) .$ Finally, the entropy or Cardy like formula for BMSFT reads as It is assumed that the partition function is slowly varying at the extremum.","$S^{0} = \\log {d(h_{L} , h_{M} )} = -{\\pi ^2\\over 3 (\\hat{\\beta }_{\\phi }^{s})^2 }\\left(C_M \\hat{\\beta }_{u}^{s}+C_L \\hat{\\beta }_{\\phi }^{s}\\right).$ To find logarithmic correction we expand the integral (REF ) around the saddle point (REF ) up to quadratic term: $d(h_{L} , h_{M} ) = e^{S^{0}} \\int d\\hat{\\beta }_{u} d\\hat{\\beta }_{\\phi } e^{\\frac{1}{2} (X^2 -Y^2)},$ where $X &= {\\pi C_M\\over 3 \\hat{\\beta }^{s}_{\\phi } A}(\\hat{\\beta }_{u} - \\hat{\\beta }^{s}_{u}) , \\qquad Y = {\\pi A\\over (\\hat{\\beta }^{s}_{\\phi })^2} \\left[ -(\\hat{\\beta }_{\\phi } - \\hat{\\beta }^{s}_{\\phi })+ {C_M \\hat{\\beta }^{s}_{\\phi }\\over 3 A^2} (\\hat{\\beta }_{u}-\\hat{\\beta }^{s}_{u})\\right],\\\\A &= \\sqrt{C_M \\hat{\\beta }^{s}_{u}+\\frac{1}{3} C_L\\hat{\\beta }^{s}_{\\phi }}.$ Using (REF ), Eq.","(REF ) takes the form $d(h_{L} , h_{M} ) = e^{S^{0}}\\left(-{\\pi ^2 C_M\\over 3(\\hat{\\beta }^s_\\phi )^3}\\right)^{-1} \\int dX dY e^{\\frac{X^2 -Y^2}{2} }.$ In the above equation the result of integration is just a number.","Thus we find correction to the entropy up to a constant as [16] $S = \\log { d(h_{L} , h_{M} ) } = -{\\pi ^2\\over 3 (\\hat{\\beta }_{\\phi }^{s})^2 }\\left(C_M \\hat{\\beta }_{u}^{s}+C_L \\hat{\\beta }_{\\phi }^{s}\\right)-3\\log \\left(-{C_M^\\frac{1}{3}\\over \\hat{\\beta }_{\\phi }^{s} }\\right)+\\text{constant}.$ The interesting point is that the logarithmically corrected term can be rewritten (up to a constant) as the derivative of the leading term with respect to $C_L$ : $S=S_0-3\\log \\left(C_M^{\\frac{1}{3}}\\dfrac{\\partial S_0}{\\partial C_L}\\right).$"],["Logarithmic Correction to Entanglement Entropy","In this section, we put together the results from sections two and three to find the logarithmic correction of BMSFT entanglement entropy.","As mentioned before, the idea is to map entangled states to thermal states and then calculate entropy.","The entropy is computed using BMSFT Cardy-like formula (REF ) which also has a logarithmic correction.","In the derivation of (REF ) the identification of coordinates (REF ) plays an essential role.","It is possible to use (REF ) for finding the degeneracy of thermal states with more generic identification of coordinates as $(\\tilde{u},\\tilde{\\phi })\\sim (\\tilde{u}+i\\bar{a},\\tilde{\\phi }-ia)\\sim (\\tilde{u}+2\\pi \\bar{b},\\tilde{\\phi }-2\\pi b)$ The coordinate change between $(\\tilde{u},\\tilde{\\phi })$ and $(\\hat{u},\\hat{\\phi })$ is a BMS transformation, $\\hat{\\phi }={\\tilde{\\phi }\\over b},\\qquad \\hat{u}={\\tilde{u} \\over b}+{\\bar{b}\\over b^2}\\tilde{\\phi },$ where $\\hat{\\beta }_\\phi ={a\\over b},\\qquad \\hat{\\beta }_u={\\bar{a}\\,b-a\\,\\bar{b}\\over b^2}.$ Thus Cardy-like formula (REF ) can be written as $S=-{\\pi ^2\\over 3}\\left(C_L{b\\over a}+C_M{\\bar{a}\\,b-a\\,\\bar{b}\\over a^2}\\right)-3\\log \\left(-C_M^\\frac{1}{3}{b\\over a}\\right)$ In order to find the logarithmic correction of BMSFT entanglement entropy, it is enough to map the entanglement entropy to a thermal entropy and then use (REF ).","As it was reviewed in section , the Rindler transformation which governs this map is determined in such a way that finally induces the thermal identification (REF ).","Comparing (REF ) to (REF ) shows that $a=\\tilde{\\beta }_\\phi ,\\qquad \\bar{a}=\\tilde{\\beta }_u.$ The values of $\\tilde{\\beta }_\\phi $ , $\\tilde{\\beta }_u$ and $b$ , $\\bar{b}$ depend on the details of the Rindler transformation.","These are given in terms of the cut-offs and the interval for which entanglement entropy is calculated.","Starting from a regulated interval in the BMSFT given by $(-{l_u\\over 2}+\\epsilon _u,-{l_\\phi \\over 2}+\\epsilon _\\phi )\\rightarrow ({l_u\\over 2}-\\epsilon _u,{l_\\phi \\over 2}-\\epsilon _\\phi ),$ the Rindler transformation yields the following results [13]: For the zero temperature BMSFT on the plane we have $a&=\\tilde{\\beta }_\\phi =-{2\\pi ^2\\over \\log {l_\\phi \\over \\epsilon _\\phi }},\\qquad \\bar{a}=\\tilde{\\beta }_u=-{\\tilde{\\beta }_\\phi ^2\\over 2\\pi ^2}\\left({l_u\\over l_\\phi }-{\\epsilon _u\\over \\epsilon _\\phi }\\right),\\\\b&=-{\\tilde{\\beta }_\\phi \\over 2\\pi ^2}\\log {l_\\phi \\over \\epsilon _\\phi },\\qquad \\bar{b}={1\\over 2\\pi ^2}\\left({\\tilde{\\beta }_\\phi l_u\\over l_\\phi }-{\\tilde{\\beta }_\\phi \\epsilon _u\\over \\epsilon _\\phi }-\\tilde{\\beta }_u\\log {l_\\phi \\over \\epsilon _\\phi }\\right),$ Then the logarithmically corrected Cardy-like formula (REF ) becomes $S_{EE}={C_L\\over 6}\\log {l_\\phi \\over \\epsilon _\\phi }+{C_M\\over 6}\\left({l_u\\over l_\\phi }-{\\epsilon _u\\over \\epsilon _\\phi }\\right)-3\\log \\left(C_M^\\frac{1}{3}\\log {l_\\phi \\over \\epsilon _\\phi }\\right)+\\text{constant}.$ The third term in the above formula is the calculated correction.","For the finite temperature BMSFT with identification $(u,\\phi )\\sim (u+i\\beta _u,\\phi -i\\beta _\\phi ),$ we can use the results of [13] and Eq.","(REF ) to write $\\nonumber S_{EE}={C_L\\over 6}\\log \\left({\\beta _\\phi \\over \\pi \\epsilon _\\phi }\\sinh {\\pi l_\\phi \\over \\beta _\\phi }\\right)&+{C_M\\over 6}{1\\over \\beta _\\phi }\\left[\\pi \\left(l_u+{\\beta _u\\over \\beta _\\phi }l_\\phi \\right)\\coth {\\pi l_\\phi \\over \\beta _\\phi }-\\beta _u\\right]-{C_M\\epsilon _u\\over 6\\,\\epsilon _\\phi }\\\\ &-3\\log \\left(C_M^{1\\over 3}\\log \\left({\\beta _\\phi \\over \\pi \\epsilon _\\phi }\\sinh {\\pi l_\\phi \\over \\beta _\\phi }\\right)\\right)+\\text{constant}.$ The forth term in the above equation is the obtained correction for finite temperature.","For the zero temperature BMSFT on the cylinder, the entanglement entropy with logarithmic correction reads as $S_{EE}={C_L\\over 6}\\log \\left({2\\over \\epsilon _\\phi }\\sin {l_\\phi \\over 2}\\right)+{C_M\\over 12}\\left(l_u\\cot {l_\\phi \\over 2}-{2\\epsilon _u\\over \\epsilon _\\phi }\\right)-3\\log \\left(C_M^{1\\over 3}\\log \\left({2\\over \\epsilon _\\phi }\\sin {l_\\phi \\over 2}\\right)\\right)+\\text{costant}.$ It is clear from all the above cases that up to a constant, the entanglement entropy formula including the logarithmic correction is given by $S_{EE}=S_0-3\\log \\left(C_M^{\\frac{1}{3}}\\dfrac{\\partial S_0}{\\partial C_L}\\right)$ This formula is also valid for the thermal entropy when $S_0$ is the Cardy-like formula.","Thus, we propose a universal form for the logarithmic correction of entanglement entropy and thermal entropy.","We expect the same common form for the CFT case.","In the next section, we propose a similar form of logarithmic correction to CFT thermal and entanglement entropy and try to find (REF ) by taking limit from CFT formula."],["Logarithmic correction of BMSFT entropy by taking limit from CFT counterpart","For a two dimensional CFT with central charges $c$ and $\\bar{c}$ and right and left temperatures $\\beta $ and $\\bar{\\beta }$ , the Cardy formula is $S_0={\\pi ^2\\over 3}\\left({c\\over \\beta }+{\\bar{c}\\over \\bar{\\beta }}\\right).$ Using AdS/CFT correspondence, this formula results in the same entropy as the outer horizon entropy of asymptotically AdS black holes in the gravity side.","The logarithmic correction to (REF ) was evaluated in [17]: $S_{log}=-{3\\over 2}\\log \\left({c^{1/3}\\over \\beta }\\right)-{3\\over 2}\\log \\left({\\bar{c}^{1/3}\\over \\bar{\\beta }}\\right)+\\text{constant}.$ Employing (REF ) and (REF ) we can write $S=S_0+S_{log}=S_0-{3\\over 2}\\log \\left(c^{1/3}\\dfrac{\\partial S_0}{\\partial c}\\right)-{3\\over 2}\\log \\left(\\bar{c}^{1/3}\\dfrac{\\partial S_0}{\\partial \\bar{c}}\\right)+\\text{constant}.$ Using Rindler method we can expect the same formula for the logarithmically corrected CFT entanglement entropy.","Thus we propose (REF ) as a universal formula that can be used for both the thermal entropy and the entanglement entropy.","It is known that taking flat space limit from the asymptotically AdS spacetimes (written in the appropriate coordinate) yields asymptotically flat spacetimes.","It is proposed in [6] that the flat space limit in the bulk corresponds to taking the ultra-relativistic limit in the boundary CFT.","In other words, BMSFT is given by taking ultra-relativistic limit from the CFT.","Starting from conformal algebra in two dimensions, one can introduce an ultra-relativistic contraction and generate BMS algebra [6].","The relation between central charges of conformal algebra and BMS algebra is given by $C_L=\\lim _{\\epsilon \\rightarrow 0}\\left(c-\\bar{c}\\right),\\qquad C_M=\\lim _{\\epsilon \\rightarrow 0}\\epsilon \\left(c+\\bar{c}\\right)$ where $\\epsilon $ is a dimensionless parameter which corresponds to $G/\\ell $ on the gravity side.","$G$ is the Newton constant and $\\ell $ is the AdS-radius.","Thus $\\epsilon \\rightarrow 0$ in the boundary corresponds to $\\ell \\rightarrow \\infty $ .","It is plausible to find all BMSFT quantities by taking limit from the CFT counterparts.","Thus one can look for the possible relation between (REF ) and (REF ).","However, assuming $S_0$ in (REF ) as the Cardy formula (REF ) and using (REF ) does not result in the Cardy-like formula (REF ).","It was shown in [23] and [24] that the appropriate formula which its limit yields the Cardy-like formula is $S_{inner}^0={\\pi ^2\\over 3}\\left({c\\over \\beta }-{\\bar{c}\\over \\bar{\\beta }}\\right).$ For taking limit we should scale the temperatures as below: $\\beta _u=\\lim _{\\epsilon \\rightarrow 0}\\left({\\beta -\\bar{\\beta }\\over 2 \\epsilon }\\right), \\qquad \\beta _\\phi =\\lim _{\\epsilon \\rightarrow 0}-\\left({\\beta +\\bar{\\beta }\\over 2 }\\right).$ Using AdS/CFT correspondence, (REF ) corresponds to the inner horizon entropy of asymptotically AdS black holes.","Thus, we expect that taking limit from (REF ) which is written for the outer horizon does not yield (REF ).","In the rest of this section, we introduce appropriate formula which taking limit from it result in (REF ).","Let us start from $S^0$ in (REF ).","When it is the thermal entropy given by the Cardy-like formula, the corresponding formula in the CFT will be (REF ).","The question is then what is the corresponding CFT formula when $S^0$ in (REF ) is not the leading order of entanglement entropy?","It is known that taking ultra-relativistic limit from the entanglement entropy of CFT is not well-defined.","For example, for a zero temperature CFT on a plane, the entanglement entropy of an interval $-{R\\over 2}(\\cosh \\kappa ,\\sinh \\kappa )\\rightarrow {R\\over 2}(\\cosh \\kappa ,\\sinh \\kappa ),$ is given by [30] $S_{EE}={c+\\bar{c}\\over 2}\\log {R\\over \\epsilon _R}-{c-\\bar{c}\\over 6}\\kappa $ where $\\epsilon _R$ is a cut-off.","Using (REF ) together with ultra-relativistic contraction $t\\rightarrow \\epsilon t$ shows that (REF ) is divergent in the $\\epsilon \\rightarrow 0$ limit.","It is proposed in [11] that BMSFT entanglement entropy can be found by taking the limit from the CFT counterpart.","In the prescription of [11] it is argued that the symmetries are enough to fix the form of the entanglement entropy.","Then they use a new contraction of conformal algebra which results in BMS algebra.","But the relation between central charges of conformal algebra and BMS algebra differs from (REF ).","Since the final algebra is the same as the ultra-relativistic contraction of [6], the author of [11] argued that their results are the entanglement entropy of BMSFT.","If we continue the logic which relates the Cardy-like formula to the limit of inner horizon entropy then we conclude that the entanglement entropy of BMSFT should be related to a formula in the CFT which is transformed to the inner horizon formula by using Rindler transformation [13].","It is not difficult to check that the formula $S_{inner}={c-\\bar{c}\\over 2}\\log {R\\over \\epsilon _R}-{c+\\bar{c}\\over 6}\\kappa $ results in the leading term of entanglement entropy of zero temperature BMSFT on the plane after taking the ultra-relativistic limit.","This formula is nothing but the inner horizon Cardy formula (REF ) if we use Rindler transformation [13].","Therefore we conclude that $S^0$ in (REF ) is given by taking limit from a formula which is related to the inner horizon of dual spacetime.","Similarly, the logarithmic correction in (REF ) is also given by taking limit from a formula which is the logarithmic correction to the inner horizon entropy.","It is not difficult to check that taking $\\epsilon \\rightarrow 0$ limit from $S_{log,inner}=-{3\\over 2}\\log \\left| c^{1/3}\\dfrac{\\partial S_0}{\\partial c}\\right|-{3\\over 2}\\log \\left| \\bar{c}^{1/3}\\dfrac{\\partial S_0}{\\partial \\bar{c}}\\right|-\\log \\epsilon ,$ results in the logarithmic term of (REF ) up to a constant.","The interpretation of (REF ) in the gravity is of importance.","As mentioned before $\\epsilon $ in the field theory corresponds to $G/\\ell $ on the bulk side.","For the BTZ black holes, we have $\\beta ={2\\pi \\ell \\over (r_++r_-)},\\qquad \\bar{\\beta }={2\\pi \\ell \\over (r_+-r_-)}.$ where $r_\\pm $ are the radii of inner and outer horizons.","Thus, using (REF ), (REF ) and (REF ) , we can find the logarithmic correction to the inner horizon entropy of BTZ black hole as $S^{BTZ}_{inner}={\\pi r_-\\over 2 G}-{3\\over 2}\\log {r_+^2-r_-^2\\over \\ell ^2}+\\text{constant},$ where we have substituted central charges as $c=\\bar{c}={3\\ell \\over 2 G}$ .","On the other hand, using (REF ), (REF ) and (REF ) we find that $S^{BTZ}_{outer}={\\pi r_+\\over 2 G}-{3\\over 2}\\log {r_+^2-r_-^2\\over \\ell ^2}-\\log {\\ell \\over G}+\\text{constant}.$ It is clear from (REF ) and (REF ) that due to the logarithmic corrections, multiplication of inner and outer horizon entropies depends not only on the angular momentum but also on the mass of BTZ.","This corrects the result of [25] which in the leading term the multiplication of inner and outer entropies is mass independent."],["Conclusion","In this paper we introduced a generic formula for the logarithmic corrections of the BMSFT thermal and the entanglement entropies.","Our derivation is based on the Rindler method which makes connections between the thermal and the entanglement entropies.","The most important application of the present work will reveal itself in the reconstruction of the bulk dynamics beyond the classical regime.","This reconstruction has been done recently in the bulk AdS case through the perturbation of the entanglement entropy [20], [21].","After generalizing the method of [20] for the flat-space holography, we will be able to derive the quantum corrections to the Einstein gravity without the cosmological constant field equations using the results of the current paper.","Most of the works in the flat-space holography can be performed by taking the flat space-limit from the AdS/CFT calculations.","One of the possible roads for finding the corrections of the BMSFT entanglement entropy formula is taking limit from the calculation of [31] which studies the one-loop bulk corrections to the Ryu-Takayanagi formula."],["Acknowledgements","The authors would like to thank S. M. Hosseini for his useful comments on the manuscript.","We are also grateful to Yousef Izadi for his comments on the revised version.","We would like to thank the referee for his/her useful comments.","This research is supported by research grant No.","600/1476 of Shahid Beheshti University, G.C.","."]],"string":"[\n [\n \"Logarithmic Correction to BMSFT Entanglement Entropy\"\n ],\n [\n \"Abstract Using Rindler method we derive the logarithmic correction to the entanglement entropy of a two dimensional BMS-invariant field theory (BMSFT).\",\n \"In particular, we present a general formula for extraction of the logarithmic corrections to both the thermal and the entanglement entropies.\",\n \"We also present a CFT formula related to the logarithmic correction of the BTZ inner horizon entropy which results in our formula after taking appropriate limit.\"\n ],\n [\n \"Introduction\",\n \"One of the lessons of the AdS/CFT correspondence [1] is that the asymptotic symmetry of the asymptotically AdS spacetimes in d+1 dimensions is the same as conformal symmetry in one dimension lower.\",\n \"One can use this idea to generalize the gauge/gravity duality beyond the AdS/CFT correspondence.\",\n \"Accordingly , in any non-AdS/non-CFT correspondence, the symmetry of the dual field theory should be the same as the asymptotic symmetry of the gravity solutions.\",\n \"The asymptotic symmetry of asymptotically flat spacetimes were known long before Maldacena's conjecture.\",\n \"Such symmetries are known as BMS, which were first found at null infinity of four-dimensional asymptotically flat spacetimes [2], and later were generalized to the three-dimensional case [3].\",\n \"Recently, Barnich and his collaborates [4] have shown that imposing just locally well-definiteness condition is enough to enhance both the translation and the rotation symmetry of the Poincare group at null infinity to an infinite-dimensional symmetry group.\",\n \"In this work BMS group refers to this infinite-dimensional group which consists of super-rotation and super-translation generators.\",\n \"In one dimension lower than gravity theory, the BMS algebra is given by an ultra-relativistic contraction of the conformal algebra.\",\n \"Thus BMS symmetry can be the symmetry of a lower dimensional field theory.\",\n \"It is proposed that the holographic dual of d+1-dimensional asymptotically flat spacetimes are ultra-relativistic d-dimensional BMS-invariant field theories (BMSFTs) [5],[6].\",\n \"In flat-space holography, BMSFT entanglement entropy and its holographic dual were first studied in [7] and the entanglement entropy of a two-dimensional BMSFT was computed by a method similar to the CFT [8].\",\n \"A holographic interpretation of the BMSFT entanglement entropy in [7] (similar to the Ryu-Takayanagi proposal in the context of AdS/CFT correspondence [9]) is given using Chern-Simons formulation of three-dimensional flat-space gravity [10].\",\n \"Other aspects of BMSFT entanglement entropy has been studied in [11],[12].\",\n \"A remarkable progress in this subject has been achieved recently in [13] where Rindler method [14] is used to derive not only the BMSFT entanglement entropy formula but also the holographic description in terms of some curves length in the bulk theory.\",\n \"The central idea of the Rindler method is to find local unitary transformations which map entangled states to the thermal states in the field theory.\",\n \"Then one can use the thermal entropy formula and find the entanglement entropy.\",\n \"Calculation of thermal entropies in BMSFTs is performed using Cardy-like formula first introduced in [15].\",\n \"Using Rindler method, the entanglement entropy is given by Cardy-like formula.\",\n \"Similar to the Cardy-formula in CFT , the saddle point approximation is used to derive this formula.\",\n \"Thus, it is possible to improve approximation and find possible corrections of this formula.\",\n \"In [16], employing a method first used for the Cardy formula in [17]We note however that the Cardy (high temperature) limit is not always reliable for extracting the black hole entropy [18].\",\n \"On the other hand, in two dimensional CFTs the logarithmic corrections are only universal in the Cardy regime.\",\n \"From the dual three dimensional gravity perspective, the universality of the correction is usually because heat kernels do not contain logarithmic terms [19].\",\n \"Thus the logarithmic correction to the BMSFT thermal and entanglement entropies are universal in the limit that BMSFT Cardy-like formula is reliable., logarithmic correction to the Cardy-like formula has been derived.\",\n \"In this paper, we use the logarithmic correction to Cardy-like formula along with Rindler method to find the logarithmic correction of BMSFT entanglement entropy.\",\n \"Similar to the leading term, this correction depends on central charges of BMS algebra, the interval of the sub-system and the cut-off.\",\n \"However, the interesting point is that we can rewrite the corrections in a universal form: $S=S_0-3\\\\log \\\\left(C_M^{\\\\frac{1}{3}}\\\\dfrac{\\\\partial S_0}{\\\\partial C_L}\\\\right)$ where $S_0$ is the leading term and $C_L$ and $C_M$ are the central charges of BMS algebra.\",\n \"$S$ can be both of the thermal entropy (which is given by the Cardy-like formula) and the entanglement entropy.\",\n \"This formula works for all BMSFTs on plane or cylinder at zero or finite temperature.\",\n \"The entanglement entropy of the boundary theory can be used to reconstruct the bulk dynamics [20], [21].\",\n \"In this view, the logarithmic correction of the entanglement entropy might be helpful to find quantum correction to the Einstein field equations.\",\n \"This paper is the first step on this road.\",\n \"The idea is to use the first law of the entanglement entropy as the analogue of the first law of (black hole) thermodynamics [20], [21], [22].\",\n \"The better understanding of BMSFT modular hamiltonian might allow us to achieve not only the classical bulk dynamics but also the quantum correction to the Einstein equation without the cosmological constant.\",\n \"One approach to study Flat/BMSFT is to take limit from the AdS/CFT calculations.\",\n \"According to the proposal of [6], the flat space limit in the gravity side corresponds to taking an ultra-relativistic limit from the CFT calculations.\",\n \"Hence one can find all BMSFT formulas by taking limit from a CFT counterpart.\",\n \"It was shown in [23] and [24] that the Cardy-like formula of BMSFT is given by taking limit from a formula in the CFT which is related to the inner horizon entropy of the gravity solution.\",\n \"If we assume the same relation for the logarithmic correction, the logarithmic term in (REF ) is related to a logarithmic correction in the entropy of the inner horizon.\",\n \"We propose a suitable logarithmic correction in the inner horizon formula which corresponds to the logarithmic term of (REF ) after taking limit.\",\n \"Using logarithmically corrected inner and outer horizon entropy formulas we can calculate their multiplication.\",\n \"The observation is that the previously known result, the multiplication being mass independent for the Einstein gravity [25] (see also [26]) , is violated in the presence of the logarithmic terms.\",\n \"The structure of this paper is as follows: In section two we review the results of [13] which uses the Rindler method to derive the BMSFT entanglement entropy and its holographic description.\",\n \"In section three we review the derivation of BMSFT Cardy-like formula and its logarithmic correction.\",\n \"In section four we put together the results of section two and three to derive the logarithmic correction of BMSFT entanglement entropy.\",\n \"Section five is devoted to the derivation of the BMSFT entanglememnt entropy formula by taking flat-space limit.\"\n ],\n [\n \"Entanglement Entropy of BMSFT using Rindler method\",\n \"One of the advantages of the Rindler method is the convenience it provides for the calculation of entanglement entropy in the context of gauge/gravity duality.\",\n \"In this view, the thermal entropy of the boundary theory is mapped to the horizon entropy of the black hole (object) in the gravity side.\",\n \"Thus, one can use it to prove Ryu-Takayanagi (RT) formula for the holographic entanglement of CFTs.\",\n \"Moreover, since many thermal properties of the gravitational systems are known for the non-AdS cases, we expect that the Rindler method gives a lot of insights to find the analogue of RT formula for the dualities beyond the AdS/CFT correspondence.\",\n \"In Rindler method, the asymptotic symmetries play an essential role.\",\n \"We are interested in BMSFTs which are proposed to be the holographic dual of asymptotically flat spacetimes.\",\n \"In our case, a Rindler transformation is of the form of BMS transformation and final coordinates are invariant under specific thermal identifications.\",\n \"These transformations act like unitary operators $U_{\\\\mathcal {R}}$ on the fields and map the reduced density matrix in the subregion $\\\\mathcal {A}$ (which we want to calculate the entanglement entropy for) to a thermal density matrix in the interval $\\\\mathcal {B}$ , $\\\\rho _{\\\\mathcal {A}} = U_{\\\\mathcal {R}}\\\\, \\\\rho _{\\\\mathcal {B}}\\\\, U_{\\\\mathcal {R}}^{-1}.$ Since the unitary transformations do not change the entropy, the thermal entropy of the subregion $\\\\mathcal {B}$ is the same as entanglement entropy of the subregion $\\\\mathcal {A}$ .\",\n \"A two-dimensional BMSFT has the following symmetry [27]: $\\\\nonumber \\\\tilde{u} &= \\\\partial _{\\\\phi }{f(\\\\phi )} u +g(\\\\phi ),\\\\\\\\\\\\tilde{\\\\phi } &= f(\\\\phi ),$ where $f(\\\\phi )$ and $g(\\\\phi )$ are arbitrary functions of the original coordinate $(u,\\\\phi )$ .\",\n \"The BMS algebra is given by using infinitesimal BMS transformation as $\\\\nonumber [L_n,L_m]&=(n-m)L_{n+m}+{C_L\\\\over 12}n(n^2-1)\\\\delta _{n+m,0},\\\\\\\\\\\\nonumber [L_n,M_m]&=(n-m)M_{n+m}+{C_M\\\\over 12}n(n^2-1)\\\\delta _{n+m,0},\\\\\\\\{[}M_n,M_m]&=0.$ where for a BMSFT on a plane with coordinates $(u,\\\\phi )$ the generators $L_n$ and $M_n$ are given by $\\\\nonumber L_{n} &= -u (n+1) \\\\phi ^n \\\\partial _{u} - \\\\phi ^{n+1} \\\\partial _{\\\\phi }\\\\\\\\M_{n} &= \\\\phi ^{n+1}\\\\partial _{u}.$ The global part of this algebra is identified with $n=0,\\\\pm 1$ .\",\n \"A Rindler transformation is of the form $\\\\tilde{x}=T(x)$ , but it should be invariant under some imaginary identification (thermal identification) of the new coordinate $\\\\tilde{x}^i\\\\sim \\\\tilde{x}^i+i\\\\tilde{\\\\beta }^i$ .\",\n \"Moreover, vectors $\\\\partial _{\\\\tilde{x}^i}$ annihilate the vacuum and hence should be written as the linear combination of the global part of the BMS algebra: $\\\\partial _{\\\\tilde{x}^i} = \\\\sum _{n=-1}^{1}{( b_{n} L_{n} +d_{n} M_{n} )}.$ Using (REF ) and (REF ) we conclude that $\\\\partial _{\\\\phi }{f(\\\\phi )} = \\\\frac{1}{Y}, \\\\qquad \\\\partial _{\\\\phi }{g(\\\\phi )} = -\\\\frac{T}{Y^2},$ where $Y &= - b_{-1} -b_{0} \\\\phi -b_{1} \\\\phi ^2,\\\\\\\\T &= d_{-1} + d_{0} \\\\phi + d_{1} \\\\phi ^2.$ Solutions to (REF ) determine vector $ \\\\partial _{\\\\tilde{x}^i}$ .\",\n \"It is assumed that entangled region is given by $\\\\mathcal {A} = \\\\lbrace (\\\\frac{-l_{\\\\phi }}{2} , \\\\frac{-l_{u}}{2}) \\\\cup (\\\\frac{l_{\\\\phi }}{2} , \\\\frac{l_{u}}{2})\\\\rbrace $ .\",\n \"We can use the following constraints to find vector $ \\\\partial _{\\\\tilde{x}^i}$ and the geometric (modular) flow $ k_t = -\\\\tilde{\\\\beta _{\\\\phi }} \\\\partial _{\\\\tilde{\\\\phi }} + \\\\tilde{\\\\beta _{u}} \\\\partial _{\\\\tilde{u}} $ : The finite interval on $\\\\mathcal {A}$ should be mapped to the infinite interval on $\\\\mathcal {B}$ , $( \\\\frac{-l_{\\\\phi }}{2} , \\\\frac{-l_u}{2} ) &\\\\rightarrow ( -\\\\infty , -\\\\infty ),\\\\\\\\( \\\\frac{l_{\\\\phi }}{2} , \\\\frac{l_u}{2} ) &\\\\rightarrow ( \\\\infty , \\\\infty ).$ The origin of the entangled interval is mapped to the origin of the thermal interval $(0, 0) \\\\rightarrow (0,0)$ The thermal interval (tilde coordinate) obeys a thermal identification of the following form $( \\\\tilde{\\\\phi } , \\\\tilde{u} ) \\\\sim ( \\\\tilde{\\\\phi } + i \\\\tilde{\\\\beta }_{\\\\phi } ,\\\\tilde{u} - i \\\\tilde{\\\\beta }_{u} ) .$ The modular flow $k_t$ vanishes on the boundary of the entangled region $k_t(\\\\partial {\\\\mathcal {A}} ) = 0 \\\\Rightarrow {\\\\left\\\\lbrace \\\\begin{array}{ll}k_t( \\\\frac{-l_\\\\phi }{2} ,\\\\frac{-l_u}{2} ) = 0,\\\\\\\\k_t( \\\\frac{l_\\\\phi }{2} ,\\\\frac{l_u}{2} ) =0\\\\end{array}\\\\right.\",\n \"}$ Using these conditions, [13] completely determines the Rindler transformation and modular flow of a BMSFT on the plane as below: $\\\\tilde{\\\\phi } &= \\\\frac{ \\\\tilde{ \\\\beta }_{\\\\phi } }{ \\\\pi } \\\\tanh ^{-1}{ \\\\frac{2 \\\\phi }{ l_{\\\\phi }} } ,\\\\\\\\\\\\tilde{u} + \\\\frac{ \\\\tilde{\\\\beta }_{u} }{ \\\\tilde{\\\\beta }_{\\\\phi } } \\\\tilde{\\\\phi } &= \\\\frac{2 \\\\tilde{\\\\beta }_{\\\\phi } ( u l_{\\\\phi } - l_{u} \\\\phi )}{\\\\pi (l_{\\\\phi }^2 - 4 \\\\phi ^2)},$ $k_t = -\\\\tilde{\\\\beta }_{\\\\phi } \\\\partial _{\\\\phi } + \\\\tilde{\\\\beta }_{u} \\\\partial _{u} =\\\\frac{- \\\\pi }{2 l_{\\\\phi }} \\\\left( \\\\left(l_{\\\\phi }^2 - 4 \\\\phi ^2\\\\right)\\\\partial _{\\\\phi } + \\\\left(l_u l_{\\\\phi } +4 \\\\frac{lu}{l_{\\\\phi }} \\\\phi ^2- 8 u \\\\phi \\\\right)\\\\partial _{u} \\\\right)$ For a BMSFT with identification of coordinates as $(\\\\tilde{u} , \\\\tilde{\\\\phi } ) \\\\sim ( \\\\tilde{u} + i \\\\bar{a} , \\\\tilde{\\\\phi } - i a) \\\\sim ( \\\\tilde{u} + 2 \\\\pi \\\\bar{b} , \\\\tilde{\\\\phi } - 2 \\\\pi b),$ The degeneracy of states is given by a Cardy-like formula [15], [13], [28] (see next section).\",\n \"$S_{\\\\bar{b}|b}(\\\\bar{a}|a) = \\\\frac{- \\\\pi ^2}{3} \\\\left(C_{L} \\\\frac{b}{a} +C_{M} \\\\frac{(\\\\bar{a} b - a \\\\bar{b} )}{a^2}\\\\right).$ Using (REF ), the entanglement entropy of a BMSFT on the plane for the interval $\\\\mathcal {A} = \\\\lbrace (\\\\frac{-l_{\\\\phi }}{2} + \\\\epsilon _{\\\\phi } , \\\\frac{-l_{u}}{2} + \\\\epsilon _{u} ) \\\\cup (\\\\frac{l_{\\\\phi }}{2} -\\\\epsilon _{\\\\phi } , \\\\frac{-l_{u}}{2} - \\\\epsilon _{\\\\phi } )\\\\rbrace $ becomes [13] $S_{EE} = \\\\frac{C_{L}}{6} \\\\log { \\\\frac{l_{\\\\phi }}{\\\\epsilon _{\\\\phi }} } + \\\\frac{C_{M} }{6} \\\\left( \\\\frac{l_{u}}{l_{\\\\phi } }- \\\\frac{\\\\epsilon _{u}}{\\\\epsilon _{\\\\phi }}\\\\right),$ where $\\\\epsilon _u$ and $\\\\epsilon _\\\\phi $ are ultraviolet cut-offs in $u$ and $\\\\phi $ coordinates.\",\n \"Similarly the entanglement entropy for finite temperature BMSFT on the cylinder has been calculated in [13].\"\n ],\n [\n \"Logarithmic Correction to Cardy-like Formula of BMSFT\",\n \"The entropy of CFT thermal states is calculated using Cardy formula.\",\n \"Using the saddle point approximation, it is possible to find a similar formula for the degeneracy of thermal states in a BMSFT [15], [13], [28].\",\n \"Due to the Rindler method, any correction to Cardy-like formula has influence on the entanglement entropy formula.\",\n \"In this section, we review the logarithmic correction to Cardy-like formula [16] and then use it to find the logarithmic correction to the entanglement entropy.\",\n \"Method of [16] is based on [17] that introduces the first order logarithmic correction to the Cardy formula (For a calculation of all order corrections see [29]) We start from the modular invariant partition function of BMSFT on a torus defined by $Z_{0}( \\\\hat{\\\\beta }_{u} | \\\\hat{\\\\beta }_{\\\\phi } ) = \\\\operatorname{Tr}e^{-\\\\hat{\\\\beta }_{u} (M_{0} - \\\\frac{C_{M}}{24}) + \\\\hat{\\\\beta }_{\\\\phi } (L_{0} -\\\\frac{C_{L}}{24}) }= e^{\\\\hat{\\\\beta }_{u} \\\\frac{C_{M}}{24} - \\\\hat{\\\\beta }_{\\\\phi } \\\\frac{C_{L}}{24} } Z( \\\\hat{\\\\beta }_{u} | \\\\hat{\\\\beta }_{\\\\phi } ),$ where $ Z( \\\\hat{\\\\beta }_{u} | \\\\hat{\\\\beta }_{\\\\phi } )$ is $Z( \\\\hat{\\\\beta }_{u} | \\\\hat{\\\\beta }_{\\\\phi } ) =\\\\operatorname{Tr}e^{-\\\\hat{\\\\beta }_{u} M_{0} + \\\\hat{\\\\beta }_{\\\\phi } L_{0}}= \\\\sum _{h_M,h_L} e^{( -\\\\hat{\\\\beta }_{u} h_{M} + \\\\hat{\\\\beta }_{\\\\phi } h_{L})} d(h_{M},h_{L}),$ and identification of torus are $(\\\\hat{u},\\\\hat{\\\\phi })\\\\sim (\\\\hat{u}+i\\\\hat{\\\\beta }_u,\\\\hat{\\\\phi }-i\\\\hat{\\\\beta }_\\\\phi )\\\\sim (\\\\hat{u},\\\\hat{\\\\phi }-2\\\\pi ).$ Here, $h_L$ and $h_M$ are respectively the eigenvalues of $L_0$ and $M_0$ .\",\n \"It is shown that the BMS modular invariant partition function satisfies [13] $Z_{0}( \\\\hat{\\\\beta }_{u} | \\\\hat{\\\\beta }_{\\\\phi } ) =Z_{0}\\\\left(-4 \\\\pi ^2 \\\\frac{\\\\hat{\\\\beta }_{u}}{\\\\hat{\\\\beta }_{\\\\phi }^2} | \\\\frac{4 \\\\pi ^2}{\\\\hat{\\\\beta }_{\\\\phi }}\\\\right).$ Plugging equation (REF ) into (REF ) gives, $Z( \\\\hat{\\\\beta }_{u} | \\\\hat{\\\\beta }_{\\\\phi } ) = e^{ \\\\hat{\\\\beta }_{u} \\\\frac{C_M}{24} - \\\\hat{\\\\beta }_{\\\\phi } \\\\frac{C_L}{24} - \\\\frac{ \\\\pi ^2 \\\\hat{\\\\beta }_{u} C_{M}}{6 \\\\hat{\\\\beta }_{\\\\phi }^2} - \\\\frac{ \\\\pi ^2 C_{L}}{6 \\\\hat{\\\\beta }_{\\\\phi }} } Z\\\\left(-4 \\\\pi ^2 \\\\frac{\\\\hat{\\\\beta }_{u}}{\\\\hat{\\\\beta }_{\\\\phi }^2} | \\\\frac{4 \\\\pi ^2}{\\\\hat{\\\\beta }_{\\\\phi }}\\\\right).$ By using the inverse Laplace transformation in last term of (REF ) we find $d(h_{L} , h_{M} ) = \\\\int d\\\\hat{\\\\beta }_{u} d\\\\hat{\\\\beta }_{\\\\phi } e^{ \\\\hat{\\\\beta }_{u} \\\\frac{C_M}{24} - \\\\hat{\\\\beta }_{\\\\phi } \\\\frac{C_L}{24} - \\\\frac{ \\\\pi ^2 \\\\hat{\\\\beta }_{u} C_{M}}{6 \\\\hat{\\\\beta }_{\\\\phi }^2} - \\\\frac{ \\\\pi ^2 C_{L}}{6 \\\\hat{\\\\beta }_{\\\\phi }} + \\\\hat{\\\\beta }_{u} h_M - \\\\hat{\\\\beta }_{\\\\phi } h_L } Z\\\\left(-4 \\\\pi ^2 \\\\frac{\\\\hat{\\\\beta }_{u}}{\\\\hat{\\\\beta }_{\\\\phi }^2} | \\\\frac{4 \\\\pi ^2}{\\\\hat{\\\\beta }_{\\\\phi }} \\\\right).$ In order to simplify (REF ), we use two approximations.\",\n \"First, we consider large charges which yields $d(h_{L} , h_{M} ) = \\\\int d\\\\hat{\\\\beta }_{u} d\\\\hat{\\\\beta }_{\\\\phi } e^{ - \\\\frac{ \\\\pi ^2 \\\\hat{\\\\beta }_{u} C_{M}}{6 \\\\hat{\\\\beta }_{\\\\phi }^2} - \\\\frac{ \\\\pi ^2 C_{L}}{6 \\\\hat{\\\\beta }_{\\\\phi }} + \\\\hat{\\\\beta }_{u} h_M - \\\\hat{\\\\beta }_{\\\\phi } h_L } Z\\\\left(-4 \\\\pi ^2 \\\\frac{\\\\hat{\\\\beta }_{u}}{\\\\hat{\\\\beta }_{\\\\phi }^2} | \\\\frac{4 \\\\pi ^2}{\\\\hat{\\\\beta }_{\\\\phi }}\\\\right).$ Then, we approximate (REF ) around the saddle point given by $(\\\\hat{\\\\beta }_{\\\\phi }^{s})^2 = \\\\frac{\\\\pi ^2 C_{M}}{6 h_{M}},\\\\qquad \\\\hat{\\\\beta }_{u}^{s} ={\\\\hat{\\\\beta }_{\\\\phi }^{s}\\\\over 2h_M\\\\,C_M} (C_{M} h_{L} - C_{L} h_{M} ) .$ Finally, the entropy or Cardy like formula for BMSFT reads as It is assumed that the partition function is slowly varying at the extremum.\",\n \"$S^{0} = \\\\log {d(h_{L} , h_{M} )} = -{\\\\pi ^2\\\\over 3 (\\\\hat{\\\\beta }_{\\\\phi }^{s})^2 }\\\\left(C_M \\\\hat{\\\\beta }_{u}^{s}+C_L \\\\hat{\\\\beta }_{\\\\phi }^{s}\\\\right).$ To find logarithmic correction we expand the integral (REF ) around the saddle point (REF ) up to quadratic term: $d(h_{L} , h_{M} ) = e^{S^{0}} \\\\int d\\\\hat{\\\\beta }_{u} d\\\\hat{\\\\beta }_{\\\\phi } e^{\\\\frac{1}{2} (X^2 -Y^2)},$ where $X &= {\\\\pi C_M\\\\over 3 \\\\hat{\\\\beta }^{s}_{\\\\phi } A}(\\\\hat{\\\\beta }_{u} - \\\\hat{\\\\beta }^{s}_{u}) , \\\\qquad Y = {\\\\pi A\\\\over (\\\\hat{\\\\beta }^{s}_{\\\\phi })^2} \\\\left[ -(\\\\hat{\\\\beta }_{\\\\phi } - \\\\hat{\\\\beta }^{s}_{\\\\phi })+ {C_M \\\\hat{\\\\beta }^{s}_{\\\\phi }\\\\over 3 A^2} (\\\\hat{\\\\beta }_{u}-\\\\hat{\\\\beta }^{s}_{u})\\\\right],\\\\\\\\A &= \\\\sqrt{C_M \\\\hat{\\\\beta }^{s}_{u}+\\\\frac{1}{3} C_L\\\\hat{\\\\beta }^{s}_{\\\\phi }}.$ Using (REF ), Eq.\",\n \"(REF ) takes the form $d(h_{L} , h_{M} ) = e^{S^{0}}\\\\left(-{\\\\pi ^2 C_M\\\\over 3(\\\\hat{\\\\beta }^s_\\\\phi )^3}\\\\right)^{-1} \\\\int dX dY e^{\\\\frac{X^2 -Y^2}{2} }.$ In the above equation the result of integration is just a number.\",\n \"Thus we find correction to the entropy up to a constant as [16] $S = \\\\log { d(h_{L} , h_{M} ) } = -{\\\\pi ^2\\\\over 3 (\\\\hat{\\\\beta }_{\\\\phi }^{s})^2 }\\\\left(C_M \\\\hat{\\\\beta }_{u}^{s}+C_L \\\\hat{\\\\beta }_{\\\\phi }^{s}\\\\right)-3\\\\log \\\\left(-{C_M^\\\\frac{1}{3}\\\\over \\\\hat{\\\\beta }_{\\\\phi }^{s} }\\\\right)+\\\\text{constant}.$ The interesting point is that the logarithmically corrected term can be rewritten (up to a constant) as the derivative of the leading term with respect to $C_L$ : $S=S_0-3\\\\log \\\\left(C_M^{\\\\frac{1}{3}}\\\\dfrac{\\\\partial S_0}{\\\\partial C_L}\\\\right).$\"\n ],\n [\n \"Logarithmic Correction to Entanglement Entropy\",\n \"In this section, we put together the results from sections two and three to find the logarithmic correction of BMSFT entanglement entropy.\",\n \"As mentioned before, the idea is to map entangled states to thermal states and then calculate entropy.\",\n \"The entropy is computed using BMSFT Cardy-like formula (REF ) which also has a logarithmic correction.\",\n \"In the derivation of (REF ) the identification of coordinates (REF ) plays an essential role.\",\n \"It is possible to use (REF ) for finding the degeneracy of thermal states with more generic identification of coordinates as $(\\\\tilde{u},\\\\tilde{\\\\phi })\\\\sim (\\\\tilde{u}+i\\\\bar{a},\\\\tilde{\\\\phi }-ia)\\\\sim (\\\\tilde{u}+2\\\\pi \\\\bar{b},\\\\tilde{\\\\phi }-2\\\\pi b)$ The coordinate change between $(\\\\tilde{u},\\\\tilde{\\\\phi })$ and $(\\\\hat{u},\\\\hat{\\\\phi })$ is a BMS transformation, $\\\\hat{\\\\phi }={\\\\tilde{\\\\phi }\\\\over b},\\\\qquad \\\\hat{u}={\\\\tilde{u} \\\\over b}+{\\\\bar{b}\\\\over b^2}\\\\tilde{\\\\phi },$ where $\\\\hat{\\\\beta }_\\\\phi ={a\\\\over b},\\\\qquad \\\\hat{\\\\beta }_u={\\\\bar{a}\\\\,b-a\\\\,\\\\bar{b}\\\\over b^2}.$ Thus Cardy-like formula (REF ) can be written as $S=-{\\\\pi ^2\\\\over 3}\\\\left(C_L{b\\\\over a}+C_M{\\\\bar{a}\\\\,b-a\\\\,\\\\bar{b}\\\\over a^2}\\\\right)-3\\\\log \\\\left(-C_M^\\\\frac{1}{3}{b\\\\over a}\\\\right)$ In order to find the logarithmic correction of BMSFT entanglement entropy, it is enough to map the entanglement entropy to a thermal entropy and then use (REF ).\",\n \"As it was reviewed in section , the Rindler transformation which governs this map is determined in such a way that finally induces the thermal identification (REF ).\",\n \"Comparing (REF ) to (REF ) shows that $a=\\\\tilde{\\\\beta }_\\\\phi ,\\\\qquad \\\\bar{a}=\\\\tilde{\\\\beta }_u.$ The values of $\\\\tilde{\\\\beta }_\\\\phi $ , $\\\\tilde{\\\\beta }_u$ and $b$ , $\\\\bar{b}$ depend on the details of the Rindler transformation.\",\n \"These are given in terms of the cut-offs and the interval for which entanglement entropy is calculated.\",\n \"Starting from a regulated interval in the BMSFT given by $(-{l_u\\\\over 2}+\\\\epsilon _u,-{l_\\\\phi \\\\over 2}+\\\\epsilon _\\\\phi )\\\\rightarrow ({l_u\\\\over 2}-\\\\epsilon _u,{l_\\\\phi \\\\over 2}-\\\\epsilon _\\\\phi ),$ the Rindler transformation yields the following results [13]: For the zero temperature BMSFT on the plane we have $a&=\\\\tilde{\\\\beta }_\\\\phi =-{2\\\\pi ^2\\\\over \\\\log {l_\\\\phi \\\\over \\\\epsilon _\\\\phi }},\\\\qquad \\\\bar{a}=\\\\tilde{\\\\beta }_u=-{\\\\tilde{\\\\beta }_\\\\phi ^2\\\\over 2\\\\pi ^2}\\\\left({l_u\\\\over l_\\\\phi }-{\\\\epsilon _u\\\\over \\\\epsilon _\\\\phi }\\\\right),\\\\\\\\b&=-{\\\\tilde{\\\\beta }_\\\\phi \\\\over 2\\\\pi ^2}\\\\log {l_\\\\phi \\\\over \\\\epsilon _\\\\phi },\\\\qquad \\\\bar{b}={1\\\\over 2\\\\pi ^2}\\\\left({\\\\tilde{\\\\beta }_\\\\phi l_u\\\\over l_\\\\phi }-{\\\\tilde{\\\\beta }_\\\\phi \\\\epsilon _u\\\\over \\\\epsilon _\\\\phi }-\\\\tilde{\\\\beta }_u\\\\log {l_\\\\phi \\\\over \\\\epsilon _\\\\phi }\\\\right),$ Then the logarithmically corrected Cardy-like formula (REF ) becomes $S_{EE}={C_L\\\\over 6}\\\\log {l_\\\\phi \\\\over \\\\epsilon _\\\\phi }+{C_M\\\\over 6}\\\\left({l_u\\\\over l_\\\\phi }-{\\\\epsilon _u\\\\over \\\\epsilon _\\\\phi }\\\\right)-3\\\\log \\\\left(C_M^\\\\frac{1}{3}\\\\log {l_\\\\phi \\\\over \\\\epsilon _\\\\phi }\\\\right)+\\\\text{constant}.$ The third term in the above formula is the calculated correction.\",\n \"For the finite temperature BMSFT with identification $(u,\\\\phi )\\\\sim (u+i\\\\beta _u,\\\\phi -i\\\\beta _\\\\phi ),$ we can use the results of [13] and Eq.\",\n \"(REF ) to write $\\\\nonumber S_{EE}={C_L\\\\over 6}\\\\log \\\\left({\\\\beta _\\\\phi \\\\over \\\\pi \\\\epsilon _\\\\phi }\\\\sinh {\\\\pi l_\\\\phi \\\\over \\\\beta _\\\\phi }\\\\right)&+{C_M\\\\over 6}{1\\\\over \\\\beta _\\\\phi }\\\\left[\\\\pi \\\\left(l_u+{\\\\beta _u\\\\over \\\\beta _\\\\phi }l_\\\\phi \\\\right)\\\\coth {\\\\pi l_\\\\phi \\\\over \\\\beta _\\\\phi }-\\\\beta _u\\\\right]-{C_M\\\\epsilon _u\\\\over 6\\\\,\\\\epsilon _\\\\phi }\\\\\\\\ &-3\\\\log \\\\left(C_M^{1\\\\over 3}\\\\log \\\\left({\\\\beta _\\\\phi \\\\over \\\\pi \\\\epsilon _\\\\phi }\\\\sinh {\\\\pi l_\\\\phi \\\\over \\\\beta _\\\\phi }\\\\right)\\\\right)+\\\\text{constant}.$ The forth term in the above equation is the obtained correction for finite temperature.\",\n \"For the zero temperature BMSFT on the cylinder, the entanglement entropy with logarithmic correction reads as $S_{EE}={C_L\\\\over 6}\\\\log \\\\left({2\\\\over \\\\epsilon _\\\\phi }\\\\sin {l_\\\\phi \\\\over 2}\\\\right)+{C_M\\\\over 12}\\\\left(l_u\\\\cot {l_\\\\phi \\\\over 2}-{2\\\\epsilon _u\\\\over \\\\epsilon _\\\\phi }\\\\right)-3\\\\log \\\\left(C_M^{1\\\\over 3}\\\\log \\\\left({2\\\\over \\\\epsilon _\\\\phi }\\\\sin {l_\\\\phi \\\\over 2}\\\\right)\\\\right)+\\\\text{costant}.$ It is clear from all the above cases that up to a constant, the entanglement entropy formula including the logarithmic correction is given by $S_{EE}=S_0-3\\\\log \\\\left(C_M^{\\\\frac{1}{3}}\\\\dfrac{\\\\partial S_0}{\\\\partial C_L}\\\\right)$ This formula is also valid for the thermal entropy when $S_0$ is the Cardy-like formula.\",\n \"Thus, we propose a universal form for the logarithmic correction of entanglement entropy and thermal entropy.\",\n \"We expect the same common form for the CFT case.\",\n \"In the next section, we propose a similar form of logarithmic correction to CFT thermal and entanglement entropy and try to find (REF ) by taking limit from CFT formula.\"\n ],\n [\n \"Logarithmic correction of BMSFT entropy by taking limit from CFT counterpart\",\n \"For a two dimensional CFT with central charges $c$ and $\\\\bar{c}$ and right and left temperatures $\\\\beta $ and $\\\\bar{\\\\beta }$ , the Cardy formula is $S_0={\\\\pi ^2\\\\over 3}\\\\left({c\\\\over \\\\beta }+{\\\\bar{c}\\\\over \\\\bar{\\\\beta }}\\\\right).$ Using AdS/CFT correspondence, this formula results in the same entropy as the outer horizon entropy of asymptotically AdS black holes in the gravity side.\",\n \"The logarithmic correction to (REF ) was evaluated in [17]: $S_{log}=-{3\\\\over 2}\\\\log \\\\left({c^{1/3}\\\\over \\\\beta }\\\\right)-{3\\\\over 2}\\\\log \\\\left({\\\\bar{c}^{1/3}\\\\over \\\\bar{\\\\beta }}\\\\right)+\\\\text{constant}.$ Employing (REF ) and (REF ) we can write $S=S_0+S_{log}=S_0-{3\\\\over 2}\\\\log \\\\left(c^{1/3}\\\\dfrac{\\\\partial S_0}{\\\\partial c}\\\\right)-{3\\\\over 2}\\\\log \\\\left(\\\\bar{c}^{1/3}\\\\dfrac{\\\\partial S_0}{\\\\partial \\\\bar{c}}\\\\right)+\\\\text{constant}.$ Using Rindler method we can expect the same formula for the logarithmically corrected CFT entanglement entropy.\",\n \"Thus we propose (REF ) as a universal formula that can be used for both the thermal entropy and the entanglement entropy.\",\n \"It is known that taking flat space limit from the asymptotically AdS spacetimes (written in the appropriate coordinate) yields asymptotically flat spacetimes.\",\n \"It is proposed in [6] that the flat space limit in the bulk corresponds to taking the ultra-relativistic limit in the boundary CFT.\",\n \"In other words, BMSFT is given by taking ultra-relativistic limit from the CFT.\",\n \"Starting from conformal algebra in two dimensions, one can introduce an ultra-relativistic contraction and generate BMS algebra [6].\",\n \"The relation between central charges of conformal algebra and BMS algebra is given by $C_L=\\\\lim _{\\\\epsilon \\\\rightarrow 0}\\\\left(c-\\\\bar{c}\\\\right),\\\\qquad C_M=\\\\lim _{\\\\epsilon \\\\rightarrow 0}\\\\epsilon \\\\left(c+\\\\bar{c}\\\\right)$ where $\\\\epsilon $ is a dimensionless parameter which corresponds to $G/\\\\ell $ on the gravity side.\",\n \"$G$ is the Newton constant and $\\\\ell $ is the AdS-radius.\",\n \"Thus $\\\\epsilon \\\\rightarrow 0$ in the boundary corresponds to $\\\\ell \\\\rightarrow \\\\infty $ .\",\n \"It is plausible to find all BMSFT quantities by taking limit from the CFT counterparts.\",\n \"Thus one can look for the possible relation between (REF ) and (REF ).\",\n \"However, assuming $S_0$ in (REF ) as the Cardy formula (REF ) and using (REF ) does not result in the Cardy-like formula (REF ).\",\n \"It was shown in [23] and [24] that the appropriate formula which its limit yields the Cardy-like formula is $S_{inner}^0={\\\\pi ^2\\\\over 3}\\\\left({c\\\\over \\\\beta }-{\\\\bar{c}\\\\over \\\\bar{\\\\beta }}\\\\right).$ For taking limit we should scale the temperatures as below: $\\\\beta _u=\\\\lim _{\\\\epsilon \\\\rightarrow 0}\\\\left({\\\\beta -\\\\bar{\\\\beta }\\\\over 2 \\\\epsilon }\\\\right), \\\\qquad \\\\beta _\\\\phi =\\\\lim _{\\\\epsilon \\\\rightarrow 0}-\\\\left({\\\\beta +\\\\bar{\\\\beta }\\\\over 2 }\\\\right).$ Using AdS/CFT correspondence, (REF ) corresponds to the inner horizon entropy of asymptotically AdS black holes.\",\n \"Thus, we expect that taking limit from (REF ) which is written for the outer horizon does not yield (REF ).\",\n \"In the rest of this section, we introduce appropriate formula which taking limit from it result in (REF ).\",\n \"Let us start from $S^0$ in (REF ).\",\n \"When it is the thermal entropy given by the Cardy-like formula, the corresponding formula in the CFT will be (REF ).\",\n \"The question is then what is the corresponding CFT formula when $S^0$ in (REF ) is not the leading order of entanglement entropy?\",\n \"It is known that taking ultra-relativistic limit from the entanglement entropy of CFT is not well-defined.\",\n \"For example, for a zero temperature CFT on a plane, the entanglement entropy of an interval $-{R\\\\over 2}(\\\\cosh \\\\kappa ,\\\\sinh \\\\kappa )\\\\rightarrow {R\\\\over 2}(\\\\cosh \\\\kappa ,\\\\sinh \\\\kappa ),$ is given by [30] $S_{EE}={c+\\\\bar{c}\\\\over 2}\\\\log {R\\\\over \\\\epsilon _R}-{c-\\\\bar{c}\\\\over 6}\\\\kappa $ where $\\\\epsilon _R$ is a cut-off.\",\n \"Using (REF ) together with ultra-relativistic contraction $t\\\\rightarrow \\\\epsilon t$ shows that (REF ) is divergent in the $\\\\epsilon \\\\rightarrow 0$ limit.\",\n \"It is proposed in [11] that BMSFT entanglement entropy can be found by taking the limit from the CFT counterpart.\",\n \"In the prescription of [11] it is argued that the symmetries are enough to fix the form of the entanglement entropy.\",\n \"Then they use a new contraction of conformal algebra which results in BMS algebra.\",\n \"But the relation between central charges of conformal algebra and BMS algebra differs from (REF ).\",\n \"Since the final algebra is the same as the ultra-relativistic contraction of [6], the author of [11] argued that their results are the entanglement entropy of BMSFT.\",\n \"If we continue the logic which relates the Cardy-like formula to the limit of inner horizon entropy then we conclude that the entanglement entropy of BMSFT should be related to a formula in the CFT which is transformed to the inner horizon formula by using Rindler transformation [13].\",\n \"It is not difficult to check that the formula $S_{inner}={c-\\\\bar{c}\\\\over 2}\\\\log {R\\\\over \\\\epsilon _R}-{c+\\\\bar{c}\\\\over 6}\\\\kappa $ results in the leading term of entanglement entropy of zero temperature BMSFT on the plane after taking the ultra-relativistic limit.\",\n \"This formula is nothing but the inner horizon Cardy formula (REF ) if we use Rindler transformation [13].\",\n \"Therefore we conclude that $S^0$ in (REF ) is given by taking limit from a formula which is related to the inner horizon of dual spacetime.\",\n \"Similarly, the logarithmic correction in (REF ) is also given by taking limit from a formula which is the logarithmic correction to the inner horizon entropy.\",\n \"It is not difficult to check that taking $\\\\epsilon \\\\rightarrow 0$ limit from $S_{log,inner}=-{3\\\\over 2}\\\\log \\\\left| c^{1/3}\\\\dfrac{\\\\partial S_0}{\\\\partial c}\\\\right|-{3\\\\over 2}\\\\log \\\\left| \\\\bar{c}^{1/3}\\\\dfrac{\\\\partial S_0}{\\\\partial \\\\bar{c}}\\\\right|-\\\\log \\\\epsilon ,$ results in the logarithmic term of (REF ) up to a constant.\",\n \"The interpretation of (REF ) in the gravity is of importance.\",\n \"As mentioned before $\\\\epsilon $ in the field theory corresponds to $G/\\\\ell $ on the bulk side.\",\n \"For the BTZ black holes, we have $\\\\beta ={2\\\\pi \\\\ell \\\\over (r_++r_-)},\\\\qquad \\\\bar{\\\\beta }={2\\\\pi \\\\ell \\\\over (r_+-r_-)}.$ where $r_\\\\pm $ are the radii of inner and outer horizons.\",\n \"Thus, using (REF ), (REF ) and (REF ) , we can find the logarithmic correction to the inner horizon entropy of BTZ black hole as $S^{BTZ}_{inner}={\\\\pi r_-\\\\over 2 G}-{3\\\\over 2}\\\\log {r_+^2-r_-^2\\\\over \\\\ell ^2}+\\\\text{constant},$ where we have substituted central charges as $c=\\\\bar{c}={3\\\\ell \\\\over 2 G}$ .\",\n \"On the other hand, using (REF ), (REF ) and (REF ) we find that $S^{BTZ}_{outer}={\\\\pi r_+\\\\over 2 G}-{3\\\\over 2}\\\\log {r_+^2-r_-^2\\\\over \\\\ell ^2}-\\\\log {\\\\ell \\\\over G}+\\\\text{constant}.$ It is clear from (REF ) and (REF ) that due to the logarithmic corrections, multiplication of inner and outer horizon entropies depends not only on the angular momentum but also on the mass of BTZ.\",\n \"This corrects the result of [25] which in the leading term the multiplication of inner and outer entropies is mass independent.\"\n ],\n [\n \"Conclusion\",\n \"In this paper we introduced a generic formula for the logarithmic corrections of the BMSFT thermal and the entanglement entropies.\",\n \"Our derivation is based on the Rindler method which makes connections between the thermal and the entanglement entropies.\",\n \"The most important application of the present work will reveal itself in the reconstruction of the bulk dynamics beyond the classical regime.\",\n \"This reconstruction has been done recently in the bulk AdS case through the perturbation of the entanglement entropy [20], [21].\",\n \"After generalizing the method of [20] for the flat-space holography, we will be able to derive the quantum corrections to the Einstein gravity without the cosmological constant field equations using the results of the current paper.\",\n \"Most of the works in the flat-space holography can be performed by taking the flat space-limit from the AdS/CFT calculations.\",\n \"One of the possible roads for finding the corrections of the BMSFT entanglement entropy formula is taking limit from the calculation of [31] which studies the one-loop bulk corrections to the Ryu-Takayanagi formula.\"\n ],\n [\n \"Acknowledgements\",\n \"The authors would like to thank S. M. Hosseini for his useful comments on the manuscript.\",\n \"We are also grateful to Yousef Izadi for his comments on the revised version.\",\n \"We would like to thank the referee for his/her useful comments.\",\n \"This research is supported by research grant No.\",\n \"600/1476 of Shahid Beheshti University, G.C.\",\n \".\"\n ]\n]"},"id":{"kind":"string","value":"1709.01804"}}},{"rowIdx":899,"cells":{"text":{"kind":"list like","value":[["The GTC exoplanet transit spectroscopy survey VIII. Flat transmission\n spectrum for the warm gas giant WASP-80"],["Abstract We set out to study the atmosphere of WASP-80b, a warm inflated gas giant with an equilibrium temperature of $\\sim$800~K, using ground-based transmission spectroscopy covering the spectral range from 520~to~910~nm.","The observations allow us to probe the existence and abundance of K and Na in WASP-80b's atmosphere, existence of high-altitude clouds, and Rayleigh-scattering in the blue end of the spectrum.","We observed two spectroscopic time series of WASP-80b transits with the OSIRIS spectrograph installed in the Gran Telescopio CANARIAS, and use the observations to estimate the planet's transmission spectrum between 520~nm and 910~nm in 20~nm-wide passbands, and around the K~I and Na~I resonance doublets in 6~nm-wide passbands.","We model three previously published broadband datasets consisting of 27 light curves jointly prior to the transmission spectroscopy analysis in order to obtain improved prior estimates for the planet's orbital parameters, average radius ratio, and stellar density.","We recover a flat transmission spectrum with no evidence of Rayleigh scattering or K~I or Na~I absorption, and obtain an improved system characterisation as a by-product of the broadband- and GTC-dataset modelling.","The transmission spectra estimated separately from the two observing runs are consistent with each other, as are the transmission spectra estimated using either a parametric or nonparametric systematics models.","The flat transmission spectrum favours an atmosphere model with high-altitude clouds over cloud-free models with stellar or sub-stellar metallicities."],["Introduction","Transmission spectroscopy allows us to probe the existence and abundance of atmospheric species in the atmospheres of transiting extrasolar planets [34], [5].","The method requires a high observing precision, which has made the space-based studies most successful in finding significant features in the transmission spectra [7], [36], [13], but the developments in instrumentation, observing techniques, and data analysis methods have also enabled transmission spectroscopy studies to be carried out successfully using ground-based telescopes.","The signal of interest – variations in the effective planetary radius as a function of wavelength – is minute, corresponding to changes of $\\sim 0.01\\%$ in the observed transit depth and $\\sim 0.1\\%$ in the effective planet-star radius ratio.","Further complications arise from possible high-altitude clouds, which can mask any atmospheric extinction features, leading to a flat transmission spectrum [21], [4], and from the fact that atmospheric extinction is not the only source of wavelength-dependent features in transmission spectra.","Both instrumental and astrophysical sources, such as host star's spots [2] and plages [26], flux contamination from a possible unresolved source, and incorrectly accounted for stellar limb darkening, can all imprint features that can be difficult to disentangle from the atmospheric signal.","Notwithstanding the complications, ground-based transmission spectroscopy has been used successfully to identify features attributed to absorption in planetary atmospheres.","Simultaneous measurements of the target star and several comparison stars – a process similar to relative photometry [3], [14] – the use of Gaussian processes have facilitated the robust modelling of systematics [15], [32], [31], and the use of Bayesian inference methods has allowed for realistic uncertainty estimation that is crucial when assessing the true significance of the identified transmission spectrum features.","We report a ground-based transmission spectroscopy study of WASP-80b [37].","We have observed spectroscopic time series of two WASP-80b transits with the OSIRIS spectrograph [33] installed in the 10.4 m Gran Telescopio CANARIAS (GTC) on La Palma, Spain.","The observations cover the spectral range from 520 to 910 nm, probing the planet atmosphere for a possible Rayleigh scattering signal in the blue end of the spectrum, and the visible-light extinction features of the K I and Na I resonance doublets at 767 nm and 589.4 nm, respectively.","WASP-80b [37], [22], [12], [38], a warm gas giant orbiting a bright (V=11.87) late-K / early-M dwarf on a 3.07 d orbit, was identified as a promising target for transmission spectroscopy from its discovery.","The planet has a low surface gravity ($M_\\mathrm {p} = 0.56\\;\\mathrm {M_{Jup}} $ , $R_\\mathrm {p} = 0.99 \\mathrm {R_{Jup}} $ , $g = 14.34$ ms$^{-2}$ , [22]), and its large radius ratio leads to $\\sim $ 3% deep transits, which, combined with the brightness of its host star, enhance our abilities to detect any possible transmission spectrum features.","WASP-80b is a warm gas giant with an equilibrium temperature $\\sim $ 800 K [37], [22].","This very likely places the planet into the pL class (no temperature inversion) in the classification by [11].","The main spectroscopic features in the visible passband for pL class planets are expected to be from Rayleigh scattering and K I and Na I resonance doublet absorption, of which K I absorption detection was recently claimed by [35] based on transmission spectroscopy analysis carried out with the FORS2 spectrograph installed in the VLT.","Table: Identifiers for WASP-80 with its coordinates and magnitudes (SIMBAD, retrieved 2017-07-04).Our study consists of two main analyses: joint analysis of 27 previously published broadband transit light curves observed in 7 passbands (broadband dataset), joint analysis of the two GTC-observed spectroscopic transit time series (transmission spectroscopy dataset), which both consist of a set of analyses with different prior assumptions and modelling approaches carried out to ensure the robustness of the final results.","The broadband dataset analysis is carried out to obtain improved estimates for the planet's broadband radius ratios, orbital parameters, and stellar density (system parameters).","The marginal parameter posteriors from the broadband dataset analysis are then used as priors in the GTC-data analysis.","The GTC transmission spectroscopy starts with a direct-modelling analysis with a flexible Gaussian-process-based systematics model to further constrain the system parameters, and the final transmission spectroscopy uses a divide-by-white approach with either a parametric or nonparametric residual systematics model.","This paper is divided roughly into three sections.","We outline the numerical methods and the generic equations for the calculation of posterior probability densities in §.","We continue in § by carrying out a detailed joint modelling of three priorly observed broadband datasets described in [37], [22], and [12].","The datasets cover 27 transit light curves observed in $g^{\\prime }$ , $r^{\\prime }$ , $i^{\\prime }$ , $I$ , $z^{\\prime }$ , $J$ , $H$ , and $K$ .","We describe the GTC observations and data reduction in § , detail the analysis in §, present the transmission spectroscopy results in §, and discuss the results in §.","Finally, we conclude the paper in §.","The raw data are publicly available from Zenodo and GTC data archive, and the whole analysis with reduced data is available from GitHub github.com/hpparvi/Parviainen-2017-WASP-80b as an easy-to-follow set of IPython notebooks and Python codes to help with the reproducibility of the study.","We use a fully Bayesian approach to transmission spectroscopy: our parameter estimates are based on the marginal posterior densities derived from a joint posterior density estimated with Markov Chain Monte Carlo (MCMC) sampling, and the marginal parameter posteriors from the broadband dataset analysis are used as priors in the GTC transmission spectroscopy analysis.","Our datasets consist of light curves observed either photometrically, or constructed from spectroscopic observations.","A light curve is modelled as a product of a baseline and transit model, where the combined model is parametrised with a parameter vector $\\theta $ .","When modelling multiple light curves jointly, the parameter vector is divided into parameters shared between all the light curves, passband-specific parameters shared between light curves observed in the same passband, observing-run-specific parameters, and light-curve-specific parameters.","Especially, the parameters defining the planetary orbit (zero epoch, orbital period, impact parameter, and stellar density) are shared between all the light curves included into the analysis, and thus the likelihoods from all the light curves contribute to the parameter posteriors.","The planet-star radius ratios and stellar limb darkening coefficients are considered passband-dependent, but observing-run-independent parameters (although this is not strictly true, since spots and plages, whether occulted or not, have an effect on the transit depths).","That is, the light curves observed in a given passband all contribute to the radius ratio and limb darkening posteriors for the passband.","Finally, the parameters defining the baseline and noise properties are considered either observing-run- or light-curve-specific, i.e., they depend both on the passband and observing run (this depends on the specific modelling approach, as detailed later).","Joint modelling leads to relatively high-dimensional models.","The number of free parameters in the analyses presented here varies from 10 to $\\sim $ 250.","However, the approach allows us to utilise the data fully.","Simultaneous modelling of different observing runs reduces our sensitivity on systematics, and simultaneous multiband analysis reduces the degeneracies between the estimated radius ratios, orbital impact parameter, and stellar limb darkening.","The parameter posteriors are estimated as a two-step process.","First, a population-based global optimisation method (Differential evolution implemented in PyDE) is used to obtain a parameter vector population that is clumped close to the global posterior maximum.","The parameter vector population is then used to initialise the emcee Markov Chain Monte Carlo (MCMC) sampler [10], [16], which is used to create a sample of parameter vectors drawn from the model posterior (see the analysis-specific sections for practical details).","The transit model uses the quadratic limb darkening formalism by [23], and is calculated using PyTransit [27].","PyTransit contains optimisations to compute a transit in multiple passbands with only a minor additional computational cost to the computation of a single passband transit, which reduces the computational burden due to the joint modelling approach.","The analyses have been carried out both with and without LDTk-based constraints on the stellar limb darkening.","Quadratic limb darkening is parametrised using the parametrisation presented in [20], which is aimed for efficient sampling of the physically allowed limb darkening coefficient space.","When the limb darkening is not constrained, we marginalise over the whole limb darkening parameter space allowed by the data.","When the limb darkening is constrained, it is done by fitting the observational data jointly with the limb darkening profiles created using the LDTk-package [28], and by marginalising over the limb darkening coefficients allowed by the stellar density profiles.","LDTk uses PHOENIX-calculated stellar atmosphere library by [18] to construct limb darkening profiles with the uncertainties in the stellar properties propagated into the uncertainties in the limb darkening profiles.","We also repeat the analyses for different systematics-modelling approaches, for separate subsets of data, and with synthethic mock data, and using the target star alone without dividing by the comparison star, to test the reliability of our approach.","Unless otherwise specified, the parameter point estimates correspond to posterior medians, and the uncertainties correspond to the central 68% posterior intervals.","We do not plot point parameter estimates (these are listed in tables), but prefer to show either the posterior distributions or limits based on central posterior intervals.","The analyses rely on Python- and Fortran-based code utilising SciPy, NumPy [39], IPython [29], Pandas [24], matplotlib [17], seaborn,$\\!$http://stanford.edu/~mwaskom/software/seaborn PyFITS,$\\!$PyFITS is a product of the Space Telescope Science Institute, which is operated by AURA for NASA and F2PY [30].","The transits were modelled with PyTransit Freely available from https://github.com/hpparvi/PyTransit [27], the limb darkening computations were carried out with LDTk $\\!$Available from https://github.com/hpparvi/ldtk [28], global optimisation was carried out with PyDE,$\\!$Available from https://github.com/hpparvi/PyDE the MCMC sampling was carried out with emcee [10], [16], and the Gaussian processes were computed using GeorgeAvailable from https://dan.iel.fm/george [1]."],["Posteriors and likelihoods","The unnormalised log posterior density for a dataset consisting of $n_\\mathrm {lc}$ light curves observed in $n_\\mathrm {pb}$ passbands is $\\ln P(\\theta |D) = \\ln P(\\theta ) + \\sum _i^{n_\\mathrm {lc}} \\ln P(\\vec{D_{\\mathrm {LC}}} _{,i}|\\theta ) +\\sum _i^{n_\\mathrm {pb}} \\ln P(\\vec{D_{\\mathrm {LD}}} _{,i}|\\theta ),$ where $\\theta $ is the parameter vector encapsulating all the model parameters, $\\ln P(\\theta )$ is the log prior, $\\vec{D_{\\mathrm {LC}}} $ are the light curves, $\\ln P(\\vec{D_{\\mathrm {LC}}} |\\theta )$ is the log likelihood for the photometry, $\\vec{D_{\\mathrm {LD}}} $ are the theoretical limb darkening profiles calculated by LDTk, and $\\ln P(\\vec{D_{\\mathrm {LD}}} |\\theta )$ is the log likelihood for the limb darkening profile.","Assuming that the uncertainties (noise) in the observations are normally distributed, we can write the log likelihood for the data $\\vec{D}$ in vector form as $\\ln P(\\vec{D}|\\theta ) = -\\frac{1}{2} \\left( n_\\mathrm {D} \\ln 2\\pi +\\ln |\\Sigma | +\\vec{r}^\\mathrm {T}\\Sigma ^{-1}\\vec{r}\\right),$ where $n_\\mathrm {D}$ is the number of datapoints, $\\vec{r}$ is the residual vector, and $\\Sigma $ is the covariance matrix.","If the noise is white (uncorrelated), the covariance matrix is diagonal, and the likelihood can be written out explicitly in scalar form as $\\ln P(\\vec{D}|\\theta ) = -\\frac{1}{2}\\left(n_\\mathrm {D}\\ln 2\\pi +\\sum _j^{n_\\mathrm {D}} \\ln \\sigma _{\\mathrm {j}}^2 + \\sum _{j=1}^{n_\\mathrm {D}}\\frac{\\vec{r}^2}{2\\sigma _{\\mathrm {j}}^2} \\right),$ where $\\sigma _{\\mathrm {j}}$ is the uncertainty of the $j$ th datapoint.","If the per-point uncertainty does not vary significantly, this equation can be simplified further into $\\ln P(\\vec{D}|\\theta ) = -\\frac{n_\\mathrm {D}}{2} \\ln 2\\pi \\sigma _{\\mathrm {i}}^2 -\\frac{1}{2} \\sum _{j=1}^{n_\\mathrm {D}}\\frac{\\vec{r}^2}{2\\sigma _{\\mathrm {j}}^2}.$ If the noise (here used to describe the leftover variation not explained by the parametric transit and baseline models) is not white, the covariance matrix will have off-diagonal elements, and the matrix needs to be inverted for the likelihood evaluation.","This is the case when the noise is presented as a Gaussian process [31], [15], [32].","The covariance matrix $\\Sigma $ in Eq.","(REF ) is now $\\Sigma = \\vec{K}(\\vec{x},\\vec{x}) + \\sigma ^2\\vec{I},$ where $\\vec{K}(\\vec{x},\\vec{x})$ is defined by a covariance function (also known as a covariance kernel).","We describe the likelihood and covariance functions separately for the broadband dataset analysis and GTC transmission spectroscopy in Sects.","and , respectively, since the two analyses use slightly different modelling approaches."],["Overview","We estimate the posterior densities for the WASP-80b orbital parameters based on the three datasets observed by [22], [37], and [12], abbreviated from herein as M14, T13, and F14, respectively.","The datasets contain 27 light curves (Fig.","REF ) observed in $g^{\\prime }$ , $r^{\\prime }$ , $i^{\\prime }$ , $I$ , $z^{\\prime }$ , $J$ , $H$ , and $K$ .","We model all the light curves jointly, and, in contrast to [22], include the M14 GROND J, H, and K light curves.","We also simplify the analysis slightly by merging the $I$ and $i^{\\prime }$ passbands.","The analysis is carried out assuming either a parametric or nonparametric systematics model (white or red noise), constant or wavelength-dependent radius ratio, and with and without LDTk to constrain the stellar limb darkening, which leads to eight separate analysis sets defined in Table REF .","The T13 and M14 datasets, as obtained from VizieR, were detrended by their original authors, and do not include other information than the observation time, flux, and error estimates.","The F14 dataset was kindly provided by the author without detrending, and included all the auxiliary information (such as the airmass, and x- and y-centroid shifts for each exposure) used in the original analysis presented in [12].","Table: Broadband analysis runs for the external datasets.","The white-noise runs include only the and datasets, while the red-noise runs also include the light curvesby .","The constant and varying radius ratios mark whether the radius ratiowas allowed to vary from passband to passband, or whether it was assumed to be wavelength-independent.Table: Marginal posterior medians from the external data broadband analysis (run ckrn_ldtkwith systematics modelled using GPs, wavelength-independent radius ratio, and limb darkeningconstrained with LDTk).","The uncertainties correspond to the central 68% posterior intervals.Table: Broadband radius ratio estimates and their uncertainties from the vkrn_ldtk run.We used the T13 and M14 datasets for an initial modelling with a parametric systematics model and white additive noise, and include the F14 dataset in the final runs using nonparametric systematics model.","The nonparametric systematics model represents the systematics (and white noise) as a Gaussian process (GP) with time as the only covariate for the T13 and M14 datasets, and with time, airmass, x-shift, and y-shift as covariates for the F14 dataset.","We do not marginalise over the GP hyperparameters in the broadband data analysis, but fix them to the values fitted from the white-noise run residuals.","As described in Sect.","REF , the parameter estimation starts with a parameter vector population that fills the prior space uniformly.","A differential evolution (DE) optimisation is used to clump the population close to the global posterior maximum (the number of DE iterations depending slightly on the run, but is usually close to 1000), after which MCMC sampling is carried out using emcee.","The sampler is run for 10$\\,$ 000 iterations, which yields 12000 independent posterior samples when using a population size of 150, thinning factor of 100, and burn-in period of 2000 iterations, where the thinning factor and burn-in period were chosen by studying the parameter chain populations and the average parameter autocorrelation lengths."],["Log posterior and likelihoods","The log posterior for the combined broadband dataset given a parameter vector $\\theta $ is $\\ln P(\\theta |D) &= \\ln P(\\theta ) \\nonumber \\\\&+ \\ln P(D_\\mathrm {T13}|\\theta ) + \\ln P(D_\\mathrm {M14}|\\theta ) +\\ln P(D_\\mathrm {F14}|\\theta ) \\\\&+ \\sum _i^{n_\\mathrm {pb}} \\ln P(\\vec{D_{\\mathrm {LD}}} _{,i}|\\theta ), \\nonumber $ where the first term is the log prior, followed by the per-dataset log likelihoods, and the last term is the sum of the LDTk-calculated log likelihoods for the limb darkening coefficients for each passband.","The exact form of the three likelihoods follows either Eq.","(REF ) or Eq.","(REF ), depending on the chosen systematics model.","The parametric (white noise) model assumes a constant average per-light-curve uncertainty (that is, the observation noise is the same for all the datapoints in a single light curve), and the kernels for the GP model are detailed below.","The light curve model consists of a product of a baseline function and a transit model with quadratic limb darkening calculated using PyTransit.","The residual vector for a single light curve in Eq.","(REF ) is $\\vec{r} = \\vec{f}_o - \\vec{f}_m(\\vec{X}, \\theta ) = \\vec{f}_o - \\mathcal {B}(\\vec{X}, \\theta ) \\,\\mathcal {T}(\\vec{t}, \\theta )$ where $\\vec{f}_o$ is the observed flux, $\\vec{f}_m$ the modelled flux, $\\mathcal {B}$ the baseline model, $\\mathcal {T}$ the transit model, $\\vec{X}$ is a matrix containing the the input parameters (covariates), $\\vec{t}$ are the mid-exposure times, and $\\theta $ is the model parameter vector."],["Gaussian process kernels","The T13 and M14 datasets do not include other auxiliary information than the mid-exposure times.","Thus, we model the noise as a Gaussian process with time as the only covariate, the kernel being a sum of an exponential kernel and a white noise term $k = A_t \\exp \\left[ - \\eta _t (t_i-t_j) \\right] + \\sigma ^2 \\delta _{ij},$ where $t$ are the time values, $A_t$ is the GP output amplitude, $\\eta _t$ the inverse time scale, $\\sigma $ the average white noise standard deviation, and $\\delta $ the Kronecker delta.","The F14 dataset includes also airmass, and per-observation x- and y-centroid estimates.","This allows us to use a slightly more complex kernel, where we have an exponential time component, squared exponential airmass component, squared exponential PSF-centroid component, and a white noise term.","The kernel becomes $k &= A_t \\exp \\left[-\\eta _t (t_i-t_j) \\right] \\nonumber \\\\& + A_a \\exp \\left[-\\eta _a (a_i-a_j)^2 \\right] \\nonumber \\\\& + A_{xy} \\exp \\left[-\\eta _x (x_i-x_j)^2 -\\eta _y (y_i-y_j)^2 \\right] \\nonumber \\\\& + \\sigma ^2\\delta _{ij},$ where $t$ , $a$ , $x$ , and $y$ are the time, airmass, x, and y estimates, respectively; $A_t$ , $A_a$ , $A_{xy}$ are the GP time, airmass and xy output amplitudes; and $\\eta _t$ , $\\eta _a$ , $\\eta _x$ , and $\\eta _y$ the time, airmass, x and y inverse time scales."],["Results","We adopt the ckrn_ldtk (passband-independent radius ratio, GP systematics, and limb darkening constrained using the LDTk) run as our final analysis, and report the stellar, orbital, and planetary parameter estimates in Tables REF and REF .","The posterior distributions are all close-to normal, as shown in Figs.","REF and REF .","The parameter estimates agree with the previous WASP-80b studies.","The broadband transmission spectrum from the vkrn_ldtk run, shown in Fig.","REF is consistent with a flat line.","The result agrees with a previous broadband transmission spectrum analysis by [38].","We used the ExoTransmit transmission spectrum modelling package to test different atmosphere scenarios, but the precision in radius ratios is not sufficient to meaningfully distinguish between any of the physically plausible scenarios.","Figure: Marginal- and joint-posteriors for the radius ratio, stellar density, and impactparameter corresponding to the final ckrn_ldtk run.","The 68% centralinterval is marked with a darker shade.Figure: Broadband radius ratio posteriors estimated by jointly modelling the three priordatasets.","The results correspond to the final vkrn_ldtk run with rednoise modelled using GPs and limb darkening constrained using LDTk.Figure: Broadband quadratic limb darkening coefficient posteriors.","The results correspond to thefinal ckrn runs with red noise modelled with GPs and limb darkening eitherunconstrained (left) or constrained using LDTk (right).The limb darkening coefficient posteriors are plotted in Fig.","REF , both with and without constraints from LDTk.","The two versions agree with each other within uncertainties.","The observed light curve, conditional model distribution (for the red noise model), and the residuals are shown in Fig.","REF .","Figure: The 27 broadband light curves observed by ,, and after the removal of the mean GP trend, theposterior model medians, and the residuals."],["Observations","We observed two WASP-80b transits simultaneously with one comparison star using the OSIRIS (Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy) spectrograph installed in the GTC (Gran Telescopio CANARIAS) on the nights starting 16 July 2014 and 25 August 2014 (observing runs R1 and R2 respectively).","The R1 observations carried from 23:30 UT till 3:41 UT and the R2 observations from 20:35 UT till 1:00 UT.","The observing conditions were good, with a seeing of $\\sim $ 0.8 in the beginning of each night, and all the observations were carried out with an airmass smaller than 1.4.","OSIRIS [6] contains two Marconi CCD42-82 2048$\\times $ 4096 pixel CCDs, which were used in the standard 2$\\times $ 2 binning mode yielding a plate scale of 0.254.","The observations were carried out using grism R1000R with a 40-wide custom-designed slit that aims to minimise the systematics related to variations in flux loss.","We chose TYC 5165-00235-1 as the comparison star, located at a distance of 6.9 from WASP-80.","The star has a similar V magnitude ($V=11.62$ ), but is slightly redder ($J=8.376$ ).","The two stars were positioned equidistantly from the optical axis close to the centre of each CCD.","The slit does not include other stars bright enough to be useful in the analysis.","The exposure time was 6 s for R1 and 5 s for R2.","While the exposure time was short, it was long enough for the comparison star to saturate during the final parts of the WASP-80b ingress during R1.","This was fixed by defocusing the instrument, but meant that the comparison star cannot be used in the reduction during this saturated period.","For R1, 81 flat fields were taken before the transit, and 66 bias frames after the transit.","For R2, 100 flat fields were taken after the transit, and 15 bias frames before the transit.","Three arc frames (Xe, HgAr, and Ne) for R1 were observed on 14 July and for R2 on the same night as the observations, after the transit."],["Data reduction and passband sets","The passband-integrated light curves are produced from the raw data using the pipeline described in [9], [8].","The pipeline carries out the basic CCD data reduction steps, calculates a 2D wavelength solution, removes the sky, and generates a reduced 1D spectrum for WASP-80 and the comparison star (Fig.","REF ).","The light curves are then generated by integrating the spectra multiplied by a transmission function defining the passband.","We created four sets of passbands (datasets) listed in Table REF .","The W dataset consists of a single light curve covering the whole usable spectrum (white light curve), the NB (narrow-band dataset) covers the whole usable spectrum in 20 nm-wide bins, and the Na I and K I datasets cover the Na and K lines, respectively, in 6 nm-wide bins.","The final white-light WASP-80 and reference star lightcurves with the relative light curve are shown in Fig.","REF .","The white-light white noise estimates for R1 and R2 are 400 and 520 ppm, respectively.","For the NB narrow-band dataset, the white noise level varies from 600 to 2100 ppm, with a median level of 720 and 870 ppm for R1 and R2, respectively.","The difference in the white noise levels is expected due to the different exposure times used in R1 and R2 (6 and 5 s, respectively).","Figure: Normalised, sky-subtracted, and wavelength-calibrated spectra for WASP-80b (dark blueline) and the comparison star (orange line).Figure: White-light light curves for the comparison star (top), WASP-80 (middle), and therelative light curve (bottom) for both observing runs.","A small number of clear outliers have beenomitted for clarity.Table: Passband sets extracted from the GTC spectroscopy."],["Systematics","The relative light curves in Fig.","REF feature systematics not corrected by division by the comparison star.","Specially, the baseline is affected by a smooth trend that cannot be modelled by a simple parametric model as a function of time (or any of the simultaneously measured auxiliary variables).","The trend causes a pre- and post-transit baseline difference of $\\sim 2\\%$ during run 1, and a $\\sim 1\\%$ difference during run 2.","A similar, smooth, nearly linear, variation as a function of the rotator angle accompanied by relatively smooth 'bumps', has been observed in previous OSIRIS transmission spectroscopy studies, and is likely caused by vignetting in the telescope pupil space [25].","Fortunately, the systematics are mainly common-mode, without significant variation across the spectrum.","Common-mode systematics can be removed by either fitting the white-light light curves with a flexible GP, which can be used to create a common-mode systematics model, or by using a divide-by-white approach.","After the common-mode correction, the residual wavelength-dependent systematics can be accounted for either with parametric or non-parametric approaches, both of which were used in our analyses."],["Overview","The GTC transmission spectroscopy covers the modelling of the white light curves and the three narrowband datasets described in Sec.","REF .","We used three modelling approaches: DIR direct modelling with flexible GP-based systematics DWW divide-by-white with parametric systematics DWR divide-by-white with GP-based systematics The first approach, direct modelling of the light curves with a flexible GP-based systematics model, was used to obtain a model for the common-mode systematics (from the white light curves), and to improve the system characterisation (from the full-spectrum 20 nm-wide dataset).","The divide-by-white (DW) approaches were then used for transmission spectroscopy, with the direct-modelling posteriors used as priors on the wavelength-independent system parameters (the motivation for this is discussed later).","The analyses were repeated with and without LDTk to assess how sensitive the results are to assumptions about limb darkening, and modelling the two nights separately and jointly, to test whether the results are consistent from night to night.","The parameter posteriors are estimated in similar fashion to the prior data analysis in Sec.","REF .","Differential evolution is used to create a parameter vector population clumped around the global posterior maximum, which used to initialise the emcee sampler.","The sampler is ran over $10\\times 15\\;000$ iterations (that is, ten sets of $15\\;000$ iterations where each set is initialised from the final state of the previous set) with 600 chains and a thinning factor of 100, which gives us a final set of 75$\\;$ 000 independent posterior samples.","The number of iterations, chains, and the thinning factor were chosen after studying the chain populations and per-parameter autocorrelation lengths.","Unlike in the broadband dataset analysis, we keep the GP hyperparameters free and marginalize over them.","This slows down the sampling process, but yields more reliable posteriors, and allows us to study how well the GP kernels represent the systematics.","The direct-modelling approach DIR reproduces the light curves as a product of a baseline and a transit model directly.","We model all the passbands for both nights in a dataset jointly, which leads to a slightly involved parameterisation, but aims to utilise the data fully.","As mentioned earlier, the model parameters can be divided into four categories: a) passband- and baseline-independent parameters, b) achromatic per-night baseline parameters, c) chromatic per-light-curve parameters, and d) chromatic parameters that should stay constant between observation runs.","The direct model parametrisation and the parameter priors are listed in Table REF .","Priors for the zero epoch, orbital period, impact parameter, and stellar density are based on the broadband dataset analysis posteriors (run ckrn_ldtk).","The $u$ and $v$ limb darkening coefficients correspond to the [20] quadratic limb darkening model parametrisation, where uniform priors from 0 to 1 lead to uninformative priors covering the physically viable values for the quadratic limb darkening priors.","The GP hyperparameters can be chosen to be night- or light-curve-dependent, but we choose to keep them independent for practical reasons.","Our approach adds three GP hyperparameters to the analysis, and requires two covariance matrix inversions per posterior evaluation (one per night).","Making the GP hyperparameters light-curve-dependent would add three free parameters to the model for each light curve, and require a covariance matrix inversion for each light curve, which would make marginalisation over the GP hyperparameters costly.","Making the GP hyperparameters night-dependent would be feasible, since the approach would add only three more parameters to the model, and would not require additional covariance matrix inversions.","However, our analyses for separate nights result with compatible GP hyperparameter posteriors, and we choose the simplest approach.","Table: Model parametrisations and priors.","𝒰(a,b)\\mathcal {U}(a,b) stands for a uniform prior from a to b, where a andb are omitted when the range is chosen to be wide enough not to affect the posteriors.","𝒩(μ,σ)\\mathcal {N}(\\mu ,\\sigma )stands for a normal prior with mean μ\\mu and standard deviation σ\\sigma .","The system parametershave normal priors with means and standard deviations corresponding to the values shown in Table .The direct model represents each light curve as a Gaussian process $f \\sim \\mathcal {N}(c_b\\;\\mathcal {T}(\\vec{t}, \\theta ), \\Sigma ),$ where $c_b$ is a baseline constant, $\\mathcal {T}$ is the transit model, and $\\Sigma $ is the covariance matrix.","We use airmass $x$ , and telescope rotator angle, $\\alpha $ , as GP input parameters (covariates), with a covariance matrix defined as a sum of a linear kernel and squared-exponential kernel, $\\Sigma (i,j) = \\left( \\frac{x_j \\cdot x_i}{\\gamma _x}\\right) + a_\\alpha ^2 \\exp \\left( \\frac{(\\alpha _j - \\alpha _i)^2}{\\gamma _\\alpha }\\right) + \\delta _{ij} \\sigma $ where $a_\\alpha $ is an output scale parameter, $\\gamma _x$ and $\\gamma _\\alpha $ are the input scale parameters, and $\\sigma $ is the average white noise.","Any residual airmass-dependent systematics should be approximately linear, and can be modelled with a linear kernel, while the possible chromatic rotator-angle dependencies are expected to be smooth, and well-modelled by a squared-exponential kernel.","We also tested a GP with time as a covariate (with a Matern kernel), but it did not affect the posteriors significantly.","Also, the power spectral density (PSD) of the residuals with the transit and the two-covariate GP mean removed is approximately constant, which suggests that the noise is white after the residual airmass and rotator-angle dependencies are accounted for.","The white noise level does not vary significantly from passband to passband in our datasets, and we choose to use a single average white noise estimate for an observing run (again, so that we do not need to invert the covariance matrix separately for each light curve).","The white noise estimate is an average of the estimates calculated separately for each light curve, which are calculated as $\\sigma = \\mathrm {std}(\\vec{f}_d) / \\sqrt{2}$ where $\\vec{f}_d = f_i - f_{i-1}$ .","That is, we estimate the white noise from the standard deviation of the light curve differentials.","The radius ratios and limb darkening coefficients yield three free parameters per passband, and the baseline constant a further free parameter per light curve.","In total, the number of free parameters for a direct model reaches 108 for the 20-nm run with 20 passbands.","The computations can still be carried out with a modern laptop in a matter of hoursPyTransit, which was used to compute the light curves, is optimised to compute a set of multiband light curves with only a minor additional cost to calculating a single passband.","for a single analysisHowever, while a single analysis can be carried out in a laptop, the Glamdring-cluster in Oxford University and the TeideHPC supercomputer in Spain were used to carry out the final analyses., but tests must be carried out to ensure that the final posterior sample gives a reliable representation of the true posterior distribution."],["Divide-by-white model with and without GP systematics","The divide-by-white (DW) approachOur approach would be more accurately named as divide-by-average, since we're dividing each dataset by the dataset mean.","allows us to remove any common-mode systematics from the dataset with the cost of increased per-passband white noise.","The residual vector $\\vec{r}$ in Eq.","(REF ) for the DW approach is $\\vec{r} = \\frac{1}{\\vec{b}(\\theta , \\mathbf {X})} \\left( \\frac{\\vec{f}}{1/n_\\mathrm {pb} \\sum \\vec{f}} - \\frac{\\mathcal {T} (\\vec{t}, \\theta )}{1/n_\\mathrm {pb} \\sum \\mathcal {T} (\\vec{t}, \\theta )} \\right), $ where $\\mathbf {f}$ is the observed flux vector, $\\mathcal {T}$ is the transit model, and $\\vec{b}$ is the residual baseline model.","That is, we divide both the observed and modelled fluxes by the values averaged over all the passbands in the dataset.","The DW approach does not only remove the common-mode systematics, but all colour-independent signals.","Effectively, the signals left in the data are due to colour-dependent systematics, changes in the radius radio, and changes in stellar limb darkening.","The system parameters, such as the orbital period and impact parameter, are poorly constrained, so we set informative priors based on the nb_ldtk direct-model analysis on them.","The average radius ratio is not well constrained either, so we set a prior on the median radius ratio based on the broadband dataset analysis.","We use median instead of mean to ensure that the prior does not affect the scaling of the transmission spectrum, as using mean would.","We repeat the DW analysis using first a parametric baseline model (DWW approach), and then a more flexible Gaussian process-based baseline model (DWR approach).","The parametric model represents the baseline for each light curve as a linear function of time and airmass $\\vec{b}_i(\\vec{t}, \\vec{x}, \\theta ) = \\theta _{i,a} + \\theta _{i,b} \\vec{t} + \\theta _{i,c} \\vec{x},$ where $\\theta _{i,a}$ , $\\theta _{i,b}$ , and $\\theta _{i,b}$ are the light curve specific baseline constant, linear time coefficient and linear airmass coefficient, respectively.","The residual noise is considered white, which makes the approach significantly faster than using GPs, but yields three free parameters per light curve to the model.","The DWR analysis uses the same GP kernel as the DIR approach, changing only the GP mean function to the one in Eq.","REF ."],["System characterisation","The direct-modelling DIR runs were used to improve the system characterisation from the broadband dataset analysis, and yield the final WASP-80b parameter estimates listed in Table REF .","The estimates correspond to the full-spectrum narrow-band dataset (NB) with LDTk.","The narrow-band run was preferred over the white-light run since the colour information allows us to mitigate the degeneracy between the average impact parameter, radius ratio, and limb darkening, leading to improved posterior estimates.","Table: Results from the final GTC characterisation run (nb_12_ldtkwith systematics modelled using GPs, wavelength-dependent radius ratio, and limb darkeningconstrained with LDTk).","The uncertainty is based on the central 68% posteriorinterval.","We do not list the radius ratio estimates, since the (common-mode)systematics are too strong for direct modelling to constrain them."],["Transmission spectroscopy","We show the transmission spectra for the NB, K I, and Na I datasets in Fig.","REF .","These results correspond to the joint DWR run with a GP systematics model and limb darkening coefficients constrainted by LDTk.","The figures show the central 68% and 99% radius ratio posterior intervals, and the point estimates are listed in Table REF .","Table: Passband centres, ranges, and radius ratio estimates for the three narrow-band datasetsfrom the final DWR approach.Comparison of the results from the two different divide-by-white approaches can be found below in Sect.","REF , and a comparison of the results obtained using only the light curves from one of the two can be found Sect.","REF .","We do not show the results for the analysis without LDTk.","Unconstrained limb darkening does not have a significant effect on the radius ratio posterior medians, but does increase the posterior width significantly due to the strong degeneracy caused by the DW approach.","Figure: The NB light curves for both observing runs and the DIR approach posterior median model(black line) as a function of the phase.","Passband centres are marked in nanometres.Figure: Transmission spectra for the NB dataset with 20 passbands spanning from 520 nm to 920 nm (top),the Na I set covering the Na I doublet (bottom left), and the K I dataset covering the K I double (bottom right).The inner boxes correspond to the central 68% posterior intervals, the outer boxes to the central 99% posteriorintervals, and Na I and K I lines are marked as black lines.","The light gray line shows an Exo-Transmit spectrumwith Na and K described in Sect.",", the black dots show the Exo-Transmitspectrum binned to the dataset binning, and the slashed horizontal line shows the transmission spectrum mean."],["Flat transmission spectrum","We adopt the results from the more conservative approach, DWR, as the final results of the analysis.","All the spectra in Fig.","REF are flat within uncertainties, in agreement with the broadband analysis, and the broadband transmission spectroscopy carried out by [38].","The NB features a single strongly deviating passband centred at 807 nm, with a significantly smaller radius ratio than the average.","The passband is likely affected by instrumental observation-geometry-dependent systematics (the deviation is evident in both observing runs, albeit slightly different), and we discuss it in more detail in Sect.","REF .","The flat transmission spectrum allows us to rule out strong Rayleigh scattering or Na I or K I absorption in WASP-80b's atmosphere, but does not justify detailed atmospheric modelling.","However, basic modelling with Exo-transmit [19] strongly favours a flat spectrum over a spectrum from an atmosphere with 0.1 or 1 solar metallicity and K and Na: a likelihood ratio test between the Exo-Transmit spectra and a flat spectrum favours the flat model with likelihood ratios around 1000-5000.","(We fit the model spectra to the estimated spectrum with a free scaling factor, and the flat spectrum is the mean of the estimated spectrum.)","Figure REF includes an Exo-Transmit spectrum with K and Na calculated assuming Solar metallicity, gas-phase chemistry, $T_\\mathrm {eq}=800$ K, $log {\\it g}= 3.18$ , $R_\\star = 0.57 R_\\odot $ , and $k=0.17$ as an example.","The DWR approach was chosen over DWW because the DWW results show greater discrepancies in the spectra estimated from R1 and R2 separately, as discussed below.","It is likely that the parametric systematics model is not flexible enough, and the unaccounted-for systematics lead to biases in the radius ratio values.","The GP kernel in the DWR approach is flexible enough to marginalize over the systematics, and still allows for a reliable radius ratio estimation (tested with two mock datasets discussed in Sect.","REF )."],["Comparison with the previous K I detection","[35] reported of a detection of strong K I absorption in WASP-80b's atmosphere based on transmission spectroscopy with the FORS2 spectrograph.","Our results disagree with the reported K I detection, as shown in Fig.","REF , but agree with the redmost part of their spectrum.","Both of our divide-by-white approaches are consistent with each other, as are the transmission spectra estimated separately from the two observing runs (as discussed below in Sects.","REF and REF ), which leads us to believe that the strong K I signal reported by [35] is due to systematics, even though the analysis described in their paper seems rigorous in all standards.","Figure: Comparison between the results from the GTC analysis and the results presented by ."],["Comparison of modelling approaches","The two divide-by-white approaches differ in their way of modelling the systematics.","The DWW approach models the systematics with parametric model as a combination of linear functions of time and airmass, and assumes that the everything else is white normally distributed noise, while the DWR approach models the systematics as a Gaussian process.","In the DWW case all systematics not corrected by the division by the dataset average light curve or modelled by the parametric model may cause biases in the radius ratio estimates.","The GP kernel in the DWR case is relatively flexible, even when we impose a linear relation on airmass, and should yield less bias-prone radius ratio estimates.","This is true especially since we're integrating over the GP hyperparameters, and do not significantly constrain them with priorsThe GP hyperparameters are actually allowed to probe the parameter space where they model time correlation rather than rotator-angle correlation, since the rotator angle itself is a slowly and smoothly varying function of time.","We compare the transmission spectra obtained with DWW and DWR approaches in Figs.","REF , REF , and REF .","The differences are small for the Na and K analyses, which can be expected due to the small wavelength range covered by the datasets.","However, the NB set analysed with DWW approach features a Rayleigh-like signal in the bluemost passbands that is not visible in the DWR results.","When analysing the two nights separately (Sect.","REF ), we can see that the Rayleigh-like signal arises from the second night, and is not significant in the first night.","However, the first night shows an increasing trend towards the red end of the spectrum, not visible in the second night spectrum.","In theory, the varying Rayleigh-like slope could be interpreted as variations in stellar activity.","However, a more likely explanation is residual wavelength-dependent systematics not accounted for properly by the parametric model.","Figure: Radius ratio posterior distributions approximated as normal distributions for the NB DWW (left) and DWR (right) runs.Figure: Radius ratio posterior distributions approximated as normal distributions for the Na DWW (left) and DWR (right) runs.Figure: Radius ratio posterior distributions approximated as normal distributions for the K DWW (left) and DWR (right) runs."],["Comparison of individual nights","We show the transmission spectra estimated separately from run 1 or run 2 either with the DWW or DWR approach in Figs REF , REF , and REF .","The results from the different nights agree with some exceptions.","Both the blue and red ends of the DWW NB spectrum differ systematically, as mentioned earlier.","the Rayleigh-like signal visible in the DWW model originates clearly only from R2, while the corresponding passbands are close or below the mean level for R1.","However the three redmost passbands are well above the mean level for R1, while R2 shows a flat spectrum in the red.","These differences disappear in DWR analysis, where the both runs give compatible flat-within-uncertainties spectrum.","Figure: Radius ratio posterior distributions approximated as normal distributions for the NB DWW (up) and DWR (below) runsusing only the data from night 1 (left) or night 2 (right).Figure: Radius ratio posterior distributions approximated as normal distributions for the Na I DWW (up) and DWR (below) runsusing only the data from night 1 (left) or night 2 (right).Figure: Radius ratio posterior distributions approximated as normal distributions for the K I DWW (up) and DWR (below) runsusing only the data from night 1 (left) or night 2 (right)."],["Outlier passband at 807 nm","A single passband centred at 807 nm in the NB dataset transmission spectrum deviates from the mean.","This signal is visible on both nights, and a detailed analysis shows that the features is smooth, but somewhat different between the nights.","We could not trace the source of the feature, but can only speculate.","First, the feature is very unlikely of astrophysical origin.","Certain astrophysical phenomena, such as non-transited star spots and flares, can affect the radius ratio estimates, but the feature would require a bright narrow-band source.","Second, the feature is not caused by the comparison star.","We repeated the DW NB analyses using absolute photometry without dividing with the comparison star (practical since the divide-by-white approach removes the common-mode systematics), but this did not affect the feature."],["Mock dataset analysis","We also carry out a sensitivity (reality) checks by using two mock datasets based on the NB dataset.","We calculate synthethic light curves using the observed time stamps and the system parameter posterior medians from the direct modelling.","The radius ratios follow a saw-tooth-like pattern for the mock dataset 1, and a transmission spectrum calculated by Exo-Transmit for the mock dataset 2, and the limb darkening coefficients are based on the theoretical values calculated using LDTk.","The baselines are modelled as sums of a linear time trend and a linear airmass trend with normally distributed random coefficients, and we add normally distributed white noise with standard deviation corresponding to the true light curve standard deviation estimates.","The radius ratio posterior densities from the mock dataset analyses agree with the true radius ratios for both DW approaches."],["Conclusions","We have carried out a transmission spectroscopy analysis for WASP-80b using two OSIRIS-observed spectroscopic time series, and a joint analysis of 27 previously observed broadband light curves to provide reliable parameter priors for the transmission spectroscopy analysis.","The OSIRIS data feature strong common-mode systematics, but these can be removed using either a divide-by-white (DW) approach or initial white-light GP modelling.","We chose the divide-by-white approach, (or, more accurately, divide-by-dataset-average) which increases the per-passband white noise levels, but removes any systematics-model dependencies from the common-mode systematics removal.","The transmission spectroscopy analyses were repeated modelling both datasets jointly and separately, with or without LDTk-constructed stellar limb darkening priors, and using either a parametric or nonparametric (GP-based) systematics model.","We chose the most conservative approach as the final analysis detailed in this paper: a DW approach accompanied with a flexible Gaussian process systematics model where the GP hyperparameters are marginalized over in the posterior sampling process.","This was motivated by significant nightly variations in the transmission spectrum obtained using the parametric systematics model.","Despite the nightly variations in the DWW analysis, the joint analyses are consistent with each other: the transmission spectrum is flat within uncertainties.","Especially, we do not detect significant Na or K absorption, and, our results do not agree with the detection of potassium by [35].","The absence of significant features does not justify detailed atmospheric modelling.","However, basic Exo-Transmit modelling favours a truly flat spectrum over atmosphere models with stellar or sub-stellar metallicity and K and Na.","Ground-based observations are prone to complex systematics, and transmission spectroscopy is carried out at the limits of what the instruments are capable of, or were designed for.","The most robust approach for ground-based transmission spectroscopy should try to account for this by repeating the observations several times, preferably with different instruments covering the same wavelength range.","Repeated observations do not only improve the final precision that can be reached, but also our capability to decouple the systematics from the minute transit depth variations.","We thank the anonymous referee for their constructive and useful comments.","HP has received support from the Leverhulme Research Project grant RPG-2012-661.","FM acknowledges the support of the French Agence Nationale de la Recherche (ANR), under the program ANR-12-BS05-0012 Exo-atmos.","The work has been supported by the Spanish MINECO grants ESP2013-48391-C4-2-R and ESP2014-57495-C2-1-R. GC acknowledges the support by the National Natural Science Foundation of China (Grant No.","11503088) and the Natural Science Foundation of Jiangsu Province (Grant No.","BK20151051).","Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma.","The authors wish to acknowledge the contribution of Teide High-Performance Computing facilities to the results of this research.","TeideHPC facilities are provided by the Instituto Tecnológico y de Energías Renovables (ITER, SA).","The authors wish to acknowledge the contribution of Glamdring computing cluster in the Subdepartment of Astrophysics, Department of Physics, University of Oxford, to the results of this research."],["DW model fit and residuals","Figures REF and REF show the DWR analysis model fit and residuals, respectively.","Figure: The NB light curves for both observing runs and the DWR approach posterior median model(black line) as a function of the phase.","Passband centres are marked in nanometres.Figure: The NB residuals from the DWR analysis.","Passband centres are marked in nanometres."]],"string":"[\n [\n \"The GTC exoplanet transit spectroscopy survey VIII. Flat transmission\\n spectrum for the warm gas giant WASP-80\"\n ],\n [\n \"Abstract We set out to study the atmosphere of WASP-80b, a warm inflated gas giant with an equilibrium temperature of $\\\\sim$800~K, using ground-based transmission spectroscopy covering the spectral range from 520~to~910~nm.\",\n \"The observations allow us to probe the existence and abundance of K and Na in WASP-80b's atmosphere, existence of high-altitude clouds, and Rayleigh-scattering in the blue end of the spectrum.\",\n \"We observed two spectroscopic time series of WASP-80b transits with the OSIRIS spectrograph installed in the Gran Telescopio CANARIAS, and use the observations to estimate the planet's transmission spectrum between 520~nm and 910~nm in 20~nm-wide passbands, and around the K~I and Na~I resonance doublets in 6~nm-wide passbands.\",\n \"We model three previously published broadband datasets consisting of 27 light curves jointly prior to the transmission spectroscopy analysis in order to obtain improved prior estimates for the planet's orbital parameters, average radius ratio, and stellar density.\",\n \"We recover a flat transmission spectrum with no evidence of Rayleigh scattering or K~I or Na~I absorption, and obtain an improved system characterisation as a by-product of the broadband- and GTC-dataset modelling.\",\n \"The transmission spectra estimated separately from the two observing runs are consistent with each other, as are the transmission spectra estimated using either a parametric or nonparametric systematics models.\",\n \"The flat transmission spectrum favours an atmosphere model with high-altitude clouds over cloud-free models with stellar or sub-stellar metallicities.\"\n ],\n [\n \"Introduction\",\n \"Transmission spectroscopy allows us to probe the existence and abundance of atmospheric species in the atmospheres of transiting extrasolar planets [34], [5].\",\n \"The method requires a high observing precision, which has made the space-based studies most successful in finding significant features in the transmission spectra [7], [36], [13], but the developments in instrumentation, observing techniques, and data analysis methods have also enabled transmission spectroscopy studies to be carried out successfully using ground-based telescopes.\",\n \"The signal of interest – variations in the effective planetary radius as a function of wavelength – is minute, corresponding to changes of $\\\\sim 0.01\\\\%$ in the observed transit depth and $\\\\sim 0.1\\\\%$ in the effective planet-star radius ratio.\",\n \"Further complications arise from possible high-altitude clouds, which can mask any atmospheric extinction features, leading to a flat transmission spectrum [21], [4], and from the fact that atmospheric extinction is not the only source of wavelength-dependent features in transmission spectra.\",\n \"Both instrumental and astrophysical sources, such as host star's spots [2] and plages [26], flux contamination from a possible unresolved source, and incorrectly accounted for stellar limb darkening, can all imprint features that can be difficult to disentangle from the atmospheric signal.\",\n \"Notwithstanding the complications, ground-based transmission spectroscopy has been used successfully to identify features attributed to absorption in planetary atmospheres.\",\n \"Simultaneous measurements of the target star and several comparison stars – a process similar to relative photometry [3], [14] – the use of Gaussian processes have facilitated the robust modelling of systematics [15], [32], [31], and the use of Bayesian inference methods has allowed for realistic uncertainty estimation that is crucial when assessing the true significance of the identified transmission spectrum features.\",\n \"We report a ground-based transmission spectroscopy study of WASP-80b [37].\",\n \"We have observed spectroscopic time series of two WASP-80b transits with the OSIRIS spectrograph [33] installed in the 10.4 m Gran Telescopio CANARIAS (GTC) on La Palma, Spain.\",\n \"The observations cover the spectral range from 520 to 910 nm, probing the planet atmosphere for a possible Rayleigh scattering signal in the blue end of the spectrum, and the visible-light extinction features of the K I and Na I resonance doublets at 767 nm and 589.4 nm, respectively.\",\n \"WASP-80b [37], [22], [12], [38], a warm gas giant orbiting a bright (V=11.87) late-K / early-M dwarf on a 3.07 d orbit, was identified as a promising target for transmission spectroscopy from its discovery.\",\n \"The planet has a low surface gravity ($M_\\\\mathrm {p} = 0.56\\\\;\\\\mathrm {M_{Jup}} $ , $R_\\\\mathrm {p} = 0.99 \\\\mathrm {R_{Jup}} $ , $g = 14.34$ ms$^{-2}$ , [22]), and its large radius ratio leads to $\\\\sim $ 3% deep transits, which, combined with the brightness of its host star, enhance our abilities to detect any possible transmission spectrum features.\",\n \"WASP-80b is a warm gas giant with an equilibrium temperature $\\\\sim $ 800 K [37], [22].\",\n \"This very likely places the planet into the pL class (no temperature inversion) in the classification by [11].\",\n \"The main spectroscopic features in the visible passband for pL class planets are expected to be from Rayleigh scattering and K I and Na I resonance doublet absorption, of which K I absorption detection was recently claimed by [35] based on transmission spectroscopy analysis carried out with the FORS2 spectrograph installed in the VLT.\",\n \"Table: Identifiers for WASP-80 with its coordinates and magnitudes (SIMBAD, retrieved 2017-07-04).Our study consists of two main analyses: joint analysis of 27 previously published broadband transit light curves observed in 7 passbands (broadband dataset), joint analysis of the two GTC-observed spectroscopic transit time series (transmission spectroscopy dataset), which both consist of a set of analyses with different prior assumptions and modelling approaches carried out to ensure the robustness of the final results.\",\n \"The broadband dataset analysis is carried out to obtain improved estimates for the planet's broadband radius ratios, orbital parameters, and stellar density (system parameters).\",\n \"The marginal parameter posteriors from the broadband dataset analysis are then used as priors in the GTC-data analysis.\",\n \"The GTC transmission spectroscopy starts with a direct-modelling analysis with a flexible Gaussian-process-based systematics model to further constrain the system parameters, and the final transmission spectroscopy uses a divide-by-white approach with either a parametric or nonparametric residual systematics model.\",\n \"This paper is divided roughly into three sections.\",\n \"We outline the numerical methods and the generic equations for the calculation of posterior probability densities in §.\",\n \"We continue in § by carrying out a detailed joint modelling of three priorly observed broadband datasets described in [37], [22], and [12].\",\n \"The datasets cover 27 transit light curves observed in $g^{\\\\prime }$ , $r^{\\\\prime }$ , $i^{\\\\prime }$ , $I$ , $z^{\\\\prime }$ , $J$ , $H$ , and $K$ .\",\n \"We describe the GTC observations and data reduction in § , detail the analysis in §, present the transmission spectroscopy results in §, and discuss the results in §.\",\n \"Finally, we conclude the paper in §.\",\n \"The raw data are publicly available from Zenodo and GTC data archive, and the whole analysis with reduced data is available from GitHub github.com/hpparvi/Parviainen-2017-WASP-80b as an easy-to-follow set of IPython notebooks and Python codes to help with the reproducibility of the study.\",\n \"We use a fully Bayesian approach to transmission spectroscopy: our parameter estimates are based on the marginal posterior densities derived from a joint posterior density estimated with Markov Chain Monte Carlo (MCMC) sampling, and the marginal parameter posteriors from the broadband dataset analysis are used as priors in the GTC transmission spectroscopy analysis.\",\n \"Our datasets consist of light curves observed either photometrically, or constructed from spectroscopic observations.\",\n \"A light curve is modelled as a product of a baseline and transit model, where the combined model is parametrised with a parameter vector $\\\\theta $ .\",\n \"When modelling multiple light curves jointly, the parameter vector is divided into parameters shared between all the light curves, passband-specific parameters shared between light curves observed in the same passband, observing-run-specific parameters, and light-curve-specific parameters.\",\n \"Especially, the parameters defining the planetary orbit (zero epoch, orbital period, impact parameter, and stellar density) are shared between all the light curves included into the analysis, and thus the likelihoods from all the light curves contribute to the parameter posteriors.\",\n \"The planet-star radius ratios and stellar limb darkening coefficients are considered passband-dependent, but observing-run-independent parameters (although this is not strictly true, since spots and plages, whether occulted or not, have an effect on the transit depths).\",\n \"That is, the light curves observed in a given passband all contribute to the radius ratio and limb darkening posteriors for the passband.\",\n \"Finally, the parameters defining the baseline and noise properties are considered either observing-run- or light-curve-specific, i.e., they depend both on the passband and observing run (this depends on the specific modelling approach, as detailed later).\",\n \"Joint modelling leads to relatively high-dimensional models.\",\n \"The number of free parameters in the analyses presented here varies from 10 to $\\\\sim $ 250.\",\n \"However, the approach allows us to utilise the data fully.\",\n \"Simultaneous modelling of different observing runs reduces our sensitivity on systematics, and simultaneous multiband analysis reduces the degeneracies between the estimated radius ratios, orbital impact parameter, and stellar limb darkening.\",\n \"The parameter posteriors are estimated as a two-step process.\",\n \"First, a population-based global optimisation method (Differential evolution implemented in PyDE) is used to obtain a parameter vector population that is clumped close to the global posterior maximum.\",\n \"The parameter vector population is then used to initialise the emcee Markov Chain Monte Carlo (MCMC) sampler [10], [16], which is used to create a sample of parameter vectors drawn from the model posterior (see the analysis-specific sections for practical details).\",\n \"The transit model uses the quadratic limb darkening formalism by [23], and is calculated using PyTransit [27].\",\n \"PyTransit contains optimisations to compute a transit in multiple passbands with only a minor additional computational cost to the computation of a single passband transit, which reduces the computational burden due to the joint modelling approach.\",\n \"The analyses have been carried out both with and without LDTk-based constraints on the stellar limb darkening.\",\n \"Quadratic limb darkening is parametrised using the parametrisation presented in [20], which is aimed for efficient sampling of the physically allowed limb darkening coefficient space.\",\n \"When the limb darkening is not constrained, we marginalise over the whole limb darkening parameter space allowed by the data.\",\n \"When the limb darkening is constrained, it is done by fitting the observational data jointly with the limb darkening profiles created using the LDTk-package [28], and by marginalising over the limb darkening coefficients allowed by the stellar density profiles.\",\n \"LDTk uses PHOENIX-calculated stellar atmosphere library by [18] to construct limb darkening profiles with the uncertainties in the stellar properties propagated into the uncertainties in the limb darkening profiles.\",\n \"We also repeat the analyses for different systematics-modelling approaches, for separate subsets of data, and with synthethic mock data, and using the target star alone without dividing by the comparison star, to test the reliability of our approach.\",\n \"Unless otherwise specified, the parameter point estimates correspond to posterior medians, and the uncertainties correspond to the central 68% posterior intervals.\",\n \"We do not plot point parameter estimates (these are listed in tables), but prefer to show either the posterior distributions or limits based on central posterior intervals.\",\n \"The analyses rely on Python- and Fortran-based code utilising SciPy, NumPy [39], IPython [29], Pandas [24], matplotlib [17], seaborn,$\\\\!$http://stanford.edu/~mwaskom/software/seaborn PyFITS,$\\\\!$PyFITS is a product of the Space Telescope Science Institute, which is operated by AURA for NASA and F2PY [30].\",\n \"The transits were modelled with PyTransit Freely available from https://github.com/hpparvi/PyTransit [27], the limb darkening computations were carried out with LDTk $\\\\!$Available from https://github.com/hpparvi/ldtk [28], global optimisation was carried out with PyDE,$\\\\!$Available from https://github.com/hpparvi/PyDE the MCMC sampling was carried out with emcee [10], [16], and the Gaussian processes were computed using GeorgeAvailable from https://dan.iel.fm/george [1].\"\n ],\n [\n \"Posteriors and likelihoods\",\n \"The unnormalised log posterior density for a dataset consisting of $n_\\\\mathrm {lc}$ light curves observed in $n_\\\\mathrm {pb}$ passbands is $\\\\ln P(\\\\theta |D) = \\\\ln P(\\\\theta ) + \\\\sum _i^{n_\\\\mathrm {lc}} \\\\ln P(\\\\vec{D_{\\\\mathrm {LC}}} _{,i}|\\\\theta ) +\\\\sum _i^{n_\\\\mathrm {pb}} \\\\ln P(\\\\vec{D_{\\\\mathrm {LD}}} _{,i}|\\\\theta ),$ where $\\\\theta $ is the parameter vector encapsulating all the model parameters, $\\\\ln P(\\\\theta )$ is the log prior, $\\\\vec{D_{\\\\mathrm {LC}}} $ are the light curves, $\\\\ln P(\\\\vec{D_{\\\\mathrm {LC}}} |\\\\theta )$ is the log likelihood for the photometry, $\\\\vec{D_{\\\\mathrm {LD}}} $ are the theoretical limb darkening profiles calculated by LDTk, and $\\\\ln P(\\\\vec{D_{\\\\mathrm {LD}}} |\\\\theta )$ is the log likelihood for the limb darkening profile.\",\n \"Assuming that the uncertainties (noise) in the observations are normally distributed, we can write the log likelihood for the data $\\\\vec{D}$ in vector form as $\\\\ln P(\\\\vec{D}|\\\\theta ) = -\\\\frac{1}{2} \\\\left( n_\\\\mathrm {D} \\\\ln 2\\\\pi +\\\\ln |\\\\Sigma | +\\\\vec{r}^\\\\mathrm {T}\\\\Sigma ^{-1}\\\\vec{r}\\\\right),$ where $n_\\\\mathrm {D}$ is the number of datapoints, $\\\\vec{r}$ is the residual vector, and $\\\\Sigma $ is the covariance matrix.\",\n \"If the noise is white (uncorrelated), the covariance matrix is diagonal, and the likelihood can be written out explicitly in scalar form as $\\\\ln P(\\\\vec{D}|\\\\theta ) = -\\\\frac{1}{2}\\\\left(n_\\\\mathrm {D}\\\\ln 2\\\\pi +\\\\sum _j^{n_\\\\mathrm {D}} \\\\ln \\\\sigma _{\\\\mathrm {j}}^2 + \\\\sum _{j=1}^{n_\\\\mathrm {D}}\\\\frac{\\\\vec{r}^2}{2\\\\sigma _{\\\\mathrm {j}}^2} \\\\right),$ where $\\\\sigma _{\\\\mathrm {j}}$ is the uncertainty of the $j$ th datapoint.\",\n \"If the per-point uncertainty does not vary significantly, this equation can be simplified further into $\\\\ln P(\\\\vec{D}|\\\\theta ) = -\\\\frac{n_\\\\mathrm {D}}{2} \\\\ln 2\\\\pi \\\\sigma _{\\\\mathrm {i}}^2 -\\\\frac{1}{2} \\\\sum _{j=1}^{n_\\\\mathrm {D}}\\\\frac{\\\\vec{r}^2}{2\\\\sigma _{\\\\mathrm {j}}^2}.$ If the noise (here used to describe the leftover variation not explained by the parametric transit and baseline models) is not white, the covariance matrix will have off-diagonal elements, and the matrix needs to be inverted for the likelihood evaluation.\",\n \"This is the case when the noise is presented as a Gaussian process [31], [15], [32].\",\n \"The covariance matrix $\\\\Sigma $ in Eq.\",\n \"(REF ) is now $\\\\Sigma = \\\\vec{K}(\\\\vec{x},\\\\vec{x}) + \\\\sigma ^2\\\\vec{I},$ where $\\\\vec{K}(\\\\vec{x},\\\\vec{x})$ is defined by a covariance function (also known as a covariance kernel).\",\n \"We describe the likelihood and covariance functions separately for the broadband dataset analysis and GTC transmission spectroscopy in Sects.\",\n \"and , respectively, since the two analyses use slightly different modelling approaches.\"\n ],\n [\n \"Overview\",\n \"We estimate the posterior densities for the WASP-80b orbital parameters based on the three datasets observed by [22], [37], and [12], abbreviated from herein as M14, T13, and F14, respectively.\",\n \"The datasets contain 27 light curves (Fig.\",\n \"REF ) observed in $g^{\\\\prime }$ , $r^{\\\\prime }$ , $i^{\\\\prime }$ , $I$ , $z^{\\\\prime }$ , $J$ , $H$ , and $K$ .\",\n \"We model all the light curves jointly, and, in contrast to [22], include the M14 GROND J, H, and K light curves.\",\n \"We also simplify the analysis slightly by merging the $I$ and $i^{\\\\prime }$ passbands.\",\n \"The analysis is carried out assuming either a parametric or nonparametric systematics model (white or red noise), constant or wavelength-dependent radius ratio, and with and without LDTk to constrain the stellar limb darkening, which leads to eight separate analysis sets defined in Table REF .\",\n \"The T13 and M14 datasets, as obtained from VizieR, were detrended by their original authors, and do not include other information than the observation time, flux, and error estimates.\",\n \"The F14 dataset was kindly provided by the author without detrending, and included all the auxiliary information (such as the airmass, and x- and y-centroid shifts for each exposure) used in the original analysis presented in [12].\",\n \"Table: Broadband analysis runs for the external datasets.\",\n \"The white-noise runs include only the and datasets, while the red-noise runs also include the light curvesby .\",\n \"The constant and varying radius ratios mark whether the radius ratiowas allowed to vary from passband to passband, or whether it was assumed to be wavelength-independent.Table: Marginal posterior medians from the external data broadband analysis (run ckrn_ldtkwith systematics modelled using GPs, wavelength-independent radius ratio, and limb darkeningconstrained with LDTk).\",\n \"The uncertainties correspond to the central 68% posterior intervals.Table: Broadband radius ratio estimates and their uncertainties from the vkrn_ldtk run.We used the T13 and M14 datasets for an initial modelling with a parametric systematics model and white additive noise, and include the F14 dataset in the final runs using nonparametric systematics model.\",\n \"The nonparametric systematics model represents the systematics (and white noise) as a Gaussian process (GP) with time as the only covariate for the T13 and M14 datasets, and with time, airmass, x-shift, and y-shift as covariates for the F14 dataset.\",\n \"We do not marginalise over the GP hyperparameters in the broadband data analysis, but fix them to the values fitted from the white-noise run residuals.\",\n \"As described in Sect.\",\n \"REF , the parameter estimation starts with a parameter vector population that fills the prior space uniformly.\",\n \"A differential evolution (DE) optimisation is used to clump the population close to the global posterior maximum (the number of DE iterations depending slightly on the run, but is usually close to 1000), after which MCMC sampling is carried out using emcee.\",\n \"The sampler is run for 10$\\\\,$ 000 iterations, which yields 12000 independent posterior samples when using a population size of 150, thinning factor of 100, and burn-in period of 2000 iterations, where the thinning factor and burn-in period were chosen by studying the parameter chain populations and the average parameter autocorrelation lengths.\"\n ],\n [\n \"Log posterior and likelihoods\",\n \"The log posterior for the combined broadband dataset given a parameter vector $\\\\theta $ is $\\\\ln P(\\\\theta |D) &= \\\\ln P(\\\\theta ) \\\\nonumber \\\\\\\\&+ \\\\ln P(D_\\\\mathrm {T13}|\\\\theta ) + \\\\ln P(D_\\\\mathrm {M14}|\\\\theta ) +\\\\ln P(D_\\\\mathrm {F14}|\\\\theta ) \\\\\\\\&+ \\\\sum _i^{n_\\\\mathrm {pb}} \\\\ln P(\\\\vec{D_{\\\\mathrm {LD}}} _{,i}|\\\\theta ), \\\\nonumber $ where the first term is the log prior, followed by the per-dataset log likelihoods, and the last term is the sum of the LDTk-calculated log likelihoods for the limb darkening coefficients for each passband.\",\n \"The exact form of the three likelihoods follows either Eq.\",\n \"(REF ) or Eq.\",\n \"(REF ), depending on the chosen systematics model.\",\n \"The parametric (white noise) model assumes a constant average per-light-curve uncertainty (that is, the observation noise is the same for all the datapoints in a single light curve), and the kernels for the GP model are detailed below.\",\n \"The light curve model consists of a product of a baseline function and a transit model with quadratic limb darkening calculated using PyTransit.\",\n \"The residual vector for a single light curve in Eq.\",\n \"(REF ) is $\\\\vec{r} = \\\\vec{f}_o - \\\\vec{f}_m(\\\\vec{X}, \\\\theta ) = \\\\vec{f}_o - \\\\mathcal {B}(\\\\vec{X}, \\\\theta ) \\\\,\\\\mathcal {T}(\\\\vec{t}, \\\\theta )$ where $\\\\vec{f}_o$ is the observed flux, $\\\\vec{f}_m$ the modelled flux, $\\\\mathcal {B}$ the baseline model, $\\\\mathcal {T}$ the transit model, $\\\\vec{X}$ is a matrix containing the the input parameters (covariates), $\\\\vec{t}$ are the mid-exposure times, and $\\\\theta $ is the model parameter vector.\"\n ],\n [\n \"Gaussian process kernels\",\n \"The T13 and M14 datasets do not include other auxiliary information than the mid-exposure times.\",\n \"Thus, we model the noise as a Gaussian process with time as the only covariate, the kernel being a sum of an exponential kernel and a white noise term $k = A_t \\\\exp \\\\left[ - \\\\eta _t (t_i-t_j) \\\\right] + \\\\sigma ^2 \\\\delta _{ij},$ where $t$ are the time values, $A_t$ is the GP output amplitude, $\\\\eta _t$ the inverse time scale, $\\\\sigma $ the average white noise standard deviation, and $\\\\delta $ the Kronecker delta.\",\n \"The F14 dataset includes also airmass, and per-observation x- and y-centroid estimates.\",\n \"This allows us to use a slightly more complex kernel, where we have an exponential time component, squared exponential airmass component, squared exponential PSF-centroid component, and a white noise term.\",\n \"The kernel becomes $k &= A_t \\\\exp \\\\left[-\\\\eta _t (t_i-t_j) \\\\right] \\\\nonumber \\\\\\\\& + A_a \\\\exp \\\\left[-\\\\eta _a (a_i-a_j)^2 \\\\right] \\\\nonumber \\\\\\\\& + A_{xy} \\\\exp \\\\left[-\\\\eta _x (x_i-x_j)^2 -\\\\eta _y (y_i-y_j)^2 \\\\right] \\\\nonumber \\\\\\\\& + \\\\sigma ^2\\\\delta _{ij},$ where $t$ , $a$ , $x$ , and $y$ are the time, airmass, x, and y estimates, respectively; $A_t$ , $A_a$ , $A_{xy}$ are the GP time, airmass and xy output amplitudes; and $\\\\eta _t$ , $\\\\eta _a$ , $\\\\eta _x$ , and $\\\\eta _y$ the time, airmass, x and y inverse time scales.\"\n ],\n [\n \"Results\",\n \"We adopt the ckrn_ldtk (passband-independent radius ratio, GP systematics, and limb darkening constrained using the LDTk) run as our final analysis, and report the stellar, orbital, and planetary parameter estimates in Tables REF and REF .\",\n \"The posterior distributions are all close-to normal, as shown in Figs.\",\n \"REF and REF .\",\n \"The parameter estimates agree with the previous WASP-80b studies.\",\n \"The broadband transmission spectrum from the vkrn_ldtk run, shown in Fig.\",\n \"REF is consistent with a flat line.\",\n \"The result agrees with a previous broadband transmission spectrum analysis by [38].\",\n \"We used the ExoTransmit transmission spectrum modelling package to test different atmosphere scenarios, but the precision in radius ratios is not sufficient to meaningfully distinguish between any of the physically plausible scenarios.\",\n \"Figure: Marginal- and joint-posteriors for the radius ratio, stellar density, and impactparameter corresponding to the final ckrn_ldtk run.\",\n \"The 68% centralinterval is marked with a darker shade.Figure: Broadband radius ratio posteriors estimated by jointly modelling the three priordatasets.\",\n \"The results correspond to the final vkrn_ldtk run with rednoise modelled using GPs and limb darkening constrained using LDTk.Figure: Broadband quadratic limb darkening coefficient posteriors.\",\n \"The results correspond to thefinal ckrn runs with red noise modelled with GPs and limb darkening eitherunconstrained (left) or constrained using LDTk (right).The limb darkening coefficient posteriors are plotted in Fig.\",\n \"REF , both with and without constraints from LDTk.\",\n \"The two versions agree with each other within uncertainties.\",\n \"The observed light curve, conditional model distribution (for the red noise model), and the residuals are shown in Fig.\",\n \"REF .\",\n \"Figure: The 27 broadband light curves observed by ,, and after the removal of the mean GP trend, theposterior model medians, and the residuals.\"\n ],\n [\n \"Observations\",\n \"We observed two WASP-80b transits simultaneously with one comparison star using the OSIRIS (Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy) spectrograph installed in the GTC (Gran Telescopio CANARIAS) on the nights starting 16 July 2014 and 25 August 2014 (observing runs R1 and R2 respectively).\",\n \"The R1 observations carried from 23:30 UT till 3:41 UT and the R2 observations from 20:35 UT till 1:00 UT.\",\n \"The observing conditions were good, with a seeing of $\\\\sim $ 0.8 in the beginning of each night, and all the observations were carried out with an airmass smaller than 1.4.\",\n \"OSIRIS [6] contains two Marconi CCD42-82 2048$\\\\times $ 4096 pixel CCDs, which were used in the standard 2$\\\\times $ 2 binning mode yielding a plate scale of 0.254.\",\n \"The observations were carried out using grism R1000R with a 40-wide custom-designed slit that aims to minimise the systematics related to variations in flux loss.\",\n \"We chose TYC 5165-00235-1 as the comparison star, located at a distance of 6.9 from WASP-80.\",\n \"The star has a similar V magnitude ($V=11.62$ ), but is slightly redder ($J=8.376$ ).\",\n \"The two stars were positioned equidistantly from the optical axis close to the centre of each CCD.\",\n \"The slit does not include other stars bright enough to be useful in the analysis.\",\n \"The exposure time was 6 s for R1 and 5 s for R2.\",\n \"While the exposure time was short, it was long enough for the comparison star to saturate during the final parts of the WASP-80b ingress during R1.\",\n \"This was fixed by defocusing the instrument, but meant that the comparison star cannot be used in the reduction during this saturated period.\",\n \"For R1, 81 flat fields were taken before the transit, and 66 bias frames after the transit.\",\n \"For R2, 100 flat fields were taken after the transit, and 15 bias frames before the transit.\",\n \"Three arc frames (Xe, HgAr, and Ne) for R1 were observed on 14 July and for R2 on the same night as the observations, after the transit.\"\n ],\n [\n \"Data reduction and passband sets\",\n \"The passband-integrated light curves are produced from the raw data using the pipeline described in [9], [8].\",\n \"The pipeline carries out the basic CCD data reduction steps, calculates a 2D wavelength solution, removes the sky, and generates a reduced 1D spectrum for WASP-80 and the comparison star (Fig.\",\n \"REF ).\",\n \"The light curves are then generated by integrating the spectra multiplied by a transmission function defining the passband.\",\n \"We created four sets of passbands (datasets) listed in Table REF .\",\n \"The W dataset consists of a single light curve covering the whole usable spectrum (white light curve), the NB (narrow-band dataset) covers the whole usable spectrum in 20 nm-wide bins, and the Na I and K I datasets cover the Na and K lines, respectively, in 6 nm-wide bins.\",\n \"The final white-light WASP-80 and reference star lightcurves with the relative light curve are shown in Fig.\",\n \"REF .\",\n \"The white-light white noise estimates for R1 and R2 are 400 and 520 ppm, respectively.\",\n \"For the NB narrow-band dataset, the white noise level varies from 600 to 2100 ppm, with a median level of 720 and 870 ppm for R1 and R2, respectively.\",\n \"The difference in the white noise levels is expected due to the different exposure times used in R1 and R2 (6 and 5 s, respectively).\",\n \"Figure: Normalised, sky-subtracted, and wavelength-calibrated spectra for WASP-80b (dark blueline) and the comparison star (orange line).Figure: White-light light curves for the comparison star (top), WASP-80 (middle), and therelative light curve (bottom) for both observing runs.\",\n \"A small number of clear outliers have beenomitted for clarity.Table: Passband sets extracted from the GTC spectroscopy.\"\n ],\n [\n \"Systematics\",\n \"The relative light curves in Fig.\",\n \"REF feature systematics not corrected by division by the comparison star.\",\n \"Specially, the baseline is affected by a smooth trend that cannot be modelled by a simple parametric model as a function of time (or any of the simultaneously measured auxiliary variables).\",\n \"The trend causes a pre- and post-transit baseline difference of $\\\\sim 2\\\\%$ during run 1, and a $\\\\sim 1\\\\%$ difference during run 2.\",\n \"A similar, smooth, nearly linear, variation as a function of the rotator angle accompanied by relatively smooth 'bumps', has been observed in previous OSIRIS transmission spectroscopy studies, and is likely caused by vignetting in the telescope pupil space [25].\",\n \"Fortunately, the systematics are mainly common-mode, without significant variation across the spectrum.\",\n \"Common-mode systematics can be removed by either fitting the white-light light curves with a flexible GP, which can be used to create a common-mode systematics model, or by using a divide-by-white approach.\",\n \"After the common-mode correction, the residual wavelength-dependent systematics can be accounted for either with parametric or non-parametric approaches, both of which were used in our analyses.\"\n ],\n [\n \"Overview\",\n \"The GTC transmission spectroscopy covers the modelling of the white light curves and the three narrowband datasets described in Sec.\",\n \"REF .\",\n \"We used three modelling approaches: DIR direct modelling with flexible GP-based systematics DWW divide-by-white with parametric systematics DWR divide-by-white with GP-based systematics The first approach, direct modelling of the light curves with a flexible GP-based systematics model, was used to obtain a model for the common-mode systematics (from the white light curves), and to improve the system characterisation (from the full-spectrum 20 nm-wide dataset).\",\n \"The divide-by-white (DW) approaches were then used for transmission spectroscopy, with the direct-modelling posteriors used as priors on the wavelength-independent system parameters (the motivation for this is discussed later).\",\n \"The analyses were repeated with and without LDTk to assess how sensitive the results are to assumptions about limb darkening, and modelling the two nights separately and jointly, to test whether the results are consistent from night to night.\",\n \"The parameter posteriors are estimated in similar fashion to the prior data analysis in Sec.\",\n \"REF .\",\n \"Differential evolution is used to create a parameter vector population clumped around the global posterior maximum, which used to initialise the emcee sampler.\",\n \"The sampler is ran over $10\\\\times 15\\\\;000$ iterations (that is, ten sets of $15\\\\;000$ iterations where each set is initialised from the final state of the previous set) with 600 chains and a thinning factor of 100, which gives us a final set of 75$\\\\;$ 000 independent posterior samples.\",\n \"The number of iterations, chains, and the thinning factor were chosen after studying the chain populations and per-parameter autocorrelation lengths.\",\n \"Unlike in the broadband dataset analysis, we keep the GP hyperparameters free and marginalize over them.\",\n \"This slows down the sampling process, but yields more reliable posteriors, and allows us to study how well the GP kernels represent the systematics.\",\n \"The direct-modelling approach DIR reproduces the light curves as a product of a baseline and a transit model directly.\",\n \"We model all the passbands for both nights in a dataset jointly, which leads to a slightly involved parameterisation, but aims to utilise the data fully.\",\n \"As mentioned earlier, the model parameters can be divided into four categories: a) passband- and baseline-independent parameters, b) achromatic per-night baseline parameters, c) chromatic per-light-curve parameters, and d) chromatic parameters that should stay constant between observation runs.\",\n \"The direct model parametrisation and the parameter priors are listed in Table REF .\",\n \"Priors for the zero epoch, orbital period, impact parameter, and stellar density are based on the broadband dataset analysis posteriors (run ckrn_ldtk).\",\n \"The $u$ and $v$ limb darkening coefficients correspond to the [20] quadratic limb darkening model parametrisation, where uniform priors from 0 to 1 lead to uninformative priors covering the physically viable values for the quadratic limb darkening priors.\",\n \"The GP hyperparameters can be chosen to be night- or light-curve-dependent, but we choose to keep them independent for practical reasons.\",\n \"Our approach adds three GP hyperparameters to the analysis, and requires two covariance matrix inversions per posterior evaluation (one per night).\",\n \"Making the GP hyperparameters light-curve-dependent would add three free parameters to the model for each light curve, and require a covariance matrix inversion for each light curve, which would make marginalisation over the GP hyperparameters costly.\",\n \"Making the GP hyperparameters night-dependent would be feasible, since the approach would add only three more parameters to the model, and would not require additional covariance matrix inversions.\",\n \"However, our analyses for separate nights result with compatible GP hyperparameter posteriors, and we choose the simplest approach.\",\n \"Table: Model parametrisations and priors.\",\n \"𝒰(a,b)\\\\mathcal {U}(a,b) stands for a uniform prior from a to b, where a andb are omitted when the range is chosen to be wide enough not to affect the posteriors.\",\n \"𝒩(μ,σ)\\\\mathcal {N}(\\\\mu ,\\\\sigma )stands for a normal prior with mean μ\\\\mu and standard deviation σ\\\\sigma .\",\n \"The system parametershave normal priors with means and standard deviations corresponding to the values shown in Table .The direct model represents each light curve as a Gaussian process $f \\\\sim \\\\mathcal {N}(c_b\\\\;\\\\mathcal {T}(\\\\vec{t}, \\\\theta ), \\\\Sigma ),$ where $c_b$ is a baseline constant, $\\\\mathcal {T}$ is the transit model, and $\\\\Sigma $ is the covariance matrix.\",\n \"We use airmass $x$ , and telescope rotator angle, $\\\\alpha $ , as GP input parameters (covariates), with a covariance matrix defined as a sum of a linear kernel and squared-exponential kernel, $\\\\Sigma (i,j) = \\\\left( \\\\frac{x_j \\\\cdot x_i}{\\\\gamma _x}\\\\right) + a_\\\\alpha ^2 \\\\exp \\\\left( \\\\frac{(\\\\alpha _j - \\\\alpha _i)^2}{\\\\gamma _\\\\alpha }\\\\right) + \\\\delta _{ij} \\\\sigma $ where $a_\\\\alpha $ is an output scale parameter, $\\\\gamma _x$ and $\\\\gamma _\\\\alpha $ are the input scale parameters, and $\\\\sigma $ is the average white noise.\",\n \"Any residual airmass-dependent systematics should be approximately linear, and can be modelled with a linear kernel, while the possible chromatic rotator-angle dependencies are expected to be smooth, and well-modelled by a squared-exponential kernel.\",\n \"We also tested a GP with time as a covariate (with a Matern kernel), but it did not affect the posteriors significantly.\",\n \"Also, the power spectral density (PSD) of the residuals with the transit and the two-covariate GP mean removed is approximately constant, which suggests that the noise is white after the residual airmass and rotator-angle dependencies are accounted for.\",\n \"The white noise level does not vary significantly from passband to passband in our datasets, and we choose to use a single average white noise estimate for an observing run (again, so that we do not need to invert the covariance matrix separately for each light curve).\",\n \"The white noise estimate is an average of the estimates calculated separately for each light curve, which are calculated as $\\\\sigma = \\\\mathrm {std}(\\\\vec{f}_d) / \\\\sqrt{2}$ where $\\\\vec{f}_d = f_i - f_{i-1}$ .\",\n \"That is, we estimate the white noise from the standard deviation of the light curve differentials.\",\n \"The radius ratios and limb darkening coefficients yield three free parameters per passband, and the baseline constant a further free parameter per light curve.\",\n \"In total, the number of free parameters for a direct model reaches 108 for the 20-nm run with 20 passbands.\",\n \"The computations can still be carried out with a modern laptop in a matter of hoursPyTransit, which was used to compute the light curves, is optimised to compute a set of multiband light curves with only a minor additional cost to calculating a single passband.\",\n \"for a single analysisHowever, while a single analysis can be carried out in a laptop, the Glamdring-cluster in Oxford University and the TeideHPC supercomputer in Spain were used to carry out the final analyses., but tests must be carried out to ensure that the final posterior sample gives a reliable representation of the true posterior distribution.\"\n ],\n [\n \"Divide-by-white model with and without GP systematics\",\n \"The divide-by-white (DW) approachOur approach would be more accurately named as divide-by-average, since we're dividing each dataset by the dataset mean.\",\n \"allows us to remove any common-mode systematics from the dataset with the cost of increased per-passband white noise.\",\n \"The residual vector $\\\\vec{r}$ in Eq.\",\n \"(REF ) for the DW approach is $\\\\vec{r} = \\\\frac{1}{\\\\vec{b}(\\\\theta , \\\\mathbf {X})} \\\\left( \\\\frac{\\\\vec{f}}{1/n_\\\\mathrm {pb} \\\\sum \\\\vec{f}} - \\\\frac{\\\\mathcal {T} (\\\\vec{t}, \\\\theta )}{1/n_\\\\mathrm {pb} \\\\sum \\\\mathcal {T} (\\\\vec{t}, \\\\theta )} \\\\right), $ where $\\\\mathbf {f}$ is the observed flux vector, $\\\\mathcal {T}$ is the transit model, and $\\\\vec{b}$ is the residual baseline model.\",\n \"That is, we divide both the observed and modelled fluxes by the values averaged over all the passbands in the dataset.\",\n \"The DW approach does not only remove the common-mode systematics, but all colour-independent signals.\",\n \"Effectively, the signals left in the data are due to colour-dependent systematics, changes in the radius radio, and changes in stellar limb darkening.\",\n \"The system parameters, such as the orbital period and impact parameter, are poorly constrained, so we set informative priors based on the nb_ldtk direct-model analysis on them.\",\n \"The average radius ratio is not well constrained either, so we set a prior on the median radius ratio based on the broadband dataset analysis.\",\n \"We use median instead of mean to ensure that the prior does not affect the scaling of the transmission spectrum, as using mean would.\",\n \"We repeat the DW analysis using first a parametric baseline model (DWW approach), and then a more flexible Gaussian process-based baseline model (DWR approach).\",\n \"The parametric model represents the baseline for each light curve as a linear function of time and airmass $\\\\vec{b}_i(\\\\vec{t}, \\\\vec{x}, \\\\theta ) = \\\\theta _{i,a} + \\\\theta _{i,b} \\\\vec{t} + \\\\theta _{i,c} \\\\vec{x},$ where $\\\\theta _{i,a}$ , $\\\\theta _{i,b}$ , and $\\\\theta _{i,b}$ are the light curve specific baseline constant, linear time coefficient and linear airmass coefficient, respectively.\",\n \"The residual noise is considered white, which makes the approach significantly faster than using GPs, but yields three free parameters per light curve to the model.\",\n \"The DWR analysis uses the same GP kernel as the DIR approach, changing only the GP mean function to the one in Eq.\",\n \"REF .\"\n ],\n [\n \"System characterisation\",\n \"The direct-modelling DIR runs were used to improve the system characterisation from the broadband dataset analysis, and yield the final WASP-80b parameter estimates listed in Table REF .\",\n \"The estimates correspond to the full-spectrum narrow-band dataset (NB) with LDTk.\",\n \"The narrow-band run was preferred over the white-light run since the colour information allows us to mitigate the degeneracy between the average impact parameter, radius ratio, and limb darkening, leading to improved posterior estimates.\",\n \"Table: Results from the final GTC characterisation run (nb_12_ldtkwith systematics modelled using GPs, wavelength-dependent radius ratio, and limb darkeningconstrained with LDTk).\",\n \"The uncertainty is based on the central 68% posteriorinterval.\",\n \"We do not list the radius ratio estimates, since the (common-mode)systematics are too strong for direct modelling to constrain them.\"\n ],\n [\n \"Transmission spectroscopy\",\n \"We show the transmission spectra for the NB, K I, and Na I datasets in Fig.\",\n \"REF .\",\n \"These results correspond to the joint DWR run with a GP systematics model and limb darkening coefficients constrainted by LDTk.\",\n \"The figures show the central 68% and 99% radius ratio posterior intervals, and the point estimates are listed in Table REF .\",\n \"Table: Passband centres, ranges, and radius ratio estimates for the three narrow-band datasetsfrom the final DWR approach.Comparison of the results from the two different divide-by-white approaches can be found below in Sect.\",\n \"REF , and a comparison of the results obtained using only the light curves from one of the two can be found Sect.\",\n \"REF .\",\n \"We do not show the results for the analysis without LDTk.\",\n \"Unconstrained limb darkening does not have a significant effect on the radius ratio posterior medians, but does increase the posterior width significantly due to the strong degeneracy caused by the DW approach.\",\n \"Figure: The NB light curves for both observing runs and the DIR approach posterior median model(black line) as a function of the phase.\",\n \"Passband centres are marked in nanometres.Figure: Transmission spectra for the NB dataset with 20 passbands spanning from 520 nm to 920 nm (top),the Na I set covering the Na I doublet (bottom left), and the K I dataset covering the K I double (bottom right).The inner boxes correspond to the central 68% posterior intervals, the outer boxes to the central 99% posteriorintervals, and Na I and K I lines are marked as black lines.\",\n \"The light gray line shows an Exo-Transmit spectrumwith Na and K described in Sect.\",\n \", the black dots show the Exo-Transmitspectrum binned to the dataset binning, and the slashed horizontal line shows the transmission spectrum mean.\"\n ],\n [\n \"Flat transmission spectrum\",\n \"We adopt the results from the more conservative approach, DWR, as the final results of the analysis.\",\n \"All the spectra in Fig.\",\n \"REF are flat within uncertainties, in agreement with the broadband analysis, and the broadband transmission spectroscopy carried out by [38].\",\n \"The NB features a single strongly deviating passband centred at 807 nm, with a significantly smaller radius ratio than the average.\",\n \"The passband is likely affected by instrumental observation-geometry-dependent systematics (the deviation is evident in both observing runs, albeit slightly different), and we discuss it in more detail in Sect.\",\n \"REF .\",\n \"The flat transmission spectrum allows us to rule out strong Rayleigh scattering or Na I or K I absorption in WASP-80b's atmosphere, but does not justify detailed atmospheric modelling.\",\n \"However, basic modelling with Exo-transmit [19] strongly favours a flat spectrum over a spectrum from an atmosphere with 0.1 or 1 solar metallicity and K and Na: a likelihood ratio test between the Exo-Transmit spectra and a flat spectrum favours the flat model with likelihood ratios around 1000-5000.\",\n \"(We fit the model spectra to the estimated spectrum with a free scaling factor, and the flat spectrum is the mean of the estimated spectrum.)\",\n \"Figure REF includes an Exo-Transmit spectrum with K and Na calculated assuming Solar metallicity, gas-phase chemistry, $T_\\\\mathrm {eq}=800$ K, $log {\\\\it g}= 3.18$ , $R_\\\\star = 0.57 R_\\\\odot $ , and $k=0.17$ as an example.\",\n \"The DWR approach was chosen over DWW because the DWW results show greater discrepancies in the spectra estimated from R1 and R2 separately, as discussed below.\",\n \"It is likely that the parametric systematics model is not flexible enough, and the unaccounted-for systematics lead to biases in the radius ratio values.\",\n \"The GP kernel in the DWR approach is flexible enough to marginalize over the systematics, and still allows for a reliable radius ratio estimation (tested with two mock datasets discussed in Sect.\",\n \"REF ).\"\n ],\n [\n \"Comparison with the previous K I detection\",\n \"[35] reported of a detection of strong K I absorption in WASP-80b's atmosphere based on transmission spectroscopy with the FORS2 spectrograph.\",\n \"Our results disagree with the reported K I detection, as shown in Fig.\",\n \"REF , but agree with the redmost part of their spectrum.\",\n \"Both of our divide-by-white approaches are consistent with each other, as are the transmission spectra estimated separately from the two observing runs (as discussed below in Sects.\",\n \"REF and REF ), which leads us to believe that the strong K I signal reported by [35] is due to systematics, even though the analysis described in their paper seems rigorous in all standards.\",\n \"Figure: Comparison between the results from the GTC analysis and the results presented by .\"\n ],\n [\n \"Comparison of modelling approaches\",\n \"The two divide-by-white approaches differ in their way of modelling the systematics.\",\n \"The DWW approach models the systematics with parametric model as a combination of linear functions of time and airmass, and assumes that the everything else is white normally distributed noise, while the DWR approach models the systematics as a Gaussian process.\",\n \"In the DWW case all systematics not corrected by the division by the dataset average light curve or modelled by the parametric model may cause biases in the radius ratio estimates.\",\n \"The GP kernel in the DWR case is relatively flexible, even when we impose a linear relation on airmass, and should yield less bias-prone radius ratio estimates.\",\n \"This is true especially since we're integrating over the GP hyperparameters, and do not significantly constrain them with priorsThe GP hyperparameters are actually allowed to probe the parameter space where they model time correlation rather than rotator-angle correlation, since the rotator angle itself is a slowly and smoothly varying function of time.\",\n \"We compare the transmission spectra obtained with DWW and DWR approaches in Figs.\",\n \"REF , REF , and REF .\",\n \"The differences are small for the Na and K analyses, which can be expected due to the small wavelength range covered by the datasets.\",\n \"However, the NB set analysed with DWW approach features a Rayleigh-like signal in the bluemost passbands that is not visible in the DWR results.\",\n \"When analysing the two nights separately (Sect.\",\n \"REF ), we can see that the Rayleigh-like signal arises from the second night, and is not significant in the first night.\",\n \"However, the first night shows an increasing trend towards the red end of the spectrum, not visible in the second night spectrum.\",\n \"In theory, the varying Rayleigh-like slope could be interpreted as variations in stellar activity.\",\n \"However, a more likely explanation is residual wavelength-dependent systematics not accounted for properly by the parametric model.\",\n \"Figure: Radius ratio posterior distributions approximated as normal distributions for the NB DWW (left) and DWR (right) runs.Figure: Radius ratio posterior distributions approximated as normal distributions for the Na DWW (left) and DWR (right) runs.Figure: Radius ratio posterior distributions approximated as normal distributions for the K DWW (left) and DWR (right) runs.\"\n ],\n [\n \"Comparison of individual nights\",\n \"We show the transmission spectra estimated separately from run 1 or run 2 either with the DWW or DWR approach in Figs REF , REF , and REF .\",\n \"The results from the different nights agree with some exceptions.\",\n \"Both the blue and red ends of the DWW NB spectrum differ systematically, as mentioned earlier.\",\n \"the Rayleigh-like signal visible in the DWW model originates clearly only from R2, while the corresponding passbands are close or below the mean level for R1.\",\n \"However the three redmost passbands are well above the mean level for R1, while R2 shows a flat spectrum in the red.\",\n \"These differences disappear in DWR analysis, where the both runs give compatible flat-within-uncertainties spectrum.\",\n \"Figure: Radius ratio posterior distributions approximated as normal distributions for the NB DWW (up) and DWR (below) runsusing only the data from night 1 (left) or night 2 (right).Figure: Radius ratio posterior distributions approximated as normal distributions for the Na I DWW (up) and DWR (below) runsusing only the data from night 1 (left) or night 2 (right).Figure: Radius ratio posterior distributions approximated as normal distributions for the K I DWW (up) and DWR (below) runsusing only the data from night 1 (left) or night 2 (right).\"\n ],\n [\n \"Outlier passband at 807 nm\",\n \"A single passband centred at 807 nm in the NB dataset transmission spectrum deviates from the mean.\",\n \"This signal is visible on both nights, and a detailed analysis shows that the features is smooth, but somewhat different between the nights.\",\n \"We could not trace the source of the feature, but can only speculate.\",\n \"First, the feature is very unlikely of astrophysical origin.\",\n \"Certain astrophysical phenomena, such as non-transited star spots and flares, can affect the radius ratio estimates, but the feature would require a bright narrow-band source.\",\n \"Second, the feature is not caused by the comparison star.\",\n \"We repeated the DW NB analyses using absolute photometry without dividing with the comparison star (practical since the divide-by-white approach removes the common-mode systematics), but this did not affect the feature.\"\n ],\n [\n \"Mock dataset analysis\",\n \"We also carry out a sensitivity (reality) checks by using two mock datasets based on the NB dataset.\",\n \"We calculate synthethic light curves using the observed time stamps and the system parameter posterior medians from the direct modelling.\",\n \"The radius ratios follow a saw-tooth-like pattern for the mock dataset 1, and a transmission spectrum calculated by Exo-Transmit for the mock dataset 2, and the limb darkening coefficients are based on the theoretical values calculated using LDTk.\",\n \"The baselines are modelled as sums of a linear time trend and a linear airmass trend with normally distributed random coefficients, and we add normally distributed white noise with standard deviation corresponding to the true light curve standard deviation estimates.\",\n \"The radius ratio posterior densities from the mock dataset analyses agree with the true radius ratios for both DW approaches.\"\n ],\n [\n \"Conclusions\",\n \"We have carried out a transmission spectroscopy analysis for WASP-80b using two OSIRIS-observed spectroscopic time series, and a joint analysis of 27 previously observed broadband light curves to provide reliable parameter priors for the transmission spectroscopy analysis.\",\n \"The OSIRIS data feature strong common-mode systematics, but these can be removed using either a divide-by-white (DW) approach or initial white-light GP modelling.\",\n \"We chose the divide-by-white approach, (or, more accurately, divide-by-dataset-average) which increases the per-passband white noise levels, but removes any systematics-model dependencies from the common-mode systematics removal.\",\n \"The transmission spectroscopy analyses were repeated modelling both datasets jointly and separately, with or without LDTk-constructed stellar limb darkening priors, and using either a parametric or nonparametric (GP-based) systematics model.\",\n \"We chose the most conservative approach as the final analysis detailed in this paper: a DW approach accompanied with a flexible Gaussian process systematics model where the GP hyperparameters are marginalized over in the posterior sampling process.\",\n \"This was motivated by significant nightly variations in the transmission spectrum obtained using the parametric systematics model.\",\n \"Despite the nightly variations in the DWW analysis, the joint analyses are consistent with each other: the transmission spectrum is flat within uncertainties.\",\n \"Especially, we do not detect significant Na or K absorption, and, our results do not agree with the detection of potassium by [35].\",\n \"The absence of significant features does not justify detailed atmospheric modelling.\",\n \"However, basic Exo-Transmit modelling favours a truly flat spectrum over atmosphere models with stellar or sub-stellar metallicity and K and Na.\",\n \"Ground-based observations are prone to complex systematics, and transmission spectroscopy is carried out at the limits of what the instruments are capable of, or were designed for.\",\n \"The most robust approach for ground-based transmission spectroscopy should try to account for this by repeating the observations several times, preferably with different instruments covering the same wavelength range.\",\n \"Repeated observations do not only improve the final precision that can be reached, but also our capability to decouple the systematics from the minute transit depth variations.\",\n \"We thank the anonymous referee for their constructive and useful comments.\",\n \"HP has received support from the Leverhulme Research Project grant RPG-2012-661.\",\n \"FM acknowledges the support of the French Agence Nationale de la Recherche (ANR), under the program ANR-12-BS05-0012 Exo-atmos.\",\n \"The work has been supported by the Spanish MINECO grants ESP2013-48391-C4-2-R and ESP2014-57495-C2-1-R. GC acknowledges the support by the National Natural Science Foundation of China (Grant No.\",\n \"11503088) and the Natural Science Foundation of Jiangsu Province (Grant No.\",\n \"BK20151051).\",\n \"Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma.\",\n \"The authors wish to acknowledge the contribution of Teide High-Performance Computing facilities to the results of this research.\",\n \"TeideHPC facilities are provided by the Instituto Tecnológico y de Energías Renovables (ITER, SA).\",\n \"The authors wish to acknowledge the contribution of Glamdring computing cluster in the Subdepartment of Astrophysics, Department of Physics, University of Oxford, to the results of this research.\"\n ],\n [\n \"DW model fit and residuals\",\n \"Figures REF and REF show the DWR analysis model fit and residuals, respectively.\",\n \"Figure: The NB light curves for both observing runs and the DWR approach posterior median model(black line) as a function of the phase.\",\n \"Passband centres are marked in nanometres.Figure: The NB residuals from the DWR analysis.\",\n \"Passband centres are marked in nanometres.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01875"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":8,"numItemsPerPage":100,"numTotalItems":1867602,"offset":800,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc1MDU0NzkxOCwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzUwNTUxNTE4LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.z4qHFMnC1bYbvKWK5_AHhLReJ35ut-areSdlcdOjpWGvNF6HguiDcQbacrfusiFz5_j49UDNKL0B1QHySxeLCQ","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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"Visibility Estimation for the CHARA/JouFLU Exozodi Survey"
],
[
"Abstract We discuss the estimation of the interferometric visibility (fringe contrast) for the exozodi survey conducted at the CHARA array with the JouFLU beam combiner.",
"We investigate the use of the statistical median to estimate the uncalibrated visibility from an ensemble of fringe exposures.",
"Under a broad range of operating conditions, numerical simulations indicate that this estimator has a smaller bias compared to other estimators.",
"We also propose an improved method for calibrating visibilities, which not only takes into account the time-interval between observations of calibrators and science targets, but also the uncertainties of the calibrators' raw visibilities.",
"We test our methods with data corresponding to stars that do not display the exozodi phenomenon.",
"The results of our tests show that the proposed method yields smaller biases and errors.",
"The relative reduction in bias and error is generally modest, but can be as high as $\\sim 20-40\\%$ for the brightest stars of the CHARA data, and statistically significant at the $95\\%$ confidence level (CL)."
],
[
"Introduction",
"Two-beam optical interferometers have measured hundreds of angular diameters as small as a fraction of a milliarcsecond, with uncertainties of a few percent e.g.",
"[13], [24], [5], [6], [31], and have furthered our understanding of stellar structure and evolution.",
"The improvements in precision of these instruments e.g.",
"[23], [9], [11], [19], [17], [26] have also allowed detecting faint ($\\sim 1\\%$ ) near-infrared (NIR) exozodiacal light emitted from the vicinity ($0.1\\,AU-10\\,AU$ ) of stellar photospheres [8], [1], [12], [2], [14], which has motivated the developments presented here.",
"These precise observations may have deep implications on our understanding of planetary system evolution and direct detection of exoplanets in the habitable zone.",
"Since NIR excesses are typically detected at the few sigma level, even modest improvements in the data analysis may have a noticeable impact on the measured exozodiacal light level and detection significance.",
"The developments presented here are motivated by the search for exozodi phenomena with optical and near-infrared interferometry, but our results apply to high accuracy visibility observations in general.",
"The main interferometric observable with a two beam interferometer is the visibility modulus, which is a measure of interference fringe contrast.",
"A consequence of the Van Cittert-Zernike theorem is that the fringe visibility modulus is related to the angular radiance distribution of the astrophysical source by a Fourier transform, that goes from angular space to baseline space, where the baseline is defined as the projected telescopes separation.",
"The visibility is measured by taking the frequency power spectrum of the fringes, and the area $S$ under the fringe frequency peakAssuming that $I(t)$ is a dimensionless fringe pattern, its power spectrum ($|\\tilde{I}(\\omega )|^2$ ) has units of time squared, and the area under the fringe peak ($S=\\int |\\tilde{I}(\\omega )|^2 d\\omega $ ) has units of time.",
"can be related to the visibility as [3] $|V|^2 = 4\\Delta \\sigma v S, $ where $v$ is the fringe scanning speed in a co-axial beam-combiner, and $\\Delta \\sigma $ is related to the optical bandwidth $\\Delta \\lambda $ and the central wavelength $\\lambda $ as $\\Delta \\sigma =\\Delta \\lambda /\\lambda ^2$ .",
"Typically, the measurable quantity for a fringe exposure is the squared visibility modulus rather than the visibility modulus.",
"Detailed descriptions of the extraction of the visibility from an interferogram can be found in [25], and [10] and [16] discuss the case of a spatially filtered co-axial combiner.",
"Ground-based interferometers are affected by atmospheric turbulence, which deforms the optical wave-front at time-scales that are typically shorter than $1\\,\\mathrm {s}$ .",
"To first order, atmospheric turbulence causes the fringe position to drift in time, which makes long exposures difficult and generally many ($\\sim 100-200$ ) short ($\\ll 1\\,\\mathrm {s}$ ) fringe exposures need to be taken.",
"The uncalibrated (raw) visibility modulus is then estimated from a set of many squared visibility moduli.",
"This paper discusses two main points: $i)$ how to estimate the uncalibrated (raw) visibility from an ensemble of fringe exposures while relaxing some assumptions about the statistical distribution of the measured raw visibilities, and $ii)$ how to calibrate the raw visibilities with a more general approach.",
"We address the uncalibrated visibility estimation problem in Section .",
"The problem of data calibration is addressed in Section .",
"In Section we test our strategies with real interferometric data."
],
[
"Estimating the uncalibrated visibility",
"The purpose of this section is to find a statistical estimator of the uncalibrated visibility with minimal bias and uncertainty, based on a sequence of $|V|^2$ measurements.",
"Typical approaches to estimate the visibility from an ensemble of fringe exposures essentially fit the data to a particular statistical distribution.",
"One approach to estimate the squared visibility is to simply take the ensemble mean and standard deviation of the squared visibilities.",
"However, the mean of $|V|^2$ is a biased estimator because the mean of squared-visibilities ($\\langle |V|^2 \\rangle $ ) is not equal to the square of the mean ($\\langle |V| \\rangle ^2$ ), the two quantities differing by the variance ($\\sigma _{|v|}^2$ ).",
"A classic approach has been to essentially average the variance subtracted squared visibilities [28], [27], assuming photon and detector noise.",
"This approach has been successfully applied by interferometrists for measuring angular separations (e.g.",
"[15]), diameters, and limb-darkening (e.g.",
"[20]).",
"Given that the visibility modulus can only take positive values, another approach is to fit the data to a log-normal distribution [29].",
"However, if data do not follow the assumed distribution, the corresponding estimator may become biased and less suitable for highly accurate measurements.",
"A simple example where assuming Gaussian distributed data can yield a biased result is in the presence of outliers in an otherwise Gaussian distribution of visibilities.",
"One example of such a probability distribution $\\mathcal {P}$ can be approximately described as $\\mathcal {P}(|V|) \\approx (1-\\epsilon ) \\mathcal {N}(\\mu ,\\sigma ) + \\epsilon \\,\\delta (|V|-\\mu -n\\sigma ),$ where $\\mathcal {N}$ is a normal distribution of mean $\\mu $ and standard deviation $\\sigma $ , and $\\delta $ is a Dirac distribution weighed by a probability $\\epsilon $ of the outlier to occur at $n$ standard deviations away from the mean $\\mu $ .",
"In this example the bias induced by the outliers on the mean of $V$ is simply $\\langle |V| \\rangle -\\mu = n\\sigma \\epsilon $ , which is greater than a standard deviation when $n\\epsilon >1$ , and can become comparable to the effects we are trying to measure.",
"A well-known method to alleviate this problem is to reject visibilities that lie far from the ensemble median, which is less sensitive to outliers, but the rejection criteria are somewhat arbitrary and we would like to avoid rejecting data as much as possible.",
"Another example of non-Gaussian data can be seen in Figure REF , where we show probability plots for two different sets of visibilities obtained at the Center for High Angular Resolution Astronomy (CHARA) [29] using the JouFLU beam-combiner (Jouvence of the Fiber-Linked Unit for Optical Recombination [26]).",
"For each set of visibilities we compare the percentiles derived from the measured visibilities to those expected from a Gaussian distribution with the same mean and standard deviation as observed in the data.",
"The left-most point of each plot is the first percentile, and the right-most point the 99th percentile.",
"The left plot of Figure REF shows data that are basically Gaussian, while the right plot shows data departing strongly from Gaussian statistics, since the distribution displays a heavy tail at low visibilities.",
"In the case of Gauss distributed data (Fig.",
"REF , left), the median and the mean are virtually indistinguishable, while in the example of non-Gaussian data, the mean is nearly 2 standard deviations away from the ensemble median.",
"Figure: From an ensemble of CHARA measured stellar visibilities, or rather |V| 2 \\sqrt{|V|^2}, for two different objects (HD 34411 on the left panel and HD 23249 on the right panel), we calculate the data percentiles shown on the vertical axis, and compare them to the percentile that would correspond to a best-fit Gauss distribution.In general, the true distribution of the data is unknown, and assuming a particular distribution may lead to incorrect estimates of the visibility.",
"A solution is to have an estimator that makes no assumptions on the statistical nature of the data."
],
[
"The Median as a Visibility Estimator",
"We propose to use the Median as an estimator of the visibility, or rather $V_{\\rm {median}}= \\sqrt{\\rm {Median}(|V|^2)}, $ The median is known to be very resilient to outliers and seems to have the smallest bias among all the other visibility estimators we have experimented with as we will show below.",
"To investigate the performance of various estimators, we consider two main sources of noise on the squared visibility: i) atmospheric wave-front distortions, which are dominated by differential-piston noise in the case of a spatially-filtered beam combiner, and ii) detector photon noise and/or background noise.",
"Piston noise can be modeled as multiplicative noise, and to illustrate the effect of differential-piston noise on the measured visibility, we first simulated interferograms affected by a time dependent Optical Path Difference (OPD) introduced by the atmosphere as described by [21] and [7].",
"The time dependent OPD was modeled as having a Kolmogorov Power Spectral Density, and a standard deviation of 4 microns during $0.5\\,\\mathrm {s}$ exposures.",
"The effect of detector noise on the squared visibility estimate of Eq.",
"1, can be modeled as the difference of the square of two independent Gaussian variables, i.e.",
"the difference between the real integrated detector-noise Power Spectral Density (PSD), and the estimated integrated detector-noise PSD.",
"Then we computed the visibility modulus for each interferogram using Eq.",
"REF , resulting in the histogram of squared visibilities shown in Figure REF .",
"The simulated piston distribution of $4\\,\\mu \\mathrm {m}$ (rms) results in a standard deviation error of $\\sim 0.003$ on the visibility.",
"In order to further simplify our simulations, we model the squared visibility distribution as $|V|^2=\\mu ^2\\left(1 + \\mathcal {N}_1\\right)^2+\\mathcal {N}_2, $ Figure: Histograms of simulated squared visibilities, with nominal μ=0.35\\mu =0.35 (shown in red).",
"The left row shows distributions only affected only by atmospheric piston with a Kolmogorov power spectral distribution.",
"The atmospheric piston simulation assumes a wind speed of 10m/s10\\,\\mathrm {m/s}, a 33m33\\,\\mathrm {m} baseline, 1m1\\,\\mathrm {m} apertures, and an OPD standard deviation of 4μm4\\,\\mu \\mathrm {m} (top panels) and 8μm8\\,\\mu \\mathrm {m} (bottom panels) for each scan of 0.5s0.5\\,\\mathrm {s} in duration (200 total scans for each histogram).",
"For the case of the 4μm4\\,\\mu \\mathrm {m} rms OPD simulation, the visibility estimators presented in the appendix give V median =0.325±0.0032V_{median}=0.325\\pm 0.0032, V mean =0.323±0.0034V_{mean}=0.323\\pm 0.0034, V lognorm =0.320±0.0031V_{lognorm}=0.320\\pm 0.0031, and V norm =0.320±0.0032V_{norm}=0.320\\pm 0.0032.",
"When the OPD approaches the JouFLU coherence length of ∼10μm\\sim 10\\mu \\mathrm {m}, the visibility becomes difficult to estimate.",
"The second column includes simulated photon (Poisson) noise added to each fringe exposure.where $\\mu $ is the true (positive) visibility, and $\\mathcal {N}_1$ , $\\mathcal {N}_2$ are zero mean random noise variables with respective uncertainty $\\sigma _1$ and $\\sigma _2$ .",
"In this simplified model, $\\mathcal {N}_1$ corresponds to atmospheric piston noise, and the term $\\mathcal {N}_2$ results from the detector noise, where we assume that the power-spectrum bias has been removed (see [22] for a discussion on the power spectrum bias).",
"Note that the mean of $|V|^2$ is a biased estimator because $\\langle |V|^2 \\rangle =\\mu ^2(1 + \\sigma _1^2)$ .",
"Assuming a particular distribution for the visibilities (e.g.",
"lognormal) will also result in a noticeable visibility bias because the data do not follow the assumed distribution perfectly.",
"Below we describe some basic cases when the median is an unbiased estimator.",
"In the particular case when $\\mathcal {N}_2\\sim 0$ (bright star and piston-limited noise), and further assuming that $|\\mu (1 + \\rm {min}(\\mathcal {N}_1))|<\\mu (1+\\rm {Median}(\\mathcal {N}_1))$ , a condition that is generally satisfied unless visibilities are very low and noisy (see Figure REF ), we have $\\rm {Median}(|V|^2)=(\\mu (1+\\rm {Median}(\\mathcal {N}_1)))^2$ .",
"This means that if the $\\mathcal {N}_1$ noise has zero median, regardless of its statistical distribution, then the median of the observed $|V|^2$ can be used as an unbiased estimate of the object's visibility, i.e.",
"$\\sqrt{\\rm {Median}(|V|^2)}=\\mu $ .",
"Similarly, in the opposite limiting case when $\\mathcal {N}_1=0$ (photon-limited noise), we get $\\rm {Median}(|V|^2)=\\mu ^2+\\rm {Median}(\\mathcal {N}_2)$ .",
"If we further assume that $\\rm {Median}(\\mathcal {N}_2)=0$ , then $\\rm {Median}(|V|^2)=\\mu ^2$ if $\\rm {Median}(\\mathcal {N}_2)=0$ , regardless of $\\mathcal {N}_2$ 's statistical distribution.",
"Figure: Left panel: Simulated distribution of measured visibilities |V||V| affected by multiplicative noise of the form (1+𝒩 1 )(1+\\mathcal {N}_1), where 𝒩 1 \\mathcal {N}_1 has a zero median.",
"Note that visibilities can extend to negative values.",
"Right panel: Corresponding distribution of the squared visibilities.",
"In the case shown here and whenever | min (|V|)|< Median (|V|)|\\rm {min}(|V|)|<\\rm {Median(|V|)}, we have simply Median (|V| 2 )= Median (|V|)\\sqrt{\\rm {Median}(|V|^2)}=\\rm {Median}(|V|).",
"The median estimator is found to have the smaller bias of the 4 estimators, and a 1σ\\sigma estimation uncertainty consistent with that of the other estimators.",
"Only the V lognorm V_{\\rm {lognorm}} estimator provides a slightly smaller uncertainty, but at the expense of a much larger bias.In general we are somewhere in between the two extreme cases discussed above, so we simulated distributions of 200 visibility points using Eq.",
"REF , taking $\\mathcal {N}_1$ as a Gaussian random variable, and $\\mathcal {N}_2$ as a $\\chi ^2$ distributed variable, with various realistic values of $\\sigma _1$ , $\\sigma _2$ .",
"Then we computed the estimation bias resulting from different visibility estimators.",
"In all cases, the relative bias is defined as $|V_{\\rm {est}}-\\mu |/\\mu $ , where $V_{\\rm {est}}$ is one the 4 different estimators described in [30], and also provided in Appendix A of this manuscript, namely: $V_{\\rm {lognorm}}$ , $V_{\\rm {norm}}$ , $V_{\\rm {median}}$ (defined in equation REF above), and $V_{\\rm {mean}}$ (a simple mean and standard deviation).",
"In Figure REF (left) we compare the bias, and find that in the context of the model described by equation REF , $V_{\\rm {median}}$ generally has the smallest bias among the estimators tested.",
"Figure: Using Equation , and assuming Gaussian noise for 𝒩 1 \\mathcal {N}_1, and χ 2 \\chi ^2 distributed noise for 𝒩 2 \\mathcal {N}_2, we compare the relative biases expected for different estimators (left panels).",
"The assumed visibility is μ=0.3\\mu =0.3 and we take a fixed σ 2 =0.02\\sigma _2=0.02 for the top-left panel, which corresponds to the effect of detector noise on a typical target star with K=3, and a fixed σ 1 =0.04\\sigma _1=0.04 (the piston noise, estimated from the simulation presented in Figure ) for the bottom-left panel.",
"The right panels shows the corresponding 1σ1\\sigma -spread for the various estimators based on an ensemble of 200 simulated scans.",
"the range of σ 2 \\sigma _2 values corresponds to a range in stellar magnitudes K=0-4.9.From the bottom-left panel of Figure REF , we note that the visibility bias, using the median estimator, remains below $\\sim 0.5\\%$ , even for the faintest stars ($\\sigma _2=10^{-4}-0.7$ , corresponding to stellar magnitudes ranging between K=0-4.9).",
"This shows that using calibrators of very similar brightness to the science target is not required when using the median estimator, and for stars brighter than $K\\approx 4.9$ .",
"In any case, in the JouFLU survey we adopt a conservative approach, and use calibrators that do not differ by more than $\\sim 1$ magnitude from the science target.",
"At some point ($\\sigma _2\\sim 0.3$ ), the uncertainties become too large for highly precise visibility measurements, as shown in the bottom-right panel of Figure REF .",
"This is actually close to the point where the detector noise becomes comparable to the signal counts.",
"The JouFLU beam combiner typically detects $\\sim 100$ coherent photons per fringe exposure for an unresolved $\\mathrm {K}\\approx 3$ star, while the background (dark) counts rms is $\\sim 10$ counts per fringe exposure.",
"Since the $\\mathcal {N}_2$ term standard deviation results from the difference between two squared Gaussians of standard deviation $\\sim 10/100$ , the resulting standard deviation for a typical K=3 star is $\\sigma _2\\approx 2\\times (10/100)^2 = 0.02$ .",
"Therefore, detector noise is the limiting source of noise when the stellar magnitude approaches $K \\sim 4.5$ ($\\sigma _2 \\sim 0.3$ ).",
"Next we investigate the uncertainty of the median, which we estimate from bootstrapping the ensemble of visibility measurements resulting from individual fringe exposures as follows: from a set of $N=150$ exposures, a “bootstrap sample” of the data is generated by randomly selecting $N$ visibilities.",
"Then $10^4$ such bootstrapped samples are generated and a median is computed for each of them.",
"Last, we estimate the $68\\%$ confidence interval by finding the 16th and 84th percentiles of the ensemble of bootstrap medians.",
"We perform analogous simulations for the other 3 estimators and compare the uncertainties derived for all 4 estimators.",
"As shown in Figure REF (right), our simulations show that the $V_{\\rm {median}}$ estimator has a very similar uncertainty to the $V_{\\rm {mean}}$ and $V_{\\rm {norm}}$ estimators, and is marginally larger than the uncertainty of $V_{\\rm {lognorm}}$ , which has the largest bias.",
"In Section we discuss the performance of $V_{\\rm {median}}$ with real interferometric data."
],
[
"Hybrid Calibration",
"The raw visibility $\\mu $ must be calibrated in order to relate it to the angular brightness distribution of the science target.",
"Optical interferometry requires calibrator stars whose angular diameters are known with sufficient precision predict their expected theoretical visibility $V_{\\rm {exp}}$ with high accuracy.",
"The science target calibrated visibility can be expressed as $V=\\mu \\, T,$ where $T$ is known as the transfer function, and defined in terms of the expected visibility of the calibrator $V_{\\rm {exp}}$ and the calibrator's measured raw visibility $\\mu _c$ : $T\\equiv \\frac{V_{\\rm {exp}}}{\\mu _c}.$ Calibrators are typically observed before and after the science target since the transfer function may vary in time, and we find the visibility transfer functions $T_1$ and $T_2$ at times $t_1$ and $t_2$ .",
"The problem we address here is how to best estimate the transfer function $T$ at the time of observation $t$ ($t_1<t<t_2$ ), which reduces to finding the weights $\\alpha _1$ and $\\alpha _2$ for the transfer functions $T_1$ and $T_2$ , i.e.",
"$V=\\mu \\,T=\\mu \\left\\lbrace \\alpha _1 T_1 +\\alpha _2 T_2 \\right\\rbrace \\frac{1}{\\alpha _1+\\alpha _2}$ One possibility to find the weights $\\alpha _1$ and $\\alpha _2$ is to assume that the transfer function varies linearly in time, and perform a linear time-interpolation to estimate the transfer function at the time of observation.",
"In this case the weights are $\\alpha _1(t)=\\left(\\frac{t_2-t}{t_1-t_2}\\right);\\,\\,\\, \\alpha _2(t)=\\left(\\frac{t-t_1}{t_1-t_2}\\right).$ Alternatively -and especially if the $T_1$ and $T_2$ estimated values differ by less than their individual error bars- it is also reasonable to assume that the transfer function did not change between $t_1$ and $t_2$ , in which case we could calculate a “weighted mean” with statistical weights $\\alpha _1=\\frac{1}{\\sigma _1^2} ;\\,\\,\\, \\alpha _2=\\frac{1}{\\sigma _2^2}.$ We propose a solution for the coefficients ($\\alpha _1$ , $\\alpha _2$ ) combining weights from the linear time interpolation and from the weighted mean: $\\boxed{ \\alpha _1(t)=\\frac{1}{\\sigma _1^2}\\left(\\frac{t_2-t}{t_1-t_2}\\right)} & \\boxed{\\alpha _2(t)=\\frac{1}{\\sigma _2^2}\\left(\\frac{t-t_1}{t_1-t_2}\\right) }.$ Note that when $\\sigma _1=\\sigma _2$ , equation REF reduces to a linear time-interpolation, and when $t=(t_2+t_1)/2$ , equation REF reduces to a weighted mean.",
"If $t$ is much closer to $t_1$ than to $t_2$ , then $T_1$ will have a much larger weight than $T_2$ , unless the uncertainty on $T_2$ ($\\sigma _2$ ) is much smaller than that on $T_1$ ($\\sigma _1$ ).",
"For computing the final uncertainty in the calibrated visibility (Eq.",
"REF ) we first take into account the statistical uncertaintiesIn the case of the JouFLU beam combiner, the statistical uncertainties of each transfer function include the effects of correlations between the two interferometric channels as described by [22] of $T_1$ and $T_2$ .",
"The statistical uncertainty $\\sigma _{\\rm {stat}}$ on the estimated calibrated visibility comes from propagating the uncertainties on $\\mu $ , $T_1$ and $T_2$ , and adding them in quadrature, i.e.",
": $\\sigma _{\\rm {stat}}^2 &=& \\left(\\frac{\\partial V}{\\partial \\mu }\\sigma _{\\mu }\\right)^2+\\left(\\frac{\\partial V}{\\partial T_1}\\sigma _1\\right)^2+\\left(\\frac{\\partial V}{\\partial T_2}\\sigma _2\\right)^2.",
"\\nonumber \\\\&=& T^2 \\sigma _{\\mu }^2 + \\left( (\\alpha _1\\sigma _1)^2 + (\\alpha _2\\sigma _2)^2\\right)\\frac{\\mu ^2}{ (\\alpha _1+\\alpha _2)^2}$ In addition we also consider the systematic uncertainty coming from the departure of $T_{1}$ from $T_{2}$ .",
"This systematic uncertainty is calculated as a weighed standard deviation, i.e.",
"$\\sigma _{\\rm {syst}}^2=\\mu ^2\\frac{\\left(T_1-T \\right)^2\\alpha _1 + \\left( T_2-T \\right)^2\\alpha _2}{\\alpha _1+\\alpha _2}.",
"$ The final uncertainty in the visibility can be estimated by adding errors in quadrature as $\\sigma _{|V|}^2=\\sigma _{\\rm {stat}}^2+\\sigma _{\\rm {syst}}^2.$ According to Eq.",
"REF , the distribution of the systematic uncertainty is a $\\chi ^2$ distribution, which can result in a high probability for the over-estimation of $\\sigma _{syst}$ .",
"The results presented above can be generalized to an arbitrary number of calibrators, or alternatively we could resort to Gaussian process estimation (as suggested by the referee), but we restrict the results presented below to the nearest neighbor approximation, i.e.",
"two calibration measurements: one before and one after the science target."
],
[
"All Stars",
"We tested the performance of the different estimators and calibration methods on a set of 166 calibrated visibility points obtained with JouFLU/CHARA between 2013 and 2015 for the exozodiacal light survey, for stars as bright as K=1.4 and as faint as K=4.7.",
"These tests were only performed on stars for which the naked star model produced a $\\chi ^2$ that is not too high (reduced $\\chi ^2$ lower than 3.0 with $\\sim $ 5 degrees of freedom) and no excess was detected (with a threshold of $3\\sigma $ ) in the star + dust model.",
"This reduces possible biases due to the presence of exozodiacal structures or stellar companions around the target stars, and still includes the majority of targets in the overall survey sample ($\\sim 70\\%$ ).",
"The goal here is to see which estimator and calibration method gives the smallest bias and uncertainty, while the full results of the exozodi survey will be presented in an upcoming paper.",
"In order to measure the bias, we compare each calibrated visibility point to a modeled photospheric visibility based on long baseline angular diameter measurements obtained by [5] with CHARA/CLASSIC.",
"If long-baseline measurements are not available, we use a surface brightness (V-K) angular diameter to model the visibility, which is acceptable for the JouFLU program since stars are mostly unresolved at the short ($\\sim 30\\,\\mathrm {m}$ ) baselines used, and angular diameter uncertainties induce negligible error at these short baselines.",
"For example a star with a uniform-disk angular diameter of $(1\\pm 0.05)\\,\\mathrm {mas}$ would introduce an uncertainty of $\\sim 0.07\\%$ in the interferometric visibility at the $33\\,\\mathrm {m}$ baseline).",
"Figure: Histogram of the absolute value of the relative bias for various estimators.",
"The vertical dotted lines represent the 68%68\\% confidence interval.",
"The confidence interval of V median V_{\\rm {median}} (shown in red) is smaller than for all other estimators considered here (V mean V_{\\rm {mean}}, V lognorm V_{\\rm {lognorm}}, and V norm V_{\\rm {norm}}).We compare the performance of various estimators ($V_{\\rm {median}}$ , $V_{\\rm {mean}}$ , $V_{\\rm {norm}}$ , $V_{\\rm {lognorm}}$ ) and calibration methods (LI: Linear time Interpolation, and HI: Hybrid Interpolation discussed in Sec.",
"), i.e.",
"we analyzed the whole data set using 8 different visibility estimation methods.",
"Figure REF shows histograms of $|V-V_{\\rm {model}}|/V_{\\rm {model}}$ , Table REF provides the median values of the $|$ bias$|$ , the median visibility error, and also shows the uncertainties ($68\\%$ CI) associated to these median values, which are calculated via bootstrapping techniques.",
"Figure REF and Table REF show that when using the $V_{\\rm {median}}$ estimator with the Hybrid interpolation calibration, hereafter $V_{\\rm {median,HI}}$ , the biases and uncertainties are generally lower.",
"We also find that their statistical dispersion is lower, i.e, that the range of observed biases and visibility uncertainties is smaller.",
"In other words, the $V_{\\rm {median,HI}}$ method provides more robust and more consistent calibration results.",
"To quantify the statistical significance of our findings, we use the following bootstrapping method: for each visibility estimation method, there is a set of 166 calibrated visibility measurements, which we resample many (1000) times to form many bootstrapped sets.",
"For each bootstrapped set we calculate the 84th and 16th percentiles of the bias and visibility error.",
"Then, to calculate the probability (confidence level CL) that $V_{\\rm {median,HI}}$ is a better estimator, we simply count the number of times that the $68\\%$ confidence interval (CI) is smaller for the $V_{\\rm {median,HI}}$ estimator.",
"Figure REF illustrates this for the bias and its uncertainty found for the $V_{\\rm {median}}$ and $V_{\\rm {mean}}$ estimators.",
"Figure: Bootstrap values for the | Bias ||\\rm {Bias}| (left panel) and its 68%68\\% CI for the V median V_{\\rm {median}} and V mean V_{\\rm {mean}} estimators (right panel).",
"The probability that | Bias ||\\rm {Bias}| is greater for V mean , LI V_{\\rm {mean,LI}} than for V median V_{\\rm {median}} is 94%94\\%, and the probability that the Confidence Interval of | Bias ||\\rm {Bias}| is greater for V mean , LI V_{\\rm {mean,LI}} than for V median V_{\\rm {median}} is 99%99\\%.",
"The distributions show that V median V_{\\rm {median}} has a smaller bias and error than V mean V_{\\rm {mean}}.Table: In the second column we report the median relative percent ||bias|| (defined as |V-V model |/V model |V-V_{\\rm {model}}|/V_{\\rm {model}}) and its bootstrapped 68%68\\% confidence level.",
"The last column shows the error for each visibility estimation method.",
"We provide values for various estimators and interpolation methods (LI: Linear time Interpolation, HI: Hybrid Interpolation).As shown in Table REF (third column), the most statistically significant improvement ($>95\\%$ CL) is in the reduction of the confidence intervals of $|V-V_{\\rm {model}}|/V_{\\rm {model}}$ .",
"Table REF shows that most of the improvement comes from using the median estimator described in Sec.",
"REF , and a slight improvement is due to the Hybrid interpolation method described in Sec.",
".",
"Table: Probability that the ||Bias|| of V median , HI V_{\\rm {median,HI}} is reduced.",
"The second column shows the probability that Median |V-V model |/V model )\\rm {Median}|V-V_{\\rm {model}}|/V_{\\rm {model}}) is smaller for the V median , HI V_{\\rm {median,HI}} estimator than for the other estimators listed in the first column.",
"The third column shows the probability, P(CI(|V median , HI -V model |/V model )P(CI (|V_{\\rm {median,HI}}-V_{\\rm {model}}|/V_{\\rm {model}}), that the 68%68\\% confidence interval of |V median , HI -V model |/V model |V_{\\rm {median,HI}}-V_{\\rm {model}}|/V_{\\rm {model}} is smaller when V median , HI V_{\\rm {median,HI}} is used.Similarly, we quantify the reduction of the visibility uncertainty resulting from the use of $V_{\\rm {median,HI}}$ .",
"As shown in Table REF , we find that the median uncertainty is significantly ($95\\%$ CL) smaller by as much as $\\sim 20\\%$ relative to the $V_{\\rm {norm}}$ and $V_{\\rm {mean}}$ estimators, and comparable to the $V_{\\rm {lognorm}}$ and $V_{\\rm {median,LI}}$ .",
"As for the confidence interval of the visibility error, we find that $V_{\\rm {median,HI}}$ does significantly better than most estimators, but not significantly better than $V_{\\rm {lognorm,HI}}$ and $V_{\\rm {median,LI}}$ .",
"Table: Probability that V median , HI V_{\\rm {median,HI}} has a smaller error than the estimators shown in the first column.",
"The second column is the probability that the median visibility error is smaller for V median , HI V_{\\rm {median,HI}} than for the other visibility estimation methods.",
"The third column is the probability that the 68%68\\% confidence interval of the visibility error is smaller for the V median , HI V_{\\rm {median,HI}}From the results of Table REF , we also note that the median (typical) bias is close to the typical visibility uncertainty.",
"This indicates that there is a global agreement of the visibility measurements with the stellar models for all visibility estimation methods.",
"However, we consider $V_{\\rm {median,HI}}$ a more robust estimator in view of the reduced dispersion of the bias and error."
],
[
"Influence of Stellar Brightness",
"Finally, we investigate the improvements as a function of stellar brightness, so we split our data set in two groups of comparable size: data points corresponding to brighter stars ($K<3.5$ ), and data corresponding to fainter stars ($K>3.5$ ).",
"For the brighter stars, we find that the $|$ bias$|$ (actually the relative bias absolute value) is $(1.74\\pm 0.16)\\%$ (for $V_{\\rm {median,HI}}$ ), while for the group of fainter stars the relative $|$ bias$|$ is $(2.3\\pm 0.39)\\%$ .",
"For the brighter stars, we find that the use of $V_{\\rm {median,HI}}$ results in a significanlty smaller $\\rm {Median}(|V-V_{\\rm {model}}|/V_{\\rm {model}})$ compared to most other estimation methods.",
"Table REF and Figure REF show that the (median) $|$ bias$|$ is reduced by at least 23% when the median estimator ($V_{\\rm {median}}$ ) is used, and Table REF shows that these improvements are statistically significant at the $95\\%$ CL.",
"Compared to what we find with the whole stellar sample, we find no additional reduction in the visibility uncertainty for the bright stars as shown in Table REF and REF .",
"For the fainter stars, the performance of $V_{\\rm {median,HI}}$ is still superior with probabilities comparable to those described in Tables REF and REF .",
"In general, special care must be taken for the calibration of the brightest stars, since the calibrator must be small enough to be considered a point-source, but of comparable brightness to the science target so that the interference fringes have similar noise characteristics.",
"These two requirements are not generally met, since bright stars typically have larger angular diameters, and generally a compromise is made between brightness, closeness in the sky, and angular diameter [4], [18].",
"However, in the Exozodi survey, we have mainly used the smallest baseline of the CHARA array ($33\\,\\mathrm {m}$ ), for which most stars remain virtually unresolved, and with transfer functions that have a weak dependence on the stellar model.",
"For example, the largest and brightest (K=2) calibrator used in our survey has a uniform angular diameter estimated at $1.71 \\pm 0.024\\,\\mathrm {mas}$ ([18] catalog).",
"This corresponds to an interferometric visibility of $0.979\\pm 0.0006$ at the $33\\,\\mathrm {m}$ baseline.",
"Even assuming a pessimistic diameter uncertainty of 5% on this worst case calibrator, the resulting visibility uncertainty is 0.7%.",
"Additionally, our results never rely on a single calibrator star, and we nominally use 3 different calibrator stars for each science target.",
"A more extended discussion of our calibrators selection will be presented in our main survey results summary paper.",
"Table: Similar to Table , but only showing results for stars with K<3.5K<3.5.Figure: Graphical representation of Table .",
"The plot shows the abs(Bias) and the error for different estimators for stars with K<3.5K<3.5.",
"The median estimator has a smaller ||bias|| than all the other estimators, and has a smaller error than smaller error than the other estimators except V lognorm V_{\\rm {lognorm}}.",
"The statistical significance of our findings is shown in Table .Table: Similar to Table , but only showing results for stars with K<3.5K<3.5.Table: Similar to Table , but only showing results for stars with K<3.5K<3.5."
],
[
"Discussion and conclusions",
"The goal of this paper was to present the data analysis strategy for the CHARA/JouFLU exozodi survey.",
"Our approach was to relax some of the assumptions made when estimating the visibilities, and to propose a more general methodology that is not only strictly valid for data ($V$ or $|V|^2$ ) affected by Gaussian errors, or errors with a priori perfectly known statistical distributions.",
"We have shown that assuming a particular statistical distribution for an ensemble of fringe exposures may lead to a biased estimation of the uncalibrated visibility.",
"Using the median as the visibility estimator is a natural choice since it is more resilient to outliers and skewed distributions.",
"We also show that the median estimator is an unbiased estimator in several limiting cases, as long as the dominant source of measurement noise has a zero median, and whatever its statistical distribution is otherwise.",
"Bootstrapping to find the errorbars of the median estimator also makes no assumptions on the statistics and is therefore quite general.",
"Our proposed method for estimating the transfer function is also more general since it reduces to the commonly used linear interpolation when the calibrator's visibilities have similar uncertainties, but gives more weight to calibrators that have smaller uncertainties.",
"We have performed tests with simulated and real data, and have concluded that the formalism implemented here yields statistically significant reductions in visibility estimation biases and uncertainties when compared to other methods.",
"Our tests with CHARA/JouFLU data show that the improvements from using the median estimator are even greater for brighter stars, namely that the visibility bias is significantly smaller, as expected from the visibility model presented in Eq.",
"REF .",
"The tests presented above have been limited to mostly unresolved stars, but our results will likely be valid for resolved stars as long as the visibility is not extremely low and noisy (i.e.",
"$|\\rm {min}(V)|<\\rm {Median(V)}$ ).",
"Our motivation is to use these data reduction strategies in an upcoming paper describing the latest results of the exozodi survey using the CHARA/JouFLU instrument.",
"But the methodology presented here applies to the estimation of interferometric visibilities in general, whether coming from single mode beam combiners or not, co-axial or multi-axial.",
"It also generally applies to the interpretation of repeated measurements with poorly known noise characteristics."
],
[
"Acknowledgments:",
"This research was supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory administered by Universities Space Research Association under contract with NASA.",
"PN and BM are grateful for support from the NASA Exoplanet Research Program element, though grant number NNN13D460T.",
"This work is based upon observations obtained with the Georgia State University Center for High Angular Resolution Astronomy Array at Mount Wilson Observatory.",
"The CHARA Array is supported by the National Science Foundation under Grant No.",
"AST-1211929.",
"Institutional support has been provided from the GSU College of Arts and Sciences and the GSU Office of the Vice President for Research and Economic Development.",
"We also thank the anonymous referee for his/her valuable comments which improved the quality of this manuscript.",
"Here we provide the definitions of the visibility estimators used throughout this paper, which are described in [30].",
"If we assume that the statistical distribution of the visibility is normal, then the following are unbiased estimators.",
"$V_{\\rm {norm}}$ is defined as $V_{\\rm {norm}}= \\left( \\frac{3\\langle |V|^2 \\rangle ^2 - \\langle |V|^4 \\rangle }{2}\\right)^{1/4},$ with the following uncertainty: $\\sigma _{\\rm {norm}}^2 = \\sqrt{\\langle |V| \\rangle ^4 - \\frac{1}{2}\\left( \\langle |V|^2 \\rangle ^2 - \\langle |V|^4 \\rangle \\right) } - \\langle |V| \\rangle ^2.$ The $V_{\\rm {lognorm}}$ estimator is defined as: $V_{\\rm {lognorm}} = \\exp \\left(\\lambda + \\frac{1}{2}\\sigma ^2\\right) ,$ where $\\lambda = \\frac{1}{4}\\rm {ln}\\frac{\\langle |V|^2 \\rangle ^4}{\\langle |V|^4 \\rangle },$ $\\sigma = \\frac{1}{4}\\rm {ln}\\frac{\\langle |V|^4 \\rangle }{\\langle |V|^2 \\rangle ^2},$ and the corresponding uncertainty is $\\sigma _{\\rm {lognorm}}^2 = \\exp {(2\\lambda + 2 \\sigma ^2)}-\\exp {(2\\lambda +\\sigma ^2)}.$"
]
] | 1709.01656 |
[
[
"Tests of Catastrophic Outlier Prediction in Empirical Photometric\n Redshift Estimation with Redshift Probability Distributions"
],
[
"Abstract We present results of using individual galaxies' redshift probability information derived from a photometric redshift (photo-z) algorithm, SPIDERz, to identify potential catastrophic outliers in photometric redshift determinations.",
"By using two test data sets comprised of COSMOS multi-band photometry spanning a wide redshift range (0<z<4) matched with reliable spectroscopic or other redshift determinations we explore the efficacy of a novel method to flag potential catastrophic outliers in an analysis which relies on accurate photometric redshifts.",
"SPIDERz is a custom support vector machine classification algorithm for photo-z analysis that naturally outputs a distribution of redshift probability information for each galaxy in addition to a discrete most probable photo-z value.",
"By applying an analytic technique with flagging criteria to identify the presence of probability distribution features characteristic of catastrophic outlier photo-z estimates, such as multiple redshift probability peaks separated by substantial redshift distances, we can flag potential catastrophic outliers in photo-z determinations.",
"We find that our proposed method can correctly flag large fractions (>50%) of the catastrophic outlier galaxies, while only flagging a small fraction (<5%) of the total non-outlier galaxies, depending on parameter choices.",
"The fraction of non-outlier galaxies flagged varies significantly with redshift and magnitude, however.",
"We examine the performance of this strategy in photo-z determinations using a range of flagging parameter values.",
"These results could potentially be useful for utilization of photometric redshifts in future large scale surveys where catastrophic outliers are particularly detrimental to the science goals."
],
[
"Introduction",
"Accurate photometric redshift estimates (photo-zs) with well constrained and understood error properties are critical for the current and coming era of large multi-band extragalactic surveys [23], [20], [8], such as the Large Synoptic Survey Telescope (LSST)http://www.lsst.org, Euclidhttp://sci.esa.int/euclid, Wide Field Infrared Survey Telescope (WFIRST)http://wfirst.gsfc.nasa.gov, Hyper-Suprime Cam (HSC)http://www.naoj.org/Projects/HSC, and Kilo-Degree Survey (KiDS)http://kids.strw.leidenuniv.nl for which precise redshift estimates will be needed for millions or billions of galaxies extending to high redshifts.",
"In particular, photometric redshift accuracy is the primary source of systematic error in weak-lensing surveys [8].",
"Works modeling the error relation between photometric ($z_{phot}$ ) and spectroscopic ($z_{spec}$ ) redshifts as a Gaussian have found that achieving less than 50% degredation in cosmological parameter uncertainties requires the bias $\\langle z_{phot} - z_{spec} \\rangle $ and scatter $\\langle (z_{phot} - z_{spec})^2 \\rangle ^{1/2}$ quantities in each redshift bin $\\Delta z=0.1$ to be constrained to roughly 0.003-0.01 [33], [23], [29] with tighter constraints when these distributions are non-Gaussian.",
"Limiting the occurrence of catastrophic outlier photo-z estimates — those galaxies whose estimated redshift differs substantially from their actual redshift — is a top priority for controlling photo-z errors.",
"In addressing this challenge we present a study directed toward a novel method to flag potential catastrophic outlier photo-z predictions through the utilization of individual galaxy redshift probability information.",
"We utilize SPIDERz [28], a custom implementation of a support vector machine classification model for photometric redshift analysis, which naturally outputs an effective redshift probability distribution for each galaxyavailable from http://spiderz.sourceforge.net with usage documentation provided there.. SPIDERz's natural output of an effective redshift probability distribution for each galaxy is not necessarily typical for empirical photo-z estimation methods (which make a predictive model based on a training set with known redshifts), but some other empirical methods which can output probability information are ArborZ [19], TPZ [13], SkyNet [10], ANNz2 [36].",
"The techniques discussed in this work should theoretically be relevant to any photo-z estimation method which provides the requisite redshift probability distribution information for individual galaxies.",
"The performance of candidate photo-z methods should ideally be demonstrated on test data that is representative of the data anticipated by future large-scale surveys.",
"In particular, some data sets, such as much of the LSST catalog, will have photometric data for optical bands only while others, such as Euclid, will have, or overlap with, infrared bands.",
"Additionally, some important data sets will span a large redshift range with many high redshift objects.",
"In order to perform an analysis on real data approximating these conditions, we use two relatively large data sets of photometry from the Cosmic Evolution Survey (COSMOS) COSMOS2015 photometric catalog [30] with known redshifts spanning the redshift range $0<z<4$ .",
"One set consists of the overlap of COSMOS photometry with spectroscopic redshifts from the 3D-HST survey performed with the Hubble Space Telescope and reported in [34] featuring 3704 galaxies, and the other consists of COSMOS photometry with previously reliably estimated (see criteria in §REF ) 30-band photometric redshifts.",
"Furthermore, in order to approximate the photometric redshift conditions of future large scale surveys, we adopt training set sizes that are much smaller than evaluation set sizes.",
"Figure: Examples of EPDFs as determined by SPIDERz for particular individual galaxies in the COSMOSx3D-HST data set described in §.",
"The top panel shows an EPDF with a singular uniform probability peak, which is typical of galaxies with accurate redshift estimates.",
"The middle panel shows a classic doubly peaked EPDF where the spectroscopic redshift is near the slightly lower peak, which is often the case for catastrophic outlier redshift estimates.",
"The bottom panel shows an EPDF without a clear probability peak, which also can be the case for catastrophic outlier redshift estimates.Photo-z methods have been traditionally divided into two categories: template-fitting and empirical methods.",
"Template-fitting methods rely on fitting galaxy photometry to template spectra evolved with redshift, typically derived using $\\chi ^2$ minimization, e.g.",
"Le Phare [3], [24], BPZ [6], HyperZ [9], zebra [16], EAZY [11], gazelle [27], and DELIGHT [32].",
"Template-fitting methods depend critically on the extent to which galaxy spectral energy distributions (SEDs) library templates adequately represent properties of observed SEDs corresponding to target galaxy populations for which one wants to estimate the redshifts; the selection of ill-fitted SED templates provides the greatest source of errors in redshift determinations with these models.",
"Some techniques for template fitting have incorporated the use of training sets of objects with known photometry and spectroscopic redshifts to better calibrate representative SED templates [7], [24], [25].",
"Figure: Reconstructed redshift distributions from a determination with SPIDERz using 1200 training galaxies compared to the actual COSMOSx3D-HST evaluation sample of 2323 galaxies.",
"Test data for the determination shown in this figure only were limited to z<2.9z < 2.9 to prevent the occurrence of unoccupied redshift bins at high redshifts.",
"Distributions are shown for the actual spectroscopic redshift, the single best-estimate (highest probability bin) photo-z, the summed EPDF, and the weighted summed EPDF.Empirical methods, which rely on training sets with known redshifts to derive a mapping from photometry to redshift, depend critically on the extent to which training galaxy populations adequately represent target galaxy populations in terms of the parameter overlap of photometric inputs and true redshift distributions.",
"Early examples empirical photo-z methods utilized relatively simple techniques to achieve such a mapping [15].",
"More recently, models that produce mappings with greater complexity utilizing machine learning have been examined, e.g.",
"artificial neural networks [17], [14], [41], [39], [12], [40], support vector machines [42], [43], [28], Gaussian process regression [44], boosted decision trees [19], random forests [13], [35], genetic algorithms [22], sparse Gaussian framework [2], nearest neighbor search [5], [4], and spectral connectivity analysis [18].",
"A review and comparison of a number of existing photo-z methods can be found in [21], [1], [37].",
"Figure: N(z)N(z) distribution for the 3704 galaxies comprising the COSMOSxHST (left) and 58622 galaxies comprising the COSMOS2015 (right) test data sets used in this analysis.Here we follow convention [21] and define “outliers” as those galaxies where $Outliers: {{\\vert z_{phot}-z_{spec} \\vert } \\over {1+z_{spec}}} > .15,$ where $z_{phot}$ and $z_{spec}$ are the estimated photo-z and actual (spectroscopically determined) redshift of the object.",
"Although there is not a standard, universal definition of “catastrophic outliers” we use a definition that is typical [8] $O_{c}: {{\\vert z_{phot}-z_{spec} \\vert }} > 1.0.$ The RMS photo-z error in a realization is given by a standard definition $\\sigma _{\\Delta z/(1+z)} \\equiv \\sqrt{ {{1} \\over {n_{gals}}} \\Sigma _{gals} \\left( {{ z_{phot}-z_{spec} } \\over {1+z_{spec}}} \\right) ^2 },$ where $n_{gals}$ is the number of galaxies in the evaluation testing set and $\\Sigma _{gals}$ represents a sum over those galaxies.",
"We also calculate the RMS error without the inclusion of outlier galaxies, referring to this quantity as the “reduced” RMS or R-RMS.",
"In § we present a summary overview of the SVM model implemented in SPIDERz and discuss the probability information produced for each galaxy.",
"In § we present a method for flagging potential catastrophic outlier photo-z estimates made by SPIDERz through the utilization of redshift probability information.",
"In § we discuss the results of testing SPIDERz on the two test data sets utilizing COSMOS multi-band photometry.",
"We present a discussion in §."
],
[
"SPIDERz and effective probability distributions",
"A full discussion of the SPIDERz algorithm, mathematical theory, and a suite of tests with various data sets and comparisons with other photo-z determination methods is available in [28].",
"Here, we will provide a brief outline of the machine learning photo-z process for context, but we primarily focus on the utilization of the naturally available probability information for each galaxy produced during photo-z evaluations with SPIDERz.",
"The general technique we propose in this work for utilizing the probability information, however, should theoretically be relevant to any photo-z estimation method which provides the requisite probability information for individual galaxies.",
"Generally speaking, machine learning photo-z codes perform two main processes: training and evaluation.",
"The output of the training process is a mapping from band magnitudes (and potentially additional information) to redshift.",
"The collection of mappings comprise a predictive model that can be used to make photo-z predictions on evaluation galaxies.",
"SPIDERz utilizes support vector classification to make photo-z predictions, where bins of redshift are assigned class labels, and photo-z estimation is performed via the solution to a multi-class classification problem.",
"SPIDERz solves the multi-class problem with a “one against one” or “pairwise coupling” approach that treats the complex multi-class problem as a series of simpler binary class problems consisting of every possible pairing of classes (in this case redshift bins).",
"Thus for a system comprised of $m$ distinct classes ($m$ redshift bins in this case), SPIDERz formulates and solves $m(m-1) \\over 2$ separate binary classification problems, choosing the more likely class (redshift bin) in each binary pairing.",
"Each instance of classification in favor of a particular redshift bin can be regarded as a `vote' for that class.",
"The entire collection of $m(m-1) \\over 2$ votes forms a distribution (see Figure REF for examples) that we call an `effective' probability distribution (EPDF) for each galaxy, with the relative probability of each redshift bin proportional to the number of times the corresponding class was chosen as the best binary solution.",
"This EPDF is not continuous, but rather is resolved to the bin-width level.",
"Discrete $z_{phot}$ estimates, if they are desired, can be obtained for each galaxy by simply taking the redshift bin with the highest number of votes.",
"Examples of actual EPDFs for individual galaxies in the COSMOSx3D-HST data set described in §REF are shown in Figure REF .",
"The top panel shows the presence a uniform singular probability peak characteristic of typical cases where $z_{phot} \\approx z_{spec}$ .",
"The middle and bottom panels show distributions with multiple peaked probabilities throughout wide redshift distances, which is a feature that is typical of many inaccurate $z_{phot}$ estimates.",
"We use the terminology “effective PDF” because of the way that all bins are used in comparison, thus artificially inflating low probability bins due to the inevitable pairwise comparisons of two low probability bins.",
"However the overall shape of the EPDFs, in regard to higher probability bins which are the only ones relevant in this analysis, approaches that of a true probability distribution.",
"To breifly illustrate how the EPDF compares to a true probability distribution function, if one desired to mitigate the effect of low probability bin inflation in the EPDFs for comparisons between the summed EPDF for all galaxies and the known $N(z)$ distribution in testing determinations, one would apply weights to the EPDFs that are proportional to the fractional population of training galaxies in each redshift bin relative to the total training galaxy population.",
"Weights are determined for each redshift bin $\\Delta z_i$ by $w_i = {{N(\\Delta z_i)} \\over {N}}$ $\\displaystyle \\sum _{i=1}^{l} w_i = 1,$ where $N = \\displaystyle \\sum _{i}^{l} N(\\Delta z_i)$ and $l$ is the number of redshift bins.",
"Weights are applied to the EPDF by $P_{w}(\\Delta z_i) = l(\\displaystyle \\sum _{i=1}^{l} P_{Eff}(\\Delta z_i)* w_i)),$ where $P_{w}(\\Delta z_i)$ is the weighted probability for some redshift bin $\\Delta z_i$ , and $P_{Eff}(\\Delta z_i)$ is the probability given by the unweighted EPDF.",
"In this way, as shown in Figure REF , we can see that there is meaningful probability information in the EPDFs and that they can be made, in aggregate, to approach a true probability distribution with weighting.",
"For the present work, however, the degree of fidelity of the EPDFs to true probability distribution functions is not important, as only the highest probabilty bins are relevant, and so no weighting is applied — the analyses in this work simply use the raw EPDFs as output by SPIDERz.",
"The reason for this is severalfold: Firstly, we would like to demonstrate the method of this work with the raw output of a machine learning classifier, for the simplest, most general situation.",
"Further, while it is the case that in the analyses here the training set and the evaluation set have practically the same redshift distribution, that is not necessarily the case for all generic photo-z evaluations going forward, so weighting the individual output galaxy probabilities by the particular redshift distribution of the training set may not be appropriate.",
"Additionally, in this work we are focusing on the utility of individual galaxy probability functions.",
"If one were to weight those functions individually by the cumulative redshift distribution of a given single training set, the amount that high probability peaks are scaled up and down would be highly dependent on the particulars of that training set, and would be different for another training set; therefore values investigated quantitatively here would be entirely training set dependent, and certain training sets would result in a weighting where no individual galaxies have high probability peaks at high redshifts.",
"We note that to produce Figure REF , due to the relatively limited population of galaxies at high redshifts in the COSMOSx3D-HST data set used in this analysis, the presence of unpopulated redshift bins at high $z$ in a training set is often unavoidable.",
"So in order to present a useful comparison between the summed EPDFs and distribution of discrete most probable estimates $N(z_{phot})$ produced in SPIDERz determinations with the actual redshift distribution $N(z_{spec})$ for this particular data set we utilized a subset of test data galaxies restricted to $z < 2.9$ , ensuring all redshift bins are populated, for this particular calculation only.",
"By default, SPIDERz chooses the most probable (commonly occurring) redshift bin as a single valued photo-z estimate for the galaxy.",
"In this analysis we use this method for discrete photo-z predictions, such as those shown in Figure REF .",
"In this work we seek a method to identify potential catastrophic outliers in such photo-z predictions.",
"SPIDERz also allows users flexibility in redshift bin size.",
"We generally find determinations have increased accuracy and precision when smaller bin sizes are used, however the optimal bin size for any determination will be dependent on the size and nature of the training set (decreasing the bin size for determinations lowers existing parameter overlap between training and evaluation sets) and can be approached via trial-and-error or approximated with the bin size introduced as an additional parameter in a grid search [28]."
],
[
"Strategy for identifying potential catastrophic outliers with EPDFs",
"To identify potential catastrophic outlier photo-z estimates we focus on the existence of individual galaxies' EPDFs displaying multiple probability peaks or, somewhat equivalently, a `weak' primary probability peak.",
"There is some ambiguity in what constitutes multiple substantial probability peaks in a galaxy's EPDF.",
"In particular, a secondary peak is more likely to be significant if it is closer in height (probability) to the primary (highest probability) peak, and also if it is located farther away in redshift from the primary peak.",
"Let us denote the ratio of the probability of a secondary peak $i$ to the primary peak in a galaxy's EPDF as $p_f = {{p_{i}} \\over {p_{max}}},$ where $p_{max}$ is the probability of the primary (highest probability) peak, and let us also denote the redshift distance between that secondary peak and the primary peak as $\\Delta {z_{peak}}$ .",
"Thus a designated minimum value for $p_f$ ($p_{f,min}$ ), and a designated minimum value for $\\Delta {z_{peak}}$ ($\\Delta {z_{peak,min}}$ ), can serve as filter values above which a multiply peaked EPDF is flagged.",
"If at least one redshift bin in an EPDF distribution satisfies both of the $\\Delta {z_{peak}} > \\Delta {z_{peak,min}}$ and $p_f > p_{f,min}$ criteria, the galaxy is flagged as a potential catastrophic outlier.",
"The optimal values for $p_{f,min}$ and $\\Delta {z_{peak,min}}$ will vary depending on factors such as the redshift range of test data and designated bin size, and the relative importance of flagging more catastrophic outliers versus avoiding spurious flaggings.",
"The simplest way to deal with flagged galaxies would be to remove them from analyses which rely on photo-zs.",
"This would, of course, remove some fraction of catastrophic outliers and other outliers, along with some fraction of non-outliers.",
"In §REF we show that the former number can be relatively high and the latter relatively low.",
"In this analysis going forward we consider flagging being somewhat equivalent to removal from consideration, while acknowledging that other strategies, such as de-weighting while not completely eliminating flagged galaxies in analyses, are possible and likely desirable in some circumstances."
],
[
"Results",
"In this section, we present the results from our study of using EPDFs to identify probable outlier and catastrophic outlier galaxy estimates as discussed in §.",
"We begin with a discussion of the two test data sets used in these photo-z analyses.",
"Next we provide results from photo-z determinations performed with SPIDERz on the test data sets — both with and without application of the EPDF outlier identification method discussed in §.",
"Metrics of performance of this method are provided for a range of values for the identification criteria, assuming here a simple removal of flagged galaxies.",
"Figure: The best discrete photo-z estimation (most probable redshift, as discussed in §) as determined by SPIDERz versus the actual redshift for the COSMOSx3D-HST data set discussed in § for a realization of the five-band (toptop) and ten-band (bottombottom) cases.",
"The catastrophic outlier identification method discussed in § was employed for these determinations with the Δz peak,min =0.6\\Delta {z_{peak,min}}=0.6 criteria and the flagged galaxies are shown by red crosses.",
"These determinations were performed with a training set consisting of 1200 galaxies chosen at random and an evaluation testing set consisting of the other 2504 galaxies.",
"A bin size of 0.1 was used.",
"Outliers in a determination are defined by equation , shown as those points lying outside of the two diagonal lines.",
"The density of points within the lines is quite high — only 2.6% of points lie outside of the lines as outliers for the ten-band case (BOTTOM) before flagging and 6.7% for the five-band case (TOP)."
],
[
"Test Data Sets",
"To obtain a data set of real galaxies with publicly available spectroscopic redshifts containing sources throughout a large redshift range including higher redshifts we use spectroscopic redshifts from the 3D-HST survey performed with the Hubble Space Telescope and reported in [34] that overlap with photometry from the COSMOS2015 photometric catalog [30] which reports photometry for over half a million objects in the COSMOS field [38].",
"For spectroscopic redshifts we use the reported “best available” redshift measurement and eliminate those flagged as having their redshift obtained from photometry or as being stars.",
"This results in a data set of 3704 galaxies, of which 383 (10.3%) have $z > 2$ and 948 (25.6%) have $ z > 1.5$ .",
"The $N(z)$ distribution for this data set is shown in Figure REF .",
"These data span an $i$ -band magnitude range from 27.05 to 18.16 with a median of 23.74.",
"In order to form an additional test set with a significantly larger number of real galaxies, we also utilize galaxies from the COSMOS2015 photometric catalog that contain particularly reliable, previously estimated photometric redshifts derived from a large number of photometric bands.",
"As the COSMOS2015 catalog provides photometry for some galaxies in up to 31 optical, infrared, and UV bands, those galaxies with (i) magnitude values for at least 30 bands of photometry, and (ii) for which the stated $\\chi ^2$ for the redshift estimate is $<1$ , and (iii) for which the stated photo-z value from the minimum $\\chi ^2$ estimate is less than 0.1 redshift away from the stated photo-z value from the peak of the pdf, can be considered to have highly reliable previous redshift estimates.",
"Applying these criteria result in a data set of 58622 galaxies spanning an $i$ band magnitude range from 27.17 to 19.00 with a median of 24.08.",
"For shorthand purposes we will refer to this set here as the “COSMOS-reliable-$z$ ” test data set.",
"The $N(z)$ distribution for this data set is also shown in Figure REF .",
"Although the COSMOS2015 catalog provides photometry in a potentially large number of optical, infrared, and UV bands, we choose to restrict our test analyses to the $u$ , $B$ , $V$ , $r$ , $i$ , $z+$ , $Y$ , $H$ , $J$ , and $Ks$ bands, and a subset of five of these bands, because with data sets approaching 30 bands of photometry, the distinction between photo-z estimation and spectroscopic redshift determination is somewhat muddled, and in any case this does not represent a realistic photometric situation for upcoming large surveys such as LSST, even for subsets which would have infrared survey overlap.",
"In the following sections we refer to test data consisting of only five optical bands ($u$ , $B$ , $r$ , $i$ , $z+$ ) as the `five-band case', which could resemble the default situation for obtaining photometric redshifts from a very large optical survey, and similarly refer to test data comprised of all ten aforementioned bands as the `ten-band case', which could resemble the situation for obtaining photometric results from a large optical survey that overlaps a large near-infrared survey.",
"For these bands we use aperture magnitudes measured in a 3” aperture.",
"The depths of the photometry for the bands are given in Table 1 of [30].",
"We have not utilized galaxies with missing photometry values in these bands — for the COSMOSx3D-HST test set the number of galaxies where this is the case is negligible, while for the COSMOS-reliable-$z$ test set applying this filter has almost no effect since this data set by definition contains 30 reliable bands of photometry.",
"Unless otherwise noted, all determinations are performed with randomly selected training and testing set populations of 1200 and 2504 galaxies, respectively for the COSMOSx3D-HST data set, and 5000 and 53622 galaxies for the COSMOS-reliable-$z$ data set.",
"Increasing the training population size beyond 1200 for the COSMOSx3D-HST data set produced only marginal improvements in photo-z accuracy.",
"For the COSMOS-reliable-$z$ data set we chose to maintain a training set to evaluation set size ratio of below 1:10 to more closely approximate the photo-z conditions of future large scale survey analyses than would be achieved with doing analyses with larger ratios.",
"We note that the galaxies in these data sets span the largest redshift range of publicly available real galaxy photo-z test data with photometry down to these magnitudes of which we are aware.",
"We also note that a significant limitation is posed on the performance accuracy of SPIDERz due to inadequate parameter overlap between training and evaluation galaxies in sparsely populated redshift regions, which, among other restrictive influences, imposes a lower limit on the redshift bin size that can be effectively used."
],
[
"Results for various parameter choices",
"Figure REF displays the estimated SPIDERz photo-z versus actual redshift for an example of typical determinations with the five-band and ten-band cases for the COSMOSx3D-HST data setdiscussed in §REF .",
"The EPDF outlier identification method discussed in § was then employed for these determinations with particular flagging parameters $\\Delta {z_{peak,min}}=0.6$ and $p_{f,min}=0.95$ .",
"Red data points indicate flagged potential catastrophic outlier estimates in these cases.",
"Estimates with the ten-band case are of course significantly better than with the five-band case.",
"To examine the influence of our proposed method for flagging potential catastrophic outliers in photo-z determinations, we performed an extensive analysis with test determinations on the five-band and ten-band cases for both the COSMOSx3D-HST and COSMOS-reliable-$z$ data sets using a range of $\\Delta {z_{peak,min}}$ values and $p_{f,min}$ values, redshift bin sizes, and training population sizes.",
"Perhaps surprisingly, we determine that appropriate values of $p_{f,min}$ are quite high, with any values below $p_{f,min} = 0.9$ resulting in an unacceptably large number of spurious flaggings.",
"We find variations in the designated value for $\\Delta {z_{peak,min}}$ greatly influence the performance of the outlier identification method, as measured by the relative numbers of correct outlier identifications versus spurious removal of non-outliers, however variations in $p_{f,min}$ produced marginal difference in the range $0.90 \\le p_{f,min} \\le 0.98$ .",
"Figure: Visualization of photo-z performance metrics from determinations performed by SPIDERz on the COSMOSx3D-HST data set discussed in § for the five photometric band case using a range of Δz peak,min \\Delta {z_{peak,min}} values and fixed p f p_{f} = 0.95, considering that all flagged galaxies would be removed from an analysis that relied on accurate photo-zs.",
"We also include the performance for the default case of no flagging on the left-most portion of the x-axis labeled “D.”.",
"The determinations were performed with a bin size of 0.1, and a training set consisting of 1200 galaxies chosen at random and an evaluation testing set consisting of the other 2504 galaxies, with results averaged over six determinations.",
"The performance metrics shown include the percentage of outliers (TOP), followed by the percentage of outliers removed (2nd from TOP), followed by the percentage of catastrophic outliers remaining (3rd from TOP), followed by the percentage of non-outliers removed (3rd from BOTTOM), followed by the percentage of catastrophic outliers removed (2nd from BOTTOM), and finally the percentage of removed galaxies that are outliers (BOTTOM).",
"The variance in performance across the six randomized realizations is indicated.Figure: Same as Figure but for the COSMOS-reliable-zz data set discussed in §.",
"The variance in performance across the six randomized realizations is indicated.",
"Results from this data set are quite similar to those from the COSMOSx3D-HST data set shown in Figure but with smaller error bars as would be expected from a much larger data set.Table: Improvements in RMS and R-RMS (defined by equation ), and the percentage of catastrophic outliers (O c O_c, defined in equation ) after flagging potential catastrophic outlier EPDFs in SPIDERz determinations on COSMOSx3D-HST test data for the five photometric band case for a range of Δz peak,min \\Delta {z_{peak,min}} and p f,min p_{f,min} values, with a redshift bin size of 0.1, assuming removal of flagged galaxies.",
"Six determinations were performed for every case, each with randomized training and evaluation testing sets consisting of 1200 and 2504 galaxies respectively, and results averaged.",
"The default case is for no flagging.",
"We also show the percentage of non-outliers (O non O_{non}) flagged.Figure: Redshift histogram of the number of outliers (left) and catastrophic outliers (right), both as defined in equations and respectively, present in one particular typical determination with the five photometric band case for the COSMOSx3DHST test data set compared to the numbers flagged through the use of the EPDF flagging method with flagging parameter values p f,min =0.95p_{f,min} = 0.95 and Δz peak,min =0.6\\Delta {z_{peak,min}} = 0.6.Figure: The percentage of non-outliers flagged through the use of the EPDF flagging method in bins of 0.1 in redshift (left) and in equally populated sextiles of ii-band magnitude from highest to lowest magnitude indexed with the median magnitude (right) for the COSMOS-reliable-zz test data set with flagging parameter values p f,min =0.95p_{f,min} = 0.95 and Δz peak,min =0.6\\Delta {z_{peak,min}} = 0.6.",
"The results here are for one particular representative determination.",
"The standard deviation from averaging over multiple determinations would be smaller than the plotting symbols in the ii-band case and small in the redshift case.",
"The redshift bins where a large fraction of non-outliers are flagged are those which are least populated in the sample generally.We also find that discrete photo-z accuracy is generally highest on this test data when using redshift bin sizes between 0.1 and 0.05; the use of larger bin sizes significantly reduced photo-z precision across all $z$ values and particularly at lower $z$ s, as expected, while the use of bin sizes less than 0.05 produced a significant number of unoccupied bins at higher redshifts and deteriorated parameter overlap between training and evaluation sets.",
"Figures REF and REF and Tables REF and REF show various performance metrics from determinations with SPIDERz using the EPDF outlier identification method on COSMOSx3D-HST test data.",
"Table REF highlights the percentage of outliers, percentage of outliers removed, percentage of removed galaxies that are outliers, percentage of non-outliers removed, percentage of catastrophic outliers removed, and finally the percentage of catastrophic outliers remaining for determinations on five-band and ten-band cases for this data set, with a range of values for $\\Delta {z_{peak,min}}$ and a fixed $p_{f,min}$ of 0.95, while figure REF provides a visual compendium of some of those quantities for the five-band case.",
"Table REF shows various metrics for several combinations of $\\Delta {z_{peak,min}}$ and $p_{f,min}$ values.",
"Figure REF shows a redshift histogram of the reduction in the number of catastrophic outliers and outliers present in a typical determination with the five-band case with one particular parameter value choice.",
"Figure REF shows performance metrics from determinations with SPIDERz using the EPDF outlier identification method on the COSMOS-reliable-z test data set.",
"Comparing Figures REF and REF it is clear that results from the two test data sets are quite similar but with significantly smaller error bars in the COSMOS-reliable-$z$ case as would be expected from a significantly larger data set.",
"We see that certain choices for $\\Delta {z_{peak,min}}$ and $p_{f,min}$ result in successfully flagging a high percentage (> 50%) of the catastrophic outliers while flagging a small percentage (2-4%) of the non-outlier galaxies.",
"On the other hand, low values of $\\Delta {z_{peak,min}}$ result in the flagging of a large percentage of the non-outlier galaxies.",
"It is also of interest to explore whether this method flags an excessive fraction of galaxies at higher redshifts and/or higher magnitudes.",
"In Figure REF we show the percentage of non-outliers flagged in bins of 0.1 in redshift (left panel) and in sextiles of $i$ -band magnitude (right panel) for the COSMOS-reliable-$z$ test data set with flagging parameter values $p_{f,min} = 0.95$ and $\\Delta {z_{peak,min}} = 0.6$ .",
"It is seen that less than 15% of non-outliers are flagged in the highest magnitude (dimmest flux) sextile but in a few of the least populated redshift bins in the sample roughly half of non-outliers are flagged.",
"This suggests that steps could be taken to mitigate this effect within certain low population redshift bins, as discussed in §."
],
[
"Discussion",
"In this work, we have considered the utilization of SPIDERz's effective redshift probability distributions for flagging likely catastrophic outlier photo-z predictions — gross mis-estimations defined by $|z_{phot} - z_{spec}| > 1$ — by considering galaxies with multiple or ill-defined peaks in photo-z probability separated by redshift.",
"We introduced a formalism with two threshold criteria: the minimum redshift separation of multiple peaks ($\\Delta {z_{peak,min}}$ ) and the minimum probability ratio of secondary probability peaks to the highest probability peak ($p_{f,min}$ ), as discussed in §, to preemptively flag potential catastrophic outlier estimates.",
"We implemented this method in SPIDERz photo-z determinations performed with real galaxy test data spanning a wide redshift range $0 < z < 4$ and utilizing limited photometric bands to estimate photometric redshift (see §REF ), testing a range of threshold values $\\Delta {z_{peak,min}}$ and $p_{f,min}$ (see §).",
"We found $\\Delta {z_{peak,min}}$ to have the greatest influence on the fraction of catastrophic outliers which were flagged, while $p_{f,min}$ was sub-dominant in this regard but most strongly correlated with flagging precision, with low values of $p_{f,min}$ leading to a higher number of non-outliers flagged.",
"Optimal values for $\\Delta {z_{peak,min}}$ and $p_{f,min}$ for any given application would result from striking an acceptable balance between more thoroughly flagging catastrophic outlier galaxies and reducing the number of spuriously flagged non-outlier galaxies.",
"We present results for a variety of choices of $\\Delta {z_{peak,min}}$ where this trade-off can be seen, particularly in Figure REF and Tables 1 and 2.",
"There are a range of options away from the lowest values of $\\Delta {z_{peak,min}}$ where the percentage of catastrophic outliers flagged is quite high and the percentage of non-outliers flagged is relatively low.",
"For all parameter choices, more non-outliers are flagged than outliers, but this is likely inevitable considering that in the default case more than 90% of the galaxies in the five-band case and 95% in the ten-band case are non-outliers.",
"We have seen that with proper choices for $\\Delta {z_{peak,min}}$ and $p_{f,min}$ EPDFs can be utilized to flag potential catastrophic outlier photo-z predictions with a high degree of overall effectiveness in determinations performed on a data set which spans a wide redshift range and contains realistic photometry in a limited number of wavebands.",
"As discussed in §, in a future large scale survey utilizing photometric redshifts, the simplest use of such flagging information would be to simply remove the flagged galaxies from science analyses in which catastrophic outlier redshift predictions are detrimental, such as weak-lensing cosmology.",
"Another simple option for utilization of flagging information could include de-weighting of potential catastrophic outliers in cosmological probes.",
"If such flagged galaxies are simply removed from analysis, there is, necessarily, a trade-off between more complete removal of actual catastrophic outliers and spurious removal of non-outliers.",
"In this work we present various options for the parameters $\\Delta {z_{peak,min}}$ and $p_{f,min}$ (discussed in §) which lead to different points on this trade-off continuum.",
"We show the various results for catastrophic outliers removed, spurious removals, and other metrics in Tables REF and REF and visualizations in Figures REF and REF .",
"It is seen that for a range of flagging parameter values a favorable ratio of total genuine catastrophic outlier flagging to spurious non-outlier flagging is obtained, for example flagging of significantly more than half of catastrophic outliers while spuriously flagging only 2-4% of non-outliers.",
"With the need to obtain precise redshift estimates satisfying photo-z error constraints for probing cosmological parameters and the abundance of galaxies that will be observed in future large photometric surveys, it may be reasonable in many cases to accept a slightly larger (although still low) percentage of overall spurious removals in exchange for maximizing the number of removed catastrophic outlier photo-z estimates.",
"It is important to note, however, that as seen in Figure REF a significant fraction (approaching half in the most dramatic cases) of non-outliers are flagged in a few of the more sparsely populated redshift bins, including some of those at higher redshifts.",
"This points toward a possible strategy beyond simple removal of flagged galaxies in these particular redshift bins in order to not lose for cosmological analyses such a large fraction of high redshift galaxies in a data set.",
"We will explore possible weighting strategies for this in a future work.",
"We do also note two crucial caveats regarding this: (1) that in this work, as mentioned in §, in order to approximate the photo-z conditions applying to future large scale surveys, we utilize much larger evaluation sets than training sets in this study.",
"Thus it is likely that by adopting a larger training to evaluation set size ratio than here, as has been done in many other photo-z studies in the literature, one could reduce the percentage of spuriously flagged non-outliers in the sparsely populated redshift bins given a similarly sized test data set.",
"Also, (2) it is likely the case that, for a given training to evaluation set size ratio and $N(z)$ distribution, there will be a lower percentage of spuriously flagged non-outliers in relatively sparsely populated redshift bins given a larger overall test data set.",
"However even with a very large training set high redshift bins will contain a higher proportion of potential catastrophic outliers and therefore spurious removals due to the degeneracy between Balmer and Lyman breaks in galaxy spectra.",
"While this analysis focused on utilization of EPDFs provided by SPIDERz, there is no reason that it should not be generalizable with analagous parameters to any photo-z estimation method which provides redshift probability distribution information for each galaxy.",
"While the parameters we used in this work to flag EPDF features, $\\Delta {z_{peak,min}}$ and $p_{f,min}$ , were effective in distinguishing likely catastrophic outliers, the optimal values of these parameters for a given purpose may be data set dependent to some extent.",
"Also other photo-z estimation codes and probability determination methods may or may not necessitate alternate parameter values and/or definitions to those employed in this work.",
"We also note that in general results in empirical photo-z estimation methods often depend on the degree of representativeness of the training set relative to the evaluation set."
]
] | 1709.01576 |
[
[
"Guarding Path Polygons with Orthogonal Visibility"
],
[
"Abstract We are interested in the problem of guarding simple orthogonal polygons with the minimum number of $ r $-guards.",
"The interior point $ p $ belongs an orthogonal polygon $ P $ is visible from $ r $-guard $ g $, if the minimum area rectangle contained $ p $ and $ q $ lies within $ P $.",
"A set of point guards in polygon $ P $ is named guard set (as denoted $ G $) if the union of visibility areas of these point guards be equal to polygon $ P $ i.e.",
"every point in $ P $ be visible from at least one point guards in $ G $.",
"For an orthogonal polygon, if dual graph of vertical decomposition is a path, it is named path polygon.",
"In this paper, we show that the problem of finding the minimum number of $ r $-guards (minimum guard set) becomes linear-time solvable in orthogonal path polygons.",
"The path polygon may have dent edges in every four orientations.",
"For this class of orthogonal polygon, the problem has been considered by Worman and Keil who described an algorithm running in $ O(n^{17} poly\\log n) $-time where $ n $ is the size of the input polygon.",
"The problem of finding minimum number of guards for simple polygon with general visibility is NP-hard, even if polygon be orthogonal.",
"Our algorithm is purely geometric and presents a new strategy for $ r $-guarding orthogonal polygons and guards can be placed everywhere in the interior and boundary of polygon."
],
[
"Introduction",
"The target of the art gallery problem is finding a set $ G $ of point guards in polygon $ P $ such that every point in $ P $ is visible from some members of $ G $ where a guard $ g $ and a point $ p $ are visible if the line-segment $ gp $ is contained in $ P $ .",
"It is shown that finding the optimum number of guards (the minimum guard set) required to cover an arbitrary simple polygon is NP-hard [18].",
"The art gallery problem is also NP-hard for orthogonal polygons and even remains NP-hard for monotone polygons [22].",
"In the orthogonal art gallery problem, it is assumed that the visibility is in orthogonal mode instead of standard line visibility.",
"In the polygon $ P $ and under orthogonal visibility(r-visibility), points $ p $ and $ q $ are visible from each other, if the minimum axis-aligned rectangle spanned by these two points is contained in $ P $ , This kind of visibility is , also, called r-visibility [20] i.e.",
"two points $ p $ and $ q $ are r-visible (orthogonally visible) from each other if the minimum area rectangle contained $ p $ and $ q $ has no intersection with the exterior of $ P $ .",
"A polygon is orthogonal if its edges are either horizontal or vertical, in every orthogonal polygon the number of vertical edges is equal to the number of horizontal ones.",
"Worman and Keil [24] studied the decomposition of orthogonal polygons into optimum number of r-star(star-shaped) sub-polygons that is equivalent to the orthogonal art gallery problem.",
"They presented a polynomial-time algorithm for the problem under r-visibility, so, they showed that the problem is polynomially solvable.",
"Their algorithm is processable in $ O(n^{17}poly \\log n) $ , hence, it is not so fast.",
"A cover of a polygon $ P $ by a set $ S $ of sub-polygons is defined such that the union of the sub-polygons in $ S $ be equal to $ P $ and the sub-polygons are required to be mutually disjoint except along their boundaries.",
"$ r $ -star is an orthogonal star-shaped polygon, and every r-star polygons are orthoconvex that will defined later.",
"Clearly, the problem of determining a minimum cover of a simple orthogonal polygon by r-stars is equivalent to determining a minimum set of r-visibility guards to guard the entire polygon i.e.",
"finding minimum covers by star-shaped sub-polygons is equivalent to finding the minimum guard set needed such that every point in the polygon is visible to some guards.",
"A linear-time ($O(n)$ -time) algorithm for covering a $ x $ -monotone orthogonal polygon with the minimum number of $ r $ -star polygons was presented by Gewali and et.",
"al.",
"[12].",
"Palios and Tzimas [21] considered the problem on class-3 orthogonal polygons without holes, i.e., orthogonal polygons that have reflex edges (dents) along at most 3 different orientations.",
"They presented an algorithm with time complexity of $ O(n+k \\log k)$ where $ k $ is the size of a minimum r-star cover(the size of output).",
"It is shown that problem is NP-hard on orthogonal polygons with holes by Beidl and Mehrabi [3].",
"A polygon is named tree polygon if dual graph of the polygon is an undirected graph in which any two nodes are connected by exactly one path(tree graph).",
"Also, They gave an algorithm for tree polygon in $ O(n) $ -time.",
"If vertex guards are only allowed (vertex guard variant), iterations of their algorithm yields an $ O(n^4) $ solution for general orthogonal polygons [6].",
"If $ S $ be a set of points in the polygon $ P $ , and every two points of $ S $ are not visible from each other, then $ S $ is called textithidden set.",
"So, If a hidden set is also a guard set, it is called hidden guard set.",
"Hoorfar and Bagheri [15] showed that finding the minimum number of guards is linear-time even under the constraint that the guards are hidden from each other, for some monotone polygons in the orthogonal art gallery problem.",
"In this paper, we study the orthogonal art gallery problem on path orthogonal polygons.",
"We take advantage of geometric properties of these polygons and we present an 1-pass $ O(n) $ -time algorithm to report the locations of a minimum-cardinality set of $ r $ -visibility guards to cover the entire polygon, where $ n $ is the number of vertices of given path polygon.",
"This is the one of the few purely geometric algorithm for this problem.",
"The first one is presented by Palios and Tzimas [21] and another one is given by Hoorfar and Bagheri [15], [14].",
"Actually, we generalize the ideas of the latter papers to yield faster algorithms for the problem on path orthogonal polygons.",
"Note that a path polygon is not tree polygon and have the dent edges in every four orientations, so the fastest known algorithm for it, have time complexity of $ O(n^{17}poly \\log n) $ and presented by Worman and Keil [24].",
"In the other word, we show that the $ r $ -guarding problem is linear-time solvable on path polygons without holes.",
"Comparing our results to the one by Worman and Keil, their algorithm works for a broader class of polygons, but is too slower.",
"In this paper, visibility means r-visibility (orthogonal visibility) and guarding is under r-visibility and also, monotonicity means $ x $ -monotonicity unless explicitly mentioned."
],
[
"Preliminaries",
"Assume $ P $ is an orthogonal polygon with $ n $ edges, the interior angles of all reflex vertices belonged to $ P $ are equal to $ \\frac{3\\pi }{2} $ and the interior angles of all convex vertices belonged to $ P $ are equal to $ \\frac{\\pi }{2} $ .",
"It is obvious that the number of reflex vertices of an orthogonal polygon with $ n $ vertices is equal to $ \\frac{n-4}{2} $ and the number of its convex vertices is equal to $ \\frac{n+4}{2} $ , exactly.",
"A decomposition of an orthogonal polygon $ P $ obtain by extending the edges of $ P $ incident to their reflex vertices until intersect the boundary.",
"Therefore, by plotting these vertical and horizontal line segments, at most $ (\\frac{n-2}{2})^2 $ rectangles are obtained, then we have a partition, such that the union of the parts of partition be equal to $ P $ and the parts be mutually disjoint except along their boundaries.",
"Every obtained rectangle part is named $ pixel $ .",
"If we assign a node to each pixel and then connect every two nodes of which their corresponding pixels are adjacent, by one edge, the created graph is called dual graph.",
"Some orthogonal polygons are named according to the type of their dual graphs.",
"An orthogonal polygon is called tree, if dual graph of the polygon is a tree and an orthogonal polygon is called $ k $ -width where its dual graph be a $ k $ -width tree.",
"If after decomposition of $ P $ , the vertices of all the pixels lie on the boundary of $ P $ , the polygon is named thin.",
"A tree polygon is a thin polygon without hole.",
"A vertical decomposition of an orthogonal polygon $ P $ with $ n $ vertices obtain by extending only the vertical edges of $ P $ incident to their reflex vertices until intersect the boundary.",
"So, after the vertical decomposition of $ P $ , at most $ \\frac{n-2}{2} $ rectangles will be obtained, this kind of partition is called vertical partition.",
"If we assign a node to each rectangle and then connect every two nodes of which their corresponding rectangles are adjacent, by one edge, the created graph is called dual graph of vertical decomposition.",
"An orthogonal polygon is called path , if dual graph of its vertical decomposition (not general decomposition) is a path, see figure REF .",
"Figure: Illumination of the vertical decomposition of a path polygon and its notations.The number of horizontal and vertical edges of an orthogonal polygon is the same.",
"If an edge $ \\epsilon _1\\in P $ has two endpoints of angle $ \\frac{\\pi }{2} $ , it is called tooth edge and if an edge $ \\epsilon \\in E $ has two endpoints of angle $ \\frac{3\\pi }{2} $ , it is called dent edge.",
"The edge direction is defined as same as the direction of its normal vector from interior to exterior of the polygon.",
"For $ i=1,2,3,4 $ , the class-$ i $ of orthogonal polygon contains the polygons which have dent edge only in $ i $ different directions [21].",
"Every orthogonal polygon has tooth edges in all four directions, every orthogonal $ x $ -monotone polygon has no dent edge in the directions perpendicular to the $ y $ -axis and every orthogonal $ y $ -monotone polygon has no dent edge in the directions perpendicular to the $ x $ -axis.",
"Not every polygon has dent edge, hence, an orthogonal polygon that has no dent edge is named orthogonally convex and sometimes it is also called orthoconvex polygon.",
"The orthoconvex polygon is both $ x $ -monotone and $ y $ -monotone i.e.",
"if polygon $ P $ is $ x $ -monotone and also $ y $ -monotone, then $ P $ is orthoconvex.",
"Assume we decompose a simple path polygon $ P $ with $ n $ edges into rectangular parts (rectangles) obtained by extending every vertical edges incident to their reflex vertices of $ P $ .",
"The dual graph $ G $ of this vertical decomposition is path which has two node of degree one.",
"The rectangles corresponding to these two nodes are called first rectangle and last rectangle.",
"Let $R =\\lbrace R_{1},R_{2},\\dots ,R_{m}\\rbrace $ be the set of rectangles, where $ m = \\frac{n-2}{2} $ , ordered from first to last rectangles according to the order of their corresponding nodes in the graph $ G $ .",
"For an illustration see figure REF , $ R_1 $ and $ R_{26} $ are first and last rectangle in this example, respectively.",
"We denote the upper horizontal edges of rectangle $ R_{i} $ by $ u_{i} $ and the lower horizontal edges of $ R_i $ by $ l_{i} $ .",
"Let consider that $ U =\\lbrace u_{1},u_{2},u_3,\\dots ,u_{m} \\rbrace $ and $ L=\\lbrace l_{1},l_{2},l_3,\\dots ,l_{m} \\rbrace $ .",
"Two consecutive rectangles are mutually disjoint except along their boundaries, hence, the intersection between every two consecutive rectangles $ R_i $ and $ R_{i+1} $ is a vertical segment that is denoted as $ s_i $ .",
"See the rectangle that is enclosed in a circle in figure REF for the illumination.",
"For every horizontal segment $ s $ the $ y $ -coordinate of every points on $ s $ is the same, so, we denote the $ y $ -coordinate of $ s $ by $ y(s) $ .",
"Similarly, For every vertical segment $ s^{\\prime } $ the $ x $ -coordinate of every points on $ s $ is the same, hence, we denote the $ x $ -coordinate of $ s^{\\prime } $ by $ x(s^{\\prime }) $ .",
"For a point $ p $ , $ y $ -coordinate and $ x $ -coordinate of $ p $ is denoted by $ y(p) $ and $ x(p) $ , respectively.",
"Without reducing generality, We assume that for every two different vertical edges $ e $ and $ e^{\\prime } $ , the $ x $ -coordinates of both of them is not same($ x(e)\\ne x(e^{\\prime }) $ ), hence, it is clear that for every $ 1 \\le j \\le m-1 $, $ y(u_{j})=y(u_{j+1}) $ or $ y(l_{j})=y(l_{j+1}) $ .",
"Also, we denote the horizontal edge of $ P $ that contains segment $ u_{k} $ by $ e(u_{k}) $ and the horizontal edge of $ P $ that contains segment $ l_{i} $ by $ e(l_{i}) $ .",
"Let the sets $ E_{U}=\\lbrace e(u_{j})|1 \\le j \\le m\\rbrace $ and $ E_{L}=\\lbrace e(l_{j})|1 \\le j \\le m\\rbrace $ be sets of horizontal edges of upper chain and lower chain of $ P $ ordered corresponding to their rectangles order.",
"In the set $ E $ of horizontal edges of $ P $ , $ e_{M} $ is called local maximum if $ e_M $ be higher than two neighbor horizontal edges ($ y(e_{M})> y(e_{M-1}) $ and $ y(e_{M})> y(e_{M+1}) $ ) and also, $ e_{m} $ is called local minimum if $ e_m $ is lower than two neighbor horizontal edges ($ y(e_{m})< y(e_{m-1}) $ and $ y(e_{m})< y(e_{m+1}) $ ).",
"If edge $ \\epsilon _1 \\in E_U $ be a local maximum, then the internal angles of its both endpoints are equal to $ \\frac{\\pi }{2} $ and if $ \\epsilon _2 \\in E_U $ be a local minimum, then the internal angles of its two endpoints are $ \\frac{3\\pi }{2} $ .",
"If edge $ \\epsilon _3 \\in E_L $ be a local minimum, then the internal angles of its both endpoints are equal to $ \\frac{\\pi }{2} $ and if $ \\epsilon _4 $ be a local maximum, then the internal angles of its two endpoints are $ \\frac{3\\pi }{2} $ .",
"If $ e( u_{M}) $ be local maximum then $ u_{M} $ is called local maximum and if $ e( u_{m}) $ be local minimum then $ u_{m} $ is called local minimum.",
"Every rectangle $ R_i $ has the height $ h_i=*{y(u_i)-y(l_i)} $ , so, in the set $ R $ of rectangles obtained by vertical decomposition, rectangle $ R_{l} $ is called local maximum if its height be greater then two adjacent rectangles ($ h_l>h_{l-1} $ and $ h_l>h_{l+1} $ ), and $ R_y $ is named local minimum if its height be less than two adjacent rectangles ($ h_y<h_{y-1} $ and $ h_y<h_{y+1} $ ).",
"Every rectangle has two adjacent rectangles except the first and last ones which are have only one adjacent rectangle.",
"Two objects $ o $ and $ o^{\\prime } $ in polygon $ P $ are defined as weak visible if every point of $ o $ is visible to some point of $ o^{\\prime } $ .",
"The interior area of polygon $ P $ , as denoted $int(P)$ , is the set of points that are bounded by $ P $ , the exterior area of $ P $ , as denoted $ext(P)$ , is the set of the nearby and far away exterior points and the boundary of $ P $ , as denoted $bound(P)$ , is the set of all points on the boundary of $ P $ .",
"Clearly, a polygon is union of $ int(P)$ and $ bound(P) $ .",
"If $ e $ be a horizontal line segment, then left endpoint of $ e $ is denoted as $ left(e) $ and right endpoints of $ e $ is denoted as $ right(e) $ .",
", also, if $ e $ be vertical, then top endpoint of $ e $ is denoted as $ top(e) $ and down endpoint of $ e $ is denoted as $ down(e) $ .",
"A star-shape polygon is a polygon $ \\rho $ that has some internal points so that the entire $ \\rho $ is straight-line visible form each of them.",
"Similarly, A $ r $ -star polygon is an orthogonal polygon $ \\varrho $ such that there exist some internal points which the entire $ \\varrho $ is orthogonally visible ($ r $ -visible) form each of them, the set of these internal points that are visible from the entire polygon is named kernel.",
"If a polygon has a kernel, we are able to cover it with only one guard.",
"Therefore, the problem of the decomposition an orthogonal polygon to the minimum number of $ r $ -star sub-polygons is equal to the problem of guarding an orthogonal polygon with the minimum number of $ r $ -guards, that is claimed in the previous section.",
"The bounding box of a set of objects is the minimum area box (rectangle) within which all the objects lie.",
"So, the bounding box of a polygon $ P $ is the axis-aligned minimum area rectangle within which all the points of $ P $ lie, for the orthogonal polygons, their bounding boxes are edge aligned, too.",
"If an $ x $ -monotone ($ y $ -monotone) orthogonal polygon has a horizontal (vertical) edge in common with its bounding box(rectangle), the polygon is called histogram, the common edge is named base.",
"If an orthoconvex polygon has an edge in common with its bounding rectangle, the polygon is named pyramid.",
"Every pyramid polygon is histogram, too.",
"An orthoconvex polygon that has two adjacent edges in common with its bounding box is called fan polygon.",
"Fan polygons are also pyramid and histogram and at least one of their vertices belongs to their kernel.",
"Base on the presented classification in paper [21], clearly, histogram belong to class-1 of orthogonal polygons, while $ x $ -monotone(or $ y $ -monotone) belong to class-2.",
"In the class-2 of orthogonal polygons, members have dent edges in two different directions, for monotone polygons these two directions are parallel.",
"This subclass of class-2 is denoted as class-2(a) and if the two directions are perpendicular, the subclass is denoted as class-2(b).",
"In the following, we prove the adapted lemma that was originally presented in [14].",
"Figure: (a)A r r -star polygon P P that has δ 1 \\delta _{1} and δ 2 \\delta _{2} .",
"Point M M is the intersection of δ 1 \\delta _{1} and δ 2 \\delta _{2} .",
"(b)The decomposition of polygon P P into four parts P 1 P_{1},P 2 P_{2},P 3 P_{3} and P 4 P_{4} which are fan polygonsAn orthogonally convex (orthoconvex) polygon $ P $ is $ r $ -star, if the leftmost and rightmost vertical edges of $ P $ are mutually weak visible and the upper and lower horizontal edges of $ P $ are mutually weak visible, too.",
"Because the leftmost and rightmost vertical edges of $ P $ are mutually weak visible, there exists a horizontal line segment $ \\delta _{1} $ which is connecting the leftmost and rightmost vertical edges of $ P $ such that lies in $ P $ .",
"Similarly, Because the upper and lower horizontal edges of $ P $ are mutually weak visible, there exists a vertical line segment $ \\delta _{2} $ which is connecting the upper and the lower horizontal edges of $ P $ such that lies in $ P $ .",
"If $ \\delta _{1} $ connects the leftmost and rightmost vertical edges of $ P $ and $ \\delta _{2} $ connects the upper and lower horizontal edges of $ P $ then they have an intersection $M$ that is contained in $ P $ .",
"$ \\delta _{1} $ and $ \\delta _{2} $ divide $P$ into 4 parts $ P_{1}$ ,$ P_{2}$ ,$ P_{3}$ and $ P_{4}$ .",
"All obtained pars are fan polygons with $M$ as their common core vertex.",
"In every part, the entire $ M $ it is in the kernel, hence, if guard $ g $ is placed in the kernel, every point in $ P $ is visible to it.",
"In the next section, we present a linear-time exact algorithm for finding the minimum guarding of orthogonal path polygons.",
"Our algorithm uses the geometric approach that is presented in our previous researches [15], [14] to improved and obtain new results for orthogonal art gallery problem.",
"Using this geometric approach instead of current graph theoretical leads to the algorithms with improved and better time complexity.",
"In this approach, we find the exact geometric positions of the point guards.",
"Therefore, some of our definitions and notations is similar to our cited papers."
],
[
"An Algorithm for Guarding Path Galleries",
"The path polygon has this property which can be divided into a number of sub-polygons, each of which can covered independently.",
"The shortest watchman route of these sub-polygons is an orthogonal straight line segment.",
"Also, for every sub-polygon we will prove that there is an optimum guard set which is all its guards are placed on the shortest watchman route of the sub-polygon.",
"Hence, we will find that optimum guard set that is located on a set of line segments and it reduces the execution time of the algorithm.",
"Besides that we will show that the visibility areas of all the guards that are located in a sub-polygon have not any effective intersections with the visibility areas of the guards that are located in another sub-polygons.",
"So, the minimum number of guards that are required for guarding path polygon will be equal to the sum of the minimum numbers of guards that are required for the obtained sub-polygons.",
"These sub-polygons are named balanced orthogonal polygon that are monotone and straight-line walkable.",
"Straight-line walkable polygon means a polygon that its shortest watchman route(path) is a line segment i.e.",
"a mobile guard can cover the entire polygon by walking back and forth on a straight route.",
"For orthogonal polygon, this concept corresponds to the concept of balanced polygon.",
"Actually, the described polygons have a area as named corridor that straight shortest path is a part of it.",
"For example, consider a histogram polygon, its base edge is a watchman route that is a part of its corridor.",
"We will find this corridor using a ray-shooting (beam throwing) method in the next subsection.",
"A path polygon is not necessarily straight walkable (or balanced), therefore, we will decompose a path polygon into the minimum number of balanced parts, then locating guards for every part, separately.",
"At the first, the path polygon belongs to class-4 of the described orthogonal classification, but after this decomposition all the obtained parts are belong to class-2, because all the dent edges of path polygon that have horizontal direction are removed after the partition."
],
[
"The Decomposition of a Path Polygon into the Balanced Parts",
"Suppose $ P $ be path polygon with $ n $ vertices that set $ R $ be its rectangle parts that are obtained after vertical decomposition and $ U $ and $ L $ be the sets of upper and lower edges of these obtained rectangles corresponding to the definitions that is explained in the previous section.",
"Two of these rectangles are sources which are have only one adjacent parts while another have exactly 2, one of source rectangle is considered as start and another as last, also corresponding to the described order.",
"The start rectangle and the general path polygon have one vertical edge in common, as denoted $\\varepsilon $ .",
"Propagate a light beam in rectilinear path perpendicular to $ \\varepsilon $ and also collinear with the $ X $ -axis.",
"Whole the light beam or a part of it passes through some members of the set $ R $ (name this subset $ R_{\\pi } $ ) and these rectangles together make a sub-polygon $ \\pi $ of $ P $ .",
"See figure REF (a).",
"The rectangles that belong to the polygon $ \\pi $ have this geometric property which $ y $ -ordinates of their upper edges are greater than $ y $ -ordinates of their lower edges i.e.",
"in polygon(sub-polygon) $ \\pi $ all the dent edges of upper chain are higher than all the dent edges of lower chain.",
"it is established that $ \\min _{u_{i} \\in \\pi } (y(u_{i}))\\ge \\max _{l_{j} \\in \\pi }(y(l_{j})) $ for every $ u_{i}$ and $l_{i} $ belongs to rectangles of $ R_{\\pi } $ .",
"Hence, there is a rectangular corridor $ \\varsigma $ which is connecting the leftmost and rightmost vertical edge of $ \\pi $ so that $ \\varsigma $ has no intersection with $ ext(\\pi ) $ and contained in $ \\pi $ .",
"If the leftmost vertical edge of sub-polygon $ \\pi $ is denoted as $ v $ , the rightmost vertical edge is denoted as $ v^{\\prime } $ , also let $ y_1=\\min _{u_{i} \\in \\pi } (y(u_{i})) $ and $ y_2=\\max _{l_{j} \\in \\pi }(y(l_{j})) $ , then $ \\varsigma $ is a axis-aligned rectangle spanned by two points with the coordinates $ (x(v), y_1) $ and $ (x(v^{\\prime }), y_2)$ .",
"Therefore, $ \\pi $ is walkable and balanced and every horizontal line segment that is connecting $ v $ and $ v^{\\prime } $ and located in $ \\varsigma $ can be its shortest watchman route.",
"After recognizing the first balanced sub-polygon(part) $ \\pi $ , we remove it from the path polygon $ P $ and iterate these operations to find next balanced parts until $ P $ is decomposed completely into balanced and monotone parts.",
"It is about the last rectangles of every obtained balanced sub-polygons(parts).",
"suppose that $ P $ is decomposed into the balanced parts (polygons) $ \\pi _{1},\\pi _{2},\\dots ,\\pi _{k} $ and let call the set of rectangles that is located in $ \\pi _i $ , $ R_{\\pi _i} $ , for every integer $ 1\\le i\\le k $ .",
"The last rectangle in $ R_{\\pi _i} $ is called cut rectangle because the intersection between $ \\pi _i $ and $ \\pi _{i+1} $ is the left edge of the cut rectangle(as denoted $ R_{cut} $ ).",
"In fact, cut rectangle $ R_{\\pi _i} $ is a border area and can either belong to the current part $ \\pi _i $ or next part $ \\pi _{i+1} $ .",
"If we want to find the path polygon $ P $ with the minimum number of guards, there may be a difference between two cases, that $ R_{cut} $ belongs to $ \\pi _i $ (case 1) or $ R_{cut} $ belongs to $ \\pi _{i+1} $ (case 2).",
"Let name previous rectangle of $ R_{cut} $ , $ R_{cut-1} $ , it is optimum that if $ R_{cut-1} $ is a local minimum, then we assign $ R_{cut} $ to $ \\pi _{i+1} $ (case 2).",
"We prove this proposition in lemma REF."
],
[
"Every cut rectangle is divided into three disjoint parts obtained by extending the horizontal edges of $ R_{cut-1} $ and $ R_{cut+1} $ incident to their common vertices with $ R_{cut} $ until intersect the boundary.",
"The parts are called as internal, middle and external parts, for an illustration see figure REF (b).",
"The part that adjacent to $ R_{cut-1} $ is called internal part, and the part that adjacent to $ R_{cut+1} $ is called external part and third one is called middle part.",
"It is necessary to place a guard in cut rectangle for covering it, because it is impossible that the interior of middle part be guarded with an $ r $ -guard that is not located in the cut rectangle $ R_{cut} $ .",
"If the previous rectangle $ R_{cut-1} $ be a local minimum, then we delete the cut rectangle from the set $ R_{\\pi _i} $ and allocate it to the set $ R_{\\pi _{i+1}} $ .",
"Using this strategy reduces the number of required guards in some cases.",
"For simplicity, we claim that: There exists a minimum cardinality guard set $ G={g_1, g_2,g_3\\dots ,g_{opt}} $ for a path polygon $ P $ so that all guards are located in the corridors.",
"In these paper, we want to find the guard set $ G $ for path polygon $ P $ that is optimum and all its guards are located in corridors $ \\varsigma _{1},\\varsigma _{2},\\dots ,\\varsigma _{k} $ of the obtained balanced sub-polygons $ \\pi _{1},\\pi _{2},\\dots ,\\pi _{k} $ that $ k $ is the minimum number of sub-polygons($ 1\\le k \\le \\lfloor \\frac{n}{4}\\rfloor $ ).",
"It is the optimum for guarding path polygon $ P $ that if $ R_{cut-1} $ is a local minimum, then we assign $ R_{cut} $ to $ \\pi _{i+1} $ instead of assigning it to $ \\pi _i $ .",
"Figure: Four different cases occur for assigning the cut rectangle R cut R_{cut} to π i \\pi _i or π i+1 \\pi _{i+1} .Suppose that after the decomposition of the path polygon $ P $ , for an integer $ i $ , $ \\pi _i $ and $ \\pi _{i+1} $ be two adjacent sub-polygons and a cut rectangle $ R_{cut} $ is located between them, as shown in figures REF .",
"Let $ \\varsigma _i $ and $ \\varsigma _{i+1} $ are corridors of $ \\pi _i $ (or $ \\pi _i \\cup R_{cut} $ ) and $ \\pi _{i+1} $ (or $ \\pi _{i+1} \\cup R_{cut} $ ), respectively.",
"The previous rectangle of $ R_{cut} $ is called $ R_{cut-1} $ and the next rectangle of $ R_{cut} $ is called $ R_{cut+1} $ .",
"As already mentioned, because the interior of middle part of cut rectangle $ R_{cut} $ is not orthogonally visible from any points of $ P-R_{cut} $ , it is necessary to place a guard $ g $ in $ R_{cut} $ for guarding it.",
"Where is the best position for this guard $ g $ ?",
"locating $ g $ in the intersection between $ R_{cut} $ and $ \\varsigma _i $ (as denoted $ R_{cut}\\cap \\varsigma _i $ ) or in the intersection between $ R_{cut} $ and $ \\varsigma _{i+1} $ (as denoted $ R_{cut}\\cap \\varsigma _{i+1} $ ) is better than anywhere else in $ R_{cut} $ .",
"If we locate $ g $ in $ R_{cut}\\cap \\varsigma _i $ , $ g $ guard $ R_{cut} $ and some rectangles before it which belong to $ \\pi _i $ and if we locate $ g $ in $ R_{cut}\\cap \\varsigma _{i+1} $ , $ g $ guard $ R_{cut} $ and some rectangles after it which belong to $ \\pi _{i+1} $ .",
"Which one lead to the minimum guarding of path polygon $ P $ ?",
"There are four different cases which are shown in figure REF .",
"In two cases (a) and (b), $ R_{cut-1} $ is a local minimum (because $ h_{cut-1}<h_{cut} $ and $ h_{cut-1}<h_{cut-2} $ ), by placing $ g $ in the area $ h_{cut} \\cap \\varsigma _i $ the rectangle $ R_{cut-1} $ is guarded but the rectangle $ R_{cut-2} $ is not guarded (completely).",
"So, certainly, there is a guard $ g^{\\prime } $ in the guard set that cover (guard) rectangle $R_{cut-2} $ and we know that the height of $ R_{cut-2} $ is higher than the height of $ R_{cut-1} $ , hence $ g^{\\prime } $ can also guard $ R_{cut-1} $ completely.",
"So, if the rectangle $ R_{cut-1} $ be local minimum, then locating $ g $ in the area $ h_{cut} \\cap \\varsigma _i $ is not useful.",
"Therefore, it is better to locate $ g $ in the area $ h_{cut} \\cap \\varsigma _{i+1} $ .",
"It happens if we assign the cut rectangle $ R_{cut} $ to $ \\pi _{i+1} $ instead of assigning it to $ \\pi _i $ (whether $ R_{cut+1} $ be local minimum or not).",
"In the case (c), $ R_{cut+1} $ is a local minimum (because $ h_{cut+1}<h_{cut} $ and $ h_{cut+1}<h_{cut+2} $ ), by placing $ g $ in the area $ h_{cut} \\cap \\varsigma _{i+1} $ the rectangles $ R_{cut} $ and $ R_{cut+1} $ are guarded but the rectangle $ R_{cut+2} $ is not guarded (completely).",
"So, certainly, there is a guard $ g^{\\prime } $ in the guard set that cover (guard) rectangle $R_{cut+2} $ and we know that the height of $ R_{cut+2} $ is higher than the height of $ R_{cut+1} $ , hence $ g^{\\prime } $ can also guard $ R_{cut+1} $ completely($ g^{\\prime } $ is located somewhere in $ \\varsigma _{i+1} $ ).",
"So, if the rectangle $ R_{cut+1} $ be local minimum, then locating $ g $ in the area $ h_{cut} \\cap \\varsigma _{i+1} $ is not useful.",
"Therefore, it is better to locate $ g $ in the area $ h_{cut} \\cap \\varsigma _{i} $ .",
"It happens if we assign the cut rectangle $ R_{cut} $ to $ \\pi _i $ instead of assigning it to $ \\pi _{i+1} $ , while $ R_{cut-1} $ is not local minimum.",
"In the case (d), both of $ R_{cut-1} $ and $ R_{cut+1} $ are not local minimum.",
"The rectangle $ R_{cut-1} $ is not local minimum and the height of $ R_{cut-1} $ is higher than the height of $ R_{cut-2} $ , so, for guarding $ R_{cut-1} $ it is necessary to place a guard in area $ (R_{cut-1}\\cup R_{cut})\\cap \\varsigma _i $ .",
"Also, the rectangle $ R_{cut+1} $ is not local minimum and the height of $ R_{cut+1} $ is higher than the height of $ R_{cut+2} $ , so, for guarding $ R_{cut+1} $ it is necessary to place a guard in area $ (R_{cut}\\cup R_{cut+1})\\cap \\varsigma _{i+1} $ .",
"Well, we do not need two guards in $ R_{cut} $ then, only for simplicity, we locate one guard in area $ R_{cut}\\cap \\varsigma _i $ and another guard in area $ R_{cut+1}\\cap \\varsigma _{i+1} $ .",
"It happens when we assign the cut rectangle $ R_{cut} $ to $ \\pi _i $ instead of assigning it to $ \\pi _{i+1} $ .",
"Figure: An illustration of the cases occur when two adjacent rectangles of R cut R_{cut} are located on the same side.In four described cases that are shown in figure REF , two adjacent rectangles $ R_{cut-1} $ and $ R_{cut+1} $ are located on different sides of $ R_{cut} $ .",
"Clearly, if the rectangles $ R_{cut-1} $ and $ R_{cut+1} $ are located in the same side, four another cases are occurred that they are similar to the previous four cases.",
"For an illumination see figure REF , the case that is shown in (b) is equal to (a) and the case that is shown in (d) is equal to (b).",
"Therefore, we do not focus on these new four cases.",
"Remember the decomposition of path polygon $ P $ into the balanced sub-polygons and suppose that we find first balanced sub-polygon of $ P $ , so, we remove it from $ P $ and iterate algorithm for $ P-\\pi $ until $ P $ is decomposed into several balanced $ x $ -monotone polygon.",
"We remove the rectangles belong to $ \\pi $ belong to $ R $ .",
"We know the members of $ R $ are ordered and labeled from 1, after removing, we relabel the remained members from 1, again, to simplify the description of the algorithm.",
"Certainly, the same processes will be occurred for $ U $ and $ L $ .",
"The number of iterations is equal to the cardinality of $ R $ (in the beginning).",
"Therefore, the time complexity of the decomposition path polygon $ P $ into balanced sub-polygons is processable in the linear-time corresponding to the size of $ P $ .",
"Now, we describe the linear-time algorithm for decomposition $ P $ into the balanced sub-polygons.",
"[] an path polygon with $ n $ vertices minimum number of balaced monotone polygons set $ min_u=u_1$ and $ max_l=l_1 $ set of rectangles $ R \\ne \\emptyset $ $ u_i>max_l $ or $ l_i<min_u $ $ i-2 = 1 $ or $ R_{i-2} $ is not local minimum $ R = R - \\lbrace R_1,R_2,\\dots ,R_{i-2},R_{i-1} \\rbrace $ $ U = U - \\lbrace u_1,u_2,\\dots ,u_{i-2},u_{i-1} \\rbrace $ $ L = L - \\lbrace l_1,l_2,\\dots ,l_{i-2},l_{i-1} \\rbrace $ $ R = R - \\lbrace R_1,R_2,\\dots ,R_{i-2}\\rbrace $ $ U = U - \\lbrace u_1,u_2,\\dots ,u_{i-2}\\rbrace $ $ L = L - \\lbrace l_1,l_2,\\dots ,l_{i-2}\\rbrace $ refresh the index of $ R $ , $ U $ and $ L $ starting with 1 reset $ min_u=u_1$ and $ max_l=l_1 $ set $min_u=\\min (min_u,u_{i})$ and $max_l=\\max (max_l,l_{i})$ The algorithm for decomposition path polygon $ P $ into the balanced sub-polygons.",
"Every balanced and monotone polygon $ \\pi $ has an axis-aligned rectangular area $ \\varsigma $ (named corridor) which is connecting the leftmost and rightmost edges of $ \\pi $ .",
"This area is also connecting the lowest dent edge of upper chain and the highest dent edge of lower chain.",
"Suppose that $ P $ is decomposed into a set of the balanced sub-polygons $ \\pi _{1},\\pi _{2},\\dots ,\\pi _{k} $ and $ \\varsigma _{1},\\varsigma _{2},\\dots ,\\varsigma _{k} $ be their corridors, respectively.",
"So, if $i \\ne j$ , There is no point in the interior of $ \\varsigma _i $ such that orthogonally visible from $ \\varsigma _j $ .",
"Due to this fact, if we optimally cover $ P $ so that all the guards are located on the corridors, guarding each of sub-polygons can be done independently i.e.",
"the minimum number of guards for guarding the entire polygon is the sum of the minimum number of guards that are necessary for every sub-polygons.",
"One addition point is about proving the explained claim.",
"To prove claim REF , we present an algorithm in the next sections and prove that its results is optimum.",
"In the previous subsection, we explained that every balanced (walkable) polygon has a rectangular area, named corridor which is the entire polygon is weak visible from it.",
"Now, we describe an algorithm to find the minimum number of guards and their positions for an orthogonal and monotone balanced polygon, such that all guards is only located in the corridor.",
"The presented algorithm in this section is the improved version of the algorithm that is already presented in our paper [15].",
"Assume that $ P $ is a balanced orthogonal monotone with $ n $ vertices, after vertical decomposition, the sets $ R $ , $ U $ , $ L $ , $ E_L $ and $ E_U $ are obtained for the polygon $ P $ according to their definitions.",
"Let $ \\varepsilon $ and $ \\varepsilon ^{\\prime } $ be the leftmost and rightmost vertical edges of $ P $ and $ e_{min} $ and $ e_{max} $ be the lowest horizontal edge of the upper chain of $ P $ and the highest horizontal edge of the lower chain of polygon $ P $ , respectively.",
"An axis-aligned rectangular area that is contained in $ P $ and spanned by points $ (x(\\varepsilon ),y(e_{min})) $ and $ (x(\\varepsilon ^{\\prime }),y(e_{max})) $ is named corridor of $ P $ .",
"The corridor of a balanced monotone polygon $ P $ is not empty and denoted as $ \\varsigma _p $ .",
"For a horizontal edge $ e $ of the polygon $ P $ , the set of every point $p\\in P$ which there is a point $ q \\in e $ such that $ pq $ is a line segment normal to $ e $ and completely inside $ P $ , is named orthogonal shadow of $ e $ , as denoted $ os_e $ (for abbreviation).",
"For the balanced monotone orthogonal $ P $ with $ n $ vertices, we present algorithm REF to find the minimum number of guards and their positions.",
"In the following, we explain the details of the algorithm and illustrate it.",
"First, we find all tooth edges of the set $ E = E_L \\cup E_U $ and call the obtained set as $ D $ .",
"For every $ d_i \\in D $ , we compute orthogonal shadow of $ d_i $ as $ os_{i} $ .",
"Let $ D=\\lbrace d_1, d_2,\\dots ,d_k\\rbrace $ and $ OS=\\lbrace os_1,os_2,\\dots ,os_k\\rbrace $ , ordered from left to right by $ x $ -coordination of their left vertical edges.",
"Every tooth edge $ t $ can be covered just with a guard which is placed in orthogonal shadow $ os_{t} $ , not anywhere else.",
"Suppose that $ P $ is a monotone orthogonal polygon and assume that the tooth edge $ t $ is guarded with $ \\gamma $ that is not placed in the shadow $ os_{t} $ , so, $ \\gamma $ is not in the $ x $ -coordinate of any points on the edge $ t $ .",
"Assume that the left and right endpoints of $ t $ are denoted as $ L_t $ and $ R_t $ , and let $ x $ -coordinate of $ L_t $ be greater than $ x $ -coordinate of $ \\gamma $ i.e.",
"$ x(L_t )> x(\\gamma )$ .",
"Clearly, $ \\gamma $ is visible to $ t $ , so, two endpoints $ R_t $ and $ L_t $ are visible to $ \\gamma $ .",
"Hence, there exists an axis-aligned rectangle spanned by the $ R_t $ and $ \\gamma $ is contained in $ P $ .",
"These two points is not in the same $ x $ -coordinate and even in the same $ y $ -coordinate, we know that every vertices of the rectangle belong to $ P $ as denoted $ R_t=(x(R_t),y(R_t)) $ , $ A=(x(R_t),y(\\gamma )) $ , $ B=(x(\\gamma ),y(R_t)) $ and $ g_d=(x(\\gamma ),y(\\gamma ))$ .",
"So, the horizontal edge $ BR_t $ is contained in $ P $ , Completely.",
"It is impossible, because $ t \\subset BR_t $ i.e.",
"if an edge of polygon be only a part of a segment which belong to the polygon, So, it is not really an edge.",
"Figure: An Illumination of the definitions, the tooth edges are shown in bold segments and all bold bordered rectangles of ς \\varsigma belong to Position Position that are the positions for guards.Hence, according to the lemma REF , at least, one guard must be placed in every orthogonal shadow of tooth edges.",
"We show in the algorithm that this number of guards is sufficient for guarding the entire polygon $ P $ and no extra guard is needed.",
"See figure REF , some orthogonal shadows of tooth edges of the upper chain may have intersection with some orthogonal shadows of tooth edged of the lower chain.",
"If it happen, we place a guard in the intersection between them to reduce the number of guards i.e.",
"if two different tooth edges $ t $ and $ t^{\\prime } $ belong to $ E_U $ and $ E_L $ , respectively, and the intersection between their orthogonal shadows is not empty (as denoted $ os_{e_1}\\cap os_{e_2}\\ne \\emptyset $ ) for guarding both of them one guard on the intersection is sufficient.",
"So, in the $ OS $ , we replace two members $ os_{t}$ and $ os_{t^{\\prime }}$ with the intersection of them($ os_{e_1}\\cap os_{e_2}$ ).",
"We know that the intersection of every 3 members of $ OS $ is empty and after these replacement the cardinality of set $ OS $ is equal to $ \\kappa \\le k $ .",
"Remember that the strategy of our guarding is placing guards in the corridor of balanced monotone orthogonal polygon $ P $ and we know the orthogonal shadow of every tooth edge of $ P $ has intersection with $ \\varsigma $ .",
"Assume that the rectangular area $Position_i=os_i \\cap \\varsigma $ (for every $ i $ between 1 and $ \\kappa $ ) and $Position=\\lbrace Position_1,Position_2,\\dots ,Position_\\kappa \\rbrace $ s.t.",
"$ (\\kappa \\le k) $ , ordered corresponding to their rectangle order.",
"Now, we know that the intersection of every 2 elements of set $ Position $ is empty.",
"The set $ Position $ is the positions for placing guards, one guard must be located in every member of $ Position $ , see figure REF again.",
"Using this strategy leads to find the positions for locating the minimum number of guards in the balanced orthogonal sub-polygon $ P $ in the linear time corresponding to number of its vertices $ n $ .",
"In algorithm REF , the set $ Position $ is the positions for the optimum guard set and the variable $ GuardNumber $ is the cardinality of the optimum guard set.",
"[] the horizontal edges of two chains of balanced monotone orthogonal polygon $ P $ with $ n $ vertices ($ E_L $ ,$ E_U $ ) the optimum number of point guards ($ GuardNumber $ ) and their positions ($ Position $ ) Set $ GuardNumber=0 $ and $ Position=\\emptyset $ Set $ e_{min}= the~lowest~horizontal~edge~of~E_U $ Set $ e_{max}= the~highest~horizontal~edge~of~E_L $ edge $ e_{i} $ belongs to $ E_L $ Interior angles of $right(e_i)$ and $ left(e_i)$ are equal to $\\frac{\\pi }{2}$ $ A_i=(x(left(e_i)),y(e_{max}) $ $ B_i=(x(right(e_i)),y(e_{min}) $ Set $ Position_i= $ rectangle spanned by $ A_i $ and $B_i $ Set $ Position_L=Position_L \\cup \\lbrace Position_i $ } $ GuardNumber++ $ ; edge $ e_{i} $ belongs to $ E_U $ Interior angles of $right(e_i)$ and $ left(e_i)$ are equal to $\\frac{\\pi }{2}$ $ A_i=(x(left(e_i)),y(e_{max}) $ $ B_i=(x(right(e_i)),y(e_{min}) $ Set $ Position_i= $ rectangle spanned by $ A_i $ and $B_i $ Set $ Position_U=Position_U \\cup \\lbrace Position_i $ } $ GuardNumber++ $ ; Merge the sorted lists $ Position_L$ and $ Position_U $ as sorted list $ Position $ .",
"horizontal segment $ position_{i} $ belongs to $ Position $ $ Position_i\\cap Position_{i+1}\\ne \\emptyset $ $ Position_i=Position_i\\cap Position_{i+1} $ $ Position=Position-\\lbrace Position_{i+1}\\rbrace $ $ GuardNumber-- $ ; Optimum guarding of a balanced monotone orthogonal polygon $ P $ with $ n $ vertices.",
"The positions of all guards are in the set $ SI $ and every elements of $ SI $ is a subset of corridor $ \\varsigma $ , so, all guards are located on corridor $ \\varsigma $ .",
"It is clear that the time complexity of the algorithm is as same as the cardinality of set $ E $ and it is linear-time according to the size of $ E $ .",
"The minimum number of guards for cover a balanced monotone orthogonal polygon $ P $ is equal to $ GuardNumber $ that is obtained by algorithm REF .",
"Suppose that $ GuardNumber $ guards is sufficient to guard the entire polygon $ P $ , using lemma REF prove that this number of guards necessary even for guarding the tooth edges of $ P $ .",
"Every area $ Position_i\\in Position $ is a subset of a $ r $ -star sub-polygon i.e.",
"if we decompose $ P $ into $ r $ -star parts(sub-polygons) then the kernels of every $ r $ -star sub-polygons has at least one point in the elements of $ Position $ , so the entire $ P $ is covered by these $ GuardNumber $ guards and their positions."
],
[
"Time Complexity of Algorithm",
"Now, we discuss about efficiency and time complexity of the whole solution and we explain that why our algorithm is processable in $ O(n) $ - time while n be the size of the input(path polygon $ P $ ).",
"Given path polygon $ P $ , for guarding $ P $ , we need to decompose the polygon into balanced parts with algorithm REF .",
"The vertical decomposition and finding optimum balanced orthogonal parts are solvable in the linear-time ($ O(n)-time $ ) because the number of rectangles is order of $ O(n)$ .",
"After that the problem is divided into subproblems which are finding minimum guard set for the obtained balanced sub-polygons(parts).",
"We use algorithm REF for guarding monotone parts, hence, subproblems is solvable in the linear-time corresponding to its size.",
"The total time of solving sub-problems is $ O(n) $ -time i.e.",
"the total number of vertices of the all obtained balanced parts is $ O(n) $ , so, algorithm REF is run in $ O(n) $ -time for all balanced parts.",
"Therefore, all computations handle in $ O(n) $ -time.",
"Finally, $ GuardNumber $ is referred to the optimum number of guards needed to cover path polygon $ P $ .",
"Therefore, we have proved the general result of the paper: There is a geometric algorithm that can find the minimum number of guards for given orthogonal path polygon $ P $ with $ n $ vertices, with r-guards in $ O(n) $ -time."
],
[
"Conclusion",
"We studied the problem of finding the minimum number of r-guards for an orthogonal path polygon.",
"This problem is a well-known version that is named orthogonal art gallery problem.",
"The total target in the orthogonal art gallery problem is finding the optimum set of r-guards $ G $ which is a set of point guards in polygon $ P $ that all points of the $ P $ are orthogonally visible from at least one r-guard in $ G $ .",
"We present an exact optimum algorithm for finding the guard set for path galleries.",
"We solved this problem in the linear time according to $ n $ where $ n $ is the number of sides of path polygon.",
"the space complexity of our algorithm is $ O(n) $ , too.",
"Many of the algorithms presented in this field are based on graph theory, but our proposed algorithm is based on geometric approach which is presented in paper [15].",
"This approach can lead to improved performance and efficiency in the algorithms.",
"We use our previous strategy [15] that was provided a purely geometric algorithm for the orthogonal art gallery problem where the galleries are monotone and extending the algorithm for path galleries.",
"Actually, we improved the time complexity of the orthogonal art gallery problem for path polygons from $ O(n^{17} poly\\log n) $ -time [24] to linear-time.",
"For the future works, we want to try to solve this problem for every simple orthogonal polygon with/without holes.",
"Both time and space complexity of our presented algorithm is order of $ O(n) $ and it is the best for these galleries."
]
] | 1709.01569 |
[
[
"Semianalytic calculation of cosmic microwave background anisotropies\n from wiggly and superconducting cosmic strings"
],
[
"Abstract We study how the presence of world-sheet currents affects the evolution of cosmic string networks, and their impact on predictions for the cosmic microwave background (CMB) anisotropies generated by these networks.",
"We provide a general description of string networks with currents and explicitly investigate in detail two physically motivated examples: wiggly and superconducting cosmic string networks.",
"By using a modified version of the CMBact code, we show quantitatively how the relevant network parameters in both of these cases influence the predicted CMB signal.",
"Our analysis suggests that previous studies have overestimated the amplitude of the anisotropies for wiggly strings.",
"For superconducting strings the amplitude of the anisotropies depends on parameters which presently are not well known - but which can be measured in future high-resolution numerical simulations."
],
[
"Introduction",
"As demonstrated in [1], symmetry breaking processes in early universe scenarios can lead to the formation of topologically stable line-like concentrations of energy, known as cosmic strings (for general reviews see [2], [3]).",
"These one-dimensional objects evolve and interact with each other, forming a cosmic string network.",
"Depending on their origin, strings can have significantly different properties and observational signatures.",
"Examples of theoretically well-motivated scenarios where the presence of cosmic strings is expected include brane inflation [4], [5], [6], [7], supersymmetric grand unified theories with hybrid inflation [8], [9], [10], [11], [12], [13] and many others [14], [15], [16], [17].",
"In most cases, cosmic strings are stable and survive to the present era, acting as fossils for these models.",
"Hence, quantitative bounds placed on string networks can lead to strong constraints on the underlying early universe model.",
"One difficulty is precisely that different models can produce strings with different properties, with varying observational predictions for the corresponding string networks.",
"Hence, in order to achieve reliable observational constraints on the underlying early universe models from cosmic string network phenomenology, one needs to develop an accurate description of cosmic string network evolution, taking into account the distinctive features of different types of cosmic strings.",
"One way to accomplish this task is through numerical simulations [18], [19].",
"This approach provides reliable results, but is currently limited by computer capabilities, especially when one tries to include non-trivial cosmic string features like world-sheet currents.",
"At present, multi-tension cosmic string networks and strings with currents are very time-consuming to model, and cannot be simulated with both high-resolution and sufficiently large dynamic range (see [20] for a recent field theory simulation of $pq$ -strings).",
"Further, numerical simulations have to be repeated for different values of cosmological and string parameters and are thus not particularly flexible for parameter determination through direct confrontation with observational data.",
"There is an alternative – and largely complementary – semi-analytic approach for the description of cosmic string network evolution based on the velocity-dependent one-scale (VOS) model [21], [22].",
"In this treatment it is much easier to add non-trivial features for cosmic strings [23], [24], [25], [26], [27], [28], [29], [30] allowing evolution over large dynamical ranges that cannot be achieved by numerical simulations.",
"However, semi-analytic descriptions involve free parameters, which can only be reliably calibrated by comparison to simulations.",
"As a result, a combination of such analytic descriptions and numerical simulations is at present the best approach for studying the evolution of cosmic string networks with non-trivial properties.",
"Among different methods for detecting observational signals from cosmic string networks, cosmic microwave background (CMB) anisotropies offer one of the most sensitive and robust probes [31].",
"Current results obtained using cosmic string network simulations [32], [33] and calibrated semi-analytic descriptions [34] yield very similar constraints for simple global cosmic strings, the current limit on the string tension being at the level of $G \\mu \\lesssim 10^{-7}$ .",
"However, as discussed above, it is the latter approach that allows us to go beyond these vanilla strings and quantitatively study the observational effects of additional properties on cosmic strings.",
"One such additional feature that we can anticipate in many cosmic string models is the presence of a worldsheet current.",
"This can be caused by a coupling between the field forming the cosmic string and other fields, by trapped charged fermion modes along the string [35] (which is common in supersymmetric models [36], [37]), by trapped vector fluxes on non-Abelian strings [38], and other specific mechanisms (for example symmetry breaking of an accidental symmetry in SU(2) stings [39]).",
"From a phenomenological point of view, the presence of such currents gives rise to effective, macroscopic properties on the string.",
"For example, small-scale structure (wiggles) on strings can be described by a specific type of current [40], [41], [42].",
"In what follows we will show quantitatively how the presence of currents on the string worldsheet can affect observational predictions for the string CMB signal, paying particular attention to the special cases of wiggly strings and superconducting strings.",
"In the case of wiggly strings this generalises and extends the work of [43], where string wiggles were taken into account through a constant free parameter $\\alpha $ .",
"In our approach we can construct the most general model for wiggly strings, leading to a full description of wiggles, including their evolution and their effect on the string equations of motion.",
"For superconducting strings, some relevant model parameters are less well known (due to the lack of numerical simulations of these models) but we are also able to provide a full description.",
"In both cases, our results will enable a more detailed and robust comparison to observations, which we leave for future work."
],
[
"String model with currents",
"In order to obtain an effective two-dimensional Lagrangian of a string-like object from a four-dimensional field theory, one usually follows the procedure of [35].",
"This coarse-graining approach unavoidably involves the loss of some of the features of the original four-dimensional description; in particular, it cannot describe key properties of superconducting strings like current saturation and supersonic wiggle propagation.",
"As a result, there is only a phenomenological approach to reproduce properties of the original four-dimensional model [44], [45].",
"On the other hand, one is often interested in averaged equations of motion and these can be the same for different Lagrangians (for an explicit example see [25]).",
"Thus, focusing on deriving the exact form of the Lagrangian is not necessarily the most productive route to obtaining accurate string network evolution.",
"Bearing in mind the subtleties described above, we consider the general form of a two-dimensional Lagrangian involving an arbitrary function of a string current.",
"First, note that a current on a two-dimensional space can be represented as a derivative of a scalar field $\\varphi $ , $J_a = \\varphi _{,a},$ where $_{,a}=\\frac{\\partial }{\\partial \\sigma ^a}$ , with $\\sigma ^a$ the coordinates on the string worldsheet (Latin indexes $a$ , $b$ run over $0,1$ ).",
"We can thus build three possible terms “living\" on the worldsheet, out of which the Lagrangian will be constructed ${\\begin{array}{c}\\text{[1]: } \\; \\varphi ^{,a} \\varphi ^{,b} \\gamma _{ab} = \\kappa , \\\\\\text{[2]: } \\; \\varepsilon ^{ac} \\varepsilon ^{bd} \\gamma _{ab} \\gamma _{cd} = \\gamma , \\\\\\text{[3]: } \\; \\varepsilon ^{ac} \\varepsilon ^{bd} \\gamma _{ab} \\varphi _{,c} \\varphi _{,d} = \\Delta ,\\end{array}}$ where $\\gamma _{ab}=g_{\\mu \\nu } x^{\\mu }_{,a} x^{\\nu }_{,b}$ is the induced metric on the string worldsheet ($g_{\\mu \\nu }$ being the background space-time metric with Greek indexes $\\mu $ , $\\nu $ corresponding to 4-dimensional space-time coordinates), $\\gamma $ is the determinant of the induced metric and $\\varepsilon ^{ab}$ is the Levi-Civita symbol in two dimensions.",
"The term $\\Delta $ is motivated by the Dirac-Born-Infeld (DBI) action for cosmic strings (relevant studies can be found in [46], [47] and [26]).",
"We would like to stress the fact that $\\kappa $ and $\\Delta $ are not independent variables.",
"However, it is beneficial to introduce them as they help to organize the equations into useful form.",
"Taking into account the three possible terms in Eq.",
"(REF ) we can write down the general form of the action generalising the Nambu-Goto action to the case of a string with current $S = - \\mu _0 \\int f(\\kappa , \\gamma , \\Delta ) \\sqrt{-\\gamma /2} d^2 \\sigma ,$ where $\\mu _0$ is a constant of dimensions $[Energy]^2$ determined by the symmetry breaking scale giving rise to string formation.",
"The arbitrary choice of the function $f(\\kappa ,\\gamma ,\\Delta )$ can break reparametrisation invariance of the generalized action of Eq.",
"(REF ).",
"In order to preserve invariance of the action under reparametrizations, the last two terms of Eq.",
"REF should be connected in the following way $f(\\kappa ,\\Delta /\\gamma )$ .",
"Hereinafter for the sake of simplicity the function $f(\\kappa ,\\Delta /\\gamma )$ in equations will be denoted just as $f$ .",
"Assuming that cosmic strings are moving in a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background with metric $ds^2 = a^2(\\tau ) \\left( d\\tau ^2 - dl^2 \\right)$ , we can build the stress-energy tensor from the action (REF ) $\\begin{split}& T^{\\mu \\nu } (y) = \\frac{\\mu _0}{\\sqrt{-g}} \\int d^2 \\sigma \\sqrt{-\\gamma } \\delta ^{(4)}(y-x(\\sigma )) \\\\& \\left( \\tilde{U} \\tilde{u}^{\\mu } \\tilde{u}^{\\nu } - \\tilde{T} \\tilde{v}^{\\mu } \\tilde{v}^{\\nu } - \\Phi (\\tilde{u}^{\\mu } \\tilde{v}^{\\nu } + \\tilde{v}^{\\mu } \\tilde{u}^{\\nu }) \\right),\\end{split}$ where $\\tilde{u}^{\\mu } = \\frac{\\sqrt{\\epsilon } \\dot{x}^{\\mu }}{(-\\gamma )^{1/4}}$ and $\\tilde{v}^{\\mu } = \\frac{ x^{\\prime \\mu }}{\\sqrt{\\epsilon } (-\\gamma )^{1/4}}$ are orthonormal timelike and spacelike vectors respectively ($\\tilde{u}^{\\mu } \\tilde{u}_{\\mu }=1$ , $\\tilde{v}^{\\mu } \\tilde{v}_{\\mu }=-1$ ), $\\epsilon = \\sqrt{\\frac{\\textbf {x}^{\\prime \\, 2}}{1-\\dot{\\textbf {x}}^2}}$ and $& \\tilde{U} = f - 2\\frac{\\partial f}{\\partial \\gamma } \\frac{\\Delta }{\\gamma } - 2 \\gamma ^{00} \\frac{\\partial f}{\\partial \\kappa } \\dot{\\varphi }^2 + 2 \\gamma ^{11} \\frac{\\partial f}{\\partial \\Delta } \\varphi ^{\\prime \\, 2} , \\\\& \\tilde{T} = f - 2\\frac{\\partial f}{\\partial \\gamma } \\frac{\\Delta }{\\gamma } - 2 \\gamma ^{11} \\frac{\\partial f}{\\partial \\kappa } \\varphi ^{\\prime \\, 2} + 2 \\gamma ^{00} \\frac{\\partial f}{\\partial \\Delta } \\dot{\\varphi }^2 , \\\\& \\Phi = \\frac{2 }{\\sqrt{-\\gamma }} \\left( - \\frac{\\partial f}{\\partial \\kappa } -\\frac{\\partial f}{\\partial \\Delta } \\right) \\varphi ^{\\prime } \\dot{\\varphi }\\,;$ here and henceforth dots and primes respectively denote time and space derivatives.",
"It is important to note that for this modification of the Lagrangian, the stress-energy tensor (REF ) has non-diagonal terms induced by the presence of the current.",
"Let us obtain the equations of motion for the action (REF ) using the definitions of $\\tilde{U}$ in (REF ), $\\tilde{T}$ in () and $\\Phi $ in ().",
"A variation of the action (REF ) with respect to $x^{\\mu }$ and $\\varphi $ gives $& \\partial _{\\tau } ( \\epsilon \\tilde{U} ) + \\frac{\\dot{a}}{a} \\epsilon \\left( \\dot{\\textbf {x}}^2 (\\tilde{U} + \\tilde{T}) + \\tilde{U} - \\tilde{T} \\right) = \\partial _{\\sigma } \\Phi , \\\\& \\ddot{\\textbf {x}} \\epsilon \\tilde{U} + \\dot{\\textbf {x}} \\epsilon \\frac{\\dot{a}}{a} \\left( 1-\\dot{\\textbf {x}}^2 \\right) \\left( \\tilde{U} + \\tilde{T} \\right) = \\nonumber \\\\& =\\partial _{\\sigma } \\left( \\frac{ \\tilde{T} }{\\epsilon } \\textbf {x}^{\\prime } \\right) + \\textbf {x}^{\\prime } \\left( 2 \\frac{\\dot{a}}{a} \\Phi + \\dot{\\Phi } \\right) + 2 \\Phi \\dot{\\textbf {x}}^{\\prime },\\\\& \\partial _{\\tau } \\left( \\left( \\frac{\\partial f}{\\partial \\kappa } + \\frac{\\partial f}{\\partial \\Delta } \\right) \\epsilon \\dot{\\varphi } \\right) = \\partial _{\\sigma } \\left( \\left( \\frac{\\partial f}{\\partial \\kappa } + \\frac{\\partial f}{\\partial \\Delta } \\right) \\frac{\\varphi ^{\\prime }}{\\epsilon } \\right).$ where we have chosen a parametrisation satisfying the transverse temporal conditions $\\dot{\\textbf {x}} \\cdot \\textbf {x}^{\\prime }=0$ and $x^0=\\tau $ .",
"As can be seen from the equations of motion (REF ) and (), string dynamics does not depend explicitly on the form of the current contribution $f(\\kappa , \\Delta /\\gamma )$ .",
"The dynamics of the string is defined completely by $\\tilde{U}$ , $\\tilde{T}$ and $\\Phi $ , which can be associated to mass per unit length and string tension.",
"Indeed, it is only the dynamics of $\\varphi $ itself – Eq.",
"() – that explicitly depends on $\\partial f/\\partial \\kappa $ and $\\partial f/\\partial \\Delta $ .",
"This provides us an alternative approach to studying string dynamics effectively, without an explicit connection between an effective Nambu-Goto-like action and the original field theory model.",
"One can instead study the behaviour of $\\tilde{U}$ , $\\tilde{T}$ and $\\Phi $ in the original four-dimensional model in the framework of field theory (as it was done for example in [48], [49], [50], [51], [52]) and then insert the dynamics of $\\tilde{U}$ , $\\tilde{T}$ and $\\Phi $ in the equations of motion (REF ) and ().",
"Additionally, we also note that one can easily generalize the equations of motion (REF )-() to include any number of uncoupled scalar fields, associated to corresponding currents.",
"In this case, we can simply rewrite the variables $\\kappa $ and $\\Delta $ as $\\kappa _i = \\gamma _{ab} \\varphi _i^{,a} \\varphi _i^{,b}, \\quad \\Delta _i = \\varepsilon ^{ac} \\varepsilon ^{bd} \\gamma _{ab} \\varphi _{i,c} \\varphi _{i,d}\\,,$ where the index $i$ runs over the number of fields.",
"There is no summation over $i$ ; if a sum over this index is to be taken it will be written explicitly.",
"Definitions (REF ), () and () in the case of multiple currents generalise to $& \\hspace{-7.22743pt} \\tilde{U}= f + 2 \\sum _i \\left( -\\gamma ^{00} \\frac{\\partial f}{\\partial \\kappa _i} \\dot{\\varphi _i}^2 + \\gamma ^{11} \\frac{\\partial f}{\\partial \\Delta _i} \\varphi _i^{\\prime \\, 2} - \\frac{\\partial f}{\\partial \\gamma } \\frac{\\Delta _i}{\\gamma } \\right), \\\\& \\hspace{-7.22743pt} \\tilde{T} = f + 2 \\sum _i \\left( -\\gamma ^{11} \\frac{\\partial f}{\\partial \\kappa _i} \\varphi _i^{\\prime \\, 2} + \\gamma ^{00} \\frac{\\partial f}{\\partial \\Delta _i} \\dot{\\varphi _i}^2 - \\frac{\\partial f}{\\partial \\gamma } \\frac{\\Delta _i}{\\gamma } \\right) , \\\\& \\Phi = \\frac{2}{\\sqrt{-\\gamma }} \\sum _i \\left(- \\frac{\\partial f}{\\partial \\kappa _i} -\\frac{\\partial f}{\\partial \\Delta _i} \\right) \\varphi _i^{\\prime } \\dot{\\varphi }_i .$ With definitions (REF )-() the form of the stress-energy tensor (REF ) and the equations of motion (REF )-() stay unchanged.",
"On the other hand, the equation of motion for the scalar field () is substituted by the set of equations $& \\partial _{\\tau } \\left( \\left( \\frac{\\partial f}{\\partial \\kappa _i} + \\frac{\\partial f}{\\partial \\Delta _i} \\right) \\epsilon \\dot{\\varphi }_i \\right) = \\partial _{\\sigma } \\left( \\left( \\frac{\\partial f}{\\partial \\kappa _i} + \\frac{\\partial f}{\\partial \\Delta _i} \\right) \\frac{\\varphi _i^{\\prime }}{\\epsilon } \\right).$ We see, therefore, that if we extend the action (REF ) to include additional scalar fields $\\varphi _i$ , the structure of the equations of motion together with the form of the general stress-energy tensor remains unchanged; we only need to add a new index $i$ to $\\kappa $ and $\\Delta $ .",
"This fact will be useful in our considerations below.",
"For now, let us diagonalise the stress-energy tensor (REF ) and define the mass per unit length and tension for these strings with currents, following Refs.",
"[40], [49] $& T^{\\mu }_{\\nu } u^{\\nu } = U \\delta ^{\\mu }_{\\nu } u^{\\nu }, \\\\& T^{\\mu }_{\\nu } v^{\\nu } = T \\delta ^{\\mu }_{\\nu } v^{\\nu }\\,.$ The new orthonormal timelike $u^\\mu $ and spacelike $v^\\mu $ vectors are eigenvectors of the stress-energy tensor (REF ) with corresponding eigenvalues $U$ (mass per unit length) and $T$ (tension).",
"These eigenvalues are related to the original $\\tilde{U}$ , $\\tilde{T}$ and $\\Phi $ in (REF )-() by $& U = \\mu _0/2 \\left( \\tilde{U} + \\tilde{T} + \\Delta \\right), \\\\& T = \\mu _0/2 \\left(\\tilde{U} + \\tilde{T} - \\Delta \\right)\\,,$ while the eigenvectors can be expressed in terms of the original $\\tilde{u}^{\\mu }$ and $\\tilde{v}^{\\mu }$ as $& u^{\\mu } = a \\tilde{u}^{\\mu } + \\sqrt{a^2 - 1} \\tilde{v}^{\\mu }, \\\\& v^{\\mu } = \\sqrt{a^2 - 1} \\tilde{u}^{\\mu } + a \\tilde{v}^{\\mu },$ with $a=\\frac{1}{2} \\left[ 1+ \\frac{\\tilde{U}-\\tilde{T}}{\\Delta } \\right]$ and $\\Delta =\\sqrt{(\\tilde{U}-\\tilde{T})^2 - 4 \\Phi ^2}\\,.$ The passage from the equations of motion of a single string segment to an effective description of a whole network of strings is done through an averaging procedure [21] leading to the VOS model for cosmic strings.",
"Following this approach, we begin by dotting equation () with vectors $\\dot{\\textbf {x}}$ and $\\textbf {x}^{\\prime }$ and using the property of our parametrization $\\dot{\\textbf {x}} \\cdot \\textbf {x}^{\\prime }=0$ to obtain $& \\dot{\\textbf {x}} \\cdot \\ddot{\\textbf {x}} \\epsilon \\tilde{U} + \\dot{\\textbf {x}}^2 \\epsilon \\frac{\\dot{a}}{a} \\left( 1-\\dot{\\textbf {x}}^2 \\right) \\left( \\tilde{U} + \\tilde{T} \\right) = \\nonumber \\\\& =\\frac{ \\tilde{T} }{\\epsilon } \\dot{\\textbf {x}} \\cdot \\textbf {x}^{\\prime \\prime } - 2 \\Phi \\ddot{\\textbf {x}} \\cdot \\textbf {x}^{\\prime }, \\\\& \\textbf {x}^{\\prime } \\cdot \\ddot{\\textbf {x}} \\epsilon \\tilde{U} - \\textbf {x}^{\\prime } \\cdot \\textbf {x}^{\\prime \\prime } \\frac{ \\tilde{T} }{\\epsilon } + 2 \\Phi \\textbf {x}^{\\prime \\prime } \\cdot \\dot{\\textbf {x}} = \\nonumber \\\\& = \\textbf {x}^{\\prime 2} \\left( 2 \\frac{\\dot{a}}{a} \\Phi + \\dot{\\Phi } + \\frac{ \\tilde{T^{\\prime }} }{\\epsilon } - \\frac{ \\tilde{T} }{\\epsilon ^{2}} \\epsilon ^{\\prime } \\right).$ Using the expression $\\frac{\\epsilon ^{\\prime }}{\\epsilon } = \\frac{ \\textbf {x}^{\\prime } \\cdot \\textbf {x}^{\\prime \\prime }}{ \\textbf {x}^{\\prime 2} } - \\frac{ \\textbf {x}^{\\prime } \\cdot \\ddot{\\textbf {x}} }{1-\\dot{\\textbf {x}}^{2}}$ we can eliminate the terms proportional to $\\epsilon ^{\\prime }$ and $\\textbf {x}^{\\prime } \\cdot \\ddot{\\textbf {x}}$ obtaining the equation ${\\begin{array}{c}\\dot{\\textbf {x}} \\cdot \\ddot{\\textbf {x}} \\epsilon \\tilde{U} + \\dot{\\textbf {x}}^2 \\epsilon \\frac{\\dot{a}}{a} \\left( 1-\\dot{\\textbf {x}}^2 \\right) \\left( \\tilde{U} + \\tilde{T} \\right) - \\frac{ \\tilde{T} }{\\epsilon } \\dot{\\textbf {x}} \\cdot \\textbf {x}^{\\prime \\prime } = \\\\= 2 \\Phi \\frac{1-\\dot{\\textbf {x}}^2}{\\tilde{U}-\\tilde{T}} \\left( \\tilde{T}^{\\prime } + \\epsilon \\left( 2 \\frac{\\dot{a}}{a} \\Phi + \\dot{\\Phi } - 2 \\Phi \\frac{\\textbf {x}^{\\prime \\prime } \\cdot \\dot{\\textbf {x}}}{ \\textbf {x}^{\\prime 2} } \\right) \\right).\\end{array}}$ We now introduce the macroscopic variables $& E = \\mu _0 a \\int \\tilde{U} \\epsilon d \\sigma \\,,\\\\& E_0 = \\mu _0 a \\int \\epsilon d \\sigma \\,,\\\\& v^2 = \\left\\langle \\dot{x}^2 \\right\\rangle \\,,$ where $\\left\\langle ... \\right\\rangle = \\frac{\\int ...\\, \\epsilon d \\sigma }{\\int \\epsilon d \\sigma }$ denotes the (energy-weighted) averaging operation.",
"These macroscopic quantities are, respectively, the total energy, the 'bare' energy (without the contribution form the current) and the Root-Mean Squared (RMS) velocity.",
"Using these definitions we proceed to average equations (REF ) and (REF ) finding $& \\dot{E} + \\frac{\\dot{a}}{a} E \\left( \\upsilon ^2 \\left( 1 + W \\right) - W \\right) = \\left\\langle \\Phi ^{\\prime }/\\epsilon \\right\\rangle E_0 , \\\\& \\dot{\\upsilon } + \\upsilon \\frac{\\dot{a}}{a} \\left( 1-\\upsilon ^2 \\right) \\left( 1 + W \\right) - (1-\\upsilon ^2) W \\frac{k(\\upsilon )}{R_c} \\nonumber = \\\\& = \\left< 2 \\frac{\\Phi }{\\tilde{U}} \\frac{1-\\dot{\\textbf {x}}^2}{1-\\tilde{T}/\\tilde{U}} \\left(\\frac{ \\tilde{T}^{\\prime }}{\\epsilon } + 2 \\frac{\\dot{a}}{a} \\Phi + \\dot{\\Phi } - 2 \\Phi \\frac{\\textbf {x}^{\\prime \\prime } \\cdot \\dot{\\textbf {x}}}{ \\textbf {x}^{\\prime 2} } \\right)\\right>.$ Here, we have defined $W=\\left\\langle \\tilde{T}/\\tilde{U} \\right\\rangle $ and introduced the average comoving radius of curvature of strings in the network, $R_c$ , and the curvature parameter, $k(\\upsilon )$ , satisfying $\\left< \\frac{\\dot{\\textbf {x}}}{\\epsilon } \\cdot \\left( \\frac{\\textbf {x}^{\\prime }}{\\epsilon } \\right)^{\\prime } \\right>= \\frac{k(\\upsilon )}{R_c} \\upsilon (1-\\upsilon ^2)$ .",
"For ordinary cosmic strings, an accurate ansatz for the curvature parameter as a function of velocity $k(\\upsilon ) = \\frac{2 \\sqrt{2}}{\\pi } (1-v^2) (1+2 \\sqrt{2}v^3) \\frac{1-8v^6}{1+8v^6} \\,,$ has been derived in [22].",
"We assume that this function stays valid for strings with currents as well.",
"Following the procedure of [21], [22], we rewrite the averaged equations of motion (REF -) in terms of more convenient macroscopic variables: the comoving characteristic length $L_c$ and the comoving correlation length $\\xi _c$ , which are related to the energies in (REF -) by the following expressions $E=\\frac{\\mu _0 V}{a^2 L_c^2}$ and $E_0=\\frac{\\mu _0 V}{a^2 \\xi _c^2}\\,,$ where $V$ is the volume over which the averaging has been performed.",
"In addition, we employ the VOS model approximation that the average radius of curvature of cosmic strings in the network is equal to the correlation length, i.e.",
"$R_c \\approx \\xi _c$ .",
"Assuming further that the averaged macroscopic quantities can be split as $\\left< \\Phi \\tilde{U} \\tilde{T} \\right> = \\left< \\Phi \\right> \\left< \\tilde{U} \\right> \\left< \\tilde{T} \\right> $ we obtain the following system of equations $& 2 \\dot{L_c} = \\frac{\\dot{a}}{a} L_c \\left( \\upsilon ^2 \\left( 1 + W \\right) - W +1 \\right) - \\frac{\\sqrt{1-\\upsilon ^2} Q_{,s}}{\\hat{U}} , \\\\& \\dot{\\upsilon } + \\upsilon \\frac{\\dot{a}}{a} \\left( 1-\\upsilon ^2 \\right) \\left( 1 + W \\right) - (1-\\upsilon ^2) W \\frac{k(\\upsilon )}{\\xi _c} \\nonumber = \\\\& \\hspace{-7.22743pt} = 2 \\frac{ Q }{\\hat{U} } \\frac{1-\\upsilon ^2}{1-W} \\left(\\sqrt{1-\\upsilon ^2}\\, \\hat{T}_{,s} + \\dot{Q} + 2 Q \\left( \\frac{\\dot{a}}{a} - \\frac{k(\\upsilon ) \\upsilon }{\\xi _c} \\right) \\right),$ where $\\left< \\Phi \\right> = Q$ , $\\left< \\dot{\\Phi } \\right> = \\dot{Q}$ , $\\hat{U}= \\left< \\tilde{U} \\right>$ , $\\hat{T}= \\left< \\tilde{T} \\right>$ , the correlation and characteristic lengths are related by $\\xi _c = L_c \\sqrt{\\hat{U}}$ and a new derivative variable $_{,s} = \\frac{\\partial }{\\partial s}$ has been introduced, corresponding to the parametrization $ds=\\sqrt{\\textbf {x}^{\\prime 2}}d\\sigma $ .",
"Equations (REF -) are the averaged macroscopic equations describing a network of cosmic strings with a current.",
"It is apparent that scaling solutions ($L_c = \\varepsilon _c \\tau , \\; \\upsilon = \\text{const}$ when $a \\propto \\tau ^n$ with $\\varepsilon _c$ and $n$ constants [21], [22]) exist if the averaged quantities $\\hat{U}$ , $\\hat{T}$ and $Q$ are appropriately restricted.",
"In particular we see from (REF )-() that scaling behaviour – typical for ordinary string networks – can arise when $\\hat{U}, \\hat{T}, Q = \\text{const}$ , while $T_{,s} \\; \\text{and} \\; Q_{,s} \\sim 1/\\tau $ .",
"Additionally, the requirement of a well-defined $\\varepsilon _c$ implies the condition $\\upsilon ^2 < \\frac{2+n(W-1)}{n(W+1)}.$ Equation (REF ) relates the rms string velocity $\\upsilon $ to the ratio $W=\\left\\langle \\tilde{T}/\\tilde{U} \\right\\rangle $ for a given expansion rate (characterisd by $n$ ) for a cosmic string network with currents.",
"These general relations will be useful when we consider the special case of a wiggly string network.",
"We now concentrate on how these modifications can influence predictions for the CMB anisotropy from cosmic strings.",
"We follow the approach of [43], [53], [54].",
"Rather than working with the full network of cosmic strings, we consider a number of straight string segments in Minkowski space that decay according to the evolution of strings in an expanding FLRW metric, and have velocities and lengths determined by the VOS model.",
"We start from the Fourier transform of the stress-energy tensor (REF ) of a single straight string segment on which the contribution from string currents has been averaged as above $\\begin{split}& \\Theta ^{\\mu \\nu } = \\mu _0 \\int _{- \\xi _0 \\tau /2}^{ \\xi _0 \\tau /2} \\biggl [ \\hat{U} \\epsilon \\dot{X}^{\\mu } \\dot{X}^{\\nu }- \\hat{T} \\frac{X^{\\prime \\mu } X^{\\prime \\nu }}{\\epsilon } - \\\\& \\quad - Q \\left(\\dot{X}^{\\mu } X^{\\prime \\nu }+ \\dot{X}^{\\nu } X^{\\prime \\mu } \\right) \\biggl ] \\text{e}^{i {\\bf k}\\cdot {\\bf X}} d \\sigma ,\\end{split}$ where the vector $X^{\\mu } = x_0^{\\mu }+\\sigma X^{\\prime \\, \\mu } + \\tau \\dot{X}^{\\mu }$ represents the straight, stick-like solution for a string moving with velocity $\\upsilon $ (so that $\\dot{X}^{\\mu } \\dot{X}_{\\mu }=1-\\upsilon ^2$ ) and with worldsheet coordinates $\\sigma $ and $\\tau $ in the transverse temporal gauge.",
"The comoving length of a string segment at conformal time $\\tau $ is $\\xi _0 \\tau $ , where $\\xi _0$ will be determined from the macroscopic evolution equations (REF -).",
"Variables $\\hat{U}$ , $\\hat{T}$ and $Q$ are constants for the straight string, as follows from the equations of motion (REF -).",
"The four-vector $x_0^{\\mu }=(1,{\\bf x_0})$ is a random location for a single string segment, while $X^{\\prime \\mu }$ and $\\dot{X}^{\\mu }$ are randomly oriented and satisfy the transverse condition $X^{\\prime }_{\\mu } \\dot{X}^{\\mu }=0$ .",
"We can choose these vectorsNote that although we work in the transverse temporal gauge we have chosen the normalization ${\\bf X}^{\\prime 2}=1$ .",
"This may seem to be inconsistent as ${\\bf X}^{\\prime 2}=\\epsilon ^2 (1-\\dot{\\bf X}^2)$ and $\\epsilon $ is evolving according to equation (REF ).",
"However, we are implicitly taking this effect into account by having the limits of the integral (REF ) be time-dependent through the time evolution of $\\xi _0$ .",
"This evolves according to the macroscopic equation (REF ), which has been derived by averaging equations (REF ) and (REF ).",
"as $& \\dot{X}^{\\mu } = \\begin{pmatrix} 1 \\\\ \\upsilon ( \\cos \\theta \\cos \\phi \\cos \\psi - \\sin \\phi \\sin \\psi ) \\\\ \\upsilon ( \\cos \\theta \\sin \\phi \\cos \\psi + \\cos \\phi \\sin \\psi ) \\\\ -\\upsilon \\sin \\theta \\cos \\psi \\end{pmatrix}, \\\\& X^{\\prime \\; \\mu } = \\begin{pmatrix} 0 \\\\ \\sin \\theta \\cos \\phi \\\\ \\sin \\theta \\sin \\phi \\\\ \\cos \\theta \\end{pmatrix}\\,.",
"\\\\ \\nonumber $ Without loss of generality we can choose the wave vector along the third axis $\\textbf {k}=k \\hat{k}_3$ and integrating over $\\sigma $ we obtain the following expressions $& \\Theta _{00} = \\frac{\\mu _0 \\hat{U}}{\\sqrt{1-v^2}} \\frac{\\sin (k X_3 \\xi _0 \\tau /2)}{k X_3/2} \\cos (\\textbf {k} \\cdot \\textbf {x}_0+k X_3 v \\tau ) , \\\\& \\Theta _{ij} = \\Theta _{00} \\biggl [ v^2 \\dot{X}_i \\dot{X}_j - \\hat{T} / \\hat{U}(1-v^2) X_i^{\\prime } X_j^{\\prime } - \\nonumber \\\\& - v Q/\\hat{U} \\left(\\dot{X}_i X^{\\prime }_j+ \\dot{X}_j X^{\\prime }_i \\right) \\biggl ],$ where the indices $i$ , $j$ run over the 3-dimensional spatial coordinates.",
"The scalar, vector and tensor components can be defined as ${\\begin{array}{c}\\Theta ^S = \\left( 2 \\Theta _{33} - \\Theta _{11} - \\Theta _{22} \\right)/2 ,\\end{array}}$ ${\\begin{array}{c}\\Theta ^V = \\Theta _{13},\\end{array}}$ ${\\begin{array}{c}\\Theta ^T=\\Theta _{12}.\\end{array}}$ Substituting (REF ) and () in (REF )-(REF ), we obtain the scalar, vector and tensor contributions for a straight string segment with stress-energy tensor (REF ), (REF ) ${\\begin{array}{c}\\frac{2 \\Theta ^S}{\\Theta _{00}} =\\biggl [ v^2 (3 \\dot{X}_3 \\dot{X}_3 - 1) - 6 v Q / \\hat{U} X_3^{\\prime } \\dot{X}_3 - \\\\-(1-v^2) \\hat{T} / \\hat{U} (3 X_3^{\\prime } X_3^{\\prime } -1) \\biggl ] ,\\end{array}}$ ${\\begin{array}{c}\\frac{\\Theta ^V}{\\Theta _{00}} =\\biggl [ v^2 \\dot{X}_1 \\dot{X}_3 - \\hat{T}/\\hat{U} (1-v^2) X_1^{\\prime } X_3^{\\prime } - \\\\- v Q / \\hat{U} \\left( X_1^{\\prime } \\dot{X}_3 + \\dot{X}_1 X_3^{\\prime } \\right) \\biggl ],\\end{array}}$ ${\\begin{array}{c}\\frac{\\Theta ^T}{\\Theta _{00}} =\\biggl [ v^2 \\dot{X}_1 \\dot{X}_2 - \\hat{T}/\\hat{U} (1-v^2) X_1^{\\prime } X_2^{\\prime } - \\\\- v Q / \\hat{U} \\left( X_1^{\\prime } \\dot{X}_2 + \\dot{X}_1 X_2^{\\prime } \\right) \\biggl ] .\\end{array}}$ Following the prescription of reference [55], we can then calculate the unequal time two-point correlators by averaging over locations, string orientations and velocity orientations of the string segment $\\left\\langle \\Theta ^{I}(k, \\tau _1) \\Theta ^{J}(k, \\tau _2) \\right\\rangle = \\frac{2 \\mu _0^2 \\mathcal {F}(\\tau _1, \\tau _2, \\xi _0 )}{16 \\pi ^3} \\int _0^{2 \\pi } d \\phi \\int _0^{\\pi } \\sin \\theta d \\theta \\int _0^{2 \\pi } d \\psi \\int _0^{2 \\pi } d \\chi \\, \\Theta ^{I}(k,\\tau _1) \\Theta ^{J}(k, \\tau _2)\\,.$ Here, the indices $I$ and $J$ correspond to the scalar, vector, tensor and “00\" components.",
"The function $\\mathcal {F}(\\tau _1, \\tau _2, \\xi _0)$ describes the string decay rate.",
"It is chosen to have the same form as for ordinary (without currents) cosmic strings [43] $\\mathcal {F} (\\tau _1, \\tau _2, \\xi _0) = \\frac{1}{\\left( \\xi _0 Max(\\tau _1, \\tau _2) \\right)^3},$ but here $\\xi _0$ is determined by the modified VOS equations (REF -).",
"The phase $\\chi = \\textbf {k} \\cdot \\textbf {x}_0 $ arises from varying over string locations ${\\bf x}_0$ (refer to equation (REF )), which we integrate over.",
"We can write the general form of the correlators as $& \\hspace{-7.22743pt} \\left\\langle \\Theta ^{I}(k, \\tau _1) \\Theta ^{J}(k, \\tau _2) \\right\\rangle = \\frac{\\mu _0^2 \\mathcal {F}(\\tau _1, \\tau _2, \\xi _0) }{k^2 (1-\\upsilon ^2)} B^{I-J}(\\tau _1,\\tau _2).$ If we are only interested in the approximation $k \\tau <1$ (superhorizon scales), we can expand $B^{I-J}$ keeping only terms that are up to $k^2$ .",
"In this case the non-zero correlators are the following $& B^{00-00}(\\tau _1,\\tau _2) \\approx \\hat{U}^2 \\xi _0^2 k^2 \\tau _1 \\tau _2, \\\\& B^{S-S} \\approx \\frac{1}{5} \\frac{B^{00-00}}{\\hat{U}^2}(\\tau _1,\\tau _2) \\times \\\\& \\biggl ( \\hat{U}^2 \\upsilon ^4 + \\hat{T} \\hat{U} \\upsilon ^2 (1-\\upsilon ^2)+ \\hat{T}^2 (1- \\upsilon ^2)^2 + 3 \\upsilon ^2 Q^2 \\biggl ), \\nonumber \\\\& B^{V-V} \\approx \\frac{1}{3} B^{S-S},\\\\& B^{T-T} \\approx \\frac{1}{3} B^{S-S}.$ In the Appendix we give exact expressions for the equal time two-point correlators $B^{I-J}(\\tau )$ and provide semi-analytic expressions for the unequal time two-point correlators valid for all (i.e.",
"from subhorizon through to superhorizon) modes $k$ .",
"Having computed the correlators (REF ), let us now assume that the cosmic string network under consideration has reached a scaling regime.",
"We can then assume that $\\xi _0$ , $\\upsilon $ together with $\\hat{U}$ , $\\hat{T}$ and $Q$ do not depend on $\\tau $ and $\\sigma $ .",
"To obtain an analytic estimate of the string-induced CMB anisotropy, let us consider the string network evolving in the matter domination epoch ($n=2$ ).",
"For this case we can use the following solution of the linearised Einstein-Boltzmann equations [53], [56] ${\\begin{array}{c}\\frac{\\delta T}{T} = -\\frac{1}{2} \\int _{\\tau _i}^{\\tau _f} d\\tau \\dot{h}_{ij} n^i n^j ,\\\\\\dot{h}_{ij} = \\dot{h}_{ij}^{S} + \\dot{h}_{ij}^{V} + \\dot{h}_{ij}^{T} ,\\end{array}}$ ${\\begin{array}{c}\\dot{h}_{ij}^{S} = -\\rho \\sum _k \\text{e}^{i \\textbf {k}\\cdot \\textbf {x}} \\int _0^{\\tau } d \\tau ^{\\prime } \\\\\\left(\\frac{1}{3} \\delta _{ij} \\left( \\frac{\\tau ^{\\prime }}{\\tau } \\right)^6 (\\Theta ^{Tr} + 2 \\Theta ^S) - k_i k_j \\left( \\frac{\\tau ^{\\prime }}{\\tau } \\right)^4 \\Theta ^S \\right), \\\\\\end{array}}$ ${\\begin{array}{c}\\dot{h}_{ij}^{V} = \\sum _k \\text{e}^{i \\textbf {k} \\cdot \\textbf {x}} \\left( \\dot{V}_{i} k_j + \\dot{V}_{j} k_i \\right),\\\\\\dot{V}_{i} = \\rho \\int _0^{\\tau } d \\tau ^{\\prime } \\left( \\frac{\\tau ^{\\prime }}{\\tau } \\right) \\Theta ^V_i,\\end{array}}$ ${\\begin{array}{c}\\dot{h}_{ij}^{T}=\\rho \\int _0^{\\tau } d \\tau ^{\\prime } k^3 \\tau ^{\\prime 4} F(k \\tau ^{\\prime },k \\tau ) \\Theta _{ij}^T, \\\\F(k \\tau ^{\\prime },k \\tau ) = G_1(k \\tau ^{\\prime }) \\dot{G}_2(k \\tau ) - G_2(k \\tau ^{\\prime }) \\dot{G}_1(k \\tau ),\\end{array}}$ where $\\rho =16 \\pi G$ , $G_1(k \\tau ) = \\frac{\\cos (k \\tau )}{(k \\tau )^2}+\\frac{\\cos (k \\tau )}{(k \\tau )^3}$ , $G_2(k \\tau ) = \\frac{\\cos (k \\tau )}{(k \\tau )^3}+\\frac{\\sin (k \\tau )}{(k \\tau )^2}$ , $\\frac{\\delta T}{T}$ are the CMB temperature fluctuations, $n^i$ is a unit vector defining the direction of CMB photons, and $\\Theta ^{Tr}$ is the trace of the Fourier transformed stress-energy tensor.",
"We can now compute the angular power spectrum $C_l$ of the CMB anisotropy using the expressions [53]: ${\\begin{array}{c}C_l^{S} = \\frac{1}{2 \\pi } \\int _0^{\\infty } k^2 dk \\\\\\left< \\int _0^{\\tau _0} d \\tau \\left(\\frac{1}{3} \\dot{h}_1 + \\dot{h}_2 \\frac{d^2}{d(k \\Delta \\tau )^2} \\right) j_l(k \\Delta \\tau ) \\right>^2,\\end{array}}$ ${\\begin{array}{c}C_l^{V}=\\frac{2}{\\pi } \\int _0^{\\infty } k^2 dk l(l+1) \\\\\\left< \\int _0^{\\tau 0} d \\tau \\dot{h}^V \\frac{d}{d(k \\Delta \\tau )} \\left( j_l(k \\Delta \\tau ) / (k \\Delta \\tau ) \\right) \\right>^2,\\end{array}}$ ${\\begin{array}{c}C_l^{T} = \\frac{1}{2 \\pi } \\int _0^{\\infty } k^2 dk \\frac{(l+2)!}{(l-2)!}",
"\\\\\\left< \\int _0^{\\tau 0} \\frac{d \\tau }{(k \\Delta \\tau )^2} \\dot{h}^T j_l(k \\Delta \\tau ) \\right>^2,\\end{array}}$ where $\\Delta \\tau = \\tau _0 - \\tau $ (with $\\tau _0$ the value of conformal time today), $j_l(k \\Delta \\tau )$ are spherical Bessel functions, and $\\dot{h}_1$ , $\\dot{h}_2$ are defined as $& \\dot{h}_1(\\tau ) = - \\rho \\int d \\tau ^{\\prime } \\left(\\frac{ \\tau ^{\\prime } }{\\tau } \\right)^6 (\\Theta ^{Tr}(\\tau ^{\\prime }) + 2 \\Theta ^S(\\tau ^{\\prime })), \\\\& \\dot{h}_2(\\tau ) = - \\rho \\int d \\tau ^{\\prime }\\left(\\frac{ \\tau ^{\\prime } }{\\tau } \\right)^4 \\Theta ^S(\\tau ^{\\prime }).$ We proceed by making a further approximation on the correlators (REF ).",
"The dominant contribution to the two-point correlator is when $\\tau _1 \\rightarrow \\tau _2$ (see for example [55]), which allows us to approximate (REF ) as ${\\begin{array}{c}\\left\\langle \\Theta ^{I}(k, \\tau _1) \\Theta ^{J}(k, \\tau _2) \\right\\rangle = \\frac{\\mu _0^2 \\mathcal {F}(\\tau _1, \\tau _2, \\xi _0) }{k^2 (1-\\upsilon ^2)} \\times \\\\B^{I-J}(\\tau _1) \\delta (\\tau _1 - \\tau _2),\\end{array}}$ where $\\delta (\\tau _1-\\tau _2)$ is Dirac delta function and $B^{I-J}(\\tau _1) = B^{I-J}(\\tau _1, \\tau _1)$ .",
"By using this form of the correlators (REF ) one can rewrite equations (REF ), (REF ) and (REF ) as $& C_l^{S} = \\frac{\\kappa ^2}{2 \\pi } \\int _0^{ \\infty } k^2 dk \\int _0^{\\tau 0} d \\tau _1 \\int _0^{\\tau 0} d \\tau _2 \\int _0^{\\tau _1} d \\tau _1^{\\prime } \\frac{f(\\tau _1^{\\prime },\\varepsilon )}{k^2 (1-v^2)} \\frac{\\tau _1^{\\prime 8}}{\\tau _1^4 \\tau _2^4} F_{sc} (\\tau _1^{\\prime }) , \\\\& C_l^{V}=\\frac{2 \\kappa ^2}{\\pi } \\int _0^{\\infty } k^2 dk l(l+1) \\int _0^{\\tau 0} d \\tau _1 \\int _0^{\\tau 0} d \\tau _2 \\frac{j_l^{\\prime }(k \\tau _1)}{k \\tau _1} \\frac{j_l^{\\prime }(k \\tau _2)}{k \\tau _2} \\int _0^{\\tau _1} d \\tau _1^{\\prime } \\frac{\\tau _1^{\\prime 8} f(\\tau _1^{\\prime }, \\xi _0)}{k^2 (1-v^2)} B^{V-V}(\\tau _1^{\\prime }), \\\\& C_l^{T} = \\frac{\\kappa ^2}{2 \\pi } \\int _0^{\\infty } k^2 dk \\frac{(l+2)!}{(l-2)!}",
"\\int _0^{\\tau 0} d \\tau _1 \\int _0^{\\tau 0} d \\tau _2 \\frac{j_l(k \\tau _1)}{(k \\tau _1)^2} \\frac{j_l(k \\tau _2)}{(k \\tau _2)^2} \\int _0^{\\tau 1} d \\tau _1^{\\prime } k^6 \\tau _1^{\\prime 8} F(\\tau _1^{\\prime }, \\tau _1) F(\\tau _2^{\\prime },\\tau _2) \\frac{f(\\tau _1^{\\prime },\\xi _0)}{k^2(1-v^2)} B^{T-T} (\\tau _1^{\\prime }),$ where $& F_{sc} = \\frac{1}{9} j_l(k \\tau _1) j_l(k \\tau _2) \\frac{\\tau _1^{\\prime 4}}{\\tau _1^2 \\tau _2^2} \\biggl ( B^{Tr-Tr}(\\tau _1^{\\prime }) + 4 B^{Tr-S}(\\tau _1^{\\prime }) + 4 B^{S-S}(\\tau _1^{\\prime }) \\biggl ) + \\nonumber \\\\& + \\frac{1}{3} \\left( j_l^{\\prime \\prime } (k \\tau _1) j_l(k \\tau _2) \\frac{\\tau ^{\\prime \\; 2}}{\\tau _2^2} + j_l^{\\prime \\prime } (k \\tau _2) j_l(k \\tau _1) \\frac{\\tau ^{\\prime \\; 2}}{\\tau _1^2} \\right) \\left( B^{Tr-S}(\\tau _1^{\\prime }) + 2 B^{S-S}(\\tau _1^{\\prime }) \\right) + j_l^{\\prime \\prime }(k \\tau _1) j_l^{\\prime \\prime } (k \\tau _2) B^{S-S}(\\tau _1^{\\prime }),$ $\\hspace{-36.135pt} \\text{with trace components: } & \\; B^{Tr-Tr}(\\tau _1^{\\prime }) =\\left[1 + v^2 - \\tilde{T} / \\tilde{U}(1-v^2) \\right]^2 B^{00-00}(\\tau _1^{\\prime }),\\\\& B^{Tr-S}(\\tau _1^{\\prime })=\\left[1 + v^2 - \\tilde{T} / \\tilde{U}(1-v^2) \\right] B^{00-S}(\\tau _1^{\\prime }).$ In the final form of equations (REF ) and (REF ) we have expressed the contribution from the “00\" component in terms of the trace component “$Tr$ \" using the relations (REF ) and (), which can be derived from (REF ) and ().",
"It should be stressed that in obtaining equations (REF ), () and () we have only used the approximation (REF ).",
"We have thus succeeded to derive full semi-analytic expressions for the scalar, vector and tensor contributions to the angular powerspectrum from cosmic strings with arbitrary currents, valid in matter domination and under the approximation (REF ).",
"In the superhorizon limit $k \\tau < 1$ considered above, the two-point correlators have the simple form (REF )-() and we can factor out from the integrals (REF )-() the key quantities characterising the cosmic string network: $\\upsilon $ , $\\xi _0$ , $\\hat{U}$ , $\\hat{T}$ and $Q$ .",
"This allows us to establish a direct connection between cosmic string network parameters and the string contribution to CMB anisotropies, valid on superhorizon scales.",
"For the vector () and tensor () contributions it is easy to see that ${\\begin{array}{c}\\hspace{-7.22743pt} C_l^{V,T} \\sim (G \\mu _0)^2 \\times \\\\\\frac{\\hat{U}^2 \\upsilon ^4 + \\hat{T} \\hat{U} \\upsilon ^2 (1-\\upsilon ^2) + \\hat{T}^2 (1-\\upsilon ^2)^2 + 3 \\upsilon ^2 Q^2 }{\\xi _0 (1-v^2)}\\,,\\end{array}}$ which agrees with the result of [55] in the limit $Q=0$ , $U=\\alpha \\mu _0$ and $T=\\mu _0/\\alpha $ .",
"The treatment of the scalar mode (REF ) is more subtle.",
"We will estimate it to leading order, using the following asymptotic form of the spherical Bessel function $j_l(x) \\sim x^l$ , valid when $0<x<<\\sqrt{l+1}$ .",
"This approximation is justified when we consider the scalar contribution at large multipole moments $l$ .",
"Since the angular power spectrum $C_l$ for cosmic string networks typically peaks at $l>500$ we can take the leading term of (REF ) as $j_l^{\\prime \\prime }(k \\tau _1) j_l^{\\prime \\prime }(k \\tau _2) B^{S-S}(\\tau _1^{\\prime })$ .",
"It follows that, in this approximation, the scalar contribution will be the same as the above approximate expressions for the vector and tensor components ${\\begin{array}{c}C_{1<<l}^{S} \\sim (G \\mu _0)^2 \\times \\\\\\frac{\\hat{U}^2 \\upsilon ^4 + \\hat{T} \\hat{U} \\upsilon ^2 (1-\\upsilon ^2) + \\hat{T}^2 (1-\\upsilon ^2)^2 + 3 \\upsilon ^2 Q^2 }{\\xi _0 (1-v^2)}.\\end{array}}$ Let us briefly summarize the results presented in this section.",
"For the action (REF ) describing a string with arbitrary current we first derived the stress-energy tensor (REF ) and obtained the (microscopic) equations of motion (REF )-(), the mass per unit length (REF ) and string tension ().",
"From these, we then developed a macroscopic VOS evolution model (REF )-() for a string network with arbitrary currents and used it to estimate analytically the CMB contribution (REF )-(REF ) from such a string network in the matter domination era.",
"All these results are determined by $\\hat{U}$ , $\\hat{T}$ and $Q$ .",
"In particular, changing the form of the function $f(\\kappa ,\\Delta /\\gamma )$ in the string action (REF ) leads to a redefinition of $\\hat{U}$ , $\\hat{T}$ and $Q$ rather than a change in the string equations of motion (REF )-().",
"In the next two sections we will consider two specific physically motivated cases: wiggly and superconducting (chiral) cosmic string networks."
],
[
"Wiggly model",
"In this section we consider the case of wiggly cosmic strings.",
"This model was developed as an effective description of small-scale structure on cosmic strings [40], [41], [42].",
"By applying a suitable phenomenological Lagrangian, the evolution of wiggly string networks was studied in [24], [57].",
"However, a great deal about the dynamics and scaling behaviour of the network can be understood by focusing on the equation of state for wiggly strings, without specifying the precise form of the Lagrangian.",
"The equation of state for wiggly strings is $\\begin{split}U T =& \\mu _0^2, \\\\U = \\mu _0 \\mu , \\; & \\; T = \\mu _0 / \\mu , \\\\\\end{split}$ or equivalently $\\begin{split}\\hat{U} \\hat{T} =& 1\\,, \\\\\\hat{U} = \\mu , \\; \\; \\hat{T} = & 1/ \\mu , \\; \\; Q=0\\,,\\end{split}$ where $\\mu $ is a dimensionless parameter quantifying the amount of wiggles on the string, and $\\mu =1$ corresponds to the usual Nambu-Goto string (the same parameter was denoted as $\\alpha $ in [43]).",
"Applying the equation of state (REF ) to the averaged equations of motion (REF ) and () we obtain $& 2 \\frac{d L_c}{d \\tau } = \\frac{\\dot{a}}{a} L_c \\left[ 1 + \\upsilon ^{2} - \\frac{1-\\upsilon ^2}{\\mu ^2} \\right], \\\\& \\frac{d\\upsilon }{d \\tau } = \\left( 1 - \\upsilon ^2 \\right) \\left[ \\frac{k(\\upsilon )}{L_c \\mu ^{5/2}} - \\frac{\\dot{a}}{a} \\upsilon \\left( 1 + \\frac{1}{\\mu ^2} \\right) \\right]\\,.$ Note that the comoving correlation length is connected to comoving characteristic length by the following relation $\\xi _c = \\sqrt{\\mu } L_c$ .",
"We can now include an energy loss term $F(\\upsilon , \\mu )$ on the right-hand side of equation (REF ) and assume scaling behaviour of the network $L_c = \\varepsilon \\tau $ (while $\\xi _c = \\xi _0 \\tau $ ).",
"Note that in this case the previously obtained constraint (REF ) has the form $v^2 < \\frac{2/n - 1 + 1 / \\mu ^2}{1+1 / \\mu ^2},$ where, in the scaling regime, $\\mu $ is a constant.",
"The expression (REF ) means that the rms string velocity has an upper limit, determined by the expansion rate $n$ and amount of wiggles $\\mu $ on the string; this is illustrated in figure REF .",
"Figure: The constraint () on the square of the rms velocity, v 2 v^2, depending on the expansion rate nn and the amount of wiggles μ\\mu .It is important to note that this restriction was obtained just by using the equation of state for wiggly cosmic strings (REF ) in our general equation (REF ).",
"This means that any Lagrangian suitable for wiggly string description (i.e.",
"any choice of $f(\\kappa , \\Delta /\\gamma )$ satisfying (REF ) for the equation of state) cannot change this relation.",
"Moreover, it is valid for any energy loss function $F(v,\\mu )$ .",
"Thus, any wiggly cosmic string network with any energy loss function of the form $F(v,\\mu )$ must satisfy the constraint (REF ).",
"To get a feeling for the size of the maximum network velocity in (REF ) we consider two limiting cases: strings without wiggles ($\\mu =1$ ) and highly wiggly strings ($\\mu \\rightarrow \\infty $ ): $& v^2 < 1/n \\qquad (\\mu =1), \\\\& v^2 < 2/n-1 \\qquad (\\mu \\rightarrow \\infty ).$ As seen from Eq.",
"(REF ), for strings without wiggles only very fast expansion rates $n$ can cause a significant restriction to the string network velocity, while for highly wiggled strings the limit () provides a severe constraint even when $n=2$ (matter domination era).",
"For wiggly strings with $\\mu =1.5$ in the matter domination era ($n=2$ ) the velocity is limited as $v^2<0.3$ , which is close to the values of rms velocities from field theory simulations [19] in the matter domination era.",
"Let us now study the full description of the wiggly cosmic string network model [24], [57] described by the action $S = \\mu _0 \\int \\omega \\sqrt{-\\gamma } d^2 \\sigma ,$ where $\\omega = \\sqrt{1-\\kappa } $ .",
"The derivation of the averaged equations of motion for this model can be found in [24], [57].",
"We will use the final system of equations in the following form (where we have omitted the term responsible for scale dependence) $& 2 \\frac{dL_c}{d \\tau } = \\frac{\\dot{a}}{a} L_c \\left[ 1 + \\upsilon ^{2} - \\frac{1-\\upsilon ^2}{\\mu ^2} \\right] + \\frac{c f_a \\upsilon }{\\sqrt{\\mu }}, \\\\& \\frac{d\\upsilon }{d \\tau } = \\left( 1 - \\upsilon ^2 \\right) \\left[ \\frac{k}{L_c \\mu ^{5/2}} - \\frac{\\dot{a}}{a} \\upsilon \\left( 1 + \\frac{1}{\\mu ^2} \\right) \\right], \\\\& \\frac{1}{\\mu } \\frac{d \\mu }{d \\tau } = \\frac{\\upsilon }{L \\sqrt{\\mu }} \\left[ k \\left( 1 - \\frac{1}{\\mu ^2} \\right) - c (f_a - f_o - S) \\right]- \\nonumber \\\\& \\qquad \\qquad \\qquad -\\frac{\\dot{a}}{a} \\left(1-\\frac{1}{\\mu ^2} \\right),$ where the three functions $f_a(\\mu )$ , $f_0(\\mu )$ and $S(\\mu )$ quantify energy loss/transfer: $& 2 \\left( \\frac{d \\xi }{dt} \\right)_{\\text{only big loops}} = c f_0(\\mu ) \\upsilon , \\\\& 2 \\left( \\frac{d L}{dt} \\right)_{\\text{all loops}} = c f_a(\\mu ) \\frac{\\upsilon L}{\\xi }, \\\\& 2 \\left( \\frac{d \\xi }{dt} \\right)_{\\text{energy transfer}} = c S(\\mu ) \\upsilon \\,.$ Here, $c$ is a constant “loop chopping\" parameter (see below), $t=\\int a d\\tau $ , and $\\xi =\\xi _c a$ , $L=L_c a$ are the physical (rather than comoving) lengthscales corresponding to $\\xi _c$ and $L_c$ .",
"The term $f_0(\\mu )$ accounts for the energy loss due to the formation of big loops.",
"Here, “big\" means that they are formed by intersections of strings separated by distances of order the correlation length $\\xi $ or by self-intersections at the scale of the radius of curvature $R \\approx \\xi $ .",
"The function $f_a(\\mu )$ describes the energy loss caused by all types of loops, and the difference $f_a(\\mu )-f_0(\\mu )$ corresponds to the energy loss by small loops only, which is driven by the presence of wiggles.",
"In order to reproduce correctly the original model without wiggles, we can use the energy loss/transfer functions as discussed in [57] $& f_0(\\mu ) = 1, \\\\& f_a(\\mu ) = 1 + \\eta \\left( 1 - \\frac{1}{\\sqrt{\\mu }} \\right), \\\\& S(\\mu ) = D(1-\\frac{1}{\\mu ^2}),$ where $D$ and $\\eta $ are constants.",
"Thus, the evolution of a wiggly string network is described by the system of ordinary differential equations (REF )-(), which, in view of equations (REF )-() includes three free constant parameters $c$ , $D$ and $\\eta $ [57]: $\\bullet $ $c$ is the “loop chopping efficiency\" parameter quantifying how much energy the network loses due to the production of ordinary loops; $\\bullet $ $\\eta $ is a parameter describing the energy loss enhancement due to the creation of small loops caused by the presence of wiggles; $\\bullet $ $D$ is a parameter quantifying the amount of energy transferred from large to small scales.",
"By making various different choices of parameters $c$ , $\\eta $ and $D$ we can explore the effects of the energy loss/transfer mechanisms described above on the evolution of the string network.",
"Note that $c$ has been measured in Abelian-Higgs and Goto-Nambu simulations to be $c=0.23\\pm 0.04$ [58], [59], but there are no such measurements for the other two parameters.",
"Let us study how these phenomenological quantities can change the prediction for the CMB anisotropy caused by wiggly cosmic string networks.",
"Figure: Evolution of the rms velocity υ\\upsilon , comoving characteristic length L c L_c and amount of wiggles μ\\mu as a function of redshift zz for wiggly cosmic string networks with different values of the parameter DD, obtained by a modified version of the CMBact code .",
"The horizontal dashed red and blue lines correspond to the usual (without wiggles; μ=1\\mu =1) scaling regimes for radiation (red shaded area) and matter domination (blue shaded area) epochs respectively.",
"Note that the horizontal (redshift) axis is depicted in a linear scale in the redshift range 0<z<10<z<1 and in a logarithmic scale for z>1z>1.Figure: CMB anisotropy for wiggly cosmic string networks obtained by a modified version of the CMBact code .",
"The panels show scalar, vector and tensor contributions (top to bottom) of the BBBB, TTTT, TETE and EEEE modes (left to right).",
"(Note there is no BBBB contribution from scalar modes.)",
"These have been computed for different values of DD with fixed η\\eta .",
"The CMBact result from with α=2\\alpha =2 is shown by the black dashed line for comparison.In order to investigate in detail the effects of string wiggles on the predicted CMB anisotropies from cosmic string networks, we implement the wiggly VOS model (REF )-() into the CMBact code [43].",
"The original code was developed so as to take into account the presence of string wiggles in the computation of the string-induced CMB anisotropy.",
"However, in the original CMBact package, wiggles were modelled by a single (constant) phenomenological parameter $\\alpha =\\mu $ modifying the effective mass per unit length and string tension at the level of the stress-energy tensor (REF ).",
"In other words, within the approximations of the original CMBact code, the amount of wiggles was not a dynamical parameter and did not influence the equations of motion, while from the wiggly VOS model we have just discussed it is clear that these effects must, in general, be present.",
"Here, we implement the full description of wiggly strings in CMBact.",
"Using the equation of state for wiggly strings (REF ) we first rewrite the stress-energy tensor (REF ) as $\\begin{split}& \\qquad \\quad T^{\\mu \\nu } (y) = \\frac{\\mu _0}{\\sqrt{-g}} \\int d^2 \\sigma \\\\& \\left( \\epsilon \\mu \\dot{\\textbf {x}}^{\\mu } \\dot{\\textbf {x}}^{\\nu } - \\frac{\\textbf {x}^{\\prime \\mu } \\textbf {x}^{\\prime \\nu }}{\\epsilon \\mu } \\right) \\delta ^{(4)}(y-x(\\sigma ))\\,,\\end{split}$ where $\\mu $ is the amount of wiggles, which is now dynamical, satisfying equation ().",
"The size of string segments is set to be equalFor an even more realistic model we could consider the strings segments to have a range of sizes and speeds picked from appropriate distributions as in [34], but here we want to focus on the effects of string wiggles only and compare to the results of the original CMBact code, which also takes all segments to have the same size and speed.",
"to the correlation length $\\xi _0 \\tau $ .",
"We also change the VOS equations of motion in CMBact to the full system (REF )-() and implement the stress-energy components (REF )-(REF ).",
"With these modifications, we achieve a full treatment of wiggly cosmic string networks in CMBact.",
"In figure REF we show our results for network evolution and in figure REF the corresponding CMB anisotropies computed in our modified version of CMBact.",
"In both figures we also show the corresponding results of the original CMBact code [43] for comparison.",
"Regarding figure REF , we note that the accuracy of CMBact is comparatively worse at low redshifts; this explains why the effects of the matter to acceleration transition seemingly become visible around redshifts of a few, while the onset of acceleration occurs below $z=1$ .",
"This point is not crucial for our analysis, since our goal is to make a comparative study of the effects of the additional degrees of freedom on the strings.",
"Moreover, these low redshifts have a relatively small effect on the overall CMB signal.",
"Nevertheless, this is an issue which should be addressed if this code is to be used for quantitative comparisons with current or forthcoming CMB data.",
"We have chosen to vary parameter $D$ , keeping $\\eta $ fixed, which allows us to cover a wide range of $\\mu $ values.",
"Fixing $D$ and increasing $\\eta $ is equivalent to decreasing the amount of wiggles and an effective change of $c$ , which is already covered from our variation of $D$ with fixed $\\eta $ .",
"It is also important to note that in order to have an attractor scaling solution when $a \\propto \\tau ^n$ the following condition must be satisfied $\\eta > \\frac{D\\left( 1 - 1/\\mu ^2 \\right) }{1-1/\\sqrt{\\mu }}.$ Physically, this means that in order to achieve a scaling solution, small scale structure should be able to lose energy (controlled by parameter $\\eta $ ) faster than it receives the energy from large scales (controlled by parameter $D$ ).",
"When the condition (REF ) is violated, energy accumulates at small scales and there is no stable scaling regime for these wiggly cosmic strings.",
"In practice, the condition (REF ) is used as a guide for estimating the range of variation of $D$ .",
"Figure REF shows how the full treatment of wiggly cosmic string networks affects the prediction for the string-induced CMB anisotropy.",
"Note that the CMB contribution is generally smaller than for ordinary cosmic strings (i.e.",
"without wiggles, $\\mu =1$ ).",
"This is mainly due to a reduction in the rms string velocity $\\upsilon $ (see figure REF ) when the amount of wiggles $\\mu $ increases.",
"In view of the observed changes to the usual CMB predictions for cosmic strings, we argue that to achieve accurate results for wiggly cosmic strings, one should study them in the framework of the complete wiggly model (REF )-() and the modified version of CMBact developed here.",
"This generally leads to a weakening of the CMB-derived constraint on the string tension $\\mu _0$ (but note that there is also a region in parameter space – for large $D$ – where the correlation length can actually become smaller than for ordinary strings, see figure REF ).",
"Note that both the evolution and CMB results from our wiggly VOS model are somewhat closer in comparison to results from Abelian-Higgs simulations (and similarly ordinary VOS results are closer to Nambu-Goto simulations).",
"It is then tempting to speculate that wiggles play a dynamical role analogous to that of the averaged field fluctuations that appear in Abelian-Higgs field theory simulations (as opposed to effective Nambu-Goto simulations).",
"This hypothesis may be investigated by direct comparisons of Abelian-Higgs and Goto-Nambu simulations with suitably high resolutions and dynamic ranges.",
"To end this section, let us return to the wiggly model but this time without referring to the specific Lagrangian (REF ).",
"We wish to study the scaling regime for wiggly strings but leaving the amount of wiggles $\\mu $ as a free parameter that we can tune.",
"Instead of varying parameter $D$ , as it was done above, we can vary $\\mu $ .",
"This approach does not require an assumption on the energy transfer function (); we only need to define how energy loss depends on the amount of wiggles ().",
"Let us now estimate how the rms velocity $\\upsilon $ and comoving characteristic length $L_c$ are related to the parameters $c$ , $\\eta $ and $\\mu $ in the scaling regime.",
"We insert the scaling solution $L_c = \\varepsilon \\tau $ , $\\upsilon =$ const to equations (REF ), () to obtain the algebraic equations $& \\varepsilon \\left(2-n \\left[ 1 + \\upsilon ^{2} - \\frac{1-\\upsilon ^2}{\\mu ^2} \\right] \\right) = \\frac{ c f_a(\\mu ) \\upsilon }{\\sqrt{\\mu }} , \\\\& \\frac{k(\\upsilon )}{\\varepsilon \\mu ^{5/2}} = n \\upsilon \\left( 1 + \\frac{1}{\\mu ^2} \\right),$ where we have included energy loss function $f_a(\\mu )$ given by ().",
"Figure: Dependence of the scaling values of the rms velocity, vv, and the comoving correlation length divided by conformal time, ε\\varepsilon , on the amount of wiggles μ\\mu for different expansion rates nn.Figure: Comparison between the behaviour of the string-induced angular power spectrum C l C_l for different amount of wiggles in our analytic approximation (solid lines) and the numerical computation using our modified CMBact code (circles).",
"The dependence on μ\\mu has been estimated analytically using equations (), () together with the equations for the scaling regime of the network ()-().",
"Using the value of μ\\mu in the matter domination era and C l C_l's for scalar (green), vector (blue) and tensor (red) components at l=700l=700 (where the sum peaks), we have obtained the C l -μC_l - \\mu dependence from the CMBact code.Despite the reduction of the equations of motion to algebraic equations (REF ) and () in the scaling regime, it is still not possible to solve them analytically, mainly due to the complicated form of the momentum parameter (REF ).",
"To study how the amount of wiggles affects the macroscopic parameters $\\upsilon $ (rms string velocities) and $\\varepsilon $ (comoving correlation length in units of conformal time) in the scaling regime we solve the system (REF )-() numerically for different expansion rates $n$ .",
"The results are shown in figure REF .",
"It is seen that the rms velocity $\\upsilon $ , as anticipated from the restriction (REF ), decreases with the growth of the amount of wiggles $\\mu $ .",
"This is also in agreement with our results for the rms velocity evolution (see figure REF ) in the dynamical wiggly model for a realistic expansion history.",
"The situation for $\\varepsilon $ is more interesting.",
"The correlation length does not increase monotonically with the amount of wiggles but has a maximum around $\\mu =1.5-1.9$ .",
"This is also in agreement to our full treatment in figure REF where we modelled string wiggles by varying parameter $D$ and took a realistic expansion history.",
"Since we have computed the velocity $\\upsilon $ and correlation length $\\xi _0 \\tau = \\sqrt{\\mu } \\varepsilon \\tau $ in the scaling regime, we can use equations (REF ) and (REF ) to estimate how the contribution to the CMB anisotropy from cosmic strings depends on the amount of string wiggles.",
"For wiggly cosmic strings the angular power spectrum $C_l$ has the following dependence (which coincides with the result in [55]) $C_l \\sim (G \\mu _0)^2 \\frac{\\mu ^4 \\upsilon ^4 + \\mu ^2 \\upsilon ^2 (1-\\upsilon ^2)+ (1-\\upsilon ^2)^2 }{\\mu ^2 \\xi _0 (1-v^2)},$ where scalar, vector and tensor components depend on string parameters in the same way.",
"We can now compare the dependence in equation (REF ) with our numerical results using our modified CMBact code.",
"By choosing the $\\mu $ value for the matter domination era and looking at the peak ($l \\approx 700$ ) of the sum of the scalar, vector and tensor contributions we plot them in comparison to the analytic estimate from (REF ) and (REF ).",
"This comparison is shown in figure REF .",
"For our approximate estimate it is seen that after a fast decrease of $C_l$ 's with growing amount of wiggles $\\mu $ , the value of $C_l$ reaches a plateau.",
"A similar behaviour is seen for vector, tensor and scalar components obtained from the full treatment using our modified CMBact code, even though the agreement is somewhat weaker for the scalar contribution.",
"These results reaffirm the approximations used to estimate the analytic dependence of $C_l$ on the string network characteristics."
],
[
"Superconducting model (chiral case)",
"Another special case of current-carrying cosmic strings of notable physical interest is the case of superconducting cosmic strings.",
"This type of strings has been studied thoroughly in the framework of field-theory [35], [48], [49], [50], [51], [60], [61].",
"In all these cases the stress-energy tensor on the string worldsheet has the following form: $T^{a}_b = \\begin{pmatrix} A + B & -C \\\\ C & A - B \\end{pmatrix}\\,.$ where $A$ arises from the field responsible for the string core formation, while $B$ and $C$ represent additional contributions due to coupling with external fields (dynamics of currents).",
"The stress-energy tensor (REF ) is written for the worldsheet metric $\\eta ^{ab} = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}$ on a 4-dimensional Minskowski spacetime background with $\\epsilon = 1$ .",
"Consider now the two-dimensional stress-energy tensor for the action (REF ), which reads $T^{a}_b = \\begin{pmatrix} \\mu _0 \\tilde{U} & - \\mu _0 \\frac{\\Phi }{\\epsilon } \\\\ \\mu _0 \\epsilon \\Phi & \\mu _0 \\tilde{T} \\end{pmatrix}.$ There is an obvious correspondence between the stress-energy tensors (REF ) and (REF ); they are in agreement if we demand the chiral condition [25], [26] $\\kappa \\rightarrow 0\\,,$ which also means $\\Delta \\rightarrow 0\\,;$ here $\\kappa $ and $\\Delta $ are defined by Eq.",
"(REF ).",
"These imply that $\\tilde{U} = 1+\\Phi $ , $\\tilde{T} = 1 - \\Phi $ .",
"In Minkowski space ($\\epsilon =1$ ) we see that $A=\\mu _0$ , $B = \\mu _0 \\Phi $ and $C=\\mu _0 \\Phi $ , so we have the condition $B=C$ .",
"In order to avoid this situation and be able to reproduce a stress-energy tensor of the form (REF ) within the Nambu-Goto approximation, we need to use at least two scalar fields.",
"It has already been demonstrated that adding any number of additional fields (REF ) together with the definitions (REF )-() keeps the evolution equations (REF )-() unchanged, replacing the scalar field equation () by the set of equations (REF ).",
"In effect, introducing additional fields makes $C$ and $B$ different in Minkowski space.",
"Indeed, when we add extra scalar fields we obtain a stress-energy tensor in the form of (REF ) with the correspondenceHere we used an assumption that all multipliers $\\frac{\\partial f}{\\partial \\Delta _i}$ are equal as well as all $\\frac{\\partial f}{\\partial \\kappa _i}$ are equal (REF ).",
"$A = \\mu _0$ , $B=2 \\mu _0 \\gamma ^{00} \\left( \\frac{\\partial f}{\\partial \\kappa } - \\frac{\\partial f}{\\partial \\Delta } \\right) \\sum _i \\dot{\\varphi _i}^2 = \\mu _0 \\Psi $ and $C = \\mu _0 \\Phi $ , where $\\Phi $ is given by equation () and we have assumed a Minkowski background.",
"These correspond to $\\begin{split}\\hat{U} = 1+\\left<\\Psi \\right>, \\; & \\; \\hat{T} = 1-\\left<\\Psi \\right>, \\; \\; Q=\\left<\\Phi \\right>.\\end{split}$ Thus, this multiple worldsheet field approach provides enough flexibility to reproduce the field-theoretical stress-energy tensor variables in (REF ) within the Nambu-Goto approximation.",
"Let us now consider the equations of motion for chiral currents.",
"We will apply our averaging procedure to the system of equations (REF ) for the currents, similarly to what we already did for first two equations (REF ) and () for the correlation length and string velocity.",
"First of all, we note that in order to have the appropriate Nambu-Goto limit for the action (REF ) when $\\varphi =0$ we need to have $f(\\kappa , \\Delta )\\xrightarrow[\\kappa \\rightarrow 0]{}1$ and additionally $\\frac{\\partial f(\\kappa , \\Delta )}{\\partial \\kappa } \\xrightarrow[\\kappa \\rightarrow 0]{} \\text{const}$ (as well as $\\frac{\\partial f(\\kappa , \\Delta )}{\\partial \\Delta } \\xrightarrow[\\Delta \\rightarrow 0]{} \\text{const}$ ).",
"These conditions allow us to make simplifications, similar to what was done in reference [26], and consider the case of conserved microscopic charges for each field $& \\epsilon \\dot{\\varphi _i}= \\phi _{i}=\\text{const} \\;, \\\\& \\varphi _i^{\\prime }= \\psi _{i}=\\text{const} \\; ,$ which leads to the additional condition $\\epsilon ^{\\prime }=0$ .",
"Furthermore, in this case we can define $\\Psi $ and $\\Phi $ as $\\Psi = \\left( \\frac{\\partial f}{\\partial \\kappa } + \\frac{\\partial f}{\\partial \\Delta } \\right) \\sum _i \\frac{\\phi _i^2}{a^2 \\textbf {x}^{\\prime 2}},$ $\\Phi = \\left( \\frac{\\partial f}{\\partial \\kappa } + \\frac{\\partial f}{\\partial \\Delta } \\right) \\sum _i \\frac{\\phi _i \\psi _i}{a^2 \\textbf {x}^{\\prime 2}},$ and (REF ) gives us $\\sum _i \\phi _i^2 = \\sum _i \\psi _i^2 .$ Expressions (REF ) and (REF ) tell us that if we use the condition of conserved microscopic charges (REF )-() we have two variables $\\Psi $ and $\\Phi $ which evolve in the same way and differ only by a multiplicative constant $\\beta $ : $\\Psi \\beta = \\Phi ,$ where $\\beta = \\frac{\\sum _i \\phi _i \\psi _i }{\\sum _i \\phi _i^2 }$ .",
"Together with (REF ), this implies that $0<\\beta <1$ .",
"By direct differentiation of equation (REF ) we obtain the following evolution equations for the field $\\Psi $ (clearly, the same equations are also obeyed by $\\Phi $ ) $& \\dot{\\Psi } + 2 \\frac{\\dot{a}}{a} \\Psi = 2 \\Psi \\frac{\\dot{\\textbf {x}} \\cdot \\textbf {x}^{\\prime \\prime }}{\\textbf {x}^{\\prime \\, 2}}, \\\\& \\Psi ^{\\prime } + 2 \\Psi \\frac{\\textbf {x}^{\\prime } \\cdot \\textbf {x}^{\\prime \\prime }}{\\textbf {x}^{\\prime \\, 2}} = 0.$ Following the approach of [26], we average the equations of motion (REF ), () and substitute the equation of state (REF ) into equations (REF ) and ().",
"This leads to the VOS model for superconducting chiral strings, taking into account energy and charge losses (for details on these loss terms see [26]) $& \\frac{dL_c}{d \\tau } = \\frac{\\dot{a}}{a} L_c \\frac{\\upsilon ^{2} + Q}{1+Q} + \\upsilon \\left( \\frac{Q s \\beta }{(1+Q)^{3/2}} + \\frac{c}{2} \\right), \\\\& \\hspace{-7.22743pt} \\frac{d\\upsilon }{d \\tau } = \\frac{1-\\upsilon ^2}{1+Q}\\left[ \\frac{k(\\upsilon ) }{L_c \\sqrt{1+Q}} \\left( 1 - Q (1+\\frac{2 s \\beta }{k(\\upsilon )}) \\right) - 2 \\frac{\\dot{a}}{a} \\upsilon \\right], \\\\& \\hspace{-7.22743pt} \\frac{d Q}{d \\tau } = 2 Q \\left( \\frac{k(\\upsilon ) \\upsilon }{L_c \\sqrt{1+Q}} - \\frac{\\dot{a}}{a}\\right) + \\frac{c \\upsilon \\left( 1- \\sqrt{1+Q} \\right)}{L_c } \\sqrt{1+Q} \\,.$ We have used the assumption $\\left\\langle \\frac{\\Psi ^{\\prime }}{\\epsilon (1+\\Psi )} \\right\\rangle = -s \\frac{\\upsilon }{R_c} \\frac{2 Q}{1+Q}$ [26] and that the correlation and characteristic lengths are related by $\\xi _c=L_c \\sqrt{1+Q}$ .",
"Therefore, our general analysis of chiral current dependence in the action (REF ) including the addition of extra worldsheet fields has not introduced significant changes in the macroscopic equations describing superconducting chiral cosmic string networks, as compared to the results in [26] (the only difference is that the constant $s$ has now been changed to $\\beta s$ ).",
"Note also that the final result does not depend explicitly on the precise form of the Lagrangian; the important physics can be encoded in the equations of state of the strings, in agreement with our previous discussion.",
"The evolution of string networks described by equations (REF )-() was carefully studied in [26].",
"It was shown that these networks have generalized scaling solutionsIn this context, by “generalized scaling solutions\" we mean that all three quantities $L_c/\\tau $ , $v$ and $Q$ approach constant non-zero values (and so the strings have non-zero charge).",
"For larger expansion rates there are also solutions with a decaying charge $Q$ for which $L_c/\\tau $ and $v$ are (non-zero) constants but $Q$ approaches zero in a power-law fashion [26].",
"These correspond to the standard linear scaling solutions of (uncharged) Nambu-Goto strings and we do not discuss them here in detail.",
"only if the following relation is satisfied $n=\\frac{2 k(\\upsilon )-c \\tilde{W}}{c+k(\\upsilon )},$ where $\\tilde{W}=\\frac{\\sqrt{1+Q_s}-1}{1+Q_s^{-1}}$ , with $Q_s$ a constant corresponding to the scaling value of the function $Q$ .",
"As we can see from equation (REF ), the expansion rate $n$ for scaling solutions (with constant charge) cannot be larger than $n \\le 2$ .",
"The maximal value of $n$ is reached when $c=0$ , while for $c=0.23$ (which we use here) we have the condition $n \\lesssim 1.6$ for scaling behaviour.",
"For expansion rates $n$ larger than the right hand side of (REF ) the charge $Q$ on the string decays.",
"It is worth noting that at the same time the asymptotic value of the charge $Q_s$ is limited from equation () to satisfy the following (see figure REF ) $Q_s < \\frac{k(\\upsilon )}{2 s \\beta + k(\\upsilon )}.$ Figure: Constraints on the possible values of the charge Q s Q_s depending on the rms velocity υ\\upsilon and parameter ss.Figure: Evolution of the rms velocity υ\\upsilon , comoving characteristic length L c L_c and charge QQ depending on redshift zz for superconducting (chiral) cosmic string networks with different initial conditions Q 0 Q_0 for the string charge, obtained by a modified version of the CMBact code .",
"The horizontal dashed red and blue lines correspond to the usual (without charge, Q=0Q=0) scaling regimes for radiation (red shaded area) and matter domination (blue shaded area) eras respectively.",
"Note that the horizontal (redshift) axis is depicted in a linear scale in the redshift range 0<z<10<z<1 and in a logarithmic scale for z>1z>1.Figure: CMB anisotropy results for superconducting (chiral) cosmic sting networks obtained by our modified version of CMBact .",
"The panels show the scalar, vector and tensor contributions (top to bottom) to the BBBB, TTTT, TETE and EEEE power spectra (left to right).",
"The calculations are done for different initial conditions of the charge Q 0 Q_0.For all other expansion rates that do not satisfy conditions (REF ) and (REF ), there is no scaling regime with nonzero $Q$ .",
"However, in all cases (even in the absence of scaling solutions) we can still evolve the network with our modified VOS model and use equations (REF )-(REF ) with the stress-energy tensor $& T^{\\mu \\nu } (y) = \\frac{\\mu _0}{\\sqrt{-g}} \\int \\sqrt{-\\gamma } \\biggl ( (1+\\Psi ) u^{\\mu } u^{\\nu } - (1-\\Psi ) v^{\\mu } v^{\\nu } - \\nonumber \\\\& - \\alpha \\Psi (u^{\\mu } v^{\\nu } + v^{\\mu } u^{\\nu }) \\biggl )\\delta ^{(4)}(y-x(\\sigma )) d^2 \\sigma $ to modify the CMBact code for a superconducting chiral cosmic string network.",
"In the absence of scaling the charge $Q$ for the cosmic string network evolution is controlled mainly by the initial condition $Q_0$ .",
"Note that, unlike the wiggly case, there are currently no numerical simulations which can provide us with benchmarks for the value of this charge.",
"Thus, by varying $Q_0$ we obtain different evolutions for cosmic superconducting string networks (see figure REF ) and their corresponding contributions to the CMB anisotropy (see figure REF ).",
"Figure: Scaling values of rms velocity, vv, and comoving correlation length divided by conformal time, ε\\varepsilon , depending on the charge QQ, for different expansion rates nn.Figure: Behaviour of C l C_l for different string charge QQ, obtained from the analytical approximation.Let us consider the network at specific values $n$ satisfying equations (REF ), (REF ).",
"For that we will use the typical scaling ansatz $L_c = \\varepsilon \\tau $ with constant $\\varepsilon $ , $\\upsilon $ and $Q$ in equations (REF )-() $& \\varepsilon = n \\varepsilon \\frac{v^{2} + Q}{1+Q} + v \\left( \\frac{Q s \\alpha }{(1+Q)^{3/2}} + \\frac{c}{2} \\right), \\\\& \\frac{k(v) }{\\varepsilon \\sqrt{1+Q}} \\left( 1 - Q (1+\\frac{2 s \\alpha }{k(v)}) \\right) = 2 n \\upsilon .$ Equations (REF ) and () describe the network evolution in the scaling regime (they will be valid only in a range of $n$ satisfying (REF )).",
"Using these equations and the analytic form of the angular power spectrum's dependence on string network parameters (REF ), we can make an estimation of $C_l$ for the network of superconducting chiral strings ${\\begin{array}{c}C_l \\sim (G \\mu _0)^2 \\frac{\\upsilon ^4 (1+3 \\beta ^2 Q^2)}{\\varepsilon (1-v^2)} + \\\\\\frac{v^2(3 Q^2 (1-\\beta ^2) + 4 \\beta Q -1)+(1-\\beta Q)^2}{\\varepsilon (1-v^2)},\\\\\\end{array}}$ where scalar, vector and tensor components depend on string parameters in the same way.",
"We can then solve the algebraic equations (REF ), () numerically for different values of $Q$ (some solutions are shown in figure REF ) and insert them in (REF ) to get an estimate of how the angular power spectrum $C_l$ depends on the value of the charge $Q$ (figure REF ).",
"In both our numerical calculations using the modified CMBact code (figures REF , REF ) and in our analytic estimates (figures REF , REF ) we observe that the string rms velocity tends to decrease as we increase the charge $Q$ .",
"The comoving correlation length in units of the conformal time $\\varepsilon $ increases for small values of the charge, but then reaches a maximum and eventually decreases for higher values of the charge $Q$ .",
"Concerning the angular power spectra $C_l$ , it should be noted that it is difficult to make extensive comparisons in the case of superconducting strings, as there is no scaling behaviour in the full range of expansion rates $n$ and we do not know which $Q$ values we should choose from our numerical results in figure REF .",
"However, it is clear that the analytic approach and numerical computation are in qualitative agreement.",
"In particular the angular power spectrum $C_l$ decreases as we increase the charge $Q$ on the string."
],
[
"Conclusions",
"There are many well-motivated scenarios in early universe physics that can leave behind relic defects in the form of cosmic strings.",
"These relics can be utilised as “fossils\" for cosmological research, helping us to obtain a better understanding of the physical processes that took place in the early universe.",
"By developing an accurate description of the evolution of cosmic string networks and using it to calculate quantitative predictions of string-induced observational signals, we can obtain strong constraints on theoretical models leading to a better understanding of early universe physics.",
"Here, we presented a detailed study of the evolution of cosmic strings with currents and demonstrated how the presence of worldsheet currents affects the predictions for the CMB anisotropy produced by cosmic string networks.",
"In section II we considered the action (REF ) describing strings with an arbitrary dependence on worldsheet currents.",
"We have described how to average the microscopic equations of motion for this model to obtain macroscopic evolution equations (without energy loss) for the string network (REF )-().",
"These describe the time evolution of the rms string velocity $\\upsilon $ and characteristic length $L$ , and depend only on three parameters $\\hat{U}$ , $\\hat{T}$ and $Q$ defining the string equation of state.",
"These same parameters, together with the network quantities $L$ and $\\upsilon $ , appear directly in the string stress-energy tensor (REF ) which seeds the string-induced CMB anisotropy.",
"This provides a direct connection between modelling string evolution and computing CMB anisotropies from cosmic string networks, which has allowed us to obtain simple analytic estimates for the dependence of the string angular power spectrum $C_l$ on macroscopic network parameters (REF )-(REF ).",
"For a more complete semi-analytic treatment of the CMB anisotropy for strings with currents, we have adapted the methodology of [34] and have provided coefficients for the relevant integrals in the Appendix.",
"In sections III and IV we considered two specific cases of strings with currents: wiggly and superconducting cosmic strings respectively.",
"In each case we computed the CMB signal numerically using appropriately modified versions of CMBact.",
"For wiggly string networks (section III) we studied the specific case when the parameter $\\kappa $ in (REF ) only carries a time dependence, $\\kappa =\\kappa (\\tau )$ .",
"We studied network dynamics using an effective action for wiggly cosmic strings, and introduced the averaged macroscopic equations into CMBact, allowing us to compute CMB anisotropies from these strings.",
"CMBact has already built in the option to study wiggly strings, but this was done through a single constant parameter.",
"Here, for the first time, we were able to take into account the time evolution of wiggles and their influence on the macroscopic equations of motion for the string network.",
"This full treatment brought important changes in modelling wiggly cosmic string networks.",
"From Fig.",
"REF we see that wiggly strings can produce a lower signal in CMB anisotropy than ordinary strings (when the other parameters are fixed), which had not been appreciated before our work.",
"We have also compared our analytic estimation (REF ) to our numerical results from the CMBact code.",
"The comparison shows that the main trend for $C_l$ (decreasing of $C_l$ as $\\mu $ increases, for multiple moments $1<<l$ ) is captured correctly.",
"We argue that for reliable constraints on wiggly string networks through the CMB signal, the evolution of string currents and its effect on string dynamics – as captured by our wiggly model – should be taken into account.",
"We point out that comparing results from our analytic wiggly string network evolution and the standard VOS model for ordinary cosmic strings, has a broad resemblance to the differences that appear between Abelian-Higgs and Nambu-Goto numerical simulations for strings.",
"In particular, increasing the amount of wiggles $\\mu $ leads to slower rms velocities and a lower contribution to the string-induced CMB anisotropy decreases.",
"This is similar to the difference between Abelian-Higgs and Nambu-Goto string networks, where the Abelian-Higgs strings tend to be slower and produce a lower CMB signal.",
"This is, at present, a speculative observation requiring further investigation to see if a more firm analogy may be established.",
"The other type of strings that were scrutinized in this work are superconducting cosmic strings.",
"It was shown that if we use the microscopic charge conservation (REF ) and the chiral condition ($\\kappa , \\Delta \\rightarrow 0$ , which appears in field theory studies of cosmic strings), we can obtain the averaged equations of motion (REF )-() without specifying the precise dependence on string currents $f(\\kappa _i,\\Delta _i/\\gamma )$ in the action (REF ).",
"This implies that the debate on the correct form of the Lagrangian for superconducting strings [45] – while important from a fundamental physics point of view – does not have a crucial impact on phenomenological descriptions based on averaging the microscopic dynamics.",
"By comparison to the work of [26], we notice that the introduction of additional currents for superconducting strings only led to the change $s \\rightarrow \\beta s$ in the macroscopic VOS model.",
"Introducing the appropriate modifications to CMBact, we have found that the string-induced CMB anisotropies tend to decrease with increasing the charge $Q$ of superconducting stings.",
"Since the charge $Q$ does not have a scaling behaviour in the full range of physically relevant expansion rates (REF ), but generally decreases with evolution, the main effect on the CMB anisotropy comes from the initial charge $Q_0$ at the moment of string formation.",
"We varied the initial charge to obtain a range of network dynamics histories and computed the corresponding CMB signal predictions.",
"Numerical simulations are needed to further quantify the relevant model parameters.",
"The approach developed here can be useful in Markov chain Monte Carlo analysis of cosmological models with cosmic strings [34].",
"It allows to obtain more accurate constraints on wiggly and superconducting string network parameters directly from CMB observations.",
"Finally, it is worth noting that the effects of the presence of currents on strings, described by our macroscopic VOS model, will also have a non-trivial impact on other observational windows for cosmic string networks, such as the stochastic gravitational wave background generated by string networks [62], [63], [64], [65], [66], [67], [68], [69], [70], [71].",
"Our results on string evolution and the methodology developed here for computing the two-point (unequal time) correlator will be useful for further studies in this direction too.",
"This work of IR is supported by the FCT fellowship (SFRH/BD/52699/2014), within the FCT PD Program PhD::SPACE (PD/00040/2012).",
"IR is grateful for the hospitality of the University of Nottingham, where part of this work was carried out.",
"The work of AA was partly supported by an Advanced Nottingham Research Fellowship at the University of Nottingham.",
"CJM is supported by an FCT Research Professorship, contract reference IF/00064/2012, funded by FCT/MCTES (Portugal) and POPH/FSE (EC).",
"We would like to thank Patrick Peter, Tom Charnock, José Pedro Vieira and colleagues from P.S.",
"for fruitful discussions and help."
],
[
"Appendix: Analytic expressions for equal-time correlators",
"As shown in [55] the integral (REF ) can be expanded in the following way ${\\begin{array}{c}\\left< \\Theta ^I(k,\\tau _1) \\Theta ^J(k,\\tau _2) \\right> = \\frac{f(\\tau _1,\\tau _2,\\xi _0) \\mu _0^2}{k^2(1-\\upsilon ^2)} \\times \\\\\\sum _{i=1}^{6} A_i^{IJ} \\left[ I_i(x_{-},\\rho ) - I_i(x_{+},\\rho ) \\right],\\end{array}}$ where $I$ , $J$ correspond to the “00\", scalar, vector and tensor components of the stress-energy tensor and the form of the six integrals $I_i$ are as given in [55].",
"The coefficients $A_i^{IJ}$ , together with the full expressions for the analytic equal time correlators $B^{IJ}$ , are listed below (where, in this Appendix, we use the definitions $\\rho = k | \\tau _1 - \\tau _2| \\upsilon $ , $x_{1,2}=k \\xi _0 \\tau _{1,2}$ , $x_{\\pm } = (x_1 \\pm x_2)/2$ ): $& A_1^{00-00} &=& &2 \\hat{U}^2, \\\\& A_i^{00-00} &=& &0, \\\\& & & &\\qquad (i=2,..,6) \\\\& A_1^{00-S} &=& &\\hat{U}( \\hat{T} + (2 \\hat{U}- \\hat{T} )v^2 ), \\\\& A_2^{00-S} &=& &- 3 \\hat{U} \\left( \\hat{T} (1-v^2) + \\hat{U} v^2 \\right)\\\\&A_3^{00-S} &=& &0\\\\&A_4^{00-S} &=& &-3 \\hat{U}^2 v^2\\\\&A_5^{00-S} &=& &3 \\hat{U}^2 v^2 \\\\&A_6^{00-S} &=& &0\\\\$ $& A_1^{S-S} &=& &\\frac{-27 \\hat{U}^2 v^4 + \\rho ^2 (\\hat{T} + (2 \\hat{U} - \\hat{T})v^2)^2}{2 \\rho ^2}\\\\& A_2^{S-S} &=& &\\frac{3 \\left( 9 \\hat{U}^2 v^4 + \\rho ^2 \\left( \\hat{T}^2 (1-v^2)^2 - \\hat{U}^2 v^4 \\right) \\right) }{2 \\rho ^2} \\\\&A_3^{S-S} &=& &-\\frac{9}{2} \\left( \\left( \\hat{U} v^2 + \\hat{T} (1-v^2)\\right)^2 - 4 v^2 Q^2 \\right) \\\\&A_4^{S-S} &=& &\\frac{3 \\hat{U} v^2 \\left( 9 \\hat{U} v^2 - \\rho ^2 (\\hat{T}(1-v^2) + 2 \\hat{U} v^2) \\right)}{\\rho ^2} \\\\&A_5^{S-S} &=& &-\\frac{3 \\hat{U} v^2 \\left(9 \\hat{U} v^2 - \\rho ^2 (\\hat{T}(1-v^2) + 2 \\hat{U} v^2) \\right)}{\\rho ^2} \\\\&A_6^{S-S} &=& &9 v^2 \\left( \\hat{U}^2 v^2 + \\hat{T} \\hat{U} (1-v^2) - 2 Q^2 ) \\right)\\\\& A_1^{V-V} &=& &\\frac{3 \\hat{U}^2 v^4 + \\rho ^2 v^2 Q^2}{\\rho ^2} \\\\& A_2^{V-V} &=& &- \\frac{3 \\hat{U}^2 v^4}{\\rho ^2} \\\\&A_3^{V-V} &=& &\\left( \\hat{U} v^2 + \\hat{T} (1-v^2) \\right)^2 - 4 v^2 Q^2 \\\\&A_4^{V-V} &=& &-\\left(6/\\rho ^2 - 1 \\right) \\hat{U}^2 v^4 - v^2 Q^2 \\\\&A_5^{V-V} &=& &\\left(6/\\rho ^2 - 1 \\right) \\hat{U}^2 v^4 + v^2 Q^2 \\\\&A_6^{V-V} &=& &-2 v^2 \\left(\\hat{U}^2 v^2 + \\hat{T} \\hat{U} (1-v^2) - 2 Q^2 \\right) \\\\& A_1^{T-T} &=& &\\frac{\\rho ^2 \\hat{T}^2 \\left( 1-v^2 \\right)^2 -3 \\hat{U}^2 v^4}{4 \\rho ^2} \\\\& A_2^{T-T} &=& &\\frac{3 \\hat{U}^2 v^4 - \\rho ^2 \\left( \\hat{T}^2 \\left( 1-v^2 \\right)^2 - \\hat{U}^2 v^4 \\right) }{4 \\rho ^2} \\\\&A_3^{T-T} &=& &-\\frac{1}{4} \\left( \\hat{U} v^2 + \\hat{T}(1-v^2) \\right)^2 + v^2 Q^2 \\\\&A_4^{T-T} &=& &\\frac{v^2 \\left( 3 \\hat{U}^2 v^2 +\\rho ^2 \\left( \\hat{T} \\hat{U}(1-v^2) + 2 Q^2 \\right) \\right)}{2 \\rho ^2} \\\\&A_5^{T-T} &=& &-\\frac{v^2 \\left( 3 \\hat{U}^2 v^2 + \\rho ^2 \\left( \\hat{T} \\hat{U}(1-v^2) + 2 Q^2 \\right) \\right)}{2 \\rho ^2} \\\\&A_6^{T-T} &=& &\\frac{v^2}{2} \\left( \\hat{U}^2 v^2 + \\hat{T} \\hat{U} (1-v^2) - 2 Q^2 \\right)$ $& B^{00-00}(\\tau ) = 2 \\hat{U}^2 (\\cos (x)-1+x Si(x)), \\nonumber \\\\& B^{00-S} = \\frac{1}{2 x} \\left( \\hat{U}(2\\hat{T}+v^2(\\hat{U}-2\\hat{T}))(x \\cos (x) + 3 \\sin (x)+x (x Si(x) - 4)) \\right), \\nonumber \\\\& B^{S-S} = \\frac{1}{16 x^3} \\biggl ( \\biggl [ 8 \\hat{T} \\hat{U} v^2 (1-v^2)(x^2-18)+8 \\hat{T}^2 (1-v^2)^2 (x^2-18)+\\hat{U}^2 v^4(11x^2-54)+288v^2 Q^2 \\biggl ] x \\cos (x) + \\nonumber \\\\&+x^3 \\biggl [ 32 \\left( 3 v^2 Q^2 - \\hat{U}^2 v^4 -\\hat{T} \\hat{U} v^2 (1-v^2)-\\hat{T}^2 (1-v^2)^2) \\right)+\\left( 11 \\hat{U}^2 v^4 + 8\\hat{T} \\hat{U} v^2 (1-v^2)+8 \\hat{T}^2 (1-v^2)^2 \\right) x Si(x) \\biggl ]- \\nonumber $ $& - 3 \\sin (x) \\biggl [ 8 \\hat{T} \\hat{U} v^2 (1-v^2)(x^2-6)+8 \\hat{T}^2 (1-v^2)^2(x^2-6)-\\hat{U}^2 v^4 (18+z^2)+96 v^2 Q^2 \\biggl ], \\nonumber \\\\& B^{V-V} = \\frac{1}{24 x^3} \\biggl ( 3 x \\cos (x) \\left[ 16 \\hat{T} (1-v^2) (\\hat{T} - (\\hat{T}-\\hat{U})v^2) + \\hat{U}^2 v^4 (6+z^2) + 4 v^2 (x^2-8) Q^2 \\right] + \\nonumber \\\\& x^3 \\left[ 16 \\hat{T} (1-v^2)(\\hat{T} - (\\hat{T}-\\hat{U})v^2) - 32 v^2 Q^2 + 3 v^2 x (\\hat{U}^2 v^2 + 4 Q^2 ) Si(x) \\right] - \\nonumber \\\\& 3 \\sin (x) \\left[ 16 \\hat{T} (1-v^2) (\\hat{T}-(\\hat{T}-\\hat{U})v^2) + \\hat{U}^2 v^4 (6-x^2) + 4 v^2 (x^2 - 8) Q^2 \\right] \\biggl ), \\nonumber \\\\& B^{T-T} = \\frac{1}{96 x^3} \\biggl ( 3 x \\cos (x) \\left[ (3 \\hat{U}^2 v^4 + 8 \\hat{T} \\hat{U} v^2 (1-v^2) +8 \\hat{T}^2 (1-v^2)^2 ) (x^2 - 2) + 16 v^2 (2+x^2) Q^2 \\right] + \\nonumber \\\\& + x^3 \\left[ 64 \\hat{T} (1-v^2) (v^2 (\\hat{T}-\\hat{U})-\\hat{T})-64 v^2 Q^2 + 3 x (3 \\hat{U}^2 v^4 + 8 \\hat{T} \\hat{U} v^2 (1-v^2) +8 \\hat{T}^2 (1-v^2)^2 +16 v^2 Q^2 ) Si(x) \\right] + \\nonumber \\\\& + 3 \\sin (x) \\left[ \\hat{U}^2 v^4 (6-5 x^2) + 8 \\hat{T} \\hat{U} v^2 (1-v^2) (2 + x^2) + 8 \\hat{T}^2 (1-v^2)^2 (2+x^2)+16 v^2 (x^2 - 2) Q^2 \\right] \\biggl ) \\nonumber .$"
]
] | 1709.01839 |
[
[
"Spoken English Intelligibility Remediation with PocketSphinx Alignment\n and Feature Extraction Improves Substantially over the State of the Art"
],
[
"Abstract We use automatic speech recognition to assess spoken English learner pronunciation based on the authentic intelligibility of the learners' spoken responses determined from support vector machine (SVM) classifier or deep learning neural network model predictions of transcription correctness.",
"Using numeric features produced by PocketSphinx alignment mode and many recognition passes searching for the substitution and deletion of each expected phoneme and insertion of unexpected phonemes in sequence, the SVM models achieve 82 percent agreement with the accuracy of Amazon Mechanical Turk crowdworker transcriptions, up from 75 percent reported by multiple independent researchers.",
"Using such features with SVM classifier probability prediction models can help computer-aided pronunciation teaching (CAPT) systems provide intelligibility remediation."
],
[
"Introduction",
"Authentic intelligibility, the ability of listeners to correctly transcribe recorded utterances, initially used for CAPT by [1] and [2], is a better measure of pronunciation assessment for spoken language learners compared to mispronunciations identified by expert pronunciation judges or panels of experts, because such mispronunciations are associated with only 16% of intelligibility problems, according to [3], who state: We investigated ... which words are likely to be misrecognized and which words are likely to be marked as pronunciation errors.",
"We found that only 16% of the variability in word-level intelligibility can be explained by the presence of obvious mispronunciations.",
"Words perceived as mispronounced remain intelligible in about half of all cases.",
"At the same time ... annotators were often unable to identify the word when listening to the audio but did not perceive it as mispronounced when presented with its transcription.",
"This substantial improvement is not yet well understood by most CAPT community.",
"Currently, expert human pronunciation judges assess student performance, often with large inter-rater variability between experts scoring the same utterances.",
"Since most formal mispronunciations do not substantially impede understanding of spoken language, automatic speech recognition CAPT systems trained to approximate the subjective assessments of judges do not perform as well as might be expected after intensive work on the issue by several hundred researchers spanning decades ([4], [5].)",
"While there are many commercial CAPT applications, there is no consensus among speech language pathologists about which of them, if any, work well ([6]).",
"In high stakes situations, systems imitating subjective assessments of human judges have, for example, prevented native English speakers and trained English language radio announcers from immigrating to Australia ([7], [8]).",
"A more technical related problem with traditional CAPT approaches is that popular pronunciation assessment metrics, primarily goodness of pronunciation (GOP) as defined by [9], are quotients with such vaguely specified denominators [10] that they tend to correlate weakly with authentic intelligibility.",
"Earlier work suffers from similar problems.",
"We are offering remediation of authentic intelligibility for English CAPT to 17zuoye.com's 30 million K-6 English language students in China, and we are deploying the same technology in the Wikimedia Foundation's Wiktionary dictionaries along with their phonetics and pronunciation articles in Wikipedia to provide free CAPT assessment and remediation exercises.",
"We are measuring which feedback choices perform the best for student proficiency outcomes, and studying the possibility of using students to provide transcriptions instead of paid crowdworkers."
],
[
"Adapting PocketSphinx for feature extraction",
"We chose to use PocketSphinx[11] system's alignment routines.",
"We tried a two-pass alignment approach over a fixed grammar by using the time endpoints from recognizing the phonemes of the expected utterance in sequence, using a finite state grammar with no alternative or optional components other than silence, defined using a JSpeech Grammar Format file.",
"The results for the first pass were discarded, because its purpose was solely to perform cepstral mean normalization for adapting to the audio characteristics of the microphone, channel, and noise.",
"We found that grammar-based alignment, which is optimized for speed instead of accuracy, resulted in less correctly predictive features than using a single pass of the alignment API functions, which are only available from the PocketSphinx C API instead of command line invocations.",
"The results of the alignment are used to select audio sub-segments of the utterance to indicate substitutions of expected phonemes, insertions of unexpected phonemes, deletions of the expected phonemes, and five physiological measures of the vocal tract, in multiple subsequent recognizer passes of each three and two adjacent phonemes at a time.",
"Figure REF illustrates the non-physiological part of this feature extraction process.",
"Figure: Feature extraction: The three phonemes ofthe word `cat' are aligned, producing durations d n d_n andacoustic scores a n a_n .",
"Then several passes of recognitionto the audio aligned to groups of three (T n T_n ) and two(D n D_n ) phonemes are used to measure phonemesubstitutions, and insertions and deletions, respectively.After alignment, we run the recognizer on each sub-segment of the audio corresponding to each three aligned phonemes in sequence, and count how soon the expected phoneme occurs in the n-best recognition results.",
"Then we run the recognizer on each sub-segment corresponding to each two adjacent phonemes in sequence, simultaneously counting how frequently the initial expected phoneme is omitted when searching for the insertion of all 39 phonemes and silence in between the two expected phones.",
"The substitution detection pass focuses on three adjacent phonemes at a time as located by the alignment routine.",
"For the audio sub-segment of each three adjacent phonemes from the alignment, we use a grammar specifying the first and last of the three as the only options on the ends, with an alternative allowing for any one phoneme (including diphthongs) in the middle.",
"The score, in the range $[0, 1]$ , represents how high the expected middle phoneme ranks in the n-best results of all the possible phonemes in between the other two.",
"We ask the recognizer for as many n-best results as possible, because sometimes a truncated grammar result (e.g., only two phonemes instead of three) result, but we often get at least 30 results from the 40 possible phonemes and silence, and sometimes get 70 results.",
"The insertion and deletion pass operates on the audio sub-segments of two adjacent phonemes at a time, using a grammar to look for the first expected phoneme in the front as the only possibility, followed by an optional alternative of any phoneme other than the expected second phoneme counting as insertions, and then followed by the expected second phoneme specified as optional to account for deletion.",
"Each time an insertion or deletion is returned in the n-best results before only the expected two phonemes are returned, the $[0, 1]$ score is reduced.",
"We also produce each phoneme's duration and the logarithm of its acoustic score from the alignment phase as features in our SVM or DNN classifier feature inputs.",
"For each phoneme, we produce: (1) a duration; (2) an acoustic score from the alignment, corresponding to the numerator of the GOP score of [9]; (3) a $[0, 1]$ score measuring phoneme substitution, and (4) a $[0, 1]$ score measuring insertions and deletions.",
"One final additional insertion and deletion measurement appears at the end of the feature vector for each word; in a multi-word phrase, that final score is shared as identical to the first insertion and deletion measurement of the next word.",
"As this article was going to press, we added five additional physiological features per phoneme, relating to place, closedness, roundedness, voicing, and the proportion of neighboring phonemes less likely.",
"([12]) We use some non-standard PocketSphinx parameters.",
"We use a frame rate of 65 frames per second instead of 100, because learners are not likely to speak very quickly.",
"We use a -topn value of 64 instead of 2.",
"This provides more accurate recognition results at the expense of longer runtime, but our feature extraction system runs in better than real time in a single thread of a 2016 Apple MacBook Air, and on user's browsers as a pocketsphinx.js adaptation in JavaScript.",
"We use a -beam parameter of $ 10^{-57} $ , a -wbeam parameter of $ 10^{-56} $ , and a -maxhmmpf value of $ -1 $ for the same reason.",
"We set -fsgusefiller to \"no\" so that optional pauses are not assumed between every word, allowing us to define words comprised of a single CMUBET phoneme without slowdown."
],
[
"Compiling featex.c with PocketSphinx",
"The C source code to perform the feature extraction, featex.c, and instructions for compiling and using it are available under the MIT open source code license at: https://github.com/jsalsman/featex"
],
[
"Using pocketsphinx.js in web browsers",
"Feature extraction can take place in web browsers' JavaScript code using the Emscripten system of compiling C to JavaScript, and audio recorded in web browsers supporting microphone input.",
"During the initialization process, the browser is checked for microphone availability and the sampling frequency at which it operates.",
"A media source stream is requested to record audio from the microphone, and connected to a recorder thread which listens or stops listening based on browser user interface events.",
"The pocketsphinx.js module is initialized inside a web worker to asynchronously call the alignment and feature extraction modules.",
"[1] Web client algorithm The user presses the 'Record' button.",
"The recorder thread starts listening.",
"The user presses the 'Stop' button.",
"The recorded audio is converted and downsampled if necessary.",
"The extracted feature vector and word is sent to the intelligibility prediction service (see sections 5.1 and 7.)",
"Assessment feedback is provided to the user.",
"The integrated code and detailed compilation instructions can be found at [13].",
"For more information and an example of an integrated web browser system, please see [14].",
"For an example of how such a system might be integrated into Wiktionary, please see [15]."
],
[
"Obtaining transcriptions of student utterances",
"We consistently obtained faster responses from Amazon Mechanical Turk when paying $0.03 per transcript compared to $0.15.",
"We believe crowdworkers prefer to do low-paying tasks because they are likely to be easier and will cause fewer problems if the work is rejected.",
"We are studying the possibility of using our English learners to provide transcriptions instead of paying crowdworkers, as bona fide listening comprehension and typing exercises suitable for assessments in their own right."
],
[
"Predicting intelligibility",
"Using nine features per phoneme as described above (but not depicted) with support vector machine classification routines from the Python Scikit-learn SVC library configured with a radial basis function kernel and probability prediction, we obtain 82% accuracy in predicting the intelligibility of about 700 basic English words in agreement with Amazon Mechanical Turk workers, using about 30 recordings per words and four transcripts per recording.",
"We have measured strong evidence that increasing the number of recordings per word and transcripts per recording can result in very substantial accuracy improvements.",
"We have obtained similar results on longer phrases.",
"Using the four features per phoneme to train a linear logistic regression model, we only get 75% accuracy, which was reported by [1] and [2] and the ETS ([3]).",
"For a client-server system to predict word intelligibility from feature vectors, please see [16]."
],
[
"Measuring the accuracy of intelligibility assessment",
"When different transcripts of the same utterance of a word show both intelligible and unintelligible results, we measure accuracy as a fraction of the best possible result.",
"For example, if the same utterance was transcribed correctly by three transcriptionists but incorrectly by a fourth, the maximum unadjusted accuracy achievable from predicting that utterance's intelligibility is 75%, so an unadjusted accuracy of 50% is adjusted to be 67%, representing the proportion of the maximum possible accuracy.",
"In practice, the probability of intelligibility is a floating point value in [0, 1], which is typically compared to a threshold, the estimated intelligibility of other words in the same phrase, or both, so the accuracy with which we can predict intelligibility by transcriptionists is used as a benchmark by which we can measure the relative utility of different prediction methods."
],
[
"Determining optimal feedback",
"We use the modeled probability of intelligibility of each word in a prompt word or phrase to help students improve their pronunciation by providing audiovisual feedback indicating which word(s) were pronounced the worst.",
"How many words to indicate were not pronounced well after each utterance is an open question.",
"For words which are not considered sufficiently intelligible, we can use the SVM classifier probability prediction models to determine which identical numerical improvement to each phoneme's non-duration features improves the probability of word intelligibility the most.",
"We can also see how increasing and decreasing each phoneme's duration improves the intelligibility of the word.",
"Such adjustments to the features derived from automatic speech recognition may be more useful as products than sums to identify the specific phoneme(s) most in need of improvement in the less unintelligible word(s).",
"Figure REF shows how we determine the phoneme-level feedback for each word.",
"Figure: Determining feedback: Adjusting the featurescores for each phoneme changes the probability of intelligibilityof the whole word.",
"The adjustments which make the best changessignal which phoneme(s) need improvement the most."
],
[
"Conclusion",
"Using PocketSphinx automatic speech recognition with improved phonetic accuracy features training SVM prediction models can help CAPT systems provide better intelligibility remediation.",
"Researchers and commercial software publishers should try to understand the reasons this technique is superior to the state of the art, and adopt it for improved CAPT outcomes."
],
[
"acknowledgments",
"We thank 17zuoye.com (China), Zzish.com (UK), Prof. Seiichi Nakagawa, the Google Open Source Programs Office, and the Wikimedia Foundation for their kind financial support, suggestions, personnel resources, and educational infrastructure."
]
] | 1709.01713 |
[
[
"Inhomogeneous shearlet coorbit spaces"
],
[
"Abstract In this paper we establish inhomogeneous coorbit spaces related to the continuous shearlet transform and the weighted Lebesgue spaces $L_{p,v}, p\\geq 1,$ for certain weights $v$.",
"We present an inhomogeneous shearlet frame for $L_2(\\mathbb{R}^d)$ which gives rise to a reproducing kernel $R_\\mathfrak{F}$ that is not contained in the space $\\mathcal{A}_{1,m_v}$.",
"To show that the inhomogeneous shearlet coorbit spaces are Banach spaces we introduce a generalization of the approach of Fornasier, Rauhut and Ullrich."
],
[
"Introduction",
"When analyzing a given signal, the decomposition of the signal into a certain set of building blocks is crucial.",
"Which kinds of building blocks to choose depends on the information that one wants to extract from the signal.",
"Very popular kinds of building blocks are wavelets, especially when dealing with signals with isolated singularities.",
"Because of its isotropic nature, the wavelet transform cannot efficiently deal with anisotropic features, therefore several extensions of this framework were proposed, among those the shearlet transform.",
"While the wavelets consist only of dilated and translated copies of a mother function, the shearlets are also sheared in each scale, thereby changing the orientation of the functions.",
"This makes them especially well suited to deal with localized directional features in a signal.",
"Indeed, it was shown in Ref.",
"[18], [14] that the shearlet transform can be used to resolve the wavefront set of a signal and in Ref.",
"[16] that the approximation of cartoon-like images with shearlets is optimally sparse.",
"Another main advantage of shearlets, which sets them apart from other such frameworks like the ridgelets [2], curvelets [1] or contourlets [8] for example, is, that the continuous shearlet transform, introduced and investigated in Ref.",
"[4], [5], [6], [15], stems from the action of a square-integrable representation of a topological group, the so-called full shearlet group ${S}$ .",
"This property makes it possible to use the abstract coorbit theory, developed by Feichtinger and Gröchenig in Ref.",
"[9], [10], [11], to define smoothness spaces related to the shearlet transform by measuring the decay of the voice transform.",
"Shearlet coorbit spaces were investigated by Dahlke et al in a series of papers.",
"[3], [4], [5], [6], [7] Since the shearlets being used to construct these spaces need to have vanishing moments, any polynomial part in a signal is ignored by the transform because for a polynomial $g$ one has $\\mathcal {SH}(f+g)(x) = \\langle {f+g},{\\psi _x} \\rangle = \\langle {f},{\\psi _x} \\rangle = \\mathcal {SH}f(x)$ .",
"This leads to the resulting shearlet coorbit spaces being homogeneous spaces.",
"However, in practice the smoothness spaces being used, for example to analyze the regularity of the solution space of an operator equation, are usually inhomogeneous.",
"Therefore, inhomogeneous smoothness spaces related to the shearlet transform are also of interest.",
"In this paper we introduce non-homogeneous shearlet coorbit spaces by using a generalization of the coorbit theory developed by Fornasier, Rauhut, Ullrich et al.",
"[12], [17], [20] Their approach uses a more general parameter space for the transform, resulting in more design flexibility.",
"Instead of the parameter space being a locally compact topological group, it is only assumed to be a locally compact topological Hausdorff space, thereby allowing the construction of inhomogeneous coorbit spaces.",
"Moreover it is needed for the reproducing kernel $R_\\mathfrak {F}$ to be integrabel, which poses difficulties in some applications.",
"For that reason we present a generalization of their approach in the sense that we only need $R_\\mathfrak {F}$ to be integrabel for parameters $q>1$ ."
],
[
"Outline",
"After giving a short overview of the main definitions and results of this generalized coorbit theory in sec:coorbittheory, we use this approach in sec:shearletcoorbit to define a new shearlet transform given by a continuous frame $\\mathfrak {F}= \\lbrace \\psi _x\\rbrace _{x\\in X}$ through the action $ \\mathcal {SH}_\\mathfrak {F}f (x) = \\langle {f},{\\psi _x} \\rangle ,\\quad x \\in X, $ where the frame is indexed by a topological Hausdorff space $X$ (without group structure).",
"We prove that an integrability condition for (integration) parameters $q>1$ on the kernel function $ R_\\mathfrak {F}: X\\times X \\rightarrow \\, (x,y) \\mapsto \\langle {\\psi _y},{\\psi _x} \\rangle $ holds so that the coorbit spaces $ \\mathcal {SC}_{\\mathfrak {F},\\tau ,p}^{r} = \\lbrace f\\, \\vert \\, \\mathcal {SH}_\\mathfrak {F}f \\in L_{p,v_{r}}(X) \\rbrace ,\\quad p\\ge 1, v_{r,n}\\text{ weight function on }X, $ classifying distributions by the decay of their transform, are well-defined Banach spaces.",
"As it turns out these spaces coincide for different $\\tau $ .",
"Furthermore we restrict ourselves to the case of odd dimensions.",
"This is due to the fact that otherwise our specific construction of the frame is not well-defined.",
"We also note that there are other approaches, not based on coorbit space theory, to develop inhomogeneous shearlet smoothness spaces.",
"In Ref.",
"[19] Labate, Mantovani and Negi used the notion of decomposition spaces to define shearlet smoothness spaces, while in Ref.",
"[21], [22] Vera applied the framework of the $\\varphi $ -transform, introduced by Frazier and Jawerth, for this purpose."
],
[
"Notation",
"We finish this section by stating a few notational conventions.",
"Throughout this paper $d \\in {N}$ with $d \\ge 2$ is the space dimension.",
"We usually treat elements $x\\in {R}^d$ as $x = (x_1,\\tilde{x})$ with $\\tilde{x} = (x_2,\\ldots ,x_d) \\in {R}^{d-1}$ .",
"For two elements $x,y\\in {R}^d$ we use the canonical inner product $ x\\cdot y = \\sum _{i=1}^d x_i y_i.",
"$ The convention ${R}^*$ is used for the set ${R}\\setminus \\lbrace 0\\rbrace $ , ${R}_+$ will denote the set of all positive real numbers and ${R}_{\\ge 0}$ the set of all non-negative real numbers.",
"For a measure space $(X, \\Sigma ,\\mu )$ with a weight function $v: X \\rightarrow (0,\\infty )$ we denote the usual (weighted) Lebesgue spaces by $L_{p,v}(X,\\mu )$ or just by $L_{p,v}$ , if the respective measure space is clear from the context, while $L_1^{\\text{loc}}(X,\\mu )$ is used for the space of locally integrable functions on $X$ .",
"The norm for the weighted Lebesgue spaces is hereby given through $\\Vert {f} \\Vert _{L_{p,v}} = \\Vert {f\\cdot v} \\Vert _{L_p}$ .",
"For the unweighted Lebesgue spaces with $v \\equiv 1$ we write $L_p(X,\\mu )$ and $L_p$ .",
"We use the Hilbert space $L_2({R}^d)$ of complex-valued, square-integrable functions on ${R}^d$ with the inner product $ \\langle {f},{g} \\rangle _{L_2({R}^d)} = \\int _{{R}^d} f(x) \\overline{g(x)}\\,\\mathrm {d}{x}.",
"$ For two functions $f,g\\in L_2({R}^d)$ the convolution product $f\\ast g$ is defined as $ (f\\ast g)(x) = \\int _{{R}^d} f(y)g(x-y)\\,\\mathrm {d}{y}.",
"$ We write ${C}^k, k \\in {N}_0$ for the space of functions $f:{R}^d \\rightarrow , for which all (classical) partial derivatives $ f$ for $${N}$ 0d, $\\vert {\\alpha } \\vert $ k$ exist and are continuous.We also use $ C0$ for the space of infinitely differentiable functions on $${R}$ d$ with compact support and $ S$ denotes the spaces of Schwartz-functions on $${R}$ d$.",
"We will use the letter $ q$ to refer to the kernel spaces $ Aq$ and $ ,$ to refer to the integrability parameters of the spaces of test functions $ H$.We denote with $ p'=pp-1$ the Hölder-dual of $ p1$.$ Concerning the Fourier transform of a function $f\\in L_1({R}^d)$ we write $\\hat{f} = \\mathcal {F}(f)$ using the convention $ \\mathcal {F}(f)(\\omega ) := \\int _{{R}^d} f(x) e^{-2\\pi i \\omega \\cdot x}\\,\\mathrm {d}{x},\\qquad \\omega \\in {R}^d, $ with the same symbol being used for the extension to functions $f\\in L_2({R}^d)$ .",
"Given a measure space $(X,\\Sigma ,\\mu )$ we say that a Banach space $Y$ of locally integrable, complex-valued functions on $X$ satisfies Condition $(Y)$ , if it is solid, i.e.",
"if from $f\\in L_1^{\\text{loc}}(X,\\mu ), g \\in Y$ with $\\vert {f} \\vert \\le \\vert {g} \\vert $ almost everywhere it follows that $f\\in Y$ with $\\Vert {f} \\vert {Y} \\Vert \\le \\Vert {g} \\vert {Y} \\Vert $ .",
"Lastly, for quantities $a$ and $b$ we write $a \\lesssim b$ if there exists a finite constant $C > 0$ so that $a \\le C\\cdot b$ , with the constant being independent of the relevant parameters."
],
[
"Generalized coorbit theory",
"In this section we give a short overview of the generalized coorbit theory.",
"We follow Ref.",
"[12], [20] in our exposition.",
"For our setting we introduce a generalization of their approach with respect to an additional integrability parameter.",
"To generalize the classical coorbit theory—which assumes a locally compact group as the underlying parameter space of the respective transform—the generalization of Fornasier and Rauhut allows for the parameter space to be of a more general nature.",
"In this case the parameter space $X$ is only assumed to be a locally compact Hausdorff space equipped with a positive Radon measure $\\mu $ .",
"In the following $\\mathcal {H}$ denotes a separable Hilbert space (the signal space), which is usually $L_2$ , and $v$ is a weight function on $X$ while $Y$ is a Banach space of equivalence classes of almost everywhere equal, complex-valued functions on $X$ .",
"We start with a set of functions $\\mathfrak {F}= \\lbrace \\psi _x\\rbrace _{x\\in X} \\subset \\mathcal {H}$ , which is indexed by the parameter space, and constitutes a tight continuous frame.",
"I.e., the map $X\\rightarrow x\\mapsto \\langle {f},{\\psi _x} \\rangle $ is measurable for each $f\\in \\mathcal {H}$ and there exists a finite constant $A > 0$ such that $A \\Vert {f} \\vert {\\mathcal {H}} \\Vert ^2 = \\int _X \\vert {\\langle {f},{\\psi _x} \\rangle } \\vert ^2\\,\\mathrm {d}\\mu (x)\\text{ for all }f\\in \\mathcal {H}.$ Based on $\\mathfrak {F}$ , a signal transform on the space $\\mathcal {H}$ is introduced in the following way.",
"Definition 2.1 Let $\\mathfrak {F}= \\lbrace \\psi _x\\rbrace _{x\\in X} \\subset \\mathcal {H}$ be a tight continuous frame.",
"Then the associated voice transform is defined as the mapping $ V_\\mathfrak {F}: \\mathcal {H}\\rightarrow L_2(X,\\mu ),\\quad f \\mapsto V_\\mathfrak {F}f $ with $ V_\\mathfrak {F}f: X \\rightarrow \\quad x \\mapsto \\langle {f},{\\psi _x} \\rangle .",
"$ The above transform is well defined due to eq:frame."
],
[
"Kernel spaces",
"In order for the resulting smoothness spaces to be well defined, conditions on the voice transform $V_\\mathfrak {F}$ and therefore conditions on $\\mathfrak {F}$ are needed.",
"In this approach the kernel function $R_\\mathfrak {F}: X\\times X \\rightarrow (x,y)\\mapsto R_\\mathfrak {F}(x,y) := V_\\mathfrak {F}\\psi _y(x) = \\langle {\\psi _y},{\\psi _x} \\rangle ,$ the reproducing kernel, is used.",
"To formulate certain conditions on this kernel function the following spaces, classifying kernel functions in terms of integrability, are used.",
"For $1\\le q\\le \\infty $ let $ \\mathcal {A}_q := \\Bigl \\lbrace K: X \\times X \\rightarrow K\\text{ is measurable}, \\Vert {K} \\vert {\\mathcal {A}_q} \\Vert < \\infty \\Bigr \\rbrace $ with $\\Vert {K} \\vert {\\mathcal {A}_q} \\Vert := \\max \\bigg \\lbrace &\\operatornamewithlimits{ess\\,sup}_{x\\in X} \\left(\\int _X \\vert {K(x,y)} \\vert ^q\\,\\mathrm {d}\\mu (y)\\right)^{1/q},\\\\& \\operatornamewithlimits{ess\\,sup}_{y\\in X} \\left(\\int _X \\vert {K(x,y)} \\vert ^q\\,\\mathrm {d}\\mu (x)\\right)^{1/q}\\bigg \\rbrace $ and the usual adaptation for $q=\\infty $ .",
"Through a weight function $v\\ge 1$ on $X$ a kernel weight function is defined via $ m_v: X \\times X \\rightarrow (0,\\infty ), (x,y) \\mapsto \\max \\biggl \\lbrace \\frac{v(x)}{v(y)},\\frac{v(y)}{v(x)}\\biggr \\rbrace .",
"$ Now the associated weighted kernel space $\\mathcal {A}_{q,{m_v}}$ is given by $ \\mathcal {A}_{q,{m_v}} := \\Bigl \\lbrace K: X \\times X \\rightarrow K\\cdot m_v \\in \\mathcal {A}_q\\Bigr \\rbrace $ where $ \\Vert {K} \\vert {\\mathcal {A}_{q,{m_v}}} \\Vert := \\Vert {K\\cdot m_v} \\vert {\\mathcal {A}_q} \\Vert .",
"$ In the following, depending on the context, $K$ will also denote the kernel operator induced by the kernel function acting on a function $F$ through $ K(F)(x) := \\int _X K(x,y) F(y)\\,\\mathrm {d}\\mu (y)\\text{ for } x \\in X.",
"$ This way a reproducing identity is established through the action of $R_\\mathfrak {F}$ , namely $R_\\mathfrak {F}(V_\\mathfrak {F}f)=V_\\mathfrak {F}f$ for all $f\\in \\mathcal {H}$ .",
"The following auxiliary Lemma for kernel operators underlines the importance of the kernel spaces $\\mathcal {A}_{q,{m_v}}$ .",
"Lemma 2.1 Let $K$ be a kernel with $K\\in \\mathcal {A}_{q,m_v}$ for all $q>1$ .",
"Then we have the continuous embeddings $K(L_{p,v}(X,\\mu )) \\hookrightarrow L_{r,v}(X,\\mu )$ for all $1< p < r \\le \\infty $ .",
"For fixed $1< p < r <\\infty $ and $g\\in L_{p,v}(X,\\mu )$ with $\\Vert {g} \\vert {L_{p,v}} \\Vert \\le 1$ arbitrary one has $\\Vert {K(g)} \\vert {L_{r,v}} \\Vert &= \\sup _{\\begin{array}{c}h\\in L_{r^{\\prime },\\frac{1}{v}} \\\\ \\Vert {h} \\vert {L_{r^{\\prime },\\frac{1}{v}}} \\Vert \\le 1\\end{array}} |\\langle {K(g)},{h} \\rangle | \\\\&\\le \\sup _{\\begin{array}{c}h\\in L_{r^{\\prime },\\frac{1}{v}} \\\\ \\Vert {h} \\vert {L_{r^{\\prime },\\frac{1}{v}}} \\Vert \\le 1\\end{array}} \\int _X \\int _X|K(x,y)g(y)h(x)|\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&=: \\sup _{\\begin{array}{c}h\\in L_{r^{\\prime },\\frac{1}{v}} \\\\ \\Vert {h} \\vert {L_{r^{\\prime },\\frac{1}{v}}} \\Vert \\le 1\\end{array}} I_{K,p,r},$ where $r^{\\prime }$ denotes the Hölder-dual of $r$ satisfying $1/r+1/r^{\\prime }=1$ .",
"For some $0<\\varepsilon <1/p-1/r$ we set $\\alpha :=r>0$ , $\\beta :=p^{\\prime }>0$ , $1/\\gamma :=1/p-1/r>0$, $a:=1/r+\\varepsilon $ , $b:=p/r$ , $c:=1/r^{\\prime }-\\varepsilon $ , $d:=r^{\\prime }/p^{\\prime }$ , $e:=1-p/r$ , $f:=r^{\\prime }/p-r^{\\prime }/r$ .",
"These choices suffice the following relations: $1/\\alpha +1/\\beta +1/\\gamma =1, \\hspace{14.22636pt} a+c&=1, \\hspace{14.22636pt} b\\alpha =p, \\hspace{14.22636pt} d\\beta =r^{\\prime }, \\hspace{14.22636pt} a\\alpha >1, \\\\b+e&=1, \\hspace{14.22636pt} e\\gamma =p, \\hspace{14.22636pt} f\\gamma =r^{\\prime }, \\hspace{14.22636pt} c\\beta >1, \\\\d+f&=1.$ By applying the three-way Young inequality, see lemma:threewayyoungsinequality, we obtain $I_{K,p,r} &\\le \\int _X\\int _X|K(x,y)m_v(x,y)|^a|f(y)v(y)|^b\\cdot |K(x,y)m_v(x,y)|^c|h(x)v(x)^{-1}|^d\\\\&\\hspace{56.9055pt}\\cdot |g(y)v(y)|^e|h(x)v(x)^{-1}|^f\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\le \\frac{1}{\\alpha }\\int _X\\int _X|K(x,y)m_v(x,y)|^{a\\alpha }|g(y)v(y)|^p\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{56.9055pt}+ \\frac{1}{\\beta }\\int _X\\int _X|K(x,y)m_v(x,y)|^{c\\beta }|h(x)v(x)^{-1}|^{r^{\\prime }}\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{56.9055pt}+ \\frac{1}{\\gamma }\\int _X\\int _X|g(y)v(y)|^p|h(x)v(x)^{-1}|^{r^{\\prime }}\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y).$ For the first summand we deduce the estimation $&\\int _X\\int _X|K(x,y)m_v(x,y)|^{a\\alpha }|g(y)v(y)|^p\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{56.9055pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{y\\in X}\\int _X|K(x,y)|^{a\\alpha }|m_v(x,y)|^{a\\alpha }\\,\\mathrm {d}\\mu (x)\\right)\\int _X|g(y)|^p|v(y)|^p\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{56.9055pt} \\le \\Vert {K} \\vert {\\mathcal {A}_{a\\alpha ,m_v}} \\Vert ^{a\\alpha }\\Vert {g} \\vert {L_{p,v}} \\Vert ^p$ and the other two summands can be treated analogously.",
"Thus we obtain $I_{K,p,r}&\\le \\frac{1}{\\alpha }\\Vert {K} \\vert {\\mathcal {A}_{a\\alpha ,m_v}} \\Vert ^{a\\alpha }\\Vert {g} \\vert {L_{p,v}} \\Vert ^p + \\frac{1}{\\beta }\\Vert {K} \\vert {\\mathcal {A}_{c\\beta ,m_v}} \\Vert ^{c\\beta }\\Vert {h} \\vert {L_{r^{\\prime },\\frac{1}{v}}} \\Vert ^{r^{\\prime }}\\\\&\\quad +\\frac{1}{\\gamma }\\Vert {g} \\vert {L_{p,v}} \\Vert ^p\\Vert {h} \\vert {L_{r^{\\prime },\\frac{1}{v}}} \\Vert ^{r^{\\prime }} \\\\&\\le \\max \\left\\lbrace 1,\\Vert {K} \\vert {\\mathcal {A}_{a\\alpha ,m_v}} \\Vert ^{a\\alpha },\\Vert {K} \\vert {\\mathcal {A}_{c\\beta ,m_v}} \\Vert ^{c\\beta }\\right\\rbrace =: C_K$ for all $g,h$ .",
"Hence, $\\Vert {K} \\vert {L_{p,v}\\rightarrow L_{r,v}} \\Vert \\le C_K$ .",
"If $1< p<r=\\infty $ and $g\\in L_p(X,\\mu )$ arbitrary, it follows with Hölder's inequality that $\\Vert {K(g)} \\vert {L_{\\infty ,v}} \\Vert &\\le \\operatornamewithlimits{ess\\,sup}_{x\\in X}\\int _X|K(x,y)m_v(x,y)|\\cdot |g(y)v(y)|\\,\\mathrm {d}\\mu (y) \\\\&\\le \\Vert {K} \\vert {\\mathcal {A}_{p^{\\prime },m_v}} \\Vert ^{p^{\\prime }}\\Vert {g} \\vert {L_{p,v}} \\Vert ^p,$ which concludes the proof.",
"Remark 1 (i) The assumptions in lemma:kernelproperty can be weakened in the sense, that we only need specific $q>1$ for the assertion to hold, but this setting is sufficient for our work.",
"(ii) The proof is similar to the proof of Schur's test, also known as the generalized Young inequality.",
"By letting $K\\in \\mathcal {A}_{1,m_v}$ and $p=r$ it follows that $1/\\gamma =0$ and $a\\alpha =c\\beta =1$ .",
"This means we only use the two-way Young inequality and we are in the setting of Schur's test, see lemma:schurstest."
],
[
"Coorbit spaces",
"Before introducing coorbit spaces the concept of signals can first be generalized from elements of the Hilbert space $\\mathcal {H}$ to a suitable space of distributions.",
"First of all, for $1\\le \\tau \\le 2$ consider the spaces $ \\mathcal {H}_{\\tau ,v}:=\\lbrace f\\in \\mathcal {H},V_\\mathfrak {F}f\\in L_{\\tau ,v}(X,\\mu )\\rbrace $ of test functions equipped with the natural norm $ \\Vert {f} \\vert {\\mathcal {H}_{\\tau ,v}} \\Vert :=\\Vert {V_\\mathfrak {F}f} \\vert {L_{\\tau ,v}} \\Vert .",
"$ First we note, that these spaces are non-empty, moreover the following Lemma holds.",
"Lemma 2.2 If $R_\\mathfrak {F}\\in \\mathcal {A}_{\\tau ,m_v}$ , then $\\mathfrak {F}\\subset \\mathcal {H}_{\\tau ,v}$ .",
"For $x\\in X$ arbitrary one has $\\Vert {\\psi _x} \\vert {\\mathcal {H}_{\\tau ,v}} \\Vert ^\\tau &= \\int _X |V_\\mathfrak {F}\\psi _x(y)|^\\tau v(y)^\\tau \\,\\mathrm {d}\\mu (y) \\\\&\\le v(x)^\\tau \\int _X |R_\\mathfrak {F}(y,x)|^\\tau m_v(y,x)^\\tau \\,\\mathrm {d}\\mu (y) \\\\&\\le v(x)^\\tau \\Vert {R_\\mathfrak {F}} \\vert {\\mathcal {A}_{\\tau ,m_v}} \\Vert ^{\\tau },$ which proves the assertion.",
"Since $\\mathfrak {F}$ establishes a frame for $\\mathcal {H}$ this means $\\mathcal {H}_{\\tau ,v}\\subset \\mathcal {H}$ is dense.",
"Moreover, the spaces $\\mathcal {H}_{\\tau ,v}$ are Banach spaces, as the following Lemma states.",
"Lemma 2.3 If $R_\\mathfrak {F}\\in \\mathcal {A}_{\\tau ^{\\prime },m_v}$ then the space $\\mathcal {H}_{\\tau ,v}$ is a Banach space.",
"Let $\\lbrace f_n\\rbrace _{n\\in {N}}\\subset \\mathcal {H}_{\\tau ,v}\\subset \\mathcal {H}$ be a Cauchy sequence, which means $\\lbrace g_n\\rbrace _{n\\in {N}}:=\\lbrace V_\\mathfrak {F}f_n\\rbrace _{n\\in {N}}$ is a Cauchy sequence in $L_{\\tau ,v}(X,\\mu )$ .",
"By the completeness of $L_{\\tau ,v}$ there exists a unique $g\\in L_{\\tau ,v}$ with $g_n\\rightarrow g$ .",
"Furthermore, by the reproducing formula it holds $R_\\mathfrak {F}(g_n)=g_n$ for all $n\\in {N}$ , which implies $R_\\mathfrak {F}(g)=g$ .",
"Then, by Hölder's inequality, for every $x\\in X$ it holds $|R_\\mathfrak {F}(g)(x)| &\\le \\int _X|R_\\mathfrak {F}(x,y)g(y)|\\,\\mathrm {d}\\mu (y) \\\\&\\le \\Vert {R_\\mathfrak {F}(x,\\cdot )} \\vert {L_{\\tau ^{\\prime },\\frac{1}{v}}} \\Vert \\cdot \\Vert {g} \\vert {L_{\\tau ,v}} \\Vert \\\\&\\le v(x)^{-1}\\Vert {R_\\mathfrak {F}} \\vert {\\mathcal {A}_{\\tau ^{\\prime },m_v}} \\Vert \\cdot \\Vert {g} \\vert {L_{\\tau ,v}} \\Vert .$ Thus, $g=R_\\mathfrak {F}(g)\\in L_\\infty $ and since $L_\\infty \\cap L_{\\tau ,v}\\subset L_2$ it follows $g\\in L_2$ .",
"Since the application of $R_\\mathfrak {F}$ is the orthogonal projection from $L_2$ onto the image of $V_\\mathfrak {F}$ there exists $f\\in \\mathcal {H}$ such that $g=V_\\mathfrak {F}f$ .",
"Moreover, $V_\\mathfrak {F}f\\in L_{\\tau ,v}$ means $f\\in \\mathcal {H}_{\\tau ,v}$ and $f_n\\rightarrow f\\in \\mathcal {H}_{\\tau ,v}$ .",
"Hence, this set of test functions leads to the Gelfand triple setting of dense embeddings $ \\mathcal {H}_{\\tau ,v} \\hookrightarrow \\mathcal {H}\\cong \\mathcal {H}^\\sim \\hookrightarrow (\\mathcal {H}_{\\tau ,v})^\\sim $ with $(\\mathcal {H}_{\\tau ,v})^\\sim $ being the canonical anti-dual space (the space of all conjugate linear, continuous functionals) of $\\mathcal {H}_{\\tau ,v}$ and this space can be interpreted as a space of distributions.",
"An element $h\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ is hereby identified with the functional $f\\rightarrow \\langle {h},{f} \\rangle $ .",
"With these embeddings it is possible to extend the notion of the voice transform in a canonical way to elements $f\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ by $V_{\\mathfrak {F},\\tau }f(x)=f(\\psi _x)$ .",
"By lemma:framesubset this is well defined.",
"With assumptions on the reproducing kernel we can prove the following nesting property.",
"Lemma 2.4 If $R_\\mathfrak {F}\\in \\mathcal {A}_{q,m_v}$ for every $q>1$ then $\\mathcal {H}_{\\sigma ,v}\\subset \\mathcal {H}_{\\tau ,v}$ and $(\\mathcal {H}_{\\tau ,v})^\\sim \\subset (\\mathcal {H}_{\\sigma ,v})^\\sim $ for all $\\sigma <\\tau $ .",
"Assume $f\\in \\mathcal {H}_{\\sigma ,v}$ , which means $f\\in \\mathcal {H}$ with $V_\\mathfrak {F}f\\in L_{\\sigma ,v}$ .",
"Since the reproducing identity holds it follows $V_\\mathfrak {F}f=R_\\mathfrak {F}(V_\\mathfrak {F}f)\\in R_\\mathfrak {F}(L_{\\sigma ,v})$ and with lemma:kernelproperty we derive $V_\\mathfrak {F}f\\in L_{\\tau ,v}$ , hence $f\\in \\mathcal {H}_{\\tau ,v}$ .",
"The second assertion is immediate.",
"For the coorbit spaces to be well defined we need the following two auxiliary Lemmas.",
"Lemma 2.5 The expression $\\Vert {V_{\\mathfrak {F},\\tau }f} \\vert {L_{\\tau ^{\\prime },\\frac{1}{v}}(X,\\mu )} \\Vert $ is an equivalent norm on $(\\mathcal {H}_{\\tau ,v})^\\sim $ , where $\\tau ^{\\prime }$ denotes the Hölder-dual of $\\tau $ .",
"First we note that $V_{\\mathfrak {F}}$ is acting as a unitary operator on $\\mathcal {H}$ , and so does $V_{\\mathfrak {F},\\tau }$ .",
"Moreover, by definition we have $V_{\\mathfrak {F},\\tau }(\\mathcal {H}_{\\tau ,v})=L_{2}\\cap L_{\\tau ,v}$ , which is dense in $L_{\\tau ,v}$ .",
"Then, by definition of the norm one has $\\Vert {F} \\vert {(\\mathcal {H}_{\\tau ,v})^\\sim } \\Vert &= \\sup _{\\begin{array}{c}h\\in \\mathcal {H}_{\\tau ,v} \\\\ \\Vert {h} \\vert {\\mathcal {H}_{\\tau ,v}} \\Vert \\le 1\\end{array}}\\vert {\\langle {F},{h} \\rangle } \\vert \\\\&= \\sup _{\\begin{array}{c}h\\in \\mathcal {H}_{\\tau ,v} \\\\ \\Vert {V_{\\mathfrak {F},\\tau }h} \\vert {L_{q,v}} \\Vert \\le 1\\end{array}}\\vert {\\langle {V_{\\mathfrak {F},\\tau }F},{V_{\\mathfrak {F},\\tau }h} \\rangle } \\vert \\\\&= \\sup _{\\begin{array}{c}H\\in V_{\\mathfrak {F},\\tau }(\\mathcal {H}_{\\tau ,v}) \\\\ \\Vert {H} \\vert {L_{q,v}} \\Vert \\le 1\\end{array}}\\vert {\\langle {V_{\\mathfrak {F},\\tau }F},{H} \\rangle } \\vert \\\\&= \\sup _{\\begin{array}{c}H\\in L_{\\tau ,v} \\\\ \\Vert {H} \\vert {L_{\\tau ,v}} \\Vert \\le 1\\end{array}}\\vert {\\langle {V_{\\mathfrak {F},\\tau }F},{H} \\rangle } \\vert \\\\&= \\Vert {V_{\\mathfrak {F},\\tau }F} \\vert {L_{\\tau ^{\\prime },\\frac{1}{v}}} \\Vert ,$ which concludes the proof.",
"Lemma 2.6 (i) For $f\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ it holds $V_{\\mathfrak {F},\\tau }f\\in L_{\\tau ^{\\prime },\\frac{1}{v}}$ and the mappings $V_{\\mathfrak {F},\\tau }:(\\mathcal {H}_{\\tau ,v})^\\sim \\rightarrow L_{\\tau ^{\\prime },\\frac{1}{v}}$ are injective.",
"(ii) The reproducing formula extends to $(\\mathcal {H}_{\\tau ,v})^\\sim $ , i.e.",
"$R_{\\mathfrak {F}}(V_{\\mathfrak {F},\\tau }f)=V_{\\mathfrak {F},\\tau }f$ for all $f\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ .",
"(iii) Conversely, if $F\\in L_{\\tau ^{\\prime },\\frac{1}{v}}$ satisfies the reproducing property $R_{\\mathfrak {F}}(F)=F$ then there exists $f\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ such that $V_{\\mathfrak {F},\\tau }f=F$ .",
"(i) The assertion follows immediately from lemma:equivalentnorm.",
"(ii) Suppose that $f\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ .",
"Since $X$ is $\\sigma $ -compact there exists a sequence of nested compact subsets $(U_n)_{n\\in {N}}$ such that $X=\\bigcup _{n\\in {N}}U_n$ .",
"Denote by $\\chi _{U_n}$ the characteristic function of $U_n$ and let $F_n:=\\chi _{U_n}V_{\\mathfrak {F},\\tau }f\\in L_2$ .",
"Obviously this series converges pointwise to $V_{\\mathfrak {F},\\tau }f$ .",
"For any $x\\in X$ we then have $R_\\mathfrak {F}(x,y)F_n(y) ={\\left\\lbrace \\begin{array}{ll}R_\\mathfrak {F}(x,y)V_{\\mathfrak {F},\\tau }f(y), & y\\in U_n,\\\\0 , & \\mbox{else},\\end{array}\\right.",
"}$ which means that $|R_\\mathfrak {F}(x,y)F_n(y)|\\le |R_\\mathfrak {F}(x,y)V_{\\mathfrak {F},\\tau }f(y)|$ for all $y\\in X$ .",
"Furthermore the expression $R_\\mathfrak {F}(x,\\cdot )V_{\\mathfrak {F},\\tau }f$ is $L_1$ -integrabel and by Hölder's inequality we obtain the estimation $\\Vert {R_\\mathfrak {F}(x,\\cdot )V_{\\mathfrak {F},\\tau }f} \\vert {L_1} \\Vert &\\le \\Vert {R_\\mathfrak {F}(x,\\cdot )} \\vert {L_{\\tau ,v}} \\Vert \\,\\Vert {V_{\\mathfrak {F},\\tau }f} \\vert {L_{\\tau ^{\\prime },\\frac{1}{v}}} \\Vert \\\\&\\le v(x)\\Vert {R_\\mathfrak {F}} \\vert {\\mathcal {A}_{\\tau ,m_v}} \\Vert \\,\\Vert {f} \\vert {(\\mathcal {H}_{\\tau ,v})^\\sim } \\Vert $ for every $x\\in X$ .",
"Since the reproducing property holds for every $F_n$ and because of Lebesgue's convergence theorem we obtain $V_{\\mathfrak {F},\\tau }f(x) &= \\lim _{n\\rightarrow \\infty }F_n(x) = \\lim _{n\\rightarrow \\infty }\\int _X R_\\mathfrak {F}(x,y)F_n(y)\\,\\mathrm {d}\\mu (y)\\\\& = \\int _X R_\\mathfrak {F}(x,y)V_{\\mathfrak {F},\\tau }f(y)\\,\\mathrm {d}\\mu (y) = R_\\mathfrak {F}(V_{\\mathfrak {F},\\tau }f)(x).$ (iii) The adjoint mapping of $V_{\\mathfrak {F},\\tau }:\\mathcal {H}_{\\tau ,v}\\rightarrow L_{\\tau ,v}$ is given by $V_{\\mathfrak {F},\\tau }^\\ast :L_{\\tau ^{\\prime },\\frac{1}{v}}\\rightarrow (\\mathcal {H}_{\\tau ,v})^\\sim , \\quad V_{\\mathfrak {F},\\tau }^\\ast F=\\int _X F(x)\\psi _x\\,\\mathrm {d}\\mu (x) \\quad \\mbox{for }F\\in L_{\\tau ^{\\prime },\\frac{1}{v}}.$ Thus for $f:=V_{\\mathfrak {F},\\tau }^\\ast F\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ it holds $F(y) = R_\\mathfrak {F}F(y) = \\int _X\\langle {\\psi _x},{\\psi _y} \\rangle F(x)\\,\\mathrm {d}\\mu (x) = V_{\\mathfrak {F},\\tau }V_{\\mathfrak {F},\\tau }^\\ast F(y) = V_{\\mathfrak {F},\\tau }f(y)$ for every $y\\in X$ .",
"Now we are ready to define the coorbit spaces.",
"Definition 2.2 The coorbit spaces of $L_{p,v}(X,\\mu )$ with respect to the frame $\\mathfrak {F}=\\lbrace \\psi _x\\rbrace _{x\\in X}$ and the integrability parameter $\\tau $ are defined as $\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v}) := \\left\\lbrace f\\in (\\mathcal {H}_{\\tau ,v})^\\sim :V_{\\mathfrak {F},\\tau }f\\in L_{p,v}(X,\\mu )\\right\\rbrace $ endowed with the natural norms $\\Vert {f} \\vert {\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})} \\Vert := \\Vert {V_{\\mathfrak {F},\\tau }f} \\vert {L_{p,v}} \\Vert .$ The following proposition is essential when dealing with coorbit spaces.",
"Proposition 2.7 Suppose that $R_\\mathfrak {F}(L_{p,v})\\subset L_{\\tau ^{\\prime },\\frac{1}{v}}$ .",
"(i) A function $F\\in L_{p,v}$ is of the form $V_{\\mathfrak {F},\\tau }f$ for some $f\\in \\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})$ if and only if $R_\\mathfrak {F}F=F$ .",
"(ii) The spaces $(\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v}),\\Vert {\\cdot } \\vert {\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})} \\Vert )$ are Banach spaces.",
"(iii) The map $V_{\\mathfrak {F},\\tau }:\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})\\rightarrow L_{p,v}$ induces an isometric isomorphism between $\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})$ and the reproducing kernel space $\\lbrace F\\in L_{p,v}:R_\\mathfrak {F}F=F\\rbrace \\subset L_{p,v}$ .",
"(i) Assume $f\\in \\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})$ , then by definition $f\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ and by lemma:hproperties ii) the reproducing identity holds.",
"Conversely, if $F\\in L_{p,v}$ satisfies $R_\\mathfrak {F}F=F$ we deduce by our assumption $F\\in L_{\\tau ^{\\prime },\\frac{1}{v}}$ .",
"lemma:hproperties iii) implies that there exists $f\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ such that $V_\\mathfrak {F}f=F$ , which shows the assertion.",
"(ii) Suppose that $\\lbrace f_n\\rbrace _{n\\in {N}}$ is a Cauchy sequence in $\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})$ implying that $F_n := V_{\\mathfrak {F},\\tau } f_n$ is a Cauchy sequence in $L_{p,v}$ .",
"By the completeness of $L_{p,v}$ this sequence convergences to an element $F\\in L_{p,v}$ .",
"By i) it holds $R_\\mathfrak {F}F_n=F_n$ for all $n\\in {N}$ and hence $R_\\mathfrak {F}F=F$ .",
"Again by i) there exists an $f\\in \\mathrm {Co}_{\\mathfrak {F},q}(L_{p,v})$ with $V_{\\mathfrak {F},\\tau }f=F$ and the completeness is shown.",
"(iii) The assertion follows with (i) and the injectivity of $V_{\\mathfrak {F},\\tau }$ .",
"Remark 2 The assumption in proposition:coproperties may appear strange, but is readily fulfilled for the following setting.",
"If we assume $R_\\mathfrak {F}\\in \\mathcal {A}_{q,m_v}$ for all $q>1$ then it follows from lemma:kernelproperty that $R_\\mathfrak {F}(L_{p,v}) \\subset L_{\\tau ^{\\prime },v} \\subset L_{\\tau ^{\\prime },\\frac{1}{v}}$ for all $1<p<\\tau ^{\\prime }<\\infty $."
],
[
"Dependency on $\\tau $ , {{formula:42f48e8f-35df-4ad2-9b62-64349083c2c3}} , {{formula:82e4fba3-35b7-43dd-af6e-a3c70c793aa5}} and {{formula:5acbf62c-886d-4789-a24c-390a21296efe}}",
"We will now discuss the dependency of the coorbit spaces on the parameters involved.",
"To this end we always assume $R_\\mathfrak {F}\\in \\mathcal {A}_{q,m_v}$ for all $q>1$ as suggested in remark:assumptionfulfilled.",
"We can obtain some nesting properties for the parameters $\\tau $ and $p$ as well as the weight $v$ .",
"Lemma 2.8 (i) For all $\\sigma <\\tau $ we have $\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})\\subset \\mathrm {Co}_{\\mathfrak {F},\\sigma }(L_{p,v})$ .",
"(ii) For all $p<r$ we have $\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})\\subset \\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{r,v})$ .",
"(iii) For two weights fulfilling $v\\le w$ we have $\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,w})\\subset \\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})$ .",
"(i) This follows immediately from lemma:nesting.",
"(ii) Assume $f\\in \\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})$ , meaning $f\\in (\\mathcal {H}_{\\tau ,v})^\\sim $ with $V_{\\mathfrak {F},\\tau }f\\in L_{p,v}$ .",
"By lemma:hproperties (ii) the reproducing identity extends to $(\\mathcal {H}_{\\tau ,v})^\\sim $ , thus $V_{\\mathfrak {F},\\tau }f=R_\\mathfrak {F}(V_{\\mathfrak {F},\\tau }f)\\in R_\\mathfrak {F}(L_{p,v})$ .",
"With lemma:kernelproperty we derive $V_{\\mathfrak {F},\\tau }f\\in L_{r,v}$ , which shows the assumption.",
"(iii) Since $L_{p,w}\\subset L_{p,v}$ the assertion holds.",
"Remark 3 Under the additional assumption $R_\\mathfrak {F}\\in \\mathcal {A}_{1,m_v}$ , the spaces $\\mathrm {Co}_{\\mathfrak {F},1}(L_{p,v})$ , which are analyzed in Ref.",
"[12], are well-defined by Schur's test, see lemma:schurstest.",
"Hence, by lemma:embeddings (i) we have the embeddings $\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})\\subset \\mathrm {Co}_{\\mathfrak {F},1}(L_{p,v})$ for all $1<\\tau \\le 2$ .",
"This is not applicable for the inhomogeneous shearlet coorbit spaces we are looking at in this paper but may be of interest for other spaces.",
"To identify conditions under which the Coorbit spaces are independent of the frame, we introduce a second Parseval frame for $\\mathcal {H}$ we denote by $\\mathfrak {G}=\\lbrace \\tilde{\\psi }_x\\rbrace _{x\\in X}$ and introduce the Gramian kernel as $G(\\mathfrak {F},\\mathfrak {G})(x,y):=\\langle {\\tilde{\\psi }_y},{\\psi _x} \\rangle .$ Then, the following holds true.",
"Proposition 2.9 Assume that $\\mathfrak {F}=\\lbrace \\psi _x\\rbrace _{x\\in X}$ and $\\mathfrak {G}=\\lbrace \\tilde{\\psi }_x\\rbrace _{x\\in X}$ are two Parseval frames for $\\mathcal {H}$ fulfilling all necessary conditions on the reproducing kernels and the corresponding Gramian kernel fulfills $G(\\mathfrak {F},\\mathfrak {G})\\in \\mathcal {A}_{1,m_v}$ .",
"Then it holds $\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})=\\mathrm {Co}_{\\mathfrak {G},\\tau }(L_{p,v})$ .",
"By expanding $V_\\mathfrak {F}$ with respect to $\\mathfrak {G}$ we obtain $V_\\mathfrak {F}f(x) = \\langle {f},{\\psi _x} \\rangle = \\int _X\\langle {f},{\\tilde{\\psi }_y} \\rangle \\langle {\\tilde{\\psi }_y},{\\psi _x} \\rangle \\,\\mathrm {d}\\mu (y) = G(\\mathfrak {F},\\mathfrak {G})(V_\\mathfrak {G}f)(x)$ and the same holds for the extended voice transform.",
"By our assumption we derive with Schur's test that $G(\\mathfrak {F},\\mathfrak {G})(L_{p,v})\\subset L_{p,v}$ and it holds $\\Vert {f} \\vert {\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v})} \\Vert \\le \\Vert {G(\\mathfrak {F},\\mathfrak {G})} \\vert {\\mathcal {A}_{1,m_v}} \\Vert \\,\\Vert {f} \\vert {\\mathrm {Co}_{\\mathfrak {G},\\tau }(L_{p,v})} \\Vert .$ The converse is shown analogously and the assertion follows."
],
[
"Shearlet coorbit spaces",
"In this section we introduce an inhomogeneous version of the shearlet transform and define smoothness spaces associated to this transform.",
"In order to accomplish this we use the generalized coorbit theory outlined in sec:coorbittheory.",
"Since our approach is based on the homogeneous shearlet transform and the resulting coorbit spaces (as treated in Ref.",
"[3], [4], [5], [6]), we start by giving a short overview of the respective theory.",
"By modifying the homogeneous shearlet transform, we then develop a new transform, given through the action of an (inhomogeneous) frame.",
"For this new transform we then show that all the necessary conditions on the reproducing kernel hold, so that we can introduce the associated coorbit spaces with respect to the (weighted) Lebesgue spaces."
],
[
"Homogeneous shearlet transform",
"To define the shearlet transform, one starts with an admissible function $\\psi \\in L_2({R}^d)$ , i.e.",
"a function satisfying the condition $c_\\psi := \\int _{{R}^d} \\frac{\\vert {\\hat{\\psi }(\\omega )} \\vert ^2}{\\vert {\\omega _1} \\vert ^d}\\,\\mathrm {d}\\omega < \\infty .$ This condition is necessary for the transform to be square-integrable.",
"The admissible function is then translated, dilated and sheared in order to change its localization, scale and orientation.",
"For a parameter $a \\in {R}^*$ let $ A_a = \\begin{pmatrix}a & 0_{d-1}^T\\\\0_{d-1} & \\operatorname{sign}(a)\\vert {a} \\vert ^{\\frac{1}{d}} I_{d-1}\\end{pmatrix} $ denote a generalized parabolic scaling matrix and for a parameter $s \\in {R}^{d-1}$ let $ S_s = \\begin{pmatrix}1 & s^T\\\\0_{d-1} & I_{d-1}\\end{pmatrix} $ denote the so-called shear matrix.",
"It is easy to see that $\\vert {\\det S_s} \\vert = 1$ and $\\vert {\\det A_a} \\vert = \\vert {a} \\vert ^{2-\\frac{1}{d}}$ .",
"Using these matrices one can then define the translated, dilated and sheared version of $\\psi $ through $ \\psi _{(a,s,t)}(x) = \\vert {\\det A_a} \\vert ^{-\\frac{1}{2}} \\psi (A_a^{-1} S_s^{-1}(x-t)).$ In the homogeneous setting, the shearlet transform is then defined through the action of a unitary, irreducible and integrable representation of the full parameter group, the so-called shearlet group ${S} = {R}^* \\times {R}^{d-1} \\times {R}^d$ with the group law $ (a,s,t) \\circ (a^{\\prime },s^{\\prime },t^{\\prime }) = (aa^{\\prime },s + \\vert {a} \\vert ^{1-\\frac{1}{d}} s^{\\prime },t + S_s A_a t^{\\prime }).",
"$ Given the mapping $\\pi : {S} \\rightarrow \\mathcal {U}(L_2({R}^d))$ with $\\pi (a,s,t)\\psi = \\psi _{(a,s,t)}$ , which can be shown to be a unitary group representation, the shearlet transform is defined as $ \\mathcal {SH}: L_2({R}^d) \\rightarrow L_2({S}),\\quad f \\mapsto \\mathcal {SH}f $ with $ \\mathcal {SH}f: {S} \\rightarrow \\quad (a,s,t) \\mapsto \\langle {f},{\\pi (a,s,t)\\psi } \\rangle _{L_2({R}^d)}.",
"$ Based on this notion of the shearlet transform Dahlke et al.",
"introduced homogeneous shearlet coorbit spaces with respect to the Lebesgue spaces by using the coorbit space theory developed by Feichtinger and Gröchenig in Ref.",
"[9], [10], [11]."
],
[
"Inhomogeneous shearlet frame",
"Similar to the wavelet approach in Ref.",
"[20] we now introduce an inhomogeneous shearlet transform by restricting the dilation parameter to a closed subset of the full parameter group, thereby only covering the higher-frequency content of a signal.",
"To analyze the polynomial and lower-frequency part a second function is introduced to construct an inhomogeneous frame of functions in $L_2({R}^d)$ as the set of building blocks for our new transform.",
"Therefore we choose the set $ X := \\Bigl ( \\lbrace \\infty \\rbrace \\times {R}^{d-1} \\times {R}^d \\Bigr ) \\cup \\Bigl ( [-1,1]^* \\times {R}^{d-1} \\times {R}^d \\Bigr ) $ as the new parameter space with “$\\infty $ ” representing an isolated point in ${R}$ and $[-1,1]^* := [-1,1]\\setminus \\lbrace 0\\rbrace $ .",
"The right-hand side of the union is the aforementioned subspace of the shearlet group ${S}$ , which is closed under the group action.",
"Obviously, this definition leads to a locally compact Hausdorff space.",
"In the following definition we introduce a measure on the parameter space so that $X$ , together with its Borel $\\sigma $ -algebra, becomes a measure space.",
"Definition 3.1 On the space $X$ a measure $\\mu $ is defined by $\\int \\limits ^{{}}_{{X}} {F(x)}\\,\\mathrm {d}{\\mu (x)} := \\int \\limits ^{{}}_{{{R}^d}} {\\!\\int \\limits ^{{}}_{{{R}^{d-1}}} {{F(\\infty ,s,t)}}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t} + \\int \\limits ^{{}}_{{{R}^d}} {\\!\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\!\\int \\limits ^{1}_{-1} {F(a,s,t)}\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}$ with $F$ being a complex-valued function on $X$ which is measurable with respect to the Borel $\\sigma $ -algebra.",
"The first summand in the definition above is composed of the point measure on ${R}$ and the Lebesgue measure on ${R}^{d-1} \\times {R}^d$ , while the second summand is the restriction of the (left) Haar measure on the shearlet group to the subset $[-1,1]^*\\times {R}^{d-1}\\times {R}^d$ .",
"Therefore it is obvious that $\\mu $ given by eq:measure is a positive Radon measure.",
"Choosing the measure space $(X,\\mathfrak {B}(X),\\mu )$ as the underlying index space, we can introduce a continuous shearlet frame.",
"Definition 3.2 Let $a \\in {R}^*$ , $s \\in {R}^{d-1}$ and $t \\in {R}^d$ .",
"Then (i) $L_t: L_2({R}^d)\\rightarrow L_2({R}^d)$ with $L_t \\psi := \\psi (\\cdot - t)$ is called the (left) translation operator, (ii) $D_{S_s}: L_2({R}^d)\\rightarrow L_2({R}^d)$ with $D_{S_s} \\psi := \\psi (S_s^{-1} \\cdot )$ is called the shearing operator, and (iii) $D_{A_a}: L_2({R}^d)\\rightarrow L_2({R}^d)$ with $D_{A_a} \\psi := \\vert {\\det A_a} \\vert ^{-\\frac{1}{2}} \\psi (A_a^{-1} \\cdot )$ is called the (anisotropic) dilation operator.",
"Using the above defined operators, we can define an inhomogeneous shearlet frame.",
"Definition 3.3 Let $\\Phi , \\Psi \\in L_2({R}^d)$ with $\\Psi $ being an admissible shearlet.",
"Then we define $\\mathfrak {F}:= \\lbrace \\psi _x \\rbrace _{x \\in X}$ with $&\\psi _{(\\infty ,s,t)} := L_t D_{S_s} \\Phi = \\Phi (S_s^{-1} (\\cdot - t))\\mbox{ and}\\\\&\\psi _{(a,s,t)} := L_t D_{S_s} D_{A_a} \\Psi = \\vert {\\det A_a} \\vert ^{-\\frac{1}{2}} \\Psi (A_a^{-1} S_s^{-1} (\\cdot - t)).$ The main theorem of this section is that $\\mathfrak {F}$ , given by eq:generator and eq:shearlet, constitutes a continuous Parseval frame under the conditions given in theorem:tightframe below so that the transform based on $\\mathfrak {F}$ is well defined.",
"To this end we need two technical results that can also be found in Ref. [6].",
"Lemma 3.1 For all $(\\alpha ,s,t) \\in X$ with $\\alpha = a$ or $\\alpha = \\infty $ and $f,\\psi \\in L_2({R}^d)$ the identity $\\langle {f},{\\psi _{(\\alpha ,s,t)}} \\rangle _{L_2({R}^d)} = (f * \\psi ^{*}_{(\\alpha ,s,0)})(t)$ holds true with $\\psi ^{*} := \\overline{\\psi (-\\cdot )}$ .",
"Lemma 3.2 Let $\\phi \\in L_2({R}^d)$ , $a \\in {R}^*$ , $s \\in {R}^{d-1}$ and $\\xi \\in {R}^d$ .",
"Then the following equations hold: (i) $\\mathcal {F}(D_{S_s} \\phi ) (\\xi ) = \\hat{\\phi }(S^T_s \\xi )$ ; (ii) $\\mathcal {F}(D_{S_s} D_{A_a} \\phi ) (\\xi ) = \\vert {\\det A_a} \\vert ^{\\frac{1}{2}} \\hat{\\phi }(A_a S^T_s \\xi )$ .",
"We now state the main theorem of this section, which identifies conditions on $\\Phi $ and $\\Psi $ for $\\mathfrak {F}$ being a continuous Parseval frame.",
"Theorem 3.3 Let $\\Psi \\in L_1({R}^d)\\cap L_2({R}^d)$ be an admissible shearlet and let $\\Phi \\in L_1({R}^d)\\cap L_2({R}^d)$ be such that $\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\frac{\\vert {\\hat{\\Phi }(y,\\sigma )} \\vert ^2}{\\vert {y} \\vert ^{d-1}}}\\,\\mathrm {d}{\\sigma } + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^{{\\vert {y} \\vert }}_{{-\\vert {y} \\vert }} {\\frac{\\vert {\\hat{\\Psi }(\\xi _1,\\tilde{\\xi })} \\vert ^2}{\\vert {\\xi _1} \\vert ^d}}\\,\\mathrm {d}{\\xi _1}}\\,\\mathrm {d}{\\tilde{\\xi }} = 1\\quad \\text{for almost every }y\\in {R}.$ Then the inhomogeneous shearlet frame $\\mathfrak {F}$ is a continuous Parseval frame of $L_2({R}^d)$ , i.e., $\\int \\limits ^{{}}_{{X}} {\\vert {\\langle {f},{\\psi _x} \\rangle } \\vert ^2}\\,\\mathrm {d}{\\mu (x)} = \\Vert {f} \\vert {L_2({R}^d)} \\Vert ^2,\\quad f\\in L_2({R}^d).$ Applying eq:measure, Fubini's and Plancherel's theorem we obtain $\\int \\limits ^{{}}_{{X}} {\\vert {\\langle {f},{\\psi _x} \\rangle } \\vert ^2}\\,\\mathrm {d}{\\mu (x)} &= \\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\vert {\\langle {f},{\\psi _{(\\infty ,s,t)}} \\rangle } \\vert ^2}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}\\\\&\\qquad \\qquad + \\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\vert {\\langle {f},{\\psi _{(a,s,t)}} \\rangle } \\vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}\\\\&= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\langle {f},{\\psi _{(\\infty ,s,t)}} \\rangle } \\vert ^2}\\,\\mathrm {d}{t}}\\,\\mathrm {d}{s}\\\\&\\qquad \\qquad + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^{1}_{-1} \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\langle {f},{\\psi _{(a,s,t)}} \\rangle } \\vert ^2}\\,\\mathrm {d}{t}\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}\\\\&= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\Vert {\\langle {f},{\\psi _{(\\infty ,s,\\cdot )}} \\rangle } \\vert {L_2({R}^d)} \\Vert ^2}\\,\\mathrm {d}{s}\\\\&\\qquad \\qquad + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\Vert {\\langle {f},{\\psi _{(a,s,\\cdot )}} \\rangle } \\vert {L_2({R}^d)} \\Vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}\\\\&= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\Vert {\\mathcal {F}(\\langle {f},{\\psi _{(\\infty ,s,\\cdot )}} \\rangle )} \\vert {L_2({R}^d)} \\Vert ^2}\\,\\mathrm {d}{s}\\\\&\\qquad \\qquad + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\Vert {\\mathcal {F}(\\langle {f},{\\psi _{(a,s,\\cdot )}} \\rangle )} \\vert {L_2({R}^d)} \\Vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}\\\\&= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\!\\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\mathcal {F}(\\langle {f},{\\psi _{(\\infty ,s,\\cdot )}} \\rangle )(t)} \\vert ^2}\\,\\mathrm {d}{t}}\\,\\mathrm {d}{s}\\\\&\\qquad \\qquad + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^{1}_{-1} \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\mathcal {F}(\\langle {f},{\\psi _{(a,s,\\cdot )}} \\rangle )(t)} \\vert ^2}\\,\\mathrm {d}{t}\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}.$ Using lemma:convolution, Fubini's theorem, the fact that $\\mathcal {F}(f * g) = \\hat{f} \\hat{g}$ and $\\vert {\\mathcal {F}(f^*)} \\vert = \\vert {\\mathcal {F}(f)} \\vert $ leads to ${\\int \\limits ^{{}}_{{X}} {\\vert {\\langle {f},{\\psi _x} \\rangle } \\vert ^2}\\,\\mathrm {d}{\\mu (x)} = \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\mathcal {F}(f * \\psi ^{*}_{(\\infty ,s,0)})(t)} \\vert ^2}\\,\\mathrm {d}{t}}\\,\\mathrm {d}{s}}\\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\mathcal {F}(f * \\psi ^{*}_{(a,s,0)})(t)} \\vert ^2}\\,\\mathrm {d}{t}\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}\\\\&= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\hat{f}(t)} \\vert ^2 \\vert {\\mathcal {F}(\\psi ^{*}_{(\\infty ,s,0)})(t)} \\vert ^2}\\,\\mathrm {d}{t}}\\,\\mathrm {d}{s}\\\\&\\qquad \\qquad \\qquad \\qquad + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\hat{f}(t)} \\vert ^2 \\vert {\\mathcal {F}(\\psi ^{*}_{(a,s,0)})(t)} \\vert ^2}\\,\\mathrm {d}{t}\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}\\\\&= \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\hat{f}(t)} \\vert ^2 \\biggl ( \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\vert {\\mathcal {F}(\\psi _{(\\infty ,s,0)}) (t)} \\vert ^2}\\,\\mathrm {d}{s} + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\vert {\\mathcal {F}(\\psi _{(a,s,0)})(t)} \\vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s} \\biggr )}\\,\\mathrm {d}{t}.$ Thus, if we can prove that $\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\vert {\\mathcal {F}(\\psi _{(\\infty ,s,0)}) (t)} \\vert ^2}\\,\\mathrm {d}{s} + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\vert {\\mathcal {F}(\\psi _{(a,s,0)})(t)} \\vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s} \\stackrel{!",
"}{=} 1$ for almost every $t\\in {R}^d$ ,the assertion follows, since then $\\int \\limits ^{{}}_{{X}} {\\vert {\\langle {f},{\\psi _x} \\rangle } \\vert ^2}\\,\\mathrm {d}{\\mu (x)} = \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\hat{f}(t)} \\vert ^2}\\,\\mathrm {d}{t} = \\Vert {\\hat{f}} \\vert {L_2({R}^d)} \\Vert ^2 = \\Vert {f} \\vert {L_2({R}^d)} \\Vert ^2.$ Hence, it remains to show eq:condtightframe.",
"Assuming that $t_1 \\ne 0$ we use lemma:fourier to obtain ${\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\vert {\\mathcal {F}(\\psi _{(\\infty ,s,0)}) (t)} \\vert ^2}\\,\\mathrm {d}{s} + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\vert {\\mathcal {F}(\\psi _{(a,s,0)})(t)} \\vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}}\\\\&= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\vert {\\mathcal {F}(D_{S_s} \\Phi ) (t)} \\vert ^2}\\,\\mathrm {d}{s} + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\vert {\\mathcal {F}(D_{S_s} D_{A_a} \\Psi ) (t)} \\vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}\\\\&= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\vert {\\hat{\\Phi }(S^T_s t)} \\vert ^2}\\,\\mathrm {d}{s} + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\vert {\\det A_a} \\vert \\vert {\\hat{\\Psi }(A_a S_s^T t)} \\vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}\\\\&= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\vert {\\hat{\\Phi }(t_1,\\tilde{t} + t_1 s)} \\vert ^2}\\,\\mathrm {d}{s} + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\vert {\\det A_a} \\vert \\vert {\\hat{\\Psi }(a t_1,\\operatorname{sign}(a) \\vert {a} \\vert ^{\\frac{1}{d}} (\\tilde{t} + t_1 s))} \\vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s},$ with $t = (t_1,\\tilde{t})^T$ , $\\tilde{t} \\in {R}^{d-1}$ .",
"Substituting $ \\sigma := \\tilde{t} + t_1 s \\mbox{ and } \\xi = (\\xi _1,\\tilde{\\xi }) := (a t_1,\\operatorname{sign}(a) \\vert {a} \\vert ^{\\frac{1}{d}} (\\tilde{t} + t_1 s)),$ we end up with ${\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\vert {\\mathcal {F}(\\psi _{(\\infty ,s,0)}) (t)} \\vert ^2}\\,\\mathrm {d}{s} + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^1_{-1} \\vert {\\mathcal {F}(\\psi _{(a,s,0)})(t)} \\vert ^2\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}}\\\\&\\hspace{56.9055pt}= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\vert {t_1} \\vert ^{-(d-1)} \\vert {\\hat{\\Phi }(t_1,\\sigma )} \\vert ^2}\\,\\mathrm {d}{\\sigma } + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^{{\\vert {t_1} \\vert }}_{{-\\vert {t_1} \\vert }} {\\vert {\\xi _1} \\vert ^{-d} \\vert {\\hat{\\Psi }(\\xi _1,\\tilde{\\xi })} \\vert ^2}\\,\\mathrm {d}{\\xi _1}}\\,\\mathrm {d}{\\tilde{\\xi }}\\\\&\\hspace{56.9055pt}= \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\frac{\\vert {\\hat{\\Phi }(t_1,\\sigma )} \\vert ^2}{\\vert {t_1} \\vert ^{d-1}}}\\,\\mathrm {d}{\\sigma } + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits ^{{\\vert {t_1} \\vert }}_{{-\\vert {t_1} \\vert }} {\\frac{\\vert {\\hat{\\Psi }(\\xi _1,\\tilde{\\xi })} \\vert ^2}{\\vert {\\xi _1} \\vert ^d}}\\,\\mathrm {d}{\\xi _1}}\\,\\mathrm {d}{\\tilde{\\xi }},$ and eq:condtightframe follows from assumption eq:assumptions.",
"Remark 4 The proof of theorem:tightframe can also be stated in a similar manner for the case of a tight frame with arbitrary frame constant $A < \\infty $ .",
"The only difference is that $\\Phi $ and $\\Psi $ have to satisfy $\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\frac{\\vert {\\hat{\\Phi }(y,\\sigma )} \\vert ^2}{\\vert {y} \\vert ^{d-1}}}\\,\\mathrm {d}{\\sigma } + \\int \\limits ^{{}}_{{{R}^{d-1}}} {\\!\\int \\limits ^{{\\vert {y} \\vert }}_{{-\\vert {y} \\vert }} {\\frac{\\vert {\\hat{\\Psi }(\\xi _1,\\tilde{\\xi })} \\vert ^2}{\\vert {\\xi _1} \\vert ^d}}\\,\\mathrm {d}{\\xi _1}}\\,\\mathrm {d}{\\tilde{\\xi }} = A\\quad \\text{for almost every }y\\in {R}$ instead of eq:assumptions.",
"Remark 5 For a given shearlet $\\Psi $ it is still necessary to show that one can satisfy condition eq:assumptions for a function $\\Phi \\in L_1({R}^d) \\cap L_2({R}^d)$ .",
"To this end we restrict ourselves to odd dimensions and we define $\\hat{\\Phi }:{R}^d\\rightarrow by\\begin{equation*}\\hat{\\Phi }(\\xi ) := \\xi _1^{\\frac{d-1}{2}}\\biggl (\\int \\limits ^{{}}_{{{R}\\setminus [-\\vert {\\xi _1} \\vert ,\\vert {\\xi _1} \\vert ]}} {\\frac{\\vert {\\hat{\\Psi }(\\omega _1,\\tilde{\\xi })} \\vert ^2}{\\vert {\\omega _1} \\vert ^d}}\\,\\mathrm {d}{\\omega _1}\\biggr )^{1/2}.\\end{equation*}It is straightforward to see that $$ fulfills {eq:assumptions}.",
"Moreover, $ L2(${R}$ d)$ is immediate and $ L1(${R}$ d)$ can be shown if $ C0(${R}$ d)$, see {example:psiphi}.$ Because of theorem:tightframe, we can now state the definition of the shearlet transform based on $\\mathfrak {F}$ .",
"Definition 3.4 Let $\\Phi ,\\Psi \\in L_2({R}^d)$ satisfy the assumptions of theorem:tightframe and let $\\mathfrak {F}= \\lbrace \\psi _x \\rbrace _{x\\in X}$ be given by definition:shearletframe.",
"Then the shearlet transform based on $\\mathfrak {F}$ is defined as $ \\mathcal {SH}_\\mathfrak {F}: L_2({R}^d) \\rightarrow L_2(X,\\mu ), f \\mapsto \\mathcal {SH}_\\mathfrak {F}f $ with $ \\mathcal {SH}_\\mathfrak {F}f: X \\rightarrow x \\mapsto \\langle {f},{\\psi _x} \\rangle .",
"$"
],
[
"Conditions on the reproducing kernel",
"The main goal of this section is to lay the foundations for the definition of the coorbit spaces $\\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v}(X,\\mu ))$ , $1\\le p<\\infty $ , $p<\\tau ^{\\prime }<\\infty $ , with $v$ being a weight function on $X$ , associated to the inhomogeneous shearlet transform introduced in the previous section.",
"To prove that these spaces are well-defined Banach spaces, we need to show that the conditions on $\\mathfrak {F}$ , as stated in sec:coorbittheory, are satisfied.",
"By remark:assumptionfulfilled it suffices to show that $R_\\mathfrak {F}\\in \\mathcal {A}_{q,m_v}$ for all $q>1$ .",
"To this end we need the following auxiliary results.",
"Lemma 3.4 Let $a,a^{\\prime } \\in [-1,1]^*,\\ s,s^{\\prime } \\in {R}^{d-1},\\ t,t^{\\prime } \\in {R}^d$ and $\\varphi _{(a,s,t)} := \\vert {\\det A_a} \\vert ^{-\\frac{1}{2}} \\Phi (A_a^{-1} S_s^{-1} (\\cdot - t))$ .",
"It follows that $\\vert {\\langle {\\psi _{(\\infty ,s,t)}},{\\psi _{(\\infty ,s^{\\prime },t^{\\prime })}} \\rangle } \\vert = \\vert {(\\mathcal {SH} \\Phi )(\\infty ,s-s^{\\prime },S_{s^{\\prime }}^{-1}(t-t^{\\prime }))} \\vert ,$ $\\vert {\\langle {\\psi _{(\\infty ,s,t)}},{\\psi _{(a^{\\prime },s^{\\prime },t^{\\prime })}} \\rangle } \\vert = \\vert {\\langle {\\Psi },{\\varphi _{(a^{\\prime -1},\\vert {a^{\\prime }} \\vert ^{\\frac{1}{d}-1}(s-s^{\\prime }),A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1}(t-t^{\\prime })}} \\rangle } \\vert ,$ $\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(\\infty ,s^{\\prime },t^{\\prime })}} \\rangle } \\vert = \\vert {(\\mathcal {SH} \\Phi )(a,s-s^{\\prime },S_{s^{\\prime }}^{-1}(t-t^{\\prime }))} \\vert ,$ $\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(a^{\\prime },s^{\\prime },t^{\\prime })}} \\rangle } \\vert = \\vert {(\\mathcal {SH} \\Psi )(aa^{\\prime -1},\\vert {a^{\\prime }} \\vert ^{\\frac{1}{d}-1}(s-s^{\\prime }),A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1} (t-t^{\\prime }))} \\vert .$ We only state the proof for eq:IP4 in detail, eq:IP1–eq:IP3 can be proven analogously.",
"By the definition of $\\psi _{(a,s,t)}$ we obtain ${\\langle {\\psi _{(a,s,t)}},{\\psi _{(a^{\\prime },s^{\\prime },t^{\\prime })}} \\rangle = \\int \\limits ^{{}}_{{{R}^d}} {\\psi _{(a,s,t)}(x) \\overline{\\psi _{(a^{\\prime },s^{\\prime },t^{\\prime })}(x)}}\\,\\mathrm {d}{x}}\\\\&\\qquad = \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\det A_a} \\vert ^{-\\frac{1}{2}}\\Psi (A_a^{-1} S_s^{-1}(x - t)) \\overline{\\vert {\\det A_{a^{\\prime }}} \\vert ^{-\\frac{1}{2}} \\Psi (A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1}(x - t^{\\prime }))}}\\,\\mathrm {d}{x},$ which, by means of the substitution $y = A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1}(x-t^{\\prime })$ , leads to ${\\langle {\\psi _{(a,s,t)}},{\\psi _{(a^{\\prime },s^{\\prime },t^{\\prime })}} \\rangle = \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\det A_{aa^{\\prime -1}}} \\vert ^{-\\frac{1}{2}} \\Psi (A_a^{-1} S_s^{-1} (S_{s^{\\prime }} A_{a^{\\prime }} y + t^{\\prime } - t)) \\overline{\\Psi (y)}}\\,\\mathrm {d}{y}}\\\\&\\qquad = \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\det A_{aa^{\\prime -1}}} \\vert ^{-\\frac{1}{2}} \\Psi (A_a^{-1} S_s^{-1} S_{s^{\\prime }} A_{a^{\\prime }}(y - (A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1} (t-t^{\\prime })))) \\overline{\\Psi (y)}}\\,\\mathrm {d}{y}\\\\&\\qquad = \\int \\limits ^{{}}_{{{R}^d}} {\\vert {\\det A_{aa^{\\prime -1}}} \\vert ^{-\\frac{1}{2}} \\Psi (A_{aa^{\\prime -1}}^{-1} S_{\\vert {a^{\\prime }} \\vert ^{\\frac{1}{d}-1} (s-s^{\\prime })}^{-1}(y - (A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1} (t-t^{\\prime })))) \\overline{\\Psi (y)}}\\,\\mathrm {d}{y}\\\\&\\qquad = \\langle {\\psi _{(aa^{\\prime -1},\\vert {a^{\\prime }} \\vert ^{\\frac{1}{d}-1}(s-s^{\\prime }),A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1} (t-t^{\\prime }))}},{\\Psi } \\rangle .$ This yields $\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(a^{\\prime },s^{\\prime },t^{\\prime })}} \\rangle } \\vert &= \\vert {\\langle {\\psi _{(aa^{\\prime -1},\\vert {a^{\\prime }} \\vert ^{\\frac{1}{d}-1}(s-s^{\\prime }),A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1} (t-t^{\\prime }))}},{\\Psi } \\rangle } \\vert \\\\&= \\vert {\\langle {\\Psi },{\\psi _{(aa^{\\prime -1},\\vert {a^{\\prime }} \\vert ^{\\frac{1}{d}-1}(s-s^{\\prime }),A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1} (t-t^{\\prime }))}} \\rangle } \\vert \\\\&= \\vert {(\\mathcal {SH} \\Psi )(aa^{\\prime -1},\\vert {a^{\\prime }} \\vert ^{\\frac{1}{d}-1}(s-s^{\\prime }),A_{a^{\\prime }}^{-1} S_{s^{\\prime }}^{-1} (t-t^{\\prime }))} \\vert .$ Using the auxiliary result above, we can prove the following lemma concerning the $\\mathcal {A}_{q,m_v}$ -Norm of $R_\\mathfrak {F}$ .",
"Lemma 3.5 Let $R_\\mathfrak {F}$ be the kernel function associated to the inhomogeneous shearlet frame as defined by eq:kernel.",
"Then for every $q$ the following identity holds: $ {\\operatornamewithlimits{ess\\,sup}_{(\\alpha ,\\sigma ,\\tau ) \\in X} \\int \\limits ^{{}}_{{X}} {\\vert {R_\\mathfrak {F}((\\alpha ,\\sigma ,\\tau ),(a,s,t))} \\vert ^q m_v((\\alpha ,\\sigma ,\\tau ),(a,s,t))^q}\\,\\mathrm {d}{\\mu (a,s,t)}} \\\\&= \\max \\Biggl \\lbrace \\operatornamewithlimits{ess\\,sup}_{(\\sigma ,\\tau )\\in {R}^{d-1}\\times {R}^d} \\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}}\\biggl ( \\max \\left\\lbrace \\frac{v(\\infty ,\\sigma ,\\tau )}{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})},\\frac{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})}{v(\\infty ,\\sigma ,\\tau )}\\right\\rbrace ^q \\vert {\\langle {\\Phi },{\\psi _{(\\infty ,s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q \\\\&\\hspace{28.45274pt}+ \\int \\limits _{-1}^{1} \\max \\left\\lbrace \\frac{v(\\infty ,\\sigma ,\\tau )}{v(a,\\tilde{\\sigma _2},\\tilde{\\tau _2})},\\frac{v(a,\\tilde{\\sigma _2},\\tilde{\\tau _2})}{v(\\infty ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\Phi },{\\psi _{(a,s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}} \\biggr )\\,\\mathrm {d}s^{\\prime }\\,\\mathrm {d}t^{\\prime },\\\\& \\operatornamewithlimits{ess\\,sup}_{(\\alpha ,\\sigma ,\\tau ) \\in [-1,1]^*\\times {R}^{d-1}\\times {R}^d} \\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}}\\biggl (\\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})},\\frac{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q \\vert {\\langle {\\Phi },{\\psi _{(\\alpha ,s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q \\\\&\\hspace{28.45274pt}+ \\int \\limits _{-\\vert {\\alpha } \\vert ^{-1}}^{\\vert {\\alpha } \\vert ^{-1}} \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\tilde{\\alpha },\\tilde{\\sigma _3},\\tilde{\\tau _3})},\\frac{v(\\tilde{\\alpha },\\tilde{\\sigma _3},\\tilde{\\tau _3})}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\Psi },{\\psi _{(a^{\\prime },s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a^{\\prime }}{\\vert {a^{\\prime }} \\vert ^{d+1}} \\biggr )\\,\\mathrm {d}s^{\\prime }\\,\\mathrm {d}t^{\\prime } \\Biggr \\rbrace $ with $\\tilde{\\sigma _1} = \\sigma -s^{\\prime }, \\tilde{\\tau _1} = \\tau - S_{\\tilde{\\sigma _1}}t^{\\prime }, \\tilde{\\sigma _2} = \\sigma +s^{\\prime }, \\tilde{\\tau _2} = \\tau + S_\\sigma t^{\\prime }, \\tilde{\\alpha } = \\alpha a^{\\prime }, \\tilde{\\sigma _3} = \\sigma + \\vert {\\alpha } \\vert ^{1-\\frac{1}{d}}s^{\\prime }, \\tilde{\\tau _3} = \\tau + S_\\sigma A_\\alpha t^{\\prime }$ .",
"Let $(\\alpha ,\\sigma ,\\tau ) \\in X$ with $\\alpha \\in \\lbrace \\infty \\rbrace \\cup [-1,1]^*$ .",
"Using eq:IP1 and eq:IP3 we obtain $\\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\infty ,s,t)},\\frac{v(\\infty ,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(\\infty ,s,t)}},{\\psi _{(\\alpha ,\\sigma ,\\tau )}} \\rangle } \\vert ^q}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}\\\\= \\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\infty ,s,t)},\\frac{v(\\infty ,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\Phi },{\\psi _{(\\alpha ,\\sigma -s,S_s^{-1}(\\tau -t))}} \\rangle } \\vert ^q}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}.$ Substituting $s^{\\prime } = \\sigma -s$ and $t^{\\prime } = S_{\\sigma -s^{\\prime }}^{-1}(\\tau -t)$ then leads to $\\begin{split}\\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\infty ,s,t)},\\frac{v(\\infty ,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(\\infty ,s,t)}},{\\psi _{(\\alpha ,\\sigma ,\\tau )}} \\rangle } \\vert ^q}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}\\\\= \\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\infty ,\\tilde{\\sigma _1},t)},\\frac{v(\\infty ,\\tilde{\\sigma _1},t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\bigl \\vert {\\bigl \\langle {\\Phi },{\\psi _{(\\alpha ,s^{\\prime },S_{\\sigma -s^{\\prime }}^{-1}(\\tau -t))}} \\bigr \\rangle } \\bigr \\vert ^q}\\,\\mathrm {d}{s^{\\prime }}}\\,\\mathrm {d}{t}\\\\= \\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})},\\frac{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\Phi },{\\psi _{(\\alpha ,s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q}\\,\\mathrm {d}{s^{\\prime }}}\\,\\mathrm {d}{t^{\\prime }}.\\end{split}$ Analogously we see that $\\begin{split}\\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits _{-1}^{1} \\max \\left\\lbrace \\frac{v(\\infty ,\\sigma ,\\tau )}{v(a,s,t)},\\frac{v(a,s,t)}{v(\\infty ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(\\infty ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}\\\\= \\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits _{-1}^{1} \\max \\left\\lbrace \\frac{v(\\infty ,\\sigma ,\\tau )}{v(a,\\tilde{\\sigma _2},\\tilde{\\tau _2})},\\frac{v(a,\\tilde{\\sigma _2},\\tilde{\\tau _2})}{v(\\infty ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\Phi },{\\psi _{(a,s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s^{\\prime }}}\\,\\mathrm {d}{t^{\\prime }},\\end{split}$ for $\\sigma \\in {R}^{d-1}$ and $\\tau \\in {R}^d$ .",
"Now let $\\alpha \\in [-1,1]^*$ .",
"Then eq:IP4 yields ${\\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits _{-1}^1 \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(a,s,t)},\\frac{v(a,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(\\alpha ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}}\\\\&= \\int \\limits _{{R}^d} \\int \\limits _{{R}^{d-1}} \\int \\limits _{-1}^1 \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(a,s,t)},\\frac{v(a,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\\\&\\hspace{113.81102pt}\\cdot \\bigl \\vert {\\bigl \\langle {\\Psi },{\\psi _{(a {\\alpha }^{-1},\\vert {\\alpha } \\vert ^{\\frac{1}{d} - 1} (s - \\sigma ),A_{\\alpha }^{-1} S_{\\sigma }^{-1} (t - \\tau ))}} \\bigr \\rangle } \\bigr \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}\\,\\mathrm {d}s\\,\\mathrm {d}t,$ which—by substituting $a^{\\prime } := a {\\alpha }^{-1}$ —leads to ${\\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits _{-1}^1 \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(a,s,t)},\\frac{v(a,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(\\alpha ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}}\\\\&= \\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}}\\int \\limits _{-\\vert {\\alpha } \\vert ^{-1}}^{\\vert {\\alpha } \\vert ^{-1}} \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\tilde{\\alpha },s,t)},\\frac{v(\\tilde{\\alpha },s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\\\&\\hspace{113.81102pt}\\cdot \\bigl \\vert {\\bigl \\langle {\\Psi },{\\psi _{(a^{\\prime },\\vert {\\alpha } \\vert ^{\\frac{1}{d} - 1} (s - \\sigma ),A_{\\alpha }^{-1} S_{\\sigma }^{-1} (t - \\tau ))}} \\bigr \\rangle } \\bigr \\vert ^q \\frac{1}{\\vert {\\alpha } \\vert ^d}\\,\\frac{\\mathrm {d}a^{\\prime }}{\\vert {a^{\\prime }} \\vert ^{d+1}}\\,\\mathrm {d}s\\,\\mathrm {d}t.$ Again, substituting with $s^{\\prime } := \\vert {\\alpha } \\vert ^{\\frac{1}{d}-1} (s - \\sigma )$ and $t^{\\prime } := A_{\\alpha }^{-1} S_{\\sigma }^{-1} (t - \\tau )$ , we get $\\begin{split}{\\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits _{-1}^1 \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(a,s,t)},\\frac{v(a,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(\\alpha ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}}\\,\\mathrm {d}{s}}\\,\\mathrm {d}{t}}\\\\&= \\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}}\\int \\limits _{-\\vert {\\alpha } \\vert ^{-1}}^{\\vert {\\alpha } \\vert ^{-1}} \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\tilde{\\alpha },\\tilde{\\sigma _3},t)},\\frac{v(\\tilde{\\alpha },\\tilde{\\sigma _3},t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\\\&\\hspace{113.81102pt}\\cdot \\bigl \\vert {\\bigl \\langle {\\Psi },{\\psi _{(a^{\\prime },s^{\\prime },A_{\\alpha }^{-1} S_{\\sigma }^{-1} (t - \\tau ))}} \\bigr \\rangle } \\bigr \\vert ^q \\vert {\\alpha } \\vert ^{\\frac{1}{d} - 2}\\,\\frac{\\mathrm {d}a^{\\prime }}{\\vert {a^{\\prime }} \\vert ^{d+1}}\\,\\mathrm {d}s^{\\prime }\\,\\mathrm {d}t\\\\&= \\int \\limits ^{{}}_{{{R}^d}} {\\int \\limits ^{{}}_{{{R}^{d-1}}} {\\int \\limits _{-\\vert {\\alpha } \\vert ^{-1}}^{\\vert {\\alpha } \\vert ^{-1}} \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\tilde{\\alpha },\\tilde{\\sigma _3},\\tilde{\\tau _3})},\\frac{v(\\tilde{\\alpha },\\tilde{\\sigma _3},\\tilde{\\tau _3})}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\Psi },{\\psi _{(a^{\\prime },s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a^{\\prime }}{\\vert {a^{\\prime }} \\vert ^{d+1}}}\\,\\mathrm {d}{s^{\\prime }}}\\,\\mathrm {d}{t^{\\prime }}.\\end{split}$ Using eq:firstint, eq:secondint, and eq:thirdint, we now have ${\\operatornamewithlimits{ess\\,sup}_{(\\alpha ,\\sigma ,\\tau ) \\in X} \\int \\limits ^{{}}_{{X}} {\\vert {R_\\mathfrak {F}((\\alpha ,\\sigma ,\\tau ),(a,s,t))} \\vert ^q m_v((\\alpha ,\\sigma ,\\tau ),(a,s,t))^q}\\,\\mathrm {d}{\\mu (a,s,t)}}\\\\&= \\operatornamewithlimits{ess\\,sup}_{(\\alpha ,\\sigma ,\\tau ) \\in X} \\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}} \\Biggl (\\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\infty ,s,t)},\\frac{v(\\infty ,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(\\infty ,s,t)}},{\\psi _{(\\alpha ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\\\&\\hspace{28.45274pt}+ \\int \\limits _{-1}^{1} \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(a,s,t)},\\frac{v(a,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(\\alpha ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}} \\Biggr )\\,\\mathrm {d}s\\,\\mathrm {d}t\\\\&= \\max \\Biggl \\lbrace \\operatornamewithlimits{ess\\,sup}_{(\\sigma ,\\tau ) \\in {R}^{d-1}\\times {R}^d} \\int \\limits _{{R}^d} \\int \\limits _{{R}^{d-1}} \\Biggl (\\max \\left\\lbrace \\frac{v(\\infty ,\\sigma ,\\tau )}{v(\\infty ,s,t)},\\frac{v(\\infty ,s,t)}{v(\\infty ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(\\infty ,s,t)}},{\\psi _{(\\infty ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\\\&\\hspace{28.45274pt}+ \\int \\limits _{-1}^{1} \\max \\left\\lbrace \\frac{v(\\infty ,\\sigma ,\\tau )}{v(a,s,t)},\\frac{v(a,s,t)}{v(\\infty ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(\\infty ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}} \\Biggr )\\,\\mathrm {d}s\\,\\mathrm {d}t ,\\\\& \\operatornamewithlimits{ess\\,sup}_{(\\alpha ,\\sigma ,\\tau ) \\in [-1,1]^*\\times {R}^{d-1}\\times {R}^d} \\int \\limits _{{R}^d} \\int \\limits _{{R}^{d-1}} \\Biggl (\\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\infty ,s,t)},\\frac{v(\\infty ,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(\\infty ,s,t)}},{\\psi _{(\\alpha ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\\\&\\hspace{28.45274pt}+ \\int \\limits _{-1}^{1} \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(a,s,t)},\\frac{v(a,s,t)}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\psi _{(a,s,t)}},{\\psi _{(\\alpha ,\\sigma ,\\tau )}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}} \\Biggr ) \\,\\mathrm {d}s\\,\\mathrm {d}t \\Biggr \\rbrace \\\\&= \\max \\Biggl \\lbrace \\operatornamewithlimits{ess\\,sup}_{(\\sigma ,\\tau )\\in {R}^{d-1}\\times {R}^d} \\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}}\\biggl ( \\max \\left\\lbrace \\frac{v(\\infty ,\\sigma ,\\tau )}{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})},\\frac{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})}{v(\\infty ,\\sigma ,\\tau )}\\right\\rbrace ^q \\vert {\\langle {\\Phi },{\\psi _{(\\infty ,s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q \\\\&\\hspace{28.45274pt}+ \\int \\limits _{-1}^{1} \\max \\left\\lbrace \\frac{v(\\infty ,\\sigma ,\\tau )}{v(a,\\tilde{\\sigma _2},\\tilde{\\tau _2})},\\frac{v(a,\\tilde{\\sigma _2},\\tilde{\\tau _2})}{v(\\infty ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\Phi },{\\psi _{(a,s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}} \\biggr )\\,\\mathrm {d}s^{\\prime }\\,\\mathrm {d}t^{\\prime },\\\\& \\operatornamewithlimits{ess\\,sup}_{(\\alpha ,\\sigma ,\\tau ) \\in [-1,1]^*\\times {R}^{d-1}\\times {R}^d} \\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}}\\biggl (\\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})},\\frac{v(\\infty ,\\tilde{\\sigma _1},\\tilde{\\tau _1})}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q \\vert {\\langle {\\Phi },{\\psi _{(\\alpha ,s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q \\\\&\\hspace{28.45274pt}+ \\int \\limits _{-\\vert {\\alpha } \\vert ^{-1}}^{\\vert {\\alpha } \\vert ^{-1}} \\max \\left\\lbrace \\frac{v(\\alpha ,\\sigma ,\\tau )}{v(\\tilde{\\alpha },\\tilde{\\sigma _3},\\tilde{\\tau _3})},\\frac{v(\\tilde{\\alpha },\\tilde{\\sigma _3},\\tilde{\\tau _3})}{v(\\alpha ,\\sigma ,\\tau )}\\right\\rbrace ^q\\vert {\\langle {\\Psi },{\\psi _{(a^{\\prime },s^{\\prime },t^{\\prime })}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a^{\\prime }}{\\vert {a^{\\prime }} \\vert ^{d+1}} \\biggr )\\,\\mathrm {d}s^{\\prime }\\,\\mathrm {d}t^{\\prime } \\Biggr \\rbrace $ We use lemma:kernel to prove $R_\\mathfrak {F}\\in \\mathcal {A}_{q,m_v}$ for certain functions $\\Phi $ and $\\Psi $ .",
"Since it is not possible to construct functions $\\Phi , \\Psi \\in L_2$ with compact support in the spatial domain satisfying the conditions in remark:choicePhi, in the following we assume $\\Psi $ to be a bandlimited Schwartz function, in particular $ \\operatorname{supp}\\hat{\\Psi } \\subseteq ([-a_1,-a_0]\\cup [a_0,a_1])\\times Q_b $ with $0 < a_0 < a_1$ and $Q_b := \\operatornamewithlimits{\\times }_{i = 1}^{d-1} [-b_i,b_i]$ for $b \\in {R}^{d-1}_+$ .",
"The function $\\Phi $ is chosen in the same way as in remark:choicePhi.",
"It follows that $\\operatorname{supp}\\hat{\\Phi }\\subseteq [-a_1,a_1] \\operatornamewithlimits{\\times }Q_b.$ As weight functions on $X$ we consider $v_{r}(\\alpha ,s,t) = v_{r}(\\alpha ) := {\\left\\lbrace \\begin{array}{ll}1, &\\alpha = \\infty ,\\\\|\\alpha |^{-r}, &\\alpha \\in [-1,1]^\\ast ,\\end{array}\\right.",
"}$ with $r\\in {R}_{\\ge 0}$ , which satisfy all necessary conditions.",
"Through simple calculations one can verify the following properties of the moderate weight $m_{v_r}$ associated with $v_{r}$ for $a,a^{\\prime } \\in [-1,1]^\\ast $ : $m_{v_{r}}(\\infty ,\\infty ) &= 1,\\\\m_{v_{r}}(a,\\infty ) &= m_{v_r}(\\infty ,a) = |a|^{-r},\\\\m_{v_{r}}(a,a^{\\prime }) &= \\max \\left\\lbrace \\frac{\\vert {a} \\vert }{\\vert {a^{\\prime }} \\vert },\\frac{\\vert {a^{\\prime }} \\vert }{\\vert {a} \\vert }\\right\\rbrace ^{-r}.$ The following technical lemma concerns support properties of $\\Phi $ and $\\Psi $ in the frequency domain, similar to Lemma 3.1, Ref. [6].",
"Lemma 3.6 Let $0<a_0<a_1$ and $b\\in {R}^{d-1}_+$ .",
"Then with $\\Psi $ and $\\Phi $ defined as above and for $a\\in {R}^\\ast $ and $s\\in {R}^{d-1}$ we have (i) $\\hat{\\Psi }\\hat{\\Psi }(A_aS_s^T\\cdot )\\lnot \\equiv 0$ implies $a\\in [-\\frac{a_1}{a_0},-\\frac{a_0}{a_1}]\\cup [\\frac{a_0}{a_1},\\frac{a_1}{a_0}]$ and $s\\in Q_{d_1}$ with $d_1:=(a_0^{-1}+a_0^{-(1+\\frac{1}{d})}a_1^{\\frac{1}{d}})b$ , (ii) Assume $\\vert {a} \\vert \\le 1$ then $\\hat{\\Phi }\\hat{\\Psi }(A_aS_s^T\\cdot )\\lnot \\equiv 0$ implies $a\\in [-1,-\\frac{a_0}{a_1}]\\cup [\\frac{a_0}{a_1},1]$ and $s\\in Q_{d_2}$ with $d_2:=(a_0^{-1}+a_0^{-(1+\\frac{1}{d})}a_1^\\frac{1}{d})b$ , (iii) $\\operatorname{supp}\\hat{\\Phi }\\hat{\\Phi }(S_s^T\\cdot )\\subseteq \\Omega _s:=\\lbrace x\\in {R}^d:|x_1|\\le a_1,\\max \\lbrace -b_i,-b_i-s_{i-1}x_1\\rbrace \\le x_i\\le \\min \\lbrace b_i,b_i-s_{i-1}x_1\\rbrace ,i=2,\\ldots d\\rbrace $ .",
"The proof of list:parampsipsi can be found in Lemma 3.1, Ref. [6].",
"To prove list:paramphipsi we assume there exists a $\\xi \\in \\operatorname{supp}\\hat{\\Phi }\\cap \\operatorname{supp}\\hat{\\Psi }(A_a S_s^T\\cdot )$ which means that $\\xi \\in \\operatorname{supp}\\hat{\\Phi } \\mbox{ and } A_a S_s^T \\xi \\in \\operatorname{supp}\\hat{\\Psi }.$ This leads to $\\vert {\\xi _1} \\vert \\le a_1, \\\\-b_i \\le \\xi _{i+1} \\le b_i, \\\\a_0 \\le \\vert {a} \\vert \\vert {\\xi _1} \\vert \\le a_1, \\\\-b_i \\vert {a} \\vert ^{-\\frac{1}{d}} - \\xi _1 s_i \\le \\xi _{i+1} \\le b_i \\vert {a} \\vert ^{-\\frac{1}{d}} - \\xi _1 s_i, $ for $i = 1,\\ldots ,d-1$ .",
"By (REF ) and () it follows that $\\vert {a} \\vert \\ge \\frac{a_0}{a_1}$ which means $a \\in [-1,-\\frac{a_0}{a_1}]\\cup [\\frac{a_0}{a_1},1]$ .",
"Using () and $\\vert {a} \\vert \\le 1$ , it follows that $a_0 \\le \\vert {a} \\vert \\vert {\\xi _1} \\vert \\le \\vert {\\xi _1} \\vert $ .",
"Also, with () and () we obtain $-b_i \\vert {a} \\vert ^{-\\frac{1}{d}}-b_i \\le \\xi _1 s_i \\le b_i \\vert {a} \\vert ^{-\\frac{1}{d}}+b_i,$ which leads to $\\vert {s_i} \\vert \\le \\vert {\\xi _1} \\vert ^{-1}( b_i \\vert {a} \\vert ^{-\\frac{1}{d}} + b_i) \\le a_0^{-1} b_i \\left(\\frac{a_0}{a_1}\\right)^{-\\frac{1}{d}} + a_0^{-1}b_i$ for $i = 1,\\ldots ,d-1$ which proves list:paramphipsi.",
"To prove list:paramphiphi we assume there exists $\\xi \\in \\operatorname{supp}\\hat{\\Phi }\\cap \\operatorname{supp}\\hat{\\Phi }(S_s^T\\cdot )$ which means that $\\xi \\in \\operatorname{supp}\\hat{\\Phi }$ and $S_s^T\\xi \\in \\operatorname{supp}\\hat{\\Phi }$ .",
"This leads to $\\vert {\\xi _1} \\vert \\le a_1,\\\\-b_i \\le \\xi _{i+1} \\le b_i,\\\\-b_i \\le \\xi _1 s_i + \\xi _{i+1} \\le b_i$ for all $i=1,\\ldots ,d-1$ , which means $\\xi \\in \\Omega _s$ .",
"The following two auxiliary Lemmas are of technical nature only and the proof of lemma:subconvolution1 is based on a draft by Steidl, Dahlke, Häuser and Teschke.",
"Lemma 3.7 For all $y,z\\in {R}$ , $\\lambda ,\\lambda ^{\\prime }>0$ and $k>1$ the following integral estimation holds true $\\int \\limits ^{{}}_{{{R}}} {(1+\\lambda |x-y|)^{-k}(1+\\lambda ^{\\prime }|x-z|)^{-k}}\\,\\mathrm {d}{x} \\lesssim \\max \\lbrace \\lambda ,\\lambda ^{\\prime }\\rbrace ^{-1}(1+\\min \\lbrace \\lambda ,\\lambda ^{\\prime }\\rbrace |y-z|)^{-k}.$ Let $y,z\\in {R}$ be arbitrary and assume without loss of generality that $\\lambda \\le \\lambda ^{\\prime }$ .",
"Assume further that $|y-z|\\le \\lambda ^{-1}$ , then $(1+\\lambda |x-y|)^{-k} \\le 1 \\le 2^k (1+\\lambda |y-z|)^{-k}$ and thus $\\int \\limits ^{{}}_{{{R}}} {(1+\\lambda |x-y|)^{-k}(1+\\lambda ^{\\prime }|x-z|)^{-k}}\\,\\mathrm {d}{x} &\\lesssim (1+\\lambda |y-z|)^{-k}\\int \\limits ^{{}}_{{{R}}} {(1+\\lambda ^{\\prime }|x-z|)^{-k}}\\,\\mathrm {d}{x} \\\\&= (1+\\lambda ^{\\prime }|y-z|)^{-k}\\frac{1}{\\lambda ^{\\prime }}\\int \\limits ^{{}}_{{{R}}} {(1+|x|)^{-k}}\\,\\mathrm {d}{x} \\\\&\\lesssim \\frac{1}{\\lambda ^{\\prime }}(1+\\lambda |y-z|)^{-k}.$ On the other hand if $|y-z|>\\lambda ^{-1}$ let $H_y$ and $H_z$ be the two half-axes containing the points $y$ and $z$ respectively, such that $H_y\\cap H_z=\\lbrace \\frac{y+z}{2}\\rbrace $ .",
"Then, for every $x\\in H_z$ it holds $|x-y|\\ge \\frac{1}{2}|y-z|$ and thus $\\int \\limits ^{{}}_{{H_z}} {(1+\\lambda |x-y|)^{-k}(1+\\lambda ^{\\prime }|x-z|)^{-k}}\\,\\mathrm {d}{x} &\\le \\left(1+\\frac{\\lambda }{2}|y-z|\\right)^{-k}\\int \\limits ^{{}}_{{H_z}} {(1+\\lambda ^{\\prime }|x-z|)^{-k}}\\,\\mathrm {d}{x} \\\\&\\lesssim (1+\\lambda |y-z|)^{-k}\\frac{1}{\\lambda ^{\\prime }}\\int \\limits ^{{}}_{{{R}}} {(1+|x|)^{-k}}\\,\\mathrm {d}{x} \\\\&\\lesssim \\frac{1}{\\lambda ^{\\prime }}(1+\\lambda |y-z|)^{-k}$ Similarily for every $x\\in H_y$ it holds $|x-z|\\ge \\frac{1}{2}|y-z|$ and since $|y-z|>\\lambda ^{-1}$ we first deduce $(1+\\lambda ^{\\prime }|x-z|)^{-k} &\\le \\left(\\frac{\\lambda ^{\\prime }}{2}|y-z|\\right)^{-k} \\lesssim \\left(\\frac{\\lambda }{\\lambda ^{\\prime }}\\right)^k(\\lambda |y-z|)^{-k}\\\\& \\lesssim \\left(\\frac{\\lambda }{\\lambda ^{\\prime }}\\right)^k(1+\\lambda |y-z|)^{-k} \\le \\frac{\\lambda }{\\lambda ^{\\prime }}(1+\\lambda |y-z|)^{-k}$ and hence we derive the estimate $\\int \\limits ^{{}}_{{H_y}} {(1+\\lambda |x-y|)^{-k}(1+\\lambda ^{\\prime }|x-z|)^{-k}}\\,\\mathrm {d}{x} &\\le \\frac{\\lambda }{\\lambda ^{\\prime }}(1+\\lambda |y-z|)^{-k}\\int \\limits ^{{}}_{{H_y}} {(1+\\lambda |x-y|)^{-k}}\\,\\mathrm {d}{x} \\\\&\\le \\frac{1}{\\lambda ^{\\prime }}(1+\\lambda |y-z|)^{-k}\\int \\limits ^{{}}_{{{R}}} {(1+|x|)^{-k}}\\,\\mathrm {d}{x} \\\\&\\lesssim \\frac{1}{\\lambda ^{\\prime }}(1+\\lambda |y-z|)^{-k}.$ Combining eq:lemma372 and eq:lemma373 thus yields $&\\int \\limits ^{{}}_{{{R}}} {(1+|x-y|)^{-k}(1+\\lambda |x-z|)^{-k}}\\,\\mathrm {d}{x}\\\\&\\qquad = \\biggl (\\int \\limits _{H_y}+\\int \\limits _{H_z}\\biggr )(1+|x-y|)^{-k}(1+\\lambda |x-z|)^{-k}\\,\\mathrm {d}x \\lesssim \\frac{1}{\\lambda ^{\\prime }}(1+\\lambda |y-z|)^{-k}$ and together with eq:lemma371 this completes the proof.",
"Lemma 3.8 For all $y,z\\in {R}^\\ast $ , $\\lambda \\ne 0$ and $k>1$ we have $&\\int \\limits ^{{}}_{{{R}}} {(1+|x|)^{-k}(1+|x-y|)^{-k}(1+|\\lambda x-z|)^{-k}}\\,\\mathrm {d}{x} \\\\&\\hspace{28.45274pt} \\lesssim (1+|y|)^{-k}\\max \\lbrace 1,|\\lambda |\\rbrace ^{-1}\\\\&\\hspace{85.35826pt}\\cdot \\left[\\left(1+\\min \\lbrace 1,|\\lambda |\\rbrace \\left|y-\\frac{z}{\\lambda }\\right|\\right)^{-k}+\\left(1+\\min \\lbrace 1,|\\lambda |\\rbrace \\left|\\frac{z}{\\lambda }\\right|\\right)^{-k}\\right].$ We use the ideas of the proof of Lemma 11.1.1, Ref.",
"[13], as well as lemma:subconvolution1 and define the set $N_y:=\\lbrace x\\in {R}:|x-y|\\le \\frac{|y|}{2}\\rbrace $ .",
"For all $x\\in N_y$ it follows that $\\vert {x} \\vert \\ge \\frac{\\vert {y} \\vert }{2}$ and thus $(1+|x|)^{-k} \\le \\left(1+\\frac{|y|}{2}\\right)^{-k} \\le 2^k(1+|y|)^{-k}.$ On the other hand if $x\\in N_y^c$ one has $(1+|x-y|)^{-k}\\le (1+\\frac{|y|}{2})^{-k}$ .",
"Hence, with lemma:subconvolution1 we can derive $&\\int \\limits ^{{}}_{{{R}}} {(1+|x|)^{-k}(1+|x-y|)^{-k}(1+|\\lambda x-z|)^{-k}}\\,\\mathrm {d}{x} \\\\&\\hspace{28.45274pt} = \\biggl (\\int \\limits _{N_y}+\\int \\limits _{N_y^c}\\biggr )(1+|x|)^{-k}(1+|x-y|)^{-k}(1+|\\lambda x-z|)^{-k}\\,\\mathrm {d}x \\\\&\\hspace{28.45274pt} \\lesssim (1+|y|)^{-k}\\int \\limits ^{{}}_{{{R}}} {(1+|x-y|)^{-k}\\left(1+|\\lambda |\\left|x-\\frac{z}{\\lambda }\\right|\\right)^{-k}}\\,\\mathrm {d}{x} \\\\&\\hspace{85.35826pt}+ (1+|y|)^{-k}\\int \\limits ^{{}}_{{{R}}} {(1+|x|)^{-k}\\left(1+|\\lambda |\\left|x-\\frac{z}{\\lambda }\\right|\\right)^{-k}}\\,\\mathrm {d}{x} \\\\&\\hspace{28.45274pt} \\lesssim (1+|y|)^{-k}\\max \\lbrace 1,|\\lambda |\\rbrace ^{-1}\\left(1+\\min \\lbrace 1,|\\lambda |\\rbrace \\left|y-\\frac{z}{\\lambda }\\right|\\right)^{-k} \\\\&\\hspace{85.35826pt}+(1+|y|)^{-k}\\max \\lbrace 1,|\\lambda |\\rbrace ^{-1}\\left(1+\\min \\lbrace 1,|\\lambda |\\rbrace \\left|\\frac{z}{\\lambda }\\right|\\right)^{-k},$ which concludes the proof.",
"Now we are able to prove that the integrability condition on the kernel function is satisfied, i.e.",
"that $R_\\mathfrak {F}\\in \\mathcal {A}_{q,{m_{v_{r}}}}$ .",
"Theorem 3.9 Let $\\Psi \\in L_1({R}^d)\\cap L_2({R}^d)$ be an admissible shearlet with $ \\operatorname{supp}\\hat{\\Psi } \\subseteq ([-a_1,-a_0]\\cup [a_0,a_1])\\times Q_b.",
"$ Let $\\Phi \\in L_1({R}^d)\\cap L_2({R}^d)$ be chosen as in remark:choicePhi so that condition eq:assumptions is satisfied for $0 < a_0 < a_1$ and $b \\in {R}_+^{d-1}$ and additionally $\\hat{\\Phi }\\in {C}^\\infty _0({R}^d)$ .",
"Then, for every $q>1$ the kernel $R_\\mathfrak {F}$ fulfills $ R_\\mathfrak {F}\\in \\mathcal {A}_{q,{m_{v_r}}}.",
"$ For $q>1$ fixed we use lemma:kernel and look at the four summands in eq:kernel independently.",
"We need to show that all summands are bounded and for that we use lemma:param.",
"Let $\\tilde{\\alpha }:=\\alpha a$ and by using lemma:param list:parampsipsi with the specific weight $v_r$ we obtain $&\\operatornamewithlimits{ess\\,sup}_{(\\alpha ,\\sigma ,\\tau )\\in X}\\int \\limits _X\\vert {R_\\mathfrak {F}((\\alpha ,\\sigma ,\\tau ),(a,s,t))} \\vert ^q m_{v_r}(\\alpha ,a)^q\\,\\mathrm {d}\\mu (a,s,t) \\\\&\\hspace{14.22636pt} =\\max \\Biggl \\lbrace \\int \\limits _{{R}^{d}}\\int \\limits _{{R}^{d-1}}\\biggl (\\vert {\\langle {\\Phi },{\\psi _{(\\infty ,s,t)}} \\rangle } \\vert ^q+\\int \\limits _{-1}^1\\vert {a} \\vert ^{-rq}\\vert {\\langle {\\Phi },{\\psi _{(a,s,t)}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}\\biggr )\\,\\mathrm {d}s\\,\\mathrm {d}t, \\\\&\\hspace{71.13188pt} \\operatornamewithlimits{ess\\,sup}_{\\alpha \\in [-1,1]^\\ast }\\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}}\\biggl (\\vert {\\alpha } \\vert ^{-rq}\\vert {\\langle {\\Phi },{\\psi _{(\\alpha ,s,t)}} \\rangle } \\vert ^q \\\\&\\hspace{99.58464pt} +\\int \\limits _{-\\vert {\\alpha } \\vert ^{-1}}^{\\vert {\\alpha } \\vert ^{-1}}\\max \\left\\lbrace \\frac{\\vert {\\alpha } \\vert }{\\vert {\\tilde{\\alpha }} \\vert },\\frac{\\vert {\\tilde{\\alpha }} \\vert }{\\vert {\\alpha } \\vert }\\right\\rbrace ^{-rq}\\vert {\\langle {\\Psi },{\\psi _{(a,s,t)}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}\\biggr )\\,\\mathrm {d}s\\,\\mathrm {d}t\\Biggr \\rbrace .",
"$ We need to show that all four summands of eq:frameinaq13 are bounded and for this we will treat the summands independently.",
"First, since $\\mathcal {F}(f^\\ast )=\\overline{\\mathcal {F}(f)}$ and $\\Phi \\ast \\psi _{(a,s,0)}^\\ast \\in L_1({R}^d)$ we obtain $\\langle {\\Phi },{\\psi _{(a,s,t)}} \\rangle = (\\Phi \\ast \\psi _{(a,s,0)}^\\ast )(t) = \\mathcal {F}^{-1}(\\mathcal {F}(\\Phi \\ast \\psi _{(a,s,0)}^\\ast ))(t) = \\mathcal {F}^{-1}(\\hat{\\Phi }\\overline{\\mathcal {F}(\\psi _{(a,s,0)})})(t)$ which leads to $\\int \\limits _{R^d}\\vert {\\langle {\\Phi },{\\psi _{(a,s,t)}} \\rangle } \\vert ^q\\,\\mathrm {d}t = \\Vert {\\mathcal {F}^{-1}(\\hat{\\Phi }\\overline{\\mathcal {F}(\\psi _{(a,s,0)})})} \\vert {L_q} \\Vert ^q.$ Applying lemma:param list:paramphipsi we see that $\\hat{\\Phi }\\mathcal {F}(\\psi _{(a,s,0)})\\equiv 0$ for all $s\\notin Q_{d_2}$ or $a\\notin [-1,-\\frac{a_0}{a_1}]\\cup [\\frac{a_0}{a_1},1]$ , which implies $ \\Vert {\\mathcal {F}^{-1}(\\hat{\\Phi }\\overline{\\mathcal {F}(\\psi _{(a,s,0)})})} \\vert {L_q} \\Vert ^q=0 $ for all $s\\notin Q_{d_2}$ or $a\\notin [-1,-\\frac{a_0}{a_1}]\\cup [\\frac{a_0}{a_1},1]$ .",
"Thus, with lemma:param list:paramphipsi we derive $&\\operatornamewithlimits{ess\\,sup}_{\\alpha \\in [-1,1]^\\ast }\\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}}\\vert {\\alpha } \\vert ^{-rq}\\vert {\\langle {\\Phi },{\\psi _{(\\alpha ,s,t)}} \\rangle } \\vert ^q\\,\\mathrm {d}s\\,\\mathrm {d}t \\\\&\\hspace{56.9055pt} =\\operatornamewithlimits{ess\\,sup}_{\\alpha \\in [-1,1]^\\ast }\\vert {\\alpha } \\vert ^{-rq}\\int \\limits _{{R}^{d-1}}\\Vert {\\mathcal {F}^{-1}(\\hat{\\Phi }\\overline{\\mathcal {F}(\\psi _{(\\alpha ,s,0)})})} \\vert {L_q} \\Vert ^q\\,\\mathrm {d}s \\\\&\\hspace{56.9055pt} =\\operatornamewithlimits{ess\\,sup}_{\\alpha \\in [-1,-\\frac{a_0}{a_1}]\\cup [\\frac{a_0}{a_1},1]}\\vert {\\alpha } \\vert ^{-rq}\\int \\limits _{Q_{d_2}}\\Vert {\\Phi \\ast \\psi _{(\\alpha ,s,0)}^\\ast } \\vert {L_q} \\Vert ^q\\,\\mathrm {d}s < \\infty .$ Using the same arguments as well as lemma:param list:parampsipsi we obtain $&\\operatornamewithlimits{ess\\,sup}_{\\alpha \\in [-1,1]^\\ast }\\int \\limits _{{R}^d}\\int \\limits _{{R}^{d-1}}\\int \\limits _{-\\vert {\\alpha } \\vert ^{-1}}^{\\vert {\\alpha } \\vert ^{-1}}\\max \\left\\lbrace \\frac{\\vert {\\alpha } \\vert }{\\vert {\\tilde{\\alpha }} \\vert },\\frac{\\vert {\\tilde{\\alpha }} \\vert }{\\vert {\\alpha } \\vert }\\right\\rbrace ^{-rq}\\vert {\\langle {\\Psi },{\\psi _{(a,s,t)}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}\\,\\mathrm {d}s\\,\\mathrm {d}t \\\\& \\le \\int \\limits _{{R}}\\max \\left\\lbrace \\vert {a} \\vert ,\\vert {a} \\vert ^{-1}\\right\\rbrace ^{-rq}\\int \\limits _{{R}^{d-1}}\\Vert {\\mathcal {F}^{-1}(\\hat{\\Phi }\\overline{\\mathcal {F}(\\psi _{(a,s,0)})})} \\vert {L_q} \\Vert ^q\\,\\mathrm {d}s\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}} \\\\& = \\biggl (\\int \\limits _{-\\frac{a_1}{a_0}}^{-\\frac{a_0}{a_1}}+\\int \\limits _{\\frac{a_0}{a_1}}^{\\frac{a_1}{a_0}}\\biggr )\\max \\left\\lbrace \\vert {a} \\vert ,\\vert {a} \\vert ^{-1}\\right\\rbrace ^{-rq}\\int \\limits _{Q_{d_1}}\\Vert {\\Phi \\ast \\psi _{(a,s,0)}^\\ast } \\vert {L_q} \\Vert ^q\\,\\mathrm {d}s\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}} < \\infty .$ Again, with analogous arguments and lemma:param list:paramphipsi it follows that $&\\int \\limits _{{R}^{d}}\\int \\limits _{{R}^{d-1}}\\int \\limits _{-1}^1\\vert {a} \\vert ^{-rq}\\vert {\\langle {\\Phi },{\\psi _{(a,s,t)}} \\rangle } \\vert ^q\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}}\\,\\mathrm {d}s\\,\\mathrm {d}t \\\\&\\hspace{56.9055pt} = \\int \\limits _{-1}^1\\vert {a} \\vert ^{-rq}\\int \\limits _{{R}^{d-1}}\\Vert {\\mathcal {F}^{-1}(\\hat{\\Psi }\\overline{\\mathcal {F}(\\psi _{(a,s,0)})})} \\vert {L_q} \\Vert ^q\\,\\mathrm {d}s\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}} \\\\&\\hspace{56.9055pt} = \\biggl (\\int \\limits _{-1}^{-\\frac{a_0}{a_1}}+\\int \\limits _{\\frac{a_0}{a_1}}^1\\biggr )\\vert {a} \\vert ^{-rq}\\int \\limits _{Q_{d_2}}\\Vert {\\Psi \\ast \\psi _{(a,s,0)}^\\ast } \\vert {L_q} \\Vert ^q\\,\\mathrm {d}s\\,\\frac{\\mathrm {d}a}{\\vert {a} \\vert ^{d+1}} < \\infty .$ For the last summand in eq:frameinaq13 we choose $q_0$ , $q_1$ positive, such that $q_0+q_1=q$ .",
"We will specify the choice at the end of the proof.",
"Then, it follows that $&\\int \\limits _{{R}^{d}}\\int \\limits _{{R}^{d-1}}\\vert {\\langle {\\Phi },{\\psi _{(\\infty ,s,t)}} \\rangle } \\vert ^q\\,\\mathrm {d}s\\,\\mathrm {d}t \\\\&\\hspace{28.45274pt} = \\int \\limits _{{R}^{d-1}}\\int \\limits _{{R}^d}\\vert {(\\Phi \\ast \\psi ^\\ast _{(\\infty ,s,0)}(t)} \\vert ^{q_0+q_1}\\,\\mathrm {d}t\\,\\mathrm {d}s \\\\&\\hspace{28.45274pt} = \\int \\limits _{{R}^{d-1}}\\int \\limits _{{R}^d}\\vert {(\\Phi \\ast \\psi ^\\ast _{(\\infty ,s,0)})(t)} \\vert ^{q_0}\\vert {\\mathcal {F}^{-1}(\\hat{\\Phi }\\overline{\\mathcal {F}(\\psi _{(\\infty ,s,0)})})(t)} \\vert ^{q_1}\\,\\mathrm {d}t\\,\\mathrm {d}s \\\\&\\hspace{28.45274pt} \\lesssim \\int \\limits _{{R}^{d-1}}\\int \\limits _{{R}^d}\\biggl (\\int \\limits _{{R}^d}\\vert {\\Phi (x)\\psi ^\\ast _{(\\infty ,s,0)}(x-t)} \\vert \\,\\mathrm {d}x\\biggr )^{q_0}\\,\\mathrm {d}t\\,\\biggl (\\int \\limits _{{R}^d}\\vert {\\hat{\\Phi }(\\omega )\\mathcal {F}\\psi ^\\ast _{(\\infty ,s,0)}(\\omega )} \\vert \\,\\mathrm {d}\\omega \\biggr )^{q_1}\\,\\mathrm {d}s \\\\&\\hspace{28.45274pt} =: \\int \\limits _{{R}^{d-1}}I_0(s) I_1(s)\\,\\mathrm {d}s.$ In the following we will treat both factors $I_0$ and $I_1$ independently.",
"$I_0(s)$ : We assume in the following $0<q_0<1$ .",
"Since $\\hat{\\Phi }\\in {C}_c^\\infty ({R}^d)$ , for every $k\\in {N}$ it follows that $|\\Phi (x)|\\lesssim (1+|x|)^{-k}$ for all $x\\in {R}^d$ with the constant depending on $k$ and $d$ .",
"Then, $I_0(s) &\\lesssim \\int \\limits _{{R}^d}\\biggl (\\int \\limits _{{R}^d}\\prod _{i=1}^d\\left[(1+|x_i+t_i|)^{-k}(1+|(S_{-s}x)_i|)^{-k}\\right]\\,\\mathrm {d}x\\biggr )\\,\\mathrm {d}t =: \\int \\limits _{{R}^3} I_s(t)^{q_0} \\,\\mathrm {d}t$ for $s\\in {R}^{d-1}$ fixed and where $(S_{-s}x)_i$ denotes the $i$ -th entry of the vector $S_{-s}x\\in {R}^d$ .",
"With this notation we intend to show $\\int \\limits _{{R}^3}I_s(t)^{q_0}\\,\\mathrm {d}t \\lesssim (1+\\Vert s\\Vert )^{1-q_0}\\int \\limits _{{R}^d}\\prod _{i=1}^d(1+|t_i|)^{-kq_0}\\,\\mathrm {d}t$ with the constant depending on $k$ and $q_0$ only.",
"For this we first show an auxiliary result for $d=3$ which we will then generalize to arbitrary dimensions.",
"To illustrate our method we differentiate between the following four cases for $s\\in {R}^2$ with $s_1,s_2\\ne 0$ .",
"Case 1: $|s_1|,|s_2|\\le 1$ .",
"With lemma:subconvolution1 and lemma:subconvolution2 we obtain $I_s(t) &\\lesssim \\int \\limits _{{R}^2}(1+|t_1+s_1x_2+s_2x_3|)^{-k}(1+|x_2+t_2|)^{-k}\\\\&\\qquad \\qquad \\cdot (1+|x_2|)^{-k}(1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}(x_2,x_3) \\\\&\\lesssim \\int \\limits _{{R}}(1+|t_2|)^{-k}(1+|-s_1t_2+t_1+s_2x_3|)^{-k}(1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}x_3 \\\\&\\hspace{28.45274pt} + \\int \\limits _{{R}}(1+|t_2|)^{-k}(1+|t_1+s_2x_3|)^{-k}(1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}x_3 \\\\&\\lesssim (1+|t_2|)^{-k}(1+|t_3|)^{-k}\\big [(1+|s_2t_3+s_1t_2+t_1|)^{-k} \\\\&\\hspace{28.45274pt}+(1+|s_1t_2-t_1|)^{-k}+(1+|s_2t_3+t_1|)^{-k}+(1+|t_1|)^{-k}\\big ].$ Case 2: $|s_1|\\le 1,|s_2|>1$ .",
"Again, with lemma:subconvolution1 and lemma:subconvolution2 we obtain $I_s(t) &\\lesssim \\int \\limits _{{R}}(1+|t_2|)^{-k}(1+|-s_1t_2+t_1+s_2x_3|)^{-k}(1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}x_3 \\\\&\\hspace{28.45274pt} + \\int \\limits _{{R}}(1+|t_2|)^{-k}(1+|t_1+s_2x_3|)^{-k}(1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}x_3 \\\\&\\lesssim |s_2|^{-1}(1+|t_2|)^{-k}(1+|t_3|)^{-k}\\big [(1+|-t_3+s_1s_2^{-1}t_2-s_2^{-1}t_1|)^{-k} \\\\&\\hspace{28.45274pt} +(1+|s_1s_2^{-1}t_2-s_2^{-1}t_1|)^{-k}+(1+|t_3+s_2^{-1}t_1|)^{-k}+(1+|s_2^{-1}t_1|)^{-k}\\big ].",
"$ Case 3: $|s_1|>1,|s_2|\\le |s_1|$ .",
"Similarily we apply lemma:subconvolution1 and lemma:subconvolution2 to derive $I_s(t) &\\lesssim \\int \\limits _{{R}^2}(1+|t_1+s_1x_2+s_2x_3|)^{-k}(1+|x_2+t_2|)^{-k}\\\\&\\qquad \\qquad \\qquad \\qquad \\cdot (1+|x_2|)^{-k}(1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}(x_2,x_3) \\\\&\\lesssim \\int \\limits _{{R}}(1+|t_2|)^{-k}|s_1|^{-1}(1+|-t_2+s_1^{-1}t_1+s_1^{-1}s_2x_3|)^{-k}\\\\&\\qquad \\qquad \\qquad \\qquad \\cdot (1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}x_3 \\\\&\\hspace{-14.22636pt} +\\int \\limits _{{R}}(1+|t_2|)^{-k}|s_1|^{-1}(1+|s_1^{-1}t_1+s_1^{-1}s_2x_3|)^{-k}(1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}x_3 \\\\&\\lesssim |s_1|^{-1}(1+|t_2|)^{-k}(1+|t_3|)^{-k}\\big [(1+|-s_1^{-1}s_2t_3-t_2+s_1^{-1}t_1|)^{-k} \\\\&\\hspace{14.22636pt} +(1+|-t_2+s_1^{-1}t_1|)^{-k}+(1+|-s_1^{-1}s_2t_3+s_1^{-1}t_1|)^{-k}+(1+|s_1^{-1}t_1|)^{-k}\\big ].$ Case 4: $|s_1|>1,|s_2|>|s_1|$ .",
"Finally we apply lemma:subconvolution1 and lemma:subconvolution2 again and conclude $I_s(t) &\\lesssim \\int \\limits _{{R}}(1+|t_2|)^{-k}|s_1|^{-1}(1+|-t_2+s_1^{-1}t_1+s_1^{-1}s_2x_3|)^{-k}\\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\cdot (1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}x_3 \\\\&\\hspace{-14.22636pt} +\\int \\limits _{{R}}(1+|t_2|)^{-k}|s_1|^{-1}(1+|s_1^{-1}t_1+s_1^{-1}s_2x_3|)^{-k}(1+|x_3+t_3|)^{-k}(1+|x_3|)^{-k}\\,\\mathrm {d}x_3 \\\\&\\lesssim |s_2|^{-1}(1+|t_2|)^{-k}(1+|t_3|)^{-k}\\big [(1+|-t_3-s_1s_2^{-1}t_2+s_2^{-1}t_1|)^{-k} \\\\&\\hspace{14.22636pt} +(1+|-s_1s_2^{-1}t_2+s_2^{-1}t_1|)^{-k}+(1+|-t_3+s_2^{-1}t_1|)^{-k}+(1+|s_2^{-1}t_1|)^{-k}\\big ].",
"$ The four cases eq:frameinaq6, eq:frameinaq7, eq:frameinaq8, eq:frameinaq9 yield the estimate $I_s(t) \\lesssim \\vert {\\det A_s^i} \\vert \\sum _{i=1}^4\\prod _{j=1}^3(1+\\vert {(A_s^i t)_j} \\vert )^{-k}$ with the Matrices $A_s^i$ , $s\\in {R}^2$ , $i=1,\\ldots ,4$ , being of the form $A_s^i =\\begin{pmatrix}\\lambda & \\mu & \\nu \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{pmatrix}\\quad \\mbox{for some } \\lambda ,\\mu ,\\nu \\in {R}\\mbox{ depending on } s_1,s_2.$ In particular it follows from the four cases that $\\vert {\\det A_s^i} \\vert = |\\lambda | = \\left\\lbrace \\begin{array}{ll}1, & |s_1|,|s_2|\\le 1, \\\\|s_2|^{-1}, & |s_1|\\le 1,|s_2|>1, \\\\|s_1|^{-1}, & |s_1|>1,|s_2|\\le |s_1|, \\\\|s_2|^{-1}, & |s_1|>1,|s_2|>|s_1|\\end{array}\\right\\rbrace = \\max \\lbrace 1,|s_1|,|s_2|\\rbrace ^{-1}.$ We now intend to show, that this result holds for arbitrary dimension.",
"To this extend we fix $d\\ge 3$ as well as $s\\in {R}^{d-1}$ with $s_i\\ne 0$ for all $i=1,\\ldots ,d-1$ and assume that there exist matrices $A_s^i$ for $1\\le i\\le 2^{d-1}$ of the form $A_s^i=\\begin{pmatrix}\\ast & \\ast & \\cdots & \\ast \\\\& 1 & & \\\\& & \\ddots & \\\\& & & 1\\end{pmatrix}$ with $\\det A_s^i = (A_s^i)_{11}=\\max \\lbrace 1,|s_1|,\\ldots ,|s_{d-1}|\\rbrace ^{-1}=\\min \\lbrace 1,|s_1|^{-1},\\ldots ,|s_{d-1}|^{-1}\\rbrace =:\\min (s)$ .",
"Assume the estimate $I_s(t) \\lesssim \\min (s)\\sum _{i=1}^{2^{d-1}}\\prod _{j=1}^d(1+|(A^i_s t)_j|)^{-k}$ holds true for fixed $d$ .",
"As shown in (REF ), this readily is the case for $d=3$ .",
"We now intend to show that the estimate (REF ) also holds for $d+1$ .",
"Then, (REF ) will hold for arbitrary dimension by full induction over the dimension.",
"To this end we fix $s\\in {R}^d$ with $s_i\\ne 0$ for all $i=1,\\ldots ,d$ and define $\\tilde{x}:=(x_1,\\ldots ,x_d)$ , $\\tilde{s}:=(s_1,\\ldots ,s_{d-1})$ , $\\tilde{t}:=(t_1,\\ldots ,t_d)$ and $u:=(t_1+s_d x_{d+1},t_2,\\ldots ,t_d)$ .",
"Then we deduce from (REF ) the estimate $I_s(t) &= \\int \\limits _{{R}^{d+1}}\\prod _{i=1}^{d+1}\\left[(1+|x_i+t_i|)^{-k}(1+|(S_{-s}x)_i|)^{-k}\\right]\\,\\mathrm {d}x \\\\&= \\int \\limits _{R}(1+|x_{d+1}+t_{d+1}|)^{-k}(1+|x_{d+1}|)^{-k}\\\\&\\hspace{56.9055pt}\\biggl (\\int \\limits _{R^d}\\prod _{i=1}^d\\left[(1+|x_i+u_i|)^{-k}(1+|(S_{-\\tilde{s}}\\tilde{x})_i|)^{-k}\\right]\\,\\mathrm {d}\\tilde{x}\\biggr )\\,\\mathrm {d}x_{d+1} \\\\&= \\int \\limits _{R}I_{\\tilde{s}}(u)(1+|x_{d+1}+t_{d+1}|)^{-k}(1+|x_{d+1}|)^{-k}\\,\\mathrm {d}x_{d+1} \\\\&\\lesssim \\min (\\tilde{s})\\sum _{i=1}^{2^{d-1}}\\prod _{j=1}^d\\int \\limits _{R}(1+|(A^i_{\\tilde{s}}u)_j|)^{-k}(1+|x_{d+1}+t_{d+1}|)^{-k}(1+|x_{d+1}|)^{-k}\\,\\mathrm {d}x_{d+1}, $ whereby we remember $(S_{-s}x)_i=x_i$ for all $i=2,\\ldots ,d+1$ and $(A^i_{\\tilde{s}}u)_j=u_j$ for all $j=2,\\ldots ,d$ .",
"Since all integrals for $j\\ne 1$ will remain unchanged we are now interested in the integrals in (REF ) for arbitrary $1\\le i\\le 2^{d-1}$ , $j=1$ and obtain with lemma:subconvolution2 $&\\int \\limits _{R}(1+|(A^i_{\\tilde{s}}u)_1|)^{-k}(1+|x_{d+1}+t_{d+1}|)^{-k}(1+|x_{d+1}|)^{-k}\\,\\mathrm {d}x_{d+1} \\\\&= \\int \\limits _{R}(1+|(A^i_{\\tilde{s}}\\tilde{t})_1+\\min (\\tilde{s})s_d x_{d+1}|)^{-k}(1+|x_{d+1}+t_{d+1}|)^{-k}(1+|x_{d+1}|)^{-k}\\,\\mathrm {d}x_{d+1} \\\\&\\lesssim \\min (\\tilde{s})(1+|t_{d+1}|)^{-k}\\max \\lbrace 1,|\\min (\\tilde{s})s_d|\\rbrace ^{-1} \\\\&\\hspace{56.9055pt}\\times \\biggl [ \\left( 1+ \\left| \\min \\lbrace 1,|\\min (\\tilde{s})s_d|\\rbrace t_{d+1} - \\frac{\\min \\lbrace 1,|\\min (\\tilde{s})s_d|\\rbrace }{|\\min (\\tilde{s})s_d|}(A_{\\tilde{s}}^i \\tilde{t})_1\\right|\\right)^{-k} \\\\&\\hspace{113.81102pt} +\\left(1+\\frac{\\min \\lbrace 1,|\\min (\\tilde{s})s_d|\\rbrace }{|\\min (\\tilde{s})s_d|}|(A_{\\tilde{s}}^i \\tilde{t})_1|\\right)^{-k}\\,\\biggr ] \\\\&= \\max \\lbrace \\min (\\tilde{s})^{-1},|s_d|\\rbrace ^{-1}\\\\&\\hspace{28.45274pt}\\cdot \\left[(1+|(B^i_s t)_{d+1}|)^{-k}(1+|(B^i_s t)_1|)^{-k}+(1+|(C^i_s t)_{d+1}|)^{-k}(1+|(C^i_s t)_1|)^{-k}\\right]$ for some matrices $B_s^i$ , $C_s^i$ of the form (REF ) where $(B_s^i)_{11}=(C_s^i)_{11}=(A_{\\tilde{s}}^i)_{11}\\left(\\frac{\\min \\lbrace 1,|\\min (\\tilde{s})s_d|\\rbrace }{|\\min (\\tilde{s})s_d|}\\right)=\\min \\lbrace |s_d|^{-1},\\min (\\tilde{s})\\rbrace =\\min (s).$ Since $\\max \\lbrace \\min (\\tilde{s})^{-1},|s_d|\\rbrace ^{-1}=\\max \\lbrace 1,|s_1|,\\ldots ,|s_d|\\rbrace ^{-1}=\\min (s)$ we derive together with (REF ) the estimate (REF ) for $d+1$ .",
"Hence, (REF ) holds true for arbitrary dimension.",
"With this at hand we return to arbitrary dimension $d$ and further deduce $\\vert {\\det A_s^i} \\vert ^{-1} = \\max \\lbrace 1,|s_1|,\\ldots ,|s_{d-1}|\\rbrace \\le 1+\\max \\lbrace |s_1|,\\ldots ,|s_{d-1}|\\rbrace \\lesssim 1+\\Vert s\\Vert .$ Now we can prove the following estimate for $0<q_0<1$ and almost every $s\\in {R}^d$ : $\\int \\limits _{{R}^3}I_s(t)^{q_0}\\,\\mathrm {d}t &\\lesssim \\vert {\\det A_s^i} \\vert ^{q_0}\\int \\limits _{{R}^d}\\bigg (\\sum _{i=1}^{2^{d-1}}\\prod _{j=1}^d(1+\\vert {(A_s^i t)_j} \\vert )^{-k}\\bigg )^{q_0}\\,\\mathrm {d}t \\\\&\\le \\vert {\\det A_s^i} \\vert ^{q_0}\\sum _{i=1}^{2^{d-1}}\\int \\limits _{{R}^d}\\prod _{j=1}^d(1+|(A_s^i t)_j|)^{-kq_0}\\,\\mathrm {d}t \\\\&\\lesssim \\vert {\\det A_s^i} \\vert ^{q_0-1}\\int \\limits _{{R}^d}\\prod _{j=1}^d(1+|t_j|)^{-kq_0}\\,\\mathrm {d}t \\\\&\\lesssim (1+\\Vert s\\Vert )^{1-q_0}\\int \\limits _{{R}^d}\\prod _{j=1}^d(1+|t_j|)^{-kq_0}\\,\\mathrm {d}t,$ which shows eq:frameinaq5.",
"$I_1(s)$ : We shall now deal with the second factor in eq:frameinaq3 for $q_1>0$ .",
"By lemma:param list:paramphiphi and the definition of $\\hat{\\Phi }$ we obtain $I_1(s)^{1/q_1} &= \\int \\limits _{{R}^d}\\vert {\\hat{\\Phi }(\\omega )\\hat{\\Phi }(S_s^T\\omega )} \\vert \\,\\mathrm {d}\\omega \\\\& \\le \\int \\limits _{\\Omega _s}\\vert {\\omega _1} \\vert ^{d-1}\\biggl (\\int \\limits ^{{}}_{{{R}}} {\\frac{\\vert {\\hat{\\Psi }(\\xi _1,\\tilde{\\omega })} \\vert ^2}{\\vert {\\xi _1} \\vert ^d}}\\,\\mathrm {d}{\\xi _1}\\biggr )^\\frac{1}{2} \\biggl (\\int \\limits ^{{}}_{{{R}}} {\\frac{\\vert {\\hat{\\Psi }(\\xi _1,\\widetilde{S_s^T\\omega })} \\vert ^2}{\\vert {\\xi _1} \\vert ^d}}\\,\\mathrm {d}{\\xi _1}\\biggr )^\\frac{1}{2}\\,\\mathrm {d}\\omega $ with $\\Omega _s=\\lbrace x\\in {R}^d:|x_1|\\le a_1,\\max \\lbrace -b_i,-b_i-s_{i-1}x_1\\rbrace \\le x_i\\le \\min \\lbrace b_i,b_i-s_{i-1}x_1\\rbrace ,i=2,\\ldots d\\rbrace $ .",
"Since $\\hat{\\Psi }$ is compactly supported and continuous, we conclude $I_1(s)^{1/q_1} \\lesssim \\int \\limits ^{{}}_{{\\Omega _s}} {|\\omega _1|^{d-1}}\\,\\mathrm {d}{\\omega }.$ In the following we assume $s>0$ componentwise, all other cases can be treated analogously by symmetry arguments.",
"Then, for any $\\omega \\in \\Omega _s$ it follows from lemma:param list:paramphiphi that $|\\omega _1|\\le 2b_is_{i-1}^{-1}$ for all $i=2,\\ldots ,d$ , hence, $|\\omega _1|\\lesssim (\\max _{i=1,\\ldots ,d-1}s_i)^{-1}=|s|_\\infty ^{-1}$.",
"Moreover, since $\\omega \\in \\operatorname{supp}\\hat{\\Phi }$ , we derive $-b_i\\le \\omega _i\\le b_i$ for all $i=1,\\ldots ,d$ .",
"We can now estimate eq:frameinaq10 in the following manner: $I_1(s)^{1/q_1} \\lesssim \\int \\limits _{|\\omega _1|\\le \\min \\lbrace b_1,|s|_\\infty ^{-1}\\rbrace }|\\omega _1|^{d-1}\\,\\mathrm {d}\\omega _1.$ Assume first that $|s|_\\infty ^{-1}\\ge b_1$ , then we have $I_1(s)^{1/q_1} \\lesssim \\int \\limits _{|\\omega _1|\\le b_1}|\\omega _1|^{d-1}\\,\\mathrm {d}\\omega _1 \\lesssim b_1^d \\lesssim (1+\\Vert s\\Vert )^{-d}.$ On the other hand if $|s|_\\infty ^{-1}<b_1$ it follows that $I_1(s)^{1/q_1} \\lesssim \\int \\limits _{|\\omega _1|\\le |s|_\\infty ^{-1}}|\\omega _1|^{d-1}\\,\\mathrm {d}\\omega _1 \\lesssim |s|_\\infty ^{-d} \\lesssim (1+\\Vert s\\Vert )^{-d}.$ In both cases we obtain $I_1(s) \\lesssim (1+\\Vert s\\Vert )^{-dq_1}.$ Plugging eq:frameinaq5 and eq:frameinaq11 into eq:frameinaq4 now yields $\\int \\limits _{{R}^{d}}\\int \\limits _{{R}^{d-1}}\\vert {\\langle {\\Phi },{\\psi _{(\\infty ,s,t)}} \\rangle } \\vert ^q\\,\\mathrm {d}s\\,\\mathrm {d}t &\\lesssim \\int \\limits _{{R}^{d-1}}I_0(s) I_1(s)\\,\\mathrm {d}s \\\\&\\lesssim \\int \\limits _{{R}^d}\\prod _{i=1}^d(1+|t_i|)^{-kq_0}\\,\\mathrm {d}t\\int \\limits _{{R}^{d-1}}(1+\\Vert s\\Vert )^{1-q_0-dq_1}\\,\\mathrm {d}s.$ For any choice of $q_0$ we can find a $k\\in {N}$ , such that the first integral in eq:frameinaq12 converges.",
"The second integral in eq:frameinaq12 is known to converge if and only if $q_0+dq_1>d$ .",
"This can be obtained by setting $q_0=\\frac{q-1}{d}$ and $q_1=\\frac{d-1}{d}q+\\frac{1}{d}$ .",
"If $q>1$ this satisfies $ q_0 + q_1 = \\frac{q-1}{d} + \\frac{d-1}{d} q + \\frac{1}{d} = \\frac{1}{d} (q-1+q(d-1)+1) = q $ and $q_0 + d q_1 &= \\frac{q-1}{d} + (d-1)q + 1 = 1 + q\\left(d-1+\\frac{1}{d}\\right)-\\frac{1}{d}\\\\&= d - \\left( d-1+\\frac{1}{d}\\right) + q \\left(d-1+\\frac{1}{d}\\right) = d + (q-1)\\left( d-1+\\frac{1}{d}\\right) > d$ and we finally conclude $\\int \\limits _{{R}^{d}}\\int \\limits _{{R}^{d-1}}\\vert {\\langle {\\Phi },{\\psi _{(\\infty ,s,t)}} \\rangle } \\vert ^q\\,\\mathrm {d}s\\,\\mathrm {d}t < \\infty .$ Altogether with eq:frameinaq1, eq:frameinaq2 and eq:frameinaq3 we have now shown that all four summands in eq:frameinaq13 are bounded and this concludes the proof.",
"At this point we intend to show that there exist functions $\\hat{\\Phi }$ satisfying the assumptions of theorem:frameinaq.",
"Indeed we will show that we can find $\\hat{\\Psi }$ so that $\\hat{\\Phi }\\in {C}_0^\\infty ({R}^d)$ .",
"Example 3.1 We fix any odd dimension $d$ .",
"Then, for $\\xi =(\\xi _1,\\tilde{\\xi })$ let $\\hat{\\Psi }(\\xi ):= \\hat{\\psi _1}(\\xi _1)\\hat{\\psi _2}(\\tilde{\\xi })$ with $ \\hat{\\psi _1}(\\xi _1) := {\\left\\lbrace \\begin{array}{ll}\\vert {\\xi _1} \\vert ^{\\frac{d}{2}}e^{\\frac{1}{(\\xi _1 - 1)(\\xi _1 - 3)}}, & 1 < \\xi _1 < 3\\\\\\vert {\\xi _1} \\vert ^\\frac{d}{2}e^{\\frac{1}{(\\xi _1 + 1)(\\xi _1 + 3)}}, & -3 < \\xi _1 < -1\\\\0, &\\text{otherwise}\\end{array}\\right.}",
"$ and $\\hat{\\psi _2} \\in {C}^\\infty _0({R}^{d-1})$ with $\\hat{\\psi }\\ge 0$ .",
"According to remark:choicePhi we set $\\hat{\\Phi }(\\xi ) &:= \\xi _1^{\\frac{d-1}{2}}\\biggl (\\int \\limits _{{R}\\setminus [-\\vert {\\xi _1} \\vert ,\\vert {\\xi _1} \\vert ]}\\frac{\\vert {\\hat{\\Psi }(\\omega _1,\\tilde{\\xi })} \\vert ^2}{\\vert {\\omega _1} \\vert ^d}\\,\\mathrm {d}\\omega _1\\biggr )^{1/2} \\\\&= \\xi _1^\\frac{d-1}{2}\\vert {\\hat{\\psi }_2(\\tilde{\\xi })} \\vert \\biggl (2\\int \\limits _{\\max \\lbrace \\vert {\\xi _1} \\vert ,1\\rbrace }^3e^{\\frac{2}{(\\omega _1 - 1)(\\omega _1 - 3)}}\\,\\mathrm {d}\\omega _1\\biggr )^{1/2} =: \\xi _1^\\frac{d-1}{2}\\vert {\\hat{\\psi }_2(\\tilde{\\xi })} \\vert \\hat{\\varphi }_1(\\xi _1)$ with $\\hat{\\Phi }(\\xi )=0$ for $\\vert {\\xi _1} \\vert >3$ .",
"Now we show that this function satisfies the required assumptions.",
"The fact that $\\hat{\\psi _1} \\in {C}^\\infty _0({R})$ and therefore $\\hat{\\Psi } \\in {C}^\\infty _0({R}^d)$ is immediately obvious.",
"With the given construction, together with remark:choicePhi, we see that the necessary condition from theorem:tightframe is satisfied, i.e.",
"the functions $\\Phi $ and $\\Psi $ constitute a Parseval frame.",
"Furthermore if we assume $\\hat{\\Phi }\\in {C}^\\infty _0({R}^d)\\subset {S}({R}^d)$ then $\\Phi \\in {S}({R}^d)\\subset L_1({R}^d)\\cap L_2({R}^d)$ and all necessary conditions on $\\Phi $ are satisfied.",
"So we need to show that $\\hat{\\Phi } \\in {C}^\\infty _0({R}^d)$ , which means that we will show that $\\hat{\\varphi _1}$ is infinitely continuously differentiable since $\\xi _1^\\frac{d-1}{2}$ is a monomial.",
"To show this we need to prove that $ \\lim \\limits _{x\\nearrow 3} \\frac{\\mathrm {d}^n}{\\mathrm {d}x^n} (\\hat{\\varphi _1}(x)) = 0 $ and $ \\lim \\limits _{x\\searrow 1} \\frac{\\mathrm {d}^n}{\\mathrm {d}x^n} (\\hat{\\varphi _1}(x)) = 0 $ for all $n\\in {N}$ .",
"Since both statements are proven in an analogous manner, we will only show the proof of the first statement and for the remainder of this example we assume $2 < x < 3$ .",
"Since we have $\\hat{\\varphi _1}(x) = (f\\circ g)(x)$ with $f(x) = \\sqrt{x}$ and $ g(x) = 2 \\int \\limits _{x}^3 e^{\\frac{2}{(\\omega - 1)(\\omega - 3)}}\\,\\mathrm {d}{\\omega }, $ we can use Faà di Bruno's formula to get a closed expression for the n-th derivative.",
"Recall that for two functions $f$ and $g$ the identity $\\frac{\\mathrm {d}^n}{\\mathrm {d}x^n}\\bigl ((f\\circ g)(x)\\bigr ) = \\sum \\limits _{k=1}^n \\frac{\\mathrm {d}^k f}{\\mathrm {d}x^k}(g(x)) B_{n,k}\\Bigl (\\frac{\\mathrm {d}g}{\\mathrm {d}x}(x),\\frac{\\mathrm {d}^2 g}{\\mathrm {d}x^2}(x),\\ldots ,\\frac{\\mathrm {d}^{(n-k+1)} g}{\\mathrm {d}x^{(n-k+1)}}(x)\\Bigr )$ holds with $B_{n,k}$ being the Bell polynomials, i.e.",
"$ B_{n,k}(x_1,x_2,\\ldots ,x_{(n-k+1)}) = \\sum \\frac{n!",
"}{j_1!\\cdots j_{(n-k+1)}!}",
"\\Bigl (\\frac{x_1}{1!",
"}\\Bigr )^{j_1}\\cdots \\Bigl (\\frac{x_{(n-k+1)}}{(n-k+1)!",
"}\\Bigr )^{j_{(n-k+1)}}.",
"$ The sum in the above expression is taken over all $(j_1,\\ldots ,j_{(n-k+1)})$ with $j_1+\\cdots +j_{(n-k+1)} = k$ and $j_1 + 2j_2 + \\cdots + (n-k+1) j_{(n-k+1)} = n$ .",
"The derivatives of the square root satisfy $\\frac{\\mathrm {d}^k f}{\\mathrm {d}x^k}(x) = c_k x^{-k+\\frac{1}{2}}$ with $c_k$ being some constant and since because of $1<x<3$ we have $\\frac{\\mathrm {d}g}{\\mathrm {d}x}(x) = -2 e^{\\frac{2}{(x-1)(x-3)}}$ this means that for all $k \\in {N}$ the derivatives of $g$ satisfy $ \\frac{\\mathrm {d}^k g}{\\mathrm {d}x^k}(x) = Q_k(x) e^{\\frac{2}{(x-1)(x-3)}} $ with $Q_k$ being some rational function without singularities in the interval $(1,3)$ .",
"Thus, using eq:faadibruno we now have $\\frac{\\mathrm {d}^n \\hat{\\varphi _1}}{\\mathrm {d}x^n}(x) &= \\sum \\limits _{k=1}^n c_k \\bigl (g(x)\\bigr )^{-k+\\frac{1}{2}} \\sum \\limits _{(j_1,\\ldots ,j_{(n-k+1)})} c_{n,k,j} \\Bigl ( Q_1(x)e^{\\frac{2}{(x-1)(x-3)}} \\Bigr )^{j_1} \\cdots \\\\&\\hspace{156.49014pt}\\cdots \\Bigl ( Q_{(n-k+1)}(x)e^{\\frac{2}{(x-1)(x-3)}} \\Bigr )^{j_{(n-k+1)}}\\\\&= \\sum \\limits _{k=1}^n R_{k,n}(x) \\bigl (g(x)\\bigr )^{-k+\\frac{1}{2}} \\bigl ( e^{\\frac{2}{(x-1)(x-3)}} \\bigr )^{k}\\\\&= \\sum \\limits _{k=1}^n \\biggl (\\frac{\\tilde{R}_{k,n}(x) \\bigl (e^{\\frac{2}{(x-1)(x-3)}}\\bigr )^{1+\\frac{1}{2k-1}}}{g(x)}\\biggr )^{k-\\frac{1}{2}}$ where $R_{k,n}$ is a rational function for every $k=1,\\ldots ,n$ possibly changing from line to line and $\\tilde{R}_{k,n}(x):= R_{k,n}(x)^{\\frac{1}{k-\\frac{1}{2}}}$ .",
"Since $ \\lim \\limits _{x\\nearrow 3} \\tilde{R}_{k,n}(x) \\bigl ( e^{\\frac{2}{(x-1)(x-3)}} \\bigr )^{1+\\frac{1}{2k-1}} = 0\\quad \\text{and}\\quad \\lim \\limits _{x\\nearrow 3} g(x) = 0 $ we use l'Hospital's rule to determine the limit of the fraction.",
"For the derivative of the numerator we obtain ${\\frac{\\mathrm {d}}{\\mathrm {d}x}\\Bigl (\\tilde{R}_{k,n}(x) \\bigl ( e^{\\frac{2}{(x-1)(x-3)}} \\bigr )^{1+\\frac{1}{2k-1}}\\Bigr )}\\\\&\\hspace{28.45274pt}= \\frac{\\mathrm {d}}{\\mathrm {d}x} \\tilde{R}_{k,n}(x) \\bigl ( e^{\\frac{2}{(x-1)(x-3)}} \\bigl )^{1+\\frac{1}{2k-1}} + \\tilde{R}_{k,n}(x) \\frac{\\mathrm {d}}{\\mathrm {d}x} \\bigl ( e^{\\frac{2}{(x-1)(x-3)}} \\bigr )^{1+\\frac{1}{2k-1}}\\\\&\\hspace{28.45274pt}= Q(x) \\bigl ( e^{\\frac{2}{(x-1)(x-3)}} \\bigr )^{1+\\frac{1}{2k-1}}$ where $Q$ is of the form $Q(x) = Q_2(x) (Q_1(x))^{\\frac{-2k+3}{2k-1}} + Q_3(x)(Q_1(x))^{\\frac{2}{2k-1}}$ with $Q_1,Q_2,Q_3$ being rational functions.",
"This, together with eq:derivg, yields $\\lim \\limits _{x\\nearrow 3} \\frac{\\frac{\\mathrm {d}}{\\mathrm {d}x}\\Bigl ( \\tilde{R}_{k,n}(x) \\bigl ( e^{\\frac{2}{(x-1)(x-3)}} \\bigr )^{1+\\frac{1}{2k-1}} \\Bigr )}{\\frac{\\mathrm {d}}{\\mathrm {d}x}\\bigl (g(x)\\bigr )} = \\lim \\limits _{x\\nearrow 3} Q(x) e^{\\frac{2}{(2k-1)((x-1)(x-3))}} = 0.$ Thus, with l'Hospital's rule we get $\\lim \\limits _{x\\nearrow 3} \\frac{\\mathrm {d}^n \\hat{\\varphi _1}}{\\mathrm {d}x^n}(x) &= \\lim \\limits _{x\\nearrow 3} \\sum \\limits _{k=1}^n \\biggl (\\frac{\\tilde{R}_{k,n}(x) \\bigl (e^{\\frac{2}{(x-1)(x-3)}}\\bigr )^{1+\\frac{1}{2k-1}}}{g(x)}\\biggr )^{k-\\frac{1}{2}}\\\\&= \\sum \\limits _{k=1}^n \\biggl ( \\lim \\limits _{x\\nearrow 3} \\frac{\\tilde{R}_{k,n}(x) \\bigl (e^{\\frac{2}{(x-1)(x-3)}}\\bigr )^{1+\\frac{1}{2k-1}}}{g(x)}\\biggr )^{k-\\frac{1}{2}}\\\\&= \\sum \\limits _{k=1}^n \\biggl ( \\lim \\limits _{x\\nearrow 3} \\frac{\\frac{\\mathrm {d}}{\\mathrm {d}x}\\Bigl (\\tilde{R}_{k,n}(x) \\bigl (e^{\\frac{2}{(x-1)(x-3)}}\\bigr )^{1+\\frac{1}{2k-1}}\\Bigr )}{\\frac{\\mathrm {d}}{\\mathrm {d}x}\\bigl (g(x)\\bigr )}\\biggr )^{k-\\frac{1}{2}} = 0.$ This proves that $\\hat{\\varphi _1} \\in {C}^{\\infty }_0({R})$ and therefore that $\\hat{\\Phi } \\in {C}^{\\infty }_0({R}^d)$ ."
],
[
"Inhomogeneous shearlet coorbit spaces",
"Now we are able to give a definition of the coorbit spaces associated to our inhomogeneous shearlet frame with respect to the weighted Lebesgue spaces $L_{p,v_r}(X,\\mu )$ .",
"Definition 3.5 Let the shearlet frame $\\mathfrak {F}$ be chosen so that it satisfies the conditions in theorem:frameinaq.",
"Then for $1\\le p<\\infty $ and $1<\\tau \\le 2$ with $p<\\tau ^{\\prime }$ the shearlet coorbit space with respect to the Lebesgue space $L_{p,v_{r}}(X,\\mu )$ is defined as $ \\mathcal {SC}^{r}_{\\mathfrak {F},\\tau ,p} := \\mathrm {Co}_{\\mathfrak {F},\\tau }(L_{p,v_r}(X,\\mu )) = \\lbrace f \\in (\\mathcal {H}_{\\tau ,v_{r}})^\\sim : \\mathcal {SH}_{\\mathfrak {F},\\tau } f \\in L_{p,v_{r}}(X,\\mu ) \\rbrace .",
"$ It is endowed with the natural norm $ \\Vert {f} \\vert {\\mathcal {SC}^{r}_{\\mathfrak {F},\\tau ,p}} \\Vert := \\Vert {\\mathcal {SH}_{\\mathfrak {F},\\tau } f} \\vert {L_{p,v_{r}}(X,\\mu )} \\Vert .",
"$ These spaces are well-defined Banach spaces, which is implied by theorem:frameinaq.",
"Theorem 3.10 With the same assumptions as in theorem:frameinaq the spaces $\\mathcal {SC}^{r}_{\\mathfrak {F},\\tau ,p}$ are well-defined Banach spaces.",
"As stated in remark:assumptionfulfilled, theorem:frameinaq and lemma:kernelproperty imply that the assumption in proposition:coproperties is fulfilled.",
"Hence, the assertion follows.",
"The following results are straightforward.",
"Lemma 3.11 Let $1< p<q<\\infty $ , $1<\\tau \\le 2$ with $p,q<\\tau ^{\\prime }$ and $0\\le r<s$ .",
"Furthermore let $\\mathfrak {F}$ and $\\mathfrak {G}$ satisfiy the conditions in theorem:frameinaq with $G(\\mathfrak {F},\\mathfrak {G})\\in \\mathcal {A}_{1,m_{v_r}}$ .",
"Then, (i) $\\mathcal {SC}^{r}_{\\mathfrak {F},\\tau ,p}\\subset \\mathcal {SC}^{r}_{\\mathfrak {F},\\tau ,q}$ , (ii) $\\mathcal {SC}^{s}_{\\mathfrak {F},\\tau ,p}\\subset \\mathcal {SC}^{r}_{\\mathfrak {F},\\tau ,p}$ , (iii) $\\mathcal {SC}^{r}_{\\mathfrak {F},\\tau ,p}=\\mathcal {SC}^{r}_{\\mathfrak {G},\\tau ,p}$ .",
"(i) and (ii) follow from lemma:embeddings (ii), (iii) is a consequence of proposition:equalityfrf.",
"Even though we introduced new integrability conditions on the kernel to obtain new spaces, these spaces are in fact one and the same, as the following proposition shows.",
"Proposition 3.12 Let $1\\le p<\\infty $ , $1<\\sigma ,\\tau \\le 2$ with $p<\\sigma ^{\\prime },\\tau ^{\\prime }$ .",
"Then, $\\mathcal {SC}^{r}_{\\mathfrak {F},\\tau ,p}=\\mathcal {SC}^{r}_{\\mathfrak {F},\\sigma ,p}$ .",
"Assume $f\\in \\mathcal {SC}^{r}_{\\mathfrak {F},\\sigma ,p}$ , i.e.",
"$f\\in (\\mathcal {H}_{\\sigma ,v_r})^\\sim $ with $\\mathcal {SH}_{\\mathfrak {F},\\sigma }f\\in L_{p,v_r}$ , by the reproducing identity and lemma:kernelproperty it holds $\\mathcal {SH}_{\\mathfrak {F},\\sigma }f=R_\\mathfrak {F}(\\mathcal {SH}_{\\mathfrak {F},\\sigma }f)\\in R_\\mathfrak {F}(L_{p,v_r})\\subset L_{\\tau ^{\\prime },v_r}\\subset L_{\\tau ^{\\prime },\\frac{1}{v_r}}$ .",
"Thus, lemma:equivalentnorm yields $f\\in (\\mathcal {H}_{\\tau ,v_r})^\\sim $ and $f\\in \\mathcal {SC}^{r}_{\\mathfrak {F},\\tau ,p}$ .",
"Equivalently the converse is shown.",
"Remark 6 With proposition:coorbitequality at hand the coorbit spaces solely depend on $p$ and not on $\\tau $ .",
"Thus it is justified to omit the parameter $\\tau $ and simply write $\\mathcal {SC}^r_{\\mathfrak {F},p} = \\lbrace f\\in (\\mathcal {H}_{\\tau ,v_r})^\\sim :\\mathcal {SH}_{\\mathfrak {F},\\tau }f\\in L_{p,v_r}(X,\\mu )\\rbrace $ for $1\\le p<\\infty $ and some $\\tau $ fulfilling $p<\\tau ^{\\prime }<\\infty $ ."
],
[
"In this appendix we will briefly discuss Young's inequality, the three-way Young's inequality and Schur's test mentioned in sec:coorbittheory.",
"Lemma 1.1 (Young's inequality) Let $a,b\\ge 0$ and $p,q>0$ with $1/p+1/q=1$ , then $ab \\le \\frac{a^p}{p}+\\frac{b^q}{q}.$ Lemma 1.2 (Three-way Young's inequality) Let $a,b,c\\ge 0$ and $p,q,r>0$ with $1/p+1/q+1/r=1$ , then $abc \\le \\frac{a^p}{p}+\\frac{b^q}{q}+\\frac{c^r}{r}.$ By applying Young's inequality twice and observing $\\frac{p^{\\prime }}{q}+\\frac{p^{\\prime }}{r}=1$ with $\\frac{1}{p}+\\frac{1}{p^{\\prime }}=1$ we obtain $abc \\le \\frac{a^p}{p} + \\frac{b^{p^{\\prime }}c^{p^{\\prime }}}{p^{\\prime }} \\le \\frac{a^p}{p} + \\frac{1}{p^{\\prime }}\\left(\\frac{(b^{p^{\\prime }})^{q/p^{\\prime }}}{q/p^{\\prime }} + \\frac{(c^{p^{\\prime }})^{r/p^{\\prime }}}{r/p^{\\prime }}\\right) = \\frac{a^p}{p}+\\frac{b^q}{q}+\\frac{c^r}{r},$ which proves the claim.",
"Lemma 1.3 (Schur's test) For a kernel $K:X\\times X\\rightarrow with $ KA1,mv$ the corresponding kernel operator fulfills $$\\Vert {K} \\vert {L_{p,v}\\rightarrow L_{p,v}} \\Vert \\le \\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert $$ for all $ 1p$.$ For $p<\\infty $ assume $f\\in L_{p,v}$ with $\\Vert {f} \\vert {L_{p,v}} \\Vert \\le 1$ , then $\\Vert {K(f)} \\vert {L_{p,v}} \\Vert &= \\sup _{\\begin{array}{c}g\\in L_{p^{\\prime },\\frac{1}{v}} \\\\ \\Vert {g} \\vert {L_{p^{\\prime },\\frac{1}{v}}} \\Vert \\le 1\\end{array}}\\langle {K(f)},{g} \\rangle \\\\&\\le \\sup _{\\begin{array}{c}g\\in L_{p^{\\prime },\\frac{1}{v}} \\\\ \\Vert {g} \\vert {L_{p^{\\prime },\\frac{1}{v}}} \\Vert \\le 1\\end{array}}\\int _X\\int _X\\vert {K(x,y)f(y)g(x)} \\vert \\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y),$ where $p^{\\prime }$ denotes the Hölder-dual of $p$ .",
"By Young's inequality we obtain $&\\int _X\\int _X\\vert {K(x,y)f(y)g(x)} \\vert \\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{56.9055pt} \\le \\frac{1}{p}\\int _X\\int _X \\vert {K(x,y)} \\vert m_v(x,y)\\cdot \\vert {f(y)} \\vert ^p v(y)^p\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{85.35826pt}+ \\frac{1}{p^{\\prime }}\\int _X\\int _X \\vert {K(x,y)} \\vert m_v(x,y)\\cdot \\vert {g(x)} \\vert ^{p^{\\prime }}\\frac{1}{v(x)^{p^{\\prime }}}\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{56.9055pt} \\le \\frac{1}{p}\\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert \\cdot \\Vert {f} \\vert {L_{p,v}} \\Vert ^p+\\frac{1}{p^{\\prime }}\\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert \\cdot \\Vert {g} \\vert {L_{p^{\\prime },\\frac{1}{v}}} \\Vert ^{p^{\\prime }}.$ Thus, $\\Vert {K(f)} \\vert {L_{p,v}\\rightarrow L_{p,v}} \\Vert \\le \\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert $ .",
"On the other hand for $p=\\infty $ and $f\\in L_{\\infty ,v}$ we have $\\Vert {K(f)} \\vert {L_{\\infty ,v}} \\Vert &\\le \\operatornamewithlimits{ess\\,sup}_{x\\in X}\\int _X\\vert {K(x,y)} \\vert m_v(x,y)\\cdot \\vert {f(y)} \\vert v(y)\\,\\mathrm {d}\\mu (y)\\\\&\\le \\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert \\cdot \\Vert {f} \\vert {L_{\\infty ,v}} \\Vert ,$ which concludes the proof.",
"In this appendix we will briefly discuss Young's inequality, the three-way Young's inequality and Schur's test mentioned in sec:coorbittheory.",
"Lemma 1.1 (Young's inequality) Let $a,b\\ge 0$ and $p,q>0$ with $1/p+1/q=1$ , then $ab \\le \\frac{a^p}{p}+\\frac{b^q}{q}.$ Lemma 1.2 (Three-way Young's inequality) Let $a,b,c\\ge 0$ and $p,q,r>0$ with $1/p+1/q+1/r=1$ , then $abc \\le \\frac{a^p}{p}+\\frac{b^q}{q}+\\frac{c^r}{r}.$ By applying Young's inequality twice and observing $\\frac{p^{\\prime }}{q}+\\frac{p^{\\prime }}{r}=1$ with $\\frac{1}{p}+\\frac{1}{p^{\\prime }}=1$ we obtain $abc \\le \\frac{a^p}{p} + \\frac{b^{p^{\\prime }}c^{p^{\\prime }}}{p^{\\prime }} \\le \\frac{a^p}{p} + \\frac{1}{p^{\\prime }}\\left(\\frac{(b^{p^{\\prime }})^{q/p^{\\prime }}}{q/p^{\\prime }} + \\frac{(c^{p^{\\prime }})^{r/p^{\\prime }}}{r/p^{\\prime }}\\right) = \\frac{a^p}{p}+\\frac{b^q}{q}+\\frac{c^r}{r},$ which proves the claim.",
"Lemma 1.3 (Schur's test) For a kernel $K:X\\times X\\rightarrow with $ KA1,mv$ the corresponding kernel operator fulfills $$\\Vert {K} \\vert {L_{p,v}\\rightarrow L_{p,v}} \\Vert \\le \\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert $$ for all $ 1p$.$ For $p<\\infty $ assume $f\\in L_{p,v}$ with $\\Vert {f} \\vert {L_{p,v}} \\Vert \\le 1$ , then $\\Vert {K(f)} \\vert {L_{p,v}} \\Vert &= \\sup _{\\begin{array}{c}g\\in L_{p^{\\prime },\\frac{1}{v}} \\\\ \\Vert {g} \\vert {L_{p^{\\prime },\\frac{1}{v}}} \\Vert \\le 1\\end{array}}\\langle {K(f)},{g} \\rangle \\\\&\\le \\sup _{\\begin{array}{c}g\\in L_{p^{\\prime },\\frac{1}{v}} \\\\ \\Vert {g} \\vert {L_{p^{\\prime },\\frac{1}{v}}} \\Vert \\le 1\\end{array}}\\int _X\\int _X\\vert {K(x,y)f(y)g(x)} \\vert \\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y),$ where $p^{\\prime }$ denotes the Hölder-dual of $p$ .",
"By Young's inequality we obtain $&\\int _X\\int _X\\vert {K(x,y)f(y)g(x)} \\vert \\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{56.9055pt} \\le \\frac{1}{p}\\int _X\\int _X \\vert {K(x,y)} \\vert m_v(x,y)\\cdot \\vert {f(y)} \\vert ^p v(y)^p\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{85.35826pt}+ \\frac{1}{p^{\\prime }}\\int _X\\int _X \\vert {K(x,y)} \\vert m_v(x,y)\\cdot \\vert {g(x)} \\vert ^{p^{\\prime }}\\frac{1}{v(x)^{p^{\\prime }}}\\,\\mathrm {d}\\mu (x)\\,\\mathrm {d}\\mu (y) \\\\&\\hspace{56.9055pt} \\le \\frac{1}{p}\\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert \\cdot \\Vert {f} \\vert {L_{p,v}} \\Vert ^p+\\frac{1}{p^{\\prime }}\\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert \\cdot \\Vert {g} \\vert {L_{p^{\\prime },\\frac{1}{v}}} \\Vert ^{p^{\\prime }}.$ Thus, $\\Vert {K(f)} \\vert {L_{p,v}\\rightarrow L_{p,v}} \\Vert \\le \\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert $ .",
"On the other hand for $p=\\infty $ and $f\\in L_{\\infty ,v}$ we have $\\Vert {K(f)} \\vert {L_{\\infty ,v}} \\Vert &\\le \\operatornamewithlimits{ess\\,sup}_{x\\in X}\\int _X\\vert {K(x,y)} \\vert m_v(x,y)\\cdot \\vert {f(y)} \\vert v(y)\\,\\mathrm {d}\\mu (y)\\\\&\\le \\Vert {K} \\vert {\\mathcal {A}_{1,m_v}} \\Vert \\cdot \\Vert {f} \\vert {L_{\\infty ,v}} \\Vert ,$ which concludes the proof."
]
] | 1709.01742 |
[
[
"Measuring the Similarity of Sentential Arguments in Dialog"
],
[
"Abstract When people converse about social or political topics, similar arguments are often paraphrased by different speakers, across many different conversations.",
"Debate websites produce curated summaries of arguments on such topics; these summaries typically consist of lists of sentences that represent frequently paraphrased propositions, or labels capturing the essence of one particular aspect of an argument, e.g.",
"Morality or Second Amendment.",
"We call these frequently paraphrased propositions ARGUMENT FACETS.",
"Like these curated sites, our goal is to induce and identify argument facets across multiple conversations, and produce summaries.",
"However, we aim to do this automatically.",
"We frame the problem as consisting of two steps: we first extract sentences that express an argument from raw social media dialogs, and then rank the extracted arguments in terms of their similarity to one another.",
"Sets of similar arguments are used to represent argument facets.",
"We show here that we can predict ARGUMENT FACET SIMILARITY with a correlation averaging 0.63 compared to a human topline averaging 0.68 over three debate topics, easily beating several reasonable baselines."
],
[
"=1 pdfinfo= Title=Measuring the Similarity of Sentential Arguments in Dialog, Author=Amita Misra, Brian Ecker, and Marilyn A. Walker, Subject=, Keywords=argument, similarity ,social media [pages=1-last]sententialArgument.pdf"
]
] | 1709.01887 |
[
[
"General Relativistic effects in the structure of massive white dwarfs"
],
[
"Abstract In this work we investigate the structure of white dwarfs using the Tolman-Oppenheimer-Volkoff equations and compare our results with those obtained from Newtonian equations of gravitation in order to put in evidence the importance of General Relativity (GR) for the structure of such stars.",
"We consider in this work for the matter inside white dwarfs two equations of state, frequently found in the literature, namely, the Chandrasekhar and Salpeter equations of state.",
"We find that using Newtonian equilibrium equations, the radii of massive white dwarfs ($M>1.3M_{\\odot}$) are overestimated in comparison with GR outcomes.",
"For a mass of $1.415M_{\\odot}$ the white dwarf radius predicted by GR is about 33\\% smaller than the Newtonian one.",
"Hence, in this case, for the surface gravity the difference between the general relativistic and Newtonian outcomes is about 65\\%.",
"We depict the general relativistic mass-radius diagrams as $M/M_{\\odot}=R/(a+bR+cR^2+dR^3+kR^4)$, where $a$, $b$, $c$ and $d$ are parameters obtained from a fitting procedure of the numerical results and $k=(2.08\\times 10^{-6}R_{\\odot})^{-1}$, being $R_{\\odot}$ the radius of the Sun in km.",
"Lastly, we point out that GR plays an important role to determine any physical quantity that depends, simultaneously, on the mass and radius of massive white dwarfs."
],
[
"Introduction",
"Massive white dwarfs (WD) were documented in [1], [2], [3], [4] reaching for the first time values nearby to the Chandrasekhar mass limit of $1.44M_{\\odot }$ [5], with $M_{\\odot }$ being the mass of the Sun.",
"Particularly, in the Extreme Ultraviolet Explorer all-sky survey (EUVE) one can find several data of massive WD.",
"The most massive WD presented in the EUVE observations have a mass of $1.41M_{\\odot }$ [4].",
"Also in Refs.",
"[2]-[3] a considerable amount of WD with masses between $1.32-1.37M_{\\odot }$ can be found.",
"Moreover, recent observations reveals the existence of some super-luminous type Ia supernovae, e.g, SN200-6gz, SN2007if, SN2009dc [6], [7], [8], [9] (for a review about this kind of astrophysical event we quote [10]).",
"Some authors suggest that possible explanations for these objects are super-Chandrasekhar WD [11], [8], [12].",
"To achieve super-Chandrasekhar WD authors of Refs.",
"[13], [14], [15], [16], [17] consider the WD in a presence of very strong magnetic fields.",
"However, those putative WD with strong magnetic fields were showed to be unstable in [18].",
"Also some attention has been driven to the description of some anomalous X-ray pulsars (AXP) and soft gamma-ray repeaters (SGR) as highly magnetized very massive rotation powered white dwarf pulsars [19], [20], [21], [22], [23], [24].",
"In view of these new discoveries it is worth to better identify how General Relativity (GR) can affect the structure of very massive WD.",
"Indeed, since the discovery of WD, the discussion about the importance of GR for these objects was established.",
"The general relativistic effects on the structure of WD was first qualitatively discussed by Kaplan [25].",
"Fifteen years later Chandrasekhar derived the instability criteria in a general relativistic framework [26].",
"Kaplan concluded that when the WD radius becomes smaller than $1.1\\times 10^3 ~{\\rm km}$ , GR would probably induces a dynamical instability in the star.",
"Furthermore, Chandrasekhar [26] concluded that general relativistic effects leads to a smaller critical central density and, consequently, it limits the value of the radius.",
"In addition, J. Cohen et al [27] studied the oscillation period of WD taking GR effects into account and they argue that general relativistic WD unlike Newtonian ones have a minimum fundamental period for a given composition.",
"In more recent works [28], [29], [30] calculations were made for general relativistic uniformly-rotating WD.",
"Summarizing those results: they showed that the rotation can have astrophysical implications to Soft-Gamma Repeaters and Anomalous X-Ray Pulsars, rotation can also uplift slightly the maximum stable mass configuration and they also showed that the spin-up and spin-down eras are different for sub and super-Chandrasekhar WD.",
"In addition, in [31] the authors compare the results of general relativistic uniformly-rotating WD with uniformly-rotating Newtonian WD.",
"In light of uniformly-rotating case the minimum rotation period is about the same in both cases (general relativistic and Newtonian) and the general relativistic maximum mass is slightly below the Newtonian maximum mass.",
"However, they also showed by the turning point method that general relativistic WD can be axis-symmetrically unstable while Newtonian WD are stable.",
"Turning to the static case, our main aim in the present work is to show that general relativistic effects are relevant for the determination of the radius of massive WD.",
"In fact, there are some works where general relativistic calculations of the mass-radius relation were made for a static model of WD [32], [33], [34], [35], [36].",
"In those works authors find that general relativistic hydrostatic equilibrium yields to a maximum mass slightly below the Chandrasekhar limiting mass.",
"Nevertheless, the role of GR for the radius of very massive WD ($M>1.3M_{\\odot }$ ) was not stressed.",
"Thus, in the next sections of this paper we address, minutely, the general relativistic effects on the mass-radius relation of massive WD, showing that GR is quite important to determine the radii of those massive WD.",
"Moreover, considering the observational data obtained for very massive WD, GR turns out to be very relevant to estimate, precisely, the WD radius and others WD properties, such as surface gravity."
],
[
"Equation of state",
"The result of Chandrasekhar mass limit is one of the most established astrophysical constraints since there is no confirmed observational data of WD with masses above $1.44M_{\\odot }$ , until now.",
"Therefore, in the present work we use the Chandrasekhar equation of state, i.e., the equation of state that describes a fully degenerate relativistic electron gas.",
"In such a model the pressure and total energy density of the fluid are given, respectively, by [37], [38] $p(k_F) = \\frac{1}{3\\pi ^2\\hbar ^3}\\int _0^{k_F}\\frac{k^4c^2}{\\sqrt{k^2c^2+m_e^2c^4}}dk,$ $\\epsilon (k_F) &=& \\rho c^2 + \\epsilon _e\\\\&=& \\frac{m_N\\mu _ek_F^3}{3\\pi ^2 \\hbar ^3}c^2 + \\frac{1}{\\pi ^2\\hbar ^3}\\int _0^{k_F}\\sqrt{k^2c^2+m_e^2c^4}k^2dk \\nonumber ,$ where $c$ is the speed of light, $m_e$ is the electron mass, $m_N$ the nucleon mass, $\\hbar $ the reduced Planck constant, $\\mu _e$ is the ratio between the nucleon number $A$ and atomic number $Z$ for ions and $k_F$ is the Fermi momentum of the electron.",
"Eq.",
"(REF ) is the isotropic electron degeneracy pressure.",
"The first and second terms in Eq.",
"(REF ) are, respectively, the energy density related to the rest mass of the ions and the electron energy density.",
"Eqs.",
"(REF ) and (REF ) can be put in a simpler form in order to favor future numerical calculations $p(x) &= \\epsilon _0 f(x), \\\\\\epsilon (x) &= \\epsilon _0 g(x),$ where $\\epsilon _0=m_ec^2/\\pi ^2\\lambda _e^3$ , with $\\lambda _e$ being the electron Compton wavelength, $x=k_F/m_ec$ the dimensionless Fermi momentum and the functions $f(x)$ and $g(x)$ using $\\mu _e=\\frac{A}{Z}=2$ are $f(x)=\\frac{1}{24}\\left[(2x^3-3x)\\sqrt{x^2+1}+3\\textrm {asinh} x\\right],$ $g(x)=1215.26x^3+\\frac{1}{8}\\left[(2x^3+x)\\sqrt{x^2+1}-\\textrm {asinh} x\\right].$ In terms of the dimensionless Fermi momentum $x$ the mass density becomes $\\rho =9.738\\times 10^5 \\mu _e x^3 ~{\\rm g/cm^3}.$ Salpeter in a seminal paper [39] have improved the above EoS by considering several corrections, such as: electrostatic corrections due to Coulomb interaction, deviations of the electron charge distribution from uniformity, inverse beta-decay process, inclusion of correlation and exchange energies.",
"Hamada & Salpeter [40] have showed, a posteriori, that the mass-radius relation of WD are modified in a nontrivial way depending on the interior composition due to those corrections applied by Salpeter."
],
[
"Newtonian case",
"We assume in our calculations that the mass configuration of the star is static and spherically symmetric.",
"In this case the pressure and the density of the fluid are functions of the radial coordinate $r$ only.",
"Hence, for the structure of a Newtonian star we have the following equilibrium equations [37] $\\frac{dp}{dr}&=-\\frac{Gm\\rho }{r^2},\\\\\\nonumber \\\\\\frac{dm}{dr}&=4\\pi r^2\\rho ,$ where $p$ denotes pressure, $G$ is the gravitational constant, $\\rho $ is the mass density, and $m$ represents the enclosed mass inside a sphere of radius $r$ ."
],
[
"Special relativistic case",
"In addition to Newtonian case one may consider special relativistic improvements.",
"For such, the mass density is replaced by the total energy density of the system (see Eq.",
"(REF )), consequently, the relativistic kinetic energy of the electrons are taken into account, such that Eqs.",
"(REF ) and () becomes $\\frac{dp}{dr}&=-\\frac{Gm(\\rho c^2+e_e)}{r^2c^2}=-\\frac{Gm\\epsilon }{r^2c^2},\\\\\\nonumber \\\\\\frac{dm}{dr}&=\\frac{4\\pi r^2(\\rho c^2+e_e)}{c^2}=\\frac{4\\pi r^2\\epsilon }{c^2}.$ Along the present paper we will refer to this case as special relativistic (SR) case."
],
[
"General relativistic case",
"To derive the hydrostatic equilibrium equations in a general relativistic framework it is used the interior Schwarzschild solution and the energy-momentum tensor of a perfect fluid [41].",
"For a detailed derivation of the general relativistic hydrostatic equilibrium equation see [42], [43], [44].",
"The Eqs.",
"(REF ) and () now reads $\\frac{dp}{dr}&=-(\\epsilon +p) \\frac{d\\phi }{dr}\\nonumber \\\\ &=-\\frac{Gm\\epsilon }{c^2r^2}\\left[1+\\frac{p}{\\epsilon }\\right]\\left[1+\\frac{4\\pi r^3p}{mc^2}\\right]\\left[1-\\frac{2Gm}{c^2r}\\right]^{-1},\\\\\\nonumber \\\\\\frac{dm}{dr}&=\\frac{4\\pi r^2\\epsilon }{c^2},$ being $e^{2\\phi }$ the temporal metric coefficient $g_{00}$ .",
"In the weak field limit $g_{00}=1+2\\Phi /c^2+\\mathcal {O}^2$ , where $\\Phi $ corresponds to the Newtonian gravitational potential.",
"So, the formal definition of the gravitational field of a static and spherically symmetric object in GR corresponds to $g_{GR}=-d\\phi /dr$ , where $\\phi $ represents the general relativistic gravitational potential.",
"After integrating the above equations the interior Schwarzschild solution is matched smoothly with the vacuum exterior Schwarzschild line element.",
"The new three terms in square brackets of the equilibrium equation (REF ) are general relativistic corrections terms.",
"From Eq.",
"(REF ) is reliable that the general relativistic effects become relevant when the star is sufficiently compact, i.e., when the factor $2Gm/c^2r$ approaches unity and when the pressure is high enough to become comparable to the energy density of the fluid, i.e., when $p/\\epsilon $ and $4\\pi r^3p/mc^2$ are comparable with the unity."
],
[
"Initial and boundary conditions",
"In this paper we use the equation of state () to solve the equilibrium equations through a forth-order Runge-Kutta method.",
"The initial conditions are $p(r=0)=p_c,\\quad \\rho (r=0)=\\rho _c \\quad {\\rm and} \\quad m(r=0)=0.$ The star's surface is reached when the pressure vanishes, consequently the energy density (or mass density) also goes to zero at the surface.",
"Therefore the boundary conditions reads $p(r=R)= 0, \\quad \\rho (r=R)=0 \\quad {\\rm and} \\quad m(r=R)=M,$ where $R$ and $M$ mean the total radius and total mass of the star, respectively.",
"Since one may use several values for the central pressure $p_c$ a family of solutions can be found for the star mass and radius."
],
[
"Comparison between Newtonian and general relativistic cases",
"Using several values of central pressure $p_c$ we construct the mass-radius and mass-central density relations for the three cases previously explained.",
"From Fig.",
"(REF ) it can be seen that the purely Newtonian case (Sect.",
"(REF )) does not have the secular instability $\\partial M/\\partial R>0$ (more details, see [26], [45], [46], [47]), while SR case presents instability when the electrons are highly relativistic.",
"This aspect is easier to see in Fig.",
"(REF ), where we highlight the region of massive WD for the mass-radius relation.",
"We also display in Fig.",
"(REF ) the observational data of the most massive white dwarf ($M=1.41M_{\\odot }\\pm 0.04$ ) found in literature [4].",
"Figure: Mass-radius relation of massive WD.",
"The curves follow the same representation as in Fig.().",
"The full blue circles mark the maximum masses.",
"The dotted red line represents the measured mass of the most massive white dwarf (M=1.41M ⊙ ±0.04M=1.41M_{\\odot }\\pm 0.04) found in literature and the shaded orange region corresponds to its estimated error.Table: Maximum mass and minimum radius for the static models of WD stars.From Fig.",
"(REF ) it is worth to note that GR does not affect greatly the maximum mass, rather it diminishes the maximum stable mass a few percents $\\sim 3\\%$ (see also Tab.",
"(REF )).",
"However, it is worthwhile to cite that the minimum radii, i.e., the radii corresponding to the predicted maximum masses, are very different.",
"For instance, the minimum radius predicted by general relativistic calculations is about three times larger than Newtonian ones (see Tab.",
"(REF )).",
"Similar results can be found in [48], [49]."
],
[
"Fixed total star mass",
"From Fig.",
"(REF ) it can also be seen that for a fixed total star mass between $1.3-1.415M_{\\odot }$ the values of radii are very sensitive depending on the case.",
"Tab.",
"(REF ) shows the calculated radii for several values of total mass from Newtonian and general relativistic cases.",
"Table: Corresponding radii to fixed total star masses in Newtonian and general relativistic cases.",
"R Newton R_{\\textrm {Newton}} means the radius predicted by Newtonian case (see Sect.",
"()), R SR R_{\\textrm {SR}} is the radius given by Sect.",
"(), R GR R_{\\textrm {GR}} is the radius in general relativistic case given in Sect.",
"() and the R NR R_{\\textrm {NR}} is the radius supplied by non-relativistic approximation, where the mass follows the relation M/M ⊙ ∝1/R 3 M/M_{\\odot } \\propto 1/R^3.Tab.",
"(REF ) presents the values of central mass density for some values of fixed total mass for several cases.",
"Table: Corresponding central mass densities to fixed total masses in Newtonian and general relativistic cases.",
"ρ C Newton \\rho _C^{\\textrm {Newton}} means the central density achieved in Newtonian case (Sect.",
"()), ρ C SR \\rho _C^{\\textrm {SR}} is the central density given by SR case Sect.",
"(), ρ C GR \\rho _C^{\\textrm {GR}} is the central density found for the general relativistic case Sect.",
"().In Fig.",
"(REF ) we show, for a fixed total mass of $M=1.415M_{\\odot }$ , the profiles of mass, gradient of pressure and energy density for the three cases.",
"We remark from Fig.",
"(REF ), that in order to obtain the same total mass in all cases the structure of the stars are very distinct.",
"From Fig.",
"(REF ) we can note that in general relativistic case the energy density in the central region of the star is larger than in Newtonian cases.",
"This effect at same time makes the WD's mass more concentrated at the star center and the pressure gradient to decay more sharply for the general relativistic calculations.",
"Figure: From top to bottom: a) mass profiles, b) energy density profiles and c) gradient of pressure profiles.",
"All profiles correspond to a fixed total star mass of M=1.415M ⊙ M=1.415M_{\\odot }.One can calculate the Newtonian gravitational field as $g_{\\textrm {Newton}}=-\\frac{Gm}{r^2},$ and the general relativistic gravitational field from (REF ) becomes $g_{\\textrm {GR}}=-\\frac{Gm}{r^2}\\left[1+\\frac{4\\pi r^3p}{mc^2}\\right]\\left[1-\\frac{2Gm}{c^2r}\\right]^{-1}.$ Fig.",
"(REF ) displays the gravitational fields of the stars with total mass $1.415M_{\\odot }$ as a function of radial coordinate, we can note that the gravitational fields are initially very different, however, outside the star the fields match each other, thus implying that the gravitational field outside the star can be regarded as Newtonian.",
"In addition, inside the star, where the gravitational fields are very different, it can be observed that there is a deviation of about $\\sim 200$ % between the correspondent highest values of the gravitational field (the minima of the curves in Fig.",
"(REF )), this is due to the very different central densities of the three cases (see Tab.",
"(REF ) and Fig.",
"(REF b)).",
"In particular, the central density $\\rho ^{{\\rm GR}}_C$ is about 4 times larger than $\\rho _C^{{\\rm Newton}}$ , for the fixed total mass of $1.415M_{\\odot }$ .",
"We also display in Fig.",
"(REF ) the gravitational potentials of WD with total mass $M=1.415M_{\\odot }$ , from we can note a fairly difference between them.",
"Figure: General relativistic and Newtonian gravitational fields as a function of radial coordinate for a fixed total star mass of 1.415M ⊙ 1.415M_{\\odot }.Figure: General relativistic and Newtonian gravitational potentials as a function of radial coordinate for a fixed total star mass of 1.415M ⊙ 1.415M_{\\odot }.Figure: General relativistic gravitational fields as a function of radial coordinate for a total mass of 1.415M ⊙ 1.415M_{\\odot }, calculated with and without correction terms.In Fig.",
"(REF ) we show for the same mass of $1.415M_{\\odot }$ the general relativistic gravitational field, calculated in three different ways: with all corrections terms, with no curvature term and without any correction term.",
"A priori, from the Fig.",
"(REF ) the correction terms seems to be not relevant as well, however, they yields to the important effect observed in the case of fixed total masses, thus allowing larger densities near to the center of the star $r<300$ km (see Fig.",
"(REF b)).",
"Henceforth, in the present work we compare just the results of the purely Newtonian case (i.e., without special relativistic corrections) with the general relativistic outcomes for the equilibrium configurations of WD.",
"For this purpose, we firstly define the quantity $\\Delta R=\\frac{R_{\\textrm {Newton}}-R_{\\textrm {GR}}}{R_{\\textrm {GR}}},$ being $R_{\\textrm {Newton}}$ and $R_{\\textrm {GR}}$ the radii given, respectively, by the Newtonian and general relativistic cases.",
"Therefore, $\\Delta R$ means the relative difference between the values of radii for fixed total masses.",
"Figure: Radius relative difference ΔR\\Delta R versus fixed total star mass.In Fig.",
"(REF ) we plot the quantity $\\Delta R$ for some values of fixed total mass between $1.3-1.415M_{\\odot }$ .",
"We can see that this quantity increases very fast when approaching $1.41M_{\\odot }$ .",
"In fact, when the mass is about $1.41M_{\\odot }$ we can see that the relative difference in radius is nearly $50\\%$ .",
"It is worth to mention that the relative difference in radius is about $37\\%$ for a mass of exactly $1.41M_{\\odot }$ , i.e., the measured mass of the white dwarf $EUVE~J~1659+440$ [4]."
],
[
"Surface Gravity",
"It is worth to study the general relativistic effects on the surface gravity of the stars since this quantity can be observationally found.",
"We calculate the surface gravity $g$ in a Newtonian framework as $g_{\\textrm {Newton}}=-\\frac{GM}{R^2}.$ To calculate the general relativistic surface gravity we use the expression given by [41], [50] $g=-\\left(\\frac{GM}{R^2}\\right)\\frac{1}{1-\\frac{2GM}{c^2R}},$ in which is merely Eq.",
"(REF ) for the case of zero pressure.",
"Using the values of Tab.",
"(REF ) we calculate the Newtonian surface gravity and general relativistic surface gravity.",
"Figure: Surface gravity versus fixed total star mass.",
"The dotted red line is the measurement of mass of the most massive white dwarf (EUVEJ1659+440EUVE~J~1659+440) found in literature and the shaded orange region is its estimated error.In Fig.",
"(REF ) is plotted the surface gravity against fixed total masses together with the observational data of the white dwarf $EUVE~J~1659+440$ .",
"It is easy to see that using Newtonian results we are sub-estimating the surface gravity of the stars in comparison with general relativistic outcomes."
],
[
"Surface gravity relative difference",
"Since the values for general relativistic surface gravity are much higher than Newtonian ones we define the quantity $\\Delta g=\\frac{g_{\\textrm {Newton}}-g_{\\textrm {GR}}}{g_{\\textrm {GR}}}.$ Figure: Relative difference between Newtonian surface gravity and general relativistic surface gravity against fixed total star mass.The relative difference $\\Delta g$ is shown in Fig.",
"(REF ).",
"It is interesting to note that in Fig.",
"(REF ), for a mass of $1.415M_{\\odot }$ , we have about 55% of relative difference for the values of surface gravity."
],
[
"Fit of the general relativistic mass-radius relation",
"Keeping in mind the importance of GR for WD, we fit the general relativistic mass-radius relation in order to obtain an analytic expression that better estimate the WD radius, rather than the non-relativistic Newtonian expression $\\frac{M}{M_{\\odot }}=2.08\\times 10^{-6} \\left(\\frac{R}{R_{\\odot }}\\right)^{-3},$ where $R_{\\odot }$ is the radius of the Sun.",
"The expression we use to fit the general relativistic curve in Fig.",
"(REF ) is given by $\\frac{M}{M_{\\odot }}=\\frac{R}{a+bR+cR^2+dR^3+kR^4},$ where $k$ is the inverse of the constant in the non-relativistic Newtonian mass-radius relation $k=(2.08\\times 10^{-6}R_{\\odot }^3)^{-1}$ .",
"The constants $a$ , $b$ , $c$ and $d$ are parameters that depend on the interior fluid EoS of the star, such that, using the EoS described by Eq.",
"() and $\\mu _e=2$ (Fig.",
"(REF )) we find $a&=&20.86~ \\textrm {km} \\nonumber \\\\b&=&0.66 \\nonumber \\\\c&=&2.48 \\times 10^{-5}~ \\textrm {km^{-1}}\\nonumber \\\\d&=&2.43\\times 10^{-9}~ \\textrm {km^{-2}}.$ Figure: Fit of the general relativistic mass-radius diagram with Eq.",
"() (red dashed line) and the non-relativistic limit (dotted orange line).Table: Values of the constants for the analytic mass-radius relations.We employed Eq.",
"(REF ) to depict analytically other mass-radius relations derived from a few EoS models, such as, the Salpeter EoS (for He, C and O stars) and $\\mu _e=2.154$ .",
"The values of fitted parameters are given in Tab.",
"(REF )."
],
[
"Concluding Remarks",
"In this paper we showed that General Relativity is very important to estimate correctly the radius of a massive WD ($M> 1.3M_{\\odot }$ ) and, consequently, to calculate the surface gravity.",
"We also showed that the minimum radii are very different within either Newtonian or general relativistic cases (about 200% at most).",
"We demonstrate that for fixed values of total mass there is a large deviation from Newtonian WD radius to general relativistic WD radius, for example, for a mass close to the value $M= 1.42M_{\\odot }$ the Newtonian radius is about 50% larger than the general relativistic one.",
"For the most massive WD found in literature $M=1.41\\pm 0.04$ [4] the Newtonian value of radius is now 37% larger than the general relativistic one (or at least 6% for a mass of $1.37M_{\\odot }$ ).",
"Due to those deviations in radius the surface gravity is expected to be 55% smaller in Newtonian case in comparison with the result from GR for a fixed total mass of about $M= 1.42M_{\\odot }$ .",
"Briefly, the GR effects produces a different correlation between surface gravity and radius, what may induce changes in the values of observational parameters.",
"In particular, if we measured the surface gravity for a massive WD that we know the mass, the correct radius obtained by GR is going to be smaller than the one we would obtain if we do not take into account general relativity, because of the different mass-radius relation of the two cases.",
"The WD structure in a general relativistic, finite temperature case was studied in [51], in which was showed that the finite temperature effects are more significant the less massive the star is.",
"The deviations arising from thermal effects are negligible for stars with $M<1.2M_{\\odot }$ .",
"On the other hand the main effects of GR appears for stars with $M>1.3M_{\\odot }$ , what turns both effects crucial for the determination of the WD mass-radius relation from observations.",
"We also found a novel analytic mass-radius relation by fitting the general relativistic mass-radius relationship obtained numerically.",
"We suggest that it can be useful to calculate other properties of the stars like magnetic dipole field, moment of inertia, gravitational red-shift and so on.",
"GAC thanks Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for the financial support.",
"The authors also acknowledge FAPESP for support under the thematic project # 2013/26258-4."
]
] | 1709.01635 |
[
[
"Estimating the epidemic risk using non-uniformly sampled contact data"
],
[
"Abstract Many datasets describing contacts in a population suffer from incompleteness due to population sampling and underreporting of contacts.",
"Data-driven simulations of spreading processes using such incomplete data lead to an underestimation of the epidemic risk, and it is therefore important to devise methods to correct this bias.",
"We focus here on a non-uniform sampling of the contacts between individuals, aimed at mimicking the results of diaries or surveys, and consider as case studies two datasets collected in different contexts.",
"We show that using surrogate data built using a method developed in the case of uniform population sampling yields an improvement with respect to the use of the sampled data but is strongly limited by the underestimation of the link density in the sampled network.",
"We put forward a second method to build surrogate data that assumes knowledge of the density of links within one of the groups forming the population.",
"We show that it gives very good results when the population is strongly structured, and discuss its limitations in the case of a population with a weaker group structure.",
"These limitations highlight the interest of measurements using wearable sensors able to yield accurate information on the structure and durations of contacts."
],
[
"Introduction",
"An increasing number of studies on epidemic spreading processes use data-driven models.",
"In particular, contact patterns between individuals are considered to play an important role in determining the possible outcome of the transmission of infectious diseases in a population [1], [2], [3].",
"Many datasets describing contacts between individuals in various contexts have thus been gathered by different research groups, using techniques ranging from surveys or diaries to wearable sensors [6], [4], [5], [7], [8], [9], [11], [10], [12], [13], [14], [15], [16], [17].",
"The resulting data are typically in the form of contact networks in which nodes represent individuals and edges represent the existence of at least one contact between the individuals linked.",
"Such network data can however be incomplete, for two main reasons.",
"On the one hand, not all individuals agree to participate to the data collection (either not answering the surveys or not willing to wear a sensor), leading to node sampling.",
"On the other hand, contacts between participating individuals might not all figure in the gathered data, so that links are missing from the data.",
"In the case of diaries for instance, each individual only remembers a fraction of his/her contacts, the longest contacts being better reported [18], [14], [19].",
"If instead the available data comes from a survey about friendship relations, it will typically miss many short encounters between people who are not friends [14].",
"Finally, even in the case of wearable sensors, some short contacts might not be detected, the actual detection of a contact might depend on its duration, depending on the sensitivity of the measuring infrastructure, and the temporal resolution may vary [20].",
"It is thus of interest to understand how the resulting data incompleteness or limited resolution affects the properties of the measured contact network [21], [22], [24], [20], how it affects the outcome of data-driven models using incomplete data [23], [24], [20], [25], and most importantly if it is possible to infer the real network structure or statistical properties from incomplete information [26], [27], [28] and/or to devise methods to correctly estimate the epidemic risk even from incomplete data [24], [29] [Note that, for some types of wearable sensors, the opposite problem of false positives, i.e., of reported contacts that are not relevant for propagation events, can also arise.",
"Here we focus on data incompleteness, but investigations of the impact of false positives would also be of clear interest].",
"To obtain such an estimation, a possibility is to try and construct surrogate datasets using only the information contained in the incomplete data such that these surrogate data, despite not being strictly equal to the original data, are “similar enough” with respect to the spreading process of interest.",
"Here, “similar enough” means that the outcomes of simulations of spread using the surrogate data should be close to the ones using the real, complete data.",
"This issue has been addressed in the case of uniform population sampling by Génois et al.",
"[24].",
"Uniform node sampling indeed maintains not only the network density, but also the whole contact matrix of densities that describes the structure of links in a population structured in distinct groups, such as classes in a school.",
"We recall that the density of a network of $N$ nodes and $E$ edges is defined as the ratio of the number of edges to the maximal possible number of edges that could exist between the nodes, i.e., $d = E / (N(N-1)/2)$ .",
"The contact matrix of densities gives for each pair of groups $X$ and $Y$ the ratio between the total number of links $E_{XY}$ between the $n_X$ individuals in $X$ and the $n_Y$ individuals in $Y$ , and the maximum number of such possible links ($n_Xn_Y$ if $X \\ne Y$ or $n_X(n_X-1)/2$ if $X=Y$ ).",
"Moreover, the sampling also maintains the temporal statistics of contacts.",
"It is thus possible to measure this contact matrix and the temporal statistics in the incomplete data and to construct surrogate data having the same statistics as the original one.",
"Spreading processes simulated using such surrogate data have been shown to reproduce well the outcome of simulations using the whole dataset [24].",
"A similar method has been shown to work well also in a case study of contact diaries collected together with data from wearable sensors [29]: although not all contacts were reported in the diaries, building surrogate data using the contact matrix measured in the diaries and publicly available statistics on contact durations made it possible to correctly estimate the outcome of simulations of spreading processes.",
"In these two studies, the density of the sampled data was however either equal (for uniform population sampling) or close (for the diaries) to the one of the original data.",
"In other cases, such as e.g.",
"the friendship survey of Ref.",
"[14], the density of the data is much smaller than the one of the contact data, and the method has indeed been shown to fail in this case [29].",
"Sampled data with smaller density than the original one occur as soon as the links between sampled nodes are not all present.",
"This is expected to be the case in particular for data coming from diaries or surveys, and it is of interest to test the limits of the reconstruction method, as a function of the sampling properties, and possibly to understand how to overcome this obstacle.",
"Here, we tackle this issue by considering incomplete contact data stemming from a non-uniform sampling procedure intended to mimic a survey procedure in which (i) not all individuals participate and (ii) the contacts of each respondent are reported with a probability depending on their duration [30].",
"We consider empirical contact datasets, resample them using this non-uniform sampling procedure and design surrogate data as described above.",
"We show that, at low sampling, the use of such surrogate data in simulations of spreading processes is not enough to estimate the epidemic risk, even if it yields an improvement with respect to the use of the raw sampled data.",
"We thus consider the case in which additional information is available for one of the groups forming the population: if one group is uniformly sampled (for instance if wearable sensors are available for this group), yielding a good estimate of its density, it is possible through a simple rescaling procedure to estimate a new contact matrix for the data and to use it to construct another surrogate dataset that yields better results.",
"By using two datasets with different structures and varying the parameters of sampling and spreading processes, we explore the efficiency and limits of both procedures.",
"We first describe the datasets and the different steps of our methodology, which are in the same spirit as Refs [24], [29].",
"Starting from a network of contacts between individuals in a population, we perform a specific resampling that leads to an incomplete dataset.",
"We describe two methods to create surrogate datasets using the statistical information contained in the incomplete data.",
"Spreading processes are then simulated on top of the incomplete and of the surrogate datasets, and their outcomes compared with the ones of simulations using the whole original contact network.",
"The procedure is summarized in Fig.",
"REF .",
"Figure: Sketch of the procedures considered in the article.",
"Weconsider a dataset describing a contact network.",
"We perform a resamplingaccording to a certain (non-uniform) sampling method.",
"We then measure a number of statistics ofthe resampled data and use these statistics to construct surrogate data.",
"We simulate spreading processeson the original, resampled and surrogate data and compare the outcomes, as given by the fraction of largeepidemics and by the whole distribution of epidemic sizes.We will use two datasets describing face-to-face contacts between individuals, collected and made publicly available by the SocioPatterns collaboration (see the SocioPatterns website http://www.sociopatterns.org/).",
"The first dataset (Thiers13) has been collected in a French high school in December 2013.",
"The resulting contact network has $N=327$ nodes representing students (divided in 9 classes corresponding to different fields of study) and $E=5818$ weighted edges (see [14] for a detailed description and analysis of this dataset), for an average link density of $2E/(N(N-1)) \\approx 0.11$ .",
"The participation rate reached $86.5\\%$ (there were overall 378 students in the 9 classes).",
"The classes are of similar sizes (see Table REF ), and most edges ($69\\%$ , accounting for $93\\%$ of the weights, i.e., of the total contact time between students) are found within classes (Fig.",
"REF ).",
"The second dataset (InVS) has been collected in the office buildings of the Institut de Veille Sanitaire (French Institute for Public Health Surveillance) in March 2015, and describes the contacts between 217 individuals divided in 12 departments.",
"The contact network has 4274 edges, i.e., a density of $\\approx 0.18$ .",
"The departments have very different sizes and participation rate, with an average participation rate of $60\\%$ (see Table REF ) and, while most contact time occurs within departments ($76\\%$ ), the corresponding fraction of edges is only $42\\%$ (Fig.",
"REF ).",
"In each dataset, an edge between two individuals corresponds to the fact that these individuals have been in contact at least once during the data collection, and the edge weight gives the total contact time between them.",
"The contact matrices of edge densities are shown in Fig.",
"REF for both datasets.",
"In the Thiers13 case, we have moreover access to a network describing friendship relations between students, obtained by a survey to which 135 of the 327 students answered.",
"This friendship network has 413 unweighted edges.",
"Figure: Density contact matrices.",
"Left: contact matrix giving the density of edges between classes during the study (Thiers13),right: contact matrix giving the density of edges between departments during the study (InVS).For each matrix, the entry at row XX and column YY is givenby the total number of links between individuals in class or department XX and individuals in class or department YY,normalized by the maximum number of observable links (n X n Y n_Xn_Y or n X (n X -1)/2n_X(n_X-1)/2 if X=YX=Y,with n X n_X the cardinality of XX).",
"These matrices give only structural information as they do not take into account edge weights.Table: Number of individuals in each class participating to the data collection in the highschool (Thiers13 dataset), percentagewith respect to the population under study and participation rates.Table: Number of individuals in each departmentparticipating to the data collection in the office buildings of InVS, percentage with respect to the population under study and participation rate.We consider the sampling method put forward in [30], called “EGOref”, based on (i) a uniform sampling of nodes and (ii) a non-uniform sampling of edges, edges with larger weights being preferentially sampled.",
"This procedure is designed to mimic a sampling of links obtained for instance through a survey on friendship relations in a population: it is inspired by the result of Ref.",
"[14] that the longest contacts measured in the Thiers13 dataset corresponded to reported friendships, while many short contacts did not.",
"In particular, we have shown in Ref.",
"[30] that the outcome of simulations of spreading processes on the friendship network of the Thiers13 dataset can be reproduced if the EGOref sampling method is applied to the contact network of the same dataset, with correctly adjusted sampling parameters.",
"More precisely, the EGOref sampling depends indeed on two parameters.",
"Starting from a weighted contact network with $N_0$ nodes, we select $N$ of these nodes (called “egos”) uniformly at random (a non-uniform selection could also be considered).",
"For each ego $i$ , each edge $i-j$ is selected with a probability equal to $p*\\frac{W_{ij}}{S_{i}}$ , with $W_{ij}$ the weight of the edge between $i$ and $j$ , $S_i = \\sum _\\ell W_{i\\ell }$ the strength of the ego node $i$ and $p$ the sampling parameter.",
"We then keep only the egos and the selected edges linking them and we remove the other edges (between egos and non-egos and between non-egos) and nodes (non-egos).",
"With this method, we end up with a tunable number of nodes $N$ and a number of edges that depends on the parameter $p$ .",
"Figure REF summarizes this process.",
"The parameter $p$ clearly has an effect on the density of the sampled network.",
"This is in contrast with the case of a uniform population sampling in which all the edges between sampled nodes are kept, and which thus conserves the network density [24].",
"Figure REF displays the ratio between the density of the EGOref sampled network and the original contact network for the two datasets used here: the density of the sampled network increases with the parameter $p$ .",
"On the other hand, the density of the sampled network does not depend on the number of sampled nodes (Figure S1 in the Supplementary Information).",
"Despite the decrease in network density caused by the EGOref sampling, it is worth noting that the contact matrices of edge densities measured in the original contact network and in the EGOref sampled network remain very similar for both datasets (see Figures S2-S4 in the Supplementary Information).",
"These matrices give, for each pair of groups in the population (here, classes or departments), the number of links between these groups normalized by the maximum possible number of such links (obtained if each member of one group is linked to all members of the other group).",
"A large similarity between contact matrices indicates that the overall structure of the network is preserved by the sampling, even if the specific values of densities are changed.",
"Figure: Sketch of the EGOref sampling process.",
"We firstselect a certain number of nodes as egos.",
"Each ego “chooses” toreport some of its links, with probability depending on theirweights.",
"A link can be selected twice if joins two egos(blue edges).",
"We then finally keep only the egos and, amongthe chosen edges, only the ones joining egos.Figure: Impact of sampling on network density.",
"Ratio between the density of the EGOref samplednetwork and the density of the whole contact network as afunction of the parameter pp for the two datasets.",
"Here thenumber NN of sampled individuals is 70%70\\% of the totalpopulation: changing this number does not change the ratio.As in Refs.",
"[24] and [29], our initial goal is to use only the information contained in the sampled data to construct surrogate contact networks that are statistically similar to the full data.",
"As already made clear in the introduction and in refs [24], [29], we emphasize again that the point is not to infer the missing links but to build a “plausible” version of these links, such that the simulations of epidemic spread on the resulting network, as described below, yield an accurate estimation of the epidemic risk.",
"We consider two distinct methods to construct such surrogate data.",
"The first method is the equivalent for static networks of the one presented in [24], [29].",
"First, the contact matrix of edge densities is measured in the incomplete data.",
"Assuming that the number of missing nodes in each group (class or department) is known, we add the missing nodes in each and we add links randomly in each group and between groups in such a way to keep the contact matrix fixed to its measured value.",
"More precisely, we first measure the density $d$ in the incomplete data as $2E/(N(N-1))$ where $N$ and $E$ are the numbers of nodes and edges in these incomplete data.",
"Knowing the number $n$ of missing nodes, we can deduce the number $e$ of additional links needed to keep the density constant when we add the $n$ missing nodes, through $d = 2(E+e)/((N+n)(N+n-1))$ .",
"We moreover transform the contact matrix of edge densities $\\rho _{XY}$ into a row-normalized contact matrix $C$ , in which the element $C_{XY} = \\rho _{XY} / \\sum _Z \\rho _{XZ}$ gives the probability for a node of group $X$ to have a link to a node of group $Y$ .",
"Then, for each missing edge, we proceed as follows: (i) we extract at random a node $i$ (among the total population of $N+n$ nodes); (ii) knowing the group $X$ that $i$ belongs to, we extract at random a target group $Y$ with probability given by $C_{XY}$ ; (iii) we draw at random a node $j$ in group $Y$ such that $i$ and $j$ are not yet linked, and we add a link between $i$ and $j$ .",
"The resulting network has the same number of nodes as the original network and the same contact matrix of densities and overall density as the sampled network.",
"Weights are finally assigned to the edges: weights are taken at random from the empirical distribution of weights of the contact network, which is known to be a robust feature of human contact patterns and does not depend on the context [15].",
"This method has been shown to yield good results when the incomplete data results from a uniform population sampling of the original contact network [24], which preserves the overall network density.",
"In Ref.",
"[29], it has also been shown to be able to create relevant surrogate data (i.e., yielding the same outcome as the original data when used in simulations of spreading processes) from contact diaries data.",
"Also in this latter case, the density of the network deduced from diaries was similar to the one of the original contact data.",
"In Ref.",
"[29] moreover, it has been shown that one can use a pool of publicly available contact duration statistics to assign weights to the edges of the surrogate data.",
"However, as the EGOref sampling method yields sampled network with densities smaller than the original data (Fig.",
"REF ) and as it is well known that density plays an important role in determining the outcome of a spreading process, we propose a second construction method, in which we assume in addition that the data includes the density of edges within one of the groups (chosen at random) in the original non-sampled network.",
"We therefore first measure the contact matrix of edge densities $\\rho _{XY}$ in the sampled data, as in the first method.",
"We then compute the ratio $f$ between the real density $\\tilde{\\rho }_{AA}$ in the original data for the group $A$ that is assumed to be known and the measured density $\\rho _{AA}$ .",
"We then rescale the average density of the graph by the factor $f$ , $d^{\\prime } = f \\times d$ and compute the new number $e^{\\prime }$ of edges to be added through $d^{\\prime } = 2(E+e^{\\prime })/((N+n)(N+n-1))$ .",
"The procedure is then the same as the previous one: we add the missing nodes and $e^{\\prime }$ links in order to preserve the normalized contact matrix $C_{XY}$ , and weights are taken from the empirical distribution of aggregated contact durations and assigned at random to edges.",
"The rationale behind the second method is that the infrastructure of wearable sensors able to measure this density could have been available for only one group in the population (or even a random fraction of one group), for instance, while only partial information from a survey is available for the other groups.",
"We then assume that the EGOref sampling method affects in a similar way all parts of the graph, so that a global rescaling of the density, which also rescales all the elements of the contact matrix of edge densities by the same factor, should yield values closer to the original data.",
"In the following, we will apply each method to contact networks sampled using the EGOref method for various values of the parameters $p$ and $N$ .",
"As Ref.",
"[30] has shown the similarity between the friendship network of the Thiers13 dataset and the outcome of the EGOref sampling applied to the Thiers13 contact network, for a specific parameter value, we will also apply these methods to construct surrogate contact data using the friendship network instead of a sampled version of the original contact network.",
"Our goal is to understand if it is possible to use incomplete datasets to estimate the epidemic risk in a population by constructing surrogate data and using them in the simulations of epidemic spread.",
"As a paradigm of epidemic process, we consider the Susceptible-Infectious-Recovered (SIR) model: in this model, nodes are initially all susceptible (S), except one in the Infectious state, chosen at random and seed of the process.",
"Each Susceptible (S) node $i$ can become infectious when in contact with an Infectious one $j$ .",
"This occurs at a rate $\\beta W_{ij} /T$ where $W_{ij}$ is the weight of the link $i-j$ and $T$ the total measurement time [31], [30] (i.e., the probability for $i$ to become infectious during a time step $dt$ is $\\beta W_{ij} dt /T$ ).",
"Infectious nodes become Recovered (R) at rate $\\mu $ and cannot be infected anymore.",
"The process ends when there are no Infectious nodes any more.",
"We perform numerical simulations of this model for each dataset on the original contact network, on the sampled networks at various values of the parameters $p$ and $N$ , and on the surrogate datasets built using the two methods described above (note that we will equivalently write “surrogate data” or “reconstructed networks” to describe the surrogate datasets).",
"For the Thiers13 case, we also perform simulations on the friendship networks and the corresponding surrogate data.",
"We also vary the ratio $\\beta /\\mu $ that modulates, in each given network, the impact of the modelled disease.",
"To quantify the epidemic risk, we measure in each simulation the epidemic size as given by the final fraction of recovered nodes.",
"We compare the distributions of epidemic sizes, the fraction of epidemics with size larger than $20\\%$ and the average size of these epidemics (the cut-off of $20\\%$ is chosen arbitrarily to distinguish between small and large epidemics; changing the value of this threshold does not alter our results).",
"We finally note that we consider here static versions of the contact networks, while the original SocioPatterns data provides temporally resolved contacts.",
"The EGOref sampling process indeed mimics a procedure yielding a static sample of the actual contact network.",
"In the context of models of infectious diseases with realistic timescales of several days, this can represent enough information to obtain an estimate of the epidemic risk, as discussed in Ref.",
"[31].",
"For faster spread, one could add to the surrogate construction method an additional step of building realistic contact timelines as in Ref [24].",
"We discuss separately the results obtained with the two datasets.",
"The Thiers13 one is indeed much more strongly “structured” than the InVS one, in the sense that the fraction of interactions occurring within each class is very high.",
"Moreover, all classes are of similar sizes and have similar link densities.",
"Classes are also arranged in groups of 2 or 3 classes corresponding to the major topic of study of their students.",
"In the InVS case, departments are of different sizes, their link densities vary more and the pattern of interactions between departments is less structured.",
"Table: Thiers13 dataset: Features of the original contact network, of the sampled one and of the surrogate data, for an EGOref sampling withp=30p=30 and N=135N=135.Table: Thiers13 dataset: Basic features of the empirical networks and of the reconstructed network (obtainedby applying the second method of construction of surrogate data to the friendship network).Table REF gives some basic features of the original, the EGOref sampled and the surrogate networks built with the two methods described above, for the EGOref parameter values yielding number of nodes and edges similar to the friendship network.",
"This corresponds to $N=135$ and $p=30$ .",
"The sampled network has a much lower density than the original contact data, smaller clustering and larger average shortest path.",
"The surrogate network built using the first method has by construction the same density, and has an even smaller clustering, while the second method yields values much closer to the original ones.",
"Moreover, the similarities between the contact matrices of the original, sampled and surrogate data all exceed $99\\%$ (see also Figure S3 of the Supplementary Information).",
"However, the fraction of intra-class links is larger in the reconstructed network than in the original one ($83\\%$ vs $69\\%$ , see Supplementary Information), while the fraction of contact durations these intra-class links carry is slightly smaller ($83\\%$ vs $93\\%$ ; note that since weights are put at random, the fraction of links and the fraction of weights they carry are the same in the reconstructed data).",
"Figures REF and REF compare the outcome of SIR simulations performed on the original contact network, on the EGOref sampled networks and on the surrogate data built using the two reconstruction methods, for various values of the sampling parameters.",
"Figure REF first displays the average size of large epidemics (i.e., the ones reaching at least $20\\%$ of the population) as a function of the spreading parameter $\\beta /\\mu $ .",
"As expected and already explored [30], simulations performed on the EGOref sampled network yield a strong underestimation of the epidemic risk with respect to results obtained with the use of the contact network, except at large $p$ and $N$ (in this case, the sampled network is almost equal to the original one, and the random assignment of weights to the links leads in fact to a slight overestimation of the epidemic risk [30]; this occurs only at unrealistically large values of $p$ ).",
"The use of surrogate data obtained with the first method improves the estimation of the epidemic risk with respect to the use of the sampled data but still leads to a clear underestimation for small and intermediate values of $p$ .",
"This is not unexpected given the reconstruction method maintains the density of the sampled network.",
"The second method, which leads to surrogate data with densities closer to the original one, allows to obtain a much better estimation of the epidemic risk.",
"Figure: Thiers13 dataset: Outcomes of SIR spreading simulations.",
"Average size of epidemics with sizeabove 20% as a function of the spreading parameter β/μ\\beta /\\mu for different values of pp and NN.The simulations are performed on contact network, EGOref sampled network and the reconstructed networksusing the two methods of reconstruction described in the text.Figure: Thiers13 dataset: Distributions of epidemic sizes for SIR spreading simulations.The simulations are performed on the contact network on the EGOref sampled network and onthe surrogate data (reconstructed networks) built using the two methods of reconstruction described in the text.The parameter of spreading β/μ\\beta /\\mu , the parameter pp and the number of sampled individuals NN are given above each plot.Figure REF focuses on the case of $N=135$ and displays the whole distributions of epidemic sizes obtained from SIR simulations for different values of the spreading parameter $\\beta /\\mu $ and of the parameter of sampling $p$ .",
"In all cases, the distributions obtained with the sampled network remain narrow and do not develop a peak at large values of epidemic sizes.",
"The distributions obtained with surrogate data are both broader but can differ strongly from each other depending on the value of $p$ .",
"At very large $p$ , the sampling procedure almost does not affect the network density, so that both methods yield very close outcomes; as discussed above, the random assignment of weights leads then to a peak at large values of the epidemic size that is shifted to values larger than with the original network.",
"For more realistic small and intermediate values of $p$ , the first method leads to distributions that are much narrower than the outcome of simulations on the original network, while the second method yields a much better agreement, albeit with a systematic small overestimation of the largest epidemic sizes.",
"We now turn to the case of the friendship network.",
"Table REF compares the main features of the contact network, the friendship network and the surrogate data obtained by the second method of reconstruction applied to the friendship network.",
"The reconstruction procedure allows to recover a density similar to the one of the contact data.",
"Moreover, the contact matrix of these three networks are very similar (similarity values of more than $98\\%$ ).",
"We however note that the fraction of within-classes links is larger in the friendship network ($75\\%$ ) than in the contact network ($69\\%$ ), and that this characteristics holds also for the surrogate data.",
"Figures REF and REF show the outcomes of epidemic spreading simulations performed on the empirical networks and on the reconstructed network.",
"While the simulations using the friendship network leads to a strong under-estimation of the epidemic risk, Figure REF shows that the use of the surrogate data yields a very good estimation of the fraction of epidemics with size above $20\\%$ and of the average epidemic size, across a large range of values of $\\beta /\\mu $ .",
"Figure REF displays the whole distributions of epidemic sizes for different values of the spreading parameter $\\beta /\\mu $ .",
"It confirms that the surrogate data yields distributions with more similar shapes to the contact network case than the friendship network.",
"However, the maximal sizes of epidemics are systematically overestimated, which might be ascribed to the larger fraction of weights on inter-class links in the surrogate data with respect to the contact network.",
"Figure: Thiers13 dataset: Outcome of SIR spreadingsimulations.",
"Fraction of epidemics with size above 20%as a function of the spreading parameter β/μ\\beta /\\mu (left)and average size of epidemic with size above 20% as afunction of the spreading parameter β/μ\\beta /\\mu (right).",
"Thesimulations are performed on the contactnetwork, on the friendship network and on surrogate data obtained by applying the second methodof construction to the friendship network.Figure: Thiers13 dataset: Distributions of epidemicsizes of SIR spreading simulations.",
"The simulations are performed on the contactnetwork, on the friendship network and on surrogate data obtained by applying the second methodof construction to the friendship network.",
"The value of β/μ\\beta /\\mu used is givenabove each plot.We now investigate the results obtained with the InVS dataset.",
"As discussed above, the structuration in departments leads to a less structured contact matrix than in the highschool case.",
"Table REF and Figure S4 of the Supplementary Information compare some characteristics of the sampled and surrogate data (with the second method of reconstruction) to the original network for $p=30$ and $N=93$ (i.e., a similar fraction of the population as used in the example of the Thiers13 dataset).",
"Here, even the second method of reconstruction leads to a network density smaller than the original one, even if much closer than for the sampled data.",
"The contact matrices of the sampled and of the surrogate data are very similar to the one of the original data (Figure S4 of the Supplementary Information), but, as in the previous case, the fraction of intra-department edges is larger in the surrogate data than in the original one ($58\\%$ versus $42\\%$ ) while the fraction of the weights these links carry is much smaller ($58\\%$ versus $76\\%$ in the original contact data).",
"Table: InVS dataset: Basic features of the contact network, of the EGOref sampled network (here withp=30p=30 and NN=40% of the total number of nodes) and of the surrogate data obtained usingthe second method of reconstruction.Figures REF and REF compare the outcomes of SIR simulations performed on the contact network, the sampled and the reconstructed networks, for various values of the sampling and spreading parameters.",
"As for the Thiers13 case, the simulations on the sampled networks strongly underestimate the epidemic risk, except obviously at large $p$ and $N$ .",
"The use of surrogate data generally improves the estimation of the epidemic risk, except at large $N$ for the first reconstruction method, as in this case almost no nodes have to be added so the network is almost unchanged by this method.",
"For small and intermediate $p$ , the second method gives better estimations of the average epidemic size while, at very large $p$ , the effect of assigning randomly weights leads as before to a slight overestimation for reconstructed networks.",
"At small $\\beta /\\mu $ and $p$ , the second method of reconstruction can even overestimate this size substantially.",
"Figure REF sheds some more light by displaying the whole distributions of epidemic sizes for several values of the spreading parameter $\\beta /\\mu $ and of the sampling parameter $p$ , for a rather small $N$ of the order of $40\\%$ of the total population.",
"While the distributions obtained with the second reconstructed networks are generally closer to the ones obtained with the original contact data than with the EGOref sampled network (which always leads to a very strong underestimation) or the first reconstruction method, a peak at very large epidemic sizes is observed for the reconstructed data, which is not present for the original contact data.",
"The maximal sizes of epidemics is thus overestimated.",
"As in the Thiers13 case, this effect is likely to be due to the fact that the amount of weights on inter-departments edges is larger in the surrogate data than in the original data, allowing for a more efficient spread across the whole population.",
"Figure: InVS dataset: Outcome of SIR spreadingsimulations.",
"Average size of epidemics with size above20% as a function of β/μ\\beta /\\mu for several values of pp and NN.",
"The simulations areperformed on the contact network on the EGOref sampled network and on the surrogate data obtainedusing two different methods of reconstruction.Figure: InVS dataset: Distributions of epidemic sizesof SIR spreading simulations.",
"The parameter of spreadingβ/μ\\beta /\\mu , the parameter pp and the number of sampledindividuals NN are given above each figure.",
"The simulationsare performed on contact network, EGOref sampled network andthe reconstructed networks using two different methods ofreconstruction.This paper positions itself in the context of the issue of data incompleteness in contact networks.",
"More specifically, since many datasets are de facto incomplete, it is important to assess how data incompleteness affects the outcome of data-driven simulations, how the resulting biases can be compensated, and how much data is needed for the simulations [24], [29], [30], [31], [32].",
"We have here considered the case of non-uniformly sampled contact data and focused on a sampling procedure designed to mimic data resulting from surveys or diaries [30].",
"This sampling procedure results in both population sampling, as not all individuals in the population are respondents, and in non-uniform link sampling, to mimic the fact that longer contacts have a larger probability to be remembered or to correspond also to friendship links.",
"We have applied this sampling procedure, called EGOref, on two datasets of contact networks in two different contexts and varied its two parameters, which determine the population participation rate and the fraction of sampled links.",
"The datasets concern populations structured in groups with very different mixing patterns: in a high school, the class structure strongly determines contacts, with more than $90\\%$ of the duration of contacts occurring within classes; in office buildings on the other hand, the population is divided into departments but the impact on the contacts is less strong.",
"As expected from previous investigations, using sampled data to run simulations of spreading processes leads to a strong underestimation of the epidemic risk, as quantified by the distribution of epidemic sizes.",
"We have therefore considered the issue of building surrogate data from the sampled data, such that simulations using the surrogate data yield a better estimation of the epidemic risk.",
"The first method we envisioned has been shown to yield good results in the context of uniform population sampling [24].",
"It is based on the fact that the uniform sampling keeps invariant the contact matrix giving the densities of links between groups in the population.",
"We have shown that the resulting surrogate data, when built from sampled data using the EGOref procedure, yields better estimations than the raw sampled data, but still yields a largely underestimated risk, since the link sampling of the EGOref procedure leads to a sampled network with a (possibly much) lower density than the original one, and the method of Ref.",
"[24] does not compensate for this bias.",
"This implies that more information is needed on the data than just the contact matrix of densities measured in the sampled data.",
"One of the simplest way to add information to the sampled data is to assume that the link density of one of the groups of the population is known: this can occur for instance if a measurement of contacts using wearable sensors is feasible only for a small subset of the population, for practical reasons, so that it is possible to correctly measure the link density for that group.",
"We therefore considered a second method of construction of surrogate data, in which we first rescaled all the elements of the contact matrix measured in the sampled data by the ratio between the known and measured densities of the \"known\" group.",
"The resulting surrogate contact network has thus a density closer to the original one.",
"In the case of a strongly structured population, such as the high school, we have obtained a strong improvement of the results and a good estimation of the epidemic risk, for a large range of realistic parameters, even if the maximum size of epidemics is slightly overestimated.",
"We have also shown that this method gives good results when applied on the data obtained from a survey asking students about their friendship relations.",
"In the case of the less-structured population (offices), we obtained a clear improvement of the epidemic risk estimation, but the probability of very large epidemics becomes strongly overestimated when using the surrogate data.",
"We have linked this overestimation with the fact that the total amount of weights carried by the inter-class or inter-department edges is larger in the surrogate than in the original data.",
"Indeed, as shown in Ref.",
"[24], intra-class and intra-department links tend to carry larger weights than inter-class and inter-department ones.",
"As weights are distributed randomly in the surrogate data, without taking this into account, the reconstruction tends to attribute more weight to the inter-groups edges with respect to the original data, which favors the spread.",
"Our results are overall two-sided.",
"On the one hand, it is remarkable that, using very little information, namely the contact matrix of the sampled data and if possible the knowledge of the density of one single group, using surrogate data instead of the raw sampled data leads to a strong improvement of the epidemic risk estimation as quantified by the fraction of large epidemics and the average size of these epidemics.",
"On the other hand, the lack of information about the precise values of the densities of links between pairs of groups, about the relative weights of intra- and inter-groups edges, as well as about the potential existence of small cohesive substructure, may lead, when the surrogate data is used, to distributions of epidemic sizes differing from the original ones, with for instance the over-estimation of the largest epidemic sizes and the presence of a peak at very large epidemic sizes.",
"This shows both that survey data can be effectively used to construct surrogate data, but also that more detailed information coming from data collection with wearable sensors is of enormous value, even if such collection concerns only a fraction of the population, as it allows (i) to have a correct estimation of the overall density, (ii) to obtain the distribution of contact durations, (iii) if enough sensors are available, to obtain a much better picture of the contact matrix and of the fraction of weights corresponding to intra- and inter-groups contacts, all elements having a role in the unfolding of spreading processes in a population.",
"Our study contributes to the discussion on the amount of details actually needed in contact data to be used in data-driven models.",
"It is worth noting some of its limitations.",
"We have focused here on one specific sampling model, while others might be of interest.",
"We argue that this procedure is particularly relevant as it mimics surveys or diaries, as described in Ref.",
"[30].",
"The procedure also assumes a uniform node sampling, while positive or negative correlations with actual contact activity might exist.",
"Additional investigations concerning such non-uniform population sampling would certainly be of interest, as well as studies on other datasets or on synthetic populations with tuneable characteristics.",
"Finally, the effect of using sampled or surrogate data for data-driven simulations of other types of processes ought to be investigated.",
"Preliminary simulations of the Susceptible-Infectious-Susceptible model (SIS), in which individuals who recover become again susceptible and can catch again the disease, show that results similar to the SIR case are obtained (not shown): the sampling leads to a strong underestimation of the epidemic risk that is compensated only partially by the first method of reconstruction; the second method gives excellent results for the very structured dataset but still leads to an underestimation of the epidemic risk for the offices dataset.",
"The issue of the performance of using surrogate data in more complex processes such as complex contagion remains open for future investigations.",
"A.B.",
"and J.F.",
"conceived and designed the study.",
"J.F.",
"performed the numerical simulations and the statistical analysis.",
"A.B.",
"and J.F.",
"wrote the manuscript.",
"The authors declare no competing financial interests.",
"Estimating the epidemic risk using non-uniformly sampled contact data: Supplementary Information Figure: Ratio between the density of the EGOref samplednetwork and the density of the whole contact network as afunction of the parameter pp and of the percentage of samplednodes for the Thiers13 dataset (left) and the InVSdataset (right).Figure: Similarity between the contact matrices of the sampled and original networks, as afunction of the parameter pp and of the percentage of samplednodes for the Thiers13 dataset (left) and the InVSdataset (right).Figure: Thiers13 dataset: Contact matrices giving the density ofedges between departments for the contact network, theEGOref network (with p=30p=30 and NN=40% of the total numberof nodes) and the reconstructed network using the secondmethod of reconstruction.",
"The similarities between the three matrices are all above 98%98\\%.Figure: InVS dataset: Contact matrices giving the density ofedges between departments for the contact network, theEGOref network (with p=30p=30 and NN=40% of the total numberof nodes) and the reconstructed network using the secondmethod of reconstruction.",
"Similarity between the matrix ofthe contact network and of the EGOref network: 93%93\\%, betweenthe matrix of the contact network and of the reconstructednetwork: 98%98\\%, between the matrix of the EGOref network andof the reconstructed network: 94%94\\%.Figure: Number of intra-class and inter-class edges for Thiers13 and InVS in the sampled network, at varying p and for N = 40%40\\%.Figure: Total weight carried by intra-class and inter-class edges for Thiers13 and InVS in the sampled network, at varying p and for N = 40%40\\%.Figure: Number of intra-class and inter-class edges for Thiers13 and InVS in the reconstructed network, at varying p and for N = 40%40\\%.The horizontal lines give the values in the original data.Figure: Total weight carried by intra-class and inter-class edges for Thiers13 and InVS in the reconstructed network, at varying p and for N = 40%40\\%.The horizontal lines give the values in the original data.Figure: Fraction of intra-class edges as a function of the sampling parameterpp and of the percentage of sampled nodes for the Thiers13 dataset (left) and the InVSdataset (right).Figure: Fraction of the total weight carried by intra-class edges in thesurrogate data as a function of the sampling parameterpp and of the percentage of sampled nodes for the Thiers13 dataset (left) and the InVSdataset (right)."
]
] | 1709.01548 |
[
[
"Thermal properties of a string bit model at large N"
],
[
"Abstract We study the finite temperature properties of a recently introduced string bit model designed to capture some features of the emergent string in the tensionless limit.",
"The model consists of a pair of bosonic and fermionic bit operators transforming in the adjoint representation of the color group SU(N).",
"Color confinement is not achieved as a dynamical effect, but instead is enforced by an explicit singlet projection.",
"At large N and finite temperature, the model has a non trivial thermodynamics.",
"In particular, there is a Hagedorn type transition at a finite temperature $T=T_H$ where the string degrees of freedom are liberated and the free energy gets a large contribution $\\sim N^{2}$ that plays the role of an order parameter.",
"For $T>T_H$, the low temperature phase becomes unstable.",
"In the new phase, the thermodynamically favoured configurations are characterized by a non-trivial gapped density of the SU(N) angles associated with the singlet projection.",
"We present an accurate algorithm for the determination of the density profile at $N=\\infty$.",
"In particular, we determine the gap endpoint at generic temperature and analytical expansions valid near the Hagedorn transition as well as at high temperature.",
"The leading order corrections are characterized by non-trivial exponents that are determined analytically and compared with explicit numerical calculations."
],
[
"Introduction and summary of results",
"Thorn's string bits models have been originally proposed as a description of superstrings where stability and causality are manifest [1], [2], [3], [4], [5].",
"In the framework of 't Hooft $1/N$ expansion and light-cone parametrization of the string, one considers the continuum limit of very long chains composed of elementary string bits transforming in the adjoint of the color gauge group $SU(N)$ .",
"When the number of bits $M$ gets large, the bit chains behave approximately like continuous strings with recovered Lorentz invariance.",
"On general grounds, this requires also the number of colors $N$ to be large.",
"For recent numerical studies at finite $M, N$ see [6].",
"The finite temperature thermodynamics of such string bit models is quite rich in the 't Hooft large $N$ limit.",
"Stringy low energy states turn out to be color singlets separated from non-singlets by an infinite gap in units of the characteristic singlet energy $\\sim 1/M$ [4], [5].",
"This means that color confinement emerges as a consequence of the dynamics.",
"Besides, the singlet subspace exhibits a Hagedorn transition [7] at infinite $N$ [8], [9] signalled by a divergence of the partition function for temperatures above a certain finite temperature $T>T_{\\rm H}$ .",
"As usual, this behaviour is generically associated with a density of states growing exponentially with energy as in the original dual resonance models [10] or modern string theory [11].",
"When string perturbation theory is identified with the 't Hooft $1/N$ expansion of string bit dynamics, the $N=\\infty $ Hagedorn transition is consistent the interpretation of $T_{\\rm H}$ in the free string as an artifact of the zero coupling limit [11] with a possible phase transition near $T_{\\rm H}$ to a phase dominated by the fundamental degrees of freedom of the emergent string theory.",
"Recently and remarkably, the Hagedorn transition of string bit models has been further clarified [12], and discovered also in simpler reduced systems where the singlet restriction is imposed from the beginning as a kinematical constraint and not as a dynamical feature [13], [14].",
"The starting point is the thermal partition function $Z = \\text{tr}\\,e^{-\\beta \\,(H+\\mu \\,M)},$ where $\\beta $ is the inverse temperature, $H$ the string bit model Hamiltonian, and $M$ is the bit number operator associated with the chemical potential $\\mu $ .",
"The partition function (REF ) is quite natural and has a simple origin from the light-cone description of the emergent string where $H = P^{-}/\\sqrt{2}$ , $\\mu \\,M = P^{+}/\\sqrt{2}$ , and thus $H+\\mu \\,M = P^{0} = (P^{+}+P^{-})/\\sqrt{2}$ [15].",
"The reduced model considered in [13], [14] is the projection of (REF ) on the subspace of singlets states with $H=0$ , i.e.",
"for the associated tensionless string, and is described by the simpler partition function $Z_{0}= \\text{tr}_{\\rm singlets}\\,x^{M}, \\qquad x = e^{-\\beta \\,\\mu }.$ Here, we shall focus on the simple model considered in [13] which consists of one pair of bosonic and fermionic string bits operators $a$ and $b$ , both transforming in the adjoint of $SU(N)$ .",
"Before singlet projection, the large $N$ limit of the string bit model describes a non-covariant subcritical light-cone string with no transverse coordinates and one Grassmann world-sheet field.",
"In general, an important feature of string bit models is that they can be formulated in a space-less fashion with emerging spatial transverse and longitudinal coordinates [5].",
"Thus, they may be regarded as a realization of 't Hooft holography [16].",
"Extensions to models with more bit species and discussion of $1/N$ corrections have been addressed in [14].",
"The bit number operator is $M = \\text{tr}(\\overline{a}\\,a+\\overline{b}\\,b)$ , where trace is in color space, and the projected partition function (REF ) can be computed by group averaging according to the analysis of [13], [14] The prefactor $(1-x)/(1+x)$ in (REF ) takes into account that the bit operators $a,b$ are traceless and hence are adjoints under $SU(N)$ .",
"$Z_{0} &= \\frac{1-x}{1+x}\\,\\int dU(\\mathbf {\\vartheta })\\,\\text{tr} (x^{M}\\,e^{i\\,G_{k}\\vartheta _{k}}) =\\frac{1-x}{1+x}\\,\\int dU(\\mathbf {\\vartheta })\\,\\prod _{1\\le k < \\ell \\le N}\\frac{1+x\\,e^{i\\,(\\vartheta _{k}-\\vartheta _{\\ell })}}{1-x\\,e^{i\\,(\\vartheta _{k}-\\vartheta _{\\ell })}} \\\\&= \\left(\\frac{1+x}{1-x}\\right)^{N-1}\\,\\int dU(\\mathbf {\\vartheta })\\,\\prod _{1\\le k < \\ell \\le N}\\frac{1+x^{2}+2\\,x\\,\\cos (\\vartheta _{k}-\\vartheta _{\\ell })}{1+x^{2}-2\\,x\\,\\cos (\\vartheta _{k}-\\vartheta _{\\ell })},$ where $G_{k}$ span the Cartan subalgebra of $U(N)$ .",
"The group integration in (REF ) is with respect to the normalized Haar measure $dU(\\mathbf {\\vartheta }) = \\frac{1}{N!\\,(2\\pi )^{N}}\\,\\int _{-\\pi }^{\\pi }d^{N}\\mathbf {\\vartheta }\\,\\prod _{1\\le k < \\ell \\le N}\\,4\\,\\sin ^{2}\\left(\\frac{\\vartheta _{k}-\\vartheta _{\\ell }}{2}\\right).$ In the 't Hooft large $N$ limit, the partition function (REF ) may be evaluated by saddle point methods.",
"The dominant saddle contribution is characterized by a continuous density of phases $\\rho (\\vartheta ; x)$ .",
"The analysis of [13], [14] shows that there exists, for $N=\\infty $ , a critical point $x_{\\rm H}=1/2$ .",
"For low temperatures $x<x_{\\rm H}$ , the stable solution of the saddle point condition is associated with a uniform constant density $\\rho (\\vartheta ; x) = 1/(2\\pi )$ and a partition function that has a finite $N\\rightarrow \\infty $ limit.",
"Instead, above the Hagedorn temperature, i.e.",
"for $x>x_{\\rm H}$ , the density $\\rho (\\vartheta ; x)$ is a non trivial function which is non zero on a finite subinterval $|\\vartheta |\\le \\vartheta _{0}(x)<\\pi $ .",
"In this gapped phase, the partition function has the leading large $N$ behaviour $\\log Z_{0} = N^{2}\\,F_{2}(x)+\\mathcal {O}(N\\,\\log N)$ where $F_{2}(x)$ is a function of the temperature growing monotonically from $F_{2}(1/2)=0$ up to $F_{2}(1) = \\log 2$ .",
"This function may be regarded as an order parameter that measures the smooth activation of the string bit degrees of freedom above the Hagedorn temperature.",
"This change of behaviour at $x=x_{\\rm H}$ is similar to what happens in the unitary matrix model transition [17] with the coupling constant of the latter being traded here by the temperature parameter $x$ .",
"Similar results have also been obtained in [18] for free adjoint $U(N)$ SYM on $S^{3}\\times \\mathbb {R}$ , see also [19].",
"More generally, in the context of AdS/CFT duality, it is an important issue to understand the thermodynamics of specific conformal theories with singlet constraint, see for instance [20], [18], [21], [22], [23] and the recent M-theory motivated study [24].",
"At temperatures above the Hagedorn transition, the precise form of the phase density profile $\\rho (\\vartheta ; x)$ is not known in analytic form, not even in the strict $N=\\infty $ limit.",
"The aim of this paper is to provide more information about this quantity and the related width $\\vartheta _{0}(x)$ .",
"To this aim, following the strategy of [18], we reconsider the solution of the partition function for $U(N)$ gauge theory on a 2d lattice at large $N$ for a broad class of single-plaquette actions found in [25].",
"We exploit it in order to cast the homogenous integral equation governing $\\rho (\\vartheta ; x)$ into an infinite dimensional linear system involving the higher (trigonometric) momenta of $\\rho $ .",
"Truncation to a finite number of modes provides an accurate algorithm for the determination of the density.",
"As we shall discuss, the outcome is not only numerical because some analytical information can be extracted from the above mentioned linear system.",
"Besides, analysis of the numerical data produced by the algorithm suggests how to extract precise analytical information from the integral equation in certain limits.",
"A summary of our results follows: For $x\\rightarrow x_{\\rm H}= 1/2$ the distribution gap closes, i.e.",
"$\\vartheta _{0}(x)\\rightarrow \\pi $ , with a correction vanishing as $\\sim (T-T_{\\rm H})^{1/4}$ , Near $x_{\\rm H}$ , the phase density approaches a Wigner semicircle law (in the variable $\\sin (\\vartheta /2)$ ).",
"$\\vartheta _{0}(x) &= \\pi -2\\,\\sqrt{2}\\,\\left(\\frac{x-x_{\\rm H}}{2}\\right)^{1/4}-\\frac{2\\sqrt{2}}{3}\\,\\left(\\frac{x-x_{\\rm H}}{2}\\right)^{3/4}+\\cdots , \\\\\\rho (\\vartheta ; x\\rightarrow x_{\\rm H}) &\\sim \\frac{1}{\\pi \\,\\sin ^{2}(\\vartheta _{0}/2)}\\,\\,\\left(\\sin ^{2}\\frac{\\vartheta _{0}}{2}-\\sin ^{2}\\frac{\\vartheta }{2}\\right)^{1/2}\\,\\cos \\frac{\\vartheta }{2}.$ At high temperature, $x\\rightarrow 1$ , the phase distribution collapses with $\\vartheta _{0}(x)\\sim T^{-1/3}$ .",
"A non uniform quadratic distribution is achieved inside $[-\\vartheta _{0}, \\vartheta _{0}]$ $\\vartheta _{0}(x\\rightarrow 1) &= \\left[6\\,\\pi \\,(1-x)\\right]^{1/3}+\\cdots , \\\\\\rho (\\vartheta ; x\\rightarrow 1) &\\sim \\frac{3}{4\\,\\vartheta _{0}^{3}}\\,\\left(\\vartheta _{0}^{2}-\\vartheta ^{2}\\right).$ The order parameter, i.e.",
"the function $F_{2}(x)$ appearing in the expansion $\\log Z_{0} = N^{2}\\,F_{2}(x) + \\cdots $ , admits the following expansions around $x=x_{\\rm H}$ and $x=1$ $F_{2}(x\\rightarrow x_{\\rm H}) = \\frac{1}{2}\\,(x-x_{\\rm H})+\\cdots ,\\qquad F_{2}(x\\rightarrow 1) = \\log 2-\\frac{3\\,(6\\pi )^{2/3}}{20}\\,(1-x)^{2/3}+\\cdots .$ The first expansion shows that $F_{2}(x)$ is linear just above $x_{\\rm H}$ as originally suggested in [12].",
"The second expansion shows the leading correction to the known infinite temperature limit $\\log 2$ .",
"The plan of the paper is the following.",
"In Sec.",
"() we present the integral equation for the phase density $\\rho (\\vartheta ; x)$ discussing first some of its features at finite $N$ .",
"Then, our proposed $N=\\infty $ self-consistent algorithm and its predictions are presented.",
"Sec.",
"() is devoted to the derivation of various analytical expansions.",
"In particular, in Sec.",
"(REF ) and (REF ) we discuss the expansion of the phase density near the Hagedorn temperature and at high temperature $x\\rightarrow 1$ .",
"The behaviour of the partition function near $x=x_{\\rm H}$ and $x=1$ is considered in Sec.",
"(REF ).",
"Conclusions and open directions are briefly discussed in a final section."
],
[
"Self-consistent determination of the density at $N=\\infty $",
"As discussed in [13], the determination of the saddle point $\\mathbf {\\vartheta }$ of (REF ) for finite $N$ amounts to finding the solution of the set of equations $\\sum _{\\ell \\ne k}\\cot \\left(\\frac{\\vartheta _{k}-\\vartheta _{\\ell }}{2}\\right)-\\frac{4\\,x\\,(1+x^{2})\\,\\sin (\\vartheta _{k}-\\vartheta _{\\ell })}{1+x^{4}-2\\,x^{2}\\,\\cos (2\\,(\\vartheta _{k}-\\vartheta _{\\ell }))}=0.$ The numerical solution of (REF ) for $N=100$ and $x>x_{\\rm H}$ is shown in the left panel of Fig.",
"(REF ) where one appreciates the opening of a gap whose width increases as $x\\rightarrow 1$ .",
"The distribution of the roots $\\mathbf {\\vartheta }$ is non trivial, i.e.",
"it is not uniform.",
"Precisely at the $N=\\infty $ Hagedorn transition point, $x=x_{\\rm H}$ , the gap closes as $N\\rightarrow \\infty $ according to the finite size scaling $\\pi -\\vartheta _{0} = \\mathcal {O}\\left(N^{-\\delta }\\right)$ with $\\delta \\simeq 1/4$ , as shown in the right panel.",
"This slow convergence of observables at increasing $N$ means that a reliable characterization of the model for $N=\\infty $ is difficult by extrapolation from finite $N$ data.",
"Besides, we are interested in analytical expansions near Hagedorn transition as well as at high temperature.",
"For these reasons, we present in the next section a self-consistent accurate treatment of the $N=\\infty $ limit that will prove itself to be more effective than finite $N$ extrapolation.",
"At $N\\rightarrow \\infty $ , the roots of (REF ) are described by a smooth density $\\rho (\\vartheta ; x)$ which is positive for $|\\vartheta |<\\vartheta _{0}(x)$ and vanishes at $\\vartheta =\\pm \\vartheta _{0}$ .",
"Taking the continuum limit of (REF ), the function $\\rho (\\vartheta ; x)$ obeys the homogeneous integral equation $& \\int _{-\\vartheta _{0}(x)}^{\\vartheta _{0}(x)} d\\vartheta \\,G(\\vartheta ^{\\prime }-\\vartheta ; x)\\,\\rho (\\vartheta ; x) = 0, \\\\& G(\\vartheta ; x) = \\cot \\left(\\frac{\\vartheta }{2}\\right)-\\frac{4\\,x\\,(1+x^{2})\\,\\sin \\vartheta }{x^{4}+1-2\\,x^{2}\\,\\cos (2\\vartheta )}.$ To solve it, we exploit the remarkably simple identity $\\frac{x\\,(1+x^{2})\\,\\sin \\vartheta }{x^{4}+1-2\\,x^{2}\\,\\cos (2\\vartheta )} = \\sum _{n=0}^{\\infty }x^{2n+1}\\,\\sin ((2n+1)\\,\\vartheta ),$ that holds in our case, i.e.",
"for $0<x<1$ and real $\\vartheta $ .",
"The expansion (REF ) allows to write (REF ) in the form $& \\int _{-\\vartheta _{0}(x)}^{\\vartheta _{0}(x)}d\\vartheta \\,\\cot \\left(\\frac{\\vartheta ^{\\prime }-\\vartheta }{2}\\right)\\,\\rho (\\vartheta ; x) =\\\\& \\qquad \\qquad 4\\,\\sum _{n=0}^{\\infty }x^{2n+1}\\,\\int _{-\\vartheta _{0}(x)}^{\\vartheta _{0}(x)}d\\vartheta \\,\\sin ((2n+1)\\,(\\vartheta ^{\\prime }-\\vartheta ))\\,\\rho (\\vartheta ; x).$ Taking into account that the density is expected to be even, $\\rho (\\vartheta ; x) = \\rho (-\\vartheta ; x)$ , we can further simplify (REF ) and obtain $& \\int _{-\\vartheta _{0}(x)}^{\\vartheta _{0}(x)}d\\vartheta \\,\\cot \\left(\\frac{\\vartheta ^{\\prime }-\\vartheta }{2}\\right)\\,\\rho (\\vartheta ; x) = \\\\& \\qquad \\qquad 4\\,\\sum _{n=0}^{\\infty }x^{2n+1}\\,\\sin ((2n+1)\\vartheta ^{\\prime })\\,\\int _{-\\vartheta _{0}(x)}^{\\vartheta _{0}(x)}d\\vartheta \\,\\cos ((2n+1)\\,\\vartheta )\\,\\rho (\\vartheta ; x).$ It is convenient to recast (REF ) in the apparently inhomogeneous form $\\int _{-\\vartheta _{0}(x)}^{\\vartheta _{0}(x)}d\\vartheta \\,\\cot \\left(\\frac{\\vartheta ^{\\prime }-\\vartheta }{2}\\right)\\,\\rho (\\vartheta ) =4\\,\\sum _{n=0}^{\\infty }\\rho _{n}\\,x^{2n+1}\\,\\sin ((2n+1)\\vartheta ^{\\prime }),$ where we have introduced the trigonometric momenta $\\rho _{n}(x) = \\int _{-\\vartheta _{0}(x)}^{\\vartheta _{0}(x)}d\\vartheta \\,\\cos ((2n+1)\\,\\vartheta )\\,\\rho (\\vartheta ;x).$ As discussed in [18], the general solution of the problem (REF ) is known and reads A self-consistent interpretation of the solution (REF ) first appeared in [18] in a different context, see also the recent application [24].",
"$\\rho (\\vartheta ) &= \\frac{1}{\\pi }\\sqrt{\\sin ^{2}\\left(\\frac{\\vartheta _{0}}{2}\\right)-\\sin ^{2}\\left(\\frac{\\vartheta }{2}\\right)}\\,\\sum _{m=1}^{\\infty }Q_{m}\\,\\cos \\left[(m-\\tfrac{1}{2})\\,\\vartheta \\right],\\\\Q_{m} &= \\mathop {\\sum _{\\ell =0}^{\\infty }}_{\\frac{m+\\ell -1}{2} = 0, 1, 2, \\dots }4\\,x^{m+\\ell }\\rho _{\\frac{m+\\ell -1}{2}}\\,P_{\\ell }(\\cos \\vartheta _{0}),$ where, for brevity, we have omitted the explicit dependence on $x$ .",
"We can now truncate the expansion (REF ) by keeping only a fixed number of terms $\\mathbf {\\rho }^{(K)} = \\lbrace \\rho _{k}\\rbrace _{k=0,\\dots , K}$ .",
"The density is thus written in terms of the finite set of quantities $\\mathbf {\\rho }^{(K)}$ .",
"Replacing the density expression into (REF ) we obtain a homogeneous linear system $\\mathcal {M}^{(K)}(x, \\vartheta _{0}^{(K)})\\,\\mathbf {\\rho }^{(K)}=0.$ Non trivial solutions exists only if $\\det \\mathcal {M}^{(K)}(x, \\vartheta _{0}^{(K)})=0,$ which is the condition that determines the approximate gap width $\\vartheta _{0}^{(K)}$ for each $x>x_{\\rm H}$ .",
"Once $\\vartheta _{0}^{(K)}$ is computed, we solve (REF ) for the eigenvector $\\mathbf {\\rho }^{(K)}$ and obtain the density from (REF ).",
"The eigenvector normalization is fixed by requiring $\\rho (\\vartheta )$ to be normalized with unit integral.",
"To appreciate the accuracy of the method, we show in Tab.",
"(REF ) the solution $\\vartheta _{0}^{(K)}(x)$ of (REF ) evaluated at various $x>x_{\\rm H}$ , and with $K$ growing from 10 to 34.",
"Table: Solution of the condition () for various x>x H x>x_{\\rm H} and increasing number ofmodes KK.",
"As a guide, we write in red the digits that change moving to the next row.The convergence appears to be exponential in $K$ although with a decreasing rate as $x\\rightarrow 1$ .",
"This is because the effect of the convergence factors $x^{2n+1}$ in (REF ) is reduced.",
"Nevertheless, still at $x=0.9$ , the accuracy is of about 6 digits for $K=34$ .",
"Working out the prediction of the above algorithm in the interval $x_{\\rm H}< x <1$ we obtain the black curve in Fig.",
"(REF ) where we plot $\\sin (\\vartheta _{0}(x)/2)$ vs. $x$ .",
"To appreciate the convergence with $N$ , we also show some sample points obtained at finite $N=20,50,100$ from the solution of (REF ).",
"The dashed curves are analytical approximations valid around $x_{\\rm H}$ and $x=1$ derived in the next section, i.e.",
"The expansion of $\\vartheta _{0}(x)$ in (REF ) is an equivalent form of (REF ).",
"$x &\\rightarrow x_{\\rm H}:\\qquad \\sin \\frac{\\vartheta _{0}(x)}{2} = 1-\\sqrt{\\frac{}{}}{x-\\frac{1}{2}}{2}-\\frac{1}{4}\\left(x-\\frac{1}{2}\\right)+\\cdots ,\\\\x &\\rightarrow 1:\\qquad \\vartheta _{0}(x) = \\left[6\\,\\pi \\,(1-x)\\right]^{1/3}+\\cdots .$ Figure: Temperature dependence of the phase gap ϑ 0 (x)\\vartheta _{0}(x).",
"The black central curve is the result ofthe N=∞N=\\infty algorithm keeping K=34K=34 modes.",
"The blue symbols show the finite NN resultsat N=20,50,100N=20, 50, 100 from the solution of ().",
"The dashed brown and red curves are the analyticalapproximations in ().As we shall discuss later, the self-consistent determination of $\\rho (\\vartheta )$ provides also analytical information near the Hagedorn transition.",
"We shall see that only the first term in (REF ) survives.",
"This shows that $\\rho (\\vartheta )$ is well described by $x \\rightarrow x_{\\rm H}:\\qquad \\rho (\\vartheta ; x) \\rightarrow \\frac{1}{\\pi \\,\\sin ^{2}(\\vartheta _{0}/2)}\\,\\left(\\sin ^{2}\\frac{\\vartheta _{0}}{2}-\\sin ^{2}\\frac{\\vartheta }{2}\\right)^{1/2}\\,\\cos \\frac{\\vartheta }{2},$ which is Wigner semi-circle law in the variable $\\sin (\\vartheta /2)$ , well known in the theory of random symmetric matrices.",
"Strictly at $x=x_{\\rm H}$ this reduces to $\\rho (\\vartheta ; 1/2) = \\frac{1}{\\pi }\\,\\cos ^{2}\\frac{\\vartheta }{2}$ .",
"For $x\\rightarrow 1$ , we have found that the phase density is very well described by a quadratic law inside its support, i.e.",
"$x \\rightarrow 1:\\qquad \\rho (\\vartheta ; x) \\rightarrow \\frac{3}{4\\,\\vartheta _{0}^{3}}\\,(\\vartheta _{0}^{2}-\\vartheta ^{2}),$ as will also be confirmed analytically in the next section.",
"The limiting forms (REF ) and (REF ) are tested in Fig.",
"(REF ).",
"In the two panels, we show the exact density profile from the self-consistent algorithm and the predictions (REF ) and (REF ) at $x=0.501$ and $x=0.99$ respectively.",
"The horizontal scale in the two panels is quite different due to the wide variation of $\\vartheta _{0}(x)$ .",
"Up to a rescaling, the gross shape of the two densities is roughly similar, although the two regimes are clearly associated with different functions (semi-circle and quadratic).",
"Figure: Temperature dependence of the phase density ρ(ϑ;x)\\rho (\\vartheta ; x).",
"Left: Just above the Hagedorntransition.",
"The black line is the density obtained by plugging in () the solution of ().The orange points are sample evaluations of () and superimpose quite well.",
"Right:Near x=1x=1.",
"Again, the black line is the result from the self-consistent algorithm, whilethe orange points are samples of ().",
"Apart from the very ends of the distribution, the agreementis very good."
],
[
"Analytical expansions",
"In this section, we derive the analytical expansions (REF ) characterizing the phase density $\\rho (\\vartheta ;x)$ and its endpoint $\\vartheta _{0}(x)$ near the Hagedorn transition and at very high temperature $x\\rightarrow 1$ ."
],
[
"Opening of the gap near the Hagedorn transition",
"The condition (REF ) may be solved perturbatively around $x=x_{\\rm H}$ .",
"It is an algebraic equation in the variables $x$ and $h = \\sin (\\vartheta _{0}^{(K)}/2)$ whose complexity increases rapidly with $K$ .",
"Just to give an example, for the almost trivial case $K=1$ we have the constraint $K=1:\\quad & 1+2 h^2 \\left(h^2-2\\right) x = 0.$ The branch starting at $(x,h) = (1/2,1)$ has the expansion $K=1:\\quad h = 1-\\left(\\frac{x-x_{\\rm H}}{2}\\right)^{1/2}-\\frac{1}{4}\\,(x-x_{\\rm H})+\\frac{3}{2}\\left(\\frac{x-x_{\\rm H}}{2}\\right)^{3/2}+\\cdots .$ For $K=2$ , the condition (REF ) is much more complicated and reads $K=2:\\quad & 1+2 h^2 \\left(h^2-2\\right) x+2 h^2 \\left(100 h^{10}-312 h^8+366 h^6-200 h^4+51 h^2-6\\right) x^3\\\\&+4 h^8\\left(25 h^8-152 h^6+288 h^4-224 h^2+64\\right) x^4 = 0.$ Expanding again around $x_{\\rm H}$ we find $K=2:\\quad h = 1-\\left(\\frac{x-x_{\\rm H}}{2}\\right)^{1/2}-\\frac{1}{4}\\,(x-x_{\\rm H})-\\frac{33}{2}\\left(\\frac{x-x_{\\rm H}}{2}\\right)^{3/2}+\\cdots .$ Repeating the procedure for increasing $K$ , one finds that the first two terms of the expansion of $h$ are independent on $K$ , $h = 1-\\left(\\frac{x-x_{\\rm H}}{2}\\right)^{1/2}-\\frac{1}{4}\\,(x-x_{\\rm H})+\\text{c}^{(K)}\\,\\left(\\frac{x-x_{\\rm H}}{2}\\right)^{3/2}+\\cdots .$ while the values of the third coefficient are $c^{(K)} = \\frac{3}{2},-\\frac{33}{2},-\\frac{199}{6},-\\frac{793}{18},-\\frac{76153}{1530},-\\frac{2484163}{47430},-\\frac{5915131}{110670},-\\frac{32551537891}{604368870},\\cdots .$ Increasing $K$ up to 30 and working with exact rational values, this sequence converges numerically to an asymptotic value that can be estimated by Wynn acceleration algorithm [26].",
"The results are quite stable and independent on the Wynn algorithm parameter and give $c^{(\\infty )} = -54.0888227$ .",
"Such a large value suggests that the expansion (REF ) could be only asymptotic, as expected near a phase transition.",
"Plugging the expansion (REF ) in the linear system (REF ) one finds that all $\\rho _{n>0}$ vanish linearly with $x-x_{\\rm H}$ .",
"This leaves the semi-circle asymptotic density that we wrote in (REF )."
],
[
"Density collapse at high temperature",
"The expansion in the high temperature regime $x\\rightarrow 1$ is more complicated and cannot be obtained from the formalism of Section () because all $\\rho _{n}$ have a non trivial limit.",
"Nevertheless, we can check consistency of the quadratic density (REF ) by studying the $x\\rightarrow 1$ limit of the integral equation (REF ).",
"This is non trivial due to the $x$ dependence of $\\vartheta _{0}(x)$ .",
"Analysis of the numerical data computed in Section () suggest that $\\vartheta _{0} = \\kappa \\, (1-x)^{1/3}+\\cdots .$ Actually, this Ansatz may be self-consistently checked in the following together with the determination of the amplitude $\\kappa $ .",
"To this aim, the density can be rescaled $\\rho (\\vartheta ) = \\frac{1}{\\vartheta _{0}}\\widetilde{\\rho }(\\vartheta /\\vartheta _{0}),\\qquad \\int _{-1}^{1}du\\, \\widetilde{\\rho }(u)=1,$ and the integral equation (REF ) can be written in the new variables $\\int _{-1}^{1}du^{\\prime }\\bigg [\\cot \\left(\\vartheta _{0}\\,\\frac{u-u^{\\prime }}{2}\\right)-\\frac{4\\,x\\,(1+x^{2})\\,\\sin (\\vartheta _{0}(u-u^{\\prime })}{x^{4}+1-2\\,x^{2}\\,\\cos (2\\,\\vartheta _{0}\\,(u-u^{\\prime }))}\\bigg ]\\,\\widetilde{\\rho }(u^{\\prime }) = 0.$ Let us denote the kernel in the integral as $G(u; x)$ , it is useful to plot it as a function of $u$ at various $x\\rightarrow 1$ with the substitution $\\vartheta _{0} \\rightarrow \\kappa \\,(1-x)^{1/3}$ .",
"This is shown in the left panel of Fig.",
"(REF ) where $\\kappa =1$ to see what is going on.",
"As $x\\rightarrow 1$ , the kernel splits into the sum of a linear background plus a $\\delta ^{\\prime }(u)$ term which is localized in a region of width $\\sim (1-x)^{2/3}$ .",
"The background part comes from the naive $x\\rightarrow 1$ expansion of $G(u,x) = \\cot \\left(\\kappa (1-x)^{1/3}\\frac{u}{2}\\right)-\\frac{4\\,x\\,(1+x^{2})\\,\\sin (\\kappa \\,(1-x)^{1/3}\\,u)}{x^{4}+1-2\\,x^{2}\\,\\cos (2\\,\\kappa \\,(1-x)^{1/3}\\,u)}.$ This gives $G(u,x) = -\\frac{1}{2}\\,\\kappa \\,u\\,(1-x)^{1/3}+\\cdots ,$ which may be used for $|u|\\gg (1-x)^{2/3}$ .",
"The second contribution comes from the integral (REF ) after a zooming associated with $u = (1-x)^{2/3}\\,\\xi $ .",
"At leading order, we get At leading order, the integration region of $\\xi $ is symmetric and we can drop all odd contributions, some of which requires a principal value definition.",
"$\\int _{-1}^{1}du^{\\prime } \\,G(u-u^{\\prime }; x)\\,\\widetilde{\\rho }(u^{\\prime }) &= -2\\,(1-x)^{1/3}\\,\\widetilde{\\rho }\\,^{\\prime }(u)\\,\\int _{-\\infty }^{\\infty }d\\xi \\,\\frac{1}{\\kappa \\,(1+\\kappa ^{2}\\,\\xi ^{2})}+\\cdots \\\\& = -\\frac{2\\,\\pi }{\\kappa ^{2}}\\,(1-x)^{1/3}\\,\\widetilde{\\rho }\\,^{\\prime }(u)+\\cdots ,$ which has indeed the form of a $\\delta ^{\\prime }(u)$ contribution in the kernel.",
"Consistency of the power $1/3$ in the $1-x$ factor in (REF ) and (REF ) is important to get a non trivial result and checks our scaling hypothesis.",
"In summary, at this order in the $x\\rightarrow 1$ expansion, the integral equation becomes simply $-\\frac{1}{2}\\,\\kappa \\,u-\\frac{2\\,\\pi }{\\kappa ^{2}}\\,\\widetilde{\\rho }\\,^{\\prime }(u)=0.$ This gives both the quadratic density and the constant $\\kappa $ in $\\vartheta _{0}(x)$ , see (REF ), $\\widetilde{\\rho }(u) = \\frac{3}{4}\\,(1-u^{2}),\\qquad \\kappa = (6\\,\\pi )^{1/3},$ in agreement with (REF ) and (REF ).",
"Figure: Detailed structure of some relevant integral kernels.",
"Left: This panel showsthe evaluation of the function GG defined in () and evaluated with κ=1\\kappa =1.",
"The plotshows that the kernel is composed of a linear background plus a singular part which is localizedin a region of width ∼(1-x) 2/3 \\sim (1-x)^{2/3} and approximates, as a distribution, a δ ' \\delta ^{\\prime } contribution.Right: This panel showsthe evaluation of the function HH defined in ().",
"Similar to the left panel, we identify in thex→1x\\rightarrow 1 limit a quadratic background plus a narrow δ\\delta like contribution fully discussed in the text."
],
[
"The transition order parameter",
"Further consistency checks of the derived aymptotic densities come from the analysis of the large $N$ behaviour of $\\log Z_{0}$ , i.e.",
"the free energy up to trivial factors.",
"The function $F_{2}(x)$ appearing as the leading term in the large $N$ expansion $\\log Z_{0} = N^{2}\\,F_{2}(x) + \\mathcal {O}(N\\log N),$ can be computed from the density $\\rho (\\vartheta ; x)$ as the double integral $F_{2}(x) = \\frac{1}{2}\\int _{-\\vartheta _{0}(x)}^{\\vartheta _{0}(x)} d\\vartheta \\,d\\vartheta ^{\\prime }\\,\\log \\bigg [4\\,\\sin ^{2}\\left(\\frac{\\vartheta -\\vartheta ^{\\prime }}{2}\\right) \\,\\frac{1+x^{2}+2\\,x\\,\\cos (\\vartheta -\\vartheta ^{\\prime })}{1+x^{2}-2\\,x\\,\\cos (\\vartheta -\\vartheta ^{\\prime })}\\bigg ]\\,\\rho (\\vartheta )\\,\\rho (\\vartheta ^{\\prime }).$ As we mentioned in the introduction, the function $F_{2}(x)$ can be regarded as an order parameter for the Hagedorn transition.",
"It is zero for $0<x<x_{\\rm H}$ and increases monotonically for $x>x_{\\rm H}$ .",
"The maximum value is attained at $x=1$ and is $F_{2}(1) = \\log 2$ .",
"This follows from the exact relation [14] $Z_{0}(x\\rightarrow 1) = \\left(\\frac{2}{1-x}\\right)^{N-1}\\frac{R_{N}}{N!",
"},$ where $R_{N}$ is the number of labeled Eulerian digraphs with $N$ nodes.",
"Basic information about this sequence may be found at the OEIS link http://oeis.org/A007080.",
"The asymptotic behaviour of $R_{N}$ has been recently computed in [14] and reads $R_{N}\\stackrel{N\\rightarrow \\infty }{\\sim } \\left(\\frac{2^{N}}{\\sqrt{\\pi N}}\\right)^{N-1}\\,e^{-1/4}\\,\\sqrt{N}\\,\\left[1+\\frac{3}{16N}+\\mathcal {O}(N^{-2})\\right],$ from which we get the term $N^{2}\\,\\log 2$ in $\\log Z_{0}$ .",
"Near the Hagedorn transition, we can evaluate $F_{2}(x)$ using the distribution (REF ).",
"Direct expansion around $x=x_{\\rm H}$ gives a leading linear behaviour $F_{2}(x) = \\text{c}\\,(x-x_{\\rm H})+\\cdots ,$ where $\\text{c}$ is a constant that is obtained from a rather involved finite double integral.",
"It can be safely extracted from the ratio $F_{2}(x_{\\rm H}+\\varepsilon )/\\varepsilon $ as $\\varepsilon \\rightarrow 0$ .",
"Using $\\varepsilon = 0-10^{-3}$ and a fit of the form $a+b\\,\\sqrt{\\varepsilon }$ , we reproduce the numerical data very well with $\\text{c}=1/2$ with a precision of one part in $10^{6}$ .",
"For this reason, we assume that this value of $\\text{c}$ is exact.",
"A rather small range of values of $\\varepsilon $ is needed suggesting again that the expansion around the Hagedorn temperature is only asymptotic.",
"This is quite reasonable in this case because $F_{2}(x)$ is certainly not analytic at $x_{\\rm H}$ – it is zero below the Hagedorn temperature and non zero above it.",
"The linear behaviour (REF ) was originally predicted in [12].",
"The computation of the leading correction to $F_{2}(x)$ for $x\\rightarrow 1$ is more tricky and, again, it is again important to analyze in details the structure of the rescaled kernel with the leading order expression for $\\vartheta _{0}$ , i.e.",
"$H(u; x) = \\frac{\\vartheta _{0}^{2}}{2}\\,\\log \\bigg [4\\,\\sin ^{2}\\left(\\vartheta _{0}\\frac{u}{2}\\right) \\,\\frac{1+x^{2}+2\\,x\\,\\cos (\\vartheta _{0}\\,u)}{1+x^{2}-2\\,x\\,\\cos (\\vartheta _{0}\\,u)}\\bigg ],\\qquad \\vartheta _{0} = \\kappa \\,(1-x)^{1/3}.$ A plot of $H(u,x)$ as a function of $u$ with $x\\rightarrow 1$ is shown in the right panel of Fig.",
"(REF ).",
"There is a naive quadratic contribution that comes from the direct expansion of $H$ , $H(u; x) = \\log 2-\\frac{(3\\,\\pi )^{2/3}}{4\\cdot 2^{1/3}}\\,u^{2}\\,(1-x)^{2/3}+\\cdots .$ Integrating over $\\vartheta $ , $\\vartheta ^{\\prime }$ in (REF ), this gives a first contribution to $F_{2}(x)$ $F_{2}^{(a)}(x) = \\log 2 -\\frac{(6\\,\\pi )^{2/3}}{20}\\,(1-x)^{2/3}.$ A second contribution comes from zooming in the region $u-u^{\\prime }\\sim (1-x)^{2/3}$ as in the previous section.",
"This gives a second $\\delta (u-u^{\\prime })$ -like contribution leading to $F_{2}^{(b)}(x) = \\frac{3}{10}(1-x)^{2/3}\\int _{-\\infty }^{\\infty }d\\xi \\log \\left(\\frac{\\kappa ^{2}\\xi ^{2}}{1+\\kappa ^{2}\\xi ^{2}}\\right)= -\\frac{3\\pi }{5\\kappa }\\,(1-x)^{2/3}.$ Summing (REF ) and (REF ), we obtain the expansion (REF ).",
"In Fig.",
"(REF ), we show the evaluation of (REF ) using the leading order density (REF ) with $\\vartheta _{0}$ as in (REF ).",
"We also show the approximation (REF ) as well as the exact numerical data points obtained in [13].",
"The agreement is remarkable despite the fact that we used the asymptotic density valid for $x\\rightarrow 1$ .",
"This shows that $F_{2}(x)$ appears to be little dependent on the fine structure of the density itself.",
"This is further confirmed by the fact that evaluation of $F_{2}(x)$ with the $x\\rightarrow x_{\\rm H}$ density or with the $x\\rightarrow 1$ one are practically indistinguishable up to $x\\simeq 0.9$ .",
"Figure: Evaluation of the order parameter F 2 (x)F_{2}(x).",
"The black curve is the result of the evaluationof () using the asymptotic quadratic density in ().",
"Blue triangles areexact numerical data points taken from .",
"Finally, the brown dashed curve is theanalytical approximation in ()."
],
[
"Conclusions",
"In this paper we have considered the large $N$ thermodynamics of a simple $SU(N)$ string bit model devised to capture the tensionless limit of the associated string.",
"The model lives in the color singlet sector and involves a projection implemented by a suitable group average, i.e.",
"integration over $U\\in SU(N)$ .",
"Dominant configurations are characterized by a non trivial density $\\rho (\\vartheta ; T)$ of the $U$ invariant phases $\\vartheta _{1}, \\dots , \\vartheta _{N}$ .",
"We have analyzed the model in the gapped phase, i.e.",
"for temperatures above the Hagedorn transition $T>T_{\\rm H}$ where $\\rho (\\vartheta ; T)$ is non zero only in the interval $|\\vartheta |\\le \\vartheta _{0}(T)<\\pi $ .",
"By means of numerical and analytical tools, we have discussed in some details the temperature dependence of the phase density $\\rho (\\vartheta ; T)$ including the gap endpoint $\\vartheta _{0}(T)$ .",
"Our results provide quantitative information about the crossover from the low to high temperature phases in the considered model.",
"It remains an open question to understand precisely which changes occur in models with more bits and if $1/N$ corrections are taken into account to resolve the phase transition.",
"The corrections we found at $N=\\infty $ contains non trivial power exponents, see (REF ) and (REF ).",
"In particular, the phase density support $[-\\vartheta _{0},\\vartheta _{0}]$ opens a gap in the $\\vartheta $ distribution of width $2\\,(\\pi -\\vartheta _{0})\\sim (T-T_{\\rm H})^{1/4}$ near the Hagedorn transition.",
"Besides, the support collapses with $\\vartheta _{0}\\sim T^{-1/3}$ for $T\\gg T_{\\rm H}$ .",
"It could be interesting to understand these relations in the context of a finite but large $N$ double scaling limit as in the Hagedorn transition for IIB string theory in an anti-de Sitter spacetime [27], [28]."
]
] | 1709.01801 |
[
[
"Predicting Melting Points by the Graovac-Pisanski Index"
],
[
"Abstract Theoretical molecular descriptors alias topological indices are a convenient means for expressing in a numerical form the chemical structure encoded in a molecular graph.",
"The structure descriptors derived from molecular graphs are widely used in Quantitative Structure-Property Relationship (QSPR) and Quantitative Structure-Activity Relationship (QSAR).",
"In this paper, we are interested in the Graovac-Pisanski index (also called modified Wiener index) introduced in 1991 by Graovac and Pisanski, which encounters beside the distances in a molecular graph also its symmetries.",
"In the QSPR analysis we first calculate the Graovac-Pisanski index for different families of hydrocarbon molecules using a simple program and then we show a correlation with the melting points of considered molecules.",
"We show that the melting points of the alkane series can be very effectively predicted by the Graovac-Pisanski index and for the rest of considered molecules (PAH's and octane isomers) the regression models are different, but we establish some correlation with the melting points for them as well."
],
[
"Theoretical molecular descriptors (also called topological indices) are graph invariants that play an important role in chemistry, pharmaceutical sciences, materials science and engineering, etc.",
"The value of a molecular descriptor must be independent of the particular characteristics of the molecular representation, such as atom numbering or labeling.",
"We model molecules of hydrocarbons by the corresponding molecular graph, where the vertices are the carbon atoms and the edges of the graph are the bonds between them.",
"One of the most investigated topological indices is the Wiener index introduced in 1947 [17].",
"This index is defined as the sum of distances between all the pairs of vertices in a molecular graph.",
"Wiener showed that the Wiener index is closely correlated with the boiling points of alkane molecules.",
"In order to take into account also the symmetries of a molecule, Graovac and Pisanski in 1991 introduced the modified Wiener index [6].",
"However, the name modified Wiener index was later used for different variations of the Wiener index and therefore, Ghorbani and Klavžar suggested the name Graovac-Pisanski index[5], which is also used in this paper.",
"Very recently, the Graovac-Pisanski index of some molecular graphs and nanostructures was extensively studied[1], [3], [7], [8], [9], [13].",
"Moreover, the closed formulas for the Graovac-Pisanski index of zig-zag nanotubes were computed[14].",
"The only known connection of the Graovac-Pisanski index with some molecular properties is the correlation with the topological efficiency [2].",
"Therefore, it was pointed out by Ghorbani and Klavžar [5] that the QSPR or QSAR analysis should be performed in order to establish correlation with some other physical or chemical properties of molecules.",
"On the other hand, there is no known molecular descriptor well correlated to the melting points of molecules, since graph-theoretical abstraction in most cases disregards many information that are relevant for the value of the melting point[12], [15].",
"The symmetries of a molecule play an important role in the process of melting[11].",
"The more symmetrical the molecules are, easier it is for them to stack together.",
"Consequently, the fewer spaces there are between them and so the melting point is expected to be higher.",
"Therefore, since the Graovac-Pisanski index considers the symmetries of a molecule, in this paper we investigate its correlation with the melting points of molecules.",
"We work with the data set of molecules proposed by the International Academy of Mathematical Chemistry, in particular alkane series, polyaromatic hydrocarbons (PAH's) and octane isomers.",
"We show that for alkane series the melting point is very well correlated with the Graovac-Pisanski index and for PAH's the correlation is a little bit weaker.",
"For octane isomers, the melting points are good correlated with the number of symmetries.",
"However, we conclude that for alkanes the size of a molecule contributes the most to its melting point.",
"A graph $G$ is an ordered pair $G = (V, E)$ of a set $V$ of vertices (also called nodes or points) together with a set $E$ of edges, which are 2-element subsets of $V$ (more information about some basic concepts in graph theory can be found in a book written by West[16]).",
"Having a molecule, if we represent atoms by vertices and bonds by edges, we obtain a molecular graph.",
"The graphs considered in this paper are all finite and connected.",
"The distance $d_G(x,y)$ between vertices $x$ and $y$ of a graph $G$ is the length of a shortest path between vertices $x$ and $y$ in $G$ (we often use $d(x,y)$ for $d_G(x,y)$ ).",
"The Wiener index of a graph $G$ is defined as $\\displaystyle {W(G) = \\frac{1}{2} \\sum _{u \\in V(G)} \\sum _{v \\in V(G)} d_G(u,v)}$ .",
"Moreover, if $S \\subseteq V(G)$ , then $\\displaystyle {W(S) = \\frac{1}{2} \\sum _{u \\in S} \\sum _{v \\in S} d_G(u,v)}$ .",
"An isomorphism of graphs $G$ and $H$ with $|E(G)|=|E(H)|$ is a bijection $f$ between the vertex sets of $G$ and $H$ , $f: V(G)\\rightarrow V(H)$ , such that for any two vertices $u$ and $v$ of $G$ it holds that if $u$ and $v$ are adjacent in $G$ then $f(u)$ and $f(v)$ are adjacent in $H$ .",
"When $G$ and $H$ are the same graph, the function $f$ is called an automorphism of $G$ .",
"The composition of two automorphisms is again an automorphism, and the set of automorphisms of a given graph $G$ , under the composition operation, forms a group ${{\\rm Aut}}(G)$ , which is called the automorphism group of the graph $G$ .",
"The Graovac-Pisanski index of a graph $G$ , $GP(G)$ , is defined as $GP(G) = \\frac{|V(G)|}{2 |{{\\rm Aut}}(G)|} \\sum _{u \\in V(G)} \\sum _{\\alpha \\in {{\\rm Aut}}(G)} d_G(u, \\alpha (u)).$ Next, we mention some important concepts of group theory.",
"If $G$ is a group and $X$ is a set, then a group action $\\phi $ of $G$ on $X$ is a function $\\phi :G \\times X \\rightarrow X$ that satisfies the following: $\\phi (e,x) = x$ for any $x \\in X$ (where $e$ is the neutral element of $G$ ) and $\\phi (gh,x)=\\phi (g,\\phi (h,x))$ for all $g,h \\in G$ and $x \\in X$ .",
"The orbit of an element $x$ in $X$ is the set of elements in $X$ to which $x$ can be moved by the elements of $G$ , i.e.",
"the set $\\lbrace \\phi (g,x) \\, | \\, g \\in G \\rbrace $ .",
"If $G$ is a graph and ${{\\rm Aut}}(G)$ the automorphism group, then $\\phi : {{\\rm Aut}}(G) \\times V(G) \\rightarrow V(G)$ , defined by $\\phi (\\alpha ,u) = \\alpha (u)$ for any $\\alpha \\in {{\\rm Aut}}(G)$ , $u \\in V(G)$ , is called the natural action of the group ${{\\rm Aut}}(G)$ on $V(G)$ .",
"It was shown by Graovac and Pisanski[6] that if $V_1, \\ldots , V_t$ are the orbits under the natural action of the group ${{\\rm Aut}}(G)$ on $V(G)$ , then $GP(G) = |V(G)| \\sum _{i=1}^t \\frac{1}{|V_i|}W(V_i).$ As an example, we calculate the Graovac-Pisanski index for one of the octane isomers, more precisely for 2-methyl-3-ethyl-pentane, see Figure REF .",
"Figure: Molecular graph GG of 2-methyl-3-ethyl-pentane.We do the calculation in two different ways.",
"First, we calculate directly by the definition.",
"There are all together 3 nontrivial (different from the identity) automorphisms of the considered molecular graph $G$ : $\\alpha _1 = (1\\ 6)(2)(3)(4)(5)(7)(8),\\ \\alpha _2 = (1)(2)(3)(4\\ 7)(5\\ 8)(6),\\ \\alpha _3 = (1\\ 6)(2)(3)(4\\ 7)(5\\ 8).$ Therefore, $GP(G) & = d(1,\\alpha _1(1)) + d(6,\\alpha _1(6)) + d(4,\\alpha _2(4)) + d(7,\\alpha _2(7)) + d(5,\\alpha _2(5)) + d(8,\\alpha _2(8)) \\\\& + d(1,\\alpha _3(1)) + d(6,\\alpha _3(6)) + d(4,\\alpha _3(4)) + d(7,\\alpha _3(7)) + d(5,\\alpha _3(5)) + d(8,\\alpha _3(8)) \\\\& = 2+2+2+2+4+4+2+2+2+2+4+4 \\\\& = 32.$ Now, we calculate the index by using orbits.",
"The vertex set is partitioned into 5 orbits $V_1=\\lbrace 1,6\\rbrace ,\\,V_2=\\lbrace 2\\rbrace ,\\,V_3=\\lbrace 3\\rbrace ,\\,V_4=\\lbrace 4,7\\rbrace ,\\,V_5=\\lbrace 5,8\\rbrace $ and the Graovac-Pisanski index of 2-methyl-3-ethyl-pentane is calculated as $ GP(G) = |V(G)| \\sum _{i=1}^5 \\frac{1}{|V_i|}W(V_i)=8\\left( \\frac{2}{2}+\\frac{2}{2}+\\frac{4}{2}\\right)=32\\,.",
"$ In this section we present an algorithm which was used to compute the Graovac-Pisanski index and the number of automorphisms of a graph.",
"The algorithm contains two special functions, i.e.",
"calculateAutomorphisms and calculateDistances.",
"Let $G$ be a graph represented by a adjacency matrix with vertices $1,2,\\ldots ,n$ .",
"The function calculateAutomorphisms determines all the automorphisms of graph $G$ and saves them in a set $A$ .",
"One possibility is to go through all the permutations of the set $\\lbrace 1,2,\\ldots , n \\rbrace $ and check whether it is an automorphism.",
"Note that for the graph automorphism problem (which is the problem of testing whether a graph has a nontrivial automorphism) it is still unknown whether it has a polynomial time algorithm or it is NP-complete[10].",
"The function calculateDistances computes the matrix $M$ from the adjacency matrix of $G$ .",
"The element $M_{i,j}$ , $i,j \\in \\lbrace 1,2,\\ldots , n \\rbrace $ of $M$ represents the distance between vertices $i$ and $j$ in $G$ .",
"Note that this algorithm is known as Floyd-Warshall algorithm [4] and has the time complexity $O(n^3)$ .",
"[H] InputInputOutputOutput $GP(G)$ and $|{\\rm Aut}(G)|$ CAcalculateAutomorphisms CDcalculateDistances $A \\leftarrow $ ($G$ ) $M \\leftarrow $ ($G$ ) $X \\leftarrow 0$ each $\\alpha \\in A$ $i=1$ to $n$ $X \\leftarrow X + M_{i,\\alpha (i)}$ $GP(G) \\leftarrow \\frac{n}{2|A|}X$ $|{\\rm Aut}(G)| \\leftarrow |A|$ The Graovac-Pisanski index We notice that the time complexity of the algorithm is not polynomial if we go through all the permutations.",
"However, for many chemical graphs the set of all the automorphisms of a graph can be easily obtained by hand.",
"In such a case, we can skip the first line of the algorithm and consequently, it becomes much more efficient.",
"In this section, we have 31 alkane molecules, which are divided into two sets.",
"The training set contains 26 molecules and the test set has 5 molecules (in Table REF the molecules in the test set are written in bold style).",
"The data show that a logarithmic function $f(x)= a \\ln x + b$ fits the best among all elementary functions.",
"After performing nonlinear regression on the training set, we obtain $a=34,196$ and $ b=68,575$ , see Figure REF .",
"Therefore, the following equation can be used to predict the melting points of the alkane series $MP = 34,\\!196 \\ln GP + 68,\\!575.$ Figure: Nonlinear regression on the training set.We use the above formula to compute the predicted melting point $\\widehat{MP}$ for every molecule from the test set, see Table REF .",
"From Table REF we can see that the average error on the test set is less than 2%.",
"Moreover, the statistics shows very good correlation since $R^2=0,\\!9847$ .",
"To conclude the section, we test the obtained formula on all the molecules from Table REF and we obtain Table REF .",
"We can see that the average error is around 4%.",
"Figure REF shows the comparison between the melting points and predicted melting points.",
"Figure: Melting points and predicted melting points for alkane molecules.In this section, we consider 20 PAH molecules.",
"In the first part, we divide these molecules into two sets (the training set contains 16 molecules and the test set has 4 molecules) and perform linear regression with respect to the Graovac-Pisanski index.",
"In the second part, we improve the correlation by performing multilinear regression with respect to the Graovac-Pisanski index, the Wiener index and the number of automorphisms.",
"The data for the PAH molecules is collected in Table REF .",
"The linear regression results in the function (see Figure REF ) $MP = 0,\\!6501 \\, GP + 10,\\!926.$ Figure: Linear regression on the training set for PAH's.We use the above formula to compute the predicted melting point $\\widehat{MP}$ for every molecule from the test set, see Table REF .",
"From Table REF we can see that the average error on the test set is around 11% and the statistics shows quite good correlation since $R^2=0,\\!8388$ .",
"To improve the correlation, we perform multilinear regression on the whole set of PAH's as described in the beginning of the section.",
"It results in the formula $MP = -46,\\!248 + 13,\\!038\\,(\\# {\\rm Aut}) + 0,\\!446\\,{GP} + 0,\\!235\\,{W}.$ The regression statistics (multiple $R$ is 0,946; $R^2$ is 0,894; adjusted $R^2$ is 0,874; and standard error is 30,665) shows better correlation than the linear regression.",
"As the last one, we consider the set of 14 octane isomers.",
"However, all together there are 18 octane isomers, but for 4 of them, the data for the melting point was unavailable.",
"We compute the Graovac-Pisanski index and the number of automorphisms for these molecules, see Table REF .",
"It turns out that the correlation between the Graovac-Pisanski index and the melting point is not that good ($R^2$ is 0,2423), but there is good correlation ($R^2$ is 0,9687) between the number of automorphisms and the melting point, see Figure REF .",
"However, we can see that the data for 2,2,3,3-tetramethylbutane is standing out.",
"If we exclude this molecule from our observation, the correlation between the number of automorphisms and the melting points becomes much weaker (close to zero), but the correlation between the Graovac-Pisanski index and the melting points becomes slightly stronger ($R^2$ is 0,4537).",
"Therefore, we can conclude that the size of an alkane molecule contributes the most to its melting point.",
"Figure: Linear regression between the number of automorphisms and the melting point for octane isomers.In recent years revived Graovac-Pisanki index was considered and calculated in different ways for some families of graphs, such as fullerene graphs or catacondensed benzenoid graphs, but almost no chemical application was known so far.",
"In this paper we use the QSPR analysis and show that this index is well correlated with the melting points of the alkane series and polyaromatic hydrocarbon molecules.",
"Beside that the connection between the number of graph automorphisms of the octane isomers and theirs melting point is established.",
"In some sense this result in similar to the seminal paper in the field of molecular descriptors [17] by Wiener from 1947, where it was shown that the boiling points of the alkane series can be predicted from the Wiener index.",
"Therefore, these results might contribute to further development in the area of obtaining the Graovac-Pisanski index for different molecular graphs.",
"The authors Matevž Črepnjak and Petra Žigert Pleteršek acknowledge the financial support from the Slovenian Research Agency, research core funding No.",
"P1-0285 and No.",
"P1-0297, respectively.",
"The author Niko Tratnik was financially supported by the Slovenian Research Agency."
]
] | 1709.01832 |
[
[
"Unveiling the physics behind the spectral variations of \"changing-look\"\n quasars with optical polarimetry"
],
[
"Abstract A handful of active galactic nuclei (AGN) have shown strong spectral variations in the optical band between epochs that are years apart.",
"The appearance or disappearance of broad emission lines in their spectra completely changes their classification.",
"Since their nucleus orientation cannot change in such short timescales another physical interpretation has to be found.",
"Several scenarios are competing to explain their changing-look nature and, for the first time, we conduct polarized radiative transfer Monte Carlo simulations for all the models.",
"We demonstrate that all interpretations have distinctive features in both total optical flux and continuum polarization such as proposed by Hutsem\\'ekers and collaborators.",
"Distinguishing between the different scenarios is thus straightforward.",
"We apply our results on the changing-look quasar J1011+5442 and confirm the conclusions found by Hutsem\\'ekers and collaborators: in this specific case, the disappearance of the broad emission lines is due to a change in accretion rate."
],
[
"Introduction",
"Active galactic nuclei (AGN) are parsec-scale powerful lighthouses that reside in the core of galaxies.",
"Their intrinsic luminosity often outshines the light from their host galaxy and such amount of radiation can only be produced by accretion onto a supermassive black hole [34], [40].",
"This central engine irradiates its close environment which, in the first half parsec, is mainly constituted of electrons and atoms.",
"Through photoionization by continuum photons, emission lines emerge from the gas that has a compact and equatorial morphology [13].",
"Submitted to Keplerian motion, this region naturally emits broadened lines.",
"However not all AGN show broad emission lines, which lead to the classification of quasarsQuasar is the historical term coined when the first quasi-stellar objects of this type have been discovered.",
"Unlike Seyfert galaxies, quasars are radio-loud, cosmological AGN situated in the distant Universe.",
"according to the presence (type-1) or absence (type-2) of those specific spectral features [37].",
"We only realized that all AGN types might be correlated by their inclination when [1] discovered broad emission lines in the polarized flux of a type-2 Seyfert galaxy.",
"It follows that type-1 AGN are seen from the polar direction, where there is no equatorial obscuration, while type-2 AGN are seen from the equatorial plane.",
"The broad line region (BLR) is thus hidden behind an opaque wall of dust for the later type.",
"The classification has several intermediate categories ranging from 1.2 to 1.8, all showing broad emission lines, with an increasing [O iii]$\\lambda $ 5007/H$\\beta $ flux ratio [33].",
"The last type, 1.9, includes AGN which only show broad H$\\alpha $ lines.",
"Once optically classified, an AGN is not expected to flicker between two extreme types, at least not in a human timescale.",
"Nevertheless, several “changing-look” AGN were discovered in the past decades.",
"Examples of those AGN are Mrk 1018 [8], NGC 1365 [36], SDSS J015957.64+003310.5 [19] or IC 751 [35].",
"Those AGN are characterized by rapid (month to years timescales) dimming/brightening in total flux, and they often show variation in the strength of their broad emission lines.",
"In several cases the broad emission disappears and an AGN changes from type-1 to type-1.9 or 2 [8].",
"The disappearance of the broad emission lines are often associated with a drop in the continuum flux, suggesting a common physical mechanism [19].",
"Several scenarios are proposed to explain the optical-ultraviolet-X-ray variations.",
"First is a possible obscuration of the central source by a large amount of dust or gas crossing the observer's line-of-sight.",
"This could be due to the motion of dense cloudlets originating from the torus [15] or a variation in the height of the circumnuclear dust region [43].",
"This variation can occur after an peak in accretion activity and the resulting intense radiation field wipes out the outer layers of dust.",
"Conversely, the dusty torus can also recover from this activity and restore its amount of dust.",
"A different interpretation was suggested by [19] based on the pioneering work of [11].",
"In this scenario, a change in ionizing flux from the central engine, related to a smaller accretion activity, may cause the broad emission features and the BLR to disappear.",
"A fainter continuum ultimately reduces the emission line intensity of the BLR since there is not enough photons to ionize the gas.",
"The BLR (and later the torus) should disintegrate at small bolometric luminosities (L$_{\\rm bol} \\le $ 5$\\times $ 10$^{39}$ M$^{2/3}_7$ , [10]) as a low accretion efficiency would not be able to sustain the required cloud flow rate [12], [11].",
"Understanding the physics behind the spectral variations of changing-look AGN is thus fundamental to probe the life cycle of galaxies, how AGN evacuate or replenish their gaseous/dust material, and what is the true morphology of the innermost AGN regions.",
"To solve these questions, we present radiative transfer simulations of the polarized optical light emerging from those different scenarios.",
"We aim to check whether optical polarimetry can unveil the true interpretation behind the type-1 to type-1.9/2 changing-look appearance of those peculiar AGN.",
"In Sect.",
"we introduce the Monte Carlo code and the models to be investigated.",
"We explore the total flux and polarimetric signal of the different models in Sect.",
"and apply our results to a specific object in Sect. .",
"We conclude in Sect.",
"about the importance of polarimetry to elegantly unveil the correct case-by-case physical interpretation."
],
[
"Modeling changing-look AGN",
"To model the optical continuum polarization expected from a changing-look AGN, we use the Monte Carlo radiative transfer code stokes [16], [26], [25].",
"This numerical tool was used to model, predict, fit and interpret the polarization signatures of large variety of sources, from exoplanets to AGN [24], [27].",
"stokes works from the near-infrared to the hard X-ray bands and simulates the random and successive interactions of light with matter under a large number of possible geometries and scales.",
"The code allows to virtually explore all kinds of geometrical configurations for an AGN, from the central supermassive black hole to extended outflows.",
"All the scattering physics is included in the code [16] and the user can register the linear and circular polarization at all polar and azimuthal viewing angles.",
"The code, in its basic version, is available on-line at: http://www.stokes-program.info/.",
"It is the same version of stokes that was used in this publication.",
"Figure: Unscaled AGN model used in this paper.",
"The central optical source(yellow star) irradiates an electron-filled broad line region(BLR, in orange) and a coplanar, more distant, dust-filledcircumnuclear region (torus, in red).",
"The source also irradiatestorus-collimated, ionized, ejection winds along the polar axis(primrose yellow).",
"Quantitative details such as sizes,half-opening angles or composition are given in the text.Our basic AGN model is presented in Fig.",
"REF (unscaled).",
"The model consists of a central point like, isotropic, mono-energetic source that irradiates $\\lambda $ 550 nm photons.",
"We selected this wavelength according to the effective wavelength midpoint for a standard V filter [3].",
"This is also the waveband where a large number of historical AGN polarization measures were achieved [4], [44].",
"The scattering-induced polarization of AGN being almost-wavelength independent in the ultraviolet, optical and near-infrared bands [31], [7], [45], our modeling is thus conservative for broadband observations.",
"Only the amount of depolarization from external sources can vary with wavelength, allowing for easier polarimetric detections in the ultraviolet waveband where the host contamination is less important [48].",
"At a distance of 0.01 pc from the continuum source, we set up the inner radius of the broad line region according to the theoretical relation between the BLR inner radius and the monochromatic flux at $\\lambda $ 550 nm for a 10$^{44}$ erg.s$^{-1}$ AGN luminosity [9].",
"The BLR geometry is a flared disk with half-opening angle of 20$^\\circ $ from the equatorial plane [26].",
"It is uniformly filled with electrons and the Keplerian motion of the BLR is fixed to 4000 km.s$^{-1}$ [32].",
"The radial optical depth in the V-band of the BLR region is set to 3 at the present stage but will vary in the photoionizing continuum dimming scenario.",
"The outer radius of the BLR is set at the inner radius of the dusty torus, physically constrained by the dust sublimation radius.",
"Following the monitoring observations made by [46] in the optical and near-infrared wave bands for a set of Seyfert-1 galaxies, we choose a torus inner radius of 0.1 pc.",
"The torus outer radius is fixed to 5 pc, following recent interferometric observations [47].",
"The half-opening angle of the circumnuclear region is set to 45$^\\circ $ and its optical depth is larger than 50 in the V-band.",
"Note that the value of the torus half-opening angle will also change in the scenario that explains the spectral variations of changing-look quasars by dust obscuration.",
"Finally, the torus half-opening angle collimates a pair of conical ejection winds along the polar axes.",
"The winds onset at a radial distance of 0.01 pc from the central source and stops at 10 pc.",
"It is filled with electrons [1] and its radial optical depth is of the order of 0.1 [26].",
"This AGN model will be used to infer the optical linear continuum polarization of changing-look AGN according to the scenarios detailed in Sect. .",
"For clarity purpose, we focus our modeling on type-1 to type-1.9/2 transitions for the remained of this paper."
],
[
"AGN polarization and ionizing continuum dimming",
"We present the results of our computations for the generic AGN model in Fig REF .",
"The top panel is the total optical flux normalized to the maximum flux observed at an inclination of 0$^\\circ $ .",
"We see that, for type-1 viewing angles (up to the torus horizon, i.e.",
"45$^\\circ $ ), the flux is almost constant but suddenly drops at types-2 orientations.",
"This transition is also visible in the middle and bottom panels, the continuum linear polarization degree and polarization position angle, respectively.",
"At type-1 views, the degree of polarization is small ($<$ 4%) and associated with a polarization angle of 90$^\\circ $ (parallel to the symmetry axis of the system).",
"The polarization degree is null at perfect pole-on inclinations due to the symmetry of the system.",
"The polarization degree then increases with inclination until the transition between type-1 and type-2 line-of-sights.",
"The polarization position angle rotates down to 0$^\\circ $ (perpendicular to the symmetry axis of the system) and the polarization degree increases up to 54% at equatorial views.",
"In the case of a dimming of the continuum source, the amount of radiation scattering of the BLR region will be smaller and the flux will drop.",
"This will result in a lower normalized flux at all inclinations and the lack of photoionizing radiation will prevent the emission of strong broad emission lines.",
"However it will have no impact on the intrinsic polarization degree and angle since they are relative quantities.",
"For this interpretation, we expect to see no variations in the (linear continuum) polarization properties of the source."
],
[
"BLR-fading scenario",
"In the case of a strong dimming (bolometric luminosity $\\le $ ∼10$^{42}$ erg.s$^{-1}$ ), the work of [11] suggests that the broad line emission region follows an evolutionary sequence that is directly related to the accretion rate of the compact source.",
"If so, the BLR becomes less dense as the accretion rate is decreasing, eventually disappearing [12].",
"To investigate this interpretation, we decrease the optical depth of the BLR from 3 to 0.01 to simulate the progressive disintegration of the region.",
"Following [12], the timescales for the BLR and torus disappearance is short, i.e a few Keplerian orbits.",
"Note that, in our modeling, we consider a strong coupling between the BLR and the electron-scattering disk predicted by [44].",
"The equatorial electron region is necessary to reproduce both the observed polarization position angle of type-1 AGN and the intrinsic polarization of broadened lines.",
"The electron disk is considered to be a continuous flow between the torus and the inner parts of the BLR ($ibid.$ ).",
"In the work of [44], the two regions were disconnected since their code only considered single scattering but there is no physical reasons for decoupling of the two regions when multiple scattering is enabled.",
"Fig.",
"REF shows the normalized total flux at $\\lambda $ 550 nm, the polarization degree and the polarization position angle of the AGN model as a function of the BLR optical depth $\\tau $ .",
"The observer's inclination is fixed at a typical type-1 inclination angle of 30$^\\circ $ with respect to the symmetry axis of the model.",
"At $\\tau $ = 3, the degree of polarization is maximum (about 2.79%) and the polarization position angle is equal to 90$^\\circ $ .",
"This corresponds to the expected polarization angle of type-1 AGN and the relatively high degree of polarization obtained is due to the optical thickness of the BLR.",
"The efficiency of the equatorial medium to intercept radiation and scatter it towards the observer is maximal.",
"The resulting polarization degree, despite being diluted by the unpolarized central source seen by transmission through the wind, is thus superior to 1%.",
"For higher densities of the BLR, multiple scattering will start to depolarize radiation.",
"However, when the BLR region starts to decrease in density due to a lower accretion rate, less electron equatorial scattering is happening.",
"The polarization degree thus decreases gradually with $\\tau $ .",
"It reaches a minimum around $\\tau $ = 0.05 where the polarization position angle rotates from 90$^\\circ $ to 0$^\\circ $ .",
"Equatorial scattering is then inefficient to produce the observed polarization angle of Seyfert-1s and the polarization degree remains below 0.15% for the lower optical depths of the BLR.",
"The observed total flux of the AGN follows the same trend as the polarization degree, decreasing by 25% when the BLR is extremely optically thin.",
"We thus see that, if the changing-look nature of the few AGN observed is due to the disappearance of the BLR, we expect a strong diminution of the total flux.",
"The polarization degree should also decrease and an orthogonal flip of the polarization position angle is expected in the optical band.",
"Figure: Variation of the optical total flux (normalized, top),optical linear polarization degree (middle) and polarizationposition angle (bottom) with respect to the gradual fadingof the BLR region.",
"The observer's inclination is set to 30 ∘ ^\\circ"
],
[
"Varying the amount of dusty obscuration",
"Fig.",
"REF presents the results of our modeling for the alternative interpretation behind the changing-look nature of AGN.",
"In this case, it is believed that the amount of dust absorption along the observer's line-of-sight is varying [30].",
"To model it, we fix the optical depth of the BLR to 3 and add a removable outer layer with an opening angle of 5$^\\circ $ on the top of the circumnuclear region that is blocking the view of a type-1.9/2 observer.",
"When the dusty layer is on the top of the torus, the half-opening angle of the circumnuclear dust material changes from 45$^\\circ $ to 50$^\\circ $ (measured from the equatorial plane).",
"Consequently, the half-opening angle of the polar winds is reduced by 5$^\\circ $ with respect to the model presented in the previous section.",
"It value is then fixed to 40$^\\circ $ .",
"The layer of dust intercepting the observer's line-of-sight can be added or removed at will (see the gray region in Fig.",
"REF ).",
"By shaving off the outer layers of the torus, we allow a type-1.9/2 observer to have a direct (type-1) view of the central AGN engine without changing the nucleus orientation.",
"This is represented by the dashed line in Fig.",
"REF .",
"We examine the model from 30 observer's inclinations equally distributed in cosine angle between 30$^\\circ $ and 60$^\\circ $ .",
"We focus on this range of inclinations as it represents the estimated transition angles between type-1 and type-2 AGN categories, slightly depending on the radio-loudness of the quasars [2], [21], [39], [23], [22].",
"Fig.",
"REF presents the results of the two scenarios: a), when the dusty layer is present on the top of the torus (in black) and b), when this layer is removed (in red).",
"The top panel shows the difference in normalized total flux between the two scenarios as a function of the observer's inclination.",
"We see that, at the inclinations where the observer's viewing angle crosses the removable layer of dust (shaded grey region on Figs.",
"REF and REF ), if there is a variable amount of obscuration, the flux drops by a factor that is inclination dependent.",
"At maximum, the flux difference is a factor 40 between the unobscured (type-1) and obscured (type-1.9/2) model.",
"The transition angle at which the two models change from type-1 to type-2 strongly depends on the presence/absence of the dusty layer.",
"At a given inclination, one can see the impact of variable obscuration onto the computed flux.",
"The smaller flux observed in the type-1.9/2 scenario is due to radiation scattered onto the polar winds, plus a minor contribution of backscattering of photons onto the torus funnel opposite to the observer's side.",
"This change in scattering geometry (from direct light to polar-scattered radiation) has a profound impact on the polarization degree and angle.",
"For a given inclination within the grey area (where variable obscuration occurs), the difference in terms of optical continuum polarization for the two scenarios is large.",
"If the line-of-sight toward the nucleus is uncovered, the polarization degree is only a couple of percent, while it rises to 10 – 20% when the torus horizon is obscured with dust.",
"The exact variation of polarization degree between the two states of the changing-look AGN depends on the inclination of the observer.",
"Nevertheless, it is always a significant change than can be as high as 20%.",
"This increase in polarization is directly due to the different paths radiation has to follow to escape the type-1 and type-1.9/2 models.",
"Additionally, the change in spectral type is systematically associated with an orthogonal rotation of the polarization position angle.",
"Figure: Results of the two scenarios described inthe text: with a dusty layer on top of the torus(black dots) and when the layer is removed(red triangles).",
"The top panel shows the normalizedtotal flux, the middle panel is the polarizationdegree, and the bottom panel is the observed polarizationposition angle for the two cases.",
"The grey box representsthe angular region obscured by the removable dusty layer."
],
[
"Discussion",
"We saw that the three interpretations (the ionizing continuum dimming, the BLR-fading and the variation of dusty obscuration scenarios) have very clear and distinctive features in total flux and polarimetry.",
"In the first case, a change in continuum radiation due to a variation in accretion rate will not change the observed polarization properties of the AGN.",
"If the accretion rate becomes insufficient to sustain the BLR (and torus), the total flux of the AGN should change by a factor $\\sim $ 25% simply because of the importance of equatorial scattering that redirect photons towards the observer.",
"With decreasing accretion rates onto the central supermassive black hole, the polarization degree is also decreasing as electron scattering inside the BLR becomes inefficient.",
"The broad line emission will appear increasingly weaker and the polarization position angle finally rotates by 90$^\\circ $ .",
"In the last case, the difference in total flux before and after a change of look is much higher.",
"Up to 98% of the central flux is absorbed by the circumnuclear dusty medium.",
"Radiation escapes from the AGN by scattering inside the polar outflows and thus carry a larger polarization degree due to the Thomson laws.",
"The polarization position angle also rotates between the two spectral states.",
"An increase of about 10 – 20% in polarization degree is expected for a changing-look quasar alternating from a type-1 to a type-2 classification."
],
[
"The case of J1011+5442",
"The three interpretations are thus distinctively different, except for their polarization position angle that might rotate in the last two cases.",
"By monitoring the flux and polarization state of a sample of changing-look AGN candidate, it would be possible to easily distinguish the correct physical behind the spectral variations.",
"However we do not have archival, large monitoring campaigns of AGN polarization yet.",
"To overcome this lack of data, [18] used a single polarization measurement as a diagnostic tool of the changing-look nature of the quasar SDSS J101152.98+544206.4 (hereafter J1011+5442).",
"The authors used the William Herschel telescope to measure the polarization of J1011+5442, a $z$ = 0.246 quasar with an absolute magnitude M$_{\\rm i}$ = -22.87 [41].",
"Between 2002 and 2015, the blue continua and broad optical emission lines of J1011+5442 have been observed to decline [38], changing the optical classification of the quasar from type 1 to type 1.9.",
"[18] observed J1011+5442 in its faint state on February 19, 2017 and, after correcting for the chromatic dependence of the half-wave plate zero-angle and instrumental polarization, found a linear polarization degree of 0.15 $\\pm $ 0.22%.",
"Knowing that the interstellar polarization towards J1011+5442 is expected to be of the order of 0.1%, [18] concluded that the polarization of the quasar is compatible with a null intrinsic polarizationThe polarization position angle of J1011+5442 could not be estimated with reasonable accuracy [18].. Based on their results and theoretical polarization arguments, the authors suggested that the quasar was seen at an inclination close to the pole and that the undetected polarization degree was a proof for the lack of photoionizing continuum.",
"The Monte Carlo simulations achieved in this paper clearly prove the correctness of their conclusion: if J1011+5442 changed its spectral state because of additional equatorial dust obscuration along the observer's line-of-sight, its intrinsic degree of polarization should be of the order of 10 – 20% and thus easily detectable against interstellar polarization.",
"Thanks to our computations, we can go two steps further.",
"First, if the polarization degree of the quasar is of the order of 0.15 $\\pm $ 0.22%, its nucleus inclination is strictly inferior to 9$^\\circ $ (see Fig.",
"REF ).",
"It is a conservative estimation of the inclination since it does not depend on the exact half-opening angle or optical depths of the model components, but rather on the axisymmetry of the unified model itself [26].",
"Second, looking at Fig.",
"3 from [38], the decrease in flux at $\\lambda $ 550 nm between the two epoch is approximatively 35%.",
"This drop in flux is similar to what we found in Sect.",
"when the BLR is fading away (25%).",
"The difference can be easily compensated by an intrinsic dimming of the continuum source and/or a modification of the geometrical configuration of the broad emission line region (the half-opening angle and optical depth of the BLR being the two critical parameters here).",
"However, this drop in flux is clearly not compatible with the 98% diminution expected in the scenario where dust clouds from the torus genuinely block radiation from the central engine."
],
[
"Polarization reverberation mapping",
"As already mentioned in [18], scattering in AGN is considered to take place up to a few parsecs.",
"Additionally, the amount of reprocessing events is inclination-dependent, such as shown in Fig.",
"REF and explained in Sect.",
"REF .",
"Hence, if the central continuum source suddenly suffers a strong dimming, polarization from the polar outflows which extend over 10 pc [5] is expected to last up to $\\sim $ 32 years after the continuum change.",
"A polarimetric echo of the past core activity will remain visible in the extended structures of the AGN, such as expected in the case of the Galactic center in the X-ray band [6], [28], [29].",
"The observed polarization properties of an AGN, if integrated over the whole structure, will be impacted.",
"Since the timescale of changes of look is of the order of a few years [20], this effect might have an impact on the proposed polarization diagnostics unless high angular resolution polarization maps are available for nearby objects (such as for the case of NGC 1068 which was observed using the Spectro-Polarimetric High-contrast Exoplanet REsearch instrument – SPHERE – on the Very Large Telescope, see [17]).",
"This is where the polarization reverberation mapping technique is the most valuable.",
"[14] demonstrated how the polarization variability in NGC 4151 can be used to probe the size and structure of scattering regions.",
"Between 1997 and 2003, the type-1 radio-quiet AGN NGC 4151 has shown variations of an order of magnitude in its optical polarized flux while its polarization position angle remained constant.",
"Since the sizes of the different scattering regions in AGN span over several orders of magnitude, the time delay we measure is thus directly related to the geometry of the system.",
"Scattering inside the BLR produces a polarization angle parallel to the radio axis of the system and its temporal delay is shorter than photons that have scattered onto the polar outflows, where the scattering-induced polarization angle is perpendicular.",
"It follows that polarization reverberation mapping can precisely locate where scattering happens.",
"In addition, if temporal changes in the wavelength dependence of polarization across broad emission lines are detected, this could imply a change in the BLR and scatterer geometry [42].",
"This effect would be enhanced for increasing nucleus inclinations and the two scenarios discussed in this paper could be, in principle, distinguished thanks to polarization reverberation mapping (Rojas Lobos et al.",
"submitted; Marin et al.",
"in prep.",
")."
],
[
"Conclusions",
"In this work, we have investigated the different interpretations behind the changing-look nature observed for a few AGN.",
"The first model predicts that the disappearance of the broad emission line and the decrease in flux are due to the vanishing of the broad emission line region, linked with a decrease of the black hole accretion rate.",
"If the accretion rate becomes too low, the BLR itself can progressively disappear since mass accretion can no longer sustain the required column densities.",
"The last model explains the same spectral changes with a variation in the obscuration of the observer's viewing angle that is grazing the torus horizon.",
"Either the outer layers of the torus are puffed-up or clouds intercept the line-of-sight.",
"For the first time, both models were investigated using radiative transfer Monte Carlo calculations and their optical flux and polarization signals were found to be distinctively different.",
"If the spectral variations are due to a lack of photionizing radiation, the flux should drop accordingly but no variation of the polarization properties are expected, since they are relative quantities and they do not depended on the amount of photons scattering inside the BLR.",
"In the case of a progressive disappearance of the BLR, a $\\sim $ 25% decrease in total flux and a polarization degree $<$ 0.1% are expected.",
"Finally, if variable obscuration is the correct scenario, the total flux should be reduced by about a factor 40 and the polarization degree should be, at least, 10% higher.",
"In the last two cases the polarization position angle should rotate by 90$^\\circ $ .",
"We applied our results to the J1011+5442 quasar investigated by [18] and found strong evidences for the correctness of their interpretation.",
"We extended the polarimetric investigation of the authors by estimating the nucleus inclination of the quasar ($<$ 9$^\\circ $ ).",
"Our computations show that the total flux variation of the quasar over a decade is also consistent with their primary conclusion: the broad emission line region of J1011+5442 switched off between 2002 and 2015.",
"The same conclusion was postulated by [19] and [20] based on specific and systematic photometric searches for changing-look quasars.",
"We, for the first time, provide detailed computations of the expected flux attenuation and optical polarization expected from all the scenarios.",
"Our paper clearly shows that distinguishing between the various physical interpretations is easy.",
"New optical polarimetric observations of AGN showing a changing-look behavior will immediately tell if the change is due to an intrinsic dimming of the ionizing continuum source, a BLR disappearance or a variation in line-of-sight obscuration.",
"Prior polarimetric measurements are not vital since the actual intrinsic polarization degree should be significant enough to tell apart the two last models (null or $<$ 0.1% polarization in the first case, high, $\\ge $ 10% polarization in the other case).",
"Archival polarimetric information about the polarization angle and degree would be beneficial to detect if the spectral change is simply due to a lack of photoionizing radiation (no variations in polarization between the two epochs) or a progressive disappearance of the BLR (decrease of polarization degree and rotation of the polarization position angle).",
"Coupling the polarimetric measurements with past photometric data would greatly facilitate the interpretation, since the change in flux level is also model-dependent.",
"We thus advocate for systematic polarimetric observations of changing-look AGN in order to fully understand their true nature.",
"The author would like to acknowledge the anonymous referee for her/his useful comments that improved the quality of the paper."
]
] | 1709.01699 |
[
[
"BranchyNet: Fast Inference via Early Exiting from Deep Neural Networks"
],
[
"Abstract Deep neural networks are state of the art methods for many learning tasks due to their ability to extract increasingly better features at each network layer.",
"However, the improved performance of additional layers in a deep network comes at the cost of added latency and energy usage in feedforward inference.",
"As networks continue to get deeper and larger, these costs become more prohibitive for real-time and energy-sensitive applications.",
"To address this issue, we present BranchyNet, a novel deep network architecture that is augmented with additional side branch classifiers.",
"The architecture allows prediction results for a large portion of test samples to exit the network early via these branches when samples can already be inferred with high confidence.",
"BranchyNet exploits the observation that features learned at an early layer of a network may often be sufficient for the classification of many data points.",
"For more difficult samples, which are expected less frequently, BranchyNet will use further or all network layers to provide the best likelihood of correct prediction.",
"We study the BranchyNet architecture using several well-known networks (LeNet, AlexNet, ResNet) and datasets (MNIST, CIFAR10) and show that it can both improve accuracy and significantly reduce the inference time of the network."
],
[
"Introduction",
"One of the reasons for the success of deep networks is their ability to learn higher level feature representations at successive nonlinear layers.",
"In recent years, advances in both hardware and learning techniques have emerged to train even deeper networks, which have improved classification performance further [4], [8].",
"The ImageNet challenge exemplifies the trend to deeper networks, as the state of the art methods have advanced from 8 layers (AlexNet), to 19 layers (VGGNet), and to 152 layers (ResNet) in the span of four years [7], [13], [20].",
"However, the progression towards deeper networks has dramatically increased the latency and energy required for feedforward inference.",
"For example, experiments that compare VGGNet to AlexNet on a Titan X GPU have shown a factor of 20x increase in runtime and power consumption for a reduction in error rate of around 4% (from 11% to 7%) [11].",
"The trade off between resource usage efficiency and prediction accuracy is even more noticeable for ResNet, the current state of the art method for the ImageNet Challenge, which has an order of magnitude more layers than VGGNet.",
"This rapid increase in runtime and power for gains in accuracy may make deeper networks less tractable in many real world scenarios, such as real-time control of radio resources for next-generation mobile networking, where latency and energy are important factors.",
"Figure: A simple BranchyNet with two branches added to the baseline (original) AlexNet.",
"The first branch has two convolutional layers and the second branch has 1 convolutional layer.",
"The “Exit” boxes denote the various exit points of BranchyNet.",
"This figure shows the general structure of BranchyNet, where each branch consists of one or more layers followed by an exit point.",
"In practice, we generally find that it is not necessary to add multiple convolutional layers at a branch in order to achieve good performance.To lessen these increasing costs, we present BranchyNet, a neural network architecture where side branches are added to the main branch, the original baseline neural network, to allow certain test samples to exit early.",
"This novel architecture exploits the observation that it is often the case that features learned at earlier stages of a deep network can correctly infer a large subset of the data population.",
"By exiting these samples with prediction at earlier stages and thus avoiding layer-by-layer processing for all layers, BranchyNet significantly reduces the runtime and energy use of inference for the majority of samples.",
"Figure REF shows how BranchyNet modifies a standard AlexNet by adding two branches with their respective exit points.",
"BranchyNet is trained by solving a joint optimization problem on the weighted sum of the loss functions associated with the exit points.",
"Once the network is trained, BranchyNet utilizes the exit points to allow the samples to exit early, thus reducing the cost of inference.",
"At each exit point, BranchyNet uses the entropy of a classification result (e.g., by softmax) as a measure of confidence in the prediction.",
"If the entropy of a test sample is below a learned threshold value, meaning that the classifier is confident in the prediction, the sample exits the network with the prediction result at this exit point, and is not processed by the higher network layers.",
"If the entropy value is above the threshold, then the classifier at this exit point is deemed not confident, and the sample continues to the next exit point in the network.",
"If the sample reaches the last exit point, which is the last layer of the baseline neural network, it always performs classification.",
"Three main contributions of this paper are: Fast Inference with Early Exit Branches: BranchyNet exits the majority of the samples at earlier exit points, thus reducing layer-by-layer weight computation and I/O costs, resulting in runtime and energy savings.",
"Regularization via Joint Optimization: BranchyNet jointly optimizes the weighted loss of all exit points.",
"Each exit point provides regularization on the others, thus preventing overfitting and improving test accuracy.",
"Mitigation of Vanishing Gradients: Early exit points provide additional and more immediate gradient signal in back propagation, resulting in more discriminative features in lower layers, thus improving accuracy."
],
[
"Background and Related Prior Work",
"LeNet-5 [15] introduced the standard convolutional neural networks (CNN) structure which is composed of stacked convolutional layers, optionally followed by contrast normalization and maxpooling, and then finally followed by one or more fully-connected layers.",
"This structure has performed well in several image tasks such as image classification.",
"AlexNet [13], VGG [20], ResNet [7] and others have expanded on this structure with their own innovative approaches to make the network deeper and larger for improved classification accuracy.",
"Due to the computational costs of deep networks, improving the efficiency of feedforward inference has been heavily studied.",
"Two such approaches are network compression and implementation optimization.",
"Network compression schemes aim to reduce the the total number of model parameters of a deep network and thus reduce the amount of computation required to perform inference.",
"Bucilua et al.",
"(2006) proposed a method of compressing a deep network into a smaller network that achieves a slightly reduced level of accuracy by retraining a smaller network on synthetic data generated from a deep network [3].",
"More recently, Han et al.",
"(2015) have proposed a pruning approach that removes network connections with small contributions [5].",
"However, while pruning approaches can significantly reduce the number of model parameters in each layer, converting that reduction into a significant speedup is difficult using standard GPU implementations due to the lack of high degrees of exploitable regularity and computation intensity in the resulting sparse connection structure [6].",
"Kim et al.",
"(2015) use a Tucker decomposition (a tensor extension of SVD) to extract shared information between convolutional layers and perform rank selection [11].",
"This approach reduces the number of network parameters, making the network more compact, at the cost of a small amount of accuracy loss.",
"These network compression methods are orthogonal to the BranchyNet approach taken in this paper, and could potentially be used in conjunction to improve inference efficiency further.",
"Implementation optimization approaches reduce the runtime of inference by making the computation algorithmically faster.",
"Vanhoucke et al.",
"(2011) explored code optimizations to speed up the execution of convolutional neural networks (CNNs) on CPUs [25].",
"Mathieu et al.",
"(2013) showed that convolution using FFT can be used to speed up training and inference for CNNs [17].",
"Recently, Lavin et al.",
"(2015) have introduced faster algorithms specifically for 3x3 convolutional filters (which are used extensively in VGGNet and ResNet) [14].",
"In contrast, BranchyNet makes modifications to the network structure to improve inference efficiency.",
"Deeper and larger models are complex and tend to overfit the data.",
"Dropout [21], L1 and L2 regularization and many other techniques have been used to regularize the network and prevent overfitting.",
"Additionally, Szegedy et al.",
"(2015) introduced the concept of adding softmax branches in the middle layers of their inception module within deep networks as a way to regularize the main network [23].",
"While also providing similar regularization functionalities, BranchyNet has a new goal of allowing early exits for test samples which can already be classified with high confidence.",
"One main challenge with (very) deep neural networks is the vanishing gradient problem.",
"Several papers have introduced ideas to mitigate this issue including normalized network initialization [4], [16] and intermediate normalization layers [10].",
"Recently, new approaches such as Highway Networks [22], ResNet [7], and Deep Networks with Stochastic Depth [9] have been studied.",
"The main idea is to add skip (shortcut) connections in between layers.",
"This skip connection is an identity function which helps propagate the gradients in the backpropagation step of neural network training.",
"Panda et al.",
"[18] propose Conditional Deep Learning (CDL) by iteratively adding linear classifiers to each convolutional layer, starting with the first layer, and monitoring the output to decide whether a sample can be exited early.",
"BranchyNet allows for more general branch network structures with additional layers at each exit point while CDL only uses a cascade of linear classifiers, one for each convolutional layer.",
"In addition, CDL does not jointly train the classifier with the original network.",
"We observed in our paper that jointly training the branch with the original network significantly improve the performance of the overall architecture when compared to CDL."
],
[
"BranchyNet",
"BranchyNet modifies the standard deep network structure by adding exit branches (also called side branches or simply branches for brevity), at certain locations throughout the network.",
"These early exit branches allow samples which can be accurately classified in early stages of the network to exit at that stage.",
"In training the classifiers at these exit branches, we also consider network regularization and mitigation of vanishing gradients in backprogation.",
"For the former, branches will provide regularization on the main branch (baseline network), and vice versa.",
"For the latter, a relatively shallower branch at a lower layer will provide more immediate gradient signal in backpropagation, resulting in discriminative features in lower layers of the main branch, thus improving its accuracy.",
"In designing the BranchyNet architecture, we address a number of considerations, including (1) locations of branch points, (2) structure of a branch (weight layers, fully-connected layers, etc.)",
"as well as its size and depth, (3) classifier at the exit point of a branch, (4) exit criteria for a branch and the associated test cost against the criteria, and (5) training of classifiers at exit points of all branches.",
"In general, this “branch” notion can be recursively applied, that is, a branch may have branches, resulting in a tree structure.",
"For simplicity, in this paper we focus a basic scenario where there are only one-level branches which do not have nested branches, meaning there are no tree branches.",
"In this paper, we describe BranchyNet with classification tasks in mind; however, the architecture is general and can also be used for other tasks such as image segmentation and object detection."
],
[
"Architecture",
"A BranchyNet network consists of an entry point and one or more exit points.",
"A branch is a subset of the network containing contiguous layers, which do not overlap other branches, followed by an exit point.",
"The main branch can be considered the baseline (original) network before side branches are added.",
"Starting from the lowest branch moving to highest branch, we number each branch and its associated exit point with increasing integers starting at one.",
"For example, the shortest path from the entry point to any exit is exit 1, as illustrated in Figure REF ."
],
[
"Training BranchyNet",
"For a classification task, the softmax cross entropy loss function is commonly used as the optimization objective.",
"Here we describe how BranchyNet uses this loss function.",
"Let $$ be a one-hot ground-truth label vector, $$ be an input sample and $\\mathcal {C}$ be the set of all possible labels.",
"The objective function can be written as L(, ;) = -1|C| cC yc yc, where = softmax() = ()cC (zc), and = fexitn(;), where $f_{\\text{exit}_n}$ is the output of the $n$ -th exit branch and $\\theta $ represents the parameters of the layers from an entry point to the exit point.",
"The design goal of each exit branch is to minimize this loss function.",
"To train the entire BranchyNet, we form a joint optimization problem as a weighted sum of the loss functions of each exit branch Lbranchynet(, ;) = n=1N wn L(exitn, ;), where $N$ is the total number of exit points.",
"Section REF discusses how one might choose weights $w_n$ .",
"The algorithm consists of two steps: the feedforward pass and the backward pass.",
"In the feedforward pass, the training data set is passed through the network, including both main and side branches, the output from the neural network at all exit points is recorded, and the error of the network is calculated.",
"In backward propagation, the error is passed back through the network and the weights are updated using gradient descent.",
"For gradient descent, we use Adam algorithm [12], though other variants of Stochastic Gradient Descent (SGD) can also be used."
],
[
"Fast Inference with BranchyNet",
"Once trained, BranchyNet can be used for fast inference by classifying samples at earlier stages in the network based on the algorithm in Figure REF .",
"If the classifier at an exit point of a branch has high confidence about correctly labeling a test sample $$ , the sample is exited and returns a predicted label early with no further computation performed by the higher branches in the network.",
"We use entropy as a measure of how confident the classifier at an exit point is about the sample.",
"Entropy is defined as entropy() = cC yc yc, where $$ is a vector containing computed probabilities for all possible class labels and $\\mathcal {C}$ is a set of all possible labels.",
"Figure: BranchyNet Fast Inference Algorithm.",
"is an input sample, is a vector where the nn-th entry T n T_n is the threshold for determining whether to exit a sample at the nn-th exit point, and NN is the number of exit points of the network.To perform fast inference on a given BranchyNet network, we follow the procedure as described in Figure REF .",
"The procedure requires $$ , a vector where the $n$ -th entry is the threshold used to determine if the input $$ should exit at the $n$ -th exit point.",
"In section REF , we discuss how these thresholds may be set.",
"The procedure begins with the lowest exit point and iterates to the highest and final exit point of the network.",
"For each exit point, the input sample is fed through the corresponding branch.",
"The procedure then calculates the softmax and entropy of the output and checks if the entropy is below the exit point threshold $T_n$ .",
"If the entropy is less than $T_n$ , the class label with the maximum score (probability) is returned.",
"Otherwise, the sample continues to the next exit point.",
"If the sample reaches the last exit point, the label with the maximum score is always returned."
],
[
"Results",
"In this section, we demonstrate the effectiveness of BranchyNet by adapting three widely studied convolutional neural networks on the image classification task: LeNet, AlexNet, and ResNet.",
"We evaluate Branchy-LeNet (B-LeNet) on the MNIST dataset and both Branchy-AlexNet (B-AlexNet) and Branchy-ResNet (B-ResNet) on the CIFAR10 data set.",
"We present evaluation results for both CPU and GPU.",
"We use a 3.0GHz CPU with 20MB L3 Cache and NVIDIA GeForce GTX TITAN X (Maxwell) 12GB GPU.",
"For simplicity, we only describe convolutional and fully-connected layers of each network.",
"Generally, these networks may also contain max pooling, non-linear activation functions (e.g., a rectified linear unit and sigmoid), normalization (e.g., local response normalization, batch normalization), and dropout.",
"For LeNet-5 [15] which consists of 3 convolutional layers and 2 fully-connected layers, we add a branch consisting of 1 convolutional layer and 1 fully-connected layer after the first convolutional layer of the main network.",
"For AlexNet [13] which consists of 5 convolutional layers and 3 fully-connected layers, we add 2 branches.",
"One branch consisting of 2 convolutional layers and 1 fully-connected layer is added after the 1st convolutional layer of the main network, and another branch consisting of 1 convolutional layer and 1 fully-connected layer is added after the 2nd convolutional layer of the main network.",
"For ResNet-110 [7] which consists of 109 convolutional layers and 1 fully-connected layer, we add 2 branches.",
"One branch consisting of 3 convolutional layers and 1 fully-connected layer is added after the 2nd convolutional layer of the main network, and the second branch consisting of 2 convolutional layers and 1 fully-connected layer is added after the 37th convolutional layer of the main network.",
"We initialize B-LeNet, B-AlexNet and B-ResNet with weights trained from LeNet, AlexNet and ResNet respectively.",
"We found the initializing each BranchyNet network with the weights trained from the baseline network improved the classification accuracy of the network by several percent over random initialization.",
"To train these networks, we use Adam algorithm with a step size ($\\alpha $ ) of 0.001 and exponential decay rates for first and second moment estimates ($\\beta _1,\\beta _2$ ) of 0.99 and 0.999 respectively.",
"Figure REF shows the GPU performance results of BranchyNet when applied to each network.",
"For all of the networks, BranchyNet outperforms the original baseline network.",
"The reported runtime is the average among all test samples.",
"B-LeNet has the largest performance gain due to a more efficient branch which achieves almost the same level of accuracy as the last exit branch.",
"For AlexNet and ResNet, we see that the performance gain is still substantial, but since more samples are required to exit at the last layer, smaller than B-LeNet.",
"The knee point denoted as the green star represents an optimal threshold point, where the accuracy of BranchyNet is comparable to the main network, but the inference is performed significantly faster.",
"For B-ResNet, the accuracy is slightly lower than the baseline.",
"A different threshold could be chosen which gives accuracy higher than ResNet but with much less savings in inference time.",
"The performance characteristics of BranchyNet running on CPU follow a similar trend to the performance of BranchyNet running on GPU.",
"Figure: The overall classification accuracy of B-AlexNet for varying entropy threshold for the first exit branch.",
"For this experiment, all samples not exited in at the first branch are exited at the final exit.",
"The entropy at a given value is the max entropy of all samples up to that point.Table REF highlights the selected knee threshold values, exit (%) and gain in speed up, for BranchyNet for each network for both CPU and GPU.",
"The $$ column denotes the threshold values for each exit branch.",
"Since the last exit branch must exit all samples, it does not require an exit threshold.",
"Therefore, for a 2-branch network, such as B-LeNet, there is a single $$ value and for a 3-branch network, such as B-AlexNet and B-ResNet, there are two $$ values.",
"Further analysis of the sensitivity of the $$ parameters is discussed in Section .",
"The Exit (%) column shows the percentage of samples exited at each branch point.",
"For all networks, we see that BranchyNet is able to exit a large percentage of the test samples before the last layer, leading to speedups in inference time.",
"B-LeNet exits 94% of samples at the first exit branch, while B-AlexNet and B-ResNet exit 65% and 41% respectively.",
"Exiting these samples early translate to CPU/GPU speedup gains of 5.4/4.7x over LeNet, 1.5/2.4x over AlexNet, and 1.9/1.9x over ResNet.",
"The branch structure for B-ResNet mimics that of B-AlexNet.",
"Table: Selected performance results for BranchyNet on the different network structures.",
"The BrachyNet rows correspond to the knee points denoted as green stars in Figure ."
],
[
"Analysis and Discussion",
"In this section, we provide additional analysis on key aspects BranchyNet."
],
[
"Hyperparameter Sensitivity",
"Two important hyperparameters of BranchyNet are the weights ${w_n}$ in joint optimization (Section REF ) and the exit thresholds $$ for the fast inference algorithm described in Figure REF .",
"When selecting the weight of each branch, we observed that giving more weight to early branches improves the accuracy of the later branches due to the added regularization.",
"On a simplified version of BranchyAlexNet with only the first and last branch, weighting the first branch with $1.0$ and the last branch with $0.3$ provides a 1% increase in classification accuracy over weighting each branch equally.",
"Giving more weight to earlier exit branches encourages more discriminative feature learning in early layers of the network and allows more samples to exit early with high confidence.",
"Figure REF shows how the choice of $$ affects the number of samples exited at the first branch point in B-AlexNet.",
"We observe that the entropy value has a distinctive knee where it rapidly becomes less confident in the test samples.",
"Thus in this case it is relatively easy to identify the knee and learn a corresponding threshold.",
"In practice, the choice of exit threshold for each exit point depends on applications and datasets.",
"The exit thresholds should be chosen such that it satisfies the inference latency requirement of an application while maintaining the required accuracy.",
"An additional hyperparameter not mentioned explicitly is the location of the branch points in the network.",
"In practice, we find the location of the first branch point depends on the difficulty of the dataset.",
"For a simpler dataset, such as MNIST, we can place a branch directly after the first layer and immediately see accurate classification.",
"For more challenging datasets, branches should be placed higher in order to still achieve strong classification performance.",
"For any additional branches, we currently place them at equidistant points throughout the network.",
"Future work will be to derive an algorithm to find the optimal placement locations of the branches automatically."
],
[
"Tuning Entropy Thresholds",
"The results shown in Figure REF provides the accuracy and runtime for a range of $$ values.",
"These $$ values show how BranchyNet trades off accuracy for faster runtime as the entropy thresholds increase.",
"However, in practice, we may want to set $$ automatically to met a specified runtime or accuracy constraint.",
"One approach is to simply screen over $$ as done here and pick a setting that satisfies the constraints.",
"We provided code used to generate the performance results which includes a method for performing this screening [24].",
"Additionally, it may be possible to use a Meta-Recognition algorithm [19], [26] to estimate the characteristics of unseen test samples and adjust $$ automatically in order to maintain a specified runtime or accuracy goal.",
"One simple approach for creating such a Meta-Recognition algorithm would be to train a small Multilayer Perceptron (MLP) for each corresponding exit point on the output softmax probability vectors $\\hat{}$ for that exit.",
"The MLP at an exit point would attempt to predict if a given sample would be correctly classified at the specific exit.",
"More generally, this approach is closely related to the open world recognition problem [2], [1], which is interested in quantifying the uncertainty of a model for a particular set of unseen or out of set test samples.",
"We can expand on the MLP approach further by using a different formulation than SoftMax, such as OpenMax [2], which attempts to quantify the uncertainty directly in the probability vector $\\hat{}$ by adding an additional uncertain class.",
"These approaches could be used to tune $$ automatically to a new test set by estimating the difficulty of the test data and adapting $$ accordingly to meet the runtime or accuracy constraints.",
"This work is outside the scope of this paper, which only provides the groundwork BranchyNet architecture, but will be explored in future work."
],
[
"Effects of Structure of Branches",
"Figure REF shows the impact on the accuracy of the last exit by adding additional convolutional layers in an earlier side branch for a modified version of B-AlexNet with only the first side branch.",
"We see that there is a optimal number of layers to improve the accuracy of the main branch, and that adding too many layers can actually harm overall accuracy.",
"In addition to convolutional layers, adding a few fully-connected layers after convolutional layers to a branch also proves helpful since this allows local and global features to combine and form more discriminative features.",
"The number of layers in a branch and the size of an exit branch should be chosen such that the overall size of the branch is less than amount of computation needed to do to exit at a later exit point.",
"Generally, we find that earlier branch points should have more layers, and later branch points should have fewer layers."
],
[
"Effects of cache",
"Since the majority of samples are exited at early branch points, the later branches are used more rarely.",
"This allows weights at these early exit branches to be cached more efficiently.",
"Figure REF shows the effect of cache based on various $$ values for B-AlexNet.",
"We see that the more aggressive $$ values have faster runtime on the CPU and also less cache miss rates.",
"One could use this insight to select a branch structure that can fits more effectively in a cache, potentially speeding up inference further.",
"Figure: The runtime and CPU cache miss rate for the B-AlexNet model as the entropy threshold is varied."
],
[
"Conclusion",
"We have proposed BranchyNet, a novel network architecture that promotes faster inference via early exits from branches.",
"Through proper branching structures and exit criteria as well as joint optimization of loss functions for all exit points, the architecture is able to leverage the insight that many test samples can be correctly classified early and therefore do not need the later network layers.",
"We have evaluated this approach on several popular network architectures and shown that BranchyNet can reduce the inference cost of deep neural networks and provide 2x-6x speed up on both CPU and GPU.",
"BranchyNet is a toolbox for researchers to use on any deep network models for fast inference.",
"BranchyNet can be used in conjunction with prior works such as network pruning and network compression [3], [5].",
"BranchyNet can be adapted to solve other types of problems such as image segmentation, and is not just limited to classification problems.",
"For future work, we plan to explore Meta-Recognition algorithms, such as OpenMax, to automatically adapt $$ to new test samples."
],
[
"Acknowledgment",
"This work is supported in part by gifts from the Intel Corporation and in part by the Naval Supply Systems Command award under the Naval Postgraduate School Agreements No.",
"N00244-15-0050 and No.",
"N00244-16-1-0018."
]
] | 1709.01686 |
[
[
"CNN-Based Projected Gradient Descent for Consistent Image Reconstruction"
],
[
"Abstract We present a new method for image reconstruction which replaces the projector in a projected gradient descent (PGD) with a convolutional neural network (CNN).",
"CNNs trained as high-dimensional (image-to-image) regressors have recently been used to efficiently solve inverse problems in imaging.",
"However, these approaches lack a feedback mechanism to enforce that the reconstructed image is consistent with the measurements.",
"This is crucial for inverse problems, and more so in biomedical imaging, where the reconstructions are used for diagnosis.",
"In our scheme, the gradient descent enforces measurement consistency, while the CNN recursively projects the solution closer to the space of desired reconstruction images.",
"We provide a formal framework to ensure that the classical PGD converges to a local minimizer of a non-convex constrained least-squares problem.",
"When the projector is replaced with a CNN, we propose a relaxed PGD, which always converges.",
"Finally, we propose a simple scheme to train a CNN to act like a projector.",
"Our experiments on sparse view Computed Tomography (CT) reconstruction for both noiseless and noisy measurements show an improvement over the total-variation (TV) method and a recent CNN-based technique."
],
[
"Introduction ",
"While medical imaging is a fairly mature area, there is recent evidence that it may still be possible to reduce the radiation dose and/or speedup the acquisition process without compromising image quality [1].",
"This can be accomplished with the help of sophisticated reconstruction algorithms that incorporate some prior knowledge (e.g., sparsity) on the class of underlying images.",
"The reconstruction task is usually formulated as an inverse problem, where the image-formation physics is modeled by an operator ${\\bf {H}}:{\\mathbb {R}}^N \\rightarrow {\\mathbb {R}}^M $ (called the forward model).",
"The measurement equation is ${\\rm \\bf y}={\\rm \\bf H}{\\rm \\bf x}+{\\rm \\bf n}\\in {\\mathbb {R}^{M}}$ , where ${\\rm \\bf x}$ is the space-domain image that we are interested in recovering and ${\\rm \\bf n}\\in {\\mathbb {R}^{M}}$ is the noise intrinsic to the acquisition process.",
"In the case of extreme imaging, the number and the quality of the measurements are both reduced as much as possible, e.g., in order to decrease either the scanning time in MRI or the radiation dose in CT.",
"Moreover, the measurements are typically very noisy due to short integration times, which calls for some form of denoising.",
"Indeed, there may be significantly fewer measurements than the number of unknowns ($M << N$ ).",
"This gives rise to an ill-posed problem in the sense that there may be an infinity of consistent images that map to the same measurements ${\\bf {y}}$ .",
"Thus, one challenge of the reconstruction algorithm is essentially to select the “best” solution among a multitude of potential candidates.",
"The available reconstruction algorithms can be broadly arranged in three categories (or generations), which represent the continued efforts of the research community to address the aforementioned challenges.",
"Here, the reconstruction is performed directly by applying a suitable linear operator.",
"This may be the backprojection (BP) ${{\\rm \\bf H}^{}{\\rm \\bf y} or the filtered backprojection (FBP){\\bf {F}}{{\\rm \\bf H}^{}{\\rm \\bf y}, where {\\bf {F}}: {\\mathbb {R}^N}\\rightarrow {\\mathbb {R}^N} is a regularized version of ({{\\rm \\bf H}^{}{\\rm \\bf H})^{-1}.\\footnote {Or, {\\bf {F}} can be a regularized version of ({\\rm \\bf H}{{\\rm \\bf H}^{})^{-1} and be applied as {{\\rm \\bf H}^{}{\\bf {F}}, as is often the case in CT.}FBP-type algorithms are fast; they provide excellent results when the number of measurements are sufficient and the noise is small \\cite {Pan2009}.However, they are not suitable for extreme imaging scenarios because they introduce artifacts intimately connected to the inversion step.",
"}}\\subsubsection {Iterative Algorithms} These algorithms avoid the shortcomings of the classical ones by solving\\begin{equation}{\\bf {x}}^*= \\operatornamewithlimits{arg\\,min}_{{\\bf {x}}} (E({\\bf {H}} {\\bf {x}}, {\\bf {y}})+\\lambda R({\\bf {x}})),\\end{equation}where E:{\\mathbb {R}^{M}}\\times {\\mathbb {R}^{M}}\\rightarrow {\\mathbb {R}}^+ is a data-fidelity term that favors solutions that are consistent with the measurements, R: {\\mathbb {R}^N}\\rightarrow {\\mathbb {R}}^+ is a suitable regularizer that encodes the prior knowledge about the image {\\bf {x}} to be reconstructed, and \\lambda \\in {\\mathbb {R}}^+ is a tradeoff parameter.",
"The quantity {\\rm \\bf y}^\\ast ={\\bf {H}}{\\bf {x}}^\\ast , where{\\rm \\bf x}^\\ast is the solution of (\\ref {iterative}), can be interpreted as the denoised version of {\\bf {y}}.Under the assumption that the functionals E and R are convex, one can show that the solution of (\\ref {iterative}) also satisfies\\begin{equation}{\\rm \\bf x}^\\ast =\\arg \\min _{{\\bf {x}} \\in {\\mathbb {R}^N}} R({\\bf {x}}) \\quad \\text{s.t.}",
"\\quad {\\bf {H}} {\\bf {x}}={\\bf {y}}_0.\\end{equation}Therefore,among all potential solutions that admit the denoised measurement {\\rm \\bf y}^\\ast , the algorithm picks the one with the least R. This shows that the quality of the reconstruction depends heavily on the prior encoder R. Generally, these priors are either handpicked (\\emph {e.g}.\\hbox{}, total variation (TV) or the \\ell _1-norm of the wavelet coefficients of the image \\cite {Bouman1993,Charbonnier1997,Lustig2007, Candes2007,Ramani2011}) or learned through a dictionary \\cite {Elad2006,Candes2011,Ravishankar2017}.",
"However, in either case, they are restricted to well-behaved functionals that can be minimized via a convex routine \\cite {Figueiredo2003,daubechies2004iterative,beck2009fast,boyd2011distributed}.This limits the type of prior knowledge that can be injected into the algorithm.",
"}}An interesting variant within the class of iterative algorithms is ``plug-and-play ADMM^{\\prime \\prime }\\cite {venkatakrishnan2013plug}.",
"We recall that ADMM is an iterative optimization technique \\cite {boyd2011distributed}, which repeatedly alternates between: (i) a linear solver that reinforces consistency w.r.t.\\hbox{} measurements; (ii) a nonlinear operation that re-injects the prior.",
"Interestingly, the effect of (ii) is akin to denoising.",
"This perspective has resulted in a scheme where an off-the-shelf denoiser is plugged into the later step \\cite {venkatakrishnan2013plug,chan2017plug,sreehari2016plug,chang2017,romano2016little}.This scheme, therefore, is more general than the optimization framework (\\ref {iterative}) but still lacks theoretical justifications.",
"In fact, there is no good understanding yet of the connection between the use of a given denoiser and the regularization it imposes.",
"}\\subsubsection {Learning-based Algorithms} Recently, there has been a surge in using deep learning to solve inverse problems in imaging~\\cite {jin2017deep,han2017deep,antholzer2017deep,wang2016accelerating,ReviewMike}, establishing new state-of-the-art results for tasks like sparse-view CT reconstruction~\\cite {jin2017deep}.",
"These approaches use the convolutional neural network (CNN) as a regressor.",
"But, rather than attempting to reconstructthe image from the measurements $y$ directly, the most successful strategies so far have been to train the CNN as a regressor between rogue initial reconstruction $A y$, where $A: RM RN$, and the final, desired reconstruction \\cite {jin2017deep,han2017deep}.",
"This initial reconstruction could be obtained by FBP ($FHy$), BP ($Hy$) or by any other linear operation.$ Once the training is complete, the reconstruction for an unseen measurement $ {\\rm \\bf y}$ is given by ${\\rm \\bf x}^* ={C\\!N\\!N_{{\\bf {{\\theta }}}^*}}({\\bf {A}} {\\rm \\bf y})$ , where ${C\\!N\\!N_{{\\bf {{\\theta }}}}}: {\\mathbb {R}^N}\\rightarrow {\\mathbb {R}^N}$ denotes the CNN as a function and ${\\bf {{\\theta }}}^*$ denotes the internal parameters of the CNN after training.",
"These schemes exploit the fact that the structure of images can be learnt from representative examples.",
"CNNs are favored because of the way they encode/represent the data in their hidden layers.",
"In this sense, a CNN can be seen as a good prior encoder.",
"Although the results reported so far are remarkable in terms of image quality, there is still some concern on whether or not they can be trusted, especially in the context of diagnostic imaging.",
"The main limitation of direct algorithms such as [19] is that they do not provide any worst case performance guarantee.",
"Moreover, even in the case of noiseless (or low noise) measurements, there is no insurance that the reconstructed image is consistent with the measurements.",
"This is not overly surprising because, unlike the iterative schemes, there is no feedback mechanism that imposes this minimal requirement.",
"Figure: (a) Block diagram of projected gradient descent using a CNN as the projector.The gradient step w.r.t.",
"the data-fidelity term E=∥𝐇𝐱-𝐲∥ 2 E=\\Vert {\\rm \\bf H}{\\rm \\bf x}-{\\rm \\bf y}\\Vert ^2, promotes consistency with the measurements and the projector forces the solution to belong to the set of desired solutions.If the CNN is only an approximate projector, the scheme may diverge.",
"(b) Block diagram of the proposed relaxed projected gradient descent.The α k \\alpha _ks are updated in such a way that the algorithm always converges (see Algorithm for more details).We propose a simple yet effective iterative scheme which tries to incorporate the advantages of the existing algorithms and side-steps their disadvantages (see Figure REF ).",
"Moreover, it outperforms the existing algorithms to reconstruct biomedical images from their sparse-view CT measurements.",
"Specifically, We initialize our reconstruction using a classical algorithm.",
"We learn a CNN that acts as a projector onto a set ${\\mathcal {S}}$ which can be intuitively thought of as the manifold of the data (e.g., biomedical images).",
"In this sense, our CNN encodes the prior knowledge of the data.",
"Its purpose is to map an input image to an output image that enjoys a more structural fit to the training data than the input image.",
"Similarly to variational methods, we iteratively alternate between minimizing the data-fidelity term and projecting the result onto the set ${\\mathcal {S}}$ by applying a suitable variant of the projected gradient descent (PGD) which ensures convergence.",
"This scheme in spirit is similar to plug-and-play ADMM but is simpler to analyze.",
"In this way, instead of performing the reconstruction by a feedbackless pipeline, we perform it by iteratively enforcing measurement consistency and injecting prior knowledge."
],
[
"Roadmap for the Paper",
"The paper is organized as follows: In Section , we discuss the formal framework that motivates our approach.",
"We mathematically justify the use of a projector onto a set as an effective strategy to solve inverse problems.",
"In Section , we present an iterative algorithm inspired from PGD.",
"It has been modified so as to converge in practical cases where the projection property is only approximate.",
"This is crucial because, although we train the CNN as a projector using a training set, there is no guarantee that it will act like a projector for an unseen data.",
"We discuss in Section a novel technique to train the CNN as a projector onto a set, especially when the training data is small (e.g., around 500 images in our case).",
"This is followed by results and conclusion."
],
[
"Related and Prior Work",
"Deep learning has shown promising results in the cases of image denoising, superresolution and deconvolution.",
"Recently, it has also been used to solve inverse problems in imaging using limited data [19], [21], [20], [22], and in compressed sensing [24].",
"However, as discussed earlier, these regression based approaches lack the feedback mechanism that can be beneficial to solve the inverse problems.",
"The other usage of deep learning is to complement the iterative algorithms.",
"This includes learning a CNN as an unrolled version of ISTA [25] and ADMM [26].",
"In [27], inverse problems including non-linear forward models are solved by partially learning the gradient descent.",
"In [28] the iterative algorithm is replaced by a recurrent neural network (RNN).",
"In all of these approaches the training depends entirely on the iterative scheme the neural network is used with.",
"On the other hand, [14], [15], [18], [17], [16] use plug-and-play ADMM that uses a denoiser which is learnt independently of the iterative scheme.",
"In [17], a generative adversarial network (GAN) trained as a projector onto a set is plugged into the plug-and-play ADMM and is used to solve arbitrary linear inverse problem.",
"However, due to the adversarial nature, the training is complicated and requires extremely large datasets (around 8 million images).",
"Performing such training would be challenging in biomedical imaging applications because of the lack of large datasets and the high-resolution of the images ($512 \\times 512$ or more).",
"Also, in many cases, the performance of [17] is worse than the regression-based deep-learning methods specialized for a given inverse problem."
],
[
"Theoretical Framework",
"The central theme of this paper is to use CNN iteratively with PGD to solve inverse problem.",
"The task of the CNN is to act like a projector.",
"It projects the input image to the space of desired images.",
"To understand why this scheme will be effective, we first analyze how using a projector onto a set, combined with gradient descent, can be helpful in solving inverse problems.",
"Proofs of all the theoretical results except Theorem REF can be found in the supplementary material."
],
[
"Notation",
"We consider the finite-dimensional Hilbert space ${\\mathbb {R}^N}$ equipped with the scalar product $\\left\\langle {\\cdot }\\, , \\,{\\cdot }\\right\\rangle $ that induces the $\\ell _2$ norm $\\left\\Vert \\cdot \\right\\Vert _2$ .",
"The spectral norm of the matrix ${\\rm \\bf H}$ , denoted by $\\left\\Vert {\\rm \\bf H}\\right\\Vert _2$ , is equal to its largest singular value.",
"For ${\\rm \\bf x}\\in {\\mathbb {R}^N}$ and $\\varepsilon >0$ , we denote by ${\\mathcal {B}}_{\\varepsilon }({\\rm \\bf x})$ the $\\ell _2$ -ball centered at ${\\rm \\bf x}$ with radius $\\varepsilon $ , i.e., ${\\mathcal {B}}_{\\varepsilon }({\\rm \\bf x})= \\left\\lbrace {\\rm \\bf z}\\in {\\mathbb {R}^N}: \\left\\Vert {\\rm \\bf z}-{\\rm \\bf x}\\right\\Vert _2\\le \\varepsilon \\right\\rbrace .$ The operator $T:{\\mathbb {R}^N}\\rightarrow {\\mathbb {R}^N}$ is Lipschitz-continuous with constant $L$ if $\\left\\Vert T({\\rm \\bf x})-T({\\rm \\bf z})\\right\\Vert _2 \\le L\\left\\Vert {\\rm \\bf x}-{\\rm \\bf z}\\right\\Vert _2,\\quad \\forall {\\rm \\bf x},{\\rm \\bf z}\\in {\\mathbb {R}^N}.$ It is contractive if it is Lipschitz-continuous with constant $L<1$ and non-expansive if $L=1$ .",
"A fixed point ${\\rm \\bf x}^*$ of $T$ (if any) satisfies $T({\\rm \\bf x}^{*})={\\rm \\bf x}^*$ .",
"Given a set ${\\mathcal {S}}\\subset {\\mathbb {R}^N}$ , a mapping $P_{{\\mathcal {S}}}:{\\mathbb {R}^N}\\rightarrow {\\mathcal {S}}$ is called a projector if it satisfies the idempotent property $P_{{\\mathcal {S}}} P_{{\\mathcal {S}}}=P_{{\\mathcal {S}}}$ .",
"It is called an orthogonal projector if $P_{{\\mathcal {S}}}({\\rm \\bf x})=\\inf _{{\\rm \\bf z}\\in {\\mathcal {S}}}\\left\\Vert {\\rm \\bf x}-{\\rm \\bf z}\\right\\Vert _2,\\quad \\forall {\\rm \\bf x}\\in {\\mathbb {R}^N}.$"
],
[
"Constrained Least Squares",
"Consider the problem of reconstructing an image ${\\rm \\bf x}\\in {\\mathbb {R}^N}$ from its noisy measurements ${\\rm \\bf y}={\\rm \\bf H}{\\rm \\bf x}+{\\rm \\bf n}$ , where ${\\rm \\bf H}\\in {\\mathbb {R}^{M\\times N}}$ is the linear forward model and ${\\rm \\bf n}\\in {\\mathbb {R}^{M}}$ is additive white Gaussian noise.",
"Our reconstruction incorporates a strong form of prior knowledge about the original image: We assume that ${\\rm \\bf x}$ must lie in some set ${\\mathcal {S}}\\subset {\\mathbb {R}^N}$ that contains all objects of interest.",
"The proposed way to make the reconstruction consistent with the measurements as well as with the prior knowledge is to solve the constrained least-squares problem $\\min _{{\\rm \\bf x}\\in {\\mathcal {S}}}\\, \\frac{1}{2}\\left\\Vert {\\rm \\bf H}{\\rm \\bf x}-{\\rm \\bf y}\\right\\Vert ^2_2.$ The condition ${{\\rm \\bf x}\\in {\\mathcal {S}}}$ in (REF ) plays the role of a regularizer.",
"If no two points in ${\\mathcal {S}}$ have the same measurements and incase ${\\rm \\bf y}$ is noiseless, then out of all the points in ${\\mathbb {R}}^N$ that are consistent with the measurement ${\\rm \\bf y}$ , (REF ) selects a unique point ${\\rm \\bf x}^*\\in {\\mathcal {S}}$ .",
"In this way, it removes the ill-posedness of the inverse problem.",
"When the measurements are noisy, (REF ) returns a point ${\\rm \\bf x}^*\\in {\\mathcal {S}}$ such that ${\\rm \\bf y}^*={\\rm \\bf H}{\\rm \\bf x}^*$ is as close as possible to ${\\rm \\bf y}$ .",
"Thus, it also denoises the measurement, where the quantity ${\\rm \\bf y}^*$ can be regarded as the denoised version of ${\\rm \\bf y}$ .",
"It is remarkable that (REF ) is a generalized formulation of the regularization schemes in (), which can be rewritten as $\\min _{{\\rm \\bf x}\\in {\\mathcal {S}}_R}\\, \\frac{1}{2}\\left\\Vert {\\rm \\bf H}{\\rm \\bf x}-{\\rm \\bf y}\\right\\Vert ^2_2,$ where ${\\mathcal {S}}_R=\\lbrace {\\rm \\bf x}\\in {\\mathbb {R}^N}:R({\\rm \\bf x}) \\le \\tau \\rbrace $ for some unique $\\tau $ dependent on the regularization parameter $\\lambda $ .",
"The point ${\\rm \\bf x}^{*}\\in {\\mathcal {S}}$ is called a local minimizer of (REF ) if $\\exists \\varepsilon >0:\\left\\Vert {\\rm \\bf H}{\\rm \\bf x}^*-{\\rm \\bf y}\\right\\Vert _2 \\le \\left\\Vert {\\rm \\bf H}{\\rm \\bf x}-{\\rm \\bf y}\\right\\Vert _2,\\forall {\\rm \\bf x}\\in {\\mathcal {S}}\\cap {\\mathcal {B}}_{\\varepsilon }({\\rm \\bf x}^*).$"
],
[
"Projected Gradient Descent",
"When ${\\mathcal {S}}$ is a closed convex set, it is well known [29] that a solution of (REF ) can be found by PGD ${\\rm \\bf x}_{k+1} &= P_{{\\mathcal {S}}} ({\\rm \\bf x}_{k} - \\gamma {{\\rm \\bf H}^{{\\rm \\bf H}}{\\rm \\bf x}_k + \\gamma {{\\rm \\bf H}^{}{\\rm \\bf y}),}where \\gamma is a step size chosen such that \\gamma < 2/\\left\\Vert {{\\rm \\bf H}^{{\\rm \\bf H}}_2.",
"This algorithm combines the orthogonal projection onto {\\mathcal {S}} with the gradient descent w.r.t.\\hbox{} the quadratic objective function (also called the Landweber update~\\cite {Landweber:1951}).",
"PGD~\\cite [Sec.",
"2.3]{Bertsekas:1999} is a subclass of the forward-backward splitting~\\cite {CombettesW:2006,CombettesP:2011}, which is known in the \\ell _1-minimization literature as Iterative Shrinkage/Thresholding Algorithms (ISTA)~\\cite {Figueiredo2003,BectBAC:2003,daubechies2004iterative}.",
"}In our problem, \\right.",
"{\\mathcal {S}} is presumably non-convex, but we propose to still use the update~(\\ref {eq:PL}) with some projector P_{{\\mathcal {S}}} that may not be orthogonal.",
"In the rest of this section, we provide sufficient conditions on the projector P_{{\\mathcal {S}}} (not on {\\mathcal {S}} itself) under which~(\\ref {eq:PL}) leads to a local minimizer of~(\\ref {prob}).",
"Similarly to the convex case, we characterize the local minimizers of~(\\ref {prob}) by the fixed points of the combined operator{\\begin{@align}{1}{-1}G_{\\gamma }({\\rm \\bf x}) = P_{{\\mathcal {S}}} ({\\rm \\bf x}- \\gamma {{\\rm \\bf H}^{{\\rm \\bf H}}{\\rm \\bf x}+ \\gamma {{\\rm \\bf H}^{}{\\rm \\bf y})}}and then show that some fixed point of that operator must be reached by the iteration {\\rm \\bf x}_{k+1}=G_{\\gamma }({\\rm \\bf x}_k) as k\\rightarrow \\infty , no matter the value of {\\rm \\bf x}_0.We first state a sufficient condition for each fixed point of G_{\\gamma } to become a local minimizer of~(\\ref {prob}).\\begin{prop}Let \\gamma >0 and P_{{\\mathcal {S}}} be such that, for all {\\rm \\bf x}\\in {\\mathbb {R}^N},{\\begin{@align}{1}{-1}\\left\\langle {{\\rm \\bf z}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle \\le 0, \\quad \\forall {\\rm \\bf z}\\in {\\mathcal {S}}\\cap {\\mathcal {B}}_{\\varepsilon }(P_{{\\mathcal {S}}}{\\rm \\bf x}),\\end{@align}}for some \\varepsilon >0.Then, any fixed point of the operator G_{\\gamma } in~(\\ref {eq:opT}) is a local minimizer of~(\\ref {prob}).",
"Furthermore, if~(\\ref {eq:local}) is satisfied globally, in the sense that{\\begin{@align}{1}{-1}\\left\\langle {{\\rm \\bf z}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle \\le 0, \\quad \\forall {\\rm \\bf x}\\in {\\mathbb {R}^N},{\\rm \\bf z}\\in {\\mathcal {S}},\\end{@align}}then any fixed point of G_{\\gamma } is a solution of~(\\ref {prob}).\\end{prop}Two remarks are in order.",
"First, (\\ref {eq:global}) is a well-known property of orthogonal projections onto closed convex sets.",
"It actually implies the convexity of {\\mathcal {S}} (see ~Proposition~\\ref {thm:convex}).",
"Second, (\\ref {eq:local}) is much more relaxed and easily achievable, for example, as stated in Proposition~\\ref {thm:union_convex}, by orthogonal projections onto unions of closed convex sets (special cases are unions of subspaces, which have found some applications in data modeling and clustering~\\cite {AldroubiT:2011}).\\begin{prop}If P_{{\\mathcal {S}}} is a projector onto {\\mathcal {S}}\\subset {\\mathbb {R}^N} that satisfies~(\\ref {eq:global}), then {\\mathcal {S}} must be convex.\\end{prop}\\end{@align}\\begin{prop}If {\\mathcal {S}} is a union of a finite number of closed convex sets in {\\mathbb {R}^N}, then the orthogonal projector P_{{\\mathcal {S}}} onto {\\mathcal {S}} satisfies~(\\ref {eq:local}).\\end{prop}}The above results suggest that, when {\\mathcal {S}} is non-convex, the best we can hope for is to find a local minimizer of~(\\ref {prob}) through a fixed point of G_{\\gamma }.",
"Theorem~\\ref {thm:fixed_point} provides a sufficient condition for PGD to converge to a unique fixed point of G_{\\gamma }.",
"}\\begin{thm}Let \\lambda _{\\max },\\lambda _{\\min } be the largest and smallest eigenvalues of {\\rm \\bf H}^{{\\rm \\bf H}, respectively.",
"If P_{{\\mathcal {S}}} satisfies~(\\ref {eq:local}) and is Lipschitz-continuous with constant L<(\\lambda _{\\max }+\\lambda _{\\min })/(\\lambda _{\\max }-\\lambda _{\\min }), then, for \\gamma =2/(\\lambda _{\\max }+\\lambda _{\\min }), the sequence \\lbrace {\\rm \\bf x}_k\\rbrace generated by~(\\ref {eq:PL}) converges to a local minimizer of~(\\ref {prob}), regardless of the initialization {\\rm \\bf x}_0.",
"}\\end{thm}It is important to note that the projector P_{{\\mathcal {S}}} can never be contractive since it preserves the distance between any two points on {\\mathcal {S}}.",
"Therefore, when {\\rm \\bf H} has a nontrivial null space, the condition L<(\\lambda _{\\max }+\\lambda _{\\min })/(\\lambda _{\\max }-\\lambda _{\\min }) of Theorem~\\ref {thm:fixed_point} is not feasible.",
"The smallest possible Lipschitz constant of P_{\\mathcal {S}} is L=1, which means P_{{\\mathcal {S}}} is non-expansive.",
"Even with this condition, it is not guaranteed that the combined operator F_{\\gamma } has a fixed point.",
"This limitation can be overcome when F_{\\gamma } is assumed to have a nonempty set of fixed points.",
"Indeed, we state in Theorem~\\ref {thm:fixed_point2} that one of them must be reached by iterating the {averaged operator} \\alpha \\operatorname{Id}+(1-\\alpha )G_{\\gamma }, where \\alpha \\in (0,1) and \\operatorname{Id} is the identity operator.\\begin{thm}Let \\lambda _{\\max } be the largest eigenvalue of {\\rm \\bf H}^{{\\rm \\bf H}.",
"If P_{{\\mathcal {S}}} satisfies~(\\ref {eq:local}) and is a non-expansive operator such that G_{\\gamma } in~(\\ref {eq:opT}) has a fixed point for some \\gamma < 2/\\lambda _{\\max }, then the sequence \\lbrace {\\rm \\bf x}_k\\rbrace generated by{\\begin{@align}{1}{-1}{\\rm \\bf x}_{k+1} = (1-\\alpha ){\\rm \\bf x}_{k} + \\alpha G_{\\gamma }({\\rm \\bf x}_k),\\end{@align}}for any \\alpha \\in (0,1),converges to a local minimizer of~(\\ref {prob}), regardless of the initialization {\\rm \\bf x}_0.",
"}\\end{thm}\\section {Relaxation with Guaranteed Convergence}Despite their elegance, Theorems~\\ref {thm:fixed_point} and~\\ref {thm:fixed_point2} are not directly useful when we construct the projector P_{{\\mathcal {S}}} by training a CNN, because it is unclear how to enforce the Lipschitz continuity of P_{{\\mathcal {S}}} on the CNN architecture.",
"Without putting any constraints on the CNN, however, we can still achieve the convergence of the reconstruction sequence by modifying PGD as described in Algorithm~\\ref {alg}; we name it relaxed projected gradient descent (RPGD).",
"In the proposed algorithm, the projector P_{{\\mathcal {S}}} is replaced with a general nonlinear operator F. We also introduce a sequence \\lbrace c_k\\rbrace that governs the rate of convergence of the algorithm and a sequence \\lbrace \\alpha _k\\rbrace of relaxation parameters that evolves with the algorithm.",
"The convergence of RPGD is guaranteed by Theorem~\\ref {thm:main}.",
"More importantly, if the nonlinear operator F is actually a projector and the relaxation parameters do not go all the way to 0, then RPGD converges to a meaningful point.$ Theorem 1 Let the input sequence $\\lbrace c_k\\rbrace $ of Algorithm REF be asymptotically upper-bounded by $C<1$ .",
"Then, the following statements hold true for the reconstruction sequence $\\lbrace {\\rm \\bf x}_k\\rbrace $ : ${\\rm \\bf x}_k\\rightarrow {\\rm \\bf x}^{*}$ as $k\\rightarrow \\infty $ , for all choices of $F$ ; if $F$ is continuous and the relaxation parameters $\\lbrace \\alpha _k\\rbrace $ are lower-bounded by $\\varepsilon >0$ , then ${\\rm \\bf x}^*$ is a fixed point of $G_{\\gamma }({\\rm \\bf x}) = F({\\rm \\bf x}- \\gamma {{\\rm \\bf H}^{{\\rm \\bf H}}{\\rm \\bf x}+ \\gamma {{\\rm \\bf H}^{}{\\rm \\bf y});}\\item [(iii)] if, in addition to \\emph {(ii)}, F is indeed a projector onto {\\mathcal {S}} that satisfies~(\\ref {eq:local}), then {\\rm \\bf x}^* is a local minimizer of~(\\ref {prob}).",
"}$ We prove Theorem REF in Appendix REF .",
"Note that the weakest statement here is (i); it always guarantees the convergence of RPGD, albeit not necessarily to a fixed point of $G_{\\gamma }$ .",
"Moreover, the assumption about the continuity of $F$ in (ii) is automatically true when $F$ is a CNN.",
"Training CNN as a projector With these theoretical foundations in place, we move on to the matter of training a CNN to act as the projector in RPGD (Algorithm REF ).",
"For any point ${\\rm \\bf x}\\in {\\mathcal {S}}$ , a projector onto ${\\mathcal {S}}$ should satisfy $P_{\\mathcal {S}}{\\rm \\bf x}= {\\rm \\bf x}$ .",
"Moreover, we want ${\\rm \\bf x}= P_{\\mathcal {S}}(\\tilde{{\\rm \\bf x}}) ,$ where $\\tilde{{\\rm \\bf x}}$ is any perturbed version of ${\\rm \\bf x}$ .",
"Given a training set, $\\lbrace {\\rm \\bf x}^1,\\ldots , {\\rm \\bf x}^Q\\rbrace $ , of points drawn from the set ${\\mathcal {S}}$ , we generate an ensemble of $N \\times Q$ perturbed points, $\\lbrace \\lbrace \\tilde{{\\rm \\bf x}}^{1,1}, \\ldots , \\tilde{{\\rm \\bf x}}^{Q,1}\\rbrace , \\ldots , \\lbrace \\tilde{{\\rm \\bf x}}^{1,N} \\ldots , \\tilde{{\\rm \\bf x}}^{Q,N} \\rbrace \\rbrace $ and train the CNN by minimizing the loss function $J({\\theta }) = \\sum _{n=1}^{N} \\underbrace{\\sum _{q=1}^Q\\left\\Vert {\\rm \\bf x}^q-C\\!N\\!N_{{\\theta }}( \\tilde{{\\rm \\bf x}}^{q,n})\\right\\Vert _2^2}_{J_n({\\theta })}.$ The optimization proceeds by stochastic gradient descent for $T$ epochs, where an epoch is defined as one pass though the training data.",
"It remains to select the perturbations that generate the ${\\rm \\bf x}^{q,n}$ .",
"Our goal here is to create a diverse set of perturbations so that the CNN does not overfit one specific type.",
"In our experiments, while training for the $t$ -th epoch, we chose $\\tilde{{\\rm \\bf x}}^{q,1} &= {\\rm \\bf x}^{q}&:&\\text{No perturbation} \\\\\\tilde{{\\rm \\bf x}}^{q,2} &= {\\bf {A}} {\\rm \\bf H}{\\rm \\bf x}^{q}&:&\\text{Specific linear perturbation}\\\\\\tilde{{\\rm \\bf x}}^{q,3} &= C\\!N\\!N_{{\\theta }_{t-1}} (\\tilde{{\\rm \\bf x}}^{q,2})&:&\\text{Dynamic non-linear perturbation}, $ where ${\\bf {A}}$ is a classical linear reconstruction algorithm like FBP or BP, ${\\rm \\bf H}$ is the forward model, and ${\\theta }_t$ are the CNN parameters after $t$ epochs.",
"We now comment on each of these perturbations in detail.",
"Keeping $\\tilde{{\\rm \\bf x}}^{q,1}$ in the training ensemble will train the CNN with the defining property of the projector: the projector maps a point in the set ${\\mathcal {S}}$ onto itself.",
"If the CNN were only trained with (REF ), it would be an autoencoder [36].",
"To understand the perturbation $\\tilde{{\\rm \\bf x}}^{q,2}$ in (), recall that ${\\bf {A}} {\\rm \\bf H}{\\rm \\bf x}^{q}$ is a classical linear reconstruction of ${\\rm \\bf x}^{q}$ from its measurement ${\\rm \\bf y}={\\rm \\bf H}{\\rm \\bf x}^q$ .",
"This perturbation is useful because we will initialize RPGD with ${\\bf {A}} {\\rm \\bf H}{\\rm \\bf x}^{q}$ .",
"Using only () for training will return the same CNN as in [19].",
"The perturbation $\\tilde{{\\rm \\bf x}}^{q,3}$ in () is the output of the CNN whose parameters ${\\theta }_t$ change with every epoch $t$ , thus it is a non-linear and dynamic (epoch-dependent) perturbation of ${\\rm \\bf x}^q$ .",
"The rationale for using () is that it greatly increases the training diversity (allowing the network to see $T$ new perturbations of each training point) without greatly increasing the total training size (only requiring an additional $Q$ gradient computations per epoch).",
"Moreover, () is in sync with the iterative scheme of RPGD, where the output of the CNN is processed with a gradient descent and is again fed into itself.",
"Architecture The architecture we use is the same as in [19], which is a U-net [37] with intrinsic skip connections among its layers and an extrinsic skip connection between the input and the output.",
"The intrinsic skip connections help to eliminate singularities during the training [38].",
"The extrinsic skip connections make this network a residual net; i.e., $C\\!N\\!N=\\operatorname{Id}+ Unet$ , where $\\operatorname{Id}$ denotes the identity operator and $Unet : {\\mathbb {R}}^N \\rightarrow {\\mathbb {R}}^N$ denotes the Unet as a function.",
"The U-net therefore actually provides the projection error (negative perturbation) that should be added to the input to get the projection.",
"Residual nets have been shown to be effective in the image recognition [39] and inverse problem cases[19].",
"While the residual net architecture does not increase the capacity or the approximation power of the CNN, it does help in learning functions that are close to an identity operator, as is the case in our setting.",
"Sequential Training Strategy We train the CNN in 3 stages.",
"In stage 1, we train it for $T_1$ epochs w.r.t.",
"the loss function $J_2$ which only uses the ensemble $\\lbrace \\tilde{{\\rm \\bf x}}^{q,2}\\rbrace $ generated by ().",
"In stage 2, we add the ensemble $\\lbrace \\tilde{{\\rm \\bf x}}^{q,3}\\rbrace $ according to () at every epoch and then train the CNN w.r.t.",
"the loss function $J_2+J_3$ ; we repeat this procedure for $T_2$ epochs.",
"Finally, in stage 3, we train the CNN for $T_3$ epochs with all the ensembles $\\lbrace \\tilde{{\\rm \\bf x}}^{q,1},\\tilde{{\\rm \\bf x}}^{q,2},\\tilde{{\\rm \\bf x}}^{q,3}\\rbrace $ to minimize the original loss function $J=J_1+J_2+J_3$ from (REF ).",
"The above sequential procedure helps speed up the training.",
"The training with $\\lbrace \\tilde{{\\rm \\bf x}}^{q,1}\\rbrace $ is initially bypassed with using the residual net, which is close to the identity operator.",
"It is only incorporated in the last few epochs of stage 3.",
"After training with only $\\lbrace \\tilde{{\\rm \\bf x}}^{q,2}\\rbrace $ in stage 1, $\\tilde{{\\rm \\bf x}}^{q,3}$ will be close to ${\\rm \\bf x}^{q}$ , since it is the output of the CNN for the input $\\tilde{{\\rm \\bf x}}^{q,2}$ .",
"This will ease the training with $\\lbrace \\tilde{{\\rm \\bf x}}^{q,3}\\rbrace $ , which is added after stage 1.",
"Experiments We validate our proposed method on the difficult case of sparse-view CT reconstruction with low dosage exposure.",
"The measurement operator ${\\bf {H}}$ is now the Radon transform.",
"It maps an image to the values of its integrals along a known set of lines [40].",
"In 2D, these measurements can be indexed by the angles and offsets of the lines and arranged in a 2D sinogram.",
"We are particularly interested in the case where the total number of measurements is smaller than the number of pixels in the reconstruction.",
"For example, we aim to reconstruct a (512 $\\times $ 512) image from 45 angles, each with 729 offsets sinogram; i.e., to reconstruct ${\\rm \\bf x}\\in {\\mathbb {R}}^{512 \\times 512}$ from ${\\rm \\bf y}\\in {\\mathbb {R}}^{45 \\times 729}$ .",
"This corresponds to about 8 times fewer measurements than the image to be reconstructed.",
"Dataset Our dataset consists of clinically realistic invivo ($512 \\times 512$ ) CT scans of human abdomen from Mayo clinic for the AAPM Low Dose CT Grand Challenge [41].",
"This data includes CT scans of 10 patients obtained using full dose.",
"We use 475 images from 9 patients for training and 25 images from the other patient for testing.",
"This is the same data used in [19].",
"These images serve as the ground truth.",
"From these images, we generate the measurements (sinograms) using the radon command in Matlab, which corresponds to the forward model ${\\rm \\bf H}$ .",
"The sinograms always have 729 offsets per view, but we vary the number of views in different experiments.",
"Our task is to reconstruct these images from their sparse-view sinograms.",
"We take 2 scenarios: 144 views and 45 views, which corresponds to $\\small {\\times }$ 5 and $\\small {\\times }$ 16 dosage reductions (assuming a full-view sinogram has 720 views).",
"The backprojection ${{\\rm \\bf H}^{} is implemented via the \\texttt {iradon} command with a normalization to satisfy the adjoint property.To make the experiments more realistic and to reduce the inverse crime, the sinograms are generated by perturbing the angles of the views slightly by adding a zero-mean additive white Gaussian noise (AWGN) with standard deviation of 0.05 degrees.",
"This creates a slight mismatch between the actual measurement process and the forward model {\\rm \\bf H}.",
"We also add various amounts of zero-mean Gaussian noise to the sinograms.",
"The SNR of the sinogram {\\rm \\bf y}+{\\rm \\bf n} is defined as\\begin{equation}\\text{SNR}({\\rm \\bf y}+{\\bf {n}},{\\rm \\bf y}) = 20 \\log _{10}\\left({\\left\\Vert {\\rm \\bf y}\\right\\Vert _2}/{\\left\\Vert {\\rm \\bf n}\\right\\Vert _2} \\right).\\end{equation}}Given the ground truth $x$, our figure of merit for the reconstructed $x*$ is the regressed SNR, given by\\begin{equation}\\text{SNR} ({\\rm \\bf x}^*,{\\rm \\bf x}) =\\arg \\max _{a, b} \\text{SNR}(a {\\bf {x}}^* +b, {\\rm \\bf x}),\\end{equation}where the scalars $ a$ and $ b$ serve to scale the data and remove any DC offset, which can greatly affect the SNR but are of little practical importance.$ Comparison Methods We compare four reconstruction methods and report the SNRs for all of them.",
"FBP is the classical direct inversion of the Radon transform ${\\rm \\bf H}$ , here implemented in Matlab by the iradon command with the ram-lak filter and linear interpolation as options.",
"TV solves $\\min _{{\\bf {x}}} \\left({\\frac{1}{2}}\\Vert {\\bf {H}}{\\bf {x}}-{\\bf {y}}\\Vert _2^2+\\lambda \\Vert {\\bf {x}}\\Vert _{\\text{T}V}\\right) \\, \\text{s.t. }",
"{\\rm \\bf x}>0.$ The optimization is carried out via ADMM [13].",
"For a given testing image the parameter $\\lambda $ is tuned so as to maximize the SNR of the reconstruction.",
"FBPconv is the deep-learning-based regression technique [19] that corresponds to a CNN trained with only the ensemble in ().",
"In the testing phase, the FBP of the measurements is fed into the trained CNN to output the reconstruction image.",
"RPGD is our proposed method which is described in Algorithm REF where the nonlinear operator $F$ is the CNN trained as a projector (as discussed in section ).",
"Training and Selection of Parameters We now describe how training and/or parameter selection occurred for the reconstruction methods.",
"FBP has no free hyperparameters.",
"For TV, we chose $\\lambda $ by a grid search through 20 values for each test image.",
"While carrying the optimization with ADMM we put the penalty term, $\\rho = \\lambda $ .",
"The rationale for this heuristic is that the soft-threshold parameter is of the same order of magnitude as the image gradients.",
"We set the number of iterations to 100, which was enough to show good empirical convergence.",
"As discussed in section , the CNNs for RPGD is trained in 3 stages, with the following configurations: $\\small {\\times }$ 5, no noise: $T_1=80$ , $T_2=49$ , $T_3=5$ .",
"$\\small {\\times }$ 16, no noise: $T_1=71$ , $T_2=41$ , $T_3=11$ .",
"We used the CNN obtained right after the first stage for FBPconv, since during this stage only the training ensemble in () was taken into account.",
"We empirically found that the training error $J_2$ converged in $T_1$ epochs of stage 1, yielding an optimal performance for FBPconv.",
"In addition, we also trained the CNNs w.r.t.",
"40 dB noise level in the measurements, by replacing the ensemble in () with $\\lbrace {\\bf {A}} {\\rm \\bf y}^q\\rbrace $ where ${\\rm \\bf y}^q={\\rm \\bf H}{\\rm \\bf x}^q+{\\rm \\bf n}$ .",
"With 20% probability, we also perturbed the views of the measurements with AWGN of 0.05 standard deviation so as to enforce robustness to model mismatch.",
"These CNNs were initialized with the ones obtained after the first stage of the noiseless training and were then trained with the following configurations: $\\small {\\times }$ 5, 40-dB noise: $T_1=35$ , $T_2=49$ , $T_3=5$ .",
"$\\small {\\times }$ 16, 40-dB noise: $T_1=32$ , $T_2=41$ , $T_3=11$ .",
"Similarly to the noiseless case, the CNNs obtained after the first and the third stage of the above training were used in FBPconv and RPGD, respectively.",
"For clarity, these variants will be referred to as FBPconv40 and $\\textbf {RPGD40}$ .",
"During the training, the learning rate was logarithmically decreased from $10^{-2}$ to $10^{-3}$ in stage 1 and kept at $10^{-3}$ for stages 2 and 3.",
"The batch size was fixed to 2, the gradient above $10^{-2}$ was clipped and the momentum was set to $0.99$ .",
"The total training time for the noiseless case was around 21.5 hours on GPU Titan X (Pascal architecture).",
"The hyper-parameters for RPGD were chosen as follows: The relaxation parameter $\\alpha _0$ was initialized with 1, the sequence $\\lbrace c_k\\rbrace $ was set to a constant $C$ where $C=0.99$ for RPGD and $C = 0.8$ for RPGD40.",
"For each noise and dosage reduction level, the only free parameter, $\\gamma $ , was tuned to give the best average SNR over 25 test images.",
"In all experiments, the gradient step was removed from the first iteration.",
"On the GPU, one iteration of RPGD takes less than 1 second.",
"The algorithm is stopped when the residual $\\Vert {\\rm \\bf x}_{k+1}-{\\rm \\bf x}_k\\Vert _2$ reaches a value less than 1, which is sufficiently small compared to the dynamic range [0, 350] of the image.",
"It takes around 1-2 minutes to reconstruct an image with RPGD.",
"Results Table: Averaged reconstruction SNRs over 25 images for 16 and 5 times dosage reductions with low measurement noise.Table: Averaged reconstruction SNRs over 25 images for 16 and 5 times dosage reductions with high measurement noise.Figure: (a) Reconstructions of a test image from noiseless measurements with 45 views (×\\small {\\times }16 dosage reduction) using FBP, TV, FBPconv, and RPGD (proposed): first row shows the full images, second row shows their zoomed versions.",
"(b) Zoomed reconstructions of the same image when the measurement is noisy with 45 dB (first row) and 40 dB (second row) SNR.",
"In these cases, FBPconv and RPGD are replaced by FBPconv40 and RPGD40, respectively.Figure: Reconstructions of a test image from noisy measurements with 144 views (×\\small {\\times }5 dosage reduction) using FBP, TV, FBPconv40, and RPGD40 (proposed).",
"First row shows the results for measurement noise with SNR = 45 dB; second row shows their zoomed version.",
"Third row shows the zoomed results for measurement noise with SNR = 35 dB.Figure: Convergence with iteration kk of RPGD for the ×\\small {\\times }16, no-noise case when C=0.99C=0.99.",
"Results are averaged over 25 test images.",
"(a) SNRs of 𝐱 k {\\rm \\bf x}_k w.r.t.",
"the ground-truth image.",
"(b) SNRs of 𝐇𝐱 k {\\rm \\bf H}{\\rm \\bf x}_k w.r.t.",
"the ground-truth sinogram.",
"(c) Evolution of the relaxation parameters α k \\alpha _k.Tables REF and REF report the results of various methods for low and high measurement noise, respectively.",
"FBPconv and RPGD are used for low noise, while FBPconv40 and RPGD40 are used for high noise.",
"The reconstruction SNRs are averaged over the 25 test images.",
"In the low-noise cases (Table REF ), the proposed method, RPGD, outperforms all the others for both $\\small {\\times }5$ and $\\small {\\times }16$ reductions.",
"FBP performs the worst but is able to retain enough information to be utilized by FBPConv and RPGD.",
"Thanks to the convexity of the iterative scheme, TV is able to perform well but tends to smooth textures and edges.",
"On the other hand, FBPConv outperforms TV.",
"However, it is outperformed by RPGD.",
"This is mainly due to the feedback mechanism in RPGD which lets RPGD use the information in the given measurements to increase the quality of the reconstruction.",
"In fact, for the $\\small {\\times }16$ , no noise case, the SNRs of the sinogram of the reconstructed images for TV, FBPconv, and RPGD are around 47 dB, 57 dB, and 62 dB, respectively.",
"This means that not only reconstruction using RPGD has better image quality but is also more reliable since it is consistent with the given noiseless measurement.",
"In the noisier cases (Table REF ), RPGD40 outperforms the other methods in low-view cases ($\\small {\\times }16$ ) and is more consistent in performance than the others in high-view ($\\small {\\times }5$ ) cases.",
"FBPconv40 substantially outperforms TV in the two scenarios with 40-dB noise measurement, over which it was actually trained.",
"However, as the level of noise deviates from 40 dB, the performance of FBPconv40 degrades significantly.",
"Surprisingly, its performances in the 45-dB cases are much worse than those in the corresponding 40-dB cases.",
"This implies that FBPConv is highly sensitive to the difference between the training and the testing conditions.",
"By contrast, RPGD40 is more robust to this difference due to its iterative correction.",
"In fact, for $\\small {\\times } 16$ case with 45-dB and 35-dB noise level, it outperforms FBPconv40 by around 3.5 dB and 6 dB, respectively.",
"Fig.",
"REF (a) illustrates the reconstructions of a test image for $\\small {\\times } 16$ case when measurement is noiseless.",
"FBP is dominated by line artifacts, while TV satisfactorily removes these but blurs the fine structures.",
"FBPConv and RPGD, on the other hand, are able to reconstruct these details.",
"The zoomed version shows that RPGD is able to reconstruct the fine details better than the others.",
"This observation remains the same when the measurement quality degrades.",
"Fig.",
"REF (b) shows the reconstructions for 45-dB and 40-dB noise levels.",
"In these scenarios, RPGD40 is significantly better than both FBPconv40 and TV.",
"Fig.",
"REF compares the reconstructions for the $ \\small {\\times } 5$ case when the noise levels are 45 dB and 35 dB.",
"It is visible that FBPconv40 results in a noisy image and TV is again blurred.",
"RPGD40 retains the fine structures and is the best performer.",
"Convergence of RPGD Figs.",
"REF (a) and (b) shows the evolution of SNR of images ${\\rm \\bf x}_k$ and their measurements ${\\bf {H}} {\\rm \\bf x}_k$ w.r.t.",
"the ground truth image and ground truth measurement, respectively.",
"Fig.",
"REF (c) shows the $\\alpha _k$ w.r.t.",
"the iteration $k$ .",
"The results are averaged over 25 test images for $\\small {\\times }16$ , no noise case and $C=0.99$ .",
"RPGD outperforms all the other methods in terms of both image quality and measurement consistency.",
"Due to the high value of the step size ($\\gamma =2 \\times 10^{-3}$ ) and the large difference ${\\rm \\bf H}{\\rm \\bf x}_k -{\\rm \\bf y}$ , the initial few iterations have large gradients resulting in the instability of the algorithm.",
"The reason is that the CNN is fed with ${\\rm \\bf x}_k -\\gamma {{\\rm \\bf H}^{}({\\rm \\bf H}{\\rm \\bf x}_k -{\\rm \\bf y}) which is drastically different from the perturbations on which it was trained.In this situation, \\alpha _k decreases steeply and stabilizes the algorithm.At convergence, \\alpha _k\\ne 0, therefore, according to Theorem~\\ref {thm:main}, {\\rm \\bf x}_{100} is the fixed point of (\\ref {eq:opF}) where F=C\\!N\\!N.",
"}$ Conclusion We have proposed a simple yet effective iterative scheme (RPGD) where one step of enforcing measurement consistency is followed by a CNN which tries to project the solution onto the set of the data that we are interested in.",
"The whole scheme is ensured to be convergent.",
"We also introduced a novel method to train a CNN that acts like a projector using a reasonably small dataset (475 images).",
"For sparse-view CT reconstruction our method outperforms the previous techniques for both noiseless and noisy measurements.",
"The proposed framework can be used to solve other inverse problems like super-resolution, deconvolution, accelerated MRI, etc.",
"This can bring more robustness and reliability to the current deep learning based techniques.",
"Proof of Theorem REF (i) Set ${\\rm \\bf r}_k={\\rm \\bf x}_{k+1}-{\\rm \\bf x}_k$ .",
"On one hand, it is clear that ${\\rm \\bf r}_k &= (1-\\alpha _k){\\rm \\bf x}_k + \\alpha _k {\\rm \\bf z}_k - {\\rm \\bf x}_{k}= \\alpha _k\\left({\\rm \\bf z}_k-{\\rm \\bf x}_k\\right).$ On the other hand, from the construction of $\\lbrace \\alpha _k\\rbrace $ , $&\\alpha _{k}\\left\\Vert {\\rm \\bf z}_{k} - {\\rm \\bf x}_{k}\\right\\Vert _2 \\le c_k\\alpha _{k-1}\\left\\Vert {\\rm \\bf z}_{k-1} - {\\rm \\bf x}_{k-1}\\right\\Vert _2\\nonumber \\\\\\Leftrightarrow &\\quad \\quad \\quad \\ \\,\\left\\Vert {\\rm \\bf r}_{k}\\right\\Vert _2 \\le c_k \\left\\Vert {\\rm \\bf r}_{k-1}\\right\\Vert _2.$ Iterating (REF ) gives $\\left\\Vert {\\rm \\bf r}_{k}\\right\\Vert _2 \\le \\left\\Vert {\\rm \\bf r}_0\\right\\Vert _2 \\prod _{i=1}^{k} c_{i}, \\quad \\forall k\\ge 1.$ We now show that $\\lbrace {\\rm \\bf x}_k\\rbrace $ is a Cauchy sequence.",
"Since $\\lbrace c_k\\rbrace $ is asymptotically upper-bounded by $C<1$ , there exists $K$ such that $c_k \\le C,\\forall k> K$ .",
"Let $m,n$ be two integers such that $m>n>K$ .",
"By using (REF ) and the triangle inequality, $\\left\\Vert {\\rm \\bf x}_m-{\\rm \\bf x}_n\\right\\Vert _2 &\\le \\sum _{k=n}^{m-1} \\left\\Vert {\\rm \\bf r}_{k}\\right\\Vert _2\\le \\left\\Vert {\\rm \\bf r}_0\\right\\Vert _2 \\prod _{i=1}^{K} c_{i} \\sum _{k=n-K}^{m-1-K} C^k\\nonumber \\\\&\\le \\left(\\left\\Vert {\\rm \\bf r}_0\\right\\Vert _2 \\prod _{i=1}^{K} c_{i}\\right) \\frac{C^{n-K}-C^{m-K}}{1-C}.$ The last inequality proves that $\\left\\Vert {\\rm \\bf x}_m-{\\rm \\bf x}_n\\right\\Vert _2\\rightarrow 0$ as $m\\rightarrow \\infty ,n\\rightarrow \\infty $ , or $\\lbrace {\\rm \\bf x}_k\\rbrace $ is a Cauchy sequence in the complete metric space ${\\mathbb {R}^N}$ .",
"As a consequence, $\\lbrace {\\rm \\bf x}_{k}\\rbrace $ must converge to some point ${\\rm \\bf x}^*\\in {\\mathbb {R}^N}$ .",
"(ii) Assume from now on that $\\lbrace \\alpha _k\\rbrace $ is lower-bounded by $\\varepsilon >0$ .",
"By definition, $\\lbrace \\alpha _k\\rbrace $ is also non-increasing and, thus, convergent to $\\alpha ^*>0$ .",
"Next, we rewrite the update of ${\\rm \\bf x}_k$ in Algorithm REF as ${\\rm \\bf x}_{k+1} = (1-\\alpha _k){\\rm \\bf x}_{k} + \\alpha _k G_{\\gamma }({\\rm \\bf x}_{k}),$ where $G_{\\gamma }$ is defined by (REF ).",
"Taking the limit of both sides of (REF ) leads to ${\\rm \\bf x}^* &= (1-\\alpha ^*){\\rm \\bf x}^* + \\alpha ^*\\lim _{k\\rightarrow \\infty }G_{\\gamma }({\\rm \\bf x}_k).$ Moreover, since the nonlinear operator $F$ is continuous, $G_{\\gamma }$ is also continuous.",
"Hence, $\\lim _{k\\rightarrow \\infty }G_{\\gamma }({\\rm \\bf x}_k) =G_{\\gamma }\\left(\\lim _{k\\rightarrow \\infty }{\\rm \\bf x}_k\\right) = G_{\\gamma }({\\rm \\bf x}^*).$ By plugging (REF ) into (REF ), we get that ${\\rm \\bf x}^*=G_{\\gamma }({\\rm \\bf x}^*)$ , which means ${\\rm \\bf x}^*$ is a fixed point of the operator $G_{\\gamma }$ .",
"(iii) Now that $F= P_{{\\mathcal {S}}}$ satisfies (), we invoke Proposition to infer that ${\\rm \\bf x}^*$ is a local minimizer of (REF ), thus completing the proof.",
"Acknowledgment The authors would like to thank Emmanuel Soubies for his helpful suggestions on training the CNN.",
"We thankfully acknowledge the support of NVIDIA Corporation which donated the Titan X GPU used in this research.",
"The authors would like to thank Dr. Cynthia McCollough, the Mayo Clinic, the American Association of Physicists in Medicine, and grants EB017095 and EB017185 from the National Institute of Biomedical Imaging and Bioengineering for giving opportunities to use real-invivo CT DICOM images.",
"Supplementary material Proof of Proposition Suppose that () is fulfilled and let ${\\rm \\bf x}^*\\in {\\mathcal {S}}$ be a fixed point of $G_{\\gamma }$ .",
"We show that ${\\rm \\bf x}^*$ is also a local minimizer of (REF ).",
"Indeed, setting ${\\rm \\bf x}={\\rm \\bf x}^*-\\gamma {{\\rm \\bf H}^{{\\rm \\bf H}}{\\rm \\bf x}^{*} + \\gamma {{\\rm \\bf H}^{}{\\rm \\bf y} leads to P_{{\\mathcal {S}}}{\\rm \\bf x}={\\rm \\bf x}^*.",
"Then, there exists \\varepsilon >0 such that, for all {\\rm \\bf z}\\in {\\mathcal {S}}\\cap {\\mathcal {B}}_{\\varepsilon }({\\rm \\bf x}^{*}),{\\begin{@align*}{1}{-1}0&\\ge \\left\\langle {{\\rm \\bf z}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle \\\\&=\\gamma \\left\\langle {{\\rm \\bf z}-{\\rm \\bf x}^*}\\, , \\,{{{\\rm \\bf H}^{}{\\rm \\bf y}-{\\rm \\bf H}^{{\\rm \\bf H}{\\rm \\bf x}^{*}}\\\\&=\\frac{\\gamma }{2}\\left(\\left\\Vert {\\rm \\bf H}{\\rm \\bf x}^*-{\\rm \\bf y}\\right\\Vert ^2_2 -\\left\\Vert {\\rm \\bf H}{\\rm \\bf z}-{\\rm \\bf y}\\right\\Vert ^2_2 + \\left\\Vert {\\rm \\bf H}{\\rm \\bf x}^*-{\\rm \\bf H}{\\rm \\bf z}\\right\\Vert ^2_2\\right).",
"}}Since \\right.\\gamma >0, the last inequality implies that{\\begin{@align*}{1}{-1}\\left\\Vert {\\rm \\bf H}{\\rm \\bf x}^*-{\\rm \\bf y}\\right\\Vert ^2_2 \\le \\left\\Vert {\\rm \\bf H}{\\rm \\bf z}-{\\rm \\bf y}\\right\\Vert ^2_2,\\quad \\forall {\\rm \\bf z}\\in {\\mathcal {S}}\\cap {\\mathcal {B}}_{\\varepsilon }({\\rm \\bf x}^{*}),\\end{@align*}}which means that {\\rm \\bf x}^{*} is a local minimizer of~(\\ref {prob}).\\end{@align*}Assume now that P_{{\\mathcal {S}}} satisfies~(\\ref {eq:global}).",
"By just removing the \\varepsilon -ball in the above argument, one easily verifies that{\\begin{@align*}{1}{-1}\\left\\Vert {\\rm \\bf H}{\\rm \\bf x}^*-{\\rm \\bf y}\\right\\Vert ^2_2 \\le \\left\\Vert {\\rm \\bf H}{\\rm \\bf z}-{\\rm \\bf y}\\right\\Vert ^2_2,\\quad \\forall {\\rm \\bf z}\\in {\\mathcal {S}},\\end{@align*}}which means that {\\rm \\bf x}^* is a solution of~(\\ref {prob}).",
"\\subsection {Proof of Proposition~\\ref {thm:convex}} We prove by contradiction.",
"Assuming that {\\mathcal {S}} is non-convex, there must exist {\\rm \\bf x}_1,{\\rm \\bf x}_2\\in {\\mathcal {S}} and \\alpha \\in (0,1) such that {\\rm \\bf x}=\\alpha {\\rm \\bf x}_1 + (1-\\alpha ){\\rm \\bf x}_2\\notin {\\mathcal {S}}.",
"Since P_{{\\mathcal {S}}}{\\rm \\bf x}\\in {\\mathcal {S}}, it must be that{\\begin{@align*}{1}{-1}0 &<\\left\\Vert {\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}\\right\\Vert ^2_2=\\left\\langle {{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle \\\\&=\\alpha \\left\\langle {{\\rm \\bf x}_1-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle \\\\&\\quad + (1-\\alpha )\\left\\langle {{\\rm \\bf x}_2-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle .\\end{@align*}}Thus, there exists i\\in \\lbrace 0,1\\rbrace such that{\\begin{@align*}{1}{-1}\\left\\langle {{\\rm \\bf x}_i-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle > 0,\\end{@align*}}which violates~(\\ref {eq:global}).",
"So, {\\mathcal {S}} is convex.",
"}\\subsection {Proof of Proposition~\\ref {thm:union_convex}} Suppose that {\\mathcal {S}}=\\bigcup _{i=1}^{n}{\\mathcal {C}}_i, where {\\mathcal {C}}_i is a closed convex set for all i=1,\\ldots ,n. The statement is trivial when n=1; assume now that n\\ge 2.",
"Let {\\rm \\bf x}\\in {\\mathbb {R}^N} and \\hat{{\\rm \\bf x}} be the orthogonal projection of {\\rm \\bf x} onto {\\mathcal {S}}.",
"Consider two cases.",
"}\\emph {Case 1}: \\hat{{\\rm \\bf x}}\\in \\bigcap _{i=1}^n{\\mathcal {C}}_i.\\\\It is then clear that{\\begin{@align*}{1}{-1}\\left\\Vert \\hat{{\\rm \\bf x}}-{\\rm \\bf x}\\right\\Vert _2 \\le \\left\\Vert {\\rm \\bf z}-{\\rm \\bf x}\\right\\Vert _2,\\forall {\\rm \\bf z}\\in {\\mathcal {C}}_i,\\forall i.\\end{@align*}}This means that \\hat{{\\rm \\bf x}} is the orthogonal projection of {\\rm \\bf x} onto each {\\mathcal {C}}_i.",
"Consequently,{\\begin{@align*}{1}{-1}\\left\\langle {{\\rm \\bf z}-\\hat{{\\rm \\bf x}}}\\, , \\,{{\\rm \\bf x}-\\hat{{\\rm \\bf x}}}\\right\\rangle \\le 0,\\forall {\\rm \\bf z}\\in {\\mathcal {C}}_i,\\forall i\\le n,\\end{@align*}}which implies that~(\\ref {eq:local}) holds true for all \\varepsilon > 0.",
"}\\emph {Case 2}: $xi=1nCi$.\\\\Without loss of generality, there exists $ m<n$ such that{\\begin{@align}{1}{-1}\\hat{{\\rm \\bf x}}\\in \\bigcap _{i=1}^m {\\mathcal {C}}_i,\\quad \\hat{{\\rm \\bf x}}\\notin \\bigcup _{i=m+1}^n {\\mathcal {C}}_i.\\end{@align}}Let $ d$ be the distance from $x$ to the set $ T=i=m+1n Ci$.",
"Since each $ Ci$ is closed, $ T$ must be closed too and, so, $ d>0$.",
"We now choose $ 0<<d$.",
"Then, $ B(x)T=$.",
"Therefore,{\\begin{@align}{1}{-1}{\\mathcal {S}}\\cap {\\mathcal {B}}_\\varepsilon (\\hat{{\\rm \\bf x}}) =\\bigcup _{i=1}^{m}\\left({\\mathcal {C}}_i\\cap {\\mathcal {B}}_\\varepsilon (\\hat{{\\rm \\bf x}})\\right) = \\bigcup _{i=1}^{m}\\tilde{{\\mathcal {C}}}_i,\\end{@align}}where $ Ci=CiB(x)$ is clearly a convex set, for all $ im$.",
"It is straightforward that $x$ is the orthogonal projection of $x$ onto the set $ i=1mCi$ and that $xi=1mCi$.",
"We are back to Case 1 and, therefore,{\\begin{@align}{1}{-1}\\left\\langle {{\\rm \\bf z}-\\hat{{\\rm \\bf x}}}\\, , \\,{{\\rm \\bf x}-\\hat{{\\rm \\bf x}}}\\right\\rangle \\le 0,\\forall {\\rm \\bf z}\\in \\tilde{{\\mathcal {C}}}_i,\\forall i\\le m.\\end{@align}}From~(\\ref {eq:intersect}) and~(\\ref {eq:case}), (\\ref {eq:local}) is fulfilled for the chosen $$.\\subsection {Proof of Theorem~\\ref {thm:fixed_point}} Let $ {i}$ denote the set of eigenvalues of $HH$.",
"We first have that, for all $xRN$,{\\begin{@align}{1}{-1}\\left\\Vert {\\rm \\bf x}- \\gamma {\\rm \\bf H}^{{\\rm \\bf H}{\\rm \\bf x}_2 \\le \\left\\Vert {\\rm \\bf I}-\\gamma {\\rm \\bf H}^{{\\rm \\bf H}_2\\left\\Vert {\\rm \\bf x}\\right\\Vert _2,}\\right.where the spectral norm of {\\rm \\bf I}-\\gamma {{\\rm \\bf H}^{{\\rm \\bf H}} is given by{\\begin{@align}{1}{-1}\\left\\Vert {\\rm \\bf I}-\\gamma {\\rm \\bf H}^{{\\rm \\bf H}_2 = \\max _{i}\\lbrace |1-\\gamma \\lambda _i|\\rbrace .",
"}\\right.On the other hand, choosing \\gamma ={2}/{(\\lambda _{\\max }+\\lambda _{\\min })} yields{\\begin{@align}{1}{-1}&\\frac{2\\lambda _{\\min }}{\\lambda _{\\max } + \\lambda _{\\min }} \\le \\gamma \\lambda _i \\le \\frac{2\\lambda _{\\max }}{\\lambda _{\\max } + \\lambda _{\\min }},\\quad \\forall i \\nonumber \\\\\\Leftrightarrow \\quad & | 1-\\gamma \\lambda _i| \\le \\frac{\\lambda _{\\max }-\\lambda _{\\min }}{\\lambda _{\\max } + \\lambda _{\\min }},\\quad \\forall i.\\end{@align}}By combining~(\\ref {eq:gradient_bound}), (\\ref {eq:spectral_norm}), and~(\\ref {eq:absolute}),{\\begin{@align}{1}{-1}\\left\\Vert {\\rm \\bf x}- \\gamma {\\rm \\bf H}^{{\\rm \\bf H}{\\rm \\bf x}_2 \\le \\frac{\\lambda _{\\max }-\\lambda _{\\min }}{\\lambda _{\\max } + \\lambda _{\\min }}\\left\\Vert {\\rm \\bf x}\\right\\Vert _2, \\quad \\forall {\\rm \\bf x}.",
"}\\right.Combining~(\\ref {eq:gradient_Lipschitz}) with the Lipschitz continuity of P_{{\\mathcal {S}}} gives{\\begin{@align}{1}{-1}&\\left\\Vert G_{\\gamma }({\\rm \\bf x}) - G_{\\gamma }({\\rm \\bf z})\\right\\Vert _2 \\le L \\left\\Vert ({\\rm \\bf x}- {\\rm \\bf z}) - \\gamma {\\rm \\bf H}^{{\\rm \\bf H}({\\rm \\bf x}-{\\rm \\bf z})_2 \\nonumber \\\\&\\quad \\le L\\,\\frac{\\lambda _{\\max }-\\lambda _{\\min }}{\\lambda _{\\max } + \\lambda _{\\min }}\\left\\Vert {\\rm \\bf x}-{\\rm \\bf z}\\right\\Vert _2,\\quad \\forall {\\rm \\bf x},\\forall {\\rm \\bf z}.",
"}\\right.Since L<(\\lambda _{\\max }+\\lambda _{\\min })/(\\lambda _{\\max }-\\lambda _{\\min }), (\\ref {eq:contractive}) implies that G_{\\gamma } is a contractive mapping.",
"By the Banach-Picard fixed point theorem~\\cite [Thm.",
"1.48]{BauschkeC:2011}, \\lbrace {\\rm \\bf x}_k\\rbrace defined by {\\rm \\bf x}_{k+1}=G_{\\gamma }({\\rm \\bf x}_{k}) converges to the unique fixed point {\\rm \\bf x}^{*} of G_{\\gamma }, for every initialization {\\rm \\bf x}_0.",
"Finally, since P_{{\\mathcal {S}}} satisfies~(\\ref {eq:local}), by Proposition~\\ref {thm:minimizer}, {\\rm \\bf x}^* is also a local minimizer of~(\\ref {prob}).\\end{@align}}\\subsection {Proof of Theorem~\\ref {thm:fixed_point2}} Again, let \\lbrace \\lambda _i\\rbrace be the set of eigenvalues of {{\\rm \\bf H}^{{\\rm \\bf H}}.",
"With \\gamma <2/\\lambda _{\\max }, one readily verifies that, \\forall {\\rm \\bf x}\\in {\\mathbb {R}^N},{\\begin{@align*}{1}{-1}\\left\\Vert {\\rm \\bf x}- \\gamma {\\rm \\bf H}^{{\\rm \\bf H}{\\rm \\bf x}_2 &\\le \\max _i\\left\\lbrace |1-\\gamma \\lambda _i|\\right\\rbrace \\cdot \\left\\Vert {\\rm \\bf x}\\right\\Vert _2\\le \\left\\Vert {\\rm \\bf x}\\right\\Vert _2.",
"}\\right.Combining this with the non-expansiveness of P_{{\\mathcal {S}}} leads to{\\begin{@align*}{1}{-1}\\left\\Vert G_{\\gamma }({\\rm \\bf x}) - G_{\\gamma }({\\rm \\bf z})\\right\\Vert _2 &\\le \\left\\Vert ({\\rm \\bf x}- {\\rm \\bf z}) - \\gamma {\\rm \\bf H}^{{\\rm \\bf H}({\\rm \\bf x}-{\\rm \\bf z})_2\\\\&\\le \\left\\Vert {\\rm \\bf x}-{\\rm \\bf z}\\right\\Vert _2, \\quad \\forall {\\rm \\bf x},{\\rm \\bf z}\\in {\\mathbb {R}^N}.",
"}\\right.Now that G_{\\gamma } is a non-expansive operator with a nonempty fixed-point set, we invoke the Krasnosel^{\\prime }skiĭ-Mann theorem~\\cite [Thm.",
"5.14]{BauschkeC:2011} to deduce that the iteration~(\\ref {eq:averaged}) must converge to a fixed point of G_{\\gamma } which is, by Proposition~\\ref {thm:minimizer}, also a local minimizer of~(\\ref {prob}).\\end{@align*}}\\end{@align*}}}\\end{@align}}\\end{@align}}}}\\right.\\end{@align}}$With these theoretical foundations in place, we move on to the matter of training a CNN to act as the projector in RPGD (Algorithm REF ).",
"For any point ${\\rm \\bf x}\\in {\\mathcal {S}}$ , a projector onto ${\\mathcal {S}}$ should satisfy $P_{\\mathcal {S}}{\\rm \\bf x}= {\\rm \\bf x}$ .",
"Moreover, we want ${\\rm \\bf x}= P_{\\mathcal {S}}(\\tilde{{\\rm \\bf x}}) ,$ where $\\tilde{{\\rm \\bf x}}$ is any perturbed version of ${\\rm \\bf x}$ .",
"Given a training set, $\\lbrace {\\rm \\bf x}^1,\\ldots , {\\rm \\bf x}^Q\\rbrace $ , of points drawn from the set ${\\mathcal {S}}$ , we generate an ensemble of $N \\times Q$ perturbed points, $\\lbrace \\lbrace \\tilde{{\\rm \\bf x}}^{1,1}, \\ldots , \\tilde{{\\rm \\bf x}}^{Q,1}\\rbrace , \\ldots , \\lbrace \\tilde{{\\rm \\bf x}}^{1,N} \\ldots , \\tilde{{\\rm \\bf x}}^{Q,N} \\rbrace \\rbrace $ and train the CNN by minimizing the loss function $J({\\theta }) = \\sum _{n=1}^{N} \\underbrace{\\sum _{q=1}^Q\\left\\Vert {\\rm \\bf x}^q-C\\!N\\!N_{{\\theta }}( \\tilde{{\\rm \\bf x}}^{q,n})\\right\\Vert _2^2}_{J_n({\\theta })}.$ The optimization proceeds by stochastic gradient descent for $T$ epochs, where an epoch is defined as one pass though the training data.",
"It remains to select the perturbations that generate the ${\\rm \\bf x}^{q,n}$ .",
"Our goal here is to create a diverse set of perturbations so that the CNN does not overfit one specific type.",
"In our experiments, while training for the $t$ -th epoch, we chose $\\tilde{{\\rm \\bf x}}^{q,1} &= {\\rm \\bf x}^{q}&:&\\text{No perturbation} \\\\\\tilde{{\\rm \\bf x}}^{q,2} &= {\\bf {A}} {\\rm \\bf H}{\\rm \\bf x}^{q}&:&\\text{Specific linear perturbation}\\\\\\tilde{{\\rm \\bf x}}^{q,3} &= C\\!N\\!N_{{\\theta }_{t-1}} (\\tilde{{\\rm \\bf x}}^{q,2})&:&\\text{Dynamic non-linear perturbation}, $ where ${\\bf {A}}$ is a classical linear reconstruction algorithm like FBP or BP, ${\\rm \\bf H}$ is the forward model, and ${\\theta }_t$ are the CNN parameters after $t$ epochs.",
"We now comment on each of these perturbations in detail.",
"Keeping $\\tilde{{\\rm \\bf x}}^{q,1}$ in the training ensemble will train the CNN with the defining property of the projector: the projector maps a point in the set ${\\mathcal {S}}$ onto itself.",
"If the CNN were only trained with (REF ), it would be an autoencoder [36].",
"To understand the perturbation $\\tilde{{\\rm \\bf x}}^{q,2}$ in (), recall that ${\\bf {A}} {\\rm \\bf H}{\\rm \\bf x}^{q}$ is a classical linear reconstruction of ${\\rm \\bf x}^{q}$ from its measurement ${\\rm \\bf y}={\\rm \\bf H}{\\rm \\bf x}^q$ .",
"This perturbation is useful because we will initialize RPGD with ${\\bf {A}} {\\rm \\bf H}{\\rm \\bf x}^{q}$ .",
"Using only () for training will return the same CNN as in [19].",
"The perturbation $\\tilde{{\\rm \\bf x}}^{q,3}$ in () is the output of the CNN whose parameters ${\\theta }_t$ change with every epoch $t$ , thus it is a non-linear and dynamic (epoch-dependent) perturbation of ${\\rm \\bf x}^q$ .",
"The rationale for using () is that it greatly increases the training diversity (allowing the network to see $T$ new perturbations of each training point) without greatly increasing the total training size (only requiring an additional $Q$ gradient computations per epoch).",
"Moreover, () is in sync with the iterative scheme of RPGD, where the output of the CNN is processed with a gradient descent and is again fed into itself."
],
[
"Architecture",
"The architecture we use is the same as in [19], which is a U-net [37] with intrinsic skip connections among its layers and an extrinsic skip connection between the input and the output.",
"The intrinsic skip connections help to eliminate singularities during the training [38].",
"The extrinsic skip connections make this network a residual net; i.e., $C\\!N\\!N=\\operatorname{Id}+ Unet$ , where $\\operatorname{Id}$ denotes the identity operator and $Unet : {\\mathbb {R}}^N \\rightarrow {\\mathbb {R}}^N$ denotes the Unet as a function.",
"The U-net therefore actually provides the projection error (negative perturbation) that should be added to the input to get the projection.",
"Residual nets have been shown to be effective in the image recognition [39] and inverse problem cases[19].",
"While the residual net architecture does not increase the capacity or the approximation power of the CNN, it does help in learning functions that are close to an identity operator, as is the case in our setting."
],
[
"Sequential Training Strategy",
"We train the CNN in 3 stages.",
"In stage 1, we train it for $T_1$ epochs w.r.t.",
"the loss function $J_2$ which only uses the ensemble $\\lbrace \\tilde{{\\rm \\bf x}}^{q,2}\\rbrace $ generated by ().",
"In stage 2, we add the ensemble $\\lbrace \\tilde{{\\rm \\bf x}}^{q,3}\\rbrace $ according to () at every epoch and then train the CNN w.r.t.",
"the loss function $J_2+J_3$ ; we repeat this procedure for $T_2$ epochs.",
"Finally, in stage 3, we train the CNN for $T_3$ epochs with all the ensembles $\\lbrace \\tilde{{\\rm \\bf x}}^{q,1},\\tilde{{\\rm \\bf x}}^{q,2},\\tilde{{\\rm \\bf x}}^{q,3}\\rbrace $ to minimize the original loss function $J=J_1+J_2+J_3$ from (REF ).",
"The above sequential procedure helps speed up the training.",
"The training with $\\lbrace \\tilde{{\\rm \\bf x}}^{q,1}\\rbrace $ is initially bypassed with using the residual net, which is close to the identity operator.",
"It is only incorporated in the last few epochs of stage 3.",
"After training with only $\\lbrace \\tilde{{\\rm \\bf x}}^{q,2}\\rbrace $ in stage 1, $\\tilde{{\\rm \\bf x}}^{q,3}$ will be close to ${\\rm \\bf x}^{q}$ , since it is the output of the CNN for the input $\\tilde{{\\rm \\bf x}}^{q,2}$ .",
"This will ease the training with $\\lbrace \\tilde{{\\rm \\bf x}}^{q,3}\\rbrace $ , which is added after stage 1."
],
[
"Experiments",
"We validate our proposed method on the difficult case of sparse-view CT reconstruction with low dosage exposure.",
"The measurement operator ${\\bf {H}}$ is now the Radon transform.",
"It maps an image to the values of its integrals along a known set of lines [40].",
"In 2D, these measurements can be indexed by the angles and offsets of the lines and arranged in a 2D sinogram.",
"We are particularly interested in the case where the total number of measurements is smaller than the number of pixels in the reconstruction.",
"For example, we aim to reconstruct a (512 $\\times $ 512) image from 45 angles, each with 729 offsets sinogram; i.e., to reconstruct ${\\rm \\bf x}\\in {\\mathbb {R}}^{512 \\times 512}$ from ${\\rm \\bf y}\\in {\\mathbb {R}}^{45 \\times 729}$ .",
"This corresponds to about 8 times fewer measurements than the image to be reconstructed."
],
[
"Dataset",
"Our dataset consists of clinically realistic invivo ($512 \\times 512$ ) CT scans of human abdomen from Mayo clinic for the AAPM Low Dose CT Grand Challenge [41].",
"This data includes CT scans of 10 patients obtained using full dose.",
"We use 475 images from 9 patients for training and 25 images from the other patient for testing.",
"This is the same data used in [19].",
"These images serve as the ground truth.",
"From these images, we generate the measurements (sinograms) using the radon command in Matlab, which corresponds to the forward model ${\\rm \\bf H}$ .",
"The sinograms always have 729 offsets per view, but we vary the number of views in different experiments.",
"Our task is to reconstruct these images from their sparse-view sinograms.",
"We take 2 scenarios: 144 views and 45 views, which corresponds to $\\small {\\times }$ 5 and $\\small {\\times }$ 16 dosage reductions (assuming a full-view sinogram has 720 views).",
"The backprojection ${{\\rm \\bf H}^{} is implemented via the \\texttt {iradon} command with a normalization to satisfy the adjoint property.To make the experiments more realistic and to reduce the inverse crime, the sinograms are generated by perturbing the angles of the views slightly by adding a zero-mean additive white Gaussian noise (AWGN) with standard deviation of 0.05 degrees.",
"This creates a slight mismatch between the actual measurement process and the forward model {\\rm \\bf H}.",
"We also add various amounts of zero-mean Gaussian noise to the sinograms.",
"The SNR of the sinogram {\\rm \\bf y}+{\\rm \\bf n} is defined as\\begin{equation}\\text{SNR}({\\rm \\bf y}+{\\bf {n}},{\\rm \\bf y}) = 20 \\log _{10}\\left({\\left\\Vert {\\rm \\bf y}\\right\\Vert _2}/{\\left\\Vert {\\rm \\bf n}\\right\\Vert _2} \\right).\\end{equation}}Given the ground truth $x$, our figure of merit for the reconstructed $x*$ is the regressed SNR, given by\\begin{equation}\\text{SNR} ({\\rm \\bf x}^*,{\\rm \\bf x}) =\\arg \\max _{a, b} \\text{SNR}(a {\\bf {x}}^* +b, {\\rm \\bf x}),\\end{equation}where the scalars $ a$ and $ b$ serve to scale the data and remove any DC offset, which can greatly affect the SNR but are of little practical importance.$"
],
[
"Comparison Methods",
"We compare four reconstruction methods and report the SNRs for all of them.",
"FBP is the classical direct inversion of the Radon transform ${\\rm \\bf H}$ , here implemented in Matlab by the iradon command with the ram-lak filter and linear interpolation as options.",
"TV solves $\\min _{{\\bf {x}}} \\left({\\frac{1}{2}}\\Vert {\\bf {H}}{\\bf {x}}-{\\bf {y}}\\Vert _2^2+\\lambda \\Vert {\\bf {x}}\\Vert _{\\text{T}V}\\right) \\, \\text{s.t. }",
"{\\rm \\bf x}>0.$ The optimization is carried out via ADMM [13].",
"For a given testing image the parameter $\\lambda $ is tuned so as to maximize the SNR of the reconstruction.",
"FBPconv is the deep-learning-based regression technique [19] that corresponds to a CNN trained with only the ensemble in ().",
"In the testing phase, the FBP of the measurements is fed into the trained CNN to output the reconstruction image.",
"RPGD is our proposed method which is described in Algorithm REF where the nonlinear operator $F$ is the CNN trained as a projector (as discussed in section )."
],
[
"Training and Selection of Parameters",
"We now describe how training and/or parameter selection occurred for the reconstruction methods.",
"FBP has no free hyperparameters.",
"For TV, we chose $\\lambda $ by a grid search through 20 values for each test image.",
"While carrying the optimization with ADMM we put the penalty term, $\\rho = \\lambda $ .",
"The rationale for this heuristic is that the soft-threshold parameter is of the same order of magnitude as the image gradients.",
"We set the number of iterations to 100, which was enough to show good empirical convergence.",
"As discussed in section , the CNNs for RPGD is trained in 3 stages, with the following configurations: $\\small {\\times }$ 5, no noise: $T_1=80$ , $T_2=49$ , $T_3=5$ .",
"$\\small {\\times }$ 16, no noise: $T_1=71$ , $T_2=41$ , $T_3=11$ .",
"We used the CNN obtained right after the first stage for FBPconv, since during this stage only the training ensemble in () was taken into account.",
"We empirically found that the training error $J_2$ converged in $T_1$ epochs of stage 1, yielding an optimal performance for FBPconv.",
"In addition, we also trained the CNNs w.r.t.",
"40 dB noise level in the measurements, by replacing the ensemble in () with $\\lbrace {\\bf {A}} {\\rm \\bf y}^q\\rbrace $ where ${\\rm \\bf y}^q={\\rm \\bf H}{\\rm \\bf x}^q+{\\rm \\bf n}$ .",
"With 20% probability, we also perturbed the views of the measurements with AWGN of 0.05 standard deviation so as to enforce robustness to model mismatch.",
"These CNNs were initialized with the ones obtained after the first stage of the noiseless training and were then trained with the following configurations: $\\small {\\times }$ 5, 40-dB noise: $T_1=35$ , $T_2=49$ , $T_3=5$ .",
"$\\small {\\times }$ 16, 40-dB noise: $T_1=32$ , $T_2=41$ , $T_3=11$ .",
"Similarly to the noiseless case, the CNNs obtained after the first and the third stage of the above training were used in FBPconv and RPGD, respectively.",
"For clarity, these variants will be referred to as FBPconv40 and $\\textbf {RPGD40}$ .",
"During the training, the learning rate was logarithmically decreased from $10^{-2}$ to $10^{-3}$ in stage 1 and kept at $10^{-3}$ for stages 2 and 3.",
"The batch size was fixed to 2, the gradient above $10^{-2}$ was clipped and the momentum was set to $0.99$ .",
"The total training time for the noiseless case was around 21.5 hours on GPU Titan X (Pascal architecture).",
"The hyper-parameters for RPGD were chosen as follows: The relaxation parameter $\\alpha _0$ was initialized with 1, the sequence $\\lbrace c_k\\rbrace $ was set to a constant $C$ where $C=0.99$ for RPGD and $C = 0.8$ for RPGD40.",
"For each noise and dosage reduction level, the only free parameter, $\\gamma $ , was tuned to give the best average SNR over 25 test images.",
"In all experiments, the gradient step was removed from the first iteration.",
"On the GPU, one iteration of RPGD takes less than 1 second.",
"The algorithm is stopped when the residual $\\Vert {\\rm \\bf x}_{k+1}-{\\rm \\bf x}_k\\Vert _2$ reaches a value less than 1, which is sufficiently small compared to the dynamic range [0, 350] of the image.",
"It takes around 1-2 minutes to reconstruct an image with RPGD."
],
[
"Results",
"Tables REF and REF report the results of various methods for low and high measurement noise, respectively.",
"FBPconv and RPGD are used for low noise, while FBPconv40 and RPGD40 are used for high noise.",
"The reconstruction SNRs are averaged over the 25 test images.",
"In the low-noise cases (Table REF ), the proposed method, RPGD, outperforms all the others for both $\\small {\\times }5$ and $\\small {\\times }16$ reductions.",
"FBP performs the worst but is able to retain enough information to be utilized by FBPConv and RPGD.",
"Thanks to the convexity of the iterative scheme, TV is able to perform well but tends to smooth textures and edges.",
"On the other hand, FBPConv outperforms TV.",
"However, it is outperformed by RPGD.",
"This is mainly due to the feedback mechanism in RPGD which lets RPGD use the information in the given measurements to increase the quality of the reconstruction.",
"In fact, for the $\\small {\\times }16$ , no noise case, the SNRs of the sinogram of the reconstructed images for TV, FBPconv, and RPGD are around 47 dB, 57 dB, and 62 dB, respectively.",
"This means that not only reconstruction using RPGD has better image quality but is also more reliable since it is consistent with the given noiseless measurement.",
"In the noisier cases (Table REF ), RPGD40 outperforms the other methods in low-view cases ($\\small {\\times }16$ ) and is more consistent in performance than the others in high-view ($\\small {\\times }5$ ) cases.",
"FBPconv40 substantially outperforms TV in the two scenarios with 40-dB noise measurement, over which it was actually trained.",
"However, as the level of noise deviates from 40 dB, the performance of FBPconv40 degrades significantly.",
"Surprisingly, its performances in the 45-dB cases are much worse than those in the corresponding 40-dB cases.",
"This implies that FBPConv is highly sensitive to the difference between the training and the testing conditions.",
"By contrast, RPGD40 is more robust to this difference due to its iterative correction.",
"In fact, for $\\small {\\times } 16$ case with 45-dB and 35-dB noise level, it outperforms FBPconv40 by around 3.5 dB and 6 dB, respectively.",
"Fig.",
"REF (a) illustrates the reconstructions of a test image for $\\small {\\times } 16$ case when measurement is noiseless.",
"FBP is dominated by line artifacts, while TV satisfactorily removes these but blurs the fine structures.",
"FBPConv and RPGD, on the other hand, are able to reconstruct these details.",
"The zoomed version shows that RPGD is able to reconstruct the fine details better than the others.",
"This observation remains the same when the measurement quality degrades.",
"Fig.",
"REF (b) shows the reconstructions for 45-dB and 40-dB noise levels.",
"In these scenarios, RPGD40 is significantly better than both FBPconv40 and TV.",
"Fig.",
"REF compares the reconstructions for the $ \\small {\\times } 5$ case when the noise levels are 45 dB and 35 dB.",
"It is visible that FBPconv40 results in a noisy image and TV is again blurred.",
"RPGD40 retains the fine structures and is the best performer."
],
[
"Convergence of RPGD",
"Figs.",
"REF (a) and (b) shows the evolution of SNR of images ${\\rm \\bf x}_k$ and their measurements ${\\bf {H}} {\\rm \\bf x}_k$ w.r.t.",
"the ground truth image and ground truth measurement, respectively.",
"Fig.",
"REF (c) shows the $\\alpha _k$ w.r.t.",
"the iteration $k$ .",
"The results are averaged over 25 test images for $\\small {\\times }16$ , no noise case and $C=0.99$ .",
"RPGD outperforms all the other methods in terms of both image quality and measurement consistency.",
"Due to the high value of the step size ($\\gamma =2 \\times 10^{-3}$ ) and the large difference ${\\rm \\bf H}{\\rm \\bf x}_k -{\\rm \\bf y}$ , the initial few iterations have large gradients resulting in the instability of the algorithm.",
"The reason is that the CNN is fed with ${\\rm \\bf x}_k -\\gamma {{\\rm \\bf H}^{}({\\rm \\bf H}{\\rm \\bf x}_k -{\\rm \\bf y}) which is drastically different from the perturbations on which it was trained.In this situation, \\alpha _k decreases steeply and stabilizes the algorithm.At convergence, \\alpha _k\\ne 0, therefore, according to Theorem~\\ref {thm:main}, {\\rm \\bf x}_{100} is the fixed point of (\\ref {eq:opF}) where F=C\\!N\\!N.",
"}$"
],
[
"Conclusion",
"We have proposed a simple yet effective iterative scheme (RPGD) where one step of enforcing measurement consistency is followed by a CNN which tries to project the solution onto the set of the data that we are interested in.",
"The whole scheme is ensured to be convergent.",
"We also introduced a novel method to train a CNN that acts like a projector using a reasonably small dataset (475 images).",
"For sparse-view CT reconstruction our method outperforms the previous techniques for both noiseless and noisy measurements.",
"The proposed framework can be used to solve other inverse problems like super-resolution, deconvolution, accelerated MRI, etc.",
"This can bring more robustness and reliability to the current deep learning based techniques."
],
[
"Proof of Theorem ",
"(i) Set ${\\rm \\bf r}_k={\\rm \\bf x}_{k+1}-{\\rm \\bf x}_k$ .",
"On one hand, it is clear that ${\\rm \\bf r}_k &= (1-\\alpha _k){\\rm \\bf x}_k + \\alpha _k {\\rm \\bf z}_k - {\\rm \\bf x}_{k}= \\alpha _k\\left({\\rm \\bf z}_k-{\\rm \\bf x}_k\\right).$ On the other hand, from the construction of $\\lbrace \\alpha _k\\rbrace $ , $&\\alpha _{k}\\left\\Vert {\\rm \\bf z}_{k} - {\\rm \\bf x}_{k}\\right\\Vert _2 \\le c_k\\alpha _{k-1}\\left\\Vert {\\rm \\bf z}_{k-1} - {\\rm \\bf x}_{k-1}\\right\\Vert _2\\nonumber \\\\\\Leftrightarrow &\\quad \\quad \\quad \\ \\,\\left\\Vert {\\rm \\bf r}_{k}\\right\\Vert _2 \\le c_k \\left\\Vert {\\rm \\bf r}_{k-1}\\right\\Vert _2.$ Iterating (REF ) gives $\\left\\Vert {\\rm \\bf r}_{k}\\right\\Vert _2 \\le \\left\\Vert {\\rm \\bf r}_0\\right\\Vert _2 \\prod _{i=1}^{k} c_{i}, \\quad \\forall k\\ge 1.$ We now show that $\\lbrace {\\rm \\bf x}_k\\rbrace $ is a Cauchy sequence.",
"Since $\\lbrace c_k\\rbrace $ is asymptotically upper-bounded by $C<1$ , there exists $K$ such that $c_k \\le C,\\forall k> K$ .",
"Let $m,n$ be two integers such that $m>n>K$ .",
"By using (REF ) and the triangle inequality, $\\left\\Vert {\\rm \\bf x}_m-{\\rm \\bf x}_n\\right\\Vert _2 &\\le \\sum _{k=n}^{m-1} \\left\\Vert {\\rm \\bf r}_{k}\\right\\Vert _2\\le \\left\\Vert {\\rm \\bf r}_0\\right\\Vert _2 \\prod _{i=1}^{K} c_{i} \\sum _{k=n-K}^{m-1-K} C^k\\nonumber \\\\&\\le \\left(\\left\\Vert {\\rm \\bf r}_0\\right\\Vert _2 \\prod _{i=1}^{K} c_{i}\\right) \\frac{C^{n-K}-C^{m-K}}{1-C}.$ The last inequality proves that $\\left\\Vert {\\rm \\bf x}_m-{\\rm \\bf x}_n\\right\\Vert _2\\rightarrow 0$ as $m\\rightarrow \\infty ,n\\rightarrow \\infty $ , or $\\lbrace {\\rm \\bf x}_k\\rbrace $ is a Cauchy sequence in the complete metric space ${\\mathbb {R}^N}$ .",
"As a consequence, $\\lbrace {\\rm \\bf x}_{k}\\rbrace $ must converge to some point ${\\rm \\bf x}^*\\in {\\mathbb {R}^N}$ .",
"(ii) Assume from now on that $\\lbrace \\alpha _k\\rbrace $ is lower-bounded by $\\varepsilon >0$ .",
"By definition, $\\lbrace \\alpha _k\\rbrace $ is also non-increasing and, thus, convergent to $\\alpha ^*>0$ .",
"Next, we rewrite the update of ${\\rm \\bf x}_k$ in Algorithm REF as ${\\rm \\bf x}_{k+1} = (1-\\alpha _k){\\rm \\bf x}_{k} + \\alpha _k G_{\\gamma }({\\rm \\bf x}_{k}),$ where $G_{\\gamma }$ is defined by (REF ).",
"Taking the limit of both sides of (REF ) leads to ${\\rm \\bf x}^* &= (1-\\alpha ^*){\\rm \\bf x}^* + \\alpha ^*\\lim _{k\\rightarrow \\infty }G_{\\gamma }({\\rm \\bf x}_k).$ Moreover, since the nonlinear operator $F$ is continuous, $G_{\\gamma }$ is also continuous.",
"Hence, $\\lim _{k\\rightarrow \\infty }G_{\\gamma }({\\rm \\bf x}_k) =G_{\\gamma }\\left(\\lim _{k\\rightarrow \\infty }{\\rm \\bf x}_k\\right) = G_{\\gamma }({\\rm \\bf x}^*).$ By plugging (REF ) into (REF ), we get that ${\\rm \\bf x}^*=G_{\\gamma }({\\rm \\bf x}^*)$ , which means ${\\rm \\bf x}^*$ is a fixed point of the operator $G_{\\gamma }$ .",
"(iii) Now that $F= P_{{\\mathcal {S}}}$ satisfies (), we invoke Proposition to infer that ${\\rm \\bf x}^*$ is a local minimizer of (REF ), thus completing the proof."
],
[
"Acknowledgment",
"The authors would like to thank Emmanuel Soubies for his helpful suggestions on training the CNN.",
"We thankfully acknowledge the support of NVIDIA Corporation which donated the Titan X GPU used in this research.",
"The authors would like to thank Dr. Cynthia McCollough, the Mayo Clinic, the American Association of Physicists in Medicine, and grants EB017095 and EB017185 from the National Institute of Biomedical Imaging and Bioengineering for giving opportunities to use real-invivo CT DICOM images."
],
[
"Proof of Proposition ",
"Suppose that () is fulfilled and let ${\\rm \\bf x}^*\\in {\\mathcal {S}}$ be a fixed point of $G_{\\gamma }$ .",
"We show that ${\\rm \\bf x}^*$ is also a local minimizer of (REF ).",
"Indeed, setting ${\\rm \\bf x}={\\rm \\bf x}^*-\\gamma {{\\rm \\bf H}^{{\\rm \\bf H}}{\\rm \\bf x}^{*} + \\gamma {{\\rm \\bf H}^{}{\\rm \\bf y} leads to P_{{\\mathcal {S}}}{\\rm \\bf x}={\\rm \\bf x}^*.",
"Then, there exists \\varepsilon >0 such that, for all {\\rm \\bf z}\\in {\\mathcal {S}}\\cap {\\mathcal {B}}_{\\varepsilon }({\\rm \\bf x}^{*}),{\\begin{@align*}{1}{-1}0&\\ge \\left\\langle {{\\rm \\bf z}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle \\\\&=\\gamma \\left\\langle {{\\rm \\bf z}-{\\rm \\bf x}^*}\\, , \\,{{{\\rm \\bf H}^{}{\\rm \\bf y}-{\\rm \\bf H}^{{\\rm \\bf H}{\\rm \\bf x}^{*}}\\\\&=\\frac{\\gamma }{2}\\left(\\left\\Vert {\\rm \\bf H}{\\rm \\bf x}^*-{\\rm \\bf y}\\right\\Vert ^2_2 -\\left\\Vert {\\rm \\bf H}{\\rm \\bf z}-{\\rm \\bf y}\\right\\Vert ^2_2 + \\left\\Vert {\\rm \\bf H}{\\rm \\bf x}^*-{\\rm \\bf H}{\\rm \\bf z}\\right\\Vert ^2_2\\right).",
"}}Since \\right.\\gamma >0, the last inequality implies that{\\begin{@align*}{1}{-1}\\left\\Vert {\\rm \\bf H}{\\rm \\bf x}^*-{\\rm \\bf y}\\right\\Vert ^2_2 \\le \\left\\Vert {\\rm \\bf H}{\\rm \\bf z}-{\\rm \\bf y}\\right\\Vert ^2_2,\\quad \\forall {\\rm \\bf z}\\in {\\mathcal {S}}\\cap {\\mathcal {B}}_{\\varepsilon }({\\rm \\bf x}^{*}),\\end{@align*}}which means that {\\rm \\bf x}^{*} is a local minimizer of~(\\ref {prob}).\\end{@align*}Assume now that P_{{\\mathcal {S}}} satisfies~(\\ref {eq:global}).",
"By just removing the \\varepsilon -ball in the above argument, one easily verifies that{\\begin{@align*}{1}{-1}\\left\\Vert {\\rm \\bf H}{\\rm \\bf x}^*-{\\rm \\bf y}\\right\\Vert ^2_2 \\le \\left\\Vert {\\rm \\bf H}{\\rm \\bf z}-{\\rm \\bf y}\\right\\Vert ^2_2,\\quad \\forall {\\rm \\bf z}\\in {\\mathcal {S}},\\end{@align*}}which means that {\\rm \\bf x}^* is a solution of~(\\ref {prob}).",
"\\subsection {Proof of Proposition~\\ref {thm:convex}} We prove by contradiction.",
"Assuming that {\\mathcal {S}} is non-convex, there must exist {\\rm \\bf x}_1,{\\rm \\bf x}_2\\in {\\mathcal {S}} and \\alpha \\in (0,1) such that {\\rm \\bf x}=\\alpha {\\rm \\bf x}_1 + (1-\\alpha ){\\rm \\bf x}_2\\notin {\\mathcal {S}}.",
"Since P_{{\\mathcal {S}}}{\\rm \\bf x}\\in {\\mathcal {S}}, it must be that{\\begin{@align*}{1}{-1}0 &<\\left\\Vert {\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}\\right\\Vert ^2_2=\\left\\langle {{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle \\\\&=\\alpha \\left\\langle {{\\rm \\bf x}_1-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle \\\\&\\quad + (1-\\alpha )\\left\\langle {{\\rm \\bf x}_2-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle .\\end{@align*}}Thus, there exists i\\in \\lbrace 0,1\\rbrace such that{\\begin{@align*}{1}{-1}\\left\\langle {{\\rm \\bf x}_i-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\, , \\,{{\\rm \\bf x}-P_{{\\mathcal {S}}}{\\rm \\bf x}}\\right\\rangle > 0,\\end{@align*}}which violates~(\\ref {eq:global}).",
"So, {\\mathcal {S}} is convex.",
"}\\subsection {Proof of Proposition~\\ref {thm:union_convex}} Suppose that {\\mathcal {S}}=\\bigcup _{i=1}^{n}{\\mathcal {C}}_i, where {\\mathcal {C}}_i is a closed convex set for all i=1,\\ldots ,n. The statement is trivial when n=1; assume now that n\\ge 2.",
"Let {\\rm \\bf x}\\in {\\mathbb {R}^N} and \\hat{{\\rm \\bf x}} be the orthogonal projection of {\\rm \\bf x} onto {\\mathcal {S}}.",
"Consider two cases.",
"}\\emph {Case 1}: \\hat{{\\rm \\bf x}}\\in \\bigcap _{i=1}^n{\\mathcal {C}}_i.\\\\It is then clear that{\\begin{@align*}{1}{-1}\\left\\Vert \\hat{{\\rm \\bf x}}-{\\rm \\bf x}\\right\\Vert _2 \\le \\left\\Vert {\\rm \\bf z}-{\\rm \\bf x}\\right\\Vert _2,\\forall {\\rm \\bf z}\\in {\\mathcal {C}}_i,\\forall i.\\end{@align*}}This means that \\hat{{\\rm \\bf x}} is the orthogonal projection of {\\rm \\bf x} onto each {\\mathcal {C}}_i.",
"Consequently,{\\begin{@align*}{1}{-1}\\left\\langle {{\\rm \\bf z}-\\hat{{\\rm \\bf x}}}\\, , \\,{{\\rm \\bf x}-\\hat{{\\rm \\bf x}}}\\right\\rangle \\le 0,\\forall {\\rm \\bf z}\\in {\\mathcal {C}}_i,\\forall i\\le n,\\end{@align*}}which implies that~(\\ref {eq:local}) holds true for all \\varepsilon > 0.",
"}\\emph {Case 2}: $xi=1nCi$.\\\\Without loss of generality, there exists $ m<n$ such that{\\begin{@align}{1}{-1}\\hat{{\\rm \\bf x}}\\in \\bigcap _{i=1}^m {\\mathcal {C}}_i,\\quad \\hat{{\\rm \\bf x}}\\notin \\bigcup _{i=m+1}^n {\\mathcal {C}}_i.\\end{@align}}Let $ d$ be the distance from $x$ to the set $ T=i=m+1n Ci$.",
"Since each $ Ci$ is closed, $ T$ must be closed too and, so, $ d>0$.",
"We now choose $ 0<<d$.",
"Then, $ B(x)T=$.",
"Therefore,{\\begin{@align}{1}{-1}{\\mathcal {S}}\\cap {\\mathcal {B}}_\\varepsilon (\\hat{{\\rm \\bf x}}) =\\bigcup _{i=1}^{m}\\left({\\mathcal {C}}_i\\cap {\\mathcal {B}}_\\varepsilon (\\hat{{\\rm \\bf x}})\\right) = \\bigcup _{i=1}^{m}\\tilde{{\\mathcal {C}}}_i,\\end{@align}}where $ Ci=CiB(x)$ is clearly a convex set, for all $ im$.",
"It is straightforward that $x$ is the orthogonal projection of $x$ onto the set $ i=1mCi$ and that $xi=1mCi$.",
"We are back to Case 1 and, therefore,{\\begin{@align}{1}{-1}\\left\\langle {{\\rm \\bf z}-\\hat{{\\rm \\bf x}}}\\, , \\,{{\\rm \\bf x}-\\hat{{\\rm \\bf x}}}\\right\\rangle \\le 0,\\forall {\\rm \\bf z}\\in \\tilde{{\\mathcal {C}}}_i,\\forall i\\le m.\\end{@align}}From~(\\ref {eq:intersect}) and~(\\ref {eq:case}), (\\ref {eq:local}) is fulfilled for the chosen $$.\\subsection {Proof of Theorem~\\ref {thm:fixed_point}} Let $ {i}$ denote the set of eigenvalues of $HH$.",
"We first have that, for all $xRN$,{\\begin{@align}{1}{-1}\\left\\Vert {\\rm \\bf x}- \\gamma {\\rm \\bf H}^{{\\rm \\bf H}{\\rm \\bf x}_2 \\le \\left\\Vert {\\rm \\bf I}-\\gamma {\\rm \\bf H}^{{\\rm \\bf H}_2\\left\\Vert {\\rm \\bf x}\\right\\Vert _2,}\\right.where the spectral norm of {\\rm \\bf I}-\\gamma {{\\rm \\bf H}^{{\\rm \\bf H}} is given by{\\begin{@align}{1}{-1}\\left\\Vert {\\rm \\bf I}-\\gamma {\\rm \\bf H}^{{\\rm \\bf H}_2 = \\max _{i}\\lbrace |1-\\gamma \\lambda _i|\\rbrace .",
"}\\right.On the other hand, choosing \\gamma ={2}/{(\\lambda _{\\max }+\\lambda _{\\min })} yields{\\begin{@align}{1}{-1}&\\frac{2\\lambda _{\\min }}{\\lambda _{\\max } + \\lambda _{\\min }} \\le \\gamma \\lambda _i \\le \\frac{2\\lambda _{\\max }}{\\lambda _{\\max } + \\lambda _{\\min }},\\quad \\forall i \\nonumber \\\\\\Leftrightarrow \\quad & | 1-\\gamma \\lambda _i| \\le \\frac{\\lambda _{\\max }-\\lambda _{\\min }}{\\lambda _{\\max } + \\lambda _{\\min }},\\quad \\forall i.\\end{@align}}By combining~(\\ref {eq:gradient_bound}), (\\ref {eq:spectral_norm}), and~(\\ref {eq:absolute}),{\\begin{@align}{1}{-1}\\left\\Vert {\\rm \\bf x}- \\gamma {\\rm \\bf H}^{{\\rm \\bf H}{\\rm \\bf x}_2 \\le \\frac{\\lambda _{\\max }-\\lambda _{\\min }}{\\lambda _{\\max } + \\lambda _{\\min }}\\left\\Vert {\\rm \\bf x}\\right\\Vert _2, \\quad \\forall {\\rm \\bf x}.",
"}\\right.Combining~(\\ref {eq:gradient_Lipschitz}) with the Lipschitz continuity of P_{{\\mathcal {S}}} gives{\\begin{@align}{1}{-1}&\\left\\Vert G_{\\gamma }({\\rm \\bf x}) - G_{\\gamma }({\\rm \\bf z})\\right\\Vert _2 \\le L \\left\\Vert ({\\rm \\bf x}- {\\rm \\bf z}) - \\gamma {\\rm \\bf H}^{{\\rm \\bf H}({\\rm \\bf x}-{\\rm \\bf z})_2 \\nonumber \\\\&\\quad \\le L\\,\\frac{\\lambda _{\\max }-\\lambda _{\\min }}{\\lambda _{\\max } + \\lambda _{\\min }}\\left\\Vert {\\rm \\bf x}-{\\rm \\bf z}\\right\\Vert _2,\\quad \\forall {\\rm \\bf x},\\forall {\\rm \\bf z}.",
"}\\right.Since L<(\\lambda _{\\max }+\\lambda _{\\min })/(\\lambda _{\\max }-\\lambda _{\\min }), (\\ref {eq:contractive}) implies that G_{\\gamma } is a contractive mapping.",
"By the Banach-Picard fixed point theorem~\\cite [Thm.",
"1.48]{BauschkeC:2011}, \\lbrace {\\rm \\bf x}_k\\rbrace defined by {\\rm \\bf x}_{k+1}=G_{\\gamma }({\\rm \\bf x}_{k}) converges to the unique fixed point {\\rm \\bf x}^{*} of G_{\\gamma }, for every initialization {\\rm \\bf x}_0.",
"Finally, since P_{{\\mathcal {S}}} satisfies~(\\ref {eq:local}), by Proposition~\\ref {thm:minimizer}, {\\rm \\bf x}^* is also a local minimizer of~(\\ref {prob}).\\end{@align}}\\subsection {Proof of Theorem~\\ref {thm:fixed_point2}} Again, let \\lbrace \\lambda _i\\rbrace be the set of eigenvalues of {{\\rm \\bf H}^{{\\rm \\bf H}}.",
"With \\gamma <2/\\lambda _{\\max }, one readily verifies that, \\forall {\\rm \\bf x}\\in {\\mathbb {R}^N},{\\begin{@align*}{1}{-1}\\left\\Vert {\\rm \\bf x}- \\gamma {\\rm \\bf H}^{{\\rm \\bf H}{\\rm \\bf x}_2 &\\le \\max _i\\left\\lbrace |1-\\gamma \\lambda _i|\\right\\rbrace \\cdot \\left\\Vert {\\rm \\bf x}\\right\\Vert _2\\le \\left\\Vert {\\rm \\bf x}\\right\\Vert _2.",
"}\\right.Combining this with the non-expansiveness of P_{{\\mathcal {S}}} leads to{\\begin{@align*}{1}{-1}\\left\\Vert G_{\\gamma }({\\rm \\bf x}) - G_{\\gamma }({\\rm \\bf z})\\right\\Vert _2 &\\le \\left\\Vert ({\\rm \\bf x}- {\\rm \\bf z}) - \\gamma {\\rm \\bf H}^{{\\rm \\bf H}({\\rm \\bf x}-{\\rm \\bf z})_2\\\\&\\le \\left\\Vert {\\rm \\bf x}-{\\rm \\bf z}\\right\\Vert _2, \\quad \\forall {\\rm \\bf x},{\\rm \\bf z}\\in {\\mathbb {R}^N}.",
"}\\right.Now that G_{\\gamma } is a non-expansive operator with a nonempty fixed-point set, we invoke the Krasnosel^{\\prime }skiĭ-Mann theorem~\\cite [Thm.",
"5.14]{BauschkeC:2011} to deduce that the iteration~(\\ref {eq:averaged}) must converge to a fixed point of G_{\\gamma } which is, by Proposition~\\ref {thm:minimizer}, also a local minimizer of~(\\ref {prob}).\\end{@align*}}\\end{@align*}}}\\end{@align}}\\end{@align}}}}\\right.\\end{@align}}$"
]
] | 1709.01809 |
[
[
"Asynchronous Stochastic Approximation Based Learning Algorithms for\n As-You-Go Deployment of Wireless Relay Networks along a Line"
],
[
"Abstract We are motivated by the need for impromptu (or as-you-go) deployment of multihop wireless networks, by human agents or robots; the agent moves along a line, makes wireless link quality measurements at regular intervals, and makes on-line placement decisions using these measurements.",
"As a first step, we have formulated such deployment along a line as a sequential decision problem.",
"In our earlier work, we proposed two possible deployment approaches: (i) the pure as-you-go approach where the deployment agent can only move forward, and (ii) the explore-forward approach where the deployment agent explores a few successive steps and then selects the best relay placement location.",
"The latter was shown to provide better performance but at the expense of more measurements and deployment time, which makes explore-forward impractical for quick deployment by an energy constrained agent such as a UAV.",
"Further, the deployment algorithm should not require prior knowledge of the parameters of the wireless propagation model.",
"In [1] we, therefore, developed learning algorithms for the explore-forward approach.",
"The current paper provides deploy-and-learn algorithms for the pure as-you-go approach.",
"We formulate the sequential relay deployment problem as an average cost Markov decision process (MDP), which trades off among power consumption, link outage probabilities, and the number of deployed relay nodes.",
"First we show structural results for the optimal policy.",
"Next, by exploiting the special structure of the optimality equation and by using the theory of asynchronous stochastic approximation, we develop two learning algorithms that asymptotically converge to the set of optimal policies as deployment progresses.",
"Numerical results show reasonably fast speed of convergence, and hence the model-free algorithms can be useful for practical, fast deployment of emergency wireless networks."
],
[
"Introduction",
"In emergency situations, such as fires in large buildings or forests, or houses in a flooded neighbourhood (without electric power and telecom infrastructure), there is a need to quickly deploy wireless networks for situation monitoring.",
"Such networks could be deployed by first responders (e.g., fire-fighters moving through a burning building [2]), or by robots (e.g., unmanned aerial vehicles (UAVs) hopping over the rooftops of flooded homes or flying over a long road [3], [4], [5]), or by forest guards along forest trails ([1]).See [6] and [7] for application of multihop wireless sensor networks in wildlife monitoring and forest fire detection.",
"[8] illustrates a future possibility where drones deploy high speed, solar-powered access points on the roofs of city buildings in order to provide high speed internet connection.",
"The drone can land on the ground or on a rooftop for link quality measurements, and can again take off.",
"Typically, such networks would have one or more base-stations, where the command and control would reside, and to which the measurements from the sensors in the field would need to be routed.",
"For example, in the case of the fire-fighting example, the base-station would be in a control truck parked outside the building.",
"Evidently, in such emergency situations, there is a need for “as-you-go” deployment algorithms as there is no time for network planning.",
"As they move through the affected area, the first-responders would need to deploy wireless relays, in order to provide routes for the wireless sensors for situation monitoring.",
"With the above motivation for quick deployment of multihop wireless networks, in our work, in the present and earlier papers ([1], [9], [10]), we have considered the particular situation of as-you-deployment of relays along a line, starting from a base-station, in order to connect a source of data (e.g., a sensor) whose location is revealed (or is itself placed) only during the deployment process.",
"Figure REF depicts our model for as-you-go deployment along a line, and also illustrates the difference between planned deployment and as-you-go deployment.",
"As-you-go deployment along a line is motivated by the need for quick deployment of relay networks along long forest trails by humans or mobile robots, and relay network deployment along a long straight road by human agents or UAVs.",
"In practice, the location of the data source would be a-priori unknown, as the deployment agent would also need to select locations at which to place the sensors.",
"Yet, as the deployment agent traverses the line, he or she (or it) has to judiciously deploy wireless relays so as to end up with a viable network connecting the data source (e.g., the sensor) to the sink.",
"In a planned approach, all possible links could be evaluated; in an as-you-go approach, however, the agent needs to make decisions based on whatever links can be evaluated as deployment progresses.",
"Motivated by the need for as-you-go deployment of wireless sensor networks (WSNs) over large terrains, such as forest trails, in our earlier work [1] we had considered the problem of multihop wireless network deployment along a line, where a single deployment agent starts from a sink node (e.g., a base-station), places relays as the agent walks along the line, and finally places a source node (e.g., a sensor) where required.",
"We formulated this problem as a measurement based sequential decision problem with an appropriate additive cost over hops.",
"In order to explore the range of possibilities, we considered two alternatives for measurement and deployment: (i) the explore-forward approach: after placing a node, the deployment agent explores several potential placement locations along the next line segment, and then decides on where to place the next node, and (ii) the pure as-you-go approach: the deployment agent only moves forward, making measurements and committing to deploying nodes as he goes.",
"As expected, in [1] we found that the explore-forward approach yields better performance (in terms of the additive per hop cost (see [1]); but, of course, this approach takes more time for the completion of deployment.",
"Hence, explore-forward is prohibitive when soldiers or robots need to quickly deploy a relay network along a forest trail or along a long road.",
"In addition, a deployment agent such as a UAV would be limited by its fuel, and it would be desirable to complete the mission as quickly as possible, without many fuel consuming manoeuvres.",
"Thus, pure as-you-go is the only option for network deployment by UAVs along long roads (see [3] for practical network deployment along a road by a UAV).",
"Further, in an emergency situation, the algorithm cannot expect to be given the parameters of the propagation environment; this gives rise to the need for deploy-and-learn algorithms.",
"In [1], although we introduced explore-forward and pure as-you-go approaches, we developed learning algorithms for explore-forward alone.",
"However, with the above motivation, our current paper fills in an important gap by proposing online learning algorithms for pure as-you-go deployment.",
"We mathematically formulate the problem of pure as-you-go deployment along a line as an optimal sequential decision problem so as to minimize the expected average cost per step, where the cost of a deployment is a linear combination of the sum transmit power, the sum outage probability and the number of relays deployed.",
"We formulate the problem as a Markov decision process (MDP) and obtain the optimal policy structure.",
"Next, we propose two learning algorithms (based on asynchronous stochastic approximation) and prove their asymptotic convergence to the optimal policy for the long-run average cost minimization problem.",
"Finally, we demonstrate the convergence rate of the learning algorithms via numerical exploration.",
"The new contributions of this paper, in relation to [1], are discussed in Section REF .",
"Figure: A line network connecting a source (e.g., a sensor)to a sink (e.g., a control centre) via relay nodes.",
"The dots in between (filled and unfilled)denote potential relay locations, and are spaced δ\\delta meters apart.The deployed network consists of three relays (dots labeled Relay 1, 2, and 3) placed at three potential locations.",
"The solidarrows show the multi-hop path from the source to the sink.",
"The unfilled dots represent locationswhere no relay was placed.",
"The dotted arrows represent some otherpossible links between pairs of potential locations.",
"In case of planned deployment, link qualities between all potentiallocation pairs need to be measured.",
"But, in as-you-go deployment, the agent only measures the qualities of link from his (or its)current location to the previouslyplaced nodes.Prior work on the problem of impromptu deployment of WSN consists of mostly heuristic algorithms validated by experimentation.",
"For example, the authors of [11] address this problem by studying experimentally the variation in indoor link quality.",
"The authors of [12] also took a similar approach.",
"The authors of [13] provide heuristics for deploying (incrementally) sensors so that a certain area is covered (e.g., self-deployment of autonomous robot teams).",
"Bao and Lee, in [14], address the problem of a group of first responders starting from a base station (e.g., a command center) and placing relay nodes while walking through a region devoid of communication infrastructure, in order to stay connected among themselves as well as with the base station.",
"Liu et al., in [2], describe a breadcrumbs system meant for firefighters operating inside a building; this paper is in similar spirit with ours, but their goal is just to maintain connection with $k$ previously placed nodes.",
"This work was later extended by them in [15] which provides a reliable multiuser breadcrumbs system.",
"However, all the above works are based on heuristic algorithms, rather than on rigorous formulations; hence they do not provide any provable performance guarantee.",
"A nice survey on rapid deployment of post-disaster networks is available in [16].",
"Sensor network deployment by UAVs have also been studied in literature (see [4], [5]).",
"In our current paper, we have formulated as-you-go deployment as an MDP, found structural results for the optimal policy, and proposed learning algorithms to solve the sequential decision problems without using any prior knowledge of the radio propagation parameters.",
"The use of MDP to formulate as-you-go deployment was first proposed by Mondal et al.",
"in [17].",
"This work was later extended by Sinha et al.",
"in [18], where the authors have provided an algorithm derived from an MDP formulation, so as to create a multi-hop wireless relay network between a sink and a source located at an unknown location, by placing relay nodes along a random lattice path.",
"However, these papers do not consider spatial variability of wireless link qualities due to shadowing, which allows them to develop deployment algorithms that place the next relay based on the distance from the previously placed relay.",
"The spatial variation of link qualities due to shadowing requires measurement-based deployment; here the deployment agent makes placement decision at a given location based on the link quality to the previously placed node.",
"Measurement-based as-you-go deployment was formulated first in [9], and was later extended in [1].",
"The authors of [1] have proposed two possible approaches for deployment along a line: (i) the pure as-you-go approach and (ii) the explore-forward approach.",
"[1] has provided MDP formulations and policy structures for both approaches; transition probabilities of the MDPs depend on the radio propagation parameters in the environment, and, in practice, these parameters are not known to the agent prior to deployment.",
"Hence, [1] also provides learning algorithms for the explore-forward approach, that converge asymptotically to the set of optimal deployment policies as more and more measurements are made in course of deployment.",
"One of these learning algorithms was used for actual network deployment (see [1] and [10]).",
"Design of a two-connected network to guard against node and link failures was discussed in [19], but it did not provide any learning algorithm.",
"We also developed, in [20], as-you-go deployment algorithms for deploying a multi-relay line network, so that the end-to-end achievable rate is maximized; but it was done for an information-theoretic, full-duplex, multi-relay channel model where the nodes carry out decode-and-forward relaying.",
"However, devices with such sophisticated relaying capability is not yet available for full commercial use.",
"On the other hand, our current paper designs deployment algorithms for networks carrying packetized data, which is common in present day wireless standards."
],
[
"Contributions of this paper, in relation to {{cite:e92aa9b2dd0aa323014e2e7401b69d7cb8ded462}}:",
"(i) New deploy-and-learn algorithms: Our current paper provides learning algorithms for the pure as-you-go approach (Algorithm REF and Algorithm REF ), whereas [1] provides learning algorithms only for explore-forward.",
"The learning algorithms are required to discover the optimal deployment policy as deployment progresses, for the situation where no prior accurate knowledge on the statistical nature of radio propagation environment is available.",
"Learning algorithms for pure as-you-go deployment is an important requirement since the pure as-you-go deployment approach is more suitable for very fast deployment over a large region.",
"In fact, the number of measurements in explore-forward deployment can be double or triple than that of pure as-you-go ([1]) for practical deployment; this makes pure as-you-go a better choice for emergency network deployment by soldiers or commandos or energy-constrained autonomous agents such as robots and UAVs.",
"Unlike [1], the learning algorithms presented in this paper make use of asynchronous stochastic approximation, where different iterates are updated at different time instants (in the learning algorithms proposed in [1], all iterates are updated when a new relay is placed).",
"We provide formal proof for the convergence of our proposed learning algorithms to the optimal deployment policy for pure as-you-go deployment; these proofs require a significant and non-trivial novel mathematical analysis (compared to [1]) in order to address many technical issues that arise in the proofs.",
"In other words, the most important contributions of the current paper w.r.t.",
"[1], are the newly proposed learning algorithms for pure as-you-go deployment and their convergence proofs, which are new to the literature and addresses the problem of very fast deployment.",
"Interestingly, one of the learning algorithms proposed in this paper exhibits a nice separation property between estimation and control, which is not present in the learning algorithms presented in [1].",
"(ii) Average cost MDP formulation: [1] formulates the pure-as-you deployment problem for a line having a random length $L \\sim Geometric(\\theta )$ (mean is $\\frac{1}{\\theta }$ ), i.e., $\\mathbb {P}(L=l)=(1-\\theta )^{l-1} \\theta $ where $ l \\in \\lbrace 1,2,\\cdots ,\\infty \\rbrace $ ; the average cost optimal policy is obtained by taking $\\theta \\rightarrow 0$ .",
"Clearly, this requires value iteration to compute the optimal policy prior to deployment.",
"This also requires the knowledge of radio propagation parameters, since they determine the transition probabilities of the MDP.",
"On the other hand, our present paper establishes the structure of the optimal policy by using the average cost optimality equation (see (REF )) with necessary modification; it turns out that such a formulation along with the special structure of the problem enables us to propose very simple learning algorithms to find the optimal policy, irrespective of whether the radio propagation parameters are known apriori or not.",
"Thus, the average cost MDP formulation is a precursor to the learning algorithms (Algorithm REF and Algorithm REF ) presented later in this paper.",
"Some new interesting properties of the value functions and the policy structure are also proved in the current paper, which were not present in [1] because the problem was formulated as discounted cost MDP in [1].",
"(iii) Additional measurements to facilitate learning: The pure-as-you go approach considered in our current paper is not exactly the same as that described in [1].",
"In [1], the agent makes a link quality measurement from the current location to the immediate previous node that he had placed.",
"On the contrary, in the pure as-you-go approach described in our present paper, the agent measures link qualities from the current location to all previously placed nodes that are located within a certain distance.",
"This is done to facilitate learning the optimal policy.",
"The exact reason behind using this variation of pure as-you-go deployment will be explained in Section REF .",
"(iv) Bidirectional traffic: In Section REF , we explain how the deployment algorithms presented in this paper can be adapted to the case where each link in the network has to carry data packets in both directions."
],
[
"Organization",
"The rest of the paper is organized as follows.",
"The system model has been described in Section .",
"MDP formulation for pure-as-you deployment has been provided in Section .",
"The learning algorithms have been proposed in Section and Section .",
"Convergence speed of the learning algorithms are demonstrated numerically in Section , after which the conclusion follows.",
"The proofs of all theorems are provided in the appendices available as supplementary material."
],
[
"System Model",
"In this section, we describe the system model assumed in this paper.",
"It has to be noted that the system model and notation used in this paper are similar in many aspects to those of [1]; a significant difference in the system model will be found in the deployment procedure as described in Section REF (deployment process), and in Section REF (bi-directional traffic).",
"The channel model (Section REF ), traffic model (Section REF ) and deployment objective (Section REF ) subsections are almost similar to the respective sections in [1]; but we describe the system model here in detail to make this paper self-contained.",
"We assume that the line (i.e., the road or the forest trail along which the network is deployed) is discretized into steps (starting from the sink), each having length $\\delta $ .",
"The points located at distances $\\lbrace k \\delta \\rbrace _{k \\in \\lbrace 1,2,3,\\cdots \\rbrace }$ are called potential locations; the agent is allowed to place nodes only at these potential locations.",
"As the single deployment agent walks along the line, at each potential location, the agent measures the link quality from the current location to the previously placed nodes that are within a certain distance from the current location; placement decisions are made based on these measurements.",
"After deployment, as shown in Figure REF , the sink is called Node 0, and the relays are enumerated as nodes $\\lbrace 1,2,3,\\cdots \\rbrace $ as we move away from the sink.",
"A link whose transmitter is Node $i$ and receiver is Node $j$ is called link $(i,j)$ ."
],
[
"Wireless Channel Model",
"We consider a wireless channel model where, for a link (i.e., a transmitter-receiver pair) with length $r$ and transmit power $\\gamma $ , the received power of a packet (say the $k$ -th packet) is given by: $P_{rcv,k}=\\gamma c \\bigg (\\frac{r}{r_0}\\bigg )^{-\\eta }H_kW $ Here $c$ is the path-loss at a reference distance $r_0$ , and $\\eta $ is the path-loss exponent.",
"The fading random variable seen by the $k$ -th packet is $H_k$ (e.g., $H_k$ is exponentially distributed for Rayleigh fading); it takes independent values over different coherent times.",
"$W$ denotes the shadowing random variable that captures the (random) spatial variation in path-loss.",
"In this paper, $W$ is assumed to take values from a set $\\mathcal {W}$ , and we denote by $g(w)$ the probability mass function or probability density function of $W$ , depending on whether $\\mathcal {W}$ is countable or uncountable.",
"We assume that the transmit power of each node comes from a discrete set, $\\mathcal {S}:=\\lbrace P_1, P_2, \\cdots , P_M \\rbrace $ , where the power levels are arranged in ascending order.",
"Shadowing becomes spatially uncorrelated if the transmitter or receiver is moved by a certain distance that depends on the sizes of the scatterers in the environment (see [21]).",
"It was shown experimentally that, in a forest-like environment, shadowing has log-normal distribution (i.e., $\\log _{10}W \\sim \\mathcal {N}(0,\\sigma ^2)$ where $\\sigma $ is the standard deviation of log-normal shadowing) and the shadowing decorrelation distance can be as small as 6 meters (see [10]).",
"In this paper, we assume that the step size $\\delta $ is chosen to be more than the shadowing decorrelation distance; this allows us to assume that the shadowing at any two different links in the network are independent.",
"The $k$ -th packet is said to see an outage in the link if $P_{rcv,k} \\le P_{rcv-min}$ , where $P_{rcv-min}$ is a threshold depending on the modulation scheme and the properties of the receiving node.",
"For example, $P_{rcv-min}$ can be chosen to be $-88$ dBm for the TelosB “motes” (see [22]), and $-97$ dBm for iWiSe motes (see [23]).",
"For a link with length $r$ , transmit power $\\gamma $ and shadowing realization $W=w$ , the outage probability is denoted by $Q_{out}(r,\\gamma ,w)$ ; it is increasing in $r$ and decreasing in $\\gamma $ , $w$ .",
"$Q_{out}(r,\\gamma ,w)=\\mathbb {P}(P_{rcv,k} \\le P_{rcv-min})$ depends on the fading statistics; if $H$ is exponentially distributed with mean 1 (i.e., for Rayleigh fading), then $Q_{out}(r,\\gamma ,w)=\\mathbb {P}( \\gamma c (\\frac{r}{r_0})^{-\\eta }wH \\le P_{rcv-min} )=1-e^{-\\frac{P_{rcv-min}(\\frac{r}{r_0})^{\\eta }}{\\gamma c w}}$ .",
"The outage probability of a randomly chosen link (with given $r$ and $\\gamma $ ) is a random variable, with the randomness coming from shadowing $W$ .",
"Outage probability can be measured by sending a large number of packets over a link and calculating the fraction of packets with received power less than $P_{rcv-min}$ ."
],
[
"Pure As-You-Go Deployment Process",
"After placing a relay, the agent starts measuring the link qualities from the next $B$ locations one by one (the value of $B$ is fixed prior to deployment).",
"At any given location, the agent uses the measurements from the current location to make a placement decision; the agent does not make measurements from all of those $B$ locations in order to place a new relay.",
"At any given location, the agent measures the link qualities from the given location to all previously placed nodes that are within $B \\delta $ distance from the current location; see Figure REF .",
"Let us assume that the agent is standing at a distance $k \\delta $ from the sink.",
"Let $\\mathcal {I}_k:=\\lbrace r \\in \\lbrace 1,2,\\cdots ,B \\rbrace : \\text{a relay was placed at a distance $(k-r)\\delta $ from the sink} \\rbrace \\rbrace $ .",
"Then, the agent at this location will measure the outage probabilities $\\lbrace Q_{out}(r,\\gamma ,w_r)\\rbrace _{\\gamma \\in \\mathcal {S}, r \\in \\mathcal {I}_k}$ ($w_r$ is the realization of shadowing in a link of length $r$ steps).",
"However, at each location, only the link quality to the immediately previous node is used to decide whether to place a relay there or to move on to the next step.",
"If the decision is to place a relay, then the agent also decides which transmit power $\\gamma \\in \\mathcal {S}$ to use at that particular node.",
"If the decision is not to place a relay, the agent moves to the next step.",
"In this process, if he reaches $B$ steps away from the previous relay, or if the source location is encountered, then he must place a node there.",
"It is important to note that, while the measurement to the immediately previous node is used to make a placement decision, other measurements made in this process provide useful information about the statistical characteristics of the radio propagation environment (more precisely, the probability distribution of $Q_{out}(r,\\gamma ,\\cdot )$ for $r \\in \\lbrace 1,2,\\cdots ,B\\rbrace , \\gamma \\in \\mathcal {S}$ ), and those measurements are used to learn the optimal deployment policy as described in Section and Section .",
"But if the radio propagation parameters (such as $\\eta $ and $\\sigma $ ) are exactly known, i.e., if the probability distribution of $Q_{out}(r,\\gamma ,\\cdot )$ is known exactly, then these additional measurements will not be required (since shadowing is i.i.d.",
"across links, these measurements will not provide any information about the link quality between the current location and the immediately previous node); this situation has been explored in Section , where measurement is made only to the previously placed relay node.",
"Choice of $B$ : In general, the choice of $B$ depends on the constraints and requirements for the deployment.",
"Large $B$ results in better performance at the expense of more measurements.",
"One can simply choose $B$ to be the largest integer such that, the probability that a randomly chosen wireless link with length $B \\delta $ respects a certain outage constraint, is larger than some pre-specified target.",
"This will make sure that the probability of obtaining a workable link is small in case the agent reaches a location that is more than $B \\delta $ distance away from the previously placed node.",
"Figure: Illustration of pure as-you-go deployment with learning for B=4B=4.",
"Herethe deployment agent has already placed Relay 1 and Relay 2, and the corresponding inter-relay distances areU 1 U_1 and U 2 U_2.",
"The placed relays use transmit powers Γ 1 \\Gamma _1 and Γ 2 \\Gamma _2,thereby achieving outage probabilities Q out (1,0) Q_{out}^{(1,0)} and Q out (2,1) Q_{out}^{(2,1)}(in the links shown by solid arrows).",
"After placing Relay 2, the agent measured the link qualities from the nextlocation to the sink, Relay 1 and Relay 2 (since B=4B=4) and the algorithm advised him not to place a node there.Then the deployment agent moved to the next location (which is at a distance of 2δ2 \\delta from Relay 2)and measured the link qualities to Relay 1 and Relay 2 (but not to the sink since B=4B=4).In this snap-shot of the deployment process, the agent isevaluating the next location at r=3δr=3 \\delta distance from Relay 2 (see the dotted arrows).",
"Since B=4B=4, the agent measures thelink qualities from the current location to both Relay 1 and Relay 2; this corresponds to ℐ 6 ={3,4}\\mathcal {I}_6=\\lbrace 3,4\\rbrace (see Section for the definition of ℐ 6 \\mathcal {I}_6), since the distances toRelay 2 and Relay 1 from the current location are 3δ3 \\delta and 4δ4 \\delta respectively.Based on these measurements, the deployment agent will decide whether to place a relay at r=3δr=3 \\delta or not, and the transmit power of the nodein case the decision is to place; if thedecision is not to place a relay here, then a relay must be placed at the next location (since B=4B=4),and the agent would be at a distance of BδB \\delta from the last placed relay (i.e., Relay 2)."
],
[
"Network Cost Minimization Objective",
"We first define the cost that we use to evaluate the performance of any deployment policy.",
"A deployment policy $\\pi $ takes as input the distance of the current location of the agent from the previous relay and the link quality to the previously placed node, and provides the placement decision for that location and transmit power (if the decision is to place a relay) as output.",
"We denote the number of relays placed up to $x$ steps from the sink by $N_x$ , and let us define $N_0 = 0$ .",
"Since deployment decisions are based on measurements of (random) outage probabilities, $\\lbrace N_x\\rbrace _{x \\ge 1}$ is a random process.",
"After the deployment is over, let us denote by $\\Gamma _i$ the transmit power used by node $i$ , and by $Q_{out}^{(i, i-1)}$ the outage probability over the link $(i, i-1)$ (see Figure REF ).",
"Note that, $\\Gamma _i$ and $Q_{out}^{(i, i-1)}$ are random variables since shadowing between various potential location pairs are random variables, whose exact realization is known only after measurement.",
"Given the measurement values (i.e., the information available to the deployment agent) and the deployment policy, one can find the exact realizations of $\\Gamma _i$ and $Q_{out}^{(i, i-1)}$ .",
"The expected cost of the deployed network up to $x \\delta $ distance is given by a sum of hop costs as follows: $\\mathbb {E}_{\\pi } ( \\sum _{i=1}^{N_x} \\Gamma _i + \\xi _{out} \\sum _{i=1}^{N_x}Q_{out}^{(i,i-1)}+ \\xi _{relay} N_x )$ which is the expectation (under policy $\\pi $ ) of a linear combination of the sum power $\\sum _{i=1}^{N_x} \\Gamma _i$ , the sum outage $\\sum _{i=1}^{N_x} Q_{out}^{(i, i-1)}$ , and the number of relays $N_x$ .",
"For small outage probabilities, the sum-outage $\\sum _{i=1}^{N_x} Q_{out}^{(i, i-1)}$ is approximately equal to the probability that a packet sent from the point $x$ to the sink encounters an outage along the path (see also Section REF for a better understanding of the outage cost in light of the traffic model).",
"The sum power $\\sum _{i=1}^{N_x} \\Gamma _i$ is proportional to the battery depletion rate in the network, in case wake-on radios are used (see [1] for a detailed discussion).",
"The multipliers $\\xi _{out} \\ge 0$ and $\\xi _{relay} \\ge 0$ capture the emphasis we place on $\\sum _{i=1}^{N_x} Q_{out}^{(i, i-1)}$ or $N_x$ .",
"A large value of $\\xi _{out}$ will aim for deployment with smaller end-to-end expected outage.",
"$\\xi _{relay}$ can be viewed as the cost of placing a relay.",
"Since the distance $L$ to the source from the sink is not known prior to deployment, we simply assume that $L=\\infty $ .",
"This assumption is practical when the distance of the source from the sink is large (e.g., deployment along a long forest trail).",
"$L=\\infty $ is also equivalent to the scenario where deployment is done serially along multiple trails in a forest, provided that the radio propagation environment in various trails are statistically identical; we deploy serially along multiple lines but use this formulation to minimize the per-step cost averaged over all the lines.",
"Next, we define the optimization problems that we seek to address in this paper."
],
[
"The Unconstrained Problem",
"We seek to solve the following problem: $\\inf _{\\pi \\in \\Pi } \\limsup _{x \\rightarrow \\infty } \\frac{\\mathbb {E}_{\\pi }\\sum _{i=1}^{N_x}(\\Gamma _i+\\xi _{out}Q_{out}^{(i,i-1)}+\\xi _{relay})}{x}$ where $\\Pi $ is the set of all possible placement policies.",
"We formulate (REF ) as an average cost MDP."
],
[
"The Constrained Problem",
"(REF ) is the relaxed version of the following constrained problem: $&& \\inf _{\\pi \\in \\Pi } \\limsup _{x \\rightarrow \\infty } \\frac{\\mathbb {E}_{\\pi }\\sum _{i=1}^{N_x}\\Gamma _i}{x} \\nonumber \\\\&s.t.& \\, \\limsup _{x \\rightarrow \\infty } \\frac{\\mathbb {E}_{\\pi }\\sum _{i=1}^{N_x}Q_{out}^{(i,i-1)}}{x} \\le \\overline{q}, \\nonumber \\\\&& \\text{and } \\limsup _{x \\rightarrow \\infty } \\frac{\\mathbb {E}_{\\pi }N_x}{x} \\le \\overline{N}$ Here we seek to minimize the mean power per step subject to constraints on the mean outage per step and the mean number of relays per step.",
"It turns out that (REF ) is the relaxed version of the constrained problem (REF ), with $\\xi _{out}$ and $\\xi _{relay}$ as the Lagrange multipliers.",
"The constrained problem can be solved by solving the unconstrained problem, under proper choice of the Lagrange multipliers.",
"The following theorem tells us how to choose the Lagrange multipliers $\\xi _{out}$ and $\\xi _{relay}$ (see [24], Theorem $4.3$ ): Theorem 1 For the constrained problem (REF ), if there exists a pair $\\xi _{out}^* \\ge 0$ , $\\xi _{relay}^* \\ge 0$ and a policy $\\pi ^*$ such that $\\pi ^*$ is the optimal policy of the unconstrained problem (REF ) under $(\\xi _{out}^*, \\xi _{relay}^*)$ , and if the constraints in (REF ) are met with equality under the policy $\\pi ^*$ , then $\\pi ^*$ is an optimal policy for the constrained problem (REF ) as well.$\\Box $"
],
[
"Traffic Model",
"Motivated by our prior work reported in [17], [18], [9], [1], we assume that the traffic in the network is so light that there is only one packet in the network at a time; this model is called the “lone packet model” (or the zero traffic model).",
"This model results in collision-free transmissions, since there are no simultaneous transmissions in the network.",
"As a result, we can easily write down the communication cost in the line network as a sum of hop costs (Section REF ).",
"It has been formally shown that network design under the lone packet model may be necessary for designing a network with positive traffic carrying capability (see [25]).",
"We can easily adapt the result of [25] to show that, for a finite line network, if a target end-to-end packet delivery probability has to be achieved under positive traffic, then it is necessary to achieve that target under lone packet traffic.",
"Now, the end-to-end packet error rate under lone packet traffic is approximately equal to the sum outage; this justifies the sum outage cost in (REF ) and the outage constraint in (REF ).",
"Network design for a given positive traffic rate is left for future research.",
"In a line network, if interference-free communication is achieved via multi-channel access and frequency reuse after several hops, then the traffic model essentially becomes lone packet.",
"There have been recent efforts to use multiple channels available in $802.15.4$ radio in WSN; see [26], [27], [28], [29].",
"The lone packet traffic model is realistic for WSNs carrying low duty cycle measurements, or just an occasional alarm packet.",
"For example, recently developed passive infra-red (PIR) sensor platforms can detect and classify human or animal intrusion ([30]); such sensors deployed in a forest generate very low data.",
"The paper [6] uses $1.1\\%$ duty cycle for a multi-hop WSN for wildlife monitoring; the sensors gather data from RFID collars tied the animals, and generate light traffic.",
"Very light traffic model is also realistic for condition monitoring/industrial telemetry applications ([31]), where infrequent measurements are taken.",
"Very light traffic model is also common in machine-to-machine communication ([32]).",
"The paper [33] illustrate sensors with small sampling rate and sampled data size; it shows several bytes per second data rate requirement for habitat monitoring.",
"We assume that data packets traverse the network in a hop-by-hop fashion, without skipping any intermediate relay.",
"Later we will explain in Section REF why we do not consider the possibility of relay skipping in this paper; the reason is increased computational complexity without a very significant gain in network performance."
],
[
"Extension to Bi-Directional Traffic Flow",
"Let us consider the situation where the traffic is still lone packet, but a packet can flow towards either direction along the line network with equal probabilities.",
"In such cases, one can define the cost of link $(i,i-1)$ as $\\Gamma _{i, forward}+\\Gamma _{i-1,reverse}+ \\xi _{out} Q_{out}^{(i,i-1, forward)}+\\xi _{out} Q_{out}^{(i-1,i, reverse)}+\\xi _{relay}$ , where $\\Gamma _{i, forward}$ is the transmit power used from node $i$ to node $(i-1)$ , and $\\Gamma _{i-1, reverse}$ is the transmit power used from node $(i-1)$ to node $i$ .",
"Similar meanings apply for the outage probabilities $Q_{out}^{(i,i-1, forward)}$ and $Q_{out}^{(i-1,i, reverse)}$ , under transmit power levels $\\Gamma _{i, forward}$ and $\\Gamma _{i-1, reverse}$ , respectively.",
"It has to be noted that the shadowing between two potential locations in forward and reverse directions, $W_{forward}$ and $W_{reverse}$ , may not necessarily be independent.",
"But the shadowing random variable pair $(W_{forward}, W_{reverse}) \\in \\mathbb {R}_{+}^2$ between two potential locations have a joint distribution, and this pair assumes independent and identically distributed (i.i.d.)",
"value in $\\mathbb {R}_{+}^2$ if either the transmitter or the receiver is moved beyond the shadowing decorrelation distance (which is smaller than the step size $\\delta $ ).",
"Hence, with this new link cost, our formulation (REF ) can easily be adapted to deploy a network carrying bi-directional traffic.",
"In the process of deployment, the agent has to measure link qualities in both forward and reverse directions in such situation.",
"The action at each step is to decide whether to place a relay; if the decision is to place a relay, then the agent also decides the transmit power levels used in that link along the forward and the reverse directions.",
"Since the design for bi-directional traffic carrying network is mathematically equivalent to the design for unidirectional traffic carrying network, we will consider only unidirectional traffic for the rest of this paper."
],
[
"Formulation for known propagation parameters",
"Throughout this section, we will assume that we seek to solve the unconstrained problem given in (REF ), and that the radio propagation parameters (such as $\\eta $ and the standard deviation $\\sigma $ for log-normal shadowing) are known prior to deployment.",
"We formulate the problem as an average cost MDP, and develop a threshold policy for deployment.",
"In the process, we also discover some interesting properties of the value function, which do not follow from the discounted cost formulation.",
"Note that, we assume throughout this section that measurement only to the immediately previous node is used to make a placement decision at any given location.",
"Measurement to more than one previous nodes will be used later in order to develop the learning algorithms."
],
[
"Markov Decision Process (MDP) Formulation",
"When the deployment agent is $r$ steps away from the previous node ($r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ ), the agent measures the outage probabilities $\\lbrace Q_{out}(r,\\gamma ,w)\\rbrace _{\\gamma \\in \\mathcal {S}}$ on the link from the current location to the previous node,Note that, for the time being, we will ignore the measurements made to other nodes from the set $\\mathcal {I}_k$ .",
"where $w$ is the realization of shadowing in that link.",
"Then the algorithm decides whether to place a relay there, and also the transmit power $\\gamma \\in \\mathcal {S}$ in case it decides to place a relay.",
"We formulate the problem as an average cost MDP with state space $\\lbrace 1,2,\\cdots ,B\\rbrace \\times \\mathcal {W}$ , where a typical state is of the form $(r,w), 1 \\le r \\le B, w \\in \\mathcal {W}$ .",
"If $r \\le B-1$ , the action is either to place a relay and select a transmit power, or not to place.",
"If $r=B$ , the only feasible action is to place and select a transmit power $\\gamma \\in \\mathcal {S}$ .",
"If a relay is placed at state $(r,w)$ and if a transmit power $\\gamma $ is chosen for it, then a hop-cost of $\\gamma +\\xi _{out}Q_{out}(r,\\gamma ,w)+\\xi _{relay}$ is incurred.We have taken $(r,w)$ as a typical state for the sake of simplicity in representation; for the channel model given by (REF ), we can also take $(r,\\lbrace Q_{out}(r,\\gamma ,w)\\rbrace _{\\gamma \\in \\mathcal {S}})$ as a typical state, since the cost of an action depends on the state $(r,w)$ only via the outage probabilities.",
"A deterministic Markov policy $\\pi $ is a sequence of mappings $\\lbrace \\mu _k\\rbrace _{k \\ge 1}$ from the state space to the action space.",
"The policy $\\pi $ is called a stationary policy if $\\mu _k=\\mu $ for all $k$ .",
"Given the state (i.e., the measurements), the policy provides the placement decision."
],
[
"Optimal Policy Based on Average Cost Optimality Equation",
"We will first derive the structure of an optimal policy based on the average cost optimality equation (ACOE).",
"Let $\\lambda ^*$ (or $\\lambda ^*(\\xi _{out},\\xi _{relay})$ ) be the optimal average cost per step for the unconstrained problem (REF ) under the pure as-you-go deployment approach, and let $v^*(r,w)$ be the differential cost for the state $(r,w)$ , where $1 \\le r \\le B$ and $w \\in \\mathcal {W}$ .",
"The average cost optimality equation for our MDP is as follows (by the theory of [34], for the case of finite $\\mathcal {W}$ , and by the theory developed in [35], when $\\mathcal {W}$ is a Borel subset of the real line): $v^*(r,w)&=&\\min \\bigg \\lbrace \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , w))+\\xi _{relay}-\\lambda ^* \\nonumber \\\\&& + \\sum _{w^{\\prime }}g(w^{\\prime })v^*(1,w^{\\prime }), -\\lambda ^*+\\sum _{w^{\\prime }}g(w^{\\prime })v^*(r+1,w^{\\prime }) \\bigg \\rbrace \\nonumber \\\\&& \\forall 1 \\le r \\le B-1 \\nonumber \\\\v^*(B,w)&=& \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , w))+\\xi _{relay}-\\lambda ^* \\nonumber \\\\&& +\\sum _{w^{\\prime }}g(w^{\\prime })v^*(1,w^{\\prime }) $ where $g(w)$ was defined (in Section REF ) to be the probability mass function or probability density function of shadowing $W$ .",
"The ACOE (REF ) can be explained as follows.",
"When the state is $(r,w)$ , the deployment agent can either place or may not place a relay.",
"If he places a relay, he will incur a stage cost of $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , w))+\\xi _{relay}$ and the next (random) state is $(1, W^{\\prime })$ , where $W^{\\prime }$ has p.m.f.",
"or p.d.f.",
"$g(w^{\\prime })$ .",
"If he does not place, then he incurs 0 cost at that step and the next state is $(r+1, W^{\\prime })$ .",
"When at state $(B,w)$ , he can only place a relay and incur a cost of $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , w))+\\xi _{relay}$ at that stage and the next (random) state is $(1,W^{\\prime })$ .",
"Note that, $\\min _{\\gamma \\in \\mathcal {S}}$ appears in the single-stage cost because choice of transmit power of the placed node is also a part of the action, and a transmit power is chosen so that the single-stage cost for a placed relay is minimized.",
"Note that, by multiplying both sides of (REF ) with $g(w)$ and taking summation over $w$ , we obtain the following: $V(r)&=&\\mathbb {E}_{W} \\min \\bigg \\lbrace \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , W))+\\xi _{relay}-\\lambda ^* \\nonumber \\\\& & + V(1), -\\lambda ^*+V(r+1) \\bigg \\rbrace \\forall 1 \\le r \\le B-1 \\nonumber \\\\V(B)&=&\\mathbb {E}_{W} \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , W))+\\xi _{relay}-\\lambda ^*+ V(1) \\nonumber \\\\$ where $V(r)=\\sum _{w}g(w)v^*(r,w) \\forall 1 \\le r \\le B$ .",
"Now, it is easy to see that if any $V(\\cdot )$ satisfies (REF ), then $V(\\cdot )+c$ for any constant number $c$ also satisfies (REF ).",
"Hence, we can put $V(1)=\\lambda ^*$ in (REF ) and obtain: $V(r)&=&\\mathbb {E}_{W} \\min \\bigg \\lbrace \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , W))+\\xi _{relay}, \\nonumber \\\\& & V(r+1)-V(1) \\bigg \\rbrace \\forall 1 \\le r \\le B-1 \\nonumber \\\\V(B)&=&\\mathbb {E}_{W} \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , W))+\\xi _{relay}$ Remark: Let $c(r,W):=\\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , W))+\\xi _{relay}$ be the (random) cost incurred if we place a relay at a distance $r$ from the previous relay.",
"(REF ) shows the criteria for optimality to be $V(r)=\\mathbb {E}_{W} \\min \\lbrace c(r,W), V(r+1)-V(1)\\rbrace $ for $r \\le B-1$ and $V(B)=\\mathbb {E}_{W} c(B,W)$ .",
"We will see in Algorithm REF that, by solving this system of (nonlinear) equations, one can find the optimal policy; there is no need to compute the differential cost for each state explicitly.",
"Also, (REF ) will be particularly useful when we develop online deploy-and-learn algorithms in later sections, using the theory of stochastic approximation.",
"Theorem 2 There exists a unique vector $\\underline{V}^*=[V^*(1), \\, V^*(2), \\, \\cdots \\, , V^*(B)]^T$ satisfying (REF ).",
"Also, $V^*(r) \\ge r V^*(1)$ for all $r \\in \\lbrace 1,2,\\cdots ,B-1\\rbrace $ and $V^*(r)$ is increasing in $r$ .",
"See Appendix ."
],
[
"Policy Structure",
"Algorithm REF specifies the optimal decision when the agent is $r$ steps away from the previously placed node and the shadowing realization from the current location to the previously placed node is $w$ .",
"[t!]",
"Input: $\\xi _{out}$ , $\\xi _{relay}$ , $\\underline{V}^*$ .",
"Output: Placement decision at each step.",
"Pre-compute: The threshold values $c_{th}(r):=V^*(r+1)-V^*(1)$ for all $1 \\le r \\le B-1$ .",
"Initialization: $r=1$ (distance from the previous node) $1 \\le r \\le B$ Measure $Q_{out}(r,\\gamma ,w) \\forall {\\gamma \\in \\mathcal {S}}$ ; $r \\le B-1$ and $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , w))+\\xi _{relay} \\le c_{th}(r)$ Place a new relay and use transmit power $\\arg \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , w))$ ; Move to next step and set $r=1$ ; $r \\le B-1$ and $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , w))+\\xi _{relay} > c_{th}(r)$ Do not place a relay and move to next step; $r=r+1$ ; Place a new relay (since $r=B$ ); Use transmit power $\\arg \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , w))$ ; Move to next step; Set $r=1$ .",
"OptAsYouGo Algorithm Theorem 3 The policy given by Algorithm REF is optimal for the unconstrained problem in (REF ).",
"The threshold $c_{th}(r)$ is increasing in $r$ .",
"From (REF ), the optimal policy is to place a relay at state $(r,w)$ if the cost of placing is less than the cost of not placing.",
"Hence, the policy structure follows from equations (REF ), (REF ) and (REF ).",
"$c_{th}(r)$ is increasing in $r$ since $V^*(r+1)$ is increasing in $r$ .",
"We denote the optimal policy given by Algorithm REF by $\\pi ^*(\\xi _{out}, \\xi _{relay})$ ."
],
[
"Some properties of the optimal cost",
"Let us consider a sub-class of stationary deployment policies (parameterized by $\\underline{V}$ , $\\xi _{out} \\ge 0$ and $\\xi _{relay} \\ge 0$ ) where $\\underline{V}^*(\\cdot )$ in Algorithm REF is replaced by any vector $\\underline{V}$ .",
"Under this sub-class of policies, let us denote by $(U_k,\\Gamma _k,Q_{out}^{(k,k-1)}), k \\ge 1,$ the sequence of inter-node distances, transmit powers and link outage probabilities (see Figure REF ).",
"Since shadowing is i.i.d.",
"across links, the deployment process probabilistically restarts after each relay placement.",
"Hence, $(U_k,\\Gamma _k,Q_{out}^{(k,k-1)}), k \\ge 1,$ is an i.i.d.",
"sequence.",
"Let $\\overline{\\Gamma }(\\underline{V},\\xi _{out},\\xi _{relay})$ , $\\overline{Q}_{out}(\\underline{V},\\xi _{out},\\xi _{relay})$ and $\\overline{U}(\\underline{V},\\xi _{out},\\xi _{relay})$ denote the mean power per link, mean outage per link and mean placement distance (in steps) respectively, under this sub-class of policies.",
"We denote by $\\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})$ , $\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})$ and $\\overline{U}^*(\\xi _{out},\\xi _{relay})$ the optimal mean power per link, the optimal mean outage per link and the optimal mean placement distance (in steps) respectively, under Algorithm REF , where $\\underline{V}^*$ is used instead of any general $\\underline{V}$ .",
"Now, the optimal mean power per step, the optimal mean outage per step, and the optimal mean number of relays per step are given by $\\frac{\\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ , $\\frac{\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ and $\\frac{1}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ (by the Renewal-Reward theorem).",
"Theorem 4 The optimal average cost per step for problem (REF ), $\\lambda ^*(\\xi _{out},\\xi _{relay})$ , is concave, increasing and Lipschitz continuous in $\\xi _{out}\\ge 0$ , $\\xi _{relay} \\ge 0$ .",
"See Appendix .",
"Theorem 5 $\\underline{V}^*=(V^*(1),V^*(2), \\cdots , V^*(B))$ is Lipschitz continuous in $(\\xi _{out},\\xi _{relay})$ .",
"See Appendix .",
"Theorem 6 For a given $\\xi _{out}$ , the mean number of relays per step under Algorithm REF , $\\frac{1}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ , decreases with $\\xi _{relay}$ .",
"Similarly, for a given $\\xi _{relay}$ , the optimal mean outage per step, $\\frac{\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ , decreases with $\\xi _{out}$ .",
"The proof is exactly same as the proof of [1]."
],
[
"A note on the objective function in (",
"Even though the deployment policy developed in this section uses only the measurements made to the immediately previous placed node in order to make a placement location, we will see in subsequent sections that measurements to all placed relay nodes located within $B$ steps from the current location of the agent will be used for on-line learning of the optimal deployment policy.",
"A question that naturally arises is whether we can do better with the additional measurements (when the propagation parameters are known and the optimal policy can be computed prior to deployment); this might require skipping some already placed relay nodes after the deployment is over.",
"The possibility of relay skipping was considered in [9]; in the current paper, we briefly describe a similar formulation in our context and explain why we rule out the possibility of relay skipping.",
"Let us consider deployment up to $x$ steps.",
"After the deployment is over, we construct a directed acyclic graph over the deployed nodes (including the sink) as follows.",
"Links are all directed edges from each node to every node with smaller index and located within a distance of $B$ steps.",
"Hence, if $i$ and $j$ are two nodes with $i>j$ and $\\sum _{k=j+1}^i U_k \\le B$ , there is a link $(i,j)$ between them.",
"Consider all directed acyclic paths from node $N_x$ to the sink over this graph.",
"Let us denote by $\\mathbf {p}$ any arbitrary directed acyclic path, and by $\\mathcal {E}(\\mathbf {p})$ the set of (directed) links of the path $\\mathbf {p}$ .",
"We also define $\\mathcal {P}_x:=\\lbrace \\mathbf {p}:(i,j) \\in \\mathcal {E}(\\mathbf {p}) \\Rightarrow N_x \\ge i>j \\ge 0, \\sum _{k=j+1}^i U_k \\le B \\rbrace $ .",
"Let us denote a generic link (edge) on this graph by $e$ , and the transmit power and outage probability on edge $e$ by $\\Gamma ^{(e)}$ and $Q_{out}^{(e)}$ .",
"Let us consider the following problem: $&& \\min _{\\pi \\in \\Pi } \\lim \\sup _{x \\rightarrow \\infty } \\nonumber \\\\&&\\frac{ \\mathbb {E}_{\\pi } \\bigg ( \\min _{\\mathbf {p} \\in \\mathcal {P}_x} \\sum _{e \\in \\mathcal {E}(\\mathbf {p})} \\bigg ( \\Gamma ^{(e)}+ \\xi _{out} Q_{out}^{(e)} \\bigg ) + \\xi _{relay} N_x \\bigg ) }{x} \\nonumber \\\\&& $ We call $\\sum _{e \\in \\mathcal {E}(\\mathbf {p})} \\bigg ( \\Gamma ^{(e)}+ \\xi _{out} Q_{out}^{(e)} \\bigg )$ the length of the path $\\mathbf {p}$ , and $\\min _{\\mathbf {p} \\in \\mathcal {P}_x} \\sum _{e \\in \\mathcal {E}(\\mathbf {p})} \\bigg ( \\Gamma ^{(e)}+ \\xi _{out} Q_{out}^{(e)} \\bigg )$ the length of the shortest path.",
"Formulation of problem (REF ) as an MDP will require as the typical state the distance of all nodes located within $B$ steps from the current location, the realization of shadowing to all these nodes (through the measured outage probabilities), and the lengths of the shortest paths from all these nodes to the sink.",
"A similar situation was considered in [9].",
"It turns out that the state space becomes very large (the number of all possible lengths of shortest paths grows to $\\infty $ as $x \\rightarrow \\infty $ , even when the set $\\mathcal {W}$ of possible values of shadowing is finite), and the policy computation becomes numerically intensive; but the numerical results of [9] show that the margin of performance improvement achieved via this formulation (instead of the formulation used earlier in this section) is not significant.",
"Hence, in this paper, we only consider formulation (REF ) and proceed with it.",
"Note that, for any given values of $\\xi _{out}$ and $\\xi _{relay}$ , the optimal policy given by Algorithm REF can be completely specified by the vector $\\underline{V}^*$ .",
"But, the computation of $\\underline{V}^*$ requires the agent to solve a system of nonlinear equations (which is computationally intensive), and these nonlinear equations can be specified only when the channel model parameters (e.g., path-loss exponent $\\eta $ and standard deviation $\\sigma $ for log-normal shadowing) are known apriori.",
"However, in practice, these parameters may not be available prior to deployment.",
"Under this situation, the deployment agent has to learn the optimal policy as deployment progresses, and use the corresponding updated policy at each step to make a placement decision.",
"In order to address this requirement, we propose an algorithm which will maintain a running estimate of $\\underline{V}^*$ , and update this estimate at each step (using new measurements made at each step).",
"Using the theory of Asynchronous Stochastic Approximation (see [36]), we show that, as the number of deployed relays goes to infinity, the running estimate converges to $\\underline{V}^*$ almost surely.",
"From (REF ) (and the notation defined immediately after (REF )), we see that the optimal $\\underline{V}^*$ is the unique real zero of the system of equations: $\\mathbb {E}_{W} \\min \\lbrace c(r,W), V(r+1)-V(1)\\rbrace -V(r)=0$ for $r \\le B-1$ and $\\mathbb {E}_{W} \\, c(B,W)-V(B)=0$ .",
"We use asynchronous stochastic approximation so that the iterates $\\lbrace \\underline{V}^{(k)}\\rbrace _{k \\ge 0}$ converge asymptotically to this unique zero."
],
[
"OptAsYouGoLearning Algorithm",
"Suppose that the deployment agent is standing $k$ steps away from the sink node.",
"At the $k$ -th step, the agent makes a placement decision and then performs a learning operation.",
"Let us recall the deployment process (see Section REF and Figure REF ) and notation: $\\mathcal {I}_k:=\\lbrace r \\in \\lbrace 1,2,\\cdots ,B \\rbrace : \\text{a relay was placed at a distance $(k-r)\\delta $ from the sink} \\rbrace \\rbrace $ .",
"For the learning operation, $\\mathcal {I}_k \\subset \\lbrace 1, \\cdots , B\\rbrace $ denotes the set of the values of $r$ for which links from the current location to the placed relay $r$ steps backwards are measured, and for which $V(r)$ is updated, when the agent is at a distance $k \\delta $ from the sink.",
"Clearly, for each $k \\ge 1$ , $\\mathcal {I}_k$ is a random set.",
"Let us denote by $\\underline{V}^{(k)}$ the estimate of $\\underline{V}^*$ after an update (i.e., a learning operation) is made at the $k$ -th step from the sink.",
"At step $k$ (after a placement decision is made), $V^{(k-1)}(r)$ for $r \\in \\mathcal {I}_k$ is updated to $V^k(r)$ , and it is not updated for $r \\notin \\mathcal {I}_k$ (which means that $V^{(k)}(r)=V^{(k-1)}(r)$ for $r \\notin \\mathcal {I}_k$ ).",
"Let us define $\\nu (r,k):=\\sum _{i=1}^k \\mathbb {I} \\lbrace r \\in \\mathcal {I}_i \\rbrace $ the number of times the estimate of $V^*(r)$ is updated up to the $k$ -th step.",
"Note that, Algorithm REF requires the agent to measure link quality only to the previous node, whereas the learning algorithm presented in this section involves link quality measurement to more than one previous nodes (unlike our prior paper [1]).",
"This is necessary because, if we make measurement only to last relay, then, depending on the initial estimate $\\underline{V}^{(0)}$ , there could arise a situation that the inter-relay distance never equals to $B$ steps in the entire deployment process, which implies that $V^{(0)}(B)$ will never be updated, thereby converging to an unintended policy.",
"Making measurements to all previously placed nodes located at distance less than $B\\delta $ from the current location ensures that $\\liminf _{k \\rightarrow \\infty }\\frac{\\nu (r,k)}{k}>0$ almost surely, which is required for the convergence proof.",
"The OptAsYouGoLearning algorithm is provided in Algorithm REF .",
"[t!]",
"Input: $\\xi _{out}$ , $\\xi _{relay}$ , and a decreasing positive sequence $\\lbrace a(n)\\rbrace _{n \\ge 1}$ such that $\\sum _{n=1}^{\\infty } a(n)=\\infty $ , $\\sum _{n=1}^{\\infty } a^2(n) < \\infty $ .",
"Output: Placement decision at each step.",
"Initialization: $r^{\\prime }=1$ (distance from the previous node), $k=1$ (distance of the current location from the sink), initial estimate $\\underline{V}^{(0)}$ .",
"$1 \\le r^{\\prime } \\le B$ Find $\\mathcal {I}_k:=\\lbrace r \\in \\lbrace 1,2,\\cdots ,B \\rbrace : \\text{relay placement at $(k-r)\\delta $ distance from sink} \\rbrace \\rbrace $ ; Find $\\nu (r,k):=\\sum _{i=1}^k \\mathbb {I} \\lbrace r \\in \\mathcal {I}_i \\rbrace \\forall r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ ; Measure $Q_{out}(r,\\gamma ,w_r) \\forall {\\gamma \\in \\mathcal {S}}, r \\in \\mathcal {I}_k$ ; $r^{\\prime } \\le B-1$ and $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r^{\\prime },\\gamma , w_{r^{\\prime }}))+\\xi _{relay} \\le -V^{(k-1)}(1)+V^{(k-1)}(r^{\\prime }+1)$ Place a new relay and use transmit power $\\arg \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r^{\\prime },\\gamma , w_{r^{\\prime }}))$ ; Do the following updates: $&& V^{(k)}(r)\\nonumber \\\\&=&V^{(k-1)}(r)+ a(\\nu (r,k)) \\mathbb {I}\\lbrace r \\in \\mathcal {I}_k\\rbrace \\bigg [ \\min \\bigg \\lbrace \\min _{\\gamma }(\\gamma + \\nonumber \\\\&& \\xi _{out} Q_{out}(r,\\gamma , w_r)) +\\xi _{relay}, -V^{(k-1)}(1) \\nonumber \\\\&& +V^{(k-1)}(r+1) \\bigg \\rbrace -V^{(k-1)}(r) \\bigg ], \\forall 1 \\le r \\le B-1 \\nonumber \\\\&& V^{(k)}(B) \\nonumber \\\\&=&V^{(k-1)}(B)+ a(\\nu (B,k)) \\mathbb {I}\\lbrace B \\in \\mathcal {I}_k\\rbrace \\bigg [ \\min _{\\gamma }(\\gamma + \\nonumber \\\\&& \\xi _{out} Q_{out}(B,\\gamma , w_B)) +\\xi _{relay}-V^{(k-1)}(B) \\bigg ]$ Move to next step and set $r^{\\prime }=1$ ; $r^{\\prime } \\le B-1$ and $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(r^{\\prime },\\gamma , w_{r^{\\prime }}))+\\xi _{relay} > -V^{(k-1)}(1)+V^{(k-1)}(r^{\\prime }+1)$ Do not place, do the same updates as (REF ); Move to next step and do $r^{\\prime }=r^{\\prime }+1$ ; Place a new relay (since $r^{\\prime }=B$ ); Use transmit power $\\arg \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , w_B))$ ; Do the same updates as (REF ); Move to next step and set $r^{\\prime }=1$.",
"k=k+1; OptAsYouGoLearning Algorithm Theorem 7 Under Algorithm REF , $V^{(k)}(r) \\rightarrow V^*(r)$ almost surely for all $1 \\le r \\le B$ .",
"See Appendix .",
"Discussion of Algorithm REF : The basic idea: From (REF ) (and the notation defined immediately after (REF )), we see that the optimal $\\underline{V}^*$ is the unique real zero of the system of equations: $\\mathbb {E}_{W} \\min \\lbrace c(r,W), V(r+1)-V(1)\\rbrace -V(r)=0$ for $r \\le B-1$ and $\\mathbb {E}_{W} \\, c(B,W)-V(B)=0$ .",
"We use asynchronous stochastic approximation so that the iterates converge asymptotically to this unique zero.",
"Asynchronous stochastic approximation: In standard stochastic approximation techniques, all iterates are updated at the same time.",
"However, the pure as-you-go deployment scheme does not allow the deployment agent to update all iterates at each step.",
"Since only a subset $\\mathcal {I}_k \\subset \\lbrace 1, \\cdots , B\\rbrace $ of iterates can be updated at step $k$ , we have to use asynchronous stochastic approximation.",
"The proof of Theorem REF exhibits a nice separation between the estimation and control.",
"In other words, the iterates will asymptotically converge to $\\underline{V}^*$ (and the policy will converge to the optimal policy) even when the placement decisions are not made according to the proposed threshold policy (but the measurement and update scheme should be unchanged); but it may not yield the optimal cost for problem (REF ) since we do not use the optimal policy at each stage.",
"However, this nice separation property will not hold in next section when we vary $\\xi _{out}$ and $\\xi _{relay}$ in order to solve the constrained problem (REF ).",
"Note that, since the state space of the MDP in Section is large (potentially infinite and even uncountable), it will not be easy to use traditional Q-learning algorithms.",
"In fact, all the state action-pairs in a Q-learning algorithm need to repeat comparably often over infinite time horizon to guarantee the desired convergence, but this may not happen in case of infinite state space (arising out of infinite $\\mathcal {W}$ ).",
"On the other hand, Algorithm REF provides a learning algorithm with provable convergence guarantee while having only $B$ number of iterates."
],
[
"OptAsYouGoAdaptiveLearning for the Constrained problem",
"In Section , we provided a deploy-and-learn algorithm for given $\\xi _{out}$ and $\\xi _{relay}$ .",
"However, Theorem REF tells us how to choose the Lagrange multipliers $\\xi _{out}$ and $\\xi _{relay}$ (if they exist) in (REF ) in order to solve the constrained problem (REF ).",
"But we need to know the radio propagation parameters (e.g., $\\eta $ and $\\sigma $ ) in order to compute a pair $(\\xi _{out}^*, \\xi _{relay}^*)$ that satisfies the condition given in Theorem REF .",
"In practice, these parameters may not be known.",
"Hence, we provide a sequential placement algorithm such that, as deployment progresses, the placement policy (updated at each step) converges to the set of optimal policies for the constrained problem (REF ).",
"We modify the OptAsYouGoLearning algorithm so that a running estimate $(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ gets updated at each step, and asymptotically converges to the set of optimal $(\\underline{V}^*(\\xi _{out},\\xi _{relay}),\\xi _{out},\\xi _{relay})$ tuples.",
"This algorithm is based on two time-scale stochastic approximation (see [37])."
],
[
"Some Useful Notation and Assumptions",
"In this subsection, we will introduce some assumptions and notation (these were provided in [1], but are repeated here for completeness).",
"Definition 1 We denote by $\\gamma ^*$ the optimal mean power per step for problem (REF ), for a given constraint pair $(\\overline{q},\\overline{N})$ .",
"The set $\\mathcal {K}(\\overline{q},\\overline{N})$ is defined as follows: $&& \\mathcal {K}(\\overline{q},\\overline{N}) := \\bigg \\lbrace (\\underline{V}^*(\\xi _{out},\\xi _{relay}),\\xi _{out},\\xi _{relay}): \\\\&& \\frac{\\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}=\\gamma ^* ,\\frac{ \\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay}) }{ \\overline{U}^*(\\xi _{out},\\xi _{relay}) } \\le \\overline{q} \\\\&& \\frac{1}{ \\overline{U}^*(\\xi _{out},\\xi _{relay})} \\le \\overline{N},\\xi _{out} \\ge 0, \\xi _{relay} \\ge 0 \\bigg \\rbrace $ $\\Box $ Note that, the pair $(\\overline{q},\\overline{N})$ can be infeasible.",
"For example, if $\\overline{N}=\\frac{1}{B}$ (i.e., inter-node distance is $B$ ) and $\\overline{q}< \\frac{\\mathbb {E}_W Q_{out}(B,P_M,W)}{B}$ ($P_M$ is the maximum available transmit power), the outage constraint cannot be satisfied while meeting the constraint on the mean number of relays per step, even by using the maximum transmit power $P_M$ .",
"$\\mathcal {K}(\\overline{q},\\overline{N})$ is empty if $(\\overline{q},\\overline{N})$ is infeasible.",
"In this paper, we assume that $\\mathcal {K}(\\overline{q},\\overline{N})$ is non-empty (i.e., $(\\overline{q},\\overline{N})$ is a feasible pair), which is true for feasible pairs of $\\mathcal {K}(\\overline{q},\\overline{N})$ : Assumption 1 $\\overline{q}$ and $\\overline{N}$ are such that there exists at least one pair $ \\xi _{out}^* \\ge 0, \\xi _{relay}^* \\ge 0$ such that $(\\underline{V}^*(\\xi _{out}^*,\\xi _{relay}^*),\\xi _{out}^*,\\xi _{relay}^*) \\in \\mathcal {K}(\\overline{q},\\overline{N})$ .$\\Box $ Assumption 2 The probability density function (p.d.f.)",
"of the shadowing random variable $W$ is continuous over $(0,\\infty )$ ; i.e., $\\mathbb {P}(W=w)=0$ for any $w \\in (0,\\infty )$ (e.g., log-normal shadowing).$\\Box $ Theorem 8 Under Assumption REF and Algorithm REF , the optimal mean power per step $\\frac{\\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ , the optimal mean placement rate $\\frac{1}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ and the optimal mean outage per step $\\frac{\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ , are continuous in $(\\xi _{out},\\xi _{relay})$ .",
"See Appendix .",
"Remark: Theorem REF implies that there is no need to do any randomization among deterministic policies (unlike [38]) in order to meet the constraints with equality."
],
[
"OptAsYouGoAdaptiveLearning Algorithm",
"The basic idea behind this algorithm (Algorithm REF ; see next page) is to vary $\\xi _{out}^{(k)}$ and $\\xi _{relay}^{(k)}$ at a much slower rate than $\\underline{V}^{(k)}$ , as if $\\xi _{out}^{(k)}$ and $\\xi _{relay}^{(k)}$ are varied in an outer loop and $\\underline{V}^{(k)}$ is varied in an inner loop.",
"If the outage in a newly created link is larger than the budgeted outage for a link with that length, then $\\xi _{out}$ is increased with the hope that subsequent links will have smaller outage; the opposite is done in case the outage in a newly created link is smaller.",
"On the other hand, if a newly created link is shorter than $\\frac{1}{\\overline{N}}$ , then $\\xi _{relay}$ is increased, otherwise it is decreased.",
"Notation in Algorithm REF : $\\Lambda _{[0,A_1]}(x)$ denotes the projection of $x$ on the interval $[0,A_1]$ .",
"Let the power, outage and link length of the new relay (if placed) at the $k$ -th step be $\\Gamma _{N_k}$ , $Q_{out}^{(N_k,N_k-1)}$ and $U_{N_k}$ (recall that $N_k$ is the number of nodes placed up to the $k$ -th step).",
"Note that, $\\mathbb {I}\\lbrace N_k=N_{k-1}+1 \\rbrace $ is the indicator that a relay is placed at the $k$ -th step.",
"[h!]",
"Input: Two positive numbers $A_1$ and $A_2$ appropriately chosen, two decreasing positive sequences $\\lbrace a(n)\\rbrace _{n \\ge 1}$ and $\\lbrace b(n)\\rbrace _{n \\ge 1}$ such that $\\sum _{n=1}^{\\infty } a(n)=\\infty $ , $\\sum _{n=1}^{\\infty } a^2(n) < \\infty $ , $\\sum _{n=1}^{\\infty } b(n)=\\infty $ , $\\sum _{n=1}^{\\infty } b^2(n) < \\infty $ and $\\lim _{n \\rightarrow \\infty }\\frac{b(\\lfloor \\frac{n}{B} \\rfloor )}{a(n)}=0$ .",
"Output: Placement decision at each step.",
"Initialization: $r^{\\prime }=1$ (distance from the previous node), $k=1$ (distance of the current location from the sink), initial estimates $\\underline{V}^{(0)}$ , $\\xi _{out}^{(0)}$ , $\\xi _{relay}^{(0)}$ .",
"$1 \\le r^{\\prime } \\le B$ Find $\\mathcal {I}_k:=\\lbrace r \\in \\lbrace 1,2,\\cdots ,B \\rbrace : \\text{relay placed at $(k-r)\\delta $ distance from sink} \\rbrace \\rbrace $ ; Find $\\nu (r,k):=\\sum _{i=1}^k \\mathbb {I} \\lbrace r \\in \\mathcal {I}_i \\rbrace \\forall r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ ; Measure $Q_{out}(r,\\gamma ,w_r) \\forall {\\gamma \\in \\mathcal {S}}, r \\in \\mathcal {I}_k$ ; $r^{\\prime } \\le B-1$ and $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out}^{(k-1)} Q_{out}(r^{\\prime },\\gamma , w_{r^{\\prime }}))+\\xi _{relay}^{(k-1)} \\le -V^{(k-1)}(1)+V^{(k-1)}(r^{\\prime }+1)$ Place a new relay and use transmit power $\\arg \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out}^{(k-1)} Q_{out}(r^{\\prime },\\gamma , w_{r^{\\prime }}))$ ; Do the following updates: $&& V^{(k)}(r)=V^{(k-1)}(r)+ a(\\nu (r,k)) \\mathbb {I}\\lbrace r \\in \\mathcal {I}_k, r <B\\rbrace \\nonumber \\\\&& \\bigg [ \\min \\bigg \\lbrace \\min _{\\gamma }(\\gamma + \\xi _{out}^{(k-1)} Q_{out}(r,\\gamma , w_r)) +\\xi _{relay}^{(k-1)}, \\nonumber \\\\&& -V^{(k-1)}(1)+V^{(k-1)}(r+1) \\bigg \\rbrace -V^{(k-1)}(r) \\bigg ] \\nonumber \\\\&& V^{(k)}(B) =V^{(k-1)}(B)+ a(\\nu (B,k)) \\mathbb {I}\\lbrace B \\in \\mathcal {I}_k\\rbrace \\nonumber \\\\&& \\bigg [ \\min _{\\gamma }(\\gamma + \\xi _{out}^{(k-1)} Q_{out}(B,\\gamma , w_B)) +\\xi _{relay}^{(k-1)} \\nonumber \\\\&& -V^{(k-1)}(B) \\bigg ] \\nonumber \\\\&& \\xi _{out}^{(k)} = \\bigg [\\xi _{out}^{(k-1)}+b({N_k}) \\mathbb {I}\\lbrace N_k=N_{k-1}+1 \\rbrace \\nonumber \\\\&& \\bigg ( Q_{out}^{(N_k,N_k-1)}-\\overline{q}U_{N_k} \\bigg ) \\bigg ]_{0}^{A_1} \\nonumber \\\\&& \\xi _{relay}^{(k)} =\\bigg [\\xi _{relay}^{(k-1)}+b({N_k}) \\mathbb {I}\\lbrace N_k=N_{k-1}+1 \\rbrace \\nonumber \\\\&& \\bigg ( 1-\\overline{N}U_{N_k} \\bigg ) \\bigg ]_{0}^{A_2}$ Move to next step and set $r^{\\prime }=1$ ; $r^{\\prime } \\le B-1$ and $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out}^{(k-1)} Q_{out}(r^{\\prime },\\gamma , w_{r^{\\prime }}))+\\xi _{relay}^{(k-1)} > -V^{(k-1)}(1)+V^{(k-1)}(r^{\\prime }+1)$ Do not place, and perform updates as in (REF ); Move to next step and set $r^{\\prime }=r^{\\prime }+1$ ; Place a new relay (since $r^{\\prime }=B$ ); Use power $\\arg \\min _{\\gamma \\in \\mathcal {S}}(\\gamma + \\xi _{out}^{(k-1)} Q_{out}(B,\\gamma , w_B))$ ; Do the same updates as (REF ); Move to next step and set $r^{\\prime }=1$.",
"k=k+1;.",
"OptAsYouGoAdaptiveLearning Theorem 9 Under Assumption REF , Assumption REF and under proper choice of $A_1$ and $A_2$ , we have $(\\underline{V}^{(k)}, \\xi _{out}^{(k)}, \\xi _{relay}^{(k)}) \\rightarrow \\mathcal {K}(\\overline{q},\\overline{N})$ almost surely for Algorithm REF .",
"See Appendix .",
"We complete the proof in four steps.",
"First, we show that the difference between $\\underline{V}^{(k)}$ and $\\underline{V}^*(\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ converges to 0 almost surely.",
"This proves the desired convergence in the faster timescale.",
"Next, we pose the slower timescale iteration as a projected stochastic approximation iteration (see [39]).",
"Next, we show that the slower timescale iteration satisfies some conditions given in [39] (see [39]).",
"Finally, we argue (using Theorem $5.3.1$ of [39]) that the slower timescale iterates converge to the set of stationary points of a suitable ordinary differential equation.",
"It is to be noted that while the proof to some extent follows the outline of the proof of [1], significantly new nontrivialities arise in our work as compared to the proof of [1].",
"For example, we had to prove the boundedness of the faster timescale iterates separately, since the asynchronous updates in the faster timescale do not allow us to mimic the proof of [1].",
"Similarly there are many other steps which require significant novel additional mathematical analysis compared to [1].",
"Hence, in this proof, we proved intermediate results wherever necessary, and skipped some steps if they follow from the proof of [1].",
"Choice of $A_1$ and $A_2$ : $A_1$ and $A_2$ need to be chosen carefully, otherwise the iterates $(\\xi _{out}^{(k)}, \\xi _{relay}^{(k)})$ may converge to undesired points on the boundary of $[0,A_1] \\times [0,A_2]$ .",
"In general, a stationary point on the boundary of $[0,A_1] \\times [0,A_2]$ may not correspond to a point in $\\mathcal {K}(\\overline{q},\\overline{N})$ .",
"Hence, we borrow a scheme from [1] for choosing $A_1$ and $A_2$ which ensures that, if $(\\xi _{out}^{\\prime },\\xi _{relay}^{\\prime })$ is a stationary point of the o.d.e., then $(\\underline{V}^*(\\xi _{out}^{\\prime },\\xi _{relay}^{\\prime }),\\xi _{out}^{\\prime },\\xi _{relay}^{\\prime }) \\in \\mathcal {K}(\\overline{q},\\overline{N})$ .",
"The number $A_1$ has to be chosen so large that, for all $u \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ , we will have $\\mathbb {P}(\\operatornamewithlimits{arg\\,min}_{\\gamma \\in \\mathcal {S}}(\\gamma +A_1 Q_{out}(u,\\gamma ,W))=P_M)>1-\\kappa $ for some small enough $\\kappa >0$ .",
"We also need the condition that $\\frac{\\overline{Q}_{out}^*(A_1,0)}{\\overline{U}^*(A_1,0)} \\le \\overline{q}$ .",
"The number $A_2$ has to be chosen so large that, for any $\\xi _{out} \\in [0,A_1]$ , we will have $\\overline{U}^*(\\xi _{out},A_2) > \\frac{1}{\\overline{N}}$ (when $\\frac{1}{\\overline{N}}<B$ ).",
"The numbers $A_1$ and $A_2$ have to be chosen so large that there exists at least one pair $(\\xi _{out}^{\\prime },\\xi _{relay}^{\\prime })$ for which $(\\underline{V}^*(\\xi _{out}^{\\prime },\\xi _{relay}^{\\prime }),\\xi _{out}^{\\prime },\\xi _{relay}^{\\prime }) \\in \\mathcal {K}(\\overline{q},\\overline{N})$ .$\\Box $ Discussion of Algorithm REF : Two timescales: The update scheme (REF ) is based on two-timescale stochastic approximation (see [37]).",
"Since $\\lim _{n \\rightarrow \\infty }\\frac{b(\\lfloor \\frac{n}{B} \\rfloor )}{a(n)}=0$ , we can say that $\\xi _{out}$ and $\\xi _{relay}$ are adapted in a slower timescale, and $\\underline{V}$ is updated in a faster timescale, as if $\\xi _{out}$ and $\\xi _{relay}$ are updated in a slow outer loop, and, $\\underline{V}$ is updated in an inner loop.",
"Structure of the iteration: The slower timescale iteration involves updating $\\xi _{out}$ and $\\xi _{relay}$ based on whether the corresponding constraints are violated in a link (after placing a relay); if a constraint is violated by a newly created link, then the corresponding Lagrange multiplier is increased to counterbalance it in subsequent node placements.",
"The goal is to meet both constraints with equality (if possible) in the long run.",
"Asymptotic behaviour of the iterates: If $\\overline{q}> \\frac{\\mathbb {E}_W Q_{out}(B,P_1,W)}{B}$ ; we will have $\\xi _{out}^{(k)} \\rightarrow 0$ ; here the policy places all the relays at the $B$ -th step and uses the smallest power $P_1$ at each node.",
"If the constraints are not feasible, then either $\\xi _{out}^{(k)} \\rightarrow A_1$ or $\\xi _{relay}^{(k)} \\rightarrow A_2$ or both happens.",
"Simulation results show that $\\mathcal {K}(\\overline{q},\\overline{N})$ has only one tuple in case the pair $(\\overline{q},\\overline{N})$ is feasible.$\\Box $"
],
[
"Asymptotic Performance of Algorithm ",
"Though Algorithm REF induces a nonstationary policy, Theorem REF states that the sequence of policies generated by Algorithm REF converges to the set of optimal stationary, deterministic policies for the constrained problem (REF ).",
"Let $\\pi _{oaygal}$ denote the (nonstationary) deployment policy induced by Algorithm REF .",
"Theorem 10 Under Assumption REF , Assumption REF and proper choice of $A_1$ and $A_2$ , we have: $&& \\limsup _{x \\rightarrow \\infty } \\frac{\\mathbb {E}_{\\pi _{oaygal}}\\sum _{i=1}^{N_x}\\Gamma _i}{x} = \\gamma ^* \\nonumber \\\\&& \\, \\limsup _{x \\rightarrow \\infty } \\frac{\\mathbb {E}_{\\pi _{oaygal}}\\sum _{i=1}^{N_x}Q_{out}^{(i,i-1)}}{x} \\le \\overline{q}, \\,\\,\\limsup _{x \\rightarrow \\infty } \\frac{\\mathbb {E}_{\\pi _{oaygal}}N_x}{x} \\le \\overline{N} \\nonumber \\\\$ The proof is similar to [1].",
"Figure: NO_CAPTIONFigure: Convergence speed of OptAsYouGoLearning (OAYGL) with the number of steps, kk.In the legends, “OAYG” refers to the values that are obtained if Algorithm is used; these are the target values for OptAsYouGoLearning."
],
[
"Convergence Speed of Learning Algorithms: A Simulation Study",
"In this section, we provide a simulation study for the convergence rate of Algorithm REF and Algorithm REF ."
],
[
"Parameter Values Used in the Simulation",
"For simulation, we consider a deployment environment similar to that considered in [1].",
"The details of the simulation environment are provided below.",
"We assume that deployment is done with iWiSe motes ([23]) equipped with 9 dBi antennas.",
"$\\mathcal {S}$ , the set of transmit power levels, is taken to be $\\lbrace -18,-7,-4,0,5\\rbrace $ dBm, which is a subset of available transmit power levels for iWiSe motes.",
"Under the channel model as given by (REF ), our measurements in a forest-like environment gave $\\eta =4.7$ and $c=10^{0.17}$ (i.e., $1.7$ dB); the experimental details can be found in [10].",
"From the statistical analysis of the measurement data, we also showed that shadowing $W$ follows log-normal distribution in such a forest-like environment; $W=10^{\\frac{Y}{10}}$ with $Y \\sim \\mathcal {N}(0, \\sigma ^2)$ , where $\\sigma =7.7$ dB was obtained from our data analysis.",
"Shadowing decorrelation distance was calculated as 6 meters; hence we consider deployment with $\\delta =20$ meter.",
"The fading turned out to be Rayleigh fading.",
"Outage is defined to be the event when a packet is received at a power level below $P_{rcv-min} = 10^{-9.7}$ mW (i.e., $-97$ dBm); for a commercial implementation of IEEE $802.15.4$ , received power $-97$ dBm results in a $2\\%$ packet loss probability for 127 byte packets for iWiSe motes (obtained from measurements).",
"We choose $B$ in the following way.",
"We define a link to be workable if it has an outage probability less than $3\\%$ .",
"$B$ is chosen to be the largest integer such that the probability of finding a workable link of length $B \\delta $ is greater than $20\\%$ , under 5 dBm transmit power.",
"For the parameters $\\eta =4.7$ and $\\sigma =7.7$ dB, and 5 dBm transmit power, $B$ turned out to be 5.",
"It is important to note that, the radio propagation parameters (e.g., $\\eta $ and $\\sigma $ ) and modeling assumptions (e.g., log-normal shadowing) are obtained and validated using field data collected via extensive measurements in a forest-like environment; the details of these experiments can be found in [10].",
"Hence, in this paper, we evaluate our algorithms only via MATLAB simulation of an environment that has radio propagation model and parameters obtained from experiments in [10].",
"This is done by generating random channel gains in MATLAB, for the wireless links that need to be measured in course of the deployment process.",
"The performance variation of OptAsYouGo algorithm with $(\\xi _{out},\\xi _{relay})$ has been demonstrated numerically in [1], which comply with Theorem REF and Theorem REF ."
],
[
"OptAsYouGoLearning for Given Multipliers",
"Here we study the rate of convergence for OptAsYouGoLearning with $\\xi _{out}=125$ , $\\xi _{relay}=2$ .",
"Let us assume that the propagation environment, in which deployment is being carried out, is characterized by the parameters given in Section REF (i.e., $\\eta =4.7$ , $\\sigma =7.7$ dB etc.).",
"The optimal average cost per step, under these parameter values, is $\\lambda ^*=V^*(1)=1.85$ (computed numerically).These values of $\\xi _{out}$ and $\\xi _{relay}$ are chosen because they can produce reasonable values of placement rate, mean power per step and mean outage per step, which can be used in practical networks.",
"However, these values are chosen only for illustration purposes, and the choice will vary depending on the requirement for deployment.",
"We numerically study the performance of the following three types of algorithms: (i) $\\eta $ and $\\sigma $ are known prior to deployment (the agent uses the fixed optimal policy with $\\xi _{relay}=2$ and $\\xi _{out}=125$ in this case), (ii) the agent has imperfect estimates of $\\eta $ and $\\sigma $ deployment, and OptAsYouGoLearning is used to update the policy as deployment progresses, and (iii) the agent has imperfect estimates of $\\eta $ and $\\sigma $ deployment, but the corresponding suboptimal policy is used along the infinite line without any update.",
"We use the abbreviations OAYGL and OAYG for OptAsYouGoLearning and Optimal Algorithm for As-You-Go deployment (i.e., Algorithm REF ), respectively.",
"Also, following the terminology in [1], we use the abbreviation FPWU for “Fixed Policy without Update.” Next, we formally explain the various cases considered in our simulations: OAYG: Here the agent knows $\\eta =4.7$ , $\\sigma =7.7$ dB prior to deployment, and uses Algorithm REF with $\\xi _{out}=125$ , $\\xi _{relay}=2$ .",
"OAYGL Case 1: Here the true $\\eta =4.7$ and $\\sigma =7.7$ dB are unknown to the deployment agent.",
"But the agent has an initial estimate $\\eta =5$ , $\\sigma =8$ dB.",
"Hence, he starts deploying using a $\\underline{V}^{(0)}$ which is optimal for these imperfect estimates of $\\eta $ and $\\sigma $ , and $\\xi _{out}=125$ , $\\xi _{relay}=2$ .",
"He updates the policy using the OptAsYouGoLearning algorithm as deployment progresses.",
"OAYGL Case 2: This is different from OAYGL Case 1 only in the aspect that here deployment starts with the optimal policy for $\\eta =4$ , $\\sigma =7$ dB.",
"FPWU Case 1: Here the true $\\eta $ and $\\sigma $ are unknown prior to deployment, and the agent has an initial estimate $\\eta =5$ , $\\sigma =8$ dB.",
"The agent computes $\\underline{V}^*$ for these imperfect initial estimates and $\\xi _{out}=125$ , $\\xi _{relay}=2$ , and uses this policy throughout the deployment process without any update.",
"This case will demonstrate the gain in performance by updating the policy under OptAsYouGoLearning, w.r.t.",
"the case where the suboptimal policy is used throughout the deployment process.",
"FPWU Case 2: It differs from FPWU Case 1 only in the aspect that here the agent has initial estimates $\\eta =4$ , $\\sigma =7$ dB.",
"For simulation of OAYGL, we chose $a(k)=\\frac{120}{k}$ .",
"We simulated 2000 independent network deployments (i.e., 2000 sample paths of the deployment process) with OptAsYouGoLearning, and estimated (by averaging over 2000 deployments) the expectation of $V^{(k)}(1)$ , mean power per step (i.e., $\\frac{\\sum _{j=1}^{N_k} \\Gamma _j}{k}$ ), mean outage per step (i.e., $\\frac{\\sum _{j=1}^{N_k} Q_{out}^{(j,j-1)}}{k}$ ) and mean placement distance (i.e., $\\frac{k}{N_k}$ ), in the part of the network between the sink node to the $k$ -th step.",
"The results are summarized in Figure REF .",
"Asymptotically the estimates are supposed to converge to the values provided by OAYG.",
"Observations: We observe that the estimate of $\\mathbb {E} (V^{(k)}(1))$ approaches the optimal cost $\\lambda ^*=V^*(1)=1.85$ (for the actual propagation parameters), as $k$ increases, and gets to within $10\\%$ of the optimal cost by the time where $k=35$ to 40 (within a distance of 800 meters), while starting with two widely different initial guesses of the propagation parameters.",
"The estimates of mean power per step, mean outage per step and mean placement distance also converges very fast to the corresponding values achieved by OAYG.",
"It also shows that, if the performance of the initial imperfect policy (FPWU) is significantly different than that of OAYG, then OptAsYouGoLearning will provide closer performance to OAYG, as compared to FPWU (see the mean placement distance plot).",
"Note that, even though Theorem REF guarantees almost sure convergence, the convergence speed will vary across sample paths.",
"But here we demonstrate speed of convergence after averaging over 2000 sample paths.",
"Figure: Convergence speed of OptAsYouGoAdaptiveLearning (OAYGAL) with the number of steps, kk.In the legends, “OAYG” refers to the values that are obtained if Algorithm is used;these are the target values for OptAsYouGoAdaptiveLearning.",
"Evolution of ξ out (k) \\xi _{out}^{(k)} and ξ relay (k) \\xi _{relay}^{(k)} are shownfor a longer time, since they converge slowly to their respective target values."
],
[
"OptAsYouGoAdaptiveLearning",
"Now we will demonstrate the performance of OptAsYouGoAdaptiveLearning (Algorithm REF ) for deployment over a finite distance under an unknown propagation environment.",
"We again assume that the true propagation parameters are given by $\\eta =4.7$ , $\\sigma =7.7$ dB.",
"For these parameters, under the choice $\\xi _{relay}=2$ and $\\xi _{out}=125$ , the optimal average cost per step will be $\\lambda ^*=1.85$ , which can be achieved by OAYG (Algorithm REF ).",
"OAYG in this case will yield a mean placement distance of $2.285$ steps, a mean outage per step of $\\frac{0.0101}{2.285}=0.0044$ , and a mean power per step of $0.423$ mW.",
"Now, let us suppose that we need to solve the constrained problem in (REF ) with the targets $\\overline{q}=0.0044$ and $\\overline{N}=\\frac{1}{2.285}$ , but the true $\\eta $ and $\\sigma $ of the environment are unknown to us.",
"Hence, we need to employ OptAsYouGoAdaptiveLearning (we use the abbreviation OAYGAL for it); as compared to OptAsYouGoLearning, we need to make an additional choice of $\\xi _{out}^{(0)}$ and $\\xi _{relay}^{(0)}$ .",
"We consider the following cases in our simulations: OAYG: This is same as in Section REF OAYGAL Case 1: Here the true $\\eta =4.7$ and $\\sigma =7.7$ dB are unknown to the deployment agent.",
"But the agent has an initial estimate $\\eta =5$ , $\\sigma =8$ dB.",
"Hence, he starts deploying using a $\\underline{V}^{(0)}$ which is optimal for these imperfect estimates of $\\eta $ and $\\sigma $ , and $\\xi _{out}^{(0)}=100$ , $\\xi _{relay}^{(0)}=3$ .",
"He updates the policy using the OptAsYouGoAdaptiveLearning algorithm as deployment progresses.",
"OAYGAL Case 2: This is same as OAYGAL Case 1, except that the agent starts deploying using a policy corresponding to the wrong initial estimate $\\eta =4$ , $\\sigma =7$ dB (under $\\xi _{out}^{(0)}=100$ , $\\xi _{relay}^{(0)}=3$ ).",
"FPWU Case 3: Here the agent uses $\\xi _{out}=100$ , $\\xi _{relay}=3$ , and uses the corresponding optimal policy for the imperfect estimates $\\eta =5$ , $\\sigma =8$ dB, throughout the deployment process.",
"FPWU Case 4: This is similar to FPWU Case 3; the only difference is that the optimal policy for the imperfect estimates $\\eta =4$ , $\\sigma =7$ dB is used throughout deployment.",
"For simulation of OAYGAL, we chose the step sizes as follows.",
"We took $a(k)=\\frac{1}{k^{0.55}}$ , $b(k)=\\frac{100}{k^{0.8}}$ for the $\\xi _{out}$ update and $b(k)=\\frac{1}{k^{0.8}}$ for the $\\xi _{relay}$ update (however, both $\\xi _{out}$ and $\\xi _{relay}$ are updated in the same timescale).",
"We simulated 2000 independent network deployments (i.e., 2000 sample paths of the deployment process) with OptAsYouGoLearning, and estimated (by averaging over 2000 deployments) the expectations of $V^{(k)}(1)$ , mean power per step, mean outage per step mean placement distance, $\\xi _{out}^{(k)}$ and $\\xi _{relay}^{(k)}$ , in the part of the network between the sink node to the $k$ -th step.",
"The results are summarized in Figure REF (see previous page).",
"Observations: Under OAYGAL Case 1 the estimates of the expectations of $V^{(2000)}(1)$ , $\\xi _{out}^{(2000)}$ , $\\xi _{relay}^{(2000)}$ , mean power per step up to the 2000th step, mean outage per step up to the 2000th step, and mean placement distance over 2000 steps are $1.8479$ , $124.89$ , $2.01$ , $0.4222$ , $0.04403$ and $2.2852$ , whereas the corresponding target values are $1.85$ , 125, 2, $0.4223$ , $0.00441$ and $2.2857$ , respectively.",
"Similarly, for OAYGAL Case 2 also, the quantities converge close to the target values.",
"In practice, the performance metrics are reasonably close to their respective target values within 100 steps (i.e., 2 kms).",
"FPWU Case 3 and FPWU Case 4 either violate some constraint or uses significantly higher per-step power compared to OAYG.",
"But, by using OptAsYouGoAdaptiveLearning, we can achieve mean power per step close to the optimal while (possibly) violating the constraints by small amount.",
"However, performance of OAYGAL is significantly closer to the target compared to FPWU.$\\Box $ The speed of convergence will depend on the choice of $a(k)$ and $b(k)$ , of $\\xi _{out}^{(0)}$ , $\\xi _{relay}^{(0)}$ and the initial estimates of $\\eta $ and $\\sigma $ .",
"However, optimizing convergence speed over step size sequences is left for future research."
],
[
"Conclusion",
"In this paper, we have formulated the problem of pure-as-you-go deployment along a line, under a very light traffic assumption.",
"The problem was formulated as an average cost MDP, and its optimal policy structure was studied analytically.",
"We also proposed two learning algorithms that asymptotically converge to the corresponding optimal policies.",
"Numerical results have been provided to illustrate the speed of convergence of the learning algorithms.",
"While this paper provides an interesting set of results, it can be extended or modified in several ways: (i) One can attempt to develop deployment algorithms for 2 dimensional regions, where multiple agents cooperate to carry out the deployment.",
"(ii) One can also attempt to develop deployment algorithms that can provide theoretical guarantees on the data rate supported by the deployed networks (instead of assuming that the traffic is lone packet).",
"(iii) The optimization of the rate of convergence for the learning algorithms by proper choice of the step sizes is also a challenging problem.",
"We leave these issues for future research endeavours.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION Supplementary Material Title: “Asynchronous Stochastic Approximation Based Learning Algorithms for As-You-Go Deployment of Wireless Relay Networks along a Line” Authors: Arpan Chattopadhyay, Avishek Ghosh, anurag Kumar Formulation for known propagation parameters Proof of Theorem REF : From (REF ), $V(B)$ is unique for fixed $\\xi _{out}$ and $\\xi _{relay}$ .",
"Hence, we can say that $V(B)$ is a continuous and decreasing function of $V(1)$ .",
"Now, let us assume that $V(r+1)$ is continuous and decreasing in $V(1)$ for some $r, 1 \\le r \\le B-1$ .",
"Let us recall (REF ) for $V(r)$ .",
"Since $V(r+1)$ is continuous and decreasing in $V(1)$ by our induction hypothesis, it is evident from (REF ) that $V(r)$ is also continuous and decreasing in $V(1)$ .",
"Proceeding in this way, we can write $V(1)=\\phi (V(1))$ where $\\phi (\\cdot )$ is continuous and decreasing in $V(1)$ .",
"But $V(1)$ is continuous and strictly increasing in $V(1)$ .",
"Hence, $V(1)=\\phi (V(1))$ has a unique fixed point $V^*(1)$ .",
"Now, from (REF ), $V(B-1)$ is unique since $V(1)=V^*(1)$ is unique and $V(B)$ is unique.",
"Proceeding backwards in this way, we can show that we have a unique $V^*(r)$ for all $r$ .",
"Now, from (REF ), we find that $V^*(r) \\le -V^*(1)+V^*(r+1)$ , i.e., $V^*(r+1) \\ge V^*(r)+V^*(1)$ for all $r \\in \\lbrace 1,2,\\cdots ,B-1\\rbrace $ .",
"Also, $V^*(1)=\\lambda ^*>0$ and it is unique.",
"This proves the second part of the theorem.",
"$\\Box $ Proof of Theorem REF : Let us denote the mean power per link, mean outage per link and mean placement distance (in steps) under a stationary policy $\\pi $ by $\\overline{\\Gamma }_{\\pi }$ , $\\overline{Q}_{out,\\pi }$ and $\\overline{U}_{\\pi }$ .",
"Then, by Renewal-Reward Theorem, we have $\\lambda ^{*}(\\xi _{out},\\xi _{relay}) =\\inf _{\\pi }\\frac{\\Gamma _{\\pi }+\\xi _{out}\\overline{Q}_{out,\\pi }+\\xi _{relay}}{\\overline{U}_{\\pi }}$ .",
"The numerator is affine and increasing in $\\xi _{out}$ and $\\xi _{relay}$ , and the denominator is independent of $\\xi _{out}$ and $\\xi _{relay}$ .",
"Hence, $\\lambda ^{*}(\\xi _{out},\\xi _{relay})$ is concave, increasing in $\\xi _{out}$ and $\\xi _{relay}$ , since the pointwise infimum of increasing affine functions of $(\\xi _{out},\\xi _{relay})$ is increasing and jointly concave in $(\\xi _{out},\\xi _{relay})$ .",
"Now, any increasing, concave function is continuous.",
"Hence, $\\lambda ^{*}(\\xi _{out},\\xi _{relay})$ is continuous in $(\\xi _{out},\\xi _{relay})$ .",
"Also, it is easy to see that $\\lambda ^{*}(\\xi _{out},\\xi _{relay})$ is Lipschitz in each argument with Lipschitz constant 1.",
"Proof of Theorem REF : By Theorem REF , $V^*(1):=\\lambda ^*$ is Lipschitz continuous in $(\\xi _{out}, \\xi _{relay})$ .",
"By (REF ), $V^*(B)$ is Lipschitz continuous in $(\\xi _{out},\\xi _{relay})$ .",
"Hence, by (REF ), $V^*(B-1)$ is also Lipschitz continuous in $(\\xi _{out}, \\xi _{relay})$ .",
"Thus, by using backward induction, we can show that $V^*(r)$ is Lipschitz continuous in $(\\xi _{out}, \\xi _{relay})$ for all $1 \\le r \\le B$ .",
"OptAsYouGoLearning: Learning with Pure As-You-Go Deployment, for Given Lagrange Multipliers Proof of Theorem REF : We can rewrite (REF ) as follows: $V^{(k)}(r)&=&V^{(k-1)}(r)+ a(\\nu (r,k)) \\mathbb {I}\\lbrace r \\in \\mathcal {I}_k\\rbrace \\bigg [ f_r(\\underline{V}^{(k-1)})+ M_k(r) \\bigg ] \\nonumber \\\\&& $ where, for all $1 \\le r \\le B-1$ $f_r(\\underline{V}^{(k-1)})&=&\\mathbb {E}_W \\bigg [ \\min \\bigg \\lbrace \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , W))+\\xi _{relay}, \\\\&& -V^{(k-1)}(1)+V^{(k-1)}(r+1) \\bigg \\rbrace -V^{(k-1)}(r) \\bigg ]$ $M_k(r)&=& \\bigg [\\min \\bigg \\lbrace \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , w_r))+\\xi _{relay}, \\\\&& -V^{(k-1)}(1)+V^{(k-1)}(r+1) \\bigg \\rbrace -V^{(k-1)}(r) \\bigg ]-f_r(\\underline{V}^{(k-1)})$ and $f_B(\\underline{V}^{(k-1)})=\\mathbb {E}_W \\bigg [ \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , W)) +\\xi _{relay} -V^{(k-1)}(B) \\bigg ]$ $M_k (B)&=&\\bigg [ \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , w_B)) +\\xi _{relay}-V^{(k-1)}(B) \\bigg ] \\\\&& -f_B(\\underline{V}^{(k-1)})$ Let $\\underline{M}_k:=(M_k(1), \\cdots , M_k(B))$ .",
"Let us denote the $\\sigma $ -field $\\mathcal {F}_k:=\\sigma (\\underline{V}_i, \\mathcal {I}_i, \\underline{M}_i, i \\le k-1)$ ; it is the information available to the deployment agent before making any decision at the $k$ -th step.",
"Clearly, the update equations fall under the category of Asynchronous Stochastic Approximation algorithms (see [36]).",
"In order to see whether $\\underline{V}^{(k)} \\rightarrow \\underline{V}^*$ almost surely, we will first check whether the five assumptions mentioned in [36] are satisfied.",
"Checking Assumption 1 of [36]: For each $r, 1 \\le r \\le B$ , $V(r)$ gets updated at least once in every $B$ steps.",
"Hence, $\\lim \\inf _{k \\rightarrow \\infty }\\frac{\\nu (r,k)}{k} \\ge \\frac{1}{B}>0 $ almost surely.",
"Hence, the assumption is satisfied.",
"Checking Assumption 2 of [36]: If we choose $\\lbrace a(k)\\rbrace _{k \\ge 1}$ to be a bounded, decreasing sequence with $\\sum _k a(k)=\\infty $ and $\\sum _k a^2(k) < \\infty $ , this condition will be satisfied.",
"Checking Assumption 3 of [36]: Not applicable to our problem since before updating $\\underline{V}^{(k)}$ the deployment agent knows $\\underline{V}^{(k-1)}$ .",
"Before checking the other two conditions, we will establish a lemma.",
"Let us consider the following system of o.d.e-s: $\\dot{V}_t (r) = \\kappa _t(r) f_r(\\underline{V}_t) \\,\\,\\, \\forall r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ where $\\kappa _t(r) \\in (0,1]$ for all $r$ and $t$ .",
"By Theorem REF , this system of o.d.e-s has an unique stationary point $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ .",
"Lemma 1 $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ is a globally asymptotically stable equilibrium for the system of o.d.e-s (REF ).",
"Also, $\\underline{V}=0$ is a globally asymptotically stable equilibrium for (REF ) when $\\gamma $ , $\\xi _{out}$ and $\\xi _{relay}$ are replaced by 0 in the definition of $f_r(\\underline{V})$ for all $r \\in \\lbrace 1,2,\\cdots ,B \\rbrace $ .",
"Note that, by Theorem REF , $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ is the unique stationary point for (REF ).",
"Now, the proof for this lemma follows from similar line of arguments as in the appendix of [40] (which uses results from [41] and [42]).",
"Checking Assumption 4 of [36]: It is easy to see that $f_r (\\underline{V})$ is Lipschitz in $\\underline{V}$ for each $r$ ; this satisfies Assumption 4(i).",
"Let us consider the ODE (REF ) with $0<\\kappa _t(r) \\le 1$ corresponds to the relative rate at which $V(r)$ is updated.",
"By Lemma REF , $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ is a globally asymptotically stable equilibrium for the system of o.d.e-s (REF ).",
"Hence, Assumption 4(ii) is satisfied.",
"Consider the functions $\\frac{f_r(c\\underline{V})}{c}, c \\ge 1$ for all $r$ .",
"Clearly, $\\lim _{c \\rightarrow \\infty } \\frac{f_r(c\\underline{V})}{c}=\\min \\lbrace 0,-V(1)+V(r+1)\\rbrace -V(r)$ for $r \\ne B$ , and $\\lim _{c \\rightarrow \\infty } \\frac{f_B(c\\underline{V})}{c}=-V(B)$ .",
"Note that $\\frac{f_r(c\\underline{V})}{c}$ for all $r$ and $\\lim _{c \\rightarrow \\infty } \\frac{f_r(c\\underline{V})}{c}$ all are continuous in $\\underline{V}$ , and $\\frac{f_r(c\\underline{V})}{c}$ is decreasing in $c$ .",
"Hence, by Theorem $7.13$ of [43], convergence of $\\frac{f_r(c\\underline{V})}{c}$ over compacts is uniform.",
"Hence, Assumption 4(iii) is satisfied.",
"Consider the ODE: $\\dot{V}_t(r)=\\kappa _t(r)(\\min \\lbrace 0,-V_t(1)+V_t(r+1)\\rbrace -V_t(r))$ for $r \\ne B$ and $\\dot{V}_t(B)=\\kappa _B(t)(-V_t(B))$ .",
"Clearly, by the second part of Lemma REF , there is a unique globally asymptotically stable equilibrium $\\underline{V}=\\underline{0}$ .",
"Hence, Assumption 4(iv) is satisfied.",
"Checking Assumption 5 of [36]: It is easy to see that, $\\lbrace \\underline{M}_k\\rbrace _{k \\ge 1}$ is a Martingale difference sequence adapted to $\\mathcal {F}_k$ .",
"Hence, Assumption 5(i) is satisfied.",
"Now, $|M_{k+1}(r)| & \\le & 2 \\bigg | \\bigg (\\min \\lbrace P_M+\\xi _{out}+\\xi _{relay},-V^{(k)}(1) \\\\&& +V^{(k)}(r+1)\\rbrace -V^{(k)}(r) \\bigg ) \\bigg |$ and $|M_{k+1}(B)| \\le \\bigg | \\bigg (P_M+\\xi _{out}+\\xi _{relay} -V^{(k)}(B) \\bigg ) \\bigg |$ Hence, $||M_{k+1}|| \\le C_0 (1+||\\underline{V}^{(k)}||)$ for some $C_0>0$ .",
"Hence, Assumption 5(ii) is satisfied.",
"Now, by [36], $\\underline{V}^{(k)} \\rightarrow \\underline{V}^*$ .$\\Box $ OptAsYouGoAdaptiveLearning with Constraints on Outage Probability and Relay Placement Rate Proof of Theorem REF Let us denote by $g(r, \\gamma ),r \\in \\lbrace 1,2,\\cdots ,B\\rbrace , \\gamma \\in \\mathcal {S}$ the joint distribution of $(U_k, \\Gamma _k)$ under Algorithm REF .",
"For the time being, let us assume that $g(r,\\gamma )$ is continuous in $(\\xi _{out},\\xi _{relay})$ .",
"Then, the mean placement distance $\\overline{U}^*(\\xi _{out},\\xi _{relay})= \\sum _{r=1}^B \\sum _{\\gamma \\in \\mathcal {S}} r g (r,\\gamma )$ , and the mean power per link $\\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})=\\sum _{r=1}^B\\sum _{\\gamma \\in \\mathcal {S}} \\gamma g (r,\\gamma )$ are both continuous in $(\\xi _{out},\\xi _{relay})$ .",
"Now, by Renewal-Reward Theorem, $\\lambda ^*(\\xi _{out}, \\xi _{relay}) = \\frac{ \\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})+\\xi _{out}\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})+\\xi _{relay} }{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ Since $\\lambda ^*(\\xi _{out}, \\xi _{relay})$ is continuous in $(\\xi _{out},\\xi _{relay})$ (by Theorem REF ), we conclude that $\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})$ is continuous in $\\xi _{out}$ and $\\xi _{relay}$ .",
"Hence, $\\frac{\\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ , $\\frac{\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ and $\\frac{1}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ are continuous in $(\\xi _{out},\\xi _{relay})$ .",
"$\\Box $ Now, the proof of the theorem is completed by the following lemma.",
"Lemma 2 Under Assumption REF , $g(r,\\gamma )$ is continuous in $(\\xi _{out},\\xi _{relay})$ .",
"We will first prove the result for $r \\le B-1$ .",
"Let us fix an $r \\in \\lbrace 1,\\cdots ,B-1\\rbrace $ and any $\\gamma \\in \\mathcal {S}$ .",
"We will only show that $g(r,\\gamma )$ is continuous in $\\xi _{out}$ ; the proof for continuity of $g(r,\\gamma )$ w.r.t.",
"$\\xi _{relay}$ will be similar.",
"Let us consider a sequence $\\lbrace \\xi _n\\rbrace _{n \\ge 1}$ such that $\\xi _n \\rightarrow \\xi _{out}$ .",
"Let us denote the joint probability distribution of $(U_k, \\Gamma _k)$ by $g_n(r,\\gamma )$ , if Algorithm REF is used with $\\xi _n$ as the cost for unit outage.",
"We will show that $\\lim _{n \\rightarrow \\infty } g_n(r,\\gamma ) \\rightarrow g(r,\\gamma )$ .",
"Define the sets $\\mathcal {E}_{r,\\gamma ^{\\prime }}=\\bigg \\lbrace w_r: \\gamma +\\xi _{out}Q_{out}(r,\\gamma ,w_r) < \\gamma ^{\\prime }+\\xi _{out}Q_{out}(r,\\gamma ^{\\prime },w_r) \\bigg \\rbrace $ and $\\mathcal {E}_u=\\bigg \\lbrace w_u: \\min _{\\gamma \\in \\mathcal {S}}(\\gamma +\\xi _{out}Q_{out}(u,\\gamma ,w_u))>-\\xi _{relay}-V^*(1)+V^*(u+1) \\bigg \\rbrace $ for all $1 \\le u \\le r$ .",
"Let us define $\\mathcal {E}=\\cap _{\\gamma ^{\\prime } \\ne \\gamma }\\mathcal {E}_{r,\\gamma ^{\\prime }}\\cap _{u \\le r-1} \\mathcal {E}_u \\cap \\overline{\\mathcal {E}_r}$ , where $\\overline{\\mathcal {E}_r}$ is the set complement of $\\mathcal {E}_r$ .",
"Now, $g(r,\\gamma )=\\mathbb {P}(\\mathcal {E})=\\mathbb {E}(\\mathbb {I}_{\\mathcal {E}})$ , where $\\mathbb {I}$ denotes the indicator function.",
"The expectation is over the joint distribution of $(W_1,W_2, \\cdots , W_r)$ (shadowing random variables from $r$ locations).",
"Now, for any $\\gamma ^{\\prime } \\ne \\gamma $ , we have $\\mathbb {P}\\bigg ( \\gamma +\\xi _{out}Q_{out}(r,\\gamma ,W_r) = \\gamma ^{\\prime }+\\xi _{out}Q_{out}(r,\\gamma ^{\\prime },W_r) \\bigg )=0$ , and $\\mathbb {P}\\bigg ( \\min _{\\gamma \\in \\mathcal {S}}(\\gamma +\\xi _{out}Q_{out}(u,\\gamma ,W_u))=-\\xi _{relay}-V^*(1)+V^*(u+1) \\bigg )=0$ for all $u \\le r$ ; these two assertions follow from Assumption REF and from the continuity of $Q_{out}(r,\\gamma ,w)$ in $w$ .",
"Hence, we can safely assume the following: $\\overline{\\mathcal {E}_{r,\\gamma ^{\\prime }}}$ has the same expression as $\\mathcal {E}_{r,\\gamma ^{\\prime }}$ except that the $<$ sign is replaced by $>$ sign.",
"$\\overline{\\mathcal {E}_u}$ has the same expression as $\\mathcal {E}_u$ except that the $>$ sign is replaced by $<$ sign.",
"Let $\\mathcal {E}_{r,\\gamma ^{\\prime }}^{(n)}$ , $\\mathcal {E}_u^{(n)}$ and $\\mathcal {E}^{(n)}$ be the sets obtained by replacing $\\xi _{out}$ by $\\xi _n$ in the expressions of the sets $\\mathcal {E}_{r,\\gamma ^{\\prime }}$ , $\\mathcal {E}_u$ and $\\mathcal {E}$ respectively (also $\\underline{V}^*$ has to be replaced by the corresponding optimal $\\underline{V}^{(n,*)}$ ).",
"Clearly, we can make similar claims for $\\mathcal {E}_{r,\\gamma ^{\\prime }}^{(n)}$ , $\\mathcal {E}_u^{(n)}$ .",
"Now, if we can show that $\\mathbb {E}(\\mathbb {I}_{\\mathcal {E}^{(n)}}) \\rightarrow \\mathbb {E}(\\mathbb {I}_{\\mathcal {E}})$ , the lemma will be proved, because $g(r,\\gamma )=\\mathbb {P}(\\mathcal {E})=\\mathbb {E}(\\mathbb {I}_{\\mathcal {E}})$ .",
"Claim 1 $\\lim _{n \\rightarrow \\infty }\\mathbb {I}_{ \\mathcal {E}_u^{(n)} } \\rightarrow \\mathbb {I}_{ \\mathcal {E}_u} $ , and $\\lim _{n \\rightarrow \\infty } \\mathbb {I}_{ \\mathcal {E}_{r,\\gamma ^{\\prime }}^{(n)} } \\rightarrow \\mathbb {I}_{ \\mathcal {E}_{r,\\gamma ^{\\prime }} } $ almost surely, for $\\gamma ^{\\prime } \\ne \\gamma $ .",
"Suppose that, for some value of $w_u$ , $\\mathbb {I}_{ \\mathcal {E}_u}(w_u)=1$ , i.e., $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma +\\xi _{out}Q_{out}(u,\\gamma ,w_u))>-\\xi _{relay}-V^*(1)+V^*(u+1)$ .",
"Now, $V^*(1)$ and $V^*(u+1)$ are continuous in $(\\xi _{out},\\xi _{relay})$ for all $1 \\le u \\le r$ (see Theorem $\\ref {theorem:V-continuous-in-xi}$ ).",
"Hence, there exists an integer $n_0$ large enough, such that for all $n > n_0$ , we have $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma +\\xi _n Q_{out}(u,\\gamma ,w_u))>-\\xi _{relay}-\\bigg ( V^{(n,*)}(1)+V^{(n,*)}(u+1) \\bigg )\\bigg |_{\\xi _{out}=\\xi _n}$ , i.e., $\\mathbb {I}_{ \\mathcal {E}_u^{(n)} } (w_u)=1$ for all $n > n_0$ .",
"Hence, $\\mathbb {I}_{ \\mathcal {E}_u^{(n)} } (w_u) \\rightarrow \\mathbb {I}_{ \\mathcal {E}_u }(w_u) $ if $\\mathbb {I}_{ \\mathcal {E}_u }(w_u)=1$ .",
"For the case $\\mathbb {I}_{ \\mathcal {E}_u } (w_u)=0$ , we can have similar arguments.",
"This proves the first part of the claim, and second part can be proved by similar arguments.",
"Now, $\\mathbb {I}_{\\mathcal {E}^{(n)}}=\\prod _{\\gamma ^{\\prime } \\ne \\gamma } \\mathbb {I}_{\\mathcal {E}_{r,\\gamma ^{\\prime }}^{(n)}}\\prod _{u \\le r-1} \\mathbb {I}_{\\mathcal {E}_u^{(n)}} \\times \\mathbb {I}_{\\overline{\\mathcal {E}_r^{(n)}}}$ .",
"By Claim REF , $\\mathbb {I}_{\\mathcal {E}^{(n)}} \\rightarrow \\mathbb {I}_{\\mathcal {E}}$ almost surely as $n \\rightarrow \\infty $ .",
"Hence, by Dominated Convergence Theorem, we have $\\mathbb {E}(\\mathbb {I}_{\\mathcal {E}^{(n)}}) \\rightarrow \\mathbb {E}(\\mathbb {I}_{\\mathcal {E}})$ .",
"We can prove the same statement for $r=B$ in a similar method; but we need to define $\\mathcal {E}=\\cap _{\\gamma ^{\\prime } \\ne \\gamma }\\mathcal {E}_{B,\\gamma ^{\\prime }}\\cap _{u \\le B-1} \\mathcal {E}_u $ .",
"Hence, the lemma is proved.",
"Figure: NO_CAPTION Proof of Theorem REF We denote the shadowing in the link between the potential locations located at distances $i \\delta $ and $j \\delta $ from the sink node, by the random variable $W_{i,j}$ .",
"The sample space $\\Omega $ is defined to be the collection of all $\\omega $ such that each $\\omega $ corresponds to a fixed realization $\\lbrace w_{i,j}: i \\ge 0, j \\ge 0, i>j, 1 \\le i-j \\le B \\rbrace $ of shadowing that could be encountered in the deployment process over infinite horizon.",
"Let $\\mathcal {F}$ be the Borel $\\sigma $ -algebra on $\\Omega $ .",
"We also define a sequence of sub-$\\sigma $ fields $\\mathcal {F}_k:=\\sigma \\bigg (W_{i,j}: i \\ge 0, j \\ge 0, k \\ge i>j, 1 \\le i-j \\le B \\bigg )$ ; $\\mathcal {F}_k$ is increasing in $k$ , and captures the history of the deployment process up to $k \\delta $ distance.",
"Let us recall the outline of the proof of Theorem REF in Section REF .",
"The Faster Time-Scale Iteration of $\\underline{V}^{(k)}$ Let us denote by $\\underline{V}^*(\\xi _{out}, \\xi _{relay})$ the value of $\\underline{V}^*$ , for given $\\xi _{out}$ and $\\xi _{relay}$ .",
"Let us also define $\\overline{a}(k):=\\max _{r \\in \\mathcal {I}_k} a ( \\nu (r,k) )$ .",
"Using the first order Taylor series expansion of the function $\\Lambda _{[0,A_1]}(\\cdot )$ , and using the fact that $\\Lambda _{[0,A_1]}(\\xi _{out}^{(k-1)})=\\xi _{out}^{(k-1)}$ (since $\\xi _{out}^{(k-1)} \\in [0,A_1]$ ), we rewrite the update equation (REF ) as (REF ).",
"Now, for the update equation for $\\xi _{relay}$ in (REF ), we can write: $&& \\lim _{\\beta \\downarrow 0} \\frac{\\Lambda _{[0,A_2]}\\bigg (\\xi _{relay}^{(k-1)} + \\beta (1-\\overline{N}U_{N_k}) \\bigg )-\\xi _{relay}^{(k-1)}}{\\beta } \\\\&=& (1-\\overline{N}U_{N_k}) \\mathbb {I} \\lbrace 0< \\xi _{relay}^{(k-1)} <A_2\\rbrace \\\\&+& (1-\\overline{N}U_{N_k})^+ \\mathbb {I} \\lbrace \\xi _{relay}^{(k-1)} =0 \\rbrace \\\\&-& (1-\\overline{N}U_{N_k})^- \\mathbb {I} \\lbrace \\xi _{relay}^{(k-1)} =A_2 \\rbrace \\\\$ where $y^+=\\max \\lbrace y,0\\rbrace $ and $y^-=-\\min \\lbrace y,0\\rbrace $ .",
"We can write similar expression for the $\\xi _{out}^{(k)}$ update.",
"Since outage probabilities and placement distances are bounded quantities, and since $N_k \\ge \\lfloor \\frac{k}{B} \\rfloor $ and $\\lim _{k \\rightarrow 0}\\frac{b(\\lfloor \\frac{k}{B} \\rfloor )}{\\overline{a}(k)}=0$ , we have: $&& \\lim _{k \\rightarrow \\infty } \\bigg (\\frac{b(N_k)}{\\overline{a}(k)} \\bigg ( \\lim _{\\beta \\downarrow 0} \\bigg (\\Lambda _{[0,A_1]}\\bigg (\\xi _{out}^{(k-1)}+ \\beta (Q_{out}^{(N_k,N_{k-1})} \\\\&& -\\overline{q}U_{N_k}) \\bigg ) -\\xi _{out}^{(k-1)}\\bigg )/ \\beta + \\frac{o(b(N_k))}{b(N_k)} \\bigg ) \\bigg )=0$ Similar claim can be made for $\\xi _{relay}$ update.",
"Lemma 3 Under Algorithm REF , the faster timescale iterates $\\lbrace \\underline{V}^{(k)}\\rbrace _{k \\ge 1}$ are almost surely bounded.",
"Note that, (REF ) combines the faster and slower timescale iterations in a single timescale where the step size is $\\overline{a}(n)$ .",
"We will now use the theory from [44] to prove this lemma.",
"Note that, the R.H.S.",
"of the faster timescale iteration in (REF ) is Lipschitz continuous in both faster and slower timescale iterates.",
"Hence, the first part of [44] is satisfied.",
"[44] can be checked, using similar arguments as in checking [36] in the proof of Theorem REF .",
"Also, $\\sum _{n=1}^{\\infty }\\overline{a}(n) \\ge \\sum _{n=1}^{\\infty }a(n)=\\infty $ and $\\sum _{n=1}^{\\infty }\\overline{a}^2(n)\\le \\sum _{n=1}^{\\infty } a^2( \\lfloor \\frac{n}{B} \\rfloor ) <\\infty $ , which satisfies [44].",
"Checking [44]: Let us consider the following set of o.d.e.",
"(similar to what we considered in the proof of Theorem REF ): $ \\dot{V}_t (r) = \\kappa _t(r) f_r(\\underline{V}_t,\\xi _{out}(t),\\xi _{relay}(t))$ for $r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ , $\\dot{\\xi }_{out}(t)=0$ and $\\dot{\\xi }_{relay}(t)=0$ (recall the interpretation of $\\kappa _t(r)$ from Appendix ).",
"Note that, $\\lim _{c \\rightarrow \\infty } \\frac{f_r(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}=\\mathbb {E}_{W}\\min \\lbrace \\xi _{out}Q_{out}(r,\\gamma ,W)+\\xi _{relay},-V(1)+V(r+1)\\rbrace -V(r)$ for $r \\ne B$ , and $\\lim _{c \\rightarrow \\infty } \\frac{f_B(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}=\\xi _{out} \\mathbb {E}_{W}Q_{out}(B,\\gamma ,W)+\\xi _{relay}-V(B)$ .",
"Note that $\\frac{f_r(c\\underline{V})}{c}$ for all $r$ and $\\lim _{c \\rightarrow \\infty } \\frac{f_r(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}$ all are continuous in $(\\underline{V}, \\xi _{out},\\xi _{relay})$ , and $\\frac{f_r(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}$ is decreasing in $c$ .",
"Hence, by Theorem $7.13$ of [43], convergence of $\\frac{f_r(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}$ over compacts is uniform.",
"Hence, one part of [44] is proved.",
"Next, by similar analysis done while checking [36] in the proof of Theorem REF (using Lemma REF ), we can verify the second part of [44].",
"Hence, using similar analysis as in [44] (adapted to the case of asynchronous stochastic approximation), we can claim that $||\\underline{V}^{(k)}|| \\le C^* (1+\\xi _{out}^{(k)}+\\xi _{relay}^{(k)})$ for all $k \\ge 1$ , for some $C^*>0$ .",
"Now, since the slower timescale iterates are bounded in our problem, the faster timescale iterates are also bounded.",
"This completes the proof of Lemma REF .",
"Lemma 4 For Algorithm REF , we have $(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)}) \\rightarrow \\lbrace (\\underline{V}^*(\\xi _{out},\\xi _{relay}),\\xi _{out},\\xi _{relay}):(\\xi _{out},\\xi _{relay}) \\in [0,A_1] \\times [0,A_2] \\rbrace $ almost surely, i.e., $\\lim _{k \\rightarrow \\infty }||\\underline{V}^{(k)}-\\underline{V}^*(\\xi _{out}^{(k)},\\xi _{relay}^{(k)})||=0$ almost surely.",
"Note that, the functions $f_r(\\underline{V},\\xi _{out},\\xi _{relay})=\\mathbb {E}_W \\bigg [ \\min \\bigg \\lbrace \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , W))+\\xi _{relay},-V(1)+V(r+1) \\bigg \\rbrace -V(r) \\bigg ]$ and $f_B(\\underline{V},\\xi _{out},\\xi _{relay})=\\mathbb {E}_W \\bigg [ \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , W))+\\xi _{relay}-V(B) \\bigg ]$ are Lipschitz continuous in all arguments (by Theorem $\\ref {theorem:V-continuous-in-xi}$ ), and the collection of o.d.e.",
"$\\dot{\\underline{V}}_r(t)=\\kappa _t(r) f_r(\\underline{V}(t),\\xi _{out},\\xi _{relay})$ for all $r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ (see [37] and the proof of Theorem REF for an interpretation of $\\kappa _t(r)$ ) has a unique globally asymptotically stable equilibrium $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ for any $\\xi _{out} \\ge 0$ , $\\xi _{relay} \\ge 0$ (see Lemma REF in the proof of Theorem REF ).",
"Also, by Theorem REF , $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ is Lipschitz continuous in $\\xi _{out}$ and $\\xi _{relay}$ .",
"On the other hand, by Lemma REF and the projection in the slower timescale, the iterates are almost surely bounded.",
"Hence, by a similar argument as in the proof [37], and by Theorem REF , $(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ converges to the internally chain transitive invariant sets of the collection of o.d.e.",
"given by $\\dot{V}_r(t)=\\kappa _t(r) f_r(\\underline{V}(t),\\xi _{out},\\xi _{relay})$ for all $r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ , $\\dot{\\xi }_{out}(t)=0$ , $\\dot{\\xi }_{relay}(t)=0$ (where $\\underline{V}(t):=\\lbrace V_1(t),V_2(t),\\cdots ,V_B(t) \\rbrace $ ).",
"Hence, $(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)}) \\rightarrow \\lbrace (\\underline{V}^*(\\xi _{out},\\xi _{relay}),\\xi _{out},\\xi _{relay}):(\\xi _{out},\\xi _{relay}) \\in [0,A_1] \\times [0,A_2] \\rbrace $ and $\\lim _{k \\rightarrow \\infty }||\\underline{V}^{(k)}-\\underline{V}^*(\\xi _{out}^{(k)},\\xi _{relay}^{(k)})||=0$ .",
"Remark: Lemma REF does not guarantee the convergence of the slower timescale iterates.",
"Figure: NO_CAPTION The slower timescale iteration We will pose the slower timescale update as a projected stochastic approximation (see [39]).",
"In order to do that and to avoid complicated notation, for the rest of this appendix we will denote by $\\underline{V}^{(k)}$ , $\\xi _{out}^{(k)}$ and $\\xi _{relay}^{(k)}$ the values of the corresponding variable after placing the $k$ -th relay and performing the update (earlier they were defined to be the iterates after a decision is made at the $k$ -th step).",
"Let us also recall the definition of the functions $\\overline{Q}_{out}(\\cdot ,\\cdot ,\\cdot )$ , $\\overline{Q}_{out}^*(\\cdot ,\\cdot )$ , $\\overline{U}(\\cdot ,\\cdot ,\\cdot )$ , $\\overline{U}^*(\\cdot ,\\cdot )$ .",
"Let us define the functions $\\overline{Q}_{out}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ and $\\overline{U}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ to be the mean link outage and mean length of the $k$ -th link that is created by Algorithm REF (using the two-timescale update) starting with $\\underline{V}^{(k-1)}$ , $\\xi _{out}^{(k-1)}$ and $\\xi _{relay}^{(k-1)}$ (which are obtained by the algorithm after placing the $(k-1)$ -st relay and and doing the learning/update operation; note that, these quantities are obtained after placing $(k-1)$ nodes and not at the $(k-1)$ -th step).",
"The difference between $\\overline{U}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ and $\\overline{U}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ can be explained as follows.",
"$\\overline{U}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ is the mean length of the $k$ -th link where no quantity is updated in the process of measurements made to create the $k$ -th link; hence, $\\overline{U}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ is the mean placement distance of a stationary policy which is similar to Algorithm REF except that $\\xi _{out}$ , $\\xi _{relay}$ and $\\underline{V}^*$ are replaced by $\\xi _{out}^{(k-1)}$ , $\\xi _{relay}^{(k-1)}$ and $\\underline{V}^{(k-1)}$ respectively.",
"On the other hand, $\\overline{U}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ is the mean length of the $k$ -th link created under Algorithm REF (with $(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ as starting parameters), where the iterates are updated at each step between placement of the $(k-1)$ -th node and the $k$ -th node.",
"Let us denote by $\\mathcal {G}$ the set $[0,A_1] \\times [0,A_2]$ , defined by the following constraints: $-\\xi _{out} \\le 0, \\xi _{out} \\le A_1, -\\xi _{relay} \\le 0, \\xi _{relay} \\le A_2$ Clearly, projection onto the set $\\mathcal {G}$ is nothing but coordinate wise projection.",
"We rewrite the slower timescale iteration in (REF ) as (REF ) (note the definitions of the functions $f_1(\\xi _{out},\\xi _{relay})$ , $f_2(\\xi _{out},\\xi _{relay})$ , $g_1(\\underline{V},\\xi _{out},\\xi _{relay})$ , $g_2(\\underline{V},\\xi _{out},\\xi _{relay})$ , $l_1(\\underline{V},\\xi _{out},\\xi _{relay})$ and $l_2(\\underline{V},\\xi _{out},\\xi _{relay})$ in (REF )).",
"The random variables $M_{1}^{(k)}$ and $M_{2}^{(k)}$ are two zero mean Martingale difference noise sequences w.r.t.",
"$\\mathcal {F}_{k-1}$ (information available up to the $(k-1)$ -st placement instant); this happens due to i.i.d.",
"shadowing across links.",
"(REF ) has the form of a projected stochastic approximation (see [39]).",
"In order to show the desired convergence of the iterates in (REF ), we will use [39]; this requires us to check five conditions from [39], which is done in the next subsection.",
"$\\Box $ Checking the five conditions from [39] We will first present a lemma that will be useful for checking one condition.",
"Lemma 5 Under Assumption REF , the quantities $\\overline{\\Gamma }(\\underline{V},\\xi _{out},\\xi _{relay})$ , $\\overline{Q}_{out}(\\underline{V},\\xi _{out},\\xi _{relay})$ and $\\overline{U}(\\underline{V},\\xi _{out},\\xi _{relay})$ are continuous in $\\underline{V}$ , $\\xi _{out}$ and $\\xi _{relay}$ .",
"The proof is similar to that of Theorem REF .",
"Now, we will check conditions $A5.1.3$ , $A5.1.4$ , $A5.1.5$ , $A5.3.1.$ and $A5.3.2$ from [39].",
"Checking Condition $A5.1.3$ : We need $f_1(\\cdot ,\\cdot )$ and $f_2(\\cdot ,\\cdot )$ to be continuous functions; this holds by Theorem REF .$\\Box $ Checking Condition $A5.1.4$ : This condition is satisfied by the choice of the sequence $\\lbrace b(k)\\rbrace _{k \\ge 1}$ .$\\Box $ Checking Condition $A5.1.5$ : This condition requires that $\\lim _{k \\rightarrow \\infty }g_1(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ , $\\lim _{k \\rightarrow \\infty }g_2(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ , $\\lim _{k \\rightarrow \\infty }l_1(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ and $\\lim _{k \\rightarrow \\infty }l_2(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ almost surely.",
"We can find a probability 1 subset of the sample space $\\Omega $ , such that for any sample path in this subset the conclusions of Lemma REF and Lemma REF hold.",
"Take one such sample path $\\omega $ .",
"By Lemma REF , for this sample path $\\omega $ , we can find a compact subset $\\mathcal {C} \\subset \\mathbb {R}^B$ such that $(\\underline{V}^{(k)}, \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})$ lies inside the compact set $\\mathcal {C} \\times [0,A_1]\\times [0,A_2]$ for all $k \\ge 1$ along this sample path.",
"By Lemma REF and the fact that continuous functions are uniformly continuous over compact sets, we can say that $\\overline{Q}_{out}(\\underline{V}, \\xi _{out}, \\xi _{relay})$ , $\\overline{\\Gamma }(\\underline{V}, \\xi _{out}, \\xi _{relay})$ and $\\overline{U}(\\underline{V}, \\xi _{out}, \\xi _{relay})$ are uniformly continuous over the compact set $\\mathcal {C} \\times [0,A_1]\\times [0,A_2]$ .",
"Now, the Euclidean distance between $(\\underline{V}^{(k)}, \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})$ and $(\\underline{V}^*(\\xi _{out}^{(k)}, \\xi _{relay}^{(k)}), \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})$ converges to 0 along the sample path $\\omega $ .",
"Hence, by uniform continuity, we can say that $\\lim _{k \\rightarrow \\infty }|\\overline{Q}_{out}(\\underline{V}^{(k)}, \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})-\\overline{Q}_{out}(\\underline{V}^*(\\xi _{out}^{(k)}, \\xi _{relay}^{(k)}), \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})|=0$ and $\\lim _{k \\rightarrow \\infty }|\\overline{U}(\\underline{V}^{(k)}, \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})-\\overline{U}(\\underline{V}^*(\\xi _{out}^{(k)}, \\xi _{relay}^{(k)}), \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})|=0$ along this sample path $\\omega $ .",
"Hence, $\\lim _{k \\rightarrow \\infty }g_1(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ and $\\lim _{k \\rightarrow \\infty }g_2(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ almost surely.",
"On the other hand, since $\\mathcal {C}$ is bounded, we can say that $\\lbrace \\underline{V}^{(k)} \\rbrace _{k \\ge 1}$ is bounded for the chosen $\\omega $ .",
"In a similar way as in the proof of Theorem REF , in case of Lemma REF we can show that $g(r,\\gamma )$ is continuous in $\\overline{V}$ , $\\xi _{out}$ and $\\xi _{relay}$ .",
"Now, between the placement of the $(k-1)$ -st relay and $k$ -th relay, at each step, $g(r,\\gamma )$ for all $r \\in \\lbrace 1,2,\\cdots ,B\\rbrace , \\gamma \\in \\mathcal {S}$ can change at most by an amount $K^* a(k-1-B)$ (for a suitable constant $K^*>0$ ), and hence we can claim that $\\lim _{k \\rightarrow \\infty }|\\overline{U}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})-\\overline{U}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})|=0$ , $\\lim _{k \\rightarrow \\infty }|\\overline{Q}_{out}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})-\\overline{Q}_{out}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})|=0$ .",
"Hence, we obtain that $\\lim _{k \\rightarrow \\infty }l_1(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})=0$ and $\\lim _{k \\rightarrow \\infty }l_2(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})=0$ .",
"Also, $g_1(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ , $g_2(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ , $l_1(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ and $l_2(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ are uniformly bounded across $k \\ge 1$ , since the outage probabilities and placement distances are bounded quantities.",
"Hence, this condition is satisfied.",
"Checking Condition $A5.3.1$ : This condition is easy to check, and done in [1].$\\Box $ Checking Condition $A5.3.2$ : This condition is easy to check, and done in [1].$\\Box $ Finishing the Proof of Theorem REF Consider the function $h(\\xi _{out},\\xi _{relay}):=\\bigg (\\frac{f_1(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})},\\frac{f_2(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})} \\bigg )=\\bigg (\\frac{\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}-\\overline{q},\\frac{1}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}-\\overline{N} \\bigg )$ and the map: $&& \\overline{\\Lambda }_{\\mathcal {G}}(h(\\xi _{out},\\xi _{relay})) \\nonumber \\\\&=& \\lim _{0<\\beta \\rightarrow 0} \\frac{\\Lambda _{\\mathcal {G}}\\bigg ((\\xi _{out},\\xi _{relay})+\\beta h(\\xi _{out},\\xi _{relay}))\\bigg )-(\\xi _{out},\\xi _{relay})}{\\beta } \\nonumber \\\\$ Lemma 6 If $(\\xi _{out},\\xi _{relay}) \\in [0,A_1] \\times [0,A_2]$ is a zero of $\\overline{\\Lambda }_{\\mathcal {G}}\\bigg (\\frac{f_1(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})},\\frac{f_2(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})} \\bigg )$ , then $(\\underline{V}^*(\\xi _{out},\\xi _{relay}),\\xi _{out},\\xi _{relay}) \\in \\mathcal {K}(\\overline{q},\\overline{N})$ , provided that $A_1$ and $A_2$ are chosen using the procedure described in Section .",
"The proof is similar to the proof of [1].",
"Now, by using similar arguments as in [1] and using [39], We can show that the iterates $(\\xi _{out}^{(k)}, \\xi _{relay}^{(k)})$ will converge almost surely to the set of stationary points of the o.d.e.",
"$(\\dot{\\xi }_{out}(t), \\dot{\\xi }_{relay}(t))=\\overline{\\Lambda }_{\\mathcal {G}}\\bigg (\\frac{f_1(\\xi _{out}(t),\\xi _{relay}(t))}{\\overline{U}^*(\\xi _{out}(t),\\xi _{relay}(t))},\\frac{f_2(\\xi _{out}(t),\\xi _{relay}(t))}{\\overline{U}^*(\\xi _{out}(t),\\xi _{relay}(t))} \\bigg )$ .",
"Using this result and using Lemma REF and Lemma REF , we obtain that $(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)}) \\rightarrow \\mathcal {K}(\\overline{q},\\overline{N})$ almost surely, where $k\\delta $ can be the distance from the sink or $k$ can be the index of a placed relay node (the result holds for both interpretations of $k$ ).",
"This completes the proof of Theorem REF .",
"$\\Box $"
],
[
"Formulation for known propagation parameters",
"Proof of Theorem REF : From (REF ), $V(B)$ is unique for fixed $\\xi _{out}$ and $\\xi _{relay}$ .",
"Hence, we can say that $V(B)$ is a continuous and decreasing function of $V(1)$ .",
"Now, let us assume that $V(r+1)$ is continuous and decreasing in $V(1)$ for some $r, 1 \\le r \\le B-1$ .",
"Let us recall (REF ) for $V(r)$ .",
"Since $V(r+1)$ is continuous and decreasing in $V(1)$ by our induction hypothesis, it is evident from (REF ) that $V(r)$ is also continuous and decreasing in $V(1)$ .",
"Proceeding in this way, we can write $V(1)=\\phi (V(1))$ where $\\phi (\\cdot )$ is continuous and decreasing in $V(1)$ .",
"But $V(1)$ is continuous and strictly increasing in $V(1)$ .",
"Hence, $V(1)=\\phi (V(1))$ has a unique fixed point $V^*(1)$ .",
"Now, from (REF ), $V(B-1)$ is unique since $V(1)=V^*(1)$ is unique and $V(B)$ is unique.",
"Proceeding backwards in this way, we can show that we have a unique $V^*(r)$ for all $r$ .",
"Now, from (REF ), we find that $V^*(r) \\le -V^*(1)+V^*(r+1)$ , i.e., $V^*(r+1) \\ge V^*(r)+V^*(1)$ for all $r \\in \\lbrace 1,2,\\cdots ,B-1\\rbrace $ .",
"Also, $V^*(1)=\\lambda ^*>0$ and it is unique.",
"This proves the second part of the theorem.",
"$\\Box $ Proof of Theorem REF : Let us denote the mean power per link, mean outage per link and mean placement distance (in steps) under a stationary policy $\\pi $ by $\\overline{\\Gamma }_{\\pi }$ , $\\overline{Q}_{out,\\pi }$ and $\\overline{U}_{\\pi }$ .",
"Then, by Renewal-Reward Theorem, we have $\\lambda ^{*}(\\xi _{out},\\xi _{relay}) =\\inf _{\\pi }\\frac{\\Gamma _{\\pi }+\\xi _{out}\\overline{Q}_{out,\\pi }+\\xi _{relay}}{\\overline{U}_{\\pi }}$ .",
"The numerator is affine and increasing in $\\xi _{out}$ and $\\xi _{relay}$ , and the denominator is independent of $\\xi _{out}$ and $\\xi _{relay}$ .",
"Hence, $\\lambda ^{*}(\\xi _{out},\\xi _{relay})$ is concave, increasing in $\\xi _{out}$ and $\\xi _{relay}$ , since the pointwise infimum of increasing affine functions of $(\\xi _{out},\\xi _{relay})$ is increasing and jointly concave in $(\\xi _{out},\\xi _{relay})$ .",
"Now, any increasing, concave function is continuous.",
"Hence, $\\lambda ^{*}(\\xi _{out},\\xi _{relay})$ is continuous in $(\\xi _{out},\\xi _{relay})$ .",
"Also, it is easy to see that $\\lambda ^{*}(\\xi _{out},\\xi _{relay})$ is Lipschitz in each argument with Lipschitz constant 1.",
"Proof of Theorem REF : By Theorem REF , $V^*(1):=\\lambda ^*$ is Lipschitz continuous in $(\\xi _{out}, \\xi _{relay})$ .",
"By (REF ), $V^*(B)$ is Lipschitz continuous in $(\\xi _{out},\\xi _{relay})$ .",
"Hence, by (REF ), $V^*(B-1)$ is also Lipschitz continuous in $(\\xi _{out}, \\xi _{relay})$ .",
"Thus, by using backward induction, we can show that $V^*(r)$ is Lipschitz continuous in $(\\xi _{out}, \\xi _{relay})$ for all $1 \\le r \\le B$ ."
],
[
"OptAsYouGoLearning: Learning with Pure As-You-Go Deployment, for Given Lagrange Multipliers",
"Proof of Theorem REF : We can rewrite (REF ) as follows: $V^{(k)}(r)&=&V^{(k-1)}(r)+ a(\\nu (r,k)) \\mathbb {I}\\lbrace r \\in \\mathcal {I}_k\\rbrace \\bigg [ f_r(\\underline{V}^{(k-1)})+ M_k(r) \\bigg ] \\nonumber \\\\&& $ where, for all $1 \\le r \\le B-1$ $f_r(\\underline{V}^{(k-1)})&=&\\mathbb {E}_W \\bigg [ \\min \\bigg \\lbrace \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , W))+\\xi _{relay}, \\\\&& -V^{(k-1)}(1)+V^{(k-1)}(r+1) \\bigg \\rbrace -V^{(k-1)}(r) \\bigg ]$ $M_k(r)&=& \\bigg [\\min \\bigg \\lbrace \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , w_r))+\\xi _{relay}, \\\\&& -V^{(k-1)}(1)+V^{(k-1)}(r+1) \\bigg \\rbrace -V^{(k-1)}(r) \\bigg ]-f_r(\\underline{V}^{(k-1)})$ and $f_B(\\underline{V}^{(k-1)})=\\mathbb {E}_W \\bigg [ \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , W)) +\\xi _{relay} -V^{(k-1)}(B) \\bigg ]$ $M_k (B)&=&\\bigg [ \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , w_B)) +\\xi _{relay}-V^{(k-1)}(B) \\bigg ] \\\\&& -f_B(\\underline{V}^{(k-1)})$ Let $\\underline{M}_k:=(M_k(1), \\cdots , M_k(B))$ .",
"Let us denote the $\\sigma $ -field $\\mathcal {F}_k:=\\sigma (\\underline{V}_i, \\mathcal {I}_i, \\underline{M}_i, i \\le k-1)$ ; it is the information available to the deployment agent before making any decision at the $k$ -th step.",
"Clearly, the update equations fall under the category of Asynchronous Stochastic Approximation algorithms (see [36]).",
"In order to see whether $\\underline{V}^{(k)} \\rightarrow \\underline{V}^*$ almost surely, we will first check whether the five assumptions mentioned in [36] are satisfied.",
"Checking Assumption 1 of [36]: For each $r, 1 \\le r \\le B$ , $V(r)$ gets updated at least once in every $B$ steps.",
"Hence, $\\lim \\inf _{k \\rightarrow \\infty }\\frac{\\nu (r,k)}{k} \\ge \\frac{1}{B}>0 $ almost surely.",
"Hence, the assumption is satisfied.",
"Checking Assumption 2 of [36]: If we choose $\\lbrace a(k)\\rbrace _{k \\ge 1}$ to be a bounded, decreasing sequence with $\\sum _k a(k)=\\infty $ and $\\sum _k a^2(k) < \\infty $ , this condition will be satisfied.",
"Checking Assumption 3 of [36]: Not applicable to our problem since before updating $\\underline{V}^{(k)}$ the deployment agent knows $\\underline{V}^{(k-1)}$ .",
"Before checking the other two conditions, we will establish a lemma.",
"Let us consider the following system of o.d.e-s: $\\dot{V}_t (r) = \\kappa _t(r) f_r(\\underline{V}_t) \\,\\,\\, \\forall r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ where $\\kappa _t(r) \\in (0,1]$ for all $r$ and $t$ .",
"By Theorem REF , this system of o.d.e-s has an unique stationary point $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ .",
"Lemma 1 $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ is a globally asymptotically stable equilibrium for the system of o.d.e-s (REF ).",
"Also, $\\underline{V}=0$ is a globally asymptotically stable equilibrium for (REF ) when $\\gamma $ , $\\xi _{out}$ and $\\xi _{relay}$ are replaced by 0 in the definition of $f_r(\\underline{V})$ for all $r \\in \\lbrace 1,2,\\cdots ,B \\rbrace $ .",
"Note that, by Theorem REF , $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ is the unique stationary point for (REF ).",
"Now, the proof for this lemma follows from similar line of arguments as in the appendix of [40] (which uses results from [41] and [42]).",
"Checking Assumption 4 of [36]: It is easy to see that $f_r (\\underline{V})$ is Lipschitz in $\\underline{V}$ for each $r$ ; this satisfies Assumption 4(i).",
"Let us consider the ODE (REF ) with $0<\\kappa _t(r) \\le 1$ corresponds to the relative rate at which $V(r)$ is updated.",
"By Lemma REF , $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ is a globally asymptotically stable equilibrium for the system of o.d.e-s (REF ).",
"Hence, Assumption 4(ii) is satisfied.",
"Consider the functions $\\frac{f_r(c\\underline{V})}{c}, c \\ge 1$ for all $r$ .",
"Clearly, $\\lim _{c \\rightarrow \\infty } \\frac{f_r(c\\underline{V})}{c}=\\min \\lbrace 0,-V(1)+V(r+1)\\rbrace -V(r)$ for $r \\ne B$ , and $\\lim _{c \\rightarrow \\infty } \\frac{f_B(c\\underline{V})}{c}=-V(B)$ .",
"Note that $\\frac{f_r(c\\underline{V})}{c}$ for all $r$ and $\\lim _{c \\rightarrow \\infty } \\frac{f_r(c\\underline{V})}{c}$ all are continuous in $\\underline{V}$ , and $\\frac{f_r(c\\underline{V})}{c}$ is decreasing in $c$ .",
"Hence, by Theorem $7.13$ of [43], convergence of $\\frac{f_r(c\\underline{V})}{c}$ over compacts is uniform.",
"Hence, Assumption 4(iii) is satisfied.",
"Consider the ODE: $\\dot{V}_t(r)=\\kappa _t(r)(\\min \\lbrace 0,-V_t(1)+V_t(r+1)\\rbrace -V_t(r))$ for $r \\ne B$ and $\\dot{V}_t(B)=\\kappa _B(t)(-V_t(B))$ .",
"Clearly, by the second part of Lemma REF , there is a unique globally asymptotically stable equilibrium $\\underline{V}=\\underline{0}$ .",
"Hence, Assumption 4(iv) is satisfied.",
"Checking Assumption 5 of [36]: It is easy to see that, $\\lbrace \\underline{M}_k\\rbrace _{k \\ge 1}$ is a Martingale difference sequence adapted to $\\mathcal {F}_k$ .",
"Hence, Assumption 5(i) is satisfied.",
"Now, $|M_{k+1}(r)| & \\le & 2 \\bigg | \\bigg (\\min \\lbrace P_M+\\xi _{out}+\\xi _{relay},-V^{(k)}(1) \\\\&& +V^{(k)}(r+1)\\rbrace -V^{(k)}(r) \\bigg ) \\bigg |$ and $|M_{k+1}(B)| \\le \\bigg | \\bigg (P_M+\\xi _{out}+\\xi _{relay} -V^{(k)}(B) \\bigg ) \\bigg |$ Hence, $||M_{k+1}|| \\le C_0 (1+||\\underline{V}^{(k)}||)$ for some $C_0>0$ .",
"Hence, Assumption 5(ii) is satisfied.",
"Now, by [36], $\\underline{V}^{(k)} \\rightarrow \\underline{V}^*$ .$\\Box $"
],
[
"Proof of Theorem ",
"Let us denote by $g(r, \\gamma ),r \\in \\lbrace 1,2,\\cdots ,B\\rbrace , \\gamma \\in \\mathcal {S}$ the joint distribution of $(U_k, \\Gamma _k)$ under Algorithm REF .",
"For the time being, let us assume that $g(r,\\gamma )$ is continuous in $(\\xi _{out},\\xi _{relay})$ .",
"Then, the mean placement distance $\\overline{U}^*(\\xi _{out},\\xi _{relay})= \\sum _{r=1}^B \\sum _{\\gamma \\in \\mathcal {S}} r g (r,\\gamma )$ , and the mean power per link $\\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})=\\sum _{r=1}^B\\sum _{\\gamma \\in \\mathcal {S}} \\gamma g (r,\\gamma )$ are both continuous in $(\\xi _{out},\\xi _{relay})$ .",
"Now, by Renewal-Reward Theorem, $\\lambda ^*(\\xi _{out}, \\xi _{relay}) = \\frac{ \\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})+\\xi _{out}\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})+\\xi _{relay} }{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ Since $\\lambda ^*(\\xi _{out}, \\xi _{relay})$ is continuous in $(\\xi _{out},\\xi _{relay})$ (by Theorem REF ), we conclude that $\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})$ is continuous in $\\xi _{out}$ and $\\xi _{relay}$ .",
"Hence, $\\frac{\\overline{\\Gamma }^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ , $\\frac{\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ and $\\frac{1}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}$ are continuous in $(\\xi _{out},\\xi _{relay})$ .",
"$\\Box $ Now, the proof of the theorem is completed by the following lemma.",
"Lemma 2 Under Assumption REF , $g(r,\\gamma )$ is continuous in $(\\xi _{out},\\xi _{relay})$ .",
"We will first prove the result for $r \\le B-1$ .",
"Let us fix an $r \\in \\lbrace 1,\\cdots ,B-1\\rbrace $ and any $\\gamma \\in \\mathcal {S}$ .",
"We will only show that $g(r,\\gamma )$ is continuous in $\\xi _{out}$ ; the proof for continuity of $g(r,\\gamma )$ w.r.t.",
"$\\xi _{relay}$ will be similar.",
"Let us consider a sequence $\\lbrace \\xi _n\\rbrace _{n \\ge 1}$ such that $\\xi _n \\rightarrow \\xi _{out}$ .",
"Let us denote the joint probability distribution of $(U_k, \\Gamma _k)$ by $g_n(r,\\gamma )$ , if Algorithm REF is used with $\\xi _n$ as the cost for unit outage.",
"We will show that $\\lim _{n \\rightarrow \\infty } g_n(r,\\gamma ) \\rightarrow g(r,\\gamma )$ .",
"Define the sets $\\mathcal {E}_{r,\\gamma ^{\\prime }}=\\bigg \\lbrace w_r: \\gamma +\\xi _{out}Q_{out}(r,\\gamma ,w_r) < \\gamma ^{\\prime }+\\xi _{out}Q_{out}(r,\\gamma ^{\\prime },w_r) \\bigg \\rbrace $ and $\\mathcal {E}_u=\\bigg \\lbrace w_u: \\min _{\\gamma \\in \\mathcal {S}}(\\gamma +\\xi _{out}Q_{out}(u,\\gamma ,w_u))>-\\xi _{relay}-V^*(1)+V^*(u+1) \\bigg \\rbrace $ for all $1 \\le u \\le r$ .",
"Let us define $\\mathcal {E}=\\cap _{\\gamma ^{\\prime } \\ne \\gamma }\\mathcal {E}_{r,\\gamma ^{\\prime }}\\cap _{u \\le r-1} \\mathcal {E}_u \\cap \\overline{\\mathcal {E}_r}$ , where $\\overline{\\mathcal {E}_r}$ is the set complement of $\\mathcal {E}_r$ .",
"Now, $g(r,\\gamma )=\\mathbb {P}(\\mathcal {E})=\\mathbb {E}(\\mathbb {I}_{\\mathcal {E}})$ , where $\\mathbb {I}$ denotes the indicator function.",
"The expectation is over the joint distribution of $(W_1,W_2, \\cdots , W_r)$ (shadowing random variables from $r$ locations).",
"Now, for any $\\gamma ^{\\prime } \\ne \\gamma $ , we have $\\mathbb {P}\\bigg ( \\gamma +\\xi _{out}Q_{out}(r,\\gamma ,W_r) = \\gamma ^{\\prime }+\\xi _{out}Q_{out}(r,\\gamma ^{\\prime },W_r) \\bigg )=0$ , and $\\mathbb {P}\\bigg ( \\min _{\\gamma \\in \\mathcal {S}}(\\gamma +\\xi _{out}Q_{out}(u,\\gamma ,W_u))=-\\xi _{relay}-V^*(1)+V^*(u+1) \\bigg )=0$ for all $u \\le r$ ; these two assertions follow from Assumption REF and from the continuity of $Q_{out}(r,\\gamma ,w)$ in $w$ .",
"Hence, we can safely assume the following: $\\overline{\\mathcal {E}_{r,\\gamma ^{\\prime }}}$ has the same expression as $\\mathcal {E}_{r,\\gamma ^{\\prime }}$ except that the $<$ sign is replaced by $>$ sign.",
"$\\overline{\\mathcal {E}_u}$ has the same expression as $\\mathcal {E}_u$ except that the $>$ sign is replaced by $<$ sign.",
"Let $\\mathcal {E}_{r,\\gamma ^{\\prime }}^{(n)}$ , $\\mathcal {E}_u^{(n)}$ and $\\mathcal {E}^{(n)}$ be the sets obtained by replacing $\\xi _{out}$ by $\\xi _n$ in the expressions of the sets $\\mathcal {E}_{r,\\gamma ^{\\prime }}$ , $\\mathcal {E}_u$ and $\\mathcal {E}$ respectively (also $\\underline{V}^*$ has to be replaced by the corresponding optimal $\\underline{V}^{(n,*)}$ ).",
"Clearly, we can make similar claims for $\\mathcal {E}_{r,\\gamma ^{\\prime }}^{(n)}$ , $\\mathcal {E}_u^{(n)}$ .",
"Now, if we can show that $\\mathbb {E}(\\mathbb {I}_{\\mathcal {E}^{(n)}}) \\rightarrow \\mathbb {E}(\\mathbb {I}_{\\mathcal {E}})$ , the lemma will be proved, because $g(r,\\gamma )=\\mathbb {P}(\\mathcal {E})=\\mathbb {E}(\\mathbb {I}_{\\mathcal {E}})$ .",
"Claim 1 $\\lim _{n \\rightarrow \\infty }\\mathbb {I}_{ \\mathcal {E}_u^{(n)} } \\rightarrow \\mathbb {I}_{ \\mathcal {E}_u} $ , and $\\lim _{n \\rightarrow \\infty } \\mathbb {I}_{ \\mathcal {E}_{r,\\gamma ^{\\prime }}^{(n)} } \\rightarrow \\mathbb {I}_{ \\mathcal {E}_{r,\\gamma ^{\\prime }} } $ almost surely, for $\\gamma ^{\\prime } \\ne \\gamma $ .",
"Suppose that, for some value of $w_u$ , $\\mathbb {I}_{ \\mathcal {E}_u}(w_u)=1$ , i.e., $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma +\\xi _{out}Q_{out}(u,\\gamma ,w_u))>-\\xi _{relay}-V^*(1)+V^*(u+1)$ .",
"Now, $V^*(1)$ and $V^*(u+1)$ are continuous in $(\\xi _{out},\\xi _{relay})$ for all $1 \\le u \\le r$ (see Theorem $\\ref {theorem:V-continuous-in-xi}$ ).",
"Hence, there exists an integer $n_0$ large enough, such that for all $n > n_0$ , we have $\\min _{\\gamma \\in \\mathcal {S}}(\\gamma +\\xi _n Q_{out}(u,\\gamma ,w_u))>-\\xi _{relay}-\\bigg ( V^{(n,*)}(1)+V^{(n,*)}(u+1) \\bigg )\\bigg |_{\\xi _{out}=\\xi _n}$ , i.e., $\\mathbb {I}_{ \\mathcal {E}_u^{(n)} } (w_u)=1$ for all $n > n_0$ .",
"Hence, $\\mathbb {I}_{ \\mathcal {E}_u^{(n)} } (w_u) \\rightarrow \\mathbb {I}_{ \\mathcal {E}_u }(w_u) $ if $\\mathbb {I}_{ \\mathcal {E}_u }(w_u)=1$ .",
"For the case $\\mathbb {I}_{ \\mathcal {E}_u } (w_u)=0$ , we can have similar arguments.",
"This proves the first part of the claim, and second part can be proved by similar arguments.",
"Now, $\\mathbb {I}_{\\mathcal {E}^{(n)}}=\\prod _{\\gamma ^{\\prime } \\ne \\gamma } \\mathbb {I}_{\\mathcal {E}_{r,\\gamma ^{\\prime }}^{(n)}}\\prod _{u \\le r-1} \\mathbb {I}_{\\mathcal {E}_u^{(n)}} \\times \\mathbb {I}_{\\overline{\\mathcal {E}_r^{(n)}}}$ .",
"By Claim REF , $\\mathbb {I}_{\\mathcal {E}^{(n)}} \\rightarrow \\mathbb {I}_{\\mathcal {E}}$ almost surely as $n \\rightarrow \\infty $ .",
"Hence, by Dominated Convergence Theorem, we have $\\mathbb {E}(\\mathbb {I}_{\\mathcal {E}^{(n)}}) \\rightarrow \\mathbb {E}(\\mathbb {I}_{\\mathcal {E}})$ .",
"We can prove the same statement for $r=B$ in a similar method; but we need to define $\\mathcal {E}=\\cap _{\\gamma ^{\\prime } \\ne \\gamma }\\mathcal {E}_{B,\\gamma ^{\\prime }}\\cap _{u \\le B-1} \\mathcal {E}_u $ .",
"Hence, the lemma is proved.",
"Figure: NO_CAPTION"
],
[
"Proof of Theorem ",
"We denote the shadowing in the link between the potential locations located at distances $i \\delta $ and $j \\delta $ from the sink node, by the random variable $W_{i,j}$ .",
"The sample space $\\Omega $ is defined to be the collection of all $\\omega $ such that each $\\omega $ corresponds to a fixed realization $\\lbrace w_{i,j}: i \\ge 0, j \\ge 0, i>j, 1 \\le i-j \\le B \\rbrace $ of shadowing that could be encountered in the deployment process over infinite horizon.",
"Let $\\mathcal {F}$ be the Borel $\\sigma $ -algebra on $\\Omega $ .",
"We also define a sequence of sub-$\\sigma $ fields $\\mathcal {F}_k:=\\sigma \\bigg (W_{i,j}: i \\ge 0, j \\ge 0, k \\ge i>j, 1 \\le i-j \\le B \\bigg )$ ; $\\mathcal {F}_k$ is increasing in $k$ , and captures the history of the deployment process up to $k \\delta $ distance.",
"Let us recall the outline of the proof of Theorem REF in Section REF ."
],
[
"Let us denote by $\\underline{V}^*(\\xi _{out}, \\xi _{relay})$ the value of $\\underline{V}^*$ , for given $\\xi _{out}$ and $\\xi _{relay}$ .",
"Let us also define $\\overline{a}(k):=\\max _{r \\in \\mathcal {I}_k} a ( \\nu (r,k) )$ .",
"Using the first order Taylor series expansion of the function $\\Lambda _{[0,A_1]}(\\cdot )$ , and using the fact that $\\Lambda _{[0,A_1]}(\\xi _{out}^{(k-1)})=\\xi _{out}^{(k-1)}$ (since $\\xi _{out}^{(k-1)} \\in [0,A_1]$ ), we rewrite the update equation (REF ) as (REF ).",
"Now, for the update equation for $\\xi _{relay}$ in (REF ), we can write: $&& \\lim _{\\beta \\downarrow 0} \\frac{\\Lambda _{[0,A_2]}\\bigg (\\xi _{relay}^{(k-1)} + \\beta (1-\\overline{N}U_{N_k}) \\bigg )-\\xi _{relay}^{(k-1)}}{\\beta } \\\\&=& (1-\\overline{N}U_{N_k}) \\mathbb {I} \\lbrace 0< \\xi _{relay}^{(k-1)} <A_2\\rbrace \\\\&+& (1-\\overline{N}U_{N_k})^+ \\mathbb {I} \\lbrace \\xi _{relay}^{(k-1)} =0 \\rbrace \\\\&-& (1-\\overline{N}U_{N_k})^- \\mathbb {I} \\lbrace \\xi _{relay}^{(k-1)} =A_2 \\rbrace \\\\$ where $y^+=\\max \\lbrace y,0\\rbrace $ and $y^-=-\\min \\lbrace y,0\\rbrace $ .",
"We can write similar expression for the $\\xi _{out}^{(k)}$ update.",
"Since outage probabilities and placement distances are bounded quantities, and since $N_k \\ge \\lfloor \\frac{k}{B} \\rfloor $ and $\\lim _{k \\rightarrow 0}\\frac{b(\\lfloor \\frac{k}{B} \\rfloor )}{\\overline{a}(k)}=0$ , we have: $&& \\lim _{k \\rightarrow \\infty } \\bigg (\\frac{b(N_k)}{\\overline{a}(k)} \\bigg ( \\lim _{\\beta \\downarrow 0} \\bigg (\\Lambda _{[0,A_1]}\\bigg (\\xi _{out}^{(k-1)}+ \\beta (Q_{out}^{(N_k,N_{k-1})} \\\\&& -\\overline{q}U_{N_k}) \\bigg ) -\\xi _{out}^{(k-1)}\\bigg )/ \\beta + \\frac{o(b(N_k))}{b(N_k)} \\bigg ) \\bigg )=0$ Similar claim can be made for $\\xi _{relay}$ update.",
"Lemma 3 Under Algorithm REF , the faster timescale iterates $\\lbrace \\underline{V}^{(k)}\\rbrace _{k \\ge 1}$ are almost surely bounded.",
"Note that, (REF ) combines the faster and slower timescale iterations in a single timescale where the step size is $\\overline{a}(n)$ .",
"We will now use the theory from [44] to prove this lemma.",
"Note that, the R.H.S.",
"of the faster timescale iteration in (REF ) is Lipschitz continuous in both faster and slower timescale iterates.",
"Hence, the first part of [44] is satisfied.",
"[44] can be checked, using similar arguments as in checking [36] in the proof of Theorem REF .",
"Also, $\\sum _{n=1}^{\\infty }\\overline{a}(n) \\ge \\sum _{n=1}^{\\infty }a(n)=\\infty $ and $\\sum _{n=1}^{\\infty }\\overline{a}^2(n)\\le \\sum _{n=1}^{\\infty } a^2( \\lfloor \\frac{n}{B} \\rfloor ) <\\infty $ , which satisfies [44].",
"Checking [44]: Let us consider the following set of o.d.e.",
"(similar to what we considered in the proof of Theorem REF ): $ \\dot{V}_t (r) = \\kappa _t(r) f_r(\\underline{V}_t,\\xi _{out}(t),\\xi _{relay}(t))$ for $r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ , $\\dot{\\xi }_{out}(t)=0$ and $\\dot{\\xi }_{relay}(t)=0$ (recall the interpretation of $\\kappa _t(r)$ from Appendix ).",
"Note that, $\\lim _{c \\rightarrow \\infty } \\frac{f_r(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}=\\mathbb {E}_{W}\\min \\lbrace \\xi _{out}Q_{out}(r,\\gamma ,W)+\\xi _{relay},-V(1)+V(r+1)\\rbrace -V(r)$ for $r \\ne B$ , and $\\lim _{c \\rightarrow \\infty } \\frac{f_B(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}=\\xi _{out} \\mathbb {E}_{W}Q_{out}(B,\\gamma ,W)+\\xi _{relay}-V(B)$ .",
"Note that $\\frac{f_r(c\\underline{V})}{c}$ for all $r$ and $\\lim _{c \\rightarrow \\infty } \\frac{f_r(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}$ all are continuous in $(\\underline{V}, \\xi _{out},\\xi _{relay})$ , and $\\frac{f_r(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}$ is decreasing in $c$ .",
"Hence, by Theorem $7.13$ of [43], convergence of $\\frac{f_r(c\\underline{V}, c \\xi _{out}, c \\xi _{relay})}{c}$ over compacts is uniform.",
"Hence, one part of [44] is proved.",
"Next, by similar analysis done while checking [36] in the proof of Theorem REF (using Lemma REF ), we can verify the second part of [44].",
"Hence, using similar analysis as in [44] (adapted to the case of asynchronous stochastic approximation), we can claim that $||\\underline{V}^{(k)}|| \\le C^* (1+\\xi _{out}^{(k)}+\\xi _{relay}^{(k)})$ for all $k \\ge 1$ , for some $C^*>0$ .",
"Now, since the slower timescale iterates are bounded in our problem, the faster timescale iterates are also bounded.",
"This completes the proof of Lemma REF .",
"Lemma 4 For Algorithm REF , we have $(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)}) \\rightarrow \\lbrace (\\underline{V}^*(\\xi _{out},\\xi _{relay}),\\xi _{out},\\xi _{relay}):(\\xi _{out},\\xi _{relay}) \\in [0,A_1] \\times [0,A_2] \\rbrace $ almost surely, i.e., $\\lim _{k \\rightarrow \\infty }||\\underline{V}^{(k)}-\\underline{V}^*(\\xi _{out}^{(k)},\\xi _{relay}^{(k)})||=0$ almost surely.",
"Note that, the functions $f_r(\\underline{V},\\xi _{out},\\xi _{relay})=\\mathbb {E}_W \\bigg [ \\min \\bigg \\lbrace \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(r,\\gamma , W))+\\xi _{relay},-V(1)+V(r+1) \\bigg \\rbrace -V(r) \\bigg ]$ and $f_B(\\underline{V},\\xi _{out},\\xi _{relay})=\\mathbb {E}_W \\bigg [ \\min _{\\gamma }(\\gamma + \\xi _{out} Q_{out}(B,\\gamma , W))+\\xi _{relay}-V(B) \\bigg ]$ are Lipschitz continuous in all arguments (by Theorem $\\ref {theorem:V-continuous-in-xi}$ ), and the collection of o.d.e.",
"$\\dot{\\underline{V}}_r(t)=\\kappa _t(r) f_r(\\underline{V}(t),\\xi _{out},\\xi _{relay})$ for all $r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ (see [37] and the proof of Theorem REF for an interpretation of $\\kappa _t(r)$ ) has a unique globally asymptotically stable equilibrium $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ for any $\\xi _{out} \\ge 0$ , $\\xi _{relay} \\ge 0$ (see Lemma REF in the proof of Theorem REF ).",
"Also, by Theorem REF , $\\underline{V}^*(\\xi _{out},\\xi _{relay})$ is Lipschitz continuous in $\\xi _{out}$ and $\\xi _{relay}$ .",
"On the other hand, by Lemma REF and the projection in the slower timescale, the iterates are almost surely bounded.",
"Hence, by a similar argument as in the proof [37], and by Theorem REF , $(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ converges to the internally chain transitive invariant sets of the collection of o.d.e.",
"given by $\\dot{V}_r(t)=\\kappa _t(r) f_r(\\underline{V}(t),\\xi _{out},\\xi _{relay})$ for all $r \\in \\lbrace 1,2,\\cdots ,B\\rbrace $ , $\\dot{\\xi }_{out}(t)=0$ , $\\dot{\\xi }_{relay}(t)=0$ (where $\\underline{V}(t):=\\lbrace V_1(t),V_2(t),\\cdots ,V_B(t) \\rbrace $ ).",
"Hence, $(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)}) \\rightarrow \\lbrace (\\underline{V}^*(\\xi _{out},\\xi _{relay}),\\xi _{out},\\xi _{relay}):(\\xi _{out},\\xi _{relay}) \\in [0,A_1] \\times [0,A_2] \\rbrace $ and $\\lim _{k \\rightarrow \\infty }||\\underline{V}^{(k)}-\\underline{V}^*(\\xi _{out}^{(k)},\\xi _{relay}^{(k)})||=0$ .",
"Remark: Lemma REF does not guarantee the convergence of the slower timescale iterates.",
"Figure: NO_CAPTION"
],
[
"We will pose the slower timescale update as a projected stochastic approximation (see [39]).",
"In order to do that and to avoid complicated notation, for the rest of this appendix we will denote by $\\underline{V}^{(k)}$ , $\\xi _{out}^{(k)}$ and $\\xi _{relay}^{(k)}$ the values of the corresponding variable after placing the $k$ -th relay and performing the update (earlier they were defined to be the iterates after a decision is made at the $k$ -th step).",
"Let us also recall the definition of the functions $\\overline{Q}_{out}(\\cdot ,\\cdot ,\\cdot )$ , $\\overline{Q}_{out}^*(\\cdot ,\\cdot )$ , $\\overline{U}(\\cdot ,\\cdot ,\\cdot )$ , $\\overline{U}^*(\\cdot ,\\cdot )$ .",
"Let us define the functions $\\overline{Q}_{out}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ and $\\overline{U}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ to be the mean link outage and mean length of the $k$ -th link that is created by Algorithm REF (using the two-timescale update) starting with $\\underline{V}^{(k-1)}$ , $\\xi _{out}^{(k-1)}$ and $\\xi _{relay}^{(k-1)}$ (which are obtained by the algorithm after placing the $(k-1)$ -st relay and and doing the learning/update operation; note that, these quantities are obtained after placing $(k-1)$ nodes and not at the $(k-1)$ -th step).",
"The difference between $\\overline{U}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ and $\\overline{U}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ can be explained as follows.",
"$\\overline{U}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ is the mean length of the $k$ -th link where no quantity is updated in the process of measurements made to create the $k$ -th link; hence, $\\overline{U}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ is the mean placement distance of a stationary policy which is similar to Algorithm REF except that $\\xi _{out}$ , $\\xi _{relay}$ and $\\underline{V}^*$ are replaced by $\\xi _{out}^{(k-1)}$ , $\\xi _{relay}^{(k-1)}$ and $\\underline{V}^{(k-1)}$ respectively.",
"On the other hand, $\\overline{U}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ is the mean length of the $k$ -th link created under Algorithm REF (with $(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})$ as starting parameters), where the iterates are updated at each step between placement of the $(k-1)$ -th node and the $k$ -th node.",
"Let us denote by $\\mathcal {G}$ the set $[0,A_1] \\times [0,A_2]$ , defined by the following constraints: $-\\xi _{out} \\le 0, \\xi _{out} \\le A_1, -\\xi _{relay} \\le 0, \\xi _{relay} \\le A_2$ Clearly, projection onto the set $\\mathcal {G}$ is nothing but coordinate wise projection.",
"We rewrite the slower timescale iteration in (REF ) as (REF ) (note the definitions of the functions $f_1(\\xi _{out},\\xi _{relay})$ , $f_2(\\xi _{out},\\xi _{relay})$ , $g_1(\\underline{V},\\xi _{out},\\xi _{relay})$ , $g_2(\\underline{V},\\xi _{out},\\xi _{relay})$ , $l_1(\\underline{V},\\xi _{out},\\xi _{relay})$ and $l_2(\\underline{V},\\xi _{out},\\xi _{relay})$ in (REF )).",
"The random variables $M_{1}^{(k)}$ and $M_{2}^{(k)}$ are two zero mean Martingale difference noise sequences w.r.t.",
"$\\mathcal {F}_{k-1}$ (information available up to the $(k-1)$ -st placement instant); this happens due to i.i.d.",
"shadowing across links.",
"(REF ) has the form of a projected stochastic approximation (see [39]).",
"In order to show the desired convergence of the iterates in (REF ), we will use [39]; this requires us to check five conditions from [39], which is done in the next subsection.",
"$\\Box $"
],
[
"We will first present a lemma that will be useful for checking one condition.",
"Lemma 5 Under Assumption REF , the quantities $\\overline{\\Gamma }(\\underline{V},\\xi _{out},\\xi _{relay})$ , $\\overline{Q}_{out}(\\underline{V},\\xi _{out},\\xi _{relay})$ and $\\overline{U}(\\underline{V},\\xi _{out},\\xi _{relay})$ are continuous in $\\underline{V}$ , $\\xi _{out}$ and $\\xi _{relay}$ .",
"The proof is similar to that of Theorem REF .",
"Now, we will check conditions $A5.1.3$ , $A5.1.4$ , $A5.1.5$ , $A5.3.1.$ and $A5.3.2$ from [39].",
"Checking Condition $A5.1.3$ : We need $f_1(\\cdot ,\\cdot )$ and $f_2(\\cdot ,\\cdot )$ to be continuous functions; this holds by Theorem REF .$\\Box $ Checking Condition $A5.1.4$ : This condition is satisfied by the choice of the sequence $\\lbrace b(k)\\rbrace _{k \\ge 1}$ .$\\Box $ Checking Condition $A5.1.5$ : This condition requires that $\\lim _{k \\rightarrow \\infty }g_1(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ , $\\lim _{k \\rightarrow \\infty }g_2(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ , $\\lim _{k \\rightarrow \\infty }l_1(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ and $\\lim _{k \\rightarrow \\infty }l_2(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ almost surely.",
"We can find a probability 1 subset of the sample space $\\Omega $ , such that for any sample path in this subset the conclusions of Lemma REF and Lemma REF hold.",
"Take one such sample path $\\omega $ .",
"By Lemma REF , for this sample path $\\omega $ , we can find a compact subset $\\mathcal {C} \\subset \\mathbb {R}^B$ such that $(\\underline{V}^{(k)}, \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})$ lies inside the compact set $\\mathcal {C} \\times [0,A_1]\\times [0,A_2]$ for all $k \\ge 1$ along this sample path.",
"By Lemma REF and the fact that continuous functions are uniformly continuous over compact sets, we can say that $\\overline{Q}_{out}(\\underline{V}, \\xi _{out}, \\xi _{relay})$ , $\\overline{\\Gamma }(\\underline{V}, \\xi _{out}, \\xi _{relay})$ and $\\overline{U}(\\underline{V}, \\xi _{out}, \\xi _{relay})$ are uniformly continuous over the compact set $\\mathcal {C} \\times [0,A_1]\\times [0,A_2]$ .",
"Now, the Euclidean distance between $(\\underline{V}^{(k)}, \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})$ and $(\\underline{V}^*(\\xi _{out}^{(k)}, \\xi _{relay}^{(k)}), \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})$ converges to 0 along the sample path $\\omega $ .",
"Hence, by uniform continuity, we can say that $\\lim _{k \\rightarrow \\infty }|\\overline{Q}_{out}(\\underline{V}^{(k)}, \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})-\\overline{Q}_{out}(\\underline{V}^*(\\xi _{out}^{(k)}, \\xi _{relay}^{(k)}), \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})|=0$ and $\\lim _{k \\rightarrow \\infty }|\\overline{U}(\\underline{V}^{(k)}, \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})-\\overline{U}(\\underline{V}^*(\\xi _{out}^{(k)}, \\xi _{relay}^{(k)}), \\xi _{out}^{(k)}, \\xi _{relay}^{(k)})|=0$ along this sample path $\\omega $ .",
"Hence, $\\lim _{k \\rightarrow \\infty }g_1(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ and $\\lim _{k \\rightarrow \\infty }g_2(\\underline{V}^{(k-1)},\\xi _{out}^{(k-1)},\\xi _{relay}^{(k-1)})=0$ almost surely.",
"On the other hand, since $\\mathcal {C}$ is bounded, we can say that $\\lbrace \\underline{V}^{(k)} \\rbrace _{k \\ge 1}$ is bounded for the chosen $\\omega $ .",
"In a similar way as in the proof of Theorem REF , in case of Lemma REF we can show that $g(r,\\gamma )$ is continuous in $\\overline{V}$ , $\\xi _{out}$ and $\\xi _{relay}$ .",
"Now, between the placement of the $(k-1)$ -st relay and $k$ -th relay, at each step, $g(r,\\gamma )$ for all $r \\in \\lbrace 1,2,\\cdots ,B\\rbrace , \\gamma \\in \\mathcal {S}$ can change at most by an amount $K^* a(k-1-B)$ (for a suitable constant $K^*>0$ ), and hence we can claim that $\\lim _{k \\rightarrow \\infty }|\\overline{U}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})-\\overline{U}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})|=0$ , $\\lim _{k \\rightarrow \\infty }|\\overline{Q}_{out}^{\\prime }(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})-\\overline{Q}_{out}(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})|=0$ .",
"Hence, we obtain that $\\lim _{k \\rightarrow \\infty }l_1(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})=0$ and $\\lim _{k \\rightarrow \\infty }l_2(\\underline{V}^{(k-1)}, \\xi _{out}^{(k-1)}, \\xi _{relay}^{(k-1)})=0$ .",
"Also, $g_1(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ , $g_2(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ , $l_1(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ and $l_2(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)})$ are uniformly bounded across $k \\ge 1$ , since the outage probabilities and placement distances are bounded quantities.",
"Hence, this condition is satisfied.",
"Checking Condition $A5.3.1$ : This condition is easy to check, and done in [1].$\\Box $ Checking Condition $A5.3.2$ : This condition is easy to check, and done in [1].$\\Box $"
],
[
"Consider the function $h(\\xi _{out},\\xi _{relay}):=\\bigg (\\frac{f_1(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})},\\frac{f_2(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})} \\bigg )=\\bigg (\\frac{\\overline{Q}_{out}^*(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}-\\overline{q},\\frac{1}{\\overline{U}^*(\\xi _{out},\\xi _{relay})}-\\overline{N} \\bigg )$ and the map: $&& \\overline{\\Lambda }_{\\mathcal {G}}(h(\\xi _{out},\\xi _{relay})) \\nonumber \\\\&=& \\lim _{0<\\beta \\rightarrow 0} \\frac{\\Lambda _{\\mathcal {G}}\\bigg ((\\xi _{out},\\xi _{relay})+\\beta h(\\xi _{out},\\xi _{relay}))\\bigg )-(\\xi _{out},\\xi _{relay})}{\\beta } \\nonumber \\\\$ Lemma 6 If $(\\xi _{out},\\xi _{relay}) \\in [0,A_1] \\times [0,A_2]$ is a zero of $\\overline{\\Lambda }_{\\mathcal {G}}\\bigg (\\frac{f_1(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})},\\frac{f_2(\\xi _{out},\\xi _{relay})}{\\overline{U}^*(\\xi _{out},\\xi _{relay})} \\bigg )$ , then $(\\underline{V}^*(\\xi _{out},\\xi _{relay}),\\xi _{out},\\xi _{relay}) \\in \\mathcal {K}(\\overline{q},\\overline{N})$ , provided that $A_1$ and $A_2$ are chosen using the procedure described in Section .",
"The proof is similar to the proof of [1].",
"Now, by using similar arguments as in [1] and using [39], We can show that the iterates $(\\xi _{out}^{(k)}, \\xi _{relay}^{(k)})$ will converge almost surely to the set of stationary points of the o.d.e.",
"$(\\dot{\\xi }_{out}(t), \\dot{\\xi }_{relay}(t))=\\overline{\\Lambda }_{\\mathcal {G}}\\bigg (\\frac{f_1(\\xi _{out}(t),\\xi _{relay}(t))}{\\overline{U}^*(\\xi _{out}(t),\\xi _{relay}(t))},\\frac{f_2(\\xi _{out}(t),\\xi _{relay}(t))}{\\overline{U}^*(\\xi _{out}(t),\\xi _{relay}(t))} \\bigg )$ .",
"Using this result and using Lemma REF and Lemma REF , we obtain that $(\\underline{V}^{(k)},\\xi _{out}^{(k)},\\xi _{relay}^{(k)}) \\rightarrow \\mathcal {K}(\\overline{q},\\overline{N})$ almost surely, where $k\\delta $ can be the distance from the sink or $k$ can be the index of a placed relay node (the result holds for both interpretations of $k$ ).",
"This completes the proof of Theorem REF .",
"$\\Box $"
]
] | 1709.01566 |
[
[
"Implementation and validation of two-phase boiling flow models in\n OpenFOAM"
],
[
"Abstract Prediction of two-phase boiling flows using the computational fluid dynamics (CFD) approach is very challenging since several sub-models for interfacial mass, momentum and energy transfer in such flows are still not well established and require further development and validation.",
"Once validating a particular model, it is important that all key parameter involved in the model are carefully verified.",
"Such verification is typically performed by separate effect tests, where one parameter at a time is compared to a measured or otherwise known value.",
"Needless to say that for complex models, which are typical for CFD applications to two-phase flow, the number of independent parameters that need to be verified can be quite high.",
"This particular feature makes the validation process of complex CFD models in open source codes very attractive, since full access to the implementation details is possible.",
"This paper is concerned with implementation and validation of two-phase boiling bubbly flow models using the OpenFOAM, open source environment.",
"The model employs the two-fluid formulation of the conservation equations with the Reynolds-averaged treatment of the turbulent terms.",
"The model consists of six conservation equations for the liquid and the vapor phase, allowing for the thermodynamic non-equilibrium and compressibility of both phases.",
"In addition, the model includes two transport equations for the turbulence kinetic energy and energy dissipation and one transport equation for the interfacial area concentration.",
"New models for wall heat partitioning as well as for the phase change terms in nucleate boiling have been implemented.",
"Sensitivity studies as well as validation of the model against measured data available in the open literature have been performed and it has been shown that a reasonable agreement between predictions and experiments has been achieved."
],
[
"Introduction",
"One of the important issues of the current and future sustainable energy systems is the efficiency and stability of heat removal due to natural or mixed convection, forced convection or boiling heat transfer.",
"In some energy systems natural heat convection is envisaged during normal operation.",
"This type of heat removal is very reliable since it doesn't depend on availability of external pumping resources, and coolant flow through the system is assured by the gravity force.",
"The drawback of the natural circulation is its inherent instability and also relatively low heat transfer efficiency.",
"Thus, in many high heat flux technologies, such as e.g.",
"nuclear reactors, the boiling heat transfer is preferred as the most efficient heat transfer mode.",
"The design of high heat flux systems requires a thorough fluid flow and heat transfer analysis in complex geometries.",
"Traditionally experimental methods have been used for these purposes in the past.",
"The drawback of such methods is their large cost and time consumption, inherently related to all required experimental work.",
"In addition, experimental methods are rather difficult to be used for a design optimization, where various geometry and/or operation condition variations are to be tested.",
"For such purposes the most efficient design and optimization approach is based on computational tools, which are able to capture the geometry details and to include the governing phenomena.",
"Currently the computational fluid dynamics (CFD) technology is widely used to design and to optimize heat transfer and fluid flow systems if single-phase flow conditions prevail.",
"For two-phase flow applications, and in particular for boiling heat transfer conditions the CFD technology is still not mature enough.",
"In particular, there is still lack of thoroughly validated and generally valid closure laws for subcooled and saturated nucleate flow boiling heat transfer, with a potential to be extended to predict the departure from nucleate boiling (DNB).",
"The major aim of this paper is to contribute with new model development and validation in this particular area using open source CFD code OpenFOAM.",
"The first model suitable for CFD applications was developed by [15], who proposed a heat flux partitioning scheme to separately deal with vapor generation, sensible heat and quenching terms in the proximity of the heated wall.",
"In the bulk bubbly flow, [8] proposed a two-equation model to predict the bubble size (and thus the interfacial area concentration) as a function of local flow conditions."
],
[
"Field equation in two-phase bubbly flow",
"The present model includes mass, linear momentum and energy conservation equations for liquid and vapor phase.",
"In addition, transport equations for the interfacial area concentration and for the turbulence are used to close the model.",
"The details of the employed governing equations are given below."
],
[
"Phase continuity equation",
"$\\Gamma _k $ means the mass gained by phase $k$ .",
"($k=l,v$ )"
],
[
"Linear momentum conservation equation",
"Here the interfacial velocity is modeled as ${ \\mathbf {U}}_{ki}={\\left\\lbrace \\begin{array}{ll}{ \\mathbf {U}}_{l} & \\text{if}\\ \\Gamma _{v}>0,\\ \\text{evaporation} \\\\{ \\mathbf {U}}_{v} & \\text{if}\\ \\Gamma _{v}<0,\\ \\text{condensation}\\end{array}\\right.",
"}$ using the upwind scheme.",
"According to the Boussinesq hypothesis, the turbulent stress strain relation is analogous to that of Newtonian fluids and consequently the effective stress appears as a function of fluid properties and velocity, which is used by [19] in OpenFOAM, ${\\mathbf {\\tau }}_k^{\\rm eff}={{\\mathbf {\\tau }}_k} + {\\mathbf {\\tau }}_k^{\\rm t}=\\rho _k\\nu _k^{\\rm eff}\\left( \\nabla { \\mathbf {U}}_{k} + \\left( \\nabla { \\mathbf {U}}_{k}\\right)^T - \\frac{2}{3}\\mathbf {I} \\nabla \\cdot { \\mathbf {U}}_{k} \\right)-\\frac{2}{3}\\mathbf {I} \\rho _k k_k$ and, $\\nu _k^{\\rm eff}=\\nu _k+\\nu _k^{\\rm t}$"
],
[
"Enthalpy equation",
"where $a_w$ refers to heated area per unit controlled volume of fluid between the wall and the liquid phase.",
"[16] discussed the mass conservation and energy conservation at the interface and first proposed the corresponding equations in two-phase flow.",
"Here we formulate the mass flux $\\Gamma _l$ from phase $v$ to phase $l$ furthermore as, $\\Gamma _{l}={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{a_i q_{li}^{\\prime \\prime }+a_iq_{vi}^{\\prime \\prime }}{h_{v}-h_{l,\\rm sat}} &\\text{condensation} \\\\\\displaystyle \\frac{a_i q_{li}^{\\prime \\prime }+a_iq_{vi}^{\\prime \\prime }}{h_{v,\\rm sat}-h_{l}} &\\text{evaporation}\\end{array}\\right.",
"}$ where the interfacial enthalpy $h_{ki}\\:(k=l,v)$ is modeled with the upwind approximation.",
"The modeling of interfacial heat transfer $a_i q_{li}^{\\prime \\prime }$ and $a_i q_{vi}^{\\prime \\prime }$ will be introduced in the following section.",
"Equation REF could be applied to the heat transfer in the bulk.",
"For those cells which are adjacent to the wall directly, we have totally different heat transfer mechanism since there are interaction among the liquid, vapor and walls.",
"Here we assume that only evaporation is allowed in those cells, which is consistent with the situation in boiling flows.",
"In those cells, the total heat transfer per unit volume to phase $l$ is given as, $q^{\\prime \\prime \\prime }_l =a_i q_{li}^{\\prime \\prime } -\\Gamma _{vl}h_{l}+a_w { {q}^{\\prime \\prime }_{lw}}$ and the total heat transfer to phase $v$ as, $q^{\\prime \\prime \\prime }_v =a_i q_{vi}^{\\prime \\prime } +\\Gamma _{vl}h_{v}+a_w { {q}^{\\prime \\prime }_{vw}}$ The energy balance in those cells could be written as $q^{\\prime \\prime \\prime }_l+q^{\\prime \\prime \\prime }_v=a_w{q}^{\\prime \\prime }_{w}$ Usually we make an assumption that in subcooled flow boiling, the temperature of the vapor phase is constant and equal to the saturation temperature.",
"In addition, we neglect a direct heating of vapor from the wall, that is: $a_w { {q}^{\\prime \\prime }_{vw}}=0$ .",
"With these assumptions it is straightforward to calculate the heat flux to each phase in cells adjacent to the heated walls.",
"Using the Fourier's law of conduction for the liquid phase, the molecular heat flux in Eqn.",
"REF can be written as, ${\\mathbf {q}}_k^{\\prime \\prime }=-\\frac{\\lambda _l}{c_{pl}}\\nabla h_l$ where $\\lambda $ and $c_p$ are respectively the thermal conductivity and the specific heat.",
"The turbulent heat flux is found as follows, ${\\mathbf {q}}_l^{\\rm t}=-\\frac{\\lambda _l^{\\rm t}}{c_{pl}}\\nabla h_l$ where the turbulent thermal conductivity is given as,, ${\\lambda _l^{\\rm t}}=\\frac{c_{pl}\\rho _l\\nu _l^{\\rm t}}{ {\\rm Pr}_l^{\\rm t}}$ where Pr$_l^{\\rm t}$ is the turbulent Prandtl number of phase $l$ .",
"A constant value of 0.9 has been chosen for Pr$_l^{\\rm t}$ in the calculations presented in this paper.",
"In OpenFOAM, equation REF of liquid phase is reorganized into a phase intensive form, $&\\frac{\\partial {h_l}}{\\partial t}+{\\mathbf {U}}_l \\cdot \\nabla {h_l} - \\nabla \\cdot (\\kappa _l^{\\rm eff}\\nabla h_l)-\\kappa _l^{\\rm eff} \\frac{\\nabla (\\beta \\rho _l)}{\\beta \\rho _l}\\cdot \\nabla h_l \\cr =&{\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{\\Gamma _{lv} h_{l,\\rm sat}- \\Gamma _{lv} h_l + a_i {q}^{\\prime \\prime }_{li} }{\\beta \\rho _l} &\\text{bulk condensation} \\\\\\displaystyle \\frac{a_i {q}^{\\prime \\prime }_{li} }{\\beta \\rho _l} &\\text{bulk evaporation} \\\\\\displaystyle \\frac{a_i {q}^{\\prime \\prime }_{li} +a_w { {q}^{\\prime \\prime }_{lw}}}{\\beta \\rho _l} &\\text{near wall cells}\\end{array}\\right.",
"}$ where, $\\kappa _l^{\\rm eff}= \\frac{\\lambda _l}{\\rho _l c_{pl}}+\\frac{\\nu _l^{\\rm t}}{{\\rm Pr}_l^{\\rm t}}$ The term $\\displaystyle \\frac{a_w { {q}^{\\prime \\prime }_{lw}}}{\\beta \\rho _l}$ on the right hand side (RHS) of Eqn.",
"REF results from the thermal boundary condition at heated walls.",
"Thus we treat this term by a gradient boundary condition in the energy transport equation.",
"In a similar manner, equation REF of the vapor phase is given as follows, $&\\frac{\\partial {h_v}}{\\partial t}+{\\mathbf {U}}_v \\cdot \\nabla {h_v} - \\nabla \\cdot (\\kappa _v^{\\rm eff}\\nabla h_v)-\\kappa _v^{\\rm eff} \\frac{\\nabla (\\alpha \\rho _v)}{\\alpha \\rho _v}\\cdot \\nabla h_v \\cr =&{\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{ a_i {q}^{\\prime \\prime }_{vi} }{\\alpha \\rho _v} &\\text{bulk condensation}\\\\\\displaystyle \\frac{\\Gamma _{vl} h_{v,\\rm sat}- \\Gamma _{vl} h_v + a_i {q}^{\\prime \\prime }_{vi} }{\\alpha \\rho _v} &\\text{bulk evaporation }\\\\\\displaystyle \\frac{ a_i {q}^{\\prime \\prime }_{vi} }{\\alpha \\rho _v} &\\text{near wall cells}\\end{array}\\right.",
"}$ where, $\\kappa _v^{\\rm eff}= \\frac{\\lambda _v}{\\rho _v c_{pv}}+\\frac{\\nu _v^{\\rm t}}{{\\rm Pr}_v^{\\rm t}}$"
],
[
"Interfacial area concentration transport equation",
"The interfacial area concentration corresponds to the area of the gas bubbles per unit volume.",
"For spherical bubbles, $a_i=\\frac{6\\alpha }{D_S}$ where $D_S$ is the bubble Sauter diameter, equal to the diameter of a sphere of an equivalent volume.",
"[8] modeled sink and source terms of the interfacial area concentration based on mechanisms of bubble-bubble and bubble-turbulent eddy random collisions, and they also introduced the effect by gas expansion, $\\frac{\\partial a_i}{\\partial t}+\\nabla \\cdot (a_i {\\mathbf {U}}_v )=\\frac{2}{3}\\frac{a_i}{\\alpha }\\left( \\frac{\\partial \\alpha }{\\partial t}+\\nabla \\cdot (\\alpha {\\mathbf {U}}_v) \\right)+ \\Phi _{\\rm BB}+ \\Phi _{\\rm BC}+\\Phi _{\\rm NUC}$ The first term on the RHS of Eqn.",
"REF refers to the contribution of phase change and expansion due to the pressure change.",
"$\\Phi _{\\rm BB}$ and $\\Phi _{\\rm BC}$ represent the bubble number variations induced by the breakup and coalescence phenomena, respectively.",
"In the [8] model, they are defined as, $\\Phi _{\\rm BC} = -\\frac{1}{3\\psi }\\left(\\frac{\\alpha }{a_i}\\right)^2\\cdot \\Gamma _C \\frac{\\alpha ^2 \\epsilon _l^{1/3}}{D_S^{11/3}(\\alpha _{\\max }-\\alpha )}\\exp \\left( -K_C \\frac{D_S^{11/3}\\rho _l^{1/2}\\epsilon _l^{1/3}}{\\sigma ^{1/2}} \\right)$ with $\\Gamma _C=0.0314$ and $K_C=1.29, \\alpha _{\\max }=0.74$ , and $\\Phi _{\\rm BB} = \\frac{1}{3\\psi }\\left(\\frac{\\alpha }{a_i}\\right)^2\\cdot \\Gamma _B \\frac{\\alpha (1-\\alpha ) \\epsilon _l^{1/3}}{D_S^{11/3}(\\alpha _{\\max }-\\alpha )}\\exp \\left( -K_B \\frac{\\sigma }{D_S^{5/3}\\rho _l\\epsilon _l^{2/3}}\\right)$ with $\\Gamma _B=0.0209$ and $K_B=1.59$ .",
"Here $\\psi =1/(36\\pi )$ for spherical bubbles.",
"$\\Phi _{\\rm NUC}$ refers to an increase of interfacial area concentration by a bubble nucleation at the heated wall.",
"[2] proposed the nucleation source term as, $\\Phi _{\\rm NUC} = \\pi d_{lo}^2 \\cdot {N^{\\prime \\prime } f a_w}$ where $d_{lo}$ is the bubble lift-off diameter, $N^{\\prime \\prime }$ the active nucleation site density, and $f$ the bubble departure frequency.",
"[26] proposed the breakup and coalescence term as, $\\Phi _{\\rm BC} = -\\frac{1}{3\\psi }\\left(\\frac{\\alpha }{a_i}\\right)^2\\cdot K_{c1}\\frac{\\alpha ^2 \\epsilon _l^{1/3}}{D_S^{11/3}}\\frac{1}{1-(\\alpha /\\alpha _{\\max })^{1/3}+K_{c2}\\alpha \\sqrt{{\\rm We}/{{\\rm We}_{cr}}}}\\exp \\left( -K_{c3} \\sqrt{\\frac{\\rm We}{{\\rm We}_{cr}}}\\right)$ where $K_{c1}$ = 2.86, $K_{c2}$ = 1.922, $K_{c3}$ = 1.017, ${\\rm We}_{cr}$ = 1.24 and $\\alpha _{\\max }$ = 0.52.",
"$\\Phi _{\\rm BB} = \\frac{1}{3\\psi }\\left(\\frac{\\alpha }{a_i}\\right)^2\\cdot K_{b1}\\frac{\\alpha (1-\\alpha ) \\epsilon _l^{1/3}}{D_S^{11/3}} \\frac{1}{1+K_{b2}(1-\\alpha )\\sqrt{{\\rm We}/{\\rm We}_{cr}}} \\exp \\left( -\\frac{{\\rm We}_{cr}}{{\\rm We}} \\right)$ where $K_{b1}=1.6$ , and $K_{b2}=0.42$ .",
"[18] proposed a $S_\\gamma $ model in which the breakup terms can be written down as, $\\Phi _{\\rm BB} = \\pi \\int _{D_{S_{cr}}}^\\infty \\frac{(2^{1/3}-1)D_S^2}{\\tau _{br}}nP{\\rm d}D_S$ Here $n=\\displaystyle \\frac{6\\alpha }{\\pi D_S^3}$ is the bubble number density.",
"$P$ represents the log-normal distribution of bubble diameter, $P=\\frac{1}{\\sqrt{2\\pi }D_S\\hat{\\sigma }} \\exp \\left( -\\frac{({\\rm ln}D_S -{\\rm ln}\\overline{D_S})^2}{2\\hat{\\sigma }^2} \\right)$ where we use $\\hat{\\sigma }=0.5$ in the current solver.",
"In the original paper, the breakup source term is modeled in two regimes: the viscous breakup regime and inertia breakup regime.",
"And the overall source term should be summed up over the two regimes.",
"However, since the mechanism is not well explained in the viscous regime, only the inertial breakup part is included in the current solver.",
"The Kolmogorov length scale $L_k$ is used to evaluate the regime that breakup takes place.",
"$L_k=\\left( \\frac{\\nu ^3}{\\epsilon } \\right)^{1/4}$ Considering that only those bubbles of big size can break, the critical size in the inertia regime becomes as follows, $D_{Scr}=(1+C_\\alpha \\alpha )\\left( \\frac{\\sigma {\\rm We}_{cr}}{2\\rho _l} \\right)^{3/5} \\epsilon ^{-2/5}$ $\\tau _{br}= 2\\pi k_{br}\\sqrt{\\frac{(3\\rho _v+2\\rho _l)D_S^3}{192\\sigma }}$ with $C_{\\alpha }=0$ and $k_{br}=0.2$ .",
"The source term from bubble coalescence is modeled as, $\\Phi _{\\rm BC} = \\pi (2^{1/3}-2)\\left(\\frac{6\\alpha }{\\pi }\\right)^2k_{coll}U_r P_{coal}D_S^{-2}$ where, $k_{coll}=\\left(\\frac{2\\pi }{15}\\right)^{1/2}$ $U_r = (\\epsilon D_S)^{1/3}$ $P_{coal} = \\frac{\\Phi _{\\rm max}}{\\pi }\\left( 1-\\frac{k_{cl,2}^2 ({\\rm We}-{\\rm We}_0)^2}{16\\Phi _{\\rm max}^2} \\right)^{1/2}$ $\\Phi _{\\rm max}=\\frac{8h_0^2\\rho _l\\sigma }{{\\rm We}_0\\mu _v^2D_S}$ with the following coefficient: $k_{cl,2}=12.7$ , ${\\rm We}_0=0.8{\\rm We}_{cr}$ and $h_0=8.3h_{cr}$ $h_{cr}=\\left( \\frac{A_HD_S}{24\\pi \\sigma } \\right)^{1/3}$ where $A_H=5.0 \\times 10^{-21}$ is the Hamaker constant."
],
[
"Turbulence of liquid phase",
"[19] proposed the standard $k-\\epsilon $ model as follows, $\\frac{\\partial (\\beta \\rho _l k_l)}{\\partial t}+\\nabla \\cdot (\\beta \\rho _l \\mathbf {U}_l k_l)=\\nabla \\cdot \\left[ \\beta \\left( \\frac{\\mu _l^{\\rm eff}}{\\sigma _k} \\right)\\nabla k_l \\right]+\\beta G-\\beta \\rho _l \\epsilon _l$ $\\frac{\\partial (\\beta \\rho _l \\epsilon _l)}{\\partial t}+\\nabla \\cdot (\\beta \\rho _l \\mathbf {U}_l \\epsilon _l)=\\nabla \\cdot \\left[ \\beta \\left( \\frac{\\mu _l^{\\rm eff}}{\\sigma _\\epsilon }\\right)\\nabla \\epsilon _l \\right]+\\frac{\\beta \\epsilon _l}{k_l}(C_{\\epsilon 1}G-C_{\\epsilon 2}\\rho _l \\epsilon _l)$ Here $G$ stands for the production of turbulent kinetic energy and is defined as, $G=2\\mu _l^{\\rm t}\\left(\\nabla \\mathbf {U}_l \\cdot {\\rm dev}(\\nabla \\mathbf {U}_l + (\\nabla \\mathbf {U}_l)^T)\\right)$ In the above model, no effect of the dispersed phase on the turbulence in the continuous phase is taken into account.",
"This deficiency is removed in the model proposed by [26], where an additional source term, representing the above-mentioned effect, is included, $\\frac{\\partial (\\beta \\rho _l k_l)}{\\partial t}+\\nabla \\cdot (\\beta \\rho _l \\mathbf {U}_l k_l)=&\\nabla \\cdot \\left[ \\beta \\left( \\frac{\\mu _l^{\\rm t}}{\\sigma _k} \\right)\\nabla k_l \\right]-\\beta \\rho _l \\epsilon _l + \\beta {\\mathbf {\\tau }}_l : \\nabla {\\mathbf {U}}_l\\cr &- (\\mathbf {M}_v^d + \\mathbf {M}_v^{vm})\\cdot (\\mathbf {U}_v - \\mathbf {U}_l)-\\sigma (\\Phi _{\\rm BC}+\\Phi _{\\rm BB})+k_{li}\\Gamma _l$ $\\frac{\\partial (\\beta \\rho _l \\epsilon _l)}{\\partial t}+\\nabla \\cdot (\\beta \\rho _l \\mathbf {U}_l \\epsilon _l)=&\\nabla \\cdot \\left[ \\beta \\left( \\frac{\\mu _l^{\\rm t}}{\\sigma _\\epsilon }\\right)\\nabla \\epsilon _l \\right]-C_{\\epsilon 2}\\beta \\rho _l\\frac{ \\epsilon _l^2}{k_l} + C_{\\epsilon 1} \\beta \\frac{ \\epsilon _l}{k_l} {\\mathbf {\\tau }}_l : \\nabla {\\mathbf {U}}_l-\\frac{2}{3} \\beta \\rho _l {\\epsilon _l} \\nabla \\cdot {\\mathbf {U}}_l\\cr &- C_{\\epsilon 3}(\\mathbf {M}_v^d + \\mathbf {M}_v^{vm})\\cdot (\\mathbf {U}_v - \\mathbf {U}_l) \\left( \\frac{\\epsilon _l}{D_S^2} \\right)^{1/3}+\\epsilon _{li}\\Gamma _l$ The liquid Reynolds stress tensor is modeled as, ${\\mathbf {\\tau }}_l = \\rho _l \\nu _l^{\\rm t} \\left( \\nabla {\\mathbf {U}}_l + (\\nabla {\\mathbf {U}}_l)^T \\right)-\\frac{2}{3}\\rho _l \\left( k_l+\\nu _l^{\\rm t} \\nabla \\cdot {\\mathbf {U}}_l \\right){\\mathbf {I}}$ The turbulent viscosity of liquid phase is given by [20] as, $\\nu _l^{\\rm t} = C_\\mu \\frac{k^2}{\\epsilon }+\\frac{1}{2}C_{\\mu b}D_S \\alpha |\\mathbf {U}_v - \\mathbf {U}_l|$ The coefficients used in this work are $\\sigma _k=1.0$ , $\\sigma _\\epsilon =1.3$ , $C_{\\epsilon 1}=1.44$ , $C_{\\epsilon 2}=1.92$ , $C_{\\epsilon 3}=0.6$ , $C_\\mu =0.09$ and $C_{\\mu b}=1.2$ ."
],
[
"Turbulence of vapor phase",
"The turbulence of vapor phase is assumed to be dependent on that of the liquid phase.",
"To this end, a turbulence response coefficient $C_t$ , defined as the ratio of the root mean square values of dispersed phase velocity, is introduced.",
"In this approach, the effective viscosity of the vapor phase is expressed as $\\nu _v^{\\rm eff}=\\nu _v+C_{t}^2\\nu _l^{\\rm t}$ In a more elaborated model, $C_t$ could be calculated as a function of local parameters, such as e.g.",
"void fraction.",
"However, in the present approach the influence of the liquid phase is neglected and $C_t$ is set equal to zero."
],
[
"Interfacial momentum transfer closure laws",
"The interfacial forces acting on a bubble are caused by the liquid which surrounds it.",
"Ignoring the effect of the change of the mean curvature on the mixture momentum source, we have, $\\mathbf {M}_v +\\mathbf {M}_l=0$ The closure relationships for the interfacial forces are expressed in terms of the following non-dimensional numbers, Eotvos number, ${\\rm Eo}=\\frac{(\\rho _l-\\rho _v)gD_S^2}{\\sigma }$ Reynolds number, ${\\rm Re}_b = \\frac{|\\mathbf {U}_v -\\mathbf {U}_l|D_S}{\\nu _l}$ ${\\rm Re}_{bm} = \\frac{\\rho _l|\\mathbf {U}_v -\\mathbf {U}_l|D_S}{\\mu _m}$ Here, $\\mu _m = \\mu _l \\left( 1-\\frac{\\alpha }{\\alpha _{\\rm max}} \\right)^{-2.5\\alpha _{\\rm max}\\mu ^\\ast }$ $\\mu ^\\ast =\\frac{\\mu _v+0.4\\mu _l}{\\mu _v+\\mu _l}$ The interfacial momentum transfer terms include different kinds of forces, each of them representing a separate physical phenomenon, including the drag force, the lift force, the wall lubrication force, the turbulent dispersion force and the virtual mass force, which constitute the total interfacial force as follows, $\\mathbf {M}_v = \\mathbf {M}_v^d+\\mathbf {M}_v^l+\\mathbf {M}_v^{wl}+\\mathbf {M}_v^{td}+\\mathbf {M}_v^{vm}$"
],
[
"Drag force",
"This force represents a resistance of the relative motion between two phases.",
"$\\mathbf {M}_v^d = -\\frac{3}{4}\\frac{C_{ds}}{D_S}\\rho _l \\alpha |\\mathbf {U}_v -\\mathbf {U}_l|(\\mathbf {U}_v -\\mathbf {U}_l)$ The following two models for the drag force coefficient are included in the current solver: [21], $C_{ds}=\\max \\left(\\frac{24}{{\\rm Re}_b}(1+0.15{\\rm Re}_b^{0.687}), 0.44 \\right)$ [10], $C_{ds}=\\max \\left(\\frac{24}{{\\rm Re}_{bm}}(1+0.15{\\rm Re}_{bm}^{0.687}), 0.44 \\right)$"
],
[
"Lift force",
"When a particle travels through the fluid with a non-uniform lateral velocity field, a lateral force will be acting between the fluid and the particle, $\\mathbf {M}_v^l = C_l \\rho _l \\alpha (\\mathbf {U}_v -\\mathbf {U}_l)\\times \\nabla \\times \\mathbf {U}_l$ In the present model the lift coefficient $C_l$ is calculated from the [24] model, $C_l ={\\left\\lbrace \\begin{array}{ll}{\\rm min}(0.288{\\rm tanh}(0.121{\\rm Re}_b),f({\\rm Eo}_d)) & {{\\rm Eo}_d < 4} \\\\f({\\rm Eo}_d)& {4<{\\rm Eo}_d < 10} \\\\-0.27& {\\rm Eo}_d > 10\\end{array}\\right.",
"}$ $f({\\rm Eo}_d)=0.001509{\\rm Eo}_d^3 -0.0159{\\rm Eo}_d^2 -0.0204{\\rm Eo}_d +0.474$ Here, ${\\rm Eo}_d= \\frac{(\\rho _l-\\rho _v)gd_h^2}{\\sigma }$ $d_h = D_S (1+0.163{\\rm Eo}^{0.757})^{1/3}$ It should be noted that the force is turned off in the cells adjacent to walls in order to avoid unexpected fluctuation of void fraction in those cells in numerical simulation."
],
[
"Wall lubrication force",
"This force was first proposed by [1] in order to explain the near wall void fraction features.",
"$\\mathbf {M}_v^{wl} =C_{w} \\rho _l \\alpha |\\mathbf {U}_r -(\\mathbf {U}_r \\cdot \\mathbf {n}_w )\\mathbf {n}_w |^2 \\mathbf {n}_w$ The following two models for the wall lubrication force coefficient are included in the current solver: [24], $C_{w} = \\frac{1}{2}C_{wl}D_S \\left( \\frac{1}{y_w^2} - \\frac{1}{(D_{pipe}-y_w)^2} \\right)$ $C_{wl} ={\\left\\lbrace \\begin{array}{ll}0.47& {{\\rm Eo} < 1} \\\\\\exp (-0.933{\\rm Eo}+0.179)& {1<{\\rm Eo}< 5} \\\\0.00599{\\rm Eo}-0.0187& {5<{\\rm Eo}< 33} \\\\0.179& {{\\rm Eo} > 33}\\end{array}\\right.",
"}$ [6], $C_{w} = C_{wl} \\max \\left( 0, \\frac{1}{C_{wd}}\\frac{1-y_w/C_{wc}D_S}{y_w(y_w/C_{wc}D_S)^{p-1}} \\right)$ It is suggested that $C_{wc}=10.0$ , $C_{wd}=6.8$ and $p=1.7$ ."
],
[
"Turbulent dispersion force",
"The turbulent dispersion force accounts for the turbulent fluctuations of the liquid phase and the effects, which the fluctuations have on the distribution of the gas phase.",
"The following models are currently included in the solver: [7], $\\mathbf {M}_v^{td} =-C_{d} \\frac{3}{4} \\frac{\\rho _l}{D_S} \\frac{\\nu _l^t}{\\sigma _\\alpha } |\\mathbf {U}_r | \\nabla \\alpha $ [3], $\\mathbf {M}_v^{td} =-C_{td} \\rho _l k_l \\nabla \\alpha $"
],
[
"Virtual mass force",
"Currently it is assumed that $C_{vm}=0.5$ ."
],
[
"Liquid-vapor interfacial heat transfer closure laws",
"[26] proposed the following model for the liquid phase interfacial heat transfer, $a_i {q}^{\\prime \\prime }_{li} ={\\left\\lbrace \\begin{array}{ll}c_{li} a_i (h_{l,\\rm sat}-h_l) & \\text{bulk} \\\\\\text{not specified} & \\text{near wall cells}\\end{array}\\right.",
"}$ and, $c_{li} = \\frac{\\lambda _l}{c_{pl}D_S}\\rm Nu$ The Nusselt number is ${\\rm Nu}={\\left\\lbrace \\begin{array}{ll}2+0.6{\\rm Re}^{0.5}{\\rm Pr}^{0.33} & \\text{if}\\ {\\rm Ja}<0,\\ \\text{condensation} \\\\\\max (\\rm Nu_1, Nu_2, Nu_3) & \\text{if}\\ {\\rm Ja}>0,\\ \\text{evaporation}\\end{array}\\right.",
"}$ where, ${\\rm Ja}= \\frac{\\rho _lc_{pl}(T_l-T_{\\rm sat})}{\\rho _vh_{fg}},\\:{\\rm Re}=\\frac{D_SU_r}{\\nu _l},\\:{\\rm Pe}=\\frac{D_SU_r}{\\kappa _l}$ ${\\rm Nu}_1= \\sqrt{\\frac{4\\rm Pe}{\\pi }},\\:{\\rm Nu}_2= \\frac{12}{\\pi }{\\rm Ja},\\:{\\rm Nu}_3= 2$ The interface to vapor heat transfer is expressed in the following manner, $a_i {q}^{\\prime \\prime }_{vi} = c_{vi} (h_{v,\\rm sat}-h_v)$ $c_{vi} =\\frac{\\alpha \\rho _v}{\\delta t}$ where $\\delta t$ is numerical time step.",
"The above equations make sure that the vapor temperature is very close to the saturation temperature."
],
[
"Subcooled nucleate boiling model",
"The wall heat transfer model for subcooled boiling flow was first proposed by [15], who partitioned the wall heat flux into three components: single phase convection, transient conduction as well as evaporation.",
"The heat transfer coefficient for each process is correlated against experiment respectively.",
"More recent work is done by [23] and they believe that the total heat flux is assumed to be additively composed of a forced convective and a nucleate boiling component."
],
[
"Single phase convective heat transfer",
"The single phase forced convection heat flux outside the influence area is calculated by [15] as, $q^{\\prime \\prime }_{c}=h_{fc}A_{1 \\Phi }(T_w-T_l)$ where $h_{fc}$ is the single phase liquid heat transfer coefficient, $A_{1 \\Phi }$ is the area fraction dominated by single phase convection, $T_w$ is wall temperature and $T_l$ is the subcooled liquid temperature.",
"The single phase forced convective heat transfer coefficient $h_{fc}$ is modeled as, $h_{fc}=\\rho _l c_{pl} \\frac{u_{\\tau }}{T^+}$ where the dimensionless temperature is modeled by [11], $T^+ =\\Pr y^+ \\exp (-\\eta )+(2.12 \\ln y^+ + \\beta _t) \\exp (-1/\\eta )$ and, $\\beta _t=(3.85{\\rm Pr}^{1/3}-1.3)^2+2.12\\ln \\Pr $ $\\eta =\\frac{0.01(\\Pr y^+)^4}{1+5{\\rm Pr}^3y^+}$ $y^+=\\frac{\\rho _lu_{\\tau }|\\mathbf {U}_l|}{\\mu _l}$ The friction velocity is coupled with $k-\\epsilon $ model, $u_{\\tau }=C_\\mu ^{0.25}\\sqrt{k};$"
],
[
"Quenching heat transfer",
"The quenching (or transient conduction) heat flux is modeled as, $q^{\\prime \\prime }_{q}=h_q A_{b}(T_w-T_l)$ where $A_{b}$ represents the bubble influenced area fraction.",
"According to [15], the bubble influenced area is determined by $A_{b}=\\min \\left[ 1, N^{\\prime \\prime } K \\left( \\frac{\\pi d^2_{lo}}{4} \\right) \\right]$ Here $K$ determines the size of the bubble influence area around the nucleation site on the surface.",
"$K=4$ is recommended by [5].",
"The quenching heat transfer coefficient is given by [5], $h_q =2\\frac{\\lambda _l}{\\sqrt{\\pi \\kappa _l t}}$ where $t=0.8/f$ represents the life span that the quenching heat flux experiences."
],
[
"Evaporation heat transfer",
"The evaporation rate is calculated as, $\\Gamma _{vl}=\\frac{\\pi }{6}d_{lo}^2\\rho _vfN^{\\prime \\prime }a_w$"
],
[
"Bubble detachment size",
"There are quite a few models to calculate the lift-off diameter & departure diameter.",
"[25] made a correlation of bubble detachment diameter which is validated with pressure from 0.1 to 17.7 Mpa, heat flux from 0.47 to 10.64 MW/m$^2$ , inlet velocity from 0.08 to 9.15 m/s, inlet subcooling from 3.0 to 86 K. [22] developed a bubble lift-off model based on force analysis.",
"Their test runs were performed at 1 bar, and the model was validated with heat flux from 60.7 to 206 kW/m$^2$ , inlet velocity from 0.487 to 0.939 m/s, inlet subcooling from 1.5 to 20 K. [13] developed a correlation against the experimental data directly, $d_{lo}=d_{\\rm ref}\\exp \\left( -\\frac{T_{\\rm sat}-T_l}{\\Delta T_{\\rm refd}} \\right)$ where the reference value could be found at [13] for certain experiment."
],
[
"Bubble detachment frequency",
"A simple estimation of the bubble departure frequency as the terminal rise velocity over the departure size is used here, [4], $f=\\sqrt{\\frac{4}{3}\\frac{(\\rho _l-\\rho _v)g}{\\rho _l d_{lo}}}$"
],
[
"Active nucleation site density",
"A few models have been implement in the current solver [17], [9], [12], [13].",
"Here the [13] model is used for the validation.",
"$N^{\\prime \\prime }=N_{\\rm ref}\\left( \\frac{T_w-T_l}{\\Delta T_{\\rm refN}} \\right)^p$ The reference value can be found in [13]."
],
[
"Liquid bulk temperature",
"Another issue arises from the bulk liquid temperature.",
"Here we used $T_{bulk}=T_w-\\frac{T^{+}_{y^{+}_{bulk}}}{T^{+}_{y^{+}_{cell}}}(T_w-T_{cell})$ which is already implemented in ANSYS CFX5.",
"The bulk temperature is obtained by setting $y^{+}_{bulk}=250$ .",
"Here the subscript cell refers to the cells adjacent to walls."
],
[
"Test cases",
"Two data sets were considered in calculations: the void fraction measurements performed by Bartolomej for subcooled boiling heat transfer to water under 45 bar pressure [14], [12] and subcooled boiling heat transfer to refrigerant R-12 performed in the DEBORA experiment [26], [13].",
"The experiment conditions used as test case are listed in Table REF .",
"Table: Selected test cases and their experiment conditionsThe tests were simulated in a quasi-two-dimensional cylindrical geometry, with 100 meshes in the axial direction and 20 meshes in the radial direction.",
"The center of the grid cell adjacent to the wall has a non-dimensional coordinate of $y^+ = 60$ in Bartolomej test and $y^+ = 100$ in DEBORA test, approximately.",
"Grid refinement study performed by [13] for the DEBORA experiment indicates that these values of $y^+$ provide grid-independent solutions.",
"The boundary condition for liquid enthalpy adopted the fixedGradient type in order to account for the applied wall heat flux into liquid (see in Eqn.",
"REF ), as, $\\nabla _f^\\perp h_l = \\frac{a_i {q}^{\\prime \\prime }_{li} +a_w { {q}^{\\prime \\prime }_{lw}}}{\\beta \\rho _l\\kappa _l^{\\rm eff}}$ The mass conservation and energy conservation over the whole pipe are carefully checked in the steady state.",
"A typical error is ${\\Delta G}/{G_{in}}=0.048\\%$ and ${\\Delta q^{\\prime \\prime }}/{q_{w}^{\\prime \\prime }}=1.6\\%$ .",
"In this test case, the following interfacial models are selected: Drag force: [10] Lift force: [24] Wall lubrication force: [24] Turbulent dispersion force: [3] Figure: Axial steady state distribution of void fraction in Bartolomej experimentFigure REF shows the comparison between the experimental and calculation results for the Bartolomej experiment, using the [26] models with $C_{td}=2.5$ .",
"Since we used a uniformly distributed temperature profile as the inlet boundary condition for the energy conservation equation, there is a discrepancy between the predicted and measured temperature in the region nearby, as shown in Fig.",
"REF .",
"However, the temperature of the bulk and at the centerline could be well predicted after the flow becomes fully developed.",
"The averaged void fraction is somehow underestimated, which may be due to several reasons.",
"Firstly, we used a two-equation interfacial area concentration model in which the condensation rate could be overestimated due to underestimated bubble size.",
"Unfortunately, the measurement of the bubble size is not available in the Bartolomej experiment, rending it difficult to evaluate the prediction of the bubble size.",
"Secondly, the underestimation of void fraction could be also related to the modeling of interfacial forces, for example, turbulent dispersion force.",
"If we have a large turbulent dispersion force that drives bubbles towards the cold bulk, the condensation could also be overestimated and results in a rather low void fraction.",
"Thirdly, the observed discrepancy could also result from the underestimation of evaporation rate, which depends on the wall heat partitioning model.",
"Figure: Comparison between the DEBORA experiment and calculation results: Radial void fractionFigure: Comparison between the DEBORA experiment and calculation results: Radial Sauter mean diameterFigure: Comparison between the DEBORA experiment and calculation results: Radial liquid temperatureFigures REF - REF show the comparison between the measured and predicted results of DEBORA experiment.",
"Two sets of breakup and coalescence models were tested in our simulation.",
"One should notice that [26] breakup and coalescence model is used together with their turbulence modeling and [18] breakup and coalescence model together with the standard $k-\\epsilon $ model.",
"In addition, the sensitivity of turbulent dispersion force coefficient was tested here.",
"The suggested value of $C_{td}$ is usually in the region [0.1, 1.0] for bubbly flow.",
"However $C_{td}=1.0$ is not sufficient enough to push the evaporation bubbles away from the surface, leading to an accumulation of void fraction near the wall, as shown in Fig.",
"REF .",
"Due to that the local void fraction close to the wall may reach too high levels (above 0.74) exceeding the limits of the applicability of the present bubbly flow model.",
"That is why we could not do the simulation with [18] model together with $C_{td}=1.0$ in case of DEB6, as shown in Fig.",
"REF .",
"In general, a quite satisfactory agreement between the measured and the calculated void fraction distribution has been obtained.",
"In particular, Fig.",
"REF reveals that significant improvement in over-all accuracy can be obtained by choosing the turbulence dispersion force coefficient in the range between 1.0 and 2.5.",
"The accuracy of prediction of bubble size is, however, not satisfactory.",
"As shown in Fig.",
"REF , the bubble size is significantly underestimated in the observation part of the test section.",
"This could be caused by underestimation of the bubble coalescence rate in this region.",
"The results indicate that more work is needed to improve the interfacial area transport models.",
"Figure REF shows a very good agreement between predicted and measured radial temperature distributions for both cases."
],
[
"Conclusion",
"A two-fluid boiling flow model has been implemented into the OpenFOAM solver and validated against the Bartolomej and the DEBORA experimental data.",
"The model includes the closure relationships for the heat transfer and phase change for bubbles moving in a subcooled liquid.",
"Bubble size is predicted from the interfacial area concentration transport equations, including the source and sink terms resulting from the bubble coalescence and breakup, nucleation at walls as well as phase change induced source term.",
"The present model has been validated against measurements performed in a vertical upward flow in a heated pipe.",
"The prediction of void fraction as well as the liquid temperature profile could be done with quite satisfactory accuracy.",
"The accuracy of prediction of the bubble size distribution is found quite low, indicating that still more work is needed to improve the interfacial area transport models."
],
[
"Acknowledgments",
"Financial supports from NORTHNET, as well as support from the Swedish National Infrastructure for Computing are gratefully acknowledged.",
"ll 2lNomenclature $A$ area fraction $a_i$ interfacial area concentration, m$^{-1}$ $C$ interfacial force coefficient $c_{li}$ heat transfer coefficient given by Eqn.",
"REF , kg$\\cdot $ m$^{-2}\\cdot $ s$^{-1}$ $c_{p}$ specific heat, J$\\cdot $ kg$^{-1}\\cdot $ K$^{-1}$ $c_{vi}$ heat transfer coefficient given by Eqn.",
"REF , kg$\\cdot $ m$^{-3}\\cdot $ s$^{-1}$ $D_S$ Sauter mean diameter, m $d$ diameter, m $d_h$ hydraulic diameter, m Eo Eotvos number $f$ bubble departure frequency, s$^{-1}$ $G$ turbulent production, kg$\\cdot $ m$^{-1}\\cdot $ s$^{-3}$ or mass flow rate, kg$\\cdot $ m$^{-2}\\cdot $ s$^{-1}$ $g$ gravity constant, m$\\cdot $ s$^{-2}$ $h$ enthalpy, J$\\cdot $ kg$^{-1}$ $h_{fc}$ single phase convective heat transfer coefficient, W$\\cdot $ m$^{-2}\\cdot $ K$^{-1}$ $h_{fg}$ latent heat, J$\\cdot $ kg$^{-1}$ $h_{q}$ quenching heat transfer coefficient, W$\\cdot $ m$^{-2}\\cdot $ K$^{-1}$ Ja Jacob number $k$ turbulent kinetic energy, m$^2\\cdot $ s$^{-2}$ $\\mathbf {M}$ interfacial momentum transfer rate, kg$\\cdot $ m$^{-2}\\cdot $ s$^{-2}$ $N^{\\prime \\prime }$ active nucleation site density, m$^{-2}$ Nu Nusselt number $\\mathbf {n}_w$ unit vector normal to wall PePéclet number Pr Prandtl number $p$ pressure, Pa $ q^{\\prime \\prime },{\\mathbf {q}}^{\\prime \\prime }$ heat flux, W$\\cdot $ m$^{-2}$ $ q^{\\prime \\prime \\prime }$ heat flow rate per unit volume, W$\\cdot $ m$^{-3}$ Re Reynolds number $T$ temperature, K $t$ time, s $\\mathbf {U}$ velocity, m$\\cdot $ s$^{-1}$ $u_\\tau $ friction velocity, m$\\cdot $ s$^{-1}$ We Weber number 2lGreek letters $\\alpha $ void fraction $\\beta $ void fraction for continuous phase $\\epsilon $ turbulent dissipation rate, m$^2\\cdot $ s$^{-3}$ $\\Gamma $ rate of phase change, kg$\\cdot $ m$^{-3}\\cdot $ s$^{-1}$ $\\kappa $ thermal diffusivity, m$^2\\cdot $ s$^{-1}$ $\\lambda $ thermal conductivity, W$\\cdot $ m$^{-1}\\cdot $ K$^{-1}$ $\\mu $ dynamic viscosity, kg$\\cdot $ m$^{-1}\\cdot $ s$^{-1}$ $\\nu $ kinematic viscosity, m$^{2}\\cdot $ s$^{-1}$ $\\psi $ factor depending on bubble shape $\\rho $ density, kg$\\cdot $ m$^{-3}$ $\\sigma $ interfacial tension, N$\\cdot $ m$^{-1}$ $\\mathbf {\\tau }$ stress tensor, N$\\cdot $ m$^{-2}$ $\\tau _w$ wall shear stress, N$\\cdot $ m$^{-2}$ 2lSuperscripts eff effective $d$ drag $l$ lift $t$ turbulence $td$ turbulent dispersion $vm$ virtual mass $w$ wall $wl$ wall lubrication 2lSubscripts $1\\Phi $ single phase BB bubble breakup BC bubble coalescence $b$ bubble $c$ convection $fc$ single phase forced convection $i$ interphase $k$ phase $l$ liquid $lo$ lift-off NUC nucleation $q$ quenching $r$ relative refreference satsaturation $v$ vapor $w$ wall"
]
] | 1709.01783 |
[
[
"On the Relationship between Ideal Cluster Points and Ideal Limit Points"
],
[
"Abstract Let $X$ be a first countable space which admits a non-trivial convergent sequence and let $\\mathcal{I}$ be an analytic P-ideal.",
"First, it is shown that the sets of $\\mathcal{I}$-limit points of all sequences in $X$ are closed if and only if $\\mathcal{I}$ is also an $F_\\sigma$-ideal.",
"Moreover, let $(x_n)$ be a sequence taking values in a Polish space without isolated points.",
"It is known that the set $A$ of its statistical limit points is an $F_\\sigma$-set, the set $B$ of its statistical cluster points is closed, and that the set $C$ of its ordinary limit points is closed, with $A\\subseteq B\\subseteq C$.",
"It is proved the sets $A$ and $B$ own some additional relationship: indeed, the set $S$ of isolated points of $B$ is contained also in $A$.",
"Conversely, if $A$ is an $F_\\sigma$-set, $B$ is a closed set with a subset $S$ of isolated points such that $B\\setminus S\\neq \\emptyset$ is regular closed, and $C$ is a closed set with $S\\subseteq A\\subseteq B\\subseteq C$, then there exists a sequence $(x_n)$ for which: $A$ is the set of its statistical limit points, $B$ is the set of its statistical cluster points, and $C$ is the set of its ordinary limit points.",
"Lastly, we discuss topological nature of the set of $\\mathcal{I}$-limit points when $\\mathcal{I}$ is neither $F_\\sigma$- nor analytic P-ideal."
],
[
"Introduction",
"The aim of this article is to establish some relationship between the set of ideal cluster points and the set of ideal limit points of a given sequence.",
"To this aim, let $\\mathcal {I}$ be an ideal on the positive integers $\\mathbf {N}$ , i.e., a collection of subsets of $\\mathbf {N}$ closed under taking finite unions and subsets.",
"It is assumed that $\\mathcal {I}$ contains the collection $\\mathrm {Fin}$ of finite subsets of $\\mathbf {N}$ and it is different from the whole power set $\\mathcal {P}(\\mathbf {N})$ .",
"Note that the family $\\mathcal {I}_0$ of subsets with zero asymptotic density, that is, $\\mathcal {I}_0:=\\left\\lbrace S\\subseteq \\mathbf {N}: \\lim _{n\\rightarrow \\infty } \\frac{|S\\cap \\lbrace 1,\\ldots ,n\\rbrace |}{n} =0\\right\\rbrace $ is an ideal.",
"Let also $x=(x_n)$ be a sequence taking values in a topological space $X$ , which will be always assumed hereafter to be Hausdorff.",
"We denote by $\\Lambda _x(\\mathcal {I})$ the set of $\\mathcal {I}$ -limit points of $x$ , that is, the set of all $\\ell \\in X$ for which $\\lim _{k\\rightarrow \\infty }x_{n_k}=\\ell ,$ for some subsequence $(x_{n_k})$ such that $\\lbrace n_k: k \\in \\mathbf {N}\\rbrace \\notin \\mathcal {I}$ .",
"In addition, let $\\Gamma _x(\\mathcal {I})$ be the set of $\\mathcal {I}$ -cluster points of $x$ , that is, the set of all $\\ell \\in X$ such that $\\lbrace n: x_n \\in U\\rbrace \\notin \\mathcal {I}$ for every neighborhood $U$ of $\\ell $ .",
"Note that $\\mathrm {L}_x:=\\Lambda _x(\\mathrm {Fin})=\\Gamma _x(\\mathrm {Fin})$ is the set of ordinary limit points of $x$ ; we also shorten $\\Lambda _x:=\\Lambda _x(\\mathcal {I}_0)$ and $\\Gamma _x:=\\Gamma _x(\\mathcal {I}_0)$ .",
"Statistical limit points and statistical cluster points (i.e., $\\mathcal {I}_{0}$ -limit points and $\\mathcal {I}_0$ -cluster points, resp.)",
"of real sequences were introduced by Fridy [10], cf.",
"also [2], [5], [11], [13], [15], [17].",
"We are going to provide in Section , under suitable assumptions on $X$ and $\\mathcal {I}$ , a characterization of the set of $\\mathcal {I}$ -limit points.",
"Recall that $\\Gamma _x(\\mathcal {I})$ is closed and contains $\\Lambda _x(\\mathcal {I})$ , see e.g.",
"[4].",
"Then, it is shown that: $\\Lambda _x(\\mathcal {I})$ is an $F_\\sigma $ -set, provided that $\\mathcal {I}$ is an analytic P-ideal (Theorem REF ); $\\Lambda _x(\\mathcal {I})$ is closed, provided that $\\mathcal {I}$ is an $F_\\sigma $ -ideal (Theorem REF ); $\\Lambda _x(\\mathcal {I})$ is closed for all $x$ if and only if $\\Lambda _x(\\mathcal {I})=\\Gamma _x(\\mathcal {I})$ for all $x$ if and only if $\\mathcal {I}$ is an $F_\\sigma $ -ideal, provided that $\\mathcal {I}$ is an analytic P-ideal (Theorem REF ); For every $F_\\sigma $ -set $A$ , there exists a sequence $x$ such that $\\Lambda _x(\\mathcal {I})=A$ , provided that $\\mathcal {I}$ is an analytic P-ideal which is not $F_\\sigma $ (Theorem REF ); Each of isolated point $\\mathcal {I}$ -cluster point is also an $\\mathcal {I}$ -limit point (Theorem REF ).",
"In addition, we provide in Section some joint converse results: Given $A\\subseteq B \\subseteq C\\subseteq \\mathbf {R}$ where $A$ is an $F_\\sigma $ -set, $B$ is non-empty regular closed, and $C$ is closed, then there exists a real sequence $x$ such that $\\Lambda _x=A$ , $\\Gamma _x=B$ , and $\\mathrm {L}_x=C$ (Theorem REF and Corollary REF ); Given non-empty closed sets $B \\subseteq C\\subseteq \\mathbf {R}$ , there exists a real sequence $x$ such that $\\Lambda _x(\\mathcal {I})=\\Gamma _x(\\mathcal {I})=B$ and $\\mathrm {L}_x=C$ , provided $\\mathcal {I}$ is an $F_\\sigma $ -ideal different from $\\mathrm {Fin}$ (Theorem REF ).",
"Lastly, it is shown in Section that: $\\Lambda _x(\\mathcal {I})$ is analytic, provided that $\\mathcal {I}$ is a co-analytic ideal (Proposition REF ); An ideal $\\mathcal {I}$ is maximal if and only if each real sequence $x$ admits at most one $\\mathcal {I}$ -limit point (Proposition REF and Corollary REF ).",
"We conclude by showing that there exists an ideal $\\mathcal {I}$ and a real sequence $x$ such that $\\Lambda _x(\\mathcal {I})$ is not an $F_\\sigma $ -set (Example REF )."
],
[
"Topological structure of $\\mathcal {I}$ -limit points",
"We recall that an ideal $\\mathcal {I}$ is said to be a P-ideal if it is $\\sigma $ -directed modulo finite, i.e., for every sequence $(A_n)$ of sets in $\\mathcal {I}$ there exists $A \\in \\mathcal {I}$ such that $A_n\\setminus A$ is finite for all $n$ ; equivalent definitions were given, e.g., in [1].",
"By identifying sets of integers with their characteristic function, we equip $\\mathcal {P}(\\mathbf {N})$ with the Cantor-space topology and therefore we can assign the topological complexity to the ideals on $\\mathbf {N}$ .",
"In particular, an ideal $\\mathcal {I}$ is analytic if it is a continuous image of a $G_\\delta $ -subset of the Cantor space.",
"Moreover, a map $\\varphi : \\mathcal {P}(\\mathbf {N}) \\rightarrow [0,\\infty ]$ is a lower semicontinuous submeasure provided that: (i) $\\varphi (\\emptyset )=0$ ; (ii) $\\varphi (A) \\le \\varphi (B)$ whenever $A\\subseteq B$ ; (iii) $\\varphi (A\\cup B) \\le \\varphi (A)+\\varphi (B)$ for all $A,B$ ; and (iv) $\\varphi (A)=\\lim _{n}\\varphi (A\\cap \\lbrace 1,\\ldots ,n\\rbrace )$ for all $A$ .",
"By a classical result of Solecki, an ideal $\\mathcal {I}$ is an analytic P-ideal if and only if there exists a lower semicontinuous submeasure $\\varphi $ such that $\\mathcal {I}=\\mathcal {I}_\\varphi :=\\lbrace A\\subseteq \\mathbf {N}: \\Vert A\\Vert _\\varphi =0\\rbrace $ and $\\varphi (\\mathbf {N})<\\infty $ , where $\\Vert A\\Vert _\\varphi :=\\lim _n \\varphi (A\\setminus \\lbrace 1,\\ldots ,n\\rbrace )$ for all $A\\subseteq \\mathbf {N}$ , see [19].",
"Note, in particular, that for every $n\\in \\mathbf {N}$ it holds $\\Vert A\\Vert _\\varphi =\\Vert A\\setminus \\lbrace 1,\\ldots ,n\\rbrace \\Vert _\\varphi .$ Hereafter, unless otherwise stated, an analytic P-ideal will be always denoted by $\\mathcal {I}_\\varphi $ , where $\\varphi $ stands for the associated lower semicontinuous submeasure as in (REF ).",
"Given a sequence $x=(x_n)$ taking values in a first countable space $X$ and an analytic P-ideal $\\mathcal {I}_\\varphi $ , define $\\mathfrak {u}(\\ell ):=\\lim _{k \\rightarrow \\infty }\\Vert \\lbrace n: x_n \\in U_k\\rbrace \\Vert _\\varphi $ for each $\\ell \\in X$ , where $(U_k)$ is a decreasing local base of neighborhoods at $\\ell $ .",
"It is easy to see that the limit in (REF ) exists and its value is independent from the choice of $(U_k)$ .",
"Lemma 2.1 The map $\\mathfrak {u}$ is upper semi-continuous.",
"In particular, the set $\\Lambda _x(\\mathcal {I}_\\varphi ,q):=\\lbrace \\ell \\in X: \\mathfrak {u}(\\ell )\\ge q\\rbrace .$ is closed for every $q>0$ .",
"We need to prove that ${U}_y:=\\lbrace \\ell \\in X: \\mathfrak {u}(\\ell )<y\\rbrace $ is open for all $y \\in \\mathbf {R}$ (hence ${U}_\\infty $ is open too).",
"Clearly, ${U}_y=\\emptyset $ if $y\\le 0$ .",
"Hence, let us suppose hereafter $y>0$ and ${U}_y\\ne \\emptyset $ .",
"Fix $\\ell \\in {U}_y$ and let $(U_k)$ be a decreasing local base of neighborhoods at $\\ell $ .",
"Then, there exists $k_0 \\in \\mathbf {N}$ such that $\\Vert \\lbrace n: x_n \\in U_k\\rbrace \\Vert _\\varphi <y$ for every $k\\ge k_0$ .",
"Fix $\\ell ^\\prime \\in U_{k_0}$ and let $(V_k)$ be a decreasing local base of neighborhoods at $\\ell ^\\prime $ .",
"Fix also $k_1 \\in \\mathbf {N}$ such that $V_{k_1} \\subseteq U_{k_0}$ .",
"It follows by the monotonicity of $\\varphi $ that $\\Vert \\lbrace n: x_n \\in V_k\\rbrace \\Vert _\\varphi \\le \\Vert \\lbrace n: x_n \\in U_{k_0}\\rbrace \\Vert _\\varphi <y$ for every $k \\ge k_1$ .",
"In particular, $\\mathfrak {u}(\\ell ^\\prime )<y$ and, by the arbitrariness of $\\ell ^\\prime $ , $U_{k_0} \\subseteq {U}_y$ .",
"At this point, we provide a useful characterization of the set $\\Lambda _x(\\mathcal {I}_\\varphi )$ (without using limits of subsequences) and we obtain, as a by-product, that it is an $F_\\sigma $ -set.",
"Theorem 2.2 Let $x$ be a sequence taking values in a first countable space $X$ and $\\mathcal {I}_\\varphi $ be an analytic P-ideal.",
"Then $\\Lambda _x(\\mathcal {I}_\\varphi )=\\lbrace \\ell \\in X: \\mathfrak {u}(\\ell )>0\\rbrace .$ In particular, $\\Lambda _x(\\mathcal {I}_\\varphi )$ is an $F_\\sigma $ -set.",
"Let us suppose that there exists $\\ell \\in \\Lambda _x(\\mathcal {I}_\\varphi )$ and let $(U_k)$ be a decreasing local base of neighborhoods at $\\ell $ .",
"Then, there exists $A\\subseteq \\mathbf {N}$ such that $\\lim _{n\\rightarrow \\infty , n \\in A} x_{n}=\\ell $ and $\\Vert A\\Vert _\\varphi >0$ .",
"At this point, note that, for each $k \\in \\mathbf {N}$ , the set $\\lbrace n\\in A: x_n \\notin U_k\\rbrace $ is finite, hence it follows by (REF ) that $\\mathfrak {u}(\\ell ) \\ge \\Vert A\\Vert _\\varphi >0$ .",
"On the other hand, suppose that there exists $\\ell \\in X$ such that $\\mathfrak {u}(\\ell )>0$ .",
"Let $(U_k)$ be a decreasing local base of neighborhoods at $\\ell $ and define $\\mathcal {A}_k:=\\lbrace n: x_n \\in U_k\\rbrace $ for each $k \\in \\mathbf {N}$ ; note that $\\mathcal {A}_k$ is infinite since $\\Vert \\mathcal {A}_k\\Vert _\\varphi \\downarrow \\mathfrak {u}(\\ell )>0$ implies $\\mathcal {A}_k\\notin \\mathcal {I}_\\varphi $ for all $k$ .",
"Set for convenience $\\theta _0:=0$ and define recursively the increasing sequence of integers $(\\theta _k)$ so that $\\theta _k$ is the smallest integer greater than both $\\theta _{k-1}$ and $\\min \\mathcal {A}_{k+1}$ such that $\\varphi (\\mathcal {A}_k \\cap (\\theta _{k-1},\\theta _k]) \\ge \\mathfrak {u}(\\ell )\\left(1-{1}{k}\\right).$ Finally, define $\\mathcal {A}:=\\bigcup _{k} \\left(\\mathcal {A}_k \\cap (\\theta _{k-1},\\theta _k]\\right).$ Since $\\theta _k \\ge k$ for all $k$ , we obtain $\\varphi (\\mathcal {A}\\setminus \\lbrace 1,\\ldots ,n\\rbrace ) \\ge \\varphi (\\mathcal {A}_{n+1} \\cap (\\theta _{n},\\theta _{n+1}]) > \\mathfrak {u}(\\ell )\\left(1-{1}{n}\\right)$ for all $n$ , hence $\\Vert \\mathcal {A}\\Vert _\\varphi \\ge \\mathfrak {u}(\\ell ) >0$ .",
"In addition, we have by construction $\\lim _{n\\rightarrow \\infty , n \\in \\mathcal {A}}x_n=\\ell $ .",
"Therefore $\\ell $ is an $\\mathcal {I}_\\varphi $ -limit point of $x$ .",
"To sum up, this proves (REF ).",
"Lastly, rewriting (REF ) as $\\Lambda _x(\\mathcal {I}_\\varphi )=\\bigcup _{n}\\Lambda _x(\\mathcal {I}_\\varphi ,{1}{n})$ and considering that each $\\Lambda _x(\\mathcal {I}_\\varphi ,{1}{n})$ is closed by Lemma REF , we conclude that $\\Lambda _x(\\mathcal {I}_\\varphi )$ is an $F_\\sigma $ -set.",
"The fact that $\\Lambda _x(\\mathcal {I}_\\varphi )$ is an $F_\\sigma $ -set already appeared in [3], although with a different argument.",
"The first result of this type was given in [13] for the case $\\mathcal {I}_\\varphi =\\mathcal {I}_0$ and $X=\\mathbf {R}$ .",
"Later, it was extended in [5] for first countable spaces.",
"However, in the proofs contained in [3], [5] it is unclear why the constructed subsequence $(x_n: n \\in \\mathcal {A})$ converges to $\\ell $ .",
"Lastly, Theorem REF generalizes, again with a different argument, [14] for the case $X$ metrizable.",
"A stronger result holds in the case that the ideal is $F_\\sigma $ .",
"We recall that, by a classical result of Mazur, an ideal $\\mathcal {I}$ is $F_\\sigma $ if and only if there exists a lower semicontinuous submeasure $\\varphi $ such that $\\mathcal {I}=\\lbrace A\\subseteq \\mathbf {N}: \\varphi (A)<\\infty \\rbrace ,$ with $\\varphi (\\mathbf {N})=\\infty $ , see [16].",
"Theorem 2.3 Let $x=(x_n)$ be a sequence taking values in a first countable space $X$ and let $\\mathcal {I}$ be an $F_\\sigma $ -ideal.",
"Then $\\Lambda _x(\\mathcal {I})=\\Gamma _x(\\mathcal {I})$ .",
"In particular, $\\Lambda _x(\\mathcal {I})$ is closed.",
"Since it is known that $\\Lambda _x(\\mathcal {I})\\subseteq \\Gamma _x(\\mathcal {I})$ , the claim is clear if $\\Gamma _x(\\mathcal {I})=\\emptyset $ .",
"Hence, let us suppose hereafter that $\\Gamma _x(\\mathcal {I})$ is non-empty.",
"Fix $\\ell \\in \\Gamma _x(\\mathcal {I})$ and let $(U_k)$ be a decreasing local base of neighborhoods at $\\ell $ .",
"Letting $\\varphi $ be a lower semicontinuous submeasure associated with $\\mathcal {I}$ as in (REF ) and considering that $\\ell $ is an $\\mathcal {I}$ -cluster point, we have $\\varphi (A_k)=\\infty $ for all $k \\in \\mathbf {N}$ , where $A_k:=\\lbrace n: x_n \\in U_k\\rbrace $ .",
"Then, set $a_0:=0$ and define an increasing sequence of integers $(a_k)$ which satisfies $\\varphi (A_k \\cap (a_{k-1},a_k]) \\ge k$ for all $k$ (note that this is possible since $\\varphi (A_k\\setminus S)=\\infty $ whenever $S$ is finite).",
"At this point, set $A:=\\bigcup _{k} A_k \\cap (a_{k-1},a_k]$ .",
"It follows by the monotonocity of $\\varphi $ that $\\varphi (A)=\\infty $ , hence $A\\notin \\mathcal {I}$ .",
"Moreover, for each $k \\in \\mathbf {N}$ , we have that $\\lbrace n \\in A: x_n \\notin U_k\\rbrace $ is finite: indeed, if $n \\in A_j \\cap (a_{j-1},a_j]$ for some $j\\ge k$ , then by construction $x_n \\in U_j$ , which is contained in $U_k$ .",
"Therefore $\\lim _{n\\rightarrow \\infty ,\\, n \\in A} x_{n} = \\ell $ , that is, $\\ell \\in \\Lambda _x(\\mathcal {I})$ .",
"Since summable ideals are $F_\\sigma $ P-ideals, see e.g.",
"[7], we obtain the following corollary which was proved in [14]: Corollary 2.4 Let $x$ be a real sequence and let $\\mathcal {I}$ be a summable ideal.",
"Then $\\Lambda _x(\\mathcal {I})$ is closed.",
"It turns out that, within the class of analytic P-ideals, the property that the set of $\\mathcal {I}$ -limit points is always closed characterizes the subclass of $F_\\sigma $ -ideals: Theorem 2.5 Let $X$ be a first countable space which admits a non-trivial convergent sequence.",
"Let also $\\mathcal {I}_\\varphi $ be an analytic P-ideal.",
"Then the following are equivalent: $\\mathcal {I}_\\varphi $ is also an $F_\\sigma $ -ideal; $\\Lambda _x(\\mathcal {I}_\\varphi )=\\Gamma _x(\\mathcal {I}_\\varphi )$ for all sequences $x$ ; $\\Lambda _x(\\mathcal {I}_\\varphi )$ is closed for all sequences $x$ ; there does not exist a partition $\\lbrace A_n: n \\in \\mathbf {N}\\rbrace $ of $\\mathbf {N}$ such that $\\Vert A_n\\Vert _\\varphi >0$ for all $n$ and $\\lim _n\\Vert \\bigcup _{k>n}A_k\\Vert _\\varphi =0$ .",
"REF $\\Rightarrow $ REF follows by Theorem REF and REF $\\Rightarrow $ REF is clear.",
"REF $\\Rightarrow $ REF By hypothesis, there exists a sequence $(\\ell _n)$ converging to $\\ell \\in X$ such that $\\ell _n \\ne \\ell $ for all $n$ .",
"Let us suppose that there exists a partition $\\lbrace A_n: n \\in \\mathbf {N}\\rbrace $ of $\\mathbf {N}$ such that $\\Vert A_n\\Vert _\\varphi >0$ for all $n$ and $\\lim _k\\Vert \\bigcup _{n\\ge k}A_n\\Vert _\\varphi =0$ .",
"Then, define the sequence $x=(x_n)$ by $x_n=\\ell _i$ for all $n \\in A_i$ .",
"Then, we have that $\\lbrace \\ell _n: n \\in \\mathbf {N}\\rbrace \\subseteq \\Lambda _x(\\mathcal {I}_\\varphi )$ .",
"On the other hand, since $X$ is first countable Hausdorff, it follows that for all $k \\in \\mathbf {N}$ there exists a neighborhood $U_k$ of $\\ell $ such that $\\textstyle \\lbrace n: x_n \\in U_k\\rbrace \\subseteq \\lbrace n: x_n=\\ell _i \\text{ for some }i\\ge k\\rbrace =\\bigcup _{n\\ge k}A_n.$ Hence, by the monotonicity of $\\varphi $ , we obtain $0<\\Vert \\lbrace n:x_n \\in U_k\\rbrace \\Vert _\\varphi \\downarrow 0$ , i.e., $\\mathfrak {u}(\\ell )=0$ , which implies, thanks to Theorem REF , that $\\ell \\notin \\Lambda _x(\\mathcal {I}_\\varphi )$ .",
"In particular, $\\mathcal {I}_\\varphi $ is not closed.",
"REF $\\Rightarrow $ REF Lastly, assume that the ideal $\\mathcal {I}_\\varphi $ is not an $F_\\sigma $ -ideal.",
"According to the proof of [19], cf.",
"also [18], this is equivalent to the existence, for each given $\\varepsilon >0$ , of some set $M\\subseteq \\mathbf {N}$ such that $0<\\Vert M\\Vert _\\varphi \\le \\varphi (M)<\\varepsilon $ .",
"This allows to define recursively a sequence of sets $(M_n)$ such that $\\Vert M_n\\Vert _\\varphi >\\sum _{k\\ge n+1} \\varphi (M_k)>0.$ for all $n$ and, in addition, $\\sum _k \\varphi (M_k)<\\varphi (\\mathbf {N})$ .",
"Then, it is claimed that there exists a partition $\\lbrace A_n: n \\in \\mathbf {N}\\rbrace $ of $\\mathbf {N}$ such that $\\Vert A_n\\Vert _\\varphi >0$ for all $n$ and $\\lim _n\\Vert \\bigcup _{k> n}A_k\\Vert _\\varphi =0$ .",
"To this aim, set $M_0:=\\mathbf {N}$ and define $A_n:=M_{n-1}\\setminus \\bigcup _{k\\ge n} M_k$ for all $n\\in \\mathbf {N}$ .",
"It follows by the subadditivity and monotonicity of $\\varphi $ that $\\textstyle \\varphi (M_{n-1} \\setminus \\lbrace 1,\\ldots ,k\\rbrace ) \\le \\varphi (A_n \\setminus \\lbrace 1,\\ldots ,k\\rbrace )+\\varphi \\left(\\bigcup _{k\\ge n}M_k\\right)$ for all $n,k \\in \\mathbf {N}$ ; hence, by the lower semicontinuity of $\\varphi $ and (REF ), $\\textstyle \\Vert A_n\\Vert _\\varphi \\ge \\Vert M_{n-1}\\Vert _\\varphi -\\varphi \\left(\\bigcup _{k\\ge n}M_k\\right)\\ge \\Vert M_{n-1}\\Vert _\\varphi -\\sum _{k\\ge n}\\varphi (M_k)>0$ for all $n \\in \\mathbf {N}$ .",
"Finally, again by the lower semicontinuity of $\\varphi $ , we get $\\textstyle \\Vert \\bigcup _{k> n}A_k\\Vert _\\varphi =\\Vert \\bigcup _{k\\ge n}M_{k}\\Vert _\\varphi \\le \\varphi \\left(\\bigcup _{k\\ge n}M_{k}\\right) \\le \\sum _{k\\ge n}\\varphi (M_{k})$ which goes to 0 as $n\\rightarrow \\infty $ .",
"This concludes the proof.",
"At this point, thanks to Theorem REF and Theorem REF , observe that, if $X$ is a first countable space which admits a non-trivial convergent sequence and $\\mathcal {I}_\\varphi $ is an analytic P-ideal which is not $F_\\sigma $ , then there exists a sequence $x$ such that $\\Lambda _x(\\mathcal {I}_\\varphi )$ is a non-closed $F_\\sigma $ -set.",
"In this case, indeed, all the $F_\\sigma $ -sets can be obtained: Theorem 2.6 Let $X$ be a first countable space where all closed sets are separable and assume that there exists a non-trivial convergent sequence.",
"Fix also an analytic P-ideal $\\mathcal {I}_\\varphi $ which is not $F_\\sigma $ and let $B\\subseteq X$ be a non-empty $F_\\sigma $ -set.",
"Then, there exists a sequence $x$ such that $\\Lambda _x(\\mathcal {I}_\\varphi )=B$ .",
"Let $(B_k)$ be a sequence of non-empty closed sets such that $\\bigcup _k B_k=B$ .",
"Let also $\\lbrace b_{k,n}: n \\in \\mathbf {N}\\rbrace $ be a countable dense subset of $B_k$ .",
"Thanks to Theorem REF , there exists a partition $\\lbrace A_n: n \\in \\mathbf {N}\\rbrace $ of $\\mathbf {N}$ such that $\\Vert A_n\\Vert _\\varphi >0$ for all $n$ and $\\lim _n \\Vert \\bigcup _{k>n}A_k\\Vert _\\varphi =0$ .",
"Moreover, for each $k \\in \\mathbf {N}$ , set $\\theta _{k,0}:=0$ and it is easily seen that there exists an increasing sequence of positive integers $(\\theta _{k,n})$ such that $\\varphi (A_k \\cap (\\theta _{k,n-1},\\theta _{k,n}]) \\ge \\frac{1}{2}\\Vert A_k \\setminus \\lbrace 1,\\ldots ,\\theta _{k,n-1}\\rbrace \\Vert _\\varphi =\\frac{1}{2}\\Vert A_k\\Vert _\\varphi $ for all $n$ .",
"Hence, setting $A_{k,n}:=A_k \\cap \\bigcup _{m \\in A_n}(\\theta _{k,m-1},\\theta _{k,m}]$ , we obtain that $\\lbrace A_{k,n}: n\\in \\mathbf {N}\\rbrace $ is a partition of $A_k$ such that $\\frac{1}{2}\\Vert A_k\\Vert _\\varphi \\le \\Vert A_{k,n}\\Vert _\\varphi \\le \\Vert A_k\\Vert _\\varphi $ for all $n,k$ .",
"At this point, let $x=(x_n)$ defined by $x_n=b_{k,m}$ whenever $n \\in A_{k,m}$ .",
"Fix $\\ell \\in B$ , then there exists $k \\in \\mathbf {N}$ such that $\\ell \\in B_k$ .",
"Let $(b_{k,r_m})$ be a sequence in $B_k$ converging to $\\ell $ .",
"Thus, set $\\tau _0:=0$ and let $(\\tau _m)$ be an increasing sequence of positive integers such that $\\varphi (A_{k,r_m} \\cap (\\tau _{m-1},\\tau _m]) \\ge \\frac{1}{2}\\Vert A_{k,r_m}\\Vert _\\varphi $ for each $m$ .",
"Setting $A:=\\bigcup _m A_{k,r_m} \\cap (\\tau _{m-1},\\tau _m]$ , it follows by construction that $\\lim _{n\\rightarrow \\infty , n \\in A}x_n=\\ell $ and $\\Vert A\\Vert _\\varphi \\ge \\frac{1}{4}\\Vert A_k\\Vert _\\varphi >0$ .",
"This shows that $B\\subseteq \\Lambda _x(\\mathcal {I}_\\varphi )$ .",
"To complete the proof, fix $\\ell \\notin B$ and let us suppose for the sake of contradiction that there exists $A\\subseteq \\mathbf {N}$ such that $\\lim _{n\\rightarrow \\infty , n \\in A}x_n=\\ell $ and $\\Vert A\\Vert _\\varphi >0$ .",
"For each $m \\in \\mathbf {N}$ , let $U_m$ be an open neighborhood of $\\ell $ which is disjoint from the closed set $B_1 \\cup \\cdots B_m$ .",
"It follows by the subadditivity and the monotonicity of $\\varphi $ that there exists a finite set $Y$ such that $\\textstyle \\Vert A\\Vert _\\varphi \\le \\Vert Y\\Vert _\\varphi + \\Vert \\lbrace n \\in A: x_n \\notin B_1 \\cup \\cdots \\cup B_m\\rbrace \\Vert _\\varphi \\le \\Vert \\bigcup _{k>m}A_k\\Vert _\\varphi .$ The claim follows by the arbitrariness of $m$ and the fact that $\\lim _m \\Vert \\bigcup _{k>m}A_k\\Vert _\\varphi =0$ .",
"Note that every analytic P-ideal without the Bolzano-Weierstrass property cannot be $F_\\sigma $ , see [8].",
"Hence Theorem REF applies to this class of ideals.",
"It was shown in [5] that if $X$ is a topological space where all closed sets are separable, then for each $F_\\sigma $ -set $A$ and closed set $B$ there exist sequences $a=(a_n)$ and $b=(b_n)$ with values in $X$ such that $\\Lambda _a=A$ and $\\Gamma _b=B$ .",
"As an application of Theorem REF , we prove that, in general, its stronger version with $a=b$ fails (e.g., there are no real sequences $x$ such that $\\Lambda _x=\\lbrace 0\\rbrace $ and $\\Gamma _x=\\lbrace 0,1\\rbrace $ ).",
"Here, a topological space $X$ is said to be locally compact if for every $x \\in X$ there exists a neighborhood $U$ of $x$ such that its closure $\\overline{U}$ is compact, cf.",
"[6].",
"Theorem 2.7 Let $x=(x_n)$ be a sequence taking values in a locally compact first countable space and fix an analytic P-ideal $\\mathcal {I}_\\varphi $ .",
"Then each isolated $\\mathcal {I}_\\varphi $ -cluster point is also an $\\mathcal {I}_\\varphi $ -limit point.",
"Let us suppose for the sake of contradiction that there exists an isolated $\\mathcal {I}_\\varphi $ -cluster point, let us say $\\ell $ , which is not an $\\mathcal {I}_\\varphi $ -limit point.",
"Let $(U_k)$ be a decreasing local base of open neighborhoods at $\\ell $ such that $\\overline{U}_1$ is compact.",
"Let also $m$ be a sufficiently large integer such that $U_m \\cap \\Gamma _x(\\mathcal {I}_\\varphi )=\\lbrace \\ell \\rbrace $ .",
"Thanks to [6] the underlying space is, in particular, regular, hence there exists an integer $r>m$ such that $\\overline{U}_r$ is a compact contained in $U_m$ .",
"In addition, since $\\ell $ is an $\\mathcal {I}_\\varphi $ -cluster point and it is not an $\\mathcal {I}_\\varphi $ -limit point, it follows by Theorem REF that $0<\\Vert \\lbrace n: x_n \\in U_k\\rbrace \\Vert _\\varphi \\downarrow \\mathfrak {u}(\\ell )=0.$ In particular, there exists $s \\in \\mathbf {N}$ such that $0<\\Vert \\lbrace n: x_n \\in U_s\\rbrace \\Vert _\\varphi <\\Vert \\lbrace n: x_n \\in U_{r}\\rbrace \\Vert _\\varphi $ .",
"Observe that $K:=\\overline{U}_{r} \\setminus U_s$ is a closed set contained in $\\overline{U}_1$ , hence it is compact.",
"By construction we have that $K \\cap \\Gamma _x(\\mathcal {I}_\\varphi )=\\emptyset $ .",
"Hence, for each $z \\in K$ , there exists an open neighborhood $V_z$ of $z$ such that $V_z \\subseteq U_m$ and $\\lbrace n: x_n \\in V_z\\rbrace \\in \\mathcal {I}_\\varphi $ , i.e., $\\Vert \\lbrace n: x_n \\in V_z\\rbrace \\Vert _\\varphi =0$ .",
"It follows that $\\bigcup _{z \\in K} V_z$ is an open cover of $K$ which is contained in $U_m$ .",
"Since $K$ is compact, there exists a finite set $\\lbrace z_1,\\ldots ,z_t\\rbrace \\subseteq K$ for which $K \\subseteq V_{z_1} \\cup \\cdots \\cup V_{z_t} \\subseteq U_m.$ At this point, by the subadditivity of $\\varphi $ , it easily follows that $\\Vert A\\cup B\\Vert _\\varphi \\le \\Vert A\\Vert _\\varphi +\\Vert B\\Vert _\\varphi $ for all $A,B\\subseteq \\mathbf {N}$ .",
"Hence we have $\\begin{split}\\Vert \\lbrace n: x_n \\in K\\rbrace \\Vert _\\varphi &\\ge \\Vert \\lbrace n: x_n \\in \\overline{U}_r\\rbrace \\Vert _\\varphi - \\Vert \\lbrace n: x_n \\in U_{s}\\rbrace \\Vert _\\varphi \\\\&\\ge \\Vert \\lbrace n: x_n \\in U_{r}\\rbrace \\Vert _\\varphi - \\Vert \\lbrace n: x_n \\in U_{s}\\rbrace \\Vert _\\varphi >0.\\end{split}$ On the other hand, it follows by (REF ) that $\\textstyle \\Vert \\lbrace n: x_n \\in K\\rbrace \\Vert _\\varphi \\le \\Vert \\lbrace n: x_n \\in \\bigcup _{i=1}^t V_{z_i}\\rbrace \\Vert _\\varphi \\le \\sum _{i=1}^t \\Vert \\lbrace n: x_n \\in V_{z_i}\\rbrace \\Vert _\\varphi =0.$ This contradiction concludes the proof.",
"The following corollary is immediate (we omit details): Corollary 2.8 Let $x$ be a real sequence for which $\\Gamma _x$ is a discrete set.",
"Then $\\Lambda _x=\\Gamma _x$ ."
],
[
"Joint Converse results",
"We provide now a kind of converse of Theorem REF , specializing to the case of the ideal $\\mathcal {I}_0$ : informally, if $B$ is a sufficiently smooth closed set and $A$ is an $F_\\sigma $ -set containing the isolated points of $B$ , then there exists a sequence $x$ such that $\\Lambda _x=A$ and $\\Gamma _x=B$ .",
"To this aim, we need some additional notation: let $\\mathrm {d}^\\star $ , $\\mathrm {d}_\\star $ , and $\\mathrm {d}$ be the upper asymptotic density, lower asymptotic density, and asymptotic density on $\\mathbf {N}$ , resp.",
"; in particular, $\\mathcal {I}_0=\\lbrace S\\subseteq \\mathbf {N}: \\mathrm {d}^\\star (S)=0\\rbrace $ .",
"Given a topological space $X$ , the interior and the closure of a subset $S\\subseteq X$ are denoted by $S^\\circ $ and $\\overline{S}$ , respectively; $S$ is said to be regular closed if $S=\\overline{S^\\circ }$ .",
"We let the Borel $\\sigma $ -algebra on $X$ be $\\mathcal {B}(X)$ .",
"A Borel probability measure $\\mu : \\mathcal {B}(X) \\rightarrow [0,1]$ is said to be strictly positive whenever $\\mu (U)>0$ for all non-empty open sets $U$ .",
"Moreover, $\\mu $ is atomless if, for each measurable set $A$ with $\\mu (A)>0$ , there exists a measurable subset $B\\subseteq A$ such that $0<\\mu (B)<\\mu (A)$ .",
"Then, a sequence $(x_n)$ taking values in $X$ is said to be $\\mu $-uniformly distributed whenever $\\mu (F) \\ge \\mathrm {d}^\\star (\\lbrace n: x_n \\in F\\rbrace )$ for all closed sets $F$ , cf.",
"[9].",
"Theorem 3.1 Let $X$ be a separable metric space and $\\mu : \\mathcal {B}(X)\\rightarrow [0,1]$ be an atomless strictly positive Borel probability measure.",
"Fix also sets $A\\subseteq B\\subseteq C\\subseteq X$ such that $A$ is an $F_\\sigma $ -set, and $B,C$ are closed sets such that: (i) $\\mu (B)>0$ , (ii) the set $S$ of isolated points of $B$ is contained in $A$ , and (iii) $B\\setminus S$ is regular closed.",
"Then there exists a sequence $x$ taking values in $X$ such that $\\Lambda _x=A,\\,\\,\\Gamma _x=B,\\,\\text{ and }\\,\\mathrm {L}_x=C.$ Set $F:=B\\setminus S$ and note that, by the separability of $X$ , $S$ at most countable.",
"In particular, $\\mu (S)=0$ , hence $\\mu (F)=\\mu (B)>0$ .",
"Let us assume for now that $A$ is non-empty.",
"Since $A$ is an $F_\\sigma $ -set, there exists a sequence $(A_k)$ of non-empty closed sets such that $\\bigcup _k A_k=A$ .",
"Considering that $X$ is (hereditarily) second countable, then every closed set is separable.",
"Hence, for each $k \\in \\mathbf {N}$ , there exists a countable set $\\lbrace a_{k,n}: n \\in \\mathbf {N}\\rbrace \\subseteq A_k$ with closure $A_k$ .",
"Considering that $F$ is a separable metric space on its own right and that the (normalized) restriction $\\mu _F$ of $\\mu $ on $F$ , that is, $\\mu _F: \\mathcal {B}(F) \\rightarrow [0,1]: Y\\mapsto \\frac{1}{\\mu (F)}\\mu (Y)$ is a Borel probability measure, it follows by [9] that there exists a $\\mu _F$ -uniformly distributed sequence $(b_n)$ which takes values in $F$ and satisfies (REF ).",
"Lastly, let $\\lbrace c_n: n \\in \\mathbf {N}\\rbrace $ be a countable dense subset of $C$ .",
"At this point, let ${C}$ be the set of non-zero integer squares and note that $\\mathrm {d}({C})=0$ .",
"For each $k \\in \\mathbf {N}$ define ${A}_k:=\\lbrace 2^kn: n\\in \\mathbf {N}\\setminus 2\\mathbf {N}\\rbrace \\setminus {C}$ and ${B}:=\\mathbf {N}\\setminus (2\\mathbf {N}\\cup {C})$ .",
"It follows by construction that $\\lbrace {A}_k: k \\in \\mathbf {N}\\rbrace \\cup \\lbrace {B}, {C}\\rbrace $ is a partition of $\\mathbf {N}$ .",
"Moreover, each ${A}_k$ admits asymptotic density and $\\textstyle \\lim _{n\\rightarrow \\infty }\\mathrm {d}\\left(\\bigcup _{k\\ge n}{A}_k\\right)=0.$ Finally, for each positive integer $k$ , let $\\lbrace {A}_{k,m}: m \\in \\mathbf {N}\\rbrace $ be the partition of ${A}_k$ defined by ${A}_{k,1}:={A}_k \\cap \\bigcup _{n \\in {A}_1 \\cup {B} \\cup {C}}[n!,(n+1)!",
")$ and ${A}_{k,m}:={A}_k \\cap \\bigcup _{n \\in {A}_m}[n!,(n+1)!",
")$ for all integers $m\\ge 2$ .",
"Then, it is easy to check that $\\mathrm {d}^\\star ({A}_{k,1})=\\mathrm {d}^\\star ({A}_{k,2})=\\cdots =\\mathrm {d}({A}_k)=2^{-k-1}.$ Hence, define the sequence $x=(x_n)$ by $x_n = \\begin{cases*}\\,a_{k,m} & if n \\in {A}_{k,m}, \\\\\\,b_{m} & if n is the m-th term of {B}, \\\\\\,c_{m} & if n is the m-th term of {C}.\\end{cases*}$ To complete the proof, let us verify that (REF ) holds true: Claim (i): $\\mathrm {L}_x=C$ .",
"Note that $x_n \\in C$ for all $n \\in \\mathbf {N}$ .",
"Since $C$ is closed by hypothesis, then $\\mathrm {L}_x \\subseteq C$ .",
"On the other hand, if $\\ell \\in C$ , then there exists a sequence $(c_n)$ taking values in $C$ converging (in the ordinary sense) to $\\ell $ .",
"It follows by the definition of $(x_n)$ that there exists a subsequence $(x_{n_k})$ converging to $\\ell $ , i.e., $C\\subseteq \\mathrm {L}_x$ .",
"Claim (ii): $\\Gamma _x=B$ .",
"Fix $\\ell \\notin B$ and let $U$ be an open neighborhood of $\\ell $ disjoint from $B$ (this is possible since, in the opposite, $\\ell $ would belong to $\\overline{B}=B$ ).",
"Then, $\\lbrace n: x_n \\in U\\rbrace \\subseteq {C}$ , which implies that $\\Gamma _x \\subseteq B$ .",
"Note that the Borel probability measure $\\mu _F$ defined in (REF ) is clearly atomless.",
"Moreover, given an open set $U\\subseteq X$ with non-empty intersection with $F$ , then $U \\cap F^\\circ \\ne \\emptyset $ : indeed, in the opposite, we would have $F^\\circ \\subseteq U^c$ , which is closed, hence $F=\\overline{F^\\circ } \\subseteq U^c$ , contradicting our hypothesis.",
"This proves that every non-empty open set $V$ (relative to $F$ ) contains a non-empty open set of $X$ .",
"Therefore $\\mu _F$ is also strictly positive.",
"With these premises, fix $\\ell \\in F$ and let $V$ be a open neighborhood of $\\ell $ (relative to $F$ ).",
"Since $(b_n)$ is $\\mu _F$ -uniformly distributed and $\\mu _F$ is strictly positive, it follows by (REF ) that $\\begin{split}0<\\mu _F(V)=1-\\mu _F(V^c)&\\le 1-\\mathrm {d}^\\star (\\lbrace n: b_n \\in V^c\\rbrace )\\\\&=\\mathrm {d}_\\star (\\lbrace n: b_n \\in V\\rbrace ) \\le \\mathrm {d}^\\star (\\lbrace n: b_n \\in V\\rbrace ).\\end{split}$ Since $\\mathrm {d}({B})={1}{2}$ , we obtain by standard properties of $\\mathrm {d}^\\star $ that $\\mathrm {d}^\\star (\\lbrace n: x_n \\in V\\rbrace ) \\ge \\mathrm {d}^\\star (\\lbrace n \\in {B}: x_n \\in V\\rbrace ) =\\frac{1}{2}\\mathrm {d}^\\star (\\lbrace n: b_n \\in V\\rbrace )>0.$ We conclude by the arbitrariness of $V$ and $\\ell $ that $F\\subseteq \\Gamma _x$ .",
"Hence, we miss only to show that $S\\subseteq \\Gamma _x$ .",
"To this aim, fix $\\ell \\in S$ , thus $\\ell $ is also an isolated point of $A$ .",
"Hence, there exist $k,m \\in \\mathbf {N}$ such that $a_{k,m}=\\ell $ .",
"We conclude that $\\mathrm {d}^\\star (\\lbrace n: x_n \\in U\\rbrace ) \\ge \\mathrm {d}^\\star (\\lbrace n: x_n=\\ell \\rbrace )\\ge \\mathrm {d}({A}_k)>0$ for each neighborhood $U$ of $\\ell $ .",
"Therefore $B=F\\cup S \\subseteq \\Gamma _x$ .",
"Claim (iii): $\\Lambda _x=A$ .",
"Fix $\\ell \\in A$ , hence there exists $k \\in \\mathbf {N}$ for which $\\ell $ belongs to the (non-empty) closed set $A_k$ .",
"Since $\\lbrace a_{k,n}: n \\in \\mathbf {N}\\rbrace $ is dense in $A_k$ , there exists a sequence $(a_{k,r_m}: m \\in \\mathbf {N})$ converging to $\\ell $ .",
"Recall that $x_n=a_{k,r_m}$ whenever $n \\in {A}_{k,r_m}$ for each $m \\in \\mathbf {N}$ .",
"Set by convenience $\\theta _0:=0$ and define recursively an increasing sequence of positive integers $(\\theta _m)$ such that $\\theta _m$ is an integer greater than $\\theta _{m-1}$ for which $\\mathrm {d}^\\star \\left({A}_{k,r_m} \\cap (\\theta _{m-1},\\theta _m]\\right) \\ge \\frac{\\mathrm {d}({A}_k)}{2}=2^{-k-2}.$ Then, setting $\\mathcal {A}:=\\bigcup _m {A}_{k,r_m} \\cap (\\theta _{m-1},\\theta _m]$ , we obtain that the subsequence $(x_n: n \\in \\mathcal {A})$ converges to $\\ell $ and $\\mathrm {d}^\\star (\\mathcal {A})>0$ .",
"In particular, $A\\subseteq \\Lambda _x$ .",
"On the other hand, it is known that $\\Lambda _x \\subseteq \\Gamma _x$ , see e.g.",
"[10].",
"If $A=B$ , it follows by Claim (ii) that $\\Lambda _x \\subseteq A$ and we are done.",
"Otherwise, fix $\\ell \\in B\\setminus A=F\\setminus A$ and let us suppose for the sake of contradiction that there exists a subsequence $(x_{n_k})$ such that $\\lim _k x_{n_k}=\\ell $ and $\\mathrm {d}^\\star (\\lbrace n_k: k \\in \\mathbf {N}\\rbrace )>0$ .",
"Fix a real $\\varepsilon >0$ .",
"Then, thanks to (REF ), there exists a sufficiently large integer $n_0$ such that $\\mathrm {d}\\left(\\bigcup _{k>n_0}{A}_k\\right) \\le \\varepsilon $ .",
"In addition, since $F$ is a metric space and $\\mu _F$ is atomless and strictly positive (see Claim (ii)), we have $\\lim _{n\\rightarrow \\infty }\\mu _F(V_n)=\\mu _F(\\lbrace \\ell \\rbrace )=0,$ where $V_n$ is the open ball (relative to $F$ ) with center $\\ell $ and radius ${1}{n}$ .",
"Hence, there exists a sufficiently large integer $m^\\prime $ such that $0<\\mu _F(V_{m^\\prime })\\le \\varepsilon $ .",
"In addition, there exists an integer $m^{\\prime \\prime }$ such that $V_{m^{\\prime \\prime }}$ is disjoint from the closed set $A_1 \\cup \\cdots \\cup A_{n_0}$ .",
"Then, set $V:=V_m$ , where $m$ is an integer greater than $\\max (m^\\prime ,m^{\\prime \\prime })$ such that $\\mu _F(V)<\\mu _F(V_{\\max (m^\\prime ,m^{\\prime \\prime })})$ .",
"In particular, by the monotonicity of $\\mu _F$ , we have $0<\\mu _F(V) \\le \\mu _F(\\overline{V}) \\le \\mu _F(V_{m^\\prime }) \\le \\varepsilon .$ At this point, observe there exists a finite set $Y$ such that $\\begin{split}\\lbrace n_k: k \\in \\mathbf {N}\\rbrace &= \\lbrace n_k: x_{n_k} \\in V\\rbrace \\cup Y \\\\&\\subseteq \\textstyle \\left(\\bigcup _{k>n_0}{A}_k\\right) \\cup \\lbrace n \\in {B}: x_n \\in V\\rbrace \\cup {C} \\cup Y.\\end{split}$ Therefore, by the subadditivity of $\\mathrm {d}^\\star $ , (REF ), and (REF ), we obtain $\\begin{split}\\mathrm {d}^\\star (\\lbrace n_k: k \\in \\mathbf {N}\\rbrace ) &\\le \\varepsilon +\\mathrm {d}^\\star (\\lbrace n \\in {B}: x_n \\in V\\rbrace )\\le \\varepsilon +\\mathrm {d}^\\star (\\lbrace n \\in {B}: b_n \\in V\\rbrace ) \\\\&\\le \\varepsilon +\\mathrm {d}^\\star (\\lbrace n \\in {B}: b_n \\in \\overline{V}\\rbrace ) \\le \\varepsilon +\\mu _F(\\overline{V}) \\le 2\\varepsilon .\\end{split}$ It follows by the arbitrariness of $\\varepsilon $ that $\\mathrm {d}(\\lbrace n_k: k \\in \\mathbf {N}\\rbrace )=0$ , i.e., $\\Lambda _x \\subseteq A$ .",
"To complete the proof, assume now that $A=\\emptyset $ .",
"In this case, note that necessarily $S=\\emptyset $ , and it is enough to replace (REF ) with $x_n = \\begin{cases*}\\,b_{n-\\lfloor \\sqrt{n}\\rfloor } & if n\\notin {C}, \\\\\\,c_{\\sqrt{n}} & if n \\in {C}.\\end{cases*}$ Then, it can be shown with a similar argument that $\\Lambda _x=\\emptyset $ , $\\Gamma _x=B$ , and $\\mathrm {L}_x=C$ .",
"It is worth noting that Theorem REF cannot be extended to the whole class of analytic P-ideals.",
"Indeed, it follows by Theorem REF that if $\\mathcal {I}$ is an $F_\\sigma $ ideal on $\\mathbf {N}$ then the set of $\\mathcal {I}$ -limit points is closed set, cf.",
"also Theorem REF below.",
"In addition, the regular closedness of $B\\setminus S$ is essential in the proof of Theorem REF .",
"On the other hand, there exist real sequences $x$ such that $\\Gamma _x$ is the Cantor set $\\mathcal {C}$ (which is a perfect set but not regular closed): Example 3.2 Given a real $r \\in [0,1)$ and an integer $b\\ge 2$ , we write $r$ in base $b$ as $\\sum _n a_n/b^{n}$ , where each $a_n$ belongs to $\\lbrace 0,1,\\ldots ,b-1\\rbrace $ and $a_{n}=\\zeta $ for all sufficiently large $n$ only if $\\zeta =0$ .",
"This representation is unique.",
"Let $x=(x_n)$ be the sequence $(0,0,1,0,\\frac{1}{2},1,0,\\frac{1}{3},\\frac{2}{3},1,\\ldots )$ .",
"This sequence is unifomly distributed in $[0,1]$ , i.e., $\\mathrm {d}(\\lbrace n: x_n \\in [a,b]\\rbrace )=b-a$ for all $0\\le a <b\\le 1$ , and $\\Gamma _x=[0,1]$ , see e.g.",
"[10].",
"Let also $T:[0,1] \\rightarrow \\mathcal {C}$ be the injection defined by $r\\mapsto T(r)$ , where if $r=\\sum _n a_n/2^{n} \\in [0,1)$ in base 2 then $T(r)=\\sum _n 2a_n/3^{n}$ in base 3, and $1\\mapsto 1$ .",
"Observe that $\\mathcal {C}\\setminus T([0,1))$ is the set of points of the type $2(1/3^{n_1}+\\cdots +1/3^{n_{k-1}})+1/3^{n_{k}}$ , for some non-negative integers $n_1 < \\cdots < n_k$ ; in particular, $\\overline{T([0,1))}=\\mathcal {C}$ .",
"Since the sequence $T(x):=(T(x_n))$ takes values in the closed set $\\mathcal {C}$ , it is clear that $\\Gamma _{T(x)} \\subseteq \\mathcal {C}$ .",
"On the other hand, fix $\\ell \\in T([0,1))$ with representation $\\sum _n 2a_n/3^{n}$ in base 3, where $a_n \\in \\lbrace 0,1\\rbrace $ for all $n$ .",
"For each $k$ , let $U_k$ be the open ball with center $\\ell $ and radius $1/3^{k}$ .",
"It follows that $\\begin{split}\\lbrace n: T(x_n) \\in U_k\\rbrace &\\supseteq \\left\\lbrace n: T(x_n) \\in \\left[\\frac{2a_1}{3}+\\cdots +\\frac{2a_k}{3^k},\\frac{2a_1}{3}+\\cdots +\\frac{2a_k}{3^k}+\\frac{1}{3^k}\\right)\\right\\rbrace \\\\&= \\left\\lbrace n: x_n \\in \\left[\\frac{a_1}{2}+\\cdots +\\frac{a_k}{2^k},\\frac{a_1}{2}+\\cdots +\\frac{a_k}{2^k}+\\frac{1}{2^k}\\right)\\right\\rbrace .\\end{split}$ Since $(x_n)$ is equidistributed, then $\\mathrm {d}^\\star (\\lbrace n: T(x_n) \\in U_k\\rbrace ) \\ge 1/2^k$ for all $k$ .",
"In particular, $\\Gamma _{T(x)}$ is a closed set containing $T([0,1))$ , therefore $\\Gamma _{T(x)}=\\mathcal {C}$ .",
"Finally, we provide a sufficient condition for the existence of an atomless strictly positive Borel probability measure: Corollary 3.3 Let $X$ be a Polish space without isolated points and fix sets $A\\subseteq B\\subseteq C\\subseteq X$ such that $A$ is an $F_\\sigma $ -set, $B \\ne \\emptyset $ is regular closed, and $C$ is closed.",
"Then, there exists a sequence $x$ taking values in $X$ which satisfies (REF ).",
"First, observe that the restriction $\\tilde{\\lambda }$ of the Lebesgue measure $\\lambda $ on the set ${I}:=(0,1)\\setminus \\mathbf {Q}$ is an atomless strictly positive Borel probability measure.",
"Thanks to [6], $X$ contains a dense subspace $D$ which is homeomorphic to $\\mathbf {R}\\setminus \\mathbf {Q}$ , which is turn is homeomorphic to ${I}$ , let us say through $\\eta : D \\rightarrow {I}$ .",
"This embedding can be used to transfer the measure $\\tilde{\\lambda }$ to the target space by setting $\\mu : \\mathcal {B}(X)\\rightarrow [0,1]: Y \\mapsto \\tilde{\\lambda }(\\eta (Y\\cap D)).$ Lastly, since $B$ is non-empty closed regular, then it has no isolated points and contains an open set $U$ of $X$ .",
"In particular, considering that $\\eta $ is an open map, we get by (REF ) that $\\mu (B) \\ge \\mu (U)=\\tilde{\\lambda }(\\eta (U\\cap D))>0$ .",
"The claim follows by Theorem REF .",
"Note that, in general, the condition $B \\ne \\emptyset $ cannot be dropped: indeed, it follows by [5] that, if $X$ is compact, then every sequence $(x_n)$ admits at least one statistical cluster point.",
"We conclude with another converse result related to ideals $\\mathcal {I}$ of the type $F_\\sigma $ (recall that, thanks to Theorem REF , every $\\mathcal {I}$ -limit point is also an $\\mathcal {I}$ -cluster point): Theorem 3.4 Let $X$ be a first countable space where all closed sets are separable and let $\\mathcal {I}\\ne \\mathrm {Fin}$ be an $F_\\sigma $ -ideal.",
"Fix also closed sets $B,C \\subseteq X$ such that $\\emptyset \\ne B\\subseteq C$ .",
"Then there exists a sequence $x$ such that $\\Lambda _x(\\mathcal {I})=\\Gamma _x(\\mathcal {I})=B$ and $\\mathrm {L}_x=C$ .",
"By hypothesis, there exists an infinite set $I \\in \\mathcal {I}$ .",
"Let $\\varphi $ be a lower semicontinuous submeasures associated to $\\mathcal {I}$ as in (REF ).",
"Let $\\lbrace b_n: n \\in \\mathbf {N}\\rbrace $ and $\\lbrace c_n: n \\in \\mathbf {N}\\rbrace $ be countable dense subsets of $B$ and $C$ , respectively.",
"In addition, set $m_0:=0$ and let $(m_k)$ be an increasing sequence of positive integers such that $\\varphi ((\\mathbf {N}\\setminus I) \\cap (m_{k-1},m_k]) \\ge k$ for all $k$ (note that this is possible since $\\varphi (\\mathbf {N}\\setminus I)=\\infty $ and $\\varphi $ is a lower semicontinuous submeasure).",
"At this point, given a partition $\\lbrace H_n: n \\in \\mathbf {N}\\rbrace $ of $\\mathbf {N}\\setminus I$ , where each $H_n$ is infinite, we set $\\textstyle M_k:=(\\mathbf {N}\\setminus I) \\cap \\bigcup _{n \\in H_k}(m_{n-1},m_n]$ for all $k \\in \\mathbf {N}$ .",
"Then, it is easily checked that $\\lbrace M_k: k \\in \\mathbf {N}\\rbrace $ is a partition of $\\mathbf {N}\\setminus I$ with $M_k \\notin \\mathcal {I}$ for all $k$ , and that the sequence $(x_n)$ defined by $x_n = \\begin{cases*}\\,b_{k} & if n\\in M_k, \\\\\\,c_{k} & if n is the k-th term of I.\\end{cases*}$ satisfies the claimed conditions.",
"In particular, Theorem REF and Theorem REF fix a gap in a result of Das [3] and provide its correct version."
],
[
"Concluding remarks",
"In this last section, we are interested in the topological nature of the set of $\\mathcal {I}$ -limit points when $\\mathcal {I}$ is neither $F_\\sigma $ - nor analytic P-ideal.",
"Let $\\mathcal {N}$ be the set of strictly increasing sequences of positive integers.",
"Then $\\mathcal {N}$ is a Polish space, since it is closed subspace of the Polish space $\\mathbf {N}^{\\mathbf {N}}$ (equipped with the product topology of the discrete topology on $\\mathbf {N}$ ).",
"Let also $x=(x_n)$ be a sequence taking values in a first countable regular space $X$ and fix an arbitrary ideal $\\mathcal {I}$ on $\\mathbf {N}$ .",
"For each $\\ell \\in X$ , let $(U_{\\ell ,m})$ be a decreasing local base of open neighborhoods at $\\ell $ .",
"Then, $\\ell $ is an $\\mathcal {I}$ -limit point of $x$ if and only if there exists a sequence $(n_k) \\in \\mathcal {N}$ such that $\\lbrace n_k:k \\in \\mathbf {N}\\rbrace \\notin \\mathcal {I}\\,\\,\\text{ and }\\,\\,\\lbrace n: x_n \\notin U_{\\ell ,m}\\rbrace \\in \\mathrm {Fin} \\,\\text{ for all }m.$ Set $\\mathcal {I}^c:=\\mathcal {P}(\\mathbf {N})\\setminus \\mathcal {I}$ and define the continuous function $\\psi : \\mathcal {N} \\rightarrow \\lbrace 0,1\\rbrace ^{\\mathbf {N}}: (n_k) \\mapsto \\chi _{\\lbrace n_k:\\, k \\in \\mathbf {N}\\rbrace },$ where $\\chi _S$ is the characteristic function of a set $S\\subseteq \\mathbf {N}$ .",
"Moreover, define $\\zeta _m: \\mathcal {N} \\times X \\rightarrow \\lbrace 0,1\\rbrace ^{\\mathbf {N}}: (n_k) \\times \\ell \\mapsto \\chi _{\\lbrace n:\\, x_n \\notin U_{\\ell ,m}\\rbrace }$ for each $m$ .",
"Hence, it easily follows by (REF ) that $\\Lambda _x(\\mathcal {I})=\\pi _X \\left(\\bigcap _m \\left(\\psi ^{-1}(I^c) \\times X \\,\\cap \\, \\zeta _m^{-1}(\\mathrm {Fin})\\right)\\right),$ where $\\pi _X: \\mathcal {N}\\times X \\rightarrow X$ stands for the projection on $X$ .",
"Proposition 4.1 Let $x=(x_n)$ be a sequence taking values in a first countable regular space $X$ and let $\\mathcal {I}$ be a co-analytic ideal.",
"Then $\\Lambda _x(\\mathcal {I})$ is analytic.",
"For each $(n_k) \\in \\mathcal {N}$ and $\\ell \\in X$ , the sections $\\zeta _m((n_k),\\cdot )$ and $\\zeta _m(\\cdot ,\\ell )$ are continuous.",
"Hence, thanks to [20], each function $\\zeta _m$ is Borel measurable.",
"Since $\\mathrm {Fin}$ is an $F_\\sigma $ -set, we obtain that each $\\zeta _m^{-1}(\\mathrm {Fin})$ is Borel.",
"Moreover, since $\\mathcal {I}$ is a co-analytic ideal and $\\psi $ is continuous, it follows that $\\psi ^{-1}(\\mathcal {I}^c)\\times X$ is an analytic subset of $\\mathcal {N}\\times X$ .",
"Therefore $\\Lambda _x(\\mathcal {I})$ is the projection on $X$ of the analytic set $\\bigcap _m \\left(\\psi ^{-1}(I^c) \\times X \\,\\cap \\, \\zeta _m^{-1}(\\mathrm {Fin})\\right)$ , which proves the claim.",
"The situation is much different for maximal ideals, i.e., ideals which are maximal with respect to inclusion.",
"In this regard, we recall if $\\mathcal {I}$ is a maximal ideal then every bounded real sequence $x$ is $\\mathcal {I}$ -convergent, i.e., there exists $\\ell \\in \\mathbf {R}$ such that $\\lbrace n: |x_n-\\ell |\\ge \\varepsilon \\rbrace \\in \\mathcal {I}$ for every $\\varepsilon >0$ , cf.",
"[12].",
"Let $B(a,r)$ the open ball with center $a$ and radius $r$ in a given metric space $(X,d)$ , and denote by $\\mathrm {diam}\\, S$ the diameter of a non-empty set $S\\subseteq X$ , namely, $\\sup _{a,b \\in S}d(a,b)$ .",
"Then, the metric space is said to be smooth if $\\lim _{k\\rightarrow \\infty }\\sup _{a \\in X} \\mathrm {diam} \\,\\overline{B(a,{1}{k})}=0.$ Note that (REF ) holds if, e.g., the closure of each open ball $B(a,r)$ coincides with the corresponding closed ball $\\lbrace b \\in X: d(a,b) \\le r\\rbrace $ .",
"Proposition 4.2 Let $x$ be a sequence taking values in a smooth compact metric space $X$ and let $\\mathcal {I}$ be a maximal ideal.",
"Then $x$ has exactly one $\\mathcal {I}$ -cluster point.",
"In particular, $\\Lambda _x(\\mathcal {I})$ is closed.",
"Since $X$ is a compact metric space, then $X$ is totally bounded, i.e., for each $\\varepsilon >0$ there exist finitely many open balls with radius $\\varepsilon $ covering $X$ .",
"Moreover, it is well known that an ideal $\\mathcal {I}$ is maximal if and only if either $A \\in \\mathcal {I}$ or $A^c \\in \\mathcal {I}$ for every $A\\subseteq \\mathbf {N}$ .",
"Hence, fix $k \\in \\mathbf {N}$ , let $\\lbrace B_{k,1},\\ldots ,B_{k,m_k}\\rbrace $ be a cover of $X$ of open balls with radius ${1}{k}$ , and define ${C}_{k,i}:=\\lbrace n: x_n \\in C_{k,i}\\rbrace $ for each $i\\le m_k$ , where $C_{k,i}:=B_{k,i}\\setminus (B_{k,1} \\cup \\cdots \\cup B_{k,i-1})$ and $B_{k,0}:=\\emptyset $ .",
"Considering that $\\lbrace {C}_{k,1},\\ldots ,{C}_{k,m_k}\\rbrace $ is a partition of $\\mathbf {N}$ , it follows by the above observations that there exists a unique $i_k \\in \\lbrace 1,\\ldots ,m_k\\rbrace $ for which ${C}_{k,i_k} \\notin \\mathcal {I}$ .",
"At this point, let $(G_k)$ be the decreasing sequence of closed sets defined by $G_k:=\\overline{C_{1,i_1} \\cap \\cdots \\cap C_{k,i_k}}$ for all $k$ .",
"Note that each $G_k$ is non-empty, the diameter of $G_k$ (which is contained in $\\overline{B_{k,i_k}}$ ) goes to 0 as $k\\rightarrow \\infty $ , and $\\lbrace n: x_n \\in G_k\\rbrace \\notin \\mathcal {I}$ for all $k$ .",
"Since $X$ is a compact metric space, then $\\bigcap _k G_k$ is a singleton $\\lbrace \\ell \\rbrace $ .",
"Considering that every open ball with center $\\ell $ contains some $G_k$ with $k$ sufficiently large, it easily follows that $\\Gamma _x(\\mathcal {I})=\\lbrace \\ell \\rbrace $ .",
"In particular, since each $\\mathcal {I}$ -limit point is also an $\\mathcal {I}$ -cluster point, we conclude that $\\Lambda _x(\\mathcal {I})$ is either empty or the singleton $\\lbrace \\ell \\rbrace $ .",
"Corollary 4.3 An ideal $\\mathcal {I}$ is maximal if and only if every real sequence $x$ has at most one $\\mathcal {I}$ -limit point.",
"First, let us assume that $\\mathcal {I}$ is a maximal ideal.",
"Let us suppose that there exists $k>0$ such that $A_k:=\\lbrace n: |x_n|>k\\rbrace \\in \\mathcal {I}$ and define a sequence $y=(y_n)$ by $y_n=k$ if $n \\in A_k$ and $y_n=x_n$ otherwise.",
"Then, it follows by [3] and Proposition REF that there exists $\\ell \\in \\mathbf {R}$ such that $\\Lambda _x(\\mathcal {I})=\\Lambda _y(\\mathcal {I})\\subseteq \\Gamma _y(\\mathcal {I})=\\lbrace \\ell \\rbrace $ .",
"Now, assume that $A_k^c\\in \\mathcal {I}$ for all $k \\in \\mathbf {N}$ .",
"Hence, letting $z=(z_n)$ be the sequence defined by $z_n=x_n$ if $n \\in A_k$ and $z_n=k$ otherwise, we obtain $\\Lambda _x(\\mathcal {I})=\\Lambda _z(\\mathcal {I}) \\subseteq \\mathrm {L}_z \\subseteq \\mathbf {R}\\setminus (-k,k).$ Therefore, it follows by the arbitrariness of $k$ that $\\Lambda _x(\\mathcal {I})=\\emptyset $ .",
"Conversely, let us assume that $\\mathcal {I}$ is not a maximal ideal.",
"Then there exists $A \\subseteq \\mathbf {N}$ such that $A\\notin \\mathcal {I}$ and $A^c \\notin \\mathcal {I}$ .",
"Then, the sequence $(x_n)$ defined by $x_n=\\chi _A(n)$ for each $n$ has two $\\mathcal {I}$ -limit points.",
"We conclude by showing that there exists an ideal $\\mathcal {I}$ and a real sequence $x$ such that $\\Lambda _x(\\mathcal {I})$ is not an $F_\\sigma $ -set.",
"Example 4.4 Fix a partition $\\lbrace P_m: m \\in \\mathbf {N}\\rbrace $ of $\\mathbf {N}$ such that each $P_m$ is infinite.",
"Then, define the ideal $\\mathcal {I}:=\\lbrace A \\subseteq \\mathbf {N}: \\lbrace m: A \\cap P_m \\notin \\mathrm {Fin}\\rbrace \\in \\mathrm {Fin}\\rbrace ,$ which corresponds to the Fubini product $\\mathrm {Fin} \\times \\mathrm {Fin}$ on $\\mathbf {N}^2$ (it is known that $\\mathcal {I}$ is a $F_{\\sigma \\delta \\sigma }$ -ideal and it is not a P-ideal).",
"Given a real sequence $x=(x_n)$ , let us denote by $x \\upharpoonright P_m$ the subsequence $(x_n: n \\in P_m)$ .",
"Hence, a real $\\ell $ is an $\\mathcal {I}$ -limit point of $x$ if and only if there exists a subsequence $(x_{n_k})$ converging to $\\ell $ such that $\\lbrace n_k: k \\in \\mathbf {N}\\rbrace \\cap P_m$ is infinite for infinitely many $m$ .",
"Moreover, for each $m$ of this type, the subsequence $(x_{n_k}) \\upharpoonright P_m$ converges to $\\ell $ .",
"It easily follows that $\\Lambda _x(\\mathcal {I})=\\bigcap _k \\bigcup _{m\\ge k} \\mathrm {L}_{x \\upharpoonright P_m}.$ (In particular, since each $\\mathrm {L}_{x \\upharpoonright P_m}$ is closed, then $\\Lambda _x(\\mathcal {I})$ is an $F_{\\sigma \\delta }$ -set.)",
"At this point, let $(q_t: t \\in \\mathbf {N})$ be the sequence $({0}{1},{1}{1},{0}{2},{1}{2},{2}{2},{0}{3},{1}{3},{2}{3},{3}{3},\\ldots )$ , where $q_t:=a_t/b_t$ for each $t$ , and note that $\\lbrace q_t: t \\in \\mathbf {N}\\rbrace =\\mathbf {Q} \\cap [0,1]$ .",
"It follows by construction that $a_t \\le b_t$ for all $t$ and $b_t = \\sqrt{2t}(1+o(1))$ as $t\\rightarrow \\infty $ .",
"In particular, if $m$ is a sufficiently large integer, then $\\min _{i \\le m:\\, q_i \\ne q_m}\\,|q_i-q_m| \\ge \\left(\\frac{1}{\\sqrt{2m}(1+o(1))}\\right)^2 > \\frac{1}{3m}.$ Lastly, for each $m \\in \\mathbf {N}$ , define the closed set $C_m:=[0,1] \\cap \\bigcap _{t\\le m}\\left(q_t-\\frac{1}{2^m},q_t+\\frac{1}{2^m}\\right)^c.$ We obtain by (REF ) that, if $m$ is sufficiently large, let us say $\\ge k_0$ , then $C_m \\cup C_{m+1} = [0,1] \\cap \\bigcap _{t\\le m}\\left(q_t-\\frac{1}{2^{m+1}},q_t+\\frac{1}{2^{m+1}}\\right)^c.$ It follows by induction that $C_m \\cup C_{m+1} \\cup \\cdots \\cup C_{m+n} = [0,1] \\cap \\bigcap _{t\\le m}\\left(q_t-\\frac{1}{2^{m+n}},q_t+\\frac{1}{2^{m+n}}\\right)^c.$ for all $n \\in \\mathbf {N}$ .",
"In particular, $\\bigcup _{m \\ge k}C_m = [0,1]\\setminus \\lbrace q_1,\\ldots ,q_k\\rbrace $ whenever $k \\ge k_0$ .",
"Let $x$ be a real sequence such that each $\\lbrace x_n: n \\in P_m\\rbrace $ is a dense subset of $C_m$ .",
"Therefore, it follows by (REF ) that $\\Lambda _x(\\mathcal {I})=\\bigcap _k \\bigcup _{m\\ge k} C_m \\subseteq \\bigcap _{k \\ge k_0} \\bigcup _{m\\ge k} C_m = \\bigcap _{k \\ge k_0} \\,[0,1]\\setminus \\lbrace q_1,\\ldots ,q_k\\rbrace =[0,1]\\setminus \\mathbf {Q}.$ On the other hand, if a rational $q_t$ belongs to $\\Lambda _x(\\mathcal {I})$ , then $q_t \\in \\bigcup _{m\\ge k} C_m$ for all $k \\in \\mathbf {N}$ , which is impossible whenever $k \\ge t$ .",
"This proves that $\\Lambda _x(\\mathcal {I})=[0,1]\\setminus \\mathbf {Q}$ , which is not an $F_\\sigma $ -set.",
"We leave as an open question to determine whether there exists a real sequence $x$ and an ideal $\\mathcal {I}$ such that $\\Lambda _x(\\mathcal {I})$ is not Borel measurable."
],
[
"Acknowledgments",
"The authors are greatful to Szymon Głąb (Łódź University of Technology, PL) for suggesting to investigate the ideal $\\mathrm {Fin} \\times \\mathrm {Fin}$ in Example REF ."
]
] | 1709.01680 |
[
[
"Non-Reciprocal Thermal Material by Spatio-Temporal Modulation"
],
[
"Abstract The thermal properties of a material with a spatio-temporal modulation in both the thermal conductivity and the mass density are studied.",
"The special configuration studied here consists of a modulation in a wave-like fashion.",
"It is found that these materials behaves, in an effective way, as materials with an internal convection-like term that provides them of non-reciprocal properties, in the sense that the flow of heat has different properties when it propagates in the same direction or in the opposite one to the modulation of the parameters.",
"An effective medium description is presented which accurately describes the modulated material, and numerical simulations supports both the non-reciprocal properties and the effective medium description.",
"It is found that these materials are promising candidates for the design of thermal diodes and other advanced devices for the control of the heat flow at all scales."
],
[
"acknowledgements",
"Work supported by the LabEx AMADEus (ANR-10- 444 LABX-42) in the framework of IdEx Bordeaux (ANR-10- 445IDEX-03-02), France."
],
[
"Supplementary Material",
"The diffusion equation for a material with parameters depending on both space and time in a wave-like fashion is given by $\\frac{\\partial }{\\partial x}\\left(\\sigma (x-v_0t)\\frac{\\partial T}{\\partial x}\\right)=\\rho (x-v_0t)\\frac{\\partial T}{\\partial t}.$ The change of variable $n=x-v_0t$ and $\\tau =t$ transform this equation in $\\frac{\\partial }{\\partial n}\\left(\\sigma (n)\\frac{\\partial T}{\\partial n}\\right)=\\rho (n)\\frac{\\partial T}{\\partial \\tau }-\\rho (n) v_0 \\frac{\\partial T}{\\partial n}$ where both $\\sigma (n)$ and $\\rho (n)$ are periodic functions of $n$ .",
"These functions can be expanded as Fourier series in the traditional way $ \\sigma (n)&=\\sum _G\\sigma _Ge^{-iGn}\\\\\\rho (n)&=\\sum _G\\rho _Ge^{-iGn} $ and the solution for the temperature field can also be expressed in the form of a Bloch function $T(n,\\tau )=e^{-ikn}e^{i\\Omega \\tau }\\phi (n)=e^{-ikn}e^{i\\Omega \\tau }\\sum _G\\phi _Ge^{-iGn}$ where we have used the property of periodicity of $\\phi (n)$ .",
"Inserting the above equations into equation (REF ) we arrive to the typical matrix equation defining the solutions $\\Omega =\\Omega (k)$ , $-\\sum _{G^{\\prime }}(k+G)\\sigma _{G-G^{\\prime }}(k+G^{\\prime })T_{G^{\\prime }}=i\\Omega \\sum _{G^{\\prime }}\\rho _{G-G^{\\prime }}T_{G^{\\prime }}+iv_0\\sum _{G^{\\prime }}\\rho _{G-G^{\\prime }}(k+G^{\\prime })T_{G^{\\prime }},$ we can now reorganize the second term of the right hand side of the above equation to group the term $\\Omega +v_0k$ , $-\\sum _{G^{\\prime }}(k+G)\\sigma _{G-G^{\\prime }}(k+G^{\\prime })T_{G^{\\prime }}=i(\\Omega +v_0k)\\sum _{G^{\\prime }}\\rho _{G-G^{\\prime }}T_{G^{\\prime }}+iv_0\\sum _{G^{\\prime }}\\rho _{G-G^{\\prime }}G^{\\prime }T_{G^{\\prime }}.$ Notice that the term $\\Omega +v_0k$ is actually the frequency in the $x-t$ frame, since the solutions of the equation in this frame are $T(n,\\tau )=e^{-ikn}e^{i\\Omega \\tau }\\phi (n)=e^{-ikx}e^{i(\\Omega +v_0 k) t}\\phi (x-v_0t)=e^{-ikx}e^{i\\omega t}\\phi (x-v_0t)$ therefore the replacement $k=k$ and $\\omega =\\Omega +v_0k$ returns the system of equations to the $x-t$ frame, giving $-\\sum _{G^{\\prime }}(k+G)\\sigma _{G-G^{\\prime }}(k+G^{\\prime })T_{G^{\\prime }}=i\\omega \\sum _{G^{\\prime }}\\rho _{G-G^{\\prime }}T_{G^{\\prime }}+iv_0\\sum _{G^{\\prime }}\\rho _{G-G^{\\prime }}G^{\\prime }T_{G^{\\prime }}.$ The above equation will allow us to define the effective parameters of the material with the spatio-temporal modulation.",
"We are interested now in the average temperature field, $T_0$ , since it can be interpreted as the macroscopic temperature, therefore we split the above equation in two terms, those for $G=0$ and those for $G\\ne 0$ , $ -k^2\\sigma _0T_0-k\\sum _{G^{\\prime }}\\sigma _{-G^{\\prime }}(k+G^{\\prime })T_{G^{\\prime }}&=i\\omega \\rho _{0}T_{0}+i\\sum _{G^{\\prime }}\\rho _{-G^{\\prime }}\\left(\\omega +v_0G^{\\prime }\\right)T_{G^{\\prime }}\\\\-(k+G)\\sigma _{G}kT_{0}-\\sum _{G^{\\prime }}(k+G)\\sigma _{G-G^{\\prime }}(k+G^{\\prime })T_{G^{\\prime }}&=i\\omega \\rho _{G}T_{0}+i\\sum _{G^{\\prime }}\\rho _{G-G^{\\prime }}\\left(\\omega +iv_0G^{\\prime }\\right)T_{G^{\\prime }} $ and we solve for $T_G$ from the second one, $T_{G^{\\prime }}=-\\sum _{G}\\chi _{G^{\\prime }G}\\left[(k+G)\\sigma _{G}k+i \\omega \\rho _{G}\\right]T_0$ with $\\chi _{GG^{\\prime }}=\\left[(k+G)\\sigma _{G-G^{\\prime }}(k+G^{\\prime })+i\\left(\\omega +v_0G^{\\prime }\\right)\\rho _{G-G^{\\prime }}\\right]^{-1}$ and we introduce it in the first one, to obtain $\\left(k^2\\sigma _0+i\\omega \\rho _{0}-\\sum _{G^{\\prime },G}\\left[k\\sigma _{-G^{\\prime }}(k+G^{\\prime })+i\\left(\\omega +v_0G^{\\prime }\\right)\\rho _{-G^{\\prime }}\\right]\\chi _{G^{\\prime }G}\\left[(k+G)\\sigma _{G}k+i \\omega \\rho _{G}\\right]\\right)T_0=0\\\\$ which can be expressed as $\\left(k^2\\sigma ^*+i\\omega \\rho ^*-i k C-i\\omega k (S+S^{\\prime })\\right)T_0=0$ with the effective parameters defined as $ \\sigma ^*(\\omega ,k)&=\\sigma _0-\\sum _{G^{\\prime },G}\\sigma _{-G^{\\prime }}(k+G^{\\prime })\\chi _{G^{\\prime }G}(k+G)\\sigma _{G}\\\\\\rho ^*(\\omega ,k)&=\\rho _0-i\\omega \\sum _{G^{\\prime },G}\\rho _{-G^{\\prime }}\\chi _{G^{\\prime }G}\\rho _{G}-iv_0 \\sum _{G^{\\prime },G}G^{\\prime }\\rho _{-G^{\\prime }}\\chi _{G^{\\prime }G}\\rho _{G}\\\\S(\\omega ,k)&=\\sum _{G^{\\prime },G}\\sigma _{-G^{\\prime }}(k+G^{\\prime })\\chi _{G^{\\prime }G}\\rho _{G}\\\\S^{\\prime }(\\omega ,k)&=\\sum _{G^{\\prime },G}\\rho _{-G^{\\prime }}\\chi _{G^{\\prime }G}\\sigma _{G}(k+G)\\\\C(\\omega ,k)&=v_0\\sum _{G^{\\prime },G}\\rho _{-G^{\\prime }}G^{\\prime }\\chi _{G^{\\prime }G}\\sigma _{G}(k+G) $ since $T_0$ is the average temperature $\\langle T\\rangle $ , and replacing $k\\rightarrow i\\partial _x$ and $\\omega \\rightarrow -i\\partial _t$ , we can propose that the wave equation for the macroscopic temperature is $\\sigma ^*\\frac{\\partial ^2 \\langle T\\rangle }{\\partial x^2}= \\rho ^*\\frac{\\partial \\langle T\\rangle }{\\partial t}+C\\frac{\\partial \\langle T\\rangle }{\\partial x}-i(S+S^{\\prime })\\frac{\\partial ^2 \\langle T\\rangle }{\\partial x\\partial t}$ The simpler case of modulation is a simple cosinus perturbation of the form $ \\sigma (x-v_0t)&=\\sigma _0\\left[1+\\Delta _\\sigma \\cos \\frac{2\\pi }{d}(x-v_0t)\\right]\\\\\\rho (x-v_0t)&=\\rho _0\\left[1+\\Delta _\\rho \\cos \\frac{2\\pi }{d}(x-v_0t)\\right] $ where the mass density and conductivity changes periodically from $\\rho _b=\\rho _0(1-\\Delta _\\rho )$ to $\\rho _a=\\rho _0(1+\\Delta _\\rho )$ and from $\\sigma _b=\\sigma _0(1-\\Delta _\\sigma )$ to $\\sigma _a=\\sigma _0(1+\\Delta _\\sigma )$ , respectively.",
"In this case we have therefore only one Fourier component different than 0, so that the $\\chi _{GG^{\\prime }}$ matrix is diagonal with elements $\\chi _{\\pm }$ given by $\\chi _\\pm =\\frac{d}{2\\pi }\\frac{d}{2\\pi \\sigma _0 \\pm i v_0d\\rho _0}$ Then it is easy to see that $ \\sigma ^*&=\\sigma _0\\left[1-\\frac{8\\pi ^2\\sigma _1^2}{4\\pi ^2\\sigma _0^2+v_0^2d^2\\rho _0^2}\\right]\\\\\\rho ^*&=\\rho _0\\left[1+\\frac{2v_0^2d^2\\rho _1^2}{4\\pi ^2\\sigma _0^2+v_0^2d^2\\rho _0^2}\\right]\\\\S&=S^{\\prime }=\\frac{2iv_0d^2\\rho _0\\rho _1\\sigma _1}{4\\pi ^2\\sigma _0^2+v_0^2d^2\\rho _0^2}\\\\C&=v_0\\frac{8\\pi ^2\\sigma _0\\sigma _1\\rho _1}{4\\pi ^2\\sigma _0^2+v_0^2d^2\\rho _0^2} $ or $ \\sigma ^*&=\\sigma _0\\left[1-\\frac{1}{2}\\frac{\\Delta _\\sigma ^2}{1+\\Gamma ^2}\\right]\\\\\\rho ^*&=\\rho _0\\left[1+\\frac{\\Gamma ^2}{2}\\frac{ \\Delta _\\rho ^2}{1+\\Gamma ^2}\\right]\\\\S&=S^{\\prime }=\\frac{\\rho _0d}{2\\pi }\\frac{\\Delta _\\rho \\Delta _\\sigma }{2}\\frac{i\\Gamma }{1+\\Gamma ^2}\\\\C&=\\frac{2\\pi \\sigma _0}{d} \\frac{\\Delta _\\rho \\Delta _\\sigma }{2}\\frac{\\Gamma }{1+\\Gamma ^2} $"
]
] | 1709.01541 |
[
[
"Simulation of Fluid Particle Cutting - Validation and Case Study"
],
[
"Abstract In this paper we present the comparison of experiments and numerical simulations for bubble cutting by a wire.",
"The air bubble is surrounded by water.",
"In the experimental setup an air bubble is injected on the bottom of a water column.",
"When the bubble rises and contacts the wire, it is separated into two daughter bubbles.",
"The flow is modeled by the incompressible Navier-Stokes equations.",
"A meshfree method is used to simulate the bubble cutting.",
"We have observed that the experimental and numerical results are in very good agreement.",
"Moreover, we have further presented simulation results for liquid with higher viscosity.",
"In this case the numerical results are close to previously published results."
],
[
"Introduction",
"Fluid particle cutting plays an important role in gas-liquid and liquid-gas contactors.",
"In gas-liquid contactors, the bubble size distribution, determining the mass transfer area, is influenced by the local hydrodynamics, but also by measuring probes such as needle probes [1], [2], [3] and mesh based conductivity probes [4].",
"The shape of the probes are mainly cylindrical, while the probe may be in flow direction but also in a rectangular angle to it.",
"The rising bubbles approach the immersed object and starts to change its shape.",
"Depending on the position of the bubble to the wire, the bubble will pass the object or be cutted in two fragments (daughter bubbles).",
"Beside the unwanted cutting at probes, a wire mesh can be used to generate smaller bubbles and homogenise the flow structure.",
"Furthermore, in liquid-gas contactors, phase separation is often a problem.",
"Demisters are then frequently used to prevent a phase slip (entrainment) of fine dispersed phase droplets in the continuous product phase.",
"A loss of the total solvent inventory within one year is reported causing costs and environmental hazards.",
"Entrainment can cause a significant reduction in separation efficiency.",
"Demisters are based on wire meshes, where the small droplets should accumulate.",
"Using an optimal design, the small droplet seperation efficiencies can be up to 99.9%.",
"Nevertheless, bigger droplets tend to break up in the rows of wires.",
"Hence, particle cutting is a frequently observed phenomena in various separation processes ranging from low viscosity to high viscosity of the continuous fluid.",
"Nevertheless, it can be hardly investigated under operation conditions due to the complex insertion of optical probes into the apparatus or the complex mesh structure e.g.",
"of the demister, but also the operation conditions as high pressure, high dispersed phase hold ups make an experimental investigation challenging.",
"In this study, we focus on the simulation of particle cutting at a single wire strengthened by experimental investigations to generate the basis for further numerical studies at complex geometries and fluid flow conditions such as demister simulations.",
"For the simulation of bubble cutting, a meshfree approach is applied.",
"It overcomes several drawbacks of classical computational fluid dynamics (CFD) methods such as Finite Element Method (FEM) , Finite Volume Method (FVM).",
"The main drawback of the classical methods (FEM, FVM) is the relatively expensive geometrical mesh/grid required to carry out the numerical computations.",
"The computational costs to generate and maintain the grid becomes particularly high for complex geometries and when the grid moves in time, as in the case of fluid particles with a dynamic interface or in case where the interface between fluids changes in time.",
"For such problems meshfree methods are appropriate.",
"Here, we use a meshfree method, based on the generalized finite difference method, called Finite Pointset Method (FPM).",
"The two phase flow is modeled by using the continuous surface force (CSF) model [8].",
"Each phase is indicated by the color of the respective particles.",
"When particles move, they carry all the information about the flow with them such as their color, density, velocity, etc.",
"The colors, densities and viscosity values of all particles remain constant during the time evolution.",
"The fluid-fluid interface is easily determined with the help of the color function [9].",
"In [13] an implementation of the CSF model within the FPM was presented to simulate surface-tension driven flows.",
"We have further extended the method to simulate wetting phenomena [11]."
],
[
"Experimental Setup",
"Bubble cutting is investigated in a Plexiglas column filled with reversed osmosis water up to a level of 10 cm.",
"The column has a width and depth of 46 mm.",
"A syringe pump (PSD/3, Hamilton) is used to inject air of known volume at the bottom of the column.",
"The injection diameter is 8 mm.",
"The schematic setup is given in Figure REF .",
"In a distance of 60 mm from the bottom, a wire is mounted in the middle of the column.",
"The wire has a diameter of 3 mm.",
"Two cameras (Imaging Solutions NX8-S2 and Os 8-S2) are mounted in an angle of 90to track the bubble motion over an image sequence, respectively over time.",
"Both cameras are triggered and allow a synchronous detection at 4000 fps and a resolution of 1600x1200 px$^2$ .",
"By tracking the bubble motion from two sides, it is possible to analyse the side movement and to detect the exact position of bubble contact with the wire.",
"Also, the bubble deformation can be analyzed in two direction and therefore leads to more precise results compared to single camera setups.",
"Figure: Sketch of the experimental setup showing the plexiglas column in the middle (blue)."
],
[
"Bubble Motion Analyses",
"For bubble motion analyses, the tool box ImageJ (https://imagej.nih.gov/ij/) is used.",
"The raw images (Fig.",
"REF a/b) are therefore binarized, followed by a watershed segmentation.",
"The tracks of the single bubble and cutted particles are analysed using the Plugin Mtrack2 (http://imagej.net/MTrack2).",
"Two particle tracks are tracked, one by each camera and reconstructed using Matlab software toolbox ( (Fig.",
"REF c) These are the basis for three dimensional reconstruction of the bubble motion.",
"The conversion of pixels to metric length is done by a afore performed calibration.",
"Matlab is used to reconstruct the bubble in a three dimensional domain.",
"The images are converted to greyscale and further to binary images.",
"Possible holes (white spots in a surrounded black bubble structure) are filled to get a better identification of the bubbles.",
"To detect the bubble position, a distance transform is performed, followed by a watershed segmentation to separate the bubble from the pipe structure.",
"Finally, the bubble size and shape is determined from each image.",
"In a next step, the basic grey scale images are again converted to binary images, followed by a watershed algorithm [5].",
"Finally, the resulting structures are transformed into 3D space.",
"Therefore, the detected structures are extruded into the third dimension resulting in overlapping structures.",
"The overlapping structures represent the bubble and the wire and are visualized in Fig.",
"REF d. Applying the assumption of an ellipsoidic structure for the bubble, results finally in Fig.",
"REF e. Figure: 3D bubble reconstruction"
],
[
"Mathematical Model ",
"We consider a one-fluid model for two immiscible fluids which are liquid and gas.",
"We model the equation of motion of these fluids by the incompressible Navier-Stokes equations, which are given in the Lagrangian form $\\frac{d{\\bf x}}{dt} &=&{\\bf v}\\\\\\nabla \\cdot {\\bf v} &= &0 \\\\\\rho \\frac{d{\\bf v}}{dt} &= &-\\nabla p + \\nabla \\cdot (2 \\mu \\tau ) + \\rho {\\bf g} + \\rho {\\bf F}_S,$ where ${\\bf v}$ is the fluid velocity, $\\rho $ is the fluid density, $p$ is the pressure, $\\tau $ is the stress tensor given by $\\tau = \\frac{1}{2}(\\nabla {\\bf v} + (\\nabla {\\bf v})^T)$ , $\\bf {g}$ is the external force and ${\\bf F}_S$ is the surface tension force.",
"The quantity ${\\bf F}_S$ is force density, which acts on the interface and its neighbor of the interface between gas and liquid.",
"We compute the surface tension force CSF model of Brackbill et al ([8]) and is given by ${\\bf F}_S = \\sigma \\kappa {\\bf n} \\delta _S,$ where, $\\sigma $ is the surface tension coefficient, ${\\bf n}$ is the unit normal vector at the interface and its neighbor, $\\kappa $ is the curvature and $\\delta _S$ is the surface delta function.",
"We note that $\\delta _S$ is quite strong in the interface and its surroundings.",
"We solve the equations (REF ) with initial and boundary conditions."
],
[
"Numerical methods",
"We solve the equations (REF ) by a meshfree Lagrangian particle method.",
"In this method, we first approximate a computational domain by discrete grid points.",
"The grids points are divided into two parts as interior and boundary particles.",
"The boundary particles approximate boundaries and we prescribe boundary conditions on them.",
"The interior particles move with the fluid velocities.",
"Particles may come very close to each other or can go far away from each other leading to very fine or very coarse approximations.",
"This problem has to be tackled carefully due to stability reasons.",
"To obtain a uniform distribution of particles in each time step one has to add or remove particles, if necessary.",
"We refer to [10] for details of such a particle management."
],
[
"Computation of the quantities in surface tension force",
"For meshfree particle methods the interfaces between fluids are easily tracked by using flags on the particles.",
"Initially, we assign different flags or color function $c$ of particles representing the corresponding fluids.",
"We define the color function $c = 1$ for fluid type 1 and $c=2$ for fluid type 2.",
"On the interface and its vicinity, the Shepard interpolation is applied for smoothing of the color functions using $\\tilde{c}({\\bf x}) = \\frac{\\sum _{i=1}^{m} w_i c_i}{\\sum _{i=1}^{m}},$ where $m$ is the number of neighbor of arbitrary particle having position ${\\bf x}$ , $c_i$ are the color values at neighboring particle $i$ and $w_i$ is the weight as a function of distance from ${\\bf x}$ to $\\bf {x}_i$ given by $w_i = w( {\\bf x}_i - {\\bf x}; h) =\\left\\lbrace \\begin{array}{l}\\exp (- \\alpha \\frac{\\Vert {\\bf x}_i - {\\bf x} \\Vert ^2 }{h^2} ),\\quad \\mbox{if } \\frac{\\Vert {\\bf x}_i - {\\bf x} \\Vert }{h} \\le 1\\\\0, \\qquad \\qquad \\quad \\quad \\quad \\quad \\mbox{else},\\end{array}\\right.$ where $ \\alpha $ is a positive constant After smoothing the color function, we compute the unit normal vector ${\\bf n} = \\frac{ \\nabla \\tilde{c}}{ | \\nabla \\tilde{c} |}.$ Finally, we compute the curvature by $\\kappa = -\\nabla \\cdot {\\vec{n}}.$ The quantity $\\delta _s$ is approximated as $\\delta _s \\approx | \\nabla \\tilde{c} |.$ Here, $\\delta _s$ is non-zero in the vicinity of the interface and vanishes far from it."
],
[
"Numerical scheme",
"We solve the Navier-Stokes equations (REF ) with the help of Chorin's projection method [7].",
"Here, the projection method is adopted in the Lagrangian meshfree particle method.",
"Consider the discrete time levels $t^n = n~dt, n = 0,1,2,\\ldots $ with time step $dt$ .",
"Let ${\\bf x}^n$ be the position of a particle at time level $n$ .",
"In the Lagrangian particle scheme we compute the new particle positions at the time level $(n+1)$ by ${\\bf x}^{n+1} = {\\bf x}^n + dt \\; {\\bf v}^n$ and then use Chorin's pressure projection scheme in new positions of particles.",
"The pressure projection scheme is divided into two steps.",
"The first step consists of computing the intermediate velocity $\\bf {v}^{*}$ with neglecting the pressure term ${\\bf v}^{*} = {\\bf v}^n + \\frac{dt}{\\rho } \\nabla \\cdot (2 \\mu \\tau ^{*} ) + dt ~{\\bf g} + \\frac{dt}{\\rho } {\\bf F}^n_S.$ Since we use the Lagrangian formulation, we do not need to handle the nonlinear convective term.",
"The second step consists of computation of pressure and the velocity at time level $(n+1)$ by solving the equation ${\\bf v}^{n+1} = {\\bf v}^{*} - dt \\; \\frac{\\nabla p ^{n+1}}{\\rho }$ where ${\\bf v}^{n+1}$ should obey the continuity equation $\\nabla \\cdot {\\bf v}^{n+1} = 0.$ We observe from the equation (REF ) that the new pressure $p^{n+1}$ is necessary in order to compute the new velocity.",
"${\\bf v}^{n+1}$ .",
"Now, we take the divergence of equation (REF ) on both sides and use of the continuity constraint (REF ), we obtain the pressure Poisson equation $\\nabla \\cdot \\left(\\frac{ \\nabla p^{n+1}}{\\rho }\\right) =\\frac{\\nabla \\cdot {\\bf v}^{*}}{dt}.$ In order to derive the boundary condition for $p$ we project the equation (REF ) on the outward unit normal vector ${\\bf n}$ at the boundary $\\Gamma $ and then we obtain the Neumann boundary condition $\\left(\\frac{\\partial p}{\\partial {\\bf n} }\\right)^{n+1} =- \\frac{\\rho }{dt} ({\\bf v}^{n+1}_{\\Gamma } - {\\bf v}^{*}_{\\Gamma }) \\cdot {\\bf n},$ where ${\\bf v}_{\\Gamma }$ is the value of ${\\bf v}$ on $\\Gamma $ .",
"Assuming ${\\bf v}\\cdot {\\bf n} = 0$ on $\\Gamma $ , we obtain $\\left(\\frac{\\partial p}{\\partial {\\bf n} }\\right)^{n+1} = 0$ on $\\Gamma $ .",
"We note that we have to approximate the spatial derivatives at each particle position as well as solve the second order elliptic problems for the velocities and the pressure.",
"The spatial derivatives at each particle position are approximated from its neighboring clouds of particles based on the weighted least squares method.",
"The weight is a function of a distance of a particle position to its neighbors.",
"We observe that in Eq.",
"REF there is a discontinues coefficient $\\mu $ inside the divergence operator since the viscosities of two liquid may have the ratio of up to 1 to 100.",
"Similarly, the density ratio also has 1 to 1000, which can be seen also in Eq.",
"REF .",
"This discontinous coefficients have to be smoothed for stable computation.",
"This is done using a similar procedure as for smoothing the color function.",
"We denote the smoothed viscosity and density by $\\tilde{\\mu }$ and $\\tilde{\\rho }$ , respectively.",
"We note that we smooth the density and viscosity while solving Eqs.",
"REF and REF , but keep them constant on each phase of particles during the entire computational time.",
"If the density and viscosity has larger ratios, we may have to iterate the smoothing 2 or 3 times.",
"Finally, Eq.",
"REF and Eq.",
"REF can be re-expressed as $u^{*} - \\frac{dt}{\\tilde{\\rho }} \\nabla \\tilde{\\mu }\\cdot \\nabla u^{*} -dt \\frac{\\tilde{\\mu }}{\\tilde{\\rho }} \\Delta u^{*}&=&u^n + dt \\; g_x +\\frac{dt}{\\rho }(\\frac{\\partial \\tilde{\\mu }}{\\partial x}\\frac{\\partial u^n}{\\partial x} +\\frac{\\partial \\tilde{\\mu }}{\\partial y}\\frac{\\partial v^n}{\\partial x} )\\\\v^{*} -\\frac{dt}{\\tilde{\\rho }} \\nabla \\tilde{\\mu }\\cdot \\nabla v^{*} -dt \\frac{\\tilde{\\mu }}{\\tilde{\\rho }} \\Delta v^{*}&= &v^n + dt \\; g_y +\\frac{dt}{\\tilde{\\rho }}(\\frac{\\partial \\tilde{\\mu }}{\\partial x}\\frac{\\partial u^n}{\\partial y} +\\frac{\\partial \\tilde{\\mu }}{\\partial y}\\frac{\\partial v^n}{\\partial y} )\\\\-\\frac{\\nabla \\tilde{\\rho }}{\\tilde{\\rho }}\\cdot \\nabla p^{n+1} + \\Delta p^{n+1} &=& \\tilde{\\rho }\\frac{\\nabla \\cdot {\\vec{v}}^{*}}{dt}.$ Note that, for constant density, the first term of Eq.",
"vanishes and we get the pressure Poisson equation.",
"Far from the interface we have $\\tilde{\\mu }= \\mu $ and $\\tilde{\\rho }= \\rho $ .",
"The momentum and pressure equations have the following general form $A \\psi + {\\bf B}\\cdot \\nabla \\psi + C \\Delta \\psi = f,$ where $A, {\\bf B}$ and $C$ are known quantities.",
"This equation is solved with Dirichlet or Neumann boundary conditions $\\psi = \\psi _{\\Gamma D} \\quad \\quad \\quad \\mbox{or} \\quad \\frac{\\partial \\psi }{\\partial \\vec{n}} = \\psi _{\\Gamma N}.",
"$ Remark: For the x component of the momentum equations we have $A=1, {\\bf B} = -\\frac{dt}{\\tilde{\\rho }}\\nabla \\tilde{\\mu }, C=-\\frac{dt}{\\tilde{\\rho }}\\tilde{\\mu }$ and $f$ is equal to the right hand side of Eq.",
"REF .",
"Similarly, for the pressure equation Eq.",
"we have $A=0, {\\bf B}=\\frac{\\nabla \\tilde{\\rho }}{\\tilde{\\rho }}, C = 1$ and $f=\\tilde{\\rho }\\frac{\\nabla \\cdot {\\bf v}^{*}}{dt}$ .",
"In the following section we describe the method of solving equations Eqs.",
"REF - REF by a meshfree particle method, called the Finite Pointset Method (FPM)."
],
[
" A meshfree particle method for general elliptic boundary value problems",
"In this subsection we describe a meshfree method for solving second order elliptic boundary value problems of type Eqs.",
"REF - REF .",
"The method will be described in a two-dimensional space.",
"The extension of the method to three-dimensional space is straightforward.",
"Let $\\Omega \\in R^2$ be the computational domain.",
"The domain $\\Omega $ is approximated by particles of positions ${\\bf x}_i, i=1,\\ldots ,N$ , which are socalled numerical grid points.",
"Consider a scaler function $\\psi ({\\bf x})$ and let $\\psi _i = \\psi ({\\bf x}_i)$ be its discrete values at particle indices $i=1,\\ldots , N$ .",
"We approximate the spatial derivatives of $\\psi ({\\bf x})$ at an arbitrary position ${\\bf x} \\in \\lbrace {\\bf x}_i, i = 1, \\ldots , N \\rbrace $ , from the values of its neighboring points.",
"We introduce a weight function $w = w({\\bf x}_i- {\\bf x}, h)$ with a compact support $h$ .",
"The value of $h$ can be $2.5$ to 3 times the initial spacing of particles such that the minimum number of neighbor is guaranteed in order to approximate the spatial derivatives.",
"But it is user defined quantity.",
"This weight function has two properties, first, it avoids the influence of the far particles and the second it reduce the unnecessary neighbors in the computational part.",
"One can consider different weight function, in this paper we consider the Gaussian weight function defined in (REF ), where $ \\alpha $ is equal to $6.25$ .",
"Let $ P({\\bf x}, h) = \\lbrace {\\bf x}_j :j=1,2,\\ldots ,m \\rbrace $ be the set of $ m $ neighboring particles of $ {\\bf x} $ in a circle of radius $h$ .",
"We note that the point ${\\bf x}$ is itself one of ${\\bf x}_j$ .",
"We consider Taylor expansions of $\\psi ({\\bf x}_i)$ around $ {\\bf x} = (x,y)$ $\\psi (x_j,y_j)= \\psi (x,y)+\\frac{\\partial \\psi }{\\partial x} (x_j - x) +\\frac{\\partial \\psi }{\\partial y} (y_j - y) +\\frac{1}{2} \\frac{\\partial ^2 \\psi }{\\partial x^2} (x_j - x)^2 +\\nonumber \\\\\\quad \\quad \\quad \\frac{\\partial ^2 \\psi }{\\partial x\\partial y} (x_j - x)(y_j-y) +\\frac{1}{2} \\frac{\\partial ^2 \\psi }{\\partial y^2} (y_j - y)^2 + e_j$ for $j = 1, \\ldots , m$ , where $ e_j $ is the residual error.",
"Let the coefficients of the Taylor expansion be denoted by $a_1 = \\psi (x,y), \\;a_2 = \\frac{\\partial \\psi }{\\partial x}, \\;a_3 = \\frac{\\partial \\psi }{\\partial y}, \\; $ $a_4 = \\frac{\\partial ^2\\psi }{\\partial x^2}, \\;a_5 = \\frac{\\partial ^2\\psi }{\\partial x\\partial y},a_6 = \\frac{\\partial ^2\\psi }{\\partial y^2}.",
"\\;$ We add the constraint that at particle position $(x,y)$ the partial differential equation (REF ) should be satisfied.",
"If the point $(x,y)$ lies on the boundary, also the boundary condition (REF ) needs to be satisfied.",
"Therefore, we add Eqs.",
"REF and REF to the $m$ equations (REF ).",
"Equations REF and REF are re-expressed as $A a_1 + B_1 a_2 + B_2 a_3 + C (a_4 + a_6 ) = f + e_{m+1}\\\\n_x a_2 + n_y a_3 = \\psi _{\\Gamma N} + e_{m+2},$ where ${\\bf B} = (B_1, B_2)$ and $n_x, n_y$ are the $x,y$ components of the unit normal vector ${\\bf n}$ on the boundary $\\Gamma $ .",
"The coefficients $a_i, i = 1,\\ldots ,6$ are the unknowns.",
"We have six unknowns and $m+1$ equations for the interior points and $m+2$ unknowns for the Neumann boundary points.",
"This means, we always need a minimum of six neighbors.",
"In general, we have more than six neighbors, so the system is overdetermined and can be written in matrix form as ${\\bf e}= M {\\bf a} - {\\bf b},$ where $M=\\left( \\begin{array}{cccccc}1 & ~dx_1 & ~dy_1 & ~\\frac{1}{2}dx^2_1 & ~dx_1 dy_1 & ~\\frac{1}{2} dy^2_1 \\\\\\vdots & \\vdots &\\vdots & \\vdots &\\vdots &\\vdots \\\\1 &~dx_m & ~dy_m & ~\\frac{1}{2}dx^2_m & ~dx_m dy_m & ~\\frac{1}{2} dy^2_m \\\\A & ~B_1 & ~B_2 & ~C & ~0 & ~C \\\\0 & ~n_x & ~n_y & ~0 &~0 &~0\\end{array} \\right),$ with the vectors given by ${ \\bf a} = \\left( a_1, a_2 , \\ldots a_6 \\right)^T , \\;{\\bf b} = \\left( \\psi _1 , \\ldots , \\psi _m, f, \\psi _N \\right)^T $ and ${ \\bf e} = \\left( e_1, \\ldots , e_m, e_{m+1}, e_{m+2} \\right)^T $ and $dx_j = x_{j} - x, \\; dy_j = y_{j}-y$ .",
"For the numerical implementation, we set $n_x = n_y = 0$ and $\\psi _{\\Gamma N} = 0$ for the interior particles.",
"For the Dirichlet boundary particles, we directly prescribe the boundary conditions, and for the Neumann boundary particles the matrix coefficients are given by Eq.",
"REF .",
"The unknowns $a_i$ are computed by minimizing a weighted error over the neighboring points.",
"Thus, we have to minimize the following quadratic form $J = \\sum _{i=1}^{m + 2} w_i e_i^2 = (M {\\bf a} - {\\bf b})^T W (M {\\bf a} - {\\bf b}),$ where $W=\\left( \\begin{array}{cccccc}w_1 & 0 & \\cdots & 0 & 0 & 0 \\\\\\vdots & \\vdots & \\cdots & \\vdots \\\\0 & 0 & \\cdots & w_m &0 & 0\\\\0 & 0 & \\cdots & 0 &1 & 0 \\\\0 & 0 & \\cdots & 0 &0 & 1\\end{array} \\right).$ The minimization of $ J $ with respect to ${\\bf a}$ formally yields ( if $M^T W M$ is nonsingular) ${\\bf a} = (M^T W M)^{-1} (M^T W) {\\bf b}.$ In Eq.",
"REF the vector $( M ^T W) { \\bf b}$ is explicitly given by $( M ^T W) { \\bf b} =\\left( \\sum _{j=1}^m w_j \\psi _j , \\;\\sum _{j=1}^m w_j dx_j \\psi _j + B_1 f + n_x \\psi _{\\Gamma N},\\right.\\nonumber \\\\ \\left.\\sum _{j=1}^m w_j dy_j \\psi _j + B_2 f + n_y \\psi _{\\Gamma N}, \\;\\frac{1}{2}\\sum _{j=1}^m w_j dx^2_j \\psi _j + C f , \\;\\right.\\nonumber \\\\ \\left.\\sum _{j=1}^m w_j dx_j dy_j \\psi _j, \\;\\frac{1}{2}\\sum _{j=1}^m w_j dy^2_j \\psi _j + C f \\;\\right)^T.$ Equating the first components on both sides of Eq.",
"REF , we get $\\psi = Q_{1} \\left(\\sum _{j=1}^m w_j \\psi _j \\right) +Q_{2} \\left( \\sum _{j=1}^m w_j dx_j \\psi _j + B_1 f + n_x \\psi _{\\Gamma N} \\right) +\\nonumber \\\\Q_{3}\\left(\\sum _{j=1}^m w_j dy_j \\psi _j + B_2 f + n_y \\psi _{\\Gamma N} \\right) +Q_{4} \\left( \\frac{1}{2}\\sum _{j=1}^m w_j dx^2_j \\psi _j + C f\\right) +\\nonumber \\\\Q_{5} \\left( \\sum _{j=1}^m w_j dx_j dy_j \\psi _j \\right) +Q_{6} \\left( \\frac{1}{2}\\sum _{j=1}^m w_j dy^2_j \\psi _j + C f\\right),$ where $Q_{1}, Q_{2}, \\ldots , Q_{6}$ are the components of the first row of the matrix $( M^T W M)^{-1}$ .",
"Rearranging the terms, we have $\\nonumber \\psi - \\sum _{j=1}^m w_j\\left( Q_{1} + Q_{2} dx_j +Q_{3} dy_j + Q_{4} \\frac{dx^2_j}{2} +Q_{5} dx_j ~dy_j + Q_{6} \\frac{dy^2_j}{2} \\right) \\psi _j =\\\\\\left( Q_{2} B_1 + Q_{3} B_2 + Q_{4} C+ Q_{6} C \\right) f +\\left(Q_{2} n_x + Q_{3} n_y \\right) \\psi _{\\Gamma N}.",
"\\quad \\quad \\quad $ We obtain the following sparse linear system of equations for the unknowns $\\psi _i, i=1,\\ldots , N$ $\\nonumber \\psi _i - \\sum _{j=1}^{m(i)} w_{i_j}\\left( Q_{1} + Q_{2} dx_{i_j} +Q_{3} dy_{i_j} + Q_{4} \\frac{dx^2_{i_j}}{2} +Q_{5} dx_{i_j} dy_{i_j} +Q_{6} \\frac{dy^2_{i_j}}{2} \\right) \\psi _{i_j} =\\\\\\left( Q_{2} B_1 + Q_{3} B_2 + Q_{4} C+ Q_{6} C \\right) f_i +\\left(Q_{2} n_x + Q_{3} n_y \\right) \\psi _{\\Gamma N_i}.",
"\\quad \\quad \\quad $ In matrix form we have $L{\\bf \\Psi } = {\\bf R},$ where ${\\bf R}$ is the right-hand side vector, ${\\bf \\Psi }$ is the unknown vector and $L$ is the sparse matrix having non-zero entries only for neighboring particles.",
"We solve the sparse system (REF ) by the Gauss-Seidel method.",
"In each time iteration the initial values of $\\psi $ for time step $n+1$ are taken as the values from previous time step $n$ .",
"While solving the equations for intermediate velocities and the pressure will require more iterations in the first few time steps.",
"After a certain number of time steps, the velocities values and the pressure values at the old time step are close to those of new time step, so the number of iterations is dramatically reduced.",
"We stop the iteration process if $\\frac{\\sum _{i=1}^N |\\psi _i^{\\tau + 1} - \\psi _i ^{(\\tau )} | }{\\sum _{i=1}^N |\\psi ^{(\\tau + 1)}_i |} < \\epsilon ,$ for $\\tau = 0, 1, 2, \\ldots $ and $ \\epsilon $ is a small positive constant and can be defined by the user."
],
[
"Validation: low viscosity",
"In a first step, we validate the simulations with the experimental results of the single bubble cutting in reversed osmosis purified water.",
"We apply the same bubble diameter from the experiments ($6.5mm$ ) and the wire diameter of $3mm$ in the simulation.",
"The viscosity of the fluid (water) is $\\mu _l = 0.001 Pa.s$ and the interfacial tension between water and air is $\\sigma =0.072 N/m$ ..",
"The density of water is $\\rho _l=998.2 kg/m^3$ and the density of air is approximated by $\\rho _g=1 kg/m^3$ and the dynamic viscosity of air is $\\mu _g=2e^{-5}$ .",
".",
"For the numerical simulations we consider a two-dimensional geometry of size $36mm\\times 63mm$ .",
"The initial bubble position has the center at $x=18mm$ and $y=10mm$ and the wire has the center at $x=19.5mm$ and $y=45mm$ as shown in Fig.",
"REF .",
"The bubble is approximated by red particles, the liquid is approximated by blue particles.",
"The white circular spot is the position of the wire.",
"We have considered the total number of boundary particles, (including 4 walls and the wire) equal to 527 and the initial number of interior particles equal to 18310.",
"The constant time step $t=5e^{-6}$ is considered.",
"Here the horizontal distance between the initial center of bubble and the wire is $d_x = 1.5mm$ .",
"In all four walls and the wire we have considered no-slip boundary conditions.",
"Initially, the velocity and pressure are equal to zero.",
"The gravitational force is ${\\bf g} = (0, -9.81) m/s^2$ .",
"The comparison between simulation and experiment is depicted in Fig.",
"REF and REF .",
"We extracted a time sequence from the experiments and the corresponding simulations, starting at 0.19 seconds simulated time to 0.27 seconds.",
"The temporal distance between each image is 0.02 s. The rising bubble approaches the wire and starts to deform.",
"There is no direct contact during this phase between the bubble and the wire.",
"Due to the non central approach to the wire, the bubble is cut in a smaller daughter bubble (right) and a larger bubble (left).",
"The larger bubble has three times the diameter of the smaller bubble.",
"The comparison of the cutting process gives a qualitatively good agreement between the experiment and the simulation.",
"Also the shape and size of the mother and daughter bubbles are qualitatively very good agreement.",
"A detailed comparison of the bubble path from experiment and simulation is shown in Fig.",
"REF .",
"The bubble position during first contact is important, which agrees well between experiment and simulation.",
"Nevertheless, the path of the larger bubble in the simulation shows after the cutting a slightly different behaviour than in the experiment.",
"In the experiment, the larger bubble moves inwards again, while the bubble in the simulation moves horizontally away from the wire, which may arise from a slight horizontally movement of the bubble in the experiment.",
"The cutting of the bubble also depends on the bubble velocity, plotted in Fig.",
"REF .",
"The bubble accelerates in the simulation and finally reaches the same end velocity that is observed in the experiment.",
"After the splitting into two daughter bubbles, the larger daughter bubble raises faster than the smaller one.",
"The experimental results are governed by higher fluctuations especially for the smaller bubble, which results from very short temporal distances between the images and fluctuations by detecting the bubble interface.",
"Neverthless, the average velocity for the smaller bubble fits with 0.12 m/s quiet well to the simulated result.",
"Figure: Initial position of bubble and liquid particles for low viscosity.Figure: Bubble cutting over time t=0.19s,0.21s,0.23st=0.19s, 0.21s, 0.23s.",
"Experiment (left) vs. simulation (right).Figure: Bubble cutting over time t=0.25s,0.27st=0.25s, 0.27s..",
"Experiment (left) vs. simulation (right).Figure: Path of the mother bubble and the two daughter bubbles.Figure: Velocities of mother and daughter bubbles."
],
[
"Case study: high viscosity",
"The computational domain is the same as in the previous section.",
"However, the position of the wire is changed.",
"The data has been taken from chapter 6 of [6].",
"The liquid has density $\\rho _l=1250 kg/m^3$ , dynamical viscosity $\\mu _l = 0.219 Pa.s$ .",
"Similarly the gas density $\\rho _g=1 kg/m^3$ and the viscosity $\\mu _g=2e^{-5} Pa.s$ .",
"The surface tension coefficient $\\sigma =0.0658 N/m$ .",
"We consider a bubble of diameter $9.14mm$ with its initial center at $(18mm, 9mm)$ .",
"We consider a wire (in 2D a circle) of diameter $3.1mm$ with different centers at $y=45mm$ and $ x=18mm, 18.5mm, 19mm$ and $19.5mm$ .",
"This means, we consider the initial distance $d_x$ between the center of the bubble and the center of the wire equal to $d_x=0mm, 0.5mm, 1mm, 1.5mm$ .",
"Fig.",
"REF shows the initial geometry with $d_x = 0mm$ .",
"The initial number of particles and the time step are the same as in the low viscosity case.",
"The initial and boundary conditions and the rest of other parameters are same as in the previous case.",
"Figure: Initial position of bubble and liquid particles with d x =0d_x = 0.In Figs.",
"REF - REF we have plotted the positions of the bubble and the wire for $d_x=0mm, 0.5mm, 1mm$ and $1.5mm$ at time $t = 0.288, 0.320, 0.352$ and $t=0.384$ seconds, respectively.",
"For $d_x = 0$ we observe the wire located in the middle of the bubble as expected.",
"When we increased the distance $d_x$ from $0.5mm$ to $1.5mm$ , we observed that the left part of the bubble is increasing and the right part becomes smaller.",
"We clearly observe that the daughter bubbles are symmetric for $d_x = 0mm$ in contrast to the other cases.",
"We further observe a small layer between the wire and the bubble.",
"After $t=0.352$ seconds we observe the cutting of the bubble, see Figs.",
"REF and REF .",
"Two daughter bubbles arise, a larger one on the left and a smaller one on the right side of the wire.",
"The overall numerical results are comparable with the results presented in [6].",
"Figure: The position of bubble with different positions of the wire at t=0.288t=0.288.Figure: The position of bubble with different positions of the wire at t=0.320t=0.320.Figure: The position of bubble with different positions of the wire at t=0.352t=0.352.Figure: The position of bubble with different positions of the wire at t=0.384t=0.384.In Fig.",
"REF we have plotted the trajectories of the mother and bubble droplets.",
"We observed that the mother droplet is cutted into two daughter bubble slightly below the wire, compare with Fig.",
"REF .",
"The trajectories are plotted up to time $t=0.5 s$ .",
"We see that when the size of the daughter bubble is increasing, it travels longer than the smaller bubbles.",
"The reason is that the rising velocity of the larger bubble is larger than the smaller ones, see Fig.",
"REF for the velocities of mother and daughter bubbles.",
"Figure: The trajectories of the mother and the daughter bubbles.",
".Figure: The rising velocities of the mother and the daughter bubbles.",
"."
],
[
"Concluding Remarks",
"The cutting of bubbles at a single tube (wire) was investigated experimentally and numerically.",
"For the simulations, a mesh free method was applied.",
"The method enables a description of the deforming interface and the hydrodynamics of bubble cutting.",
"For a first validation, we compared the solver to experimental data using the system air and water.",
"A suffiently good agreement could be found in regard to bubble shape, bubble movement and cutting process itself.",
"To study the effect of higher viscosity and bubble position, a case study was done.",
"One observes that the inital position of the bubble to the wire has a high impact on the final daughter bubble size ratio.",
"A centric approach of the bubble to the wire leads to a cutting of the bubble in two equally sized daughter bubbles.",
"By increasing the inital distance to the wire, the daughter-bubble size ratio increases and the deviation between the velocities of daughter bubbles increases.",
"Also, the movement of the bubbles directly behind the wire changes.",
"While the bubbles split behind the wire at the centric approach, with increasing unsymmetry, the bubbles start to move inwards after the initial separation.",
"In future, further studies with overlapping wires are planned."
],
[
"Acknowledgment",
"This work is supported by the German research foundation, DFG grant KL 1105/27-1 and by RTG GrK 1932 “Stochastic Models for Innovations in the Engineering Sciences”, project area P1."
]
] | 1709.01729 |
[
[
"On the approximation of the boundary layers for the controllability\n problem of nonlinear singularly perturbed systems"
],
[
"Abstract A new systematic approach to the construction of approximate solutions to a class of nonlinear singularly perturbed feedback control systems using the boundary layer functions especially with regard to the possible occurrence of the boundary layers is proposed.",
"For example, problems with feedback control, such as the steady-states of the thermostats, where the controllers add or remove heat, depending upon the temperature registered in another place of the heated bar, can be interpreted with a second-order ordinary differential equation subject to a nonlocal three--point boundary condition.",
"The $O(\\epsilon)$ accurate approximation of behavior of these nonlinear systems in terms of the exponentially small boundary layer functions is given.",
"At the end of this paper, we formulate the unsolved controllability problem for nonlinear systems."
],
[
"Motivation and introduction",
"In various fields of science and engineering, systems with two-time-scale dynamics are often investigated.",
"In state space, such systems are commonly modeled using the mathematical framework of singular perturbations, with a small parameter, say $\\epsilon $ , determining the degree of separation between the \"slow\" and \"fast\" channels of the system.",
"Singularly perturbed systems (SPS) can also occur due to the presence of small \"parasitic\" parameters, armature inductance in a common model for most DC motors, small time constants, etc.",
"Singular perturbation problems arise also in heat transfer problem with large Peclet numbers (we often assume $\\epsilon $ to be small in order to diminish the effect of diffusion ([23]), Navier-Stokes flows with large Reynolds numbers, chemical reactor theory, aerodynamics, control of reaction-diffusion processes ([8], [20]), quantum mechanics ([1]), optimal control ([24]), for example.",
"The literature on control of nonlinear SPS is extensive, at least starting with the pioneering work of P. Kokotovic et al.",
"nearly 30 years ago ([18]) and continuing to the present including authors such as Z. Artstein ([2], [3]), V. Gaitsgory ([4], [12], [13]), etc (see, e.g.",
"[6], [7], [10], [15], [22] and the references therein)."
],
[
"Problem formulation",
"In this paper, we will consider the nonlinear singularly perturbed feedback control system without an outer disturbance of the form $y^{\\prime }(t)&=&w(t) \\\\\\epsilon w^{\\prime }(t)&=&-ky(t)+f\\left(u(t),y(t)\\right)\\\\v(t)&=&g(y(t))$ with the required nonlocal boundary conditions $v(t_i)=v(t_m)=v(t_f),\\quad t_i<t_m<t_f,$ where $\\epsilon >0$ is a small perturbation parameter, $[y,w]^T$ is the state vector, $v(t)$ is the measured output, $u(t)$ is the input control, $k<0$ is a constant and $g$ is a monotone increasing (decreasing) function on $\\mathbb {R}$ .",
"The state and control variables are not constrained by any boundaries, initial time $t_i$ and final time $t_f$ are fixed and $y(t_i),$ $y(t_f)$ are free.",
"Such boundary value problems can arise in the study of the steady–states of a heated bar with the thermostats, where the controllers at $t=t_i$ and $t=t_f$ maintain a temperature according to the temperature detected by a sensor at $t=t_m.$ In this case, we consider a uniform bar of length $t_f-t_i$ with non-uniform temperature lying on the $t$ -axis from $t=t_i$ to $t=t_f.$ The parameter $\\epsilon $ represents the thermal diffusivity.",
"Different from [5], in this paper we will not assume that $y(t_i)$ and $y(t_f)$ are fixed and moreover we investigate three-point boundary value problem.",
"There have been some papers considered the multi-point boundary value problems in the literature (see, e.g.",
"[14], [16], [17], [28]) by applying the well known coincidence degree theory and Schauder fixed point theorem or the method of lower and upper solutions.",
"However, there have been fewer papers considered the three–point boundary value problems for SPS without the derivative in the boundary conditions.",
"Recently, in the paper [19], it has been studied the nonlinear system of the form $\\epsilon ^2y^{\\prime \\prime }=f(t,y,y^{\\prime }),$ $0<t<1$ subject to the boundary conditions $y(0)=0,$ $y(1)=py(\\tau ),$ $0<\\tau <1$ and $p<1,$ where the assumption $p<1$ was crucial for proving the main result.",
"One of the typical behaviors of SPS is the boundary layer phenomenon: the solutions vary rapidly within very thin layer regions near the boundary.",
"The novelty of our approach lies in the introduction of the exponentially small boundary layer functions into the analysis of nonlocal boundary value problems and approximation of their solutions.",
"The situation in the case of nonlocal boundary value problem is complicated by the fact that there is an inner point in the boundary conditions, in contrast to the \"standard\" boundary conditions as the Dirichlet problem, Neumann problem, Robin problem, periodic boundary value problem ([9], [11]), for example.",
"In the problem considered, there does not exist a positive solution $\\tilde{\\zeta }_\\epsilon $ of differential equation $\\epsilon y^{\\prime \\prime }-my=0,$ $m>0,$ $\\epsilon >0$ (that is, $\\tilde{\\zeta }_\\epsilon $ is convex) such that $\\tilde{\\zeta }_\\epsilon (t_m)-\\tilde{\\zeta }_\\epsilon (t_i)=\\eta (t_m)-\\eta (t_i)>0$ and $\\tilde{\\zeta }_\\epsilon (t)\\rightarrow 0^+$ for $t\\in (t_i,t_f]$ and $\\epsilon \\rightarrow 0^+,$ which could be used to solve this problem by the method of lower and upper solutions and consequently, to approximate the solutions.",
"The application of convex functions is essential for composing the appropriate barrier functions for two-endpoint boundary conditions, see, e.g.",
"[9].",
"The following assumptions will be made throughout the paper.",
"A1.",
"For limiting problem (in () letting $\\epsilon \\rightarrow 0^+$ ) $ky=f\\left(u(t),y\\right)$ there exists $C^2$ function $\\eta =\\eta (t)$ (that is, $\\eta $ is continuous up to second derivative) such that $k\\eta (t)=f\\left(u(t),\\eta (t)\\right)$ on $[t_i,t_f].$ Denote ${H}(\\eta )=\\left\\lbrace (t,y);\\quad t_i\\le t\\le t_f, \\vert y-\\eta (t)\\vert <d(t)\\right\\rbrace ,$ where $d(t)$ is the positive continuous function on $[ t_i,t_f]$ such that $d(t) = \\left\\lbrace \\begin{array}{ll}\\vert \\eta (t_m)-\\eta (t_i)\\vert +\\delta & \\textrm {for t_i\\le t\\le t_i+\\frac{\\delta }{2}}\\\\\\delta & \\textrm {for t_i+\\delta \\le t\\le t_f-\\delta ,}\\\\\\vert \\eta (t_f)-\\eta (t_m)\\vert +\\delta & \\textrm {for t_f-\\frac{\\delta }{2}\\le t\\le t_f}\\end{array} \\right.$ $\\delta $ is a small positive constant.",
"A2.",
"The function $f\\in C^1({H}(\\eta ))$ satisfies the condition $\\left|\\frac{\\partial f(u(t),y)}{\\partial y}\\right|\\le \\lambda <-k \\quad \\mathrm {for\\ every}\\quad (t,y)\\in {H}(\\eta ).$ The assumption (A2) means that the linearization of SPS (REF ), () in a neighbourhood of the set $[\\eta (t),0],$ $t\\in [t_i,t_f],$ as a set of critical points, has no eigenvalues on the imaginary axis.",
"In this paper, we characterize the dynamics for slow variable $y$ in a neighborhood of $\\eta (t)$ for sufficiently small values of the singular perturbation parameter $\\epsilon $ and $t\\in [t_i,t_f].$ Especially, we focus our attention on the appearance of boundary layers.",
"Moreover, we give the $O(\\epsilon )$ accurate approximation of $y$ on $[t_i,t_f].$ Obviously, $y$ is a solution of boundary value problem $\\epsilon y^{\\prime \\prime }(t)+ky(t)=f\\left(u(t),y(t)\\right)$ $y(t_i)=y(t_m)=y(t_f),\\quad t_i<t_m<t_f.$ Recently in [27] we have shown that the solutions of (REF ), (REF ), in general, start with fast transient ($\\left|w_\\epsilon (t_i)\\right|\\rightarrow \\infty $ ) of $y_\\epsilon (t)$ from $y_\\epsilon (t_i)$ to $\\eta (t),$ which is the so–called boundary layer phenomenon, and after decay of this transient they remain close to $\\eta (t)$ with an arising new fast transient of $y_\\epsilon (t)$ from $\\eta (t)$ to $y_\\epsilon (t_f)$ ($\\left|w_\\epsilon (t_f)\\right|\\rightarrow \\infty $ ).",
"Boundary layers are formed due to the nonuniform convergence of the exact solution $y_\\epsilon $ to the degenerate solution $\\eta $ in the neighborhood of the ends $t_i$ and $t_f$ of the considered interval."
],
[
"Behavior of SPS for $\\epsilon \\rightarrow 0^+$",
"Theorem 1 (compare with [27], Theorem 2.1) Under the assumptions (A1) and (A2) there exists $\\epsilon _0 $ such that for every $\\epsilon \\in (0,\\epsilon _0]$ and for every input control $u$ the SPS (REF ), (REF ) has in ${H}(\\eta )$ an unique realization, $y_\\epsilon ,$ satisfying the inequality $-\\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t)-\\hat{\\zeta }_\\epsilon (t)-C\\epsilon \\le y_\\epsilon (t)-(\\eta (t)+\\zeta _\\epsilon (t))\\le \\hat{\\zeta }_\\epsilon (t)+ C\\epsilon $ for $\\eta (t_m)-\\eta (t_i)\\ge 0$ and $-\\hat{\\zeta }_\\epsilon (t)-C\\epsilon \\le y_\\epsilon (t)-(\\eta (t)+\\zeta _\\epsilon (t))\\le \\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t)+\\hat{\\zeta }_\\epsilon (t)+ C\\epsilon $ for $\\eta (t_m)-\\eta (t_i)\\le 0$ on $[ t_i,t_f]$ where $\\zeta _\\epsilon (t) & = & \\frac{\\eta (t_m)-\\eta (t_i)}{D}\\cdot \\Big ( e^{\\sqrt{\\frac{m}{\\epsilon }}(t_f-t)}-e^{\\sqrt{\\frac{m}{\\epsilon }}(t-t_f)} \\\\&+&e^{\\sqrt{\\frac{m}{\\epsilon }}(t-t_m)}-e^{\\sqrt{\\frac{m}{\\epsilon }}(t_m-t)}\\Big ),\\\\\\hat{\\zeta }_\\epsilon (t) & = &\\frac{\\vert \\eta (t_f)-\\eta (t_m)\\vert }{D}\\cdot \\Big ( e^{\\sqrt{\\frac{m}{\\epsilon }}(t-t_i)}-e^{\\sqrt{\\frac{m}{\\epsilon }}(t_i-t)} \\\\&+&e^{\\sqrt{\\frac{m}{\\epsilon }}(t_m-t)}-e^{\\sqrt{\\frac{m}{\\epsilon }}(t-t_m)}\\Big ),\\\\D & = &\\left(e^{\\sqrt{\\frac{m}{\\epsilon }}(t_f-t_i)}+e^{\\sqrt{\\frac{m}{\\epsilon }}(t_m-t_f)}+e^{\\sqrt{\\frac{m}{\\epsilon }}(t_i-t_m)}\\right)\\\\&-&\\left(e^{\\sqrt{\\frac{m}{\\epsilon }}(t_i-t_f)}+e^{\\sqrt{\\frac{m}{\\epsilon }}(t_f-t_m)}+e^{\\sqrt{\\frac{m}{\\epsilon }}(t_m-t_i)}\\right),$ $m=-k-\\lambda ,$ $C=\\frac{1}{m}\\max \\left\\lbrace \\left|\\eta ^{\\prime \\prime } (t)\\right|; t\\in [ t_i,t_f]\\right\\rbrace $ and the positive function $\\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t)&=&\\frac{\\lambda \\vert \\eta (t_m)-\\eta (t_i)\\vert }{\\sqrt{m\\epsilon }}\\cdot \\left[-{ O}(1)\\frac{\\zeta _\\epsilon (t)}{(\\eta (t_m)-\\eta (t_i))}\\right.",
"\\\\&+&\\left.",
"{ O}\\left(e^{\\sqrt{\\frac{m}{\\epsilon }}(t_i-t_m)}\\right)\\frac{\\hat{\\zeta }_\\epsilon (t)}{\\vert \\eta (t_f)-\\eta (t_m)\\vert }+t{ O}\\left(e^{\\sqrt{\\frac{m}{\\epsilon }}\\chi (t)}\\right)\\right],$ $\\chi (t)<0$ for $t\\in (t_i,t_f]$ and $\\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t_i)=\\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t_m).$ We write $s(\\epsilon )={O}(r(\\epsilon ))$ when $0<\\lim \\limits _{\\epsilon \\rightarrow 0^+ }\\left|\\frac{s(\\epsilon )}{ r(\\epsilon )}\\right|<\\infty .$ The function $\\zeta _{\\epsilon }(t)$ satisfies $\\epsilon \\zeta ^{\\prime \\prime }_{\\epsilon }-m\\zeta _{\\epsilon }=0,$ $\\zeta _{\\epsilon }(t_m)-\\zeta _{\\epsilon }(t_i)=-(\\eta (t_m)-\\eta (t_i)),$ $\\zeta _{\\epsilon }(t_f)-\\zeta _{\\epsilon }(t_m)=0,$ $\\zeta _{\\epsilon }(t)\\ge 0$ $(\\le 0)$ is decreasing (increasing) for $t_i\\le t\\le \\frac{t_f+t_m}{2}$ and increasing (decreasing) for $\\frac{t_f+t_m}{2}\\le t\\le t_f$ if $\\eta (t_m)-\\eta (t_i)\\ge 0$ $(\\le 0),$ $\\zeta _{\\epsilon }(t)$ converges uniformly to 0 for $\\epsilon \\rightarrow 0^+$ on every compact subset of $(t_i, t_f],$ $\\zeta _{\\epsilon }(t)=(\\eta (t_m)-\\eta (t_i)){O}\\left(e^{\\sqrt{\\frac{m}{\\epsilon }}\\chi (t)}\\right)$ where $\\chi (t)=t_i-t$ for $t_i\\le t\\le \\frac{t_f+t_m}{2}$ and $\\chi (t)=t-t_f+t_i-t_m$ for $\\frac{t_f+t_m}{2}<t\\le t_f.$ The function $\\hat{\\zeta }_{\\epsilon }(t)$ satisfies $\\epsilon \\hat{\\zeta }_\\epsilon ^{\\prime \\prime }-m\\hat{\\zeta }_\\epsilon =0,$ $\\hat{\\zeta }_{\\epsilon }(t_m)-\\hat{\\zeta }_{\\epsilon }(t_i)=0,$ $\\hat{\\zeta }_{\\epsilon }(t_f)-\\hat{\\zeta }_{\\epsilon }(t_m)=\\vert \\eta (t_f)-\\eta (t_m)\\vert ,$ $\\hat{\\zeta }_{\\epsilon }(t)\\ge 0$ is decreasing for $t_i\\le t\\le \\frac{t_i+t_m}{2}$ and increasing for $\\frac{t_i+t_m}{2}\\le t\\le t_f$ , $\\hat{\\zeta }_{\\epsilon }(t)$ converges uniformly to 0 for $\\epsilon \\rightarrow 0^+$ on every compact subset of $[t_i, t_f),$ $\\hat{\\zeta }_{\\epsilon }(t)=\\vert \\eta (t_f)-\\eta (t_m)\\vert {O}\\left(e^{\\sqrt{\\frac{m}{\\epsilon }}\\hat{\\chi }(t)}\\right)$ where $\\hat{\\chi }(t)=t-t_f$ for $\\frac{t_i+t_m}{2}\\le t\\le t_f$ and $\\hat{\\chi }(t)=t_m-t_f+t_i-t$ for $t_i\\le t<\\frac{t_i+t_m}{2}.$ The correction function $\\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t)=-\\frac{\\left(\\psi _\\epsilon (t_i)-\\psi _\\epsilon (t_m)\\right)}{(\\eta (t_m)-\\eta (t_i))}\\zeta _\\epsilon (t)+\\frac{\\left(\\psi _\\epsilon (t_m)-\\psi _\\epsilon (t_f)\\right)}{\\vert \\eta (t_f)-\\eta (t_m)\\vert }\\hat{\\zeta }_\\epsilon (t)+\\psi _\\epsilon (t)$ where $\\psi _\\epsilon (t)&=&\\frac{\\lambda \\vert \\eta (t_m)-\\eta (t_i)\\vert }{D\\sqrt{m\\epsilon }}t\\Big (e^{\\sqrt{\\frac{m}{\\epsilon }}(t_f-t)}+e^{\\sqrt{\\frac{m}{\\epsilon }}(t-t_f)} \\\\&-&e^{\\sqrt{\\frac{m}{\\epsilon }}(t_m-t)}-e^{\\sqrt{\\frac{m}{\\epsilon }}(t-t_m)}\\Big )$ converges uniformly to $0^+$ on $[t_i,t_f]$ for $\\epsilon \\rightarrow 0^+.$ Theorem REF implies that $y_\\epsilon (t)=\\eta (t)+O(\\epsilon )$ on every compact subset of $(t_i,t_f)$ and $\\lim \\limits _{\\epsilon \\rightarrow 0^+ }y_\\epsilon (t_i)=\\lim \\limits _{\\epsilon \\rightarrow 0^+ }y_\\epsilon (t_f)=\\lim \\limits _{\\epsilon \\rightarrow 0^+ }y_\\epsilon (t_m)=\\eta (t_m).$ Consequently, $\\lim \\limits _{\\epsilon \\rightarrow 0^+ }g\\left(y_\\epsilon (t_i)\\right)=\\lim \\limits _{\\epsilon \\rightarrow 0^+ }g\\left(y_\\epsilon (t_f)\\right)=\\lim \\limits _{\\epsilon \\rightarrow 0^+ }g\\left(y_\\epsilon (t_m)\\right)=g\\left(\\eta (t_m)\\right).$ Due to the assumption that $g$ is strictly monotone, the boundary layer effect occurs at the point $t_i$ or/and $t_f$ in the case when $\\eta (t_i)\\ne \\eta (t_m)$ or/and $\\eta (t_f)\\ne \\eta (t_m).$"
],
[
"Approximation of realization of SPS",
"The application of numerical methods may give rise to difficulties when the singular perturbation parameter $\\epsilon $ tends to zero, especially in the nonlinear case.",
"Then the mesh needs to be refined substantially to grasp the solution within the boundary layers (piecewise uniform mesh of Shishkin-type; see, e.g.",
"[21], [25] and the references therein).",
"The advantage of our approach is that we have to solve only on the parameter $\\epsilon $ independent limiting problem $ky=f\\left(u(t),y\\right),$ see the assumption (A1).",
"Then a singular perturbation method is applied to obtain an approximate solution of SPS (REF ), (REF ) composed of a solution $\\eta $ of reduced problem, small constant and two boundary layer functions to recover the lost nonlocal boundary conditions in the degeneration process.",
"We use the linear combination of the functions $\\eta (t), \\zeta _{\\epsilon }(t)$ and $\\hat{\\zeta }_{\\epsilon }(t)$ to approximate the exact solution of SPS (REF ), (REF ) by the following way.",
"For $\\eta \\left(t_f\\right)-\\eta \\left(t_m\\right)\\le 0$ we define the approximate realization $\\tilde{y}_\\epsilon (t)$ of SPS (REF ), (REF ) by $\\tilde{y}_\\epsilon (t)=\\eta (t)+\\zeta _\\epsilon (t)+\\hat{\\zeta }_\\epsilon (t)+C\\epsilon $ and analogously, for $\\eta \\left(t_f\\right)-\\eta \\left(t_m\\right)\\ge 0$ we define $\\tilde{y}_\\epsilon (t)=\\eta (t)+\\zeta _\\epsilon (t)-\\hat{\\zeta }_\\epsilon (t)-C\\epsilon $ where the $\\epsilon -$ independent constant $C$ is defined in Theorem REF .",
"It is not difficult to verify that $\\tilde{y}_\\epsilon (t)$ satisfies the boundary conditions (REF ) and $\\lim \\limits _{\\epsilon \\rightarrow 0^+ }\\tilde{y}_\\epsilon (t_i)=\\eta (t_m)=\\lim \\limits _{\\epsilon \\rightarrow 0^+ }\\tilde{y}_\\epsilon (t_f).$ Further, for $\\eta \\left(t_f\\right)-\\eta \\left(t_m\\right)\\le 0$ and $\\eta \\left(t_m\\right)-\\eta \\left(t_i\\right)\\le 0$ we obtain the inequality $-\\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t)\\le \\tilde{y}_\\epsilon (t)-y_\\epsilon (t)\\le 2\\hat{\\zeta }_\\epsilon (t)+2C\\epsilon ,$ for $\\eta \\left(t_f\\right)-\\eta \\left(t_m\\right)\\ge 0$ and $\\eta \\left(t_m\\right)-\\eta \\left(t_i\\right)\\ge 0$ $-\\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t)\\le y_\\epsilon (t)-\\tilde{y}_\\epsilon (t)\\le 2\\hat{\\zeta }_\\epsilon (t)+2C\\epsilon ,$ for $\\eta \\left(t_f\\right)-\\eta \\left(t_m\\right)\\le 0$ and $\\eta \\left(t_m\\right)-\\eta \\left(t_i\\right)\\ge 0$ $0\\le \\tilde{y}_\\epsilon (t)-y_\\epsilon (t)\\le \\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t)+2\\hat{\\zeta }_\\epsilon (t)+2C\\epsilon ,$ for $\\eta \\left(t_f\\right)-\\eta \\left(t_m\\right)\\ge 0$ and $\\eta \\left(t_m\\right)-\\eta \\left(t_i\\right)\\le 0$ $0\\le y_\\epsilon (t)-\\tilde{y}_\\epsilon (t)\\le \\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t)+2\\hat{\\zeta }_\\epsilon (t)+2C\\epsilon .$ The right sides of the inequalities (REF )–(REF ) are $O(\\epsilon )$ on every compact subset of $[ t_i, t_f).$ On the other hand, taking into consideration the facts that $\\tilde{y}_\\epsilon (t_i)=\\tilde{y}_\\epsilon (t_f),$ $y_\\epsilon (t_i)=y_\\epsilon (t_f)$ and monotonicity of the functions $\\zeta ^{\\mathrm {(corr)}}_{\\epsilon }(t)+2\\hat{\\zeta }_\\epsilon (t)+2C\\epsilon $ and $2\\hat{\\zeta }_\\epsilon (t)+2C\\epsilon $ with respect to the variable $t$ in a left neighbourhood of $t_f$ for small $\\epsilon ,$ we have $\\left|y_\\epsilon (t)-\\tilde{y}_\\epsilon (t)\\right|\\le O(\\epsilon )$ on $[ t_i, t_f],$ that is, $\\tilde{y}_\\epsilon (t)$ is $O(\\epsilon )$ accurate approximation of exact solution $y_\\epsilon (t)$ of (REF ), (REF ) on the whole interval $[ t_i, t_f].$ We also see that $\\left|\\tilde{w}_\\epsilon (t_i)\\right|\\rightarrow \\infty $ and $\\left|\\tilde{w}_\\epsilon (t_f)\\right|\\rightarrow \\infty $ for $\\epsilon \\rightarrow 0^+,$ where $\\tilde{w}_\\epsilon \\equiv \\tilde{y}^{\\prime }_\\epsilon .$ Thus, $\\tilde{y}_\\epsilon (t)$ is a good approximation of the boundary layers arising in the endpoints of the considered interval $[ t_i, t_f].$ We remark that in the special case when $C=0,$ that is, if $\\eta $ is a first-degree polynomial function or a piecewise linear function (in the second case a small generalization of Theorem REF is needed) we obtain the exponential convergence rate of $\\tilde{y}_\\epsilon $ to $y_\\epsilon $ on $[ t_i, t_f]$ for $\\epsilon \\rightarrow 0^+.$ We remind, that $\\tilde{y}_\\epsilon (t)=\\eta (t)$ is not an appropriate approximation of $ y_\\epsilon (t)$ because do not respect the possible appearance of boundary layers.",
"Consider SPS with quadratic nonlinearity of the form $\\epsilon y^{\\prime \\prime }+ky=y^2+u(t),\\quad k<0,\\quad u\\in C^2\\left([t_i,t_f]\\right)$ with the boundary conditions (REF ).",
"The assumptions of Theorem REF are satisfied if and only if there exists $\\lambda >0$ such that $\\frac{1}{4}\\left(k^2-(\\lambda -k)^2\\right)&<& u(t)<\\frac{1}{4}\\left(k^2-(\\lambda +k)^2\\right)\\quad \\mathrm {on}\\quad [ t_i,t_f]\\\\\\left|u(t_m)-u(t_i)\\right|&<&\\frac{1}{8}\\left(\\lambda -k-\\iota (t_i)\\right)\\left(\\iota (t_i)+\\iota (t_m)\\right)\\\\\\left|u(t_f)-u(t_m)\\right|&<&\\frac{1}{8}\\left(\\lambda -k-\\iota (t_f)\\right)\\left(\\iota (t_f)+\\iota (t_m)\\right)\\\\\\left|u(t_m)-u(t_i)\\right|&<&\\frac{1}{8}\\left(\\lambda +k+\\iota (t_i)\\right)\\left(\\iota (t_i)+\\iota (t_m)\\right)\\\\\\left|u(t_f)-u(t_m)\\right|&<&\\frac{1}{8}\\left(\\lambda +k+\\iota (t_f)\\right)\\left(\\iota (t_f)+\\iota (t_m)\\right),$ where $\\iota (t)=\\sqrt{k^2-4u(t)}.$ For an illustrative example let we consider the problem (REF ), (REF ) with $k=-2,$ $u(t)=t,$ $t_i=0,$ $t_f=1/2,$ $t_m=1/4$ and $g=\\mathrm {id}.$ It is not difficult to verify that the solution $\\eta (t)=-1+\\sqrt{1-t}$ of reduced problem satisfies the conditions (REF )–() for every $\\lambda \\in \\left(\\frac{2}{\\sqrt{2}+\\sqrt{3}}+2-\\sqrt{2},2\\right).$ Thus, on the basis of Theorem REF , there exists $\\epsilon _0=\\epsilon _0(\\lambda )$ such that for every $\\epsilon \\in (0,\\epsilon _0]$ the problem $\\epsilon y^{\\prime \\prime }-2y=y^2+t,$ (REF ) has in ${H}(\\eta )$ the unique solution which is $O(\\epsilon )$ close to the approximate solution (REF ) on $[ t_i, t_f]$ (Fig.",
"REF ), that is, to the function $\\tilde{y}_\\epsilon (t)=-1+\\sqrt{1-t}+\\zeta _\\epsilon (t)+\\hat{\\zeta }_\\epsilon (t)+\\epsilon \\left[(2-\\lambda )\\sqrt{2}\\right]^{-1}.$ Figure: Boundary layer phenomenon for solution of singularly perturbed problem ϵy '' -2y=y 2 +t,\\epsilon y^{\\prime \\prime }-2y=y^2+t, y(0)=y(1/4)=y(1/2)y(0)=y(1/4)=y(1/2) (the solid line) with ϵ=0.0001.\\epsilon =0.0001.",
"The dotted and dashed lines represent the approximate solution y ˜ ϵ (t)\\tilde{y}_\\epsilon (t) (with λ=1.6\\lambda =1.6) and solution of reduced problem, the function η(t)=-1+1-t,\\eta (t)=-1+\\sqrt{1-t}, respectivelyIn the context of previous analysis of the steady–state solutions of 1-D heat transfer equation, it would be interesting to investigate the occurrence of boundary layers for $\\epsilon \\rightarrow 0^+$ of perturbed, non-stationary 1-D heat transfer equation, written in the usual form as $\\frac{\\partial y}{\\partial t}=\\epsilon \\frac{\\partial ^2 y}{\\partial x^2}+ky-f\\left(u(x),y\\right)$ subject to the nonlocal boundary conditions $v(x_i,t)=v(x_m,t)=v(x_f,t),\\quad x_i<x_m<x_f,\\quad t\\in [0,\\infty ),$ where $v(x,t)=g(y(x,t)).$ The solution $y_\\epsilon (x,t)$ represents the temperature at point $x$ of the heated bar in the time $t,$ $x\\in [ x_i,x_f],$ $t\\in [ 0,\\infty ).$ For the initial value problems, the numerical analysis of non-stationary reaction-diffusion systems shows on the presence of boundary layer phenomenon (see, e.g.",
"[26])."
],
[
"Feedback control of semilinear SPS",
"In this section we consider SPS (REF ), (REF ), () with $\\epsilon w^{\\prime }(t)=-ky(t)+f(y(t))+u(t).$ Let $\\left|f^{\\prime }(y)\\right|\\le \\lambda <-k$ for $y\\in \\mathbb {R}.$ Moreover, assume that $g\\in C^1$ and $g_{-1}\\in C^2$ on $\\mathbb {R}$ where $g_{-1}$ denotes an inverse function for $g.$ Now, if $v^0\\in C^2\\left([t_i,t_f]\\right)$ is desired output of SPS (REF ), (REF ), () satisfying (REF ) then it is easy to verify that an adequate feedback control input $u^0$ to obtain close $v^0$ output is $u^0(t)=kg_{-1}\\left(v^0(t)\\right)-f\\left(g_{-1}\\left(v^0(t)\\right)\\right).$ Hence $\\eta ^0(t)=g_{-1}\\left(v^0(t)\\right)$ and an observable realization $g\\left(y^0_\\epsilon \\right)$ of system (REF ), (REF ), () with the boundary condition (REF ) is $O(\\epsilon )$ close to the $g\\left(\\tilde{y}^0_\\epsilon (t)\\right).$ Indeed, as follows from the Lagrange Theorem and (REF )–(REF ), $\\left|g\\left(y^0_\\epsilon (t)\\right)-g\\left(\\tilde{y}^0_\\epsilon (t)\\right)\\right|&\\le & \\mu \\left|y^0_\\epsilon (t)-\\eta ^0(t) \\right|\\\\&\\le & \\mu \\frac{\\epsilon }{m}\\max \\left\\lbrace \\left|\\eta ^{0^{\\prime \\prime }}(t)\\right|; t\\in [ t_i,t_f]\\right\\rbrace $ where $\\mu =\\max \\left\\lbrace \\left|g^{\\prime }(y)\\right|; (t,y)\\in H\\left(\\eta ^0\\right)\\right\\rbrace .$"
],
[
"Unsolved controllability problem ",
"Consider the dynamical model described by singularly perturbed differential equation $\\epsilon y^{\\prime \\prime }(t)+\\frac{1}{2}\\tilde{f}\\left(u(t),y(t)\\right)=0,$ where $\\tilde{f}=2(ky-f)\\in C\\left(\\mathbb {R}^{2}\\right)$ (see (REF )), $u\\in C\\left([0,t_f]\\right)$ is a continuous control input and $0<\\epsilon <<1$ is a singular perturbation parameter.",
"Let $\\tilde{f}\\ne 0,$ and without loss of generality we will assume that $\\tilde{f}>0$ and $t_i=0.$ In this case, the reduced problem $\\tilde{f}\\left(u(t),y(t)\\right)=0$ does not have a solution $\\eta $ (Assumption (A1)), which was the crucial assumption to prove Theorem REF .",
"Denote by $\\lbrace t^*_{i,\\epsilon }\\rbrace $ the set of turning points in $(0,t_m)$ of exact solutions $y_\\epsilon $ for problem (REF ) satisfying $y_\\epsilon (0)=y_\\epsilon (t_m),$ that is, $y_\\epsilon ^{\\prime }(t^*_{i,\\epsilon })=0$ and $y_\\epsilon ^{\\prime \\prime }(t^*_{i,\\epsilon })\\ne 0.$ For the problems considered in the previous sections, the turning points are determined for small $\\epsilon $ with sufficient precision by the turning points of the solution $\\eta $ of reduced problem.",
"Obviously, for (REF ) there is only one turning point $t^*_\\epsilon $ of the solution $y_\\epsilon $ on $[0,t_f],$ and in $t^*_\\epsilon $ acquires its local and global maximum on $[0,t_m]$ and it is possible to steer the control system (REF ) from the state $y_\\epsilon (0)$ to the state $y_\\epsilon (t_m),$ $0<t_m<t_f,$ satisfying $y_\\epsilon (0)=y_\\epsilon (t_m)$ with an arbitrary second boundary condition and for every small $\\epsilon .$ Now we will analyze the location of this turning point.",
"Let consider a special case of (REF ) when $\\tilde{f}\\left(u(t),y(t)\\right)\\equiv \\tilde{f}\\left(u_0,y(t)\\right),$ that is, the nonlinear mathematical model $\\epsilon y^{\\prime \\prime }(t)+\\frac{1}{2} \\tilde{f}\\left(u_0,y(t)\\right)=0,$ with the initial conditions $y_\\epsilon (0)=y_{0,\\epsilon },$ $y^{\\prime }_\\epsilon (0)=y_{1,\\epsilon },$ where $y_{0,\\epsilon },y_{1,\\epsilon }$ are the arbitrary real numbers.",
"Obviously, $y_{1,\\epsilon }>0,$ because in the case $y_{1,\\epsilon }\\le 0$ the solution $y_\\epsilon $ of (REF ) satisfying $y_\\epsilon (0)=y_\\epsilon (t_m)$ has a local minimum at some $t_0\\in (0,t_m)$ with $y^{\\prime \\prime }_\\epsilon (t_0)\\ge 0$ which contradicts to the assumption on positivity of the function $\\tilde{f}.$ Denote by $\\tilde{F}_{u_0}$ the antiderivative of $\\tilde{f}(u_0,y),$ that is, $\\tilde{F}_{u_0}=\\int \\tilde{f}(u_0,y)\\mathrm {d}y.$ The function $\\tilde{F}_{u_0}$ is strictly increasing and by $\\tilde{F}^{-1}_{u_0}$ we denote an inverse function to $\\tilde{F}_{u_0}.$ Integrating the differential equation (REF ) we have $\\epsilon (y^{\\prime }_\\epsilon (t))^2+\\tilde{F}_{u_0}(y(t))=\\epsilon y_{1,\\epsilon }^2+\\tilde{F}_{u_0}(y_{0,\\epsilon }).$ Now applying the standard methods we obtain that for every $t\\in [0,t_f],$ $y_\\epsilon (t)$ is an unique root of the equation $\\pm 2\\epsilon \\int \\limits _{\\sqrt{\\epsilon y_{1,\\epsilon }^2-\\int \\limits _{y_{0,\\epsilon }}^{y_\\epsilon (t)}\\tilde{f}(u_0,s)\\mathrm {d}s}}^{\\sqrt{\\epsilon y_{1,\\epsilon }^2}}\\left[\\tilde{f}\\left(\\tilde{F}^{-1}_{u_0}\\left(\\tilde{F}_{u_0}(y_{0,\\epsilon })+\\epsilon y_{1,\\epsilon }^2-z^2\\right)\\right) \\right]^{-1}\\mathrm {d}z=t,$ where the sign $+(-)$ on the subintervals of $[0,t_f]$ with $y^{\\prime }_\\epsilon \\ge 0$ ($y^{\\prime }_\\epsilon <0$ ), that is, for $t\\in (0,t^*_\\epsilon ]$ ($t\\in (t^*_\\epsilon ,t_f]$ ) is considered, respectively.",
"Taking into consideration that $y_\\epsilon ^{\\prime }(t^*_\\epsilon )=0$ we have $\\tilde{F}_{u_0}(y(t^*_\\epsilon ))=\\epsilon y_{1,\\epsilon }^2+\\tilde{F}_{u_0}(y_{0,\\epsilon }).$ Thus for computation of the turning point we obtain from (REF ) the equation $2\\epsilon \\int \\limits _{0}^{\\sqrt{\\epsilon y_{1,\\epsilon }^2}}\\left[\\tilde{f}\\left(\\tilde{F}^{-1}_{u_0}\\left(\\tilde{F}_{u_0}(y_{0,\\epsilon })+\\epsilon y_{1,\\epsilon }^2-z^2\\right)\\right) \\right]^{-1}\\mathrm {d}z=t^*_\\epsilon .$ To illustrate this theory, let us consider (REF ) with $\\tilde{f}\\left(u_0,y(t)\\right)=e^y.$ The solution of initial problem is $y_\\epsilon (t)=\\mathop {ln}\\nolimits \\left[c_1-c_1\\left(\\frac{e^{\\mp \\frac{\\sqrt{c_1}}{\\epsilon }(t+c_2)}-1}{e^{\\mp \\frac{\\sqrt{c_1}}{\\epsilon }(t+c_2)}+1}\\right)^2\\right],$ where the sign $-(+)$ on the subintervals of $[0,t_f]$ with $y^{\\prime }_\\epsilon \\ge 0$ ($y^{\\prime }_\\epsilon <0$ ) holds, respectively.",
"The constants $c_1,$ $c_2$ are $c_1=\\epsilon y_{1,\\epsilon }^2+\\tilde{F}_{u_0}(y_{0,\\epsilon }),\\quad c_2=-\\frac{\\epsilon }{\\sqrt{c_1}}\\mathop {ln}\\nolimits \\frac{\\sqrt{c_1}+\\sqrt{\\epsilon }y_{1,\\epsilon }}{\\sqrt{c_1}-\\sqrt{\\epsilon }y_{1,\\epsilon }}.$ From (REF ) we have $y_\\epsilon (t^*_\\epsilon )=\\mathop {ln}\\nolimits c_1.$ Thus, as follows from (REF ), $t^*_\\epsilon +c_2=0$ and we obtain $t^*_\\epsilon =\\frac{\\epsilon }{\\sqrt{c_1}}\\mathop {ln}\\nolimits \\frac{\\sqrt{c_1}+\\sqrt{\\epsilon }y_{1,\\epsilon }}{\\sqrt{c_1}-\\sqrt{\\epsilon }y_{1,\\epsilon }}.$ On the other hand, from (REF ), equating $y_\\epsilon (0)$ and $y_\\epsilon (t_m)$ we get $2c_2+t_m=0.$ Comparing this with (REF ) we obtain $t^*_\\epsilon =\\frac{t_m}{2}.$ The following questions arise in this context: (i) Where is located the turning point $t^*_\\epsilon $ for nonlinear singularly perturbed system (REF ) with $\\tilde{f}>0$ subject to required boundary condition $y_\\epsilon (0)=y_\\epsilon (t_m),$ $0<t_m<t_f$ in general?",
"Does have the position independent of singular perturbation parameter $\\epsilon $ ?",
"(ii) Can be controlled a location of turning point by using an appropriate control signal $u$ ?"
],
[
"Acknowledgments",
"I would like to express our gratitude to the referees for all the valuable and constructive comments."
]
] | 1709.01707 |
[
[
"From subKautz digraphs to cyclic Kautz digraphs"
],
[
"Abstract The Kautz digraphs $K(d,\\ell)$ are a well-known family of dense digraphs, widely studied as a good model for interconnection networks.",
"Closely related to these, the cyclic Kautz digraphs $CK(d,\\ell)$ were recently introduced by B\\\"ohmov\\'a, Huemer and the author, and some of its distance-related parameters were fixed.",
"In this paper we propose a new approach to the cyclic Kautz digraphs by introducing the family of the subKautz digraphs $sK(d,\\ell)$, from where the cyclic Kautz digraphs can be obtained as line digraphs.",
"This allows us to give exact formulas for the distance between any two vertices of both $sK(d,\\ell)$ and $CK(d,\\ell)$.",
"Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices.",
"Using these parameters, we also prove that $sK(d,\\ell)$ and $CK(d,\\ell)$ are maximally vertex-connected and super-edge-connected.",
"Whereas $K(d,\\ell)$ are optimal with respect to the diameter, we show that $sK(d,\\ell)$ and $CK(d,\\ell)$ are optimal with respect to the mean distance, whose exact values are given for both families when $\\ell=3$.",
"Finally, we provide a lower bound on the girth of $CK(d,\\ell)$ and $sK(d,\\ell)$."
],
[
"Introduction",
"Originally, the Kautz digraphs were introduced by Kautz [9] in 1968.",
"They have many applications, for example, they are useful as network topologies for connecting processors.",
"The Kautz digraphs have the smallest diameter among all digraphs with their number of vertices and degree.",
"The cyclic Kautz digraphs $CK(d,\\ell )$ were recently introduced by Böhmová, Huemer and the author [2], [3], as subdigraphs with special symmetries of the Kautz digraphs $K(d,\\ell )$ , see for example Fiol, Yebra and Alegre [7].",
"In contrast with these, the set of vertices of the cyclic Kautz digraphs is invariant under cyclic permutations of the sequences representing them.",
"Thus, apart from their possible applications in interconnection networks, the cyclic Kautz digraphs $CK(d,\\ell )$ could be relevant in coding theory, because they are related to cyclic codes.",
"A linear code $C$ of length $\\ell $ is called cyclic if, for every codeword $c=(c_1,\\ldots ,c_\\ell )$ , the codeword $(c_\\ell ,c_1,\\ldots ,c_{\\ell -1})$ is also in $C$ .",
"This cyclic permutation allows to identify codewords with polynomials.",
"For more information about cyclic codes and coding theory, see Van Lint [10] (Chapter 6).",
"With respect to other properties of the cyclic Kautz digraphs $CK(d,\\ell )$ , their number of vertices follows sequences that have several interpretations.",
"For example, for $d=2$ (that is, 3 different symbols) and $\\ell =2,3,\\ldots $ , the number of vertices follows the sequence $6,6,18,30,66,\\ldots $ According to the On-Line Encyclopedia of Integer Sequences [12], this is the sequence A092297.",
"For $d=3$ (4 different symbols) and $\\ell =2,3,\\ldots $ , we get the sequence $12,24,84,240,732,\\ldots $ corresponding to A226493 and A218034 in [12].",
"In this paper we give an alternative definition of $CK(d,\\ell )$ , by introducing the family of the subKautz digraphs $sK(d,\\ell )$ , from where the cyclic Kautz digraphs can be obtained as line digraphs.",
"We present the exact formula of the distance between any two vertices of $sK(d,\\ell )$ and $CK(d,\\ell )$ .",
"This allows us to compute the diameter and the semigirth of both families, also providing an efficient routing algorithm to find the shortest path between any pair of vertices.",
"Using these parameters, we also prove that $sK(d,\\ell )$ and $CK(d,\\ell )$ are maximally vertex-connected and super-edge-connected.",
"Whereas $K(d,\\ell )$ are optimal with respect to the diameter, we show that $sK(d,\\ell )$ and $CK(d,\\ell )$ are optimal with respect to the mean distance, whose exact values are given for both families when $\\ell =3$ .",
"Finally, we provide a lower bound on the girth of $sK(d,\\ell )$ and $CK(d,\\ell )$ ."
],
[
"Notation",
"We consider simple digraphs (or directed graphs) without loops or multiple arcs, and we follow the usual notation for them.",
"That is, a digraph $G=(V,E)$ consists of a (finite) set $V=V(G)$ of vertices and a set $E=E(G)$ of arcs (directed edges) between vertices of $G$ .",
"If $a=(u,v)$ is an arc between vertices $u$ and $v$ , then the vertex $u$ is adjacent to the vertex $v$ , and the vertex $v$ is adjacent from $u$ .",
"Let $\\Gamma ^+(v)$ and $\\Gamma ^-(v)$ denote the set of vertices adjacent from and to the vertex $v$ , respectively.",
"Their cardinalities are the out-degree $\\delta ^+(v)=|\\Gamma ^+(v)|$ of the vertex $v$ , and the in-degree $\\delta ^-(v)=|\\Gamma ^-(v)|$ of the vertex $v$ .",
"A digraph $G$ is called $d$-out-regular if $\\delta ^+(v)=d$ , $d$-in-regular if $\\delta ^-(v)=d$ , and $d$-regular if $\\delta ^+(v)=\\delta ^-(v)=d$ , for all $v\\in V$ .",
"The minimum degree $\\delta =\\delta (G)$ of $G$ is the minimum over all the in-degrees and out-degrees of the vertices of $G$ .",
"A digon is a directed cycle on 2 vertices.",
"For other notation, and unless otherwise stated, we follow the book by Bang-Jensen and Gutin [1].",
"In the line digraph $L(G)$ of a digraph $G$ , each vertex represents an arc of $G$ , $V(L(G))=\\lbrace uv: (u,v)\\in E(G)\\rbrace $ , and a vertex $uv$ is adjacent to a vertex $wz$ when $v=w$ , that is, when in $G$ the arc $(u,v)$ is adjacent to the arc $(w,z)$ : $u\\rightarrow v(=w)\\rightarrow z$ .",
"Fiol and Lladó defined in [6] the partial line digraph $PL(G)$ of a digraph $G$ , where some (but not necessarily all, as in the line digraph $L(G)$ ) of the arcs in $G$ become vertices in $PL(G)$ .",
"Let $E^{\\prime }\\subseteq E$ be a subset of arcs which are incident to all vertices of $G$ , that is, $\\lbrace v:(u,v)\\in E^{\\prime }\\rbrace =V$ .",
"A digraph $PL(G)$ is said to be a partial line digraph of $G$ if its vertices represent the arcs of $E^{\\prime }$ , that is, $V(PL(G))=\\lbrace uv:(u,v)\\in E^{\\prime }\\rbrace $ , and a vertex $uv$ is adjacent to the vertices $v^{\\prime }w$ , for each $w\\in \\Gamma _G^+(v)$ , where $v^{\\prime }=\\left\\lbrace \\begin{array}{ll}v & \\mbox{if } vw\\in V(PL(G)), \\\\\\mbox{any other vertex of } \\Gamma _G^-(w) \\mbox{ such that } v^{\\prime }w\\in V(PL(G))& \\mbox{otherwise}.\\end{array}\\right.$ A digraph $G$ is strongly connected when, for any pair of vertices $x,y\\in V$ , there always exists an $x\\rightarrow y$ path, that is, a path from the vertex $x$ to the vertex $y$ .",
"The strong connectivity $\\kappa =\\kappa (G)$ (or strong vertex-connectivity) of $G$ is the smallest number of vertices whose deletion results in a digraph that is either not strongly connected or trivial.",
"Analogously, the strong arc-connectivity $\\lambda =\\lambda (G)$ of $G$ is the smallest number of arcs whose deletion results in a not strongly connected digraph.",
"Since we only deal with strong connectivities, from now on we are going to refer to them simply as connectivities.",
"Now we only consider connected digraphs, so $\\delta \\ge 1$ .",
"It is known that $\\kappa \\le \\lambda \\le \\delta $ , see Geller and Harary [8].",
"A digraph $G$ is maximally connected when $\\kappa =\\lambda =\\delta $ .",
"If $G$ is a maximally arc-connected digraph $(\\lambda =\\delta )$ , then any set of arcs adjacent from [to] a vertex $x$ with out-degree [in-degree] $\\delta $ is a minimum order arc-disconnecting set.",
"Similarly, if $G$ is a maximally vertex-connected digraph $(\\kappa =\\delta )$ , the set of vertices adjacent from [to] $x$ is a minimum order vertex-disconnecting set.",
"In this context, these arc or vertex sets are called trivial.",
"Note that the deletion of any trivial set isolates a vertex of in-degree or out-degree $\\delta $ .",
"A digraph $G$ is super-$\\kappa $ if every minimum vertex-disconnecting set is trivial.",
"Analogously, $G$ is super-$\\lambda $ is all its minimum arc-disconnecting sets are trivial.",
"If $G$ is super-$\\kappa $ , then $\\kappa =\\delta $ , and if $G$ is super-$\\lambda $ , then $\\lambda =\\delta $ .",
"In general, the converses are not true.",
"We say that a digraph is weakly antipodal when every vertex $u$ has exactly one vertex $v$ at maximum distance (the diameter), and it is antipodal when simultaneously $u$ and $v$ are at maximum distance from each other.",
"For instance, the directed cycle $C_n$ is weakly antipodal, whereas the symmetric directed cycle $C_n^*$ with even $n$ is antipodal."
],
[
"The semigirth",
"We recall the definition of the semigirth: For a given digraph $G$ , let $\\gamma =\\gamma (G)$ , for $1\\le \\gamma \\le D$ , where $D$ is the diameter, be the greatest integer such that for any two (not necessarily different) vertices $x,y\\in V$ , $(a)$ if ${\\rm dist}(x,y)<\\gamma $ , then the shortest $x\\rightarrow y$ path is unique, and there is no an $x\\rightarrow y$ path of length ${\\rm dist}(x,y)+1$ ; $(b)$ if ${\\rm dist}(x,y)=\\gamma $ , then there is only one shortest $x\\rightarrow y$ path.",
"Note that $\\gamma $ is well defined when $G$ has no loops.",
"In [5], Fàbrega and Fiol proved that, if a digraph $G$ (different from a directed cycle) has semigirth $\\gamma $ , then its line digraph $L(G)$ has semigirth $\\gamma +1$ .",
"The diameter also has the same behaviour, that is, if the diameter of $G$ is $D$ , then its line digraph $L(G)$ has diameter $D+1$ .",
"We also recall two results from Fàbrega and Fiol [5] on the connectivities and superconnectivities.",
"Theorem 1 ([5]) Let $G=(V,E)$ be a loopless digraph with minimum degree $\\delta >1$ , semigirth $\\gamma $ , diameter $D$ and connectivities $\\lambda $ and $\\kappa $ .",
"$(a)$ If $D\\le 2\\gamma $ , then $\\lambda =\\delta $ .",
"$(b)$ If $D\\le 2\\gamma -1$ , then $\\kappa =\\delta $ .",
"Theorem 2 ([5]) Let $G=(V,E)$ be a loopless digraph with minimum degree $\\delta \\ge 3$ , semigirth $\\gamma $ , and diameter $D$ .",
"$(a)$ If $D\\le 2\\gamma $ , then $G$ is super-$\\lambda $ .",
"$(b)$ If $D\\le 2\\gamma -2$ , then $G$ is super-$\\kappa $ ."
],
[
"Moore digraphs with respect to the diameter and the mean distance",
"The Moore bound on the number of vertices for digraphs with diameter $D$ and maximum degree $\\Delta $ is $N(\\Delta ,D)=\\frac{\\Delta ^{D+1}-1}{\\Delta -1}$ for $\\Delta >1$ and $N(1,D)=D+1$ .",
"Notice that $N\\sim O(\\Delta ^{D})$ .",
"The digraphs that attain the Moore bound $N(\\Delta ,D)$ are called Moore digraphs.",
"The only Moore digraphs are the directed cycles on $D+1$ vertices and the complete digraphs on $\\Delta +1$ vertices.",
"For $D>1$ and $\\Delta >1$ , there are no Moore digraphs.",
"For more information, see the survey by Miller and Širaň [11].",
"The mean distance corresponding to a digraph attaining the Moore bound is given in the following result.",
"As the only Moore digraphs are the directed cycles and the complete digraphs, this bound gives an idea of how close is a digraph (with diameter $D$ and maximum degree $\\Delta $ ) of being a Moore digraph.",
"Lemma 1 The mean distance $\\overline{\\partial }(\\Delta ,D)$ of a digraph with diameter $D$ and maximum degree $\\Delta $ attaining the Moore bound would be $\\overline{\\partial }(\\Delta ,D)=\\frac{D\\Delta ^{D+2}-(1+D)\\Delta ^{D+1}+\\Delta }{\\Delta ^{D+2}-\\Delta ^{D+1}-\\Delta +1}.$ We compute $\\overline{\\partial }(\\Delta ,D)$ taking into account that the maximum number of vertices at distance $k$ is $\\Delta ^k$ .",
"$\\overline{\\partial }(\\Delta ,D) &=& \\frac{1}{N(\\Delta ,D)}\\sum _{k=0}^D k\\Delta ^k=\\frac{\\Delta }{N(\\Delta ,D)}\\sum _{k=0}^D k\\Delta ^{k-1}= \\frac{\\Delta }{N(\\Delta ,D)}\\left(\\sum _{k=0}^D \\Delta ^{k}\\right)^{\\prime }\\\\&=& \\frac{\\Delta }{N(\\Delta ,D)} \\left(\\frac{\\Delta ^{D+1}-1}{\\Delta -1}\\right)^{\\prime }= \\frac{D\\Delta ^{D+2}-(1+D)\\Delta ^{D+1}+\\Delta }{\\Delta ^{D+2}-\\Delta ^{D+1}-\\Delta +1}.$ We can define a digraph as optimal with respect to the diameter (the maximum delay in a message transmission), but also with respect to the mean distance (the average delay in a message transmission).",
"So, we can say that a digraph is optimal when, if $N$ is of the order of $\\Delta ^k$ , then its mean distance is of the order of $k$ , that is, when $\\overline{\\partial }\\sim O(\\log _{\\Delta }N)$ ."
],
[
"Kautz-like digraphs",
"The Kautz $K(d,\\ell )$ , the subKautz $sK(d,\\ell )$ , the cyclic Kautz $CK(d,\\ell )$ , and the modified cyclic Kautz $MCK(d,\\ell )$ digraphs have vertices represented by words on an alphabet, and adjacencies between vertices correspond to shifts of the words.",
"In these Kautz-like digraphs a path $\\mbox{$x$}\\rightarrow \\mbox{$y$}$ corresponds to a sequence beginning with $\\mbox{$x$}=x_1x_2\\ldots x_{\\ell }$ and finishing with $\\mbox{$y$}=y_1y_2\\ldots y_{\\ell }$ , where every subsequence of length $\\ell $ corresponds to a vertex of the corresponding digraph."
],
[
"Kautz and subKautz digraphs",
"Next, we recall the definitions of the Kautz $K(d,\\ell )$ , and we define a new family of Kautz-like digraphs called the subKautz digraphs $sK(d,\\ell )$ .",
"See examples of both in Figure REF .",
"A Kautz digraph $K(d,\\ell )$ has the vertices $x_1x_2\\ldots x_{\\ell }$ , where $x_i\\in \\mathbb {Z}_{d+1}$ , with $x_{i}\\ne x_{i+1}$ for $i=1,\\ldots ,\\ell -1$ , and adjacencies $x_1x_2\\ldots x_{\\ell }\\quad \\rightarrow \\quad x_2x_3\\ldots x_{\\ell } y,\\qquad y\\ne x_{\\ell }.$ Given integers $d$ and $\\ell $ , with $d,\\ell \\ge 2$ , a subKautz digraph $sK(d,\\ell )$ has set of vertices $V=\\lbrace x_1x_2\\ldots x_{\\ell }:x_i\\ne x_{i+1},\\ i=1,\\ldots ,\\ell -1\\rbrace \\subset \\mathbb {Z}_{d+1}^{\\ell }$ , and adjacencies $x_1x_2\\ldots x_{\\ell }\\quad \\rightarrow \\quad x_2\\ldots x_{\\ell }x_{\\ell +1}, \\qquad x_{\\ell +1}\\ne x_1,x_{\\ell }.$ Hence, the subKautz digraph $sK(d,\\ell )$ has $d^{\\ell }+d^{\\ell -1}$ vertices, as the Kautz digraph $K(d,\\ell )$ .",
"Besides, the out-degree of a vertex $x_1x_2\\ldots x_{\\ell }$ is $d$ if $x_1=x_{\\ell }$ , and $d-1$ otherwise.",
"In particular, the subKautz digraph $sK(d,2)$ is $(d-1)$ -regular and can be obtained from the Kautz digraph $K(d,2)$ by removing all its arcs forming a digon.",
"Note that the subKautz digraph $sK(d,\\ell )$ is a subdigraph of the Kautz digraph $K(d,\\ell )$ ."
],
[
"Cyclic Kautz and modified cyclic Kautz digraphs",
"Next, we recall the definitions of the cyclic Kautz digraphs $CK(d,\\ell )$ and the modified cyclic Kautz digraphs $MCK(d,\\ell )$ .",
"See an example of both in Figure REF .",
"A cyclic Kautz digraph $CK(d,\\ell )$ has the vertices $x_1x_2\\ldots x_{\\ell }$ , where $x_i\\in \\mathbb {Z}_{d+1}$ , with $x_{i}\\ne x_{i+1}$ for $i=1,\\ldots ,\\ell -1$ , and $x_{\\ell }\\ne x_1$ , and adjacencies $x_1x_2\\ldots x_{\\ell }\\quad \\rightarrow \\quad x_2x_3\\ldots x_{\\ell } y,\\qquad y\\ne x_2,x_{\\ell }.$ Note that the cyclic Kautz digraphs $CK(d,\\ell )$ are subdigraphs of the Kautz digraph $K(d,\\ell )$ .",
"It was proved in [3] that when $d=2$ the cyclic Kautz digraphs $CK(2,\\ell )$ are not connected (except for the case $\\ell =4$ ), and when $\\ell =2$ the cyclic Kautz digraphs $CK(d,2)$ coincide with the Kautz digraphs $K(d,2)$ .",
"Recall that the diameter of the Kautz digraphs is optimal, that is, for a fixed out-degree $d$ and number of vertices $(d+1)d^{\\ell -1}$ , the Kautz digraph $K(d,\\ell )$ has the smallest diameter $(D=\\ell )$ among all digraphs with $(d+1) d^{\\ell -1}$ vertices and degree $d$ (see, for example, Miller and Širáň [11]).",
"Since the diameter of the cyclic Kautz digraphs $CK(d,\\ell )$ is greater than the diameter of the Kautz digraphs $K(d,\\ell )$ , in [4] we constructed the modified cyclic Kautz digraphs $MCK(d,\\ell )$ by adding some arcs to $CK(d,\\ell )$ , in order to obtain the same diameter as $K(d,\\ell )$ , without increasing the maximum degree.",
"In a cyclic Kautz digraph $CK(d,\\ell )$ , a vertex labeled with $a_2\\ldots a_{\\ell +1}$ is forbidden if $a_2=a_{\\ell +1}$ .",
"For each label, we replace the first symbol $a_2$ by one of the possible symbols $a_2^{\\prime }$ such that now $a_2^{\\prime }\\ne a_3, a_{\\ell +1}$ (so $a_2^{\\prime }\\ldots a_{\\ell +1}$ represents a vertex).",
"Then, we add arcs from the vertex $a_1\\ldots a_{\\ell }$ to the vertex $a_2^{\\prime }\\ldots a_{\\ell +1}$ , with $a_1\\ne a_{\\ell }$ and $a_2^{\\prime }\\ne a_3,a_{\\ell +1}$ .",
"Note that $CK(d,\\ell )$ and $MCK(d,\\ell )$ have the same vertices, because we only add arcs to $CK(d,\\ell )$ to obtain $MCK(d,\\ell )$ .",
"Lemma 2 $(a)$ The cyclic Kautz digraph $CK(d,\\ell )$ is the line digraph of the subKautz digraph $sK(d,\\ell -1)$ , that is, $CK(d,\\ell )=L(sK(d,\\ell -1))$ .",
"$(b)$ The modified cyclic Kautz digraph $MCK(d,\\ell )$ is the partial line digraph of the Kautz digraph $K(d,\\ell -1)$ , that is, $MCK(d,\\ell )=PL(K(d,\\ell -1))$ .",
"$(a)$ From (REF ) we can write the arcs $(x_1x_2\\ldots x_{\\ell -1},x_2\\ldots x_{\\ell -1}x_{\\ell })$ of $sK(d,\\ell -1)$ as $x_1x_2\\ldots x_{\\ell -1}x_{\\ell }$ with $x_i\\ne x_{i+1}$ and $x_1\\ne x_{\\ell }$ , which corresponds to the vertices of $CK(d,\\ell )$ .",
"Moreover, two arcs are adjacent in $sK(d,\\ell -1)$ if $x_1x_2\\ldots x_{\\ell } \\quad \\rightarrow \\quad x_2\\ldots x_{\\ell }x_{\\ell +1},$ where $x_1\\ne x_{\\ell }$ , as required for the vertices of $CK(d,\\ell )$ .",
"$(b)$ This was proved in [4].",
"In taking the partial line digraph, it suffices to consider only the arcs in $K(d,\\ell -1)$ that are also in $sK(d,\\ell -1)$ .",
"By using spectral techniques, the order $n_{d,\\ell }$ of a cyclic Kautz digraph $CK(d,\\ell )$ was given in [2], [3].",
"Here we use a combinatorial proof of this result.",
"Proposition 1 The order $n_{d,\\ell }$ of a cyclic Kautz digraph $CK(d,\\ell )$ (that coincide with the size of the subKautz digraph $sK(d,\\ell -1)$ ) is $n_{d,1}=d+1$ and $n_{d,\\ell }=d^{\\ell }+(-1)^{\\ell }d\\qquad \\mbox{for $\\ell \\ge 2$}.$ The number $N_{d,\\ell }$ of sequences $x_1x_2\\ldots x_{\\ell }$ with $x_i\\ne x_{i+1}$ for $i=1,\\ldots ,\\ell -1$ (vertices of $K(d,\\ell )$ ) is $d^{\\ell }+d^{\\ell -1}$ .",
"Then, to compute $n_{d,\\ell }$ , we must subtract from $N_{d,\\ell }$ the number $n^{\\prime }_{d,\\ell }$ of sequences $x_1x_2\\ldots x_{\\ell }$ such that $x_1=x_{\\ell }$ .",
"But this is the same as the number of sequences $x_2\\ldots x_{\\ell }$ with $x_2\\ne x_{\\ell }$ and $x_i\\ne x_{i+1}$ for $i=2,\\ldots ,\\ell -1$ , which is $n_{d,\\ell -1}$ .",
"Consequently, we get the recurrence $n_{d,\\ell }=d^{\\ell }+d^{\\ell -1}-n_{d,\\ell -1}\\qquad \\mbox{for $\\ell \\ge 3$}.$ Thus, (REF ) follows by applying recursively (REF ) and using that $n_{d,2}=d^2+d$ .",
"In the following result we prove a way of finding an $sK(d,\\ell )$ a from the Kautz digraphs $K(d,\\ell )$ .",
"We use the cyclic Kautz digraphs $CK(d,\\ell )$ in the proof.",
"Lemma 3 The subKautz digraphs $sK(d,\\ell )$ can be obtained from the Kautz digraphs $K(d,\\ell )$ by removing all the arcs of the closed walks of length $\\ell $ in the complete symmetric digraph $K^*_{d+1}$ .",
"From their definition, the subKautz digraphs $sK(d,\\ell )$ are obtained from $K(d,\\ell )$ by removing the arcs of the form $x_1x_2\\ldots x_{\\ell }\\rightarrow x_2\\ldots x_{\\ell }x_{1},$ which correspond to the vertices $x_1x_2\\ldots x_{\\ell }x_{1}$ of $K(d,\\ell +1)$ , which in turn correspond to the closed walks of length $\\ell $ in the complete symmetric digraph $K^*_{d+1}$ .",
"A simple property of symmetry shared by all the Kautz-like digraphs is the following.",
"The converse digraph is obtained by changing the direction of all the arcs in the original digraph.",
"Lemma 4 The Kautz digraphs $K(d,\\ell )$ , the subKautz digraphs $sK(d,\\ell )$ , and the cyclic Kautz digraphs $CK(d,\\ell )$ are isomorphic to their converses.",
"Since the mapping $\\Psi (x_1x_2\\ldots x_{\\ell })=x_{\\ell }\\ldots x_2x_1$ satisfies $\\Psi (\\Gamma ^+(\\lbrace x_1x_2\\ldots x_{\\ell }\\rbrace ))&=&\\Psi (\\lbrace x_2x_3\\ldots x_{\\ell }y\\ : y\\in \\mathbb {Z}_{d+1}, y\\ne x_{\\ell }\\rbrace )\\\\&=& \\lbrace y x_{\\ell }\\ldots x_3x_2: y\\in \\mathbb {Z}_{d+1}, y\\ne x_{\\ell }\\rbrace \\\\&=& \\Gamma ^- (\\lbrace x_{\\ell }\\ldots x_2x_1\\rbrace ) = \\Gamma ^- (\\Psi (\\lbrace x_1x_2\\ldots x_{\\ell }\\rbrace )),$ where in the case of $CK(d,\\ell )$ also $y\\ne x_{2}$ , it is an isomorphism between every of such digraphs and its converse."
],
[
"Routing, distances and girth in $CK(d,\\ell )$",
"In this section, we only need to consider the cases with $d\\ge 3$ and $\\ell \\ge 3$ because, as said in the Introduction, when $d=2$ the cyclic Kautz digraphs $CK(2,\\ell )$ are not connected (except for the case $\\ell =4$ ), and when $\\ell =2$ , the cyclic Kautz digraphs $CK(d,2)$ coincide with the Kautz digraphs $K(d,2)$ .",
"We begin the study of the routing and distance in $CK(d,\\ell )$ with the case $d,\\ell \\ge 4$ and, afterwards, we deal with the case $d=3$ or $\\ell =3$ ."
],
[
"Routing and distances when $d,\\ell \\ge 4$",
"For simplicity, and without loss of generality, we fix the length $\\ell $ of the sequences, for instance, assume that we are dealing with the cyclic Kautz digraph $CK(d,7)$ on the alphabet $\\mathbb {Z}_{d+1}=\\lbrace 0,1,\\ldots ,d\\rbrace $ with $d\\ge 4$ .",
"Let us consider two generic vertices: $\\begin{array}{ccccccccc}\\mbox{$x$}&=& x_1&x_2&x_3&x_4&x_5&x_6&x_7, \\\\\\mbox{$y$}&=& y_1&y_2&y_3&y_4&y_5&y_6&y_7,\\end{array}$ and the extended sequence of $\\mbox{$x$}$ , that is, $\\begin{array}{ccccccccccccccc}\\tilde{\\mbox{$x$}}& = & x_1&x_2&x_3&x_4&x_5&x_6&x_7&\\overline{x_2}&\\overline{x_3}&\\overline{x_4}&\\overline{x_5}&\\overline{x_6}&\\overline{x_7},\\end{array}$ where $\\overline{x_i}\\in \\mathbb {Z}_{d+1}$ and $\\overline{x_i}\\ne x_i$ .",
"(Note that we also can interpret $\\tilde{\\mbox{$x$}}$ as a set of sequences of length $2\\ell -1$ .)",
"Then, to find the distance ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})$ , we compute the intersection $\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}$ , which is the maximum subsequence of $\\tilde{\\mbox{$x$}}$ that coincides with the initial subsequence of $\\mbox{$y$}$ .",
"Analogously, the intersection $\\mbox{$x$}\\sqcap \\mbox{$y$}$ is the maximum final subsequence of $\\mbox{$x$}$ that coincides with the initial subsequence of $\\mbox{$y$}$ .",
"According to the length of such a subsequence, we distinguish three cases: $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|> \\ell -1$ ($\\Rightarrow \\ell -1 \\ge |\\mbox{$x$}\\sqcap \\mbox{$y$}|\\ge 1$ ): For instance, suppose that $|\\mbox{$x$}\\sqcap \\mbox{$y$}|=4$ , so that we have the coincidence pattern: $\\begin{array}{ccccccccccccc}x_1&x_2&x_3&x_4&x_5&x_6&x_7&\\overline{x_2}&\\overline{x_3}&\\overline{x_4}&\\overline{x_5}&\\overline{x_6}&\\overline{x_7}\\\\& & & y_1&y_2&y_3&y_4&y_5&y_6&y_7&& &\\end{array}$ where $y_i=x_{i+3}$ for $i=1,\\ldots ,4$ , and $(a1)$ $y_5\\ne x_2$ and $y_5\\ne y_4=x_7$ , $(a2)$ $y_6\\ne x_3,y_5$ , $(a3)$ $y_7\\ne x_4=y_1$ and $y_7\\ne y_6$ .",
"Then, the only shortest path from $\\mbox{$x$}$ to $\\mbox{$y$}$ is $\\mbox{$x$}\\!=\\!x_1x_2x_3y_1y_2y_3y_4\\ \\rightarrow \\ x_2x_3y_1y_2y_3y_4y_5\\ \\rightarrow \\ x_3y_1y_2y_3y_4y_5y_6\\ \\rightarrow \\ y_1y_2y_3y_4y_5y_6y_7\\!=\\!\\mbox{$y$}.$ Hence, in this case, ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})=3$ and, in general, ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})=\\ell -|\\mbox{$x$}\\sqcap \\mbox{$y$}|\\le \\ell -1.$ $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|=\\ell - 1$ : If $y_1\\ne x_7$ , we reason as in case $(a)$ and we get ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})= \\ell $ .",
"Otherwise, if $y_1=x_7$ , the sequence $x_2x_3\\ldots x_7y_1$ does not correspond to any vertex.",
"Then, we have to consider the `second largest' intersection satisfying the next case $(c)$ : $1\\le |\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|<\\ell -1$ .",
"(Since $\\ell \\ge 4$ , we prove later that this is always possible.)",
"Thus, we get ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})=2\\ell -1-|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|$ .",
"Note that the number of vertices at distance $\\ell $ is of the order of $d^{\\ell }$ , which also corresponds to the optimal mean distance.",
"$1\\le |\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|<\\ell -1$ : Suppose, for instance, that $|\\widetilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|=3$ .",
"$\\begin{array}{ccccccccccccccccc}x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & \\overline{x_2} & \\overline{x_3} &\\overline{x_4} & \\overline{x_5} & \\overline{x_6} & \\overline{x_7} & & & & \\\\& & & & & & &\\overline{y_4} &\\overline{y_5} &\\overline{y_6} & y_1 & y_2 & y_3 & y_4 & y_5 & y_6 & y_7 \\\\& & & & & & = &z_1 &z_2 &z_3& y_1 & y_2 & y_3 & y_4 & y_5 & y_6 & y_7\\end{array}$ where $(c1)$ $z_1\\ne x_7,x_2,y_4$ , $(c2)$ $z_2\\ne z_1,x_3,y_5$ , $(c3)$ $z_3\\ne z_2,x_4,y_6,y_1$ , $(c4)$ $y_1\\ne z_3,x_5$ .",
"Then, ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})=10$ and, in general, ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})=2\\ell -1-|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|\\le 2\\ell -2.$ Now we are ready to prove the following result.",
"Theorem 3 The diameter of the cyclic Kautz digraph $CK(d,\\ell )$ with $d,\\ell \\ge 4$ is $D=2\\ell -2$ .",
"First, we claim that $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|\\ge 1$ .",
"Indeed, on the contrary, we would have that $y_1=x_7\\ne x_6$ and $y_2\\ne y_1=x_7$ .",
"Consequently, $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|\\ge 2$ , a contradiction.",
"Then, if $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|= 1$ , we are in case $(c)$ .",
"Otherwise, from the above reasoning, we have at least an intersection $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|=2<\\ell -1$ , as $\\ell \\ge 4$ , and case $(c)$ applies again.",
"Finally, the existence of two vertices $\\mbox{$x$}$ and $\\mbox{$y$}$ at maximum distance is as follows.",
"We have two cases: If $\\ell $ is even, consider the vertices $\\mbox{$x$}=1010\\ldots 1012$ and $\\mbox{$y$}=0202\\ldots 02$ .",
"If $\\ell $ is odd, consider the vertices $\\mbox{$x$}=0101\\ldots 012$ and $\\mbox{$y$}=0202\\ldots 021$ .",
"Then, in both cases it is easily checked that $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|=1$ and, hence, ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})=2\\ell -2$ .",
"Fiol, Yebra, and Alegre [7] proved that if the diameter of any digraph (different from a directed cycle) is $D$ , then the diameter of its line digraph is $D+1$ .",
"Since $CK(d,\\ell )$ are the line digraphs of the subKautz digraphs $sK(d,\\ell -1)$ , the diameter of the former is one unit more than the latter.",
"Corollary 1 The diameter of the subKautz digraph $sK(d,\\ell )$ with $d\\ge 4$ and $\\ell \\ge 3$ is $2\\ell -1$ ."
],
[
"Routing and distances when $d=3$ or {{formula:1943a53b-d79a-47bd-ad8c-5fdebb6cf14a}}",
"Looking at the case $(c3)$ above, if $d=3$ and all the elements $z_2, x_4, y_6, y_1$ are different, then $z_3$ has no possible value.",
"Analogously, if $\\ell =3$ , there must exist two vertices $\\mbox{$x$}=x_1x_2x_3$ and $\\mbox{$y$}=y_1y_2y_3$ , such that $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|=2$ (not smaller than $\\ell -1$ ), and with $y_1=x_3$ .",
"Thus, neither of the strategies in the above cases $(c)$ and $(b)$ can be applied.",
"However, the following reasoning shows that we always can find a path of length $2\\ell -1$ .",
"First, we deal with the case $d=3$ , where for simplicity we assume that $\\ell =5$ .",
"We reason as if $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|=0$ : $\\begin{array}{cccccccccccccc}x_1 & x_2 & x_3 & x_4 & x_5 & \\overline{x_2} & \\overline{x_3} & \\overline{x_4} & \\overline{x_5} & & & & & \\\\& & & & & \\overline{y_1} & \\overline{y_2} & \\overline{y_3} & \\overline{y_4} & y_1 & y_2 & y_3 & y_4 & y_5 \\\\& & & & = & z_1 & z_2 & z_3 & z_4 & y_1 & y_2 & y_3 & y_4 & y_5\\end{array}$ where we would need the following conditions: $(d1)$ $z_1\\ne x_2,x_5,y,1$ , $(d2)$ $z_2\\ne z_1,x_3,y_2$ , $(d3)$ $z_3\\ne z_2,x_4,y_3$ , $(d4)$ $z_4\\ne z_3,x_5,y_4,y-1$ .",
"If $d\\ge 4$ (for $\\ell =3$ ), this conditions can always be fulfilled, and the required path is guaranteed.",
"If $d=3$ , and either $y_1=x_2$ , or $y_2=x_3$ , or $y_3=x_4$ , or $y_4=x_5$ , or $y_1=x_5$ , then there is always a possible choice of $z_1,z_2,z_3$ and $z_4$ in $\\mathbb {Z}_4$ .",
"Consequently, ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})\\le 9$ .",
"Otherwise, if $y_i\\ne x_{i+1}$ for $i=1,\\ldots ,4$ and $y_1\\ne x_5$ , we can reason as if $|\\tilde{\\mbox{$x$}}\\sqcap \\mbox{$y$}|=4(=\\ell -1)$ .",
"In this case, the path from $\\mbox{$x$}$ to $\\mbox{$y$}$ is: $\\mbox{$x$}=x_1 x_2 x_3 x_4 x_5 \\ \\rightarrow \\ x_2 x_3 x_4 x_5y_1\\ \\rightarrow \\ x_3 x_4 x_5y_1y_2 \\ \\rightarrow \\ \\cdots \\ \\rightarrow \\ y_1 y_2 y_3 y_4 y_5=\\mbox{$y$},$ which implies that ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})\\le 5$ .",
"Thus, in any case, ${\\rm dist}(\\mbox{$x$},\\mbox{$y$})\\le 2\\ell -1.$ This leads to the following result.",
"Proposition 2 $(i)$ The diameter of the cyclic Kautz digraphs $CK(3,\\ell )$ with $\\ell \\ne 4$ and that of $CK(d,\\ell )$ with $\\ell =3$ is $2\\ell -1$ .",
"$(ii)$ The diameter of the cyclic Kautz digraph $CK(3,4)$ is $2\\ell -2=6$ .",
"$(i)$ We only need to exhibit two vertices at distance $2\\ell -1$ .",
"For $CK(3,\\ell )$ with $\\ell \\ge 5$ , when $\\ell $ is odd, we can take the vertices $\\mbox{$x$}={0 1 0 1 {\\ldots }0 1 2}$ and $\\mbox{$y$}= {2 1 0 1 0 {\\ldots } 1 0}$ .",
"When $\\ell $ is even, two vertices at maximum distance are $\\mbox{$x$}={1 0 2 0 2 0 {\\ldots } 2 0 1 2}$ and $\\mbox{$y$}={2 1 3 0 2 0 2 {\\ldots } 0 2 0 1 0}$ .",
"In both cases, it was proved that these vertices are at maximum distance in [3].",
"The case of the cyclic Kautz digraph $CK(3,3)$ , shown in Figure REF $(b)$ , can be easily checked to have diameter $2\\ell -1=5$ , for instance, the vertices at maximum distance from 012 are 210 and 213.",
"In general, for $CK(d,3)$ , we show that two vertices at maximum distance 5 are $\\mbox{$x$}=x_1 x_2 x_3$ and $\\mbox{$y$}=x_3 x_2 y_3$ as follows.",
"If this distance were 2, then we would get the sequence $x_1 x_2 x_3x_2y_3$ , but $x_2 x_3x_2$ is not a vertex of $CK(d,3)$ .",
"If this distance were 3, then we would get the sequence $x_1 x_2 x_3x_3x_2y_3$ , but $x_2 x_3x_3$ is not a vertex of $CK(d,3)$ .",
"If this distance were 4, then we would get the sequence $x_1 x_2 x_3y_1x_3x_2y_3$ , but $x_3 y_1x_3$ is not a vertex of $CK(d,3)$ .",
"Then, the distance is 5, with the sequence $x_1 x_2 x_3y_1y_2x_3x_2y_3$ .",
"$(ii)$ The cyclic Kautz digraph $CK(3,4)$ on 84 vertices with labels $x_1x_2x_3x_4$ , $x_i\\in \\mathbb {Z}_4$ , is the line digraph of the subKautz digraph $sK(3,3)$ shown in Figure REF $(a)$ .",
"Then, since $sK(3,3)$ has diameter 5, we conclude that $CK(3,4)$ has diameter 6, as claimed.",
"Corollary 2 $(i)$ The diameter of the subKautz digraphs $sK(d,\\ell )$ with either $d=3$ and $\\ell \\ge 4$ or $d\\ge 3$ and $\\ell =2$ is $2\\ell $ .",
"$(ii)$ The diameter of the subKautz digraph $sK(3,3)$ is $2\\ell -1=5$ .",
"See Figure REF for a summary of the diameters of $sK(d,\\ell )$ and $CK(d,\\ell )$ .",
"Figure: Summary of the diameters of sK(d,ℓ)sK(d,\\ell ) and CK(d,ℓ)CK(d,\\ell ), depending on the values of dd and ℓ\\ell ."
],
[
"The girth",
"Now we give a lower bound on the girth of a cyclic Kautz digraph $CK(d,\\ell )$ .",
"Lemma 5 The girth $g$ of the cyclic Kautz digraph $CK(d,\\ell )$ is at least the minimum positive integer $k$ such that $\\ell $ is not congruent with $1\\ (\\emph {mod}\\ k)$ .",
"A cycle of minimum length $g$ , rooted to a vertex $\\mbox{$x$}$ , corresponds to a path from $\\mbox{$x$}$ to $\\mbox{$x$}$ of the same length.",
"This means that the maximum length of the (nontrivial) intersection $\\mbox{$x$}\\sqcap \\mbox{$x$}$ is $\\ell -g$ .",
"For instance, with $\\ell =7$ and $g=4$ we would have the intersection pattern $\\begin{array}{ccccccccccccc}x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & \\overline{x_2} & \\overline{x_3} & \\overline{x_4} & \\overline{x_5}& \\overline{x_6} & \\overline{x_7}\\\\& & & & x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7.",
"& &\\end{array}$ Then, in general, this means that the sequence representing $\\mbox{$x$}$ is periodic: $x_i=x_{i+g}$ for every $i=1,2,\\ldots ,\\ell -g$ .",
"Now, if $\\ell \\equiv r\\ (\\textrm {mod}\\ g)$ , then $x_{\\ell }=x_r$ , which is possible if $r\\ne 1$ , and in this case the cycle would be $\\mbox{$x$}&= & x_1x_2\\ldots x_g\\ldots x_1x_2\\ldots x_g x_1x_2\\ldots x_{r}\\\\&\\rightarrow & x_2\\ldots x_g\\ldots x_1x_2\\ldots x_g x_1x_2\\ldots x_{r} x_{r+1}\\\\&\\rightarrow & \\cdots \\ \\rightarrow \\ x_{g-r+1}\\ldots x_g \\ldots x_1x_2\\ldots x_g x_1x_2\\ldots x_{r}x_{r+1}\\ldots x_{g}\\\\&\\rightarrow & x_{g-r+2}\\ldots x_g \\ldots x_1x_2\\ldots x_g x_1x_2\\ldots x_{r}x_{r+1}\\ldots x_{g}x_1\\\\&\\rightarrow & \\cdots \\ \\rightarrow \\ x_1x_2\\ldots x_{g}\\ldots x_1x_2\\ldots x_{r}x_{r+1}\\ldots x_{g}x_1\\ldots x_r=\\mbox{$x$}.$ This completes the proof.",
"Note that the girth reaches the bound when there exists a vertex $\\mbox{$x$}$ that satisfies the cases $(a)$ , $(b)$ , $(c)$ or $(d)$ (given at the beginning of this section) for the existence of a path of length $g$ from $\\mbox{$x$}$ to $\\mbox{$y$}=\\mbox{$x$}$ .",
"In particular, this is fulfilled if $d$ is large enough.",
"As an example, if $\\ell =13$ Lemma REF gives $g\\ge 5$ .",
"However, a possible vertex $\\mbox{$x$}$ only exists for $d\\ge 4$ .",
"Indeed, assume that $\\mbox{$x$}=x_1x_2x_3x_4x_5x_1x_2x_3x_4x_5x_1x_2x_3$ , where $x_i\\in \\mathbb {Z}_4$ for $i=1,\\ldots ,5$ .",
"Since $x_2\\ne x_1$ and $x_3\\ne x_2,x_1$ , we can take, without loss of generality $\\mbox{$x$}=012x_4x_5012x_4x_5 012$ .",
"Then, a path of length $g=5$ from $\\mbox{$x$}$ to $\\mbox{$x$}$ should be $\\mbox{$x$}&=&012x_4x_5012x_4x_5 012 \\ \\rightarrow \\ 12x_4x_5012x_4x_5 012x_4 \\ \\rightarrow \\ 2x_4x_5012x_4x_5 012x_4x_5\\\\&&x_4x_5012x_4x_5 012x_4x_5 0 \\ \\rightarrow \\ x_5012x_4x_5 012x_4x_5 01 \\ \\rightarrow \\ 012x_4x_5012x_4x_5 012\\ =\\ \\mbox{$x$}.$ Therefore, since $x_4\\ne 2,1$ and $0\\ne x_4$ , then $x_4=3$ .",
"Moreover, since $x_5\\ne x_4,2$ , $0\\ne x_5$ , and $1\\ne x_5$ , then $x_5\\notin \\lbrace 0,1,2,3\\rbrace $ , which is a contradiction.",
"In fact, when $d=3$ , it turns out that $CK(3,13)$ has girth $g=7$ , for example, with the vertex $\\mbox{$x$}=0120123012012$ .",
"A direct consequence of this result is that there exist cyclic Kautz digraphs with arbitrarily large girth.",
"Indeed, if $\\ell =\\textrm {lcm}(2,3,\\ldots ,n)+1$ , we have that $\\ell =1\\ (\\textrm {mod}\\ i)$ for every $i=2,3,\\ldots ,n$ .",
"Then, according to Lemma REF , $CK(d,\\ell )$ must have girth $g>n$ .",
"It is known that if a digraph $G$ has girth $g$ , then its line digraph $L(G)$ also has girth $g$ , see Fàbrega and Fiol [5].",
"Since $L(sK(d,\\ell ))=CK(d,\\ell +1)$ , both digraphs have the same girth."
],
[
"Connectivity and superconnectivity",
"It is well-known that the Kautz digraphs $K(d,\\ell )$ have maximal (edge- and vertex-) connectivities (see Fàbrega and Fiol [5]).",
"The following result shows that this is also the case for the other Kautz-like digraph studied here, see Figure REF for a summary.",
"Proposition 3 $(i)$ The subKautz digraph $sK(d,\\ell )$ with $d\\ge 3$ and $\\ell \\ge 2$ is super-$\\lambda $ .",
"$(ii)$ The subKautz digraph $sK(d,\\ell )$ with either $d=\\ell =3$ , or $d\\ge 4$ and $\\ell \\ge 3$ , is maximally vertex-connected.",
"$(iii)$ The cyclic Kautz digraph $CK(d,\\ell )$ with $d\\ge 3$ and $\\ell \\ge 3$ is super-$\\lambda $ .",
"$(iv)$ The cyclic Kautz digraph $CK(d,\\ell )$ with either $d=3$ and $\\ell =4$ , or $d,\\ell \\ge 4$ , is super-$\\kappa $ .",
"$(iv)$ The cyclic Kautz digraph $CK(d,\\ell )$ with either $d=3$ and $\\ell \\ne 4$ , or $d\\ge 4$ and $\\ell =3$ , is maximally vertex-connected.",
"Since both $sK(d,\\ell )$ and $CK(d,\\ell )$ are subdigraphs of $K(d,\\ell )$ , with semigirth $\\ell $ (see Fàbrega and Fiol [5]), then the semigirths of these digraphs are at least $\\ell $ .",
"Hence, by using that the diameters of $sK(d,\\ell )$ and $CK(d,\\ell )$ are given in Theorem REF , Proposition REF , and Corollaries REF and REF , the result follows from Theorems REF and REF .",
"Figure: Summary of the connectivities of sK(d,ℓ)sK(d,\\ell ) and CK(d,ℓ)CK(d,\\ell ), depending on the values of dd and ℓ\\ell ."
],
[
"Cyclic Kautz digraphs $CK(d,3)$ with {{formula:be6f4d76-933d-4d37-8222-4a01344e4c62}}",
"The cyclic Kautz digraphs $CK(d,3)$ with $d\\ge 3$ have some special properties that, in general, are not shared with $CK(d,\\ell )$ with $\\ell >3$ .",
"These properties are listed in the following result.",
"Lemma 6 The cyclic Kautz digraphs $CK(d,3)$ with $d\\ge 3$ satisfy the following properties: $(a)$ $(d-1)$ -regular.",
"$(b)$ Number of vertices: $N=d^3-d$ , number of arcs: $m=(d+1)d(d-1)^2$ .",
"$(c)$ Diameter: $2\\ell -1=5$ .",
"$(d)$ $CK(d,3)$ are the line digraphs of the subKautz digraphs $sK(d,2)$ , which are obtained from the Kautz digraphs $K(d,2)$ by removing the arcs of the digons.",
"$(e)$ Vertex-transitive.",
"$(f)$ Eulerian and Hamiltonian.",
"$(a)$ , $(b)$ , $(c)$ and $(d)$ come from the properties of general $CK(d,\\ell )$ .",
"$(e)$ Since $sK(d,2)$ (with $d\\ge 3$ ) are vertex-transitive and arc-transitive, their line digraphs $CK(d,3)$ are vertex-transitive.",
"$(f)$ $sK(d,2)$ and $CK(d,3)$ with $d\\ge 3$ are Eulerian, because they are $(d-1)$ -regular.",
"Since $sK(d,2)$ (with $d\\ge 3$ ) are Eulerian, their line digraphs $CK(d,3)$ are Hamiltonian."
],
[
"Mean distance",
"As said before, $CK(d,\\ell )$ are asymptotically optimal with respect to the mean distance.",
"Now, we give the exact formulas for the mean distance of $sK(d,2)$ and $CK(d,3)$ with $d\\ge 3$ .",
"Let $n$ and $N$ be the numbers of vertices of $sK(d,2)$ and $CK(d,3)$ , respectively.",
"Lemma 7 $(a)$ The mean distance of the antipodal subKautz digraph $sK(d,2)$ with $d\\ge 3$ is $\\overline{\\partial ^*}=\\frac{2d^2+3d-1}{d^2+d}.$ $(b)$ The mean distance of the cyclic Kautz digraph $CK(d,3)$ with $d\\ge 3$ is $\\overline{\\partial }=\\frac{3d^3+d^2-5d-2}{d^3-d}.$ Since $CK(d,3)$ (and also $sK(2,2)$ ) with $d\\ge 3$ is vertex-transitive, we can compute the number of vertices from any given vertex.",
"First, we fix the distance layers in $sK(2,2)$ .",
"Thus, in Table REF , we give the numbers $n_k(u,v)$ of vertices at distance $k=0,1,\\ldots ,4$ from vertex $u=01$ to vertex $v\\in \\lbrace 01,1x,\\ldots ,10\\rbrace $ .",
"Table: Numbers of vertices vv at distance kk from u=01u=01.Then, the total numbers $n_i=n_i(u)$ of vertices at distance $i=0,1,\\ldots ,4$ from $u$ turn out to be $n_0 =1, \\quad n_1 =d-1, \\quad n_2 =(d-1)^2, \\quad n_3 = 2(d-1), \\quad n_4 = 1,$ with $n=n_0+n_1+\\cdots +n_4=d^2+d$ , and showing that $sK(2,2)$ is antipodal.",
"Now we use again that $CK(d,3)$ is the line digraph of $sK(d,2)$ to conclude that, in the former, the numbers $N_i$ of vertices at distance $i=0,1,\\ldots ,5$ from a given vertex, say 201, are $&& N_0 = n_0 =1, \\quad N_1 = n_1 =d-1, \\quad N_2 = (d-1)n_1 =(d-1)^2, \\quad N_3 = (d-1)n_2-1\\\\&& =(d-1)^3-1, \\quad N_4 = (d-1)n_3 =2(d-1)^2, \\quad N_5 = (d-1)n_4 =d-1,$ satisfying $N=N_0+N_1+\\cdots +N_5=d^3-d$ , as requested.",
"Note that in $N_3=(d-1)n_2-1$ we subtract one unit due to the presence in $sK(d,2)$ of the cycle of length 3: $20\\rightarrow 01\\rightarrow 12 \\rightarrow 20$ .",
"Then, the mean distances of $CK(d,3)$ with $d\\ge 3$ are, respectively, $\\displaystyle \\overline{\\partial ^*}=\\frac{1}{n}\\sum _{k=0}^4 kn_k$ , and $\\displaystyle \\overline{\\partial }=\\frac{1}{N}\\sum _{k=0}^5 kN_k$ , which gives the results.",
"Observe that, since $CK(d,3)$ is the line digraph of $sK(d,2)$ , the respective mean distance satisfies the inequality $\\overline{\\delta }<\\overline{\\delta ^*}$ , in concordance with the results by Fiol, Yebra, and Alegre [7].",
"Also, note that the mean distances of $sK(d,2)$ and $CK(d,3)$ , with $d\\ge 3$ , tend, respectively, to 2 and 3 for large degree $d-1$ , that is, they are asymptotically optimal."
]
] | 1709.01882 |
[
[
"Hilbert space operators with compatible off-diagonal corners"
],
[
"Abstract Given a complex, separable Hilbert space $\\mathcal{H}$, we characterize those operators for which $\\| P T (I-P) \\| = \\| (I-P) T P \\|$ for all orthogonal projections $P$ on $\\mathcal{H}$.",
"When $\\mathcal{H}$ is finite-dimensional, we also obtain a complete characterization of those operators for which $\\mathrm{rank}\\, (I-P) T P = \\mathrm{rank}\\, P T (I-P)$ for all orthogonal projections $P$.",
"When $\\mathcal{H}$ is infinite-dimensional, we show that any operator with the latter property is normal, and its spectrum is contained in either a line or a circle in the complex plane."
],
[
"Let $\\mathcal {H}$ be a complex, separable Hilbert space.",
"By $\\mathcal {B}( \\mathcal {H})$ , we denote the algebra of bounded, linear operators on $\\mathcal {H}$ .",
"If $\\dim \\, \\mathcal {H}= n < \\infty $ , then we identify $\\mathcal {H}$ with $\\mathbb {C}^n$ and $\\mathcal {B}( \\mathcal {H})$ with $\\mathbb {M}_n(\\mathbb {C})$ .",
"One of the most important open problems in operator theory is the Invariant Subspace Problem, which asks whether or not every bounded, linear operator $T$ acting on a complex, infinite-dimensional, separable Hilbert space $\\mathcal {H}$ admits a non-trivial invariant subspace; that is, a closed subspace $\\mathcal {M}\\notin \\lbrace \\lbrace 0 \\rbrace , \\mathcal {H}\\rbrace $ for which $T \\mathcal {M}\\subseteq \\mathcal {M}$ .",
"We say that an operator $T \\in \\mathcal {B}( \\mathcal {H})$ is (orthogonally) reductive if for each orthogonal projection $P \\in \\mathcal {B}( \\mathcal {H})$ , the condition $P T (I-P) = 0$ implies that $(I-P) T P = 0$ .",
"The Reductive Operator Conjecture is the assertion that every reductive operator is normal.",
"It was shown by Dyer, Pederson and Procelli [9] that the Invariant Subspace Problem admits a positive solution if and only if the Reductive Operator Conjecture is true.",
"Our goal in this paper is to study two variants of orthogonal reductivity.",
"Let $T \\in \\mathcal {B}( \\mathcal {H})$ and $P \\in \\mathcal {B}( \\mathcal {H})$ be an orthogonal projection.",
"We refer to the operator $P^\\perp T P: P\\mathcal {H}\\rightarrow P^\\perp \\mathcal {H}$ as an off-diagonal corner of $T$ .",
"Relative to the decomposition $\\mathcal {H}= P \\mathcal {H}\\oplus P^\\perp \\mathcal {H}$ , we may write $T=\\left[{\\begin{matrix}A & B \\\\ C & D\\end{matrix}}\\right] $ .",
"We refer to the block-entries of such block-matrices via their geographic positions: NW, NE, SE, SW, and the NE and the SW block-entries are examples of the off-diagonal corners.",
"In the work below, we shall be interested in two phenomena: firstly, when the operator norm of $B (= B_P)$ coincides with the operator norm of $C (=C_P)$ for all projections $P$ , and secondly, when the rank of $B$ coincides with the rank of $C$ for all projections $P$ .",
"Clearly, any operator which satisfies one of these two conditions is orthogonally reductive.",
"An example is given in Section below to show that the converse to this statement is false.",
"In the case of normal matrices, some related work has been done by Bhatia and Choi [5].",
"For instance, if the dimension of the space is $2 n < \\infty $ , and if $P$ is a projection of rank $n$ , it is a consequence of the fact that the Euclidean norm of the $k^{th}$ column of a normal matrix coincides with that of the $k^{th}$ row for all $k$ that the Hilbert-Schmidt (or Frobenius) norm of $B$ always equals that of $C$ .",
"Further, they show that $\\Vert B \\Vert \\le \\sqrt{n} \\Vert C \\Vert $ , and that equality can be achieved for some normal matrix $T \\in \\mathbb {M}_{2n}(\\mathbb {C})$ and some projection $P$ of rank $n$ if and only if $n \\le 3$ ."
],
[
"Definition.",
"Let $T \\in \\mathcal {B}( \\mathcal {H})$ .",
"We say that $T$ has the common norm property (property (CN)) if for any projection $P \\in \\mathcal {B}( \\mathcal {H})$ we have that $\\Vert P T P^\\perp \\Vert = \\Vert P^\\perp T P \\Vert .",
"$ We denote by $\\mathfrak {G_{norm}}$ the set of operators with property (CN).",
"We say that $T$ has the common rank property (property (CR)) if for any projection $P \\in \\mathcal {B}( \\mathcal {H})$ we have that $\\mathrm {rank}\\, P T P^\\perp = \\mathrm {rank}\\, P^\\perp T P. $ We denote by $\\mathfrak {G_{rank}}$ the set of operators with property (CR).",
"As we shall see, our results depend upon whether or not $\\mathcal {H}$ is finite-dimensional.",
"When the Hilbert space is finite-dimensional and of dimension at least four, then we shall show that the set of operators satisfying property (CN) coincides with the set of operators satisfying property (CR), and that this consists of those operators which are scalar translates of scalar multiples of hermitian (or of unitary) operators.",
"(See Theorem REF below.)",
"In the infinite-dimensional setting, we obtain a complete characterization of those operators satisfying property (CN).",
"Again, any scalar translate of a scalar multiple of a hermitian operator will suffice.",
"This time, however, the unitary operators involved must have essential spectrum contained in only half of a circle.",
"(See Theorem REF below.)",
"The problem of characterizing those operators acting on an infinite-dimensional Hilbert space which enjoy property (CR) is much more delicate.",
"We are able to demonstrate that any operator $T$ satisfying property (CR) must once again be a scalar translate of a scalar multiple of a hermitian (or of a unitary) operator.",
"In particular, such operators are normal.",
"However, an obstruction occurs in that it is not the case that every unitary operator has property (CR).",
"Indeed, as is well-known (see Section for an example) – not every unitary operator is reductive."
],
[
"We shall need some standard notation and definitions in what follows.",
"If $T=\\left[{\\begin{matrix}A & B \\\\ C & D\\end{matrix}}\\right]$ is a block-matrix in $\\mathbb {M}_n(\\mathbb {C})$ , and $A$ is invertible, then the matrix $D-CA^{-1}B$ is said to be the Schur complement of $A$ in $T$ and is denoted by $T|A$ .",
"In such a case $T$ is invertible if and only if $T|A$ is, and when this happens, the SE block-corner $\\left(T^{-1}\\right)_{_{SE}}$ of $T^{-1}$ is $(T|A)^{-1}$ .",
"Furthermore: $\\left(T^{-1}\\right)_{_{SW}}=-(T|A)^{-1}CA^{-1}\\ \\text{ and }\\ \\left(T^{-1}\\right)_{_{NE}}=-A^{-1}B\\,(T|A)^{-1}.$ Similarly, if $B$ is invertible then $C-DB^{-1}A$ is the Schur complement $T|B$ of $B$ in $T$ , and $T$ is invertible if and only if $T|B$ is, in which case $\\left(T^{-1}\\right)_{_{NE}}=(T|B)^{-1}.$ Corresponding statements and concepts apply to $C$ and $D$ as well.",
"As always, $\\mathbb {T}= \\lbrace z \\in \\mathbb {C}: |z| = 1\\rbrace $ .",
"A subset of $ is \\textbf {circlinear} if it is contained in a circle or a straight line.",
"By $$\\mathcal {K}( \\mathcal {H})$$, we denote the closed, two-sided ideal of compact operators in $$\\mathcal {B}( \\mathcal {H})$$, and $ : $\\mathcal {B}( \\mathcal {H})$$\\mathcal {B}( \\mathcal {H})$ /$\\mathcal {K}( \\mathcal {H})$$ denotes the canonical map from $$\\mathcal {B}( \\mathcal {H})$$ into the \\textbf {Calkin algebra} $$\\mathcal {B}( \\mathcal {H})$ /$\\mathcal {K}( \\mathcal {H})$$.The \\textbf {essential spectrum} $ e(T)$ of $ T $\\mathcal {B}( \\mathcal {H})$$ is the spectrum of $ (T)$ in the Calkin algebra $$\\mathcal {B}( \\mathcal {H})$ /$\\mathcal {K}( \\mathcal {H})$$, and we say that $ T$ is a \\textbf {Fredholm operator} if $ 0 e(T)$.",
"The \\textbf {Fredholm domain} of $ T$ is $ F(T) = $\\mathbb {C}$ e(T)$.",
"We say that $ T$ is a \\textbf {semi-Fredholm operator} if $ (T)$ is either left or right invertible in $$\\mathcal {B}( \\mathcal {H})$ /$\\mathcal {K}( \\mathcal {H})$$, and define the \\textbf {semi-Fredholm domain} of $ T$ to be $ sF(T) = { $\\mathbb {C}$ : (T-I) is semi-Fredholm}$.",
"The complement of $ sF(T)$ is called the \\textbf {left-right essential spectrum} of $ T$ and is denoted by $ r e(T)$.",
"If $ T$ is semi-Fredholm, we define the \\textbf {index} of $ T$ to be $ ind T = $\\mathrm {nul}$ T - $\\mathrm {nul}$ T* $\\mathbb {Z}$ {-, }$.",
"When $ T$ is Fredholm, we have that $ ind T $\\mathbb {Z}$$.$ We say that $T$ is triangular if there exists an orthonormal basis $\\lbrace e_n\\rbrace _{n=1}^\\infty $ for $\\mathcal {H}$ such that the matrix $[T] = [t_{i, j}]$ for $T$ relative to this basis (i.e.",
"$t_{i, j} = \\langle T e_j, e_i \\rangle $ ) satisfies $t_{i, j} = 0$ for all $i > j$ .",
"The operator is said to be quasitriangular if it is of the form $T = T_0 + K$ , where $T_0$ is triangular and $K$ is compact.",
"It was shown by Apostol, Foiaş, and Voiculescu [2] that $T$ is quasitriangular if and only if $\\mathrm {ind}\\, (T-\\lambda I) \\ge 0$ whenever $\\lambda \\in \\varrho _{sF}(T)$ .",
"Finally, $T$ is biquasitriangular if each of $T$ and $T^*$ is quasitriangular, i.e.",
"if and only if $\\mathrm {ind}\\, (T-\\lambda I) = 0$ for all $\\lambda \\in \\varrho _{sF}(T)$ .",
"Recall also that if $T \\in \\mathcal {B}( \\mathcal {H})$ , then $|T| = (T^* T)^{1/2}$ denotes the absolute value of $T$ .",
"A unitary operator $U \\in \\mathcal {B}( \\mathcal {H})$ is said to be absolutely continuous if the spectral measure for $U$ is absolutely continuous with respect to Lebesgue measure restricted to $\\sigma (U)$ , while $U$ is said to be singular if the spectral measure of $U$ is singular with respect to Lebesgue measure restricted to $\\sigma (U)$ .",
"These notions will only be used in Section ."
],
[
"We begin with a few simple remarks.",
"Although the proofs are rather elementary, we shall list these in the form of a Proposition so as to be able to more easily refer to them later.",
"The proofs are left to the reader."
],
[
"Proposition.",
"Suppose that $R, T \\in \\mathcal {B}( \\mathcal {H})$ and that $R$ has property (CR) and $T$ has property (CN).",
"For all $\\lambda , \\mu \\in \\mathbb {C}$ , we have that $\\lambda I + \\mu R$ and $R^*$ have property (CR), while $\\lambda I + \\mu T$ and $T^*$ have property (CN).",
"Suppose that $\\mathcal {H}= \\mathcal {H}_1 \\oplus \\mathcal {H}_2$ .",
"If there exist $A \\in \\mathcal {B}(\\mathcal {H}_1)$ and $D \\in \\mathcal {B}(\\mathcal {H}_2)$ such that $R = A \\oplus D$ , then $A$ and $D$ both have property (CR).",
"If there exist $A \\in \\mathcal {B}(\\mathcal {H}_1)$ and $D \\in \\mathcal {B}(\\mathcal {H}_2)$ such that $T = A \\oplus D$ , then $A$ and $D$ both have property (CN).",
"If $V \\in \\mathcal {B}( \\mathcal {H})$ is unitary, then $V^* R V$ has property (CR) and $V^* T V$ has property (CN).",
"If $L = L^* \\in \\mathcal {B}( \\mathcal {H})$ , then $L$ has both property (CR) and property (CN).",
"In the case of property (CN), we also observe the following.",
"For $T \\in \\mathcal {B}( \\mathcal {H})$ , let us denote by $\\mathcal {U}(T)$ the unitary orbit of $T$ , i.e.",
"$\\mathcal {U}(T) = \\lbrace V^* T V: V \\in \\mathcal {B}( \\mathcal {H})\\mbox{ unitary} \\rbrace $ .",
"Recall that two operators $S$ and $T$ are said to be approximately unitarily equivalent if $S \\in \\overline{\\mathcal {U}(T)}$ (equivalently, $T \\in \\overline{\\mathcal {U}(S)}$ ).",
"The proofs of the following assertions are elementary and are left to the reader."
],
[
"Proposition.",
"The set $\\mathfrak {G_{norm}}$ of operators with property (CN) is closed.",
"If $T \\in \\mathcal {B}( \\mathcal {H})$ has property (CN) and there exists $S \\in \\overline{\\mathcal {U}(T)}$ of the form $S = A \\oplus D$ , then $A, D$ have property (CN).",
"The following remark, while innocuous in appearance, is actually the key to a number of calculations below."
],
[
"Remark.",
"Let $U \\in \\mathcal {B}( \\mathcal {H})$ be a unitary operator and $P \\in \\mathcal {B}( \\mathcal {H})$ be a projection.",
"Write $U = \\left[{\\begin{matrix}A & B \\\\ C & D\\end{matrix}}\\right]$ relative to $\\mathcal {H}= P \\mathcal {H}\\oplus P^\\perp \\mathcal {H}$ .",
"The fact that $U$ is unitary implies that $I = A A^* + B B^* = A^* A + C^*C. $ Thus $B B^* = I - A A^*$ and $C^* C = I - A^* A.$ It follows that $\\Vert B \\Vert ^2 = \\Vert B B^* \\Vert = 1 - \\min \\lbrace \\lambda : \\lambda \\in \\sigma (A A^*) \\rbrace , $ and similarly $\\Vert C \\Vert ^2 = \\Vert C^* C \\Vert = 1 - \\min \\lbrace \\mu : \\mu \\in \\sigma (A^*A)\\rbrace .",
"$ However, it is a standard fact that $\\sigma (A A^*) \\cup \\lbrace 0 \\rbrace = \\sigma (A^* A) \\cup \\lbrace 0\\rbrace $ , and thus the only way that we can have $\\Vert B \\Vert \\ne \\Vert C \\Vert $ is if either $0 \\in \\sigma (A A^*)$ but $0 \\notin \\sigma (A^*A)$ , or $0 \\in \\sigma (A^* A)$ but $0 \\notin \\sigma (A A^*)$ .",
"This argument demonstrates the rather interesting fact that if $U = \\left[{\\begin{matrix}A & B \\\\ C & D\\end{matrix}}\\right]$ is a unitary operator and $\\Vert B \\Vert \\ne \\Vert C \\Vert $ , then $\\min (\\Vert B \\Vert , \\Vert C \\Vert ) < 1 = \\max (\\Vert B \\Vert , \\Vert C \\Vert ).$ In particular, if $U = \\left[{\\begin{matrix}A & B \\\\ C & D\\end{matrix}}\\right]$ is a unitary such that $\\Vert B \\Vert < \\Vert C \\Vert (=1)$ , then every unitary $U^{\\prime }$ close enough to $U$ has the form $\\left[{\\begin{matrix}A^{\\prime } & B^{\\prime } \\\\ C^{\\prime } & D^{\\prime }\\end{matrix}}\\right]$ where $\\Vert B^{\\prime } \\Vert < \\Vert C^{\\prime } \\Vert \\mathbf {=1},$ which is remarkable."
],
[
"We now turn to the case where the Hilbert space under consideration is finite-dimensional (and complex)."
],
[
"Proposition.",
"Let $n \\ge 2$ be an integer and $T \\in \\mathbb {M}_n(\\mathbb {C})$ .",
"If $T$ has property (CN) or property (CR), then $T$ is normal.",
"Proof.",
"This is an easy consequence of the fact that given any $T \\in \\mathbb {M}_n(\\mathbb {C})$ , there exists an orthonormal basis with respect to which the matrix of $T$ is upper triangular.",
"Either property clearly implies that the matrix of $T$ is in fact diagonal with respect to this basis, and hence that $T$ is normal.",
"$\\Box $"
],
[
"Proposition.",
"Let $n \\ge 2$ , and let $U \\in \\mathbb {M}_n(\\mathbb {C})$ be a unitary operator.",
"Then $U$ has both property (CN) and property (CR).",
"Proof.",
"Let $P \\in \\mathbb {M}_n(\\mathbb {C})$ be a projection, and relative to the decomposition $\\mathbb {C}^n = P \\mathbb {C}^n \\oplus P^\\perp \\mathbb {C}^n$ , let us write $U = \\begin{bmatrix} A &B \\\\ C & D \\end{bmatrix}.",
"$ As noted in Remark REF , since $U$ is unitary, we have that $B B^* = I - A A^*$ and $C^* C = I - A^*A$ .",
"Observe, however, that in the finite-dimensional setting we have that $A^* A$ is unitarily equivalent to $A A^*$ , and thus $B B^*$ is unitarily equivalent to $C^* C$ .",
"Thus $\\Vert B \\Vert = \\Vert C \\Vert $ , and $\\mathrm {rank}\\, B = \\mathrm {rank}\\, B B^* = \\mathrm {rank}\\, C^* C = \\mathrm {rank}\\, C$ .",
"$\\Box $ Combining this with Proposition REF (a), we obtain:"
],
[
"Proposition.",
"Let $T \\in \\mathbb {M}_n(\\mathbb {C})$ .",
"If $T$ is either hermitian or unitary, then for all $\\lambda , \\mu \\in \\mathbb {C}$ , $\\lambda I + \\mu T$ has both property (CN) and property (CR).",
"Our goal is to prove that if $T \\in \\mathbb {M}_n(\\mathbb {C})$ has either property (CN) or property (CR), then it is of the form $\\lambda I + \\mu X$ where $X$ is either hermitian or unitary."
],
[
"Remark.",
"The common link between these two cases is the geometry of the set of eigenvalues of $T$ .",
"If $T$ is normal, then $T = \\lambda I + \\mu V$ where $V$ is unitary if and only if all of the eigenvalues of $T$ lie on a common circle.",
"If $T$ is normal, then $T = \\lambda I + \\mu L$ where $L = L^*$ if and only if all of the eigenvalues of $T$ lie on a common line.",
"That is to say, the union of these two sets of operators is precisely the class of normal operators whose spectra are circlinear.",
"Given a matrix $B \\in \\mathbb {M}_{n, m}(\\mathbb {C})$ , we denote by $\\Vert B \\Vert _2 = \\mathrm {tr} (B^* B)^{1/2}$ the Hilbert-Schmidt (or Fröbenius) norm of $B$ ."
],
[
"Proposition.",
"Let $k, \\ell \\ge 1$ be integers, and suppose that $T = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix}$ is a normal operator in $\\mathcal {B}(\\mathbb {C}^k \\oplus \\mathbb {C}^\\ell )$ .",
"Then $\\Vert B \\Vert _2 = \\Vert C \\Vert _2.",
"$ Proof.",
"The fact that $T$ is normal implies that $A A^* + B B^* = A^* A + C^* C$ .",
"Using the fact that the trace is linear and that $\\mathrm {tr} (X Y) = \\mathrm {tr} (Y X)$ for all $X \\in \\mathbb {M}_{k, \\ell }(\\mathbb {C})$ , $Y \\in \\mathbb {M}_{\\ell , k}(\\mathbb {C})$ , we see that $\\Vert B \\Vert _2 = \\mathrm {tr}(B B^*) = \\mathrm {tr}(C^* C) = \\Vert C \\Vert _2.",
"$ $\\Box $ We begin by considering the exceptional cases where the dimension of the underlying Hilbert space is too small to allow anything interesting to happen."
],
[
"Proposition.",
"If $2 \\le n \\le 3$ , and let $T \\in \\mathbb {M}_n(\\mathbb {C})$ .",
"The following are equivalent: $T$ is normal.",
"$T$ has property (CN).",
"$T$ has property (CR).",
"Proof.",
"By Proposition REF , both (b) and (c) imply (a).",
"Conversely, if $T \\in \\mathbb {M}_n(\\mathbb {C})$ is normal and $0 \\ne P \\ne I$ is a projection in $\\mathbb {M}_n(\\mathbb {C})$ , then $P T P^\\perp $ and $P^\\perp T P$ both have rank at most one.",
"From this and from Proposition REF , we find that $\\Vert P T P^\\perp \\Vert = \\Vert P T P^\\perp \\Vert _2 = \\Vert P^\\perp T P \\Vert _2 = \\Vert P^\\perp T P \\Vert .",
"$ Thus $T$ has property (CN); that is, (a) implies (b).",
"This also shows that $P T P^\\perp $ and $P^\\perp T P$ either both have rank 0 or both have rank 1.",
"Hence (a) implies (c) as well.",
"$\\Box $ For the remainder of this section, we shall assume that the dimension $n$ of the underlying Hilbert space is at least 4."
],
[
"Remark.",
"Let us now show that the problem of characterizing which operators in $\\mathbb {M}_n(\\mathbb {C})$ have property (CN) (resp.",
"property (CR)) reduces to the case where $n= 4$ .",
"Of course, by Proposition REF , we may restrict our attention to normal operators.",
"Let $n > 4$ , and suppose that $T \\in \\mathbb {M}_n(\\mathbb {C})$ is normal.",
"As observed in Remark REF , if all of the eigenvalues of $T$ are either co-linear or co-circular (i.e.",
"all lie on the same circle), then there exist $\\alpha , \\beta \\in \\mathbb {C}$ and either a hermitian operator $L$ or a unitary operator $V$ such that $T = \\alpha I + \\beta L$ , or $T = \\alpha I + \\beta V$ .",
"Either way, by Proposition REF , $T$ has property (CN) and property (CR).",
"Conversely, suppose that $T$ has property (CN) (resp.",
"$T$ has property (CR)), and suppose we know that every $X \\in \\mathbb {M}_4(\\mathbb {C})$ with property (CN) (resp.",
"with property (CR)) has eigenvalues that are either co-linear or co-circular.",
"Given any $\\lbrace \\theta _1, \\theta _2, \\theta _3, \\theta _4\\rbrace $ in $\\sigma (T)$ , we can write $T = R \\oplus Y$ , where $R$ is a normal operator in $\\mathbb {M}_4(\\mathbb {C})$ with $\\sigma (R) = \\lbrace \\theta _1, \\theta _2, \\theta _3, \\theta _4 \\rbrace $ .",
"By Proposition REF (b), $R$ has property (CN) (resp.",
"$R$ has property (CR)).",
"It follows from our hypothesis that $\\lbrace \\theta _1, \\theta _2, \\theta _3, \\theta _4\\rbrace $ are either co-linear or co-circular.",
"Since this is true for an arbitrary collection of four elements from $\\sigma (T)$ , we conclude that all of the eigenvalues of $T$ are either co-linear or co-circular.",
"As before, this implies the existence of $\\alpha , \\beta \\in \\mathbb {C}$ and either a hermitian operator $L$ or a unitary operator $V$ such that $T = \\alpha I + \\beta L$ , or $T = \\alpha I + \\beta V$ .",
"We now concentrate on proving that a $4 \\times 4$ matrix $T$ has property (CN) (resp.",
"property (CR)) if and only if the eigenvalues of $T$ are either co-linear or co-circular."
],
[
"Lemma.",
"Let $X, Y \\in \\mathbb {M}_2(\\mathbb {C})$ and suppose that $\\Vert X \\Vert _2 = \\Vert Y \\Vert _2$ .",
"The following are equivalent: $\\Vert X \\Vert = \\Vert Y \\Vert $ ; $\\mathrm {tr}( (X^* X)^2) = \\mathrm {tr}( (Y^* Y)^2)$ ; $| \\det (X)| = | \\det (Y) |$ .",
"Proof.",
"Again, since the Fröbenius norm, the operator norm, and the trace functional are all invariant under unitary conjugation, we may assume without loss of generality that $X^* X$ and $Y^* Y$ are not only positive but diagonal, say $X^* X = \\begin{bmatrix} x_1 & 0 \\\\ 0 & x_2 \\end{bmatrix}, \\ \\ \\ \\ \\ Y^* Y = \\begin{bmatrix} y_1 & 0 \\\\ 0 & y_2 \\end{bmatrix}, $ with $0 \\le x_1, x_2, y_1, y_2$ .",
"The hypothesis that $\\Vert X \\Vert _2 = \\Vert Y \\Vert _2$ is the statement that $\\varrho = x_1 + x_2 = y_1 + y_2$ .",
"implies (b).",
"Suppose that $\\Vert X \\Vert = \\Vert Y \\Vert $ .",
"Then $\\Vert X \\Vert ^2 = \\Vert Y \\Vert ^2$ and so $\\max \\lbrace x_1, x_2 \\rbrace = \\max \\lbrace y_1, y_2 \\rbrace $ .",
"By reindexing if necessary, we may assume that $x_1 = y_1$ .",
"But we have also assumed that $x_1 + x_2 = y_1 + y_2$ , and so $x_2 = y_2$ .",
"It follows that $\\mathrm {tr} ( (X^* X)^2) = x_1^2 + x_2^2 = y_1^2 + y_2^2 = \\mathrm {tr} ((Y^*Y)^{2}).",
"$ implies (c).",
"Our current hypotheses are that $x_1 + x_2 = y_1 + y_2$ and that $x_1^2 + x_2^2 = y_1^2 + y_2^2$ .",
"Thus $|\\det (Y)|^2&= \\det (Y^* Y)\\\\&= y_1 y_2 \\\\&= \\frac{1}{2} \\left((y_1 + y_2)^2 - (y_1^2 + y_2^2)\\right)\\\\&= \\frac{1}{2} \\left((x_1 + x_2)^2 - (x_1^2 + x_2^2)\\right)\\\\&= x_1 x_2\\\\&= \\det (X^* X)\\\\&= |\\det (X)|^2,$ from which (c) follows.",
"implies (a).",
"Suppose that $| \\det (X)| = |\\det (Y)|$ .",
"Then, as we have just computed, $x_1 x_2 = | \\det (X)|^2 = |\\det (Y)|^2 = y_1 y_2$ .",
"But then $x_1 + x_2 = y_1 + y_2$ and $x_1 x_2 = y_1 y_2$ together imply that $\\lbrace x_1, x_2 \\rbrace = \\lbrace y_1, y_2 \\rbrace $ .",
"In particular, $\\Vert X \\Vert ^2 = \\Vert X^* X \\Vert = \\max \\lbrace x_1, x_2 \\rbrace = \\max \\lbrace y_1, y_2 \\rbrace = \\Vert Y^* Y \\Vert = \\Vert Y\\Vert ^2.",
"$ This completes the proof.",
"$\\Box $"
],
[
"Theorem.",
"Suppose that $T$ is an invertible normal block-matrix in $\\mathbb {M}_{4}(\\mathbb {C})$ with $2\\times 2$ blocks, and the off-diagonal corners of $T$ have equal rank (respectively equal operator norm), then the same is true for the off-diagonal corners of $T^{-1}$ .",
"Proof.",
"Let us start with the case of equal ranks, and employ a proof by contradiction, supposing that $\\displaystyle T=\\left[{\\begin{matrix}A & B \\\\ C & D\\end{matrix}}\\right]$ is an invertible normal matrix with $\\displaystyle T^{-1}=\\left[{\\begin{matrix}A^{\\prime } & B^{\\prime } \\\\ C^{\\prime } & D^{\\prime }\\end{matrix}}\\right]$ , and $\\mathrm {rank}\\, {B}=\\mathrm {rank}\\, {C}, \\text{ but } \\mathrm {rank}\\, {B^{\\prime }}\\ne \\mathrm {rank}\\, {C^{\\prime }}.$ Since $T^{-1}$ is normal, every invariant subspace of $T^{-1}$ is reducing, and so if either ${B^{\\prime }}$ or ${C^{\\prime }}$ is zero then both ${B^{\\prime }}$ and ${C^{\\prime }}$ are zero, contradicting our hypothesis.",
"Hence we may assume that one of $B^{\\prime }$ and $C^{\\prime }$ has rank 1 and the other has rank equal to 2.",
"Passing to $T^{{*}}$ if necessary, we can assume without loss of generality that $\\mathrm {rank}\\, {C^{\\prime }}=1<2=\\mathrm {rank}\\, {B^{\\prime }}.$ In particular, $B^{\\prime }$ is invertible, as is $T^{-1}$ , and therefore $\\displaystyle B=\\left(T^{-1}|B^{\\prime }\\right)^{-1}$ , so that $B$ is invertible.",
"Consequently $C$ has rank 2 and is invertible.",
"Hence $\\displaystyle C^{\\prime }=\\left(T|C\\right)^{-1}$ , and therefore $C^{\\prime }$ is invertible, i.e.",
"has rank 2, equal to that of $B^{\\prime }$ , contradicting our hypothesis.",
"Next let us deal with the case of equal operator norms.",
"Let us suppose that $\\left\\Vert {B} \\right\\Vert =\\left\\Vert {C} \\right\\Vert $ , or equivalently, by Lemma REF , that $|\\det {B}|=|\\det {C}|$ .",
"First, let us treat the case “$A$ is invertible”.",
"In this case $\\det {C^{\\prime }}=\\det {\\left(-(T|A)^{-1}CA^{-1}\\right)}=\\frac{\\det {C}}{\\det {(T|A)}\\det {A}}$ and $\\det {B^{\\prime }}=\\det {\\left(-A^{-1}B\\,(T|A)^{-1}\\right)}=\\frac{\\det {B}}{\\det {(T|A)}\\det {A}},$ so that $\\det {C^{\\prime }}=\\det {B^{\\prime }}$ , and therefore $\\left\\Vert {B^{\\prime }} \\right\\Vert =\\left\\Vert {C^{\\prime }} \\right\\Vert $ , again by Lemma REF .",
"Now, the remaining case is “$A$ is not invertible”.",
"In this case there is a sequence $\\left[\\alpha _{k}\\right]_{k\\in \\mathbb {N}}$ convergent to zero and such that each $\\alpha _{k}$ is neither an eigenvalue of $T$ , nor of $A$ .",
"Applying the already settled case “A is invertible” to each (invertible and normal) $T-\\alpha _{k}\\, I$ , we can conclude that for each $k$ the off-diagonal corners of $\\left(T-\\alpha _{k}\\, I\\right)^{-1}$ have equal norms.",
"Yet $\\lim _{k \\rightarrow \\infty } \\Vert (T-\\alpha _k I)^{-1} - T^{-1} \\Vert = 0$ , and therefore the off-diagonal corners of $\\left(T-\\alpha _{k}\\, I\\right)^{-1}$ converge to those of $T^{-1}$ , showing that the latter have equal norms as well, and the proof is complete.",
"$\\Box $"
],
[
"Corollary.",
"Suppose that $T$ is a normal block-matrix in $\\mathbb {M}_{4}(\\mathbb {C})$ with $2\\times 2$ blocks such that a Möbius map $M(z)=\\frac{az+b}{cz+d}$ is finite at all eigenvalues of $T$ .",
"If the off-diagonal corners of $T$ have equal rank (respectively, equal operator norm), then the same is true for the off-diagonal corners of $M(T)$ .",
"Proof.",
"The claim is obviously true if $c=0\\ne d$ .",
"Let us consider the case $c\\ne 0.$ In this case, $\\frac{-d}{c}$ is not an eigenvalue of $T$ , and $M(z)=\\frac{a}{c}+\\left(\\frac{b-\\frac{ad}{c}}{c}\\right)\\cdot \\frac{1}{z+\\frac{d}{c}}.$ If the off-diagonal corners of $T$ have equal rank (respectively, equal operator norm), then the same is true for $T+\\frac{d}{c}I$ .",
"Then, by Theorem REF , the off-diagonal corners of $\\left(T+\\frac{d}{c}I\\right)^{-1}$ have equal rank (respectively, equal operator norm), and so the same can be said about the off-diagonal corners of $M(T)$ .",
"$\\Box $"
],
[
"Corollary.",
"If $T\\in \\mathbb {M}_{4}($ has property (CN) or if $T$ has property (CR), then $T$ is normal and $M(T)$ has the same property for any Möbius map $M$ that is finite on the spectrum of $T$ .",
"Proof.",
"This is the consequence of Proposition REF , Corollary REF and the standard analytic functional-calculus fact that $M\\left(U^{^{*}}TU\\right)=U^{^{*}}M(T)U.$ $\\Box $"
],
[
"Proposition.",
"If $T$ is a normal block-matrix in $\\mathbb {M}_{4}(\\mathbb {C})$ with $2\\times 2$ blocks, and the spectrum of $T$ is $\\lbrace 0,1,2, \\delta \\rbrace $ , where $\\delta \\notin \\mathbb {R}$ , then there exists a unitary block-matrix $U$ in $\\mathbb {M}_{4}(\\mathbb {C})$ such that, with respect to the $2\\times 2$ block partitioning, $\\mathrm {rank}\\, \\left(U^{^{*}}TU\\right)_{_{NE}}<2=\\mathrm {rank}\\, {\\left(U^{^{*}}TU\\right)_{_{SW}}}.$ Proof.",
"Every complex number $\\delta $ other than 2 can be expressed as $2-8/(6+\\beta )$ for a unique $\\beta \\ne -6$ .",
"Furthermore, $\\delta $ is real exactly when $\\beta $ is real.",
"Hence, after applying a unitary similarity we can assume without loss of generality that $T=\\left(\\begin{array}{cccc}0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 2 & 0 \\\\0 & 0 & 0 & 2-\\frac{8}{\\beta +6}\\\\\\end{array}\\right)$ The unitary $U$ shall be the product $VW$ of the unitaries $V=\\left(\\begin{array}{cccc}\\sigma & 0 & \\gamma & 0 \\\\0 & \\gamma & 0 & \\sigma \\\\\\gamma & 0 & -\\sigma & 0 \\\\0 & \\sigma & 0 & -\\gamma \\\\\\end{array}\\right)\\ \\ \\ \\ \\text{and}\\ \\ \\ \\ W=\\left(\\begin{array}{cccc}\\frac{1}{\\sqrt{2}} &\\frac{e^{i \\theta }}{\\sqrt{2}} & 0 & 0 \\\\\\frac{1}{\\sqrt{2}} &-\\frac{e^{i \\theta }}{\\sqrt{2}} & 0 & 0 \\\\0 & 0 & \\frac{1}{\\sqrt{2}} &\\frac{1}{\\sqrt{2}} \\\\0 & 0 & \\frac{1}{\\sqrt{2}} &-\\frac{1}{\\sqrt{2}} \\\\\\end{array}\\right),$ where $\\sigma , \\gamma $ and $\\theta $ , to be specified later, are subject to the conditions: $\\sigma \\ne \\gamma ,\\ \\ \\sigma ^{2}+\\gamma ^{2}=1,\\ \\ 0<\\sigma ,\\ \\ 0<\\gamma , \\ \\ \\ \\text{and}\\ \\ \\ \\theta \\text{ is not an integer multiple of } \\pi .$ A direct calculation shows that $\\left(U^{^{*}}TU\\right)_{_{NE}}=\\left(\\begin{array}{cc}-\\frac{(3 \\beta +10) \\gamma \\sigma }{2 (\\beta +6)} &-\\frac{4 \\gamma ^2}{\\beta +6}-\\frac{1}{2} e^{i\\theta } \\sigma ^2 \\\\-\\frac{1}{2} e^{-i \\theta }\\gamma ^2-\\frac{4 \\sigma ^2}{\\beta +6} & -\\frac{(3 \\beta +10)\\gamma \\sigma }{2 (\\beta +6)}\\\\\\end{array}\\right)$ and $\\left(U^{^{*}}TU\\right)_{_{SW}}=\\left(\\begin{array}{cc}-\\frac{(3 \\beta +10) \\gamma \\sigma }{2 (\\beta +6)} &-\\frac{1}{2} e^{i \\theta }\\gamma ^2-\\frac{4 \\sigma ^2}{\\beta +6} \\\\-\\frac{4 \\gamma ^2}{\\beta +6}-\\frac{1}{2} e^{-i\\theta } \\sigma ^2 &-\\frac{(3 \\beta +10) \\gamma \\sigma }{2 (\\beta +6)} \\\\\\end{array}\\right),$ and that $\\det {\\left(U^{^{*}}TU\\right)_{_{NE}}}&=\\left(\\frac{-2\\gamma ^2\\sigma ^2}{\\beta +6}\\right)\\left(\\frac{1}{\\frac{e^{i \\theta }\\sigma ^2}{\\gamma ^2}} +\\frac{e^{i \\theta }\\sigma ^2}{\\gamma ^2}-\\beta \\right),$ while $\\det {\\left(U^{^{*}}TU\\right)_{_{SW}}}=\\left(\\frac{-2}{\\beta +6}\\right)\\left(\\gamma ^4 e^{i \\theta }+e^{-i \\theta } \\sigma ^4-\\gamma ^2 \\sigma ^2\\beta \\right).$ Now one can see that $\\det {\\left(U^{^{*}}TU\\right)_{_{NE}}}-\\det {\\left(U^{^{*}}TU\\right)_{_{SW}}}=\\frac{4 i \\left(\\gamma ^4-\\sigma ^4\\right) \\sin (\\theta )}{\\beta +6}\\ne 0,$ because of the conditions that we have imposed on $\\sigma , \\gamma $ and $\\theta $ .",
"Note that $\\displaystyle \\frac{e^{i \\theta }\\sigma ^2}{\\gamma ^2}$ can take on any non-real complex value even when $\\sigma , \\gamma $ are restricted to be distinct positive numbers whose squares add up to 1, and $\\theta $ is not an integer multiple of $\\pi $ .",
"It is also easy to see that the equation $\\displaystyle \\zeta +\\frac{1}{\\zeta }=\\beta $ has a complex solution for $\\zeta $ , and since $\\beta \\notin \\mathbb {R}$ , the solution cannot be real.",
"It follows that there exist $\\sigma , \\gamma $ and $\\theta $ satisfying the conditions: $\\sigma \\ne \\gamma ,\\ \\ \\sigma ^{2}+\\gamma ^{2}=1,\\ \\ 0<\\sigma ,\\ \\ 0<\\gamma , \\ \\ \\ \\text{and}\\ \\ \\ \\theta \\text{ is not an integer multiple of } \\pi ,$ as well as the condition $\\frac{1}{\\frac{e^{i \\theta }\\sigma ^2}{\\gamma ^2}} +\\frac{e^{i \\theta }\\sigma ^2}{\\gamma ^2}=\\beta .$ These are the $\\sigma , \\gamma $ and $\\theta $ that we use in the construction of $U$ , and it is now clear that such a $U$ is the one we seek, since for this $U$ : $0=\\det {\\left(U^{^{*}}TU\\right)_{_{NE}}}\\ne \\det {\\left(U^{^{*}}TU\\right)_{_{SW}}}.$ $\\Box $"
],
[
"Corollary.",
"If $T\\in \\mathbb {M}_{4}($ has property (CN) or if $T$ has property (CR), then $T$ is normal and the spectrum of $T$ is circlinear.",
"Proof.",
"Such $T$ has to be normal by Proposition REF .",
"To verify the rest of the claims we proceed by contradiction.",
"Suppose that the eigenvalues $\\lambda _{0}, \\lambda _{1}, \\lambda _{2}, \\lambda _{3}$ of $T$ are not circlinear.",
"Then they are all distinct.",
"By the spectral mapping theorem, given a Möbius map $M$ that is finite on the spectrum of $T$ , the eigenvalues of $M(T)$ are the images of the eigenvalues of $T$ under $M$ .",
"It is well-known that Möbius maps take circlines to circlines, and exhibit sharp three-fold transitivity.",
"In particular there is a unique Möbius map ${M}_{_{o}}$ such that ${M}_{_{o}}(\\lambda _{i})=i,\\ \\text{ for } i=0,1,2.$ If ${M}_{_{o}}$ sends $\\lambda _{3}$ to $z_{_{o}}$ that is a real number or “$\\infty $ ”, then the inverse of ${M}_{_{o}}$ sends $0,1,2, z_{_{o}}$ to $\\lambda _{0}, \\lambda _{1}, \\lambda _{2}, \\lambda _{3}$ , indicating that the latter set is part of the image (under ${M}_{_{o}}$ ) of the extended real line, and hence must be circlinear, contrary to our hypothesis.",
"Therefore ${M}_{_{o}}$ sends $\\lambda _{3}$ to some complex non-real number $\\delta $ .",
"Applying Corollary REF and Proposition REF to ${M}_{_{o}}(T)$ yields a contradiction, and the proof is complete.",
"$\\Box $ By combining Remark REF and Corollary REF , we obtain the main theorem of this section."
],
[
"Theorem.",
"Let $n \\ge 4$ be an integer and $T \\in \\mathbb {M}_n(\\mathbb {C})$ .",
"The following are equivalent.",
"$T$ has property (CN).",
"$T$ has property (CR).",
"One of the following holds.",
"There exist $\\lambda , \\mu \\in \\mathbb {C}$ and $V \\in \\mathbb {M}_n(\\mathbb {C})$ unitary such that $T = \\lambda I + \\mu V$ .",
"There exist $\\lambda , \\mu \\in \\mathbb {C}$ and $L = L^*\\in \\mathbb {M}_n(\\mathbb {C})$ such that $T = \\lambda I +\\mu L$ ."
],
[
"There is also an alternative proof for Theorem REF that does not involve Möbius maps, and while we have chosen not to include it here, we will gladly share it with an interested reader.",
"Clearly the invariance of property (CN) and property (CR) under Möbius maps (as in Corollary REF ) can be inferred from Theorem REF ."
],
[
"Let us now consider the case where the underlying Hilbert space is infinite-dimensional and separable.",
"We begin by studying operators with property (CN).",
"In the finite-dimensional setting, we saw that any such operator is normal.",
"While this is also true in the infinite-dimensional setting, the proof is rather different.",
"Recall that an operator $T \\in \\mathcal {B}( \\mathcal {H})$ is said to be strongly reductive if, whenever $(P_n)_{n=1}^\\infty $ is a sequence of orthogonal projections such that $\\lim _n \\Vert P_n^\\perp T P_n \\Vert = 0$ , it follows that $\\lim _n \\Vert P_n T P_n^\\perp \\Vert = 0$ (or equivalently, $\\lim _n \\Vert P_n T - T P_n \\Vert = 0$ ).",
"Let us say that a compact set $\\Omega \\subseteq \\mathbb {C}$ is Lavrentiev if it has empty interior and if $\\mathbb {C}\\setminus \\Omega $ is connected."
],
[
"Proposition.",
"Let $\\mathcal {H}$ be an infinite-dimensional, separable Hilbert space.",
"If $T \\in \\mathcal {B}( \\mathcal {H})$ has property (CN), then $T$ is normal and has Lavrentiev spectrum.",
"Proof.",
"It is an immediate consequence of the definition that if $T$ has property (CN), then $T$ is strongly reductive.",
"It was shown by Harrison [12] that any strongly reductive operator has Lavrentiev spectrum.",
"Apostol, Foiaş and Voiculescu [3] showed that any strongly reductive operator is normal.",
"It is also easy to obtain the normality of $T$ which enjoys property (CN) directly.",
"Suppose that $T \\in \\mathcal {B}( \\mathcal {H})$ has property (CN), and let $e \\in \\mathcal {H}$ be an arbitrary vector of norm one.",
"Then $P_e (x) = \\langle x, e \\rangle e$ , $x \\in \\mathcal {H}$ defines a rank one projection.",
"By hypothesis, $\\Vert P_e^\\perp T^* P_e \\Vert = \\Vert P_e T P_e^\\perp \\Vert = \\Vert P_e^\\perp T P_e \\Vert $ .",
"Now $\\langle T T^* e, e \\rangle = \\Vert T^* e \\Vert ^2&= \\Vert P_e T^* e \\Vert ^2 + \\Vert P_e^\\perp T^* e \\Vert ^2 \\\\&= \\Vert P_e T^* P_e e \\Vert ^2 + \\Vert P_e^\\perp T^* P_e e \\Vert ^2 \\\\&= \\Vert P_e T^* P_e \\Vert ^2 + \\Vert P_e^\\perp T^* P_e \\Vert ^2.$ Similarly, $\\langle T^* T e, e \\rangle = \\Vert P_e T P_e\\Vert ^2 + \\Vert P_e^\\perp T P_e\\Vert ^2.",
"$ Now $\\Vert P_e T P_e \\Vert = \\Vert P_e T^* P_e \\Vert , $ and combining this with the fact that $\\Vert P_e^\\perp T^* P_e \\Vert = \\Vert P_e^\\perp T P_e \\Vert $ from above, we see that $\\langle T T^* e, e \\rangle = \\langle T^* T e, e \\rangle .",
"$ Since $e$ was an arbitrary norm-one vector, we conclude that $T T^* = T^* T$ ; i.e.",
"that $T$ is normal.",
"$\\Box $"
],
[
"In Section , we noted that if $L = L^* \\in \\mathcal {B}( \\mathcal {H})$ and $\\lambda , \\mu \\in \\mathbb {C}$ , then $\\lambda I + \\mu L$ has property (CN).",
"Although in the finite-dimensional setting every unitary operator $V$ also has property (CN), this is no longer true in the infinite-dimensional setting, as the following counterexample shows.",
"Let $\\lbrace e_n \\rbrace _{n \\in \\mathbb {Z}}$ be an orthonormal basis for our Hilbert space $\\mathcal {H}$ , and consider the bilateral shift operator $W$ determined by $W e_n = e_{n-1}$ , $n \\in \\mathbb {Z}$ .",
"Then $W$ is unitary and $\\sigma (W) = \\mathbb {T}$ .",
"By Proposition REF , $W$ does not have property (CN).",
"This can be seen directly as well.",
"If $P$ is the orthogonal projection of $\\mathcal {H}$ onto $\\mathcal {M}= \\overline{\\mathrm {span}} \\lbrace e_n: n \\le 0\\rbrace $ , then $\\mathcal {M}$ is invariant for $W$ , so that $P^\\perp W P = 0$ .",
"However, $0 \\ne e_0 = W e_1 = P W P^\\perp e_1$ , so that $P W P^\\perp \\ne 0$ .",
"A fortiori, $W$ has neither property (CN) nor property (CR).",
"Furthermore, since the set of operators having property (CN) is clearly (norm-)closed, no operator close enough to $W$ has property (CN).",
"For $n \\ge 3$ , let $C_n \\in \\mathbb {M}_n(\\mathbb {C})$ denote an $n$ -cycle; that is, there is an orthonormal basis $\\lbrace e_k\\rbrace _{k=1}^n$ of $\\mathbb {C}^n$ , such that $C_n e_k = e_{k+1}$ , $1 \\le k \\le n-1$ , and $C_n e_n = e_1$ .",
"It follows easily from the results in [7] that there exists a sequence $(V_n)_{n=1}^\\infty $ of unitary operators with $V_n \\simeq C_n \\otimes I$ for all $n \\ge 1$ such that $\\lim _n V_n = W$ .",
"Thus $V_n$ does not have property (CN) for all sufficiently large $n$ .",
"That is, for sufficiently large $n$ , $C_n \\otimes I$ fails to have property (CN), despite the fact that $C_n \\in \\mathbb {M}_n(\\mathbb {C})$ has property (CN) by Proposition REF .",
"In fact, as we shall soon see, $C_n \\otimes I$ fails to have property (CN) for all $n \\ge 3$ ."
],
[
" Let us recall that the numerical range of $T \\in \\mathcal {B}( \\mathcal {H})$ is the set $W(T) = \\lbrace \\langle T x, x \\rangle : \\Vert x \\Vert = 1 \\rbrace $ , and the numerical radius of $T$ is $w(T) = \\sup \\lbrace |\\lambda |: \\lambda \\in W(T) \\rbrace $ .",
"It is known that the numerical range of an operator is always a convex set (this is the classical Toeplitz-Hausdorff Theorem), and that the closure of the numerical range of $T$ always contains $\\sigma (T)$ (see, for example, Problem 214 of [11]).",
"In trying to characterize operators with property (CN), we know that we may restrict our attention to normal operators.",
"For a normal $M$ , it is known that $\\overline{W(M)} = \\mathrm {co} (\\sigma (M))$ , that is, the closure of the numerical range of $M$ is the convex hull of the spectrum of $M$ .",
"If $\\sigma (M)$ happens to be finite, then (by the Spectral Theorem) $\\sigma (M)$ consists of the eigenvalues of $M$ , and these belong to the numerical range of $M$ , as does their convex hull.",
"Hence in such a case $W(M)=\\mathrm {co} (\\sigma (M))$ .",
"The numerical radius defines a norm on $\\mathcal {B}( \\mathcal {H})$ which is equivalent to the operator norm, because $\\frac{1}{2} \\Vert T \\Vert \\le w(T) \\le \\Vert T \\Vert $ for all $T \\in \\mathcal {B}( \\mathcal {H})$ .",
"(See, e.g.",
"[11], Chapter 22 for all of these results.)",
"A state on $\\mathcal {B}( \\mathcal {H})/\\mathcal {K}( \\mathcal {H})$ is a positive linear functional of norm one.",
"For $T \\in \\mathcal {B}( \\mathcal {H})$ , the essential numerical range $W_e(T)$ of $T$ is the set $\\lbrace \\varphi (\\pi (T)): \\varphi \\mbox{ is a state on } \\mathcal {B}( \\mathcal {H})/\\mathcal {K}( \\mathcal {H})\\rbrace $ .",
"It is known that $W_e(T)$ is closed and convex, and so it follows that $W_e(T)=\\mathrm {co} (\\sigma _{e}(T))$ , whenever $T$ is normal."
],
[
"Theorem.",
"(Fillmore-Stampfli-Williams; Theorem 5.1 of [10]) For $T \\in \\mathcal {B}( \\mathcal {H})$ , the following conditions are equivalent: $0 \\in W_e(T)$ .",
"There exists an orthonormal sequence $(e_n)_{n=1}^\\infty $ in $\\mathcal {H}$ such that $\\lim _n \\langle T e_n, e_n \\rangle = 0$ .",
"$0 \\in \\cap \\lbrace \\overline{W(T+F)} : F \\mbox{ is of finite rank}\\rbrace $ .",
"From this it easily follows that $W_e(T) = \\cap \\lbrace \\overline{W(T+F)} : F \\mbox{ is of finite rank}\\rbrace $ ."
],
[
"If $R \\in \\mathcal {B}(\\mathcal {K})$ where $\\mathcal {K}\\subseteq \\mathcal {H}$ is a subspace of $\\mathcal {H}$ , then $T$ is said to be a dilation of $R$ if, relative to the decomposition $\\mathcal {H}= \\mathcal {K}\\oplus \\mathcal {K}^\\perp $ , we may write $T = \\begin{bmatrix} R & B \\\\ C & D \\end{bmatrix} $ for some choice of $B, C$ and $D$ .",
"Recall that if $\\lbrace e_n \\rbrace _{n=1}^\\infty $ is an orthonormal basis for $\\mathcal {H}$ , then the unilateral forward shift on $\\mathcal {H}$ is the operator $S \\in \\mathcal {B}( \\mathcal {H})$ satisfying $S e_n = e_{n+1}$ for all $n \\ge 1$ .",
"The following result of Choi and Li will be useful."
],
[
"Theorem.",
"(Choi-Li; Theorem 4.3 of [6]) Suppose that $A \\in \\mathcal {B}( \\mathcal {H})$ , and $T \\in \\mathbb {M}_3(\\mathbb {C})$ has a non-trivial reducing subspace.",
"Then $A$ has a dilation that is unitarily equivalent to $T \\otimes I$ if and only if $W(A) \\subseteq W(T)$ ."
],
[
"Theorem.",
"Suppose that $V \\in \\mathbb {M}_3(\\mathbb {C})$ is a unitary operator and that 0 lies in the interior of $W(V)$ .",
"Then $V \\otimes I$ does not have property (CN).",
"Proof.",
"Clearly every unitary operator $V$ in $\\mathbb {M}_3(\\mathbb {C})$ has a non-trivial reducing subspace.",
"Note also that if $S \\in \\mathcal {B}( \\mathcal {H})$ is the unilateral forward shift, $\\mathrm {spr}(S) \\le \\Vert S \\Vert = 1$ .",
"(In fact, $\\mathrm {spr}(S) = 1$ , but that is not important here.)",
"Thus $\\mathrm {spr} (\\varepsilon S) \\le \\varepsilon $ for all $\\varepsilon > 0$ .",
"Since 0 lies in the interior of $W(V)$ , there exists $\\varepsilon _0 > 0$ such that $W(\\varepsilon _0 S) \\subseteq W(V)$ .",
"Let $A = \\varepsilon _0 S$ .",
"By Theorem REF above, we may write $V \\otimes I \\simeq \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix}.",
"$ Let $P = S S^*$ , so that $P$ is an orthogonal projection of co-rank one (i.e.",
"the rank of $(I-P)$ is equal to one).",
"In particular, $\\sigma (P) = \\lbrace 0, 1 \\rbrace $ .",
"Then, since $V \\otimes I$ is unitary, we have that $B B^* = I - A A^* = I - \\varepsilon _0^2 P \\\\C^* C = I - A^* A = I - \\varepsilon _0^2 I.$ It follows that $\\Vert B \\Vert ^2 = \\Vert B B^* \\Vert = \\Vert I - \\varepsilon _0^2 P \\Vert = 1, $ while $\\Vert C \\Vert ^2 = \\Vert C^* C \\Vert = \\Vert I - \\varepsilon _0^2 I \\Vert = 1-\\varepsilon _0^2.",
"$ In particular, $\\Vert B \\Vert \\ne \\Vert C \\Vert $ , so that $V \\otimes I$ does not have property (CN).",
"$\\Box $ It follows from Theorem REF that $C_n \\otimes I$ does not have property (CN) for $n \\ge 3$ , since in such a case 0 lies in the interior of $W(C_n)$ ."
],
[
"Corollary.",
"Suppose that $U \\in \\mathcal {B}( \\mathcal {H})$ is a unitary operator that has property (CN).",
"Then 0 does not lie in the interior of $W_e(U)$ .",
"Proof.",
"We prove a contrapositive implication.",
"Suppose that 0 lies in the interior of the essential numerical range of $U$ , that is in the interior of the convex hull of the essential spectrum of $U$ .",
"Then there exist $\\alpha , \\beta $ and $\\gamma $ in the essential spectrum of $U$ such that 0 lies in the interior of the convex hull of $\\lbrace \\alpha , \\beta , \\gamma \\rbrace $ .",
"Let $V \\in \\mathbb {M}_3(\\mathbb {C})$ be a unitary operator with spectrum $\\lbrace \\alpha , \\beta , \\gamma \\rbrace $ .",
"Then, as noted in Section REF , $W(V)$ is closed and $W(V) = \\mathrm {co} \\lbrace \\alpha , \\beta , \\gamma \\rbrace $ .",
"Hence 0 lies in the interior of $W(V)$ .",
"By Theorem REF , $V \\otimes I$ does not have property (CN).",
"Since $\\alpha , \\beta $ and $\\gamma $ lie in the essential spectrum of the normal operator $U$ , $U$ is approximately unitarily equivalent to $U \\oplus (V \\otimes I)$ .",
"(This is a consequence of the Weyl-von Neumann-Berg Theorem for normal operators – see, e.g.",
"Theorem II.4.4 of [8] – and can also be deduced from the results of [7].)",
"It now follows from Proposition REF (b) that $U$ does not have property (CN).",
"$\\Box $"
],
[
"Theorem.",
"Suppose that $U \\in \\mathcal {B}( \\mathcal {H})$ is a unitary operator and 0 does not lie in $W_e(U)$ .",
"Then $U$ has property (CN).",
"Proof.",
"We prove a contrapositive implication.",
"Let $U$ be a unitary operator which fails to have property (CN).",
"Then there exists a projection $P \\in \\mathcal {B}( \\mathcal {H})$ such that with respect to the decomposition $\\mathcal {H}= P\\mathcal {H}\\oplus P^\\perp \\mathcal {H}$ , we may write $U = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} $ where $\\Vert B \\Vert \\ne \\Vert C \\Vert $ .",
"As noted in Remark REF , this can only happen if one of the following holds: either $0 \\in \\sigma (A A^*) $ but $0 \\notin \\sigma (A^* A)$ , $0 \\in \\sigma (A^* A)$ but $0 \\notin \\sigma (A A^*)$ .",
"Since an operator $T$ has property (CN) if and only if $T^*$ has property (CN), by replacing $U$ by $U^*$ if necessary (which does not affect the conclusion, as $0 \\in W_e(U)$ if and only if $0 \\in W_e(U^*)$ ), we may assume without loss of generality that $0 \\in \\sigma (A A^*)$ but $0 \\notin \\sigma (A^* A)$ .",
"In particular, $A$ is not invertible.",
"Since $A^* A$ is invertible, we see that $A$ is bounded below.",
"Hence $\\mathrm {ran}\\, A$ is closed and $\\mathrm {nul}\\, A = 0$ .",
"Therefore $A$ is semi-Fredholm.",
"If $\\mathrm {nul}\\, A^* = 0$ , then $\\mathrm {ran}\\, A = P\\mathcal {H}$ , so that $A$ is invertible, which is a contradiction.",
"Thus $\\mathrm {nul}\\, A^* > 0$ , and so $\\mathrm {ind}\\, A < 0$ .",
"Since $\\mathrm {ind}\\, (A+F) = \\mathrm {ind}\\, A < 0$ for all finite-rank operators $F \\in \\mathcal {B}(P \\mathcal {H})$ , and since $\\sigma (T) \\subseteq \\overline{W(T)}$ for all operators $T$ , we may apply Theorem REF of Fillmore, Stampfli and Williams to obtain: $0&\\in \\cap \\lbrace \\sigma (A+F) : F \\in \\mathcal {B}(P \\mathcal {H}), \\ F \\mbox{ finite-rank}\\rbrace \\\\&\\subseteq \\cap \\lbrace \\overline{W(A+F)}: F \\in \\mathcal {B}(P \\mathcal {H}),\\ F \\mbox{ finite-rank} \\rbrace \\\\&= W_e(A),$ Since $W_e(A) \\subseteq W_e(U)$ , the result follows.",
"$\\Box $"
],
[
"Theorem.",
"Suppose that $U \\in \\mathcal {B}( \\mathcal {H})$ is unitary and that 0 lies on the boundary of $W_e(U)$ .",
"Then $U$ has property (CN).",
"Proof.",
"The hypotheses of the theorem imply that $\\sigma _e(U)$ lies on a closed half-circle of $\\mathbb {T}$ , and includes two diametrically opposite points.",
"By multiplying $U$ by an appropriate $\\mu \\in \\mathbb {T}$ (which does not affect the conclusion of the Theorem), we may assume without loss of generality that $\\sigma _e(U) \\subseteq \\mathbb {T}\\cap \\lbrace z\\in \\mathbb {C}: \\mathrm {Re}{z} \\ge 0\\rbrace $ , and that $\\lbrace i, -i \\rbrace \\subseteq \\sigma _e(U)$ .",
"For each $n \\ge 1$ , let $\\mathcal {C}_n = \\lbrace z \\in \\mathbb {T}: z= e^{i \\theta }, \\frac{\\pi }{2} - \\frac{1}{n} \\le \\theta \\le \\frac{\\pi }{2} + \\frac{1}{n}\\rbrace $ , and let $Q_n$ be the spectral projection for $U$ corresponding to $\\mathcal {C}_n$ .",
"Then $U = X_n + Y_n$ , where $X_n = U Q_n^\\perp $ , and where $Y_n = U Q_n$ is a unitary with $\\sigma (Y_n) \\subseteq \\mathcal {C}_n$ .",
"Set $V_n = X_n + e^{i (\\frac{\\pi }{2}-\\frac{1}{n})} Q_n$ .",
"It is reasonably straightforward to check that $\\Vert U - V_n \\Vert = \\Vert Y_n - e^{i (\\frac{\\pi }{2}-\\frac{1}{n})} Q_n \\Vert \\le \\frac{4 \\pi }{n}$ and that $\\sigma _e(V_n) \\subseteq \\Omega _n = \\lbrace z= e^{i \\theta } \\in \\mathbb {T}: -\\frac{\\pi }{2} \\le \\theta \\le \\frac{\\pi }{2} - \\frac{1}{n} \\rbrace $ .",
"Thus 0 does not lie in the closed, convex hull of $\\sigma _e(V_n)$ , and in particular, 0 does not lie in $W_{e}(V_n)$ for any $n \\ge 1$ .",
"By Theorem REF , $V_n$ has property (CN).",
"But as we saw in Proposition REF , the set $\\mathfrak {G_{norm}}$ of operators with property (CN) is closed, and thus $U$ has property (CN).",
"$\\Box $ Combining these results, and keeping in mind that $W_e(T)=\\mathrm {co} (\\sigma _{e}(T))$ for normal $T$ , we obtain the following."
],
[
"Corollary.",
"The following are equivalent for a unitary operator $U\\in \\mathcal {B}( \\mathcal {H})$ .",
"$U$ has property (CN).",
"0 does not lie in the interior of $W_e(U)$ .",
"There exists a half-circle $\\mathcal {C}$ of $\\mathbb {T}$ such that $\\sigma _e(U) \\subseteq \\mathcal {C}$ (i.e.",
"there exists $\\mu \\in \\mathbb {T}$ such that $\\sigma _e(U) \\subseteq \\mathbb {T}\\cap \\lbrace \\mu z \\in \\mathbb {C}: \\mathrm {Re}(z) \\ge 0\\rbrace )$ .",
"We are now ready to state and prove the main theorem of this section."
],
[
"Theorem.",
"Let $T \\in \\mathcal {B}( \\mathcal {H})$ .",
"The following conditions are equivalent.",
"$T$ has property (CN).",
"One of the following holds.",
"There exist $\\lambda , \\mu \\in \\mathbb {C}$ and $L=L^* \\in \\mathcal {B}( \\mathcal {H})$ such that $T = \\lambda I + \\mu L$ .",
"There exist $\\lambda , \\mu \\in \\mathbb {C}$ with $\\mu \\ne 0$ and a unitary operator $U \\in \\mathcal {B}( \\mathcal {H})$ with $\\sigma _e(U) \\subseteq \\mathbb {T}\\cap \\lbrace z \\in \\mathbb {C}: \\mathrm {Re} (z) \\ge 0\\rbrace $ such that $T = \\lambda I + \\mu U$ .",
"Proof.",
"Suppose first that (a) holds, and recall that this implies that $T$ is normal.",
"If $\\sigma (T)$ has at most three points, then those points are either co-linear or co-circular.",
"It is routine to check from this that $T$ is either of the form of (b) (i), or there exist $\\lambda , \\mu \\in \\mathbb {C}$ with $\\mu \\ne 0$ and a unitary $U\\in \\mathcal {B}( \\mathcal {H})$ such that $T= \\lambda I +\\mu U$ .",
"In this case, since $\\mu \\ne 0$ , $T$ has property (CN) if and only if $U$ has property (CN).",
"But then $U$ has property (CN) by our hypothesis on $T$ , and so the spectral conditions on $U$ follow from Corollary REF .",
"Thus we suppose that $\\sigma (T)$ has cardinality at least 4, and we let $\\lbrace \\alpha , \\beta , \\gamma , \\delta \\rbrace $ be four distinct points in $\\sigma (T)$ .",
"By Theorem II.4.4 in [8], $T$ is approximately unitarily equivalent to an operator of the form $A \\oplus D$ , where $D = \\mathrm {diag} (\\alpha , \\beta , \\gamma , \\delta ) \\in \\mathbb {M}_4(\\mathbb {C})$ .",
"By Proposition REF (b), $D$ has property (CN).",
"By Corollary REF , the eigenvalues of $D$ are either co-linear or co-circular.",
"Since this is true for any choice of four distinct points of $\\sigma (T)$ , we see that $\\sigma (T)$ is either contained in a line – in which case it is easily seen that there exist $\\lambda , \\mu $ and $L$ as in (b) (i) such that $T = \\lambda I + \\mu L$ , or $\\sigma (T)$ lies on a proper circle, i.e.",
"there exist $\\lambda , \\mu \\in \\mathbb {C}$ with $\\mu \\ne 0$ and a unitary $U\\in \\mathcal {B}( \\mathcal {H})$ such that $T= \\lambda I +\\mu U$ .",
"We argue as in the previous paragraph to obtain the spectral conditions on $U$ .",
"Suppose next that (b) holds.",
"If there exist $\\lambda , \\mu \\in \\mathbb {C}$ and $L=L^* \\in \\mathcal {B}( \\mathcal {H})$ such that $T = \\lambda I + \\mu L$ , then $T$ has property (CN) by Proposition REF (d).",
"If there exist $\\lambda , \\mu \\in \\mathbb {C}$ with $\\mu \\ne 0$ and a unitary operator $U \\in \\mathcal {B}( \\mathcal {H})$ with $\\sigma _e(U) \\subseteq \\mathbb {T}\\cap \\lbrace z \\in \\mathbb {C}: \\mathrm {Re} (z) \\ge 0\\rbrace $ such that $T = \\lambda I + \\mu U$ .",
"Then $T$ has property (CN) if and only if $U$ has property (CN) by Proposition REF (a).",
"But $U$ has property (CN) by Corollary REF , whence $T$ has property (CN).",
"$\\Box $"
],
[
" It is natural to consider a weakening of the property (CN) obtained by restricting our attention to the finite-rank projections $P\\in \\mathcal {B}( \\mathcal {H})$ in the case when $\\mathcal {H}$ is infinite-dimensional.",
"As the reader can easily check, the results and proofs presented in this section readily demonstrate that such a “weakening” of the property (CN) is in fact equivalent to the original property (CN)."
],
[
"We next turn our attention to the study of operators with property (CR), acting on an infinite-dimensional Hilbert space.",
"Although we have not been able to obtain a complete classification of such operators, we will mention a number of interesting facts.",
"We recall from above that an operator $X \\in \\mathcal {B}( \\mathcal {H})$ is said to be (orthogonally) reductive if for each projection $P \\in \\mathcal {B}( \\mathcal {H})$ , the condition $P T P^\\perp = 0$ implies that $P^\\perp T P = 0$ .",
"It is clear that if $X$ has property (CR), then $X$ must be reductive.",
"It should be noted that not every normal operator is reductive.",
"Sarason [18] has shown that a normal operator $N$ is reductive if and only if $N^*$ lies in the weak operator topology closure of the set of polynomials in $N$ .",
"As a concrete example, let $W \\in \\mathcal {B}( \\mathcal {H})$ be the bilateral shift; i.e.",
"let $\\lbrace e_n \\rbrace _{n\\in \\mathbb {Z}}$ be an orthonormal basis for $\\mathcal {H}$ and let $W$ be defined by $W e_n = e_{n-1}$ for all $n \\in \\mathbb {Z}$ .",
"It is well-known that $W$ is unitary with $\\sigma (W) = \\mathbb {T}$ .",
"If $P$ is the orthogonal projection onto $\\mathcal {M}= \\overline{\\mathrm {span}} \\lbrace e_n : n \\le 0\\rbrace $ , then clearly $\\mathcal {M}$ is invariant for $W$ – so $\\mathrm {rank}\\, P^\\perp W P = 0$ , but it is easily verified that $P W P^\\perp $ has rank 1.",
"Thus $W$ fails to be reductive.",
"The condition that an operator have property (CR) is strictly stronger than asking that it be orthogonally reductive – see Example REF below.",
"It is worth observing that there is one inherent weakness in the definition of orthogonally reductive operators: it is entirely possible that there might exist an operator with no non-trivial closed, invariant subspace, in which case the operator is reductive for trivial reasons.",
"On the other hand, it was shown by Popov and Tcaciuc [16] that given any operator $T$ acting on an infinite-dimensional, complex, separable Hilbert space $\\mathcal {H}$ , there exists an orthogonal projection $P$ of infinite rank and co-rank such that $\\mathrm {rank}\\, P T P^\\perp \\le 1$ .",
"(Their result actually holds for operators acting on reflexive Banach spaces and beyond, but we do not require that here.)",
"As such, property (CR) always has significance for Hilbert space operators.",
"We begin with some observations regarding the general class ${\\mathfrak {G}_{\\mathfrak {rank}}}$ of operators with property (CR)."
],
[
"Proposition.",
"Suppose that $T \\in \\mathcal {B}( \\mathcal {H})$ has property (CR).",
"Then $T$ is biquasitriangular; that is, $\\mathrm {ind}\\, (T-\\lambda I) = 0$ for all $\\lambda \\in \\varrho _{sF}(T)$ .",
"Proof.",
"It is clear that if $\\lambda \\in \\mathbb {C}$ , then $T - \\lambda I$ and $(T-\\lambda I)^*$ also have property (CR).",
"Suppose that $\\lambda \\in \\varrho _{sF}(T)$ and that $\\mathrm {ind}(T-\\lambda I) \\ne 0$ .",
"By considering $(T-\\lambda I)^*$ if necessary, we may assume that $\\mathrm {ind}\\, (T-\\lambda I) > 0$ (it is possibly infinite).",
"Thus $\\mathrm {nul}\\, (T-\\lambda I) > \\mathrm {nul}\\, (T-\\lambda I)^*$ .",
"Write $\\mathcal {H}= \\ker \\, (T-\\lambda I) \\oplus (\\ker \\, (T-\\lambda I))^\\perp $ , and write $(T-\\lambda I) = \\begin{bmatrix} 0 & B \\\\ 0 & D \\end{bmatrix} $ relative to this decomposition.",
"If $B = 0$ , then $(T-\\lambda I)^* = \\begin{bmatrix} 0 & 0 \\\\ 0 & D^* \\end{bmatrix}$ , showing that $\\mathrm {nul}\\, (T-\\lambda I)^* \\ge \\mathrm {nul}\\, (T-\\lambda I)$ , a contradiction.",
"But then $B \\ne 0$ implies that $T-\\lambda I$ does not have property (CR), and hence neither does $T$ .",
"The contrapositive is the statement that if $T$ has property (CR), then $T$ is biquastriangular.",
"$\\Box $"
],
[
"It is clear that if $T$ has property (CR) and $\\lambda $ is an eigenvalue for $T$ , then it is a reducing eigenvalue for $T$ ; that is, we may write $T \\simeq \\lambda Q \\oplus T_0$ , where $Q$ is an orthogonal projection and $\\lambda $ is no longer an eigenvalue for $T_0$ (though it may be an approximate eigenvalue for $T_0$ ).",
"By repeating this for each of the eigenvalues of $T$ , this allows us to write $T \\simeq M \\oplus T_4$ , where $M$ is a diagonal operator whose eigenvalues are precisely the eigenvalues of $T$ , and where $T_4$ has no eigenvalues.",
"Since direct summands of operators with property (CR) still have property (CR), it follows from the results of Section that all of the spectrum of $M$ is either co-linear or co-circular.",
"That is, the eigenvalues of any operator $T$ with property (CR) are either co-linear or co-circular, and they are reducing eigenvalues for $T$ .",
"Our next goal is to prove that every operator which satisfies property (CR) is normal with circlinear spectrum.",
"We shall accomplish this through a sequence of lemmas.",
"It is worth noting that we shall not invoke the full strength of the property (CR) hypothesis.",
"Indeed, for the next few results, we only require a weaker form of property (CR) that requires that $T$ be reductive and that if $P$ is a projection for which $\\mathrm {rank}\\, P^\\perp T P = 1$ , then $\\mathrm {rank}\\, P T P^\\perp = 1$ .",
"It can in fact be shown that Proposition REF also holds under this weaker hypothesis, though we shall not need that here."
],
[
"Proposition.",
"Let $T \\in \\mathcal {B}( \\mathcal {H})$ and suppose that $T$ has property (CR).",
"Then there exist $\\alpha , \\beta , \\gamma $ and $\\delta \\in \\mathbb {C}$ , not all equal to zero, and an operator $F \\in \\mathcal {B}( \\mathcal {H})$ of rank at most three such that $\\alpha I + \\beta T + \\gamma T^* + \\delta T^* T + F = 0.",
"$ Proof.",
"Fix $0 \\ne \\xi \\in \\mathcal {H}$ .",
"We first claim that the set $S_\\xi = \\lbrace \\xi , T \\xi , T^* \\xi , T^* T \\xi \\rbrace $ is linearly dependent.",
"Let $\\mathcal {M}_\\xi = \\mathrm {span}\\, \\lbrace \\xi , T \\xi \\rbrace $ .",
"If $\\dim \\, \\mathcal {M}_\\xi = 1$ , then clearly $\\lbrace \\xi , T \\xi \\rbrace $ is linearly dependent, whence $S_\\xi $ is linearly dependent and we are done.",
"Suppose therefore that $\\dim \\, \\mathcal {M}_\\xi = 2$ and let $P_\\xi $ denote the orthogonal projection of $\\mathcal {H}$ onto $\\mathcal {M}_\\xi $ .",
"Note that $T \\xi \\in \\mathcal {M}_\\xi $ implies that $\\mathrm {rank}\\, P_\\xi ^\\perp T P_\\xi \\in \\lbrace 0, 1\\rbrace $ .",
"From our hypothesis, $\\mathrm {rank}\\, P_\\xi ^\\perp T^* P_\\xi \\in \\lbrace 0, 1\\rbrace $ .",
"But then $\\dim \\, (P_\\xi \\mathcal {H}+ P_\\xi ^\\perp T^* P_\\xi \\mathcal {H})&\\le \\dim \\, (P_\\xi \\mathcal {H}) + \\dim \\, (P_\\xi ^\\perp T^* P_\\xi \\mathcal {H}) \\\\&\\le 2 + 1 =3.$ Since $S_\\xi \\subseteq P_\\xi \\mathcal {H}+P_\\xi ^\\perp T^* P_\\xi \\mathcal {H}$ , our claim follows.",
"As $0 \\ne \\xi \\in \\mathcal {H}$ was arbitrary, we see that the set $\\lbrace I, T, T^*, T^* T \\rbrace $ is locally linearly dependent in the sense of [1], [4] and [14].",
"By Theorem 2 of [4], there exist $\\alpha , \\beta , \\gamma ,$ and $\\delta \\in \\mathbb {C}$ , not all equal to zero, such that $\\mathrm {rank}\\, (\\alpha I + \\beta T + \\gamma T^* + \\delta T^* T) \\le 3.",
"$ This clearly implies the statement of the theorem.",
"$\\Box $ We begin by dealing with the case where $\\delta $ above is equal to zero."
],
[
"Lemma.",
"Let $\\mathcal {H}$ be a complex Hilbert space and suppose that $T \\in \\mathcal {B}( \\mathcal {H})$ .",
"If there exist complex numbers $\\alpha , \\beta $ and $\\gamma $ , not all equal to zero, and $F \\in \\mathcal {B}( \\mathcal {H})$ of rank at most $m < \\frac{1}{2} \\, \\dim \\, \\mathcal {H}$ such that $\\alpha I + \\beta T + \\gamma T^* + F = 0, $ then there exist a hermitian operator $R$ , a finite-rank operator $L$ of rank at most $2 m$ , and $\\mu , \\lambda \\in \\mathbb {C}$ such that $T = \\lambda (R + L) + \\mu I.",
"$ Proof.",
"Case 1.",
"Suppose that $\\gamma = 0$ .",
"In this case, we have that $\\alpha I + \\beta T + F = 0$ .",
"If $\\beta = 0$ , then the fact that $F$ has finite rank $\\mathrm {rank}\\, F = m < \\dim \\, \\mathcal {H}= \\mathrm {rank}\\, I$ implies that $\\alpha = 0 (= \\beta = \\gamma )$ , contradicting our hypothesis.",
"Hence $\\beta \\ne 0$ .",
"But then $T = - \\alpha \\beta ^{-1} I - \\beta ^{-1} F. $ If $\\alpha = 0$ , then $T = - \\beta ^{-1} F = -\\beta ^{-1} (0 + F) + 0 I $ expresses $T$ in the desired form.",
"If $\\alpha \\ne 0$ , then writing $T = - \\alpha \\beta ^{-1} (I - \\alpha ^{-1} F) + 0 I $ expresses $T$ in the desired form.",
"Case 2.",
"Suppose that $\\beta = 0$ .",
"Then $\\alpha I + \\gamma T^* + F = 0$ , and arguing as before, $\\gamma \\ne 0$ .",
"Thus $T ^* = - \\alpha \\gamma ^{-1} I - \\gamma ^{-1} F$ .",
"If $\\alpha = 0$ , then $T^* = -\\gamma ^{-1} (0 + F) + 0 I $ means that $T = -\\overline{\\gamma }^{-1} (0 + F^*) + 0 I $ expresses $T$ in the desired form.",
"If $\\alpha \\ne 0$ , then writing $T^* = - \\alpha \\gamma ^{-1} (I - \\alpha ^{-1} F) + 0 I $ means that $T = -\\overline{\\alpha } \\overline{\\gamma }^{-1} (I - \\overline{\\alpha }^{-1} F^*) + 0 I $ expresses $T$ in the desired form.",
"Case 3.",
"Suppose that $\\beta \\ne 0 \\ne \\gamma $ .",
"We have that $\\alpha I + \\beta T + \\gamma T^* + F = 0$ , whence $\\overline{\\alpha } I + \\overline{\\beta } T^* + \\overline{\\gamma } T + F^* = 0$ .",
"Set $\\varrho = (\\alpha + \\overline{\\alpha })$ , $\\theta = \\beta + \\overline{\\gamma }$ and $F_0 = F + F^*$ .",
"Adding the two previous equations involving $T$ yields: $\\varrho I + \\theta T + \\overline{\\theta } T^* + F_0 = 0, $ and $\\mathrm {rank}\\, F_0 \\le 2 \\ \\mathrm {rank}\\, F \\le 2 m < \\dim \\, \\mathcal {H}.$ Subcase 3.A.",
"$\\theta = 0$ .",
"If $\\theta = 0$ , then $\\varrho I + F_0 = 0$ , combined with the fact that $\\dim \\, \\mathcal {H}> \\mathrm {rank}\\, F_0$ implies that $\\varrho = 0 = F_0$ .",
"That is, $\\alpha \\in i \\mathbb {R}$ and $\\gamma = -\\overline{\\beta } \\ne 0$ , so that $\\alpha I + \\beta T - \\overline{\\beta } T^* + F = 0.",
"$ Let $A = \\beta T + \\frac{\\alpha }{2} I$ , and let $A = R+ i B$ be the Cartesian decomposition of $A$ , so that $R$ and $B$ are hermitian.",
"The above equation shows that $0 = (A - A^*) + F = 2 i B + F$ , and thus $B$ has finite rank at most $m$ and $T = \\beta ^{-1} (R + i B) - \\frac{\\alpha \\beta ^{-1}}{2} I $ expresses $T$ in the desired form.",
"Subcase 3.B.",
"$\\theta \\ne 0$ .",
"We have $\\varrho I + \\theta T + \\overline{\\theta } T^* + F_0 = 0, $ where $\\mathrm {rank}\\, F_0 \\le 2 m$ and $\\varrho \\in \\mathbb {R}$ .",
"Let $\\displaystyle \\kappa = \\frac{\\varrho i}{2 | \\theta |^2} \\in i \\mathbb {R}$ , and $A = (\\overline{\\theta } i)^{-1} T - \\kappa I$ .",
"Then $T = \\overline{\\theta } i(A + \\kappa I)$ and our equation $\\varrho I + \\theta T + \\overline{\\theta } T^* + F_0 = 0$ implies that $0 = |\\theta |^2 i (A - A^*) + F_0.",
"$ In particular, $A - A^*$ has rank at most $2 m$ .",
"Again, we write $A = R + i B$ where $R = (A+A^*)/2$ and $B = (A-A^*)/2i$ .",
"Then $B$ has rank at most $\\mathrm {rank}\\, F_0 \\le 2m$ and $T = \\overline{\\theta } i(R+i B) + \\overline{\\theta } i \\kappa I $ expresses $T$ in the desired form.",
"$\\Box $"
],
[
"Lemma.",
"Let $\\mathcal {H}$ be a complex Hilbert space and suppose that $T \\in \\mathcal {B}( \\mathcal {H})$ satisfies $\\mathrm {rank}\\, (\\alpha I + \\beta T + \\gamma T^* + \\delta T^* T) \\le 3 $ for some $\\alpha , \\beta , \\gamma , \\delta \\in \\mathbb {C}$ , where $\\delta \\ne 0$ .",
"Then there exist a unitary operator $V$ , a finite-rank operator $L$ of rank at most 12, and $\\mu , \\lambda \\in \\mathbb {C}$ such that $T = \\lambda (V+ L) + \\mu I.",
"$ Proof.",
"It is clear that there is no loss of generality in assuming that $\\delta = \\frac{1}{2}$ .",
"Choose $F \\in \\mathcal {B}( \\mathcal {H})$ with $\\mathrm {rank}\\, F \\le 3$ such that $\\alpha I + \\beta T + \\gamma T^* + \\frac{1}{2} T^* T + F = 0.$ This trivially implies that $\\overline{\\alpha } I + \\overline{\\gamma } T + \\overline{\\beta } T^* + \\frac{1}{2} T^* T + F^* = 0$ .",
"As before, we set $\\varrho = \\alpha + \\overline{\\alpha }$ , $\\theta = \\beta + \\overline{\\gamma }$ and $F_0 = F + F^*$ .",
"Then $\\varrho I + \\theta T + \\overline{\\theta } T + T^* T + F_0 = 0.",
"$ A routine calculation shows that $(\\varrho - |\\theta |^2) I + (T + \\overline{\\theta }I)^*(T+ \\overline{\\theta } I) + F_0 = 0.",
"$ Of course, $(T + \\overline{\\theta }I)^*(T+ \\overline{\\theta } I) \\ge 0$ , and so - by considering this equation modulo the compact operators, we conclude that $|\\theta |^2 - \\varrho \\ge 0$ .",
"We again consider two cases.",
"Case 1.",
"$| \\theta |^2 = \\varrho $ .",
"Then $(T + \\overline{\\theta }I)^*(T+ \\overline{\\theta } I) = -F_0.",
"$ But then $|T + \\overline{\\theta } I|$ has finite rank, so that $G= T + \\overline{\\theta } I$ has finite rank.",
"That is, $T = -\\overline{\\theta } (I - \\overline{\\theta }^{-1} G) + 0 I $ expresses $T$ in the desired form.",
"Case 2.",
"$| \\theta |^2 > \\varrho $ .",
"Set $V_0 = (|\\theta |^2 - \\varrho )^{-1/2} (T + \\overline{\\theta } I)$ and $F_2= (|\\theta |^2 - \\varrho )^{-1} F_0$ .",
"Then $V_0^* V_0&= (|\\theta |^2 - \\varrho )^{-1} (T+\\overline{\\theta } I)^* (T + \\overline{\\theta }I) \\\\&= (|\\theta |^2 - \\varrho )^{-1} ( (|\\theta |^2-\\varrho )I - F_0) \\\\&= I - F_2.$ That is, $\\pi (V_0)$ is an isometry in the Calkin algebra.",
"In particular, $V_0$ is semi-Fredholm, and therefore $(T+\\overline{\\theta } I)$ is semi-Fredholm.",
"But $T + \\overline{\\theta } I$ has property (CR), so $T + \\overline{\\theta } I$ is biquasitriangular, by Proposition REF .",
"Hence $V_0$ is Fredholm with index 0.",
"Using the polar decomposition and the fact that $V_0$ has index 0, we may find a unitary operator $U$ such that $V_0 = U |V_0| = U (I-F_2)^{1/2}$ .",
"Thus $U - V_0$ is of finite rank, and $V_0 = U - (U - V_0) = (|\\theta |^2 - \\varrho )^{-1/2} (T + \\overline{\\theta } I).",
"$ In other words, $T = (|\\theta |^2 - \\varrho )^{1/2} (U + (V_0-U)) - \\overline{\\theta } I $ again expresses $T$ in the desired form.",
"$\\Box $"
],
[
"Proposition.",
"Let $\\mathcal {H}$ be an infinite-dimensional complex Hilbert space and $F \\in \\mathcal {B}( \\mathcal {H})$ be a finite-rank operator.",
"Suppose that $V \\in \\mathcal {B}( \\mathcal {H})$ is unitary.",
"If $W = V + F$ has property (CR), then $W$ is normal and $\\sigma (W)$ is circlinear.",
"Suppose that $R \\in \\mathcal {B}( \\mathcal {H})$ is hermitian.",
"If $L = R+F$ has property (CR), then $R$ is normal and $\\sigma (L)$ is circlinear.",
"Proof.",
"It is obvious that $I - W^* W$ is of finite rank.",
"By Corollary 6.17 of [17], $W$ must have a non-trivial invariant subspace, which must – by virtue of property (CR) – in fact be an orthogonally reducing subspace for $W$ .",
"Thus we may write $W \\simeq W_1 \\oplus W_2$ , and it is clear that each $W_k$ must satisfy property (CR) (by Proposition REF ) and be of the form $V_k + F_k$ for some unitary operator $V_k$ and some finite-rank operator $F_k$ , $k = 1,2$ .",
"At least one of these summands acts on an infinite-dimensional space, and thus we may again apply Theorem 6.17 of [17] to find non-trivial invariant – hence reducing – subspaces for that summand.",
"Repeating this process, we see that for any $n \\ge 1$ , we can find $n$ summands $X_{n, 1}, X_{n,2}, \\ldots ,$ $X_{n, n}$ of $W$ such that $W = X_{n,1} \\oplus X_{n,2} \\oplus \\cdots \\oplus X_{n,n}.",
"$ Furthermore, a moments' thought will convince the reader that at most $\\mathrm {rank}\\, F$ of these summands can fail to be unitary themselves, and hence when $n > \\mathrm {rank}\\, F$ , at least one of the $X_{n, k}$ 's is a unitary operator.",
"Let $\\ \\ \\ \\ \\ \\ \\ \\ \\mathcal {J}= \\lbrace (U, \\mathcal {M}): U \\mbox{ is a unitary direct summand of } W \\mbox{ acting on the subspace } \\mathcal {M}\\mbox{ of } \\mathcal {H}\\rbrace .",
"$ The above paragraph shows that $\\mathcal {J}$ is non-empty.",
"Partially order $\\mathcal {J}$ by setting $(U_1, \\mathcal {M}_1) \\le (U_2, \\mathcal {M}_2)$ if $\\mathcal {M}_1 \\le \\mathcal {M}_2$ .",
"(Note that this automatically implies that $U_1$ is a direct summand of $U_2$ .)",
"If $\\mathcal {C}= \\lbrace (U_\\nu , \\mathcal {M}_\\nu ) : \\nu \\in \\Gamma \\rbrace $ is a chain in $\\mathcal {C}$ , then by setting $\\mathcal {M}= \\overline{\\cup _{\\nu \\in \\Gamma } \\mathcal {M}_\\nu }$ , we see that $\\mathcal {M}$ is a reducing subspace for $W$ (as each $\\mathcal {M}_\\nu $ is), and $U = W|_{\\mathcal {M}}$ is unitary (since it is clearly unitary on the dense submanifold $\\cup _{\\nu \\in \\Gamma } \\mathcal {M}_\\nu $ of $\\mathcal {M}$ ).",
"It follows from Zorn's Lemma that $\\mathcal {J}$ admits a maximal element $(U_0, \\mathcal {M}_0)$ .",
"If $\\mathcal {M}_0^\\perp $ is infinite-dimensional, then the argument of the first two paragraphs can be used to show that $W|_{\\mathcal {M}_0^\\perp }$ admits a unitary direct summand, contradicting the maximality of $(U_0, \\mathcal {M}_0)$ .",
"Thus $m= \\dim \\, \\mathcal {M}_0^\\perp < \\infty $ .",
"Write $W = U_0 \\oplus Y$ , where $Y$ acts on $\\mathcal {M}_0^\\perp $ , and note that $Y$ has property (CR).",
"We may view $Y$ as an element of $\\mathbb {M}_m(\\mathbb {C})$ , so that $Y$ can be upper triangularized with respect to some orthonormal basis.",
"The fact that $Y$ has property (CR) implies that it is reductive, and is therefore normal.",
"This forces $W$ to be normal as well.",
"There remains to show that $\\sigma (W)$ is circlinear.",
"Note that if $\\sigma (W)$ is finite, then all elements of $\\sigma (W)$ are eigenvalues, and so $\\sigma (W)$ is circlinear by the comments of Section REF .",
"Hence we may assume that $\\sigma (W)$ is infinite, which is equivalent to assuming that $\\sigma (U_0)$ is infinite.",
"In this case, we shall prove that $W$ is unitary.",
"We argue by contradiction.",
"Suppose otherwise, and let $\\tau \\in \\sigma (Y)$ with $|\\tau | \\ne 1$ .",
"Let $\\mathcal {N}\\subseteq \\ker \\, (W - \\tau I) \\subseteq \\mathcal {M}_0^\\perp $ be a one-dimensional subspace.",
"We see that the operator $Z = U_0 \\oplus \\tau $ , being a direct summand of $W$ , also satisfies property (CR).",
"With respect to the decomposition $\\mathcal {M}_0 \\oplus \\mathcal {N}$ , we may write $Z = \\begin{bmatrix} U_0 & 0 \\\\ 0 & \\tau \\end{bmatrix}.",
"$ Let $x \\in \\mathcal {M}_0$ be a unit vector such that $\\lbrace x, U_0 x, U_0^2 x\\rbrace $ is linearly independent.",
"Such a vector must exist, otherwise $U_0$ is boundedly locally linearly dependent, which – by Kaplansky's Theorem [13], Lemma 14 – implies that $U_0$ is algebraic, and therefore has finite spectrum, a contradiction of our current assumption.",
"Thus $\\lbrace U_0^* x, x, U_0 x \\rbrace $ is again linearly independent, as $U_0$ is unitary.",
"We shall now find vectors $y$ and $z$ in $\\mathcal {M}_0 \\oplus \\mathcal {N}$ such that $\\mathcal {E}_1 = \\lbrace y, z, Zy, Zz \\rbrace $ is linearly independent, but $\\mathcal {E}_2 = \\lbrace y, z, Z^*y, Z^* z \\rbrace $ is not.",
"This will yield the desired contradiction, by implying that $(I-P)Z P$ and $(I-P)Z^* P$ have ranks two and one respectively.",
"Let $y = \\begin{bmatrix} x \\\\ 1 \\end{bmatrix}$ and $z = \\begin{bmatrix} U_0 x \\\\\\xi \\end{bmatrix}$ , with $\\xi \\in \\mathbb {C}$ to be determined shortly.",
"(Here, we have identified $\\mathcal {N}$ with $\\mathbb {C}$ .)",
"Now $\\mathcal {E}_1 = \\left\\lbrace \\begin{bmatrix} x \\\\ 1 \\end{bmatrix}, \\begin{bmatrix} U_0 x \\\\\\xi \\end{bmatrix}, \\begin{bmatrix} U_0 x \\\\ \\tau \\end{bmatrix}, \\begin{bmatrix} U_0^2 x \\\\ \\tau \\xi \\end{bmatrix} \\right\\rbrace , $ and $\\mathcal {E}_2 = \\left\\lbrace \\begin{bmatrix} x \\\\ 1 \\end{bmatrix}, \\begin{bmatrix} U_0 x \\\\\\xi \\end{bmatrix}, \\begin{bmatrix} U_0^* x \\\\ \\overline{\\tau } \\end{bmatrix}, \\begin{bmatrix} x \\\\ \\overline{\\tau } \\xi \\end{bmatrix} \\right\\rbrace .",
"$ Let $\\xi = \\tau $ .",
"Then $\\mathcal {E}_1$ is linearly dependent, but $\\mathcal {E}_2$ is not, because $\\overline{\\tau } \\xi = |\\tau |^2 \\ne 1$ .",
"The proof of this result is similar.",
"We may use Corollary 6.15 of [17] to assert that if $L - L^*$ has finite rank, then $L$ has a non-trivial invariant subspace, which is again orthogonally reducing by our hypothesis that $L$ satisfies property (CR).",
"One then looks for a maximal hermitian direct summand, and separately argues the cases where that summand has finite or infinite spectrum.",
"The details are left to the reader.",
"$\\Box $"
],
[
"Theorem.",
"Let $\\mathcal {H}$ be an infinite-dimensional, complex Hilbert space, and let $T \\in \\mathcal {B}( \\mathcal {H})$ .",
"If $T$ satisfies property (CR), then there exist $\\lambda , \\mu \\in \\mathbb {C}$ and $A \\in \\mathcal {B}( \\mathcal {H})$ with $A$ either selfadjoint or an orthogonally reductive unitary operator such that $T = \\lambda A + \\mu I$ .",
"In particular, if $T$ satisfies property (CR), then $T$ is normal with circlinear spectrum.",
"Proof.",
"By combining Lemma REF and Lemma REF , we can assume without loss of generality that $T = X + F$ , where $F$ is of finite rank and $X$ is either selfadjoint or unitary.",
"Either way, by Proposition REF , we see that $T$ is normal with circlinear spectrum.",
"From this it is easy to verify that $T$ is of the form $\\lambda A + \\mu I$ for some $\\lambda , \\mu \\in \\mathbb {C}$ with $A$ either selfadjoint or unitary.",
"The fact that $T$ is orthogonally reductive implies that $A$ is as well.",
"(This last argument is superfluous when considering the case where $A$ is selfadjoint.)",
"$\\Box $"
],
[
"Example.",
"We mention in passing that property (CR) is a strictly stronger condition than that of being orthogonally reductive.",
"Indeed, suppose that $N \\in \\mathcal {B}( \\mathcal {H})$ is a normal operator with $\\sigma (N) = \\lbrace 1, 2, 3, 4+i\\rbrace $ .",
"Thus the eigenvalues of $N$ are neither co-linear nor co-circular, and so $N$ does not have property (CR), by Corollary REF .",
"However, then $N$ is orthogonally reductive, as $N^*$ is a polynomial function of $N$ , combined with Sarason's result [18]."
],
[
"It would be interesting to know whether or not the converse of Theorem REF holds.",
"Indeed, suppose that $N \\in \\mathcal {B}( \\mathcal {H})$ is normal and has co-linear spectrum.",
"Arguing as before, we have that there exist scalars $\\lambda , \\mu \\in \\mathbb {C}$ and a hermitian operator $L$ such that $N = \\lambda I + \\mu L$ .",
"It is routine to verify that $N$ has property (CR).",
"For normal operators with co-circular spectrum, the problem is a bit more complicated."
],
[
"Proposition.",
"Let $U \\in \\mathcal {B}( \\mathcal {H})$ be unitary and suppose that $\\sigma (U) \\ne \\mathbb {T}$ .",
"Then $U$ has property (CR).",
"Proof.",
"Let $0 \\ne P \\ne I$ be a projection in $\\mathcal {B}( \\mathcal {H})$ , and relative to $\\mathcal {H}= P\\mathcal {H}\\oplus P^\\perp \\mathcal {H}$ , write $U = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix}.",
"$ Our goal is to show that $\\mathrm {rank}\\, B = \\mathrm {rank}\\, C$ .",
"As always, we have $B B^* &= I - A A^* \\\\C^* C &= I - A^* A.$ If $B$ and $C$ are both of infinite rank, then there is nothing to prove.",
"Thus we may suppose that either $B$ or $C$ is of finite rank.",
"Now, since $U$ has property (CR) if and only if $U^*$ has property (CR), we may suppose – by taking adjoints if necessary – that $\\mathrm {rank}\\, C$ is of finite rank and that $\\mathrm {rank}\\, C \\le \\mathrm {rank}\\, B$ .",
"Case 1.",
"$B$ is compact.",
"Then $A$ and $D$ are essentially unitary.",
"Since $\\sigma _e(A) \\subseteq \\sigma _e(U) \\ne \\mathbb {T}$ , it follows that $\\mathrm {ind}\\, A = 0$ .",
"(That is, in order for $A$ to have non-zero index, 0 must lie in a bounded component of $\\mathbb {C}\\setminus \\sigma _e(A)$ , of which there are none.)",
"Write $A = V |A|$ , and note that as $\\mathrm {ind}\\, A = 0$ , we may assume without loss of generality that $V$ is unitary.",
"Thus $A A^* = V |A| |A| V^* = V (A^* A) V^*$ .",
"That is, $A A^*$ and $A^* A$ are unitarily equivalent.",
"But then $B B^*$ and $C^* C$ are unitarily equivalent, whence $\\mathrm {rank}\\, B = \\mathrm {rank}\\, B B^* = \\mathrm {rank}\\, C^* C = \\mathrm {rank}\\, C$ .",
"Case 2.",
"$B$ is not compact.",
"We shall show that under the hypothesis that $\\sigma (U) \\ne \\mathbb {T}$ , this cannot happen.",
"Indeed, the equation $C^* C = I - A^*A$ with $\\mathrm {rank}\\, C < \\infty $ implies that $1 = \\pi (I) = \\pi (A)^* \\pi (A).",
"$ Thus $\\pi (A)$ is a partial isometry in the Calkin algebra $\\mathcal {B}( \\mathcal {H})/\\mathcal {K}( \\mathcal {H})$ , which implies that $\\pi (A) \\pi (A)^*$ is a projection.",
"The fact that $B$ is not compact, combined with the fact that $B B^* = I - A A^*$ shows that $1 = \\pi (I) \\ne \\pi (A) \\pi (A)^*.",
"$ Thus $\\pi (A)$ is not unitary.",
"Choose a projection $R \\in \\mathcal {B}( \\mathcal {H})$ such that $\\pi (R) = \\pi (A) \\pi (A)^*$ .",
"By Lemma V.6.4 of [8], there exists a partial isometry $W \\in \\mathcal {B}( \\mathcal {H})$ such that $W = R W$ and $\\pi (W)=\\pi (A)$ .",
"Moreover, by that same result, the integer $\\xi =\\mathrm {rank}\\, (I - W^* W) - \\mathrm {rank}\\, (R - W W^*)$ is defined independent of the choice of $W$ .",
"In our case, $\\mathrm {rank}\\, (I - W^* W) < \\infty $ while $\\mathrm {rank}\\, (I-R) = \\infty $ and $\\mathrm {rank}\\, (R- W W^*) < \\infty $ .",
"Hence $\\mathrm {rank}\\, (I - W W^*) = \\infty $ .",
"Thus we have that $W$ is a partial isometry with initial space $W^* W \\mathcal {H}$ , and final space $W \\mathcal {H}$ , and $\\dim \\, (W^* W \\mathcal {H})^\\perp < \\infty $ ; and $\\dim \\, (W \\mathcal {H})^\\perp = \\infty $ .",
"It is routine to produce a partial isometry $W_0$ with initial space $(W^* W \\mathcal {H})^\\perp < \\infty $ and final space contained in $(W \\mathcal {H})^\\perp $ , and to verify that $V = W + W_0$ is an isometry on $\\mathcal {H}$ .",
"By the Wold Decomposition, $V$ is unitarily equivalent to $S^{(\\kappa )} \\oplus Y$ , where $S$ denotes the unilateral forward shift, $Y$ is a unitary operator, and $\\kappa \\in \\mathbb {N}\\cup \\lbrace 0, \\infty \\rbrace $ .",
"If $\\kappa = 0$ , then $V$ is unitary.",
"But then $\\pi (V)= \\pi (W)= \\pi (A)$ is also unitary, a contradiction.",
"Thus $\\kappa \\ne 0$ .",
"But then $\\sigma _e(A) = \\sigma _e(V) \\supseteq \\sigma _e(S) = \\mathbb {T}$ .",
"On the other hand, it is not too hard to show that $\\partial (\\sigma _e(A)) \\subseteq \\sigma _{\\ell r e}(A) \\subseteq \\sigma _e(U)$ .",
"(For example, by the Corollary to Theorem 4.3 of [10], there exists a compact operator $K_1 \\in \\mathcal {B}(P \\mathcal {H})$ such that $A + K_1 = \\begin{bmatrix} \\lambda I & 0 \\\\ 0 & A_4 \\end{bmatrix}$ with respect to the decomposition $P \\mathcal {H}= \\mathcal {M}\\oplus (P\\mathcal {H}\\ominus \\mathcal {M})$ , for an appropriate subspace $\\mathcal {M}\\subseteq P \\mathcal {H}$ satisfying $\\dim \\, \\mathcal {M}= \\dim \\, (P \\mathcal {H}\\ominus \\mathcal {M}) = \\infty $ .",
"Letting $K = K_1 \\oplus 0$ yields that $U + K = \\begin{bmatrix} \\lambda I & 0 & B_1 \\\\ 0 & A_4 & B_2 \\\\ 0 & 0 & D \\end{bmatrix}$ .",
"Thus $\\lambda \\in \\sigma _e(U+K) = \\sigma _e(K)$ .)",
"But $\\partial (\\sigma _e(A)) = \\mathbb {T}$ , which contradicts our hypothesis that $\\sigma (U) \\ne \\mathbb {T}$ .",
"This shows that the case where $C$ is of finite rank and $B$ is not compact cannot happen, and completes the proof.",
"$\\Box $ Having seen that the bilateral shift $W$ is a unitary operator with $\\sigma (W) = \\mathbb {T}$ which is not reductive, we now show that there exists a unitary operator whose spectrum is the unit circle $\\mathbb {T}$ , but which nonetheless has property (CR).",
"Before embarking upon the proof of this, we first require a result due to Wu and Takahashi [20].",
"Recall that if $X \\in \\mathcal {B}( \\mathcal {H})$ is an operator, then we define the defect indices of $X$ to be $d_X &= \\dim (\\overline{\\mathrm {ran}}(I-X^* X)^{1/2}) \\mbox{ and } \\\\d_{X^*} &= \\dim (\\overline{\\mathrm {ran}} (I-X X^*)^{1/2}).$"
],
[
"Proposition.",
"(Wu-Takahashi; Theorem 3.5 of [20]) Let $X \\in \\mathcal {B}( \\mathcal {H})$ be a contraction and suppose that $d_X \\ne d_{X^*}$ .",
"Then $X$ does not admit a singular unitary dilation."
],
[
"Proposition.",
"Let $ (d_n)_{n=1}^\\infty $ be a sequence in $\\mathbb {T}$ and let $V = \\mathrm {diag}\\, (d_n)_{n=1}^\\infty $ be a corresponding diagonal unitary operator in $\\mathcal {B}( \\mathcal {H})$ .",
"Then $V$ has property (CR).",
"Proof.",
"Suppose that we can find a projection $P \\in \\mathcal {B}( \\mathcal {H})$ such that with respect to the decomposition $\\mathcal {H}= P \\mathcal {H}\\oplus P^\\perp \\mathcal {H}$ we may write $V = \\begin{bmatrix} A & B \\\\ C &D \\end{bmatrix} $ with $\\mathrm {rank}\\, C < \\mathrm {rank}\\, B$ .",
"(In particular, we must have $\\mathrm {rank}\\, C < \\infty $ ).",
"Then $B B^* &= I - A A^* \\mbox{ and }\\\\C^* C &= I - A^* A$ have different ranks.",
"Since $C$ is of finite rank, $A$ is essentially isometric and thus is a semi-Fredholm operator - in particular, both $A$ and $A^*$ have closed range.",
"Also, $\\mathrm {rank}\\, C = \\mathrm {rank}\\, C^* C &= \\mathrm {rank}\\, (I - A A^*) = d_{A^*} < \\infty , \\mbox{ while } \\\\\\mathrm {rank}\\, B = \\mathrm {rank}\\, B B^* &= \\mathrm {rank}\\, (I- A^* A) = d_{A}.$ In other words, $A$ is a contraction and the defect indices of $A$ are unequal.",
"By Proposition REF above, $A$ does not admit unitary dilation.",
"But $U$ is diagonal, and is therefore a singular unitary dilation of $A$ , which is obviously a contradiction.",
"$\\Box $ If, in Proposition REF we choose $\\lbrace d_n\\rbrace _n$ to be dense in $\\mathbb {T}$ , we immediately obtain the following consequence:"
],
[
"Corollary.",
"There exists a unitary operator $V$ with $\\sigma (V)= \\mathbb {T}$ which has property (CR)."
],
[
"Wermer [19] has shown that a unitary operator $U$ fails to be reductive if and only if Lebesgue measure is absolutely continuous with respect to the spectral measure $\\mu $ for $U$ .",
"Since any operator with property (CR) is necessarily reductive, this provides a measure-theoretic obstruction to property (CR) for unitary operators.",
"Another consequence of the above analysis is that it proves that the set $\\mathfrak {G}_{\\mathfrak {rank}}$ of operators with property (CR) is not closed.",
"Indeed, it follows easily from [7] that the bilateral shift $W$ is a limit of unitary operators $V_n$ such that $\\sigma (V_n) \\ne \\mathbb {T}$ .",
"(The $V_n$ 's can in fact be chosen to be unitary operators with spectrum $\\Gamma _n = \\lbrace e^{2 \\pi i \\theta }: 0 \\le \\theta \\le 1 - \\frac{1}{n} \\rbrace $ .)",
"As we saw in Proposition REF , each $V_n$ has property (CR), but $W = \\lim _n V_n$ does not.",
"Alternatively, the Weyl-von Neumann-Berg Theorem (see, e.g., [8], Theorem II.4.4) shows that there exists a sequence $(W_n)_{n=1}^\\infty $ of diagonal unitary operators such that $\\sigma (W_n) = \\mathbb {T}$ for all $n \\ge 1$ , such that $W = \\lim _n W_n$ .",
"We now investigate a consequence of property (CR) which relates to cyclic subspaces for operators."
],
[
"Proposition.",
"Suppose that $T$ is reductive.",
"Then $T$ and $T^*$ have the same cyclic subspaces.",
"In particular, if $T$ has property (CR), then $T$ and $T^*$ have the same cyclic subspaces.",
"Proof.",
"Suppose that $0 \\ne \\mathcal {M}\\subseteq \\mathcal {H}$ is a cyclic subspace for $T$ , and let $0 \\ne x \\in \\mathcal {M}$ be a cyclic vector for $T$ in $\\mathcal {M}$ , so that $\\mathcal {M}= \\overline{\\mathrm {span}} \\lbrace x, T x, T^2 x, \\ldots \\rbrace $ .",
"If $P$ is the orthogonal projection of $\\mathcal {H}$ onto $\\mathcal {M}$ , then $P^\\perp T P = 0$ , so by reductivity, $P T P^\\perp = 0$ , which implies that $\\mathcal {M}$ is invariant for $T^*$ .",
"Now let $\\mathcal {N}= \\overline{\\mathrm {span}}\\, \\lbrace x, T^*x, (T^*)^2 x, \\ldots \\rbrace $ be the cyclic subspace for $T^*$ generated by $x$ .",
"Since $x \\in \\mathcal {M}$ and $\\mathcal {M}$ is invariant for $T^*$ , we see that $\\mathcal {N}\\subseteq \\mathcal {M}$ .",
"Also, as $T^*$ is also reductive, the argument of the first paragraph shows that $x \\in \\mathcal {N}$ is invariant for $T$ .",
"But then $\\mathcal {N}\\supseteq \\mathcal {M}$ , whence $\\mathcal {N}= \\mathcal {M}$ , completing the proof.",
"$\\Box $ There exists a variant of this result which is somewhat interesting."
],
[
"Proposition.",
"Let $T \\in \\mathcal {B}( \\mathcal {H})$ and suppose that for each orthogonal projection $P \\in \\mathcal {B}( \\mathcal {H})$ , the off-diagonal corner $P^\\perp T P$ has rank one if and only if $P T P^\\perp $ has rank one.",
"A subspace $\\mathcal {M}$ of $\\mathcal {H}$ of dimension at least 3 is cyclic for $T$ if and only if it is cyclic for $T^*$ .",
"Proof.",
"Given $T$ as in the statement of the Proposition, it is clear that $T^*$ also has this property.",
"The argument used to prove Proposition REF shows that it suffices to show that the cyclic subspace $\\mathcal {M}= \\overline{\\mathrm {span}} \\lbrace x, Tx, T^2 x, \\ldots \\rbrace $ for $T$ generated by a non-zero vector $x$ is invariant for $T^*$ .",
"We consider first the case where $\\mathcal {M}$ is infinite-dimensional, as it is the easier of the two.",
"For each $n\\ge 1$ , let $P_n$ denote the orthogonal projection of $\\mathcal {H}$ onto $\\overline{\\mathrm {span}} \\lbrace x, Tx, T^2 x, \\ldots , T^n x \\rbrace $ , and let $P$ denote the orthogonal projection of $\\mathcal {H}$ onto $\\mathcal {M}$ .",
"It is clear that $(P_n)_{n=1}^\\infty $ is an increasing sequence which converges in the strong operator topology to $P$ .",
"An easy calculation then shows that the sequence $(P_n^\\perp T P_n)_{n=1}^\\infty $ converges in the strong operator topology to $P^\\perp T P$ .",
"As $x$ is a cyclic vector for $T$ in $\\mathcal {M}$ , we have that $\\mathrm {rank}\\, (P_n^\\perp T P_n) = 1$ for all $n \\ge 1$ , and our hypothesis then asserts that $\\mathrm {rank}\\, (P_n^\\perp T^* P_n) = 1$ for all $n \\ge 1$ .",
"But rank is lower-semicontinuous with respect to the strong operator topology, and thus $\\mathrm {rank}\\, P^\\perp T^* P \\le 1$ .",
"If $\\mathrm {rank}\\, P^\\perp T^* P = 1$ , then the hypothesis on $T$ implies that $\\mathrm {rank}\\, P^\\perp T P = 1$ , contradicting the fact that $\\mathcal {M}$ is invariant for $T$ .",
"Hence $P^\\perp T^* P = 0$ , proving that $\\mathcal {M}$ is invariant for $T^*$ .",
"Next we suppose that $\\mathcal {M}$ is finite-dimensional with $\\dim \\, \\mathcal {M}= N \\ge 3$ , and we find a cyclic vector $x$ for $T$ so that $\\mathcal {M}=\\mathrm {span} \\lbrace x, Tx, T^2 x, \\ldots , T^{N-1} x\\rbrace $ .",
"Let $\\lbrace e_1, e_2, \\ldots , e_N\\rbrace $ be the orthonormal basis obtained from $\\lbrace x, Tx, T^2 x, \\ldots , T^{N-1} x\\rbrace $ by applying the Gram-Schmidt process, so that $\\mathrm {span} \\lbrace e_1, e_2, \\ldots , e_k\\rbrace = \\mathrm {span} \\lbrace x, Tx, \\ldots , T^{k-1} x \\rbrace $ for $1 \\le k \\le N$ .",
"Let $Q_k$ denote the orthogonal projection of $\\mathcal {H}$ onto $\\mathbb {C}e_k$ , $1 \\le k$ , and define $P_k = Q_1 + Q_2 + \\cdots + Q_k$ , $1 \\le k \\le N$ .",
"Finally, extend $\\lbrace e_k\\rbrace _{k=1}^N$ to an orthonormal basis $\\lbrace e_k \\rbrace _{k=1}^\\infty $ for $\\mathcal {H}$ .",
"Note that the fact that $x$ is cyclic for $\\mathcal {M}$ , combined with our hypothesis, implies that $\\mathrm {rank}\\, P_k^\\perp T P_k = 1 = \\mathrm {rank}\\, P_k T P_k^\\perp $ , $1 \\le k \\le N-1$ .",
"Moreover, $\\mathcal {M}$ is invariant for $T$ , whence $P_N^\\perp T P_N = P^\\perp T P = 0$ .",
"By hypothesis, $\\mathrm {rank}\\, P T P^\\perp \\ne 1$ .",
"But $P T P^\\perp &= P_N T P_N^\\perp \\\\&= P_{N-1} T P_{N-1}^\\perp P_N^\\perp + Q_N T P_N^\\perp ,$ so that $\\mathrm {rank}\\, P T P^\\perp \\le \\mathrm {rank}\\, P_{N-1} T P_{N-1}^\\perp + \\mathrm {rank}\\, Q_N T P_N^\\perp \\le 1 + 1 = 2.$ Thus $\\mathrm {rank}\\, P T P^\\perp \\in \\lbrace 0, 2\\rbrace $ , and our goal is to show that $\\mathrm {rank}\\, P T P^\\perp \\ne 2$ .",
"Suppose, to the contrary, that $\\mathrm {rank}\\, P T P^\\perp = 2$ .",
"It follows that $\\mathrm {rank}\\, P_{N-1} T P_{N-1}^\\perp = 1 = \\mathrm {rank}\\, Q_N T P_N^\\perp $ .",
"Thus there exists $1 \\le k \\le N-1$ such that $Q_k T P_N^\\perp \\ne 0$ , and $\\mathrm {rank}\\, (Q_k + Q_N) T P^\\perp =\\mathrm {rank}\\, (Q_k + Q_N) T P_N^\\perp = 2$ .",
"Case 1.",
"$k = N-1$ .",
"Let us reorder the basis for $P \\mathcal {H}$ as $\\lbrace e_{N-1}, e_N, e_1, e_2, \\ldots , e_{N-2} \\rbrace $ .",
"The matrix for $T$ relative to $ P \\mathcal {H}\\oplus P^\\perp \\mathcal {H}$ is: $[T] = \\begin{bmatrix} t_{N-1, N-1} & t_{N-1, N} & t_{N-1, 1} & t_{N-1, 2} & \\ldots & t_{N-1, N-3} & t_{N-1, N-2} & Q_{N-1} T P^\\perp \\\\t_{N, N-1} & t_{N, N} & t_{N, 1} & t_{N, 2} & \\ldots & t_{N, N-3} & t_{N, N-2} & Q_{N} T P^\\perp \\\\\\vdots & & & &\\ldots & & \\vdots \\\\t_{N-3, N-1} & t_{N-3, N} & t_{N-3, 1} & t_{N-3, 2} & \\ldots & t_{N-3, N-3} & t_{N-3, N-2} & Q_{N-3} T P^\\perp \\\\t_{N-2, N-1} & t_{N-2, N} & t_{N-2, 1} & t_{N-2, 2} & \\ldots & t_{N-2, N-3} & t_{N-2, N-2} & Q_{N-2} T P^\\perp \\\\0 & 0 & 0 & 0 & \\ldots & 0 & 0 & P^\\perp T P^\\perp \\end{bmatrix}$ Let $R = P - Q_{N-2}$ .",
"Since $t_{N-2, N-3} \\ne 0$ (as $x$ is a cyclic vector for $\\mathcal {M}$ ), it follows that $\\mathrm {rank}\\, R^\\perp T R = 1$ .",
"Thus $\\mathrm {rank}\\, \\begin{bmatrix} t_{N-1, N-2} & Q_{N-1} T P^\\perp \\\\ t_{N, N-2} & Q_N T P^\\perp \\\\ \\vdots & \\vdots \\\\ t_{N-3, N-2} & Q_{N-3} T P^\\perp \\end{bmatrix} = \\mathrm {rank}\\, R T R^\\perp = 1, $ and so $\\mathrm {rank}\\, (Q_{N-1} + Q_N) T P^\\perp = \\mathrm {rank} \\begin{bmatrix} Q_{N-1} T P^\\perp \\\\ Q_N T P^\\perp \\end{bmatrix} \\le 1$ , a contradiction.",
"Thus in this case, $P T P^\\perp = 0$ , so $\\mathcal {M}$ is invariant for $T^*$ .",
"Case 2.",
"$1 \\le k < N-1$ .",
"This time we reorder the basis for $P \\mathcal {H}$ as $\\lbrace e_k, e_{k+2}, \\ldots , e_{N}, e_1, \\ldots , e_{k-1}, e_{k+1} \\rbrace $ .",
"The matrix for $T$ relative to $P \\mathcal {H}\\oplus P^\\perp \\mathcal {H}$ is then: $[T] = \\begin{bmatrix} t_{k,k} & t_{k, k+2} & \\ldots & t_{k, N} & t_{k, 1} & \\ldots & t_{k, k-1} &t_{k, k+1} & Q_{k} T P_N^\\perp \\\\t_{k+2,k} & t_{k+2, k+2} & \\ldots & t_{k+2, N} & t_{k+2, 1} & \\ldots & t_{k+2, k-1} &t_{k+2, k+1} & Q_{k+2} T P_N^\\perp \\\\\\vdots & & & &\\ldots & & \\vdots \\\\t_{k-1,k} & t_{k-1, k+2} & \\ldots & t_{k-1, N} & t_{k-1, 1} & \\ldots & t_{k-1, k-1} &t_{k-1, k+1} & Q_{k-1} T P_N^\\perp \\\\t_{k+1,k} & t_{k+1, k+2} & \\ldots & t_{k+1, N} & t_{k+1, 1} & \\ldots & t_{k+1, k-1} &t_{k+1, k+1} & Q_{k+1} T P_N^\\perp \\\\0 & 0 & 0 & 0 & 0 & \\ldots & 0 & 0 & P_N^\\perp T P_N^\\perp \\end{bmatrix}$ Let $R = P - Q_{k+1}$ .",
"Since $t_{k+1, k} \\ne 0$ (as $x$ is a cyclic vector for $\\mathcal {M}$ ), it follows that $\\mathrm {rank}\\, R^\\perp T R = 1$ .",
"By hypothesis, $\\mathrm {rank}\\, R T R^\\perp = 1$ .",
"Thus $\\mathrm {rank}\\, (Q_k +Q_N) T P^\\perp =\\mathrm {rank}\\, (Q_k + Q_N) [ R T R^\\perp ] P^\\perp \\le \\mathrm {rank}\\, R T R^\\perp = 1, $ a contradiction.",
"Thus in this case as well, $P T P^\\perp = 0$ , so $\\mathcal {M}$ is invariant for $T^*$ .",
"The remainder of the proof is identical to the second paragraph of the proof of Proposition REF .",
"$\\Box $"
],
[
"In [12], Ken Harrison introduced the notion of essentially reductive operators: we say that $T \\in \\mathcal {B}( \\mathcal {H})$ is essentially reductive if for each projection $P$ we have that $P T P^\\perp $ compact if and only if $P^\\perp T P$ is compact.",
"(One can view this as $\\pi (T)$ having property (CR) in the Calkin algebra.)",
"In the paper [15] (Theorem 2), Moore shows that every essentially reductive operator $T$ is essentially normal – i.e.",
"$\\pi (T)$ is normal in the Calkin algebra.",
"Earlier, Harrison ([12], Theorem 4.5) had characterized all essentially normal operators which are essentially reductive.",
"Combining these results, one obtains the following."
],
[
"Theorem.",
"(Moore; Corollary 1 of [15]) Let $T \\in \\mathcal {B}( \\mathcal {H})$ .",
"The following are equivalent.",
"$T$ is essentially reductive.",
"$T$ is essentially normal and $\\sigma _e(T)$ is Lavrentiev.",
"The next result is a simple consequence of Moore's Theorem together with Theorem REF and Proposition REF ."
],
[
"Corollary.",
"Let $T \\in \\mathcal {B}( \\mathcal {H})$ .",
"The following are equivalent.",
"$T$ is essentially reductive and has property (CR).",
"One of the following holds.",
"There exist $\\lambda , \\mu \\in \\mathbb {C}$ and a hermitian operator $R$ such that $T = \\lambda R + \\mu I$ .",
"There exist $\\lambda , \\mu \\in \\mathbb {C}$ and a unitary operator $V$ with $\\sigma (V) \\ne \\mathbb {T}$ such that $T = \\lambda V + \\mu I$ ."
]
] | 1709.01840 |
[
[
"One-loop effective actions and higher spins. II"
],
[
"Abstract In this paper we continue and improve the analysis of the effective actions obtained by integrating out a scalar and a fermion field coupled to external symmetric sources, started in the previous paper.",
"The first subject we study is the geometrization of the results obtained there, that is we express them in terms of covariant Jacobi tensors.",
"The second subject concerns the treatment of tadpoles and seagull terms in order to implement off-shell covariance in the initial model.",
"The last and by far largest part of the paper is a repository of results concerning all two point correlators (including mixed ones) of symmetric currents of any spin up to 5 and in any dimensions between 3 and 6.",
"In the massless case we also provide formulas for any spin in any dimension."
],
[
"Introduction",
"This paper is a follow-up of [1].",
"In that paper we analyzed the two-point functions of conserved currents of two models (a free scalar and a free Dirac fermion model coupled to diverse backgrounds) in various dimensions.",
"For a background, represented by a completely symmetric field, the two-point function of the current minimally coupled to it is the basic ingredient of its (quadratic) effective action (EA).",
"We found in [1] that the effective action for any background field obtained in this way is based on the corresponding linearized Fronsdal kinetic operator, [2], in the nonlocal form introduced by Francia and Sagnotti, [3].",
"In view of constructing a covariant action for a completely symmetric tensor field, this result is promising.",
"It suggests that integrating out scalar or fermion fields (or any other field by which one can form conserved currents) may be a useful way to analyze the dynamics of higher spin fields.",
"But of course what we have done in [1] is only the beginning.",
"The crucial next step is the calculation of the three-point functions of conserved currents, the analysis of the (lowest order) interaction terms in the effective actions and their consistency with covariance.",
"Before arriving at the three-point functions, it is however necessary to improve our analysis of the quadratic EA.",
"In fact in the course of our research we realized that it inevitably branches out in different directions.",
"At the same time, in [1], several aspects and questions were left behind .",
"In this and a subsequent paper we would like to cover as thoroughly as possible any aspect of the quadratic EA's.",
"The first issue is the geometrization (at the linear level) of our results in [1].",
"They were expressed there mostly in terms of a projection operator, which is very convenient in that context because it automatically ensures conservation.",
"But, in this way, the geometrical content of the resulting equations of motion or the EA remains implicit.",
"Now the formulation of our results in terms of geometrical objects is essential, if our target is to arrive at covariant EA's.",
"One first aim of this paper will be to geometrize the results of [1].",
"We will do it in terms of Jacobi tensors.",
"A second related important point is related to local subtractions.",
"In [1] we found several violations of the Ward Identities induced by the conservation of the initial current (which induces a gauge invariance of the relevant minimal coupling).",
"Such violations consist of local terms, so that it is rather elementary to recover conservation by subtracting local counterterms from the EA.",
"There is nothing special in this, it is a very ordinary procedure.",
"The interesting point is that it is in general not necessary to do it, because the perturbative field theory formalism already automatically takes care of covariance, provided one takes into account not only the two-point bubble diagrams but also other diagrams such as tadpole and seagull ones.",
"Now, from a practical point of view it is much easier to subtract easily identifiable local counterterms, than calculating additional diagrams to guarantee conservation.",
"The latter could appear as an academic exercise for spin 1 and 2, where we already know the covariant form of the minimal coupling.",
"But, it is important to show that dimensional regularization, which we use, is giving manifestly covariant expressions (without subtractions by hand).",
"For spin 3 and higher it may be a very useful and even necessary calculation.",
"The reason is that seagull diagrams are related to terms in the initial action that do not belong to the minimal model we start with (a scalar or fermion field minimally coupled to a background field).",
"Conservation (without subtractions) requires the presence of such additional terms and constraints not only their form but also their coefficients.",
"It is clear, that when we consider higher spin backgrounds, this remark may be used in order to determine additional action terms, as well as conditions for their coefficients.",
"This goes in the direction of constructing an initial off shell covariant model, an important target in itself and a necessary step in the construction of a covariant EA.",
"The third important issue is represented by mixed two-point correlators.",
"In [1] we have considered only two-point functions of each current with itself.",
"Of course this provides basic information about the relevant EA.",
"However higher spin theories are known to be consistent only if they encompass an infinite number of fields (although in 3d consistent theories may exist with a finite number of fields).",
"It is obvious that this requires not only the knowledge of the two-point correlator of each higher spin current with itself, but also of any two currents (mixed correlators) coupled to fields that may enter the action.",
"This part has the structure of a repository of results about the two point correlators of symmetric currents of spin up to 5 in dimension $3\\le d\\le 6$ for both the massive scalar and fermion theory.",
"In 3d we also consider the odd parity sector which emerges from the parity-breaking fermion mass term, and we find a nice generalization of Pope and Townsend's Chern-Simons-like action in the case when different higher-spin fields are taken into consideration.",
"In this paper we will deal with these three issues.",
"Other topics, such as the discussion of the ambiguities inherent in the choice of the conserved currents in the initial matter model, will be included in a subsequent article.",
"This is a good point to mention that our research is indebted to several preexisting works, in particular with [4], [5] as far as the inspiration is concerned, with [6], [7], [8], [9], [10], [11], [12], [13] as far as the methods are concerned and with [3], [14], [15], [16], [17], [18], [19] for HS theories.",
"Other papers of ours, related to the present one are, beside [1], [20], [21].",
"The paper is organized as follows.",
"In the next section we show how to geometrize the results of [1] and of this paper, that is how to express them in terms of Jacobi tensors.",
"In section 3 we discuss the issue of tadpole and seagull terms and how they guarantee covariance without subtractions in the case of spin 1 and 2.",
"Section 4 forms the bulk of the paper.",
"After an explanatory introduction we list all possible conserved two-point correlators for currents up to spin 5, including the mixed ones.",
"This part of the paper is intended as a source book.",
"It contains the complete correlators as well as their UV and IR expansions.",
"Several results were already contained in [1].",
"We have left them here for completeness.",
"Finally, section 5 is devoted to some conclusions."
],
[
"Geometry in effective actions",
"The construction of interacting quantum field theories with massless higher spin ($s>2$ ) fields still poses an interesting theoretical problem.",
"On the one hand, there are different \"no-go\" theorems putting serious constraints on such theories, especially in flat space-time.",
"On the other hand, we have significant higher spin results: free fields can be constructed in the same manner as in lower spin cases (see, e.g.",
"[22]); a few cubic interaction terms have been constructed in the literature (see [15], [16]); most notably, a full consistent covariant HS theory in AdS background has been constructed by Vasilev and collaborators [23].",
"In our previous paper, [1], we remarked that free lower spin field theories possess conserved higher spin currents which simply \"beg\" to be coupled to higher spin fields.",
"Therefore such simple models seem to be a useful tool to study higher spin theories.",
"A basic ingredient of the approach in [1] is the connection between the on shell conservation of the initial free field theory current and the gauge invariance of the minimal coupling term with the higher spin field, which induces a gauge invariance of the linearized higher spin EA (or covariance of the corresponding equation of motion).",
"In [1] this invariance was left somewhat implicit.",
"There is, however, a way to make it explicit, by expressing the results in terms of covariant `geometric' tensors constructed out of the symmetric higher spin fields.",
"In this section we would like to make connection with such a geometrization program.",
"In the sequel we first introduce well-known definitions and properties about higher spin tensors, their linearized eom's and their possible geometrical formulations.",
"Then we show how to use this material to express the results obtained in [1], [20] and in this paper in a geometric language.",
"Differences between lower spin ($s\\le 2$ ) and higher spin theories emerge already at the level of classical free field theories.",
"The simplest way to construct a free theory of higher spin field is provided by the Fronsdal equation, [2], [24], [25]: $\\mathcal {F} \\equiv \\Box \\varphi - \\partial \\, \\partial \\cdot \\varphi + \\partial ^2\\varphi ^{\\prime } = 0$ where the spin-$s$ field is described by the completely symmetric rank-$s$ tensor field $\\varphi \\equiv \\varphi _{\\mu _1\\cdots \\mu _s}$ .",
"In this expression standard HS conventions from [3], [17], [18] are assumed.Conventions assume symmetrization over free indices with minimal number of terms and without any symmetry factors.",
"Also, a prime denotes contraction of a pair of indices, so, e.g., $\\varphi ^{\\prime } \\equiv \\varphi _{\\mu _1\\cdots \\mu _{s-2}} = \\eta ^{\\mu _{s-1}\\mu _s}\\varphi _{\\mu _1\\cdots \\mu _s}$ is a completely symmetric rank-$(s-2)$ tensor field.",
"The Fronsdal equation (REF ) is invariant under local transformations parametrised by traceless completely symmetric rank-$(s-1)$ tensor fields $\\Lambda \\equiv \\Lambda _{\\mu _1\\cdots \\mu _{s-1}}$ $\\delta \\varphi = \\partial \\Lambda $ with $\\Lambda ^{\\prime } = 0$ .",
"While this gauge symmetry guarantees that the field propagates only free spin-$s$ excitations, we see that for $s\\ge 3$ the gauge symmetry is constrained to trace-free parameters $\\Lambda $ .",
"One can rewrite the Fronsdal equation in an unconstrained form by introducing a rank-$(s-3)$ compensator field $\\alpha $ transforming on (unconstrained) gauge transformations (REF ) as $\\delta \\alpha = \\Lambda ^{\\prime }$ , in the following way $\\mathcal {F} = \\partial ^3 \\alpha $ This equation is invariant under the unconstrained gauge transformations (REF ) because the variation of $\\alpha $ exactly cancels the variation of the Fronsdal tensor.",
"Most important, the system $\\varphi ,\\alpha $ can be cast in a (local) Lagrangian form.",
"By the partial gauge fixing condition $\\alpha =0$ one obtains the original Fronsdal's equation (REF ).",
"The generalization $F^{(n)}$ of the Fronsdal differential operator, which is gauge invariant for $n$ large enough, is given in terms of the recursive equation $F^{(n+1)}=F^{(n)}+\\frac{1}{(n+1)(2n+1)} \\frac{\\partial ^2}{\\square }{F^{(n)}}^{\\prime } -\\frac{1}{n+1} \\frac{\\partial }{\\square } \\partial \\cdot F^{(n)}$ with $F^{(0)}= \\square \\varphi $ .",
"So, in particular, $F^{(1)} \\equiv F= \\square \\varphi -\\partial \\partial \\cdot \\varphi +\\partial ^2\\varphi ^{\\prime }$ is the original Fronsdal operator.",
"However, the connection with our results cannot be in terms of the tensor $F^{(n)}$ , because the latter does not satisfy a conservation law, while our results are conserved two-point functions.",
"To make the connection one constructs the Einstein-like tensor $G^{(n)} = \\sum _{p=0}^n (-1)^p \\frac{(n-p)!",
"}{2^p n!}",
"\\,\\eta ^p\\,F^{(n)[p]}$ where the superscript in square bracket denotes the number of time $F^{(n)}$ has been traced, and $\\eta $ is the Minkowski metric.",
"The association of $\\varphi $ with the spin $s$ is as follows: $\\left\\lbrace \\begin{matrix} s= 2n &\\quad s\\quad {\\rm even}\\\\s= 2n-1 &\\quad s \\quad {\\rm odd}\\end{matrix}\\right.\\nonumber $ The $G^{(n)}$ tensor is divergenceless $\\partial \\cdot G^{(n)}=0 $ The free (unconstrained) linearized equations of motion for $\\varphi $ are $G^{(n)}=0$ Once again, it can be shown that such an equation can be cast in local Lagrangian form, provided one introduces auxiliary fields (compensators).",
"$G^{(n)}$ are the objects that can be directly connected with the LHS of (REF ) below."
],
[
"Geometrization in terms of Jacobi tensors",
"In [1] all the two-point correlators and corresponding effective actions are presented in momentum space and expressed in terms of the projector $\\pi ^{(k)}_{\\mu \\nu }= \\eta _{\\mu \\nu } - \\frac{k_\\mu k_\\nu }{k^2}$ Applied to $k^\\nu $ gives 0, so any two-point function expressed in terms of it alone is conserved.",
"We showed that any conserved correlator for spin $s$ can be written in terms of the following structures: $\\tilde{A}^{(s)}_0(k,n_1,n_2)&=& (n_1 \\!\\cdot \\!\\pi ^{(k)}\\!\\cdot \\!n_2)^s\\\\\\tilde{A}^{(s)}_1(k,n_1,n_2)&=& (n_1 \\!\\cdot \\!\\pi ^{(k)}\\!\\cdot \\!n_2)^{s-2} (n_1 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_1)(n_2\\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)\\\\\\ldots && \\ldots \\ldots \\nonumber \\\\\\tilde{A}^{(s)}_l(k,n_1,n_2) &=& (n_1 \\!\\cdot \\!\\pi ^{(k)}\\!\\cdot \\!n_2)^{s-2l} (n_1 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_1)^l(n_2 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)^l\\\\\\ldots && \\ldots \\ldots \\nonumber $ where $n_1,n_2$ are generic polarization vectors, and $n_1\\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2= n_1^\\mu \\pi ^{(k)}_{\\mu \\nu }n_2^\\nu $ .",
"There are $\\lfloor s/2 \\rfloor $ independent such terms.",
"The generic term in the final formulas are combinations of $\\tilde{A}^{(s)}_l(k,n_1,n_2)$ with numerical coefficients $a_l$ , say $\\tilde{E}^{(s)}(k,n_1,n_2)= \\sum _{l=0}^{\\lfloor s/2 \\rfloor } a_l \\tilde{A}^{(s)}_l(k,n_1,n_2)$ preceded by a function of $f(|k|,m)$ and the mass $m$This function can be expanded in series of $m/|k|$ or $|k|/m$ near the IR and UV, respectively, which gives the tomographic expansions considered in [1].",
"The latter clearly show that the structures of the two-point functions (and corresponding linearized EA's) are determined by the unique Fronsdal operator appropriate for the given source, although, generally, the operator appears in a nonlocal form and in different gauges.",
"In this paper we consider only these operators and disregard the function $f$ ..",
"Eq.",
"(REF ) can be easily translated into a corresponding differential operator by Fourier anti-transforming $E^{(s)}(\\partial ,n_1,n_2)= \\sum _{l=0}^{\\lfloor s/2 \\rfloor } a_lA^{(s)}_l(\\partial ,n_1,n_2)$ These are the types of differential operators that appear in the EA's acting on the spin $s$ field $\\varphi _{\\mu _1\\ldots \\mu _s}$ .",
"The corresponding eom will take the following form.",
"Set $(m^s \\cdot \\varphi )=\\frac{1}{s!}",
"m_{\\mu _1\\ldots \\mu _s}\\varphi ^{\\mu _1\\ldots \\mu _s}$ and $n_1=n\\equiv \\lbrace n_\\mu \\rbrace $ , $n_2= {\\partial }_m \\equiv \\left\\lbrace \\frac{\\partial }{\\partial m_\\mu }\\right\\rbrace $ .",
"The eom are $\\frac{1}{s!}",
"(\\partial _n)^s E^{(s)}(\\partial ,n,\\partial _m)(m^s \\cdot \\varphi )=0$ multiplied by a function of $k^2$ and $m$ .",
"The purpose of this section is to rewrite the equations such as (REF ) in the geometrical form of [3].",
"To this end let us introduce the symbol of $G^{(n)}$ , $\\tilde{\\cal G}^{(n)}(k,n_1,n_2)$ , as follows.",
"First we saturate all its $s$ naked indices of $G^{(n)}$ with $n_1$ polarizations, then we Fourier transform it and replace the Fourier transform of $\\varphi $ , $\\tilde{\\varphi }$ , with a symmetric tensor made out of the product of $s$ polarizations $n_2$ .",
"Finally we define ${\\cal G}^{(n)}\\equiv \\frac{1}{s!}",
"\\,(\\partial _n)^s {\\cal G}^{(n)}(\\partial ,n,\\partial _m)\\, (m^s \\cdot \\varphi )$ Then the connection between (REF ) and (REF ) is given by $\\frac{1}{k^2}\\, \\tilde{\\cal G}^{(n)}(k,n_1,n_2)= \\sum _{l=0}^{\\lfloor s/2 \\rfloor }(-1)^l \\left( \\begin{matrix} \\lfloor s/2 \\rfloor \\\\ l\\end{matrix} \\right)\\tilde{A}_l^{(s)}(k, n_1,n_2), $ which corresponds to a particular choice of the coefficients $a_l$ in (REF ).",
"Of course we are interested not only in the relation (REF ), but in expressing all the $\\tilde{A}_l^{(s)}(k, n_1,n_2)$ in terms of the $\\tilde{\\cal G}^{(n)}(k,n_1,n_2)$ .",
"To do so we have to take the successive traces of (REF ).",
"We have, for instance ${\\tilde{\\cal G}}^{(n)}{}^{\\prime } = -2\\lfloor s/2 \\rfloor (2\\lfloor s/2 \\rfloor +D-4) {\\tilde{\\cal G}^{(n-1)}}\\,(n_2 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)$ In general ${\\tilde{\\cal G}^{(n)[p]}} = (-2)^p \\frac{(2\\lfloor s/2 \\rfloor +D-4)!!",
"(\\lfloor s/2 \\rfloor )!",
"}{(2\\lfloor s/2 \\rfloor +D-2p-4)!!",
"(\\lfloor s/2\\rfloor -p)!",
"}{\\tilde{\\cal G}^{(n-p)}}\\,(n_2 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)^p$ and ${\\tilde{\\cal G}^{(n)[n]}} = (-2)^n \\frac{(2\\lfloor s/2 \\rfloor +D-4)!!",
"(\\lfloor s/2 \\rfloor )!}{(D-4)!!",
"}{\\tilde{\\cal G}^{(0)}}\\,(n_2 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)^n$ for $s$ even, with ${\\tilde{\\cal G}^{(0)}}={k^2}$ , and ${\\tilde{\\cal G}^{(n)[n-1]}} = (-2)^{n-1} \\frac{(2\\lfloor s/2 \\rfloor +D-4)!!",
"(\\lfloor s/2 \\rfloor )!}{(D-4)!!",
"}{\\tilde{\\cal G}^{(1)}}\\,(n_2 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)^{n-1}$ for $s$ odd, with ${\\tilde{\\cal G}^{(1)}}={k^2}(n_1 \\!\\cdot \\!\\pi ^{(k)}\\!\\cdot \\!n_2) $ .",
"Now, using (REF ), one can write $(n_1 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)^s &\\equiv & \\tilde{A}_0^{(s)}(k,n_1,n_2)=\\frac{1}{k^2}\\tilde{\\cal G}^{(n)}(k, n_1,n_2) \\\\&+& \\sum _{l=0}^{\\lfloor s/2 \\rfloor -1} (-1)^l \\left( \\begin{matrix} \\lfloor s/2 \\rfloor \\\\ l+1\\end{matrix} \\right)(n_1 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_1)^{l+1}(n_1 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)^{s-2l-2} (n_2 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)^{l+1} \\nonumber $ for even $s$ , and a similar expression for odd $s$ .",
"Now the strategy consists in repeating the same step for the second line in (REF ), by using (REF ) and successively (REF ).",
"The end result is ${k^2} (n_1 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)^s = \\sum _{p=0}^{\\lfloor s/2 \\rfloor }\\left( -\\frac{1}{2} \\right)^p\\frac{(2\\lfloor s/2 \\rfloor +D-2p-4)!!}{p!",
"(2\\lfloor s/2 \\rfloor +D-4)!!",
"}(n_1 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_1)^p\\, \\tilde{\\cal G}^{(n)[p]}(k,n_1,n_2)\\nonumber \\\\ $ In a similar way one can obtain $&&{k^2} (n_1 \\!\\cdot \\!\\pi ^{(k)}\\!\\cdot \\!n_2)^{s-2l} (n_1 \\!\\cdot \\!\\pi ^{(k)}\\!\\cdot \\!n_1)^l(n_2 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_2)^l\\\\&&=\\frac{1}{\\left(\\begin{matrix} \\lfloor s/2 \\rfloor \\\\ l \\end{matrix} \\right)} \\sum _{p=l}^{\\lfloor s/2 \\rfloor }\\left( -\\frac{1}{2} \\right)^p\\left( \\begin{matrix} p\\\\ l \\end{matrix} \\right)\\frac{(2\\lfloor s/2 \\rfloor +D-2p-4)!!}{p!",
"(2\\lfloor s/2 \\rfloor +D-4)!!",
"}(n_1 \\!\\cdot \\!",
"\\pi ^{(k)}\\!\\cdot \\!n_1)^p\\, \\tilde{\\cal G}^{(n)[p]}(k,n_1,n_2)\\nonumber $ In conclusion any expression of the type (REF ), i.e.",
"any conserved structure, can be expressed in terms of the generalized Einstein symbols $\\tilde{\\cal G}^{(n)}(k, n_1,n_2)$ and its traces.",
"Thus any EA (or any eom) we obtain from our models, by integrating out matter, can be expressed in terms of the generalized Einstein tensor $G^{(n)}$ and its traces preceded by a function of $\\square $ and the mass $m$ of the model, with suitable multiples of the operator $\\eta ^{\\mu \\nu }- \\frac{\\partial ^\\mu \\partial ^\\nu }{\\square } \\nonumber $ acting on the traces.",
"Using (REF ) one can replace the dependence on ${G}^{(n)}$ of such expressions with the dependence on ${F}^{(n)}$ .",
"The geometrization program can be completed by introducing the Jacobi tensors $R_{\\mu _1,\\ldots \\mu _s\\nu _1\\ldots \\nu _s}$ (one of the possible generalizations of the 4d Riemann tensor, [26], [27]) by means of $\\frac{1}{(s!",
")^2} (m^s\\!\\cdot \\!",
"R^{(s)} \\!\\cdot \\!n^s) = \\sum _{l=0}^s \\frac{(-1)^l}{s!(s-l)!l!}",
"(m\\!\\cdot \\!\\partial )^{s-l} (n\\!\\cdot \\!\\partial )^{l}(m^l\\!\\cdot \\!",
"\\varphi \\!\\cdot \\!n^{s-l})$ The tensors $R^{(s)}$ are connected to the $F^{(n)}$ as follows: $F^{(n)}= {\\left\\lbrace \\begin{array}{ll} \\frac{1}{\\square ^{n-1}} R^{(s)[n]} &\\quad s=2n\\\\\\frac{1}{\\square ^{n-1}}\\partial \\!\\cdot \\!",
"R^{(s)[n-1]} & \\quad s=2n-1\\end{array}\\right.}",
"$ where the traces in square brackets refer to the first set of indices.",
"In this way we can express any EA or any eom in terms of $R^{(s)}$ and traces (in the second set of indices) thereof.",
"Since above we have referred to [3], we feel that, to end this section, it is opportune for us to clarify the context in which our results are derived and point out the differences with the spirit of [3], [17], [18].",
"In these papers the initial purpose was to write down a generalization of the Fronsdal equations for higher spin in such a way as to avoid the constraints needed in the original formulation of [2].",
"The authors of [3] chose to sacrifice locality in favor of an unconstrained gauge symmetry.",
"The typical (linearized) non-local equation of motion one obtains in this way is (REF ).",
"As we have already pointed out, it can be shown that such an equation can be cast in Lagrangian form, provided one introduces auxiliary fields (compensators).",
"Therefore one can say that the nonlocality of (REF ) is a gauge artifact, with no physical implication.",
"However equations of motion invariant under unrestricted gauge symmetry are far from unique.",
"There actually exist several families of them depending on arbitrary parameters (by the way, this is evident by reversing the argument above and starting from the generic operator (REF ), instead of the completely fixed one (REF )).",
"These are all equally valid as long as the field $\\varphi $ is considered in isolation and the linearized eom is the free one, (REF ).",
"However, if the spin $s$ system is minimally coupled to a conserved current the question arises as to whether the propagating degrees of freedom are the truly physical ones, i.e.",
"those corresponding to the appropriate little group representation for massless fields.",
"The authors of [17], [18] were able to prove that there exist only one choice for the Einstein-like tensor which is Lagrangian and satisfies such a physicality condition.",
"Such `physical' Einstein tensors do not correspond, in general, to the kinetic operators we find in our effective action in section below.",
"This is not surprising, as our main goal is covariance: our purpose is to arrive at a covariant EA with respect to a completely unfolded gauge symmetry.",
"In a logical development the next step will be to introduce auxiliary fields to eliminate nonlocalities.",
"Following this we would need to gauge-fix the action and introduce appropriate ghosts to produce the physical propagators.",
"At that point would the problem handled by [17], [18] come to the surface.",
"However, we would like to recall that our immediate prospect is to construct the linearized covariant EA in preparation for the analysis of the three-point function."
],
[
"Tadpoles, seagulls and conservation",
"In this section we wish to illustrate the role of tadpole and seagull diagrams in implementing conservation in two-point correlators.",
"In [1], in order to evaluate the two point correlators of conserved currents we computed only the bubble diagrams formed by two internal scalar or fermion lines and two vertices.",
"In this way we found several violations of the relevant Ward identities.",
"Such violations consist of local terms, so that it was rather elementary to recover conservation by subtracting local counterterms from the EA.",
"However it is in general not necessary to do this, because the perturbative field theory formalism already automatically takes care of it provided one takes into account not only the two-point bubble diagrams but also other diagrams such as tadpole and seagull ones, [28], [29].",
"Although this is a rather well-known fact, we would like to show it in detail here for spin 1 and 2 as a guide for the more challenging higher spin cases.",
"The reason is that seagull diagrams reflect the presence in the initial action of additional terms, additional with respect to the minimal couplings (symbolically $\\int j\\varphi $ ), which are on-shell covariant, but off-shell non-covariant.",
"One of the crucial steps in our program is clearly implementing off-shell gauge covariance of the initial models, that is adding to the minimal couplings in the relevant actions the terms that render them off-shell covariant, at least to the lowest order in a perturbative approach to the gauge symmetry.",
"We know such additional terms exactly in the case of spin 1 and spin 2 (because we know the full covariant action), but not yet for higher spins.",
"In the latter cases, however, we can implement off-shell current conservation by satisfying the corresponding two-point function Ward identity.",
"In turn this requires considering tadpoles and seagull terms.",
"The latter, in particular, originate from the above additional terms, which in this way may hopefully be identifiedAn approach related to ours is outlined in [9].",
"It is based on Weyl quantization.",
"Its main advantage is that it provides a full quantum action and quantum symmetry for the initial scalar model.",
"It will be interesting to compare the two approaches.. Hereafter in this section we work out the cases of spin 1 and spin 2 in any dimension (in 3d for the odd parity part) in detail, showing the role of tadpole and seagull terms in the Ward identities for two-point functions of spin 1 and 2 respectively, and their origin in the various terms of the initial actions.",
"We keep the derivation at a pedagogical level and, for completeness, we analyze the full structure of the relevant two-point functions and, in particular, their IR and UV expansions, as well as their contributions to the EA's.",
"Starting from the generating function $Z[a]=e^{i W[a]}=\\int D\\psi D \\bar{\\psi } e^{i (S_0 + S_{int}[a])}$ where $a$ is the external higher spin field, we will compute the effective action for the external source fields up to the quadratic order: $i\\,W[a]&=&i\\,W[0] +\\int d^dx\\,a_{\\mu _{1}\\ldots \\mu _{s}}(x)\\Theta ^{\\mu _{1} \\ldots \\mu _{s}}(x)\\nonumber \\\\&&+\\frac{1}{2!}",
"\\int d^dx d^dy\\,a_{\\mu _{1}\\ldots \\mu _{s}}(x) a_{\\nu _{1}\\ldots \\nu _{s}}(y)T^{\\mu _{1} \\ldots \\mu _{s}\\nu _{1}\\ldots \\nu _{s}}(x,y)+\\ldots $ where $\\Theta ^{\\mu _{1} \\ldots \\mu _{s}}(x)=\\frac{\\delta \\left(i\\,W[a]\\right)}{\\delta a_{\\mu _{1}\\ldots \\mu _{s}}(x)}\\Big {|}_{a=0}$ is a tadpole (1-point function) and $T^{\\mu _{1} \\ldots \\mu _{s} \\nu _{1}\\ldots \\nu _{s}}(x, y)=\\frac{\\delta ^2\\left(i\\,W[a]\\right)}{\\delta a_{\\mu _{1}\\ldots \\mu _{s}}(x) \\delta a_{\\nu _{1}\\ldots \\nu _{s}}(y)}\\Big {|}_{a=0}$ is a 2-point function.",
"Using Feynman diagrams we wish to compute the 2-point function including not only the bubble diagram (as in [1]) but also tadpoles and seagulls.",
"The one-loop 1-pt correlator for the external field is (up to the linear order): $\\langle \\!\\langle J^{\\mu _{1} \\ldots \\mu _{s}}(x)\\rangle \\!\\rangle &=& \\frac{\\delta W[a] }{\\delta a_{\\mu _{1}\\ldots \\mu _{s}}(x)}\\nonumber \\\\&=&-i\\left(\\Theta ^{\\mu _{1} \\ldots \\mu _{s}}(x) + \\int d^dy \\, a_{\\nu _{1}\\ldots \\nu _{s}}(y)T^{\\mu _{1} \\ldots \\mu _{s}\\nu _{1}\\ldots \\nu _{s}}(x,y)+\\ldots \\right)$ External spin $s$ fields $a_{\\mu _1\\ldots \\mu _s}$ are in particular $\\text{Spin 1}\\quad a_\\mu & = & A_\\mu \\quad \\text{ gauge field}\\\\\\text{Spin 2}\\quad a_{\\mu \\nu }& = & h_{\\mu \\nu } \\quad \\text{ graviton field} \\nonumber $ We will need one-loop conservation which for spin 1 reads $\\partial _\\mu \\langle \\!\\langle J^{\\mu }(x) \\rangle \\!",
"\\rangle =0 $ Ward identity for the two-point function in momentum space can be written as $k_\\mu \\tilde{T}^{\\mu \\nu }(k)=0$ Furthermore, for spin 2, the energy-momentum tensor is defined with $\\langle \\!\\langle T^{\\mu \\nu }(x) \\rangle \\!",
"\\rangle =\\frac{2}{\\sqrt{g}}\\frac{\\delta W}{\\delta h_{\\mu \\nu }(x)}$ .",
"The full conservation law of the energy-momentum tensor is $\\nabla _\\mu \\langle \\!\\langle T^{\\mu \\mu }(x) \\rangle \\!",
"\\rangle =0 $ Hence, the Ward identity for one-point function is $\\partial _\\mu \\Theta ^{\\mu \\mu }(x) =0$ while for two-point correlator we have $\\partial _{\\mu } T^{\\mu \\mu \\nu \\nu }(x,y)&=&\\frac{1}{2} \\eta ^{\\nu \\nu }\\delta (x-y)\\partial _\\mu \\Theta ^{\\mu \\mu }(x)+\\frac{1}{2}\\Theta ^{\\nu \\nu }(x)\\partial ^\\mu \\delta (x-y)\\nonumber \\\\&&- \\partial _\\mu \\left(\\delta \\left(x-y\\right)\\Theta ^{\\mu \\nu }\\left(x\\right)\\right)\\eta ^{\\mu \\nu }$ As we will see, the tadpole contribution is $\\tilde{\\Theta }^{\\mu \\mu }(k)=\\tilde{\\Theta }\\, \\eta ^{\\mu \\mu }$ where $\\tilde{\\Theta }$ is a constant.",
"The Ward identity in momentum space is now $k_\\mu \\tilde{T}^{\\mu \\mu \\nu \\nu } (k) =\\left[ - k^{\\nu } \\eta ^{\\mu \\nu }+\\frac{1}{2} k^\\mu \\eta ^{\\nu \\nu }\\right]\\tilde{\\Theta }$"
],
[
"Fermions - spin 1",
"The action for the theory of fermions interacting with gauge field can be written as $S=\\int \\text{d}x \\left[\\bar{\\psi }\\left(i\\gamma ^\\mu D_\\mu -m\\right)\\psi \\right]$ where $D_\\mu =\\partial _\\mu -i\\,A_\\mu $ .",
"There is one fermion-fermion-photon vertex $V^\\mu _{ffp}&:& i\\gamma ^\\mu $ In the case of fermions coupled to gauge field the tadpole diagram vanishes, while the seagull is zero because the theory is linear in the gauge field.",
"The only contribution we get from the 2-pt correlator ((11.7) from [1]) which in the momentum space reads $\\tilde{T}^{\\mu \\nu }(k)&=&\\frac{2^{-d+\\lfloor \\frac{d}{2}\\rfloor } \\, i\\, \\pi ^{-\\frac{d}{2}} m^{d-2}}{4m^2-k^2}\\Gamma \\left(1-\\frac{d}{2}\\right)\\nonumber \\\\&&\\times \\left(-4m^2 +\\, _2F_1\\left[1,1-\\frac{d}{2};\\frac{3}{2};\\frac{k^2}{4m^2}\\right](4m^2+(d-2)k^2)\\right)\\pi ^{\\mu \\nu }$ where $\\pi ^{\\mu \\nu }=\\eta ^{\\mu \\nu }-\\frac{k^\\mu k^\\nu }{k^2}$ is the projector.",
"Since the 2-point correlator can be expressed in terms of the projector, it satisfies Ward identity (REF ) We can expand the two-point correlator in the IR region $\\tilde{T}^{\\mu \\nu }(k) &=&-2^{1-d+\\lfloor \\frac{d}{2}\\rfloor } i\\, m^{d-2} \\pi ^{-\\frac{d}{2}}\\sum _{n=1}^\\infty \\frac{ n\\, m^{-2n} \\Gamma \\left(2n-\\frac{d}{2}\\right)}{2^{n}(2n+1)!!",
"}k^{2n}\\pi ^{\\mu \\nu }$ Using the Fourier transform of (REF ) in the one-loop 1-point function (REF ) we get $\\langle \\!\\langle J^\\mu (x)\\rangle \\!\\rangle =2^{1-d+\\lfloor \\frac{d}{2}\\rfloor }m^{d-2} \\pi ^{-\\frac{d}{2}}\\sum _{n=1}^\\infty \\frac{(-1)^n n\\, m^{-2n}\\Gamma \\left(2n-\\frac{d}{2}\\right)}{2^n (2n+1)!!",
"}\\Box ^{n-1}\\partial _\\nu F^{\\mu \\nu }$ The one-loop 1-point correlator satisfies (REF ) Using the same expansion in the IR (REF ) for the effective action (REF ) we obtain $W&=&2^{-1-d+\\lfloor \\frac{d}{2}\\rfloor } m^{d-2} \\pi ^{-\\frac{d}{2}}\\sum _{n=1}^\\infty \\frac{(-1)^n n\\, m^{-2n} \\Gamma \\left(2n-\\frac{d}{2}\\right)}{2^n (2n+1)!!",
"}\\int \\text{d}^dxF_{\\mu \\nu }\\Box ^{n-1}F^{\\mu \\nu }\\nonumber \\\\&\\stackrel{\\text{IR}}{=}&-\\frac{2^{-2-d+\\lfloor \\frac{d}{2}\\rfloor }}{3}\\, m^{d-4}\\pi ^{-\\frac{d}{2}}\\Gamma \\left(2-\\frac{d}{2}\\right)\\int \\text{d}^dxF_{\\mu \\nu }F^{\\mu \\nu }$ So, in the IR region (large m) we get the Maxwell action.",
"Furthermore, the dominating term in the UV $\\left(O(m^0)\\right)$ of (REF ) corresponds to the massless case (B.2) from [1] $\\tilde{T}^{\\mu \\nu }(k)\\stackrel{UV}{=}-\\frac{2^{2-2d+\\lfloor \\frac{d}{2}\\rfloor } \\, \\pi ^{\\frac{3}{2}-\\frac{d}{2}}(d-2)}{\\left(-1+e^{i\\pi d}\\right)\\Gamma \\left(\\frac{d+1}{2}\\right)}(k^2)^{\\frac{d}{2}-1}\\pi ^{\\mu \\nu }$ The effective action in the UV is then $W\\stackrel{UV}{=} \\frac{(-1)^{\\frac{d}{2}}2^{1-2d+\\lfloor \\frac{d}{2}\\rfloor } \\,\\pi ^{\\frac{3}{2}-\\frac{d}{2}}(d-2)}{\\left(-1+e^{i\\pi d}\\right)\\Gamma \\left(\\frac{d+1}{2}\\right)}F^{\\mu \\nu }\\Box ^{\\frac{d}{2}-2}F_{\\mu \\nu }$ For the analysis of the odd parity correlators we will restrict ourselves to $d=3$ .",
"The odd part of the two-point correlator is non-vanishing only in $3d$ and it is given by $\\tilde{T}^{\\mu \\nu }_{o}(k)=\\frac{m}{2\\pi k}\\text{ArcCoth}\\left(\\frac{2m}{k}\\right) \\epsilon ^{\\mu \\nu \\lambda }k_\\lambda $ The expansion of (REF ) in the IR reads $\\tilde{T}^{\\mu \\nu }_{o}(k)=\\frac{1}{\\pi }\\sum _{n=0}^\\infty \\frac{k^{2n}m^{-2n}}{2^{2(n+1)}(2n+1)}\\epsilon ^{\\mu \\nu \\lambda }k_\\lambda $ Using the IR expansion in (REF ), the odd part of the one-loop 1-point correlator is now $\\langle \\!\\langle J^\\mu (x)\\rangle \\!\\rangle =\\frac{1}{\\pi }\\sum _{n=0}^\\infty \\frac{(-1)^n m^{-2n}}{2^{2n+3}(2n+1)}\\epsilon ^{\\mu \\nu \\lambda }\\Box ^nF_{\\lambda \\nu }$ and just like the even parity part satisfies (REF ).",
"The effective action in the IR (the dominating term) $W&\\stackrel{IR}{=}&\\frac{1}{8\\pi }\\epsilon ^{\\mu \\nu \\lambda }\\int d^3x\\,A_\\mu \\partial _\\nu A_\\lambda +\\ldots $ corresponds to Chern-Simons term in 3d $S_{CS}=\\frac{1}{8\\pi }\\int d^3x\\,Tr\\left(A \\wedge d A+\\frac{2}{3} A\\wedge A\\wedge A\\right)$"
],
[
"Scalars - spin 1",
"The action in the scalar QED model is $S=\\int \\text{d}^dx\\left[D_\\mu \\varphi ^\\dagger D^\\mu \\varphi -m^2\\varphi ^\\dagger \\varphi \\right]$ where $D_\\mu =\\partial _\\mu -i \\,A_\\mu $ .",
"The full covariant action is $S=\\int \\text{d}x\\left[\\partial _\\mu \\varphi ^\\dagger \\partial ^\\mu \\varphi +i\\,A_\\mu \\left(\\varphi ^\\dagger \\partial ^\\mu \\varphi -\\partial ^\\mu \\varphi ^\\dagger \\varphi \\right)+A_\\mu A^\\mu \\varphi ^\\dagger \\varphi -m^2\\varphi ^\\dagger \\varphi \\right]$ In the scalar model the scalar-scalar-photon vertex is $V^\\mu _{ssp}(p,p^{\\prime })&:& -i(p+p^{\\prime })^\\mu $ and we also have scalar-scalar-photon-photon vertex (coming from $\\int d^d xA^\\mu A_\\mu \\varphi ^\\dagger \\varphi $ term in Lagrangian) $V^{\\mu \\nu }_{sspp}(p,p^{\\prime }) &:& 2i\\eta ^{\\mu \\nu }$ The two-point function for the massive scalar in any dimension $d$ for spin $s=1$ is $\\tilde{T}^{\\mu \\nu }(k)=- 2^{1-d}\\, i\\, \\pi ^{-d/2}m^{d-2} \\Gamma \\left(1-\\frac{d}{2}\\right)\\left( \\,\\,{_2F_1}\\left[1,1-\\frac{d}{2};\\frac{3}{2};\\frac{k^2}{4m^2}\\right]\\pi ^{\\mu \\nu }+\\frac{k^\\mu k^\\nu }{k^2}\\right)$ which has a non-conserved part.",
"However, since the theory is quadratic in the external photon field $A$ we also have a seagull diagram (which is obtained by joining with a unique a fermion line the two fermion legs of the vertex (REF )) for which we obtain $\\tilde{T}_{(s)}^{\\mu \\nu }(k) = 2^{1-d}i \\pi ^{-\\frac{d}{2}}m^{d-2}\\Gamma \\left(1-\\frac{d}{2}\\right)\\eta ^{\\mu \\nu } $ After combining (REF ) and (REF ) we can write down the full 2-point function $\\tilde{T}^{\\mu \\nu }(k) = 2^{1-d}i\\pi ^{-\\frac{d}{2}}m^{d-2}\\Gamma \\left(1-\\frac{d}{2}\\right)\\left(1-{ _2F_1} \\left[1,1-\\frac{d}{2};\\frac{3}{2};\\frac{k^2}{4 m^2}\\right]\\right)\\pi ^{\\mu \\nu },$ which is conserved.",
"Expanding the two-point function (REF ) in the IR gives $\\tilde{T}^{\\mu \\nu }(k)& =&-2^{-d} i\\, m^{d-4} \\pi ^{-\\frac{d}{2}}\\sum _{n=0}^\\infty \\frac{ \\, m^{-2n} \\Gamma \\left(2+n-\\frac{d}{2}\\right)}{2^{n}(2n+3)!!",
"}k^{2n+2}\\pi ^{\\mu \\nu }$ Using the IR expansion together with (REF ), the one-loop 1-point function (REF ) now reads $\\langle \\!\\langle J^\\mu \\rangle \\!\\rangle =-2^{-d} m^{d-4} \\pi ^{-\\frac{d}{2}}\\sum _{n=0}^\\infty \\frac{(-1)^{n} \\, m^{-2n} \\Gamma \\left(2+n-\\frac{d}{2}\\right)}{2^n (2n+3)!!",
"}\\Box ^{n}\\partial _\\nu F^{\\mu \\nu }$ On the other hand, the dominating term of the effective action in the IR region is $W\\stackrel{\\text{IR}}{=}-\\frac{2^{-d}}{3} m^{d-4} \\pi ^{-\\frac{d}{2}}\\Gamma \\left(2-\\frac{d}{2}\\right)\\int \\text{d}^dx F_{\\mu \\nu }F^{\\mu \\nu }$ In the IR (for large mass m) we get the Maxwell action.",
"The leading order term of the expansion in the UV (term $m^0$ corresponds to (B.13) from [1]) $\\tilde{T}^{\\mu \\nu }(k) \\stackrel{UV}{=} -\\frac{2^{3-2d} \\, \\pi ^{\\frac{3}{2}-\\frac{d}{2}}(k^2)^{\\frac{d}{2}-1}}{\\left(-1+e^{i\\pi d}\\right)\\Gamma \\left(\\frac{d+1}{2}\\right)}\\pi ^{\\mu \\nu }$ Hence, the effective action in the UV is $W\\stackrel{UV}{=} -i\\frac{(-1)^{\\frac{d}{2}}2^{3-2d} \\, \\pi ^{\\frac{3}{2}-\\frac{d}{2}}}{\\left(-1+e^{i\\pi d}\\right)\\Gamma \\left(\\frac{d+1}{2}\\right)}\\int d^dxF^{\\mu \\nu }\\Box ^{\\frac{d}{2}-2}F_{\\mu \\nu }$"
],
[
"Fermions - spin 2",
"Let us consider the free fermion theory in a generic dimension $d$ $S&=& \\int d^d x \\, \\sqrt{|g|} \\left[\\, i\\overline{\\psi } E_a^m\\gamma ^a\\left(\\partial _m +\\frac{1}{2} \\Omega _m \\right)\\psi -m\\bar{\\psi }\\psi \\right]$ where $E^m_a$ is the inverse vierbein.",
"From now on we will set $g_{\\mu \\nu }=\\eta _{\\mu \\nu }+ h_{\\mu \\nu }$ .",
"Using the following expansions $g^{\\mu \\nu }= \\eta ^{\\mu \\nu }-h^{\\mu \\nu } +(h^2)^{\\mu \\nu }+\\ldots , &\\qquad &\\sqrt{|g|}= 1+\\frac{1}{2} h+\\frac{1}{8} h^2 -\\frac{1}{4} h^{\\mu \\nu }h_{\\mu \\nu }+\\ldots ,\\nonumber \\\\e_a^\\mu = \\delta _a^\\mu -\\frac{1}{2} h_a^\\mu +\\frac{3}{8} (h^2)_a^\\mu +\\ldots , &\\qquad &e^a_\\mu = \\delta ^a_\\mu +\\frac{1}{2} h^a_\\mu -\\frac{1}{8} (h^2)^a_\\mu +\\ldots $ we can expand the parity even part of the action (REF ) in powers of $h$ : $S_e&=& \\int d^dx \\, \\Big {[} \\frac{i}{2} \\overline{\\psi }\\gamma ^m{\\stackrel{\\leftrightarrow }{\\partial }}_m \\psi -m\\bar{\\psi }\\psi +\\frac{1}{2} h_\\mu ^\\mu \\left( \\frac{i}{2} \\overline{\\psi }\\gamma ^m{\\stackrel{\\leftrightarrow }{\\partial }}_m \\psi -m\\bar{\\psi }\\psi \\right) -\\frac{i}{4}\\overline{\\psi }h^m_a\\gamma ^a {\\stackrel{\\leftrightarrow }{\\partial }}_m \\psi \\nonumber \\\\&&\\qquad \\quad + \\frac{1}{ 8} \\left( (h_\\mu ^\\mu )^2-2 h_\\mu ^\\nu h_\\nu ^\\mu \\right)\\left(\\frac{i}{2} \\overline{\\psi }\\gamma ^m{\\stackrel{\\leftrightarrow }{\\partial }}_m \\psi -m\\bar{\\psi }\\psi \\right)\\nonumber \\\\&&\\qquad \\quad -\\frac{i}{8} h^\\mu _\\mu \\overline{\\psi }h^m_a\\gamma ^a {\\stackrel{\\leftrightarrow }{\\partial }}_m \\psi +\\frac{3i}{16}\\overline{\\psi } (h^2)^m_a \\gamma ^a {\\stackrel{\\leftrightarrow }{\\partial }}_m\\psi +\\ldots \\Big {]} $ There is one fermion-fermion-graviton vertexWe use the convention according to which two repeated identical indices represent a symmetrized couple of indices, and so on.",
": $V^{\\mu \\mu }_{ffh}(p,p^{\\prime })&:& -\\frac{i}{4} (p+p^{\\prime })^\\mu \\gamma ^\\mu +\\frac{i}{4}\\eta ^{\\mu \\mu } ({p}+{p^{\\prime }}-2m) $ and one vertex with two fermions and two gravitons: $V^{\\mu \\mu \\nu \\nu }_{ffhh}(p,p^{\\prime })&:&\\frac{ 3i}{16} \\left( (p+p^{\\prime })^\\mu \\gamma ^{\\nu }\\eta ^{\\mu \\nu } + (p+p^{\\prime })^{\\nu } \\gamma ^{\\mu } \\eta ^{\\mu \\nu }\\right)\\nonumber \\\\&&+\\frac{i}{8} ({p}+{p^{\\prime }}-2m) \\left( \\eta ^{\\mu \\mu } \\eta ^{\\nu \\nu }-2\\eta ^{\\mu \\nu } \\eta ^{\\mu \\nu }\\right)\\nonumber \\\\&&-\\frac{i}{8}\\left((p+p^{\\prime })^\\mu \\gamma ^\\mu \\eta ^{\\nu \\nu }+(p+p^{\\prime })^{\\nu }\\gamma ^{\\nu }\\eta ^{\\mu \\mu }\\right)$ We can also expand the odd parity part of the action (the latter contains a part proportional to the completely antisymmetric symbol).",
"We will restrict ourselves to 3d because only in this case can we get a non-vanishing contribution to the effective action and 1-point correlator.",
"$S_o&=& \\frac{1}{16}\\int d^3x \\,\\epsilon ^{abc} \\partial _a h_{b\\sigma }h_c^\\sigma \\overline{\\psi } \\psi $ The relevant vertex with two fermions and two gravitons is $V^{\\mu \\mu \\nu \\nu }_{\\epsilon , ffhh}&:&\\frac{1}{16} \\, \\eta ^{\\mu \\nu }\\epsilon ^{\\mu \\nu \\lambda } \\,(k-k^{\\prime })_\\lambda $"
],
[
"Even parity part",
"The tadpole contribution is now $\\tilde{\\Theta }^{\\mu \\mu }(k)= -2^{-2-d+\\lfloor \\frac{d}{2}\\rfloor }\\,i\\,m^d \\pi ^{\\frac{d}{2}}\\Gamma \\left(-\\frac{d}{2}\\right) \\eta ^{\\mu \\mu }=\\tilde{\\Theta }\\, \\eta ^{\\mu \\mu }$ where $\\tilde{\\Theta }$ is a constant.",
"Since the theory of gravity is non-linear we have a contribution from the seagull term, which can be written as $\\tilde{T}^{\\mu \\mu \\nu \\nu }_{(s)}(k) = 2^{-3-d+\\lfloor \\frac{d}{2}\\rfloor }\\, i\\,m^d \\pi ^{\\frac{d}{2}}\\Gamma \\left(-\\frac{d}{2}\\right) \\left(3\\eta ^{\\mu \\nu }\\eta ^{\\mu \\nu }-2 \\eta ^{\\mu \\mu }\\eta ^{\\nu \\nu }\\right)$ The bubble diagram contributes two parts, the transverse (conserved) part, $\\tilde{T}^{\\mu \\mu \\nu \\nu }_t(k)&=&-\\frac{1}{d(d+1)k^2} 2^{-2-d+\\lfloor \\frac{d}{2}\\rfloor }\\, i\\,m^d\\pi ^{\\frac{d}{2}}\\Gamma \\left(1-\\frac{d}{2}\\right)\\nonumber \\\\ &&\\left[\\left(-8m^2+(d+1)k^2+{_2F_1}\\left[1,-\\frac{d}{2},\\frac{1}{2},\\frac{k^2}{4m^2}\\right](8m^2+(d-1)k^2)\\right)\\pi ^{\\mu \\nu }\\pi ^{\\mu \\nu } \\right.",
"\\nonumber \\\\&& \\left.",
"+\\left(-4m^2+(d+1)k^2+{_2F_1}\\left[1,-\\frac{d}{2},\\frac{1}{2},\\frac{k^2}{4m^2}\\right](4m^2-k^2)\\right)\\pi ^{\\mu \\mu }\\pi ^{\\nu \\nu }\\right]$ whose expansion in the IR is $\\tilde{T}^{\\mu \\mu \\nu \\nu }_t(k)=-2^{-3-d+\\lfloor \\frac{d}{2}\\rfloor } i\\, m^d \\pi ^{-\\frac{d}{2}}\\sum _{n=1}^\\infty \\frac{ m^{-2n} \\Gamma \\left(n-\\frac{d}{2}\\right)}{2^{n}(2n+1)!!",
"}k^{2n}\\left((2n-1)\\pi ^{\\mu \\nu }\\pi ^{\\mu \\nu }-\\pi ^{\\mu \\mu }\\pi ^{\\nu \\nu }\\right),$ and the non-transverse (non-conserved) part ${\\tilde{T}^{\\mu \\mu \\nu \\nu }_{nt}}(k)&=& -2^{-3-d+\\lfloor \\frac{d}{2}\\rfloor }\\,i\\,m^d\\pi ^{\\frac{d}{2}}\\Gamma \\left(-\\frac{d}{2}\\right)\\left(\\eta ^{\\mu \\nu }\\eta ^{\\mu \\nu }- \\eta ^{\\mu \\mu }\\eta ^{\\nu \\nu }\\right).$ Taking formulas (REF ), (REF ), (REF ) and (REF ) and substituting them in (REF ) we can see that the Ward identity is satisfied for any dimension $d$ .",
"The one-loop 1-point function (energy-momentum tensor) defined as $\\langle \\!\\langle T^{\\mu \\nu }(x) \\rangle \\!",
"\\rangle =\\frac{2}{\\sqrt{g}}\\frac{\\delta W}{\\delta h_{\\mu \\nu }(x)}$ now becomes $\\langle \\!\\langle T^{\\mu \\mu }(x) \\rangle \\!",
"\\rangle &=&-2^{-1-d+\\lfloor \\frac{d}{2}\\rfloor }\\,m^d \\pi ^{-\\frac{d}{2}}\\left[\\Gamma \\left(-\\frac{d}{2}\\right)g^{\\mu \\mu }+\\sum _{n=1}^\\infty \\frac{(-1)^n\\, m^{-2n} \\Gamma \\left(n-\\frac{d}{2}\\right)}{2^{n+1}(2n+1)!!",
"}\\right.\\nonumber \\\\&&\\left.",
"\\times \\left((2n-1)\\Box ^{n-1}G^{\\mu \\mu }+(n-1)\\Box ^{n-2}(\\eta ^{\\mu \\mu }\\Box -\\partial ^\\mu \\partial ^\\mu )R\\right)\\right]+O(h^2)$ where $G_{\\mu \\mu }=R_{\\mu \\mu }-\\frac{1}{2} \\eta _{\\mu \\mu }R$ is the Einstein tensor.",
"The energy-momentum tensor is clearly divergence free (REF ).",
"For the effective action in the IR we obtain (in the even parity sector) $W &\\stackrel{\\text{IR}}{=}&- 2^{-1-d+\\lfloor \\frac{d}{2}\\rfloor }m^d\\pi ^{-\\frac{d}{2}} \\int \\text{d}^dx \\sqrt{g}\\times \\left[\\Gamma \\left(-\\frac{d}{2}\\right)- \\frac{\\Gamma \\left(1-\\frac{d}{2}\\right)}{24m^2} R \\right.\\nonumber \\\\&&\\left.",
"- \\frac{\\Gamma \\left(2-\\frac{d}{2}\\right)}{80m^4}\\left(R_{\\mu \\nu \\lambda \\rho }R^{\\mu \\nu \\lambda \\rho }-2R_{\\mu \\nu }R^{\\mu \\nu }+\\frac{1}{3}R^2\\right)+\\ldots \\right]+O(h^3)$ The divergent part of the effective action for $d=4$ (i.e.",
"$d=4+\\varepsilon $ ) is $W &\\stackrel{\\text{IR}}{=}& \\frac{1}{8\\pi ^2\\varepsilon } \\int \\text{d}^4x \\sqrt{g} \\left(m^4+ \\frac{1}{12}m^{2}R- \\frac{1}{40}\\mathcal {W}^2+\\ldots \\right)+O(h^3)$ The first term is a cosmological constant term and the second is the linearized Einstein-Hilbert action.",
"The third term ($m^0$ term) is the Weyl density $\\mathcal {W}^2=R_{\\mu \\nu \\lambda \\rho }R^{\\mu \\nu \\lambda \\rho }-2R_{\\mu \\nu }R^{\\mu \\nu }+\\frac{1}{3} R^2$ (conformal invariant in 4d).",
"The dominating term in the UV ($O(m^0)$ term corresponds to (B.3) from [1]) of the transverse part $\\tilde{T}_{t}^{\\mu \\mu \\nu \\nu }(k)$ is $\\tilde{T}_t^{\\mu \\mu \\nu \\nu }(k)\\stackrel{UV}{=} \\frac{2^{-3-2d+\\lfloor \\frac{d}{2}\\rfloor } \\, \\pi ^{\\frac{3}{2}-\\frac{d}{2}}(k^2)^{\\frac{d}{2}}}{\\left(-1+e^{i\\pi d}\\right)\\Gamma \\left(\\frac{d+3}{2}\\right)}((d-1)\\pi ^{\\mu \\nu }\\pi ^{\\mu \\nu }-\\pi ^{\\mu \\mu }\\pi ^{\\nu \\nu })$ The effective action in the UV is then $W &\\stackrel{UV}{=}&(-1)^{\\frac{d}{2}} \\frac{2^{-4-2d+\\lfloor \\frac{d}{2}\\rfloor }\\pi ^{\\frac{3}{2}-\\frac{d}{2}}}{(-1+e^{i\\pi d})\\Gamma \\left(\\frac{d+3}{2}\\right)} \\int \\text{d}^dx \\sqrt{g}\\left[(d-4)R_{\\mu \\nu \\lambda \\rho }\\Box ^{\\frac{d}{2}-2}R^{\\mu \\nu \\lambda \\rho }\\right.\\nonumber \\\\&&\\left.",
"+6 \\left(R_{\\mu \\nu \\lambda \\rho }\\Box ^{\\frac{d}{2}-2}R^{\\mu \\nu \\lambda \\rho }-2R_{\\mu \\nu }\\Box ^{\\frac{d}{2} -2}R^{\\mu \\nu }+\\frac{1}{3}R\\Box ^{\\frac{d}{2} -2}R\\right)+\\ldots \\right]+O(h^3)$"
],
[
"Odd parity part",
"In 3d the contribution from the seagull diagram with vertex (REF ) becomes $\\tilde{T}^{\\mu \\mu \\nu \\nu }_{(s,o)}(k)=-\\frac{m^2}{16\\pi }\\eta ^{\\mu \\nu }\\epsilon ^{\\mu \\nu \\lambda }k_\\lambda $ The odd part of the two-point correlator is non-vanishing only in $3d$ (the vertex is (REF )).",
"The transverse part can be written as $\\tilde{T}^{\\mu \\mu \\nu \\nu }_{t,o}(k)&=&-\\frac{m}{64\\pi k}\\left((k^2-4m^2)\\text{ArcCoth}\\left(\\frac{2m}{k}\\right)+2mk\\right)\\pi ^{\\mu \\nu }\\epsilon ^{\\mu \\nu \\lambda }k_\\lambda $ and the expansion of $\\tilde{T}^{\\mu \\mu \\nu \\nu }_{t,o}(k)$ in the IR is $\\tilde{T}^{\\mu \\mu \\nu \\nu }_{t,o}(k)=-\\frac{1}{64\\pi }\\sum _{n=0}^\\infty \\frac{k^{2(n+1)}m^{-2n}}{4^{2n}(4(n+1)^2-1)}\\pi ^{\\mu \\nu } \\epsilon ^{\\mu \\nu \\lambda }k_\\lambda $ The odd non-transverse part readsIn the notation from the previous paper $\\pi ^{\\mu \\nu } \\epsilon ^{\\mu \\nu \\lambda }k_\\lambda $ corresponds to $(n_1\\cdot \\pi \\cdot n_2)\\epsilon (k\\cdot n_1\\cdot n_2)$ $\\tilde{T}^{\\mu \\mu \\nu \\nu }_{nt,o}(k)&=&\\frac{m^2}{16\\pi }\\eta ^{\\mu \\nu }\\epsilon ^{\\mu \\nu \\lambda }k_\\lambda $ and can be canceled by the seagull contribution (REF ).",
"So, only the transverse odd part remains.",
"The odd part of the one-loop 1-pt function (energy-momentum tensor) $\\langle \\!\\langle T^{\\mu \\mu }(x) \\rangle \\!",
"\\rangle =\\frac{1}{32\\pi }\\sum _{n=0}^\\infty \\frac{(-1)^n\\, m^{-2n} }{4^{2n}(4(n+1)^2-1)}\\Box ^{n}C^{\\mu \\mu }$ where $C^{\\mu \\mu }$ is linearized the Cotton tensor (REF ).",
"The effective action in the IR (the dominating term) $W\\stackrel{IR}{=}-\\frac{1}{384\\pi }\\epsilon ^{\\mu \\nu \\lambda }\\int d^3x\\,h_{\\nu \\nu }\\left(\\partial _\\lambda \\partial ^\\mu \\partial ^\\nu h_{\\mu \\mu }-\\partial _\\lambda \\Box h_\\mu ^\\nu \\right)+O(h^3)$ corresponds to gravitational Chern-Simons term in 3d $S_{gCS}=\\frac{1}{192\\pi }\\epsilon ^{\\mu \\nu \\lambda }\\int d^3x\\,\\left(\\partial _\\mu \\omega _\\nu {}^{ab}\\omega _\\lambda ^{ b a}+\\frac{2}{3}\\omega _{\\mu a}{}^b\\omega _{\\nu b}{}^c\\omega _{\\lambda c}{}^a \\right)$"
],
[
"Scalars - spin 2",
"Let us consider the action of a scalar field $\\varphi $ in a curved space ($g_{\\mu \\nu }=\\eta _{\\mu \\nu }+h_{\\mu \\nu }$ ) with a scalar curvature coupling $S=\\int d^dx\\sqrt{g}\\left(g^{\\mu \\nu }\\partial _\\mu \\varphi ^{\\dagger }\\partial _\\nu \\varphi -m^2\\varphi ^{\\dagger }\\varphi +\\xi R \\varphi ^{\\dagger }\\varphi \\right)$ Let us redefine $\\phi =g^{\\frac{1}{4}}\\varphi $ .",
"The expansion of the action in the external field h is $S&=&\\int d^dx\\left[\\eta ^{\\mu \\nu }\\partial _\\mu \\phi ^{\\dagger }\\partial _\\nu \\phi -m^2\\phi ^{\\dagger }\\phi + h^{\\mu \\nu }\\left(\\frac{1}{4}\\phi ^{\\dagger }\\stackrel{\\leftrightarrow }{\\partial }_\\mu \\stackrel{\\leftrightarrow }{\\partial }_\\nu \\phi +\\left(\\xi -\\frac{1}{4}\\right)(\\partial _\\mu \\partial _\\nu -\\Box \\eta _{\\mu \\nu })\\phi ^{\\dagger }\\phi \\right)\\right.\\nonumber \\\\&&\\left.",
"+h^{\\mu \\sigma }h_\\sigma ^\\nu \\partial _\\mu \\phi ^{\\dagger }\\partial _\\nu \\phi +\\frac{1}{16}h\\Box h\\phi ^{\\dagger }\\phi +\\left(-\\frac{\\xi }{4}+\\frac{1}{8}\\right)\\partial _\\mu h \\partial ^\\mu h\\phi ^{\\dagger }\\phi -2\\xi h^{\\mu \\nu }\\partial _\\nu \\partial _\\lambda h_\\mu ^\\lambda \\phi ^{\\dagger }\\phi \\right.\\nonumber \\\\&&\\left.+ \\xi h^{\\mu \\nu }\\Box h_{\\mu \\nu }\\phi ^{\\dagger }\\phi -\\xi \\partial _\\nu h^{\\mu \\nu }\\partial _\\lambda h_\\mu ^\\lambda \\phi ^{\\dagger }\\phi +\\frac{3}{4}\\xi \\partial _\\lambda h_{\\mu \\nu }\\partial ^\\lambda h^{\\mu \\nu }\\phi ^{\\dagger }\\phi -\\frac{1}{2}\\xi \\partial _\\lambda h^{\\mu \\nu }\\partial _\\nu h_\\mu ^\\lambda \\phi ^{\\dagger }\\phi \\right.\\nonumber \\\\&&\\left.\\left(\\xi -\\frac{1}{4}\\right) h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu h\\phi ^{\\dagger }\\phi + \\left(\\xi -\\frac{1}{4}\\right)\\partial _\\mu h\\partial _\\nu h^{\\mu \\nu }\\phi ^{\\dagger }\\phi \\right]$ The scalar-scalar-graviton vertex is: $V^{\\mu \\mu }_{ssh}(p,p^{\\prime })&:& -\\frac{i}{4} (p^\\mu +p^{\\prime \\mu })^2 -i\\left(\\xi -\\frac{1}{4}\\right)\\left((p^{\\prime \\mu }-p^\\mu )^2 - \\eta ^{\\mu \\mu } (p^{\\prime }-p)^2\\right) $ and there is a vertex with two scalars and two gravitons: $V^{\\mu \\mu \\nu \\nu }_{sshh}(p,p^{\\prime },k,k^{\\prime })&:&i\\eta ^{\\mu \\nu }\\left( p^{\\prime \\mu }p^\\nu +p^\\mu p^{\\prime \\nu }\\right)-i\\Big {[}\\left(\\xi -\\frac{1}{4}\\right)\\left(\\eta ^{\\mu \\mu }k^\\nu k^\\nu +\\eta ^{\\nu \\nu }k^\\mu k^\\mu \\right)\\nonumber \\\\&&+2\\left(\\xi \\eta ^{\\mu \\nu }\\eta ^{\\mu \\nu }+\\frac{1}{16}\\eta ^{\\mu \\mu }\\eta ^{\\nu \\nu }\\right)k^2-4\\xi \\eta ^{\\mu \\nu }k^{\\mu }k^{\\nu }\\Big {]}\\nonumber \\\\&&- i\\Big {[}\\left(\\left(\\frac{1}{4}-\\frac{\\xi }{2}\\right)\\eta ^{\\mu \\mu }\\eta ^{\\nu \\nu }+\\frac{3}{2}\\xi \\eta ^{\\mu \\nu }\\eta ^{\\mu \\nu }\\right)k\\cdot k^{\\prime } \\\\&&+\\left(\\xi -\\frac{1}{4}\\right)\\left(\\eta ^{\\mu \\mu }k^\\nu k^{\\prime \\nu }+\\eta ^{\\nu \\nu }k^\\mu k^{\\prime \\mu }\\right)-2\\xi \\eta ^{\\mu \\nu }k^\\mu k^{\\prime \\nu }-\\xi \\eta ^{\\mu \\nu } k^\\nu k^{\\prime \\mu }\\Big {]}\\nonumber $ The result for the tadpole diagram is $\\tilde{\\Theta }^{\\mu \\mu }= 2^{-d-1} \\,i\\, \\pi ^{-d/2} m^d \\Gamma \\left(-\\frac{d}{2}\\right) \\,\\eta ^{\\mu \\mu }$ while the contribution from the seagull term is $\\tilde{T}_{(s)}^{\\mu \\mu \\nu \\nu }(k)&=&-2^{-4-d} \\,i \\pi ^{-d/2} m^{d-2} \\Gamma \\left(-\\frac{d}{2}\\right) \\nonumber \\\\&&\\times \\left(d k^2 (1-4 \\xi ) \\eta ^{\\mu \\mu } \\eta ^{\\nu \\nu }+4 \\eta ^{\\mu \\nu }\\eta ^{\\mu \\nu } \\left(4 m^2-d k^2 \\xi \\right)+8 d \\xi \\eta ^{\\mu \\nu }k^{\\mu } k^{\\nu }\\right)$ Furthermore, the transverse part of the bubble diagram reads $\\tilde{T}_{t}^{\\mu \\mu \\nu \\nu }(k)&=&-\\frac{1}{3 d \\left(d^2-1\\right) k^4}i2^{-d-2} e^{-\\frac{1}{2} i \\pi d} \\pi ^{-d/2} (-m^2)^{d/2} m^{-2} \\Gamma \\left(1-\\frac{d}{2}\\right)\\nonumber \\\\&& \\Big {[} \\Big {(}12 \\left(d^2-1\\right) k^4 m^2 \\left(8 \\xi ^2-8 \\xi +1\\right)+d\\left(d^2-1\\right) k^6 \\left(24 \\xi ^2-1\\right)\\nonumber \\\\&&+24 d k^2 m^4 (3-8 \\xi ) -192 k^2 m^4 \\xi +96m^6 \\nonumber \\\\&&+ \\left(-6 k^4 m^2 \\left(d^2(1-4 \\xi )^2+d (8 \\xi -2)-2 \\left(8 \\xi ^2-8 \\xi +1\\right)\\right)\\right.\\nonumber \\\\&&\\left.+24 k^2 m^4 (d (8 \\xi -2)+8\\xi )-96 m^6\\right)\\, _2F_1\\left[1,-\\frac{d}{2};-\\frac{1}{2};\\frac{k^2}{4m^2}\\right]\\Big {)}\\pi ^{\\mu \\mu } \\pi ^{\\nu \\nu }\\nonumber \\\\&& +\\left(-12 d^2 k^4 m^2+d \\left(d^2-1\\right) k^6+48 d k^2 m^4-96 k^2m^4+12k^4 m^2+192 m^6\\right.\\nonumber \\\\&&\\left.-12 m^2 \\left(k^2-4 m^2\\right)^2 \\,_2F_1\\left[1,-\\frac{d}{2};-\\frac{1}{2};\\frac{k^2}{4 m^2}\\right]\\right)\\pi ^{\\mu \\nu }\\pi ^{\\mu \\nu }\\Big {]}$ The expansion of the transverse part $\\tilde{T}_{t}^{\\mu \\mu \\nu \\nu }(k)$ in the IR is $\\tilde{T}_{t}^{\\mu \\mu \\nu \\nu }(k)&=& 2^{-3-d} i\\, m^{d-4} \\pi ^{-\\frac{d}{2}}k^4\\sum _{n=0}^\\infty \\frac{ \\, m^{-2n}\\Gamma \\left(2+n-\\frac{d}{2}\\right)}{2^{n}\\,(2n+5)!!",
"}k^{2n}\\nonumber \\\\&&\\times \\left(\\pi ^{\\mu \\nu }\\pi ^{\\mu \\nu }+\\frac{a(n,\\xi )}{2}\\pi ^{\\mu \\mu } \\pi ^{\\nu \\nu }\\right)$ where $a(n,\\xi )$ is a constant $a(n,\\xi )=(2n+5)(2n+3)(4\\xi -1)^2+2(2n+5)(4\\xi -1)+1$ The non-transverse part of the bubble diagram is $\\tilde{T}_{nt}^{\\mu \\mu \\nu \\nu }(k)&=&\\frac{2^{-4-d}}{3} \\,i\\,\\pi ^{-d/2}m^{d-2} \\Gamma \\left(-\\frac{d}{2}\\right) \\nonumber \\\\&&\\left(\\eta ^{\\mu \\nu }\\eta ^{\\mu \\nu } \\left(24 m^2-2 d k^2\\right)+4 \\,d\\,\\eta ^{\\mu \\nu } k^{\\mu } k^{\\nu }+ 2 d\\, (6 \\xi -1) \\eta ^{\\mu \\mu } k^{\\nu }k^\\nu \\right.\\nonumber \\\\&&\\left.+\\eta ^{\\nu \\nu }\\left(\\eta ^{\\mu \\mu } \\left(d k^2 (5-24 \\xi )+12 m^2\\right)+2 d (6 \\xi -1)k^{\\mu }k^\\mu \\right) \\right)$ The seagull diagram and the non-transverse part of 2-pt function together give $\\tilde{T}_{(s)}^{\\mu \\mu \\nu \\nu }(k)+\\tilde{T}_{nt}^{\\mu \\mu \\nu \\nu }(k)&=&- 2^{-d-2}i \\pi ^{-d/2} m^d \\Gamma \\left(-\\frac{d}{2}\\right) \\left(2 \\eta ^{\\mu \\nu }\\eta ^{\\mu \\nu }-\\eta ^{\\mu \\mu } \\eta ^{\\nu \\nu }\\right)\\\\&&+ 2^{-d-1} i \\pi ^{-d/2} m^{d-2} \\left(\\xi -\\frac{1}{6}\\right)\\Gamma \\left(1-\\frac{d}{2}\\right) k^2\\left(\\pi ^{\\mu \\nu }\\pi ^{\\mu \\nu }-\\pi ^{\\mu \\mu } \\pi ^{\\nu \\nu }\\right)\\nonumber $ Taking formulas (REF ), (REF ), (REF ) and (REF ) and substituting them in (REF ) we can see that the Ward identity is satisfied for any dimension $d$ .",
"The one-loop 1-point correlator $\\langle \\!\\langle T^{\\mu \\mu }(x) \\rangle \\!",
"\\rangle &=&-2^{-d}\\,m^d\\pi ^{-\\frac{d}{2}}\\left[\\Gamma \\left(-\\frac{d}{2}\\right)g^{\\mu \\mu }-\\frac{2\\Gamma \\left(1-\\frac{d}{2}\\right)}{m^2}\\left(\\xi -\\frac{1}{6}\\right)G^{\\mu \\mu } \\right.\\nonumber \\\\&&\\left.+\\sum _{n=2}^\\infty \\frac{(-1)^n\\, m^{-2n} \\Gamma \\left(n-\\frac{d}{2}\\right)}{2^{n} (2n+1)!!",
"}\\Box ^{n-2}\\right.\\nonumber \\\\&&\\left.\\times \\left(-2\\Box G^{\\mu \\mu }+\\left(1-\\frac{a(n,\\xi )}{2}\\right)(\\eta ^{\\mu \\mu }\\Box -\\partial ^\\mu \\partial ^\\mu )R\\right)\\right]+O(h^2)$ satisfies (REF ).",
"For the effective action in the IR we obtain $W [h] &\\stackrel{\\text{IR}}{=}&2^{-d}m^d\\pi ^{-\\frac{d}{2}}\\int d^dx\\sqrt{g}\\left[\\Gamma \\left(-\\frac{d}{2}\\right)-\\frac{\\Gamma \\left(1-\\frac{d}{2}\\right)}{2m^2}\\left(\\xi -\\frac{1}{6}\\right)R\\right.\\nonumber \\\\&&\\left.",
"+\\frac{\\Gamma \\left(2-\\frac{d}{2}\\right)}{120m^4}\\left(R_{\\mu \\nu \\lambda \\rho }R^{\\mu \\nu \\lambda \\rho }+\\frac{a(0,\\xi )}{2}R^2\\right)+\\ldots \\right]+O(h^3)$ For $\\xi =\\frac{1}{6}$ (the conformal value) the third term in the expansion is proportional to $&\\propto & m^{d-4} \\int d^dx\\sqrt{g}\\left(R_{\\mu \\nu \\lambda \\rho }R^{\\mu \\nu \\lambda \\rho }-\\frac{1}{3} R^2\\right)$ We can use the Gauss-Bonnet theorem $R_{\\mu \\nu \\lambda \\rho }R^{\\mu \\nu \\lambda \\rho }-4R_{\\mu \\nu }R^{\\mu \\nu }+R^2=\\text{totalderivative}$ to write the divergent part of the effective action in $d=4$ as a Weyl square density $W &\\stackrel{\\text{IR}}{=}& -\\frac{1}{16\\pi ^2\\varepsilon } \\int d^4x\\sqrt{g} \\left(m^4+\\frac{1}{30}\\mathcal {W}^2\\right)+O(h^3)$ In the massless case ($m^0$ is the dominating term in the UV) we have $\\tilde{T}^{\\mu \\mu \\nu \\nu }(k)&\\stackrel{\\text{UV}}{=}&\\frac{2^{-1-2d} \\, \\pi ^{\\frac{3}{2}-\\frac{d}{2}}(k^2)^{\\frac{d}{2}}}{\\left(-1+e^{i\\pi d}\\right)\\Gamma \\left(\\frac{d+3}{2}\\right)}\\left(\\pi ^{\\mu \\nu }\\pi ^{\\mu \\nu }+\\frac{b(d,\\xi )}{2}\\pi ^{\\mu \\mu }\\pi ^{\\nu \\nu }\\right)$ where $b(d,\\xi )=(d^2-1)(4\\xi -1)^2+2(d+1)(4\\xi -1)+1$ The effective action in the UV now becomes $W &\\stackrel{UV}{=}&(-1)^{\\frac{d}{2}} \\frac{2^{-2-2d+\\lfloor \\frac{d}{2}\\rfloor }\\pi ^{\\frac{3}{2}-\\frac{d}{2}}}{(-1+e^{i\\pi d})\\Gamma \\left(\\frac{d+3}{2}\\right)} \\int \\text{d}^dx \\, \\left(R^{\\mu \\nu \\lambda \\rho }\\Box ^{\\frac{d}{2}-2}R_{\\mu \\nu \\lambda \\rho }+\\frac{b(d,\\xi )}{2} R\\Box ^{\\frac{d}{2} -2}R\\right)$ After we use (REF ) and put $\\xi =\\frac{1}{6}$ in 4d we will again get the Weyl square density $W &\\stackrel{UV}{=}& \\int \\text{d}^dx \\, \\mathcal {W}^2$ This last section of the paper is a systematic collection of results concerning all types of two-point correlators, including the mixed ones, for symmetric currents of of spin up to 5 and in dimension $3\\le d\\le 6$ .",
"It also contains results concerning the correlators of currents of any spin and in any dimensions, in the case of massless models, for which it is possible to write down very compact formulas.",
"The two point amplitudes in question for fermion and of scalar currents for spins up to 5, are schematically denoted as follows: $\\tilde{T}_{{\\mu _1\\ldots \\mu _{s_1}}{\\nu _1\\ldots \\nu _{s_2}}}(k) \\equiv \\langle {\\tilde{J}}_{\\mu _1\\ldots \\mu _{s_1}}(-k) {\\tilde{J}}_{\\nu _1\\ldots \\nu _{s_2}}(k) \\rangle \\,,$ Scalar and fermion currents are given by ${\\tilde{T}}^\\text{s}_{\\mu _1\\ldots \\mu _s}=i^s \\varphi ^\\dagger \\left({\\stackrel{\\leftrightarrow }{\\partial }}_\\mu \\right)^{s}\\varphi \\,,\\quad {\\tilde{T}}^\\text{f}_{\\mu _1\\ldots \\mu _s}=i^{s-1} \\bar{\\psi }\\gamma _\\mu \\left({\\stackrel{\\leftrightarrow }{\\partial }}_\\mu \\right)^{s-1}\\psi $ (For fermions in case $s=0$ we use ${\\tilde{T}}^\\text{f}_{s=0}=\\bar{\\psi }\\psi $ .)",
"These currents will be henceforth referred to as simple currents.",
"In the fermionic case the two point correlator is ${\\tilde{T}}^\\textrm {f}_{\\mu _1 \\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2}}(k)& =&-\\int \\frac{d^dp}{(2\\pi )^d}\\, {\\rm Tr}\\left( \\frac{i}{{p}-m} \\gamma _\\sigma \\frac{{ i}}{{p}-{k}-m} \\gamma _\\tau \\right)V^\\sigma _{\\mu _1\\ldots \\mu _{s_1}}V^\\tau _{\\nu _1\\ldots \\nu _{s_2}}$ whereas in the scalar case it is ${\\tilde{T}}^\\textrm {s}_{\\mu _1 \\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2}}(k)& =&\\int \\frac{d^dp}{(2\\pi )^d}\\,\\frac{1}{(p^2-m^2)((p-k)^2-m^2)}V_{\\mu _1\\ldots \\mu _{s_1}}V_{\\nu _1\\ldots \\nu _{s_2}} $ with the Feynman vertices for fermions and scalars respectively $V^\\sigma _{\\mu _1\\ldots \\mu _s} = i \\, \\delta ^\\sigma _\\mu \\, (2 p_\\mu - k_\\mu )^{s-1}\\,,\\quad V_{\\mu _1\\ldots \\mu _s} = i \\, (2 p_\\mu - k_\\mu )^{s} $ To label the correlators we often suppress writing indices and add the number of space-time dimensions in the subscript on the left hand side.",
"Additionally, when $s_1, s_2 \\ne 0$ , we split the amplitudes in the transverse and the non-transverse part, so for the correlator of e.g.",
"fermionic spin-$s_1$ and spin-$s_2$ currents in $d$ dimensions we write: ${\\tilde{T}}^\\textrm {f}_{s_1,s_2,d} ={\\tilde{T}}^\\textrm {f,t}_{s_1,s_2,d} +{\\tilde{T}}^\\textrm {f,nt}_{s_1,s_2,d}$ There is no preferred way to do the splitting in (REF ) because one can always add some transverse quantity to ${\\tilde{T}}^\\textrm {t}$ and subtract the same quantity from ${\\tilde{T}}^\\textrm {nt}$ .",
"However, it always happens that the non-transverse part can be chosen to be a polynomial in $k$ and $m$ (i.e.",
"local).",
"Here, we always make this choice so that the non-transverse part is local.",
"After this choice is made there is still some remaining freedom in the splitting into the transverse and the non-transverse part in (REF ), nevertheless the quantities we define below do not depend on this remaining freedom.",
"One such quantity is ${\\tilde{T}}^\\textrm {f,UV-IR}_{s_1,s_2,d}$ , the difference between the UV and the IR expansions in the shortly explained sense.",
"Since, as explained above, the non-transverse part is always local the non-transverse parts of UV and IR are the same and therefore cancel so that only transverse parts remain in the expression for ${\\tilde{T}}^\\textrm {f,UV-IR}_{s_1,s_2,d}$ ${\\tilde{T}}^\\textrm {f,UV-IR}_{s_1,s_2,d} ={\\tilde{T}}^\\textrm {f,UV}_{s_1,s_2,d} -{\\tilde{T}}^\\textrm {f,IR}_{(0)s_1,s_2,d}$ where the UV and IR expansions are denoted by ${\\tilde{T}}^\\textrm {f,t,UV}_{s_1,s_2,d}$ and ${\\tilde{T}}^\\textrm {f,t,IR}_{s_1,s_2,d}$ respectively, and $ {\\tilde{T}}^\\textrm {f,IR}_{(0)s_1,s_2,d}$ is the part of the IR expansion of order ${\\cal O}(m^n)$ with $n\\ge 0$ .",
"Another such quantities are the divergences of the correlators: $\\left(k \\cdot {\\tilde{T}}^\\textrm {f}_{s_1,s_2,d}\\right)_{\\mu _2 \\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2}} = k^{\\mu _1}\\left({\\tilde{T}}^\\textrm {f,nt}_{s_1,s_2,d}\\right)_{\\mu _1 \\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2}} \\nonumber \\\\\\left( {\\tilde{T}}^\\textrm {f}_{s_1,s_2,d} \\cdot k \\right)_{\\mu _1 \\ldots \\mu _{s_1}\\nu _2\\ldots \\nu _{s_2}}=k^{\\nu _1}\\left({\\tilde{T}}^\\textrm {f,nt}_{s_1,s_2,d}\\right)_{\\mu _1 \\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2}}$ The definitions (REF ), (REF ), (REF ) are analogous in the scalar case.",
"Before listing the results for the massive models, it is worth to show some general formulas (for any spin and any dimension) that it was possible to obtain for the massless case.",
"(We recall that the results for the massless cases correspond to the dominant term in the UV expansion of the massive case.)",
"In addition some general formulas are easy to write in terms of particular linear combination of the previous currents which become traceless in the massless case.",
"These “traceless” versions of the currents can be defined in the following way: ${\\tilde{T}}^{\\textrm {st}}_{\\mu _1\\ldots \\mu _s}=\\sum _{l=0}^{\\lfloor \\frac{s}{2}\\rfloor }a^{{\\textrm {s}}}_{s,l}\\left(\\Box \\pi _{\\mu \\mu }\\right)^l{\\tilde{T}}^{\\textrm {s}}_{\\mu _1\\ldots \\mu _{s-2l}}\\,, \\quad {\\tilde{T}}^{\\textrm {ft}}_{\\mu _1\\ldots \\mu _s}=\\sum _{l=0}^{\\lfloor \\frac{s-1}{2}\\rfloor }{a^{\\textrm {f}}_{s,l}}\\left(\\Box \\pi _{\\mu \\mu }\\right)^l{\\tilde{T}}^{\\textrm {f}}_{\\mu _1\\ldots \\mu _{s-2l}}$ where $a^{{\\textrm {s}}}_{s,l}=\\frac{{(-1)^l}s!\\,\\Gamma \\left(s+\\frac{d-3}{2}-l\\right)}{2^{2l}{l!",
"}(s-2l)!\\,\\Gamma \\left(s+\\frac{d-3}{2}\\right)}\\, ,\\quad { a^{\\textrm {f}}_{s,l}=\\frac{(-1)^l(s-1)!\\,\\Gamma \\left(s+\\frac{d-3}{2}-l\\right)}{2^{2l}l!",
"(s-2l-1)!\\,\\Gamma \\left(s+\\frac{d-3}{2}\\right)}}$ It is easy to see that amplitudes for two general spins $s_1$ and $s_2$ for the “traceless” currents can be written as linear combinations of the amplitudes (REF ) and (REF ) of the “simple” currents (REF ) ${\\tilde{T}}^\\text{st}_{\\mu _1\\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2}} =\\sum _{l=0}^{\\lfloor \\frac{s_1}{2}\\rfloor }\\sum _{k=0}^{\\lfloor \\frac{s_2}{2}\\rfloor }a^{{\\textrm {s}}}_{s_1,l} a^{{\\textrm {s}}}_{s_2,k} \\left(k^2\\eta _{\\mu \\mu }-k^2_\\mu \\right)^l \\left(k^2\\eta _{\\nu \\nu }-k^2_\\nu \\right)^k {\\tilde{T}}^\\text{s}_{\\mu _1\\ldots \\mu _{s_1-2l}\\nu _1\\ldots \\nu _{s_2-2k}}\\nonumber $ ${\\tilde{T}}^\\text{ft}_{\\mu _1\\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2}} =\\sum _{l=0}^{\\lfloor \\frac{s_1-1}{2}\\rfloor }\\sum _{k=0}^{\\lfloor \\frac{s_2-1}{2}\\rfloor }a^{{\\textrm {f}}}_{s_1,l} a^{{\\textrm {f}}}_{s_2,k}\\left(k^2\\eta _{\\mu \\mu }-k^2_\\mu \\right)^l \\left(k^2\\eta _{\\nu \\nu }-k^2_\\nu \\right)^k{\\tilde{T}}^\\text{f}_{\\mu _1\\ldots \\mu _{s_1-2l}\\nu _1\\ldots \\nu _{s_2-2k}}\\nonumber $ The result for the traceless currents in the massless limit is ${\\tilde{T}}^\\text{st,massless}_{\\mu _1\\ldots \\mu _{s}\\nu _1\\ldots \\nu _{s}} &=&(-1)^s \\frac{ 2^{4-2d-s} \\pi ^{\\frac{3}{2}-\\frac{d}{2}}s!",
"\\left(k^2\\right)^{\\frac{d}{2}+s-2}}{ \\left(-1+e^{i \\pi d}\\right) \\Gamma \\left(\\frac{d+2s-1}{2}\\right)}\\sum _{l=0}^{\\lfloor \\frac{s}{2}\\rfloor } {a_{s,l}^{\\textrm {s}}}\\pi ^l_{\\mu \\mu }\\pi ^l_{\\nu \\nu }\\pi _{\\mu \\nu }^{s-2l}\\\\&=&(-1)^s \\frac{ 2^{4-2d-s} \\pi ^{\\frac{3}{2}-\\frac{d}{2}} s!\\left(k^2\\right)^{\\frac{d}{2}+s-2}}{ \\left(-1+e^{i \\pi d}\\right) \\Gamma \\left(\\frac{d+2s-1}{2}\\right)}\\pi ^s_{\\mu \\nu }\\,\\,{_2 F_1}\\left(\\frac{1-s}{2},-\\frac{s}{2},\\frac{5-d-2s}{2},\\frac{\\pi _{\\mu \\mu }\\pi _{\\nu \\nu }}{\\pi _{\\mu \\nu }^2}\\right)\\nonumber $ We note that for traceless currents mixed spin terms are zero i.e.",
"the result vanishes for spin ${s_1} \\ne s_2$ .",
"For simple currents this is not the case and the general expression for spin $s_1\\times s_2$ , ${s_2 \\geqslant s_1}$ is ${\\tilde{T}}^\\text{s,massless}_{\\mu _1\\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2}} &=&(-1)^{\\frac{s_1+s_2}{2}} \\frac{\\left(2\\lfloor \\frac{s_2+1}{2}\\rfloor -1\\right)!",
"!\\left(2\\lfloor \\frac{s_2+1}{2}\\rfloor \\right)!!",
"2^{4-2d-\\frac{s_1+s_2}{2}} \\pi ^{\\frac{3}{2}-\\frac{d}{2}} \\left(k^2\\right)^{\\frac{d+s_1+s_2}{2}-2}}{\\left(2\\lfloor \\frac{s_2}{2}\\rfloor -2\\lfloor \\frac{s_1}{2}\\rfloor \\right)!!",
"\\left(-1+e^{i \\pi d}\\right) \\Gamma \\left(\\frac{d+s_1+s_2-1}{2}\\right)}\\nonumber \\\\&&\\times \\pi ^{\\frac{s_2-s_1}{2}}_{\\nu \\nu }\\sum _{l=0}^{\\lfloor \\frac{{s_1}}{2}\\rfloor }\\frac{s_1!(s_2-s_1)!!}{2^{\\frac{l(l+1)}{2}}(s_1-2l)!(s_2-s_1+2l)!!",
"}\\pi ^l_{\\mu \\mu }\\pi ^l_{\\nu \\nu }\\pi _{\\mu \\nu }^{{s_1}-2l}$ For fermions in the massless limit it also happens that only the diagonal ($s_1=s_2\\equiv s$ and $s>0$ ) amplitudes survive for the traceless currents ${\\tilde{T}}^\\text{ft,massless,even}_{\\mu _1\\ldots \\mu _{s}\\nu _1\\ldots \\nu _{s}} & =(-1)^s \\frac{ 2^{3-2d-s+\\lfloor \\frac{d}{2} \\rfloor } \\pi ^{\\frac{3}{2}-\\frac{d}{2}} (s-1)!",
"(d-3+s)\\left(k^2\\right)^{\\frac{d}{2}+s-2}}{ \\left(-1+e^{i \\pi d}\\right) \\Gamma \\left(\\frac{d+2s-1}{2}\\right)}\\sum _{l=0}^{\\lfloor \\frac{s}{2}\\rfloor } {a_{s,l}^{\\textrm {s}}}\\pi ^l_{\\mu \\mu }\\pi ^l_{\\nu \\nu }\\pi _{\\mu \\nu }^{s-2l}\\nonumber \\\\& = (-1)^s \\frac{ 2^{3-2d-s+\\lfloor \\frac{d}{2} \\rfloor } \\pi ^{\\frac{3}{2}-\\frac{d}{2}}(s-1)!",
"(d-3+s) \\left(k^2\\right)^{\\frac{d}{2}+s-2}}{ \\left(-1+e^{i \\pi d}\\right) \\Gamma \\left(\\frac{d+2s-1}{2}\\right)}\\nonumber \\\\& \\quad \\quad \\times \\pi ^s_{\\mu \\nu }\\,\\,{_2 F_1}\\left(\\frac{1-s}{2},-\\frac{s}{2},\\frac{5-d-2s}{2},\\frac{\\pi _{\\mu \\mu }\\pi _{\\nu \\nu }}{\\pi _{\\mu \\nu }^2}\\right)$ The formula above is valid for $d\\ge 4$ and for the even part in $d=3$ .",
"For the odd part in $d=3$ we obtain for traceless currents, for the dominant term in the UV, a general expression for spin $s_1\\times s_2$ , ${s_2 \\geqslant s_1}$ , ${s_1>0,\\, s_2>0}$ ${\\tilde{T}}^{\\text{ft,UV dominant,odd}}_{\\mu _1\\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2};3D}&=&(-1)^{\\frac{s_1+s_2}{2}} \\frac{i m k^{s_1+s_2-3}}{2^{s_2+1}}\\pi _{\\nu \\nu }^{\\frac{s_2-s_1}{2}}\\sum _{l=0}^{{\\lfloor \\frac{s_1-1}{2}\\rfloor }}\\frac{(-1)^l\\Gamma \\left(s_1-l\\right)}{2^{2l} l!\\Gamma \\left(s_1-2l\\right)}\\pi ^l_{\\mu \\mu }\\pi ^l_{\\nu \\nu }\\pi _{\\mu \\nu }^{s_1-2l-1}\\epsilon _{\\sigma \\mu \\nu }k^\\sigma \\nonumber \\\\&=&(-1)^{\\frac{s_1+s_2}{2}} \\frac{i m k^{s_1+s_2-3}}{2^{s_2+1}}\\pi _{\\nu \\nu }^{\\frac{s_2-s_1}{2}}\\pi ^{s_1-1}_{\\mu \\nu }\\nonumber \\\\&& \\quad \\times \\,\\,{_2 F_1}\\left(\\frac{1-s_1}{2},{1}-\\frac{s_1}{2},1-s_1,\\frac{\\pi _{\\mu \\mu }\\pi _{\\nu \\nu }}{\\pi _{\\mu \\nu }^2}\\right)\\epsilon _{\\sigma \\mu \\nu }k^\\sigma $ In Appendix we show that this formula is a straightforward generalization of the linearized action proposed long ago by Pope and Townsend, [30], for conformal higher spin fields.",
"In the case of simple currents we instead get ${\\tilde{T}}^{\\text{f,UV dominant,odd}}_{\\mu _1\\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2};3D} &=&(-1)^{\\frac{s_1+s_2}{2}} \\frac{\\left(2\\lfloor \\frac{s_2-1}{2}\\rfloor \\right)!",
"!\\left(s_1+s_2-2\\lfloor \\frac{s_1-1}{2}\\rfloor -3\\right)!!",
"m k^{s_1+s_2-3}}{2^2(s_1+s_2-2)!!",
"{\\left(2\\lfloor \\frac{s_2}{2}\\rfloor -2\\lfloor \\frac{s_1}{2}\\rfloor \\right)!!}",
"}\\\\&&\\times \\pi ^{\\frac{s_2-s_1}{2}}_{\\nu \\nu }\\epsilon _{\\sigma \\mu \\nu }k^\\sigma \\sum _{l=0}^{{\\lfloor \\frac{s_1-1}{2}\\rfloor }}\\frac{(s_1-1)!(s_2-s_1)!!}{2^{\\frac{l(l+1)}{2}}(s_1-2l-1)!(s_2-s_1+2l)!!",
"}\\pi ^l_{\\mu \\mu }\\pi ^l_{\\nu \\nu }\\pi _{\\mu \\nu }^{{s_1}-2l-1}\\nonumber $ In the case of simple currents it is possible to write the formula for the IR expansion of the transverse part: ${\\tilde{T}}^{\\text{f,t,IR,odd}}_{\\mu _1\\ldots \\mu _{s_1}\\nu _1\\ldots \\nu _{s_2};3D} &=&(-1)^{\\frac{s_1+s_2}{2}-1} \\frac{\\left(2\\lfloor \\frac{s_2-1}{2}\\rfloor \\right)!",
"!\\left(s_1+s_2-2\\lfloor \\frac{s_1-1}{2}\\rfloor -3\\right)!!",
"k^{s_1+s_2-2}}{2^2\\pi (s_1+s_2-1)!!",
"{\\left(2\\lfloor \\frac{s_2}{2}\\rfloor -2\\lfloor \\frac{s_1}{2}\\rfloor \\right)!!}",
"}\\\\&&\\times \\pi ^{\\frac{s_2-s_1}{2}}_{\\nu \\nu }\\epsilon _{\\sigma \\mu \\nu }k^\\sigma \\sum _{l=0}^{{\\lfloor \\frac{s_1-1}{2}\\rfloor }}\\frac{(s_1-1)!(s_2-s_1)!!}{2^{\\frac{l(l+1)}{2}}(s_1-2l-1)!(s_2-s_1+2l)!!",
"}\\pi ^l_{\\mu \\mu }\\pi ^l_{\\nu \\nu }\\pi _{\\mu \\nu }^{{s_1}-2l-1}\\nonumber $ In the rest of the section we list the results for the massive case.",
"The results are given for $d=3,4,5,6$ and spin $s\\le 5$ .",
"For even $d$ , we use $d \\rightarrow d + \\varepsilon $ and expand around $\\varepsilon $ .",
"For odd $d$ this is not necessary.",
"It is convenient to use the following shorthand notation $L_n &= \\frac{2}{\\varepsilon } + \\log \\left(\\frac{m^2}{4 \\pi }\\right)+\\gamma -\\sum _{k=1}^{n}\\frac{1}{k}$ as well as $K &= \\log \\left( -\\frac{k^2}{m^2} \\right)\\nonumber \\\\P &= \\frac{2}{\\varepsilon } + \\log \\left(-\\frac{k^2}{4 \\pi }\\right)+\\gamma $ We see that there is a relationship $P = K + L_0$ Furthermore we define $T &= -\\frac{2 i \\coth ^{-1}\\left(\\frac{2 m}{k}\\right)}{\\pi } \\nonumber \\\\S &= \\sqrt{4 m^2-k^2} \\csc ^{-1}\\left(\\frac{2 m}{k}\\right)$ It turns out that $T$ is useful in even dimensions $d$ and $S$ is useful in odd.",
"The branches of the functions $T$ and $S$ are chosen such that the IR and UV expansions are $T& \\quad \\stackrel{IR}{=}\\quad -\\frac{i k}{\\pi m}-\\frac{i k^3}{12 \\pi m^3}-\\frac{i k^5}{80 \\pi m^5} + \\ldots \\nonumber \\\\S& \\quad \\stackrel{IR}{=}\\quad k - \\frac{k^3}{ 12 m^2} - \\frac{k^5}{120 m^4} + \\ldots $ and $T& \\quad \\stackrel{UV}{=}\\quad 1-\\frac{4 i m}{\\pi k}-\\frac{16 i m^3}{3 \\pi k^3}-\\frac{64 i m^5}{5 \\pi k^5} + \\ldots \\nonumber \\\\S& \\quad \\stackrel{UV}{=}\\quad \\frac{k K}{2}-\\frac{m^2 \\left(1+K\\right)}{k}+\\frac{m^4 \\left(1-2 K\\right)}{2 k^3}+\\frac{m^6 \\left(5-6 K\\right)}{3 k^5} + \\ldots $ In the results for UV-IR which follow, the difference is shown for the terms containing the powers of $m$ and $k$ that “overlap” in UV and IR in sense that those powers appear both in UV and in IR expansions.",
"The rest, i.e.",
"the UV expansion that does not overlap with the IR, is denoted by ellipses.",
"The following results are organized as follows: first come the ones for the scalar model, sec.4.1-4.3, then those of the fermion models, sec.",
"4.4-4.6.",
"Sections 4.1 and 4.4 contain the full transverse analytic expressions of the correlators.",
"Sections 4.2 and 4.5 contain the UV and IR expansions of the latter, as well as the above-mentioned UV-IR expressions.",
"Sections 4.3. and 4.6 are devoted to the non-transverse local parts of the correlators.",
"These are the non-transverse expressions which have not been already discussed in the previous section and should be eliminated by the use of tadpole and seagull terms.",
"The method to obtain the results below has been explained in [1] and is largely based on the approach of Davydychev and collaborators, [31], see [32]."
],
[
"Scalar amplitudes",
"Scalars, spin 0 x 0, dimension 3: $\\tilde{T}_{0,0;3\\text{D}}^{\\text{s}} & = -\\frac{ T}{8}\\frac{1}{k} $ Scalars, spin 0 x 0, dimension 4: $\\tilde{T}_{0,0;4\\text{D}}^{\\text{s}} & = \\frac{i }{8 \\pi ^2}\\left(1-\\frac{L_0}{2}\\right)-\\frac{i S}{8 \\pi ^2}\\frac{1}{k} $ Scalars, spin 0 x 0, dimension 5: $\\tilde{T}_{0,0;5\\text{D}}^{\\text{s}} & = -\\frac{i }{32 \\pi ^2} m+\\frac{ T}{32 \\pi } \\left(-\\frac{1}{4} k+\\frac{m^2}{k}\\right) $ Scalars, spin 0 x 0, dimension 6: $\\tilde{T}_{0,0;6\\text{D}}^{\\text{s}} & = \\frac{i }{16 \\pi ^3} \\left( \\left(\\frac{1}{9}-\\frac{L_0}{24}\\right) k^2+ \\left(-\\frac{7}{12}+\\frac{L_0}{4}\\right) m^2\\right)+\\frac{i S}{48 \\pi ^3} \\left(-\\frac{1}{4} k+\\frac{m^2}{k}\\right) $ Scalars, spin 0 x 2, dimension 3: $\\tilde{T}_{0,2;3\\text{D}}^{\\text{s,t}} & = k^2 \\pi _{\\nu \\nu } \\left(-\\frac{i }{4 \\pi }\\frac{ m}{k^2}+ T \\left(\\frac{1}{16}\\frac{1}{k}-\\frac{1}{4}\\frac{ m^2}{k^3}\\right)\\right) $ $\\tilde{T}_{0,2;3\\text{D}}^{\\text{s,nt}} & = \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } m\\right) $ Scalars, spin 0 x 2, dimension 4: $\\tilde{T}_{0,2;4\\text{D}}^{\\text{s,t}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{6 \\pi ^2} \\left( \\left(-\\frac{1}{3}+\\frac{L_0}{8}\\right)+\\frac{m^2}{k^2}\\right)+\\frac{i S}{6 \\pi ^2} \\left(\\frac{1}{4}\\frac{1}{k}-\\frac{ m^2}{k^3}\\right)\\right) $ $\\tilde{T}_{0,2;4\\text{D}}^{\\text{s,nt}} & = \\eta _{\\nu \\nu } \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right) $ Scalars, spin 0 x 2, dimension 5: $\\tilde{T}_{0,2;5\\text{D}}^{\\text{s,t}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{32 \\pi ^2} \\left(\\frac{1}{4} m+\\frac{m^3}{k^2}\\right)+\\frac{ T}{32 \\pi } \\left(\\frac{1}{16} k-\\frac{1}{2}\\frac{ m^2}{k}+\\frac{m^4}{k^3}\\right)\\right) $ $\\tilde{T}_{0,2;5\\text{D}}^{\\text{s,nt}} & = \\eta _{\\nu \\nu } \\left(-\\frac{i }{12 \\pi ^2} m^3\\right) $ Scalars, spin 0 x 2, dimension 6: $\\tilde{T}_{0,2;6\\text{D}}^{\\text{s,t}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^3} \\left( \\left(-\\frac{23}{1200}+\\frac{L_0}{160}\\right) k^2+ \\left(\\frac{43}{240}-\\frac{L_0}{16}\\right) m^2-\\frac{1}{5}\\frac{ m^4}{k^2}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{60 \\pi ^3} \\left(\\frac{1}{16} k-\\frac{1}{2}\\frac{ m^2}{k}+\\frac{m^4}{k^3}\\right)\\right) $ $\\tilde{T}_{0,2;6\\text{D}}^{\\text{s,nt}} & = \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right) $ Scalars, spin 0 x 4, dimension 3: $\\tilde{T}_{0,4;3\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi } \\left(\\frac{5 }{4}\\frac{ m}{k^2}-3 \\frac{ m^3}{k^4}\\right)+ T \\left(-\\frac{3 }{64}\\frac{1}{k}+\\frac{3 }{8}\\frac{ m^2}{k^3}-\\frac{3 }{4}\\frac{ m^4}{k^5}\\right)\\right) $ $\\tilde{T}_{0,4;3\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi } m\\right)+\\eta _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(- k^2 m+4 m^3\\right)\\right) $ Scalars, spin 0 x 4, dimension 4: $\\tilde{T}_{0,4;4\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{23}{120}-\\frac{L_0}{16}\\right)-\\frac{7 }{6}\\frac{ m^2}{k^2}+2 \\frac{ m^4}{k^4}\\right)+\\frac{i S}{5 \\pi ^2} \\left(-\\frac{1}{8}\\frac{1}{k}+\\frac{m^2}{k^3}-2 \\frac{ m^4}{k^5}\\right)\\right) $ $\\tilde{T}_{0,4;4\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i L_1}{4 \\pi ^2} m^2\\right)+\\eta _{\\nu \\nu }^2 \\left(\\frac{i }{8 \\pi ^2} \\left( L_1 k^2 m^2-3 L_2 m^4\\right)\\right) $ Scalars, spin 0 x 4, dimension 5: $\\tilde{T}_{0,4;5\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{8 \\pi ^2} \\left(-\\frac{1}{32} m-\\frac{1}{3}\\frac{ m^3}{k^2}+\\frac{1}{2}\\frac{ m^5}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(-\\frac{1}{64} k+\\frac{3 }{16}\\frac{ m^2}{k}-\\frac{3 }{4}\\frac{ m^4}{k^3}+\\frac{m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{0,4;5\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i }{6 \\pi ^2} m^3\\right)+\\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left(\\frac{1}{12} k^2 m^3-\\frac{1}{5} m^5\\right)\\right) $ Scalars, spin 0 x 4, dimension 6: $\\tilde{T}_{0,4;6\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{11}{14700}-\\frac{L_0}{4480}\\right) k^2+ \\left(-\\frac{337}{33600}+\\frac{L_0}{320}\\right) m^2+\\frac{1}{42}\\frac{ m^4}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{35}\\frac{ m^6}{k^4}\\right)+\\frac{i S}{35 \\pi ^3} \\left(-\\frac{1}{64} k+\\frac{3 }{16}\\frac{ m^2}{k}-\\frac{3 }{4}\\frac{ m^4}{k^3}+\\frac{m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{0,4;6\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{32 \\pi ^3} m^4\\right)+\\eta _{\\nu \\nu }^2 \\left(\\frac{i }{32 \\pi ^3} \\left(-\\frac{ L_2}{2} k^2 m^4+ L_3 m^6\\right)\\right) $ Scalars, spin 1 x 1, dimension 3: $\\tilde{T}_{1,1;3\\text{D}}^{\\text{s,t}} & = k^2 \\pi _{\\mu \\nu } \\left(-\\frac{i }{4 \\pi }\\frac{ m}{k^2}+ T \\left(\\frac{1}{16}\\frac{1}{k}-\\frac{1}{4}\\frac{ m^2}{k^3}\\right)\\right) $ $\\tilde{T}_{1,1;3\\text{D}}^{\\text{s,nt}} & = \\eta _{\\mu \\nu } \\left(\\frac{i }{2 \\pi } m\\right) $ Scalars, spin 1 x 1, dimension 4: $\\tilde{T}_{1,1;4\\text{D}}^{\\text{s,t}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{6 \\pi ^2} \\left( \\left(-\\frac{1}{3}+\\frac{L_0}{8}\\right)+\\frac{m^2}{k^2}\\right)+\\frac{i S}{6 \\pi ^2} \\left(\\frac{1}{4}\\frac{1}{k}-\\frac{ m^2}{k^3}\\right)\\right) $ $\\tilde{T}_{1,1;4\\text{D}}^{\\text{s,nt}} & = \\eta _{\\mu \\nu } \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right) $ Scalars, spin 1 x 1, dimension 5: $\\tilde{T}_{1,1;5\\text{D}}^{\\text{s,t}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{32 \\pi ^2} \\left(\\frac{1}{4} m+\\frac{m^3}{k^2}\\right)+\\frac{ T}{32 \\pi } \\left(\\frac{1}{16} k-\\frac{1}{2}\\frac{ m^2}{k}+\\frac{m^4}{k^3}\\right)\\right) $ $\\tilde{T}_{1,1;5\\text{D}}^{\\text{s,nt}} & = \\eta _{\\mu \\nu } \\left(-\\frac{i }{12 \\pi ^2} m^3\\right) $ Scalars, spin 1 x 1, dimension 6: $\\tilde{T}_{1,1;6\\text{D}}^{\\text{s,t}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{12 \\pi ^3} \\left( \\left(-\\frac{23}{1200}+\\frac{L_0}{160}\\right) k^2+ \\left(\\frac{43}{240}-\\frac{L_0}{16}\\right) m^2-\\frac{1}{5}\\frac{ m^4}{k^2}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{60 \\pi ^3} \\left(\\frac{1}{16} k-\\frac{1}{2}\\frac{ m^2}{k}+\\frac{m^4}{k^3}\\right)\\right) $ $\\tilde{T}_{1,1;6\\text{D}}^{\\text{s,nt}} & = \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right) $ Scalars, spin 1 x 3, dimension 3: $\\tilde{T}_{1,3;3\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{4 \\pi } \\left(\\frac{5 }{4}\\frac{ m}{k^2}-3 \\frac{ m^3}{k^4}\\right)+ T \\left(-\\frac{3 }{64}\\frac{1}{k}+\\frac{3 }{8}\\frac{ m^2}{k^3}-\\frac{3 }{4}\\frac{ m^4}{k^5}\\right)\\right) $ $\\tilde{T}_{1,3;3\\text{D}}^{\\text{s,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{2 \\pi } m\\right)+\\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(- k^2 m+4 m^3\\right)\\right) $ Scalars, spin 1 x 3, dimension 4: $\\tilde{T}_{1,3;4\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{23}{120}-\\frac{L_0}{16}\\right)-\\frac{7 }{6}\\frac{ m^2}{k^2}+2 \\frac{ m^4}{k^4}\\right)+\\frac{i S}{5 \\pi ^2} \\left(-\\frac{1}{8}\\frac{1}{k}+\\frac{m^2}{k^3}-2 \\frac{ m^4}{k^5}\\right)\\right) $ $\\tilde{T}_{1,3;4\\text{D}}^{\\text{s,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+\\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{8 \\pi ^2} \\left( L_1 k^2 m^2-3 L_2 m^4\\right)\\right) $ Scalars, spin 1 x 3, dimension 5: $\\tilde{T}_{1,3;5\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{8 \\pi ^2} \\left(-\\frac{1}{32} m-\\frac{1}{3}\\frac{ m^3}{k^2}+\\frac{1}{2}\\frac{ m^5}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(-\\frac{1}{64} k+\\frac{3 }{16}\\frac{ m^2}{k}-\\frac{3 }{4}\\frac{ m^4}{k^3}+\\frac{m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{1,3;5\\text{D}}^{\\text{s,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+\\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(\\frac{1}{12} k^2 m^3-\\frac{1}{5} m^5\\right)\\right) $ Scalars, spin 1 x 3, dimension 6: $\\tilde{T}_{1,3;6\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{11}{14700}-\\frac{L_0}{4480}\\right) k^2+ \\left(-\\frac{337}{33600}+\\frac{L_0}{320}\\right) m^2+\\frac{1}{42}\\frac{ m^4}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{35}\\frac{ m^6}{k^4}\\right)+\\frac{i S}{35 \\pi ^3} \\left(-\\frac{1}{64} k+\\frac{3 }{16}\\frac{ m^2}{k}-\\frac{3 }{4}\\frac{ m^4}{k^3}+\\frac{m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{1,3;6\\text{D}}^{\\text{s,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+\\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{32 \\pi ^3} \\left(-\\frac{ L_2}{2} k^2 m^4+ L_3 m^6\\right)\\right) $ Scalars, spin 1 x 5, dimension 3: $\\tilde{T}_{1,5;3\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{\\pi } \\left(-\\frac{11 }{32}\\frac{ m}{k^2}+\\frac{5 }{3}\\frac{ m^3}{k^4}-\\frac{5 }{2}\\frac{ m^5}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{5 }{128}\\frac{1}{k}-\\frac{15 }{32}\\frac{ m^2}{k^3}+\\frac{15 }{8}\\frac{ m^4}{k^5}-\\frac{5 }{2}\\frac{ m^6}{k^7}\\right)\\right) $ $\\tilde{T}_{1,5;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi } m\\right)+k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(-\\frac{1}{2} k^2 m+\\frac{10 }{3} m^3\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi } \\left(-3 k^2 m+20 m^3\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(\\frac{1}{2} k^4 m-\\frac{10 }{3} k^2 m^3+8 m^5\\right)\\right) $ Scalars, spin 1 x 5, dimension 4: $\\tilde{T}_{1,5;4\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(-\\frac{22}{105}+\\frac{L_0}{16}\\right)+\\frac{29 }{15}\\frac{ m^2}{k^2}-\\frac{20 }{3}\\frac{ m^4}{k^4}+8 \\frac{ m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{7 \\pi ^2} \\left(\\frac{1}{8}\\frac{1}{k}-\\frac{3 }{2}\\frac{ m^2}{k^3}+6 \\frac{ m^4}{k^5}-8 \\frac{ m^6}{k^7}\\right)\\right) $ $\\tilde{T}_{1,5;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{i L_1}{4 \\pi ^2} m^2\\right)+k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{8 \\pi ^2} \\left( L_1 k^2 m^2-5 L_2 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left( L_1 k^2 m^2-5 L_2 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left(-\\frac{ L_1}{2} k^4 m^2+\\frac{5 L_2}{2} k^2 m^4-5 L_3 m^6\\right)\\right) $ Scalars, spin 1 x 5, dimension 5: $\\tilde{T}_{1,5;5\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{32 \\pi ^2} \\left(\\frac{5 }{64} m+\\frac{73 }{48}\\frac{ m^3}{k^2}-\\frac{55 }{12}\\frac{ m^5}{k^4}+5 \\frac{ m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{32 \\pi } \\left(\\frac{5 }{256} k-\\frac{5 }{16}\\frac{ m^2}{k}+\\frac{15 }{8}\\frac{ m^4}{k^3}-5 \\frac{ m^6}{k^5}+5 \\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{1,5;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{i }{6 \\pi ^2} m^3\\right)+k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{1}{4} k^2 m^3- m^5\\right)\\right)+k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{1}{2} k^2 m^3-2 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{1}{12} k^4 m^3+\\frac{1}{3} k^2 m^5-\\frac{4 }{7} m^7\\right)\\right) $ Scalars, spin 1 x 5, dimension 6: $\\tilde{T}_{1,5;6\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{7 \\pi ^3} \\left( \\left(-\\frac{563}{181440}+\\frac{L_0}{1152}\\right) k^2+ \\left(\\frac{1091}{20160}-\\frac{L_0}{64}\\right) m^2-\\frac{1}{5}\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{13 }{27}\\frac{ m^6}{k^4}-\\frac{4 }{9}\\frac{ m^8}{k^6}\\right)+\\frac{i S}{21 \\pi ^3} \\left(\\frac{1}{192} k-\\frac{1}{12}\\frac{ m^2}{k}+\\frac{1}{2}\\frac{ m^4}{k^3}-\\frac{4 }{3}\\frac{ m^6}{k^5}+\\frac{4 }{3}\\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{1,5;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{32 \\pi ^3} m^4\\right)+k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{32 \\pi ^3} \\left(-\\frac{ L_2}{2} k^2 m^4+\\frac{5 L_3}{3} m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{16 \\pi ^3} \\left(-\\frac{ L_2}{2} k^2 m^4+\\frac{5 L_3}{3} m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{32 \\pi ^3} \\left(\\frac{ L_2}{2} k^4 m^4-\\frac{5 L_3}{3} k^2 m^6+\\frac{5 L_4}{2} m^8\\right)\\right) $ Scalars, spin 2 x 2, dimension 3: $\\tilde{T}_{2,2;3\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(\\frac{3 }{4}\\frac{ m}{k^2}-\\frac{ m^3}{k^4}\\right)+ T \\left(-\\frac{1}{32}\\frac{1}{k}+\\frac{1}{4}\\frac{ m^2}{k^3}-\\frac{1}{2}\\frac{ m^4}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi } \\left(-\\frac{1}{4}\\frac{ m}{k^2}-\\frac{ m^3}{k^4}\\right)+ T \\left(-\\frac{1}{64}\\frac{1}{k}+\\frac{1}{8}\\frac{ m^2}{k^3}-\\frac{1}{4}\\frac{ m^4}{k^5}\\right)\\right) $ $\\tilde{T}_{2,2;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\left(\\frac{i }{\\pi } m\\right)+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{2 i }{3 \\pi } m^3\\right)+\\eta _{\\mu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(-\\frac{1}{2} k^2 m+\\frac{4 }{3} m^3\\right)\\right) $ Scalars, spin 2 x 2, dimension 4: $\\tilde{T}_{2,2;4\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{3 \\pi ^2} \\left( \\left(\\frac{23}{300}-\\frac{L_0}{40}\\right)- \\left(\\frac{41}{120}+\\frac{L_0}{8}\\right)\\frac{ m^2}{k^2}+\\frac{4 }{5}\\frac{ m^4}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{15 \\pi ^2} \\left(-\\frac{1}{4}\\frac{1}{k}+2 \\frac{ m^2}{k^3}-4 \\frac{ m^4}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left( \\left(\\frac{23}{600}-\\frac{L_0}{80}\\right)+ \\left(-\\frac{43}{120}+\\frac{L_0}{8}\\right)\\frac{ m^2}{k^2}+\\frac{2 }{5}\\frac{ m^4}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{15 \\pi ^2} \\left(-\\frac{1}{8}\\frac{1}{k}+\\frac{m^2}{k^3}-2 \\frac{ m^4}{k^5}\\right)\\right) $ $\\tilde{T}_{2,2;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\left(-\\frac{i L_1}{4 \\pi ^2} m^2\\right)+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_2}{8 \\pi ^2} m^4\\right)+\\eta _{\\mu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{ L_1}{2} k^2 m^2- L_2 m^4\\right)\\right) $ Scalars, spin 2 x 2, dimension 5: $\\tilde{T}_{2,2;5\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{6 \\pi ^2} \\left(-\\frac{1}{64} m-\\frac{1}{3}\\frac{ m^3}{k^2}+\\frac{1}{4}\\frac{ m^5}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{8 \\pi } \\left(-\\frac{1}{192} k+\\frac{1}{16}\\frac{ m^2}{k}-\\frac{1}{4}\\frac{ m^4}{k^3}+\\frac{1}{3}\\frac{ m^6}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{24 \\pi ^2} \\left(-\\frac{1}{32} m+\\frac{1}{3}\\frac{ m^3}{k^2}+\\frac{1}{2}\\frac{ m^5}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(-\\frac{1}{192} k+\\frac{1}{16}\\frac{ m^2}{k}-\\frac{1}{4}\\frac{ m^4}{k^3}+\\frac{1}{3}\\frac{ m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{2,2;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\left(-\\frac{i }{6 \\pi ^2} m^3\\right)+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i }{15 \\pi ^2} m^5\\right)+\\eta _{\\mu \\nu }^2 \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{1}{4} k^2 m^3-\\frac{2 }{5} m^5\\right)\\right) $ Scalars, spin 2 x 2, dimension 6: $\\tilde{T}_{2,2;6\\text{D}}^{\\text{s,t}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{3 \\pi ^3} \\left( \\left(\\frac{11}{7350}-\\frac{L_0}{2240}\\right) k^2+ \\left(-\\frac{337}{16800}+\\frac{L_0}{160}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{65}{2688}+\\frac{L_0}{64}\\right)\\frac{ m^4}{k^2}-\\frac{2 }{35}\\frac{ m^6}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^3} \\left(-\\frac{1}{96} k+\\frac{1}{8}\\frac{ m^2}{k}-\\frac{1}{2}\\frac{ m^4}{k^3}+\\frac{2 }{3}\\frac{ m^6}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^3} \\left( \\left(\\frac{11}{14700}-\\frac{L_0}{4480}\\right) k^2+ \\left(-\\frac{337}{33600}+\\frac{L_0}{320}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{127}{2688}-\\frac{L_0}{64}\\right)\\frac{ m^4}{k^2}-\\frac{1}{35}\\frac{ m^6}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^3} \\left(-\\frac{1}{192} k+\\frac{1}{16}\\frac{ m^2}{k}-\\frac{1}{4}\\frac{ m^4}{k^3}+\\frac{1}{3}\\frac{ m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{2,2;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{32 \\pi ^3} m^4\\right)+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{96 \\pi ^3} m^6\\right)+\\eta _{\\mu \\nu }^2 \\left(\\frac{i }{16 \\pi ^3} \\left(-\\frac{ L_2}{4} k^2 m^4+\\frac{ L_3}{3} m^6\\right)\\right) $ Scalars, spin 2 x 4, dimension 3: $\\tilde{T}_{2,4;3\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(-\\frac{3 }{8}\\frac{ m}{k^2}+\\frac{4 }{3}\\frac{ m^3}{k^4}-2 \\frac{ m^5}{k^6}\\right)+ T \\left(\\frac{1}{32}\\frac{1}{k}-\\frac{3 }{8}\\frac{ m^2}{k^3}+\\frac{3 }{2}\\frac{ m^4}{k^5}-2 \\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(\\frac{1}{32}\\frac{ m}{k^2}+\\frac{1}{3}\\frac{ m^3}{k^4}-\\frac{1}{2}\\frac{ m^5}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{1}{128}\\frac{1}{k}-\\frac{3 }{32}\\frac{ m^2}{k^3}+\\frac{3 }{8}\\frac{ m^4}{k^5}-\\frac{1}{2}\\frac{ m^6}{k^7}\\right)\\right) $ $\\tilde{T}_{2,4;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{i }{\\pi } m\\right)+k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{4 i }{3 \\pi } m^3\\right)+k_{\\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{2 i }{3 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi } \\left(-3 k^2 m+16 m^3\\right)\\right)+k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(-\\frac{1}{2} k^2 m+\\frac{8 }{3} m^3\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(-\\frac{2 }{3} k^2 m^3+\\frac{8 }{5} m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(\\frac{1}{2} k^4 m-\\frac{8 }{3} k^2 m^3+\\frac{32 }{5} m^5\\right)\\right) $ Scalars, spin 2 x 4, dimension 4: $\\tilde{T}_{2,4;4\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{88}{3675}+\\frac{L_0}{140}\\right)+ \\left(\\frac{823}{4200}+\\frac{L_0}{40}\\right)\\frac{ m^2}{k^2}-\\frac{16 }{21}\\frac{ m^4}{k^4}+\\frac{32 }{35}\\frac{ m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^2} \\left(\\frac{1}{2}\\frac{1}{k}-6 \\frac{ m^2}{k^3}+24 \\frac{ m^4}{k^5}-32 \\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{22}{3675}+\\frac{L_0}{560}\\right)+ \\left(\\frac{337}{4200}-\\frac{L_0}{40}\\right)\\frac{ m^2}{k^2}-\\frac{4 }{21}\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{8 }{35}\\frac{ m^6}{k^6}\\right)+\\frac{i S}{35 \\pi ^2} \\left(\\frac{1}{8}\\frac{1}{k}-\\frac{3 }{2}\\frac{ m^2}{k^3}+6 \\frac{ m^4}{k^5}-8 \\frac{ m^6}{k^7}\\right)\\right) $ $\\tilde{T}_{2,4;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{i L_1}{4 \\pi ^2} m^2\\right)+k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_2}{8 \\pi ^2} m^4\\right)+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left( L_1 k^2 m^2-4 L_2 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{ L_1}{4} k^2 m^2- L_2 m^4\\right)\\right)+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{ L_2}{2} k^2 m^4- L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left(-\\frac{ L_1}{4} k^4 m^2+ L_2 k^2 m^4-2 L_3 m^6\\right)\\right) $ Scalars, spin 2 x 4, dimension 5: $\\tilde{T}_{2,4;5\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{8 \\pi ^2} \\left(\\frac{1}{64} m+\\frac{7 }{16}\\frac{ m^3}{k^2}-\\frac{11 }{12}\\frac{ m^5}{k^4}+\\frac{m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{8 \\pi } \\left(\\frac{1}{256} k-\\frac{1}{16}\\frac{ m^2}{k}+\\frac{3 }{8}\\frac{ m^4}{k^3}-\\frac{ m^6}{k^5}+\\frac{m^8}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{32 \\pi ^2} \\left(\\frac{1}{64} m-\\frac{11 }{48}\\frac{ m^3}{k^2}-\\frac{11 }{12}\\frac{ m^5}{k^4}+\\frac{m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{32 \\pi } \\left(\\frac{1}{256} k-\\frac{1}{16}\\frac{ m^2}{k}+\\frac{3 }{8}\\frac{ m^4}{k^3}-\\frac{ m^6}{k^5}+\\frac{m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{2,4;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{i }{6 \\pi ^2} m^3\\right)+k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{15 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 \\eta _{\\nu \\nu }^2 \\left(-\\frac{i }{15 \\pi ^2} m^5\\right)+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{1}{2} k^2 m^3-\\frac{8 }{5} m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{1}{4} k^2 m^3-\\frac{4 }{5} m^5\\right)\\right)+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left(\\frac{1}{3} k^2 m^5-\\frac{4 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{1}{12} k^4 m^3+\\frac{4 }{15} k^2 m^5-\\frac{16 }{35} m^7\\right)\\right) $ Scalars, spin 2 x 4, dimension 6: $\\tilde{T}_{2,4;6\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(-\\frac{563}{317520}+\\frac{L_0}{2016}\\right) k^2+ \\left(\\frac{1091}{35280}-\\frac{L_0}{112}\\right) m^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{407}{4480}+\\frac{L_0}{64}\\right)\\frac{ m^4}{k^2}+\\frac{52 }{189}\\frac{ m^6}{k^4}-\\frac{16 }{63}\\frac{ m^8}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{105 \\pi ^3} \\left(\\frac{1}{48} k-\\frac{1}{3}\\frac{ m^2}{k}+2 \\frac{ m^4}{k^3}-\\frac{16 }{3}\\frac{ m^6}{k^5}+\\frac{16 }{3}\\frac{ m^8}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(-\\frac{563}{1270080}+\\frac{L_0}{8064}\\right) k^2+ \\left(\\frac{1091}{141120}-\\frac{L_0}{448}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{233}{4480}+\\frac{L_0}{64}\\right)\\frac{ m^4}{k^2}+\\frac{13 }{189}\\frac{ m^6}{k^4}-\\frac{4 }{63}\\frac{ m^8}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{105 \\pi ^3} \\left(\\frac{1}{192} k-\\frac{1}{12}\\frac{ m^2}{k}+\\frac{1}{2}\\frac{ m^4}{k^3}-\\frac{4 }{3}\\frac{ m^6}{k^5}+\\frac{4 }{3}\\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{2,4;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{32 \\pi ^3} m^4\\right)+k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{48 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_3}{96 \\pi ^3} m^6\\right)+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^3} \\left(-\\frac{ L_2}{8} k^2 m^4+\\frac{ L_3}{3} m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{8 \\pi ^3} \\left(-\\frac{ L_2}{8} k^2 m^4+\\frac{ L_3}{3} m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{32 \\pi ^3} \\left(-\\frac{ L_3}{3} k^2 m^6+\\frac{ L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{8 \\pi ^3} \\left(\\frac{ L_2}{8} k^4 m^4-\\frac{ L_3}{3} k^2 m^6+\\frac{ L_4}{2} m^8\\right)\\right) $ Scalars, spin 3 x 3, dimension 3: $\\tilde{T}_{3,3;3\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi } \\left(-\\frac{7 }{16}\\frac{ m}{k^2}+\\frac{2 }{3}\\frac{ m^3}{k^4}-\\frac{ m^5}{k^6}\\right)+ T \\left(\\frac{1}{64}\\frac{1}{k}-\\frac{3 }{16}\\frac{ m^2}{k^3}+\\frac{3 }{4}\\frac{ m^4}{k^5}-\\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(\\frac{3 }{16}\\frac{ m}{k^2}+2 \\frac{ m^3}{k^4}-3 \\frac{ m^5}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{3 }{128}\\frac{1}{k}-\\frac{9 }{32}\\frac{ m^2}{k^3}+\\frac{9 }{8}\\frac{ m^4}{k^5}-\\frac{3 }{2}\\frac{ m^6}{k^7}\\right)\\right) $ $\\tilde{T}_{3,3;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(\\frac{3 i }{2 \\pi } m\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{2 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(-3 k^2 m+8 m^3\\right)\\right)+\\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi } \\left(-10 k^2 m^3+24 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi } \\left(\\frac{1}{2} k^4 m-\\frac{4 }{3} k^2 m^3+\\frac{16 }{5} m^5\\right)\\right) $ Scalars, spin 3 x 3, dimension 4: $\\tilde{T}_{3,3;4\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{44}{3675}+\\frac{L_0}{280}\\right)+ \\left(\\frac{149}{4200}+\\frac{3 L_0}{40}\\right)\\frac{ m^2}{k^2}-\\frac{8 }{21}\\frac{ m^4}{k^4}+\\frac{16 }{35}\\frac{ m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^2} \\left(\\frac{1}{4}\\frac{1}{k}-3 \\frac{ m^2}{k^3}+12 \\frac{ m^4}{k^5}-16 \\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{22}{1225}+\\frac{3 L_0}{560}\\right)+ \\left(\\frac{337}{1400}-\\frac{3 L_0}{40}\\right)\\frac{ m^2}{k^2}-\\frac{4 }{7}\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{24 }{35}\\frac{ m^6}{k^6}\\right)+\\frac{i S}{35 \\pi ^2} \\left(\\frac{3 }{8}\\frac{1}{k}-\\frac{9 }{2}\\frac{ m^2}{k^3}+18 \\frac{ m^4}{k^5}-24 \\frac{ m^6}{k^7}\\right)\\right) $ $\\tilde{T}_{3,3;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(-\\frac{3 i L_1}{8 \\pi ^2} m^2\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{3 i L_2}{8 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{3 L_1}{2} k^2 m^2-3 L_2 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{3 L_2}{2} k^2 m^4-3 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{2 \\pi ^2} \\left(-\\frac{ L_1}{4} k^4 m^2+\\frac{ L_2}{2} k^2 m^4- L_3 m^6\\right)\\right) $ Scalars, spin 3 x 3, dimension 5: $\\tilde{T}_{3,3;5\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{16 \\pi ^2} \\left(\\frac{1}{64} m+\\frac{53 }{48}\\frac{ m^3}{k^2}-\\frac{11 }{12}\\frac{ m^5}{k^4}+\\frac{m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(\\frac{1}{256} k-\\frac{1}{16}\\frac{ m^2}{k}+\\frac{3 }{8}\\frac{ m^4}{k^3}-\\frac{ m^6}{k^5}+\\frac{m^8}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{32 \\pi ^2} \\left(\\frac{3 }{64} m-\\frac{11 }{16}\\frac{ m^3}{k^2}-\\frac{11 }{4}\\frac{ m^5}{k^4}+3 \\frac{ m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{32 \\pi } \\left(\\frac{3 }{256} k-\\frac{3 }{16}\\frac{ m^2}{k}+\\frac{9 }{8}\\frac{ m^4}{k^3}-3 \\frac{ m^6}{k^5}+3 \\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{3,3;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(-\\frac{i }{4 \\pi ^2} m^3\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left(\\frac{1}{4} k^2 m^3-\\frac{2 }{5} m^5\\right)\\right)+\\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(k^2 m^5-\\frac{12 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{1}{12} k^4 m^3+\\frac{2 }{15} k^2 m^5-\\frac{8 }{35} m^7\\right)\\right) $ Scalars, spin 3 x 3, dimension 6: $\\tilde{T}_{3,3;6\\text{D}}^{\\text{s,t}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(-\\frac{563}{635040}+\\frac{L_0}{4032}\\right) k^2+ \\left(\\frac{1091}{70560}-\\frac{L_0}{224}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{59}{4480}-\\frac{3 L_0}{64}\\right)\\frac{ m^4}{k^2}+\\frac{26 }{189}\\frac{ m^6}{k^4}-\\frac{8 }{63}\\frac{ m^8}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{105 \\pi ^3} \\left(\\frac{1}{96} k-\\frac{1}{6}\\frac{ m^2}{k}+\\frac{m^4}{k^3}-\\frac{8 }{3}\\frac{ m^6}{k^5}+\\frac{8 }{3}\\frac{ m^8}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(-\\frac{563}{423360}+\\frac{L_0}{2688}\\right) k^2+ \\left(\\frac{1091}{47040}-\\frac{3 L_0}{448}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{699}{4480}+\\frac{3 L_0}{64}\\right)\\frac{ m^4}{k^2}+\\frac{13 }{63}\\frac{ m^6}{k^4}-\\frac{4 }{21}\\frac{ m^8}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^3} \\left(\\frac{1}{192} k-\\frac{1}{12}\\frac{ m^2}{k}+\\frac{1}{2}\\frac{ m^4}{k^3}-\\frac{4 }{3}\\frac{ m^6}{k^5}+\\frac{4 }{3}\\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{3,3;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(\\frac{3 i L_2}{64 \\pi ^3} m^4\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_3}{32 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{16 \\pi ^3} \\left(-\\frac{3 L_2}{4} k^2 m^4+ L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{32 \\pi ^3} \\left(- L_3 k^2 m^6+\\frac{3 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{16 \\pi ^3} \\left(\\frac{ L_2}{4} k^4 m^4-\\frac{ L_3}{3} k^2 m^6+\\frac{ L_4}{2} m^8\\right)\\right) $ Scalars, spin 3 x 5, dimension 3: $\\tilde{T}_{3,5;3\\text{D}}^{\\text{s,t}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi } \\left(\\frac{27 }{16}\\frac{ m}{k^2}-\\frac{73 }{12}\\frac{ m^3}{k^4}+\\frac{55 }{3}\\frac{ m^5}{k^6}-20 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{5 }{256}\\frac{1}{k}+\\frac{5 }{16}\\frac{ m^2}{k^3}-\\frac{15 }{8}\\frac{ m^4}{k^5}+5 \\frac{ m^6}{k^7}-5 \\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi } \\left(-\\frac{15 }{64}\\frac{ m}{k^2}-\\frac{73 }{16}\\frac{ m^3}{k^4}+\\frac{55 }{4}\\frac{ m^5}{k^6}-15 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{15 }{1024}\\frac{1}{k}+\\frac{15 }{64}\\frac{ m^2}{k^3}-\\frac{45 }{32}\\frac{ m^4}{k^5}+\\frac{15 }{4}\\frac{ m^6}{k^7}-\\frac{15 }{4}\\frac{ m^8}{k^9}\\right)\\right) $ $\\tilde{T}_{3,5;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{3 i }{2 \\pi } m\\right)+k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{4 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{2 i }{\\pi } m^3\\right)+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(-3 k^2 m+24 m^3\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(-3 k^2 m+16 m^3\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{\\pi } \\left(-2 k^2 m^3+8 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(-4 k^2 m^3+16 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(3 k^4 m-16 k^2 m^3+64 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi } \\left(\\frac{1}{2} k^4 m-\\frac{8 }{3} k^2 m^3+\\frac{32 }{3} m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi } \\left(14 k^4 m^3-56 k^2 m^5+96 m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(-\\frac{1}{2} k^6 m+\\frac{8 }{3} k^4 m^3-\\frac{32 }{3} k^2 m^5+\\frac{128 }{7} m^7\\right)\\right) $ Scalars, spin 3 x 5, dimension 4: $\\tilde{T}_{3,5;4\\text{D}}^{\\text{s,t}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(\\frac{563}{5670}-\\frac{L_0}{36}\\right)- \\left(\\frac{2159}{2520}+\\frac{3 L_0}{8}\\right)\\frac{ m^2}{k^2}+\\frac{32 }{5}\\frac{ m^4}{k^4}-\\frac{416 }{27}\\frac{ m^6}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{128 }{9}\\frac{ m^8}{k^8}\\right)+\\frac{i S}{21 \\pi ^2} \\left(-\\frac{1}{6}\\frac{1}{k}+\\frac{8 }{3}\\frac{ m^2}{k^3}-16 \\frac{ m^4}{k^5}+\\frac{128 }{3}\\frac{ m^6}{k^7}-\\frac{128 }{3}\\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(\\frac{563}{7560}-\\frac{L_0}{48}\\right)+ \\left(-\\frac{1091}{840}+\\frac{3 L_0}{8}\\right)\\frac{ m^2}{k^2}+\\frac{24 }{5}\\frac{ m^4}{k^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{104 }{9}\\frac{ m^6}{k^6}+\\frac{32 }{3}\\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{7 \\pi ^2} \\left(-\\frac{1}{24}\\frac{1}{k}+\\frac{2 }{3}\\frac{ m^2}{k^3}-4 \\frac{ m^4}{k^5}+\\frac{32 }{3}\\frac{ m^6}{k^7}-\\frac{32 }{3}\\frac{ m^8}{k^9}\\right)\\right) $ $\\tilde{T}_{3,5;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{3 i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{3 i L_2}{8 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{3 L_1}{2} k^2 m^2-9 L_2 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{3 L_1}{4} k^2 m^2-3 L_2 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{3 L_2}{2} k^2 m^4-5 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{3 L_2}{2} k^2 m^4-5 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left(-\\frac{3 L_1}{4} k^4 m^2+3 L_2 k^2 m^4-10 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{ L_1}{8} k^4 m^2+\\frac{ L_2}{2} k^2 m^4-\\frac{5 L_3}{3} m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left(-\\frac{3 L_2}{2} k^4 m^4+5 L_3 k^2 m^6-\\frac{15 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(\\frac{ L_1}{8} k^6 m^2-\\frac{ L_2}{2} k^4 m^4+\\frac{5 L_3}{3} k^2 m^6-\\frac{5 L_4}{2} m^8\\right)\\right) $ Scalars, spin 3 x 5, dimension 5: $\\tilde{T}_{3,5;5\\text{D}}^{\\text{s,t}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{1}{1024} m-\\frac{25 }{384}\\frac{ m^3}{k^2}+\\frac{2 }{15}\\frac{ m^5}{k^4}-\\frac{7 }{24}\\frac{ m^7}{k^6}+\\frac{1}{4}\\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{4 \\pi } \\left(-\\frac{1}{1024} k+\\frac{5 }{256}\\frac{ m^2}{k}-\\frac{5 }{32}\\frac{ m^4}{k^3}+\\frac{5 }{8}\\frac{ m^6}{k^5}-\\frac{5 }{4}\\frac{ m^8}{k^7}+\\frac{m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(-\\frac{3 }{2048} m+\\frac{7 }{256}\\frac{ m^3}{k^2}+\\frac{1}{5}\\frac{ m^5}{k^4}-\\frac{7 }{16}\\frac{ m^7}{k^6}+\\frac{3 }{8}\\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(-\\frac{3 }{1024} k+\\frac{15 }{256}\\frac{ m^2}{k}-\\frac{15 }{32}\\frac{ m^4}{k^3}+\\frac{15 }{8}\\frac{ m^6}{k^5}-\\frac{15 }{4}\\frac{ m^8}{k^7}+3 \\frac{ m^{10}}{k^9}\\right)\\right) $ $\\tilde{T}_{3,5;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{i }{4 \\pi ^2} m^3\\right)+k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(\\frac{1}{4} k^2 m^3-\\frac{6 }{5} m^5\\right)\\right)+k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left(\\frac{1}{4} k^2 m^3-\\frac{4 }{5} m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{\\pi ^2} \\left(\\frac{1}{5} k^2 m^5-\\frac{4 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(\\frac{2 }{5} k^2 m^5-\\frac{8 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{1}{4} k^4 m^3+\\frac{4 }{5} k^2 m^5-\\frac{16 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi ^2} \\left(-\\frac{1}{4} k^4 m^3+\\frac{4 }{5} k^2 m^5-\\frac{16 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{1}{5} k^4 m^5+\\frac{4 }{7} k^2 m^7-\\frac{16 }{21} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{1}{4} k^6 m^3-\\frac{4 }{5} k^4 m^5+\\frac{16 }{7} k^2 m^7-\\frac{64 }{21} m^9\\right)\\right) $ Scalars, spin 3 x 5, dimension 6: $\\tilde{T}_{3,5;6\\text{D}}^{\\text{s,t}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^3} \\left( \\left(\\frac{1627}{1372140}-\\frac{L_0}{3168}\\right) k^2+ \\left(-\\frac{12701}{498960}+\\frac{L_0}{144}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{25903}{443520}+\\frac{3 L_0}{64}\\right)\\frac{ m^4}{k^2}-\\frac{692 }{1485}\\frac{ m^6}{k^4}+\\frac{256 }{297}\\frac{ m^8}{k^6}-\\frac{64 }{99}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{693 \\pi ^3} \\left(-\\frac{1}{16} k+\\frac{5 }{4}\\frac{ m^2}{k}-10 \\frac{ m^4}{k^3}+40 \\frac{ m^6}{k^5}-80 \\frac{ m^8}{k^7}+64 \\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi ^3} \\left( \\left(\\frac{1627}{1829520}-\\frac{L_0}{4224}\\right) k^2+ \\left(-\\frac{12701}{665280}+\\frac{L_0}{192}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{24667}{147840}-\\frac{3 L_0}{64}\\right)\\frac{ m^4}{k^2}-\\frac{173 }{495}\\frac{ m^6}{k^4}+\\frac{64 }{99}\\frac{ m^8}{k^6}-\\frac{16 }{33}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{231 \\pi ^3} \\left(-\\frac{1}{64} k+\\frac{5 }{16}\\frac{ m^2}{k}-\\frac{5 }{2}\\frac{ m^4}{k^3}+10 \\frac{ m^6}{k^5}-20 \\frac{ m^8}{k^7}+16 \\frac{ m^{10}}{k^9}\\right)\\right) $ $\\tilde{T}_{3,5;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{3 i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{16 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i L_3}{32 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{16 \\pi ^3} \\left(-\\frac{3 L_2}{4} k^2 m^4+3 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{8 \\pi ^3} \\left(-\\frac{3 L_2}{8} k^2 m^4+ L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{32 \\pi ^3} \\left(- L_3 k^2 m^6+\\frac{5 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{16 \\pi ^3} \\left(- L_3 k^2 m^6+\\frac{5 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{8 \\pi ^3} \\left(\\frac{3 L_2}{8} k^4 m^4- L_3 k^2 m^6+\\frac{5 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{8 \\pi ^3} \\left(\\frac{ L_2}{8} k^4 m^4-\\frac{ L_3}{3} k^2 m^6+\\frac{5 L_4}{6} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{32 \\pi ^3} \\left( L_3 k^4 m^6-\\frac{5 L_4}{2} k^2 m^8+3 L_5 m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{8 \\pi ^3} \\left(-\\frac{ L_2}{8} k^6 m^4+\\frac{ L_3}{3} k^4 m^6-\\frac{5 L_4}{6} k^2 m^8+ L_5 m^{10}\\right)\\right) $ Scalars, spin 4 x 4, dimension 3: $\\tilde{T}_{4,4;3\\text{D}}^{\\text{s,t}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{2 \\pi } \\left(\\frac{15 }{16}\\frac{ m}{k^2}-\\frac{5 }{12}\\frac{ m^3}{k^4}+\\frac{11 }{3}\\frac{ m^5}{k^6}-4 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{1}{128}\\frac{1}{k}+\\frac{1}{8}\\frac{ m^2}{k^3}-\\frac{3 }{4}\\frac{ m^4}{k^5}+2 \\frac{ m^6}{k^7}-2 \\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(-\\frac{3 }{16}\\frac{ m}{k^2}-\\frac{21 }{4}\\frac{ m^3}{k^4}+11 \\frac{ m^5}{k^6}-12 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{3 }{128}\\frac{1}{k}+\\frac{3 }{8}\\frac{ m^2}{k^3}-\\frac{9 }{4}\\frac{ m^4}{k^5}+6 \\frac{ m^6}{k^7}-6 \\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi } \\left(-\\frac{3 }{64}\\frac{ m}{k^2}+\\frac{11 }{16}\\frac{ m^3}{k^4}+\\frac{11 }{4}\\frac{ m^5}{k^6}-3 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{3 }{1024}\\frac{1}{k}+\\frac{3 }{64}\\frac{ m^2}{k^3}-\\frac{9 }{32}\\frac{ m^4}{k^5}+\\frac{3 }{4}\\frac{ m^6}{k^7}-\\frac{3 }{4}\\frac{ m^8}{k^9}\\right)\\right) $ $\\tilde{T}_{4,4;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{2 i }{\\pi } m\\right)+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{4 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{8 i }{\\pi } m^3\\right)+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(-3 k^2 m+8 m^3\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{16 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi } \\left(-40 k^2 m^3+128 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{5 \\pi } \\left(-20 k^2 m^3+64 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi } \\left(6 k^4 m-8 k^2 m^3+\\frac{256 }{5} m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi } \\left(-8 k^2 m^5+\\frac{96 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi } \\left(20 k^4 m^3-64 k^2 m^5+\\frac{768 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi } \\left(-\\frac{1}{2} k^6 m+\\frac{2 }{3} k^4 m^3-\\frac{64 }{15} k^2 m^5+\\frac{256 }{35} m^7\\right)\\right) $ Scalars, spin 4 x 4, dimension 4: $\\tilde{T}_{4,4;4\\text{D}}^{\\text{s,t}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{563}{19845}-\\frac{L_0}{126}\\right)+ \\left(\\frac{2297}{17640}-\\frac{27 L_0}{56}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1339}{560}-\\frac{3 L_0}{8}\\right)\\frac{ m^4}{k^4}-\\frac{832 }{189}\\frac{ m^6}{k^6}+\\frac{256 }{63}\\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{105 \\pi ^2} \\left(-\\frac{1}{3}\\frac{1}{k}+\\frac{16 }{3}\\frac{ m^2}{k^3}-32 \\frac{ m^4}{k^5}+\\frac{256 }{3}\\frac{ m^6}{k^7}-\\frac{256 }{3}\\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{563}{6615}-\\frac{L_0}{42}\\right)+ \\left(-\\frac{1091}{735}+\\frac{3 L_0}{7}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1221}{280}+\\frac{3 L_0}{4}\\right)\\frac{ m^4}{k^4}-\\frac{832 }{63}\\frac{ m^6}{k^6}+\\frac{256 }{21}\\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^2} \\left(-\\frac{1}{3}\\frac{1}{k}+\\frac{16 }{3}\\frac{ m^2}{k^3}-32 \\frac{ m^4}{k^5}+\\frac{256 }{3}\\frac{ m^6}{k^7}-\\frac{256 }{3}\\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{563}{52920}-\\frac{L_0}{336}\\right)+ \\left(-\\frac{1091}{5880}+\\frac{3 L_0}{56}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{699}{560}-\\frac{3 L_0}{8}\\right)\\frac{ m^4}{k^4}-\\frac{104 }{63}\\frac{ m^6}{k^6}+\\frac{32 }{21}\\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^2} \\left(-\\frac{1}{24}\\frac{1}{k}+\\frac{2 }{3}\\frac{ m^2}{k^3}-4 \\frac{ m^4}{k^5}+\\frac{32 }{3}\\frac{ m^6}{k^7}-\\frac{32 }{3}\\frac{ m^8}{k^9}\\right)\\right) $ $\\tilde{T}_{4,4;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{i L_1}{2 \\pi ^2} m^2\\right)+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{3 i L_2}{2 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{3 L_1}{2} k^2 m^2-3 L_2 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{i L_3}{2 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left(3 L_2 k^2 m^4-8 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{4 \\pi ^2} \\left(3 L_2 k^2 m^4-8 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{ L_1}{2} k^4 m^2+\\frac{ L_2}{2} k^2 m^4-\\frac{8 L_3}{3} m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left( L_3 k^2 m^6-\\frac{3 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(-3 L_2 k^4 m^4+8 L_3 k^2 m^6-12 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^2} \\left(\\frac{ L_1}{8} k^6 m^2-\\frac{ L_2}{8} k^4 m^4+\\frac{2 L_3}{3} k^2 m^6- L_4 m^8\\right)\\right) $ Scalars, spin 4 x 4, dimension 5: $\\tilde{T}_{4,4;5\\text{D}}^{\\text{s,t}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{1}{512} m-\\frac{73 }{192}\\frac{ m^3}{k^2}+\\frac{1}{15}\\frac{ m^5}{k^4}-\\frac{7 }{12}\\frac{ m^7}{k^6}+\\frac{1}{2}\\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{2 \\pi } \\left(-\\frac{1}{5120} k+\\frac{1}{256}\\frac{ m^2}{k}-\\frac{1}{32}\\frac{ m^4}{k^3}+\\frac{1}{8}\\frac{ m^6}{k^5}-\\frac{1}{4}\\frac{ m^8}{k^7}+\\frac{1}{5}\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{3 }{512} m+\\frac{7 }{64}\\frac{ m^3}{k^2}+\\frac{6 }{5}\\frac{ m^5}{k^4}-\\frac{7 }{4}\\frac{ m^7}{k^6}+\\frac{3 }{2}\\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{2 \\pi } \\left(-\\frac{3 }{5120} k+\\frac{3 }{256}\\frac{ m^2}{k}-\\frac{3 }{32}\\frac{ m^4}{k^3}+\\frac{3 }{8}\\frac{ m^6}{k^5}-\\frac{3 }{4}\\frac{ m^8}{k^7}+\\frac{3 }{5}\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{10 \\pi ^2} \\left(-\\frac{3 }{2048} m+\\frac{7 }{256}\\frac{ m^3}{k^2}-\\frac{1}{5}\\frac{ m^5}{k^4}-\\frac{7 }{16}\\frac{ m^7}{k^6}+\\frac{3 }{8}\\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(-\\frac{3 }{5120} k+\\frac{3 }{256}\\frac{ m^2}{k}-\\frac{3 }{32}\\frac{ m^4}{k^3}+\\frac{3 }{8}\\frac{ m^6}{k^5}-\\frac{3 }{4}\\frac{ m^8}{k^7}+\\frac{3 }{5}\\frac{ m^{10}}{k^9}\\right)\\right) $ $\\tilde{T}_{4,4;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{i }{3 \\pi ^2} m^3\\right)+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{4 i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left(\\frac{1}{2} k^2 m^3-\\frac{4 }{5} m^5\\right)\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{8 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(4 k^2 m^5-\\frac{64 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{5 \\pi ^2} \\left(2 k^2 m^5-\\frac{32 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi ^2} \\left(- k^4 m^3+\\frac{4 }{5} k^2 m^5-\\frac{128 }{35} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{35 \\pi ^2} \\left(4 k^2 m^7-\\frac{16 }{3} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(-2 k^4 m^5+\\frac{32 }{7} k^2 m^7-\\frac{128 }{21} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{1}{4} k^6 m^3-\\frac{1}{5} k^4 m^5+\\frac{32 }{35} k^2 m^7-\\frac{128 }{105} m^9\\right)\\right) $ Scalars, spin 4 x 4, dimension 6: $\\tilde{T}_{4,4;6\\text{D}}^{\\text{s,t}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(\\frac{1627}{4802490}-\\frac{L_0}{11088}\\right) k^2+ \\left(-\\frac{12701}{1746360}+\\frac{L_0}{504}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{166489}{3104640}+\\frac{27 L_0}{448}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{126691}{665280}+\\frac{L_0}{32}\\right)\\frac{ m^6}{k^4}+\\frac{512 }{2079}\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{128 }{693}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{693 \\pi ^3} \\left(-\\frac{1}{40} k+\\frac{1}{2}\\frac{ m^2}{k}-4 \\frac{ m^4}{k^3}+16 \\frac{ m^6}{k^5}-32 \\frac{ m^8}{k^7}+\\frac{128 }{5}\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(\\frac{1627}{1600830}-\\frac{L_0}{3696}\\right) k^2+ \\left(-\\frac{12701}{582120}+\\frac{L_0}{168}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{24667}{129360}-\\frac{3 L_0}{56}\\right)\\frac{ m^4}{k^2}- \\left(\\frac{31583}{110880}+\\frac{L_0}{16}\\right)\\frac{ m^6}{k^4}+\\frac{512 }{693}\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{128 }{231}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{231 \\pi ^3} \\left(-\\frac{1}{40} k+\\frac{1}{2}\\frac{ m^2}{k}-4 \\frac{ m^4}{k^3}+16 \\frac{ m^6}{k^5}-32 \\frac{ m^8}{k^7}+\\frac{128 }{5}\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(\\frac{1627}{12806640}-\\frac{L_0}{29568}\\right) k^2+ \\left(-\\frac{12701}{4656960}+\\frac{L_0}{1344}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{24667}{1034880}-\\frac{3 L_0}{448}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{23777}{221760}+\\frac{L_0}{32}\\right)\\frac{ m^6}{k^4}+\\frac{64 }{693}\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{16 }{231}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{231 \\pi ^3} \\left(-\\frac{1}{320} k+\\frac{1}{16}\\frac{ m^2}{k}-\\frac{1}{2}\\frac{ m^4}{k^3}+2 \\frac{ m^6}{k^5}-4 \\frac{ m^8}{k^7}+\\frac{16 }{5}\\frac{ m^{10}}{k^9}\\right)\\right) $ $\\tilde{T}_{4,4;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{16 \\pi ^3} m^4\\right)+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{16 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_3}{8 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{8 \\pi ^3} \\left(-\\frac{3 L_2}{4} k^2 m^4+ L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i L_4}{32 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^3} \\left(-\\frac{ L_3}{2} k^2 m^6+ L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{8 \\pi ^3} \\left(-\\frac{ L_3}{2} k^2 m^6+ L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{2 \\pi ^3} \\left(\\frac{ L_2}{8} k^4 m^4-\\frac{ L_3}{12} k^2 m^6+\\frac{ L_4}{3} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{32 \\pi ^3} \\left(-\\frac{ L_4}{2} k^2 m^8+\\frac{3 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^3} \\left(\\frac{ L_3}{4} k^4 m^6-\\frac{ L_4}{2} k^2 m^8+\\frac{3 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{4 \\pi ^3} \\left(-\\frac{ L_2}{16} k^6 m^4+\\frac{ L_3}{24} k^4 m^6-\\frac{ L_4}{6} k^2 m^8+\\frac{ L_5}{5} m^{10}\\right)\\right) $ Scalars, spin 5 x 5, dimension 3: $\\tilde{T}_{5,5;3\\text{D}}^{\\text{s,t}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi } \\left(-\\frac{31 }{64}\\frac{ m}{k^2}-\\frac{23 }{24}\\frac{ m^3}{k^4}-\\frac{32 }{15}\\frac{ m^5}{k^6}+\\frac{14 }{3}\\frac{ m^7}{k^8}-4 \\frac{ m^9}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{1}{256}\\frac{1}{k}-\\frac{5 }{64}\\frac{ m^2}{k^3}+\\frac{5 }{8}\\frac{ m^4}{k^5}-\\frac{5 }{2}\\frac{ m^6}{k^7}+5 \\frac{ m^8}{k^9}-4 \\frac{ m^{10}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(\\frac{5 }{64}\\frac{ m}{k^2}+\\frac{125 }{24}\\frac{ m^3}{k^4}-\\frac{32 }{3}\\frac{ m^5}{k^6}+\\frac{70 }{3}\\frac{ m^7}{k^8}-20 \\frac{ m^9}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{5 }{256}\\frac{1}{k}-\\frac{25 }{64}\\frac{ m^2}{k^3}+\\frac{25 }{8}\\frac{ m^4}{k^5}-\\frac{25 }{2}\\frac{ m^6}{k^7}+25 \\frac{ m^8}{k^9}-20 \\frac{ m^{10}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(\\frac{15 }{256}\\frac{ m}{k^2}-\\frac{35 }{32}\\frac{ m^3}{k^4}-8 \\frac{ m^5}{k^6}+\\frac{35 }{2}\\frac{ m^7}{k^8}-15 \\frac{ m^9}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{15 }{2048}\\frac{1}{k}-\\frac{75 }{512}\\frac{ m^2}{k^3}+\\frac{75 }{64}\\frac{ m^4}{k^5}-\\frac{75 }{16}\\frac{ m^6}{k^7}+\\frac{75 }{8}\\frac{ m^8}{k^9}-\\frac{15 }{2}\\frac{ m^{10}}{k^{11}}\\right)\\right) $ $\\tilde{T}_{5,5;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^4 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{5 i }{2 \\pi } m\\right)+k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{20 i }{3 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^4 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{20 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{3 \\pi } \\left(-15 k^2 m+40 m^3\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{16 i }{\\pi } m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^4 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{8 i }{\\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(-20 k^2 m^3+96 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{\\pi } \\left(-20 k^2 m^3+64 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi } \\left(5 k^4 m+64 m^5\\right)\\right)+k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi } \\left(-56 k^2 m^5+160 m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{7 \\pi } \\left(-112 k^2 m^5+320 m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{7 \\pi } \\left(140 k^4 m^3-448 k^2 m^5+1280 m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3+k_{\\mu }^2 \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{3 \\pi } \\left(20 k^4 m^3-64 k^2 m^5+\\frac{1280 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi } \\left(-\\frac{5 }{2} k^6 m-\\frac{10 }{3} k^4 m^3-\\frac{64 }{3} k^2 m^5+\\frac{1280 }{21} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi } \\left(56 k^4 m^5-160 k^2 m^7+\\frac{640 }{3} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi } \\left(-20 k^6 m^3+64 k^4 m^5-\\frac{1280 }{7} k^2 m^7+\\frac{5120 }{21} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi } \\left(\\frac{1}{2} k^8 m+\\frac{2 }{3} k^6 m^3+\\frac{64 }{15} k^4 m^5-\\frac{256 }{21} k^2 m^7+\\frac{1024 }{63} m^9\\right)\\right) $ Scalars, spin 5 x 5, dimension 4: $\\tilde{T}_{5,5;4\\text{D}}^{\\text{s,t}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(-\\frac{6508}{343035}+\\frac{L_0}{198}\\right)+ \\left(-\\frac{116687}{249480}+\\frac{55 L_0}{72}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{270101}{55440}+\\frac{15 L_0}{8}\\right)\\frac{ m^4}{k^4}+\\frac{11072 }{1485}\\frac{ m^6}{k^6}-\\frac{4096 }{297}\\frac{ m^8}{k^8}+\\frac{1024 }{99}\\frac{ m^{10}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{693 \\pi ^2} \\left(\\frac{1}{k}-20 \\frac{ m^2}{k^3}+160 \\frac{ m^4}{k^5}-640 \\frac{ m^6}{k^7}+1280 \\frac{ m^8}{k^9}-1024 \\frac{ m^{10}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(-\\frac{6508}{68607}+\\frac{5 L_0}{198}\\right)+ \\left(\\frac{12701}{6237}-\\frac{5 L_0}{9}\\right)\\frac{ m^2}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{25903}{5544}+\\frac{15 L_0}{4}\\right)\\frac{ m^4}{k^4}+\\frac{11072 }{297}\\frac{ m^6}{k^6}-\\frac{20480 }{297}\\frac{ m^8}{k^8}+\\frac{5120 }{99}\\frac{ m^{10}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{693 \\pi ^2} \\left(5 \\frac{1}{k}-100 \\frac{ m^2}{k^3}+800 \\frac{ m^4}{k^5}-3200 \\frac{ m^6}{k^7}+6400 \\frac{ m^8}{k^9}-5120 \\frac{ m^{10}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(-\\frac{1627}{45738}+\\frac{5 L_0}{528}\\right)+ \\left(\\frac{12701}{16632}-\\frac{5 L_0}{24}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{24667}{3696}+\\frac{15 L_0}{8}\\right)\\frac{ m^4}{k^4}+\\frac{1384 }{99}\\frac{ m^6}{k^6}-\\frac{2560 }{99}\\frac{ m^8}{k^8}+\\frac{640 }{33}\\frac{ m^{10}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{231 \\pi ^2} \\left(\\frac{5 }{8}\\frac{1}{k}-\\frac{25 }{2}\\frac{ m^2}{k^3}+100 \\frac{ m^4}{k^5}-400 \\frac{ m^6}{k^7}+800 \\frac{ m^8}{k^9}-640 \\frac{ m^{10}}{k^{11}}\\right)\\right) $ $\\tilde{T}_{5,5;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^4 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{5 i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{5 i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^4 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{15 i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{5 L_1}{2} k^2 m^2-5 L_2 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{5 i L_3}{2 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^4 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{5 i L_3}{4 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(15 L_2 k^2 m^4-60 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{4 \\pi ^2} \\left(15 L_2 k^2 m^4-40 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{4 \\pi ^2} \\left(-5 L_1 k^4 m^2-40 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left(5 L_3 k^2 m^6-\\frac{25 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{2 \\pi ^2} \\left(5 L_3 k^2 m^6-\\frac{25 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(-15 L_2 k^4 m^4+40 L_3 k^2 m^6-100 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3+k_{\\mu }^2 \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{5 L_2}{4} k^4 m^4+\\frac{10 L_3}{3} k^2 m^6-\\frac{25 L_4}{3} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^2} \\left(\\frac{5 L_1}{8} k^6 m^2+\\frac{5 L_2}{8} k^4 m^4+\\frac{10 L_3}{3} k^2 m^6-\\frac{25 L_4}{3} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left(-5 L_3 k^4 m^6+\\frac{25 L_4}{2} k^2 m^8-15 L_5 m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(\\frac{5 L_2}{4} k^6 m^4-\\frac{10 L_3}{3} k^4 m^6+\\frac{25 L_4}{3} k^2 m^8-10 L_5 m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{ L_1}{8} k^8 m^2-\\frac{ L_2}{8} k^6 m^4-\\frac{2 L_3}{3} k^4 m^6+\\frac{5 L_4}{3} k^2 m^8-2 L_5 m^{10}\\right)\\right) $ Scalars, spin 5 x 5, dimension 5: $\\tilde{T}_{5,5;5\\text{D}}^{\\text{s,t}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{1}{3072} m+\\frac{367 }{2304}\\frac{ m^3}{k^2}+\\frac{97 }{480}\\frac{ m^5}{k^4}+\\frac{11 }{40}\\frac{ m^7}{k^6}-\\frac{17 }{36}\\frac{ m^9}{k^8}+\\frac{1}{3}\\frac{ m^{11}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{2 \\pi } \\left(\\frac{1}{12288} k-\\frac{1}{512}\\frac{ m^2}{k}+\\frac{5 }{256}\\frac{ m^4}{k^3}-\\frac{5 }{48}\\frac{ m^6}{k^5}+\\frac{5 }{16}\\frac{ m^8}{k^7}-\\frac{1}{2}\\frac{ m^{10}}{k^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3}\\frac{ m^{12}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{5 }{3072} m-\\frac{85 }{2304}\\frac{ m^3}{k^2}-\\frac{95 }{96}\\frac{ m^5}{k^4}+\\frac{11 }{8}\\frac{ m^7}{k^6}-\\frac{85 }{36}\\frac{ m^9}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{5 }{3}\\frac{ m^{11}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{2 \\pi } \\left(\\frac{5 }{12288} k-\\frac{5 }{512}\\frac{ m^2}{k}+\\frac{25 }{256}\\frac{ m^4}{k^3}-\\frac{25 }{48}\\frac{ m^6}{k^5}+\\frac{25 }{16}\\frac{ m^8}{k^7}-\\frac{5 }{2}\\frac{ m^{10}}{k^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{5 }{3}\\frac{ m^{12}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{16 \\pi ^2} \\left(\\frac{5 }{1024} m-\\frac{85 }{768}\\frac{ m^3}{k^2}+\\frac{33 }{32}\\frac{ m^5}{k^4}+\\frac{33 }{8}\\frac{ m^7}{k^6}-\\frac{85 }{12}\\frac{ m^9}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ 5 \\frac{ m^{11}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(\\frac{5 }{4096} k-\\frac{15 }{512}\\frac{ m^2}{k}+\\frac{75 }{256}\\frac{ m^4}{k^3}-\\frac{25 }{16}\\frac{ m^6}{k^5}+\\frac{75 }{16}\\frac{ m^8}{k^7}-\\frac{15 }{2}\\frac{ m^{10}}{k^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ 5 \\frac{ m^{12}}{k^{11}}\\right)\\right) $ $\\tilde{T}_{5,5;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^4 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{5 i }{12 \\pi ^2} m^3\\right)+k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{3 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^4 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{2 i }{\\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{5 }{2} k^2 m^3-4 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{8 i }{7 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^4 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{4 i }{7 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left(14 k^2 m^5-48 m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{7 \\pi ^2} \\left(14 k^2 m^5-32 m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{5 }{6} k^4 m^3-\\frac{32 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi ^2} \\left(4 k^2 m^7-\\frac{80 }{9} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{7 \\pi ^2} \\left(8 k^2 m^7-\\frac{160 }{9} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left(-14 k^4 m^5+32 k^2 m^7-\\frac{640 }{9} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3+k_{\\mu }^2 \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{3 \\pi ^2} \\left(-2 k^4 m^5+\\frac{32 }{7} k^2 m^7-\\frac{640 }{63} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{5 }{4} k^6 m^3+k^4 m^5+\\frac{32 }{7} k^2 m^7-\\frac{640 }{63} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi ^2} \\left(-4 k^4 m^7+\\frac{80 }{9} k^2 m^9-\\frac{320 }{33} m^{11}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left(2 k^6 m^5-\\frac{32 }{7} k^4 m^7+\\frac{640 }{63} k^2 m^9-\\frac{2560 }{231} m^{11}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^5 \\left(\\frac{i }{3 \\pi ^2} \\left(-\\frac{1}{4} k^8 m^3-\\frac{1}{5} k^6 m^5-\\frac{32 }{35} k^4 m^7+\\frac{128 }{63} k^2 m^9-\\frac{512 }{231} m^{11}\\right)\\right) $ Scalars, spin 5 x 5, dimension 6: $\\tilde{T}_{5,5;6\\text{D}}^{\\text{s,t}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{7 \\pi ^3} \\left( \\left(-\\frac{88069}{463783320}+\\frac{L_0}{20592}\\right) k^2+ \\left(\\frac{172673}{35675640}-\\frac{L_0}{792}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{5813737}{51891840}-\\frac{55 L_0}{576}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{3782563}{8648640}-\\frac{5 L_0}{32}\\right)\\frac{ m^6}{k^4}-\\frac{736 }{1755}\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{2432 }{3861}\\frac{ m^{10}}{k^8}-\\frac{512 }{1287}\\frac{ m^{12}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{3003 \\pi ^3} \\left(\\frac{1}{24} k-\\frac{ m^2}{k}+10 \\frac{ m^4}{k^3}-\\frac{160 }{3}\\frac{ m^6}{k^5}+160 \\frac{ m^8}{k^7}-256 \\frac{ m^{10}}{k^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{512 }{3}\\frac{ m^{12}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^3} \\left( \\left(-\\frac{88069}{92756664}+\\frac{5 L_0}{20592}\\right) k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{172673}{7135128}-\\frac{5 L_0}{792}\\right) m^2+ \\left(-\\frac{337471}{1297296}+\\frac{5 L_0}{72}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{157049}{864864}+\\frac{5 L_0}{16}\\right)\\frac{ m^6}{k^4}-\\frac{736 }{351}\\frac{ m^8}{k^6}+\\frac{12160 }{3861}\\frac{ m^{10}}{k^8}-\\frac{2560 }{1287}\\frac{ m^{12}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{3003 \\pi ^3} \\left(\\frac{5 }{24} k-5 \\frac{ m^2}{k}+50 \\frac{ m^4}{k^3}-\\frac{800 }{3}\\frac{ m^6}{k^5}+800 \\frac{ m^8}{k^7}-1280 \\frac{ m^{10}}{k^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{2560 }{3}\\frac{ m^{12}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi ^3} \\left( \\left(-\\frac{88069}{247351104}+\\frac{5 L_0}{54912}\\right) k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{172673}{19027008}-\\frac{5 L_0}{2112}\\right) m^2+ \\left(-\\frac{337471}{3459456}+\\frac{5 L_0}{192}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{328301}{576576}-\\frac{5 L_0}{32}\\right)\\frac{ m^6}{k^4}-\\frac{92 }{117}\\frac{ m^8}{k^6}+\\frac{1520 }{1287}\\frac{ m^{10}}{k^8}-\\frac{320 }{429}\\frac{ m^{12}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{1001 \\pi ^3} \\left(\\frac{5 }{192} k-\\frac{5 }{8}\\frac{ m^2}{k}+\\frac{25 }{4}\\frac{ m^4}{k^3}-\\frac{100 }{3}\\frac{ m^6}{k^5}+100 \\frac{ m^8}{k^7}-160 \\frac{ m^{10}}{k^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{320 }{3}\\frac{ m^{12}}{k^{11}}\\right)\\right) $ $\\tilde{T}_{5,5;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^4 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{5 i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{5 i L_3}{48 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^4 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{5 i L_3}{16 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{i }{8 \\pi ^3} \\left(-\\frac{5 L_2}{4} k^2 m^4+\\frac{5 L_3}{3} m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{5 i L_4}{32 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^4 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{5 i L_4}{64 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{16 \\pi ^3} \\left(-5 L_3 k^2 m^6+15 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{8 \\pi ^3} \\left(-\\frac{5 L_3}{2} k^2 m^6+5 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{8 \\pi ^3} \\left(\\frac{5 L_2}{4} k^4 m^4+5 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{32 \\pi ^3} \\left(-\\frac{5 L_4}{2} k^2 m^8+5 L_5 m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{16 \\pi ^3} \\left(-\\frac{5 L_4}{2} k^2 m^8+5 L_5 m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^3} \\left(\\frac{5 L_3}{4} k^4 m^6-\\frac{5 L_4}{2} k^2 m^8+5 L_5 m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3+k_{\\mu }^2 \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{12 \\pi ^3} \\left(\\frac{5 L_3}{4} k^4 m^6-\\frac{5 L_4}{2} k^2 m^8+5 L_5 m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{4 \\pi ^3} \\left(-\\frac{5 L_2}{16} k^6 m^4-\\frac{5 L_3}{24} k^4 m^6-\\frac{5 L_4}{6} k^2 m^8+\\frac{5 L_5}{3} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{32 \\pi ^3} \\left(\\frac{5 L_4}{2} k^4 m^8-5 L_5 k^2 m^{10}+5 L_6 m^{12}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^3} \\left(-\\frac{5 L_3}{4} k^6 m^6+\\frac{5 L_4}{2} k^4 m^8-5 L_5 k^2 m^{10}+5 L_6 m^{12}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^5 \\left(\\frac{i }{4 \\pi ^3} \\left(\\frac{ L_2}{16} k^8 m^4+\\frac{ L_3}{24} k^6 m^6+\\frac{ L_4}{6} k^4 m^8-\\frac{ L_5}{3} k^2 m^{10}+\\frac{ L_6}{3} m^{12}\\right)\\right) $"
],
[
"Expansions in UV and IR for scalars",
"Scalars, spin 0 x 0, dimension 3: $\\tilde{T}_{0,0;3\\text{D}}^{\\text{s,UV}} & = -\\frac{1}{8}\\frac{1}{k}+\\frac{i }{2 \\pi }\\frac{ m}{k^2}+\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{8 i }{5 \\pi }\\frac{ m^5}{k^6}+\\frac{32 i }{7 \\pi }\\frac{ m^7}{k^8}+\\frac{128 i }{9 \\pi }\\frac{ m^9}{k^{10}}+\\frac{512 i }{11 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots $ $\\tilde{T}_{0,0;3\\text{D}}^{\\text{s,IR}} & = \\frac{i }{8 \\pi } \\left(\\frac{1}{m}+\\frac{1}{12}\\frac{ k^2}{m^3}+\\frac{1}{80}\\frac{ k^4}{m^5}+\\frac{1}{448}\\frac{ k^6}{m^7}+\\frac{1}{2304}\\frac{ k^8}{m^9}+\\frac{1}{11264}\\frac{ k^{10}}{m^{11}}+\\ldots \\right) $ $\\tilde{T}_{0,0;3\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 0 x 0, dimension 4: $\\tilde{T}_{0,0;4\\text{D}}^{\\text{s,UV}} & = \\frac{i }{4 \\pi ^2} \\left( \\left(\\frac{1}{2}-\\frac{P}{4}\\right)+ \\left(\\frac{1}{2}+\\frac{K}{2}\\right)\\frac{ m^2}{k^2}+ \\left(-\\frac{1}{4}+\\frac{K}{2}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{5}{6}+K\\right)\\frac{ m^6}{k^6}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\left(-\\frac{59}{24}+\\frac{5 K}{2}\\right)\\frac{ m^8}{k^8}+ \\left(-\\frac{449}{60}+7 K\\right)\\frac{ m^{10}}{k^{10}}+ \\left(-\\frac{1417}{60}+21 K\\right)\\frac{ m^{12}}{k^{12}}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\ldots \\right) $ $\\tilde{T}_{0,0;4\\text{D}}^{\\text{s,IR}} & = \\frac{i }{16 \\pi ^2} \\left(- L_0+\\frac{1}{6}\\frac{ k^2}{m^2}+\\frac{1}{60}\\frac{ k^4}{m^4}+\\frac{1}{420}\\frac{ k^6}{m^6}+\\frac{1}{2520}\\frac{ k^8}{m^8}+\\frac{1}{13860}\\frac{ k^{10}}{m^{10}}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{1}{72072}\\frac{ k^{12}}{m^{12}}+\\ldots \\right) $ $\\tilde{T}_{0,0;4\\text{D}}^{\\text{s,UV-IR}} & = \\frac{i }{8 \\pi ^2}\\left(1-\\frac{K}{2}\\right)+\\ldots $ Scalars, spin 0 x 0, dimension 5: $\\tilde{T}_{0,0;5\\text{D}}^{\\text{s,UV}} & = \\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{128} k+\\frac{ \\pi }{32}\\frac{ m^2}{k}-\\frac{i }{12}\\frac{ m^3}{k^2}-\\frac{i }{15}\\frac{ m^5}{k^4}-\\frac{4 i }{35}\\frac{ m^7}{k^6}-\\frac{16 i }{63}\\frac{ m^9}{k^8}-\\frac{64 i }{99}\\frac{ m^{11}}{k^{10}}+\\ldots \\right) $ $\\tilde{T}_{0,0;5\\text{D}}^{\\text{s,IR}} & = \\frac{i }{16 \\pi ^2} \\left(- m+\\frac{1}{12}\\frac{ k^2}{m}+\\frac{1}{240}\\frac{ k^4}{m^3}+\\frac{1}{2240}\\frac{ k^6}{m^5}+\\frac{1}{16128}\\frac{ k^8}{m^7}+\\frac{1}{101376}\\frac{ k^{10}}{m^9}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{1}{585728}\\frac{ k^{12}}{m^{11}}+\\ldots \\right) $ $\\tilde{T}_{0,0;5\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 0 x 0, dimension 6: $\\tilde{T}_{0,0;6\\text{D}}^{\\text{s,UV}} & = \\frac{i }{16 \\pi ^3} \\left( \\left(\\frac{1}{9}-\\frac{P}{24}\\right) k^2+ \\left(-\\frac{1}{2}+\\frac{P}{4}\\right) m^2- \\left(\\frac{3}{8}+\\frac{K}{4}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{36}-\\frac{K}{6}\\right)\\frac{ m^6}{k^4}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\left(\\frac{7}{48}-\\frac{K}{4}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{47}{120}-\\frac{K}{2}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{379}{360}-\\frac{7 K}{6}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right) $ $\\tilde{T}_{0,0;6\\text{D}}^{\\text{s,IR}} & = \\frac{i }{64 \\pi ^3} \\left( \\left(-1+L_0\\right) m^2-\\frac{ L_0}{6} k^2+\\frac{1}{60}\\frac{ k^4}{m^2}+\\frac{1}{840}\\frac{ k^6}{m^4}+\\frac{1}{7560}\\frac{ k^8}{m^6}+\\frac{1}{55440}\\frac{ k^{10}}{m^8}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{1}{360360}\\frac{ k^{12}}{m^{10}}+\\frac{1}{2162160}\\frac{ k^{14}}{m^{12}}+\\ldots \\right) $ $\\tilde{T}_{0,0;6\\text{D}}^{\\text{s,UV-IR}} & = \\frac{i }{16 \\pi ^3} \\left( \\left(\\frac{1}{9}-\\frac{K}{24}\\right) k^2+ \\left(-\\frac{1}{4}+\\frac{K}{4}\\right) m^2\\right)+\\ldots $ Scalars, spin 0 x 2, dimension 3: $\\tilde{T}_{0,2;3\\text{D}}^{\\text{s,t,UV}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{16}\\frac{1}{k}-\\frac{i }{2 \\pi }\\frac{ m}{k^2}-\\frac{1}{4}\\frac{ m^2}{k^3}+\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{8 i }{15 \\pi }\\frac{ m^5}{k^6}+\\frac{32 i }{35 \\pi }\\frac{ m^7}{k^8}+\\frac{128 i }{63 \\pi }\\frac{ m^9}{k^{10}}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{512 i }{99 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{0,2;3\\text{D}}^{\\text{s,t,IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{8 \\pi } \\left(-\\frac{1}{3}\\frac{1}{m}-\\frac{1}{60}\\frac{ k^2}{m^3}-\\frac{1}{560}\\frac{ k^4}{m^5}-\\frac{1}{4032}\\frac{ k^6}{m^7}-\\frac{1}{25344}\\frac{ k^8}{m^9}-\\frac{1}{146432}\\frac{ k^{10}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right) $ $\\tilde{T}_{0,2;3\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 0 x 2, dimension 4: $\\tilde{T}_{0,2;4\\text{D}}^{\\text{s,t,UV}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left( \\left(-\\frac{1}{9}+\\frac{P}{24}\\right)+ \\left(\\frac{1}{4}-\\frac{K}{4}\\right)\\frac{ m^2}{k^2}+ \\left(\\frac{3}{8}+\\frac{K}{4}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{36}+\\frac{K}{6}\\right)\\frac{ m^6}{k^6}+ \\left(-\\frac{7}{48}+\\frac{K}{4}\\right)\\frac{ m^8}{k^8}+ \\left(-\\frac{47}{120}+\\frac{K}{2}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{379}{360}+\\frac{7 K}{6}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;4\\text{D}}^{\\text{s,t,IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{48 \\pi ^2} \\left( L_0-\\frac{1}{10}\\frac{ k^2}{m^2}-\\frac{1}{140}\\frac{ k^4}{m^4}-\\frac{1}{1260}\\frac{ k^6}{m^6}-\\frac{1}{9240}\\frac{ k^8}{m^8}-\\frac{1}{60060}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{360360}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;4\\text{D}}^{\\text{s,UV-IR}} & = k^2 \\pi _{\\nu \\nu } \\frac{i }{6 \\pi ^2}\\left(-\\frac{1}{3}+\\frac{K}{8}\\right)+\\ldots $ Scalars, spin 0 x 2, dimension 5: $\\tilde{T}_{0,2;5\\text{D}}^{\\text{s,t,UV}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{512} k-\\frac{ \\pi }{64}\\frac{ m^2}{k}+\\frac{i }{12}\\frac{ m^3}{k^2}+\\frac{ \\pi }{32}\\frac{ m^4}{k^3}-\\frac{i }{15}\\frac{ m^5}{k^4}-\\frac{4 i }{105}\\frac{ m^7}{k^6}-\\frac{16 i }{315}\\frac{ m^9}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{64 i }{693}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;5\\text{D}}^{\\text{s,t,IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{48 \\pi ^2} \\left(m-\\frac{1}{20}\\frac{ k^2}{m}-\\frac{1}{560}\\frac{ k^4}{m^3}-\\frac{1}{6720}\\frac{ k^6}{m^5}-\\frac{1}{59136}\\frac{ k^8}{m^7}-\\frac{1}{439296}\\frac{ k^{10}}{m^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{2928640}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;5\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 0 x 2, dimension 6: $\\tilde{T}_{0,2;6\\text{D}}^{\\text{s,t,UV}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{8 \\pi ^3} \\left( \\left(-\\frac{23}{1800}+\\frac{P}{240}\\right) k^2+ \\left(\\frac{1}{9}-\\frac{P}{24}\\right) m^2+ \\left(-\\frac{1}{16}+\\frac{K}{8}\\right)\\frac{ m^4}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{11}{72}+\\frac{K}{12}\\right)\\frac{ m^6}{k^4}- \\left(\\frac{1}{288}+\\frac{K}{24}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{23}{1200}-\\frac{K}{20}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{37}{720}-\\frac{K}{12}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;6\\text{D}}^{\\text{s,t,IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{192 \\pi ^3} \\left( \\left(1-L_0\\right) m^2+\\frac{ L_0}{10} k^2-\\frac{1}{140}\\frac{ k^4}{m^2}-\\frac{1}{2520}\\frac{ k^6}{m^4}-\\frac{1}{27720}\\frac{ k^8}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{240240}\\frac{ k^{10}}{m^8}-\\frac{1}{1801800}\\frac{ k^{12}}{m^{10}}-\\frac{1}{12252240}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;6\\text{D}}^{\\text{s,UV-IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{192 \\pi ^3} \\left( \\left(-\\frac{23}{75}+\\frac{K}{10}\\right) k^2+ \\left(\\frac{5}{3}-K\\right) m^2\\right)\\right)+\\ldots $ Scalars, spin 0 x 4, dimension 3: $\\tilde{T}_{0,4;3\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(-\\frac{3 }{64}\\frac{1}{k}+\\frac{i }{2 \\pi }\\frac{ m}{k^2}+\\frac{3 }{8}\\frac{ m^2}{k^3}-\\frac{2 i }{\\pi }\\frac{ m^3}{k^4}-\\frac{3 }{4}\\frac{ m^4}{k^5}+\\frac{8 i }{5 \\pi }\\frac{ m^5}{k^6}+\\frac{32 i }{35 \\pi }\\frac{ m^7}{k^8}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{128 i }{105 \\pi }\\frac{ m^9}{k^{10}}+\\frac{512 i }{231 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{0,4;3\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{8 \\pi } \\left(\\frac{1}{5}\\frac{1}{m}+\\frac{1}{140}\\frac{ k^2}{m^3}+\\frac{1}{1680}\\frac{ k^4}{m^5}+\\frac{1}{14784}\\frac{ k^6}{m^7}+\\frac{1}{109824}\\frac{ k^8}{m^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{732160}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;3\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 0 x 4, dimension 4: $\\tilde{T}_{0,4;4\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left( \\left(\\frac{23}{150}-\\frac{P}{20}\\right)+ \\left(-\\frac{5}{6}+\\frac{K}{2}\\right)\\frac{ m^2}{k^2}+ \\left(\\frac{3}{4}-\\frac{3 K}{2}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{11}{6}+K\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{1}{24}+\\frac{K}{2}\\right)\\frac{ m^8}{k^8}+ \\left(-\\frac{23}{100}+\\frac{3 K}{5}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{37}{60}+K\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;4\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{80 \\pi ^2} \\left(- L_0+\\frac{1}{14}\\frac{ k^2}{m^2}+\\frac{1}{252}\\frac{ k^4}{m^4}+\\frac{1}{2772}\\frac{ k^6}{m^6}+\\frac{1}{24024}\\frac{ k^8}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{180180}\\frac{ k^{10}}{m^{10}}+\\frac{1}{1225224}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;4\\text{D}}^{\\text{s,UV-IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\frac{i }{40 \\pi ^2}\\left(\\frac{23}{15}-\\frac{K}{2}\\right)+\\ldots $ Scalars, spin 0 x 4, dimension 5: $\\tilde{T}_{0,4;5\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{1024} k+\\frac{3 \\pi }{256}\\frac{ m^2}{k}-\\frac{i }{12}\\frac{ m^3}{k^2}-\\frac{3 \\pi }{64}\\frac{ m^4}{k^3}+\\frac{i }{5}\\frac{ m^5}{k^4}+\\frac{ \\pi }{16}\\frac{ m^6}{k^5}-\\frac{4 i }{35}\\frac{ m^7}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{16 i }{315}\\frac{ m^9}{k^8}-\\frac{64 i }{1155}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;5\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{16 \\pi ^2} \\left(-\\frac{1}{5} m+\\frac{1}{140}\\frac{ k^2}{m}+\\frac{1}{5040}\\frac{ k^4}{m^3}+\\frac{1}{73920}\\frac{ k^6}{m^5}+\\frac{1}{768768}\\frac{ k^8}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{6589440}\\frac{ k^{10}}{m^9}+\\frac{1}{49786880}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;5\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 0 x 4, dimension 6: $\\tilde{T}_{0,4;6\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^3} \\left( \\left(\\frac{11}{3675}-\\frac{P}{1120}\\right) k^2+ \\left(-\\frac{23}{600}+\\frac{P}{80}\\right) m^2+ \\left(\\frac{7}{96}-\\frac{K}{16}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{48}+\\frac{K}{8}\\right)\\frac{ m^6}{k^4}- \\left(\\frac{25}{192}+\\frac{K}{16}\\right)\\frac{ m^8}{k^6}- \\left(\\frac{17}{2400}+\\frac{K}{40}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{13}{2400}-\\frac{K}{40}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;6\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{320 \\pi ^3} \\left( \\left(-1+L_0\\right) m^2-\\frac{ L_0}{14} k^2+\\frac{1}{252}\\frac{ k^4}{m^2}+\\frac{1}{5544}\\frac{ k^6}{m^4}+\\frac{1}{72072}\\frac{ k^8}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{720720}\\frac{ k^{10}}{m^8}+\\frac{1}{6126120}\\frac{ k^{12}}{m^{10}}+\\frac{1}{46558512}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;6\\text{D}}^{\\text{s,UV-IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{20 \\pi ^3} \\left( \\left(\\frac{11}{735}-\\frac{K}{224}\\right) k^2+ \\left(-\\frac{31}{240}+\\frac{K}{16}\\right) m^2\\right)\\right)+\\ldots $ Scalars, spin 1 x 1, dimension 3: $\\tilde{T}_{1,1;3\\text{D}}^{\\text{s,t,UV}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{1}{16}\\frac{1}{k}-\\frac{i }{2 \\pi }\\frac{ m}{k^2}-\\frac{1}{4}\\frac{ m^2}{k^3}+\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{8 i }{15 \\pi }\\frac{ m^5}{k^6}+\\frac{32 i }{35 \\pi }\\frac{ m^7}{k^8}+\\frac{128 i }{63 \\pi }\\frac{ m^9}{k^{10}}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{512 i }{99 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{1,1;3\\text{D}}^{\\text{s,t,IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{8 \\pi } \\left(-\\frac{1}{3}\\frac{1}{m}-\\frac{1}{60}\\frac{ k^2}{m^3}-\\frac{1}{560}\\frac{ k^4}{m^5}-\\frac{1}{4032}\\frac{ k^6}{m^7}-\\frac{1}{25344}\\frac{ k^8}{m^9}-\\frac{1}{146432}\\frac{ k^{10}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right) $ $\\tilde{T}_{1,1;3\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 1 x 1, dimension 4: $\\tilde{T}_{1,1;4\\text{D}}^{\\text{s,t,UV}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left( \\left(-\\frac{1}{9}+\\frac{P}{24}\\right)+ \\left(\\frac{1}{4}-\\frac{K}{4}\\right)\\frac{ m^2}{k^2}+ \\left(\\frac{3}{8}+\\frac{K}{4}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{36}+\\frac{K}{6}\\right)\\frac{ m^6}{k^6}+ \\left(-\\frac{7}{48}+\\frac{K}{4}\\right)\\frac{ m^8}{k^8}+ \\left(-\\frac{47}{120}+\\frac{K}{2}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{379}{360}+\\frac{7 K}{6}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;4\\text{D}}^{\\text{s,t,IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{48 \\pi ^2} \\left( L_0-\\frac{1}{10}\\frac{ k^2}{m^2}-\\frac{1}{140}\\frac{ k^4}{m^4}-\\frac{1}{1260}\\frac{ k^6}{m^6}-\\frac{1}{9240}\\frac{ k^8}{m^8}-\\frac{1}{60060}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{360360}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;4\\text{D}}^{\\text{s,UV-IR}} & = k^2 \\pi _{\\mu \\nu } \\frac{i }{6 \\pi ^2}\\left(-\\frac{1}{3}+\\frac{K}{8}\\right)+\\ldots $ Scalars, spin 1 x 1, dimension 5: $\\tilde{T}_{1,1;5\\text{D}}^{\\text{s,t,UV}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{512} k-\\frac{ \\pi }{64}\\frac{ m^2}{k}+\\frac{i }{12}\\frac{ m^3}{k^2}+\\frac{ \\pi }{32}\\frac{ m^4}{k^3}-\\frac{i }{15}\\frac{ m^5}{k^4}-\\frac{4 i }{105}\\frac{ m^7}{k^6}-\\frac{16 i }{315}\\frac{ m^9}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{64 i }{693}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;5\\text{D}}^{\\text{s,t,IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{48 \\pi ^2} \\left(m-\\frac{1}{20}\\frac{ k^2}{m}-\\frac{1}{560}\\frac{ k^4}{m^3}-\\frac{1}{6720}\\frac{ k^6}{m^5}-\\frac{1}{59136}\\frac{ k^8}{m^7}-\\frac{1}{439296}\\frac{ k^{10}}{m^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{2928640}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;5\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 1 x 1, dimension 6: $\\tilde{T}_{1,1;6\\text{D}}^{\\text{s,t,UV}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{8 \\pi ^3} \\left( \\left(-\\frac{23}{1800}+\\frac{P}{240}\\right) k^2+ \\left(\\frac{1}{9}-\\frac{P}{24}\\right) m^2+ \\left(-\\frac{1}{16}+\\frac{K}{8}\\right)\\frac{ m^4}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{11}{72}+\\frac{K}{12}\\right)\\frac{ m^6}{k^4}- \\left(\\frac{1}{288}+\\frac{K}{24}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{23}{1200}-\\frac{K}{20}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{37}{720}-\\frac{K}{12}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;6\\text{D}}^{\\text{s,t,IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{192 \\pi ^3} \\left( \\left(1-L_0\\right) m^2+\\frac{ L_0}{10} k^2-\\frac{1}{140}\\frac{ k^4}{m^2}-\\frac{1}{2520}\\frac{ k^6}{m^4}-\\frac{1}{27720}\\frac{ k^8}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{240240}\\frac{ k^{10}}{m^8}-\\frac{1}{1801800}\\frac{ k^{12}}{m^{10}}-\\frac{1}{12252240}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;6\\text{D}}^{\\text{s,UV-IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{192 \\pi ^3} \\left( \\left(-\\frac{23}{75}+\\frac{K}{10}\\right) k^2+ \\left(\\frac{5}{3}-K\\right) m^2\\right)\\right)+\\ldots $ Scalars, spin 1 x 3, dimension 3: $\\tilde{T}_{1,3;3\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(-\\frac{3 }{64}\\frac{1}{k}+\\frac{i }{2 \\pi }\\frac{ m}{k^2}+\\frac{3 }{8}\\frac{ m^2}{k^3}-\\frac{2 i }{\\pi }\\frac{ m^3}{k^4}-\\frac{3 }{4}\\frac{ m^4}{k^5}+\\frac{8 i }{5 \\pi }\\frac{ m^5}{k^6}+\\frac{32 i }{35 \\pi }\\frac{ m^7}{k^8}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{128 i }{105 \\pi }\\frac{ m^9}{k^{10}}+\\frac{512 i }{231 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{1,3;3\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{8 \\pi } \\left(\\frac{1}{5}\\frac{1}{m}+\\frac{1}{140}\\frac{ k^2}{m^3}+\\frac{1}{1680}\\frac{ k^4}{m^5}+\\frac{1}{14784}\\frac{ k^6}{m^7}+\\frac{1}{109824}\\frac{ k^8}{m^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{732160}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;3\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 1 x 3, dimension 4: $\\tilde{T}_{1,3;4\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left( \\left(\\frac{23}{150}-\\frac{P}{20}\\right)+ \\left(-\\frac{5}{6}+\\frac{K}{2}\\right)\\frac{ m^2}{k^2}+ \\left(\\frac{3}{4}-\\frac{3 K}{2}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{11}{6}+K\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{1}{24}+\\frac{K}{2}\\right)\\frac{ m^8}{k^8}+ \\left(-\\frac{23}{100}+\\frac{3 K}{5}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{37}{60}+K\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;4\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{80 \\pi ^2} \\left(- L_0+\\frac{1}{14}\\frac{ k^2}{m^2}+\\frac{1}{252}\\frac{ k^4}{m^4}+\\frac{1}{2772}\\frac{ k^6}{m^6}+\\frac{1}{24024}\\frac{ k^8}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{180180}\\frac{ k^{10}}{m^{10}}+\\frac{1}{1225224}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;4\\text{D}}^{\\text{s,UV-IR}} & = k^4 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\frac{i }{40 \\pi ^2}\\left(\\frac{23}{15}-\\frac{K}{2}\\right)+\\ldots $ Scalars, spin 1 x 3, dimension 5: $\\tilde{T}_{1,3;5\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{1024} k+\\frac{3 \\pi }{256}\\frac{ m^2}{k}-\\frac{i }{12}\\frac{ m^3}{k^2}-\\frac{3 \\pi }{64}\\frac{ m^4}{k^3}+\\frac{i }{5}\\frac{ m^5}{k^4}+\\frac{ \\pi }{16}\\frac{ m^6}{k^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{4 i }{35}\\frac{ m^7}{k^6}-\\frac{16 i }{315}\\frac{ m^9}{k^8}-\\frac{64 i }{1155}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;5\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{16 \\pi ^2} \\left(-\\frac{1}{5} m+\\frac{1}{140}\\frac{ k^2}{m}+\\frac{1}{5040}\\frac{ k^4}{m^3}+\\frac{1}{73920}\\frac{ k^6}{m^5}+\\frac{1}{768768}\\frac{ k^8}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{6589440}\\frac{ k^{10}}{m^9}+\\frac{1}{49786880}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;5\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 1 x 3, dimension 6: $\\tilde{T}_{1,3;6\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{4 \\pi ^3} \\left( \\left(\\frac{11}{3675}-\\frac{P}{1120}\\right) k^2+ \\left(-\\frac{23}{600}+\\frac{P}{80}\\right) m^2+ \\left(\\frac{7}{96}-\\frac{K}{16}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{48}+\\frac{K}{8}\\right)\\frac{ m^6}{k^4}- \\left(\\frac{25}{192}+\\frac{K}{16}\\right)\\frac{ m^8}{k^6}- \\left(\\frac{17}{2400}+\\frac{K}{40}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{13}{2400}-\\frac{K}{40}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;6\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{320 \\pi ^3} \\left( \\left(-1+L_0\\right) m^2-\\frac{ L_0}{14} k^2+\\frac{1}{252}\\frac{ k^4}{m^2}+\\frac{1}{5544}\\frac{ k^6}{m^4}+\\frac{1}{72072}\\frac{ k^8}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{720720}\\frac{ k^{10}}{m^8}+\\frac{1}{6126120}\\frac{ k^{12}}{m^{10}}+\\frac{1}{46558512}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;6\\text{D}}^{\\text{s,UV-IR}} & = k^4 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{20 \\pi ^3} \\left( \\left(\\frac{11}{735}-\\frac{K}{224}\\right) k^2+ \\left(-\\frac{31}{240}+\\frac{K}{16}\\right) m^2\\right)\\right)+\\ldots $ Scalars, spin 1 x 5, dimension 3: $\\tilde{T}_{1,5;3\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{5 }{128}\\frac{1}{k}-\\frac{i }{2 \\pi }\\frac{ m}{k^2}-\\frac{15 }{32}\\frac{ m^2}{k^3}+\\frac{10 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{15 }{8}\\frac{ m^4}{k^5}-\\frac{8 i }{\\pi }\\frac{ m^5}{k^6}-\\frac{5 }{2}\\frac{ m^6}{k^7}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{32 i }{7 \\pi }\\frac{ m^7}{k^8}+\\frac{128 i }{63 \\pi }\\frac{ m^9}{k^{10}}+\\frac{512 i }{231 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{1,5;3\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{8 \\pi } \\left(-\\frac{1}{7}\\frac{1}{m}-\\frac{1}{252}\\frac{ k^2}{m^3}-\\frac{1}{3696}\\frac{ k^4}{m^5}-\\frac{5 }{192192}\\frac{ k^6}{m^7}-\\frac{1}{329472}\\frac{ k^8}{m^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{2489344}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;3\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 1 x 5, dimension 4: $\\tilde{T}_{1,5;4\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{22}{735}+\\frac{P}{112}\\right)+ \\left(\\frac{31}{120}-\\frac{K}{8}\\right)\\frac{ m^2}{k^2}+ \\left(-\\frac{35}{48}+\\frac{5 K}{8}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{5}{24}-\\frac{5 K}{4}\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{125}{96}+\\frac{5 K}{8}\\right)\\frac{ m^8}{k^8}+ \\left(\\frac{17}{240}+\\frac{K}{4}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{13}{240}+\\frac{K}{4}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;4\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{112 \\pi ^2} \\left( L_0-\\frac{1}{18}\\frac{ k^2}{m^2}-\\frac{1}{396}\\frac{ k^4}{m^4}-\\frac{1}{5148}\\frac{ k^6}{m^6}-\\frac{1}{51480}\\frac{ k^8}{m^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{437580}\\frac{ k^{10}}{m^{10}}-\\frac{1}{3325608}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;4\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\frac{i }{7 \\pi ^2}\\left(-\\frac{22}{105}+\\frac{K}{16}\\right)+\\ldots $ Scalars, spin 1 x 5, dimension 5: $\\tilde{T}_{1,5;5\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{5 \\pi }{8192} k-\\frac{5 \\pi }{512}\\frac{ m^2}{k}+\\frac{i }{12}\\frac{ m^3}{k^2}+\\frac{15 \\pi }{256}\\frac{ m^4}{k^3}-\\frac{i }{3}\\frac{ m^5}{k^4}-\\frac{5 \\pi }{32}\\frac{ m^6}{k^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{4 i }{7}\\frac{ m^7}{k^6}+\\frac{5 \\pi }{32}\\frac{ m^8}{k^7}-\\frac{16 i }{63}\\frac{ m^9}{k^8}-\\frac{64 i }{693}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;5\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{16 \\pi ^2} \\left(\\frac{1}{7} m-\\frac{1}{252}\\frac{ k^2}{m}-\\frac{1}{11088}\\frac{ k^4}{m^3}-\\frac{1}{192192}\\frac{ k^6}{m^5}-\\frac{1}{2306304}\\frac{ k^8}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{22404096}\\frac{ k^{10}}{m^9}-\\frac{1}{189190144}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;5\\text{D}}^{\\text{s,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Scalars, spin 1 x 5, dimension 6: $\\tilde{T}_{1,5;6\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{2 \\pi ^3} \\left( \\left(-\\frac{563}{635040}+\\frac{P}{4032}\\right) k^2+ \\left(\\frac{11}{735}-\\frac{P}{224}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{47}{960}+\\frac{K}{32}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{25}{288}-\\frac{5 K}{48}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{5}{384}+\\frac{5 K}{32}\\right)\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{137}{960}+\\frac{K}{16}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{3}{320}+\\frac{K}{48}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;6\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{448 \\pi ^3} \\left( \\left(1-L_0\\right) m^2+\\frac{ L_0}{18} k^2-\\frac{1}{396}\\frac{ k^4}{m^2}-\\frac{1}{10296}\\frac{ k^6}{m^4}-\\frac{1}{154440}\\frac{ k^8}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{1750320}\\frac{ k^{10}}{m^8}-\\frac{1}{16628040}\\frac{ k^{12}}{m^{10}}-\\frac{1}{139675536}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;6\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{448 \\pi ^3} \\left( \\left(-\\frac{563}{2835}+\\frac{K}{18}\\right) k^2+ \\left(\\frac{247}{105}-K\\right) m^2\\right)\\right)+\\ldots $ Scalars, spin 2 x 2, dimension 3: $\\tilde{T}_{2,2;3\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(-\\frac{1}{32}\\frac{1}{k}+\\frac{i }{2 \\pi }\\frac{ m}{k^2}+\\frac{1}{4}\\frac{ m^2}{k^3}-\\frac{4 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{1}{2}\\frac{ m^4}{k^5}+\\frac{16 i }{15 \\pi }\\frac{ m^5}{k^6}+\\frac{64 i }{105 \\pi }\\frac{ m^7}{k^8}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{256 i }{315 \\pi }\\frac{ m^9}{k^{10}}+\\frac{1024 i }{693 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(-\\frac{1}{64}\\frac{1}{k}+\\frac{1}{8}\\frac{ m^2}{k^3}-\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{1}{4}\\frac{ m^4}{k^5}+\\frac{8 i }{15 \\pi }\\frac{ m^5}{k^6}+\\frac{32 i }{105 \\pi }\\frac{ m^7}{k^8}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{128 i }{315 \\pi }\\frac{ m^9}{k^{10}}+\\frac{512 i }{693 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{2,2;3\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{6 \\pi } \\left(\\frac{m}{k^2}+\\frac{1}{10}\\frac{1}{m}+\\frac{1}{280}\\frac{ k^2}{m^3}+\\frac{1}{3360}\\frac{ k^4}{m^5}+\\frac{1}{29568}\\frac{ k^6}{m^7}+\\frac{1}{219648}\\frac{ k^8}{m^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{1464320}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{6 \\pi } \\left(-\\frac{ m}{k^2}+\\frac{1}{20}\\frac{1}{m}+\\frac{1}{560}\\frac{ k^2}{m^3}+\\frac{1}{6720}\\frac{ k^4}{m^5}+\\frac{1}{59136}\\frac{ k^6}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{439296}\\frac{ k^8}{m^9}+\\frac{1}{2928640}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;3\\text{D}}^{\\text{s,UV-IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{3 \\pi }\\frac{ m}{k^2}\\right)+\\ldots $ Scalars, spin 2 x 2, dimension 4: $\\tilde{T}_{2,2;4\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left( \\left(\\frac{23}{450}-\\frac{P}{60}\\right)- \\left(\\frac{7}{36}-\\frac{P}{6}+\\frac{L_0}{4}\\right)\\frac{ m^2}{k^2}+ \\left(\\frac{1}{4}-\\frac{K}{2}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{11}{18}+\\frac{K}{3}\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{1}{72}+\\frac{K}{6}\\right)\\frac{ m^8}{k^8}+ \\left(-\\frac{23}{300}+\\frac{K}{5}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{37}{180}+\\frac{K}{3}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{23}{1800}-\\frac{P}{240}\\right)+ \\left(-\\frac{1}{9}+\\frac{P}{24}\\right)\\frac{ m^2}{k^2}+ \\left(\\frac{1}{16}-\\frac{K}{8}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{11}{72}+\\frac{K}{12}\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{1}{288}+\\frac{K}{24}\\right)\\frac{ m^8}{k^8}+ \\left(-\\frac{23}{1200}+\\frac{K}{20}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{37}{720}+\\frac{K}{12}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;4\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{24 \\pi ^2} \\left( \\left(1-L_0\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{5}+\\frac{1}{70}\\frac{ k^2}{m^2}+\\frac{1}{1260}\\frac{ k^4}{m^4}+\\frac{1}{13860}\\frac{ k^6}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{120120}\\frac{ k^8}{m^8}+\\frac{1}{900900}\\frac{ k^{10}}{m^{10}}+\\frac{1}{6126120}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{24 \\pi ^2} \\left( \\left(-1+L_0\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{10}+\\frac{1}{140}\\frac{ k^2}{m^2}+\\frac{1}{2520}\\frac{ k^4}{m^4}+\\frac{1}{27720}\\frac{ k^6}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{240240}\\frac{ k^8}{m^8}+\\frac{1}{1801800}\\frac{ k^{10}}{m^{10}}+\\frac{1}{12252240}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;4\\text{D}}^{\\text{s,UV-IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{12 \\pi ^2} \\left( \\left(\\frac{23}{75}-\\frac{K}{10}\\right)+ \\left(-\\frac{5}{3}+K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{24 \\pi ^2} \\left( \\left(\\frac{23}{75}-\\frac{K}{10}\\right)+ \\left(-\\frac{5}{3}+K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\ldots $ Scalars, spin 2 x 2, dimension 5: $\\tilde{T}_{2,2;5\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{1536} k+\\frac{ \\pi }{128}\\frac{ m^2}{k}-\\frac{i }{12}\\frac{ m^3}{k^2}-\\frac{ \\pi }{32}\\frac{ m^4}{k^3}+\\frac{2 i }{15}\\frac{ m^5}{k^4}+\\frac{ \\pi }{24}\\frac{ m^6}{k^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{8 i }{105}\\frac{ m^7}{k^6}-\\frac{32 i }{945}\\frac{ m^9}{k^8}-\\frac{128 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{3072} k+\\frac{ \\pi }{256}\\frac{ m^2}{k}-\\frac{ \\pi }{64}\\frac{ m^4}{k^3}+\\frac{i }{15}\\frac{ m^5}{k^4}+\\frac{ \\pi }{48}\\frac{ m^6}{k^5}-\\frac{4 i }{105}\\frac{ m^7}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{16 i }{945}\\frac{ m^9}{k^8}-\\frac{64 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;5\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{12 \\pi ^2} \\left(-\\frac{1}{3}\\frac{ m^3}{k^2}-\\frac{1}{10} m+\\frac{1}{280}\\frac{ k^2}{m}+\\frac{1}{10080}\\frac{ k^4}{m^3}+\\frac{1}{147840}\\frac{ k^6}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{1537536}\\frac{ k^8}{m^7}+\\frac{1}{13178880}\\frac{ k^{10}}{m^9}+\\frac{1}{99573760}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^2} \\left(\\frac{1}{3}\\frac{ m^3}{k^2}-\\frac{1}{20} m+\\frac{1}{560}\\frac{ k^2}{m}+\\frac{1}{20160}\\frac{ k^4}{m^3}+\\frac{1}{295680}\\frac{ k^6}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3075072}\\frac{ k^8}{m^7}+\\frac{1}{26357760}\\frac{ k^{10}}{m^9}+\\frac{1}{199147520}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;5\\text{D}}^{\\text{s,UV-IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(-\\frac{i }{18 \\pi ^2}\\frac{ m^3}{k^2}\\right)+\\ldots $ Scalars, spin 2 x 2, dimension 6: $\\tilde{T}_{2,2;6\\text{D}}^{\\text{s,t,UV}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi ^3} \\left( \\left(\\frac{11}{11025}-\\frac{P}{3360}\\right) k^2+ \\left(-\\frac{23}{1800}+\\frac{P}{240}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{5}{576}-\\frac{P}{48}+\\frac{L_0}{32}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{144}+\\frac{K}{24}\\right)\\frac{ m^6}{k^4}- \\left(\\frac{25}{576}+\\frac{K}{48}\\right)\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{17}{7200}+\\frac{K}{120}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{13}{7200}-\\frac{K}{120}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^3} \\left( \\left(\\frac{11}{3675}-\\frac{P}{1120}\\right) k^2+ \\left(-\\frac{23}{600}+\\frac{P}{80}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1}{6}-\\frac{P}{16}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{48}+\\frac{K}{8}\\right)\\frac{ m^6}{k^4}- \\left(\\frac{25}{192}+\\frac{K}{16}\\right)\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{17}{2400}+\\frac{K}{40}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{13}{2400}-\\frac{K}{40}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;6\\text{D}}^{\\text{s,t,IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{32 \\pi ^3} \\left( \\left(-\\frac{1}{4}+\\frac{L_0}{6}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{15}+\\frac{L_0}{15}\\right) m^2-\\frac{ L_0}{210} k^2+\\frac{1}{3780}\\frac{ k^4}{m^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{83160}\\frac{ k^6}{m^4}+\\frac{1}{1081080}\\frac{ k^8}{m^6}+\\frac{1}{10810800}\\frac{ k^{10}}{m^8}+\\frac{1}{91891800}\\frac{ k^{12}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{698377680}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{64 \\pi ^3} \\left( \\left(\\frac{1}{2}-\\frac{L_0}{3}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{15}+\\frac{L_0}{15}\\right) m^2-\\frac{ L_0}{210} k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3780}\\frac{ k^4}{m^2}+\\frac{1}{83160}\\frac{ k^6}{m^4}+\\frac{1}{1081080}\\frac{ k^8}{m^6}+\\frac{1}{10810800}\\frac{ k^{10}}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{91891800}\\frac{ k^{12}}{m^{10}}+\\frac{1}{698377680}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;6\\text{D}}^{\\text{s,UV-IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{6 \\pi ^3} \\left( \\left(\\frac{11}{3675}-\\frac{K}{1120}\\right) k^2+ \\left(-\\frac{31}{1200}+\\frac{K}{80}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{7}{96}-\\frac{K}{16}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^3} \\left( \\left(\\frac{11}{3675}-\\frac{K}{1120}\\right) k^2+ \\left(-\\frac{31}{1200}+\\frac{K}{80}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{7}{96}-\\frac{K}{16}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\ldots $ Scalars, spin 2 x 4, dimension 3: $\\tilde{T}_{2,4;3\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{32}\\frac{1}{k}-\\frac{i }{2 \\pi }\\frac{ m}{k^2}-\\frac{3 }{8}\\frac{ m^2}{k^3}+\\frac{8 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{3 }{2}\\frac{ m^4}{k^5}-\\frac{32 i }{5 \\pi }\\frac{ m^5}{k^6}-2 \\frac{ m^6}{k^7}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{128 i }{35 \\pi }\\frac{ m^7}{k^8}+\\frac{512 i }{315 \\pi }\\frac{ m^9}{k^{10}}+\\frac{2048 i }{1155 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{128}\\frac{1}{k}-\\frac{3 }{32}\\frac{ m^2}{k^3}+\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{3 }{8}\\frac{ m^4}{k^5}-\\frac{8 i }{5 \\pi }\\frac{ m^5}{k^6}-\\frac{1}{2}\\frac{ m^6}{k^7}+\\frac{32 i }{35 \\pi }\\frac{ m^7}{k^8}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{128 i }{315 \\pi }\\frac{ m^9}{k^{10}}+\\frac{512 i }{1155 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{2,4;3\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(-\\frac{1}{5}\\frac{ m}{k^2}-\\frac{1}{35}\\frac{1}{m}-\\frac{1}{1260}\\frac{ k^2}{m^3}-\\frac{1}{18480}\\frac{ k^4}{m^5}-\\frac{1}{192192}\\frac{ k^6}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{1647360}\\frac{ k^8}{m^9}-\\frac{1}{12446720}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(\\frac{1}{5}\\frac{ m}{k^2}-\\frac{1}{140}\\frac{1}{m}-\\frac{1}{5040}\\frac{ k^2}{m^3}-\\frac{1}{73920}\\frac{ k^4}{m^5}-\\frac{1}{768768}\\frac{ k^6}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{6589440}\\frac{ k^8}{m^9}-\\frac{1}{49786880}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;3\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(-\\frac{2 i }{5 \\pi }\\frac{ m}{k^2}\\right)+\\ldots $ Scalars, spin 2 x 4, dimension 4: $\\tilde{T}_{2,4;4\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{88}{3675}+\\frac{P}{140}\\right)+ \\left(\\frac{109}{600}-\\frac{P}{10}+\\frac{L_0}{8}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{7}{12}+\\frac{K}{2}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{1}{6}-K\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{25}{24}+\\frac{K}{2}\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{17}{300}+\\frac{K}{5}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(-\\frac{13}{300}+\\frac{K}{5}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{22}{3675}+\\frac{P}{560}\\right)+ \\left(\\frac{23}{300}-\\frac{P}{40}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{7}{48}+\\frac{K}{8}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{1}{24}-\\frac{K}{4}\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{25}{96}+\\frac{K}{8}\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{17}{1200}+\\frac{K}{20}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(-\\frac{13}{1200}+\\frac{K}{20}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;4\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{20 \\pi ^2} \\left( \\left(-\\frac{1}{2}+\\frac{L_0}{2}\\right)\\frac{ m^2}{k^2}+\\frac{ L_0}{7}-\\frac{1}{126}\\frac{ k^2}{m^2}-\\frac{1}{2772}\\frac{ k^4}{m^4}-\\frac{1}{36036}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{360360}\\frac{ k^8}{m^8}-\\frac{1}{3063060}\\frac{ k^{10}}{m^{10}}-\\frac{1}{23279256}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{40 \\pi ^2} \\left( \\left(1-L_0\\right)\\frac{ m^2}{k^2}+\\frac{ L_0}{14}-\\frac{1}{252}\\frac{ k^2}{m^2}-\\frac{1}{5544}\\frac{ k^4}{m^4}-\\frac{1}{72072}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{720720}\\frac{ k^8}{m^8}-\\frac{1}{6126120}\\frac{ k^{10}}{m^{10}}-\\frac{1}{46558512}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;4\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{88}{735}+\\frac{K}{28}\\right)+ \\left(\\frac{31}{30}-\\frac{K}{2}\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{22}{735}+\\frac{K}{112}\\right)+ \\left(\\frac{31}{120}-\\frac{K}{8}\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\ldots $ Scalars, spin 2 x 4, dimension 5: $\\tilde{T}_{2,4;5\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{2048} k-\\frac{ \\pi }{128}\\frac{ m^2}{k}+\\frac{i }{12}\\frac{ m^3}{k^2}+\\frac{3 \\pi }{64}\\frac{ m^4}{k^3}-\\frac{4 i }{15}\\frac{ m^5}{k^4}-\\frac{ \\pi }{8}\\frac{ m^6}{k^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{16 i }{35}\\frac{ m^7}{k^6}+\\frac{ \\pi }{8}\\frac{ m^8}{k^7}-\\frac{64 i }{315}\\frac{ m^9}{k^8}-\\frac{256 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{8192} k-\\frac{ \\pi }{512}\\frac{ m^2}{k}+\\frac{3 \\pi }{256}\\frac{ m^4}{k^3}-\\frac{i }{15}\\frac{ m^5}{k^4}-\\frac{ \\pi }{32}\\frac{ m^6}{k^5}+\\frac{4 i }{35}\\frac{ m^7}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{ \\pi }{32}\\frac{ m^8}{k^7}-\\frac{16 i }{315}\\frac{ m^9}{k^8}-\\frac{64 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;5\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{20 \\pi ^2} \\left(\\frac{1}{3}\\frac{ m^3}{k^2}+\\frac{1}{7} m-\\frac{1}{252}\\frac{ k^2}{m}-\\frac{1}{11088}\\frac{ k^4}{m^3}-\\frac{1}{192192}\\frac{ k^6}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{2306304}\\frac{ k^8}{m^7}-\\frac{1}{22404096}\\frac{ k^{10}}{m^9}-\\frac{1}{189190144}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{20 \\pi ^2} \\left(-\\frac{1}{3}\\frac{ m^3}{k^2}+\\frac{1}{28} m-\\frac{1}{1008}\\frac{ k^2}{m}-\\frac{1}{44352}\\frac{ k^4}{m^3}-\\frac{1}{768768}\\frac{ k^6}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{9225216}\\frac{ k^8}{m^7}-\\frac{1}{89616384}\\frac{ k^{10}}{m^9}-\\frac{1}{756760576}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;5\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{15 \\pi ^2}\\frac{ m^3}{k^2}\\right)+\\ldots $ Scalars, spin 2 x 4, dimension 6: $\\tilde{T}_{2,4;6\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{563}{1587600}+\\frac{P}{10080}\\right) k^2+ \\left(\\frac{22}{3675}-\\frac{P}{560}\\right) m^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{143}{9600}-\\frac{P}{80}+\\frac{L_0}{64}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{5}{144}-\\frac{K}{24}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{1}{192}+\\frac{K}{16}\\right)\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{137}{2400}+\\frac{K}{40}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{3}{800}+\\frac{K}{120}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^3} \\left( \\left(-\\frac{563}{3175200}+\\frac{P}{20160}\\right) k^2+ \\left(\\frac{11}{3675}-\\frac{P}{1120}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{23}{1200}+\\frac{P}{160}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{5}{288}-\\frac{K}{48}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{1}{384}+\\frac{K}{32}\\right)\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{137}{4800}+\\frac{K}{80}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{3}{1600}+\\frac{K}{240}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;6\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{80 \\pi ^3} \\left( \\left(\\frac{3}{8}-\\frac{L_0}{4}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{7}-\\frac{L_0}{7}\\right) m^2+\\frac{ L_0}{126} k^2-\\frac{1}{2772}\\frac{ k^4}{m^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{72072}\\frac{ k^6}{m^4}-\\frac{1}{1081080}\\frac{ k^8}{m^6}-\\frac{1}{12252240}\\frac{ k^{10}}{m^8}-\\frac{1}{116396280}\\frac{ k^{12}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{977728752}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{320 \\pi ^3} \\left( \\left(-\\frac{3}{2}+L_0\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{7}-\\frac{L_0}{7}\\right) m^2+\\frac{ L_0}{126} k^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{2772}\\frac{ k^4}{m^2}-\\frac{1}{72072}\\frac{ k^6}{m^4}-\\frac{1}{1081080}\\frac{ k^8}{m^6}-\\frac{1}{12252240}\\frac{ k^{10}}{m^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{116396280}\\frac{ k^{12}}{m^{10}}-\\frac{1}{977728752}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;6\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{80 \\pi ^3} \\left( \\left(-\\frac{563}{19845}+\\frac{K}{126}\\right) k^2+ \\left(\\frac{247}{735}-\\frac{K}{7}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{47}{30}+K\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{320 \\pi ^3} \\left( \\left(-\\frac{563}{19845}+\\frac{K}{126}\\right) k^2+ \\left(\\frac{247}{735}-\\frac{K}{7}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{47}{30}+K\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\ldots $ Scalars, spin 3 x 3, dimension 3: $\\tilde{T}_{3,3;3\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{1}{64}\\frac{1}{k}-\\frac{i }{2 \\pi }\\frac{ m}{k^2}-\\frac{3 }{16}\\frac{ m^2}{k^3}+\\frac{4 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{3 }{4}\\frac{ m^4}{k^5}-\\frac{16 i }{5 \\pi }\\frac{ m^5}{k^6}-\\frac{ m^6}{k^7}+\\frac{64 i }{35 \\pi }\\frac{ m^7}{k^8}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{256 i }{315 \\pi }\\frac{ m^9}{k^{10}}+\\frac{1024 i }{1155 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{3 }{128}\\frac{1}{k}-\\frac{9 }{32}\\frac{ m^2}{k^3}+\\frac{2 i }{\\pi }\\frac{ m^3}{k^4}+\\frac{9 }{8}\\frac{ m^4}{k^5}-\\frac{24 i }{5 \\pi }\\frac{ m^5}{k^6}-\\frac{3 }{2}\\frac{ m^6}{k^7}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{96 i }{35 \\pi }\\frac{ m^7}{k^8}+\\frac{128 i }{105 \\pi }\\frac{ m^9}{k^{10}}+\\frac{512 i }{385 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{3,3;3\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{2 \\pi } \\left(-\\frac{3 }{5}\\frac{ m}{k^2}-\\frac{1}{70}\\frac{1}{m}-\\frac{1}{2520}\\frac{ k^2}{m^3}-\\frac{1}{36960}\\frac{ k^4}{m^5}-\\frac{1}{384384}\\frac{ k^6}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{3294720}\\frac{ k^8}{m^9}-\\frac{1}{24893440}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(\\frac{3 }{5}\\frac{ m}{k^2}-\\frac{3 }{140}\\frac{1}{m}-\\frac{1}{1680}\\frac{ k^2}{m^3}-\\frac{1}{24640}\\frac{ k^4}{m^5}-\\frac{1}{256256}\\frac{ k^6}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{2196480}\\frac{ k^8}{m^9}-\\frac{3 }{49786880}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;3\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(-\\frac{i }{5 \\pi }\\frac{ m}{k^2}\\right)+\\ldots $ Scalars, spin 3 x 3, dimension 4: $\\tilde{T}_{3,3;4\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{44}{3675}+\\frac{P}{280}\\right)+ \\left(\\frac{17}{600}-\\frac{P}{20}+\\frac{L_0}{8}\\right)\\frac{ m^2}{k^2}+ \\left(-\\frac{7}{24}+\\frac{K}{4}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1}{12}-\\frac{K}{2}\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{25}{48}+\\frac{K}{4}\\right)\\frac{ m^8}{k^8}+ \\left(\\frac{17}{600}+\\frac{K}{10}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{13}{600}+\\frac{K}{10}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{22}{1225}+\\frac{3 P}{560}\\right)+ \\left(\\frac{23}{100}-\\frac{3 P}{40}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{7}{16}+\\frac{3 K}{8}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{1}{8}-\\frac{3 K}{4}\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{25}{32}+\\frac{3 K}{8}\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{17}{400}+\\frac{3 K}{20}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(-\\frac{13}{400}+\\frac{3 K}{20}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;4\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{40 \\pi ^2} \\left( \\left(-3+3 L_0\\right)\\frac{ m^2}{k^2}+\\frac{ L_0}{7}-\\frac{1}{126}\\frac{ k^2}{m^2}-\\frac{1}{2772}\\frac{ k^4}{m^4}-\\frac{1}{36036}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{360360}\\frac{ k^8}{m^8}-\\frac{1}{3063060}\\frac{ k^{10}}{m^{10}}-\\frac{1}{23279256}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{40 \\pi ^2} \\left( \\left(3-3 L_0\\right)\\frac{ m^2}{k^2}+\\frac{3 L_0}{14}-\\frac{1}{84}\\frac{ k^2}{m^2}-\\frac{1}{1848}\\frac{ k^4}{m^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{24024}\\frac{ k^6}{m^6}-\\frac{1}{240240}\\frac{ k^8}{m^8}-\\frac{1}{2042040}\\frac{ k^{10}}{m^{10}}-\\frac{1}{15519504}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;4\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{44}{735}+\\frac{K}{56}\\right)+ \\left(\\frac{31}{60}-\\frac{K}{4}\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{22}{245}+\\frac{3 K}{112}\\right)+ \\left(\\frac{31}{40}-\\frac{3 K}{8}\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\ldots $ Scalars, spin 3 x 3, dimension 5: $\\tilde{T}_{3,3;5\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{4096} k-\\frac{ \\pi }{256}\\frac{ m^2}{k}+\\frac{i }{12}\\frac{ m^3}{k^2}+\\frac{3 \\pi }{128}\\frac{ m^4}{k^3}-\\frac{2 i }{15}\\frac{ m^5}{k^4}-\\frac{ \\pi }{16}\\frac{ m^6}{k^5}+\\frac{8 i }{35}\\frac{ m^7}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{ \\pi }{16}\\frac{ m^8}{k^7}-\\frac{32 i }{315}\\frac{ m^9}{k^8}-\\frac{128 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{3 \\pi }{8192} k-\\frac{3 \\pi }{512}\\frac{ m^2}{k}+\\frac{9 \\pi }{256}\\frac{ m^4}{k^3}-\\frac{i }{5}\\frac{ m^5}{k^4}-\\frac{3 \\pi }{32}\\frac{ m^6}{k^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{12 i }{35}\\frac{ m^7}{k^6}+\\frac{3 \\pi }{32}\\frac{ m^8}{k^7}-\\frac{16 i }{105}\\frac{ m^9}{k^8}-\\frac{64 i }{1155}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;5\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{20 \\pi ^2} \\left(\\frac{m^3}{k^2}+\\frac{1}{14} m-\\frac{1}{504}\\frac{ k^2}{m}-\\frac{1}{22176}\\frac{ k^4}{m^3}-\\frac{1}{384384}\\frac{ k^6}{m^5}-\\frac{1}{4612608}\\frac{ k^8}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{44808192}\\frac{ k^{10}}{m^9}-\\frac{1}{378380288}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{20 \\pi ^2} \\left(-\\frac{ m^3}{k^2}+\\frac{3 }{28} m-\\frac{1}{336}\\frac{ k^2}{m}-\\frac{1}{14784}\\frac{ k^4}{m^3}-\\frac{1}{256256}\\frac{ k^6}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{3075072}\\frac{ k^8}{m^7}-\\frac{1}{29872128}\\frac{ k^{10}}{m^9}-\\frac{3 }{756760576}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;5\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{30 \\pi ^2}\\frac{ m^3}{k^2}\\right)+\\ldots $ Scalars, spin 3 x 3, dimension 6: $\\tilde{T}_{3,3;6\\text{D}}^{\\text{s,t,UV}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{563}{3175200}+\\frac{P}{20160}\\right) k^2+ \\left(\\frac{11}{3675}-\\frac{P}{1120}\\right) m^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(-\\frac{41}{9600}-\\frac{P}{160}+\\frac{L_0}{64}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{5}{288}-\\frac{K}{48}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{1}{384}+\\frac{K}{32}\\right)\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{137}{4800}+\\frac{K}{80}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{3}{1600}+\\frac{K}{240}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^3} \\left( \\left(-\\frac{563}{1058400}+\\frac{P}{6720}\\right) k^2+ \\left(\\frac{11}{1225}-\\frac{3 P}{1120}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{23}{400}+\\frac{3 P}{160}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{5}{96}-\\frac{K}{16}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{1}{128}+\\frac{3 K}{32}\\right)\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{137}{1600}+\\frac{3 K}{80}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{9}{1600}+\\frac{K}{80}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;6\\text{D}}^{\\text{s,t,IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{160 \\pi ^3} \\left( \\left(\\frac{9}{4}-\\frac{3 L_0}{2}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{7}-\\frac{L_0}{7}\\right) m^2+\\frac{ L_0}{126} k^2-\\frac{1}{2772}\\frac{ k^4}{m^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{72072}\\frac{ k^6}{m^4}-\\frac{1}{1081080}\\frac{ k^8}{m^6}-\\frac{1}{12252240}\\frac{ k^{10}}{m^8}-\\frac{1}{116396280}\\frac{ k^{12}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{977728752}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{320 \\pi ^3} \\left( \\left(-\\frac{9}{2}+3 L_0\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{3}{7}-\\frac{3 L_0}{7}\\right) m^2+\\frac{ L_0}{42} k^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{924}\\frac{ k^4}{m^2}-\\frac{1}{24024}\\frac{ k^6}{m^4}-\\frac{1}{360360}\\frac{ k^8}{m^6}-\\frac{1}{4084080}\\frac{ k^{10}}{m^8}-\\frac{1}{38798760}\\frac{ k^{12}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{325909584}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;6\\text{D}}^{\\text{s,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{160 \\pi ^3} \\left( \\left(-\\frac{563}{19845}+\\frac{K}{126}\\right) k^2+ \\left(\\frac{247}{735}-\\frac{K}{7}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{47}{30}+K\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{320 \\pi ^3} \\left( \\left(-\\frac{563}{6615}+\\frac{K}{42}\\right) k^2+ \\left(\\frac{247}{245}-\\frac{3 K}{7}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{47}{10}+3 K\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\ldots $ Scalars, spin 3 x 5, dimension 3: $\\tilde{T}_{3,5;3\\text{D}}^{\\text{s,t,UV}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(-\\frac{5 }{256}\\frac{1}{k}+\\frac{i }{2 \\pi }\\frac{ m}{k^2}+\\frac{5 }{16}\\frac{ m^2}{k^3}-\\frac{8 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{15 }{8}\\frac{ m^4}{k^5}+\\frac{32 i }{3 \\pi }\\frac{ m^5}{k^6}+5 \\frac{ m^6}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{128 i }{7 \\pi }\\frac{ m^7}{k^8}-5 \\frac{ m^8}{k^9}+\\frac{512 i }{63 \\pi }\\frac{ m^9}{k^{10}}+\\frac{2048 i }{693 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(-\\frac{15 }{1024}\\frac{1}{k}+\\frac{15 }{64}\\frac{ m^2}{k^3}-\\frac{2 i }{\\pi }\\frac{ m^3}{k^4}-\\frac{45 }{32}\\frac{ m^4}{k^5}+\\frac{8 i }{\\pi }\\frac{ m^5}{k^6}+\\frac{15 }{4}\\frac{ m^6}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{96 i }{7 \\pi }\\frac{ m^7}{k^8}-\\frac{15 }{4}\\frac{ m^8}{k^9}+\\frac{128 i }{21 \\pi }\\frac{ m^9}{k^{10}}+\\frac{512 i }{231 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{3,5;3\\text{D}}^{\\text{s,t,IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(\\frac{3 }{7}\\frac{ m}{k^2}+\\frac{1}{63}\\frac{1}{m}+\\frac{1}{2772}\\frac{ k^2}{m^3}+\\frac{1}{48048}\\frac{ k^4}{m^5}+\\frac{1}{576576}\\frac{ k^6}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{5601024}\\frac{ k^8}{m^9}+\\frac{1}{47297536}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(-\\frac{3 }{7}\\frac{ m}{k^2}+\\frac{1}{84}\\frac{1}{m}+\\frac{1}{3696}\\frac{ k^2}{m^3}+\\frac{1}{64064}\\frac{ k^4}{m^5}+\\frac{1}{768768}\\frac{ k^6}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{7468032}\\frac{ k^8}{m^9}+\\frac{3 }{189190144}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;3\\text{D}}^{\\text{s,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{2 i }{7 \\pi }\\frac{ m}{k^2}\\right)+\\ldots $ Scalars, spin 3 x 5, dimension 4: $\\tilde{T}_{3,5;4\\text{D}}^{\\text{s,t,UV}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{563}{39690}-\\frac{P}{252}\\right)- \\left(\\frac{673}{5880}-\\frac{P}{14}+\\frac{L_0}{8}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{47}{60}-\\frac{K}{2}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{25}{18}+\\frac{5 K}{3}\\right)\\frac{ m^6}{k^6}- \\left(\\frac{5}{24}+\\frac{5 K}{2}\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{137}{60}+K\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{3}{20}+\\frac{K}{3}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{563}{52920}-\\frac{P}{336}\\right)+ \\left(-\\frac{44}{245}+\\frac{3 P}{56}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{47}{80}-\\frac{3 K}{8}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{25}{24}+\\frac{5 K}{4}\\right)\\frac{ m^6}{k^6}- \\left(\\frac{5}{32}+\\frac{15 K}{8}\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{137}{80}+\\frac{3 K}{4}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{9}{80}+\\frac{K}{4}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;4\\text{D}}^{\\text{s,t,IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{28 \\pi ^2} \\left( \\left(\\frac{3}{2}-\\frac{3 L_0}{2}\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{9}+\\frac{1}{198}\\frac{ k^2}{m^2}+\\frac{1}{5148}\\frac{ k^4}{m^4}+\\frac{1}{77220}\\frac{ k^6}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{875160}\\frac{ k^8}{m^8}+\\frac{1}{8314020}\\frac{ k^{10}}{m^{10}}+\\frac{1}{69837768}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{56 \\pi ^2} \\left( \\left(-3+3 L_0\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{6}+\\frac{1}{132}\\frac{ k^2}{m^2}+\\frac{1}{3432}\\frac{ k^4}{m^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{51480}\\frac{ k^6}{m^6}+\\frac{1}{583440}\\frac{ k^8}{m^8}+\\frac{1}{5542680}\\frac{ k^{10}}{m^{10}}+\\frac{1}{46558512}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;4\\text{D}}^{\\text{s,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{14 \\pi ^2} \\left( \\left(\\frac{563}{2835}-\\frac{K}{18}\\right)+ \\left(-\\frac{247}{105}+K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{56 \\pi ^2} \\left( \\left(\\frac{563}{945}-\\frac{K}{6}\\right)+ \\left(-\\frac{247}{35}+3 K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\ldots $ Scalars, spin 3 x 5, dimension 5: $\\tilde{T}_{3,5;5\\text{D}}^{\\text{s,t,UV}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{4096} k+\\frac{5 \\pi }{1024}\\frac{ m^2}{k}-\\frac{i }{12}\\frac{ m^3}{k^2}-\\frac{5 \\pi }{128}\\frac{ m^4}{k^3}+\\frac{4 i }{15}\\frac{ m^5}{k^4}+\\frac{5 \\pi }{32}\\frac{ m^6}{k^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{16 i }{21}\\frac{ m^7}{k^6}-\\frac{5 \\pi }{16}\\frac{ m^8}{k^7}+\\frac{64 i }{63}\\frac{ m^9}{k^8}+\\frac{ \\pi }{4}\\frac{ m^{10}}{k^9}-\\frac{256 i }{693}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{3 \\pi }{16384} k+\\frac{15 \\pi }{4096}\\frac{ m^2}{k}-\\frac{15 \\pi }{512}\\frac{ m^4}{k^3}+\\frac{i }{5}\\frac{ m^5}{k^4}+\\frac{15 \\pi }{128}\\frac{ m^6}{k^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{4 i }{7}\\frac{ m^7}{k^6}-\\frac{15 \\pi }{64}\\frac{ m^8}{k^7}+\\frac{16 i }{21}\\frac{ m^9}{k^8}+\\frac{3 \\pi }{16}\\frac{ m^{10}}{k^9}-\\frac{64 i }{231}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;5\\text{D}}^{\\text{s,t,IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(-\\frac{1}{7}\\frac{ m^3}{k^2}-\\frac{1}{63} m+\\frac{1}{2772}\\frac{ k^2}{m}+\\frac{1}{144144}\\frac{ k^4}{m^3}+\\frac{1}{2882880}\\frac{ k^6}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{39207168}\\frac{ k^8}{m^7}+\\frac{1}{425677824}\\frac{ k^{10}}{m^9}+\\frac{1}{3972993024}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{1}{7}\\frac{ m^3}{k^2}-\\frac{1}{84} m+\\frac{1}{3696}\\frac{ k^2}{m}+\\frac{1}{192192}\\frac{ k^4}{m^3}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3843840}\\frac{ k^6}{m^5}+\\frac{1}{52276224}\\frac{ k^8}{m^7}+\\frac{1}{567570432}\\frac{ k^{10}}{m^9}+\\frac{1}{5297324032}\\frac{ k^{12}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right) $ $\\tilde{T}_{3,5;5\\text{D}}^{\\text{s,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(-\\frac{i }{21 \\pi ^2}\\frac{ m^3}{k^2}\\right)+\\ldots $ Scalars, spin 3 x 5, dimension 6: $\\tilde{T}_{3,5;6\\text{D}}^{\\text{s,t,UV}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^3} \\left( \\left(\\frac{1627}{2401245}-\\frac{P}{5544}\\right) k^2+ \\left(-\\frac{563}{39690}+\\frac{P}{252}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{611}{23520}-\\frac{P}{28}+\\frac{L_0}{16}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{37}{180}+\\frac{K}{6}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{35}{144}-\\frac{5 K}{12}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{17}{120}+\\frac{K}{2}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{49}{120}+\\frac{K}{6}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^3} \\left( \\left(\\frac{1627}{6403320}-\\frac{P}{14784}\\right) k^2+ \\left(-\\frac{563}{105840}+\\frac{P}{672}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{11}{245}-\\frac{3 P}{224}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{37}{480}+\\frac{K}{16}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{35}{384}-\\frac{5 K}{32}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{17}{320}+\\frac{3 K}{16}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{49}{320}+\\frac{K}{16}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;6\\text{D}}^{\\text{s,t,IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{112 \\pi ^3} \\left( \\left(-\\frac{9}{8}+\\frac{3 L_0}{4}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{9}+\\frac{L_0}{9}\\right) m^2-\\frac{ L_0}{198} k^2+\\frac{1}{5148}\\frac{ k^4}{m^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{154440}\\frac{ k^6}{m^4}+\\frac{1}{2625480}\\frac{ k^8}{m^6}+\\frac{1}{33256080}\\frac{ k^{10}}{m^8}+\\frac{1}{349188840}\\frac{ k^{12}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3212537328}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{448 \\pi ^3} \\left( \\left(\\frac{9}{2}-3 L_0\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{3}+\\frac{L_0}{3}\\right) m^2-\\frac{ L_0}{66} k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{1716}\\frac{ k^4}{m^2}+\\frac{1}{51480}\\frac{ k^6}{m^4}+\\frac{1}{875160}\\frac{ k^8}{m^6}+\\frac{1}{11085360}\\frac{ k^{10}}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{116396280}\\frac{ k^{12}}{m^{10}}+\\frac{1}{1070845776}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;6\\text{D}}^{\\text{s,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{28 \\pi ^3} \\left( \\left(\\frac{1627}{343035}-\\frac{K}{792}\\right) k^2+ \\left(-\\frac{811}{11340}+\\frac{K}{36}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{389}{840}-\\frac{K}{4}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{112 \\pi ^3} \\left( \\left(\\frac{1627}{114345}-\\frac{K}{264}\\right) k^2+ \\left(-\\frac{811}{3780}+\\frac{K}{12}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{389}{280}-\\frac{3 K}{4}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\ldots $ Scalars, spin 4 x 4, dimension 3: $\\tilde{T}_{4,4;3\\text{D}}^{\\text{s,t,UV}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(-\\frac{1}{128}\\frac{1}{k}+\\frac{i }{2 \\pi }\\frac{ m}{k^2}+\\frac{1}{8}\\frac{ m^2}{k^3}-\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{3 }{4}\\frac{ m^4}{k^5}+\\frac{64 i }{15 \\pi }\\frac{ m^5}{k^6}+2 \\frac{ m^6}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{256 i }{35 \\pi }\\frac{ m^7}{k^8}-2 \\frac{ m^8}{k^9}+\\frac{1024 i }{315 \\pi }\\frac{ m^9}{k^{10}}+\\frac{4096 i }{3465 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(-\\frac{3 }{128}\\frac{1}{k}+\\frac{3 }{8}\\frac{ m^2}{k^3}-\\frac{4 i }{\\pi }\\frac{ m^3}{k^4}-\\frac{9 }{4}\\frac{ m^4}{k^5}+\\frac{64 i }{5 \\pi }\\frac{ m^5}{k^6}+6 \\frac{ m^6}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{768 i }{35 \\pi }\\frac{ m^7}{k^8}-6 \\frac{ m^8}{k^9}+\\frac{1024 i }{105 \\pi }\\frac{ m^9}{k^{10}}+\\frac{4096 i }{1155 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(-\\frac{3 }{1024}\\frac{1}{k}+\\frac{3 }{64}\\frac{ m^2}{k^3}-\\frac{9 }{32}\\frac{ m^4}{k^5}+\\frac{8 i }{5 \\pi }\\frac{ m^5}{k^6}+\\frac{3 }{4}\\frac{ m^6}{k^7}-\\frac{96 i }{35 \\pi }\\frac{ m^7}{k^8}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{3 }{4}\\frac{ m^8}{k^9}+\\frac{128 i }{105 \\pi }\\frac{ m^9}{k^{10}}+\\frac{512 i }{1155 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{4,4;3\\text{D}}^{\\text{s,t,IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{5 \\pi } \\left(2 \\frac{ m^3}{k^4}+\\frac{27 }{14}\\frac{ m}{k^2}+\\frac{1}{63}\\frac{1}{m}+\\frac{1}{2772}\\frac{ k^2}{m^3}+\\frac{1}{48048}\\frac{ k^4}{m^5}+\\frac{1}{576576}\\frac{ k^6}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{5601024}\\frac{ k^8}{m^9}+\\frac{1}{47297536}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi } \\left(-4 \\frac{ m^3}{k^4}-\\frac{12 }{7}\\frac{ m}{k^2}+\\frac{1}{21}\\frac{1}{m}+\\frac{1}{924}\\frac{ k^2}{m^3}+\\frac{1}{16016}\\frac{ k^4}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{192192}\\frac{ k^6}{m^7}+\\frac{1}{1867008}\\frac{ k^8}{m^9}+\\frac{3 }{47297536}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi } \\left(2 \\frac{ m^3}{k^4}-\\frac{3 }{14}\\frac{ m}{k^2}+\\frac{1}{168}\\frac{1}{m}+\\frac{1}{7392}\\frac{ k^2}{m^3}+\\frac{1}{128128}\\frac{ k^4}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{1537536}\\frac{ k^6}{m^7}+\\frac{1}{14936064}\\frac{ k^8}{m^9}+\\frac{3 }{378380288}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;3\\text{D}}^{\\text{s,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{5 \\pi } \\left(\\frac{4 }{7}\\frac{ m}{k^2}-\\frac{16 }{3}\\frac{ m^3}{k^4}\\right)\\right)+k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(-\\frac{16 i }{5 \\pi }\\frac{ m^3}{k^4}\\right)+\\ldots $ Scalars, spin 4 x 4, dimension 4: $\\tilde{T}_{4,4;4\\text{D}}^{\\text{s,t,UV}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{563}{99225}-\\frac{P}{630}\\right)- \\left(-\\frac{859}{29400}-\\frac{P}{35}+\\frac{L_0}{8}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{511}{1200}-\\frac{P}{5}+\\frac{L_0}{8}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{5}{9}+\\frac{2 K}{3}\\right)\\frac{ m^6}{k^6}- \\left(\\frac{1}{12}+K\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{137}{150}+\\frac{2 K}{5}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{3}{50}+\\frac{2 K}{15}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{563}{33075}-\\frac{P}{210}\\right)+ \\left(-\\frac{352}{1225}+\\frac{3 P}{35}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{143}{200}-\\frac{3 P}{5}+\\frac{3 L_0}{4}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{5}{3}+2 K\\right)\\frac{ m^6}{k^6}- \\left(\\frac{1}{4}+3 K\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{137}{50}+\\frac{6 K}{5}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{9}{50}+\\frac{2 K}{5}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{563}{264600}-\\frac{P}{1680}\\right)+ \\left(-\\frac{44}{1225}+\\frac{3 P}{280}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{23}{100}-\\frac{3 P}{40}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{5}{24}+\\frac{K}{4}\\right)\\frac{ m^6}{k^6}- \\left(\\frac{1}{32}+\\frac{3 K}{8}\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{137}{400}+\\frac{3 K}{20}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{9}{400}+\\frac{K}{20}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;4\\text{D}}^{\\text{s,t,IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{10 \\pi ^2} \\left( \\left(\\frac{9}{8}-\\frac{3 L_0}{4}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{27}{28}-\\frac{27 L_0}{28}\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{63}+\\frac{1}{1386}\\frac{ k^2}{m^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{36036}\\frac{ k^4}{m^4}+\\frac{1}{540540}\\frac{ k^6}{m^6}+\\frac{1}{6126120}\\frac{ k^8}{m^8}+\\frac{1}{58198140}\\frac{ k^{10}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{488864376}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{9}{8}+\\frac{3 L_0}{4}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{3}{7}+\\frac{3 L_0}{7}\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{42}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{924}\\frac{ k^2}{m^2}+\\frac{1}{24024}\\frac{ k^4}{m^4}+\\frac{1}{360360}\\frac{ k^6}{m^6}+\\frac{1}{4084080}\\frac{ k^8}{m^8}+\\frac{1}{38798760}\\frac{ k^{10}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{325909584}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{40 \\pi ^2} \\left( \\left(\\frac{9}{2}-3 L_0\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{3}{7}+\\frac{3 L_0}{7}\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{42}+\\frac{1}{924}\\frac{ k^2}{m^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{24024}\\frac{ k^4}{m^4}+\\frac{1}{360360}\\frac{ k^6}{m^6}+\\frac{1}{4084080}\\frac{ k^8}{m^8}+\\frac{1}{38798760}\\frac{ k^{10}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{325909584}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;4\\text{D}}^{\\text{s,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{563}{19845}-\\frac{K}{126}\\right)+ \\left(-\\frac{247}{735}+\\frac{K}{7}\\right)\\frac{ m^2}{k^2}+ \\left(\\frac{47}{30}-K\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{563}{6615}-\\frac{K}{42}\\right)+ \\left(-\\frac{247}{245}+\\frac{3 K}{7}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{47}{10}-3 K\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{40 \\pi ^2} \\left( \\left(\\frac{563}{6615}-\\frac{K}{42}\\right)+ \\left(-\\frac{247}{245}+\\frac{3 K}{7}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{47}{10}-3 K\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\ldots $ Scalars, spin 4 x 4, dimension 5: $\\tilde{T}_{4,4;5\\text{D}}^{\\text{s,t,UV}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{10240} k+\\frac{ \\pi }{512}\\frac{ m^2}{k}-\\frac{i }{12}\\frac{ m^3}{k^2}-\\frac{ \\pi }{64}\\frac{ m^4}{k^3}+\\frac{i }{15}\\frac{ m^5}{k^4}+\\frac{ \\pi }{16}\\frac{ m^6}{k^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{32 i }{105}\\frac{ m^7}{k^6}-\\frac{ \\pi }{8}\\frac{ m^8}{k^7}+\\frac{128 i }{315}\\frac{ m^9}{k^8}+\\frac{ \\pi }{10}\\frac{ m^{10}}{k^9}-\\frac{512 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{3 \\pi }{10240} k+\\frac{3 \\pi }{512}\\frac{ m^2}{k}-\\frac{3 \\pi }{64}\\frac{ m^4}{k^3}+\\frac{2 i }{5}\\frac{ m^5}{k^4}+\\frac{3 \\pi }{16}\\frac{ m^6}{k^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{32 i }{35}\\frac{ m^7}{k^6}-\\frac{3 \\pi }{8}\\frac{ m^8}{k^7}+\\frac{128 i }{105}\\frac{ m^9}{k^8}+\\frac{3 \\pi }{10}\\frac{ m^{10}}{k^9}-\\frac{512 i }{1155}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{3 \\pi }{81920} k+\\frac{3 \\pi }{4096}\\frac{ m^2}{k}-\\frac{3 \\pi }{512}\\frac{ m^4}{k^3}+\\frac{3 \\pi }{128}\\frac{ m^6}{k^5}-\\frac{4 i }{35}\\frac{ m^7}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{3 \\pi }{64}\\frac{ m^8}{k^7}+\\frac{16 i }{105}\\frac{ m^9}{k^8}+\\frac{3 \\pi }{80}\\frac{ m^{10}}{k^9}-\\frac{64 i }{1155}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;5\\text{D}}^{\\text{s,t,IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{1}{5}\\frac{ m^5}{k^4}-\\frac{9 }{28}\\frac{ m^3}{k^2}-\\frac{1}{126} m+\\frac{1}{5544}\\frac{ k^2}{m}+\\frac{1}{288288}\\frac{ k^4}{m^3}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{5765760}\\frac{ k^6}{m^5}+\\frac{1}{78414336}\\frac{ k^8}{m^7}+\\frac{1}{851355648}\\frac{ k^{10}}{m^9}+\\frac{1}{7945986048}\\frac{ k^{12}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(\\frac{2 }{5}\\frac{ m^5}{k^4}+\\frac{2 }{7}\\frac{ m^3}{k^2}-\\frac{1}{42} m+\\frac{1}{1848}\\frac{ k^2}{m}+\\frac{1}{96096}\\frac{ k^4}{m^3}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{1921920}\\frac{ k^6}{m^5}+\\frac{1}{26138112}\\frac{ k^8}{m^7}+\\frac{1}{283785216}\\frac{ k^{10}}{m^9}+\\frac{1}{2648662016}\\frac{ k^{12}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{1}{5}\\frac{ m^5}{k^4}+\\frac{1}{28}\\frac{ m^3}{k^2}-\\frac{1}{336} m+\\frac{1}{14784}\\frac{ k^2}{m}+\\frac{1}{768768}\\frac{ k^4}{m^3}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{15375360}\\frac{ k^6}{m^5}+\\frac{1}{209104896}\\frac{ k^8}{m^7}+\\frac{1}{2270281728}\\frac{ k^{10}}{m^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{21189296128}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;5\\text{D}}^{\\text{s,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{15 \\pi ^2} \\left(-\\frac{2 }{7}\\frac{ m^3}{k^2}+\\frac{8 }{5}\\frac{ m^5}{k^4}\\right)\\right)+k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{8 i }{25 \\pi ^2}\\frac{ m^5}{k^4}\\right)+\\ldots $ Scalars, spin 4 x 4, dimension 6: $\\tilde{T}_{4,4;6\\text{D}}^{\\text{s,t,UV}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{2 \\pi ^3} \\left( \\left(\\frac{1627}{12006225}-\\frac{P}{27720}\\right) k^2+ \\left(-\\frac{563}{198450}+\\frac{P}{1260}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{5393}{235200}-\\frac{P}{140}+\\frac{L_0}{32}\\right)\\frac{ m^4}{k^2}- \\left(\\frac{461}{7200}-\\frac{P}{30}+\\frac{L_0}{48}\\right)\\frac{ m^6}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{7}{144}-\\frac{K}{12}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{17}{600}+\\frac{K}{10}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{49}{600}+\\frac{K}{30}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{1627}{8004150}-\\frac{P}{18480}\\right) k^2+ \\left(-\\frac{563}{132300}+\\frac{P}{840}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{44}{1225}-\\frac{3 P}{280}\\right)\\frac{ m^4}{k^2}- \\left(\\frac{31}{800}-\\frac{P}{20}+\\frac{L_0}{16}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{7}{96}-\\frac{K}{8}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{17}{400}+\\frac{3 K}{20}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{49}{400}+\\frac{K}{20}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^3} \\left( \\left(\\frac{1627}{32016600}-\\frac{P}{73920}\\right) k^2+ \\left(-\\frac{563}{529200}+\\frac{P}{3360}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{11}{1225}-\\frac{3 P}{1120}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{23}{600}+\\frac{P}{80}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{7}{384}-\\frac{K}{32}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{17}{1600}+\\frac{3 K}{80}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{49}{1600}+\\frac{K}{80}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;6\\text{D}}^{\\text{s,t,IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{40 \\pi ^3} \\left( \\left(-\\frac{11}{24}+\\frac{L_0}{4}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{81}{112}+\\frac{27 L_0}{56}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{63}+\\frac{L_0}{63}\\right) m^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{ L_0}{1386} k^2+\\frac{1}{36036}\\frac{ k^4}{m^2}+\\frac{1}{1081080}\\frac{ k^6}{m^4}+\\frac{1}{18378360}\\frac{ k^8}{m^6}+\\frac{1}{232792560}\\frac{ k^{10}}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{2444321880}\\frac{ k^{12}}{m^{10}}+\\frac{1}{22487761296}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{40 \\pi ^3} \\left( \\left(\\frac{11}{12}-\\frac{L_0}{2}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{9}{14}-\\frac{3 L_0}{7}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{21}+\\frac{L_0}{21}\\right) m^2-\\frac{ L_0}{462} k^2+\\frac{1}{12012}\\frac{ k^4}{m^2}+\\frac{1}{360360}\\frac{ k^6}{m^4}+\\frac{1}{6126120}\\frac{ k^8}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{77597520}\\frac{ k^{10}}{m^8}+\\frac{1}{814773960}\\frac{ k^{12}}{m^{10}}+\\frac{1}{7495920432}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{160 \\pi ^3} \\left( \\left(-\\frac{11}{6}+L_0\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{9}{28}-\\frac{3 L_0}{14}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{42}+\\frac{L_0}{42}\\right) m^2-\\frac{ L_0}{924} k^2+\\frac{1}{24024}\\frac{ k^4}{m^2}+\\frac{1}{720720}\\frac{ k^6}{m^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{12252240}\\frac{ k^8}{m^6}+\\frac{1}{155195040}\\frac{ k^{10}}{m^8}+\\frac{1}{1629547920}\\frac{ k^{12}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{14991840864}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;6\\text{D}}^{\\text{s,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{10 \\pi ^3} \\left( \\left(\\frac{1627}{2401245}-\\frac{K}{5544}\\right) k^2+ \\left(-\\frac{811}{79380}+\\frac{K}{252}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{389}{5880}-\\frac{K}{28}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{37}{180}+\\frac{K}{6}\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{10 \\pi ^3} \\left( \\left(\\frac{1627}{800415}-\\frac{K}{1848}\\right) k^2+ \\left(-\\frac{811}{26460}+\\frac{K}{84}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{389}{1960}-\\frac{3 K}{28}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{37}{60}+\\frac{K}{2}\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{80 \\pi ^3} \\left( \\left(\\frac{1627}{800415}-\\frac{K}{1848}\\right) k^2+ \\left(-\\frac{811}{26460}+\\frac{K}{84}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{389}{1960}-\\frac{3 K}{28}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{37}{60}+\\frac{K}{2}\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\ldots $ Scalars, spin 5 x 5, dimension 3: $\\tilde{T}_{5,5;3\\text{D}}^{\\text{s,t,UV}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{1}{256}\\frac{1}{k}-\\frac{i }{2 \\pi }\\frac{ m}{k^2}-\\frac{5 }{64}\\frac{ m^2}{k^3}-\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{5 }{8}\\frac{ m^4}{k^5}-\\frac{64 i }{15 \\pi }\\frac{ m^5}{k^6}-\\frac{5 }{2}\\frac{ m^6}{k^7}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{256 i }{21 \\pi }\\frac{ m^7}{k^8}+5 \\frac{ m^8}{k^9}-\\frac{1024 i }{63 \\pi }\\frac{ m^9}{k^{10}}-4 \\frac{ m^{10}}{k^{11}}+\\frac{4096 i }{693 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{5 }{256}\\frac{1}{k}-\\frac{25 }{64}\\frac{ m^2}{k^3}+\\frac{20 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{25 }{8}\\frac{ m^4}{k^5}-\\frac{64 i }{3 \\pi }\\frac{ m^5}{k^6}-\\frac{25 }{2}\\frac{ m^6}{k^7}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{1280 i }{21 \\pi }\\frac{ m^7}{k^8}+25 \\frac{ m^8}{k^9}-\\frac{5120 i }{63 \\pi }\\frac{ m^9}{k^{10}}-20 \\frac{ m^{10}}{k^{11}}+\\frac{20480 i }{693 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{15 }{2048}\\frac{1}{k}-\\frac{75 }{512}\\frac{ m^2}{k^3}+\\frac{75 }{64}\\frac{ m^4}{k^5}-\\frac{8 i }{\\pi }\\frac{ m^5}{k^6}-\\frac{75 }{16}\\frac{ m^6}{k^7}+\\frac{160 i }{7 \\pi }\\frac{ m^7}{k^8}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{75 }{8}\\frac{ m^8}{k^9}-\\frac{640 i }{21 \\pi }\\frac{ m^9}{k^{10}}-\\frac{15 }{2}\\frac{ m^{10}}{k^{11}}+\\frac{2560 i }{231 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{5,5;3\\text{D}}^{\\text{s,t,IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi } \\left(-\\frac{10 }{7}\\frac{ m^3}{k^4}-\\frac{55 }{126}\\frac{ m}{k^2}-\\frac{1}{693}\\frac{1}{m}-\\frac{1}{36036}\\frac{ k^2}{m^3}-\\frac{1}{720720}\\frac{ k^4}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{9801792}\\frac{ k^6}{m^7}-\\frac{1}{106419456}\\frac{ k^8}{m^9}-\\frac{1}{993248256}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(\\frac{20 }{7}\\frac{ m^3}{k^4}+\\frac{20 }{63}\\frac{ m}{k^2}-\\frac{5 }{693}\\frac{1}{m}-\\frac{5 }{36036}\\frac{ k^2}{m^3}-\\frac{1}{144144}\\frac{ k^4}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{5 }{9801792}\\frac{ k^6}{m^7}-\\frac{5 }{106419456}\\frac{ k^8}{m^9}-\\frac{5 }{993248256}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(-\\frac{10 }{7}\\frac{ m^3}{k^4}+\\frac{5 }{42}\\frac{ m}{k^2}-\\frac{5 }{1848}\\frac{1}{m}-\\frac{5 }{96096}\\frac{ k^2}{m^3}-\\frac{1}{384384}\\frac{ k^4}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{5 }{26138112}\\frac{ k^6}{m^7}-\\frac{5 }{283785216}\\frac{ k^8}{m^9}-\\frac{5 }{2648662016}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;3\\text{D}}^{\\text{s,UV-IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{21 \\pi } \\left(-\\frac{4 }{3}\\frac{ m}{k^2}+16 \\frac{ m^3}{k^4}\\right)\\right)+k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{80 i }{21 \\pi }\\frac{ m^3}{k^4}\\right)+\\ldots $ Scalars, spin 5 x 5, dimension 4: $\\tilde{T}_{5,5;4\\text{D}}^{\\text{s,t,UV}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{6508}{2401245}+\\frac{P}{1386}\\right)+ \\left(-\\frac{10837}{158760}-\\frac{P}{63}+\\frac{L_0}{8}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{7837}{11760}+\\frac{P}{7}+\\frac{L_0}{8}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{37}{45}-\\frac{2 K}{3}\\right)\\frac{ m^6}{k^6}+ \\left(-\\frac{35}{36}+\\frac{5 K}{3}\\right)\\frac{ m^8}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{17}{30}+2 K\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{49}{30}+\\frac{2 K}{3}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{6508}{480249}+\\frac{5 P}{1386}\\right)+ \\left(\\frac{1126}{3969}-\\frac{5 P}{63}\\right)\\frac{ m^2}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{611}{1176}-\\frac{5 P}{7}+\\frac{5 L_0}{4}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{37}{9}-\\frac{10 K}{3}\\right)\\frac{ m^6}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{175}{36}+\\frac{25 K}{3}\\right)\\frac{ m^8}{k^8}- \\left(\\frac{17}{6}+10 K\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{49}{6}+\\frac{10 K}{3}\\right)\\frac{ m^{12}}{k^{12}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{1627}{320166}+\\frac{5 P}{3696}\\right)+ \\left(\\frac{563}{5292}-\\frac{5 P}{168}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{44}{49}+\\frac{15 P}{56}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{37}{24}-\\frac{5 K}{4}\\right)\\frac{ m^6}{k^6}+ \\left(-\\frac{175}{96}+\\frac{25 K}{8}\\right)\\frac{ m^8}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{17}{16}+\\frac{15 K}{4}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{49}{16}+\\frac{5 K}{4}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;4\\text{D}}^{\\text{s,t,IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{14 \\pi ^2} \\left( \\left(-\\frac{45}{8}+\\frac{15 L_0}{4}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{55}{36}+\\frac{55 L_0}{36}\\right)\\frac{ m^2}{k^2}+\\frac{ L_0}{99}-\\frac{1}{2574}\\frac{ k^2}{m^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{77220}\\frac{ k^4}{m^4}-\\frac{1}{1312740}\\frac{ k^6}{m^6}-\\frac{1}{16628040}\\frac{ k^8}{m^8}-\\frac{1}{174594420}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{1606268664}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(\\frac{45}{8}-\\frac{15 L_0}{4}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{5}{9}-\\frac{5 L_0}{9}\\right)\\frac{ m^2}{k^2}+\\frac{5 L_0}{198}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{5 }{5148}\\frac{ k^2}{m^2}-\\frac{1}{30888}\\frac{ k^4}{m^4}-\\frac{1}{525096}\\frac{ k^6}{m^6}-\\frac{1}{6651216}\\frac{ k^8}{m^8}-\\frac{1}{69837768}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{5 }{3212537328}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{56 \\pi ^2} \\left( \\left(-\\frac{45}{2}+15 L_0\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{5}{3}-\\frac{5 L_0}{3}\\right)\\frac{ m^2}{k^2}+\\frac{5 L_0}{66}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{5 }{1716}\\frac{ k^2}{m^2}-\\frac{1}{10296}\\frac{ k^4}{m^4}-\\frac{1}{175032}\\frac{ k^6}{m^6}-\\frac{1}{2217072}\\frac{ k^8}{m^8}-\\frac{1}{23279256}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{5 }{1070845776}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;4\\text{D}}^{\\text{s,UV-IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(-\\frac{6508}{343035}+\\frac{K}{198}\\right)+ \\left(\\frac{811}{2835}-\\frac{K}{9}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{389}{210}+K\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(-\\frac{6508}{68607}+\\frac{5 K}{198}\\right)+ \\left(\\frac{811}{567}-\\frac{5 K}{9}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{389}{42}+5 K\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{14 \\pi ^2} \\left( \\left(-\\frac{1627}{22869}+\\frac{5 K}{264}\\right)+ \\left(\\frac{811}{756}-\\frac{5 K}{12}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{389}{56}+\\frac{15 K}{4}\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\ldots $ Scalars, spin 5 x 5, dimension 5: $\\tilde{T}_{5,5;5\\text{D}}^{\\text{s,t,UV}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{24576} k-\\frac{ \\pi }{1024}\\frac{ m^2}{k}+\\frac{i }{12}\\frac{ m^3}{k^2}+\\frac{5 \\pi }{512}\\frac{ m^4}{k^3}+\\frac{i }{15}\\frac{ m^5}{k^4}-\\frac{5 \\pi }{96}\\frac{ m^6}{k^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{32 i }{105}\\frac{ m^7}{k^6}+\\frac{5 \\pi }{32}\\frac{ m^8}{k^7}-\\frac{128 i }{189}\\frac{ m^9}{k^8}-\\frac{ \\pi }{4}\\frac{ m^{10}}{k^9}+\\frac{512 i }{693}\\frac{ m^{11}}{k^{10}}+\\frac{ \\pi }{6}\\frac{ m^{12}}{k^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{5 \\pi }{24576} k-\\frac{5 \\pi }{1024}\\frac{ m^2}{k}+\\frac{25 \\pi }{512}\\frac{ m^4}{k^3}-\\frac{2 i }{3}\\frac{ m^5}{k^4}-\\frac{25 \\pi }{96}\\frac{ m^6}{k^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{32 i }{21}\\frac{ m^7}{k^6}+\\frac{25 \\pi }{32}\\frac{ m^8}{k^7}-\\frac{640 i }{189}\\frac{ m^9}{k^8}-\\frac{5 \\pi }{4}\\frac{ m^{10}}{k^9}+\\frac{2560 i }{693}\\frac{ m^{11}}{k^{10}}+\\frac{5 \\pi }{6}\\frac{ m^{12}}{k^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(\\frac{5 \\pi }{65536} k-\\frac{15 \\pi }{8192}\\frac{ m^2}{k}+\\frac{75 \\pi }{4096}\\frac{ m^4}{k^3}-\\frac{25 \\pi }{256}\\frac{ m^6}{k^5}+\\frac{4 i }{7}\\frac{ m^7}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{75 \\pi }{256}\\frac{ m^8}{k^7}-\\frac{80 i }{63}\\frac{ m^9}{k^8}-\\frac{15 \\pi }{32}\\frac{ m^{10}}{k^9}+\\frac{320 i }{231}\\frac{ m^{11}}{k^{10}}+\\frac{5 \\pi }{16}\\frac{ m^{12}}{k^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;5\\text{D}}^{\\text{s,t,IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{7 \\pi ^2} \\left(\\frac{m^5}{k^4}+\\frac{55 }{108}\\frac{ m^3}{k^2}+\\frac{1}{198} m-\\frac{1}{10296}\\frac{ k^2}{m}-\\frac{1}{617760}\\frac{ k^4}{m^3}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{14002560}\\frac{ k^6}{m^5}-\\frac{1}{212838912}\\frac{ k^8}{m^7}-\\frac{1}{2554066944}\\frac{ k^{10}}{m^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{26108239872}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left(-2 \\frac{ m^5}{k^4}-\\frac{10 }{27}\\frac{ m^3}{k^2}+\\frac{5 }{198} m-\\frac{5 }{10296}\\frac{ k^2}{m}-\\frac{1}{123552}\\frac{ k^4}{m^3}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{2800512}\\frac{ k^6}{m^5}-\\frac{5 }{212838912}\\frac{ k^8}{m^7}-\\frac{5 }{2554066944}\\frac{ k^{10}}{m^9}-\\frac{5 }{26108239872}\\frac{ k^{12}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi ^2} \\left(\\frac{m^5}{k^4}-\\frac{5 }{36}\\frac{ m^3}{k^2}+\\frac{5 }{528} m-\\frac{5 }{27456}\\frac{ k^2}{m}-\\frac{1}{329472}\\frac{ k^4}{m^3}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{7468032}\\frac{ k^6}{m^5}-\\frac{5 }{567570432}\\frac{ k^8}{m^7}-\\frac{5 }{6810845184}\\frac{ k^{10}}{m^9}-\\frac{5 }{69621972992}\\frac{ k^{12}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right) $ $\\tilde{T}_{5,5;5\\text{D}}^{\\text{s,UV-IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{21 \\pi ^2} \\left(\\frac{2 }{9}\\frac{ m^3}{k^2}-\\frac{8 }{5}\\frac{ m^5}{k^4}\\right)\\right)+k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(-\\frac{8 i }{21 \\pi ^2}\\frac{ m^5}{k^4}\\right)+\\ldots $ Scalars, spin 5 x 5, dimension 6: $\\tilde{T}_{5,5;6\\text{D}}^{\\text{s,t,UV}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{88069}{3246483240}+\\frac{P}{144144}\\right) k^2+ \\left(\\frac{1627}{2401245}-\\frac{P}{5544}\\right) m^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(-\\frac{41519}{2540160}-\\frac{P}{504}+\\frac{L_0}{64}\\right)\\frac{ m^4}{k^2}- \\left(-\\frac{8327}{141120}+\\frac{P}{84}+\\frac{L_0}{96}\\right)\\frac{ m^6}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{59}{1440}+\\frac{K}{24}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{23}{720}-\\frac{K}{12}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{3}{80}+\\frac{K}{12}\\right)\\frac{ m^{12}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{88069}{649296648}+\\frac{5 P}{144144}\\right) k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1627}{480249}-\\frac{5 P}{5544}\\right) m^2+ \\left(-\\frac{563}{15876}+\\frac{5 P}{504}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{121}{14112}-\\frac{5 P}{84}+\\frac{5 L_0}{48}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{59}{288}+\\frac{5 K}{24}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{23}{144}-\\frac{5 K}{12}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{3}{16}+\\frac{5 K}{12}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{88069}{1731457728}+\\frac{5 P}{384384}\\right) k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1627}{1280664}-\\frac{5 P}{14784}\\right) m^2+ \\left(-\\frac{563}{42336}+\\frac{5 P}{1344}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{11}{147}-\\frac{5 P}{224}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{59}{768}+\\frac{5 K}{64}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{23}{384}-\\frac{5 K}{32}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{9}{128}+\\frac{5 K}{32}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;6\\text{D}}^{\\text{s,t,IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{56 \\pi ^3} \\left( \\left(\\frac{55}{24}-\\frac{5 L_0}{4}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{55}{48}-\\frac{55 L_0}{72}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{99}-\\frac{L_0}{99}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{ L_0}{2574} k^2-\\frac{1}{77220}\\frac{ k^4}{m^2}-\\frac{1}{2625480}\\frac{ k^6}{m^4}-\\frac{1}{49884120}\\frac{ k^8}{m^6}-\\frac{1}{698377680}\\frac{ k^{10}}{m^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{8031343320}\\frac{ k^{12}}{m^{10}}-\\frac{1}{80313433200}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{56 \\pi ^3} \\left( \\left(-\\frac{55}{12}+\\frac{5 L_0}{2}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{5}{6}+\\frac{5 L_0}{9}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{5}{99}-\\frac{5 L_0}{99}\\right) m^2+\\frac{5 L_0}{2574} k^2-\\frac{1}{15444}\\frac{ k^4}{m^2}-\\frac{1}{525096}\\frac{ k^6}{m^4}-\\frac{1}{9976824}\\frac{ k^8}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{139675536}\\frac{ k^{10}}{m^8}-\\frac{1}{1606268664}\\frac{ k^{12}}{m^{10}}-\\frac{1}{16062686640}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{224 \\pi ^3} \\left( \\left(\\frac{55}{6}-5 L_0\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{5}{4}+\\frac{5 L_0}{6}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{5}{66}-\\frac{5 L_0}{66}\\right) m^2+\\frac{5 L_0}{1716} k^2-\\frac{1}{10296}\\frac{ k^4}{m^2}-\\frac{1}{350064}\\frac{ k^6}{m^4}-\\frac{1}{6651216}\\frac{ k^8}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{93117024}\\frac{ k^{10}}{m^8}-\\frac{1}{1070845776}\\frac{ k^{12}}{m^{10}}-\\frac{1}{10708457760}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;6\\text{D}}^{\\text{s,UV-IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{84 \\pi ^3} \\left( \\left(-\\frac{88069}{38648610}+\\frac{K}{1716}\\right) k^2+ \\left(\\frac{9551}{228690}-\\frac{K}{66}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1307}{3780}+\\frac{K}{6}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{319}{210}-K\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{84 \\pi ^3} \\left( \\left(-\\frac{88069}{7729722}+\\frac{5 K}{1716}\\right) k^2+ \\left(\\frac{9551}{45738}-\\frac{5 K}{66}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1307}{756}+\\frac{5 K}{6}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{319}{42}-5 K\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{224 \\pi ^3} \\left( \\left(-\\frac{88069}{7729722}+\\frac{5 K}{1716}\\right) k^2+ \\left(\\frac{9551}{45738}-\\frac{5 K}{66}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1307}{756}+\\frac{5 K}{6}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{319}{42}-5 K\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\ldots $"
],
[
"Divergences of the scalar amplitudes",
"Scalars, spin 0 x 2: $\\tilde{T}_{0,2;3\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\nu } \\left(\\frac{i }{2 \\pi } m\\right) $ $\\tilde{T}_{0,2;4\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\nu } \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right) $ $\\tilde{T}_{0,2;5\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\nu } \\left(-\\frac{i }{12 \\pi ^2} m^3\\right) $ $\\tilde{T}_{0,2;6\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\nu } \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right) $ Scalars, spin 0 x 4: $\\tilde{T}_{0,4;3\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\nu }^3 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{2 i }{\\pi } m^3\\right) $ $\\tilde{T}_{0,4;4\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\nu }^3 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{8 \\pi ^2} m^4\\right) $ $\\tilde{T}_{0,4;5\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\nu }^3 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i }{5 \\pi ^2} m^5\\right) $ $\\tilde{T}_{0,4;6\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\nu }^3 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{32 \\pi ^3} m^6\\right) $ Scalars, spin 1 x 1: $k\\cdot \\tilde{T}_{1,1;3\\text{D}}^{\\text{s,nt}} & = k_{\\nu } \\left(\\frac{i }{2 \\pi } m\\right) $ $k\\cdot \\tilde{T}_{1,1;4\\text{D}}^{\\text{s,nt}} & = k_{\\nu } \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right) $ $k\\cdot \\tilde{T}_{1,1;5\\text{D}}^{\\text{s,nt}} & = k_{\\nu } \\left(-\\frac{i }{12 \\pi ^2} m^3\\right) $ $k\\cdot \\tilde{T}_{1,1;6\\text{D}}^{\\text{s,nt}} & = k_{\\nu } \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right) $ Scalars, spin 1 x 3: $k\\cdot \\tilde{T}_{1,3;3\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^3 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{2 i }{\\pi } m^3\\right) $ $\\tilde{T}_{1,3;3\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^2 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu } \\eta _{\\nu \\nu } \\left(\\frac{2 i }{3 \\pi } m^3\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(\\frac{4 i }{3 \\pi } m^3\\right) $ $k\\cdot \\tilde{T}_{1,3;4\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^3 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{8 \\pi ^2} m^4\\right) $ $\\tilde{T}_{1,3;4\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^2 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_2}{8 \\pi ^2} m^4\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(-\\frac{i L_2}{4 \\pi ^2} m^4\\right) $ $k\\cdot \\tilde{T}_{1,3;5\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^3 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i }{5 \\pi ^2} m^5\\right) $ $\\tilde{T}_{1,3;5\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^2 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i }{15 \\pi ^2} m^5\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(-\\frac{2 i }{15 \\pi ^2} m^5\\right) $ $k\\cdot \\tilde{T}_{1,3;6\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^3 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{32 \\pi ^3} m^6\\right) $ $\\tilde{T}_{1,3;6\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^2 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{96 \\pi ^3} m^6\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(\\frac{i L_3}{48 \\pi ^3} m^6\\right) $ Scalars, spin 1 x 5: $k\\cdot \\tilde{T}_{1,5;3\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^5 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{20 i }{3 \\pi } m^3\\right)+k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{8 i }{\\pi } m^5\\right) $ $\\tilde{T}_{1,5;3\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^4 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{4 i }{\\pi } m^3\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{8 i }{3 \\pi } m^3\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{8 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{32 i }{5 \\pi } m^5\\right) $ $k\\cdot \\tilde{T}_{1,5;4\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^5 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{5 i L_2}{4 \\pi ^2} m^4\\right)+k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{5 i L_3}{4 \\pi ^2} m^6\\right) $ $\\tilde{T}_{1,5;4\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^4 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{i L_2}{2 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_3}{4 \\pi ^2} m^6\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{\\pi ^2} m^6\\right) $ $k\\cdot \\tilde{T}_{1,5;5\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^5 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{3 \\pi ^2} m^5\\right)+k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{4 i }{7 \\pi ^2} m^7\\right) $ $\\tilde{T}_{1,5;5\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^4 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{4 i }{15 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{4 i }{35 \\pi ^2} m^7\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{16 i }{35 \\pi ^2} m^7\\right) $ $k\\cdot \\tilde{T}_{1,5;6\\text{D}}^{\\text{s,nt}} & = k_{\\nu }^5 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{5 i L_3}{48 \\pi ^3} m^6\\right)+k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{5 i L_4}{64 \\pi ^3} m^8\\right) $ $\\tilde{T}_{1,5;6\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^4 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{16 \\pi ^3} m^6\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{i L_3}{24 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_4}{64 \\pi ^3} m^8\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i L_4}{16 \\pi ^3} m^8\\right) $ Scalars, spin 2 x 2: $k\\cdot \\tilde{T}_{2,2;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^2 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu } \\eta _{\\nu \\nu } \\left(\\frac{2 i }{3 \\pi } m^3\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(\\frac{4 i }{3 \\pi } m^3\\right) $ $k\\cdot \\tilde{T}_{2,2;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^2 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_2}{8 \\pi ^2} m^4\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(-\\frac{i L_2}{4 \\pi ^2} m^4\\right) $ $k\\cdot \\tilde{T}_{2,2;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^2 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i }{15 \\pi ^2} m^5\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(-\\frac{2 i }{15 \\pi ^2} m^5\\right) $ $k\\cdot \\tilde{T}_{2,2;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^2 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{96 \\pi ^3} m^6\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(\\frac{i L_3}{48 \\pi ^3} m^6\\right) $ Scalars, spin 2 x 4: $k\\cdot \\tilde{T}_{2,4;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^4 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{4 i }{\\pi } m^3\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{8 i }{3 \\pi } m^3\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{8 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{32 i }{5 \\pi } m^5\\right) $ $\\tilde{T}_{2,4;3\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu }^2 k_{\\nu }^3 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(\\frac{2 i }{3 \\pi } m^3\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{2 i }{\\pi } m^3\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(\\frac{4 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{8 i }{5 \\pi } m^5\\right)+ \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{16 i }{5 \\pi } m^5\\right) $ $k\\cdot \\tilde{T}_{2,4;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^4 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{i L_2}{2 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_3}{4 \\pi ^2} m^6\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{\\pi ^2} m^6\\right) $ $\\tilde{T}_{2,4;4\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu }^2 k_{\\nu }^3 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(-\\frac{i L_2}{8 \\pi ^2} m^4\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{8 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{4 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{2 \\pi ^2} m^6\\right) $ $k\\cdot \\tilde{T}_{2,4;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^4 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{4 i }{15 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{4 i }{35 \\pi ^2} m^7\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{16 i }{35 \\pi ^2} m^7\\right) $ $\\tilde{T}_{2,4;5\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu }^2 k_{\\nu }^3 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(-\\frac{i }{15 \\pi ^2} m^5\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{4 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{8 i }{35 \\pi ^2} m^7\\right) $ $k\\cdot \\tilde{T}_{2,4;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu } k_{\\nu }^4 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{16 \\pi ^3} m^6\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{i L_3}{24 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_4}{64 \\pi ^3} m^8\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i L_4}{16 \\pi ^3} m^8\\right) $ $\\tilde{T}_{2,4;6\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu }^2 k_{\\nu }^3 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(\\frac{i L_3}{96 \\pi ^3} m^6\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{32 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(\\frac{i L_3}{16 \\pi ^3} m^6\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_4}{64 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_4}{32 \\pi ^3} m^8\\right) $ Scalars, spin 3 x 3: $k\\cdot \\tilde{T}_{3,3;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^3 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(\\frac{2 i }{3 \\pi } m^3\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{2 i }{\\pi } m^3\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(\\frac{4 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{8 i }{5 \\pi } m^5\\right)+ \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{16 i }{5 \\pi } m^5\\right) $ $k\\cdot \\tilde{T}_{3,3;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^3 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(-\\frac{i L_2}{8 \\pi ^2} m^4\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{8 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{4 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{2 \\pi ^2} m^6\\right) $ $k\\cdot \\tilde{T}_{3,3;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^3 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(-\\frac{i }{15 \\pi ^2} m^5\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{4 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{8 i }{35 \\pi ^2} m^7\\right) $ $k\\cdot \\tilde{T}_{3,3;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^3 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(\\frac{i L_3}{96 \\pi ^3} m^6\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{32 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(\\frac{i L_3}{16 \\pi ^3} m^6\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_4}{64 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_4}{32 \\pi ^3} m^8\\right) $ Scalars, spin 3 x 5: $k\\cdot \\tilde{T}_{3,5;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^5 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(\\frac{2 i }{3 \\pi } m^3\\right)+ \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{20 i }{3 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{16 i }{3 \\pi } m^5\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{8 i }{\\pi } m^5\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{32 i }{\\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{32 i }{3 \\pi } m^5\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{32 i }{7 \\pi } m^7\\right)+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{64 i }{7 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{128 i }{7 \\pi } m^7\\right) $ $\\tilde{T}_{3,5;3\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu }^3 k_{\\nu }^4 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(\\frac{2 i }{\\pi } m^3\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{4 i }{\\pi } m^3\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{8 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{48 i }{5 \\pi } m^5\\right)+k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(\\frac{8 i }{5 \\pi } m^5\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(\\frac{32 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{96 i }{5 \\pi } m^5\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{96 i }{35 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{384 i }{35 \\pi } m^7\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{256 i }{35 \\pi } m^7\\right) $ $k\\cdot \\tilde{T}_{3,5;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^5 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(-\\frac{i L_2}{8 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{5 i L_2}{4 \\pi ^2} m^4\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{5 i L_3}{6 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{5 i L_3}{4 \\pi ^2} m^6\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{5 i L_3}{\\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(-\\frac{5 i L_3}{3 \\pi ^2} m^6\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{5 i L_4}{8 \\pi ^2} m^8\\right)+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{5 i L_4}{4 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{5 i L_4}{2 \\pi ^2} m^8\\right) $ $\\tilde{T}_{3,5;4\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu }^3 k_{\\nu }^4 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(-\\frac{3 i L_2}{8 \\pi ^2} m^4\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{3 i L_2}{2 \\pi ^2} m^4\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_3}{2 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_3}{4 \\pi ^2} m^6\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{\\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{3 i L_3}{\\pi ^2} m^6\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{3 i L_4}{8 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{3 i L_4}{2 \\pi ^2} m^8\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(-\\frac{i L_4}{\\pi ^2} m^8\\right) $ $k\\cdot \\tilde{T}_{3,5;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^5 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(-\\frac{i }{15 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{2 i }{3 \\pi ^2} m^5\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{8 i }{21 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{4 i }{7 \\pi ^2} m^7\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{16 i }{7 \\pi ^2} m^7\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(-\\frac{16 i }{21 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{16 i }{63 \\pi ^2} m^9\\right)+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{32 i }{63 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{64 i }{63 \\pi ^2} m^9\\right) $ $\\tilde{T}_{3,5;5\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu }^3 k_{\\nu }^4 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(-\\frac{i }{5 \\pi ^2} m^5\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{4 i }{5 \\pi ^2} m^5\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{24 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(-\\frac{4 i }{35 \\pi ^2} m^7\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(-\\frac{16 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{48 i }{35 \\pi ^2} m^7\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{16 i }{105 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{64 i }{105 \\pi ^2} m^9\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(-\\frac{128 i }{315 \\pi ^2} m^9\\right) $ $k\\cdot \\tilde{T}_{3,5;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^2 k_{\\nu }^5 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(\\frac{i L_3}{96 \\pi ^3} m^6\\right)+ \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{5 i L_3}{48 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{5 i L_4}{96 \\pi ^3} m^8\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{5 i L_4}{64 \\pi ^3} m^8\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{5 i L_4}{16 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{5 i L_4}{48 \\pi ^3} m^8\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_5}{32 \\pi ^3} m^{10}\\right)+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_5}{16 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_5}{8 \\pi ^3} m^{10}\\right) $ $\\tilde{T}_{3,5;6\\text{D}}^{\\text{s,nt}}\\cdot k & = k_{\\mu }^3 k_{\\nu }^4 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(\\frac{i L_3}{32 \\pi ^3} m^6\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{16 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{i L_3}{8 \\pi ^3} m^6\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{3 i L_4}{32 \\pi ^3} m^8\\right)+k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_4}{64 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(\\frac{i L_4}{16 \\pi ^3} m^8\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{3 i L_4}{16 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{3 i L_5}{160 \\pi ^3} m^{10}\\right)+ \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{3 i L_5}{40 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i L_5}{20 \\pi ^3} m^{10}\\right) $ Scalars, spin 4 x 4: $k\\cdot \\tilde{T}_{4,4;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^4 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(\\frac{2 i }{\\pi } m^3\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{4 i }{\\pi } m^3\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{8 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{48 i }{5 \\pi } m^5\\right)+k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(\\frac{8 i }{5 \\pi } m^5\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(\\frac{32 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{96 i }{5 \\pi } m^5\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{96 i }{35 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{384 i }{35 \\pi } m^7\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{256 i }{35 \\pi } m^7\\right) $ $k\\cdot \\tilde{T}_{4,4;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^4 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(-\\frac{3 i L_2}{8 \\pi ^2} m^4\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{3 i L_2}{2 \\pi ^2} m^4\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_3}{2 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_3}{4 \\pi ^2} m^6\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{\\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{3 i L_3}{\\pi ^2} m^6\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{3 i L_4}{8 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{3 i L_4}{2 \\pi ^2} m^8\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(-\\frac{i L_4}{\\pi ^2} m^8\\right) $ $k\\cdot \\tilde{T}_{4,4;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^4 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(-\\frac{i }{5 \\pi ^2} m^5\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{4 i }{5 \\pi ^2} m^5\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{24 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(-\\frac{4 i }{35 \\pi ^2} m^7\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(-\\frac{16 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{48 i }{35 \\pi ^2} m^7\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{16 i }{105 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{64 i }{105 \\pi ^2} m^9\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(-\\frac{128 i }{315 \\pi ^2} m^9\\right) $ $k\\cdot \\tilde{T}_{4,4;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^3 k_{\\nu }^4 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(\\frac{i L_3}{32 \\pi ^3} m^6\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{16 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{i L_3}{8 \\pi ^3} m^6\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{3 i L_4}{32 \\pi ^3} m^8\\right)+k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_4}{64 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(\\frac{i L_4}{16 \\pi ^3} m^8\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{3 i L_4}{16 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{3 i L_5}{160 \\pi ^3} m^{10}\\right)+ \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{3 i L_5}{40 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i L_5}{20 \\pi ^3} m^{10}\\right) $ Scalars, spin 5 x 5: $k\\cdot \\tilde{T}_{5,5;3\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^4 k_{\\nu }^5 \\left(\\frac{i }{2 \\pi } m\\right)+k_{\\mu }^2 k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(\\frac{4 i }{\\pi } m^3\\right)+k_{\\mu }^4 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{20 i }{3 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{40 i }{3 \\pi } m^3\\right)+k_{\\nu }^5 \\eta _{\\mu \\mu }^2 \\left(\\frac{8 i }{5 \\pi } m^5\\right)+k_{\\mu }^4 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{8 i }{\\pi } m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{32 i }{\\pi } m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{64 i }{\\pi } m^5\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu } \\left(\\frac{64 i }{7 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{192 i }{7 \\pi } m^7\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{768 i }{7 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{128 i }{7 \\pi } m^7\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\left(\\frac{256 i }{7 \\pi } m^7\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{512 i }{7 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{128 i }{21 \\pi } m^9\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{512 i }{21 \\pi } m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{1024 i }{21 \\pi } m^9\\right)+k_{\\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{2048 i }{63 \\pi } m^9\\right)+k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{1024 i }{63 \\pi } m^9\\right) $ $k\\cdot \\tilde{T}_{5,5;4\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^4 k_{\\nu }^5 \\left(-\\frac{i L_1}{8 \\pi ^2} m^2\\right)+k_{\\mu }^2 k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+k_{\\mu }^4 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{5 i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{5 i L_2}{2 \\pi ^2} m^4\\right)+k_{\\nu }^5 \\eta _{\\mu \\mu }^2 \\left(-\\frac{i L_3}{4 \\pi ^2} m^6\\right)+k_{\\mu }^4 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{5 i L_3}{4 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{5 i L_3}{\\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{10 i L_3}{\\pi ^2} m^6\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{5 i L_4}{4 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{15 i L_4}{4 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{15 i L_4}{\\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{5 i L_4}{2 \\pi ^2} m^8\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\left(-\\frac{5 i L_4}{\\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(-\\frac{10 i L_4}{\\pi ^2} m^8\\right)+k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(-\\frac{3 i L_5}{4 \\pi ^2} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{3 i L_5}{\\pi ^2} m^{10}\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{6 i L_5}{\\pi ^2} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{4 i L_5}{\\pi ^2} m^{10}\\right)+k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(-\\frac{2 i L_5}{\\pi ^2} m^{10}\\right) $ $k\\cdot \\tilde{T}_{5,5;5\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^4 k_{\\nu }^5 \\left(-\\frac{i }{12 \\pi ^2} m^3\\right)+k_{\\mu }^2 k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+k_{\\mu }^4 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{3 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{4 i }{3 \\pi ^2} m^5\\right)+k_{\\nu }^5 \\eta _{\\mu \\mu }^2 \\left(-\\frac{4 i }{35 \\pi ^2} m^7\\right)+k_{\\mu }^4 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{4 i }{7 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{16 i }{7 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{32 i }{7 \\pi ^2} m^7\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{32 i }{63 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{32 i }{21 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{128 i }{21 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{64 i }{63 \\pi ^2} m^9\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\left(-\\frac{128 i }{63 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(-\\frac{256 i }{63 \\pi ^2} m^9\\right)+k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(-\\frac{64 i }{231 \\pi ^2} m^{11}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{256 i }{231 \\pi ^2} m^{11}\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{512 i }{231 \\pi ^2} m^{11}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{1024 i }{693 \\pi ^2} m^{11}\\right)+k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(-\\frac{512 i }{693 \\pi ^2} m^{11}\\right) $ $k\\cdot \\tilde{T}_{5,5;6\\text{D}}^{\\text{s,nt}} & = k_{\\mu }^4 k_{\\nu }^5 \\left(\\frac{i L_2}{64 \\pi ^3} m^4\\right)+k_{\\mu }^2 k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(\\frac{i L_3}{16 \\pi ^3} m^6\\right)+k_{\\mu }^4 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{5 i L_3}{48 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{5 i L_3}{24 \\pi ^3} m^6\\right)+k_{\\nu }^5 \\eta _{\\mu \\mu }^2 \\left(\\frac{i L_4}{64 \\pi ^3} m^8\\right)+k_{\\mu }^4 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{5 i L_4}{64 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{5 i L_4}{16 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{5 i L_4}{8 \\pi ^3} m^8\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_5}{16 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{3 i L_5}{16 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{3 i L_5}{4 \\pi ^3} m^{10}\\right)+k_{\\mu }^3 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_5}{8 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\left(\\frac{i L_5}{4 \\pi ^3} m^{10}\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i L_5}{2 \\pi ^3} m^{10}\\right)+k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_6}{32 \\pi ^3} m^{12}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_6}{8 \\pi ^3} m^{12}\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_6}{4 \\pi ^3} m^{12}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i L_6}{6 \\pi ^3} m^{12}\\right)+k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{i L_6}{12 \\pi ^3} m^{12}\\right) $"
],
[
"Fermion amplitudes",
"Fermions, spin 0 x 0, dimension 3: $\\tilde{T}_{0,0;3\\text{D}}^{\\text{f}} & = -\\frac{i }{2 \\pi } m+ T \\left(-\\frac{1}{8} k+\\frac{1}{2}\\frac{ m^2}{k}\\right) $ Fermions, spin 0 x 0, dimension 4: $\\tilde{T}_{0,0;4\\text{D}}^{\\text{f}} & = \\frac{i }{4 \\pi ^2} \\left( \\left(1-\\frac{L_0}{2}\\right) k^2+ \\left(-5+3 L_0\\right) m^2\\right)+\\frac{i S}{4 \\pi ^2} \\left(- k+4 \\frac{ m^2}{k}\\right) $ Fermions, spin 0 x 0, dimension 5: $\\tilde{T}_{0,0;5\\text{D}}^{\\text{f}} & = \\frac{i }{4 \\pi ^2} \\left(-\\frac{1}{4} k^2 m+\\frac{5 }{3} m^3\\right)+\\frac{ T}{4 \\pi } \\left(-\\frac{1}{16} k^3+\\frac{1}{2} k m^2-\\frac{ m^4}{k}\\right) $ Fermions, spin 0 x 0, dimension 6: $\\tilde{T}_{0,0;6\\text{D}}^{\\text{f}} & = \\frac{i }{4 \\pi ^3} \\left( \\left(\\frac{1}{9}-\\frac{L_0}{24}\\right) k^4+ \\left(-\\frac{37}{36}+\\frac{5 L_0}{12}\\right) k^2 m^2+ \\left(\\frac{65}{24}-\\frac{5 L_0}{4}\\right) m^4\\right)+\\nonumber \\\\ & \\quad + \\frac{i S}{3 \\pi ^3} \\left(-\\frac{1}{16} k^3+\\frac{1}{2} k m^2-\\frac{ m^4}{k}\\right) $ Fermions, spin 0 x 2, dimension 3: $\\tilde{T}_{0,2;3\\text{D}}^{\\text{f,t}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi }\\frac{ m^2}{k^2}+ T \\left(-\\frac{1}{8}\\frac{ m}{k}+\\frac{1}{2}\\frac{ m^3}{k^3}\\right)\\right) $ $\\tilde{T}_{0,2;3\\text{D}}^{\\text{f,nt}} & = \\eta _{\\nu \\nu } \\left(-\\frac{i }{\\pi } m^2\\right) $ Fermions, spin 0 x 2, dimension 4: $\\tilde{T}_{0,2;4\\text{D}}^{\\text{f,t}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left( \\left(\\frac{2}{3}-\\frac{L_0}{4}\\right) m-2 \\frac{ m^3}{k^2}\\right)+\\frac{i S}{3 \\pi ^2} \\left(-\\frac{1}{2}\\frac{ m}{k}+2 \\frac{ m^3}{k^3}\\right)\\right) $ $\\tilde{T}_{0,2;4\\text{D}}^{\\text{f,nt}} & = \\eta _{\\nu \\nu } \\left(\\frac{i L_1}{2 \\pi ^2} m^3\\right) $ Fermions, spin 0 x 2, dimension 5: $\\tilde{T}_{0,2;5\\text{D}}^{\\text{f,t}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{8 \\pi ^2} \\left(-\\frac{1}{4} m^2-\\frac{ m^4}{k^2}\\right)+\\frac{ T}{8 \\pi } \\left(-\\frac{1}{16} k m+\\frac{1}{2}\\frac{ m^3}{k}-\\frac{ m^5}{k^3}\\right)\\right) $ $\\tilde{T}_{0,2;5\\text{D}}^{\\text{f,nt}} & = \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} m^4\\right) $ Fermions, spin 0 x 2, dimension 6: $\\tilde{T}_{0,2;6\\text{D}}^{\\text{f,t}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^3} \\left( \\left(\\frac{23}{600}-\\frac{L_0}{80}\\right) k^2 m+ \\left(-\\frac{43}{120}+\\frac{L_0}{8}\\right) m^3+\\frac{2 }{5}\\frac{ m^5}{k^2}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{15 \\pi ^3} \\left(-\\frac{1}{8} k m+\\frac{m^3}{k}-2 \\frac{ m^5}{k^3}\\right)\\right) $ $\\tilde{T}_{0,2;6\\text{D}}^{\\text{f,nt}} & = \\eta _{\\nu \\nu } \\left(-\\frac{i L_2}{8 \\pi ^3} m^5\\right) $ Fermions, spin 0 x 4, dimension 3: $\\tilde{T}_{0,4;3\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(-\\frac{5 }{4}\\frac{ m^2}{k^2}+3 \\frac{ m^4}{k^4}\\right)+ T \\left(\\frac{3 }{32}\\frac{ m}{k}-\\frac{3 }{4}\\frac{ m^3}{k^3}+\\frac{3 }{2}\\frac{ m^5}{k^5}\\right)\\right) $ $\\tilde{T}_{0,4;3\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{\\pi } m^2\\right)+\\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(k^2 m^2-4 m^4\\right)\\right) $ Fermions, spin 0 x 4, dimension 4: $\\tilde{T}_{0,4;4\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{23}{30}+\\frac{L_0}{4}\\right) m+\\frac{14 }{3}\\frac{ m^3}{k^2}-8 \\frac{ m^5}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{5 \\pi ^2} \\left(\\frac{1}{2}\\frac{ m}{k}-4 \\frac{ m^3}{k^3}+8 \\frac{ m^5}{k^5}\\right)\\right) $ $\\tilde{T}_{0,4;4\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i L_1}{\\pi ^2} m^3\\right)+\\eta _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(- L_1 k^2 m^3+3 L_2 m^5\\right)\\right) $ Fermions, spin 0 x 4, dimension 5: $\\tilde{T}_{0,4;5\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{1}{32} m^2+\\frac{1}{3}\\frac{ m^4}{k^2}-\\frac{1}{2}\\frac{ m^6}{k^4}\\right)+\\frac{ T}{4 \\pi } \\left(\\frac{1}{64} k m-\\frac{3 }{16}\\frac{ m^3}{k}+\\frac{3 }{4}\\frac{ m^5}{k^3}-\\frac{ m^7}{k^5}\\right)\\right) $ $\\tilde{T}_{0,4;5\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{2 i }{3 \\pi ^2} m^4\\right)+\\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{1}{3} k^2 m^4+\\frac{4 }{5} m^6\\right)\\right) $ Fermions, spin 0 x 4, dimension 6: $\\tilde{T}_{0,4;6\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{22}{3675}+\\frac{L_0}{560}\\right) k^2 m+ \\left(\\frac{337}{4200}-\\frac{L_0}{40}\\right) m^3-\\frac{4 }{21}\\frac{ m^5}{k^2}+\\frac{8 }{35}\\frac{ m^7}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^3} \\left(\\frac{1}{8} k m-\\frac{3 }{2}\\frac{ m^3}{k}+6 \\frac{ m^5}{k^3}-8 \\frac{ m^7}{k^5}\\right)\\right) $ $\\tilde{T}_{0,4;6\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i L_2}{4 \\pi ^3} m^5\\right)+\\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^3} \\left(\\frac{ L_2}{2} k^2 m^5- L_3 m^7\\right)\\right) $ Fermions, spin 1 x 1, dimension 3: $\\tilde{T}_{1,1;3\\text{D}}^{\\text{f,t}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{4 \\pi }\\frac{ m}{k^2}+ T \\left(\\frac{1}{16}\\frac{1}{k}+\\frac{1}{4}\\frac{ m^2}{k^3}\\right)\\right)+(k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{i T}{4}\\frac{ m}{k}\\right) $ $\\tilde{T}_{1,1;3\\text{D}}^{\\text{f,nt}} & = 0 $ Fermions, spin 1 x 1, dimension 4: $\\tilde{T}_{1,1;4\\text{D}}^{\\text{f,t}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left( \\left(-\\frac{5}{12}+\\frac{L_0}{4}\\right)-\\frac{ m^2}{k^2}\\right)+\\frac{i S}{3 \\pi ^2} \\left(\\frac{1}{2}\\frac{1}{k}+\\frac{m^2}{k^3}\\right)\\right) $ $\\tilde{T}_{1,1;4\\text{D}}^{\\text{f,nt}} & = 0 $ Fermions, spin 1 x 1, dimension 5: $\\tilde{T}_{1,1;5\\text{D}}^{\\text{f,t}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{16 \\pi ^2} \\left(\\frac{3 }{4} m-\\frac{ m^3}{k^2}\\right)+\\frac{ T}{16 \\pi } \\left(\\frac{3 }{16} k-\\frac{1}{2}\\frac{ m^2}{k}-\\frac{ m^4}{k^3}\\right)\\right) $ $\\tilde{T}_{1,1;5\\text{D}}^{\\text{f,nt}} & = 0 $ Fermions, spin 1 x 1, dimension 6: $\\tilde{T}_{1,1;6\\text{D}}^{\\text{f,t}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{3 \\pi ^3} \\left( \\left(-\\frac{77}{1200}+\\frac{L_0}{40}\\right) k^2+ \\left(\\frac{31}{120}-\\frac{L_0}{8}\\right) m^2+\\frac{1}{5}\\frac{ m^4}{k^2}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{5 \\pi ^3} \\left(\\frac{1}{12} k-\\frac{1}{4}\\frac{ m^2}{k}-\\frac{1}{3}\\frac{ m^4}{k^3}\\right)\\right) $ $\\tilde{T}_{1,1;6\\text{D}}^{\\text{f,nt}} & = 0 $ Fermions, spin 1 x 3, dimension 3: $\\tilde{T}_{1,3;3\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi } \\left(-\\frac{1}{4}\\frac{ m}{k^2}+3 \\frac{ m^3}{k^4}\\right)+ T \\left(-\\frac{1}{64}\\frac{1}{k}-\\frac{1}{8}\\frac{ m^2}{k^3}+\\frac{3 }{4}\\frac{ m^4}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{1}{2 \\pi }\\frac{ m^2}{k^2}+i T \\left(\\frac{1}{8}\\frac{ m}{k}-\\frac{1}{2}\\frac{ m^3}{k^3}\\right)\\right) $ $\\tilde{T}_{1,3;3\\text{D}}^{\\text{f,nt}} & = \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{4 i }{3 \\pi } m^3\\right)+(k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{1}{\\pi } m^2\\right) $ Fermions, spin 1 x 3, dimension 4: $\\tilde{T}_{1,3;4\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{31}{180}-\\frac{L_0}{12}\\right)+\\frac{2 }{3}\\frac{ m^2}{k^2}-4 \\frac{ m^4}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{5 \\pi ^2} \\left(-\\frac{1}{6}\\frac{1}{k}-\\frac{1}{3}\\frac{ m^2}{k^3}+4 \\frac{ m^4}{k^5}\\right)\\right) $ $\\tilde{T}_{1,3;4\\text{D}}^{\\text{f,nt}} & = \\eta _{\\nu \\nu } \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{2 \\pi ^2} m^4\\right) $ Fermions, spin 1 x 3, dimension 5: $\\tilde{T}_{1,3;5\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{8 \\pi ^2} \\left(-\\frac{1}{16} m+\\frac{1}{6}\\frac{ m^3}{k^2}-\\frac{ m^5}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{8 \\pi } \\left(-\\frac{1}{64} k+\\frac{1}{16}\\frac{ m^2}{k}+\\frac{1}{4}\\frac{ m^4}{k^3}-\\frac{ m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{1,3;5\\text{D}}^{\\text{f,nt}} & = \\eta _{\\nu \\nu } \\eta _{\\mu \\nu } \\left(\\frac{4 i }{15 \\pi ^2} m^5\\right) $ Fermions, spin 1 x 3, dimension 6: $\\tilde{T}_{1,3;6\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(\\frac{599}{35280}-\\frac{L_0}{168}\\right) k^2+ \\left(-\\frac{247}{2520}+\\frac{L_0}{24}\\right) m^2-\\frac{1}{7}\\frac{ m^4}{k^2}+\\frac{4 }{7}\\frac{ m^6}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{7 \\pi ^3} \\left(-\\frac{1}{60} k+\\frac{1}{12}\\frac{ m^2}{k}+\\frac{2 }{15}\\frac{ m^4}{k^3}-\\frac{4 }{5}\\frac{ m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{1,3;6\\text{D}}^{\\text{f,nt}} & = \\eta _{\\nu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{12 \\pi ^3} m^6\\right) $ Fermions, spin 1 x 5, dimension 3: $\\tilde{T}_{1,5;3\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(\\frac{1}{16}\\frac{ m}{k^2}-\\frac{11 }{6}\\frac{ m^3}{k^4}+5 \\frac{ m^5}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{1}{128}\\frac{1}{k}+\\frac{3 }{32}\\frac{ m^2}{k^3}-\\frac{9 }{8}\\frac{ m^4}{k^5}+\\frac{5 }{2}\\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{2 \\pi } \\left(-\\frac{5 }{4}\\frac{ m^2}{k^2}+3 \\frac{ m^4}{k^4}\\right)+i T \\left(-\\frac{3 }{32}\\frac{ m}{k}+\\frac{3 }{4}\\frac{ m^3}{k^3}-\\frac{3 }{2}\\frac{ m^5}{k^5}\\right)\\right) $ $\\tilde{T}_{1,5;3\\text{D}}^{\\text{f,nt}} & = k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{4 i }{3 \\pi } m^3\\right)+k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{8 i }{3 \\pi } m^3\\right)+\\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(\\frac{4 }{3} k^2 m^3-\\frac{32 }{5} m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{2 }{\\pi } m^2\\right)+(k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi } \\left(k^2 m^2-4 m^4\\right)\\right) $ Fermions, spin 1 x 5, dimension 4: $\\tilde{T}_{1,5;4\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(-\\frac{247}{2100}+\\frac{L_0}{20}\\right)-\\frac{3 }{5}\\frac{ m^2}{k^2}+\\frac{116 }{15}\\frac{ m^4}{k^4}-16 \\frac{ m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{7 \\pi ^2} \\left(\\frac{1}{10}\\frac{1}{k}+\\frac{1}{5}\\frac{ m^2}{k^3}-\\frac{32 }{5}\\frac{ m^4}{k^5}+16 \\frac{ m^6}{k^7}\\right)\\right) $ $\\tilde{T}_{1,5;4\\text{D}}^{\\text{f,nt}} & = k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_2}{2 \\pi ^2} m^4\\right)+k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{\\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(- L_2 k^2 m^4+4 L_3 m^6\\right)\\right) $ Fermions, spin 1 x 5, dimension 5: $\\tilde{T}_{1,5;5\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{16 \\pi ^2} \\left(\\frac{3 }{64} m-\\frac{3 }{16}\\frac{ m^3}{k^2}+\\frac{31 }{12}\\frac{ m^5}{k^4}-5 \\frac{ m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(\\frac{3 }{256} k-\\frac{1}{16}\\frac{ m^2}{k}-\\frac{3 }{8}\\frac{ m^4}{k^3}+3 \\frac{ m^6}{k^5}-5 \\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{1,5;5\\text{D}}^{\\text{f,nt}} & = k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{4 i }{15 \\pi ^2} m^5\\right)+k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{8 i }{15 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{4 }{3} k^2 m^5+\\frac{32 }{7} m^7\\right)\\right) $ Fermions, spin 1 x 5, dimension 6: $\\tilde{T}_{1,5;6\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{7 \\pi ^3} \\left( \\left(-\\frac{1937}{226800}+\\frac{L_0}{360}\\right) k^2+ \\left(\\frac{811}{12600}-\\frac{L_0}{40}\\right) m^2+\\frac{2 }{15}\\frac{ m^4}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{152 }{135}\\frac{ m^6}{k^4}+\\frac{16 }{9}\\frac{ m^8}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{3 \\pi ^3} \\left(\\frac{1}{420} k-\\frac{1}{60}\\frac{ m^2}{k}-\\frac{1}{35}\\frac{ m^4}{k^3}+\\frac{44 }{105}\\frac{ m^6}{k^5}-\\frac{16 }{21}\\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{1,5;6\\text{D}}^{\\text{f,nt}} & = k_{\\mu } k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_3}{12 \\pi ^3} m^6\\right)+k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{6 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^3} \\left(\\frac{ L_3}{3} k^2 m^6- L_4 m^8\\right)\\right) $ Fermions, spin 2 x 2, dimension 3: $\\tilde{T}_{2,2;3\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(-\\frac{1}{4}\\frac{ m}{k^2}+\\frac{m^3}{k^4}\\right)+ T \\left(-\\frac{1}{32}\\frac{1}{k}+\\frac{1}{2}\\frac{ m^4}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi } \\left(\\frac{1}{4}\\frac{ m}{k^2}+\\frac{m^3}{k^4}\\right)+ T \\left(\\frac{1}{64}\\frac{1}{k}-\\frac{1}{8}\\frac{ m^2}{k^3}+\\frac{1}{4}\\frac{ m^4}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{1}{2 \\pi }\\frac{ m^2}{k^2}+i T \\left(\\frac{1}{8}\\frac{ m}{k}-\\frac{1}{2}\\frac{ m^3}{k^3}\\right)\\right) $ $\\tilde{T}_{2,2;3\\text{D}}^{\\text{f,nt}} & = \\left(\\eta _{\\mu \\nu }^2+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{2 i }{3 \\pi } m^3\\right)+(k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{1}{\\pi } m^2\\right) $ Fermions, spin 2 x 2, dimension 4: $\\tilde{T}_{2,2;4\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{3}{50}-\\frac{L_0}{40}\\right)+ \\left(-\\frac{19}{180}+\\frac{L_0}{12}\\right)\\frac{ m^2}{k^2}-\\frac{8 }{15}\\frac{ m^4}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{5 \\pi ^2} \\left(-\\frac{1}{4}\\frac{1}{k}+\\frac{1}{3}\\frac{ m^2}{k^3}+\\frac{8 }{3}\\frac{ m^4}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left( \\left(-\\frac{23}{300}+\\frac{L_0}{40}\\right)+ \\left(\\frac{43}{60}-\\frac{L_0}{4}\\right)\\frac{ m^2}{k^2}-\\frac{4 }{5}\\frac{ m^4}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{15 \\pi ^2} \\left(\\frac{1}{4}\\frac{1}{k}-2 \\frac{ m^2}{k^3}+4 \\frac{ m^4}{k^5}\\right)\\right) $ $\\tilde{T}_{2,2;4\\text{D}}^{\\text{f,nt}} & = \\left(\\eta _{\\mu \\nu }^2+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_2}{4 \\pi ^2} m^4\\right) $ Fermions, spin 2 x 2, dimension 5: $\\tilde{T}_{2,2;5\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{12 \\pi ^2} \\left(-\\frac{1}{8} m+\\frac{7 }{12}\\frac{ m^3}{k^2}-\\frac{ m^5}{k^4}\\right)+\\frac{ T}{4 \\pi } \\left(-\\frac{1}{96} k+\\frac{1}{16}\\frac{ m^2}{k}-\\frac{1}{3}\\frac{ m^6}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^2} \\left(\\frac{1}{32} m-\\frac{1}{3}\\frac{ m^3}{k^2}-\\frac{1}{2}\\frac{ m^5}{k^4}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{8 \\pi } \\left(\\frac{1}{192} k-\\frac{1}{16}\\frac{ m^2}{k}+\\frac{1}{4}\\frac{ m^4}{k^3}-\\frac{1}{3}\\frac{ m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{2,2;5\\text{D}}^{\\text{f,nt}} & = \\left(\\eta _{\\mu \\nu }^2+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{2 i }{15 \\pi ^2} m^5\\right) $ Fermions, spin 2 x 2, dimension 6: $\\tilde{T}_{2,2;6\\text{D}}^{\\text{f,t}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{31}{7056}-\\frac{L_0}{672}\\right) k^2+ \\left(-\\frac{277}{8400}+\\frac{L_0}{80}\\right) m^2+ \\left(\\frac{347}{10080}-\\frac{L_0}{48}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{8 }{105}\\frac{ m^6}{k^4}\\right)+\\frac{i S}{21 \\pi ^3} \\left(-\\frac{1}{16} k+\\frac{2 }{5}\\frac{ m^2}{k}-\\frac{1}{5}\\frac{ m^4}{k^3}-\\frac{8 }{5}\\frac{ m^6}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^3} \\left( \\left(-\\frac{11}{3675}+\\frac{L_0}{1120}\\right) k^2+ \\left(\\frac{337}{8400}-\\frac{L_0}{80}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{127}{672}+\\frac{L_0}{16}\\right)\\frac{ m^4}{k^2}+\\frac{4 }{35}\\frac{ m^6}{k^4}\\right)+\\frac{i S}{35 \\pi ^3} \\left(\\frac{1}{48} k-\\frac{1}{4}\\frac{ m^2}{k}+\\frac{m^4}{k^3}-\\frac{4 }{3}\\frac{ m^6}{k^5}\\right)\\right) $ $\\tilde{T}_{2,2;6\\text{D}}^{\\text{f,nt}} & = \\left(\\eta _{\\mu \\nu }^2+\\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{24 \\pi ^3} m^6\\right) $ Fermions, spin 2 x 4, dimension 3: $\\tilde{T}_{2,4;3\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi } \\left(\\frac{1}{4}\\frac{ m}{k^2}-\\frac{7 }{3}\\frac{ m^3}{k^4}+8 \\frac{ m^5}{k^6}\\right)+ T \\left(\\frac{1}{64}\\frac{1}{k}-\\frac{3 }{4}\\frac{ m^4}{k^5}+2 \\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(-\\frac{1}{32}\\frac{ m}{k^2}-\\frac{1}{3}\\frac{ m^3}{k^4}+\\frac{1}{2}\\frac{ m^5}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{1}{128}\\frac{1}{k}+\\frac{3 }{32}\\frac{ m^2}{k^3}-\\frac{3 }{8}\\frac{ m^4}{k^5}+\\frac{1}{2}\\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{1}{2 \\pi } \\left(-\\frac{5 }{4}\\frac{ m^2}{k^2}+3 \\frac{ m^4}{k^4}\\right)+i T \\left(-\\frac{3 }{32}\\frac{ m}{k}+\\frac{3 }{4}\\frac{ m^3}{k^3}-\\frac{3 }{2}\\frac{ m^5}{k^5}\\right)\\right) $ $\\tilde{T}_{2,4;3\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{4 i }{3 \\pi } m^3\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{2 i }{3 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(\\frac{2 }{3} k^2 m^3-\\frac{8 }{5} m^5\\right)\\right)+\\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(\\frac{2 }{3} k^2 m^3-\\frac{24 }{5} m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{1}{\\pi } m^2\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{1}{\\pi } \\left(k^2 m^2-4 m^4\\right)\\right) $ Fermions, spin 2 x 4, dimension 4: $\\tilde{T}_{2,4;4\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{141}{980}+\\frac{3 L_0}{56}\\right)+ \\left(\\frac{157}{420}-\\frac{L_0}{4}\\right)\\frac{ m^2}{k^2}+\\frac{76 }{21}\\frac{ m^4}{k^4}-\\frac{64 }{7}\\frac{ m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{7 \\pi ^2} \\left(\\frac{3 }{20}\\frac{1}{k}-\\frac{2 }{5}\\frac{ m^2}{k^3}-4 \\frac{ m^4}{k^5}+\\frac{64 }{5}\\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{44}{3675}-\\frac{L_0}{280}\\right)+ \\left(-\\frac{337}{2100}+\\frac{L_0}{20}\\right)\\frac{ m^2}{k^2}+\\frac{8 }{21}\\frac{ m^4}{k^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{16 }{35}\\frac{ m^6}{k^6}\\right)+\\frac{i S}{35 \\pi ^2} \\left(-\\frac{1}{4}\\frac{1}{k}+3 \\frac{ m^2}{k^3}-12 \\frac{ m^4}{k^5}+16 \\frac{ m^6}{k^7}\\right)\\right) $ $\\tilde{T}_{2,4;4\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_2}{2 \\pi ^2} m^4\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(-\\frac{ L_2}{2} k^2 m^4+ L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left(-\\frac{ L_2}{2} k^2 m^4+3 L_3 m^6\\right)\\right) $ Fermions, spin 2 x 4, dimension 5: $\\tilde{T}_{2,4;5\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{1}{64} m-\\frac{5 }{48}\\frac{ m^3}{k^2}+\\frac{5 }{12}\\frac{ m^5}{k^4}-\\frac{ m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{4 \\pi } \\left(\\frac{1}{256} k-\\frac{1}{32}\\frac{ m^2}{k}+\\frac{1}{2}\\frac{ m^6}{k^5}-\\frac{ m^8}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{16 \\pi ^2} \\left(-\\frac{1}{64} m+\\frac{11 }{48}\\frac{ m^3}{k^2}+\\frac{11 }{12}\\frac{ m^5}{k^4}-\\frac{ m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(-\\frac{1}{256} k+\\frac{1}{16}\\frac{ m^2}{k}-\\frac{3 }{8}\\frac{ m^4}{k^3}+\\frac{m^6}{k^5}-\\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{2,4;5\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{4 i }{15 \\pi ^2} m^5\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{2 i }{15 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{2 }{3} k^2 m^5+\\frac{8 }{7} m^7\\right)\\right)+\\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{2 }{3} k^2 m^5+\\frac{24 }{7} m^7\\right)\\right) $ Fermions, spin 2 x 4, dimension 6: $\\tilde{T}_{2,4;6\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{25}{15876}+\\frac{L_0}{2016}\\right) k^2+ \\left(\\frac{2713}{176400}-\\frac{3 L_0}{560}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{379}{16800}+\\frac{L_0}{80}\\right)\\frac{ m^4}{k^2}-\\frac{20 }{189}\\frac{ m^6}{k^4}+\\frac{64 }{315}\\frac{ m^8}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{3 \\pi ^3} \\left(\\frac{1}{336} k-\\frac{11 }{420}\\frac{ m^2}{k}+\\frac{1}{35}\\frac{ m^4}{k^3}+\\frac{4 }{15}\\frac{ m^6}{k^5}-\\frac{64 }{105}\\frac{ m^8}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(\\frac{563}{317520}-\\frac{L_0}{2016}\\right) k^2+ \\left(-\\frac{1091}{35280}+\\frac{L_0}{112}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{233}{1120}-\\frac{L_0}{16}\\right)\\frac{ m^4}{k^2}-\\frac{52 }{189}\\frac{ m^6}{k^4}+\\frac{16 }{63}\\frac{ m^8}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{105 \\pi ^3} \\left(-\\frac{1}{48} k+\\frac{1}{3}\\frac{ m^2}{k}-2 \\frac{ m^4}{k^3}+\\frac{16 }{3}\\frac{ m^6}{k^5}-\\frac{16 }{3}\\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{2,4;6\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{12 \\pi ^3} m^6\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{i L_3}{24 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{8 \\pi ^3} \\left(\\frac{ L_3}{3} k^2 m^6-\\frac{ L_4}{2} m^8\\right)\\right)+\\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{8 \\pi ^3} \\left(\\frac{ L_3}{3} k^2 m^6-\\frac{3 L_4}{2} m^8\\right)\\right) $ Fermions, spin 3 x 3, dimension 3: $\\tilde{T}_{3,3;3\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{2 \\pi } \\left(\\frac{1}{8}\\frac{ m}{k^2}-\\frac{1}{3}\\frac{ m^3}{k^4}+2 \\frac{ m^5}{k^6}\\right)+ T \\left(\\frac{1}{64}\\frac{1}{k}-\\frac{1}{16}\\frac{ m^2}{k^3}-\\frac{1}{4}\\frac{ m^4}{k^5}+\\frac{m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(-\\frac{1}{16}\\frac{ m}{k^2}-\\frac{3 }{2}\\frac{ m^3}{k^4}+3 \\frac{ m^5}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{1}{128}\\frac{1}{k}+\\frac{5 }{32}\\frac{ m^2}{k^3}-\\frac{7 }{8}\\frac{ m^4}{k^5}+\\frac{3 }{2}\\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^2 \\left(\\frac{1}{4 \\pi } \\left(-3 \\frac{ m^2}{k^2}+4 \\frac{ m^4}{k^4}\\right)+i T \\left(-\\frac{1}{16}\\frac{ m}{k}+\\frac{1}{2}\\frac{ m^3}{k^3}-\\frac{ m^5}{k^5}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{1}{2 \\pi } \\left(\\frac{1}{4}\\frac{ m^2}{k^2}+\\frac{m^4}{k^4}\\right)+i T \\left(-\\frac{1}{32}\\frac{ m}{k}+\\frac{1}{4}\\frac{ m^3}{k^3}-\\frac{1}{2}\\frac{ m^5}{k^5}\\right)\\right) $ $\\tilde{T}_{3,3;3\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{4 i }{3 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi } \\left(4 k^2 m^3-\\frac{64 }{5} m^5\\right)\\right)+\\eta _{\\mu \\nu }^3 \\left(-\\frac{32 i }{15 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{2 }{\\pi } m^2\\right)+(k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{4 }{3 \\pi } m^4\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{1}{3 \\pi } \\left(3 k^2 m^2-8 m^4\\right)\\right) $ Fermions, spin 3 x 3, dimension 4: $\\tilde{T}_{3,3;4\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{599}{4410}+\\frac{L_0}{21}\\right)+ \\left(\\frac{247}{315}-\\frac{L_0}{3}\\right)\\frac{ m^2}{k^2}+\\frac{8 }{7}\\frac{ m^4}{k^4}-\\frac{32 }{7}\\frac{ m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{7 \\pi ^2} \\left(\\frac{2 }{15}\\frac{1}{k}-\\frac{2 }{3}\\frac{ m^2}{k^3}-\\frac{16 }{15}\\frac{ m^4}{k^5}+\\frac{32 }{5}\\frac{ m^6}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{457}{8820}-\\frac{L_0}{84}\\right)+ \\left(-\\frac{382}{315}+\\frac{L_0}{3}\\right)\\frac{ m^2}{k^2}+\\frac{92 }{21}\\frac{ m^4}{k^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{48 }{7}\\frac{ m^6}{k^6}\\right)+\\frac{i S}{7 \\pi ^2} \\left(-\\frac{1}{30}\\frac{1}{k}+\\frac{13 }{15}\\frac{ m^2}{k^3}-\\frac{16 }{3}\\frac{ m^4}{k^5}+\\frac{48 }{5}\\frac{ m^6}{k^7}\\right)\\right) $ $\\tilde{T}_{3,3;4\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_2}{2 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{ L_2}{2} k^2 m^4+\\frac{4 L_3}{3} m^6\\right)\\right)+\\eta _{\\mu \\nu }^3 \\left(\\frac{2 i L_3}{3 \\pi ^2} m^6\\right) $ Fermions, spin 3 x 3, dimension 5: $\\tilde{T}_{3,3;5\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{8 \\pi ^2} \\left(\\frac{5 }{192} m-\\frac{31 }{144}\\frac{ m^3}{k^2}+\\frac{1}{4}\\frac{ m^5}{k^4}-\\frac{ m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{8 \\pi } \\left(\\frac{5 }{768} k-\\frac{1}{16}\\frac{ m^2}{k}+\\frac{1}{8}\\frac{ m^4}{k^3}+\\frac{1}{3}\\frac{ m^6}{k^5}-\\frac{ m^8}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{16 \\pi ^2} \\left(-\\frac{1}{192} m+\\frac{35 }{144}\\frac{ m^3}{k^2}+\\frac{25 }{12}\\frac{ m^5}{k^4}-3 \\frac{ m^7}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{16 \\pi } \\left(-\\frac{1}{768} k+\\frac{1}{16}\\frac{ m^2}{k}-\\frac{5 }{8}\\frac{ m^4}{k^3}+\\frac{7 }{3}\\frac{ m^6}{k^5}-3 \\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{3,3;5\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{4 i }{15 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{15 \\pi ^2} \\left(-4 k^2 m^5+\\frac{64 }{7} m^7\\right)\\right)+\\eta _{\\mu \\nu }^3 \\left(\\frac{32 i }{105 \\pi ^2} m^7\\right) $ Fermions, spin 3 x 3, dimension 6: $\\tilde{T}_{3,3;6\\text{D}}^{\\text{f,t}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{15 \\pi ^3} \\left( \\left(-\\frac{1021}{52920}+\\frac{L_0}{168}\\right) k^2+ \\left(\\frac{317}{1470}-\\frac{L_0}{14}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{77}{120}+\\frac{L_0}{4}\\right)\\frac{ m^4}{k^2}-\\frac{32 }{63}\\frac{ m^6}{k^4}+\\frac{32 }{21}\\frac{ m^8}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{21 \\pi ^3} \\left(\\frac{1}{60} k-\\frac{1}{6}\\frac{ m^2}{k}+\\frac{2 }{5}\\frac{ m^4}{k^3}+\\frac{8 }{15}\\frac{ m^6}{k^5}-\\frac{32 }{15}\\frac{ m^8}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^3} \\left(\\frac{1}{5040} k^2+ \\left(-\\frac{457}{29400}+\\frac{L_0}{280}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{779}{4200}-\\frac{L_0}{20}\\right)\\frac{ m^4}{k^2}-\\frac{8 }{21}\\frac{ m^6}{k^4}+\\frac{16 }{35}\\frac{ m^8}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^3} \\left(\\frac{1}{12}\\frac{ m^2}{k}-\\frac{ m^4}{k^3}+4 \\frac{ m^6}{k^5}-\\frac{16 }{3}\\frac{ m^8}{k^7}\\right)\\right) $ $\\tilde{T}_{3,3;6\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{12 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{6 \\pi ^3} \\left(\\frac{ L_3}{2} k^2 m^6- L_4 m^8\\right)\\right)+\\eta _{\\mu \\nu }^3 \\left(-\\frac{i L_4}{12 \\pi ^3} m^8\\right) $ Fermions, spin 3 x 5, dimension 3: $\\tilde{T}_{3,5;3\\text{D}}^{\\text{f,t}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi } \\left(-\\frac{3 }{16}\\frac{ m}{k^2}+\\frac{3 }{4}\\frac{ m^3}{k^4}-\\frac{31 }{3}\\frac{ m^5}{k^6}+20 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{3 }{256}\\frac{1}{k}+\\frac{1}{16}\\frac{ m^2}{k^3}+\\frac{3 }{8}\\frac{ m^4}{k^5}-3 \\frac{ m^6}{k^7}+5 \\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi } \\left(\\frac{7 }{64}\\frac{ m}{k^2}+\\frac{155 }{48}\\frac{ m^3}{k^4}-\\frac{47 }{4}\\frac{ m^5}{k^6}+15 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{7 }{1024}\\frac{1}{k}-\\frac{9 }{64}\\frac{ m^2}{k^3}+\\frac{33 }{32}\\frac{ m^4}{k^5}-\\frac{13 }{4}\\frac{ m^6}{k^7}+\\frac{15 }{4}\\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi } \\left(\\frac{3 }{4}\\frac{ m^2}{k^2}-\\frac{8 }{3}\\frac{ m^4}{k^4}+4 \\frac{ m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ i T \\left(\\frac{1}{16}\\frac{ m}{k}-\\frac{3 }{4}\\frac{ m^3}{k^3}+3 \\frac{ m^5}{k^5}-4 \\frac{ m^7}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi } \\left(-\\frac{1}{16}\\frac{ m^2}{k^2}-\\frac{2 }{3}\\frac{ m^4}{k^4}+\\frac{m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ i T \\left(\\frac{1}{64}\\frac{ m}{k}-\\frac{3 }{16}\\frac{ m^3}{k^3}+\\frac{3 }{4}\\frac{ m^5}{k^5}-\\frac{ m^7}{k^7}\\right)\\right) $ $\\tilde{T}_{3,5;3\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{8 i }{3 \\pi } m^3\\right)+ \\left(k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{4 i }{3 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{\\pi } \\left(\\frac{4 }{3} k^2 m^3-\\frac{32 }{5} m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(\\frac{8 }{3} k^2 m^3-\\frac{64 }{5} m^5\\right)\\right)+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{64 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(-\\frac{64 i }{15 \\pi } m^5\\right)+\\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi } \\left(-\\frac{4 }{3} k^4 m^3+\\frac{32 }{5} k^2 m^5-\\frac{64 }{5} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi } \\left(\\frac{64 }{3} k^2 m^5-\\frac{512 }{7} m^7\\right)\\right)+k_{\\mu }^2 k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{1}{\\pi } m^2\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{2 }{\\pi } m^2\\right)+k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{8 }{3 \\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{4 }{3 \\pi } m^4\\right)+k_{\\mu } k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{1}{3 \\pi } \\left(6 k^2 m^2-32 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{1}{3 \\pi } \\left(3 k^2 m^2-16 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi } \\left(\\frac{4 }{3} k^2 m^4-\\frac{16 }{5} m^6\\right)\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{1}{\\pi } \\left(- k^4 m^2+\\frac{16 }{3} k^2 m^4-\\frac{64 }{5} m^6\\right)\\right) $ Fermions, spin 3 x 5, dimension 4: $\\tilde{T}_{3,5;4\\text{D}}^{\\text{f,t}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(\\frac{1937}{14175}-\\frac{2 L_0}{45}\\right)+ \\left(-\\frac{1622}{1575}+\\frac{2 L_0}{5}\\right)\\frac{ m^2}{k^2}-\\frac{32 }{15}\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{2432 }{135}\\frac{ m^6}{k^6}-\\frac{256 }{9}\\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{3 \\pi ^2} \\left(-\\frac{4 }{105}\\frac{1}{k}+\\frac{4 }{15}\\frac{ m^2}{k^3}+\\frac{16 }{35}\\frac{ m^4}{k^5}-\\frac{704 }{105}\\frac{ m^6}{k^7}+\\frac{256 }{21}\\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{1231}{132300}+\\frac{L_0}{420}\\right)+ \\left(\\frac{258}{1225}-\\frac{2 L_0}{35}\\right)\\frac{ m^2}{k^2}-\\frac{104 }{105}\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{128 }{45}\\frac{ m^6}{k^6}-\\frac{64 }{21}\\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{\\pi ^2} \\left(\\frac{1}{210}\\frac{1}{k}-\\frac{11 }{105}\\frac{ m^2}{k^3}+\\frac{4 }{5}\\frac{ m^4}{k^5}-\\frac{272 }{105}\\frac{ m^6}{k^7}+\\frac{64 }{21}\\frac{ m^8}{k^9}\\right)\\right) $ $\\tilde{T}_{3,5;4\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_2}{\\pi ^2} m^4\\right)+ \\left(k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i L_2}{2 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{2 \\pi ^2} \\left(- L_2 k^2 m^4+4 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(- L_2 k^2 m^4+4 L_3 m^6\\right)\\right)+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{4 i L_3}{\\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{4 i L_3}{3 \\pi ^2} m^6\\right)+\\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left( L_2 k^4 m^4-4 L_3 k^2 m^6+7 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left(-4 L_3 k^2 m^6+12 L_4 m^8\\right)\\right) $ Fermions, spin 3 x 5, dimension 5: $\\tilde{T}_{3,5;5\\text{D}}^{\\text{f,t}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{1}{512} m+\\frac{1}{48}\\frac{ m^3}{k^2}-\\frac{3 }{80}\\frac{ m^5}{k^4}+\\frac{1}{3}\\frac{ m^7}{k^6}-\\frac{1}{2}\\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{2 \\pi } \\left(-\\frac{1}{1024} k+\\frac{3 }{256}\\frac{ m^2}{k}-\\frac{1}{32}\\frac{ m^4}{k^3}-\\frac{1}{8}\\frac{ m^6}{k^5}+\\frac{3 }{4}\\frac{ m^8}{k^7}-\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{8 \\pi ^2} \\left(\\frac{1}{256} m-\\frac{5 }{48}\\frac{ m^3}{k^2}-\\frac{137 }{120}\\frac{ m^5}{k^4}+3 \\frac{ m^7}{k^6}-3 \\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{8 \\pi } \\left(\\frac{1}{1024} k-\\frac{7 }{256}\\frac{ m^2}{k}+\\frac{9 }{32}\\frac{ m^4}{k^3}-\\frac{11 }{8}\\frac{ m^6}{k^5}+\\frac{13 }{4}\\frac{ m^8}{k^7}-3 \\frac{ m^{10}}{k^9}\\right)\\right) $ $\\tilde{T}_{3,5;5\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{8 i }{15 \\pi ^2} m^5\\right)+ \\left(k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{4 i }{15 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{4 }{3} k^2 m^5+\\frac{32 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{8 }{3} k^2 m^5+\\frac{64 }{7} m^7\\right)\\right)+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{64 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{64 i }{105 \\pi ^2} m^7\\right)+\\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left(\\frac{4 }{3} k^4 m^5-\\frac{32 }{7} k^2 m^7+\\frac{64 }{9} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{105 \\pi ^2} \\left(-64 k^2 m^7+\\frac{512 }{3} m^9\\right)\\right) $ Fermions, spin 3 x 5, dimension 6: $\\tilde{T}_{3,5;6\\text{D}}^{\\text{f,t}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{7 \\pi ^3} \\left( \\left(\\frac{11861}{2286900}-\\frac{L_0}{660}\\right) k^2+ \\left(-\\frac{11126}{155925}+\\frac{L_0}{45}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{19067}{69300}-\\frac{L_0}{10}\\right)\\frac{ m^4}{k^2}+\\frac{32 }{99}\\frac{ m^6}{k^4}-\\frac{3008 }{1485}\\frac{ m^8}{k^6}+\\frac{256 }{99}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{33 \\pi ^3} \\left(-\\frac{1}{70} k+\\frac{19 }{105}\\frac{ m^2}{k}-\\frac{64 }{105}\\frac{ m^4}{k^3}-\\frac{32 }{35}\\frac{ m^6}{k^5}+\\frac{128 }{15}\\frac{ m^8}{k^7}-\\frac{256 }{21}\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{7 \\pi ^3} \\left( \\left(-\\frac{29497}{27442800}+\\frac{L_0}{3960}\\right) k^2+ \\left(\\frac{13751}{415800}-\\frac{L_0}{120}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{8689}{23100}+\\frac{L_0}{10}\\right)\\frac{ m^4}{k^2}+\\frac{1504 }{1485}\\frac{ m^6}{k^4}-\\frac{368 }{165}\\frac{ m^8}{k^6}+\\frac{64 }{33}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{33 \\pi ^3} \\left(\\frac{1}{420} k-\\frac{31 }{420}\\frac{ m^2}{k}+\\frac{4 }{5}\\frac{ m^4}{k^3}-\\frac{424 }{105}\\frac{ m^6}{k^5}+\\frac{1024 }{105}\\frac{ m^8}{k^7}-\\frac{64 }{7}\\frac{ m^{10}}{k^9}\\right)\\right) $ $\\tilde{T}_{3,5;6\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{6 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{i L_3}{12 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{4 \\pi ^3} \\left(\\frac{ L_3}{3} k^2 m^6- L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^3} \\left(\\frac{ L_3}{3} k^2 m^6- L_4 m^8\\right)\\right)+k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i L_4}{2 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(-\\frac{i L_4}{6 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^3} \\left(-\\frac{ L_3}{3} k^4 m^6+ L_4 k^2 m^8-\\frac{7 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left(\\frac{ L_4}{6} k^2 m^8-\\frac{2 L_5}{5} m^{10}\\right)\\right) $ Fermions, spin 4 x 4, dimension 3: $\\tilde{T}_{4,4;3\\text{D}}^{\\text{f,t}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{2 \\pi } \\left(-\\frac{1}{16}\\frac{ m}{k^2}-\\frac{11 }{12}\\frac{ m^3}{k^4}-\\frac{5 }{3}\\frac{ m^5}{k^6}+4 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{1}{128}\\frac{1}{k}+\\frac{1}{16}\\frac{ m^2}{k^3}-\\frac{ m^6}{k^7}+2 \\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{8 \\pi } \\left(13 \\frac{ m^3}{k^4}-32 \\frac{ m^5}{k^6}+48 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{3 }{32}\\frac{ m^2}{k^3}+\\frac{9 }{8}\\frac{ m^4}{k^5}-\\frac{9 }{2}\\frac{ m^6}{k^7}+6 \\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi } \\left(\\frac{3 }{64}\\frac{ m}{k^2}-\\frac{11 }{16}\\frac{ m^3}{k^4}-\\frac{11 }{4}\\frac{ m^5}{k^6}+3 \\frac{ m^7}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{3 }{1024}\\frac{1}{k}-\\frac{3 }{64}\\frac{ m^2}{k^3}+\\frac{9 }{32}\\frac{ m^4}{k^5}-\\frac{3 }{4}\\frac{ m^6}{k^7}+\\frac{3 }{4}\\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^3 \\left(\\frac{1}{\\pi } \\left(\\frac{7 }{8}\\frac{ m^2}{k^2}-\\frac{4 }{3}\\frac{ m^4}{k^4}+2 \\frac{ m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ i T \\left(\\frac{1}{32}\\frac{ m}{k}-\\frac{3 }{8}\\frac{ m^3}{k^3}+\\frac{3 }{2}\\frac{ m^5}{k^5}-2 \\frac{ m^7}{k^7}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{1}{16 \\pi } \\left(-3 \\frac{ m^2}{k^2}-32 \\frac{ m^4}{k^4}+48 \\frac{ m^6}{k^6}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ i T \\left(\\frac{3 }{64}\\frac{ m}{k}-\\frac{9 }{16}\\frac{ m^3}{k^3}+\\frac{9 }{4}\\frac{ m^5}{k^5}-3 \\frac{ m^7}{k^7}\\right)\\right) $ $\\tilde{T}_{4,4;3\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{\\pi } m^3\\right)+ \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{4 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{2 i }{\\pi } m^3\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{16 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(4 k^2 m^3-16 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{\\pi } \\left(2 k^2 m^3-8 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi } \\left(-8 k^2 m^3-\\frac{64 }{5} m^5\\right)\\right)+\\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi } \\left(8 k^2 m^5-\\frac{96 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi } \\left(-10 k^4 m^3+40 k^2 m^5-96 m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi } \\left(\\frac{2 }{3} k^4 m^3+\\frac{16 }{15} k^2 m^5-\\frac{192 }{35} m^7\\right)\\right)+k_{\\mu }^2 k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{3 }{\\pi } m^2\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu } k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }+k_{\\mu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{4 }{\\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{1}{\\pi } \\left(3 k^2 m^2-8 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{1}{5 \\pi } \\left(20 k^2 m^4-48 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{1}{\\pi } \\left(- k^4 m^2+\\frac{8 }{3} k^2 m^4-\\frac{32 }{5} m^6\\right)\\right) $ Fermions, spin 4 x 4, dimension 4: $\\tilde{T}_{4,4;4\\text{D}}^{\\text{f,t}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{50}{3969}-\\frac{L_0}{252}\\right)+ \\left(-\\frac{2713}{22050}+\\frac{3 L_0}{70}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{817}{4200}+\\frac{3 L_0}{20}\\right)\\frac{ m^4}{k^4}+\\frac{160 }{189}\\frac{ m^6}{k^6}-\\frac{512 }{315}\\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{3 \\pi ^2} \\left(-\\frac{1}{42}\\frac{1}{k}+\\frac{22 }{105}\\frac{ m^2}{k^3}-\\frac{8 }{35}\\frac{ m^4}{k^5}-\\frac{32 }{15}\\frac{ m^6}{k^7}+\\frac{512 }{105}\\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{62}{6615}-\\frac{L_0}{168}\\right)+ \\left(\\frac{1651}{2940}-\\frac{3 L_0}{28}\\right)\\frac{ m^2}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{421}{140}+\\frac{3 L_0}{2}\\right)\\frac{ m^4}{k^4}+\\frac{176 }{9}\\frac{ m^6}{k^6}-\\frac{512 }{21}\\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{\\pi ^2} \\left(-\\frac{1}{420}\\frac{1}{k}-\\frac{1}{21}\\frac{ m^2}{k^3}+\\frac{4 }{5}\\frac{ m^4}{k^5}-\\frac{368 }{105}\\frac{ m^6}{k^7}+\\frac{512 }{105}\\frac{ m^8}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{563}{26460}+\\frac{L_0}{168}\\right)+ \\left(\\frac{1091}{2940}-\\frac{3 L_0}{28}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{699}{280}+\\frac{3 L_0}{4}\\right)\\frac{ m^4}{k^4}+\\frac{208 }{63}\\frac{ m^6}{k^6}-\\frac{64 }{21}\\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{35 \\pi ^2} \\left(\\frac{1}{12}\\frac{1}{k}-\\frac{4 }{3}\\frac{ m^2}{k^3}+8 \\frac{ m^4}{k^5}-\\frac{64 }{3}\\frac{ m^6}{k^7}+\\frac{64 }{3}\\frac{ m^8}{k^9}\\right)\\right) $ $\\tilde{T}_{4,4;4\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+ \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{3 i L_2}{2 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(-\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i L_3}{\\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left(-3 L_2 k^2 m^4+10 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{2 \\pi ^2} \\left(-\\frac{3 L_2}{2} k^2 m^4+5 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi ^2} \\left(3 L_2 k^2 m^4+4 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(- L_3 k^2 m^6+\\frac{3 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{3 L_2}{2} k^4 m^4-5 L_3 k^2 m^6+\\frac{21 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{ L_2}{4} k^4 m^4-\\frac{ L_3}{3} k^2 m^6+\\frac{3 L_4}{2} m^8\\right)\\right) $ Fermions, spin 4 x 4, dimension 5: $\\tilde{T}_{4,4;5\\text{D}}^{\\text{f,t}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{3 }{512} m+\\frac{9 }{128}\\frac{ m^3}{k^2}+\\frac{211 }{480}\\frac{ m^5}{k^4}+\\frac{13 }{24}\\frac{ m^7}{k^6}-\\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{\\pi } \\left(-\\frac{3 }{10240} k+\\frac{1}{256}\\frac{ m^2}{k}-\\frac{1}{64}\\frac{ m^4}{k^3}+\\frac{1}{8}\\frac{ m^8}{k^7}-\\frac{1}{5}\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{3 }{1024} m-\\frac{1}{256}\\frac{ m^3}{k^2}-\\frac{493 }{320}\\frac{ m^5}{k^4}+\\frac{41 }{16}\\frac{ m^7}{k^6}-3 \\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{\\pi } \\left(-\\frac{3 }{20480} k+\\frac{3 }{128}\\frac{ m^4}{k^3}-\\frac{3 }{16}\\frac{ m^6}{k^5}+\\frac{9 }{16}\\frac{ m^8}{k^7}-\\frac{3 }{5}\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left(\\frac{3 }{2048} m-\\frac{7 }{256}\\frac{ m^3}{k^2}+\\frac{1}{5}\\frac{ m^5}{k^4}+\\frac{7 }{16}\\frac{ m^7}{k^6}-\\frac{3 }{8}\\frac{ m^9}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{8 \\pi } \\left(\\frac{3 }{5120} k-\\frac{3 }{256}\\frac{ m^2}{k}+\\frac{3 }{32}\\frac{ m^4}{k^3}-\\frac{3 }{8}\\frac{ m^6}{k^5}+\\frac{3 }{4}\\frac{ m^8}{k^7}-\\frac{3 }{5}\\frac{ m^{10}}{k^9}\\right)\\right) $ $\\tilde{T}_{4,4;5\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{2 i }{5 \\pi ^2} m^5\\right)+ \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{4 i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(-\\frac{2 i }{5 \\pi ^2} m^5\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{16 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{4 }{5} k^2 m^5+\\frac{16 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{2 }{5} k^2 m^5+\\frac{8 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{15 \\pi ^2} \\left(8 k^2 m^5+\\frac{64 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{35 \\pi ^2} \\left(-8 k^2 m^7+\\frac{32 }{3} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(\\frac{2 }{5} k^4 m^5-\\frac{8 }{7} k^2 m^7+\\frac{32 }{15} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{15 \\pi ^2} \\left(-2 k^4 m^5-\\frac{16 }{7} k^2 m^7+\\frac{64 }{7} m^9\\right)\\right) $ Fermions, spin 4 x 4, dimension 6: $\\tilde{T}_{4,4;6\\text{D}}^{\\text{f,t}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{859}{1960200}-\\frac{L_0}{7920}\\right) k^2+ \\left(-\\frac{11441}{1746360}+\\frac{L_0}{504}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{62801}{1940400}-\\frac{3 L_0}{280}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{35171}{831600}-\\frac{L_0}{40}\\right)\\frac{ m^6}{k^4}-\\frac{992 }{10395}\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{512 }{3465}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{55 \\pi ^3} \\left(-\\frac{1}{72} k+\\frac{4 }{21}\\frac{ m^2}{k}-\\frac{52 }{63}\\frac{ m^4}{k^3}+\\frac{32 }{63}\\frac{ m^6}{k^5}+\\frac{32 }{7}\\frac{ m^8}{k^7}-\\frac{512 }{63}\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{5353}{21344400}-\\frac{L_0}{12320}\\right) k^2+ \\left(-\\frac{6401}{5821200}+\\frac{L_0}{1680}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{35867}{1293600}+\\frac{3 L_0}{560}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{967}{15400}+\\frac{L_0}{20}\\right)\\frac{ m^6}{k^4}-\\frac{304 }{693}\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{512 }{1155}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{55 \\pi ^3} \\left(-\\frac{1}{112} k+\\frac{1}{21}\\frac{ m^2}{k}+\\frac{2 }{3}\\frac{ m^4}{k^3}-\\frac{48 }{7}\\frac{ m^6}{k^5}+\\frac{464 }{21}\\frac{ m^8}{k^7}-\\frac{512 }{21}\\frac{ m^{10}}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(-\\frac{1627}{3201660}+\\frac{L_0}{7392}\\right) k^2+ \\left(\\frac{12701}{1164240}-\\frac{L_0}{336}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{24667}{258720}+\\frac{3 L_0}{112}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{23777}{55440}-\\frac{L_0}{8}\\right)\\frac{ m^6}{k^4}-\\frac{256 }{693}\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{64 }{231}\\frac{ m^{10}}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{231 \\pi ^3} \\left(\\frac{1}{80} k-\\frac{1}{4}\\frac{ m^2}{k}+2 \\frac{ m^4}{k^3}-8 \\frac{ m^6}{k^5}+16 \\frac{ m^8}{k^7}-\\frac{64 }{5}\\frac{ m^{10}}{k^9}\\right)\\right) $ $\\tilde{T}_{4,4;6\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{8 \\pi ^3} m^6\\right)+ \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{4 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^2 \\left(\\frac{i L_3}{8 \\pi ^3} m^6\\right)+ \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{i L_4}{8 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^3} \\left( L_3 k^2 m^6-\\frac{5 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{8 \\pi ^3} \\left( L_3 k^2 m^6-\\frac{5 L_4}{2} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{6 \\pi ^3} \\left(- L_3 k^2 m^6- L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{8 \\pi ^3} \\left(\\frac{ L_4}{2} k^2 m^8-\\frac{3 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{8 \\pi ^3} \\left(- L_3 k^4 m^6+\\frac{5 L_4}{2} k^2 m^8-\\frac{21 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{4 \\pi ^3} \\left(\\frac{ L_3}{6} k^4 m^6+\\frac{ L_4}{6} k^2 m^8-\\frac{3 L_5}{5} m^{10}\\right)\\right) $ Fermions, spin 5 x 5, dimension 3: $\\tilde{T}_{5,5;3\\text{D}}^{\\text{f,t}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi } \\left(\\frac{1}{64}\\frac{ m}{k^2}+\\frac{7 }{6}\\frac{ m^3}{k^4}+\\frac{3 }{10}\\frac{ m^5}{k^6}-\\frac{8 }{3}\\frac{ m^7}{k^8}+4 \\frac{ m^9}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{1}{256}\\frac{1}{k}-\\frac{3 }{64}\\frac{ m^2}{k^3}+\\frac{1}{8}\\frac{ m^4}{k^5}+\\frac{1}{2}\\frac{ m^6}{k^7}-3 \\frac{ m^8}{k^9}+4 \\frac{ m^{10}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(\\frac{1}{64}\\frac{ m}{k^2}-\\frac{31 }{12}\\frac{ m^3}{k^4}+\\frac{31 }{6}\\frac{ m^5}{k^6}-\\frac{52 }{3}\\frac{ m^7}{k^8}+20 \\frac{ m^9}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(\\frac{1}{256}\\frac{1}{k}+\\frac{1}{64}\\frac{ m^2}{k^3}-\\frac{7 }{8}\\frac{ m^4}{k^5}+\\frac{13 }{2}\\frac{ m^6}{k^7}-19 \\frac{ m^8}{k^9}+20 \\frac{ m^{10}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(-\\frac{9 }{256}\\frac{ m}{k^2}+\\frac{3 }{4}\\frac{ m^3}{k^4}+\\frac{53 }{8}\\frac{ m^5}{k^6}-16 \\frac{ m^7}{k^8}+15 \\frac{ m^9}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ T \\left(-\\frac{9 }{2048}\\frac{1}{k}+\\frac{51 }{512}\\frac{ m^2}{k^3}-\\frac{57 }{64}\\frac{ m^4}{k^5}+\\frac{63 }{16}\\frac{ m^6}{k^7}-\\frac{69 }{8}\\frac{ m^8}{k^9}+\\frac{15 }{2}\\frac{ m^{10}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^4 \\left(\\frac{1}{\\pi } \\left(-\\frac{15 }{16}\\frac{ m^2}{k^2}+\\frac{5 }{12}\\frac{ m^4}{k^4}-\\frac{11 }{3}\\frac{ m^6}{k^6}+4 \\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ i T \\left(-\\frac{1}{64}\\frac{ m}{k}+\\frac{1}{4}\\frac{ m^3}{k^3}-\\frac{3 }{2}\\frac{ m^5}{k^5}+4 \\frac{ m^7}{k^7}-4 \\frac{ m^9}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{4 \\pi } \\left(\\frac{3 }{4}\\frac{ m^2}{k^2}+21 \\frac{ m^4}{k^4}-44 \\frac{ m^6}{k^6}+48 \\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ i T \\left(-\\frac{3 }{64}\\frac{ m}{k}+\\frac{3 }{4}\\frac{ m^3}{k^3}-\\frac{9 }{2}\\frac{ m^5}{k^5}+12 \\frac{ m^7}{k^7}-12 \\frac{ m^9}{k^9}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{2 \\pi } \\left(\\frac{3 }{64}\\frac{ m^2}{k^2}-\\frac{11 }{16}\\frac{ m^4}{k^4}-\\frac{11 }{4}\\frac{ m^6}{k^6}+3 \\frac{ m^8}{k^8}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ i T \\left(-\\frac{3 }{512}\\frac{ m}{k}+\\frac{3 }{32}\\frac{ m^3}{k^3}-\\frac{9 }{16}\\frac{ m^5}{k^5}+\\frac{3 }{2}\\frac{ m^7}{k^7}-\\frac{3 }{2}\\frac{ m^9}{k^9}\\right)\\right) $ $\\tilde{T}_{5,5;3\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{8 i }{3 \\pi } m^3\\right)+ \\left(k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^4 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{8 i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{16 i }{3 \\pi } m^3\\right)+ \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{64 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^4 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{32 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi } \\left(40 k^2 m^3-256 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{5 \\pi } \\left(40 k^2 m^3-128 m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi } \\left(-32 k^2 m^3-\\frac{128 }{5} m^5\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi } \\left(32 k^2 m^5-\\frac{704 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{5 \\pi } \\left(64 k^2 m^5-\\frac{1408 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi } \\left(-40 k^4 m^3+128 k^2 m^5-\\frac{4096 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3+k_{\\mu }^2 \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{3 \\pi } \\left(-8 k^4 m^3+\\frac{128 }{5} k^2 m^5-\\frac{4096 }{35} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{3 \\pi } \\left(20 k^4 m^3-\\frac{512 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi } \\left(-32 k^4 m^5+\\frac{704 }{7} k^2 m^7-\\frac{1024 }{7} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi } \\left(8 k^6 m^3-\\frac{128 }{5} k^4 m^5+\\frac{4096 }{35} k^2 m^7-\\frac{22528 }{105} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^5 \\left(\\frac{i }{3 \\pi } \\left(-4 k^6 m^3+\\frac{512 }{35} k^2 m^7-\\frac{4096 }{105} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{4 }{\\pi } m^2\\right)+k_{\\mu }^2 k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{8 }{\\pi } m^4\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^3 k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{16 }{\\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{1}{\\pi } \\left(6 k^2 m^2-16 m^4\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{32 }{5 \\pi } m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{1}{5 \\pi } \\left(80 k^2 m^4-256 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{1}{5 \\pi } \\left(40 k^2 m^4-128 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{1}{3 \\pi } \\left(-12 k^4 m^2+16 k^2 m^4-\\frac{512 }{5} m^6\\right)\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{1}{5 \\pi } \\left(16 k^2 m^6-\\frac{192 }{7} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{1}{5 \\pi } \\left(-40 k^4 m^4+128 k^2 m^6-\\frac{1536 }{7} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{1}{\\pi } \\left(k^6 m^2-\\frac{4 }{3} k^4 m^4+\\frac{128 }{15} k^2 m^6-\\frac{512 }{35} m^8\\right)\\right) $ Fermions, spin 5 x 5, dimension 4: $\\tilde{T}_{5,5;4\\text{D}}^{\\text{f,t}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{7 \\pi ^2} \\left( \\left(-\\frac{23722}{571725}+\\frac{2 L_0}{165}\\right)+ \\left(\\frac{89008}{155925}-\\frac{8 L_0}{45}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{211289}{69300}-\\frac{27 L_0}{10}\\right)\\frac{ m^4}{k^4}-\\frac{256 }{99}\\frac{ m^6}{k^6}+\\frac{24064 }{1485}\\frac{ m^8}{k^8}-\\frac{2048 }{99}\\frac{ m^{10}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{33 \\pi ^2} \\left(\\frac{4 }{35}\\frac{1}{k}-\\frac{152 }{105}\\frac{ m^2}{k^3}+\\frac{512 }{105}\\frac{ m^4}{k^5}+\\frac{256 }{35}\\frac{ m^6}{k^7}-\\frac{1024 }{15}\\frac{ m^8}{k^9}+\\frac{2048 }{21}\\frac{ m^{10}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{83338}{12006225}+\\frac{8 L_0}{3465}\\right)+ \\left(\\frac{13004}{1091475}-\\frac{4 L_0}{315}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{99289}{242550}+\\frac{27 L_0}{35}\\right)\\frac{ m^4}{k^4}-\\frac{55808 }{10395}\\frac{ m^6}{k^6}+\\frac{22016 }{1485}\\frac{ m^8}{k^8}-\\frac{10240 }{693}\\frac{ m^{10}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{99 \\pi ^2} \\left(\\frac{16 }{35}\\frac{1}{k}-\\frac{8 }{5}\\frac{ m^2}{k^3}-\\frac{1664 }{35}\\frac{ m^4}{k^5}+\\frac{15104 }{35}\\frac{ m^6}{k^7}-\\frac{47104 }{35}\\frac{ m^8}{k^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{10240 }{7}\\frac{ m^{10}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{13511}{2286900}-\\frac{L_0}{660}\\right)+ \\left(-\\frac{52379}{363825}+\\frac{4 L_0}{105}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{227603}{161700}-\\frac{27 L_0}{70}\\right)\\frac{ m^4}{k^4}-\\frac{11552 }{3465}\\frac{ m^6}{k^6}+\\frac{23488 }{3465}\\frac{ m^8}{k^8}-\\frac{1280 }{231}\\frac{ m^{10}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{11 \\pi ^2} \\left(-\\frac{1}{30}\\frac{1}{k}+\\frac{27 }{35}\\frac{ m^2}{k^3}-\\frac{736 }{105}\\frac{ m^4}{k^5}+\\frac{3296 }{105}\\frac{ m^6}{k^7}-\\frac{2432 }{35}\\frac{ m^8}{k^9}+\\frac{1280 }{21}\\frac{ m^{10}}{k^{11}}\\right)\\right) $ $\\tilde{T}_{5,5;4\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{\\pi ^2} m^4\\right)+ \\left(k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^4 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{3 i L_2}{\\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(-\\frac{2 i L_2}{\\pi ^2} m^4\\right)+ \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{4 i L_3}{\\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^4 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{2 i L_3}{\\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(-3 L_2 k^2 m^4+16 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{\\pi ^2} \\left(-3 L_2 k^2 m^4+8 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi ^2} \\left(12 L_2 k^2 m^4+8 L_3 m^6\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(-4 L_3 k^2 m^6+11 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{\\pi ^2} \\left(-4 L_3 k^2 m^6+11 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(3 L_2 k^4 m^4-8 L_3 k^2 m^6+32 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3+k_{\\mu }^2 \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{3 \\pi ^2} \\left(3 L_2 k^4 m^4-8 L_3 k^2 m^6+32 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{5 L_2}{2} k^4 m^4+\\frac{20 L_4}{3} m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left(2 L_3 k^4 m^6-\\frac{11 L_4}{2} k^2 m^8+\\frac{36 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(- L_2 k^6 m^4+\\frac{8 L_3}{3} k^4 m^6-\\frac{32 L_4}{3} k^2 m^8+\\frac{88 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi ^2} \\left(\\frac{ L_2}{2} k^6 m^4-\\frac{4 L_4}{3} k^2 m^8+\\frac{16 L_5}{5} m^{10}\\right)\\right) $ Fermions, spin 5 x 5, dimension 5: $\\tilde{T}_{5,5;5\\text{D}}^{\\text{f,t}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi ^2} \\left(\\frac{7 }{15360} m-\\frac{83 }{11520}\\frac{ m^3}{k^2}-\\frac{183 }{800}\\frac{ m^5}{k^4}-\\frac{1}{24}\\frac{ m^7}{k^6}+\\frac{49 }{180}\\frac{ m^9}{k^8}-\\frac{1}{3}\\frac{ m^{11}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{\\pi } \\left(\\frac{7 }{61440} k-\\frac{1}{512}\\frac{ m^2}{k}+\\frac{3 }{256}\\frac{ m^4}{k^3}-\\frac{1}{48}\\frac{ m^6}{k^5}-\\frac{1}{16}\\frac{ m^8}{k^7}+\\frac{3 }{10}\\frac{ m^{10}}{k^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{3}\\frac{ m^{12}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(\\frac{11 }{15360} m-\\frac{79 }{11520}\\frac{ m^3}{k^2}+\\frac{1223 }{2400}\\frac{ m^5}{k^4}-\\frac{27 }{40}\\frac{ m^7}{k^6}+\\frac{317 }{180}\\frac{ m^9}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{5 }{3}\\frac{ m^{11}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{\\pi } \\left(\\frac{11 }{61440} k-\\frac{1}{512}\\frac{ m^2}{k}-\\frac{1}{256}\\frac{ m^4}{k^3}+\\frac{7 }{48}\\frac{ m^6}{k^5}-\\frac{13 }{16}\\frac{ m^8}{k^7}+\\frac{19 }{10}\\frac{ m^{10}}{k^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{5 }{3}\\frac{ m^{12}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{8 \\pi ^2} \\left(-\\frac{13 }{5120} m+\\frac{257 }{3840}\\frac{ m^3}{k^2}-\\frac{569 }{800}\\frac{ m^5}{k^4}-\\frac{137 }{40}\\frac{ m^7}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{389 }{60}\\frac{ m^9}{k^8}-5 \\frac{ m^{11}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{ T}{8 \\pi } \\left(-\\frac{13 }{20480} k+\\frac{9 }{512}\\frac{ m^2}{k}-\\frac{51 }{256}\\frac{ m^4}{k^3}+\\frac{19 }{16}\\frac{ m^6}{k^5}-\\frac{63 }{16}\\frac{ m^8}{k^7}+\\frac{69 }{10}\\frac{ m^{10}}{k^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- 5 \\frac{ m^{12}}{k^{11}}\\right)\\right) $ $\\tilde{T}_{5,5;5\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{8 i }{15 \\pi ^2} m^5\\right)+ \\left(k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^4 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{8 i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(-\\frac{16 i }{15 \\pi ^2} m^5\\right)+ \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{64 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^4 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{32 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(-8 k^2 m^5+\\frac{256 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{5 \\pi ^2} \\left(-8 k^2 m^5+\\frac{128 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{15 \\pi ^2} \\left(32 k^2 m^5+\\frac{128 }{7} m^7\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{35 \\pi ^2} \\left(-32 k^2 m^7+\\frac{704 }{9} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{35 \\pi ^2} \\left(-64 k^2 m^7+\\frac{1408 }{9} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(8 k^4 m^5-\\frac{128 }{7} k^2 m^7+\\frac{4096 }{63} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3+k_{\\mu }^2 \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{15 \\pi ^2} \\left(8 k^4 m^5-\\frac{128 }{7} k^2 m^7+\\frac{4096 }{63} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{3 \\pi ^2} \\left(-4 k^4 m^5+\\frac{512 }{63} m^9\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{35 \\pi ^2} \\left(32 k^4 m^7-\\frac{704 }{9} k^2 m^9+\\frac{1024 }{11} m^{11}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{15 \\pi ^2} \\left(-8 k^6 m^5+\\frac{128 }{7} k^4 m^7-\\frac{4096 }{63} k^2 m^9+\\frac{2048 }{21} m^{11}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^5 \\left(\\frac{i }{15 \\pi ^2} \\left(4 k^6 m^5-\\frac{512 }{63} k^2 m^9+\\frac{4096 }{231} m^{11}\\right)\\right) $ Fermions, spin 5 x 5, dimension 6: $\\tilde{T}_{5,5;6\\text{D}}^{\\text{f,t}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{659507}{4058104050}+\\frac{2 L_0}{45045}\\right) k^2+ \\left(\\frac{158813}{52026975}-\\frac{L_0}{1155}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{299671}{14189175}+\\frac{2 L_0}{315}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{3691967}{37837800}+\\frac{9 L_0}{140}\\right)\\frac{ m^6}{k^4}+\\frac{128 }{3003}\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{4096 }{19305}\\frac{ m^{10}}{k^8}+\\frac{2048 }{9009}\\frac{ m^{12}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{39 \\pi ^3} \\left(\\frac{4 }{1155} k-\\frac{2 }{33}\\frac{ m^2}{k}+\\frac{8 }{21}\\frac{ m^4}{k^3}-\\frac{64 }{77}\\frac{ m^6}{k^5}-\\frac{256 }{231}\\frac{ m^8}{k^7}+\\frac{8704 }{1155}\\frac{ m^{10}}{k^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{2048 }{231}\\frac{ m^{12}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{156833}{579729150}+\\frac{L_0}{12870}\\right) k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{565952}{156080925}-\\frac{4 L_0}{3465}\\right) m^2+ \\left(-\\frac{10061}{2579850}+\\frac{L_0}{315}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{2347967}{18918900}-\\frac{9 L_0}{70}\\right)\\frac{ m^6}{k^4}+\\frac{82048 }{135135}\\frac{ m^8}{k^6}-\\frac{183296 }{135135}\\frac{ m^{10}}{k^8}+\\frac{10240 }{9009}\\frac{ m^{12}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{13 \\pi ^3} \\left(\\frac{1}{495} k-\\frac{2 }{77}\\frac{ m^2}{k}+\\frac{8 }{231}\\frac{ m^4}{k^3}+\\frac{64 }{63}\\frac{ m^6}{k^5}-\\frac{512 }{77}\\frac{ m^8}{k^7}+\\frac{18944 }{1155}\\frac{ m^{10}}{k^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{10240 }{693}\\frac{ m^{12}}{k^{11}}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{367291}{3607203600}-\\frac{L_0}{40040}\\right) k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{178613}{59459400}+\\frac{L_0}{1320}\\right) m^2+ \\left(\\frac{346921}{9459450}-\\frac{L_0}{105}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{91573}{382200}+\\frac{9 L_0}{140}\\right)\\frac{ m^6}{k^4}+\\frac{2416 }{6435}\\frac{ m^8}{k^6}-\\frac{27904 }{45045}\\frac{ m^{10}}{k^8}+\\frac{1280 }{3003}\\frac{ m^{12}}{k^{10}}\\right)+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{i S}{143 \\pi ^3} \\left(-\\frac{1}{140} k+\\frac{17 }{84}\\frac{ m^2}{k}-\\frac{7 }{3}\\frac{ m^4}{k^3}+\\frac{296 }{21}\\frac{ m^6}{k^5}-\\frac{992 }{21}\\frac{ m^8}{k^7}+\\frac{8768 }{105}\\frac{ m^{10}}{k^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1280 }{21}\\frac{ m^{12}}{k^{11}}\\right)\\right) $ $\\tilde{T}_{5,5;6\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{6 \\pi ^3} m^6\\right)+ \\left(k_{\\mu }^2 k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^4 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{2 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^3 \\eta _{\\mu \\nu }^2 \\left(\\frac{i L_3}{3 \\pi ^3} m^6\\right)+ \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{i L_4}{2 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^4 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^4 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(-\\frac{i L_4}{4 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^3} \\left( L_3 k^2 m^6-4 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2+k_{\\mu }^3 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{2 \\pi ^3} \\left( L_3 k^2 m^6-2 L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi ^3} \\left(-2 L_3 k^2 m^6- L_4 m^8\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^3} \\left( L_4 k^2 m^8-\\frac{11 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2\\right) \\left(\\frac{i }{2 \\pi ^3} \\left( L_4 k^2 m^8-\\frac{11 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left(-\\frac{ L_3}{2} k^4 m^6+ L_4 k^2 m^8-\\frac{16 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3+k_{\\mu }^2 \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{3 \\pi ^3} \\left(-\\frac{ L_3}{2} k^4 m^6+ L_4 k^2 m^8-\\frac{16 L_5}{5} m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{i }{3 \\pi ^3} \\left(\\frac{5 L_3}{4} k^4 m^6-2 L_5 m^{10}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left(-\\frac{ L_4}{4} k^4 m^8+\\frac{11 L_5}{20} k^2 m^{10}-\\frac{3 L_6}{5} m^{12}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^3} \\left(\\frac{ L_3}{2} k^6 m^6- L_4 k^4 m^8+\\frac{16 L_5}{5} k^2 m^{10}-\\frac{22 L_6}{5} m^{12}\\right)\\right)+\\nonumber \\\\ & \\quad + \\eta _{\\mu \\nu }^5 \\left(\\frac{i }{3 \\pi ^3} \\left(-\\frac{ L_3}{4} k^6 m^6+\\frac{2 L_5}{5} k^2 m^{10}-\\frac{4 L_6}{5} m^{12}\\right)\\right) $"
],
[
"Expansions in UV and IR for fermions",
"Fermions, spin 0 x 0, dimension 3: $\\tilde{T}_{0,0;3\\text{D}}^{\\text{f,UV}} & = -\\frac{1}{8} k+\\frac{1}{2}\\frac{ m^2}{k}-\\frac{4 i }{3 \\pi }\\frac{ m^3}{k^2}-\\frac{16 i }{15 \\pi }\\frac{ m^5}{k^4}-\\frac{64 i }{35 \\pi }\\frac{ m^7}{k^6}-\\frac{256 i }{63 \\pi }\\frac{ m^9}{k^8}-\\frac{1024 i }{99 \\pi }\\frac{ m^{11}}{k^{10}}+\\ldots $ $\\tilde{T}_{0,0;3\\text{D}}^{\\text{f,IR}} & = \\frac{i }{4 \\pi } \\left(-4 m+\\frac{1}{3}\\frac{ k^2}{m}+\\frac{1}{60}\\frac{ k^4}{m^3}+\\frac{1}{560}\\frac{ k^6}{m^5}+\\frac{1}{4032}\\frac{ k^8}{m^7}+\\frac{1}{25344}\\frac{ k^{10}}{m^9}+\\frac{1}{146432}\\frac{ k^{12}}{m^{11}}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\ldots \\right) $ $\\tilde{T}_{0,0;3\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 0 x 0, dimension 4: $\\tilde{T}_{0,0;4\\text{D}}^{\\text{f,UV}} & = \\frac{i }{2 \\pi ^2} \\left( \\left(\\frac{1}{2}-\\frac{P}{4}\\right) k^2+ \\left(-2+\\frac{3 P}{2}\\right) m^2- \\left(\\frac{9}{4}+\\frac{3 K}{2}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{6}-K\\right)\\frac{ m^6}{k^4}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\left(\\frac{7}{8}-\\frac{3 K}{2}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{47}{20}-3 K\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{379}{60}-7 K\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right) $ $\\tilde{T}_{0,0;4\\text{D}}^{\\text{f,IR}} & = \\frac{i }{4 \\pi ^2} \\left( \\left(-1+3 L_0\\right) m^2- \\left(\\frac{1}{3}+\\frac{L_0}{2}\\right) k^2+\\frac{1}{20}\\frac{ k^4}{m^2}+\\frac{1}{280}\\frac{ k^6}{m^4}+\\frac{1}{2520}\\frac{ k^8}{m^6}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{1}{18480}\\frac{ k^{10}}{m^8}+\\frac{1}{120120}\\frac{ k^{12}}{m^{10}}+\\frac{1}{720720}\\frac{ k^{14}}{m^{12}}+\\ldots \\right) $ $\\tilde{T}_{0,0;4\\text{D}}^{\\text{f,UV-IR}} & = \\frac{i }{\\pi ^2} \\left( \\left(\\frac{1}{3}-\\frac{K}{8}\\right) k^2+ \\left(-\\frac{3}{4}+\\frac{3 K}{4}\\right) m^2\\right)+\\ldots $ Fermions, spin 0 x 0, dimension 5: $\\tilde{T}_{0,0;5\\text{D}}^{\\text{f,UV}} & = \\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{64} k^3+\\frac{ \\pi }{8} k m^2-\\frac{ \\pi }{4}\\frac{ m^4}{k}+\\frac{8 i }{15}\\frac{ m^5}{k^2}+\\frac{32 i }{105}\\frac{ m^7}{k^4}+\\frac{128 i }{315}\\frac{ m^9}{k^6}+\\frac{512 i }{693}\\frac{ m^{11}}{k^8}+\\ldots \\right) $ $\\tilde{T}_{0,0;5\\text{D}}^{\\text{f,IR}} & = \\frac{i }{3 \\pi ^2} \\left(2 m^3-\\frac{1}{2} k^2 m+\\frac{1}{40}\\frac{ k^4}{m}+\\frac{1}{1120}\\frac{ k^6}{m^3}+\\frac{1}{13440}\\frac{ k^8}{m^5}+\\frac{1}{118272}\\frac{ k^{10}}{m^7}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{1}{878592}\\frac{ k^{12}}{m^9}+\\frac{1}{5857280}\\frac{ k^{14}}{m^{11}}+\\ldots \\right) $ $\\tilde{T}_{0,0;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 0 x 0, dimension 6: $\\tilde{T}_{0,0;6\\text{D}}^{\\text{f,UV}} & = \\frac{i }{2 \\pi ^3} \\left( \\left(\\frac{1}{18}-\\frac{P}{48}\\right) k^4+ \\left(-\\frac{17}{36}+\\frac{5 P}{24}\\right) k^2 m^2+ \\left(1-\\frac{5 P}{8}\\right) m^4+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\left(\\frac{55}{72}+\\frac{5 K}{12}\\right)\\frac{ m^6}{k^2}+ \\left(\\frac{5}{288}+\\frac{5 K}{24}\\right)\\frac{ m^8}{k^4}+ \\left(-\\frac{23}{240}+\\frac{K}{4}\\right)\\frac{ m^{10}}{k^6}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\left(-\\frac{37}{144}+\\frac{5 K}{12}\\right)\\frac{ m^{12}}{k^8}+\\ldots \\right) $ $\\tilde{T}_{0,0;6\\text{D}}^{\\text{f,IR}} & = \\frac{i }{16 \\pi ^3} \\left( \\left(\\frac{11}{2}-5 L_0\\right) m^4+ \\left(-1+\\frac{5 L_0}{3}\\right) k^2 m^2- \\left(\\frac{1}{15}+\\frac{L_0}{6}\\right) k^4+\\frac{1}{84}\\frac{ k^6}{m^2}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{1}{1512}\\frac{ k^8}{m^4}+\\frac{1}{16632}\\frac{ k^{10}}{m^6}+\\frac{1}{144144}\\frac{ k^{12}}{m^8}+\\frac{1}{1081080}\\frac{ k^{14}}{m^{10}}+\\frac{1}{7351344}\\frac{ k^{16}}{m^{12}}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\ldots \\right) $ $\\tilde{T}_{0,0;6\\text{D}}^{\\text{f,UV-IR}} & = \\frac{i }{16 \\pi ^3} \\left( \\left(\\frac{23}{45}-\\frac{K}{6}\\right) k^4+ \\left(-\\frac{25}{9}+\\frac{5 K}{3}\\right) k^2 m^2+ \\left(\\frac{5}{2}-5 K\\right) m^4\\right)+\\ldots $ Fermions, spin 0 x 2, dimension 3: $\\tilde{T}_{0,2;3\\text{D}}^{\\text{f,t,UV}} & = k^2 \\pi _{\\nu \\nu } \\left(-\\frac{1}{8}\\frac{ m}{k}+\\frac{i }{\\pi }\\frac{ m^2}{k^2}+\\frac{1}{2}\\frac{ m^3}{k^3}-\\frac{4 i }{3 \\pi }\\frac{ m^4}{k^4}-\\frac{16 i }{15 \\pi }\\frac{ m^6}{k^6}-\\frac{64 i }{35 \\pi }\\frac{ m^8}{k^8}-\\frac{256 i }{63 \\pi }\\frac{ m^{10}}{k^{10}}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{1024 i }{99 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{0,2;3\\text{D}}^{\\text{f,t,IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi } \\left(\\frac{1}{3}+\\frac{1}{60}\\frac{ k^2}{m^2}+\\frac{1}{560}\\frac{ k^4}{m^4}+\\frac{1}{4032}\\frac{ k^6}{m^6}+\\frac{1}{25344}\\frac{ k^8}{m^8}+\\frac{1}{146432}\\frac{ k^{10}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{798720}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;3\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 0 x 2, dimension 4: $\\tilde{T}_{0,2;4\\text{D}}^{\\text{f,t,UV}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{2}{9}-\\frac{P}{12}\\right) m+ \\left(-\\frac{1}{2}+\\frac{K}{2}\\right)\\frac{ m^3}{k^2}- \\left(\\frac{3}{4}+\\frac{K}{2}\\right)\\frac{ m^5}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1}{18}-\\frac{K}{3}\\right)\\frac{ m^7}{k^6}+ \\left(\\frac{7}{24}-\\frac{K}{2}\\right)\\frac{ m^9}{k^8}+ \\left(\\frac{47}{60}-K\\right)\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;4\\text{D}}^{\\text{f,t,IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^2} \\left(- L_0 m+\\frac{1}{10}\\frac{ k^2}{m}+\\frac{1}{140}\\frac{ k^4}{m^3}+\\frac{1}{1260}\\frac{ k^6}{m^5}+\\frac{1}{9240}\\frac{ k^8}{m^7}+\\frac{1}{60060}\\frac{ k^{10}}{m^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{360360}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;4\\text{D}}^{\\text{f,UV-IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2}\\left(\\frac{2}{3}-\\frac{K}{4}\\right) m\\right)+\\ldots $ Fermions, spin 0 x 2, dimension 5: $\\tilde{T}_{0,2;5\\text{D}}^{\\text{f,t,UV}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{128} k m+\\frac{ \\pi }{16}\\frac{ m^3}{k}-\\frac{i }{3}\\frac{ m^4}{k^2}-\\frac{ \\pi }{8}\\frac{ m^5}{k^3}+\\frac{4 i }{15}\\frac{ m^6}{k^4}+\\frac{16 i }{105}\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{64 i }{315}\\frac{ m^{10}}{k^8}+\\frac{256 i }{693}\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;5\\text{D}}^{\\text{f,t,IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^2} \\left(- m^2+\\frac{1}{20} k^2+\\frac{1}{560}\\frac{ k^4}{m^2}+\\frac{1}{6720}\\frac{ k^6}{m^4}+\\frac{1}{59136}\\frac{ k^8}{m^6}+\\frac{1}{439296}\\frac{ k^{10}}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{2928640}\\frac{ k^{12}}{m^{10}}+\\frac{1}{18104320}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 0 x 2, dimension 6: $\\tilde{T}_{0,2;6\\text{D}}^{\\text{f,t,UV}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{23}{1800}-\\frac{P}{240}\\right) k^2 m+ \\left(-\\frac{1}{9}+\\frac{P}{24}\\right) m^3+ \\left(\\frac{1}{16}-\\frac{K}{8}\\right)\\frac{ m^5}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{11}{72}+\\frac{K}{12}\\right)\\frac{ m^7}{k^4}+ \\left(\\frac{1}{288}+\\frac{K}{24}\\right)\\frac{ m^9}{k^6}+ \\left(-\\frac{23}{1200}+\\frac{K}{20}\\right)\\frac{ m^{11}}{k^8}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;6\\text{D}}^{\\text{f,t,IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{24 \\pi ^3} \\left( \\left(-1+L_0\\right) m^3-\\frac{ L_0}{10} k^2 m+\\frac{1}{140}\\frac{ k^4}{m}+\\frac{1}{2520}\\frac{ k^6}{m^3}+\\frac{1}{27720}\\frac{ k^8}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{240240}\\frac{ k^{10}}{m^7}+\\frac{1}{1801800}\\frac{ k^{12}}{m^9}+\\frac{1}{12252240}\\frac{ k^{14}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,2;6\\text{D}}^{\\text{f,UV-IR}} & = k^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{24 \\pi ^3} \\left( \\left(\\frac{23}{75}-\\frac{K}{10}\\right) k^2 m+ \\left(-\\frac{5}{3}+K\\right) m^3\\right)\\right)+\\ldots $ Fermions, spin 0 x 4, dimension 3: $\\tilde{T}_{0,4;3\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{3 }{32}\\frac{ m}{k}-\\frac{i }{\\pi }\\frac{ m^2}{k^2}-\\frac{3 }{4}\\frac{ m^3}{k^3}+\\frac{4 i }{\\pi }\\frac{ m^4}{k^4}+\\frac{3 }{2}\\frac{ m^5}{k^5}-\\frac{16 i }{5 \\pi }\\frac{ m^6}{k^6}-\\frac{64 i }{35 \\pi }\\frac{ m^8}{k^8}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{256 i }{105 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{1024 i }{231 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{0,4;3\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi } \\left(-\\frac{1}{5}-\\frac{1}{140}\\frac{ k^2}{m^2}-\\frac{1}{1680}\\frac{ k^4}{m^4}-\\frac{1}{14784}\\frac{ k^6}{m^6}-\\frac{1}{109824}\\frac{ k^8}{m^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{732160}\\frac{ k^{10}}{m^{10}}-\\frac{1}{4526080}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;3\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 0 x 4, dimension 4: $\\tilde{T}_{0,4;4\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{23}{150}+\\frac{P}{20}\\right) m+ \\left(\\frac{5}{6}-\\frac{K}{2}\\right)\\frac{ m^3}{k^2}+ \\left(-\\frac{3}{4}+\\frac{3 K}{2}\\right)\\frac{ m^5}{k^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{11}{6}+K\\right)\\frac{ m^7}{k^6}- \\left(\\frac{1}{24}+\\frac{K}{2}\\right)\\frac{ m^9}{k^8}+ \\left(\\frac{23}{100}-\\frac{3 K}{5}\\right)\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;4\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{20 \\pi ^2} \\left( L_0 m-\\frac{1}{14}\\frac{ k^2}{m}-\\frac{1}{252}\\frac{ k^4}{m^3}-\\frac{1}{2772}\\frac{ k^6}{m^5}-\\frac{1}{24024}\\frac{ k^8}{m^7}-\\frac{1}{180180}\\frac{ k^{10}}{m^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{1225224}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;4\\text{D}}^{\\text{f,UV-IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{10 \\pi ^2}\\left(-\\frac{23}{15}+\\frac{K}{2}\\right) m\\right)+\\ldots $ Fermions, spin 0 x 4, dimension 5: $\\tilde{T}_{0,4;5\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{256} k m-\\frac{3 \\pi }{64}\\frac{ m^3}{k}+\\frac{i }{3}\\frac{ m^4}{k^2}+\\frac{3 \\pi }{16}\\frac{ m^5}{k^3}-\\frac{4 i }{5}\\frac{ m^6}{k^4}-\\frac{ \\pi }{4}\\frac{ m^7}{k^5}+\\frac{16 i }{35}\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{64 i }{315}\\frac{ m^{10}}{k^8}+\\frac{256 i }{1155}\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;5\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{1}{5} m^2-\\frac{1}{140} k^2-\\frac{1}{5040}\\frac{ k^4}{m^2}-\\frac{1}{73920}\\frac{ k^6}{m^4}-\\frac{1}{768768}\\frac{ k^8}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{6589440}\\frac{ k^{10}}{m^8}-\\frac{1}{49786880}\\frac{ k^{12}}{m^{10}}-\\frac{1}{343982080}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 0 x 4, dimension 6: $\\tilde{T}_{0,4;6\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{22}{3675}+\\frac{P}{560}\\right) k^2 m+ \\left(\\frac{23}{300}-\\frac{P}{40}\\right) m^3+ \\left(-\\frac{7}{48}+\\frac{K}{8}\\right)\\frac{ m^5}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1}{24}-\\frac{K}{4}\\right)\\frac{ m^7}{k^4}+ \\left(\\frac{25}{96}+\\frac{K}{8}\\right)\\frac{ m^9}{k^6}+ \\left(\\frac{17}{1200}+\\frac{K}{20}\\right)\\frac{ m^{11}}{k^8}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;6\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{40 \\pi ^3} \\left( \\left(1-L_0\\right) m^3+\\frac{ L_0}{14} k^2 m-\\frac{1}{252}\\frac{ k^4}{m}-\\frac{1}{5544}\\frac{ k^6}{m^3}-\\frac{1}{72072}\\frac{ k^8}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{720720}\\frac{ k^{10}}{m^7}-\\frac{1}{6126120}\\frac{ k^{12}}{m^9}-\\frac{1}{46558512}\\frac{ k^{14}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{0,4;6\\text{D}}^{\\text{f,UV-IR}} & = k^4 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(-\\frac{22}{735}+\\frac{K}{112}\\right) k^2 m+ \\left(\\frac{31}{120}-\\frac{K}{8}\\right) m^3\\right)\\right)+\\ldots $ Fermions, spin 1 x 1, dimension 3: $\\tilde{T}_{1,1;3\\text{D}}^{\\text{f,t,UV}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{1}{16}\\frac{1}{k}+\\frac{1}{4}\\frac{ m^2}{k^3}-\\frac{4 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{32 i }{15 \\pi }\\frac{ m^5}{k^6}-\\frac{192 i }{35 \\pi }\\frac{ m^7}{k^8}-\\frac{1024 i }{63 \\pi }\\frac{ m^9}{k^{10}}-\\frac{5120 i }{99 \\pi }\\frac{ m^{11}}{k^{12}}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\ldots \\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{i }{4}\\frac{ m}{k}-\\frac{1}{\\pi }\\frac{ m^2}{k^2}-\\frac{4 }{3 \\pi }\\frac{ m^4}{k^4}-\\frac{16 }{5 \\pi }\\frac{ m^6}{k^6}-\\frac{64 }{7 \\pi }\\frac{ m^8}{k^8}-\\frac{256 }{9 \\pi }\\frac{ m^{10}}{k^{10}}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{1024 }{11 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{1,1;3\\text{D}}^{\\text{f,t,IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{4 \\pi } \\left(-\\frac{1}{3}\\frac{1}{m}-\\frac{1}{30}\\frac{ k^2}{m^3}-\\frac{3 }{560}\\frac{ k^4}{m^5}-\\frac{1}{1008}\\frac{ k^6}{m^7}-\\frac{5 }{25344}\\frac{ k^8}{m^9}-\\frac{3 }{73216}\\frac{ k^{10}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + (k\\cdot \\epsilon )_{\\mu \\nu } \\left(\\frac{1}{4 \\pi } \\left(- 1-\\frac{1}{12}\\frac{ k^2}{m^2}-\\frac{1}{80}\\frac{ k^4}{m^4}-\\frac{1}{448}\\frac{ k^6}{m^6}-\\frac{1}{2304}\\frac{ k^8}{m^8}-\\frac{1}{11264}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{53248}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;3\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 1 x 1, dimension 4: $\\tilde{T}_{1,1;4\\text{D}}^{\\text{f,t,UV}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{5}{36}+\\frac{P}{12}\\right)-\\frac{1}{2}\\frac{ m^2}{k^2}- \\left(\\frac{1}{4}+\\frac{K}{2}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{4}{9}-\\frac{2 K}{3}\\right)\\frac{ m^6}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{11}{8}-\\frac{3 K}{2}\\right)\\frac{ m^8}{k^8}+ \\left(\\frac{62}{15}-4 K\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{463}{36}-\\frac{35 K}{3}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;4\\text{D}}^{\\text{f,t,IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{ L_0}{3}-\\frac{1}{15}\\frac{ k^2}{m^2}-\\frac{1}{140}\\frac{ k^4}{m^4}-\\frac{1}{945}\\frac{ k^6}{m^6}-\\frac{1}{5544}\\frac{ k^8}{m^8}-\\frac{1}{30030}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{154440}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;4\\text{D}}^{\\text{f,UV-IR}} & = k^2 \\pi _{\\mu \\nu } \\frac{i }{12 \\pi ^2}\\left(-\\frac{5}{3}+K\\right)+\\ldots $ Fermions, spin 1 x 1, dimension 5: $\\tilde{T}_{1,1;5\\text{D}}^{\\text{f,t,UV}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{3 \\pi }{256} k-\\frac{ \\pi }{32}\\frac{ m^2}{k}-\\frac{ \\pi }{16}\\frac{ m^4}{k^3}+\\frac{4 i }{15}\\frac{ m^5}{k^4}+\\frac{32 i }{105}\\frac{ m^7}{k^6}+\\frac{64 i }{105}\\frac{ m^9}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1024 i }{693}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;5\\text{D}}^{\\text{f,t,IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{1}{3} m-\\frac{1}{30}\\frac{ k^2}{m}-\\frac{1}{560}\\frac{ k^4}{m^3}-\\frac{1}{5040}\\frac{ k^6}{m^5}-\\frac{5 }{177408}\\frac{ k^8}{m^7}-\\frac{1}{219648}\\frac{ k^{10}}{m^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{7 }{8785920}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 1 x 1, dimension 6: $\\tilde{T}_{1,1;6\\text{D}}^{\\text{f,t,UV}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{77}{3600}+\\frac{P}{120}\\right) k^2+ \\left(\\frac{5}{72}-\\frac{P}{24}\\right) m^2+\\frac{1}{8}\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{5}{72}+\\frac{K}{12}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{5}{144}+\\frac{K}{12}\\right)\\frac{ m^8}{k^6}+ \\left(-\\frac{43}{400}+\\frac{3 K}{20}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{13}{45}+\\frac{K}{3}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;6\\text{D}}^{\\text{f,t,IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{8 \\pi ^3} \\left( \\left(\\frac{1}{3}-\\frac{L_0}{3}\\right) m^2+\\frac{ L_0}{15} k^2-\\frac{1}{140}\\frac{ k^4}{m^2}-\\frac{1}{1890}\\frac{ k^6}{m^4}-\\frac{1}{16632}\\frac{ k^8}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{120120}\\frac{ k^{10}}{m^8}-\\frac{1}{772200}\\frac{ k^{12}}{m^{10}}-\\frac{1}{4594590}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,1;6\\text{D}}^{\\text{f,UV-IR}} & = k^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{12 \\pi ^3} \\left( \\left(-\\frac{77}{300}+\\frac{K}{10}\\right) k^2+ \\left(\\frac{1}{3}-\\frac{K}{2}\\right) m^2\\right)\\right)+\\ldots $ Fermions, spin 1 x 3, dimension 3: $\\tilde{T}_{1,3;3\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(-\\frac{1}{64}\\frac{1}{k}-\\frac{1}{8}\\frac{ m^2}{k^3}+\\frac{4 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{3 }{4}\\frac{ m^4}{k^5}-\\frac{32 i }{15 \\pi }\\frac{ m^5}{k^6}-\\frac{64 i }{35 \\pi }\\frac{ m^7}{k^8}-\\frac{1024 i }{315 \\pi }\\frac{ m^9}{k^{10}}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{5120 i }{693 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{8}\\frac{ m}{k}+\\frac{1}{\\pi }\\frac{ m^2}{k^2}-\\frac{i }{2}\\frac{ m^3}{k^3}-\\frac{4 }{3 \\pi }\\frac{ m^4}{k^4}-\\frac{16 }{15 \\pi }\\frac{ m^6}{k^6}-\\frac{64 }{35 \\pi }\\frac{ m^8}{k^8}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{256 }{63 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{1024 }{99 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{1,3;3\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi } \\left(\\frac{1}{15}\\frac{1}{m}+\\frac{1}{210}\\frac{ k^2}{m^3}+\\frac{1}{1680}\\frac{ k^4}{m^5}+\\frac{1}{11088}\\frac{ k^6}{m^7}+\\frac{5 }{329472}\\frac{ k^8}{m^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{366080}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{1}{4 \\pi } \\left(\\frac{1}{3}+\\frac{1}{60}\\frac{ k^2}{m^2}+\\frac{1}{560}\\frac{ k^4}{m^4}+\\frac{1}{4032}\\frac{ k^6}{m^6}+\\frac{1}{25344}\\frac{ k^8}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{146432}\\frac{ k^{10}}{m^{10}}+\\frac{1}{798720}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;3\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 1 x 3, dimension 4: $\\tilde{T}_{1,3;4\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{31}{900}-\\frac{P}{60}\\right)+\\frac{1}{6}\\frac{ m^2}{k^2}+ \\left(-\\frac{3}{4}+\\frac{K}{2}\\right)\\frac{ m^4}{k^4}- \\left(\\frac{8}{9}+\\frac{2 K}{3}\\right)\\frac{ m^6}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1}{8}-\\frac{K}{2}\\right)\\frac{ m^8}{k^8}+ \\left(\\frac{38}{75}-\\frac{4 K}{5}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{49}{36}-\\frac{5 K}{3}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;4\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{12 \\pi ^2} \\left(-\\frac{ L_0}{5}+\\frac{1}{35}\\frac{ k^2}{m^2}+\\frac{1}{420}\\frac{ k^4}{m^4}+\\frac{1}{3465}\\frac{ k^6}{m^6}+\\frac{1}{24024}\\frac{ k^8}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{150150}\\frac{ k^{10}}{m^{10}}+\\frac{1}{875160}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;4\\text{D}}^{\\text{f,UV-IR}} & = k^4 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\frac{i }{60 \\pi ^2}\\left(\\frac{31}{15}-K\\right)+\\ldots $ Fermions, spin 1 x 3, dimension 5: $\\tilde{T}_{1,3;5\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{512} k+\\frac{ \\pi }{128}\\frac{ m^2}{k}+\\frac{ \\pi }{32}\\frac{ m^4}{k^3}-\\frac{4 i }{15}\\frac{ m^5}{k^4}-\\frac{ \\pi }{8}\\frac{ m^6}{k^5}+\\frac{32 i }{105}\\frac{ m^7}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{64 i }{315}\\frac{ m^9}{k^8}+\\frac{1024 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;5\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{12 \\pi ^2} \\left(-\\frac{1}{5} m+\\frac{1}{70}\\frac{ k^2}{m}+\\frac{1}{1680}\\frac{ k^4}{m^3}+\\frac{1}{18480}\\frac{ k^6}{m^5}+\\frac{5 }{768768}\\frac{ k^8}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{1098240}\\frac{ k^{10}}{m^9}+\\frac{7 }{49786880}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 1 x 3, dimension 6: $\\tilde{T}_{1,3;6\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{599}{176400}-\\frac{P}{840}\\right) k^2+ \\left(-\\frac{31}{1800}+\\frac{P}{120}\\right) m^2-\\frac{1}{24}\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{7}{72}-\\frac{K}{12}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{19}{144}+\\frac{K}{12}\\right)\\frac{ m^8}{k^6}+ \\left(-\\frac{1}{400}+\\frac{K}{20}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{7}{225}+\\frac{K}{15}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;6\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\nu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{24 \\pi ^3} \\left( \\left(-\\frac{1}{5}+\\frac{L_0}{5}\\right) m^2-\\frac{ L_0}{35} k^2+\\frac{1}{420}\\frac{ k^4}{m^2}+\\frac{1}{6930}\\frac{ k^6}{m^4}+\\frac{1}{72072}\\frac{ k^8}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{600600}\\frac{ k^{10}}{m^8}+\\frac{1}{4375800}\\frac{ k^{12}}{m^{10}}+\\frac{1}{29099070}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,3;6\\text{D}}^{\\text{f,UV-IR}} & = k^4 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{15 \\pi ^3} \\left( \\left(\\frac{599}{11760}-\\frac{K}{56}\\right) k^2+ \\left(-\\frac{2}{15}+\\frac{K}{8}\\right) m^2\\right)\\right)+\\ldots $ Fermions, spin 1 x 5, dimension 3: $\\tilde{T}_{1,5;3\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{128}\\frac{1}{k}+\\frac{3 }{32}\\frac{ m^2}{k^3}-\\frac{4 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{9 }{8}\\frac{ m^4}{k^5}+\\frac{32 i }{5 \\pi }\\frac{ m^5}{k^6}+\\frac{5 }{2}\\frac{ m^6}{k^7}-\\frac{192 i }{35 \\pi }\\frac{ m^7}{k^8}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{1024 i }{315 \\pi }\\frac{ m^9}{k^{10}}-\\frac{1024 i }{231 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(-\\frac{3 i }{32}\\frac{ m}{k}-\\frac{1}{\\pi }\\frac{ m^2}{k^2}+\\frac{3 i }{4}\\frac{ m^3}{k^3}+\\frac{4 }{\\pi }\\frac{ m^4}{k^4}-\\frac{3 i }{2}\\frac{ m^5}{k^5}-\\frac{16 }{5 \\pi }\\frac{ m^6}{k^6}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{64 }{35 \\pi }\\frac{ m^8}{k^8}-\\frac{256 }{105 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{1024 }{231 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{1,5;3\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{4 \\pi } \\left(-\\frac{1}{35}\\frac{1}{m}-\\frac{1}{630}\\frac{ k^2}{m^3}-\\frac{1}{6160}\\frac{ k^4}{m^5}-\\frac{1}{48048}\\frac{ k^6}{m^7}-\\frac{1}{329472}\\frac{ k^8}{m^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{3 }{6223360}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{4 \\pi } \\left(-\\frac{1}{5}-\\frac{1}{140}\\frac{ k^2}{m^2}-\\frac{1}{1680}\\frac{ k^4}{m^4}-\\frac{1}{14784}\\frac{ k^6}{m^6}-\\frac{1}{109824}\\frac{ k^8}{m^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{732160}\\frac{ k^{10}}{m^{10}}-\\frac{1}{4526080}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;3\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 1 x 5, dimension 4: $\\tilde{T}_{1,5;4\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{247}{14700}+\\frac{P}{140}\\right)-\\frac{1}{10}\\frac{ m^2}{k^2}+ \\left(\\frac{13}{12}-\\frac{K}{2}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{4}{3}+2 K\\right)\\frac{ m^6}{k^6}- \\left(\\frac{21}{8}+\\frac{3 K}{2}\\right)\\frac{ m^8}{k^8}- \\left(\\frac{2}{75}+\\frac{4 K}{5}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{5}{12}-K\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;4\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{20 \\pi ^2} \\left(\\frac{ L_0}{7}-\\frac{1}{63}\\frac{ k^2}{m^2}-\\frac{1}{924}\\frac{ k^4}{m^4}-\\frac{1}{9009}\\frac{ k^6}{m^6}-\\frac{1}{72072}\\frac{ k^8}{m^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{510510}\\frac{ k^{10}}{m^{10}}-\\frac{1}{3325608}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;4\\text{D}}^{\\text{f,UV-IR}} & = k^6 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\frac{i }{140 \\pi ^2}\\left(-\\frac{247}{105}+K\\right)+\\ldots $ Fermions, spin 1 x 5, dimension 5: $\\tilde{T}_{1,5;5\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{3 \\pi }{4096} k-\\frac{ \\pi }{256}\\frac{ m^2}{k}-\\frac{3 \\pi }{128}\\frac{ m^4}{k^3}+\\frac{4 i }{15}\\frac{ m^5}{k^4}+\\frac{3 \\pi }{16}\\frac{ m^6}{k^5}-\\frac{32 i }{35}\\frac{ m^7}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{5 \\pi }{16}\\frac{ m^8}{k^7}+\\frac{64 i }{105}\\frac{ m^9}{k^8}+\\frac{1024 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;5\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left(\\frac{1}{35} m-\\frac{1}{630}\\frac{ k^2}{m}-\\frac{1}{18480}\\frac{ k^4}{m^3}-\\frac{1}{240240}\\frac{ k^6}{m^5}-\\frac{1}{2306304}\\frac{ k^8}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{18670080}\\frac{ k^{10}}{m^9}-\\frac{7 }{945950720}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 1 x 5, dimension 6: $\\tilde{T}_{1,5;6\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{1937}{1587600}+\\frac{P}{2520}\\right) k^2+ \\left(\\frac{247}{29400}-\\frac{P}{280}\\right) m^2+\\frac{1}{40}\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{11}{72}+\\frac{K}{12}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{5}{48}-\\frac{K}{4}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{117}{400}+\\frac{3 K}{20}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1}{75}+\\frac{K}{15}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{1,5;6\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\nu \\nu }^2 \\pi _{\\mu \\nu } \\left(\\frac{i }{40 \\pi ^3} \\left( \\left(\\frac{1}{7}-\\frac{L_0}{7}\\right) m^2+\\frac{ L_0}{63} k^2-\\frac{1}{924}\\frac{ k^4}{m^2}-\\frac{1}{18018}\\frac{ k^6}{m^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{216216}\\frac{ k^8}{m^6}-\\frac{1}{2042040}\\frac{ k^{10}}{m^8}-\\frac{1}{16628040}\\frac{ k^{12}}{m^{10}}-\\frac{1}{122216094}\\frac{ k^{14}}{m^{12}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right) $ $\\tilde{T}_{1,5;6\\text{D}}^{\\text{f,UV-IR}} & = k^6 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{140 \\pi ^3} \\left( \\left(-\\frac{1937}{11340}+\\frac{K}{18}\\right) k^2+ \\left(\\frac{71}{105}-\\frac{K}{2}\\right) m^2\\right)\\right)+\\ldots $ Fermions, spin 2 x 2, dimension 3: $\\tilde{T}_{2,2;3\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(-\\frac{1}{32}\\frac{1}{k}+\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{1}{2}\\frac{ m^4}{k^5}-\\frac{8 i }{5 \\pi }\\frac{ m^5}{k^6}-\\frac{32 i }{21 \\pi }\\frac{ m^7}{k^8}-\\frac{128 i }{45 \\pi }\\frac{ m^9}{k^{10}}-\\frac{512 i }{77 \\pi }\\frac{ m^{11}}{k^{12}}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\ldots \\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{1}{64}\\frac{1}{k}-\\frac{1}{8}\\frac{ m^2}{k^3}+\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{1}{4}\\frac{ m^4}{k^5}-\\frac{8 i }{15 \\pi }\\frac{ m^5}{k^6}-\\frac{32 i }{105 \\pi }\\frac{ m^7}{k^8}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{128 i }{315 \\pi }\\frac{ m^9}{k^{10}}-\\frac{512 i }{693 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{i }{8}\\frac{ m}{k}+\\frac{1}{\\pi }\\frac{ m^2}{k^2}-\\frac{i }{2}\\frac{ m^3}{k^3}-\\frac{4 }{3 \\pi }\\frac{ m^4}{k^4}-\\frac{16 }{15 \\pi }\\frac{ m^6}{k^6}-\\frac{64 }{35 \\pi }\\frac{ m^8}{k^8}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{256 }{63 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{1024 }{99 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{2,2;3\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(-\\frac{1}{3}\\frac{ m}{k^2}+\\frac{1}{20}\\frac{1}{m}+\\frac{1}{336}\\frac{ k^2}{m^3}+\\frac{1}{2880}\\frac{ k^4}{m^5}+\\frac{1}{19712}\\frac{ k^6}{m^7}+\\frac{1}{119808}\\frac{ k^8}{m^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{675840}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{6 \\pi } \\left(\\frac{m}{k^2}-\\frac{1}{20}\\frac{1}{m}-\\frac{1}{560}\\frac{ k^2}{m^3}-\\frac{1}{6720}\\frac{ k^4}{m^5}-\\frac{1}{59136}\\frac{ k^6}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{439296}\\frac{ k^8}{m^9}-\\frac{1}{2928640}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu } \\left(\\frac{1}{4 \\pi } \\left(\\frac{1}{3}+\\frac{1}{60}\\frac{ k^2}{m^2}+\\frac{1}{560}\\frac{ k^4}{m^4}+\\frac{1}{4032}\\frac{ k^6}{m^6}+\\frac{1}{25344}\\frac{ k^8}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{146432}\\frac{ k^{10}}{m^{10}}+\\frac{1}{798720}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;3\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 2 x 2, dimension 4: $\\tilde{T}_{2,2;4\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left( \\left(\\frac{3}{25}-\\frac{P}{20}\\right)+ \\left(-\\frac{1}{9}+\\frac{P}{6}\\right)\\frac{ m^2}{k^2}+ \\left(-\\frac{5}{4}+\\frac{K}{2}\\right)\\frac{ m^4}{k^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{7}{6}+K\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{19}{72}-\\frac{5 K}{6}\\right)\\frac{ m^8}{k^8}+ \\left(\\frac{281}{300}-\\frac{7 K}{5}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{151}{60}-3 K\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{23}{900}+\\frac{P}{120}\\right)+ \\left(\\frac{2}{9}-\\frac{P}{12}\\right)\\frac{ m^2}{k^2}+ \\left(-\\frac{1}{8}+\\frac{K}{4}\\right)\\frac{ m^4}{k^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{11}{36}+\\frac{K}{6}\\right)\\frac{ m^6}{k^6}- \\left(\\frac{1}{144}+\\frac{K}{12}\\right)\\frac{ m^8}{k^8}+ \\left(\\frac{23}{600}-\\frac{K}{10}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{37}{360}-\\frac{K}{6}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;4\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{4 \\pi ^2} \\left( \\left(-\\frac{1}{3}+\\frac{L_0}{3}\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{10}+\\frac{1}{84}\\frac{ k^2}{m^2}+\\frac{1}{1080}\\frac{ k^4}{m^4}+\\frac{1}{9240}\\frac{ k^6}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{65520}\\frac{ k^8}{m^8}+\\frac{1}{415800}\\frac{ k^{10}}{m^{10}}+\\frac{1}{2450448}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^2} \\left( \\left(1-L_0\\right)\\frac{ m^2}{k^2}+\\frac{ L_0}{10}-\\frac{1}{140}\\frac{ k^2}{m^2}-\\frac{1}{2520}\\frac{ k^4}{m^4}-\\frac{1}{27720}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{240240}\\frac{ k^8}{m^8}-\\frac{1}{1801800}\\frac{ k^{10}}{m^{10}}-\\frac{1}{12252240}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;4\\text{D}}^{\\text{f,UV-IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left( \\left(\\frac{3}{25}-\\frac{K}{20}\\right)+ \\left(\\frac{1}{18}+\\frac{K}{6}\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{12 \\pi ^2} \\left( \\left(-\\frac{23}{75}+\\frac{K}{10}\\right)+ \\left(\\frac{5}{3}-K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\ldots $ Fermions, spin 2 x 2, dimension 5: $\\tilde{T}_{2,2;5\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{384} k+\\frac{ \\pi }{64}\\frac{ m^2}{k}-\\frac{2 i }{15}\\frac{ m^5}{k^4}-\\frac{ \\pi }{12}\\frac{ m^6}{k^5}+\\frac{8 i }{35}\\frac{ m^7}{k^6}+\\frac{32 i }{189}\\frac{ m^9}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{128 i }{495}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{1536} k-\\frac{ \\pi }{128}\\frac{ m^2}{k}+\\frac{ \\pi }{32}\\frac{ m^4}{k^3}-\\frac{2 i }{15}\\frac{ m^5}{k^4}-\\frac{ \\pi }{24}\\frac{ m^6}{k^5}+\\frac{8 i }{105}\\frac{ m^7}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{32 i }{945}\\frac{ m^9}{k^8}+\\frac{128 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;5\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{2 \\pi ^2} \\left(\\frac{1}{9}\\frac{ m^3}{k^2}-\\frac{1}{20} m+\\frac{1}{336}\\frac{ k^2}{m}+\\frac{1}{8640}\\frac{ k^4}{m^3}+\\frac{1}{98560}\\frac{ k^6}{m^5}+\\frac{1}{838656}\\frac{ k^8}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{6082560}\\frac{ k^{10}}{m^9}+\\frac{1}{39829504}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{6 \\pi ^2} \\left(-\\frac{1}{3}\\frac{ m^3}{k^2}+\\frac{1}{20} m-\\frac{1}{560}\\frac{ k^2}{m}-\\frac{1}{20160}\\frac{ k^4}{m^3}-\\frac{1}{295680}\\frac{ k^6}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{3075072}\\frac{ k^8}{m^7}-\\frac{1}{26357760}\\frac{ k^{10}}{m^9}-\\frac{1}{199147520}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 2 x 2, dimension 6: $\\tilde{T}_{2,2;6\\text{D}}^{\\text{f,t,UV}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{4 \\pi ^3} \\left( \\left(\\frac{31}{1764}-\\frac{P}{168}\\right) k^2+ \\left(-\\frac{3}{25}+\\frac{P}{20}\\right) m^2+ \\left(\\frac{1}{18}-\\frac{P}{12}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{13}{36}-\\frac{K}{6}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{17}{48}+\\frac{K}{4}\\right)\\frac{ m^8}{k^6}+ \\left(-\\frac{7}{360}+\\frac{K}{6}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{211}{1800}+\\frac{7 K}{30}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^3} \\left( \\left(-\\frac{11}{3675}+\\frac{P}{1120}\\right) k^2+ \\left(\\frac{23}{600}-\\frac{P}{80}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{6}+\\frac{P}{16}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{48}-\\frac{K}{8}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{25}{192}+\\frac{K}{16}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{17}{2400}+\\frac{K}{40}\\right)\\frac{ m^{10}}{k^8}+ \\left(-\\frac{13}{2400}+\\frac{K}{40}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;6\\text{D}}^{\\text{f,t,IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{16 \\pi ^3} \\left( \\left(\\frac{1}{2}-\\frac{L_0}{3}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{5}+\\frac{L_0}{5}\\right) m^2-\\frac{ L_0}{42} k^2+\\frac{1}{540}\\frac{ k^4}{m^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{9240}\\frac{ k^6}{m^4}+\\frac{1}{98280}\\frac{ k^8}{m^6}+\\frac{1}{831600}\\frac{ k^{10}}{m^8}+\\frac{1}{6126120}\\frac{ k^{12}}{m^{10}}+\\frac{1}{41081040}\\frac{ k^{14}}{m^{12}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{16 \\pi ^3} \\left( \\left(-\\frac{1}{2}+\\frac{L_0}{3}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{15}-\\frac{L_0}{15}\\right) m^2+\\frac{ L_0}{210} k^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{3780}\\frac{ k^4}{m^2}-\\frac{1}{83160}\\frac{ k^6}{m^4}-\\frac{1}{1081080}\\frac{ k^8}{m^6}-\\frac{1}{10810800}\\frac{ k^{10}}{m^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{91891800}\\frac{ k^{12}}{m^{10}}-\\frac{1}{698377680}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,2;6\\text{D}}^{\\text{f,UV-IR}} & = k^4 \\pi _{\\mu \\nu }^2 \\left(\\frac{i }{16 \\pi ^3} \\left( \\left(\\frac{31}{441}-\\frac{K}{42}\\right) k^2+ \\left(-\\frac{7}{25}+\\frac{K}{5}\\right) m^2- \\left(\\frac{5}{18}+\\frac{K}{3}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^3} \\left( \\left(-\\frac{11}{3675}+\\frac{K}{1120}\\right) k^2+ \\left(\\frac{31}{1200}-\\frac{K}{80}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{7}{96}+\\frac{K}{16}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\ldots $ Fermions, spin 2 x 4, dimension 3: $\\tilde{T}_{2,4;3\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{64}\\frac{1}{k}-\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{3 }{4}\\frac{ m^4}{k^5}+\\frac{24 i }{5 \\pi }\\frac{ m^5}{k^6}+2 \\frac{ m^6}{k^7}-\\frac{32 i }{7 \\pi }\\frac{ m^7}{k^8}-\\frac{128 i }{45 \\pi }\\frac{ m^9}{k^{10}}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{1536 i }{385 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(-\\frac{1}{128}\\frac{1}{k}+\\frac{3 }{32}\\frac{ m^2}{k^3}-\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{3 }{8}\\frac{ m^4}{k^5}+\\frac{8 i }{5 \\pi }\\frac{ m^5}{k^6}+\\frac{1}{2}\\frac{ m^6}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{32 i }{35 \\pi }\\frac{ m^7}{k^8}-\\frac{128 i }{315 \\pi }\\frac{ m^9}{k^{10}}-\\frac{512 i }{1155 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(-\\frac{3 i }{32}\\frac{ m}{k}-\\frac{1}{\\pi }\\frac{ m^2}{k^2}+\\frac{3 i }{4}\\frac{ m^3}{k^3}+\\frac{4 }{\\pi }\\frac{ m^4}{k^4}-\\frac{3 i }{2}\\frac{ m^5}{k^5}-\\frac{16 }{5 \\pi }\\frac{ m^6}{k^6}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{64 }{35 \\pi }\\frac{ m^8}{k^8}-\\frac{256 }{105 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{1024 }{231 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{2,4;3\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi } \\left(\\frac{1}{5}\\frac{ m}{k^2}-\\frac{3 }{140}\\frac{1}{m}-\\frac{1}{1008}\\frac{ k^2}{m^3}-\\frac{1}{10560}\\frac{ k^4}{m^5}-\\frac{3 }{256256}\\frac{ k^6}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{599040}\\frac{ k^8}{m^9}-\\frac{1}{3829760}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{2 \\pi } \\left(-\\frac{1}{5}\\frac{ m}{k^2}+\\frac{1}{140}\\frac{1}{m}+\\frac{1}{5040}\\frac{ k^2}{m^3}+\\frac{1}{73920}\\frac{ k^4}{m^5}+\\frac{1}{768768}\\frac{ k^6}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{6589440}\\frac{ k^8}{m^9}+\\frac{1}{49786880}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{1}{4 \\pi } \\left(-\\frac{1}{5}-\\frac{1}{140}\\frac{ k^2}{m^2}-\\frac{1}{1680}\\frac{ k^4}{m^4}-\\frac{1}{14784}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{109824}\\frac{ k^8}{m^8}-\\frac{1}{732160}\\frac{ k^{10}}{m^{10}}-\\frac{1}{4526080}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;3\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 2 x 4, dimension 4: $\\tilde{T}_{2,4;4\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{141}{4900}+\\frac{3 P}{280}\\right)+ \\left(\\frac{4}{75}-\\frac{P}{20}\\right)\\frac{ m^2}{k^2}+ \\left(\\frac{19}{24}-\\frac{K}{4}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{5}{4}+\\frac{3 K}{2}\\right)\\frac{ m^6}{k^6}- \\left(\\frac{101}{48}+\\frac{5 K}{4}\\right)\\frac{ m^8}{k^8}+ \\left(\\frac{1}{600}-\\frac{7 K}{10}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{79}{200}-\\frac{9 K}{10}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{44}{3675}-\\frac{P}{280}\\right)+ \\left(-\\frac{23}{150}+\\frac{P}{20}\\right)\\frac{ m^2}{k^2}+ \\left(\\frac{7}{24}-\\frac{K}{4}\\right)\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{12}+\\frac{K}{2}\\right)\\frac{ m^6}{k^6}- \\left(\\frac{25}{48}+\\frac{K}{4}\\right)\\frac{ m^8}{k^8}- \\left(\\frac{17}{600}+\\frac{K}{10}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{13}{600}-\\frac{K}{10}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;4\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^2} \\left( \\left(\\frac{1}{5}-\\frac{L_0}{5}\\right)\\frac{ m^2}{k^2}+\\frac{3 L_0}{70}-\\frac{1}{252}\\frac{ k^2}{m^2}-\\frac{1}{3960}\\frac{ k^4}{m^4}-\\frac{1}{40040}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{327600}\\frac{ k^8}{m^8}-\\frac{1}{2356200}\\frac{ k^{10}}{m^{10}}-\\frac{1}{15519504}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{20 \\pi ^2} \\left( \\left(-1+L_0\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{14}+\\frac{1}{252}\\frac{ k^2}{m^2}+\\frac{1}{5544}\\frac{ k^4}{m^4}+\\frac{1}{72072}\\frac{ k^6}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{720720}\\frac{ k^8}{m^8}+\\frac{1}{6126120}\\frac{ k^{10}}{m^{10}}+\\frac{1}{46558512}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;4\\text{D}}^{\\text{f,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{20 \\pi ^2} \\left( \\left(-\\frac{141}{245}+\\frac{3 K}{14}\\right)+ \\left(\\frac{1}{15}-K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{44}{735}-\\frac{K}{56}\\right)+ \\left(-\\frac{31}{60}+\\frac{K}{4}\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\ldots $ Fermions, spin 2 x 4, dimension 5: $\\tilde{T}_{2,4;5\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{1024} k-\\frac{ \\pi }{128}\\frac{ m^2}{k}+\\frac{2 i }{15}\\frac{ m^5}{k^4}+\\frac{ \\pi }{8}\\frac{ m^6}{k^5}-\\frac{24 i }{35}\\frac{ m^7}{k^6}-\\frac{ \\pi }{4}\\frac{ m^8}{k^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{32 i }{63}\\frac{ m^9}{k^8}+\\frac{128 i }{495}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{4096} k+\\frac{ \\pi }{256}\\frac{ m^2}{k}-\\frac{3 \\pi }{128}\\frac{ m^4}{k^3}+\\frac{2 i }{15}\\frac{ m^5}{k^4}+\\frac{ \\pi }{16}\\frac{ m^6}{k^5}-\\frac{8 i }{35}\\frac{ m^7}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{ \\pi }{16}\\frac{ m^8}{k^7}+\\frac{32 i }{315}\\frac{ m^9}{k^8}+\\frac{128 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;5\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left(-\\frac{1}{15}\\frac{ m^3}{k^2}+\\frac{3 }{140} m-\\frac{1}{1008}\\frac{ k^2}{m}-\\frac{1}{31680}\\frac{ k^4}{m^3}-\\frac{3 }{1281280}\\frac{ k^6}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{4193280}\\frac{ k^8}{m^7}-\\frac{1}{34467840}\\frac{ k^{10}}{m^9}-\\frac{3 }{756760576}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{10 \\pi ^2} \\left(\\frac{1}{3}\\frac{ m^3}{k^2}-\\frac{1}{28} m+\\frac{1}{1008}\\frac{ k^2}{m}+\\frac{1}{44352}\\frac{ k^4}{m^3}+\\frac{1}{768768}\\frac{ k^6}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{9225216}\\frac{ k^8}{m^7}+\\frac{1}{89616384}\\frac{ k^{10}}{m^9}+\\frac{1}{756760576}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 2 x 4, dimension 6: $\\tilde{T}_{2,4;6\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{25}{15876}+\\frac{P}{2016}\\right) k^2+ \\left(\\frac{141}{9800}-\\frac{3 P}{560}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{75}+\\frac{P}{80}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{17}{144}+\\frac{K}{24}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{7}{64}-\\frac{3 K}{16}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{113}{480}+\\frac{K}{8}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{23}{2400}+\\frac{7 K}{120}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{563}{1587600}-\\frac{P}{10080}\\right) k^2+ \\left(-\\frac{22}{3675}+\\frac{P}{560}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{23}{600}-\\frac{P}{80}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{5}{144}+\\frac{K}{24}\\right)\\frac{ m^6}{k^4}- \\left(\\frac{1}{192}+\\frac{K}{16}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{137}{2400}+\\frac{K}{40}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{3}{800}+\\frac{K}{120}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;6\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{16 \\pi ^3} \\left( \\left(-\\frac{3}{10}+\\frac{L_0}{5}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{3}{35}-\\frac{3 L_0}{35}\\right) m^2+\\frac{ L_0}{126} k^2-\\frac{1}{1980}\\frac{ k^4}{m^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{40040}\\frac{ k^6}{m^4}-\\frac{1}{491400}\\frac{ k^8}{m^6}-\\frac{1}{4712400}\\frac{ k^{10}}{m^8}-\\frac{1}{38798760}\\frac{ k^{12}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{287567280}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{80 \\pi ^3} \\left( \\left(\\frac{3}{2}-L_0\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{7}+\\frac{L_0}{7}\\right) m^2-\\frac{ L_0}{126} k^2+\\frac{1}{2772}\\frac{ k^4}{m^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{72072}\\frac{ k^6}{m^4}+\\frac{1}{1081080}\\frac{ k^8}{m^6}+\\frac{1}{12252240}\\frac{ k^{10}}{m^8}+\\frac{1}{116396280}\\frac{ k^{12}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{977728752}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{2,4;6\\text{D}}^{\\text{f,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^3} \\left( \\left(-\\frac{25}{3969}+\\frac{K}{504}\\right) k^2+ \\left(\\frac{177}{4900}-\\frac{3 K}{140}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{13}{600}+\\frac{K}{20}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{80 \\pi ^3} \\left( \\left(\\frac{563}{19845}-\\frac{K}{126}\\right) k^2+ \\left(-\\frac{247}{735}+\\frac{K}{7}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{47}{30}-K\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\ldots $ Fermions, spin 3 x 3, dimension 3: $\\tilde{T}_{3,3;3\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{1}{64}\\frac{1}{k}-\\frac{1}{16}\\frac{ m^2}{k^3}-\\frac{1}{4}\\frac{ m^4}{k^5}+\\frac{32 i }{15 \\pi }\\frac{ m^5}{k^6}+\\frac{m^6}{k^7}-\\frac{256 i }{105 \\pi }\\frac{ m^7}{k^8}-\\frac{512 i }{315 \\pi }\\frac{ m^9}{k^{10}}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{8192 i }{3465 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(-\\frac{1}{128}\\frac{1}{k}+\\frac{5 }{32}\\frac{ m^2}{k^3}-\\frac{4 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{7 }{8}\\frac{ m^4}{k^5}+\\frac{64 i }{15 \\pi }\\frac{ m^5}{k^6}+\\frac{3 }{2}\\frac{ m^6}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{64 i }{21 \\pi }\\frac{ m^7}{k^8}-\\frac{512 i }{315 \\pi }\\frac{ m^9}{k^{10}}-\\frac{1024 i }{495 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^2 \\left(-\\frac{i }{16}\\frac{ m}{k}-\\frac{1}{\\pi }\\frac{ m^2}{k^2}+\\frac{i }{2}\\frac{ m^3}{k^3}+\\frac{8 }{3 \\pi }\\frac{ m^4}{k^4}-i \\frac{ m^5}{k^5}-\\frac{32 }{15 \\pi }\\frac{ m^6}{k^6}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{128 }{105 \\pi }\\frac{ m^8}{k^8}-\\frac{512 }{315 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{2048 }{693 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(-\\frac{i }{32}\\frac{ m}{k}+\\frac{i }{4}\\frac{ m^3}{k^3}+\\frac{4 }{3 \\pi }\\frac{ m^4}{k^4}-\\frac{i }{2}\\frac{ m^5}{k^5}-\\frac{16 }{15 \\pi }\\frac{ m^6}{k^6}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{64 }{105 \\pi }\\frac{ m^8}{k^8}-\\frac{256 }{315 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{1024 }{693 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{3,3;3\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi } \\left(\\frac{2 }{5}\\frac{ m}{k^2}-\\frac{1}{35}\\frac{1}{m}-\\frac{1}{840}\\frac{ k^2}{m^3}-\\frac{1}{9240}\\frac{ k^4}{m^5}-\\frac{5 }{384384}\\frac{ k^6}{m^7}-\\frac{1}{549120}\\frac{ k^8}{m^9}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{7 }{24893440}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi } \\left(-\\frac{2 }{5}\\frac{ m}{k^2}+\\frac{1}{140}\\frac{1}{m}-\\frac{1}{73920}\\frac{ k^4}{m^5}-\\frac{1}{384384}\\frac{ k^6}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{2196480}\\frac{ k^8}{m^9}-\\frac{1}{12446720}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^2 \\left(\\frac{1}{3 \\pi } \\left(-\\frac{ m^2}{k^2}-\\frac{1}{10}-\\frac{1}{280}\\frac{ k^2}{m^2}-\\frac{1}{3360}\\frac{ k^4}{m^4}-\\frac{1}{29568}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{219648}\\frac{ k^8}{m^8}-\\frac{1}{1464320}\\frac{ k^{10}}{m^{10}}-\\frac{1}{9052160}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\nu \\nu } \\left(\\frac{1}{3 \\pi } \\left(\\frac{m^2}{k^2}-\\frac{1}{20}-\\frac{1}{560}\\frac{ k^2}{m^2}-\\frac{1}{6720}\\frac{ k^4}{m^4}-\\frac{1}{59136}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{439296}\\frac{ k^8}{m^8}-\\frac{1}{2928640}\\frac{ k^{10}}{m^{10}}-\\frac{1}{18104320}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;3\\text{D}}^{\\text{f,UV-IR}} & = k^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^2 \\left(-\\frac{2 }{3 \\pi }\\frac{ m^2}{k^2}\\right)+\\ldots $ Fermions, spin 3 x 3, dimension 4: $\\tilde{T}_{3,3;4\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{599}{22050}+\\frac{P}{105}\\right)+ \\left(\\frac{31}{225}-\\frac{P}{15}\\right)\\frac{ m^2}{k^2}+\\frac{1}{3}\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{7}{9}+\\frac{2 K}{3}\\right)\\frac{ m^6}{k^6}- \\left(\\frac{19}{18}+\\frac{2 K}{3}\\right)\\frac{ m^8}{k^8}+ \\left(\\frac{1}{50}-\\frac{2 K}{5}\\right)\\frac{ m^{10}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{56}{225}-\\frac{8 K}{15}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{457}{44100}-\\frac{P}{420}\\right)+ \\left(-\\frac{107}{450}+\\frac{P}{15}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{3}{4}-\\frac{K}{2}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{5}{9}+\\frac{4 K}{3}\\right)\\frac{ m^6}{k^6}- \\left(\\frac{113}{72}+\\frac{5 K}{6}\\right)\\frac{ m^8}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{7}{150}+\\frac{2 K}{5}\\right)\\frac{ m^{10}}{k^{10}}+ \\left(\\frac{151}{900}-\\frac{7 K}{15}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;4\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi ^2} \\left( \\left(\\frac{1}{5}-\\frac{L_0}{5}\\right)\\frac{ m^2}{k^2}+\\frac{ L_0}{35}-\\frac{1}{420}\\frac{ k^2}{m^2}-\\frac{1}{6930}\\frac{ k^4}{m^4}-\\frac{1}{72072}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{600600}\\frac{ k^8}{m^8}-\\frac{1}{4375800}\\frac{ k^{10}}{m^{10}}-\\frac{1}{29099070}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left( \\left(-\\frac{1}{5}+\\frac{L_0}{5}\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{140}-\\frac{1}{55440}\\frac{ k^4}{m^4}-\\frac{1}{360360}\\frac{ k^6}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{2402400}\\frac{ k^8}{m^8}-\\frac{1}{15315300}\\frac{ k^{10}}{m^{10}}-\\frac{1}{93117024}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;4\\text{D}}^{\\text{f,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{15 \\pi ^2} \\left( \\left(-\\frac{599}{1470}+\\frac{K}{7}\\right)+ \\left(\\frac{16}{15}-K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{15 \\pi ^2} \\left( \\left(\\frac{457}{2940}-\\frac{K}{28}\\right)+ \\left(-\\frac{77}{30}+K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\ldots $ Fermions, spin 3 x 3, dimension 5: $\\tilde{T}_{3,3;5\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{1}{\\pi ^2} \\left(\\frac{5 \\pi }{6144} k-\\frac{ \\pi }{128}\\frac{ m^2}{k}+\\frac{ \\pi }{64}\\frac{ m^4}{k^3}+\\frac{ \\pi }{24}\\frac{ m^6}{k^5}-\\frac{32 i }{105}\\frac{ m^7}{k^6}-\\frac{ \\pi }{8}\\frac{ m^8}{k^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{256 i }{945}\\frac{ m^9}{k^8}+\\frac{512 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{12288} k+\\frac{ \\pi }{256}\\frac{ m^2}{k}-\\frac{5 \\pi }{128}\\frac{ m^4}{k^3}+\\frac{4 i }{15}\\frac{ m^5}{k^4}+\\frac{7 \\pi }{48}\\frac{ m^6}{k^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{64 i }{105}\\frac{ m^7}{k^6}-\\frac{3 \\pi }{16}\\frac{ m^8}{k^7}+\\frac{64 i }{189}\\frac{ m^9}{k^8}+\\frac{512 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;5\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi ^2} \\left(-\\frac{2 }{15}\\frac{ m^3}{k^2}+\\frac{1}{35} m-\\frac{1}{840}\\frac{ k^2}{m}-\\frac{1}{27720}\\frac{ k^4}{m^3}-\\frac{1}{384384}\\frac{ k^6}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{3843840}\\frac{ k^8}{m^7}-\\frac{7 }{224040960}\\frac{ k^{10}}{m^9}-\\frac{1}{236487680}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{3 \\pi ^2} \\left(\\frac{2 }{15}\\frac{ m^3}{k^2}-\\frac{1}{140} m-\\frac{1}{221760}\\frac{ k^4}{m^3}-\\frac{1}{1921920}\\frac{ k^6}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{15375360}\\frac{ k^8}{m^7}-\\frac{1}{112020480}\\frac{ k^{10}}{m^9}-\\frac{1}{756760576}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 3 x 3, dimension 6: $\\tilde{T}_{3,3;6\\text{D}}^{\\text{f,t,UV}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{3 \\pi ^3} \\left( \\left(-\\frac{1021}{264600}+\\frac{P}{840}\\right) k^2+ \\left(\\frac{599}{14700}-\\frac{P}{70}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{31}{300}+\\frac{P}{20}\\right)\\frac{ m^4}{k^2}-\\frac{1}{6}\\frac{ m^6}{k^4}+ \\left(\\frac{11}{48}-\\frac{K}{4}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{107}{300}+\\frac{K}{5}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{7}{600}+\\frac{K}{10}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{6 \\pi ^3} \\left(\\frac{1}{2520} k^2+ \\left(-\\frac{457}{14700}+\\frac{P}{140}\\right) m^2+ \\left(\\frac{107}{300}-\\frac{P}{10}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{7}{12}+\\frac{K}{2}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{1}{6}-K\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{25}{24}+\\frac{K}{2}\\right)\\frac{ m^{10}}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{17}{300}+\\frac{K}{5}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;6\\text{D}}^{\\text{f,t,IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{2 \\pi ^3} \\left( \\left(-\\frac{1}{20}+\\frac{L_0}{30}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{105}-\\frac{L_0}{105}\\right) m^2+\\frac{ L_0}{1260} k^2-\\frac{1}{20790}\\frac{ k^4}{m^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{432432}\\frac{ k^6}{m^4}-\\frac{1}{5405400}\\frac{ k^8}{m^6}-\\frac{1}{52509600}\\frac{ k^{10}}{m^8}-\\frac{1}{436486050}\\frac{ k^{12}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{3259095840}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{20 \\pi ^3} \\left( \\left(\\frac{1}{2}-\\frac{L_0}{3}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{42}+\\frac{L_0}{42}\\right) m^2-\\frac{1}{16632}\\frac{ k^4}{m^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{216216}\\frac{ k^6}{m^4}-\\frac{1}{2162160}\\frac{ k^8}{m^6}-\\frac{1}{18378360}\\frac{ k^{10}}{m^8}-\\frac{1}{139675536}\\frac{ k^{12}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{977728752}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,3;6\\text{D}}^{\\text{f,UV-IR}} & = k^6 \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{30 \\pi ^3} \\left( \\left(-\\frac{1021}{26460}+\\frac{K}{84}\\right) k^2+ \\left(\\frac{389}{1470}-\\frac{K}{7}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{17}{60}+\\frac{K}{2}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^6 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{i }{15 \\pi ^3} \\left( \\left(-\\frac{44}{735}+\\frac{K}{56}\\right) m^2+ \\left(\\frac{31}{60}-\\frac{K}{4}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\ldots $ Fermions, spin 3 x 5, dimension 3: $\\tilde{T}_{3,5;3\\text{D}}^{\\text{f,t,UV}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(-\\frac{3 }{256}\\frac{1}{k}+\\frac{1}{16}\\frac{ m^2}{k^3}+\\frac{3 }{8}\\frac{ m^4}{k^5}-\\frac{64 i }{15 \\pi }\\frac{ m^5}{k^6}-3 \\frac{ m^6}{k^7}+\\frac{512 i }{35 \\pi }\\frac{ m^7}{k^8}+5 \\frac{ m^8}{k^9}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{1024 i }{105 \\pi }\\frac{ m^9}{k^{10}}-\\frac{16384 i }{3465 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{7 }{1024}\\frac{1}{k}-\\frac{9 }{64}\\frac{ m^2}{k^3}+\\frac{4 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{33 }{32}\\frac{ m^4}{k^5}-\\frac{32 i }{5 \\pi }\\frac{ m^5}{k^6}-\\frac{13 }{4}\\frac{ m^6}{k^7}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{64 i }{5 \\pi }\\frac{ m^7}{k^8}+\\frac{15 }{4}\\frac{ m^8}{k^9}-\\frac{2048 i }{315 \\pi }\\frac{ m^9}{k^{10}}-\\frac{1024 i }{385 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{16}\\frac{ m}{k}+\\frac{1}{\\pi }\\frac{ m^2}{k^2}-\\frac{3 i }{4}\\frac{ m^3}{k^3}-\\frac{16 }{3 \\pi }\\frac{ m^4}{k^4}+3 i \\frac{ m^5}{k^5}+\\frac{64 }{5 \\pi }\\frac{ m^6}{k^6}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- 4 i \\frac{ m^7}{k^7}-\\frac{256 }{35 \\pi }\\frac{ m^8}{k^8}-\\frac{1024 }{315 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{4096 }{1155 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{64}\\frac{ m}{k}-\\frac{3 i }{16}\\frac{ m^3}{k^3}-\\frac{4 }{3 \\pi }\\frac{ m^4}{k^4}+\\frac{3 i }{4}\\frac{ m^5}{k^5}+\\frac{16 }{5 \\pi }\\frac{ m^6}{k^6}-i \\frac{ m^7}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{64 }{35 \\pi }\\frac{ m^8}{k^8}-\\frac{256 }{315 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{1024 }{1155 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{3,5;3\\text{D}}^{\\text{f,t,IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(-\\frac{4 }{35}\\frac{ m}{k^2}+\\frac{2 }{315}\\frac{1}{m}+\\frac{1}{4620}\\frac{ k^2}{m^3}+\\frac{1}{60060}\\frac{ k^4}{m^5}+\\frac{1}{576576}\\frac{ k^6}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{4667520}\\frac{ k^8}{m^9}+\\frac{7 }{236487680}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi } \\left(\\frac{4 }{7}\\frac{ m}{k^2}-\\frac{1}{84}\\frac{1}{m}-\\frac{1}{5544}\\frac{ k^2}{m^3}-\\frac{1}{192192}\\frac{ k^4}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{22404096}\\frac{ k^8}{m^9}+\\frac{1}{94595072}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi } \\left(\\frac{1}{5}\\frac{ m^2}{k^2}+\\frac{1}{35}+\\frac{1}{1260}\\frac{ k^2}{m^2}+\\frac{1}{18480}\\frac{ k^4}{m^4}+\\frac{1}{192192}\\frac{ k^6}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{1647360}\\frac{ k^8}{m^8}+\\frac{1}{12446720}\\frac{ k^{10}}{m^{10}}+\\frac{1}{85995520}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi } \\left(-\\frac{1}{5}\\frac{ m^2}{k^2}+\\frac{1}{140}+\\frac{1}{5040}\\frac{ k^2}{m^2}+\\frac{1}{73920}\\frac{ k^4}{m^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{768768}\\frac{ k^6}{m^6}+\\frac{1}{6589440}\\frac{ k^8}{m^8}+\\frac{1}{49786880}\\frac{ k^{10}}{m^{10}}+\\frac{1}{343982080}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;3\\text{D}}^{\\text{f,UV-IR}} & = k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{4 }{5 \\pi }\\frac{ m^2}{k^2}\\right)+\\ldots $ Fermions, spin 3 x 5, dimension 4: $\\tilde{T}_{3,5;4\\text{D}}^{\\text{f,t,UV}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{1937}{99225}-\\frac{2 P}{315}\\right)+ \\left(-\\frac{494}{3675}+\\frac{2 P}{35}\\right)\\frac{ m^2}{k^2}-\\frac{2 }{5}\\frac{ m^4}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{22}{9}-\\frac{4 K}{3}\\right)\\frac{ m^6}{k^6}+ \\left(-\\frac{5}{3}+4 K\\right)\\frac{ m^8}{k^8}- \\left(\\frac{117}{25}+\\frac{12 K}{5}\\right)\\frac{ m^{10}}{k^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{16}{75}+\\frac{16 K}{15}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{1231}{132300}+\\frac{P}{420}\\right)+ \\left(\\frac{1513}{7350}-\\frac{2 P}{35}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{53}{60}+\\frac{K}{2}\\right)\\frac{ m^4}{k^4}+ (2-2 K)\\frac{ m^6}{k^6}+ \\left(-\\frac{5}{24}+\\frac{7 K}{2}\\right)\\frac{ m^8}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{259}{75}+\\frac{8 K}{5}\\right)\\frac{ m^{10}}{k^{10}}- \\left(\\frac{61}{300}+\\frac{3 K}{5}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;4\\text{D}}^{\\text{f,t,IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{2}{7}+\\frac{2 L_0}{7}\\right)\\frac{ m^2}{k^2}-\\frac{2 L_0}{63}+\\frac{1}{462}\\frac{ k^2}{m^2}+\\frac{1}{9009}\\frac{ k^4}{m^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{108108}\\frac{ k^6}{m^6}+\\frac{1}{1021020}\\frac{ k^8}{m^8}+\\frac{1}{8314020}\\frac{ k^{10}}{m^{10}}+\\frac{1}{61108047}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{35 \\pi ^2} \\left( \\left(2-2 L_0\\right)\\frac{ m^2}{k^2}+\\frac{ L_0}{12}-\\frac{1}{396}\\frac{ k^2}{m^2}-\\frac{1}{20592}\\frac{ k^4}{m^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3500640}\\frac{ k^8}{m^8}+\\frac{1}{16628040}\\frac{ k^{10}}{m^{10}}+\\frac{1}{93117024}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;4\\text{D}}^{\\text{f,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{35 \\pi ^2} \\left( \\left(\\frac{1937}{2835}-\\frac{2 K}{9}\\right)+ \\left(-\\frac{284}{105}+2 K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{35 \\pi ^2} \\left( \\left(-\\frac{1231}{3780}+\\frac{K}{12}\\right)+ \\left(\\frac{1093}{210}-2 K\\right)\\frac{ m^2}{k^2}\\right)\\right)+\\ldots $ Fermions, spin 3 x 5, dimension 5: $\\tilde{T}_{3,5;5\\text{D}}^{\\text{f,t,UV}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{ \\pi }{2048} k+\\frac{3 \\pi }{512}\\frac{ m^2}{k}-\\frac{ \\pi }{64}\\frac{ m^4}{k^3}-\\frac{ \\pi }{16}\\frac{ m^6}{k^5}+\\frac{64 i }{105}\\frac{ m^7}{k^6}+\\frac{3 \\pi }{8}\\frac{ m^8}{k^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{512 i }{315}\\frac{ m^9}{k^8}-\\frac{ \\pi }{2}\\frac{ m^{10}}{k^9}+\\frac{1024 i }{1155}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(\\frac{ \\pi }{8192} k-\\frac{7 \\pi }{2048}\\frac{ m^2}{k}+\\frac{9 \\pi }{256}\\frac{ m^4}{k^3}-\\frac{4 i }{15}\\frac{ m^5}{k^4}-\\frac{11 \\pi }{64}\\frac{ m^6}{k^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{32 i }{35}\\frac{ m^7}{k^6}+\\frac{13 \\pi }{32}\\frac{ m^8}{k^7}-\\frac{64 i }{45}\\frac{ m^9}{k^8}-\\frac{3 \\pi }{8}\\frac{ m^{10}}{k^9}+\\frac{2048 i }{3465}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;5\\text{D}}^{\\text{f,t,IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{15 \\pi ^2} \\left(\\frac{4 }{7}\\frac{ m^3}{k^2}-\\frac{2 }{21} m+\\frac{1}{308}\\frac{ k^2}{m}+\\frac{1}{12012}\\frac{ k^4}{m^3}+\\frac{1}{192192}\\frac{ k^6}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{2178176}\\frac{ k^8}{m^7}+\\frac{7 }{141892608}\\frac{ k^{10}}{m^9}+\\frac{1}{165541376}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{4 }{21}\\frac{ m^3}{k^2}+\\frac{1}{84} m-\\frac{1}{5544}\\frac{ k^2}{m}-\\frac{1}{576576}\\frac{ k^4}{m^3}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{156828672}\\frac{ k^8}{m^7}+\\frac{1}{851355648}\\frac{ k^{10}}{m^9}+\\frac{1}{5297324032}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;5\\text{D}}^{\\text{f,UV-IR}} & = \\ldots \\textrm {(i.e.\\ no overlap)} $ Fermions, spin 3 x 5, dimension 6: $\\tilde{T}_{3,5;6\\text{D}}^{\\text{f,t,UV}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{11861}{16008300}-\\frac{P}{4620}\\right) k^2+ \\left(-\\frac{1937}{198450}+\\frac{P}{315}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{247}{7350}-\\frac{P}{70}\\right)\\frac{ m^4}{k^2}+\\frac{1}{15}\\frac{ m^6}{k^4}+ \\left(-\\frac{19}{72}+\\frac{K}{6}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{13}{150}-\\frac{2 K}{5}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{127}{300}+\\frac{K}{5}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{29497}{192099600}+\\frac{P}{27720}\\right) k^2+ \\left(\\frac{1231}{264600}-\\frac{P}{840}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1513}{29400}+\\frac{P}{70}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{43}{360}-\\frac{K}{12}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{3}{16}+\\frac{K}{4}\\right)\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{59}{1200}+\\frac{7 K}{20}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{31}{100}+\\frac{2 K}{15}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;6\\text{D}}^{\\text{f,t,IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(\\frac{3}{28}-\\frac{L_0}{14}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{63}+\\frac{L_0}{63}\\right) m^2-\\frac{ L_0}{924} k^2+\\frac{1}{18018}\\frac{ k^4}{m^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{432432}\\frac{ k^6}{m^4}+\\frac{1}{6126120}\\frac{ k^8}{m^6}+\\frac{1}{66512160}\\frac{ k^{10}}{m^8}+\\frac{1}{611080470}\\frac{ k^{12}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{4997280288}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{70 \\pi ^3} \\left( \\left(-\\frac{3}{2}+L_0\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{12}-\\frac{L_0}{12}\\right) m^2+\\frac{ L_0}{396} k^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{20592}\\frac{ k^4}{m^2}+\\frac{1}{10501920}\\frac{ k^8}{m^6}+\\frac{1}{66512160}\\frac{ k^{10}}{m^8}+\\frac{1}{465585120}\\frac{ k^{12}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3212537328}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{3,5;6\\text{D}}^{\\text{f,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{35 \\pi ^3} \\left( \\left(\\frac{11861}{457380}-\\frac{K}{132}\\right) k^2+ \\left(-\\frac{1307}{5670}+\\frac{K}{9}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{179}{420}-\\frac{K}{2}\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{70 \\pi ^3} \\left( \\left(-\\frac{29497}{2744280}+\\frac{K}{396}-\\frac{\\log (2)}{198}-\\frac{\\log (\\pi )}{396}+\\right.\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\quad \\left.\\left.\\left.+ \\frac{1}{396} \\log (4 \\pi )\\right) k^2+ \\left(\\frac{229}{945}-\\frac{K}{12}\\right) m^2+ \\left(-\\frac{883}{420}+K\\right)\\frac{ m^4}{k^2}\\right)\\right)+\\ldots $ Fermions, spin 4 x 4, dimension 3: $\\tilde{T}_{4,4;3\\text{D}}^{\\text{f,t,UV}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(-\\frac{1}{128}\\frac{1}{k}+\\frac{1}{16}\\frac{ m^2}{k^3}-\\frac{2 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{16 i }{15 \\pi }\\frac{ m^5}{k^6}-\\frac{ m^6}{k^7}+\\frac{192 i }{35 \\pi }\\frac{ m^7}{k^8}+2 \\frac{ m^8}{k^9}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{256 i }{63 \\pi }\\frac{ m^9}{k^{10}}-\\frac{1024 i }{495 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(-\\frac{3 }{32}\\frac{ m^2}{k^3}+\\frac{2 i }{\\pi }\\frac{ m^3}{k^4}+\\frac{9 }{8}\\frac{ m^4}{k^5}-\\frac{8 i }{\\pi }\\frac{ m^5}{k^6}-\\frac{9 }{2}\\frac{ m^6}{k^7}+\\frac{96 i }{5 \\pi }\\frac{ m^7}{k^8}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ 6 \\frac{ m^8}{k^9}-\\frac{384 i }{35 \\pi }\\frac{ m^9}{k^{10}}-\\frac{512 i }{105 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{3 }{1024}\\frac{1}{k}-\\frac{3 }{64}\\frac{ m^2}{k^3}+\\frac{9 }{32}\\frac{ m^4}{k^5}-\\frac{8 i }{5 \\pi }\\frac{ m^5}{k^6}-\\frac{3 }{4}\\frac{ m^6}{k^7}+\\frac{96 i }{35 \\pi }\\frac{ m^7}{k^8}+\\frac{3 }{4}\\frac{ m^8}{k^9}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{128 i }{105 \\pi }\\frac{ m^9}{k^{10}}-\\frac{512 i }{1155 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^3 \\left(\\frac{i }{32}\\frac{ m}{k}+\\frac{1}{\\pi }\\frac{ m^2}{k^2}-\\frac{3 i }{8}\\frac{ m^3}{k^3}-\\frac{8 }{3 \\pi }\\frac{ m^4}{k^4}+\\frac{3 i }{2}\\frac{ m^5}{k^5}+\\frac{32 }{5 \\pi }\\frac{ m^6}{k^6}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- 2 i \\frac{ m^7}{k^7}-\\frac{128 }{35 \\pi }\\frac{ m^8}{k^8}-\\frac{512 }{315 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{2048 }{1155 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{3 i }{64}\\frac{ m}{k}-\\frac{9 i }{16}\\frac{ m^3}{k^3}-\\frac{4 }{\\pi }\\frac{ m^4}{k^4}+\\frac{9 i }{4}\\frac{ m^5}{k^5}+\\frac{48 }{5 \\pi }\\frac{ m^6}{k^6}-3 i \\frac{ m^7}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{192 }{35 \\pi }\\frac{ m^8}{k^8}-\\frac{256 }{105 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{1024 }{385 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{4,4;3\\text{D}}^{\\text{f,t,IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi } \\left(-\\frac{2 }{5}\\frac{ m^3}{k^4}-\\frac{3 }{35}\\frac{ m}{k^2}+\\frac{1}{252}\\frac{1}{m}+\\frac{1}{7920}\\frac{ k^2}{m^3}+\\frac{3 }{320320}\\frac{ k^4}{m^5}+\\frac{1}{1048320}\\frac{ k^6}{m^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{8616960}\\frac{ k^8}{m^9}+\\frac{3 }{189190144}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi } \\left(\\frac{4 }{5}\\frac{ m^3}{k^4}+\\frac{3 }{70}\\frac{ m}{k^2}+\\frac{1}{840}\\frac{1}{m}+\\frac{1}{12320}\\frac{ k^2}{m^3}+\\frac{1}{128128}\\frac{ k^4}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{1098240}\\frac{ k^6}{m^7}+\\frac{3 }{24893440}\\frac{ k^8}{m^9}+\\frac{3 }{171991040}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi } \\left(-2 \\frac{ m^3}{k^4}+\\frac{3 }{14}\\frac{ m}{k^2}-\\frac{1}{168}\\frac{1}{m}-\\frac{1}{7392}\\frac{ k^2}{m^3}-\\frac{1}{128128}\\frac{ k^4}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{1537536}\\frac{ k^6}{m^7}-\\frac{1}{14936064}\\frac{ k^8}{m^9}-\\frac{3 }{378380288}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^3 \\left(\\frac{1}{\\pi } \\left(\\frac{3 }{5}\\frac{ m^2}{k^2}+\\frac{1}{70}+\\frac{1}{2520}\\frac{ k^2}{m^2}+\\frac{1}{36960}\\frac{ k^4}{m^4}+\\frac{1}{384384}\\frac{ k^6}{m^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3294720}\\frac{ k^8}{m^8}+\\frac{1}{24893440}\\frac{ k^{10}}{m^{10}}+\\frac{1}{171991040}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\mu \\nu } \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi } \\left(-\\frac{3 }{5}\\frac{ m^2}{k^2}+\\frac{3 }{140}+\\frac{1}{1680}\\frac{ k^2}{m^2}+\\frac{1}{24640}\\frac{ k^4}{m^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{256256}\\frac{ k^6}{m^6}+\\frac{1}{2196480}\\frac{ k^8}{m^8}+\\frac{3 }{49786880}\\frac{ k^{10}}{m^{10}}+\\frac{3 }{343982080}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;3\\text{D}}^{\\text{f,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(-\\frac{4 i }{15 \\pi }\\frac{ m^3}{k^4}\\right)+k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{6 i }{5 \\pi }\\frac{ m^3}{k^4}\\right)+k^6 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^3 \\left(\\frac{2 }{5 \\pi }\\frac{ m^2}{k^2}\\right)+\\ldots $ Fermions, spin 4 x 4, dimension 4: $\\tilde{T}_{4,4;4\\text{D}}^{\\text{f,t,UV}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{50}{3969}-\\frac{P}{252}\\right)+ \\left(-\\frac{141}{1225}+\\frac{3 P}{70}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{161}{600}-\\frac{P}{10}+\\frac{L_0}{4}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{17}{18}-\\frac{K}{3}\\right)\\frac{ m^6}{k^6}+ \\left(-\\frac{7}{8}+\\frac{3 K}{2}\\right)\\frac{ m^8}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{113}{60}+K\\right)\\frac{ m^{10}}{k^{10}}- \\left(\\frac{23}{300}+\\frac{7 K}{15}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{62}{33075}-\\frac{P}{840}\\right)+ \\left(\\frac{281}{2450}-\\frac{3 P}{140}\\right)\\frac{ m^2}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{111}{200}-\\frac{9 P}{20}+\\frac{3 L_0}{4}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{37}{12}-\\frac{5 K}{2}\\right)\\frac{ m^6}{k^6}+ \\left(-\\frac{17}{16}+\\frac{21 K}{4}\\right)\\frac{ m^8}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{1113}{200}+\\frac{27 K}{10}\\right)\\frac{ m^{10}}{k^{10}}- \\left(\\frac{59}{200}+\\frac{11 K}{10}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{563}{132300}+\\frac{P}{840}\\right)+ \\left(\\frac{88}{1225}-\\frac{3 P}{140}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{23}{50}+\\frac{3 P}{20}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{5}{12}-\\frac{K}{2}\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{1}{16}+\\frac{3 K}{4}\\right)\\frac{ m^8}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{137}{200}+\\frac{3 K}{10}\\right)\\frac{ m^{10}}{k^{10}}- \\left(\\frac{9}{200}+\\frac{K}{10}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;4\\text{D}}^{\\text{f,t,IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{2 \\pi ^2} \\left( \\left(-\\frac{9}{20}+\\frac{3 L_0}{10}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{3}{35}+\\frac{3 L_0}{35}\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{126}+\\frac{1}{1980}\\frac{ k^2}{m^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{40040}\\frac{ k^4}{m^4}+\\frac{1}{491400}\\frac{ k^6}{m^6}+\\frac{1}{4712400}\\frac{ k^8}{m^8}+\\frac{1}{38798760}\\frac{ k^{10}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{287567280}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{2 \\pi ^2} \\left( \\left(\\frac{9}{10}-\\frac{3 L_0}{5}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{3}{70}-\\frac{3 L_0}{70}\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{420}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3080}\\frac{ k^2}{m^2}+\\frac{1}{48048}\\frac{ k^4}{m^4}+\\frac{1}{514800}\\frac{ k^6}{m^6}+\\frac{3 }{13613600}\\frac{ k^8}{m^8}+\\frac{1}{35271600}\\frac{ k^{10}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{250699680}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{20 \\pi ^2} \\left( \\left(-\\frac{9}{2}+3 L_0\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{3}{7}-\\frac{3 L_0}{7}\\right)\\frac{ m^2}{k^2}+\\frac{ L_0}{42}-\\frac{1}{924}\\frac{ k^2}{m^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{24024}\\frac{ k^4}{m^4}-\\frac{1}{360360}\\frac{ k^6}{m^6}-\\frac{1}{4084080}\\frac{ k^8}{m^8}-\\frac{1}{38798760}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{325909584}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;4\\text{D}}^{\\text{f,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{50}{3969}-\\frac{K}{252}\\right)+ \\left(-\\frac{177}{2450}+\\frac{3 K}{70}\\right)\\frac{ m^2}{k^2}- \\left(\\frac{13}{300}+\\frac{K}{10}\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{62}{6615}-\\frac{K}{168}\\right)+ \\left(\\frac{457}{980}-\\frac{3 K}{28}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{201}{40}+\\frac{9 K}{4}\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{20 \\pi ^2} \\left( \\left(-\\frac{563}{6615}+\\frac{K}{42}\\right)+ \\left(\\frac{247}{245}-\\frac{3 K}{7}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{47}{10}+3 K\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\ldots $ Fermions, spin 4 x 4, dimension 5: $\\tilde{T}_{4,4;5\\text{D}}^{\\text{f,t,UV}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{3 \\pi }{10240} k+\\frac{ \\pi }{256}\\frac{ m^2}{k}-\\frac{ \\pi }{64}\\frac{ m^4}{k^3}+\\frac{2 i }{15}\\frac{ m^5}{k^4}+\\frac{16 i }{105}\\frac{ m^7}{k^6}+\\frac{ \\pi }{8}\\frac{ m^8}{k^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{64 i }{105}\\frac{ m^9}{k^8}-\\frac{ \\pi }{5}\\frac{ m^{10}}{k^9}+\\frac{256 i }{693}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{3 \\pi }{20480} k+\\frac{3 \\pi }{128}\\frac{ m^4}{k^3}-\\frac{2 i }{5}\\frac{ m^5}{k^4}-\\frac{3 \\pi }{16}\\frac{ m^6}{k^5}+\\frac{8 i }{7}\\frac{ m^7}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{9 \\pi }{16}\\frac{ m^8}{k^7}-\\frac{32 i }{15}\\frac{ m^9}{k^8}-\\frac{3 \\pi }{5}\\frac{ m^{10}}{k^9}+\\frac{384 i }{385}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(\\frac{3 \\pi }{40960} k-\\frac{3 \\pi }{2048}\\frac{ m^2}{k}+\\frac{3 \\pi }{256}\\frac{ m^4}{k^3}-\\frac{3 \\pi }{64}\\frac{ m^6}{k^5}+\\frac{8 i }{35}\\frac{ m^7}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{3 \\pi }{32}\\frac{ m^8}{k^7}-\\frac{32 i }{105}\\frac{ m^9}{k^8}-\\frac{3 \\pi }{40}\\frac{ m^{10}}{k^9}+\\frac{128 i }{1155}\\frac{ m^{11}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;5\\text{D}}^{\\text{f,t,IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^2} \\left(\\frac{2 }{25}\\frac{ m^5}{k^4}+\\frac{1}{35}\\frac{ m^3}{k^2}-\\frac{1}{252} m+\\frac{1}{7920}\\frac{ k^2}{m}+\\frac{1}{320320}\\frac{ k^4}{m^3}+\\frac{1}{5241600}\\frac{ k^6}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{60318720}\\frac{ k^8}{m^7}+\\frac{1}{567570432}\\frac{ k^{10}}{m^9}+\\frac{1}{4674109440}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left(-\\frac{4 }{25}\\frac{ m^5}{k^4}-\\frac{1}{70}\\frac{ m^3}{k^2}-\\frac{1}{840} m+\\frac{1}{12320}\\frac{ k^2}{m}+\\frac{1}{384384}\\frac{ k^4}{m^3}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{5491200}\\frac{ k^6}{m^5}+\\frac{3 }{174254080}\\frac{ k^8}{m^7}+\\frac{1}{515973120}\\frac{ k^{10}}{m^9}+\\frac{1}{4074864640}\\frac{ k^{12}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left(\\frac{2 }{5}\\frac{ m^5}{k^4}-\\frac{1}{14}\\frac{ m^3}{k^2}+\\frac{1}{168} m-\\frac{1}{7392}\\frac{ k^2}{m}-\\frac{1}{384384}\\frac{ k^4}{m^3}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{7687680}\\frac{ k^6}{m^5}-\\frac{1}{104552448}\\frac{ k^8}{m^7}-\\frac{1}{1135140864}\\frac{ k^{10}}{m^9}-\\frac{1}{10594648064}\\frac{ k^{12}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right) $ $\\tilde{T}_{4,4;5\\text{D}}^{\\text{f,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{4 i }{75 \\pi ^2}\\frac{ m^5}{k^4}\\right)+k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(-\\frac{6 i }{25 \\pi ^2}\\frac{ m^5}{k^4}\\right)+\\ldots $ Fermions, spin 4 x 4, dimension 6: $\\tilde{T}_{4,4;6\\text{D}}^{\\text{f,t,UV}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{859}{1960200}-\\frac{P}{7920}\\right) k^2+ \\left(-\\frac{25}{3969}+\\frac{P}{504}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{141}{4900}-\\frac{3 P}{280}\\right)\\frac{ m^4}{k^2}- \\left(-\\frac{211}{3600}-\\frac{P}{60}+\\frac{L_0}{24}\\right)\\frac{ m^6}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{31}{288}+\\frac{K}{24}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{23}{400}-\\frac{3 K}{20}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{41}{240}+\\frac{K}{12}\\right)\\frac{ m^{12}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{5353}{21344400}-\\frac{P}{12320}\\right) k^2+ \\left(-\\frac{31}{33075}+\\frac{P}{1680}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{281}{9800}+\\frac{3 P}{560}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{61}{1200}-\\frac{3 P}{40}+\\frac{L_0}{8}\\right)\\frac{ m^6}{k^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{59}{192}+\\frac{5 K}{16}\\right)\\frac{ m^8}{k^6}+ \\left(\\frac{1}{800}-\\frac{21 K}{40}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{401}{800}+\\frac{9 K}{40}\\right)\\frac{ m^{12}}{k^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{1627}{16008300}+\\frac{P}{36960}\\right) k^2+ \\left(\\frac{563}{264600}-\\frac{P}{1680}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{22}{1225}+\\frac{3 P}{560}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{23}{300}-\\frac{P}{40}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{7}{192}+\\frac{K}{16}\\right)\\frac{ m^8}{k^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{17}{800}+\\frac{3 K}{40}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{49}{800}+\\frac{K}{40}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;6\\text{D}}^{\\text{f,t,IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{8 \\pi ^3} \\left( \\left(\\frac{11}{30}-\\frac{L_0}{5}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{9}{70}-\\frac{3 L_0}{35}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1}{63}+\\frac{L_0}{63}\\right) m^2-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{ L_0}{990} k^2+\\frac{1}{20020}\\frac{ k^4}{m^2}+\\frac{1}{491400}\\frac{ k^6}{m^4}+\\frac{1}{7068600}\\frac{ k^8}{m^6}+\\frac{1}{77597520}\\frac{ k^{10}}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{718918200}\\frac{ k^{12}}{m^{10}}+\\frac{1}{5917831920}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{4 \\pi ^3} \\left( \\left(-\\frac{11}{30}+\\frac{L_0}{5}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{9}{280}+\\frac{3 L_0}{140}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{420}+\\frac{L_0}{420}\\right) m^2-\\frac{ L_0}{3080} k^2+\\frac{1}{48048}\\frac{ k^4}{m^2}+\\frac{1}{1029600}\\frac{ k^6}{m^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{13613600}\\frac{ k^8}{m^6}+\\frac{1}{141086400}\\frac{ k^{10}}{m^8}+\\frac{1}{1253498400}\\frac{ k^{12}}{m^{10}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{9994560576}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{40 \\pi ^3} \\left( \\left(\\frac{11}{6}-L_0\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{9}{28}+\\frac{3 L_0}{14}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{1}{42}-\\frac{L_0}{42}\\right) m^2+\\frac{ L_0}{924} k^2-\\frac{1}{24024}\\frac{ k^4}{m^2}-\\frac{1}{720720}\\frac{ k^6}{m^4}-\\frac{1}{12252240}\\frac{ k^8}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{155195040}\\frac{ k^{10}}{m^8}-\\frac{1}{1629547920}\\frac{ k^{12}}{m^{10}}-\\frac{1}{14991840864}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{4,4;6\\text{D}}^{\\text{f,UV-IR}} & = k^8 \\pi _{\\mu \\nu }^4 \\left(\\frac{i }{4 \\pi ^3} \\left( \\left(\\frac{859}{490050}-\\frac{K}{1980}\\right) k^2+ \\left(-\\frac{137}{7938}+\\frac{K}{126}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{249}{4900}-\\frac{3 K}{70}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{23}{450}+\\frac{K}{15}\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{i }{40 \\pi ^3} \\left( \\left(\\frac{5353}{533610}-\\frac{K}{308}\\right) k^2+ \\left(-\\frac{181}{13230}+\\frac{K}{42}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{809}{980}+\\frac{3 K}{14}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{57}{10}-3 K\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{20 \\pi ^3} \\left( \\left(-\\frac{1627}{800415}+\\frac{K}{1848}\\right) k^2+ \\left(\\frac{811}{26460}-\\frac{K}{84}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{389}{1960}+\\frac{3 K}{28}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{37}{60}-\\frac{K}{2}\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\ldots $ Fermions, spin 5 x 5, dimension 3: $\\tilde{T}_{5,5;3\\text{D}}^{\\text{f,t,UV}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{1}{256}\\frac{1}{k}-\\frac{3 }{64}\\frac{ m^2}{k^3}+\\frac{4 i }{3 \\pi }\\frac{ m^3}{k^4}+\\frac{1}{8}\\frac{ m^4}{k^5}+\\frac{1}{2}\\frac{ m^6}{k^7}-\\frac{512 i }{105 \\pi }\\frac{ m^7}{k^8}-3 \\frac{ m^8}{k^9}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{4096 i }{315 \\pi }\\frac{ m^9}{k^{10}}+4 \\frac{ m^{10}}{k^{11}}-\\frac{8192 i }{1155 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{1}{256}\\frac{1}{k}+\\frac{1}{64}\\frac{ m^2}{k^3}-\\frac{8 i }{3 \\pi }\\frac{ m^3}{k^4}-\\frac{7 }{8}\\frac{ m^4}{k^5}+\\frac{128 i }{15 \\pi }\\frac{ m^5}{k^6}+\\frac{13 }{2}\\frac{ m^6}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{4096 i }{105 \\pi }\\frac{ m^7}{k^8}-19 \\frac{ m^8}{k^9}+\\frac{22528 i }{315 \\pi }\\frac{ m^9}{k^{10}}+20 \\frac{ m^{10}}{k^{11}}-\\frac{16384 i }{495 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(-\\frac{9 }{2048}\\frac{1}{k}+\\frac{51 }{512}\\frac{ m^2}{k^3}-\\frac{57 }{64}\\frac{ m^4}{k^5}+\\frac{32 i }{5 \\pi }\\frac{ m^5}{k^6}+\\frac{63 }{16}\\frac{ m^6}{k^7}-\\right.\\nonumber \\\\ & \\quad \\quad \\left.- \\frac{704 i }{35 \\pi }\\frac{ m^7}{k^8}-\\frac{69 }{8}\\frac{ m^8}{k^9}+\\frac{1024 i }{35 \\pi }\\frac{ m^9}{k^{10}}+\\frac{15 }{2}\\frac{ m^{10}}{k^{11}}-\\frac{13312 i }{1155 \\pi }\\frac{ m^{11}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^4 \\left(-\\frac{i }{64}\\frac{ m}{k}-\\frac{1}{\\pi }\\frac{ m^2}{k^2}+\\frac{i }{4}\\frac{ m^3}{k^3}+\\frac{4 }{3 \\pi }\\frac{ m^4}{k^4}-\\frac{3 i }{2}\\frac{ m^5}{k^5}-\\frac{128 }{15 \\pi }\\frac{ m^6}{k^6}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ 4 i \\frac{ m^7}{k^7}+\\frac{512 }{35 \\pi }\\frac{ m^8}{k^8}-4 i \\frac{ m^9}{k^9}-\\frac{2048 }{315 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{8192 }{3465 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(-\\frac{3 i }{64}\\frac{ m}{k}+\\frac{3 i }{4}\\frac{ m^3}{k^3}+\\frac{8 }{\\pi }\\frac{ m^4}{k^4}-\\frac{9 i }{2}\\frac{ m^5}{k^5}-\\frac{128 }{5 \\pi }\\frac{ m^6}{k^6}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ 12 i \\frac{ m^7}{k^7}+\\frac{1536 }{35 \\pi }\\frac{ m^8}{k^8}-12 i \\frac{ m^9}{k^9}-\\frac{2048 }{105 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{8192 }{1155 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(-\\frac{3 i }{512}\\frac{ m}{k}+\\frac{3 i }{32}\\frac{ m^3}{k^3}-\\frac{9 i }{16}\\frac{ m^5}{k^5}-\\frac{16 }{5 \\pi }\\frac{ m^6}{k^6}+\\frac{3 i }{2}\\frac{ m^7}{k^7}+\\right.\\nonumber \\\\ & \\quad \\quad \\left.+ \\frac{192 }{35 \\pi }\\frac{ m^8}{k^8}-\\frac{3 i }{2}\\frac{ m^9}{k^9}-\\frac{256 }{105 \\pi }\\frac{ m^{10}}{k^{10}}-\\frac{1024 }{1155 \\pi }\\frac{ m^{12}}{k^{12}}+\\ldots \\right) $ $\\tilde{T}_{5,5;3\\text{D}}^{\\text{f,t,IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{5 \\pi } \\left(\\frac{36 }{7}\\frac{ m^3}{k^4}+\\frac{16 }{63}\\frac{ m}{k^2}-\\frac{2 }{231}\\frac{1}{m}-\\frac{2 }{9009}\\frac{ k^2}{m^3}-\\frac{1}{72072}\\frac{ k^4}{m^5}-\\frac{1}{816816}\\frac{ k^6}{m^7}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{7 }{53209728}\\frac{ k^8}{m^9}-\\frac{1}{62078016}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi } \\left(-\\frac{72 }{7}\\frac{ m^3}{k^4}+\\frac{8 }{63}\\frac{ m}{k^2}-\\frac{8 }{693}\\frac{1}{m}-\\frac{1}{2574}\\frac{ k^2}{m^3}-\\frac{1}{36036}\\frac{ k^4}{m^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{376992}\\frac{ k^6}{m^7}-\\frac{1}{3325608}\\frac{ k^8}{m^9}-\\frac{1}{26138112}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi } \\left(\\frac{36 }{7}\\frac{ m^3}{k^4}-\\frac{8 }{21}\\frac{ m}{k^2}+\\frac{1}{132}\\frac{1}{m}+\\frac{1}{8008}\\frac{ k^2}{m^3}+\\frac{1}{192192}\\frac{ k^4}{m^5}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{3267264}\\frac{ k^6}{m^7}+\\frac{1}{47297536}\\frac{ k^8}{m^9}+\\frac{1}{662165504}\\frac{ k^{10}}{m^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^4 \\left(\\frac{1}{5 \\pi } \\left(-4 \\frac{ m^4}{k^4}-\\frac{27 }{7}\\frac{ m^2}{k^2}-\\frac{2 }{63}-\\frac{1}{1386}\\frac{ k^2}{m^2}-\\frac{1}{24024}\\frac{ k^4}{m^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{288288}\\frac{ k^6}{m^6}-\\frac{1}{2800512}\\frac{ k^8}{m^8}-\\frac{1}{23648768}\\frac{ k^{10}}{m^{10}}-\\frac{1}{180590592}\\frac{ k^{12}}{m^{12}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{1}{5 \\pi } \\left(8 \\frac{ m^4}{k^4}+\\frac{24 }{7}\\frac{ m^2}{k^2}-\\frac{2 }{21}-\\frac{1}{462}\\frac{ k^2}{m^2}-\\frac{1}{8008}\\frac{ k^4}{m^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{96096}\\frac{ k^6}{m^6}-\\frac{1}{933504}\\frac{ k^8}{m^8}-\\frac{3 }{23648768}\\frac{ k^{10}}{m^{10}}-\\frac{1}{60196864}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu }^2 \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{5 \\pi } \\left(-4 \\frac{ m^4}{k^4}+\\frac{3 }{7}\\frac{ m^2}{k^2}-\\frac{1}{84}-\\frac{1}{3696}\\frac{ k^2}{m^2}-\\frac{1}{64064}\\frac{ k^4}{m^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{768768}\\frac{ k^6}{m^6}-\\frac{1}{7468032}\\frac{ k^8}{m^8}-\\frac{3 }{189190144}\\frac{ k^{10}}{m^{10}}-\\frac{1}{481574912}\\frac{ k^{12}}{m^{12}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right) $ $\\tilde{T}_{5,5;3\\text{D}}^{\\text{f,UV-IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{32 i }{105 \\pi }\\frac{ m^3}{k^4}\\right)+k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(-\\frac{64 i }{105 \\pi }\\frac{ m^3}{k^4}\\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\nu }^4 \\left(\\frac{1}{5 \\pi } \\left(-\\frac{8 }{7}\\frac{ m^2}{k^2}+\\frac{32 }{3}\\frac{ m^4}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^8 (k\\cdot \\epsilon )_{\\mu \\nu } \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^2 \\pi _{\\nu \\nu } \\left(\\frac{32 }{5 \\pi }\\frac{ m^4}{k^4}\\right)+\\ldots $ Fermions, spin 5 x 5, dimension 4: $\\tilde{T}_{5,5;4\\text{D}}^{\\text{f,t,UV}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{23722}{4002075}+\\frac{2 P}{1155}\\right)+ \\left(\\frac{7748}{99225}-\\frac{8 P}{315}\\right)\\frac{ m^2}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(-\\frac{7073}{14700}-\\frac{4 P}{35}+\\frac{L_0}{2}\\right)\\frac{ m^4}{k^4}-\\frac{8 }{15}\\frac{ m^6}{k^6}+ \\left(\\frac{19}{9}-\\frac{4 K}{3}\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{52}{75}+\\frac{16 K}{5}\\right)\\frac{ m^{10}}{k^{10}}- \\left(\\frac{254}{75}+\\frac{8 K}{5}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^2} \\left( \\left(-\\frac{83338}{12006225}+\\frac{8 P}{3465}\\right)+ \\left(\\frac{724}{99225}-\\frac{4 P}{315}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{2873}{7350}-\\frac{8 P}{35}+L_0\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{44}{9}+\\frac{8 K}{3}\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{92}{9}-\\frac{32 K}{3}\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{14}{75}+\\frac{88 K}{5}\\right)\\frac{ m^{10}}{k^{10}}- \\left(\\frac{1252}{75}+\\frac{112 K}{15}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^2} \\left( \\left(\\frac{13511}{2286900}-\\frac{P}{660}\\right)+ \\left(-\\frac{9323}{66150}+\\frac{4 P}{105}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{3273}{2450}-\\frac{27 P}{70}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{8}{3}+2 K\\right)\\frac{ m^6}{k^6}+ \\left(\\frac{89}{24}-\\frac{11 K}{2}\\right)\\frac{ m^8}{k^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{36}{25}+\\frac{36 K}{5}\\right)\\frac{ m^{10}}{k^{10}}- \\left(\\frac{617}{100}+\\frac{13 K}{5}\\right)\\frac{ m^{12}}{k^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;4\\text{D}}^{\\text{f,t,IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{81}{28}-\\frac{27 L_0}{14}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{8}{63}-\\frac{8 L_0}{63}\\right)\\frac{ m^2}{k^2}+\\frac{2 L_0}{231}-\\frac{4 }{9009}\\frac{ k^2}{m^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{54054}\\frac{ k^4}{m^4}-\\frac{1}{765765}\\frac{ k^6}{m^6}-\\frac{1}{8314020}\\frac{ k^8}{m^8}-\\frac{4 }{305540235}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{624660036}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(-\\frac{81}{14}+\\frac{27 L_0}{7}\\right)\\frac{ m^4}{k^4}+ \\left(\\frac{4}{63}-\\frac{4 L_0}{63}\\right)\\frac{ m^2}{k^2}+\\frac{8 L_0}{693}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{1287}\\frac{ k^2}{m^2}-\\frac{1}{27027}\\frac{ k^4}{m^4}-\\frac{1}{353430}\\frac{ k^6}{m^6}-\\frac{4 }{14549535}\\frac{ k^8}{m^8}-\\frac{1}{32162130}\\frac{ k^{10}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{255542742}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{81}{28}-\\frac{27 L_0}{14}\\right)\\frac{ m^4}{k^4}+ \\left(-\\frac{4}{21}+\\frac{4 L_0}{21}\\right)\\frac{ m^2}{k^2}-\\frac{ L_0}{132}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{4004}\\frac{ k^2}{m^2}+\\frac{1}{144144}\\frac{ k^4}{m^4}+\\frac{1}{3063060}\\frac{ k^6}{m^6}+\\frac{1}{51731680}\\frac{ k^8}{m^8}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{814773960}\\frac{ k^{10}}{m^{10}}+\\frac{1}{14991840864}\\frac{ k^{12}}{m^{12}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;4\\text{D}}^{\\text{f,UV-IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{35 \\pi ^2} \\left( \\left(-\\frac{23722}{114345}+\\frac{2 K}{33}\\right)+ \\left(\\frac{5228}{2835}-\\frac{8 K}{9}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{358}{105}+4 K\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{35 \\pi ^2} \\left( \\left(-\\frac{83338}{343035}+\\frac{8 K}{99}\\right)- \\left(\\frac{536}{2835}+\\frac{4 K}{9}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{2816}{105}-8 K\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left( \\left(\\frac{13511}{457380}-\\frac{K}{132}\\right)+ \\left(-\\frac{6803}{13230}+\\frac{4 K}{21}\\right)\\frac{ m^2}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{3711}{980}-\\frac{27 K}{14}\\right)\\frac{ m^4}{k^4}\\right)\\right)+\\ldots $ Fermions, spin 5 x 5, dimension 5: $\\tilde{T}_{5,5;5\\text{D}}^{\\text{f,t,UV}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{1}{\\pi ^2} \\left(\\frac{7 \\pi }{61440} k-\\frac{ \\pi }{512}\\frac{ m^2}{k}+\\frac{3 \\pi }{256}\\frac{ m^4}{k^3}-\\frac{4 i }{15}\\frac{ m^5}{k^4}-\\frac{ \\pi }{48}\\frac{ m^6}{k^5}-\\frac{ \\pi }{16}\\frac{ m^8}{k^7}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{512 i }{945}\\frac{ m^9}{k^8}+\\frac{3 \\pi }{10}\\frac{ m^{10}}{k^9}-\\frac{4096 i }{3465}\\frac{ m^{11}}{k^{10}}-\\frac{ \\pi }{3}\\frac{ m^{12}}{k^{11}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{1}{\\pi ^2} \\left(\\frac{11 \\pi }{61440} k-\\frac{ \\pi }{512}\\frac{ m^2}{k}-\\frac{ \\pi }{256}\\frac{ m^4}{k^3}+\\frac{8 i }{15}\\frac{ m^5}{k^4}+\\frac{7 \\pi }{48}\\frac{ m^6}{k^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{128 i }{105}\\frac{ m^7}{k^6}-\\frac{13 \\pi }{16}\\frac{ m^8}{k^7}+\\frac{4096 i }{945}\\frac{ m^9}{k^8}+\\frac{19 \\pi }{10}\\frac{ m^{10}}{k^9}-\\frac{2048 i }{315}\\frac{ m^{11}}{k^{10}}-\\frac{5 \\pi }{3}\\frac{ m^{12}}{k^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{1}{\\pi ^2} \\left(-\\frac{13 \\pi }{163840} k+\\frac{9 \\pi }{4096}\\frac{ m^2}{k}-\\frac{51 \\pi }{2048}\\frac{ m^4}{k^3}+\\frac{19 \\pi }{128}\\frac{ m^6}{k^5}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{32 i }{35}\\frac{ m^7}{k^6}-\\frac{63 \\pi }{128}\\frac{ m^8}{k^7}+\\frac{704 i }{315}\\frac{ m^9}{k^8}+\\frac{69 \\pi }{80}\\frac{ m^{10}}{k^9}-\\frac{1024 i }{385}\\frac{ m^{11}}{k^{10}}-\\frac{5 \\pi }{8}\\frac{ m^{12}}{k^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right) $ $\\tilde{T}_{5,5;5\\text{D}}^{\\text{f,t,IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{36 }{35}\\frac{ m^5}{k^4}-\\frac{16 }{189}\\frac{ m^3}{k^2}+\\frac{2 }{231} m-\\frac{2 }{9009}\\frac{ k^2}{m}-\\frac{1}{216216}\\frac{ k^4}{m^3}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{4084080}\\frac{ k^6}{m^5}-\\frac{1}{53209728}\\frac{ k^8}{m^7}-\\frac{1}{558702144}\\frac{ k^{10}}{m^9}-\\frac{3 }{15229806592}\\frac{ k^{12}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^2} \\left(\\frac{72 }{35}\\frac{ m^5}{k^4}-\\frac{8 }{189}\\frac{ m^3}{k^2}+\\frac{8 }{693} m-\\frac{1}{2574}\\frac{ k^2}{m}-\\frac{1}{108108}\\frac{ k^4}{m^3}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{1884960}\\frac{ k^6}{m^5}-\\frac{1}{23279256}\\frac{ k^8}{m^7}-\\frac{1}{235243008}\\frac{ k^{10}}{m^9}-\\frac{1}{2076791808}\\frac{ k^{12}}{m^{11}}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^2} \\left(-\\frac{36 }{35}\\frac{ m^5}{k^4}+\\frac{8 }{63}\\frac{ m^3}{k^2}-\\frac{1}{132} m+\\frac{1}{8008}\\frac{ k^2}{m}+\\frac{1}{576576}\\frac{ k^4}{m^3}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{16336320}\\frac{ k^6}{m^5}+\\frac{1}{331082752}\\frac{ k^8}{m^7}+\\frac{1}{5959489536}\\frac{ k^{10}}{m^9}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{121838452736}\\frac{ k^{12}}{m^{11}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;5\\text{D}}^{\\text{f,UV-IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(-\\frac{32 i }{525 \\pi ^2}\\frac{ m^5}{k^4}\\right)+k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{64 i }{525 \\pi ^2}\\frac{ m^5}{k^4}\\right)+\\ldots $ Fermions, spin 5 x 5, dimension 6: $\\tilde{T}_{5,5;6\\text{D}}^{\\text{f,t,UV}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{3 \\pi ^3} \\left( \\left(-\\frac{659507}{1352701350}+\\frac{2 P}{15015}\\right) k^2+ \\left(\\frac{11861}{1334025}-\\frac{P}{385}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1937}{33075}+\\frac{2 P}{105}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{9523}{29400}-\\frac{2 P}{35}+\\frac{L_0}{4}\\right)\\frac{ m^6}{k^4}+\\frac{1}{5}\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{83}{150}+\\frac{2 K}{5}\\right)\\frac{ m^{10}}{k^8}+ \\left(\\frac{1}{25}-\\frac{4 K}{5}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{\\pi ^3} \\left( \\left(-\\frac{156833}{579729150}+\\frac{P}{12870}\\right) k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{41669}{12006225}-\\frac{4 P}{3465}\\right) m^2+ \\left(-\\frac{181}{99225}+\\frac{P}{315}\\right)\\frac{ m^4}{k^2}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(-\\frac{5323}{44100}-\\frac{4 P}{105}+\\frac{L_0}{6}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{19}{36}-\\frac{K}{3}\\right)\\frac{ m^8}{k^6}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{182}{225}+\\frac{16 K}{15}\\right)\\frac{ m^{10}}{k^8}- \\left(\\frac{13}{50}+\\frac{22 K}{15}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{\\pi ^3} \\left( \\left(\\frac{367291}{3607203600}-\\frac{P}{40040}\\right) k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{13511}{4573800}+\\frac{P}{1320}\\right) m^2+ \\left(\\frac{9323}{264600}-\\frac{P}{105}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1091}{4900}+\\frac{9 P}{140}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{13}{48}-\\frac{K}{4}\\right)\\frac{ m^8}{k^6}+ \\left(-\\frac{313}{1200}+\\frac{11 K}{20}\\right)\\frac{ m^{10}}{k^8}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\left(\\frac{11}{50}+\\frac{3 K}{5}\\right)\\frac{ m^{12}}{k^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;6\\text{D}}^{\\text{f,t,IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(-\\frac{33}{56}+\\frac{9 L_0}{28}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{1}{21}+\\frac{2 L_0}{63}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{1}{231}-\\frac{L_0}{231}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{2 L_0}{9009} k^2-\\frac{1}{108108}\\frac{ k^4}{m^2}-\\frac{1}{3063060}\\frac{ k^6}{m^4}-\\frac{1}{49884120}\\frac{ k^8}{m^6}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{611080470}\\frac{ k^{10}}{m^8}-\\frac{1}{6246600360}\\frac{ k^{12}}{m^{10}}-\\frac{1}{56219403240}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(\\frac{33}{28}-\\frac{9 L_0}{14}\\right)\\frac{ m^6}{k^4}+ \\left(-\\frac{1}{42}+\\frac{L_0}{63}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{4}{693}-\\frac{4 L_0}{693}\\right) m^2+\\frac{ L_0}{2574} k^2-\\frac{1}{54054}\\frac{ k^4}{m^2}-\\frac{1}{1413720}\\frac{ k^6}{m^4}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{2 }{43648605}\\frac{ k^8}{m^6}-\\frac{1}{257297040}\\frac{ k^{10}}{m^8}-\\frac{1}{2555427420}\\frac{ k^{12}}{m^{10}}-\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.- \\frac{1}{22487761296}\\frac{ k^{14}}{m^{12}}+\\ldots \\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(-\\frac{33}{56}+\\frac{9 L_0}{28}\\right)\\frac{ m^6}{k^4}+ \\left(\\frac{1}{14}-\\frac{L_0}{21}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1}{264}+\\frac{L_0}{264}\\right) m^2-\\frac{ L_0}{8008} k^2+\\frac{1}{288288}\\frac{ k^4}{m^2}+\\frac{1}{12252240}\\frac{ k^6}{m^4}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\frac{1}{310390080}\\frac{ k^8}{m^6}+\\frac{1}{6518191680}\\frac{ k^{10}}{m^8}+\\frac{1}{149918408640}\\frac{ k^{12}}{m^{10}}+\\ldots \\right)\\right) $ $\\tilde{T}_{5,5;6\\text{D}}^{\\text{f,UV-IR}} & = k^{10} \\pi _{\\mu \\nu }^5 \\left(\\frac{i }{105 \\pi ^3} \\left( \\left(-\\frac{659507}{38648610}+\\frac{2 K}{429}\\right) k^2+ \\left(\\frac{8396}{38115}-\\frac{K}{11}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{992}{945}+\\frac{2 K}{3}\\right)\\frac{ m^4}{k^2}+ \\left(\\frac{109}{105}-2 K\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu } \\pi _{\\mu \\nu }^3 \\pi _{\\nu \\nu } \\left(\\frac{i }{15 \\pi ^3} \\left( \\left(-\\frac{156833}{38648610}+\\frac{K}{858}\\right) k^2+ \\left(\\frac{27809}{800415}-\\frac{4 K}{231}\\right) m^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(\\frac{583}{13230}+\\frac{K}{21}\\right)\\frac{ m^4}{k^2}+ \\left(-\\frac{1268}{735}+\\frac{4 K}{7}\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\nonumber \\\\ & \\quad + k^{10} \\pi _{\\mu \\mu }^2 \\pi _{\\mu \\nu } \\pi _{\\nu \\nu }^2 \\left(\\frac{i }{5 \\pi ^3} \\left( \\left(\\frac{367291}{721440720}-\\frac{K}{8008}\\right) k^2+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{5023}{457380}+\\frac{K}{264}\\right) m^2+ \\left(\\frac{5543}{52920}-\\frac{K}{21}\\right)\\frac{ m^4}{k^2}+\\right.\\right.\\nonumber \\\\ & \\quad \\quad \\quad \\left.\\left.+ \\left(-\\frac{1027}{1960}+\\frac{9 K}{28}\\right)\\frac{ m^6}{k^4}\\right)\\right)+\\ldots $"
],
[
"Divergences of the fermion amplitudes",
"Fermions, spin 0 x 2: $\\tilde{T}_{0,2;3\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu } \\left(-\\frac{i }{\\pi } m^2\\right) $ $\\tilde{T}_{0,2;4\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu } \\left(\\frac{i L_1}{2 \\pi ^2} m^3\\right) $ $\\tilde{T}_{0,2;5\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu } \\left(\\frac{i }{3 \\pi ^2} m^4\\right) $ $\\tilde{T}_{0,2;6\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu } \\left(-\\frac{i L_2}{8 \\pi ^3} m^5\\right) $ Fermions, spin 0 x 4: $\\tilde{T}_{0,4;3\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu }^3 \\left(-\\frac{i }{\\pi } m^2\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{4 i }{\\pi } m^4\\right) $ $\\tilde{T}_{0,4;4\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu }^3 \\left(\\frac{i L_1}{2 \\pi ^2} m^3\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{3 i L_2}{2 \\pi ^2} m^5\\right) $ $\\tilde{T}_{0,4;5\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu }^3 \\left(\\frac{i }{3 \\pi ^2} m^4\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{4 i }{5 \\pi ^2} m^6\\right) $ $\\tilde{T}_{0,4;6\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu }^3 \\left(-\\frac{i L_2}{8 \\pi ^3} m^5\\right)+k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{4 \\pi ^3} m^7\\right) $ Fermions, spin 1 x 1: $k\\cdot \\tilde{T}_{1,1;3\\text{D}}^{\\text{f,nt}} & = 0 $ $k\\cdot \\tilde{T}_{1,1;4\\text{D}}^{\\text{f,nt}} & = 0 $ $k\\cdot \\tilde{T}_{1,1;5\\text{D}}^{\\text{f,nt}} & = 0 $ $k\\cdot \\tilde{T}_{1,1;6\\text{D}}^{\\text{f,nt}} & = 0 $ Fermions, spin 1 x 3: $k\\cdot \\tilde{T}_{1,3;3\\text{D}}^{\\text{f,nt}} & = k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{4 i }{3 \\pi } m^3\\right) $ $\\tilde{T}_{1,3;3\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } \\eta _{\\nu \\nu } \\left(-\\frac{4 i }{9 \\pi } m^3\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(-\\frac{8 i }{9 \\pi } m^3\\right)+k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{2 }{3 \\pi } m^2\\right) $ $k\\cdot \\tilde{T}_{1,3;4\\text{D}}^{\\text{f,nt}} & = k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{2 \\pi ^2} m^4\\right) $ $\\tilde{T}_{1,3;4\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{6 \\pi ^2} m^4\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{3 \\pi ^2} m^4\\right) $ $k\\cdot \\tilde{T}_{1,3;5\\text{D}}^{\\text{f,nt}} & = k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{4 i }{15 \\pi ^2} m^5\\right) $ $\\tilde{T}_{1,3;5\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } \\eta _{\\nu \\nu } \\left(\\frac{4 i }{45 \\pi ^2} m^5\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(\\frac{8 i }{45 \\pi ^2} m^5\\right) $ $k\\cdot \\tilde{T}_{1,3;6\\text{D}}^{\\text{f,nt}} & = k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{12 \\pi ^3} m^6\\right) $ $\\tilde{T}_{1,3;6\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{36 \\pi ^3} m^6\\right)+k_{\\nu } \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{18 \\pi ^3} m^6\\right) $ Fermions, spin 1 x 5: $k\\cdot \\tilde{T}_{1,5;3\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{8 i }{3 \\pi } m^3\\right)+k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{32 i }{5 \\pi } m^5\\right) $ $\\tilde{T}_{1,5;3\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{8 i }{5 \\pi } m^3\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{16 i }{15 \\pi } m^3\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{32 i }{25 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{128 i }{25 \\pi } m^5\\right)+k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{4 }{5 \\pi } m^2\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{16 }{5 \\pi } m^4\\right) $ $k\\cdot \\tilde{T}_{1,5;4\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{\\pi ^2} m^4\\right)+k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{2 i L_3}{\\pi ^2} m^6\\right) $ $\\tilde{T}_{1,5;4\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{3 i L_2}{5 \\pi ^2} m^4\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{2 i L_2}{5 \\pi ^2} m^4\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{2 i L_3}{5 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{8 i L_3}{5 \\pi ^2} m^6\\right) $ $k\\cdot \\tilde{T}_{1,5;5\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{8 i }{15 \\pi ^2} m^5\\right)+k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{32 i }{35 \\pi ^2} m^7\\right) $ $\\tilde{T}_{1,5;5\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{8 i }{25 \\pi ^2} m^5\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{16 i }{75 \\pi ^2} m^5\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{32 i }{175 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{128 i }{175 \\pi ^2} m^7\\right) $ $k\\cdot \\tilde{T}_{1,5;6\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{6 \\pi ^3} m^6\\right)+k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_4}{4 \\pi ^3} m^8\\right) $ $\\tilde{T}_{1,5;6\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{10 \\pi ^3} m^6\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{15 \\pi ^3} m^6\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_4}{20 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_4}{5 \\pi ^3} m^8\\right) $ Fermions, spin 2 x 2: $k\\cdot \\tilde{T}_{2,2;3\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu } \\eta _{\\mu \\nu }+k_{\\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{2 i }{3 \\pi } m^3\\right)+k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{1}{2 \\pi } m^2\\right) $ $k\\cdot \\tilde{T}_{2,2;4\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu } \\eta _{\\mu \\nu }+k_{\\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i L_2}{4 \\pi ^2} m^4\\right) $ $k\\cdot \\tilde{T}_{2,2;5\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu } \\eta _{\\mu \\nu }+k_{\\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{2 i }{15 \\pi ^2} m^5\\right) $ $k\\cdot \\tilde{T}_{2,2;6\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\nu } \\eta _{\\mu \\nu }+k_{\\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{24 \\pi ^3} m^6\\right) $ Fermions, spin 2 x 4: $k\\cdot \\tilde{T}_{2,4;3\\text{D}}^{\\text{f,nt}} & = k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{\\pi } m^3\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{2 i }{3 \\pi } m^3\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{8 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{24 i }{5 \\pi } m^5\\right)+k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{1}{2 \\pi } m^2\\right)+k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{2 }{\\pi } m^4\\right) $ $\\tilde{T}_{2,4;3\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(-\\frac{2 i }{3 \\pi } m^3\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i }{\\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{8 i }{5 \\pi } m^5\\right)+ \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{12 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{3 }{4 \\pi } m^2\\right)+k_{\\mu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{1}{\\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{2 }{\\pi } m^4\\right) $ $k\\cdot \\tilde{T}_{2,4;4\\text{D}}^{\\text{f,nt}} & = k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{4 \\pi ^2} m^4\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_3}{2 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{3 i L_3}{2 \\pi ^2} m^6\\right) $ $\\tilde{T}_{2,4;4\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(\\frac{i L_2}{4 \\pi ^2} m^4\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{3 i L_2}{8 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{i L_3}{2 \\pi ^2} m^6\\right)+ \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{3 i L_3}{4 \\pi ^2} m^6\\right) $ $k\\cdot \\tilde{T}_{2,4;5\\text{D}}^{\\text{f,nt}} & = k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{2 i }{5 \\pi ^2} m^5\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{2 i }{15 \\pi ^2} m^5\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{8 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{24 i }{35 \\pi ^2} m^7\\right) $ $\\tilde{T}_{2,4;5\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(\\frac{2 i }{15 \\pi ^2} m^5\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{i }{5 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{8 i }{35 \\pi ^2} m^7\\right)+ \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{12 i }{35 \\pi ^2} m^7\\right) $ $k\\cdot \\tilde{T}_{2,4;6\\text{D}}^{\\text{f,nt}} & = k_{\\mu } k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{8 \\pi ^3} m^6\\right)+k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{24 \\pi ^3} m^6\\right)+k_{\\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_4}{16 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{3 i L_4}{16 \\pi ^3} m^8\\right) $ $\\tilde{T}_{2,4;6\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(-\\frac{i L_3}{24 \\pi ^3} m^6\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_3}{16 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_4}{16 \\pi ^3} m^8\\right)+ \\left(k_{\\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{3 i L_4}{32 \\pi ^3} m^8\\right) $ Fermions, spin 3 x 3: $k\\cdot \\tilde{T}_{3,3;3\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(-\\frac{4 i }{9 \\pi } m^3\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{4 i }{3 \\pi } m^3\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(-\\frac{8 i }{9 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{64 i }{45 \\pi } m^5\\right)+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{128 i }{45 \\pi } m^5\\right)+k_{\\nu } \\eta _{\\mu \\nu }^2 \\left(-\\frac{32 i }{15 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{2 }{3 \\pi } m^2\\right)+k_{\\mu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{8 }{9 \\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{16 }{9 \\pi } m^4\\right) $ $k\\cdot \\tilde{T}_{3,3;4\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(\\frac{i L_2}{6 \\pi ^2} m^4\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{2 \\pi ^2} m^4\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{3 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{4 i L_3}{9 \\pi ^2} m^6\\right)+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{8 i L_3}{9 \\pi ^2} m^6\\right)+k_{\\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{2 i L_3}{3 \\pi ^2} m^6\\right) $ $k\\cdot \\tilde{T}_{3,3;5\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(\\frac{4 i }{45 \\pi ^2} m^5\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(\\frac{4 i }{15 \\pi ^2} m^5\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(\\frac{8 i }{45 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{64 i }{315 \\pi ^2} m^7\\right)+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{128 i }{315 \\pi ^2} m^7\\right)+k_{\\nu } \\eta _{\\mu \\nu }^2 \\left(\\frac{32 i }{105 \\pi ^2} m^7\\right) $ $k\\cdot \\tilde{T}_{3,3;6\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^3 \\eta _{\\mu \\mu } \\left(-\\frac{i L_3}{36 \\pi ^3} m^6\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{12 \\pi ^3} m^6\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{18 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_4}{18 \\pi ^3} m^8\\right)+k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{i L_4}{9 \\pi ^3} m^8\\right)+k_{\\nu } \\eta _{\\mu \\nu }^2 \\left(-\\frac{i L_4}{12 \\pi ^3} m^8\\right) $ Fermions, spin 3 x 5: $k\\cdot \\tilde{T}_{3,5;3\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(-\\frac{4 i }{9 \\pi } m^3\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{8 i }{3 \\pi } m^3\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{8 i }{9 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{32 i }{5 \\pi } m^5\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{256 i }{15 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^3 \\eta _{\\mu \\nu }^2+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{64 i }{15 \\pi } m^5\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{64 i }{15 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{128 i }{15 \\pi } m^7\\right)+k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{512 i }{35 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{2 }{3 \\pi } m^2\\right)+k_{\\mu } k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{16 }{3 \\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{32 }{9 \\pi } m^4\\right)+k_{\\mu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{32 }{15 \\pi } m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{128 }{15 \\pi } m^6\\right) $ $\\tilde{T}_{3,5;3\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(-\\frac{4 i }{3 \\pi } m^3\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{8 i }{5 \\pi } m^3\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{16 i }{15 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(-\\frac{32 i }{25 \\pi } m^5\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(-\\frac{128 i }{25 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{256 i }{25 \\pi } m^5\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{192 i }{25 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{64 i }{25 \\pi } m^7\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{256 i }{25 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{1536 i }{175 \\pi } m^7\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(-\\frac{1024 i }{175 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{4 }{5 \\pi } m^2\\right)+k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\left(-\\frac{16 }{15 \\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{16 }{5 \\pi } m^4\\right)+k_{\\mu } k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{32 }{5 \\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{64 }{25 \\pi } m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{128 }{25 \\pi } m^6\\right) $ $k\\cdot \\tilde{T}_{3,5;4\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(\\frac{i L_2}{6 \\pi ^2} m^4\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{\\pi ^2} m^4\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{i L_2}{3 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{2 i L_3}{\\pi ^2} m^6\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{16 i L_3}{3 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^3 \\eta _{\\mu \\nu }^2+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{4 i L_3}{3 \\pi ^2} m^6\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{7 i L_4}{6 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{7 i L_4}{3 \\pi ^2} m^8\\right)+k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{4 i L_4}{\\pi ^2} m^8\\right) $ $\\tilde{T}_{3,5;4\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(\\frac{i L_2}{2 \\pi ^2} m^4\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{3 i L_2}{5 \\pi ^2} m^4\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{2 i L_2}{5 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(\\frac{2 i L_3}{5 \\pi ^2} m^6\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(\\frac{8 i L_3}{5 \\pi ^2} m^6\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{16 i L_3}{5 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{12 i L_3}{5 \\pi ^2} m^6\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{7 i L_4}{10 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{14 i L_4}{5 \\pi ^2} m^8\\right)+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{12 i L_4}{5 \\pi ^2} m^8\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{8 i L_4}{5 \\pi ^2} m^8\\right) $ $k\\cdot \\tilde{T}_{3,5;5\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(\\frac{4 i }{45 \\pi ^2} m^5\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{8 i }{15 \\pi ^2} m^5\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{8 i }{45 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{32 i }{35 \\pi ^2} m^7\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{256 i }{105 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^3 \\eta _{\\mu \\nu }^2+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{64 i }{105 \\pi ^2} m^7\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{64 i }{135 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{128 i }{135 \\pi ^2} m^9\\right)+k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{512 i }{315 \\pi ^2} m^9\\right) $ $\\tilde{T}_{3,5;5\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(\\frac{4 i }{15 \\pi ^2} m^5\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{8 i }{25 \\pi ^2} m^5\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(\\frac{16 i }{75 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(\\frac{32 i }{175 \\pi ^2} m^7\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(\\frac{128 i }{175 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{256 i }{175 \\pi ^2} m^7\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{192 i }{175 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{64 i }{225 \\pi ^2} m^9\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{256 i }{225 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{512 i }{525 \\pi ^2} m^9\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{1024 i }{1575 \\pi ^2} m^9\\right) $ $k\\cdot \\tilde{T}_{3,5;6\\text{D}}^{\\text{f,nt}} & = k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(-\\frac{i L_3}{36 \\pi ^3} m^6\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{6 \\pi ^3} m^6\\right)+k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{18 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_4}{4 \\pi ^3} m^8\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{2 i L_4}{3 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu }^3 \\eta _{\\mu \\nu }^2+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{i L_4}{6 \\pi ^3} m^8\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{7 i L_5}{60 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{7 i L_5}{30 \\pi ^3} m^{10}\\right)+k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{2 i L_5}{5 \\pi ^3} m^{10}\\right) $ $\\tilde{T}_{3,5;6\\text{D}}^{\\text{f,nt}}\\cdot k & = k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\left(-\\frac{i L_3}{12 \\pi ^3} m^6\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{10 \\pi ^3} m^6\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{15 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_4}{20 \\pi ^3} m^8\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(-\\frac{i L_4}{5 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{2 i L_4}{5 \\pi ^3} m^8\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{3 i L_4}{10 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{7 i L_5}{100 \\pi ^3} m^{10}\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{7 i L_5}{25 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{6 i L_5}{25 \\pi ^3} m^{10}\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(-\\frac{4 i L_5}{25 \\pi ^3} m^{10}\\right) $ Fermions, spin 4 x 4: $k\\cdot \\tilde{T}_{4,4;3\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }\\right) \\left(-\\frac{i }{\\pi } m^3\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{2 i }{\\pi } m^3\\right)+k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(-\\frac{8 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(-\\frac{4 i }{\\pi } m^5\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{12 i }{\\pi } m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{36 i }{5 \\pi } m^5\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{96 i }{35 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{48 i }{5 \\pi } m^7\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(-\\frac{192 i }{35 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{3 }{4 \\pi } m^2\\right)+k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\left(-\\frac{1}{\\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{3 }{\\pi } m^4\\right)+k_{\\mu } k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{6 }{\\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{12 }{5 \\pi } m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2+k_{\\mu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{24 }{5 \\pi } m^6\\right) $ $k\\cdot \\tilde{T}_{4,4;4\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }\\right) \\left(\\frac{3 i L_2}{8 \\pi ^2} m^4\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{3 i L_2}{4 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(\\frac{i L_3}{2 \\pi ^2} m^6\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(\\frac{5 i L_3}{4 \\pi ^2} m^6\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{15 i L_3}{4 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{9 i L_3}{4 \\pi ^2} m^6\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{3 i L_4}{4 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{21 i L_4}{8 \\pi ^2} m^8\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{3 i L_4}{2 \\pi ^2} m^8\\right) $ $k\\cdot \\tilde{T}_{4,4;5\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }\\right) \\left(\\frac{i }{5 \\pi ^2} m^5\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{2 i }{5 \\pi ^2} m^5\\right)+k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(\\frac{8 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(\\frac{4 i }{7 \\pi ^2} m^7\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{12 i }{7 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(\\frac{36 i }{35 \\pi ^2} m^7\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{32 i }{105 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(\\frac{16 i }{15 \\pi ^2} m^9\\right)+k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(\\frac{64 i }{105 \\pi ^2} m^9\\right) $ $k\\cdot \\tilde{T}_{4,4;6\\text{D}}^{\\text{f,nt}} & = \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }\\right) \\left(-\\frac{i L_3}{16 \\pi ^3} m^6\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{8 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_4}{16 \\pi ^3} m^8\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(-\\frac{5 i L_4}{32 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{15 i L_4}{32 \\pi ^3} m^8\\right)+ \\left(k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^2+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{9 i L_4}{32 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{3 i L_5}{40 \\pi ^3} m^{10}\\right)+ \\left(k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{21 i L_5}{80 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu }^3 \\left(-\\frac{3 i L_5}{20 \\pi ^3} m^{10}\\right) $ Fermions, spin 5 x 5: $k\\cdot \\tilde{T}_{5,5;3\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^2 k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(-\\frac{8 i }{5 \\pi } m^3\\right)+k_{\\mu }^4 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{8 i }{3 \\pi } m^3\\right)+k_{\\mu }^3 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{16 i }{15 \\pi } m^3\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^5 \\eta _{\\mu \\mu }^2 \\left(-\\frac{32 i }{25 \\pi } m^5\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{512 i }{25 \\pi } m^5\\right)+k_{\\mu }^4 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{32 i }{5 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }^2\\right) \\left(-\\frac{384 i }{25 \\pi } m^5\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{768 i }{25 \\pi } m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{1408 i }{175 \\pi } m^7\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{4224 i }{175 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{13824 i }{175 \\pi } m^7\\right)+k_{\\mu }^3 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{2816 i }{175 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\left(-\\frac{4096 i }{175 \\pi } m^7\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{12288 i }{175 \\pi } m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(-\\frac{6144 i }{175 \\pi } m^7\\right)+k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(-\\frac{1024 i }{175 \\pi } m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{4096 i }{175 \\pi } m^9\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{22528 i }{525 \\pi } m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{45056 i }{1575 \\pi } m^9\\right)+k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(-\\frac{4096 i }{315 \\pi } m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\left(-\\frac{4 }{5 \\pi } m^2\\right)+k_{\\mu } k_{\\nu }^4 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\left(-\\frac{16 }{5 \\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{32 }{5 \\pi } m^4\\right)+k_{\\mu }^2 k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\left(-\\frac{64 }{5 \\pi } m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{384 }{25 \\pi } m^6\\right)+k_{\\mu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{64 }{25 \\pi } m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\left(-\\frac{256 }{25 \\pi } m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^2 (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2+k_{\\mu }^2 k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }\\right) \\left(-\\frac{768 }{25 \\pi } m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{768 }{175 \\pi } m^8\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }+k_{\\mu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu }\\right) \\left(-\\frac{3072 }{175 \\pi } m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } (k\\cdot \\epsilon )_{\\mu \\nu } \\eta _{\\mu \\nu }^3 \\left(-\\frac{2048 }{175 \\pi } m^8\\right) $ $k\\cdot \\tilde{T}_{5,5;4\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^2 k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(\\frac{3 i L_2}{5 \\pi ^2} m^4\\right)+k_{\\mu }^4 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{i L_2}{\\pi ^2} m^4\\right)+k_{\\mu }^3 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{2 i L_2}{5 \\pi ^2} m^4\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^5 \\eta _{\\mu \\mu }^2 \\left(\\frac{2 i L_3}{5 \\pi ^2} m^6\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{32 i L_3}{5 \\pi ^2} m^6\\right)+k_{\\mu }^4 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{2 i L_3}{\\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }^2\\right) \\left(\\frac{24 i L_3}{5 \\pi ^2} m^6\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{48 i L_3}{5 \\pi ^2} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu } \\left(\\frac{11 i L_4}{5 \\pi ^2} m^8\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{33 i L_4}{5 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{108 i L_4}{5 \\pi ^2} m^8\\right)+k_{\\mu }^3 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{22 i L_4}{5 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\left(\\frac{32 i L_4}{5 \\pi ^2} m^8\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{96 i L_4}{5 \\pi ^2} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{48 i L_4}{5 \\pi ^2} m^8\\right)+k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{36 i L_5}{25 \\pi ^2} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{144 i L_5}{25 \\pi ^2} m^{10}\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{264 i L_5}{25 \\pi ^2} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{176 i L_5}{25 \\pi ^2} m^{10}\\right)+k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{16 i L_5}{5 \\pi ^2} m^{10}\\right) $ $k\\cdot \\tilde{T}_{5,5;5\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^2 k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(\\frac{8 i }{25 \\pi ^2} m^5\\right)+k_{\\mu }^4 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{8 i }{15 \\pi ^2} m^5\\right)+k_{\\mu }^3 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(\\frac{16 i }{75 \\pi ^2} m^5\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^5 \\eta _{\\mu \\mu }^2 \\left(\\frac{32 i }{175 \\pi ^2} m^7\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(\\frac{512 i }{175 \\pi ^2} m^7\\right)+k_{\\mu }^4 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{32 i }{35 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }^2\\right) \\left(\\frac{384 i }{175 \\pi ^2} m^7\\right)+k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{768 i }{175 \\pi ^2} m^7\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu } \\left(\\frac{1408 i }{1575 \\pi ^2} m^9\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(\\frac{1408 i }{525 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(\\frac{1536 i }{175 \\pi ^2} m^9\\right)+k_{\\mu }^3 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{2816 i }{1575 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\left(\\frac{4096 i }{1575 \\pi ^2} m^9\\right)+k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{4096 i }{525 \\pi ^2} m^9\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(\\frac{2048 i }{525 \\pi ^2} m^9\\right)+k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(\\frac{1024 i }{1925 \\pi ^2} m^{11}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(\\frac{4096 i }{1925 \\pi ^2} m^{11}\\right)+k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(\\frac{2048 i }{525 \\pi ^2} m^{11}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(\\frac{4096 i }{1575 \\pi ^2} m^{11}\\right)+k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(\\frac{4096 i }{3465 \\pi ^2} m^{11}\\right) $ $k\\cdot \\tilde{T}_{5,5;6\\text{D}}^{\\text{f,nt}} & = k_{\\mu }^2 k_{\\nu }^5 \\eta _{\\mu \\mu } \\left(-\\frac{i L_3}{10 \\pi ^3} m^6\\right)+k_{\\mu }^4 k_{\\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{i L_3}{6 \\pi ^3} m^6\\right)+k_{\\mu }^3 k_{\\nu }^4 \\eta _{\\mu \\nu } \\left(-\\frac{i L_3}{15 \\pi ^3} m^6\\right)+\\nonumber \\\\ & \\quad + k_{\\nu }^5 \\eta _{\\mu \\mu }^2 \\left(-\\frac{i L_4}{20 \\pi ^3} m^8\\right)+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\nu \\nu } \\left(-\\frac{4 i L_4}{5 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^4 k_{\\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{i L_4}{4 \\pi ^3} m^8\\right)+ \\left(k_{\\mu } k_{\\nu }^4 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }+k_{\\mu }^2 k_{\\nu }^3 \\eta _{\\mu \\nu }^2\\right) \\left(-\\frac{3 i L_4}{5 \\pi ^3} m^8\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 k_{\\nu }^2 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{6 i L_4}{5 \\pi ^3} m^8\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{11 i L_5}{50 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{33 i L_5}{50 \\pi ^3} m^{10}\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu } \\left(-\\frac{54 i L_5}{25 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^3 \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{11 i L_5}{25 \\pi ^3} m^{10}\\right)+k_{\\nu }^3 \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\left(-\\frac{16 i L_5}{25 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\mu }^2 k_{\\nu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{48 i L_5}{25 \\pi ^3} m^{10}\\right)+k_{\\mu } k_{\\nu }^2 \\eta _{\\mu \\nu }^3 \\left(-\\frac{24 i L_5}{25 \\pi ^3} m^{10}\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu }^2 \\eta _{\\nu \\nu }^2 \\left(-\\frac{3 i L_6}{25 \\pi ^3} m^{12}\\right)+k_{\\mu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu } \\eta _{\\nu \\nu }^2 \\left(-\\frac{12 i L_6}{25 \\pi ^3} m^{12}\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\mu } \\eta _{\\mu \\nu }^2 \\eta _{\\nu \\nu } \\left(-\\frac{22 i L_6}{25 \\pi ^3} m^{12}\\right)+k_{\\mu } \\eta _{\\mu \\nu }^3 \\eta _{\\nu \\nu } \\left(-\\frac{44 i L_6}{75 \\pi ^3} m^{12}\\right)+\\nonumber \\\\ & \\quad + k_{\\nu } \\eta _{\\mu \\nu }^4 \\left(-\\frac{4 i L_6}{15 \\pi ^3} m^{12}\\right) $"
],
[
"Conclusion",
"We finally sum up our major results.",
"In this paper we have pursued further the program started in [1], considering in particular the quadratic part of the effective action.",
"First of all, we have discussed the relevant issue of the geometric interpretation of the obtained effective actions.",
"The basic outcome of [1] was that, upon considering on-shell conserved higher spin currents, the corresponding effective actions inherit an off-shell gauge invariance once a finite number of local counterterms are subtracted.",
"In particular our (linearized) gauge invariance involves unconstrained fully symmetric parameters and is the same as the one considered in [3], [17], [18], [19].",
"We are therefore naturally led to the problem of expressing our results in the geometric language of [26], [27].",
"This is done in full generality in section .",
"Another relevant issue is whether it is possible to construct a gauge invariant effective action without the subtraction of ad hoc non-invariant counterterms.",
"The answer to this question of course requires the choice of a specific regularization scheme.",
"We decided to work in dimensional regularization, which turns out to be particularly convenient for the lower spin cases.",
"In fact, in section we have explictly shown that for spin 1 and spin 2 gauge fields it is generally possible to introduce additional local terms allowed by covariance (involving the spin 0 current) such that the effective action is gauge invariant with no non-covariant subtractions needed thanks to the tadpole and seagull diagrams entering the Ward Identities.",
"This is no surprise, as in these cases we already know fully off-shell covariant versions of QED and gravity coupled to ordinary scalar and spin $1/2$ matter.",
"Nevertheless our explict computations work as a promising test for higher spin gauge fields.",
"Related questions are whether this whole procedure depends on the choice of the on-shell conserved current that is minimally coupled in the first place and whether it can be restricted to a finite number of higher spin currents.",
"In the cases of spin 1 and 2 we checked there is no need to introduce higher spin currents and and off-shell invariant couplings between the matter and gauge sectors can be obtained for any choice of on-shell conserved Noether currents.",
"A particularly interesting choice for the currents and the couplings for the spin 2 case in $d=4$ is the one for which the dominating term in the UV expansion is the Weyl density ((REF ), (REF )), corresponding to the emerging conformal symmetry in the massless case.",
"This term is also found in the corresponding IR expansion ((REF ), (REF )).",
"The case of higher spin gauge fields, in particular as far as the choice of different currents is concerned, will be treated in details in a subsequent work.",
"The major task of this paper was the completion of the construction of the quadratic part of the higher spin effective action started in [1], [20].",
"We did it in section presenting all two point correlators of symmetric currents of any spin up to 5 and in any dimension between 3 and 6.",
"We also spelled out UV and IR expansions, finding compact formulae for dominating terms in the two limits.",
"In particular, we also included the mixed ones which were not considered in our previous work.",
"The results of section show that these terms turn out to have the usual structure found for the diagonal ones, i.e.",
"the sum of a nonlocal transverse part and a local longitudinal one.",
"We expect that their presence is crucial when one tries to test the invariance of the effective action beyond the lowest order in the gauge fields as well as when one tries to introduce tadpoles and seagulls in the Ward Identities.",
"In $d=3$ odd-parity kinetic terms are present when spin $1/2$ matter is integrated out.",
"We find that for the traceless currents considered in section the UV limit coincides with an interesting generalization of the conformal higher spin action found in [30].",
"Although made up of the same invariant building blocks, the kinetic terms appearing in this section do not coincide with the ones found in [3], [17] by coupling the gauge theory to matter and considering the analysis of the propagating degrees of freedom.",
"The fundamental reason is that our effective actions are obtained after a subtraction procedure that is required to give invariant terms, but still allows for a wide class of possible choices.",
"A proper discussion of the propagating degrees of freedom in our case would require the removal of nonlocalities by introducing compensating fields and the consideration of physical propagators after proper gauge fixing.",
"However, the next logical step is the computation of three-point functions which would provide an insight in gauge invariance beyond the lowest order and therefore prepare the ground for the construction of fully covariant effective actions.",
"Acknowledgements.",
"This research has been supported by the Croatian Science Foundation under the project No.",
"8946 and by the University of Rijeka under the research support No. 13.12.1.4.05.",
"Appendices"
],
[
"Spin 2 - expansions",
"Here, we list several useful expansions of geometrical quantities in terms of spin 2 field $h_{\\mu \\nu }$ .",
"Riemann tensor $R_{\\mu \\nu \\lambda \\rho }=-\\frac{1}{2} \\partial _\\mu \\partial _\\lambda h_{\\nu \\rho }+\\frac{1}{2} \\partial _\\mu \\partial _\\rho h_{\\nu \\lambda }+\\frac{1}{2}\\partial _\\nu \\partial _\\lambda h_{\\mu \\rho }-\\frac{1}{2} \\partial _\\nu \\partial _\\rho h_{\\mu \\lambda }$ Ricci tensor $R_{\\mu \\nu }=-\\frac{1}{2} \\partial _\\mu \\partial _\\nu h+\\frac{1}{2} \\partial _\\mu \\partial _\\lambda h_\\nu ^\\lambda +\\frac{1}{2} \\partial _\\nu \\partial _\\lambda h_\\mu ^\\lambda -\\frac{1}{2}\\Box h_{\\mu \\nu }$ Ricci scalar $R&=&-\\Box h+\\partial _\\mu \\partial _\\nu h^{\\mu \\nu }\\nonumber \\\\&&+h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu h-2h^{\\mu \\nu }\\partial _\\nu \\partial _\\lambda h_\\mu ^\\lambda -\\frac{1}{4}\\partial _\\nu h\\partial ^\\nu h- \\partial _\\nu h^{\\mu \\nu }\\partial _\\lambda h_\\mu ^\\lambda + \\partial _\\mu h\\partial _\\nu h^{\\mu \\nu }+h^{\\mu \\nu }\\Box h_{\\mu \\nu }\\nonumber \\\\&&-\\frac{1}{2}\\partial _\\lambda h^{\\mu \\nu }\\partial _\\nu h_\\mu ^\\lambda +\\frac{3}{4}\\partial _\\lambda h_{\\mu \\nu }\\partial ^\\lambda h^{\\mu \\nu }$ Einstein-Hilbert $\\sqrt{g}R=-\\Box h+\\partial _\\mu \\partial _\\nu h^{\\mu \\nu }+\\frac{1}{2}\\left(h^{\\mu \\nu }\\Box h_{\\mu \\nu }+2h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu h-h\\Box h-2h^{\\mu \\lambda }\\partial _\\mu \\partial _\\lambda h_\\nu ^\\lambda \\right)$ $\\sqrt{g} R & \\quad \\leftrightarrow \\quad k^2 \\pi _{\\mu \\mu } h^{\\mu \\mu }+h^{\\mu \\mu } \\left(-\\frac{1}{4} k^2 \\left(\\pi _{\\mu \\nu }^2-\\pi _{\\mu \\mu }\\pi _{\\nu \\nu }\\right)\\right)h^{\\nu \\nu }$ Riemann squared $R^2_{\\mu \\nu \\lambda \\rho }=h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\partial _\\lambda \\partial _\\rho h^{\\lambda \\rho }-2h^{\\mu \\nu }\\partial _\\mu \\partial _\\lambda \\Box h_\\nu ^\\lambda +h^{\\mu \\nu }\\Box ^2 h_{\\mu \\nu }$ $\\sqrt{g} R_{\\kappa \\lambda \\rho \\sigma }R^{\\kappa \\lambda \\rho \\sigma } & \\quad \\leftrightarrow \\quad h^{\\mu \\mu }\\left(k^4 \\pi _{\\mu \\nu }^2\\right)h^{\\nu \\nu }$ Ricci tensor squared $R^2_{\\mu \\nu }=\\frac{1}{4} h^{\\mu \\nu }\\Box ^2 h_{\\mu \\nu }-\\frac{1}{2}h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\Box h- \\frac{1}{2}h^{\\mu \\nu }\\partial _\\mu \\partial _\\lambda \\Box h_\\nu ^\\lambda +\\frac{1}{2}h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\partial _\\lambda \\partial _\\rho h^{\\lambda \\rho }+\\frac{1}{4} h\\Box ^2 h$ $\\sqrt{g} R_{\\kappa \\lambda }R^{\\kappa \\lambda } & \\quad \\leftrightarrow \\quad h^{\\mu \\mu }\\left(\\frac{1}{4}k^4 \\left(\\pi _{\\mu \\nu }^2+\\pi _{\\mu \\mu }\\pi _{\\nu \\nu }\\right)\\right)h^{\\nu \\nu }$ Ricci scalar squared $R^2= h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\partial _\\lambda \\partial _\\rho h^{\\lambda \\rho }-2h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\Box h+h\\Box ^2 h$ $\\sqrt{g} R^2 & \\quad \\leftrightarrow \\quad h^{\\mu \\mu }\\left(k^4 \\pi _{\\mu \\mu }\\pi _{\\nu \\nu }\\right)h^{\\nu \\nu }$ Weyl density $W_{\\mu \\nu \\rho \\sigma } &= R_{\\mu \\nu \\rho \\sigma } -\\frac{1}{d-2} \\left(R_{\\mu \\rho }g_{\\nu \\sigma }-R_{\\mu \\sigma }g_{\\nu \\rho }-R_{\\nu \\rho }g_{\\mu \\sigma }+R_{\\nu \\sigma }g_{\\mu \\rho }\\right)+\\\\\\nonumber & \\quad +\\frac{R}{(d-1)(d-2)} \\left(g_{\\mu \\rho }g_{\\nu \\sigma }-g_{\\mu \\sigma }g_{\\nu \\rho } \\right)$ $W_{\\mu \\nu \\rho \\sigma }W^{\\mu \\nu \\rho \\sigma } &=R_{\\mu \\nu \\rho \\sigma }R^{\\mu \\nu \\rho \\sigma } -\\frac{4}{d-2} R_{\\mu \\nu }R^{\\mu \\nu }+\\frac{2}{d^2-3d+2} R^2$ $\\sqrt{g} W_{\\kappa \\lambda \\rho \\sigma }W^{\\kappa \\lambda \\rho \\sigma } & \\quad \\leftrightarrow \\quad h^{\\mu \\mu }\\left(\\frac{d-3}{d-2}k^4 \\pi _{\\mu \\nu }^2-\\frac{d-3}{(d-1)(d-2)}k^4 \\pi _{\\mu \\mu } \\pi _{\\nu \\nu }\\right)h^{\\nu \\nu }$ Weyl density for $d=4$ $\\mathcal {W}^2&=&R_{\\mu \\nu \\lambda \\rho }^2-2R_{\\mu \\nu }^2+\\frac{1}{3} R^2\\\\&=&\\frac{1}{2} h^{\\mu \\nu }\\Box ^2 h_{\\mu \\nu }-h^{\\mu \\nu }\\partial _\\mu \\partial _\\lambda \\Box h_\\nu ^\\lambda +\\frac{1}{3}h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\Box h +\\frac{1}{3}h^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\partial _\\lambda \\partial _\\rho h^{\\lambda \\rho }-\\frac{1}{6} h\\Box ^2 h\\nonumber $ Cotton tensor for $d=3$ $C_{\\mu \\nu }=\\epsilon _\\mu {}^{\\tau \\rho }\\partial _\\tau \\left(R_{\\rho \\nu }-\\frac{1}{d-1}g_{\\nu \\rho }R\\right)=\\frac{1}{2} \\epsilon _\\mu {}^{\\rho \\tau }\\partial _\\tau \\left(\\Box h_{\\nu \\rho }-\\partial _\\lambda \\partial _\\nu h_\\rho ^\\lambda \\right)$"
],
[
"Higher spin traceless actions",
"In this appendix we will review the parity-odd actions that are expected in 3D for conformal higher spin in Minkowski background, showing that they coincide with the UV limits of the amplitudes considered in sections REF , REF , REFIt is understood the UV limit should be carried out as described in [21], introducing an ad hoc flavor index..",
"The action considered in [30] can be easily generalized to the case when the quadratic terms involve fields of different spins, namely $I_{s_1,s_2} & = & \\frac{1}{2}\\int d^{3}x\\sum _{r=0}^{s_1-1}\\binom{2s_1}{2r+1}\\overset{(s_1)}{h}^{\\alpha _{1}\\ldots \\alpha _{2s_1}}\\left(\\square \\right)^{r}\\partial _{\\alpha _{1}}\\phantom{}^{\\beta _{1}}\\ldots \\partial _{\\alpha _{2s_1-2r-1}}\\phantom{}^{\\beta _{2s_1-2r-1}}\\nonumber \\\\& & \\partial ^{\\beta _{2s_1+1}\\beta _{2s_1+2}}\\ldots \\partial ^{\\beta _{2s_2-1}\\beta _{2s_2}}\\overset{(s_2)}{h}_{\\beta _{1}\\ldots \\beta _{2s_1-2r-1}\\alpha _{2s_1-2r}\\ldots \\alpha _{2s_1}\\beta _{2s_1+1}\\ldots \\beta _{2s_2}},$ where we assume $s_1\\le s_2$ and $\\partial _{\\alpha }\\phantom{}^{\\beta }=\\left(\\gamma ^{\\mu }\\right)_{\\alpha }\\phantom{}^{\\beta }\\partial _{\\mu }$ .",
"We define $\\left(\\gamma _{\\mu }\\right)_{\\alpha \\beta }=\\varepsilon _{\\beta \\gamma }\\left(\\gamma _{\\mu }\\right)_{\\alpha }\\phantom{}^{\\gamma }$ and $\\left(\\gamma _{\\mu }\\right)^{\\alpha \\beta }=\\varepsilon ^{\\alpha \\gamma }\\left(\\gamma _{\\mu }\\right)_{\\gamma }\\phantom{}^{\\beta }$ , in agreement with the conventions of Wess and Bagger.",
"Going from the spinor notation $h^{\\alpha _{1}\\ldots \\alpha _{2s}}=h^{\\mu _{1}\\ldots \\mu _{s}}\\left(\\gamma _{\\mu _{1}}\\right)^{\\alpha _{1}\\alpha _{2}}\\ldots \\left(\\gamma _{\\mu _{s}}\\right)^{\\alpha _{2s-1}\\alpha _{2s}}$ to the standard tensor one, we get $I_{s_1,s_2} & = & \\frac{1}{2}\\int d^{3}x\\sum _{r=0}^{s_1-1}\\binom{2s_1}{2r+1}\\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\left(\\gamma _{\\mu _{1}}\\right)^{\\alpha _{1}\\alpha _{2}}\\ldots \\left(\\gamma _{\\mu _{s_1}}\\right)^{\\alpha _{2s_1-1}\\alpha _{2s_1}}\\left(\\square \\right)^{r}\\\\& & \\partial _{\\alpha _{1}}\\phantom{}^{\\beta _{1}}\\ldots \\partial _{\\alpha _{2s_1-2r-1}}\\phantom{}^{\\beta _{2s_1-2r-1}}\\delta _{\\alpha _{2s_1-2r}}^{\\beta _{2s_1-2r}}\\ldots \\delta _{\\alpha _{2s_1}}^{\\beta _{2s_1}} \\partial ^{\\beta _{2s_1+1}\\beta _{2s_1+2}}\\ldots \\partial ^{\\beta _{2s_2-1}\\beta _{2s_2}}\\\\& & \\left(\\gamma _{\\nu _{1}}\\right)_{\\beta _{1}\\beta _{2}}\\ldots \\left(\\gamma _{\\nu _{s_2}}\\right)_{\\beta _{2s_2-1}\\beta _{2s_2}}\\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s_2}}\\\\& = & \\frac{1}{2}\\int d^{3}x\\sum _{r=0}^{s_1-1}\\binom{2s_1}{2r+1}\\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\left(\\square \\right)^{r}\\left[\\left(\\gamma _{\\mu _{1}}\\right)^{\\alpha _{1}\\alpha _{2}}\\partial _{\\alpha _{1}}\\phantom{}^{\\beta _{1}}\\left(\\gamma _{\\nu _{1}}\\right)_{\\beta _{1}\\beta _{2}}\\partial _{\\alpha _{2}}\\phantom{}^{\\beta _{2}}\\right]\\ldots \\\\& & \\ldots \\left[\\left(\\gamma _{\\mu _{s_1-r-1}}\\right)^{\\alpha _{2s_1-2r-3}\\alpha _{2s_1-2r-2}}\\partial _{\\alpha _{2s_1-2r-3}}\\phantom{}^{\\beta _{2s_1-2r-3}}\\right.\\nonumber \\\\&& \\qquad \\left.\\left(\\gamma _{\\nu _{s_1-r-1}}\\right)_{\\beta _{2s_1-2r-3}\\beta _{2s_1-2r-2}}\\partial _{\\alpha _{2s_1-2r-2}}\\phantom{}^{\\beta _{2s_1-2r-2}}\\right]\\\\& & \\left[\\left(\\gamma _{\\mu _{s_1-r}}\\right)^{\\alpha _{2s_1-2r-1}\\alpha _{2s_1-2r}}\\partial _{\\alpha _{2s_1-2r-1}}\\phantom{}^{\\beta _{2s_1-2r-1}}\\left(\\gamma _{\\nu _{s_1-r}}\\right)_{\\beta _{2s_1-2r-1}\\beta _{2s_1-2r}}\\delta _{\\alpha _{2s_1-2r}}^{\\beta _{2s_1-2r}}\\right]\\\\& & \\left[\\left(\\gamma _{\\mu _{s_1-r+1}}\\right)^{\\alpha _{2s_1-2r+1}\\alpha _{2s_1-2r+2}}\\left(\\gamma _{\\nu _{s-r+1}}\\right)_{\\beta _{2s-2r+1}\\beta _{2s-2r+2}}\\delta _{\\alpha _{2s-2r+1}}^{\\beta _{2s_1-2r+1}}\\delta _{\\alpha _{2s_1-2r+2}}^{\\beta _{2s_1-2r+2}}\\right]\\ldots \\\\& & \\ldots \\left[\\left(\\gamma _{\\mu _{s_1}}\\right)^{\\alpha _{2s_1-1}\\alpha _{2s_1}}\\left(\\gamma _{\\nu _{s_1}}\\right)_{\\beta _{2s_1-1}\\beta _{2s_1}}\\delta _{\\alpha _{2s_1-1}}^{\\beta _{2s_1-1}}\\delta _{\\alpha _{2s_1}}^{\\beta _{2s_1}}\\right]\\\\& & \\left[\\partial ^{\\beta _{2s_1+1}\\beta _{2s_1+2}}\\left(\\gamma _{\\nu _{s_1+1}}\\right)_{\\beta _{2s_1+1}\\beta _{2s_1+2}}\\right]\\ldots \\left[\\partial ^{\\beta _{2s_2-1}\\beta _{2s_2}}\\left(\\gamma _{\\nu _{s_2}}\\right)_{\\beta _{2s_2-1}\\beta _{2s_2}}\\right]\\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s}}\\,,$ where, keeping in mind that in $D=3$ $\\left(\\gamma _\\mu \\right)_{\\alpha \\beta }=\\left(\\gamma _\\mu \\right)_{\\beta \\alpha }$ , we can easily recognize the traces $Tr\\left[\\gamma _{\\mu _{i}}\\left(\\gamma \\centerdot \\partial \\right)\\gamma _{\\nu _{i}}\\left(\\gamma \\centerdot \\partial \\right)\\right]& = &\\left[\\left(\\gamma _{\\mu _{i}}\\right)^{\\alpha _{2i-1}\\alpha _{2i}}\\partial _{\\alpha _{2i-1}}\\phantom{}^{\\beta _{2i-1}}\\left(\\gamma _{\\nu _{i}}\\right)_{\\beta _{2i-1}\\beta _{2i}}\\partial _{\\alpha _{2i}}\\phantom{}^{\\beta _{2i}}\\right] ,\\\\-Tr\\left[\\gamma _{\\mu _{s_1-r}}\\left(\\gamma \\centerdot \\partial \\right)\\gamma _{\\nu _{s_1-r}}\\right]& = &\\left[\\left(\\gamma _{\\mu _{s_1-r}}\\right)^{\\alpha _{2s_1-2r-1}\\alpha _{2s_1-2r}}\\partial _{\\alpha _{2s_1-2r-1}}\\phantom{}^{\\beta _{2s_1-2r-1}}\\right.",
"\\nonumber \\\\ && \\quad \\left.\\left(\\gamma _{\\nu _{s_1-r}}\\right)_{\\beta _{2s_1-2r-1}\\beta _{2s_1-2r}}\\delta _{\\alpha _{2s_1-2r}}^{\\beta _{2s_1-2r}}\\right],\\\\-Tr\\left[\\gamma _{\\mu _{j}}\\gamma _{\\nu _{j}}\\right]& = &\\left[\\left(\\gamma _{\\mu _{j}}\\right)^{\\alpha _{2j-1}\\alpha _{2j}}\\left(\\gamma _{\\nu _{j}}\\right)_{\\beta _{2j-1}\\beta _{2j}}\\delta _{\\alpha _{2j-1}}^{\\beta _{2j-1}}\\delta _{\\alpha _{2j}}^{\\beta _{2j}}\\right] ,\\\\-Tr\\left[\\left(\\gamma \\centerdot \\partial \\right)\\gamma _{\\nu _{k}}\\right] & = & \\left[\\partial ^{\\beta _{2k-1}\\beta _{2k}}\\left(\\gamma _{\\nu _{k}}\\right)_{\\beta _{2k-1}\\beta _{2k}}\\right]\\,.$ In order to evaluate these traces, we need the following rules $Tr\\left[\\gamma _{\\mu }\\gamma _{\\nu }\\right] & = & 2\\eta _{\\mu \\nu }\\,,\\\\Tr\\left[\\gamma _{\\mu }\\gamma _{\\nu }\\gamma _{\\rho }\\right] & = & -2\\varepsilon _{\\mu \\nu \\rho }\\,,\\\\Tr\\left[\\gamma _{\\mu }\\gamma _{\\nu }\\gamma _{\\rho }\\gamma _{\\sigma }\\right] & = & 2\\left(\\eta _{\\mu \\nu }\\eta _{\\rho \\sigma }-\\eta _{\\mu \\rho }\\eta _{\\nu \\sigma }+\\eta _{\\mu \\sigma }\\eta _{\\nu \\rho }\\right)\\,,$ which imply $Tr\\left[\\left(\\gamma \\centerdot \\partial \\right)\\gamma _{\\nu _{k}}\\right] & = & 2\\partial _{\\nu _{k}}\\,,\\nonumber \\\\Tr\\left[\\gamma _{\\mu _{s-r}}\\left(\\gamma \\centerdot \\partial \\right)\\gamma _{\\nu _{s-r}}\\right] & = & 2\\varepsilon _{\\mu _{s-r}\\nu _{s-r}\\rho }\\partial ^{\\rho }\\,,\\nonumber \\\\Tr\\left[\\gamma _{\\mu _{i}}\\left(\\gamma \\centerdot \\partial \\right)\\gamma _{\\nu _{i}}\\left(\\gamma \\centerdot \\partial \\right)\\right] & = & 2\\left(2\\partial _{\\mu _{i}}\\partial _{\\nu _{i}}-\\eta _{\\mu _{i}\\nu _{i}}\\square \\right)\\,.\\nonumber $ We can therefore rewrite the action $I_{s_1,s_2}$ as $I_{s_1,s_2} & = & \\frac{1}{2}\\left(-2\\right)^{s_2}\\int d^{3}x\\sum _{r=0}^{s_1-1}\\binom{2s_1}{2r+1}\\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\left(\\square \\right)^{r}\\left(\\eta _{\\mu _{1}\\nu _{1}}\\square -2\\partial _{\\mu _{1}}\\partial _{\\nu _{1}}\\right)\\ldots \\nonumber \\\\& & \\ldots \\left(\\eta _{\\mu _{s_1-r-1}\\nu _{s_1-r-1}}\\square -2\\partial _{\\mu _{s_1-r-1}}\\partial _{\\nu _{s_1-r-1}}\\right)\\varepsilon _{\\mu _{s_1-r}\\nu _{s_1-r}\\rho }\\partial ^{\\rho }\\nonumber \\\\& & \\eta _{\\mu _{s_1-r+1}\\nu _{s_1-r+1}}\\ldots \\eta _{\\mu _{s_1}\\nu _{s_1}}\\partial _{\\nu _{s_1+1}}\\ldots \\partial _{\\nu _{s_2}}\\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s_2}}\\nonumber \\\\& = & -\\left(-2\\right)^{s_2-1}\\int d^{3}x\\sum _{r=0}^{s_1-1}\\binom{2s_1}{2r+1}\\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\left(\\square \\right)^{r}\\varepsilon _{\\mu _{1}\\nu _{1}\\rho }\\partial ^{\\rho }\\eta _{\\mu _{2}\\nu _{2}}\\ldots \\eta _{\\mu _{r+1}\\nu _{r+1}}\\nonumber \\\\& & \\left(\\eta _{\\mu _{r+2}\\nu _{r+2}}\\square -2\\partial _{\\mu _{r+2}}\\partial _{\\nu _{r+2}}\\right)\\ldots \\left(\\eta _{\\mu _{s_1}\\nu _{s_1}}\\square -2\\partial _{\\mu _{s_1}}\\partial _{\\nu _{s_1}}\\right)\\nonumber \\\\& &\\partial _{\\nu _{s_1+1}}\\ldots \\partial _{\\nu _{s_2}} \\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s_2}}\\,.",
"$ By elementary manipulations, one can write the action (REF ) as a triple summation $I_{s_1,s_2}& = & -\\left(-2\\right)^{s_2-1}\\int d^{3}x \\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\varepsilon _{\\mu _{1}\\nu _{1}\\rho }\\partial ^{\\rho }\\sum _{r=0}^{s_1-1}\\sum _{j=0}^{s_1-r-1}\\sum _{k=0}^{s_1-j-1}\\nonumber \\\\& & \\left(-2\\right)^{j}\\binom{2s_1}{2r+1}\\binom{s_1-r-1}{j}\\binom{s_1-j-1}{k}\\left(\\eta _{\\mu _{2}\\nu _{2}}\\square -\\partial _{\\mu _{2}}\\partial _{\\nu _{2}}\\right)\\ldots \\left(\\eta _{\\mu _{k+1}\\nu _{k+1}}\\square -\\partial _{\\mu _{k+1}}\\partial _{\\nu _{k+1}}\\right)\\nonumber \\\\& &\\left(\\partial _{\\mu _{k+2}}\\partial _{\\nu _{k+2}}\\right)\\ldots \\left(\\partial _{\\mu _{s_1}}\\partial _{\\nu _{s_1}}\\right)\\partial _{\\nu _{s_1+1}}\\ldots \\partial _{\\nu _{s_2}} \\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s_2}}\\,.$ We can now switch the order of summation $I_{s_1,s_2}& = & -\\left(-2\\right)^{s_2-1}\\int d^{3}x \\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\varepsilon _{\\mu _{1}\\nu _{1}\\rho }\\partial ^{\\rho }\\sum _{k=0}^{s_1-1}\\left(-\\right)^{s_1-k-1}\\sum _{j=0}^{s_1-k-1}\\sum _{r=0}^{s_1-j-1}\\nonumber \\\\& & \\left(-2\\right)^{j}\\binom{2s_1}{2r+1}\\binom{s_1-r-1}{j}\\binom{s_1-j-1}{k}\\left(\\eta _{\\mu _{2}\\nu _{2}}\\square -\\partial _{\\mu _{2}}\\partial _{\\nu _{2}}\\right)\\ldots \\left(\\eta _{\\mu _{k+1}\\nu _{k+1}}\\square -\\partial _{\\mu _{k+1}}\\partial _{\\nu _{k+1}}\\right)\\nonumber \\\\& &\\left(-\\partial _{\\mu _{k+2}}\\partial _{\\nu _{k+2}}\\right)\\ldots \\left(-\\partial _{\\mu _{s_1}}\\partial _{\\nu _{s_1}}\\right)\\partial _{\\nu _{s_1+1}}\\ldots \\partial _{\\nu _{s_2}} \\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s_2}}\\,,$ and perform two summations using the tables or Mathematica $&&\\sum _{j=0}^{s_1-k-1}\\sum _{r=0}^{s_1-j-1}\\left(-2\\right)^{j}\\binom{2s_1}{2r+1}\\binom{s_1-r-1}{j}\\binom{s_1-j-1}{k}\\nonumber \\\\&& \\qquad = 2^{-1+2s_1}\\binom{s_1-1}{k}\\phantom{}_{2}F_{1}\\left(\\frac{1}{2}-s_1,1+k-s_1,1-2s_1;2\\right)\\,,$ thus obtaining $I_{s_1,s_2}& = & -\\left(-2\\right)^{s_2-1}2^{-1+2s_1}\\int d^{3}x \\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\varepsilon _{\\mu _{1}\\nu _{1}\\rho }\\partial ^{\\rho }\\sum _{k=0}^{s_1-1}\\left(-\\right)^{s_1-k-1}\\binom{s_1-1}{k}\\nonumber \\\\& &\\phantom{}_{2}F_{1}\\left(\\frac{1}{2}-s_1,1+k-s_1,1-2s_1;2\\right) \\left(\\eta _{\\mu _{2}\\nu _{2}}\\square -\\partial _{\\mu _{2}}\\partial _{\\nu _{2}}\\right)\\ldots \\left(\\eta _{\\mu _{k+1}\\nu _{k+1}}\\square -\\partial _{\\mu _{k+1}}\\partial _{\\nu _{k+1}}\\right)\\nonumber \\\\& &\\left(-\\partial _{\\mu _{k+2}}\\partial _{\\nu _{k+2}}\\right)\\ldots \\left(-\\partial _{\\mu _{s_1}}\\partial _{\\nu _{s_1}}\\right)\\partial _{\\nu _{s_1+1}}\\ldots \\partial _{\\nu _{s_2}} \\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s_2}}\\,,$ or equivalently $I_{s_1,s_2}& = & -\\left(-2\\right)^{s_2-1}2^{-1+2s_1}\\int d^{3}x \\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\varepsilon _{\\mu _{1}\\nu _{1}\\rho }\\partial ^{\\rho }\\sum _{k=0}^{s_1-1}\\left(-\\right)^{k}\\binom{s_1-1}{s_1-1-k}\\nonumber \\\\& &\\phantom{}_{2}F_{1}\\left(-k,\\frac{1}{2}-s_1,1-2s_1;2\\right)\\left(-\\partial _{\\mu _{2}}\\partial _{\\nu _{2}}\\right)\\ldots \\left(-\\partial _{\\mu _{k+1}}\\partial _{\\nu _{k+1}}\\right)\\nonumber \\\\& & \\left(\\eta _{\\mu _{k+2}\\nu _{k+2}}\\square -\\partial _{\\mu _{k+2}}\\partial _{\\nu _{k+2}}\\right)\\ldots \\left(\\eta _{\\mu _{s_1}\\nu _{s_1}}\\square -\\partial _{\\mu _{s_1}}\\partial _{\\nu _{s_1}}\\right)\\partial _{\\nu _{s_1+1}}\\ldots \\partial _{\\nu _{s_2}} \\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s_2}}\\,.$ Using the recursion relation $\\phantom{}_{2}F_{1}\\left(-k,\\frac{1}{2}-s,1-2s;2\\right)=\\frac{2-2s+k}{k+1}\\phantom{}_{2}F_{1}\\left(-k-2,\\frac{1}{2}-s,1-2s;2\\right)$ and the starting values $\\phantom{}_{2}F_{1}\\left(0,\\frac{1}{2}-s,1-2s;2\\right)=1$ and $\\phantom{}_{2}F_{1}\\left(-1,\\frac{1}{2}-s,1-2s;2\\right)=0$ , one finds $\\phantom{}_{2}F_{1}\\left(-2j-1,\\frac{1}{2}-s,1-2s;2\\right) & = & 0\\,,\\\\\\phantom{}_{2}F_{1}\\left(-2j,\\frac{1}{2}-s,1-2s;2\\right) & = & \\frac{\\Gamma \\left(2j+1\\right)\\Gamma \\left(1-2s\\right)}{\\Gamma \\left(2j-2s+1\\right)}P_{k}^{\\left(-2s,s-2j-\\frac{1}{2}\\right)}\\\\& = & \\frac{\\Gamma \\left(2j+1\\right)\\Gamma \\left(1-2s\\right)}{\\Gamma \\left(2j-2s+1\\right)}\\binom{j-s-\\frac{1}{2}}{j}\\\\& = & \\frac{\\left(-1\\right)^{j}}{2^{2j}}\\frac{\\Gamma \\left(2j\\right)\\Gamma \\left(s-j\\right)}{\\Gamma \\left(s\\right)\\Gamma (j)}\\,.$ The summation is therefore only over even $k=2j$ and, using the fact that $h$ 's are traceless and the convenient notation of projectors $\\pi _{\\mu \\nu }=\\eta _{\\mu \\nu }-\\frac{\\partial _\\mu \\partial _\\nu }{\\square }$ .",
"we can write down $I_{s_1,s_2}& = & -\\left(-2\\right)^{s_2-1}2^{-1+2s_1}\\int d^{3}x \\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\varepsilon _{\\mu _{1}\\nu _{1}\\rho }\\partial ^{\\rho }\\square ^{s_1-1}\\sum _{j=0}^{\\left[\\frac{s_1-1}{2}\\right]}\\left(-\\right)^{j}\\frac{1}{2^{2j}}\\binom{s_1-1-j}{j}\\nonumber \\\\& & \\pi _{\\mu _2 \\mu _3}\\ldots \\pi _{\\mu _{2j} \\mu _{2j+1}}\\pi _{\\nu _2 \\nu _3}\\ldots \\pi _{\\nu _{2j} \\nu _{2j+1}}\\pi _{\\mu _{2j+2} \\nu _{2j+2}}\\ldots \\pi _{\\mu _{s_1} \\nu _{s_1}}\\partial _{\\nu _{s_1+1}}\\ldots \\partial _{\\nu _{s_2}} \\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s_2}}\\,.$ One can go to momentum representation by the substitutions $\\overset{(s)}{h}(x)=\\int \\frac{d^{3}k}{(2\\pi )^3} e^{-i k x}\\overset{(s)}{h}(k) $ , $I_{s_1,s_2}& = & -\\left(-2\\right)^{s_2-1}2^{-1+2s_1}(-i)^{s_2-s_1+1}\\\\&&\\quad \\int d^{3}k \\overset{(s_1)}{h}^{\\mu _{1}\\ldots \\mu _{s_1}}\\varepsilon _{\\mu _{1}\\nu _{1}\\rho }k^{\\rho }(-k^2)^{s_1-1}\\sum _{j=0}^{\\left[\\frac{s_1-1}{2}\\right]}\\left(-\\right)^{j}\\frac{1}{2^{2j}}\\binom{s_1-1-j}{j}\\nonumber \\\\&&\\quad \\quad \\pi _{\\mu _2 \\mu _3}\\ldots \\pi _{\\mu _{2j} \\mu _{2j+1}}\\pi _{\\nu _2 \\nu _3}\\ldots \\pi _{\\nu _{2j} \\nu _{2j+1}}\\pi _{\\mu _{2j+2} \\nu _{2j+2}}\\ldots \\pi _{\\mu _{s_1} \\nu _{s_1}}k_{\\nu _{s_1+1}}\\ldots k_{\\nu _{s_2}} \\overset{(s_2)}{h}^{\\nu _{1}\\ldots \\nu _{s_2}}\\,,\\nonumber $ which corresponds to the amplitude (REF ) up to an overall constant.",
"This follows from the fact that $h$ is traceless and $s_2-s_1$ is even so we can substitute $k_{\\nu _{s_1+1}}\\ldots k_{\\nu _{s_2}}$ with $\\pi $ 's."
]
] | 1709.01738 |
[
[
"Improving Landmark Localization with Semi-Supervised Learning"
],
[
"Abstract We present two techniques to improve landmark localization in images from partially annotated datasets.",
"Our primary goal is to leverage the common situation where precise landmark locations are only provided for a small data subset, but where class labels for classification or regression tasks related to the landmarks are more abundantly available.",
"First, we propose the framework of sequential multitasking and explore it here through an architecture for landmark localization where training with class labels acts as an auxiliary signal to guide the landmark localization on unlabeled data.",
"A key aspect of our approach is that errors can be backpropagated through a complete landmark localization model.",
"Second, we propose and explore an unsupervised learning technique for landmark localization based on having a model predict equivariant landmarks with respect to transformations applied to the image.",
"We show that these techniques, improve landmark prediction considerably and can learn effective detectors even when only a small fraction of the dataset has landmark labels.",
"We present results on two toy datasets and four real datasets, with hands and faces, and report new state-of-the-art on two datasets in the wild, e.g.",
"with only 5\\% of labeled images we outperform previous state-of-the-art trained on the AFLW dataset."
],
[
"[Supplementary Information for Improving Landmark Localization with Semi-Supervised Learning ] In table REF we compare with other models on MTFL dataset which provides 5 landmarks on facial images: eye-centers, nose tip, mouth corners.",
"We follow the same protocol as for comparison, where we use train and valid sets of 9,000 and 1,000 images, respectively.",
"We test our model on AFLW and AFW subsets, with 29,995 and 337 images, that were re-annotated with 5 landmarks.",
"For the $L+A$ case we use the head-pose which is categorized into one of the five cases: right profile, right, frontal, left, left profile.",
"Other attribute labels, e.g.",
"gender and wearing glasses, cannot be determined from such few landmarks and therefore are not useful in our proposed semi-supervised learning of landmarks.",
"Table: Results on MTFL test sets for 100% labelled dataThe impact of an attribute on the landmark in sequential training depends on the amount of informational overlap between the attribute and the landmarks.",
"We suggest to measure the normalized mutual information adjusted to randomness (Adjusted Mutual Information (AMI)), as a selection heuristic, prior to applying our method.",
"AMI ranges from 0 to 1 and indicates the fraction of statistical overlap.",
"We compute for each attribute its AMI with all landmark coordinates.",
"On Multi-PIE we got AMI(x;y) = 0.045, indicating a low mutual information between coordinates x and y.",
"We therefore compute AMI for attribute (A) and every landmark (as x,y pair) by discretizing every variable uniformly under assumption of coordinate independence: AMI(A;x,y) = AMI(A;x) + AMI(A;y).",
"Every variable is uniformly discretized to have 20 levels at most.",
"Finally we measure averaged mutual information between an attribute and the set of landmarks as $\\frac{1}{N \\times L} \\sum _{n \\in N} \\sum _{x_n \\in X_n, y_n \\in Y_n} AMI(A_n;x_n) + AMI(A_n;y_n)$ where $N$ and $L$ indicate the number of samples and landmarks, $X_n$ and $Y_n$ indicate the set of $x$ and $y$ landmark coordinates per sample $n$ .",
"In Table REF we observe that hand gesture labels and head pose regression are among the most effective attributes for our method.",
"There is little mutual information between wearing glasses and landmarks, indicating lack of usefulness of this attribute for our semi-supervised setting.",
"Table: Mutual Information between all landmarks and each attributeThe attributes that are mostly useful yield a high accuracy, or low error, if we just train a neural network that takes only ground truth landmarks as input and predicts the attribute.",
"This indicates that by relying only on landmarks we can get high accuracy for those attributes.",
"In Table REF we compare the attribute prediction accuracy from the proposed Seq-MT model with a case when we do such prediction from GT landmarks.",
"Prediction from GT landmarks always outperforms the one of Seq-MT.",
"This indicates that in our semi-supervised setting, where we have few labelled landmarks, by improving the predicted locations of landmarks, both attribute and landmarks error would reduce.",
"Table: Attribute classification accuracy (MultiPIE, HGR1)—higher is better—or prediction error (AFLW)—lower is better—from GT & estimated landmarks.",
"Heatmap-MT(L) and Seq-MT(L) have the same architectures but use different loss functions (softmax vs. soft-argmax).",
"RCN(L) and RCN+(L) also only differ in their loss function.",
"When comparing these models in Tables , , , , and soft-argmax outperforms soft-max.",
"To further examine these two losses we replace soft-max with soft-argmax in Heatmap-MT and show the results in Table REF .",
"Comparing the results in Table REF with Tables and , we observe improved performance of landmark localization using soft-armgax.",
"In soft-max the model cannot be more accurate than the number of elements in the grid, since soft-max does a classification over the pixels.",
"However, in soft-argmax the model can regress to any real number and hence can get more accurate results.",
"We believe this is the reason behind its better performance.",
"Table: Results on Heatmap-MT (L+A) comparing soft-max with soft-argmax.",
"Although the focus of this paper is on improving landmark localization, in order to observe the impact of each multi-tasking approach on the attribute classification accuracy, we report the classification results on emotion in Table REF and on camera in Table REF .",
"Results show that the classification accuracy improves by providing more labeled landmarks, despite having the number of (image, class label) pairs unchanged.",
"It indicates that improving landmark localization can directly impact the classification accuracy.",
"Landmarks are especially more helpful in emotion classification.",
"On camera classification, the improvement is small and all models are getting high accuracy.",
"Another observation is that Heatmap-MT performs better on classification tasks compared to the other two multi-tasking approaches.",
"We believe this is due to passing more high-level features from the image to the attribute classification network compared to Seq-MT.",
"However, this model is performing worse than Seq-MT on landmark localization.",
"The Seq-MT model benefits from the landmark bottleneck to improve its landmark localization accuracy.",
"In Tables REF and REF by adding the ELT cost the classification accuracy improves (in addition to landmarks) indicating the improved performance in landmark localization can enhance classification performance.",
"Figure REF provides further localization examples on Multi-PIE dataset.",
"In Table REF we show classification accuracy obtained using different multi-tasking techniques.",
"Similar to the Multi-PIE dataset, we observe increased accuracy by providing more labeled landmarks, showing the classification would benefit directly from landmarks.",
"Also similar to Multi-PIE, we observe better classification accuracy with Heatmap-MT.",
"Comparing Seq-MT models, we observe improved classification accuracy by using the ELT cost.",
"It demonstrates the impact of this component on both landmark localization and classification accuracy.",
"Figure REF provides further landmark localization examples on hands dataset.",
"In Figure REF we show the architecture of RCN ${+}{}$ used for 300W and AFLW datasets.",
"In Figure REF we illustrate further samples from 300W dataset.",
"The samples show the improved accuracy obtained in both Seq-MT and RCN ${+}{}$ by using the ELT loss.",
"In Table REF we show pose estimation error using different percentage of labelled data for RCN${+}{}$ (L+ELT+A) model and compare the results to a model trained to estimate pose from GT landmarks.",
"All models get close results compared to GT model indicating RCN${+}{}$ (L+ELT+A) can do a reliable pose estimation using a small set of labelled landmarks.",
"Figure REF shows some samples on AFLW test set.",
"Table: Emotion classification accuracy on Multi-PIE test set.",
"In percent; higher is better.Table: Camera classification accuracy on Multi-PIE test set.",
"In percent; higher is better.Figure: Extra examples of our model predictions on Multi-PIE test set.We observe close predictions by 1) and 2) indicating the effectiveness of our proposed ELT cost even with only a small amount of labeled landmarks.",
"Comparison between 3) and 4) shows the improvement obtained with both the ELT loss and the sequential multitasking architecture when using a small percentage of labeled landmarks.",
"Note that the model trained with ELT loss preserves better the joint distribution over the landmarks even with a small number of labeled landmarks.",
"The last two examples show examples with high errors.",
"Best viewed in color with zoom.Figure: Extra examples of our model predictions on the HGR1 , test set.",
"GT represents ground-trust annotations, while numbers 100, 50, and 20 indicate the percentage of the training set with labeled landmarks.",
"Results are computed with Seq-MT (L+ELT+A) model (denoted *) and Seq-MT (L).",
"Examples illustrate improvement of the landmark prediction by using the class label and the ELT cost in addition to the labeled landmarks.",
"The last three examples on the bottom row show examples with high errors.",
"Best viewed in color with zoom.Figure: The ReCombinator Networks (RCN) architecture used for experiments on 300W dataset.",
"P indicates a pooling layer.",
"All pooling layers have stride of 2.",
"C indicates a convolutional layer.",
"The number written below C indicates the convolution kernel size.",
"All convolutions have stride of 1.",
"U indicates an upsampling layer, where each feature map is upsampled to the next (bigger) feature map resolution.",
"K indicates concatenation, where the upsampled features are concatenated with features of the same resolution before a pooling is applied to them.",
"The dashed arrows indicate the feature maps are carried forward for concatenation.",
"The solid arrows following each other, e.g.",
"P, C, indicate the order of independent operations that are applied.",
"The number written above feature maps in n@w×hn@w \\times h format indicate number of feature maps nn and the width ww and height hh of the feature maps.",
"On AFLW, we use 70 feature maps per layer (instead of 64) and we get two levels coarser to get to 1×11 \\times 1 resolution (instead of 5×55 \\times 5).",
"On both datasets we shoud β=100\\beta =100 for soft-argmax layer.Figure: Extra examples of our model predictions on 300W test-set.",
"The first two columns depict examples where all models get accurate predictions, The next 5 columns illustrate the improved accuracy obtained by using ELT loss in two different architectures (Seq-MT and RCN).",
"The last two columns show difficult examples where error is high.",
"The rectangles indicate the regions that landmarks are mostly affected.",
"The green and red dots show ground truth (GT) and model predictions (MP), respectively.",
"The yellow lines show the error by connecting GT and MP.",
"Note that the ELT loss improves predictions in both architectures.",
"Best viewed in color with zoom.Figure: Extra examples of our model predictions on the AFLW test set.",
"Comparing the first and second rows shows the improvement obtained by using ELT+A with only 1% of labelled landmarks.",
"Note the model trained using ELT+A preserves better the distribution over the landmarks.",
"The last two columns in the bottom row show samples with high error on small percentage of labelled landmaks, which is due to extreme rotation.",
"The bottom row shows the prediction using L+ELT+A on the entire set of labelled landmarks, which gets the best results.",
"The green and red dots show ground truth (GT) and model predictions (MP), respectively.",
"The yellow lines show the error by connecting GT and MP.",
"Best viewed in color with zoom.Table: Classification accuracy on hands test set.",
"In percent; higher is better.Table: Pose degree estimation error on AFLW test set, as average of yaw, pitch, roll values.",
"lower is better.Table: Architecture details for Comm-MT Model on Blocks dataset.Table: Architecture details for Heatmap-MT Model on Blocks datasets.The architecture details of Seq-MT model on different datasets can be seen in Tables REF , REF and REF .",
"Architecture details of Comm-MT and Heatmap-MT for Blocks dataset are shown in Tables REF and REF .",
"For other dataset, the kernel size and the number of feature maps for conv layers and the number of units for FC layers change similar to Seq-MT model on those datasets.",
"Table: Architecture details of Seq-MT model used for Shapes and Blocks datasets.Each conv layer has three values as w×h×nw \\times h \\times n indicating width (w), height (h) of kernel and the number of feature maps (n) of the convolutional layer.",
"SAME indicates the input map is padded with zeros such that input and output maps have the same resolution.Table: Architecture details of Seq-MT model used for Hands and Multi-PIE datasets.Table: Architecture details of Seq-MT model used for 300W datasets."
]
] | 1709.01591 |
[
[
"Estimation of a Low-rank Topic-Based Model for Information Cascades"
],
[
"Abstract We consider the problem of estimating the latent structure of a social network based on the observed information diffusion events, or cascades, where the observations for a given cascade consist of only the timestamps of infection for infected nodes but not the source of the infection.",
"Most of the existing work on this problem has focused on estimating a diffusion matrix without any structural assumptions on it.",
"In this paper, we propose a novel model based on the intuition that an information is more likely to propagate among two nodes if they are interested in similar topics which are also prominent in the information content.",
"In particular, our model endows each node with an influence vector (which measures how authoritative the node is on each topic) and a receptivity vector (which measures how susceptible the node is for each topic).",
"We show how this node-topic structure can be estimated from the observed cascades, and prove the consistency of the estimator.",
"Experiments on synthetic and real data demonstrate the improved performance and better interpretability of our model compared to existing state-of-the-art methods."
],
[
"Introduction",
"The spread of information in online web or social networks, the propagation of diseases among people, as well as the diffusion of culture among countries are all examples of information diffusion processes or cascades.",
"In many of the applications, it is common to observe the spread of a cascade, but not the underlying network structure that facilitates the spread.",
"For example, marketing data sets capture the times of purchase of products by consumers, but not whether the consumer was influenced by a recommendation of a friend or an advertisement on TV; we can observe when a person falls ill, but we cannot observe who infected him/her.",
"In all these settings, we can observe the propagation of information but cannot observe the way they propagate.",
"There is a vast literature on recovering the underlying network structure based on the observations of information diffusion.",
"A network is represented by a diffusion matrix that characterizes connection between nodes, that is, the diffusion matrix gives weight/strength of the arcs between all ordered pairs of vertices.",
"[15] propose a continuous time diffusion model and formulate the problem of recovering the underlying network diffusion matrix by maximizing the log-likelihood function.",
"The model of [15] imposes no structure among nodes and allows for arbitrary diffusion matrices.",
"As a modification of this basic model, [7] consider a more sophisticated topic-sensitive model where each information cascade is associated with a topic distribution on several different topics.",
"Each topic is associated with a distinct diffusion matrix and the diffusion matrix for a specific cascade is a weighted sum of these diffusion matrices with the weights given by the topic distribution of the cascade.",
"This model can capture our intuition that news on certain topics (e.g., information technology) may spread much faster and broader than some others (e.g., military).",
"However, since the diffusion matrix for each topic can be arbitrary, the model fails to capture the intuition that nodes have intrinsic topics of interest.",
"In this paper, we propose a novel mathematical model that incorporates the node-specific topics of interest.",
"Throughout the paper we use the diffusion of news among people as an example of cascades for illustrative purposes.",
"An item of news is usually focused on one or a few topics (e.g., entertainment, foreign policy, health), and is more likely to propagate between two people if both of them are interested in these same topics.",
"Furthermore, a news item is more likely to be shared from node 1 to node 2 if node 1 is influential/authoritative in the topic, and node 2 is receptive/susceptible to the topic.",
"Our proposed mathematical model is able to capture this intuition.",
"We show how this node-topic structure (influence and receptivity) can be estimated based on observed cascades with a theoretical guarantee.",
"Finally, on the flip side, after obtaining such a network structure, we can then use this structure to assign a topic distribution to a new cascade.",
"For example, an unknown disease can be classified by looking at its propagation behavior.",
"To the best of our knowledge, this is the first paper to leverage users' interests for recovering the underlying network structure from observed information cascades.",
"Theoretically, we prove that our proposed algorithm converges linearly to the true model parameters up to statistical error; experimentally, we demonstrate the scalability of our model to large networks, robustness to overfitting, and better performance compared to existing state-of-the-art methods on both synthetic and real data.",
"While existing algorithms output a large graph representing the underlying network structure, our algorithm outputs the topic interest of each node, which provides better interpretability.",
"This structure can then be used to predict future diffusions, or for customer segmentation based on interests.",
"It can also be applied to build recommendation systems, and for marketing applications such as targeted advertising, which is impossible for existing works.",
"A conference version of this paper was presented in the 2017 IEEE International Conference on Data Mining (ICDM) series [44].",
"Compared to the conference version, in this paper we extend the results in the following ways: (1) we introduce a new penalization method and a new algorithm in Section ; (2) we build theoretical result for our proposed algorithm in Section ; (3) we discuss several variants and applications of our model in Section ; (4) we evaluate the performance of our algorithm on a new dataset in Section REF ."
],
[
"Related Work",
"A large body of literature exists on recovery of latent network structure based on observed information diffusion cascades [25], [19].",
"See [20] for a survey.",
"[41] introduce a Generalized Linear Cascade Model for discrete time.",
"Alternative approaches to analysis of discrete time networks have been considered in [8], [27], [26], [34], [38], [10].",
"In this paper we focus on network inference under the continuous-time diffusion model introduced in [15], where the authors formulate the network recovery problem as a convex program and propose an efficient algorithm ($\\textbf {NetRate}$ ) to recover the diffusion matrix.",
"In a follow-up work, [14] look at the problem of finding the best $K$ edge graph of the network.",
"They show that this problem is NP-hard and develop NetInf algorithm that can find a near-optimal set of $K$ directed edges.",
"[16] consider a dynamic network inference problem, where it is assumed that there is an unobserved dynamic network that changes over time and propose InfoPath algorithm to recover the dynamic network.",
"[5] relax the restriction that the transmission function should have a specific form, and propose KernelCascade algorithm that can infer the transmission function automatically from the data.",
"It allows each pair of nodes to have a different type of transmission model and hence better captures the heterogeneous influence among nodes.",
"[49] use multi-dimensional Hawkes processes to capture the temporal patterns of nodes behaviors.",
"By optimizing the nuclear and $\\ell _1$ norm simultaneously, ADM4 algorithm recovers the network structure that is both low-rank and sparse.",
"[36] consider external influence in the model: information can reach a node via the links of the social network or through the influence of external sources.",
"[35] further assume interaction among cascades: competing cascades decrease each other's probability of spreading, while cooperating cascades help each other in being adopted throughout the network.",
"[17] prove a lower bound on the number of cascades needed to recover the whole network structure correctly.",
"[21] combine Hawkes processes and topic modeling to simultaneously reason about the information diffusion pathways and the topics of the observed text-based cascades.",
"Other related works also include [2], [32], [6], [13], [24], [48].",
"The work most closely related to ours is [7], where the authors propose a topic-sensitive model that modifies the basic model of [15] to allow cascades with different topics to have different diffusion rates.",
"However, this topic-sensitive model still fails to account for the interaction between nodes and topics."
],
[
"Organization of the Paper",
"In Section we briefly review the basic continuous-time diffusion network model introduced in [15] and the topic-sensitive model introduced in [7].",
"We propose our influence-receptivity model in Section .",
"Section details two optimization algorithms.",
"Section provides theoretical results for the proposed algorithm.",
"In Section we discuss extensions of our model.",
"Sections and present experimental results on synthetic dataset and two real world datasets, respectively.",
"We conclude in Section ."
],
[
"Notation",
"We use $p$ to denote the number of nodes in a network and $K$ to denote the number of topics.",
"The number of observed cascades is denoted as $n$ .",
"We use subscripts $i, j \\in \\lbrace 1, \\ldots , p\\rbrace $ to index nodes; $k \\in \\lbrace 1,\\ldots , K\\rbrace $ to index topics; and $c$ to index each cascade.",
"For any matrix $A$ , we use $\\Vert A\\Vert _2$ and $\\Vert A\\Vert _F$ to denote the matrix spectral norm and Frobenius norm, respectively.",
"Moreover, $\\Vert A\\Vert _0 = \\big |(i,j): A_{ij} \\ne 0\\big |$ denotes the number of nonzero components of a matrix.",
"The operation $[A]_+$ keeps only nonnegative values of $A$ and puts zero in place of negative values.",
"For a nonnegative matrix $A$ , the operation $\\text{Hard}(A, s)$ keeps only the $s$ largest components of $A$ and zeros out the rest of the entries.",
"We use $S = \\text{supp}(A) = {(i,j): A_{ij} \\ne 0}$ to denote the support set of matrix $A$ (with an analogous definition for a vector).",
"For any matrix $A$ and support set $S$ , we denote $[A]_S$ as the matrix that takes the same value as $A$ on $S$ , and zero elsewhere.",
"For any matrices $A$ and $B$ , denote $\\langle A, B \\rangle = (A^\\top B)$ as the matrix inner product and $\\langle A, B \\rangle _S = \\big ([A]_S^\\top \\cdot [B]_S \\big )$ as the inner product on the support $S$ only."
],
[
"Background",
"We briefly review the basic continuous time diffusion network model introduced in [15] in Section REF .",
"The topic-sensitive model introduced as a modification of the basic model in [7] is reviewed in Section REF ."
],
[
"Network structure and cascade generating process.",
"The model of [15] assumes that the underlying network is composed of $p$ nodes and uses a non-negative diffusion matrix $A = \\lbrace \\alpha _{ji}\\rbrace $ to parameterize the edges among them.",
"The parameter $\\alpha _{ji}$ measures the transmission rate from $j$ to $i$ , where a larger $\\alpha _{ji}$ means stronger connection from $j$ to $i$ .",
"The absence of $j \\rightarrow i$ edge is denoted by $\\alpha _{ji} = 0$ .",
"For every node $i$ , self infection is not considered and $\\alpha _{ii} = 0$ .",
"A cascade based on the model and network here is generated in the following way.",
"At the beginning, with time 0, one of the $p$ nodes is infected as a source node.",
"When a node $j$ is infected, it samples a time at which it infects other uninfected nodes it is connected to.",
"The transmission time $\\tau _{ji}$ from node $j$ to $i$ follows a random distribution with a density $\\ell (\\tau ;\\alpha _{ji})$ for $\\tau \\ge 0$ (this density is called the transmission function/kernel).",
"A node $i$ is infected the first time one of the nodes which can reach $i$ infects it.",
"After being infected, node $i$ becomes a new source and begins to infect other nodes by following the same procedure and sampling the transmission times to other uninfected nodes that it can reach.",
"The model assumes an observation window of length $T$ time units since the infection of the source node; nodes that are not infected until time $T$ are regarded as uninfected.",
"We write $\\ell (t_i \\mid t_j;\\alpha _{ji}) = \\ell (t_i-t_j;\\alpha _{ji})$ to indicate the density that $i$ is infected by $j$ at time $t_i$ given that $j$ is infected at time $t_j$ , parameterized by $\\alpha _{ji}$ .",
"The transmission times of each infection are assumed to be independent, and a node remains infected in the whole process once it is infected."
],
[
"Data.",
"In order to fit parameters of the model above, we assume that there are $n$ independent cascades denoted by the set $C^n = \\lbrace t^1, \\ldots , t^n\\rbrace $ .",
"A cascade $c$ is represented by $ t^c$ , which is a $p$ -dimensional vector $ t^c = (t_1^c, \\ldots , t_p^c)$ indicating the time of infection of the $p$ nodes; $t_i^c \\in [0, T^c] \\bigcup \\lbrace \\infty \\rbrace $ with $T^c$ being the observation window for the cascade $c$ .",
"Although not necessary, for notational simplicity we assume $T^c = T$ for all the cascades.",
"For an infected node, only the first infected time is recorded even if it is infected by multiple neighbors.",
"For the source node $i$ , $t_i^c = 0$ , while node uninfected up to time $T$ we use the convention $t_i^c = \\infty $ ."
],
[
"Likelihood function.",
"The likelihood function of an observed cascade $t$ is given by $\\begin{aligned}\\ell ( t; A) = \\prod \\limits _{t_i \\le T} \\prod \\limits _{t_m > T} S(T\\mid t_i;\\alpha _{im}) \\times \\prod \\limits _{k:t_k < t_i}S(t_i\\mid t_k;\\alpha _{ki}) \\sum _{j:t_j<t_i}H(t_i \\mid t_j;\\alpha _{ji}) ,\\end{aligned}$ where $S(t_i\\mid t_j;\\alpha _{ji}) = 1 - \\int _{t_j}^{t_i} \\ell (t-t_j;\\alpha _{ji})\\,dt$ is the survival function and $H(t_i|t_j;\\alpha _{ji}) = \\ell (t_i-t_j;\\alpha _{ji}) / S(t_i|t_j;\\alpha _{ji})$ is the hazard function [15].",
"Note that the likelihood function consists of two probabilities.",
"The first one is the probability that an uninfected node “survives” given its infected neighbors; the second one is the density that an infected node is infected at the specific observed time.",
"The transmission function affects the behavior of a cascade.",
"Several commonly used transmission functions are exponential, Rayleigh, and power-law distributions [15].",
"For exponential transmission, the diffusion rate reaches its maximum value at the beginning and then decreases exponentially.",
"Because of this property, it can be used to model information diffusion on internet or a social network, since (breaking) news usually spread among people immediately, while with time a story gradually becomes unpopular.",
"The exponential transmission function is given by $\\ell (\\tau ;\\alpha _{ji}) = \\alpha _{ji} \\cdot \\exp (-\\alpha _{ji}\\tau )$ for $\\tau \\ge 0$ and $\\ell (\\tau ;\\alpha _{ji}) = 0$ otherwise.",
"We then have $S(t+\\tau \\mid t;\\alpha _{ji}) = \\exp (-\\alpha _{ji}\\tau )$ and $H(t+\\tau \\mid t;\\alpha _{ji}) = \\alpha _{ji}$ .",
"As a different example, with the Rayleigh transmission function the diffusion rate is small at the beginning; it then rises to a peak and then drops.",
"It can be used to model citation networks, since it usually takes some time to publish a new paper and cite the previous paper.",
"New papers then gradually become known by researchers.",
"The Rayleigh transmission function is given as $\\ell (\\tau ;\\alpha _{ji}) = \\alpha _{ji} \\tau \\cdot \\exp \\Big (-\\frac{1}{2}\\alpha _{ji}\\tau ^2\\Big )$ for $\\tau \\ge 0$ and $\\ell (\\tau ;\\alpha _{ji}) = 0$ otherwise.",
"We then have $S(t+\\tau \\mid t;\\alpha _{ji}) = \\exp (-\\frac{1}{2}\\alpha _{ji}\\tau ^2)$ and $H(t+\\tau \\mid t;\\alpha _{ji}) = \\alpha _{ji}\\tau $ .",
"We will use these two transmission functions in Section for modeling information diffusion on internet and in citation networks, respectively."
],
[
"Optimization problem.",
"The unknown parameter is the diffusion matrix $A$ , which can be estimated by maximizing the likelihood $\\begin{aligned}& \\mathop {\\text{minimize}}_{\\alpha _{ji}} \\quad -\\frac{1}{n} \\sum _{c\\in C^n} \\log \\, {\\ell ( t^c; A)} \\\\& \\text{subject to} \\quad \\alpha _{ji} \\ge 0 , j \\ne i.\\end{aligned}$ A nice property of the above optimization program is that it can be further separated into $p$ independent subproblems involving individual columns of $A$ .",
"Specifically, the $i^{th}$ subproblem is to infer the incoming edges into the node $i$ $\\begin{aligned}& \\mathop {\\text{minimize}}_{\\alpha _i} \\quad \\, \\phi ( \\alpha _i) \\\\& \\text{subject to} \\,\\,\\,\\,\\, \\alpha _{ji} \\ge 0 , j \\ne i,\\end{aligned}$ where the parameter $\\alpha _i = \\lbrace \\alpha _{ji} \\mid j=1, \\ldots , N, j \\ne i\\rbrace $ denotes the $i^{th}$ column of $A$ and the objective function is $\\phi ( \\alpha _i) = -\\frac{1}{n}\\sum _{c\\in C^n} \\phi _i( t^c ;\\alpha _i),$ with $\\phi _i(\\cdot ; \\alpha _i)$ denoting the likelihood function for one cascade.",
"For example, for the exponential transmission function, we have $\\phi _i( {t}; {\\alpha }_i) = \\text{log} \\Bigg ( \\sum _{j:t_j<t_i} \\alpha _{ji} \\Bigg ) - \\sum _{j:t_j<t_i} \\alpha _{ji}(t_i-t_j)$ for an infected node, and $\\phi _i( {t}; {\\alpha }_i) = - \\sum _{j:t_j<T} \\alpha _{ji}(T-t_j)$ for an uninfected node.",
"See [15] for more details.",
"The problem (REF ) is convex in $ \\alpha _i$ and can be solved by a standard gradient-based algorithm.",
"The linear terms in (REF ) and (REF ) act as an $\\ell _1$ penalty on the unknown parameter and automatically encourage sparse solutions.",
"Nonetheless, adding an explicit $\\ell _1$ penalty can further improve results.",
"[17] propose to solve the following regularized optimization problem $\\begin{aligned}& \\mathop {\\text{minimize}}_{\\alpha _i} \\quad \\phi ( \\alpha _i) + \\lambda \\Vert \\alpha _i\\Vert _1 \\\\& \\text{subject to} \\quad \\alpha _{ji} \\ge 0 , j \\ne i,\\end{aligned}$ using a proximal gradient algorithm [39]."
],
[
"Topic-sensitive model",
"The basic model described above makes an unrealistic assumption that each cascade spreads based on the same diffusion matrix $A$ .",
"However, for example, posts on information technology usually spread much faster than those on economy and military.",
"[7] extend the basic model to incorporate this phenomena.",
"Their topic-sensitive model assumes that there are in total $K$ topics, and each cascade can be represented as a topic vector in the canonical $K$ -dimensional simplex, in which each component is the weight of a topic: $ {m}^c := (m_1^c,...,m_K^c)^{\\top }$ with $\\sum _k m_k^c = 1$ and $m_k^c \\in [0,1]$ .",
"Each topic $k$ is assumed to have its own diffusion matrix $A^k= \\left\\lbrace \\alpha _{ji}^k \\right\\rbrace $ , and the diffusion matrix of the cascade $A^c = \\left\\lbrace \\alpha _{ji}^c \\right\\rbrace $ is the weighted sum of the $K$ matrices: $\\alpha _{ji}^c = \\sum _{k=1}^K \\alpha _{ji}^k m_k^c .$ In this way, the diffusion matrix $A^c$ can be different for different cascades.",
"For each cascade $c$ , the propagation model remains the same as the basic model described in the previous section, but with the diffusion matrix $A^c$ given in (REF ).",
"The unknown parameters $A^1, \\ldots , A^K$ can be estimated by maximizing the regularized log-likelihood.",
"[7] use a group lasso type penalty and solve the following regularized optimization problem $\\begin{aligned}& \\mathop {\\text{minimize}}_{\\alpha _{ji}^k} \\quad -\\frac{1}{n}\\sum _{c\\in C^n} \\phi _i\\Big ( t^c ; \\big \\lbrace \\alpha _{ji}^c \\big \\rbrace _{j=1}^p \\Big ) + \\lambda \\sum _j \\Vert \\alpha _{ji}\\Vert _2 \\\\& \\text{subject to} \\quad \\alpha _{ji}^c = \\sum _{k=1}^K \\alpha _{ji}^k m_k^c, \\\\& \\qquad \\qquad \\quad \\,\\, \\alpha _{ji}^k \\ge 0 , \\, j \\ne i,\\end{aligned}$ with a proximal gradient based block coordinate descent algorithm."
],
[
"An Influence-Receptivity based Topic-sensitive Model",
"In this section we describe our proposed influence-receptivity model.",
"Our motivation for proposing a new model for information diffusion stems from the observation that the two models discussed in Section to not impose any structural assumptions on $A$ or $A^k$ other than nonnegativity and sparsity.",
"However, in real world applications we observe node-topic interactions in the diffusion network.",
"For example, different social media outlets usually focus on different topics, like information technology, economy or military.",
"If the main focus of a media outlet is on information technology, then it is more likely to publish or cite news with that topic.",
"Here the topics of interest of a media outlet imparts the network structure.",
"As another example, in a university, students may be interested in different academic subjects, may have different music preferences, or follow different sports.",
"In this way it is expected that students who share the same or similar areas of interest may have much stronger connections.",
"Here the areas of interest among students impart the structure to the diffusion network.",
"Finally, in the context of epidemiology, people usually have different immune systems, and a disease such as flu, usually tends to infect some specific people, while leaving others uninfected.",
"It is very likely that the infected people (by a specific disease) may have similar immune system, and therefore tend to become contagious together.",
"Here the types of immune system among people imparts the structure.",
"Taking this intuition into account, we build on the topic-sensitive diffusion model of [7] by imposing a node-topic interaction.",
"This interaction corresponds to the structural assumption on the cascade diffusion matrix $A^c$ for each cascade $c$ .",
"As before, a cascade $c$ is represented by its weight on $K$ topics ($K \\ll p$ ): $ m^c = (m_1^c, m_2^c, \\ldots , m_K^c)^{\\top } $ , with $\\sum _k m_k^c = 1$ and $m_k^c \\in [0,1]$ .",
"Each node is parameterized by its “interest” in each of these $K$ topics as a $K$ dimensional (row) vector.",
"Stacking them together, the “interest” of all the $p$ nodes form a $p \\times K$ dimensional matrix.",
"To describe such structure, we propose two node-topic matrices $B_1, B_2 \\in \\mathbb {R}^{p \\times K}$ , where $B_1$ measures how much a node can infect others (the influence matrix) and $B_2$ measures how much a node can be infected by others (the receptivity matrix).",
"We use $b_{ik}^1$ and $b_{ik}^2$ to denote the elements on $i^{th}$ row and $k^{th}$ column of $B_1$ and $B_2$ , respectively.",
"A large $b_{ik}^1$ means that node $i$ tends to infect others on topic $k$ ; while a large $b_{ik}^2$ means that node $i$ tends to be infected by others on topic $k$ .",
"These two matrices model the observation that, in general, the behaviors of infecting others and being infected by others are usually different.",
"For example, suppose a media outlet $i$ has many experts in a topic $k$ , then it will publish many authoritative articles on this topic.",
"These articles are likely to be well-cited by others and therefore it has a large $b_{ik}^1$ .",
"However, its $b_{ik}^2$ may not be large, because $i$ has experts in topic $k$ and does not need to cite too many other news outlets on topic $k$ .",
"On the other hand, if a media outlet $i$ is only interested in topic $k$ but does not have many experts, then it will have a small $b_{ik}^1$ and a large $b_{ik}^2$ .",
"For a specific cascade $c$ on topic $k$ , there will be an edge $j \\rightarrow i$ if and only if node $j$ tends to infect others on topic $k$ (large $b_{jk}^1$ ) and node $i$ tends to be infected by others on topic $k$ (large $b_{ik}^2$ ).",
"For a cascade $c$ with the topic-weight $ m^c$ , the diffusion parameter $\\alpha _{ji}^c$ is modeled as $\\alpha _{ji}^c = \\sum _{k=1}^K b_{jk}^1 \\cdot m_k^c \\cdot b_{ik}^2.$ The diffusion matrix for a cascade $c$ can be then represented as $A^c = B_1 \\cdot M^c \\cdot B_2^{\\top } = \\sum _{k=1}^K m_k^c \\cdot b_k^1 {b_k^2}^\\top ,$ where $M^c = (m^c)$ is a diagonal matrix representing the topic weight and $B_j = [b^j_1, \\ldots , b^j_K]$ with $b^j_k$ denoting the $k^{\\text{th}}$ column of $B_j$ , $j=1,2$ .",
"In a case where one does not consider self infection, we can modify the diffusion matrix for a cascade $c$ as $A^c = B_1 M^c B_2^{\\top } - \\text{diag}(B_1 M^c B_2^{\\top } ).$ Under the model in (REF ), the matrix $M^c$ is known for each cascade $c \\in C^n$ , and the unknown parameters are $B_1$ and $B_2$ only.",
"The topic weights can be obtained from a topic model, such as latent Dirichlet allocation [1], as long as we are given the text information of each cascade, for example, the main text in a website or abstract/keywords of a paper.",
"The number of topics $K$ is user specified or can be estimated from data [22].",
"The extension to a setting with an unknown topic distribution $M^c$ is discussed in Section REF .",
"With a known topic distribution $M^c$ , our model has $2pK$ parameters.",
"Compared to the basic model, which has $p^2$ parameters, and the topic-sensitive model, which has $p^2K$ parameters, we observe that our proposed model has much fewer parameters since, usually, we have $K \\ll p$ .",
"Based on (REF ), our model can be viewed as a special case of the topic-sensitive model where each topic diffusion matrix $A^k$ is assumed to be of rank 1.",
"A natural generalization of our model is to relax the constraint and consider topic diffusion matrices of higher rank, which would correspond to several influence and receptivity vectors affecting the diffusion together."
],
[
"Estimation",
"In this section we develop an estimation procedure for parameters of the model described in the last section.",
"In Section REF and REF we reparameterize the problem and introduce regularization terms in order to guarantee unique solution to estimation procedure.",
"We then propose efficient algorithms to solve the regularized problem in Section REF ."
],
[
"Reparameterization",
"The negative log-likelihood function for our model is easily obtained by plugging the parametrization of a diffusion matrix in (REF ) into the original problem (REF ).",
"Specifically, the objective function we would like to minimize is given by $f(B_1, B_2) = -\\frac{1}{n} \\sum _{c\\in C^n} \\log {\\ell \\big ( t^c; B_1 M^c B_2^\\top \\big )}.$ Unfortunately, this objective function is not separable in each column of $B_1, B_2$ , so we have to deal with entire matrices.",
"Based on (REF ), recall that the diffusion matrix $A^c$ can be viewed as a weighted sum of $K$ rank-1 matrices.",
"Let $\\Theta _k = b_k^1 {b_k^2}^\\top $ and denote the collection of these rank-1 matrices as $\\Theta = (\\Theta _1, \\ldots , \\Theta _K)$ .",
"With some abuse of notation, the objective function $f(\\cdot )$ in (REF ) can be rewritten as $f(\\Theta ) = f(\\Theta _1, \\ldots , \\Theta _K)= -\\frac{1}{n} \\sum _{c\\in C^n} \\log {\\ell \\bigg ( t^c; \\sum _{k=1}^K m_k^c \\cdot \\Theta _k \\bigg )}.$ Note that since that $\\log \\ell (\\cdot )$ is convex and $A^c$ is linear in $\\Theta _k$ , the objective function $f(\\Theta )$ is convex in $\\Theta $ when we ignore the rank-1 one constraint on $\\Theta _k$ ."
],
[
"Parameter estimation",
"To simplify the notation, we use $f(\\cdot )$ to denote the objective function in (REF ) or (REF ), regardless of the parameterization as $B_1,B_2$ or $\\Theta $ .",
"From the parameterization $\\Theta _k = b_k^1 {b_k^2}^\\top $ , it is clear that if we multiply $b_k^1$ by a constant $\\gamma $ and multiply $b_k^2$ by $1/\\gamma $ , the matrix $\\Theta _k$ and the objective function (REF ) remain unchanged.",
"In particular, we see that the problem is not identifiable if parameterized by $B_1,B_2$ .",
"To solve this issues we add regularization.",
"A reasonable and straightforward choice of regularization is the $\\ell _1$ norm regularization on $B_1$ and $B_2$ .",
"We define the following norm $g_1(B_1, B_2) = \\big \\Vert B_1 + B_2\\big \\Vert _{1,1} \\triangleq \\sum _{i, k} b^1_{ik} + b^2_{ik}$ and the regularized objective becomes $f_1(B_1, B_2) = -\\frac{1}{n} \\sum _{c\\in C^n} \\log {\\ell \\big ( t^c; B_1 M^c B_2^\\top \\big )} + \\lambda \\cdot g_1(B_1, B_2),$ where $\\lambda $ is a tuning parameter.",
"With this regularization, if we focus on the $k^{\\text{th}}$ column, then the term we would like to minimize is $\\gamma \\Vert b^1_k\\Vert _1 + \\frac{1}{\\gamma } \\Vert b^2_k\\Vert _1.$ Clearly, in order to minimize (REF ) we should select $\\gamma $ such that the two terms in (REF ) are equal.",
"This means that, at the optimum, the column sums of $B_1$ and $B_2$ are equal.",
"We therefore avoid the scaling issue by adding the $\\ell _1$ norm penalty.",
"An alternative choice of the regularizer is motivated by the literature on matrix factorization [23], [42], [40], [11], [47].",
"In a matrix factorization problem, the parameter matrix $X$ is assumed to be low-rank, which can be explicitly represented as $X = UT^\\top $ where $X \\in ^{p \\times p}$ , $U, V \\in ^{p \\times r}$ , and $r$ is the rank of $X$ .",
"Similar to our problem, this formulation is also not identifiable.",
"By adding the regularization term $\\Vert UU^\\top - VV^\\top \\Vert _F^2$ , the singular values of $U$ and $V$ are the same at the optimum [50], [47], [40], [45].",
"Motivated by this approach, we consider the following regularization term $g_2(B_1,B_2) = \\frac{1}{4} \\cdot \\sum _{k=1}^K \\Big ( \\big \\Vert b^1_k\\big \\Vert _2^2 - \\big \\Vert b^2_k \\big \\Vert _2^2 \\Big )^2,$ which arises from viewing our problem as a matrix factorization problem with rank-1 matrices.",
"The regularized objective function is therefore given by $f_2(B_1, B_2) = -\\frac{1}{n} \\sum _{c\\in C^n} \\log {\\ell \\big ( t^c; B_1 M^c B_2^\\top \\big )} + \\lambda \\cdot g_2(B_1,B_2).$ Note that for this regularization penalty, at the minimum, we have that $g_2(B_1,B_2) = 0$ and that the $\\ell _2$ -norm of the columns of $B_1$ and $B_2$ are equal.",
"Furthermore, we can pick any positive regularization penalty $\\lambda $ .",
"In summary, both regularizers $g_1(\\cdot )$ and $g_2(\\cdot )$ force the columns of $B_1$ and $B_2$ to be balanced.",
"At optimum the columns will have the same $\\ell _1$ norm if $g_1$ is used and the same $\\ell _2$ norm if $g_2$ is used.",
"As a result, for each topic $k$ , the total magnitudes of “influence” and “receptivity” are the same.",
"In particular, a regularizer enforces the conservation law that the total amount of output should be equal to the total amount of input.",
"The $\\ell _1$ norm regularizer induces a biased sparse solution.",
"In contrast, the regularizer $g_2$ neither introduces bias nor encourages a sparse solution.",
"Since in real world applications each node is usually interested in only a few topics, the two matrices $B_1, B_2$ are assumed to be sparse, as we state in the next section.",
"Taking this into account, if the regularizer $g_2$ is used, we need to threshold the estimator to obtain a sparse solution.",
"Based on our experience, both regularizers provide good estimators for $B_1$ and $B_2$ .",
"In conclusion, the optimization problem that we are going to solve is $\\begin{aligned}&\\mathop {\\text{minimize}}_{B_1,B_2} \\quad -\\frac{1}{n} \\sum _{c\\in C^n} \\log {\\ell \\big ( t^c; B_1 M^c B_2^\\top \\big )} + \\lambda \\cdot g(B_1, B_2) \\\\& \\text{subject to} \\quad B_1, B_2 \\ge 0,\\end{aligned}$ where the regularization $g(\\cdot )$ is either $g_1(\\cdot )$ , defined in (REF ), or $g_2(\\cdot )$ , defined in (REF )."
],
[
"Optimization algorithm",
"While the optimization program (REF ) is convex in the diffusion matrix $A$ , the proposed problem (REF ) is nonconvex in $B_1, B_2$ .",
"Our model for a diffusion matrix (REF ) is bilinear and, as a result, the problem (REF ) is a biconvex problem in $B_1$ and $B_2$ , that is, the problem is convex in $B_1$ and $B_2$ , but not jointly convex.",
"[18] provide a survey of methods for minimizing biconvex functions.",
"In general, there are no efficient algorithms for finding the global minimum of a biconvex problem.",
"[9] propose a global optimization algorithm, which alternately solves primal and relaxed dual problem.",
"This algorithm is guaranteed to find the global minimum, but the time complexity is usually exponential.",
"For our problem, we choose to develop a gradient-based algorithm.",
"For the regularizer $g_1$ , since the $\\ell _1$ norm is non-smooth, we develop a proximal gradient descent algorithm [39]; for the regularizer $g_2$ , we use an iterative hard thresholding algorithm [45].",
"[tb] Proximal gradient descent for (REF ) with regularizer $g_1(\\cdot )$ Initialize $B_1^{(0)}$ , $B_2^{(0)}$ $tolerance > \\epsilon $ $B_1^{(t+1)} = \\Big [B_1^{(t)} - \\eta \\nabla _{B_1} f\\big (B_1^{(t)}, B_2^{(t)}\\big ) - \\lambda \\eta \\Big ]_+$ $B_2^{(t+1)} = \\Big [B_2^{(t)} - \\eta \\nabla _{B_2} f\\big (B_1^{(t)}, B_2^{(t)}\\big ) - \\lambda \\eta \\Big ]_+$ [tb] Gradient descent with hard thresholding for (REF ) with regularizer $g_2(\\cdot )$ Initialize $B_1^{(0)}$ , $B_2^{(0)}$ $tolerance > \\epsilon $ $B_1^{(t+0.5)} = \\Big [B_1^{(t)} - \\eta \\cdot \\nabla _{B_1} f\\big (B_1^{(t)}, B_2^{(t)}\\big ) - \\eta \\cdot \\nabla _{B_1} g_2\\big (B_1^{(t)}, B_2^{(t)}\\big )\\Big ]_+$ $B_1^{(t+1)} = \\text{Hard}\\big (B_1^{(t+0.5)}, s\\big )$ $B_2^{(t+0.5)} = \\Big [B_2^{(t)} - \\eta \\cdot \\nabla _{B_2} f\\big (B_1^{(t)}, B_2^{(t)}\\big ) - \\eta \\cdot \\nabla _{B_2} g_2\\big (B_1^{(t)}, B_2^{(t)}\\big )\\Big ]_+$ $B_2^{(t+1)} = \\text{Hard}\\big (B_2^{(t+0.5)}, s\\big )$ Since the optimization problem (REF ) is nonconvex, we need to carefully initialize the iterates $B_1^{(0)}, B_2^{(0)}$ for both algorithms.",
"We find the initial iterates by minimizing the objective function $f(\\Theta )$ , defined in (REF ), without the rank-1 constraint.",
"As discussed earlier, the objective function $f(\\Theta )$ is convex in $\\Theta $ and can be minimized by, for example, the gradient descent algorithm.",
"After obtaining the minimizer $\\widehat{\\Theta }= (\\widehat{\\Theta }_1, \\ldots , \\widehat{\\Theta }_K)$ , we find the best rank-1 approximation of each $\\widehat{\\Theta }_k$ .",
"According to the Eckart-Young-Mirsky theorem, the best rank-1 approximation is obtained by the singular value decomposition (SVD) by keeping the largest singular value and corresponding singular vectors.",
"Specifically, suppose the leading term of SVD for $\\widehat{\\Theta }_k$ is denoted as $\\sigma _ku_kv_k^\\top $ for each $k$ , then the initial values are given by $B_1^{(0)} = \\text{Hard}\\big ( [u_1\\sigma _1^{1/2}, \\ldots , u_K\\sigma _K^{1/2}] , s \\big )$ and $B_2^{(0)} = \\text{Hard}\\big ( [v_1\\sigma _1^{1/2}, \\ldots , v_K\\sigma _K^{1/2}] , s \\big )$ .",
"Starting from $B_1^{(0)}$ , $B_2^{(0)}$ , we apply one of the two gradient-based algorithms described in Algorithm REF and Algorithm REF , until convergence to a pre-specified tolerance level $\\epsilon $ is reached The code is available at http://home.uchicago.edu/~ming93/research.html..",
"The gradient $\\nabla _B f(B_1, B_2)$ can be calculated by the chain rule.",
"The specific form depends on the transmission function used.",
"In practice, the tuning parameters $\\lambda $ and $s$ can be selected by cross-validation.",
"To further accelerate the algorithm one can use the stochastic gradient descent algorithm."
],
[
"Theoretical results",
"In this section we establish main theoretical results.",
"Since the objective function is nonconvex in $B_1, B_2$ , proving theoretical result based on the $\\ell _1$ norm penalization is not straightforward.",
"For example, without the restricted strong convexity assumption, the usual analysis applied to nonconvex M-estimators [33] does not apply to our model.",
"Therefore, to make headway on our problem, we focus on the optimization problem with the regularizer $g_2$ and leverage tools that have been used in analyzing matrix factorization problems [23], [42], [40], [11], [47].",
"Compared to these works which focus on recovering one rank-$K$ matrix, our goal is to recover $K$ rank-1 matrices.",
"Let $B_1^*, B_2^*$ denote the true influence and receptivity matrices; the corresponding rank-1 matrices are given by $\\Theta _k^* = {b^1_k}^* {b^2_k}^{*\\top }$ , for each topic $k$ .",
"We start by stating assumptions under which the theory is developed.",
"The first assumption states that the parameter matrices are sparse.",
"Each column of the true influence and receptivity matrices are assumed to be sparse with $\\Vert b_k^{1*}\\Vert _0 = \\Vert b_k^{2*}\\Vert _0 = s^*$ , where $\\Vert b\\Vert _0 = \\big |j: b_{j} \\ne 0\\big |$ denotes the number of nonzero components of a vector.",
"The above assumption can be generalized in a straightforward way to allow different columns to have different levels of sparsity.",
"The next assumption puts regularity conditions on the Hessian matrix of the objective function.",
"First, we recall the Hessian matrix corresponding to the objective function $\\phi ( \\alpha )$ in (REF ) for the basic cascade model.",
"For a cascade $c$ , the Hessian matrix is given by $(\\alpha ) = D(\\alpha ) + X(t^c; \\alpha ) \\cdot X(t^c; \\alpha )^\\top ,$ where $D(\\alpha )$ is a diagonal matrix, $X(t^c; \\alpha ) = h(t^c; \\alpha )^{-1} \\nabla _\\alpha h(t^c; \\alpha ),$ with $h(t; \\alpha ) = {\\left\\lbrace \\begin{array}{ll}\\sum _{j: t_j < t_i} H(t_i | t_j; \\alpha _{ji}) & \\mbox{if } t_i < T, \\\\0 & \\mbox{otherwise},\\end{array}\\right.",
"}$ and $H(t_i | t_j; \\alpha _{ji})$ is the hazard function defined in Section REF .",
"Recalling that $\\alpha \\in ^p$ denotes the $i^{\\text{th}}$ column of $A$ , we have that $(\\alpha ) \\in ^{p \\times p}$ .",
"Both $D(\\alpha )$ and $X(t^c; \\alpha )$ are simple for the common transmission functions.",
"For example, for exponential transmission, we have that $D(\\alpha ) = 0$ is the all zero matrix and $\\big [X(t^c; \\alpha )\\big ]_j ={\\left\\lbrace \\begin{array}{ll}\\Big (\\sum _{\\ell : t_\\ell < t_i} \\alpha _{\\ell i}\\Big )^{-1} & \\mbox{if } t_j < t_i \\\\0 & \\mbox{otherwise.}\\end{array}\\right.",
"}$ See [17] for more details.",
"Let $[\\Theta _k]_i \\in ^{p}$ denote the $i^{\\text{th}}$ column of $\\Theta _k$ and let $\\Theta ^{[i]} = \\Big [[\\Theta _1]_i, [\\Theta _2]_i, \\ldots , [\\Theta _K]_i\\Big ] \\in ^{p \\times K}$ be the collection of $K$ such columns.",
"Since $A^c = \\sum _{k} m_k^c \\cdot \\Theta _k$ , we have that the $i^{\\text{th}}$ column of $A^c$ is a linear combination of $\\Theta ^{[i]}$ .",
"Therefore, the Hessian matrix of $f(\\Theta )$ with respect to $\\Theta ^{[i]}$ is a quadratic form of the Hessian matrices defined in (REF ).",
"For a specific cascade $c$ , denote the transformation matrix as $P^c =\\begin{bmatrix}m_1^c \\cdot I_p & m_2^c \\cdot I_p & \\ldots & m_K^c \\cdot I_p\\end{bmatrix}\\in ^{p \\times pK}.$ Then we have $\\alpha _i^c = P^c \\cdot \\Theta ^{[i]}$ , where $\\alpha _i^c$ denotes the $i^{\\text{th}}$ column of $A^c$ .",
"Using the chain rule, we obtain that the Hessian matrix of $f(\\Theta )$ with respect to $\\Theta ^{[i]}$ for one specific cascade $c$ is given by $H^c \\big (\\Theta ^{[i]}\\big ) = {P^c}^\\top \\cdot (\\alpha _i^c) \\cdot P^c \\in ^{pK \\times pK}.$ The Hessian matrix of the objective function $f(\\Theta )$ with respect to $\\Theta ^{[i]}$ is now given as $H(\\Theta ^{[i]}) = \\frac{1}{n} \\sum _{c} H^c(\\Theta ^{[i]}).$ We make the following assumption on the Hessian matrix.",
"There exist constants $\\mu , L > 0$ , so that $\\mu \\cdot I_{pK} \\preceq H(\\Theta ^{[i]}) \\preceq L \\cdot I_{pK}$ hold uniformly for any $i \\in \\lbrace 1, \\ldots , p\\rbrace $ .",
"The optimization problem (REF ), used to find the diffusion matrix $A$ for the basic cascade model, is separable across columns of $A$ as shown in (REF ).",
"Similarly, the objective function $f(\\Theta )$ is separable across $\\Theta ^{[i]}$ , if we ignore the rank-1 constraint.",
"As a result, the Hessian matrix of $f(\\Theta )$ with respect to $\\Theta $ , is (after an appropriate permutation of rows and columns) a block diagonal matrix in $^{p^2K \\times p^2K}$ with each block given by $H(\\Theta ^{[i]}) \\in ^{p \\times p}$ .",
"Therefore, Assumption ensures that $f(\\Theta )$ is strongly convex and smooth in $\\Theta $ .",
"The upper bound in Assumption is easy to satisfy.",
"The lower bound ensures that the problem is identifiable.",
"The Hessian matrix depends in a non-trivial way on the network structure, diffusion process, and the topic distributions.",
"Without the influence-receptivity structure, [17] establish conditions for the basic cascade model under which we can recover the network structure consistently from the observed cascades.",
"The conditions require that the behavior of connected nodes are reasonably similar among the cascades, but not deterministically related; and also that connected nodes should get infected together more often than non-connected nodes.",
"Assumption is also related to the setting in [46], who consider the squared loss, where the condition ensures that the topic distribution among the $n$ cascades is not too highly correlated, since otherwise we cannot distinguish them.",
"In our setting, Assumption is a combination of the two cases: we require that the network structure, diffusion process, and the topic distributions interact in a way to make the problem is identifiable.",
"We refer the readers to [17] and [46] for additional discussion."
],
[
"Subspace distance.",
"Since the factorization is not unique, as discussed earlier, we will measure convergence of algorithms using the subspace distance.",
"Define the set of $r$ -dimensional orthogonal matrices as $\\mathcal {O}(r) = \\lbrace O \\in ^{r}: O^\\top O = O O^\\top = I_r \\rbrace .$ Suppose $X^* \\in ^{p \\times p}$ is a rank-$r$ matrix that can be decomposed as $X^* = {U^*}{V^*}^\\top $ with $U^*, V^* \\in ^{p \\times r}$ and $\\sigma _i(U^*) = \\sigma _i(V^*)$ where $\\sigma _i(U)$ denotes the $i^{\\rm {th}}$ singular value of $U$ .",
"Let $X = U V^\\top $ be an estimator of $X^*$ .",
"The subspace distance between $X$ and $X^*$ is measured as $\\min _{O \\in \\mathcal {O}(r)} \\Big \\lbrace \\Vert U - U^* O \\Vert _F^2 + \\Vert V - V^* O \\Vert _F^2 \\Big \\rbrace .$ The above formula measures the distance between matrices up to an orthogonal rotation.",
"For our problem, the matrices $\\Theta _k$ are constrained to be rank-1, and the only possible rotation is given by $o = \\pm 1$ .",
"Moreover, since $B_1, B_2 \\ge 0$ are nonnegative, the negative rotation is eliminated.",
"As a result, the subspace distance for our problem reduces to the usual Euclidean distance.",
"Let $B = [B_1, B_2]$ and $B^* = [B_1^*, B_2^*]$ , then the “subspace distance” between $B$ and $B^*$ is defined as $d^2(B, B^*)= \\min _{o_k \\in \\lbrace \\pm 1\\rbrace }\\sum _{k=1}^K \\big \\Vert b_k^1 - {b_k^1}^*o_k\\big \\Vert _2^2 + \\big \\Vert b_k^2 - {b_k^2}^*o_k\\big \\Vert _2^2= \\big \\Vert B_1 - B_1^*\\big \\Vert _F^2 + \\big \\Vert B_2 - B_2^*\\big \\Vert _F^2.$"
],
[
"Statistical error.",
"The notion of the statistical error measures how good our estimator can be.",
"In a statistical estimation problem with noisy observations, even the best estimator can only be an approximation to the true parameter.",
"The statistical error measures how well does the best estimator estimates the true unknown parameter.",
"For a general statistical estimation problem, the statistical error is usually defined as the norm of the gradient of the objective function evaluated at the true parameter.",
"For our problem, since we have rank-1 and sparsity constraints, we define the statistical error as $e_{\\text{stat}} = \\sup _{\\Delta \\in \\Omega (s)} \\, \\big \\langle \\nabla _{\\Theta } f(\\Theta ^*), \\Delta \\big \\rangle ,$ where the set $\\Omega (s)$ is defined as $\\Omega (s) = \\big \\lbrace \\Delta : \\Delta = [\\Delta _1, \\ldots , \\Delta _K], \\Delta _k \\in ^{p \\times p}, {\\rm rank}(\\Delta _k) = 2, \\Vert \\Delta _k\\Vert _0 = 2s^2, \\Vert \\Delta \\Vert _F = 1 \\big \\rbrace .$ The statistical error depends on the network structure, diffusion process, and the topic distributions, and it scales as $n^{-1/2}$ with the sample size.",
"With these preliminaries, we are ready to state the main theoretical results for our proposed algorithm.",
"Our first results quantifying the accuracy of the initialization step.",
"Let $\\widehat{\\Theta }= \\arg \\min _{\\Theta }\\ f(\\Theta )$ be the unconstrained minimizer of $f(\\Theta )$ .",
"Suppose Assumption is satisfied, and we set $s = c \\cdot s^*$ in Algorithm REF for some constant $c > 1$ .",
"We have $\\big \\Vert \\widehat{\\Theta }- \\Theta ^* \\big \\Vert _F^2 \\le \\frac{2}{\\mu } \\big \\Vert \\nabla f(\\Theta ^*) \\big \\Vert _F.$ Furthermore, $d^2 \\big ( B^{(0)}, B^* \\big )\\le \\frac{80 \\xi ^2 K \\big \\Vert \\nabla f(\\Theta ^*) \\big \\Vert _F^2}{\\mu ^2 \\sigma ^*},$ where $\\xi $ is defined as $\\xi ^2 = 1 + \\frac{2}{\\sqrt{c-1}}$ and $\\sigma ^* = \\min _k \\Vert \\Theta _k^*\\Vert _2$ .",
"The upper bound obtained in (REF ) and (REF ) can be viewed as a statistical error for the problem without rank-1 constraints.",
"As a statistical error, the upper bound naturally scales with the sample size as $n^{-1/2}$ .",
"With a large enough sample size, the initial point will be within the radius of convergence to the true parameter such that $d^2\\big ( B^{(0)}, B^* \\big ) \\le { \\frac{1}{4} \\gamma \\sigma ^* } \\cdot \\min \\Big \\lbrace 1, \\frac{1}{4(\\mu +L )} \\Big \\rbrace ,$ where $\\gamma = \\min \\lbrace 1, \\mu L/(\\mu + L)\\rbrace $ .",
"This enables us to prove the following result.",
"Suppose Assumptions and are satisfied.",
"Furthermore, suppose the sample size $n$ is large enough so that (REF ) holds and $e_{\\rm stat}^2 \\le \\frac{1-\\beta }{3\\eta K \\xi ^2} \\cdot \\frac{\\mu L }{\\mu + L }\\cdot { \\frac{1}{4} \\gamma \\sigma ^* } \\cdot \\min \\Big \\lbrace 1, \\frac{1}{4(\\mu +L )} \\Big \\rbrace .$ Then the iterates obtained by Algorithm REF , with $s = c \\cdot s^*$ , $c>1$ , and the step size $\\eta \\le \\frac{1}{8\\Vert B^{(0)}\\Vert _2^2} \\cdot \\min \\Big \\lbrace \\frac{K}{2(\\mu +L )}, 1\\Big \\rbrace ,$ satisfy $d^2 \\Big ( B^{(T)}, B^* \\Big ) \\le \\beta ^T \\cdot d^2 \\Big ( B^{(0)}, B^* \\Big ) + \\frac{C}{1-\\beta }\\cdot e^2_{\\rm {stat}},$ where $\\beta < 1$ and $C$ is a constant.",
"Theorem REF establishes convergence of iterates produced by properly initialized Algorithm REF .",
"The first term in (REF ) corresponds to the optimization error, which decreases exponentially with the number of iterations, while the second term corresponds to the unavoidable statistical error.",
"In particular, Theorem REF shows linear convergence of the iterates up to statistical error, which depends on the network structure, diffusion process, and the topic distributions.",
"Note that the condition on $e_{\\rm stat}$ is not stringent, since in the case that it is not satisfied, then already the initial point $B^{(0)}$ is accurate enough.",
"Proofs of Theorem REF and REF are given in Appendix."
],
[
"Some variants and extensions",
"In this section we discuss several variants and application specific extensions of the proposed model.",
"Section REF considers the extension where in addition to the influence and susceptibility to topics, information propagation is further regulated by a friendship network.",
"Section REF discusses how we can use the $B_1$ and $B_2$ matrices to estimate the topic distribution of a new cascade for which we do not have the topic distribution apriori.",
"Section REF discusses how estimated matrices $B_1$ and $B_2$ can serve as embedding of the nodes.",
"Finally, in Section REF we consider estimation of $B_1, B_2$ in the setting where the topic distributions of cascades are unknown."
],
[
"Cascades regulated by friendship networks",
"We have used news and media outlets as our running example so far and have assumed that each node can influence any other node.",
"However, in social networks, a user can only see the news or tweets published by their friends or those she chooses to follow.",
"If two users do not know each other, then even if they are interested in similar topics, they still cannot “infect” each others.",
"Considering this we can modify our model in the following way: $A^c = B_1M^cB_2^{\\top } \\otimes F,$ where $\\otimes $ denotes element-wise multiplication.",
"$F \\in \\lbrace 0,1\\rbrace ^{p \\times p}$ is a known matrix indicating whether two nodes are “friends” ($f_{ji}=1$ ) or not ($f_{ji}=0$ ).",
"The modified optimization problem is a straightforward extension of (REF ) obtained by replacing the expression for $A^c$ with the new model (REF ).",
"The only thing that changes in Algorithms REF and REF is the gradient calculation.",
"As a further modification, we can allow for numeric values in $F$ .",
"Here we again have $f_{ji} = 0$ if node $j$ and $i$ are not friends; when node $j$ and $i$ are friends, the value $f_{ji} > 0$ measures how strong the friendship is.",
"A larger value means a stronger friendship, and hence node $j$ could infect node $i$ in a shorter period of time.",
"Under this setting, we assume knowledge of whether $f_{ji}$ is 0 or not, but not the actual value of $f_{ji}$ when it is non-zero.",
"This modification is useful in dealing with information diffusion over a social network where we know whether two nodes are friends or not, but we do not know how strong the friendship is.",
"We then have to estimate $B = [B_1, B_2]$ and $F$ jointly, resulting in a more difficult optimization problem.",
"A practical estimation procedure is to alternately optimize $B$ and $F$ .",
"With a fixed $F$ , the optimization problem for $B$ can be solved using Algorithm REF or REF , except for an additional element-wise multiplication with $F$ when calculating gradient.",
"With a fixed $B$ , the optimization problem in $F$ is convex and, therefore, can be solved by any gradient-based iterative algorithm."
],
[
"Estimating topic distribution $m^c$",
"Up to now we have assumed that each topic distribution $M^c = (m^c)$ is known.",
"However, once $B_1,B_2$ have been estimated, we can use them to classify a new cascade $c$ by recovering its topic-weight vector $ m^c$ .",
"For example, if an unknown disease becomes prevalent among people, then we may be able to determine the type of this new disease and identify the vulnerable population of nodes.",
"Moreover, with estimated $B_1$ and $B_2$ , we can recalculate the topic distribution of all the cascades used to fit the model.",
"By comparing the estimated distribution with the topic distribution of the cascades we can find the ones where the two topic distributions differ a lot.",
"These cascades are potentially “outliers” or have abnormal propagation behavior and should be further investigated.",
"The maximum likelihood optimization problem for estimating the topic distribution $ m^c$ is: $\\begin{aligned}& \\mathop {\\text{minimize}}_{m_k^c} \\quad -\\log {\\ell \\big ( t^c;B_1M^cB_2^{\\top } \\big )} \\\\& \\text{subject to} \\quad \\sum _k m_k^c = 1, \\\\& \\, \\, \\, \\quad \\qquad \\qquad 0 \\le m_k^c \\le 1.",
"\\\\\\end{aligned}$ This problem is easier to solve than (REF ) since $A^c = B_1M^cB_2^{\\top } $ is linear in $M^c$ and therefore the problem is convex in $M^c$ .",
"The constraint $\\sum _{k} m^c_k = 1$ and $0 \\le m_k^c \\le 1$ can be incorporated in a projected gradient descent method, where in each iteration we apply gradient descent update on $M^c$ and project it to the simplex."
],
[
"Interpreting node-topic matrices $B_1$ and {{formula:d4f28bcf-69f9-4722-a088-39832fb3dc3e}}",
"While throughout the paper we have used the diffusion of news as a running example, our model and the notion of “topic” is much more broadly applicable.",
"As discussed before it can represent features capturing susceptibility to diseases, as well as, geographic position, nationality, etc.",
"In addition to the ability to forecast future information cascades, the influence-receptivity matrices $B_1$ and $B_2$ can also find other uses.",
"For example, we can use the rows of $B_2$ to learn about the interests of users and for customer segmentation.",
"In epidemiology, we can learn about the vulnerability of population to different diseases, and allocate resources accordingly.",
"The rows of $B_1,B_2$ act as a natural embedding of users in $\\mathbb {R}^{2K}$ and thus define a similarity metric, which can be used to cluster the nodes or build recommender systems.",
"In Section illustrate how to use this embedding to cluster and visualize nodes.",
"The influence-receptivity structure is thus naturally related to graph embedding.",
"See [3] for a recent comprehensive survey of graph embedding.",
"As a closely related work in graph embedding literature, [4] propose a model which also embeds nodes into $\\mathbb {R}^{2K}$ .",
"Compared to their model, our model allows for interaction of embedding (influence and receptivity) vectors and the topic information, resulting in more interpretable topics.",
"Moreover, our model has flexibility to choose the transmission function based on different applications and comes with theoretical results on convergence rate and error analysis.",
"For example, as will be shown in Section , for information propagation on the internet (e.g.",
"media outlets citing articles, Facebook and Twitter users sharing posts), we can choose the exponential transmission function; for the citation network, the Raleigh transmission function is a more appropriate choice."
],
[
"When topic distribution is unknown",
"Throughout the paper we assume that the topic distribution $M^c$ is known for each cascade.",
"For example, the topic distribution can be calculated by Topic Modeling [1] with the text information of each cascade.",
"Alternatively it can come from the knowledge of domain experts.",
"However, in many applications domain experts or textual information may be unavailable.",
"Even if such resources are available, the topic distribution obtained from Topic Modeling may be inaccurate or intractable in practice.",
"In this case we must learn the topic distribution and the influence-receptivity structure together.",
"For this problem, our observations constitute of the timestamps for each cascade as usual, and the variables to be optimized are $B = [B_1, B_2]$ and $M^c$ for each cascade $c$ .",
"A practical algorithm is to alternately optimize on $B$ and $M^c$ – with a fixed $M^c$ , we follow Algorithm REF or REF to update $B$ ; with a fixed $B$ , we follow (REF ) to update $M^c$ on each $c$ .",
"The two procedures are repeated until convergence.",
"Theoretical analysis of this alternating minimization algorithm under the log-likelihood in (REF ) is beyond the scope of the paper.",
"For a simpler objective functions, such as the $\\ell _2$ loss, the theoretical analysis is tractable and the output of the alternating minimization algorithm (the estimated $B$ and $M$ ) can be shown to converge to the true value up to the statistical error in both $B$ and $M$ .",
"Specifically, we denote $M^*$ as the true topic distribution and $f(\\Theta , M)$ as the loss function defined in (REF ).",
"Denote the statistical error defined in (REF ) as $e_{\\text{stat}, B}$ and similarly define the statistical error on the topic distribution $M$ as $\\begin{aligned}e_{\\text{stat}, M}^2 &= \\sum _{c \\in C^n} \\sum _{k=1}^K \\Big [ \\nabla _{m^c_{k}} \\, f(\\Theta ^*, M^*) \\Big ]^2.\\end{aligned}$ Denote $B^{[t]}$ and $M^{[t]}$ as the output of the alternating minimization algorithm at iteration $t$ .",
"Under some additional mild assumptions, after one iterate of the alternating minimization algorithm we have the contraction on $B$ as $d^2 \\big ( B^{[t+1]}, B^* \\big ) \\le C_1 \\cdot e^2_{{\\rm {stat}}, B} + \\beta _1 \\cdot d^2\\big (M^{[t]}, M^*\\big ),$ for some constant $C_1$ and $\\beta _1 < 1$ .",
"Similarly, after one iterate of the alternating minimization algorithm we have the contraction on $M$ as $d^2\\big (M^{[t+1]}, M^*\\big ) \\le C_2 \\cdot e_{{\\rm {stat}}, M}^2 + \\beta _2 \\cdot d^2 \\big ( B^{[t]}, B^* \\big ),$ for some constant $C_2$ and $\\beta _2 < 1$ .",
"Combining these two inequalities, after $T$ iterations of the alternative minimization algorithm we get $\\begin{aligned}d^2\\big (B^{[T]},B^*\\big ) &+ d^2\\big (M^{[T]}, M^*\\big ) \\le C_0 ( e_{{\\rm stat}, M}^2 + e_{{\\rm stat}, B}^2 ) + \\beta _0^T \\Big [ d^2\\big (B^{[0]},B^*\\big ) + d^2\\big (M^{[0]}, M^*\\big ) \\Big ],\\end{aligned}$ for some constant $\\beta _0 = \\max \\lbrace \\beta _1, \\beta _2\\rbrace < 1$ .",
"This shows that the iterates of the alternating minimization algorithm converge linearly to the true values up to statistical error.",
"We refer the readers to Section 5 of [46] for more details."
],
[
"Synthetic Datasets",
"In this section we demonstrate the effectiveness of our model on synthetic datasets.",
"Since several existing algorithms are based on the $\\ell _1$ norm regularization, for fair comparison, we focus on our proposed Algorithm REF ."
],
[
"Estimation accuracy",
"We first evaluate our model on a synthetic dataset and compare the predictive power of the estimated model with that of Netrate and TopicCascade.",
"In simulation we set $p=200$ nodes, $K = 10$ topics.",
"We generate the true matrices $B_1$ and $B_2$ row by row.",
"For each row, we randomly pick 2-3 topics and assign a random number Unif$(0.8, 1.8) \\cdot \\zeta $ , where $\\zeta = 3$ with probability 0.3 and $\\zeta = 1$ with probability 0.7.",
"We make 30% of the values 3 times larger to capture the large variability in interests.",
"All other values are set to be 0 and we scale $B_1$ and $B_2$ to have the same column sum.",
"To generate cascades, we randomly choose a node $j$ as the source.",
"The $j^{th}$ row of $B_1$ describes the “topic distribution” of node $j$ on infecting others.",
"Therefore we sample a $K$ dimensional topic distribution $ m^c$ from Dir($b^1_{j,:}$ ), where $b^1_{j,:}$ is the $j^{th}$ row of $B_1$ and Dir($\\cdot $ ) is Dirichlet distribution.",
"According to our model (REF ), the diffusion matrix of this cascade is $A^c = B_1 M^c B_2^{\\top }$ .",
"The rest of the cascade propagation follows the description in Section REF .",
"For experiments we use exponential transmission function as in (REF ).",
"The diffusion process continues until either the overall time exceeds the observation window $T = 1$ , or there are no nodes reachable from the currently infected nodes.",
"We record the first infection time for each node.",
"Figure: Comparison of our method with Netrate and TopicCascadeWe vary the number of cascades $n \\in \\lbrace 300, 500, 1000, 2000, 5000, 10000\\rbrace $ .",
"For all three models, we fit the model on a training dataset and choose the regularization parameter $\\lambda $ on a validation dataset.",
"Each setting of $n$ is repeated 5 times and we report the average value.",
"We consider two metrics to compare our model with NetRate [15] and TopicCascade [7]: (1) We generate independent $n = 5000$ test data and calculate negative log-likelihood function on test data for the three models.",
"A good model should be able to generalize well and hence should have small negative log-likelihood.",
"From Figure REF we see that, when the sample size is small, both Netrate and TopicCascade have large negative log-likelihood on test dataset; while our model generalizes much better.",
"When sample size increases, NetRate still has large negative log-likelihood because it fails to consider the topic structure; TopicCascade behaves more and more closer to our model, which is as expected, since our model is a special case of the the topic-sensitive model.",
"However, our model requires substantially fewer parameters.",
"(2) We calculate the true diffusion matrix $A^k$ for each topic $k$ based on our model: $A^k = B_1M_{(k)}B_2^{\\top } $ where $M_{(k)}$ is diagonal matrix with 0 on all diagonal elements but 1 on location $k$ .",
"We also generate the estimated $\\widehat{A}^k$ from the three models as follows: for our model we use the estimated $\\widehat{B}_1$ and $\\widehat{B}_2$ ; for TopicCascade model the $\\widehat{A}^k$ is estimated directly as a parameter of the mode; for Netrate we use the estimated $\\widehat{A}$ as the common topic diffusion matrix for each topic $k$ .",
"Finally, we compare the estimation error of the three models: ${\\rm error} = \\frac{1}{K} \\sum _{k=1}^K \\frac{\\Vert \\widehat{A}^k - A^k\\Vert }{ \\Vert A^k\\Vert }$ .",
"From Figure REF we see that both Netrate and TopicCascade have large estimation error even if we have many samples; while our model has much smaller estimation error."
],
[
"Running time",
"We next compare the running times of the three methods.",
"For fair comparison, for each method we set the step size, initialization, penalty $\\lambda $ , and tolerance level to be the same.",
"Also one third of the samples are generated by each model.",
"For our model we follow the data generation procedure as described before; for TopicCascade, for each topic $k$ , we randomly select 5% of the components of $A^k$ to be nonzero, and these nonzero values are set as before as Unif$(0.8, 1.8) \\cdot \\zeta $ , where $\\zeta = 3$ with probability 0.3 and $\\zeta = 1$ with probability 0.7; for Netrate, we again randomly select 5% of the components of $A$ to be nonzero with values Unif$(0.8, 1.8) \\cdot \\zeta $ , and we randomly assign topic distributions.",
"We run the three methods on 12 kernels.",
"For Netrate and TopicCascade, since they are separable in each column, we run 12 columns in parallel; for our method, we calculate the gradient in parallel.",
"We use our Algorithm REF for our method and the proximal gradient algorithm for the other two methods, as suggested in [17].",
"We fix a baseline model size $n=500,p=50,K=10$ , and set a free parameter $\\xi $ .",
"For $\\xi = \\lbrace 1,2,5,8\\rbrace $ , each time we increase $n,p$ by a factor of $\\xi $ and record the running time (in seconds) of each method.",
"Table REF summarizes the results based on 5 replications in each setting.",
"We can see that Netrate is the fastest because it does not consider the topic distribution.",
"When $p$ becomes large, our algorithm is faster than TopicCascade and is of the same order as Netrate.",
"This demonstrates that although our model is not separable in each column, it can still deal with large networks.",
"Table: Running time comparison (in sec)"
],
[
"Real World Dataset",
"In this section we evaluate our model on two real world datasets.",
"We again focus on our proposed Algorithm REF ."
],
[
"Memetracker Dataset",
"The first dataset is the MemeTracker dataset [30]Data available at http://www.memetracker.org/data.html.",
"This dataset contains 172 million news articles and blog posts from 1 million online sources over a period of one year from September 1, 2008 till August 31, 2009.",
"Since the use of hyperlinks to refer to the source of information is relatively rare in mainstream media, the authors use the MemeTracker methodology [28] to extract more than 343 million short textual phrases.",
"After aggregating different textual variants of the same phrase, we consider each phrase cluster as a separate cascade $c$ .",
"Since all documents are time stamped, a cascade $c$ is simply a set of time-stamps when websites first mentioned a phrase in the phrase cluster $c$ .",
"Also since the diffusion rate of information on the internet usually reaches its peak when the information first comes out and decays rapidly, we use exponential transmission function here.",
"For our experiments we use the top 500 media sites and blogs with the largest 5000 cascades (phrase clusters).",
"For each website we record the time when they first mention a phrase in the particular phrase cluster.",
"We set the number of topic $K$ to be 10, and perform Topic Modeling to extract 10 most popular topics.",
"We choose the regularization parameter $\\lambda $ based on a hold-out validation set, and then use our Algorithm REF to estimate the two node-topic matrices.",
"The two matrices and the key words of the 10 topics are given in Tables REF ($B_1$ ) and Table REF ($B_2$ ).",
"The keywords of the 10 topics are shown at the head of each table; the first column is the url of the website.",
"The websites above the center line in each table are the most popular websites.",
"We have also hand-picked some less popular websites below the center line whose url suggest that they focus on specific topics, for example politics, business, sports, etc.",
"The top websites are mostly web portals and they broadly post and cite news in many topics.",
"Therefore to demonstrate that our model does extract some meaningful information, we select less popular websites below the center line and hope we can correctly extract the topics of interest of these specific websites.",
"From the two tables we can see that in general the influence matrix $B_1$ is much sparser than the receptivity matrix $B_2$ , which means that websites tend to post news and blogs in many topics but only a few of them will be cited by others.",
"The websites we hand pick are not as active as the top websites.",
"Therefore the values for these websites are much smaller.",
"For the top websites we only display entries which are above the threshold of 0.1, and leave smaller entries blank in the two tables; for the hand selected websites, only 0 values are left blank.",
"From the two tables we see that our model performs quite well on those specific websites.",
"For example the political websites have a large value on topic 4 (election); the business and economics websites have large value on topic 3 (economy), etc.",
"Those “as expected” large values are shown in boldface in order to highlight them.",
"We then visualize the estimated $B_1$ and $B_2$ using t-SNE algorithm [43] to see whether nodes are clustered with respect to a set of topics, and whether the clusters in $B_1$ correspond to the ones in $B_2$ .",
"In $B_1$ and $B_2$ , each row is a 10 dimensional vector corresponding to a website.",
"We use t-SNE algorithm to give each website a location in a two-dimensional map and the scatter plot of $B_1$ and $B_2$ are given in Figure REF and Figure REF .",
"From the two figures we see that these points do not form clear clusters, which means most of the websites are in general interested in many of the topics and they do not differ too much from each other.",
"We can see clearer clusters in the next example.",
"Figure: Scatter plot of B 1 B_1 and B 2 B_2 using t-SNE algorithm, for Memetracker datasetFinally we check the performance of our method on about 1500 test cascades and compare with Netrate and TopicCascade.",
"Since the number of parameters are different for the three models, besides negative log-likelihood, we also use AIC and BIC as our metrics.",
"Table REF summarizes the results.",
"The first column shows the names of the three methods and the following columns are the averaged negative log-likelihood on train set, averaged negative log-likelihood on test set, number of total parameters, number of nonzero parameters, AIC and BIC on test set calculated using the negative log-likelihood on test set (third column) and the number of nonzero parameters (fifth column).",
"From the table we see that our model has the largest negative log-likelihood on train set, and one reason for that is that our model have fewest parameters.",
"However, we can see that both Netrate and TopicCascade are overfitting, while our method can generalize to test set with little overfitting.",
"Our method uses much fewer parameters but has comparable negative log-likelihood on test, and also our method has the smallest AIC and BIC value.",
"Table: Comparison of the 3 methods on test cascades for Memetracker datasetTable: The influence matrix B 1 B_1 for Memetracker datasetTable: The receptivity matrix B 2 B_2 for Memetracker dataset"
],
[
"Arxiv Citation Dataset",
"The second dataset is the ArXiv high-energy physics theory citation network dataset [29], [12]Data available at http://snap.stanford.edu/data/cit-HepTh.html.",
"This dataset includes all papers published in ArXiv high-energy physics theory section from 1992 to 2003.",
"We treat each author as a node and each publication as a cascade.",
"For our experiments we use the top 500 authors with the largest 5000 cascades.",
"For each author we record the time when they first cite a particular paper.",
"Since it usually takes some time to publish papers we use rayleigh transmission function here.",
"We set the number of topic $K$ to be 6, and perform Topic Modeling on the abstracts of each paper to extract 6 most popular topics.",
"We then use our Algorithm REF to estimate the two node-topic matrices.",
"The two matrices and the key words of the 6 topics are given in Tables REF ($B_1$ ) and Table REF ($B_2$ ).",
"Again the keywords of the 6 topics are shown at the head of each table and the first column is the name of the author.",
"We compare the learnt topics to the research interests listed by the authors in their website and we find that our model is able to discover the research topics of the authors accurately.",
"For example Arkady Tseytlin reports string theory, quantum field theory and gauge theory; Shin'ichi Nojiri reports field theory; Burt A. Ovrut reports gauge theory; Amihay Hanany reports string theory; Ashoke Sen reports string theory and black holes as their research areas in their webpages.",
"Moreover, Ashok Das has papers in supergravity, supersymmetry, string theory, and algebras; Ian Kogan has papers in string theory and boundary states; Gregory Moore has papers in algebras and non-commutativity.",
"These are all successfully captured by our method.",
"We then again visualize the estimated $B_1$ and $B_2$ using t-SNE algorithm for which the scatter plots are shown in Figures REF .",
"Here we see distinct patterns in the two figures.",
"Figure REF shows 6 “petals\" corresponding to the authors interested in 6 topics, while the points in the center corresponds to the authors who have small influence on all the 6 topics.",
"We therefore apply $K$ -Means algorithm to get 7 clusters for the influence matrix $B_1$ as shown in Figure REF (each color corresponds to one cluster), and then plot receptivity matrix $B_2$ in Figure REF using these colors.",
"We see that although Figure REF also shows several clusters, the patterns are clearly different from Figure REF .",
"This demonstrates the necessity of having different influence matrix $B_1$ and receptivity matrix $B_2$ in our model.",
"Figure: Scatter plot of B 1 B_1 and B 2 B_2 using t-SNE algorithm, for Citation datasetFinally we check the performance of our method on about 1200 test cascades and compare with Netrate and TopicCascade.",
"Table REF summarizes the results.",
"Similar as before, although Netrate and TopicCascade have smaller negative log-likelihood on train data, our method has the best performance on test data with significantly less parameters and little overfitting.",
"So again we see that our model works quite well on this citation dataset.",
"Table: Comparison of the 3 methods on test cascades for citation datasetTable: The influence matrix B 1 B_1 for citation datasetTable: The receptivity matrix B 2 B_2 for citation dataset"
],
[
"Conclusion",
"The majority of work on information diffusion has focused on recovering the diffusion matrix while ignoring the structure among nodes.",
"In this paper, we propose an influence-receptivity model that takes the structure among nodes into consideration.",
"We develop two efficient algorithms and prove that the iterates of the algorithm converge linearly to the true value up to a statistical error.",
"Experimentally, we demonstrate that our model performs well in both synthetic and real data, and produces a more interpretable model.",
"There are several interesting research threads we plan to pursue.",
"In terms of modeling, an interesting future direction would be to allow each cascade to have a different propagation rate.",
"In our current model, two cascades with the same topic distribution will have the same diffusion behavior.",
"In real world, we expect some information to be intrinsically more interesting and hence spread much faster.",
"Another extension would be allowing dynamic influence-receptivity matrices over time.",
"Finally, all existing work on network structure recovery from cascades assumes that the first node observed to be infected is the source of the diffusion.",
"In many scenarios, the source may be latent and directly infect many nodes.",
"Extending our model to incorporate this feature is work in progress."
],
[
"Acknowledgments",
"This work is partially supported by an IBM Corporation Faculty Research Fund at the University of Chicago Booth School of Business.",
"This work was completed in part with resources provided by the University of Chicago Research Computing Center."
],
[
"Proof of Theorem ",
"Since $f(\\Theta )$ is strongly convex in $\\Theta $ , we have $f(\\widehat{\\Theta }) - f(\\Theta ^*) - \\big \\langle \\nabla f(\\Theta ^*), \\widehat{\\Theta }- \\Theta ^* \\big \\rangle \\ge \\frac{\\mu }{2} \\big \\Vert \\widehat{\\Theta }- \\Theta ^* \\big \\Vert _F^2.$ On the other hand, since $\\widehat{\\Theta }$ is the global minimum, we have $f(\\widehat{\\Theta }) \\le f(\\Theta ^*).$ Combining the above two inequalities, we obtain $\\frac{\\mu }{2}\\big \\Vert \\widehat{\\Theta }- \\Theta ^* \\big \\Vert _F^2 \\le - \\big \\langle \\nabla f(\\Theta ^*), \\widehat{\\Theta }- \\Theta ^* \\big \\rangle \\le \\big \\Vert \\nabla f(\\Theta ^*) \\big \\Vert _F \\cdot \\big \\Vert \\widehat{\\Theta }- \\Theta ^* \\big \\Vert _F$ and $\\big \\Vert \\widehat{\\Theta }- \\Theta ^* \\big \\Vert _F \\le \\frac{2}{\\mu } \\big \\Vert \\nabla f(\\Theta ^*) \\big \\Vert _F.$ This shows that for any $k$ , we have $\\big \\Vert \\widehat{\\Theta }_k - \\Theta _k^* \\big \\Vert _F \\le \\frac{2}{\\mu } \\big \\Vert \\nabla f(\\Theta ^*) \\big \\Vert _F.$ According to the construction of the initialization point, the rank-1 SVD of $\\Theta _k$ is given by $\\sigma _k u_k v_k^\\top $ .",
"Since it is the best rank-1 approximation of $\\widehat{\\Theta }_k$ , we have that $\\big \\Vert \\sigma _k u_k v_k^\\top - \\widehat{\\Theta }_k \\big \\Vert _F \\le \\big \\Vert \\widehat{\\Theta }_k - \\Theta _k^* \\big \\Vert _F.$ By the triangular inequality $\\big \\Vert \\sigma _k u_k v_k^\\top - \\Theta _k^* \\big \\Vert _F \\le \\big \\Vert \\sigma _k u_k v_k^\\top - \\widehat{\\Theta }_k \\big \\Vert _F + \\big \\Vert \\widehat{\\Theta }_k - \\Theta _k^* \\big \\Vert _F \\le 2 \\big \\Vert \\widehat{\\Theta }_k - \\Theta _k^* \\big \\Vert _F \\le \\frac{4}{\\mu } \\big \\Vert \\nabla f(\\Theta ^*) \\big \\Vert _F.$ Then by Lemma 5.14 in [42] we have $\\big \\Vert {b_k^1}^{(0)} - {b_k^1}^* \\big \\Vert _2^2 + \\big \\Vert {b_k^2}^{(0)} - {b_k^2}^* \\big \\Vert _2^2 \\le \\frac{2}{\\sqrt{2} - 1} \\cdot \\frac{\\big \\Vert \\sigma _k u_k v_k^\\top - \\Theta _k^* \\big \\Vert _F^2}{\\Vert \\Theta _k^*\\Vert _2}.$ Let $\\sigma ^* = \\min _k \\Vert \\Theta _k^*\\Vert _2$ .",
"Using Lemma 3.3 in [31], we have the following upper bound on the initialization $B^{(0)} = \\big [ B_1^{(0)}, B_2^{(0)} \\big ]$ , $d^2 \\big ( B^{(0)}, B^* \\big ) \\le \\xi ^2 \\cdot \\frac{2K}{\\sqrt{2} - 1} \\cdot \\frac{16 \\big \\Vert \\nabla f(\\Theta ^*) \\big \\Vert _F^2}{\\mu ^2 \\sigma ^*}\\le \\frac{80 \\xi ^2 K \\big \\Vert \\nabla f(\\Theta ^*) \\big \\Vert _F^2}{\\mu ^2 \\sigma ^*},$ where $\\xi $ is defined as $\\xi ^2 = 1 + \\frac{2}{\\sqrt{c-1}}$ with $c$ set as $s = cs^*$ as in Theorem REF ."
],
[
"Proof of Theorem ",
"The key part of the proof is to quantify the estimation error after one iteration.",
"We then iteratively apply this error bound.",
"For notation simplicity, we omit the superscript indicating the iteration number $t$ when quantifying the iteration error.",
"We denote the current iterate as $B = [B_1, B_2]$ and the next iterate as $B^+ = [B_1^+, B_2^+]$ .",
"Recall that the true values are given by $B^* = [B_1^*, B_2^*]$ with columns given by ${b_k^1}^*, {b_k^2}^*$ .",
"The $k^{\\text{th}}$ columns of $B_1, B_2, B_1^+, B_2^+$ are denoted as $b_k^1, b_k^2, b_k^{1+}, b_k^{2+}$ .",
"We use $b_k$ and $b_k^+$ to denote $b_k = [b_k^1, b_k^2]$ and $b_k^+ = [b_k^{1+}, b_k^{2+}]$ .",
"According to the update rule given in Algorithm REF , we have $B_1^+ & = \\text{Hard} \\Big ( B_1 - \\eta \\cdot \\nabla _{B_1} f\\big (B_1, B_2\\big ) - \\eta \\cdot \\nabla _{B_1} g\\big (B_1, B_2\\big ), s \\Big ), \\\\B_2^+ & = \\text{Hard} \\Big ( B_2 - \\eta \\cdot \\nabla _{B_2} f\\big (B_1, B_2\\big ) - \\eta \\cdot \\nabla _{B_2} g\\big (B_1, B_2\\big ), s \\Big ),$ with the regularization term $ g(B_1,B_2) = \\frac{1}{4} \\cdot \\sum _{k=1}^K \\Big ( \\big \\Vert b^1_k\\big \\Vert _2^2 - \\big \\Vert b^2_k \\big \\Vert _2^2 \\Big )^2 $ given in (REF ).",
"Note that, since the true values $B_1^*, B_2^*$ are nonnegative and the negative values only make the estimation accuracy worse, we can safely ignore the operation $[B]_+$ in the theoretical analysis.",
"Moreover, when quantifying the estimation error after one iteration, we assume that the current estimate $B$ is not too far away from the true value $B^*$ in that $d^2(B,B^*) \\le { \\frac{1}{4} \\gamma \\sigma ^* } \\cdot \\min \\Big \\lbrace 1, \\frac{1}{4(\\mu +L )} \\Big \\rbrace ,$ where $\\gamma = \\min \\lbrace 1, \\mu L/(\\mu + L)\\rbrace $ and $\\sigma ^* = \\min _k \\Vert \\Theta _k^*\\Vert _2$ .",
"This upper bound (REF ) is satisfied for $B^{(0)}$ when the sample size is large enough, as assumed in (REF ).",
"In the proof, we will show that (REF ) is also satisfied in each iteration of Algorithm REF .",
"Therefore we can recursively apply the estimation error bound for one iteration.",
"Let $S_1 = \\text{supp}(B_1) \\cup \\text{supp}(B_1^+) \\cup \\text{supp}(B_1^*)~\\text{ and }~S_2 = \\text{supp}(B_2) \\cup \\text{supp}(B_2^+) \\cup \\text{supp}(B_2^*)$ denote the nonzero positions of the current iterate, next iterate, and the true value.",
"Similarly, let $S_{1k} = \\text{supp}(b^1_k) \\cup \\text{supp}(b_k^{1+}) \\cup \\text{supp}(b_k^{1*})\\text{ and }S_{2k} = \\text{supp}(b^2_k) \\cup \\text{supp}(b_k^{2+}) \\cup \\text{supp}(b_k^{2*})$ capture the support for the $k^{\\text{th}}$ column.",
"With this notation, we have $\\begin{aligned}d^2(B^+, B^*) &= \\big \\Vert B_1^+ - B_1^*\\big \\Vert _F^2 + \\big \\Vert B_2^+ - B_2^*\\big \\Vert _F^2 \\\\&\\le \\xi ^2 \\Big ( \\big \\Vert B_1 - B_1^* - \\eta \\cdot \\big [\\nabla _{B_1} f\\big (B_1, B_2\\big ) + \\nabla _{B_1} g\\big (B_1, B_2\\big ) \\big ]_{S_1} \\big \\Vert _F^2 \\\\& \\qquad \\qquad \\qquad + \\big \\Vert B_2 - B_2^* - \\eta \\cdot \\big [\\nabla _{B_2} f\\big (B_1, B_2\\big ) + \\nabla _{B_2} g\\big (B_1, B_2\\big ) \\big ]_{S_2} \\big \\Vert _F^2 \\Big ) \\\\& \\le \\xi ^2 \\Big ( d^2(B, B^*) - 2\\eta \\cdot \\big \\langle \\nabla _{B} f\\big (B\\big ) + \\nabla _{B} g\\big (B\\big ), B - B^* \\big \\rangle _{S_1 \\cup S_2} \\\\& \\qquad \\qquad \\qquad + \\eta ^2 \\cdot \\big \\Vert \\big [\\nabla _{B} f\\big (B\\big ) + \\nabla _{B} g\\big (B\\big ) \\big ]_{S_1 \\cup S_2} \\big \\Vert _F^2 \\Big ) \\\\& \\le \\xi ^2 \\Big ( d^2(B, B^*) - 2\\eta \\cdot \\big \\langle \\nabla _{B} f\\big (B\\big ) + \\nabla _{B} g\\big (B\\big ), B - B^* \\big \\rangle _{S_1 \\cup S_2} \\\\& \\qquad \\qquad \\qquad + 2\\eta ^2 \\cdot \\big \\Vert \\big [\\nabla _{B} f\\big (B\\big ) \\big ]_{S_1 \\cup S_2} \\big \\Vert _F^2 + 2\\eta ^2 \\cdot \\big \\Vert \\big [\\nabla _{B} g\\big (B\\big ) \\big ]_{S_1 \\cup S_2} \\big \\Vert _F^2 \\Big ),\\end{aligned}$ where the first inequality follows from Lemma 3.3 of [31] and $\\xi $ is defined as $\\xi ^2 = 1 + \\frac{2}{\\sqrt{c-1}}$ with $c$ set as $s = cs^*$ .",
"Different from the existing work on matrix factorization that focuses on recovery of a single rank-$K$ matrix, in our model, we have $K$ rank-1 matrices.",
"Therefore we have to deal with each column of $B_1$ and $B_2$ separately.",
"With some abuse of notation, we denote $f_k(b_k) = f_k(b_k^1, b_k^2) = f_k(\\Theta _k) = f(\\Theta _1, \\ldots , \\Theta _k, \\ldots , \\Theta _K)$ as a function of the $k^{\\text{th}}$ columns of $B_1, B_2$ , with all the other columns fixed.",
"The gradient of $f_k(\\Theta _k)$ with respect to $b_k^1$ is then given by $\\nabla f_k(\\Theta _k)\\cdot b_k^2$ .",
"Similarly, we denote $g_k(b_k) = g_k(b_k^1, b_k^2) = \\frac{1}{4} \\Big ( \\big \\Vert b^1_k\\big \\Vert _2^2 - \\big \\Vert b^2_k \\big \\Vert _2^2 \\Big )^2,$ such that $g(B_1,B_2) = \\sum _{k=1}^K g_k(b_k)$ .",
"We first deal the terms involving regularization $g(\\cdot )$ in (REF ).",
"Denote $\\Delta b_k = \\big \\Vert b^1_k\\big \\Vert _2^2 - \\big \\Vert b^2_k \\big \\Vert _2^2$ , so that $g_k(b_k) = \\frac{1}{4} (\\Delta b_k)^2$ .",
"Then $\\Big \\Vert \\big [ \\nabla _{B} g\\big (B\\big ) \\big ]_{S_1 \\cup S_2} \\Big \\Vert _F^2 \\le \\sum _{k=1}^K \\Vert \\nabla g_k(b_k) \\Vert _F^2 \\le \\sum _{k=1}^K (\\Delta b_k) ^2 \\cdot \\Vert b_k \\Vert _2^2\\le \\Vert B\\Vert _2^2 \\cdot \\sum _{k=1}^K (\\Delta b_k) ^2.$ Equation (36) in the proof of Lemma B.1 in [40] gives us $\\big \\langle \\nabla _{B} g\\big (B\\big ), B - B^* \\big \\rangle _{S_1 \\cup S_2}\\ge \\sum _{k=1}^K \\Big [ \\frac{5}{8} (\\Delta b_k) ^2 - \\frac{1}{2} \\Delta b_k \\cdot \\Vert b_k - b_k^* \\Vert _2^2 \\Big ].$ We then bound the two terms in (REF ).",
"For the first term, we have $\\begin{aligned}(\\Delta b_k)^2 &\\ge \\big \\Vert b_k^1 {b_k^1}^\\top - {b_k^1}^* {{b_k^1}^*}^\\top \\big \\Vert _F^2 + \\big \\Vert b_k^2 {b_k^2}^\\top - {b_k^2}^* {{b_k^2}^*}^\\top \\big \\Vert _F^2- 2 \\big \\Vert b_k^1 {b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\big \\Vert _F^2 \\\\&\\ge \\gamma \\cdot \\Big ( \\big \\Vert b_k^1 {b_k^1}^\\top - {b_k^1}^* {{b_k^1}^*}^\\top \\big \\Vert _F^2 + \\big \\Vert b_k^2 {b_k^2}^\\top - {b_k^2}^* {{b_k^2}^*}^\\top \\big \\Vert _F^2 + 2 \\big \\Vert b_k^1 {b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\big \\Vert _F^2 \\Big ) \\\\& \\qquad \\qquad - \\frac{4\\mu L}{\\mu + L} \\big \\Vert b_k^1 {b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\big \\Vert _F^2 \\\\& \\ge \\frac{3}{2} \\gamma \\big \\Vert \\Theta _k^* \\big \\Vert _2 \\cdot \\Big ( \\big \\Vert b_k^1 - {b_k^1}^* \\big \\Vert _2^2 + \\big \\Vert b_k^2 - {b_k^2}^* \\big \\Vert _2^2 \\Big )- \\frac{4\\mu L}{\\mu + L} \\big \\Vert b_k^1 {b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\big \\Vert _F^2,\\end{aligned}$ where the last inequality follows from Lemma 5.1 in [42], and $\\gamma = \\min \\lbrace 1, \\mu L/(\\mu + L)\\rbrace $ as before.",
"For the second term in (REF ), recall that the current iterate satisfies the condition (REF ), so that $\\begin{aligned}\\frac{1}{2} \\Delta b_k \\cdot \\Vert b_k - b_k^* \\Vert _2^2& \\le \\frac{1}{2} \\Delta b_k \\cdot \\Vert b_k - b_k^* \\Vert _2 \\cdot \\sqrt{ \\frac{1}{4} \\gamma \\sigma ^* }\\\\&\\le \\frac{1}{16} \\gamma \\sigma ^* \\cdot \\Vert b_k - b_k^* \\Vert _2^2 + \\frac{1}{4} (\\Delta b_k)^2.\\end{aligned}$ Plugging (REF ) and (REF ) into (REF ) and summing over $k$ , we obtain $\\begin{aligned}\\big \\langle \\nabla _{B} g\\big (B\\big ), B - B^* \\big \\rangle _{S_1 \\cup S_2}&\\ge \\frac{3}{8} \\sum _{k=1}^K (\\Delta b_k) ^2 - \\frac{1}{16} \\sum _{k=1}^K \\gamma \\sigma ^* \\cdot \\Vert b_k - b_k^* \\Vert _2^2 \\\\&= \\frac{1}{4} \\sum _{k=1}^K (\\Delta b_k) ^2 + \\frac{1}{8} \\sum _{k=1}^K (\\Delta b_k) ^2 - \\frac{1}{16} \\gamma \\sigma ^* \\cdot d^2(B, B^*) \\\\&\\ge \\frac{1}{8} \\gamma \\sigma ^* d^2(B, B^*) - \\frac{\\mu L}{2(\\mu + L)} \\big \\Vert b_k^1 {b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\big \\Vert _F^2 + \\frac{1}{4} \\sum _{k=1}^K (\\Delta b_k) ^2.\\end{aligned}$ Together with (REF ), we obtain $\\begin{aligned}&-2\\eta \\big \\langle \\nabla _{B} g\\big (B\\big ), B - B^* \\big \\rangle _{S_1 \\cup S_2} + 2\\eta ^2 \\Big \\Vert \\big [ \\nabla _{B} g\\big (B\\big ) \\big ]_{S_1 \\cup S_2} \\Big \\Vert _F^2 \\\\&\\qquad \\le -\\frac{1}{4} \\eta \\gamma \\sigma ^* d^2(B, B^*) + \\eta \\frac{\\mu L}{\\mu + L} \\big \\Vert b_k^1 {b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\big \\Vert _F^2 + \\Big ( 2\\eta ^2 \\Vert B\\Vert _2^2 - \\frac{1}{2} \\eta \\Big ) \\sum _{k=1}^K (\\Delta b_k) ^2.\\end{aligned}$ Next, we upper bound the terms in (REF ) involving the objective function $f(\\cdot )$ .",
"For the inner product term, for each $k$ , we have $\\begin{aligned}&\\Big \\langle [ \\nabla f_k(b_k^1 {b_k^2}^\\top ) \\cdot b_k^2 ]_{S_1}, b_k^1 - {b_k^1}^* \\Big \\rangle + \\Big \\langle [ \\nabla f_k(b_k^1 {b_k^2}^\\top ) \\cdot b_k^1 ]_{S_2}, b_k^2 - {b_k^2}^* \\Big \\rangle \\\\&= \\Big \\langle \\nabla f_k(b_k^1 {b_k^2}^\\top ), (b_k^1 - {b_k^1}^*){b_k^2}^\\top + b_k^1 ({b_k^2 - {b_k^2}^*})^\\top \\Big \\rangle _{S_{1k}, S_{2k}} \\\\&= \\Big \\langle \\nabla f_k(b_k^1 {b_k^2}^\\top ), (b_k^1 - {b_k^1}^*)({b_k^2 - {b_k^2}^*})^\\top + b_k^1{b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\Big \\rangle _{S_{1k}, S_{2k}} \\\\&= \\Big \\langle \\nabla f_k(b_k^1 {b_k^2}^\\top ), (b_k^1 - {b_k^1}^*)({b_k^2 - {b_k^2}^*})^\\top \\Big \\rangle _{S_{1k}, S_{2k}}+ \\Big \\langle \\nabla f_k(b_k^1 {b_k^2}^\\top ), b_k^1{b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\Big \\rangle _{S_{1k}, S_{2k}} \\\\&= \\underbrace{\\Big \\langle \\nabla f_k(b_k^1 {b_k^2}^\\top ), (b_k^1 - {b_k^1}^*)({b_k^2 - {b_k^2}^*})^\\top \\Big \\rangle _{S_{1k}, S_{2k}} }_{W_{1k}}+ \\underbrace{ \\Big \\langle \\nabla f_k( {b_k^1}^* {{b_k^2}^*}^\\top ), b_k^1{b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\Big \\rangle _{S_{1k}, S_{2k}} }_{W_{2k}} \\\\& \\qquad \\qquad \\qquad + \\underbrace{ \\Big \\langle \\nabla f_k(b_k^1 {b_k^2}^\\top ) - \\nabla f_k( {b_k^1}^* {{b_k^2}^*}^\\top ), b_k^1{b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\Big \\rangle _{S_{1k}, S_{2k}} }_{W_{3k}}.\\end{aligned}$ For the term $W_{3k}$ , Theorem 2.1.11 of [37] gives $\\begin{aligned}W_{3k} & \\ge \\frac{\\mu L}{\\mu + L }\\cdot \\Big \\Vert {b_k^1 {b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top }\\Big \\Vert _F^2+ \\frac{1}{\\mu +L }\\cdot \\Big \\Vert {\\nabla f( b_k^1 {b_k^2}^\\top ) - \\nabla f( {b_k^1}^* {{b_k^2}^*}^\\top )}_{S_{1k},S_{2k}}\\Big \\Vert _F^2.\\end{aligned}$ For the term $W_{2k}$ , according to the definition of the statistical error in (REF ), we have $\\begin{aligned}\\sum _{k=1}^K W_{2k} &\\ge -e_{\\rm stat}\\cdot \\sum _{k=1}^K \\Big \\Vert b_k^1{b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top \\Big \\Vert _F \\\\&\\ge - \\frac{K}{2}\\frac{\\mu + L }{\\mu L}e_{\\rm stat}^2 -\\frac{1}{2}\\frac{\\mu L}{\\mu + L } \\sum _{k=1}^K \\Big \\Vert { b_k^1{b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top }\\Big \\Vert _F^2.\\end{aligned}$ For the term $W_{1k}$ , $\\begin{aligned}\\sum _{k=1}^K W_{1k} &= \\sum _{k=1}^K\\Big \\langle \\nabla f_k( {b_k^1}^* {{b_k^2}^*}^\\top ), (b_k^1 - {b_k^1}^*)({b_k^2 - {b_k^2}^*})^\\top \\Big \\rangle _{S_{1k}, S_{2k}}\\\\&\\qquad \\qquad \\qquad + \\Big \\langle \\nabla f_k(b_k^1 {b_k^2}^\\top ) - \\nabla f_k( {b_k^1}^* {{b_k^2}^*}^\\top ), (b_k^1 - {b_k^1}^*)({b_k^2 - {b_k^2}^*})^\\top \\Big \\rangle _{S_{1k}, S_{2k}} \\\\& \\ge - {e_{\\rm stat} + \\sum _{k=1}^K \\Big \\Vert {\\nabla f_k(b_k^1 {b_k^2}^\\top ) - \\nabla f_k( {b_k^1}^* {{b_k^2}^*}^\\top )}_{S_{1k},S_{2k}}\\Big \\Vert _F}\\cdot d^2(B, B^*) \\\\& \\ge - {e_{\\rm stat} + \\sum _{k=1}^K\\Big \\Vert {\\nabla f_k(b_k^1 {b_k^2}^\\top ) - \\nabla f_k( {b_k^1}^* {{b_k^2}^*}^\\top )}_{S_{1k},S_{2k}}\\Big \\Vert _F}\\sqrt{\\frac{\\gamma \\sigma ^*}{16(\\mu +L )}} d(B, B^*) \\\\& \\ge -\\frac{K}{2(\\mu +L )}\\cdot {e_{\\rm stat}^2+ \\sum _{k=1}^K \\Big \\Vert {\\nabla f_k(b_k^1 {b_k^2}^\\top ) - \\nabla f_k( {b_k^1}^* {{b_k^2}^*}^\\top )}_{S_{1k},S_{2k}}\\Big \\Vert _F^2} \\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - \\frac{1}{16} \\gamma \\sigma ^*\\cdot d^2(B, B^*),\\end{aligned}$ where we use the fact that $d(B,B^*)$ satisfies (REF ), $\\big \\Vert (b_k^1 - {b_k^1}^*)({b_k^2 - {b_k^2}^*})^\\top \\big \\Vert _F \\le \\big \\Vert b_k^1 - {b_k^1}^*\\big \\Vert _F \\big \\Vert {b_k^2 - {b_k^2}^*}\\big \\Vert _F \\le \\big \\Vert b_k^1 - {b_k^1}^*\\big \\Vert _F^2 + \\big \\Vert {b_k^2 - {b_k^2}^*} \\big \\Vert _F^2,$ and that their summation is $d^2(B, B^*)$ .",
"For the term in (REF ) involving square of $f(\\cdot )$ , we have $\\Big \\Vert \\big [\\nabla _{B} f\\big (B\\big ) \\big ]_{S_1 \\cup S_2} \\Big \\Vert _F^2 \\le 4\\cdot \\Big ( \\sum _{k=1}^K\\Big \\Vert {\\nabla f(b_k^1 {b_k^2}^\\top )-\\nabla f({b_k^1}^* {{b_k^2}^*}^\\top )}_{S_1, S_2}\\Big \\Vert _F^2 +e_{\\text{stat}}^2\\Big ) \\cdot \\Vert B\\Vert _{2}^2.$ Combining (REF ), (REF ), (REF ), and (REF ), we obtain $\\begin{aligned}&-2\\eta \\big \\langle \\nabla _{B} f\\big (B\\big ), B - B^* \\big \\rangle _{S_1 \\cup S_2} + \\eta ^2 \\Big \\Vert \\big [\\nabla _{B} f\\big (B\\big ) \\big ]_{S_1 \\cup S_2} \\Big \\Vert _F^2 \\\\&\\le e_{\\text{stat}}^2 \\cdot \\Big ( 8\\Vert B\\Vert _2^2\\eta ^2 + \\frac{K(\\mu +L)}{\\mu L} \\eta + \\frac{K}{\\mu +L} \\eta \\Big ) \\\\& \\qquad \\qquad - \\frac{\\mu L }{\\mu + L } \\eta \\sum _{k=1}^K \\Big \\Vert { b_k^1{b_k^2}^\\top - {b_k^1}^* {{b_k^2}^*}^\\top }\\Big \\Vert _F^2 + \\frac{1}{8} \\gamma \\sigma ^*\\eta \\cdot d^2(B, B^*) \\\\&\\qquad \\qquad + \\Big (8\\eta ^2 \\Vert B\\Vert _2^2 - \\frac{K\\eta }{\\mu + L} \\Big ) \\sum _{k=1}^K \\Big \\Vert {\\nabla f_k(b_k^1 {b_k^2}^\\top ) - \\nabla f_k( {b_k^1}^* {{b_k^2}^*}^\\top )}_{S_{1k},S_{2k}}\\Big \\Vert _F^2.\\end{aligned}$ Plugging (REF ) and (REF ) into (REF ), we obtain $\\begin{aligned}d^2(B^+, B^*) &=\\xi ^2 {1 - \\frac{1}{4} \\gamma \\sigma ^*\\eta } \\cdot d^2(B, B^*) + \\xi ^2 \\Big ( 2\\eta ^2 \\Vert B\\Vert _2^2 - \\frac{1}{2} \\eta \\Big ) \\sum _{k=1}^K (\\Delta b_k) ^2 \\\\& \\qquad +\\xi ^2 {8\\eta ^2 \\cdot \\Vert B\\Vert _2^2 - \\frac{K\\eta }{\\mu + L}}\\cdot \\sum _{k=1}^K \\Big \\Vert {\\nabla f_k(b_k^1 {b_k^2}^\\top ) - \\nabla f_k( {b_k^1}^* {{b_k^2}^*}^\\top )}_{S_{1k},S_{2k}}\\Big \\Vert _F^2 \\\\&\\qquad +\\xi ^2 {\\frac{K(\\mu +L)}{\\mu L} \\eta +\\frac{K\\eta }{\\mu + L} + 8 \\eta ^2 \\cdot \\Vert B\\Vert _2^2}\\cdot e_{\\rm stat}^2.\\end{aligned}$ When the step size satisfies $\\eta \\le \\frac{1}{4\\Vert B\\Vert _2^2} \\cdot \\min \\Big \\lbrace \\frac{K}{2(\\mu +L )}, 1\\Big \\rbrace ,$ the second and third terms in (REF ) are non-positive.",
"Therefore, we can upper bound them with 0 to obtain $d^2 \\Big ( B^{(t+1)} , B^* \\Big ) \\le \\beta \\cdot d^2 \\Big ( B^{(t)}, B^* \\Big ) + 3\\eta K \\xi ^2 \\cdot \\frac{\\mu + L }{\\mu L } \\cdot e_{{\\rm stat}}^2 ,$ with the contraction value $\\beta = \\xi ^2 \\Big ( 1 - \\frac{1}{4} \\gamma \\sigma ^*\\eta \\Big ) < 1.$ From (REF ) we see that $\\beta $ is a multiplication of two terms.",
"The first term $\\xi ^2 = 1 + \\frac{2}{\\sqrt{c-1}}$ is slightly larger than 1, while the second term is smaller than 1.",
"In order to guarantee that $\\beta < 1$ , we should choose a conservative hard thresholding parameter (recall that $s = c \\cdot s^*$ ), so that $\\xi ^2$ is close to 1.",
"In practice, we observe that $\\beta < 1$ for a large range of hard thresholding parameters.",
"Notice that without the hard thresholding step, we are guaranteed to have $\\beta < 1$ .",
"In order to iteratively apply the error bound (REF ), we need to show that the condition (REF ) is satisfied in each iteration.",
"A sufficient condition is to require $e_{\\rm stat}^2 \\le \\frac{1-\\beta }{3\\eta K \\xi ^2} \\cdot \\frac{\\mu L }{\\mu + L }\\cdot { \\frac{1}{4} \\gamma \\sigma ^* } \\cdot \\min \\Big \\lbrace 1, \\frac{1}{4(\\mu +L )} \\Big \\rbrace .$ It is straightforward to verify that (REF ) and (REF ) imply that the next iterate also satisfies the condition (REF ).",
"To justify the condition (REF ), consider the case where the condition (REF ) is violated.",
"Together with (REF ), this shows that $d^2(B,B^*) \\le C \\cdot e_{\\rm stat}^2$ , which means that the current iterate is already optimal.",
"Therefore, we can assume (REF ) and then (REF ) is satisfied for all the iterations.",
"With the error bound (REF ) we can complete the proof.",
"For a large enough sample size, the initial point $B^{(0)}$ satisfies (REF ).",
"The proof above shows that (REF ) is satisfied with $t = 0$ .",
"The condition (REF ) ensures that the next iterate $B^{(1)}$ also satisfies (REF ).",
"Iterating the argument, we obtain $d^2 \\Big ( B^{(T)}, B^* \\Big ) \\le \\beta ^T \\cdot d^2 \\Big ( B^{(0)}, B^* \\Big ) + \\frac{3\\eta K\\xi ^2}{1-\\beta }\\cdot \\frac{\\mu + L }{\\mu L}\\cdot e_{{\\rm stat}}^2,$ which shows that the iterates of Algorithm REF converge linearly to the true value up to a statistical error.",
"Finally, it remains to provide an upper bound on the step size (REF ) that is independent of the norm of the value in each iterate $\\Vert B\\Vert _2$ , as given in (REF ).",
"This can be established as in the proof of Lemma 4 in [45].",
"The proof is now complete."
]
] | 1709.01919 |
[
[
"Electromagnetic trace anomaly in a generalized linear sigma model"
],
[
"Abstract We build the electromagnetic trace anomaly effective term for a generalized linear sigma model with two chiral nonets, one with a quark-antiquark structure, the other one with a four quark content.",
"In the leading order of this framework, we study the decays into two photons of the lowest isosinglet scalar mesons.",
"We find that the direct inclusion of underlying mixing among two- and four-quark components in the trace anomaly term is essential in order for the model prediction to agree with the available experimental data on decay width of $f_0(980)$ to two photons.",
"Consequently, this sets a lower bound of 0.5 KeV on the decay with of $f_0(500)$ to two photons."
],
[
"Introduction",
"The inverted mass spectrum of the low lying scalar mesons with respect to the pseudoscalar and vector ones is a long standing low energy QCD puzzle [1] for which various solutions have been proposed [2]-[26] almost all of them dealing with the particular quark substructure of the scalar mesons.",
"In a series of papers [27]-[32] we proposed and studied in detail a generalized linear sigma model with two chiral nonets, one with a two quark substructure the other one with a four quark content.",
"In this framework the physical scalar states were found to have a significant admixture of two- and four-quark components, with those below 1 GeV generally containing a larger four-quark component compared to those above 1 GeV.",
"The generalized linear sigma model described in [27]-[32] contained, besides the relevant terms pertaining to mass and interactions, also a term that mocks up the gluon axial anomaly.",
"An extra term corresponding to the electromagnetic axial anomaly was further introduced in [30] where the decays of the pseudoscalar mesons to two photons were computed and studied with a good agreement with the experimental data.",
"It seems then natural to extend this picture to include also the trace anomaly and analyze the decays of scalar mesons to two photons in the same context.",
"In section II we briefly present our generalized linear sigma model followed by a derivation of the relevant term in the Lagrangian that leads to the correct electromagnetic trace anomaly in section III.",
"In section IV we give our numerical computation for the decay of $f_0(500)$ and $f_0(980)$ to two photons and discuss the results."
],
[
"Generalized linear sigma model",
"The model of interest is a generalized linear sigma model with two chiral nonets, one with a quark-antiquark structure $M$ , the other one with a four quark structure $M^{\\prime }$ : $&&M=S+i\\Phi \\nonumber \\\\&&M^{\\prime }=S^{\\prime }+i\\Phi ^{\\prime },$ where $S$ and $S^{\\prime }$ represent the scalar nonets and $\\Phi $ and $\\Phi ^{\\prime }$ the pseudoscalar nonets.",
"The matrices $M$ and $M^{\\prime }$ transform in the same way under $SU(3)_L \\times SU(3)_R$ but have different $U(1)_A$ transformation properties.",
"The Lagrangian has the content: ${\\cal L}=-\\frac{1}{2}{\\rm Tr}[D_{\\mu }MD^{\\mu }M^{\\dagger }]-\\frac{1}{2}{\\rm Tr}[D_{\\mu }M^{\\prime }D^{\\mu }M^{\\prime \\dagger }]-V_0(M,M^{\\prime })-V_{SB},$ where, $&&D_{\\mu }M=\\partial _{\\mu }M-ieQMA_{\\mu }+ieMQA_{\\mu }\\nonumber \\\\&&D^{\\mu }M^{\\dagger }=\\partial ^{\\mu }M^{\\dagger }+ieM^{\\dagger }QA^{\\mu }-ieQM^{\\dagger }A^{\\mu },$ and $Q={\\rm diag}(\\frac{2}{3}, -\\frac{1}{3}, -\\frac{1}{3})$ .",
"Here in the leading order of the model which corresponds to retaining only terms with no more than eight quark and antiquark lines, $V_0&=&-c_2{\\rm Tr}[MM^{\\dagger }]+c_4{\\rm Tr}[MM^{\\dagger }MM^{\\dagger }]+d_2{\\rm Tr}[M^{\\prime }M^{\\prime \\dagger }]+e_3(\\epsilon _{abc}\\epsilon ^{def}M^a_dM^b_eM^{\\prime c}_f+h.c.",
")+\\nonumber \\\\&&c_3\\left[\\gamma _1\\ln \\left(\\frac{\\det M}{\\det M^{\\dagger }}\\right)+(1-\\gamma _1)\\ln \\left(\\frac{{\\rm Tr}(MM^{\\prime \\dagger })}{{\\rm Tr}(M^{\\prime }M^{\\dagger })}\\right)\\right]^2.$ The potential is invariant under $U(3)_L \\times U(3)_R$ with the exception of the last term which breaks $U(1)_A$ .",
"The symmetry breaking term has the form: $V_{SB}=-2 {\\rm Tr}[AS]$ where $A={\\rm diag}(A_1,A_2,A_3)$ is a matrix proportional to the three light quark masses.",
"The model allows for two-quark condensates, $\\alpha _a=\\langle S_a^a \\rangle $ as well as four-quark condensates $\\beta _a=\\langle {S^{\\prime }}_a^a \\rangle $ .",
"Here we assume [33] isotopic spin symmetry so $A_1$ =$A_2$ and: $\\alpha _1 = \\alpha _2 \\ne \\alpha _3, \\hspace{56.9055pt}\\beta _1 = \\beta _2 \\ne \\beta _3$ We also need the “minimum\" conditions, $\\left< \\frac{\\partial V_0}{\\partial S}\\right> + \\left< \\frac{\\partial V_{SB}}{\\partial S}\\right>=0,\\quad \\quad \\left< \\frac{\\partial V_0}{\\partial S^{\\prime }}\\right>=0.$ There are twelve parameters describing the Lagrangian and the vacuum.",
"These include the six coupling constants given in Eq.",
"(REF ), the two quark mass parameters, ($A_1=A_2,A_3$ ) and the four vacuum parameters ($\\alpha _1=\\alpha _2,\\alpha _3,\\beta _1=\\beta _2,\\beta _3$ ).",
"The four minimum equations reduce the number of needed input parameters to eight.",
"The details of numerical work for solving this system is given in [32], and for the readers convenience a summary is given in Appendix A.",
"The fields of interest are the neutral $I=0$ scalar mesons: $&&f_a=\\frac{S_1^1+S_2^2}{\\sqrt{2}}\\nonumber \\\\&&f_b=S^3_3\\nonumber \\\\&&f_c=\\frac{S^{1\\prime }_1+S^{2\\prime }_2}{\\sqrt{2}}\\nonumber \\\\&&f_d=S^{3\\prime }_3.$ The scalars mix with each other within their group and form the physical states: $\\left(\\begin{array}{c}f_1\\\\f_2\\\\f_3\\\\f_4\\end{array}\\right)=L_0^{-1}\\left(\\begin{array}{c}f_a\\\\f_b\\\\f_c\\\\f_d\\end{array}\\right)$ Here $L_0$ is the rotation matrix and depends on the model inputs.",
"Based on the fit in Ref.",
"[32] the first two physical states are: $&&f_1=f_0(500)\\nonumber \\\\&&f_2=f_0(980)$ The experimental candidates for the remaining two states predicted by the model ($f_3$ and $f_4$ ) are $f_0(1370)$ , $f_0(1500)$ and $f_0(1710)$ .",
"However, the exact identification requires inclusion of a scalar glueball which, for simplicity, was not included in the present order of the model.",
"In this work our main focus is on $f_0(500)$ and $f_0(980)$ ."
],
[
"The trace anomaly term",
"The electromagnetic trace anomaly has the expression: $\\theta ^{\\mu }_{\\mu }=\\partial ^{\\mu }D_{\\mu }=-\\frac{\\beta (e)}{2e}F^{\\mu \\nu }F_{\\mu \\nu },$ where $\\theta ^{\\mu }_{\\mu }$ is the trace of the energy momentum tensor, $D_{\\mu }$ is the dilatation current, $e$ is the electric charge and $\\beta (e)$ is the corresponding beta function.",
"Eq.",
"(REF ) only displays the contribution to trace anomaly due to electromagnetic group which is relevant for the present work.",
"Note that the full trace anomaly also contains contributions from the gluon fields and has the expression: $\\theta ^{\\mu }_{\\mu }=\\partial ^{\\mu }D_{\\mu }=-\\frac{\\beta (e)}{2e}F^{\\mu \\nu }F_{\\mu \\nu }-\\frac{\\beta (g_3)}{2g_3}G^{a\\mu \\nu }G^a_{\\mu \\nu },$ where $g_3$ is the strong coupling constant and $G^{a\\mu \\nu }$ is the gluon tensor.",
"We apply the method introduced in [33]-[35] where for an arbitrary Lagrangian with fields $\\eta _A$ of mass dimension 1 and $\\xi _A$ with mass dimension 4, ${\\cal L}=-\\frac{1}{2}\\sum _A\\partial ^{\\mu }\\eta _A\\partial _{\\mu }\\eta _A-V(\\eta _A,\\xi _A),$ the improved energy momentum tensor is defined as: $\\theta _{\\mu \\nu }=\\delta _{\\mu \\nu }{\\cal L}+\\sum _A\\partial _{\\mu }\\eta _A\\partial _{\\nu }\\eta _A-\\frac{1}{6}\\sum _A(\\partial _{\\mu }\\partial _{\\nu }-\\delta _{\\mu \\nu }\\Box )\\eta _A^2$ Here the fields $\\eta _A$ and $\\xi _A$ transform under the scale transformation: $&&\\delta \\eta _A=\\eta _A+x_{\\mu }\\partial ^{\\mu }\\eta _A\\nonumber \\\\&&\\delta \\xi _A=4\\xi _A+x_{\\mu }\\partial ^{\\mu }\\xi _A$ The trace the energy momentum tensor can be written then as, $\\theta ^{\\mu }_{\\mu }=\\partial _{\\mu }(x^{\\mu }{\\cal L})-\\delta {\\cal L},$ which can be computed to be, $\\theta ^{\\mu }_{\\mu }=\\sum _A(4\\xi _A\\frac{\\partial V}{\\partial \\xi _A}+\\eta _A\\frac{\\partial V}{\\partial \\eta _A})-4V.$ We shall use the expression in Eq.",
"(REF ) to derive a suitable effective term that mocks up the electromagnetic anomaly.",
"Using this approach, it can be shown that the term ${\\cal L}_s= bF^{\\mu \\nu }F_{\\mu \\nu }\\left\\lbrace \\tau _1\\left[\\ln \\left(\\frac{\\det {M}}{\\Lambda ^3}\\right)+\\ln \\left(\\frac{\\det {M^{\\dagger }}}{\\Lambda ^3}\\right)\\right] +\\tau _2\\left[\\ln \\left(\\frac{{\\rm Tr}MM^{\\prime \\dagger }}{\\Lambda ^2}\\right)+\\ln \\left(\\frac{{\\rm Tr}M^{\\prime }M^{\\dagger }}{\\Lambda ^2}\\right)\\right]\\right\\rbrace $ satisfies the anomaly in Eq.",
"(REF ) provided that the dimensionless coefficients $\\tau _1$ and $\\tau _2$ satisfy the constraint $6 \\tau _1 + 4 \\tau _2=1$ .",
"Here in calculations the square of the electromagnetic tensor is assimilated to a scalar field of mass dimension 4.",
"The term in Eq.",
"(REF ) is chiral and $U(1)_A$ invariant, constructed by analogy with the axial anomaly and is minimal.",
"It can be however expanded to include other possible contributions with higher orders of $\\Lambda $ (which is expected to be associated with $\\Lambda _{QCD}$ ).",
"By applying Eq.",
"(REF ) and requiring that Eq.",
"(REF ) is satisfied we determine $b=\\frac{e^2}{12\\pi ^2}$ where we used $\\beta (e)=\\frac{1}{6\\pi ^2}e^3$ .",
"In order to determine the coupling of the physical scalars, we expand the terms in the curly brackets in Eq.",
"(REF ) around the vacuum expectation values of $S$ and $S^{\\prime }$ to show that these terms are equal to: $\\left\\lbrace \\cdots \\right\\rbrace =2\\tau _1\\left[\\frac{1}{\\alpha _1}(S^1_1+S^2_2)+\\frac{1}{\\alpha _3}S^3_3\\right]+\\frac{2\\tau _2}{2\\alpha _1\\beta _1+\\alpha _3\\beta _3}\\left[\\alpha _1(S^{1\\prime }_1+S^{2\\prime }_2)+\\alpha _3S^{3\\prime }_3+\\beta _1(S^1_1+S^2_2)+\\beta _3S^3_3\\right].$ Then the coupling of the physical states with the two photons can be read off easily as: $F_{fi} = 4 b \\left[\\tau _1 \\left({{\\sqrt{2}}\\over \\alpha _1} (L_0)_{1i} + {1\\over \\alpha _3} (L_0)_{2i}\\right)+{\\tau _2\\over {2 \\alpha _1 \\beta _1 + \\alpha _3 \\beta _3}}\\left({\\sqrt{2}} \\beta _1 (L_0)_{1i} + \\beta _3 (L_0)_{2i}+ \\alpha _1 {\\sqrt{2}} (L_0)_{3i} + \\alpha _3 (L_0)_{4i}\\right)\\right]$ where $i=1 \\cdots 4$ corresponds to the four isosinglet states (where in this work only the first two are of our interest).",
"The amplitude of decaying to two photons are: $A_i(f_i\\rightarrow \\gamma \\gamma )=-F_{f_i}(k_{1\\mu }\\epsilon _{1\\nu }-k_{1\\nu }\\epsilon _{1\\mu })(k_{2\\mu }\\epsilon _{2\\nu }-k_{2\\nu }\\epsilon _{2\\mu }),$ where $k_1$ , $k_2$ are the photon momenta and $\\epsilon _1$ , $\\epsilon _2$ are the photon polarizations.",
"The decay width is given by: $\\Gamma (f_i\\rightarrow \\gamma \\gamma )=F_i^2\\frac{m_{f_i}^3}{16\\pi }$ where $m_{f_i}$ is the mass of the meson $f_i$ ."
],
[
"Decay rates and discussion",
"As stated previously, our focus in this paper is on the two photon decays of $f_0(500)$ and $f_0(980)$ .",
"For $f_0(980)$ the experimental value of $\\Gamma \\left[f_0(980)\\rightarrow \\gamma \\gamma \\right] = 0.31^{+0.05}_{-0.04}$ KeV is listed in PDG [1].",
"Our model prediction for this decay width is found from Eqs.",
"(REF ) and (REF ) with rotation matrices $L_0$ imported from the prior work [32].",
"In this estimate the main model uncertainties stem from two of the experimental inputs ($m[\\pi (1300)]$ and $A_3/A_1$ ) used in [32] to fix the model parameters.",
"In addition, the two new parameters $\\tau _1$ and $\\tau _2$ in the trace anomaly in Eq.",
"(REF ) are a priori unknown, and therefore, after the constraint $6 \\tau _1 + 4 \\tau _2 = 1$ is considered, one of them still remains undetermined and needs to be varied (we choose to run $\\tau _2$ because it measures the direct effect of chiral nonet mixing on the anomaly term in (REF )).",
"The result is shown in Fig.",
"REF versus $\\tau _2$ with the error bars representing the uncertainties due to variation of $m[\\pi (1300)]=1.2-1.4$ and $A_3/A_1=27-30$ .",
"The two horizontal lines give the experimental bounds [1] discussed above.",
"It can be clearly seen that with small $\\tau _2$ (which measures the contribution of chiral mixing between nonets $M$ and $M^{\\prime }$ ) the model predictions do not overlap with the experimental values.",
"This is very consistent with other observations within this model where it is found that chiral mixing is essential for understanding the global properties of scalar mesons [32].",
"It is seen that for values of $\\tau _2 \\ge 0.7$ and $\\tau _2 \\le -0.8$ the model predictions overlap with experiment.",
"Figure: Partial decay width to two photon (KeV) of f 2 f_2 [or f 0 (980f_0(980)] vs τ 2 \\tau _2 predicted by the leading order of the generalized linear sigma model.",
"The error bars represent the uncertainty of the model in its leading order and the circles represent the average predictions at each value of τ 2 \\tau _2 (the uncertainties shown stem from the uncertainties of m[π(1300)]m[\\pi (1300)] in the range 1.2-1.4 and of the ratio A 3 /A 1 A_3/A_1 varied in the range of 27 to 30).",
"The two parallel lines show the experimental range for this decay reported in PDG .",
"Overlap with experiment becomes possible for τ 2 ≥0.7\\tau _2 \\ge 0.7 and τ 2 ≤-0.8\\tau _2 \\le -0.8.Similarly, the prediction for the two-photon decay width of $f_0(500)$ is given in Fig.",
"REF .",
"Considering the acceptable ranges of $\\tau _2 \\ge 0.7$ and $\\tau _2 \\le -0.8$ , the prediction of this decay width shows the lower bound of approximately 0.5 KeV (occurring around $\\tau _2\\approx -0.8 $ ).",
"This can be compared with other estimates in the literature such as $1.2\\pm 0.04$ KeV [37] or $10\\pm 6$ KeV [38].",
"In [39] the authors made a thorough amplitude analysis of the experimental data for $\\gamma \\gamma \\rightarrow \\pi ^+\\pi ^-$ to find the position of the sigma pole at $0.441-i0.272$ which corresponds to two scenarios for a decay width of $f_0(500)$ to two photons of $3.1\\pm 0.5$ KeV or $2.4\\pm 0.4$ KeV.",
"More recently performing a similar analysis Dai and Pennington [40] found this decay width to be $2.05\\pm 0.21$ KeV.",
"They also found $\\Gamma (f_0(980))\\rightarrow \\gamma \\gamma =0.32 \\pm 0.05$ KeV.",
"The decay widths of the low lying scalar mesons were analyzed in the literature from the perspective that they proceed mainly through pions and kaons loops [41], [42], [43], [44].",
"It is generally hypothesized that because of this assumption it is hard for these decays to be relevant for the quark substructure of the scalar mesons [1].",
"In summary, the lower bound of 0.5 KeV for the decay width of $f_0(500)$ to two photons obtained in this analysis (within the leading order of the generalized linear sigma model) is qualitatively consistent with other estimates [37], [38], [39], [40].",
"A more precise prediction is expected when higher order effects are taken into account.",
"Within the current approach it was also shown that chiral nonet mixing is an essential ingredient in understanding this decay width in which direct inclusion of mixing in modeling the trace anomaly is needed.",
"This last point further supports the importance of underlying mixing among two- and four-quark components in exploring the spectroscopy of light scalar mesons.",
"Figure: Partial decay width to two photon (KeV) of f 1 f_1 [or f 0 (500f_0(500)] vs τ 2 \\tau _2 predicted by the leading order of the generalized linear sigma model.",
"The error bars represent the uncertainty of the model in its leading order and the circles represent the average predictions at each value of τ 2 \\tau _2 (the uncertainties shown stem from the uncertainties of m[π(1300)]m[\\pi (1300)] in the range 1.2-1.4 and of the ratio A 3 /A 1 A_3/A_1 varied in the range of 27 to 30).Comparing with Fig.",
", the acceptable region is τ 2 ≥0.7\\tau _2 \\ge 0.7 and τ 2 ≤-0.8\\tau _2 \\le -0.8 which sets a lower bound of approximately 0.5 KeV on the decay with of f 0 (500)f_0(500) to two photons."
],
[
"Acknowledgments",
"A.H.F.",
"gratefully acknowledges the support of College of Arts and Sciences of SUNY Poly in the Fall 2017 semester."
],
[
"Brief review of the Numerical analysis for model parameters and rotation matrices",
"In this appendix we give a summary of numerical determination of the eight independent Lagrangian parameters of Eqs.",
"(REF ) and (REF ).",
"Five of these eight are determined from the following masses together with the pion decay constant: $m[a_0(980)] &=& 980 \\pm 20\\, {\\rm MeV}\\nonumber \\\\ m[a_0(1450)] &=& 1474 \\pm 19\\, {\\rm MeV}\\nonumber \\\\m[\\pi (1300)] &=& 1300 \\pm 100\\, {\\rm MeV}\\nonumber \\\\m_\\pi &=& 137 \\, {\\rm MeV}\\nonumber \\\\F_\\pi &=& 131 \\, {\\rm MeV}$ Since $m[\\pi (1300)]$ has a large uncertainty, the Lagrangian parameters would depend on on the choice of this experimental input.",
"The sixth input is taken as the light “quark mass ratio\" $A_3/A_1$ , which are varied over its appropriate range (in this work we use 27-30).",
"The remaining two parameters ($c_3$ and $\\gamma _1$ ) only affect the isosinglet pseudoscalars (whose properties also depend on the ten parameters discussed above).",
"However, there are several choices for determination of these two parameters depending on how the four isosinglet pseudoscalars predicted in this model are matched to many experimental candidates below 2 GeV.",
"The two lightest predicted by the model ($\\eta _1$ and $\\eta _2$ ) are identified with $\\eta (547)$ and $\\eta ^{\\prime }(958)$ with masses: $m^{\\rm exp.",
"}[\\eta (547)] &=& 547.853 \\pm 0.024\\, {\\rm MeV},\\nonumber \\\\m^{\\rm exp.",
"}[\\eta ^{\\prime } (958)] &=& 957.78 \\pm 0.06\\, {\\rm MeV}.$ For the two heavier ones ($\\eta _3$ and $\\eta _4$ ), there are six ways that they can be identified with the four experimental candidates above 1 GeV: $\\eta (1295)$ , $\\eta (1405)$ , $\\eta (1475)$ , and $\\eta (1760)$ with masses, $m^{\\rm exp.",
"}[\\eta (1295)] &=& 1294 \\pm 4\\, {\\rm MeV},\\nonumber \\\\m^{\\rm exp.",
"}[\\eta (1405)] &=& 1409.8 \\pm 2.4 \\,{\\rm MeV},\\nonumber \\\\m^{\\rm exp.",
"}[\\eta (1475)] &=& 1476 \\pm 4\\, {\\rm MeV},\\nonumber \\\\m^{\\rm exp.",
"}[\\eta (1760)] &=& 1756 \\pm 9 \\,{\\rm MeV}.$ This leads to six scenarios considered in detail in [32].",
"The two experimental inputs for determination of the two parameters $c_3$ and $\\gamma _1$ are taken to be Tr$M_\\eta ^2$ and det$M_\\eta ^2$ , i.e.",
"${\\rm Tr}\\, \\left( M^2_\\eta \\right) &=&{\\rm Tr}\\, \\left( {M^2_\\eta } \\right)_{\\rm exp},\\nonumber \\\\{\\rm det}\\, \\left( M^2_\\eta \\right) &=&{\\rm det}\\, \\left( {M^2_\\eta } \\right)_{\\rm exp}.$ Moreover, for each of the six scenarios, $\\gamma _1$ is found from a quadratic equation, and as a result, there are altogether twelve possibilities for determination of $\\gamma _1$ and $c_3$ .",
"Since only Tr and det of experimental masses are imposed for each of these twelve possibilities, the resulting $\\gamma _1$ and $c_3$ do not necessarily recover the exact individual experimental masses, therefore the best overall agreement between the predicted masses (for each of the twelve possibilities) were examined in [32].",
"Quantitatively, the goodness of each solution was measured by the smallness of the following quantity: $\\chi _{sl} =\\sum _{k=1}^4{{\\left| m^{\\rm theo.",
"}_{sl}(\\eta _k) -m^{\\rm exp.",
"}_{s}(\\eta _k)\\right|}\\over m^{\\rm exp.",
"}_{s}(\\eta _k)},$ in which $s$ corresponds to the scenario (i.e.",
"$s= 1 \\cdots 6$ ) and $l$ corresponds to the solution number (i.e.",
"$l=$ I, II).",
"The quantity $\\chi _{sl}\\times 100$ gives the overall percent discrepancy between our theoretical prediction and experiment.",
"For the six scenarios and the two solutions for each scenario, $\\chi _{sl}$ was analyzed in ref.",
"[32].",
"For the third scenario (corresponding to identification of $\\eta _3$ and $\\eta _4$ with experimental candidates $\\eta (1295)$ and $\\eta (1760)$ ) and solution I the best agreement with the mass spectrum of the eta system was obtained (i.e.",
"$\\chi _{3\\rm {I}}$ was the smallest).",
"Furthermore, all six scenarios were examined in the analysis of $\\eta ^{\\prime }\\rightarrow \\eta \\pi \\pi $ decay in [45] and it was found that the best overall result (both for the partial decay width of $\\eta ^{\\prime }\\rightarrow \\eta \\pi \\pi $ as well as the energy dependence of its squared decay amplitude) is obtained for scenario “3I” consistent with the analysis of ref.",
"[32].",
"In this work, we use the result of “3I” scenario.",
"The numerical values for the rotation matrix $L_0$ defined in (REF ) can be consequently determined.",
"Since two of the model inputs $A_3/A_1$ and $m[\\pi (1300)]$ have large uncertainties, the numerical values of these rotation matrices naturally have some dependencies on these two inputs.",
"Table REF gives numerical values of $L_0$ for three values of $m[\\pi (1300)]$ and three values of $A_3/A_1$ .",
"Table: Rotation matrix L 0 L_0 for several values of A 3 /A 1 A_3/A_1 and m[π(1300)]m[\\pi (1300)]."
]
] | 1709.01834 |
[
[
"Experimental signatures of the quantum nature of radiation reaction in\n the field of an ultra-intense laser"
],
[
"Abstract The description of the dynamics of an electron in an external electromagnetic field of arbitrary intensity is one of the most fundamental outstanding problems in electrodynamics.",
"Remarkably, to date there is no unanimously accepted theoretical solution for ultra-high intensities and little or no experimental data.",
"The basic challenge is the inclusion of the self-interaction of the electron with the field emitted by the electron itself - the so-called radiation reaction force.",
"We report here on the experimental evidence of strong radiation reaction, in an all-optical experiment, during the propagation of highly relativistic electrons (maximum energy exceeding 2 GeV) through the field of an ultra-intense laser (peak intensity of $4\\times10^{20}$ W/cm$^2$).",
"In their own rest frame, the highest energy electrons experience an electric field as high as one quarter of the critical field of quantum electrodynamics and are seen to lose up to 30% of their kinetic energy during the propagation through the laser field.",
"The experimental data show signatures of quantum effects in the electron dynamics in the external laser field, potentially showing departures from the constant cross field approximation."
],
[
"INTRODUCTION",
"In the realm of classical electrodynamics, the problem of radiation reaction (RR) is satisfactorily described by the Landau-Lifshitz (LL) equation [1], which has been theoretically demonstrated to be the self-consistent classical equation of motion for a charged particle [1], [2].",
"However, when the electron experiences extremely intense fields the LL equation may no longer be assumed valid [3].",
"A full quantum description is thus required and this is currently the subject of active theoretical research (see, for instance, Refs.",
"[3], [4], [5], [6], [7], [8], [9], [10]).",
"Purely quantum effects can be triggered in these conditions, including the stochastic nature of photon emission [5], [6], a hard cut-off in the maximum energy of the emitted photons [9], and pair production [10].",
"Besides the intrinsic fundamental interest in investigating this regime in laboratory experiments, RR is often invoked to explain the radiative properties of powerful astrophysical objects, such as pulsars and quasars [11], [12].",
"A detailed characterisation of RR is also important for a correct description of high-field experiments using the next generation of multi-petawatt laser facilities, such as the Extreme Light Infrastructure [13], [14], Apollon [15], Vulcan 20PW [16], and XCELS [17] where focussed intensities exceeding $10^{23}$ W/cm$^2$ are expected.",
"The LL equation is obtained assuming that the electromagnetic field in the rest frame of the electron is much smaller than the classical critical field $F_0 = 4\\pi \\epsilon _0 m_e^2 c^4 / e^3 \\approx 1.8\\times 10^{20}\\;\\text{V/m}$ [1] and constant over distances of the order of the classical electron radius $r_0 = e^2 / 4\\pi \\epsilon _0 m_e c^2 \\approx 2.8\\times 10^{-15}\\;\\text{m}$ .",
"These conditions are automatically satisfied in classical electrodynamics since quantum effects are negligible as long as the rest frame fields are much smaller than the critical field of Quantum Electrodynamics (QED) $F_{cr} = \\alpha F_0 \\approx 1.3\\times 10^{18}\\;\\text{V/m}$ $\\ll $ $F_0$ [9] and remain constant over distances of the order of the reduced Compton wavelength $\\lambda _C = r_0 / \\alpha \\approx 3.9\\times 10^{-13}\\;\\text{m}$ $\\gg $ $r_0$ ($\\alpha \\approx $ 1/137 is the fine structure constant).",
"An electric field with amplitude of the order of the critical field $F_{cr}$ is able to impart an energy of the order of $mc^2$ to an electron over a length of the order of $\\lambda _C$ .",
"If the amplitude of the laser field in the rest frame of the electron is of the order of $F_{cr}$ , the quantum recoil undergone by the electron when it emits a photon is thus not negligible [10].",
"Also, if the laser wavelength in the rest frame of the electron is of the order of $\\lambda _C$ , then already the absorption of a single laser photon would impart to the electron a recoil comparable with its rest energy.",
"Even for GeV electrons with Lorentz factor $\\gamma _e\\gtrsim $ 2000, the micron-scale wavelength of typical high-power laser systems ($\\lambda _L\\approx 0.8 - 1 \\mu $ m) implies that the only relevant condition on classicality is on the laser field amplitude $F_L$ , which, for a plane wave, can be expressed by stating that the quantum parameter $\\chi \\approx (1-\\cos \\theta ) \\gamma _e F_L/ F_{cr}$ has to be much smaller than unity.",
"Here $\\theta $ is the angle between the laser propagation direction and the electron momentum in the laboratory frame.",
"Thus the validity of the LL approach can be expected to break down when quantum effects on the electron's motion become important, i.e., when $\\chi $ becomes a sizeable fraction of unity.",
"In the intense fields that can be created by modern-day lasers, one must also account for the possibility of multiple laser-photons being absorbed and resulting in the emission of a single high-energy photon by the electron.",
"For each photon formation length the number of absorbed photons per electron is of the order of the laser dimensionless amplitude $a_0 = e F_L \\lambda _L/ 2\\pi m_e c^2$ [10].",
"Available lasers can now easily reach $a_0\\gg 1$ , thus allowing for experimental investigations of this strong-field quantum regime.",
"The multi-GeV electrons available at accelerator laboratories world-wide would provide an excellent basis for RR studies in the non-linear and quantum regime, but are rarely available concurrently with ultra-intense lasers.",
"The development of compact laser-driven wakefield accelerators (LWFA) [18] provides a well-suited alternative, since it allows GeV electron beams to be generated directly at high power laser laboratories capable of achieving field strengths of $a_0\\gg 1$ [19], [20], [21].",
"The plausibility of such an experimental approach is evidenced by the observation of non-linearities in Compton scattering in previous experimental campaigns [22], [23], [24], motivating the study reported here.",
"To date, only one laser-based experimental campaign has reached a sizeable fraction of the Schwinger field in the rest frame of an electron ($\\chi \\approx 0.2 $ ) [25], [26].",
"Whilst these experiments gave evidence of non-linearities in Compton scattering [25] and generation of electron-positron pairs [26], no measurements were performed to directly assess the level of RR in the spectrum of the scattered electron beam.",
"Moreover, despite the high field achieved in the electron rest frame, the relatively low intensity of the scattering laser ($a_0\\approx 0.3-0.4$ ) implies that single photon absorption was the dominant absorption mechanism in the electron dynamics in the field.",
"In other words, non-linearities only occurred perturbatively; the relative strength of the emission of the $n^{th}$ harmonic scales as $a_0^{2n}$ , implying that non-linear Compton scattering was strongly suppressed.",
"In our experimental configuration, a much higher laser intensity ($a_0\\simeq 10$ ) allowed a strongly non-linear regime of RR to be accessed (i.e., multi-photon absorption even within a single photon formation length).",
"We report here on substantial energy loss (up to 30%) experienced by a laser-driven multi-GeV electron beam (maximum Lorentz factor $\\gamma _e > 4\\times 10^3$ ) [27] during its propagation through the focus of a high-intensity laser (dimensionless amplitude $a_0\\approx 10$ ).",
"A stable regime of laser-driven electron acceleration, obtained using gas-cell targets, allowed us to directly compare the spectrum of the electrons before and after the interaction with the laser.",
"This provides a detailed test of different models of radiation reaction in an electric field that is a sizeable fraction (up to 25%) of the Schwinger field, distinguishing these results from others recently published in the literature [28].",
"Best agreement with the experimental data is found for a semi-classical model that weights the LL equation with the ratio between the quantum and classical synchrotron emission spectrum (coefficient of determination $R^2=96$ %, against $R^2=87$ % for the LL), indicating the emergence of quantum effects in the electron dynamics.",
"A residual mismatch between the semi-classical model and the experimental data at low energies could be explained by a potential departure from the realm of validity of the constant-cross-field-approximation (CCFA), an approximation commonly used in modelling the quantum emission of an electron in an external electromagnetic field."
],
[
"EXPERIMENTAL SETUP",
"The experimental set-up is shown schematically in Fig.",
"REF a.",
"One of the twin laser beams of the Astra Gemini laser system (Driver Laser in Fig.",
"REF a), was focussed at the entrance of a helium-filled gas-cell in order to accelerate a multi-GeV electron beam, via the laser wakefield acceleration mechanism [18], [27].",
"The gas-cell was operated at a backing pressure of 60 mbar that, once fully ionised, corresponds to an electron density of $2 \\times 10^{18}$ cm$^{-3}$ .",
"The laser with a pulse duration of ${42(3)}{fs}$ was focussed using an $f/40$ spherical mirror down to a focal spot with Full-Width-Half-Maxima (FWHM), along the two axis, of $\\sigma _x ={59(2)}{}$ and $\\sigma _y= {67(2)}{}$ containing ${9}{J}$ (normalised intensity of $a_0\\approx 1.7$ ).",
"The laser-driven wakefields in the plasma accelerated the electron beam in the blow-out regime [18], producing stable beams with a broad energy spectrum exceeding 2 GeV ($\\gamma _e \\approx 4\\times 10^3$ ) [27].",
"The electron spectra were recorded by a magnetic spectrometer consisting of a 15 cm long dipole magnet with a peak magnetic field of 1.0 T and a LANEX scintillator screen placed 2m away from the gas-cell.",
"The minimum electron energy recorded on the LANEX screen in this configuration was 350 MeV and its energy resolution is of the order of $\\delta E / E \\approx 5\\%$ for an electron energy of 1.5 GeV.",
"The electron beam source size can be estimated to be $D_e\\le {1}{}$ , as deduced by rescaling the size of typical betatron sources in similar conditions [29].",
"The energy-dependent beam divergence was determined by measuring the beam width perpendicular to the direction of dispersion on the electron spectrometer screen 2 m downstream from the gas cell.",
"For electron energies exceeding 1 GeV, the divergence is measured to be $\\theta _e=(0.70\\pm 0.05)$ mrad.",
"Even though this gives in principle only the divergence along one of the transverse dimensions of the beam, the regime of laser-wakefield we are operating in generates accelerating fields with a radially symmetric distribution [18].",
"This in turn results in cylindrically symmetric electron beams, as confirmed by our analysis [30].",
"The detailed energy-dependent divergence measured in the experiment was used as in input for the numerical simulations discussed later in the article.",
"Measurements of the pointing fluctuation of the laser-driven electron beam indicate, as an average over 100 consecutive shots, an approximately Gaussian distribution (confidence of 95% from the Kolmogorov-Smirnov test) centred on the laser propagation axis with a standard deviation of ${3.2(8)}{mrad}$ [30].",
"The use of a gas-cell target, instead of a gas-jet reported elsewhere [28] for similar experimental conditions, results in better shot-to-shot stability in the electron spectrum [31], [32], with the maximum energy of the electrons closely related to the energy of the drive laser, as discussed in the next section.",
"Moreover, it allowed much higher electron energies to be reached and, therefore, a much higher fraction of the Schwinger field in the electron rest frame.",
"The second laser beam (Scattering Laser in Fig.",
"REF a) was focused, using an $f/2$ off-axis parabola with a concentric $f/7$ hole (energy loss of 10%), 1 cm downstream of the exit of the gas-cell exactly counter-propagating with respect to the laser-wakefield accelerated electron beam.",
"On-shot measurements of the laser temporal profile using a Frequency Resolved Optical Gating (FROG) device indicate a Gaussian distribution with a duration of ${42(3)}{fs}$ .",
"The energy contained in the laser after compression was measured, for each shot, by integrating the beam near-field on a camera that was previously absolutely calibrated against an energy meter, giving a value of ${8.8(7)}{J}$ .",
"The radial distribution of the laser intensity at focus is shown in Fig.",
"REF b. and it arises from an average of ten consecutive measurements at low power (spatial resolution of the detector of ${0.2}{}$ /pixel).",
"Independent measurements of the intensity profile at low-power and full-power indicate a broadening of the focal spot radius of the order of 10% in the latter case [33].",
"This effect is taken into account in the computed transverse laser field distribution shown in Fig.",
"REF c. The scattering and driver laser are linearly polarised along perpendicular axes (horizontal and vertical, respectively) in order to further reduce risks of back-propagation of the lasers in the amplification chains.",
"However, numerical simulations show that the particular polarisation axes used in the experiment is virtually irrelevant in determining the energy loss experienced by the electrons.",
"Both lasers are generated from the same oscillator and synchronised using a spectral interferometry technique discussed in Ref.",
"[34] and already used in a similar experimental setup [22].",
"This system had a temporal resolution of approximately 40 fs.",
"Due to the inherent lag of the laser-accelerated electron beam in respect to the driver laser, the scattering laser has defocussed for approximately 64 fs before interacting with the electrons [18], [27].",
"At this time delay, the scattering laser has a rather flat profile, with a peak $a_0$ of the order of 10 and a full width half maximum of ${7}{}$ (see Fig.",
"REF .c).",
"The energy contained in the Compton-generated $\\gamma $ -ray beam was measured using a 5 cm thick caesium-iodide (CsI) scintillator placed, on-axis, 4m downstream of the electron-laser interaction point.",
"The transverse diameter of each scintillation rod is 5mm, implying an angular resolution of the order of 1.25 mrad.",
"The energy deposited on the scintillator, modelled with FLUKA [35] simulations, is almost linear in the range 10-400 MeV and best fitted ($R^2$ =95%) by: $E_{DEP} = 2.08\\times 10^{-2} E_{INC}+ 0.68$ with $E_{DEP}$ and $E_{INC}$ the deposited energy and the energy of the incident photon, respectively."
],
[
"ELECTRON-LASER OVERLAP AND STABILITY",
"One of the main measurables to experimentally assess the amount of RR experienced by the electron beam is the change in spectral energy density from a typical reference electron spectrum to the spectrum of the scattered electrons.",
"In our experiment, the laser-driven electron beams [27] were obtained in a stable regime where their spectral shape was a reproducible function of the input laser energy (Fig.",
"REF ), unlike results recently reported using a gas-jet target [28].",
"In Fig.",
"REF .a, we show the correlation between the energy of the laser driving the wakefield and the cut-off energy of the accelerated electron beam.",
"The cut-off energy is defined as the energy at which the beam spectral intensity falls down to 10% of its peak value.",
"The empty squares depict shots with the scattering laser off with a linear fit represented by the dashed blue line.",
"The vast majority of these shots fall within 1$\\sigma $ (68% confidence, darker blue band in the figure) with all of them still within a 2$\\sigma $ band (95% confidence, lighter blue band in the figure).",
"The colour-coded circles depict instead shots with the scattering laser on.",
"The colour of each circle represents the total energy of the photon beam emitted via Compton scattering, as recorded by the CsI scintillator, normalised by the total kinetic energy in the recorded electron beam (kinetic energy exceeding 350 MeV, lower limit of the magnetic spectrometer).",
"As discussed above, the energies of both the driver and scattering laser were measured live on each shot, allowing to clearly identify suitable reference shots (scattering laser off) for each shot with the scattering laser on.",
"The intrinsic shot-to-shot pointing fluctuations of LWFA beams [30] results in a statistical fluctuation of the spatial overlap of the laser spot with the electron beam.",
"To discern between shots of poor and good overlap we use the energy contained in the Compton $\\gamma $ -ray beam generated during the interaction, an established method for this class of experiments (see, for instance, Ref.",
"[25]).",
"The total energy emitted via Compton scattering scales as $E_{ph} \\propto \\int a_0\\gamma _e^2N_e(a_0)\\, da_0$ , with $N_e(a_0)$ the number of electrons interacting with a field of amplitude $a_0$ [36].",
"Whilst the CsI detector did not allow for the extraction of the spectral distribution of the photon beam, the signal recorded is proportional to the total energy contained in the Compton-scattered photon beam, allowing us to discern between shots with best overlap (and, therefore, both higher energy loss in the electron beam and high photon yield) from those with poorer overlap.",
"This is exemplified in Fig.",
"REF a, where the total photon yield recorded on the CsI detector is plotted against the percentage of energy loss experienced by the electron beam.",
"The data appear to follow a linear trend, which is also reproduced by numerical simulations assuming different transverse misalignments of the electron beam in respect to the main axis of the scattering laser.",
"These simulations are performed using a semi-classical model of radiation reaction since, as will be discussed in the following, this is the model the best reproduces our experimental data.",
"This correlation allows us to distinguish between shots with good overlap (labelled c and d in Fig.",
"REF a) from shots with poor overlap (such as shot labelled b in Fig.",
"REF a).",
"Indeed, shots with relatively low photon yield all fall within the 2$\\sigma $ band (lighter blue band) of the linear dependence of the electron beam cut-off energy on the energy of the driver laser.",
"On the other hand, the two shots with the brightest photon signal (labelled with d and c in Fig.",
"REF a) both fall outside the 2$\\sigma $ band, implying that the probability of them being just the result of a random fluctuation is smaller than 0.2%.",
"This places high confidence that a measurement of a lower electron energy is directly related to the occurrence of strong RR.",
"In the following we will then focus on three exemplary laser shots: shot labelled as d in Fig.REF a, a good candidate for best overlap, shot c as a a good candidate for a slight misalignment between the scattering laser and the electron beam, and shot b as a good candidate for poor overlap and, therefore negligible RR.",
"For each of these shots, we have selected the spectra of the primary electron beam whose driver laser energy falls within $±0.5$ J (grey bands in Fig.",
"REF a) of that of the shot under interest, as reference spectra.",
"The associated spectral densities are plotted in Figs.",
"REF b, REF c, and REF d. For each of these frames, the thin red lines represent single shot spectral densities, thick black lines represent the average, and the associated bands represent one standard deviation.",
"As one can see, within each energy band of the driver laser energy, the electron spectral densities were remarkably stable, justifying their use as reference electron spectra for each event with the scattering laser on.",
"In the following, our analysis will be based on single-electron spectra normalised by dividing the measured spectrum by the overall number of electrons with energy exceeding 350 MeV, in order to eliminate shot-to-shot fluctuations in the total electron number without affecting the spectral shape of the beam."
],
[
"ELECTRON ENERGY LOSS: EXPERIMENTAL RESULTS",
"We will now focus our attention only on shots where the CsI detector indicates best overlap between the high-energy component of the electron beam and the scattering laser (shots c and d in Fig.",
"REF a).",
"A comparison between the measured spectral energy density of the initial (scattering laser off) and scattered (scattering laser on) electron beam for conditions of best overlap (shot d in Fig.",
"REF a) is shown in Fig.",
"REF d. The corresponding single-shot spectral energy densities and the associated uncertainties for the reference electron beams are shown in Fig.",
"REF d and exhibit a spectral profile that decreases with energy up to 2 GeV, with a clear spectral peak at approximately 1.2 GeV.",
"The spectral energy density of the electrons after the interaction with the scattering laser beam (red line in Fig.",
"REF d) not only shows a reduction in the cut-off energy but also a significant change in spectral shape, with virtually no electrons with an energy exceeding 1.6 GeV.",
"Moreover, the local maximum in the spectrum is now shifted down to an energy of approximately 1 GeV and there is clear accumulation of electrons at lower energies, suggesting a net energy loss for the highest energy electrons of the order of 30%.",
"On the other hand, a comparison between the scattered and reference electron spectral density for a shot with lower yield (labelled as c in Fig.",
"REF .a) clearly evidences a lower amount of energy loss (of the order of 20%, frame REF .c), whereas a typical shot with even lower photon yield shows virtually no loss in the electron energy (frame REF .b).",
"As a first remark, it is interesting to note that the overall electron energy loss, observed for conditions of best overlap, is slightly lower than a classical estimate based on the LL equation.",
"For our experiment, we can assume a plane wave with a Gaussian temporal field profile given by $\\exp (-\\varphi ^2/\\sigma _{\\varphi }^2)$ , where $\\varphi =\\omega _L(t-z/c)$ is the laser phase, $\\omega _L$ is the laser angular frequency, and $\\sigma _{\\varphi }=\\omega _Lt_L/\\sqrt{2\\log 2}$ .",
"Here $t_L$ represents the FWHM of the laser intensity.",
"In this case, and assuming $\\gamma _e\\gg a_0$ , the analytical solution of the LL equation [37], provides: $\\frac{\\Delta \\gamma _e}{\\gamma _e} \\approx \\frac{\\sqrt{\\pi /\\log 2}\\tau _0t_L\\omega _L^2\\gamma _ea_0^2/2}{1+\\sqrt{\\pi /\\log 2}\\tau _0t_L\\omega _L^2\\gamma _ea_0^2/2},$ with $\\tau _0 = 2 r_0 / 3c \\approx 6.3\\times 10^{-24}$ s, $t_L = 42\\pm 3$ fs the laser duration, and $\\omega _L=2.4\\times 10^{15}$ rad/s the laser carrier frequency (see also Ref.",
"[38], where there $t_L$ corresponds to $\\sigma _{\\varphi }/\\omega _L$ in our notation).",
"For $\\gamma _e=4000$ and $a_0=10$ , the LL equation predicts an energy loss of about 40%, slightly higher than the experimental findings.",
"We observe that under the present experimental conditions (ultra-relativistic electrons with $\\gamma _e\\gg a_0$ and initially counter-propagating with respect to the laser field) it is possible to approximate $\\gamma _e\\approx \\gamma _e(1-v_{e,z}/c)/2$ , with $v_{e,z}\\approx -c$ being the electron velocity along the propagation direction of the laser field, and thus use directly Eqs.",
"(8) and (9) in [37] to estimate the relative energy loss.",
"However, in order to provide a more detailed comparison with the different theoretical models of RR, an extensive series of simulations were performed assuming different radiation reaction models and will be discussed in the next section."
],
[
"ELECTRON ENERGY LOSS: COMPARISON WITH THEORY",
"A quantitative comparison between the experimental data and different theoretical models of RR is shown in Fig.",
"REF .",
"Here, the normalised experimental spectral energy density of the scattered electrons in conditions of best overlap are compared with the corresponding theoretical curves obtained by simulating the effect of the scattering laser on reference spectra using different models and both a multi-particle code and a Particle-In-Cell (PIC) code.",
"For each frame in the figure, the error bands of the multi-particle code correspond to the uncertainties in the reference electron spectra as well as uncertainties in the intensity of the scattering laser measured for each shot ($\\Delta a_0/a_0 \\simeq 4$ %).",
"The multi-particle code assumes a beam of $10^7$ electrons generated by sampling first from the experimental electron beam spectrum and then from the energy-dependent divergence, assumed to follow a Gaussian distribution with zero mean and Full Width Half Maximum (FWHM) extracted from the experimental data.",
"The electron three dimensional momentum was then calculated from the sampled electron energy and from the two sampled divergence angles.",
"In order to account for the free electron propagation from the gas-cell, the initial transverse electron spatial distribution was obtained assuming ballistic propagation of the electrons over 1 cm from a point-like source.",
"The longitudinal distribution of the electron beam was assumed to be Gaussian with ${12}{}$ FWHM, i.e.",
"${40}{fs}$ duration.",
"The transverse laser pulse field profile was instead obtained by fitting the experimental transverse profile (see Fig.",
"REF b) with the linear superposition of two Gaussian pulses.",
"Each Gaussian pulse was accurately modelled by including terms up to the fifth order in the diffraction angle.",
"The resulting peak amplitude of the laser field at the focus was $a_0 \\approx 22.5$ with approximately ${2.5}{}$ FWHM of the transverse intensity profile.",
"The laser pulse temporal profile was Gaussian with ${42}{fs}$ duration FMHM of the laser pulse intensity.",
"Since the accelerated electrons lag behind the laser pulse, the head-on collision between the peak of the scattering laser and the peak of the electron beam was set to occur 64fs after the scattering laser pulse reached the focus.",
"This results in both a reduction of the maximal laser field at the interaction from $a_0 \\approx 22.5$ to $a_0 \\approx 10$ , and into an increased diameter (FWHM of the intensity) from ${2.5}{}$ to about ${6.9}{}$ (see Fig.",
"REF .c).",
"These simulations were performed assuming different models, associated with different degrees of approximation in modelling RR.",
"A perturbative method (PT, shown in Fig.",
"REF a), the Landau-Lifshitz equation (LL, shown in Fig.",
"REF b), a semi-classical model (SC, shown in Fig.",
"REF c), and a quantum electro-dynamic model (QED, shown in Fig.",
"REF d).",
"A discussion of the results predicted by each model is given below.",
"The PT is routinely used for modelling particle acceleration and transport in synchrotrons [39].",
"In this case, the electron trajectory in the field is calculated classically using the Lorentz force and the corresponding emitted energy is calculated assuming the relativistic Larmor formula.",
"In this model, the electron energy loss is only accounted for by subtracting the total energy emitted by each electron after the propagation in the field.",
"This model effectively ignores radiation-radiation effects during the propagation of the electron inside the beam.",
"The model significantly fails in reproducing the experimental data for energies approximately below 1.4 GeV as it greatly overestimates the energy loss.",
"This is to be expected, since this model does not account for the continuous energy loss by the electron due to radiation throughout the electron propagation in the laser field and therefore predicts a higher emission of radiation.",
"The predictions of the LL model are shown in Fig.",
"REF b.",
"It must be noted here that we neglect the term in the equation containing the derivatives of the electromagnetic field [40], since it is negligibly small in our experimental regime and it averages out to zero for a plane-wave pulse [37].",
"The LL equation is able to reproduce the experimental data more closely, if compared to the PT model, resulting in an overall coefficient of determination $R^2=87$ %.",
"However, this model appears to over-estimate the energy loss experienced by the electron beam.",
"Even though the experimental data does not allow us to draw a definite conclusion in this regard, a slight overestimate of the energy loss is to be expected due to the non-negligible value of the quantum parameter $\\chi $ in this experiment since, strictly speaking, the LL is valid only under the assumption of $\\chi \\ll 1$ .",
"For non-negligible $\\chi $ , the LL overestimates the energy loss experienced by the electrons, which results in a spectral peak that is significantly down-shifted if compared with that of the experimental data ($0.78\\pm 0.05$ GeV against 0.96 GeV in the experiment).",
"This is because the LL is a purely classical model, with no upper bound in the frequency of the emitted radiation and with continuous emission.",
"In reality, each electron cannot emit a photon with an energy exceeding its kinetic energy, effectively introducing a sharp cut-off in the spectrum of the emitted radiation [10].",
"This cut-off reduces the total amount of radiation that each electron can emit, thus resulting in a lower energy loss.",
"This effect of a hard quantum cut-off can be phenomenologically included by multiplying the radiation reaction force in the LL equation by a “weighting” function $g(\\chi )=I_{\\text{Q}}/I_{C}$ [41], where $I_{\\text{Q}}$ is the quantum radiation intensity: $I_{\\text{Q}} = \\frac{e^2 m_e^2}{3 \\sqrt{3} \\pi \\hbar ^2} \\int _0^{\\infty }{\\frac{u(4u^2+5u+4)}{(1+u)^4} \\text{K}_{2/3}\\left(\\frac{2u}{3\\chi }\\right)du}$ and $I_{C}$ = $2 e^2 m_e^2 \\chi ^2 / 3 \\hbar ^2$ is the classical radiation intensity (see Eqs.",
"(4.50) and (4.52) in Ref. [42]).",
"In our simulations, the following interpolation formula is employed: $g(\\chi ) \\approx \\frac{1}{[1 + 4.8 (1 + \\chi ) \\ln (1 + 1.7 \\chi ) + 2.44 \\chi ^2]^{2/3}}$ which approximates the function $g(\\chi )$ with accuracy better than 2% for arbitrary $\\chi $ (see Eqs.",
"(4.57) in Ref. [42]).",
"With this weighting function, the known classical overestimate of the total emitted energy with respect to the more accurate quantum expression is then avoided.",
"However, in this “semi-classical” model the emission of radiation is still included as a “classical” continuous process, i.e., the quantum stochastic nature of photon emission is ignored.",
"Moreover, we point out that the used expression of $I_{\\text{Q}}$ is derived within the so-called local-constant-crossed field approximation, as described in more detail below.",
"A comparison between the predictions of this model and the experimental results is shown in Fig.",
"REF c. This semi-classical model is able to closely reproduce the experimental data, with an overall coefficient of determination $R^2 = 96$ %.",
"Indeed, there is agreement for almost all energies, with only a slight deviation around the spectral peak, that is located by the SC model at $0.90\\pm 0.03$ GeV and it corresponds to 0.96 GeV in the experiment.",
"However, deviations from the SC model are almost all within $1\\sigma $ , and all well within the $2\\sigma $ level.",
"This agreement is significantly better than the one obtained assuming a purely classical model based on the LL ($R^2 =$ 87%).",
"This improved agreement of the semi-classical LL model compared to the unmodified LL provides a preliminary indication of the onset of quantum effects under the conditions of the experiment.",
"Finally, a comparison between the experimentally measured spectrum of the scattered electrons and numerical calculations based on a multi-particle QED code (green curve) is shown in Fig.",
"REF d. In this model, the stochastic photon emission was calculated for arbitrary electron and photon energies, under the constant-cross-field-approximation.",
"Each electron was propagated according to the Lorentz equation between two consecutive photon emission events [43].",
"This model is, within the uncertainties of the experiment, able to reproduce the general features of the experimental data.",
"However, there still is a non-negligible mismatch, especially in the shape of the spectral energy density.",
"This mismatch results in a coefficient of determination that is slightly lower ($R^2=92$ %) than the semiclassical case.",
"In order to rule out collective effects in the electron beam as a possible source for this mismatch, 3-dimensional PIC simulations using the code EPOCH [44] have also been carried out.",
"For these simulations, the laser and electron bunch simulated were the same as in the multi-particle simulations.",
"The spatial domain extended over ${78.7}{}$ in the direction of laser propagation (discretised over 1020 cells) and ${40}{}$ in each of the transverse directions (discretised over 920 cells).",
"The collision between the laser pulse and electron bunch occurred 64 fs after the laser pulse reached focus.",
"The electron bunch was represented by $1.5\\times 10^7$ macro-particles using third-order particle weighting.",
"The data required to reproduce the PIC simulation results is available in Ref.",
"[45].",
"Indeed, the PIC and the multi-particle QED model yield very similar results confirming that collective effects are negligible in our experimental conditions (see Fig.",
"REF .d).",
"A possible explanation of this residual mismatch shown by the SC and QED models is a limited validity of the constant-cross-field-approximation (CCFA) for our experimental parameters.",
"This approximation is used to calculate the function $g(\\chi )$ in the SC model and the probabilities of photon emission in the QED model.",
"The main assumption is that the photon emission is instantaneous or, equivalently, that the formation time of each emitted photon is much smaller than the time where the laser field changes significantly.",
"This allows one to assume a static electromagnetic field during the photon formation process.",
"In order for the CCFA to be valid, we then need that the typical temporal variation of the laser field is much longer than the photon formation time, a reasonable assumption for ultra-intense fields (dimensionless laser amplitude $a_0$ greatly exceeding 1).",
"However, this condition is not necessarily met in our experimental conditions where a peak dimensionless amplitude of $a_0\\simeq 10$ was reached.",
"The coherence time $\\tau _{COH}$ of the photon in an electric field of magnitude $F_L$ can be estimated as [10]: $\\tau _{COH}\\sim \\frac{F_{cr}}{F_L}\\frac{\\hbar }{mc^2} = \\frac{1}{a_0\\omega _L},$ where $\\omega _L$ is the laser frequency.",
"On the other hand the typical temporal variation of the laser electric field is of the order of a quarter of the laser period, i.e., the time it takes the laser electric field to go from zero to its peak value: $\\tau _{LASER} \\simeq 0.6$ fs.",
"Due to the Gaussian temporal profile of the laser intensity, the electron experiences an increasing intensity during its transit through the laser field, resulting in photon formation lengths that are a significant fraction of the typical timescale over which the electric field oscillates.",
"This fractions are of the same order as $1/a_0$ , which is not negligible through the laser envelope in our experiment.",
"The CCFA used to obtain radiation reaction in the SC model might then not be strictly valid in our experiment.",
"Indeed, assuming the CCFA for a temporally varying electromagnetic field results in overestimating the energy loss of the electron beam [46], as confirmed by the lower electron energy predicted by the SC when compared with our experimental data.",
"This mismatch is even larger if a QED model based on stochastic photon emission is considered since, in this case, also the photon emission probability relies on the CCFA.",
"In this respect, our experiment suggests that stochasticity effects, which are included in the quantum model but not in the semi-classical model, are less important than effects beyond the CCFA.",
"These preliminary results motivate study of high-field quantum electrodynamics beyond the CCFA, an area of theoretical research that has only recently started to be investigated (see, for instance, [47], [46]).",
"We have performed a series of simulations, assuming a semi-classical model of RR, in order to check whether a weaker electron energy loss might be attributed to an unaccounted slight transverse misalignment between the electron beam momenta and the direction of propagation of the scattering laser.",
"As an example, a shot with a weaker energy loss (labelled with c in Fig.",
"REF .a) is well reproduced by the semi-classical calculations if an impact parameter of 5 $\\mu $ m is assumed (see Supplementary Material).",
"However, a full parametric study of the transverse misalignment has not been able to compensate the residual mismatch between theoretical models and experimental data shown in Fig.",
"REF .",
"As a concluding remark, we must further emphasize that additional potential sources of mismatch might be identified in an incomplete knowledge of the local properties of the laser field, such as its phase content and longitudinal distribution of its intensity.",
"For precise QED testing, these are quantities that must be accurately determined in the focus of a high intensity laser, an extremely challenging task currently subject of active research towards the construction of the next generation of ultra-high intensity laser facilities."
],
[
"CONCLUSIONS AND OUTLOOK",
"In conclusion, we report on the experimental detection of strong radiation reaction in an all-optical experiment.",
"The experimental data give clear evidence of significant energy loss ($>30\\%$ ) of ultra-relativistic electrons during their interaction with an ultra-intense laser field.",
"In their own rest frame, the highest energy electrons experience an electric field as high as one quarter of the critical field of quantum electrodynamics.",
"The experimental data is best theoretically modelled by taking into account radiation reaction occurring during the propagation of the electrons through the laser field, and best agreement is found for the semi-classical correction of the Landau-Lifshitz equation.",
"The experiment provides a preliminary indication of the limited validity of the constant-cross-field-approximation for our experimental parameters.",
"In order to precisely determine these effects in this class of experiments, several routes can be followed, including fine characterisation of the local properties of the laser fields, improved spectral and pointing stability of the electron beam, and narrower energy spectra of the primary electron beam."
],
[
"Acknowledgements:",
"G. Sarri and M. Zepf wish to acknowledge support from the Engineering and Physical Sciences Research Council (EPSRC), UK (grant numbers: EP/P010059/1 and EP/N027175/1).",
"CPR, JMC and SPDM acknowledge support from EPSRC (grant numbers: EP/M018156/1 and EP/M018555/1).",
"KP, JMC, EG, SPDM and ZN acknowledge funding from STFC (ST/J002062/1 and ST/P002021/1).",
"AT and KB acknowledge support from the US NSF CAREER Award 1054164 and AT, KB, and KK from the US DOD grant W911NF1610044 and US DOE grant DE-NA0002372.",
"All the authors acknowledge the technical support from the Central Laser Facility."
]
] | 1709.01861 |
[
[
"Clustering of Data with Missing Entries using Non-convex Fusion\n Penalties"
],
[
"Abstract The presence of missing entries in data often creates challenges for pattern recognition algorithms.",
"Traditional algorithms for clustering data assume that all the feature values are known for every data point.",
"We propose a method to cluster data in the presence of missing information.",
"Unlike conventional clustering techniques where every feature is known for each point, our algorithm can handle cases where a few feature values are unknown for every point.",
"For this more challenging problem, we provide theoretical guarantees for clustering using a $\\ell_0$ fusion penalty based optimization problem.",
"Furthermore, we propose an algorithm to solve a relaxation of this problem using saturating non-convex fusion penalties.",
"It is observed that this algorithm produces solutions that degrade gradually with an increase in the fraction of missing feature values.",
"We demonstrate the utility of the proposed method using a simulated dataset, the Wine dataset and also an under-sampled cardiac MRI dataset.",
"It is shown that the proposed method is a promising clustering technique for datasets with large fractions of missing entries."
],
[
"Introduction",
"Clustering is an exploratory data analysis technique that is widely used to discover natural groupings in large datasets, where no labeled or pre-classified samples are available apriori.",
"Specifically, it assigns an object to a group if it is similar to other objects within the group, while being dissimilar to objects in other groups.",
"Example applications include analysis of gene experssion data, image segmentation, identification of lexemes in handwritten text, search result grouping and recommender systems [1].",
"A wide variety of clustering methods have been introduced over the years; see [2] for a review of classical methods.",
"However, there is no consensus on a particular clustering technique that works well for all tasks, and there are pros and cons to most existing algorithms.",
"The common clustering techniques such as k-means [3], k-medians [4] and spectral clustering [5] are implemented using the Lloyd's algorithm which is non-convex and thus sensitive to initialization.",
"Recently, linear programming and semi-definite programming based convex relaxations of the k-means and k-medians algorithms have been introduced [6] to overcome the dependence on initialization.",
"Unlike the Lloyd's algorithm, these relaxations can provide a certificate of optimality.",
"However, all of the above mentioned techniques require apriori knowledge of the desired number of clusters.",
"Hierarchical clustering methods [7], which produce easily interpretable and visualizable clustering results for a varying number of clusters, have been introduced to overcome the above challenge.",
"A drawback of [7] is its sensitivity to initial guess and perturbations in the data.",
"The more recent convex clustering technique (also known as sum-of-norms clustering) [8] retains the advantages of hierarchical clustering, while being invariant to initialization, and producing a unique clustering path.",
"Theoretical guarantees for successful clustering using the convex-clustering technique are also available [9].",
"Most of the above clustering algorithms cannot be directly applied to real-life datasets, where a large fraction of samples are missing.",
"For example, gene expression data often contains missing entries due to image corruption, fabrication errors or contaminants [10], rendering gene cluster analysis difficult.",
"Likewise, large databases used by recommender systems (e.g Netflix) usually have a huge amount of missing data, which makes pattern discovery challenging [11].",
"The presence of missing responses in surveys [12] and failing imaging sensors in astronomy [13] are reported to make the analysis in these applications challenging.",
"Several approaches were introduced to extend clustering to missing-data applications.",
"For example, a partially observed dataset can be converted to a fully observed one using either deletion or imputation [14].",
"Deletion involves removal of variables with missing entries, while imputation tries to estimate the missing values and then performs clustering on the completed dataset.",
"An extension of the weighted sum-of-norms algorithm (originally introduced for fully sampled data [8]) has been proposed where the weights are estimated from the data points by using some imputation techniques on the missing entries [15].",
"Kernel-based methods for clustering have also been extended to deal with missing entries by replacing Euclidean distances with partial distances [16], [17].",
"A majorize minimize algorithm was introduced to solve for the cluster-centres and cluster memberships in [18], which offers proven reduction in cost with iteration.",
"In [19] and [20] the data points are assumed to lie on a mixture of $K$ distributions, where $K$ is known.",
"The algorithms then alternate between the maximum likelihood estimation of the distribution parameters and the missing entries.",
"A challenge with these algorithms is the lack of theoretical guarantees for successful clustering in the presence of missing entries.",
"In contrast, there has been a lot of work in recent years on matrix completion for different data models.",
"Algorithms along with theoretical guarantees have been proposed for low-rank matrix completion [21] and subspace clustering from data with missing entries [22], [23].",
"However, these algorithms and their theoretical guarantees cannot be trivially extended to the problem of clustering in the presence of missing entries.",
"The main focus of this paper is to introduce an algorithm for the clustering of data with missing entries and to theoretically analyze the conditions needed for perfect clustering in the presence of missing data.",
"The proposed algorithm is inspired by the sum-of-norms clustering technique [8]; it is formulated as an optimization problem, where an auxiliary variable assigned to each data point is an estimate of the centre of the cluster to which that point belongs.",
"A fusion penalty is used to enforce equality between many of these auxiliary variables.",
"Since we have experimentally observed that non-convex fusion penalties provide superior clustering performance, we focus on the analysis of clustering using a $\\ell _0$ fusion penalty in the presence of missing entries, for an arbitrary number of clusters.",
"The analysis reveals that perfect clustering is guaranteed with high probability, provided the number of measured entries (probability of sampling) is high enough; the required number of measured entries depends on several parameters including intra-cluster variance and inter-cluster distance.",
"We observe that the required number of entries is critically dependent on coherence, which is a measure of the concentration of inter cluster differences in the feature space.",
"Specifically, if the clustering of the points is determined only by a very small subset of all the available features, then the clustering becomes quite unstable if those particular feature values are unknown for some points.",
"Other factors which influence the clustering technique are the number of features, number of clusters and total number of points.",
"We also extend the theoretical analysis to the case without missing entries.",
"The analysis in this setting shows improved bounds when a uniform random distribution of points in their respective clusters is considered, compared to the worst case analysis considered in the missing-data setting.",
"We expect that improved bounds can also be derived for the case with missing data when a uniform random distribution is considered.",
"We also propose an algorithm to solve a relaxation of the above $\\ell _0$ penalty based clustering problem, using non-convex saturating fusion penalties.",
"The algorithm is demonstrated on a simulated dataset with different fractions of missing entries and cluster separations.",
"We observe that the algorithm is stable with changing fractions of missing entries, and the clustering performance degrades gradually with an increase in the number of missing entries.",
"We also demonstrate the algorithm on clustering of the Wine dataset [24] and reconstruction of a dynamic cardiac MRI dataset from few Fourier measurements.",
"We consider the clustering of points drawn from one of $K$ distinct clusters $C_1, C_2, \\ldots , C_K$ .",
"We denote the center of the clusters by $\\mathbf {c}_1, \\mathbf {c}_2, \\ldots , \\mathbf {c}_K \\in \\mathbb {R}^P$ .",
"For simplicity, we assume that there are $M$ points in each of the clusters.",
"The individual points in the $k^{\\rm th}$ cluster are modelled as: $\\mathbf {z}_k(m) = \\mathbf {c}_k + \\mathbf {n}_{k}(m); ~~m=1,..,M, ~k=1,\\ldots ,K$ Here, $\\mathbf {n}_{k}(m)$ is the noise or the variation of $\\mathbf {z}_k(m)$ from the cluster center $\\mathbf {c}_k$ .",
"The set of input points $\\lbrace \\mathbf {x}_i\\rbrace ,i=1,..,KM$ is obtained as a random permutation of the points $\\lbrace \\mathbf {z}_k(m)\\rbrace $ .",
"The objective of a clustering algorithm is to estimate the cluster labels, denoted by $\\mathcal {C}(\\mathbf {x}_i)$ for $i = 1,..,KM$ .",
"The sum-of-norms (SON) method is a recently proposed convex clustering algorithm [8].",
"Here, a surrogate variable $\\mathbf {u}_i$ is introduced for each point $\\mathbf {x}_i$ , which is an estimate of the centre of the cluster to which $\\mathbf {x}_i$ belongs.",
"As an example, let $K=2$ and $M=5$ .",
"Without loss of generality, let us assume that $\\mathbf {x}_1, \\mathbf {x}_2, \\ldots , \\mathbf {x}_5$ belong to $\\mathcal {C}_1$ and $\\mathbf {x}_6, \\mathbf {x}_7, \\ldots , \\mathbf {x}_{10}$ belong to $\\mathcal {C}_2$ .",
"Then, we expect to arrive at the solution: $\\mathbf {u}_1= \\mathbf {u}_2 = \\ldots = \\mathbf {u}_5= \\mathbf {c}_1$ and $\\mathbf {u}_6= \\mathbf {u}_7 = \\ldots = \\mathbf {u}_{10} = \\mathbf {c}_2$ .",
"In order to find the optimal $\\lbrace \\mathbf {u}_i^*\\rbrace $ , the following optimization problem is solved: $\\lbrace \\mathbf {u}_i^*\\rbrace = \\arg \\min _{\\lbrace \\mathbf {u}_i\\rbrace } \\sum _{i=1}^{KM}\\Vert \\mathbf {x}_i - \\mathbf {u}_i\\Vert _2^2 + \\lambda \\sum _{i=1}^{KM} \\sum _{j=1}^{KM} \\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _{p}$ The fusion penalty ($\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _{p}$ ) can be enforced using different $\\ell _p$ norms, out of which the $\\ell _1$ , $\\ell _2$ and $\\ell _\\infty $ norms have been used in literature [8].",
"The use of sparsity promoting fusion penalties encourages sparse differences $\\mathbf {u}_i-\\mathbf {u}_j$ , which facilitates the clustering of the points $\\lbrace \\mathbf {u}_i\\rbrace $ .",
"For an appropriately chosen $\\lambda $ , the $\\mathbf {u}_i$ 's corresponding to $\\mathbf {x}_i$ 's from the same cluster converge to the same point.",
"The main benefit of this convex scheme over classical clustering algorithms is the convergence of the algorithm to the global minimum.",
"The above optimization problem can be solved efficiently using the Alternating Direction Method of Multipliers (ADMM) algorithm and the Alternating Minimization Algorithm (AMA) [25].",
"Truncated $\\ell _1$ and $\\ell _2$ norms have also been used recently in the fusion penalty, resulting in non-convex optimization problems [26].",
"It has been shown that these penalties provide superior performance to the traditional convex penalties.",
"Convergence to local minimum using an iterative algorithm has also been guaranteed in the non-convex setting.",
"The sum-of-norms algorithm has also been used as a visualization and exploratory tool to discover patterns in datasets [15].",
"Clusterpath diagrams are a common way to visualize the data.",
"This involves plotting the solution path as a function of the regularization parameter $\\lambda $ .",
"For a very small value of $\\lambda $ , the solution is given by: $\\mathbf {u}_i^* = \\mathbf {x}_i$ , i.e.",
"each point forms its individual cluster.",
"For a very large value of $\\lambda $ , the solution is given by: $\\mathbf {u}_i^* = c$ , i.e.",
"every point belongs to the same cluster.",
"For intermediate values of $\\lambda $ , more interesting behaviour is seen as various $\\lbrace \\mathbf {u}_i\\rbrace $ merge and reveal the cluster structure of the data.",
"In this paper, we extend the algorithm to account for missing entries in the data.",
"We present theoretical guarantees for clustering with and without missing entries using an $\\ell _0$ fusion penalty.",
"Next, we approximate the $\\ell _0$ penalty by non-convex saturating penalties, and solve the resulting relaxed optimization problem using an iterative reweighted least squares (IRLS) strategy [27].",
"The proposed algorithm is shown to perform clustering correctly in the presence of large fractions of missing entries.",
"Figure: Central Assumptions: (a) and (b) illustrate different instances where points belonging to ℝ 2 \\mathbb {R}^2 are to be separated into 3 different clusters (denoted by the colours red, green and blue).",
"Assumptions A.1 and A.2 related to cluster separation and cluster size respectively, are illustrated in both (a) and (b).",
"The importance of assumption A.3 related to feature concentration can also be appreciated by comparing (a) and (b).",
"In (a), points in the red and blue clusters cannot be distinguished solely on the basis of feature 1, while the red and green clusters cannot be distinguished solely on the basis of feature 2.",
"Thus, it is difficult to correctly cluster these points if either of the feature values is unknown.",
"In (b), due to low coherence (as assumed in A.3), this problem does not arise."
],
[
"Central Assumptions",
"We make the following assumptions (illustrated in Fig REF ), which are key to the successful clustering of the points: A.1: Cluster separation: Points from different clusters are separated by $\\delta >0$ in the $\\ell _2$ sense, i.e: $\\min _{\\lbrace m,n\\rbrace }\\Vert \\mathbf {z}_{k}(m) -\\mathbf {z}_{l}(n)\\Vert _{2} \\ge \\delta ; ~\\forall \\; k\\ne l$ We require $\\delta >0$ for the clusters to be non-overlapping.",
"A high $\\delta $ corresponds to well separated clusters.",
"A.2: Cluster size: The maximum separation of points within any cluster in the $\\ell _{\\infty }$ sense is $\\epsilon \\ge 0$ , i.e: $\\max _{\\lbrace m,n\\rbrace }\\Vert \\mathbf {z}_{k}(m) - \\mathbf {z}_{k}(n)\\Vert _{\\infty } = \\epsilon ; ~\\forall k=1,\\ldots ,K$ Thus, the $k^{\\rm th}$ cluster is contained within a cube of size $\\epsilon $ , with center $\\mathbf {c}_k$ .",
"A.3: Feature concentration: The coherence of a vector $\\mathbf {y} \\in \\mathbb {R}^P$ is defined as [21]: $\\mu (\\mathbf {y}) = \\frac{P\\Vert \\mathbf {y}\\Vert _{\\infty }^2}{\\Vert \\mathbf {y}\\Vert _2^2}$ By definition: $1 \\le \\mu (\\mathbf {y}) \\le P$ .",
"Intuitively, a vector with a high coherence has a few large values and several small ones.",
"Specifically, if $\\mu (\\mathbf {y}) = P$ , then $\\mathbf {y}$ has only 1 non-zero value.",
"In contrast, if $\\mu (\\mathbf {y}) = 1$ , then all the entries of $\\mathbf {y}$ are equal.",
"We bound the coherence of the difference between points from different clusters as: $\\max _{\\lbrace m,n\\rbrace }\\mu (\\mathbf {z}_k(m) - \\mathbf {z}_l(n)) \\le \\mu _0; ~\\forall \\; k\\ne l$ $\\mu _0$ is indicative of the difficulty of the clustering problem in the presence of missing data.",
"If $\\mu _0=P$ , then two clusters differ only a single feature, suggesting that it is difficult to assign the correct cluster to a point if this feature is not sampled.",
"The best case scenario is $\\mu _0=1$ , when all the features are equally important.",
"In general, cluster recovery from missing data becomes challenging with increasing $\\mu _0$ .",
"The quantity $\\kappa = \\frac{\\epsilon \\sqrt{P}}{\\delta }$ is a measure of the difficulty of the clustering problem.",
"Small values of $\\kappa $ suggest large inter-cluster separation compared to the cluster size; the recovery of such well-defined clusters is expected to be easier than the case with large $\\kappa $ values.",
"Note the $\\ell _2$ norm is used in the definition of $\\delta $ , while the $\\ell _{\\infty }$ norm is used to define $\\epsilon $ .",
"If $\\delta = \\epsilon \\sqrt{P}$ , then $\\kappa = 1$ ; this value of $\\kappa $ is of special importance since $\\kappa < 1$ is a requirement for successful recovery in our main results.",
"We study the problem of clustering the points $\\lbrace \\mathbf {x}_i\\rbrace $ in the presence of entries missing uniformly at random.",
"We arrange the points $\\lbrace \\mathbf {x}_i\\rbrace $ as columns of a matrix $\\mathbf {X}$ .",
"The rows of the matrix are referred to as features.",
"We assume that each entry of $\\mathbf {X}$ is observed with probability $p_0$ .",
"The entries measured in the $i^{th}$ column are denoted by: $\\mathbf {y}_i = \\mathbf {S}_i\\, \\mathbf {x}_i, ~~ i=1,..,KM$ where $\\mathbf {S}_i$ is the sampling matrix, formed by selecting rows of the identity matrix.",
"We consider solving the following optimization problem to obtain the cluster memberships from data with missing entries: $\\begin{split}\\lbrace \\mathbf {u}_i^{*}\\rbrace = & \\min _{\\lbrace \\mathbf {u}_i\\rbrace } \\sum _{i=1}^{KM}\\sum _{j=1}^{KM}\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _{2,0}\\\\ & \\mbox{ s.t } \\Vert \\mathbf {S}_i\\;(\\mathbf {x}_i - \\mathbf {u}_i)\\Vert _\\infty \\le {\\frac{\\epsilon }{2}}, i\\in \\lbrace 1 \\ldots KM\\rbrace \\end{split}$ We use the above constrained formulation rather than the unconstrained formulation in (REF ) to avoid the dependence on $\\lambda $ .",
"The $\\ell _{2,0}$ norm is defined as: $\\Vert \\mathbf {x}\\Vert _{2,0} = {\\left\\lbrace \\begin{array}{ll}0 &, \\text{if $\\Vert \\mathbf {x}\\Vert _2 = 0$}\\\\1 &, \\text{otherwise}\\end{array}\\right.",
"}$ Similar to the SON scheme (REF ), we expect that all $\\mathbf {u}_i$ 's that correspond to $\\mathbf {x}_i$ in the same cluster are equal, while $\\mathbf {u}_i$ 's from different clusters are not equal.",
"We consider the cluster recovery to be successful when there are no mis-classifications.",
"We claim that the above algorithm can successfully recover the clusters with high probability when: The clusters are well separated (i.e, low $\\kappa = \\frac{\\epsilon \\sqrt{P}}{\\delta })$ ).",
"The sampling probability $p_0$ is sufficiently high.",
"The coherence $\\mu _0$ is small.",
"Before moving on to a formal statement and proof of this result, we consider a simple special case to illustrate the approach.",
"In order to aid the reader in following the results, all the important symbols used in the paper have been summarized in Table REF .",
"Table: Notations used"
],
[
"Noiseless Clusters with Missing Entries",
"We consider the simple case where all the points belonging to the same cluster are identical.",
"Thus every cluster is \"noiseless\", and we have: $\\epsilon =0$ and hence $\\kappa = 0$ .",
"The optimization problem (REF ) now reduces to: $\\begin{split}\\lbrace \\mathbf {u}_i^{*}\\rbrace = & \\min _{\\lbrace \\mathbf {u}_i\\rbrace } \\sum _{i=1}^{KM}\\sum _{j=1}^{KM}\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _{2,0}\\\\ & \\mbox{ s.t } \\mathbf {S}_i\\,\\mathbf {x}_i = \\mathbf {S}_i\\, \\mathbf {u}_i, i\\in \\lbrace 1 \\ldots KM\\rbrace \\end{split}$ Next, we state a few results for this special case in order to provide some intuition about the problem.",
"The results are not stated with mathematical rigour and are not accompanied by proofs.",
"In the next sub-section, when we consider the general case, we will provide lemmas and theorems (with proofs in the appendix), which generalize the results stated here.",
"Specifically, Lemmas REF , REF , REF and Theorem REF generalize Results REF , REF , REF and REF respectively.",
"We will first consider the data consistency constraint in (REF ) and determine possible feasible solutions.",
"We observe that all the points in any specified cluster can share a centre without violating the data consistency constraint: Result 2.1 Consider any two points $\\mathbf {x}_1 $ and $\\mathbf {x}_2 $ from the same cluster.",
"A solution $\\mathbf {u}$ exists for the following equations: $\\mathbf {S}_i\\,\\mathbf {x}_i &=& \\mathbf {S}_i\\, \\mathbf {u}; ~~i=1,2$ with probability 1.",
"The proof for the above result is trivial in this special case, since all points in the same cluster are the same.",
"We now consider two points from different clusters.",
"Result 2.2 Consider two points $\\mathbf {x}_1 $ and $\\mathbf {x}_2 $ from different clusters.",
"A solution $\\mathbf {u}$ exists for the following equations: $\\mathbf {S}_i\\,\\mathbf {x}_i = \\mathbf {S}_i\\, \\mathbf {u}; ~~i=1,2$ with low probability, when the sampling probability $p_0$ is high and coherence $\\mu _0$ is low.",
"By definition, $\\mathbf {S}_1 = \\mathbf {S}_{\\mathcal {I}_1}$ and $\\mathbf {S}_2 = \\mathbf {S}_{\\mathcal {I}_2}$ , where $\\mathcal {I}_1$ and $\\mathcal {I}_2$ are the index sets of the features that are sampled (not missing) in $\\mathbf {x}_1$ and $\\mathbf {x}_2$ respectively.",
"We observe that (REF ) can be satisfied, iff: $\\mathbf {S}_{\\mathcal {I}_1 \\cap \\mathcal {I}_2} (\\mathbf {x}_1 - \\mathbf {x}_2) = \\mathbf {0}$ which implies that the features of $\\mathbf {x}_1$ and $\\mathbf {x}_2$ are the same on the index set $\\mathcal {I}_1 \\cap \\mathcal {I}_2$ .",
"If the probability of sampling $p_0$ is sufficiently high, then the number of samples at commonly observed locations: $|\\mathcal {I}_1 \\cap \\mathcal {I}_2| = q$ will be high, with high probability.",
"If the coherence $\\mu _0$ defined in assumption A3 is low, then with high probability the vector $\\mathbf {x}_1 - \\mathbf {x}_2$ does not have $q$ entries that are equal to 0.",
"In other words, the cluster memberships are not determined by only a few features.",
"Thus, for a small value of $\\mu _0$ and high $p_0$ , we can ensure that (REF ) occurs with very low probability.",
"We now generalize the above result to obtain the following: Result 2.3 Assume that $\\lbrace \\mathbf {x}_i: i\\in \\mathcal {I}, |\\mathcal {I}|= M\\rbrace $ is a set of points chosen randomly from multiple clusters (not all are from the same cluster).",
"A solution $\\mathbf {u}$ exists for the following equations: $\\mathbf {S}_i\\,\\mathbf {x}_i = \\mathbf {S}_i\\, \\mathbf {u}; ~\\forall i \\in \\mathcal {I}$ with low probability, when the sampling probability $p_0$ is high and coherence $\\mu _0$ is low.",
"The key message of the above result is that large mis-classified clusters are highly unlikely.",
"We will show that all feasible solutions containing small mis-classified clusters are associated with higher cost than the correct solution.",
"Thus, we can conclude that the algorithm recovers the ground truth solution with high probability, as summarized by the following result.",
"Result 2.4 The optimization problem (REF ) results in the ground-truth clustering with a high probability if the sampling probability $p_0$ is high and the coherence $\\mu _0$ is low."
],
[
"Noisy Clusters with Missing Entries",
"We will now consider the general case of noisy clusters with missing entries, and will determine the conditions required for (REF ) to yield successful recovery of clusters.",
"The reasoning behind the proof in the general case is similar to that for the special case discussed in the previous sub-section.",
"Before proceeding to the statement of the lemmas and theorems, we define the following quantities: Upper bound for probability that two points have less than $\\frac{p_0^2P}{2}$ commonly observed locations: $\\gamma _0 (\\frac{e}{2})^{-\\frac{p_0^2P}{2}}$ Given that two points from different clusters have more than $\\frac{p_0^2P}{2}$ commonly observed locations, upper bound for probability that they can yield the same $\\mathbf {u}$ without violating the constraints in (REF ): $\\delta _0 e^{-\\frac{p_0^2P(1-\\kappa ^2)^2}{\\mu _0^2}}$ Upper bound for probability that two points from different clusters can yield the same $\\mathbf {u}$ without violating the constraints in (REF ): $\\beta _0 1-(1-\\delta _0)(1-\\gamma _0)$ Upper bound for failure probability of (REF ): $\\eta _0 \\sum _{\\lbrace m_j\\rbrace \\in \\mathcal {S}}\\left[\\beta _0^{\\frac{1}{2}(M^2-\\sum _j {m_j^2})} \\prod _j {M \\atopwithdelims ()m_j}\\right]$ where $\\mathcal {S}$ is the set of all sets of positive integers $\\lbrace m_j\\rbrace $ such that: $2 \\le \\mathcal {U}(\\lbrace m_j\\rbrace ) \\le K$ and $\\sum _j m_j = M$ .",
"Here, the function $\\mathcal {U}$ counts the number of non-zero elements in a set.",
"For example, if $K=2$ then $\\mathcal {S}$ contains all sets of 2 positive integers $\\lbrace m_1, m_2\\rbrace $ , such that $m_1 + m_2 = M$ .",
"Thus, $\\mathcal {S} = \\lbrace \\lbrace 1,M-1\\rbrace , \\lbrace 2,M-2\\rbrace , \\lbrace 3,M-3\\rbrace , \\ldots , \\lbrace M-1,1\\rbrace \\rbrace $ and (REF ) reduces to: $\\eta _0 = \\sum _{i=1}^{M-1}\\left[\\beta _0^{i(M-i)} {M \\atopwithdelims ()i}^2\\right]$ We note that the expression for $\\eta _0$ is quite involved.",
"Hence, to provide some intuition, we simplify this expression for the special case where there are only two clusters.",
"Under the assumption that $\\log \\beta _0 \\le \\frac{1}{M-1} + \\frac{2}{M-2}\\log \\frac{1}{M-1}$ , it can be shown that $\\eta _0$ is upper-bounded as: $\\begin{split}\\eta _0 & = \\sum _{i=1}^{M-1}\\left[\\beta _0^{i(M-i)} {M \\atopwithdelims ()i}^2\\right]\\\\ &\\le M^3 \\beta _0^{M-1} \\\\ &\\eta _{0,{\\rm approx}}\\end{split}$ The above upper bound is derived in Appendix .",
"We now state the results for clustering with missing entries in the general noisy case.",
"The following two lemmas are generalizations of Results REF and REF to the noisy case.",
"Lemma 2.1 Consider any two points $\\mathbf {x}_1 $ and $\\mathbf {x}_2 $ from the same cluster.",
"A solution $\\mathbf {u}$ exists for the following equations: $\\Vert \\mathbf {S}_i\\,(\\mathbf {x}_i-\\mathbf {u})\\Vert _{\\infty } &\\le & {\\frac{\\epsilon }{2}}; ~ ~i=1,2$ with probability 1.",
"The proof of this lemma is in Appendix .",
"Lemma 2.2 Consider any two points $\\mathbf {x}_1$ and $\\mathbf {x}_2$ from different clusters, and assume that $\\kappa <1$ .",
"A solution $\\mathbf {u}$ exists for the following equations: $\\Vert \\mathbf {S}_i\\,(\\mathbf {x}_i-\\mathbf {u})\\Vert _{\\infty } &\\le & {\\frac{\\epsilon }{2}}; ~ ~i=1,2$ with probability less than $\\beta _0$ .",
"The proof of this lemma is in Appendix .",
"We note that $\\beta _0$ decreases with a decrease in $\\kappa $ .",
"A small $\\epsilon $ implies less variability within clusters and a large $\\delta $ implies well-separated clusters, together resulting in a low value of $\\kappa $ .",
"Both these characteristics are desirable for clustering and result in a low value of $\\beta _0$ .",
"This lemma also demonstrates that the coherence assumption is important in ensuring that the sampled entries are sufficient to distinguish between a pair of points from different clusters.",
"As a result, $\\beta _0$ decreases with a decrease in the value of $\\mu _0$ .",
"As expected, we also observe that $\\beta _0$ decreases with an increase in $p_0$ .",
"The above result can be generalized to consider a large number of points from multiple clusters.",
"If we choose $M$ points such that not all of them belong to the same cluster, then it can be shown that with high probability, they cannot share the same $\\mathbf {u}$ without violating the constraints in (REF ).",
"This idea (a generalization of Result REF ) is expressed in the following lemma: Lemma 2.3 Assume that $\\lbrace \\mathbf {x}_i: i\\in \\mathcal {I}, |\\mathcal {I}|= M\\rbrace $ is a set of points chosen randomly from multiple clusters (not all are from the same cluster).",
"If $\\kappa <1$ , a solution $\\mathbf {u}$ does not exist for the following equations: $\\Vert \\mathbf {S}_i\\,(\\mathbf {x}_i-\\mathbf {u})\\Vert _{\\infty } \\le {\\frac{\\epsilon }{2}}; ~ ~\\forall i \\in \\mathcal {I}$ with probability exceeding $1 - \\eta _0$ .",
"The proof of this lemma is in Appendix .",
"We note here, that for a low value of $\\beta _0$ and a high value of $M$ (number of points in each cluster), we will arrive at a very low value of $\\eta _0$ .",
"Using Lemmas REF , REF and REF , we now move on to our main result which is a generalization of Result REF : Theorem 2.4 If $\\kappa <1$ , the solution to the optimization problem (REF ) is identical to the ground-truth clustering with probability exceeding $1 - \\eta _0$ .",
"The proof of the above theorem is in Appendix .",
"The reasoning follows from Lemma REF .",
"It is shown in the proof that all solutions with cluster sizes smaller than $M$ are associated with a higher cost than the ground-truth solution."
],
[
"Clusters without Missing Entries",
"We now study the case where there are no missing entries.",
"In this special case, optimization problem (REF ) reduces to: $\\begin{split}\\lbrace \\mathbf {u}_i^{*}\\rbrace = & \\min _{\\lbrace \\mathbf {u}_i\\rbrace } \\sum _{i=1}^{KM}\\sum _{j=1}^{KM}\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _{2,0}\\\\ & \\mbox{ s.t } \\Vert \\mathbf {x}_i - \\mathbf {u}_i\\Vert _{\\infty } \\le \\frac{\\epsilon }{2}, ~i\\in \\lbrace 1 \\ldots KM\\rbrace \\end{split}$ We have the following theorem guaranteeing successful recovery for clusters without missing entries: Theorem 2.5 If $\\kappa <1$ , the solution to the optimization problem (REF ) is identical to the ground-truth clustering.",
"The proof for the above Theorem is in Appendix .",
"We note that the above result does not consider any particular distribution of the points in each cluster.",
"Instead, if we consider that the points in each cluster are sampled from certain particular probability distributions such as the uniform random distribution, then a larger $\\kappa $ is sufficient to ensure success with high probability.",
"In the general case where no such distribution is assumed, we cannot make a probabilistic argument, and a smaller $\\kappa $ is required.",
"We now consider a special case, where the noise $\\mathbf {n}_k(m)$ is a zero mean uniform random variable $\\sim U(-\\epsilon /2,\\epsilon /2)$ .",
"Thus, the points within each cluster are uniformly distributed in a cube of side $\\epsilon $ .",
"We note that $\\delta $ is now a random variable, and thus instead of using the constant $\\kappa = \\frac{\\epsilon \\sqrt{P}}{\\delta }$ (as in previous lemmas), we define the following constant: $\\kappa ^{\\prime }=\\frac{\\epsilon \\sqrt{P}}{c}$ where $c$ is defined as the minimum separation between the centres of any 2 clusters in the dataset: $\\min _{\\lbrace k,l\\rbrace }\\Vert \\mathbf {c}_{k} -\\mathbf {c}_{l}\\Vert _{2} \\ge c; ~\\forall \\; k\\ne l$ We also define the following quantity: $\\beta _1=e^{-\\frac{P(1-\\frac{5}{6}\\kappa ^{\\prime 2})^2}{8 \\kappa ^{\\prime 2}}}$ We arrive at the following result for two points in different clusters: Lemma 2.6 Let $\\kappa ^{\\prime } < \\sqrt{\\frac{6}{5}}$ .",
"If the points in each cluster follow a uniform random distribution, then for two points $\\mathbf {x}_1$ and $\\mathbf {x}_2$ belonging to different clusters, a solution $\\mathbf {u}$ exists for the following equations: $\\Vert \\mathbf {x}_i-\\mathbf {u}\\Vert _{\\infty } &\\le & {\\frac{\\epsilon }{2}}; ~ ~i=1,2$ with probability less than $\\beta _1$ .",
"The proof for the above lemma is in Appendix .",
"This implies that for $\\kappa ^{\\prime } <\\sqrt{\\frac{6}{5}}$ , two points from different clusters cannot be misclassified to a single cluster with high probability.",
"As $\\eta _0$ is expressed in terms of $\\beta _0$ in (REF ), we can also express $\\eta _1$ in terms of $\\beta _1$ .",
"We get the following guarantee for perfect clustering: Theorem 2.7 If the points in each cluster follow a uniform random distribution and $\\kappa ^{\\prime } < \\sqrt{\\frac{6}{5}}$ , then the solution to the optimization problem (REF ) is identical to the ground-truth clustering with probability exceeding $1 - \\eta _1$ .",
"Note that $\\kappa = \\kappa ^{\\prime }\\frac{c}{\\delta }$ .",
"Thus, the above result allows for values $\\kappa >1$ .",
"Our results show that if we do not consider the distribution of the points, then we arrive at the bound $\\kappa < 1$ with and without missing entries, as seen from Theorems REF and REF respectively.",
"A uniform random distribution can also be assumed in the case of missing entries.",
"Similar to Theorem REF , we expect an improved bound for the case with missing entries as well."
],
[
"Constrained formulation",
"We propose to solve a relaxation of the optimization problem (REF ), which is more computationally feasible.",
"The relaxed problem is given by: $\\begin{split}\\lbrace \\mathbf {u}_i^{*}\\rbrace = & \\min _{\\lbrace \\mathbf {u}_i\\rbrace } \\sum _{i=1}^{KM}\\sum _{j=1}^{KM}\\phi \\left(\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _2\\right)\\\\ & \\mbox{ s.t } \\Vert \\mathbf {S}_i(\\mathbf {x}_i - \\mathbf {u}_i)\\Vert _\\infty \\le {\\frac{\\epsilon }{2}}, i\\in \\lbrace 1 \\ldots KM\\rbrace \\end{split}$ where $\\phi $ is a function approximating the $\\ell _{0}$ norm.",
"Some examples of such functions are: $\\ell _p$ norm: $\\phi (x) = |x|^p$ , for some $0<p<1$ .",
"$H_1$ penalty: $\\phi (x) = 1-e^{-\\frac{x^2}{2\\sigma ^2}}$ .",
"These functions approximate the $\\ell _0$ penalty more accurately for lower values of $p$ and $\\sigma $ , as illustrated in Fig REF .",
"We reformulate the problem using a majorize-minimize strategy.",
"Specifically, by majorizing the penalty $\\phi $ using a quadratic surrogate functional, we obtain: Figure: Different penalty functions φ\\phi .",
"(a) The ℓ 0 \\ell _0 norm (b) The ℓ p \\ell _p penalty function which is non-convex for 0<p<10<p<1 and convex for p=1p=1 (c) The H 1 H_1 penalty function.",
"The ℓ p \\ell _p and H 1 H_1 penalties closely approximate the ℓ 0 \\ell _0 norm for low values of pp and σ\\sigma respectively.$\\phi (x) \\le w(x) x^2 + d$ where $w(x) = \\frac{\\phi ^{^{\\prime }}(x)}{2x}$ , and $d$ is a constant.",
"For the two penalties considered here, we obtain the weights as: $\\ell _p$ norm: $w(x) = (\\frac{2}{p}x^{(2-p)}+\\alpha )^{-1} $ .",
"The infinitesimally small $\\alpha $ term is introduced to deal with situations where $x=0$ .",
"For non-zero $x$ , we get the expression $w(x) \\approx \\frac{p}{2}x^{p-2}$ .",
"$H_1$ penalty: $w(x) = \\frac{1}{2\\sigma ^2}e^{-\\frac{x^2}{2\\sigma ^2}}$ .",
"We can now state the majorize-minimize formulation for problem (REF ) as: $\\begin{split}\\lbrace \\mathbf {u}_i^*, w_{ij}^*\\rbrace = &\\arg \\min _{\\lbrace \\mathbf {u}_i,w_{ij}\\rbrace } \\sum _{i=1}^{KM} \\sum _{j=1}^{KM} w_{ij}~\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _2^2\\\\ &\\mbox{ s.t } \\Vert \\mathbf {S}_i(\\mathbf {x}_i - \\mathbf {u}_i)\\Vert _\\infty \\le \\;{\\frac{\\epsilon }{2}}, i\\in \\lbrace 1 \\ldots KM\\rbrace \\end{split}$ where the constant $d$ has been ignored.",
"In order to solve problem (REF ), we alternate between two sub-problems till convergence.",
"At the $n^{th}$ iteration, these sub-problems are given by: $w_{ij}^{(n)} = \\frac{\\phi ^{^{\\prime }}\\left(\\Vert \\mathbf {u}_i^{(n-1)} - \\mathbf {u}_j^{(n-1)}\\Vert _2\\right)}{2\\Vert \\mathbf {u}_i^{(n-1)} - \\mathbf {u}_j^{(n-1)}\\Vert _2}$ $\\begin{split}\\lbrace \\mathbf {u}_i^{(n)}\\rbrace = & \\arg \\min _{\\lbrace \\mathbf {u}_i\\rbrace } \\sum _{i=1}^{KM} \\sum _{j=1}^{KM} w_{ij}^{(n)}\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _2^2\\\\ &\\mbox{ s.t } \\Vert \\mathbf {S}_i(\\mathbf {x}_i - \\mathbf {u}_i)\\Vert _\\infty \\le \\;{\\frac{\\epsilon }{2}}, i\\in \\lbrace 1 \\ldots KM\\rbrace \\end{split}$ Figure: Comparison of different penalties.",
"We show here the 2 most significant principal components of the solutions obtained using the IRLS algorithm.",
"(a) It can be seen that the ℓ 1 \\ell _{1} penalty is unable to cluster the points even though the clusters are well-separated.",
"(b) The ℓ 0.1 \\ell _{0.1} penalty is able to cluster the points correctly.",
"However, the cluster-centres are not correctly estimated.",
"(c) The H 1 H_1 penalty correctly clusters the points and also gives a good estimate of the centres."
],
[
"Unconstrained formulation",
"For larger datasets, it might be computationally intensive to solve the constrained problem.",
"In this case, we propose to solve the following unconstrained problem: $\\lbrace \\mathbf {u}_i^*\\rbrace = \\arg \\min _{\\lbrace \\mathbf {u}_i\\rbrace } \\sum _{i=1}^{KM}\\Vert \\mathbf {S}_i(\\mathbf {u}_i - \\mathbf {x}_i)\\Vert _2^2 + \\lambda \\sum _{i=1}^{KM} \\sum _{j=1}^{KM} \\phi (\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _2)$ As before, we can state the majorize-minimize formulation for problem (REF ) as: $\\begin{split}\\lbrace \\mathbf {u}_i^*, w_{ij}^*\\rbrace = \\arg \\min _{\\lbrace \\mathbf {u}_i,w_{ij}\\rbrace }\\sum _{i=1}^{KM}\\Vert &\\mathbf {S}_i(\\mathbf {u}_i - \\mathbf {x}_i)\\Vert _2^2\\\\ + & \\lambda \\sum _{i=1}^{KM}\\sum _{j=1}^{KM} w_{ij}\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _2^2\\end{split}$ In order to solve the problem (REF ), we alternate between two sub-problems till convergence.",
"The 1st sub-problem is the same as (REF ).",
"The 2nd sub-problem is given by: $\\begin{split}\\lbrace \\mathbf {u}_i^{(n)}\\rbrace = \\arg \\min _{\\lbrace \\mathbf {u}_i\\rbrace } \\sum _{i=1}^{KM}\\Vert &\\mathbf {S}_i(\\mathbf {u}_i - \\mathbf {x}_i)\\Vert _2^2\\\\ + &\\lambda \\sum _{i=1}^{KM} \\sum _{j=1}^{KM} w_{ij}^{(n)}\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _2^2\\end{split}$"
],
[
"Comparison of penalties",
"We compare the performance of different penalties when used as a surrogate for the $\\ell _0$ norm.",
"For this purpose, we use a simulated dataset with points in $\\mathbb {R}^{50}$ belonging to 3 well-separated clusters, with 200 points in each cluster.",
"For this particular experiment, we considered $\\mathbf {x}_1, \\mathbf {x}_2,\\ldots , \\mathbf {x}_{200} \\in \\mathcal {C}_1$ , $\\mathbf {x}_{201}, \\mathbf {x}_{202},\\ldots , \\mathbf {x}_{400} \\in \\mathcal {C}_2$ and $\\mathbf {x}_{401}, \\mathbf {x}_{402},\\ldots , \\mathbf {x}_{600} \\in \\mathcal {C}_3$ .",
"We do not consider the presence of missing entries for this experiment.",
"We solve problem (REF ) to cluster the points using the $\\ell _{1}$ , $\\ell _{p}$ (for $p=0.1$ ) and $H_1$ (for $\\sigma =0.5$ ) penalties.",
"The results are shown in Fig REF .",
"Only for the purpose of visualization, we take a PCA of the data matrix $\\mathbf {X} \\in \\mathbb {R}^{50\\times 600}$ and retain the 2 most significant principal components to get a matrix of points $\\in \\mathbb {R}^{2\\times 600}$ .",
"These points are plotted in the figure, with red, blue and green representing points from different clusters.",
"We similarly obtain the 2 most significant components of the estimated centres and plot the resulting points in black.",
"In (b) and (c), we note that $\\mathbf {u}_1^* = \\mathbf {u}_2^* = \\ldots = \\mathbf {u}_{200}^*$ , $\\mathbf {u}_{201}^*= \\mathbf {u}_{202}^*=\\ldots = \\mathbf {u}_{400}^*$ and $\\mathbf {u}_{401}^*=\\mathbf {u}_{402}^*=\\ldots = \\mathbf {u}_{600}^*$ .",
"Thus, the $\\ell _p$ penalty and the $H_1$ penalty are able to correctly cluster the points.",
"This behaviour is not seen in (a).",
"Thus it is concluded that the convex $\\ell _{1}$ penalty is unable to cluster the points.",
"The cluster-centres estimated using the $\\ell _{p}$ penalty are inaccurate.",
"The $H_1$ penalty out-performs the other two penalties and accurately estimates the cluster-centres.",
"We can explain this behaviour intuitively by observing the plots in Fig REF .",
"The $\\ell _{1}$ norm penalizes differences between all pairs of points.",
"The $\\ell _{0.1}$ semi-norm penalizes differences between points that are close.",
"Due to the saturating nature of the penalty, it does not heavily penalize differences between points that are further away.",
"The same is true for the $H_1$ penalty.",
"However, we note that the $H_1$ penalty saturates to 1 very quickly, similar to the $\\ell _0$ norm.",
"This behaviour is missing for the $\\ell _{0.1}$ penalty.",
"For this reason, it is seen that the $\\ell _{0.1}$ penalty also penalizes inter-cluster distances (unlike the $H_1$ penalty), and shrinks the distance between the estimated centres of different clusters.",
"Figure: Study of Theoretical Guarantees.",
"The quantities γ 0 ,δ 0 \\gamma _0, \\delta _0 and β 0 \\beta _0 defined in Section are studied in (a), (b) and (c) respectively.",
"In (b) and (c), P=50P=50 and μ 0 =1.5\\mu _0=1.5 are assumed.",
"β 0 \\beta _0 gives the probability that 2 points from different clusters can share a centre.",
"As expected, this value decreases with increase in p 0 p_0 and decrease in κ\\kappa .",
"Considering K=2K=2 clusters, a lower bound for the probability of successful clustering (1-η 0 )(1-\\eta _0) using the proposed algorithm is shown in (d) for different values of κ\\kappa .",
"The approximate values (1-η 0, approx )(1-\\eta _{0,{\\rm approx}}) computed using () are shown in (e).Figure: Experimental results for probability of success.",
"Guarantees are shown for a simulated dataset with K=2K=2 clusters.",
"The clustering was performed using () with an H 1 H_1 penalty and partial distance based initialization.",
"For (a) and (b) it is assumed that κ=0.39\\kappa = 0.39 and μ 0 =2.3\\mu _0 = 2.3.",
"(a) shows the experimentally obtained probability of success of clustering for clusters with points from a uniform random distribution.",
"(b) shows the theoretical lower bound for the probability of success.",
"(c) shows the experimentally obtained probability of success for a more challenging dataset with κ=1.15\\kappa = 1.15 and μ 0 =13.2\\mu _0 = 13.2.",
"Note that we do not have theoretical guarantees for this case, since our analysis assumes that κ<1\\kappa < 1."
],
[
"Initialization Strategies",
"Our experiments emphasize the need for a good initialization of the weights $w_{ij}$ for convergence to the correct cluster centre estimates.",
"This dependence on the initial value arises from the non-convexity of the optimization problem.",
"We consider two different strategies for initializing the weights: Partial Distances: Consider a pair of points $\\mathbf {x}_1, \\mathbf {x}_2$ observed by sampling matrices $\\mathbf {S}_1 = \\mathbf {S}_{\\mathcal {I}_1}$ and $\\mathbf {S}_2 = \\mathbf {S}_{\\mathcal {I}_2}$ respectively.",
"Let the set of common indices be $\\omega \\mathcal {I}_1 \\cap \\mathcal {I}_2$ .",
"We define the partial distance as $\\Vert \\mathbf {y}_\\omega \\Vert = \\sqrt{\\frac{P}{|\\omega |}}\\Vert \\mathbf {x}_{1\\omega } - \\mathbf {x}_{2\\omega }\\Vert $ , where $\\mathbf {x}_{i\\omega }$ represents the set of entries of $\\mathbf {x}_i$ restricted to the index set $\\omega $ .",
"Instead of the actual distances which are not available, the partial distances $\\Vert \\mathbf {y}_\\omega \\Vert $ can be used for computing the weights.",
"Imputation Methods: The weights can be computed from estimates $\\lbrace \\mathbf {u}_i^{(0)}\\rbrace $ , where: $\\mathbf {u}_i^{(0)} = \\mathbf {S}_i \\mathbf {x}_i + (\\mathbf {I}- \\mathbf {S}_i) \\mathbf {m}$ Here $\\mathbf {m}$ is a constant vector, specific to the imputation technique.",
"The zero-filling technique corresponds to $\\mathbf {m} = \\mathbf {0}$ .",
"Better estimation techniques can be derived where the $j^{th}$ row of $\\mathbf {m}$ can be set to the mean of all measured values in the $j^{th}$ row of $\\mathbf {X}$ .",
"We will observe experimentally that for a good approximation of the initial weights $\\mathbf {W}^{(0)}$ , we get the correct clustering.",
"Conversely, the clustering fails for a bad initial guess.",
"Our experiments demonstrate the superiority of a partial distance based initialization strategy over a zero-filled initialization.",
"We study the proposed theoretical guarantees for Theorem REF for different settings.",
"We also test the proposed algorithm on simulated and real datasets.",
"The simulations are used to study the performance of the algorithm with change in parameters such as fraction of missing entries, number of points to be clustered etc.",
"We also study the effect of different initialization techniques on the algorithm performance.",
"We demonstrate the algorithm on the publicly available Wine dataset [24], and use the algorithm to reconstruct a dataset of under-sampled cardiac MR images."
],
[
"Study of Theoretical Guarantees",
"We observe the behaviour of the quantities $\\gamma _0, \\delta _0, \\beta _0, \\eta _0$ and $\\eta _{0,{\\rm approx}}$ (defined in section REF ) as a function of parameters $p_0, P, \\kappa $ and $M$ .",
"Fig REF shows a few plots that illustrate the change in these quantities as the different parameters are varied.",
"$\\gamma _0$ is an upper bound for the probability that a pair of points have $< \\frac{p_0^2P}{2}$ entries observed at common locations.",
"In Fig REF (a), the change in $\\gamma _0$ is shown as a function of $p_0$ for different values of $P$ .",
"In subsequent plots, we fix $P=50$ and $\\mu _0=1.5$ .",
"$\\delta _0$ is an upper bound for the probability that a pair of points from different clusters can share a common centre, given that $\\ge \\frac{p_0^2P}{2}$ entries are observed at common locations.",
"In Fig REF (b), the change in $\\delta _0$ is shown as a function of $p_0$ for different values of $\\kappa $ .",
"In Fig REF (c), the behaviour of $\\beta _0 = 1-(1-\\gamma _0)(1- \\delta _0)$ is shown, which is the probability mentioned in Lemma REF .",
"We consider the two cluster setting, (i.e.",
"$K=2$ ) for subsequent plots.",
"$\\eta _0$ is the probability of failure of the clustering algorithm (REF ).",
"In (d) and (e), plots are shown for $(1-\\eta _0)$ and $(1-\\eta _{0,{\\rm approx}})$ as a function of $p_0$ for different values of $\\kappa $ and $M$ .",
"Here, $\\eta _{0,{\\rm approx}}$ is an upper bound for $\\eta _0$ computed using (REF ).",
"As expected, the probability of success of the clustering algorithm increases with increase in $p_0$ and $M$ and decrease in $\\kappa $ ."
],
[
"Clustering of Simulated Data",
"We simulated datasets with $K=2$ disjoint clusters in $\\mathbb {R}^{50}$ with a varying number of points per cluster ($M=6,12,25,50,100$ ).",
"The points in each cluster follow a uniform random distribution.",
"We study the probability of success of the $H_1$ penalty based constrained clustering algorithm (with partial-distance based initialization) as a function of $\\kappa $ , $M$ and $p_0$ .",
"For a particular set of parameters the experiment was conducted 20 times to compute the probability of success of the algorithm.",
"Between these 20 trials, the cluster-centers remain the same, while the points sampled from these clusters are different and the locations of the missing entries are different.",
"Fig REF (a) shows the result for datasets with $\\kappa = 0.39$ and $\\mu _0 = 2.3$ .",
"The theoretical guarantees for successfully clustering the dataset are shown in (b).",
"Note that the theoretical guarantees do not assume that the points are taken from a uniform random distribution.",
"Also, the theoretical bounds assume that we are solving the original problem using a $\\ell _0$ norm, whereas the experimental results were generated for the $H_1$ penalty.",
"Our theoretical guarantees hold for $\\kappa < 1$ .",
"However, we demonstrate in (c) that even for the more challenging case where $\\kappa = 1.15$ and $\\mu _0 = 13.2$ , our clustering algorithm is successful.",
"Note that we do not have theoretical guarantees for this case.",
"However, by assuming a uniform random distribution on the points, we expect that we can get better theoretical guarantees (similar to Theorem REF for the case without missing entries).",
"Clustering results with $K=3$ simulated clusters are shown in Fig REF .",
"We simulated Dataset-1 with $K=3$ disjoint clusters in $\\mathbb {R}^{50}$ and $M=200$ points in each cluster.",
"In order to generate this dataset, 3 cluster centres in $\\mathbb {R}^{50}$ were chosen from a uniform random distribution.",
"The distances between the 3 pairs of cluster-centres are $3.5$ , $2.8$ and $3.3$ units respectively.",
"For each of these 3 cluster centres, 200 noisy instances were generated by adding zero-mean white Gaussian noise of variance 0.1.",
"The dataset was sub-sampled with varying fractions of missing entries ($p_0=1,0.9,0.8,\\ldots ,0.3,0.2$ ).",
"The locations of the missing entries were chosen uniformly at random from the full data matrix.",
"We also generate Dataset-2 by halving the distance between the cluster centres, while keeping the intra-cluster variance fixed.",
"We test both the constrained (REF ) and unconstrained (REF ) formulations of our algorithm on these datasets.",
"Both the proposed initialization techniques for the IRLS algorithm (i.e.",
"zero-filling and partial-distance) are also tested here.",
"Since the points lie in $\\mathbb {R}^{50}$ , we take a PCA of the points and their estimated centres (similar to Fig REF ) and plot the 2 most significant components.",
"The 3 colours distinguish the points according to their ground-truth clusters.",
"Each point $\\mathbf {x}_i$ is joined to its centre estimate $\\mathbf {u}_i^*$ by a line.",
"As expected, we observe that the clustering algorithms are more stable with fewer missing entries.",
"We also note that the results are quite sensitive to the initialization technique.",
"We observe that the partial distance based initialization technique out-performs the zero-filled initialization.",
"The unconstrained algorithm with partial distance-based initialization shows superior performance to the alternative schemes.",
"Thus, we use this scheme for subsequent experiments on real datasets."
],
[
"Clustering of Wine Dataset",
"We apply the clustering algorithm to the Wine dataset [24].",
"The data consists of the results of a chemical analysis of wines from 3 different cultivars.",
"Each data point has $P=13$ features.",
"The 3 clusters have 59, 71 and 48 points respectively, resulting in a total of 178 data points.",
"We created a dataset without outliers by retaining only $M=40$ points per cluster, resulting in a total of 120 data points.",
"We under-sampled these datasets using uniform random sampling with different fractions of missing entries.",
"The results are displayed in Fig REF using the PCA technique as explained in the previous sub-section.",
"It is seen that the clustering is quite stable and degrades gradually with increasing fractions of missing entries."
],
[
"Cardiac MR Image Reconstruction",
"We apply the proposed algorithm to the reconstruction of a cardiac MR image time series.",
"In MRI, samples are collected in the Fourier domain.",
"However, due to the slow nature of the acquisition, only a small fraction of the Fourier samples can be acquired in each time frame.",
"The goal of image reconstruction is to recover the image series from the incomplete Fourier observations.",
"In the case of cardiac MRI, the different images in the time series appear in clusters determined by the cardiac and respiratory phase.",
"Thus, the proposed algorithm can be applied to the image reconstruction problem.",
"The cardiac data was acquired on a Siemens Aera MRI scanner at the University of Iowa.",
"The subject was asked to breathe freely, and 10 radial lines of Fourier data was acquired to reconstruct each image frame.",
"Fourier data corresponding to 1000 frames was acquired and the image series was reconstructed using the proposed unconstrained algorithm.",
"We performed spectral clustering [5] on the reconstructed images to form 20 clusters.",
"A few reconstructed frames belonging to 2 different clusters are illustrated in Fig REF .",
"The images displayed have minimal artefacts and are of diagnostic quality.",
"Figure: Clustering results in simulated datasets.",
"The H 1 H_1 penalty is used to cluster two datasets with varying fractions of missing entries.",
"Both the constrained and unconstrained formulation results are presented with different initialization techniques (zero-filled and partial-distance based).",
"We show here the 2 most significant principal components of the solutions.",
"The original points {𝐱 i }\\lbrace \\mathbf {x}_i\\rbrace are connected to their cluster centre estimates {𝐮 i }\\lbrace \\mathbf {u}_i\\rbrace by lines.",
"Inter-cluster distances in Dataset 2 are half of those in Dataset 1, while intra-cluster distances remain the same.",
"Consequently, Dataset 1 performs better at a higher fraction of missing entries.",
"For the unconstrained clustering formulation with partial-distance based initialization, the cluster centre estimates are relatively stable with varying fractions of missing entries."
],
[
"Discussion",
"We have proposed a technique to cluster points when some of the feature values of all the points are unknown.",
"We theoretically studied the performance of an algorithm that minimizes an $\\ell _0$ fusion penalty subject to certain constraints relating to consistency with the known features.",
"We concluded that under favourable clustering conditions, such as well-separated clusters with low intra-cluster variance, the proposed method performs the correct clustering even in the presence of missing entries.",
"However, since the problem is NP-hard, we propose to use other penalties that approximate the $\\ell _0$ norm.",
"We observe experimentally that the $H_1$ penalty is a good surrogate for the $\\ell _0$ norm.",
"This non-convex saturating penalty is shown to perform better in the clustering task than previously used convex norms and penalties.",
"We describe an IRLS based strategy to solve the relaxed problem using the surrogate penalty.",
"Our theoretical analysis reveals the various factors that determine whether the points will be clustered correctly in the presence of missing entries.",
"It is obvious that the performance degrades with the decrease in the fraction of sampled entries ($p_0$ ).",
"Moreover, it is shown that the difference between points from different clusters should have low coherence ($\\mu _0$ ).",
"This means that the expected clustering should not be dependent on only a few features of the points.",
"Intuitively, if the points in different clusters can be distinguished by only 1 or 2 features, then a point missing these particular feature values cannot be clustered correctly.",
"Moreover, we note that a high number of points per cluster ($M$ ), high number of features ($P$ ) and a low number of clusters ($K$ ) make the data less sensitive to missing entries.",
"Finally, well-separated clusters with low intra-cluster variance (resulting in low values of $\\kappa $ ) are desirable for correct clustering.",
"Our experimental results show great promise for the proposed technique.",
"In particular, for the simulated data, we note that the cluster-centre estimates degrade gradually with increase in the fraction of missing entries.",
"Depending on the characteristics of the data such as number of points and cluster separation distance, the clustering algorithm fails at some particular fraction of missing entries.",
"We also show the importance of a good initialization for the IRLS algorithm, and our proposed initialization technique using partial distances is shown to work very well.",
"Figure: Clustering on Wine dataset.",
"The H 1 H_1 penalty is used to cluster the Wine datasets with varying fractions of missing entries.Figure: Cardiac MRI reconstruction results.",
"The images were reconstructed from highly under-sampled Fourier data using the unconstrained formulation.",
"A sampling mask for 1 particular frame is shown in (a), along with the Fourier data for that frame in (b).",
"The missing Fourier entries were filled with zeros and an inverse Fourier Transform was taken to get the corrupted image in (c).",
"The clustering algorithm was applied to this data and the resulting images were clustered into 20 clusters using spectral clustering.",
"(d) shows some reconstructed images from 2 different clusters.The proposed algorithm performs well on the MR image reconstruction task, resulting in images with minimal artefacts and diagnostic quality.",
"It is to be noted that the MRI images are reconstructed satisfactorily from very few Fourier samples.",
"In this case the fraction of observed samples is around $5\\%$ .",
"However, we see that the simulated datasets and the Wine datasets cannot be clustered at such a high fraction of missing samples.",
"The fundamental difference between the MRI dataset and the other datasets is the coherence $\\mu _0$ .",
"For the MRI data, we acquire Fourier samples.",
"Since we know that the low frequency samples are important for image reconstruction, the MRI scanner acquires more low frequency samples.",
"This is a case where high coherence is helpful in clustering.",
"However, for the simulated and Wine data, we do not know apriori which features are more important.",
"In any case the sampling pattern is random, and as predicted by theory, it is more useful to have low coherence.",
"The conclusion is that if the sampling pattern is within our control, it is useful to have high coherence if the relative importance of the different features is known apriori.",
"If this is unknown, then random sampling is preferred and it is useful to have low coherence.",
"Our future work will focus on deriving guarantees for the case of high $\\mu _0$ when the locations of the important features are known with some confidence, and the sampling pattern can be adapted accordingly.",
"Our theory assumes well-separated clusters and does not consider the presence of any outliers.",
"Theoretical and experimental analysis for the clustering performance in the presence of outliers needs to be investigated.",
"Improving the algorithm performance in the presence of outliers is a direction for future work.",
"Moreover, we have shown improved bounds for the clustering success in the absence of missing entries when the points within a cluster are assumed to follow a uniform random distribution.",
"We expect this trend to also hold for the case with missing entries.",
"This case will be analyzed in future work."
],
[
"Conclusion",
"We propose a clustering technique for data in the presence of missing entries.",
"We prove theoretically that a constrained $\\ell _0$ norm minimization problem recovers the clustering correctly even in the presence of missing entries.",
"An efficient algorithm that solves a relaxation of the above problem is presented next.",
"It is demonstrated that the cluster centre estimates obtained using the proposed algorithm degrade gradually with an increase in the number of missing entries.",
"The algorithm is also used to cluster the Wine dataset and reconstruct MRI images from under-sampled Fourier data.",
"The presented theory and results demonstrate the utility of the proposed algorithm in clustering data when some of the feature values of the data are unknown."
],
[
"Proof of Lemma ",
"Since $\\mathbf {x}_1 $ and $\\mathbf {x}_2 $ are in the same cluster, $\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _{\\infty } \\le \\epsilon $ .",
"For all the points in this particular cluster, let the $p^{th}$ feature be bounded as: $f^p_{min} \\le \\mathbf {x}(p) \\le f^p_{max}$ .",
"Then we can construct a vector $\\mathbf {u}$ , such that $\\mathbf {u}(p) = \\frac{1}{2}(f^p_{min}+f^p_{max})$ .",
"Now, since $f^p_{max}-f^p_{min} \\le \\epsilon $ , the following condition will be satisfied for this particular choice of $\\mathbf {u}$ : $\\Vert \\mathbf {x}_i-\\mathbf {u}\\Vert _{\\infty } &\\le & {\\frac{\\epsilon }{2}}; ~ ~i=1,2$ From this, it follows trivially that the following will also hold: $\\Vert \\mathbf {S}_i\\,(\\mathbf {x}_i-\\mathbf {u})\\Vert _{\\infty } &\\le & {\\frac{\\epsilon }{2}}; ~ ~i=1,2$"
],
[
"Lemma ",
"Lemma 8.1 Consider any pair of points $\\mathbf {x}_1, \\mathbf {x}_2 \\in \\mathbb {R}^P$ observed by sampling matrices $\\mathbf {S}_1 = \\mathbf {S}_{\\mathcal {I}_1}$ and $\\mathbf {S}_2 = \\mathbf {S}_{\\mathcal {I}_2}$ , respectively.",
"We assume the set of common indices ($\\omega \\mathcal {I}_1 \\cap \\mathcal {I}_2$ ) to be of size $q = |\\mathcal {I}_1 \\cap \\mathcal {I}_2|$ .",
"Then, for some $0< t < \\frac{q}{P}$ , the following result holds true regarding the partial distance $\\Vert \\mathbf {y}_\\omega \\Vert _2 = \\Vert \\mathbf {S}_{\\mathcal {I}_1\\cap \\mathcal {I}_2}\\left(\\mathbf {x}_{1}-\\mathbf {x}_{2}\\right)\\Vert _2$ : $\\mathbb {P}\\left(\\Vert \\mathbf {y}_\\omega \\Vert _2^2 \\le \\left(\\frac{q}{P}-t\\right)\\Vert \\mathbf {y}\\Vert _2^2\\right) \\le e^{-\\frac{2t^2P^2}{q\\mu _0^2}}$ We use some ideas for bounding partial distances from Lemma 3 of [22].",
"We rewrite the partial distance $\\Vert \\mathbf {y}_\\omega \\Vert _2^2$ as the sum of $q$ variables drawn uniformly at random from $\\lbrace y_1^2, y_2^2, \\ldots , y_P^2\\rbrace $ .",
"By replacing a particular variable in the summation by another one, the value of the sum changes by at most $\\Vert \\mathbf {y}\\Vert _{\\infty }^2$ .",
"Applying McDiarmid's Inequality, we get: $\\mathbb {P}\\left(E(\\Vert \\mathbf {y}_{\\omega }\\Vert _2^2) - \\Vert \\mathbf {y}_\\omega \\Vert _2^2 \\ge c\\right) \\le e^{-\\frac{2c^2}{\\sum _{i=1}^{q}\\Vert \\mathbf {y}\\Vert _\\infty ^4}} = e^{-\\frac{2c^2}{q\\Vert \\mathbf {y}\\Vert _{\\infty }^4}}$ From our assumptions, we have $E(\\Vert \\mathbf {y}_{\\omega }\\Vert _2^2) = \\frac{q}{P}\\Vert \\mathbf {y}\\Vert _2^2$ .",
"We also have $\\frac{\\Vert \\mathbf {y}\\Vert _2^2}{\\Vert \\mathbf {y}\\Vert _{\\infty }^2} \\ge \\frac{P}{\\mu _0}$ by (REF ).",
"We now substitute $c = t\\Vert \\mathbf {y}\\Vert _2^2$ , where $0 < t < \\frac{q}{P}$ .",
"Using the results above, we simplify expression (REF ) as: $\\begin{split}\\mathbb {P} \\left(\\Vert \\mathbf {y}_\\omega \\Vert _2^2 \\le \\left(\\frac{q}{P}-t\\right)\\Vert \\mathbf {y}\\Vert _2^2 \\right) &\\le e^{-\\frac{2t^2\\Vert \\mathbf {y}\\Vert _2^4}{q\\Vert \\mathbf {y}\\Vert _{\\infty }^4}}\\\\&\\le e^{-\\frac{2t^2P^2}{q\\mu _0^2}}\\\\\\end{split}$"
],
[
"Proof of Lemma ",
"We will use proof by contradiction.",
"Specifically, we consider two points $\\mathbf {x}_1$ and $\\mathbf {x}_2$ belonging to different clusters and assume that there exists a point $\\mathbf {u}$ that satisfies: $\\Vert \\mathbf {S}_i\\,(\\mathbf {x}_i-\\mathbf {u})\\Vert _{\\infty } &\\le & {\\frac{\\epsilon }{2}}; i=1,2$ We now show that the above assumption is violated with high probability.",
"Following the notation of Lemma REF , we denote the difference between the vectors by $\\mathbf {y}=\\mathbf {x}_1-\\mathbf {x}_2$ and the partial distances by: $\\Vert \\mathbf {y}_\\omega \\Vert _2 = \\Vert \\mathbf {S}_{\\mathcal {I}_1 \\cap \\mathcal {I}_2}~\\left(\\mathbf {x}_{1}-\\mathbf {x}_{2}\\right)\\Vert _2$ Using (REF ) and applying triangle inequality, we obtain $\\Vert \\mathbf {y}_{\\omega }\\Vert _{\\infty } \\le {\\epsilon }$ , which translates to $\\Vert \\mathbf {y}_{\\omega }\\Vert _{2} \\le \\epsilon \\sqrt{q}$ , where $q = |\\mathcal {I}_1 \\cap \\mathcal {I}_2|$ is the number of commonly observed locations.",
"We need to show that with high probability, the partial distances satisfy: $\\Vert \\mathbf {y}_{\\omega }\\Vert _2^2 > \\epsilon ^2q$ which will contradict (REF ).",
"We first focus on finding a lower bound for $q$ .",
"Using the Chernoff bound and setting $\\mathbb {E}(q) = p_0^2\\,P$ , we have: $\\mathbb {P}\\left(q \\ge \\frac{p_0^2P}{2}\\right) > 1-\\gamma _0$ where $\\gamma _0 = (\\frac{e}{2})^{-\\frac{p_0^2P}{2}}$ .",
"Thus, we can assume that $q \\ge \\frac{p_0^2P}{2}$ with high probability.",
"Using Lemma REF , we have the following result for the partial distances: $\\mathbb {P}\\left(\\Vert \\mathbf {y}_\\omega \\Vert _2^2 \\le \\left(\\frac{q}{P}-t\\right)\\Vert \\mathbf {y}\\Vert _2^2\\right) \\le e^{-\\frac{2t^2P^2}{q\\mu _0^2}}$ Since $\\mathbf {x}_1$ and $\\mathbf {x}_2$ are in different clusters, we have $\\Vert \\mathbf {y}\\Vert _2 \\ge \\delta $ .",
"We will now determine the value of $t$ for which the above upper bound will equal the RHS of (REF ): $\\left(\\frac{q}{P}-t\\right)\\Vert \\mathbf {y}\\Vert _2^2 = \\epsilon ^2q$ or equivalently: $t=\\frac{q}{P}-\\frac{\\epsilon ^2q}{\\Vert \\mathbf {y}\\Vert _2^2} \\ge \\frac{q}{P}-\\frac{\\epsilon ^2q}{\\delta ^2} = \\frac{q}{P}(1-\\kappa ^2)$ Since $t > 0$ , we require $\\kappa < 1$ , where $\\kappa = \\frac{\\epsilon \\sqrt{P}}{\\delta }$ .",
"Using the above, we get the following bound if we assume that $q \\ge \\frac{p_0^2P}{2}$ : $\\frac{t^2}{q} \\ge \\frac{q}{P^2}(1-\\kappa ^2)^2 \\ge \\frac{p_0^2}{2P}(1-\\kappa ^2)^2$ We now obtain the following probability bound for any $q \\ge \\frac{p_0^2P}{2}$ : $\\begin{split}\\mathbb {P}\\left(\\Vert \\mathbf {y}_\\omega \\Vert ^2 > \\epsilon ^2q \\right) & \\ge 1-e^{-\\frac{2t^2P^2}{q\\mu _0^2}}\\\\ & \\ge 1-e^{-\\frac{p_0^2P(1-\\kappa ^2)^2}{\\mu _0^2}}\\\\ & = 1- \\delta _0\\end{split}$ Combining (REF ) and (REF ), the probability for (REF ) to hold is $\\le 1 - (1-\\gamma _0)(1-\\delta _0) = \\beta _0$ ."
],
[
"Proof of Lemma ",
"We construct a graph where each point $\\mathbf {x}_i$ is represented by a node.",
"Lemma REF implies that a pair of points belonging to the same cluster can yield the same $\\mathbf {u}$ in a feasible solution with probability 1.",
"Hence, we will assume that there exists an edge between two nodes from the same cluster with probability 1.",
"Lemma REF indicates that a pair of points belonging to different clusters can yield the same $\\mathbf {u}$ in a feasible solution with a low probability of $\\beta _0$ .",
"We will assume that there exists an edge between two nodes from different clusters with probability $\\beta _0$ .",
"We will now evaluate the probability that there exists a fully-connected sub-graph of size $M$ , where all the nodes have not been taken from the same cluster.",
"We will follow a methodology similar to [28], which gives an expression for the probability distribution of the maximal clique (i.e.",
"largest fully connected sub-graph) size in a random graph.",
"Unlike the proof in [28], in our graph every edge is not present with equal probability.",
"We define the following random variables: $t $ Size of the largest fully connected sub-graph containing nodes from more than 1 cluster $n $ Number of $M$ membered complete sub-graphs containing nodes from more than 1 cluster Our graph can have an $M$ membered clique iff $n$ is non-zero.",
"Thus, we have: $\\mathbb {P} \\left(t \\ge M \\right) = \\mathbb {P} \\left(n \\ne 0 \\right)$ Since the distribution of $n$ is restricted only to the non-negative integers, it can be seen that: $\\mathbb {P} \\left(n\\ne 0\\right) \\le E(n)$ Combining the above 2 results, we get: $\\mathbb {P} \\left(t \\ge M \\right) \\le E(n)$ Let us consider the formation of a particular clique of size $M$ using $m_1, m_2, \\ldots , m_K$ nodes from clusters $C_1, C_2, \\ldots , C_K$ respectively such that $\\sum _{j=1}^{K}m_j = M$ , and at least 2 of the variables $\\lbrace m_j\\rbrace $ are non-zero.",
"The number of ways to choose such a collection of nodes is: $\\prod _j {M \\atopwithdelims ()m_j}$ .",
"In order to form a solution $\\lbrace m_j\\rbrace $ , we need $\\frac{1}{2}(M^2-\\sum _j {m_j^2})$ inter-cluster edges to be present.",
"We recall that each of these edges is present with probability $\\beta _0$ .",
"Thus, the probability that such a collection of nodes forms a clique is $\\beta _0^{\\frac{1}{2}(M^2-\\sum _j {m_j^2})}$ .",
"This gives the following result: $E(N) = \\sum _{\\lbrace m_j\\rbrace \\in \\mathcal {S}} \\beta _0^{\\frac{1}{2}(M^2-\\sum _j {m_j^2})} \\prod _j {M \\atopwithdelims ()m_j} = \\eta _0$ where $\\mathcal {S}$ is the set of all sets of positive integers $\\lbrace m_j\\rbrace $ such that: $2 \\le \\mathcal {U}(\\lbrace m_j\\rbrace ) \\le K$ and $\\sum _j m_j = M$ .",
"Here, the function $\\mathcal {U}$ counts the number of non-zero elements in a set.",
"Thus, we have: $\\mathbb {P} \\left(t\\ge M \\right) \\le \\eta _0$ This proves that with probability $\\ge 1-\\eta _0$ , a set of points of cardinality $\\ge M$ not all belonging to the same cluster cannot all have equal cluster-centre estimates."
],
[
"Proof of Theorem ",
"Lemma REF indicates that fully connected original clusters with size $M$ are likely with probability 1, while Lemma REF shows that the size of misclassified large clusters cannot exceed $M-1$ with very high probability.",
"These results enable us to re-express the optimization problem (REF ) as a simpler maximization problem.",
"We will then show that with high probability, any feasible solution other than the ground-truth solution results in a cost higher than the ground-truth solution.",
"Let a candidate solution have $k$ groups of sizes $M_1, M_2,\\ldots , M_k$ respectively.",
"The centre estimates for all points within a group are equal.",
"These are different from the centre estimates of other groups.",
"Without loss of generality, we will assume that at most $K$ of these groups each have points belonging to only a single ground-truth cluster, i.e.",
"they are \"pure\".",
"The rest of the clusters in the candidate solution are \"mixed\" clusters.",
"If we have a candidate solution with greater than $K$ pure clusters, then they can always be merged to form $K$ pure clusters; the merged solution will always result in a lower cost.",
"The objective function in (REF ) can thus be rewritten as: $\\begin{split}\\sum _{i=1}^{KM}\\sum _{j=1}^{KM}\\Vert \\mathbf {u}_i - \\mathbf {u}_j\\Vert _{2,0}& = \\sum _{i=1}^{k} M_i (KM-M_i) \\\\& = K^2M^2 - \\sum _{i=1}^{k} M_i^2\\end{split}$ Since we assume that the first $K$ clusters are pure, therefore they have a size $0 \\le M_i \\le M$ , $i=1,\\ldots , K$ .",
"The remaining clusters are mixed and have size $\\le M-1$ with probability $\\ge 1-\\eta _0$ .",
"Hence, we have the constraints $0 \\le M_i \\le (M-1)$ , $i=K+1,\\ldots , k$ .",
"We also have a constraint on the total number of points, i.e.",
"$\\sum _{i=1}^k M_i = KM$ .",
"Thus, the problem (REF ) can be rewritten as the constrained optimization problem: $\\begin{split}\\lbrace M_i^*,k^*\\rbrace = & \\max _{\\lbrace M_i\\rbrace ,k}\\sum _{i=1}^{k} M_i^2 \\\\\\mbox{ s.t. }",
"& 0 \\le M_i \\le M, i=1,\\ldots , K \\\\& 0 \\le M_i \\le M-1, i=K+1,\\ldots ,k \\\\& \\sum _{i=1}^k M_i = KM\\end{split}$ Note that we cannot have $k < K$ , with probability $\\ge 1-\\eta _0$ , since that involves a solution with cluster size $> M$ .",
"We can evaluate the best solution $\\lbrace M_i^*\\rbrace $ for each possible value of $k$ in the range $K \\le k \\le MK$ .",
"Then we can compare these solutions to get the solution with the highest cost.",
"We note that the feasible region is a polyhedron and the objective function is convex.",
"Thus, for each value of $k$ , we only need to check the cost at the vertices of the polyhedron formed by the constraints, since the cost at all other points in the feasible region will be lower.",
"The vertex points are formed by picking $k-1$ out of the $k$ box constraints and setting $M_i$ to be equal to one of the 2 possible extremal values.",
"We note that all the vertex points have either $K$ or $K+1$ non-zero values.",
"As a simple example, if we choose $M=10$ and $K=4$ , then the vertex points of the polyhedron (corresponding to different solutions $\\lbrace M_i\\rbrace )$ are given by all possible permutations of the following: $(10,10,10,10,0,0 \\ldots 0)$ : 4 clusters $(10,10,10,0,1,9,0 \\ldots 0)$ : 5 clusters $(10,10,0,0,2,9,9,0 \\ldots 0)$ : 5 clusters $(10,0,0,0,3,9,9,9,0 \\ldots 0)$ : 5 clusters $(0,0,0,0,4,9,9,9,9,0 \\ldots 0)$ : 5 clusters In the general case the vertices are given by permutations of the following: $(M,M,\\ldots ,M,0,0 \\ldots 0)$ : $K$ clusters $(M,M,\\ldots ,0,0,1,M-1,0 \\ldots 0)$ : $K+1$ clusters $(M,M,\\ldots ,0,0,2,M-1,M-1 \\ldots 0)$ : $K+1$ clusters ... $(0,0, \\ldots 0,K,M-1,M-1 \\ldots M-1, 0)$ : $K+1$ clusters Now, it is easily checked that the 1st candidate solution in the list (which is also the ground-truth solution) has the maximum cost.",
"Mixed clusters with size $> M-1$ cannot be formed with probability $> 1 - \\eta _0$ .",
"Thus, with the same probability, the solution to the optimization problem (REF ) is identical to the ground-truth clustering.",
"This concludes the proof of the theorem."
],
[
"Upper Bound for $\\eta _0$ in the 2-cluster case",
"We introduce the following notation: $F(i) = i(M-i)\\log \\beta _0$ , for $i \\in [1, M-1]$ .",
"$G(i) = 2[\\log \\Gamma (M+1) - \\log \\Gamma (i+1) - \\log \\Gamma (M-i+1)]$ , for $i \\in [1, M-1]$ where $\\Gamma $ is the Gamma function.",
"We note that both the functions $F$ and $G$ are symmetric about $i = \\frac{M}{2}$ , and have unique minimum and maximum respectively for $i = \\frac{M}{2}$ .",
"We will show that the maximum for the function $F + G$ is achieved at the points $i=1,M-1$ .",
"We note that: $G^{\\prime }(i) = -2[\\Psi (i+1) - \\Psi (M-i+1)]$ where $\\Psi $ is the digamma function, defined as the log derivative of the $\\Gamma $ function.",
"We now use the expansion: $\\Psi (i+1) = \\log i + \\frac{1}{2i}$ Substituting, we get: $G^{\\prime }(i) = - 2\\left[\\log \\frac{i}{M-i} + \\frac{M-2i}{2i(M-i)}\\right]$ We also have: $F^{\\prime }(i) = (M-2i)\\log \\beta _0$ Adding, we get: $\\begin{split}F^{\\prime }(i) + G^{\\prime }(i) = (M-2i) (&\\log \\beta _0 - \\frac{1}{i(M-i)})\\\\& -2\\log \\frac{i}{(M-i)})\\end{split}$ Now, in order to ensure that $F^{\\prime }(i) + G^{\\prime }(i) \\le 0$ , we have to arrive at conditions such that: $\\log \\beta _0 \\le \\frac{1}{i(M-i)} + \\frac{2}{M-2i}\\log \\frac{i}{M-i}$ Since the RHS is monotonically increasing in the interval $i \\in [1, \\frac{M}{2}-1]$ the above condition reduces to: $\\log \\beta _0 \\le \\frac{1}{M-1} + \\frac{2}{M-2}\\log \\frac{1}{M-1}$ Under the above condition, for all $i \\in [1, \\frac{M}{2}]$ : $F^{\\prime }(i) + G^{\\prime }(i) \\le 0$ Thus, the function $F + G$ reaches its maxima at the extremal points given by $i=1,M-1$ .",
"For positive integer values of $i$ , i.e.",
"$i \\in \\lbrace 1, 2, \\ldots , M-1\\rbrace $ : $F(i) + G(i) = \\log [\\beta _0^{i(M-i)} {M \\atopwithdelims ()i}^2]$ Thus, the function $\\beta _0^{i(M-i)} {M \\atopwithdelims ()i}^2$ also reaches its maxima at $i=1,M-1$ .",
"This maximum value is given by: $\\beta _0^{M-1}M^2$ .",
"This gives the following upper bound for $\\eta _0$ : $\\begin{split}\\eta _0 & \\le \\sum _{i=1}^{M-1} [\\beta _0^{M-1} M^2] \\\\ & = M^2(M-1)\\beta _0^{M-1} \\\\ & \\le M^3 \\beta _0^{M-1} \\\\ & = \\eta _{0,{\\rm approx}}\\end{split}$"
],
[
"Proof of Theorem ",
"We consider any two points $\\mathbf {x}_1$ and $\\mathbf {x}_2$ that are in different clusters.",
"Let us assume that there exists some $\\mathbf {u}$ satisfying the data consistency constraint: $\\Vert \\mathbf {x}_i - \\mathbf {u}\\Vert _{\\infty } \\le \\epsilon /2, ~~i=1,2.$ Using the triangle inequality, we have $\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _{\\infty } \\le \\epsilon $ and consequently, $\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _{2} \\le \\epsilon \\sqrt{P}$ .",
"However, if we have a large inter-cluster separation $\\delta > \\epsilon \\sqrt{P}$ , then this is not possible.",
"Thus, if $\\delta > \\epsilon \\sqrt{P}$ , then points in different clusters cannot be misclassified to a single cluster.",
"Among all feasible solutions, clearly the solution to problem (REF ) with the minimum cost is the one where all points in the same cluster merge to the same $\\mathbf {u}$ .",
"Thus, $\\kappa < 1$ ensures that we will have the correct clustering."
],
[
"Proof of Lemma ",
"The idea is similar to that in Theorem REF .",
"We will show that with high probability two points $\\mathbf {x}_1$ and $\\mathbf {x}_2$ that are in different clusters satisfy $\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _2 > \\epsilon \\sqrt{P}$ with high probability, which implies that (REF ) is violated.",
"Let points in $C_1$ and $C_2$ follow uniform random distributions in $\\mathbb {R}^P$ with centres $\\mathbf {c}_1$ and $\\mathbf {c}_2$ respectively.",
"The expected distance between $\\mathbf {x}_1 \\in \\mathcal {C}_1$ and $\\mathbf {x}_2 \\in \\mathcal {C}_2$ is given by: $\\begin{split}E(\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _2^2) &= \\frac{1}{\\epsilon ^{2}}\\sum _{p=1}^P\\int _{\\mathbf {c}_1^p-\\frac{\\epsilon }{2}}^{\\mathbf {c}_1^p+\\frac{\\epsilon }{2}}\\int _{\\mathbf {c}_2^p-\\frac{\\epsilon }{2}}^{\\mathbf {c}_2^p+\\frac{\\epsilon }{2}}(\\mathbf {x}_1^p - \\mathbf {x}_2^p )^2 d\\mathbf {x}_1^p d\\mathbf {x}_2^p \\\\ &= \\Vert \\mathbf {c}_1 - \\mathbf {c}_2\\Vert _2^2 + \\frac{P}{6}\\epsilon ^2\\\\&= c_{12}^2 + \\frac{P}{6}\\epsilon ^2\\end{split}$ where $\\mathbf {c}_i^p$ and $\\mathbf {x}_i^p$ are the $p^{th}$ features of $\\mathbf {c}_i$ and $\\mathbf {x}_i$ respectively, and $c_12 = \\Vert \\mathbf {c}_1 - \\mathbf {c}_2\\Vert _2$ .",
"Let $c_i = |\\mathbf {c}_1^i - \\mathbf {c}_2^i|$ , for $i = 1,2,\\ldots ,P$ .",
"Using Mcdiarmid's inequality: $\\begin{split}& \\mathbb {P} \\left(\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _2^2 \\le E(\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _2^2) - t \\right) \\\\ & \\le e^{-\\frac{2t^2}{\\sum _{i=1}^P|(c_i+\\epsilon )^2-(c_i-\\epsilon )^2|^2}} \\\\ & = e^{-\\frac{t^2}{8\\epsilon ^2c_{12}^2}}\\end{split}$ Let $t = E(\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _2^2) - P\\epsilon ^2$ .",
"Then we have: $\\mathbb {P} \\left(\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _2 \\le \\epsilon \\sqrt{P} \\right) \\le e^{-\\frac{(c_{12}^2-\\frac{5P}{6}\\epsilon ^2)^2}{8\\epsilon ^2c_{12}^2}}$ We note that the RHS above is a decreasing function of $c_{12}$ .",
"Thus, we consider some $c \\le c_{12}$ , such that $c$ is the minimum distance between any 2 cluster centres in the dataset.",
"We then have the following bound: $\\mathbb {P} \\left(\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _2 \\le \\epsilon \\sqrt{P} \\right) \\le e^{-\\frac{(c^2-\\frac{5P}{6}\\epsilon ^2)^2}{8\\epsilon ^2c^2}}$ To ensure $t>0$ , we require: $c > \\sqrt{\\frac{5P}{6}}\\epsilon $ , or equivalently, $\\kappa ^{\\prime } = \\frac{\\epsilon \\sqrt{P}}{c} < \\sqrt{\\frac{6}{5}}$ .",
"We now get the probability bound: $\\mathbb {P} \\left(\\Vert \\mathbf {x}_1 - \\mathbf {x}_2\\Vert _2 \\le \\epsilon \\sqrt{P}\\right) \\le e^{-\\frac{P(1-\\frac{5}{6}\\kappa ^{\\prime 2})^2}{8 \\kappa ^{\\prime 2}}} = \\beta _1$ Thus, (REF ) is violated with probability exceeding $1-\\beta _1$ ."
]
] | 1709.01870 |
[
[
"360 Panorama Cloning on Sphere"
],
[
"Abstract In this paper, we address a novel problem of cloning a patch of the source spherical panoramic image to the target spherical panoramic image, which we call 360 panorama cloning.",
"Considering the sphere geometry constraint embedded in spherical panoramic images, we develop a coordinate-based method that directly clones in the spherical domain.",
"Our method neither differentiates the polar regions and equatorial regions, nor identifies the boundaries in the unrolled planar-formatted panorama.",
"We discuss in depth two unique issues in panorama cloning, i.e.",
"preserving the patch's orientation, and handling the large-patch cloning (covering over 180 field of view) which may suffer from discoloration artifacts.",
"As experimental results demonstrate, our method is able to get visually pleasing cloning results and achieve real time cloning performance."
],
[
"Introduction",
"Panorama, a wide-angle view representation of a physical scene, has a long history dated back to the 18th century.",
"Rather than being artistic works in early days, panoramic images have became more prevalent in recent years due to the progress of image stitching techniques and spherical imaging system, for displaying landscapes, street side views, or astronomic sceneries.",
"Nowadays, enormous panoramas are publicly available via multiple well-known navigation websites, including Google Street View, Microsoft Bing Map, 360 Cities, etc.",
"The public panoramas usually capture the whole field of view in all directions around the photographer, and most of which also cover up and down views.",
"Due to the embedded sphere geometry, for viewing convenience a 360$^\\circ $ panorama is commonly represented in the unrolled latitude-longitude planar format.",
"It shows all the views of the panorama in a way natural to viewers, often with the horizon aligned to the equatorial line in the sphere, yet subjective to increasing distortions near two poles.",
"Recently there are a number of panorama-based applications emerging, including online street-level virtual navigation , , city-scale change detection , scene recognition and view detection , floor-plan construction etc.",
"These applications utilize the content in a panorama without image editing or modification.",
"In terms of panoramic image editing, Schröder and Sweldens addressed the problem of sharpening and smoothing 360$^\\circ $ panoramas via spherical wavelets.",
"Bülow used spherical harmonic functions to smooth 3D surfaces, which can be regarded as panoramas.",
"Kazhdan and Hoppe proposed a panorama enhancement method based on a metric-aware solver.",
"Zhu et al.",
"proposed a panorama completion method, in which the hole region is filled after it has been warped to 2D plane.",
"In this paper, we address the problem of image cloning for 360$^\\circ $ panoramas, which clones a patch of the source panorama to the target panorama.",
"Traditional methods for planar image cloning typically solve a Poisson equation with Dirichlet boundary condition defined by the target image , .",
"The fundamental machinery is to construct a Laplacian membrane that smoothly interpolates the difference between the source and target images along the boundary across the entire cloned region.",
"Later on, an approximation approach based on mean value coordinates was originally proposed in .",
"Because of its advantages in terms of speed, memory footprint, and parallelizability, the coordinate-based method has been extended in several works , , .",
"Considering 360$^\\circ $ panorama cloning, an intuitive user interface is to perform the cloning over the latitude-longitude planar format.",
"However, planar image cloning methods are not directly applicable.",
"One major difficulty lies in that we should guarantee the cloned patch to have consistent geometric deformations with the target image wherever the cloned patch is pasted.",
"It is noted that panorama is subjective to the sphere geometry constraint.",
"Hence, we treat the cloned patch as a spherical polygon, and extend the coordinate-based cloning technique to the spherical domain.",
"Our work has three contributions: We are the first to study 360$^\\circ $ panorama cloning, a new problem belonging to panorama editing, and get pleasing cloning results.",
"We explain how to preserve the orientation of cloned patches, which is an inherent problem of panorama cloning, with a two-step rotation estimation method.",
"We remove discoloration artifacts when cloned patches cover over 180$^\\circ $ filed-of-view by a splitting-based method, which is based on our new deduction of spherical mean value coordinates.",
"By providing wide fields of view far beyond planar images, 360$^\\circ $ panoramas have been successfully applied in a number of recent new applications.",
"For instance, online street-level virtual navigation , synthesizes new panoramas at intermediate views in-between pre-captured ones.",
"City-scale change detection estimates architectural structure changes between a panorama collection and a maintained cadastral 3D model.",
"Xiao et al.",
"proposed to recognize image scene and view direction based on a large panorama dataset.",
"Micusik and Kosecka constructed 3D city model from street-view panoramic sequences.",
"There also exist several works on panorama smoothing and sharpening, using spherical wavelets , spherical harmonics functions , and metric-aware solver .",
"Zhu et al.",
"proposed a panorama completion method for street view images , which needs to warp the hole region to 2D plane first.",
"In this paper, we focus on panorama cloning, a new problem that has not been studied yet to the best of our knowledge."
],
[
"Image Cloning",
"Traditional image cloning methods , , eventually solve a large sparse linear system defined by the Poisson equation.",
"Solving the Poisson equation for large cloned patches is a computing and memory intensive task.",
"Although some acceleration algorithms were proposed, they are still time consuming for real-time applications.",
"Coordinate-based techniques , offer practical alternatives for image cloning.",
"These methods diffuse the differences along the patch's boundary to the interior region by interpolation without solving Poisson equations.",
"The widely used diffusion weights are mean value coordinates and harmonic coordinates .",
"Coordinate-based methods have nice properties, including high speed, small memory footprint and ease of parallelization.",
"They have recently been extended to video composition and stereoscopic image cloning .",
"Our panorama cloning belongs to the coordinate-based technique."
],
[
"Mean Value Coordinates",
"The mean value coordinate motivated by mean value theorem has important properties, such as smoothness, linear independence, and refinability.",
"It can be used to approximate a harmonic-like solution to the boundary interpolation problem, and has many applications in computer graphics , such as Phong shading for arbitrary polygons, image warping and image cloning .",
"Many extensions have been made to planar mean value coordinate.",
"Ju et al.",
"generalized mean value coordinates from closed 2D polygons to closed triangular meshes, which are continuous and smooth on the interior of meshes.",
"Langer et al.",
"proposed spherical mean value coordinate (SMVC), which gives the coordinates for point $\\mathbf {v}$ located on the unit sphere with respect to a spherical polygon.",
"By replacing the linear precision property of planar coordinates by a requirement in terms of center of mass, introduced a method for defining and computing barycentric coordinates with respect to polygons on general surfaces.",
"Panozzo et al.",
"presented an efficient method for computing and inverting weighted averages on surfaces for point on surfaces.",
"Our panorama cloning uses spherical mean value coordinates as diffusion weights because of the embedded sphere geometry.",
"Let $S={S}^2$ , $T={S}^2$ be the domain of source and target panoramas, $g:S\\rightarrow R, f^*:T\\rightarrow R$ be the intensities of source and target image.",
"Assume that we want to seamlessly clone the source patch $\\Omega _{s}\\subset S$ to the target patch $\\Omega _{t}\\subset T$ .",
"Similar to the planar image cloning presented by Farbman et al.",
", the intensity of the point $\\mathbf {v}\\in \\Omega _{t}$ will be given by $f(\\mathbf {v})=g(\\mathbf {v})+\\sum _{i=1}^{n}\\lambda _{i}(\\mathbf {v})(f^*-g)(\\mathbf {v}_{i}),$ where $\\lambda _{i}(\\mathbf {v})$ is the spherical mean value coordinates of $\\mathbf {v}$ with respect to the boundary of patch $\\Omega _{t}$ , whose vertices are $\\mathrm {P}=\\lbrace \\mathbf {v}_{1},\\ldots ,\\mathbf {v}_{n}\\rbrace $ .",
"According to our deduction in Appendix and , $\\lambda _{i}(\\mathbf {v})$ is given by $\\lambda _{i}(\\mathbf {v})=\\frac{(\\tan \\frac{\\alpha _{i-1}}{2}+\\tan \\frac{\\alpha _{i}}{2})/\\sin \\theta _{i}}{\\sum \\nolimits _{j}\\cot \\theta _{j}(\\tan \\frac{\\alpha _{j-1}}{2}+\\tan \\frac{\\alpha _{j}}{2})},$ where $\\theta _{i}$ is the angle between $\\mathbf {v}$ and $\\mathbf {v}_{i}$ ; $\\alpha _{i}$ is the signed angle between vectors $\\mathbf {v}\\times \\mathbf {v}_{i}$ and $\\mathbf {v}\\times \\mathbf {v}_{i+1}$ (see Figure REF ).",
"Because the mean value interpolant is very smooth away from the boundary of the cloned region, we construct an adaptive mesh for the spherical polygon as in to avoid much of the computation.",
"Figure: Notation for the deduction of the spherical mean value coordinates, see Appendix A for more details.It seems that spherical image cloning is quite similar to planar image cloning.",
"However, we find that it suffers from two unique problems.",
"First, we need to preserve the orientation of the cloned patch after transformation so as to avoid weird appearance.",
"Second, cloning large patches may result in discoloration artifacts, which damage the appearance of the cloned patches seriously.",
"In the next, we will discuss these two issues in details."
],
[
"Preserving Cloned Patch Orientation",
"Panoramas are usually captured with the horizon aligned with the equatorial line.",
"To compute the intensity difference $f^*-g$ in Equation REF , we need to determine the boundary of the target patch according to the source patch.",
"In planar image cloning, the target patch can be computed by translating the source patch with offset $\\mathbf {v}_{s}-\\mathbf {v}_{t}$ , where $\\mathbf {v}_{s}$ is the datum point, e.g.",
"the centroid of the source patch, and $\\mathbf {v}_{t}$ is the specified target cloning position.",
"In spherical image cloning, since $\\mathbf {v}_{s}$ and $\\mathbf {v}_{t}$ are both on the surface of sphere, the transformation between $\\mathbf {v}_{s}$ and $\\mathbf {v}_{t}$ now becomes 3D rotation.",
"A naive method to compute the 3D rotation is using the axis-angle representation, which is given by Rodriguez' rotation formula $\\mathcal {R}(\\mathbf {u},\\theta ) = I\\cos \\theta +\\sin \\theta [\\mathbf {u}]_{\\times }+(1-\\cos \\theta )\\mathbf {u}\\otimes \\mathbf {u},$ where $\\mathbf {u}=\\mathbf {v}_{s} \\times \\mathbf {v}_{t}$ is the rotation axis and $\\theta =\\arccos (\\mathbf {v}_{s} \\cdot \\mathbf {v}_{t})$ is the rotation angle.",
"However this method may greatly change the orientation of cloned patch, leading to unexpected appearance as shown in Figure REF (c).",
"Figure: To estimate the rotation between 𝐯 s \\mathbf {v}_{s} and 𝐯 t \\mathbf {v}_{t}, we first rotate 𝐯 s \\mathbf {v}_{s} to make it have the same azimuthal angle with 𝐯 t \\mathbf {v}_{t}, then rotate the intermediate point 𝐯 s ' \\mathbf {v}_{s}{^{\\prime }} to 𝐯 t \\mathbf {v}_{t}.Here, we apply a two-step rotation estimation method to preserve the orientation of cloned patch (illustrated in Figure REF ).",
"Assume the spherical coordinates of $\\mathbf {v}_{s}$ and $\\mathbf {v}_{t}$ are $(\\phi _{s}, \\theta _{s})$ and $(\\phi _{t}, \\theta _{t})$ respectively, where $\\phi _{s}$ and $\\phi _{t}$ are azimuthal angles, $\\theta _{s}$ and $\\theta _{t}$ are polar angles.",
"We first rotate $\\mathbf {v}_{s}$ around $z$ -axis by $\\phi _{t}-\\phi _{s}$ to make it have the same azimuthal angle with $\\mathbf {v}_{t}$ .",
"This rotation is denoted as $R_{1}$ , given by $R_{1}=\\left[\\begin{array}{ccc}\\cos (\\phi _{t}-\\phi _{s}) & -\\sin (\\phi _{t}-\\phi _{s}) & 0\\\\\\sin (\\phi _{t}-\\phi _{s}) & \\cos (\\phi _{t}-\\phi _{s}) & 0\\\\0 & 0 & 1\\\\\\end{array}\\right].$ The intermediate rotated point has the spherical coordinate $(\\phi _{t}, \\theta _{s})$ and is denoted as $\\mathbf {v}_{s}{^{\\prime }}=R_{1}\\mathbf {v}_{s}$ .",
"The second step is to transform $\\mathbf {v}_{s}{^{\\prime }}$ to $\\mathbf {v}_{t}$ .",
"This can be accomplished by rotating $\\mathbf {v}_{s}{^{\\prime }}$ around the normal $\\mathbf {u}{^{\\prime }}$ of blue-colored plane by $\\theta _{t}-\\theta _{s}$ , where $\\mathbf {u}{^{\\prime }}=\\mathbf {v}_{s}{^{\\prime }}\\times [0\\ 0\\ 1]^{T}.$ This rotation is represented by Rodriguez' rotation formula in Equation REF as $R_{2}=\\mathcal {R}(\\mathbf {u}{^{\\prime }},\\theta _{t}-\\theta _{s}).$ Hence, the final rotation $R$ between $\\mathbf {v}_s$ and $\\mathbf {v}_t$ is the combination of $R_{1}$ and $R_{2}$ , given by $R=R_{2}R_{1}.$ The cloning results with rotation matrix computed by different methods are shown in Figure REF .",
"We can see that the orientation of the parterre is well preserved by the two-step rotation estimation method.",
"Figure: The cloning results using different methods to compute the rotation matrix: (a) the source panorama and selected cloning patch; (b) the target panorama; (c) cloning result using naive method; (d) cloning result using two-step estimation method."
],
[
"Large Patch Cloning",
"The basic cloning method works well for small cloning patches, however it inevitably suffers from discoloration artifacts when we want to clone a quite large patch (see the example shown in Figure REF (a)).",
"The main reason is that the spherical mean value coordinate given by Equation REF “overflows”, when the angle $\\theta _{i}$ between point $\\mathbf {v}$ and boundary point $\\mathbf {v}_{i}$ is more than 180$^\\circ $ .",
"The problem of discoloration artifacts is unique to spherical panorama cloning.",
"Although some previous works , , , observed a similar problem, the artifacts they tackled are caused by large color differences or texture differences, which is not discussed in our paper.",
"Figure: Discoloration artifacts.",
"(a) Discoloration artifacts when the cloned patch covers more than 180 ∘ ^\\circ .",
"(b) The artifacts can be removed with splitting-based cloning.Figure: Compared with the central pixels (the second row), the component of the spherical mean value coordinates for the pixels near the ends of the patch may have large absolute values.",
"The horizontal axis represents the indices of pixels on the cloned patch boundary.",
"See text for details."
],
[
"Analyzing Decoloration Artifacts",
"From Figure REF we notice that discolored pixels appear near two ends of the cloned patch, while central pixels have much less artifacts.",
"By further examining the discolored pixels, we find their spherical mean value coordinates (SMVC) have rather large magnitudes.",
"Although we currently have no way to prove this analytically, we try to give an explanation by examining the computation of SMVC.",
"To facilitate the analysis, we re-deduce SMVC by applying the stereographic projection in Appendix A.",
"Here, we only recall critical relations.",
"Let $\\bar{\\lambda }_i$ be planar mean value coordinate of one point with respect to the projected polygon, $\\tilde{\\lambda }_i$ denotes intermediate coordinate, and $\\lambda _i$ denotes the final spherical mean value coordinate with respect to the spherical polygon.",
"From the deduction, the intermediate coordinate $\\tilde{\\lambda }_i$ is the scaling of $\\bar{\\lambda }_i$ (Equation REF ), i.e.",
"$\\tilde{\\lambda }_i=\\frac{2}{1+\\cos \\theta _{i}}\\bar{\\lambda }_i.$ The spherical mean value coordinates $\\lambda _i$ is related to $\\tilde{\\lambda }_i$ (Equation REF ) by $\\lambda _i=\\frac{\\tilde{\\lambda }_{i}}{2-\\sum _{i}\\tilde{\\lambda }_{i}}.$ Figure REF demonstrates the coordinate values for one central pixel and one discolored pixel (both from Figure REF (a)).",
"For the discolored pixel (the first row of Figure REF ), there is one angle $\\theta _i$ (between $\\mathbf {v}$ and boundary point $\\mathbf {v}_i$ ) very close to $\\pi $ .",
"Due to Equation REF , the corresponding $\\tilde{\\lambda }_i$ is abnormally enlarged as shown in Figure REF (c).",
"In Equation REF , the sum $\\sum _{i}\\tilde{\\lambda }_{i}$ would be close to 2, e.g.",
"2.0191 for this exampler discolored pixel.",
"This makes $\\lambda _i$ be negative and have large magnitude as shown in Figure REF (d).",
"On the contrary, for the central pixel (the bottom row in Figure REF ), the angle $\\theta _i$ is far less than $\\pi $ .",
"The curve of coordinate $\\tilde{\\lambda }_i$ has a similar shape with that of $\\bar{\\lambda }_i$ .",
"As a result, $\\lambda _i$ is the scaling of $\\tilde{\\lambda }_i$ with small factor, and has normal magnitude.",
"For the example in Figure REF (a), the cloned region has a field of view around 300$^\\circ $ , and hence the discoloration artifacts seem to appear almost everywhere."
],
[
"Splitting-based Cloning",
"Since no decoloration artifacts occur when $\\theta _i$ (the angle between the interpolated point and boundary points) is less than 180$^\\circ $ , we think about the possibility to constrain the field of view of cloning patches.",
"Encouraged by the fact that the central points in large cloning patches can get normal appearance, we develop a simple solution, by splitting the large cloning patch into two small patches, each of which has less than 180$^\\circ $ .",
"In the following, we first derive the modified spherical mean value coordiantes.",
"Figure: The definition of sub regions and boundaries for splitting-based cloning.Suppose the adaptive mesh constructed in the cloned patch is $\\Omega $ , it is split into two sub-regions $\\Omega _{1}$ and $\\Omega _{2}$ along a path $\\mathrm {P}_{c}$ (its selection will be discussed in Section REF ), as illustrated in Figure REF .",
"Let $\\mathrm {P}$ denote the boundary of $\\Omega $ , and $\\mathrm {P}_{1}=\\partial \\Omega _{1}$ , $\\mathrm {P}_{2}=\\partial \\Omega _{2}$ denote the boundaries of the two sub-regions.",
"Then the splitting path can be represented as $\\mathrm {P}_{c}=\\mathrm {P}_{1}\\cap \\mathrm {P}_{2}$ .",
"The vertices of the adaptive mesh are separated into two sets $A=\\lbrace \\mathbf {v}|\\mathbf {v}\\in \\Omega _{1}\\rbrace $ and $B=\\lbrace \\mathbf {v}|\\mathbf {v}\\in \\Omega _{2}\\rbrace $ .",
"For clarity, we introduce two auxiliary variables $\\mathrm {P}_{l}=\\mathrm {P}_{1}-\\mathrm {P}_{2}$ and $\\mathrm {P}_{r}=\\mathrm {P}-\\mathrm {P}_{l}$ , which correspond to the sky blue line and the green line in Figure REF .",
"For the vertices $\\mathbf {v}\\in A\\cap B$ , i.e.",
"the vertices on the splitting path $\\mathrm {P}_{c}$ , their spherical mean value coordinates can be directly computed with respect to the original spherical polygon boundary $\\mathrm {P}$ , i.e.",
"$\\mathbf {v}=\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}}\\lambda _{\\mathbf {v}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i},\\ \\mathrm {if}\\ \\mathbf {v}\\in \\mathrm {P}_{c},$ where $\\lambda _{\\mathbf {v}\\_\\mathbf {v}_{i}}$ is the spherical mean value coordinate of $\\mathbf {v}$ corresponding to $\\mathbf {v}_{i}$ .",
"For the vertices $\\mathbf {v}\\in A-B$ , i.e.",
"the rest vertices in $\\Omega _{1}$ , we first compute their spherical mean value coordinates with respect to $\\mathrm {P}_{1}$ and get $\\mathbf {v}=\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}_{1}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i},\\ \\mathrm {if}\\ \\mathbf {v}\\in A-B.$ Because $\\mathrm {P}_{1}=\\mathrm {P}_{c}\\cup \\mathrm {P}_{l}$ , Equation REF can be written as $\\mathbf {v}=\\sum \\limits _{\\mathbf {v}_{j}\\in \\mathrm {P}_{c}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{j}}\\mathbf {v}_{j}+\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}_{l}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i}.$ By substituting $\\mathbf {v}_{j}\\in \\mathrm {P}_{c}$ with Equation REF , the above equation becomes $\\begin{aligned}\\mathbf {v}=\\sum \\limits _{\\mathbf {v}_{j}\\in \\mathrm {P}_{c}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{j}}\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}}\\lambda _{\\mathbf {v}_{j}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i}+\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}_{l}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i}\\\\=\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}}\\sum \\limits _{\\mathbf {v}_{j}\\in \\mathrm {P}_{c}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{j}}\\lambda _{\\mathbf {v}_{j}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i}+\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}_{l}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i}.\\end{aligned}$ Since $\\mathrm {P}=\\mathrm {P}_{l}\\cup \\mathrm {P}_{r}$ , we further rewrite Equation REF as $\\begin{split}\\mathbf {v}=\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}_{l}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i}+\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}_{l}}\\sum \\limits _{\\mathbf {v}_{j}\\in \\mathrm {P}_{c}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{j}}\\lambda _{\\mathbf {v}_{j}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i}\\\\+\\sum \\limits _{\\mathbf {v}_{i}\\in \\mathrm {P}_{r}}\\sum \\limits _{\\mathbf {v}_{j}\\in \\mathrm {P}_{c}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{j}}\\lambda _{\\mathbf {v}_{j}\\_\\mathbf {v}_{i}}\\mathbf {v}_{i}.\\end{split}$ Then the coordinates for vertices $\\mathbf {v}\\in A-B$ with respect to the original spherical polygon boundary $\\mathrm {P}$ can be expressed as $\\lambda _{\\mathbf {v}\\_\\mathbf {v}_{i}}=\\left\\lbrace \\begin{array}{lr}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{i}} + \\sum \\limits _{\\mathbf {v}_{j}\\in \\mathrm {P}_{c}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{j}}\\lambda _{\\mathbf {v}_{j}\\_\\mathbf {v}_{i}},&\\mathrm {if}\\ \\mathbf {v}_{i}\\in \\mathrm {P}_{l}; \\\\\\\\\\sum \\limits _{\\mathbf {v}_{j}\\in \\mathrm {P}_{c}}\\bar{\\lambda }_{\\mathbf {v}\\_\\mathbf {v}_{j}}\\lambda _{\\mathbf {v}_{j}\\_\\mathbf {v}_{i}}, &\\mathrm {if}\\ \\mathbf {v}_{i}\\in \\mathrm {P}_{r}.",
"\\\\\\end{array}\\right.$ These coordinates are used as the weights to diffuse intensity difference when cloning the patch.",
"A similar way can be taken to compute the coordinates for the vertices $\\mathbf {v}\\in B-A$ .",
"It is worthy pointing out that the coordinates computed by the splitting-based method are different from those computed by the basic method in Section .",
"We think the reason is that when computing with respect to different polygons, the tangent and the reciprocal of the distance, both of which are nonlinear functions, will get different values."
],
[
"Splitting Path",
"The key point for the splitting-based cloning to work is breaking the large patch into two small patches, each covering less than 180$^\\circ $ .",
"Because the source patches are selected on the unrolled spherical images, they unlikely cover more than 180$^\\circ $ vertically.",
"Accordingly, a safe way to get two reasonable patches is vertically splitting the spherical polygon in the middle.",
"Figure: Compared with the planar image cloning, our method can preserve the shape of the cloned object: (top) the source panorama and selected cloning regions, (middle) the cloning results using planar image cloning method , (below) the cloning results using our method.To be specific, we first find the median of the azimuthal angles of the boundary vertices.",
"Based on it, a great circle that passes through $z$ -axis is constructed (the green curve in Figure REF ).",
"We assume that the great circle and the spherical polygon have only two intersection points.",
"Next, we find the boundary vertices nearest to each of the intersection points, and compute the splitting path (marked as the red curve in Figure REF ) as the shortest path of mesh points connecting the two boundary vertices.",
"In addition to this splitting path generation method, we have tested another two solutions.",
"As will be discussed in Section REF , our current method is easy to compute and able to generate consistently pleasing results."
],
[
"Experimental results",
"In the experiments, we use images from SUN360 panorama database .",
"We first compare the cloning results of our method with those generated by planar image cloning, then we show the results generated with and without the splitting when the cloned region covers more than 180$^\\circ $ .",
"After discussing the performance of the splitting-based cloning, we give our timing statistics.",
"Figure: Compared with the planar image cloning , our method does not suffer from the boundary problem when the cloning position gets close to the boundaries of the panorama."
],
[
"Comparison with Planar Image Cloning",
"Figure REF shows the results by our method and the planar image cloning .",
"Although it is hard to distinguish which results are better in the unfolded latitude-longitude panoramic format, it becomes clearer after mapping onto the sphere.",
"In Figure REF (a), we replace the lamp of a room with another one.",
"In the result of the planar image cloning method, the original circular lamp becomes elliptical and the lamp decoration is stretched.",
"On the contrast, our method preserves the lamp shape as well as the decoration.",
"A similar performance is observed in Figure REF (b), in which the flowerpot in the cloning result of the planar method is extruded, while our result does not suffer from the problem.",
"Figure: When the cloned patch covers more than 180 ∘ ^\\circ (a), the cloning directly using spherical mean value coordinates will cause discoloration artifacts (b).",
"Our splitting-based method can deal with this problem (c).Another advantage of our method is that it can naturally circumvent the boundary problem due to the spherical mapping.",
"For the example in Figure REF (a), we clone the inscriptions on the source image to the wall of the target image.",
"When the cloning position gets close to the left boundary of the target spherical image, the cloned region can automatically appear in the right part of the target image with our method.",
"Although the same effect can be achieved by simple tricks in the planar image cloning, it can not deal with the case when the cloning position gets close to the top or bottom boundary of the target image.",
"As shown in Figure REF (b), the planar image cloning results in a much distorted bird in the spherical domain.",
"The main reason is that the left and right boundary of the panorama should be coincided and represent a meridian, while the top and the bottom boundary represent the north and south pole of the sphere respectively."
],
[
"Large Patch Cloning",
"Figure REF gives some examples of cloned regions covering more than 180$^\\circ $ field-of-view.",
"On the first row, we clone a long building between two images.",
"For the result generated by the panorama cloning without splitting, the two wings of the cloned building are discolored and darker than the main part.",
"This is because the target image has higher intensities than the source image, and after we diffuse positive intensity differences with negative weights whose magnitudes are large (see Section REF ), the intensity becomes very small.",
"For the result generated by the cloning with splitting, the intensity of the cloned region is more smooth, and gives more pleasant appearance.",
"The second row gives another example, in which the building in the source image is placed on the platform in the target image.",
"Contrasted with the first example, the two ends of this building are brighter than the central part if cloning without splitting.",
"This is due to the fact that the intensity of the target image in this example is lower than that of the source image.",
"As the cloned region is more complicated, if we directly clone the selected patch to the target image, the cloned region will conflict with the surrounding background of the target image.",
"To make them match well, we compute a matte from the source patch, and remove the unwanted part by modulating the cloned region with the matte.",
"The third row in Figure REF gives one more example that also uses matte modulation.",
"Figure: The splitting paths and corresponding cloning results: (a) the source and target panoramas, (b) the splitting path from the method given in Section , (c) the splitting path computed by performing PCA on the boundary of the spherical polygon (Alternative 1) and (d) the splitting path computed by performing PCA on the boundary of the projected spherical polygon (Alternative 2)."
],
[
"Comparing Different Splitting Path Solutions",
"In the experiments, we observe that the computed coordinates are not equal for different splitting paths.",
"Besides the splitting path generation method given in Section REF , we consider another two possible solutions and make comparison in the following.",
"Alternative 1 performs the PCA on the vertices of the spherical polygon $\\mathrm {P}$ .",
"The first principal component is treated as the direction of the spherical polygon, and induces a great circle, whose normal vector is the principal component.",
"Given the great circle, the splitting path then can be computed as the method given in Section REF .",
"Alternative 2 performs the PCA on the projected spherical polygon $\\bar{\\mathrm {P}}$ , and the principal component is used to determine a great circle, based on which the splitting path is computed.",
"The generated splitting paths and their corresponding cloning results are show in Figure REF .",
"We find when the range of the angles that the spherical polygon covers is considerably large, Alternative 1 may give inappropriate splitting path.",
"As demonstrated in Figure REF (c), one sub-region after splitting still covers more than 180$^\\circ $ , and so the cloned result suffers from discoloration artifacts.",
"On the other hand, Alternative 2 works well even for the cloned patches that are not horizontal.",
"In our experience, the slanted objects that cover more than 180$^\\circ $ are rare.",
"Therefore we choose the method given in Section REF , which avoid the PCA operation and is more timing efficient."
],
[
"Timing Performance",
"We test the timing performance on a desktop installed with Intel(R) Core(TM) i5¨C3330 CPU @ 3.00GHz and NVIDIA GeForce GT 640.",
"Our current implementation constructs the adaptive mesh, computes the spherical mean value coordinate, and evaluates the membrane at each mesh vertex on the CPU.",
"The interpolation of pixels within each triangle of the mesh is performed on the GPU by exploiting the hardware rasterization ability.",
"Table: Performance statistics for panorama cloning: the times for splitting-based cloning are in bold type.The timing is recorded as two parts: the preprocessing time, which correspond to the steps of adaptive mesh construction, spherical mean value coordinates computation and membrane evaluation; and the cloning time, which corresponds to pixel color interpolation.",
"For a specified cloning patch, the preprocessing procedure is carried out only once for different target cloning positions.",
"Table REF gives the running time with respect to different number of boundary pixels, adaptive mesh vertices and cloned pixels.",
"For the 3rd to 5th cases, which clone the patches that cover more than 180$^\\circ $ , we record the time for cloning without and with splitting (bold type).",
"From the table we can see that as the number of boundary pixels and adaptive mesh vertices increases, more time is required for the preprocessing procedure.",
"Although the cloning time increases as the cloned regions become larger, this part of time is relatively short.",
"Another fact is that cloning with splitting takes much more preprocessing time than that without splitting, which seriously affects the timing performance.",
"Note that because panoramas are oversampled near poles, if we clone a patch near the equator of the source panorama to the polar regions of the target panorama, the cloned patch can be obviously blurred.",
"Supersampling is used to deal with this problem, and Table REF shows the cloning times for different supersampling configurations when there are about 510 boundary pixels and 1970 adaptive mesh vertices.",
"The cloning time without supersampling is the same to that of the 2nd case in Table REF , in which the test cases are performed without supersampling.",
"Table REF shows that the cloning time for 2$\\times $ 2 supersampling is approximate to that without supersampling.",
"When more samples are used, the cloning time is increased accordingly.",
"Although it needs about 5ms to render a frame for 16$\\times $ 16 supersampling, it can still clone the patch in real time.",
"Actually for quality, the results shown in this paper are generated using this configuration.",
"Table: Performance statistics for different supersampling configuration"
],
[
"Conclusion",
"In this paper, we propose a coordinate based algorithm to solve the problem of 360$^\\circ $ panorama cloning, which has not been studied previously.",
"In our work, we re-derive spherical mean value coordinates (SMVCs), and use SMVCs as the weights to achieve panorama cloning.",
"To preserve the orientation of the cloned patch, we develop a two-step rotation computation method.",
"A splitting-based method is proposed to remove discoloration artifacts for the case when cloning large patches cover more than 180$^\\circ $ filed-of-view.",
"With the proposed method, we can get satisfactory results and achieve real time cloning performance.",
"As one future improvement, we would like to port the computation of SMVC and the evaluation of mean value coordinate membrane to the GPU."
],
[
"Spherical Mean Value Coordinates",
"The basic idea to compute spherical mean value coordinates of point $\\mathbf {v}$ is first projecting the spherical polygon $\\mathrm {P}$ to the tangent plane of unit sphere at $\\mathbf {v}$ to get a planar polygon $\\bar{\\mathrm {P}}$ , then computing planar mean value coordinates of $\\mathbf {v}$ respect to $\\bar{\\mathrm {P}}$ , followed by an anisotropic scaling.",
"To project the spherical polygon, Langer et al.",
"use gnomonic projection, which would project two antipodal points to the same point on the tangent plane.",
"In our work, we use the generalized stereographic projection to deduce the spherical mean value coordinates.",
"The main advantage of stereographic projection over gnomonic projection is that except for the projection point, it can project the entire sphere onto a plane without introducing overlapping.",
"Particularly, this new deduction is helpful to analysis the decoloration problem.",
"As illustrated in Figure REF , we first move the origin of the 3D coordinate system from the sphere center to the projection point, which is the antipodal point of $\\mathbf {v}$ and noted as $\\dot{\\mathbf {v}}=-\\mathbf {v}.$ In the translated coordinate system, the boundary vertices $\\mathbf {v}_{i}$ of spherical polygon, its projected counterpart $\\bar{\\mathbf {v}}_{i}$ and point $\\mathbf {v}$ are represented by $\\mathbf {v}^{\\prime }_{i}=\\mathbf {v}_{i}-\\dot{\\mathbf {v}}, \\bar{\\mathbf {v}}^{\\prime }_{i}=\\bar{\\mathbf {v}}_{i}-\\dot{\\mathbf {v}}, \\mathbf {v}^{\\prime }=\\mathbf {v}-\\dot{\\mathbf {v}}.$ Following 2D mean value coordinate definition, the planar mean value coordinates of $\\mathbf {v}^{\\prime }$ with respect to the projected polygon $\\bar{\\mathrm {P}}^{\\prime }=\\lbrace \\bar{\\mathbf {v}}^{\\prime }_{1},\\ldots ,\\bar{\\mathbf {v}}^{\\prime }_{n}\\rbrace $ are given by $\\bar{\\lambda }_{i} = \\frac{w_{i}}{\\sum _{j}w_{j}}, \\; w_{i} = \\frac{\\tan \\frac{\\alpha _{i-1}}{2}+\\tan \\frac{\\alpha _{i}}{2}}{d_{i}},$ where $\\alpha _{i}$ is the signed angle between vectors $\\mathbf {v}^{\\prime }\\times \\mathbf {v}^{\\prime }_{i}$ and $\\mathbf {v}^{\\prime }\\times \\mathbf {v}^{\\prime }_{i+1}$ , and the distance $d_{i}=\\Vert \\bar{\\mathbf {v}}^{\\prime }_{i}-\\mathbf {v}^{\\prime }\\Vert $ as shown in Figure REF .",
"Because each $\\bar{\\mathbf {v}}^{\\prime }_{i}$ is a scaling of $\\mathbf {v}^{\\prime }_{i}$ , the coordinate $\\tilde{\\lambda }_{i}$ of $\\mathbf {v}^{\\prime }$ with respect to the spherical polygon $\\mathrm {P}^{\\prime }=\\lbrace \\mathbf {v}^{\\prime }_{1},\\ldots ,\\mathbf {v}^{\\prime }_{n}\\rbrace $ (in the translated coordinate system) can be constructed from $\\bar{\\lambda }_{i}$ , i.e., $\\tilde{\\lambda }_{i}=\\frac{\\Vert \\bar{\\mathbf {v}}^{\\prime }_{i}\\Vert }{\\Vert \\mathbf {v}^{\\prime }_{i}\\Vert }\\bar{\\lambda }_{i}=\\frac{2}{1+\\cos \\theta _{i}}\\bar{\\lambda }_{i},$ where $\\theta _{i}$ is the angle between $\\mathbf {v}$ and $\\mathbf {v}_{i}$ .",
"What's more, the coordinate $\\tilde{\\lambda }_{i}$ defined in Equation REF satisfies $\\sum \\nolimits _{i}\\tilde{\\lambda }_{i}\\mathbf {v}^{\\prime }_{i}=\\mathbf {v}^{\\prime }.$ Note that the current computation is operated in the translated coordinate system (originated at $-\\mathbf {v}$ ), while our goal is computing the spherical mean value coordinate $\\lambda _{i}$ of point $\\mathbf {v}$ with respect to the spherical polygon $\\mathrm {P}$ (in the original coordinate system centered at sphere center), which should satisfy $\\sum \\nolimits _{i}\\lambda _{i}\\mathbf {v}_{i}=\\mathbf {v}.$ Substituting $\\mathbf {v}^{\\prime }_{i}$ and $\\mathbf {v}^{\\prime }$ with Equation REF , Equation REF becomes $\\sum \\nolimits _{i}\\tilde{\\lambda }_{i}(\\mathbf {v}_{i}+\\mathbf {v})=2\\mathbf {v},$ which can be further rewritten as $\\sum \\nolimits _{i}\\tilde{\\lambda }_{i}\\mathbf {v}_{i}=(2-\\sum \\nolimits _{i}\\tilde{\\lambda }_{i})\\mathbf {v}.$ By comparing Equation REF with the above equation, we can get $\\lambda _{i} = \\frac{\\tilde{\\lambda }_{i}}{2-\\sum _{i}\\tilde{\\lambda }_{i}}.$ By using Equation REF , REF and REF in addition to some trigonometrics identities, the spherical mean value coordinates will have the final form as $\\lambda _{i}(\\mathbf {v})=\\frac{(\\tan \\frac{\\alpha _{i-1}}{2}+\\tan \\frac{\\alpha _{i}}{2})/\\sin \\theta _{i}}{\\sum \\nolimits _{j}\\cot \\theta _{j}(\\tan \\frac{\\alpha _{j-1}}{2}+\\tan \\frac{\\alpha _{j}}{2})}.$"
]
] | 1709.01638 |
[
[
"Estimating the outcome of spreading processes on networks with\n incomplete information: a mesoscale approach"
],
[
"Abstract Recent advances in data collection have facilitated the access to time-resolved human proximity data that can conveniently be represented as temporal networks of contacts between individuals.",
"While this type of data is fundamental to investigate how information or diseases propagate in a population, it often suffers from incompleteness, which possibly leads to biased conclusions.",
"A major challenge is thus to estimate the outcome of spreading processes occurring on temporal networks built from partial information.",
"To cope with this problem, we devise an approach based on Non-negative Tensor Factorization (NTF) -- a dimensionality reduction technique from multi-linear algebra.",
"The key idea is to learn a low-dimensional representation of the temporal network built from partial information, to adapt it to take into account temporal and structural heterogeneity properties known to be crucial for spreading processes occurring on networks, and to construct in this way a surrogate network similar to the complete original network.",
"To test our method, we consider several human-proximity networks, on which we simulate a loss of data.",
"Using our approach on the resulting partial networks, we build a surrogate version of the complete network for each.",
"We then compare the outcome of a spreading process on the complete networks (non altered by a loss of data) and on the surrogate networks.",
"We observe that the epidemic sizes obtained using the surrogate networks are in good agreement with those measured on the complete networks.",
"Finally, we propose an extension of our framework when additional data sources are available to cope with the missing data problem."
],
[
"Introduction",
"The advances made in data collection technologies have led to a wealth of high-resolution time-resolved data.",
"Mobile sensing devices, social networking applications and wearable sensors have indeed significantly contributed to monitor social interactions and physical proximity of individuals in time [1], [2], [3], [4], [5], [6].",
"Such fine-grained data monitoring is crucial for a deeper study of human proximity dynamics described by complex temporal networks – in which links are drawn between nodes representing individuals when they are in close-range [7] – and their interplay with contagion processes.",
"Physical proximity interactions play indeed a fundamental role in conveying information or in the spread of diseases [8], [9], [10], [11].",
"They can thus inform our understanding of how messages or infectious diseases such as flu-like illnesses propagate among individuals.",
"However, despite the efforts made to increase the accuracy in the data collection, relational data often suffer from incompleteness resulting in missing links in empirical networks [12].",
"This lack of information can arise for several reasons: limited participation during surveys, incomplete records (diary-based, device-based) [13], [14], [15], [16], and technical issues occurring during the data collection process.",
"In the case of smartphone sensing such as in [3], [5], proximity might be undetected during some time windows.",
"For instance, people might turn off their Bluetooth or Communication Data Records might not provide access to the vicinity of individuals to a cell tower at each time, possibly leading to undetected co-presence events.",
"In the case of self-reporting of sexual relationships, individuals might choose not to disclose all of their partners, which leads to biases when focusing on the spread of sexually transmitted disease [17].",
"Data incompleteness can affect the measured properties and structure of temporal networks [18], [19] and can reflect on the simulated evolutions of contagion processes, leading to inaccurate conclusions [20], [21].",
"The investigation of information or disease propagation processes on networks built from such data must thus be undertaken carefully.",
"Several approaches have been put forward to cope with missing links in networks [22].",
"These methods include: distributional models, which estimate the likelihood of the presence of a link on the basis of the observed links and nodes attributes [23], hierarchical structure methods [24], stochastic block models [25], and expectation maximization methods [26], which try to extract the connectivity patterns in the available part of the network to infer and complete the unknown part.",
"However, the goal of these methods is the exact recovery of single links, a complicated task that becomes nearly impossible to achieve when a large amount of data is missing.",
"Moreover, this might actually not be necessary if the goal is to estimate the global outcome of a contagion process at the population level and not the risk concerning a specific individual.",
"Other approaches have been developed to specifically estimate important properties of epidemic spread and information cascade without trying to recover the original network [27], [20], [15], [28], [29].",
"These methods are however either process specific or rely on the existence of known mesocale structures (such as groups in the population) in the network, together with the knowledge of the structure to which each population member belongs.",
"Here, we propose a self-contained approach that does not rely on such a priori knowledge.",
"As in [20], [15], the aim of this work is not to recover the exact links which are missing in the data but instead to build a surrogate version of the network of interactions.",
"The main difference with these previous works, that is a building block of our approach, is that we uncover in the incomplete data mesoscale structures that involve nodes affected by missing activity.",
"To study the network at the mesoscale level, we take advantage of tensor decomposition techniques, already applied in several fields [30], to extract both the topological and temporal properties of the network [31].",
"An important aspect of our method is that it does not rely on the availability of metadata or external information on the nodes, and that it allows us to recover fundamental properties of the studied temporal network, such as the temporal activity of nodes with partial information (i.e., the temporal evolution of their number of contacts).",
"We use this approach to build a surrogate version of the network that yields a correct estimate of the outcome of a simulated spreading process occurring on the network.",
"Furthermore, we extend our method to take advantage of other sources of information that might be available, such as information deriving from subsidiary data sources.",
"The paper is organized as follows: in Section we present the notations used in the paper; in section we describe the problem statement; in Section we explain the method that we developed to carry out the study; in Section we report the results achieved by our approach in the study of several temporal human proximity networks; in Section we discuss the performance and limitations of the method and future research directions."
],
[
"Notations",
"The following notations are used throughout the present paper.",
"Lower-case letters denote scalar variables, e.g., $t$ , capital letters denote defined constants, e.g., $T$ , and boldface lower-case letters denote vectors, e.g., $\\mathbf {t}$ .",
"Matrices are denoted by boldface capital letters, e.g., $\\mathbf {T}$ , where the $i$ -th column of a matrix $\\mathbf {T}$ is $\\mathbf {t}_i$ and the $\\left(i,j\\right)$ -entry is $t_{ij}$ .",
"Third order tensors are denoted by bold calligraphic letters, e.g., ${T}$ , whose $\\left(i,j,k\\right)$ -entry is $t_{ijk}$ .",
"The tensor product is denoted by $\\circ $ , the Hadamard (element-wise) product by $\\ast $ , the Kronecker product by $\\otimes $ , and the outer product by $\\cdot $ .",
"$\\Vert \\cdot \\Vert _F$ stands for the Frobenius norm."
],
[
"Problem Statement",
"The purpose of the present work is the development of a methodology able to reproduce the outcome of contagion processes on temporal networks of human interactions, starting from incomplete information on these networks.",
"We consider a scenario in which part of the activity of a fraction of the network nodes (i.e.",
"part of their interactions over time) is missing in the data.",
"To provide an estimate of the outcome of spreading processes on the network, we do not try to recover the exact missing links and interaction events of these nodes.",
"Our method aims instead at building a surrogate version of the complete network by taking advantage of the only information that is available.",
"As we will describe in details in Sec.",
"REF and Sec.",
"REF , this information can be related either only to the partial network of human contacts or to richer data than can be used as a proxy for human proximity.",
"For instance, we could have access both to the partial temporal contact network and to the approximated location of individuals provided by smartphones, through the GPS, Bluetooth or WiFi signals: such additional information can conveniently be represented as a temporal bipartite network between individuals and locations, in which a link is drawn between an individual and a location when the individual is detected in that location.",
"We will show how such information can be integrated in our framework.",
"The underlying assumption of our method is that we can leverage the mesoscale properties of the partial network, such as the presence of correlations in the node activities, to build a surrogate version of the complete network.",
"To extract these mesoscale properties from the incomplete data we rely on the Non-negative Tensor Factorization (NTF) technique [30], .",
"In particular, our method is based on a NTF framework handling missing values [33], [34].",
"By applying the NTF on a tensor representing a temporal network we can indeed identify groups of nodes having similar connectivity patterns and whose links have similar activation times.",
"Each of these groups can be seen as a sub-network.",
"By studying the structure and the temporal activity of each sub-network, we can infer the properties of the nodes whose activity is partially missing.",
"However, standard tensor decomposition techniques such as NTF are based on the assumption that noise in the data follows a Gaussian distribution.",
"Thus, even if the data are intrinsically heterogeneous, as is typically the case in human contact data [35], the decomposition will provide a low-rank approximation in which these heterogeneous characteristics are not preserved [36].",
"As a result, the NTF tends to approximate the network in a way that makes the properties of the sub-networks extracted more homogeneous than in the original network.",
"This trend towards homogeneity changes the temporal and structural properties of the network such as the distribution of the number of contacts per link (i.e., the number of activation times of each link) in the aggregated network, and thus has to be carefully taken into account.",
"Indeed, heterogeneity, both in the aggregated number or duration of contacts per link and in temporal properties (e.g., broad inter-event time distributions) have been shown to play a key role in spreading processes occurring over static and temporal networks [37], [38], [39], [40], [41], [42], [43].",
"Since we are interested in recovering the outcome of such processes, we need to preserve the heterogeneity properties of the network.",
"Thus, instead of directly using the approximated network provided by the NTF, we will use the information about the nodes belonging to each sub-network and their related activity patterns as a guide to build a surrogate network.",
"In this surrogate network we will reintroduce the heterogeneity properties as described below.",
"Our method to build a surrogate network from partial information and recover the outcome of contagion processes is thus divided in two main steps: .",
"The extraction of mesoscale structures from the partial temporal network through NTF adapted to handle incomplete information; .",
"the construction of a surrogate network, in which we use both the information provided by step REF and the heterogeneity properties of the network."
],
[
"Extracting mesoscale structures from a partial temporal network",
"Three-way tensors (i.e.",
"three-dimensional arrays) are natural representations of temporal networks: given an undirected temporal network, composed by $N$ nodes and $k =1,\\dots ,K$ time intervals, we can represent its snapshots $G_k = (V,E_k)$ , which have $|V|=N$ nodes and set of links $E_k$ , by $K$ adjacency matrices $\\mathbf {M}_k\\in \\mathbb {R}^{N\\times N}$ of the form: $\\mathbf {M}_k ={\\left\\lbrace \\begin{array}{ll}m_{ij} = 1 \\;\\; \\mbox{if} \\;\\;\\left(i,j\\right) \\in E_k\\,, \\\\m_{ij} = 0 \\;\\; \\mbox{otherwise}\\,.\\end{array}\\right.",
"}$ These adjacency matrices form the slices of a tensor ${T}\\in \\mathbb {R}^{I\\times J\\times K}$ , with here $I=J=N$ .",
"In the case of missing data, zero values in the tensor can either correspond to no activity or to undetected activity.",
"We however need to factorize only the part of the tensor that corresponds either to measured activity or to actual inactivity (absence of contact).",
"To this aim, the zero entries that correspond to possibly undetected activities have to be masked in the tensor.",
"If a node with partial information has no activity at all measured during a given snapshot (in a given slice of the tensor), we consider that the related zero entries in the tensor ${T}$ might correspond to possibly undetected activities, i.e., possibly missing contacts.",
"On the other hand, if a node, for which we know that only partial information is present, has at least one contact measured in a slice, we assume that no information was lost at all in that tensor slice because of that node.",
"In other terms, we assume for each node and each time slice that either all or none of the activity recorded by that node in that time slice is present in the data.",
"This would be for instance the case if a sensor measuring proximity is turned off or if the GPS coordinates of a mobile user are not collected during a time window: all the proximity information concerning this sensor in this time window is then lost.",
"However, if the measuring device is not turned off, all the proximity relationships with other devices that are also on during that time window are present in the incomplete data.",
"We thus introduce a binary tensor ${W}$ , of the same size of ${T}$ , whose entries are defined as $w_{ijk} ={\\left\\lbrace \\begin{array}{ll}0 \\;\\; \\mbox{if} \\;\\; \\mbox{$i$ or $j$ has no activity in the time window $k$,}\\; \\\\1 \\;\\; \\mbox{otherwise.}",
"\\,\\end{array}\\right.",
"}$ This tensor is used to mask the part of the tensor ${T}$ that might be linked to possibly undetected activity.",
"The approximation of the masked tensor $ {T}\\ast {W}$ consists in minimizing the following cost function with non-negative constraints [34]: fw(,A, B, C) = W(T - Rr=1r arbr cr)2F , where $R$ is the rank of the approximation and is hereinafter called the number of components.",
"The vectors $\\mathbf {a}_r$ , $\\mathbf {b}_r$ , and $\\mathbf {c}_r$ with $r\\in \\left[1,R\\right]$ form respectively the factor matrices $\\mathbf {A}\\in \\mathbb {R}^{I\\times R}$ , $\\mathbf {B}\\in \\mathbb {R}^{J\\times R}$ , $\\mathbf {C}\\in \\mathbb {R}^{K\\times R}$ .",
"Each tuple of vectors $(\\mathbf {a}_r,\\mathbf {b}_r, \\mathbf {c}_r)$ describes a mesocale strucure.",
"As we explain below, $\\mathbf {a}_r$ and $\\mathbf {b}_r$ indicate which nodes and links participate to the mesoscale structure $r$ , while $\\mathbf {c}_r$ describes the temporal activity of the structure $r$ .",
"For the sake of readability, the cost function is re-written in the following form: fw(,A, B, C) = W(T - ; A, B, C)2F .",
"Using the mask ${W}$ amounts to approximating the tensor ${T}$ based only on information of which we are certain: we are approximating only the part of the tensor composed of elements that are either 1 corresponding to measured events or 0 corresponding to real absence of contact, without taking into account all the 0 values that might correspond to undetected activity.",
"In other terms, minimizing the cost function $f_w$ corresponds to finding the best approximation of the non-masked part of the tensor by a sum of components corresponding each to a mesoscale structure.",
"The analysis of $ \\mathbf {A}, \\mathbf {B}$ and $\\mathbf {C}$ yields then information on which link is involved in which mesocale structure, including in particular the links involving nodes with incomplete information.",
"Several methods are available to estimate the factor matrices $\\mathbf {A}, \\mathbf {B}$ and $\\mathbf {C}$ [44], [45], and details are given in Section ."
],
[
"Extracting mesoscale structures from coupled temporal networks",
"Here, we propose an extension of the mesoscale structure detection method to the case in which we have access to richer information – not expressible in one single temporal network.",
"The integration of such information might help to better recover the missing entries in the tensor describing the temporal contact network: for instance, if we have access, in addition to the partial contact network, to the location of individuals involved in the contacts, the latter can contribute to recover the missing information on contacts.",
"To integrate such additional data, we propose to use the so-called Joint Non-negative Tensor Factorization (JNTF) [46] that makes it possible to decompose multiple temporal networks at once in a coupled manner (in practice, we will consider the partial temporal contact network and a position network evolving with time).",
"Let us consider the general case of $S$ different temporal networks, each represented by a tensor ${T}_s$ , with possibly different dimensions as they might represent different types of information.",
"We can approximate them in a coupled way by computing the following minimization problem with non-negative constraints: minSs=1Ts - s; As, Bs, Cs 2F s.t.",
"s, As, Bs, Cs0 .",
"Different couplings can be considered by imposing that some of the factor matrices $\\mathbf {A}_s, \\mathbf {B}_s, \\mathbf {C}_s$ in the equation are equal for different values of $s$ .",
"The idea behind the introduction of this coupling is that different networks can provide partially redundant information, and that this redundancy is relevant for recovering missing entries.",
"For instance, if we have access to the locations of nodes but only to a partial information concerning their contacts, we can couple the decomposition of the tensor representing the partial contact network to the decomposition of the tensor representing the time evolution of the location of nodes, and gain in this way information on the possible contacts over time.",
"In practice, we would impose $\\mathbf {A}_1=\\mathbf {A}_2$ and $\\mathbf {C}_1=\\mathbf {C}_2$ , which correspond respectively to imposing the same nodes' memberships and the same activity timeline for each mesoscale structure in the resulting approximations of the contact and location tensors.",
"The joint factorization of tensors – including one with missing information ${T}_{s^{\\prime }}$ –, can be adapted to handle missing values in the same way as the NTF: $\\min &\\Big ( \\alpha _{s^{\\prime }}\\Vert {W}\\ast \\left({T}_{s^{\\prime }} - \\llbracket \\lambda _{s^{\\prime }}; \\mathbf {A}_{s^{\\prime }}, \\mathbf {B}_{s^{\\prime }}, \\mathbf {C}_{s^{\\prime }}\\rrbracket \\right)\\Vert ^2_F \\\\+ & \\sum ^S_{{s=1 \\\\ s\\ne s^{\\prime }}}\\alpha _{s}\\Vert {T}_s-\\llbracket \\lambda _s; \\mathbf {A}_s, \\mathbf {B}_s, \\mathbf {C}_s\\rrbracket \\Vert ^2_F \\Big )\\\\&\\mbox{s.t.",
"}\\;\\; \\lambda _s, \\mathbf {A}_s, \\mathbf {B}_s, \\mathbf {C}_s\\ge 0 \\;\\forall s\\nonumber \\,.$ To solve this minimization problem we adapted the active-set-like method with Karush-Kuhn-Tucker optimality conditions [47], [44] (see Section for details).",
"In the following, we set all the $\\alpha _s$ parameters equal to 1.",
"We note however that they could be used to tune the relevance of the information provided by each tensor.",
"As an example, by setting $\\alpha _{s^{\\prime }}>\\alpha _{s}$ , $\\forall s \\ne s^{\\prime }$ we would give more importance to the information provided by the network with partial information than to the one given by the subsidiary data."
],
[
"Surrogate network",
"Both types of factorization, either from a partial network or from coupled temporal networks, provide approximations ${T}_{app}$ of the temporal networks as sums of components, each corresponding to a mesoscale structure.",
"Each mesoscale structure is fully described by the respective columns of $\\mathbf {A}$ , $\\mathbf {B}$ and $\\mathbf {C}$ and can be seen as a sub-network composed of links with similar temporal properties, as we now describe.",
"To analyze the mesoscale structures revealed by the factorization step and interpret them as sub-networks, we estimate the memberships of the links using $\\mathbf {A}$ and $\\mathbf {B}$ and we determine the activation times by binarizing $\\mathbf {C}$ .",
"First, the membership weight of each link $\\left(i,j\\right)$ to the $r$ -th component is given by the $\\left(i,j\\right)^{th}$ element of the following Kronecker product: $\\mathbf {a}_r\\otimes \\mathbf {b}_r = \\mathbf {a}_r\\cdot \\mathbf {b}^T_r .$ As we consider undirected networks, the actual membership of each link is symmetrized in the following way: $\\frac{\\mathbf {a}_r\\cdot \\mathbf {b}^T_r+ \\mathbf {b}_r\\cdot \\mathbf {a}^T_r}{2} .$ We consider that the links with the largest membership weights, composing $95\\%$ of the total sum of the squared memberships, belong to the component $r$ (note that, depending on the membership values, a link could belong to more than one component).",
"Then, to detect the times in which each mesoscale structure is active, we consider the matrix $\\mathbf {C}$ , which summarizes the temporal activity of each component: by using the Otsu method [48], a common way to perform binary thresholding, we transform the temporal activity of each component in its binary version (1 if it is active, 0 otherwise).",
"These two steps allow us to determine for each mesoscale structure when it is active and which links it involves.",
"For each structure, we then select the links involving at least one node with partial information and the times in which its activity was potentially lost (in the sense that no activity of that node was recorded at all for this particular time), and we add the corresponding elements of the structure to the partial network ${T}$ to create the surrogate tensor ${T}_{surrogate}$ .",
"The rationale is that the factorization has allowed us to determine, for the links for which only partial information is available, to which mesoscale structures they belong.",
"Using the activity timelines of the mesoscale structures, we thus reconstruct the missing parts of the activity timelines of these links.",
"Moreover, the binarization step enables to reintroduce the heterogeneity in the network.",
"As mentioned previously, the properties of the sub-networks extracted tend to be more homogenous, in particular the distribution of weights of each sub-network is more homogenous than in the real data.",
"Indeed, the factorization finds groups of links that have strongly correlated activity timelines and approximates them as fully correlated.",
"As a result, the distribution of the link weights in the network described by ${T}_{app}$ – the number of times in which each link is active – becomes more homogeneous and the activity tends to be less bursty than in the empirical network.",
"By binarizing the approximation in the way described above we are removing links which were associated to small weights and increasing weights of links with the highest weights in each component of the approximated case.",
"This contributes to make the overall distribution of weights more heterogenous.",
"When the JNTF is used, an additional step turns out to be relevant.",
"The JNTF indeed yields components influenced by both the activity in the partial proximity network and the subsidiary temporal network (such as a network linking individuals and locations).",
"Thus, the JNTF will approximate the network in a way based also on co-presence events of individuals.",
"As co-presence is a necessary but not sufficient condition for two individuals to be in contact, the links added when creating ${T}_{app}$ are less likely to correspond exactly to contacts that were missing than in the NTF case.",
"Moreover, for the same reason the number of links in ${T}_{app}$ and their weights might be greater than in the original data and they might present less bursty activity patterns.",
"As mentioned in REF , the weight of the information provided by the second tensor can of course be tuned in the JNTF by adding coefficients so that the components extracted by the factorization are more representative of the network with partial information than of the other one.",
"The investigation of this possibility is however outside the scope of this paper.",
"The procedure of binarization of ${T}_{app}$ described above needs thus to be completed to make the properties of the network more heterogeneous.",
"In order to do this, we have to discard some of the times in which the links involving nodes with partial information are active in ${T}_{app}$ to compensate for the overestimation and get closer to the empirical network.",
"The key idea here is to zero out some elements in ${T}_{surrogate}$ to recover a weight distribution comparable to the empirical one, measured on the links involving only nodes with no missing information.",
"We pick from this distribution a number of weights equal to the number of links having partial information, for which we have to adjust the weights, and we assign them at random to these links.",
"Finally, we compare for each link the new value to its weight in ${T}_{surrogate}$ : if the new weight is smaller than the old one, we erase at random parts of the link activity that are present in ${T}_{surrogate}$ , until we reach the new weight; if instead the new weight is larger than the old one, we do not act on that link's activity.",
"The reason why we can rely on the weight distribution measured on the partial network is due to its robustness to sampling, as shown in [4], [15], [20], [27] for various sampling procedures and in the Supplementary Information S.REF for the sampling considered here.",
"After the reassignment of weights, the resulting tensor ${T}_{surrogate}$ is used to approximate the whole temporal network of contacts and perform simulations of spreading processes."
],
[
"Results",
"We apply the method described in the previous sections to three temporal human proximity networks.",
"The data were collected by the SocioPatterns collaboration (www.sociopatterns.org) in two conferences in Italy (HT09) and France (SFHH), and in a primary school in France (LSCH).",
"In each case, the proximity of individuals in the network was measured with wearable sensors able to capture face-to-face contacts occurring in a $1-2$ -meter range with a 20-second time resolution.",
"As for the purpose of the present paper we do not need such a high-resolution, we aggregate the data to a 15-minute resolution.",
"The HT09 dataset was collected during the ACM Hypertext 2009 conference [49].",
"The resulting temporal network has $N=113$ nodes and $K=237$ snapshots.",
"The SFHH dataset was collected during the conference of the Société Française d'Hygiène Hospitalière, yielding a temporal network with $N=417$ nodes and $K=129$ time snapshots [4].",
"Finally, the LSCH dataset was collected in a primary school in Lyon [50].",
"The resulting network has $N=241$ nodes and $K=130$ snapshots.",
"We consider all the snapshots as unweighted networks (meaning that each element of the tensor is either 0 or 1).",
"We simulate on each dataset a loss of data determined by a fraction of nodes $p_{nodes} \\in \\left[0.1, 0.2, 0.4\\right]$ for which we zero out the activity occurring during the first half of the total timespan, i.e., the information concerning each of these nodes is lost over a fraction $p_{times}=0.5$ of the temporal snapshots.",
"In practice, for each dataset we start from the complete tensor ${\\hat{T}}$ and we create a tensor with partial information ${T}$ by erasing, in the first half of the temporal slices of ${\\hat{T}}$ , the elements related to $N p_{nodes}$ chosen at random.",
"To test the limits of our method, we also considered cases where either all the nodes lost some information or where a fraction of the nodes lost all their activity.",
"More details on the cases dealt with are provided below."
],
[
"Approximated network",
"For each dataset and for each data loss scenario, we performed the approximation of ${T}$ , i.e., the minimization of Eq.",
"(REF ), with a number of components selected using the so called Core Consistency Diagnostic [51] as a guide (see Sec.",
"for details).",
"We perform 20 decompositions in each case, varying the initial conditions, and we use the one with the highest core consistency value.",
"Moreover, for each value of $p_{nodes}$ we repeat the procedure on 10 different sets of nodes with missing information chosen at random, in order to evaluate the variations in the performance of our method.",
"From the factorization of the tensors with missing information ${T}_{HT09}$ , ${T}_{SFHH}$ , and ${T}_{LSCH}$ , we recover a first approximated version of the complete network ${T}_{app}$ for each dataset.",
"We evaluate the results provided by the decomposition by computing the Pearson's coefficient between the complete and approximated temporal activities of the nodes for which only partial information is available in ${T}$ .",
"The correlation is measured only on the part of the activity that is missing in the partial data.",
"We report the correlation coefficients found in Table REF and we show in the Supplementary Information S.REF a representative example of the temporal activities of these nodes with partial information in the complete, partial, and approximated networks.",
"The correlation coefficients shown are measured on 10 different sets of nodes with missing information for each dataset.",
"The ranges of Pearson's coefficient indicate positive moderate to high correlation between the complete and approximated node activity, indicating a good recovery of the node temporal activities.",
"Let us notice that by construction the approximation through the factorization relies on the existence of correlated activity patterns and consequently performs better in the presence of strong such patterns.",
"This explains why the temporal activities are better recovered (larger Pearson correlation coefficients) in the LSCH case than in HT09 and SFHH.",
"Indeed, in schools the schedule is quite constrained and all the students of a given class have highly correlated activity timelines, leading to stronger correlated activity patterns than in conferences, during which attendees are free to move and interact with different people at different times.",
"Table: Range of values of the Pearson's coefficient computed by comparing the original and approximated node activity for each node inthe different sets considered.",
"The table also reports the median value obtained.",
"The p-values are lower than 10 -3 10^{-3}."
],
[
"Surrogate network",
"As illustrated in the previous paragraphs, the approximation step achieves good results to recover single node temporal activities by using NTF handling missing information.",
"However, as discussed above, heterogeneity properties of the link weights are crucial in determining the outcome of spreading processes.",
"We show indeed in the Supplementary Information S.REF that simulating a spreading process directly on an approximated network ${T}_{app}$ without the binarization step results in a strong underestimation of the original outcome of the spreading process.",
"We thus apply the procedure described above and build a surrogate temporal network ${T}_{surrogate}$ relevant to estimate the outcome of spreading processes.",
"For the JNTF, for each link involving nodes with partial information, we assign a weight extracted from the distribution of weights measured on the links with no missing information, and, if needed, we remove a part of the link's activity present in ${T}_{surrogate}$ but not in ${T}$ until we match this weight."
],
[
"Estimate of the outcome of spreading processes",
"To evaluate the performance of the method in estimating the outcome of spreading processes on the network, we simulate susceptible-infected-recovered (SIR) processes (see details in Sec.",
"REF ) over three temporal networks for each dataset: the complete, the partial, and the surrogate network.",
"In each case we run multiple simulations for each set of values ($\\beta $ , $\\mu $ ) of the infection and recovery probabilities per unit time.",
"We focus on the couples of probabilities $\\left(\\beta ,\\mu \\right)$ that satisfy the following criterion.",
"The spreading has to be finished within the timespan of the dataset, and the epidemic size, defined as the final fraction of recovered individuals, has to be greater than $20\\%$ and lower than $80\\%$ (the selection is based on the median of the epidemic size).",
"These conditions ensure to avoid the selection of parameters such that the simulations either never reach a significant epidemic size and/or are too slow with respect to the total timespan of the network.",
"The limit for the final number of recovered individuals to $80\\%$ of the entire population prevents the selection of parameters leading to too fast spreading that are far from realistic conditions.",
"For each selected pair of parameter values $\\beta $ and $\\mu $ , we compute the distribution of the epidemic sizes in the three cases (here called “Complete\", “Partial\", “Surrogate\").",
"As we simulate a loss of data by considering 10 different sets of randomly chosen nodes with partial information, we report the median distribution of the epidemic size on the simulations over these 10 cases as well as the 25-th and 75-th percentiles.",
"We first consider the case in which the only available information is the partial temporal network ${T}$ , and then a case in which an additional source of information is available, namely the location of individuals over time."
],
[
"Using only the partial temporal network",
"In Fig.",
"REF we report the epidemic size distributions obtained for the LSCH dataset, for two couples of selected spreading parameters: a) $\\beta = 0.3$ , $\\mu = 0.3$ and b) $\\beta = 0.15$ , $\\mu = 0.25$ and for different fractions of nodes $p_{nodes}$ with partial information.",
"The figure shows that our method yields a good approximation of the distribution obtained in the original case, while the one obtained by simply simulating the SIR process on the network with partial information strongly underestimates the epidemic size.",
"This underestimation becomes stronger as $p_{nodes}$ increases while the estimation given by the surrogate network remains of good quality.",
"Figure: Distributions of epidemic sizes computed in the complete, partial, and surrogate cases for the LSCH dataset.",
"Each panel corresponds to onevalue of the fraction p nodes p_{nodes} of nodes with partial information and one couple of spreading parameters: a) β=0.3\\beta = 0.3 and μ=0.3\\mu = 0.3; b) β=0.15\\beta = 0.15 and μ=0.25\\mu = 0.25.For the partial and surrogate cases, the symbols andlines show the median distribution of the epidemic size computed from the results relative to the 10 different sets of nodes with partial information,while the shaded area is delimited by the 25-th and 75-th percentiles.Figure: Distributions of epidemic sizes computed in the complete, partial, and surrogate cases for the HT09 dataset.",
"Each panel corresponds to onevalue of the fraction p nodes p_{nodes} of nodes with partial information and one couple ofspreading parameters: a) β=0.60\\beta = 0.60 and μ=0.10\\mu = 0.10; b) β=0.25\\beta = 0.25 and μ=0.05\\mu = 0.05.The symbols and lines show the median distribution of the epidemic size computed from the results relative to the 10 different sets of nodes with partial information, while the shaded area is delimited by the 25-th and 75-th percentiles.Figure: Distributions of epidemic sizes computed in the complete, partial, and surrogate cases for the SFHH dataset.",
"Each panel corresponds to onevalue of the fraction p nodes p_{nodes} of nodes with partial information and one couple ofspreading parameters: a) β=0.3\\beta = 0.3 and μ=0.10\\mu = 0.10; b) β=0.25\\beta = 0.25 and μ=0.08\\mu = 0.08.For the partial and surrogate cases, the symbols andlines show the median distribution of the epidemic size computed from the results relative to the 10 different sets of nodes with partial information, while the shaded area is delimited by the 25-th and 75-th percentiles.We obtain similar results for the HT09 and SFHH datasets, for which we report the results of numerical simulations of the SIR process for various values of the spreading parameters ($\\beta $ ,$\\mu $ ) and of $p_{nodes}$ in Fig.s REF and REF .",
"In all cases, using only the partial temporal network in the numerical simulations of the SIR process leads to a clear underestimation of the epidemic sizes; this underestimation becomes worse as $p_{nodes}$ increases.",
"The simulations performed on the surrogate network are systematically much closer to the ones based on the complete information, leading thus to a strong improvement in the prediction of the epidemic risk.",
"For the SFHH case however, we observe that the agreement becomes worse when $p_{nodes}$ increases.",
"This is due to the fact that during large conferences attendees tend to follow the schedule less rigorously than in small ones and move around and engage in contacts more freely and in a more random way, thus leading to less correlated activity patterns.",
"In such a case, auxiliary data could be useful, as in the case we describe in the next section."
],
[
"Using both the partial network and an additional proxy",
"By construction, the method based on the NTF cannot handle extreme cases such as missing information for all nodes at the same time or activity fully missing for some nodes (as considered in [20]).",
"To address this limitation, we propose an extension of our method that makes it possible to take advantage of additional information that can be used as a proxy for human proximity.",
"To this aim, we consider the JNTF method described in Sec.",
"REF .",
"We test this extended method on the LSCH dataset, for which we have access to the approximate position of individuals in time.",
"There are indeed 15 locations in the school: 10 classes, the cafeteria, the playground, two staircases and a control room.",
"The resulting bipartite temporal network relating individuals and locations is composed by $N = 241$ nodes representing individuals, 15 nodes representing locations, and 130 temporal snapshots: the tensor representing this additional information has dimensions $I = 241$ , $J = 15$ and $K = 130$ .",
"Let us notice that due to the temporal resolution selected, a node might appear in several locations in the same snapshot.",
"To compare the methods of construction of surrogate data based on the JNTF and on the NTF decompositions, we simulated a loss of data on the contact network for a fraction of nodes $p_{nodes} = 0.2$ and for several fractions of the timespan $p_{times}=\\left[0.6,0.8,1\\right]$ selected consecutively on the temporal activity of the nodes.",
"Here, the set of nodes with partial activity is the same for all values of $p_{times}$ , so that we can compare the outcomes of the SIR process and the impact of incrementally removing larger fractions of the temporal activity of the same nodes.",
"We build surrogate data using the methods based on the NTF and on the JNTF decompositions, performed respectively on the partial contact networks and on the joint partial contact and location networks.",
"For the JNTF, we impose a coupling on the first and third dimensions (i.e., we impose the equality of the matrices obtained in the decompositions of the two tensors, for $\\mathbf {A}$ on the one hand and for $\\mathbf {C}$ on the other hand), as the two networks have the same nodes related to individuals and the same snapshots in time.",
"The choice of this coupling relies on the reasonable assumption that co-located individuals are more likely to be in contact.",
"Figure: Results achieved by the surrogate construction methods using either the NTF or the JNTF.Each subfigure corresponds to a fixed fraction of nodes p nodes =0.2p_{nodes} = 0.2 withincreasing loss of information for these nodes: p times =0.6,0.8,1p_{times}=0.6,0.8,1.The JNTF is computed on the two tensors representing the temporal social network of the LSCH dataset and the related temporal location network.Here, we report the epidemic size distributions of the complete, partial and surrogate networksfor SIR processes with two different couples of infection/recovery probabilities: a) β=0.3\\beta = 0.3 and μ=0.3\\mu = 0.3, b) β=0.15\\beta = 0.15 and μ=0.25\\mu = 0.25.In the last panels of both a) and b), as p times =1p_{times}=1 no information can be gained on the nodeswith missing data using simply NTF, so the results of the NTF-based method coincide with the ones obtained by simulating the SIR over the network with partial information.Figure: Results achieved by the surrogate construction method using the JNTF.",
"Each subfigure corresponds to the case in which p nodes =1p_{nodes} = 1 and p times =0.5p_{times}=0.5.",
"The JNTF is computed on the two tensors representing the temporal human proximity network of the LSCH dataset and the related temporal location network.Here, we report the results for two different couples of infection/recovery probabilities: a) β=0.3\\beta = 0.3 and μ=0.3\\mu = 0.3, b) β=0.15\\beta = 0.15 and μ=0.25\\mu = 0.25.The symbols and lines show the median distributions of epidemic sizes computed over the 10 random generations of the missing times,while the shaded area is delimited by the 25-th and 75-th percentiles.In Fig.",
"REF , we display the results obtained for $p_{nodes} = 0.2$ and $p_{times} = 0.6, 0.8,1$ .",
"The different subfigures in a) and b) correspond to SIR processes with the following infection and recovery probabilities: $\\beta = 0.3$ , $\\mu = 0.3$ , and $\\beta = 0.15$ and $\\mu = 0.25$ .",
"As in the previous cases, the distributions of epidemic sizes computed on the network with partial information show a clear underestimation of the number of individuals infected.",
"For $p_{times}=0.6$ and $p_{times}=0.8$ , the results achieved with both procedures based on the NTF and the JNTF are in agreement and give a better estimation of the distributions of epidemic sizes obtained in the original network than using the incomplete data.",
"Finally, we report in the rightmost panels of each subfigure the extreme case for which no information about $20\\%$ of nodes is present in the partial contact network ($p_{times}=1$ ).",
"In this case, as no information is present at all for the selected nodes, no correlated activity pattern concerning them can be inferred by the NTF decomposition, which thus cannot help recover any information on these nodes.",
"Using JNTF proves then helpful and the surrogate network built using this method, which combines contact and location information, yields a distribution of epidemic size closer to the one obtained on the complete network.",
"This shows that the external information provided by the approximated location of individuals in the school helps to infer possible correlations in the contact activity, even for nodes for which no contact activity was initially known.",
"We note however that the epidemics sizes remain underestimated with respect to the complete network.",
"Finally, we consider a case in which all nodes had some activity missing.",
"We simulate a scenario in which all the nodes have lost half of their activity: for each node, we zero out its activity for half of the times taken at random among the times it was active.",
"In that way each node has a different set of times during which its activity is erased.",
"We apply the method based on the JNTF in such scenarios with 10 random generations of the missing times for each node.",
"The epidemic size distributions obtained are represented on Fig.",
"REF together with the one measured with the complete network.",
"While the distributions measured on the partial networks strongly differ from those obtained with the complete network, the approach based on the JNTF yields a much better estimation of the real epidemic size distribution."
],
[
"Discussion",
"In this work, we have proposed a new approach to face the problem of estimating the outcome of spreading processes on temporal human proximity networks built from incomplete information.",
"Our method leverages the existent correlations in the observed activity of the nodes to recover the contact properties of nodes whose activity is partially missing.",
"To this aim, we rely on tensor decomposition techniques able to extract the mesoscale properties of temporal networks.",
"In practice, the methodology we put forward follows two main steps: (i) the extraction of structures from the partial network through tensor decomposition and (ii) the construction of a surrogate network that is used in numerical simulations to estimate the outcome of spreading processes.",
"In the first step we use NTF handling missing values to decompose the partial networks into a sum of components, each describing a mesoscale structure.",
"In this step, we take into account the fact that the tensor describing the temporal contact network is based on incomplete information, and the decomposition is thus performed using only the part of the tensor composed of known contacts or known non-contacts.",
"This leads to a good approximation of the temporal activity of the nodes with partial information.",
"In the second step, we determine which links belong to each mesoscale structure, and use the binarized activity timeline of each structure to fill in the unknown part of the timeline of each link with missing information.",
"This is enough to obtain a surrogate network comparable to the empirical one when we use the NTF procedure.",
"However, the JNTF might overestimate the activity of the links with missing activity and leads to a distribution of weights (number of contacts per pair of individuals) that is more homogeneous than in the original case.",
"As the heterogeneity properties of contact networks are well known to have a crucial role in determining the outcome of spreading processes, we adjust the weight distribution of the links for which only partial information was available: this step is made possible by the robustness under sampling of the contact network weight distribution.",
"We can thus rely on the weight distribution measured on those links for which no information is missing in the incomplete network to extract at random values and assign them to the links with missing information.",
"We tested our method on three different datasets describing face-to-face contacts between individuals in different contexts (two conferences and a primary school), represented as temporal human proximity networks, on which we simulated a loss of data, namely a loss of information concerning a fraction of the individuals, each for a fraction of the total timeline.",
"We have shown that the method is able to well estimate the outcome of a spreading process, even when half of the activity is missing for $40\\%$ of the nodes, while the epidemic sizes are strongly underestimated when we simulate the process on the networks with incomplete information.",
"The performance of the method based on the NTF however decreases when the amount of missing information drastically increases.",
"In particular, it is not applicable if no information at all is available for some of the nodes.",
"To deal with this issue, we have proposed an adaptation of the method based on the joint factorization of multiple tensors (JNTF).",
"The JNTF is indeed a natural extension that allows to integrate information encoded in multiple networks.",
"It is particularly adequate if the information available concerns on the one hand the contacts of individuals and on the other hand their (approximate) location, encoded in a bipartite temporal network in which a link is drawn between a node representing an individual and a node representing a location when the individual is detected in that location.",
"The joint factorization of the tensors representing the partial temporal network of contacts and the temporal network encoding positions, constrained to extract mesoscale structures with the same nodes and the same activity timelines in both tensors, can then help to recover the missing information about the contacts.",
"We have tested this alternative method in a dataset for which both contacts and approximated positions of nodes in time are available.",
"We have shown that it yields results similar to the NTF when both methods can be applied.",
"When information about some nodes is completely lost, i.e., when the method based on NTF alone cannot be used, using JNTF allows us to recover part of the missing information and yields a distribution of epidemic sizes closer to the original than when using the partial network in the simulations.",
"It is also the case when as much as $50\\%$ of the contact activity of each node has been lost in the data.",
"The methodology we have presented manages to cope with missing information in various contexts.",
"Importantly, and contrarily to other methods, it does not rely on the availability of metadata or any knowledge on the structure of the population into groups (such as classes in a school) as in [20]: the NTF or JNTF decompositions are indeed able to extract the effective structure (both in groups and temporal) of the temporal network of interactions and to assign each individual to one or multiple mesoscale structures, each with its activity timeline.",
"Some limitations of our method stem from the tensor decomposition itself.",
"For instance, if a whole group of correlated links or nodes is missing in the data, it will not be uncovered by the NTF.",
"In such an unfavorable case, the JNTF method can compensate the lack of correlated activity in the incomplete contact network by relying on auxiliary data when available.",
"Moreover, when the temporal network of contacts lacks structure or when the nodes with partial information behave in a random way, i.e., do not exhibit any activity correlation with other nodes, neither decomposition (NTF nor JNTF) might be able to clearly assign them to any mesoscale structure and thus to determine their activity timeline.",
"A way to cope with this issue might be to attribute some random activity (based for instance on the average behaviour) to those nodes with missing information that are not found in any mesoscale structure.",
"Another limitation, which can be easily overcome though, is that for the JNTF we rely on the distribution of weights measured in the partial network.",
"Indeed, if the remaining information is not representative enough (too much information missing), the heterogeneity properties of the surrogate network might be affected as the weights assigned to the links with missing information are taken among those measured.",
"We could however take advantage of the known robustness of the weight distributions in different contexts [35], [15] to use publicly available weight distributions collected in other contexts.",
"Finally, a natural direction for future research is the extension of our technique to infer mesoscale properties of human proximity networks even when no direct information on contacts is available, but only proxies such as approximate locations of individuals have been collected."
],
[
"Methodology",
"Here we describe some technical details regarding the different steps of our procedure."
],
[
"NTF Rank Selection",
"The selection of the number of components $R$ for the decomposition is guided by the Core Consistency Diagnostic [51], which estimates to which extent the PARAFAC model $\\llbracket \\lambda ; \\mathbf {A}, \\mathbf {B}, \\mathbf {C}\\rrbracket $ with a given rank $r$ (i.e., with a sum of $r$ components) is appropriate to represent the data.",
"The core consistency is a measure that has 100 as an upper bound and values above 50 are usually considered acceptable.",
"Here, we computed the core consistency values between the tensor with partial information and its approximation for $r\\in \\left[2,\\dots ,R_{max}\\right]$ , for 5 realizations of the optimization procedure starting from different initial conditions for $\\mathbf {A}$ , $\\mathbf {B}$ and $\\mathbf {C}$ for each value of $r$ .",
"We select a rank $R = R_{cc} - 1$ , where $R_{cc}$ is the smallest rank for which the core consistency value for each of the 5 realizations is lower than 85.",
"This threshold is selected to ensure an approximation as faithful as possible to the original tensor.",
"For the JNTF case, we use as a hint for the number of components to be selected, the one obtained with the NTF on the temporal contact network with partial information.",
"It is worth noting that, when the amount of available information in the data decreases, the value of $R$ determined by the core consistency can vary.",
"This is due to the fact that by erasing an increasing percentage of node activities the correlated activity patterns are increasingly perturbed and might be destroyed: a smaller number of components (sub-networks composed by links having a correlated activity in time) is then detected."
],
[
"JNTF Computation",
"To integrate data from multiple sources we use the Joint Non-negative Tensor Factorization (JNTF), described in Sec.",
"REF .",
"To this aim, we adapt the Alternating large-scale Non-negativity-constrained Least Squares (ANLS) framework and in particular the way to compute the Karush-Kuhn-Tucker (KKT) optimality conditions in a Block Principal Pivoting (BPP) framework [47].",
"Here, we illustrate the adaptation details to solve Eq.",
"(REF ) in the case of two data sources that are coupled in two dimensions (case study in Sec.",
"), i.e.",
"$S=2$ , $\\mathbf {A} = \\mathbf {A}_1 = \\mathbf {A}_2$ , and $\\mathbf {C} = \\mathbf {C}_1 = \\mathbf {C}_2$ , such that Eq.",
"(REF ) becomes: ( 12T1-1;A,B1,C2F + 12122 .",
"+.",
"12T2-2;A,B2,C2F + 22222 ) s.t.",
"1,2,A,B1,2,C0.",
"Note that we added here the regularization terms $\\frac{\\alpha _{1,2}}{2}\\left\\Vert \\lambda _{1,2}\\right\\Vert ^2_F$ , which are sparsity penalties.",
"To solve the problem through the BPP algorithm we rewrite the minimization problem as a multiple right hand side problem of the form: $\\min \\left\\Vert \\mathbf {V}\\mathbf {X}-\\mathbf {W}\\right\\Vert ^2_F\\;,$ and solve it for each of the factor matrices (here we include $\\lambda _{1,2}$ in the factor matrices).",
"To this aim, we start by solving the problem for the factor matrices $\\mathbf {B}_1$ and $\\mathbf {B}_2$ which are respectively present in the first and third term of Eq.",
"(REF ).",
"With respect to these factor matrices we can rewrite the equation by using the 2-mode matricization [30] of ${T}_1$ and ${T}_2$ , which leads to the following approximations: T1,(2)B11(CA)T T2,(2)B22(CA)T, where 1=diag(1,1,...,1,R) 2=diag(2,1,...,2,R).",
"We can rewrite the approximations as TT1,(2)(CA)1BT1 TT2,(2)(CA)2BT2, where $\\Lambda ^T_{1,2} = \\Lambda _{1,2}$ , and thus BT1-11(CA)TT1,(2) BT2-12(CA)TT2,(2).",
"By using the following property of the Khatri-Rao product, for which $\\left(\\mathbf {C}\\odot \\mathbf {A}\\right)^{\\dagger } = \\left(\\mathbf {C}^T\\mathbf {C}\\ast \\mathbf {A}^T\\mathbf {A}\\right)^{\\dagger }\\left(\\mathbf {C}\\odot \\mathbf {A}\\right)^T$ we can write the approximation as 1(CTCATA)BT1(CA)TTT1,(2), 2(CTCATA)BT2(CA)TTT2,(2).",
"The related subproblems of Eq.",
"(REF ) for $\\mathbf {B}_1$ and $\\mathbf {B}_2$ are then reduced to: B1121(CTCATA)BT1-(CA)TTT1,(2)2F , B2122(CTCATA)BT2-(CA)TTT2,(2)2F.",
"Finally, the subproblems can be respectively written in the form of Eq.",
"(REF ) by assigning V = 1(CTCATA),X = BT1 and W = 2(CA)TTT1,(2), V = (CTCATA),X = BT2 and W = (CA)TXT2,(2).",
"The solution to the subproblems is now straightforward as illustrated in [47].",
"We now need to rewrite the subproblems for the factors $\\mathbf {A}$ and $\\mathbf {C}$ that are present in both first and third terms of Eq.",
"(REF ).",
"We show the procedure for $\\mathbf {A}$ (which is analogous for $\\mathbf {C}$ ).",
"First, we write the minimization problem by using the 1-mode matricization of ${T}_1$ and ${T}_2$ (3-mode for $\\mathbf {C}$ ): A( 12(CB1)1AT-TT1,(1)2F .",
"+ .",
"12(CB2)2AT-TT2,(1)2F ).",
"By following the procedure shown above, we can write the problem in the following way: A( 121(CTCBT1B1)AT-(CB1)TTT1,(1)2F + .",
"+ .",
"122(CTCBT2B2)AT-(CB2)TTT2,(1)2F ), in which we can assign V1 = 1CTC+BT1B1, W1 = (CB1)TTT1,(1), V2 = 2CTC+BT2B2, W2 = (CB2)TTT2,(1), andX = AT, leading to $f\\left(\\mathbf {X}\\right) = \\min _{\\mathbf {X}}\\left(\\frac{1}{2}\\left\\Vert \\mathbf {V}_1\\mathbf {X}-\\mathbf {W}_1\\right\\Vert ^2_F + \\frac{1}{2}\\left\\Vert \\mathbf {V}_2\\mathbf {X}-\\mathbf {W}_2\\right\\Vert ^2_F \\right)\\;.$ To solve the problem in Eq.",
"(REF ) we need to adapt the KKT conditions, which result to be f(X) = (VT1V1+VT2V2)V1,2X-(VT1W1 + VT2W2)W1,2 f(X)0,f(X)TX=0,X0, whose solution is given by solving $\\mathbf {X}^T\\mathbf {V}^T_{1,2}-\\mathbf {W}^T_{1,2} = 0\\;.$ Since Eq.",
"(REF ) includes regularization terms, we have to adapt the KKT conditions for the minimization problem with respect to $\\lambda _1$ and $\\lambda _2$ .",
"We show the procedure for $\\lambda _1$ , which is analogous for $\\lambda _2$ .",
"We consider the cost function built from the terms in which $\\lambda _1$ is involved (the first and second in Eq.",
"(REF )): $f_{\\lambda _1} = \\frac{1}{2}\\left\\Vert {X}_1-\\llbracket \\lambda _1; \\mathbf {A},\\mathbf {B}_1,\\mathbf {C}\\rrbracket \\right\\Vert ^2_F + \\frac{\\alpha _1}{2}\\left\\Vert \\lambda _1\\right\\Vert ^2_2\\;,$ and we rewrite it through the vectorization: $f_{\\lambda _1} = \\frac{1}{2}\\left\\Vert \\mbox{vec}\\left(\\mathbf {C}\\odot \\mathbf {B}\\odot \\mathbf {A}\\right)\\lambda _1-\\mbox{vec}\\left({X}\\right)\\right\\Vert ^2_F + \\frac{\\alpha _1}{2}\\left\\Vert \\lambda _1\\right\\Vert ^2_2\\;.$ Minimizing the cost function $f_{\\lambda _1}$ is equivalent to minimize $f^{\\prime }_{\\lambda _1}$ , obtained by incorporating the regularization term as follows: $f^{\\prime }_{\\lambda _1} = \\frac{1}{2}\\left\\Vert \\underbrace{\\binom{\\mbox{vec}\\left(\\mathbf {C}\\odot \\mathbf {B}_1\\odot \\mathbf {A}\\right)}{\\sqrt{\\alpha _1}}}_{\\mathbf {V}}\\underbrace{\\lambda _1}_{\\mathbf {x}}-\\underbrace{\\binom{\\mbox{vec}\\left({X}\\right)}{0}}_{\\mathbf {w}}\\right\\Vert ^2_F\\;;$ The solution to the minimization of $f^{\\prime }_{\\lambda _1}$ follows from [47]."
],
[
"Otsu Method",
"The Otsu method [48] is commonly used in image processing to recover the different levels of grey in pixels.",
"The same idea can be used on the temporal activities of the components by thinking of them as images composed by $K$ pixels of different values (level of activation).",
"In particular, the method for 2-dimensional functions assumes that the function given as an input contains values that follow a bimodal distribution.",
"The method computes then the optimal threshold, defined by minimizing the intra-class variance and by maximizing the inter-class variance.",
"The resulting optimal threshold can be used to divide the values of the function into two groups.",
"Here, we use the threshold given by the Otsu method to define whenever a component is active or not.",
"This is done by applying the Otsu method on the temporal activity of each component.",
"The values above the threshold correspond to the temporal activation of the component, while the values below the threshold correspond to the times in which the component is inactive."
],
[
"SIR processes",
"To simulate how an infectious disease propagates in a population, and thus how the related dynamical process spreads over the network, we run an SIR process over the network.",
"The SIR model assumes that each individual in the population can be in one of three states: Susceptible, Infectious, Recovered.",
"The propagation of the disease starts from a single individual, who is in the Infectious state (the seed of the process) while all others are initially susceptible.",
"At each timestep, each susceptible individual in contact with an infectious one has a probability $\\beta $ to become infectious.",
"Each infectious individual recovers with probability $\\mu $ per timestep.",
"For simulation purposes, the first node to be infected is chosen among the nodes that are not in the set of those with partial information.",
"For each network and each set of values of the parameters $\\beta $ and $\\mu $ , we run 1000 simulations of the spreading process.",
"In particular, the parameter space we consider a priori is given by all the couples of probabilities $\\left(\\beta ,\\mu \\right)$ , with $\\beta ,\\mu $ in the range $\\left[0.001,1\\right]$ .",
"We select the suitable couples of parameters according to the conditions described in Sec.",
"REF and we show in the main text the results corresponding to two representative examples of parameter couples for each dataset.",
"Results related to other couples of parameters are similar to the ones shown in the figures."
],
[
"Approximated network",
"By decomposing a network affected by missing information through the Non-negative Tensor Factorization (NTF) we are able to reconstruct some of the structural and temporal properties observed in the complete network.",
"One of these characteristics is the overall contact activity of the nodes.",
"In particular, given a node whose information is partially missing, we are able to reconstruct its overall activity in time, i.e., the number of contacts related to that node at each time.",
"We have indeed shown that the activities in the complete network and in the approximated one are significantly correlated and have high Pearson correlation coefficients.",
"In Fig.",
"REF , we report a representative example for the overall temporal activity of a single node in the complete, partial, and approximated network for each dataset: LSCH, HT09, and SFHH.",
"As we consider a scenario in which the missing activity concerns, in each case, the first half of the timeline, the activity of the nodes with missing information is 0 in the partial data, while it shows non-trivial patterns in the complete network, which are partially recovered in the approximated network.",
"Figure: Temporal activity: example of the total number of contacts in time of a node for each dataset and case study.We report the temporal activity in the complete network, in the one with partial information, and in the one obtained approximated by the NTF method.We show here the temporal activity only for the time steps in which we lost the information about the contacts of the node, correspondingin our scenario to the first half of the timeline.We have shown that by approximating a network with partial information via NTF we are able to recover some of its properties, such as the overall temporal activity of the nodes.",
"However, as discussed in the main text, this information is not enough to obtain outcomes of a spreading process close to the ones obtained on the complete network, because the approximation tends to make the distributions of weights more homogeneous than the empirical one.",
"We illustrate this fact by showing in Fig.",
"REF the distributions of epidemic sizes of an SIR process simulated on the complete, partial and approximated networks for the LSCH dataset, for different spreading parameters ($\\beta = 0.30$ and $\\mu = 0.30$ and $\\beta = 0.15$ and $\\mu = 0.25$ ), different fractions of nodes missing ($0.1,0.2,0.4$ ) and fraction of times with missing data $p_{times} = 0.5$ .",
"Here, we reported the results only for the LSCH dataset, as it is the one characterized by highly correlated activity patterns and thus the one for which the approximations obtained through the NTF are closer to the complete case.",
"In the primary school indeed, students' activity is determined by the school schedule and they are divided in classes.",
"This makes their activity highly correlated as their contacts are more homogeneous during the daily class activities.",
"However, as we can see from Fig.",
"REF , even if the approximated network is close to the complete one in terms of the overall activity patterns displayed, this is not the case for the outcome of the spreading process.",
"The epidemic sizes in the case of the partial and approximated networks are far from the complete network case.",
"This strong underestimation is due by the fact that when we are approximating the network via NTF, the method tends to approximate the activity patterns of the nodes in the same component as fully correlated, thus connecting the nodes in the same component and rendering the network more homogeneous.",
"Figure: Outcome of an SIR process: over the complete network, the partial network and the network approximated through the NTF for the LSCH dataset.",
"Each panel corresponds to one fraction p nodes p_{nodes} of nodes with partial information and one couple of spreading parameters: a) β=0.30\\beta = 0.30 and μ=0.30\\mu = 0.30; b) β=0.15\\beta = 0.15 and μ=0.25\\mu = 0.25.",
"The lines show the median distribution of the epidemic size computed from the results relative to 10 different sets of nodeswith missing information, while the shaded area is delimited by the 25-th and 75-th percentiles.For the JNTF case, in order to obtain outcomes of SIR processes close to the case of the complete network, we have to reintroduce the heterogeneity properties into the approximated network, and use the resulting surrogate network to simulate the SIR process.",
"We reintroduce the heterogeneity properties by reassigning the weights, i.e., the total number of contacts, to the links involving nodes with partial information.",
"In particular, we measure the weight distribution on the available part of the partial network.",
"Then, we use this distribution to pick at random a weight that will be reassigned to the link whose activity was approximated via the JNTF.",
"We rely on such a process as the weight distribution of a network is robust to various sampling procedures, as shown in Fig.",
"REF for the sampling considered in this paper: for each dataset, we compare the weight distribution of the complete network with the corresponding weight distributions computed on the links of partial networks not involving any nodes with missing information, with $p_{nodes}\\in \\left[0.1,0.2,0.4\\right]$ .",
"By construction, these distributions do not depend on $p_{times}$ .",
"The results clearly show that the weight distributions measured in the partial and complete networks are consistent, meaning that, even in the case in which we miss almost half of the links in the partial network, we are able to compute a weight distribution similar to the one of the complete network.",
"Figure: Weight distributions in the three datasets for the complete network and the partial network with increasing percentage of nodes with missinginformation: p nodes ∈0.1,0.2,0.4p_{nodes}\\in \\left[0.1,0.2,0.4\\right] and p times =0.5p_{times}=0.5."
]
] | 1709.01806 |
[
[
"Nonzero-sum games of optimal stopping and generalised Nash equilibrium"
],
[
"Abstract In the nonzero-sum setting, we establish a connection between Nash equilibria in games of optimal stopping (Dynkin games) and generalised Nash equilibrium problems (GNEP).",
"In the Dynkin game this reveals novel equilibria of threshold type and of more complex types, and leads to novel uniqueness and stability results."
],
[
"Introduction",
"In this paper we establish a connection between Nash equilibria in two different types of game.",
"The first type is the two-player, nonzero-sum Dynkin game of optimal stopping (for general background on optimal stopping problems the reader is referred to [26]).",
"Player $i \\in \\lbrace 1,2\\rbrace $ chooses a stopping time $\\tau _i$ for a strong Markov process $X=(X_t)_{t \\ge 0}$ defined on the interval $(x_\\ell , x_r)$ .",
"Reward functions $f_i$ , $g_i$ , $h_i$ are given and the reward or payoff to player $i$ is $\\begin{split}\\mathcal {J}_{i}(\\tau _{1},\\tau _{2}) f_{i}(X_{\\tau _{i}}){1}_{\\lbrace \\tau _{i} < \\tau _{-i}\\rbrace } + g_{i}(X_{\\tau _{-i}}){1}_{\\lbrace \\tau _{-i} < \\tau _{i}\\rbrace } + h_{i}(X_{\\tau _{i}}){1}_{\\lbrace \\tau _{i} = \\tau _{-i}\\rbrace },\\end{split}$ where for each player $i \\in \\lbrace 1,2\\rbrace $ the subscript $-i$ denotes the other player.",
"In this context equilibrium strategies $(\\tau _1, \\tau _2)$ of the form $\\tau _1 = \\inf \\lbrace t \\ge 0: X_t \\le \\ell \\rbrace \\quad \\text{ and } \\quad \\tau _2 = \\inf \\lbrace t \\ge 0: X_t \\ge r\\rbrace ,$ for constants $\\ell , r \\in (x_\\ell ,x_r)$ with $\\ell < r$ , are referred to as threshold-type equilibria.",
"A recent example is in [11], in which the thresholds $\\ell $ , $r$ are drawn from the disjoint strategy spaces $\\mathcal {S}_1$ and $\\mathcal {S}_2$ respectively where $\\mathcal {S}_1:=[x_\\ell ,a], \\qquad \\mathcal {S}_2:=[b,x_r],$ for some constants $a,b$ with $x_\\ell <a<b<x_r$ .",
"The second type of game is a deterministic generalised game [15] (or abstract economy [1]) with $n \\ge 2$ players, where $n$ will depend on the structure of the equilibrium studied in the Dynkin game.",
"Since the examination of all cases $n \\ge 2$ is reserved for future work, however, we focus on $n=2$ and simply provide an example with $n=3$ .",
"The connection yields novel equilibria in the Dynkin game.",
"This novelty is threefold.",
"Firstly the reward functions are not required to be differentiable.",
"Secondly we obtain novel equilibria of threshold type, since both cases $a<b$ and $a \\ge b$ are permitted.",
"Thirdly, while threshold-type equilibria correspond to the case $n=2$ , the cases $n>2$ yield equilibria with more complex structures.",
"To the best of our knowledge, these complex equilibrium structures in the Dynkin game have not been previously studied.",
"In the threshold-type case, we obtain the uniqueness of the equilibria among Markovian strategies, and results about their local and global stability."
],
[
"Background",
"The structure of Nash equilibria in nonzero-sum Dynkin games has recently been investigated in [3] and [11], where sufficient conditions for the existence and uniqueness of threshold-type equilibria are obtained.",
"A key difference between the case $n=2$ of the present paper and the latter work is that there, the functions $f_i$ in (REF ) are twice differentiable and have unique points of inflexion $a$ and $b$ respectively with $a < b$ , conditions which may all be relaxed in the present approach.",
"Appendix contains remarks on the inclusion of time discounting, and the use of other Markov processes $X$ , in our setup.",
"Our results on stability relate to an iterative approximation scheme for Nash equilibria, which has been previously studied outside the Markovian framework in [17] and, in the Markovian framework, in [6], [9], [19] and [25].",
"In [19] it is assumed that $f_{i} = g_{i}$ and in [6], [9] and [25] a condition related to superharmonicity is imposed for the $g_{i}$ .",
"The latter conditions ensure monotone convergence over the iteration, whereas the approach via stability in Section does not rely on monotonicity.",
"The special case of zero-sum Dynkin games, in which $f_i=-g_{-i}$ and $h_i=-h_{-i}$ , has received particular attention in the literature.",
"Thorough analyses of the zero-sum game for a large class of driving Markov processes can be found in [14] and [27] and in that context Assumption REF , which relates the game to a war of attrition [16], is sufficient for the existence of a Nash equilibrium among pure strategies.",
"We adopt the same setting, as is also common in the nonzero-sum context (see for example [6], [9], [25]).",
"Assumption 1 For $i = 1,2$ the functions $f_{i}$ , $g_{i}$ and $h_{i}$ are bounded and continuous on $[x_\\ell , x_r]$ , and satisfy $f_{i} \\le h_{i} \\le g_{i}$ .",
"In Remark REF we discuss how Assumption REF can be weakened without affecting the main results."
],
[
"Preliminaries",
"In this section we recall necessary background on subprocesses, superharmonic and quasi-concave functions, which should be familiar."
],
[
"Subprocesses of a Brownian motion",
"Let $W = (W_{t})_{t \\ge 0}$ be a one-dimensional standard Brownian motion defined on a filtered probability space $(\\Omega ,\\mathcal {F},\\mathbb {F}=(\\mathcal {F}_{t})_{t \\ge 0},\\hat{{P}})$ , where $\\mathbb {F}$ is the universally completed filtration [7].",
"We will write the probability measure as $\\hat{{P}}^{x}$ in the case $\\hat{{P}}(\\lbrace W_0=x\\rbrace ) = 1$ , and denote the expectation operator with respect to $\\hat{{P}}^{x}$ by $\\hat{{E}}^{x}$ .",
"From $W$ we derive subprocesses in the sense of [7].",
"More precisely, for each subset $E$ of $[0,1]$ we define a subprocess $X^E = X = (X_{t})_{t \\ge 0}$ such that $X$ and its (almost surely finite) lifetime $\\zeta $ satisfy, $\\zeta &=& \\inf \\lbrace t \\ge 0 \\colon W_{t} \\notin E\\rbrace , \\\\X_t &=&{\\left\\lbrace \\begin{array}{ll}W_t, & 0 \\le t < \\zeta , \\\\\\Delta , & t \\ge \\zeta .\\end{array}\\right.",
"}$ Here $\\Delta $ is a cemetery state and the state space of $X$ is $E$ equipped with its Borel sigma-algebra $\\mathcal {B}(E)$ and augmented by $\\Delta $ .",
"We set $\\phi (\\Delta ) = 0$ for every measurable function $\\phi $ on $E_{\\Delta } E \\cup \\Delta $ , showing in Section REF below that this choice involves no loss of generality.",
"For $x \\in E$ let ${P}^{x}$ and ${E}^{x}$ denote the probability measure and expectation operator corresponding to $X$ when ${P}^{x}(\\lbrace X_{0} = x\\rbrace ) = 1$ .",
"For every measurable function $\\phi $ vanishing outside $E$ and every $t \\ge 0$ we have [7]: ${E}^{x}[\\phi (X_{t})] = \\hat{{E}}^{x}[\\phi (W_{t}){1}_{\\lbrace t < \\zeta \\rbrace }].$ For each measurable set $A$ we write the associated first entrance time of $X$ as $D_{A} \\inf \\lbrace t \\ge 0 \\colon X_{t} \\in A\\rbrace = \\inf \\lbrace t > 0 \\colon X_{t} \\in A\\rbrace \\qquad \\text{ a.s., }$ where we take $\\inf \\emptyset = \\zeta $ (the second equality follows since every point is regular for Brownian motion, see for example [24])."
],
[
"Superharmonic functions",
"We will use the fact that value functions of various optimal stopping problems for these subprocesses are superharmonic (see e.g.",
"[10] and Proposition REF ).",
"Let $E \\subseteq [0,1]$ , $A \\in \\mathcal {B}(E_{\\Delta })$ , and write $\\mathcal {T}$ for the set of all $\\mathbb {F}$ -stopping times with values in $\\mathbb {R}_+ \\cup \\lbrace \\infty \\rbrace $ .",
"Definition 1 A measurable function $\\phi \\colon E_\\Delta \\rightarrow \\mathbb {R}$ is said to be superharmonic (resp.",
"harmonic) on $A$ if for every $x \\in E$ and $\\tau \\in \\mathcal {T}$ : $\\phi (x) \\ge \\text{(resp.",
"$ = $)}\\,\\, {E}^{x}[\\phi (X_{\\tau \\wedge D_{A^{c}}})].$ A measurable function $\\phi \\colon E_\\Delta \\rightarrow \\mathbb {R}$ is said to be subharmonic on $A$ if $-\\phi $ is superharmonic on $A$ , and harmonic on $A$ if it is both superharmonic and subharmonic on $A$ .",
"If $A = E$ then the term superharmonic, subharmonic, or harmonic is used as appropriate.",
"Using the strong Markov property, one can show that (see [27] for details): if $\\phi $ is superharmonic then $\\phi (X_{\\nu }) \\ge {E}^{x}[\\phi (X_{\\rho }) \\vert \\mathcal {F}_{\\nu }], \\text{a.s.}\\,\\,\\forall \\, \\rho ,\\nu \\in \\mathcal {T} \\,\\,\\text{such that}\\,\\, \\nu \\le \\rho .$ In other words, $\\phi $ is a superharmonic function if and only if $(\\phi (X_{t}))_{t \\ge 0}$ is a strong supermartingale with respect to $\\mathbb {F}$ .",
"Here the qualifier `strong' refers to the extension of the supermartingale property to stopping times.",
"In our setup the superharmonic functions on $E$ are also equivalent to the strongly supermedian functions on $E$ (see for example [9], [20] and [23]), as follows.",
"Taking $A = E$ and $\\tau = \\zeta $ in Definition REF , the convention $\\phi (\\Delta ) = 0$ implies that the superharmonic functions $\\phi $ on $E$ are non-negative and are therefore strongly supermedian.",
"Moreover, since $X$ is a subprocess of Brownian motion, superharmonic (respectively subharmonic and harmonic) functions are concave (resp.",
"convex, linear) on convex subsets of $E$ (see [10]).",
"We will make repeated use of the following transformation.",
"Definition 2 Given $A \\in \\mathcal {B}(E_{\\Delta })$ and a bounded measurable function $\\phi \\colon E_\\Delta \\rightarrow \\mathbb {R}$ , and recalling the first entrance time defined in (REF ), define $\\phi _{A} \\colon E_\\Delta \\rightarrow \\mathbb {R}$ by $\\phi _{A}(x) {E}^{x}\\left[\\phi (X_{D_{A}})\\right].$ It is not difficult to show (using the strong Markov property) that for any measurable function $\\phi $ , the function $\\phi _{A}$ is harmonic on $A^{c}$ , and is superharmonic if $\\phi $ is superharmonic.",
"Moreover, it is continuous whenever $\\phi $ is continuous and $A$ is closed in $E$ [30]."
],
[
"Quasi-concavity",
"For use in the existence results below, we recall the definition and some properties of quasi-concave functions (see e.g.",
"[8]).",
"The extended real line will be denoted by $\\bar{\\mathbb {R}} = [-\\infty ,+\\infty ]$ .",
"Definition 3 Let $\\mathcal {D} \\subseteq \\mathbb {R}$ be convex.",
"A function $F \\colon \\mathcal {D} \\rightarrow \\bar{\\mathbb {R}}$ is said to be quasi-concave if for every $\\alpha \\in \\mathbb {R}$ the superlevel sets $L^{+}_{\\alpha }$ defined by $L^{+}_{\\alpha } = \\left\\lbrace x \\in \\mathcal {D} \\colon F(x) \\ge \\alpha \\right\\rbrace $ are convex.",
"If the same statement holds but with the sets $\\left\\lbrace x \\in \\mathcal {D} \\colon F(x) > \\alpha \\right\\rbrace $ then $F$ is said to be strictly quasi-concave.",
"A function $F$ is said to be (strictly) quasi-convex on a convex domain $\\mathcal {D}$ if and only if $-F$ is (strictly) quasi-concave.",
"All concave functions are quasi-concave.",
"Moreover a function $F \\colon \\mathcal {D} \\rightarrow \\bar{\\mathbb {R}}$ is quasi-concave if and only if $\\mathcal {D}$ is convex and for any $x_{1},x_{2} \\in \\mathcal {D}$ and $0 \\le \\theta \\le 1$ we have $F(\\theta x_{1} + (1-\\theta )x_{2}) \\ge \\min (F(x_{1}),F(x_{2})).$ If (REF ) holds with strict inequality then $F$ is strictly quasi-concave.",
"The remainder of this paper is organised as follows.",
"In Section the two game settings are presented and connected.",
"Useful alternative expressions for the expected payoffs in the Dynkin game, as optimal stopping problems for subprocesses, are developed in Section , and our main existence and uniqueness results follow in Sections and .",
"Finally, in Section we present an extension for a more complex equilibrium structure."
],
[
"Generalised Nash equilibrium",
"In the $n$ -player generalised game each player's set of available strategies, or feasible strategy space, depends on the strategies chosen by the other $n-1$ players.",
"The case $n=2$ is as follows.",
"Player $i \\in \\lbrace 1,2\\rbrace $ has a strategy space $\\mathcal {S}_{i}$ and a set-valued map $K_{i} \\colon \\mathcal {S}_{-i} \\rightrightarrows \\mathcal {S}_{i}$ determining their feasible strategy space.",
"Denoting a generic strategy for player $i$ by $s_i$ , a strategy pair $(s_1,s_2)$ is then feasible if $s_i \\in K_i(s_{-i})$ for $i=1,2$ .",
"Setting $\\mathcal {S}_{1} = [0,a]$ and $\\mathcal {S}_{2} = [b,1]$ , the pair of mappings $K_{1} \\colon [b,1] \\rightrightarrows [0,a]$ and $K_{2} \\colon [0,a] \\rightrightarrows [b,1]$ will be given by $\\begin{split}K_{1}(y) & = [0,y \\wedge a],\\\\K_{2}(x) & = [x \\vee b,1],\\end{split}$ where $a$ and $b$ are given constants lying in the interval $(0,1)$ .",
"That is, the feasible strategy pairs are given by the convex, compact set $\\mathcal {C} = \\lbrace (x,y) \\in [0,a] \\times [b,1] \\colon x \\le y\\rbrace .$ This choice of $\\mathcal {C}$ will be appropriate for equilibria of the threshold form (REF ) in the Dynkin game.",
"(The set $\\mathcal {C}$ will be modified in Section below, where an example of a more complex equilibrium is studied).",
"Writing $U_i: \\mathcal {C} \\rightarrow \\bar{\\mathbb {R}}$ for the utility function of player $i$ , the generalised Nash equilibrium problem is then given by: Definition 4 (GNEP, $n=2$ ) Find $s^{*} = (s_{1}^{*},s_{2}^{*}) \\in \\mathcal {C}$ which is a Nash equilibrium, that is: ${\\left\\lbrace \\begin{array}{ll}U_1(s^{*}) = \\sup \\limits _{(s_1,s_2^*) \\in \\mathcal {C}} U_1(s_1,s_2^*), \\\\U_2(s^{*}) = \\sup \\limits _{(s_1^*,s_2) \\in \\mathcal {C}} U_2(s_1^*,s_2).\\end{array}\\right.",
"}$ In the proofs below it will be convenient to write $\\mathcal {S} := \\mathcal {S}_{1} \\times \\mathcal {S}_{2}$ .",
"We will also make use of the following definition: Definition 5 Let $s=(s_1,s_2, \\ldots , s_n) \\in \\mathbb {R}^n$ and $w \\in \\mathbb {R}$ .",
"Then for each $i \\in \\lbrace 1,\\ldots ,n\\rbrace $ we will write $(w,s_{-i})$ for the vector $s$ modified by replacing its $i$ th entry with $w$ ."
],
[
"Optimal stopping",
"We also consider a Dynkin game in which two players observe the Brownian motion subprocess $X$ of Section REF .",
"Each player can stop the game and receive a reward (which may be positive or negative) depending on the process value and on who stopped the game first.",
"More precisely we consider only pure strategies: that is, each player $i \\in \\lbrace 1,2\\rbrace $ chooses a stopping time $\\tau _i$ lying in $\\mathcal {T}$ as their strategy.",
"Let $f_{i}$ , $g_{i}$ and $h_{i}$ be real-valued reward functions on $E$ which respectively determine the reward to player $i$ from stopping first, second, or at the same time as the other player.",
"For convenience we will refer to the $f_i$ as the leader reward functions and to the $g_i$ as the follower reward functions.",
"Given a pair of strategies $(\\tau _{1},\\tau _{2})$ and recalling the payoff defined in (REF ), we denote the expected payoff to player $i$ by $M^{x}_{i}(\\tau _{1},\\tau _{2}) = {E}^{x}\\left[\\mathcal {J}_{i}(\\tau _{1},\\tau _{2})\\right].$ The problem of finding a Nash equilibrium for this Dynkin game is then: Definition 6 (DP) Find a pair $(\\tau ^{*}_{1},\\tau ^{*}_{2}) \\in \\mathcal {T} \\times \\mathcal {T}$ such that for every $x \\in E$ we have: ${\\left\\lbrace \\begin{array}{ll}M^{x}_{1}(\\tau ^{*}_{1},\\tau ^{*}_{2}) = \\sup \\limits _{\\tau _{1} \\in \\mathcal {T}}M^{x}_{1}(\\tau _{1},\\tau ^{*}_{2}) \\\\M^{x}_{2}(\\tau ^{*}_{1},\\tau ^{*}_{2}) = \\sup \\limits _{\\tau _{2} \\in \\mathcal {T}}M^{x}_{2}(\\tau ^{*}_{1},\\tau _{2}).\\end{array}\\right.",
"}$ If $\\tau ^{*}_{1} = D_{S_{1}}$ and $\\tau ^{*}_{2} = D_{S_{2}}$ with $S_{1}, S_{2} \\in \\mathcal {B}(E_{\\Delta })$ , then the Nash equilibrium $(D_{S_{1}},D_{S_{2}})$ is said to be Markovian."
],
[
"Linking the games",
"We now present the link between the games in the case $n=2$ and $E = (0,1)$ , which is the setting used in the rest of the paper (with the exception of Section , where $n=3$ ).",
"The idea is that threshold-type solutions to the DP can be characterised by the slopes $U_1(x,y)$ and $U_2(x,y)$ of certain secant lines.",
"This gives nothing else than a deterministic game, which may be studied in the above generalised setting in order to discover additional novel equilibria.",
"We will close this section by illustrating that this link between the DP and GNEP does not preserve the zero-sum property."
],
[
"Construction of utility functions for the GNEP",
"For $(x,y) \\in [0,1]^2$ we define $\\begin{split}U_{1}(x,y) & = {\\left\\lbrace \\begin{array}{ll}\\frac{f_{1}(x) - g_{1,[y,1]}(x)}{y - x},& x < y,\\\\-\\infty ,& \\text{otherwise},\\end{array}\\right.",
"}\\\\U_{2}(x,y) & = {\\left\\lbrace \\begin{array}{ll}\\frac{f_{2}(y) - g_{2,[0,x]}(y)}{y - x},& x < y,\\\\-\\infty ,& \\text{otherwise},\\end{array}\\right.",
"}\\end{split}$ where for $A \\in \\mathcal {B}(E_{\\Delta })$ , the function $g_{i,A}$ is obtained by taking $\\phi =g_i$ in Definition REF .",
"To ensure that these utility functions are continuous and bounded above on $\\mathcal {C}$ we strengthen Assumption REF to: Assumption 1' If $b \\le a$ then $g_{i} > f_{i}$ on $[b,a]$ for $i = 1,2$ .",
"We now record comments on this choice of utility functions in the GNEP: (i) The rationale for the form (REF ) of $U_1$ and $U_2$ is as follows.",
"Lemma REF below will confirm that in equilibrium, player 1's strategy is characterised by an optimal stopping problem with obstacle $f_{1} - g_{1,[r,1]}$ , whose geometry determines the solution (in the sense of [10], for example).",
"In particular we show in Theorem REF that, for threshold strategies, the function $U_1$ characterises the solution.",
"Similar comments of course apply to player 2.",
"(ii) The GNEP characterisation does not assume smoothness but, if the reward functions are differentiable, then the double smooth fit condition (that is, the differentiability of the players' equilibrium payoffs across the thresholds $\\ell $ and $r$ respectively) follows as a corollary.",
"(iii) Later, in Section , we show how additional functions $U_i$ may be added to characterise more complex equilibria than the threshold type, leading to GNEPs with more than two players."
],
[
"Remark on the zero-sum property",
"It is interesting to note that the zero-sum property in the DP does not imply the same for the GNEP and vice versa.",
"Suppose that the GNEP (REF ) has zero sum: that is, $\\sum _{i=1}^{2}U_{i}(x,y) = 0,\\quad \\forall \\, (x,y) \\in \\mathcal {S}.$ By Definition REF the functions $g_{1,[y,1]}$ and $g_{2,[0,x]}$ are given by: g1,[y,1](x) = {ll g1(y)xy, x [0,y) g1(x), x [y,1], .",
"g2,[0,x](y) = {ll g2(y), y [0,x] g2(x)1-y1-x, y (x,1], .",
"and we recall that $f_{1}(0) = g_{2}(0) = g_{1}(1) = f_{2}(1) = 0$ .",
"Then considering separately the case $x=0$ , $y \\in [b,1]$ in (REF ) and the case $y=1$ , $x \\in [0,a]$ , we conclude that $f_{1}(x) = f_{2}(y) = 0,\\,\\forall \\, (x,y) \\in \\mathcal {S}$ .",
"Then in the DP, any nonzero choice of the reward functions $g_i$ satisfying Assumption REF results in a game with $f_i \\ne -g_{-i}$ and hence is nonzero sum.",
"On the other hand, suppose that $a < b$ and consider the zero-sum DP with reward functions f1(x) = {ll x(a-x), x [0,a] (1-x)(a-x),x (a,1], .",
"g1(x) = {ll x(b-x), x [0,b) (1-x)(b-x),x [b,1], .",
"f2 = -g1, g2 = -f1, h1 = -h2.",
"Then for $(x,y) \\in \\mathcal {S}$ the sum of the payoffs in the GNEP is $\\sum _{i=1}^{2}U_{i}(x,y) = x\\left(\\frac{a-x}{y-x}\\right)\\left(1+\\frac{1-y}{1-x}\\right) - \\left(\\frac{(1-y)(b-y)}{y-x}\\right)\\left(\\frac{y+x}{y}\\right),$ which is strictly positive for $(x,y) \\in \\lbrace 0,a\\rbrace \\times (b,1)$ , and so the GNEP is not zero sum."
],
[
"Optimal stopping of a subprocess",
"In this section we provide three equivalent expressions for expected payoffs in the Dynkin game, as optimal stopping problems for subprocesses.",
"These will be used repeatedly to establish the existence and uniqueness results of Sections and ."
],
[
"Preliminaries",
"We begin by confirming that without loss of generality all reward functions may be set equal to zero on $\\Delta $ .",
"Suppose instead that the reward is to be nonzero on $\\Delta $ .",
"This could be accommodated by taking the following modified form for the expected payoffs: $M^{x}_{E}(\\tau ,\\sigma ) = {} & \\hat{{E}}^{x}\\bigl [\\bigl \\lbrace f(W_{\\tau }){1}_{\\lbrace \\tau < \\sigma \\rbrace } + g(W_{\\sigma }){1}_{\\lbrace \\tau > \\sigma \\rbrace } + h(W_{\\sigma }){1}_{\\lbrace \\tau = \\sigma \\rbrace }\\bigr \\rbrace {1}_{\\lbrace (\\tau \\wedge \\sigma ) < D_{E^{c}}\\rbrace }\\bigr ] \\\\& + \\hat{{E}}^{x}\\bigl [H(W_{D_{E^{c}}}){1}_{\\lbrace (\\tau \\wedge \\sigma ) \\ge D_{E^{c}}\\rbrace }\\bigr ],$ where $\\tau ,\\sigma $ denote the players' stopping times and $H$ specifies the reward received at the boundaries of $E$ .",
"This is for example the approach taken in [3], where it is assumed that $f(x) = g(x) = H(x),\\, x \\in E^{c}$ .",
"Note also that by construction only the values of $H$ on $E^c$ are relevant.",
"Now taking $\\phi =H$ and $A=E^{c}$ in Definition REF and using the strong Markov property we can show that, MxE(,) - HEc(x) = Ex[ {[f - HEc](W)1{< } + [g - HEc](W)1{> } + [h - HEc](W)1{= }}1{() < DEc}] = Ex[ {[f - HEc](X)1{< } + [g - HEc](X)1{> } + [h - HEc](X)1{= }}1{() < DEc}], where the second equality comes from (REF ) above.",
"The right-hand side of (REF ) is equal to the expected payoff (REF ) when the reward functions $f$ , $g$ , $h$ and $H$ are taken to be $\\tilde{f} = f - H_{E^{c}}$ , $\\tilde{g} = g - H_{E^{c}}$ , $\\tilde{h} = h - H_{E^{c}}$ and $\\tilde{H} \\equiv 0$ respectively.",
"It may therefore be assumed without loss of generality in the proofs below that the reward functions are zero on $\\Delta $ (and indeed on $E^c$ ).",
"We will be interested in optimally stopping the subprocess $X^{A^{c}}$ .",
"For this, define the set of stopping times $\\mathcal {T}_{0,D_{A}} \\lbrace \\tau \\in \\mathcal {T} \\colon 0 \\le \\tau \\le D_{A}\\rbrace $ .",
"The proof of the following useful result can be found in, for example, [4] and [14]: Proposition 3.1 For $A \\in \\mathcal {B}(E_\\Delta )$ and functions $f$ , $g$ and $h$ satisfying Assumption REF , the map $x \\mapsto \\check{V}(x)\\sup _{\\tau \\in \\mathcal {T}}{E}^{x}\\left[f(X_{\\tau }){1}_{\\lbrace \\tau < D_A\\rbrace } + g(X_{D_A}){1}_{\\lbrace D_A < \\tau \\rbrace } + h(X_{D_A}){1}_{\\lbrace \\tau = D_A\\rbrace }\\right],$ is measurable and satisfies: $\\forall \\rho \\in \\mathcal {T}_{0,D_{A}} \\colon {E}^{x}[\\check{V}(X_{\\rho })] \\le \\check{V}(x) \\quad \\forall x \\in E.$ In other words, $x \\mapsto \\check{V}(x)$ is superharmonic on $A^{c}$ ."
],
[
"Single player problem",
"Suppose that in the Dynkin game, the strategy of player $-i$ is specified by a set $A \\in \\mathcal {B}(E_\\Delta )$ on which that player stops.",
"The next lemma expresses the resulting optimisation problem for player $i$ in terms of optimal stopping problems of different kinds for the subprocess $X^{A^{c}}$ .",
"Lemma 3.2 For $x \\in E$ consider the problems VA(x) TMx(,DA), VA(x) TMx(,DA), VA(x) TMx(,DA), where for $\\tau \\in \\mathcal {T}$ we have Mx(,DA) Ex[ f(X)1{< DA} + g(XDA)1{DA < } + h(XDA)1{= DA}], Mx(,DA) Ex[f(X)1{< DA} + g(XDA)1{DA}], Mx(,DA) Ex[ {f - gA}(X)1{< DA}], and $f$ , $g$ and $h$ are functions satisfying Assumption REF .",
"Then, recalling Definition REF , we have VA(x) = VA(x) = gA(x) + VA(x).",
"Let $\\tau \\in \\mathcal {T}$ , $x \\in E$ be arbitrary.",
"We have $\\bar{M}^{x}(\\tau ,D_{A}) \\ge M^{x}(\\tau ,D_{A})$ and therefore $\\bar{V}^{A}(x) \\ge V^{A}(x) $ .",
"To show the reverse inequality, first recall that $x \\mapsto V^{A}(x)$ is measurable.",
"By assumption we have $V^A \\ge f$ on $E$ , so that $V^{A}(X_{\\tau }){1}_{\\lbrace \\tau < D_{A}\\rbrace } \\ge f(X_{\\tau }){1}_{\\lbrace \\tau < D_{A}\\rbrace }$ a.s., while from the strong Markov property we have $V^{A}(X_{D_{A}}) = g(X_{D_{A}})$ a.s..",
"It follows from (REF ) and superharmonicity that $\\bar{M}^{x}(\\tau ,D_{A}) \\le {E}^{x}\\bigl [V^{A}(X_{\\tau \\wedge D_{A}})\\bigr ] \\le V^{A}(x),$ and taking the supremum over $\\tau $ we have $\\bar{V}^{A}(x) = V^{A}(x)$ .",
"Finally, recalling Definition REF we have $\\bar{M}(\\tau ,D_{A}) - g_{A}(x) = {E}^{x}\\bigl [\\bigl \\lbrace f - g_{A}\\bigr \\rbrace (X_{\\tau }){1}_{\\lbrace \\tau < D_{A}\\rbrace }\\bigr ].$ Remark 3.3 It follows from (REF ) that $V^A(x) = f(x) \\iff \\tilde{V}^A(x) = f(x) - g_{A}(x).$ That is, defining the stopping region to be the subset of $A^c$ on which the obstacle equals the value function, the optimal stopping problems $V^A(x)$ and $\\tilde{V}^A(x)$ have identical stopping regions.",
"An easy consequence is that if $x \\in A^c$ lies in either stopping region then $f(x) \\ge g_{A}(x)$ , and that if $f \\le g_{A}$ on $A^{c}$ then $\\tau = D_A$ is optimal in (REF )."
],
[
"Existence of equilibria",
"In this section we exploit the link between the games to show, firstly, that the existence of a solution to the GNEP with utility functions given by (REF ) implies the existence of a threshold-type solution to the DP (Theorems REF and REF ).",
"These results are then applied to show the existence of novel Nash equilibria in the DP.",
"More precisely we will show that the following condition on the geometry of the reward functions is sufficient for the existence of an equilibrium: Condition G1.",
"There exist points $a \\in (0,1) \\text{ and } b \\in (0,1) \\text{ such that}$ (i) f1 is concave on [0,a] and is convex on [a,1] (ii) f2 is convex on [0,b] and is concave on [b,1] (iii) If b a, then fi < gi on [b,a] for i = 1,2.",
"The case $a > b$ is novel when compared with the existing literature.",
"It is interesting to note that in the case $a \\le b$ , which is analysed in [3] and [11], the generalised problem (REF ) reduces to a classical game (that is, where each player's strategy space does not depend on the other player's chosen strategy).",
"This is because the dependency between the players' strategies (which is specified by the choice of $\\mathcal {C}$ ) allows some additional control on the equilibria, which is required when $a>b$ .",
"The case when at least one of the functions $f_i$ is not differentiable is also novel."
],
[
"Preliminaries",
"A solution to the GNEP is known to exist under the following condition (see for example [1] and [15]): Condition U.",
"For each fixed $s_2 \\in \\mathcal {S}_{2}$ , the mapping $s_{1} \\mapsto U_{1}(s_1,s_2)$ is quasi-concave on $K_1(s_2)$ .",
"For each fixed $s_1 \\in \\mathcal {S}_1$ , the mapping $s_{2} \\mapsto U_{2}(s_1,s_2)$ is quasi-concave on $K_2(s_1)$ .",
"The utility functions $s \\mapsto U_{i}(s)$ for $i = 1,2$ are continuous in $s = (s_{1},s_{2})$ .",
"For convenience we record the necessary argument here: Lemma 4.1 Suppose Assumption 1' and Condition U hold.",
"Then there exists a solution $\\bigl (s^{*}_{1},s^{*}_{2}\\bigr ) \\in \\mathcal {C}$ to the GNEP (REF ) satisfying $s_{1}^{*} < s_{2}^{*}$ .",
"For $i = 1,2$ the correspondence $K_{i}$ is compact and convex valued.",
"Furthermore, using the notion of continuity for set-valued maps in [28], we can confirm that $K_{1}$ and $K_{2}$ are continuous.",
"Under the present hypotheses, $U_{i}$ is continuous on $\\mathcal {S}$ and has the quasi-concavity property specified in Condition U.",
"Therefore by Lemma 2.5 in [1], there exists a solution $s^{*}$ to (REF ).",
"From the construction (REF ), this solution must satisfy $s_{1}^{*} < s_{2}^{*}$ .",
"Remark 4.2 For possible extensions of Lemma REF see also [15], [18] and references therein.",
"Before presenting the main result of this section we need the following fact: Lemma 4.3 Suppose $\\mathcal {D} \\subseteq \\mathbb {R}$ is convex, $f \\colon \\mathcal {D} \\rightarrow \\bar{\\mathbb {R}}$ is (strictly) concave, and $\\varphi \\colon \\mathcal {D} \\rightarrow (0,\\infty )$ is linear.",
"Then the function $\\frac{f}{\\varphi } \\colon \\mathcal {D} \\rightarrow \\bar{\\mathbb {R}}$ is (strictly) quasi-concave.",
"In the case of concavity, for each $\\alpha \\in \\mathbb {R}$ define a function $F_{\\alpha } \\colon \\mathcal {D} \\rightarrow \\bar{\\mathbb {R}}$ by $F_{\\alpha }(x) = f(x) - \\alpha \\varphi (x)$ .",
"This function is concave on $\\mathcal {D}$ , and therefore quasi-concave, which means the superlevel set $\\left\\lbrace x \\in \\mathcal {D} \\colon F_{\\alpha }(x) \\ge 0 \\right\\rbrace $ is convex for every $\\alpha \\in \\mathbb {R}$ .",
"The function $\\frac{f}{\\varphi }$ is quasi-concave on $\\mathcal {D}$ since for every $\\alpha \\in \\mathbb {R}$ , $\\left\\lbrace x \\in \\mathcal {D} \\colon \\left(\\tfrac{f}{\\varphi }\\right)(x) \\ge \\alpha \\right\\rbrace = \\left\\lbrace x \\in \\mathcal {D} \\colon f(x) \\ge \\alpha \\varphi (x) \\right\\rbrace = \\left\\lbrace x \\in \\mathcal {D} \\colon F_{\\alpha }(x) \\ge 0 \\right\\rbrace .$ The proof for strictly concave $f$ follows in the same way."
],
[
"Existence results",
"The GNEP is used in this section to establish the existence of equilibria in the DP.",
"This both allows $a>b$ in Condition G1 and avoids the need for smoothness assumptions.",
"Theorem 4.4 Under Condition G1, there exists a pair $(\\ell _{*},r_{*}) \\in [0,a] \\times [b,1]$ such that $(D_{[0,\\ell _{*}]},D_{[r_{*},1]})$ is a solution to the DP.",
"We begin by noting that for each $r \\in [0,1]$ and $\\ell \\in [0,1]$ , x [0,r) U1(x,r) x [0,a] U1(x,r), x (,1] U2(,x) x [b,1] U2(,x).",
"For $r \\in (a,1]$ , eq.",
"(REF ) follows from the convexity of $f_{1} - g_{1,[r,1]}$ on $[a,r]$ and the fact that $f_{1}(r) \\le g_{1}(r) = g_{1,[r,1]}(r)$ : f1(x) - g1,[r,1](x)r - x f1(a) - g1,[r,1](a)r - a + (f1(r) - g1,[r,1](r)r - a)(x-ar-x) f1(a) - g1,[r,1](a)r - a, x (a,r).",
"Similar reasoning establishes (REF ).",
"Using Condition G1 and Lemma REF , we can verify the hypotheses of Lemma REF and assert the existence of a pair $\\bigl (\\ell ,r\\bigr ) \\in [0,a] \\times [b,1]$ with $\\ell < r$ such that ${\\left\\lbrace \\begin{array}{ll}U_{1}(x,r) \\le U_{1}(\\ell ,r),\\quad \\forall x \\in [0,r \\wedge a], \\\\U_{2}(\\ell ,y) \\le U_{2}(\\ell ,r),\\quad \\forall y \\in [\\ell \\vee b,1].\\end{array}\\right.",
"}$ The pair $(\\ell ,r)$ that satisfies (REF ) therefore also satisfies U1(x,r) U1(,r), x [0,r), U2(,y) U2(,r), y (,1], and the result follows from the next theorem.",
"Theorem 4.5 For every $r \\in [b,1]$ , a point $\\ell _{r} \\in [0,a]$ with $\\ell _{r} < r$ satisfies (REF ) if and only if $V_{1}^{[r,1]}(x) \\sup \\limits _{\\tau _{1} \\in \\mathcal {T}}M^{x}_{1}(\\tau _{1},D_{[r,1]}) = M^{x}_{1}(D_{[0,\\ell _{r}]},D_{[r,1]}),\\quad \\forall x \\in [0,1].$ Similarly, for every $\\ell \\in [0,a]$ , a point $r_{\\ell } \\in [b,1]$ with $\\ell < r_{\\ell }$ satisfies (REF ) if and only if $V_{2}^{[0,\\ell ]}(x) \\sup \\limits _{\\tau _{2} \\in \\mathcal {T}}M^{x}_{2}(D_{[0,\\ell ]},\\tau _{2}) = M^{x}_{2}(D_{[0,\\ell ]},D_{[r_{\\ell },1]}),\\quad \\forall x \\in [0,1].$ We only show (REF )$\\iff $ (REF ), since (REF )$\\iff $ (REF ) follows by similar arguments.",
"Let $r \\in [b,1]$ and $\\ell _{r} \\in [0,a]$ with $\\ell _{r} < r$ be given.",
"We will make repeated use of the function $u_{r}(x) M^{x}_{1}(D_{[0,\\ell _{r}]},D_{[r,1]}) - g_{1,[r,1]}(x) ={\\left\\lbrace \\begin{array}{ll}f_{1}(x) - g_{1,[r,1]}(x),& x \\in [0,\\ell _{r}), \\\\\\left(f_{1}(\\ell _{r}) - g_{1,[r,1]}(\\ell _{r})\\right)\\frac{r-x}{r-\\ell _{r}},& x \\in [\\ell _{r},r), \\\\0,& x \\in [r,1],\\end{array}\\right.",
"}$ where the middle line is a straightforward consequence of the identities in Appendix and the fact that, for $x \\in [0,r]$ , we have g1(r)x-rr-r - g1,[r,1](x) = g1(r)(x-rr-r - xr) = g1(r)(r(x-r) - x(r-r)r(r-r)) = -g1(r)(rr)(r-xr-r) = -g1,[r,1](r)r-xr-r. Sufficiency ($$ ).",
"Suppose that (REF ) is satisfied.",
"Substituting this in (REF ), dividing both sides of (REF ) by $r-x$ (when $x<r$ ), and using the definition (REF ) of $U_{1}$ , we obtain $\\frac{V_{1}^{[r,1]}(x) - g_{1,[r,1]}(x)}{r - x} = {\\left\\lbrace \\begin{array}{ll}U_{1}(x,r),& \\forall x \\le \\ell _{r} \\\\U_{1}(\\ell _{r},r),&\\forall \\ell _{r} < x < r.\\end{array}\\right.",
"}$ It is easy to see that $V_{1}^{[r,1]}(r) = g_{1}(r) = g_{1,[r,1]}(r)$ and $V_{1}^{[r,1]}(x) \\ge f_{1}(x)$ for all $x \\in [0,r]$ .",
"Therefore when $x \\in (\\ell _r,r)$ we have $U_{1}(\\ell _r,r) \\ge U_{1}(x,r).$ To treat the case $x \\in [0,\\ell _{r}]$ , note from Lemma REF that $x \\mapsto V_{1}^{[r,1]}(x) - g_{1,[r,1]}(x)$ is the value function of an optimal stopping problem for a subprocess as in, for example, [10] and, as such, is non-negative and superharmonic in $(0,r)$ .",
"For $0 \\le x < y \\le 1$ define $\\tau _{x,y} = D_{\\lbrace x\\rbrace } \\wedge D_{\\lbrace y\\rbrace }$ .",
"Using superharmonicity and the fact that $X$ is a positively recurrent diffusion, for every $0 \\le x \\le \\ell _{r}$ we have, V1[r,1](r) - g1,[r,1](r) Er[V1[r,1](Xx,r) - g1,[r,1](Xx,r)] = (V1[r,1](x) - g1,[r,1](x))Er[1{D{x} < D{r}}] = (V1[r,1](x) - g1,[r,1](x))r-rr-x.",
"Since for all $0 \\le x \\le \\ell _{r}$ we have $V_{1}^{[r,1]}(x) = f_{1}(x)$ , (REF ) gives $U_{1}(x,r) \\le U_{1}(\\ell _r,r), \\quad \\forall x \\in [0,\\ell _{r}],$ establishing (REF ) with $\\ell =\\ell _r$ .",
"Necessity ($\\Rightarrow $ ).",
"Suppose that the pair $(\\ell _r, r)$ satisfies (REF ) with $\\ell =\\ell _r$ .",
"We will establish (REF ) by showing that $u_r(x)=V_{1}^{[r,1]}(x) - g_{1,[r,1]}(x),\\quad \\forall x \\in [0,1].$ By construction (REF ) holds for $x \\in [r,1]$ , and so we restrict attention to the domain $[0,r]$ .",
"By Lemma REF it is sufficient to show that $u_r$ is the value function of the optimal stopping problem on $[0,r]$ with the obstacle $\\vartheta f_{1} - g_{1,[r,1]}$ .",
"Therefore using Proposition 3.2 in [10], it is enough to show that $u_{r}$ is the smallest non-negative concave majorant of $\\vartheta $ on $[0,r]$ .",
"The majorant property on $[\\ell _r,r)$ follows from (REF ), which gives $f_{1}(x) - g_{1,[r,1]}(x) \\le \\left(f_{1}(\\ell _{r}) - g_{1,[r,1]}(\\ell _{r})\\right)\\left(\\frac{r-x}{r-\\ell _{r}}\\right),\\forall x \\in [0,r],$ and the majorant property at $x=r$ follows from recalling that $f_{1}(r) \\le g_{1}(r)$ .",
"For nonnegativity we first recall that the reward functions are null at the boundaries, so taking $x=0$ in (REF ) gives $0 \\le f_{1}(\\ell _{r}) - g_{1,[r,1]}(\\ell _{r}) = u_r(\\ell _r)$ .",
"Combining this with the fact that $u_r$ equals the obstacle on $[0,\\ell _r]$ , and hence is concave there, establishes nonnegativity.",
"For concavity we note that $u_r$ is a straight line on $[\\ell _r,r]$ , so it remains only to consider any $x_{1} \\in [0,\\ell _{r})$ and $x_{2} \\in (\\ell _{r},r]$ .",
"Then we have x2 - rx2-x1ur(x1) + r-x1x2-x1ur(x2) = x2 - rx2-x1[f1(x1) - g1,[r,1](x1)] + r-x1x2-x1(f1(r) - g1,[r,1](r))(r-x2r-r) x2 - rx2-x1(f1(r) - g1,[r,1](r))(r-x1r-r) + r-x1x2-x1(f1(r) - g1,[r,1](r))(r-x2r-r) = f1(r) - g1,[r,1](r) = ur(r), where the inequality follows from (REF ).",
"Finally, since $u_r$ equals the obstacle on $[0,\\ell _r]$ and is a straight line on $[\\ell _r,r]$ , it is smaller than any other nonnegative concave majorant on $[0,r]$ .",
"Remark 4.6 Theorem REF remains valid under Condition G1 and the following condition which is weaker than Assumption REF : $f_{i} \\le g_{i}$ on $\\mathcal {S}_{-i}$ ."
],
[
"Stability and uniqueness results",
"In this section we exploit the above connection to obtain additional novel results for Nash equilibria in the DP.",
"We define a concept of stability and provide a sufficient condition under which it holds locally (Corollary ), showing in Theorem that this condition always holds in the particular case of zero-sum Dynkin games.",
"By establishing global stability, Theorem provides sufficient conditions for uniqueness of the threshold-type equilibrium of Theorem REF among the Markovian strategies.",
"Finally, Theorem transfers another uniqueness result for the GNEP to the DP."
],
[
"Policy iteration",
"We will apply the Gauss-Seidel policy iteration or tâtonnement process [16], [5] to the GNEP.",
"This iteration scheme has previously been used for Dynkin games in [9] and [19] and, outside the Markovian framework, in [17].",
"Throughout Section , for convenience we will assume the following Condition G1', rather than G1: Condition G1'.",
"Condition G1 holds, with: $a < b$ , strict convexity and strict concavity, $f_{i}, g_{i} \\in C^{2}[0,1]$ , and For all $(x,y) \\in [0,a] \\times [b,1]$ there exists $(\\hat{x},\\hat{y}) \\in (0,a]\\times [b,1)$ with $f_{1}(\\hat{x}) > g_{1}(y)\\cdot \\frac{\\hat{x}}{y}$ and $f_{2}(\\hat{y}) > g_{2}(x)\\cdot \\frac{1-\\hat{y}}{1-x}$ .",
"We emphasise that these assumptions are for ease of exposition.",
"Parts 1) and 3) imply that the GNEP utility functions are finite and smooth on $\\mathcal {S}$ , which is convenient for the transfer of results from generalised games.",
"Part 2) says that $f_{1}$ is strictly concave on $[0,a]$ and strictly convex on $[a,1]$ , and $f_{2}$ is strictly convex on $[0,b]$ and strictly concave on $[b,1]$ .",
"This ensures that iteration (i) below is well defined.",
"Part 4) removes the need to consider the points 0 and 1 as candidate thresholds, which is convenient since the principle of smooth fit (used below) may break down there.",
"Recalling the equality (REF ), this is straightforward to see from (REF ), (REF ) and (REF )–(REF ).",
"Part 4) similarly ensures that threshold-type equilibria have their thresholds in $(0,1)$ and not at either boundary 0 or 1.",
"Taking $\\ell ^{(1)} \\in [0,a]$ , we consider the following two iteration schemes: In the GNEP: taking $r^{(1)} = \\operatornamewithlimits{arg\\,max}_{y \\in [b,1]}U_{2}(\\ell ^{(1)},y)$ , for $n \\ge 2$ define $\\begin{split}\\ell ^{(n)} & = \\operatornamewithlimits{arg\\,max}_{x \\in [0,a]}U_{1}(x,r^{(n-1)}), \\quad r^{(n)} = \\operatornamewithlimits{arg\\,max}_{y \\in [b,1]}U_{2}(\\ell ^{(n)},y).",
"\\\\\\end{split}$ In the DP: taking $A_{1} = [0,\\ell ^{(1)}]$ , for $n \\ge 1$ define $\\begin{split}(i)&\\quad V_{2n}(x) = \\sup \\limits _{\\tau }{E}^{x}\\bigl [f_{2}(X_{\\tau }){1}_{\\lbrace \\tau < D_{A_{2n-1}}\\rbrace } + g_{2}(X_{D_{A_{2n-1}}}){1}_{\\lbrace \\tau \\ge D_{A_{2n-1}}\\rbrace }\\bigr ], \\\\(ii)&\\quad A_{2n} = \\lbrace x \\in [0,1] \\setminus A_{2n-1} \\colon V_{2n}(x) = f_{2}(x)\\rbrace , \\\\(iii)&\\quad V_{2n+1}(x) = \\sup \\limits _{\\tau }{E}^{x}\\bigl [f_{1}(X_{\\tau }){1}_{\\lbrace \\tau < D_{A_{2n}}\\rbrace } + g_{1}(X_{D_{A_{2n}}}){1}_{\\lbrace \\tau \\ge D_{A_{2n}}\\rbrace }\\bigr ], \\\\(iv)&\\quad A_{2n+1} = \\lbrace x \\in [0,1] \\setminus A_{2n} \\colon V_{2n+1}(x) = f_{1}(x)\\rbrace .\\end{split}$ We will call a solution $s^{*}=(\\ell ^*,r^*)$ to the GNEP (REF ) globally stable if for any $\\ell ^{(1)} \\in [0,a]$ the iteration (REF ) satisfies $\\ell ^{(n)} \\rightarrow \\ell ^*$ and $r^{(n)} \\rightarrow r^*$ , and locally stable if this convergence holds only for $\\ell ^{(1)}$ in a neighbourhood of $\\ell ^*$ .",
"Similarly we call a threshold-type solution $s^{\\prime }=(D_{[0,\\ell ^{\\prime }]},D_{[r^{\\prime },1]})$ to the DP (REF ) globally stable if for any $\\ell ^{(1)} \\in [0,a]$ the iteration (REF ) satisfies $\\begin{split}\\liminf _{n \\rightarrow \\infty }A_{2n-1} & = \\limsup _{n \\rightarrow \\infty }A_{2n-1} = [0,\\ell ^{\\prime }],\\\\\\liminf _{n \\rightarrow \\infty }A_{2n} & = \\limsup _{n \\rightarrow \\infty }A_{2n} = [r^{\\prime },1],\\end{split}$ and locally stable if convergence holds only for $\\ell ^{(1)}$ in a neighbourhood of $\\ell ^{\\prime }$ .",
"The study of appropriate generalised games with $n>2$ players yields equilibria for the two-player Dynkin problem of Definition REF with more complex structures than the threshold type which has been previously studied.",
"The systematic study of all cases $n>2$ is beyond the scope of this paper and so in this section we provide an example with $n=3$ .",
"This example uses the following relaxation of Condition G1, under which the reward function $f_1$ has an additional convex portion: Condition G2.",
"There exist points $a_{1}$ and $a_{2}$ with $0 < a_{1} \\le a_{2} < b < 1$ such that: (i) f1 is convex on [0,a1], concave on [a1,a2] and convex on [a2,1], (ii) f2 is convex on [0,b] and concave on [b,1].",
"Define sets $\\hat{\\mathcal {S}}_{1} = \\hat{\\mathcal {S}}_{2} = [a_{1},a_{2}]$ , $\\hat{\\mathcal {S}}_{3} = [b,1]$ and $\\hat{\\mathcal {S}} = \\prod _{i=1}^{3}\\hat{\\mathcal {S}}_{i}$ .",
"Let the utility functions $\\hat{U}_{i} \\colon [0,1]^3 \\rightarrow \\bar{\\mathbb {R}}$ , $i \\in \\lbrace 1,2,3\\rbrace $ be defined by $\\begin{split}\\hat{U}_{1}(x,y,z) = \\frac{f_{1}(x) - g_{1,[z,1]}(x)}{x}, \\\\\\hat{U}_{2}(x,y,z) = \\frac{f_{1}(y) - g_{1,[z,1]}(y)}{z - y}, \\\\\\hat{U}_{3}(x,y,z) = \\frac{f_{2}(z) - g_{2,[0,y]}(z)}{z - y},\\end{split}$ (taking $\\hat{U}_{2}(x,y,z)=\\hat{U}_{3}(x,y,z)=-\\infty $ if $y \\ge z$ ).",
"Define the players' feasible strategy spaces by the set-valued maps $\\hat{K}_{i} \\colon \\hat{\\mathcal {S}}_{-i} \\rightrightarrows \\hat{\\mathcal {S}}_{i}$ , where $\\hat{K}_{1}(y,z) = [a_{1},y \\wedge a_{2}], \\hat{K}_{2}(x,z) = [x \\vee a_{1},a_{2}], \\hat{K}_{3}(x,y) = [b,1],$ so that the feasible strategy triples belong to the convex, compact set $\\hat{\\mathcal {C}}$ defined by $\\hat{\\mathcal {C}} = \\lbrace (x,y,z) \\in [a_{1},a_{2}] \\times [a_{1},a_{2}] \\times [b,1] \\colon x \\le y\\rbrace .$ Theorem 5.1 Suppose that the DP satisfies Condition G2.",
"Then: there exists $s^{*} = (\\ell ^{1},\\ell ^{2},r) \\in \\hat{\\mathcal {C}}$ with $\\hat{U}_{i}(s^{*}) = \\sup \\limits _{(s_{i},s_{-i}^{*}) \\in \\hat{\\mathcal {C}}}\\hat{U}_{i}(s_{i},s_{-i}^{*}), i \\in \\lbrace 1,2,3\\rbrace ,$ a solution $s^{*} = (\\ell ^{1},\\ell ^{2},r) \\in \\hat{\\mathcal {C}}$ to (REF ) satisfies $\\hat{U}_{2}(s^{*}) \\ge 0$ if and only if $(D_{[\\ell ^{1},\\ell ^{2}]},D_{[r,1]})$ is a Nash equilibrium for the DP.",
"Part (a) follows as in the proof of Lemma REF .",
"For part (b), we claim that the pair $(\\ell ^{1},\\ell ^{2})$ solves the following problem: Problem: Find two points $\\ell ^{1}, \\ell ^{2}$ satisfying $\\begin{split}i) & \\quad a_{1} \\le \\ell ^{1} \\le \\ell ^{2} \\le a_{2}, \\\\ii) & \\quad \\hat{U}_{1}(x,\\ell ^{2},r)\\le \\hat{U}_{1}(\\ell ^{1},\\ell ^{2},r),\\quad \\forall x \\in (0,r), \\\\iii) & \\quad \\hat{U}_{2}(\\ell ^{1},y,r)\\le \\hat{U}_{2}(\\ell ^{1},\\ell ^{2},r),\\quad \\forall y \\in [0,r).\\end{split}\\qquad \\mathrm {(P)}$ To establish part iii) note that the function $y \\mapsto f_{1}(y) - g_{1,[r,1]}(y)$ is zero at $y=0$ , convex for $y \\in [0,a_{1}]$ , concave for $y \\in [a_{1},a_{2}]$ , convex for $y \\in [a_{2},r]$ , nonnegative at $y=\\ell ^{2}$ and negative at $y=r$ .",
"It is then a straightforward exercise in convex analysis, similar to that in the proof of Theorem REF , to show that the maximum of the function $y \\mapsto \\hat{U}_{2}(\\ell ^{1},y,r)$ on $[0,r)$ must be attained at a point in $[a_1,a_2]$ .",
"Taking $i=2$ in (REF ) then establishes the claim.",
"Part ii) follows similarly.",
"The necessity and sufficiency claim for the Nash equilibrium in stopping strategies then follows by applying Propositions REF and REF in the Appendix."
],
[
"Other Markov processes and discounting",
"Let $X = (X_{t})_{t \\ge 0}$ be a continuous strong Markov process defined on an interval $E = (\\ell ,r)$ .",
"Suppose that the rewards in the DP are discounted by a factor $\\lambda \\ge 0$ , so that (REF ) becomes $\\begin{split}\\mathcal {J}_{i}(\\tau _{1},\\tau _{2}) e^{-\\lambda (\\tau _{i} \\wedge \\tau _{-i})}\\lbrace f_{i}(X_{\\tau _{i}}){1}_{\\lbrace \\tau _{i} < \\tau _{-i}\\rbrace } + g_{i}(X_{\\tau _{-i}}){1}_{\\lbrace \\tau _{-i} < \\tau _{i}\\rbrace } + h_{i}(X_{\\tau _{i}}){1}_{\\lbrace \\tau _{i} = \\tau _{-i}\\rbrace }\\rbrace , \\\\i \\in \\lbrace 1,2\\rbrace .\\end{split}\\qquad \\mathrm {(1.1^{\\prime })}$ Lemma REF has a straightforward extension to the case $\\lambda > 0$ .",
"Extending the concept of superharmonic functions in Definition REF , we say that a measurable function $\\phi \\colon E_\\Delta \\rightarrow \\mathbb {R}$ is $\\lambda $ - superharmonic on a set $A \\in \\mathcal {B}(E_{\\Delta })$ if for every $x \\in E$ and $\\tau \\in \\mathcal {T}$ , $\\phi (x) \\ge {E}^{x}[e^{-\\lambda (\\tau \\wedge D_{A^{c}})}\\phi (X_{\\tau \\wedge D_{A^{c}}})].$ The function $\\phi _A$ introduced in Definition REF is given more generally by, $\\phi _{A}(x) {E}^{x}\\left[e^{-\\lambda D_{A}}\\phi (X_{D_{A}})\\right].$ It was noted in Section REF that $\\phi _A$ is continuous when $\\lambda = 0$ , $g$ is continuous and $A$ is closed, since $X=X^E$ is a subprocess of a Brownian motion.",
"This same property, which is important for ensuring that the obstacle in problem (REF ) is continuous, also holds for $\\lambda \\ge 0$ when $X$ is a more general diffusion with strictly positive diffusion coefficient [30].",
"Furthermore, when $X$ is a subprocess of a regular diffusion $Z = (Z_t)_{t \\ge 0}$ , the results in Sections – hold under an appropriate modification of Condition G1.",
"We now briefly discuss this extension when $Z$ satisfies the stochastic differential equation, ${d}Z_{t} = \\mu (Z_{t}){d}t + \\sigma (Z_{t}){d}W_{t},$ where $W = (W_{t})_{t \\ge 0}$ is a standard Brownian motion, $\\mu \\colon E \\rightarrow \\mathbb {R}$ and $\\sigma \\colon E \\rightarrow (0,\\infty )$ are measurable functions, $\\mu $ bounded and $\\sigma $ continuous, satisfying the following condition: for every $x \\in E$ , $\\int _{x-\\varepsilon }^{x+\\varepsilon }\\frac{1 + |\\mu (y)|}{\\sigma ^{2}(y)}{d}y < \\infty \\text{for some } \\varepsilon > 0.$ Let $\\mathcal {G} = \\frac{1}{2}\\sigma ^{2}(\\cdot )\\frac{{{d}}^{2}}{{d}x} + \\mu (\\cdot )\\frac{d}{{d}x}$ denote the infinitesimal generator corresponding to $Z$ ."
],
[
"Undiscounted rewards",
"For the case $\\lambda = 0$ , we first recall from [10] that there is a continuous increasing function $S$ on $E$ , the scale function, which satisfies $\\mathcal {G}S(\\cdot ) \\equiv 0$ .",
"Let $\\tilde{\\ell } = S(\\ell )$ , $\\tilde{r} = S(r)$ and $\\tilde{X} = (\\tilde{X}_{t})_{t \\ge 0}$ be a Brownian motion on $\\tilde{E} = (\\tilde{\\ell },\\tilde{r})$ .",
"Then, it follows from Proposition 3.3 of [10] that the DP corresponding to the process $X$ and rewards $f_i$ , $g_{i}$ and $h_{i}$ on $E$ can be studied by an equivalent DP corresponding to $\\tilde{X}$ with reward functions $\\tilde{f}_{i}(\\cdot ) = f_i(S^{-1}(\\cdot ))$ , $\\tilde{g}_{i}(\\cdot ) = g_i(S^{-1}(\\cdot ))$ , $\\tilde{h}_{i}(\\cdot ) = h_i(S^{-1}(\\cdot ))$ on $\\tilde{E}$ ."
],
[
"Discounted rewards",
"For the case $\\lambda > 0$ , we first let $\\psi ^{\\lambda }$ and $\\phi ^{\\lambda }$ denote the fundamental solutions to the diffusion generator equation $\\mathcal {G}w = \\lambda w$ , where $\\psi ^{\\lambda }$ is strictly increasing and $\\phi ^{\\lambda }$ is strictly decreasing [10].",
"Let $F(\\cdot ) = \\frac{\\psi ^{\\lambda }(\\cdot )}{\\phi ^{\\lambda }(\\cdot )}$ , $\\tilde{\\ell } = F(\\ell )$ , $\\tilde{r} = F(r)$ and $\\tilde{X} = (\\tilde{X}_{t})_{t \\ge 0}$ be a Brownian motion on $\\tilde{E} = (\\tilde{\\ell },\\tilde{r})$ .",
"Then, it follows from Proposition 4.3 of [10] that the DP corresponding to the process $X$ and rewards $f_i$ , $g_{i}$ and $h_{i}$ on $E$ discounted by $\\lambda > 0$ can be studied by an equivalent DP corresponding to $\\tilde{X}$ with reward functions $\\tilde{f}_{i}(\\cdot ) = \\frac{f_i}{\\phi ^{\\lambda }}(F^{-1}(\\cdot ))$ , $\\tilde{g}_{i}(\\cdot ) = \\frac{g_i}{\\phi ^{\\lambda }}(F^{-1}(\\cdot ))$ , $\\tilde{h}_{i}(\\cdot ) = \\frac{h_i}{\\phi ^{\\lambda }}(F^{-1}(\\cdot ))$ on $\\tilde{E}$ without discounting."
],
[
"Expected payoffs for threshold strategies",
"If players 1 and 2 use the strategies $D_{[0,\\ell ]}$ and $D_{[r,1]}$ respectively, where $0 \\le \\ell < r \\le 1$ , then the expected payoff $M^{x}_{1}(D_{[0,\\ell ]},D_{[r,1]})$ for player 1 (cf.",
"(REF )) satisfies, Mx1(D[0,],D[r,1]) = Ex[f1(XD[0,])1{D[0,] < D[r,1]} + g1(XD[r,1])1{D[r,1] < D[0,]}] + Ex[h1(XD[0,])1{D[0,] = D[r,1]}] = {ll f1(x), x [0,] f1()Px({D[0,] < D[r,1]}) + g1(r)Px({D[0,] > D[r,1]}), x (,r) g1(x), x [r,1] .",
"= {ll f1(x), x [0,] f1()r - xr - + g1(r)x - r - , x (,r) g1(x), x [r,1] .",
"Analogously, the expected payoff $M^{x}_{2}(D_{[0,\\ell ]},D_{[r,1]})$ for player 2 satisfies, $M^{x}_{2}(D_{[0,\\ell ]},D_{[r,1]}) = {\\left\\lbrace \\begin{array}{ll}g_{2}(x),& \\forall x \\in [0,\\ell ] \\\\g_{2}(\\ell )\\cdot \\frac{r - x}{r - \\ell } + f_{2}(r)\\cdot \\frac{x - \\ell }{r - \\ell },& \\forall x \\in (\\ell ,r) \\\\f_{2}(x),& \\forall x \\in [r,1].\\end{array}\\right.",
"}$"
],
[
"Derivatives of utility functions",
"Throughout this section we assume Condition G1' holds.",
"We first provide general formulas for the first and second partial derivatives of a utility function $U(x,y)$ which is of the form $U(x,y) = \\frac{F(x,y)}{y-x}$ .",
"xU(x,y) = xF(x,y)(y-x) + F(x,y)(y-x)2, yU(x,y) = yF(x,y)(y-x) - F(x,y)(y-x)2 xxU(x,y) = xxF(x,y)(y-x)2 + 2[xF(x,y)(y-x) + F(x,y)](y-x)3 yyU(x,y) = yyF(x,y)(y-x)2 - 2[yF(x,y)(y-x) - F(x,y)](y-x)3 xyU(x,y) = xyF(x,y)(y-x) + xF(x,y) + yF(x,y)(y-x)2 - 2[xF(x,y)(y-x) + F(x,y)](y-x)3 = xyF(x,y)(y-x) - yF(x,y) - xF(x,y)(y-x)2 + 2[yF(x,y)(y-x) - F(x,y)](y-x)3 Using equation (REF ) for the utility functions gives the following expressions for their partial derivatives, xU1(x,y) = f1(x) + f1'(x)(y-x) - g1(y)(y-x)2, yU2(x,y) = g2(x) + f2'(y)(y - x) - f2(y) (y-x)2 xxU1(x,y) = f1”(x)(y-x)2 + 2[f1(x) + f1'(x)(y-x) - g1(y)](y-x)3 yyU2(x,y) = f2”(y)(y-x)2 + 2[f2(y) - f2'(y)(y-x) - g2(x)](y-x)3 xyU1(x,y) = 2[g1(y) - f1(x)] - (f1'(x)+g1'(y))(y-x)(y-x)3 xyU2(x,y) = 2[g2(x)- f2(y)] + (g2'(x) + f2'(y))(y-x)(y-x)3"
],
[
"A verification theorem",
"Proposition D.1 Under Condition G2 and given $r \\in (a_{2},1]$ , $(\\ell ^{1},\\ell ^{2})$ is a solution to Problem (REF ) if and only if $V_{1}^{[r,1]}(x) \\sup \\limits _{\\tau _{1} \\in \\mathcal {T}}M^{x}_{1}(\\tau _{1},D_{[r,1]}) = M^{x}_{1}(D_{[\\ell ^{1},\\ell ^{2}]},D_{[r,1]}),\\quad \\forall x \\in [0,1].$ The arguments are more or less the same as those establishing Theorem REF .",
"For the sake of brevity we therefore only show the proof of necessity (Problem (REF ) $\\Rightarrow $ (REF )).",
"Define $u_{r}$ on $[0,1]$ by, ur(x) = Mx1(D[1,2],D[r,1]) - g1,[r,1](x) = {ll (f1(1) - g1,[r,1](1))x1, x [0,1), f1(x) - g1,[r,1](x), x [1,2), (f1(2) - g1,[r,1](2))r-xr-2, x [2,r), 0, x [r,1].",
".",
"Suppose $(\\ell ^{1},\\ell ^{2})$ is a solution to Problem (REF ).",
"Similarly to Theorem REF , we will prove (REF ) by showing that $u_{r}$ is the smallest non-negative concave majorant of $f_{1} - g_{1,[r,1]}$ on $[0,r]$ .",
"Initially we will analyse $u_r$ separately on $[0,\\ell ^1]$ and $[\\ell ^1,\\ell ^2]$ .",
"Observe firstly that the function $f_{1} - g_{1,[r,1]}$ is nonnegative when evaluated at the points $\\ell ^1$ and $\\ell ^2$ and hence, by concavity, on $[\\ell ^1,\\ell ^2]$ .",
"Recalling (REF ), this follows from (REF ), since $f_{1}(0) = g_{1,[r,1]}(0)$ and so $f_{1}(\\ell ^{2}) - g_{1,[r,1]}(\\ell ^{2}) \\ge 0$ .",
"Also $f_{1}(x) - g_{1,[r,1]}(x) \\le \\left(f_{1}(\\ell ^{1}) - g_{1,[r,1]}(\\ell ^{1})\\right)\\frac{x}{\\ell ^{1}}, \\forall x \\in (0,r),$ and taking $x = \\ell ^{2}$ shows that $f_{1}(\\ell ^{1}) - g_{1,[r,1]}(\\ell ^{1}) \\ge 0$ .",
"Therefore $u_{r}$ is a non-negative majorant of $f_{1} - g_{1,[r,1]}$ on $[0,\\ell ^{1}]$ .",
"This is also true on $[\\ell ^{1},r]$ , since $f_{1}(r) \\le g_{1}(r)$ and so $f_{1}(x) - g_{1,[r,1]}(x) \\le \\left(f_{1}(\\ell ^{2}) - g_{1,[r,1]}(\\ell ^{2})\\right)\\left(\\frac{r-x}{r-\\ell ^{2}}\\right),\\forall x \\in [0,r].$ Concavity holds for $u_{r}$ on the three intervals $[0,\\ell ^{1}]$ , $[\\ell ^{1},\\ell ^{2}]$ and $[\\ell ^{2},r]$ separately and, arguing as in the proof of Theorem REF , we can show that $u_{r}$ is continuous and concave on the entire interval $[0,r]$ , completing the proof.",
"Proposition D.2 Under Condition G2, for every $\\ell ^{1},\\ell ^{2}$ satisfying $0 < \\ell ^{1} \\le \\ell ^{2} < b$ , a point $r \\in [b,1]$ satisfies (REF ) with $\\ell = \\ell ^{2}$ and $U_2=\\hat{U}_3$ if and only if $V_{2}^{[\\ell ^{1},\\ell ^{2}]}(x) \\sup \\limits _{\\tau _{2} \\in \\mathcal {T}}M^{x}_{2}(D_{[\\ell ^{1},\\ell ^{2}]},\\tau _{2}) = M^{x}_{2}(D_{[\\ell ^{1},\\ell ^{2}]},D_{[r,1]}),\\quad \\forall x \\in [0,1].$ By Lemma REF it is sufficient merely to consider the optimal stopping problem on the set $[0,\\ell ^1] \\cup [\\ell ^2,1]$ with obstacle $f_{2} - g_{2,[\\ell ^1,\\ell ^2]}$ , and we will only sketch the solution.",
"Note that since $f_{2} \\le g_{2}$ it is clearly suboptimal to stop in $[\\ell ^{1},\\ell ^{2}]$ .",
"From Dynkin's formula it is also suboptimal to stop on $[0,\\ell ^{1}]$ , since $f_{2} - g_{2,[\\ell ^{1},\\ell ^{2}]}$ is convex there and $f_{2}(x) - g_{2,[\\ell ^{1},\\ell ^{2}]}(x)\\le 0$ for $x \\in \\lbrace 0,\\ell ^{1}\\rbrace $ .",
"The solution is nontrivial only on $(\\ell ^{2},1]$ , where the arguments used for Theorem REF are sufficient to complete the proof."
]
] | 1709.01905 |
[
[
"Extended Nonlocal Games from Quantum-Classical Games"
],
[
"Abstract Several variants of nonlocal games have been considered in the study of quantum entanglement and nonlocality.",
"This paper concerns two of these variants, called quantum-classical games and extended nonlocal games.",
"We give a construction of an extended nonlocal game from any quantum-classical game that allows one to translate certain facts concerning quantum- classical games to extended nonlocal games.",
"In particular, based on work of Regev and Vidick, we conclude that there exist extended nonlocal games for which no finite-dimensional entangled strategy can be optimal.",
"While this conclusion is a direct consequence of recent work of Slofstra, who proved a stronger, analogous result for ordinary (non-extended) nonlocal games, the proof based on our construction is considerably simpler, and the construction itself might potentially have other applications in the study of entanglement and nonlocality."
],
[
"Introduction",
"Various abstract notions of games have been considered in the study of entanglement and nonlocality [5], [1], [9], [11], [14], [15], [3], [12], [17], [23], [7], [6], [20], [13].",
"For instance, in a nonlocal game, two cooperating players (Alice and Bob) engage in an interaction with a third party (known as the referee) [5].",
"The referee randomly chooses a pair of questions $(x,y)$ according to a known distribution.",
"Alice receives $x$ , Bob receives $y$ , and without communicating with one another, Alice must respond with an answer $a$ and Bob with an answer $b$ .",
"The referee then evaluates a predicate $P(a,b|x,y)$ to determine whether Alice and Bob win or lose.",
"It is a well-known consequence of earlier work in theoretical physics [2], [16], [4] that entanglement shared between Alice and Bob can allow them to outperform all purely classical strategies for some nonlocal games.",
"(Nonlocal games were also previously studied in theoretical computer science, in [19] for instance, although generally not by this name and without deference to entanglement or quantum information, but rather as an abstraction of one-round, two-player classical interactive proof systems.)",
"In a nonlocal game, the referee is classical; it is only the players Alice and Bob that potentially manipulate quantum information.",
"Some generalizations of nonlocal games in which quantum information is exchanged in some way between the players and the referee include ones studied in [3], [12], [17], [23], [6], [20], [13].",
"In this paper we consider two such generalizations: quantum-classical games and extended nonlocal games.",
"1.",
"Quantum-classical games.",
"Quantum-classical games, or QC games for short, differ from nonlocal games in that the referee begins the game by preparing a tripartite quantum state, then sends one part of it to each player and keeps the third part for itself.",
"(This step replaces the generation of a classical question pair $(x,y)$ in an ordinary nonlocal game.)",
"The players respond with classical answers $a$ and $b$ as before, and finally the referee determines whether the players win or lose by measuring its part of the original quantum state it initially prepared.",
"(This step replaces the evaluation of a predicate $P(a,b|x,y)$ in an ordinary nonlocal game.)",
"Games of this form, with slight variations from the general class just described, were considered by Buscemi [3] and Regev and Vidick [20].",
"2.",
"Extended nonlocal games.",
"In an extended nonlocal game, Alice and Bob first present the referee with a quantum system of a fixed size, initialized as Alice and Bob choose, and possibly entangled with systems held by Alice and Bob.",
"(This initialization step generalizes the sharing of entanglement between Alice and Bob in an ordinary nonlocal game, allowing them to give a part of this shared state to the referee.)",
"The game then proceeds much like an ordinary nonlocal game: the referee chooses a pair of (classical) questions $(x,y)$ according to a known distribution, sends $x$ to Alice and $y$ to Bob, and receives a classical answer $a$ from Alice and $b$ from Bob.",
"Finally, to determine whether or not Alice and Bob win, the referee performs a binary-valued measurement, depending on $x$ , $y$ , $a$ , and $b$ , on the system initially sent to it by Alice and Bob.",
"(This measurement replaces the evaluation of the predicate $P(a,b|x,y)$ in an ordinary nonlocal game.)",
"Games of this form, again with a slight variation from the general class just described, were considered by Fritz [12], who called them bipartite steering games.",
"Extended nonlocal games represent a game-based formulation of the phenomenon of tripartite steering investigated in [8], [21].",
"(The clash in nomenclature reflects one's view of the referee's role either as a non-player in a game or as a participant in an experiment.)",
"Extended nonlocal games were so-named and studied in [13], as a means to unify nonlocal games with the monogamy-of-entanglement games introduced in [23].",
"Regev and Vidick [20] proved that certain QC games have the following peculiar property: if Alice and Bob make use of an entangled state of two finite-dimensional quantum systems, initially shared between them, they can never achieve perfect optimality: it is always possible for them to do better (meaning that they win with a strictly larger probability) using some different shared entangled state on two larger quantum systems.",
"Thus, it is only in the limit, as the local dimensions of their shared entangled states goes to infinity, that they can approach an optimal performance in these specific examples of games.",
"A similar result was established earlier for analogues of nonlocal games for which both the questions and answers are quantum [17], and a recent breakthrough result of Slofstra [22] has established a similar result for nonlocal games in which both the questions and answers are classical.",
"In this paper we describe a construction through which any QC game can be transformed into an extended nonlocal game, in such a way that basic properties associated with entangled strategies for the QC game are inherited by the extended nonlocal game.",
"In particular, by applying this construction to the QC games identified by Regev and Vidick, we obtain extended nonlocal games that cannot be played with perfect optimality by Alice and Bob using an entangled state on finite-dimensional systems.",
"In the language of quantum steering, this yields a tripartite steering inequality for which a maximal violation requires infinite-dimensional quantum systems.",
"While Slofstra's result subsumes this result, insofar as nonlocal games are special cases of extended nonlocal games in which the referee's quantum system is a trivial one-dimensional system, our proof is considerably simpler.",
"Moreover, this ability to transform from quantum-classical games to extended nonlocal games might potentially find utility in related settings."
],
[
"Definitions",
"We begin with precise definitions of the two classes of games considered in this paper, which are QC games and extended nonlocal games.",
"In addition, we formalize the notions of entangled strategies for these games along with their associated values, which represent the probabilities that the strategies lead to a win for Alice and Bob.",
"The reader is assumed to be familiar with standard notions of quantum information, as described in [18] and [25], for instance.",
"We will generally follow the terminology and notational conventions of [24].",
"For example, a register $\\textsf {X}$ is an abstract quantum system described by a finite-dimensional complex Hilbert space $\\mathcal {X}$ having a fixed standard basis $\\lbrace \\vert {0.5mu}1 {0.5mu}\\rangle ,\\ldots ,\\vert {0.5mu}n {0.5mu}\\rangle \\rbrace $ (for some positive integer $n$ ); the sets $\\mathrm {L}(\\mathcal {X})$ , $\\mathrm {Pos}(\\mathcal {X})$ , $\\mathrm {D}(\\mathcal {X})$ , and $\\mathrm {U}(\\mathcal {X})$ denote the set of all linear operators, positive semidefinite operators, density operators, and unitary operators (respectively) acting on such a space $\\mathcal {X}$ ; we write $X^{\\ast }$ , $\\overline{X}$ , and $X^{}$ to refer to the adjoint, entry-wise complex conjugate, and transpose of an operator $X$ (with respect to the standard basis in the case of the entry-wise complex conjugate and transpose); and $\\langle X , Y\\rangle = \\operatorname{Tr}(X^{\\ast } Y)$ denotes the Hilbert-Schmidt inner product of operators $X$ and $Y$ ."
],
[
"Extended nonlocal games",
"An extended nonlocal game is specified by the following objects: $\\bullet $ A probability distribution $\\pi :X\\times Y \\rightarrow [0,1]$ , for finite and nonempty sets $X$ and $Y$ .",
"$\\bullet $ A collection of measurement operators $\\lbrace P_{a,b,x,y}\\,:\\,a\\in A,\\; b\\in B,\\; x\\in X,\\; y\\in Y\\rbrace \\subset \\mathrm {Pos}(\\mathcal {R})$ , where $A$ and $B$ are finite and nonempty sets and $\\mathcal {R}$ is the space corresponding to a register $\\textsf {R}$ .",
"From the referee's perspective, such a game is played as follows: 1.",
"Alice and Bob present the referee with the register $\\textsf {R}$ , which has been initialized in a state of Alice and Bob's choosing.",
"(The register $\\textsf {R}$ might, for instance, be entangled with systems possessed by Alice and Bob.)",
"2.",
"The referee randomly generates a pair $(x,y) \\in X\\times Y$ according to the distribution $\\pi $ , and then sends $x$ to Alice and $y$ to Bob.",
"Alice responds with $a\\in A$ and Bob responds with $b\\in B$ .",
"3.",
"The referee measures $\\textsf {R}$ with respect to the binary-valued measurement $\\lbrace P_{a,b,x,y},\\, I-P_{a,b,x,y}\\rbrace $ .",
"The outcome corresponding to the measurement operator $P_{a,b,x,y}$ indicates that Alice and Bob win, while the other measurement result indicates that they lose.",
"There are various classes of strategies that may be considered for Alice and Bob in an extended nonlocal game, including unentangled strategies, entangled strategies (or standard quantum strategies), and commuting measurement strategies [13].",
"(Additional classes of strategies, such as no-signaling strategies, can also be defined.)",
"In this paper we will only consider entangled strategies, in which Alice and Bob begin the game in possession of finite-dimensional quantum systems that have been initialized as they choose.",
"They may then measure these systems in order to obtain answers to the referee's questions.",
"In more precise terms, an entangled strategy for an extended nonlocal game, specified by $\\pi :X\\times Y \\rightarrow [0,1]$ and $\\lbrace P_{a,b,x,y}\\,:\\,a\\in A,\\, b\\in B,\\, x\\in X,\\, y\\in Y\\rbrace \\subset \\mathrm {Pos}(\\mathcal {R})$ as above, consists of these objects: 1.",
"A state $\\sigma \\in \\mathrm {D}(\\mathcal {U}\\otimes \\mathcal {R}\\otimes $ , for $\\mathcal {U}$ being the space corresponding to a register $\\textsf {U}$ held by Alice and $ being the spacecorresponding to a register $V$ held by Bob.This state represents Alice and Bob^{\\prime }s initialization of the triple$ (U,R,V)$ immediately before $R$ is sent to thereferee.\\item [2.",
"]A measurement $ {Axa : aA}Pos(U)$ for each $ xX$,performed by Alice when she receives the question $ x$, and ameasurement $ {Byb : bB}Pos($ for each $ yY$,performed by Bob when he receives the question $ y$.$ When Alice and Bob utilize such a strategy, their winning probability $p$ may be expressed as $p =\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\(a,b) \\in A\\times B\\\\\\end{array}}\\pi (x,y)\\bigl \\langle A^x_a \\otimes P_{a,b,x,y} \\otimes B^y_b, \\sigma \\bigr \\rangle .$ The entangled value of an extended nonlocal game represents the supremum of the winning probabilities, taken over all entangled strategies.",
"If $H$ is the name assigned to an extended nonlocal game having a specification as above, then we write $\\omega ^{\\ast }_N(H)$ to denote the maximum winning probability taken over all entangled strategies for which $\\dim (\\mathcal {U}\\otimes \\le N$ , so that the entangled value of $H$ is $\\omega ^{\\ast }(H) = \\lim _{N\\rightarrow \\infty }\\omega ^{\\ast }_N(H).$ Quantum-classical games A quantum-classical game (or QC game, for short) is specified by the following objects: $\\bullet $ A state $\\rho \\in \\mathrm {D}(\\mathcal {X}\\otimes \\mathcal {S}\\otimes \\mathcal {Y})$ of a triple of registers $(\\textsf {X},\\textsf {S},\\textsf {Y})$ .",
"$\\bullet $ A collection of measurement operators $\\lbrace Q_{a,b}\\,:\\,a \\in A,\\; b\\in B\\rbrace \\subset \\mathrm {Pos}(\\mathcal {S})$ , for finite and nonempty sets $A$ and $B$ .",
"From the referee's perspective, such a game is played as follows: 1.",
"The referee prepares $(\\textsf {X},\\textsf {S},\\textsf {Y})$ in the state $\\rho $ , then sends $\\textsf {X}$ to Alice and $\\textsf {Y}$ to Bob.",
"2.",
"Alice responds with $a\\in A$ and Bob responds with $b\\in B$ .",
"3.",
"The referee measures $\\textsf {S}$ with respect to the binary-valued measurement $\\lbrace Q_{a,b},\\, I-Q_{a,b}\\rbrace $ .",
"The outcome corresponding to the measurement operator $Q_{a,b}$ indicates that Alice and Bob win, while the other measurement result indicates that they lose.",
"Similar to extended nonlocal games, one may consider various classes of strategies for QC games.",
"Again, we will consider only entangled strategies, in which Alice and Bob begin the game in possession of finite-dimensional quantum systems initialized as they choose.",
"More precisely, an entangled strategy for a QC game, specified by $\\rho \\in \\mathrm {D}(\\mathcal {X}\\otimes \\mathcal {S}\\otimes \\mathcal {Y})$ and $\\lbrace Q_{a,b}\\,:\\,a \\in A,\\; b\\in B\\rbrace \\subset \\mathrm {Pos}(\\mathcal {S})$ as above, consists of these objects: 1.",
"A state $\\sigma \\in \\mathrm {D}(\\mathcal {U}\\otimes $ , for $\\mathcal {U}$ being the space corresponding to a register $\\textsf {U}$ held by Alice and $ being the spacecorresponding to a register $V$ held by Bob.\\item [2.",
"]A measurement $ {Aa : aA}Pos(UX)$ for Alice,performed on the pair $ (U,X)$ after she receives $X$ fromthe referee, and a measurement $ {Bb : bB}Pos(Y$for Bob, performed on the pair $ (Y,V)$ after he receives$Y$ from the referee.$ The winning probability of such a strategy may be expressed as $p = \\sum _{(a,b)\\in A\\times B}\\bigl \\langle A_a \\otimes Q_{a,b} \\otimes B_b, W(\\sigma \\otimes \\rho ) W^{\\ast } \\bigr \\rangle ,$ where $W$ is the unitary operator that corresponds to the natural re-ordering of registers consistent with each of the tensor product operators $A_a \\otimes Q_{a,b} \\otimes B_b$ (i.e., the permutation $(\\textsf {U},\\textsf {V},\\textsf {X},\\textsf {S},\\textsf {Y})\\mapsto (\\textsf {U},\\textsf {X},\\textsf {S},\\textsf {Y},\\textsf {V})$ ).",
"Construction and analysis In this section we will describe a construction of an extended nonlocal game from any given QC game, and analyze the relationship between the constructed extended nonlocal game and the original QC game.",
"Construction Suppose that a QC game $G$ , specified by a state $\\rho \\in \\mathrm {D}(\\mathcal {X}\\otimes \\mathcal {S}\\otimes \\mathcal {Y})$ and a collection of measurement operators $\\lbrace Q_{a,b}\\,:\\,a \\in A,\\; b\\in B\\rbrace \\subset \\mathrm {Pos}(\\mathcal {S})$ , is given.",
"We construct an extended nonlocal game $H$ as follows: 1.",
"Let $n = \\dim (\\mathcal {X})$ and $m = \\dim (\\mathcal {Y})$ , let $X = \\bigl \\lbrace 1,\\ldots ,n^2\\bigr \\rbrace \\quad \\text{and}\\quad Y = \\bigl \\lbrace 1,\\ldots ,m^2\\bigr \\rbrace ,$ and let $\\pi :X\\times Y\\rightarrow [0,1]$ be the uniform probability distribution on these sets, so that $\\pi (x,y) = n^{-2}m^{-2}$ for every $x\\in X$ and $y\\in Y$ .",
"2.",
"Let $\\textsf {R} = (\\textsf {X},\\textsf {Y})$ , define $\\xi = \\operatorname{Tr}_{\\mathcal {S}}(\\rho ) \\quad \\text{and}\\quad \\xi _{a,b} =\\operatorname{Tr}_{\\mathcal {S}}\\bigl [\\bigl (I_{\\mathcal {X}}\\otimes Q_{a,b} \\otimes I_{\\mathcal {Y}}\\bigr )\\rho \\bigr ]$ for each $a\\in A$ and $b\\in B$ , let $\\bigl \\lbrace U_1,\\ldots ,U_{n^2}\\bigr \\rbrace \\subset \\mathrm {U}(\\mathcal {X})\\quad \\text{and}\\quad \\bigl \\lbrace V_1,\\ldots ,V_{m^2}\\bigr \\rbrace \\subset \\mathrm {U}(\\mathcal {Y})$ be orthogonal bases of unitary operators (such as the discrete Weyl operators, described in [10] for instance), and let $P_{a,b,x,y} = I_{\\mathcal {X}}\\otimes I_{\\mathcal {Y}}- (U_x \\otimes V_y)(\\xi ^{} - \\xi _{a,b}^{})(U_x \\otimes V_y)^{\\ast }$ for every $a\\in A$ , $b\\in B$ , $x\\in X$ , and $y\\in Y$ .",
"One may observe that $P_{a,b,x,y}$ is indeed a measurement operator for each $a\\in A$ , $b\\in B$ , $x\\in X$ , and $y\\in Y$ , meaning that $0 \\le P_{a,b,x,y} \\le I_{\\mathcal {X}}\\otimes I_{\\mathcal {Y}}$ , by virtue of the fact that $0 \\le \\xi _{a,b} \\le \\xi \\le I$ for every $a\\in A$ and $b\\in B$ .",
"The basic intuition behind this construction is as follows.",
"In the QC game $G$ , the referee sends $\\textsf {X}$ to Alice and $\\textsf {Y}$ to Bob, but in the extended nonlocal game $H$ it is Alice and Bob that give $\\textsf {X}$ and $\\textsf {Y}$ to the referee.",
"To simulate, within the game $H$ , the sort of transmission that occurs in $G$ , it is natural to consider teleportation—for if Alice provided the referee with the register $\\textsf {X}$ in a state maximally entangled with a register of her own, and Bob did likewise with $\\textsf {Y}$ , then the referee could effectively teleport a copy of $\\textsf {X}$ to Alice and a copy of $\\textsf {Y}$ to Bob.",
"Now, in an extended nonlocal game, the referee cannot actually perform teleportation in this way: the question pair $(x,y)$ needs to be randomly generated, independent of the state of the registers $(\\textsf {X},\\textsf {Y})$ .",
"For this reason the game $H$ is based on a form of post-selected teleportation, where $x$ and $y$ are chosen randomly, and then later compared with hypothetical measurement results that would be obtained if the referee were to perform teleportation.",
"The details of the construction above result from a combination of this idea together with algebraic simplifications.",
"Game values It is not immediate that the construction above should necessarily translate the basic properties of the game $G$ to the game $H$ ; Alice and Bob are free to behave as they choose, which is not necessarily consistent with the intuitive description of the game $H$ based on teleportation suggested above.",
"An analysis does, however, reveal that the construction works as one would hope (and perhaps expect).",
"In particular, we will prove two bounds on the value of the extended nonlocal game $H$ constructed from a QC game $G$ as described above: $\\omega ^{\\ast }_{nmN}(H) \\ge 1 - \\frac{1 - \\omega ^{\\ast }_N(G)}{nm}\\quad \\text{and}\\quad \\omega ^{\\ast }_{N}(H) \\le 1 - \\frac{1 - \\omega ^{\\ast }_{nmN}(G)}{nm},$ for every positive integer $N$ .",
"This implies that $\\omega ^{\\ast }(H) = 1 - \\frac{1 - \\omega ^{\\ast }(G)}{nm}.$ Moreover, $H$ inherits the same limiting behavior of $G$ with respect to entangled strategies, meaning that if $\\omega ^{\\ast }_N(G) < \\omega ^{\\ast }(G)$ for all $N\\in \\mathbb {N}$ , then $\\omega ^{\\ast }_N(H) < \\omega ^{\\ast }(H)$ for all $N\\in \\mathbb {N}$ as well.",
"We will begin with the first inequality in (REF ).",
"Assume that an arbitrary strategy for Alice and Bob in the game $G$ is fixed: Alice and Bob make use of a shared entangled state $\\sigma \\in \\mathrm {D}(\\mathcal {U}\\otimes $ , where $\\dim (\\mathcal {U}\\otimes \\le N$ , and their measurements are given by $\\lbrace A_a\\,:\\,a\\in A\\rbrace \\subset \\mathrm {Pos}(\\mathcal {U}\\otimes \\mathcal {X})\\quad \\text{and}\\quad \\lbrace B_b\\,:\\,b\\in B\\rbrace \\subset \\mathrm {Pos}(\\mathcal {Y}\\otimes ,$ respectively.",
"The winning probability of this strategy in the game $G$ may be expressed as $p = \\sum _{(a,b)\\in A\\times B}\\bigl \\langle A_a \\otimes Q_{a,b} \\otimes B_b, W (\\sigma \\otimes \\rho ) W^{\\ast } \\bigr \\rangle ,$ as was mentioned above, while the losing probability equals $q = \\sum _{(a,b)\\in A\\times B}\\bigl \\langle A_a \\otimes (I- Q_{a,b})\\otimes B_b, W(\\sigma \\otimes \\rho )W^{\\ast } \\bigr \\rangle = 1-p.$ We adapt this strategy to obtain one for $H$ as follows: 1.",
"Alice will hold a register $\\textsf {X}^{\\prime }$ , representing a copy of $\\textsf {X}$ , and Bob will hold $\\textsf {Y}^{\\prime }$ , representing a copy of $\\textsf {Y}$ .",
"The initial state of the register pairs $(\\textsf {X}^{\\prime },\\textsf {X})$ and $(\\textsf {Y}^{\\prime },\\textsf {Y})$ are to be the canonical maximally entangled states $\\vert {0.5mu}\\psi {0.5mu}\\rangle = \\frac{1}{\\sqrt{n}} \\sum _{j = 1}^n \\vert {0.5mu}j {0.5mu}\\rangle \\vert {0.5mu}j {0.5mu}\\rangle \\quad \\text{and}\\quad \\vert {0.5mu}\\phi {0.5mu}\\rangle = \\frac{1}{\\sqrt{m}} \\sum _{k = 1}^m \\vert {0.5mu}k {0.5mu}\\rangle \\vert {0.5mu}k {0.5mu}\\rangle ,$ respectively, where $n$ and $m$ are the dimensions of the spaces corresponding to the registers $\\textsf {X}$ and $\\textsf {Y}$ .",
"In addition, Alice holds the register $\\textsf {U}$ and Bob holds the register $\\textsf {V}$ , with $(\\textsf {U},\\textsf {V})$ being prepared in the same shared entangled state $\\sigma $ that is used in the strategy for $G$ .",
"2.",
"Upon receiving the question $x\\in X$ from the referee, Alice performs the unitary operation $\\overline{U_x}$ on $\\textsf {X}^{\\prime }$ , then measures $(\\textsf {U},\\textsf {X}^{\\prime })$ with respect to the measurement $\\lbrace A_a\\,:\\,a\\in A\\rbrace $ to obtain an answer $a\\in A$ .",
"Similarly, upon receiving $y\\in Y$ from the referee, Bob performs $\\overline{V_y}$ on $\\textsf {Y}^{\\prime }$ , then measures $(\\textsf {Y}^{\\prime },\\textsf {V})$ with respect to $\\lbrace B_b\\,:\\,b\\in B\\rbrace $ to obtain an answer $b\\in B$ .",
"The performance of this strategy can be analyzed by first ignoring the specific initialization of the registers described in step 1, and defining a measurement $\\lbrace R_0,R_1\\rbrace $ that determines, for an arbitrary initialization of these registers, whether Alice and Bob win or lose by behaving as described in step 2.",
"In particular, the measurement $\\lbrace R_0,R_1\\rbrace $ is defined on the register tuple $(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {X},\\textsf {Y},\\textsf {Y}^{\\prime },\\textsf {V})$ , the measurement operator $R_0$ corresponds to a losing outcome, and $R_1$ corresponding to a winning outcome.",
"These operators may be described as follows: $\\begin{aligned}R_0 & =\\frac{1}{n^2 m^2}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\[0.3mm](a,b) \\in A\\times B\\end{array}}(I_{\\mathcal {U}} \\otimes U_x^{}) A_a (I_{\\mathcal {U}} \\otimes \\overline{U_x})\\otimes (I_{\\mathcal {X}\\otimes \\mathcal {Y}} - P_{a,b,x,y})\\otimes (V_y^{} \\otimes I_{) B_b (\\overline{V_y}\\otimes I_{)\\\\R_1 & =\\frac{1}{n^2 m^2}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\[0.3mm](a,b) \\in A\\times B\\end{array}}(I_{\\mathcal {U}} \\otimes U_x^{}) A_a (I_{\\mathcal {U}} \\otimes \\overline{U_x})\\otimes P_{a,b,x,y}\\otimes (V_y^{} \\otimes I_{) B_b (\\overline{V_y}\\otimes I_{)= I- R_0.",
"}}}Now we may consider the initialization of the registers described in step~1.For an arbitrary choice of operators X\\in \\mathrm {L}(\\mathcal {U}) andY\\in \\mathrm {L}( we have\\begin{equation}\\bigl \\langle R_0, X \\otimes \\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert \\otimes \\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert \\otimes Y \\bigr \\rangle = \\sum _{(a,b) \\in A\\times B}\\bigl \\langle A_a \\otimes (\\xi ^{} - \\xi ^{}_{a,b}) \\otimes B_b, X \\otimes \\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert \\otimes \\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert \\otimes Y \\bigr \\rangle ,\\end{equation}by virtue of the fact that\\bigl ( \\overline{U_x} \\otimes U_x\\bigr ) \\vert {0.5mu}\\psi {0.5mu}\\rangle = \\vert {0.5mu}\\psi {0.5mu}\\rangle and \\bigl ( \\overline{V_y} \\otimes V_y\\bigr ) \\vert {0.5mu}\\phi {0.5mu}\\rangle = \\vert {0.5mu}\\phi {0.5mu}\\rangle for every x\\in X and y\\in Y.Further simplifying this expression, one obtains\\begin{equation}\\begin{aligned}\\sum _{(a,b) \\in A\\times B}\\bigl \\langle A_a \\otimes (\\xi ^{} - \\xi ^{}_{a,b}) \\otimes B_b, X \\otimes \\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert \\otimes \\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert \\otimes Y \\bigr \\rangle \\hspace{-142.26378pt}\\\\& = \\frac{1}{nm}\\sum _{(a,b) \\in A\\times B}\\bigl \\langle A_a \\otimes B_b, X \\otimes (\\xi -\\xi _{a,b}) \\otimes Y \\bigr \\rangle \\\\& = \\frac{1}{nm}\\sum _{(a,b) \\in A\\times B}\\bigl \\langle A_a \\otimes (I- Q_{a,b}) \\otimes B_b, X \\otimes \\rho \\otimes Y \\bigr \\rangle .\\end{aligned}\\end{equation}By expressing the initial state \\sigma of (\\textsf {U},\\textsf {V}) as\\sigma = \\sum _i X_i \\otimes Y_i and making use of the bilinearity ofthe above expression in X and Y, one finds that the losingprobability of Alice and Bob^{\\prime }s strategy for H is equal toq/(nm), for q being the losing probability (\\ref {eq:losing-in-G}) for theiroriginal strategy for G.}Optimizing over all strategies for G that make use of an initial shared statehaving total dimension at most N yields the required inequality\\begin{equation}\\omega ^{\\ast }_{nmN}(H) \\ge 1 - \\frac{1 - \\omega ^{\\ast }_N(G)}{nm}.\\end{equation}\\end{aligned}Next we will prove the second inequality in (\\ref {eq:two-bounds-omega-H}).Assume that an arbitrary strategy for Alice and Bob in the extended nonlocalgame H constructed from G is fixed: the strategy consists of an initialstate \\sigma \\in \\mathrm {D}(\\mathcal {U}\\otimes (\\mathcal {X}\\otimes \\mathcal {Y})\\otimes for the registers(\\textsf {U},(\\textsf {X},\\textsf {Y}),\\textsf {V}), where \\dim (\\mathcal {U}\\otimes \\le N, alongwith measurements\\begin{equation}\\bigl \\lbrace A^x_a\\,:\\,a\\in A\\bigr \\rbrace \\subset \\mathrm {Pos}(\\mathcal {U})\\quad \\text{and}\\quad \\bigl \\lbrace B^y_b\\,:\\,b\\in B\\bigr \\rbrace \\subset \\mathrm {Pos}(\\end{equation}for Alice and Bob, respectively, for each x\\in X and y \\in Y.The winning probability of this strategy may be expressed as\\begin{equation}p = \\frac{1}{n^2 m^2}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\(a,b) \\in A\\times B\\\\\\end{array}}\\Bigl \\langle A^x_a \\otimes P_{a,b,x,y} \\otimes B^y_b, \\sigma \\Bigr \\rangle \\end{equation}while the losing probability is\\begin{equation}q = \\frac{1}{n^2 m^2}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\(a,b) \\in A\\times B\\\\\\end{array}}\\Bigl \\langle A^x_a \\otimes (I- P_{a,b,x,y}) \\otimes B^y_b, \\sigma \\Bigr \\rangle = 1-p.\\end{equation}We adapt this strategy to give one for G as follows:{\\begin{list}{}{}\\item [1.",
"]Let \\textsf {X}^{\\prime } and \\textsf {Y}^{\\prime } represent copies of the registers \\textsf {X} and\\textsf {Y}.Alice and Bob will initially share the registers(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {Y}^{\\prime },\\textsf {V}) initialized to the state\\overline{\\sigma }, with Alice holding (\\textsf {U},\\textsf {X}^{\\prime }) and Bob holding(\\textsf {Y}^{\\prime },\\textsf {V}).\\item [2.",
"]Upon receiving \\textsf {X} from the referee, Alice first measures the pair(\\textsf {X}^{\\prime },\\textsf {X}) with respect to the basis\\bigl \\lbrace (I\\otimes U_x^{\\ast })\\vert {0.5mu}\\psi {0.5mu}\\rangle \\,:\\,x\\in X\\bigr \\rbrace .For whichever outcome x\\in X she obtains, she then measures \\textsf {U} withrespect to the measurement\\begin{equation}\\Bigl \\lbrace \\overline{A^x_a}\\,:\\,a\\in A\\Bigr \\rbrace \\subset \\mathrm {Pos}(\\mathcal {U})\\end{equation}to obtain an outcome a\\in A.Bob does likewise, first measuring (\\textsf {Y}^{\\prime },\\textsf {Y}) with respect to thebasis \\bigl \\lbrace (I\\otimes V_y^{\\ast })\\vert {0.5mu}\\phi {0.5mu}\\rangle \\,:\\,y\\in Y\\bigr \\rbrace ,and then measuring \\textsf {V} with respect to the measurement\\begin{equation}\\Bigl \\lbrace \\overline{B^y_b}\\,:\\,b\\in B\\Bigr \\rbrace \\subset \\mathrm {Pos}(\\end{equation}for whichever outcome y\\in Y is obtained.\\end{list}}$ Now let us consider the probability with which this strategy wins in $G$ .",
"The state of the registers $(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {X},\\textsf {S},\\textsf {Y},\\textsf {Y}^{\\prime },\\textsf {V})$ immediately after the referee sends $\\textsf {X}$ to Alice and $\\textsf {Y}$ to Bob is given by $W ( \\overline{\\sigma } \\otimes \\rho ) W^{\\ast },$ where $W$ is a unitary operator that corresponds to a permutation of registers: $(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {Y}^{\\prime },\\textsf {V},\\textsf {X},\\textsf {S},\\textsf {Y}) \\mapsto (\\textsf {U},\\textsf {X}^{\\prime },\\textsf {X},\\textsf {S},\\textsf {Y},\\textsf {Y}^{\\prime },\\textsf {V}).$ We may define a measurement $\\lbrace R_0,R_1\\rbrace $ on the register tuple $(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {X},\\textsf {S},\\textsf {Y},\\textsf {Y}^{\\prime },\\textsf {V})$ representing the outcome of the game, with $R_0$ corresponding to a losing outcome and $R_1$ corresponding to a winning outcome.",
"We have $\\begin{aligned}R_0 & =\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\[0.3mm](a,b) \\in A\\times B\\end{array}}\\overline{A^x_a} \\otimes (I\\otimes U_x^{\\ast })\\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert (I\\otimes U_x)\\otimes (I- Q_{a,b}) \\otimes (V_y^{\\ast }\\otimes I)\\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert (V_y\\otimes I)\\otimes \\overline{B^y_b}\\\\R_1 & =\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\[0.3mm](a,b) \\in A\\times B\\end{array}}\\overline{A^x_a} \\otimes (I\\otimes U_x^{\\ast })\\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert (I\\otimes U_x)\\otimes Q_{a,b} \\otimes (V_y^{\\ast }\\otimes I)\\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert (V_y\\otimes I)\\otimes \\overline{B^y_b}.\\end{aligned}$ Simplifying expressions for the probability that Alice and Bob lose yields $\\bigl \\langle R_0, W ( \\overline{\\sigma } \\otimes \\rho ) W^{\\ast } \\bigr \\rangle = \\frac{1}{nm}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\(a,b) \\in A\\times B\\\\\\end{array}}\\Bigl \\langle A^x_a \\otimes (I- P_{a,b,x,y}) \\otimes B^y_b, \\sigma \\Bigr \\rangle = nm q,$ for $q$ being the losing probability (REF ) for their original strategy for $H$ .",
"Optimizing over all strategies for $H$ that make use of an initial shared state for which Alice and Bob's total dimension is at most $N$ yields the inequality $\\omega ^{\\ast }_{N}(H) \\le 1 - \\frac{1 - \\omega ^{\\ast }_{nmN}(G)}{nm}.$ Discussion As was mentioned in the introduction, Regev and Vidick [20] have identified examples of QC games for which Alice and Bob can never achieve optimality by using a finite-dimensional entangled strategy.",
"To be more precise, they prove that there exists a QC gameTheir games fall into a category of QC games that they call quantum XOR games, in which $A = B = \\lbrace 0,1\\rbrace $ and only the parity $a\\oplus b$ of Alice and Bob's answers is relevant to the referee's determination of whether they win or lose.",
"$G$ (and in fact a family of such games) for which it holds that $\\omega ^{\\ast }_N(G) < 1$ for all $N\\in \\mathbb {N}$ , while $\\omega ^{\\ast }(G) = 1$ .",
"By applying our construction to any such game, we obtain an extended nonlocal game $H$ with the property that $\\omega ^{\\ast }_N(H) < 1$ for all $N\\in \\mathbb {N}$ , while $\\omega ^{\\ast }(H) = 1$ .",
"In greater detail, by taking the simplest known example of a QC game $G$ with the property just described, and applying our construction (along with minor simplifications), one obtains an extended nonlocal game as follows: 1.",
"Let $\\mathcal {X}= \\mathcal {Y}= \\mathbb {C}^3$ and let $U_1,\\ldots ,U_9$ be the discrete Weyl operators acting on $\\mathbb {C}^3$ .",
"Also define $\\begin{aligned}\\vert {0.5mu}\\gamma _0 {0.5mu}\\rangle & = \\frac{1}{\\sqrt{2}}\\vert {0.5mu}0 {0.5mu}\\rangle \\vert {0.5mu}0 {0.5mu}\\rangle + \\frac{1}{2}\\vert {0.5mu}1 {0.5mu}\\rangle \\vert {0.5mu}1 {0.5mu}\\rangle + \\frac{1}{2}\\vert {0.5mu}2 {0.5mu}\\rangle \\vert {0.5mu}2 {0.5mu}\\rangle ,\\\\\\vert {0.5mu}\\gamma _1 {0.5mu}\\rangle & = \\frac{1}{\\sqrt{2}}\\vert {0.5mu}0 {0.5mu}\\rangle \\vert {0.5mu}0 {0.5mu}\\rangle - \\frac{1}{2}\\vert {0.5mu}1 {0.5mu}\\rangle \\vert {0.5mu}1 {0.5mu}\\rangle - \\frac{1}{2}\\vert {0.5mu}2 {0.5mu}\\rangle \\vert {0.5mu}2 {0.5mu}\\rangle .\\end{aligned}$ 2.",
"Alice and Bob give a pair of registers $(\\textsf {X},\\textsf {Y})$ to the referee, initialized as they choose.",
"The referee randomly chooses $x,y\\in \\lbrace 1,\\ldots ,9\\rbrace $ uniformly and independently at random, then sends $x$ to Alice and $y$ to Bob.",
"Alice and Bob respond with binary values $a,b\\in \\lbrace 0,1\\rbrace $ , respectively.",
"3.",
"The referee computes $c = a\\oplus b$ , then measures the pair $(\\textsf {X},\\textsf {Y})$ with respect to the measurement $\\bigl \\lbrace I_{\\mathcal {X}}\\otimes I_{\\mathcal {Y}} - (U_x \\otimes U_y)\\vert {0.5mu}\\gamma _c {0.5mu}\\rangle \\langle {0.5mu}\\gamma _c {0.5mu}\\vert (U_x\\otimes U_y)^{\\ast },\\;(U_x \\otimes U_y)\\vert {0.5mu}\\gamma _c {0.5mu}\\rangle \\langle {0.5mu}\\gamma _c {0.5mu}\\vert (U_x \\otimes U_y)^{\\ast }\\bigr \\rbrace .$ The first outcome represents a win for Alice and Bob, and the second a loss.",
"(Note that here we have scaled the losing measurement operator by a factor of two in comparison to what is described in the construction, which has the effect of doubling the losing probability for every strategy of Alice and Bob.)",
"Assuming Alice and Bob initially entangle the pair $(\\textsf {X},\\textsf {Y})$ with finite-dimensional registers of their own, they can never win the game with certainty, but they can approach certainty by using increasingly large systems.",
"Acknowledgments This work was partially funded by Canada's NSERC.",
"We thank Marco Piani, William Slofstra, and Thomas Vidick for helpful comments and discussions."
],
[
"Construction and analysis",
"In this section we will describe a construction of an extended nonlocal game from any given QC game, and analyze the relationship between the constructed extended nonlocal game and the original QC game."
],
[
"Construction",
"Suppose that a QC game $G$ , specified by a state $\\rho \\in \\mathrm {D}(\\mathcal {X}\\otimes \\mathcal {S}\\otimes \\mathcal {Y})$ and a collection of measurement operators $\\lbrace Q_{a,b}\\,:\\,a \\in A,\\; b\\in B\\rbrace \\subset \\mathrm {Pos}(\\mathcal {S})$ , is given.",
"We construct an extended nonlocal game $H$ as follows: 1.",
"Let $n = \\dim (\\mathcal {X})$ and $m = \\dim (\\mathcal {Y})$ , let $X = \\bigl \\lbrace 1,\\ldots ,n^2\\bigr \\rbrace \\quad \\text{and}\\quad Y = \\bigl \\lbrace 1,\\ldots ,m^2\\bigr \\rbrace ,$ and let $\\pi :X\\times Y\\rightarrow [0,1]$ be the uniform probability distribution on these sets, so that $\\pi (x,y) = n^{-2}m^{-2}$ for every $x\\in X$ and $y\\in Y$ .",
"2.",
"Let $\\textsf {R} = (\\textsf {X},\\textsf {Y})$ , define $\\xi = \\operatorname{Tr}_{\\mathcal {S}}(\\rho ) \\quad \\text{and}\\quad \\xi _{a,b} =\\operatorname{Tr}_{\\mathcal {S}}\\bigl [\\bigl (I_{\\mathcal {X}}\\otimes Q_{a,b} \\otimes I_{\\mathcal {Y}}\\bigr )\\rho \\bigr ]$ for each $a\\in A$ and $b\\in B$ , let $\\bigl \\lbrace U_1,\\ldots ,U_{n^2}\\bigr \\rbrace \\subset \\mathrm {U}(\\mathcal {X})\\quad \\text{and}\\quad \\bigl \\lbrace V_1,\\ldots ,V_{m^2}\\bigr \\rbrace \\subset \\mathrm {U}(\\mathcal {Y})$ be orthogonal bases of unitary operators (such as the discrete Weyl operators, described in [10] for instance), and let $P_{a,b,x,y} = I_{\\mathcal {X}}\\otimes I_{\\mathcal {Y}}- (U_x \\otimes V_y)(\\xi ^{} - \\xi _{a,b}^{})(U_x \\otimes V_y)^{\\ast }$ for every $a\\in A$ , $b\\in B$ , $x\\in X$ , and $y\\in Y$ .",
"One may observe that $P_{a,b,x,y}$ is indeed a measurement operator for each $a\\in A$ , $b\\in B$ , $x\\in X$ , and $y\\in Y$ , meaning that $0 \\le P_{a,b,x,y} \\le I_{\\mathcal {X}}\\otimes I_{\\mathcal {Y}}$ , by virtue of the fact that $0 \\le \\xi _{a,b} \\le \\xi \\le I$ for every $a\\in A$ and $b\\in B$ .",
"The basic intuition behind this construction is as follows.",
"In the QC game $G$ , the referee sends $\\textsf {X}$ to Alice and $\\textsf {Y}$ to Bob, but in the extended nonlocal game $H$ it is Alice and Bob that give $\\textsf {X}$ and $\\textsf {Y}$ to the referee.",
"To simulate, within the game $H$ , the sort of transmission that occurs in $G$ , it is natural to consider teleportation—for if Alice provided the referee with the register $\\textsf {X}$ in a state maximally entangled with a register of her own, and Bob did likewise with $\\textsf {Y}$ , then the referee could effectively teleport a copy of $\\textsf {X}$ to Alice and a copy of $\\textsf {Y}$ to Bob.",
"Now, in an extended nonlocal game, the referee cannot actually perform teleportation in this way: the question pair $(x,y)$ needs to be randomly generated, independent of the state of the registers $(\\textsf {X},\\textsf {Y})$ .",
"For this reason the game $H$ is based on a form of post-selected teleportation, where $x$ and $y$ are chosen randomly, and then later compared with hypothetical measurement results that would be obtained if the referee were to perform teleportation.",
"The details of the construction above result from a combination of this idea together with algebraic simplifications."
],
[
"Game values",
"It is not immediate that the construction above should necessarily translate the basic properties of the game $G$ to the game $H$ ; Alice and Bob are free to behave as they choose, which is not necessarily consistent with the intuitive description of the game $H$ based on teleportation suggested above.",
"An analysis does, however, reveal that the construction works as one would hope (and perhaps expect).",
"In particular, we will prove two bounds on the value of the extended nonlocal game $H$ constructed from a QC game $G$ as described above: $\\omega ^{\\ast }_{nmN}(H) \\ge 1 - \\frac{1 - \\omega ^{\\ast }_N(G)}{nm}\\quad \\text{and}\\quad \\omega ^{\\ast }_{N}(H) \\le 1 - \\frac{1 - \\omega ^{\\ast }_{nmN}(G)}{nm},$ for every positive integer $N$ .",
"This implies that $\\omega ^{\\ast }(H) = 1 - \\frac{1 - \\omega ^{\\ast }(G)}{nm}.$ Moreover, $H$ inherits the same limiting behavior of $G$ with respect to entangled strategies, meaning that if $\\omega ^{\\ast }_N(G) < \\omega ^{\\ast }(G)$ for all $N\\in \\mathbb {N}$ , then $\\omega ^{\\ast }_N(H) < \\omega ^{\\ast }(H)$ for all $N\\in \\mathbb {N}$ as well.",
"We will begin with the first inequality in (REF ).",
"Assume that an arbitrary strategy for Alice and Bob in the game $G$ is fixed: Alice and Bob make use of a shared entangled state $\\sigma \\in \\mathrm {D}(\\mathcal {U}\\otimes $ , where $\\dim (\\mathcal {U}\\otimes \\le N$ , and their measurements are given by $\\lbrace A_a\\,:\\,a\\in A\\rbrace \\subset \\mathrm {Pos}(\\mathcal {U}\\otimes \\mathcal {X})\\quad \\text{and}\\quad \\lbrace B_b\\,:\\,b\\in B\\rbrace \\subset \\mathrm {Pos}(\\mathcal {Y}\\otimes ,$ respectively.",
"The winning probability of this strategy in the game $G$ may be expressed as $p = \\sum _{(a,b)\\in A\\times B}\\bigl \\langle A_a \\otimes Q_{a,b} \\otimes B_b, W (\\sigma \\otimes \\rho ) W^{\\ast } \\bigr \\rangle ,$ as was mentioned above, while the losing probability equals $q = \\sum _{(a,b)\\in A\\times B}\\bigl \\langle A_a \\otimes (I- Q_{a,b})\\otimes B_b, W(\\sigma \\otimes \\rho )W^{\\ast } \\bigr \\rangle = 1-p.$ We adapt this strategy to obtain one for $H$ as follows: 1.",
"Alice will hold a register $\\textsf {X}^{\\prime }$ , representing a copy of $\\textsf {X}$ , and Bob will hold $\\textsf {Y}^{\\prime }$ , representing a copy of $\\textsf {Y}$ .",
"The initial state of the register pairs $(\\textsf {X}^{\\prime },\\textsf {X})$ and $(\\textsf {Y}^{\\prime },\\textsf {Y})$ are to be the canonical maximally entangled states $\\vert {0.5mu}\\psi {0.5mu}\\rangle = \\frac{1}{\\sqrt{n}} \\sum _{j = 1}^n \\vert {0.5mu}j {0.5mu}\\rangle \\vert {0.5mu}j {0.5mu}\\rangle \\quad \\text{and}\\quad \\vert {0.5mu}\\phi {0.5mu}\\rangle = \\frac{1}{\\sqrt{m}} \\sum _{k = 1}^m \\vert {0.5mu}k {0.5mu}\\rangle \\vert {0.5mu}k {0.5mu}\\rangle ,$ respectively, where $n$ and $m$ are the dimensions of the spaces corresponding to the registers $\\textsf {X}$ and $\\textsf {Y}$ .",
"In addition, Alice holds the register $\\textsf {U}$ and Bob holds the register $\\textsf {V}$ , with $(\\textsf {U},\\textsf {V})$ being prepared in the same shared entangled state $\\sigma $ that is used in the strategy for $G$ .",
"2.",
"Upon receiving the question $x\\in X$ from the referee, Alice performs the unitary operation $\\overline{U_x}$ on $\\textsf {X}^{\\prime }$ , then measures $(\\textsf {U},\\textsf {X}^{\\prime })$ with respect to the measurement $\\lbrace A_a\\,:\\,a\\in A\\rbrace $ to obtain an answer $a\\in A$ .",
"Similarly, upon receiving $y\\in Y$ from the referee, Bob performs $\\overline{V_y}$ on $\\textsf {Y}^{\\prime }$ , then measures $(\\textsf {Y}^{\\prime },\\textsf {V})$ with respect to $\\lbrace B_b\\,:\\,b\\in B\\rbrace $ to obtain an answer $b\\in B$ .",
"The performance of this strategy can be analyzed by first ignoring the specific initialization of the registers described in step 1, and defining a measurement $\\lbrace R_0,R_1\\rbrace $ that determines, for an arbitrary initialization of these registers, whether Alice and Bob win or lose by behaving as described in step 2.",
"In particular, the measurement $\\lbrace R_0,R_1\\rbrace $ is defined on the register tuple $(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {X},\\textsf {Y},\\textsf {Y}^{\\prime },\\textsf {V})$ , the measurement operator $R_0$ corresponds to a losing outcome, and $R_1$ corresponding to a winning outcome.",
"These operators may be described as follows: $\\begin{aligned}R_0 & =\\frac{1}{n^2 m^2}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\[0.3mm](a,b) \\in A\\times B\\end{array}}(I_{\\mathcal {U}} \\otimes U_x^{}) A_a (I_{\\mathcal {U}} \\otimes \\overline{U_x})\\otimes (I_{\\mathcal {X}\\otimes \\mathcal {Y}} - P_{a,b,x,y})\\otimes (V_y^{} \\otimes I_{) B_b (\\overline{V_y}\\otimes I_{)\\\\R_1 & =\\frac{1}{n^2 m^2}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\[0.3mm](a,b) \\in A\\times B\\end{array}}(I_{\\mathcal {U}} \\otimes U_x^{}) A_a (I_{\\mathcal {U}} \\otimes \\overline{U_x})\\otimes P_{a,b,x,y}\\otimes (V_y^{} \\otimes I_{) B_b (\\overline{V_y}\\otimes I_{)= I- R_0.",
"}}}Now we may consider the initialization of the registers described in step~1.For an arbitrary choice of operators X\\in \\mathrm {L}(\\mathcal {U}) andY\\in \\mathrm {L}( we have\\begin{equation}\\bigl \\langle R_0, X \\otimes \\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert \\otimes \\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert \\otimes Y \\bigr \\rangle = \\sum _{(a,b) \\in A\\times B}\\bigl \\langle A_a \\otimes (\\xi ^{} - \\xi ^{}_{a,b}) \\otimes B_b, X \\otimes \\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert \\otimes \\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert \\otimes Y \\bigr \\rangle ,\\end{equation}by virtue of the fact that\\bigl ( \\overline{U_x} \\otimes U_x\\bigr ) \\vert {0.5mu}\\psi {0.5mu}\\rangle = \\vert {0.5mu}\\psi {0.5mu}\\rangle and \\bigl ( \\overline{V_y} \\otimes V_y\\bigr ) \\vert {0.5mu}\\phi {0.5mu}\\rangle = \\vert {0.5mu}\\phi {0.5mu}\\rangle for every x\\in X and y\\in Y.Further simplifying this expression, one obtains\\begin{equation}\\begin{aligned}\\sum _{(a,b) \\in A\\times B}\\bigl \\langle A_a \\otimes (\\xi ^{} - \\xi ^{}_{a,b}) \\otimes B_b, X \\otimes \\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert \\otimes \\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert \\otimes Y \\bigr \\rangle \\hspace{-142.26378pt}\\\\& = \\frac{1}{nm}\\sum _{(a,b) \\in A\\times B}\\bigl \\langle A_a \\otimes B_b, X \\otimes (\\xi -\\xi _{a,b}) \\otimes Y \\bigr \\rangle \\\\& = \\frac{1}{nm}\\sum _{(a,b) \\in A\\times B}\\bigl \\langle A_a \\otimes (I- Q_{a,b}) \\otimes B_b, X \\otimes \\rho \\otimes Y \\bigr \\rangle .\\end{aligned}\\end{equation}By expressing the initial state \\sigma of (\\textsf {U},\\textsf {V}) as\\sigma = \\sum _i X_i \\otimes Y_i and making use of the bilinearity ofthe above expression in X and Y, one finds that the losingprobability of Alice and Bob^{\\prime }s strategy for H is equal toq/(nm), for q being the losing probability (\\ref {eq:losing-in-G}) for theiroriginal strategy for G.}Optimizing over all strategies for G that make use of an initial shared statehaving total dimension at most N yields the required inequality\\begin{equation}\\omega ^{\\ast }_{nmN}(H) \\ge 1 - \\frac{1 - \\omega ^{\\ast }_N(G)}{nm}.\\end{equation}\\end{aligned}Next we will prove the second inequality in (\\ref {eq:two-bounds-omega-H}).Assume that an arbitrary strategy for Alice and Bob in the extended nonlocalgame H constructed from G is fixed: the strategy consists of an initialstate \\sigma \\in \\mathrm {D}(\\mathcal {U}\\otimes (\\mathcal {X}\\otimes \\mathcal {Y})\\otimes for the registers(\\textsf {U},(\\textsf {X},\\textsf {Y}),\\textsf {V}), where \\dim (\\mathcal {U}\\otimes \\le N, alongwith measurements\\begin{equation}\\bigl \\lbrace A^x_a\\,:\\,a\\in A\\bigr \\rbrace \\subset \\mathrm {Pos}(\\mathcal {U})\\quad \\text{and}\\quad \\bigl \\lbrace B^y_b\\,:\\,b\\in B\\bigr \\rbrace \\subset \\mathrm {Pos}(\\end{equation}for Alice and Bob, respectively, for each x\\in X and y \\in Y.The winning probability of this strategy may be expressed as\\begin{equation}p = \\frac{1}{n^2 m^2}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\(a,b) \\in A\\times B\\\\\\end{array}}\\Bigl \\langle A^x_a \\otimes P_{a,b,x,y} \\otimes B^y_b, \\sigma \\Bigr \\rangle \\end{equation}while the losing probability is\\begin{equation}q = \\frac{1}{n^2 m^2}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\(a,b) \\in A\\times B\\\\\\end{array}}\\Bigl \\langle A^x_a \\otimes (I- P_{a,b,x,y}) \\otimes B^y_b, \\sigma \\Bigr \\rangle = 1-p.\\end{equation}We adapt this strategy to give one for G as follows:{\\begin{list}{}{}\\item [1.",
"]Let \\textsf {X}^{\\prime } and \\textsf {Y}^{\\prime } represent copies of the registers \\textsf {X} and\\textsf {Y}.Alice and Bob will initially share the registers(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {Y}^{\\prime },\\textsf {V}) initialized to the state\\overline{\\sigma }, with Alice holding (\\textsf {U},\\textsf {X}^{\\prime }) and Bob holding(\\textsf {Y}^{\\prime },\\textsf {V}).\\item [2.",
"]Upon receiving \\textsf {X} from the referee, Alice first measures the pair(\\textsf {X}^{\\prime },\\textsf {X}) with respect to the basis\\bigl \\lbrace (I\\otimes U_x^{\\ast })\\vert {0.5mu}\\psi {0.5mu}\\rangle \\,:\\,x\\in X\\bigr \\rbrace .For whichever outcome x\\in X she obtains, she then measures \\textsf {U} withrespect to the measurement\\begin{equation}\\Bigl \\lbrace \\overline{A^x_a}\\,:\\,a\\in A\\Bigr \\rbrace \\subset \\mathrm {Pos}(\\mathcal {U})\\end{equation}to obtain an outcome a\\in A.Bob does likewise, first measuring (\\textsf {Y}^{\\prime },\\textsf {Y}) with respect to thebasis \\bigl \\lbrace (I\\otimes V_y^{\\ast })\\vert {0.5mu}\\phi {0.5mu}\\rangle \\,:\\,y\\in Y\\bigr \\rbrace ,and then measuring \\textsf {V} with respect to the measurement\\begin{equation}\\Bigl \\lbrace \\overline{B^y_b}\\,:\\,b\\in B\\Bigr \\rbrace \\subset \\mathrm {Pos}(\\end{equation}for whichever outcome y\\in Y is obtained.\\end{list}}$ Now let us consider the probability with which this strategy wins in $G$ .",
"The state of the registers $(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {X},\\textsf {S},\\textsf {Y},\\textsf {Y}^{\\prime },\\textsf {V})$ immediately after the referee sends $\\textsf {X}$ to Alice and $\\textsf {Y}$ to Bob is given by $W ( \\overline{\\sigma } \\otimes \\rho ) W^{\\ast },$ where $W$ is a unitary operator that corresponds to a permutation of registers: $(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {Y}^{\\prime },\\textsf {V},\\textsf {X},\\textsf {S},\\textsf {Y}) \\mapsto (\\textsf {U},\\textsf {X}^{\\prime },\\textsf {X},\\textsf {S},\\textsf {Y},\\textsf {Y}^{\\prime },\\textsf {V}).$ We may define a measurement $\\lbrace R_0,R_1\\rbrace $ on the register tuple $(\\textsf {U},\\textsf {X}^{\\prime },\\textsf {X},\\textsf {S},\\textsf {Y},\\textsf {Y}^{\\prime },\\textsf {V})$ representing the outcome of the game, with $R_0$ corresponding to a losing outcome and $R_1$ corresponding to a winning outcome.",
"We have $\\begin{aligned}R_0 & =\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\[0.3mm](a,b) \\in A\\times B\\end{array}}\\overline{A^x_a} \\otimes (I\\otimes U_x^{\\ast })\\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert (I\\otimes U_x)\\otimes (I- Q_{a,b}) \\otimes (V_y^{\\ast }\\otimes I)\\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert (V_y\\otimes I)\\otimes \\overline{B^y_b}\\\\R_1 & =\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\[0.3mm](a,b) \\in A\\times B\\end{array}}\\overline{A^x_a} \\otimes (I\\otimes U_x^{\\ast })\\vert {0.5mu}\\psi {0.5mu}\\rangle \\langle {0.5mu}\\psi {0.5mu}\\vert (I\\otimes U_x)\\otimes Q_{a,b} \\otimes (V_y^{\\ast }\\otimes I)\\vert {0.5mu}\\phi {0.5mu}\\rangle \\langle {0.5mu}\\phi {0.5mu}\\vert (V_y\\otimes I)\\otimes \\overline{B^y_b}.\\end{aligned}$ Simplifying expressions for the probability that Alice and Bob lose yields $\\bigl \\langle R_0, W ( \\overline{\\sigma } \\otimes \\rho ) W^{\\ast } \\bigr \\rangle = \\frac{1}{nm}\\sum _{\\begin{array}{c}(x,y) \\in X\\times Y\\\\(a,b) \\in A\\times B\\\\\\end{array}}\\Bigl \\langle A^x_a \\otimes (I- P_{a,b,x,y}) \\otimes B^y_b, \\sigma \\Bigr \\rangle = nm q,$ for $q$ being the losing probability (REF ) for their original strategy for $H$ .",
"Optimizing over all strategies for $H$ that make use of an initial shared state for which Alice and Bob's total dimension is at most $N$ yields the inequality $\\omega ^{\\ast }_{N}(H) \\le 1 - \\frac{1 - \\omega ^{\\ast }_{nmN}(G)}{nm}.$"
],
[
"Discussion",
"As was mentioned in the introduction, Regev and Vidick [20] have identified examples of QC games for which Alice and Bob can never achieve optimality by using a finite-dimensional entangled strategy.",
"To be more precise, they prove that there exists a QC gameTheir games fall into a category of QC games that they call quantum XOR games, in which $A = B = \\lbrace 0,1\\rbrace $ and only the parity $a\\oplus b$ of Alice and Bob's answers is relevant to the referee's determination of whether they win or lose.",
"$G$ (and in fact a family of such games) for which it holds that $\\omega ^{\\ast }_N(G) < 1$ for all $N\\in \\mathbb {N}$ , while $\\omega ^{\\ast }(G) = 1$ .",
"By applying our construction to any such game, we obtain an extended nonlocal game $H$ with the property that $\\omega ^{\\ast }_N(H) < 1$ for all $N\\in \\mathbb {N}$ , while $\\omega ^{\\ast }(H) = 1$ .",
"In greater detail, by taking the simplest known example of a QC game $G$ with the property just described, and applying our construction (along with minor simplifications), one obtains an extended nonlocal game as follows: 1.",
"Let $\\mathcal {X}= \\mathcal {Y}= \\mathbb {C}^3$ and let $U_1,\\ldots ,U_9$ be the discrete Weyl operators acting on $\\mathbb {C}^3$ .",
"Also define $\\begin{aligned}\\vert {0.5mu}\\gamma _0 {0.5mu}\\rangle & = \\frac{1}{\\sqrt{2}}\\vert {0.5mu}0 {0.5mu}\\rangle \\vert {0.5mu}0 {0.5mu}\\rangle + \\frac{1}{2}\\vert {0.5mu}1 {0.5mu}\\rangle \\vert {0.5mu}1 {0.5mu}\\rangle + \\frac{1}{2}\\vert {0.5mu}2 {0.5mu}\\rangle \\vert {0.5mu}2 {0.5mu}\\rangle ,\\\\\\vert {0.5mu}\\gamma _1 {0.5mu}\\rangle & = \\frac{1}{\\sqrt{2}}\\vert {0.5mu}0 {0.5mu}\\rangle \\vert {0.5mu}0 {0.5mu}\\rangle - \\frac{1}{2}\\vert {0.5mu}1 {0.5mu}\\rangle \\vert {0.5mu}1 {0.5mu}\\rangle - \\frac{1}{2}\\vert {0.5mu}2 {0.5mu}\\rangle \\vert {0.5mu}2 {0.5mu}\\rangle .\\end{aligned}$ 2.",
"Alice and Bob give a pair of registers $(\\textsf {X},\\textsf {Y})$ to the referee, initialized as they choose.",
"The referee randomly chooses $x,y\\in \\lbrace 1,\\ldots ,9\\rbrace $ uniformly and independently at random, then sends $x$ to Alice and $y$ to Bob.",
"Alice and Bob respond with binary values $a,b\\in \\lbrace 0,1\\rbrace $ , respectively.",
"3.",
"The referee computes $c = a\\oplus b$ , then measures the pair $(\\textsf {X},\\textsf {Y})$ with respect to the measurement $\\bigl \\lbrace I_{\\mathcal {X}}\\otimes I_{\\mathcal {Y}} - (U_x \\otimes U_y)\\vert {0.5mu}\\gamma _c {0.5mu}\\rangle \\langle {0.5mu}\\gamma _c {0.5mu}\\vert (U_x\\otimes U_y)^{\\ast },\\;(U_x \\otimes U_y)\\vert {0.5mu}\\gamma _c {0.5mu}\\rangle \\langle {0.5mu}\\gamma _c {0.5mu}\\vert (U_x \\otimes U_y)^{\\ast }\\bigr \\rbrace .$ The first outcome represents a win for Alice and Bob, and the second a loss.",
"(Note that here we have scaled the losing measurement operator by a factor of two in comparison to what is described in the construction, which has the effect of doubling the losing probability for every strategy of Alice and Bob.)",
"Assuming Alice and Bob initially entangle the pair $(\\textsf {X},\\textsf {Y})$ with finite-dimensional registers of their own, they can never win the game with certainty, but they can approach certainty by using increasingly large systems."
],
[
"Acknowledgments",
"This work was partially funded by Canada's NSERC.",
"We thank Marco Piani, William Slofstra, and Thomas Vidick for helpful comments and discussions."
]
] | 1709.01837 |
[
[
"Preparation of entangled states of microwave photons in a hybrid system\n via electro-optic effect"
],
[
"Abstract We propose to realize the two-mode continuous-variable entanglement of microwave photons in an electro-optic system, consisting of two superconducting microwave resonators and one or two optical cavities filled with certain electro-optic medium.",
"The cascaded and parallel schemes realize such entanglement via coherent control on the dynamics of the system, while the dissipative dynamical scheme utilizes the reservoir-engineering approach and exploits the optical dissipation as a useful resource.",
"We show that, for all the schemes, the amount of entanglement is determined by the ratio of the effective coupling strengths of \"beam-splitter\" and \"two-mode squeezing\" interactions, rather than their amplitudes."
],
[
"Introduction",
"With the development of quantum information, microwave radiation has become an important tool to couple different quantum systems due to its frequency range covering many types of qubits [1].",
"It can couple different qubits to realize quantum computation [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], and deterministic logic operations [12], [13].",
"Also people employ microwaves to trap charged particles for quantum information processing [14], [15], [16], [17].",
"Unlike optical photons, it is hard to entangle microwave photons through nonlinear optical methods with optical crystals.",
"It is thus appealing to propose alternative approaches to generate entangled microwave photons.",
"Many different approaches have been explored in different systems.",
"For instance, some theoretical works have presented the dissipation-based approach in electro-mechanical systems [18], the coherent-control-based approach through excitations of cavity Bogoliubov modes in optomechanical systems[19] and electro-mechanical systems[18], as well as the schemes utilizing solid-state superconducting circuits [20], [21], [22], [23], [24], [25], [26].",
"In previous works, it has been demonstrated that the electro-optic coupling has the same form as optomechanical[27] and electro-mechanical couplings[18].",
"So all previous considered effects can in principle be observed in electro-optic systems [28].",
"But there are several challenges of optomechanical and electro-mechanical systems to cool down their nano-/micro- mechanical oscillators.",
"First, due to the low frequencies of mechanical oscillators, the required environment temperature is ultra low.",
"Moreover, there's a limitation of physical cooling as demonstrated in [29] that mechanical oscillators can not be cooled down to their ground states under the influence of cavity frequency noise.",
"Besides cooling processes, the quality factors of high-frequency mechanical oscillators are relatively low.",
"However, we can avoid those drawbacks of micro mechanical oscillators through electro-optic systems.",
"Auxiliary modes in electro-optic systems are optical modes, which can be regard as vacuum states at experimental temperatures.",
"In addition, with the well-developed fabrication of optical cavities and superconducting circuits, it is convenient to get optical cavities with desired quality factors as well as low-dissipation superconducting microwave resonators to prepare high-quality entanglement states.",
"In this work, inspired by optomechanical systems[29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40] and other hybrid quantum systems[18], [28], [41], [27], we propose an electro-optic system comprising two separated superconducting microwave resonators and one or two auxiliary optical cavities, which are filled with certain electro-optic medium.",
"With this system, we provide three schemes to entangle these two microwave resonators via electro-optic effect: (i) cascaded scheme; (ii) parallel scheme; (iii) dissipative dynamical scheme.",
"The underlying physics for both cascaded and parallel schemes is the coherent control over their systems to realize the Bogoliubov modes consisting of the two microwave modes, while the last scheme is based on quantum reservoir engineering, which exploits the dissipations of two optical cavities as useful resources to entangle microwave photons.",
"For each scheme, we've worked out the analytic solutions and numerical simulations.",
"In Section , we also talk about the experimental feasibility of all schemes.",
"Especially, Eq.",
"(REF )-Eq.",
"() shows that the temperature dependence of the entanglement degree for the dissipative dynamical scheme is steerable.",
"It's modulated by the decay rate of both optical cavities and superconducting microwave resonators, as well as the ratio of effective coupling strengths.",
"Therefore, high quality entanglement can be realized through choosing cavities and resonators of optimized quality factors."
],
[
"The cascaded scheme",
"As shown in Fig.",
"REF , the hybrid quantum system we considered composes of two microwave resonators with frequency $\\omega _{b1}$ (LC1) and $\\omega _{b2}$ (LC2), and an optical cavity of frequency $\\omega _{a1}$ , inside which a kind of electro-optic medium(EOM) such as KDP is filled.",
"These two resonators are coupled to the optical cavity through electro-optic effect, but have no direct interaction with each other.",
"It is known that the effect of \"beam-splitter\" interaction is to exchange quantum states between two modes, and that of \"two-mode squeezing\" interaction is to get both modes entangled.",
"Therefore, one straightforward approach to entangle the microwave photons in these separated resonators is to drive the optical cavity of this system with suitably detuned lasers in a cascaded way as shown in Fig.",
"REF : (i) to set LC1 in the red-detuned regime; (ii) to set LC2 in the blue-detuned regime; (iii) to set LC1 in the red-detuned again.",
"Then at the final moment, the Bogoliubov modes composed of the two resonator modes only will be excited.",
"The detailed steps of this scheme are as the following.",
"We drive the optical cavity with different lasers in different periods: (i) $0<t<T_1$ , using the laser of frequency $\\omega _{L1}$ ; (ii) $T_1<t<T_2$ , using the laser of frequency $\\omega _{L2}$ ; (iii) $T_2<t<T_1+T_2$ , we use the laser of frequency $\\omega _{L1}$ , again.",
"We assume $\\omega _{L1}-\\omega _{a1}=-\\omega _{b1},~\\omega _{L2}-\\omega _{a1}=\\omega _{b2}$ to guarantee that the microwave resonators are in the suitable detuned regimes.",
"Through the rotating-wave approximation, in each period there is only one microwave mode interacting with the optical mode.",
"In other words, we set LC1 to be in the red-detuned regime and LC2 to be in the blue-detuned regime when they interact with the optical cavity, and both of them are isolated when they are far detuned from the optical cavity.",
"Figure: The setup of both cascaded and parallel schemes.",
"In the cascade scheme, the optical cavity is driven by a laser of frequency ω L1 \\omega _{L1} or ω L2 \\omega _{L2} for different periods, while in the parallel scheme, we impose both of these driving lasers at the same time.Figure: A diagram of the process of the cascaded scheme, and a 1 a_1, b 1 b_1, b 2 b_2 are annihilation operators of optical mode and two microwave modes, respectively.",
"When 0<t<T 1 0<t<T_1 or T 2 <t<T 1 +T 2 T_2<t<T_1+T_2, LC1 and the optical cavity exchange quantum states with each other, and during T 1 <t<T 2 T_1<t<T_2, modes of optical cavity and LC2 get entangled.As shown in [28], when we only consider one mode of the optical cavity, the interaction Hamiltonian in each period is: $H_{I,i}^C=-\\hbar g_i a_1^{\\dag } a_1 (b_i+b_i^{\\dag }),~\\\\$ $g_i=\\frac{\\omega _{a1} n^3 r_0}{2d} \\sqrt{\\frac{\\hbar \\omega _{i}}{2C_i}},~$ where $n$ , $r_0$ , and $d$ are the refractive index, electro-optic coefficient and height of the medium respectively.",
"$C_i$ refers to the capacitance of the $i^{th}$ resonator.",
"In the first period, the driven term in the total Hamiltonian is that: $H_d^C=i \\hbar [E_0 a_1^{\\dag } e^{-i \\omega _{L1} t}-E_0^* a_1 e^{i \\omega _{L1} t}],$ where $E_0$ is the complex amplitude of the driving laser.",
"If we choose a rotating frame with frequency $\\omega _{L1}$ respect to the optical mode $a_1$ , the total Hamiltonian then becomes: $H_1^C=-\\hbar \\Delta a_1^{\\dag }a_1+\\hbar \\omega _{b1} b_1^{\\dag }b_1+\\hbar \\omega _{b2} b_2^{\\dag }b_2\\\\-\\hbar g_1 a_1^{\\dag }a_1(b_1^{\\dag }+b_1)+i\\hbar (E_0 a_1^{\\dag }-E_0^*a_1),~$ where $\\Delta =\\omega _{L1}-\\omega _{a1}$ .",
"In Eq.",
"(REF ), we have assumed that the driving laser is strong enough.",
"Therefore, it is a good approximation to linearize the above Hamiltonian through replacing the optical annihilate operator $a_1$ by the sum of its stable mean value $\\bar{a}$ and its fluctuation term $\\delta a$ .",
"The interaction term between the optical mode and LC2 mode has been eliminated by the rotating wave approximation.",
"Employing Heisenberg equation, the zero order and linear terms are eliminated, so we only consider the quadratic terms.",
"Then Eq.",
"(REF ) becomes: $H^C_{1,eff}=-\\hbar \\Delta \\delta a_1^{\\dag }\\delta a_1+\\hbar \\omega _{b1} a_1^{\\dag }b_1+\\hbar \\omega _{b2} b_2^{\\dag }b_2\\\\-\\hbar \\bar{a}_1 g_1(\\delta a_1^{\\dag } b_1+\\delta a_1 b_1^{\\dag }).~$ The effective coupling strength in the first period is $\\bar{a}_1g_1$ , which can also be modulated by the power of the driving lasers.",
"For simplicity we introduce the non-dimension time: $\\tau =\\bar{a}_1g_1t,~\\tau _i=\\bar{a}_1g_1T_i,~i=1,2$ .",
"Then in the interaction picture, the Heisenberg equations for all three modes can be written as: $\\frac{d}{d\\tau }\\left[\\begin{array}{ccc}\\delta a_1(\\tau )\\\\b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{array}\\right]=\\left[\\begin{array}{ccc}0&i&0\\\\i&0&0\\\\0&0&0\\end{array}\\right]\\left[\\begin{array}{ccc}\\delta a_1(0)\\\\b_1(0)\\\\b_2^{\\dag }(0)\\end{array}\\right].$ The solution of these equations is straightforward: $\\left[\\begin{array}{ccc}\\delta a_1(\\tau )\\\\b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{array}\\right]=\\left[\\begin{array}{ccc}\\cos (\\tau )&i\\sin (\\tau )&0\\\\i\\sin (\\tau )&\\cos (\\tau )&0\\\\0&0&1\\end{array}\\right]\\left[\\begin{array}{ccc}\\delta a_1(0)\\\\b_1(0)\\\\b_2^{\\dag }(0)\\end{array}\\right].$ Similarly, in the second period we have: $\\left[\\begin{array}{ccc}\\delta a_1(\\tau )\\\\b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{array}\\right]=\\left[\\begin{array}{ccc}\\cosh (r\\Delta \\tau _1)&0&i\\sinh (r\\Delta \\tau _1)\\\\0&1&0\\\\-i\\sinh (r\\Delta \\tau _1)&0&\\cosh (r\\Delta \\tau _1)\\end{array}\\right]\\left[\\begin{array}{ccc}\\delta a_1(\\tau _1)\\\\b_1(\\tau _1)\\\\b_2^{\\dag }(\\tau _1)\\end{array}\\right],$ where $r=\\bar{a}_2g_2/\\bar{a}_1g_1$ is the ratio of the coupling strength between the blue-detuned and red-detuned type interactions, and $\\Delta \\tau _1=\\tau -\\tau _1$ .",
"As for the third period: $\\left[\\begin{array}{ccc}\\delta a_1(\\tau )\\\\b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{array}\\right]=\\left[\\begin{array}{ccc}\\cos (\\Delta \\tau _2)&i\\sin (\\Delta \\tau _2)&0\\\\i\\sin (\\Delta \\tau _2)&\\cos (\\Delta \\tau _2)&0\\\\0&0&1\\end{array}\\right]\\left[\\begin{array}{ccc}\\delta a_1(\\tau _2)\\\\b_1(\\tau _2)\\\\b_2^{\\dag }(\\tau _2)\\end{array}\\right],$ where $\\Delta \\tau _2=\\tau -\\tau _2$ .",
"Now we can get the final state of the system at $\\tau =\\tau _1+\\tau _2$ $\\left[\\begin{array}{ccc}\\delta a_1(\\tau _1+\\tau _2)\\\\b_1(\\tau _1+\\tau _2)\\\\b_2^{\\dag }(\\tau _1+\\tau _2)\\end{array}\\right]=M_1M_2M_1\\left[\\begin{array}{ccc}\\delta a_1(0)\\\\b_1(0)\\\\b_2^{\\dag }(0)\\end{array}\\right],\\\\\\\\M_1=\\left[\\begin{array}{ccc}\\cos (\\tau _1)&i\\sin (\\tau _1)&0\\\\i\\sin (\\tau _1)&\\cos (\\tau _1)&0\\\\0&0&1\\end{array}\\right],\\\\\\\\M_2=\\left[\\begin{array}{ccc}\\cosh [r(\\tau _2-\\tau _1)]&0&i\\sinh [r(\\tau _2-\\tau _1)]\\\\0&1&0\\\\-i\\sinh [r(\\tau _2-\\tau _1)&0&\\cosh [r(\\tau _2-\\tau _1)]\\end{array}\\right].\\\\$ If $\\cos (\\tau _1)=0$ , at $\\tau =\\tau _1+\\tau _2$ the evolution matrix in Eq.",
"(REF ) becomes: $M_1M_2M_1=\\left[\\begin{array}{ccc}-1&0&0\\\\0&-\\cosh [r(\\tau _2-\\tau _1)]&-\\sinh [r(\\tau _2-\\tau _1)]\\\\0&\\sinh [r(\\tau _2-\\tau _1)]&\\cosh [r(\\tau _2-\\tau _1)]\\end{array}\\right].$ Eq.",
"(REF ) shows at the instant $\\tau =\\tau _1+\\tau _2$ , the optical mode decouples with the two $LC$ modes, and the two $LC$ modes form the Bogoliubov modes.",
"Moreover, in the case of initial vacuum states, the Bogoliubov modes turn to two-mode squeezed vacuum state [18]."
],
[
"Parallel scheme",
"In the parallel scheme, we want to set LC1 in the red-detuned regime, meanwhile LC2 is in the blue-detuned regime.",
"Therefore, we need to apply two driving lasers of suitable frequencies simultaneously.",
"The total Hamiltonian can be expressed as: $H=H_0+H_I+H_d^{\\prime },\\\\H_0=\\hbar \\omega _{a1}a_1^{\\dag }a_1+\\sum _{i=1}^{2}{\\hbar \\omega _{bi}b_i^{\\dag }b_i},\\\\H_I=-\\sum _{j=1}^{2}{\\hbar g_j a_1^{\\dag }a_1(b_j+b_j^{\\dag })},\\\\H_d^{\\prime }=\\hbar \\sum _{j=1}^{2}(-1)^{j}E_j(a_1^{\\dag }e^{-i\\omega _{Lj}t}+a_1e^{i\\omega _{Lj}t}).$ Here we apply driving signals with real amplitudes $E_j(j=1,~2)$ and initial phases $\\phi _1=-\\frac{\\pi }{2},~\\phi _2=\\frac{\\pi }{2}$ .",
"Then we can use similar approach as used in [42] to simplify the Hamiltonian of our system.",
"In the interaction picture, the total Hamiltonian becomes: $H^{\\prime }=-\\sum _{j=1}^{2}{\\hbar g_j a_1^{\\dag }a_1(b_je^{-i\\omega _{bj}t}+b_j^{\\dag }e^{i\\omega _{bj}t})}\\\\+\\hbar \\sum _{j=1}^{2}(-1)^{j}E_j\\left[a_1^{\\dag }e^{(-1)^{j-1}i\\Delta _jt}+a_1 e^{(-1)^j i\\Delta _jt}\\right],~$ where $\\Delta _1=\\omega _{a1}-\\omega _{L1}$ , $\\Delta _2=\\omega _{L2}-\\omega _{a1}$ .",
"To engineer our desired coupling, we need an unitary transformation with the following defined unitary operator: $U=T\\exp \\left\\lbrace -i\\int _0^t{\\sum _{j=1}^{2}(-1)^{j}E_j\\left[a_1^{\\dag }e^{(-1)^{j-1}i\\Delta _jt}+a_1 e^{(-1)^j i\\Delta _jt}\\right]}d t\\right\\rbrace ,$ where $T$ is the time-ordering operator.",
"Unlike the cascade scheme requiring extreme strong lasers, the parallel scheme requires lasers with relatively weak intensities: $E_j/\\Delta _j<<1$ such that we can keep the leading term of $E_j/\\Delta _j$ .",
"Eq.",
"(REF ) then becomes, $\\begin{align}H^P&=U^{\\dag }\\left(H^{\\prime }-i\\hbar \\frac{\\partial }{\\partial t}\\right)U\\\\&=-\\sum _{k=1}^{2}\\hbar g_k \\Big \\lbrace a_1^{\\dag }a_1+\\sum _{j=1}^2\\frac{E_j}{\\Delta _j}\\left[a^{\\dag }(e^{(-1)^{j+1}i\\Delta _jt}-1)+\\textmd {H.c.}\\right]\\Big \\rbrace \\\\&\\times (b_ke^{-i\\omega _{bk}t}+b_k^{\\dag }e^{i\\omega _{bk}t}).\\end{align}$ Through setting $\\Delta _1=\\omega _{b1},\\Delta _2=\\omega _{b2}$ and using the rotating-wave approximation, $H^P$ turns to: $H^P_{eff}=-\\hbar g_1\\frac{E_1}{\\Delta _1}(a_1^{\\dag }b_1+a_1b_1^{\\dag })-\\hbar g_2 \\frac{E_2}{\\Delta _2}(a_1^{\\dag }b_2^{\\dag }+a_1b_2).~$ Eq.",
"(REF ) yields the Langevin equation of this system, $\\frac{d}{d\\tau }\\left[\\begin{array}{ccc}a_1\\\\b_1\\\\b_2^{\\dag }\\end{array}\\right]=\\left[\\begin{array}{ccc}-k_0&i&ir\\\\i&-k_1&0\\\\-ir&0&-k_2\\end{array}\\right]\\left[\\begin{array}{ccc}a_1\\\\b_1\\\\b_2^{\\dag }\\end{array}\\right]+\\left[\\begin{array}{ccc}f_0\\\\f_1\\\\f_2^{\\dag }\\end{array}\\right],~$ where $\\tau $ and $r$ are now defined by $\\tau =g_1\\frac{E_1}{\\Delta _1}t$ , $r=\\frac{E_2\\Delta _1}{E_1\\Delta _2}$ .",
"$\\lbrace k_i,i=0,1,2\\rbrace $ are non-dimensional decay rates defined by $k_i=\\frac{\\Gamma _i\\Delta _1}{2E_1g_1}$ and $\\lbrace \\Gamma _i,i=1,2,3\\rbrace $ are those decay rates of each mode in Eq.",
"(REF ); $f_i=\\frac{F_i\\Delta _1}{g_1E_1}$ with $\\lbrace F_i,i=0,1,2\\rbrace $ the noise operators of each modes.",
"According to [43], $\\langle F_i^{\\dag }(t)F_j(t^{\\prime })\\rangle _R=\\Gamma _i n_{i,th}\\delta _{ij}\\delta (t-t^{\\prime })$ , it is straightforward that: $\\langle f_i^{\\dag }(\\tau )f_j(\\tau ^{\\prime })\\rangle _R=2k_i n_{i,th}\\delta _{ij}\\delta (\\tau -\\tau ^{\\prime }).~$ We solve the Heisenberg equation and Langevin equation for the non-dissipative and dissipative cases, respectively.",
"(i)If $k_i=0,i=0,1,2$ , Eq.",
"(REF ) converts into a homogeneous equation.",
"The time evolution of the system is: $\\left[\\begin{array}{ccc}a_1(\\tau )\\\\b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{array}\\right]=M^P\\left[\\begin{array}{ccc}a_1(0)\\\\b_1(0)\\\\b_2^{\\dag }(0)\\end{array}\\right],$ $M^P=\\left[\\begin{array}{ccc}\\cos (\\sqrt{1-r^2}\\tau )&i\\frac{\\sin (\\sqrt{1-r^2}\\tau )}{\\sqrt{1-r^2}}&ir\\frac{\\sin (\\sqrt{1-r^2}\\tau )}{\\sqrt{1-r^2}}\\\\i\\frac{\\sin (\\sqrt{1-r^2}\\tau )}{\\sqrt{1-r^2}}&\\frac{\\cos (\\sqrt{1-r^2}\\tau )-r^2}{1-r^2}&\\frac{(\\cos (\\sqrt{1-r^2}\\tau )-1)r}{1-r^2}\\\\-ir\\frac{\\sin (\\sqrt{1-r^2}\\tau )}{\\sqrt{1-r^2}}&\\frac{(1-\\cos (\\sqrt{1-r^2}\\tau ))r}{1-r^2}&\\frac{1-r^2\\cos (\\sqrt{1-r^2}\\tau )}{1-r^2}\\end{array}\\right],~$ (ii) If $k_i\\ne 0,i=0,1,2$ , the time evolution of this system can be solved using the theory of linear differential equations.",
"We assume that: $c=\\left[\\begin{array}{ccc}-k_0&i&ir\\\\i&-k_1&0\\\\-ir&0&-k_2\\end{array}\\right].$ The eigenvalues and corresponding column eigenvectors of matrix $c$ are $\\lbrace \\lambda _i,\\vec{u}_i,~i=1,2,3\\rbrace $ , and the time evolution of such system can be written as: $\\left[\\begin{array}{ccc}a_1(\\tau )\\\\b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{array}\\right]=X(\\tau )\\left[\\begin{array}{ccc}a_1(0)\\\\b_1(0)\\\\b_2^{\\dag }(0)\\end{array}\\right]\\\\+\\int _0^{\\tau }X(\\tau -\\tau ^{\\prime })\\left[\\begin{array}{ccc}f_0(\\tau ^{\\prime })\\\\f_1(\\tau ^{\\prime })\\\\f_2^{\\dagger }(\\tau ^{\\prime })\\end{array}\\right]d\\tau ^{\\prime },\\\\\\\\X(\\tau )=[e^{\\lambda _1\\tau }\\vec{u}_1,e^{\\lambda _2\\tau }\\vec{u}_2,e^{\\lambda _3\\tau }\\vec{u}_3][\\vec{u}_1,\\vec{u}_2,\\vec{u}_3]^{-1}.$ It is not necessary for us to get exact analytic solutions in this case, so we can get numerical solutions from the expressions above.",
"As for the non-dissipative case, when $\\sqrt{1-r^2}\\tau =\\pi $ , Eq.",
"(REF ) becomes: $M^P=\\left[\\begin{array}{ccc}-1&0&0\\\\0&-\\frac{1+r^2}{1-r^2}&-\\frac{2r}{1-r^2}\\\\0&\\frac{2r}{1-r^2}&\\frac{1+r^2}{1-r^2}\\end{array}\\right].~$ Eq.",
"(REF ) indicates that at the instant $T_{\\pi }=\\pi /\\sqrt{1-r^2}$ , the optical mode decouples from the dynamics of the system.",
"We assume $\\cosh (\\xi )=(1+r^2)/(1-r^2),~\\sinh (\\xi )=2r/(1-r^2)$ and introduce the operator $S=e^{\\xi [b_1(0)b_2(0)-b_1^{\\dag }(0)b_2^{\\dag }(0)]}$ , then $b_1(T_{\\pi })=-Sb_1(0)S^{\\dag },~b_2^{\\dag }(T_{\\pi })=Sb_2^{\\dag }(0)S^{\\dag }$ , indicating the two superconducting microwave resonators are prepared in the Bogoliubov modes.",
"If the initial states of the superconducting microwave resonators are vacuum states, they will be prepared in the two-mode squeezed state with the squeezing parameter $\\xi =\\tanh ^{-1}[2r/(1+r^2)]$ at the moment $T_{\\pi }$ .",
"Fig.$\\ref {fg2}$ shows the time evolution of the photon numbers.",
"For the non-dissipative case shown in Fig.$\\ref {fg2}$ (a), at the instant $T_{\\pi }$ the photon number of the optical cavity drops to 0 and the photon numbers of the superconducting microwave resonators become equal.",
"This is in accordance with the conclusion that at that moment the optical mode decouples from the dynamics of the system and the microwave modes get entangled.",
"From Fig.$\\ref {fg2}$ (b), we can see that the periodic fluctuations of photon numbers are impeded by the dissipations.",
"As a result, the photon number of the optical cavity can't decrease to 0, reflecting that some photons of microwave modes still interact with the remaining photons of optical modes.",
"Therefore, when the effect of dissipations is notable, there is no such instant as $T_{\\pi }$ that the photons of the two microwave modes can be entangled completely fig/ Figure: The time evolution of fluctuating photon number for different values of the non-dimensional decay rate k i k_i.",
"N a1 N_{a1}, N b1 N_{b1} and N b2 N_{b2} are photon numbers for optical cavity, LC1 and LC2, respectively.",
"(a) r=0.5,k 0 =k 1 =k 2 =0r=0.5,~k_0=k_1=k_2=0, (b) r=0.5,k 0 =k 1 =k 2 =0.1r=0.5,k_0=k_1=k_2=0.1.",
"The initial condition for both (a) and (b) is N a1 (0)=n 0,th =0,N b1 (0)=N b2 (0)=n 1,th =n 2,th =0.1N_{a1}(0)=n_{0,th}=0,~N_{b1}(0)=N_{b2}(0)=n_{1,th}=n_{2,th}=0.1.",
"In (a) the ideal case, at each instant nπn\\pi , the photon number of optical mode goes to zero and those numbers of LC1 and LC2 become equal, which is in accordance with the decoupling of optical mode and the entanglement of two superconducting microwave resonators.",
"From (b), we can find that even for optical cavity, the photon number can not be zero at steady state.",
"In order to investigate the entanglement of modes $b_1$ and $b_2$ , we need the total variance $V=\\langle (\\Delta u)^2+(\\Delta v)^2\\rangle $ of EPR-like operators $u=x_1+x_2,~v=p_1-p_2$ , with $x_i=(b_i+b_i^{\\dag })/\\sqrt{2}$ and $p_i=-i(b_i-b_i^{\\dag })/\\sqrt{2}$ , $i=1,2$ [18].",
"The two-mode Gaussian state is entangled if and only if $V<2$ [18], [44].",
"Especially, for the two-mode squeezed vacuum state, $V=2e^{-2\\xi }$ .",
"We explore the effects of the dissipation and the initial thermal conditions to the entanglement of this system as illustrated in Fig.$\\ref {fg3}$ .",
"In Fig.$\\ref {fg3}$ (a), we change the initial thermal condition of each mode, and find that at the neighborhood of $T_{\\pi }$ the differences of all curves disappear.",
"We've already known that at the instant $T_{\\pi }$ , the two microwave modes are entangled.",
"Therefore, the entanglement of this system is insensitive to the initial thermal conditions.",
"But in Fig.$\\ref {fg3}$ (b), when we modulate the decay rates of all modes, the total variances vary greatly with them.",
"Thus, the low-decay condition should be satisfied in order to get better entanglements.",
"fig/ Figure: The total variance V.S.",
"phase for different values of parameters: (a) without dissipation, r=1-10 -3 ,n 1,th =n 2,th =n th r=1-10^{-3},~n_{1,th}=n_{2,th}=n_{th}, and n th =0,0.1,1n_{th}=0,~0.1,~1; (b) with dissipation, r=1-10 -3 r=1-10^{-3}, n th =0n_{th}=0, k 0 =k 1 =k 2 =kk_0=k_1=k_2=k, and k=0.001,0.01,0.1k=0.001,~0.01,~0.1."
],
[
"Dissipative dynamical scheme",
"The system in the parallel scheme is very sensitive to dissipations as shown in Fig.REF , and so does the cascaded scheme.",
"This indicates in previous schemes, we need to attenuate the dissipation as much as possible.",
"Thus, in many cases, high-Q cavities or resonators are necessary.",
"But we can also prepare our target states with low-Q optical cavities(\"bad cavities\").",
"In the dissipative dynamical scheme, large decay rates of the optical modes are required due to the fact that the dissipative effects of the optical modes here are treated as a useful resources.",
"Such schemes in other hybrid quantum systems have been explored previously[18].",
"However, in our system, what we use is the optical thermal noise, where the mean photon number at thermal equilibrium is $n_{0,th}\\approx 0$ .",
"Therefore, through the electro-optic system we can get more ideal two-mode squeezed vacuum states at the same temperature, compared with the optomechanical systems.",
"To realize our scheme, it is necessary to put LC1 and LC2 in both red- and -blue detuned regimes at the same time.",
"One approach is to add another optical cavity of frequency $\\omega _{a2}$ satisfying $|\\omega _{a2}/\\omega _{a1}-1|<<1$ paralleled to the previous one as shown in Fig.REF .",
"Through modulating the parameters in Eq.",
"(REF ), the coupling strengths of the \"beam-splitter and \"two-mode squeezing\" interactions in the second optical cavity can keep the same as the first one.",
"fig/ Figure: The setup for the dissipative dynamical scheme.",
"Two optical cavities are filled by electro-optic media and modulated by both LC1 and LC2 via the electro-optic effect.",
"LC1 is in the red-detuned regime with respect to the previous optical cavity and blue-detuned regime to the second one, but LC2 is just on the contrary.",
"The ideal situation for this scheme is $k_0>>1>>k_1$ , where $k_0$ stands for the non-dimensional decay rate of two optical cavities while $k_1$ denotes the non-dimensional decay rate of two microwave resonators.",
"Therefore, we can ignore the dissipation of two microwave modes, Then the Langevin equation of this system becomes: $\\frac{d}{d\\tau }\\left[\\begin{matrix}a_1(\\tau )\\\\a_2^{\\dag }(\\tau )\\\\b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{matrix}\\right]=\\left[\\begin{matrix}-k_0&0&i&ir\\\\0&-k_0&-ir&-r\\\\i&ir&0&0\\\\-ir&-i&0&0\\end{matrix}\\right]\\left[\\begin{matrix}a_1(\\tau )\\\\a_2^{\\dag }(\\tau )\\\\b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{matrix}\\right]+\\left[\\begin{matrix}f_{a1}(\\tau )\\\\f_{a2}^{\\dag }(\\tau )\\\\0\\\\0\\end{matrix}\\right].~$ Here the definition of all the non-dimensional variables has the same form as that in the parallel scheme.",
"In the ideal situation, we can make the adiabatic approximations to the optical modes: $a_1(\\tau )=\\frac{i}{k_0}(b_1(\\tau )+rb_2^{\\dag }(\\tau ))+\\frac{f_{a1}(\\tau )}{k_0},~\\\\a_2^{\\dag }(\\tau )=-\\frac{i}{k_0}(rb_1(\\tau )+b_2^{\\dag }(\\tau ))+\\frac{f_{a2}^{\\dag }(\\tau )}{k_0}.~$ Inserting Eq.",
"(REF ) and Eq.",
"() into Eq.",
"(REF ) yields the relationship between $b_1$ and $b_2^{\\dag }$ : $\\frac{d}{d\\tau }\\left[\\begin{matrix}b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{matrix}\\right]=\\frac{1}{k_0}\\left[\\begin{matrix}r^2-1&0\\\\0&r^2-1\\end{matrix}\\right]\\left[\\begin{matrix}b_1(\\tau )\\\\b_2^{\\dag }(\\tau )\\end{matrix}\\right]\\\\+\\frac{i}{k_0}\\left[\\begin{matrix}f_{a1}(\\tau )+rf_{a2}^{\\dag }(\\tau )\\\\-rf_{a1}(\\tau )-f_{a2}^{\\dag }(\\tau )\\end{matrix}\\right].$ We introduce some new variables and operators to simplify our expressions.",
"$x=\\frac{(1-r^2)\\tau }{k_0},~x^{\\prime }=\\frac{(1-r^2)\\tau ^{\\prime }}{k_0},~\\\\\\tilde{f}_{ai}(x)=\\frac{f_{ai}(x)}{\\sqrt{2k_0}},~i=1,2,~\\\\\\tilde{S}(x,\\varsigma )=\\exp \\left[\\varsigma (\\tilde{f}_{a1}(x)\\tilde{f}_{a2}(x)-\\tilde{f}_{a1}^{\\dag }(x)\\tilde{f}_{a2}^{\\dag }(x))\\right],~$ where $\\varsigma =\\arctan \\left[r\\right]$ is the squeezing parameter of optical modes, and by the definition of $\\tilde{f}_{ai}$ , we can get $\\langle \\tilde{f}_{ai}(x)\\tilde{f}_{aj}^{\\dag }(x^{\\prime })\\rangle _R=\\delta _{ij}\\delta (x-x^{\\prime })$ .",
"With Eq.",
"(REF )-(), the solution of Eq.",
"(REF ) can be expressed as: $b_1(x)=e^{-x}b_1(0)+\\sqrt{2}i\\int _0^xe^{-(x-x^{\\prime })}\\tilde{S}(x^{\\prime })\\tilde{f}_{a1}(x^{\\prime })\\tilde{S}^{\\dag }(x^{\\prime })dx^{\\prime },~\\\\b_2^{\\dag }(x)=e^{-x}b_2^{\\dag }(0)-\\sqrt{2}i\\int _0^xe^{-(x-x^{\\prime })}\\tilde{S}(x^{\\prime })\\tilde{f}_{a2}^{\\dag }(x^{\\prime })\\tilde{S}^{\\dag }(x^{\\prime })dx^{\\prime }.~$ From Eq.",
"(REF )-(), we can find the stable condition for this system is $r<1$ , under which as $x\\rightarrow \\infty $ the system will converge to the final state of the Bogoliubov modes composed of optical modes.",
"As in the parallel scheme, we calculate the total variance of EPR-like operators composed of $b_1$ and $b_2^{\\dag }$ in the stable case, $V=\\lim \\limits _{x\\rightarrow \\infty }{\\frac{2(1-r)}{1+r}(1-e^{-2x})=\\frac{2(1-r)}{1+r}}$ .",
"That is exactly the total variance of the ideal two-mode vacuum state $2e^{-2\\varsigma }$ with squeezing parameter $\\varsigma =\\arctan [r]$ .",
"Thus, under the assumption that $k_0\\gg 1\\gg k_1$ , no mater what the initial condition is, the two microwave modes will finally evolve to the two-mode squeezed vacuum state, definitely.",
"Let's consider this scheme in a more general case, where we only make adiabatic approximations to optical modes.",
"Then such total variance becomes $V=\\Sigma _{i=1}^2 e^{-2(x+k_i\\tau )}(2n_{th,i}+1)+\\frac{(1-r)^2+(n_{th,i}+1)k_0k_i}{1-r^2+k_0k_i}\\times [1-e^{-2(x+k_i\\tau )}]$ .",
"The effective decay rate in Eq.",
"(REF ) varies inversely with the decay rate of optical modes, and that explains why we need the optical cavity with a large decay rate.",
"In Fig.$\\ref {fg5}$ , we present the time evolution of the total variance under different decay rates of microwave modes as well as the result of an ideal two-mode squeezed vacuum state.",
"The initial conditions are chosen as the ground states for the optical cavities and the thermal states for the two microwave modes.",
"From this figure, we know that when the decay rates of microwave modes are much smaller than the effective decay rate, at steady state we can prepare nearly ideal two-mode vacuum squeezed states.",
"fig/ Figure: Plot of the total variance VV V.S.",
"scaled time xx under different decay conditions, with parameters r=0.5r=0.5, n b,th =0.01n_{b,th}=0.01, k 0 =10k_0=10, together with the result for an ideal two-mode squeezed state.",
"The effective decay rate of the subsystem formed by two superconducting microwave resonators in Eq.",
"() is γ=2(1-r 2 ) k 0 =0.15\\gamma =\\frac{2(1-r^2)}{k_0}=0.15."
],
[
"Experimental Feasibility",
"We now talk about the experimental parameters.",
"For the cascaded scheme, it is feasible to take the capacitances and inductances of $LC$ resonators as $C_1=C_2=$ 40fF, $L_1=$ 70nH, $L_2=$ 25nH.",
"Similar to that electro-optic system reported by Mankei Tsang[28], we can take the electro-optic coefficient $n^3r_0\\approx $ 300pm/V, the resonance frequency of optical cavity $\\omega _{a1}\\approx 2\\pi \\times 200$ THz, the decay rate of superconducting microwave resonators $\\Gamma _i\\approx 2\\pi \\times 1$ KHz, $i=1,~2$ , and assume the distance between two planes of each capacitor $d\\approx 10\\mu $ m. The pump power is able to reach $P=10$ mW[45].",
"Thus, the coefficients $g_i$ given by Eq.",
"(REF ) can reach $g_1\\approx 2\\pi \\times 15$ KHz, $g_2\\approx 2\\pi \\times 19$ KHz, and $\\omega _{b1}\\approx 2\\pi \\times 3$ GHz, $\\omega _{b2}\\approx 2\\pi \\times 5$ GHz.",
"In the \"overcoupled\" case, we can also work out the mean photon number of optical cavity caused by external pump $\\bar{n}_{cav,i}$ through the following equation[29] $\\bar{n}_{cav,i}=\\frac{\\Gamma }{\\Delta _i^2+\\left(\\Gamma /2\\right)^2}\\frac{P}{\\hbar \\omega _{a1}},$ where $\\Gamma $ is the total loss rate of optical cavity, which is dominated by external loss rate of the cavity.",
"For the Q-factor of optical cavities can reach $10^8$ , we have $\\Gamma _0\\approx 2\\pi \\times 0.3$ MHz.",
"In our case we choose $\\Delta _i=\\omega _{bi}$ , and get $\\bar{n}_{cav,1}\\approx 400,\\bar{n}_{cav,2}\\approx 144$ .",
"Then the effective electro-optic coupling strength can reach $\\sqrt{\\bar{n}_{cav,1}}g_1\\approx 2\\pi \\times 0.3$ MHz, $\\sqrt{\\bar{n}_{cav,2}}g_2\\approx 2\\pi \\times 0.23$ MHz.",
"If we set the time for \"two-mode squeezed\" interaction is $T_2\\sim 1.6\\mu $ s, the operation time for generating target states will be $T_c=\\pi /\\left(\\bar{a}_1g_1\\right)+T_2\\sim 3.2\\mu $ s in the cascaded scheme.",
"As for parallel and dissipative dynamical schemes, we assume the pump power is relatively low, i.e.",
"$P=10\\mu $ W, the distance $d=5\\mu $ m, the capacitances and inductances of superconducting microwave resonators $C_1=C_2=4$ fF, $L_1=700$ nH, $L_2=250$ nH, $\\Gamma _1=\\Gamma _2\\approx 2\\pi \\times 1$ KHz, and use the optical cavity with resonance frequency $2\\pi \\times 1500$ THz$(\\lambda \\approx 200nm)$ to ensure the validity of the expansion applied in both schemes.",
"In our case, $\\Gamma \\ll \\Delta _i$ .",
"Then the amplitudes of driving lasers in both schemes can be expressed as $E_i=\\sqrt{\\bar{n}_{cav,inew}}\\omega _{bi},~i=1,~2$ .",
"Therefore, the operation time for the parallel scheme to generate target states is $T_p=\\pi /\\left[g_1\\sqrt{\\bar{n}_{cav,1new}-\\bar{n}_{cav,2new}}\\right]\\sim 3.8\\mu $ s, and the time for reaching stationary state of the dissipative dynamical scheme $T_d=\\Gamma /\\left[2g_1^2\\left(\\bar{n}_{cav,1new}-\\bar{n}_{cav,2new}\\right)\\right]\\sim 1.3\\mu $ s. These times are much shorter than the photon lifetime in superconducting microwave resonators.",
"Further more, if it is allowed to realize large inductance, the effective electro-optic coupling strength will exceed $2\\pi \\times 1$ MHz.",
"For example, we take $L=63\\mu $ H and keep other parameters the same as those in the cascaded scheme, its effective electro-optic coupling strength will reach $2\\pi \\times $ 1.6MHz and the optical loss rate can be ignored for simplicity.",
"We are also concerned about the entanglement properties of systems in each scheme.",
"From Eq.",
"(REF ), we can see the squeezed parameter of the cascaded scheme in ideal case is determined by $\\zeta =r\\left(\\tau _2-\\tau _1\\right)$ .",
"Obviously, $\\zeta $ will increase with the increase of $\\tau _2$ .",
"But effects of dissipation and decoherence also grow when $\\tau _2$ increases.",
"Therefore, people need to strike a balance between both aspects.",
"We assume the temperature is approximately 100mK.",
"In the previous experimental parameters in the cascaded scheme, the thermal photon numbers are $n_{th,1}\\approx 0.3,~ n_{th,2}\\approx 0.1$ .",
"Through numerical simulation shown in Fig.REF , we can find the optimized scaled time $\\tau _2$ is 2.43 or equally $T_2\\approx 1.3\\mu $ s, and the minimum of total variance is approximately 1.56.",
"We find that the total variance is insensitive to the environment temperature when it is below 1K, but greatly relies on scaled decay rates of all modes $k_i=\\Gamma _i/\\left(2\\bar{a}_1g_1\\right),~i=0,~1,~2$ .",
"Thus, we can reset those related parameters to improve the quality of target states.",
"For example, when we change capacities and inductances to $C_1=C_2=1$ fF, $L_1=360$ nH, $L_2=350$ nH, at the same temperature, the minimal variance drops to $V\\approx 0.77$ .",
"Figure: Plot of the total variance VV V.S.",
"scaled time τ 2 \\tau _2 under above experimental parameters.",
"And we can see that the optimized scaled time is τ 2 =2.43\\tau _2=2.43, or equally T 2 ≈1.3μT_2\\approx 1.3\\mu s. In the parallel scheme, the total variance is greatly affected by not only scaled decay rates, but also the environment temperature.",
"With those experimental parameters used for discussing operation times of this scheme, scaled decay rates are $k_0\\approx 0.9,~k_1\\approx k_2\\approx 0.003$ .",
"At the temperature $T=100$ mK, the lowest variance is 0.66, but at the temperature $T=1$ K, the entanglement will be destroyed.",
"In order to discuss the total variance of the dissipative dynamical scheme, it is useful to simplify the expression of its stable variance as following: $V=\\frac{2\\left(1-r\\right)/\\left(1+r\\right)+2\\alpha \\left(n_{th,1}+n_{th,2}+2\\right)}{1+2\\alpha },\\\\\\alpha =k_i/\\gamma ,~i=1,2,\\\\\\gamma =2\\left(1-r^2\\right)/k_0.$ Where we've assumed two superconducting resonators have same the decay rate $\\Gamma _1=\\Gamma _2$ , and $\\gamma $ is the effective decay rate of the system.",
"We choose $\\Gamma _0=2\\pi \\times 30$ MHz, $\\Gamma _1=\\Gamma _2\\approx 2\\pi \\times 1.44$ KHz and keep other parameters same as the parallel scheme.",
"At the same temperature $T=100$ mK, the total variance can decrease to 0.3."
],
[
"Conclusions",
"To conclude, we propose three schemes to generate the entanglement of microwave photons with an electro-optic system, in which two superconducting microwave resonators are coupled by one or two optical cavities through electro-optic effect.",
"The first two schemes are based on coherent control over the system to realize Bogoliubov modes consisting of two microwave modes while the last scheme is based on dissipative dynamics engineering, which exploits the thermal noises of two optical cavities as useful resources to entangle microwave modes.",
"Compared to previous works, our electro-optic system can generate more ideal two-mode squeezed states in principle.",
"These schemes based on the electro-optic system may have novel applications in quantum information processing."
],
[
"Acknowledgments",
"This work is supported by the NSFC under Grant No.",
"11474227 and the Fundamental Research Funds for the Central Universities."
]
] | 1709.01856 |
[
[
"Dispersion of the solar magnetic flux in undisturbed photosphere as\n derived from SDO/HMI data"
],
[
"Abstract To explore the magnetic flux dispersion in the undisturbed solar photosphere, magnetograms acquired by Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamic Observatory (SDO) were utilized.",
"Two areas, a coronal hole area (CH) and an area of super-granulation pattern, SG, were analyzed.",
"We explored the displacement and separation spectra and the behavior of the turbulent diffusion coefficient, $K$.",
"The displacement and separation spectra are very similar to each other.",
"Small magnetic elements (of size 3-100 squared pixels and the detection threshold of 20 Mx sm$^{-2}$) in both CH and SG areas disperse in the same way and they are more mobile than the large elements (of size 20-400 squared pixels and the detection threshold of 130 Mx sm$^{-2}$).",
"The regime of super-diffusivity is found for small elements ($\\gamma \\approx 1.3 $ and $K$ growing from $\\sim$100 to $\\sim$ 300 km$^2$ s$^{-1}$).",
"Large elements in the CH area are scanty and show super-diffusion with $\\gamma \\approx 1.2$ and $K$ = (62-96) km$^2$ s$^{-1}$ on rather narrow range of 500-2200 km.",
"Large elements in the SG area demonstrate two ranges of linearity and two diffusivity regimes: sub-diffusivity on scales (900-2500) km with $\\gamma=0.88$ and $K$ decreasing from $\\sim$130 to $\\sim$100 km$^2$ s$^{-1}$, and super-diffusivity on scales (2500-4800) km with $\\gamma \\approx 1.3$ and $K$ growing from $\\sim$140 to $\\sim$200 km$^2$ s$^{-1}$.",
"Comparison of our results with the previously published shows that there is a tendency of saturation of the diffusion coefficient on large scales, i.e., the turbulent regime of super-diffusivity is gradually replaced by normal diffusion."
],
[
"Introduction",
"Solar magnetic fields are the key signature of solar activity, the major contributor to the near-Earth space weather and they as well control the total solar irradiance (TSI).",
"A closer look at the solar photosphere reveals chaos of continuously renewing mixed-polarity magnetic elements spanning all spatial scales down to the resolution limits of modern instrumentation.",
"Flux transport models strive to describe the ceaselessly changing distribution of the magnetic flux driven by convection, differential rotation and meridional flows.",
"Particular attention is paid to the dispersal of the magnetic flux.",
"For example, how does the eroded magnetic field of a sunspot or a pore move through the magnetic network from the near solar equator to the poles?",
"What are parameters of this process?",
"Magnetic flux dispersal is usually described in terms of turbulent magnetic diffusivity.",
"The corresponding diffusion coefficient characterizes mobility of magnetic elements and is a free parameter in the existing models of solar dynamo and magnetic flux transport, and it eventually affects reconstructed values of TSI.",
"Turbulent magnetic diffusivity - key parameter of the models - is the most poorly constrained parameter theoretically and observationally.",
"The competition between diffusion and advection processes determines the solar cycle memory and thus affects the prediction of an oncoming cycle [41].",
"The flux transport models usually adopt a constant value of the diffusion coefficient, which substantially differs from one model to another: from 5 km$^2$ s$^-1$ [15] to 600 km$^2$ s$^-1$ [14], [36].",
"Solar observations produce values below 300-350 km$^2$ s$^-1$ [31], [34], [23], which strongly depend on the characteristic spatial and temporal scales of the utilized data.",
"Accurate measurements of the diffusion coefficient are critical for further progress.",
"Understanding of the diffusivity scale dependence is needed to calibrate the diffusivity profiles in theoretical dynamo models [6], [28], [16], [27].",
"Hinode data allowed to detect ubiquitous transverse and fine mixed polarity fields [20], [38], [39], [11], [12], which led to an idea that an additional mechanism of magnetic field generation, such as local turbulent dynamos [25] should be at work.",
"It is a tremendously difficult task to observationally prove the existence of the local turbulent dynamo [33] and it was only indirectly argued [21], [3].",
"MHD-simulations of solar magnetoconvection can help here because of the possibility to monitor the accumulated magnetic energy and to estimate the efficiency of local dynamo effects.",
"So far they provide an affirmative answer - the local turbulent dynamo seems to work [32], [5], [35], [26].",
"In these models the diffusion is not input directly.",
"However, it can be inferred from simulations and then compared to observations, which is vitally important for further progress in the field.",
"Granular and supergranular turbulent magnetic diffusion is also considered to be a constant in the flux transport equations [37].",
"Integration of the equations over many cycles allows to derive the time dependence of the total photospheric flux.",
"The total flux, in turn, largely determines the TSI and is thus needed to reconstruct the solar radiative input to the Earth climate, as well as for future predictions [7].",
"The commonly accepted mechanism for transporting the magnetic flux over the solar surface on small scales is random walk, or, normal diffusion, when the mean-squared displacement of flow tracers varies with time, $\\tau $ , as $\\langle (\\Delta l)^2 \\rangle = 4K\\tau \\sim \\tau ^{\\gamma }$ , where $K$ is the diffusion coefficient (a scalar), and $\\gamma =1$ , e.g., [24].",
"Generally, when index $\\gamma $ deviates from unity, diffusion is called anomalous diffusion.",
"More specifically, a regime with $\\gamma >1$ is called super-diffusive, while $\\gamma <1$ , indicates sub-diffusive.",
"Parameters $\\langle (\\Delta l)^2 \\rangle $ and $\\gamma $ , generally derived from observations, allow us to determine the diffusion coefficient as a function of time and spatial scales [2].",
"Before the Hinode-era, it was acknowledged that the diffusion coefficient can sometimes vary depending on the data quality and scale of interest (see a review by [4]).",
"Recent researches based on the new-generation of solar instrumentation [2], [8], [9], [10], [17], [40], [13], strongly suggests that the diffusion coefficient is not constant and varies in direct proportion to the spatial and time scales suggesting the turbulent regime of super-diffusivity in the photosphere.",
"Magnetic bright points are frequently used as tracers of the magnetic flux [17], [40], [13], and only in the series of publications by [8], [9], [10] dispersal of magnetic flux elements were considered.",
"A considerable scattering in magnitudes of the observed index $\\gamma $ can be stated.",
"Thus, for the weakest magnetic environment of quiet sun, [13] on the basis of 121 tracers report $\\gamma =1.9 \\pm 0.7$ for the time scale interval of approximately (200-1000) s; [17] from 851 tracers found $\\gamma = 1.2 \\pm 0.2$ for the very short time intervals of approximately 4-100 s; [40] argue that index $\\gamma $ , as measured on scales below 300 s, decreases from 1.7 to 1.3 as the magnetic field increases from 100 to 450 G. [9] analyzing displacement spectra of magnetic elements in the interior of a supergranula found $\\gamma = 1.44$ on scales of approximately 100-10000 s (in the spatial domain, this corresponds to approximately 150-4000 km, see Fig 3 in [9].",
"For supergranula boundary elements, they found a break on scales of about 600 km with $\\gamma = 1.27$ below and 1.08 above.",
"Authors suggest that the lower diffusivity in the network areas facilitate the amplification of the magnetic field therein.",
"In [10] an attempt to make a step from displacement spectra to pair separation spectra (or, simply, separation spectra) was undertaken successfully.",
"A displacement spectrum technique (utilized in the above mentioned publications) operates with displacements of individual tracer from the start point of its trajectory, so that both processes, turbulent motions and large-scale advection, contribute into this spectrum.",
"To reduce the influence of advection and estimate the turbulent diffusivity, the pair separation technique [24], [19] can be applied.",
"Here, distances between two tracers at consecutive moments are calculated.",
"Since large-scale advection is expected to effect both tracers equally, it is eliminated.",
"The separation spectra for the solar photosphere were reported for the first time by [18].",
"Nearly the same value $\\gamma \\approx 1.47-1.49$ was found on scales 10-500 s for all studied magnetic environments: a coronal hole, a quiet sun area, and an active region plage.",
"Note, that for the same data sets, the index $\\gamma $ , as derived from the displacement spectra, was different and increases from the AR plage ($\\gamma =1.48$ ) area to the QS ($\\gamma =1.53$ ) and to the CH ($\\gamma =1.67$ ) [2].",
"The observed similarity of separation spectra versus individuality of displacement spectra Lepreti and co-authors explained by possible influence of the detailed structure of the velocity field on single tracers dispersal, whereas pair dispersal reflects the diffusivity in the inertial range of turbulence.",
"Therefore, a comparison between the displacement and separation spectra can provide information on the properties of the dispersal mechanism.",
"This kind of comparison is one of the aims of this study.",
"Note, that when comparing the results of [9] with those of [10], one can conclude that the displacement spectrum for the intranetwork on scales 100-10000 s with $\\gamma = 1.44$ is more shallow than the corresponding separation spectrum with $\\gamma = 1.55$ , which does not agree with the above mentioned tendency inferred from the publications of [2] and [18].",
"As we see, the majority of recent studies on the turbulent regime refer to rather small scales: the time intervals below $10^4$ s (approximately 3 hours) and spatial intervals below 1 mega-meter (Mm).",
"Meanwhile, as it was mentioned above, there exists a vital need of the diffusivity properties in a broad range of scales.",
"The seeing-free non-stop data acquired by the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamic Observatory (SDO, [29]) offer a good opportunity to extent the scale interval and analyze the magnetic flux dispersal on scales up to a day and tens of mega-meters.",
"In the present study, this opportunity is used for vast areas of undisturbed photosphere outside active regions.",
"Both displacement and separation spectra on scales 1000-4$\\times 10^4$ s (up to 11 hours) were analyzed for a weakest magnetic environment, a coronal hole, and a typical supergranula pattern.",
"Our two data sets consist of magnetogram series obtained with SDO/HMI instrument.",
"The line-of-sight hmi.M-720s magnetograms were taken in the FeI 6173.3 A spectral line with the spatial resolution of 1\" (the pixel size of 0.5\") and cadence of 12 min [30] and noise level of about 6 Mx sm$^{-2}$ [22].",
"Two regions of interest were selected (Figure REF ): an area inside a low-latitude coronal hole (hereinafter CH-area) which crossed the central meridian on January 3, 2016 at approximately 17:12 UT, and an area of decayed active region remains, a typical supergranula pattern (hereinafter SG-area) culminated around December 1, 2015 at 8:36 UT.",
"To avoid the projection effect influence, only one day long intervals from culmination were considered for both cases.",
"For the CH-area, we selected the magnitograms recorded from January 2, 17:12 UT to January 4, 17:00 UT, 2016; for the SG-area, the interval of investigation was November 30, 8:36 UT - December 2, 8:24 UT, 2015.",
"The size of the CH-area was restricted by the boundaries of the coronal hole and consisted 638$\\times $ 636 pixels, or approximately 230$\\times $ 230 Mm.",
"The SG-area covered 758$\\times $ 788 pixels, or 275$\\times $ 286 Mm.",
"The magnetograms taken near the time of culmination are shown in Figure 2.",
"Each data set of 240 magnetograms was carefully aligned using a sub-pixel alignment code based on the fast Fourier transform.",
"Figure: Two areas (inside the boxes) analysed in the present study as visible on SDO/AIA 193 A images.",
"Left - data for a coronal hole on the disk center (CH-area) recorded on January 3 at 17:12 UT, 2016; right - data for SG-area recorded on December 1 at 8:36 UT, 2015.Figure: Examples of SDO/HMI LOS magnetograms for the CH-area (left) and SG-area (right) recorded at the times depicted in Fig.1.",
"The magnetograms are scaled between -200 (black, negative magnetic polarity) and 200 (white, positive magnetic polarity) Mx sm -2 ^{-2}.",
"The FOV is 320\"×\\times 320\" (the SG-area is slightly cropped to fit the size of the CH-area)."
],
[
"Detection and tracking of magnetic flux concentrations",
"To detect magnetic elements and calculate their trajectories, we applied the modified feature detection and tracking code elaborated by [1], [2] for tracking photospheric magnetic bright points.",
"In this study, we used the absolute value of the magnetic field as input.",
"The thresholding technique was applied to obtain a mask of magnetic elements.",
"To count the weakest observed elements and at the same time to mitigate an influence of noise, we choose the threshold of $th=20$ Mx sm$^{-2}$ , which corresponds to the triple noise level.",
"A range of sizes of detected elements was selected between 3 and 100 square pixels.",
"Each detected element was labeled, the barycenter $(x_c,y_c)$ and equivalent diameter $d$ were calculated, and its counterpart on the consecutive magnetogram (if any) was found.",
"When in the current image, inside the radius of $d/2$ around the $(x_c, y_c)$ pixel, we find a pixel labeled as a center of a magnetic element in the next image, we assign the two found objects to be the same magnetic element visible on two consecutive images.",
"When an element merged with another one, or did not appear on tree consecutive magnetograms, the tracking was terminated.",
"The procedure gives us trajectories of elements.",
"As for the coronal hole area, the above procedure results in Set 1 data, i.e., small magnetic elements trajectories inside the CH area (parameters of our data sets are listed in Tables REF and REF ).",
"Table: Analyzed data setsTable: Calculated parametersTo explore dispersion of large magnetic elements, higher values of the threshold and size were selected.",
"Our experience shows that the best choice to detect magnetic elements forming the super-granula boundaries, i.e., network (NW) ensemble, is the threshold of 130 Mx sm$^{-2}$ and the size range of 20-400 square pixels.",
"For the CH area, this procedure gives us the Set 2 data (see Table REF for the calculated parameters).",
"Correspondingly, for the SG-area of well-pronounced network pattern (see Fig.",
"REF ), we obtained Set 3 for small elements and Set 4 for large elements.",
"The later represents the majority of the network elements nested on the boundaries of super-granules.",
"Parameters of data sets are in Tables REF and REF ."
],
[
"Calculation of the displacement and separation spectra",
"To analyze the diffusive properties of tracers in a turbulent flow, the Lagrangian approach [24] is usually applied [2], [18], [8], [9], [10].",
"Here, the position of a tracer along its trajectory is measured at discrete moments $t_0, t_1, ... t_i,...t_N$ , where $t_0$ is the moment when the tracer was detected for the first time.",
"Then, the time intervals $\\tau _i=t_i - t_0$ , from the starting moment, $t_0$ , to the current moment, $t_i$ , are calculated for all tracers.",
"Note that the time moments $t_i$ correspond to times when solar data were recorded.",
"The next step is to compute the spatial displacements, $(\\Delta l)_{i}$ , of an individual $j$ -th tracer as a function of time, $\\tau _i$ .",
"After that, we calculate the average (over all tracers) displacement for each $\\tau _i$ to produce the average squared displacements (the displacement spectrum): $\\langle (\\Delta l)^{2}(\\tau _i)\\rangle =\\langle |{\\bf X}_{j}(0)-{\\bf X}_{j}(\\tau _i)|^{2}\\rangle ,$ where ${\\bf X}_{j}(0)=(x_{0},y_{0})$ and ${\\bf X}_{j}(\\tau _i)=(x_{\\tau _i},y_{\\tau _i})$ are coordinates of the $j$ -th tracer at the moment of its first detection and $\\tau _i$ seconds later, respectively.",
"Both processes, advection and turbulent diffusion, contribute into this spectrum named hereinafter as displacement spectrum.",
"There are ways to significantly reduce the influence of advection and estimate the turbulent diffusivity.",
"We utilize for that a widely accepted pair separation technique [24], [18] keeping in mind that the two-particle dispersion reflects the diffusivity properties arising from the inertial range of turbulence.",
"Here, displacements are computed as distances between two tracers at consecutive moments.",
"The pair separation spectrum can be calculated as $\\langle (\\Delta l)^{2}(\\tau _i)\\rangle =\\langle (|{\\bf X}_{j}(t_i)-{\\bf X}_{k}(t_i)| - |l_0^{jk}|)^{2}\\rangle ,$ where $j$ and $k$ denote two tracers, $t_0$ is the first moment when both tracers are first detected with $l_0^{jk}$ distance between them, and $\\tau _i = t_i - t_0$ .",
"Hereinafter we refer to this kind of spectrum as separation spectrum.",
"The power index, $\\gamma $ , of the spectrum is defined as $\\langle (\\Delta l)^{2}(\\tau )\\rangle \\sim \\tau ^{\\gamma }$ and is determined as the slope of the spectra over a range of $\\tau $ .",
"When considering a displacement spectrum, the index will be noted as $\\gamma _d$ , and, correspondingly, as $\\gamma _s$ for a separation spectrum.",
"If the spectrum that we seek is indeed a power law with index $\\gamma $ , and the photospheric plasma is in the turbulence state, then the dependence of the diffusion coefficient, $K$ , from the spatial and time scales can be derived [24], [19].",
"Changes of $K$ with scales, in turn, will help us shed light on the diffusive regime.",
"An expression for the turbulent diffusion coefficient is $K(\\tau )=\\frac{1}{2D}\\frac{d}{d\\tau }\\langle (\\Delta l)^{2}(\\tau )\\rangle ,$ where $D$ is equal to 2 (3) for diffusion over a surface (volume).",
"Here, $\\langle (\\Delta l)^{2}(\\tau )\\rangle $ is the observed spectrum (Eqs.",
"REF , REF ), which can be approximated on a given range of scales as (Abramenko et al.",
"2011): $\\langle (\\Delta l)^{2}(\\tau )\\rangle =c\\tau ^{\\gamma },$ where $c=10^{y_{sect}}$ .",
"Values of $\\gamma $ and $y_{sect}$ can be derived from the best linear fit to the spectral data points plotted in a double-logarithmic plot.",
"Then, accepting that for the diffusion over the solar surface $D$ equals 2, an expression for the diffusion coefficient was obtained in Abramenko et al.",
"(2011): $K(\\tau )=\\frac{c\\gamma }{4}\\tau ^{\\gamma -1}.$ When $\\tau $ is excluded from Eqs.",
"REF and REF , a relationship between the diffusion coefficient and the spatial scale can be written as (Abramenko et al.",
"2011): $K(\\Delta l)=\\frac{c\\gamma }{4}((\\Delta l)^{2}/c)^{(\\gamma -1)/\\gamma }.$ Eqs.REF - REF show that index $\\gamma <1$ leads to an inverse dependence of the diffusion coefficient on the spatial, $\\Delta l$ , and temporal, $\\tau $ , scales (sub-diffusion).",
"Whereas conditions with $\\gamma >1$ , cause $K$ to be directly proportional to scales (super-diffusion).",
"For each of the four data sets, we calculated two types of spectra: the displacement spectrum (Eq.",
"REF ) and the separation spectrum (Eq.",
"REF ).",
"Various combinations to compare them to each other are presented in Figs.",
"REF - REF .",
"Hereinafter the parameters for the displacement (separation) spectrum are marked with the subscriber $d$ ($s$ ).",
"Does the dispersion of small magnetic elements differ from that of large magnetic elements?",
"To answer, in Figure REF the spectra for small elements are overplotted by the spectra for large elements.",
"The spectra differ significantly.",
"For the coronal hole area, CH (left panels), the spectra for small elements (Set 1) are above the spectra for large elements (Set 2) on all scales.",
"This implies that the small-scale (mainly intranetwork) magnetic elements in a coronal hole are more mobile than those forming the large-element subset, which supposedly forms the super-granula boundary skeleton.",
"The spectral indices for Set 1 can be defined in a broad range of time scales: (720-39600) s, which corresponds to 0.45-6.05 Mm for the separation spectrum and 0.46-6.67 Mm for the displacement spectrum, according to Eq.",
"REF .",
"The indices are $\\gamma _d=1.33 $ and $\\gamma _s=1.30$ , see Table REF .",
"At the same time, rather low statistics for Set 2 data (only 212 elements) did not allow us to calculate the indicies for the same linear range.",
"Here, values $\\gamma _d=1.26 $ and $\\gamma _s=1.18$ were retrieved for the narrow range of (1440-15100) s, see Table REF .",
"Analyzing the SG-area (right panels in Figure REF ), we found that Set 3 (small elements) reveals a broad linear range of 720-39600 s of super-diffusivity with $\\gamma \\approx 1.35$ , whereas an ensemble of large elements (Set 4) shows two patterns of linearity: on scales 1400-15100 s we observe the sub-diffusion with $\\gamma \\approx 0.88$ and a decreasing diffusion coefficient (see Table 2), and on scales 15100-39600 s the super-diffusion regime is visible with $\\gamma \\approx 1.25-1.33$ .",
"The change of the regime occurs on linear scales of approximately 2.5-3 Mm (see Table REF for the linear range).",
"Besides, on time scales lower than approximately 5000-8000 s, the large elements are more mobile than the small elements, and the opposite picture is observed on higher time scales.",
"We might conclude that large magnetic elements that compound the super-granular boundaries disperse in a different way than the small magnetic elements.",
"Figure: Displacement and separation spectra for small magnetic elements (Sets 1 and 3) compared for those for large magnetic elements (Sets 2 and 4).",
"In the right bottom frame, the dashed line shows the best linear fit to the separation spectrum for Set 4 on scales (1440-15100)s with γ=0.88\\gamma = 0.88, and the dash-dot-dash line shows the best linear fit on scales (15100-39600) s with γ=1.33\\gamma = 1.33.Figure: Spectra for the CH area (Sets 1 and 2) compared to the spectra for the SG area (Sets 3 and 4).Figure: The displacement spectra compared to the separation spectra.",
"Notations are the same as in Fig.",
".In Figure REF , the spectra for the CH area are compared with their counterparts for the SG area.",
"The left panels show that for both displacement and separation spectra, the small elements in the CH area disperse exactly in the same way as they do in the SG area.",
"The difference appears when we consider the dispersion of large elements which form mainly the network skeleton.",
"On all scales, large elements in the CH area being rather scanty (only 212 events) move slower than that in the SG area which are much more numerous (1602 events).",
"On time scales below 15100 s, we observe sub-diffusivity in the SG area and super-diffusivity in the CH area.",
"At the same time, in spite of \"accelerated\" dispersion (super-diffusivity), the CH large elements (Set 2) display the lower magnitudes of the diffusion coefficient (for example, $K_s $ varies from 62 to 96 km$^2$ s$^{-1}$ ) as comparing to that for the SG large elements (Set 4, $K_s $ decreases from 127 to 95 km$^2$ s$^{-1}$ , see Table REF ).",
"Figure REF demonstrates that the displacement spectra are very close to the separation spectra for all data sets (the correspondence is slightly weaker only for the low-statics Set 2).",
"This implies that on time scales below $4\\cdot 10^4$ s , or $\\sim $ 11 hours and spatial scales below $\\sim $ 6 Mm, large-scale, quasi-regular patterns of the photospheric horizontal velocity field do not affect the magnetic flux dispersion.",
"In Figures REF and REF the spectra obtained in this study are overplotted with the previously published data.",
"A general tendency is well pronounced: The spectrum becomes more shallow as the scale increases, i.e., the index $\\gamma $ reduces and the regime of well-developed super-diffusivity tends to become closer to the normal diffusion on larger scales.",
"The data reported in [9] are in a good agreements with ours (see Fig.",
"REF ): the transition between the NST- and HMI-spectra is well covered by the Hinode data.",
"The diffusion coefficients as derived from the separation spectra are presented in Figure REF along with the similar data from the NST-observations reported by [18].",
"The coefficient increases with scales for all data sets (except Set 4), however, the growth rate becomes slower on larger scales.",
"An abrupt break on scales between the two instruments coverage ($\\sim $ 500-700 s and $\\sim $ 500 km) might be artificial when the Hinode data are taken into account (see Fig REF ).",
"Anyway, a fair agreement between tiny magnetic bright points (NST data set) and small HMI-magnetic elements (Sets 1 and 3) is noticeable.",
"However, large magnetic elements (Sets 2 and 4, double lines in Fig.",
"REF ) demonstrate the significantly lower values of $K$ , which implies the suppressed flux dispersion inside the supergranula boundaries relative to the intergranular zones.",
"Figure: Displacement spectra obtained using the data from different instruments.",
"Triangles and green circles denote the NST/BBSO results for quiet sun and coronal hole regions, respectively (from , (reproduced by permission of the AAS).",
"Solid black and green lines schematically represent the result of (reproduced by permission of the AAS) for the network (NW) and intranetwork (IN) ares, respectively.",
"Black and blue dots show the HMI-spectra obtained in the present study for small elements in the SG and CH areas (Sets 1 and 3), respectively, whereas the red dots represent the spectrum from large elements in the SG area (Set 4).Figure: Separation spectra for the same data sets as shown in Fig.",
".",
"NST/BBSO spectra are from the paper by (reproduced by permission of the AAS).Figure: Left panel: diffusion coefficient as a function of temporal scale.",
"Rhight panel: diffusion coefficient as a function of spatial scale."
],
[
"Conclusions",
"Utilizing the HMI 720 s line-of-sight magnetograms for two regions in the undisturbed photosphere (a coronal hole area and an area of a decayed active region, i.e., a super-granulation pattern), we explored the behavior of the turbulent diffusion coefficient on time scales of approximately 1000-40000 s and spatial scales of approximately 500-6000 km.",
"We analyzed separately the dispersion of small and large magnetic elements.",
"We came to the following inferences.",
"- Displacement and separation spectra are very similar to each other for all analyzed data sets, which allows us to suggest that possible influence of large-scale velocity patterns is negligible for the magnetic flux dispersion on scales of interest and, therefore, the inertial range turbulence is explored.",
"- Small magnetic elements in both CH and SG areas disperse in the same way and they are more mobile than the large ones.",
"The regime of super-diffusivity is found for them ($\\gamma \\approx 1.3 $ and K growths from $\\sim $ 100 to $\\sim $ 300 km$^2$ s$^{-1}$ ).",
"Thus, the hypothesis suggested for the first time in Schrijver et al.",
"(1996) is confirmed by modern observations: large magnetic elements are indeed less mobile than the small ones.",
"- Large magnetic elements in both CH and SG areas disperse slower than the small elements.",
"In the CH area they are scanty and show super-diffusion with $\\gamma \\approx 1.2$ and $K_s = (62-96)$ km$^2$ s$^{-1}$ on rather narrow scale range of 500-2200 km.",
"Large elements of the SG area demonstrate a band in the spectra and, as a consequence, two ranges of linearity and two diffusivity regimes: the sub-diffusivity on scales (900-2500) km with $\\gamma =0.88$ and $K$ decreasing from $\\sim $ 130 to $\\sim $ 100 km$^2$ s$^{-1}$ , and the super-diffusivity on scales (2500-4800) km with $\\gamma \\approx 1.3$ and $K$ growing from $\\sim $ 140 to $\\sim $ 200 km$^2$ s$^{-1}$ .",
"The observed here sub-diffusion for large magnetic elements on small scales in the SG-area, on the contrary to super-diffusivity in the CH-area on the same scales, can be interpreted as follows.",
"We might suggest that widely scattered large elements in the CH hardly form any rigid skeleton of super-granulation, instead they rather freely disperse in the similar regime of super-diffusion as the neighbor small elements do, however with lower coefficients of diffusion.",
"A different situation we observe inside the SG area.",
"Here, the skeleton of network seems to play a role of some constrain factor preventing the \"accelerated\" dispersion in a super-diffusivity way.",
"Here, magnetic elements are forced to reduce their capability to displace while they walk inside the boundary of SG (on larger scales the super-diffusivity regime is restored).",
"Deep roots and possible inter-connectivity of magnetic flux tubes forming the SG skeleton might be in favor of the observed peculiarities of the flux dispersion.",
"This inference qualitatively agrees with results and conclusions of [9]: the regime of super-diffusivity becomes closer to normal diffusion on scales $l> 1500$ km and $\\tau > 2000$ s creating thus more favorable conditions for accumulation of magnetic flux at the boundaries of super-granules.",
"Comparison of our results with the previously published shows that there is a tendency of saturation of the diffusion coefficient on large scales, i.e., the turbulent regime of super-diffusivity gradually ceases so that normal diffusion with a constant value of $K \\approx $ 500 km$^2$ s$^{-1}$ might be observed on time scales longer than a day.",
"We presume that only strong and large magnetic elements (capable to survive so long) can be a subject of the expected random walk.",
"However, hardly the used here technique can be applied directly to explore the flux dispersion on such large scales because the basic assumption on the passive nature of the magnetic flux tracers in the turbulent flow might not be applicable on these scales [9]."
],
[
"Acknowledgements",
"SDO is a mission for NASA Living With a Star (LWS) program.",
"The SDO/HMI data were provided by the Joint Science Operation Center (JSOC).",
"The study was supported in part by the Russian Foundation for Basic Research projects 16-02-00221 A, 17-02-00049 and 17-52-53203."
]
] | 1709.01724 |
[
[
"$\\mathbb{Z}_2$ topological insulator analog for vortices in an\n interacting bosonic quantum fluid"
],
[
"Abstract $\\mathbb{Z}_2$ topological insulators for photons and in general bosons cannot be strictly implemented because of the lack of symmetry-protected pseudospins.",
"We show that the required protection can be provided by the real-space topological excitation of an interacting quantum fluid: quantum vortex.",
"We consider a Bose-Einstein Condensate at the $\\Gamma$ point of the Brillouin zone of a quantum valley Hall system based on two staggered honeycomb lattices.",
"We demonstrate the existence of a coupling between the winding number of a vortex and the valley of the bulk Bloch band.",
"This leads to chiral vortex propagation at the zigzag interface between two regions of inverted staggering, where the winding-valley coupling provides true topological protection against backscattering, contrary to the interface states of the non-interacting Hamiltonian.",
"This configuration is an analog of a $\\mathbb{Z}_2$ topological insulator for quantum vortices."
],
[
"Supplemental material",
"In this supplemental material, we present additional details on the derivation of results of the main text.",
"We discuss the winding-valley coupling and the velocity of a vortex at an interface.",
"Finally, we comment the supplemental video files.",
"The calculation of the Fourier transform $\\widetilde{\\psi }(\\mathbf {k})$ from the main text is carried out as follows.",
"In the TB approximation, $\\psi (\\mathbf {r})$ is defined only in discrete points in space, and the integration is replaced by summation.",
"Studying the core only, we take into account only the 3 atoms of the $A$ type of the central hexagon.",
"This gives the following sum: $\\widetilde{\\psi }_p \\left( \\mathbf {k} \\right) &=& {e^{i\\left( {0 - \\left( {{k_x},{k_y}} \\right)\\left( {0,0} \\right)} \\right)}} + {e^{i\\left( {\\frac{{2\\pi }}{3}p - \\left( {{k_x},{k_y}} \\right)\\left( {\\frac{{3a}}{2},\\frac{{a\\sqrt{3} }}{2}} \\right)} \\right)}}\\\\ \\nonumber &+& {e^{i\\left( {\\frac{{4\\pi }}{3}p - \\left( {{k_x},{k_y}} \\right)\\left( {0,a\\sqrt{3} } \\right)} \\right)}}$ where $p=\\pm 1$ is the vortex winding.",
"This expression can be rewritten as $\\widetilde{\\psi }_p \\left( \\mathbf {k} \\right) &=& 1 + {e^{i\\left( {\\frac{{2\\pi }}{3}p - \\frac{3}{2}a{k_x} - \\frac{{\\sqrt{3} }}{2}a{k_y}} \\right)}}\\\\ \\nonumber &+& {e^{i\\left( {\\frac{{4\\pi }}{3}p - \\sqrt{3} a{k_y}} \\right)}}$ To simplify the expressions, let us define the arguments of the two exponents as separate variables: $\\eta _p={\\frac{{2\\pi }}{3}p - \\frac{3}{2}a{k_x} - \\frac{{\\sqrt{3} }}{2}a{k_y}}$ $\\zeta _p={\\frac{{4\\pi }}{3}p - \\sqrt{3} a{k_y}}$ We can then find the position of the maximal probability density in the reciprocal space $|\\psi (\\mathbf {k})|^2$ , which writes (by separating the real and imaginary parts): ${\\left| {\\widetilde{\\psi }_p \\left( \\mathbf {k} \\right)} \\right|^2} &=& 1 + {\\cos ^2}\\eta _p + {\\cos ^2}\\zeta _p + 2\\cos \\eta _p + 2\\cos \\zeta _p \\\\ \\nonumber &+& 2\\cos \\eta _p \\cos \\zeta _p + {\\sin ^2}\\eta _p + {\\sin ^2}\\zeta _p + 2\\sin \\eta _p \\sin \\zeta _p$ which can be simplified to $\\left|\\widetilde{\\psi }_p\\left(\\mathbf {k}\\right)\\right|^2=3+2\\left(\\cos \\eta _p+\\cos \\zeta _p+2\\cos \\eta _p\\cos \\zeta _p\\right)$ The maximal value of this expression is achieved when both $\\eta _p=2\\pi \\nu $ and $\\zeta _p=2\\pi \\mu $ , where $\\nu $ and $\\mu $ are integer numbers.",
"From the latter, taking for example $\\nu =0$ , it is easy to obtain, for $p=1$ , $k_y=K$ (where $K=4\\pi /3\\sqrt{3}a$ ), and $k_x=0$ , and for $p=-1$ , $k_y=-K$ and $k_x=0$ ."
],
[
"Vortex velocity",
"We have studied how the vortex velocity depends on the parameters of the system in order to check that the propagation along the interface is not linked with the well-known vortex rolling effect.",
"First, let us see that the vortex really follows the interface, and its core is located exactly within the unit cell, which separates the two inverted materials.",
"Figure SREF shows a snapshot of the phase of the wavefunction with a vortex.",
"A $2\\pi $ phase jump line is clearly visible, and the core of the vortex is located at the end of this line.",
"The rotation direction of the vortex is shown with a red arrow, and the green arrow indicates the propagation direction of the vortex along the interface (white dashed line).",
"We see that the edge of the phase jump line is within the unit cell located at the interface.",
"Figure: Contour plot of the potential (black line) and the phase of the vortex (in color).",
"Red arrow shows the rotation direction, green arrow shows the propagation direction of the vortex.One might think that the vortex is simply rolling along the interface, like a wheel, converting rotation into propagation.",
"The characteristic distance at which the density can vary in the condensate is given by the healing length $\\xi $ and therefore the center of the vortex in this \"rolling wheel\" image has to be located at a distance $\\xi $ from the wall, which allows to find the speed of rotation of the particles where they meet with the wall (and therefore the vortex propagation speed) using the expression $v=\\frac{\\hbar }{m}\\frac{1}{r}$ where one takes $r=\\xi $ , which gives simply that the vortex propagates with a velocity roughly equal to the speed of sound in the condensate $v=\\sqrt{\\alpha n/m}$ .",
"In this model, one could therefore expect a pronounced dependence of the vortex propagation velocity on the particle density.",
"Another alternative could be that the vortex simply propagates with the group velocity of linear states at the interface, which can be calculated from the dispersion, as discussed in the main text.",
"Figure SREF compares the predictions of these models as a function of interaction energy $\\alpha n$ with numerical results (black squares).",
"Clearly, the simple predictions of the two naive models (red circles for rolling effect and black dashed line for the linear group velocity) strongly deviate from numerics.",
"The model of the rolling wheel (red dots) predicts a dependence on the density which is not observed at all (the interaction energy changes by a factor 5, and there is no significant change of the vortex velocity).",
"The group velocity of the interface states strongly overestimates the real vortex propagation speed (also by a factor 5).",
"Figure: Vortex velocity from numerical calculations and its estimation by different models.To calculate the vortex velocity, we analyze the currents that take place within its core (concentrated in a given valley because of winding-valley coupling).",
"In the bulk, the valley states are not propagating, but rotating, because the 3 quantum-mechanical current terms between the 3 pillars of the same type which have different phases (0, $2\\pi /3$ , $4\\pi /3$ ) exactly compensate each other, as these are three identical vectors rotated at 120 degrees.",
"Indeed, $j = \\frac{{n\\hbar }}{m}\\nabla \\varphi $ where $n$ is the particle density, and therefore, to calculate current in the tight-binding approach we need to consider only pillars with nonzero density and take into account the phase difference between each pair.",
"At the interface the situation changes, as can be seen in Fig.",
"4(a) of the main text.",
"The $A$ pillars on the left of the interface are not large pillars (with lower energy) but small pillars (with higher energy), and therefore, the 3 current terms (blue arrows) do not have the same prefactor.",
"The phase differences are the same, but the density on the pillars on the left of the interface is smaller (it is not zero as it would be in the bulk, because the presence of the interface mixes the Bloch states), and therefore the current term marked as a dashed line has a smaller magnitude than the other two.",
"This results in a net current pointing upwards, and this is what leads to the propagative nature of the interface states.",
"The total current reads ${\\bf {j}} = {{\\bf {j}}_1} + {{\\bf {j}}_2} + {{\\bf {j}}_3}.$ Assuming that the density on the $A$ pillars on the right of the interface is $n$ and the density on the $A$ pillars on the left of the interface is $n^{\\prime }$ , we can write the magnitude of the current terms as: ${j_{1,2}} = \\frac{n+n^{\\prime }}{2}\\frac{{\\hbar }}{m}\\frac{{2\\pi }}{{3\\sqrt{3} a}}$ and ${j_{3}} = \\frac{n^{\\prime }\\hbar }{m}\\frac{{2\\pi }}{{3\\sqrt{3} a}}$ The orientation of the vectors makes that the $X$ projection of $j_3$ is 0, while the $X$ projections of $j_1$ and $j_2$ are opposite, and so they compensate each other.",
"The $Y$ projections give: ${j_Y} = \\frac{1}{2}\\left(j_1+j_2\\right) - {j_3}$ which finally gives ${j_Y} = \\frac{{n - n^{\\prime }}}{2}\\frac{\\hbar }{m}\\frac{{2\\pi }}{{3\\sqrt{3} a}}$ Without the interface, $n=n^{\\prime }$ and $\\mathbf {j}=0$ , as expected.",
"The presence of the interface makes $n^{\\prime }<n$ .",
"If we consider an isolated problem of two pillars with coupling $J$ and energy splitting $\\Delta $ (which determines the gap in the bulk TMD analog), we can estimate $n^{\\prime }$ as $n^{\\prime } = \\frac{2n}{{1 + {{\\left( {\\Delta + \\sqrt{{\\Delta ^2} + 4{J^2}} } \\right)}^2}/4{J^2}}}$ which finally gives the expression for the group velocity of the main text, because $2\\pi \\hbar /m/3\\sqrt{3}a$ is simply an estimate of the group velocity $v_g$ in terms of the tight-binding parameters.",
"We can also calculate an approximated expression, assuming that $\\Delta \\ll J$ , $n^{\\prime } = n\\left( {1 - \\frac{\\Delta }{{2J}}} \\right)$ which gives for the net velocity along the interface ${v_Y} \\approx \\frac{\\Delta }{{2J}}\\frac{\\hbar }{m}\\frac{{2\\pi }}{{3\\sqrt{3} a}}$ The corresponding calculated velocity shown in Fig.",
"S2 by a solid black line corresponds well to the numerical results, contrary to the predictions of the simple models.",
"In the opposite limit of very large $\\Delta $ , $n^{\\prime } = n\\frac{{2{J^2}}}{{{\\Delta ^2}}}$ and ${v_Y} \\approx \\left( {1 - \\frac{{2{J^2}}}{{{\\Delta ^2}}}} \\right)\\frac{\\hbar }{m}\\frac{{2\\pi }}{{3\\sqrt{3} a}}$ This expression also increases with the increase of $\\Delta $ .",
"It is interesting to see that this expression is bounded from above by a limiting value, which cannot be exceeded by changing $\\Delta $ (but only by changing $J$ , which affects $m$ ).",
"In the supplemental video file vortexdefect.avi (also available at https://www.youtube.com/watch?v=PNsDF5xUvH4), we show the temporal evolution of the spatial density distribution of the condensate $|\\psi (\\mathbf {r},t)|^2$ , obtained by direct solution of the Gross-Pitaevskii equation with $U(\\mathbf {r})$ being the lattice potential, without the tight-binding approximation.",
"The snapshots from this movie are shown in Fig.",
"3 of the main text.",
"The vortex is attached to one side of the interface and propagates along it, passing around two corners and a defect.",
"A second movie linwp.avi (also available at https://www.youtube.com/watch?v=M7nbL5i9l44) demonstrates that a linear Gauss-Laguerre wavepacket with a non-zero angular momentum does not at all exhibit the same behavior as the vortex in an interacting condensate: the wavepacket is unstable and expands rapidly, preventing the observer to keep trace of the propagation of its center.",
"The features of the interacting BEC maintaining the vortex are therefore crucial for the results obtained in the main text."
]
] | 1709.01830 |
[
[
"Multi-color image compression-encryption algorithm based on chaotic\n system and fuzzy transform"
],
[
"Abstract In this paper an algorithm for multi-color image compression-encryption is introduced.",
"For compression step fuzzy transform based on exponential b-spline function is used.",
"In encryption step, a novel combination chaotic system based on Sine and Tent systems is proposed.",
"Also in the encryption algorithm, 3D shift based on chaotic system is introduced.",
"The simulation results and security analysis show that the proposed algorithm is secure and efficient."
],
[
"Introduction",
" In recent years with the development of the information transmission, fast and secure transmission have become important subject.",
"Various type of methods for compression-encryption and encryption have been studied in [1], [2], [3], [4], [5], [6].",
"Also multi- image encryption have been studied in [7], [8].",
"In this work, in the first step we use the fuzzy transform for image compression step.",
"Lossy image compression and reconstruction basis on fuzzy transform has been proposed in [9], [10], [11].",
"For fuzzy partition in fuzzy transform method, exponential b-spline function is used, more details about this function can be found in [12], [13].",
"In the next step, for encryption algorithm, a combination chaotic system is introduced.",
"This system is introduced in [14].",
"In this system for all values of $r\\in (0,4]$ the Lyapunov exponent is positive.",
"Also the combination chaotic system have uniform distribution over output range.",
"In the encryption algorithm, by using chaotic system, we define two-dimensional block matrix shift and three-dimensional matrix shift.",
"Then by using these matrices, images are scrambled.",
"The organization of this paper is as follows: In Section 2, image compression and reconstruction by using fuzzy transform is explained.",
"In Section 3, we introduce the chaotic system and the color image encryption algorithm.",
"Experimental results and algorithm analysis are given in Section 4.",
"A summary is given at the end of the paper in Section 5."
],
[
"Fuzzy transform based on exponential b-spline",
"In this section, we describe fuzzy transform based on exponential b-spline.",
"The basis of exponential b-spline has been proposed in [12], [13].",
"For non-decreasing sequence of knots as $x_1\\le x_2\\le \\cdots \\le x_n$ , exponential B-splines of order 2 are defined as follows $B^2_{0}(x):=\\left\\lbrace \\begin{array}{ll}\\frac{sinh(\\rho _{1}(x_{2}-x))}{sinh(\\rho _{1} h_{1})},&x\\in [x_{1},x_{2}],\\\\\\\\0,&~otherwise,\\\\\\end{array}\\right.$ $B^2_{n}(x):=\\left\\lbrace \\begin{array}{ll}\\frac{sinh(\\rho _{n-1} (x-x_{n-1}))}{sinh(\\rho _{n-1} h_{n-1})},&x\\in [x_{n-1},x_{n}],\\\\\\\\0,&~otherwise.\\\\\\end{array}\\right.$ Also for $i=1,\\cdots ,n-1$ , we define $B^2_{i}(x):=\\left\\lbrace \\begin{array}{ll}\\frac{sinh(\\rho _{i-1}(x-x_{i-1}))}{sinh(\\rho _{i-1} h_{i-1})},&x\\in [x_{i-1},x_{i}],\\\\\\\\\\frac{sinh(\\rho _{i}(x_{i+1}-x))}{sinh(\\rho _{i} h_{i})},&x\\in [x_{i},x_{i+1}],\\\\\\\\0,&~otherwise,\\\\\\end{array}\\right.$ where $\\rho _i~(i=1,\\ldots ,n-1)$ are tension parameters and $h_i:=x_{i+1}-x_{i}$ .",
"The tension parameter in b-spline function plays a important role in the slope of the function (see Fig.1).",
"Also exponential B-splines of arbitrary order can be found as follows $B^k_j(x):=\\frac{\\int ^x_{x_j}B^{k-1}_j(y) dy}{\\sigma ^{k-1}_j}-\\frac{\\int ^x_{x_{j+1}}B^{k-1}_{j+1}(y) dy}{\\sigma ^{k-1}_{j+1}},~k=3,4,\\ldots ,$ where $\\sigma ^{k-1}_{j}:=\\int ^{x_{j+k-1}}_{x_j}B^{k-1}_{j}(y)dy.$ Figure: Comparisons between exponentialb-spline for different values of tension parameters, red lines for ρ i =0.001\\rho _i=0.001 and green lines for ρ i =5,(i=1,⋯,5)\\rho _i=5,~(i=1,\\cdots ,5).The first study in fuzzy transform is introduced by Irina Perfilieva [9].",
"According to [9], Fuzzy partition and F-transforms (Fuzzy transforms) are defined as follows.",
"Definition 2.1 [9] Let $a=x_1<\\cdots <x_n=b, (n\\ge 2)$ be fixed nodes within $[a,b]$ .",
"We say that fuzzy sets $A_1,\\cdots ,A_n$ , identified with their memberhip functions $A_1(x),\\cdots ,A_n(x)$ , defined on $[a,b]$ , form a fuzzy partition of $[a,b]$ , if the fulfill the following conditions for $k=1,\\cdots ,n$ : 1-$A_k [a,b]\\longrightarrow [0,1], A_k(x_k)=1;$ 2-$A_k(x)=0$ if $x\\notin (x_{k-1},x_{k+1});$ 3-$A_k$ is continuous; 4-$A_k,k=2,\\cdots ,n$ , strictly increases on $[x_{k-1},x_{k}]$ and $A_k,k=1,\\cdots ,n-1$ , strictly decreases on $[x_{k},x_{k+1}]$ 5-for all $x\\in [a,b]$ , $\\sum ^{n}_{k=1}A_k(x)=1$ .",
"For a function $f$ be given at nodes $(p_i,q_i)\\in [a,b]\\times [c,d],~i=1,\\cdots ,N,~j=1,\\cdots ,M$ , discrete F-transform is defined as $&F_{kl}=\\frac{\\sum _{j=1}^{M}\\sum _{i=1}^{N}f_l(p_i,q_j)A_k(p_i)C_l(q_j)}{\\sum _{j=1}^{M}\\sum _{i=1}^{N}A_k(p_i)C_l(q_j)},~k=1,\\ldots ,n,~l=1,\\ldots ,m,$ where $A_1,\\cdots A_n, C_1,\\cdots ,C_m$ ( $n<N, m<M$ ) are fuzzy partitions of $[a,b]$ and $[c,d]$ , respectively.",
"Also the inverse of F-transform is defined as $f^{F}_{nm}(x,y)=\\sum ^n_{k=1}\\sum ^m_{l=1}F_{kl}A_k(x)C_l(y).$ We can easily prove that $\\lbrace B^2_i\\rbrace ^n_i$ is satisfied in the Definition REF .",
"In compression step we consider $B^2_i$ as $A_i$ and $C_i$ , then we use fuzzy transform for image compression."
],
[
"Compression-encryption algorithm",
"It is known that the histogram of the Logistic Tent system is not flat enough.",
"For solving this problem we add weights and functions in the Logistic Tent system or the Sine Tent system.",
"For details, the reader can see [14].",
"In this paper we consider a combination chaotic system based on Sine and Tent systems as follows $X_{n+1}:=\\left\\lbrace \\begin{array}{ll}\\omega _{1}f_{1}\\circ F(r,X_n)+\\alpha _{1}g_{1}(rX_n)+\\xi _{1}\\frac{(\\beta _{1}-r)X_n}{2}~mod~1,~when~X_n<0.5,\\\\\\\\\\omega _{2}f_{2}\\circ F(r,X_n)+\\alpha _{2}g_{2}(rX_n)+\\xi _{2}\\frac{(\\beta _{2}-r)(1-X_n)}{2}~mod~1,~when~X_n\\ge 0.5,\\\\\\end{array}\\right.$ where $\\omega _{1}=\\omega _{2}=1,\\alpha _{1}=\\alpha _{2}=1,\\xi _{1}=7,\\xi _{2}=15,$ $\\beta _{1}=40, \\beta _{2}=20, f_{1}=\\cos , f_{2}=\\tan , g_{1}=\\tan , g_{2}=x,F(r,x_n)=r\\sin (\\pi x_n)/4$ .",
"The Lyapunov exponent, Bifurcation, Histogram and Cobweb plots of combination chaotic system are given in figure REF .",
"From this figure we can see that for all values of $r$ , the Lyapunov exponent is positive.",
"The Cobweb plots show chaotic behavior.",
"Also this figure show that the chaotic system has uniform distribution over output range.",
"The following definitions are used in proposed algorithm.",
"Figure: (a) Lyapunov exponent plot, (b) Histogram plot, (c) Cobweb plot,(d) Bifurcation diagram.Definition 3.1 For an arbitrary vector $w\\in R^{n m}$ , matrix $\\Delta (w)=(\\delta _{i,j})\\in R^{n\\times m}$ is defined as follows $\\delta (i,j):=w(i+(j-1)n),~i=1,\\cdots ,n,~j=1,\\cdots ,m.$ Definition 3.2 For an arbitrary vector $w\\in R^{3 n m }$ , matrix $\\Lambda (w)=(\\lambda _{i,j,l})\\in R^{n\\times m\\times 3}$ is defined as follows $\\lambda (i,j,l):=w(i+(j-1)n+(l-1)nm),~i=1,\\cdots ,n,~j=1,\\cdots ,m,~l=1,2,3.$ Definition 3.3 Consider matrix $A=(a_{i,j})\\in R^{n\\times m}$ such that for all $(i,j)$ and $(k,l)$ with $(i,j)\\ne (k,l)$ , we have $a_{i,j}\\ne a_{k,l}$ .",
"Suppose matrix $B =(b_{i,j})\\in R^{n\\times m}$ that for all $(i,j)$ , $b_{i,j}=a_{k,l}$ for some $(k,l)$ .",
"For a block matrix $C=(C_{i,j})$ we define a block matrix $\\Upsilon ^2_{A\\rightarrow B}(C)=\\big (D_{i,j}\\big )$ , where $D_{i,j}=C_{k,l}$ .",
"Definition 3.4 Consider matrix $A=(a_{i,j,l})\\in R^{n\\times m \\times 3}$ such that for all $(i,j,d)$ and $(k,l,s)$ with $(i,j,d)\\ne (k,l,s)$ , we have $a_{i,j,d}\\ne a_{k,l,s}$ .",
"Suppose matrix $B =(b_{i,j,d})\\in R^{n\\times m\\times 3}$ that for all $(i,j,d)$ , $b_{i,j,d}=a_{k,l,s}$ for some $(k,l,s)$ .",
"For a matrix $C=(c_{i,j,k})$ we define a matrix $\\Upsilon ^2_{A\\rightarrow B}(C)=\\big (d_{i,j,k}\\big )$ , where $d_{i,j,k}= c_{k,l,s}$ .",
"For above definitions we can say that $\\Upsilon ^l_{ B\\rightarrow A}\\big (\\Upsilon ^l_{ A\\rightarrow B}(C)\\big )=\\Upsilon ^l_{ A\\rightarrow B}\\big (\\Upsilon ^l_{B\\rightarrow A}(C)\\big )=C,~~l=2,3.$ In this section we assume that we have a set of N images as $I^1,\\cdots ,I^N$ .",
"Also let the size of all the images are $n\\times m\\times 3$ .",
"In the following algorithm $\\mu (x_0,r,n)$ represents a vector as $(x_1,\\cdots ,x_{n})$ where $x_i (i=1,\\cdots ,n)$ are defined by using (REF ) with $x_0$ .",
"The compression-encryption algorithm is written in the following steps.",
"Figure: The step 1 and step 2 process of the proposed algorithm.Step 1.",
"In the first step input images are converted into R(red), G(green), B(blue) component matrix.",
"Then we use exponential b-spline fuzzy transform for each component part.",
"We assume that the size of compressed image is $nc\\times mc\\times 3$ .",
"In this step compressed images are considered as $CI^{i} (i=1,\\cdots ,N)$ .",
"Step 2.",
"For $i=1,\\cdots ,N$ , $CI^{i}$ is divided into $\\eta ^2$ parts as $CI^{i}_{j}~(i=1,\\cdots ,N,~j=1,\\cdots ,\\eta ^2)$ , where $\\eta :=\\left\\lbrace \\begin{array}{ll}2,&~when~N~is~1,~2 ~or~3,\\\\\\\\\\lfloor \\sqrt{N} \\rfloor ,&~otherwise.\\\\\\end{array}\\right.$ Then we consider a block matrix as $\\Theta =(CI^i_{j})$ (see figure REF ).",
"Step 3. a) $v^l~(l=1,2,3)$ are defined as $v^l=\\mu (x^l_0,r_l,N\\eta ^2)$ with $x^l_0:=\\frac{\\sum _{i,j}{CI^{1}(:,:,l)}}{nc\\times mc\\times 255}~(l=1,2,3)$ .",
"b) $Sv^l:=sort(v^l)~(l=1,2,3)$ (smallest to largest).",
"c) $A^l$ and $P^l$ are considered as $A^l=\\Delta (V^l),~P^l=\\Delta (SV^l)~(l=1,2,3)$ .",
"d) $C$ is defined as $C(:,:,l):=\\Upsilon ^2_{A^l\\rightarrow P^l}(\\Theta (:,:,l))~(l=1,2,3).$ Step 4. a) To generate different keys in each iteration, we consider $x^0$ as random number in $(0,1)$ then we define $v^0$ as $\\mu (x^0_0,r_0,N\\times nc\\times mc)$ .",
"b) $Sv^0:=sort(v^0)$ (smallest to largest).",
"c) Consider $A^0=\\Lambda (v^0),~P^0=\\Lambda (Sv^0)$ .",
"d) By using $xor$ operation and $\\Upsilon ^3_{A^0\\rightarrow P^0}$ , matrix $EI$ is determined by $EI:=\\Upsilon ^3_{A^0\\rightarrow P^0}(C)\\oplus A.$ e) The final encrypted image is found as follows $EI=circshift(EI,[\\rho _1~\\rho _2~\\rho _2]),$ where $circshift(A,r)$ circularly shifts the values in array $A$ by $r$ positions and $\\rho _l=\\lfloor x_l\\times 10^2\\rfloor ,~l=1,2,3$ ."
],
[
"Decryption algorithm",
"The decryption algorithm is the inverse process of the encryption algorithm.",
"In the first step by using Algorithm REF , compressed images are found.",
"Then by using inverse of the F-transform we can find images.",
"InputInput OutputOutput $\\lbrace CI^i\\rbrace ^{N}_{i=1}$ (Compressed images), l=1:3 $V^l:=\\mu (x_l,r_l,N\\eta ^2);$ $A^l:=\\Delta (V^l), P^{l}:=\\Delta (sort(V^l));$ $\\rho _l=\\lfloor x_l\\times 10^2\\rfloor ;$ $V^0:=\\mu (x_0,r_0,N\\times nc\\times mc);$ $A^0:=\\Lambda (V^0), P^{0}:=\\Lambda (sort(V^0));$ $EI=circshift(EI,[\\rho _1~\\rho _2~\\rho _3]);$ $C=\\Upsilon ^3_{P^0\\rightarrow A^0}(EI)\\oplus A;$ l=1:3 $\\Theta (:,:,l):=\\Upsilon ^2_{P^l\\rightarrow A^l}(C(:,:,l));$ Decryption Algorithm"
],
[
"Experimental results",
"In this section, as plain images, we use color images “Baboon, Lena, Peppers and House” (the $256\\times 256$ images with 256 grey levels) as $I^1,I^2,I^3$ and $I^4$ respectively.",
"In compression step $0.1$ is used as tension parameter.",
"Also we consider $nc=mc=100$ .",
"Fig.",
"4 shows original images and compressed images by using exponential b-spline fuzzy transform.",
"Also simulation results for encryption algorithm are shown in Fig.",
"5.",
"In this section, we consider the secret keys as $r_0=3.02,r_1=r_2=r_3=2$ .",
"Also for simulation results, we use Matlab R2014b programming.",
"Figure: (a-b-c-d) Original images, (e-f-g-h) Compressed images.Figure: (a-e-i) R, G, B components of the step 2 process of the proposed algorithm,(b-f-j) Histograms of R, G, B components of the step 2 process of the proposed algorithm,(c-g-k), R, G, B components of the encryptedimage, (d-h-l) Histograms of R, G, B components of the encryptedimage."
],
[
"Key analysis",
"In the encryption algorithm, the security keys of the proposed algorithm are composed of eight parameters.",
"In the image encryption algorithm, if we use $10^{15}$ as the precision, the key space is almost $10^{120}$ , and this space is sufficiently large to resist the brute force attack [15].",
"In the next step, we study key sensitivity.",
"We change $r_0$ as $r_0=3.02+10^{-15}$ .",
"Fig.",
"6. shows decryption images by using changed key.",
"By using this figure, we can see that the original images cannot be reconstructed.",
"Then, we can see that the proposed algorithm has high key sensitivity.",
"Also in Step 4 of encryption algorithm, a random number is used as encryption key.",
"Therefore we can generate different keys in each iteration.",
"We run the encryption algorithm twice and by using pixel-to-pixel the difference between the two images are illustrate in Fig.",
"7.",
"It can be seen in Fig.",
"7, two encrypted images are different.",
"Therefore the proposed algorithm is able to resist the chosen-plain text attack.",
"Figure: Decrypted images by usingchanged key.Figure: (a-d-g-j) The first encrypted image andhistogram of R, G, B components , (b-e-h-k) The second encrypted image andhistogram of R, G, B components, (c-f-i-l) The pixel-to-pixeldifference and histogram of R, G, B components."
],
[
"Noise and Data loss attacks",
"Appropriate encryption algorithm should resist the data loss and noise attacks.",
"In the data loss attack, some image data disappears.",
"To simulate this attack, in Fig.",
"8, we remove $100\\times 100$ of encrypted image.",
"Also in the noise attack, noise is added to the encrypted image.",
"An appropriate encryption algorithm should not increase the amount of these noises.",
"To simulate noise attack, Gaussian noise with zero-mean and $var=0.2$ is added to encrypted image.",
"The simulation results are given in Fig.s 8-9.",
"In these figures we can see that the reconstructed images contain most of original visual in formation and we can recognize the original images.",
"Figure: (a) Encrypted image, (b) Cropped attack image, (c-d-e-f)Decryption results.Figure: (a) Encrypted image, (b) Noise attack image withvar=0.2, (c-d-e-f)Decryption results."
],
[
"Statistical analysis",
"Some statistical analysis as correlation values, information entropy are studied in this section.",
"In the appropriate encryption algorithm, the correlation values between adjacent pixels are close to zero.",
"The correlation coefficients of adjacent pixels in the plain images and encrypted images are given in Table 1 and Fig.s 10-11.",
"By using numerical results, we can say that the correlation coefficients of plain images are close to one, and the correlation coefficients of the encrypted image are close to zero.",
"The correlation values are found by using formula [16] $C_{xy}=\\frac{E[(x-\\mu _x)(y-\\mu _y)]}{\\sigma _x \\sigma _y},$ where $E[\\cdot ]$ denotes the expectation value, $\\mu $ is the mean value and $\\sigma $ represents standard deviation.",
"In the next step we study histogram analysis.",
"The evaluation of the robustness of an encrypted image is studied by using histogram analysis.",
"The histograms of ciphered image are shown in Fig.",
"5.",
"The histograms show that distribution of pixel values in the encrypted image is close to uniform distribution.",
"Also chi-square test outputs are given in Table 2.",
"By using chi-square test, we accept that the data have a uniform distribution at level 0.05.",
"Also unpredictability of information can be studied by using unpredictability of information the information entropy.",
"The information entropy is defined as follows [17] $H(k)=-\\sum ^{w-1}_{i=0}P(k_i)\\log _2P(k_i),$ where $w$ is the gray level and $P(\\cdot )$ denotes the probability of symbol.",
"The ideal value of the information entropy is 8.",
"Numerical results for the information entropy are given in Table 3.",
"From Table 3, we can say that results for the encryption algorithm are very close to the ideal value.",
"From the above discussion, we can conclude that the proposed algorithm has stronger ability to withstand statistical attacks.",
"Table: Correlation coefficient values in the plain images and the cipher image.Table: Chi-square outputs for ciphered image.Table: Information entropy, NPCR and UACI results."
],
[
"Sensitivity analysis",
"NPCR (number of pixels change rate) and UACI (unified average change intensity) are calculated as $&NPCR=\\frac{\\sum _{i,j}D(i,j)}{m\\times n}\\times 100\\%,\\\\&UACI=\\frac{1}{m\\times n}\\big [ \\frac{\\sum _{i,J}|C_1(i,j)-C_2(i,j)|}{255}\\big ]\\times 100\\%,$ where $&D(i,j):=\\left\\lbrace \\begin{array}{ll}1,&when~C_1(i,j)\\ne C_2(i,j),\\\\\\\\0,&when~C_1(i,j)= C_2(i,j).\\\\\\end{array}\\right.$ In above formulae, $C_1(i,j)$ and $C_2(i,j)$ are denoted the cipher image before and after one pixel of the plain image is changed.",
"The sensitive to the changing of plain image can be studied using NPCR and UACI.",
"The best value for NPCR is $100\\%$ .",
"Also the ideal value for UACI is $33.\\overline{33}\\%$ .",
"In this step, $I_1(10,50,2), I_2(20,30,3), I_3(60,35,1)$ and $I_4(75,34,2)$ are changed to 0.",
"The simulation results in Table 3 are close to ideal values.",
"Therefore, we can say that the differential attack is impossible on proposed algorithm.",
"Figure: Correlation of neighbourhood pixels at different directions of Θ\\Theta .Figure: Correlation of neighbourhood pixels at different directions of the encrypted image."
],
[
"Conclusion",
"In this paper, by using fuzzy transform and combination chaotic system, we have introduced a multi-color image compression-encryption algorithm.",
"In fuzzy transform, exponential b-spline function is used as fuzzy partition.",
"Simulation results shown that the proposed encryption algorithm can effectively resist differential, statistical, noise, data loss, chosen-plain text attacks."
]
] | 1709.01597 |
[
[
"Symmetric Variational Autoencoder and Connections to Adversarial\n Learning"
],
[
"Abstract A new form of the variational autoencoder (VAE) is proposed, based on the symmetric Kullback-Leibler divergence.",
"It is demonstrated that learning of the resulting symmetric VAE (sVAE) has close connections to previously developed adversarial-learning methods.",
"This relationship helps unify the previously distinct techniques of VAE and adversarially learning, and provides insights that allow us to ameliorate shortcomings with some previously developed adversarial methods.",
"In addition to an analysis that motivates and explains the sVAE, an extensive set of experiments validate the utility of the approach."
],
[
"Introduction",
"Generative models that are descriptive of data have been widely employed in statistics and machine learning.",
"Factor models (FMs) represent one commonly used generative model [24], and mixtures of FMs have been employed to account for more-general data distributions [7].",
"These models typically have latent variables (e.g., factor scores) that are inferred given observed data; the latent variables are often used for a down-stream goal, such as classification [4].",
"After training, such models are useful for inference tasks given subsequent observed data.",
"However, when one draws from such models, by drawing latent variables from the prior and pushing them through the model to synthesize data, the synthetic data typically do not appear to be realistic.",
"This suggests that while these models may be useful for analyzing observed data in terms of inferred latent variables, they are also capable of describing a large set of data that do not appear to be real.",
"The generative adversarial network (GAN) [9] represents a significant recent advance toward development of generative models that are capable of synthesizing realistic data.",
"Such models also employ latent variables, drawn from a simple distribution analogous to the aforementioned prior, and these random variables are fed through a (deep) neural network.",
"The neural network acts as a functional transformation of the original random variables, yielding a model capable of representing sophisticated distributions.",
"Adversarial learning discourages the network from yielding synthetic data that are unrealistic, from the perspective of a learned neural-network-based classifier.",
"However, GANs are notoriously difficult to train, and multiple generalizations and techniques have been developed to improve learning performance [22], for example Wasserstein GAN (WGAN) [1], [2] and energy-based GAN (EB-GAN) [28].",
"While the original GAN and variants were capable of synthesizing highly realistic data (e.g., images), the models lacked the ability to infer the latent variables given observed data.",
"This limitation has been mitigated recently by methods like adversarial learned inference (ALI) [5], and related approaches.",
"However, ALI appears to be inadequate from the standpoint of inference, in that, given observed data and associated inferred latent variables, the subsequently synthesized data often do not look particularly close to the original data.",
"The variational autoencoder (VAE) [14] is a class of generative models that precedes GAN.",
"VAE learning is based on optimizing a variational lower bound, connected to inferring an approximate posterior distribution on latent variables; such learning is typically not performed in an adversarial manner.",
"VAEs have been demonstrated to be effective models for inferring latent variables, in that the reconstructed data do typically look like the original data, albeit in a blurry manner [5].",
"The form of the VAE has been generalized recently, in terms of the adversarial variational Bayesian (AVB) framework [17].",
"This model yields general forms of encoders and decoders, but it is based on the original variational Bayesian (VB) formulation.",
"The original VB framework yields a lower bound on the log likelihood of the observed data, and therefore model learning is connected to maximum-likelihood (ML) approaches.",
"From the perspective of designing generative models, it has been recognized recently that ML-based learning has limitations [1]: such learning tends to yield models that match observed data, but also have a high probability of generating unrealistic synthetic data.",
"The original VAE employs the Kullback-Leibler divergence to constitute the variational lower bound.",
"As is well known, the KL distance metric is asymmetric.",
"We demonstrate that this asymmetry encourages design of decoders (generators) that often yield unrealistic synthetic data when the latent variables are drawn from the prior.",
"From a different but related perspective, the encoder infers latent variables (across all training data) that only encompass a subset of the prior.",
"As demonstrated below, these limitations of the encoder and decoder within conventional VAE learning are intertwined.",
"We consequently propose a new symmetric VAE (sVAE), based on a symmetric form of the KL divergence and associated variational bound.",
"The proposed sVAE is learned using an approach related to that employed in the AVB [17], but in a new manner connected to the symmetric variational bound.",
"Analysis of the sVAE demonstrates that it has close connections to ALI [5], WGAN [2] and to the original GAN [9] framework; in fact, ALI is recovered exactly, as a special case of the proposed sVAE.",
"This provides a new and explicit linkage between the VAE (after it is made symmetric) and a wide class of adversarially trained generative models.",
"Additionally, with this insight, we are able to ameliorate much of the aforementioned limitations of ALI, from the perspective of data reconstruction.",
"In addition to analyzing properties of the sVAE, we demonstrate excellent performance on an extensive set of experiments."
],
[
"Background",
"Assume observed data samples $x\\sim q(x)$ , where $q(x)$ is the true and unknown distribution we wish to approximate.",
"Consider $p_{{\\theta }} (x|z)$ , a model with parameters ${\\theta }$ and latent code $z$ .",
"With prior $p(z)$ on the codes, the modeled generative process is $x\\sim p_\\theta (x|z)$ , with $z\\sim p(z)$ .",
"We may marginalize out the latent codes, and hence the model is $x\\sim p_{{\\theta }}(x)=\\int dzp_{{\\theta }}(x|z)p(z)$ .",
"To learn ${\\theta }$ , we typically seek to maximize the expected log likelihood: $\\hat{{\\theta }}=\\mbox{argmax}_{{\\theta }} ~\\mathbb {E}_{q(x)}\\log p_{{\\theta }}(x)$ , where one typically invokes the approximation $\\mathbb {E}_{q(x)}\\log p_{{\\theta }}(x)\\approx \\frac{1}{N}\\sum _{n=1}^N \\log p_{{\\theta }}(x_n)$ assuming $N$ iid observed samples $\\lbrace x_n\\rbrace _{n=1,N}$ .",
"It is typically intractable to evaluate $p_{{\\theta }}(x)$ directly, as $\\int dzp_{{\\theta }}(x|z)p(z)$ generally doesn't have a closed form.",
"Consequently, a typical approach is to consider a model $q_{{\\phi }}(z|x)$ for the posterior of the latent code $z$ given observed $x$ , characterized by parameters ${\\phi }$ .",
"Distribution $q_{{\\phi }}(z|x)$ is often termed an encoder, and $p_{\\theta }(x|z)$ is a decoder [14]; both are here stochastic, vis-à-vis their deterministic counterparts associated with a traditional autoencoder [26].",
"Consider the variational expression $\\mathcal {L}_x({\\theta },{\\phi })=\\mathbb {E}_{q(x)}\\mathbb {E}_{q_{{\\phi }}(z|x)} \\log \\big [\\frac{p_{{\\theta }}(x|z)p(z)}{q_{{\\phi }}(z|x)}\\big ]$ In practice the expectation wrt $x\\sim q(x)$ is evaluated via sampling, assuming $N$ observed samples $\\lbrace x_n\\rbrace _{n=1,N}$ .",
"One typically must also utilize sampling from $q_{{\\phi }}(z|x)$ to evaluate the corresponding expectation in (REF ).",
"Learning is effected as $(\\hat{{\\theta }},\\hat{{\\phi }})=\\mbox{argmax}_{{\\theta },{\\phi }} ~\\mathcal {L}_x({\\theta },{\\phi })$ , and a model so learned is termed a variational autoencoder (VAE) [14].",
"It is well known that $\\mathcal {L}_x({\\theta },{\\phi })=\\mathbb {E}_{q(x)}[\\log p_{{\\theta }}(x)-\\mbox{KL}(q_{{\\phi }}(z|x)\\Vert p_{{\\theta }}(z|x))]\\le \\mathbb {E}_{q(x)}[\\log p_{{\\theta }}(x)]$ .",
"Alternatively, the variational expression may be represented as $\\mathcal {L}_x({\\theta },{\\phi })=-\\mbox{KL}(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(x,z))+C_x$ where $q_{{\\phi }}(x,z)=q(x)q_{{\\phi }}(z|x)$ , $p_{{\\theta }}(x,z)=p(z)p_{{\\theta }}(x|z)$ and $C_x=\\mathbb {E}_{q(x)}\\log q(x)$ .",
"One may readily show that $&&\\mbox{KL}(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(z,z))\\nonumber \\\\&=&\\mathbb {E}_{q(x)}\\mbox{KL}(q_{{\\phi }}(z|x)\\Vert p_{{\\theta }}(z|x))+\\mbox{KL}(q(x)\\Vert p_{{\\theta }}(x))\\\\&=&\\mathbb {E}_{q_{{\\phi }}(z)}\\mbox{KL}(q_{{\\phi }}(x|z)\\Vert p_{{\\theta }}(x|z))+\\mbox{KL}(q_{{\\phi }}(z)\\Vert p(z))$ where $q_{{\\phi }}(z)=\\int q(x)q_{{\\phi }}(z|x)dx$ .",
"To maximize $\\mathcal {L}_x({\\theta },{\\phi })$ , we seek minimization of $\\mbox{KL}(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(z,z))$ .",
"Hence, from (REF ) the goal is to align $p_{{\\theta }}(x)$ with $q(x)$ , while from () the goal is to align $q_{{\\phi }}(z)$ with $p(z)$ .",
"The other terms seek to match the respective conditional distributions.",
"All of these conditions are implied by minimizing $\\mbox{KL}(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(z,z))$ .",
"However, the KL divergence is asymmetric, which yields limitations wrt the learned model."
],
[
"Limitations of the VAE",
"The support $\\mathcal {S}^{\\epsilon }_{p(z)}$ of a distribution $p(z)$ is defined as the member of the set $\\lbrace \\tilde{\\mathcal {S}}^{\\epsilon }_{p(z)}: \\int _{\\tilde{\\mathcal {S}}^{\\epsilon }_{p(z)}}p(z)dz=1-\\epsilon \\rbrace $ with minimum size $\\Vert \\tilde{\\mathcal {S}}^{\\epsilon }_{p(z)}\\Vert \\triangleq \\int _{\\tilde{\\mathcal {S}}^{\\epsilon }_{p(z)}}dz$ .",
"We are typically interested in $\\epsilon \\rightarrow 0^+$ .",
"For notational convenience we replace $\\mathcal {S}^{\\epsilon }_{p(z)}$ with $\\mathcal {S}_{p(z)}$ , with the understanding $\\epsilon $ is small.",
"We also define $\\mathcal {S}_{p(z)_{-}}$ as the largest set for which $\\int _{\\mathcal {S}_{p(z)_{-}}}p(z)dz=\\epsilon $ , and hence $\\int _{\\mathcal {S}_{p(z)}}p(z)dz+ \\int _{\\mathcal {S}_{p(z)_{-}}}p(z)dz=1$ .",
"For simplicity of exposition, we assume ${\\mathcal {S}_{p(z)}}$ and ${\\mathcal {S}_{p(z)_{-}}}$ are unique; the meaning of the subsequent analysis is unaffected by this assumption.",
"Consider $-\\mbox{KL}(q(x)\\Vert p_{{\\theta }}(x))=\\mathbb {E}_{q(x)}\\log p_{{\\theta }}(x)-C_x$ , which from (REF ) and (REF ) we seek to make large when learning ${\\theta }$ .",
"The following discussion borrows insights from [2], although that analysis was different, in that it was not placed within the context of the VAE.",
"Since $\\int _{\\mathcal {S}_{q(x)_{-}}}q(x)\\log p_{{\\theta }}(x)dx\\approx 0$ , $\\mathbb {E}_{q(x)}\\log p_{{\\theta }}(x)\\approx \\int _{\\mathcal {S}_{q(x)}} q(x)\\log p_{{\\theta }}(x)dx$ , and $\\mathcal {S}_{q(x)} = (\\mathcal {S}_{q(x)}\\cap \\mathcal {S}_{p_{{\\theta }}(x)}) \\cup (\\mathcal {S}_{q(x)}\\cap \\mathcal {S}_{p_{{\\theta }}(x)_{-}})$ .",
"If $\\mathcal {S}_{q(x)}\\cap \\mathcal {S}_{p_{{{\\theta }}}(x)_{-}}\\ne \\emptyset $ , there is a strong (negative) penalty introduced by $\\int _{\\mathcal {S}_{q(x)}\\cap \\mathcal {S}_{p_{{\\theta }}(x)_{-}}} q(x)\\log p_{{\\theta }}(x)dx$ , and therefore maximization of $\\mathbb {E}_{q(x)}\\log p_{{\\theta }}(x)$ encourages $\\mathcal {S}_{q(x)}\\cap \\mathcal {S}_{p_{{{\\theta }}}(x)_{-}}=\\emptyset $ .",
"By contrast, there is not a substantial penalty to $\\mathcal {S}_{q(x)_{-}}\\cap \\mathcal {S}_{p_{{{\\theta }}}(x)}\\ne \\emptyset $ .",
"Summarizing these conditions, the goal of maximizing $-\\mbox{KL}(q(x)\\Vert p_{{\\theta }}(x))$ encourages $\\mathcal {S}_{q(x)}\\subset \\mathcal {S}_{p_{{\\theta }}(x)}$ .",
"This implies that $p_{{\\theta }}(x)$ can synthesize all $x$ that may be drawn from $q(x)$ , but additionally there is (often) high probability that $p_{{\\theta }}(x)$ will synthesize $x$ that will not be drawn from $q(x)$ .",
"Similarly, $-\\mbox{KL}(q_{{\\phi }}(z)\\Vert p(z))=h(q_{{\\phi }}(z))+\\mathbb {E}_{q_{{\\phi }}(z)}\\log p(z)$ encourages $\\mathcal {S}_{q_{{\\phi }}(z)}\\subset \\mathcal {S}_{p(z)}$ , and the commensurate goal of increasing differential entropy $h(q_{{\\phi }}(z))=-\\mathbb {E}_{q_{{\\phi }}(z)}\\log q_{{\\phi }}(z)$ encourages that $\\mathcal {S}_{q_{{\\phi }}(z)}\\cap \\mathcal {S}_{p(z)}$ be as large as possible.",
"Hence, the goal of large $-\\mbox{KL}(q(x)\\Vert p_{{\\theta }}(x))$ and $-\\mbox{KL}(q_{{\\phi }}(z)\\Vert p(z))$ are saying the same thing, from different perspectives: ($i$ ) seeking large $-\\mbox{KL}(q(x)\\Vert p_{{\\theta }}(x))$ implies that there is a high probability that $x$ drawn from $p_{{\\theta }}(x)$ will be different from those drawn from $q(x)$ , and ($ii$ ) large $-\\mbox{KL}(q_{{\\phi }}(z)\\Vert p(z))$ implies that $z$ drawn from $p(z)$ are likely to be different from those drawn from $q_{{\\phi }}(z)$ , with $z\\in \\lbrace \\mathcal {S}_{p(z)}\\cap \\mathcal {S}_{q_{{\\phi }}(z)_{-}}\\rbrace $ responsible for the $x$ that are inconsistent with $q(x)$ .",
"These properties are summarized in Fig.",
"REF .",
"Figure: Characteristics of the encoder and decoder of the conventional VAE ℒ x \\mathcal {L}_x, for which the support of the distributions satisfy 𝒮 q(x) ⊂𝒮 p θ (x) \\mathcal {S}_{q(x)}\\subset \\mathcal {S}_{p_{{\\theta }}(x)} and 𝒮 q φ (z) ⊂𝒮 p(z) \\mathcal {S}_{q_{{\\phi }}(z)}\\subset \\mathcal {S}_{p(z)}, implying that the generative model p θ (x)p_{{\\theta }}(x) has a high probability of generating unrealistic draws.Figure: Characteristics of the new VAE expression, ℒ z \\mathcal {L}_z.Considering the remaining terms in (REF ) and (), and using similar logic on $-\\mathbb {E}_{q(x)}\\mbox{KL}(q_{{\\phi }}(z|x)\\Vert p_{{\\theta }}(z|x))=h(q_{{\\phi }}(z|x))+\\mathbb {E}_{q(x)}\\mathbb {E}_{q_{{\\phi }}(z|x)}\\log p_{{\\theta }}(z|x)$ , the model encourages $\\mathcal {S}_{q_{{\\phi }}(z|x)}\\subset \\mathcal {S}_{p_{{\\theta }}(z|x)}$ .",
"From $-\\mathbb {E}_{q_{{\\phi }}(z)}\\mbox{KL}(q_{{\\phi }}(x|z)\\Vert p_{{\\theta }}(x|z))=h(q_{{\\phi }}(x|z))+\\mathbb {E}_{q_{{\\phi }}(z)}\\mathbb {E}_{q_{{\\phi }}(x|z)}\\log p_{{\\theta }}(x|z)$ , the model also encourages $\\mathcal {S}_{q_{{\\phi }}(x|z)}\\subset \\mathcal {S}_{p_{{\\theta }}(x|z)}$ .",
"The differential entropies $h(q_{{\\phi }}(z|x))$ and $h(q_{{\\phi }}(x|z))$ encourage that $\\mathcal {S}_{q_{{\\phi }}(z|x)}\\cap \\mathcal {S}_{p_{{\\theta }}(z|x)}$ and $\\mathcal {S}_{q_{{\\phi }}(x|z)}\\cap \\mathcal {S}_{p_{{\\theta }}(x|z)}$ be as large as possible.",
"Since $\\mathcal {S}_{q_{{\\phi }}(z|x)}\\subset \\mathcal {S}_{p_{{\\theta }}(z|x)}$ , it is anticipated that $q_{{\\phi }}(z|x)$ will under-estimate the variance of $p_{{\\theta }}(x|z)$ , as is common with the variational approximation to the posterior [3]."
],
[
"Symmetric KL divergence",
"Consider the new variational expression $\\mathcal {L}_z({\\theta },{\\phi })&=&\\mathbb {E}_{p(z)}\\mathbb {E}_{p_{{\\theta }}(x|z)} \\log \\big [\\frac{q_{{\\phi }}(z|x)q(x)}{p_{{\\theta }}(x|z)}\\big ]\\\\&=&-\\mbox{KL}(p_{{\\theta }}(x,z)\\Vert q_{{\\phi }}(x,z))+C_z$ where $C_z=-h(p(z))$ .",
"Using logic analogous to that applied to $\\mathcal {L}_x$ , maximization of $\\mathcal {L}_z$ encourages distribution supports reflected in Fig.",
"REF .",
"Defining $\\mathcal {L}_{xz}({\\theta },{\\phi })=\\mathcal {L}_x({\\theta },{\\phi })+\\mathcal {L}_z({\\theta },{\\phi })$ , we have $\\mathcal {L}_{xz}({\\theta },{\\phi })=-\\mbox{KL}_s(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(x,z))+K$ where $K=C_x+C_z$ , and the symmetric KL divergence is $\\mbox{KL}_s(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(x,z))\\triangleq \\mbox{KL}(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(x,z))+\\mbox{KL}(p_{{\\theta }}(x,z)\\Vert q_{{\\phi }}(x,z))$ .",
"Maximization of $\\mathcal {L}_{xz}({\\theta },{\\phi })$ seeks minimizing $\\mbox{KL}_s(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(x,z))$ , which simultaneously imposes the conditions summarized in Figs.",
"REF and REF .",
"One may show that $&&\\mbox{KL}_s(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(x,z))=\\mathbb {E}_{p(z)}\\mbox{KL}(p_{{\\theta }}(x|z)\\Vert q_{{\\phi }}(x|z))\\nonumber \\\\&&~+\\mathbb {E}_{q_{{\\phi }}(z)}\\mbox{KL}(q_{{\\phi }}(x|z)\\Vert p_{{\\theta }}(x|z))+\\mbox{KL}_s(p(z)\\Vert q_{{\\phi }}(z))\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\\mathbb {E}_{p_{{\\theta }}(x)}\\mbox{KL}(p_{{\\theta }}(z|x)\\Vert q_{{\\phi }}(z|x))\\nonumber \\\\&&~+\\mathbb {E}_{q(x)}\\mbox{KL}(q_{{\\phi }}(z|x)\\Vert p_{{\\theta }}(z|x))+\\mbox{KL}_s(p_{{\\theta }}(x)\\Vert q(x))$ Considering the representation in (), the goal of small $\\mbox{KL}_s(p_{{\\theta }}(x)\\Vert q(x))$ encourages $\\mathcal {S}_{q(x)}\\subset \\mathcal {S}_{p_{{\\theta }}(x)}$ and $\\mathcal {S}_{p_{{\\theta }}(x)}\\subset \\mathcal {S}_{q(x)}$ , and hence that $\\mathcal {S}_{q(x)}=\\mathcal {S}_{p_{{{\\theta }}}(x)}$ .",
"Further, since $-\\mbox{KL}_s(p_{{\\theta }}(x)\\Vert q(x))=\\mathbb {E}_{q(x)}\\log p_{{\\theta }}(x)+\\mathbb {E}_{p_{{\\theta }}(x)}\\log q(x)+h(p_{{\\theta }}(x))-C_x$ , maximization of $-\\mbox{KL}_s(p_{{\\theta }}(x)\\Vert q(x))$ seeks to minimize the cross-entropy between $q(x)$ and $p_{{\\theta }}(x)$ , encouraging a complete matching of the distributions $q(x)$ and $p_{{\\theta }}(x)$ , not just shared support.",
"From (REF ), a match is simultaneously encouraged between $p(z)$ and $q_{{\\phi }}(z)$ .",
"Further, the respective conditional distributions are also encouraged to match."
],
[
"Adversarial solution",
"Assuming fixed $({\\theta },{\\phi })$ , and using logic analogous to Proposition 1 in [17], we consider $g({\\psi })&=&\\mathbb {E}_{p_{{\\phi }}(x,z)} \\log (1-\\sigma (f_{{\\psi }}(x,z))\\nonumber \\\\ &+&\\mathbb {E}_{p_{{\\theta }}(x,z)} \\log \\sigma (f_{{\\psi }}(x,z)) $ where $\\sigma (\\zeta )=1/(1+\\exp (-\\zeta ))$ .",
"The scalar function $f_{{\\psi }}(x,z)$ is represented by a deep neural network with parameters ${\\psi }$ , and network inputs $(x,z)$ .",
"For fixed $({\\theta },{\\phi })$ , the parameters ${\\psi }^*$ that maximize $g({\\psi })$ yield $f_{{\\psi }^*}(x,z)=\\log p_{{\\theta }}(x,z)-\\log q_{{\\phi }}(x,z)$ and hence $\\mathcal {L}_x({\\theta },{\\phi })&=&\\mathbb {E}_{q_{{\\phi }}(x,z)} f_{{\\psi }^*}(x,z)+C_x\\\\\\mathcal {L}_z({\\theta },{\\phi })&=&-\\mathbb {E}_{p_{{\\theta }}(x,z)} f_{{\\psi }^*}(x,z)+C_z$ Hence, to optimize $\\mathcal {L}_{xz}({\\theta },{\\phi })$ we consider the cost function $\\ell ({\\theta },{\\phi };{\\psi }^*)&=&\\mathbb {E}_{q_{{\\phi }}(x,z)} f_{{\\psi }^*}(x,z)\\nonumber \\\\&&~-\\mathbb {E}_{p_{{\\theta }}(x,z)} f_{{\\psi }^*}(x,z)$ Assuming (REF ) holds, we have $\\ell ({\\theta },{\\phi };{\\psi }^*)=-\\mbox{KL}_s(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(x,z))\\le 0$ and the goal is to achieve $\\ell ({\\theta },{\\phi };{\\psi }^*)=0$ through joint optimization of $({\\theta },{\\phi };{\\psi }^*)$ .",
"Model learning consists of alternating between (REF ) and (REF ), maximizing (REF ) wrt ${\\psi }$ with $({\\theta },{\\phi })$ fixed, and maximizing (REF ) wrt $({\\theta },{\\phi })$ with ${\\psi }$ fixed.",
"The expectations in (REF ) and (REF ) are approximated by averaging over samples, and therefore to implement this solution we need only be able to sample from $p_{{\\theta }}(x|z)$ and $q_{{\\phi }}(z|x)$ , and we do not require explicit forms for these distributions.",
"For example, a draw from $q_{{\\phi }}(z|x)$ may be constituted as $z=h_{{\\phi }}(x,{\\epsilon })$ , where $h_{{\\phi }}(x,{\\epsilon })$ is implemented as a neural network with parameters ${\\phi }$ and ${\\epsilon }\\sim \\mathcal {N}(0,{\\bf I})$ ."
],
[
"Interpretation in terms of LRT statistic",
"In (REF ) a classifier is designed to distinguish between samples $(x,z)$ drawn from $p_{{\\theta }}(x,z)=p(z)p_{{\\theta }}(x|z)$ and from $q_{{\\phi }}(x,z)=q(x)q_{{\\phi }}(z|x)$ .",
"Implicit in that expression is that there is equal probability that either of these distributions are selected for drawing $(x,z)$ , i.e., that $(x,z)\\sim [p_{{\\theta }}(x,z)+q_{{\\phi }}(x,z)]/2$ .",
"Under this assumption, given observed $(x,z)$ , the probability of it being drawn from $p_{{\\theta }}(x,z)$ is $p_{{\\theta }}(x,z)/(p_{{\\theta }}(x,z)+q_{{\\phi }}(x,z))$ , and the probability of it being drawn from $q_{{\\phi }}(x,z)$ is $q_{{\\phi }}(x,z)/(p_{{\\theta }}(x,z)+q_{{\\phi }}(x,z))$ [9].",
"Since the denominator $p_{{\\theta }}(x,z)+q_{{\\phi }}(x,z)$ is shared by these distributions, and assuming function $p_{{\\theta }}(x,z)/q_{{\\phi }}(x,z)$ is known, an observed $(x,z)$ is inferred as being drawn from the underlying distributions as $\\mbox{if}~p_{{\\theta }}(x,z)/q_{{\\phi }}(x,z)>1,~~(x,z)\\rightarrow p_{{\\theta }}(x,z)\\\\\\mbox{if}~p_{{\\theta }}(x,z)/q_{{\\phi }}(x,z)<1,~~(x,z)\\rightarrow q_{{\\phi }}(x,z)$ This is the well-known likelihood ratio test (LRT) [25], and is reflected by (REF ).",
"We have therefore derived a learning procedure based on the log-LRT, as reflected in (REF ).",
"The solution is “adversarial,” in the sense that when optimizing $({\\theta },{\\phi })$ the objective in (REF ) seeks to “fool” the LRT test statistic, while for fixed $({\\theta },{\\phi })$ maximization of (REF ) wrt ${\\psi }$ corresponds to updating the LRT.",
"This adversarial solution comes as a natural consequence of symmetrizing the traditional VAE learning procedure."
],
[
"Adversarially Learned Inference",
"The adversarially learned inference (ALI) [5] framework seeks to learn both an encoder and decoder, like the approach proposed above, and is based on optimizing $(\\hat{{\\theta }},\\hat{{\\phi }})&=&\\mbox{argmin}_{{\\theta },{\\phi }}\\max _{{\\psi }}\\lbrace \\mathbb {E}_{p_{{\\theta }}(x,z)}\\log \\sigma (f_{{\\psi }}(x,z))\\nonumber \\\\&~&+ \\mathbb {E}_{q_{{\\phi }}(x,z)}\\log (1-\\sigma (f_{{\\psi }}(x,z)))\\rbrace $ This has similarities to the proposed approach, in that the term $\\max _{{\\psi }}\\mathbb {E}_{p_{{\\theta }}(x,z)}\\log \\sigma (f_{{\\psi }}(x,z))+ \\mathbb {E}_{q_{{\\phi }}(x,z)}\\log (1-\\sigma (f_{{\\psi }}(x,z)))$ is identical to our maximization of (REF ) wrt ${\\psi }$ .",
"However, in the proposed approach, rather than directly then optimizing wrt $({\\theta },{\\phi })$ , as in (REF ), in (REF ) the result from this term is used to define $f_{{\\psi }^*}(x,z)$ , which is then employed in (REF ) to subsequently optimize over $({\\theta },{\\phi })$ .",
"Note that $\\log \\sigma (\\cdot )$ is a monotonically increasing function, and therefore we may replace (REF ) as $\\ell ^\\prime ({\\theta },{\\phi };{\\psi }^*)&=&\\mathbb {E}_{q_{{\\phi }}(x,z)} \\log \\sigma (f_{{\\psi }^*}(x,z))\\nonumber \\\\&&+ \\mathbb {E}_{p_{{\\theta }}(x,z)} \\log \\sigma (-f_{{\\psi }^*}(x,z))$ and note $\\sigma (-f_{{\\psi }^*}(x,z;{\\theta },{\\phi }))=1-\\sigma (f_{{\\psi }^*}(x,z;{\\theta },{\\phi }))$ .",
"Maximizing (REF ) wrt $({\\theta },{\\phi })$ with fixed ${\\psi }^*$ corresponds to the minimization wrt $({\\theta },{\\phi })$ reflected in (REF ).",
"Hence, the proposed approach is exactly ALI, if in (REF ) we replace $\\pm f_{{\\psi }^*}$ with $\\log \\sigma (\\pm f_{{\\psi }^*})$ ."
],
[
"Original GAN",
"The proposed approach assumed both a decoder $p_{{\\theta }}(x|z)$ and an encoder $p_{{\\phi }}(z|x)$ , and we considered the symmetric $\\mbox{KL}_s(q_{{\\phi }}(x,z)\\Vert p_{{\\theta }}(x,z))$ .",
"We now simplify the model for the case in which we only have a decoder, and the synthesized data are drawn $x\\sim p_{{\\theta }}(x|z)$ with $z\\sim p(z)$ , and we wish to learn ${\\theta }$ such that data synthesized in this manner match observed data $x\\sim q(x)$ .",
"Consider the symmetric $\\mbox{KL}_s(q(x)\\Vert p_{{\\theta }}(x))&=&\\mathbb {E}_{p(z)}\\mathbb {E}_{p_{{\\theta }}(x|z)}f_{{\\psi }^*}(x)\\nonumber \\\\&&-\\mathbb {E}_{q(x)}f_{{\\psi }^*}(x)$ where for fixed ${\\theta }$ $f_{{\\psi }^*}(x)=\\log (p_{{\\theta }}(x)/q(x))$ We consider a simplified form of (REF ), specifically $g({\\psi })&=&\\mathbb {E}_{p(z)}\\mathbb {E}_{p_{{\\theta }}(x|z)} \\log \\sigma (f_{{\\psi }}(x))\\nonumber \\\\&~&+\\mathbb {E}_{q(x)} \\log (1-\\sigma (f_{{\\psi }}(x))$ which we seek to maximize wrt ${\\psi }$ with fixed ${\\theta }$ , with optimal solution as in (REF ).",
"We optimize ${\\theta }$ seeking to maximize $-\\mbox{KL}_s(q(x)\\Vert p_{{\\theta }}(x))$ , as $\\mbox{argmax}_{{\\theta }}~\\ell ({\\theta };{\\psi }^*)$ where $\\ell ({\\theta };{\\psi }^*)=\\mathbb {E}_{q(x)} f_{{\\psi }^*}(x)- \\mathbb {E}_{p_{{\\theta }}(x,z)} f_{{\\psi }^*}(x) $ with $\\mathbb {E}_{q(x)} f_{{\\psi }^*}(x)$ independent of the update parameter ${\\theta }$ .",
"We observe that in seeking to maximize $\\ell ({\\theta };{\\psi }^*)$ , parameters ${\\theta }$ are updated as to “fool” the log-LRT $\\log [{q(x)}/{p_{{\\theta }}(x)}]$ .",
"Learning consists of iteratively updating ${\\psi }$ by maximizing $g({\\psi })$ and updating ${\\theta }$ by maximizing $\\ell ({\\theta };{\\psi }^*)$ .",
"Recall that $\\log \\sigma (\\cdot )$ is a monotonically increasing function, and therefore we may replace (REF ) as $\\ell ^\\prime ({\\theta };{\\psi }^*)=\\mathbb {E}_{p_{{\\theta }}(x,z)} \\log \\sigma (-f_{{\\psi }^*}(x))$ Using the same logic as discussed above in the context of ALI, maximizing $\\ell ^\\prime ({\\theta };{\\psi }^*)$ wrt ${\\theta }$ may be replaced by minimization, by transforming $\\sigma (\\mu )\\rightarrow \\sigma (-\\mu )$ .",
"With this simple modification, minimizing the modified (REF ) wrt ${\\theta }$ and maximizing (REF ) wrt ${\\psi }$ , we exactly recover the original GAN [9], for the special (but common) case of a sigmoidal discriminator."
],
[
"Wasserstein GAN",
"The Wasserstein GAN (WGAN) [2] setup is represented as ${{\\theta }}=\\mbox{argmin}_{{\\theta }}\\max _{{\\psi }}\\lbrace \\mathbb {E}_{q(x)} f_{{\\psi }}(x)-\\mathbb {E}_{p_{{\\theta }}(x,z)} f_{{\\psi }}(x)\\rbrace $ where $f_{{\\psi }}(x)$ must be a 1-Lipschitz function.",
"Typically $f_{{\\psi }}(x)$ is represented by a neural network with parameters ${\\psi }$ , with parameter clipping or $\\ell _2$ regularization on the weights (to constrain the amplitude of $f_{{\\psi }}(x)$ ).",
"Note that WGAN is closely related to (REF ), but in WGAN $f_{{\\psi }}(x)$ doesn't make an explicit connection to the underlying likelihood ratio, as in (REF ).",
"It is believed that the current paper is the first to consider symmetric variational learning, introducing $\\mathcal {L}_z$ , from which we have made explicit connections to previously developed adversarial-learning methods.",
"Previous efforts have been made to match $q_{{\\phi }}(z)$ to $p(z)$ , which is a consequence of the proposed symmetric VAE (sVAE).",
"For example, [16] introduced a modification to the original VAE formulation, but it loses connection to the variational lower bound [17]."
],
[
"Amelioration of vanishing gradients",
"As discussed in [2], a key distinction between the WGAN framework in (REF ) and the original GAN [9] is that the latter uses a binary discriminator to distinguish real and synthesized data; the $f_{{\\psi }}(x)$ in WGAN is a 1-Lipschitz function, rather than an explicit discriminator.",
"A challenge with GAN is that as the discriminator gets better at distinguishing real and synthetic data, the gradients wrt the discriminator parameters vanish, and learning is undermined.",
"The WGAN was designed to ameliorate this problem [2].",
"From the discussion in Section REF , we note that the key distinction between the proposed sVAE and ALI is that the latter uses a binary discriminator to distinguish $(x,z)$ manifested via the generator from $(x,z)$ manifested via the encoder.",
"By contrast, the sVAE uses a log-LRT, rather than a binary classifier, with it inferred in an adversarial manner.",
"ALI is therefore undermined by vanishing gradients as the binary discriminator gets better, with this avoided by sVAE.",
"The sVAE brings the same intuition associated with WGAN (addressing vanishing gradients) to a generalized VAE framework, with a generator and a decoder; WGAN only considers a generator.",
"Further, as discussed in Section REF , unlike WGAN, which requires gradient clipping or other forms of regularization to approximate 1-Lipschitz functions, in the proposed sVAE the $f_{{\\psi }}(x,z)$ arises naturally from the symmetrized VAE and we do not require imposition of Lipschitz conditions.",
"As discussed in Section , this simplification has yielded robustness in implementation."
],
[
"Model Augmentation ",
"A significant limitation of the original ALI setup is an inability to accurately reconstruct observed data via the process $x\\rightarrow z\\rightarrow \\hat{x}$ [5].",
"With the proposed sVAE, which is intimately connected to ALI, we may readily address this shortcoming.",
"The variational expressions discussed above may be written as $\\mathcal {L}_x=\\mathbb {E}_{q_{{\\phi }}(x,z)}\\log p_{{\\theta }}(x|z)-\\mathbb {E}_{q(x)}\\mbox{KL}(q_{{\\phi }}(z|x)\\Vert p(z))$ and $\\mathcal {L}_z=\\mathbb {E}_{p_{{\\theta }}(x,z)}\\log p_{{\\phi }}(z|x)-\\mathbb {E}_{p(z)}\\mbox{KL}(p_{{\\theta }}(x|z)\\Vert q(x))$ .",
"In both of these expressions, the first term to the right of the equality enforces model fit, and the second term penalizes the posterior distribution for individual data samples for being dissimilar from the prior (i.e., penalizes $q_{{\\phi }}(z|x)$ from being dissimilar from $p(z)$ , and likewise wrt $p_{{\\theta }}(x|z)$ and $q(x)$ ).",
"The proposed sVAE encourages the cumulative distributions $q_{{\\phi }}(z)$ and $p_{{\\theta }}(x)$ to match $p(z)$ and $q(x)$ , respectively.",
"By simultaneously encouraging more peaked $q_{{\\phi }}(z|x)$ and $p_{{\\theta }}(x|z)$ , we anticipate better “cycle consistency” [29] and hence more accurate reconstructions.",
"To encourage $q_{{\\phi }}(z|x)$ that are more peaked in the space of $z$ for individual $x$ , and also to consider more peaked $p_{{\\theta }}(x|z)$ , we may augment the variational expressions as $\\mathcal {L}_x^\\prime &=&(\\lambda +1)\\mathbb {E}_{q_{{\\phi }}(x,z)}\\log p_{{\\theta }}(x|z)\\nonumber \\\\&~&-\\mathbb {E}_{q(x)}\\mbox{KL}(q_{{\\phi }}(z|x)\\Vert p(z))\\\\\\mathcal {L}_z^\\prime &=&(\\lambda +1)\\mathbb {E}_{p_{{\\theta }}(x,z)}\\log p_{{\\phi }}(z|x)\\nonumber \\\\&~&-\\mathbb {E}_{p(z)}\\mbox{KL}(p_{{\\theta }}(x|z)\\Vert q(x)) $ where $\\lambda \\ge 0$ .",
"For $\\lambda =0$ the original variational expressions are retained, and for $\\lambda >0$ , $q_{{\\phi }}(z|x)$ and $p_{{\\theta }}(x|z)$ are allowed to diverge more from $p(z)$ and $q(x)$ , respectively, while placing more emphasis on the data-fit terms.",
"Defining $\\mathcal {L}_{xz}^\\prime =\\mathcal {L}_x^\\prime +\\mathcal {L}_z^\\prime $ , we have $\\mathcal {L}_{xz}^\\prime =\\mathcal {L}_{xz}&+&\\lambda [\\mathbb {E}_{q_{{\\phi }}(x,z)}\\log p_{{\\theta }}(x|z)\\nonumber \\\\&~&~+\\mathbb {E}_{p_{{\\theta }}(x,z)}\\log p_{{\\phi }}(z|x)]$ Model learning is the same as discussed in Sec.",
"REF , with the modification $\\ell ^\\prime ({\\theta },{\\phi };{\\psi }^*)=&\\mathbb {E}_{q_{{\\phi }}(x,z)} [f_{{\\psi }^*}(x,z)+\\lambda \\log p_{{\\theta }}(x|z)]\\nonumber \\\\-&\\mathbb {E}_{p_{{\\theta }}(x,z)} [f_{{\\psi }^*}(x,z)-\\lambda \\log p_{{\\phi }}(z|x)]$ A disadvantage of this approach is that it requires explicit forms for $p_{{\\theta }}(x|z)$ and $p_{{\\phi }}(z|x)$ , while the setup in Sec.",
"REF only requires the ability to sample from these distributions.",
"We can now make a connection to additional related work, particularly [19], which considered a similar setup to (REF ) and (), for the special case of $\\lambda =1$ .",
"While [19] had a similar idea of using a symmetrized VAE, they didn't make the theoretical justification presented in Section .",
"Further, and more importantly, the way in which learning was performed in [19] is distinct from that applied here, in that [19] required an additional adversarial learning step, increasing implementation complexity.",
"Consequently, [19] did not use adversarial learning to approximate the log-LRT, and therefore it cannot make the explicit connection to ALI and WGAN that were made in Sections REF and REF , respectively."
],
[
"Experiments",
"In addition to evaluating our model on a toy dataset, we consider MNIST, CelebA and CIFAR-10 for both reconstruction and generation tasks.",
"As done for the model ALI with Cross Entropy regularization (ALICE) [15], we also add the augmentation term ($\\lambda > 0$ as discussed in Sec. )",
"to sVAE as a regularizer, and denote the new model as sVAE-r. More specifically, we show the results based on the two models: $i$ ) sVAE: the model is developed in Sec.",
"to optimize $g({\\psi })$ in (REF ) and $\\ell ({\\theta },{\\phi };{\\psi }^*)$ in (REF ).",
"$ii$ ) sVAE-r: the model is sVAE with regularization term to optimize $g({\\psi })$ in (REF ) and $\\ell ^\\prime ({\\theta },{\\phi };{\\psi }^*)$ in (REF ).",
"The quantitative evaluation is based on the mean square error (MSE) of reconstructions, log-likelihood calculated via the annealed importance sampling (AIS) [27], and inception score (IS) [22].",
"All parameters are initialized with Xavier [8] and optimized using Adam [13] with learning rate of 0.0001.",
"No dataset-specific tuning or regularization, other than dropout [23], is performed.",
"The architectures for the encoder, decoder and discriminator are detailed in the Appendix.",
"All experimental results were performed on a single NVIDIA TITAN X GPU.",
"Figure: sVAE results on toy dataset.",
"Top: Inception Score for ALI and sVAE with λ=0,0.01,0.1\\lambda = 0, 0.01, 0.1.",
"Bottom: Mean Squared Error (MSE)."
],
[
"Toy Data",
"In order to show the robustness and stability of our model, we test sVAE and sVAE-r on a toy dataset designed in the same manner as the one in ALICE [15].",
"In this dataset, the true distribution of data $x$ is a two-dimensional Gaussian mixture model with five components.",
"The latent code $z$ is a standard Gaussian distribution $\\mathcal {N}(0,1)$ .",
"To perform the test, we consider using different values of $\\lambda $ for both sVAE-r and ALICE.",
"For each $\\lambda $ , 576 experiments with different choices of architecture and hyper-parameters are conducted.",
"In all experiments, we use mean square error (MSE) and inception score (IS) to evaluate the performance of the two models.",
"Figure REF shows the histogram results for each model.",
"As we can see, both ALICE and sVAE-r are able to reconstruct images when $\\lambda = 0.1$ , while sVAE-r provides better overall inception score.",
"Figure: sVAE results on MNIST.",
"(a) and (b) are generated sample images by sVAE and sVAE-r, respectively.",
"(c) is reconstructed images by sVAE-r: in each block, column one is ground-truth and column two is reconstructed images.",
"Note that λ\\lambda is set to 0.10.1 for sVAE-r."
],
[
"MNIST",
"The results of image generation and reconstruction for sVAE, as applied to the MNIST dataset, are shown in Figure REF .",
"By adding the regularization term, sVAE overcomes the limitation of image reconstruction in ALI.",
"The log-likelihood of sVAE shown in Table REF is calculated using the annealed importance sampling method on the binarized MNIST dataset, as proposed in [27].",
"Note that in order to compare the model performance on binarized data, the output of the decoder is considered as a Bernoulli distribution instead of the Gaussian approach from the original paper.",
"Our model achieves -79.26 nats, outperforming normalizing flow (-85.1 nats) while also being competitive to the state-of-the-art result (-79.2 nats).",
"In addition, sVAE is able to provide compelling generated images, outperforming GAN [9] and WGAN-GP [11] based on the inception scores.",
"Table: Quantitative Results on MNIST.",
"†\\dagger is calculated using AIS.",
"‡\\ddagger is reported in .Figure: CelebA reconstruction results.",
"Left column: The ground truth.",
"Middle block: sVAE-r reconstruction.",
"Right block: ALICE reconstruction.",
"λ=0,0.1,1\\lambda =0,0.1,1 and 10 from left to right in each block."
],
[
"CelebA ",
"We evaluate sVAE on the CelebA dataset and compare the results with ALI.",
"In experiments we note that for high-dimensional data like the CelebA, ALICE [15] shows a trade-off between reconstruction and generation, while sVAE-r does not have this issue.",
"If the regularization term is not included in ALI, the reconstructed images do not match the original images.",
"On the other hand, when the regularization term is added, ALI is capable of reconstructing images but the generated images are flawed.",
"In comparison, sVAE-r does well in both generation and reconstruction with different values of $\\lambda $ .",
"The results for both sVAE and ALI are shown in Figure REF and REF .",
"Generally speaking, adding the augmentation term as shown in (REF ) should encourage more peaked $q_{{\\phi }}(z|x)$ and $p_{{\\theta }}(x|z)$ .",
"Nevertheless, ALICE fails in the inference process and performs more like an autoencoder.",
"This is due to the fact that the discriminator becomes too sensitive to the regularization term.",
"On the other hand, by using the symmetric KL (REF ) as the cost function, we are able to alleviate this issue, which makes sVAE-r a more stable model than ALICE.",
"This is because sVAE updates the generator using the discriminator output, before the sigmoid, a non-linear transformation on the discriminator output scale.",
"Figure: sVAE-r and ALICE CIFAR quantitative evaluation with different values of λ\\lambda .",
"Left: IS for generation; Right: MSE for reconstruction.",
"The result is the average of multiple tests."
],
[
"CIFAR-10",
"The trade-off of ALICE [15] mentioned in Sec.",
"REF is also manifested in the results for the CIFAR-10 dataset.",
"In Figure REF , we show quantitative results in terms of inception score and mean squared error of sVAE-r and ALICE with different values of $\\lambda $ .",
"As can be seen, both models are able to reconstruct images when $\\lambda $ increases.",
"However, when $\\lambda $ is larger than $10^{-3}$ , we observe a decrease in the inception score of ALICE, in which the model fails to generate images.",
"Table: Unsupervised Inception Score on CIFAR-10The CIFAR-10 dataset is also used to evaluate the generation ability of our model.",
"The quantitative results, i.e., the inception scores, are listed in Table REF .",
"Our model shows improved performance on image generation compared to ALI and DCGAN.",
"Note that sVAE also gets comparable result as WGAN-GP [11] achieves.",
"This can be interpreted using the similarity between (REF ) and (REF ) as summarized in the Sec.",
".",
"The generated images are shown in Figure REF .",
"More results are in the Appendix.",
"Figure: sVAE CIFAR results on image generation and reconstruction."
],
[
"Conclusions",
"We present the symmetric variational autoencoder (sVAE), a novel framework which can match the joint distribution of data and latent code using the symmetric Kullback-Leibler divergence.",
"The experiment results show the advantages of sVAE, in which it not only overcomes the missing mode problem [10], but also is very stable to train.",
"With excellent performance in image generation and reconstruction, we will apply sVAE on semi-supervised learning tasks and conditional generation tasks in future work.",
"Morever, because the latent code $z$ can be treated as data from a different domain, i.e., images [29], [12] or text [6], we can also apply sVAE to domain transfer tasks.",
"Appendix"
]
] | 1709.01846 |
[
[
"The 2-Hessian and sextactic points on plane algebraic curves"
],
[
"Abstract In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated 2-Hessian curve that intersects it in its sextactic points.",
"In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the 2-Hessian.",
"In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the 2-Hessian.",
"Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve."
],
[
"Introduction",
"Let $C=V(F)$ be an algebraic curve of geometric genus $g$ and degree $d$ , given by a polynomial $F \\in x,y,z]_d$ , in the projective plane $\\mathbb {P}^2$ over $.",
"In standard terms, a point $ p$ on an irreducible curve $ C$ is called singular if all the partial derivaties of $ F$ vanish, and smooth otherwise.",
"Given two curves $ C$ and $ C'$ and a point $ p$ in the intersection, let $ (C C')p$ denote the intersection multiplicity of $ C$ and $ C'$ at $ p$.",
"Moreover, for any point $ p C$, let $ mp$ denote its multiplicity, i.e.",
"the intersection multiplicity at $ p$ of $ C$ and a generic line.$ Now, given an irreducible curve $C$ and fixed $n \\in \\mathbb {N}$ , consider curves, not necessarily irreducible, of degree $n$ in $\\mathbb {P}^2$ .",
"With $r(n)=\\frac{1}{2}n(n+3)$ and $nd \\ge r(n)$ , for every smooth point $p \\in C$ there exists a curve of degree $n$ such that the local intersection multiplicity is equal to or bigger than $r(n)$ , referred to as an osculating curve to $C$ at $p$ , see [1].",
"A smooth point where the intersection multiplicity between a curve of degree $n$ and $C$ is strictly bigger than $r(n)$ is referred to as an $n$ -Weierstrass point.",
"In this case, the curve of degree $n$ is called a hyperosculating curve.",
"For $n=1$ this comes down to tangent lines to $C$ , for each smooth point $p$ denoted by $T_p$ and given by the linear polynomial $xF_x(p)+yF_y(p)+zF_z(p)$ , with the property that $l_p=(T_p \\cdot C)_p\\ge 2$ .",
"Thus, the smooth 1-Weierstrass points on $C$ , for which $l_p>2$ , are nothing but the inflection points.",
"In particular, if $l_p=3$ , $p$ is called a simple inflection point, and, in general, the order of an inflection point is given by $v_p=l_p-2$ .",
"The main focus of this paper is the case $n=2$ .",
"For every smooth point $p$ on a curve $C$ of degree $d \\ge 3$ there exists a unique osculating curve of degree 2, denoted by $O_p$ , with $c_p=(O_p \\cdot C)_p\\ge 5$ [4].",
"In particular, we look at 2-Weierstrass points, where $c_p>5$ , which includes inflection points and sextactic points, first studied for curves of arbitrary high degree by Cayley in [5].",
"In a broader context, Weierstrass points of curves in any projective space with respect to a linear system $Q$ have been intensively studied, both classically in the case of smooth curves, and more recently for singular curves [2], [9], [15], [16], [24], [25].",
"One consequence of this research is that for a singular plane curve and a linear system $Q$ , the $Q$ -Weierstrass points include both the smooth $Q$ -Weierstrass points and the singular points [2].",
"Back in $\\mathbb {P}^2$ , in the case $n=1$ it is well known that the Hessian curve to $C$ , of degree $3(d-2)$ , given by the polynomial $H=H_1(F)=\\begin{vmatrix}F_{xx} & F_{xy} & F_{xz} \\\\F_{yx} & F_{yy} & F_{yz} \\\\F_{zx} & F_{zy} & F_{zz} \\\\\\end{vmatrix},$ intersects $C$ in its inflection points and singular points, i.e.",
"its 1-Weierstrass points.",
"In [5] Cayley presents a curve with similar properties: a curve of degree $12d-27$ that intersects $C$ in its sextactic points, higher order inflection points, and its singular points.",
"The first main result of this article is a correction of Cayley's defining polynomial for such a curve, referred to as the 2-Hessian to $C$ .",
"See sec:2hessian for notation.",
"[The 2-Hessian] Let $C=V(F)$ be a plane curve of degree $d \\ge 3$ , with $H$ the defining polynomial of the Hessian curve to $C$ .",
"Then there exists a curve of degree $12d-27$ given as the zero set of $H_2=H_2(F) = & \\; (12d^2-54d+57)H \\operatorname{Jac}(F,H,\\Omega _{\\bar{H}})\\\\& \\;+ (d-2)(12d-27)H \\operatorname{Jac}(F,H,\\Omega _{\\bar{F}}) \\\\& \\;-\\mathbf {20}(d-2)^2 \\operatorname{Jac}(F,H,\\Psi ),$ such that the intersection points between $C$ and this curve are the singular points, the higher order inflection points, and the sextactic points of $C$ .",
"Abusing notation we refer to both the Hessian curve and its defining polynomial as $H$ , and similarly we write $H_2$ for both the 2-Hessian curve and its defining polynomial.",
"Observe that the curve $V(H \\cdot H_2)$ intersects $C$ in its 2-Weierstrass points.",
"The situation is more complex for $n \\ge 3$ and for curves in higher dimensional spaces.",
"We refer to [8] for modern results on higher Hessians to smooth plane curves and, in higher dimensional spaces, generalized Hessians to smooth curves that are complete intersections.",
"To complete the picture for $n=2$ , we turn to another angle.",
"Instead of studying sextactic points on a curve using its defining polynomial and the 2-Hessian, we count the number of sextactic points using simple invariants of the curve, its inflection points, and its singular points.",
"Recall that singular points on plane curves exist in many different shapes.",
"Moreover, a singular point can be described by its multiplicity, delta invariant and number of branches.",
"In particular, a singular point is called a cusp if it is unibranched, and up to topological type, a cusp $p$ can be described by its multiplicity sequence, $\\overline{m}_p$ , the ordered sequence of multiplicities of the points above $p$ in the partial minimal embedded resolution of $C$ at $p$ , see [3].",
"Curves with only cusp singularities, cuspidal curves, have been thoroughly studied the last 25 years, see [20], [21] for an overview.",
"The research has been motivated by both classification problems and the connections such curves have to problems in the theory of open surfaces.",
"We mention briefly the Flenner–Zaidenberg rigidity conjecture, the Orevkov–Piontkowski conjecture, and the recently proved Coolidge–Nagata conjecture [14].",
"The second main result in this article is a formula for the number of sextactic points on cuspidal curves, for which the invariants involved are easily accessible.",
"Indeed, for a cusp $p$ with multiplicity $m_p$ , there exists a unique line $T_p$ such that $l_p=(T_p \\cdot C)_p>m_p$ , referred to as the tangent line to $C$ at $p$ .",
"Similarly, applicable only to cusps where $l_p=2m_p$ , there exists a, not necessarily unique, conic $O_p$ with $c_p=(O_p \\cdot C)_p>2m_p$ and $c_p \\ne 3m_p,4m_p$ , referred to as an osculating conic to $C$ at $p$ , see lem:oscint.",
"[Sextactic point formula] Let $C$ be a cuspidal curve of genus $g$ and degree $d \\ge 3$ .",
"Let $I$ denote the set of inflection points and cusps on $C$ where $l_p \\ne 2m_p$ , and let $J$ denote the set of cusps on $C$ where $l_p=2m_p$ .",
"Then the number of sextactic points $s$ on $C$ , counted with multiplicity, is given by $s = 6(2d + 5g - 5) - \\sum _I (4m_p + 4l_p - 15)-\\sum _J(10m_p+c_p-15).$ Although the formula in thm:SPF is a new restriction for cuspidal curves, it is superseded by other well known results.",
"The formula in thm:SPF has a classical touch and resembles the Plücker formula for inflection points; see [3] for a formula for curves with any kind of singularities, attributed to Weierstrass and Noether.",
"Not surprisingly, there exist similar formulas for the number of sextacic points on plane curves, subject to different restrictions, in the literature.",
"A formula for smooth curves is explicitly stated by Thorbergsson and Umehara in [28], and a formula for curves with certain singularties is given by Coolidge, attributed to Cayley, in [7].",
"Both of these formulas coincide with the formula in thm:SPF under the appropriate restrictions and generalizations.",
"Moreover, as briefly commented by Coolidge in [7], it is possible to find a formula for the number of sextactic points valid for any plane curve.",
"Indeed, the formula in thm:SPF can be directly extended to curves with multibranched singularities by dealing with one branch at the time.",
"We do not state this formula explicitly, but we discuss some of the necessary generalizations in rem:gensing2.",
"Additionally, note that a method to compute the number of sextactic points on a plane cuve with any kind of singularities has been presented, with a very similar approach to ours, by Perkinson in [24], and that a formula would also follow from such calculations.",
"For higher $n$ , a formula for the number of $n$ -Weierstrass points would require a stratification of the Weierstrass points and special treatments of each subgroup, as in the case of $I$ and $J$ in thm:SPF.",
"As described in [24], this would require close inspection of many points on the curve.",
"Hence, a precise formulation of such a general formula is not within the scope of this article.",
"This article has the following structure.",
"In sec:2hessian we fix Cayley's polynomial for the 2-Hessian to the one given in thm:2hessian, and we explore some curves and their sextactic points using this tool.",
"In sec:sextactic we prove the formula from thm:SPF for the number of sextactic points on cuspidal curves, and we apply this formula to examples.",
"Moreover, we derive a corollary which ties the sextactic point formula to the 2-Hessian.",
"In sec:rational we take a closer look at rational curves.",
"In this case, the osculating conic to a curve at a smooth point can be calculated directly from the parametrization.",
"Moreover, we show that the zeros of the Wronski determinant of the 2nd Veronese embedding of the curve correspond to its 2-Weierstrass points.",
"The figures in this article are made with GeoGebra [13], computations are done with Maple [17], and program code can be found in [18]."
],
[
"The 2-Hessian curve",
"A fair bit of notation is necessary to present the defining polynomial of the 2-Hessian to a curve.",
"For the convenience of the reader, we stay close to Cayley's original formulations from [4], [5] and begin this section by recalling the essential objects.",
"With $C$ and $F$ as before, and $p$ a point on the curve, let $DF_p(x,y,z) & = x F_x(p) + y F_y(p) + z F_z(p), \\\\D^2F_p(x,y,z) & = x^2 F_{xx}(p) + y^2 F_{yy}(p) + z^2 F_{zz}(p) \\\\ & \\quad + 2xy F_{xy}(p) + 2xz F_{xz}(p) + 2yz F_{yz}(p).$ Moreover, with $\\operatorname{Hess}(F)$ denoting the Hessian matrix of $F$ , write $\\operatorname{Hess}(F)=\\begin{bmatrix}a & h & g \\\\h & b & f \\\\g & f & c \\\\\\end{bmatrix},$ where as usual $a = F_{xx},\\; b = F_{yy},\\; c = F_{zz},\\; f = F_{yz},\\; g = F_{xz},\\; h = F_{xy},$ and $H=\\det \\operatorname{Hess}(F)$ .",
"Similarly, for the Hessian matrix of $H$ , write $\\operatorname{Hess}(H)=\\begin{bmatrix}a^{\\prime } & h^{\\prime } & g^{\\prime } \\\\h^{\\prime } & b^{\\prime } & f^{\\prime } \\\\g^{\\prime } & f^{\\prime } & c^{\\prime } \\\\\\end{bmatrix},$ so that the elements denote the second order partial derivatives of $H$ , $a^{\\prime } = H_{xx},\\; b^{\\prime } = H_{yy},\\; c^{\\prime } = H_{zz},\\; f^{\\prime } = H_{yz},\\; g^{\\prime } = H_{xz},\\; h^{\\prime } = H_{xy}.$ Following the pattern, the elements of the adjoint matrix $\\operatorname{Hess}(F)^{\\operatorname{adj}}$ are assigned the notation $\\operatorname{Hess}(F)^{\\operatorname{adj}}=\\begin{bmatrix}\\mathcal {A}& \\mathcal {H}& \\mathcal {G}\\\\\\mathcal {H}& \\mathcal {B}& \\mathcal {F}\\\\\\mathcal {G}& \\mathcal {F}& \\mathcal {C}\\\\\\end{bmatrix},$ where $\\begin{array}{c}\\mathcal {A}= bc-f^2,\\quad \\mathcal {B}= ac-g^2,\\quad \\mathcal {C}= ab - h^2, \\\\ \\mathcal {F}= hg-af,\\quad \\mathcal {G}= hf-bg,\\quad \\mathcal {H}= fg-hc.\\end{array}$ Furthermore, put $\\Omega & = (\\mathcal {A},\\mathcal {B},\\mathcal {C},\\mathcal {F},\\mathcal {G},\\mathcal {H}) \\cdot (a^{\\prime },b^{\\prime },c^{\\prime },2f^{\\prime },2g^{\\prime },2h^{\\prime }),$ which also can be expressed as the trace of the product of the adjoint matrix of the Hessian matrix of $F$ and the Hessian matrix of $H$ : $\\Omega =\\operatorname{tr}\\left(\\operatorname{Hess}(F)^{\\operatorname{adj}} \\cdot \\operatorname{Hess}(H)\\right).$ Additionally, let $\\partial _x\\Omega _{\\bar{H}} & = (\\mathcal {A}_x,\\mathcal {B}_x,\\mathcal {C}_x,\\mathcal {F}_x,\\mathcal {G}_x,\\mathcal {H}_x)\\cdot (a^{\\prime },b^{\\prime },c^{\\prime },2f^{\\prime },2g^{\\prime },2h^{\\prime }), \\\\\\partial _x\\Omega _{\\bar{F}} & = (\\mathcal {A},\\mathcal {B},\\mathcal {C},\\mathcal {F},\\mathcal {G},\\mathcal {H})\\cdot (a^{\\prime }_x,b^{\\prime }_x,c^{\\prime }_x,2f^{\\prime }_x,2g^{\\prime }_x,2h^{\\prime }_x).$ Observe that the partial derivative of $\\Omega $ with respect to $x$ can be written $\\Omega _x = \\partial _x\\Omega _{\\bar{H}} +\\partial _x\\Omega _{\\bar{F}},$ and note that none of the terms on the right hand side is an actual derivative.",
"Naturally, similar expressions can be found for $\\Omega _y$ and $\\Omega _z$ by replacing $x$ with $y$ and $z$ in the above.",
"Thus, although they are not true Jacobi determinants, we use Cayley's notation and write $\\operatorname{Jac}(F,H,\\Omega _{\\bar{H}})=\\begin{vmatrix}F_x & F_y & F_z \\\\H_x & H_y & H_z \\\\\\partial _x \\Omega _{\\bar{H}} & \\partial _y \\Omega _{\\bar{H}} & \\partial _z \\Omega _{\\bar{H}} \\\\\\end{vmatrix}\\;$ and $\\operatorname{Jac}(F,H,\\Omega _{\\bar{F}})=\\begin{vmatrix}F_x & F_y & F_z \\\\H_x & H_y & H_z \\\\\\partial _x \\Omega _{\\bar{F}} & \\partial _y \\Omega _{\\bar{F}} & \\partial _z \\Omega _{\\bar{F}} \\\\\\end{vmatrix}.$ Moreover, put $\\Psi = (\\mathcal {A},\\mathcal {B},\\mathcal {C},\\mathcal {F},\\mathcal {G},\\mathcal {H}) \\cdot (H_x^2,H_y^2,H_z^2,2H_{y}H_{z},2H_{x}H_{z},2H_{x}H_{y}),$ or equivalently, in matrix form, $\\Psi & = -\\begin{vmatrix}0 & H_x & H_y & H_z \\\\H_x & a & h & g \\\\H_y & h & b & f \\\\H_z & g & f & c\\end{vmatrix}.$ Lastly, this time an actual Jacobi determinant, let $\\operatorname{Jac}(F,H,\\Psi )$ denote the determinant $\\operatorname{Jac}(F,H,\\Psi ) = \\begin{vmatrix}F_x & F_y & F_z \\\\H_x & H_y & H_z \\\\\\Psi _x & \\Psi _y & \\Psi _z \\\\\\end{vmatrix}.$"
],
[
"The osculating conic",
"For completion, before we move on to the 2-Hessian, we present the osculating conic to a curve at a smooth point, give a formal definition of a sextactic point, and present an example.",
"For any smooth point $p$ on a curve $C$ of degree $d \\ge 3$ there exists a unique osculating curve of degree 2.",
"In the case of inflection points, this curve is the double tangent line.",
"For a smooth point that is not an inflection point, the osculating conic was first presented by Cayley in [4].",
"[[4]] Let $C$ be a plane curve of degree $d \\ge 3$ given by a polynomial $F$ .",
"If $p$ is a point on $C$ that is neither singular nor an inflection point, then, with $9H^3\\Lambda = -3 \\Omega H + 4 \\Psi $ , the osculating conic $O_p$ to $C$ at $p$ is given by $D^2F_p - \\left(\\tfrac{2}{3} \\frac{1}{H(p)}{DH}_p + \\Lambda (p) DF_p\\right)DF_p=0.$ On a plane curve $C$ , a smooth point $p$ that is not an inflection point is called a sextactic point if $c_p=(O_p \\cdot C)_p \\ge 6.$ With $s_p = c_p - 5$ , a sextactic point $p$ is said to be of order $s_p$ , or $s_p$ -sextactic.",
"A first example of a sextactic point on a curve can be found on a cubic.",
"Let $C$ be the cubic curve with a nodal singularity given by $F=y^2z-x^3-x^2z.$ Pick a smooth point on $C$ , say $p_1=(\\tfrac{-4}{5} : \\tfrac{4}{5\\sqrt{5}} : 1)$ .",
"The osculating conic $O_{p_1}$ , for which $(O_{p_1} \\cdot C)_{p_1}=5$ , is shown in fig:osccon.",
"Its defining polynomial can be directly computed with the formula in thm:osccon and [18], $1125x^2+625y^2+64z^2+400\\sqrt{5}yz+1200xz+350\\sqrt{5}xy=0.$ Now, pick $p_2=(-1:0:1)$ .",
"The osculating conic $O_{p_2}$ can be computed as above, $2x^2+y^2+z^2+3xz=0.$ Note that $(O_{p_2} \\cdot C)_{p_2}=6$ , hence $p_2$ is a sextactic point and $O_{p_2}$ a hyperosculating conic, see fig:sextactic.",
"Figure: The nodal cubic from ex:nodalcubic with the osculating conic O p 1 O_{p_1} at the smooth point p 1 p_1.Figure: The nodal cubic from ex:nodalcubic with the hyperosculating conic O p 2 O_{p_2} at the sextactic point p 2 p_2.For a thorough investigation of a nodal cubic with respect to its sextactic points using elementary methods, we refer to the Appendix by Sakai in [27] by Tono.",
"Perhaps somewhat surprisingly, a sextactic point on such a curve plays an important role in the construction of a particularly interesting series of unicuspidal curves, first studied by Orevkov in [23].",
"From the perspective of abelian groups on curves, in ex:nodalcubic it can be observed directly that $p_2=(-1:0:1)$ is a sextactic point.",
"Indeed, the curve is given on Weierstrass form, and the vertical line $x+z=0$ is the tangent line to $C$ at $p_2$ , passing through the identity element and inflection point $(0:1:0)$ .",
"Thus, $p_2$ is a 2-torsion point for the abelian group structure on the smooth part of $C$ , and it follows that $p_2$ is a sextactic point."
],
[
"The correct 2-Hessian",
"An incorrect defining polynomial of the 2-Hessian to a curve $C$ is given by Cayley in [5]: $& (12d^2-54d+57)H \\operatorname{Jac}(F,H,\\Omega _{\\bar{H}})\\\\& + (d-2)(12d-27)H \\operatorname{Jac}(F,H,\\Omega _{\\bar{F}}) \\\\& - \\mathbf {40}(d-2)^2 \\operatorname{Jac}(F,H,\\Psi ).$ Cayley's proof of this result is based on restrictions that arise when more than five points in the intersection between a conic and the curve $C$ coalesce, and it is mostly correct.",
"There is, however, an elementary computational error in the proof that affects the coefficient of the last term and makes the polynomial invalid.",
"See thm:2hessian for the correct defining polynomial of the 2-Hessian to a curve.",
"The mistake is neither corrected in the version of the paper [5] published in The collected mathematical papers of Arthur Cayley [6], nor in later works that cite the result, see [8] and [26].",
"Cayley's mistake occurs in Section 19 on p. 553 in [5] as he attempts to simplify an expression that he has obtained for the 2-Hessian in Section 10 on p. 550.",
"In the simplification Cayley introduces, in Section 18 on pp.",
"552–553, an expression $W$ and correctly states that $W H\\partial \\Omega - 5\\Omega \\partial H = \\frac{-3}{4d-9}\\vartheta \\operatorname{Jac}(F,\\Omega ,H) - \\frac{5d-9}{4d-9}\\partial (\\Omega H),$ where $\\partial = (F_y\\nu -F_z\\mu )\\partial _x + (F_z\\lambda -F_x\\nu )\\partial _y + (F_x\\mu -F_y\\lambda )\\partial _z$ and $\\vartheta = \\lambda x+\\mu y+\\nu z$ , with $\\lambda ,\\mu $ , and $\\nu $ arbitrary constants.",
"Moreover, in Section 19, Cayley states that $\\Psi \\partial H = \\frac{{\\bf {\\frac{1}{2}}}}{4d-9} \\vartheta \\operatorname{Jac}(F,\\Psi ,H) + \\frac{\\frac{3}{2}(d-2)}{4d-9}H\\partial \\Psi ,$ and subsequently calculates $\\begin{split}9HW + {\\bf {40}}\\Psi \\partial H = & -\\frac{9(5d-9)}{4d-9}H\\partial (\\Omega H) + \\frac{60(d-2)}{4d-9}H\\partial \\Psi \\\\& + \\frac{\\vartheta }{4d-9}\\left[-27H\\operatorname{Jac}(F,\\Omega ,H) + {\\bf {40}} \\operatorname{Jac}(F,\\Psi , H)\\right],\\end{split}$ where he makes the mistake of forgetting to multiply 40 by the $\\frac{1}{2}$ in the first term of $\\Psi \\partial H$ .",
"The correct calculations yield $\\begin{split}9HW+{\\bf {40}}\\Psi \\partial H = & -\\frac{9(5d-9)}{4d-9}H\\partial (\\Omega H) + \\frac{60(d-2)}{4d-9}H\\partial \\Psi \\\\ & + \\frac{\\vartheta }{4d-9}\\left[-27H\\operatorname{Jac}(F,\\Omega ,H) + {\\bf {20}} \\operatorname{Jac}(F,\\Psi , H)\\right],\\end{split}$ where the coefficient of $\\operatorname{Jac}(F,\\Psi ,H)$ in the parenthesis is 20 as opposed to 40 in Equation (REF ).",
"Following Cayley's remaining calculations in Sections 20–25 on pp.",
"553–556 in [5], using the expression in eq3:2Hes-4, leads to the polynomial in thm:2hessian.",
"Note that Cayley's result is stated for smooth curves in [5].",
"However, the proof is based on local considerations, hence the corrected polynomial identifies the higher order inflection points and sextactic points on singular curves as well.",
"Moreover, each term in the polynomial $H_2(F)$ involves a determinant with the partial derivatives of $F$ in one row, so the 2-Hessian certainly contains the singular points of $C$ .",
"Let $C$ be the curve given by the defining polynomial $F = x^4-x^3y+y^3z.$ This curve has one cusp with multiplicity sequence $\\overline{m}_p=[3]$ and two simple inflection points.",
"The defining polynomial for the 2-Hessian, computed with [18], is $-2^7\\cdot 3^{11}\\cdot 5\\cdot 7\\cdot y^{18}\\cdot \\left(4x-y\\right)\\cdot \\left(14x^2-7xy+2y^2\\right)=0.$ The intersection points of $H_2$ and $C$ are $p_1 &= (0:0:1), \\\\p_2 &= \\left(\\tfrac{64}{3}:\\tfrac{256}{3}:1\\right), \\\\p_3 &= \\left(\\tfrac{49}{24}+i\\tfrac{77\\sqrt{7}}{24}:\\tfrac{-637}{48}+i\\tfrac{343\\sqrt{7}}{48}:1\\right), \\\\p_4 &= \\left(\\tfrac{49}{24}-i\\tfrac{77\\sqrt{7}}{24}:\\tfrac{-637}{48}-i\\tfrac{343\\sqrt{7}}{48}:1\\right).$ The point $p_1$ is the cusp, while $p_2$ , $p_3$ and $p_4$ are sextactic points.",
"With [18] we compute the osculating conics for the latter and check with Maple [17] that $(O_{p_2} \\cdot C)_{p_2}=(O_{p_3} \\cdot C)_{p_3} = (O_{p_4}\\cdot C)_{p_4} = 6.$ Note that Cayley's original formula for the 2-Hessian identifies $p_2$ as a sextactic point, but not $p_3$ and $p_4$ .",
"A complete overview of this curve in terms of singularities, inflection points and sextactic points, and intersections with associated curves, can be found in tab:QuarticC1B.",
"Table: Invariants and intersections for the rational cuspidal quartic in ex:QuarticC1B."
],
[
"Sextactic point formulas",
"In this section we prove the formula for the number of sextactic points on a cuspidal curve in thm:SPF, and we apply it to examples.",
"Moreover, we state a corollary to the formula that reflects the intersection of $C$ with its Hessian and 2-Hessian."
],
[
"Proof of the sextactic point formula",
"The key ingredient in our proof is a generalized Plücker formula by Ballico and Gatto in [2].",
"To each $Q$ -Weierstrass point $p$ on a curve $C$ it is possible to assign a so-called $Q$ -Weierstrass weight $w_p(Q)$ .",
"On the other hand, the sum of the Weierstrass weights can be computed through a generalization of the Brill–Segre formula.",
"In our situation the result can be stated as follows.",
"[[2]] Let $C$ be a projective, irreducible, cuspidal curve of geometric genus $g$ , let $Q$ be a complete linear system of degree $\\deg Q$ and dimension $r$ , and let $w_p(Q)$ denote the Weierstrass weight of $C$ at $p$ with respect to $Q$ .",
"Then $\\sum _{p\\in C} w_p(Q) = (r+1)(\\deg Q + rg - r).$ In the case of plane curves, to compute the $n$ -Weierstrass weight $w_p(n)$ of a point $p \\in C$ with respect to a complete linear system of curves of degree $n$ and dimension $r$ , we use a technique by Notari from [22].",
"Assuming that $C$ is cuspidal, Notari's algorithm reduces to, for each point $p\\in C$ , finding curves $C_0,\\ldots ,C_r$ of degree $n$ such that the intersection multiplicities at $p$ are distinct.",
"Subsequently, with $h_i = (C \\cdot C_i)_p$ , the $n$ -Weierstrass weight of a unibranched point $p$ can be expressed as $w_p(n) = \\sum _{i=0}^r (h_i - i).$ Note that the integers $h_i$ do not constitute an ordered sequence.",
"The last important ingredient in the proof of thm:SPF is lem:oscint, which uses the Puiseux parametrization of $C$ at $p$ to determine $h_i$ .",
"In the remainder of this article we omit the index $p$ for the local invariants when only one point is discussed.",
"Let $p$ be a smooth point or a cusp of multiplicity $m$ on a plane curve $C$ of degree $d \\ge 3$ .",
"With $T_p$ the tangent line at $p$ , let $l=(T_p \\cdot C)_p$ .",
"If $l\\ne 2m$ , then a curve of degree 2 intersects $C$ at $p$ with one of the following intersection multiplicities: $h_0 = 0,\\; h_1 = m,\\; h_2 = l,\\; h_3 = 2m,\\; h_4 = m+l, \\; h_5 = 2l.$ If $l =2m$ , then a curve of degree 2 intersects $C$ at $p$ with one of the following intersection multiplicities: $h_0 = 0,\\; h_1 = m,\\; h_2 = 2m,\\; h_3 = 3m,\\; h_4 = 4m,\\; h_5=c,$ where $c$ is subject to the restrictions $ c > 2m \\text{ and } c \\ne 3m,4m.$ In particular, a curve of degree 2 that intersects $C$ at $p$ with intersection multiplicity $c$ is irreducible.",
"If $p$ is a cusp, such a curve is not necessarily unique, but $c$ is uniquely determined.",
"Since $p$ is unibranched, the Puiseux parametrization of $C$ at $p$ can be given by $(t^m : at^l +\\cdots : 1),$ where $a\\ne 0$ , and “$\\cdots $ ” denotes higher order terms in $t$ [10].",
"The case $l\\ne 2m$ : We choose the standard basis for plane curves of degree 2, i.e.",
"$x^2,\\; y^2,\\; z^2,\\; yz,\\; xz,\\; xy,$ and substitute the Puiseux parametrization of $C$ at $p$ into this basis.",
"This gives $x^2=t^{2m},\\quad y^2=a^2t^{2l} + \\cdots ,\\quad z^2= 1,$ $yz= at^l + \\cdots ,\\quad xz= t^m,\\quad xy= at^{m+l} + \\cdots .$ By assumption $l \\ne 2m$ , hence the basis elements represent curves with distinct intersection multiplicities at $p$ , and taking the order of $t$ in each element provides the desired values.",
"No other order is possible to obtain by any linear combination of the basis elements.",
"The case $l=2m$ : Observe that since $d \\ge 3$ , the Puiseux parametrization has the form $(t^m:at^{2m}+a_bt^{b}+\\cdots :1)$ for some non-zero constants $a$ and $a_b$ , with $b > 2m$ , and we may assume that there are no terms of order between $2m$ and $b$ in the $y$ -coordinate.",
"Then $x^2=t^{2m},\\quad y^2= a^2t^{4m} + 2a\\cdot a_b t^{2m+b}+ a_b^2t^{2b}+\\cdots ,\\quad z^2=1,$ $yz=at^{2m} + a_bt^b + \\cdots ,\\quad xz= t^m,\\quad xy= at^{3m} + a_bt^{b+m}+ \\cdots .$ Since $l = 2m$ , two of the orders of $t$ in the basis elements are equal to $2m$ , and $h_0=0,\\; h_1=m,\\; h_2=2m,\\; h_3=3m,\\; h_4=4m.$ The remaining intersection multiplicity, $h_5=c$ , can be found by computing the order of $t$ in all possible linear combinations of the basis elements.",
"There are three main cases to consider.",
"First, assume that $b \\ne 3m,4m$ .",
"If $4m<b$ , then $yz-ax^2=0$ is the unique conic with order $c=b$ in $t$ .",
"If $3m<b<4m$ , then any member of the family of conics $yz-ax^2+k_1y^2=0,$ with $k_1 \\in , has order $ c=b$ in $ t$.",
"If $ 2m<b<3m$, then any member of the family of conics $$yz-ax^2+k_1y^2+k_2xy=0,$$ with $ k1,k2 , has order $c=b$ in $t$ .",
"For the remaining two cases, $b=4m$ and $b=3m$ , we need to ensure the existence of $c$ .",
"Indeed, if $b=km$ for some $k$ , then since $p$ is unibranched, there exists a term with non-zero coefficient in the $y$ -coordinate of the Puiseux parametrization such that the order of $t$ in this term is bigger than $b$ , and $m$ is not a factor of this order.",
"Thus, a linear combination of the basis elements will provide a curve of degree 2 that intersects $C$ at $p$ with intersection multiplicity $c$ for some $c>b>2m$ and $c \\ne 3m,4m$ .",
"Thus, secondly, if $b=4m$ , then $yz-ax^2-\\frac{a_b}{a^2}y^2=0$ is the unique conic with order $c$ in $t$ for a $c > b=4m$ .",
"Third, if $b=3m$ and $c < 4m$ , then any member of the family $yz-ax^2-\\frac{a_b}{a}xy+k_1y^2=0,$ with $k_1 \\in , has order $ c$ in $ t$.",
"If $ 4m<c$, then $$yz-ax^2-\\frac{a_b}{a}xy+\\frac{a_b^2}{a^3}y^2=0,$$ is the unique conic with order $ c$ in $ t$.$ In each of the above cases, the curves are irreducible and $c$ is uniquely determined, with $c>2m$ and $c \\ne 3m,4m$ .",
"Motivated by the notion of tangent line at a cusp and lem:oscint, we give the following definition.",
"An osculating conic at a cusp $p$ for which $l=2m$ is a conic that intersects $C$ at $p$ with intersection multiplicity $c$ , for a $c > 2m$ and $c \\ne 3m,4m$ .",
"Let $C$ be a cuspidal curve of geometric genus $g$ , and let $Q$ be the complete linear system of curves of degree 2 on $C$ , with $\\deg Q=2d$ and $r=\\dim Q = 5$ .",
"Hence, the right hand side of the formula in prop:BG34 reads $6(2d+5g-5).$ For the left hand side of the formula, to compute the 2-Weierstrass weight of a point $p$ , we proceed by considering the set of points for which $l \\ne 2m$ and the set of points for which $l=2m$ , respectively.",
"The case $l\\ne 2m$ : This case includes the inflection points and some of the cusps, and coincides with the set $I$ .",
"Inserting the values from lem:oscint into eq:wp-used yields $\\begin{split}w_p(2) & = \\sum _{i=0}^5 (h_i - i) \\\\& = 4m + 4l - 15.\\end{split}$ The case $l=2m$ : Note that this case includes all smooth points that are not inflection points and some of the cusps; the latter points constitute the set $J$ .",
"As above, lem:oscint provides the possible intersections, and we gather $h_0 = 0,\\; h_1 = m,\\; h_2 = 2m,\\; h_3 = 3m,\\; h_4 = 4m,\\; h_5=c.$ Thus, the 2-Weierstrass weight of $p$ is $w_p(2) = 10m+c-15.$ Note that if $p$ is smooth, then $w_p(2)$ is equal to its order as a sextactic point, $w_p(2)=c-5=s_p$ .",
"The formula is valid even when $p$ is not sextactic, as in this case $c=5$ , or equivalently, $w_p(2)=0$ .",
"Hence, the total number of sextactic points on $C$ , counted with multiplicity, is $s = \\sum s_p$ .",
"Putting all this together while isolating $s$ , we get $s = 6(2d + 5g - 5) - \\sum _{I} (4m_p + 4l_p - 15) - \\sum _{J} (10m_p + c_p - 15).$ A sextactic point formula for curves with arbitrary singularities follows from thm:SPF by decomposing all singularities down to their irreducible branches, and subsequently calculating the Weierstrass weights for each branch separately, see [22] and [24].",
"In fact, the formula reads the same as in the cuspidal case, except that in this case $I$ denotes the set of inflection points and the branches of singular points where $l\\ne 2m$ , and $J$ denotes the branches where $l=2m$ .",
"In other words, both cusps, inflection points, and sextactic points might hide in branches of singular points with multiple branches, and these must be accounted for."
],
[
"A lemma and examples",
"The essential ingredients depending on $C$ in the sextactic point formula are the degree $d$ and genus $g$ , and $m_p$ , $l_p$ , and $c_p$ for its inflection points and cusps.",
"Finding these numbers usually requires less heavy calculations than applying the 2-Hessian directly.",
"Moreover, we have the following lemma, which shows that $l_p$ and $c_p$ for a cusp $p$ sometimes can be determined merely by the degree of $C$ and the multiplicity sequence $\\overline{m}_p$ .",
"Let $p$ be a cusp with multiplicity sequence $\\overline{m}=[m,m_1, \\ldots , 1]$ on a plane curve $C$ of degree $d\\ge 3$ .",
"Then $d \\ge l=km + m_k \\ge m+m_1$ for some $k \\ge 1$ , with $m = m_1 = \\ldots = m_{k-1}$ .",
"Moreover, if $l=2m$ , $2d \\ge c=km + m_k > 2m$ for some $k \\ge 2$ , with $m = m_1 = \\ldots = m_{k-1}$ , and $c \\ne 3m,4m$ .",
"In the first case, since $l$ is the intersection multiplicity of a curve and a line at a point, it follows from Bézout's theorem that $d \\ge l$ .",
"Moreover, by [11], we have $l=km + m_k$ for some $k \\ge 1$ , with $m = m_1 = \\ldots = m_{k-1}$ , from which we derive the last inequality.",
"In the second case, since $c$ is the intersection multiplicity of a curve and a conic at a point, it follows from Bézout's theorem that $c \\le 2d$ .",
"Additionally, by lem:oscint, the curve $O_p$ is irreducible, so as above, [11] ensures that $c=km + m_k$ for some $k \\ge 1$ , with $m = m_1 = \\ldots = m_{k-1}$ .",
"Again by lem:oscint, $c > 2m$ and $c \\ne 3m,4m$ , and the result follows.",
"In ex:QuinticC3B,ex:binomial we consider two cuspidal curves and use the formula in thm:SPF to compute the number of sextactic points on these curves.",
"Note that these curves have the same degrees and singularities; they are equisingularly equivalent.",
"However, the curves are not projectively equivalent, and they have different numbers of inflection points and sextactic points, see rem:3536.",
"We revisit these curves in ex:QuinticC3B2,ex:binomial2.",
"Let $C$ be the cuspidal quintic given by $F = y^5+2x^2y^2z - x^3z^2-xy^4.$ By explicit calculations with appropriate associated curves it can be shown that this curve has two cusps; $p_1$ with multiplicity sequence $[3,2]$ and $p_2$ with multiplicity sequence $[2_2]$ .",
"Additionally, it has one simple inflection point, $p_3$ , and two sextactic points, $p_4$ and $p_5$ , see tab:QuinticC3B.",
"On the other hand, it is possible to compute the number of sextactic points directly using thm:SPF.",
"For $p_1$ with multiplicity sequence $[3,2]$ , it follows from the first part of lem:boundc that $l=3+2=5\\ne 6= 2m$ , so $w_{p_1}(2)=4\\cdot 3 + 4 \\cdot 5 - 15=17.$ For $p_2$ with multiplicity sequence $[2_2]$ , lem:boundc does not determine $l$ , but it can be calculated directly from the defining polynomials of $C$ and $T_{p_2}$ that $l=4=2m$ .",
"By the second part of lem:boundc we have that $c=2+2+1=5$ , so $w_{p_2}(2)=10\\cdot 2+5-15=10.$ For the inflection point $p_3$ , $m=1$ and $l=3$ , so $w_{p_3}(2)=4\\cdot 1+4\\cdot 3-15=1.$ Thus, the number of sextactic points on $C$ is $s = & \\;6 \\cdot (2\\cdot 5 + 5 \\cdot 0 - 5) - 17-10-1\\\\= & \\;2.$ Table: Invariants and intersections for the curve in ex:QuinticC3B.Let $C$ be the rational cuspidal quintic given by $F=x^3z^2-y^5.$ This curve has two cusps, $p_1$ with multiplicity sequence $[3,2]$ and $p_2$ with multiplicity sequence $[2_2]$ , and no inflection points.",
"See tab:QuinticC3A.",
"For both these cusps we have that $l=5 \\ne 2m$ , hence $w_{p_1}(2)=4 \\cdot 3+4 \\cdot 5-15=17,$ and $w_{p_2}(2)=4 \\cdot 2+4 \\cdot 5-15=13.$ Thus, the number of sextactic points on $C$ is $s = & \\;6 \\cdot (2\\cdot 5 + 5 \\cdot 0 - 5) - 17-13\\\\= & \\;0.$ Table: Invariants and intersections for the curve in ex:binomial.Note that the curves from ex:QuinticC3B,ex:binomial both belong to the family of equisingular curves given by $V\\left(y^5-x(x z-\\lambda y^2)^2\\right),\\quad \\lambda \\in $ with $\\lambda =1$ and $\\lambda =0$ , respectively.",
"Indeed, for $\\lambda \\ne 0$ , the curves in the family are algebraically equivalent to the curve in ex:QuinticC3B.",
"In this case, the intersection multiplicity of the curve and its tangent at the cusp with multiplicity sequence $[2_2]$ is equal to 4, while in ex:binomial, where $\\lambda =0$ , it jumps to 5.",
"The difference in the intersection multiplicities leads to different Weierstrass weights.",
"This gives room for smooth Weierstrass points when $\\lambda \\ne 0$ , while there are no smooth Weierstrass points when $\\lambda = 0$ , see also rem:binomial."
],
[
"A corollary that ties things together",
"As a corollary to thm:SPF, we state a formula that reflects the intersection of a curve of degree $d$ with its 2-Hessian of degree $12d-27$ .",
"Let $C$ be a cuspidal curve of genus $g$ and degree $d \\ge 3$ , and let $\\delta _p$ denote the delta invariant of a singular point $p$ .",
"Then with notation as in thm:SPF, the following equations hold: $d(12d - 27)+3d(d-2) &= s + 30 \\sum \\delta _p + \\sum _I (4m_p+4l_p-15) + \\sum _J(10m_p+c_p-15),\\\\d(12d - 27) &= s + 24 \\sum \\delta _p + \\sum _I (3m_p+3l_p-12) + \\sum _J(7m_p+c_p-12).$ Recall that in the case of cusps, $\\delta _{p}$ can be calculated from the multiplicity sequence $\\overline{m}_p$ , $ \\delta _{p}= \\sum \\frac{m_{i}(m_{i}-1)}{2},$ where $m_i$ is the $i$ th element in $\\overline{m}_p$ .",
"Before we prove cor:SPF, note that the two formulas could be interpreted as an application of Bézout's theorem to $C$ and its Hessian and 2-Hessian; the terms $3d(d-2)$ and $d(12d-27)$ are simply the product of the respective degrees.",
"The remaining terms are local in nature, and we claim in con:local that these terms reflect a natural geometrical interpretation.",
"We have verified that the conjecture holds for all rational cuspidal curves of degree 4 and 5, see [18].",
"Note that a similar result is proved for the Hessian curve of a cuspidal curve, see [21].",
"The intersection multiplicity $(H_2 \\cdot C)_p$ of a cuspidal curve $C$ and its 2-Hessian curve $H_2$ at a point $p$ is determined by the multiplicity $m$ , the delta invariant $\\delta $ , and the intersection multiplicity with the tangent $l$ or the intersection multiplicity with an osculating conic $c$ .",
"If $p$ is a point on $C$ such that $l\\ne 2m$ , then $(H_2 \\cdot C)_p=24\\delta +3m+3l-12.$ If $p$ is a point on $C$ such that $l = 2m$ , then $(H_2 \\cdot C)_p=24\\delta +7m+c-12.$ By substituting Clebsch' formula for the genus of a plane curve, see [12], $g=\\frac{(d-1)(d-2)}{2}-\\sum \\delta _p,$ into the formula from thm:SPF, we infer that $s = 15d^2 -33d - 30 \\sum \\delta _p -\\sum _I (4m_p + 4l_p - 15)-\\sum _J(10m_p+c_p-15),$ which can be rewritten as the first formula.",
"Moreover, the inflection point formula for cuspidal curves, explicitly stated in [21], reads $v=3d(d-2)-\\sum (6 \\delta _p + m_p+l_p-3),$ where $v$ denotes the number of inflection points counted with multiplicity, and where the sum is taken over all cusps on $C$ .",
"This formula can be rewritten as $0=3d^2-6d-6\\sum \\delta _p - \\sum _{I \\cup J}(m_p+l_p-3).$ By subtracting eq:inflpt from eq:sextclebsch and sorting terms, we reach the second formula."
],
[
"Sextactic points on rational curves",
"In this section we assume that $C$ is a rational plane curve, i.e.",
"$g=0$ and $C$ can be given by a parametrization $\\varphi (s,t)= (\\varphi _0(s,t) : \\varphi _1(s,t) : \\varphi _2(s,t)), \\; \\text{for } (s:t) \\in \\mathbb {P}^1.$ Properties of this parametrization can be exploited to find key information about the 2-Weierstrass points of a rational curve in a natural way.",
"The results in thm:ratosccon,thm:ratww build upon standard tools for studying Weierstrass points and hyperosculating spaces, see [1], [19], [24], [25], as well as results from the previous sections.",
"Indeed, the classical flavour of the statements indicate that the results are well known.",
"However, we have failed to find a suitable reference, and include the results and their proofs for completion.",
"First in this section, we state a corollary to thm:SPF for rational cuspidal curves, which is obtained by setting $g=0$ .",
"With notation as in thm:SPF, the number of sextactic points $s$ , counted with multiplicity, on a rational cuspidal curve of degree $d \\ge 3$ is given by $s = 6(2d - 5) - \\sum _I (4m_p + 4l_p - 15) -\\sum _J(10m_p+c_p-15).$"
],
[
"The osculating conic for rational curves",
"For a smooth point $p$ on a rational curve, it is possible to compute the osculating curve of degree 2 directly from the parametrization.",
"Let $C$ be a rational plane curve given by a parametrization $\\varphi (s,t)$ , and let $\\omega (s,t)$ be the determinant $\\omega (s,t)=\\begin{vmatrix}x^2 & y^2 & z^2 & yz & xz & xy \\\\[2pt]\\frac{\\partial ^4 (\\varphi _0^2)}{\\partial s^4} & \\frac{\\partial ^4 (\\varphi _1^2)}{\\partial s^4} & \\frac{\\partial ^4 (\\varphi _2^2)}{\\partial s^4} & \\frac{\\partial ^4 (\\varphi _1\\varphi _2)}{\\partial s^4} & \\frac{\\partial ^4 (\\varphi _0\\varphi _2)}{\\partial s^4} & \\frac{\\partial ^4 (\\varphi _0\\varphi _1)}{\\partial s^4} \\\\[5pt]\\frac{\\partial ^4 (\\varphi _0^2)}{\\partial s^3 \\partial t} & \\frac{\\partial ^4 (\\varphi _1^2)}{\\partial s^3 \\partial t} & \\frac{\\partial ^4 (\\varphi _2^2)}{\\partial s^3 \\partial t} & \\frac{\\partial ^4 (\\varphi _1\\varphi _2)}{\\partial s^3 \\partial t} & \\frac{\\partial ^4 (\\varphi _0\\varphi _2)}{\\partial s^3 \\partial t} & \\frac{\\partial ^4 (\\varphi _0\\varphi _1)}{\\partial s^3 \\partial t} \\\\[5pt]\\frac{\\partial ^4 (\\varphi _0^2)}{\\partial s^2 \\partial t^2} & \\frac{\\partial ^4 (\\varphi _1^2)}{\\partial s^2 \\partial t^2} & \\frac{\\partial ^4 (\\varphi _2^2)}{\\partial s^2 \\partial t^2} & \\frac{\\partial ^4 (\\varphi _1\\varphi _2)}{\\partial s^2 \\partial t^2} & \\frac{\\partial ^4 (\\varphi _0\\varphi _2)}{\\partial s^2 \\partial t^2} & \\frac{\\partial ^4 (\\varphi _0\\varphi _1)}{\\partial s^2 \\partial t^2} \\\\[5pt]\\frac{\\partial ^4 (\\varphi _0^2)}{\\partial s \\partial t^3} & \\frac{\\partial ^4 (\\varphi _1^2)}{\\partial s \\partial t^3} & \\frac{\\partial ^4 (\\varphi _2^2)}{\\partial s \\partial t^3} & \\frac{\\partial ^4 (\\varphi _1\\varphi _2)}{\\partial s \\partial t^3} & \\frac{\\partial ^4 (\\varphi _0\\varphi _2)}{\\partial s \\partial t^3} & \\frac{\\partial ^4 (\\varphi _0\\varphi _1)}{\\partial s \\partial t^3} \\\\[5pt]\\frac{\\partial ^4 (\\varphi _0^2)}{\\partial t^4} & \\frac{\\partial ^4 (\\varphi _1^2)}{\\partial t^4} & \\frac{\\partial ^4 (\\varphi _2^2)}{\\partial t^4} & \\frac{\\partial ^4 (\\varphi _1\\varphi _2)}{\\partial t^4} & \\frac{\\partial ^4 (\\varphi _0\\varphi _2)}{\\partial t^4} & \\frac{\\partial ^4 (\\varphi _0\\varphi _1)}{\\partial t^4}\\end{vmatrix}.$ Then, for a smooth point $p=\\varphi (s_0,t_0)$ , the polynomial $\\omega (s_0,t_0) \\in x,y,z]_2$ is the defining polynomial of the osculating curve of degree 2 to $C$ at $p$ .",
"Let $v_2(C) \\subset \\mathbb {P}^5$ denote the image of $C$ under the 2nd Veronese embedding of $\\mathbb {P}^2$ to $\\mathbb {P}^5$ , such that $v_2(C)(s,t)=(\\varphi _0^2:\\varphi _1^2:\\varphi _2^2:\\varphi _1\\varphi _2:\\varphi _0\\varphi _2:\\varphi _0\\varphi _1).$ Now, consider the determinant $\\tilde{\\omega }(s,t)$ , where in the first row of $\\omega (s,t)$ the standard basis of plane conics is substituted with the coordinates of $\\mathbb {P}^5$ .",
"For a smooth point $v_2(C)(s_0,t_0)$ , the linear polynomial $\\tilde{\\omega }(s_0,t_0)$ defines a unique osculating hyperplane to $v_2(C)$ in $\\mathbb {P}^5$ , and this hyperplane corresponds to the osculating curve of degree 2 to $C$ at $p=\\varphi (s_0,t_0)$ , with defining polynomial $\\omega (s_0,t_0)$ .",
"Note that for an inflection point, $\\omega (s_0,t_0)$ is reducible and equals the (double) tangent.",
"For a smooth point that is not an inflection point, $\\omega (s_0,t_0)$ and Cayley's osculating conic from thm:osccon coincide by uniqueness.",
"The nodal cubic from ex:nodalcubic can be given by the parametrization $\\varphi (s,t) = \\left(st^2-s^3 : t^3-s^2t : s^3\\right).$ At a smooth point $\\varphi (s_0,t_0)$ the osculating curve of degree 2 has defining polynomial $\\omega (s_0,t_0)$ equal to $&(2s_0^{10}+5s_0^8t_0^2+60s_0^6t_0^4+45s_0^4t_0^6)x^2+(s_0^{10}+10s_0^8t_0^2+5s_0^6t_0^4)y^2\\\\+\\;&(s_0^{10}-5s_0^8t_0^2+10s_0^6t_0^4-10s_0^4t_0^6+5s_0^2t_0^8-t_0^{10})z^2-8 (5s_0^7t_0^3+6s_0^5t_0^5+5s_0^3t_0^7)yz\\\\+\\;&(3s_0^{10}+70s_0^6t_0^4+40s_0^4t_0^6+15s_0^2t_0^8)xz -8(5s_0^7t_0^3+3s_0^5t_0^5)xy.$ Evaluating this expression at the point $p = (-1:0:1)=\\varphi (1,0)$ gives the same defining polynomial for $O_p$ as before, $2x^2+y^2+z^2+3xz=0.$"
],
[
"The Weierstrass weight",
"For rational curves, not necessarily cuspidal, information about its 2-Weierstrass points can be found by direct computation and inspection of the zeros of a homogeneous determinantal polynomial.",
"Let $C$ be a rational plane curve with parametrization $\\varphi (s,t)$ , and let $\\xi (s,t)$ denote the Wronski determinant $\\xi (s,t)=\\begin{vmatrix}\\frac{\\partial ^5 (\\varphi _0^2)}{\\partial s^5} & \\frac{\\partial ^5 (\\varphi _1^2)}{\\partial s^5} & \\frac{\\partial ^5 (\\varphi _2^2)}{\\partial s^5} & \\frac{\\partial ^5 (\\varphi _1\\varphi _2)}{\\partial s^5} & \\frac{\\partial ^5 (\\varphi _0\\varphi _2)}{\\partial s^5} & \\frac{\\partial ^5 (\\varphi _0\\varphi _1)}{\\partial s^5} \\\\[5pt]\\frac{\\partial ^5 (\\varphi _0^2)}{\\partial s^4 \\partial t} & \\frac{\\partial ^5 (\\varphi _1^2)}{\\partial s^4 \\partial t} & \\frac{\\partial ^5 (\\varphi _2^2)}{\\partial s^4 \\partial t} & \\frac{\\partial ^5 (\\varphi _1\\varphi _2)}{\\partial s^4 \\partial t} & \\frac{\\partial ^5 (\\varphi _0\\varphi _2)}{\\partial s^4 \\partial t} & \\frac{\\partial ^5 (\\varphi _0\\varphi _1)}{\\partial s^4 \\partial t} \\\\[5pt]\\frac{\\partial ^5 (\\varphi _0^2)}{\\partial s^3 \\partial t^2} & \\frac{\\partial ^5 (\\varphi _1^2)}{\\partial s^3 \\partial t^2} & \\frac{\\partial ^5 (\\varphi _2^2)}{\\partial s^3 \\partial t^2} & \\frac{\\partial ^5 (\\varphi _1\\varphi _2)}{\\partial s^3 \\partial t^2} & \\frac{\\partial ^5 (\\varphi _0\\varphi _2)}{\\partial s^3 \\partial t^2} & \\frac{\\partial ^5 (\\varphi _0\\varphi _1)}{\\partial s^3 \\partial t^2} \\\\[5pt]\\frac{\\partial ^5 (\\varphi _0^2)}{\\partial s^2 \\partial t^3} & \\frac{\\partial ^5 (\\varphi _1^2)}{\\partial s^2 \\partial t^3} & \\frac{\\partial ^5 (\\varphi _2^2)}{\\partial s^2 \\partial t^3} & \\frac{\\partial ^5 (\\varphi _1\\varphi _2)}{\\partial s^2 \\partial t^3} & \\frac{\\partial ^5 (\\varphi _0\\varphi _2)}{\\partial s^2 \\partial t^3} & \\frac{\\partial ^5 (\\varphi _0\\varphi _1)}{\\partial s^2 \\partial t^3} \\\\[5pt]\\frac{\\partial ^5 (\\varphi _0^2)}{\\partial s \\partial t^4} & \\frac{\\partial ^5 (\\varphi _1^2)}{\\partial s \\partial t^4} & \\frac{\\partial ^5 (\\varphi _2^2)}{\\partial s \\partial t^4} & \\frac{\\partial ^5 (\\varphi _1\\varphi _2)}{\\partial s \\partial t^4} & \\frac{\\partial ^5 (\\varphi _0\\varphi _2)}{\\partial s \\partial t^4} & \\frac{\\partial ^5 (\\varphi _0\\varphi _1)}{\\partial s \\partial t^4} \\\\[5pt]\\frac{\\partial ^5 (\\varphi _0^2)}{\\partial t^5} & \\frac{\\partial ^5 (\\varphi _1^2)}{\\partial t^5} & \\frac{\\partial ^5 (\\varphi _2^2)}{\\partial t^5} & \\frac{\\partial ^5 (\\varphi _1\\varphi _2)}{\\partial t^5} & \\frac{\\partial ^5 (\\varphi _0\\varphi _2)}{\\partial t^5} & \\frac{\\partial ^5 (\\varphi _0\\varphi _1)}{\\partial t^5}\\end{vmatrix}.$ Moreover, let $(s_i:t_i)$ denote the distinct zeros of $\\xi (s,t)$ , with $i \\le 6(2d-5)$ .",
"Then the points $p_i=\\varphi (s_i,t_i)$ are 2-Weierstrass points on $C$ , and the 2-Weierstrass weight $w_{p_i}(2)$ is equal to the order of the zero of $\\xi (s,t)$ corresponding to $(s_i:t_i)$ .",
"Note that the zeros in thm:ratww correspond to all smooth 2-Weierstrass points on $C$ , but only the singular points with $w_p(2) > 0$ , i.e.",
"singular points where at least one branch is a cusp, an inflection point or a sextactic point.",
"First observe that whenever $\\xi (s,t)$ vanishes, the corresponding point on $v_2(C)$ is either singular, or there exists a hyperplane in $\\mathbb {P}^5$ that is hyperosculating to $v_2(C)$ .",
"As before, this hyperplane corresponds to a hyperosculating curve of degree 2 with respect to a point $p$ on $C$ in $\\mathbb {P}^2$ , hence determining an inflection point or a sextactic point.",
"For a smooth curve, [19] ensures that the multiplicity of a zero of $\\xi (s,t)$ equals the 2-Weierstrass weight of the corresponding point.",
"Note that this is the same as the flattening points of the Veronese embedding, as described in [1].",
"This takes care of the smooth points.",
"Alternatively, the below analysis for singular points can be applied to smooth points.",
"In the case of singular points, we consider each branch separately, and perform a local computation.",
"So choose one branch and perform a linear transformation on $C$ so that the chosen branch of $p$ corresponds to the parameter values $(s:t)=(1:0)$ , and so that its tangent is $y=0$ .",
"Moreover, by abuse of notation, observe that $\\xi (1,t)=\\xi (t)=\\begin{vmatrix}v_2(C)(t) \\\\v_2(C)^{\\prime }(t) \\\\v_2(C)^{\\prime \\prime }(t) \\\\v_2(C)^{3}(t) \\\\v_2(C)^{4}(t) \\\\v_2(C)^{5}(t) \\\\\\end{vmatrix}.$ Assume first that the chosen branch can be parametrized by $(t^m:at^l+\\ldots :1)$ , with $a \\ne 0$ and $l \\ne 2m$ .",
"Substituting this into $\\xi (t)$ , computing the determinant, and comparing with the proof of thm:SPF in pf:SPF, it follows that $\\operatorname{ord}_t \\xi (t) = 4m+4l-15= \\sum _{i=0}^5(h_i-i).$ If $l = 2m$ , first transform the branch of $C$ at $p$ so that it is given by the parametrization $(t^m:at^{2m}+\\ldots :1)$ , where $a \\ne 0$ .",
"Subsequently, by applying the Veronese embedding, consider the curve $\\rho (t)=(t^{2m}:a^2t^{4m}+\\ldots :1:t^m:at^{2m}+\\ldots :at^{3m}+\\ldots )$ in $\\mathbb {P}^5$ , which by a suitable linear transformation in $\\mathbb {P}^5$ can be given by the parametrization $\\sigma (t)=(t^{2m}:a^2t^{4m}+\\ldots :1:t^m:a_ct^{c}+\\ldots :at^{3m}+\\ldots ),$ for an $a_c \\ne 0$ , and $c>2m$ , $c \\ne 3m,4m$ , see lem:oscint.",
"Then, for a parametrized curve $\\psi $ in $\\mathbb {P}^5$ , consider the determinant $W_{\\psi }(t)=\\begin{vmatrix}\\psi (t) \\\\\\psi ^{\\prime }(t)\\\\\\psi ^{\\prime \\prime }(t)\\\\\\psi ^{(3)}(t)\\\\\\psi ^{(4)}(t)\\\\\\psi ^{(5)}(t)\\\\\\end{vmatrix}.$ Straightforward computations give that the order of $t$ in $W_{\\sigma }(t)$ is $10m+c-15$ .",
"Now, $\\operatorname{ord}_tW_{\\sigma }(t)=\\operatorname{ord}_tW_{\\rho }(t)=\\operatorname{ord}_t\\xi (t)$ , hence $\\operatorname{ord}_t\\xi (t)=10m+c-15$ .",
"Moreover, notice that in this case the inverse image of the hyperplane $x_4=0$ under the linear transformation in $\\mathbb {P}^5$ and the Veronese embedding corresponds to a conic in $\\mathbb {P}^2$ that intersects the branch of $C$ at $p$ with intersection multiplicity $h_5=c$ .",
"Hence, we have that $\\operatorname{ord}_t \\xi (t) = \\sum _{i=0}^5(h_i-i)$ .",
"Performing a similar analysis on all branches of $C$ at $p$ , and summing up, we reach $w_p(2)$ .",
"Observe that the polynomial $\\xi $ is homogeneous in $s$ and $t$ of degree $6(2d - 5)$ .",
"Since the 2-Weierstrass weights add up to this number, thm:ratww provides another proof of cor:rational-SPF.",
"We now revisit ex:QuinticC3B,ex:binomial, and compute the 2-Weierstrass points and weights using the Wronski determinant in thm:ratww.",
"Let $C$ be the rational cuspidal quintic from ex:QuinticC3B, with parametrization $\\varphi (s,t)=(s^5 : s^3t^2 : st^4+t^5).$ Computing the Wronski determinant gives $\\xi (s,t) = -2^{24}\\cdot 3^{12}\\cdot 5^2\\cdot 7^4\\cdot s^{17}t^{10}(192s^3 + 1680s^2t + 5275st^2 + 5250t^3).$ The cusps $p_1$ and $p_2$ correspond to the parameters $(0:1)$ and $(1:0)$ , respectively, while the inflection point $p_3$ and the sextactic points $p_4$ and $p_5$ correspond to zeros of $192s^3 + 1680s^2t + 5275st^2 + 5250t^3$ .",
"Determining the order of the corresponding zeros, we conclude, as in ex:QuinticC3B, that $w_{p_1}(2)=17$ , $w_{p_2}(2)=10$ , and $w_{p_3}(2)=w_{p_4}(2)=w_{p_5}(2)=1$ .",
"Let $C$ be the rational cuspidal quintic from ex:binomial, with parametrization $\\varphi (s,t)=(s^5 : s^3t^2 : t^5).$ Computing the Wronski determinant gives $\\xi (s,t) = -2^{25}\\cdot 3^{13}\\cdot 5^{5}\\cdot 7^{5} \\cdot s^{17}t^{13}.$ The cusp $p_1$ corresponds to the parameter value $(0:1)$ , and the exponent of $s$ gives the 2-Weierstrass weight, $w_{p_1}(2)=17$ .",
"The cusp $p_2$ corresponds to $(1:0)$ , thus $w_{p_2}(2)=13$ .",
"Note that the curve from ex:binomial,ex:binomial2 is an example of a cuspidal curve $C_{m,l}$ of degree $l$ with defining polynomial $F=x^mz^{l-m}-y^l.$ For any $l,m \\in \\mathbb {N}$ , with $m<l$ and $\\gcd (m,l)=1$ , the curve $C_{m,l}$ is cuspidal; bicuspidal when $1 <m <l-1$ , and unicuspidal with an inflection point otherwise.",
"It can be shown that these curves have no other $n$ -Weierstrass points for $1 \\le n < l$ .",
"A proof in a more general setting can be found in [2], but, additionally, it is possible to construct a direct proof of this claim with methods from the present article, see [18]."
],
[
"Acknowledgements",
"This article is based on results from the master's thesis of the first author [18], supervised by the second author.",
"We would like to thank Geir Ellingsrud for his senior supervision and support of this paper, Kristian Ranestad for interesting discussions, and Ragni Piene, Georg Muntingh, the editor and the referee for valuable comments.",
"Moreover, we want to express our gratitude to Live Rasmussen and the Science Library at the University of Oslo for making this project possible and for still having a printed edition of The collected mathematical papers of Arthur Cayley in-house and available on a Saturday morning."
]
] | 1709.01698 |
[
[
"Theoretical modeling of Comptonized X-ray spectra of super-Eddington\n accretion flow: origin of hard excess in Ultraluminous X-Ray Sources"
],
[
"Abstract X-ray continuum spectra of super-Eddington accretion flow are studied by means of Monte Carlo radiative transfer simulations based on the radiation hydrodynamic simulation data, in which both of thermal and bulk Compton scatterings are taken into account.",
"We compare the calculated spectra of accretion flow around black holes with masses of $M_{\\rm BH} = 10, 10^2, 10^3$, and $10^4 M_\\odot$ for a fixed mass injection rate (from the computational boundary at $10^3 r_{\\rm s}$) of $10^3 L_{\\rm Edd}/c^2$ (with $r_{\\rm s}$, $L_{\\rm Edd}$, and $c$ being the Schwarzschild radius, the Eddington luminosity, and the speed of light, respectively).",
"The soft X-ray spectra exhibit mass dependence in accordance with the standard-disk relation; the maximum surface temperature is scaled as $T \\propto M_{\\rm BH}^{-1/4}$.",
"The spectra in the hard X-ray bands, by contrast, look quite similar among different models, if we normalize the radiation luminosity by $M_{\\rm BH}$.",
"This reflects that the hard component is created by thermal and bulk Compton scattering of soft photons originating from an accretion flow in the over-heated and/or funnel regions, the temperatures of which have no mass dependence.",
"The hard X-ray spectra can be reproduced by a Wien spectrum with temperature of $T\\sim 3$ keV accompanied by a hard excess at photon energy above several keV.",
"The excess spectrum can be well fitted with a power law with a photon index of $\\Gamma \\sim 3$.",
"This feature is in good agreement with that of the recent NuSTAR observations of ULX (Ultra-Luminous X-ray sources)."
],
[
"Introduction",
"It had been long believed that the classical Eddington limit can never be exceeded, as long as objects steadily shine by accreting environmental gas.",
"Recently, however, there are growing evidences, which point the existence of astronomical objects shining at super-Eddington luminosities.",
"Super-Eddington accretion flow is the flow, in which the accretion rate onto a central object exceeds the classical limit which gives rise to super-Eddington luminosities; that is, $\\dot{M} > \\dot{M}_{\\rm Edd}/\\eta &\\equiv & L_{\\rm Edd}/(\\eta c^{2})\\nonumber \\\\&\\simeq & 10^{19} ({\\rm g~s}^{-1}) (\\eta /0.1)^{-1}(M_{\\rm BH}/10 M_\\odot ),$ where $L_{\\rm Edd}$ is the Eddington luminosity, $\\eta (\\sim 0.1)$ is the energy conversion efficiency, $M_{\\rm BH}$ is the black hole mass, and $c$ is the speed of light.",
"One of the best candidates of the super-Eddington accretors is Ultra-Luminous X-ray sources (ULXs), bright off-nuclear compact X-ray sources with luminosities of $10^{39}$ –$10^{41}$ erg s$^{-1}$ , found in nearby galaxies.",
"There are two major scenarios explaining the high luminosities of ULXs: sub-Eddington accretion to intermediate-mass black holes (e.g., [13]; [18]), and super-Eddington accretion to stellar mass black holes (e.g., [38]; [11]).",
"The discovery of X-ray pulses in M82 X-2 by [1] was striking in this respect, since it demonstrates that the central star in M82 X-2 should be a neutron star and is thus less massive than $3 M_\\odot $ .",
"Super-Eddington accretion should, hence, be required to explain its high luminosity over $10^{40}$ erg s$^{-1}$ .",
"The discovery such an object called a ULX pulsar provides a good support for the existence of super-Eddington accretion in the universe.",
"Two other ULX pulsars were discovered afterward (NGC7793 P13, [3]; NGC5907 ULX, [6]).",
"We should note that ULXs are not the only super-Eddington accretors but there exist other sites, where super-Eddington accretion is going on.",
"Some of microquasars are suspected to have super-Eddington accretion flow (see, e.g., [2]) and some quasars at high (cosmological) redshifts are another example.",
"[20] discovered the quasars at the redshift, $z=7.085$ .",
"Its black hole mass is estimated to exceed $10^{9}M_{\\odot }$ .",
"If this supermassive black hole had grown up from a population III remnant (a black hole with $M_{\\rm BH}\\sim 10$ –$10^{3}M_{\\odot }$ ), super-Eddington accretion flows is inevitable, since otherwise the growth time to produce the observed supermassive black hole exceeds the age of the universe.",
"The inflow/outflow gas dynamics and emission properties of super-Eddington accretion have been extensively studied in this decade by means of multi-dimensional radiation hydrodynamic (RHD) simulations, being pioneered by [24] (see also [25], [26] for radiation-MHD simulations).",
"More recently, general relativistic (GR) RHD simulations have been performed (key52 [31], [33]; [15]; [35]).",
"They all demonstrated that significant fraction of inflow material is blown away by strong radiation-pressure force in forms of optically Thomson-thick, moderately high-temperature ($\\sim $ several keV) disk wind.",
"[7] have reconfirmed such a structure by modified RHD simulations.",
"The emergent spectra should thus suffer Comptonization effects by the wind.",
"[10] were the first to calculate such Comptonized spectra of super-Eddington accretion flows and have shown that the theoretical spectra can well fit the typical ULX spectra in the X-ray band (key1 [9], [10]).",
"They also notice that not only thermal Comptonization but also bulk Comptonization by wind motion plays a crucial role in spectral formation.",
"Quite recently [22] solved the GR radiative transfer problem by post-processing GR radiation magnetohydrodynamic (RMHD) simulations and obtained similar results.",
"The presence of massive outflow was also pointed out from the observational points of view by [4] and [17].",
"We here make more extensive study of Comptonized spectra expected from super-Eddington accretion flow for a variety of black hole masses and see how the spectra depend on the black hole mass and the viewing angle.",
"So far the theoretical efforts have been focused on clarifying how the spectra depend on mass accretion rate, while the black hole mass dependence of that flows has been poorly investigated.",
"In addition, we pay particular attention to the detailed spectral shape in the hard X-ray ranges.",
"Now is a good time to perform such study, since thanks to NuSTAR we now have rich data of hard X-ray emission spectra of ULXs with good resolution.",
"The organization of the paper is as follows: we first explain our model and methods of calculations in section 2 and then show our results in section 3.",
"The final section is devoted to discussion and conclusions.",
"In the present study, we perform spectral calculations based on the two-dimensional (2D) radiation hydrodynanmic (RHD) simulation data, in which both of bulk and thermal Comptonization are taken into account [28].",
"This 2D RHD code solves the axisymmetric two-dimensional radiation hydrodynamic equations in the spherical coordinates.",
"The flux-limited diffusion approximation is adopted ([12]; [36]), and general relativistic effects are incorporated by adopting the pseudo-Newtonian potential [27].",
"We also adopt the $\\alpha $ viscosity prescription [34] and we set $\\alpha =0.1$ .",
"Basic equations and numerical methods are the same as those in key1 ([9], [10]).",
"Simulation settings are roughly the same as those in [10].",
"The computational domain of the radiation hydrodynamic simulations is described by $r_{\\rm in}=2r_{\\rm s}\\le r \\le r_{\\rm out}=1000r_{\\rm s}$ , and $0 \\le \\theta \\le \\pi /2$ .",
"Here $r_{\\rm s}=2GM_{\\rm BH}/c^{2}$ is the Schwarzschild radius with $G$ being the gravitational constant.",
"Grid points are distributed according to the radial and azimuthal coordinates and each grid spacing is $\\triangle \\log _{10} r = (\\log _{10} r_{\\rm out}-\\log _{10} r_{\\rm in})/N_{r}$ and $\\triangle \\cos \\theta =1/N_{\\theta }$ , respectively, where the numbers of grid points are taken to be $(N_{r},N_{\\theta })=(192,192)$ throughout the present study.",
"We start calculations with an empty space but for numerical reasons we initially put hot, rarefied, and an optically thin atmosphere.",
"Mass is injected continuously with a constant rate of $\\dot{M}_{\\rm input}=10^{3}L_{\\rm Edd}/c^{2}$ through the outer boundary at $r=r_{\\rm out}$ and $0.45\\pi \\le \\theta \\le 0.5\\pi $ .",
"We set that the injected matter has an angular momentum corresponding to the Keplerian angular momentum at $r=300r_{\\rm s}$ .",
"The matter can go out freely through the boundary $r=r_{\\rm out}, 0\\le \\theta \\le 0.45\\pi $ , and the absorbing boundary condition is adopted at $r=r_{\\rm in}$ .",
"The black hole mass is a free parameter and is set to be $M_{\\rm BH}=10^{1}M_{\\odot },10^{2}M_{\\odot },10^{3}M_{\\odot }$ , and $10^{4}M_{\\odot }$ .",
"(The previous study by [10] solved only the case with $M_{\\rm BH}=10M_{\\odot }$ .)"
],
[
"Monte Carlo Calculations of Radiative Transfer",
"The code which we use for the spectral calculation is that developed by [10].",
"This three dimensional Monte Carlo simulation code (hereafter 3D MC code) solves the three-dimensional radiative transfer by means of the Monte Carlo method by post-processing the simulation data produced by the 2D RHD code described above.",
"Before spectral calculations we time average gas mass density, temperature, and velocity at each grid point for every $0.25 M_{\\rm BH}/M_{\\odot }$ sec.",
"We only use the data after accretion flows become quasi-steady.",
"The 2D RHD simulations data given in terms of the spherical coordinates are converted to the data in the Cartesian coordinates by interpolation.",
"The computational domain of the radiative transfer calculations is set by $-300r_{\\rm s}\\le x,y,z \\le 300r_{\\rm s}$ and the numbers of the grid points are $(N_{x},N_{y},N_{z})=(160,160,160)$ .",
"In the quasi-steady accretion regime, the accretion rate onto the central black hole, which we calculate by summing up mass passing through the inner boundary at $r=r_{\\rm in}$ , is approximately $\\dot{M}\\sim 200L_{\\rm Edd}/c^{2}$ .",
"Since the mass input rate through the outer boundary is $10^3 L_{\\rm Edd}/c^2$ , about 80% of input material is wandering or being drawn away out of the computational domain.",
"The photosphere is set up $\\tau _{\\rm eff}(\\nu )=\\sqrt{3\\tau _{\\rm a}(\\nu )(\\tau _{\\rm a}(\\nu )+\\tau _{\\rm e})}=10$ calculated along $z$ -axis direction from $z_{\\rm max}=300r_{\\rm s}$ to $-z_{\\rm max}$ (with $\\tau _{\\rm a}$ and $\\tau _{\\rm e}$ being the absorption optical depth and the electron scattering optical depth, respectively).",
"Here, we only consider bremsstrahlung absorption for evaluating $\\tau _{\\rm a}$ .",
"At each frequency bin the seed photons, of which the number is $6\\times 10^5$ , are generated within a $\\tau _{\\rm eff}(\\nu )<10$ region.",
"In this region the generation point of each photon is selected such that the generated photon number per unit volume is proportional to the local bremsstrahlung emissivity.",
"We consider free-free absorption and bulk & thermal Comptonization effects to describe the interaction between photons and matter.",
"The black hole swallows photons which go through $r<2r_{\\rm s}$ .",
"The spectrum frequency setting is $\\nu _{\\rm min}=10^{14} {\\rm Hz}<\\nu <\\nu _{\\rm max}=10^{21} {\\rm Hz}$ with $\\triangle \\log _{10} \\nu = (\\log _{10}\\nu _{\\rm max}-\\log _{10}\\nu _{\\rm min})/100$ )."
],
[
"Overall flow structure",
"We first overview the flow structure by inspecting the 2D density and temperature distributions of each model.",
"The top four panels in Figure REF show the time-averaged density contours for the black hole mass of $M_{\\rm BH}=10^{1},10^{2},10^{3}$ , and $10^{4}M_{\\odot }$ from the left to the right, respectively.",
"We find a quite similar color contour pattern in every top panel, as long as the length scale and the density are normalized by $r_{\\rm s} (\\propto M_{\\rm BH})$ and by $\\rho (\\propto M_{\\rm BH}^{-1})$ , respectively.",
"Common features to all the top panels are that the flow consists of a high-density inflow part (or a disk) around the equatorial plane (indicated by the red-to-yellow color) and a low-density outflow part (or a funnel region) around the rotational axis (indicated by the blue color).",
"Note that the former disk part can be well modeled by a geometrically thick accretion disk.",
"The density dependence on $M_{\\rm BH}$ can be understood in the following way: Let us scale other relevant quantities in terms of the black hole mass; e.g., velocity by free-fall velocity, $v_{\\rm ff}=c(r/r_{\\rm s})^{-1}\\propto M_{\\rm BH}^{0}$ , luminosity by $L_{\\rm Edd}\\propto M_{\\rm BH}$ , and mass accretion rate by ${\\dot{M}}_{\\rm Edd}=L_{\\rm Edd}/c^2 \\propto M_{\\rm BH}$ .",
"From the continuity equation, we have $r^{2}\\rho v_{r} \\propto \\dot{M}$ ($\\propto M_{\\rm BH}$ ).",
"We finally obtain $\\rho \\propto M_{\\rm BH}^{-1}$ .",
"Figure: Enlarged snapshot of gas temperature contours close to the black hole at t=6.04t=6.04sfor model with M BH =10M ⊙ M_{\\rm BH}=10M_{\\odot }.",
"Overlaid are arrows which represent the direction ofgas motion.",
"(Note that the lengths of the arrows are not in scale.",
")The three key regions are separated by the blue dash lines:accretion flow around the equatorial plane (indicated by `disk'),funnel region near the polar axis (indicated by `funnel'),and the over-heated region (indicated by `over-heat').The location of the funnel wall is also indicated in this figure (by `wall').The bottom four panels in Figure REF show time-averaged gas temperature distribution.",
"Again, all the four panels look similar except in the inflow region around the equatorial plane, where we find higher temperatures in the left panels than in the right panels.",
"This difference can be easily understood, since the temperature of accretion disk is proportional to $M_{\\rm BH}^{-1/4}$ (detailed explanation will be provided later).",
"While the temperature in other regions, especially in the funnel region, is insensitive to the black hole mass.",
"For readers' convenience we specify in figure REF the precise locations of the three key regions: disk, funnel, and over-heated region.",
"This figure is an enlarged snapshot of the temperature contours and corresponds to the bottom left panel of figure REF , although the panels in figure REF are not snapshots but time-averaged pictures.",
"It is important to note that an abrupt heating occurs where the inflow material collides with the funnel wall.",
"The highest temperature part (with $T\\sim 10^{8}$ K) is produced near the black hole because of abrupt heating and it is this region plays an essential role in hard photon production [10].",
"The disk and funnel regions are distinguished in terms of the directions of the gas motion, temperature, and the density (or optical depth).",
"Thomson optical depth of the funnel region is: $\\tau _{\\rm e}&=& \\int ^{z_{\\rm max}}_{z_{\\rm wall}}\\sigma _{\\rm T}\\rho dz \\sim 1.$ Here, $\\sigma _{\\rm T}$ is Thomson cross section, $z_{\\rm max}$ is the boundary of computational box, and $z_{\\rm wall}$ is the $z$ -position of funnel wall.",
"The opening angle of funnel is roughly $\\theta _{\\rm open}\\sim 20^{\\circ }$ –$25^{\\circ }$ .",
"Figure: The gas temperature depending on radius (θ=45 ∘ \\theta =45^{\\circ } from z-axis) for models with M BH =10 1 ,10 2 ,10 3 ,10 4 M ⊙ M_{\\rm BH}=10^{1},10^{2},10^{3},10^{4}M_{\\odot }.Different mass dependences of the two regions can be understood as follows: To demonstrate that the disk temperature at a fixed radius does have a mass dependence of $T_{\\rm disk} \\propto M_{\\rm BH}^{-1/4}$ , we have checked the radial profiles of the disk temperature by using the simulation data (e.g.",
"figureREF ), confirming the relationship of $T_{\\rm disk} \\propto M_{\\rm BH}^{-1/4} \\times (r/r_{\\rm s})^{-1/2}$ .",
"This mass dependence is the same as that of the standard disk, while the radial dependence is not.",
"We can simply understand this results in the following way ([8], equation 10.22): The surface temperature of a standard-type disk at a fixed $r/r_{\\rm s}$ obeys $\\sigma T_{\\rm disk}^4 &\\sim & \\frac{3}{8\\pi }\\frac{GM\\dot{M}}{r^3} \\propto M_{\\rm BH}^{-1} \\left(\\frac{r}{r_{\\rm s}}\\right)^{-3}, $ leading to $T_{\\rm disk} \\propto M_{\\rm BH}^{-1/4}(r/r_{\\rm s})^{-3/4}$ , as long as we assume ${\\dot{M}}\\propto M_{\\rm BH}$ .",
"In the supercritical flow, by contrast, the temperature profile of the disk no longer depends on the mass accretion rate.",
"Then, the disk temperature at a fixed $r/r_{\\rm s}$ obeys $\\sigma T_{\\rm disk}^4 &\\sim & \\frac{L_{\\rm Edd}}{2\\pi r^2} \\propto M_{\\rm BH}^{-1} \\left(\\frac{r}{r_{\\rm s}}\\right)^{-2},$ leading to $T_{\\rm disk} \\propto M_{\\rm BH}^{-1/4} (r/r_{\\rm s})^{-1/2}$ .",
"Therefore, the supercritical disk has the same mass dependence as that of the standard disk [Eq.",
"(REF )].",
"In the high-temperature funnel region ($5r_{\\rm s}\\lesssim r\\lesssim 100r_{s}$ and $0^{\\circ }\\le \\theta \\lesssim 20^{\\circ }$ ), on the other hand, radiative viscous heating balances with Compton cooling.",
"That is, by equating $\\Phi _{\\rm vis}&=&\\eta \\left( r\\frac{\\partial \\Omega }{\\partial r}\\right)^{2}\\sim \\alpha E_{0}\\Omega _{\\rm K}\\left(\\frac{\\Omega }{\\Omega _{\\rm K}}\\right)^{2}\\left(\\frac{\\partial \\ln \\Omega }{\\partial \\ln r} \\right)^{2},$ and $\\Gamma _{\\rm comp}&\\sim &4\\sigma _{\\rm T}c \\frac{k_{\\rm B}T_{\\rm funnel}}{m_{\\rm e}c^{2}}\\left(\\frac{\\rho }{m_{\\rm p}}\\right)E_{0},$ we have $k_{\\rm B}T_{\\rm funnel}&\\sim &\\frac{\\alpha }{2}m_{\\rm e}c^{2}\\frac{m_{\\rm p}}{\\rho \\sigma _{\\rm T}r_{\\rm s}}\\left(\\frac{v_{\\phi }}{c}\\right)^{2}\\left(\\frac{r}{r_{\\rm s}}\\right)^{-\\frac{1}{2}}\\left(\\frac{\\partial \\ln \\Omega }{\\partial \\ln r} \\right)^{2}.$ We hence see that the right-hand side has no mass dependence, so does the funnel temperature.",
"Numerically, we find $\\sim 10^7$ K from equation (REF ) and this temperature is in good agreement with that of the funnel wall located along the line of $\\sim 20^\\circ $ from the rotation axis (see figure REF ).",
"The gas temperature within the funnel is by a factor of several to ten times greater than the above estimation.",
"This is probably because the gas within the funnel had been heated when passing through the over-heated region.",
"We should also point out that the heating process within the funnel may not be so accurately described by the $\\alpha $ viscosity prescription.",
"This point will be improved by future MHD simulations.",
"The temperature in the over-heated region is about $T_{\\rm heat} \\sim 10^8$ K regardless of the black hole mass (see figure REF and also REF ), as is expected from the relation that kinetic energy is converted to internal energy; that is, from $\\frac{1}{2}\\rho v^{2} \\propto (\\gamma -1)\\frac{\\rho k_{\\rm B}T_{\\rm heat}}{\\mu m_{\\rm p}},$ we have $k_{\\rm B}T_{\\rm heat}\\propto \\frac{\\mu m_{\\rm p}v^{2}}{2(\\gamma -1)} \\propto M_{\\rm BH}^0,$ Note that the inflow velocity $v$ is independent black hole mass.",
"In reality the temperature of the over-heated region is much lower than this simple estimation because of significant Compton cooling by soft photons emerging from the underlying accretion flow (with temperature of $T_{\\rm disk} \\sim 10^7$ K for $M_{\\rm BH}=10M_{\\odot }$ )."
],
[
"Emergent radiation spectra",
"Figure REF shows total spectra viewed by a distant face-on observer and its dependence on the black hole mass: $M_{\\rm BH}=10^{1},10^{2},10^{3}$ , and $10^{4}M_{\\odot }$ .",
"Here, by a face-observer we mean an observer viewing from the angle between ${\\rm i}=0^{\\circ } - 10^{\\circ }$ from the rotation axis.",
"There are some noteworthy features seen in figure REF .",
"First, the larger black hole mass is, the brighter becomes the flow.",
"This is because we fixed the mass accretion rate in terms of $M_{\\rm BH}$ and, hence, the luminosity of super-Eddington accretion flows is roughly proportional to the Eddington luminosity, $\\nu L_{\\nu }\\sim L_{\\rm Edd}\\propto M_{\\rm BH}$ .",
"Second, the overall spectral shape in the hard X-ray bands looks quite similar, if we normalize the radiation intensity by $M_{\\rm BH}$ .",
"In other words, the peak frequency of the hard component remains the same.",
"This is a direct consequence that the temperature of the hard X-ray emitting regions is insensitive to the black hole mass; $T_{\\rm funnel}\\propto M_{\\rm BH}^{0}$ (see the previous subsection).",
"The spectral rollover at $h\\nu _{\\rm peak}\\sim 7$ keV can be understood, if the photons from the disk region is once over-heated and is then Compton cooled by the gas at the funnel wall (with $\\sim 10^7$ K).",
"Finally, the soft X-ray spectra have a mass dependence.",
"That is, the higher the black hole mass is, the higher becomes the soft X-ray luminosity, and the lower becomes the spectral break between the Rayleigh-Jeans part and the flat spectrum part ($h\\nu _{\\rm break}\\propto k_{\\rm B}T_{\\rm disk}\\propto M_{\\rm BH}^{-1/4}$ ).",
"The luminosity in the Rayleigh-Jeans range obeys $\\nu L_{\\nu }\\propto M_{\\rm BH}^{7/4}$ , since from $T(R)&\\equiv &T_{0}\\left(\\frac{R}{r_{\\rm s}}\\right)^{-p} \\quad {\\rm with~} T_{0}\\propto M_{\\rm BH}^{-1/4},$ we find $F_{\\nu }&\\propto &\\int _{R_{\\rm in}}^{R_{\\rm out}} 2\\pi R B_{\\nu }dR\\\\&=&\\frac{4\\pi \\nu ^{2}k_{\\rm B}T_{0}r_{\\rm s}^{2}}{c^{2}}\\int _{R_{\\rm in}/r_{\\rm s}}^{R_{\\rm out}/r_{\\rm s}}\\left(\\frac{R}{r_{\\rm s}}\\right)^{1-p} d\\left(\\frac{R}{r_{\\rm s}}\\right)\\\\&\\propto &T_{0}r_{\\rm s}^{2} \\propto M_{\\rm BH}^{7/4}~~~~~~~~~~({\\rm for}~ h\\nu \\ll k_{\\rm B}T_{0})$ (Here, $R_{\\rm in}$ and $R_{\\rm out}$ are the radii of the inner and outer edges of the accretion disk, respectively.",
"and we assumed that both radii are scaled in terms of $r_{\\rm s}$ .)",
"This dependence is in good agreement with the simulation results shown in figure REF .",
"For these reasons the spectrum is roughly the summation of the high- and low-temperature blackbody emissions.",
"Figure REF displays decomposition of the total emission spectra according to the regions where photons are originally generated.",
"Here we divide the total flow region into the three: the inner region ($2r_{\\rm s}<R<10r_{\\rm s}$ ), the middle region ($10r_{\\rm s}<R<100r_{\\rm s}$ ), and the outer region ($R>100r_{\\rm s}$ ).",
"Here, $R$ is the radial coordinate in the cylindrical coordinates $R=\\sqrt{x^{2}+y^{2}}$ .",
"From this figure we understand that the observable spectrum is mainly composed by direct soft photons from accretion disk and hard photons which are Compton up-scattered near the black hole.",
"But we see a hard X-ray rollover at around 7 keV which arises due to the Compton down-scattering in low temperature outflow [10].",
"This issue (regarding the origin of hard X-rays) will be discussed in more details in the next section.",
"The bottom four panels in Figure REF show the viewing-angle dependence of the emergent spectra for each model.",
"The larger the viewing angle is, the weaker the hard X-ray flux becomes.",
"This is because an observer viewing from large angles is hard to see directly funnel near the black hole ($z\\lesssim 30r_{\\rm s}$ ).We can thus explain the very soft X-ray spectra of ULSs (Ultra-Luminous Supersoft sources; see [5], [23])."
],
[
"Brief summary",
"In the present study we elucidate the spectral properties of the super-Eddington accretion flow and outflow by means of the three dimensional Monte Carlo radiation transfer simulation based on the global 2D RHD simulation data.",
"The purpose of the present study is two-fold: (1) to extend the study by [10] for a variety of black hole masses, and (2) to calculate more accurate hard X-ray spectra.",
"Regarding the first issue, we have seen remarkably similar flow structure and overall spectral properties, especially in the hard energy bands.",
"The detailed inspection of the hard spectral component (the second issue) is made possible by increasing photons used in the Monte Carlo simulations.",
"We find a significant excess over the exponential rollover above several keV.",
"This should be a result of complex, multiple Compton scattering of photons within the over-heated region, as well as in the funnel region.",
"We admit the limitations of the present RHD simulations, since MHD processes are not properly solved there.",
"Definitely, we need radiation-MHD simulation in future work to calculate more precisely flow temperature and radiation spectra.",
"In the following subsections we make more detailed study how photons finally acquire high energy by multiple Compton scatterings, interacting with high-temperature or high-velocity plasmas."
],
[
"How is the power-law component constructed?",
"In figure REF we have already seen the excess component in the hard X-ray energy range (above several keV).",
"One possible origin for creating power-law photons is unsaturated Comptonization (e.g.",
"[29]), since we estimate the electron scattering optical depth to be $\\tau _{\\rm es}\\gg 1$ because of very high density, whereas the Compton $y$ -parameter [$y\\equiv (4k_{\\rm B}T/m_{\\rm e}c^2)\\max (\\tau _{\\rm es},\\tau _{\\rm es}^2$ )] moderately exceeds unity (around several to 10).",
"In order to understand the origin of hard photons we check the trajectories and energy variation histories of the hard photons.",
"We set two conditions to select hard photons; (1) their final energy should exceed 10 keV and (2) they should eventually be observed by a distant face-on observer.",
"The results are summarized in Figures REF – REF for the case with $M_{\\rm BH} = 10 M_\\odot $ (see also Table 1).",
"From these we confirm the following facts: These hard photons were generated as soft photons with $\\sim 1$ keV within the inflow region around the equatorial plane; $R\\lesssim 10r_{s}$ and $z\\lesssim 5r_{s}$ .",
"After the generation these photons travel around the black hole for a while, being scattered many, many times.",
"The photons then go into the over-heated region (see figure REF in section ).",
"In this region, optical depth and electron temperature are $\\tau _{\\rm e}\\approx 10$ and $k_{\\rm B}T_{\\rm e}\\approx 8.6$ keV, respectively, while the Compton $y$ parameter is very large; $y \\approx 7$ .",
"The photons are thus Compton up-scattering many times within the over-heated region so that they can acquire large energy to become hard photons.",
"These hard photons then eventually reach the foot-point of the funnel and enter the funnel region, where gas is accelerated upward by radiation-pressure force and thus has a large radial velocity, up to $\\beta =v/c\\sim 0.2$ , near the funnel wall.",
"Within the funnel the hard photons experience Compton up- and down-scatterings.",
"After passing through the funnel wall, some photons escape to directly reach the observer, while some others are reflected by the wall, return to the funnel region and are Compton down-scattered there again, and finally goes out of the funnel region to reach the observer as soft photons.",
"To summarize, the hard photon production processes are too complex to simply describe.",
"The observed hard power-law spectra are formed as a consequence of such complex matter-photon interactions.",
"As [10] claimed, not only the thermal Comptonizaion but also the bulk Comptonization play important roles in the formation of hard X-ray spectra (see their fig.",
"5).",
"The speed of the outflow that photons encounter is around $\\beta \\equiv v/c\\sim 0.2$ (see table REF ).",
"Since the photon energy amplification factor is from [28] $\\frac{\\triangle (h\\nu )}{h\\nu }&=&\\frac{\\frac{4}{3}\\beta ^{2}\\gamma ^{2}m_{\\rm e}c^{2}-h\\nu }{m_{\\rm e}c^{2}}.$ we estimate that the photon energy can acquire energy at most around $h\\nu \\sim \\frac{4}{3}\\beta ^{2}\\gamma ^{2}m_{\\rm e}c^{2}\\sim 30 (\\beta /0.2)^2$ keV.",
"For $\\beta =0.1$ -– $0.3$ we find $h\\nu \\simeq 8 - 70$ keV."
],
[
"The effects of iterated temperature, magnetic field, and general relativity.",
"We do not recompute the temperature in their simulations using the Monte Carlo code, although this is sometimes implemented by others using similar methods (e.g.",
"[22]).",
"Here, we calculate what the “correct” temperature would be, if we would use the calculated radiation spectra as a source of Compton cooling.",
"For this purpose, we check the temperature of the photosphere at around $(R, z)=(11 r_{\\rm s}, 32 r_{\\rm s})$ , from which hard X-ray ($> 10$ keV) photons mainly come from, finding that the mean photon energy is $<h\\nu >\\sim 14.9$ keV.",
"This energy corresponds to the radiation temperature of $k_{\\rm B}T_{\\rm rad}^{1} = <h\\nu >/4 \\sim 3.7$ keV.",
"On the other hand, the gas temperature found in the original RHD simulation is $k_{\\rm B}T_{\\rm gas}^{0} \\sim 1.2$ keV.",
"Therefore, we have $T_{\\rm gas}^{0} < T_{\\rm rad}^{1}$ , meaning that gas should be Compton heated by radiation.",
"However, we should point that bulk Compton is dominant over the thermal Compton even for this recalculated temperature; i.e., $v_{\\rm thermal}/c \\sim \\sqrt{3k_{\\rm B}T_{\\rm gas}^{1} / m_{\\rm e} c^2} \\sim 0.147$ (assuming $k_{\\rm B}T_{\\rm gas}^{1}\\sim k_{\\rm B}T_{\\rm rad}^{1}$ ), while the bulk velocity is $v_{\\rm bulk}/c \\sim 0.2$ .",
"We can thus safely conclude that the iterated temperature can hardly affect the hard X-ray emission properties discussed in the present paper.",
"We consider why recomputation of the temperature is not so critical in our simulations unlike the simulations by [22].",
"[22] mentioned that the gas temperature (before the recomputation) is very high around the rotation axis since the Compton-cooling is less effective.",
"The cause of that is the deficit of the radiation induced by the M1-closure method.",
"The M1 method tends to suppress the radiation energy density around the axis via the artificial centrifugal shocks (key511 [30]; [32]) Thus, after recomputation, the high gas temperature ($\\sim 10^{9}$ K) is significantly reduced to about $10^{8}$ K by the effective Compton cooling.",
"In contrast, artificial reduction of the radiation energy density around the axis does not occur in the FLD approximation method that is employed in the present work.",
"Hence, the gas temperature near the polar axis may be already low, about $10^8$ K. In addition, the relatively low gas temperature in the region far from the polar axis is nearly the same as those reported by [22].",
"In this region, the gas temperature does not change so much by the recomputaion.",
"For these reason, our spectra look quite similar to those in [22].",
"The differences in methods (MHD versus non-MHD, general relativity versus no general relativity) are surprisingly small.",
"To check if this is true, we first estimate the effects of the gravitational redshift.",
"The spectrum is reproduced by multiplying the redshift to the photon energy at only last scattering position.",
"The redshift is very important near the black hole, but in that place, photons are many Compton scattered.",
"Then it seems to be sufficient to consider the redshift at the last scattering position.",
"There is no significant difference between spectrum, e.g.",
"${\\rm sed(no~redshift)/sed(redshift)} \\sim 0.98$ – $1.2$ each photon energy.",
"This is because the location of the last scattering surface of the hard X-ray is quite far from the black hole.",
"For this reason, we conclude that our spectrum (non-GR) is almost same as [22] (GR).",
"We next check how important magnetic effects are.",
"It is important in this context to point out the inequality of radiative-pressure $>$ gas-pressure $>$ magnetic-pressure in the funnel region, which was first pointed out by radiation-MHD simulations by [26].",
"The effects of the magnetic field seem to be weak and spectrum is almost same between MHD and non-MHD.",
"However, magnetic dissipation could be more enhanced above or near the photosphere, rather than in the equatorial region.",
"It is thus necessary to check if this is the case in future radiation-MHD simulations.",
"We further estimate the effects of cyclotron emission and absorption as follows: There are two regions in which magnetic field is relatively strong: the funnel region and the region very close to the black hole.",
"In the funnel region, first of all, the magnetic field strength is $B \\sim 10^{5}$ G [26].",
"The gas velocity is up to $v < 0.5c$ , and the Lorentz factor is $\\gamma \\sim 1$ .",
"The typical frequency of cyclotron is $\\nu _{g}=eB/(2\\pi m_{\\rm e} c) \\sim 3 \\times 10^{11}$ Hz, $\\nu _{\\rm crit} =3 \\gamma ^2 \\nu _{g} \\sin \\alpha /2 \\sim \\nu _{g} \\sin \\alpha < \\nu _{g}$ .",
"In this frequency $\\nu _{\\rm crit}$ , the cyclotron process does not emit and absorb X-ray.",
"In the region near the black hole ($R<10r_{\\rm s}, z<5r_{\\rm s}$ ), secondly, the magnetic field is $B\\sim 10^{8}$ G [26].",
"In the same way, frequency is $\\nu _{g} \\sim 3\\times 10^{14}$ Hz, and so the cyclotron process does not absorb X-ray, and emit radio (or far-infrared) wave.",
"The seed photon generated near the black hole in the disk are observed as hard X-ray through the Compton effect.",
"But cyclotron radio emission does not become this seed photon because there are a lot of the seed photons by free-free emission ($\\sim 1$ keV) from the disk.",
"For these reasons, our spectrum (non-MHD) looks like that by [22] (MHD).",
"It is, of course, possible that the flow structure might be somewhat altered, if we would incorporate the MHD and GR effects.",
"Especially, the effects of MHD and GR would be important in magnetically arrested disks (MADs), and MADs around highly spinning BH, respectively.",
"(We note that we have studied non-MADs around non-rotating BHs in this paper).",
"For example, [16] carried out 3D GR-RMHD simulations of super-critically accreting MADs and found that the cyclo-synchrotron cooling is dominant in the jet and corona region.",
"[22] found that the SEDs of MADs around a highly spinning BH differ from those of non-MADs or MADs around a non-spinning BH, because the speed of jets in MADs with highly spinning BH is higher than the others and that results in the decrease of the optical depth in the jet and leads high frequency photons emitted from the plasma near the BH to be observable.",
"It is our future work to calculate radiation spectra based on the GR-RMHD simulations."
],
[
"Application to the ULXs",
"In figure REF we show the observable photon energy spectrum which is to be observed by a face-on observer for the case with $M=10 M_{\\odot }$ .",
"We find a significant excess over an exponential rollover in the hard X-ray range.",
"This is quite reminiscent of the recent NuSTAR observations of ULXs.",
"For example, [37] reported a hard X-ray excess in the NuSTAR spectrum of Holmberg II X-1, one of the most extensively studied ULXs.",
"They performed spectral fitting, finding that this excess component can be best-fit by a power-law component with the photon index of $\\Gamma =3.1^{+0.3}_{-1.2}$ .",
"They discuss that likely origin of this power-law tail is Comptonization of soft photons by a hot (or even non-thermal) coronal plasma.",
"To check to what extent our theoretical spectra can reproduce the observed ones, we perform similar spectral fittings to the theoretical one by using the same spectral models excluding ${\\rm TBABS}$ (i.e., the model which X-ray is absorbed by interstellar medium) as those used by [37].",
"The results are illustrated in figure REF .",
"It clearly shows that the theoretical spectrum can nicely be represented by two blackbody-like components with a power-law tail (see also table 2 for the best-fit parameters).",
"The most remarkable is the power-law photon index, $\\Gamma \\sim 3$ , which is in good agreement with the NuSTAR observation of Ho II X-1 [$\\Gamma = 3.1~(+0.3/-1.2)$ ].",
"The temperature of the higher temperature blackbody is $\\sim 3$ keV, somewhat higher than the observed one [$1.8~(+0.7/-0.3)$ keV].",
"The temperature of the lower temperature one ($\\sim 1.0$ keV) is, on the other hand, significantly higher than the observed one [$0.20~(+0.03/-0.04)$ keV], But this disagreement should not be taken seriously, since our computational box is not large enough to resolve the accretion flow structure at large radii, from where soft X-ray emission originates.",
"We thus focus out discussion to the hard X-ray properties from the super-Eddington flow in the present study.",
"We expect that future larger-box simulations will improve this discrepancy.",
"We may thus safely conclude that the success in reproducing the observed hard excess spectrum provides good support to the super-Eddington scenario for ULXs."
],
[
"The effects of the photosphere",
"[10] carried out the calculations for the other choices of $\\tau _{\\rm eff}=1,3,5,10,20$ for the locations of the photosphere, but confirmed that the spectrum for the photosphere $\\tau _{\\rm eff}=10$ is essentially the same as that obtained for the deeper photosphere $\\tau _{\\rm eff} >10$ .",
"We also recalculate the spectra for the case of different photosphere, i.e., $\\tau _{\\rm eff}=5,20,30$ .",
"The spectrum looks very similar and the fitting results are quite similar; now the best-fit parameter at $\\tau _{\\rm eff}=30$ is $\\Gamma = 3.1(+1.3/-2.1) $ , in good agreement with the case with $\\tau _{\\rm eff}=10$ (see tableREF ).",
"For these reason, it is essentially to set photosphere $\\tau _{\\rm eff}=10$ .",
"This work is partially supported by JSPS Grant-in-Aid for Scientific Research (C) (17K05383 S. M.; 15K05036 K.O.).",
"Numerical computations were mainly carried out on Cray XC30 at Center for Computational Astrophysics, National Astronomical Observatory of Japan.",
"This research was also supported by MEXT as“Priority Issue on Post-K computer” (Elucidation of the Fundamental Laws and Evolution of the Universe) and JICFuS."
]
] | 1709.01531 |
[
[
"Origin of Charge Separation at Organic Photovoltaic Heterojunctions: A\n Mesoscale Quantum Mechanical View"
],
[
"Abstract The high efficiency of charge generation within organic photovoltaic blends apparently contrasts with the strong \"classical\" attraction between newly formed electron-hole pairs.",
"Several factors have been identified as possible facilitators of charge dissociation, such as quantum mechanical coherence and delocalization, structural and energetic disorder, built-in electric fields, nanoscale intermixing of the donor and acceptor components of the blends.",
"Our mesoscale quantum-chemical model allows an unbiased assessment of their relative importance, through excited-state calculations on systems containing thousands of donor and acceptor sites.",
"The results on several model heterojunctions confirm that the classical model severely overestimates the binding energy of the electron-hole pairs, produced by vertical excitation from the electronic ground state.",
"Using physically sensible parameters for the individual materials, we find that the quantum mechanical energy difference between the lowest interfacial charge transfer states and the fully separated electron and hole is of the order of the thermal energy."
],
[
"Theoretical details",
"Our effective two-orbital quantum chemical model[1] is based on the second-quantized Hamiltonian: $\\hat{H} = \\sum _{i,j=1}^{2M}\\sum _{\\sigma =\\alpha }^{\\beta } h_{ij} a_{i\\sigma }^\\dagger a_{j\\sigma } +\\frac{1}{2} \\sum _{i,j,k,l=1}^{2M} \\sum _{\\sigma ,\\tau =\\alpha }^{\\beta }c_{ikjl} a_{i\\sigma }^\\dagger a_{j\\tau }^\\dagger a_{l\\tau }a_{k\\sigma } $ where $M$ is the total number of sites and $a_{i\\sigma }^\\dagger $ and $a_{j\\tau }$ are electron creation and annihilation operators satisfying the usual Fermion anticommutation rules.",
"[2], [3] Indices $i,j,k,l$ run over the orbitals (the HOMO and LUMO on site $r$ are $2r\\!-\\!1$ and $2r$ , respectively), $\\sigma $ and $\\tau $ over the possible spin states.",
"The orbitals $\\phi _k$ , which provide the basis for the second-quantized Hamiltonian, are assumed to be orthonormal: $\\int \\!",
"\\phi _{i}(\\mathbf {r}) \\phi _{j}(\\mathbf {r}) d\\mathbf {r} = \\delta _{ij} .$ The one-electron ($h_{ij}$ ) and two-electron integrals ($c_{ikjl}$ ) which form the Hamiltonian are formally defined as: $h_{ij} \\!\\!&=& \\!\\!",
"\\int \\!",
"\\phi _{i}(\\mathbf {r}) \\left[ -\\frac{1}{2}\\nabla ^2 - \\sum _{p=1}^M \\frac{Z_p^{\\mathrm {eff}}}{|\\mathbf {r}-\\mathbf {R}_p|} \\right] \\phi _{j}(\\mathbf {r}) d\\mathbf {r} \\\\c_{ikjl} \\!\\!&=& \\!\\!",
"\\int \\!\\int \\!\\frac{\\phi _{i}(\\mathbf {r}_1) \\phi _{k}(\\mathbf {r}_1) \\phi _{j}(\\mathbf {r}_2) \\phi _{l}(\\mathbf {r}_2) }{|\\mathbf {r}_1-\\mathbf {r}_2|}d\\mathbf {r}_1 d\\mathbf {r}_2 \\, , $ where $\\mathbf {R}_p$ is the location of site $p$ .",
"We compute the electronic energies and wavefunctions of the model systems with a modified version of the GAMESS-US code.",
"[4] Within the program, each site is treated as a helium atom with a double-$\\zeta $ basis.",
"This ensures that that all the arrays are dimensioned correctly.",
"Our version of the code bypasses the usual ab initio evaluation of the one- and two-electron integrals from the orbitals, replacing them by semiempirical values chosen to account for the essential physics of a system.",
"This approach is closely analogous in spirit to the early semiempirical theories of molecular electronic structure and to the Hubbard model of solid-state physics.",
"The remainder of our code is essentially identical to the standard version of GAMESS-US.",
"One ground-state (HF) plus excited-state (CIS) calculation on one of our systems takes about 15 minutes, on a single-processor workstation.",
"The use of GAMESS-US also allows us to leverage on the full range of quantum chemical methods implemented in it.",
"For example, in addition to the HF and CIS calculations[5] presented within the paper, it is possible to perform ground- and excited-state calculations based on the coupled-cluster ansatz.",
"[6] The on-site parameters of the Hamiltonian for a molecule $r$ are related to its ionization energy ($IE_r$ ), electron affinity ($EA_r$ ), singlet excitation ($SX_r$ ) and triplet excitation ($RX_r$ ): $\\left[ \\begin{array}{c}IE_r \\\\ EA_r \\\\ SX_r \\\\ TX_r\\end{array} \\right] =\\left[ \\begin{array}{rrrr}-1 & 0 & -1 & 0 \\\\0 & -1 & -2 & 1 \\\\-1 & 1 & 0 & 1 \\\\-1 & 1 & 0 & -1\\end{array} \\right]\\left[ \\begin{array}{c}\\epsilon ^H_{r} \\\\ \\epsilon ^L_{r} \\\\ c^C_{r} \\\\ c^X_{r}\\end{array} \\right] $ where $\\epsilon ^H_{r}$ and $\\epsilon ^L_{r}$ are the HOMO and LUMO energies of site $r$ , while $c^C_{r}$ and $c^X_{r}$ are short-hand notations for the two-electron integrals describing on-site Coulomb and exchange interactions ($c^C_{r} = c_{2r-1,2r-1,2r-1,2r-1}\\simeq c_{2r-1,2r-1,2r,2r}\\simeq c_{2r,2r,2r,2r}$ and $c^X_{r}=c_{2r-1,2r,2r-1,2r}$ ).",
"Equating the three on-site Coulomb integrals is a resonable and convenient approximation, which could be easily avoided if it were necessary to reproduce some additional single-site energies (e.g., the singly excited states of the cation and anion, or the doubly excited state of the neutral molecule).",
"Inverting Eq.",
"(REF ) we obtain the Hamiltonian parameters as a function of the energies: $\\left[ \\begin{array}{c}\\epsilon ^H_{r} \\\\ \\epsilon ^L_{r} \\\\ c^C_{r} \\\\ c^X_{r}\\end{array} \\right] =\\left[ \\begin{array}{rrrr}-2 & 1 & 0 & 1 \\\\-2 & 1 & 1/2 & 3/2 \\\\1 & -1 & 0 & -1 \\\\0 & 0 & 1/2 & -1/2\\end{array} \\right]\\left[ \\begin{array}{c}IE_r \\\\ EA_r \\\\ SX_r \\\\ TX_r\\end{array} \\right] .",
"$ The Coulomb integrals $c^C_{r}$ provide a rough estimate of the spatial extent of the orbitals associated with a site.",
"Let us assume that one electron within an orbital produces a Gaussian charge distribution: $\\rho _r(\\mathbf {r}) = - \\phi _r^2(\\mathbf {r}) = - \\left(\\frac{2\\alpha _r}{\\pi }\\right)^{3/2} e^{-2\\alpha _r|\\mathbf {r}-\\mathbf {R}_r|^2} $ Its self-repulsion integral is: $c^C_{r} = \\int \\int \\frac{\\rho _r(\\mathbf {r}_1)\\rho _r(\\mathbf {r}_2)}{|\\mathbf {r}_1-\\mathbf {r}_2|}d\\mathbf {r}_1d\\mathbf {r}_2= \\sqrt{\\frac{4\\alpha _r}{\\pi }} \\, .",
"$ Reversing this equation, we obtain the standard deviation $\\sigma _r$ from the integral: $\\sigma _r = \\sqrt{\\frac{3}{4\\alpha _r}} = \\sqrt{\\frac{3}{\\pi }}\\frac{1}{c^C_{r}} \\, .",
"$ Using the Coulomb integrals resulting from Eq.",
"(REF ) and the gas-phase data of C$_{70}$[7], [8], [9] and pentacene[10], [11], [12], [13] (see also Table 1 within the main manuscript), we obtain $\\sigma _r=0.434$ nm for the former and $\\sigma _r=0.402$ nm for the latter.",
"These values correspond to roughly one half of the sites' diameters.",
"The ionization energies and electron affinities which we use in our calculations are derived from the experimental gas phase ones, after adjusting for polarization effects.",
"Our estimate of this correction is based on the Born formula for the solvation free energy of a spherical charge $\\pm e$ of diameter $D_0=1.0$ nm (i.e., the nearest-neighbour distance) inside a dielectric with relative permittivity $\\varepsilon _r=3.5$ : $\\Delta G_{\\mathrm {Born}} = - \\frac{e^2}{D_0} \\left( 1-\\frac{1}{\\varepsilon _r}\\right) \\, .",
"$ This lowers the energies of both the anion and cation states, decreasing the $IE$ and increasing the $AE$ by about 1 eV (see again Table 1 within the main manucript).",
"We now consider the Hamiltonian for many interacting sites.",
"First of all, the orbital energies on one site are shifted by the interaction with the cores of the other sites.",
"We assume that the charge distribution of one core is the positive, doubly charged version of its HOMO density (see Eq.",
"(REF )): $\\rho ^{\\mathrm {core}}_r(\\mathbf {r}) = +2\\phi _r^2(\\mathbf {r}) = +2 \\left(\\frac{2\\alpha _r}{\\pi }\\right)^{3/2} e^{-2\\alpha _r|\\mathbf {r}-\\mathbf {R}_r|^2} $ The on-site elements of the one-electron Hamiltonian become: $h_{2r-1,2r-1} &=& \\epsilon _r^H + w_{2r-1} - \\sum _{p\\ne r}^M \\frac{ 2 \\mathrm {erf}\\!\\left( \\mu _{rp}R_{rp} \\right)}{\\varepsilon _r R_{rp} } \\\\h_{2r,2r} &=& \\epsilon _r^L + w_{2r} - \\sum _{p\\ne r}^M \\frac{ 2 \\mathrm {erf}\\!\\left( \\mu _{rp}R_{rp} \\right)}{\\varepsilon _r R_{rp} } \\\\h_{2r-1,2r} &=& 0$ where erf(x) is the error function and $\\mu _{rp}=\\sqrt{2\\alpha _r\\alpha _p/(\\alpha _r+\\alpha _p)}$ .",
"The $w$ 's in Eqs.",
"(REF ) and () are perturbations to the site energies, which model the effect of \"diagonal\" or \"energetic\" disorder.",
"We draw these numbers from Gaussian distributions of width $\\sigma _w=0.08$ eV, assumed for simplicity to be identical for both D or A materials.",
"Next, we assume that the inter-site elements of the one-electron Hamiltonian decay exponentially with the distance $R_{ij}$ : $h_{ij}=t_{ij} e^{-(R_{ij}-D_0)/\\Delta } .",
"$ Here $t_{ij}$ represents the coupling between two orbitals at the nearest-neighbour distance $D_0$ , and $\\Delta $ determines the decay of the couplings with increasing separation.",
"A controlled degree of \"off-diagonal\" disorder can be introduced by assuming that the $t_{ij}$ 's are drawn from suitable distributions.",
"We assume these to be Gaussians.",
"In principle, both their averages and their widths may depend on the materials and on the orbital types.",
"In the absence of further information, coming for example from detailed atomistic models of the individual materials and their interfaces, we assume that all the couplings are symmetrically distributed around a zero average with a standard deviation $\\sigma _t$ =0.08 eV (identical to $\\sigma _w$ ).",
"Furthermore, we take $\\Delta =0.35$ nm.",
"The inter-site electron-electron repulsions are represented by two-electron integrals.",
"All three- and four-center integrals are neglected, and we retain only the two-center integrals which represent the classical repulsion between two Gaussian charge clouds: $c_{ikjl} = \\delta _{ik} \\delta _{jl} \\frac{ \\mathrm {erf}\\!\\left( \\mu _{ij}R_{ij} \\right) }{\\varepsilon _r R_{ij} } .",
"$ Thanks to this zero-overlap approximation, the number of two-electron integrals to be processed in a calculation is substantially smaller than in an ab initio calculation with a comparable basis set.",
"The dipole integrals, which are necessary to compute the dipole moments and to study the effect of an external electric field, are approximated in a way consistent with the zero-overlap approximation: $\\mathbf {\\mu }_{ij} = \\int \\!",
"\\phi _{i}(\\mathbf {r}) \\mathbf {r} \\phi _{j}(\\mathbf {r}) d\\mathbf {r}\\approx \\delta _{ij} \\mathbf {R}_i$ where $\\mathbf {R}_i$ is the location of orbital $i$ .",
"Note that the dipole integral between the HOMO and LUMO on the same site is also zero, in this approximation.",
"The site charges are obtained by conventional Mulliken or Löwdin populational analyses, the two being equivalent in the case of orthogonal orbitals.",
"[2] With the dipole integrals of Eq.",
"(REF ), the quantum mechanical dipole moments, calculated as expectation values of the ground or excited state wavefunctions, coincide with the \"classical\" ones for a set of point charges $q_k^S$ at the sites with coordinates $\\mathbf {R}_k$ (the $S$ superscript identifies a state): $\\mathbf {\\mu }^S = \\sum _{k=1}^M q_k^S \\mathbf {R}_k .",
"$ Figure: Illustration and denomination of the three-dimensional model heterojunctions.",
"Donor sites are depicted in red, acceptor sites in blue.Figure S.1 shows the structure of all the three-dimensional model heterojunctions.",
"The simplest one (N0T0) is a bilayer with a planar interface between the D and A sites, orthogonal to the $z$ axis.",
"We have also studied the effect of variations on this basic system, introducing some interpenetration between the two phases in the form of a \"comb\" morphology.",
"The systems have been named according to the thickness of the mixing region (T, in number of D:A layers) and to the number of pillars (N)."
]
] | 1709.01611 |
[
[
"Information-theoretic analysis of the directional influence between\n cellular processes"
],
[
"Abstract Inferring the directionality of interactions between cellular processes is a major challenge in systems biology.",
"Time-lagged correlations allow to discriminate between alternative models, but they still rely on assumed underlying interactions.",
"Here, we use the transfer entropy (TE), an information-theoretic quantity that quantifies the directional influence between fluctuating variables in a model-free way.",
"We present a theoretical approach to compute the transfer entropy, even when the noise has an extrinsic component or in the presence of feedback.",
"We re-analyze the experimental data from Kiviet et al.",
"(2014) where fluctuations in gene expression of metabolic enzymes and growth rate have been measured in single cells of E. coli.",
"We confirm the formerly detected modes between growth and gene expression, while prescribing more stringent conditions on the structure of noise sources.",
"We furthermore point out practical requirements in terms of length of time series and sampling time which must be satisfied in order to infer optimally transfer entropy from times series of fluctuations."
],
[
"Abstract",
"Inferring the directionality of interactions between cellular processes is a major challenge in systems biology.",
"Time-lagged correlations allow to discriminate between alternative models, but they still rely on assumed underlying interactions.",
"Here, we use the transfer entropy (TE), an information-theoretic quantity that quantifies the directional influence between fluctuating variables in a model-free way.",
"We present a theoretical approach to compute the transfer entropy, even when the noise has an extrinsic component or in the presence of feedback.",
"We re-analyze the experimental data from Kiviet et al.",
"(2014) where fluctuations in gene expression of metabolic enzymes and growth rate have been measured in single cells of E. coli.",
"We confirm the formerly detected modes between growth and gene expression, while prescribing more stringent conditions on the structure of noise sources.",
"We furthermore point out practical requirements in terms of length of time series and sampling time which must be satisfied in order to infer optimally transfer entropy from times series of fluctuations.",
"Quantifying information exchange between variables is a general goal in many studies of biological systems because the complexity of such systems prohibits mechanistic bottom-up approaches.",
"Several statistical methods have been proposed to exploit either the specific dependence of the covariances between input and output variables with respect to a perturbation applied to the network [1], or the information contained in 3-point correlations [2].",
"These methods are potentially well suited for datasets obtained from destructive measurements, such as RNA sequencing or immunohistochemistry.",
"However, none of these methods exploits the information contained in time-lagged statistics, which is provided for instance by non-destructive measurements obtained from time-lapse microscopy of single cells.",
"Such experimental data should be quite relevant to understand functional relationships since they merely reflect the time delays present in the dynamics of the system.",
"Time-delayed cross-correlations between gene expression fluctuations have indeed been shown to discriminate between several mechanistic models of well characterized genetic networks [3].",
"However, such methods become difficult to interpret in the presence of feedback.",
"This situation is illustrated in reference [4] where the fluctuations in the growth rate and in the expression level of metabolic enzymes have been measured as a function of time by tracking single cells of E. coli with time-lapse microscopy.",
"The interplay between these variables has been characterized using cross-correlations as proposed in [3].",
"To circumvent the difficulty of discriminating between many complex and poorly parametrized metabolic models, the authors reduced functional relations to effective linear responses with a postulated form of effective couplings.",
"In the present work, we instead use a time-lagged and information-based method to analyze the interplay between the two fluctuating variables.",
"A crucial feature in this method is that it is model-free and it is able to disentangle the two directions of influence between the two variables, unlike the cross-correlations discussed above.",
"This type of approach was first proposed by Granger [5] in the field of econometrics and found applications in a broader area.",
"More recently, transfer entropy [6], which is a non-linear extension of Granger causality, has become a popular information-theoretic measure to infer directional relationships between jointly dependent processes [7].",
"It has been successfully applied to various biomedical time series (see for instance [8]) and used extensively in the field of neurobiology, as shown in Ref.",
"[9] and in references therein.",
"This is the tool that will be used in this work.",
"The plan of this paper is as follows.",
"We first introduce two measures of information dynamics, transfer entropy (TE) and information flow (IF).",
"We then illustrate our numerical method on a well controlled case, namely a simple linear Langevin model, and show that we can properly estimate these quantities from the generated time series.",
"We then analyze experimental data on the fluctuations of metabolism of E. coli taken from Ref. [4].",
"We provide analytical expressions for the transfer entropy and information flow rates for the model proposed in that reference.",
"After identifying a divergence in one TE rate as the sampling time goes to zero, we introduce a simplified model which is free of divergences while still being compatible with the experimental data.",
"We conclude that the inference of information-theoretic dynamical quantities can be helpful to build physically sound models of the various noise components present in chemical networks.",
"Unlike the mutual information $I(X:Y)$ that only quantifies the amount of information exchanged between two random variables $X$ and $Y$ as defined in the section on Methods, the transfer entropy (TE) is an asymmetric measure that can discriminate between a source and a target [6].",
"Consider two sampled time series $\\lbrace ..x_{i-1},x_i,x_{i+1}..\\rbrace $ and $\\lbrace ..y_{i-1},y_i,y_{i+1}..\\rbrace $ , where $i$ is the discrete time index, generated by a source process $X$ and a target process $Y$ .",
"The transfer entropy $T_{X \\rightarrow Y}$ from $X$ to $Y$ is a conditional, history-dependent mutual information defined as $T_{X \\rightarrow Y}&=\\sum P(y_{i+1},y_i^{(k)},x_i^{(l)})\\ln \\frac{P(y_{i+1}|y_i^{(k)},x_i^{(l)})}{P(y_{i+1}|y_i^{(k)})}, \\nonumber \\\\&=\\sum _i \\: [ H(y_{i+1}|y_i^{(k)})-H(y_{i+1}|y_i^{(k)},x_i^{(l)})]$ where $y_i^{(k)}=\\lbrace y_{i-k+1},\\cdots , y_i\\rbrace $ and $x_i^{(l)}=\\lbrace x_{i-l+1},\\cdots , x_i\\rbrace $ denote two blocks of past values of $Y$ and $X$ of length $k$ and $l$ respectively, $P(y_{i+1},y_i^{(k)},x_i^{(l)})$ is the joint probability of observing $y_{i+1}, y_i^{(k)},x_i^{(l)}$ , and $P(y_{i+1}|y_i^{(k)},x_i^{(l)}), P(y_{i+1}|y_i^{(k)})$ are conditional probabilities.",
"In the second line, $H(.\\vert .",
")$ denotes the conditional Shannon entropy (see Section on Methods for definition).",
"In the first equation, the summation is taken over all possible values of the random variables $y_{i+1},y_i^{(k)},x_i^{(l)}$ and over all values of the time index $i$ .",
"To put it in simple terms, $T_{X\\rightarrow Y}$ quantifies the information contained from the past of $X$ about the future of $Y$ , which the past of $Y$ did not already provide [7], [8].",
"Therefore, it should be regarded as a measure of predictability rather than a measure of causality between two time-series [10].",
"For instance, when $x_i^{(l)}$ does not bring new information on $y_{i+1}$ , then $P(y_{i+1}|y_i^{(k)},x_i^{(l)})=P(y_{i+1}|y_i^{(k)})$ and the transfer entropy vanishes because the prediction on $y_{i+1}$ is not improved.",
"With a similar definition for $T_{Y\\rightarrow X}$ , one can define the net variation of transfer entropy from $X$ to $Y$ as $\\Delta T_{X\\rightarrow Y}\\equiv T_{X\\rightarrow Y}-T_{Y\\rightarrow X}$ .",
"The sign of $\\Delta T_{X\\rightarrow Y}$ informs on the directionality of the information transfer.",
"The statistics required for properly evaluating the transfer entropy rapidly increases with $k$ and $l$ , which in practice prohibits the use of large values of $k$ and $l$ .",
"The most accessible case thus corresponds to $k=l=1$ , which we denote hereafter as $\\overline{T}_{X\\rightarrow Y}$ .",
"This quantity is then simply defined as $\\overline{T}_{X\\rightarrow Y}=\\sum _i \\big [H(y_{i+1} | y_i)- H(y_{i+1} | y_i,x_i)\\big ],$ When the dynamics of the joint process $\\lbrace X,Y \\rbrace $ is Markovian, one has $P(y_{i+1}|y_i^{(k)},x_i^{(l)})=P(y_{i+1}| y_i,x_i)$ and since $H(y_{i+1}|y_i^{(k)}) \\le H(y_{i+1}| y_i)$ one has $\\overline{T}_{X\\rightarrow Y} \\ge T_{X\\rightarrow Y}$ (see Ref.",
"[11]).",
"Therefore, $\\overline{T}_{X\\rightarrow Y}$ represents an upper bound on the transfer entropy.",
"In the case of stationary time series, which is the regime we consider in this work, it is natural to also introduce the TE rate ${\\overline{\\cal T}}_{X\\rightarrow Y}&= \\lim _{\\tau \\rightarrow 0} \\frac{H(y_{t+\\tau } | y_t)- H(y_{t+\\tau } | x_t,y_t) }{\\tau }\\nonumber \\\\&=\\lim _{\\tau \\rightarrow 0} \\frac{I(y_{t+\\tau }:y_t,x_t)- I(y_{t+\\tau }:y_t) }{\\tau }\\ ,$ where the continuous time variable $t$ replaces the discrete index $i$ .",
"In practice ${\\overline{\\cal T}}_{X\\rightarrow Y} \\simeq {\\overline{T}}_{X\\rightarrow Y}/\\tau $ , but only for sufficiently small time step $\\tau $ .",
"The most direct strategy to evaluate Eq.",
"(REF ) would be to construct empirical estimators of the probabilities from histograms of the data.",
"Although this procedure works well for evaluating other quantities, for instance the entropy production in small stochastic systems [12], it completely fails in the case of transfer entropy.",
"Indeed, such a method leads to a non-zero TE even between uncorrelated signals, due to strong biases in standard estimators based on data binning.",
"In order to overcome this problem, we used the Kraskov-Stögbauer-Grassberger (KSG) estimator which does not rely on binning, as implemented in the software package JIDT (Java Information Dynamics Toolkit) [13].",
"Using estimators of this kind is particularly important for variables that take continuous values.",
"In the following, the inference method will be applied to time series generated by diffusion processes.",
"It will then be interesting to compare the TE rate ${\\overline{\\cal T}}_{X\\rightarrow Y}$ to another measure of information dynamics, the so-called information flow [14], [15], [16] (also dubbed learning rate in the context of sensory systems [17], [11]), which is defined as the time-shifted mutual information [18] ${\\cal I}^{flow}_{X \\rightarrow Y}=\\lim _{\\tau \\rightarrow 0} \\frac{I(y_t:x_t)- I(y_t:x_{t+\\tau }) }{\\tau }\\ .$ In the special case where the two processes $X$ and $Y$ experience independent noises (the system is then called bipartite) [15], one has the inequality ${\\cal I}^{flow}_{X \\rightarrow Y} \\le {\\cal T}_{X \\rightarrow Y}$ [17], which in turn implies that ${\\cal I}^{flow}_{X \\rightarrow Y} \\le \\overline{\\cal T}_{X \\rightarrow Y}$ when the joint process is Markovian.",
"Observing a violation of this inequality is thus a strong indication that the noises on $X$ and $Y$ are correlated.",
"As will be seen later, this is indeed the situation in biochemical networks, due the presence of the so-called extrinsic noise generated by the stochasticity in the cell and in the cell environment [19] which acts on all chemical reactions within the cell, and thus induces correlations.",
"In order to benchmark our inference method and perform a rigorous test in a controlled setting, we first applied it on times series generated by a simple model for which the transfer entropy and the information flow can be computed analytically.",
"The data were obtained by simulating the two coupled Langevin equations $m\\dot{v} &= -\\gamma v-a y+\\xi , \\nonumber \\\\\\tau _r\\dot{y} &= v-y+\\eta $ that describe the dynamics of a particle of mass $m$ subjected to a velocity-dependent feedback that damps thermal fluctuation [20], [21], [16] (in these equations, the dependence of the variables on the time $t$ is implicit).",
"Here, $\\xi (t)$ is the noise generated by the thermal environment with viscous damping $\\gamma $ and temperature $T$ , while $\\eta (t)$ is the noise associated with the measurement of the particle's velocity $v(t)$ .",
"The two noises are independent and Gaussian with zero-mean and variances $\\left< \\xi (t)\\xi (t^{\\prime })\\right>=2\\gamma k_BT\\delta (t-t^{\\prime })$ and $\\left< \\eta (t)\\eta (t^{\\prime })\\right>=\\sigma ^2\\delta (t-t^{\\prime })$ .",
"$a$ is the feedback gain and $\\tau _r$ is a time constant.",
"Figure: Transfer entropy T Y→V T_{Y\\rightarrow V} for the feedback model governed by Eqs.",
"() as a function ofthe noise intensity σ 2 \\sigma ^2 for k=1k=1 (blue circles), k=3k=3 (green circles) and k=5k=5 (red circles).",
"The parameter llpresent in the definition of Eq.",
"() is fixed to 1.The lower red (resp.",
"upper blue) solid line represents the value of T Y→V T_{Y\\rightarrow V} (resp.",
"T ¯ Y→V \\overline{T}_{Y\\rightarrow V}) obtained by multiplying the theoretical rate 𝒯 Y→V {\\cal T}_{Y\\rightarrow V} (resp.",
"𝒯 ¯ Y→V \\overline{\\cal T}_{Y\\rightarrow V}) given by Eq.",
"() (resp.",
"Eq.",
"() by the sampling time τ=10 -3 \\tau =10^{-3}.The parameters of the model areT=5T=5, γ=m=1\\gamma =m=1, τ r =0.1\\tau _r=0.1, and a=8a=8.The two Langevin equations were numerically integrated with the standard Heun's method [22] using a time step $\\Delta t=10^{-3}$ , and the transfer entropy in the steady state was estimated from 100 time series of duration $t=2000$ with a sampling time (i.e., the time between two consecutive data points) $\\tau = \\Delta t$ .",
"We first checked that the TE in the direction $Y \\rightarrow V$ does vanish in the absence of feedback, i.e.",
"for $a=0$ , whereas it is non-zero as soon as $a>0$ .",
"We then tested the influence of the measurement error $\\sigma ^2$ for a fixed value of the gain $a$ .",
"As can be seen in Fig REF , $T_{V \\rightarrow Y}$ diverges as $\\sigma ^2 \\rightarrow 0$ , a feature that will play an important role in our discussion of the model for the metabolic network.",
"In the figure, the color of the symbols correspond to three different values of the parameter $k$ which represents the history length in the definition of the transfer entropy (see Eq.",
"(REF )).",
"One can see that the estimates of $T_{V \\rightarrow Y}$ for $k=1$ are in very good agreement with the theoretical prediction for $\\overline{T}_{V \\rightarrow Y}$ (upper solid line).",
"Moreover, the estimates decrease as $k$ is increased from 1 to 5, and one can reasonably expect that the theoretical value of $T_{V \\rightarrow Y}$ (lower solid line) computed in Ref.",
"[16] and given by Eq.",
"(REF ) in the section on Methods would be reached in the limit $k \\rightarrow \\infty $ .",
"Finally, by estimating the information flow and the transfer entropy, we checked that inequality (REF ) holds, as a result of the independence of the two noises $\\xi $ and $\\eta $ (see section on Methods).",
"We are now in position to analyze the fluctuations in the metabolism of E. coli at the single cell level obtained in Ref.",
"[4] using the information-theoretic notions introduced and tested in the previous section.",
"Since there are a multitude of reactions and interactions involved in the metabolism of E. coli, a complete mechanistic description is not feasible, and our model-free inference method has a crucial advantage.",
"In Ref.",
"[4], the length of the cells was recorded as a function of time using image analysis, and the growth rate was then obtained by fitting this data over subparts of the cell cycle.",
"In the same experiment, the fluorescence level of GFP, which is co-expressed with growth enzymes LacY and LacZ was recorded.",
"Three set of experiments were carried out corresponding to three levels of an inducer IPTG: low, intermediate and high.",
"Figure: Pedigree tree representing the evolution of the colony of E. coli.",
"studied in Ref.",
".",
"The splittingof the branches corresponds to cell division events, each colored point is associated to a measurement of a single celland the colors represent the growth rates as shown in the bar in the lower part of the figure.The two time series have a branching structure due to the various lineages, which all start from a single mother cell as shown in Fig REF .",
"The experimental data thus come in the form of a large ensemble of short times series which represent a record of all the cell cycles.",
"There are about $\\sim 3000$ time series, with 2 to 8 measurement points in each of them which are represented as colored points in Fig REF .",
"In order to correctly estimate the transfer entropy from such data, we have analyzed the multiple time series as independent realizations of the same underlying stochastic process.",
"For the present analysis, we fix the history length parameters $k$ and $l$ to the value $k=l=1$ , which means that we focus on $\\overline{T}$ rather than $T$ .",
"We infer the values of $\\overline{T}$ in the two directions, from growth (denoted $\\mu $ ) to gene expression (denoted $E$ ) and vice versa.",
"The results obtained for the three concentrations of IPTG are represented in Table REF .",
"The negative value of $\\overline{T}_{\\mu \\rightarrow E}$ which is found in the intermediate case is due to the numerical inference method and should be regarded as a value which cannot be distinguished from zero.",
"Table: Inferred values of the transfer entropies in the directions E→μE\\rightarrow \\mu andμ→E\\mu \\rightarrow E, and the difference ΔT ¯ E→μ =T ¯ E→μ -T ¯ μ→E \\Delta \\overline{T}_{E\\rightarrow \\mu }=\\overline{T}_{E\\rightarrow \\mu }-\\overline{T}_{\\mu \\rightarrow E}for low, medium and high concentrations of IPTG based on the data of ref.",
".",
"The TE are given in nats.Based on this analysis, we conclude that the influence between the variables is directed primarily from enzyme expression to growth in the low and intermediate IPTG experiments, while it mainly proceeds in the reverse direction in the high IPTG experiment.",
"Such results are in line with the conclusions of Ref.",
"[4] based on the measured asymmetry of the time-lagged cross-correlations.",
"Moreover, the present analysis provides an estimate of the influence between the two variables separately in the two directions from $E$ to $\\mu $ and from $\\mu $ to $E$ .",
"In particular, we observe for the low experiment that the values of TE in the two directions are of same order of magnitude, whereas in the intermediate experiment the TE from $E$ to $\\mu $ is larger, a feature which could not have been guessed from measured time delays.",
"We now turn to the analysis of the model proposed in Ref.",
"[4] to account for the experimental data.",
"The question we ask is whether the model correctly reproduces the above results for the transfer entropies, in particular the change in the sign of $\\Delta \\overline{T}_{E\\rightarrow \\mu }$ for the high concentration of IPTG.",
"The central equation of the model describes the production of the enzyme as $\\dot{E} = p-\\mu \\cdot E,$ where $E$ is the enzyme concentration, $p$ its production rate, and $\\mu $ the rate of increase in cell volume.",
"Although the function $p$ is typically non-linear, its precise expression is irrelevant because (REF ) is linearized around the stationary point defined by the mean values $E=E_0$ and $\\mu =\\mu _0$ .",
"This linearization then yields $\\delta \\dot{E} = \\delta p - \\delta \\mu E_0 - \\mu _0 \\delta E, \\,$ in terms of perturbed variables $\\delta X(t)=X(t)-X_0$ , where $X_0$ denotes the mean of $X$ .",
"The model of Ref.",
"[4] is essentially phenomenological in nature because it approximates the noises as Gaussian processes.",
"Although this approximation is often done in this field, it may not always hold since fluctuations due to low copy numbers are generally not Gaussian [23].",
"In any case, the model contains three Gaussian noises: $N_G$ is a common component while $N_E$ and $N_\\mu $ are component specific to $E$ and $\\mu $ .",
"These noises are assumed to be independent Ornstein-Uhlenbeck noises with zero mean and autocorrelation functions $\\langle N_i(t)N_i(t^{\\prime })\\rangle =\\eta _i^2e^{-\\beta _i\\vert t-t^{\\prime }\\vert }$ ($i=E,\\mu ,G$ ).",
"As commonly done, the three Ornstein-Uhlenbeck noises are generated by the auxiliary equations $\\dot{N}_i=-\\beta _i N_i+\\xi _i, \\,$ where the $\\xi _i^{\\prime }s$ are zero-mean Gaussian white noises satisfying $\\langle \\xi _i(t)\\xi _j(t^{\\prime })\\rangle =\\theta _i^2\\delta (t-t^{\\prime })\\delta _{ij}$ with $\\theta _i=\\eta _i\\sqrt{2\\beta _i}$ .",
"Introducing the constant logarithmic gains $T_{XY}$ that represent how a variable $X$ responds to the fluctuations of a source $Y$ , the equations of the model read [4] $\\frac{\\delta p}{E_0 \\mu _0} &= T_{EE} \\frac{\\delta E}{E_0} + T_{E G} N_G + N_E, \\nonumber \\\\\\frac{\\delta \\mu }{\\mu _0} &= T_{\\mu E} \\frac{\\delta E}{E_0} + T_{\\mu G} N_G + N_\\mu ,$ where specifically $T_{E\\mu }=-1$ and $T_{\\mu G}=1$ .",
"Then, eliminating $\\delta p$ from Eqs.",
"(REF ) and (REF ), one obtains the coupled equations $\\dot{x}&=\\mu _0\\big [(T_{EE}-1)x+T_{E\\mu }y+T_{EG}N_G+N_E\\big ]\\nonumber \\\\y&=T_{\\mu E}x+T_{\\mu G} N_G+N_{\\mu },$ where we have defined the reduced variables $x=\\delta E/E_0$ , $y=\\delta \\mu /\\mu _0$ .",
"We stress that $N_G$ is an extrinsic noise that affects both the enzyme concentration and the growth rate, whereas $N_E$ (resp.",
"$N_{\\mu }$ ) is an intrinsic noise that only affects $E$ (resp.",
"$\\mu $ ).",
"Note that the two effective noises $T_{EG}N_G+N_E$ and $T_{\\mu G}N_G+N_{\\mu }$ acting on $\\dot{x}$ and $y$ are colored and correlated, which makes the present model more complicated than most stochastic models studied in the current literature.",
"In fact, since we are mainly interested in the information exchanged between $x$ and $y$ , it is convenient to replace one of the noises, say $N_G$ , by the dynamical variable $y$ .",
"Differentiating the second equation in Eq.",
"(REF ), using Eq.",
"(REF ) and performing some simple manipulations, one then obtains a new set of equations for the four random variables $x,y, u\\equiv N_E,v\\equiv N_{\\mu }$ : $\\dot{x}&=a_1x+a_2 u +a_3v + a_4y\\nonumber \\\\\\dot{y}&=b_1x+b_2 u +b_3v + b_4y+\\xi _{y}\\nonumber \\\\\\dot{u}&=-\\beta _E u+\\xi _E\\nonumber \\\\\\dot{v}&=-\\beta _{\\mu } v+\\xi _{\\mu }\\ ,$ where the coefficients $a_j$ and $b_j$ ($j=1...4$ ) are defined by Eqs.",
"(REF ) in the section on Methods and $\\xi _{y}=\\xi _{\\mu }+\\xi _G$ is a new white noise satisfying $\\langle \\xi _{y}(t)\\xi _{y}(t^{\\prime })\\rangle =(\\theta _{\\mu }^2+\\theta _G^2)\\delta (t-t^{\\prime })$ and $\\langle \\xi _{y}(t)\\xi _{\\mu }(t^{\\prime })\\rangle =\\theta _{\\mu }^2\\delta (t-t^{\\prime })$ .",
"The calculation of the transfer entropy rate ${\\overline{\\cal T}}_{X\\rightarrow Y}$ (which coincides with ${\\overline{\\cal T}}_{E\\rightarrow \\mu }$ since the TE is invariant under the change of variables from $E$ to $x$ and $\\mu $ to $y$ ) is detailed in the section on Methods, together with the calculation of the information flows.",
"The final expression reads ${\\overline{\\cal T}}_{X\\rightarrow Y}&=\\frac{1}{4(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)}\\int dx\\:dy\\: p(x,y)\\big [\\bar{g}_{y}^2(x,y)-{\\bar{\\bar{g}}}_{y}^2(y)\\big ]\\,$ where $p(x,y)$ is the steady state probability distribution and the functions $\\bar{g}_{y}$ and ${\\bar{\\bar{g}}}_{y}$ are defined in Eqs.",
"(REF ) and (REF ), respectively.",
"This result agrees with that obtained in Refs.",
"[11], [18] and in [24] in special cases.",
"In Table REF , we show the results of the analysis of the time series generated by Eqs.",
"(REF ) using our numerical inference method with a sampling time $\\tau =1$ min (equal to the time step $\\Delta t$ used to numerically integrate the model).",
"One can see that the estimates of ${\\overline{\\cal T}}_{E\\rightarrow \\mu }$ are in good agreement with the predictions of Eq.",
"(REF ), with the values of the model parameters taken from Table S1 in Ref. [4].",
"Note that the negative number given by the inference method in the high IPTG experiment signals that the actual value of ${\\overline{\\cal T}}_{E\\rightarrow \\mu }$ cannot be distinguished from zero, which is indeed the theoretical prediction.",
"In contrast, the estimated and theoretical results for ${\\overline{\\cal T}}_{\\mu \\rightarrow E}$ do not agree, as the inference method yields finite values in all cases whereas the theoretical values diverge.",
"Table: Comparison between the theoretical values of the transfer entropy rates 𝒯 ¯ E→μ \\overline{\\cal T}_{E\\rightarrow \\mu } and 𝒯 ¯ μ→E \\overline{\\cal T}_{\\mu \\rightarrow E} for the modelof Ref.",
"and the values inferred from simulation data.Averages are taken over 100 times series of duration 10 6 10^6 min, sampled every 1 min.Figure: Transfer entropy rates 𝒯 ¯ E→μ {\\overline{\\cal T}}_{E \\rightarrow \\mu } and 𝒯 ¯ μ→E {\\overline{\\cal T}}_{\\mu \\rightarrow E} in the low IPTG experiment:(a) Original model of Ref.",
"(b) Modified model where N E N_E is a white noise.",
"The symbols are the estimates from theinference method when varying the sampling time τ\\tau , and the solid lines are the theoretical predictions from Eq.",
"() in (a)and from Eqs.",
"() in (b).",
"Note that 𝒯 ¯ μ→E {\\overline{\\cal T}}_{\\mu \\rightarrow E} diverges as τ\\tau goes to zero in (a) but not (b).This behavior is due to the absence of a white noise source directly affecting the dynamical evolution of $x$ in the set of Eqs.",
"(REF ).",
"Indeed, as pointed out in Ref.",
"[6] and also observed above in Fig REF , a TE rate diverges when the coupling between the variables is deterministic.",
"In the model of Ref.",
"[4], this feature can be traced back to the fact that the noise $N_E$ affecting the enzyme concentration is colored with a finite relaxation time $\\beta _E^{-1}$ .",
"Therefore, when taking the limit $\\tau \\rightarrow 0$ in Eq.",
"(REF ), one explores a time interval $\\tau <\\beta _E^{-1}$ where $N_E$ is not really random.",
"This is illustrated in Fig REF a that corresponds to the low IPTG experiment: we see that the estimate of ${\\overline{\\cal T}}_{\\mu \\rightarrow E}$ with the inference method is indeed diverging when the sampling time $\\tau $ approaches zero.",
"On the other hand, as expected, ${\\overline{\\cal T}}_{E \\rightarrow \\mu }$ remains finite and the points nicely lie on the plateau determined by Eq.",
"(REF ).",
"The obvious and simplest way to cure this undesirable feature of the original model is to treat $N_E$ as a purely white noise, which amounts to taking the limit $\\beta _E^{-1} \\rightarrow 0$ .",
"In fact, it is noticeable that the values of $\\beta _E^{-1}$ extracted from the fit of the correlation functions in Ref.",
"[4] (resp.",
"$\\beta _E^{-1}=10.7,9.9$ and $8.15$ min for the low, intermediate, and high IPTG concentrations) are significantly smaller than the time steps $\\tau _{exp}$ used for collecting the data (resp.",
"$\\tau _{exp}=28, 20$ and $15.8$ min).",
"Therefore, it is clear that the experimental data are not precise enough to decide whether $N_E$ is colored or not.",
"This issue does not arise for the other relaxation times in the model, $\\beta _{\\mu }^{-1}=\\beta _G^{-1}$ and $\\mu _0^{-1}$ , which are much longer (at least for the low and intermediate IPTG concentrations), and can be correctly extracted from the experimental data.",
"We thus propose to modify the model of Ref.",
"[4] by describing $N_E$ as a Gaussian white noise with variance $\\langle N_E(t)N_E(t^{\\prime })\\rangle =2D_E\\delta (t-t^{\\prime })$ and the same intensity as the colored noise in the original model, i.e.",
"$D_E=\\eta _E^2/\\beta _E$ (which yields $D_E\\approx 0.188 h,0.100h,0.031h$ for the three IPTG concentrations).",
"Unsurprisingly, this modification does not affect the auto and cross-correlation functions used to fit the data, as shown in Fig REF (see also section on Methods for a detailed calculation).",
"On the other hand, the values of ${\\overline{\\cal T}}_{E\\rightarrow \\mu }$ are changed (compare Tables REF and REF ) and, more importantly, ${\\overline{\\cal T}}_{\\mu \\rightarrow E}$ , given by Eq.",
"(REF ) is now finite.",
"As a result, the model predicts that the difference $\\Delta {\\overline{\\cal T}}_{E\\rightarrow \\mu }={\\overline{\\cal T}}_{E\\rightarrow \\mu }-{\\overline{\\cal T}}_{\\mu \\rightarrow E}$ is positive at low and intermediate IPTG concentrations and becomes negative at high concentration, which is in agreement with the direct analysis of the experimental data in Table REF .",
"In contrast, $\\Delta {\\overline{ \\cal T}}_{E\\rightarrow \\mu }$ was always negative in the original model as ${\\overline{\\cal T}}_{\\mu \\rightarrow E}$ is infinite.",
"Figure: (a) Autocorrelation function R μμ (τ)R_{\\mu \\mu }(\\tau ) for the three IPTG concentrations.Black lines: original model of Ref.",
", red circles: simplified model where N E N_E is a white noise.",
"(b) Same as (a) for R EE (τ)R_{EE}(\\tau ).",
"(c) Same as (a) for R Eμ (τ)R_{E\\mu }(\\tau )Table: Theoretical values of the transfer entropy rates 𝒯 ¯ E→μ \\overline{\\cal T}_{E\\rightarrow \\mu } and 𝒯 ¯ μ→E \\overline{\\cal T}_{\\mu \\rightarrow E} and theirdifference in the modified model.This new behavior of the TE rates is also manifest when the inference method is applied to the time series generated by the model and the sampling time $\\tau $ is varied.",
"As observed in Fig REF b, the inferred value of ${\\overline{\\cal T}}_{\\mu \\rightarrow E}$ no longer diverges as $\\tau \\rightarrow 0$ (compare the vertical scale with that in Fig REF a).",
"The estimates of ${\\overline{\\cal T}}_{E\\rightarrow \\mu }$ and ${\\overline{\\cal T}}_{\\mu \\rightarrow E}$ are also in good agreement with the theoretical predictions, except for the shortest value of $\\tau $ which is equal to the time step $\\Delta t=1$ min used to numerically integrate the equations.",
"It worth mentioning, however, that the error bars increase as $\\tau $ is decreased.",
"While the change in the sign of $\\Delta {\\overline{\\cal T}}_{E\\rightarrow \\mu }$ is now confirmed by the model, which is the main outcome of our analysis, one may also wonder whether the numerical values in Table REF are recovered.",
"This requires to multiply the rates in Table REF by the experimental sampling times $\\tau _{exp}$ which are different in each experiment, as indicated above.",
"One then observes significant discrepancies for the low and intermediate IPTG experiments.",
"We believe that the problem arises from the presence of many short time series in the set of experimental data.",
"This is a important issue that needs to be examined in more detail since it may be difficult to obtain long time series in practice.",
"Figure: Inferred values of Δ𝒯 ¯ E→μ \\Delta {\\cal \\overline{T}}_{E\\rightarrow \\mu } for the low IPTG experiment as a function of the length NN of the time seriesgenerated by the modified model.",
"Panels (a) and (b) correspond to sampling times τ=6\\tau =6 min and τ=1\\tau =1 min, respectively.Δ𝒯 ¯ E→μ (∞)\\Delta {\\cal \\overline{T}}_{E\\rightarrow \\mu }(\\infty ) is the exact asymptotic value.To this aim, we have studied the convergence of the estimates of $\\Delta {\\cal \\overline{T}}_{E\\rightarrow \\mu }$ to the exact asymptotic value as a function of $N$ , the length of the time series generated by the model in the stationary regime.",
"As shown in Fig REF , the convergence with $N$ is slow, which means that one can make significant errors in the estimation of $\\Delta {\\cal \\overline{T}}_{E\\rightarrow \\mu }$ if $N$ is small.",
"On the other hand, the convergence can be greatly facilitated by choosing a value of the sampling time which is not too short (but of course shorter than the equilibration time of the system), for instance $\\tau =6$ min instead of 1 min in the case considered in Fig REF .",
"The important observation is that the sign of $\\Delta {\\cal \\overline{T}}_{E\\rightarrow \\mu }$ is then correctly inferred even with $N\\approx 1000$ .",
"In contrast, with $\\tau =1$ min, this is only possible for much longer series, typically $N\\approx 50000$ .",
"This is an encouraging indication for experimental studies, as the overall acquisition time of the data can be significantly reduced.",
"Finally, we briefly comment on the results for the information flows ${\\cal I}^{flow}_{E\\rightarrow \\mu }$ and ${\\cal I}^{flow}_{\\mu \\rightarrow E}$ .",
"As already pointed out, the fact that the noises acting on the two random variables are correlated invalidates inequality (REF ).",
"This is indeed what is observed in Table REF .",
"It is also noticeable that ${\\cal I}^{flow}_{E\\rightarrow \\mu }\\ne -{\\cal I}^{flow}_{\\mu \\rightarrow E}$ , except in the high IPTG experiment where $T_{\\mu E}=0$ .",
"Table: Comparison between the theoretical values of the TE rates and the information flows for the modified modeland the values inferred from simulation data (all quantities are expressed in h -1 ^{-1}).",
"The analysis was performed with a sampling τ=6\\tau =6 min and 100 time series of 10 6 10^6 points.A challenge when studying any biochemical network is to properly identify the direction of information.",
"In this work, using the notion of transfer entropy, we have characterized the directed flow of information between the single cell growth rate and the gene expression, using a method that goes beyond what could be obtained from correlation functions, or from other inference techniques which do not exploit dynamical information.",
"Another crucial challenge in the field is to properly model the various noise components.",
"It turns out that biological systems are generally non-bipartite due the presence of an extrinsic component in the noise.",
"The present work provides on the one hand analytical expressions for the magnitude of the transfer entropy (or at least an upper bound on it) and of the information flow when the system is not bipartite, and, on the other hand a numerical method to infer the TE in all cases.",
"Furthermore, we have shown that one can correctly infer the sign of the TE difference even with short time series by properly choosing the sampling time (see Ref.",
"[25] for more details on the dependence of TE on the sampling time).",
"To conclude, we would like to emphasize that the transfer entropy is a general tool to identify variables which are relevant for time series prediction [26].",
"As such, the method has a lot of potential beyond the particular application covered in this paper: Predicting the current or future state of the environment by sensing it is an adaptation strategy followed by biological systems which can be understood using information-theoretic concepts [27], [11].",
"Similarly, during evolution, biological systems accumulate information from their environment, process it and use it quasi-optimally to increase their own fitness [28], [29].",
"In this context, transfer entropy-based methods have the potential to identify the directional interactions in co-evolution processes, which could be for instance the genomic evolution of a virus compared to that of its antigenes [30].",
"With the recent advances in high-throughput techniques and experimental evolution, we might soon be able to predict reliably the evolution of biological systems [31], and without doubt tools of information theory will play a key role in these advances.",
"In this section, we provide a detailed analysis of the information-theoretic quantities for the various models considered in this paper.",
"The section is organized as follows: Basic information-theoretic measures Transfer entropy and information flow in the feedback cooling model Transfer entropy rates and information flows in the model of Ref.",
"[4] for a metabolic network Transfer entropy rates and information flows in the modified model for the metabolic network Below we briefly recall some definitions and properties of the information-theoretic measures.",
"A fundamental quantity is the Shannon entropy which quantifies the uncertainty associated with the measurement $x$ of a random variable $X$ : $H(X)=-\\sum _{x } P(x) \\ln P(x),$ where $P(x)$ is the probability that event $x$ is realized, given an ensemble of possible outcomes.",
"With this convention, the entropy is measured in nats.",
"Similarly, for two random variables $X$ and $Y$ , one defines the joint Shannon entropy $H(X,Y)=-\\sum _{x,y} P(x,y) \\ln P(x,y),$ and the conditional Shannon entropy $H(X | Y)=-\\sum _{x,y} P(x,y) \\ln P(x | y)\\ ,$ where $P(x,y)$ and $P(x\\vert y)$ are joint and conditional probability distribution functions, respectively.",
"The mutual information $I(X:Y)$ is then a symmetric measure defined as $I(X:Y) &= \\sum _{x,y} P(x,y) \\ln \\frac{P(x,y)}{P(x)P(y)}, \\nonumber \\\\&= H(X) - H(X | Y) \\nonumber \\\\&= H(Y)-H(Y\\vert X)\\ ,$ which quantifies the reduction of the uncertainty about $X$ (resp.",
"$Y$ ) resulting from the knowledge of the value of $Y$ (resp$X$ ).",
"The more strongly $X$ and $Y$ are correlated, the larger $I(X:Y)$ is.",
"These notions can be readily extended to random processes $X= \\lbrace X_i\\rbrace $ and $Y= \\lbrace Y_i\\rbrace $ viewed as collections of individual random variables sorted by an integer time index $i$ .",
"The mutual information between the ordered time series $ \\lbrace x_i\\rbrace $ and $\\lbrace y_i\\rbrace $ , realizations of $X$ and $Y$ , is then defined as $I(X: Y) = I(Y:X) \\equiv \\sum _{ \\lbrace x_i, y_i \\rbrace } P(x_i,y_i) \\ln \\frac{P(x_{i}, y_i)}{P(x_{i})P(y_i)}\\ ,$ and characterizes the undirected information exchanged between the two processes.",
"The conditional mutual information is defined similarly.",
"In contrast, the transfer entropy $T_{X \\rightarrow Y}$ is a information-theoretic measure that is both asymmetric and dynamic as it captures the amount of information that a source process $X$ provides about the next state of a target process $Y$ .",
"More precisely, as defined by Eq.",
"(1) in the introduction, $T_{X \\rightarrow Y}=\\sum _i \\: [ I(Y_{i+1}:X_i^{(l)},Y_i^{(k)})-I(Y_{i+1}:Y_i^{(k)})],$ where $k$ and $l$ define the lengths of the process histories, i.e., $Y_i^{(k)}=\\lbrace Y_{i-k+1},\\cdots , Y_i\\rbrace $ and $X_i^{(l)}=\\lbrace X_{i-l+1},\\cdots , X_i\\rbrace $ .",
"In this work, we have focused on a history length of 1 (i.e.",
"$k=l=1$ ) and denoted the corresponding TE by $\\overline{T}_{X \\rightarrow Y}$ .",
"Hence, ${\\overline{T}}_{X \\rightarrow Y}=\\sum _i [ H(Y_{i+1}|Y_i)-H(Y_{i+1}|X_i,Y_i)]$ , which is an upper bound to $T_{X \\rightarrow Y}(k,l)$ for $l=1$ when the joint process $\\lbrace X,Y \\rbrace $ obeys a Markovian dynamics [11].",
"On the other hand, the information flow from $X$ to $Y$ is defined as the time-shifted mutual information ${\\cal I}^{flow}_{X \\rightarrow Y}=\\sum _i [I(Y_i : X_i) - I (Y_i:X_{i+1})],$ and informs on the reduction of uncertainty in $Y_i$ when knowing about $X_{i+1}$ as compared to what we had with $X_i$ only.",
"In practice, ${\\cal I}^{flow}_{X \\rightarrow Y}$ can be obtained by shifting in time one time series with respect to the other one.",
"Contrary to the transfer entropy which is always a positive quantity, the information flow ${\\cal I}^{flow}_{X \\rightarrow Y}$ may be negative or positive, depending on whether $X$ sends information to $Y$ (or $X$ gains control of $Y$ ), or $Y$ sends information to $X$ (or $X$ looses control over $Y$ ).",
"In a bipartite system one has ${\\cal I}^{flow}_{X \\rightarrow Y} = - {\\cal I}^{flow}_{Y \\rightarrow X}$ in the stationary regime.",
"This is no longer true when the system is non-bipartite.",
"We first recall the theoretical expressions of the transfer entropy rates and the information flows for the feedback-cooling model described by Eqs.",
"(REF ).",
"These quantities were computed in Ref. [16].",
"The transfer entropy rates in the stationary state are given by ${\\cal T}_{V \\rightarrow Y} &= \\frac{\\gamma }{2m}\\left(\\sqrt{1+\\frac{2T}{\\gamma \\sigma ^2}}-1\\right) \\nonumber \\\\{\\cal T}_{Y \\rightarrow V} &= \\frac{1}{2\\tau _r}\\left(\\sqrt{1+\\frac{a^2\\sigma ^2}{2\\gamma T}}-1\\right).$ Note that $2T/(\\gamma \\sigma ^2)$ is the signal-to-noise ratio that quantifies the relative size of the measurement accuracy to the thermal diffusion of the velocity.",
"Accordingly, the TE rate $ {\\cal T}_{V \\rightarrow Y}$ diverges when the control is deterministic.",
"The information flow ${\\cal I}^{flow}_{V \\rightarrow Y}$ is given by ${\\cal I}^{flow}_{V \\rightarrow Y}= \\frac{\\gamma }{m} \\left( \\frac{T \\langle y^2\\rangle }{m \\vert {\\bf \\Sigma }\\vert } - 1 \\right)$ where $\\vert {\\bf \\Sigma }\\vert $ is the determinant of the covariance matrix.",
"The analytical expressions of the elements of the matrix, $\\langle v^2\\rangle ,\\langle y^2\\rangle $ and $\\langle v y\\rangle $ , are given by Eqs.",
"(A2) in Ref. [16].",
"In contrast with $ {\\cal T}_{V \\rightarrow Y}$ , the information flow ${\\cal I}^{flow}_{V \\rightarrow Y}$ remains finite as the noise intensity vanishes.",
"The upper bounds to the transfer entropies (see Eq.",
"(2)) were computed in Ref.",
"[24] in the general case of coupled linear Langevin equations.",
"For the feedback cooling model, one obtains ${\\overline{\\cal T}}_{V \\rightarrow Y} &= \\frac{1}{2\\sigma ^2\\langle y^2\\rangle }\\vert {\\bf \\Sigma }\\vert \\nonumber \\\\{\\overline{\\cal T}}_{Y \\rightarrow V} &= \\frac{a^2}{4\\gamma k_BT\\langle v^2\\rangle }\\vert {\\bf \\Sigma }\\vert \\ .$ As shown in Fig REF , the estimate of the transfer entropy obtained by the inference method is in good agreement with the theoretical value (we stress that the figure shows the rates multiplied by the sampling time $\\tau =10^{-3}$ ).",
"In Fig REF , we also obtain satisfactory agreement between inferred value of the information flow ${\\cal I}^{flow}_{V \\rightarrow Y}$ and theoretical value, when representing these quantities against the noise intensity $\\sigma ^2$ .",
"These results of this figure confirm the inequalities ${\\cal I}^{flow}_{V \\rightarrow Y}\\le {\\cal T}_{V \\rightarrow Y}\\le {\\overline{\\cal T}}_{V \\rightarrow Y}$ .",
"Figure: 𝒯 V→Y ,𝒯 ¯ V→Y {\\cal T}_{V \\rightarrow Y}, {\\overline{\\cal T}}_{V \\rightarrow Y} and ℐ V→Y flow {\\cal I}^{flow}_{V \\rightarrow Y} as a function of the noise intensity σ 2 \\sigma ^2.",
"The parameters of the model are T=5,γ=m=1,τ r =0.1T = 5,\\gamma = m = 1, \\tau _r =0.1 and a=-0.7a=-0.7.We first compute the stationary probability distributions (pdfs) associated with Eqs.",
"(REF ) were the coefficients $a_j$ and $b_j$ are given by $a_1&=-[\\mu _E +\\mu _0 T_{\\mu E}(T_{EG}-1)]\\nonumber \\\\a_2&=\\mu _0\\nonumber \\\\a_3&=-\\mu _0 T_{EG}\\nonumber \\\\a_4&=\\mu _0(T_{EG}-1)\\nonumber \\\\b_1&=T_{\\mu E}[\\beta _G-\\mu _E -\\mu _0 T_{\\mu E}(T_{EG}-1)]\\nonumber \\\\b_2&=\\mu _0T_{\\mu E}\\nonumber \\\\b_3&=\\beta _G-\\beta _{\\mu }-\\mu _0 T_{\\mu E}T_{EG}\\nonumber \\\\b_4&=\\mu _0 T_{\\mu E}(T_{EG}-1)-\\beta _G \\ .$ We recall that $\\mu _E=\\mu _0(1+T_{\\mu E}-T_{EE})$ sets the timescale of $E$ -fluctuations [4].",
"Since Eqs.",
"(REF ) describe a set of coupled Markovian Ornstein-Uhlenbeck processes, the stationary pdf $p_{xuvy}(x,u,v,y)$ is Gaussian and given by $p_{xuvy}(x,u,v,y)=\\frac{1}{(2\\pi )^2\\sqrt{\\vert {\\bf \\Sigma } \\vert }} e^{-\\frac{1}{2}(x,u,v,y).",
"{\\bf \\Sigma }^{-1}.",
"(x,u,v,y)^T}\\ ,$ where ${\\bf \\Sigma }$ is the covariance matrix which obeys the Lyapunov equation [32] ${\\bf A}{\\bf \\Sigma }+{\\bf \\Sigma }{\\bf A}^T=2{\\bf D}\\ ,$ where ${\\bf A}=\\left(\\begin{array}{cccc}-a_1&-a_2 & -a_3&-a_4 \\\\0&\\beta _E &0 &0 \\\\0&0 &\\beta _{\\mu }& 0 \\\\-b_1&-b_2 &-b_3 &-b_4\\end{array}\\right), \\, \\, \\, \\rm {and } \\, \\, \\,{\\bf D}=\\left(\\begin{array}{cccc}0&0& 0&0 \\\\0&\\beta _E\\eta _E^2 &0 &0 \\\\0&0 &\\beta _{\\mu }\\eta _{\\mu }^2&\\beta _{\\mu }\\eta _{\\mu }^2 \\\\0&0 & \\beta _{\\mu }\\eta _{\\mu }^2& \\beta _G\\eta _G^2 + \\beta _{\\mu }\\eta _{\\mu }^2\\end{array}\\right) \\ .$ The solution of Eq.",
"(REF ) reads $\\sigma _{11}&=\\frac{\\mu _0^2}{\\mu _E}\\Big [\\frac{\\eta _E^2}{\\mu _E+\\beta _E}+\\frac{\\eta _{\\mu }^2}{\\mu _E+\\beta _{\\mu }}+\\frac{(T_{EG}-1)^2}{\\mu _E+\\beta _G}\\eta _G^2\\Big ]\\nonumber \\\\\\sigma _{12}&=\\sigma _{21}=\\frac{\\mu _0}{\\mu _E+\\beta _E}\\eta _E^2\\nonumber \\\\\\sigma _{13}&=\\sigma _{31}= \\frac{-\\mu _0}{\\mu _E+\\beta _{\\mu }}\\eta _{\\mu }^2\\nonumber \\\\\\sigma _{14}&=\\sigma _{41}=\\frac{\\mu _0}{\\mu _E}\\Big [\\frac{\\mu _0T_{\\mu E}}{\\mu _E+\\beta _E}\\eta _E^2+\\frac{(\\mu _0T_{\\mu E}-\\mu _E)}{\\mu _E+\\beta _{\\mu }}\\eta _{\\mu }^2 \\nonumber \\\\&+\\frac{(T_{EG}-1)\\big [\\mu _0T_{\\mu E}(T_{EG}-1)+\\mu _E\\big ]}{\\mu _E+\\beta _G}\\eta _G^2\\Big ] \\nonumber \\\\\\sigma _{22}&=\\eta _E^2\\nonumber \\\\\\sigma _{23}&=0\\nonumber \\\\\\sigma _{24}&=\\sigma _{42}=\\frac{\\mu _0T_{\\mu E}}{ \\mu _E+\\beta _E} \\eta _E^2\\nonumber \\\\\\sigma _{33}&=\\eta _{\\mu }^2\\nonumber \\\\\\sigma _{34}&=\\sigma _{43}=\\frac{\\mu _E+\\beta _{\\mu }-\\mu _0T_{\\mu E}}{\\mu _E+\\beta _{\\mu }}\\eta _{\\mu }^2\\nonumber \\\\\\sigma _{44}&=\\frac{\\mu _0^2T_{\\mu E}^2}{ \\mu _E(\\mu _E+\\beta _E)}\\eta _E^2+\\frac{\\big [(\\mu _0T_{\\mu E}-\\mu _E)^2+ \\mu _E\\beta _{\\mu }\\big ]}{ \\mu _E(\\mu _E+\\beta _{\\mu })}\\eta _{\\mu }^2 \\nonumber \\\\&+\\frac{\\mu _0^2T_{\\mu E}^2(T_{EG}-1)^2+\\mu _E\\big [\\mu _E+\\beta _G\\big ]}{\\mu _E(\\mu _E+\\beta _{G})}\\eta _G^2 \\nonumber \\\\&+\\frac{2\\mu _0T_{\\mu E}(T_{EG}-1)\\big ]}{\\mu _E(\\mu _E+\\beta _{G})} \\eta _G^2$ From this we can compute all marginal pdfs, in particular $p_{xy}(x,y)&=\\frac{1}{2\\pi \\sqrt{\\sigma _{11}\\sigma _{44}-\\sigma _{14}^2}}e^{-\\frac{1}{2}\\frac{\\sigma _{44}x^2-2\\sigma _{14}xy+\\sigma _{11}y^2}{\\sigma _{11}\\sigma _{44}-\\sigma _{14}^2}}\\ ,$ and $p_{x}(x)&=\\frac{1}{\\sqrt{2\\pi \\sigma _{11}}}e^{-\\frac{x^2}{2\\sigma _{11}}}\\nonumber \\\\p_{y}(y)&=\\frac{1}{\\sqrt{2\\pi \\sigma _{44}}}e^{-\\frac{y^2}{2\\sigma _{44}}}\\ .$ As an illustration, the steady-state pdf $p(\\mu )=\\frac{1}{\\mu _0}p_y(y=\\frac{\\mu -\\mu _0}{\\mu _0})$ is plotted in Fig REF for the three different IPTG concentrations (low, intermediate, and high).",
"The agreement with the experimental curves displayed in Fig 1d of Ref.",
"[4] is satisfactory.",
"Figure: Steady-state probability distribution of the growth rate for the three IPTG concentrations:low (black), intermediate (red), high (blue).For completeness, we also quote the expressions of $R_{pp}(0)$ and $R_{p\\mu }(0)$ (properly normalized) obtained from the definition $\\delta p/(\\mu _0 E_0)=\\delta \\dot{E}/(\\mu _0 E_0)+\\delta \\mu /\\mu _0+\\delta E/E_0=(T_{EE}-T_{EG}T_{\\mu E})x+u-T_{EG}(v-y)$ : $R_{pp}(0)&=(T_{EE}-T_{EG}T_{\\mu E})^2\\sigma _{11}+\\sigma _{22}+T_{EG}^2(\\sigma _{33}+\\sigma _{44}) \\nonumber \\\\&+2(T_{EE}-T_{EG}T_{\\mu E})[\\sigma _{12}+T_{EG}(\\sigma _{14}-\\sigma _{13})] \\nonumber \\\\&+2T_{EG}\\sigma _{24}-2T_{EG}^2\\sigma _{34} \\\\R_{p\\mu }(0)&=\\frac{(T_{EE}-T_{EG}T_{\\mu E})\\sigma _{14}+\\sigma _{24}+T_{EG}(\\sigma _{44}-\\sigma _{34})}{\\sqrt{R_{pp}(0)R_{\\mu \\mu }(0)}}$ with $R_{\\mu \\mu }(0)=\\sigma _{44}$ .",
"The correlation functions $R_{\\mu \\mu }(\\tau )$ , $R_{EE}(\\tau )$ , and $R_{E\\mu }(\\tau )$ , obtained by taking the inverse Fourier transform of Eqs.",
"(6) in the Supplementary Information of [4] are plotted in Fig REF .",
"In passing, we correct a few misprints in these equations: i) The correct expression of $R_{\\mu \\mu }(\\tau )$ is obtained by replacing $A_E(\\tau )$ by $R_{EE}(\\tau )$ in the first term of Eq.",
"(12) in the Supplementary Information of [4].",
"ii) Eq.",
"REF corresponds to $R_{E \\mu }(\\tau )$ and not to $R_{\\mu E}(\\tau )=R_{E \\mu }(-\\tau )$ .",
"Eq.",
"(8) then gives the correct expression of $R_{E\\mu }(\\tau )$ (and not of $R_{\\mu E}(\\tau )$ ) provided the function $A_X(\\tau )$ defined in Eq.",
"(10) is altered.",
"For $\\tau \\ge 0$ , one should have $A_X(\\tau )=\\theta _X^2\\frac{\\mu _0}{2\\beta _X(\\beta _X+\\mu _E)}e^{-\\beta _X t}\\ .$ We now address the computation of the conditional probabilities $p_{x^{\\prime } y^{\\prime }}^y(y,t+\\tau \\vert x^{\\prime },y^{\\prime },t)$ and $p_{y^{\\prime }}^y( y,t+\\tau \\vert y^{\\prime },t)$ at first order in $\\tau $ .",
"This will allow us to obtain the expressions of the upper bounds to the transfer entropy rates defined by ${\\overline{\\cal T}}_{X\\rightarrow Y}&=\\lim _{\\tau \\rightarrow 0}\\frac{I[y_{t+\\tau }:x_t,y_t]-I[y_{t+\\tau }:y_{t}]}{\\tau }\\nonumber \\\\{\\overline{\\cal T}}_{Y\\rightarrow X}&=\\lim _{\\tau \\rightarrow 0}\\frac{I[x_{t+\\tau }:x_t,y_t]-I[x_{t+\\tau }:x_{t}]}{\\tau }\\ ,$ where $I$ is the mutual information, for instance $I[y_{t+\\tau }:x_t,y_t]=\\int dy\\:dx^{\\prime }\\:dy^{\\prime }\\:p_{x^{\\prime }y^{\\prime }}^{y}(y,t+\\tau ;x^{\\prime },y^{\\prime },t)\\ln [p_{x^{\\prime }y^{\\prime }}^{y}(y,t+\\tau ;x^{\\prime },y^{\\prime },t)/[p_{y}(y)p_{xy}(x^{\\prime },y^{\\prime })]$ in the steady state (where $p_{xy}(x^{\\prime },y^{\\prime })$ and $p_y(y)$ become time independent pdfs).",
"Therefore, ${\\overline{\\cal T}}_{X\\rightarrow Y}&=\\lim _{\\tau \\rightarrow 0}\\frac{1}{\\tau } \\int dy\\:dx^{\\prime }\\:dy^{\\prime }\\:p_{x^{\\prime }y^{\\prime }}^{y}(y,t+\\tau ;x^{\\prime },y^{\\prime },t)\\nonumber \\\\\\ &\\times \\ln \\frac{p_{x^{\\prime }y^{\\prime }}^{y}(y,t+\\tau \\vert x^{\\prime },y^{\\prime },t)}{p_{y^{\\prime }}^y(y,t+\\tau \\vert y^{\\prime },t)}\\nonumber \\\\{\\overline{\\cal T}}_{Y\\rightarrow X}&=\\lim _{\\tau \\rightarrow 0}\\frac{1}{\\tau }\\int dy\\:dx^{\\prime }\\:dy^{\\prime }\\:p_{x^{\\prime }y^{\\prime }}^{x}(x,t+\\tau ; x^{\\prime },y^{\\prime },t) \\nonumber \\\\&\\times \\ln \\frac{p_{x^{\\prime }y^{\\prime }}^{x}(x,t+\\tau \\vert x^{\\prime },y^{\\prime },t)}{p_{x^{\\prime }}^x(x,t+\\tau \\vert x^{\\prime },t)}\\ .$ Note that the actual transfer entropy rates are defined as ${\\cal T}_{X\\rightarrow Y}&=\\lim _{\\tau \\rightarrow 0}\\frac{I[y_{t+\\tau }:x_t,\\lbrace y_{t^{\\prime }}\\rbrace _{t^{\\prime }\\le t}]-I[y_{t+\\tau }:\\lbrace y_{t^{\\prime }}\\rbrace _{t^{\\prime }\\le t}]}{\\tau } \\nonumber \\\\{\\cal T}_{Y\\rightarrow X}&=\\lim _{\\tau \\rightarrow 0}\\frac{I[x_{t+\\tau }:\\lbrace x_{t^{\\prime }}\\rbrace _{t^{\\prime }\\le t},y_t]-I[x_{t+\\tau }:\\lbrace x_{t^{\\prime }}\\rbrace _{t^{\\prime }\\le t}]}{\\tau } \\ .$ where $\\lbrace x_{t^{\\prime }}\\rbrace _{t^{\\prime }\\le t}$ and $\\lbrace y_{t^{\\prime }}\\rbrace _{t^{\\prime }\\le t}$ denote the full trajectories of $x_t$ and $y_t$ in the time interval $[0,t]$ .",
"Since the present model is not bipartite, the calculation of these quantities is a nontrivial task that is left aside.",
"The two-time distributions $p_{x^{\\prime }y^{\\prime }}^{y}(y,t+\\tau ;x^{\\prime },y^{\\prime },t)$ and $p_{x^{\\prime }y^{\\prime }}^{x}(x,t+\\tau ;x^{\\prime },y^{\\prime },t)$ are given by $p_{x^{\\prime }y^{\\prime }}^{y}(y,t+\\tau ;x^{\\prime },y^{\\prime },t)&= \\int dx\\:du\\:dv\\:du^{\\prime }\\:dv^{\\prime }\\:p_{{\\bf z}^{\\prime }}^{{\\bf z}}({\\bf z},t+\\tau \\vert {\\bf z}^{\\prime },t)p_{xuvy}({\\bf z}^{\\prime })\\nonumber \\\\p_{x^{\\prime }y^{\\prime }}^{x}(x,t+\\tau ;x^{\\prime },y^{\\prime },t)&= \\int dy\\:du\\:dv\\:du^{\\prime }\\:dv^{\\prime }\\:p_{{\\bf z}^{\\prime }}^{{\\bf z}}({\\bf z},t+\\tau \\vert {\\bf z}^{\\prime },t)p_{xuvy}({\\bf z}^{\\prime })$ where $p_{{\\bf z}^{\\prime }}^{{\\bf z}}({\\bf z},t+\\tau \\vert {\\bf z}^{\\prime },t)$ is the transition probability from the state ${\\bf z}^{\\prime }=(x^{\\prime },u^{\\prime },v^{\\prime },y^{\\prime })$ at time $t$ to the state ${\\bf z}=(x,u,v,y)$ at time $t+\\tau $ .",
"From the definition of the Fokker-Planck operator ${\\cal L}_{FP}$ associated with the 4-dimensional diffusion process described by Eqs.",
"REF , the transition probability for small times is given by [32] $&p_{{\\bf z}^{\\prime }}^{{\\bf z}}({\\bf z},t+\\tau \\vert {\\bf z}^{\\prime },t)=[1+\\tau {\\cal L}_{FP}({\\bf z},t)+{\\cal O}(\\tau ^2)]\\delta ({\\bf z}-{\\bf z}^{\\prime })\\nonumber \\\\&=\\delta ({\\bf z}-{\\bf z}^{\\prime })-\\tau \\sum _{i=1}^4 \\partial _{z_i}\\big [g_i({\\bf z}^{\\prime })-\\sum _j\\frac{\\theta _{i,j}^2}{2}\\partial _{z_j}\\big ]\\delta ({\\bf z}-{\\bf z}^{\\prime })$ where $g_i({\\bf z})$ is the drift coefficient in the equation for $z_i$ (with $z_1=x,z_2=u,z_3=v,z_4=y$ ), $\\theta _{2,2}=\\theta _{E},\\theta _{3,3}=\\theta _{3,4}=\\theta _{\\mu },\\theta _{4,4}=\\sqrt{\\theta _{\\mu }^2+\\theta _G^2}$ and all other $\\theta _{i,j}$ being equal to 0.",
"Let us first consider the calculation of ${\\overline{\\cal T}}_{X\\rightarrow Y}$ .",
"By integrating $p_{{\\bf z}^{\\prime }}^{{\\bf z}}({\\bf z},t+\\tau \\vert {\\bf z}^{\\prime },t)$ over $x$ , $u$ , and $v$ , we readily obtain $p_{{\\bf z}^{\\prime }}^{y}(y,t+\\tau \\vert {\\bf z}^{\\prime },t)=\\delta (y-y^{\\prime })-\\tau \\partial _{y}\\big [g_y({\\bf z}^{\\prime })-\\beta _{\\mu }\\eta _{\\mu }^2\\partial _{v}-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\partial _y\\big ]\\delta (y-y^{\\prime })+{\\cal O}(\\tau ^2)$ where the terms involving $\\partial _x,\\partial _u,\\partial _v$ cancel due to natural boundary conditions.",
"Hence, $p_{{\\bf z}^{\\prime }}^y(y &,t+\\tau ; {\\bf z}^{\\prime },t)=p_{{\\bf z}^{\\prime }}^{y}(y,t+\\tau \\vert {\\bf z}^{\\prime },t)p_{xuvy}({\\bf z}^{\\prime })\\nonumber \\\\&=\\delta (y-y^{\\prime })p({\\bf z}^{\\prime })-\\tau p_{xuvy}({\\bf z}^{\\prime }) \\times \\nonumber \\\\& \\partial _{y} \\big [g_y({\\bf z}^{\\prime })-\\beta _{\\mu }\\eta _{\\mu }^2\\partial _{v}-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\partial _y\\big ]\\delta (y-y^{\\prime }) ,$ which yields $p_{x^{\\prime }y^{\\prime }}^y(&y,t+\\tau ; x^{\\prime },y^{\\prime },t)=\\delta (y-y^{\\prime })p_{xy}(x^{\\prime },y^{\\prime }) -\\tau p_{xy}(x^{\\prime },y^{\\prime })\\partial _{y}\\big [\\bar{g}_y(x^{\\prime },y^{\\prime }) \\nonumber \\\\&-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\partial _y\\big ]\\delta (y-y^{\\prime }).$ after integration over $u^{\\prime }$ and $v^{\\prime }$ , where we have defined the averaged drift coefficient $\\bar{g}_{y}(x,y)=\\frac{1}{p_{xy}(x,y)}\\int du\\:dv\\:g_y({\\bf z})p_{xuvy}({\\bf z})\\ .$ We thus finally obtain $p_{x^{\\prime }y^{\\prime }}^y(y,&t+\\tau \\vert x^{\\prime },y^{\\prime },t)=\\delta (y-y^{\\prime })-\\tau \\partial _{y}\\big [\\bar{g}_y(x^{\\prime },y^{\\prime }) \\nonumber \\\\&-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\partial _y\\big ]\\delta (y-y^{\\prime })+{\\cal O}(\\tau ^2)\\ .$ Similarly, by also integrating $p_{{\\bf z}^{\\prime }}^y(y,t+\\tau ; x^{\\prime },y^{\\prime },t)$ over $x^{\\prime }$ , we obtain $p_{y^{\\prime }}^y(y,t &+\\tau \\vert y^{\\prime },t)=\\delta (y-y^{\\prime })-\\tau \\partial _{y}\\big [{\\bar{\\bar{g}}}_y(y^{\\prime })-(\\beta _{\\mu }\\eta _{\\mu }^2 \\nonumber \\\\&+\\beta _G \\eta _G^2)\\partial _y\\big ] \\delta (y-y^{\\prime })+{\\cal O}(\\tau ^2)\\ .$ where ${\\bar{\\bar{g}}}_{y}(y)&=\\frac{1}{p_{y}(y)}\\int dx\\:du\\:dv\\: g_y({\\bf z})p_{xuvy}({\\bf z})\\nonumber \\\\&=\\frac{1}{p_{y}(y)}\\int dx\\:{\\bar{g}}_y(x,y)p_{xy}(x,y)\\ .$ Due to the linearity of Eqs.",
"(REF ) and the Gaussian character of the pfds, one simply has $\\bar{g}_{y}(x,y)=ax+by$ and ${\\bar{\\bar{g}}}_{y}(y)=cy$ , where $a,b,c$ are complicated functions of the model parameters which we do not display here.",
"Eq.",
"(REF ) (resp.",
"Eq.",
"(REF )) merely shows that $p_{x^{\\prime }y^{\\prime }}^y(y,t+\\tau \\vert x^{\\prime },y^{\\prime },t)$ (resp.",
"$p_{y^{\\prime }}^y(y,t+\\tau \\vert y^{\\prime },t)$ ) at the lowest order in $\\tau $ is identical to the transition probability associated with an Ornstein-Uhlenbeck process with drift coefficient $\\bar{g}_{y}(x,y)$ (resp.",
"${\\bar{\\bar{g}}}_{y}(y)$ ) and diffusion coefficient $\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2$ .",
"To proceed further, it is then convenient to use to the Fourier integral representation of the $\\delta $ function and re-express $p_{x^{\\prime }y^{\\prime }}^y(y,t+\\tau \\vert x^{\\prime },y^{\\prime },t)$ and $p_{y^{\\prime }}^y(y,t+\\tau \\vert y^{\\prime },t)$ for small times as $p_{x^{\\prime }y^{\\prime }}^y(y,t+\\tau \\vert x^{\\prime },y^{\\prime },t)=\\frac{1}{2\\sqrt{\\pi (\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\tau }}e^{-\\frac{1}{4(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\tau }[y-y^{\\prime }- \\tau \\bar{g}_{y}(x^{\\prime },y^{\\prime })]^2}$ and $p_{y^{\\prime }}^y(y,t+\\tau \\vert y^{\\prime },t)=\\frac{1}{2\\sqrt{\\pi (\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\tau }}e^{-\\frac{1}{4(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\tau }[y-y^{\\prime }- \\tau {\\bar{\\bar{g}}}_{y}(y^{\\prime })]^2} \\ .$ up to corrections of the order $\\tau ^2$ [32].",
"This leads to $\\ln \\frac{p_{x^{\\prime }y^{\\prime }}^y(y,t+\\tau \\vert x^{\\prime },y^{\\prime },t)}{p_{y^{\\prime }}^y(y,t+\\tau \\vert y^{\\prime },t)}&=\\frac{1}{4(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)}\\big [2(y-y^{\\prime })-\\tau [\\bar{g}_{y}(x^{\\prime },y^{\\prime })+{\\bar{\\bar{g}}}_{y}(y^{\\prime })]\\big ] \\nonumber \\\\& \\times \\big [\\bar{g}_{y}(x^{\\prime },y^{\\prime })-{\\bar{\\bar{g}}}_{y}(y^{\\prime })\\big ]\\ ,$ and from Eq.",
"(REF ) and the definition of the transfer entropy rate [Eq.",
"(REF )], $4(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2){\\overline{\\cal T}}_{X\\rightarrow Y}&=\\lim _{\\tau \\rightarrow 0} \\frac{1}{\\tau }\\int dy\\:dx^{\\prime }\\:dy^{\\prime } p_{x^{\\prime }y^{\\prime }}^y(y,t+\\tau ;x^{\\prime },y^{\\prime },t)\\big [2(y-y^{\\prime }) \\nonumber \\\\&-\\tau [\\bar{g}_{y}(x^{\\prime },y^{\\prime })+{\\bar{\\bar{g}}}_{y}(y^{\\prime })]\\big ]\\big [\\bar{g}_{y}(x^{\\prime },y^{\\prime })-{\\bar{\\bar{g}}}_{y}(y^{\\prime })\\big ]\\nonumber \\\\&=\\lim _{\\tau \\rightarrow 0} \\frac{1}{\\tau }\\int dy\\:dx^{\\prime }\\:dy^{\\prime } p_{xy}(x^{\\prime },y^{\\prime })\\Big [\\delta (y-y^{\\prime })-\\tau \\partial _{y}[\\bar{g}_y(x^{\\prime },y^{\\prime }) \\nonumber \\\\&-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\partial _y]\\delta (y-y^{\\prime })\\Big ]\\nonumber \\\\&\\times \\big [2(y-y^{\\prime })-\\tau [\\bar{g}_{y}(x^{\\prime },y^{\\prime })+{\\bar{\\bar{g}}}_{y}(y^{\\prime })]\\big ]\\big [\\bar{g}_{y}(x^{\\prime },y^{\\prime })-{\\bar{\\bar{g}}}_{y}(y^{\\prime })\\big ]$ We then use $\\int dy\\: (y-y^{\\prime })\\Big [\\delta (y-y^{\\prime })-\\tau \\partial _{y}[\\bar{g}_y(x^{\\prime },y^{\\prime })-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\partial _y]\\delta (y-y^{\\prime })\\Big ]=\\tau \\bar{g}_y(x^{\\prime },y^{\\prime })\\ ,$ and $\\int dx^{\\prime }\\:p_{xy}(x^{\\prime },y^{\\prime })\\bar{g}_y(x^{\\prime },y^{\\prime })&=p_y(y^{\\prime }){\\bar{\\bar{g}}}_{y}(y^{\\prime })=\\int dx^{\\prime } \\: p_{xy}(x^{\\prime },y^{\\prime }){\\bar{\\bar{g}}}_{y}(y^{\\prime })\\ ,$ to finally arrive at Eq.",
"(REF ), namely ${\\overline{\\cal T}}_{X\\rightarrow Y}&=\\frac{1}{4(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)}\\int dx\\:dy\\: p_{xy}(x,y)\\big [\\bar{g}_{y}^2(x,y)-{\\bar{\\bar{g}}}_{y}^2(y)\\big ]\\ .$ A similar expression can be found in Ref.",
"[11] (see Eq.",
"(A.31) in that reference).",
"Note also that the result given in Ref.",
"[24] is obtained as a special case.",
"Inserting into Eq.",
"(REF ) the values of the parameters given in Table S1 of Ref.",
"[4], we obtain the values given in Table 2.",
"Note that ${\\overline{\\cal T}}_{E\\rightarrow \\mu }=0$ for the high IPTG concentration because $T_{\\mu E}=0$ , and therefore $\\mu (t)$ no longer depends on $E(t)$ as can be seen from Eq.",
"(REF ).",
"There is no need to detail the calculation of ${\\overline{\\cal T}}_{\\mu \\rightarrow E}$ (i.e.",
"${\\overline{\\cal T}}_{Y\\rightarrow X}$ ) because it goes along the same line, with $y$ replaced by $x$ .",
"The crucial difference is that there is no white noise acting on $\\dot{x}$ .",
"Therefore, the denominator in Eq.",
"(REF ), which is the variance of the noise $\\xi _y$ , is replaced by 0.",
"This implies that ${\\overline{\\cal T}}_{\\mu \\rightarrow E}$ is infinite.",
"The information flows ${\\cal I}^{flow}_{X\\rightarrow Y}$ and ${\\cal I}^{flow}_{Y\\rightarrow X}$ are derived from the time-shifted mutual informations $I[x_{t+\\tau }:y_t]$ and $I[y_{t+\\tau }:x_t]$ .",
"Specifically, ${\\cal I}^{flow}_{X\\rightarrow Y}&=\\lim _{\\tau \\rightarrow 0} \\frac{I[x_t:y_t]-I[x_{t+\\tau }:y_t]}{\\tau }\\nonumber \\\\{\\cal I}^{flow}_{Y\\rightarrow X}&=\\lim _{\\tau \\rightarrow 0} \\frac{I[y_t:x_t]-I[y_{t+\\tau }:x_t]}{\\tau }\\ .$ Let us first consider the second flow ${\\cal I}^{flow}_{Y\\rightarrow X}$ which requires the knowledge of $p_{x^{\\prime }}^y(y,t+\\tau ;x^{\\prime }, t)$ whose expression is obtained by integrating Eq.",
"(REF ) over $x^{\\prime }$ .",
"This yields $p_{x^{\\prime }}^y(y,t+\\tau ; x^{\\prime },t)=p_{xy}(x^{\\prime },y)-\\tau \\partial _{y}\\big [\\bar{g}_y(x^{\\prime },y) \\nonumber \\\\-(\\beta _{\\mu } \\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\partial _y \\big ]p_{xy}(x^{\\prime },y)+{\\cal O}(\\tau ^2)\\ .$ Hence $I[y_{t+\\tau }:x_t]&=\\int dx^{\\prime }\\:dy\\: p_{x^{\\prime }}^y(y,t+\\tau ; x^{\\prime },t)\\nonumber \\\\&\\hspace{28.45274pt}\\times \\ln \\frac{p_{x^{\\prime }}^y(y,t+\\tau ; x^{\\prime },t)}{p_y(y)p_x(x^{\\prime })}\\nonumber \\\\&=I[y_t:x_t]-\\tau \\int dx\\:dy\\: \\partial _{y}\\big [\\bar{g}_y(x,y) \\nonumber \\\\&-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\partial _y \\big ]p_{xy}(x,y)\\ln \\frac{p_{xy}(x,y)}{p_y(y)p_x(x)}.$ We finally obtain $&{\\cal I}^{flow}_{Y\\rightarrow X}=\\int dx\\:dy\\: \\partial _{y}\\big [\\bar{g}_y(x,y)p_{xy}(x,y) \\nonumber \\\\&-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\partial _yp_{xy}(x,y)\\big ]\\ln \\frac{p_{xy}(x,y)}{p_y(y)p_x(x)}\\ .$ A similar calculation yields ${\\cal I}^{flow}_{X\\rightarrow Y}&=\\int dx\\:dy\\: \\partial _{x}\\big [\\bar{g}_x(x,y)p_{xy}(x,y)\\big ]\\ln \\frac{p_{xy}(x,y)}{p_y(y)p_x(x)}\\ ,$ where $\\bar{g}_{x}(x,y)=\\frac{1}{p_{xy}(x,y)}\\int du\\:dv\\:g_x({\\bf z})p_{xuvy}({\\bf z})$ is an averaged drift coefficient.",
"Contrary to the case of the transfer entropy rate ${\\overline{\\cal T}}_{Y\\rightarrow X}$ , the absence of a white noise acting on $\\dot{x}$ does not lead to an infinite result for ${\\cal I}^{flow}_{Y\\rightarrow X}$ .",
"In fact, one has the symmetry relation ${\\cal I}^{flow}_{X\\rightarrow Y}=-{\\cal I}^{flow}_{Y\\rightarrow X}\\ ,$ which is readily obtained by noting that $p_{xy}(x,y)$ , the stationary solution of the Fokker-Planck equation, satisfies the equation $\\partial _x[& \\bar{g}_{x}(x,y)p_{xy}(x,y)]+\\partial _y[\\bar{g}_{y}(x,y)p_{xy}(x,y)] \\nonumber \\\\&-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2)\\frac{\\partial ^2 }{\\partial y^2}p_{xy}(x,y)=0 \\ .$ Inserting the numerical values of the parameters given in Table S1 of Ref.",
"[4], we obtain the values given in Table REF below.",
"Interestingly, $ {\\cal I}^{flow}_{E \\rightarrow \\mu }$ decreases as the IPTG concentration increases and that it becomes negative at high concentration.",
"Table: Theoretical values of ℐ X→Y flow =-ℐ Y→X flow {\\cal I}^{flow}_{X\\rightarrow Y}=-{\\cal I}^{flow}_{Y\\rightarrow X} in the original model of Ref.",
"We now repeat the above calculations for the modified model where $N_E$ is treated as a white noise.",
"Eliminating again the variable $w$ (i.e.",
"$N_G$ ) in favor of $y$ , the new set of equations that describe the stochastic dynamics and replace Eqs.",
"REF reads $\\dot{x}&=-\\big [\\mu _E +\\mu _0 T_{\\mu E}(T_{EG}-1)\\big ]x-\\mu _0 T_{EG}v \\nonumber \\\\&+\\mu _0(T_{EG}-1)y+\\xi _x \\nonumber \\\\\\dot{v}&=-\\beta _{\\mu } v+\\xi _{\\mu }\\nonumber \\\\\\dot{y}&=T_{\\mu E}\\big [\\beta _G-\\mu _E -\\mu _0 T_{\\mu E}(T_{EG}-1)\\big ]x+\\big [\\beta _G-\\beta _{\\mu } \\nonumber \\\\&-\\mu _0 T_{\\mu E}T_{EG}\\big ]v+\\big [\\mu _0 T_{\\mu E}(T_{EG}-1)-\\beta _G\\big ]y+\\widetilde{\\xi }_{y}\\ ,$ where we have defined the white noises $\\xi _x=\\mu _0N_E$ and $\\widetilde{\\xi }_{y}=\\xi _y+T_{\\mu E}\\xi _{x}$ satisfying $\\langle \\xi _x(t)\\xi _x(t^{\\prime })\\rangle =2D_E\\mu _0^2\\delta (t-t^{\\prime })$ and $\\langle \\widetilde{\\xi }_{y}(t)\\widetilde{\\xi }_y(t^{\\prime })\\rangle =(\\theta _{\\mu }^2+\\theta _G^2+2D_E\\mu _0^2T_{\\mu E}^2)\\delta (t-t^{\\prime })$ , respectively.",
"These two noises are correlated, with $\\langle \\xi _x(t)\\widetilde{\\xi }_y(t^{\\prime })\\rangle =2D_E\\mu _0^2T_{\\mu E}\\delta (t-t^{\\prime })$ .",
"The pdfs and the correlation functions can be computed as before.",
"In fact, it is clear that this simply amounts to taking the limit $\\beta _E \\rightarrow \\infty $ with $D_E=\\eta _E^2/\\beta _E$ finite in the previous equations (for instance in Eqs.",
"(REF ) for the covariances).",
"The new correlation functions are plotted in Fig REF .",
"As expected, they are almost indistinguishable from those obtained with the original model and they fit the experimental data just as well (this of course is also true for the pdfs).",
"Much more interesting are the results for the transfer entropy rates and the information flows.",
"Again, there is no need to repeat the calculations as they follow the same lines as before.",
"We now obtain ${\\overline{\\cal T}}_{X\\rightarrow Y}&=\\frac{1}{4(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2+D_E\\mu _0^2T_{\\mu E}^2)} \\times \\nonumber \\\\&\\int dx\\:dy\\: p_{xy}(x,y)\\big [\\bar{g}_{y}^2(x,y)-{\\bar{\\bar{g}}}_{y}^2(y)\\big ]\\\\{\\overline{\\cal T}}_{Y\\rightarrow X}&=\\frac{1}{4D_E\\mu _0^2}\\int dx\\:dy\\: p_{xy}(x,y)\\big [\\bar{g}_{x}^2(x,y)-{\\bar{\\bar{g}}}_{x}^2(x)\\big ]\\ ,$ where $\\bar{g}_{x}(x,y)&=\\frac{1}{p_{xy}(x,y)}\\int du\\:g_x(x,v,y)p_{xvy}(x,v,y)\\\\\\bar{g}_{y}(x,y)&=\\frac{1}{p_{xy}(x,y)}\\int du\\:g_y(x,v,y)p_{xvy}(x,v,y)\\ ,$ and ${\\bar{\\bar{g}}}_{x}(x)&=\\frac{1}{p_{x}(x)}\\int dy\\:{\\bar{g}}_x(x,y)p_{xy}(x,y)\\\\{\\bar{\\bar{g}}}_{y}(y)&=\\frac{1}{p_{y}(y)}\\int dx\\:{\\bar{g}}_y(x,y)p_{xy}(x,y)\\ .$ (Again, $g_x(x,v,y)$ and $g_y(x,v,y)$ denote the drift coefficients in Eqs.",
"(REF )).",
"The crucial difference with the results for the original model is that ${\\overline{\\cal T}}_{Y\\rightarrow X}$ is now finite.",
"Similarly, we have $\\dot{I}^{flow}_{X\\rightarrow Y}&=\\int dx\\:dy\\: \\partial _{x}\\big [\\bar{g}_x(x,y)p_{xy}(x,y) \\nonumber \\\\&-D_E\\mu _0^2\\partial _xp_{xy}(x,y)\\big ]\\ln \\frac{p_{xy}(x,y)}{p_y(y)p_x(x)}\\\\\\dot{I}^{flow}_{Y\\rightarrow X}&=\\int dx\\:dy\\: \\partial _{y}\\big [\\bar{g}_y(x,y)p_{xy}(x,y) \\nonumber \\\\&-(\\beta _{\\mu }\\eta _{\\mu }^2+\\beta _G \\eta _G^2+D_E\\mu _0^2T_{\\mu E}^2)\\partial _yp_{xy}(x,y)\\big ] \\nonumber \\\\&\\times \\ln \\frac{p_{xy}(x,y)}{p_y(y)p_x(x)}\\ .$ The numerical values of ${\\overline{\\cal T}}_{E\\rightarrow \\mu }$ and ${\\overline{\\cal T}}_{\\mu \\rightarrow E}$ are given in Table REF .",
"For completeness, we also compare these values with the estimates obtained by the inference method in Table REF .",
"We see that satisfactory results are obtained by properly choosing the sampling time $\\tau $ .",
"This is also true for the information flows ${\\cal I}^{flow}_{E\\rightarrow \\mu }$ and ${\\cal I}^{flow}_{\\mu \\rightarrow E}$ .",
"It is worth noting that the symmetry relation $\\dot{I}^{flow}_{E\\rightarrow \\mu }=-\\dot{I}^{flow}_{\\mu \\rightarrow E}$ no longer holds, except for the high IPTG concentration (as $T_{\\mu E}=0$ ).",
"This contrasts with the preceding case where $N_E$ was modeled by an Ornstein-Uhlenbeck noise.",
"We also observe that the information flows are not always smaller than the transfer entropy rates, contrary to what occurs in bipartite systems.",
"Therefore, the concept of a \"sensory capacity\" as introduced in Ref.",
"[11] is here ineffective.",
"We acknowledge J. Lizier for many insightful comments regarding the numerical evaluation of transfer entropies, and L. Peliti for stimulating discussions.",
"S.L.",
"thanks the Institute of Complex Systems (ISC-PIF), the Region Ile-de-France, and the Labex CelTisPhyBio (No.",
"ANR-10- LBX-0038) part of the IDEX PSL (No.",
"ANR-10-IDEX-0001-02 PSL) for financial support."
]
] | 1709.01746 |
[
[
"Progress-Space Tradeoffs in Single-Writer Memory Implementations"
],
[
"Abstract Most algorithms designed for shared-memory distributed systems assume the single-writer multi-reader (SWMR) setting where each process is provided with a unique register readable by all.",
"In a system where computation is performed by a bounded number n of processes coming from a very large (possibly unbounded) set of potential participants, the assumption of a SWMR memory is no longer reasonable.",
"If only a bounded number of multi-writer multi-reader (MWMR) registers are provided, we cannot rely on an a priori assignment of processes to registers.",
"In this setting, simulating SWMR memory, or equivalently, ensuring stable writing (i.e., every written value persists in the memory), is desirable.",
"In this paper, we propose a SWMR simulation that adapts the number of MWMR registers used to the desired progress condition.",
"For any given k from 1 to n, we present an algorithm that uses only n+k-1 registers to simulate a k-lock-free SWMR memory.",
"We also give a matching lower bound of n+1 registers required for the case of 2-lock-freedom, which supports our conjectures that the algorithm is space-optimal.",
"Our lower bound holds for the strictly weaker progress condition of 2-obstruction-freedom, which suggests that the space complexity for k-obstruction-free and k-lock-free SWMR simulations might coincide."
],
[
"Introduction",
"We consider a distributed computing model in which at most $n$ participating processes communicate via reading and writing to a shared memory.",
"The participating processes come from a possibly unbounded set of potential participants: each process has a unique identifier (IP address, RFID, MAC address, etc.)",
"which we, without loss of generality, assume to be an integer value.",
"Given that processes do not have an a priori knowledge of the participating set, it is natural to assume that they can only compare their identifiers to establish their relative order, otherwise they essentially run the same algorithm [14].",
"This model is therefore called comparison-based [2].",
"In the comparison-based model with bounded shared memory, we cannot assume that the processes are provided with a prior assignment of processes to distinct registers.",
"The only suitable assumption, as is the case for anonymous systems [16], is that processes have access to multi-writer multi-reader registers (MWMR).",
"In this paper, we study the space complexity of comparison-based implementations of an abstract single-writer multi-reader (SWMR) memory.",
"The abstract SWMR memory allows each participating process to write to a private abstract memory location and to read from the abstract memory locations of participating processes.",
"The SWMR abstraction can be further used to build higher-level abstractions, such as renaming [2] and atomic snapshot [1].",
"To implement an SWMR memory, we need to ensure that every write performed by a participating process on its abstract SWMR register is persistent: every future abstract read must see the written value, as long as it has not been replace by a more recent persistent write.",
"To achieve persistence in a MWMR system, the emulated abstract write may have to update multiple base MWMR registers in order to ensure that its value is not overwritten by other processes.",
"A natural question arises: How many base MWMR registers do we need?",
"In this paper, we show that the answer depends on the desired progress condition.",
"It is immediate that $n$ registers are required for a lock-free implementation, i.e., we want to ensure that at least one correct process makes progress.",
"Indeed, any algorithm using $n-1$ or less registers can be brought into the situation where every base register is covered, i.e., a process is about to execute a write operation on it [4].",
"If we let the remaining process $p_i$ complete a new abstract write operation, the other $n-1$ processes may destroy the written value by making a block write on the covered registers (each covering process performs its pending write operation).",
"Thus, the value written by $p_i$ is “lost”: no future read would find it.",
"It has been recently shown that $n$ base registers are not only necessary, but also sufficient for a lock-free implementation [8].",
"A wait-free SWMR memory implementation that guarantees progress to every correct process can be achieved with $2n-1$ registers [8].",
"The two extremes, lock-freedom and wait-freedom, suggest an intriguing question: is there a dependency between the amount of progress the implementation provides and its space complexity: if processes are guaranteed more progress, do they need more base registers?"
],
[
"Contributions.",
"In this paper, we give an evidence of such a dependency.",
"Using novel covering-based arguments, we show that any 2-obstruction-free algorithm requires $n+1$ base MWMR registers.",
"Recall that $k$ -obstruction-freedom requires that every correct process makes progress under the condition that at most $k$ processes are correct [15].",
"The stronger property of $k$ -lock-freedom [5] additionally guarantees that if more than $k$ processes are correct, then at least $k$ out of them make progress.",
"We also provide, for any $k=1,\\ldots , n$ , a $k$ -lock-free SWMR memory implementation that uses only $n+k-1$ base registers.",
"Our lower bound and the algorithm suggest the following: Conjecture 1 It is impossible to implement a $k$ -obstruction-free SWMR memory in the $n$ -process comparison-based model using $n+k-2$ MWMR registers.",
"An interesting implication of our results is that 2-lock-free and 2-obstruction-free SWMR implementations have the same optimal space complexity.",
"Given that $n$ -obstruction-freedom and $n$ -lock-freedom coincide with wait-freedom, we expect that, for all $k=1,\\ldots ,n$ , $k$ -obstruction-free and $k$ -lock-free (and all progress conditions in between [5]) require the same number $n+k-1$ of base MWMR registers.",
"Curiously, our results highlight a contrast between complexity and computability, as we know that certain problems, e.g., consensus, can be solved in an obstruction-free way, but not in a lock-free way [11].",
"Jayanti, Tan and Toueg [12] gave linear lower bounds on the space complexity of implementing a large class of perturbable objects (such as CAS and counters).",
"For atomic-snapshot algorithms, Fatourou, Ellen and Ruppert [10] showed that there is a tradeoff between the time and space complexities, both in the anonymous and the non-anonymous cases.",
"Zhu [17] showed that $n-1$ MWMR registers are required for obstruction-free consensus.",
"Delporte et al.",
"[9] studied the space complexity of anonymous $k$ -set agreement using MWMR registers, and showed a dependency between space complexity and progress conditions.",
"In particular, they provide a lower bound of $n-k+m$ MWMR registers to solve anonymous repeated $k$ -set agreement in the $m$ -obstruction-free way, for $k<m$ .",
"Delporte et al.",
"[7] showed that obstruction-free $k$ -set agreement can be solved in the $n$ -process comparison-based model using $2(n-k) + 1$ registers.",
"This upper bound was later improved to $n-k+m$ for the progress condition of $m$ -obstruction-freedom ($m\\le k$ ) by Bouzid, Raynal and Sutra [3].",
"In particular, their algorithm uses less than $n$ registers when $m<k$ .",
"To our knowledge, the only lower bound on the space-complexity of implementing an SWMR memory has been given by Delporte et al.",
"[8] who showed that lock-free comparison-based implementations require $n$ registers.",
"Delporte et al.",
"[8] proposed two SWMR memory implementations: a lock-free one, using $n$ registers, and a wait-free one, using $2n-1$ registers.",
"These algorithms are used in [6] to implement a uniform SWMR memory, i.e., assuming no prior knowledge on the number of participating processes.",
"Assuming that $p$ processes participate, the algorithms use $3p+1$ and $4p$ registers for, respectively, lock-freedom and wait-freedom.",
"The paper is organized as follows.",
"Section defines the system model and states the problem.",
"Section presents a $k$ -lock-free SWMR memory implementation.",
"Section shows that a 2-obstruction-free SWMR memory implementation requires $n+1$ MWMR registers and hence that our algorithm is optimal for $k=2$ .",
"Section concludes the paper with implications and open questions."
],
[
"Model",
"We consider the asynchronous shared-memory model, in which a bounded number $n>1$ of asynchronous crash-prone processes communicate by applying read and write operations to a bounded number $m$ of base atomic multi-writer multi-reader atomic registers.",
"An atomic register $i$ can be accessed with two memory operations: $\\textit {write}(i,v)$ that replaces the content of the register with value $v$ , and $\\textit {read}(i)$ that returns its content.",
"The processes are provided with unique identifiers from an unbounded name space.",
"Without loss of generality, we assume that the name space is the set of positive integers."
],
[
"States, configurations and executions",
"An algorithm assigned to each process is a (possibly non-deterministic) automaton that accepts high-level operation requests as an application input.",
"In each state, the process is poised to perform a step, i.e., a read or write operations on base registers.",
"Once the step is performed, the process changes its state according to the result the step operation, possibly non-deterministically and possibly to a step corresponding to another high-level operation.",
"A configuration, or system state, consists of the state of all processes and the content of all MWMR registers.",
"In the initial configurations, all processes are in their initial states, and all registers carry initial values.",
"We say that a step $e$ by a process $p$ is applicable to a configuration $C$ , if $e$ is the pending step of $p$ in $C$ , and we denote $Ce$ the configuration reached from $C$ after $p$ performed $e$ .",
"A sequence of steps $e_1,e_2,\\ldots $ is applicable to $C$ , if $e_1$ is applicable to $C$ , $e_2$ is applicable to $Ce_1$ , etc.",
"A (possibly infinite) sequence of steps applicable to a configuration $C$ is called an execution from $C$.",
"A configuration $C$ is said to be reachable from a configuration $C^{\\prime }$ , and denoted $C\\in \\mathit {Reach}(C^{\\prime })$ , if there exists a finite execution $\\alpha $ applicable to $C^{\\prime }$ , such that $C=C^{\\prime }\\alpha $ .",
"If omitted, the starting configuration is the initial configuration, and is denoted as $C\\in \\mathit {Reach}$ .",
"Processes that take at least one step of the algorithm are called participating.",
"A process is called correct in a given (infinite) execution if it takes infinitely many steps in that execution.",
"Let $\\textit {Correct}(\\alpha )$ denote the set of correct processes in the execution $\\alpha $ ."
],
[
"Comparison-based algorithms",
"We assume that the processes are allowed to use their identifiers only to compare them with the identifiers of other processes: the outputs of the algorithm only depend on the inputs, the relative order of the identifiers of the participating processes, and the schedule of their steps.",
"Formally, we say that an algorithm is comparison-based, if, for each possible execution $\\alpha $ , by replacing the identifiers of participating processes with new ones preserving their relative order, we obtain a valid execution of the algorithm.",
"Notice that the assumption does not preclude using the identifiers in communication primitives, it only ensures that decisions taken in the algorithm's run are taken only based on the identifiers relative order.",
"In this model, $m$ MWMR registers can be used to implement a wait-free $m$ -component multi-writer atomic-snapshot memory [1].",
"The memory exports operations $\\textit {Update}(i,v)$ (updating position $i$ of the memory with value $v$ ) and $\\textit {Snapshot}()$ (atomically returning the contents of the memory).",
"In the comparison-based atomic-snapshot implementation, easily derived from the original one [1], $\\textit {Update}(i,v)$ writes only once, to register $i$ , and $\\textit {Snapshot}()$ is read-only.",
"For convenience, in our upper-bound algorithm we are going to use atomic snapshots instead of read-write registers."
],
[
"SWMR memory",
"A single-writer multi-reader (SWMR) memory exports two operations: $\\textit {Write}()$ that takes a value as a parameter and $\\textit {Collect}()$ that returns a multi-set of values.",
"It is guaranteed that, in every execution, there exists a reading map $\\pi $ that associates each complete Collect operation $C$ , returning a multi-set $V=\\lbrace v_1,\\ldots ,v_s\\rbrace $ , with a set of $s$ Write operations $\\lbrace w_1,\\ldots ,w_s\\rbrace $ performed, respectively, by distinct processes $p_1,\\dots ,p_s$ such that: The set $\\lbrace p_1,\\ldots ,p_s\\rbrace $ contains all processes that completed at least one write operation before the invocation of $C$ ; For each $i=1,\\ldots ,s$ , $w_i$ is either the last write operation of process $p_i$ preceding the invocation of $C$ or a write operation of $p_i$ concurrent with $C$ .",
"Note that our definition does not guarantee atomicity of SWMR operations.",
"Moreover, we do not require that processes are allocated with a unique MWMR register that can be used as a single writer register.",
"Instead, we simply require that processes are able to simulate the use of single writer registers through implementing the SWMR memory.",
"Intuitively, a collect operation can be seen as a sequence of reads on regular registers [13], each associated with a distinct participating process.",
"Such a collect object can be easily transformed into a single-writer atomic snapshot abstraction [1]."
],
[
"Progress conditions",
"In this paper we focus on two families of progress conditions, both generalizing the wait-free progress condition, namely $k$ -lock-freedom and $k$ -obstruction-freedom.",
"An execution $\\alpha $ satisfies the property of $k$ -lock-freedom [5] (for $k\\in \\lbrace 1,\\dots , n\\rbrace $ ) if at least $\\min (k,\\textit {Correct}(\\alpha ))$ correct processes make progress in it, i.e., complete infinitely many high-level operations (in our case, Writes and Collects).",
"The special case of $n$ -lock-freedom is called wait-freedom.",
"The property of $k$ -obstruction-freedom [11], [15] requires that every correct process makes progress, under the condition that there are at most $k$ correct processes.",
"(If more than $k$ processes are correct, no progress is guaranteed.)",
"In particular, $k$ -lock-freedom is a stronger requirement than $k$ -obstruction-freedom (strictly stronger for $1\\le k< n$ ).",
"Indeed, both require that every correct process makes progress when there are at most $k$ correct processes, but $k$ -lock-freedom additionally requires that some progress is made even if there are more than $k$ correct processes."
],
[
"Upper bound: k-lock-free SWMR memory with n+k-1 registers",
"Consider a full-information algorithm in which every process alternates atomic snapshots and updates, where each update performed by a process incorporates the result of its preceding snapshot.",
"Every value written to a register will persist (i.e., will be present in the result of every subsequent snapshot), unless there is another process poised to write to that register.",
"The pigeonhole principle implies that $k$ processes can cover at most $k$ distinct registers at the same time.",
"Thus, if, at a given point of a run, a value is present in $n$ registers, then the value will persist.",
"This observation implies a simple $n$ -register lock-free SWMR implementation in which a high-level Write operation alternates snapshots and updates of all registers, one by one in the round-robin fashion, until the written high-level value is present in all $n$ registers.",
"A high-level Collect operation can simply return the set of the most recent values (defined using monotonically growing sequence numbers) returned by a snapshot operation.",
"The wait-free SWMR memory implementation in [8] using $2n-1$ registers follows the $n$ -register lock-free algorithm but, roughly, for each participating process, replaces register $n$ with register $n-1+\\textit {pos}$ , where $\\textit {pos}$ is the rank of the process among the currently observed participants.",
"This way, there is a time after which every participating process has a dedicated register to write, and each value it writes will persist.",
"In particular, every value it writes will be seen by all processes and will eventually be propagated to the $n-1$ first registers.",
"To implement a $k$ -lock-free SWMR memory using $n+k-1$ registers, a process should determine, in a dynamic fashion, to which out of the last $k-1$ registers to write.",
"In our algorithm, by default, a Write operation only uses the first $n$ registers, but if a process observes that its value is absent from some registers in the snapshot (some of its previous writes have been overwritten by other processes), it uses extra registers to propagate its value.",
"The number of these extra registers depends on how many other processes have been observed making progress.",
"$k$ -lock-free SWMR implementation using $n+k-1$ MWMR registers.",
"WriteWrite(v):End Write CollectCollect():End Collect Dodowhile $\\mathit {View}:$ list of triples of type $(\\mathit {ValueType},\\mathit {IdType},\\mathbb {N})$ , initially set to $\\emptyset $ $\\mathit {opCounter} \\in \\mathbb {N}$ , initially set to 0 $\\textit {Write}$ $\\mathit {ActiveProcs} = \\lbrace \\mathit {id}\\rbrace $ $\\mathit {View} = \\mathit {View}\\cup (v,\\mathit {id},\\mathit {opCounter})$ $\\mathit {WritePos} = 0$ $\\mathit {WritePosMax} = n$ $|\\lbrace m\\in \\lbrace 1, \\dots , n+k-1\\rbrace , (v,\\mathit {id},\\mathit {opCounter}) \\in \\mathit {Snap}[m]\\rbrace |< n$ $\\mathit {Snap} = \\mathit {MEM}.\\mathit {snapshot}()$ $\\mathit {ActiveProcs} = \\mathit {ActiveProcs} \\cup \\lbrace \\mathit {pid}: \\exists (\\_,\\mathit {pid},c)\\in \\mathit {Snap}, \\forall (\\_,\\mathit {pid},c^{\\prime })\\in \\mathit {View}, c>c^{\\prime }\\rbrace $ $\\mathit {View} = \\mathit {View}\\cup \\mathit {Snap}$ $\\mathit {Update}(\\mathit {MEM}[\\mathit {WritePos}],\\mathit {View})$ $\\mathit {WritePos} = \\mathit {WritePos} + 1 \\mathit {}\\pmod {\\mathit {WritePosMax}}$ $\\mathit {WritePosMax} = min(n+|\\mathit {ActiveProcs}|-1,n+k-1)$ $\\mathit {opCounter}=\\mathit {opCounter}+1$ $\\textit {Collect}$ $\\mathit {Reads} = \\mathit {MEM}.\\mathit {snapshot}()$ $ V = \\emptyset $ $pid$ such that $(\\_,pid,\\_)\\in \\mathit {Reads}$ $V = V\\cup \\lbrace v\\rbrace $ with $v$ such that $(v,\\mathit {pid},\\max \\lbrace c\\in \\mathbb {N},(\\_,\\mathit {pid},c)\\in \\mathit {Reads}\\rbrace )\\in \\mathit {Reads}$ Return $V$"
],
[
"Overview of the algorithm",
"Our $k$ -lock-free SWMR implementation, which uses $n+k-1$ base MWMR registers, is presented in Algorithm .",
"In a Write operation, the process adds the operation to be performed to its local view (line ).",
"The process then attempts to add its local view, together with the outcome of a snapshot, to each of the first $\\mathit {WritePosMax}$ , initially $n$ , registers (lines –).",
"At each loop, $\\mathit {WritePosMax}$ is set to the smaller value between the number of processes observed as concurrently active and the number of registers available (line ).",
"The writing process continues to do so until its Write operation value is present in at least $n$ registers (line ).",
"In this algorithm, the $k-1$ extra registers are used according to the liveness observed by blocked processses.",
"In order to be allowed to use the last register, a process must fail to complete its write while observing at least $k-1$ other processes completing their own.",
"This ensures that when a process access this last register, a $k^{th}$ process is able to be observed by processes completing operations and thus will be helped to eventually complete.",
"The Collect operation is rather straightforward.",
"It simply takes a snapshot of the memory and, for each participating process observed in the memory, it returns its most recent value (selected using associated sequence numbers, line )."
],
[
"Safety",
"At a high level, the safety of Algorithm relies on the following property of register content stability: Lemma 1 Let, at some point of a run of the algorithm, value $(v,id,c)$ be present in some register $r$ and such that no process is poised to execute an update on $r$ (i.e., no process is between taking the snapshot of $\\textit {MEM}$ (line ) and the update of $r$ (line )), then at all subsequent times $(v,id,c) \\in r$ , i.e., the value is present in the set of values stored in $r$ .",
"Suppose that at time $\\tau $ , a register $R$ contains $(v,id,c)$ and no process is poised to execute an update on $R$ .",
"Suppose, by contradiction, that $R$ does not contain it at some time $\\tau ^{\\prime }>\\tau $ .",
"Let $\\tau _{min}$ , $\\tau _{min}>\\tau $ , be the smallest time such that $(v,id,c)$ is not in $R$ .",
"Therefore, a write must have been performed on $R$ , by some process $q$ , at time $\\tau _{min}$ with a view which does not contain $(v,id,c)$ .",
"Such a write can only be performed at line , with a view including the last snapshot of $MEM$ performed by $q$ at line .",
"Process $q$ must have performed this snapshot on $R$ at some $\\tau _R<\\tau $ as $(v,id,c)$ is present in $R$ between times $\\tau $ and $\\tau _{min}$ and as $\\tau _R<\\tau _{min}$ .",
"Thus $q$ is poised to write on $R$ at time $\\tau $ — a contradiction.",
"The persistence of the values in a specific uncovered register (Lemma REF ) can be used to show the persistence of the value of a completed Write operation in $\\textit {MEM}$ : lemmapersWrite If process $p$ returns from a Write operation $(v,id(p),c)$ at time $\\tau $ , then for any time $\\tau ^{\\prime }\\ge \\tau $ there is a register containing $(v,id(p),c)$ .",
"Before returning from its Write operation, $p$ takes a snapshot of $\\textit {MEM}$ at some time $\\tau _S$ , $\\tau _S<\\tau $ (line ), which returns a view of the memory in which at least $n$ registers contain the triplet $(v,id(p),c)$ .",
"As $p$ is taking a snapshot at time $\\tau _S$ , at most $n-1$ processes can be poised to perform an update on some register at time $\\tau _S$ .",
"As a process can be poised to perform an update on at most one register, there can be at most $n-1$ distinct registers covered at time $\\tau _S$ .",
"Therefore, at time $\\tau _S$ , there is at least one uncovered register containing $(v,id(p),c)$ , let us call it $r$ .",
"By Lemma REF , $(v,id(p),c)$ will be present in $r$ at any time $\\tau ^{\\prime }>\\tau _S$ , and thus, any time $\\tau ^{\\prime }>\\tau $ .",
"With Lemma REF , we can derive the safety of our SWMR memory implementation (Section REF ): Theorem 1 Algorithm safely implements an SWMR memory.",
"It can be easily observed that a triplet $(v,id,c)$ corresponds to a unique Write operation of a value $v$ , performed by the process with identifier $id$ .",
"Therefore, a Collect operation returns a set of values proposed by Write operations from distinct processes, and thus the map $\\pi $ is well-defined.",
"By Lemma REF , the value $(v,id,c)$ corresponding to a Write operation completed at time $\\tau $ is present in some register $r$ for any time $\\tau ^{\\prime }>\\tau $ .",
"Thus, the set of values resulting from any snapshot operation performed after time $\\tau $ contains $(v,id,c)$ .",
"Hence, for any complete Collect operation $C$ , $\\pi (C)$ contains a value for every process which completed a Write operation before $C$ was invoked.",
"Also, as each value returned by a Collect is the value observed associated to the greatest sequence number for a given process, it comes from the last completed Write or from a concurrent one."
],
[
"Progress",
"We will show, by induction on $k$ , that Algorithm satisfies $k$ -lock-freedom.",
"We first show, as in [8], that Write operations of Algorithm are 1-lock-free: Lemma 2 Write operations in Algorithm satisfy 1-lock-freedom.",
"Suppose, by way of contradiction, that Write operations do not satisfy 1-lock-freedom.",
"Eventually, all $n$ first registers are infinitely often updated only by correct processes unsuccessfully trying to complete a Write operation.",
"Thus, eventually each of the $n$ first registers contain the value from one of these incomplete Write operations.",
"As there are at most $n-1$ covered registers when a snapshot is taken, one of these value is eventually permanently present in some register (Lemma REF ).",
"This value is then eventually contained in the local view of every correct process, and thus, will eventually be present in every update of all the $n$ first registers.",
"The correct process with this Write value must therefore eventually pass the test on line and, thus, complete its Write operation — a contradiction.",
"The induction step relies primarily on the helping mechanism.",
"This mechanism guarantees that a process making progress eventually ensures that the processes it observes as having a pending operation also make progress (the mechanism is similar to the one of the wait-free SWMR memory implementation of [8]): Lemma 3 If a process $q$ performing infinitely many operations sees $(v,id(p),c)$ , and if $p$ is correct, then $p$ eventually completes its $c^{th}$ Write operation.",
"By Lemma REF , if process $q$ returns from a Write operation with value $(v,id(q),c^{\\prime })$ at time $\\tau $ , then for any time $\\tau ^{\\prime }\\ge \\tau $ there is a register containing $(v,id(q),c^{\\prime })$ .",
"But note that $(v,id(q),c^{\\prime })$ is written to a register only associated with $q$ 's local view.",
"Thus, as $q$ completes an infinite number of Write operations, each local view of $q$ will eventually be forever present in some register, in particular $(v,id(p),c)$ .",
"Thus $(v,id(p),c)$ is eventually observed in every snapshot taken by correct processes, and, therefore, included in their local view.",
"This implies that it will eventually be present in every register written infinitely often, in particular in the first $n$ registers.",
"As $p$ is correct, it eventually sees $(v,id(p),c)$ in $n$ registers for the test at line and, thus, completes its corresponding $c^{th}$ Write operation.",
"By the base case provided by Lemma REF and Lemma REF , we have: Lemma 4 Write operations in Algorithm satisfy $k$ -lock-freedom.",
"We proceed by induction on $k$ , starting with the base case of $k=1$ (Lemma REF ).",
"Suppose that Write operations satisfy $\\ell $ -lock-freedom for some $\\ell < k$ .",
"Consider a run in which at least $\\ell +1$ processes are correct, but only $\\ell $ of them make progress (if such a run doesn't exist, the algorithm satisfies $(\\ell +1)$ -lock-freedom).",
"In this run, at least one correct process is eventually blocked in a Write operation.",
"According to Lemma REF , the $\\ell $ processes performing infinitely many Write operations eventually do not observe new values written by other processes.",
"By the algorithm, these processes eventually never write to the last $k-\\ell >0$ registers.",
"A correct process that never completes a Write operation will execute the while loop (lines –) infinitely many times, and thus, will infinitely often take a snapshot and update its local view (line ).",
"In particular, it will eventually observe a new Write operation performed by each of the $\\ell $ processes completing infinitely many Write operations.",
"It will then eventually include at least $\\ell +1$ processes in its set of active processes (i.e., the $\\ell $ processes performing infinitely many Write operations and itself).",
"It will therefore eventually write to the $(n+\\ell )^{th}$ register infinitely often.",
"In the considered run, this register is written infinitely often only by correct processes which do not complete new Write operations.",
"The value from at least one of such process will then be observed by the $\\ell $ processes making progress.",
"By Lemma REF , this process will eventually complete its Write operation — a contradiction.",
"Collect operations in Algorithm clearly satisfy wait-freedom as there are no loops and MWMR snapshot operations are wait-free.",
"Thus Lemma REF and the wait-freedom of Collect operations imply that: Theorem 2 Algorithm is a $k$ -lock-free implementation of an SWMR memory for $n$ processes using $n+k-1$ MWMR registers."
],
[
"Lower bound: impossibility of 2-obstruction-free SWMR memory implementations with n MWMR registers",
"The algorithm in Section gives an upper bound of $n+k-1$ on the number of MWMR registers required to implement an SWMR memory satisfying the $k$ -lock-free progress condition in the comparison-based model.",
"In this section, we present a lower bound on the number of MWMR registers required in order to provide a 2-obstruction-free, and hence also a 2-lock-free, SWMR memory implementation."
],
[
"Overview of the lower bound",
"Our proof relies on the concepts of covering and indistinguishability.",
"A register is covered at a given point of a run if there is at least one process poised to write to it (we say that the process covers the register).",
"Hence, a covered register cannot be used to ensure persistence of written data: by awakening the covering process, the adversarial scheduler can overwrite it.",
"This property alone can be used to show that $n$ registers are required for an obstruction-free (and hence also for a 1-lock-free) SWMR memory implementation [4], but not to obtain a lower bound of more than $n$ shared resources as there is always one which remains uncovered.",
"Indistinguishability captures bounds on the knowledge that a process has of the rest of the system.",
"Two system states are indistinguishable for a process if it has the same local state in both states and if the shared memory includes the same content.",
"Thus, in an SWMR memory implementation, a Write operation can safely terminate only if, in all indistinguishable states, its value is present in a register that is not covered (by a process unaware of that value).",
"In our proof, we work with a composed notion of covering and indistinguishability.",
"The idea is to show that there is a large set of reachable system states, indistinguishable to a given process $p$ , in which different sets of registers are covered.",
"Intuitively, if a set of registers is covered in one of these indistinguishable states, $p$ must necessarily write to a register outside of this set in order to complete a new Write operation.",
"Hence, if such indistinguishable states exist for all register subsets, then $p$ must write its value to all registers.",
"To perform infinitely many high-level Write operations, $p$ must then write infinitely often to all available registers.",
"But then any other process $p^{\\prime }$ taking steps can be masked by the execution of $p$ (i.e., any write $p^{\\prime }$ makes to a MWMR register can be scheduled to be overwritten by $p$ ).",
"This way we establish that no 2-obstruction free implementation exists, as it requires that at least two processes must be able to make progress concurrently."
],
[
"Preliminaries",
"Assume, by contradiction, that there exists a 2-obstruction-free SWMR implementation using only $n$ registers.",
"To establish a contradiction, we consider a set of runs by a fixed set $\\Pi $ of $n$ processes in which every process performs infinitely many Write operations with monotonically increasing arguments.",
"Let $\\mathcal {R}$ denote the set of $n$ available registers."
],
[
"Indistinguishability",
"A configuration $C$ is said to be indistinguishable from a configuration $C^{\\prime }$ for a set of processes $P$ , if the content of all registers and the states of all processes in $P$ are identical in $C$ and $C^{\\prime }$ .",
"Given a set of configurations $\\cal D$ , let $I({\\cal D},P)$ denote that any two configurations from $\\mathcal {D}$ are indistinguishable for $P$ .",
"We say that an execution is $P$ -only, for a set of processes $P$ , if it consists only of steps by processes in $P$ .",
"We say that a set of processes $P$ is hidden in an execution $\\alpha $ if all writes in $\\alpha $ performed by processes in $P$ are overwritten by some processes not in $P$ , without any read performed by processes not from $P$ in between.",
"Given a sequence of steps $\\alpha $ and a set of processes $P$ , let $\\alpha |_P$ be the sub-sequence of $\\alpha $ containing only the steps from processes in $P$ .",
"Let us denote as $\\mathcal {D}\\alpha $ the set of all configurations reached by applying $\\alpha $ to all configurations in $\\mathcal {D}$ (note that $\\alpha $ must be applicable to all configurations in $\\mathcal {D}$ ).",
"Observation 1 If a $P$ -only execution $\\alpha $ is applicable to a configuration $C$ from a set of configurations $\\cal D$ indistinguishable for $P$ , i.e., $C\\in {\\cal D}$ and $I({\\cal D},P)$ , then $\\alpha $ is applicable to any configuration $C^{\\prime }\\in {\\cal D}$ , and it maintains the indistinguishability of configurations for $P$ , i.e., $I(\\mathcal {D}\\alpha ,P)$ .",
"A similar observation can be made concerning hidden executions: Observation 2 Given an execution $\\alpha $ applicable to $C$ , with $C$ from a set of configurations $\\cal D$ indistinguishable for P. If processes in $\\Pi \\setminus P$ are hidden in $\\alpha $ , then $\\alpha |_P$ is applicable to any $C^{\\prime }\\in {\\cal D}$ , and $I((\\mathcal {D}\\alpha |_P)\\cup \\lbrace C\\alpha \\rbrace ,P)$ .",
"We say that a set of processes $P$ covers a set of registers $R$ in some configuration $C$ , if for each register $r\\in R$ , there is a process $p\\in P$ such that the next step of $p$ in $C$ is a write on $r$ (the predicate is denoted $\\mathit {Cover}(R,P,C)$ ).",
"Our lower bound result relies on a concept that we call confusion.",
"We say that a set of processes $P$ are confused on a set of registers $S$ in a set of reachable configurations $\\cal D$ , denoted $\\mathit {Confused}(P,S,\\cal D)$ , if and only if: $I(\\mathcal {D},P)$ .",
"$|S|+|P|=n+1$ .",
"For any process $p\\in \\Pi \\setminus P$ , there exist two registers $r_p,r_p^{\\prime }\\in S$ such that, for any configuration $D\\in \\cal D$ , there exists $D^{\\prime }\\in \\cal D$ , such that $p$ covers $r_p$ in $D$ and $r_p^{\\prime }$ in $D^{\\prime }$ , or vice versa, and $D$ and $D^{\\prime }$ are indistinguishable to all other processes: $\\forall p\\in \\Pi \\setminus P,\\exists r_p,r_p^{\\prime } \\in S,\\forall D\\in {\\cal D},\\exists r \\in \\lbrace r_p,r_p^{\\prime }\\rbrace :$ $\\mathit {Cover}(\\lbrace r\\rbrace ,\\lbrace p\\rbrace ,D)\\wedge (\\exists D^{\\prime }\\in {\\cal D}, I(\\lbrace D,D^{\\prime }\\rbrace ,\\Pi \\setminus \\lbrace p\\rbrace )\\wedge \\mathit {Cover}(\\lbrace r_p,r_p^{\\prime }\\rbrace \\setminus \\lbrace r\\rbrace ,\\lbrace p\\rbrace ,D^{\\prime })){}.$ For any strict subset $R$ of $S$ , there exists $D\\in \\mathcal {D}$ such that $R$ is covered by $\\Pi \\setminus P$ in $D$ : $\\forall R\\subsetneq S, \\exists D\\in {\\cal D}: \\mathit {Cover}(R,\\Pi \\setminus P, D){}.$ Intuitively, processes in $P$ are confused on $S$ in $\\cal D$ , if $\\cal D$ is a set of indistinguishable configurations for $P$ , such that any strict subset of $S$ is covered by $\\Pi \\setminus P$ in some configuration of $\\cal D$ (Conditions 1 and 4).",
"We require that as much processes are confused as possible (Condition 2).",
"Additionally, the property must hold for a set of configurations $\\mathcal {D}$ in which processes not in $P$ may cover only one out of 2 given registers, and may be cover them independently of other processes states in $\\mathcal {D}$ (Condition 3).",
"Figure: Processes {p 5 ,p 6 ,p 7 ,p 8 }\\lbrace p_5,p_6,p_7,p_8\\rbrace are confused onregisters {r 1 ,r 2 ,r 3 ,r 4 ,r 5 }\\lbrace r_1,r_2,r_3,r_4,r_5\\rbrace ; an example of a possiblecovering is given on the right.In Figure REF , we give an example of a confusing set of configuration $\\cal D$ for 8 processes and 8 registers.",
"Processes $\\lbrace p_5,p_6,p_7,p_8\\rbrace $ are confused on registers $\\lbrace r_1,r_2,r_3,r_4,r_5\\rbrace $ .",
"Registers are represented as nodes, and pairs of registers that a process might be covering are represented as edges.",
"The set of indistinguishable configurations $\\cal D$ for $\\lbrace p_5,p_6,p_7,p_8\\rbrace $ are defined via composition of states for $p_1$ , $p_2$ , $p_3$ and $p_4$ in which they, respectively, cover registers in $\\lbrace r_1,r_2\\rbrace $ , $\\lbrace r_2,r_3\\rbrace $ , $\\lbrace r_2,r_4\\rbrace $ and $\\lbrace r_4,r_5\\rbrace $ .",
"An example of a covering of $\\lbrace r_1,r_2,r_3,r_5\\rbrace $ for some particular execution is presented on the right side of Figure REF .",
"First, we are going to provide an alternative property for Condition 4 of the definition of $\\mathit {Confused}(P,S,\\mathcal {D})$ .",
"The idea is that, given $(P,S,\\mathcal {D})$ satisfying Conditions 1, 2 and 3, Condition 4 is satisfied if and only if the graph induced by the sets of registers that may be covered by processes in $\\Pi \\setminus P$ (as represented in Figure REF ) forms a connected component over $S$ .",
"More formally, that Condition 4 is satisfied if and only if, for any partition of $S$ into two non-empty subsets $S_1$ and $S_2$ , there is a process in $\\Pi \\setminus P$ for which the set of two registers it may be covering in $\\cal D$ intersects with both $S_1$ and $S_2$ : Lemma 5 $\\forall P\\subseteq \\Pi , \\forall S\\subseteq \\mathcal {R},\\forall {\\cal D}\\subseteq \\mathit {Reach}$ satisfying Conditions 1, 2 and 3 of the confusion definition, we have $\\forall R\\subsetneq S, \\exists D\\in {\\cal D}: \\mathit {Cover}(R,\\Pi \\setminus P, D)$ if and only if: $\\forall S_1,S_2\\subseteq S, (S_1\\ne \\emptyset \\wedge S_2\\ne \\emptyset \\wedge S_1\\cup S_2= S\\wedge S_1\\cap S_2 =\\emptyset ):$ $\\exists r_1\\in S_1, r_2\\in S_2, p\\in \\Pi \\setminus P, D_1,D_2\\in {\\cal D}:(\\mathit {Cover}(\\lbrace r_1\\rbrace ,\\lbrace p\\rbrace ,D_1)\\wedge \\mathit {Cover}\\left((\\lbrace r_2\\rbrace ,\\lbrace p\\rbrace ,D_2)\\right){}.$ Let us fix some $P\\subseteq \\Pi $ , $S\\subseteq \\mathcal {R}$ , and ${\\cal D}\\subseteq \\mathit {Reach}$ satisfying Conditions 1, 2 and 3 of the confusion definition.",
"First, let us assume that Condition 4 is also satisfied and consider any partition of $S$ into non-empty subsets $S_1$ and $S_2$ (i.e., $S_1\\ne \\emptyset $ , $S_2\\ne \\emptyset $ , $S_1\\cap S_2=\\emptyset $ and $S_1\\cup S_2=S$ ).",
"Assume now that there does not exist any process $p\\in \\Pi \\setminus P$ such that $p$ might be covering a register from $S_1$ or a register from $S_2$ in $\\cal D$ .",
"This implies that processes in $\\Pi \\setminus P$ can be partitioned into two subsets $Q_1$ and $Q_2$ (with $Q_1\\cap Q_2=\\emptyset $ and $Q_1\\cup Q_2=\\Pi \\setminus P$ ) such that processes in $Q_1$ , respectively $Q_2$ , may cover registers from $S_1$ , respectively $S_2$ , in $\\cal D$ .",
"By construction of the partitions, we have $|Q_1|+|Q_2|= |\\Pi \\setminus P|$ and $|S_1|+|S_2|=|S|$ .",
"Using the fact that Condition 2 is satisfied by $P$ and $S$ we obtain from $|S|+|P|= n+1$ that $|S_1|+|S_2|+(n-(|Q_1|+|Q_2|))=n+1$ , and thus, that $|S_1|+|S_2|=|Q_1|+|Q_2|+1$ .",
"This implies that either $|Q_1|<|S_1|$ or $|Q_2|<|S_2|$ , w.l.o.g., let $|Q_1|<|R_1|$ .",
"Now consider $r\\in S_2$ , $S\\setminus \\lbrace r\\rbrace $ is a strict subset of $S$ , and therefore Condition 4 implies that there exists $D\\in \\mathcal {D}$ such that $\\mathit {Cover}(S\\setminus \\lbrace r\\rbrace ,\\Pi \\setminus P, D)$ .",
"As registers in $S_1$ can only be covered by processes from $Q_1$ , then we have $\\mathit {Cover}(S_1,Q_1, D)$ .",
"Recall that, by the pigeonhole principle, a set of processes cannot cover more registers than processes it contains.",
"But $|Q_1|<|R_1|$ — a contradiction.",
"Now let us assume that given any partition of $S$ into non-empty subsets $S_1$ and $S_2$ , there exists a process $p\\in \\Pi \\setminus P$ such that $p$ might be covering a register in $S_1$ or a register in $S_2$ in $\\cal D$ .",
"Let us show that any strict subset $R$ of $S$ is covered in some configuration from $\\mathcal {D}$ and, hence, that Condition 4 is satisfied.",
"This is done by inductevely restricting the set of configurations from $\\mathcal {D}$ , by selecting the maximal subset in which some process from $\\Pi \\setminus P$ may cover only one register.",
"The idea is to select a process which may cover only one not-yet covered register in $R$ , and to select the subset in which this process covers this register.",
"Let $S_0$ be a non-empty subset of $S$ .",
"Let $p_0$ be a process from $\\Pi \\setminus P$ which might be covering a register $r_0$ in $S_0$ or a register $r_0^{\\prime }$ in $S\\setminus S_0$ in $\\cal D$ .",
"Let us assume that such a process exists and consider $\\mathcal {D}_0$ to be the subset of $\\mathcal {D}$ including all configurations in which $p_0$ is covering $r_0^{\\prime }$ .",
"Now let $S_1=S_0\\cup \\lbrace r_0^{\\prime }\\rbrace $ and repeat this procedure using $S_1$ to select some $p_1$ and compute ${\\cal D}_1$ , etc... As long as a process can be selected satisfying the condition, the sets $S_i$ keep increasing with $i$ .",
"Consider the round $j$ at which the procedure fails to find such a process.",
"This implies that there is no process which might be covering a register from either $S_j$ or $S\\setminus S_j$ in ${\\cal D}_{j-1}$ .",
"Note that by construction ${\\cal D}_{j-1}$ is a non-empty subset of ${\\cal D}$ .",
"If $S_j\\ne S$ , then $S_j$ and $S\\setminus S_j$ forms a partition of $S$ into two non-empty subsets.",
"Thus, by assumption, there exists a process $q$ which might be covering a register $r_q$ in $S_j$ or a register $r_q^{\\prime }$ in $S\\setminus S_j$ in $\\cal D$ .",
"Consider some configuration $D\\in {\\cal D}_{j-1}$ .",
"According to Condition 3 of the confusion definition, as $D\\in {\\cal D}$ , $q$ is covering either $r_q$ or $r_q^{\\prime }$ in $D$ , and there exists a configuration $D^{\\prime }$ in which $q$ is covering the other register in $\\lbrace r_q,r_q^{\\prime }\\rbrace $ , relatively to $D$ , and such that $D$ and $D^{\\prime }$ are indistinguishable to all other processes.",
"As $I(\\lbrace D,D^{\\prime }\\rbrace ,\\Pi \\setminus \\lbrace q\\rbrace )$ , if $D$ was kept in some restriction of $\\mathcal {D}_{i-1}$ towards $\\mathcal {D}_i$ , then $D^{\\prime }$ was also kept unless $q$ was the corresponding selected process $p_i$ .",
"But if $q$ was selected in an earlier iteration, both $r_q$ and $r_q^{\\prime }$ would be included in $S_j$ .",
"Thus $D$ and $D^{\\prime }$ belong to ${\\cal D}_{j-1}$ and therefore $q$ is a valid selection for $p_j$ .",
"This contradiction implies that therefore $S_j=S$ .",
"By construction, all registers in $S_j\\setminus S_0$ are covered in all configurations in $\\mathcal {D}_{j-1}$ .",
"As this is true for any non-empty $S_0$ and as $S_j=S$ , any strict subset of $S$ is covered in some configuration of $\\mathcal {D}$ and therefore Condition 4 is satisfied.",
"Lemma REF can be used to show that, given any confusion for some $P$ distinct from $\\Pi $ , $S$ and $\\mathcal {D}$ , we can identify a process $p\\in \\Pi \\setminus P$ and a register $r\\in S$ such that $P\\cup \\lbrace p\\rbrace $ is confused on $S\\setminus \\lbrace r\\rbrace $ for a subset $\\cal D^{\\prime }$ of $\\cal D$ which includes any given $C\\in \\mathcal {D}$ : Lemma 6 Given $P\\subsetneq \\Pi $ , $S\\subseteq \\mathcal {R}$ , ${\\cal D}\\subseteq \\mathit {Reach}$ : $\\mathit {Confused}(P,S,{\\cal D}) \\Rightarrow $ $\\exists p\\in \\Pi \\setminus P,\\exists r\\in S,\\forall C\\in \\mathcal {D},\\exists \\mathcal {D}^{\\prime }\\subseteq \\mathcal {D}:(C\\in \\mathcal {D}^{\\prime })\\wedge \\mathit {Confused}(P\\cup \\lbrace p\\rbrace ,S\\setminus \\lbrace r\\rbrace ,\\mathcal {D}^{\\prime })$ .",
"According to Condition 3, a process may be covering exactly two registers from $S$ in $\\mathcal {D}$ , thus, the sum over $S$ of how many distinct processes may cover each register equals to $2|\\Pi \\setminus P|=2(n-|P|)$ .",
"Note that any register $r\\in S$ may be covered by at least by one process in $\\Pi \\setminus P$ in $\\cal D$ as any strict subset of $S$ may be covered (Condition 4).",
"Therefore, there exists a register $r_c\\in S$ which can be covered by a single process $p_c\\in \\Pi \\setminus P$ in $\\cal D$ .",
"Indeed, if all registers in $S$ might be covered by two distinct processes, then, the sum over $S$ of how many distinct processes may cover each register (equal to $2(n-|P|)$ ), would be greater than or equal to $2|S|$ , or $n-|P|<|S|$ as $|P|+|S|=n+1$ (Condition 2).",
"Let $\\mathcal {D}_c$ be the subset of $\\mathcal {D}$ which includes all configurations in $\\mathcal {D}$ that are indistinguishable to $p_c$ from any configuration $C\\in \\mathcal {D}$ .",
"Let us show that $Confused(P\\cup \\lbrace p_c\\rbrace ,S\\setminus \\lbrace r_c\\rbrace ,{\\cal D}_c)$ .",
"Condition 1 holds as by construction all configurations in ${\\cal D}_c$ are indistinguishable to $p_c$ and as they are indistinguishable to all processes in $P$ , since ${\\cal D}_c$ is a subset of $\\cal D$ .",
"It is immediate, as we remove a register from $S$ and add a process to $P$ , that Condition 2 holds.",
"Now consider any process $p\\in \\Pi \\setminus (P\\cup \\lbrace p_c\\rbrace )$ and any configuration $D\\in \\mathcal {D}_c$ .",
"As $p\\in \\Pi \\setminus P$ and $D\\in \\mathcal {D}$ , Condition 3 of the confusion definition implies that that there exists $D^{\\prime }\\in \\mathcal {D}$ , $I(\\lbrace D,D^{\\prime }\\rbrace ,\\Pi \\setminus \\lbrace p\\rbrace )$ , such that $p$ covers $r_p$ and $r_p^{\\prime }$ in $D$ and $D^{\\prime }$ respectively (or vice-versa).",
"Since $I(\\lbrace D,D^{\\prime }\\rbrace ,\\Pi \\setminus \\lbrace p\\rbrace )$ , $D^{\\prime }\\in \\mathcal {D}_c$ , and since $p_c$ is the only process which may cover $r_c$ in $\\mathcal {D}$ , $r_p$ and $r_p^{\\prime }$ belong to $S\\setminus \\lbrace r_c\\rbrace $ .",
"Thus Condition 3 is verified for $P\\cup \\lbrace p\\rbrace $ , $S\\setminus \\lbrace r_c\\rbrace $ and $\\mathcal {D}_c$ .",
"Lastly, let us consider some partition of $S\\setminus \\lbrace r_c\\rbrace $ into two non-empty subsets $S_1$ and $S_2$ .",
"Both ($S_1\\cup \\lbrace r_c\\rbrace $ ,$S_2$ ) and ($S_1$ ,$S_2\\cup \\lbrace r_c\\rbrace $ ) form a partition of $S$ in two non-empty subsets.",
"Thus, as $\\mathit {Confused}(P,S,{\\cal D})$ , we can apply Lemma REF and obtain that $\\exists p_1,p_2\\in \\Pi \\setminus P$ such that $p_1$ , respectively $p_2$ , might cover registers from either $S_1\\cup \\lbrace r_c\\rbrace $ or $S_2$ , respectively either $S_1$ or $S_2\\cup \\lbrace r_c\\rbrace $ , in $\\cal D$ .",
"It follows that $p_c$ cannot be both $p_1$ and $p_2$ as $p_c$ might cover only two registers in $\\cal D$ , one of which is $r_c$ .",
"Thus, depending whether the other register belongs to $S_1$ or $S_2$ , $p_1$ or $p_2$ is disctinct from $p_c$ .",
"W.l.o.g, assume that $p_1\\ne p_c$ .",
"As $p_c$ is the only process which may be covering $r_c$ , this implies that $p_1$ might be covering a register from either $S_1$ or $S_2$ .",
"Furthermore, since Conditions 1, 2 and 3 applies to $P\\cup \\lbrace p\\rbrace $ , $S\\setminus \\lbrace r_c\\rbrace $ and $\\mathcal {D}_c$ , we can apply Lemma REF to obtain that Condition 4 is also verified.",
"We now show that the characterization can be used to increase the number of registers that processes are confused on, by decreasing the number of confused processes: Lemma 7 Let $P\\subsetneq \\Pi $ , $S\\subseteq \\mathcal {R}$ and $\\mathcal {D}\\subseteq \\mathit {Reach}$ such that $\\mathit {Confused}(P,S,\\mathcal {D})$ .",
"Given $C\\in \\mathcal {D}$ , if $\\exists p\\in P, r_1\\in S, r_2\\in \\mathcal {R}\\setminus S$ and if there exist $P$ -only executions $\\alpha _1$ and $\\alpha _2$ which are applicable to $C$ , and such that $I(\\lbrace C\\alpha _1,C\\alpha _2\\rbrace ,\\Pi \\setminus \\lbrace p\\rbrace )$ , $Cover(\\lbrace r_1\\rbrace ,\\lbrace p\\rbrace ,C\\alpha _1)$ and $Cover(\\lbrace r_2\\rbrace ,\\lbrace p\\rbrace ,C\\alpha _2)$ , then we have $\\mathit {Confused}(P\\setminus \\lbrace p\\rbrace ,S\\cup \\lbrace r_2\\rbrace ,(\\mathcal {D}\\alpha _1)\\cup (\\mathcal {D}\\alpha _2))$ .",
"Following Observation REF , as $\\alpha _1$ and $\\alpha _2$ are $P$ -only, and as $\\mathcal {D}$ satisfies Condition 1 for $P$ , $(\\mathcal {D}\\alpha _1)\\cup (\\mathcal {D}\\alpha _2)$ satisfies Condition 1 for $P\\setminus \\lbrace p\\rbrace $ .",
"Condition 2 trivially holds for $P\\setminus \\lbrace p\\rbrace $ and $S\\cup \\lbrace r\\rbrace $ as it holds for $P$ and $S$ and we remove a process from $P$ and add a register to $S$ .",
"Condition 3 is satisfied for all processes in $\\Pi \\setminus P$ and configurations in $(\\mathcal {D}\\alpha _1)\\cup (\\mathcal {D}\\alpha _2)$ as $\\alpha _1$ and $\\alpha _2$ are $P$ -only, and as $\\mathcal {D}$ satisfies Condition 3 for any process in $\\Pi \\setminus P$ .",
"Moreover, as configurations in $\\mathcal {D}$ are indistinguishable to $p\\in P$ , $p$ may only cover $r_1$ if $D\\in \\mathcal {D}\\alpha _1$ and cover $r_2$ if $D\\in \\mathcal {D}\\alpha _2$ .",
"But as given any $D\\in \\mathcal {D}$ we have $I(\\lbrace D\\alpha _1,D\\alpha _2\\rbrace ,\\Pi \\setminus \\lbrace p\\rbrace )$ , Condition 3 is also satisfied for $p$ .",
"Since Conditions 1, 2 and 3 are satisfied, we can apply Lemma REF and obtain that $\\mathit {Confused}(P\\setminus \\lbrace p\\rbrace ,S\\cup \\lbrace r_2\\rbrace ,(\\mathcal {D}\\alpha _1)\\cup (\\mathcal {D}\\alpha _2))$ .",
"Indeed, a partition of two non-empty subsets of $S\\cup \\lbrace r_2\\rbrace $ can be reduced, unless the partition is $(S,\\lbrace r_2\\rbrace )$ , to a partition of two non-empty subsets of $S$ .",
"In this case, Lemma REF can be applied for $P$ ,$S$ and $\\mathcal {D}$ , which provides us, for any partition of $S$ , with a process that may cover in $\\mathcal {D}$ a register from either set of the partition.",
"As $\\alpha _1$ is $P$ -only, it still holds for $\\mathcal {D}\\alpha _1$ .",
"For the partition $(S,\\lbrace r_2\\rbrace )$ , $p$ may cover either $r_1\\in S$ or $r_2$ in $(\\mathcal {D}\\alpha _1)\\cup (\\mathcal {D}\\alpha _2)$ ."
],
[
"The lower bound",
"To establish our lower bound, we show that there is a set of reachable configuration $\\mathcal {D}$ in which there is a process confused on all $n$ registers.",
"Intuitively, we proceed by induction on the number of “confusing” registers.",
"For the base case, we show that the initial configuration can lead to a confusion of all but one process on two registers: Lemma 8 $\\exists \\mathcal {D}\\in \\mathit {Reach},\\exists p\\in \\Pi ,\\exists S\\subseteq \\mathcal {R}: \\mathit {Confused}(\\Pi \\setminus \\lbrace p\\rbrace ,S,\\mathcal {D}){}.$ Consider any two processes $p_1$ and $p_2$ .",
"Since the algorithm is comparison-based, the first write the two processes perform in a solo execution is on the same register, let us call it $r$ .",
"Let $p_1$ execute solo until it is about to write to $r$ and then do the same with $p_2$ , let $C$ be the resulting configuration.",
"Consider the execution $\\alpha $ from $C$ in which $p_1$ executes until it is poised to write to a register $r^{\\prime }\\ne r$ and then $p_2$ executes its pending write on $r$ .",
"This execution is valid as $p_1$ must eventually write to an uncovered register.",
"We obtain $\\mathit {Confused}(\\Pi \\setminus \\lbrace p_1\\rbrace ,\\lbrace r,r^{\\prime }\\rbrace ,\\lbrace C\\alpha ,C\\alpha |_{\\lbrace p_2\\rbrace }\\rbrace )$ .",
"Indeed, as $p_1$ is hidden in $\\alpha $ , following Observation REF , we have $I(\\lbrace C\\alpha ,C\\alpha |_{\\lbrace p_2\\rbrace }\\rbrace ,\\Pi \\setminus \\lbrace p_1\\rbrace )$ (Condition 1).",
"We have $|\\Pi \\setminus \\lbrace p_1\\rbrace |+|\\lbrace r,r^{\\prime }\\rbrace |=n+1$ (Condition 2).",
"As $p_1$ covers $r^{\\prime }$ in $C\\alpha $ and $p_2$ covers $r$ in $C\\alpha |_{\\lbrace p_2\\rbrace }$ , we have Condition 4.",
"Condition 3 directly follows from Conditions 1 and 4 in this setting.",
"We now prove our inductive step.",
"Given a set of configurations in which a set of processes, $P\\ne \\Pi $ , is confused on a set of registers, $S\\ne \\mathcal {R}$ , we can obtain a set of configurations in which a set $P^{\\prime }$ of processes are confused on a set $S^{\\prime }$ of strictly more than $|S|$ registers: Lemma 9 $\\exists {\\cal D}\\subseteq \\mathit {Reach}, P\\subsetneq \\Pi ,S\\subsetneq \\mathcal {R}: \\mathit {Confused}(P,S,{\\cal D})$ $\\Rightarrow \\exists {\\cal D^{\\prime }}\\subseteq \\mathit {Reach}, P^{\\prime }\\subseteq \\Pi , S^{\\prime }\\subseteq \\mathcal {R}, S\\subsetneq S^{\\prime }:\\mathit {Confused}(P^{\\prime },S^{\\prime },{\\cal D^{\\prime }})$ .",
"Given $\\mathit {Confused}(P,S,\\mathcal {D})$ , consider $C\\in \\mathcal {D}$ such that exactly $|S|-1$ registers in $S$ are covered by processes in $\\Pi \\setminus P$ .",
"Then we can reach a configuration in which all registers not in $S$ are covered by processes in $P$ .",
"Indeed, when executed solo starting from $C$ , a process must eventually write to a register that is not covered in $C$ .",
"Thus, it must eventually write either to a register in $\\mathcal {R}\\setminus S$ or to the uncovered register in $S$ .",
"Recall that, as $|S|+|P|=n+1$ , we have $|\\mathcal {R}\\setminus S|=|P|-1$ .",
"Thus, by concatenating solo executions of processes in $P$ until they are poised to write to uncovered registers, we reach a configuration $C\\alpha $ in which all registers are covered.",
"Let $p$ be the process in $P$ covering a register from $S$ in $C\\alpha $ .",
"Note that, as $\\alpha $ is $P$ -only, we have $\\mathit {Confused}(P,S,\\mathcal {D}\\alpha )$ .",
"Thus: $\\mathit {Cover}(\\mathcal {R}\\setminus S,P\\setminus \\lbrace p\\rbrace ,C\\alpha )\\wedge \\mathit {Confused}(P,S,\\mathcal {D}\\alpha ){}.$ Now from this set of configurations, we are going to build a new one in which $P$ is confused on two distinct sets of registers.",
"By Lemma REF , there exist $p_c\\in \\Pi \\setminus P$ and $r\\in S$ such that for any $C^{\\prime }\\in \\mathcal {D}$ we have $\\mathit {Confused}(P\\cup \\lbrace p_c\\rbrace ,S\\setminus \\lbrace r\\rbrace ,\\mathcal {D}^{\\prime })$ with $C^{\\prime }\\in \\mathcal {D}^{\\prime }$ .",
"Let us select $C^{\\prime }\\in \\mathcal {D}$ to be a configuration in which $p_c$ covers $r_c\\in S\\setminus \\lbrace r\\rbrace $ (Since we have $\\mathit {Confused}(P,S,\\mathcal {D})$ , $p_c\\in P$ may cover two registers from $S$ in $\\mathcal {D}$ and so at most one can be $r$ ).",
"If $p$ is executed solo from $C^{\\prime }\\alpha $ , it must write infinitely often to all registers in $S$ to ensure that it writes to an uncovered register.",
"Hence, in a $\\lbrace p,p_c\\rbrace $ -only execution from $C^{\\prime }\\alpha $ , $p_c$ can be hidden for arbitrarly many steps as long as $p_c$ does not write to a register outside of $S$ .",
"But, as the algorithm satisfies 2-obstruction-freedom, $p_c$ must eventually write to a register outside of $S$ in such an execution.",
"Consider the $\\lbrace p,p_c\\rbrace $ -only execution $\\beta $ from $C^{\\prime }\\alpha $ in which $p_c$ is hidden and such that $p_c$ executes until it is poised to write to some register $r^{\\prime }\\in \\mathcal {R}\\setminus S$ .",
"Thus, we get two configurations $C^{\\prime }\\alpha \\beta $ and $C^{\\prime }\\alpha \\beta |_{\\lbrace p\\rbrace }$ , indistinguishable to all processes but $p_c$ , in which $p_c$ covers, respectively, $r^{\\prime }\\in \\mathcal {R}\\setminus S$ and $r_c\\in S$ .",
"Thus, the conditions of Lemma REF hold for $\\mathcal {D}^{\\prime }$ , $p_c$ , $\\alpha \\beta $ and $\\alpha \\beta |_{\\lbrace p\\rbrace }$ and so we obtain $\\mathit {Confused}(P,(S\\cup \\lbrace r^{\\prime }\\rbrace )\\setminus \\lbrace r\\rbrace , (\\mathcal {D}^{\\prime }\\alpha \\beta )\\cup (\\mathcal {D}^{\\prime }\\alpha \\beta |_{\\lbrace p\\rbrace }))$ .",
"As $\\beta $ is $\\lbrace p,p_c\\rbrace $ -only and $p_c$ is hidden in it, we have: $\\mathit {Cover}(\\mathcal {R}\\setminus S,P\\setminus \\lbrace p\\rbrace ,C\\alpha \\beta )\\wedge \\mathit {Confused}(P,S,\\mathcal {D}\\alpha \\beta |_{\\lbrace p\\rbrace })\\wedge $ $\\mathit {Confused}(P,S\\cup \\lbrace r^{\\prime }\\rbrace \\setminus \\lbrace r\\rbrace ,(\\mathcal {D}^{\\prime }\\alpha \\beta )\\cup (\\mathcal {D}^{\\prime }\\alpha \\beta |_{\\lbrace p\\rbrace })){}.$ Moreover, all configurations in the formula above are indistinguishable to processes in $P$ , since $\\mathcal {D}^{\\prime }\\subseteq \\mathcal {D}$ , $I(\\mathcal {D},P)$ , $\\alpha \\beta $ is $P\\cup \\lbrace p_c\\rbrace $ -only and $p_c$ is hidden in it (Observation REF ).",
"Let $p^{\\prime }$ be the process from $P$ that covers $r^{\\prime }$ in $C\\alpha \\beta $ .",
"According to $p$ or $p^{\\prime }$ , every proper subset of $S$ or $S\\cup \\lbrace r^{\\prime }\\rbrace \\setminus \\lbrace r\\rbrace $ may be covered in the current configuration by $\\Pi \\setminus (P\\cup \\lbrace p,p\\rbrace )$ and all other registers covered by $P\\setminus \\lbrace p,p^{\\prime }\\rbrace $ .",
"Thus, from $C\\alpha \\beta $ , to complete a Write operation, $p$ or $p^{\\prime }$ must write to all registers in one of the sets $S$ , $S\\cup \\lbrace r^{\\prime }\\rbrace \\setminus \\lbrace r\\rbrace $ or $\\lbrace r,r^{\\prime }\\rbrace $ .",
"Consider any $\\lbrace p,p^{\\prime }\\rbrace $ -only extension of $C\\alpha \\beta $ .",
"If one of $\\lbrace p,p^{\\prime }\\rbrace $ covers a register in $S\\setminus \\lbrace r\\rbrace $ , $r$ or $r^{\\prime }$ , then the other process, in any solo extension, must write respectively to all registers in $\\lbrace r,r^{\\prime }\\rbrace $ , $S$ or $(S\\cup \\lbrace r^{\\prime }\\rbrace )\\setminus \\lbrace r\\rbrace $ .",
"In particular, since $p^{\\prime }$ covers $r^{\\prime }$ in $C\\alpha \\beta $ , $p$ running solo from $C\\alpha \\beta $ must eventually cover a register in $S\\setminus \\lbrace r\\rbrace $ (Note that $S\\setminus \\lbrace r\\rbrace \\ne \\emptyset $ , since $|P|<n$ and $|P|+|S|=n+1$ ).",
"Then $p^{\\prime }$ executing solo afterwards must write to $r$ and $r^{\\prime }$ .",
"Let us stop $p^{\\prime }$ when it covers a register $r^{\\prime \\prime }\\ne r$ for the last time before writing to $r$ .",
"Let $\\gamma $ be the resulting execution, and $E=C\\alpha \\beta \\gamma $ be the resulting configuration.",
"Let $\\mathcal {E}$ and $\\mathcal {E}^{\\prime }$ denote the sets of configurations indistinguishable from $E$ to $P$ defined as $\\mathcal {D}\\alpha \\beta |_{\\lbrace p\\rbrace }\\gamma $ and $(\\mathcal {D}^{\\prime }\\alpha \\beta \\gamma )\\cup (\\mathcal {D}^{\\prime }\\alpha \\beta |_{\\lbrace p\\rbrace }\\gamma )$ respectively.",
"Note that as $\\gamma $ is $P$ -only, we still have $\\mathit {Confused}(P,S,\\mathcal {E})$ and $\\mathit {Confused}(P,S\\cup \\lbrace r^{\\prime }\\rbrace \\setminus \\lbrace r\\rbrace ,\\mathcal {E}^{\\prime })$ .",
"Now the following two cases are possible: $r^{\\prime \\prime }\\notin S\\cup \\lbrace r^{\\prime }\\rbrace $ : In this case, we let $p$ continue until it is poised to write on $r$ , and then, we let the process from $P\\setminus \\lbrace p,p^{\\prime }\\rbrace $ which covers $r^{\\prime \\prime }$ to proceed to its pending write on $r^{\\prime \\prime }$ .",
"Let $\\delta $ be this $P$ -only execution from $E$ in which $p^{\\prime }$ is hidden.",
"As $p^{\\prime }$ covers $r\\in S$ in $E\\delta $ and $r^{\\prime \\prime }\\in \\mathcal {R}\\setminus S$ in $E\\delta |_{P\\setminus \\lbrace p^{\\prime }\\rbrace }$ , as $I(\\lbrace E\\delta ,E\\delta |_{P\\setminus \\lbrace p^{\\prime }\\rbrace }\\rbrace ,\\Pi \\setminus \\lbrace p^{\\prime }\\rbrace )$ , and as $\\mathit {Confused}(P,S,\\mathcal {E})$ , we can apply Lemma REF and obtain $\\mathit {Confused}(P\\setminus \\lbrace p^{\\prime }\\rbrace ,S\\cup \\lbrace r^{\\prime \\prime }\\rbrace ,(\\mathcal {E}\\delta )\\cup (\\mathcal {E}\\delta |_{P\\setminus \\lbrace p^{\\prime }\\rbrace }))$ .",
"$r^{\\prime \\prime }\\in S\\cup \\lbrace r^{\\prime }\\rbrace $ , and so $r^{\\prime \\prime }\\in (S\\cup \\lbrace r^{\\prime }\\rbrace )\\setminus \\lbrace r\\rbrace $ : Then we have the following sub-cases: Some step performed by $p$ in its solo execution from $E$ makes $p^{\\prime }$ to choose a register other than $r$ to perform its next write in its solo extension.",
"Clearly, this step of $p$ is a write.",
"From the configuration in which $p$ is poised to execute this “critical” write, let $p^{\\prime }$ run solo until it is poised to write to $r$ and then let $p$ complete its pending write.",
"Let $E\\delta $ be the resulting configuration.",
"Now consider the execution in which $p$ completes its “critical” write, then $p^{\\prime }$ runs solo until it covers a register $r^{\\prime \\prime \\prime }\\ne r$ .",
"Let $E\\delta ^{\\prime }$ be the resulting configuration.",
"Note that as the states of the memory in $E\\delta $ and $E\\delta ^{\\prime }$ are identical, we have $I(\\lbrace E\\delta ,E\\delta ^{\\prime }\\rbrace ,\\Pi \\setminus \\lbrace p^{\\prime }\\rbrace )$ .",
"Note that $\\delta $ and $\\delta ^{\\prime }$ are $P$ -only executions, and that $p^{\\prime }$ covers $r$ in $E\\delta $ and $r^{\\prime \\prime \\prime }$ in $E\\delta ^{\\prime }$ .",
"If $r^{\\prime \\prime \\prime }\\in S$ , as we have $\\mathit {Confused}(P,(S\\cup \\lbrace r^{\\prime }\\rbrace )\\setminus \\lbrace r\\rbrace , \\mathcal {E}^{\\prime })$ , applying Lemma REF , we obtain $\\mathit {Confused}(P\\setminus \\lbrace p^{\\prime }\\rbrace ,(S\\cup \\lbrace r^{\\prime }\\rbrace ),(\\mathcal {E}^{\\prime }\\delta )\\cup (\\mathcal {E}^{\\prime }\\delta ^{\\prime }))$ .",
"If $r^{\\prime \\prime \\prime }\\in \\mathcal {R}\\setminus S$ , as we have $\\mathit {Confused}(P,S,\\mathcal {E})$ , applying Lemma REF , we obtain $\\mathit {Confused}(P\\setminus \\lbrace p^{\\prime }\\rbrace ,(S\\cup \\lbrace r^{\\prime \\prime \\prime }\\rbrace ),(\\mathcal {E}\\delta )\\cup (\\mathcal {E}\\delta ^{\\prime }))$ .",
"Otherwise, no write of $p$ is “critical”, and we let it run from $E$ until it covers $r$ (recall that, as $p^{\\prime }$ covers $r^{\\prime \\prime }\\in (S\\cup \\lbrace r^{\\prime }\\rbrace )\\setminus \\lbrace r\\rbrace $ , $p$ must eventually write to all registers in $S$ or $\\lbrace r,r^{\\prime }\\rbrace $ and, thus, to $r$ ).",
"Let then $p^{\\prime }$ run until it covers $r$ , as $p$ , and let $\\delta $ be this execution.",
"From $E\\delta $ , let $p^{\\prime }$ run until it becomes poised to write to a register $r^{\\prime \\prime \\prime }\\ne r$ , and then let $p$ perform its pending write on $r$ .",
"Let $\\lambda $ be this extension.",
"Note that as $p^{\\prime }$ is hidden in $\\lambda $ , we have $I(\\lbrace E\\delta \\lambda , E\\delta \\lambda |_{\\lbrace p^{\\prime }\\rbrace }\\rbrace ,\\Pi \\setminus \\lbrace p\\rbrace )$ .",
"Note also that $\\delta \\lambda $ and $\\delta \\lambda |_{\\lbrace p\\rbrace }$ are $P$ -only executions such that $p^{\\prime }$ covers $r$ in $E\\delta \\lambda |_{\\lbrace p\\rbrace }$ and covers $r^{\\prime \\prime \\prime }$ in $E\\delta \\lambda $ .",
"If $r^{\\prime \\prime \\prime }\\in S$ , as we have $\\mathit {Confused}(P,(S\\cup \\lbrace r^{\\prime }\\rbrace )\\setminus \\lbrace r\\rbrace , \\mathcal {E}^{\\prime })$ , applying Lemma REF , we obtain $\\mathit {Confused}(P\\setminus \\lbrace p^{\\prime }\\rbrace ,(S\\cup \\lbrace r^{\\prime }\\rbrace ),(\\mathcal {E}^{\\prime }\\delta \\lambda )\\cup (\\mathcal {E}^{\\prime }\\delta \\lambda |_{\\lbrace p\\rbrace }))$ .",
"If $r^{\\prime \\prime \\prime }\\in \\mathcal {R}\\setminus S$ , as we have $\\mathit {Confused}(P,S,\\mathcal {E})$ , applying Lemma REF , we obtain $\\mathit {Confused}(P\\setminus \\lbrace p^{\\prime }\\rbrace ,(S\\cup \\lbrace r^{\\prime \\prime \\prime }\\rbrace ),(\\mathcal {E}\\delta \\lambda )\\cup (\\mathcal {E}\\delta \\lambda |_{\\lbrace p\\rbrace }))$ .",
"Our lower bound directly follows from Lemmata REF and REF : Theorem 3 Any $n$ -process comparison-based 2-obstruction-free SWMR memory implementation requires $n+1$ MWMR registers.",
"By contradiction, suppose that an $n$ -register algorithm exists.",
"We show, by induction, that there is a reachable configuration in which a process is confused on all registers.",
"Lemma REF shows that there exists a reachable configuration in which $n-1$ processes are confused on two registers.",
"We can therefore apply Lemma REF and obtain a configuration with a confusion with strictly more registers.",
"By induction, there exist then a set of configurations $\\mathcal {D}$ and $p\\in \\Pi $ such that $\\mathit {Confused}(\\lbrace p\\rbrace ,\\mathcal {R},\\mathcal {D})$ .",
"Thus, any strict subset of $\\mathcal {R}$ is covered by the remaining $n-1$ processes in some configuration in the (indistinguishable for $p$ ) set of configurations $\\mathcal {D}$ .",
"But $p$ may complete a Write operation if and only its write value is present in a register which is not covered (by a process not aware of the value) in any of the configurations indistinguishable to $p$ .",
"Therefore, in an infinite solo execution, $p$ must write infinitely often to all registers.",
"But then, any arbitrarily long execution by any other process can be hidden by incorporating sufficiently many steps of $p$ , violating 2-obstruction-freedom—a contradiction."
],
[
"Concluding remarks",
"This paper shows that the optimal space complexity of SWMR implementations depends on the desired progress condition: lock-free algorithms trivially require $n$ registers, while 2-obstruction-free ones (and, thus, also 2-lock-free ones) require $n+1$ registers.",
"We also extend the upper bound to $k$ -lock-freedom, for all $k=1,\\ldots ,n$ , by presenting a $k$ -lock-free SWMR implementation using $n+k-1$ registers.",
"A natural conjecture is that the algorithm is optimal, i.e., no such algorithm exists for $n+k-2$ registers for all $k=1,\\ldots ,n$ .",
"Since for $k=1$ , 2 and $n$ , $k$ -obstruction-freedom and $k$ -lock-freedom impose the same space complexity, it also appears natural to expect that this is also true for all $k=1,\\ldots ,n$ .",
"An interesting corollary to our results is that to implement a 2-obstruction-free SWMR memory we need strictly more space than to implement a 1-lock-free one.",
"But the two properties are, in general, incomparable: a 2-solo run in which only one process makes progress satisfies 1-lock-freedom, but not 2-obstruction-freedom, and a run in which 3 or more processes are correct but no progress is made satisfies 2-obstruction-freedom, but not 1-lock-freedom.",
"The relative costs of incomparable progress properties, e.g., in the $(\\ell ,k)$ -freedom spectrum [5], are yet to be understood.",
"An SWMR memory can be viewed as a stable-set abstraction with a conventional put/get interface: every participating process can put values to the set and get the set's content, and every get operation returns the values previously put.",
"For the stable-set abstraction, we can extend our results to the anonymous setting, where processes are not provided with unique identifiers.",
"Indeed, we claim that the same algorithm may apply to the stable-set abstraction for anonymous systems when the number of participating processes $n$ is known.",
"But the question of whether an adaptive solution exists (expressed differently, a solution that does not assume any upper bound on the number of participating processes) for anonymous systems remains open."
]
] | 1709.01879 |
[
[
"The Magnetic Field Distribution in Strongly Magnetized Neutron Stars"
],
[
"Abstract In this work, we expand on a previously reported realistic calculation of the magnetic field profile for the equation of state inside strongly magnetized neutron stars.",
"In addition to showing that magnetic fields increase quadratically with increasing baryon chemical potential of magnetized matter (instead of exponentially, as previously assumed), we show here that the magnetic field increase with baryon number density is more complex and harder to model.",
"We do so by the analysis of several different realistic models for the microscopic description of matter in the star (including hadronic, hybrid and quark models) combined with general relativistic solutions by solving Einstein-Maxwell's field equations in a self-consistent way for stars endowed with a poloidal magnetic field."
],
[
"Introduction",
"In order to study the effects of magnetic fields in the equation of state of neutron stars, a profile for the strength of the field as a function of chemical potential or density must be provided.",
"If one has access to a code that calculates general relativistic solutions in the presence of a magnetic field by solving Einstein-Maxwell's field equations in a self-consistent way, this can be easily achieved.",
"Alternatively, ad hoc profiles for the magnetic field have been provided and used by the nuclear physics community for the past two decades.",
"The first of these ad hoc profiles was suggested in Ref.",
"[1] $B^*(n_B/n_0)=B_{\\rm {surf}}+B_0\\left[1-e^{-\\beta (n_B/n_0)^\\gamma }\\right],$ with typical choices of constants $\\beta =0.01$ and $\\gamma =3$ .",
"In this case, the magnetic field increases exponentially from a value $B_{\\rm {surf}}$ at zero density to a value $B_{\\rm {surf}}+B_0$ at asymptotically high densities.",
"This profile was subsequently used in approximately one hundred publications, among which the most cited ones are [14], [15], [2], [6].",
"An improvement over this formulation assumes a field profile as a function of baryon chemical potential $\\mu _B$ ([6]).",
"Although the second ansatz does not suffer from spurious jumps in the strength of the magnetic field in the presence of first order phase transitions (such as the deconfinement to quark matter), it is still not correct.",
"As already pointed out in Ref.",
"[13], ad hoc formulas for magnetic field profiles in neutron stars such as Eq.",
"REF do not fulfill Maxwell's equations and, therefore, are incorrect.",
"In this work, we calculate the magnetic field distribution in the polar stellar direction and translate it to be a function of microscopic thermodynamical quantities, the baryon chemical potential and for the first time the baryon number density.",
"In order to do so, the macroscopic structure of the star must be obtained from the solution of Einstein-Maxwell equations.",
"Only in this way, can we ensure that the magnetic field profile in a star respects the Einstein-Maxwell field equations.",
"In order to make our analysis as general as possible, we make use of three equations of state for the microscopic description of neutron stars with different matter composition: hadronic, hybrid and quark.",
"They represent state-of-the-art approaches that fulfill current nuclear and astrophysical constraints, such as the prediction of massive stars."
],
[
"Formalism",
"The first model was obtained from Refs.",
"[8], [9] and it will be referred to as “G-model”.",
"It is a hadronic model that simulates many-body forces among nucleons by non-linear self-couplings that come from a field dependence of the interactions.",
"The second model was obtained from Refs.",
"[7], [6] and it will be referred to as “D-model\".",
"It includes nucleons, hyperons and quarks in a self-consistent approach and reproduces chiral symmetry restoration and deconfinement at high densities.",
"The third model was obtained from Ref.",
"[11] and it will be referred to as “H-model\".",
"It is a version of the three-flavor NJL model that includes a repulsive vector-isoscalar interaction, which is crucial for the description of astrophysical data.",
"For the general-relativistic formalism to describe the macroscopic features of magnetic neutron stars, we use the LORENE C++ class library for numerical relativity ([4], [3]), which determines equilibrium configurations by solving the Einstein-Maxwell's field equations in spherical polar coordinates assuming a poloidal magnetic field configuration.",
"In this approach, the field is produced self-consistently by a macroscopic current, which is a function of the stellar radius, polar angle theta, and dipole magnetic moment $\\mu $ for each equation of state.",
"The dipole magnetic moments shown in this work were chosen to reproduce a distribution with a central stellar magnetic field close to the upper limit of the code $\\sim 10^{18}$ G and one to reproduce a surface magnetic field of $\\sim 10^{15}$ G, the maximum value observed on the surface of a star ([12]).",
"Figure: (Color online) Magnetic field profile in the polar direction in a M B =2.2M_B=2.2 M ⊙ _\\odot star as a function of baryon chemical potential obtained for the three equation of state models R, D and H. Each of these profiles are shown for dipole magnetic moments μ=3×10 32 \\mu =3\\times 10^{32} Am 2 ^2 (curves on the top) and μ=1×10 30 \\mu =1\\times 10^{30} Am 2 ^2 (curves on the bottom).We calculate the equation of state within the microscopic models without magnetic field effects, as we have already shown in Refs.",
"[5], [10] that they do not affect significantly the stellar magnetic field distribution.",
"Then, in a second step, through the solution of Einstein's equations coupled with Maxwell's equations, we determine the magnetic field profile in an individual star, and then translate that to a field profile for the microscopic equation of state of each model.",
"Later, we discuss a generalization to one profile by averaging the results from the different models."
],
[
"Results",
"Figure REF shows the magnetic field distribution in the stellar polar direction for a $M_B=2.2$ M$_\\odot $ star translated into the microscopic quantity baryon chemical potential.",
"See Ref.",
"[5] for figures showing the magnetic field as a function of stellar radius.",
"The top curves of the figure are magnetic field profiles in the stellar polar direction for a higher dipole magnetic moment, while the bottom curves are profiles for a lower value of the dipole magnetic moment.",
"The main conclusion from this figure is that different equation of state models show different magnetic field strengths, but the respective profiles have approximately the same shape (when taking into account the logarithmic scale).",
"The shape of the profiles obtained from the solution of Einstein-Maxwell's equations is well fit by a quadratic polynomial (and not exponential function, as suggested by ad hoc profiles).",
"This allows us to fit one profile using the average of the different equation of state models.",
"It depends only on the baryon chemical potential $\\mu _B$ and on the value chosen for the dipole magnetic moment $\\mu $ $B^*(\\mu _B)=\\frac{(a + b \\mu _B + c \\mu _B^2)}{B_c^2} \\ \\mu ,$ with $\\mu _B$ given in MeV and $\\mu $ in Am$^2$ in order to produce $B^*$ in units of the critical field for the electron $B_c=4.414\\times 10^{13}$ G. The coefficients $a$ , $b$ , and $c$ given in Table REF .",
"Figure: (Color online) Same as Fig.",
"but for a M B =1.6M_B=1.6 M ⊙ _\\odot star with dipole magnetic moments μ=2×10 32 \\mu =2\\times 10^{32} Am 2 ^2 (curves on the top) and μ=1×10 30 \\mu =1\\times 10^{30} Am 2 ^2 (curves on the bottom).",
"Some of the curves overlap.In Fig.",
"REF , we repeat the calculations for a $M_B=1.6$ M$_\\odot $ star again for different dipole magnetic moments.",
"Once more, each magnetic field profile has the same shape (when taking into account the logarithmic scale).",
"In this case, the parameters of the profile fit in Eq.",
"(REF ) are again given in Table REF , where from the values of the parameter “c\" it can be seen that the profiles for a larger star give on average a slightly more linear fit.",
"Note that for a less massive and less compact stars, all equations of state that contain baryons reproduce very similar results.",
"This stems from the fact that they were fitted to reproduce nuclear physics constraints and the central densities in such stars do not reach values much larger than nuclear saturation density.",
"For a detailed comparison between the results from Figs.",
"REF and REF and ad hoc exponential profiles, see Ref. [5].",
"Clearly, none of the ad hoc exponential profiles coincide with our results (except maybe for one point), even the ad hoc profiles that were chosen to match our field strengths on the surface of the star and at asymptotically high chemical potentials.",
"Table: NO_CAPTIONNext, we focus on the discussion of magnetic field distributions as a function of baryon number density.",
"This is shown is Figs.",
"REF and REF for stars with different baryon masses (and in each figure for different dipole magnetic moments).",
"It can immediately be seen that all the curves have different shapes.",
"In Fig.",
"REF , it can be clearly seen that the curves for the “G\" EoS model looks quadratic, while the others are better fit by a quartic polynomial (with completely different coefficients for the “D\" and “H\" EoS models).",
"More specific, for the “D\" hybrid model, the change in slope in the curves exemplifies the change in degrees of freedom going from a pure hadronic phase to a phase containing a mixture of hadrons and quarks with increasing quark content.",
"For the “H\" quark model, the number density drops more sharply close to the star surface.",
"Figure: (Color online) Magnetic field profile in the polar direction in a M B =2.2M_B=2.2 M ⊙ _\\odot star as a function of baryon number density obtained for the three equation of state models R, D and H. Each of these profiles are shown for dipole magnetic moments μ=3×10 32 \\mu =3\\times 10^{32} Am 2 ^2 (curves on the top) and μ=1×10 30 \\mu =1\\times 10^{30} Am 2 ^2 (curves on the bottom).In any case, we do not provide a numerical fit for the magnetic field distribution in the polar direction as a function of baryon number density, as it is evident that this would be model dependent (as far as different degrees of freedom are taken into account) and the fit would have completely different coefficients for each model EoS.",
"Note that in the case of a first order phase transition without the inclusion of a mixed phase (not shown in this work), the difference among models would be even more extreme.",
"As already discussed in Ref.",
"[5], we do not provide profiles for the magnetic field strength in the stellar equatorial direction, as those are more complicated and current dependent.",
"Figure: (Color online) Same as Fig.",
"but for a M B =1.6M_B=1.6 M ⊙ _\\odot star with dipole magnetic moments μ=2×10 32 \\mu =2\\times 10^{32} Am 2 ^2 (curves on the top) and μ=1×10 30 \\mu =1\\times 10^{30} Am 2 ^2 (curves on the bottom)."
],
[
"Conclusions",
"In this work, we provide a numerical fit that allows one to include a magnetic field profile respective to the stellar polar direction in any equation of state in a simple way.",
"This will allow analyses of magnetic field effects in specific models studying, for example, changes in stiffness, changes in population, phase transitions, temperature, transport properties, etc.",
"A further inclusion of the obtained equations of state in a symmetric static isotropic solution for Einstein's equations (TOV [17], [16]) to obtain macroscopic star properties is not a realistic approach when dealing with strong magnetic fields ([10]).",
"This is because the magnetic field distribution is different and more complicated in other directions of the star and the pure magnetic field contribution would have to be added in an isotropic manner, being either positive or negative.",
"In reality, this contribution has different signs in different directions and, therefore, requires a more advanced formalism (such as the one used in this work) which solves Einstein-Maxwell's field equations self-consistently.",
"This numerical fit for the magnetic field strength was given as a function of baryon chemical potential and is, to a large extent, model independent.",
"This would not be the case for a numerical fit as a function of the baryon number density, in which case the shape of the resulting curves depends substantially on the model.",
"In this work, we have used three very different state-of-the-art equation of state models, built with different assumptions and including different degrees of freedom.",
"They were combined with the solutions of the Einstein-Maxwell's equations in a self-consistent way, in order to provide a formula to calculate how the magnetic field varies with baryon chemical potential, depending only on the dipole magnetic moment of choice and the stellar baryonic mass.",
"The resulting fit is quadratic in form and not exponential as previously assumed.",
"This result is particularly important because it shows that a star with a surface magnetic field of $10^{15}$ G cannot reach a central one of $10^{18}$ G, as previously assumed.",
"Our fit is presented for the two most relevant types of neutron stars, with gravitational masses around 2 and $1.4$ M$_\\odot $ .",
"The authors acknowledge support from NewCompStar COST Action MP1304 and from the LOEWE program HIC for FAIR.",
"Work partially financed by CNPq under grants 308828/2013-5 (R.L.S.F) and 307458/2013-0 (S.S.A)."
]
] | 1709.01914 |
[
[
"Generic conformally flat hypersurfaces in $\\mathbb{R}^4$"
],
[
"Abstract In this paper, we study generic conformally flat hypersurfaces in the Euclidean $4$-space $\\mathbb{R}^4$ using the framework of M\\\"{o}bius geometry.",
"First, we classify locally the generic conformally flat hypersurfaces with closed M\\\"obius form under the M\\\"obius transformation group of $\\mathbb{R}^4$.",
"Such examples come from cones, cylinders, or rotational hypersurfaces over the surfaces with constant Gaussian curvature in $3$-spheres, Euclidean $3$-spaces, or hyperbolic $3$-spaces, respectively.",
"Second, we investigate the global behavior of the generic conformally flat hypersurface and give some integral formulas about these hypersurfaces."
],
[
"Introduction",
"A Riemannian manifold $(M^n, g)$ is conformally flat, if every point has a neighborhood which is conformal to an open set in the Euclidean space $\\mathbb {R}^n$ .",
"A hypersurface of the Euclidean space $\\mathbb {R}^{n+1}$ is said to be conformally flat if so it is with respect to the induced metric.",
"The dimension of the hypersurface seems to play an important role in the study of conformally flat hypersurfaces.",
"For $n\\ge 4$ , the immersed hypersurface $f: M^n \\rightarrow \\mathbb {R}^{n+1}$ is conformally flat if and only if at least $n-1$ of the principal curvatures coincide at each point by the result of Cartan-Schouten ([1],[10]).",
"Cartan-Schouten's result is no longer true for three dimensional hypersurfaces.",
"Lancaster ([6]) gave some examples of conformally flat hypersurfaces in $\\mathbb {R}^4$ having three different principal curvatures.",
"For $n=2$ , the existence of isothermal coordinates means that any Riemannian surface is conformally flat.",
"A conformally flat hypersurface $f:M^3\\rightarrow \\mathbb {R}^4$ in $\\mathbb {R}^4$ is said to be generic, if the second fundamental form has three distinct eigenvalues everywhere on $M^3$ .",
"Standard example of generic conformally flat hypersurface comes from cone, cylinder, or rotational hypersurface over a surface with constant Gaussian curvature in 3-sphere $\\mathbb {S}^3$ , Euclidean 3-space $\\mathbb {R}^3$ , or hyperbolic 3-space $\\mathbb {H}^3$ , respectively.",
"The (local) classification of these hypersurfaces is far from complete.",
"However, several partial classification results of generic conformally flat hypersurfaces were given in [2], [3], [4],[5], [7],[11],[12] and [13].",
"It is known that the conformal transformation group of $\\mathbb {R}^n$ is isomorphic to its Möbius transformation group if $n\\ge 3$ .",
"As conformal invariant objects, generic conformally flat hypersurfaces are investigated in this paper using the framework of Möbius geometry.",
"If an immersed hypersurface in $\\mathbb {R}^{n+1}$ has not any umbilical point, then we can define the so-called Möbius metric on the hypersurface, which is invariant under Möbius transformations [14].",
"Together with another quadratic form (called the Möbius second fundamental form) they form a complete system of invariants for hypersurfaces $(dim\\ge 3)$ in Möbius geometry [14].",
"Other important Möbius invariants of the hypersurface are the Möbius form and the Blaschke tensor.",
"First, we find that the standard examples of generic conformally flat hypersurface has closed Möbius form, and vice versa.",
"Theorem 1.1 Let $f:M^3\\rightarrow \\mathbb {R}^{4}$ be a generic conformally flat hypersurface.",
"The Möbius form is closed if and only if the hypersurface $f$ is locally Möbius equivalent to one of the following hypersurfaces in $\\mathbb {R}^{4}$ : $(1)$ a cylinder over a surface in $\\mathbb {R}^3$ with constant Gaussian curvature, $(2)$ a cone over a surface in $\\mathbb {S}^3$ with constant Gaussian curvature, $(3)$ a rotational hypersurface over a surface in $\\mathbb {H}^3$ with constant Gaussian curvature.",
"Second, we investigate the global behavior of compact generic conformally flat hypersurfaces by the Möbius invariants.",
"Let $(M^n,g)$ be a Riemannian manifold.",
"$K(P)$ denotes the sectional curvature of sectional plane $P (\\in \\wedge ^2TM^n)$ .",
"We call the sectional curvature $K(P)$ have sign if $K(P)\\ge 0$ for all $P\\in \\wedge ^2TM^n$ , or $K(P)\\le 0$ for all $P\\in \\wedge ^2TM^n$ .",
"Theorem 1.2 Let $f:M^3\\rightarrow \\mathbb {R}^{4}$ be a generic conformally flat hypersurface.",
"If the hypersurface $M^3$ is compact, then the sectional curvature of the Möbius metric can not have sign.",
"Theorem 1.3 Let $f:M^3\\rightarrow \\mathbb {R}^{4}$ be a generic conformally flat hypersurface.",
"If the hypersurface $M^3$ is compact, then $\\int _{M^3}\\lbrace |\\tilde{A}|^2+\\frac{1}{3}R^2-|Ric|^2-\\frac{2}{27}\\rbrace dv_g=0,$ where $\\tilde{A}:=A-\\frac{1}{3}tr(A)g$ denotes the trace-free Blaschke tensor, $|Ric|$ denotes the norm of the Ricci curvature of $g$ , and $R$ denotes the scalar curvature of $g$ .",
"Corollary 1.1 Let $f:M^3\\rightarrow \\mathbb {R}^{4}$ be a generic conformally flat hypersurface.",
"If the hypersurface $M^3$ is compact, then $\\int _{M^3}\\lbrace |\\tilde{A}|^2-\\frac{2}{27}\\rbrace dv_g>0,$ where $\\tilde{A}:=A-\\frac{1}{3}tr(A)g$ denotes the trace-free Blaschke tensor.",
"The paper is organized as follows.",
"In section 2, we review the elementary facts about Möbius geometry of hypersurfaces in $\\mathbb {R}^{n+1}$ .",
"In section 3, we investigate local behavior of generic conformally flat hypersurfaces in $\\mathbb {R}^4$ and prove Theorem REF .",
"In section 4, we investigate global behavior of generic conformally flat hypersurfaces in $\\mathbb {R}^4$ and prove Theorem REF and Theorem REF ."
],
[
"Möbius invariants of hypersurfaces in $\\mathbb {R}^{n+1}$",
"In [14], Wang has defined Möbius invariants of submanifolds in $\\mathbb {S}^{n+1}$ and given a congruent theorem of hypersurfaces in $\\mathbb {S}^{n+1}$ .",
"In this section, we define Möbius invariants and give a congruent theorem of hypersurfaces in $\\mathbb {R}^{n+1}$ in the same way in [14].",
"For details we refer to $\\cite {liu},\\cite {w}$ .",
"Let $\\mathbb {R}^{n+3}_1$ be the Lorentz space, i.e., $\\mathbb {R}^{n+3}$ with inner product $<\\cdot ,\\cdot >$ defined by $<x,y>=-x_0y_0+x_1y_1+\\cdots +x_{n+2}y_{n+2},$ for $x=(x_0,x_1,\\cdots ,x_{n+2}), y=(y_0,y_1,\\cdots ,y_{n+2})\\in \\mathbb {R}^{n+3}$ .",
"Let $f:M^{n}\\rightarrow \\mathbb {R}^{n+1}$ be a hypersurface without umbilical points and assume that $\\lbrace e_i\\rbrace $ is an orthonormal basis with respect to the induced metric $I=df\\cdot df$ with $\\lbrace \\theta _i\\rbrace $ the dual basis.",
"Let $II=\\sum _{ij}h_{ij}\\theta _i\\theta _j$ and $H=\\sum _i\\frac{h_{ii}}{n}$ be the second fundamental form and the mean curvature of $f$ , respectively.",
"We define the Möbius position vector $Y: M^n\\rightarrow \\mathbb {R}^{n+3}_1$ of $f$ by $Y=\\rho \\left(\\frac{1+|f|^2}{2},\\frac{1-|f|^2}{2},f\\right)~,~~\\rho ^2=\\frac{n}{n-1}(|II|^2-nH^2).$ Theorem 2.1 [14] Two hypersurfaces $f,\\bar{f}: M^n\\rightarrow \\mathbb {R}^{n+1}$ are Möbius equivalent if and only if there exists $T$ in the Lorentz group $O(n+2,1)$ such that $\\bar{Y}=YT.$ It follows immediately from Theorem 2.1 that $g=<dY,dY>=\\rho ^2df\\cdot df$ is a Möbius invariant, called the Möbius metric of $f$ .",
"Let $\\Delta $ be the Laplacian with respect to $g$ .",
"Define $N=-\\frac{1}{n}\\Delta Y-\\frac{1}{2n^2}<\\Delta Y,\\Delta Y>Y,$ which satisfies $<Y,Y>=0=<N,N>, ~~<N,Y>=1.$ Let $\\lbrace E_1,\\cdots ,E_n\\rbrace $ be a local orthonormal basis for $(M^n,g)$ with dual basis $\\lbrace \\omega _1,\\cdots ,\\omega _n\\rbrace $ .",
"Write $Y_i=E_i(Y)$ .",
"Then we have $<Y_i,Y>=<Y_i,N>=0, ~<Y_i,Y_j>=\\delta _{ij}, ~~1\\le i,j\\le n.$ Let $\\xi $ be the mean curvature sphere of $f$ written as $\\xi =\\left(\\frac{1+|f|^2}{2}H+f\\cdot e_{n+1},\\frac{1-|f|^2}{2}H-f\\cdot e_{n+1},Hf+e_{n+1}\\right)~,$ where $e_{n+1}$ is the unit normal vector field of $f$ in $\\mathbb {R}^{n+1}$ .",
"Thus $\\lbrace Y,N,Y_1,\\cdots ,Y_n,\\xi \\rbrace $ forms a moving frame in $\\mathbb {R}^{n+3}_1$ along $M^n$ .",
"We will use the following range of indices in this section: $1\\le i,j,k\\le n$ .",
"We can write the structure equations as following: $&&dY=\\sum _iY_i\\omega _i,\\\\&&dN=\\sum _{ij}A_{ij}\\omega _iY_j+\\sum _iC_i\\omega _i\\xi ,\\\\&&dY_i=-\\sum _jA_{ij}\\omega _jY-\\omega _iN+\\sum _j\\omega _{ij}Y_j+\\sum _jB_{ij}\\omega _j\\xi ,\\\\&&d\\xi =-\\sum _iC_i\\omega _iY-\\sum _{ij}\\omega _iB_{ij}Y_j,$ where $\\omega _{ij}$ is the connection form of the Möbius metric $g$ and $\\omega _{ij}+\\omega _{ji}=0$ .",
"The tensors ${\\bf A}=\\sum _{ij}A_{ij}\\omega _i\\otimes \\omega _j,~~{\\bf B}=\\sum _{ij}B_{ij}\\omega _i\\otimes \\omega _j,~~{\\bf C}=\\sum _iC_i\\omega _i$ are called the Blaschke tensor, the Möbius second fundamental form and the Möbius form of $f$ , respectively.",
"The covariant derivative of $C_i, A_{ij}, B_{ij}$ are defined by $&&\\sum _jC_{i,j}\\omega _j=dC_i+\\sum _jC_j\\omega _{ji},\\\\&&\\sum _kA_{ij,k}\\omega _k=dA_{ij}+\\sum _kA_{ik}\\omega _{kj}+\\sum _kA_{kj}\\omega _{ki},\\\\&&\\sum _kB_{ij,k}\\omega _k=dB_{ij}+\\sum _kB_{ik}\\omega _{kj}+\\sum _kB_{kj}\\omega _{ki}.$ The integrability conditions for the structure equations are given by $&&A_{ij,k}-A_{ik,j}=B_{ik}C_j-B_{ij}C_k,\\\\&&C_{i,j}-C_{j,i}=\\sum _k(B_{ik}A_{kj}-B_{jk}A_{ki}),\\\\&&B_{ij,k}-B_{ik,j}=\\delta _{ij}C_k-\\delta _{ik}C_j,\\\\&&R_{ijkl}=B_{ik}B_{jl}-B_{il}B_{jk}+\\delta _{ik}A_{jl}+\\delta _{jl}A_{ik}-\\delta _{il}A_{jk}-\\delta _{jk}A_{il},\\\\&&R_{ij}:=\\sum _kR_{ikjk}=-\\sum _kB_{ik}B_{kj}+(tr{\\bf A})\\delta _{ij}+(n-2)A_{ij},\\\\&&\\sum _iB_{ii}=0, ~~\\sum _{ij}(B_{ij})^2=\\frac{n-1}{n}, ~~tr{\\bf A}=\\sum _iA_{ii}=\\frac{1}{2n}(1+\\frac{n}{n-1}R),$ Here $R_{ijkl}$ denotes the curvature tensor of $g$ , and $R=\\sum _{ij}R_{ijij}$ is the Möbius scalar curvature.",
"We know that all coefficients in the structure equations are determined by $\\lbrace g, {\\bf B}\\rbrace $ when $n\\ge 3$ .",
"Thus we have Theorem 2.2 $\\cite {w}$ Two hypersurfaces $f: M^n\\rightarrow \\mathbb {R}^{n+1}$ and $\\bar{f}:M^n\\rightarrow \\mathbb {R}^{n+1} (n\\ge 3)$ are Möbius equivalent if and only if there exists a diffeomorphism $\\varphi : M^n\\rightarrow M^n$ which preserves the Möbius metric and the Möbius second fundamental form.",
"By equation (), we have $dC=0\\Leftrightarrow \\sum _k(B_{ik}A_{kj}-B_{jk}A_{ki})=0.$ For the second covariant derivative of $B_{ij}$ defined by $dB_{ij,k}+\\sum _mB_{mj,k}\\omega _{mi}+\\sum _mB_{im,k}\\omega _{mj}+\\sum _mB_{ij,m}\\omega _{mk}=\\sum _mB_{ij,km}\\omega _m,$ we have the following Ricci identities $B_{ij,kl}-B_{ij,lk}=\\sum _mB_{mj}R_{mikl}+\\sum _mB_{im}R_{mjkl}.$ We call eigenvalues of $(B_{ij})$ as Möbius principal curvatures of $f$ .",
"Clearly the number of distinct Möbius principal curvatures is the same as that of its distinct Euclidean principal curvatures.",
"Let $k_1,\\cdots ,k_n$ be the principal curvatures of $f$ , and $\\lbrace \\lambda _1,\\cdots ,\\lambda _n\\rbrace $ the corresponding Möbius principal curvatures, then the curvature sphere of principal curvature $k_i$ is $\\xi _i=\\lambda _iY+\\xi =\\left(\\frac{1+|f|^2}{2}k_i+f\\cdot e_{n+1},\\frac{1-|f|^2}{2}k_i-f\\cdot e_{n+1},k_if+e_{n+1}\\right)~.$ Note that $k_i=0$ if and only if, $<\\xi _i,(1,-1,0,\\cdots ,0)>=0.$ This means that the curvature sphere of principal curvature $k_i$ is a hyperplane in $\\mathbb {R}^{n+1}$ ."
],
[
"Generic conformally flat hypersurfaces in $\\mathbb {R}^4$",
"In this section, we give some local properties of the Möbius invariants of generic conformally flat hypersurfaces in $\\mathbb {R}^4$ .",
"Let $(M^n,g)$ be an n-dimensional Riemannian manifold, and $\\lbrace e_1,\\cdots e_n\\rbrace $ be a local orthonormal frame field on $(M^n,g)$ , and $\\lbrace \\omega _1,\\cdots ,\\omega _n\\rbrace $ its dual coframe field.",
"The Weyl conformal tensor $W=\\sum _{ijkl}W_{ijkl}\\omega _i\\otimes \\omega _j\\otimes \\omega _k\\otimes \\omega _l$ and the Schouten tensor $S=\\sum _{ij}S_{ij}\\omega _i\\otimes \\omega _j$ of $(M^n,g)$ are defined by, respectively, $\\begin{split}&W_{ijkl}=R_{ijkl}-\\frac{1}{n-2}\\lbrace R_{ik}\\delta _{jl}-R_{jk}\\delta _{il}+\\delta _{ik}R_{jl}-\\delta _{jk}R_{il}-\\frac{R}{(n-1)}(\\delta _{ik} \\delta _{jl} -\\delta _{jk} \\delta _{il})\\rbrace ,\\\\&S_{ij}=R_{ij}-\\frac{R}{2(n-1)}\\delta _{ij},\\end{split}$ where $R_{ij}$ denotes the Ricci curvature and $R$ the scalar curvature of $(M^n,g)$ .",
"A result of Weyl states that a Riemannian manifold $(M^n,g)$ of dimension $n (\\ge 4)$ is conformally flat if and only if the Weyl conformal tensor vanishes, and a Riemannian manifold $(M^n,g)$ of dimension 3 is conformally flat if and only if the Schouten tensor is a Codazzi tensor (i.e., $S_{ij,k}=S_{ik,j}$ ).",
"Using the Weyl's result, we can prove the following lemma (or see [15]), Lemma 3.1 [15] A Riemannian product $(M_1,g_1)\\times (M_2,g_2)=(M_1\\times M_2, g_1+g_2)$ is conformally flat if and only if either (1) $(M_i,g_i)$ is one dimensional curve, and $(M_j,g_j), (i\\ne j)$ is a space form, or (2) $(M_1,g_1)$ and $(M_2,g_2)$ are space forms of dimension at least two, with non-zero opposite curvatures.",
"For hypersurfaces in $\\mathbb {R}^{n+1}$ , when $n\\ge 4$ , it is well-known from the Cartan-Schouten that a hypersurface $f: M^n\\rightarrow \\mathbb {R}^{n+1}$ is conformally flat if and only if at least $n-1$ of the principal curvatures coincide at each point.",
"But Cartan-Schouten's result is no longer true in dimension 3, since there exist generic conformally flat hypersurfaces.",
"Let $f: M^3\\rightarrow \\mathbb {R}^4$ be a generic hypersurface.",
"We choose an orthonormal basis $\\lbrace E_{1},E_2,E_{3}\\rbrace $ with respect to the Möbius metric $g$ such that $(B_{ij})=diag\\lbrace b_1, b_2, b_3\\rbrace ,~~~b_1< b_2< b_3.$ Let $\\lbrace \\omega _1,\\omega _2,\\omega _3\\rbrace $ be the dual of $\\lbrace E_{1},E_2,E_{3}\\rbrace $ .",
"The conformal fundamental forms of $f$ are defined by $\\Theta _1=\\sqrt{(b_3-b_1)(b_2-b_1)}\\omega _1,~\\Theta _2=\\sqrt{(b_3-b_2)(b_2-b_1)}\\omega _2,~\\Theta _3=\\sqrt{(b_3-b_1)(b_3-b_2)}\\omega _3.$ Using the equation () and (), the Schouten tensor of $f$ is $S=\\sum _{ij}(-\\sum _lB_{il}B_{lj}+A_{ij}+\\frac{1}{6}\\delta _{ij})\\omega _i\\wedge \\omega _j.$ Thus $S_{ij,k}=-\\sum _l(B_{il,k}B_{lj}+B_{il}B_{lj,k})+A_{ij,k}.$ If the hypersurface $f$ is conformally flat, then $S_{ij,k}=S_{ik,j}$ .",
"Combining the equations (REF ) and (REF ), we obtain the following equation $b_kB_{ik,j}-b_jB_{ij,k}=2(B_{ij}C_k-B_{ik}C_j).$ Using the equation (), we have the following equations, $\\begin{split}&B_{12,3}=B_{13,2}=0,\\\\&B_{ij,i}=\\frac{3b_i}{b_j-b_i}C_j,~~B_{ii,j}=\\frac{b_i-b_k}{b_j-b_i}C_j, ~~i\\ne j, j\\ne k, i\\ne k.\\end{split}$ Using $dB_{ij}+\\sum _kB_{kj}\\omega _{ki}+\\sum _kB_{ik}\\omega _{kj}=\\sum _kB_{ij,k}\\omega _k$ and (REF ), we get $\\omega _{ij}=\\sum _k\\frac{B_{ij,k}}{b_i-b_j}\\omega _k=\\frac{B_{ij,i}}{b_i-b_j}\\omega _i+\\frac{B_{ij,j}}{b_i-b_j}\\omega _j.$ The following lemma is trivial by the equation (REF ) and (REF ), (or see [3],[13]).",
"Lemma 3.2 Let $M^3\\rightarrow \\mathbb {R}^4$ be a generic hypersurface.",
"The following are equivalent: (1), the hypersurface is conformally flat; (2), the schouten tensor is a Codazzi tensor; (3), the conformal fundamental forms $\\Theta _1,\\Theta _2,\\Theta _3$ are closed.",
"Next, we give the standard examples of generic conformally flat hypersurfaces in $\\mathbb {R}^4$ .",
"Example 3.1 Let $u: M^2\\longrightarrow \\mathbb {R}^3$ be an immersed surface.",
"We define the cylinder over $u$ in $\\mathbb {R}^4$ as $f=(id,u):\\mathbb {R}^1\\times M^2\\longrightarrow \\mathbb {R}^1 \\times \\mathbb {R}^3=\\mathbb {R}^4,~~~~f(t,y)=(t,u(x)),$ where $id:\\mathbb {R}^1\\longrightarrow \\mathbb {R}^1$ is the identity map.",
"The first fundamental form $I$ and the second fundamental form $II$ of the cylinder $f$ are, respectively, $I=I_{\\mathbb {R}^1}+I_u, \\;\\; II=II_u,$ where $I_u,II_u$ are the first and second fundamental forms of $u$ , respectively, and $I_{\\mathbb {R}^1}$ denotes the standard metric of $\\mathbb {R}^1$ .",
"Let $\\lbrace k_1,k_2\\rbrace $ be principal curvatures of surface $u$ .",
"Obviously the principal curvatures of hypersurface $f$ are $\\lbrace 0,k_1,k_2\\rbrace .$ The Möbius metric $g$ of hypersurface $f$ is $g=\\rho ^2I=\\frac{n}{n-1}(|II|^2-nH^2)I=\\left(4H_u^2-3K_u\\right)(I_{\\mathbb {R}^1}+I_u),$ where $H_u,K_u$ are the mean curvature of $u$ and Gaussian curvature of $u$ , respectively.",
"Therefore combining Lemma REF we have the following result.",
"Proposition 3.1 Let $f:M^3\\rightarrow \\mathbb {R}^4$ be a cylinder over a surface $u: M^2\\rightarrow \\mathbb {R}^3$ , then the cylinder $f$ is conformally flat if and only if the surface $u$ is of constant Gaussian curvature.",
"Example 3.2 Let $u:M^2\\longrightarrow \\mathbb {S}^3\\subset \\mathbb {R}^4$ be an immersed surface.",
"We define the cone over $u$ in $\\mathbb {R}^4$ as $f:\\mathbb {R}^+\\times M^2\\longrightarrow \\mathbb {R}^4,~~~~f(t,x)=tu(x).$ The first and second fundamental forms of the cone $f$ are, respectively, $I=I_{\\mathbb {R}^1}+t^2I_u, \\;\\; II=t~II_u,$ where $I_u,II_u,I_{\\mathbb {R}^1}$ are understood as before.",
"Let $\\lbrace k_1,k_2\\rbrace $ be principal curvatures of surface $u$ .",
"The principal curvatures of hypersurface $f$ are $\\lbrace 0,\\frac{1}{t}k_1,\\frac{1}{t}k_2\\rbrace .$ The Möbius metric $g$ of hypersurface $f$ is $g=\\frac{1}{t^2}\\left[4H_u^2-3(K_u-1)\\right](I_{\\mathbb {R}^1}+t^2I_u)=\\left[4H_u^2-3(K_u-1)\\right](I_{\\mathbb {H}^1}+I_u),$ where $H_u,K_u$ are the mean curvature and Gaussian curvature of $u$ , respectively, $I_{\\mathbb {H}^1}=\\frac{dt^2}{t^2}$ .",
"Therefore combining Lemma REF we have the following result.",
"Proposition 3.2 Let $f:M^3\\rightarrow \\mathbb {R}^4$ be a cone over a surface $u: M^2\\rightarrow \\mathbb {S}^3$ , then the cone $f$ is conformally flat if and only if the surface $u$ is of constant Gaussian curvature.",
"Example 3.3 Let $\\mathbb {R}^3_+=\\lbrace (x_1,x_2,x_3)\\in \\mathbb {R}^3|x_3>0\\rbrace $ be the upper half-space endowed with the standard hyperbolic metric $ds^2=\\frac{1}{x_3^2}[dx_1^2+dx_2^2+dx_3^2]~.$ Let $u=(x_1,x_2,x_3):M^2\\longrightarrow \\mathbb {R}^3_+$ be an immersed surface.",
"We define rotational hypersurface over $u$ in $\\mathbb {R}^4$ as $f:\\mathbb {S}^1\\times M^2\\longrightarrow \\mathbb {R}^4,~~~f(\\phi ,x_1,x_2,x_3)=(x_1,x_2,x_3\\phi ),$ where $\\phi :\\mathbb {S}^1\\longrightarrow \\mathbb {S}^1$ is the unit circle.",
"The first fundamental form and the second fundamental form of $u$ is, respectively, $\\begin{split}&I_u=\\frac{1}{x_3^2}(dx_1\\cdot dx_1+dx_2\\cdot dx_2+dx_3\\cdot dx_3),\\\\&II_u=\\frac{1}{x_3^2}(dx_1\\cdot d\\eta _1+dx_2\\cdot d\\eta _2+dx_3\\cdot d\\eta _3)-\\frac{\\eta _3}{x_3}I_u.\\end{split}$ The first and the second fundamental forms of $f$ is, respectively, $I=dx\\cdot dx=x_3^2(I_{\\mathbb {S}^1}+I_u),~~II=x_3II_u-\\eta _3I_u-\\eta _3I_{\\mathbb {S}^1}.$ Let $\\lbrace k_1,k_2\\rbrace $ be principal curvatures of $u$ .",
"Then principal curvatures of hypersurface $f$ are $\\lbrace \\frac{-\\eta _3}{x_3^2},\\frac{k_1}{x_3}-\\frac{\\eta _3}{x_3^2}~,~\\frac{k_2}{x_3}-\\frac{\\eta _3}{x_3^2}\\rbrace .$ Thus the Möbius metric of the rotational hypersurface $f$ is $g=\\rho ^2I=\\left[4H_u^2-3(K_u+1)\\right](I_{\\mathbb {S}^1}+I_u),$ where $H_u,K_u$ are the mean curvature and Gaussian curvature of $u$ , respectively.",
"Therefore combining Lemma REF we have the following result, Proposition 3.3 Let $f:M^3\\rightarrow \\mathbb {R}^4$ be a rotational hypersurface over a surface $u: M^2\\rightarrow \\mathbb {R}^3_+$ , then the hypersurface $f$ is conformally flat if and only if the surface $u$ is of constant Gaussian curvature.",
"Proposition 3.4 Let $f: M^3\\rightarrow \\mathbb {R}^4$ be one of generic conformally flat hypersurfaces given by above three examples (REF ) (REF ) (REF ).",
"Then the Möbius form is closed $(i.e., dC=0)$ .",
"The Möbius metric $g$ in above three examples (REF ) (REF ) (REF ) can be unified in a single formula: $g=\\left[4H_u^2-3(K_u-\\epsilon )\\right](ds^2+I_u)=\\rho ^2(ds^2+I_u),$ where $I_u$ , $K_u$ , and $H_u$ are the induced metric, Gaussian curvature, and mean curvature of $u:M^2\\rightarrow N^3(\\epsilon )$ , respectively.",
"Let $\\lbrace e_2,e_3\\rbrace $ be a local orthonormal basis on $TM^2$ with respect to $I_u$ , consisting of unit principal vectors of $u$ and $e_1=\\frac{\\partial }{\\partial s}$ .",
"Then $\\lbrace e_1,e_2,e_3\\rbrace $ is an orthonormal basis for $T(I\\times M^2)$ with respect to $ds^2+I_u$ .",
"Let $\\tilde{R}_{ijkl}$ denote the curvature tensor of the metric $ds^2+I_u$ , and $R_{ijkl}$ the curvature tensor for $g=\\rho ^2[ds^2+I_u]$ .",
"From Yau's paper [15], we have $\\begin{split}&R_{ijij}=\\rho ^2\\tilde{R}_{ijij}+\\rho \\rho _{ii}+\\rho \\rho _{jj}-|\\nabla \\rho |^2, i\\ne j\\\\&R_{ijik}=\\rho ^2\\tilde{R}_{ijik}+\\rho \\rho _{jk},~~ when~ \\lbrace i,j,k\\rbrace ~ are~ distinct,\\\\\\end{split}$ which implies that $(B_{ij})=diag(b_1,b_2,b_3)$ and $(A_{ij})=diag(a_1,a_2,a_3)$ under the local orthonormal basis $\\lbrace \\rho ^{-1}e_1,\\rho ^{-1}e_2,\\rho ^{-1}e_3\\rbrace $ by the equation ().",
"Thus $dC=0$ by the equation (REF ).",
"Next, we prove Theorem REF .",
"From Proposition REF , we prove another hand of Theorem REF and we assume $dC=0$ .",
"From the equation (REF ), under the orthonormal basis $\\lbrace E_{1},E_2,E_{3}\\rbrace $ in (REF ) we find $(A_{ij})=diag\\lbrace a_1, a_2, a_3\\rbrace .$ The equations (REF ) and (REF ) imply that $R_{ijik}=A_{jk}=0,~~j\\ne k,$ by the equation ().",
"From the definition of $B_{ij,kl}$ and (), (REF ) and (REF ), we have $\\begin{split}&B_{23,31}=\\frac{3b_2B_{33,1}-3b_3B_{22,1}}{(b_3-b_2)^2}C_2+\\frac{3b_3}{b_2-b_3}[C_{2,1}-\\frac{B_{12,1}}{b_1-b_2}C_1]+B_{31,3}\\frac{B_{12,1}}{b_1-b_2},\\\\&B_{23,13}=(B_{22,1}-B_{33,1})\\frac{B_{23,3}}{b_2-b_3}+(B_{11,2}-B_{33,2})\\frac{B_{13,3}}{b_1-b_3},\\end{split}$ Using Ricci identity $B_{23,31}-B_{23,13}=(b_3-b_2)R_{2313}=0$ and (REF ), we have $b_3C_{1,2}=\\frac{2b_1^2+b_2b_3}{(b_2-b_1)(b_1-b_3)}C_1C_2=-C_1C_2.$ Similarly we have $b_1C_{2,3}=-C_2C_3,~~ b_2C_{1,3}=-C_1C_3.$ Therefore $b_kC_{i,j}=-C_iC_j,i\\ne j,i\\ne k,k\\ne j.$ Now we define $\\lbrace C_{i,jk}\\rbrace $ given by $dC_{i,j}+\\sum _mC_{m,j}\\omega _{mi}+\\sum _mC_{i,m}\\omega _{mj}=\\sum _mC_{i,jm}\\omega _m.$ Let $\\lbrace i,j,k\\rbrace $ be distinct.",
"Taking derivative for (REF ) along $E_k$ and invoking (REF ) and (REF ), we get $\\begin{split}&B_{kk,k}C_{i,j}+b_k[C_{i,jk}-C_{k,j}\\frac{B_{ki,k}}{b_k-b_i}-C_{i,k}\\frac{B_{kj,k}}{b_k-b_j}]\\\\&=-C_i[C_{j,k}-C_k\\frac{B_{jk,k}}{b_k-b_j}]-C_j[C_{i,k}-C_k\\frac{B_{ik,k}}{b_k-b_i}].\\end{split}$ If $b_1b_2b_3=0$ , we can assume that $b_1=0$ , which implies that $b_2=-b_3=\\sqrt{\\frac{1}{3}}$ by the equation ().",
"Using (REF ), we have $C_1=C_2=C_3=0$ and $B_{ij,k}=0$ .",
"Next we assume that $b_1b_2b_3\\ne 0$ .",
"Because $b_1^2+b_2^2+b_3^2=\\frac{2}{3},B_{ij,j}=B_{jj,i}-C_i$ , from (REF ) and (REF ) we have $b_kC_{i,jk}=-\\frac{4}{3}\\frac{C_iC_jC_k}{b_ib_jb_k}=-\\frac{4}{3}\\frac{C_1C_2C_3}{b_1b_2b_3}.$ Since $C_{i,jk}=C_{j,ik}=C_{k,ij}$ and $b_i\\ne b_j,i\\ne j$ , from (REF ) we get $C_{i,jk}=C_{j,ik}=C_{k,ij}=0,~~C_1C_2C_3=0.$ We can assume that $C_1=0$ , then $\\omega _{12}=\\frac{B_{12,1}}{b_1-b_2}\\omega _1,~\\omega _{13}=\\frac{B_{13,1}}{b_1-b_3}\\omega _1,~\\omega _{23}=\\frac{B_{23,2}}{b_2-b_3}\\omega _2+\\frac{B_{23,3}}{b_2-b_3}\\omega _3.$ From (REF ), combining $d\\omega _{12}-\\omega _{13}\\wedge \\omega _{32}=\\frac{-1}{2}\\sum _{kl}R_{12kl}\\omega _k\\wedge \\omega _l,$ we obtain $\\begin{split}&\\frac{-1}{2}\\sum _{kl}R_{12kl}\\omega _k\\wedge \\omega _l=d(\\frac{B_{12,1}}{b_1-b_2})\\wedge \\omega _1\\\\&+[(\\frac{B_{12,1}}{b_1-b_2})^2+\\frac{B_{13,1}B_{23,2}}{(b_1-b_3)(b_2-b_3)}]\\omega _1\\wedge \\omega _2+\\frac{B_{13,1}}{b_1-b_3}[\\frac{B_{12,1}}{b_1-b_2}+\\frac{B_{23,3}}{b_2-b_3}]\\omega _1\\wedge \\omega _3,\\end{split}$ which implies that $\\begin{split}&E_3(\\frac{B_{12,1}}{b_1-b_2})-\\frac{B_{13,1}}{b_1-b_3}[\\frac{B_{12,1}}{b_1-b_2}+\\frac{B_{23,3}}{b_2-b_3}]=0,\\\\&E_2(\\frac{B_{12,1}}{b_1-b_2})-[(\\frac{B_{12,1}}{b_1-b_2})^2+\\frac{B_{13,1}B_{23,2}}{(b_1-b_3)(b_2-b_3)}]=R_{1212}=b_1b_2+a_1+a_2.\\end{split}$ Similarly we have $\\begin{split}&E_2(\\frac{B_{13,1}}{b_1-b_3})-\\frac{B_{12,1}}{b_1-b_2}[\\frac{B_{13,1}}{b_1-b_3}-\\frac{B_{23,2}}{b_2-b_3}]=0,\\\\&E_3(\\frac{B_{13,1}}{b_1-b_3})-[(\\frac{B_{13,1}}{b_1-b_3})^2-\\frac{B_{12,1}B_{23,3}}{(b_1-b_2)(b_2-b_3)}]=R_{1313}=b_1b_3+a_1+a_3.\\end{split}$ Under the local basis above, $\\lbrace Y,N,Y_1,Y_2,Y_3,\\xi \\rbrace $ forms a moving frame in $\\mathbb {R}^6_1$ along $M^3$ .",
"We define $\\begin{split}&F=b_1Y+\\xi ,\\;\\; X_2=\\frac{B_{12,1}}{b_1-b_2}Y+Y_2,\\;\\;X_3=\\frac{B_{13,1}}{b_1-b_3}Y+Y_3,\\\\&T=a_1Y+N-\\frac{B_{12,1}}{b_1-b_2}Y_2-\\frac{B_{13,1}}{b_1-b_3}Y_3-b_1\\xi ,\\\\&Q=2a_1+b_1^2+(\\frac{B_{12,1}}{b_1-b_2})^2+(\\frac{B_{13,1}}{b_1-b_3})^2.\\end{split}$ Clearly $F$ is the curvature sphere of principal curvature $k_1$ .",
"And $\\begin{split}&<F,X_2>=<F,X_3>=<F,T>=<F,Y_1>=0,\\\\&<T,X_2>=<T,X_3>=<T,Y_1>=<X_2,X_3>=0,\\\\&<F,F>=<X_2,X_2>=<X_3,X_3>=1,~ <T,T>=Q.\\end{split}$ From structure equation of the hypersurface and (REF ), we get $E_1(F)=0,~~E_2(F)=(b_1-b_2)X_2, ~~ E_3(F)=(b_1-b_3)X_3.$ Thus curvature sphere $F$ induces a surface $\\tilde{M}=M^3/L$ in the de-Sitter space $\\mathbb {S}^5_1$ $F:\\tilde{M}=M^3/L\\longrightarrow \\mathbb {S}^5_1,$ where fibers $L$ are integral submanifolds of distribution $D=span\\lbrace E_1\\rbrace $ .",
"We define $V=span\\lbrace T,Y_1\\rbrace .$ Clearly we have $F\\bot V.$ Using (REF ), (REF ) and (REF ), we can get that $\\begin{split}&E_1(Y_1)=-T,~~E_2(Y_1)=0,~~E_3(Y_1)=0,\\\\&E_1(T)=QY_1,~~E_2(T)=\\frac{B_{12,1}}{b_1-b_2}T,~~E_3(T)=\\frac{B_{13,1}}{b_1-b_3}T.\\end{split}$ This implies that subspace $V$ is parallel along $M^3$ .",
"Similarly we have $E_1(Q)=0, E_2(Q)=2\\frac{B_{12,1}}{b_1-b_2}Q, E_3(Q)=2\\frac{B_{13,1}}{b_1-b_3}Q.$ Regarding (REF ) as a linear first-order differential equation for $Q$ , we see that $Q\\equiv 0$ or $Q\\ne 0$ on an connected manifold $M^n$ .",
"Therefore there are three possibilities for the induced metric on the fixed subspace $V\\subset \\mathbb {R}^6_1$ .",
"Case 1, $Q=0$ , then $<T,T>=Q=0$ , therefore $V$ is endowed with a degenerate inner product.",
"By (REF ), $T$ determines a fixed light-like direction in $\\mathbb {R}^6_1$ .",
"Up to a Möbius transformation, we may take to be $T=\\lambda (1,-1,0,0,0,0), ~~\\lambda \\in C^{\\infty }(M^3).$ Since $V$ is a fixed degenerate subspace in $\\mathbb {R}^6_1$ , we can find a space-like vector $v$ such that $V=Span\\lbrace e=(1,-1,0,0,0,0), v\\rbrace $ and $<e,F>=<v,F>=0$ .",
"We interpret the geometry of the hypersurface $f:M^3\\rightarrow \\mathbb {R}^4$ as below: 1) $v$ determines a fixed hyperplane $\\Sigma $ in $\\mathbb {R}^4$ because of $<T,v>=0$ .",
"2) $F$ is a two parameter family of hyperplanes orthogonal to the fixed hyperplane $\\Sigma $ in $\\mathbb {R}^4$ .",
"Therefore $f(M^3)$ , as the envelope of this family of hyperplanes $F$ , is clearly a cylinder over a surface $\\tilde{M}\\subset \\mathbb {R}^3$ .",
"Case 2, $Q<0$ , then $<T,T>$ is negative, and $V$ is a Lorentz subspace in $\\mathbb {R}^6_1$ .",
"Up to a Möbius transformation, we can assume that $V=Span\\lbrace T,Y_1\\rbrace =Span\\lbrace p_0=(1,1,0,0,0,0),p_1=(1,-1,0,0,0,0)\\rbrace .$ Using the stereographic projection, $p_0, p_1$ correspond to the origin $O$ and the point at infinity $\\infty $ of $\\mathbb {R}^4$ , respectively.",
"Since $F\\perp V$ , $F$ is a two parameter family of hyperplanes (passing $O$ and $\\infty $ ).",
"Therefore $f(M^3)$ , as the envelope of this family of hyperplanes $F$ , is clearly a cone (with vertex $O$ ) over a surface $\\tilde{M}\\subset \\mathbb {S}^3$ .",
"Case 3, $Q>0$ , then $<T,T>$ is positive, and $V$ is a space-like subspace in $\\mathbb {R}^6_1$ .",
"Up to a Möbius transformation, we can assume that $V=Span\\lbrace T,Y_1\\rbrace =Span\\lbrace (0,0,1,0,0,0),(0,0,0,1,0,0)\\rbrace =\\mathbb {R}^2.$ Thus $V$ is a fixed two dimensional plane $\\mathbb {R}^2\\subset \\mathbb {R}^4$ , and $F$ is a two parameter family of hyper-sphere orthogonal to this fixed plane $\\mathbb {R}^2$ with centers locating on it.",
"Thus $F$ envelopes a rotational hypersurface $f(M^3)$ (over a surface $\\tilde{M}\\subset \\mathbb {R}^3_+$ ).",
"From Case 1, Case 2, Case 3, we prove that if $dC=0$ , then the hypersurface is Möbius equivalent to one of the standard examples of generic conformally flat hypersurface.",
"thus we complete the proof of Theorem REF ."
],
[
"Global behavior of the generic conformally flat hypersurface",
"Let $f: M^3\\rightarrow \\mathbb {R}^4$ be a generic conformally flat hypersurface.",
"We say that the pair $(U,\\omega )$ is admissible if (1), $U$ is an open subset of $M^3$ , (2), $\\omega =(\\omega _1,\\omega _2,\\omega _3)$ is a orthonormal co-frame field on $U$ with respect to the Möbius metric $g$ , (3), $\\omega _1\\wedge \\omega _2\\wedge \\omega _3=dv_g$ , (4), $B=\\sum _ib_i\\omega _i\\otimes \\omega _i$ .",
"Denote by $F=(E_1,E_2,E_3)$ the dual frame field of $\\omega $ .",
"Then it is easily-seen that, $(U,\\omega )$ is admissible if and only if $E_i$ is an unit principal vector associated to $b_i$ for each $1\\le i\\le 3$ , and $\\lbrace E_1,E_2,E_3\\rbrace $ is an oriented basis associated to the orientation of $M^3$ .",
"Denote by $\\lbrace \\omega _{ij}\\rbrace $ the connection form with respect to $(U,\\omega _1,\\omega _2,\\omega _3)$ .",
"Thus under the admissible frame field $\\lbrace E_1,E_2,E_3\\rbrace $ , $(B_{ij})=diag\\lbrace b_1, b_2, b_3\\rbrace .$ Now we introduce two 2-forms on $M^3$ : for every admissible co-frame field $(U,\\omega )$ , set $\\begin{split}&\\Phi =\\omega _{12}\\wedge \\omega _3+\\omega _{23}\\wedge \\omega _1+\\omega _{31}\\wedge \\omega _2,\\\\&\\Psi =(b_1-b_2)^2\\omega _{12}\\wedge \\omega _3+(b_2-b_3)^2\\omega _{23}\\wedge \\omega _1+(b_1-b_3)^2\\omega _{31}\\wedge \\omega _2.\\end{split}$ If $(U,\\omega )$ and $(\\tilde{U},\\tilde{\\omega })$ are both admissible co-frame fields with $U\\cap \\tilde{U}\\ne \\emptyset $ .",
"Then on $U\\cap \\tilde{U}$ , $\\omega _i=\\epsilon _i\\tilde{\\omega }_i, ~~\\omega _{ij}=\\epsilon _i\\epsilon _j\\tilde{\\omega }_{ij}$ for every $1\\le i\\le 3$ , where $\\epsilon _i=1$ or $-1$ and $\\epsilon _1\\epsilon _2\\epsilon _3=1$ .",
"Thus $\\omega _{12}\\wedge \\omega _3=\\tilde{\\omega }_{12}\\wedge \\tilde{\\omega }_3,~~\\omega _{23}\\wedge \\omega _1=\\tilde{\\omega }_{23}\\wedge \\tilde{\\omega }_1,~~\\omega _{31}\\wedge \\omega _2=\\tilde{\\omega }_{31}\\wedge \\tilde{\\omega }_2.$ Therefore the 2-forms $\\Phi , \\Psi $ are well-defined on $M^3$ .",
"Combining $d\\omega _{ij}-\\sum _k\\omega _{ik}\\wedge \\omega _{kj}=\\frac{-1}{2}\\sum _{kl}R_{ijkl}\\omega _k\\wedge \\omega _l$ , $d\\omega _i=\\sum _k\\omega _{ik}\\wedge \\omega _k$ and the equation (REF ), (REF ), we get $\\begin{split}&d(\\omega _{12}\\wedge \\omega _3)=-R_{1212}\\omega _1\\wedge \\omega _2\\wedge \\omega _3\\\\&+[\\frac{-9b_1b_2C_3^2}{(b_1-b_3)^2(b_2-b_3)^2}+\\frac{9b_2b_3C_1^2}{(b_1-b_2)^2(b_1-b_3)^2}+\\frac{9b_1b_3C_2^2}{(b_1-b_2)^2(b_2-b_3)^2}]\\omega _1\\wedge \\omega _2\\wedge \\omega _3.\\end{split}$ Similarly we can compute $d(\\omega _{23}\\wedge \\omega _1)$ and $d(\\omega _{31}\\wedge \\omega _2)$ .",
"Therefore we have $\\begin{split}&d\\Phi =[\\frac{9b_1b_2C_3^2}{(b_1-b_3)^2(b_2-b_3)^2}-R_{1212}-R_{1313}-R_{2323}]\\omega _1\\wedge \\omega _2\\wedge \\omega _3\\\\&+[\\frac{9b_2b_3C_1^2}{(b_1-b_2)^2(b_1-b_3)^2}+\\frac{9b_1b_3C_2^2}{(b_1-b_2)^2(b_2-b_3)^2}]\\omega _1\\wedge \\omega _2\\wedge \\omega _3.\\end{split}$ Using $db_i=\\sum _kB_{ii,k}\\omega _k$ and the same computation as $d\\Phi $ , we can obtain $\\begin{split}&d\\Psi =-[(b_1-b_2)^2R_{1212}+(b_1-b_3)^2R_{1313}+(b_2-b_3)^2R_{2323}]dv_g\\\\&+[\\frac{18b_1b_2C_3^2}{(b_1-b_3)^2(b_2-b_3)^2}+\\frac{18b_2b_3C_1^2}{(b_1-b_2)^2(b_1-b_3)^2}+\\frac{18b_1b_3C_2^2}{(b_1-b_2)^2(b_2-b_3)^2}]dv_g,\\end{split}$ where $dv_g=\\omega _1\\wedge \\omega _2\\wedge \\omega _3$ .",
"Combining the equation (REF ) and the equation (REF ), we have $2d\\Phi -d\\Psi =\\lbrace [(b_1-b_2)^2-2]R_{1212}+[(b_2-b_3)^2-2]R_{2323}+[(b_1-b_3)^2-2]R_{1313}\\rbrace dv_g.$ If $M^3$ is compact, the equation (REF ) implies that $\\int _{M^3}\\lbrace [(b_1-b_2)^2-2]R_{1212}+[(b_2-b_3)^2-2]R_{2323}+[(b_1-b_3)^2-2]R_{1313}\\rbrace dv_g=0.$ From $b_1+b_2+b_3=0$ and $b_1^2+b_2^2+b_3^2=\\frac{2}{3}$ , we can derive that $(b_1-b_2)^2+(b_2-b_3)^2+(b_1-b_3)^2=2.$ which implies that $(b_1-b_2)^2-2<0,~~(b_2-b_3)^2-2<0,~~(b_1-b_3)^2-2<0.$ Now we assume that the sectional curvature of $M^3$ with respect to the Möbius metric have sign, for example, the sectional curvature is nonnegative.",
"The equation (REF ) and (REF ) imply that the sectional curvature vanishes, i.e., $R_{1212}=R_{2323}=R_{1313}=0.$ In [8], authors classify the hypersurfaces $f: M^3\\rightarrow \\mathbb {R}^{4}$ with constant Möbius sectional curvature, which are non-compact.",
"This is a contradiction, thus we finish the proof of Theorem REF .",
"Using the equation () and the equation (), we have $\\begin{split}&[(b_1-b_2)^2-2]R_{1212}+[(b_2-b_3)^2-2]R_{2323}+[(b_1-b_3)^2-2]R_{1313}\\\\&=\\frac{2}{9}-\\frac{10}{3}tr(A)+3[b_1^2a_1+b_2^2a_2+b_3^2a_3].\\end{split}$ On the other hand, the equation () implies that $|Ric|^2=\\frac{2}{9}+5tr(A)^2+|A|^2-\\frac{4}{3}tr(A)-2[b_1^2a_1+b_2^2a_2+b_3^2a_3],$ where $|Ric|$ denote the norm of the Ricci curvature.",
"Combining the equation (REF ) and the equation (REF ), we can derive that $\\begin{split}&[(b_1-b_2)^2-2]R_{1212}+[(b_2-b_3)^2-2]R_{2323}+[(b_1-b_3)^2-2]R_{1313}\\\\&=\\frac{5}{9}-\\frac{16}{3}tr(A)+\\frac{15}{2}tr(A)^2+\\frac{3}{2}|A|^2-\\frac{3}{2}|Ric|^2.\\end{split}$ Let $\\tilde{A}:=A-\\frac{1}{3}tr(A)g$ denote the trace-free Blaschke tensor, then $|\\tilde{A}|^2=|A|^2-\\frac{1}{3}tr(A)^2$ .",
"Thus from the equation (REF ), we have $\\begin{split}&[(b_1-b_2)^2-2]R_{1212}+[(b_2-b_3)^2-2]R_{2323}+[(b_1-b_3)^2-2]R_{1313}\\\\&=\\frac{3}{2}|\\tilde{A}|^2-\\frac{3}{2}[|Ric|^2-\\frac{1}{3}R^2]-\\frac{1}{9}.\\end{split}$ Now if the hypersurface $M^3$ is compact, then $\\int _{M^3}\\lbrace |\\tilde{A}|^2+\\frac{1}{3}R^2-|Ric|^2-\\frac{2}{27}\\rbrace dv_g=0.$ Therefore we finish the proof of Theorem REF .",
"Since $R=tr(Ric)$ , we have $\\frac{1}{3}R^2-|Ric|^2\\le 0$ on $M^3$ .",
"if $\\frac{1}{3}R^2-|Ric|^2\\equiv 0$ , then the sectional curvature $K=0$ , and there is a contradiction by the results in [8].",
"Thus Corollary REF is proved."
]
] | 1709.01657 |
[
[
"Nonequilibrium fluctuations and enhanced diffusion of a driven particle\n in a dense environment"
],
[
"Abstract We study the diffusion of a tracer particle driven out-of-equilibrium by an external force and traveling in a dense environment of arbitrary density.",
"The system evolves on a discrete lattice and its stochastic dynamics is described by a master equation.",
"Relying on a decoupling approximation that goes beyond the naive mean-field treatment of the problem, we calculate the fluctuations of the position of the tracer around its mean value on a lattice of arbitrary dimension, and with different boundary conditions.",
"We reveal intrinsically nonequilibrium effects, such as enhanced diffusivity of the tracer induced both by the crowding interactions and the external driving.",
"We finally consider the high-density and low-density limits of the model and show that our approximation scheme becomes exact in these limits."
],
[
"Operators acting on the distribution $P$ in the master equation [Eq. (1)]",
"We denote by $X$ the position of the tracer and $\\eta _{r} \\in \\lbrace 0,1\\rbrace $ the occupation number at site $r$ .",
"$P(X,\\eta ;t)$ is the probability to find the tracer at position $X$ with the lattice in configuration $\\eta \\equiv \\lbrace \\eta _{r}\\rbrace $ .",
"$\\eta ^{r,\\mu }$ is the configuration obtained from $\\eta $ by exchanging the occupation numbers of sites $r$ and $r+ e_\\mu $ .",
"$2d\\tau ^*\\partial _t P(X,\\eta ;t) =&&\\underbrace{ \\sum _{\\mu =1}^d\\sum _{{r}\\ne X-e_\\mu ,X} \\left[ P(X,\\eta ^{{r},\\mu };t)-P(X,\\eta ;t)\\right]}_{\\equiv \\mathcal {L}_\\text{bath} P}\\nonumber \\\\&&\\underbrace{+\\frac{2d\\tau ^*}{\\tau }\\sum _{\\mu }p_\\mu \\left[\\left(1-\\eta _{X} \\right)P(X-e_{\\mu },\\eta ;t) -\\left(1-\\eta _{X+e_{\\mu }}\\right)P(X,\\eta ;t)\\right]}_{\\equiv \\mathcal {L}_\\text{TP} P}$"
],
[
"Evolution equation for the mean displacement of the tracer",
"Multiplying Eq.",
"(REF ) by $(X\\cdot e_1)$ and summing over all configurations $X$ and $\\eta $ , we find $\\frac{\\mathrm {d}\\left\\langle X_t \\right\\rangle }{\\mathrm {d}t} = \\frac{\\sigma }{\\tau } \\lbrace p_1 [1-k_{e_1}(t)]-p_{-1} [1-k_{e_{-1}}(t)] \\rbrace .$ Noticing that $\\lim _{|r|\\rightarrow \\infty } k_{r} = \\rho $ , we define the quantities $h_{r} = k_{r}-\\rho $ which are shown to obey Eq.",
"(6) from the main text.",
"Finally, the velocity of the tracer in the stationary limit is defined as $V\\equiv \\lim _{t\\rightarrow \\infty } \\frac{\\mathrm {d}\\left\\langle X_t \\right\\rangle }{\\mathrm {d}t}$ ."
],
[
"Derivation of Eqs. (6) and (7)",
"Starting from the master equation and using the decoupling approximations [Eqs.",
"(4) and (5)], we find the equations satisfied by the density profiles $h_{r}$ : ${r} \\notin \\lbrace {\\bf 0},\\pm {e_1},\\ldots ,\\pm {e_d}\\rbrace $ $2d\\tau ^*\\partial _t h_{r} = \\widetilde{L} h_{r},$ ${r} \\in \\lbrace \\pm {e_1},\\ldots ,\\pm {e_d}\\rbrace $ $2d\\tau ^*\\partial _t h_{e_{\\nu }} = \\widetilde{L} h_{e_{\\nu }} + \\rho (A_\\nu - A_{-\\nu }),$ and the cross-correlation functions $\\widetilde{g}_{r}$ [31]: for $r\\notin \\lbrace \\mathbf {0},e_{\\pm 1},\\dots ,e_{\\pm d} \\rbrace $ : $2d\\tau ^*\\partial _t\\widetilde{g}_{r}=&\\widetilde{L}\\widetilde{g}_{r}+\\frac{2d\\tau ^*}{\\tau }\\sigma \\left\\lbrace p_1(1-\\rho -h_1)\\nabla _1 h_{r}-p_{-1}(1-\\rho -h_{-1})\\nabla _{-1} h_{r}\\right\\rbrace -\\frac{2d\\tau ^*}{\\tau }\\sum _\\mu p_\\mu \\widetilde{g}_{\\mu }\\nabla _\\mu h_{r},$ for $r=e_\\nu $ with $\\nu \\ne \\pm 1$ : $2d\\tau ^*\\partial _t\\widetilde{g}_{\\nu } = & (\\widetilde{L}+A_\\nu )\\widetilde{g}_{\\nu }+\\frac{2d\\tau ^*}{\\tau }\\sigma \\left\\lbrace p_1(1-\\rho -h_{1})\\nabla _1 h_{\\nu }-p_{-1}(1-\\rho -h_{{-1}})\\nabla _{-1} h_{\\nu }\\right\\rbrace \\nonumber \\\\&-\\frac{2d\\tau ^*}{\\tau }\\sum _{\\mu }p_\\mu \\widetilde{g}_{e_\\mu }\\nabla _\\mu h_{e_\\nu } -\\frac{2d\\tau ^*}{\\tau } \\left[p_\\nu (\\rho +h_\\nu )\\widetilde{g}_{\\nu }-p_{-\\nu } \\rho \\widetilde{g}_{\\nu }\\right],$ for $r=e_1$ : $2d\\tau ^*\\partial _t\\widetilde{g}_{1} = & (\\widetilde{L}+A_1)\\widetilde{g}_{\\nu }+\\frac{2d\\tau ^*}{\\tau }\\sigma \\left\\lbrace p_1(1-\\rho -h_{1})\\nabla _1 h_{1}-p_{-1}(1-\\rho -h_{{-1}})(\\nabla _{-1} h_{1}+\\rho )\\right\\rbrace \\nonumber \\\\&-\\frac{2d\\tau ^*}{\\tau }\\sum _{\\mu }p_\\mu \\widetilde{g}_{e_\\mu }\\nabla _\\mu h_{1} -\\frac{2d\\tau ^*}{\\tau } \\left[p_1(\\rho +h_1)\\widetilde{g}_{1}-p_{-1} \\rho \\widetilde{g}_{-1}\\right],$ for $r=e_{-1}$ : $2d\\tau ^*\\partial _t\\widetilde{g}_{-1} = & (\\widetilde{L}+A_{-1})\\widetilde{g}_{-1}+\\frac{2d\\tau ^*}{\\tau }\\sigma \\left\\lbrace p_1(1-\\rho -h_{1})(\\nabla _1 h_{-1}-\\rho )-p_{-1}(1-\\rho -h_{{-1}})\\nabla _{-1} h_{-1}\\right\\rbrace \\nonumber \\\\&-\\frac{2d\\tau ^*}{\\tau }\\sum _{\\mu }p_\\mu \\widetilde{g}_{e_\\mu }\\nabla _\\mu h_{-1} -\\frac{2d\\tau ^*}{\\tau } \\left[p_{-1}(\\rho +h_{-1})\\widetilde{g}_{-1}-p_{1} \\rho \\widetilde{g}_{1}\\right], $ where we defined the operator $\\widetilde{L} \\equiv \\sum _\\mu A_\\mu \\nabla _\\mu $ and the quantities $A_\\mu \\equiv 1+\\frac{2d\\tau ^*}{\\tau }p_\\mu (1-k_{e_\\mu })$ .",
"We will define for simplicity the operators $2d\\tau ^*\\partial _t\\widetilde{g}_{r} & \\equiv \\mathcal {L}(r),\\\\2d\\tau ^*\\partial _t\\widetilde{g}_{\\nu } & \\equiv \\mathcal {L}^{\\prime }(\\nu ).$ Eqs.",
"(REF )-(REF ) can be solved by introducing the auxiliary variable $w=(w_1,\\dots ,w_d)$ and defining the generating functions $H(w;t) &=&\\begin{dcases}\\sum _{r_1=-\\infty }^\\infty \\sum _{r_2,\\dots , r_d=0}^{L-1} h_{r}(t) \\prod _{j=1}^{d} {w_j}^{r_j} & \\text{for a generalized capillary}, \\\\\\sum _{r_1,\\dots ,r_d=-\\infty }^\\infty h_{r}(t) \\prod _{j=1}^{d} {w_j}^{r_j} & \\text{for an infinitely extended lattice},\\end{dcases} \\\\G(w;t) &=&\\begin{dcases}\\sum _{r_1=-\\infty }^\\infty \\sum _{r_2,\\dots , r_d=0}^{L-1}\\widetilde{g}_{r} \\prod _{j=1}^{d} {w_j}^{r_j} & \\text{for a generalized capillary}, \\\\\\sum _{r_1,\\dots ,r_d=-\\infty }^\\infty \\widetilde{g}_{r} \\prod _{j=1}^{d} {w_j}^{r_j} & \\text{for an infinitely extended lattice}.\\end{dcases}$ Multiplying Eqs.",
"(REF ) and (REF ) by $\\prod _{j=1}^d {w_j}^{r_j}$ , summing over all lattice sites and using the boundary conditions [Eqs.",
"(REF ), (REF ), (REF ) and (REF )], we find that $H(w;t)$ and $G(w;t)$ are the solutions of the differential equations $2d\\tau ^* \\partial _t H(w;t) &=& \\left[ \\frac{A_1}{w_1}+A_{-1}w_1 + A_2\\sum _{j=2}^d\\left(\\frac{1}{w_j} + w_j \\right) - \\mathcal {A} \\right]H(w;t)+K(w;t),\\\\2d\\tau ^* \\partial _t G(w;t) &= & \\left[\\frac{A_1}{w_1}+{A_{-1}}{w_1}+A_2\\sum _{\\mu } w_{|\\mu |}^{\\mathrm {sgn}(\\mu )}-\\mathcal {A}\\right] G(w;t) \\nonumber \\\\& &+ \\frac{2d\\tau ^*}{\\tau } \\sigma \\left[p_1(1-\\rho -h_1)\\left( \\frac{1}{w_1}-1\\right)-p_{-1}(1-\\rho -h_{-1})(w_1-1)\\right]H(w;t) \\nonumber \\\\&&- \\frac{2 d \\tau ^*}{\\tau } \\left[ \\sum _{\\mu } p_\\mu \\widetilde{g}_\\mu \\left(\\frac{1}{w_{|\\mu |}^{\\mathrm {sgn}(\\mu )}}-1\\right) \\right]H(w;t)-\\mathcal {L}_0+\\sum _{\\mu } w_{|\\mu |}^{\\mathrm {sgn}(\\mu )} \\left[\\mathcal {L}^{\\prime }(\\mu )-\\mathcal {L}(\\mu )\\right], $ with $\\mathcal {A} = A_1+A_{-1} +2(d-1)A_2$ and $K(w;t)&\\equiv & A_1 (w_1-1)h_{1}+A_{-1} \\left(\\frac{1}{w_1}-1\\right)h_{-1}\\nonumber \\\\&+&A_2 \\sum _{j=2}^d\\left[ (w_j-1)h_j +\\left(\\frac{1}{w_j}-1\\right)h_{-j} \\right]+ \\rho [A_1 -A_{-1} ] \\left( w_1-\\frac{1}{w_1}\\right),\\\\\\mathcal {L}_0&=& \\sum _{\\mu } A_\\mu \\widetilde{g}_\\mu + \\frac{2d\\tau ^*}{\\tau } \\sigma \\left[p_1(1-\\rho -h_1)-p_{-1}(1-\\rho -h_{-1})\\right]-\\frac{2d\\tau ^*}{\\tau } \\sum _{\\mu } \\widetilde{g}_\\mu h_\\mu ,$ and where we used the symmetry relation $A_{\\pm 2} =\\dots A_{\\pm d}=A_2$ .",
"In the stationary limit, we find $H(w)&=&\\frac{K(w)}{\\mathcal {A}}\\frac{1}{1-\\lambda (w)} ,\\\\G(w)&=&\\frac{J_1(w)K(w)}{\\mathcal {A}^2[1-\\lambda (w)]^2}+\\frac{J_0(w)}{\\mathcal {A}[1-\\lambda (w)]},$ with $\\lambda (w)=\\frac{A_1}{\\mathcal {A}} \\frac{1}{w_1} +\\frac{A_{-1}}{\\mathcal {A}} {w_1} + \\frac{A_2}{\\mathcal {A}}\\sum _{j=2}^d \\left( \\frac{1}{w_j}+w_j\\right)$ and $J_0(w) \\equiv & \\sum _{\\mu } \\left( w_{|\\mu |}^{\\mathrm {sgn}(\\mu )}-1 \\right) \\left(A_\\mu -\\frac{2d\\tau ^*}{\\tau }p_\\mu h_\\mu \\right) \\widetilde{g}_\\mu -\\frac{2d\\tau ^*}{\\tau } \\rho \\left( w_1-\\frac{1}{w_1}\\right)(p_1\\widetilde{g}_1-p_{-1}\\widetilde{g}_{-1}) \\nonumber \\\\&+\\frac{2d\\tau ^*}{\\tau } \\sigma \\left[p_{-1}(1-\\rho -h_{-1})\\left( \\rho {w_1}-h_{-1}\\right)-p_{1}(1-\\rho -h_{1})(\\frac{\\rho }{w_1}-h_1)\\right], \\\\J_1(w) \\equiv &\\frac{2d\\tau ^*}{\\tau } \\left\\lbrace \\sigma \\left[p_1(1-\\rho -h_1)\\left( \\frac{1}{w_1}-1\\right)-p_{-1}(1-\\rho -h_{-1})(w_1-1)\\right] -\\sum _{\\mu } \\widetilde{g}_\\mu \\left(\\frac{1}{w_{|\\mu |}^{\\mathrm {sgn}(\\mu )}}-1\\right) \\right\\rbrace .",
"$ Finally, expressing the generating function variables $w_1,\\dots ,w_d$ in terms of the Fourier variables: $w_1=\\mathrm {e}^{\\mathrm {i}q_1}$ and $w_j=\\mathrm {e}^{\\frac{2\\mathrm {i}\\pi k_j}{L}}$ ($j\\ge 2$ ) for a generalized capillary, $w_j=\\mathrm {e}^{\\mathrm {i}q_j}$ ($0\\le j \\le d$ ) for an infinitely extended lattice, one can compute the inverse Fourier transform of Eqs.",
"(REF ) and () in order to retrieve the equations satisfied by $h_{r}$ and $\\widetilde{g}_{r}$ presented in the main text [Eqs.",
"(6) and (7)]."
],
[
"Expression of the generating functions $\\widehat{\\mathcal {P}}$",
"with the structure factors $\\lambda (q_1,\\dots ,q_d) & = & \\frac{A_1}{\\mathcal {A}} \\mathrm {e}^{-\\mathrm {i}q_1 }+ \\frac{A_{-1}}{\\mathcal {A}} \\mathrm {e}^{\\mathrm {i}q_{-1} }+\\frac{2A_2}{\\mathcal {A}}\\sum _{j=2}^d \\cos q_j ,\\\\\\lambda (q,k_2,\\dots ,q_d) & = & \\frac{A_1}{\\mathcal {A}} \\mathrm {e}^{-\\mathrm {i}q }+\\frac{A_{-1}}{\\mathcal {A}} \\mathrm {e}^{\\mathrm {i}q }+ \\frac{2A_2}{\\mathcal {A}}\\sum _{j=2}^d \\cos \\left( \\frac{2\\pi k_j}{L} \\right) .$ Using the definition of the functions $\\mathcal {F}_{r} = \\lim _{\\xi \\rightarrow 1} \\widehat{\\mathcal {P}}(r|\\mathbf {0};\\xi )$ and $\\mathcal {G}_{r} = \\lim _{\\xi \\rightarrow 1} \\frac{\\partial }{\\partial \\lambda }\\widehat{\\mathcal {P}}(r|\\mathbf {0};\\xi )$ , we find $\\mathcal {F}_{r}&=&\\begin{dcases}\\frac{1}{(2\\pi )^d}\\int _{[-\\pi ,\\pi ]^d} \\mathrm {d}q_1\\dots \\mathrm {d}q_d \\frac{\\prod _{j=1}^d \\mathrm {e}^{-\\mathrm {i}r_j q_j}}{1- \\lambda (q_1,\\dots ,q_d)}& \\text{for an infinitely extended lattice},\\\\\\frac{1}{L^{d-1}} \\sum _{k_2,\\dots ,k_d=0}^{L-1} \\frac{1}{2\\pi }\\int _{-\\pi }^\\pi \\mathrm {d}q \\frac{\\mathrm {e}^{-\\mathrm {i}r_1 q} \\prod _{j=2}^d \\mathrm {e}^{-2\\mathrm {i}\\pi r_j k_j/L}}{1- \\lambda (q,k_2,\\dots ,k_d)} & \\text{for a generalized capillary},\\end{dcases}\\\\\\mathcal {G}_{r}&=&\\begin{dcases}\\frac{1}{(2\\pi )^d}\\int _{[-\\pi ,\\pi ]^d} \\mathrm {d}q_1\\dots \\mathrm {d}q_d \\frac{\\prod _{j=1}^d \\mathrm {e}^{-\\mathrm {i}r_j q_j}}{[1- \\lambda (q_1,\\dots ,q_d)]^2}& \\text{for an infinitely extended lattice},\\\\\\frac{1}{L^{d-1}} \\sum _{k_2,\\dots ,k_d=0}^{L-1} \\frac{1}{2\\pi }\\int _{-\\pi }^\\pi \\mathrm {d}q \\frac{\\mathrm {e}^{-\\mathrm {i}r_1 q} \\prod _{j=2}^d \\mathrm {e}^{-2\\mathrm {i}\\pi r_j k_j/L}}{[1- \\lambda (q,k_2,\\dots ,k_d)]^2} & \\text{for a generalized capillary},\\end{dcases}$ where $\\lambda $ is given by Eq.",
"(REF ) for an infinite lattice and by Eq.",
"() for a generalized capillary.",
"These expressions of $\\mathcal {F}_{r}$ and $\\mathcal {G}_{r}$ are to be used to solve numerically Eqs.",
"(5) and (6) from the main text and to compute the velocity and diffusion coefficient of the tracer particle."
],
[
"Transient regime – derivation of Eq. (9)",
"We study the limit where $\\rho \\rightarrow 1$ is taken first, and $t\\rightarrow \\infty $ ultimately, which allows us to calculate the transient regime that precedes the ultimate diffusive regime [Eq.",
"(9)].",
"To this purpose, we start from the differential equation satisfied by the generating function $G(w;t)$ [Eq.",
"()].",
"In the limit $\\rho \\rightarrow 1$ , at leading order, this equation reduces to $2d\\tau ^* \\partial _t G(w;t) = \\left[\\sum _{\\mu } w_{|\\mu |}^{\\mathrm {sgn}(\\mu )}-2d\\right] G(w;t) +J_0(w;t).$ Defining the Laplace transform of any time-dependent function $\\psi (t)$ as $\\widehat{\\psi }(s)= \\int _0^\\infty \\mathrm {e}^{-st}\\psi (t) \\mathrm {d}t$ , we find $\\widehat{G}(w;s) = \\frac{1}{2d(1+\\tau ^*s)} \\frac{\\widehat{J}_0(w;s)}{1-\\frac{1}{1+\\tau ^*s}\\Lambda (w)}, $ where we define $\\mathcal {E}_{r} \\equiv \\begin{dcases}\\frac{1}{L^{d-1}} \\sum _{k_2,\\dots ,k_d=0}^{L-1} \\frac{1}{2\\pi }\\int _{-\\pi }^\\pi \\mathrm {d}q \\frac{e^{-\\mathrm {i}r_1q} \\prod _{j=2}^d e^{-2\\mathrm {i}\\pi r_j k_j/L}}{1- \\frac{1}{1+\\tau ^* s}\\Lambda (q,k_2,\\dots ,k_d)} & \\text{for a generalized capillary}, \\\\\\frac{1}{(2\\pi )^d} \\int _{[-\\pi ,\\pi ]^d} \\mathrm {d}q_1 \\dots \\mathrm {d}q_d \\frac{ \\prod _{j=1}^d e^{-\\mathrm {i}q_i r_i}}{1- \\frac{1}{1+\\tau ^* s}\\Lambda (q_1,\\dots ,q_d)} & \\text{otherwise}.\\end{dcases}$ and $\\Lambda (w) \\equiv \\frac{1}{2 d} \\frac{1}{w_1} +\\frac{1}{2d} {w_1} + \\frac{1}{2d}\\sum _{j=2}^d \\left( \\frac{1}{w_j}+w_j\\right)$ Recalling the definitions of $G$ [Eq.",
"()] and $J_0$ [Eq.",
"(REF )], specifying the generating function variable $w$ in terms of Fourier variables (see Section ), and computing the inverse Fourier transform of Eq.",
"(REF ), one gets $\\widehat{\\widetilde{g}}_{r}(s)=&\\sum _{\\nu }\\widehat{\\widetilde{g}}_{\\nu }(s) \\nabla _{-\\nu } \\mathcal {E}_{r} + \\frac{2d\\tau ^*}{\\tau }[p_1\\widehat{\\widetilde{g}}_{1}(s)-p_{-1}\\widehat{\\widetilde{g}}_{-1}(s)](\\nabla _1-\\nabla _{-1})\\mathcal {E}_{r} \\nonumber \\\\&-\\frac{2d\\tau ^*}{\\tau }\\left\\lbrace p_1\\left[\\frac{1-\\rho }{s}-\\widehat{h}_1(s)\\right](\\nabla _1+1)-p_{-1}\\left[\\frac{1-\\rho }{s}-\\widehat{h}_{-1}(s)\\right](\\nabla _{-1}+1) \\right\\rbrace \\mathcal {E}_{r}.$ Evaluating this equation for $r=e_1$ , $e_{-1}$ and $e_2$ , we get a system of three equations that can be solved in order to get $\\widetilde{g}_1$ , $\\widetilde{g}_{-1}$ and $\\widetilde{g}_2$ .",
"We find that the limit $s\\rightarrow 0$ of the quantities $\\mathcal {E}_{r}$ are identical to the limit $\\xi \\rightarrow 1$ of the propagators $\\widehat{P}$ .",
"Consequently, we find the following sytem: $2d \\widehat{\\widetilde{g}}_1(s) &=& \\widehat{\\widetilde{g}}_1(s) \\nabla _{-1}\\widehat{P}_{e_1}+\\widehat{\\widetilde{g}}_{-1}(s) \\nabla _{1}\\widehat{P}_{e_1}+2(d-1)\\widehat{\\widetilde{g}}_{2}(s)\\nabla _2\\widehat{P}_{e_1} +\\frac{2d\\tau ^*}{\\tau }[p_1\\widehat{\\widetilde{g}}_1(s)-p_{-1}\\widehat{\\widetilde{g}}_{-1}(s)]\\left[\\nabla _1\\widehat{P}_{e_1} - \\nabla _{-1}\\widehat{P}_{e_1}\\right] \\nonumber \\\\&&-\\frac{2d\\tau ^*}{\\tau }\\sigma \\left\\lbrace p_1\\left[\\frac{1-\\rho }{s}-\\widehat{h}_{1}(s)\\right]\\widehat{P}_{2e_{1}}-p_{-1}\\left[\\frac{1-\\rho }{s}-\\widehat{h}_{-1}(s)\\right]\\widehat{P}_{\\mathbf {0}}\\right\\rbrace ,\\\\2d \\widehat{\\widetilde{g}}_{-1}(s) &=& \\widehat{\\widetilde{g}}_1(s) \\nabla _{-1}\\widehat{P}_{e_{-1}}+\\widehat{\\widetilde{g}}_{-1}(s) \\nabla _{1}\\widehat{P}_{e_{-1}}+2(d-1)\\widehat{\\widetilde{g}}_{2}(s)\\nabla _2\\widehat{P}_{e_{-1}}+\\frac{2d\\tau ^*}{\\tau }[p_1\\widehat{\\widetilde{g}}_1(s)-p_{-1}\\widehat{\\widetilde{g}}_{-1}(s)]\\left[\\nabla _1\\widehat{P}_{e_{-1}} - \\nabla _{-1}\\widehat{P}_{e_{-1}}\\right]\\nonumber \\\\&&-\\frac{2d\\tau ^*}{\\tau }\\sigma \\left\\lbrace p_1\\left[\\frac{1-\\rho }{s}-\\widehat{h}_{1}(s)\\right]\\widehat{P}_{\\mathbf {0}}-p_{-1}\\left[\\frac{1-\\rho }{s}-\\widehat{h}_{-1}(s)\\right]\\widehat{P}_{2e_1}\\right\\rbrace , \\\\2d \\widehat{\\widetilde{g}}_{2}(s) &=& \\widehat{\\widetilde{g}}_1(s) \\nabla _{-1}\\widehat{P}_{e_{2}}+\\widehat{\\widetilde{g}}_{-1}(s) \\nabla _{1}\\widehat{P}_{e_{2}}+\\widehat{\\widetilde{g}}_{2}(s)\\left[\\nabla _{-2}\\widehat{P}_{e_{2}}+\\nabla _{2}\\widehat{P}_{e_{2}}\\right]+2(d-2)\\widehat{\\widetilde{g}}_2(s)\\nabla _3\\widehat{P}_{e_2}\\nonumber \\\\&&+\\frac{2d\\tau ^*}{\\tau }(p_1\\widehat{\\widetilde{g}}_1(s)-p_{-1}\\widehat{\\widetilde{g}}_{-1}(s))\\left[\\nabla _1\\widehat{P}_{e_{2}} - \\nabla _{-1}\\widehat{P}_{e_{2}}\\right]\\nonumber \\\\&&-\\frac{2d\\tau ^*}{\\tau }\\sigma \\left\\lbrace p_1\\left[\\frac{1-\\rho }{s}-\\widehat{h}_{1}(s)\\right]\\widehat{P}_{e_{2}+e_1}-p_{-1}\\left[\\frac{1-\\rho }{s}-\\widehat{h}_{-1}(s)\\right]\\widehat{P}_{e_2+e_{-1}}\\right\\rbrace .$ where $\\widehat{P}(r|r_0;\\xi )$ is the generating function associated to the propagator of a symmetric random walk on the considered lattice, starting from $r_0$ and arriving at $r$ .",
"We write for convenience $\\nabla _{\\nu }\\widehat{P}_{r} = \\lim _{\\xi \\rightarrow 1} \\nabla _{\\nu }\\widehat{P}(r|\\mathbf {0};\\xi )$ .",
"To simplify the system of Eqs.",
"(REF )-() a bit further, we also use the following symmetry properties on the quantities $\\widehat{P}$ [32]: $\\widehat{P}(r|r_0;\\xi )&=&\\widehat{P}(r-r_0|\\mathbf {0};\\xi ), \\\\\\widehat{P}\\left( \\sum _{j=1}^d r_je_j |\\mathbf {0};\\xi \\right)& =& \\widehat{P}\\left(-r_i e_i+\\sum _{\\begin{array}{c}{j=1}\\\\{j\\ne i}\\end{array}}^d r_je_j|\\mathbf {0};\\xi \\right).",
"$ The first relation originates from the translational invariance of the lattice, the second one from the symmetry of the random walk described by $\\widehat{P}$ .",
"We also use the relations $\\widehat{P}(\\mathbf {0}|\\mathbf {0},\\xi ) & =& 1+\\frac{1}{d}\\left[\\widehat{P}(e_1|\\mathbf {0},\\xi )+(d-1) \\widehat{P}(e_2|\\mathbf {0},\\xi ) \\right] ,\\\\\\widehat{P}(e_1|\\mathbf {0},\\xi ) & =& \\frac{1}{2d}\\left[\\widehat{P}(\\mathbf {0}|\\mathbf {0},\\xi )+\\widehat{P}(2e_1|\\mathbf {0},\\xi )+2(d-1) \\widehat{P}(e_1+e_2|\\mathbf {0},\\xi ) \\right], \\\\\\widehat{P}(e_2|\\mathbf {0},\\xi ) & =& \\frac{1}{2d}\\left[\\widehat{P}(\\mathbf {0}|\\mathbf {0},\\xi )+\\widehat{P}(2e_2|\\mathbf {0},\\xi )+\\widehat{P}(2e_1|\\mathbf {0},\\xi )+2(d-2) \\widehat{P}(e_2+e_3|\\mathbf {0},\\xi ) \\right], $ that are obtained from the generic relation $\\widehat{P}(r|r_0;\\xi )=\\delta _{r,r_0}+\\frac{\\xi }{2d} \\sum _{\\mu } \\widehat{P}(r|r_0 +e_\\mu ;\\xi )$ [32].",
"We introduce the quantities $\\alpha =\\lim _{\\xi \\rightarrow 1 }[\\widehat{P}(\\mathbf {0}|\\mathbf {0};\\xi )-\\widehat{P}(2e_1|\\mathbf {0};\\xi )]$ and $\\beta =\\lim _{\\xi \\rightarrow 1 }[\\widehat{P}(\\mathbf {0}|\\mathbf {0};\\xi )-\\widehat{P}(2e_1|\\mathbf {0};\\xi )]$ .",
"At leading order when $s \\rightarrow 0$ , one can replace the propagators $\\widehat{P}_{r}$ by $G_0(1-s)$ , where the function $G_0$ is defined as the leading order term of the expansion of the propagators $\\widehat{P}(r|r_0;\\xi )$ when $\\xi \\rightarrow 1$ : $\\widehat{P}(r|r_0;\\xi ) \\underset{\\xi \\rightarrow 1}{=}G_0(\\xi ) +\\mathcal {O}(1).$ We deduce that at leading order when $s \\rightarrow 0$ , the system of Eqs.",
"(REF )-() may be written as $\\mathbf {M}(\\delta =0) \\begin{pmatrix}\\widehat{\\widetilde{g}}_1(s) \\\\\\widehat{\\widetilde{g}}_{-1}(s) \\\\\\widehat{\\widetilde{g}}_2(s)\\end{pmatrix} = 2d\\tau ^* (1-\\rho ) \\frac{p_1-p_{-1}}{1+\\frac{2d\\alpha }{2d-\\alpha } \\frac{\\tau ^*}{\\tau }(p_1+p_{-1})} \\frac{G_0(1-s)}{s}\\begin{pmatrix}1 \\\\1 \\\\1\\end{pmatrix},$ where $\\mathbf {M}(\\delta )$ is defined as $\\mathbf {M}=\\begin{pmatrix}-\\left[2dp_1\\frac{\\tau ^*}{\\tau }(\\alpha +2\\delta )+2d-\\beta -\\delta \\right]& 2dp_{-1}\\frac{\\tau ^*}{\\tau }(\\alpha +2\\delta )-\\alpha +\\beta -\\delta & \\alpha -2\\beta \\\\2dp_1\\frac{\\tau ^*}{\\tau }(\\alpha -2\\delta )-\\alpha +\\beta +\\delta & -\\left[2dp_{-1}\\frac{\\tau ^*}{\\tau }(\\alpha -2\\delta )+2d-\\beta +\\delta \\right]& \\alpha -2\\beta \\\\\\frac{1}{2(d-1)} \\left[ -8\\delta p_1 \\frac{\\tau ^*}{\\tau } (d-1)-2\\beta (d+1) \\right.",
"& \\frac{1}{2(d-1)} \\left[ -8\\delta p_1 \\frac{\\tau ^*}{\\tau } (d-1)-2\\beta (d+1) \\right.",
"& \\frac{1}{d-1} \\left[2\\beta (d+1)-2d^2-\\alpha \\right] \\\\\\hfill +2\\delta (d-1)+\\alpha -2\\beta +2d \\Big ] & -2\\delta (d-1)+\\alpha -2\\beta +2d \\Big ] &\\end{pmatrix},$ with $\\delta = d/L^{d-1}$ .",
"We finally obtain the expressions of $\\widehat{\\widetilde{g}}_1(s)$ and $\\widehat{\\widetilde{g}}_{-1}(s)$ in the limit $s\\rightarrow 0$ : $\\widehat{\\widetilde{g}}_{\\pm 1}(s) \\underset{s\\rightarrow 0}{\\sim } (1-\\rho ) \\frac{\\sigma \\tau ^*}{\\tau } \\frac{(p_1-p_{-1})(2d-\\alpha )\\left(\\alpha -2d-\\frac{4d\\alpha \\tau ^*}{\\tau } p_{\\mp 1}\\right)}{\\left[2d-\\alpha +\\frac{2d\\alpha \\tau ^*}{\\tau }(p_1+p_{-1})\\right]^2} \\frac{G_0(1-s)}{s}$ The evolution equations for $h_{r}(t)$ [31] can be treated in a similar fashion, to find $\\mathbf {M} \\begin{pmatrix}s \\widehat{h}_1(s) \\\\s\\widehat{h}_{-1}(s) \\\\s\\widehat{h}_2(s)\\end{pmatrix} = 2d\\frac{\\tau ^*}{\\tau } (1-\\rho ) (p_1-p_{-1}) \\begin{pmatrix}-\\alpha \\\\\\alpha \\\\0\\end{pmatrix},$ and $\\lim _{\\rho \\rightarrow 1} \\frac{\\widehat{h}_{1}(s)}{1-\\rho } \\underset{s\\rightarrow 0}{=} \\pm \\frac{1}{s} \\frac{\\frac{2d\\alpha \\tau ^*}{\\tau }(p_1-p_{-1})}{\\frac{2d\\alpha \\tau ^*}{\\tau }(p_1+p_{-1})+2d-\\alpha } .$ Recalling the expression of the second cumulant [Eq.",
"(3)] and considering its Laplace transform, we get $\\mathcal {L}\\left[\\frac{\\mathrm {d}}{\\mathrm {d}t} \\sigma ^2_X(t) \\right](s) \\underset{s\\rightarrow 0}{\\sim } 2(1-\\rho )\\frac{\\tau ^*\\sigma ^2}{\\tau }\\left[\\frac{p_1-p_{-1}}{1+\\frac{2d\\alpha }{2d-\\alpha }(p_1+p_{-1})}\\right]^2 \\frac{G_0(1-s)}{s}.$ Using the following expression of the functions $G_0(\\xi )$ [32], [35] $G_0(\\xi ) \\underset{\\xi \\rightarrow 1}{\\sim }\\begin{dcases}\\frac{\\sqrt{d/2}}{L^{d-1}\\sqrt{1-\\xi }} & \\text{for a capillary of dimension $d$}, \\\\\\frac{1}{\\pi } \\ln \\frac{1}{1-\\xi } & \\text{for a two-dimensional lattice},\\end{dcases}$ and taking the inverse Laplace transform of Eq.",
"(REF ), we finally derive the result presented in the main text [Eq.",
"(9)]."
],
[
"Stationary state – derivation of Eq. (10)",
"In the limit where $\\rho \\rightarrow 1$ , at leading order in $(1-\\rho )$ , and in the stationary state we find that Eq.",
"(7) reduces to $2d\\widetilde{g}_{r} =&& \\left\\lbrace \\sum _\\mu \\widetilde{g}_\\mu \\nabla _{-\\mu } +\\frac{2d\\tau ^*}{\\tau }(p_1\\widetilde{g}_{1}-p_{-1}\\widetilde{g}_{-1})(\\nabla _1-\\nabla _{-1}) \\right.\\nonumber \\\\&& \\left.- \\frac{2d\\tau ^*}{\\tau } \\sigma \\left\\lbrace p_1(1-\\rho -h_{1})(\\nabla _1+1)-p_{-1}(1-\\rho -h_{-1})(\\nabla _{-1}+1)]\\right\\rbrace \\right\\rbrace \\mathcal {F}_{r}.",
"$ In this limit, evaluating Eq.",
"(REF ) for $r=e_1,e_{-1}$ and $e_2$ , we find that $\\widetilde{g}_1$ , $\\widetilde{g}_{-1}$ and $\\widetilde{g}_2$ are the solutions of the linear system $2d \\widetilde{g}_1 &=& \\widetilde{g}_1 \\nabla _{-1}\\mathcal {F}_{e_1}+\\widetilde{g}_{-1} \\nabla _{1}\\mathcal {F}_{e_1}+2(d-1)\\widetilde{g}_{2}\\nabla _2\\mathcal {F}_{e_1}+\\frac{2d\\tau ^*}{\\tau }(p_1\\widetilde{g}_1-p_{-1}\\widetilde{g}_{-1})\\left[\\nabla _1\\mathcal {F}_{e_1} - \\nabla _{-1}\\mathcal {F}_{e_1}\\right] \\nonumber \\\\&&-\\frac{2d\\tau ^*}{\\tau }\\sigma [p_1(1-\\rho -h_1)\\mathcal {F}_{2e_{1}}-p_{-1}(1-\\rho -h_{-1})\\mathcal {F}_{\\mathbf {0}}],\\\\2d \\widetilde{g}_{-1} &=& \\widetilde{g}_1 \\nabla _{-1}\\mathcal {F}_{e_{-1}}+\\widetilde{g}_{-1} \\nabla _{1}\\mathcal {F}_{e_{-1}}+2(d-1)\\widetilde{g}_{2}\\nabla _2\\mathcal {F}_{e_{-1}}+\\frac{2d\\tau ^*}{\\tau }(p_1\\widetilde{g}_1-p_{-1}\\widetilde{g}_{-1})\\left[\\nabla _1\\mathcal {F}_{e_{-1}} - \\nabla _{-1}\\mathcal {F}_{e_{-1}}\\right]\\nonumber \\\\&&-\\frac{2d\\tau ^*}{\\tau }\\sigma [p_1(1-\\rho -h_1)\\mathcal {F}_{\\mathbf {0}}-p_{-1}(1-\\rho -h_{-1})\\mathcal {F}_{2e_{-1}}],\\\\2d \\widetilde{g}_{2} &=& \\widetilde{g}_1 \\nabla _{-1}\\mathcal {F}_{e_{2}}+\\widetilde{g}_{-1} \\nabla _{1}\\mathcal {F}_{e_{2}}+\\widetilde{g}_{2}\\left[\\nabla _{-2}\\mathcal {F}_{e_{2}}+\\nabla _{2}\\mathcal {F}_{e_{2}}\\right]+2(d-2)\\widetilde{g}_2\\nabla _3\\mathcal {F}_{e_2} \\nonumber \\\\&&+\\frac{2d\\tau ^*}{\\tau }(p_1\\widetilde{g}_1-p_{-1}\\widetilde{g}_{-1})\\left[\\nabla _1\\mathcal {F}_{e_{2}} - \\nabla _{-1}\\mathcal {F}_{e_{2}}\\right] \\nonumber \\\\&&-\\frac{2d\\tau ^*}{\\tau }\\sigma [p_1(1-\\rho -h_1)\\mathcal {F}_{e_2+e_1}-p_{-1}(1-\\rho -h_{-1})\\mathcal {F}_{e_2+e_{-1}}].$ In what follows, we study the $\\rho \\rightarrow 1$ limit of these equations in the two situations where the lattice is a generalized capillary, and where it is a two-dimensional lattice."
],
[
"Generalized capillaries",
"We will use the relation $\\lim _{\\rho _0 \\rightarrow 0} \\nabla _\\nu \\mathcal {F}_{r} = \\lim _{\\xi \\rightarrow 1} \\left[ \\widehat{P}(r+e_\\nu |\\mathbf {0};\\xi ) - \\widehat{P}(r|\\mathbf {0};\\xi ) \\right]+\\delta _\\nu .$ with $\\delta _\\nu = {\\left\\lbrace \\begin{array}{ll}-\\frac{d}{L^{d-1}} & \\text{if $\\nu =1$}, \\\\\\frac{d}{L^{d-1}} & \\text{if $\\nu =-1$}, \\\\0& \\text{otherwise}.\\end{array}\\right.",
"}$ Contrary to the equation verified by the quantities $(h_1,h_{-1},h_2)$ [Eq.",
"(6)] which only involves differences of the functions $\\mathcal {F}_{r}$ , the system (REF )-() also involves functions $\\mathcal {F}_{r}$ alone, which diverge when $\\rho \\rightarrow 1$ .",
"In the case of generalized capillaries, we find that $\\mathcal {F}_{r} \\underset{\\rho \\rightarrow 1}{\\sim }\\frac{1}{L^{d-1} V} \\equiv \\mathcal {G}(V),$ where $V$ is the velocity of the tracer [39], [30]: $V \\underset{\\rho \\rightarrow 1}{\\sim } \\frac{\\sigma }{\\tau } \\left[ p_1(\\rho -1-h_1)-p_{-1}(\\rho -1-h_{-1}) \\right],$ and vanishes when $\\rho \\rightarrow 1$ .",
"Using Eqs.",
"(REF ) and (REF ), we simplify Eqs.",
"(REF )-().",
"With the usual symmetry properties on the quantities $\\widehat{P}$ [Eqs.",
"(REF )-()] as well as the relations (REF )-(), we rewrite Eqs.",
"(REF )-() in terms of the propagators $\\widehat{P}(\\mathbf {0}|\\mathbf {0};\\xi )$ , $\\widehat{P}(2e_1|\\mathbf {0};\\xi )$ and $\\widehat{P}(e_1|\\mathbf {0};\\xi )$ only.",
"We finally show that $\\widetilde{g}_1$ , $\\widetilde{g}_{-1}$ and $\\widetilde{g}_2$ are the solutions of the linear system $\\mathbf {M}\\begin{pmatrix}\\widetilde{g}_1 \\\\\\widetilde{g}_{-1} \\\\\\widetilde{g}_2\\end{pmatrix} = 2d\\tau ^* V (1-\\rho ) (p_1-p_{-1}) \\mathcal {G}(V)\\begin{pmatrix}1 \\\\1 \\\\1\\end{pmatrix},$ where $\\mathbf {M}$ is defined as previously [Eq.",
"(REF )].",
"At leading order in $\\rho \\rightarrow 1$ , this system has the following solutions for $\\widetilde{g}_{\\pm 1}$ : $\\widetilde{g}_{\\pm 1} \\underset{\\rho \\rightarrow 1}{\\sim } \\frac{\\sigma \\tau ^*}{L^{d-1}} \\frac{\\alpha -2d-\\frac{4d\\alpha \\tau ^*}{\\tau }p_{\\mp 1}}{\\left[2d-\\alpha +\\frac{2d\\alpha \\tau ^*}{\\tau }(p_1+p_{-1})+\\frac{4d^2}{L^{d-1}}\\frac{\\tau ^*}{\\tau }(p_1-p_{-1})\\right]^2}.$ With a similar treatment of Eq.",
"(6), we find that $h_1$ , $h_{-1}$ and $h_2$ are the solution of the linear set of equations $\\mathbf {M}\\begin{pmatrix}h_1 \\\\h_{-1} \\\\h_2\\end{pmatrix} = 2d \\frac{\\tau ^*}{\\tau } (1-\\rho ) (p_1-p_{-1})\\begin{pmatrix}-\\alpha -2\\delta \\\\\\alpha -2\\delta \\\\-2\\delta \\end{pmatrix}.$ We deduce the expressions of $h_{\\pm 1}$ : $h_{\\pm 1} = \\pm (1-\\rho ) \\frac{\\frac{2d\\tau ^*}{\\tau }(\\alpha +\\frac{2d}{L^{d-1}})(p_1-p_{-1})}{\\frac{2d\\tau ^*}{\\tau }\\alpha (p_1+p_{-1})+\\frac{4d^2}{L^{d-1}}\\frac{\\tau ^*}{\\tau }(p_1-p_{-1})+ 2d-\\alpha }$ Using Eq.",
"(3), this yields the result presented in the main text [Eq.",
"(10)]."
],
[
"Two-dimensional lattice",
"Extending the relation (REF ) to an infinite two-dimensional lattice by taking $d=2$ and $L\\rightarrow \\infty $ , we get: $\\lim _{\\rho _0 \\rightarrow 0} \\nabla _\\nu \\mathcal {F}_{r} = \\lim _{\\xi \\rightarrow 1} \\left[ \\widehat{P}(r+e_\\nu |\\mathbf {0};\\xi ) - \\widehat{P}(r|\\mathbf {0};\\xi ) \\right].$ We can also show that, for the case of a two dimensional lattice, $\\mathcal {F}_{r} \\underset{\\rho _0 \\rightarrow 0}{\\sim }\\frac{2}{\\pi } \\ln \\frac{1}{V},$ where $V$ is the velocity of the TP [Eq.",
"(REF )].",
"Using again the symmetry relations presented above [Eqs.",
"(REF )-()], we find that $\\widetilde{g}_1$ , $\\widetilde{g}_{-1}$ and $\\widetilde{g}_2$ are the solutions of the system (REF ), where we take $\\mathcal {G}(V)=\\frac{2}{\\pi } \\ln \\frac{1}{V}$ .",
"We obtain the following solutions of the system at leading order in $\\rho \\rightarrow 1$ : $\\widetilde{g}_{\\pm 1} \\underset{\\rho \\rightarrow 1}{\\sim } (1-\\rho ) \\frac{\\tau ^*}{\\tau }\\sigma \\frac{(\\alpha -4)(p_1-p_{-1})\\left(8\\alpha \\frac{\\tau ^*}{\\tau }+4-\\alpha \\right)}{4-\\alpha +4\\alpha \\frac{\\tau ^*}{\\tau }(p_1+p_{-1})} \\frac{2}{\\pi } \\ln \\frac{1}{V}$ and deduce the expression of the diffusion coefficient presented in the main text in Eq.",
"(10)."
],
[
"Low-density expansion",
"We start from the expression of the diffusion coefficient in terms of the density profiles $k_{r}$ and the cross-correlation functions $\\widetilde{g}_{r}$ [Eq.",
"(3)].",
"In the $\\rho \\rightarrow 0$ limit, we define the functions $v_{r}$ and $u_{r}$ as follows: $k_{r} \\underset{\\rho \\rightarrow 0}{=} (1+v_{r})\\rho + \\mathcal {O}(\\rho ^2) \\qquad \\qquad ; \\qquad \\qquad \\widetilde{g}_{r} \\underset{\\rho \\rightarrow 0}{=} u_{r}\\rho + \\mathcal {O}(\\rho ^2) ,$ so that equation diffusion coefficient reads $D \\underset{\\rho \\rightarrow 0}{=} \\frac{\\sigma ^2}{2 d \\tau } (p_1+p_{-1}) - \\frac{\\sigma }{2 d \\tau } \\rho \\lbrace p_1 [\\sigma (1+v_1)+2 u_1]+p_{-1}[\\sigma (1+v_{-1})-2 u_{-1}] \\rbrace + \\mathcal {O}(\\rho ^2) .$ The functions $v_{\\nu }$ have been studied before in the low-density limit [30], and are the solutions of the system: $\\widetilde{D} \\widetilde{v} = (p_1-p_{-1}) \\widetilde{\\mathcal {F}},$ with $\\widetilde{v}= \\begin{pmatrix}v_1 \\\\v_{-1} \\\\v_2\\end{pmatrix}\\qquad \\qquad ;\\qquad \\qquad \\widetilde{\\mathcal {F}}= \\begin{pmatrix}(\\nabla _1-\\nabla _{-1}) \\mathcal {F}_{e_1} \\\\(\\nabla _1-\\nabla _{-1}) \\mathcal {F}_{e_{-1}} \\\\(\\nabla _1-\\nabla _{-1}) \\mathcal {F}_{e_2}\\end{pmatrix},$ and $\\widetilde{D}=\\frac{1}{2d \\left( 1+\\frac{\\tau ^*}{\\tau } \\right)} \\begin{pmatrix}\\left( 1 + \\frac{2d\\tau ^*}{\\tau } p_{1}\\right) \\nabla _{-1} \\mathcal {F}_{e_1} -1& \\left( 1 + \\frac{2d\\tau ^*}{\\tau } p_{-1}\\right)\\nabla _{1} \\mathcal {F}_{e_1} & 2 \\left( 1 + \\frac{2d\\tau ^*}{\\tau } p_{2}\\right)\\nabla _{2} \\mathcal {F}_{e_1} \\\\\\left( 1 + \\frac{2d\\tau ^*}{\\tau } p_{1}\\right) \\nabla _{-1} \\mathcal {F}_{e_{-1}} & \\left( 1 + \\frac{2d\\tau ^*}{\\tau } p_{-1}\\right)\\nabla _{1} \\mathcal {F}_{e_{-1}}-1 & 2 \\left( 1 + \\frac{2d\\tau ^*}{\\tau } p_{2}\\right) \\nabla _{2} \\mathcal {F}_{e_{-1}} \\\\\\left( 1 + \\frac{2d\\tau ^*}{\\tau } p_{1}\\right)\\nabla _{-1} \\mathcal {F}_{e_2} & \\left( 1 + \\frac{2d\\tau ^*}{\\tau } p_{-1}\\right)\\nabla _{1} \\mathcal {F}_{e_2} & \\left( 1 + \\frac{2d\\tau ^*}{\\tau } p_{2}\\right) (\\nabla _{2} +\\nabla _{-2}) \\mathcal {F}_{e_2} -1\\end{pmatrix}.$ In order to calculate the functions $u_{r}$ , we start from Eq.",
"(7), and use the following expansions at leading order in $\\rho $ : $\\mathcal {A} \\sim 2d \\left(1+\\frac{\\tau ^*}{\\tau }\\right) \\quad ;\\quad {A}_\\nu \\sim 1+\\frac{2d\\tau ^*}{\\tau }p_\\nu \\quad ;\\quad h_\\mu \\sim v_\\mu \\rho .$ We then find, at leading order in $\\rho $ , the equation reduces to $u_{r} = &\\frac{1}{2d\\left(1+\\frac{\\tau ^*}{\\tau }\\right)} \\left\\lbrace \\sum _{\\mu } \\left(1+\\frac{2 d \\tau ^*}{\\tau }p_\\mu \\right) u_\\mu \\nabla _{-\\mu } \\mathcal {F}_{r} - \\frac{2d\\tau ^*}{\\tau } \\sigma [p_1(\\nabla _1+1+v_1)-p_{-1}(\\nabla _{-1}+1+v_{-1})] \\mathcal {F}_{r} \\right\\rbrace \\nonumber \\\\&+ \\frac{\\sigma \\frac{\\tau ^*}{\\tau }}{2d \\left(1+\\frac{\\tau ^*}{\\tau }\\right)^2} \\left\\lbrace \\sum _\\mu \\left(1+\\frac{2 d \\tau ^*}{\\tau }p_\\mu \\right) v_\\mu \\nabla _{-\\mu } - \\frac{2d\\tau ^*}{\\tau } (p_1-p_{-1}) (\\nabla _1-\\nabla _{-1}) \\right\\rbrace (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{r}.$ Bringing together the terms involving $u_{r}$ on the one hand and $v_{r}$ on the other hand, we evaluate this equation for $r=e_1$ , $e_{-1}$ and $e_2$ and obtain a closed set of three equations that we recast under the matrix form $\\widetilde{A}\\widetilde{u}=\\widetilde{C}-\\widetilde{B}\\widetilde{v}$ with $\\widetilde{u}= \\begin{pmatrix}u_1 \\\\u_{-1} \\\\u_2\\end{pmatrix}$ and $\\widetilde{A}= \\frac{1}{2d\\left(1+\\frac{\\tau ^*}{\\tau }\\right)} \\begin{pmatrix}\\left(1+\\frac{2d\\tau ^*}{\\tau }p_{1}\\right) \\nabla _{-1} \\mathcal {F}_{e_1} -1& \\left(1+\\frac{2d\\tau ^*}{\\tau }p_{-1}\\right) \\nabla _{1} \\mathcal {F}_{e_1} & 2 \\left(1+\\frac{2d\\tau ^*}{\\tau }p_{2}\\right) \\nabla _{2} \\mathcal {F}_{e_1} \\\\\\left(1+\\frac{2d\\tau ^*}{\\tau }p_{1}\\right) \\nabla _{-1} \\mathcal {F}_{e_{-1}} & \\left(1+\\frac{2d\\tau ^*}{\\tau }p_{-1}\\right) \\nabla _{1} \\mathcal {F}_{e_{-1}}-1 & 2\\left(1+\\frac{2d\\tau ^*}{\\tau }p_{2}\\right) \\nabla _{2} \\mathcal {F}_{e_{-1}} \\\\\\left(1+\\frac{2d\\tau ^*}{\\tau }p_{1}\\right) \\nabla _{-1} \\mathcal {F}_{e_2} & \\left(1+\\frac{2d\\tau ^*}{\\tau }p_{-1}\\right)\\nabla _{1} \\mathcal {F}_{e_2} &\\left(1+\\frac{2d\\tau ^*}{\\tau }p_{2}\\right) (\\nabla _{2} +\\nabla _{-2}) \\mathcal {F}_{e_2} -1\\end{pmatrix},$ $\\widetilde{B}= \\frac{\\sigma \\frac{\\tau ^*}{\\tau }}{1+\\frac{\\tau ^*}{\\tau }}\\begin{pmatrix}\\frac{1+\\frac{2d\\tau ^*}{\\tau }p_{1}}{2d\\left( 1+ \\frac{\\tau ^*}{\\tau } \\right)} \\nabla _{-1} (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{e_1} -p_1 \\mathcal {F}_{e_1}& \\frac{1+\\frac{2d\\tau ^*}{\\tau }p_{-1}}{2d\\left( 1+ \\frac{\\tau ^*}{\\tau } \\right)} \\nabla _{1} (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{e_1} +p_{-1} \\mathcal {F}_{e_1}&2\\frac{1+\\frac{2d\\tau ^*}{\\tau }p_{2}}{2d\\left( 1+ \\frac{\\tau ^*}{\\tau } \\right)} \\nabla _{2} (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{e_1} \\\\\\frac{1+\\frac{2d\\tau ^*}{\\tau }p_{1}}{2d\\left( 1+ \\frac{\\tau ^*}{\\tau } \\right)} \\nabla _{-1} (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{e_{-1}} -p_1 \\mathcal {F}_{e_{-1}}& \\frac{1+\\frac{2d\\tau ^*}{\\tau }p_{-1}}{2d\\left( 1+ \\frac{\\tau ^*}{\\tau } \\right)} \\nabla _{1} (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{e_{-1}} +p_{-1} \\mathcal {F}_{e_{-1}}&2\\frac{1+\\frac{2d\\tau ^*}{\\tau }p_{2}}{2d\\left( 1+ \\frac{\\tau ^*}{\\tau } \\right)} \\nabla _{2} (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{e_{-1}} \\\\\\frac{1+\\frac{2d\\tau ^*}{\\tau }p_{1}}{2d\\left( 1+ \\frac{\\tau ^*}{\\tau } \\right)} \\nabla _{-1} (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{e_2} -p_1 \\mathcal {F}_{e_2}& \\frac{1+\\frac{2d\\tau ^*}{\\tau }p_{-1}}{2d\\left( 1+ \\frac{\\tau ^*}{\\tau } \\right)} \\nabla _{1} (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{e_2} +p_{-1} \\mathcal {F}_{e_2}&\\frac{1+\\frac{2d\\tau ^*}{\\tau }p_{2}}{2d\\left( 1+ \\frac{\\tau ^*}{\\tau } \\right)} (\\nabla _{2}+\\nabla _{-2}) (p_1\\nabla _1-p_{-1}\\nabla _{-1}) \\mathcal {G}_{e_2} \\\\\\end{pmatrix},$ $\\widetilde{C}=\\begin{pmatrix}\\frac{\\frac{\\tau ^*}{\\tau }}{1+ \\frac{\\tau ^*}{\\tau } } \\sigma [p_1 (\\nabla _1+1)-p_{-1}(\\nabla _{-1}+1)] \\mathcal {F}_{e_1} +\\left( \\frac{\\frac{\\tau ^*}{\\tau }}{1+ \\frac{\\tau ^*}{\\tau } }\\right)^2 \\sigma (p_1-p_{-1}) (\\nabla _1-\\nabla _{-1}) (p_1 \\nabla _1 -p_{-1} \\nabla _{-1}) \\mathcal {G}_{e_1} \\\\\\frac{\\frac{\\tau ^*}{\\tau }}{1+ \\frac{\\tau ^*}{\\tau } } \\sigma [p_1 (\\nabla _1+1)-p_{-1}(\\nabla _{-1}+1)] \\mathcal {F}_{e_{-1}} +\\left( \\frac{\\frac{\\tau ^*}{\\tau }}{1+ \\frac{\\tau ^*}{\\tau } }\\right)^2 \\sigma (p_1-p_{-1}) (\\nabla _1-\\nabla _{-1}) (p_1 \\nabla _1 -p_{-1} \\nabla _{-1}) \\mathcal {G}_{e_{-1}}\\\\\\frac{\\frac{\\tau ^*}{\\tau }}{1+ \\frac{\\tau ^*}{\\tau } } \\sigma [p_1 (\\nabla _1+1)-p_{-1}(\\nabla _{-1}+1)] \\mathcal {F}_{e_2} +\\left( \\frac{\\frac{\\tau ^*}{\\tau }}{1+ \\frac{\\tau ^*}{\\tau } }\\right)^2 \\sigma (p_1-p_{-1}) (\\nabla _1-\\nabla _{-1}) (p_1 \\nabla _1 -p_{-1} \\nabla _{-1}) \\mathcal {G}_{e_2}\\\\\\end{pmatrix}.$ It is easy to take the limit of fixed obstacles in these equations ($\\tau ^*\\rightarrow \\infty $ ) to retrieve the results concerning the Lorentz gas.",
"In this particular limit, we notice $\\lim _{\\tau ^*\\rightarrow \\infty } \\widetilde{A}=\\widetilde{D}$ .",
"Finally, $u_1$ and $u_{-1}$ are obtained with $\\begin{pmatrix}u_1 \\\\u_{-1} \\\\u_2\\end{pmatrix} = \\widetilde{A}^{-1}(\\widetilde{C}-\\widetilde{B}\\widetilde{v}),$ and are used to calculate the diffusion coefficient using Eq.",
"(REF )."
]
] | 1709.01767 |
[
[
"Optical and radio properties of extragalactic radio sources with\n recurrent jet activity"
],
[
"Abstract We present a sample of 74 radio sources with recurrent jet activity.",
"The sample consists of 67 galaxies, 2 quasars and 5 unidentified sources, selected from the published data or are newly recognized.",
"The sample's redshift range is 0.002 < z < 0.7 and the size of inner and outer structures varies from 0.02 to 4248 kpc.",
"We analyse the optical and radio properties of the sample and compare them with the characteristics of ordinary one-off FRII radio sources.",
"With the help of stellar population modelling, we derive black hole masses and stellar masses of host galaxies of 35 restarting radio sources, finding that the black hole masses in restarting radio sources are comparable to those of typical single-cycle FRII radio sources.",
"The obtained median values of log M$_{BH}$ are 8.58 and 8.62 M$_{\\odot}$ Unlike the black hole masses, the stellar masses in restarting radio sources tend to be smaller than in the FRII sources.",
"Although the stellar populations of the hosts of recurrent activity sources are dominated by old stars, a significant fraction of young stars can be observed as well.",
"Based on the Sloan Digital Sky Survey photometric observations, we also analyse the morphology of the host galaxies and obtained significantly smaller concentration indices for the restarting radio sources when compared to the classical FRII hosts.",
"This effect can be interpreted as a result of frequent merger events in the history of host galaxies of restarting radio sources."
],
[
"Introduction",
"The classic extragalactic radio sources have been investigated for much over a half of century; for instance, [4] found an optical counterpart for the Cygnus A source as long ago as 1954).",
"The most widely assumed underlying model of radio sources comprises a fast rotating supermassive black hole (SMBH) accompanied by an accretion disk.",
"The transfer of disk energy to the external lobes is accomplished through powerful relativistic jets [80].",
"The jet activity phases are rather brief ($\\sim $ 10$^8$ yrs) in comparison with the entire life time of the host galaxy ($\\sim $ 10$^{10}$ yrs).",
"However, the activity of a jet can restart after a period of silence.",
"This phenomenon depends strictly on the physical conditions in the vicinity of the central SMBH.",
"The presence of sources of peculiar character, classified neither as FRI nor FRII ([30]), could indicate a recurring AGN activity.",
"The active galactic nuclei in general can be classified into a number of separate categories; among others, the radio galaxies with two or more pairs of lobes extending from the core along the same axis, called double-double radio sources (DDRS; [82]).",
"This group is rather sparse, with just about 45 objects recognized to date (for a review and references, see [73], [66]).",
"There are DDRSs of much different sizes, from merely scores of parsecs up to over one megaparsec.",
"Besides the axial symmetric sources there is another type of recurrent activity radio galaxies.",
"These are objects exhibiting large-scale diffuse relic radio emission that is due to an earlier cycle of activity.",
"Such relics can be seen around compact powerful young radio sources (e.g.",
"[7]).",
"A separate group are so-called `X-shaped' radio sources, which are also regarded as recurrent radio sources (e.g.",
"[61]; [75]).",
"We are not going to analyse these sources here, as their optical properties were studied already by [62], [63].",
"The fact that in some radio galaxies two or even three pairs of lobes can be observed implies that the time required for the jet flow to cease is shorter than that for the outer lobes to fade.",
"Thus, as the energy transported by the jets is able to last through the quiescent phase within their extended lobes, the radio galaxies can hold memory of the previous AGN activity.",
"[44] determined the duration of the inactive period for some DDRSs to be of the order of several million years.",
"Interestingly enough, this time is too long to be explained in the frame of the currently existing theoretical models that describe properties of an AGN (e.g.",
"accretion disk instability with the idle time of about 10$^4$ years; [28]).",
"On the other hand, it is too short for a new disk to form.",
"Therefore, alternative theories have been formulated.",
"The interruptions in the jet production mechanism could have been possibly caused by refuelling of the central engine.",
"Liu et al.",
"[51], [53], [52] suggested that this kind of objects may arise due to interactions of SMBH binaries with the accretion disk.",
"In such a picture, the secondary black hole (BH) drifts towards the centre, disrupting the inner parts of the accretion disk.",
"After the both black holes merge, the gap widens, effectively stopping the process of jet formation.",
"Such a behaviour could be expected in the case of the blazar OJ287 [94], which possesses two SMBHs, the smaller of which orbits the larger companion, surrounded by an accretion disk, with a period of about 11 years.",
"The jet activity may start again at a later time, triggered by new matter flowing into the inner region.",
"Consequently, a DDRS is created.",
"The accepted belief that the multiple pair of lobes are produced by recurrent jet activity was questioned by [98], who noted that such objects could also be produced by random emergence of radio plasma from an inhomogeneous, porous ISM.",
"However, in this scenario we cannot expect an ideally symmetric interstellar medium (ISM) on both sides of the central AGN.",
"This in turn would make it difficult to explain the existence of DDRSs with linearly symmetric inner lobes.",
"The AGNs with repeated jet activity could affect both the parent galaxy and the diffuse extended radio structure.",
"Therefore, some properties of the objects with multiple activity cycles can be different from the radio galaxies with single activity.",
"[98] found that there is a considerable transfer of energy and momentum to the ISM, and that the jets with powers of $10^{43}$ – $10^{46}$ erg s$^{-1}$ can inhibit star formation in the galactic core.",
"This would influence the evolution of the galaxy.",
"Therefore, the repeated jet activity could have implications for feedback processes of the parent galaxy.",
"[93] postulated that recurrent activity influences the linear size of radio sources.",
"Therefore, at least some of the giant radio galaxies could become quite large because their central AGNs experienced several jet activity cycles.",
"The jets of restarted cycles move not in the original intergalactic medium (IGM) but in a less dense environment of the primordial lobes, which allows them to reach further distances in a shorter time.",
"Nevertheless, the mechanism underlying formation of the inner lobes is not yet fully understood.",
"There are two plausible models suggested in the literature that could explain this.",
"In the first model, called the `classical FRII model', the inner lobes are formed in the same way as the lobes of typical `single-cycle' radio galaxies.",
"The inner lobes propagate through a denser medium than could be expected from purely synchrotron-emitting plasma, because thermally emitting material is drawn into the outer lobes of DDRSs during their growth and quiescent phases [41].",
"The second model, called the `bow-shock model', assumes that the inner lobes are created by reaccelerating of the outer cocoon particles at the bow-shock created by almost ballistically moving jet heads [15], [16], [72].",
"Both the `classical FRII' and the `bow-shock' models may need to be employed simultaneously to fully describe the dynamics and the structure of inner lobes.",
"Surprisingly enough, only recently it was realised that the jet activity can be a multiple phenomenon.",
"The main reason for this is that such objects are not too frequent among radio galaxies.",
"Furthermore, it is not to easy to recognize a restarting radio source, since the radio structures usually come from separate activity cycles and have significantly different surface brightnesses, angular extents, and spectral profiles.",
"The ideal instance of a DDRS would be an object with bright detached pairs of lobes but most often the structures produced by consecutive cycles merge together.",
"[68] selected a class of radio galaxies, which possess spectra that steepen outwards, from the core to the outer edge of lobes.",
"Most of them are narrow-angle-tail (NAT) or wide-angle-tail (WAT) radio galaxies, while within these two types of objects we can still find sources with multiple jet activity.",
"In order to distinguish them and understand the cycles and the phases of interruption of the jet flow, it is crucial to determine ages of charged particles in different regions within the radio lobes.",
"In order to fully recognize the recurrent activity phenomenon a statistical study of an extended sample is essential.",
"In this paper, we present the first systematic study of extragalactic radio sources with structures showing recurrent jet activity.",
"We analyse the optical and radio properties of a large sample of restarting radio sources and compare them with characteristics of ordinary FRII radio galaxies, in order to learn more about the causes underlying episodic jet activity.",
"In the next section, we briefly describe the source samples used for the analysis.",
"The observations, data reduction, and processing are described in the third section, while in the fourth section we review and discuss the results.",
"The conclusions are presented at the end of the paper.",
"Throughout the paper, a flat vacuum-dominated universe with $\\Omega $$\\rm _{m}$ =0.27, $\\Omega $$\\rm _{\\Lambda }$ =0.73 and $H$$\\rm _{0}$ =71 km s$^{-1}$ Mpc$^{-1}$ is assumed."
],
[
"The sample",
"The recent studies by [73] and [66] significantly extend the number of sources which show multiple jet activity.",
"It was noticed that multi-cycle jet activity is not just specific to giant objects, but is manifested also by smaller radio galaxies (with outer structures having sizes of only several hundred kiloparsecs).",
"Nevertheless, such objects are not too well studied and still a number of physical parameters, e.g.",
"the duration of an inactive phase, need better estimates.",
"Our sample of restarting radio sources contains 74 objects, including 67 galaxies, 2 quasars, and 5 unidentified sources.",
"The selection was based mostly on the published data, but we also included new sources, recognized recently by the authors of this paper.",
"The redshifts for our sample objects range from 0.002 to 0.7.",
"The only exception is J2107$+$ 2331, whose redshift is as high as z=2.48.",
"The radio structures show a wide range of projected linear sizes – from 0.02 to 876 kpc for the inner to 48 to 4248 kpc for the outer lobes.",
"In our sample there are 8 newly recognized restarting radio sources: J0042-0613, J0504+3806, J0914+1006, J0924+0602, J1004+5434, J1021+1216, J1520-0546, J1528+0544.",
"All of them were identified using maps from the 1.4 GHz Faint Images of the Radio Sky at Twenty centimeters (FIRST; [9]) survey and the NRAO VLA Sky Survey (NVSS; [27]).",
"The redshifts of these objects are within the range from z=0.04 to z=0.31.",
"A half of the newly recognized restarting radio sources are giants with projected linear sizes larger than 0.7 Mpc (taking into account the sizes of the outer radio lobes).",
"In most of the newly discovered sources the inner radio lobes have medium linear sizes, but there is one source J1528+0544 with a very compact (of 15 kpc) inner structure as well as one source J1021+1216 with very extended (876 kpc) inner doubles.",
"In our sample, we consider also two X-shaped radio galaxies, i.e.",
"J0009+1244 and J1513+2607.",
"While we do not analyse this type of sources in this paper in general, these two sources are of peculiar type.",
"Besides the usual X-shape morphology, they possess an additional pair of lobes in their central region.",
"The sample presented in this paper is not complete, yet it is quite representative of the class.",
"It covers a wide range of physical parameters, thus it is sufficiently large and diverse to allow for performing an analysis of their fundamental properties.",
"There are a lot of candidates in the literature for a restarting radio source with structures that still have to be confirmed as multi-episodic (e.g.",
"[66]).",
"Most of them either do not have optical identifications, or the resolution of the available radio maps is not sufficient to properly reveal their structure.",
"We also expect a number of new multi-cycle source detections in the next few years from deep survey observations with new low-frequency radio telescopes (e.g.",
"the Low Frequency Array; [96], the Long Wavelength Array; [29], and the Murchison Widefield Array; [54]).",
"We divided the radio sources from our sample according to their radio morphology into two classes: typical double-double radio sources, where two pairs of radio lobes are clearly visible (class A) and radio sources with prominent inner structure, surrounded by diffuse outer structures (class B).",
"We classified 63 radio sources studied in this paper as class A objects, while only 11 belong to the class B.",
"The restarting radio sources and their basic properties are listed in Table REF , where the newly recognized radio sources are denoted with boldface.",
"The radio maps are presented in Appendix A – class A radio sources and Appendix B – class B.",
"As a comparison sample, we used the FRII-type radio galaxies from [46], where the optical and radio data were provided.",
"This sample contains 401 FRII radio galaxies with a wide range of radio powers and sizes of radio structures.",
"Ten of the FRII radio galaxies from the comparison sample show a double-double radio structure, and therefore were not used.",
"Instead, they were included in the sample of restarting sources.",
"As a result, the final comparison sample consists of 391 sources.",
"Radio sources from both the samples are presented in Figure REF , where the relation between the distribution of the 1.4 GHz total radio luminosity and the redshift is plotted.",
"The source with exceptionally high redshift (J2107$+$ 2331) is not shown in this figure.",
"Figure: Sample of recurrent jet activity radio sources (black open and filled circles) and the comparison sample (blue asterisks) on the 1.4 GHz total radioluminosity–redshift plane.",
"The recurrent radio sources with available spectra are marked as open black circles."
],
[
"Observational data",
"In our analysis of recurrent jet activity sources we use both optical and radio data."
],
[
"Optical evidence",
"The optical data are taken mainly from the SDSS Data Release 10 (DR10; [1]).",
"The photometric data were available for 53 radio sources and the spectroscopic data were available for 30 objects.",
"Spectroscopic data of further four objects (J0301+3512, J1651+0459, J1844+4533, J2048+0701) were taken from the archives of the Telescopio Nazionale Galileo (TNG) and the 5-meter Hale telescope at the Palomar Observatory.",
"The observations and reduction procedures are described in [36] and [21].",
"Additional spectra for six objects were obtained with our dedicated spectroscopic observations using the South African Astronomical Observatory (SAAO) telescopes and the William Herschel Telescope (WHT).",
"Details of the WHT observations and data reduction are presented below.",
"Spectra of J1520$-$ 0546 and J1528$+$ 0544 were obtained with the 1.9-m SAAO telescope equipped with the Cassegrain Grating Spectrograph.",
"The optical spectra and description of observations are provided in [56].",
"WHT spectra: Spectral observations of four DDRGs, J0042$-$ 0613, J1835$+$ 6204, J2223$-$ 0206, and J2345$-$ 0449 were carried out with the WHT at the Roque de los Muchachos Observatory in La Palma, Spain.",
"The observations were obtained in the service mode on July 29th, 2009.",
"The long-slit spectra were taken using the Intermediate Dispersion Spectrograph and Imaging System Double-Armed Spectrograph, which permit simultaneous observations in both blue and red filters.",
"The slit of a width of 2was centred on the nuclei of the sources and positioned along the jet axis of the radio galaxy.",
"The integration time was split into several exposures to reduce the influence of cosmic rays.",
"Spectra of two spectrophotometric standards, i.e.",
"SP2011+065 and SP0031-124 were obtained for a proper intensity calibration.",
"Arc lamp spectra were taken before and/or after target observations to allow for an accurate wavelength calibration.",
"The average seeing during the observing run was about 1.2.",
"The WHT spectra of these restarting radio galaxies have not been published to date and we present them in Figure REF .",
"Figure: The optical WHT spectra of J0042--0613, J1835++6204, J2223--0206, and J2345--0449 restarting radio galaxies.The WHT data were reduced using the IRAF NOAO packageshttp://iraf.noao.edu.",
"Reduction steps were performed separately for each spectral range.",
"The master bias frame was created by averaging all the bias frames obtained during the observing night and subtracted from the science frames.",
"The master flat field frame was also created and all 2D-science frames were corrected for flat field.",
"Then, the cosmic rays were removed from the science exposures.",
"Wavelength calibration was performed using ArNe lamp exposures and verified by using sky lines.",
"Next, a correction for optical distortion was applied.",
"The contribution from the sky was determined from the sky regions at both sides of the resulting spectrum and subtracted.",
"The 1D-spectra extraction was performed using the APALL task.",
"The scientific exposures were flux-calibrated using exposures of the suitable spectrophotometric standard stars.",
"In order to obtain the resulting spectra with a better S/N ratio, all the calibrated one-dimensional spectra of a given galaxy were combined.",
"The spectroscopic data of good quality are available only for 37 galaxies of the 74 restarting radio sources.",
"Despite that, the galaxies with spectroscopic data are representative for the whole sample.",
"In Figure REF we marked the galaxies with available optical spectra.",
"It can be seen that the distribution of galaxies with spectra is uniform and spans a wide range of redshifts and total radio luminosities.",
"In Figure REF we plotted the distributions of SDSS r magnitudes for both the samples.",
"The sample of FRII radio galaxies contains only objects with optical spectra and is limited to r band magnitudes brighter than 22.",
"The distribution for restarting radio galaxies is more uniform and optical spectra are available for the brighter objects in sample, but the range of magnitudes for the galaxies with spectra is similar as for the comparison sample.",
"Figure: SDSS r magnitude distribution for FRII radio galaxies (top) and restarting radio galaxies (bottom).",
"The radio galaxies with available optical spectra are plotted as a solid grey boxes, both for the restarting sample and the comparison one."
],
[
"Radio maps",
"In our radio analysis we measured the angular sizes and flux densities of inner and outer radio lobes using mainly maps from the 1.4 GHz FIRST and the NVSS radio surveys.",
"For a few sources for which no data were available in NVSS/FIRST, or for which it was not possible to obtain precise measurements of angular sizes and flux densities using NVSS/FIRST, we took the relevant parameters from the literature given in Table REF ."
],
[
"Data analysis",
"The optical spectra of restarting radio sources have been used to measure the basic properties of the hosts of AGNs.",
"We applied the stellar population synthesis code STARLIGHThttp://www.astro.ufsc.br/starlightst [23] to model the observed spectrum.",
"Stellar continuum was fitted basing on the superposition of 150 stellar spectra templates (extracted from the evolutionary synthesis models of [18]) with various ages and metallicities.",
"Apart from continuum fitting, the STARLIGHT modelling gives the information about such parameters like star formation, chemical enrichment, and velocity dispersion.",
"To determine BH masses, we used the M$_{BH}$ - $\\sigma _*$ method.",
"It is based on a tight correlation between the BH mass and the velocity dispersion of the stars in the galactic bulge ($\\sigma _*$ ; [31], [33]), which is described by the relation $log\\rm (\\frac{M_{BH}}{M_{\\odot }})=\\alpha +\\beta log\\rm (\\frac{\\sigma _*}{\\rm 200 km s^{-1}})$ where the constants are $\\alpha $ = 8.13$\\pm $ 0.05 and $\\beta $ = 5.13$\\pm $ 0.34 ([35]).",
"For the two quasars (J0741$+$ 3112 and J0935$+$ 0204) we used their BH mass estimates based on the H$_\\beta $ mass scaling relation given by [85].",
"The radio luminosity of the inner ($P\\rm _{in}$ ) and outer ($P\\rm _{out}$ ) radio lobes was calculated using the following formula given by [17] $log P(WHz^{-1})=log S(mJy)-(1+\\alpha ) \\cdot log(1+z)$ $\\hspace{85.35826pt}+2log(D_L(Mpc))+17.08$ where $\\alpha $ is the spectral indexthroughout the paper we use the convention $S_{\\nu }\\sim \\nu ^{\\alpha }$ and $D_L$ is a luminosity distance.",
"The flux densities of the inner ($S_{\\rm in}$ ) and outer lobes ($S_{\\rm out}$ ) of individual sources were measured in the available maps and then listed in Table REF .",
"Following e.g.",
"[43], we adopted a typical spectral index value of $\\alpha =-0.75$ for the inner and $\\alpha =-1$ for the outer radio structures."
],
[
"Results",
"In this section, we compare some of the basic characteristics (radio morphology, physical parameters of host galaxy) of restarting radio galaxies with those of typical radio sources using the optical, radio and infrared data described above."
],
[
"Optical properties",
"Using the Starlight Synthesis Code, we modelled the stellar continuum in spectra of 35 galaxies from our sample (except for two quasars) for which spectroscopic data were available.",
"We obtained the information about a mixture of stellar populations present in a individual galaxies.",
"It can be expressed by the light-fraction population vector $x_j$ , which gives the percentage fraction of a galaxy light (luminosity) that comes from stars with a given age and metallicity ([22]).",
"According to [23], we derived the light-weighted average age $\\left<\\log t^{\\star }\\right>_L$ and the mass-weighted average age $\\left<\\log t^{\\star }\\right>_M$ , which are defined as: $\\left<\\log t^{\\star }\\right>_L=\\sum _{j=1}^{N} x_j \\log t_j \\hspace{14.22636pt} and \\hspace{14.22636pt} \\left<\\log t^{\\star }\\right>_M=\\sum _{j=1}^{N} \\mu _j \\log t_j$ respectively, where $x_j$ is the modelled light-fraction population vector, and $\\mu _j$ is the mass-fraction population vector obtained by using the model light-to-mass ratios.",
"By definition, the two weighted average ages are sensitive to different stellar populations: $\\left<\\log t^{\\star }\\right>_L$ is sensitive to the presence of young stellar populations, while $\\left<\\log t^{\\star }\\right>_M$ is sensitive to the less luminous and older stellar populations.",
"In Figures REF and REF , we compared the average ages $\\left<\\log t^{\\star }\\right>_L$ and $\\left<\\log t^{\\star }\\right>_M$ of the restarting radio galaxies and the comparison sample of typical FRII radio galaxies.",
"The observed distribution of the light-weighted age is slightly different for the restarting radio galaxies and FRIIs.",
"The $\\left<\\log t^{\\star }\\right>_L$ distribution for restarting radio galaxies peaks in the range 9.7 – 9.8, as for the comparison sample, but there is a large fraction of younger stellar populations as well.",
"The Kolmogorov-Smirnov two-sample test (K–S test) was performed to check whether the two samples reveal the same distribution.",
"We obtained the probability value p=0.001, indicating that the sample of restarting radio sources and the comparison sample have statistically different distributions of $\\left<\\log t^{\\star }\\right>_L$ .",
"In the case of mass-weighted age, the observed distributions are similar.",
"The K–S test for $\\left<\\log t^{\\star }\\right>_M$ returns a probability of 0.17 that the restarting and comparison sample radio sources have indistinguishable distributions.",
"The results indicate that while the stellar populations in the restarting radio galaxies do include old stars, similarly as in the FRII radio galaxies, they also show a considerable amount of young stars.",
"Figure: Distribution of light-weighted average age for all 35 restarting radio galaxies for which spectroscopic data are available (bottom panel), and single-cycle FRII radio galaxies (top panel).",
"The normalised number, N norm _{norm}, is a number of radio galaxies in a particular bin divided by the total number of sources in the entire sample.Figure: Distribution of mass-weighted average age for restarting radio galaxies (bottom panel), and for single-cycle FRII radio galaxies (top panel).In Figures REF and REF we plotted the galaxy mass M$^*$ (the mass of stars obtained with STARLIGHT modelling) as a function of the BH mass and the 1.4 GHz radio luminosity, respectively.",
"It can be seen that the restarting radio galaxies have lower masses of the host galaxy when compared with the FRII radio sources.",
"This may suggest that a larger fraction of galaxy mass in restarting sources is cumulated in gas and/or dust.",
"Since in elliptical galaxies the amount of gas and dust is not significant when compared to spiral galaxies, its presence can be an evidence of merger events occurring in the host galaxies of sources with restarting jet activity.",
"Figure: BH mass M BH \\rm _{BH} versus galaxy mass M * ^*.",
"The restarting radio galaxies are denoted as black dots and the FRII radio galaxies from the comparison sample as blue asterisks.Figure: Relation between radio luminosity and the galaxy masses M * ^* of the restarting and FRII galaxies.",
"The inner radio structures are marked as black dots, the outer structures as red dots, and the FRII radio galaxies from the comparison sample are marked as blue stars.",
"This labelling is used throughout further all similar figures in this paper.A very useful parameter for morphological classification of galaxies is the concentration index CI, defined as the ratio of two radii R90/R50, where R90 and R50 are the radii enclosing 90% and 50% of the r-band Petrosian flux, respectively.",
"According to [65], the CI value lower than 2.86 corresponds to late type galaxies and CI of more than 2.86 – to the early-type galaxies.",
"In Figure REF we present the distribution of concentration index for the samples of restarting and FRII radio galaxies.",
"It can be seen that the radio sources with recurrent activity tend to have smaller CI values than the radio galaxies from the comparison sample.",
"The difference in distribution of CI indices for restarting and comparison sample radio sources is statistically significant.",
"The K–S test returns a probability of 0.00004 with the maximal deviation between cumulative distributions of 0.34, which indicates that the two considered samples come from different distributions.",
"This can suggest that recurrent activity is observed more often in disturbed/amorphous galaxies.",
"Figure: Distribution of concentration indices (CIs) for the estarting radio sources and the comparison sample.Moreover, in almost all ($\\sim $ 89%) the available spectra of restarting radio sources the emission lines were visible.",
"Using our measurements of the line flux we constructed a diagnostic diagram proposed by [6] and plotted the ratios [OIII]/H$_\\beta $ and [NII]/H$_\\alpha $ .",
"In Figure REF we present diagnostic diagrams for the recurrent radio sources and the comparison sample of FRII galaxies.",
"The diagram used only emission lines with the signal-to-noise ratio of S/N$>$ 3.",
"There are 11 sources having [OIII]/H$_\\beta $$>$ 3 in our sample, which represents a high ionisation level.",
"The mean value of the [OIII]/H$_\\beta $ ratio for these sources is very high ($\\sim $ 6.6).",
"The remaining galaxies have lower [OIII]/H$_\\beta $ ratios with the mean value of 2.1.",
"The restarting radio galaxies are located in the same area on the diagram as the FRII sources, which can suggest that the host galaxies of both types of radio sources are similar.",
"There are no restarting radio sources near the solid line in Figure REF separating the HII galaxies from AGNs.",
"According to [89], the objects lying away from this line have emission lines excited mostly by an AGN source.",
"There are also no objects in the bottom part of the diagram, where the retired galaxies, i.e.",
"galaxies that have ceased forming stars and are ionized only by their old stellar populations [90], are located.",
"Figure: Diagnostic diagram: [OIII]//H β _\\beta vs. [NII]//H α _\\alpha for restarting radio galaxies (black dots) and FRII comparison sample (blue asterisks).",
"The solid line indicates the division between AGNs and star-forming galaxies according to .",
"In plotting the diagram we used only emission lines with S/N>>3.We also checked the relation between the radio luminosity and the luminosity of the [OIII] emission line.",
"It was postulated by [70] and [8] that the positive correlation can be an evidence for the physical coupling of processes that supply energy both to the emission line regions and to the extended radio structure.",
"The correlations between optical and radio properties are generally explained by the illumination model in which the gas responsible for line emission is photoionized by UV photons coming from the central AGN.",
"We plotted radio luminosity (separately for the inner and outer radio lobes) against the luminosity of the [OIII] emission line.",
"The P–L[OIII] relation is shown in Figure REF .",
"The linear fits to the data points are given by the following relations: $\\log L_{[OIII]}=(0.50\\pm 0.14)\\cdot \\log P_{in}-(4.87\\pm 3.36)$ $\\log L_{[OIII]}=(0.57\\pm 0.14)\\cdot \\log P_{out}-(7.03\\pm 3.62)$ and for the comparison sample: $\\log L_{[OIII]}=(0.93\\pm 0.08)\\cdot \\log P_{tot}-(16.6\\pm 2.1)$ The correlation coefficients for the inner radio structure, outer radio structure and the comparison sample are 0.54, 0.58 and 0.68, respectively.",
"Figure: Radio luminosity at 1.4 GHz versus [OIII]5007 line luminosity.",
"For the restarting radio sources the radio luminosities of inner (black dots) and outer (red dots) radio lobes are plotted, while for the comparison sample (blue asterisks) the total radio luminosities are shown.",
"The black and red line corresponds to the best fit for inner and outer radio lobes respectively, while the blue line corresponds to the best fit obtained for the FRII radio galaxies by .For the H$_\\alpha $ line we obtained the following relations: $\\log L_{H_{\\alpha }}=(0.38\\pm 0.12)\\cdot \\log P_{in}-(2.05\\pm 2.99)$ $\\log L_{H_{\\alpha }}=(0.46\\pm 0.13)\\cdot \\log P_{out}-(4.29\\pm 3.27)$ and for comparison sample: $\\log L_{H_{\\alpha }}=(0.80\\pm 0.06)\\cdot \\log P_{tot}-(13.4\\pm 1.7)$ The correlation coefficients for the inner radio structure, the outer radio structure and the comparison sample are 0.53, 0.59 and 0.73, respectively.",
"The slopes of the fitted lines of the inner and outer radio lobes are slightly flatter than those obtained for the FRII radio galaxies.",
"Figure: Radio luminosity at 1.4 GHz versus H α _\\alpha line luminosity.",
"The symbols are as in Figure ."
],
[
"Radio properties",
"In Figure REF we plotted the luminosity–linear size relation (P–D diagram) for the sample of restarting and single-cycle FRII radio sources.",
"Most restarting radio sources occupy the same region as the sources from the comparison sample.",
"It can be also observed that the outer doubles tend to be larger than the lobes of FRII radio galaxies.",
"This could be an observational bias resulting from the fact that it is easier to detect restarting radio sources of larger size.",
"On the P–D diagram there are also few very compact (D$<$ 1kpc) restarting radio sources with high radio luminosities (log P$>$ 26[W/Hz]).",
"According to the dynamical and luminosity evolution models of radio galaxies, three distinct evolutionary phases are to be expected.",
"In the first one, when the lobes are still expanding within the host galaxy, the radio luminosity increases with the source size, which stops as soon as the synchrotron losses become dominant (at the size of about 1 kpc).",
"Beyond this point, the radio luminosity steadily decreases with the increasing source size (up to $\\sim $ 100 kpc) and finally enters the phase of sharply decreasing of luminosity when the inverse Compton losses, resulting from the CMB energy density, dominate the synchrotron losses ([42], [2]).",
"Figure: Luminosity–linear size diagrams.",
"For the comparison sample we used total radio luminosity and for the restarting sources we plotted luminosity both for the inner (P in _{in}) and the outer (P out _{out}) structures.As shown in Figure REF , there is a significant correlation between luminosity of the outer and inner lobes.",
"The correlation coefficient of a linear fit is 0.59.",
"Figure: Relation between 1.4 GHz radio luminosity of the inner and the outer radio lobes.In our sample of restarting radio galaxies we found 14 sources, whose inner lobes are more luminous than the outer lobes (i.e.",
"J0041$+$ 3224, J0111$+$ 3906, J0301$+$ 3512, J0301$+$ 3512, J0741$+$ 3112, J0821$+$ 2117, J0840$+$ 2949, J0910$+$ 0345, J0914$+$ 1006, J0943$-$ 0819, J1021$+$ 1216, J1247$+$ 6723 J1352$+$ 3126 and J1844$+$ 4533).",
"This group of sources is very peculiar and we denoted them by an asterisk in Table REF .",
"Most of them have very compact inner radio structures, but some sources have larger inner lobes (over 100 kpc).",
"It can also be seen that the smaller is the ratio of luminosities (logP$_{out}$ /logP$_{in}$ ), the smaller inner radio structure is observed (in log scales; Fig.",
"REF ).",
"Such a trend is found particularly for the luminosity ratios of less than one.",
"For these sources we obtained positive correlation with a coefficient equal to 0.73.",
"The higher luminosity of the inner lobes when compared to that of the outer lobes is explained by [74] as a result of more efficient radio emission in the early phase of evolution of the inner lobes when they expand within a dense interstellar medium.",
"As the source expands and traverses a more diluted medium, the ratio P$_{out}$ /P$_{in}$ usually increases with the size of inner lobes before approaching values of the order of unity.",
"[82] postulated the inverse correlation between P$_{out}$ /P$_{in}$ and D$_{in}$ (basing on studies of 7 DDRSs that have the inner doubles larger than 90 kpc).",
"Our results do not confirm this trend but are in agreement with the results obtained by [74], who found that P$_{out}$ /P$_{in}$$<$ 1 in the smallest($<$ 1kpc) inner doubles.",
"They also concluded that the inverse correlation postulated by [82] has a reduced level of significance.",
"According to [66], there could be only an upper envelope to this diagram, suggesting an inverse relation.",
"The small number of sources on the left side of Figure REF can be caused by the selection effects.",
"Resolving very compact inner doubles that are smaller than 10 kpc is possible only with high-resolution radio observations.",
"Figure: Ratio of 1.4 GHz radio luminosity of outer and inner lobes against the projected linear size of inner radio lobes.The correlation coefficient is R=0.54.Furthermore, we also found that nearly half of the restarting radio galaxies have brighter inner and outer radio lobes on the same side of the host galaxy (i.e.",
"out of 41 radio galaxies for which the inner and outer lobes are well detached, the brighter inner and outer lobes are on the same side in 18 galaxies, and 23 galaxies show brighter inner and outer lobes on the opposite sides of the host galaxy).",
"A similar trend is observed, when the length of the radio lobes is considered – 22 and 19 sources, respectively.",
"These findings can be explained as resulting from the combination of orientation of a radio source and an activity intermission to have occurred between the active periods.",
"Only in an ideal case of symmetric jets and isotropic properties of the ambient medium, one could expect the same radio luminosity and equal lengths of both lobes.",
"In reality, the orientation of radio sources in space is random and only some of them are aligned in the sky plane.",
"Due to the latter, we expect Doppler effects to be responsible for the brightness differences.",
"Moreover, the light-travel time effect, consisting in that radiation from the far-side lobe arrives at the observer significantly later due to its longer distance than from the near-side lobe, should be taken into account (see [60]).",
"In Figure REF we plotted the radio luminosity against the mass of the black hole.",
"The distribution of sources in the P–M$\\rm _{BH}$ plane is much similar for both the samples.",
"We also compared the distributions of the black hole mass for the FRII sample, restarting radio galaxies sample, and X-shaped radio sources studied by [62], [63].",
"All the distributions are presented in Figure REF .",
"In the case of FRII radio galaxies, the distribution of the black hole mass is very symmetric with a peak value of $log($ M$\\rm _{BH}/M_{\\odot })$ ranging from 8.6 to 8.8 and the median value of 8.58.",
"For the restarting radio galaxies the M$\\rm _{BH}$ distribution does not have any pronounced maximum.",
"However, the median value of $log($ M$\\rm _{BH}/M_{\\odot })$ is 8.61, much the same as those of typical FRII sources.",
"The distribution for the X-shaped radio galaxies is nearly symmetric around the peak value, which is similar to the one of the restarting radio sources.",
"The median value of $log($ M$\\rm _{BH}/M_{\\odot })$ for the X-shaped galaxies of $\\sim $ 8.3 is smaller than for the other two samples.",
"Figure: Relation between BH mass and radio luminosity at 1.4 GHz.Figure: BH mass distribution for FRII (top panel), restarting radio galaxies (middle panel) and X-shaped radio sources (bottom panel)."
],
[
"Infrared properties",
"Almost all the recurrent activity radio galaxies discussed in this paper were detected by the Wide-Field Infrared Survey Explorer (WISE; [100]).",
"In Figure REF we plotted the colour-colour diagram, where the vertical axis W1-W2 corresponds to the magnitude difference between the 3.4 and the 4.6 $\\mu $ m bands and the horizontal axis W2-W3 corresponds to the magnitude difference between the 4.6 and the 12 $\\mu $ m bands.",
"We plotted the WISE colours only for 49 restarting and 388 FRII radio sources, for which all magnitudes were above the detection limit.",
"In Figure REF we marked our sources in the diagram showing the regions where the different classes of the WISE-detected sources are located ([100]).",
"The vertical dotted line, which, according to [100], divides elliptical and spiral galaxies, has a WISE W2-W3 colour value of +1.5 magnitude.",
"The most powerful AGNs lie above the W1-W2 colour level of +0.6 magnitude.",
"It can be clearly seen that most of the restarting radio sources (67%) are located in the regions where spiral galaxies, being typically ISM-abundant, reside.",
"In the comparison sample less than half (41%) of FRII radio sources are in that region.",
"Also most of the studied radio sources have W1-W2 colours lower than 0.5 and according to [38], for galaxies with W1-W2 $<$ 0.5 the main source of radiation is the stellar and ionised gas emission.",
"The location of radio sources in this region was explained by [3] as a consequence of small Eddington ratios.",
"In such galaxies, the mid-infrared colours are not dominated by the AGN, while they are contaminated by the host.",
"Therefore, the colours generally originate from stars and emission from the cold dust of star formation.",
"Figure: Colour-colour diagram for restarting and FRII radio sources.",
"The coloured areas represent different classes of sources and the dotted line (W2-W3=1.5) shows a proposed division between elliptical and spiral galaxies (according to )."
],
[
"Conclusions",
"We compiled and presented the largest (to date) sample of 74 restarting radio sources, using optical, radio, and infrared data to determine their physical properties and compare them with a sample of typical FRII radio galaxies with single activity.",
"We came to the following conclusions: (i) The black hole masses of radio sources with recurrent activity are similar to those observed in FRII radio galaxies.",
"(ii) Recurrent and typical radio sources show different compositions of stellar populations.",
"The hosts of restarting radio sources contain a larger amount of young stars.",
"(iii) The total mass of stars in the host galaxies of the recurrent activity sources is, on average, less than that in the FRII hosts.",
"The concentration index for the restarting radio sources tends to be slightly lower than that of the FRII sources, which indicates that they can have hosts with more disturbed morphologies.",
"Both the facts could be the evidence that the restarting radio sources are more common in galaxies after mergers.",
"(iv) Emission lines are visible in almost all (89%) the available spectra of recurrent jet activity hosts.",
"Basing on the diagnostic diagram, the emission lines are excited by an AGN-like driven process.",
"(v) There is a significant correlation between the luminosity of the emission line and the radio luminosity of the inner and outer radio lobes.",
"The correlation is on the same level for the H$_\\alpha $ as well as for [OIII] line.",
"(vi) There is a strong correlation between the radio luminosity of the inner and outer lobes.",
"(vii) In the case of 13 restarting radio galaxies, the inner lobes are more luminous than the outer ones.",
"This can be seen not only in very compact ($<$ 1kpc) inner doubles, but in much more extended ones as well, with linear sizes of few hundred kiloparsecs.",
"(viii) Almost half of restarting radio sources have brighter/larger inner and outer lobes on the opposite sides of host galaxy.",
"(ix) The infrared WISE colour-colour diagram shows that the 67% hosts of recurrent jet activity radio galaxies reside in the region typical for spiral galaxies or other dusty, late-type galaxies with some ongoing star formation, while only 41% of FRII radio sources are found in that region."
],
[
"Acknowledgements",
"We thank the anonymous referee for her/his very valuable comments.",
"The work is supported by Polish NSC grant DEC-2013/09/B/ST9/00599.",
"l l l l c c c c c c c c c c l Radio sources with evidence of recurrent activity.",
"Source IAU REC$\\rm _{J2000.0}$ DEC$\\rm _{J2000.0}$ Opt.",
"Red- $l_{in}$$l_{in}$ $l_{out}$$l_{out}$$S_{in}$$S_{out}$$logM_{BH}$ Class Ref.",
"Name Name $\\rm ^{h}$ $\\rm ^{m}$ $\\rm ^{s}$ $$ ' ” Id.",
"shift kpc arcsec kpc arcsec mJy mJy M$\\odot $ A$/$ B Cmt.",
"(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) 15lTable REF continued Source IAU REC DEC Opt.",
"Red- $l_{in}$$l_{in}$ $l_{out}$$l_{out}$$S_{in}$$S_{out}$$logM_{BH}$ Class Ref.",
"Name Name J2000 J2000 Id.",
"shift kpc arcsec kpc arcsec mJy mJy M$\\odot $ A$/$ B Comment (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) J0009$+$ 1244 4C12.03 00 09 52.60 $+$ 12 44 04.64 G 0.156 114.9 43 553.1 207 100 907 – A 1,f J0037$+$ 1319 3C16 00 37 44.57 $+$ 13 19 55.00 G 0.405 96.91 18 419.9 78 35 1765 – A 1,2 J0041$+$ 3224$^\\star $ 00 41 46.12 $+$ 32 24 52.65 G (0.45) 172 30 974 170 525 409 – A 3 J0042$-$ 0613 00 42 46.85 $-$ 06 13 52.92 G 0.123 518.1 237.1 753.6 344.9 690 1140 9.23 A 4,a,e J0104$-$ 6609 01 04 21.26 $-$ 66 09 17.30 G (1.19) 66.7 8 750 90 0.37 5.22 – A 5 J0111$+$ 3906$^\\star $ B0108+388 01 11 37.32 $+$ 39 06 28.10 G 0.6685 0.07 0.01 126.1 18 519 8 – B?",
"6,b,c J0116$-$ 4722 01 16 25.04 $-$ 47 22 41.60 G 0.146 455 180 1441 570 260 2640 – A 7 J0301$+$ 3512$^\\star $ 4C 34.09 03 01 42.37 $+$ 35 12 20.68 G 0.0165 0.66 2 233.7 706 1800 119 8.26 B 2,i J0301$+$ 3550$^\\star $ 4C 35.06 03 01 51.50 $+$ 35 50 30.00 G 0.0463 32.29 36 403.7 450 492 258 – A 8 J0303$+$ 1626 3C76.1 03 03 15.02 $+$ 16 26 19.06 G 0.0325 21.8 34.1 107.1 167.4 450.2 2602 – B 1 J0351$-$ 2744 PKS0349-27 03 51 35.76 $-$ 27 44 34.70 G 0.0662 251.8 200.8 437.8 349.1 2636 3009 – A 9 J0504$+$ 3806 3C134 05 04 42.19 $+$ 38 06 11.40 G – – 115.6 – 166.1 1485 7862 – A a J0709$-$ 3601 PKS0707-35 07 09 14.09 $-$ 36 01 21.80 G 0.218 627.1 179.4 1720 492 480 1444 – A 10,11 J0741$+$ 3112$^\\star $ B2 0738+31 07 41 10.70 $+$ 31 12 00.22 Q 0.632 0.03 0.005 478.5 70 2188 38 9.37 A 12,b,h J0746$+$ 4526 07 46 17.92 $+$ 45 26 34.46 G 0.5502 95 15.2 640 100.1 24.2 191.6 9.96 A 13 J0804$+$ 5809 08 04 42.79 $+$ 58 09 34.94 S – – 21.6 – 106.3 58.6 192.1 – A 13 J0821$+$ 2117$^\\star $ B0818+214 08 21 07.50 $+$ 21 17 02.87 G 0.418 2.7 0.5 209.8 38.24 148 46.4 – A 14 J0840$+$ 2949$^\\star $ 4C29.30 08 40 02.36 $+$ 29 49 02.63 G 0.0647 39.3 32 533.3 434.6 446.7 216.9 8.32 B 15 J0847$+$ 3147 IC2402 08 47 59.04 $+$ 31 47 08.37 G 0.0674 203 159.3 362.1 284.2 173.9 1303 9.28 A 16 J0855$+$ 4204 08 55 49.15 $+$ 42 04 20.11 G (0.279) 35.3 8.4 545.9 130 18.8 155.7 – A 13 J0910$+$ 0345$^\\star $ 09 10 59.10 $+$ 03 45 31.68 G (0.588) 42.3 6.4 218.8 33.1 50.7 53.4 – A 13 J0914$+$ 1006$^\\star $ 09 14 19.53 $+$ 10 06 40.59 G 0.308 216.2 48 1709 379.7 252.5 129.1 8.56 A a J0921$+$ 4538 3C219 09 21 08.61 $+$ 45 38 57.36 G 0.174 70.1 24 438.5 150 90 8046 8.43 A 17,18,19 J0924$+$ 0602 09 24 49.04 $+$ 06 02 42.80 G 0.231 80.8 22.1 424 116 4.8 90 8.85 A a J0927$+$ 2932 09 27 44.88 $+$ 29 32 32.30 S – – 24 – 115 19 17 – A 8 J0927$+$ 3510 09 27 50.59 $+$ 35 10 50.73 G (0.55) 575.5 90 2206 345 3 96 – A 20 J0929$+$ 4146 09 29 10.66 $+$ 41 46 45.59 G 0.365 655.6 130 1876 372 64 99 – A 21,d,g J0935$+$ 0204 4C02.27 09 35 18.19 $+$ 02 04 15.54 Q 0.6491 69.9 10.1 498 71.96 230.3 547.6 9.56 A 22 J0943$-$ 0819$^\\star $ B0941-080 09 43 36.94 $-$ 08 19 30.81 G 0.228 0.18 0.05 72.34 20 3232 26 – B?",
"23,h J1004$+$ 5434 10 04 51.83 $+$ 54 34 04.29 G 0.047 55.2 60.6 694.2 762.9 90 110 8.72 A 24,a J1006$+$ 3454 3C236 10 06 01.73 $+$ 34 54 10.52 G 0.101 1.8 1 4248 2310 2500 3300 8.70 A 25,26,27 J1021$+$ 1216$^\\star $ 10 21 24.21 $+$ 12 17 05.44 G 0.129 876.4 385 1865 819.4 67.3 55 8.56 A 4,a J1039$+$ 0536 10 39 28.21 $+$ 05 36 13.62 G 0.35 81.5 16.6 488.7 99.6 54.9 594.9 – A 13 J1103$+$ 0636 11 03 13.29 $+$ 06 36 16.00 G 0.4406 66.3 11.7 548.8 96.9 13.2 79.9 8.24 A 13 J1158$+$ 2621 4C26.35 11 58 20.13 $+$ 26 21 12.07 G 0.112 139 69 483.8 240 67 962 7.96 A 13,28 J1159$+$ 5820 11 59 05.68 $+$ 58 20 35.57 G 0.054 23.2 22.4 348.2 335.8 5.3 319.1 8.60 A 29 J1208$+$ 0821 12 08 56.78 $+$ 08 21 38.57 G 0.5841 111.3 16.9 648.9 98.5 2 51.9 9.27 A 13 J1238$+$ 1602 12 38 21.20 $+$ 16 02 41.42 S – – 40.8 – 115.6 8.9 56.5 – A 13 J1242$+$ 3838 12 42 36.82 $+$ 38 38 06.15 G 0.408 308.3 57 735.5 136 8 24 8.72 A 30 J1247$+$ 6723$^\\star $ 12 47 33.31 $+$ 67 23 16.46 G 0.107 0.019 0.01 1196 618 260 126 8.63 A 31,32 J1325$-$ 4301 Cen A 13 25 27.62 $-$ 43 01 08.81 G 0.0018 21.3 67 266.4$^j$ 1800$^j$ 28$\\cdot 10^4$ 52$\\cdot 10^4$ – A 33,34 533 14400 96$\\cdot 10^4$ J1326$+$ 1924 13 26 13.67 $+$ 19 24 23.75 G 0.1762 26 8.8 150.6 51 6 81.8 8.93 A 13 J1328$+$ 2752 13 28 48.45 $+$ 27 52 27.81 G 0.0911 97.3 58 220.8 131.7 27.9 219.9 7.82 A 13 J1344$-$ 0030 13 44 46.92 $-$ 00 30 09.31 G 0.5801 85.4 13 631 96.1 20.1 48.7 8.73 A 13 J1352$+$ 3126 3C293 13 52 17.88 $+$ 31 26 46.49 G 0.045 1.1 1.2 179.5 204.2 3703 1209 8.15 A 35 J1407$+$ 5132 4C51.31 14 07 18.48 $+$ 51 32 04.63 G (0.324) 84.8 18.2 707.5 151.8 9.2 646.2 – A 13 J1409$-$ 0302 14 09 48.85 $-$ 03 02 32.53 G 0.1378 52.9 22 308 128 4 48 8.61 A 36,d 1373 570 112 J1443$+$ 5201 3C303 14 43 02.75 $+$ 52 01 37.23 G 0.1412 81.9 33.3 90.9 37 1500 935 8.07 B 1 J1453$+$ 3308 4C33.33 14 53 02.86 $+$ 33 08 42.40 G 0.249 158.5 41 1299 336 34 426 9.02 A 30,37 J1500$+$ 1542 15 00 55.18 $+$ 15 42 40.56 G (0.456) 124.2 21.5 480.7 83.2 11.1 17.8 – A 13 J1504$+$ 2600 3C310 15 04 57.12 $+$ 26 00 58.46 G 0.0538 152.1 147.1 255.4 247 1547 5846 8.29 B 38 J1513$+$ 2607 3C315 15 13 40.05 $+$ 26 07 30.46 G 0.1083 5.9 3 262 134 350.2 3967 9.98 A 39,f J1516$+$ 0701 3C317 15 16 44.48 $+$ 07 01 17.83 G 0.0345 0.05 0.08 50.85 75 353 5191 8.33 B?",
"40,41 J1520$-$ 0546 15 20 13.29 $-$ 05 46 27.01 G 0.0601 118.8 103.7 1513 1320 16.5 132.8 9.14 A 4,a J1521$+$ 5214 15 21 05.90 $+$ 52 14 39.91 G (0.537) 61.9 9.8 396 62 8.5 18.5 – A 13 J1528$+$ 0544 15 28 04.95 $+$ 05 44 28.18 G 0.0401 15.0 19.2 645.2 824 32.2 379.3 7.53 A 4,a J1534$+$ 1016 15 34 18.63 $+$ 10 16 47.54 G 0.1333 164.6 70.3 507.1 216.6 43 280 – A 28 J1538$-$ 0242 15 38 41.31 $-$ 02 42 05.51 G (0.575) 58.2 8.9 517.1 79.1 8.3 75.2 – A 13 J1545$+$ 5047 15 45 17.20 $+$ 50 47 53.94 G 0.4309 61.5 11 361.1 64.6 14.3 96.9 8.39 A 13 J1548$-$ 3216 15 48 58.05 $-$ 32 16 57.60 G 0.108 312.2 160 961.8 493 78 1722 – A 42,43 J1604$+$ 3438 16 04 45.89 $+$ 34 38 16.53 G 0.282 211.6 50 846.4 200 5 146 – A 20 J1605$+$ 0711 16 05 13.74 $+$ 07 11 52.56 G 0.3112 344.1 75.9 538.6 118.8 55.3 212.6 8.36 A 13 J1627$+$ 2906 16 27 54.63 $+$ 29 06 20.01 G (0.722) 73.8 10.2 697.4 96.4 10.4 112.9 – A 13 J1628$+$ 3933 3C338 16 28 38.24 $+$ 39 33 04.55 G 0.0304 7.8 13 48 80 224 3380 9.01 B 7,44 J1649$+$ 4133 16 49 28.32 $+$ 41 33 41.61 S – – 6.7 – 39.7 2.4 39 – A 13 J1651$+$ 0459 3C348 16 51 08.15 $+$ 04 59 33.32 G 0.1550 212.6 80 319.5 120.2 9.07 B?",
"45 J1705$+$ 3940 17 05 17.83 $+$ 39 40 29.25 G (0.701) 305.3 42.7 592.7 82.9 35.2 70.7 – A 13 J1706$+$ 4340 17 06 25.44 $+$ 43 40 40.16 S (0.525) 194 31.8 687.2 110.1 79.3 69.3 – A 13,46 J1835$+$ 6204 18 35 10.92 $+$ 62 04 08.14 G 0.519 372.3 60 1378 222 200 604 – A 30 J1844$+$ 4533$^\\star $ 3C388 18 44 02.40 $+$ 45 33 29.70 G 0.0917 67.48 40 84.35 50 3362 2205 9.47 B 47,48 J2048$+$ 0701 3C424 20 48 12.03 $+$ 07 01 17.48 G 0.127 22.5 10 49.2 21.9 625 1816 – A 49 J2107$+$ 2331 4C 23.56 21 07 15.08 $+$ 23 31 43.71 G 2.483 457 55.7 492.2 60 397 – A 50 J2223$-$ 0206 3C445 22 23 49.54 $-$ 02 06 12.90 G 0.056 130 121 612 570 55 5260 8.12 A 51,52 J2345$-$ 0449 23 45 32.71 $-$ 04 49 25.32 G 0.0757 338.9 238.9 1454 1025 29 148.4 – A 4,53 Column designation: (1) – source name, (2) – other name, (3), (4) – right ascension and declination (J2000), (5) – optical identification (G – galaxy, Q – quasar, S – unidentified source), (6) – redshift, (7) and (9) – projected linear size of the inner and outer lobes, respectively, (8) and (10) – apparent angular size of the inner and outer lobes respectively, (11) and (12) – flux densities at 1.4 GHz of the inner and outer lobes, respectively, (13) – mass of the central black hole, (14) – class of radio structure (A or B), (15) – references.",
"The redshift given in brackets corresponds to the Sloan Digital Sky Survey (SDSS) photometric redshift.",
"For two objects, J1409$-$ 0302 and J1325$-$ 4301, we wrote two values of l$\\rm _{out}$ and S$\\rm _{out}$ , which correspond to the length and flux density of the middle and outer lobes, respectively.",
"The bolded names of sources correspond to newly recognized restarting radio galaxies.",
"Notes: (a) a new DDRS; (b) flux density measured at 1.5 GHz; (c) This source is counted as radio relic associated with a giga-hertz peaked (GPS) source by [88] and by [87].",
"However, the optical DSS map (see the attached figure in the Appendix) reveals a weak extended emission in the vicinity of the extended diffuse relic.",
"Therefore, in our opinion, the association is questionable.",
"Deep optical observation are necessary.",
"; (d) triple-double radio galaxy; (e) the structure should be confirmed; the flux density of 1120 mJy is for the entire source; (f) X-shape radio source; (g) there are two nearby galaxies; identification consistent with FIRST is REC: 09$\\rm ^h$ 29$\\rm ^m$ 10$$ 38 DEC: $+$ 41$$ 46$^{\\prime }$ 44$$ 5 (J2000.0); (h) GHz Peaked Spectrum radio source; (i) Compact Steep Spectrum radio source; (j) Size of one side middle lobe; ($^\\star $ ) – restarting radio galaxies with inner lobes more luminous than the outer ones.",
"References: (1) [50], (2) [86], (3) [74], (4) [56], (5) [78] (6) [7], (7) [76], (8) [87], (9) [64], (10) [93], (11) [79], (12) [83], (13) [66], (14) [59], (15) [39], (16) [34], (17) [69], (18) [14], (19) [24], (20) [55], (21) [15], (22) [40], (23) [88], (24) [84], (25) [99], (26) [91], (27) [81], (28) [67], (29) [47], (30) [82], (31) [57], (32) [12], (33) [20], (34) [25], (35) [13], (36) [37], (37) [45], (38) [95], (39) [75], (40) [97], (41) [101], (42) [77], (43) [72], (44) [32], (45) [26], (46) [58], (47) [19], (48) [71], (49) [10], (50) [11], (51) [48], (52) [49], (53) [5]."
],
[
"Radio maps of restarting radio sources – class A",
"This appendix presents radio maps of restarting radio sources, which are of typical double-double radio morphology – class A sources.",
"Most of the radio maps were obtained with the Very Large Array at 1.4 GHz.",
"Maps of J0041$+$ 3224, J0351$-$ 2744, J0927$+$ 3510, J1648$-$ 3218, J1604$+$ 3438, J2107$+$ 2331 radio sources show radio contours at 4.8 GHz, and of J0037$+$ 1319, 2048$+$ 0701 at 8.4 GHz.",
"The map of J0104$-$ 6609 was taken from [78].",
"Three radio maps were plotted using data from other radio telescopes.",
"The 330 MHz map of J0116$-$ 4722 was taken by Giant Metrewave Radio Telescope, the 1.4 GHz map of inner structure of J0821$+$ 2117 is from the EVN/MERLIN measurement, and the 4.9 GHz inner structure of J1247$+$ 6723 comes from VLBI.",
"In all the cases the radio contours are overlaid onto the R-band optical image from Digital Sky Survey (DSS).",
"Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTION"
],
[
"Radio maps of restarting radio sources – class B",
"This appendix presents the radio maps of restarting radio sources with prominent inner structures surrounded by diffuse outer structures – class B.",
"All of the radio maps presented here were obtained with the Very Large Array at 1.4 GHz.",
"The map of J0301$+$ 3512 was taken from [86].",
"In all the maps radio contours are overlaid onto the R-band optical image from Digital Sky Survey (DSS).",
"Figure: NO_CAPTIONFigure: NO_CAPTION"
]
] | 1709.01802 |
[
[
"On the Diophantine Equation $p^x + p^y = z^{2n}$"
],
[
"Abstract In an earlier paper, Tatong and Suvarnamani explores the Diophantine equation $p^x + p^y = z^2$ for a prime number $p$.",
"In that paper they find some solutions to the equation for $p=2, 3$.",
"In this paper, we look at a general version of this equation and solve it completely."
],
[
"Introduction",
"Diophantine equations of the form $a^x + b^y = z^2$ have been studied in recent papers by various authors.",
"In this paper we use elementary methods to completely solve the equation $p^x + p^y = z^{2n}$ in non-negative integers where $p$ is a prime number."
],
[
"Preliminaries",
"Theorem 2.1 The Diophantine equation $a^x - b^y = 1$ has a unique solution in integers $a, b, x,$ and $y$ with min$\\lbrace a, b, x, y \\rbrace > 1$ .",
"This solution is given by $(3, 2, 2, 3)$ .",
"This is the famous Catalan's Conjecture (which can be also called Mihailescu's Theorem) that was proven by Mihailescu.",
"For the proof see [2].",
"Lemma 2.2 For an odd prime $p > 3$ , the Diophantine equation $p^x + 1 = z^2$ has no non-negative integer solutions.",
"On the contrary, suppose that there are non-negative integer solutions $(p, x, z)$ .",
"If $x = 0$ , then $z^2 = 2$ which is not solvable in integers.",
"Therefore, $x \\ge 1$ .",
"Since $p > 3$ , therefore, $z^2 = p^x + 1 \\ge 5^1 + 1 = 6$ .",
"Thus, $z \\ge 3$ .",
"We rewrite the equation as $z^2 - p^x = 1$ .",
"Since $p > 3$ , hence by Theorem REF , $x = 1$ .",
"This means $z^2 = 1 + p$ or $(z-1)(z+1) = p$ , which is impossible as $p$ is a prime number and $z \\ge 3$ .",
"Hence, when $p > 3$ , the equation $p^x + 1 = z^2$ has no non-negative integer solutions."
],
[
"Main Result",
"We now present our main result.",
"Theorem 3.1 If $p$ is a prime number and $n \\ge 1$ is an integer, then the Diophantine equation $p^x + p^y = z^{2n}$ has the following set of solutions $(x, y, z)$ in non-negative integers.",
"For $n=1$ , $(x, y, z) ={\\left\\lbrace \\begin{array}{ll}(2s+3, 2s, 3 \\cdot 2^s), (2s, 2s+3, 3 \\cdot 2^s), (2s+1, 2s+1, 2^{s+1}), & p = 2, \\text{ and } s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\\\(2s+1, 2s, 2 \\cdot 3^s), (2s, 2s+1, 2 \\cdot 3^s), & p = 3, \\text{ and } s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\\\\\text{ No solutions}, & p > 3\\end{array}\\right.",
"}$ For $n>1$ , $(x, y, z) ={\\left\\lbrace \\begin{array}{ll}(2s+1, 2s+1, 2^{(s+1)/n}), & p = 2, s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\text{ and } s+1 \\equiv 0 \\text{ mod } n \\\\\\text{ No solutions}, & p \\ge 3\\end{array}\\right.",
"}$ The proof of this theorem combines the proofs of the next two theorems.",
"Theorem 3.2 If $p$ is a prime number then the Diophantine equation $p^x + p^y = z^2$ has the following set of solutions $(x, y, z)$ in non-negative integers.",
"$(x, y, z) ={\\left\\lbrace \\begin{array}{ll}(2s+3, 2s, 3 \\cdot 2^s), (2s, 2s+3, 3 \\cdot 2^s), (2s+1, 2s+1, 2^{s+1}), & p = 2, \\text{ and } s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\\\(2s+1, 2s, 2 \\cdot 3^s), (2s, 2s+1, 2 \\cdot 3^s), & p = 3, \\text{ and } s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\\\\\text{ No solutions}, & p > 3\\end{array}\\right.",
"}$ This is the case $n= 1$ of Theorem REF .",
"We consider several cases.",
"Case 1 : $x = y$ .",
"In this case, $p^x + p^y = 2p^x = z^2$ Thus $z = 2^{1/2}p^{x/2}$ .",
"As $p$ is a prime number and $z$ is a non-negative integer, therefore, $p = 2$ and $x$ is an odd non-negative integer.",
"Thus, when $x = 2s+1$ for any non-negative integer $s$ , we have $z = 2^{s+1}$ .",
"Hence, when $y=x$ , the solution set $(x, y, z)$ consists of 3-tuples of the form $(2s+1, 2s+1, 2^{s+1})$ for any $s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace $ .",
"Case 2 : $x \\ne y$ and $x<y$ .",
"In this case, $p^x(1+p^{y-x}) = z^2$ , which implies that $p \\mid z$ .",
"Let $e$ be the highest power of $p$ that divides $z$ , i.e., $p^e \\mid z$ but $p^{e+1} z$ .",
"Suppose $z = p^e k$ with $p k$ .",
"We then have, $p^x(1+p^{y-x}) = (p^e k )^2 = p^{2e}k^2$ Since $p$ is a prime number and $p k$ , therefore, $p^x = p^{2e}$ which implies $x = 2e$ .",
"We then have, $k^2 = 1 + p^{y-x} = 1 + p^{y-2e}$ We now divide this problem into several cases.",
"Case 2.1 : $p = 2$ and $y-2e = 1$ .",
"In this case, $k^2 = 1 + 2^{y-2e} = 1+ 2 = 3$ This has no integer solutions in $k$ , hence the Diophantine equation $p^x + p^y = z^2$ has no non-negative integer solutions when $p=2$ and $y-x = 1$ .",
"Case 2.2 : $p = 2$ and $y-2e > 1$ .",
"In this case, since $p=2$ and $y-2e > 1$ , therefore, $k^2 = 1 + 2^{y-2e} \\ge 5$ .",
"Therefore, $k \\ge 3$ .",
"Rewriting Equation (REF ) we have, $k^2 = 1 + 2^{y-2e}$ $k^2 - 2^{y-2e} = 1$ As min$\\lbrace k, 2, 2, y-2e \\rbrace > 1$ , therefore, Equation (REF ) has a unique solution $(k, y-2e)$ given by $(3, 3)$ by Theorem REF .",
"Thus, we have $x = 2e, y = 2e+3$ and $z = 2^e \\cdot 3$ .",
"Hence, when $p = 2$ and $y-x = y - 2e >1$ , the solution set $(x, y, z)$ of Equation (REF ) consists of 3-tuples of the form $(2s, 2s+3, 3 \\cdot 2^s)$ for any $s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace $ .",
"Case 2.3 : $p=3$ and $y-2e = 1$ .",
"In this case we have, $k^2 = 1 + 3^{y-2e} = 1 + 3 = 4$ This implies $k = 2$ .",
"Thus, $x=2e, y = 2e+1$ and $z = 2 \\cdot 3^e$ .",
"Thus when $p=3$ and $y-x = y - 2e =1$ , the solution set $(x, y, z)$ of Equation (REF ) consists of 3-tuples of the form $(2s, 2s+1, 2 \\cdot 3^s)$ for any $s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace $ .",
"Case 2.4 : $p = 3$ and $y-2e > 1$ .",
"In this case, since $p=3$ and $y-2e > 1$ , therefore, $k^2 = 1 + 3^{y-2e} \\ge 10$ .",
"Therefore, $k \\ge 4$ .",
"Rewriting Equation (REF ) we have, $k^2 = 1 + 3^{y-2e}$ $k^2 - 3^{y-2e} = 1$ As min$\\lbrace k, 2, 3, y-2e \\rbrace > 1$ , therefore, Equation (REF ) does not have any non-negative integer solutions by Theorem REF .",
"Thus when $p=3$ and $y-x = y - 2e > 1$ , Equation (REF ) has no solutions in non-negative integers.",
"Case 2.5 : $p >3$ .",
"Looking at Equation (REF ), $1+p^{y-2e} = k^2$ The equation has no solutions in non-negative integers by Lemma REF .",
"Thus, when $p>3$ , Equation (REF ) has no solutions in non-negative integers.",
"Thus, taking Cases 2.1-2.5 into account we see that the set of solutions $(x, y, z)$ of Equation (REF ) when $x < y$ is given by, $(x, y, z) ={\\left\\lbrace \\begin{array}{ll}(2s, 2s+3, 3 \\cdot 2^s), (2s+1, 2s+1, 2^{s+1}), & p = 2, \\text{ and } s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\\\(2s, 2s+1, 2 \\cdot 3^s), & p = 3, \\text{ and } s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\\\\\text{ No solutions}, & p > 3\\end{array}\\right.",
"}$ Case 3 : $x \\ne y$ and $x>y$ .",
"This case is similar to Case 2 and its various cases.",
"The only difference results in switching the values of $x$ and $y$ in our final answer.",
"Thus the set of solutions $(x, y, z)$ of Equation (REF ) when $x > y$ is given by, $(x, y, z) ={\\left\\lbrace \\begin{array}{ll}(2s+3, 2s, 3 \\cdot 2^s), (2s+1, 2s+1, 2^{s+1}), & p = 2, \\text{ and } s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\\\(2s+1, 2s, 2 \\cdot 3^s), & p = 3, \\text{ and } s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\\\\\text{ No solutions}, & p > 3\\end{array}\\right.",
"}$ This concludes the proof.",
"Now we consider the case $n > 1$ .",
"Theorem 3.3 If $p$ is a prime number and $n > 1$ is an integer, then the Diophantine equation $p^x + p^y = z^{2n}$ has the following set of solutions $(x, y, z)$ in non-negative integers.",
"$(x, y, z) ={\\left\\lbrace \\begin{array}{ll}(2s+1, 2s+1, 2^{(s+1)/n}), & p = 2, s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace \\text{ and } s+1 \\equiv 0 \\text{ mod } n \\\\\\text{ No solutions}, & p \\ge 3\\end{array}\\right.",
"}$ Case 1: $p = 2$ .",
"Suppose that there exists non-negative integers $x, y$ and $z$ that satisfies Equation (REF ).",
"Let $w=z^n$ .",
"In that case $(x, y, z^n)$ is a solution in non-negative integers of the equation $2^x + 2^y = w^2$ By Theorem REF the only solutions $(x, y, w)$ to Equation (REF ) are $(2s+3, 2s, 3 \\cdot 2^s), (2s, 2s+3, 3 \\cdot 2^s), (2s+1, 2s+1, 2^{s+1})$ for $s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace $ .",
"This means either $w = 3 \\cdot 2^s$ or $w = 2^{s+1}$ .",
"Case 1.1 : $w = 3 \\cdot 2^s$ .",
"This is not possible as $w = z^n = 3 \\cdot 2^s$ which is not solvable in integers $z$ for $n > 1$ .",
"Case 1.2 : $w = 2^{s+1}$ .",
"This means $z^n = 2^{s+1}$ or $z = 2^{(s+1)/n}$ .",
"If $n \\mid (s+1)$ , then $z$ is an integer and we have solutions $(2s+1, 2s+1, 2^{(s+1)/n})$ for Equation (REF ).",
"Case 2 : $p \\ge 3$ .",
"We proceed similarly to Case 1.",
"Suppose that there exists non-negative integers $x, y$ and $z$ that satisfies Equation (REF ).",
"Let $w=z^n$ .",
"In that case $(x, y, z^n)$ is a solution in non-negative integers of the equation $p^x + p^y = w^2$ Case 2.1 : $p = 3$ .",
"By Theorem REF , the only solutions $(x, y, w)$ to Equation (REF ) are $(2s+1, 2s, 2 \\cdot 3^s), (2s, 2s+1, 2 \\cdot 3^s)$ for $s \\in \\mathbb {N} \\cup \\lbrace 0\\rbrace $ .",
"This means $w = 2 \\cdot 3^s$ for both solutions.",
"However, $z^n = 2 \\cdot 3^s$ doesn't have any solutions in integers $z$ for $n > 1$ .",
"Therefore Equation (REF ) has no solutions in non-negative integers.",
"Case 2.2 : $p >3$ .",
"By Theorem REF , Equation (REF ) has no solutions $(x, y, w)$ in non-negative integers.",
"Hence Equation (REF ) has no solutions $(x, y, z)$ in non-negative integers.",
"Combining the results from Case 1 and Case 2 we see the results of Theorem REF ."
],
[
"Conclusion",
"We see that the Diophantine equation $p^x + p^y = z^{2n}$ has infinitely many solutions when $n=1$ and $p=2$ or $p=3$ .",
"The equation has no solutions when $p>3$ .",
"It also has infinitely many solutions when $n > 1$ for $p=2$ .",
"However, the equation does not have any solutions when $p \\ge 3$ .",
"All these solutions are given in the statement of Theorem REF ."
]
] | 1709.01814 |
[
[
"Anomalous diffusion analysis of the lifting events in the event-chain\n Monte Carlo for the classical XY models"
],
[
"Abstract We introduce a novel random walk model that emerges in the event-chain Monte Carlo (ECMC) of spin systems.",
"In the ECMC, the lifting variable specifying the spin to be updated changes its value to one of its interacting neighbor spins.",
"This movement can be regarded as a random walk in a random environment with a feedback.",
"We investigate this random walk numerically in the case of the classical XY model in 1,2, and 3 dimensions to find that it is superdiffusive near the critical point of the underlying spin system.",
"It is suggested that the performance improvement of the ECMC is related to this anomalous behavior."
],
[
"Introduction",
"The Monte Carlo algorithm (MC) is well-used method in statistical mechanics.",
"The most well-known and important class is the local Metropolis MC (LMC) [1] which keeps the detailed balance condition.",
"In the LMC, however, there are well-known difficulties, for example, the critical slowing down near the critical point.",
"Therefore it is important to go beyond the detailed balance for improvement of the MCs.",
"Recently MCs with the broken detailed balance condition are proposed.",
"The examples include event-chain Monte Carlo algorithm (ECMC) [2], geometric allocation algorithm[3], and skew detailed balance algorithm [4], [5].",
"They are very interesting studies that show improvement in the efficiency of the MC.",
"We will analyze the ECMC as a stochastic process in this letter.",
"The ECMC was initially introduced for hard sphere systems and then was extended to classical continuous spin systems [6], [7], [8].",
"It consists of the factorized Metropolis filter, an additional degree of freedom called the lifting variable, and rejection-free algorithm [9].",
"It does not satisfy the detailed balance condition but satisfies the global balance condition [10].",
"In Ref.",
"[6], the ECMC was applied to the $XY$ model and it was shown that the ECMC relaxes more rapidly than the LMC.",
"It is desirable to understand whether and how the ECMC is efficient for various systems.",
"This motivated us to study the stochastic dynamics of this algorithm as a first step.",
"In this letter, we will define the lifting variable random walk as that of the movement of the lifting variable that specifies the site to be updated.",
"The lifting variable hops to another site with a probability depending on the spin variables which interacts with the spin at the current site.",
"We believe that studying this walk will help one to understand the dynamics of the ECMC.",
"Moreover, this walk is interesting and is worth studying in its own right.",
"It is a novel random walk in a random environment with a feedback [11], [12].",
"We will investigate this random walk numerically in the case of the classical $XY$ model to find that it is superdiffusive near the critical point."
],
[
"The event-chain Monte Carlo",
"The ECMC is rejection-free because at rejection the update of an alternative variable is automatically accepted (an `event') in such a way that the global balance is kept [6].",
"First, we consider a spin system and the update of a configuration $a$ to $b$ by the LMC.",
"The acceptance probability for the Metropolis filter is defined by $p^\\text{Met} (a \\rightarrow b) = \\min \\left[1, \\exp (-\\beta \\Delta E) \\right],$ where $\\Delta E = E^b - E^a$ is the energy change and $\\beta $ is the inverse temperature.",
"In the continuous spin systems with a pairwise interaction, we can transform eq.",
"(REF ) into $p^\\text{Met} (a \\rightarrow b) = \\min \\left[1, \\prod _{\\langle i, j \\rangle } \\exp \\left( -\\beta \\Delta E_{ij}\\right) \\right],$ where $\\Delta E_{ij} = E_{ij}^b - E_{ij}^a$ is the pair energy change.",
"The update process satisfies the detailed balance condition $\\pi _a p^\\text{Met}( a \\rightarrow b) = \\pi _b p^\\text{Met}(b \\rightarrow a),$ where $\\pi $ is the Boltzmann weight $\\pi _* = \\exp (-\\beta E_*)$ .",
"In the ECMC, we design the update process where a spin variable receives persistent infinitesimal updates.",
"Because this violates the detailed balance condition, we seek to recover the global balance condition.",
"We employ the factorized Metropolis filter [10] $p^\\text{fact} ( a \\rightarrow b) = \\prod _{\\langle i, j \\rangle } \\min [ 1, \\exp (-\\beta \\Delta E_{ij}) ].$ It can factorize all individual pair energies.",
"The physical configuration $a$ and $b$ are the extended to include lifting variable $k$ .",
"It represents the spin currently rotated.",
"Under the factorized Metropolis filter and infinitesimal rotations, the rejection is judged for each interacting spin $\\ell $ independently and the first rejection pair $(k,\\ell )$ is determined uniquely.",
"When the first rejection occurs, we change the value of the lifting variable from $k$ to $\\ell $ .",
"Then we say that a lifting event occurs.",
"For concreteness, we describe the ECMC method for the $XY$ model in detail.",
"The classical $XY$ model is one of the simplest continuous spin models in statistical mechanics, which is defined by the energy function $E^a =\\sum _{\\langle i,j \\rangle } E_{ij}=-J \\sum _{\\langle i,j \\rangle } \\mathbf {s}_i \\cdot \\mathbf {s}_j = -J \\sum _{\\langle i,j \\rangle } \\cos (\\theta _i - \\theta _j),$ where $E_{ij}$ is the pair energy, $J$ is the coupling constant, $\\mathbf {s}_*$ is the two-component unit vector, and $\\theta _*$ is the rotation angle of $\\mathbf {s}_*$ .",
"The notation $\\langle i,j\\rangle $ means all the pairs of nearest-neighbor spins.",
"In two-dimensional square lattice case, this model has the Kosterlitz-Thouless transition [13] at $\\beta = 1.11996(6)$ [14].",
"In three-dimensional cubic case, this model has the second order phase transition at $\\beta = 0.454166$ [15].",
"For infinitesimal rotation toward the event angle $\\theta _{k, \\text{event}}$ , we introduce the event-driven approach [9], [10].",
"In order to determine $\\theta _{k, \\text{event}}$ , it is necessary to calculate the increase of the pair energy $\\Delta E_{k}(\\ell )$ of each pair $(k,\\ell )$ $\\Delta E_{k}(\\ell ) = -\\frac{1}{\\beta }\\log \\gamma _{k\\ell },$ where $\\gamma _{k\\ell }$ means a random number uniformly distributed between 0 and 1.",
"The increase and the event angle $\\theta _{k, \\text{event}}$ is related by $\\Delta E_{k}(\\ell ) = \\int ^{\\theta _{k, \\text{event}}} _{\\theta _{k, \\text{current}}} \\max \\left( 0, \\frac{d E_{k\\ell }}{d \\theta _k} \\right) d \\theta _k.$ We solve eq.",
"(REF ) to determine the event angle $\\theta _{k, \\text{event}}$ for each $\\ell $ interacting with $k$ .",
"Then we choose the $\\ell $ which gives the minimal angle.",
"For more detail, see Refs.",
"[7], [8]."
],
[
"Lifting variable random walk",
"In the ECMC of a general spin system with pairwise interactions on a regular or random lattice, the lifting variable $k$ changes its value to one of the spins $\\lbrace \\ell \\rbrace $ interacting with $k$ .",
"Thus a movement of the lifting variable can be regarded as a random walk (see fig.",
"REF ).",
"Its transition probability depends on $\\mathbf {s}_k$ and $\\mathbf {s}_\\ell $ 's.",
"We call this random walk lifting variable random walk.",
"This random walk is regarded as a special case of the random walk in a random environment[16].",
"Namely, the environment consists of the spin variables which are not independent each other.",
"Moreover, it is a random walk in the random environment with feedback [11], [12] because the spin at $k$ changes its value depending on the movement of the lifting variable.",
"For the comparison to the above random walk with feedback, we formally define a random walk in a random environment without feedback to the environment spins.",
"In the walk without feedback, given a spin configuration, the lifting variable shall stochastically move to a neighbor site according to exactly the same rule as the walk with feedback.",
"After that, the current spin $\\theta _k$ shall not be updated.",
"Therefore it is a random walk in a quenched random environment and is useless for the measurement of thermodynamic quantities of the spin system.",
"It turns out, however, to be userful for the understanding of the walk with feedback.",
"There are many possible choices of the time unit for the random walk.",
"In our analysis, time $t$ is defined to be the number of lifting events.",
"Therefore at every $t$ , the walker moves to another site and never stays at the same site.",
"Figure: A brief description of our viewpoint.",
"Arrows mean the current spin configuration.",
"The red arrow is the updated spin specified by the lifting variable.",
"The green arrow is the first spin causing the rejection (lifting event).A random walk can be characterized by the anomalous diffusion exponent $d_\\mathrm {w}$ of the expectation value of mean square displacement (MSD) $\\langle X(t)^2 \\rangle {= D} t^{2/d_\\mathrm {w}},$ where $X(t)$ means displacement from $X(0)=0$ , $D$ is the diffusion coefficient and $t$ is the time.",
"We classify diffusion into three classes with the value of $d_\\mathrm {w}$ : superdiffusion $(d_\\mathrm {w} < 2)$ , normal diffusion $(d_\\mathrm {w} = 2)$ , and subdiffusion $(d_\\mathrm {w} > 2)$ [17], [18], [19].",
"The Brownian motion is classified to normal.",
"Super and subdiffusion are called anomalous diffusion."
],
[
"Results",
"We analyze behaviors of the lifting variable random walk of the classical $XY$ model.",
"We consider the system on the square and the cubic lattices with periodic boundary condition with $N = L^d$ spins, where $L$ is the length of lattice and $d=1,2,3$ is the dimension of the model.",
"In our results, the coupling constant $J$ is set to 1.",
"The results are obtained from samples consisting of $1000, 1000,$ and 2000 random walks of length $50 \\times N$ measured by events, for $d=1,2$ , and 3, respectively.",
"We do not reset the random walk at a chain length measured by the sum of rotated angle.",
"The initial spin configuration is equilibrated at a given inverse temperature $\\beta $ before the start.",
"The data are obtained in the range $\\beta _\\mathrm {min}\\le \\beta \\le \\beta _\\mathrm {max}$ with the constant interval $\\Delta \\beta $ where $(\\beta _\\mathrm {min},\\beta _\\mathrm {max},\\Delta \\beta )=(0.500,8.000,0.500)$ , $(0.050,2.000,0.025)$ , $(0.0125,1.200,0.0125)$ for $d=1,2$ , and 3, respectively.",
"In fig.",
"REF , we show $\\log \\langle X^2 (t)/t \\rangle \\text{-} \\log t$ plots in the range of $\\beta $ 's.",
"These plots clearly show the difference of $d_\\mathrm {w}$ (slope of curves).",
"For fig.",
"REF (b) and (c), the slope of curves increase near the critical point.",
"The plot indicates that the lifting variable random walk becomes superdiffusive near the critical point.",
"We show the dependences of quntities of interest on the temperature $T=1/\\beta $ and especially their behaviors near the critical point $T=T_\\mathrm {c}$ in fig.",
"REF with feedback and in fig.",
"REF without feedback for each dimension.",
"Figure: Lifting variable random walk with feedback.",
"Results for log(〈X 2 (t)〉/t)\\log (\\langle X^2 (t) \\rangle /t) as a function of logt\\log t for (a) 1D, (b) 2D, and (c) 3D lattices.Each curve corresponds to MSD at each β\\beta .The lattice sizes are (a) L=8192L = 8192, (b) L=512L = 512, and (c) L=64L = 64, respectively.Figure: Results for the random walk with feedback.The left, middle and right columns are results of 1D, 2D and 3D XYXY models, respectively.Plots of the anomalous diffusion exponent d w d_\\mathrm {w}, the diffusion coefficient DD, the return probability, and the cover time t cover t_\\text{cover} are arranged from the top row to the bottom.The dashed lines shows the critical point T=T c T=T_\\mathrm {c} and the purple lines in the plot of d w d_\\mathrm {w} corresponds to the normal diffusion.Figure: Results of the random walk without feedback.",
"Plots are arranged in the same manner as fig. .",
"The cover time t cover t_\\text{cover} is undefined for 1D because the 1D walk without feedback never reaches the visit rate 1/21/2 in our simulation time.Figure: The size dependence of the minimum value min T d w \\min _T d_\\mathrm {w} for the random walk with feedback in 2D and 3D cases.",
"It is suggested that the values linearly converge at the infinite system size."
],
[
"Anomalous diffusion near the critical point",
"To confirm the superdiffusion observed in fig.",
"REF , we estimate $d_\\mathrm {w}$ and $D$ at each $T$ by the least square method as figs.",
"REF (a)–(f).",
"All the cases have $d_\\mathrm {w}$ larger than the ballistic movement and less than the normal diffusion.",
"In the cases of 2D and 3D with feedback, the $d_\\mathrm {w}$ -$T$ plots have peaks near $T=T_\\mathrm {c}$ and it becomes sharper with the system size increases.",
"The size dependence of $\\min _T d_\\mathrm {w}$ is shown in fig.",
"REF for sizes including those in fig.",
"REF .",
"If the linear fit $\\min _T d_{\\mathrm {w}}(L)=a\\times \\frac{1}{L} + b$ is adopted, we obtain $\\min _T d_{\\mathrm {w}}(\\infty )=1.20(1)$ for 2D and $1.64(1)$ for 3D.",
"It may be interesting to speculate that this speedup of diffusion cancels out the critical slowdown to give the efficiency of the ECMC.",
"There are differences in the way peaks are formed in the cases of 2D and 3D.",
"This can be considered to arise from the difference in nature of the phase transition.",
"In the case of 1D with feedback, $d_\\mathrm {w}$ has a peak at $T\\rightarrow +0$ in consistent with the interpretation $T_\\mathrm {c} = 0$ .",
"To tell whether this anomalous behavior is a consequence of the (quenched) random environment alone or the feedback in the ECMC is essential, we have performed the same measurement for the random walk without feedback defined above eq.",
"(REF ).",
"Figs.",
"REF (a)–(f) show that it behaves differently from that with feedback.",
"Namely, it is subdiffusive for 1D.",
"For the 3D case, it is superdiffusive and $d_\\mathrm {w}$ has a peak at $T=T_\\mathrm {c}$ .",
"For the 2D case, most surprisingly, the lifting variable random walk without feedback switches from subdiffusion to superdiffusion near $T=T_\\mathrm {c}$ .",
"We also note that it has smaller finite size effect than the 1D and 2D cases.",
"In all cases except the 1D case, the coefficient $D$ in eq.",
"(REF ) shows a behavior similar to $d_\\mathrm {w}$ while $D$ increases with the system size in the 1D case.",
"The difference may be due to the local nature of the 1D random walks described below."
],
[
"Return probability and cover time",
"We measure the return probability to have insight into the origin of the anomalous diffusion.",
"Here, the return probability is defined as the ratio of steps where the lifting variable has the same value at $t$ and $t+2$ in the time series.",
"We obtain the return probabilities as (g)–(i) of figs.",
"REF and REF .",
"Interestingly, there is almost no size dependence.",
"The result differs from that of Markov symmetric walk which means that the lifting variable random walk has the nature of persistent random walk.",
"We note that also the systems near the criticality do not have a singularity of the return probability whereas that for the system without feedback may be singular at the critical point.",
"The fact that return probability with feedback at small $T$ is smaller than that of the symmetric walk can be explained as follows.",
"Imagine that the lifting variable changes its value from $k$ to a neighbor $\\ell $ .",
"The spin $k$ is rotated from $\\theta _k$ to $\\theta _k^{\\prime }=\\theta _\\ell +\\Delta \\theta $ .",
"This $\\Delta \\theta \\in [0,\\pi )$ is small at small $T$ due to eq.",
"(REF ).",
"Then, at the next step, the spin $k$ , a neighbor of $\\ell $ , has less chance to be the first to meet the event among other interacting spins due to eq.",
"(REF ).",
"The fact that return probability at small $T$ without feedback in low dimensions is larger than that of the symmetric walk can be explained as follows.",
"The lifting variable behaves following exactly the same probability every time the variable visits a site because the spin configuration never changes.",
"At small $T$ , the probabilistic movement becomes almost deterministic.",
"Therefore there is a finite probability that the variable is trapped in a small region.",
"The simplest example of the 1D case is a pair of sites $k, k+1$ where the variable repeats the movement $k\\rightarrow k+1 \\rightarrow k \\rightarrow \\cdots $ for a long time.",
"Furthermore, we investigated the visit rate which is the rate of sites visited at least once by the lifting variable.",
"We define `cover time' $t_\\text{cover}$ as the time when the increasing visit rate reaches $1/2$ .",
"As shown in fig.",
"REF (j)–(l) and fig.",
"REF (j),(k), the time heavily depends on the situation and the temperature $T$ .",
"As $T$ increases, $t_\\text{cover}$ increases in the case with feedback, while it decreases in the case without feedback.",
"In the 3D cases with or without feedback, the slope changes at the critical point almost discontinuously.",
"In the 2D case without feedback on large lattices, the time seems to diverge toward the critical point.",
"We consider that the feedback to spin configuration enhances diffusion of lifting variable because the difference between the results of with and without feedback suggests that a lifting variable with feedback is not be trapped in a domain.",
"The behavior of return probability and cover time is consistent with the anomalous behavior of random walks."
],
[
"Discussions and Conclusions",
"We have defined the lifting variable random walk in the ECMC of spin systems and have investigated the case of the classical $XY$ model numerically.",
"We have shown that it becomes superdiffusion near the critical point.",
"This could explain the rapid mixing and the efficiency of the ECMC even at the critical point and it is consistent with arguments in Refs.",
"[6], [7].",
"Lifting variable random walk could be useful for searching a critical point of an unsolved spin model.",
"One could find it by just locating the parameter range where the lifting variable random walk becomes superdiffusive.",
"Dynamics of the ECMC algorithm has also been investigated in Ref. [8].",
"They have considered the joint probability distribution of the lifting variable and the spin configuration and have obtained the master equation it obeys.",
"If the spin configuration was integrated out in their equation, we should obtain the Fokker-Plank equation that describes the lifting variable random walk with the feedback.",
"The present model has persistent nature.",
"The probability of a movement depends on the previous movement.",
"It is known that the persistent random walk becomes normal diffusion in the long time limit[20].",
"Therefore being persistent alone does not explain our result.",
"To confirm this observation, we run an additional simulation of the persistent random walk, the second order Markov process, whose position $x(t+1)$ is equal to $x(t-1)$ with the return probability $r$ defined by the data in fig.",
"REF (g)–(i) and is equal to other possible sites at probability $(1-r)/(2d-1)$ .",
"Such walk does not indeed have the anomalous diffusion exponent $d_\\mathrm {w}$ of lifting variable random walk with or without feedback.",
"Super and subdiffusion can arise in random walks with a long jump[21], [22] or waiting time with a continuous distribution[23].",
"It seems unlikely that the anomalous diffusion is explained in these frameworks.",
"subdiffusion emerges when there are obstacles or binding sites [24], [25], [26].",
"In our model, some spin regions could behave as obstacles, however, our results can not be fully explained by the above theory because the walker in our system can become superdiffusive.",
"In the lifting variable random walk, the current step has a correlation with the step far before because the current spin configuration is constructed by the past movements.",
"In this respect, the random walk in an environment with feedback is closely related to the random walk with memory[12] but has more degrees of freedom: the environment continuous spin variables.",
"It has been reported that random walk with memory can be subdiffusive or superdiffusive[27].",
"The ECMC is being developed and applied to various models including classical Heisenberg model, the $\\mathrm {O}(n)$ models and others[28], [7], [29].",
"It would be interesting to investigate the lifting variable random walk of these models and to see how it is related to the underlying critical phenomena.",
"This is left for future works.",
"We are grateful to Shinji Iida and Junta Matsukidaira for discussions.",
"We also thank an anonymous referee for bringing our attention to the anomalous diffusion exponent for inifinite size system."
]
] | 1709.01665 |
[
[
"Opening the Black Box of Financial AI with CLEAR-Trade: A CLass-Enhanced\n Attentive Response Approach for Explaining and Visualizing Deep\n Learning-Driven Stock Market Prediction"
],
[
"Abstract Deep learning has been shown to outperform traditional machine learning algorithms across a wide range of problem domains.",
"However, current deep learning algorithms have been criticized as uninterpretable \"black-boxes\" which cannot explain their decision making processes.",
"This is a major shortcoming that prevents the widespread application of deep learning to domains with regulatory processes such as finance.",
"As such, industries such as finance have to rely on traditional models like decision trees that are much more interpretable but less effective than deep learning for complex problems.",
"In this paper, we propose CLEAR-Trade, a novel financial AI visualization framework for deep learning-driven stock market prediction that mitigates the interpretability issue of deep learning methods.",
"In particular, CLEAR-Trade provides a effective way to visualize and explain decisions made by deep stock market prediction models.",
"We show the efficacy of CLEAR-Trade in enhancing the interpretability of stock market prediction by conducting experiments based on S&P 500 stock index prediction.",
"The results demonstrate that CLEAR-Trade can provide significant insight into the decision-making process of deep learning-driven financial models, particularly for regulatory processes, thus improving their potential uptake in the financial industry."
],
[
"Introduction",
"Do machine learning algorithms need to be explainable?",
"This is an important question in today's world where machine learning algorithms, especially those based on deep learning are being used at a wide range of tasks and have shown tremendous efficacy in performing these tasks.",
"Deep learning has touted as being very disruptive to many sectors, particularly the finance sector.",
"However, deep learning, to large extent, have essentially been unexplainable \"black boxes\", with no clear explanation as to how they reach particular decisions [6].",
"This is a major hindrance to the widespread adoption of deep learning in industries like finance, where regulations are very tight.",
"In such industries with strict regulatory processes, the AI models used are required to be transparent, interpretable, and explainable.",
"Many experts in these sectors believe that relying on such 'black box' methods is a growing problem that is already very relevant due to regulatory processes in these sectors, and it is going to be increasingly more relevant in the future.",
"For example, in finance, law requires companies to explain the reason behind every decision to its perspective customer [3].",
"As such, current approaches for leveraging deep learning are not feasible such in these scenarios.",
"The limitation of deep learning in terms of transparency and interpretability have forced industries dealing with regulatory scenarios to use comparatively simple machine learning algorithms such as linear or logistic regression, decision trees, or ensemble methods such as random forests which are significantly more explainable and quite effective in simple cases.",
"However, as the complexity of the problem increases, which is very true in finance, deep learning algorithms have been shown to outperform such traditional algorithms by a wide margin across a wide range of problem domains [2].",
"As such, strategies for explaining the decisions made by deep learning algorithms are highly desired to enable their widespread use in sectors that have strong regulatory processes.",
"More recently, a number of methods were proposed to mitigate this issue of interpretability and transparency in deep learning.",
"For example, Zeiler & Fergus [7] proposed the formation of a parallel deconvolutional network to peer into different units of the network.",
"Ribeiro [4] introduced a method to build trust in models that are locally accurate, i.e., it is correct near the input data sample.",
"Selvaraju et.al.",
"[5] proposed a method called Grad-CAM that enables users to discern \"strong\" networks from the weaker ones.",
"While promising, all of the aforementioned approaches are restricted to identifying regions of interest and their influence in the decision made by the deep neural network only, thus restricting their utility for gaining a more detailed understanding of the decision process.",
"To address this issue, Kumar et.",
"al.",
"[1] recently proposed a CLass Enhanced Attentive Response (CLEAR) approach that not only identifies attentive regions of interest and their influence on the decision made, but more important provides the dominant classes associated with the attentive regions of interest.",
"This additional information about the dominant classes and their influence on the decision making progress leads to a higher degree of human interpretability, which makes it very well suited for scenarios that necessitate regulatory processes such as in finance.",
"Motivated by this, in this paper, we propose CLEAR-Trade, a CLass Enhanced Attentive Response approach to explaining and visualizing deep learning-driven stock market prediction.",
"In particular, CLEAR-Trade is designed in this paper to provide detailed explanations for the prediction decisions made by a deep learning-driven binary stock market prediction network, as shown in Fig.",
"REF .",
"Our aim is to create a powerful tool for peering into the minds of these otherwise uninterpretable 'black box' financial AI models to better visualize and understand why they are making the decisions the way they do.",
"Doing this will have a tremendous impact on day-to-day work of financial analysts in helping them better understand these deep learning-driven financial AI models, thus potentially enabling the widespread adoption of transparent financial AI.",
"Figure: Two different scenarios for stock market prediction using deep learning-driven financial AI models: a) Stock market prediction without interpretability, b)interpretable stock market prediction via CLEAR-Trade.",
"The proposed CLEAR-Trade visualization framework improves financial model interpretability by providing effective visual interpretations of the decision-making process.",
"CLEAR-Trade allows for the visualization of i) the attentive time windows responsible for stock market prediction decisions based on the financial AI model (marked in red and green), ii) their level of contribution to the stock market prediction decision (in this case, stock market index rise (green) or stock market index fall (red)), as well as iii) the dominant state (rise or fall) associated with each attentive time window.",
"This visualization enables financial analysts to better understand the rationale behind the stock market prediction decisions made based by the deep learning-driven financial AI model.Figure: An overview of the proposed CLEAR-Trade visualization framework for explaining and visualizing deep learning-driven stock market prediction.",
"As an illustrative case in this paper, for predicting whether a stock market index will rise or fall (two states), the index's past 30 days of trade data is fed into the deep learning-driven financial AI model and individual attentive response maps are computed for each state (stock market index rise or stock market index down) based on the last layer of the deep learning-driven financial AI model.",
"Based on this set of attentive response maps, two different maps are computed: 1) a dominant attentive response map, which shows the level of contribution of each time point to the decision-making process, and 2) a dominant state attentive map, which shows the dominant state associated with each time point influencing the decision-making process.",
"Finally, the dominant attentive response map and the dominant attentive state map are combined to produce the final CLEAR-Trade visualization, thus enabling the financial analyst to visualize the factors leveraged by the deep learning-driven financial AI model in predicting whether the stock market index will rise (green) or fall (red)."
],
[
"Methodology",
"With the goal of enabling transparent and interpretable deep learning-driven stock market prediction, the proposed CLEAR-Trade visualization framework presents the financial analyst with the following information pertaining to the decision-making process: the attentive time windows responsible for the decision made by the financial AI model; the attentive levels at these attentive time windows so that their level of influence over the decision made by the financial AI model can be understood; and the dominant state (in this paper, stock market index rise or fall) associated with these attentive time windows so that we can better understand why a decision was made.",
"The procedure for obtaining the CLEAR-Trade visualization for stock market prediction (in this case, predicting stock market indices but can also be applied to individual stocks) is shown in Fig.",
"REF and can be explained as follows.",
"First, a forward pass with a time-series input of historical trade information about a particular stock market index (in this case, an index's 30 days worth of open, close, highs, lows, and trade volumes) is performed through the deep learning-driven financial AI model and a stock market prediction decision output is obtained.",
"To create a CLEAR-Trade visualization associated with this particular stock market prediction decision, we first compute a set of individual response maps $\\left\\lbrace R(\\underline{x}\\vert s)|1 \\le s \\le K\\right\\rbrace $ , where $K$ is the total number of states present for stock market prediction (in this case, there are two states: stock market index rise and fall).",
"The deep learning-driven financial AI model is set up such that it contains similar number of kernels in the last layer as the number of states.",
"To elaborate the process, first consider the response for all the kernels at the the last layer $l$ of the financial AI model which can be calculated as: $\\hat{h}_{l} = \\sum _{k=1}^K z_{k,l} * w_{k,l} .$ where $*$ denotes the convolution operation.",
"To calculate the response of last layer in the input domain, we can extend this formulation for response of the specific kernel $s\\epsilon \\lbrace 1...K\\rbrace $ of the deep learning-driven financial AI model with Un-pooling layer $P^{\\prime }$ as: ${R(\\underline{x}\\vert s)} = H_{1}P^{\\prime }_{1}H_{2}P^{\\prime }_{2} ....H_{L-1}P^{\\prime }_{L-1}H_{L}^s z_{L}.$ where $H$ denotes the combined operation of convolutional and summation, for notation brevity.",
"$H_{L}^s$ represents the convolution matrix operation in which the kernel weights $w_{L}$ are all zero except that at the $s$th time point.",
"Given the set of individual attentive response maps, we then compute the dominant attentive state map, $\\hat{S}(\\underline{x})$ , by finding the state that maximizes the attentive response level, $R(\\underline{x}\\vert s)$ , across all states: $\\vspace{-5.69046pt}\\hat{S}(\\underline{x}) = \\operatornamewithlimits{argmax}\\limits _{s} {R(\\underline{x} \\vert s)} .$ Given the dominant attentive state map, $\\hat{S}(\\underline{x})$ , we can now compute the dominant attentive response map, $D_{\\hat{S}}(\\underline{x})$ , by selecting the attentive response level at a particular time point based on the identified dominant state, which can be expressed as follows: $D_{\\hat{S}}(\\underline{x}) = R(\\underline{x}\\vert \\hat{S}) .$ To form the final CLEAR-Trade visualization, we map the dominant state attentive map and the dominant attentive response map in the HSV (S in HSV is indicated as S' below to avoid confusion with state $S$ ) color space as follows: $\\begin{split}H & = F(\\hat{S}(\\underline{x})) ,\\ \\\\S^{\\prime } & = 1 ,\\ \\\\V & = D_{\\hat{S}}(\\underline{x}) .\\ \\end{split}$ where $F(.",
")$ is the color map dictionary that assigns an individual color to each dominant attentive state, $s$ .",
"Fig.",
"REF shows an example of the CLEAR-Trade visualization.",
"Figure: Architecture of the deep convolutional financial AI model used for stock market prediction process.",
"The financial AI model is embedded inthe CLEAR-Trade visualization process, which augments a set of fully convolutionallayers, a leaky rectified linear unit layer, global average pooling (GAP)and a softmax layer at the end of the model for the learning process."
],
[
"Experiments and Results",
"This section explains the experimental setup, the deep learning-driven financial AI model built for performing binary stock market prediction on the S&P 500 stock market index, and the experimental results for obtaining the efficacy of the CLEAR-Trade visualization in creating interpretable and transparent deep learning-driven financial AI models for stock market prediction.",
"Figure: Correctly and mis-classified binary stock market prediction for predicting the S&P 500 stock market index as visualized and explained using CLEAR-Trade for a deep learning-driven financial AI model trained for experimental purposes.",
"For both cases, the CLEAR-Trade visualizations (left) pinpoint the attentive time windows that are responsible for the particular stock market prediction decision made by the financial AI model (green for index rise and red for index fall).",
"In the CLEAR-Trade visualizations, the thickness of color lines indicate the influence of the attentive time windows on the prediction decision output.",
"On the right hand side, the time-wise stock data information sheet (open, high, low, close, and trade volume) of one of the graphs is shown in the descending order from the day on which decision is being made.",
"Here also, the attentive levels (transparency of color) and the dominant state (green or red) associated with these attentive time windows is shown so that we can better understand why a decision was made."
],
[
"Experimental Setup",
"For training purposes, we selected the last three years worth of trade data of the S&P 500 stock market index to train a deep convolutional neural network, shown in Fig.",
"REF , as the deep learning-driven AI financial model used in this study.",
"For preparing this data for training the financial model, we divided the data into 30-day time segments and treated the state (index rise or index fall) on the 31st day as '1' if the index was higher than previous day or '0' if the index was lower.",
"We used 90% of the data as training set and consider 10% for evaluation purposes.",
"The trained deep learning-driven financial AI model achieved a prediction accuracy of 61.22%, though it is important to note that the focus of this paper is on the ability to visualize and understanding the decision-making process of the financial AI model, not in attaining the best possible accuracy, and therefore improving the accuracy of the reference model are plans for future work."
],
[
"Stock Market Prediction Results",
"To present the effectiveness of the CLEAR-Trade visualization (as explained in Section ) to enable interpretable deep learning-driven financial AI, we used the trained stock market predictive model and obtained the CLEAR-Trade visualization results as shown in Fig.",
"REF .",
"In Fig.",
"REF , for both cases (correct and wrong predictions), it can be clearly observed from the CLEAR-Trade visualizations which time windows are most crucial to the decision-making process of the financial AI model for reaching a particular stock market prediction.",
"Specifically, in the correctly predicted cases, it can be observed that the deep learning-driven financial AI model primarily leveraged the past four days of trade data for correctly predicting whether the S&P 500 stock market index will rise or fall.",
"This is intuitive as the past few days are more likely to have a major effect on the index's behavior compared to data from a couple of weeks back.",
"In the case whether the financial AI model gets the stock market prediction incorrect, we can observe that the model primarily leverages trade data for nearly 3 weeks ago to making its decision.",
"Another observation that can be made in both cases is that in the cases where the stock market prediction is correct, the deep learning-driven financial AI model leverages only open, high and low values to make a decision.",
"This trend if leveraging only open, high, and low values is observed across the majority of correct predictions made by financial AI model.",
"This is again intuitive as unless there is a significant change in trade volume, knowledge of trade volume generally does not have a significant impact on either index rise or fall.",
"Conversely, it can be seen that when making incorrect decisions, the financial AI model strongly takes into account the trade volume as well, which is not a strong predictive feature as indicate above and as such can incorrectly influence its decisions.",
"Finally, it can be observed that the confidence of the stock market prediction in this case is low when it makes incorrect predictions, as indicated by the color transparency in the data-sheet, while in correct predictions the confidence of the stock market prediction is high.",
"Hence, based on the above mentioned observations, it is evident that CLEAR-Trade visualization not only provides a justification for particular stock market prediction decision output, it can also provide considerable insights that financial analysts can taken into account while making trading decisions."
],
[
"Conclusion",
"In this paper, we proposed CLEAR-Trade, a visualization framework that provides insight into the minds of deep learning-driven financial AI models used for stock market prediction by visualizing and explaining the decision-making process of the model.",
"Experiments pertaining to stock market prediction for the S&P 500 index showed that CLEAR-Trade visualization leads to a higher degree of human interpretability and transparency in predictions made using deep learning-driven financial AI models, hence paving a way for their use in regulatory settings.",
"The proposed visualization approach has tremendous potential to create industry-wide effect by facilitating the use of state-of-the art deep learning models for areas in finance that are under significant regulations."
],
[
"Acknowledgments",
"This research has been supported by Canada Research Chairs programs, Natural Sciences Engineering Research Council of Canada (NSERC), and Canada Foundation for Innovation (CFI)."
]
] | 1709.01574 |
[
[
"Knowledge Transfer Between Artificial Intelligence Systems"
],
[
"Abstract We consider the fundamental question: how a legacy \"student\" Artificial Intelligent (AI) system could learn from a legacy \"teacher\" AI system or a human expert without complete re-training and, most importantly, without requiring significant computational resources.",
"Here \"learning\" is understood as an ability of one system to mimic responses of the other and vice-versa.",
"We call such learning an Artificial Intelligence knowledge transfer.",
"We show that if internal variables of the \"student\" Artificial Intelligent system have the structure of an $n$-dimensional topological vector space and $n$ is sufficiently high then, with probability close to one, the required knowledge transfer can be implemented by simple cascades of linear functionals.",
"In particular, for $n$ sufficiently large, with probability close to one, the \"student\" system can successfully and non-iteratively learn $k\\ll n$ new examples from the \"teacher\" (or correct the same number of mistakes) at the cost of two additional inner products.",
"The concept is illustrated with an example of knowledge transfer from a pre-trained convolutional neural network to a simple linear classifier with HOG features."
],
[
"Introduction",
"Knowledge transfer between Artificial Intelligent systems has been the subject of extensive discussion in the literature for more than two decades [1], [2], [3], [4].",
"State-of-the art approach to date is to use, or salvage, parts of the “teacher” AI system in the “student” AI followed by re-training of the “student” [5], [6].",
"Alternatives to AI salvaging include model compression [7], knowledge distillation [8], and privileged information [9].",
"These approaches demonstrated substantial success in improving generalization capabilities of AIs as well as in reducing computational overheads [10], in cases of knowledge transfer from larger AI to the smaller one.",
"Notwithstanding, however, which of the above strategies is followed, their implementation often requires either significant resources including large training sets and power needed for training, or access to privileged information that may not necessarily be available to end-users.",
"Thus new frameworks and approaches are needed.",
"In this contribution we provide new framework for automated, fast, and non-destructive process of knowledge spreading across AI systems of varying architectures.",
"In this framework, knowledge transfer is accomplished by means of Knowledge Transfer Units comprising of mere linear functionals and/or their small cascades.",
"Main mathematical ideas are rooted in measure concentration [11], [12], [13], [14], [15] and stochastic separation theorems [16] revealing peculiar properties of random sets in high dimensions.",
"We generalize some of the latter results here and show how these generalizations can be employed to build simple one-shot Knowledge Transfer algorithms between heterogeneous AI systems whose state may be represented by elements of linear vector space of sufficiently high dimension.",
"Once knowledge has been transferred from one AI to another, the approach also allows to “unlearn” new knowledge without the need to store a complete copy of the “student” AI is created prior to learning.",
"We expect that the proposed framework may pave way for fully functional new phenomenon – Nursery of AI systems in which AIs quickly learn from each other whilst keeping their pre-existing skills largely intact.",
"The paper is organized as follows.",
"Section contains mathematical background needed to justify the proposed knowledge transfer algorithms.",
"In Section we present two algorithms for transferring knowledge between a pair of AI systems in which one operates as a teacher and the other functions as a student.",
"Section illustrates the approach with examples, and Section concludes the paper."
],
[
"Mathematical background",
"Let the set $\\mathcal {M}=\\lbrace x_1,\\dots ,x_M\\rbrace $ be an i.i.d.",
"sample from a distribution in $\\mathbb {R}^n$ .",
"Pick another set $\\mathcal {Y}=\\lbrace x_{M+1},\\dots ,x_{M+k}\\rbrace $ from the same distribution at random.",
"What is the probability that there is a linear functional separating $\\mathcal {Y}$ from $\\mathcal {M}$ ?",
"Below we provide three $k$ -tuple separation theorems: for an equidistribution in $B_n(1)$ (Theorem REF and REF ) and for a product probability measure with bounded support (Theorem REF ).",
"These two special cases cover or, indeed, approximate broad range of practically relevant situations including e.g.",
"Gaussian distributions (reduce asymptotically to equidistribution in $B_n(1)$ for $n$ large enough) and data vectors in which each attribute is a numerical and independent random variable.",
"Consider the case when the underlying probability distribution is an equidistribution in the unit ball $B_n(1)$ , and suppose that $\\mathcal {M}=\\lbrace x_1,\\dots ,x_M\\rbrace $ and $\\mathcal {Y}=\\lbrace x_{M+1},\\dots ,x_{M+k}\\rbrace $ are i.i.d.",
"samples from this distribution.",
"We are interested in determining the probability $\\mathcal {P}_1(\\mathcal {M},\\mathcal {Y})$ that there exists a linear functional $l$ separating $\\mathcal {M}$ and $\\mathcal {Y}$ .",
"An estimate of this probability is provided in the following theorem Theorem 1 Let $\\mathcal {M}=\\lbrace x_1,\\dots ,x_M\\rbrace $ and $\\mathcal {Y}=\\lbrace x_{M+1},\\dots ,x_{M+k}\\rbrace $ be i.i.d.",
"samples from the equidisribution in $B_n(1)$ .",
"Then $\\begin{split}{\\mathcal {P}}_1(\\mathcal {M},\\mathcal {Y})& \\ge \\max _{\\delta ,\\varepsilon } \\ (1-(1-\\varepsilon )^n)^{k} \\prod _{m=1}^{k-1} \\left(1-m \\left(1-\\delta ^2\\right)^{\\frac{n}{2}}\\right) \\left(1 -\\frac{\\Delta (\\varepsilon ,\\delta ,k)^\\frac{n}{2}}{2}\\right)^{M} \\\\\\Delta (\\varepsilon ,\\delta ,k)&= 1-\\left[\\frac{(1-\\varepsilon )\\sqrt{1-(k-1)\\delta ^2}}{\\sqrt{k}}-(k-1)^{\\frac{1}{2}}\\delta \\right]^2\\\\& \\mathrm {Subject} \\ \\mathrm { to:}\\\\& \\delta , \\varepsilon \\in (0,1)\\\\& 1-(k-1)\\delta ^2 \\ge 0\\\\& (k-1)(1-\\delta ^2)^{\\frac{n}{2}}\\le 1\\\\& \\frac{(1-\\varepsilon )\\sqrt{1-(k-1)\\delta ^2}}{\\sqrt{k}}-(k-1)^{\\frac{1}{2}}\\delta \\ge 0.\\end{split}$ Proof of Theorem REF.",
"Given that elements in the set $\\mathcal {Y}$ are independent, the probability $p_1$ that $\\mathcal {Y} \\subset B_n(1)\\setminus B_n(1-\\varepsilon )$ is $p_1=(1-(1-\\varepsilon )^n)^k.$ Consider an auxiliary set $\\hat{\\mathcal {Y}}=\\left\\lbrace \\hat{x}_{i}\\in \\mathbb {R}^n \\ | \\ \\hat{x}_i=(1-\\varepsilon )\\frac{x_{M+i}}{\\Vert x_{M+i}\\Vert }, \\ i=1,\\dots ,k \\right\\rbrace .$ Vectors $\\hat{x}_i\\in \\hat{\\mathcal {Y}}$ belong to the sphere of radius $1-\\varepsilon $ centred at the origin (see Figure REF , (b)).",
"Figure: Illustration to the proof of Theorem .",
"Panel (a) shows x M+1 x_{M+1}, x M+2 x_{M+2} and x M+3 x_{M+3} in the set B n (1)∖B n (1-ε)B_n(1)\\setminus B_n(1-\\varepsilon ).",
"Panel (b) shows x ^ 1 \\hat{x}_1, x ^ 2 \\hat{x}_2, and x ^ 3 \\hat{x}_3 on the sphere S n-1 (1-ε)S_{n-1}(1-\\varepsilon ).",
"Panel (c): construction of h 3 h_3.",
"Note that ∥h 3 ∥=∥x ^ 3 ∥(1-2δ 2 ) 1/2 =(1-ε)(1-2δ 2 ) 1/2 \\Vert h_3\\Vert =\\Vert \\hat{x}_3\\Vert (1-2\\delta ^2)^{1/2}=(1-\\varepsilon )(1-2\\delta ^2)^{1/2}.",
"Panel (d) shows simplex formed by orthogonal vectors h ^ 1 ,h ^ 2 ,h ^ 3 \\hat{h}_1,\\hat{h}_2,\\hat{h}_3.",
"Panel (e) illustrates derivation of functionals ll and l 0 l_0.According to [17] (proof of Proposition 3 and estimate (26)), the probability $p_2$ that for a given a given $\\delta \\in (0,1)$ all elements of $\\hat{\\mathcal {Y}}$ are pair-wise $\\delta /(1-\\varepsilon )$ -orthogonal, i.e.",
"$\\left|\\cos \\left(\\hat{x}_{i},\\hat{x}_{j} \\right)\\rangle \\right| \\le \\frac{\\delta }{1-\\varepsilon } \\ \\mbox{for all} \\ i,j\\in \\lbrace 1,\\dots ,k\\rbrace , \\ i\\ne j,$ can be estimated from below as: $p_2 \\ge p_1 \\prod _{m=1}^{k-1} \\left(1-m \\left(1-\\delta ^2\\right)^{\\frac{n}{2}}\\right) = (1-(1-\\varepsilon )^n)^k \\prod _{m=1}^{k-1} \\left(1-m \\left(1-\\delta ^2\\right)^{\\frac{n}{2}}\\right).$ for $(k-1)(1-\\delta ^2)^{\\frac{n}{2}}\\le 1$ .",
"Suppose now that (REF ) holds true.",
"Let $\\delta $ be chosen so that $1-(k-1)\\delta ^2\\ge 0$ .",
"If this is the case than there exists a set of $k$ pair-wise orthogonal vectors $\\mathcal {H}=\\lbrace h_1,h_2,\\dots ,h_{k}\\rbrace , \\ \\langle h_i,h_j\\rangle =0, \\ i,j\\in \\lbrace 1,\\dots ,k\\rbrace , \\ i\\ne j,$ such that (Figure REF , (c)) $\\Vert \\hat{x}_i-h_i\\Vert \\le (i-1)^{\\frac{1}{2}}\\delta , \\ \\Vert h_i\\Vert =(1-\\varepsilon )(1-(i-1)\\delta ^2)^{\\frac{1}{2}}, \\ \\mbox{for all} \\ i\\in \\lbrace 1,\\dots ,k\\rbrace .$ Finally, consider the set $\\hat{\\mathcal {H}}=\\left\\lbrace \\hat{h}_{i}\\in \\mathbb {R}^n \\ | \\ \\hat{h}_i=(1-\\varepsilon )(1-(k-1)\\delta ^2)^{\\frac{1}{2}}\\frac{h_{i}}{\\Vert h_{i}\\Vert }, \\ i=1,\\dots ,k \\right\\rbrace $ The set $\\hat{\\mathcal {H}}$ belongs to the sphere of radius $(1-(k-1)\\delta ^2)^{\\frac{1}{2}}$ , and its $k$ elements are vertices of the corresponding $k-1$ -simplex in $\\mathbb {R}^n$ (Figure REF , (d)).",
"Consider the functional: $l(x)=\\left\\langle \\frac{\\bar{h}}{\\Vert \\bar{h}\\Vert }, x \\right\\rangle - \\frac{(1-\\varepsilon )\\sqrt{1-(k-1)\\delta ^2}}{\\sqrt{k}}, \\ \\bar{h}=\\frac{1}{k}\\sum _{i=1}^{k} \\hat{h}_i.$ Recall that if $e_1,\\dots ,e_k$ are orthonormal vectors in $\\mathbb {R}^n$ then $\\Vert e_1 + e_2 + \\cdots + e_k \\Vert ^2 = k$ .",
"Hence $\\left\\Vert \\sum _{i=1}^{k} \\hat{h}_i \\right\\Vert =\\sqrt{k}(1-\\varepsilon )\\sqrt{1-(k-1)\\delta ^2}$ , and we can conclude that $l(\\hat{h}_i)=0$ and $l(h_i)\\ge 0$ for all $i=1,\\dots ,k$ .",
"According to (REF ), $\\Vert \\hat{x}_i -h_i\\Vert \\le (k-1)^{\\frac{1}{2}}\\delta $ for all $i=1,\\dots ,k$ .",
"Therefore the functional $l_0(x)=l(x)+(k-1)^{\\frac{1}{2}}\\delta = \\left\\langle \\frac{\\bar{h}}{\\Vert \\bar{h}\\Vert }, x \\right\\rangle - \\left(\\frac{(1-\\varepsilon )\\sqrt{1-(k-1)\\delta ^2}}{\\sqrt{k}}-(k-1)^{\\frac{1}{2}}\\delta \\right)$ satisfies the following condition: $l_0(\\hat{x}_i)\\ge 0$ and $l_0({x}_{M+i})\\ge 0$ for all $i=1,\\dots ,k$ .",
"This is illustrated with Figure REF , (e).",
"The functional $l_0$ partitions the unit ball $B_n(1)$ into the union of two disjoint sets: the spherical cap $\\mathcal {C}$ $\\mathcal {C}=\\lbrace x\\in B_n(1) \\ | l_0(x)\\ge 0 \\rbrace $ and its complement in $B_n(1)$ , $B_n(1)\\setminus \\mathcal {C}$ .",
"The volume $\\mathcal {V}$ of the cap $\\mathcal {C}$ can be estimated from above as $\\begin{split}\\mathcal {V}(\\mathcal {C})&\\le \\frac{\\Delta (\\varepsilon ,\\delta ,k)^\\frac{n}{2}}{2}, \\\\\\Delta (\\varepsilon ,\\delta ,k)&=1-\\left[\\frac{(1-\\varepsilon )\\sqrt{1-(k-1)\\delta ^2}}{\\sqrt{k}}-(k-1)^{\\frac{1}{2}}\\delta \\right]^2.\\end{split}$ Hence the probability $p_3$ that $l_0({x}_i)<0$ for all $x_i\\in \\mathcal {M}$ can be estimated from below as $p_3\\ge \\left(1 - \\frac{\\Delta (\\varepsilon ,\\delta ,k)^\\frac{n}{2}}{2}\\right)^{M}.$ Therefore, for fixed $\\varepsilon ,\\delta \\in (0,1)$ chosen so that $\\frac{(1-\\varepsilon )\\sqrt{1-(k-1)\\delta ^2}}{\\sqrt{k}}-(k-1)^{\\frac{1}{2}}\\delta \\ge 0$ , the probability $p_4(\\varepsilon ,\\delta )$ that $\\mathcal {M}$ can be separated from $\\mathcal {Y}$ by the functional $l_0$ can be estimated from below as: $p_4(\\varepsilon ,\\delta ) \\ge (1-(1-\\varepsilon )^n)^{k} \\prod _{m=1}^{k-1} \\left(1-m \\left(1-\\delta ^2\\right)^{\\frac{n}{2}}\\right) \\left(1 - \\frac{\\Delta (\\varepsilon ,\\delta ,k)^\\frac{n}{2}}{2}\\right)^{M}.$ Given that this estimate holds for all feasible values of $\\varepsilon ,\\delta $ , statement (REF ) follows.",
"$\\square $ Figure REF shows how estimate (REF ) of the probability $\\mathcal {P}_{1}(\\mathcal {M},\\mathcal {Y})$ behaves, as a function of $|\\mathcal {Y}|$ for fixed $M$ and $n$ .",
"As one can see from this figure, when $k$ exceeds some critical value ($k=9$ in this specific case), the lower bound estimate (REF ) of the probability $\\mathcal {P}_{1}(\\mathcal {M},\\mathcal {Y})$ drops.",
"This is not surprising since the bound (REF ) is a) based on rough, $L_\\infty $ -like, estimates, and b) these estimates are derived for just one class of separating functionals $l_0(x)$ .",
"Furthermore, no prior pre-processing and/or clustering was assumed for the $\\mathcal {Y}$ .",
"An alternative estimate that allows us to account for possible clustering in the set $\\mathcal {Y}$ is presented in Theorem REF .",
"Figure: Estimate () of 𝒫 1 (ℳ,𝒴)\\mathcal {P}_{1}(\\mathcal {M},\\mathcal {Y}) as a function of kk for n=2000n=2000 and M=10 5 M=10^5.Theorem 2 Let $\\mathcal {M}=\\lbrace x_1,\\dots ,x_M\\rbrace $ and $\\mathcal {Y}=\\lbrace x_{M+1},\\dots ,x_{M+k}\\rbrace $ be i.i.d.",
"samples from the equidistribution in $B_n(1)$ .",
"Let $\\mathcal {Y}_c=\\lbrace x_{M+r_1},\\dots ,x_{M+r_m}\\rbrace $ be a subset of $m$ elements from $\\mathcal {Y}$ such that $\\beta _2 (m-1) \\le \\sum _{r_j,\\ r_j\\ne r_i} \\langle x_{M+r_i}, x_{M+r_j}\\rangle \\le \\beta _1 (m-1) \\ \\mbox{for all} \\ i=1,\\dots ,m.$ Then $\\begin{split}{\\mathcal {P}}_1(\\mathcal {M},\\mathcal {Y}_c)& \\ge \\max _{\\varepsilon \\in (0,1)} (1-(1-\\varepsilon )^n)^k \\left(1 -\\frac{\\Delta (\\varepsilon ,m)^\\frac{n}{2}}{2}\\right)^{M} \\\\\\Delta (\\varepsilon ,m)&=1- \\frac{1}{m}\\left(\\frac{(1-\\varepsilon )^2 + \\beta _2 (m-1)}{\\sqrt{1+(m-1)\\beta _1}}\\right)^2\\\\& \\mathrm {Subject} \\ \\mathrm { to:}\\\\& (1-\\varepsilon )^2 + \\beta _2 (m-1) > 0 \\\\& 1+(m-1)\\beta _1 >0.\\end{split}$ Proof of Theorem REF.",
"Consider the set $\\mathcal {Y}$ .",
"Observe that $\\Vert x_{M+i}\\Vert \\ge 1-\\varepsilon $ , $\\varepsilon \\in (0,1)$ , for all $i=1,\\dots ,k$ , with probability $p_1= (1-(1-\\varepsilon )^n)^k$ .",
"Consider now the vector $\\bar{y}$ $\\bar{y}=\\frac{1}{m}\\sum _{i=1}^{m} x_{M+r_i},$ and evaluate the following inner products $\\left\\langle \\frac{\\bar{y}}{\\Vert \\bar{y}\\Vert }, x_{M+r_i} \\right\\rangle =\\frac{1}{m \\Vert \\bar{y}\\Vert } \\left(\\langle x_{M+r_i},x_{M+r_i} \\rangle + \\sum _{r_j, \\ j\\ne i} \\langle x_{M+r_i},x_{M+r_j} \\rangle \\right), \\ i=1,\\dots ,m.$ According to assumption (REF ), with probability $p_1$ , $\\left\\langle \\frac{\\bar{y}}{\\Vert \\bar{y}\\Vert }, x_{M+r_i} \\right\\rangle \\ge \\frac{1}{m \\Vert \\bar{y}\\Vert } \\left((1-\\varepsilon )^2 + \\beta _2 (m-1) \\right)$ and, respectively, $\\frac{1}{m}\\left(1+(m-1)\\beta _1 \\right)\\ge \\langle \\bar{y},\\bar{y}\\rangle \\ge \\frac{1}{m}\\left((1-\\varepsilon )^2 + \\beta _2 (m-1)\\right)$ Let $(1-\\varepsilon )^2 + \\beta _2(m-1) > 0$ and $(1-\\varepsilon )^2 + \\beta _1(m-1) > 0$ .",
"Consider the functional $l_0(x)=\\left\\langle \\frac{\\bar{y}}{\\Vert \\bar{y}\\Vert }, x \\right\\rangle - \\frac{1}{\\sqrt{m}}\\left(\\frac{(1-\\varepsilon )^2 + \\beta _2 (m-1)}{\\sqrt{1+(m-1)\\beta _1}}\\right).$ It is clear that $l_0(x_{M+r_i})\\ge 0$ for all $i=1,\\dots ,m$ by the way the functional is constructed.",
"The functional $l_0(x)$ partitions the ball $B_n(1)$ into two sets: the set $\\mathcal {C}$ defined as in (REF ) and its complement, $B_n(1)\\setminus \\mathcal {C}$ .",
"The volume $\\mathcal {V}$ of the set $\\mathcal {C}$ is bounded from above as $\\mathcal {V}(\\mathcal {C})\\le \\frac{\\Delta (\\varepsilon ,m)^{\\frac{n}{2}}}{2}$ where $\\Delta (\\varepsilon ,m)=1- \\frac{1}{m}\\left(\\frac{(1-\\varepsilon )^2 + \\beta _2 (m-1)}{\\sqrt{1+\\beta _1 (m-1)}}\\right)^2.$ Estimate (REF ) now follows.",
"$\\square $ Figure: Estimate () of 𝒫 1 (ℳ,𝒴)\\mathcal {P}_{1}(\\mathcal {M},\\mathcal {Y}) as a function of kk for n=2000n=2000 and M=10 5 M=10^5.",
"Red stars correspond to β 1 =0.5\\beta _1=0.5, β 2 =0\\beta _2=0.",
"Blue triangles stand for β 1 =0.5\\beta _1=0.5, β 2 =0.05\\beta _2=0.05, and black circles stand for β 1 =0.5\\beta _1=0.5, β 2 =0.07\\beta _2=0.07.Examples of estimates (REF ) for various parameter settings are shown in Fig.",
"REF .",
"As one can see, in absence of pair-wise strictly positive correlation assumption, $\\beta _1=0$ , the estimate's behavior, as a function of $k$ , is similar to that of (REF ).",
"However, presence of moderate pair-wise positive correlation results in significant boosts to the values of $\\mathcal {P}_1$ .",
"Remark 1 Estimates (REF ), (REF ) for the probability $P_1(\\mathcal {M},\\mathcal {Y})$ that follow from Theorems REF , REF assume that the underlying probability distribution is an equidistribution in $B_n(1)$ .",
"They can, however, be generalized to equidistribuions in ellipsoids and Gaussian distributions (cf.",
"[18]).",
"Note that proofs of Theorems REF , REF are constructive.",
"Not only they provide estimates from below of the probability that two random i.i.d.",
"drawn samples from $B_n(1)$ are linearly separable, but also they present the corresponding separating functionals explicitly as (REF ) and (REF ), respectively.",
"The latter functionals are similar to Fisher linear discriminants.",
"Whilst having explicit separation functionals is an obvious advantage from practical view point, the estimates that are associated with such functionals do not account for more flexible alternatives.",
"In what follows we present a generalization of the above results that accounts for such a possibility as well as extends applicability of the approach to samples from product distributions.",
"The results are provided in Theorem REF .",
"Theorem 3 Consider the linear space $E=\\mathrm {span} \\lbrace x_j-x_{M+1} \\ | \\ j=M+2,\\dots , M+k\\rbrace $ , let the cardinality $|\\mathcal {Y}|=k$ of the set $\\mathcal {Y}$ be smaller than $n$ .",
"Consider the quotient space $\\mathbb {R}^n / E$ .",
"Let $Q(x)$ be a representation of $x\\in \\mathbb {R}^n$ in $\\mathbb {R}^n / E$ , and let the coordinates of $Q(x_{i})$ , $i=1,\\dots ,M+1$ be independent random variables i.i.d.",
"sampled from a product distribution in a unit cube with variances $\\sigma _j>\\sigma _0>0$ , $1 \\le j\\le n-k+1$ .",
"Then for $M\\le \\frac{\\vartheta }{3} \\exp \\left(\\frac{(n-k+1)\\sigma _0^4}{2}\\right)-1$ with probability $p>1-\\vartheta $ there is a linear functional separating $\\mathcal {Y}$ and $\\mathcal {M}$ .",
"Proof of Theorem REF.",
"Observe that, in the quotient space $\\mathbb {R}^n / E$ , elements of the set $\\mathcal {Y}=\\lbrace x_{M+1},x_{M+1}+(x_{M+2}-x_{M+1}),\\dots ,x_{M+1}+(x_{M+k}-x_{M+1})\\rbrace $ are vectors whose coordinates coincide with that of the quotient representation of $x_{M+1}$ .",
"This means that the quotient representation of $\\mathcal {Y}$ consists of a single element, $Q(x_{M+1})$ .",
"Furthermore, dimension of $\\mathbb {R}^n/E$ is $n-k+1$ .",
"Let $R_0^2=\\sum _{i=1}^{n-k+1}\\sigma _i^2$ and $\\bar{Q}(x)=\\mathbb {E}(Q(x))$ .",
"According to Theorem 2 and Corollary 2 from [16], for $\\vartheta \\in (0,1)$ and $M$ satisfying $M\\le \\frac{\\vartheta }{3} \\exp \\left(\\frac{(n-k+1)\\sigma _0^4}{2}\\right) - 1,$ with probability $p>1-\\vartheta $ the following inequalities hold: $\\frac{1}{2} \\le \\frac{\\Vert Q(x_{j})-\\bar{Q}(x)\\Vert ^2}{R_0^2} \\le \\frac{3}{2}, \\ \\left\\langle \\frac{Q(x_{i})-\\bar{Q}(x)}{R_0}, \\frac{Q(x_{M+1})-\\bar{Q}(x)}{\\Vert Q(x_{M+1})-\\bar{Q}(x)\\Vert } \\right\\rangle < \\frac{1}{\\sqrt{2}}$ for all $i,j$ , $i\\ne M+1$ .",
"This implies that the functional $\\ell _0(x)= \\left\\langle \\frac{Q(x)-\\bar{Q}(x)}{R_0}, \\frac{Q(x_{M+1})-\\bar{Q}(x)}{\\Vert Q(x_{M+1})-\\bar{Q}(x)\\Vert } \\right\\rangle - \\frac{1}{\\sqrt{2}}$ separates $\\mathcal {M}$ and $\\mathcal {Y}$ with probability $p > 1-\\vartheta $ .",
"$\\square $"
],
[
"AI Knowledge Transfer Framework",
"In this section we show how Theorems REF , REF and REF can be applied for developing a novel one-shot AI knowledge transfer framework.",
"We will focus on the case of transfer knowledge between two AI systems, a teacher AI and a student AI, in which input-output behaviour of the student AI is evaluated by the teacher AI.",
"In this setting, assignment of AI roles, i.e.",
"student or teaching, is beyond the scope of this manuscript.",
"The roles are supposed to be pre-determined or otherwise chosen arbitrarily."
],
[
"General setup",
"Consider two AI systems, a student AI, denoted as $\\mathrm {AI}_s$ , and a teacher AI, demoted as $\\mathrm {AI}_t$ .",
"These legacy AI systems process some input signals, produce internal representations of the input and return some outputs.",
"We further assume that some relevant information about the input, internal signals, and outputs of $\\mathrm {AI}_s$ can be combined into a common object, $x$ , representing, but not necessarily defining, the state of $\\mathrm {AI}_s$ .",
"The objects $x$ are assumed to be elements of $\\mathbb {R}^n$ .",
"Over a period of activity system $\\mathrm {AI}_s$ generates a set $\\mathcal {S}$ of objects $x$ .",
"Exact composition of the set $\\mathcal {S}$ could depend on a task at hand.",
"For example, if $\\mathrm {AI}_s$ is an image classifier, we may be interested only in a particular subset of $\\mathrm {AI}_s$ input-output data related to images of a certain known class.",
"Relevant inputs and outputs of $\\mathrm {AI}_s$ corresponding to objects in $\\mathcal {S}$ are then evaluated by the teacher, $\\mathrm {AI}_t$ .",
"If $\\mathrm {AI}_s$ outputs differ to that of $\\mathrm {AI}_t$ for the same input then an error is registered in the system.",
"Objects $x\\in \\mathcal {S}$ associated with errors are combined into the set $\\mathcal {Y}$ .",
"The procedure gives rise to two disjoint sets: $\\mathcal {M}=\\mathcal {S}\\setminus \\mathcal {Y}, \\ \\mathcal {M}=\\lbrace x_1,\\dots ,x_{M}\\rbrace $ and $\\mathcal {Y}=\\lbrace x_{M+1},\\dots ,x_{M+k}\\rbrace .$ Figure: AI Knowledge transfer diagram.",
"AI s AI_s produces a set of its state representations, 𝒮\\mathcal {S}.",
"The representations are labelled by AI t AI_t into the set of correct responses, ℳ\\mathcal {M}, and the set of errors, 𝒴\\mathcal {Y}.",
"The student system, AI s AI_s, is then augmented by an additional “corrector” eliminating these errors.A diagram schematically representing the process is shown in Fig.",
"REF .",
"The knowledge transfer task is to “teach” $\\mathrm {AI}_s$ so that with a) $\\mathrm {AI}_s$ does not make such errors b) existing competencies of $\\mathrm {AI}_s$ on the set of inputs corresponding to internal states $x\\in \\mathcal {M}$ are retained, and c) knowledge transfer from $\\mathrm {AI}_t$ to $\\mathrm {AI}_s$ is reversible in the sense that $\\mathrm {AI}_s$ can “unlearn” new knowledge by modifying just a fraction of its parameters, if required.",
"Two algorithms for achieving such transfer knowledge are provided below."
],
[
"Knowledge Transfer Algorithms",
"Our first algorithm, Algorithm REF , considers cases when Auxiliary Knowledge Transfer Units, i.e.",
"functional additions to existing student $\\mathrm {AI}_s$ , are single linear functionals.",
"The second algorithm, Algorithm REF , extends Auxiliary Knowledge Transfer Units to two-layer cascades of linear functionals.",
"Single-functional AI Knowledge Transfer Pre-processing Centering.",
"For the given set $\\mathcal {S}$ , determine the set average, $\\bar{x}(\\mathcal {S})$ , and generate sets $\\mathcal {S}_c$ $\\begin{array}{ll}{\\mathcal {S}_c}&=\\lbrace x\\in \\mathbb {R}^n \\ | x=\\xi -\\bar{x}(\\mathcal {S}), \\ \\xi \\in \\mathcal {S}\\rbrace , \\\\{\\mathcal {Y}_c}&=\\lbrace x\\in \\mathbb {R}^n \\ | x=\\xi -\\bar{x}(\\mathcal {S}), \\ \\xi \\in \\mathcal {Y}\\rbrace .\\end{array}$ Regularization.",
"Determine covariance matrices $\\mathrm {Cov}(\\mathcal {S}_c)$ , $\\mathrm {Cov}(\\mathcal {S}_c\\setminus \\mathcal {Y}_c)$ of the sets $\\mathcal {S}_c$ and $\\mathcal {S}_c\\setminus \\mathcal {Y}_c$ .",
"Let $\\lambda _i(\\mathrm {Cov}(\\mathcal {S}_c))$ , $\\lambda _i(\\mathrm {Cov}(\\mathcal {S}_c\\setminus \\mathcal {Y}_c))$ be their corresponding eigenvalues, and $h_1, \\dots , h_n$ be the eigenvectors of $\\mathrm {Cov}(\\mathcal {S}_c)$ .",
"If some of $\\lambda _i(\\mathrm {Cov}(\\mathcal {S}_c))$ , $\\lambda _i(\\mathrm {Cov}(\\mathcal {S}_c\\setminus \\mathcal {Y}_c))$ are zero or if the ratio $ \\frac{\\max _i \\lbrace \\lambda _i(\\Sigma (\\mathcal {S}_c))\\rbrace }{\\min _i \\lbrace \\lambda _i(\\Sigma (S_c))\\rbrace }$ is too large, project $\\mathcal {S}_c$ and $\\mathcal {Y}_c$ onto appropriately chosen set of $m<n$ eigenvectors, $h_{n-m+1},\\dots ,h_n$ : $\\begin{array}{ll}{\\mathcal {S}_r}&=\\lbrace x\\in \\mathbb {R}^n \\ | x=H^T \\xi , \\ \\xi \\in \\mathcal {S}_c\\rbrace , \\\\{\\mathcal {Y}_r}&=\\lbrace x\\in \\mathbb {R}^n \\ | x=H^T \\xi , \\ \\xi \\in \\mathcal {Y}_c\\rbrace ,\\end{array}$ where $H=\\left(h_{n-m+1} \\cdots h_n\\right)$ is the matrix comprising of $m$ significant principal components of $\\mathcal {S}_c$ .",
"Whitening.",
"For the centered and regularized dataset $\\mathcal {S}_r$ , derive its covariance matrix, $\\mathrm {Cov}(\\mathcal {S}_r)$ , and generate whitened sets $\\begin{array}{ll}{\\mathcal {S}_w}&=\\lbrace x\\in \\mathbb {R}^m \\ | x=\\mathrm {Cov}(\\mathcal {S}_r)^{-\\frac{1}{2}} \\xi , \\ \\xi \\in \\mathcal {S}_r\\rbrace ,\\\\{\\mathcal {Y}_w}&=\\lbrace x\\in \\mathbb {R}^m \\ | x=\\mathrm {Cov}(\\mathcal {S}_r)^{-\\frac{1}{2}} \\xi , \\ \\xi \\in \\mathcal {Y}_r\\rbrace ,\\end{array}$ Knowledge transfer Clustering.",
"Pick $p\\ge 1$ , $p\\le k$ , $p\\in \\mathbb {N}$ , and partition the set $\\mathcal {Y}_w$ into $p$ clusters $\\mathcal {Y}_{w,1},\\dots \\mathcal {Y}_{w,p}$ so that elements of these clusters are, on average, pairwise positively correlated.",
"That is there are $\\beta _{1} \\ge \\beta _{2} > 0$ such that: $\\beta _2(|\\mathcal {Y}_{w,i}|-1)\\le \\sum _{\\xi \\in \\mathcal {Y}_{w,i}\\setminus \\lbrace x\\rbrace } \\langle \\xi ,x\\rangle \\le \\beta _1(|\\mathcal {Y}_{w,i}|-1) \\ \\mbox{for any} \\ x\\in \\mathcal {Y}_{w,i}$ Construction of Auxiliary Knowledge Units.",
"For each cluster $\\mathcal {Y}_{w,i}$ , $i=1,\\dots ,p$ , construct separating linear functionals $\\ell _i$ : $\\begin{array}{ll}\\ell _i(x)&=\\left\\langle \\frac{w_i}{\\Vert w_i\\Vert },x\\right\\rangle - c_i,\\\\w_i&=\\left(\\mathrm {Cov}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}) + \\mathrm {Cov}(\\mathcal {Y}_{w,i}) \\right)^{-1} \\left(\\bar{x}(\\mathcal {Y}_{w,i}) - \\bar{x}(\\mathcal {S}_w\\setminus \\mathcal {Y}_{w,i}) \\right)\\end{array}$ where $\\bar{x}(\\mathcal {Y}_{w,i})$ , $\\bar{x}(\\mathcal {S}_w\\setminus \\mathcal {Y}_{w,i})$ are the averages of $\\mathcal {Y}_{w,i}$ and $\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}$ , respectively, and $c_i$ is chosen as $c_i=\\min _{\\xi \\in \\mathcal {Y}_{w,i}} \\left\\langle \\frac{w_i}{\\Vert w_i\\Vert },\\xi \\right\\rangle $ .",
"Integration.",
"Integrate Auxiliary Knowledge Units into decision-making pathways of $\\mathrm {AI}_s$ .",
"If, for an $x$ generated by an input to $\\mathrm {AI}_s$ , any of $\\ell _i(x)\\ge 0$ then report $x$ accordingly (swap labels, report as an error etc.)",
"The algorithms comprise of two general stages, pre-processing stage and knowledge transfer stage.",
"The purpose of the pre-processing stage is to regularize and “sphere” the data.",
"This operation brings the setup close to the one considered in statements of Theorems REF , REF .",
"The knowledge transfer stage constructs Auxiliary Knowledge Transfer Units in a way that is very similar to the argument presenteed in the proofs of Theorems REF and REF .",
"Indeed, if $|\\mathcal {Y}_{w,i}|\\ll |\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}|$ then the term $\\left(\\mathrm {Cov}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}) + \\mathrm {Cov}(\\mathcal {Y}_{w,i}) \\right)^{-1}$ is close to identity matrix, and the functionals $\\ell _i$ are good approximations of (REF ).",
"In this setting, one might expect that performance of the knowledge transfer stage would be also closely aligned with the corresponding estimates (REF ), (REF ).",
"Remark 2 Note that the regularization step in the pre-processing stage ensures that the matrix $\\mathrm {Cov}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}) + \\mathrm {Cov}(\\mathcal {Y}_{w,i})$ is non-singular.",
"Indeed, consider $\\begin{array}{ll}&\\mathrm {Cov}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i})= \\frac{1}{|\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}|} \\sum _{x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}} (x - \\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i})) (x - \\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}))^{T}\\\\&= \\frac{1}{|\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}|} \\left( \\sum _{x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w}} (x - \\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i})) (x - \\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}))^{T} \\right.",
"+ \\\\& \\left.\\sum _{x\\in \\mathcal {Y}_w \\setminus \\mathcal {Y}_{w,i}} (x - \\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i})) (x - \\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}))^{T} \\right).\\end{array}$ Denoting $d=\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i})-\\bar{x}(\\mathcal {S}_w\\setminus \\mathcal {Y}_{w})$ and rearranging the sum below as $\\begin{array}{ll}&\\sum _{x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w}} (x - \\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i})) (x - \\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}))^{T}= \\\\&\\sum _{x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w}} (x-\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w})+d) (x-\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w})+d)^{T}=\\\\& \\sum _{x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w}} (x-\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w})) (x-\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w}))^{T} + \\\\&2 d \\sum _{x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w}} (x-\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w}))^{T} + |x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w}| d d^{T}\\\\&= \\sum _{x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w}} (x-\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w})) (x-\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w}))^{T} + |x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w}| d d^{T}\\end{array}$ we obtain that $\\mathrm {Cov}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i})$ is non-singular as long as the sum $\\sum _{x\\in \\mathcal {S}_w \\setminus \\mathcal {Y}_{w}} (x-\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w})) (x-\\bar{x}(\\mathcal {S}_w \\setminus \\mathcal {Y}_{w}))^{T}$ is non-singular.",
"The latter property, however, is guaranteed by the regularization step in Algorithm REF .",
"Remark 3 Clustering at Step 2.a can be achieved by classical $k$ -means algorithms [19] or any other method (see e.g.",
"[20]) that would group elements of $\\mathcal {Y}_w$ into clusters according to spatial proximity.",
"Remark 4 Auxiliary Knowledge Transfer Units in Step 2.b of Algorithm REF are derived in accordance with standard Fisher linear discriminant formalism.",
"This, however, need not be the case, and other methods such as e.g.",
"support vector machines [21] could be employed for this purpose there.",
"It is worth mentioning, however, that support vector machines might be prone to overfitting [22] and their training often involves iterative procedures such as e.g.",
"sequential quadratic minimization [23].",
"Furthermore, instead of the sets $\\mathcal {Y}_{w,i}$ , $\\mathcal {S}_w \\setminus \\mathcal {Y}_{w,i}$ one could use a somewhat more aggressive division: $\\mathcal {Y}_{w,i}$ and $\\mathcal {S}_w \\setminus \\mathcal {Y}_{w}$ , respectively.",
"Depending on configuration of samples $\\mathcal {S}$ and $\\mathcal {Y}$ , Algorithm REF may occasionally create knowledge transfer units, $\\ell _i$ , that are “filtering” errors too aggressively.",
"That is some $x\\in \\mathcal {S}_w\\setminus \\mathcal {Y}_w$ may accidentally trigger non-negative response, $\\ell _i(x)\\ge 0$ , and as a result of this their corresponding inputs to $\\mathrm {A}_s$ could be ignored or mishandled.",
"To mitigate this, one can increase the number of clusters and knowledge transfer units, respectively.",
"This will increase the probability of successful separation and hence alleviate the issue.",
"On the other hand, if increasing the number of knowledge transfer units is not desirable for some reason, then two-functional units could be a feasible remedy.",
"Algorithm REF presents a procedure for such an improved AI Knowledge Transfer.",
"Two-functional AI Knowledge Transfer Pre-processing.",
"Do as in Step 1 in Algorithm REF Knowledge Transfer Clustering.",
"Do as in Step 2.a in Algorithm REF Construction of Auxiliary Knowledge Units.",
"[1] Do as in Step 2.b in Algorithm REF .",
"At the end of this step first-stage functionals $\\ell _i$ , $i=1,\\dots ,p$ will be derived.",
"For each set $\\mathcal {Y}_{w,i}$ , $i=1,\\dots ,p$ , evaluate the functionals $\\ell _i$ for all $x\\in \\mathcal {S}_w\\setminus \\mathcal {Y}_{w,i}$ and identify elements $x$ such that $\\ell _i(x)\\ge 0$ and $x\\in \\mathcal {S}_w\\setminus \\mathcal {Y}_w$ (incorrect error assignment).",
"Let $\\mathcal {Y}_{e,i}$ be the set containing such elements $x$ .",
"If (there is an $i\\in \\lbrace 1,\\dots ,p\\rbrace $ such that $|\\mathcal {Y}_{e,i}| +|\\mathcal {Y}_{w,i}|>m$ ) then increment the value of $p$ : $p\\leftarrow p+1$ , and return to Step 2.a.",
"If (all sets $\\mathcal {Y}_{e,i}$ are empty) then proceed to Step 2.c.",
"For each pair of $\\ell _i$ and $\\mathcal {Y}_{w,i}\\cup \\mathcal {Y}_{e,i}$ with $\\mathcal {Y}_{e,i}$ not empty, project orthogonally sets $\\mathcal {Y}_{w,i}$ and $\\mathcal {Y}_{e,i}$ onto the hyperplane $\\ell _i(x)=0$ and form the sets $\\mathcal {L}_i(\\mathcal {Y}_{w,i})$ and $\\mathcal {L}_i(\\mathcal {Y}_{e,i})$ : $\\begin{array}{ll}\\mathcal {L}_i(\\mathcal {Y}_{w,i})&=\\left\\lbrace x\\in \\mathbb {R}^m \\ | \\ x=\\left(I_m - \\frac{w_i w_i^T}{\\Vert w_i\\Vert ^2}\\right)\\xi + \\frac{c_i w_i}{\\Vert w_i\\Vert }, \\ \\xi \\in \\mathcal {Y}_{w,i}\\right\\rbrace ,\\\\\\mathcal {L}_i(\\mathcal {Y}_{e,i})&=\\left\\lbrace x\\in \\mathbb {R}^m \\ | \\ x=\\left(I_m - \\frac{w_i w_i^T}{\\Vert w_i\\Vert ^2}\\right)\\xi + \\frac{c_i w_i}{\\Vert w_i\\Vert }, \\ \\xi \\in \\mathcal {Y}_{e,i}\\right\\rbrace .\\end{array}$ Construct a linear functional $\\ell _{2,i}$ separating $\\mathcal {L}_i(\\mathcal {Y}_{w,i})$ from $\\mathcal {L}_i(\\mathcal {Y}_{e,i})$ so that $\\ell _{2,i}(x)\\ge 0$ for all $x\\in \\mathcal {Y}_{w,i}$ and $\\ell _{2,i}(x)< 0$ for all $x\\in \\mathcal {Y}_{e,i}$ .",
"Integration.",
"Integrate Auxiliary Knowledge Units into decision-making pathways of $\\mathrm {AI}_s$ .",
"If, for an $x$ generated by an input to $\\mathrm {AI}_s$ , any of the predicates $(\\ell _i(x)\\ge 0)\\wedge (\\ell _{2,i}(x)\\ge 0)$ hold true then report $x$ accordingly (swap labels, report as an error etc.).",
"In what follows we illustrate the approach as well as the application of the proposed Knowledge Transfer algorithms in a relevant problem of a computer vision system design for pedestrian detection in live video streams."
],
[
"Example",
"Let $AI_s$ and $AI_t$ be two systems developed, e.g.",
"for the purposes of pedestrian detection in live video streams.",
"Technological progress in embedded systems and availability of platforms such as e.g.",
"Nvidia Jetson TX2 made hadrware deployment of such AI systems at the edge of computer vision processing pipelines feasible.",
"These AI systems, however, lack computational power to run state-of-the-art large scale object detection solutions such as e.g.",
"ResNet [24] in real-time.",
"Here we demonstrate that to compensate for this lack of power, AI Knowledge Transfer can be successfully employed.",
"In particular, we suggest that the edge-based system is “taught” by the state-of-the-art teacher in a non-iterative and near-real time way.",
"Since our building blocks are linear functionals, such learning will not lead to significant computational overheads.",
"At the same time, as we will show later, the proposed AI Knowledge Transfer will result in a major boost to the system's performance in the conditions of the experiment."
],
[
"Definition of $AI_s$ and {{formula:cbc156c0-5ae4-4351-869f-5e1a85b71fcc}} and rationale",
"In our experiments, the teacher AI, $AI_t$ , was modeled by a deep Convolutional Network, ResNet 18 [24] with circa 11M trainable parameters.",
"The network was trained on a “teacher” dataset comprised of $5.2$ M non-pedestrian (negatives), and 600K pedestrian (positives) images.",
"The student AI, $AI_s$ , was modelled by a linear classifier with HOG features [25] and 2016 trainable parameters.",
"The values of these parameters were the result of $AI_s$ training on a “student” dataset, a sub-sample of the “teacher” dataset comprising of 55K positives and 130K negatives, respectively.",
"This choice of $AI_s$ and $AI_t$ systems enabled us to emulate interaction between edge-based AIs and their more powerful counterparts that could be deployed on larger servers or computational clouds.",
"Moreover, to make the experiment more realistic, we assumed that internal states of both systems are inaccessible for direct observation.",
"To generate sets $\\mathcal {S}$ and $\\mathcal {Y}$ required in Algorithms REF and REF we augmented system $AI_s$ with an external generator of HOG features of the same dimension.",
"We assumed, however, that covariance matrices of positives and negatives from the “student” dataset are available for the purposes of knowledge transfer.",
"A diagram representing this setup is shown in Figure REF .",
"Figure: Knowledge transfer diagram between ResNet and HOG-SVM object detectorsA candidate image is evaluated by two systems simultaneously as well as by a HOG features generator.",
"The latter generates 2016 dimensional vectors of HOGs and stores these vectors in the set $\\mathcal {S}$ .",
"If outputs of $AI_s$ and $AI_t$ do not match the corresponding feature vector is added to the set $\\mathcal {Y}$ ."
],
[
"Error types",
"In this experiment we consider and address two types of errors: false positives (Type I errors) and false negatives (Type II errors).",
"The error types were determined as follows.",
"An error is deemed as false positive if $AI_s$ reported presence of a correctly sized full-figure image of pedestrian in a given image patch whereas no such object was there.",
"Similarly, an error is deemed as false negative if a pedestrian was present in the given image patch but $AI_s$ did not report it there.",
"In our setting, evaluation of an image patch by $AI_t$ (ResNet) took $0.01$ sec on Nvidia K80 which was several orders slower than that of $AI_s$ (linear HOG-based classifier).",
"Whilst such behavior was expected, this imposed technical limitations on the process of mitigating errors of Type II.",
"Each frame from our testing video produced 400K image patches to test.",
"Evaluation of all these candidates by our chosen $AI_t$ is prohibitive computationally.",
"To overcome this technical difficulty we tested only a limited subset of image proposals with regards to these error type.",
"To get a computationally viable number of proposals for false negative testing, we increased sensitivity of the HOG-based classifier by lowering its detection threshold from 0 to $-0.3$ .",
"This way our linear classifier with lowered threshold acted as a filter letting through more true positives at the expense of large number of false positives.",
"In this operational mode, Knowledge Transfer Unit were tasked to separate true positives from negatives in accordance with object labels supplied by $AI_t$ ."
],
[
"Datasets",
"The approach was tested on two benchmark videos: LINTHESCHER sequence [26] created by ETHZ and comprised of 1208 frames and NOTTINGHAM video [27] containing 435 frames of live footage taken with an action camera.",
"In what follows we will refer to these videos as ETHZ and NOTTINGHAM videos, respectively.",
"ETHZ video contains complete images of 8435 pedestrians, whereas NOTTINGHAM video has 4039 full-figure images of pedestrians."
],
[
"Results",
"Performance and application of Algorithms REF , REF for NOTTINGHAM and ETHZ videos are summarized in Fig.",
"REF and REF .",
"Each curves in these figures is produced by varying the values of decision-making threshold in the HOG-based linear classifier.",
"Red circles in Figure REF show true positives as a function of false positives for the original linear classifier based on HOG features.",
"Parameters of the classifier were set in accordance with Fisher linear discriminant formulae.",
"Blue stars correspond to $AI_s$ after Algorithm REF was applied to mitigate errors of Type I in the system.",
"The value of $p$ (number of clusters) in the algorithm was set to be equal to 5.",
"Green triangles illustrate application of Algorithm REF for the same error type.",
"Here Algorithm REF was slightly modified so that the resulting Knowledge Transfer Unit had only one functional $\\ell _2$ .",
"This was due to the low number of errors reaching stage two of the algorithm.",
"Black squares correspond to $AI_s$ after application of Algorithm REF (error Type I) followed by application of Algorithm REF to mitigate errors of Type II.",
"Figure: True positives as a function of false positives for NOTTINGHAM video.Figure REF shows performance of the algorithms for ETHZ sequence.",
"Red circles show performance of the original $AI_s$ , green triangles correspond to $AI_s$ supplemented with Knowledge Transfer Units derived using Algorithm REF for errors of Type I.",
"Black squares correspond to subsequent application of Algorithm REF dealing with errors of Type II.",
"Figure: True positives as a function of false positives for ETHZ video.In all these cases, supplementing $AI_s$ with Knowledge Transfer Units constructed with the help of Algorithms REF , REF for both error types resulted in significant boost to $AI_s$ performance.",
"Observe that in both cases application of Algorithm REF to address errors of Type II has led to noticeable increases of numbers of false positives in the system at the beginning of the curves.",
"Manual inspection of these false positives revealed that these errors are exclusively due mistakes of $AI_t$ itself.",
"For the sake of illustration, these errors for NOTTINGHAM video are shown in Fig.",
"REF .",
"These errors contain genuine false positives (images 12, 23-27) as well as mismatches by size (e.g.",
"1-7), and look-alikes (images 8,11,13,15-17).",
"Figure: False Positives induced by the teacher AI, AI t AI_t."
],
[
"Conclusion",
"In this work we proposed a framework for instantaneous knowledge transfer between AI systems whose internal state used for decision-making can be described by elements of a high-dimensional vector space.",
"The framework enables development of non-iterative algorithms for knowledge spreading between legacy AI systems with heterogeneous non-identical architectures and varying computing capabilities.",
"Feasibility of the framework was illustrated with an example of knowledge transfer between two AI systems for automated pedestrian detection in video streams.",
"In the basis of the proposed knowledge transfer framework are separation theorems (Theorem REF – REF ) stating peculiar properties of large but finite random samples in high dimension.",
"According to these results, $k<n$ random i.i.d.",
"elements can be separated form $M\\gg n$ randomly selected elements i.i.d.",
"sampled from the same distribution by few linear functionals, with high probability.",
"The theorems are proved for equidistributions in a ball and in a cube.",
"The results can be trivially generalized to equidistributions in ellipsoids and Gaussian distributions.",
"Generalizations to other meaningful distributions is the subject of our future work."
],
[
"Acknowledgments",
"The work was supported by Innovate UK Technology Strategy Board (Knowledge Transfer Partnership grants KTP009890 and KTP010522)."
]
] | 1709.01547 |
[
[
"An accelerated proximal iterative hard thresholding method for $\\ell_0$\n minimization"
],
[
"Abstract In this paper, we consider a non-convex problem which is the sum of $\\ell_0$-norm and a convex smooth function under box constraint.",
"We propose one proximal iterative hard thresholding type method with extrapolation step used for acceleration and establish its global convergence results.",
"In detail, the sequence generated by the proposed method globally converges to a local minimizer of the objective function.",
"Finally, we conduct numerical experiments to show the proposed method's effectiveness on comparison with some other efficient methods."
],
[
"Introduction",
"In modern science and technology, signal and image processing problems have many important applications, for example, compressive sensing, machine learning and medical imaging.",
"Signal and image processing problems can be often formulated as the following inverse problem $A(x)+\\epsilon =b,$ where $A$ is some linear or non-linear operator, $b$ is the observation data, $\\epsilon $ is some observation error and $x$ is the vector we wanted.",
"Problem (REF ) is usually ill-posed, thus solving (REF ) is non-trivial.",
"To overcome this difficulty, the prior sparsity of the signals or images is usually considered.",
"One often used minimization model is formulated as $\\min _{ x\\in X}f(x)+g(x)$ where $f(x)$ is the data fidelity term related to equation (REF ), $g(x)$ is some regularization term to promote $x$ 's sparsity, and $X\\subseteq \\mathbb {R}^n$ is some convex constraint set.",
"A natural idea for sparsity promotion is taking $g(x)=\\lambda \\Vert x\\Vert _0$ where $\\lambda >0$ is some regularization parameter and the notation $\\Vert x\\Vert _0$ , $x$ 's $\\ell _0$ norm, denotes the number of $x$ 's nonzero elements.",
"It is well-known that finding the global minimizer of $\\ell _0$ regularization problem is NP hard.",
"And it is hard to develop convergent, efficient and tractable method since $\\ell _0$ - norm is non-convex and discontinuous.",
"That is also a reason why the $\\ell _1$ convex relaxation model $\\min _{x\\in X}f(x)+\\lambda \\Vert x\\Vert _1$ are largely adopted.",
"However, $\\ell _0$ regularization problem still has some advantages over $\\ell _1$ regularization problem.",
"For example, $\\ell _1$ regularization problem may fail to recover sparse solutions for some very ill-posed inverse problems and non-Gaussian noise corruption [28].",
"Compared with $\\ell _1$ regularization problem, $\\ell _0$ regularization problem can directly recover sparser solutions.",
"Moreover, the continuity of the soft thresholding operator, $\\mathcal {S}_{\\lambda }(c)=\\arg \\min \\lambda \\Vert x\\Vert _1+\\frac{1}{2}\\Vert x-c\\Vert ^2=\\mbox{sign}(c)\\max \\lbrace |c|-\\lambda ,0\\rbrace $ used for solving $\\ell _1$ regularization problem, may yield loss of contrast and eroded signal peaks since all the coefficients are deduced.",
"In statistical learning, it is also well known that $\\ell _1$ solution is a biased estimator .",
"In many applications, $\\ell _0$ regularization achieves better sparse solution than $\\ell _1$ regularization, for example [13], [15], [30].",
"Thus we consider the following $\\ell _0$ regularization problem $\\min _{x\\in X} \\lambda \\Vert x\\Vert _0+f(x),$ and devote to design and discuss an efficient method with simple structure.",
"Analogue to the proximal forward-backward splitting (PFBS) method [19], [23], [14] for convex problems (REF ), a proximal iterative hard thresholding (PIHT) method is used in many works to solve $\\ell _0$ regularization problem (REF ) when $X=\\mathbb {R}^n$ .",
"Its convergence and convergence rate have been studied in [13], [6], [7], [2], [10], [21], [29] under different assumptions.",
"Typically, under the assumption that $f(x)$ has Lipschitz continuous gradient, it obtains the next iterative point by solving a subproblem which contains a linearization term of $f(x)$ at current iteration point $x^k$ and a proximal term.",
"In detail, the PIHT method is given as PIHT Algorithm Choose parameters $\\mu >0, \\lambda >0$ , starting point $x_0$ ; compute the Lipschitz constant $L$ of $\\nabla f(x)$ ; let $k=0$ .",
"while the stopping criterion does not hold, compute $x^{k+1}\\in \\arg \\min _{x \\in \\mathbb {R}^n}\\lambda \\Vert x\\Vert _0+\\frac{L}{2}\\Vert x-x^k+\\frac{1}{L}\\nabla f(x^k)\\Vert ^2+\\frac{\\mu }{2}\\Vert x-x^k\\Vert ^2$ $k=k+1$ end(while) As well known, the step (REF ) can be given by $x^{k+1}\\in \\mathcal {H}_{\\sqrt{\\frac{2\\lambda }{L+\\mu }}} (x^k-\\frac{1}{L+\\mu }\\nabla f(x^k)),$ where $\\mathcal {H}_{\\gamma }(\\cdot )$ is the hard thresholding operator, a set-valued componentwise operator, defined as $(\\mathcal {H}_{\\gamma }(c))_i=\\left\\lbrace \\begin{array}{ll}\\lbrace c_i\\rbrace , &\\mbox{if}\\; |c_i|>\\gamma \\\\\\lbrace 0,c_i\\rbrace &\\mbox{if}\\; |c_i|=\\gamma \\\\\\lbrace 0\\rbrace , &\\mbox{if}\\; |c_i|< \\gamma \\end{array}\\right.$ where $c_i$ denotes the $i$ th component of vector $c$ .",
"Accelerated PFBS methods have been extensively considered for solving problem (REF ) with convex $f,g$ .",
"For instance, in [5], [26], [25], [3], extrapolation steps are utilized to achieve a convergence complexity of $O(1/k^2)$ (even $o(1/k^2)$ [3]) in terms of objective value error.",
"Similar to the accelerated technique used for accelerated proximal gradient (APG) method for convex cases, we will propose one accelerated PIHT method for $\\ell _0$ minimization using extrapolation and provide its convergence results.",
"On solving non-convex problems, many algorithms, such as inertial forward-backward method [11] (IFB), monotone accelerated proximal gradient method [18] (mAPG), and non-monotone APG method [18] (nmAPG), are proposed to accelerate the convergence of the usual PFBS method.",
"In [4], an extrapolated proximal iterative hard-thresholding (EPIHT) algorithm is proposed to accelerate the PIHT for $\\ell _0$ minimization.",
"The convergence of the above mentioned algorithms are usually build upon Kurdyka-Łojasiewicz (KL) property (for details, one can see [27], [8], [9], [17], [1], [20])) of objective function.",
"In this paper, we will design an extrapolated proximal algorithm for $\\ell _0$ optimization and tackle the convergence analysis directly without using the tool of KL property.",
"The global convergence to a local minimizer of the proposed algorithm is established purely based on the convexity of $f$ and the property of $\\ell _0$ function.",
"Compared to EPIHT and some other algorithms, one advantage of our proposed scheme is that a small amount of function and gradient evaluation are involved at each iteration.",
"The setting of parameters are relatively simple compared to some other related algorithms.",
"Finally, numerical experiments also show the effectiveness of the proposed algorithm.",
"A detail presentation of the related algorithms and comparison will be present in Section .",
"The rest of the paper is organized as follows.",
"In section , we introduce the proposed algorithm and establish its convergence results.",
"In section , we will give a discussion on our method and the comparison to other state-of-the-art methods.",
"In section , we conduct experiments to show our method's numerical performance and efficiency.",
"We first introduce some notations, concepts and results that will be used in this paper.",
"For any $x\\in \\mathbb {R}^n$ , $x_i$ represents $x$ 's $i$ -th component.",
"Given any index set $I\\subseteq \\lbrace 1,2,\\ldots ,n\\rbrace $ , we let $C_I:=\\lbrace x\\in \\mathbb {R}^n: x_i=0\\text{ for all }i\\in I\\rbrace ;$ conversely, given any $x\\in \\mathbb {R}^n$ , we define the zero element index set of a vector $x\\in \\mathbb {R}^n$ as $I(x):=\\lbrace i:x_i=0\\rbrace .$ The projection operator defined on a set $C\\subseteq \\mathbb {R}^n$ is denoted by $P_{C}(x)=\\arg \\min _{z\\in C}\\frac{1}{2}\\Vert z-x\\Vert ^2.$ $P_{C}(\\cdot )$ is continuous, namely $\\lim _{k\\rightarrow +\\infty }P_{C}(x^k)=P_{C}(\\lim _{k\\rightarrow +\\infty }x^k)$ if $\\lim _{k\\rightarrow +\\infty }x^k$ exists.",
"For any $x\\in \\mathbb {R}$ , $\\Vert x\\Vert _0=0$ if $x=0$ ; otherwise $\\Vert x\\Vert _0=1$ .",
"Then for any positive integer $n$ and $y\\in \\mathbb {R}^n$ , $\\Vert y\\Vert _0=\\sum _{i=1}^n\\Vert y_i\\Vert _0$ denotes the number of $y$ 's nonzero elements.",
"Definition 2.1 A mapping $T:\\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ is said to be $L_T$ -Lipschitz continuous on the set $X\\subseteq \\mathbb {R}^n$ if there exists $L_T>0$ such that $\\Vert T(x)-T(y)\\Vert \\le L_T\\Vert x-y\\Vert ,\\quad \\forall x, y\\in X.$ Definition 2.2 Let $f:\\mathbb {R}^n \\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ be a closed proper convex function, then the subdifferential of $f$ at $x$ is defined by $\\partial f(x):=\\lbrace s\\in \\mathbb {R}^n: f(y)\\ge f(x)+\\langle s,y-x\\rangle ,\\;\\forall y\\in \\mathbb {R}^n\\rbrace .$ And each element $s\\in \\partial f(x)$ is called a subgradient of $f$ at point $x$ .",
"Moreover, if $f$ is continuous differentiable, $\\partial f(x)=\\lbrace \\nabla f(x)\\rbrace $ .",
"Lemma 2.3 [5] $f: \\mathbb {R}^n\\rightarrow \\mathbb {R}$ is continuous differentiable.",
"If $\\nabla f(x)$ is L-Lipschitz continuous, the following inequality holds $f(x)-f(y)\\le \\langle \\nabla f(y),x-y\\rangle +\\frac{L}{2}\\Vert x-y\\Vert ^2, \\;\\; \\forall x,y \\in \\mathbb {R}^n.$ Lemma 2.4 [5] Denoting $B_{L}(y)=(I+\\partial g/L)^{-1}(y-\\nabla f(y)/L).$ where $g:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ is a proper closed convex function, $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ is convex smooth and $\\nabla f$ is $L$ -Lipschitz continuous.",
"Letting $h:=f+g$ , for any $x,y\\in \\mathbb {R}^n$ , the following inequality holds $h(x)-h(B_L(y))\\ge \\frac{L}{2}\\Vert B_{L}(y)-y\\Vert ^2+L\\langle y-x,B_L(y)-y\\rangle .$"
],
[
"Model and algorithm",
"In this paper, we consider the following minimization problem $\\min _{x\\in \\mathbb {R}^n} H(x):=\\lambda \\Vert x\\Vert _0+f(x)+\\delta _X(x),$ where $X=\\lbrace x\\in \\mathbb {R}^n:l\\le x\\le u \\rbrace $ ($l,u$ can be vectors), and the indicator function $\\delta _X(x)=\\left\\lbrace \\begin{array}{ll}0, & \\text{ if } x\\in X;\\\\+\\infty , & \\text{ otherwise}.\\end{array}\\right.$ Remark 1 If $f(x)$ is coercive, one can take $l=-\\infty ,u=+\\infty $ , all the results in this paper still hold.",
"If the original problem is unconstrained and $f(x)$ isn't coercive, one can take the elements of $l$ very small and the elements of $u$ very large, for example $l=\\lbrace -10^{12}\\rbrace ^n, u=-l$ .",
"Remark 2 Here we use the uniform parameter $\\lambda \\Vert x\\Vert _0$ instead of the weighted $\\Vert \\mathbf {\\lambda }\\cdot x\\Vert _0:=\\sum _{i=1}^n\\mathbf {\\lambda }_i\\Vert x_i\\Vert _0$ for the simplicity of notation, while all the results can be easily extended to the weighted case.",
"Throughout this paper, our assumption on problem (REF ) is Assumption A: $f$ is convex differentiable and bounded from below on set $X$ ; $\\nabla f$ is $L$ -Lipschitz continuous on set $X$ .",
"For solving problem (REF ), we propose the following extrapolated type method.",
"Algorithm 1 Choose parameters $\\mu >0, \\lambda >0$ and a sequence of extrapolation weights $0<\\omega _k\\le \\omega <1$ ; compute the Lipschitz constant $L$ of $\\nabla f(x)$ ; choose starting point $x^{-1}=x^0$ ; let $k=0$ .",
"while the stopping criterion does not hold Let $y^{k+1}_i=\\left\\lbrace \\begin{array}{ll}x^{k}_i,&i\\notin I(x^k)\\\\x^{k}_i+\\omega _k(x^{k}_i-x^{k-1}_i),&i\\in I(x^k)\\end{array}\\right.$ if $\\langle y^{k+1}-x^{k}, \\nabla f(y^{k+1})\\rangle >0$ or $y^{k+1}\\notin X$ $y^{k+1}=x^{k}$ end(if) $x^{k+1}\\in \\arg \\min _{x \\in X}\\lambda \\Vert x\\Vert _0+\\frac{L}{2}\\Vert x-y^{k+1}+\\frac{1}{L}\\nabla f(y^{k+1})\\Vert ^2+\\frac{\\mu }{2}\\Vert x-y^{k+1}\\Vert ^2$ $k=k+1$ end(while) During the iteration, we assume that the support of $x^k$ is more accurate than that of $x^{k-1}$ .",
"The extrapolation is only performed in the subspace $C_{I(x^k)}$ .",
"The gradient information is used to determine whether the extrapolation step will be accepted.",
"In fact, if $\\langle y^{k+1}-x^{k}, \\nabla f(y^{k+1})\\rangle \\le 0$ , owing to the monotonicity of $\\nabla f$ (namely $\\langle y-x, \\nabla f(y)-\\nabla f(x)\\rangle \\ge 0$ ), we can get $\\langle y^{k+1}-x^{k}, \\nabla f(x^{k})\\rangle \\le 0$ ; then $y^{k+1}-x^{k}$ is a decreasing direction at point $x^{k}$ for function $f(x)+\\lambda \\Vert x\\Vert _0$ in subspace $C_{I(x^k)}$ and hence we think it is worth doing extrapolation; Otherwise we reset $y^{k+1}=x^k$ .",
"And we using $\\nabla f(y^{k+1})$ rather than $\\nabla f(x^{k})$ to reduce the amount of computation because $\\nabla f(y^{k+1})$ is used to evaluate the next iteration point $x^{k+1}$ .",
"Remark 3 In the numerical experiment, one can take an appropriate selection of parameters $\\omega _k$ such that $y^{k+1}$ is always in the set $X$ ."
],
[
"Convergence analysis",
"In this section, we present the convergence results of Algorithm 1.",
"Firstly we give some properties about the solutions of the subproblem (REF ) and show that $I(x^k)$ , the zero element index set of iteration sequence $x^k$ , changes finitely often.",
"For the subproblem (REF ), it has separable structure since $X$ is a box constraint.",
"So we just need discussing the property of the following problem's solution $\\arg \\min _{\\tilde{x}\\in \\mathbb {R}} h(\\tilde{x}):=\\delta _{ \\tilde{X}}(\\tilde{x})+\\lambda \\Vert \\tilde{x}\\Vert _0+\\frac{1}{2}(\\tilde{x}-c)^2$ where $\\tilde{X}:=\\lbrace \\tilde{x}\\in \\mathbb {R}:\\tilde{l}\\le \\tilde{x}\\le \\tilde{u}\\rbrace $ .",
"In fact, the minimum point of function $\\delta _{\\tilde{X}}(\\tilde{x})+\\lambda +\\frac{1}{2}(\\tilde{x}-c)^2$ is $P_{\\tilde{X}}(c)$ and it has different function value only at zero point compared with $h(\\tilde{x})$ .",
"When $P_{\\tilde{X}}(c)\\ne 0$ , we only need to compare the function value $h(0)$ and $h(P_{\\tilde{X}}(c))$ to get the solution.",
"In detail, For the case $0\\notin \\tilde{X}$ , the solution point is certainly $P_{\\tilde{X}}(c)$ .",
"For the case $0\\in \\tilde{X}$ , $h(P_{\\tilde{X}}(c))-h(0)=\\lambda +\\frac{1}{2}(P_{\\tilde{X}}(c))^2-cP_{\\tilde{X}}(c)$ .",
"If $c\\in \\tilde{X}$ , the solution point is $\\mathcal {H}_{\\sqrt{2\\lambda }}(P_{\\tilde{X}}(c))$ since $h(P_{\\tilde{X}}(c))-h(0)=\\lambda -\\frac{1}{2}c^2$ ; If $c>\\tilde{u}$ , the solution is obtained by comparing $h(0)$ and $h(\\tilde{u})$ ; If $c<\\tilde{l}$ , the solution is obtained by comparing $h(0)$ and $h(\\tilde{l})$ .",
"In either case above, the solution point $\\tilde{x}^*$ satisfies $|\\tilde{x}^*|\\ge \\min (\\lbrace |l|,|u|,\\sqrt{2\\lambda }\\rbrace /\\lbrace 0\\rbrace ^C)$ if it is not zero, where $\\lbrace 0\\rbrace ^C$ denotes the complement of set {0}.",
"Then we have the following results.",
"Lemma 2.5 Let $H(x)$ be the objective function defined in (REF ), and $\\lbrace x_k\\rbrace _{k=0}^\\infty $ be the sequence generated by Algorithm 1.",
"If the extrapolation weight $\\omega _k$ satisfies $0\\le \\omega _k\\le \\omega <1$ , then $\\lbrace H(x^k)\\rbrace _{k=0}^{+\\infty }$ is non-increasing; $\\sum _{k=1}^\\infty \\Vert x^k-y^{k}\\Vert ^2<\\infty $ , $\\Vert x^k-y^{k}\\Vert ^2\\rightarrow 0$ ; $I(x^k)$ changes only finitely often; $\\sum _{k=1}^\\infty \\Vert x^k-x^{k-1}\\Vert ^2<\\infty $ , $\\Vert x^k-x^{k-1}\\Vert ^2\\rightarrow 0$ .",
"1.",
"Since $\\nabla f(x)$ is $L$ -Lipschitz continuous, from Lemma 2.3, we have $f(x^{k+1})-f(y^{k+1})\\le \\langle \\nabla f(y^{k+1}),x^{k+1}-y^{k+1}\\rangle +\\frac{L}{2}\\Vert x^{k+1}-y^{k+1}\\Vert ^2.$ It is clear that $y^{k+1}\\in X$ .",
"Then from Algorithm 1, we have $&&\\lambda \\Vert x^{k+1}\\Vert _0+\\frac{L}{2}\\Vert x^{k+1}-y^{k+1}+\\frac{\\nabla f(y^{k+1})}{L}\\Vert ^2+\\frac{\\mu }{2}\\Vert x^{k+1}-y^{k+1}\\Vert ^2\\\\&\\le & \\lambda \\Vert y^{k+1}\\Vert _0+\\frac{L}{2}\\Vert y^{k+1}-y^{k+1}+\\frac{\\nabla f(y^{k+1})}{L}\\Vert ^2.\\\\$ By summing up the above two inequalities and using the fact $\\Vert y^{k+1}\\Vert _0\\le \\Vert x^k\\Vert _0$ , $f(y^{k+1})-f(x^k)\\le \\langle y^{k+1}-x^{k}, \\nabla f(y^{k+1})\\rangle \\le 0$ , we have $H(x^{k+1})+\\frac{\\mu }{2}\\Vert x^{k+1}-y^{k+1}\\Vert ^2\\le H(x^k).$ It is obvious that $\\lbrace H(x^k)\\rbrace _{k=0}^{+\\infty }$ is non-increasing.",
"2.",
"Summing the inequality (REF ) over $k=0,\\ldots ,n-1$ , we have $H(x^n)+\\frac{\\mu }{2}\\sum _{k=0}^n\\Vert x^k-y^{k}\\Vert ^2\\le H(x^0).$ So $\\lbrace \\sum _{k=0}^n\\Vert x^k-y^{k}\\Vert ^2\\rbrace $ has upper bound since $H(x)$ has lower bound on $X$ .",
"Then $\\sum _{k=0}^\\infty \\Vert x^k-y^{k}\\Vert ^2<\\infty $ and $\\Vert x^k-y^{k}\\Vert \\rightarrow 0$ .",
"3.",
"From the implementation of iteration $k$ in Algorithm 1 and the discussion about the property of problem (REF )'s solution , we have $|x^k_i|\\ge ls:=\\min (\\lbrace |l_j|,|u_j|,\\sqrt{\\frac{2\\lambda }{L+\\mu }},j=1,\\cdots ,n\\rbrace \\cap \\lbrace 0\\rbrace ^C), \\mbox{ for any } i\\notin I(x^k).$ Hence, we have $\\Vert x^k-y^{k}\\Vert \\ge ls>0$ if $I(x^k)^C\\nsubseteq I(y^{k})^C$ .",
"From $\\Vert x^k-y^{k}\\Vert \\rightarrow 0$ and $I(y^k)^C\\subseteq I(x^{k-1})^C$ , it is easy to see that $I(x^k)^C\\subseteq I(x^{k-1})^C$ always hold if $k$ is sufficiently large.",
"Then $I(x^k)$ must change only finitely often.",
"4.",
"Assume that $I(x^k)=I(x^{k+1})$ for any $k\\ge k_0$ .",
"From the subproblem (REF ), we have $&&\\lambda \\Vert x^{k+1}\\Vert _0+\\frac{L}{2}\\Vert x^{k+1}-y^{k+1}+\\frac{\\nabla f(y^{k+1})}{L}\\Vert ^2+\\frac{\\mu }{2}\\Vert x^{k+1}-y^{k+1}\\Vert ^2\\\\&\\le & \\lambda \\Vert x\\Vert _0+\\frac{L}{2}\\Vert x-y^{k+1}+\\frac{\\nabla f(y^{k+1})}{L}\\Vert ^2+\\frac{\\mu }{2}\\Vert x-y^{k+1}\\Vert ^2\\\\$ for any $x\\in C:=C_{I(x^k)}\\cap X$ and $k\\ge k_0$ .",
"So we have $x^{k+1}\\in \\arg \\min _{x \\in C} Q(x;y^{k+1}):=\\frac{L}{2}\\Vert x-y^{k+1}+\\frac{1}{L}\\nabla f(y^{k+1})\\Vert ^2+\\frac{\\mu }{2}\\Vert x-y^{k+1}\\Vert ^2,\\;\\; k\\ge k_0.$ From the optimality condition we have $0\\in \\partial \\delta _C(x^{k+1})+\\nabla Q(x^{k+1};y^{k+1})$ , namely $-\\nabla Q(x^{k+1};y^{k+1})\\in \\partial \\delta _C(x^{k+1})$ .",
"Hence for any $x\\in C$ , the following inequality holds $0\\ge \\langle -\\nabla Q(x^{k+1};y^{k+1}),x-x^{k+1}\\rangle .$ Using the above inequality and the strong convexity of $Q(x;y^{k+1})$ with modulus $L+\\mu $ , we have $&Q(x^{k};y^{k+1})&\\ge Q(x^{k+1};y^{k+1})+\\langle \\nabla Q(x^{k+1};y^{k+1}),x^k-x^{k+1}\\rangle +\\frac{L+\\mu }{2}\\Vert x^{k+1}-x^k\\Vert ^2\\\\&&\\ge Q(x^{k+1};y^{k+1})+\\frac{L+\\mu }{2}\\Vert x^{k+1}-x^k\\Vert ^2.$ Combining the above inequality and (REF ), we obtain $&&f(x^{k+1})+\\frac{\\mu }{2}\\Vert x^{k+1}-y^{k+1}\\Vert ^2+\\frac{L+\\mu }{2}\\Vert x^{k+1}-x^{k}\\Vert ^2\\\\&\\le & f(y^{k+1})+\\langle \\nabla f(y^{k+1}), x^k-y^{k+1}\\rangle + \\frac{L+\\mu }{2}\\Vert y^{k+1}-x^{k}\\Vert ^2\\\\&\\le & f(x^k)+\\frac{(L+\\mu )\\omega _k^2}{2}\\Vert x^{k-1}-x^{k}\\Vert ^2.$ Summing the above inequality over $k=1,\\ldots ,n,\\ldots $ , we have $&\\sum _{k=0}^\\infty \\frac{(L+\\mu )(1-\\omega ^2)}{2}\\Vert x^k-x^{k-1}\\Vert ^2&\\le \\sum _{k=0}^\\infty \\frac{(L+\\mu )(1-\\omega _k^2)}{2}\\Vert x^k-x^{k-1}\\Vert ^2\\\\&& \\le f(x^0)-\\min _{x\\in X} f(x)<\\infty .$ Then $\\sum _{k=1}^\\infty \\Vert x^k-x^{k-1}\\Vert ^2<\\infty $ and $\\Vert x^k-x^{k-1}\\Vert ^2\\rightarrow 0$ .",
"In the following, we establish the convergence of $\\lbrace x^k\\rbrace _{k=0}^\\infty $ .",
"Theorem 2.6 Let $H(x)$ be the objective function defined in (REF ), and $\\lbrace x^k\\rbrace _{k=0}^\\infty $ be the sequence generated by Algorithm 1, then $x^k$ is bounded; any cluster point of $\\lbrace x^k\\rbrace _{k=0}^\\infty $ is a local minimizer of $H(x)$ ; $H(x^k)\\rightarrow H(x^*)$ where $x^*$ is a cluster point of $\\lbrace x^k\\rbrace _{k=0}^\\infty $ ; if $\\omega _k\\equiv \\omega \\in (0,1)$ , $x^k$ is convergent.",
"1.",
"It is clear that $x^k\\in X$ is bounded.",
"2.",
"Assume that $x^*$ is a cluster point of $\\lbrace x^k\\rbrace _{k=0}^\\infty $ and the subsequence $x^{k_j}$ converging to $x^*$ .",
"From Lemma REF , $\\Vert x^k-x^{k-1}\\Vert \\rightarrow 0$ and $I(x^k)$ changes only finitely often.",
"So we have $\\Vert x^k-y^{k+1}\\Vert \\rightarrow 0$ and there exists $k_0$ such that for any $k\\ge k_0$ , $I(y^{k+1})=I(x^k)=I(x^{k+1})=I(x^*)$ .",
"From (REF ), we have $x^{k+1}=P_{C_{I(x^*)}\\cap X}(y^{k+1}-\\frac{\\nabla f(y^{k+1})}{L+\\mu }), \\;\\; k\\ge k_0.$ Letting $k$ be equal to $k_j$ and $j$ tend to infinity, from the continuity of projection operator, we obtain $&x^*=P_{C_{I(x^*)}\\cap X}(x^*-\\frac{\\nabla f(x^*)}{L+\\mu })$ Since $X$ is a box constraint, we have $x^*_i=P_{C_{I(x^*_i)}\\cap [l_i,u_i]}(x^*_i-\\frac{(\\nabla f(x^*))_i}{L+\\mu }).$ From the proof of Lemma REF , if $x^*_i\\ne 0$ , $C_{I(x^*_i)}=\\mathbb {R}$ and $x^*_i=P_{[l_i,u_i]}(x^*_i-\\frac{(\\nabla f(x^*))_i}{L+\\mu })$ .",
"From the property of projection operator, we have $(x_i-x^*_i)(x^*_i-\\frac{(\\nabla f(x^*))_i}{L+\\mu }-x^*_i)\\le 0 $ , namely $(x_i-x^*_i)(\\nabla f(x^*))_i\\ge 0$ , for any $x\\in X$ .",
"Denote $U:=\\lbrace \\Delta x:\\Delta x+x^*\\in X, \\Vert \\Delta x\\Vert _{\\infty }<\\min _{i\\in I(x^*)}\\lbrace \\frac{\\lambda }{|(\\nabla f(x^*))_i|}\\rbrace ; |\\Delta x_i|<|x^*_i|,\\forall i\\notin I(x^*) \\rbrace ,$ Then for any $\\Delta x\\in U$ , we have $\\sum _{i\\notin I(x^*)}\\lambda \\Vert x^*_i+\\Delta x_i\\Vert _0=\\sum _{i\\notin I(x^*)}\\lambda \\Vert x^*_i\\Vert _0,\\;\\forall i\\notin I(x^*)$ since $|\\Delta x_i|<|x^*_i|$ ; $\\Delta x_i(\\nabla f(x^*))_i\\ge 0,\\;\\forall i\\notin I(x^*)$ since $x^*+\\Delta x\\in X$ ; $\\lambda \\Vert \\Delta x_i\\Vert _0+(\\nabla f(x^*))_i\\Delta x_i\\ge 0,\\; i\\in I(x^*)$ since $\\Vert \\Delta x\\Vert _{\\infty }<\\min _{i\\in I(x^*)}\\lbrace \\lambda /|(\\nabla f(x^*))_i|\\rbrace .$ Furthermore if $\\Delta x_i\\ne 0$ , it is clear that $\\lambda \\Vert \\Delta x_i\\Vert _0+(\\nabla f(x^*))_i\\Delta x_i>0$ .",
"From the above conclusions, for any $\\Delta x \\in U $ , we have $&H(x^*+\\Delta x)-H(x^*)&=\\lambda \\Vert x^*+\\Delta x\\Vert _0-\\lambda \\Vert x^*\\Vert _0+f(x^*+\\Delta x)-f(x^*) \\\\&& \\ge \\sum _{i\\in I(x^*)}\\lambda \\Vert \\Delta x_i\\Vert _0+\\langle \\nabla f(x^*), \\Delta x\\rangle \\\\&&=\\sum _{i\\in I(x^*)}(\\lambda \\Vert \\Delta x_i\\Vert _0+(\\nabla f(x^*))_i\\Delta x_i)+\\sum _{i\\notin I(x^*)}\\Delta x_i(\\nabla f(x^*))_i\\\\&&\\ge 0$ It is clear that $x^*+U$ is a neighborhood of $x^*$ .",
"So $x^*$ is a local minimizer of objective function $H(x)$ .",
"3.",
"From the inequality (REF ), $H(x^{k+1})$ is non-increasing.",
"$f(x)$ is bounded from below on $X$ , so we have $H(x^k)$ is convergent.",
"Furthermore, $H(x^{k})\\rightarrow H(x^*)$ since $I(x^k)=I(x^*)$ when $k\\ge k_0$ and $H(x^{k_j})\\rightarrow H(x^*)$ .",
"4.",
"As we have known, for any $k\\ge k_0$ and $x\\in C:=C_{I(x^*)}\\cap X$ , we have $&x^{k+1}&=P_C(y^{k+1}-\\frac{\\nabla f(y^{k+1})}{L+\\mu }),$ and $I(y^{k+1})=I(x^k)=I(x^{k+1})=I(x^*)$ .",
"Then using Lemma REF , for any $x\\in C$ , we can obtain $f(x^{k+1})\\le f(x)+(L+\\mu )\\langle x-y^{k+1}, x^{k+1}-y^{k+1}\\rangle -\\frac{L+\\mu }{2}\\Vert x^{k+1}-y^{k+1}\\Vert ^2,k\\ge k_0.$ Setting $x=x^*$ and $x=x^{k}$ respectively, we have $f(x^{k+1})\\le f(x^*)+(L+\\mu )\\langle x^*-y^{k+1}, x^{k+1}-y^{k+1}\\rangle -\\frac{L+\\mu }{2}\\Vert x^{k+1}-y^{k+1}\\Vert ^2,k\\ge k_0,$ $f(x^{k+1})\\le f(x^{k})+(L+\\mu )\\langle x^k-y^{k+1}, x^{k+1}-y^{k+1}\\rangle -\\frac{L+\\mu }{2}\\Vert x^{k+1}-y^{k+1}\\Vert ^2,k\\ge k_0.$ Note that either $y^{k+1}=x^{k}+\\omega (x^{k}-x^{k-1})$ or $y^{k+1}=x^k$ , the above inequality always holds.",
"Case 1: there exists $k_1>k_0$ such that $y^{k+1}=x^{k}+\\omega (x^{k}-x^{k-1})$ for any $k\\ge k_1$ .",
"Multiplying the inequality (REF ) by $1-\\omega $ and (REF ) by $\\omega $ , then adding the two resulting inequalities, and using the fact $f(x^*)\\le f(x^k)$ , we obtain, for any $k\\ge k_1$ , $&&\\frac{2}{ L+\\mu }(f(x^{k+1})-f(x^k))\\\\&\\le &2\\langle (1-\\omega )x^*+\\omega x^k-y^{k+1}, x^{k+1}-y^{k+1}\\rangle -\\Vert x^{k+1}-y^{k+1}\\Vert ^2\\\\&=&\\Vert y^{k+1}-\\omega x^{k}-(1-\\omega )x^*\\Vert ^2-\\Vert x^{k+1}-\\omega x^k-(1-\\omega )x^*\\Vert ^2\\\\&=&\\Vert x^{k}-\\omega x^{k-1}-(1-\\omega )x^*\\Vert ^2-\\Vert x^{k+1}-\\omega x^k-(1-\\omega )x^*\\Vert ^2.\\\\$ This implies $\\lbrace \\frac{2}{ L+\\mu }f(x^{k})+\\Vert x^{k}-\\omega x^{k-1}-(1-\\omega )x^*\\Vert ^2\\rbrace _{k\\ge k_1}$ is a non-increasing sequence.",
"So it is convergent.",
"Noting that $x^*$ is a cluster point of $\\lbrace x^k\\rbrace _{k=0}^{+\\infty }$ , $f(x^k)\\rightarrow f(x^*)$ and $x^{k}-x^{k-1}\\rightarrow 0$ , we can obtain $\\Vert x^{k}-x^*\\Vert ^2\\rightarrow 0$ .",
"Case 2: for any $k_1>k_0$ , there exists $k>k_1$ such that $y^{k+1}=x^{k}$ .",
"For simplicity, denote $\\sigma _n:=\\frac{2}{L+\\mu }f(x^{n})$ and $\\rho _n:=\\Vert x^{n}-x^{n-1}\\Vert ^2$ .",
"If $y^{n+1}=x^{n}$ , from the inequality (REF ), we obtain $0\\le \\frac{2}{ L+\\mu }(f(x^{n+1})-f(x^*))\\le \\Vert x^n-x^*\\Vert ^2-\\Vert x^{n+1}-x^*\\Vert ^2.$ Combing it with inequality (REF ), we have $\\sigma _{n+1}+\\rho _{n+1}+(1-\\omega )^2\\Vert x^{n+1}-x^*\\Vert ^2\\le \\sigma _n+\\omega ^2\\rho _n+(1-\\omega )^2\\Vert x^{n}-x^*\\Vert ^2.$ If $y^{n+1}=x^{n}+\\omega (x^{n}-x^{n-1})$ , from the discussion in case 1, we have $\\sigma _{n+1}+\\Vert x^{n+1}-\\omega x^n-(1-\\omega )x^*\\Vert ^2\\le \\sigma _{n}+\\Vert x^{n}-\\omega x^{n-1}-(1-\\omega )x^*\\Vert ^2.$ Without loss of generality, we assume that $&&y^{k_0+1}=x^{k_0}, \\cdots , y^{k_1}=x^{k_1-1},\\\\&&y^{k_1+1}=x^{k_1}+\\omega (x^{k_1}-x^{k_1-1}), \\cdots , y^{k_2}=x^{k_2-1}+\\omega (x^{k_2-1}-x^{k_2-2}),\\\\&&y^{k_2+1}=x^{k_2}, \\cdots , y^{k_3}=x^{k_3-1},\\\\$ and this happens again and again.",
"So we just need discuss for $k_1\\le k\\le k_3-1$ .",
"Form the inequality (REF ), (REF ) and (REF ), we can obtain $\\sigma _{k_0+1}+\\rho _{k_0+1}+(1-\\omega )^2\\Vert x^{k_0+1}-x^*\\Vert ^2\\le \\sigma _{k_0}+\\omega ^2\\rho _{k_0}+(1-\\omega )^2\\Vert x^{k_0}-x^*\\Vert ^2$ $\\vdots $ $\\sigma _{k_1}+\\rho _{k_1}+(1-\\omega )^2\\Vert x^{k_1}-x^*\\Vert ^2\\le \\sigma _{k_1-1}+\\omega ^2\\rho _{k_1-1}+(1-\\omega )^2\\Vert x^{k_1-1}-x^*\\Vert ^2$ $\\sigma _{k_1+1}+\\Vert x^{k_1+1}-\\omega x^{k_1}-(1-\\omega )x^*\\Vert ^2\\le \\sigma _{k_1}+\\Vert x^{k_1}-\\omega x^{k_1-1}-(1-\\omega )x^*\\Vert ^2$ $\\vdots $ $\\sigma _{k_2}+\\Vert x^{k_2}-\\omega x^{k_2-1}-(1-\\omega )x^*\\Vert ^2\\le \\sigma _{k_2-1}+\\Vert x^{k_2-1}-\\omega x^{k_2-2}-(1-\\omega )x^*\\Vert ^2$ $\\sigma _{k_2+1}+\\rho _{k_2+1}+(1-\\omega )^2\\Vert x^{k_2+1}-x^*\\Vert ^2\\le \\sigma _{k_2}+\\omega ^2\\rho _{k_2}+(1-\\omega )^2\\Vert x^{k_2}-x^*\\Vert ^2$ $\\vdots $ $\\sigma _{k_3}+\\rho _{k_3}+(1-\\omega )^2\\Vert x^{k_3}-x^*\\Vert ^2\\le \\sigma _{k_3-1}+\\omega ^2\\rho _{k_3-1}+(1-\\omega )^2\\Vert x^{k_3-1}-x^*\\Vert ^2$ We denote the terms on the right side of the above inequalities as sequence $\\lbrace u^k\\rbrace _{k\\ge k_0}^{k_3-1}$ .",
"It's clear that $\\lbrace u^k\\rbrace _{k\\ge k_0}^{k_1-1}$ , $\\lbrace u^k\\rbrace _{k\\ge k_1}^{k_2-1}$ and $\\lbrace u^k\\rbrace _{k\\ge k_2}^{k_3-1}$ is non-increasing.",
"For the following situation $\\sigma _{k_2}+\\Vert x^{k_2}-\\omega x^{k_2-1}-(1-\\omega )x^*\\Vert ^2\\le \\sigma _{k_2-1}+\\Vert x^{k_2-1}-\\omega x^{k_2-2}-(1-\\omega )x^*\\Vert ^2$ $\\sigma _{k_2+1}+\\rho _{k_2+1}+(1-\\omega )^2\\Vert x^{k_2+1}-x^*\\Vert ^2\\le \\sigma _{k_2}+\\omega ^2\\rho _{k_2}+(1-\\omega )^2\\Vert x^{k_2}-x^*\\Vert ^2,$ if $\\langle x^{k_2}-x^{k_2-1}, x^{k_2}-x^*\\rangle \\ge 0 $ , then $&u^{k_2-1}&\\ge \\sigma _{k_2}+\\Vert x^{k_2}-\\omega x^{k_2-1}-(1-\\omega )x^*\\Vert ^2\\\\&&\\ge \\sigma _{k_2}+\\omega ^2\\Vert x^{k_2}-x^{k_2-1}\\Vert ^2+(1-\\omega )^2\\Vert x^{k_2}-x^*\\Vert ^2\\\\&&= u^{k_2}$ and the sequence $\\lbrace u^k\\rbrace _{k\\ge k_1}^{k_3-1}$ is non-increasing; otherwise $&\\Vert x^{k_2-1}-x^*\\Vert ^2&=\\Vert x^{k_2-1}-x^{k_2}\\Vert ^2+\\Vert x^{k_2}-x^*\\Vert ^2-2\\langle x^{k_2}-x^{k_2-1}, x^{k_2}-x^*\\rangle \\\\&&\\ge \\Vert x^{k_2}-x^*\\Vert ^2,$ combing it with inequality (REF ), we have $\\sigma _{k_2}+\\rho _{k_2}+(1-\\omega )^2\\Vert x^{k_2}-x^*\\Vert ^2\\le \\sigma _{k_2-1}+\\omega ^2\\rho _{k_2-1}+(1-\\omega )^2\\Vert x^{k_2-1}-x^*\\Vert ^2,$ then we redefine $u^{k_2-1}:=\\sigma _{k_2-1}+\\omega ^2\\rho _{k_2-1}+(1-\\omega )^2\\Vert x^{k_2-1}-x^*\\Vert ^2$ and hence $u^{k_2-1}\\ge u^{k_2}$ , repeating the above process for $\\lbrace u^k\\rbrace _{k\\ge k_2-1}^{k_1}$ and redefine $u^k$ if necessary, we can obtain a non-increasing sequence $\\lbrace u^k\\rbrace _{k\\ge k_1}^{k_2-1}$ .",
"If $u^{k_1}$ isn't redefined, the following situation happens $\\sigma _{k_1}+\\rho _{k_1}+(1-\\omega )^2\\Vert x^{k_1}-x^*\\Vert ^2\\le \\sigma _{k_1-1}+\\omega ^2\\rho _{k_1-1}+(1-\\omega )^2\\Vert x^{k_1-1}-x^*\\Vert ^2:=u^{k_1-1}$ $\\sigma _{k_1+1}+\\Vert x^{k_1+1}-\\omega x^{k_1}-(1-\\omega )x^*\\Vert ^2\\le \\sigma _{k_1}+\\Vert x^{k_1}-\\omega x^{k_1-1}-(1-\\omega )x^*\\Vert ^2:=u^{k_1}.$ Noting that $\\Vert x^{k_1}-x^*\\Vert ^2\\le \\Vert x^{k_1-1}-x^*\\Vert ^2$ , hence $&u^{k_1}&= \\sigma _{k_1}+\\omega \\Vert x^{k_1}-x^{k_1-1}\\Vert ^2+(1-\\omega )\\Vert x^{k_1}-x^*\\Vert ^2-\\omega (1-\\omega )\\Vert x^{k_1-1}-x^*\\Vert ^2\\\\&&\\le \\sigma _{k_1}+\\Vert x^{k_1}-x^{k_1-1}\\Vert ^2+(1-\\omega )^2\\Vert x^{k_1}-x^*\\Vert ^2\\\\&&\\le \\sigma _{k_1-1}+\\omega ^2\\rho _{k_1-1}+(1-\\omega )^2\\Vert x^{k_1-1}-x^*\\Vert ^2\\\\&&=u^{k_1-1};$ otherwise, the following situation happens $\\sigma _{k_1}+\\rho _{k_1}+(1-\\omega )^2\\Vert x^{k_1}-x^*\\Vert ^2\\le \\sigma _{k_1-1}+\\omega ^2\\rho _{k_1-1}+(1-\\omega )^2\\Vert x^{k_1-1}-x^*\\Vert ^2:=u^{k_1-1}$ $\\sigma _{k_1+1}+\\rho _{k_1+1}+(1-\\omega )^2\\Vert x^{k_1+1}-x^*\\Vert ^2\\le \\sigma _{k_1}+\\omega ^2\\rho _{k_1}+(1-\\omega )^2\\Vert x^{k_1}-x^*\\Vert ^2:=u^{k_1}$ and it's clear that $u^{k_1-1}\\ge u^{k_1}$ .",
"In summary, we can obtain a non-increasing sequence $\\lbrace u^k\\rbrace _{k\\ge k_0}^{k_3-1}$ where $u^k=\\sigma _{k}+\\omega ^2\\rho _{k}+(1-\\omega )^2\\Vert x^{k}-x^*\\Vert ^2$ or $u^k=\\sigma _{k}+\\Vert x^{k}-\\omega x^{k-1}-(1-\\omega )x^*\\Vert ^2$ .",
"Repeating this process, we finally obtain a non-increasing sequence $\\lbrace u^k\\rbrace _{k\\ge k_0}^{+\\infty }$ .",
"So it's convergent.",
"Combining the fact $x^*$ is a cluster point of $\\lbrace x^k\\rbrace _{k=0}^{+\\infty }$ , $\\sigma _k\\rightarrow \\frac{2}{L+\\mu }f(x^{*})$ and $x^{k}-x^{k-1}\\rightarrow 0$ , we can obtain that $\\Vert x^k-x^*\\Vert \\rightarrow 0$ .",
"Recently, some extrapolation type methods were proposed for $\\ell _0$ regularization problem or more general non-convex problems.",
"In particular, the inertial forward-backward (IFB) method [11] for solving problem (REF )(both $f(x)$ and $g(x)$ can be non-convex) uses Bregman distance.",
"Under Kurdyka-Łojasiewicz property theoretical framework, the sequence generated by IFB method converges to a critical point when $f+g$ is coercive.",
"When we take the Bregman distance function as $\\Vert \\cdot \\Vert ^2/2$ , IFB method is the algorithm proposed by [22] while $g(x)$ needs to be convex.",
"If we apply IFB method to $\\ell _0$ regularization problem (REF ), the iterative scheme is $&&y^{k+1}=x^{k}+2\\beta _k(x^k-x^{k-1})\\\\&&x^{k+1}\\in \\arg \\min _{x \\in X}\\lambda \\Vert x\\Vert _0+\\frac{1}{4\\alpha _k}\\Vert x-x^{k}+2\\alpha _k\\nabla f(x^{k})\\Vert ^2+\\frac{1}{4\\alpha _k}\\Vert x-y^{k+1}\\Vert ^2$ where $\\alpha _k,\\beta _k>0$ satisfy $&&0<\\underline{\\alpha }\\le \\alpha _k\\le \\overline{\\alpha },0<\\beta _k\\le \\beta \\\\ &&1>\\overline{\\alpha }L+2\\beta \\frac{\\overline{\\alpha }}{\\underline{\\alpha }}.",
"$ for some $\\overline{\\alpha },\\underline{\\alpha },\\beta >0$ and $\\nabla f$ 's Lipschitz constant $L$ .",
"It is easy to see that when $\\beta _k\\equiv 0$ , IFB becomes PIHT.",
"Usually a larger $\\alpha _k$ leads to a faster convergence.",
"However, the above inequality () implies that a larger $\\alpha _k$ leads to a small $\\beta _k$ , thus the extrapolation step will have small effect on the speed of IFB method.",
"In other words, one cannot have both of large $\\alpha _k$ and $\\beta _k$ .",
"This limits the acceleration effect of IFB method against PIHT method.",
"The extrapolated PIHT (EPIHT) method [4] is proposed for solving $\\min _{x\\in \\mathbb {R}^n} H(x):= \\lambda \\Vert x\\Vert _0+f(x)+\\frac{t}{2}\\Vert x\\Vert ^2,$ where $f$ is convex and $\\nabla f$ is Lipschitz continuous, could have both large step size and large extrapolated step size.",
"Its iterative scheme takes the form $&&y^{k+1}=x^{k}+\\omega _k(x^{k}-x^{k-1})\\\\&&\\mbox{if } H(y^{k+1})>H(x^{k}), \\;\\; \\mbox{reset } y^{k+1}=x^k\\\\&&x^{k+1}\\in \\arg \\min _{x \\in \\mathbb {R}^n}\\lambda \\Vert x\\Vert _0+\\frac{L+t}{2}\\Vert x-y^{k+1}+\\frac{\\nabla f(y^{k+1})}{L+t}\\Vert ^2+\\frac{\\mu }{2}\\Vert x-y^{k+1}\\Vert ^2$ where $\\mu >0$ , $0<\\omega _k\\le \\omega <1$ .",
"It is similar with the IFB method except that the linearization is performed at $y^{k+1}$ instead of $x^k$ and the setting for parameters is also different.",
"Under Kurdyka-Łojasiewicz property theoretical framework, the sequence generated by EPIHT method globally converges to a local minimizer of $H(x)$ .",
"For more general problem (REF ) (both $f(x)$ and $g(x)$ can be non-convex), [18] proposed the following monotone accelerated proximal gradient (mAPG) method $&&y^k=x^k+\\frac{t_{k-1}}{t_k}(z^k-x^k)+\\frac{t_{k-1}-1}{t_k}(x^k-x^{k-1})\\\\&&z^{k+1}=\\arg \\min _zg(z)+\\frac{1}{2\\alpha _y}\\Vert z-y^k+\\alpha _y\\nabla f(y^k)\\Vert ^2\\\\(\\mbox{mAPG})\\quad &&v^{k+1}=\\arg \\min _vg(v)+\\frac{1}{2\\alpha _x}\\Vert v-x^k+\\alpha _x\\nabla f(x^k)\\Vert ^2\\\\&&t_{k+1}=\\frac{\\sqrt{1+4t_k^2}+1}{2}\\\\&&x^{k+1}=\\left\\lbrace \\begin{array}{cl}z^{k+1},& \\mbox{ if }f(z^{k+1})+g(z^{k+1})\\le f(v^{k+1})+g(v^{k+1}) \\\\v^{k+1},& \\mbox{ otherwise}\\end{array}\\right.$ When $f(x), g(x)$ are convex, mAPG has $O(1/k^2)$ convergence rate; otherwise, any cluster point of iteration sequence is a critical point of $f(x)+g(x)$ .",
"Based on mAPG, [18] also proposed a non-monotone APG(nmAPG) for saving the computation cost in each step.",
"Denote $q_1=1,c_1=f(x_1)+g(x_1);q_{k+1}=\\eta q_k+1, c_{k+1}=\\frac{\\eta q_kc_k+f(x^{k+1})+g(x^{k+1})}{q_{k+1}}.$ If $f(z^{k+1})\\le c_k-\\delta \\Vert x^{k+1}-y^k\\Vert ^2$ , nmAPG gets the next iteration point by $x^{k+1}=z^{k+1}$ , otherwise, it gets the next iteration point same with mAPG.",
"Moreover, [24] proposed an inertial proximal alternating linearized minimization (iPALM) method for solving problem $\\min _{x,y}s(x)+q(x,y)+r(y).$ The iterative sequence has global convergence.",
"If $r(y)\\equiv 0$ , $s(x)=\\lambda \\Vert x\\Vert _0$ and $q(x,y)=f(x)$ , the above problem reduces to (REF ), and iPALM is simplified as $&& y^k=x^k+\\alpha _k(x^k-x^{k-1})\\\\&& z^{k}=x^k+\\beta _k(x^k-x^{k-1})\\\\&& x^{k+1}=\\arg \\min _{x\\in X} \\lambda \\Vert x\\Vert _0+\\frac{1}{2\\tau _k}\\Vert x-y^k+\\tau _k\\nabla f(z^k)\\Vert ^2$ If the objective function satisfies Kurdyka-Łojasiewicz property, the iteration sequence has global convergence, but the extrapolated step length $\\alpha _k, \\beta _k$ , and the proximal parameter $\\tau _k$ need to satisfy an equation.",
"Table: Comparisons of IFB, EPIHT, mAPG, iPALM and our method.",
"Parameters refer to α k ,β k \\alpha _k, \\beta _k in IFB, ω k ,μ\\omega _k, \\mu in EPIHT, α x ,α y \\alpha _x, \\alpha _y in mAPG, α k ,β k ,τ k \\alpha _k, \\beta _k, \\tau _k in iPALM, ω k ,μ\\omega _k, \\mu in our method and BC represents box constraint.",
"NCf represents the number of computation of ff and NCGf represents the number of computation of ∇f\\nabla f during one iteration.In Table REF , we summarize some differences of the above mentioned algorithms and our method.",
"All the methods require $f$ being differentiable and $\\nabla f$ being Lipschitz continuous and this is not stated again in the table.",
"We point out that: our method's global convergence analysis does not rely on KL property; the constraint conditions of parameters of EPIHT, mAPG and our method are relatively simpler compared to other methods; IFB and iPALM methods need the least amount of computation per iteration, but the conditions on the algorithm parameters are more complex or restricted, which could increase the total number of iteartions; compared with EPIHT and mAPG, our method need less computation cost for one iteration; This can cost less computation time when the total iteration number is fixed."
],
[
"Numerical Implementation",
"In this section, we will show some numerical results of Algorithm 1 on $\\ell _0$ minimization problems (REF ), and compare with the results of PIHT, IFB, mAPG, nmAPG, EPIHT methods.",
"All the experiments are conducted in MATLAB using a desktop computer equipped with a $4.0$ GHz 8-core AMD processor and 16GB memory."
],
[
"Compressive sensing",
"We first test the algorithms on a standard sparse signal reconstruction problem in compressive sensing [12].",
"The goal is to reconstruct a sparse signal from a set of noisy linear measurements.",
"The following $\\ell _0$ regularization formulation can be considered $\\min _{x\\in X }\\frac{1}{2}\\Vert Ax-b\\Vert ^2+\\lambda \\Vert x\\Vert _0$ where $A\\in \\mathbb {R}^{m\\times n}$ is a data matrix, $b\\in \\mathbb {R}^m$ is an observation vector, and $X=\\lbrace x\\in \\mathbb {R}^n|-10^{10}\\le x\\le 10^{10}\\rbrace $ .",
"We set $f(x)=\\Vert Ax-b\\Vert ^2/2$ and then the Lipschitz constant of $\\nabla f(x)$ is $L=\\lambda _{\\max } (A^\\top A)$ , where $\\lambda _{\\max } (A^\\top A)$ denotes the maximum eigenvalue of $A^\\top A$ .",
"For this experiment, the data matrix $A\\in \\mathbb {R}^{m\\times n}$ is a Gaussian random matrix and the columns of $A$ are normalized to have $\\ell _2$ norm of 1.",
"We set $m=3000$ and test on different size of $n$ and sparsity level $s$ of the unknown signal.",
"For each choice of $(n,s)$ , we generate the true signal $\\bar{x}\\in \\mathbb {R}^n$ containing $s$ randomly placed $\\pm 1$ spikes.",
"The observed data $b\\in \\mathbb {R}^m$ is generated by $b=Ax+\\eta ,$ where $\\eta $ is a white Gaussian noise of variance $0.05$ .",
"And for each pair of $(n,s)$ , we run our experiment 50 times to guarantee that the result is independent of any particular realization of the random matrix and true signal $\\bar{x}$ .",
"For all the methods, the stopping criteria is commonly set to be $\\frac{\\Vert x_k-x_{k-1}\\Vert }{\\mbox{max}\\lbrace 1,\\Vert x_k\\Vert \\rbrace }<10^{-5},$ and the initial point is obtained by FISTA[5] for $\\ell _1$ minimization (where the initial point is $x_0=A^Tb$ , and the stopping criteria is $\\frac{\\Vert x_k-x_{k-1}\\Vert }{\\mbox{max}\\lbrace 1,\\Vert x_k\\Vert \\rbrace }<10^{-2}$ , the regularization parameter $\\lambda =0.1$ ) and the corresponding iteration number and running time are added in the final results.",
"All the parameters are chosen according to empirically the lowest relative error $\\frac{\\Vert x-\\bar{x}\\Vert }{\\Vert \\bar{x}\\Vert }$ .",
"In detail, we choose $\\ell _0$ regularization $\\lambda _1=0.3$ ; choose $\\mu =10^{-6}$ , $\\omega _k=0.99$ for Algorithm 1 and EPIHT method; choose $\\beta _k=10^{-6}$ , $\\alpha _k=(0.999999-2\\beta _k)/L$ for IFB method; choose $\\alpha _x=\\alpha _y=1/(L+10^{-6})$ for mAPG and nmAPG method, moreover choose $\\eta =0.8$ for nmAPG method.",
"For each algorithm and each choice of $(n,s)$ of the solution $\\bar{x}$ , we conduct 50 experiments and record the average runtime, the average relative error $\\frac{\\Vert x-\\bar{x}\\Vert }{\\Vert \\bar{x}\\Vert }$ to the original signal $\\bar{x}$ , the average number of iteration the algorithm needed and their standard variance.",
"In fact, we find that the approximate solutions' $\\ell _0$ -norm of all the methods are always equal to the true $s$ and we do not list them in the tables.",
"The average relative errors of all the methods are very close.",
"In fact, if the results are rounded up to 4 decimal digits, the results are the same as present in Table REF £¬especially all the results if PIHT and IFB methods.",
"The average number of iterations and average runtime are listed in Table REF , REF respectively to compare the convergence speed of different methods.",
"We observe that: in term of average number of iterations (Table REF ), mAPG method , especially EPIHT and our method, have obvious accelerating effect compared to PIHT; recall that the amounts of computation for each step of the algorithms are different; although EPIHT and our method have fewer, similar iteration number, our method obviously has less runtime (Table REF ); the stability of all the methods is comparable, by the standard variance result present in Table REF , REF .",
"In Table REF , our method has less runtime compared to EPIHT as it requires less computation of gradient function (NCGf) per iteration.",
"The average total number of computation of gradient function is recorded in Table REF based on 20 times of experiments.",
"It can be observed that if the step (REF ) occurs in every iteration, the total NCGf should be two times the number of iterations.",
"In fact, Table REF demonstrates that the restart step (REF ) occurs in a low rate.",
"Thus the extrapolation contributes to the reduction of computation and the number of iterations.",
"Table: Results of the average and the standard variance of therelative error.Table: Results of the average and the standard variance of iterations number.Table: Results of the average and the standard variance of theruntime.Table: Results of the average total NCGf.",
"NCGf represents the number of computation of ∇f\\nabla f."
],
[
"Logistic Regression",
"Given a set of training data $(x_i,y_i),i=1,\\cdots ,N$ , where the input $x_i\\in \\mathbb {R}^n$ , and the output $y_i\\in \\lbrace 1,-1\\rbrace $ .",
"We wish to find a classffication rule from the training data, so that when given a new input $x$ , we can assign a class $y$ from $\\lbrace 1,-1\\rbrace $ to it.",
"For this example, we consider the following sparse logistic regression model using $\\ell _0$ regularization $\\min _{(u,v)\\in X} \\frac{1}{N}\\sum _{i=1}^N\\log (1+\\exp (-y_i(u^Tx_i+v)))+\\lambda \\Vert u\\Vert _0$ where $X=[-10^{10},10^{10}]^{n+1}$ .",
"The data set gisette used for our numerical experiment is taken from [16].",
"The train set contains 6000 samples of 5000 dimensions, and the test set contains 1000 samples of 5000 dimensions.",
"For all the methods, the stopping criteria is commonly set to be $\\Vert x_k-x_{k-1}\\Vert _{\\infty }<5\\times 10^{-4},$ and the initial point is obtained by FISTA[5] for $\\ell _1$ minimization (where the initial point is $zeros(n+1,1)$ , and the stopping criteria is $\\Vert x_k-x_{k-1}\\Vert _{\\infty }<0.02$ , the regularization parameter $\\lambda =0.001$ ).",
"All the parameters are chosen according to accuracy.",
"In detail, we choose the penalty parameter $\\lambda =0.00005$ ; choose $\\mu =10^{-6}$ , $\\omega _k=0.99$ for Algorithm 1 and EPIHT method; choose $\\beta _k=10^{-6}$ , $\\alpha _k=(0.999999-2\\beta _k)/L$ for IFB method; choose $\\alpha _x=\\alpha _y=1/(L+10^{-6})$ for mAPG and nmAPG method, moreover choose $\\eta =0.6$ for nmAPG method.",
"Table: Numerical results of logistic regression.The results are listed in Table REF .",
"We can see that the results of Algorithm 1 and EPIHT are better than other three methods in the sense of iterations number, runtime and accuracy.",
"Although the iteration number of Algorithm 1 is bigger than EPIHT's, their runtime is close, and the accuracy is comparable."
],
[
"Conclusions and perspectives",
"In this paper, we proposed one proximal iterative hard thresholding type method–Algorithm 1, for solving the $\\ell _0$ regularized problem.",
"We provide some convergence analysis for the proposed method.",
"We further show in some numerical experiments that, the algorithm 1 is faster than PIHT, IFB, mAPG, nmAPG and EPIHT or comparable with EPIHT."
],
[
"Acknowledgements",
"This work was partially supported by NSFC (No.",
"9133010 2), the Young Top-notch Talent program of China, 973 program (No.",
"2015CB856004)."
]
] | 1709.01668 |
[
[
"$XY$ model with higher-order exchange"
],
[
"Abstract An $XY$ model, generalized by inclusion of up to an infinite number of higher-order pairwise interactions with an exponentially decreasing strength, is studied by spin-wave theory and Monte Carlo simulations.",
"At low temperatures the model displays a quasi-long-range order phase characterized by an algebraically decaying correlation function with the exponent $\\eta = T/[2 \\pi J(p,\\alpha)]$, nonlinearly dependent on the parameters $p$ and $\\alpha$ that control the number of the higher-order terms and and the decay rate of their intensity, respectively.",
"At higher temperatures the system shows a crossover from the continuous Berezinskii-Kosterlitz-Thouless to the first-order transition for the parameter values corresponding to a highly nonlinear shape of the potential well.",
"The role of of topological excitations (vortices) in changing the nature of the transition is discussed."
],
[
"Introduction",
"Mermin-Wagner theorem [1], [2] prevents any spontaneous breakdown of continuous symmetries for 2D systems with short-range interactions, such as a standard XY model.",
"Nevertheless, it does not prevent a topological Berezinskii-Kosterlitz-Thouless (BKT) phase transition, due to the vortex-antivortex pairs unbinding [3], [4], to a quasi-long-range-order (QLRO) phase characterized by a power-law decaying correlation function.",
"Several modifications and generalizations of the XY model have been proposed, mostly by including higher-order terms to the Hamiltonian, motivated theoretically (critical properties and universality) as well as experimentally (modeling of some systems, such as liquid crystals [5], [7], superfluid A phase of $^3{\\rm He}$ [6], and high-temperature cuprate superconductors [8]).",
"Inclusion of a biquadratic term, i.e., the system with the Hamiltonian ${\\mathcal {H}}=-J_1\\sum _{\\langle i,j \\rangle }\\cos (\\phi _{i,j})-J_2\\sum _{\\langle i,j \\rangle }\\cos (2\\phi _{i,j})$ , has been shown [5], [6], [9], [10], [11], [12] to lead to the separation of the dipole phase at lower and the quadrupole phase at higher temperature, for sufficiently large biquadratic coupling.",
"The order-disorder phase transition was determined to belong to the BKT universality class, while the dipole-quadrupole phase transition had the Ising character.",
"Recent series of studies [13], [14], [15] revealed that the model, in which the biquadratic term was generalized to a nematiclike coupling of the order $q>2$ , i.e., ${\\mathcal {H}}=-J_1\\sum _{\\langle i,j \\rangle }\\cos (\\phi _{i,j})-(1-J_1)\\sum _{\\langle i,j \\rangle }\\cos (q\\phi _{i,j})$ and $0 \\le J_1 \\le 1$ , leads to a qualitatively different phase diagram for $q>3$ , with additional ordered phases originating from the competition between the ferromagnetic and pseudonematic couplings and includes phase transitions belonging to the 2D Potts, Ising, or BKT universality classes.",
"Further generalization, motivated by orientational transitions in liquid crystals, lead to taking the $k$ -th order Legendre polynomials of the dipole term, i.e., the Hamiltonian ${\\mathcal {H}}=-\\sum _{\\langle i,j \\rangle }P_k(\\cos (\\phi _{i,j}))$ .",
"With the increasing value of $k$ , one may expect a qualitative change in the nature of the transition.",
"In particular, a rigorous proof has been provided that the transition becomes first order for large enough values of $k$ in models with $O(n)$ symmetry for $n \\ge 2$ [16], [17].",
"Nevertheless, for $O(2)$ case the studied values of $k=2$ and 4 indicated that the behavior is always described by the BKT-like transition, just like in the standard XY model [18], [19].",
"This is in contrast to the $O(3)$ case, in which a strong first-order phase transition was observed for $k=4$ [20], [21].",
"Another non-linear model [22], [23], [24], [25], [26], the potential shape of which can be controlled by a single parameter $p^2$ , in the form ${\\mathcal {H}}=2J\\sum _{\\langle i,j \\rangle }(1-[\\cos ^2(\\phi _{i,j}/2)]^{p^2})$ , was introduced in effort to enable tuning its properties between the standard XY model belonging to the BKT universality and the $q$ -state Potts model, which for large $q$ shows a first-order phase transition.",
"Indeed, for large $p$ (proportional to the Potts $q$ ), such a model has been shown to undergo a first-order phase transition.",
"In the present study we introduce a generalized XY model that takes into account effects of up to an infinite number of higher-order (multipolar) terms with an exponentially vanishing influence.",
"In spite of belonging to the same universality class (having same symmetry of the order parameter and same lattice dimensionality) as the standard XY model, we demonstrate that the model can display either the BKT or the first-order phase transition from the QLRO to the paramagnetic phase, depending on the parameters that control the degree of nonlinearity of the potential."
],
[
"Model",
"The considered model assumes only nearest-neighbor pairwise ferromagnetic interactions with the potential $H_{i,j}(p,\\alpha )=-\\sum _{k=1}^{p}J_{k}\\cos ^{k}\\phi _{i,j},$ where $\\phi _{i,j}=\\phi _{i}-\\phi _{j}$ is an angle between the nearest-neighbor spins and the respective exchange interactions decay as $J_k=\\alpha ^{-k}$ , where $\\alpha >1$ .",
"For an infinite number of the higher-order terms, i.e., $p \\rightarrow \\infty $ , the Hamiltonian reduces to ${\\mathcal {H}}(\\alpha )=J(\\alpha )\\sum _{\\langle i,j \\rangle }H_{i,j}(\\alpha )=-J(\\alpha )\\sum _{\\langle i,j \\rangle }\\frac{\\cos \\phi _{i,j}}{\\alpha -\\cos \\phi _{i,j}},$ where $\\langle i,j \\rangle $ denotes the sum over nearest-neighbor spins and $J(\\alpha )=\\alpha -1$ is an exchange interaction parameter chosen to normalize the weights $J_{k}$ (scaling them so they add up to 1).",
"For a finite number of the multipolar interaction terms, the system Hamiltonian can be expressed as ${\\mathcal {H}}(p,\\alpha )=J(p,\\alpha )\\sum _{\\langle i,j \\rangle }H_{i,j}(p,\\alpha )=-J(p,\\alpha )\\sum _{\\langle i,j \\rangle }\\frac{\\cos \\phi _{i,j}\\Big [1-\\Big (\\frac{\\cos \\phi _{i,j}}{\\alpha }\\Big )^{p}\\Big ]}{\\alpha -\\cos \\phi _{i,j}},$ where $J(p,\\alpha )=(\\alpha -1)/(1-\\alpha ^{-p})$ .",
"Thus, while in the case of $p \\rightarrow \\infty $ there is only one parameter, $\\alpha $ , if the sum is truncated there are two parameters, $\\alpha $ and $p$ , that can be used to change the shape of the respective potentials through changing the number of the higher-order terms and/or their weights.",
"The shapes of the potentials in both cases are shown in Fig.",
"REF , for different values of the parameters $\\alpha $ and $p$ .",
"The case with $p \\rightarrow \\infty $ [Fig.",
"REF ] reduces to the conventional XY model when the interaction terms decay extremely fast, i.e., for $\\alpha \\rightarrow \\infty $ , with the potential acquiring a cosine form.",
"With the decrease in $\\alpha $ , the potential well gets narrower with a width tending to zero as $\\alpha \\rightarrow 1$ .",
"In the model with a finite $p$ , a similar effect on the potential shape can be observed by increasing the number of the higher-order interaction terms, for sufficiently small values of $\\alpha $ [Fig.",
"REF ].",
"In this case, in the limit of $p \\rightarrow \\infty $ the width of the potential well will depend on the value of $\\alpha $ , as shown in Fig.",
"REF .",
"It is worth noticing that for the case of a finite $p$ and a small $\\alpha $ , one can also observe a local minimum at $\\phi =\\pm \\pi $ (see Fig.",
"REF ).",
"The latter is apparently related to the presence of the nematic term, the interaction strength of which is the second largest and for $\\alpha \\rightarrow 1$ it becomes comparable with the bilinear one.",
"Therefore, care should be exercised when selecting the MC method particularly in the case of the presence of higher-order interactions with comparable strengths, when for $p>2$ even multiple local minima may develop, in order to prevent getting stuck in one of those especially at low temperatures.",
"Figure: (Color online) Potential functions of the cases of (a) p→∞p \\rightarrow \\infty for several values of α\\alpha and (b) a fixed α=1.01\\alpha =1.01 and various values of pp."
],
[
"Spin wave approximation",
"Let us consider a large scale asymptotic behavior of the two-point correlation function $g(x_1 - x_2) \\equiv \\langle \\cos (\\phi (x_1) - \\phi (x_2) )\\rangle =Re \\langle \\exp i \\lbrace \\phi (x_1) - \\phi (x_2)\\rbrace \\rangle $ in the model defined through the more general form of the Hamiltonian, given by Eq.",
"(REF ).",
"Let $x$ be the coordinate vector of $i$ -th spin, and $a$ be the lattice vector.",
"At low temperatures one can assume smoothness of the field $\\phi (x)$ , and thus we may put $\\phi (x+a) - \\phi (x) = (a \\cdot \\nabla ) \\phi (x) + \\mathcal {O}( a^2)$ .",
"Having expanded Hamiltonian up to the second order in $a$ , we find the low temperature approximation ${\\mathcal {H}}^{\\rm sw} = J ^{\\rm sw} \\sum _{x} \\sum _{a} \\frac{1}{2} \\lbrace (a \\cdot \\nabla ) \\phi (x) \\rbrace ^2 = J ^{\\rm sw} \\sum _{x} a^2 \\frac{1}{2} \\lbrace \\nabla \\phi (x) \\rbrace ^2 \\rightarrow \\frac{J ^{\\rm sw}}{2} \\int d^2 x \\lbrace \\nabla \\phi (x) \\rbrace ^2,$ where $J ^{\\rm sw} = \\alpha /(\\alpha -1) - p/(\\alpha ^p -1)$ .",
"We now see, that an asymptotic expression for the correlation function $g(x_1 - x_2)$ can be easily deduced by the Gaussian integration over all possible field configurations $g(x_1 - x_2) = &\\int \\prod _x d \\phi (x) \\exp \\left( -\\frac{J ^{\\rm sw}}{2} \\int d^2 x \\lbrace \\nabla \\phi (x) \\rbrace ^2 + i \\lbrace \\phi (x_1) - \\phi (x_2)\\rbrace \\right) =\\\\=&\\exp \\left( -\\frac{1}{J ^{\\rm sw}} \\int \\frac{d^2 k}{(2 \\pi )^2}\\frac{1 - \\exp ( i k (x_1 - x_2) ) }{k^2} \\right) = \\exp \\left(-\\frac{1}{2 \\pi J ^{\\rm sw}} \\ln \\frac{e^{\\gamma }|x_1 - x_2|}{2 a} \\right),$ where $\\gamma $ is the Euler-Mascheroni constant and the momentum integral has to be regularized in the ultra-violet region $0 \\le |k| \\lesssim 1/a$ .",
"As a result, large distance $|x_1-x_2| >> |a|$ power asymptotics reads as $\\langle \\cos (\\phi (x_1) - \\phi (x_2)) \\rangle \\sim \\left( \\frac{a}{|x_1 - x_2|}\\right)^{\\eta ^{\\rm sw}},$ where the corresponding exponent $\\eta ^{\\rm sw} = T/(2 \\pi J ^{\\rm sw})$ .",
"We note that the resulting form of the correlation function exponent is also applicable to the specific case of the well studied bilinear-biquadratic model [5], [6], [9], [10], [11], [12], with $p=2$ and $\\alpha =J_1/J_2$ ."
],
[
"Monte Carlo",
"We employ Monte Carlo (MC) simulations with the standard Metropolis dynamics for spin systems on a square lattice of a linear size $L$ , imposing the periodic boundary conditions.",
"For thermal averaging we take $N_{MC}$ MC sweeps after discarding another $N_0=0.2\\times N_{MC}$ MC sweeps for thermalization.",
"To obtain temperature dependencies of various thermodynamic quantities the simulations start in the paramagnetic phase at sufficiently high temperatures $T$ (measured in units $J/k_B$ , where $k_B$ is the Boltzmann constant), and then proceed to lower temperatures with the step $\\Delta T$ .",
"To maintain the system close to the equilibrium, at each $T-\\Delta T$ simulations are initialized using the last configuration obtained at $T$ .",
"Close to the phase transition points we also perform finite-size scaling (FSS) analysis by using the reweighting techniques [27], [28], in order to identify the order and the universality class of the transition.",
"Since in the criticality the integrated autocorrelation time $\\tau $ is expected to dramatically increase, we make sure that sufficiently long simulation times are taken especially for larger lattice sizes.",
"For reliable estimation of statistical errors we employed the $\\Gamma $ -method [29], that focuses on the explicit determination of the relevant autocorrelation functions and times, and gives more certain error estimates than for example the binning techniques.",
"Typical values of the parameters are $L=24-72$ , $N_{MC}=2 \\times 10^5$ MC sweeps, and $\\Delta T=0.025$ , for the standard MC simulations, and up to $N_{MC}=10^7$ MC sweeps, for the reweighting.",
"We avoided using larger lattice sizes, as tunneling times between the coexisting phases at first-order transitions can become enormous (see the inset of Fig.",
"REF ).",
"We calculated the following quantities: the internal energy per spin $e=\\langle {\\mathcal {H}} \\rangle /L^2$ , the specific heat per site $c$ $c=\\frac{\\langle {\\mathcal {H}}^{2} \\rangle - \\langle {\\mathcal {H}} \\rangle ^{2}}{L^2T^{2}},$ the magnetization $m=\\langle M \\rangle /L^2=\\left\\langle \\Big |\\sum _{j}\\exp (i\\phi _j)\\Big |\\right\\rangle /L^2,$ the magnetic susceptibility $\\chi = \\frac{\\langle M^{2} \\rangle - \\langle M \\rangle ^{2}}{L^2T},$ and the fourth-order magnetic Binder cumulant $U$ $U = 1-\\frac{\\langle M^{4}\\rangle }{3\\langle M^{2}\\rangle ^{2}}.$ At the standard BKT to the paramagnetic phase transition the magnetization (susceptibility) is expected to vanish (diverge) as power law, characterized by the exponent $\\eta =1/4$ .",
"The latter can be estimated by FSS of the respective quantities, as follows $m(L) \\propto L^{-\\eta /2},$ and $\\chi (L) \\propto L^{2-\\eta }.$ On the other hand, if the transition is of first order, then the internal energy $e$ and the magnetization $m$ will show a discontinuous behavior, the thermodynamic functions like the susceptibility $\\chi $ are supposed to scale with volume, i.e., $\\chi (L) \\propto L^{2}$ , and the Binder cumulant is expected to plunge to negative values [30].",
"A proper order parameter for the algebraic BKT phase is the helicity modulus $\\Upsilon $ (or spin wave stiffness) [31], [32], [33], which quantifies the resistance of the systems to a twist in the boundary conditions.",
"It is defined as the second derivative of the free energy density of the system with respect to the twist $\\tau $ along one boundary axis, which, for example, for the present XY model with the Hamiltonian (REF ) results in the following expression $\\Upsilon = \\frac{1}{L^2}\\sum _{\\langle i,j \\rangle _x} \\frac{(\\alpha -1) \\alpha [2 \\alpha \\cos \\phi _{i,j} + \\cos (2\\phi _{i,j}) - 3]}{2(\\alpha -\\cos \\phi _{i,j})^3} - \\frac{\\beta }{L^2}\\Big [\\sum _{\\langle i,j \\rangle _x} \\frac{(\\alpha -1) \\alpha \\sin \\phi _{i,j}}{(\\alpha -\\cos \\phi _{i,j})^2}\\Big ]^2,$ where the summation $\\sum _{\\langle i,j \\rangle _x}$ is taken over the nearest neighbors along the direction of the twist.",
"In order to directly study the topological excitations (defects) we evaluate a defect density $\\rho $ .",
"Let us recall that a vortex (antivortex) is a topological defect which corresponds to the spin angle change by $2\\pi $ $(-2\\pi )$ going around a closed contour enclosing the excitation core.",
"In the MC simulation they are identified by summation of the angles between adjacent four spins on each square plaquette for each equilibrium configuration.",
"Thus, the summation equal to $2\\pi $ , $-2\\pi $ and 0 means that in the plaquette there is a vortex, antivortex and no topological defect, respectively We allow for a small deviation from these values due to numerical errors.. Then the defect density $\\rho $ is obtained as a thermodynamic average of the absolute value of the vorticity (taking into consideration both vortices and antivortices) summed over the entire lattice and normalized by the system volume $L^2$ ."
],
[
"Low-temperature behavior",
"The spin-wave approximation predicts the existence of the QLRO phase characterized by a power-law decaying correlation function, given by Eq.",
"REF .",
"The exponent $\\eta ^{\\rm sw}$ is formally similar to that of the standard XY model $\\eta ^{\\rm sw}_{\\rm XY}$ , i.e., linearly dependent on the temperature, however, through the interaction $J^{\\rm sw}$ it is also nonlinearly dependent on the parameters $p$ and $\\alpha $ .",
"The reduced exponent $\\eta ^{\\rm sw}/\\eta ^{\\rm sw}_{\\rm XY}=J^{\\rm sw}_{\\rm XY}/J^{\\rm sw}$ as a function of the parameters $p$ and $\\alpha $ is depicted in Fig.",
"REF .",
"One can notice that inclusion of just a few higher-order interaction terms causes a drastic drop of the exponent, followed by a leveling off if their couplings relative to the bilinear term are very small, i.e., for larger $\\alpha $ .",
"On the other hand, if the interactions at the higher-order terms are comparable with the bilinear one, i.e., for $\\alpha \\rightarrow 1$ , the exponent is further decreased with inclusion of more and more terms.",
"We also confront the spin-wave theory exponents $\\eta ^{\\rm sw}$ with those obtained from MC simulations, for selected parameter values.",
"In Fig.",
"REF we show temperature dependencies of both $\\eta ^{\\rm sw}$ and $\\eta ^{\\rm mc}$ , for two cases of $(\\alpha ,p)=(2,2)$ and $(\\alpha ,p)=(2,\\infty )$ .",
"As expected, the correspondence is very good at low temperatures but for $T \\gtrsim 0.15$ the spin-wave approximation apparently underestimates the exponent values."
],
[
"Infinite series model",
"The effect of a varying parameter $\\alpha $ on magnetic and thermodynamic properties of the model can be observed in Fig.",
"REF , in which temperature dependencies of the internal energy, the magnetization and the helicity modulus are plotted for various values of $\\alpha $ and a fixed value of $L=24$ .",
"For $\\alpha =2$ , the effect of the higher-order terms in the Hamiltonian is almost negligible and the behavior of all the quantities resembles that of the standard XY model.",
"Namely, they show a smooth variation in the vicinity of the transition point, as expected for the BKT transition.",
"With decreasing $\\alpha $ the effect of the higher-order terms becomes more pronounced and makes changes of the quantities at the transition more dramatic.",
"In particular, as $\\alpha $ approaches the limiting value of one, all start showing an apparently discontinuous behavior, typical for a first-order phase transition.",
"In order to confirm that the observed behavior indeed corresponds to the crossover from the continuous to the first-order transition, next we study the character of the energy distribution and perform a FSS analysis in the concerned region of the parameter space.",
"In Fig.",
"REF we present the results for $\\alpha =1.03$ (a,b) and $\\alpha =1.02$ (c,d).",
"In the left panels, the plots of the energy histograms for different sizes $L$ are reweighted to the temperature at which both peaks are of equal height.",
"In both cases, the plots indicate a bimodal distribution that is characteristic for a discontinuous first-order transition.",
"Nevertheless, there is a significant difference between them.",
"We note that at the first-order transition as $L$ increases the heights of the peaks are expected to increase at the cost of the dip (barrier) between them, that should tend to zero and the distance between the peaks should approach a finite value, corresponding to the latent heat released at the discontinuous transition.",
"This is exactly what we witness in the case of $\\alpha =1.02$ [Fig.",
"REF ], however, the behavior for $\\alpha =1.03$ is quite different.",
"Namely, from Fig.",
"REF we can see that with the increasing lattice size the height of the peaks virtually does not change, the dip between them does not get deeper and it becomes narrower as the peaks continue to move towards each other.",
"Thus we believe that the observed double-peak structure for $\\alpha =1.03$ is just a finite-size effect and in the thermodynamic limit it will vanish.",
"We note that such a pseudo-first-order behavior was also observed in some other systems, such as the 4-state Potts and $J_1-J_2$ Ising models [34].",
"Figure: (Color online) Energy histograms and FSS analysis for (a,b) α=1.03\\alpha =1.03 and (c,d) α=1.02\\alpha =1.02.",
"The histograms are reweighted to the temperatures at which the peaks are of equal height.",
"The insets in the right panels show the respective Binder cumulants.",
"The inset in panel (c) demonstrates huge tunneling times for larger sizes, e. g., for L=96L=96 they are of the order of 10 6 10^6 MCS.The above conjecture is furthermore corroborated by FSS analysis and the behavior of the Binder cumulant.",
"In particular, for the case of $\\alpha =1.03$ the FSS relations [Eqs.",
"REF and REF ] give the estimate of the exponent $\\eta $ in accordance with the value $1/4$ expected for a standard BKT phase transition [Fig.",
"REF ], while for $\\alpha =1.02$ the magnetic susceptibility scales with volume [Fig.REF ], as it should be in the case of a first-order transition.",
"A smooth variation of the Binder cumulant within positive values in the former case and an abrupt descent to negative values in the latter case (see insets) provide additional evidence for such a scenario.",
"The crossover to the first-order behavior can be understood by elucidation of the role of the topological defects in a varying potential shape, tuned by the parameter $\\alpha $ .",
"In Fig.",
"REF we present temperature dependencies of the defect density $\\rho $ , for selected values of $\\alpha $ .",
"It is evident that at the transition temperature from the BKT to the paramagnetic phase $\\rho $ anomalously increases.",
"The increase becomes particularly dramatic (resembling a jump) for the values of $\\alpha $ close to one.",
"A sudden increase of the defects at the transition for $\\alpha =1.01$ is illustrated in the insets of Fig.",
"REF .",
"The snapshot in the lower panel is taken just below the transition temperature and shows just a few vortex-antivortex pairs.",
"The snapshot in the upper panel, taken just above the transition point, shows a great number of dissociated vortices (white squares) and antivortices (black squares).",
"Figure: (Color online) (a) The defect density ρ\\rho as a function of temperature, for several values of α\\alpha .",
"The insets show typical snapshots just below (lower panel) and just above (upper panel) the transition point, depicting vortices (white squares) and antivortices (black squares), for α=1.01\\alpha =1.01.",
"(b) ρ\\rho as a function of α\\alpha , for three values of TT and two values of LL.It is also interesting to study the behavior of topological excitations with the parameter $\\alpha $ .",
"In Fig.",
"REF we show dependences of the defect density $\\rho $ on $\\alpha $ , for selected temperatures $T=0.5,0.7$ and $0.9$ .",
"One can notice a sharp increase of the defect density as $\\alpha \\rightarrow 1$ (note the semi-logarithmic scale), which seems to approach a common saturation value of $\\rho _s=1/3$ (dotted line).",
"Two sets of curves obtained for two different $L=24$ and 32 that almost collapse on each other demonstrate that the behavior is practically independent of the lattice size.",
"Similar behavior has also been reported for the modified XY model, introduced by Domany et al.",
"[22], and explained in the later studies [23], [26].",
"The abrupt increase of the defects, resulting in a first-order transition, is related to the shape of the potential well.",
"Namely, for certain values of the parameter the well becomes very narrow which suppresses formation of defect pairs at low temperatures and thus facilitates their dramatic proliferation at the transition point.",
"We believe that similar mechanism is responsible for the crossover to the first-order transition also in the present model.",
"The nonlinearity of the potential well is controlled by the parameter $\\alpha $ and, as shown in Fig.",
"REF , for the values close to one it becomes narrow enough to lead to the discontinuous phase transition.",
"We note that besides the integer vortices studied above, it is reasonable to assume also the presence of various fractional vortices, resulting from the higher-order terms.",
"Since our model involves a large number of them we did not attempt to evaluate all their individual densities.",
"Nevertheless, in Fig.",
"REF one can see that in the temperature dependencies of the helicity modulus there are no anomalies, such as, for example, in Ref.",
"[15], except the one related to the transition to the paramagnetic state.",
"This fact along with the behavior of other evaluated quantities, indicates that the integer and fractional vortices unbind at the same temperature corresponding to the transition point between the BKT and paramagnetic phases.",
"Figure: (Color online) Phase boundary as a function of the parameter α\\alpha , separating the BKT and paramagnetic (P) phases.",
"The (pseudo)transition temperatures are obtained from maxima of the specific heat curves, for L=24L=24.",
"The filled symbols represent the first-order transition points and the dashed line the transition temperature of the standard XY model.Finally, the approximate phase diagram in $T-\\alpha $ parameter plane is depicted in Fig.",
"REF .",
"Rough estimates of (pseudo)transition temperatures are obtained as positions of maxima of the specific heat curves from several independent MC runs, for $L=24$ Similar values could be obtained by considering positions of maxima of the magnetic susceptibility instead of the specific heat..",
"The filled circles represent the first-order transition points at $\\alpha =1.01$ and $1.02$ , and the dashed line shows the transition temperature of the standard XY model, which is expected to be recovered in the limit of $\\alpha \\rightarrow \\infty $ .",
"We note that these pseudo-transition temperatures slightly overestimate the true thermodynamic limit values [see, e.g., Figs.",
"REF and REF ].",
"Overall, the decreasing $\\alpha $ shifts the transition temperature from the paramagnetic (P) to the BKT phase to lower values and eventually also changes the nature of the transition to the first-order one."
],
[
"Truncated series model",
"Above we demonstrated that the first-order transition is a result of the increased influence of higher-order terms.",
"Next, we will be interested in whether their infinite number is an indispensable ingredient for the first-order character of the transition or it can also persist when only a finite number of the terms is considered.",
"We showed that for $p \\rightarrow \\infty $ the first-order transition exists if $\\alpha \\gtrsim 1$ .",
"On the other hand, the case of $p=2$ is well know to show the standard BKT transition for any value of $\\alpha $ [5].",
"Therefore, for a fixed $\\alpha \\gtrsim 1$ one can expect a crossover between the two regimes at some value of $p_c$ .",
"Figure: (Color online) Energy histograms and FSS analysis for α=1.01\\alpha =1.01 and (a,b) p=50p=50 and (c,d) p=100p=100.",
"The histograms are reweighted to the temperatures at which the peaks are of equal height.",
"The insets show the respective Binder cumulants.In Fig.",
"REF we present the behavior at the transition for the cases of $p=50$ [Figs.",
"REF , REF ] and $p=100$ [Figs.",
"REF , REF ], at the value of $\\alpha =1.01$ .",
"The respective features are very similar to those observed in Fig.",
"REF , for the infinite $p$ case with $\\alpha =1.03$ and $\\alpha =1.02$ , respectively.",
"Namely, for $\\alpha =1.01$ and $p=50$ , all the measured quantities point to the continuous transition belonging to the BKT universality class, while for $\\alpha =1.01$ and $p=100$ , the transition is clearly of the first order.",
"Therefore, for $\\alpha =1.01$ the crossover value can be very roughly estimated as $50 < p_c < 100$ ."
],
[
"Summary",
"We employed spin-wave theory and Monte Carlo simulations to study effects of inclusion of higher-order nearest-neighbor pairwise interactions with an exponentially decreasing intensity, $J_k=\\alpha ^{-k}$ , where $\\alpha >1$ and $k=2,\\ldots , p$ , to the standard XY model.",
"At low temperatures, the spin wave theory predicts a quasi-long-range order phase characterized by an algebraically decaying correlation function with the exponent $\\eta ^{\\rm sw} = T/(2 \\pi J^{\\rm sw})$ , where $J ^{\\rm sw} = \\alpha /(\\alpha -1) - p/(\\alpha ^p -1)$ .",
"At higher temperatures, we showed that, in spite of belonging to the same universality class as the standard XY model, the studied generalized model can display qualitatively different behaviors, depending on the parameters $p$ and $\\alpha $ that control the degree of nonlinearity.",
"In particular, for a relatively small number of the higher-order terms $p$ and relatively fast decay of $J_k$ , the critical behavior is qualitatively similar to that of the XY model, i.e., the system shows the Berezinskii-Kosterlitz-Thouless transition to the paramagnetic phase.",
"Nevertheless, for $\\alpha \\rightarrow 1$ and $p$ large enough (not necessarily infinite), i.e., the parameters values corresponding to a highly nonlinear shape of the potential well, the transition changes to the first order.",
"We demonstrated that the change of the transition order can be related to the behavior of topological excitations (vortices).",
"Namely, in the parameter region where the potential well becomes very narrow the formation of vortex pairs at low temperatures becomes suppressed which facilitates their abrupt, discontinuous increase at the transition point.",
"This work was supported by the Scientific Grant Agency of Ministry of Education of Slovak Republic (Grant No.",
"1/0331/15) and the scientific grants of Slovak Research and Development Agency provided under contract No.",
"APVV-0132-11 and No.",
"APVV-14-0073."
]
] | 1709.01715 |
[
[
"Optimal Number of Transmit Antennas for Secrecy Enhancement in Massive\n MIMOME Channels"
],
[
"Abstract This paper studies the impact of transmit antenna selection on the secrecy performance of massive MIMO wiretap channels.",
"We consider a scenario in which a multi-antenna transmitter selects a subset of transmit antennas with the strongest channel gains.",
"Confidential messages are then transmitted to a multi-antenna legitimate receiver while the channel is being overheard by a multi-antenna eavesdropper.",
"For this setup, we approximate the distribution of the instantaneous secrecy rate in the large-system limit.",
"The approximation enables us to investigate the optimal number of selected antennas which maximizes the asymptotic secrecy throughput of the system.",
"We show that increasing the number of selected antennas enhances the secrecy performance of the system up to some optimal value, and that further growth in the number of selected antennas has a destructive effect.",
"Using the large-system approximation, we obtain the optimal number of selected antennas analytically for various scenarios.",
"Our numerical investigations show an accurate match between simulations and the analytic results even for not so large dimensions."
],
[
"Introduction",
"Massive mimo systems have been identified as a key technology for the next generation of wireless communication systems [1].",
"Consequently, physical layer security of these systems has gained significant attentions in recent years [2].",
"The main premise in physical layer security is to exploit the inherent characteristics of wireless channels.",
"The pioneering work of Wyner [3], considered the wiretap channel as a basic model for secure transmission and demonstrated that secrecy is obtained as long as the legitimate receiver observes a better channel than the eavesdropper.",
"Provisioning secrecy for mimo wiretap channels, also referred to as mimome channels, was then studied in the literature widely [4], [5].",
"The fact of using multiple antennas in these systems can significantly improve the secrecy performance by means of focusing the transmission beam to the legitimate receiver.",
"This technique in massive mimo systems effectively provides secrecy due to the large number of antennas, and makes them robust against passive eavesdropping [2].",
"The performance gains in mimo systems are mainly obtained at the expense of elevated complexity and cost.",
"In fact, the growth in the number of antennas, increases both the computational complexity and rf-cost significantly.",
"Consequently, addressing solutions to alleviate these issues has become a topic of interest.",
"Antenna selection is introduced as a possible solution which reduces the overall rf-cost, as well as the computational complexity, without significant performance loss [6].",
"This solution was recently further raised up in the context of massive mimo systems, due to the large dimensions of these systems [7], [8].",
"Although antenna selection vanishes the robustness of massive mimo systems against passive eavesdropping, it still can enhance the secrecy performance compared to conventional mimo systems.",
"For single tas, the problem of secure transmission was initially studied in [9].",
"The authors in [9] considered a scenario in which a multi-antenna transmitter with a single rf-chain intends to send confidential messages to a single-antenna legitimate receiver while a single-antenna eavesdropper is overhearing the channel.",
"The study was later extended in [10] to the case of multi-antenna eavesdropper where the authors showed that, similar to the case with a single-antenna eavesdropper, the secrecy outage probability improves when the number of antennas at the transmitter side increases.",
"[11] investigated the tas for secure transmission in the general mimome wiretap channel assuming that a single rf-chain available at the transmitter.",
"The problem was further studied in [12] for Nakagami-$m$ fading considering the single tas.",
"The impacts of the imperfect channel estimation and antenna correlation were also investigated in [13] and [11].",
"In contrast to the single tas, the secrecy performance of massive mimome channels under the multiple tas has not been yet addressed in the literature.",
"In this case, increasing the number of transmit antennas is beneficial to both the legitimate receiver and eavesdropper simultaneously, and therefore, its effect on the overall secrecy performance is not clear."
],
[
"Contributions",
"In this paper, we study the impact of multiple tas on the secrecy performance of massive mimome channels.",
"In particular, we consider a general mimome setup with a large number of transmit antennas in which a subset of antennas with the strongest channel gains is selected.",
"We show that under the given tas protocol, increasing the number of selected antennas up to an optimal value can enhance the secrecy performance of the system.",
"The impact, however, can be destructive if the number of the selected antennas exceeds the optimal value.",
"Throughout the paper the following notations are adopted.",
"Scalars, vectors and matrices are represented with non-bold, bold lower case and bold upper case letters, respectively.",
"$\\mathbf {H}^{\\mathsf {H}}$ indicates the Hermitian of $\\mathbf {H}$ , and $\\mathbf {I}_N$ is the $N\\times N$ identity matrix.",
"The determinant of $\\mathbf {H}$ and Euclidean norm of ${x}$ are denoted by $\\vert \\mathbf {H} \\vert $ and $\\Vert {x} \\Vert $ .",
"$\\log \\left(\\cdot \\right)$ and $\\mathrm {ln}\\hspace*{1.42262pt} \\left(\\cdot \\right)$ indicate the binary and natural logarithm respectively, and $\\mathbf {1}_{\\left\\lbrace \\cdot \\right\\rbrace }$ denotes the indicator function.",
"$\\mathsf {E}\\hspace{0.56905pt}\\left\\lbrace \\cdot \\right\\rbrace $ is statistical expectation, and $\\mathrm {Q}(x)= \\int _{x}^{\\infty } \\phi (x) dx$ represents the standard $\\mathrm {Q}$ -function where $\\phi (x)$ is the zero-mean and unit-variance Gaussian probability density function."
],
[
"Problem Formulation",
"Consider a Gaussian mimome channel in which the transmitter, legitimate receiver and eavesdropper are equipped with $N_\\mathrm {t}$ , $N_\\mathrm {r}$ and $N_\\mathrm {e}$ antennas, respectively.",
"At each transmit interval, the transmitter encodes its messages into the codeword ${x}_{N_\\mathrm {t}\\times 1}$ and sends it to the legitimate receiver.",
"The received signal is given by the vector ${y}_{N_\\mathrm {r}\\times 1}$ which reads ${y}=\\sqrt{\\rho _\\mathrm {m}} \\ \\mathbf {H}_\\mathrm {m}{x}+ {n}_\\mathrm {m}.",
"$ Here, $\\rho _\\mathrm {m}$ denotes the average snr at each receive antenna, ${n}_{\\mathrm {m}}$ is circularly symmetric complex Gaussian noise with zero-mean and unit-variance, i.e., ${n}_\\mathrm {m}\\sim \\mathcal {CN}\\left( \\mathbf {0}, \\mathbf {I} \\right)$ and $\\mathbf {H}_\\mathrm {m}$ is an ${N_\\mathrm {r}\\times N_\\mathrm {t}}$ iid unit-variance quasi-static Rayleigh fading channel matrix and is referred to as the main channel.",
"The ea- vesdropper overhears ${x}$ and receives ${z}_{N_\\mathrm {e}\\times 1}$ given by ${z}=\\sqrt{\\rho _\\mathrm {e}} \\ \\mathbf {H}_\\mathrm {e}\\hspace{0.56905pt} {x}+ {n}_\\mathrm {e}, $ where $\\rho _\\mathrm {e}$ is the average snr at each of the eavesdropper antennas, ${n}_\\mathrm {e}\\sim \\mathcal {CN}\\left( \\mathbf {0}, \\mathbf {I} \\right)$ , and $\\mathbf {H}_\\mathrm {e}$ identifies an $N_\\mathrm {e}\\times N_\\mathrm {t}$ iid unit-variance quasi-static Rayleigh fading channel matrix between the transmitter and eavesdropper.",
"We denote it as the eavesdropper channel.",
"The legitimate receiver and eavesdropper have the csi of their channels.",
"At the transmit side, however, the csi of the channels is not necessarily available.",
"Moreover, the main and eave- sdropper channels are supposed to be statistically independent."
],
[
"tas Protocol",
"Let the $N_\\mathrm {r}\\times 1$ vector $\\mathbf {h}_{\\mathrm {m}j}$ denote the $j$ th column vector of $\\mathbf {H}_\\mathrm {m}$ for $j\\in \\left\\lbrace 1, \\ldots , N_\\mathrm {t}\\right\\rbrace $ .",
"Represent the index set of order statistics from the arranging of vector norms $\\Vert \\mathbf {h}_{\\mathrm {m}j} \\Vert $ in decreasing order by ${W}\\left\\lbrace w_1, \\ldots , w_{N_\\mathrm {t}} \\right\\rbrace $ , i.e., $\\Vert \\mathbf {h}_{\\mathrm {m}w_1} \\Vert \\ge \\Vert \\mathbf {h}_{\\mathrm {m}w_2} \\Vert \\ge \\cdots \\ge \\Vert \\mathbf {h}_{\\mathrm {m}w_{N_\\mathrm {t}}} \\Vert .",
"$ At each transmission interval, the transmitter selects $L_\\mathrm {t}$ transmit antennas via the tas protocol $\\mathcal {S}$ as follows: The ordered index set ${W}$ is determined at the transmitter and legitimate receiver.",
"At the transmit side, the task is done either by supposing the transmitter to estimate the channel gains itself, or assuming ${W}$ to be evaluated by the legitimate receiver and given to the transmitter through a rate-limited return channel.",
"The transmitter then selects the $L_\\mathrm {t}$ antennas which correspond to the index subset ${W}_\\mathcal {S}\\left\\lbrace w_1, \\ldots , w_{L_\\mathrm {t}}\\right\\rbrace $ and transmits over them with equal average power.",
"Remark Considering the task of determining ${W}$ , the transmitter, even in the absence of a return channel, need not acquire the complete csi.",
"In fact, as ${W}$ is determined via the ordering in (REF ), the transmitter only needs to estimate the channel norms.",
"The task which can be done at the prior uplink stage simply by attaching rf power meters at each transmit antenna, and requires a significantly reduced time interval compared to the case of complete csi estimation [14]."
],
[
"Achievable Secrecy Rate",
"For the mimome channel specified in (REF ) and (REF ), the instantaneous secrecy rate is expressed as [4], [5] $\\mathcal {R}_\\mathrm {s}=[\\mathcal {R}_\\mathrm {m}-\\mathcal {R}_\\mathrm {e}]^+$ where $[x]^+=\\max \\left\\lbrace 0,x\\right\\rbrace $ .",
"Here, $\\mathcal {R}_\\mathrm {m}$ denotes the achievable rate over the main channel which is determined as $\\mathcal {R}_\\mathrm {m}=\\log \\vert \\mathbf {I}+\\rho _\\mathrm {m}\\mathbf {H}_\\mathrm {m}\\mathbf {Q}\\mathbf {H}_\\mathrm {m}^\\mathsf {H} \\vert $ and $\\mathcal {R}_\\mathrm {e}$ is the achievable rate over the eavesdropper channel given by $\\mathcal {R}_\\mathrm {e}=\\log \\vert \\mathbf {I}+\\rho _\\mathrm {e}\\mathbf {H}_\\mathrm {e}\\mathbf {Q}\\mathbf {H}_\\mathrm {e}^\\mathsf {H} \\vert $ with $\\mathbf {Q}_{N_t \\times N_t}$ being the diagonal power allocation matrix.",
"Co- nsidering the tas protocol $\\mathcal {S}$ , $\\mathbf {Q}$ reads $[\\mathbf {Q}]_{ww} = \\left\\lbrace \\begin{array}{ll}1 & w\\in {W}_\\mathcal {S}\\\\0 & w \\notin {W}_\\mathcal {S}\\end{array}\\right.$ in which the average transmit power on each selected antenna is set to be one.",
"Consequently, $\\mathcal {R}_\\mathrm {m}$ and $\\mathcal {R}_\\mathrm {e}$ reduce to $\\mathcal {R}_\\mathrm {m}&=\\hspace*{0.28453pt}\\log \\vert \\mathbf {I}+\\rho _\\mathrm {m}\\tilde{\\mathbf {H}}_\\mathrm {m}^\\mathsf {H}\\tilde{\\mathbf {H}}_{\\mathrm {m}} \\vert \\\\\\mathcal {R}_\\mathrm {e}&=\\log \\hspace*{0.28453pt}\\vert \\mathbf {I}+\\rho _\\mathrm {e}\\tilde{\\mathbf {H}}_\\mathrm {e}^\\mathsf {H}\\tilde{\\mathbf {H}}_{\\mathrm {e}} \\vert $ where $\\tilde{\\mathbf {H}}_\\mathrm {m}$ and $\\tilde{\\mathbf {H}}_\\mathrm {e}$ are $N_\\mathrm {r}\\times L_\\mathrm {t}$ matrices denoting the effective main and eavesdropper channels respectively.",
"The effective channels are constructed from $\\mathbf {H}_\\mathrm {m}$ and $\\mathbf {H}_\\mathrm {e}$ by collecting the columns which correspond to the selected antennas.",
"Substituting in (REF ), the maximum achievable secrecy rate reads $\\mathcal {R}_\\mathrm {s}\\left( \\mathcal {S} \\right)=\\left[ \\log \\frac{\\vert \\mathbf {I}+{\\rho _m}\\tilde{\\mathbf {H}}_\\mathrm {m}^\\mathsf {H}\\tilde{\\mathbf {H}}_\\mathrm {m} \\vert }{\\vert \\mathbf {I}+{\\rho _\\mathrm {e}}\\tilde{\\mathbf {H}}_\\mathrm {e}^\\mathsf {H}\\tilde{\\mathbf {H}}_\\mathrm {e} \\vert }\\right]^+ $ where the argument $\\mathcal {S}$ is written to indicate the dependency of $\\mathcal {R}_\\mathrm {s}\\left( \\mathcal {S} \\right)$ on the tas protocol.",
"Using $\\mathcal {R}_\\mathrm {s}$ in (REF ), different secrecy measures for the system is defined based on the eavesdropper's status.",
"When the csi of the eavesdropper channel is available at the transmit side, (REF ) determines the instantaneous achievable secrecy rate, and thus, its expected value determines the ergodic secrecy rate.",
"For a passive eavesdropper, the probability of $\\mathcal {R}_\\mathrm {s}$ being less than $\\mathcal {R}_\\mathrm {Out}$ evaluates the outage probability which represents probability of having information leakage when the transmitter has set the secrecy rate to $\\mathcal {R}_\\mathrm {Out}$ [15]."
],
[
"Large-System Secrecy Performance of tas",
"In this section, we investigate the secrecy performance of the mimome wiretap channel under the tas protocol $\\mathcal {S}$ considering the following two cases: The eavesdropper is equipped with significantly fewer receive antennas compared to the number of selected antennas, i.e., $N_\\mathrm {e}\\ll L_\\mathrm {t}$ .",
"The number of eavesdropper antennas grows large faster than the number of selected antennas, i.e., $N_\\mathrm {e}\\gg L_\\mathrm {t}$ .",
"Case REF can be seen as a scenario in cellular networks with the eavesdropper being a regular user terminal.",
"Moreover, Case REF describes a scenario in which the eavesdropper is a sophisticated terminal, such as portable stations.",
"Proposition REF approximates the distribution of the maximum achievable secrecy rate in the large-system limit, i.e., $N_\\mathrm {t}\\uparrow \\infty $ , for both Cases REF and REF .",
"Proposition 1 Consider the tas protocol $\\mathcal {S}$ , and let $\\eta _t&=N_\\mathrm {r}\\left[ L_\\mathrm {t} + N_\\mathrm {t} f_{N_\\mathrm {r} +1}(u) \\right] \\\\\\sigma _t^2&=\\left(L_\\mathrm {t}u-\\eta _t \\right)^2 \\left(\\frac{1}{L_\\mathrm {t}}-\\frac{1}{N_\\mathrm {t}} \\right) - \\frac{\\eta _t^2}{L_\\mathrm {t}}+ \\Xi _t $ for some non-negative real $u$ which satisfies $\\int _u^\\infty f_{N_\\mathrm {r}}(x) \\mathrm {d} x= \\frac{L_\\mathrm {t}}{N_\\mathrm {t}},$ $\\Xi _t$ which is given by $\\Xi _tN_\\mathrm {r} \\left(N_\\mathrm {r}+1\\right) \\left[ L_\\mathrm {t}+N_\\mathrm {t} f_{N_\\mathrm {r}+1}(u)+N_\\mathrm {t} f_{N_\\mathrm {r}+2}(u) \\right],$ and $f_{N_\\mathrm {r}}(\\cdot )$ which represents the chi-square probability density function with $2N_\\mathrm {r}$ degrees of freedom and mean $N_\\mathrm {r}$ , i.e., $f_{N_\\mathrm {r}}(x)= \\frac{1}{(N_\\mathrm {r}-1)!",
"}{\\left\\lbrace \\begin{array}{ll}x^{N_\\mathrm {r}-1} e^{-x} , & \\text{if}\\ x \\ge 0 \\\\0, & \\text{if}\\ x < 0.\\end{array}\\right.",
"}$ Define $L_\\mathrm {m}$ $\\min \\left\\lbrace L_\\mathrm {t},N_\\mathrm {r}\\right\\rbrace $ , $M_\\mathrm {m}\\max \\left\\lbrace L_\\mathrm {t},N_\\mathrm {r}\\right\\rbrace $ , $L_\\mathrm {e}\\min \\left\\lbrace L_\\mathrm {t},N_\\mathrm {e}\\right\\rbrace $ and $M_\\mathrm {e}\\max \\left\\lbrace L_\\mathrm {t},N_\\mathrm {e}\\right\\rbrace $ .",
"As $N_\\mathrm {t}$ grows large, the distribution of $\\mathcal {R}_\\mathrm {s}(\\mathcal {S})$ for both Cases REF and REF can be approximated by the distribution of $\\mathcal {R}_\\mathrm {asy}(\\mathcal {S})\\left[ \\mathcal {R}^\\star \\right]^+$ in which $\\mathcal {R}^\\star $ is a Gaussian random variable with mean $\\eta $ and variance $\\sigma ^2$ given by (REF ) and (REF ) on the top of the next page.",
"Figure: NO_CAPTIONProof Sketch To evaluate the large-system distribution of $\\mathcal {R}_\\mathrm {s}(\\mathcal {S})$ , one needs to determine the asymptotic distribution of $\\mathcal {R}_\\mathrm {m}$ and $\\mathcal {R}_\\mathrm {e}$ given by (REF ) and (REF ) respectively.",
"Using the results from [8] and [16], the distribution of $\\mathcal {R}_\\mathrm {m}$ and $\\mathcal {R}_\\mathrm {e}$ in the large system limits for the both cases REF and REF can be approximated by Gaussian distributions.",
"Noting the fact that the main and eavesdropper channels are independent, the random variable $\\mathcal {R}^\\star \\mathcal {R}_\\mathrm {m}-\\mathcal {R}_\\mathrm {e}, $ in the large limit, can then be approximated with a Gaussian random variable whose variance and mean is determined in terms of the variances and means of $\\mathcal {R}_\\mathrm {m}$ and $\\mathcal {R}_\\mathrm {e}$ .",
"Finally by substituting in (REF ), the proof is concluded.",
"The detailed derivations are given in the appendix.",
"Considering (REF ), one can observe that the variations of the secrecy rate vanish as the dimensions of the system increase.",
"In fact, as $N_\\mathrm {t}$ grows large $\\eta _t$ grows proportionally, and therefore, the first term in (REF ) tends to zero.",
"Moreover, as $L_\\mathrm {e}/ M_\\mathrm {e}$ in our setup is considered to be significantly small, the two other terms can be neglected further.",
"Consequently, in the large limit $\\sigma $ converges to zero; the observation which could be intuitively predicted, due to the fact that the both main and eavesdropper channels harden in the large limit.",
"The mean value $\\eta $ , however, does not necessarily increase in the large-system limit, since it is given as the difference of two terms which can both asymptotically grow large.",
"The latter observation indicates that increasing the number of selected antennas for this setup does not necessarily improve the secrecy rate.",
"We discuss this argument later in Section .",
"At this point, we employ Proposition REF to evaluate the ergodic secrecy rate and secrecy outage probability."
],
[
"Ergodic Secrecy Rate",
"For scenarios in which the csi of the eavesdropper channel is available at the transmit side, the instantaneous secrecy rate is achievable in each transmission interval.",
"Therefore, the secrecy performance is characterized by the ergodic secrecy rate which is given by taking the expectation of $\\mathcal {R}_\\mathrm {s}\\left( \\mathcal {S} \\right)$ .",
"Using Proposition REF , the ergodic secrecy rate for our setup in the large limit is approximated as $\\mathcal {R}_\\mathrm {Erg}\\left( \\mathcal {S} \\right) &\\approx \\mathsf {E}\\hspace{0.56905pt}\\left\\lbrace \\mathcal {R}_\\mathrm {asy}\\left( \\mathcal {S} \\right)\\right\\rbrace \\\\&=\\mathsf {E}\\hspace{0.56905pt}\\left\\lbrace \\left[\\mathcal {R}^\\star \\right]^+\\right\\rbrace \\\\&= \\sigma \\hspace*{1.42262pt} \\phi \\left( \\xi \\right)+ \\eta \\hspace*{1.42262pt}\\mathrm {Q}\\left( -\\xi \\right).",
"$ where $\\xi \\dfrac{\\eta }{\\sigma }$ .",
"Using the inequality $\\mathrm {Q}(x) < \\dfrac{\\phi \\left( x \\right)}{x}$ for $x>0$ , we conclude $\\mathcal {R}_\\mathrm {Erg}\\left( \\mathcal {S} \\right) > \\eta $ for $\\xi >0$ ; the bound is tight when $\\xi $ is large enough.",
"Fig.",
"REF illustrates the accuracy of the approximations, as well as the tightness of the lower-bound in (REF ).",
"The figure has been plotted for $N_\\mathrm {r}=N_\\mathrm {e}=2$ and $L_\\mathrm {t}=8$ considering 16 transmit antennas and $\\rho _\\mathrm {e}=-5$ dB.",
"As the figure shows, except for the interval of $\\rho _\\mathrm {m}$ in which $\\eta $ is close to zero, the lower-bound in (REF ) perfectly matches $\\mathcal {R}_\\mathrm {Erg}(\\mathcal {S})$ .",
"This observation is due to fact that the variance in the large limit tends to zero rapidly, and thus, the factor $\\xi $ grows significantly large even for finite values of $\\eta $ ; therefore, $\\mathbf {Q}(-\\xi )\\approx 1-\\xi ^{-1} \\phi (\\xi )$ , and the ergodic secrecy rate is approximated with $\\eta $ accurately.",
"Despite the initial assumptions on the system dimensions taken in Sections and , one observes that the approximation accurately tracks the simulations even in not so large dimensions.",
"Figure: The ergodic secrecy rate in terms of the main channel's snr.",
"The curves have been plotted for N r =N e =2N_\\mathrm {r}=N_\\mathrm {e}=2, L T =8L_\\mathrm {T}=8, N t =16N_\\mathrm {t}=16 and ρ e =-5\\rho _\\mathrm {e}=-5 dB.",
"ℛ Erg (𝒮)\\mathcal {R}_\\mathrm {Erg}(\\mathcal {S}) in () and the numerical simulations are sketched by a black line and red circles respectively.",
"The blue line indicate [η] + [\\eta ]^+ for η\\eta given in ().",
"As it shows, the approximation tracks the simulations with high accuracy even for finite dimensions."
],
[
"Secrecy Outage Probability",
"When the eavesdropper is passively overhearing the channel, the csi of the eavesdropper channel is not known at the transmitter, and thus, the secrecy rate in (REF ) is not achievable.",
"The performance in this case is described by the secrecy outage probability which for a given rate $\\mathcal {R}_\\mathrm {Out}\\ge 0$ is defined as [15] $\\mathcal {P}_\\mathrm {Out}\\left( \\mathcal {R}_\\mathrm {Out} \\right) = \\Pr \\left\\lbrace \\mathcal {R}_s \\left( \\mathcal {S} \\right) < \\mathcal {R}_\\mathrm {Out}\\right\\rbrace .",
"$ The interpretation of this secrecy measure is as follows: As the transmitter does not have the csi of the eavesdropper channel, it sets the secrecy rate to be $\\mathcal {R}_\\mathrm {Out}$ in all transmission intervals.",
"This setting implicitly imposes a primary assumption on the quality of the eavesdropper channel.",
"In this case, the outage probability in (REF ) determines the probability of eavesdropper channel having better quality than the primary assumption, or equivalently, the percentage of intervals in which the eavesdropper can decode transmitted codewords at least partially.",
"Using the large-system approximation in Proposition REF , the outage probability in the large limit reads $\\mathcal {P}_\\mathrm {Out}\\left( \\mathcal {R}_\\mathrm {Out} \\right) &\\approx \\Pr \\left\\lbrace \\mathcal {R}_\\mathrm {asy}\\left( \\mathcal {S} \\right) \\le \\mathcal {R}_\\mathrm {Out}\\right\\rbrace \\\\&= 1- \\mathrm {Q}\\left( \\frac{\\mathcal {R}_\\mathrm {Out}- \\eta }{\\sigma } \\right).",
"$ Figure: NO_CAPTION"
],
[
"Secrecy Enhancement via tas",
"Considering either the ergodic secrecy rate or the secrecy outage probability, the secrecy performance of the system in the large limit is mainly specified by $\\eta $ .",
"In contrast to $\\sigma $ which tends to zero in the asymptotic regime, the factor $\\eta $ , for a given number of receive and eavesdropper antennas, can either grow, vanish or tend to some constant in the large-system limit depending on the number of selected antennas.",
"To illustrate the point further, we have plotted in Fig.",
"REF the ergodic secrecy rate of the system as a function of $L_\\mathrm {t}$ , for some given numbers of receive and eavesdropper antennas considering both the large-system approximation given by (REF ), and numerical simulations.",
"The curves have been sketched for $\\rho _\\mathrm {m}=0$ dB and $\\rho _\\mathrm {e}=-10$ dB.",
"As the figure shows, for the considered setups, the ergodic secrecy rate meets its maximum at some values of $L_\\mathrm {t}$ which are significantly smaller than $N_\\mathrm {t}$ .",
"The observation which expresses that the tas in these scenarios, not only benefits in terms of rf-cost reduction, but also enhances the secrecy performance of the system.",
"The intuition behind this behavior comes from the fact that the growth in the number of selected antennas improves both the main and eavesdropper channels.",
"In this case as Fig.",
"REF indicates, the improvement from the legitimate receiver's point of view dominates the overall secrecy performance of the system up to a certain number of selected antennas.",
"By increasing $L_\\mathrm {t}$ further, the improvement at the eavesdropper side starts to dominate which results in the performance deficit at larger values of $L_\\mathrm {t}$ .",
"Figure: The ergodic secrecy rate as a function of L t L_\\mathrm {t} for three different cases considering N t =128N_\\mathrm {t}=128, ρ m =0\\rho _\\mathrm {m}=0 dB and ρ e =-10\\rho _\\mathrm {e}=-10 dB.",
"The approximation in () and numerical simulations are denoted by solid lines and circles respectively.",
"As it shows, the values of L t L_\\mathrm {t} in which ℛ Erg (𝒮)\\mathcal {R}_\\mathrm {Erg}(\\mathcal {S}) meets the maximum is significantly smaller than N t N_\\mathrm {t}."
],
[
"Optimal Number of Selected Antennas",
"Considering the illustrated behavior of the system, a smart choice of $L_\\mathrm {t}$ can significantly improve the overall system throughput at no cost.",
"In this case, the results given in Section can be employed, in order to find the optimal number of selected transmit antennas.",
"More precisely, using (REF ) and (REF ), one can determine the ergodic secrecy rate and secrecy outage probability in (REF ) and (REF ) as functions of $L_\\mathrm {t}$ whose maximizers are found either analytic or via a linear search.",
"We investigate this problem further through the following examples by considering the ergodic secrecy rate as the measure.",
"Same discussions can also be considered for the secrecy outage probability Example 1 (Single-antenna receivers) Consider the case in which the legitimate receiver, as well as the eavesdropper, is equipped with a single receive antenna, i.e., $N_\\mathrm {r}=N_\\mathrm {e}=1$ .",
"This can be seen as a scenario in which both the legitimate receiver and eavesdropper are handheld devices.",
"Considering the ergodic secrecy rate as the performance measure, we are interested in determining the optimal number of selected antennas which maximizes $\\mathcal {R}_\\mathrm {Erg}\\left( \\mathcal {S} \\right)$ .",
"Let us denote the optimal number of selected antennas by $L^\\star _\\mathrm {t}$ .",
"In order to employ the results in Section , we assume at this point that $L_\\mathrm {t}^\\star \\gg 1$ ; we later show that the assumption holds for the large limit of $N_\\mathrm {t}$ .",
"By substituting in (REF ) and (REF ), $\\eta $ and $\\sigma ^2$ are determined as $\\eta &=\\log \\left( \\frac{1+\\rho _\\mathrm {m}L_\\mathrm {t}\\left( 1+ \\mathrm {ln}\\hspace*{1.42262pt} N_\\mathrm {t}L_\\mathrm {t}^{-1} \\right)}{1+\\rho _\\mathrm {e}L_\\mathrm {t}} \\right)\\\\\\sigma ^2&= \\left[ \\frac{\\rho _\\mathrm {m}^2 \\hspace*{2.84526pt} L_\\mathrm {t}\\left( 2- L_\\mathrm {t}N_\\mathrm {t}^{-1} \\right)}{\\left( 1+\\rho _\\mathrm {e}L_\\mathrm {t} \\right)^2} + \\frac{L_\\mathrm {t}\\rho _\\mathrm {e}^2}{(1+\\rho _\\mathrm {e}L_\\mathrm {t})^2} \\right] \\log ^2 e.$ By taking the limit $N_\\mathrm {t}\\uparrow \\infty $ and considering $L^\\star _\\mathrm {t}\\gg 1$ , it is concluded that $\\xi $ is large, and therefore, $\\mathcal {R}_\\mathrm {Erg}\\left( \\mathcal {S} \\right)\\approx \\eta $ .",
"To find $L^\\star _\\mathrm {t}$ , we define the function $f(x) \\log \\left( \\frac{1+\\rho _\\mathrm {m}x+\\rho _\\mathrm {m}x \\hspace*{1.42262pt} \\mathrm {ln}\\hspace*{1.42262pt} N_\\mathrm {t}x^{-1} }{1+\\rho _\\mathrm {e}x} \\right)$ over the real axis.",
"It is then straightforward to show that, for $x\\in [1,N_\\mathrm {t}]$ , $f^{\\prime \\prime }(x) \\le 0$ , and thus, one can conclude that $L^\\star _\\mathrm {t}$ is an integer close to the maximizer of $f(\\cdot )$ .",
"Consequently, $L^\\star _\\mathrm {t}$ can be approximated as $L^\\star _\\mathrm {t}\\approx \\lfloor x^\\star \\rceil $ where $x^\\star $ satisfies $\\rho _\\mathrm {e}x^\\star + \\mathrm {ln}\\hspace*{1.42262pt} x^\\star + \\frac{\\rho _\\mathrm {e}}{\\rho _\\mathrm {m}}= \\mathrm {ln}\\hspace*{1.42262pt} N_\\mathrm {t} .",
"$ As $x^\\star $ grows large proportional to $N_\\mathrm {t}$ , $L_\\mathrm {t}^\\star $ grows accordingly, and therefore, the initial assumption of $L_\\mathrm {t}^\\star \\gg 1$ in the large-system limit holds.",
"Moreover, by reducing $\\rho _\\mathrm {e}$ to zero in the fixed point equation (REF ), $L_\\mathrm {t}^\\star =N_\\mathrm {t}$ which agrees with the fact that in the absence of the eavesdropper, the ergodic rate is a monotonically increasing function of $L_\\mathrm {t}$ .",
"Figure: ℛ Erg (𝒮)\\mathcal {R}_\\mathrm {Erg}(\\mathcal {S}) in Example in terms of the number of selected antennas.",
"The solid line and the circles show the approximation given by () and numerical simulations respectively considering N t =128N_\\mathrm {t}=128, ρ m =0\\rho _\\mathrm {m}=0 dB and ρ e =-10\\rho _\\mathrm {e}=-10 dB.",
"As it is observed, L t ☆ =18L_\\mathrm {t}^\\star =18 is suggested by both the approximation and simulation results.Fig.",
"REF shows the ergodic secrecy rate as a function of $L_\\mathrm {t}$ for $\\rho _\\mathrm {e}=-10$ dB and 128 transmit antennas.",
"By solving the fixed point equation in (REF ), $x^\\star =18.4$ is obtained which results in $L_\\mathrm {t}^\\star =18$ .",
"The result which is confirmed by numerical simulations as well.",
"Figure: NO_CAPTIONExample 2 (Multi-antenna eavesdropper) As another example, we consider a scenario with a single antenna legitimate receiver whose channel is being overheard by a sophisticated muti-antenna terminal, i.e., $N_\\mathrm {r}=1$ and $N_\\mathrm {e}$ growing large.",
"Similar to Example REF , let us take the ergodic secrecy rate as the performance measure.",
"For this case, $\\eta $ and $\\sigma ^2$ in (REF ) and (REF ) reduce to $\\eta &=\\log \\left( \\frac{1+\\rho _\\mathrm {m}L_\\mathrm {t}\\left( 1+ \\mathrm {ln}\\hspace*{1.42262pt} N_\\mathrm {t}L_\\mathrm {t}^{-1} \\right)}{(1+\\rho _\\mathrm {e}N_\\mathrm {e})^{L_\\mathrm {t}}} \\right) \\\\\\sigma ^2&= \\frac{\\rho _\\mathrm {m}^2 \\log ^2 e \\hspace*{2.84526pt} L_\\mathrm {t}\\left( 2- L_\\mathrm {t}N_\\mathrm {t}^{-1} \\right)}{\\left( 1+\\rho _\\mathrm {e}L_\\mathrm {t} \\right)^2} + \\frac{\\log ^2 e}{L_\\mathrm {t}}.",
"$ In contrast to Example REF , the ratio $\\xi $ in this case does not necessarily take large values, and consequently, $\\mathcal {R}_\\mathrm {Erg}\\left( \\mathcal {S} \\right)$ is not approximated by $\\eta $ .",
"Therefore, we define $c(\\cdot )$ over ${R}$ as $c(x) s(x)\\phi \\left( h(x) \\right) + f(x) \\mathrm {Q}\\left( -h(x) \\right)$ where $f(x)$ and $[s(x)]^2$ are obtained by replacing $L_\\mathrm {t}$ with $x$ in (REF ) and () respectively, and $h(x)[s(x)]^{-1}f(x)$ .",
"With similar lines of inference the optimal number of selected antennas is approximated as $L^\\star _\\mathrm {t}\\approx \\lfloor x^\\star \\rceil $ with $x^\\star $ satisfying $\\hspace*{-5.69054pt}\\phi \\left( h(x^\\star ) \\right) \\left\\lbrace 2f(x^\\star )h^{\\prime }(x^\\star )\\hspace*{-1.99168pt}-\\hspace*{-1.99168pt}s^{\\prime }(x^\\star )\\right\\rbrace \\hspace*{-2.84526pt} = \\hspace*{-2.84526pt} f^{\\prime }(x^\\star ) \\mathrm {Q}\\left( -h(x^\\star ) \\right).",
"$ In Fig.",
"REF , $\\mathcal {R}_\\mathrm {Erg}(\\mathcal {S})$ is sketched in terms of the number of selected antennas for $N_\\mathrm {e}=16$ , $\\rho _\\mathrm {e}=-25$ dB and 128 transmit antennas.",
"From the fixed point equation in (REF ), the extreme point of the curve occurs at $x^\\star =13.7$ which recovers $L_\\mathrm {t}^\\star =14$ given by the simulation results.",
"Figure: The ergodic secrecy rate in Example vs. L t L_\\mathrm {t} for the case with N e =16N_\\mathrm {e}=16, N t =128N_\\mathrm {t}=128, ρ m =0\\rho _\\mathrm {m}=0 dB and ρ e =-25\\rho _\\mathrm {e}=-25 dB.",
"The large-system approximation and numerical simulations are denoted by the solid line and circles respectively, and both suggest L t ☆ =14L_\\mathrm {t}^\\star =14."
],
[
"Conclusion",
"In this paper, we have characterized the effect of multiple tas on the secrecy performance of a massive mimome channel by considering the ergodic secrecy rate and secrecy outage probability.",
"For this setup, the instantaneous secrecy rate has been approximated in terms of a Gaussian random variable when the number of transmit antennas grows large.",
"This approximation enabled us to show that, for several scenarios, the secrecy performance of the system is maximized at some optimal number of selected antennas.",
"Our numerical simulations showed that the optimal number of transmit antennas is accurately approximated for finite dimensions."
],
[
"Appendix: Derivation of Proposition ",
"We start by determining the large-system distribution of $\\mathcal {R}_\\mathrm {m}$ given in (REF ).",
"In [8], it has been shown that the distribution of the input-output mutual information of a Gaussian mimo channel, under some constraints, is approximated sufficiently close to its exact value in terms of the random variables $\\mathrm {Tr} \\lbrace \\mathbf {J} \\rbrace $ and $\\mathrm {Tr} \\lbrace \\mathbf {J}^2 \\rbrace $ where $\\mathbf {J}\\mathbf {H}^\\mathsf {H}\\mathbf {H}$ .",
"Under the tas protocol $\\mathcal {S}$ , $\\mathrm {Tr} \\lbrace \\mathbf {J} \\rbrace $ represents the sum of $L_\\mathrm {t}$ first order statistics which at the large limit of $N_t$ converges in distribution to a Gaussian random variable whose mean and variance are given by (REF ) and (), respectively [8].",
"Using some properties of random matrices, the large-system distribution of $\\mathcal {R}_\\mathrm {m}$ is then approximated as in [8] with a Gaussian distribution whose mean and variance are given in terms of $\\eta _t$ and $\\sigma _t^2$ .",
"The next step is to evaluate the distribution of $\\mathcal {R}_\\mathrm {e}$ .",
"Noting that the main and eavesdropper channels are independent, it is concluded that the tas protocol $\\mathcal {S}$ performs as a random selection protocol from the eavesdropper's point of view.",
"Therefore, $\\mathrm {Tr} \\lbrace \\mathbf {J} \\rbrace $ and $\\mathrm {Tr} \\lbrace \\mathbf {J}^2 \\rbrace $ in both the cases REF and REF can be determined explicitly as functions of independent Gaussian random variables.",
"By substituting in [8] and taking the limit of $N_\\mathrm {t}$ growing large, the large-system distribution of $\\mathcal {R}_\\mathrm {e}$ can be approximated with a Gaussian distribution whose mean and variance are respectively given by $\\hspace*{-5.69054pt} \\eta _\\mathrm {e}&= L_\\mathrm {e}\\log \\left( 1+\\rho _\\mathrm {e}M_\\mathrm {e} \\right)\\\\\\hspace*{-5.69054pt}\\sigma _\\mathrm {e}^2 &= \\left( \\mathbf {1}_{\\left\\lbrace N_\\mathrm {e}> L_\\mathrm {t}\\right\\rbrace }\\frac{L_\\mathrm {e}}{M_\\mathrm {e}}\\hspace*{-1.99168pt} +\\hspace*{-1.99168pt}\\mathbf {1}_{\\left\\lbrace N_\\mathrm {e}< L_\\mathrm {t}\\right\\rbrace } \\frac{L_\\mathrm {e}M_\\mathrm {e}\\rho ^2_\\mathrm {e}}{\\left(1+\\rho _\\mathrm {e}M_\\mathrm {e}\\right)^2 } \\right) \\log ^2 e$ which recovers the asymptotic results for iid Gaussian fading channels given in [16].",
"Using the independency of the main and eavesdropper channels, $\\mathcal {R}^\\star $ in (REF ) can be approximated in the large system limit with a Gaussian random variable whose mean and variance are as in (REF ) and (REF ).",
"mimo[MIMO]Multiple-Input Multiple-Output mimome[MIMOME]Multiple-Input Multiple-Output Multiple-Eavesdropper csi[CSI]Channel State Information awgn[AWGN]Additive White Gaussian Noise iid[i.i.d.",
"]independent and identically distributed ut[UT]User Terminal bs[BS]Base Station tas[TAS]Transmit Antenna Selection lse[LSE]Least Squared Error rhs[r.h.s.",
"]right hand side lhs[l.h.s.",
"]left hand side wrt[w.r.t.",
"]with respect to rs[RS]Replica Symmetry rsb[RSB]Replica Symmetry Breaking papr[PAPR]Peak-to-Average Power Ratio rzf[RZF]Regularized Zero Forcing snr[SNR]Signal-to-Noise Ratio rf[RF]Radio Frequency mf[MF]Match Filtering"
]
] | 1709.01868 |
[
[
"Magnetic properties of Co doped Nb clusters"
],
[
"Abstract From magnetic deflection experiments on isolated Co doped Nb clusters we made the interesting observation of some clusters being magnetic, while others appear to be non-magnetic.",
"There are in principle two explanations for this behavior.",
"Either the local moment at the Co site is completely quenched or it is screened by the delocalized electrons of the cluster, i.e.",
"the Kondo effect.",
"In order to reveal the physical origin, we conducted a combined theoretical and experimental investigation.",
"First, we established the ground state geometry of the clusters by comparing the experimental vibrational spectra with those obtained from a density functional theory study.",
"Then, we performed an analyses based on the Anderson impurity model.",
"It appears that the non-magnetic clusters are due to a complete quenching of the local Co moment and not due to the Kondo effect.",
"In addition, the magnetic behavior of the clusters can be understood from an inspection of their electronic structure.",
"Here magnetism is favored when the effective hybridization around the chemical potential is small, while the absence of magnetism is signalled by a large effective hybridization around the chemical potential."
],
[
"Introduction",
"Electronic correlations constitute the basis of condensed matter physics and are responsible for the enormous wealth of phenomena found in solids, such as (high-$T_{c}$ ) superconductivity[1], charge- and spin-ordering[2] and fluctuations[3], colossal magnetoresistance[4], metal-insulator transition[5], half-metallicity[6], quantum Hall effect[7], heavy fermion behavior[8], etc.",
"Reducing the size, however, leads to an extreme sensitivity of these properties to the atomic arrangement, shape, and the effects of the environment.",
"The understanding and control of these size-driven processes is therefore crucial to maintain the pace of developments in nanoscience.",
"In this miniaturization trend, the ultimate limit is represented by atomic clusters.",
"Such clusters are particles composed of a countable number of atoms, from the diatomic limit up to some thousands or tens of thousands of atoms.",
"Quantum confinement effects entirely govern the behavior of matter in this size regime.",
"The discretized electronic levels lead to sudden changes of the cluster properties, for example when changing the cluster size on an atom-by-atom basis.",
"In the semiconductor technology there is already interest in systems with discrete energy spectra, for example quantum wells[9] and quantum dots[10].",
"Obviously the consideration of doped instead of pure clusters offers an even broader playground for technological applications.",
"However, doped clusters are also very interesting from a fundamental point of view.",
"For example, it is well known that already for a single magnetic impurity in a non-magnetic metallic host interesting phenomena like Friedel oscillations[11] and the Kondo effect[12] can occur.",
"How or would such effects be present in clusters?",
"Furthermore, the case of a single magnetic impurity embedded in a discrete host like a cluster offers a sensitive probe of studying the dependence of the local magnetic moment on the details of the discrete energy spectrum.",
"This could lead to valuable insight in quenching and/or Kondo screening mechanisms.",
"More precisely, the formation of the atomic magnetic moment is trivially described by the Hund's rules in the case of an isolated atom, but this process is far from trivial in the case of an atom embedded in an interacting host.",
"Recently, the magnetic moment of a single magnetic impurity in a discrete host was investigated by means of the Anderson impurity model.",
"[13] One of the things found, was that on average the local moment grows with increasing host band gap (HOMO-LUMO gap).",
"Here on average should be understood as the local moment averaged over a number of random configurations of the discrete host energy levels for a fixed host band gap.",
"Then, based on this investigation of the Anderson impurity model, the experimentally observed magnetic moments of Cr doped Au clusters were successfully explained.",
"[14] For example, it was found that the size of the measured local moment follows the trend of the calculated band gap of the host.",
"In this work we present a comprehensive study of the mechanisms governing the formation of magnetic moments in Co doped Nb clusters.",
"From magnetic deflection experiments we made the interesting observation that some clusters are strongly magnetic, while others are completely non-magnetic.",
"Note that in contrast for the Cr doped Au clusters all measured clusters were found to be magnetic.",
"There are two possibilities for the absence of magnetism in the Nb$_{x}$ Co clusters.",
"Either the local Co moment is completely quenched or it is screened by the delocalized electrons of the cluster, i.e.",
"the Kondo effect.",
"From the theoretical perspective, the difficulty in explaining the observed magnetic behavior is in the treatment of the electronic correlations.",
"Since it is not clear from the beginning whether correlations effects are weak, intermediate or strong, it is difficult to decide which theoretical approach is suitable.",
"One could expect correlations to be stronger in small clusters than in their bulk counterparts due to a stronger localization of the wave-functions.",
"On the other hand, for the clusters less screening channels are present, which could lead to an almost constant Coulomb interaction throughout the cluster [15].",
"This would render correlations effects to be unimportant.",
"As mentioned, a priori the importance of correlations effects is not known for Nb$_{x}$ Co clusters.",
"Therefore, we conducted a combined theoretical and experimental investigation.",
"First, we made a comparison of the experimental vibrational spectra with those obtained from a density functional theory (DFT) study.",
"This serves two purposes.",
"It provides the ground state geometry of the clusters.",
"Further, due to the dependence of the vibrational spectrum on the magnetic moment, the performance of DFT in predicting the magnetic moments can be investigated.",
"Then, in order to obtain a physical understanding of the experimentally observed magnetic behavior, we performed an analyses based on the Anderson impurity model.",
"From this analyses it is observed that the absence of a magnetic moment is due to a complete quenching of the Co moment and not the Kondo effect.",
"In addition, the magnetic behavior of the Nb$_{x}$ Co clusters can be understood from an inspection of their electronic structure.",
"Here magnetism is favored when the effective hybridization around the chemical potential is small, while the absence of magnetism is signaled by a large effective hybridization around the chemical potential.",
"Co doped and pure Nb clusters have already been the topic of interest in earlier works.",
"One of the most relevant experimental works on pure Nb clusters is the electric deflection experiment, which showed that cold clusters may attain an anomalous component with very large electric dipole moments.",
"[16] In contrast, the room-temperature measurements showed normal metallic polarizabilities.",
"Further, magnetic deflection experiments on pure Nb clusters showed that at very low temperatures the clusters with an odd number of atoms deflect due to a single unpaired spin that is uncoupled from the cluster lattice.",
"In contrast, at high temperatures deflections do not take place, as in the cluster the unpaired spin becomes coupled to the lattice.",
"[17] Far-infrared absorption spectra of small neutral and cationic Nb clusters combined with DFT calculations have revealed their geometries.",
"[18] Compared to pure Nb clusters, not many works focused on Co doped Nb clusters.",
"Experimentally an anion photoelectron spectroscopy study is performed, which showed that the addition of the Co atom for small Nb clusters induces bulk-like behavior, i.e.",
"closing of the band gap.",
"[19] From the theoretical side a computational study based on DFT addressed the geometric and magnetic properties finding that Nb$_{7}$ Co has no net magnetic moment, which means that the 6 $\\mu _{B}$ coming from the Co atom is completely quenched.",
"[20] The disadvantage of this purely theoretical work is the lack of an experimental confirmation, which is another reason for conducting a combined experimental and theoretical study.",
"The rest of this paper is organized as follows.",
"In Section we first present our magnetic deflection experiments.",
"Then, in Section the experimental vibrational spectra are compared with those obtained from density functional theory calculations.",
"Based on the ground state geometries obtained from this comparison, we perform a discussion based on the Anderson impurity model in Section to address the presence or absence of magnetic moments in Nb$_{x}$ Co clusters.",
"Finally, in Section we present our conclusions.",
"The magnetic moments of the Nb$_{x}$ Co clusters were obtained by means of a Stern-Gerlach setup.",
"[21] This setup consists mainly of three parts: the source, the magnet and the position sensitive time of flight mass spectrometer (PSTOFMS).",
"The source is of Milani-de Heer type.",
"[22] The clusters are produced in the source chamber by ablation of a Nb$_x$ Co$_y$ (x = 95, y = 5 %) rod due to a Nd:YAG laser producing 532 nm light.",
"More precisely, this laser is focused on the rod, which is inside a cavity of a tuneable volume.",
"The cavity is connected to a pulsed valve, responsible to introduce pulses of helium, which is the carrier gas, i.e.",
"it is responsible for the transport of the clusters across the setup.",
"The cavity is also coupled to a nozzle.",
"Due to a pressure gradient across the nozzle, the clusters expand supersonically.",
"The actual creation and cooling of the clusters takes place inside the cavity.",
"In our setup the source can be cooled down to 20 K due to a cold head.",
"Once the cluster beam has left the cavity, it crosses a conical skimmer of 1 mm width.",
"After the clusters are skimmed they reach a chopper, which has two purposes: cluster selection and measurement of their velocity.",
"Then, after the chopper there are two slits to narrow the beam in both the horizontal and vertical direction.",
"After the slits, the cluster beam reaches the magnet, i.e.",
"a 2 wire Rabi design electromagnet.",
"[23] The magnet produces an inhomogeneous magnetic field that can reach a maximum strength of 2.4 T and gradient of 350 T/mm.",
"The spins of the cluster are aligned by the magnetic field, while the cluster is deflected due to the gradient in the field.",
"For the calibration of the magnet aluminium atoms were chosen, since they are easy to produce and their magnetic properties are well known ($\\mu _B$ =1/3$\\mu _B$ , J=1/2, m$_{J}$ =$\\pm $ 1/2).",
"After the magnet the clusters have to travel 1 m before they reach the PSTOFMS.",
"In order to detect the clusters, they are ionized by an excimer laser producing an ultraviolet beam of 193 nm.",
"The ionized clusters can then be directed by the electric fields of the PSTOFMS plates towards the micro channel plate (MCP) where they are detected.",
"After the detection of the cluster, its time-of-flight is known.",
"This time-of-flight linearly depends on the deflection of the cluster, where the proportionality constant is obtained from another calibration.",
"For this calibration, a narrow slit is placed in the path of the excimer beam.",
"Then, by moving the slit, the time-of-flight can be determined for each corresponding slit position, i.e.",
"the position where the clusters will be ionized, describing the correlation between time-of-flight and deflection.",
"Since the determined proportionality constant is known to scale as the square root of the mass, it is only necessary to perform the calibration for one specific cluster size.",
"From the measurement of the deflection $x$ via the time-of-flight, the mass $m$ of the cluster and its velocity $v$ by means of the chopper, the average magnetic moment is determined from $\\begin{aligned}\\langle \\mu \\rangle =\\frac{xmv\\textsuperscript {2}}{KB}.\\end{aligned}$ Here $B$ is the magnetic field strength and $K$ is a constant that depends on the setup.",
"This constant includes the gradient of the magnet, which is determined from the calibration by the aluminium atoms.",
"During the measurements we observed three different deflection (or equivalently average magnetic moment via Eq.",
"REF ) profiles, see Fig.",
"REF (a), (b) and (c).",
"The method we used to obtain the actual value of the deflection, depends on the observed deflection profile.",
"In case of a deflection profile centered at zero (Fig.",
"REF (b)) we used a Gaussian fit, where the position of the peak of the Gaussian corresponds to the 'deflection'.",
"For the double sided deflection profile (Fig.",
"REF (c)) three peaks can be observed.",
"Here the peak at 0 is due to the spin-relaxation by the coupling to the lattice, which can be in principle reduced by increasing the carrier gas pressure.",
"Therefore, the deflection should be determined from the peaks away from zero.",
"For this purpose we used three Gaussians to fit the profile, where the deflection is determined from the peak position of the Gaussians not located at zero.",
"Finally, for single sided deflection (Fig.",
"REF (a)) the profile is in general asymmetric, which makes a fit by a Gaussian inappropriate.",
"In this case we take the position of the average of the peak as the deflection.",
"In Eq.",
"REF , the quantity $\\langle \\mu \\rangle $ corresponds to the measured average magnetic moment of the clusters.",
"However, not for all the three observed deflection profiles, this corresponds to the actual magnetic moment of the cluster.",
"While it does for no deflection and double sided deflection, it does not for single sided deflection.",
"Obviously, no deflection corresponds to a non-magnetic cluster.",
"Double sided deflection corresponds to atomic like clusters, i.e.",
"clusters where the magnetic moment can freely rotate.",
"In contrast, single sided deflection corresponds to superparamagnetic clusters with the magnetic moment coupled to the lattice.",
"Then, due to the presence of a finite temperature, not all clusters have their magnetic moment aligned with the magnetic field.",
"Therefore, the measured average magnetic moment needs to be related to the actual magnetic moment of the cluster.",
"For isolated clusters this is typically done by the Langevin-Debye function.",
"In the limit of a small magnetic field this leads to the following relation $\\begin{aligned}M =\\frac{1}{3}\\frac{\\langle \\mu \\rangle ^{2}B}{k_{B}T},\\end{aligned}$ where $M$ is the magnetic moment of the cluster, $T$ the temperature and $k_{B}$ the constant of Boltzman."
],
[
"Results: Magnetic moments",
"The results of the Stern-Gerlach experiments performed on the Co doped Nb clusters at a temperature of 25 K are presented in Fig.",
"REF .",
"Here Fig REF (a), (b) and (c) correspond to the three typical deflection profiles that were observed, and in Fig.",
"REF (d) the measured magnetic moments in $\\mu _{B}$ are presented as function of cluster size with $n$ corresponding to the number of host (Nb) atoms.",
"The black and red lines in the deflection profiles correspond respectively to the situation without and with a magnetic field.",
"The applied magnetic field was 2.4 T except for Nb$_{3}$ Co, where 1 T was used.",
"Figure: The top three figures contain the different deflection profiles observed for the Co doped Nb clusters.",
"(a) Single sided deflection profile for Nb 3 _{3}Co, which indicates a superparamagnetic cluster.",
"(b) Profile of Nb 5 _{5}Co showing no deflection, which corresponds to a non-magnetic cluster.",
"(c) Two sided deflection profile for Nb 10 _{10}Co, which refers to an atomic-like cluster.",
"All deflection profiles were measured at a temperature of 25K.",
"Further, the black and red lines correspond to the situations without and with a 2.4 T magnetic field.",
"The only exception is in (a), where the magnetic field was 1 T. For the x-axis of (a), (b) and (c) the deflection is converted to the averaged magnetic moment via Eq. .",
"In the bottom figure indicated with (d), the magnetic moment as function of cluster size is presented with nn corresponding to the number of host (Nb) atoms.",
"Here the error bars are computed from the uncertainty in the velocity of the cluster and the magnetic field.Fig.",
"REF (a) for Nb$_{3}$ Co shows a typical single sided deflection profile indicating superparamagnetic behavior.",
"Other clusters showing a single sided deflection profile were Nb$_{4}$ Co, Nb$_{6}$ Co, Nb$_{9}$ Co, Nb$_{11}$ Co, Nb$_{12}$ Co and Nb$_{13}$ Co. Then, Fig.",
"REF (b) for Nb$_{5}$ Co presents the situation with no deflection, which corresponds to a non-magnetic cluster.",
"The other clusters showing no deflection were Nb$_{7}$ Co, Nb$_{15}$ Co and Nb$_{17}$ Co.",
"The last observed deflection profile is depicted in Fig.",
"REF (c), where for Nb$_{10}$ Co an example of a two sided deflection is given, which refers to an atomic-like cluster.",
"This profile is characterized by 3 peaks, 2 peaks at $\\pm $ 1$\\mu _B$ and an additional peak at 0$\\mu _B$ .",
"Two sided deflection was also observed for all clusters containing an odd number of atoms (i.e.",
"with $n$ even) and with $n>=14$ .",
"From the magnetic moments as function of cluster size presented in Fig.",
"REF (d) it seems that the clusters can be divided into two regions.",
"For clusters with $n>=14$ the magnetic to non-magnetic behavior appears to be exactly determined by having an odd or even number of atoms in the cluster.",
"An odd number of atoms in the cluster corresponds to the situation of at least one unpaired electron and thus at least a moment of 1 $\\mu _{B}$ .",
"For an even number of atoms, all the electrons can be paired.",
"Note that the magnetic behavior of pure Nb clusters was indeed explained in this way.",
"[17] Then, there is the regime of clusters with $n<14$ , where the magnetic behavior clearly cannot be explained due the presence or absence of a single unpaired electron.",
"In this region strong fluctuations in the magnetic moment can be observed by just adding or removing a single Nb atom.",
"For example, Nb$_{4}$ Co is strongly magnetic, while Nb$_{5}$ Co is completely non-magnetic.",
"Then, again adding just one Nb atom leads to Nb$_{6}$ Co which is again strongly magnetic.",
"On the other hand Nb$_{7}$ Co is again non-magnetic.",
"It can also be observed that there is no cluster with a magnetic moment larger than that of an isolated Co atom.",
"An isolated Co atom has 7 3$d$ electrons leading to a total moment of 6$\\mu _{B}$ , where both the spin and orbital moment contribute 3$\\mu _{B}$ .",
"This indicates that the Co atom is not very effective in inducing magnetic moments in the Nb host.",
"Further, it is interesting that Nb$_{3}$ Co, Nb$_{4}$ Co and Nb$_{6}$ Co have a magnetic moment very close to that of an isolated Co atom.",
"Assuming that all this magnetic moment is at the Co site, would mean that we have a situation where both the spin and orbital moment are almost unquenched."
],
[
"Vibrational spectra: geometric and magnetic structure",
"In this section we perform a comparison of the experimental vibrational spectra with those obtained from a DFT study.",
"This serves two purposes.",
"First, due to the dependence of the vibrational spectrum on the magnetic moment, the performance of DFT in predicting the magnetic moments can be investigated.",
"Second, it provides the ground state geometry of the clusters.",
"These ground state geometries are required as an input in Section to obtain a physical understanding of the observed magnetic behavior in Section ."
],
[
"Experimental details",
"In order to record the vibrational spectra we coupled our cluster setup to the Free Electron Laser For Intra Cavity Experiments (FELICE).",
"Below a brief description of the experimental setup is given and for more details the reader is referred to Ref. Jalink2015.",
"The clusters are produced in an ablation-type cluster source in a growth channel filled by a helium carrier gas prior to ablation of a Nb$_x$ Co$_y$ (x = 95, y = 5 %) rod by a Nd:YAG laser (532 nm).",
"The temperature of the extension tube, which is attached to the cluster source for better cluster thermalization, is 77 K. After expansion in the source chamber, the mixture of clusters and carrier gas is skimmed.",
"This results in the formation of a molecular beam that is shaped by a slit with a width of 0.45 mm .",
"The interaction between the IR light and the molecular beam takes place in the center of the extraction region of the REToF mass spectrometer with a 35$^\\circ $ angle between the two beams.",
"The clusters are ionized by a frequency doubled dye laser with a photon energy of 5.4 eV entering the extraction region at a $\\sim $ 90$^\\circ $ angle with respect to the cluster beam.",
"Then, the IR pulse energies calculated inside the FELICE cavity range between 0.2 and 0.6 J over the IR scans.",
"The IR pulse consists of a 9 $\\mu $ s long train of micro-pulses with 1 ns time delay between them.",
"The experiment operates at twice the FELICE frequency which allows to record a signal with (I$_{IR+UV}(\\omega )$ ) and without (I$_{UV}$ ) IR radiation in a shot-to-shot manner.",
"The experimental IR curves are presented in terms of gain spectra (G($\\omega $ )) calculated as $\\begin{aligned}G(\\omega ) =\\frac{I_{\\textsc {ir+uv}}(\\omega )-I_{\\textsc {uv}}}{I_{\\textsc {uv}}},\\end{aligned}$ at an IR frequency $\\omega $ , and are IR power corrected."
],
[
"Computational details",
"For the calculation of the vibrational spectra we employed the DFT implementation of the Vienna ab initio simulation package (VASP).",
"[25] The projector augmented wave (PAW) method[26], [27] in combination with the Perdew, Burke, and Ernzerhof (PBE) functional is used.",
"[28] For all cluster sizes we searched for the lowest-energy geometries by using a genetic algorithm (GA)[29] in combination with DFT.",
"The details of the used method can be found in Ref. Logemann2015.",
"In addition, we also considered conformations previously reported in the literature (Nb3Co, Nb4Co, Nb5Co, Nb6Co, Nb7Co)[20] and re-optimized the mentioned structures.",
"For some clusters the GA results were equal to those already found literature, while for other clusters additional geometries lower in energy were obtained (see Sections REF -REF for details).",
"Further, for the PAWs an energy cutoff of 4293 eV is used.",
"All forces were minimized below $10^{-3}$ eV/Å.",
"In order to eliminate inter-cluster interactions, the clusters were placed in a cubic periodic box with 16 Å dimensions.",
"For the calculations, a single k point ($\\Gamma $ ) is used."
],
[
"Results: Geometric and magnetic structure",
"Below the calculated geometries of the clusters are presented by a stick model, i.e.",
"the clusters are presented by connected sticks.",
"Here green correspond to Nb and gold to Co. Further, to facilitate the comparison of the experimental and calculated results, the experimental spectra are shown with black squares accompanied by a three-point adjacent average (blue line).",
"The gray dashed line indicates the IR power corrected experimental spectrum.",
"The calculated harmonic vibrational frequencies (vertical sticks) are convoluted with a 15 $\\text{cm}^{-1}$ FWHM Gaussian line shape function.",
"All frequencies for the structures presented in this work are unscaled and the energies contain the zero-point vibrational energies (ZPVE).",
"Finally, the insets of the figures below show the energy as a function of magnetization for the presented geometries with respect to that of the ground state."
],
[
"Nb3Co",
"For Nb3Co a trigonal pyramid is found with three different magnetic states.",
"Here the Nb-Nb and Nb-Co distances differ slightly between the magnetic configurations.",
"Figure: Experimental (panel (a), squares) and calculated ((b)-(d)) IR spectra of Nb3Co.",
"The blue line is three-point adjacent average of the experimental data.",
"The gray dashed line indicates the IR power corrected spectrum.The calculated discrete vibrational frequencies (orange vertical lines) are convolutedwith a 15 cm -1 \\text{cm}^{-1} FWHM Gaussian line shape function (orange).",
"For the geometries green and gold are used for Nb and Co respectively.",
"The inset graph shows the energy as function of the magnetization for the different magnetic states.In Fig.",
"REF (b)-(d) the corresponding geometries are shown.",
"The magnetic $M=2$ $\\mu _{\\text{B}}$ (3,1)A geometry is lowest in energy, whereas (3,1)B and (3,1)C are 0.14 eV and 0.25 eV higher in energy respectively.",
"Note that geometry (3,1)A has been reported previously also as the ground state in Ref. Li2014.",
"The symmetry point group depends on the magnetization, with $C_{3v}$ for (3,1)A and $C_{s}$ for (3,1)B and (3,1)C. This difference in symmetry clearly results in significant differences in the vibrational spectra.",
"Fig.",
"REF shows that the vibrational spectrum of (3,1)A with modes at 224, 228 and 356 $\\text{cm}^{-1}$ provides the best match to the experimental modes at 212 and 328 $\\text{cm}^{-1}$ and also resolves the internal structure of the band at 212 $\\text{cm}^{-1}$ .",
"The vibrational spectra of (3,1)B and (3,1)C contain vibrational modes in the range 125-220 $\\text{cm}^{-1}$ where no experimental modes are observed.",
"Therefore, geometry (3,1)A in the $M=2$ $\\mu _{\\text{B}}$ state corresponds to the ground state of Nb3Co."
],
[
"Nb4Co",
"In the experimental spectrum of Nb4Co presented in Fig.",
"REF (a), at least four modes can be distinguished, at 150, 230, 255 and 325 $\\text{cm}^{-1}$ .",
"The three geometries lowest in energy are shown in Fig.",
"REF (b)-(d).",
"Geometry (4,1)A with $M=3$ $\\mu _{\\text{B}}$ is the lowest in energy and has $C_{3v}$ point group symmetry.",
"Figure: Experimental (panel (a)) and calculated ((b)-(d)) vibrational spectra of Nb4Co.",
"The insets show the energy as function of magnetization for each geometry.Geometry (4,1)A consists of a triganol bi-pyramid, where Nb and Co are the axial atoms.",
"In contrast, in geometry (4,1)B the Co atom is part of the equatorial triangle.",
"For geometry (4,1)B the $M=1$ $\\mu _{\\text{B}}$ state is the lowest in energy and is 0.18 eV higher compared to the lowest of (4,1)A.",
"Note that both (4,1)A and (4,1)B are previously reported in Ref.",
"Li2014, where (4,1)A with $M=3$ $\\mu _{\\text{B}}$ was also found to be the lowest in energy.",
"The vibrational spectrum of (4,1)A with $M=3$ $\\mu _{\\text{B}}$ consists of two large modes at 145 and 238 $\\text{cm}^{-1}$ and smaller modes at 173, 278 and 342 $\\text{cm}^{-1}$ , and matches very well to the experimental spectrum.",
"The vibrational spectrum of (4,1)A with $M=3$ $\\mu _{\\text{B}}$ is the only spectrum with two major modes around 150 and 230 $\\text{cm}^{-1}$ .",
"Therefore, we assign geometry (4,1)A with $M=3$ $\\mu _{\\text{B}}$ to be the ground state of Nb4Co."
],
[
"Nb5Co",
"In Fig.",
"REF (b)-(d) the three geometries found to be lowest in energy for Nb5Co are presented.",
"Geometry (5,1)A consists of a dimer-capped rhombus with $C_{s}$ point group symmetry for all considered magnetic states and has been previously reported in Ref.",
"Li2014 to be the lowest in energy for the $M=4$ $\\mu _{\\text{B}}$ state.",
"We also find geometry (5,1)A in the $M=4$ $\\mu _{\\text{B}}$ state to be the lowest in energy, although the $M=2$ $\\mu _{\\text{B}}$ state is only 0.03 eV higher in energy.",
"Geometries (5,1)B and (5,1)C both consist of a distorted Nb5 bi-pyramid with one of the faces of the bi-pyramid capped by the Co atom.",
"Geometries (5,1)B and (5,1)C differ in the distance of the Co atom to the bi-pyramid.",
"Figure: Experimental (panel (a)) and calculated ((b)-(d)) vibrational spectra of Nb5Co.Whereas for the (5,1)B geometry the $M=2$ $\\mu _{\\text{B}}$ state is the lowest in energy, the (5,1)C geometry has a non-magnetic ground state which is 0.37 eV higher in energy compared to (5,1)A.",
"The experimental spectrum of Nb5Co in Fig.",
"REF (a) shows three major bands at 170, 205 and 250 $\\text{cm}^{-1}$ , where the internal structure of the band at 205 $\\text{cm}^{-1}$ indicates at least a second mode at 220 $\\text{cm}^{-1}$ .",
"A smaller vibrational mode is present at 275 $\\text{cm}^{-1}$ .",
"If the calculated spectra of Fig.",
"REF (b)-(d) are compared to that of Fig.",
"REF (a), both (5,1) A $M=4$ $\\mu _{\\text{B}}$ and (5,1)C $M=0$ $\\mu _{\\text{B}}$ can only partially explain the experimental spectrum.",
"Whereas (5,1) $M=4$ $\\mu _{\\text{B}}$ resembles the experimental spectrum below 230 $\\text{cm}^{-1}$ , the modes at 236 and 250 $\\text{cm}^{-1}$ are not present in the calculated spectrum.",
"Due to the similar vibrational spectrum and the low difference in energy between (5,1)A $M=2$ $\\mu _{\\text{B}}$ and (5,1)A $M=4$ $\\mu _{\\text{B}}$ , the former can also not be excluded based on IR vibrational spectroscopy.",
"The vibrational spectrum of (5,1)C $M=0$ $\\mu _{\\text{B}}$ agrees for the modes above 250 $\\text{cm}^{-1}$ , but deviates significantly in the relative IR absorption intensities between modes compared to the experimentally observed gain.",
"Therefore, the IR gain spectrum of Nb5Co might by due to the geometry (5,1)A with $M=2$ $\\mu _{\\text{B}}$ or $M=4$ $\\mu _{\\text{B}}$ , or geometry(5,1)C $M=0$ $\\mu _{\\text{B}}$ .",
"However, due to the finite temperature at which the experiment is performed, the vibrational spectrum might also be due to a combination of different geometries and magnetic states.",
"On the other hand, the magnetic deflection experiments (see Section ) were performed at a lower temperature than the vibrational experiments and strictly found Nb$_{5}$ Co to be non-magnetic.",
"Therefore, the (5,1)C geometry corresponding to the $M=0$ $\\mu _{\\text{B}}$ state is ascribed to be the ground state."
],
[
"Nb6Co",
"The two geometries that were found to be the lowest in energy for Nb6Co are shown in Fig.",
"REF (b)-(c).",
"Here geometry (6,1)A consists of a distorted pentagon with both sides capped with a single Nb atom.",
"Geometry (6,1)A in the $M=3$ $\\mu _{\\text{B}}$ state is obtained as the lowest in energy.",
"Figure: Experimental (panel (a)) and calculated ((b)-(d)) vibrational spectra of Nb6Co.",
"The IR absorption intensity of (6,1)A M=3M=3 μ B \\mu _{\\text{B}} and M=5M=5 μ B \\mu _{\\text{B}} are enhanced by a factor of 5 and 2 respectively to increase visibility.All magnetic states of the (3,1)A geometry have a $C_{1}$ point group symmetry.",
"Geometry (6,1)B consists of two stacked Nb3 triangles, where the top triangle is capped with a Co atom.",
"For this geometry the $M=1$ $\\mu _{\\text{B}}$ state is the lowest in energy and has a $C_{3v}$ point group symmetry.",
"The experimental IR spectrum of Nb6Co is shown in Fig.",
"REF (a) and contains a dominant mode at 270 $\\text{cm}^{-1}$ and two smaller modes at 200 and 220 $\\text{cm}^{-1}$ .",
"The vibrational spectrum of (6,1)B $M=1$ $\\mu _{\\text{B}}$ provides the best match to the experimental spectrum with a single dominant mode at 256 $\\text{cm}^{-1}$ and several smaller modes constituting two bands at 190 and 210 $\\text{cm}^{-1}$ .",
"In the vibrational spectrum of (6,1)A the bands at 220 and 264 $\\text{cm}^{-1}$ have similar IR absorption intensities, which is in disagreement with the experimentally observed relative difference between these bands.",
"All other geometries have significant vibrational modes below 190 $\\text{cm}^{-1}$ where experimentally no modes are observed.",
"Therefore, the (6,1)B geometry with the $M=1$ $\\mu _{\\text{B}}$ state is the ground state of Nb6Co."
],
[
"Nb7Co",
"The experimental spectrum of Nb7Co in Fig.",
"REF (a) shows a clear band at 260 $\\text{cm}^{-1}$ .",
"The three geometries lowest in energy are shown in Fig.",
"REF (b)-(d), where all geometries have either a symmetry plane or no symmetry at all.",
"Geometry (7,1)A consists of a bicapped distorted pentagon, where one of the faces is capped with a Co atom.",
"Figure: Experimental (panel (a)) and calculated ((b)-(d)) vibrational spectra of Nb7Co.The lowest magnetic state of geometry (7,1)A has a magnetic moment of $M=0$ $\\mu _{\\text{B}}$ .",
"This (7,1)A geometry has been previously reported in Ref.",
"Li2014 to be also the lowest in energy.",
"The (7,1)B geometry is formed by a Nb4 square capped on one side by a Co atom and the other side by a Nb3 triangle.",
"Geometry (7,1)C is described by a bipyramid containing a Coatom at one of the tops and a single face of each pyramid is capped by a Nb atom.",
"In contrast to geometry (7,1)A, the magnetic ground states of geometries (7,1)B and (7,1)C are magnetic with $M=2$ $\\mu _{\\text{B}}$ .",
"Due to the reduction in symmetry, the vibrational spectrum of all geometries contain many vibrational modes.",
"The single experimental band at 260 $\\text{cm}^{-1}$ can be both explained by (7,1)A $M=0$ $\\mu _{\\text{B}}$ and (7,1)B.",
"However, the non-magnetic ground state of geometry (7,1)B is in very good agreement with experiment.",
"Therefore, this geometry is assigned to be the ground state."
],
[
"Nb9Co",
"Fig.",
"REF (a) shows the IR gain spectrum of Nb9Co.",
"Although this figure is not very well resolved, at least bands at 205, 240 and 280 $\\text{cm}^{-1}$ can be identified.",
"In Fig.",
"REF (b)-(d) the three Nb9Co geometries that were found to be the lowest in energy are presented.",
"Figure: Experimental (panel (a)) and calculated ((b)-(d)) vibrational spectra of Nb9Co.Here geometry (9,1)A consists of a Nb4 rhombus stacked with a Nb5 pentagon capped by a Co atom.",
"Note that geometry (9,1)A is distorted such that only a mirror plane symmetry remains.",
"The geometry indicated by (9,1)B consists of two stacked Nb4 squares, where the two open faces are capped by a Nb and Co atom.",
"The (9,1)C geometry is best described (yet poorly) by a distorted hexagon with a Nb in the center and a Co atom occupying a corner, and capped by a Nb3 triangle.",
"Here for the geometry (9,1)C in the states $M=2$ and $M=0$ $\\mu _{\\text{B}}$ there is no symmetry, while in the $M=4$ $\\mu _{\\text{B}}$ state there is only a mirror plane.",
"For geometry (9,1)C the $M=2$ $\\mu _{\\text{B}}$ state is found to be the lowest in energy, while the $M=0$ $\\mu _{\\text{B}}$ state is 0.12 eV higher in energy.",
"If the calculated vibrational spectra of Fig.",
"REF (b)-(d) are compared to the experimental spectrum, geometry (9,1)C with $M=2$ and $M=0$ provide the best match with dominant bands around 205 and 285 $\\text{cm}^{-1}$ and an intermediate mode in-between.",
"Therefore, the ground state of the Nb9Co cluster is described by the (9,1)C geometry."
],
[
"Comparison with magnetic deflection results",
"It is interesting to compare the magnetic moments obtained from the magnetic deflection experiments described in Section with those obtained above from an inspection of the vibrational spectra.",
"In Table REF the second column contains the magnetic moments obtained from the best match of the calculated DFT vibrational spectra compared to experiment.",
"For some clusters multiple magnetic moments are given, because for them it was not clear which vibrational spectrum matches the best with experiment.",
"The third column corresponds to the magnetic moments observed in the magnetic deflection experiments (see Fig.",
"REF ).",
"Except for Nb$_{5}$ Co it appears that the magnetic moments predicted by the magnetic deflection experiments are substantionally larger.",
"Part of this difference is due to not taking into account the orbital contribution to the magnetic moment within the DFT calculations.",
"However, even if we would have considered them, it is well known that orbital moments can be highly underestimated in DFT especially for clusters.",
"[31] Another possible reason for the difference in magnetic moments observed in Table REF , is an underestimation of the spin contribution within DFT.",
"Unfortunately, for Nb$_{5}$ Co and Nb$_{9}$ Co we cannot be conclusive about the magnetic moment obtained from an inspection of the vibrational spectra.",
"For Nb$_{5}$ Co the zero magnetic moment would be in agreement with the magnetic deflection experiment, but this state is 0.37 eV higher in energy than the calculated ground state.",
"Note that for Nb$_{6}$ Co and Nb$_{7}$ Co the best match of the calculated spectrum with experiment was also for a state higher in energy than the ground state, respectively 0.38 and 0.16 eV.",
"On the other hand for Nb$_{3}$ Co and Nb$_{4}$ Co the spectrum calculated for the ground state provided the best match with experiment.",
"For Nb$_{9}$ Co the state with a magnetic moment of 2 $\\mu _{B}$ would be the closest to the result of the magnetic deflection experiment.",
"Here the state with a moment of 2 $\\mu _{B}$ is 0.27 eV higher in energy than the ground state.",
"Table: Here the second column corresponds to the magnetic moments obtained from the best match of the calculated DFT vibrational spectra with respect to experiment.",
"The third column contains the magnetic moments obtained from the magnetic deflection experiments presented in Section .In this section the physical origin is explained of the magnetic behavior obtained from the magnetic deflection experiments presented in Section .",
"For example, it will be understood why some clusters are strongly magnetic, while others are non-magnetic.",
"For this purpose an analyses based on the Anderson impurity is performed, where the ground state geometries obtained in Section are required as an input."
],
[
"Theoretical background",
"There are two possible explanations for some Nb$_{x}$ Co clusters being non-magnetic.",
"It can be non-magnetic, because interactions of the Co atom with the Nb$_{x}$ host destroy the local moment at the Co site.",
"More precisely, there is a competition between Jahn-Teller distortion working against the formation of a magnetic moment and the exchange interaction between Nb and Co preferring the existence of a magnetic moment.",
"Another possibility is that the local moment at the Co site is screened by the delocalized electrons in the cluster, i.e.",
"the Kondo effect.",
"For both mechanisms it is crucial to understand physically when a local moment is formed on the Co site.",
"In case of a magnetic (transition-metal) impurity resolved in a metallic non-magnetic host this is well established within the celebrated Anderson impurity model, ${\\begin{array}{c}H=\\sum _{k,\\sigma }\\epsilon _{k\\sigma }c_{k\\sigma }^{\\dagger }c_{k\\sigma }\\\\+\\sum _{\\sigma }E_{d\\sigma }d_{\\sigma }^{\\dagger }d_{\\sigma }+U n_{d\\uparrow }n_{d\\downarrow }\\\\+\\sum _{k,\\sigma }V\\Big (d_{\\sigma }^{\\dagger }c_{k\\sigma }+c_{k\\sigma }^{\\dagger }d_{\\sigma }\\Big ).\\end{array}}$ Here $E_{d\\sigma }$ is the single-particle impurity energy level and $U$ is the onsite Coulomb repulsion between the impurity states.",
"Further, the dispersion of the non-interacting electronic bath is given by $\\epsilon _{k\\sigma }$ .",
"The coupling between the impurity and bath states is described by $V$ .",
"Within this model the formation of a local moment depends on a delicate interplay between the onsite Coulomb interaction, the coupling strength between the impurity and bath states, the position of the bare impurity level (or equivalently the filling) and the positions of the bath energy levels (the dispersion).",
"Within the static mean-field approximation the criterion for a local moment to exist is $U/\\Gamma >\\pi $ .",
"Here $2\\Gamma =\\pi V^{2} \\rho (E_{F})$ is the effective hybridization, i.e.",
"broadening of the impurity $E_{d}$ level, where $\\rho (E_{F})$ is the density of impurity states at the Fermi level.",
"From this criterion it is clear that a large onsite Coulomb interaction and small coupling between the impurity and bath are favorable for a local moment to exist.",
"It is well known that Kondo physics occurs for the model described by Eq.",
"REF at half-filling and in the limit where the hybridization can be treated perturbatively.",
"More precisely, it can be shown that in this regime the virtual spin-flip scatterings of the bath electrons against the local impurity moment are the dominant processes occuring in the system.",
"At low enough temperatures, below the Kondo temperature, they start to screen the local moment.",
"For half-filling and by treating the hybridization perturbatively, the Kondo temperature $T_{K}$ can be estimated via ${\\begin{array}{c}T_{L}=U\\Bigg (\\frac{\\Gamma }{2U}\\Bigg )^{1/2}\\exp \\Bigg [\\frac{-\\pi |E_{d}||E_{d}+U|}{2U\\Gamma }\\Bigg ],\\end{array}}$ where the Kondo temperature is equal to $T_{K}=0.041T_{L}$ [32] The Kondo effect for very small systems has been been the subject of study already for several decades, e.g.",
"for quantum dots.",
"Theoretically, the Kondo effect was predicted to take place in quantum dots.",
"[33], [34], [35] A few years later experiments confirmed these predictions.",
"[36], [37] Although less studied within the Anderson impurity model, the situation of a magnetic impurity resolved in a semi-conductor or equivalently a bulk host with a band gap has also been addressed [13], [38].",
"It has been demonstrated that a local magnetic moment on the impurity is stabilized by the introduction of a band gap.",
"In more detail a local moment can be formed even when the criterion above is not satisfied.",
"Furthermore, the magnitude of the local moment increases with increasing band gap.",
"In Ref.",
"Lau2014 the investigation of the Anderson impurity model for an impurity in a gapped host is extended to the situation of a finite sized host.",
"Interestingly, it was found that on average the local moment grows with increasing band gap (HOMO-LUMO gap).",
"Here on average should be understood as the local moment averaged over a number of random configurations of the discrete host energy levels for a fixed band gap.",
"Further, it has been shown that in the regimes, where $V\\ll E_{g}$ or $V\\gg E_{g}$ , the magnitude of the local moment merely depends on the size of the band gap ($E_{g}$ ) and not on the exact positions of the discrete energy levels of the host.",
"Namely, for $V\\ll E_{g}$ the effect of the hybridization is small no matter what the exact arrangement of the host energy levels is, while for $V\\gg E_{g}$ the impurity level hybridizes with all host levels anyway.",
"However, for the regime in between, $V\\sim E_{g}$ , the local moment strongly depends on the exact positions of the host energy levels.",
"In Ref.",
"Lau2015 these findings were successfully used to interpret the experimentally observed magnetic moments of Au$_{x}$ Cr clusters.",
"For example, the trend of the Au$_{x}$ host band gap was found to exactly follow that of the magnetic moment of the Au$_{x}$ Cr clusters."
],
[
"Computational details",
"In this work we performed for the Nb$_{x}$ Co clusters an analyses based on the Anderson impurity model in the same spirit as in Ref. Lau2015.",
"For this purpose the density functional theory (DFT) [39], [40] is employed within the full-potential linear muffin-tin orbital method [41].",
"The local density approximation (LDA) exchange-correlation functional is used in the formulation of Perdew and Wang [42].",
"For the Nb atoms the main valence basis functions were 4d, 5s and 5p states, while 4s and 4p states were treated as pseudo-core in a second energy set [41].",
"In case of Co, the 3s and 3p states were treated as pseudo-core, and the 3d, 4s and 4p states as the main valence states.",
"In all calculations the valence states were treated scalar relativistically (without spin-orbit coupling).",
"Since the employed DFT code works in k-space, a supercell approach was used.",
"A large unit cell of at least 14-Å dimensions was used in order to prevent the interaction between clusters of different unit cells.",
"In these calculations the $\\Gamma $ point was the only k-point considered.",
"The geometry of the clusters is obtained from the comparison of the experimental and DFT vibrational spectra performed in Section .",
"More precisely, the ground state geometries (3,1)A M=3, (4,1)A M=3, (5,1)C M=0, (6,1)B M=1, (7,1)B M=0 and (9,1)B M=2 are taken.",
"Note that for Nb$_{9}$ Co the structure with C$_{4v}$ symmetry is chosen.",
"Namely for a magnetic cluster the Jahn-Teller distortion should be counteracted by the exchange interaction between Nb and Co.",
"The effective onsite Coulomb repulsion $U$ between the 3d electrons of the Co impurity is obtained from DFT calculations in conjunction with the random phase approximation (RPA) within the full-potential linearized augmented plane wave (FLAPW) method [43].",
"All these calculations are performed with the GGA functional as formulated by Perdew, Burke and Ernzerhof [28].",
"Here a large unit cell of at least 12-Å dimensions is used and also only the $\\Gamma $ point is considered.",
"Further, the plane wave cuttoff is 4.0 Bohr$^{-1}$ .",
"The actual RPA calculations are performed with the SPEX code, which uses the DFT calculations as an input [44].",
"The SPEX code uses the Wannier90 library to construct the maximally localized Wannier functions [45], [46].",
"For this construction five 3d states and one 4s state are used for the Co atom."
],
[
"Results: Anderson impurity model",
"Table REF presents for each Nb$_{x}$ Co cluster its characteristic parameters related to the Anderson impurity model.",
"The center of gravity of the Co 3d projected density of states $E_{d}$ and its weighted standard deviation $\\Gamma $ are shown.",
"Also are shown, the band gap (HOMO-LUMO gap) $E_{g}$ of the bare Nb$_{x}$ host for the geometry it has in the full Nb$_{x}$ Co cluster and the effective onsite Coulomb interaction $U$ between the Co 3d electrons.",
"Although Eq.",
"REF is strictly speaking only valid for an impurity in a non-magnetic metallic host at half-filling in the limit of small hybridization, we employed it to obtain a rough estimate of the Kondo temperature $T_{K}$ for the Nb$_{x}$ Co clusters.",
"For convenience also the experimentally observed magnetic moment (see Fig.",
"REF (d)) is presented in the last column.",
"As can be observed, the impurity energy level $E_{d}$ and its broadening $2\\Gamma $ are more or less constant as a function of cluster size.",
"On the other hand, the band gap of the bare Nb$_{x}$ host strongly fluctuates as function of cluster size, while the effective onsite Coulomb repulsion slowly decreases as function of cluster size.",
"Table: The Co impurity energy level E d E_{d}, broadening of the impurity level 2Γ2\\Gamma , energy gap E g E_{g} (HOMO-LUMO gap) of the bare Nb x _{x} host and the effective onsite Coulomb interaction UU between the Co impurity 3d electrons within RPA for different Nb x _{x}Co clusters.",
"The sixth column contains a rough estimate of the Kondo temperature T K T_{K} obtained from Eq. .",
"For convenience also the experimentally observed total magnetic moment in μ B \\mu _{B} is presented in the last column.As naively expected from Refs.",
"Lau2014,Lau2015, the magnitude of the local Co moment should follow the trend of the band gap of the isolated host as a function of cluster size.",
"In other words a small band gap is expected for the clusters with zero magnetic moment, while a larger band gap is expected for the magnetic clusters.",
"It is clear that this expectation is not verified by the results in Table REF .",
"For example, magnetic Nb$_{3}$ Co and Nb$_{6}$ Co have a very small band gap compared with the non-magnetic Nb$_{5}$ Co and Nb$_{7}$ Co clusters.",
"It is also interesting to have an inspection of the criterion for the existence of a local moment in the Anderson impurity model.",
"In case of an impurity with degenerate orbitals the criterion stated above is slightly modified into $(U+4J)/\\Gamma >\\pi $ , where $J$ is the Hund exchange coupling between the impurity electrons.",
"Even when the contribution of $J$ is neglected, it is clear from Table REF that the criterion is satisfied for all clusters.",
"It was already known from Ref.",
"Lau2015 that a magnetic impurity moment can occur even when the criterion above is not satisfied.",
"However, it appears that the other way around is also possible, i.e.",
"there is no magnetic moment even when the criterion is satisfied.",
"Only considering the band gap of the bare host did not provide an explanation for some Nb$_{x}$ Co clusters being magnetic and others non-magnetic.",
"On the other hand for Au$_{x}$ Cr it perfectly predicted the magnetic moment as function of cluster size.",
"The reason is that the Au$_{x}$ host is inert, i.e.",
"there is only a small coupling between the Cr impurity states and Au$_{x}$ host states.",
"Therefore, Au$_{x}$ clusters can be considered to be in the regime $V\\ll E_{g}$ , where the size of the local moment solely depends on the band gap of the host and not on the exact positions of its energy levels.",
"This is also apparent from the observation that the local moment of the Cr impurity is barely quenched in the Au$_{x}$ Cr clusters.",
"Contrary for the Nb$_{x}$ Co clusters the magnetic moment strongly fluctuates as function of cluster size, which hints in the direction that we are in the regime $V\\sim E_{g}$ .",
"Unfortunately, this cannot be directly verified from the parameters presented in Table REF .",
"Namely, $\\Gamma $ corresponds to the effective hybridization in which both $V$ and the density of states of the host are involved.",
"However, indirectly one could argue that the Nb$_{x}$ Co clusters are in the $V\\sim E_{g}$ regime.",
"From Ref.",
"Lau2014 it is know that for $V\\gg E_{g}$ the impurity moment is almost completely quenched, while for $V\\ll E_{g}$ the moment should follow the size of the band gap.",
"Since neither of the two is in agreemen with the results of Table REF , it is expected that the Nb$_{x}$ Co clusters are in the $V\\sim E_{g}$ regime.",
"In the $V\\sim E_{g}$ regime the exact positions of the host energy levels are known to be important.",
"It would be helpful to be a bit more specific and to have a feeling for which host energy levels are important.",
"For example, intuitively one would expect only host states within a range of about $V$ around the Fermi level (chemical potential) to be important.",
"In order to verify this expectation we investigated the Anderson impurity model for an impurity with a single orbital coupled to 6 spin degenerate bath states.",
"The impurity energy level and onsite Coulomb repulsion were chosen such that the single and double occupied isolated impurity states are symmetric around the chemical potential, e.g.",
"$E_{d}=-1$ and $U=3$ .",
"Further, a total occupation (impurity plus bath) of 7 electrons was considered.",
"The Anderson impurity model was solved exactly via exact diagonalization.",
"Note that in Ref.",
"Lau2014 a tight binding approximation was employed.",
"In Table REF the influence of different arrangements of 3 occupied and 3 unoccupied (occupied and unoccupied refers to the bare bath situation) spin degenerate host states on the impurity magnetic moment is presented.",
"For all calculations $V=0.1$ is taken.",
"The columns 2 to 7 correspond to the positions of the spin degenerate occupied and unoccupied host states, column 8 contains the band gap and the last column the magnetic moment on the impurity.",
"From this table it is clear that indeed only host states within a range of $V$ are important in terms of the magnitude of the impurity magnetic moment.",
"For example, a comparison of the first 5 calculations shows this.",
"Also a comparison of the calculations 3, 8, 9 and 10 clearly indicates this.",
"Another (trivial) observation can be made from calculations 4, 6 and 7.",
"For these calculations the band gap is the same and the only difference is in the positions of the HOMO and LUMO levels with respect to the chemical potential.",
"It appears that these exact positions are unimportant as long as the band gap is fixed.",
"Finally, from calculations 3, 8, 9 and 10 it can also be concluded that not only the band gap itself, but also the number of states (density of states) involved is important.",
"Table: The impurity magnetic moment (last column) for different arrangements of the occupied (columns 2 to 4) and unoccupied (columns 5 to 7) spin degenerate host states.",
"The column with E g E_{g} contains the band gap (HOMO-LUMO gap).Since the coupling strength $V$ , the band gap and host density of states are important for the impurity magnetic moment, it would be natural to study the hybridization function corresponding to the Co 3d electrons.",
"Namely, the imaginary part of the hybridization function is proportional to the coupling strength $V$ squared and the host density of states.",
"Furthermore, in the regime $V\\sim E_{g}$ the influence of the coupling of the impurity with the host cannot be considered as a (small) perturbation like in Au$_{x}$ Cr.",
"This coupling is already taken into account explicitly within the hybridization function.",
"For details on how the hybridization function projected on the Co 3d states is obtained, the reader is referred to Ref.",
"[47] In short the Nb$_{x}$ Co cluster is first calculated self-consistently within DFT.",
"Then, from the obtained Kohn-Sham eigenstates and energies, the corresponding Green's function is constructed.",
"Next, this Green's function is projected on the 3d states.",
"This projected Green's function $G_{mm^{\\prime }}(E)$ and the hybridization function of the Co 3d states $\\Delta _{mm^{\\prime }}(E)$ are related by ${\\begin{array}{c}G_{mm^{\\prime }}(E)=\\Bigg [ E - \\epsilon _{mm^{\\prime }}+\\mu - \\Delta _{mm^{\\prime }}(\\omega )\\Bigg ], \\\\\\text{with }\\Delta _{mm^{\\prime }}(E)=\\sum _{k} \\frac{V^{*}_{km}V_{km^{\\prime }}}{E-\\epsilon _{k}+\\mu }.\\end{array}}$ Here, $E$ is the energy, $V_{km}$ represent the coupling strength of the impurity state $m$ with bath (host) state $k$ , $\\epsilon _{mm^{\\prime }}$ is obtained from the local projection of the DFT Kohn-Sham Hamiltonian and $\\epsilon _{k}$ corresponds to the energies of the bath states.",
"From the expression of the hybridization function in terms of the coupling strengths and bath energy levels, it is clear that different choices of them can lead to the same hybridization function and thus Anderson impurity problem.",
"Therefore, unless the $V_{km}$ matrix elements are computed directly, it is hard to explicitly determine whether Nb$_{x}$ Co corresponds to the $V\\sim E_{g}$ regime.",
"However, this determination is not necessary to understand the physical origin of the presence or absence of magnetism in the Nb$_{x}$ Co clusters.",
"Figure: The imaginary part of the hybridization function for the Co 3d electrons for the different Nb x _{x}Co clusters.From the discussions above we know that the HOMO-LUMO gap, the density of states at the HOMO and LUMO levels, the coupling $V$ between the impurity and host states, and the onsite Coulomb repulsion $U$ are important for the impurity magnetic moment.",
"The first three are captured by the (imaginary part of the) hybridization function.",
"Therefore, in Fig.",
"REF the imaginary part of the total (trace of $\\Delta _{mm^{\\prime }}(E)$ ) hybridization function for the Co 3d states is shown for the different Nb$_{x}$ Co clusters.",
"From this figure an estimate can be made of the coupling strength $V$ .",
"Assuming that the peak of Nb$_{3}$ Co at -0.25 eV is due to the coupling with only one bath state, would require a $V$ of about 0.37 eV.",
"Therefore, the hybridization function is only plotted roughly in this range around the chemical potential (zero energy).",
"From the model calculations presented in Table REF it is expected that a small HOMO-LUMO gap and large hybridization around the HOMO and LUMO levels is unfavourable for a magnetic moment.",
"A discussion solely based on the hybridization functions of Fig.",
"REF is complicated by the fact that the onsite Coulomb repulsion is not constant over the range of clusters investigated.",
"However, for two clusters differing only by one Nb atom in size the difference in the onsite Coulomb interaction is small.",
"Therefore, in the following the hybridization functions will be compared cluster for cluster.",
"From Fig.",
"REF it appears that Nb$_{3}$ Co has a much stronger hybridization around the chemical potential (zero energy) than Nb$_{4}$ Co. More precisely for Nb$_{3}$ Co there is a peak at about -0.25 eV and 0.1 eV, while Nb$_{4}$ Co has a peak at about -0.5 eV and a very tiny one at 0.05 eV.",
"Since the gap between the peaks is larger and the total height of the peaks is smaller for Nb$_{4}$ Co, a larger magnetic moment is expected for Nb$_{4}$ Co compared to Nb$_{3}$ Co.",
"This is comfirmed by the magnetic deflection experiment (see last column of Table REF and Fig.",
"REF ).",
"By going from magnetic Nb$_{4}$ Co to non-magnetic Nb$_{5}$ Co, it is clear that there is a huge increase of hybridization around the chemical potential.",
"Therefore, in addition with a smaller onsite Coulomb interaction it is indeed expected that Nb$_{5}$ Co has a much smaller tendency to be magnetic than Nb$_{4}$ Co (and Nb$_{3}$ Co).",
"Then, by going from non-magnetic Nb$_{5}$ Co to magnetic Nb$_{6}$ Co, there is a huge decrease of hybridization around the chemical potential.",
"More precisely, there is a huge increase from about 0.15 eV to 1.0 eV in the separation between the first peak below and above the chemical potential.",
"Thus, in accordance with experiment Nb$_{6}$ Co is expected to have a larger tendency to be magnetic than Nb$_{5}$ Co. Next, magnetic Nb$_{6}$ Co and non-magnetic Nb$_{7}$ Co will be compared.",
"As expected the hybridization around the chemical potential is larger for Nb$_{7}$ Co than for Nb$_{6}$ Co. Interestingly, Nb$_{7}$ Co has a similar hybridization around the chemical potential as Nb$_{3}$ Co.",
"However, Nb$_{3}$ Co has an onsite Coulomb interaction which is 1.4 eV larger than for Nb$_{7}$ Co.",
"Finally, non-magnetic Nb$_{7}$ Co and magnetic Nb$_{9}$ Co are compared.",
"Although Nb$_{9}$ Co has a quite large peak at about -0.15 eV, the difference between the first peak below and above the chemical potential is much larger.",
"Therefore, the effective hybridization around the chemical potential is as expected smaller for Nb$_{9}$ Co than for Nb$_{7}$ Co. To conclude, for Nb$_{3}$ Co to Nb$_{7}$ Co and Nb$_{9}$ Co the effective hybridization around the chemical potential is in agreement with the experimentally observed magnetic behavior.",
"Above we performed an analysis based on the Anderson impurity model in order to explain the experimentally observed magnetic behavior.",
"From an inspection of the hybridization function and the onsite Coulomb repulsion a trend in agreement with experiment could be predicted.",
"However, based on these observations it cannot be explained whether the moment is (completely) quenched or Kondo screened.",
"Therefore, we made an estimate of the Kondo temperature for the clusters from Eq.",
"REF , which are presented in the sixth column of Table REF .",
"In case the non-magnetic clusters occur due to a complete Kondo screening, higher Kondo temperatures are expected for the non-magnetic clusters than for the magnetic clusters.",
"From Table REF it can be observed that the results are not in accordance with this expectation.",
"For example, the highest Kondo temperatures are observed for magnetic Nb$_{3}$ Co and Nb$_{4}$ Co. Further, non-magnetic Nb$_{5}$ Co and Nb$_{7}$ Co have a smaller Kondo temperature than magnetic Nb$_{6}$ Co.",
"In addition we searched for signatures of the Kondo effect in the Nb$_{x}$ Co clusters from the experimental side.",
"For this purpose the temperature dependence of the magnetic deflection experiments was investigated.",
"In case of the Kondo effect it is expected that by approaching the Kondo temperature from below the screening of the local Co moment reduces.",
"An inspection of Table REF shows that Nb$_{5}$ Co has a Kondo temperature of 68K and Nb$_{7}$ Co of 37K.",
"However, even for temperatures up to 70K both clusters still appeared to be strictly non-magnetic.",
"These results indeed indicate that the Kondo effect is not responsible for Nb$_{5}$ Co and Nb$_{7}$ Co to appear non-magnetic."
],
[
"Conclusion",
"In this work we performed magnetic deflection experiments on Co doped Nb clusters from which we made the interesting observation that some clusters are strongly magnetic, while others are non-magnetic.",
"Further, it appeared that the magnetic behavior of the clusters could be divided into two regimes.",
"For Nb$_{x}$ Co clusters with $x>=14$ , the magnetic to non-magnetic behavior is exactly determined by having an odd or even number of atoms in the cluster, i.e.",
"having an unpaired electron or not.",
"Note that this behavior was also observed for pure Nb clusters.",
"Then, in the region $x<14$ strong fluctuations in the magnetic moment as function of cluster size are observed in contradiction with the odd/even behavior described above.",
"There are in principle two possible explanations for some clusters being non-magnetic.",
"Either the local moment at the Co site is completely quenched or it is screened by the delocalized electrons of the cluster, i.e.",
"the Kondo effect.",
"In order to reveal the physical origin, we conducted a combined theoretical and experimental investigation.",
"First, we made a comparison of the experimental vibrational spectra with those obtained from a DFT study.",
"This served two purposes.",
"It provides the ground state geometry of the clusters.",
"Further, due to the dependence of the vibrational spectrum on the magnetic moment, the performance of DFT in predicting the magnetic moments can be investigated.",
"We found that not for all clusters it could be determined which calculated vibrational spectrum has the best agreement with experiment.",
"However, for those it could, we found that the DFT magnetic moments were considerably smaller than those obtained from the magnetic deflection experiments.",
"This is due to a neglect of the orbital moments in our DFT calculations and underestimation of the spin moments within DFT.",
"Second, with the obtained ground state structures as an input we performed an analyses based on the Anderson impurity model.",
"It appears that the non-magnetic clusters are due to a complete quenching of the local Co moment and not due to the Kondo effect.",
"In addition, the magnetic behavior of the Nb$_{x}$ Co clusters can be understood from an inspection of their electronic structure.",
"Here magnetism is favored when the effective hybridization around the chemical potential is small, while the absence of magnetism is signalled by a large effective hybridization around the chemical potential."
],
[
"Acknowledgments",
"The Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) and SURFsara are acknowledged for the usage of the LISA supercomputer and their support.",
"L.P. and M.I.K.",
"acknowledges a support by European ResearchCouncil (ERC) Grant No.",
"338957."
]
] | 1709.01534 |
[
[
"Language Modeling by Clustering with Word Embeddings for Text\n Readability Assessment"
],
[
"Abstract We present a clustering-based language model using word embeddings for text readability prediction.",
"Presumably, an Euclidean semantic space hypothesis holds true for word embeddings whose training is done by observing word co-occurrences.",
"We argue that clustering with word embeddings in the metric space should yield feature representations in a higher semantic space appropriate for text regression.",
"Also, by representing features in terms of histograms, our approach can naturally address documents of varying lengths.",
"An empirical evaluation using the Common Core Standards corpus reveals that the features formed on our clustering-based language model significantly improve the previously known results for the same corpus in readability prediction.",
"We also evaluate the task of sentence matching based on semantic relatedness using the Wiki-SimpleWiki corpus and find that our features lead to superior matching performance."
],
[
"Introduction",
"Predicting reading difficulty of a document is an enduring problem in natural language processing (NLP).",
"Approaches based on shallow-length features of text date back to 1940s [9].",
"Remarkably, they are still being used and extended with more sophisticated techniques today.",
"In this paper, we use word embeddings to compose semantic features that are presumably beneficial for assessing text readability.",
"Encouraged by the recent literature in applying language models for better prediction, we aim to build a clustering-based language model using word vectors learned from corpora.",
"The resulting model is expected to reveal semantics at a higher level than word embeddings and provide discriminative features for text regression.",
"As pioneering work in text difficulty prediction, Flesch [9] explored on shallow-length features computed by averaging the number of words per sentence and the number of syllables per word.",
"The intent was to capture sentence complexity with the number of words, and word complexity with the number of syllables.",
"Chall [5] claimed the reading difficulty as a linear function of shallow-length features.",
"Kincaid [13] introduced a linear weighting scheme that became the most common measure of reading difficulty based on shallow-length features.",
"More sophisticated algorithms that measure semantics by word frequency counts and syntax from sentence length [22] and language modeling [6] have shown significant performance gains over classical methods.",
"Modern approaches treat text difficulty prediction as a discriminative task.",
"Schwarm et al.",
"[21] presented text regression based on support vector machine (SVM).",
"Peterson et al.",
"[20] used both SVM classification and regression for improvement.",
"NLP researchers went beyond the shallow features and looked into learning complex lexical and grammatical features.",
"Flor et al.",
"[10] proposed an algorithm that measures lexical complexity from word usage.",
"Vajjala et al.",
"[24] formulated semantic and syntactic complexity features from language modeling, which resulted some improvement.",
"Class-based language models, trained on the conditional probability of a word given the classes of its previous words, were commonly used in the literature [23], [3].",
"Brown clustering [4], a popular class-based language model, can learn hierarchical clusters of words by maximizing the mutual information of word bigrams.",
"Our text learning is founded on word embeddings.",
"Bengio et al.",
"[1] proposed an early neural embedding framework.",
"Mikolov et al.",
"[18] introduced the Skip-gram model for efficient training with large unstructured text, and Paragraph Vector [14] and character $n$ -grams [2], all of which we use for our implementation in this paper, followed on.",
"Most word embedding algorithms build on the distributional hypothesis [11] that word co-occurrences imply similar meaning and context.",
"Word embeddings span a high-dimensional semantic space where the Euclidean distance between word vectors measures their semantic dissimilarity [12].",
"Under the Euclidean semantic space hypothesis, we argue that clustering of word vectors should unveil a clustering-based language model.",
"In particular, we propose two clustering methods to construct language models using Brown clustering and $K$ -means with word vectors.",
"Our methods are language-independent and data-driven, and we have empirically validated their superior performance in text readability assessment.",
"Specifically, our experiment on the Common Core Standards corpus reveal that the language model learned by $K$ -means significantly improves readability prediction against contemporary approaches using the lexical and syntactic features.",
"In another experiment with the Wiki-SimpleWiki corpus, we show that our features can correctly identify sentence pairs of the similar meaning but written in different vocabulary and grammatical structure.",
"For text with easy readability, difference in reading difficulty is resulted from different document length, sentence structure, and word usage.",
"For documents at higher reading levels, however, features with richer linguistic context about domain, grammar, and style are known to be more relevant.",
"For example, based on shallow features, “To be or not to be, that is the question” would likely be considered easier than “I went to the store and bought bread, eggs, and bacon, brought them home, and made a sandwich.” Therefore, we need to capture all semantic, lexical, and grammatical features for distinguishing documents at all levels.",
"We organize the rest of this paper as follows.",
"In Section 2, we describe our approach centered around neural word embedding and probabilistic language modeling.",
"We will explain each component of our approach in detail.",
"Section 3 presents our experimental methodology for evaluation.",
"We will also discuss the empirical results.",
"Section 4 concludes the paper.",
"Figure: Our system pipeline"
],
[
"Approach",
"We review embedding schemes, clustering algorithms, and regression method used in the paper, and describe our overall pipeline."
],
[
"Word embeddings",
"Skip-gram.",
"Mikolov et al.",
"[18] proposed the Skip-gram method based on a neural network that maximizes $\\small \\frac{1}{T} \\sum _{t=1}^{T} \\sum _{-c \\le j \\le c, j\\ne 0} \\log \\,p(w_{t+j}|w_t)$ where the training word sequence $w_1, w_2,\\dots , w_T$ has a length $T$ .",
"With $w_t$ as the center word, $c$ is the training context window.",
"The conditional probability can be computed with the softmax function $p(w_{t+j}|w_t) = \\frac{e^{s(w_{t},w_{t+j})}}{\\sum _{w^{\\prime }}e^{s(w_{t},w^{\\prime })}}$ with the scoring function $s(w_{t},w_{t+j}) = \\mathbf {v}_{w_t}^\\top \\cdot \\mathbf {v}_{w_{t+j}}$ .",
"The embedding $\\mathbf {v}_{w_t}$ is a vector representation of the word $w_t$ .",
"Bag of character $n$ -grams.",
"Bojanowski et al.",
"[2] proposed an embedding method by representing each word as the sum of the vector representations of its character $n$ -grams.",
"To capture the internal structure of words, a different scoring function is introduced $\\small s(w_t,w_{t+j}) = \\sum _{g \\in G_{w_t}} \\mathbf {z}_g^\\top \\cdot \\mathbf {v}_{w_{t+j}}.$ Here, $G_{w_t}$ is the set of $n$ -grams in $w_t$ .",
"A vector representation $\\mathbf {z}_g$ is associated to each $n$ -gram $g$ .",
"This approach has an advantage in representing unseen or rare words in corpus.",
"If the training corpus is small, character $n$ -grams can outperform the Skip-gram (of words) approach."
],
[
"Paragraph embedding",
"Distributed bag-of-words.",
"While Skip-gram and character $n$ -grams can embed a word into a high-dimensional vector space, we eventually need to compute a feature vector for the whole document.",
"Le et al.",
"[15] introduced Paragraph Vector that learns a fixed-length vector representation for variable-length text such as sentences and paragraphs.",
"The distributed bag-of-words version of Paragraph Vector has the same architecture as the Skip-gram model except that the input word vector is replaced by a paragraph token."
],
[
"Clustering",
"Brown clustering.",
"Brown et al.",
"[4] introduced a hierarchical clustering algorithm that maximizes the mutual information of word bigrams.",
"The probability for a set of words $w_1, w_2,\\dots , w_T$ can be written as $\\small \\prod ^{T}_{t=1}p(w_{t}|C(w_{t}))\\,p(C(w_{t})|C(w_{t-1}))$ where $C(\\cdot )$ is a function that maps a word to its class, and $C(w_0)$ is a special start state.",
"Brown clustering hierarchically merges clusters to maximize the quality of $C$ .",
"The quality is maximized when mutual information between all bigram classes are maximized.",
"Although Brown clustering is commonly used, a major drawback is its limitation to learn only bigram statistics.",
"K-means.",
"Because word embeddings span a semantic space, clusters of word embeddings should give a higher semantic space.",
"We perform $K$ -means on word embeddings.",
"The resulting clusters are word classes grouped in semantic similarity under the Euclidean metric constraint.",
"Given word embeddings $\\mathbf {v}_{w_1}, \\mathbf {v}_{w_2},\\dots , \\mathbf {v}_{w_T}$ learned from a corpus, we find the cluster membership for a word $w_t$ as $\\small k_{\\mathbf {v}_{w_t}} = \\arg \\min _j \\Vert \\mathbf {c}^{(j)} - \\mathbf {v}_{w_t} \\Vert ^2_2$ where $\\mathbf {c}^{(j)}$ is the $j$ th cluster centroid."
],
[
"Regression",
"We consider linear support vector machine (SVM) regression $\\small \\min \\frac{1}{2} \\left\\Vert \\mathbf {w} \\right\\Vert ^2 \\\\ \\mbox{s.t.",
"}~-\\epsilon \\le y^{(i)} - (\\mathbf {w}\\cdot \\mathbf {x}^{(i)} + b) \\le \\epsilon ~~\\forall i$ where the regressed estimate $\\mathbf {w}^\\top \\cdot \\mathbf {x}^{(i)} + b$ for $i$ th input $\\mathbf {x}^{(i)}$ is optimized to be bound within an error margin $\\epsilon $ from the ground-truth label $y^{(i)}$ .",
"SVM trains a bias term $b$ to better compensate regression errors along the weight vector $\\mathbf {w}$ .",
"We train SVM regression using feature vectors formed on word embedding and clustering to predict the readability score."
],
[
"Pipeline",
"Figure REF depicts our prediction pipeline using word clusters precomputed by K-means on word embeddings.",
"When a document of an unknown readability level arrives, we preprocess tokenized text input and compute word vectors using trained word embeddings.",
"We compute cluster membership on word vectors, followed by average pooling.",
"For cluster membership, we perform the 1-of-$K$ hard assignment for each word in the document.",
"Then we compute the histogram of cluster membership.",
"By representing features in terms of histograms our approach can naturally address documents of varying lengths.",
"After some post-processing (e.g., unit-normalization), we regress the readability level.",
"Table: Baseline regression resultsTable: Clustering-based language model resultsTable: Performance comparison summary"
],
[
"Experimental Evaluation",
"Following Vajjala et al.",
"[24], we evaluate readability level prediction with the Common Core Standards corpus [7] and sentence matching with the Wiki-SimpleWiki corpus [25]."
],
[
"Common Core Standards Corpus",
"This corpus of 168 English excerpts are available as the Appendix B of the Common Core Standards reading initiative of the US education system.",
"Each text excerpt is labeled with a level in five grade bands (2-3, 4-5, 6-8, 9-10, 11+) as established by educational experts.",
"Grade levels 2.5, 4.5, 7, 9.5, and 11.5 are used as ground-truth labels.",
"We cut the corpus into train and test sets in an uniformly random 80-20 split, resulting 136 documents for training and 32 for test.",
"Evaluation metric.",
"For fair comparison with other work, we adopt Spearman's rank correlation and Pearson correlation computed between the ground-truth label and regressed value.",
"Preprocessing.",
"We convert all characters to lowercase, strip punctuations, and remove extra whitespace, URLs, currency, numbers, and stopwords using the NLTK Stopwords Corpus [17].",
"Features.",
"There are two levels of features.",
"At the word-vector level, we perform weighted average pooling of word embeddings to compose per-document feature vector.",
"We have tried tf-idf and uniform weighting schemes.",
"Brown clustering of words yields the word-vector level features as well.",
"On the contrary, $K$ -means clustering of word vectors yields higher-level features in terms of cluster structures.",
"For Brown and $K$ -means, we replace each word in a document with its numeric cluster ID and compute the histogram of cluster membership as per-document feature vector.",
"For histogram computing, we consider binary (on/off) and traditional bin counts.",
"Word and paragraph embeddings.",
"We use word2vec for the Skip-gram word embeddings.",
"We have first tried out the wiki and ap-news pretrained word2vec models.",
"Eventually, we use TensorFlow to train word2vec model from the Common Core Standards corpus.",
"We have optimized the word-vector dimension hyperparameter between 32 and 300.",
"We use fastText for character $n$ -gram word embeddings.",
"Similar to our word2vec experiment, we have tried the wiki and ap-news pretrained models for fastText before training our own.",
"While training, we use the negative sampling loss function with word-vector dimensions 32 to 300 and context window size of 5.",
"We use doc2vec that implements Paragraph Vector.",
"We have not trained our own doc2vec model and opted for the wiki and ap-news pretrained doc2vec models.",
"Brown clustering.",
"We use an open-source implementation by Liang et al. [16].",
"We have fine-tuned the number of cluster hyperparameter by varying between 10 and 200.",
"K-means clustering.",
"After embedding all words in each document, we run $K$ -means.",
"We fine-tune $K$ within 10 to 200.",
"SVM regression.",
"We use LIBLINEAR [8] for SVM regression and configure as the $\\ell _2$ -regularized $\\ell _2$ -loss linear solver with unit bias.",
"The SVM complexity hyperparameter is optimized between $10^{-5}$ and 1.",
"Our choice of linear SVM is made after also trying out a single-layer perceptron neural network regression with the number of neurons in 0.1x to 1x the feature vector dimension.",
"Results and discussion.",
"Our baseline results with pretrained models are shown in Table REF .",
"Bag-of-words performs poorly, and word2vec performs better than doc2vec.",
"We suspect that the benefit of doc2vec is not realized on this corpus due to its limited length.",
"We find fastText superior over word2vec and doc2vec.",
"Pretrained wiki outperforms ap-news.",
"We only report wiki results.",
"Table REF presents results on clustering-based language models: Brown clustering on words and $K$ -means on trained word vectors using the corpus.",
"Presented correlation values are for binary (inside parenthesis) and traditional bin counts.",
"While binary counters could be robust against ambiguities resulting from repeated texts in a document, this advantage is not present in the corpus we use here.",
"Brown clustering on words has similar performance to baseline embedding schemes.",
"The comparable performances are expected, because both Brown clustering and the baseline embedding schemes are performed on the raw words.",
"We can improve performance further with $K$ -means clustering on word vectors.",
"Rather than training word vector models on wiki, training with the Common Core Standards corpus improves the correlation.",
"fastText with $K$ -means works the best.",
"Table REF presents a summary that compares performances of our approach and the previous work.",
"Flor et al.",
"[10] implemented prediction scheme based on lexical tightness and compared their method against baselines such as text length and Flesch-Kincaid [13] in Pearson correlation.",
"Nelson et al.",
"[19] wrote a summary of commercial softwares' performances in Spearman correlation.",
"Most recently, Vajjala et al.",
"[24] implemented a scheme that uses lexical, syntactic, and psycholinguistic features.",
"Our highest correlation for Spearman is 0.83, and 0.82 for Pearson, both of which are better than the best case reported by the previous work."
],
[
"Wiki-SimpleWiki Corpus",
"We demonstrate our features derived from clustering of word embeddings are effective in another application concerning sentence matching.",
"The corpus for this application consists of 108,016 aligned sentence pairs of the same meaning drawn from (ordinary) Wikipedia and Simple Wikipedia.http://simple.wikipedia.org Simple Wikipedia uses basic vocabulary and less complex grammar to make the content of Wikipedia accessible to audiences of all reading skills.",
"Task and metric.",
"We evaluate whether or not the feature vector for an ordinary sentence formed by the proposed feature scheme can correctly predict its counterpart sentence.",
"We sample 1,000 sentence pairs.",
"Among all 1,000 pairs, we compute the probability $P_N$ that ordinary sentences and their simple counterparts are $N$ nearest neighbors in the semantic space.",
"We vary $N = 1$ to 4.",
"Features.",
"We use our best feature scheme, word embedding by fastText and $K$ -means, found in Section 3.1.",
"To compute sentence embedding, we average-pool all word embeddings in the sentence.",
"Results and discussion.",
"As Table REF shows, using only the nearest neighbor, we already achieve $P_N = 0.959$ ; as $N$ grows, we can contain different sentences of the same meaning with probability approaching 1.",
"This implies that despite differences in grammatical structure and word usage, when underlying semantics are shared between two sentences, they are mapped closely each other in the feature space.",
"Table: Average probability P N P_N that a Wiki sentence and its SimpleWiki counterpart are within the NNth nearest neighbors in the semantic feature vector space"
],
[
"Conclusion",
"Word vectors learned on neural embedding exhibit linguistic regularities and patterns explicitly.",
"In this paper, we have introduced a regression framework on clustering-based language model using word embeddings for automatic text readability prediction.",
"Our experiments with the Common Core Standards corpus demonstrate that features derived by clustering word embeddings are superior to classical shallow-length, bag-of-words, and other advanced features previously attempted on the corpus.",
"We have further evaluated our approach on sentence matching using the Wiki-SimpleWiki corpus and showed that our method can effectively capture semantics even when sentences are written with different vocabulary and grammatical structures.",
"For future work, we plan to continue our experiments with more diverse languages and larger datasets."
],
[
"Acknowledgments",
"This work is supported by the MIT Lincoln Laboratory Lincoln Scholars Program and in part by gifts from the Intel Corporation and the Naval Supply Systems Command award under the Naval Postgraduate School Agreements No.",
"N00244-15-0050 and No.",
"N00244-16-1-0018."
]
] | 1709.01888 |
[
[
"A deterministic mathematical model for the spread of two rumors"
],
[
"Abstract In this paper we propose a deterministic mathematical model that attempts to explain the propagation of a rumor using SIRS type epidemiological models with temporary immunity and nonlinear incidence rate.",
"In particular, we speculate about the dissemination of information when the so-called \"complex networks\" are used.",
"The effect of introducing a second rumor, inspired by a vaccination model, in the same population of individuals, which will try to counteract the effect of the original rumor, is studied.",
"That is a situation that occurs frequently in communities, when a rumor is counteracted by a contrary information or news, which behaves in the same way as a rumor.",
"Furthermore, qualitative analysis and numerical experimentation of the dynamic model are performed.",
"We corroborate that the dynamics of spreading rumors show similar behavior to that found in the dynamics of an infectious disease."
],
[
"Introduction and Preliminaries",
"Information spreads in ways that resemble the transmission dynamics of viruses.",
"In order to help emphasize the similarity to epidemic models, the customary variable names in epidemiology will be employed throughout.",
"The medium through which the rumors are transmitted is also an important factor in the dynamics of spreading.",
"Actually, computers are constantly used and are great means for transmitting information as well as rumors.",
"This led to consider a particular form of epidemic models, which here we prefer to call the propagation rumor (PR) model.",
"At first glance it seems that the PR model is an example of logistical behavior, because, initially, only a small fraction of the population may know some rumor.",
"These individuals pass the rumor on to their neighbors.",
"The cycle repeats and soon there are so many people who have heard the rumor that it becomes difficult to find someone who has not heard it.",
"The number of people hearing the rumor for the first time begins to level off creating an S-shaped curve [1].",
"However, the phenomenon of the spread of rumors is somewhat more complex.",
"The standard PR model was introduced in 1965 by Daley and Kendall [2] (DK model).",
"Also, the modeling of rumor propagation has been proposed by other authors such as Rapoport (1948) and Bartholomew (1967) (see [3] and references therein), and Zanette (2001) [4].",
"These approaches are based on stochastic processes.",
"In this paper we propose a deterministic approach, which draws from the SIRS epidemiological models along with the introduction of a second rumor (of different nature than the first) in the susceptible population.",
"We formulate our proposal as a compartmental model [5], with the population being divided into compartments and with assumptions about the nature and time rate of transfer from one compartment to another.",
"We describe a model for propagation of rumors, acting on a sufficiently rapid time scale such that demographic effects of a population may be ignored.",
"Our model is concerned with the spread of a rumor through individuals of a given population, with a total population size of $n$ individuals.",
"An instructive approach is to treat the problem by qualitative methods.",
"We start with the classical Kermack-McKendrick epidemic model (KM model) [6] $ \\left\\lbrace \\begin{array}{lll}\\frac{ds(t)}{dt} & = & -\\beta s(t)i(t), \\\\[1.5mm]\\frac{di(t)}{dt} & = & \\beta s(t)i(t) - \\alpha i(t), \\\\[1.5mm]\\frac{dr(t)}{dt} & = & \\alpha i(t),\\end{array}\\right.$ where $s(t)$ , $i(t)$ , and $r(t)$ denote, respectively, the numbers of susceptible, infectious, and removed individuals at time $t$ , and the parameters $\\beta >0$ and $\\alpha >0$ are known as the infection and removal rates.",
"This model, also called SIR model, forms the basis of all epidemiological models, which virtually omits the population of parasites (population of rumors) from direct consideration and assumes that the sizes of the compartments are large enough that the mixing of members is homogeneous.",
"It is based on the following assumptions: an average member of the population makes sufficient contact to transmit the rumor to $\\beta n$ others per unit time (mass action law), those infected by a rumor leave the infective class at $\\alpha i$ rate per unit time, and the time scale of the rumor is much faster than the time scale of births and deaths, so that demographic effects on the population may be ignored and the population remains constant (i.e., equal to $n=s+i+r$ ).",
"We need to add to the KM model supplementary conditions: $s(t_0)=s_0>0$ , $i(t_0)=i_0>0$ , and $r(t_0)=0$ .",
"Next, we shall carry out this procedure on a slightly more general case, allowing for a loss of immunity that causes recovered individuals to become susceptible again (i.e., temporary immunity).",
"In our case, individuals which have previously heard the rumor, and were no longer interested in spreading it to other individuals (i.e., population of type $r$ ), suddenly, for some reason, become willing to spread rumors immediately.",
"It will be assumed that this takes place at a rate proportional to the population of type $r$ .",
"Thus the equations become $\\left\\lbrace \\begin{array}{lll}\\frac{ds(t)}{dt} & = & -\\beta s(t)i(t) + \\gamma r(t), \\\\[1.5mm]\\frac{di(t)}{dt} & = & \\beta s(t)i(t) - \\alpha i(t), \\\\[1.5mm]\\frac{dr(t)}{dt} & = & \\alpha i(t) - \\gamma r(t),\\end{array}\\right.$ with a proportional rate $\\gamma $ of loss of immunity.",
"This model is called an SIRS model since removed individuals can return to class $s$ (i.e., a rate of transfer from $r$ to $s$ is added to an SIR model [5]).",
"Since $n^{\\prime }=(s+i+r)^{\\prime }=0$ , the total population size $n$ is constant and $r=n-s-i$ .",
"So, we may express the previous model by the system $ \\left\\lbrace \\begin{array}{lll}\\frac{ds(t)}{dt} & = & -\\beta s(t)i(t) + \\gamma (n-i(t)-s(t)), \\\\[1.5mm]\\frac{di(t)}{dt} & = & \\beta s(t)i(t) - \\alpha i(t).\\end{array}\\right.$ However, more complicated compartmental structures are possible; for example, there are SEIR and SEIS models, with an exposed period between being infected (i.e., to hear the rumor) and becoming infective (i.e., spreading the rumor) [5].",
"Clearly, from system (REF ) there is a rumor-free equilibrium corresponding to $i=0$ , $s=n$ .",
"In this equilibrium the whole population is healthy (but susceptible) and the rumor eventually disappears.",
"The other stable state is $s=\\alpha /\\beta $ , $i=\\gamma (n - s)/(\\alpha + \\gamma )$ .",
"Here, for $i$ to be positive, necessarily, $n > s$ .",
"Since $s=\\alpha /\\beta $ , the rumor remains in the population provided the total population $n$ exceeds $\\alpha /\\beta $ .",
"This is, whenever ${\\cal R}_0 = n\\beta / \\alpha > 1$ , where ${\\cal R}_0$ has been called the basic reproduction number (see, for example, [7]).",
"This important threshold effect was discovered by Kermack and McKendrick (i.e., population must be “large enough\" for a disease to become endemic) [6].",
"Since removal rate from the infective class is $\\alpha $ (in units of 1/time), the average period of infectivity is $1/\\alpha $ .",
"Thus $\\beta /\\alpha $ is the fraction of the population that comes into contact with an infective individual during the period of infectiousness [8].",
"The equilibrium $i=\\gamma (n - s)/(\\alpha + \\gamma )$ , which corresponds to $s=\\alpha /\\beta $ , is called an endemic equilibrium.",
"The basic reproduction number for a disease is the number of secondary infections produced by an infected individual in a population of susceptible individuals.",
"It is a known fact that the basic reproduction number ${\\cal R}_0$ is one of the most used thresholds in epidemic theory [7].",
"It allows us to determine whether there is a rumor epidemic.",
"Note that the matrix of the linearization of (REF ) at an equilibrium $(s,i)$ is $\\left(\\begin{array}{cc}-(\\beta i + \\gamma ) & -(\\beta s + \\gamma ) \\\\\\beta i & \\beta s - \\alpha \\\\\\end{array}\\right).$ By observing the sign structure at the disease-free equilibrium this matrix has negative trace and positive determinant if and only if $\\beta n < \\alpha $ , or ${\\cal R}_0 < 1$ .",
"At an endemic equilibrium, the sign structure shows that the matrix also has negative trace and positive determinant (see [5] for details).",
"In conclusion, the rumor-free equilibrium is asymptotically stable if and only if ${\\cal R}_0 < 1$ and the endemic equilibrium, which exists if and only if ${\\cal R}_0 > 1$ , is always asymptotically stable, so that the rumor persists.",
"Even if the endemic equilibrium is unstable, the instability commonly arises from a bifurcation and the infection (rumor) still persists but in an oscillatory manner [9].",
"More specifically, as ${\\cal R}_0$ increases through 1 there is an exchange of stability (bifurcation) between the rumor-free equilibrium and the endemic equilibrium.",
"This transition is called a forward bifurcation.",
"It has been noted (see references in [9]) that in epidemic models with multiple interaction mechanisms it is possible to have a very different bifurcation behavior at ${\\cal R}_0 = 1$ .",
"There may be multiple positive endemic equilibria for values of ${\\cal R}_0 < 1$ , resulting in the so-called backward bifurcation at ${\\cal R}_0 = 1$ .",
"For details see [9].",
"Although at the beginning of a rumor outbreak there is a very small number of affected individuals, and the transmission of rumors is a stochastic event depending on the pattern of contacts between members of the population, we seek to define a deterministic model that simulates the rumor phenomenon in the near future.",
"In the following sections of this paper, we assume that we are in a rumor epidemic situation (i.e., a compartmental model), which follows a disease outbreak that probably was modeled (initially) by a stochastic process (i.e., a branching process [5]).",
"Finally, it is clear that contacts do not necessarily transmit the rumor.",
"For each contact between infected and susceptible individuals, there is a probability that the infection will actually be transmitted.",
"We assume that there is a mean probability $T$ , called the transmissibility [5], of transmission of rumor.",
"The transmissibility depends on the rate of contacts, the probability that a contact will transmit rumor, the duration time of the rumor, and the susceptibility.",
"However, to simplify the model, in this article we will assume that all contacts transmit rumor (i.e., $T=1$ ).",
"This paper is organized as follows.",
"In Section 2, we make a brief presentation on some known mathematical models for the spread of rumors.",
"The proposed deterministic PR model is presented in Section 3.",
"Finally, in Section 4 we present some concluding remarks."
],
[
"Some known PR models",
"Rumors can be viewed as an “infection of the mind\" [10].",
"Therefore, the rumor propagation problem can be studied with many modeling techniques used in the study of epidemics.",
"Some of these approaches include deterministic models, stochastic models and complex networks."
],
[
"Classical epidemic models of rumor spreading",
"The DK model [2] assumes that individuals in the network are categorized into three groups: ignorants ($s$ ), spreaders ($i$ ), and stiflers ($r$ ), such that $s$ represents people who are ignorant of the rumor, $i$ is people who actively spread the rumor, and $r$ is people who have heard the rumor, but no longer are interested in spreading it.",
"(Sometimes here, we will use these terms interchangeably.)",
"The rumor is diffused through the population by pairwise contacts between spreaders and others in the population.",
"Any spreader involved in a pairwise meeting attempts to contaminate the other individual with the rumor, following the law of mass action.",
"If this other individual is an ignorant, it becomes a spreader.",
"If those involved in the meeting are a spreader and a stifler, either one or both of those, learn that the rumor is known and decide not to tell the rumor anymore, therefore turning into stiflers.",
"In the model proposed by Maki and Thompson [11] (MK model), the rumor is spread by direct contact of the spreaders with others in the population.",
"In addition, when a spreader contacts another spreader, only the initiating spreader becomes a stifler.",
"If we introduce the normalized variables $s/n$ and $i/n$ (denoted with the same symbol), and since $n=s+i+r$ , the proposed model is as follows $ \\left\\lbrace \\begin{array}{lll}\\frac{ds}{dt} & = & -\\beta si, \\\\[1.5mm]\\frac{di}{dt} & = & \\beta si - \\nu i^2 - \\nu i(1-i-s)=(\\beta + \\nu )si - \\nu i.\\end{array}\\right.$ Compared with the simple SIR model (REF ), we see that the only difference is that we have a factor $\\beta + \\nu $ in the second equation instead of $\\beta $ .",
"It is clear also that the ignorant population is decreasing, nonnegative since $s,i\\ge 0$ and $ds/dt \\le 0$ .",
"Since ${\\cal R}_0=(\\beta + \\nu )/ \\nu $ , ${\\cal R}_0>1$ if and only if $\\beta /\\nu >0$ .",
"I.e., there will be a rumor epidemic even for arbitrarily small rate parameters.",
"Recently [12], for the epidemic model, summarize some of the deterministic and stochastic threshold theory, illustrate how to calculate the stochastic thresholds, and derive some new relationships between the deterministic and stochastic thresholds.",
"We think that this approach can be applied to the PR model."
],
[
"Complex networks",
"The so-called “complex networks\" have non-trivial topological features and can vary from technological networks to social networks and biological networks [13].",
"The previous models assume a homogeneously mixing population and do not take into account the topology of the underlying social interaction networks along which rumors spread.",
"While such simple models may adequately describe the spreading process in small scale social networks, via the word-of-mouth, they become highly inadequate when applied to the spreading of rumors in large social interaction networks or by the Internet, which involves millions of nodes.",
"The topology of such large social networks shows highly complex connectivity patterns [10].",
"In 2004 Moreno et al.",
"[14] derived the mean-field equations characterizing the dynamics of a rumor process that takes place on top of complex heterogeneous networks.",
"These equations were solved numerically by means of a stochastic approach.",
"Then, they studied the spreading process for random scale-free networks.",
"Rumor spreading forms the basis for an important class of communication protocols, called gossip algorithms, which are used for large-scale information dissemination on the Internet, and in peer-to-peer file sharing applications [10].",
"In [10], Nekovee et al.",
"made several contributions to the study of rumor dynamics on complex social networks.",
"They introduce a new model that unifies the MK model with the SIR model of epidemics, and has both of these models as its limiting cases.",
"They also describe a formulation of this model on networks in terms of Interacting Markov Chains (IMC), and use this framework to derive, from first-principles, mean-field equations for the dynamics of rumor spreading on complex networks with arbitrary degree correlations.",
"More recently, Cheng et al.",
"[15] considered an online social site consisting of $n$ individuals which can be subdivided into three classes including ignorants ($s$ ), spreaders ($i$ ), and stiflers ($r$ ).",
"They introduced a stochastic epidemic model of the rumor diffusion.",
"The model assumes three status transitions in the model: from ignorants to spreaders, from spreaders to stiflers with contacts, and the spreaders to stiflers spontaneously.",
"Unlike previous rumor diffusion models, this model treats the infectious probability as a variable, which can be seen as a function of strength of ties."
],
[
"Deterministic vs. stochastic models",
"In this section the advantages and disadvantages of each type of model are pointed out.",
"It is interesting to note that most models describing epidemic spreading are deterministic because they require less data, are relatively easy to set up, and because the computer software to study them is widely available and user-friendly.",
"“In fact, models often identify behaviors that are unclear in experimental data-often because data are not reproducible and the number of data points is limited and subject to errors in measurement\" [5].",
"The dynamical behavior of deterministic models, also known as compartmental models, attempt to describe and explain what happens on the average at the population scale.",
"They fit well large populations and are now well understood, so that deterministic models are generally the chosen strategy to apply when a new problem is posed.",
"On the other hand, stochastic models are based on probabilities rather than definite rates.",
"There are much more complex models, such as those considered above in the study of complex networks, which generally involve stochastic procedures and provide much more insight into an individual-level modeling; however, they can be laborious to set up and need many simulations to yield useful predictions.",
"Here we propose a (non-stochastic) deterministic mathematical model for the spread of rumors, and we leave for future work the adaptation of our approach to the derivation of relationships between deterministic and stochastic thresholds [12]."
],
[
"Deterministic PR models",
"Thompson et al.",
"[16] started from the same framework as DK model and applied a deterministic approach; however, they considered elements such as the heterogeneity in the susceptible and spreader classes, as well as transmission, on the spread of a rumor $-$ aspects not considered in the DK model.",
"So they defined the passive and active people to be those who do not have many contacts and those who have many contacts, respectively.",
"In this Section, a different approach, but also based on a deterministic mathematical model, is introduced."
],
[
"PR model with an exponentially distributed period of temporary immunity",
"From the SIRS model (REF ), we will investigate the effect of introducing a second rumor in the same population of individuals, which will try to counteract the effect of the original rumor.",
"A situation that occurs frequently in communities, when a rumor is counteracted by a contrary information or news, which behaves in the same way as a rumor.",
"The proposed model is inspired by the model of vaccination [5], which we add to a SIRS-type model (see (REF )).",
"In a way that we hope that vaccination (second rumor) reduces susceptibility to disease (i.e., the first rumor).",
"From the study carried out by Brauer and Castillo-Chavez in [9] (see also [17], [18]) to the case of a SIS model with vaccination, we propose here an adaptation to the problem of spreading rumors.",
"More specifically, we propose an SIRS-type model defined by system (REF ), which assumes an exponentially distributed period of temporary immunity, after which the removed individuals (stiflers) can return to the susceptible class (ignorants).",
"Furthermore, as in [9], we will consider the inclusion of a new class of individuals, the vaccinated ones ($v$ ).",
"For simplicity, just as in epidemic models, we do not include births, natural deaths, and disease deaths [9], so that the total population size may be taken as constant.",
"We will add the assumption that a fraction $\\phi $ of the susceptible class knows the second rumor (i.e., it is vaccinated).",
"Also, in this model we assume that individuals who know the second rumor have a susceptibility to first rumor reduced by a factor of $\\sigma $ ($0\\le \\sigma \\le 1$ ), where $\\sigma =0$ corresponds to the situation where the second rumor counteracted the effect of the first rumor $100\\%$ , and $\\sigma =1$ corresponds to the situation where the second rumor had no effect on the first one.",
"Thus, the PR model is expressed as $ \\left\\lbrace \\begin{array}{lll}\\frac{ds(t)}{dt} & = & -\\beta s(t)i(t) - \\phi s(t) + \\gamma (n - i(t) - s(t)), \\\\[1.5mm]\\frac{di(t)}{dt} & = & \\beta s(t)i(t) + \\sigma \\beta v(t)i(t) - \\alpha i(t), \\\\[1.5mm]\\frac{dv(t)}{dt} & = & \\phi s(t) - \\sigma \\beta v(t)i(t),\\end{array}\\right.$ subject to the initial conditions prescribe $s(0)$ , $i(0)$ , $v(0)$ , with $s(0)+i(0)+v(0)=n$ , and where $\\gamma $ , as in (REF ), represents the proportional rate of loss of immunity.",
"The definition of the parameters involved can be seen in Table REF .",
"Table: Parameters definition for the PR model.Since $n=s+i+v$ is constant, $s=n-i-v$ , and the system (REF ) is equivalent to the system $ \\left\\lbrace \\begin{array}{lll}\\frac{di(t)}{dt} & = & \\beta (n - i(t) - (1-\\sigma )v(t))i(t) - \\alpha i(t), \\\\[1.5mm]\\frac{dv(t)}{dt} & = & \\phi (n - i(t)) - \\sigma \\beta v(t)i(t) - \\phi v(t).\\end{array}\\right.$ This is the basic model which we will study closely.",
"Note that if all ignorants heard the second rumor immediately ($\\phi \\rightarrow \\infty $ ), there will be no ignorant individuals who have not been vaccinated, $n=i+v$ , and the model (REF ) is equivalent to $ \\frac{di(t)}{dt} = \\sigma \\beta i(t)(n - i(t)) - \\alpha i(t),$ which is an SIS model with basic reproductive number ${\\cal R}_0^* = \\sigma \\beta n/ \\alpha = \\sigma {\\cal R}_0 \\le {\\cal R}_0$ , where ${\\cal R}_0$ is the basic reproductive number of the model (REF ).",
"But (REF ) is also a logistic equation, which is easy to solve analytically or qualitatively.",
"It coincides with the simple model proposed in [1] (at the beginning of Section , paragraph two, it was briefly discussed).",
"As it is usual in epidemiological models, the rumor-free equilibrium corresponds to $i = 0$ (so necessarily $v = n$ ), and is stable or unstable depending on the basic reproductive number [5]."
],
[
"Stability analysis",
"Following the strategy proposed in [9] to the case of a SIS model with vaccination, we consider the parameters $\\alpha $ , $\\beta $ , and $\\sigma $ as fixed and analyze the effect of varying $\\phi $ (since in practice this parameter is the one most easily controlled [9]).",
"We denote by ${\\cal R}(\\phi )$ the basic reproductive number of the model defined by (REF ).",
"Note that equilibria of this model are solutions of following system $\\beta (n - i(t) - (1 - \\sigma )v(t))i(t) & = & \\alpha i(t) , \\\\\\phi (n - i(t)) & = & \\sigma \\beta v(t)i(t) + \\phi v(t).$ Observe that if $i= 0$ , then the first equation is satisfied and from the second we obtain $v=n$ , which corresponds to the rumor-free equilibrium.",
"Now, the matrix of the linearization of (REF ) at an equilibrium $(i ,v )$ and whose determinant we want to estimate is given by $\\left( \\begin{array}{cc}-2 \\beta i - (1 - \\sigma )\\beta v - \\alpha + \\beta n & -(1 - \\sigma )\\beta i \\\\-(\\phi + \\sigma \\beta v) & -(\\phi + \\sigma \\beta i)\\end{array}\\right).$ And at the rumor-free equilibrium the matrix is $\\left( \\begin{array}{cc}-(1 - \\sigma )\\beta n - \\alpha + \\beta n & 0 \\\\-(\\phi + \\sigma \\beta n) & -\\phi \\end{array}\\right),$ which has negative eigenvalues (implying the asymptotic stability of the rumor-free equilibrium) if and only if $-(1 - \\sigma )\\beta n + \\beta n < \\alpha $ , which is equivalent to $\\sigma \\beta n < \\alpha $ or to ${\\cal R}(\\phi ) = \\sigma \\beta n/ \\alpha = \\sigma {\\cal R}_0 < 1$ .",
"Also note that ${\\cal R}(\\phi ) = \\left\\lbrace \\begin{array}{ll}\\; {\\cal R}_0, & \\rm { if } \\; \\phi =0 \\; (\\rm {there \\; is \\; no \\; a \\; second \\; rumor}) \\\\\\sigma {\\cal R}_0, & \\rm { if } \\; \\phi >0 \\; (\\rm {so}\\; {\\cal R}(\\phi ) < {\\cal R}_0 )\\end{array}\\right.$ So ${\\cal R}(\\phi ) \\le {\\cal R}_0 $ .",
"Following a similar analysis as in [9], we assume first that $0< \\sigma < 1$ to get endemic equilibrium points from $ \\beta (n - i - (1 - \\sigma )v) & = & \\alpha , \\\\ \\phi (n - i) & = & \\sigma \\beta vi + \\phi v.$ We will do this by trying to find the unknown $v$ from the first equation of (REF ) and substitute into the second equation to get an equation of the form $ ai^2 + bi + c = 0, \\; {\\rm where} \\; a= \\sigma \\beta ,\\; b=\\sigma (\\phi + \\alpha - \\beta n) \\;\\; {\\rm and} \\;\\;c=(\\alpha \\phi / \\beta ) - \\sigma \\phi n .$ Note that if ${\\cal R}(\\phi ) < 1$ , ${\\cal R}(\\phi ) = 1$ , or ${\\cal R}(\\phi ) > 1$ , then $c>0$ , $c=0$ , or $c<0$ , respectively.",
"Also, it is clear that if ${\\cal R}(\\phi ) > 1$ , then there is a unique positive root of (REF ) and thus there is a unique endemic equilibrium.",
"Now, if ${\\cal R}(\\phi ) = 1$ ($c=0$ ), there is a unique nonzero solution of equation (REF ) $i=-b/a$ , which is positive if and only if $b=\\sigma (\\phi - \\alpha - \\beta n) <0$ .",
"In this case there is a positive endemic equilibrium.",
"Since equilibria depend continuously on $\\phi $ there must then be an interval to the left of ${\\cal R}(\\phi ) = 1$ in which there are two possible equilibria; these are $i=(-b \\pm \\sqrt{b^2 - 4ac})/2a $ .",
"This shows that the system (REF ) has a backward bifurcation (see Section ) at ${\\cal R}(\\phi ) = 1$ if and only if $b<0$ and $\\beta $ is such that $c=0$ .",
"So, if $c>0$ and $b \\ge 0$ , there are no positive solutions of (REF ), therefore there are no endemic equilibria.",
"Equation (REF ) has two positive solutions, corresponding to two endemic equilibria, if and only if $c>0$ (${\\cal R}(\\phi ) < 1$ ) and $b<0$ , $b^2 > 4ac$ , or $b < -2 \\sqrt{ac}$ .",
"If $b = -2 \\sqrt{ac}$ , there is one positive solution $i=-b/2a$ of (REF ).",
"It can also be proved that such backward bifurcation is not possible for an SI model (case $\\alpha =0$ ) [9].",
"If $\\sigma = 0$ , from the equilibrium conditions (REF )-() we cannot infer anything conclusive.",
"However, from the more general epidemiological approach of [9], it follows that there is a unique endemic equilibrium if ${\\cal R}(\\phi ) > 1$ and there cannot be an endemic equilibrium if ${\\cal R}(\\phi ) < 1$ .",
"Therefore, in this case, it is not possible to have a backward bifurcation at ${\\cal R}(\\phi ) = 1$ .",
"Alternatively, if ${\\cal R}(\\phi ) = 1$ , $c = 0$ , and $\\sigma \\phi \\beta n = \\alpha \\phi $ .",
"Also, condition $b <0$ is equivalent to $\\sigma (\\phi + \\alpha ) < \\sigma \\beta n$ .",
"From whence it follows that $(\\sigma \\phi )^2 < \\alpha \\phi \\sigma (1 - \\sigma )$ .",
"Thus, a backward bifurcation occurs at ${\\cal R}(\\phi ) = 1$ if and only if this expression is satisfied.",
"Therefore, a backward bifurcation is not possible if $\\sigma = 0$ or $\\sigma = 1$ , or if $\\alpha = 0$ (i.e., for an SI model).",
"As ${\\cal R}(\\phi )=(\\sigma n/\\alpha )\\beta $ , for a glimpse of how the bifurcation curve is in the $({\\cal R}(\\phi ),i)$ -plane, we can consider $ \\beta $ as the independent variable and all other parameters as constants.",
"Differentiating implicitly with respect to $\\beta $ the equation (REF ) we obtain $(2ai + b)\\frac{di}{d\\beta } = \\sigma i(n-i) + \\frac{\\alpha \\phi }{ \\beta ^2}.$ From the equilibrium condition () we infer that $i\\le n$ , which means that the bifurcation curve has positive slope at equilibrium values with $2ai + b >0$ and negative slope at equilibrium values with $2ai + b < 0$ .",
"If there is not a backward bifurcation at ${\\cal R}(\\phi ) = 1$ , then the unique endemic equilibrium for ${\\cal R}(\\phi ) > 1$ satisfies $2ai + b = \\sqrt{b^2 - 4ac} >0$ , and the bifurcation curve has positive slope at all points where $i > 0$ .",
"The bifurcation curve is illustrated in Figure REF , using parameter values shown in Table REF ($\\beta $ being non-constant).",
"If there is a backward bifurcation at ${\\cal R}(\\phi ) = 1$ , then there is an interval on which there are two endemic equilibria given by $2ai+b = \\pm \\sqrt{b^2-4ac}$ , and the bifurcation curve has negative slope at the smaller of these and positive slope at the larger of these (see Figure REF ).",
"Figure: Forward bifurcation.",
"As ℛ(φ){\\cal R}(\\phi ) is a constant multiple of β\\beta , we can think of β\\beta as the independent variable in the bifurcation curve with the other parameters as constant .Note that from the matrix of the linearization of (REF ) at an equilibrium $(i ,v)$ and using the equilibrium conditions (REF )-(), we can get the matrix at an endemic equilibrium $(i ,v)$ ; this is $\\left( \\begin{array}{cc}- \\beta i & -(1- \\sigma )\\beta i \\\\-(\\phi + \\sigma \\beta v) & -(\\phi + \\sigma \\beta i)\\end{array}\\right).$ This matrix has negative trace and determinant equal to $\\beta i (2ai + b)$ .",
"So, if $2ai + b > 0$ (i.e., if the bifurcation curve has positive slope), then the determinant is positive and the equilibrium is asymptotically stable.",
"If $2ai + b < 0$ , the determinant is negative and the equilibrium is unstable (i.e., it is a saddle point).",
"So that an endemic equilibrium of PR model (REF ) is (locally) asymptotically stable if and only if it corresponds to a point on the bifurcation curve at which the curve is increasing.",
"For details of a similar analysis, see [9].",
"Figure: Backward bifurcation."
],
[
"Parameter estimation and numerical simulations",
"Clearly, the parameters may vary depending on the type of person, the situation in which the rumor is transmitted, and the quality of the rumor itself.",
"We have, basically, two types of rumor which determine the population classes involved: susceptibles or ignorants, infectious individuals or spreaders, and those infected with a second rumor or vaccinated individuals which, without being spreaders, will seek to counteract the effects of the first rumor.",
"Let us also remember that those individuals who lost interest in the original rumor (i.e., the removed individuals or stiflers) have temporary immunity (SIRS model).",
"So that these, along with those who know the second rumor (vaccinated), but that are susceptible to becoming contaminated with the first rumor (reduced by a factor of $\\sigma $ ), are also part of the susceptible population (ignorants).",
"In our model, which involves temporary immunity and two kinds of rumors, we propose to model the interaction between individuals on the Internet and social networks.",
"This choice is justified by some statistical information available in Internet.",
"This is: Almost half the world's population have internet access and nearly one-third of the world's population now uses social media.",
"The average user checks his social network at least twice a day.",
"Social networks are the number 1 activity on Internet.",
"80% of social media users interact in some way with other users or profiles.",
"On average, active users have 200 to 400 friends and receive more messages than they send.",
"Following the methodology of estimating parameter values proposed in [16], we considered working with these means of transmission because we can get updated information on their use and there is a lot of data online.",
"We consulted the information from websites [19] and [20] to approximate the number of users of the means of transmission; and thus, to estimate how many interactions occur on average, which allowed us to approximate the values of the parameters that we need.",
"See Table REF .",
"Table: Parameter values estimated for the PR model.Figure REF shows, in the same coordinate plane, the curves corresponding to ignorants or susceptibles ($s(t)$ ), infectives or spreaders of the first rumor ($i(t)$ ), and spreaders of the second rumor or vaccinated individuals ($v( t)$ ), for different $\\sigma $ parameter values.",
"Observe the typical behavior of infectives, according to which the number of infected individuals increases sharply and then slowly decreases until it disappears.",
"We also note regarding the behavior of susceptible individuals that during the time of spreading the rumor, its population decreases, but it does not disappear, instead it maintains a slight but steady increase.",
"Finally, for values of $\\sigma $ equal to $0.2$ and $0.0$ (Figures REF (a) and REF (b) respectively) the second rumor, as time passes, increases rapidly at first and then more slowly but steadily, thus counteracting the effect of the first rumor.",
"If $\\sigma $ takes a value close to 1 (for example, 0.7 or 0.9, see Figures REF (c) and REF (d)), this situation is reversed.",
"Figure: Behavior simulations for the PR model () for different σ\\sigma parameter values: (a) UsingTable .",
"(b) Case σ=0\\sigma =0, which corresponds to the situation where the second rumor counteracted the effect of the first rumor 100%100\\%.",
"(c) Using σ=0.7\\sigma = 0.7.",
"(d) Case σ=0.9\\sigma = 0.9, which corresponds to the situation where the second rumor “almost\" had no effect on the first one."
],
[
"An alternative simple model for the propagation of two conflicting rumors",
"Now, we consider heterogeneity in behavior, specifically contact rates.",
"We will describe the case of two conflicting rumors spreading over a population of individuals of size $n$ .",
"The proposed model is based on the vaccination model described in [5], [9], under the assumption that vaccination (the second rumor) reduces susceptibility to the first rumor.",
"Our PR model requires considering two sub-populations, the first one knows the first rumor and the other knows the second rumor.",
"For clarity, as in the case of the vaccination model, we assume that a fraction of the population knows the second rumor (i.e., individuals who have been vaccinated) prior to the spreading of the first rumor, which produces the first rumor-infected individuals, who are not necessarily infective.",
"We will describe a PR model with compartmental SIR structure in both sub-populations.",
"For simplicity, as before, we do not include births, natural deaths, and disease-related deaths.",
"The two sub-populations have constant total sizes $n_1$ and $n_2$ , and $n_1+n_2=n$ .",
"We assume mass-action contact with contact rates $\\beta _1$ , $\\beta _2$ and recovery rates $\\alpha _1$ , $\\alpha _2$ , respectively.",
"We assume also that those individuals who know both rumors have reduced their ability to spread the first rumor by a factor of $\\eta $ .",
"As in Section REF , we also assume that individuals who know the second rumor have susceptibility to first rumor reduced by a factor $\\sigma $ ($0\\le \\sigma \\le 1$ ).",
"The proposed PR model is as follows: $ \\left\\lbrace \\begin{array}{lll}\\frac{ds_1(t)}{dt} & = & -\\beta _1 s_1(t)i_1(t) -\\beta _1 s_1(t)\\eta i_2(t) = -\\beta _1 s_1(t)[i_1(t) + \\eta i_2(t)], \\\\[1.5mm]\\frac{di_1(t)}{dt} & = & \\beta _1 s_1(t)i_1(t) + \\beta _1 s_1(t)\\eta i_2(t) - \\alpha _1 i_1(t) =\\beta _1 s_1(t)[i_1(t) + \\eta i_2(t)]- \\alpha _1 i_1(t), \\\\[1.5mm]\\frac{ds_2(t)}{dt} & = & - \\sigma \\beta _2 s_2(t)i_1(t) - \\sigma \\beta _2s_2(t)\\eta i_2(t) =- \\sigma \\beta _2 s_2(t)[i_1(t)+\\eta i_2(t)], \\\\[1.5mm]\\frac{di_2(t)}{dt} & = & \\sigma \\beta _2 s_2(t)i_1(t) - \\sigma \\beta _2s_2(t)\\eta i_2(t) - \\alpha _2 i_2(t) =\\sigma \\beta _2 s_2(t)[i_1(t)+\\eta i_2(t)] - \\alpha _2 i_2(t),\\end{array}\\right.$ where $s_1(t)$ , $i_1(t)$ , $s_2(t)$ , and $i_2(t)$ denote the number of first rumor susceptibles, the number of first rumor infectives, the number of second rumor susceptibles, and the number of second rumor infectives at time $t$ , respectively.",
"Likewise, $s_1(0)$ , $i_1(0)$ , $s_2(0)$ , and $i_2(0)$ are the number of susceptibles and infectives at the initial time $t= 0$ , with $s_1(0)+i_1(0) = n_1$ and $s_2(0)+i_2(0) = n_2$ .",
"In this case, the infection by the first rumor is beginning in a population which is not fully ignorant.",
"So we should speak of the control reproduction number ${\\cal R}_c$ rather than the basic reproduction number [9].",
"Through the introduction of the next generation matrix with large domain concept for system (REF ) at the disease-free equilibrium it is relatively easy to see that ${\\cal R}_c = \\frac{\\beta _1 n_1}{\\alpha _1 } + \\frac{\\eta \\sigma \\beta _2 n_2}{\\alpha _2}$ (see [9] for details).",
"We note that the basic reproduction number of the disease is the sum of the reproduction numbers for each group [7].",
"So it should follow that the disease-free equilibrium (from the first rumor) is asymptotically stable if the reproduction number is less than 1 and unstable if it is greater than 1."
],
[
"Selecting the dataset",
"We selected two rumors, from which the primary rumor to be studied is “earth is flat\" in contrast to “earth is round\".",
"This topic was chosen because it is widely known that the Earth's shape is round, however, there are many internet users that have been spreading out the theory that the Earth is flat.",
"This started first as a joke but shortly after, many people began supporting conspiracy theories about the flatness of the Earth.",
"The measured data was taken from the Google Trends open source platform, which allows any user to look up the statistics of a search term using the Google engine throughout time.",
"The dataset we considered for this work has the daily search percent from the terms “earth is flat\", “earth is round\" and “earth is\" from February 18th, 2017 to March 17th 2017.",
"The latter was taken into account so that we could study the rumors involving the interest on the Earth's features.",
"Finally, “earth is flat\" and “earth is round\" search percent curves are relative to the “earth is\" search.",
"Numerical simulations were performed using both the PR model with an exponentially distributed period of temporary immunity (original PR model) and the alternative simple model for the propagation of two conflicting rumors (alternative PR model).",
"Since the parameters of the proposed models are bounded, MATLAB's fminsearch function could not be used, because it was designed to find local minima of unconstrained multivariable functions [21].",
"Instead, we used the MATLAB's fmincon function, which is primarily used to solve nonlinear bounded optimization problems [22].",
"This MATLAB function has many implemented algorithms, such as: “Interior-Point Optimization\", “SQP and SQP-Legacy Optimization\", “Active-Set Optimization\", and “Trust-Region-Reflective Optimization\" [22].",
"By default fmincon will try to run the latter algorithm, but it requires the gradient of the objective function (which is very hard to calculate).",
"Instead, it runs the “active-set\" algorithm described in [23].",
"The initial point for the spreaders of the first rumor was $i(0)=0.45$ (according to the “earth is flat\" rumor) and for the second rumor, $v(0)=0.03$ (according to the number of users that searched “earth is round\" at the beginning of the spread of “earth is flat\").",
"Figure REF shows the original PR model's behavior.",
"Figure: Behavior simulations for the original PR model (Section ).For the initial parameters $\\alpha (0) = 0.3$ , $\\beta (0) = 0.9$ , $\\phi (0) = 0.5$ , and $\\sigma (0)=0.0$ , the estimated parameters obtained for the simplified model (REF ) are: $ \\alpha ^* = 0.58,\\quad \\beta ^* = 1.0, \\quad \\phi ^* = 1.0, \\quad \\sigma ^*=0.0.",
"$ These parameters suggest that the rumor is very easy to spread, however, the recovery rate is very fast because everybody knows that the “earth is round\".",
"Also, this vaccine is strong enough to counteract the “earth is flat\" rumor in 100%.",
"However, as it is seen in Figure REF , the disease has not completely disappeared, contrary to the real data.",
"A better fitting for this situation is provided by the following alternative PR model.",
"For the initial parameters $\\beta _1(0) = 0.9$ , $\\alpha _1(0) = 0.9$ , $\\beta _2(0) = 0.5$ , $\\alpha _2(0) = 0.5$ , $\\eta (0)=0.5$ , and $\\sigma (0)=0.5$ , the estimated parameters obtained for the alternative model (REF ) are: $ \\beta _1^* = 0.1954, \\quad \\alpha _1^* = 0.4455,\\quad \\beta _2^* = 0.2825, \\quad \\alpha _2^* = 0.0, \\quad \\eta ^* = 0.8428, \\quad \\sigma ^*=0.2825.",
"$ Figure: Behavior simulations for the alternative PR model (Section ).As it is seen in Figure REF (a), the model fits accurately the data.",
"The second rumor susceptible and infective curves are represented separately in Figure REF (b) to make them easier to visualize.",
"Finally, the SSE of the original PR model was $0.0398$ , whereas the SSE of the alternative PR model was $0.0293$ , so the latter in fact offers a more accurate fitting to the real data."
],
[
"Final remarks",
"In this paper different possible behaviors in the dynamics of the rumor spreading has been studied.",
"In Section REF we proposed a deterministic PR model and detected relevant conditions, derived from local stability analysis of the rumor-free equilibrium and the rumor-endemic equilibrium.",
"An important fact is that the endemic equilibrium is not asymptotically stable for all values of the parameters involved.",
"However, this model may have solutions that behave periodically [5].",
"This means that the spread of a rumor still persists but in an oscillatory manner with possible variations in the number of “contagions\" as well as long periods of recurrence, and this makes its study and control difficult.",
"A relevant problem to address is to identify situations of unstable endemic equilibrium for the spread of rumors.",
"A backward bifurcation at $R_0=1$ makes the disease control more difficult.",
"We corroborate the plausible fact that a possible measure to control the spread of a rumor could be to reduce the susceptibility of the population by spreading a second rumor (vaccination), which seeks to counteract the effect of the first one (the original infection), thus reducing the susceptibility of the population to the original rumor.",
"Another measure could be to carry out a program of reeducation to “convince\" people not to express agreement with the rumor; alternatively, this strategy could also be seen as the introduction in the population a second rumor that tries to counteract the effect of the first one.",
"In addition, in Section REF we proposed an alternative simple model, which provided a better fit to the data than did the original PR model in the numerical example considered in Section REF .",
"In the case of more than two rumors, we can assume that we are concerned by only one of these rumors and the rest are considered as contrary rumors.",
"Hence, the problem can be reduced to two conflicting rumors spreading over the same population, one for the first rumor and the other englobing all the adversary rumors.",
"Thus, the results of this work can be extended to the case of more than two rumors.",
"As far as we know, the approach outlined here, using and adapting a compartment model to study the problem of spreading rumors, has not been stated before.",
"This PR model confirms the parallelism between the study of the spread of a rumor and epidemiological study of the spread of an infectious disease.",
"We think that it will be relatively easy to find direct applications for the results here considered.",
"Acknowledgements: This research was partially supported by the Decanato de Investigación y Desarrollo (DID) at USB."
]
] | 1709.01726 |
[
[
"Unsupervised Generative Modeling Using Matrix Product States"
],
[
"Abstract Generative modeling, which learns joint probability distribution from data and generates samples according to it, is an important task in machine learning and artificial intelligence.",
"Inspired by probabilistic interpretation of quantum physics, we propose a generative model using matrix product states, which is a tensor network originally proposed for describing (particularly one-dimensional) entangled quantum states.",
"Our model enjoys efficient learning analogous to the density matrix renormalization group method, which allows dynamically adjusting dimensions of the tensors and offers an efficient direct sampling approach for generative tasks.",
"We apply our method to generative modeling of several standard datasets including the Bars and Stripes, random binary patterns and the MNIST handwritten digits to illustrate the abilities, features and drawbacks of our model over popular generative models such as Hopfield model, Boltzmann machines and generative adversarial networks.",
"Our work sheds light on many interesting directions of future exploration on the development of quantum-inspired algorithms for unsupervised machine learning, which are promisingly possible to be realized on quantum devices."
],
[
"Introduction",
"Generative modeling, a typical example of unsupervised learning that makes use of huge amount of unlabeled data, lies in the heart of rapid development of modern machine learning techniques [1].",
"Different from discriminative tasks such as pattern recognition, the goal of generative modeling is to model the probability distribution of data and thus be able to generate new samples according to the distribution.",
"At the research frontier of generative modeling, it is used for finding good data representation and dealing with tasks with missing data.",
"Popular generative machine learning models include Boltzmann Machines (BM) [2], [3] and their generalizations [4], variational autoencoders (VAE) [5], autoregressive models [6], [7], normalizing flows [8], [9], [10], and generative adversarial networks (GAN) [11].",
"For generative model design, one tries to balance the representational power and efficiency of learning and sampling.",
"There is a long history of the interplay between generative modeling and statistical physics.",
"Some celebrated models, such as Hopfield model [12] and Boltzmann machine [2], [3], are closely related to the Ising model and its inverse version which learns couplings in the model based on given training configurations [13], [14].",
"The task of generative modeling also shares similarities with quantum physics in the sense that both of them try to model probability distributions in an immense space.",
"Precisely speaking, it is the wavefunctions that are modeled in quantum physics, and probability distributions are given by their squared norm according to Born's statistical interpretation.",
"Modeling probability distributions in this way is fundamentally different from the traditional statistical physics perspective.",
"Hence we may refer probability models which exploit quantum state representations as “Born Machines”.",
"Various ansatz have been developed to express quantum states, such as the variational Monte Carlo [15], the tensor network (TN) states and recently artificial neural networks [16] .",
"In fact physical systems like quantum circuits are also promising candidates for implementing Born Machines.",
"In the past decades, tensor network states and algorithms have been shown to be an incredibly potent tool set for modeling many-body quantum states [17], [18].",
"The success of TN description can be theoretically justified from a quantum information perspective [19], [20].",
"In parallel to quantum physics applications, tensor decomposition and tensor networks have also been applied in a broader context by the machine learning community for feature extraction, dimensionality reduction and analyzing the expressibility of deep neural networks [21], [22], [23], [24], [25], [26].",
"In particular, matrix product state (MPS) is a kind of TN where the tensors are arranged in a one-dimensional geometry [27].",
"The same representation is referred as tensor train decomposition in the applied math community [28].",
"Despite its simple structure, MPS can represent a large number of quantum states extremely well.",
"MPS representation of ground states has been proven to be efficient for one-dimensional gapped local Hamiltonian [29].",
"In practice, optimization schemes for MPS such as density-matrix renormalization group (DMRG) [30] have been successful even for some quantum systems in higher dimension [31].",
"Some recent works extended the application of MPS to machine learning tasks like pattern recognition [32], classification [33] and language modeling [34].",
"Efforts also drew connection between Boltzmann Machines and tensor networks [35].",
"In this paper, building on the connection between unsupervised generative modeling and quantum physics, we employ MPS as a model to learn probability distribution of given data with an algorithm which resembles DMRG [30].",
"Compared with statistical-physics based models such as the Hopfield model [12] and the inverse Ising model, MPS exhibits much stronger ability of learning, which adaptively grows by increasing bond dimensions of the MPS.",
"The MPS model also enjoys a direct sampling method [36] much more efficient than the Boltzmann machines, which require Markov Chain Monte Carlo (MCMC) process for data generation.",
"When compared with popular generative models such as GAN, our model offers a more efficient way to reconstruct and denoise from an initial (noisy) input using the direct sampling algorithm, as opposed to GAN where mapping a noisy image to its input is not straightforward.",
"The rest of the paper is organized as follow.",
"In Sec.",
"we present our model, training algorithm and direct sampling method.",
"In Sec.",
"we apply our model to three datasets: Bars-and-stripes for a proof-of-principle demonstration, random binary patterns for capacity illustration and the MNIST handwritten digits for showing the generalization ability of the MPS model in unsupervised tasks such as reconstructionof images.",
"Finally, Sec discusses future prospects of the generative modeling using more general tensor networks and quantum circuits."
],
[
"MPS for Unsupervised Learning ",
"The goal of unsupervised generative modeling is to model the joint probability distribution of given data.",
"With the trained model, one can then generate new samples from the learned probability distribution.",
"Generative modeling finds wide applications such as dimensional reduction, feature detection, clustering and recommendation systems [37].",
"In this paper, we consider a data set $\\mathcal {T}$ consisting of binary strings $v\\in \\mathcal {V} = \\lbrace 0, 1\\rbrace ^{\\otimes N}$ , which are potentially repeated and can be mapped to basis vectors of a Hilbert space of dimension $2^N$ .",
"The probabilistic interpretation of quantum mechanics [38] naturally suggests modeling data distribution with a quantum state.",
"Suppose we encode the probability distribution into a quantum wavefunction $\\Psi (v)$ , measurement will collapse it and generate a result $v = (v_1, v_2, \\cdots , v_N)$ with a probability proportional to $|\\Psi (v)|^2$ Inspired by the generative aspects of quantum mechanics, we represent the model probability distribution by $\\mathbb {P}(v) =\\frac{|\\Psi (v)|^{2}}{Z},$ where $Z = \\sum _{v\\in \\mathcal {V}}|\\Psi (v)|^{2}$ is the normalization factor.",
"We also refer it as the partition function to draw an analogy with the energy based models [39].",
"In general the wavefunction $\\Psi ({v})$ can be complex valued, but in this work we restrict it to be real valued.",
"Representing probability density using square of a function was also put forward by former works [32], [40], [41].",
"These approaches ensure the positivity of probability and naturally admit a quantum mechanical interpretation."
],
[
"Matrix Product States",
"Quantum physicists and chemists have developed many efficient classical representations of quantum wavefunctions.",
"A number of these developed representations and algorithms can be adopted for efficient probabilistic modeling.",
"Here, we parametrize the wave function using MPS: $\\Psi ({v_1,v_2,\\cdots ,v_N})={\\mathrm {Tr}}\\left(A^{(1)v_1}A^{(2)v_2}\\cdots A^{(N)v_N}\\right),$ where each $A^{(k)v_{k}}$ is a $\\mathcal {D}_{k-1}$ by $\\mathcal {D}_k$ matrix, and $\\mathcal {D}_0=\\mathcal {D}_N$ is demanded to close the trace.",
"For the the case considered here, there are $2\\sum _{k=1}^{N}\\mathcal {D}_{k-1}\\mathcal {D}_{k}$ parameters on the right-hand-side of Eq.",
"(REF ).",
"The representational power of MPS is related to Von Neumann entanglement entropy of the quantum state, which is defined as $S=-{\\mathrm {Tr}}(\\rho _{A} \\ln \\rho _{A})$ .",
"Here we divide the variables into two groups $v = (v_A, v_B)$ and $\\rho _{A} = \\sum _{v_{B}} \\Psi (v_{A}, v_{B}) \\Psi ( v^{\\prime }_{A}, v_{B})$ is the reduced density matrix of a subsystem.",
"The entanglement entropy sets a lower bound for the bond dimension at the division $S\\le \\ln (\\mathcal {D}_{k})$ .",
"Any probability distribution of a $N$ -bit system can be described by an MPS as long as its bond dimensions are free from any restriction.",
"The inductive bias using MPS with limited bond dimensions comes from dropping off the minor components of entanglement spectrum.",
"Therefore as the bond dimension increases, an MPS enhances its ability of parameterizing complicated functions.",
"See [17] and [18] for recent reviews on MPS and its applications on quantum many-body systems.",
"In practice, it is convenient to use MPS with $\\mathcal {D}_{0}=\\mathcal {D}_N=1$ and consequently reduce the left and right most matrices to vectors [30].",
"In this case, Eq.",
"(REF ) reads schematically $\\begin{minipage}[c]{0.33}\\centering \\includegraphics [page=1]{TN.pdf}\\end{minipage}=\\begin{minipage}[c]{0.47}\\centering \\includegraphics [page=2]{TN.pdf}\\end{minipage}.$ Here the blocks denote the tensors and the connected lines indicate tensor contraction over virtual indices.",
"The dangling vertical bonds denote physical indices.",
"We refer to [17], [18] for an introduction to these graphical notations of TN.",
"Henceforth, we shall present formulae with more intuitive graphical notations wherever possible.",
"The MPS representation has gauge degrees of freedom, which allows one to restrict the tensors with canonical conditions.",
"We remark that in our setting of generative modeling, the canonical form significantly benefits computing the exact partition function $Z$ .",
"More details about the canonical condition and the calculation of $Z$ can be found in Appendix ."
],
[
"Learning MPS from Data",
"Once the MPS form of wavefunction $\\Psi (v)$ is chosen, learning can be achieved by adjusting parameters of the wave function such that the distribution represented by Born's rule Eq.",
"(REF ) is as close as possible to the data distribution.",
"A standard learning method is called Maximum Likelihood Estimation which defines a (negative) log-likelihood function and optimizes it by adjusting the parameters of the model.",
"In our case, the negative log-likelihood (NLL) is defined as $\\mathcal {L} = -\\frac{1}{|\\mathcal {T}|}\\sum _{v\\in \\mathcal {T}} \\ln ~\\mathbb {P}(v),$ where $|\\mathcal {T}|$ denotes the size of the training set.",
"Minimizing the NLL reduces the dissimilarity between the model probability distribution $\\mathbb {P}(v)$ and the empirical distribution defined by the training set.",
"It is well-known that minimizing $\\mathcal {L}$ is equivalent to minimizing the Kullback-Leibler divergence between the two distributions [42].",
"Armed with canonical form, we are able to differentiate the negative log-likelihood (REF ) with respect to the components of an order-4 tensor $A^{(k,k+1)}$ , which is obtained by contracting two adjacent tensors $A^{(k)}$ and $A^{(k+1)}$ .",
"The gradient reads $\\frac{\\partial \\mathcal {L}}{\\partial A^{(k,k+1)w_kw_{k+1}}_{i_{k-1}i_{k+1}}} = \\frac{ Z^{\\prime }}{Z} -\\frac{2}{|\\mathcal {T}|} \\sum _{v\\in \\mathcal {T}} \\frac{\\Psi ^{\\prime }(v )}{\\Psi (v)} ,$ where $\\Psi ^{\\prime }(v )$ denotes the derivative of the MPS with respect to the tensor element of $A^{(k,k+1)}$ , and $Z^{\\prime } = 2\\sum _{v\\in \\mathcal {V}} \\Psi ^{\\prime }(v )\\Psi (v ) $ .",
"Note that although $Z$ and $Z^{\\prime }$ involve summations over an exponentially large number of terms, they are tractable in the MPS model via efficient contraction schemes [17].",
"In particular, if the MPS is in the mixed-canonical form [17], $Z^{\\prime }$ can be significantly simplified to $Z^{\\prime }=2 A^{(k,k+1)w_kw_{k+1}}_{i_{k-1}i_{k+1}}$ .",
"The calculation of the gradient, as well as variant techniques in gradient descent such as stochastic gradient descent (SGD) and adaptive learning rate, are detailed in Appendix .",
"After gradient descent, the merged order-4 tensor is decomposed into two order-3 tensors, and then the procedure is repeated for each pair of adjacent tensors.",
"The derived algorithm is quite similar to the celebrated DMRG method with two-site update, which allows us to adjust dynamically the bond dimensions during the optimization and to allocate computational resources to the important bonds which represent essential features of data.",
"However we emphasize that there are key differences between our algorithm and DMRG: The loss function of classic DMRG method is usually the energy, while our loss function, the averaged NLL (REF ), is a function of data.",
"With a huge amount of data, the landscape of the loss function is typically very complicated so that modern optimizers developed in the machine learning community, such as stochastic gradient descent and learning rate adapting techniques [43], are important to our algorithm.",
"Since the ultimate goal of learning is optimizing the performance on the test data, we do not really need to find the optimal parameters minimizing the loss on the training data.",
"One usually stops training before reaching the actual minima to prevent overfitting.",
"Our algorithm is data-oriented.",
"It is straightforward to parallelize over the samples since the operations applied to them are identical and independent.",
"In fact, it is a common practice in modern deep learning framework to parallelize over this so-called \"batch\" dimension [37].",
"As a concrete example, the GPU implementation of our algorithm is at least 100 times faster than the CPU implementation on the full MNIST dataset."
],
[
"Generative Sampling",
"After training, samples can be generated independently according to Eq.",
"(REF ).",
"In other popular generative models, especially the energy based model such as restricted Boltzmann machine (RBM) [3], generating new samples is often accomplished by running MCMC from an initial configuration, due to the intractability of the partition function.",
"In our model, one convenience is that the partition function can be exactly computed with complexity linear in system size.",
"Our model enjoys a direct sampling method which generates a sample bit by bit from one end of the MPS to the other [36].",
"The detailed generating process is as follow: It starts from one end, say the $N$ -th bit.",
"One directly samples this bit from the marginal probability $\\mathbb {P}(v_N)=\\sum _{v_{1},v_{2},\\ldots ,v_{N-1}}\\mathbb {P}(v) $ .",
"It is clear that this can be easily performed if we have gauged all the tensors except $A^{(N)}$ to be left-canonical because $\\mathbb {P}(v_N)=|x^{v_{N}}|^{2}/Z$ , where we define $x_{i_{N-1}}^{v_{N}}=A^{(N)v_{N}}_{i_{N-1}}$ and the normalization factor reads $Z=\\sum _{v_{N}\\in \\lbrace 0,1\\rbrace }|x^{v_{N}}|^2$ .",
"Given the value of the $N$ -th bit, one can then move on to sample the $(N-1)$ -th bit.",
"More generally, given the bit values $v_k, v_{k+1},\\cdots , v_N$ , the $(k-1)$ -th bit is sampled according to the conditional probability $\\mathbb {P}(v_{k-1} | v_{k}, v_{k+1},\\ldots ,v_N) = \\frac{\\mathbb {P}(v_{k-1}, v_{k}, \\ldots ,v_N)}{\\mathbb {P}(v_{k}, v_{k+1} \\ldots ,v_N)}.",
"$ As a result of the canonical condition, the marginal probability can be simply expressed as $\\mathbb {P}(v_{k}, v_{k+1}, \\ldots ,v_N)&= | x^{v_{k},v_{k+1},\\ldots ,v_{N}} |^{2}/Z.",
"$ $x_{i_{k-1}}^{v_{k},v_{k+1},\\ldots ,v_{N}} = \\sum _{i_{k},i_{k+1},\\cdots , i_{N-1}} A^{(k)v_k}_{i_{k-1}i_{k}}A^{(k+1)v_{k+1}}_{i_{k}i_{k+1}}\\cdots A^{(N)v_N}_{i_{N-1}}$ has been settled since the $k$ -th bit is sampled.",
"Schematically, its squared norm reads $| x^{v_{k},v_{k+1},\\ldots ,v_{N}}|^{2} =\\begin{minipage}[c]{0.31}\\centering \\includegraphics [page=15]{TN.pdf}\\end{minipage}.$ Multiplying the matrix $A^{(k-1)v_{k-1}}$ from the left, and calculating the squared norm of the resulting vector $x_{i_{k-2}}^{v_{k-1},v_{k},\\ldots ,v_{N}}=\\sum _{i_{k-1}}A^{(k-1)v_{k-1}}_{i_{k-2}i_{k-1}}x_{i_{k-1}}^{v_{k},v_{k-1},\\ldots ,v_{N}}$ , one obtains $\\mathbb {P}(v_{k-1}, v_{k}, ...,v_N) = |x^{v_{k-1},v_{k},\\ldots ,v_{N}}|^2/Z.",
"$ Combining (REF , REF ) one can compute the conditional probability (REF ) and sample the bit $v_{k-1}$ accordingly.",
"In this way, all the bit values are successively drawn from the conditional probabilities given all the bits on the right.",
"This procedure gives a sample strictly obeying the probability distribution of the MPS.",
"This sampling approach is not limited to generating samples from scratch in a sequential order.",
"It is also capable of inference tasks when part of the bits are given.",
"In that case, the canonicalization trick may not help greatly if there is a segment of unknown bits sitting between given bits.",
"Nevertheless, the marginal probabilities are still tractable because one can also contract ladder-shaped TN efficiently [17], [18].",
"As what will be shown in Sec.",
", given these flexibilities of the sampling approach, MPS-based probabilistic modeling can be applied to image reconstruction and denoising."
],
[
"Features of the model and algorithms",
"We highlight several salient features of the MPS generative model and compare it to other popular generative models.",
"Most significantly, MPS has an explicit tractable probability density, while still allows efficient learning and inference.",
"For a system sized $N$ , with prescribed maximal bond dimension $\\mathcal {D}_{\\mathrm {max}}$ , the complexity of training on a dataset of size $|\\mathcal {T}|$ is $\\mathcal {O}(|\\mathcal {T}| N \\mathcal {D}_\\mathrm {max}^{3})$ .",
"The scaling of generative sampling from a canonical MPS is $\\mathcal {O}( N \\mathcal {D}_\\mathrm {max}^{2})$ if all the bits to be sampled are connected to the boundaries, otherwise given some segments the conditional sampling scales as $\\mathcal {O}( N \\mathcal {D}_\\mathrm {max}^{3})$ ."
],
[
"Theoretical Understanding of the Expressive Power",
"The expressibility of MPS was intensively studied in the context of quantum physics.",
"The bond dimensions of MPS put an upper bound on its ability of capturing entanglement entropy.",
"These solid theoretical understandings of the representational power of MPS [17], [18] makes it an appealing model for generative tasks.",
"Considering the success of MPS for quantum systems, we expect a polynomial scaling of the computational resources for datasets with short-range correlations.",
"Treating dataset of two dimensional images using MPS is analogous to the application of DMRG to two dimensional quantum systems [31].",
"Although in principle an exact representation of the image dataset may require exponentially large bond dimensions as the image resolution increases, at computationally affordable bond dimensions the MPS may already serve as a good approximation which captures dominant features of the distribution."
],
[
"Adaptive Adjustment of Expressibility",
"Performing optimizations for the two-site tensor instead of for each tensor individually, allows one to dynamically adjust the bond dimensions during the learning process.",
"Since for realistic datasets the required bond dimensions are likely to be inhomogeneous, adjusting them dynamically allocates computational resources in an optimal manner.",
"This situation will be illustrated clearly using the MNIST data set in Sec.",
"REF , and in Fig.",
"REF .",
"Adjustment of the bond dimensions follows the distribution of singular values in (REF ), which is related to the low entanglement inductive bias of the MPS representation.",
"Adaptive adjustment of MPS is advantageous compared to most other generative models.",
"Because in most cases, the architecture (which is the main limiting factor of the expressibility of the model) is fixed during the learning procedure, only the parameters are tuned.",
"By adaptively tuning the bond dimensions, the representational power of MPS can grow as it gets more acquainted with the training data.",
"In this sense, adaptive adjustment of expressibility is analogous to the structural learning of probabilistic graphical models, which is, however, a challenging task due to discreteness of the structural information."
],
[
"Efficient Computation of Exact Gradients and Log-likelihood",
"Another advantage of MPS compared to the standard energy based model is that training can be done with high efficiency.",
"The two terms contributing to the gradient (REF ) are analogous to the negative and positive phases in the training of energy based models [39], where the visible variables are unclamped and clamped respectively.",
"In the energy based models, such as RBM, typical evaluation of the first term requires approximated MCMC sampling [44], or sophisticated mean-field approximations e.g.",
"Thouless-Anderson-Palmer equations [45].",
"Fortunately, the normalization factor and its gradient can be calculated exactly and straightforwardly for MPS.",
"The exact evaluation of gradients guarantees the associated stochastic gradient descent unbiased.",
"In addition to efficiency in computing gradients, the unbiased estimate of the log-likelihood and its gradients benefits significantly when compared with classic generative models such as RBM, where the gradients are approximated due to the intractability of partition function.",
"First, with MPS we can optimize the NLL directly, while with RBM the approximate algorithms such as Contrastive Divergence (CD) is essentially optimizing a loss function other than NLL.",
"This results in a fact that some region of configuration space could never be considered during training RBM and a subsequently poor performance on e.g.",
"denoising and reconstruction.",
"Second, with MPS we can monitor the training process easily using exact NLL instead of other quantities such as reconstruction error or pseudo-lilelihood for RBM, which introduce bias to monitoring [46]."
],
[
"Efficient Direct Sampling",
"The approach introduced in Sec.",
"REF allows direct sampling from the learned probability distribution.",
"This completely avoids the slowing mixing problem in the MCMC sampling of energy based models.",
"MCMC randomly flip the bits and compare the probability ratios for accepting and rejecting the samples.",
"However, the random walks in the state space can get stuck in a local minimum, which may bring unexpected fluctuations of long time correlation to the samples.",
"Sometimes this raises issues to the samplings.",
"As a concrete example, consider the case where all training samples are exactly memorized by both MPS and RBM.",
"This is to say that NLL of both models are exactly $\\ln |\\mathcal {T}|$ , and only training samples have finite probability in both models.",
"While other samples, even with only one bit different, have zero probability.",
"It is easy to check that our MPS model can generate samples which is identical to one of the training samples using approach introduced in Sec.",
"REF .",
"However, RBM will not work at all in generating samples, as there is no direction that MCMC could follow for increasing the probability of samplings.",
"It is known that when graphical models have an appropriate structure (such as a chain or a tree), the inference can be done efficiently [47], [48], while these structural constraints also limit the application of graphical models with intractable partition functions.",
"The MPS model, however, enjoys both the advantages of efficient direct sampling and a tractable partition function.",
"The sampling algorithm is formally similar to the ones of autoregressive models [6], [7] though, being able to dynamically adjust its expressibility makes the MPS a more flexible generative model.",
"Unlike GAN [11] or VAE [5], the MPS can explicitly gives tractable probability, which may enable more unsupervised learning tasks.",
"Moreover, the sampling in MPS works with arbitrary prior information of samples, such as fixed bits, which supports applications like image reconstruction and denoising.",
"We note that this offers an advantage over the popular GAN, which easily maps a random vector in the latent space to the image space, but having difficulties in the reverse direction — mapping a vector in the images space to the latent space as a prior information to sampling."
],
[
"Applications ",
"In this section, to demonstrate the ability and features of the MPS generative modeling, we apply it to several standard datasets.",
"As a proof of principle, we first apply our method to the toy data set of Bars and Stripes, where some properties of our model can be characterized analytically.",
"Then we train MPS as an associative memory to learn random binary patterns to study properties such as capacity and length dependences.",
"Finally we test our model on the Modified National Institute of Standards and Technology database (MNIST) to illustrate its generalization ability for generating and reconstructing images of handwritten digits.",
"The code of these experiments have been posted at https://github.com/congzlwag/UnsupGenModbyMPS."
],
[
"Bars and Stripes ",
"Bars and Stripes (BS) [50] is a data set containing $4\\times 4$ binary images.",
"Each image has either four-pixel-length vertical bars or horizontal stripes, but not both.",
"In total there are 30 different images in the dataset out of all $2^{16}$ possible ones, as shown in Fig.",
"REF .",
"These images appear with equal probability in the dataset.This toy problem allows a detailed analysis and reveals key characteristics of the MPS probabilistic model.",
"Figure: (a) The Bars and Stripes dataset.",
"(b) Ordering of the pixels when transforming the image into one dimensional vector.",
"The numbers between pixels indicate the bond dimensions of the well-trained MPS.To use MPS for modeling, we unfold the $4\\times 4$ images into one dimensional vectors as shown in Fig.",
"REF .",
"After being trained over 4 loops of batch gradient descent training the cost function converges to its minimum value, which equals to the Shannon entropy of the BS dataset $\\mathcal {S}=\\ln (30)$ , within an accuracy of $1\\times 10^{-10}$ .",
"Here what the MPS has accomplished is memorizing the thirty images rigidly, by increasing the probability of the instances appeared in the dataset, and suppressing the probability of not-shown instances towards zero.",
"We have checked that the result is insensitive to the choice of hyperparameters.",
"The bond dimensions of the learned MPS have been annotated in Fig.",
"REF .",
"It is clear that part of the symmetry of the data set has been preserved.",
"For instance, the 180° rotation around the center or the transposition of the second and the third rows would change neither the data set nor the bond dimension distribution.",
"Open boundary condition results in the decrease of bond dimensions at both ends.",
"In fact when conducting SVD at bond $k$ , there are at most $2^{\\min (k,N-k)}$ non-zero singular values because the two parts linked by bond $k$ have their Hilbert spaces of dimension $2^{k}, 2^{N-k}$ .",
"In addition, the turnings bonds have slightly smaller bond dimension ($\\mathcal {D}_{4}=\\mathcal {D}_{8}=\\mathcal {D}_{12}=15$ ) than others inside the second row and the third row, which can be explained qualitatively as these bonds carrying less entanglement than the bonds in the bulk.",
"One can directly write down the exact “quantum wave function” of the BS dataset, which has finite and uniform amplitudes for the training images and zero amplitude for other images.",
"For division on each bond, one can construct the reduced density matrix whose eigenvalues are the square of the singular values.",
"Analyzed in this way, it is confirmed that the trained MPS achieves the minimal number of required bond dimension to exactly describe the BS dataset.",
"We have generated $N_\\mathrm {s}=10^6$ independent samples from the learned MPS.",
"All these samples are training images shown in Fig.",
"REF .",
"Carrying out likelihood ratio test [51], we got the log-likelihood ratio statistic $G^2=2N_\\mathrm {s} D_\\mathrm {KL}(\\lbrace \\frac{n_j}{N_\\mathrm {s}}\\rbrace ||\\lbrace p_j\\rbrace )=22.0$ , equivalently $D_\\mathrm {KL}(\\lbrace \\frac{n_j}{N_\\mathrm {s}}\\rbrace ||\\lbrace p_j\\rbrace )=1.10\\times 10^{-5}$ .",
"The reason for adopting this statistic is that it is asymptotically $\\chi ^2$ -distributed [51].",
"The $p$ -value of this test is $0.820$ , which indicates a high probability that the uniform distribution holds true for the sampling outcomes.",
"Note that $D_\\mathrm {KL}(\\lbrace \\frac{n_j}{N_\\mathrm {s}}\\rbrace ||\\lbrace p_j\\rbrace )$ quantifies the deviation from the expected distribution to the sampling outcomes, so it reflects the performance of sampling method rather than merely the training performance.",
"In contrast to our model, for energy based models one typically has to resort to MCMC method for sampling new patterns.",
"It suffers from slow mixing problem since various patterns in the BS dataset differs substantially and it requires many MCMC steps to obtain one independent pattern."
],
[
"Random patterns",
"Capacity represents how much about data could be learned by the model.",
"Usually it is evaluated using randomly generated patterns as data.",
"For the classic Hopfield model [12] with pairwise interactions given by Hebb's rule among $N\\rightarrow \\infty $ variables, it has been shown [52] that in the low-temperature region at the thermodynamic limit there is the retrieval phase where at most $|\\mathcal {T}|_c=0.14N$ random binary patterns could be remembered.",
"In this sense, each sample generated by the model has a large overlap with one of the training pattern.",
"If the number of patterns in the Hopfield model is larger than $|\\mathcal {T}|_c$ , the model would enter the spin glass state where samples generated by the model are not correlated with any training pattern.",
"Thanks to the tractable evaluation of the partition function ${Z}$ in MPS, we are able to evaluate exactly the likelihood of every training pattern.",
"Thus the capability of the model can be easily characterized by the mean negative log-likelihood $\\mathcal {L}$ .",
"In this section we focus on the behavior of $\\mathcal {L}$ with varying number of training samples and varying system sizes.",
"In Fig.",
"REF we plot $\\mathcal {L}$ as a function of number of patterns used for training for several maximal bond dimension $\\mathcal {D}_{\\mathrm {max}}$ .",
"The figure shows that we obtain $\\mathcal {L}=\\ln |\\mathcal {T}|$ for training set no larger than $\\mathcal {D}_{\\mathrm {max}}$ .",
"As what has been shown in the previous section, this means that all training patterns are remembered exactly.",
"As the number of training patterns increases, MPS with a fixed $\\mathcal {D}_{\\mathrm {max}}$ will eventually fail in remembering exactly all the training patterns, resulting to $\\mathcal {L} > \\ln |\\mathcal {T}|$ .",
"In this regime generations of the model usually deviate from training patterns (as illustrated in Fig.",
"REF on the MNIST dataset).",
"We notice that with $|\\mathcal {T}|$ increasing, the curves in the figure deviate from $\\ln |\\mathcal {T}|$ continuously.",
"We note this is very different from the Hopfield model where the overlap between the generation and training samples changes abruptly due to the first order transition from the retrieval phase to spin glass phase.",
"Figure: NLL averaged as a function of: (a) number of random patterns used for training, with system size N=20N=20.",
"(b) system size NN, trained using |𝒯|=100|\\mathcal {T}|=100 random patterns.In both (a) and (b), different symbols correspond to different values of maximal bond dimension 𝒟 max \\mathcal {D}_{\\mathrm {max}}.Each data point is averaged over 10 random instances (i.e.",
"sets of random patterns), error bars are also plotted, although they are much smaller than symbol size.",
"The black dashed lines in figures denote ℒ=ln|𝒯|\\mathcal {L}=\\ln |\\mathcal {T}|.Fig.",
"REF also shows that a larger $\\mathcal {D}_{\\mathrm {max}}$ enables MPS to remember exactly more patterns, and produce smaller $\\mathcal {L}$ with the number of patterns $|\\mathcal {T}|$ fixed.",
"This is quite natural because enlarging $\\mathcal {D}_{\\mathrm {max}}$ amounts to the increase of parameter number of the model, hence enhances the capacity of the model.",
"In principle if $\\mathcal {D}_{\\mathrm {max}}=\\infty $ our model has infinite capacity, since arbitrary quantum states can be decomposed into MPS [17].",
"Clearly this is an advantage of our model over the Hopfield model and inverse Ising model [14], whose maximal model capacity is proportional to system size.",
"Careful readers may complain that the inverse Ising model is not the correct model to compare with, because its variation with hidden variables, i.e.",
"Boltzmann machines, do have infinite representation power.",
"Indeed, increasing the bond dimensions in MPS has similar effects to increasing the number of hidden variables in other generative models.",
"In Fig.",
"REF we plot $\\mathcal {L}$ as a function of system size $N$ , trained on $|\\mathcal {T}|=100$ random patterns.",
"As shown in the figure that with $\\mathcal {D}_{\\mathrm {max}}$ fixed $\\mathcal {L}$ increases linearly with system size $N$ , which indicates that our model gives worse memory capability with a larger system size.",
"This is due to the fact that keeping joint-distribution of variables becomes more and more difficult for MPS when the number of variables increases, especially for long-range correlated data.",
"This is a drawback of our model when compared with fully pairwise-connected models such as the inverse Ising model, which is able to capture long-distance correlations of the training data easily.",
"Fortunately Fig.",
"REF also shows that the decay of memory capability with system size can be compensated by increasing $\\mathcal {D}_{\\mathrm {max}}$ ."
],
[
"MNIST dataset of handwritten digits",
"In this subsection we perform experiments on the MNIST dataset [53].",
"In preparation we turn the grayscale images into binary numbers by threshold binarization and flattened the images row by row into a vector.",
"For the purpose of unsupervised generative modeling we do not need the labels of the digits.",
"Here we further test the capacity of the MPS for this larger-scale and more meaningful dataset.",
"Then we investigate its generalization ability via examining its performance on a separated test set, which is crucial for generative modeling."
],
[
"Model Capacity",
"Having chosen $|\\mathcal {T}|=1000$ MNIST images, we train the MPS with different maximal bond dimensions $\\mathcal {D}_{\\mathrm {max}}$ , as shown in Fig.",
"REF .",
"As $\\mathcal {D}_{\\mathrm {max}}$ increases, the final $\\mathcal {L}$ decreases to its minimum $\\ln |\\mathcal {T}|$ , and the images generated become more and more clear.",
"It is interesting that with a relatively small maximum bond dimension, e.g.",
"$\\mathcal {D}_{\\mathrm {max}}=100$ , some crucial features show up, though some of the images were not as clear as the original ones.",
"For instance the hooks and the loops that partly resembled to “2”, “3” and “9” emerge.",
"These clear characters of handwritten digits illustrate that the MPS has learned many “prototypes”.",
"Similar feature-to-prototype transition in pattern recognitions could also be observed by using a many-body interaction in the Hopfield model, or equivalently using a higher-order rectified polynomial activation function in the deep neural networks [54].",
"It is remarkable that in our model this can be achieved by simply adjusting the maximum bond dimension of the MPS.",
"Next we train another model with the restriction of $\\mathcal {D}_{\\mathrm {max}}=800$ .",
"The NLL on the training dataset reach $16.8$ , and many bonds have reached maximal dimension $\\mathcal {D}_{\\mathrm {max}}$ .",
"Fig.",
"REF shows the distribution of bond dimensions.",
"Large bond dimensions concentrated in the center of the image, where the variation of the pixels is complex.",
"The bond dimensions around the top and bottom edge of the image remain small, because those pixels are always inactivated in the images.",
"They carry no information and has no correlations with the remaining part of the image.",
"Remarkably, although the pixels on the left and right edges are also white, they also have large bond dimensions because these bonds learn to mediate the correlations between the rows of the images.",
"Figure: Bond dimensions of the MPS trained with |𝒯|=1000|\\mathcal {T}|=1000 MNIST samples, constrained to 𝒟 max =800\\mathcal {D}_{\\mathrm {max}}=800.",
"Final average NLL reaches16.816.8.",
"Each pixel in this figure corresponds to bond dimension of the right leg of the tensor associated to the identical coordinate in the original image.Figure: (a) Images generated from the same MPS as in Fig. .",
"(b) Original images randomly selected from the training set.The samples directly generated after training are shown in Fig.",
"REF .",
"We also show a few original samples from the training set in Fig.",
"REF for comparison.",
"Although many of the generated images cannot be recognized as digits, some aspects of the result are worth mentioning.",
"Firstly, the MPS learned to leave margins blank, which is the most obvious common feature in MNIST database.",
"Secondly, the activated pixels compose pen strokes that can be extracted from the digits.",
"Finally, a few of the samples could already be recognized as digits.",
"Unlike the discriminative learning task carried out in [32], it seems we need to use much larger bond dimensions to achieve a good performance in the unsupervised task.",
"We postulate the reason to be that in the classification task, local features of an image are sufficient for predicting the label.Thus MPS is not required to remember longer-range correlation between pixels.",
"For generative modeling, however, it is necessary because learning the joint distribution from the data consists of (but not limited to) learning two-point correlations between pairs of variables that could be far from each other.",
"Figure: Image reconstruction from partial images by direct sampling with the same MPS as in Fig. .",
"(a,b) Restoration of images in Fig.",
"which are selected from the training set .",
"(c,d) Reconstruction of 16 images chosen from the test set.",
"The test set contains images from the MNIST database that were not used for training.",
"The given parts are in black and the reconstructed parts are in yellow.",
"The reconstructed parts are: 12 columns from either (a,c) the left or the right, and (b,d) the top or the bottom.With the MPS restricted to $\\mathcal {D}_{\\mathrm {max}}=800$ and trained with 1000, we carry out image restoration experiments.",
"As shown in Fig.",
"REF we remove part of the images in Fig.",
"REF and then reconstruct the removed pixels (in yellow) using conditional direct sampling.",
"For column reconstruction, its performance is remarkable.",
"The reconstructed images in Fig.",
"REF are almost identical to the original ones in Fig.",
"REF .",
"On the other hand, for row reconstruction in Fig.",
"REF , it makes interesting but reasonable deviations.",
"For instance, the rightmost in the first row, an “1” has been bent to “7”."
],
[
"Generalization Ability",
"In a glimpse of its generalization ability, we also tried reconstructing MNIST images other than the training images, as shown in Fig.",
"REF , REF .",
"These results indicate that the MPS has learned crucial features of the dataset, rather than merely memorizing the training instances.",
"In fact, even as early as only 11 loops trained, the MPS could perform column reconstruction with similar image quality, but its row reconstruction performance was much worse than that trained over 251 loops.",
"It is reflected that the MPS has learned about short range patterns within each row earlier than those with long range correlations between different rows, since the images have been flattened into a one dimensional vector row by row.",
"To further illustrate our model's generalization ability, in Fig.",
"REF we plotted $\\mathcal {L}$ for the same $10^4$ test images after training on different numbers of images.",
"To save computing time we worked on rescaled images of size $14\\times 14$ .",
"The rescaling has also been adopted by past works, and it is shown that the classification on the rescaled images is still comparable with those obtained using other popular methods [32].",
"For different $|\\mathcal {T}|$ , $\\mathcal {L}$ for training images always decrease monotonically to different minima, and with a fixed $\\mathcal {D}_{\\mathrm {max}}$ it is easier for the MPS to fit fewer training images.",
"The $\\mathcal {L}$ for test images, however, behaves quite differently: for $|\\mathcal {T}|=10^{3}$ , test $\\mathcal {L}$ decreases to about $40.26$ then starts climbing quickly, while for $|\\mathcal {T}|=10^4$ the test $\\mathcal {L}$ decreases to $33.65$ then increases slowly to $34.18$ .",
"For $|\\mathcal {T}|=6\\times 10^4$ , test $\\mathcal {L}$ kept decreasing in 75 loops.",
"The behavior shown in Fig.",
"REF is quite typical in machine learning problems.",
"When training data is not enough, the model quickly overfits the training data, giving worse and worse generalization to the unseen test data.",
"An extreme example is that if our model is able to decrease training $\\mathcal {L}$ to $\\ln |\\mathcal {T}|$ , i.e.",
"completely overfits the training data, then all other images, even the images with only one pixel different from one of the training images, have zero probability in the model hence $\\mathcal {L}=\\infty $ .",
"We also observe that the best test NLL decreases as training set volume enlarges, which means the tendency of memorizing is constrained and that of generalization is enhanced.",
"The histograms of log-likelihoods for all training and test images are shown in Fig.",
"REF .",
"Notice that if the model just memorized some of the images and ignored the others, the histograms would be bi-modal.",
"It is not the case, as shown in the figure, where all distributions are centered around.",
"This indicates that the model learns all images well rather than concentrates on some images while completely ignoring the others.",
"In the bottom panel we show the detailed $\\mathcal {L}$ histogram by categories.",
"For some digits, such as “1” and “9”, the difference between training and test log-likelihood distribution is insignificant, which suggests that the model has particularly great generalization ability to these images.",
"Figure: Evolution of the average negative log-likelihood ℒ\\mathcal {L} for both training images (blue, bottom lines) and 10 4 10^4 test images (red, top lines) during training.",
"From left to right, number of images in the training set |𝒯||\\mathcal {T}| are 10 3 ,10 4 10^3, 10^4, and 6×10 4 6\\times 10^4 respectively.Figure: (Top) Distribution of -lnp-\\ln p of 60000 training images and 10000 test images given by a trained MPS with 𝒟 max =500\\mathcal {D}_{\\textrm {max}}=500.",
"The training negative log likelihood ℒ train =24.2\\mathcal {L}_{\\textrm {train}}=24.2, and the test ℒ test =30.3\\mathcal {L}_{\\textrm {test}}=30.3.",
"(Bottom) Distributions for each digit."
],
[
"Summary and Outlook",
"We have presented a tensor-network-based unsupervised model, which aims at modeling the probability distribution of samples in given unlabeled data.",
"The probabilistic model is structured as a matrix product state, which brings several advantages as discussed in Sec.",
"REF , such as adaptive and efficient learning and direct sampling.",
"Since we use square of the TN states to represent probability, the sign is redundant for probabilistic modeling besides the gauge freedom of MPS.",
"It is likely that during the optimization MPS develops different signs for different configurations.",
"The sign variation may unnecessarily increase the entanglement in MPS and therefore the bond dimensions [55].",
"However, restricting the sign of MPS may also impair the expressibility of the model.",
"One probable approach to obtain a low entanglement representation is adding a penalty term in the target function, for instance, a term proportional to Rényi entanglement entropy as in our further work on quantum tomography [56].",
"In light of these discussions, we would like point to future research on the differences and connections of MPS with non-negative matrix entries [57] and the probabilistic graphical models such as the hidden Markov model.",
"Binary data modeling links closely to quantum many-body systems with spin-$1/2$ constituents and could be straightforwardly generalized for higher dimensional data.",
"One can also follow [32], [58] to use a local feature map to lift continuous variables to a spinor space for continuous data modeling.",
"The ability and efficiency of this approach may also depend on the specific way of performing the mapping, so in terms of continuous input there are still a lot to be explored on this algorithm.",
"Moreover, for colored images one can encode the RGB values to three physical legs of each MPS tensor.",
"Similar to using MPS for studying two-dimensional quantum lattice problems [31], modeling images with MPS faces the problem of introducing long range correlations for some neighboring pixels in two dimension.",
"An obvious generalization of the present approach is to use more expressive TN with more complex structures.",
"In particular, the projected entangled pair states (PEPS) [59] is particularly suitable for images, because it takes care of correlation between pixels in two-dimension.",
"Similar to the studies of quantum systems in 2D, however, this advantage of PEPS is partially compensated by the difficulty of contracting the network and the lose of convenient canonical forms.",
"Exact contraction of a PEPS is $\\#$ P hard [60].",
"Nevertheless, one can employ tensor renormalization group methods for approximated contraction of PEPS [61], [62], [63], [64].",
"Thus, it remains to be seen whether judicious combination of these techniques really brings a better performance to generative modeling.",
"In the end, we would like to remark that perhaps the most exciting feature of quantum-inspired generative models is the possibility of being implemented by quantum devices [65], rather than merely being simulated in classical computers.",
"In that way, neither the large bond dimension nor the high computational complexity of tensor contraction, would be a problem.",
"The tensor network representation of probability may facilitate quantum generative modeling because some of the tensor network states can be prepared efficiently on a quantum computer [66], [67].",
"We thank Liang-Zhu Mu, Hong-Ye Hu, Song Cheng, Jing Chen, Wei Li, Zhengzhi Sun and E. Miles Stoudenmire for inspiring discussions.",
"We also acknowledge suggestions from anonymous reviewers.",
"P.Z.",
"acknowledges Swarm Club workshop on “Geometry, Complex Network and Machine Learning” sponsored by Kai Feng Foundation in 2016.",
"L.W.",
"is supported by the Ministry of Science and Technology of China under the Grant No.",
"2016YFA0300603 and National Natural Science Foundation of China under the Grant No.",
"11774398.",
"J.W.",
"is supported by National Training Program of Innovation for Undergraduates of China.",
"P.Z.",
"is supported by Key Research Program of Frontier Sciences，CAS，Grant No.",
"QYZDB-SSW-SYS032 and Project 11747601 of National Natural Science Foundation of China.",
"Part of the computation was carried out at the High Performance Computational Cluster of ITP, CAS."
],
[
"Canonical conditions for MPS and computation of the partition function",
"The MPS representation has gauge degrees of freedom, which means that the state is invariant after inserting identity $I = MM^{-1}$ on each bond ($M$ can be different on each bond).",
"Exploiting the gauge degrees of freedom, one can bring the MPS into its canonical form: for example, the tensor $A^{(k)}$ is called left-canonical if it satisfies $\\sum _{v_{k}\\in \\lbrace 0, 1\\rbrace } \\left( A^{(k)v_{k}} \\right)^{\\dagger }A^{(k)v_{k}} = I $ .",
"In diagrammatic notation, the left-canonical condition reads $\\begin{minipage}[c]{0.2}\\centering \\includegraphics [page=3]{TN.pdf}\\end{minipage}=\\begin{minipage}[c]{0.1}\\centering \\includegraphics [page=4]{TN.pdf}\\end{minipage}$ The right-canonical condition is defined analogously.",
"Canonicalization of each tensor can be done locally and only involves the single tensor at consideration [17], [18].",
"Each tensor in the MPS can be in a different canonical form.",
"For example, given a specific site $k$ , one can conduct gauge transformation to make all the tensors on the left, $\\lbrace A^{(i)}| i=1,2,\\cdots , k-1\\rbrace $ , left-canonical and tensors on the right, $\\lbrace A^{(i)}| i=k+1,k+2,\\cdots , N\\rbrace $ , right-canonical, while leaving $A^{(k)}$ neither left-canonical nor right-canonical.",
"This is called mixed-canonical form of the MPS [17].",
"The normalization of the MPS is particularly easy to compute in the canonical from.",
"In the graphical notation, it reads $Z =\\begin{minipage}[c]{0.35}\\centering \\includegraphics [page=5]{TN.pdf}\\end{minipage}=\\begin{minipage}[c]{0.154}\\centering \\includegraphics [page=6]{TN.pdf}\\end{minipage}.$ We note that even if the MPS is not in the canonical form, its normalization factor $Z$ can be still computed efficiently if one pays attention to the order of contraction [17], [18]."
],
[
"DMRG-like Gradient Descent algorithm for learning",
"A standard way of minimization of the cost function (REF ) is done by performing the gradient descent algorithm on the MPS tensor elements.",
"Crucially, our method allows dynamical adjustment of the bond dimension during the optimization, thus being able to allocate resources to the spatial regions where correlations among the physical variables are stronger.",
"Initially, we set the MPS with random tensors with small bond dimensions.",
"For example, all the bond dimension are set to $\\mathcal {D}_k=2$ except those on the boundaries Setting $\\mathcal {D}_k=1$ for all bonds makes the bond dimension difficult to grow in the initial training phase.",
"Since the rank of two site tensor is $1\\times 2\\times 2\\times 1$ and the number of nonzero singular value is at most 2, which is likely to be truncated back to $\\mathcal {D}_k=1$ with small cutoff.. We then carry out the canonicalization procedure so that all the tensors except the rightmost one $A^{(N)}$ are left-canonical.",
"Then, we sweep through the matrices back and forth to tune the elements of the tensors, i.e.",
"the parameters of the MPS.",
"The procedure is similar to the DMRG algorithm with two-site update where one optimizes two adjacent tensors at a time [30].",
"At each step, we firstly merge two adjacent tensors into an order-4 tensor, $\\begin{minipage}[c]{0.4}\\centering \\includegraphics [page=7]{TN.pdf}\\end{minipage}=\\begin{minipage}[c]{0.286}\\centering \\includegraphics [page=8]{TN.pdf}\\end{minipage},$ followed by adjusting its elements in order to decrease the cost function $\\mathcal {L} = \\ln Z -\\frac{1}{|\\mathcal {T}|}\\sum _{ v \\in \\mathcal {T}}\\ln |\\Psi (v )|^2 $ .",
"It is straight forward to check that its gradient with respect to an element of the tensor (REF ) reads $\\frac{\\partial \\mathcal {L}}{\\partial A^{(k,k+1)w_kw_{k+1}}_{i_{k-1}i_{k+1}}} = \\frac{ Z^{\\prime }}{Z} -\\frac{2}{|\\mathcal {T}|} \\sum _{v\\in \\mathcal {T}} \\frac{\\Psi ^{\\prime }(v )}{\\Psi (v)} ,$ where $\\Psi ^{\\prime }(v )$ denotes the derivative of the MPS with respect to the tensor (REF ), and $Z^{\\prime } = 2\\sum _{v\\in \\mathcal {V}} \\Psi ^{\\prime }(v )\\Psi (v ) $ .",
"In diagram language, they read $ \\Psi ^{\\prime }(v ) &=\\begin{minipage}[c]{0.6}\\centering \\includegraphics [page=9]{TN.pdf}\\end{minipage}\\\\\\frac{Z^{\\prime }}{2} &=\\begin{minipage}[c]{0.6}\\centering \\includegraphics [page=10]{TN.pdf}\\end{minipage} \\nonumber \\\\& =\\begin{minipage}[c]{0.3}\\centering \\includegraphics [page=11]{TN.pdf}\\end{minipage}$ The direct vertical connections of $w_k, v_k$ and $w_{k+1}, v_{k+1}$ in (REF ) stand for Kronecker delta functions $\\delta _{w_{k}v_{k}}$ and $\\delta _{w_{k+1}v_{k+1}}$ respectively, meaning that only those input data with pattern $v_{k}v_{k+1}$ contribute to the gradient with respect to the tensor elements $A^{(k,k+1)v_{k}v_{k+1}}$ .",
"Note that although $Z$ and $Z^{\\prime }$ involve summations over an exponentially large number of terms, they are tractable in MPS via efficient contraction schemes [17].",
"In particular, if the MPS is in the mixed canonical form, the computation only involves local manipulations illustrated in ().",
"Next, we carry out gradient descent to update the components of the merged tensor.",
"The update is flexible and is open to various gradient descent techniques.",
"Firstly, stochastic gradient descent is considerable.",
"Instead of averaging the gradient over the whole dataset, the second term of the gradient (REF ) can be estimated by a randomly chosen mini-batches of samples, where the size of the mini-batch $m_\\mathrm {batch}$ plays a role of hyperparameter in the training.",
"Secondly, on a specific contracted tensor one can conduct several steps of gradient descent.",
"Note that although the local update of $A^{(k,k+1)}$ does not change its environment, the shifting of $A^{(k,k+1)}$ makes a difference between $n_\\mathrm {des}$ steps of update with learning rate $\\eta $ and one update step with $\\eta ^{\\prime }=n_\\mathrm {des}\\times \\eta $ .",
"Thirdly, especially when several steps are conducted on each contracted tensor, the learning rate (the ratio of the update to the gradient) can be adaptively tuned by meta-algorithms that such as RMSProp and Adam [43].",
"In practice it is observed that sometimes the gradients become very small while it is not in the vicinity of any local minimum of the landscape.",
"In that case a plateau or a saddle point may have been encountered, and we simply increase the learning rate so that the norm of the update is a function of the dimensions of the contracted tensor.",
"After updating the order-4 tensor (REF ), it is decomposed by unfolding the tensor to a matrix, subsequently applying singular value decomposition (SVD), and finally unfolding obtained two matrices back to two order-3 tensors.",
"$\\begin{minipage}[c]{0.286}\\centering \\includegraphics [page=8]{TN.pdf}\\end{minipage}&=\\begin{minipage}[c]{0.4}\\centering \\includegraphics [page=12]{TN.pdf}\\end{minipage}\\nonumber \\\\&=\\begin{minipage}[c]{0.4}\\centering \\includegraphics [page=13]{TN.pdf}\\end{minipage}\\nonumber \\\\&\\approx \\begin{minipage}[c]{0.4}\\centering \\includegraphics [page=7]{TN.pdf}\\end{minipage}, $ where $U, V$ are unitary matrices and $\\Lambda $ is a diagonal matrix containing singular values on the diagonal.",
"The number of non-vanishing singular values will generally increase compared to the original value in Eq.",
"(REF ) because the MPS observes correlations in the data and try to capture them.",
"We truncate those singular values whose ratios to the largest one are smaller than a prescribed hyperparameter cutoff $\\epsilon _\\mathrm {cut}$ , along with their corresponding row vectors and column vectors deleted in $U$ and $V^\\dagger $ .",
"If the next bond to train on is the $(k+1)$ -th bond on the right, take $A^{(k)} = U$ so that it is left-canonical, and consequently $A^{(k+1)}= \\Lambda V^\\dagger $ .",
"While if the MPS is about to be trained on the $(k-1)$ -th bond, analogously $A^{(k+1)}=V^\\dagger $ will be right-canonical and $A^{(k)}=U\\Lambda $ .",
"This keeps the MPS in mixed-canonical form.",
"The whole training process consists of many loops.",
"In each loop the training starts from the rightmost bond (between $A^{(N-1)}$ and $A^{(N)}$ ) and sweeps to the leftmost $A^{(1)}$ , then back to the rightmost."
]
] | 1709.01662 |
[
[
"Mean-field theory of Bayesian clustering"
],
[
"Abstract We show that model-based Bayesian clustering, the probabilistically most systematic approach to the partitioning of data, can be mapped into a statistical physics problem for a gas of particles, and as a result becomes amenable to a detailed quantitative analysis.",
"A central role in the resulting statistical physics framework is played by an entropy function.",
"We demonstrate that there is a relevant parameter regime where mean-field analysis of this function is exact, and that, under natural assumptions, the lowest entropy state of the hypothetical gas corresponds to the optimal clustering of data.",
"The byproduct of our analysis is a simple but effective clustering algorithm, which infers both the most plausible number of clusters in the data and the corresponding partitions.",
"Describing Bayesian clustering in statistical mechanical terms is found to be natural and surprisingly effective."
],
[
"Introduction",
"The need for clustering analysis in scientific data exploration has grown significantly in recent years, due to the emergence of large high-dimensional datasets in areas such as high energy physics, astrophysics, biology and post-genome medicine.",
"The aim of clustering analysis is to allocate similar data items, such as stars [1], galaxies [2], bacterial communities [3], or amino-acid sequences [4], to the same category (or `cluster') in an unsupervised way.",
"Inferring the true number of clusters reliably is crucial for the discovery of new data categories.",
"Most current clustering methods, such as [5], [6], [7], make no assumptions about the data distribution, and are based on heuristic measures of similarity.",
"Some allow for estimation of the number of clusters, but use empirical approaches to do so and ad-hoc evaluation criteria tested on benchmark datasets.",
"Model-based clustering assumes that each data point comes from one of a postulated number of populations with known distributions.",
"The archetypal example is the Gaussian Mixture Model (GMM) [5], which assumes Gaussian distributions.",
"In such models Maximum likelihood (ML) inference is typically used to find data partitions [8], but this is prone to overfitting [5].",
"The number of clusters $K$ is found upon adding a `penalty' term to the log-likelihood function, such as AIC or BIC [8], sometimes with conflicting results [2].",
"Bayesian inference of GMM-generated data cures overfitting and provides a systematic way to find $K$ [5].",
"However, computing the posteriors is analytically intractable, and one tends to resort to either variational mean-field approximation [5] or computationally intensive MCMC [9].",
"A more general model-based Bayesian clustering protocol (SPD) was introduced in [10].",
"Unlike GMM, it uses priors on the partitions to compute a maximum a posteriori probability (MAP) estimate of the data partitioning.",
"Both SPD and GMM Bayesian methods are usually evaluated by clustering synthetic and benchmark real-world data.",
"This is not satisfactory; one would prefer our knowledge and our confidence in clustering outcomes to be based on more than empirical tests.",
"As a first step in this direction, in this paper we use statistical physics to study model-based Bayesian clustering.",
"This strategy was used in the past to study optimization problems, see e.g.",
"[11], and clustering [12], [13], but not Bayesian clustering.",
"Starting from the SPD model, we show that data partition inference can be formulated in terms of a quantity that can be seen as the entropy of a gas of a particles (data-points), distributed over $K$ reservoirs (clusters).",
"In the regime of a large number of particles we derive a mean-field theory to describe this gas, and show that its lowest entropy state corresponds to the optimal MAP clustering of data."
],
[
"Model of data and Bayesian clustering",
"Let us assume that we observe the sample $\\mathbf {X}=\\lbrace \\mathbf {x}_1, \\ldots , \\mathbf {x}_N\\rbrace $ , with $\\mathbf {x}_i \\in \\mathbb {R}^d$ for all $i$ , from the distribution $p\\left(\\mathbf {X}\\vert \\mathbf {\\Theta },\\Pi \\right)=\\prod _{\\mu =1}^{\\vert \\Pi \\vert }\\prod _{i_\\mu \\in S_\\mu } p(\\mathbf {x}_{i_\\mu }\\vert \\theta _{\\!\\mu }).$ This distribution is generated by the set (or `partition') $\\Pi \\!=\\!\\lbrace S_1,\\ldots ,S_{\\vert \\Pi \\vert }\\rbrace $ , with disjunct index sets (or `clusters') $S_\\mu \\!\\ne \\!\\emptyset $ , such that $S_\\mu \\!\\cap \\!",
"S_\\nu \\!=\\!\\emptyset $ for $\\mu \\ne \\nu $ and $\\cup _{\\mu =1}^{\\vert \\Pi \\vert } S_\\mu \\!=\\!",
"[N]$ with $[N]\\!=\\!\\lbrace 1,\\ldots ,N\\rbrace $ .",
"Any partition of data into $K$ clusters can be specified by binary `cluster allocation' variables $c_{i\\mu }\\!=\\!",
"{1}\\left[i\\in S_\\mu \\right]$ , where $i\\!\\in \\!",
"[N]$ and $\\mu \\!\\in \\!",
"[K]$ , forming an $N\\!\\times \\!",
"K$ partitioning matrix $\\mathbf {c}$ .",
"This matrix satisfies by construction the following constraints: $\\sum _{\\mu =1}^Kc_{i\\mu }\\!=\\!1$ for all $i\\!\\in \\!",
"[N]$ , and $\\sum _{i=1}^N c_{i\\mu }\\!\\ge \\!1$ for all $\\mu \\!\\in \\!",
"[K]$ .",
"Conversely, any $N\\times K$ matrix $\\mathbf {c}\\!\\in \\!\\lbrace 0,1\\rbrace ^{NK}$ with binary entries that satisfies these constraints induces a partition $\\Pi (\\mathbf {c})\\!=\\!\\lbrace S_1(\\mathbf {c}),\\ldots ,S_K(\\mathbf {c})\\rbrace $ of cardinality $K$ .",
"If we also know the prior distributions of model parameters, $p(\\theta _{\\!\\mu })$ , $p(\\mathbf {c}\\vert K)$ and $p(K)$ , we can use Bayes' theorem (see Appendix for details) to derive the posterior distribution $p(\\mathbf {c},K\\vert \\mathbf {X})=\\frac{\\mathrm {e}^{-N\\hat{F}_N\\left(\\mathbf {c},\\, \\mathbf {X}\\right)} p(\\mathbf {c}\\vert K) p(K) }{\\sum _{\\tilde{K}=1}^N\\!p(\\tilde{K})\\!\\sum _{ \\tilde{\\mathbf {c}} }\\mathrm {e}^{-N\\hat{F}_N\\left(\\tilde{\\mathbf {c}},\\, \\mathbf {X}\\right)} p( \\tilde{\\mathbf {c}} \\vert \\tilde{K})},$ in which $\\hat{F}_N(\\mathbf {c},\\, \\mathbf {X})&=& -\\frac{1}{N}\\log \\left\\langle \\mathrm {e}^{\\sum _{\\mu =1}^K \\sum _{i=1}^N c_{i\\mu }\\log p\\left(\\mathbf {x}_{i}\\vert \\theta _{\\!\\mu }\\right)}\\right\\rangle _{\\!\\mathbf {\\Theta }},$ with $\\langle f(\\mathbf {\\Theta })\\rangle _{\\mathbf {\\Theta }}\\!=\\!\\int \\!",
"\\big [\\prod _{\\mu =1}^K p(\\theta _{\\!\\mu })\\,\\mathrm {d}\\theta _{\\!\\mu }\\big ]f(\\mathbf {\\Theta })$ .",
"Expression (REF ) can be used to infer the most probable partition $\\Pi $ for each data sample.",
"First, for each $K\\in [N]$ one computes $\\hat{\\mathbf {c}}\\,\\vert K&=&\\mathrm {argmax}_{\\mathbf {c}}\\big \\lbrace \\mathrm {e}^{-N\\hat{F}_N(\\mathbf {c},\\, \\mathbf {X})} p(\\mathbf {c}\\vert K)\\big \\rbrace .$ Then one uses (REF ) to determine the estimate $\\hat{\\Pi }$ of $\\Pi $ : $\\hat{\\Pi }&=&\\mathrm {argmax}_{ \\hat{\\mathbf {c}}\\, \\vert K}\\big \\lbrace \\mathrm {e}^{-N\\hat{F}_N(\\hat{\\mathbf {c}},\\, \\mathbf {X})} p( \\hat{\\mathbf {c}} \\vert K) p\\left(K\\right)\\big \\rbrace .$ Clearly, a key role in our formulae is played by the function (REF ), which can be seen as an entropy of a gas of $N$ `particles' (the data-points) distributed over $K$ `reservoirs' (clusters).",
"The particles can move from one reservoir to another; $c_{i\\mu }$ tells us if particle $i$ is in reservoir $\\mu $ , and the coordinates $\\mathbf {x}_i$ act as a `quenched' disorder [14].",
"We are then interested in the minimum entropy state $\\mathrm {argmin}_{\\mathbf {c}} \\hat{F}_N(\\mathbf {c},\\, \\mathbf {X})$ ."
],
[
"Mean-field analysis of Bayesian clustering",
"Let us first consider the case where the cluster parameters are known.",
"In this case the parameter prior $p(\\theta _{\\!\\mu })$ is a delta function, and (REF ) hence becomes $\\hat{F}_N(\\mathbf {c},\\mathbf {X})&=&\\!-\\!\\sum _{\\mu =1}^K \\frac{M_\\mu (\\mathbf {c})}{N}\\!\\!\\int \\!\\mathrm {d}\\mathbf {x}~ \\hat{Q}_\\mu (\\mathbf {x}\\vert \\mathbf {c},\\mathbf {X})\\log p(\\mathbf {x}\\vert \\theta _{\\!\\mu }),~~~~$ which is now written in terms of the number of particles in cluster $\\mu $ , $M_\\mu (\\mathbf {c})=\\sum _{i=1}^N\\!c_{i\\mu }$ , and the density of particles in cluster $\\mu $ , defined as $\\hat{Q}_\\mu (\\mathbf {x}\\vert \\mathbf {c},\\,\\mathbf {X})&=&\\frac{1}{ M_\\mu (\\mathbf {c})}\\sum _{i=1}^N\\!c_{i\\mu } \\delta (\\mathbf {x}-\\mathbf {x}_{i}).$ Suppose there are $L$ distributions $q_\\nu (\\mathbf {x})$ , such that for each $\\nu $ we find $N_\\nu $ particles with $\\mathbf {x}_i$ sampled from $q_\\nu (\\mathbf {x})$ , with $\\sum _{\\nu =1}^LN_\\nu =N$ and $\\lim _{N\\rightarrow \\infty }N_\\nu /N=\\gamma (\\nu )$ .",
"For large $N$ the density (REF ) will then typically converge to $Q_\\mu (\\mathbf {x})&=&\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\, q_\\nu (\\mathbf {x}).$ Here $\\alpha (\\nu \\vert \\mu )\\!=\\!\\alpha (\\nu ,\\mu )/\\alpha (\\mu )$ is a conditional probability, defined by $\\alpha (\\mu )\\!=\\!\\lim _{N\\rightarrow \\infty } M_\\mu (\\mathbf {c})/N$ and $\\alpha (\\nu ,\\mu ) \\!=\\!\\lim _{N\\rightarrow \\infty }M_{\\nu ,\\mu }(\\mathbf {c})/N$ , where $M_\\mu (\\mathbf {c})$ is the number of particles in cluster $\\mu $ and $M_{\\nu ,\\mu }(\\mathbf {c})\\!=\\!\\sum _{i_{\\nu \\in S_\\mu (\\mathbf {c})}} {1}\\left[ \\mathbf {x}_{i_{\\nu }} \\!\\sim \\!q_\\nu (\\mathbf {x})\\right]$ is the number of those particles drawn from the distribution $q_\\nu (\\mathbf {x})$ that are allocated by $\\mathbf {c}$ to cluster $\\mu $ .",
"Clearly $\\sum _{\\mu \\le K}\\alpha (\\nu ,\\mu )\\!=\\!\\gamma (\\nu )$ , $\\sum _{\\nu \\le L} \\alpha (\\nu ,\\mu )\\!=\\!",
"\\alpha (\\mu )>0$ and $\\sum _{\\nu \\le L}\\sum _{\\mu \\le K} \\alpha (\\nu ,\\mu )\\!=\\!1$ .",
"If (REF ) holds for $N\\rightarrow \\infty $ , then $\\hat{F}_N(\\mathbf {c},\\,\\mathbf {X})$ will for $N\\rightarrow \\infty $ converge to $F( \\alpha )&=& \\sum _{\\mu =1}^K \\sum _{\\nu =1}^L \\alpha (\\nu , \\mu ) D(q_\\nu \\vert \\vert p_\\mu )+ \\sum _{\\nu =1}^L \\gamma (\\nu )H(q_\\nu ).~~~$ Here $D(q_\\nu \\vert \\vert p_\\mu )$ is the Kullback-Leibler distance between $q_\\nu (\\mathbf {x})$ and $p(\\mathbf {x}\\vert \\theta _{\\!\\mu })$ , and $H(q_\\nu )$ is a differential entropy [15].",
"The transparent and intuitive result (REF ) can be seen as a mean-field (MF) theory of $\\hat{F}_N(\\mathbf {c},\\, \\mathbf {X})$ (see Appendix for details).",
"The $L\\times K$ matrix $ \\alpha $ , with entries $\\alpha (\\nu ,\\mu )$ , acts as order parameter.",
"More generally one would have $P(F)=\\int \\!\\mathrm {d} \\alpha ~ P( \\alpha )\\, \\delta (F \\!- \\!",
"F( \\alpha ))$ , where $P( \\alpha )&=&\\lim _{N\\rightarrow \\infty }\\sum _{\\mathbf {c}, \\tilde{\\mathbf {c}}}\\,p(\\mathbf {c}\\vert K) \\,q(\\tilde{\\mathbf {c}}\\vert L)\\prod _{\\mu =1}^K\\prod _{\\nu =1}^L\\delta \\Big [\\alpha (\\nu , \\mu )- \\frac{1}{N}\\sum _{i=1}^N \\tilde{c}_{i\\nu } c_{i\\mu } \\Big ]\\!.$ Here $p(\\mathbf {c}\\vert K)$ and $q(\\tilde{\\mathbf {c}}\\vert L)$ are the assumed and the `true' distributions of partitions, respectively.",
"We can limit ourselves to working with expression (REF ), as opposed to the more involved (REF ), if $P( \\alpha )$ is a delta function.",
"We are interested in the state $ \\alpha $ for which the function $F( \\alpha )$ is minimal.",
"Firstly, from $D(q_\\nu \\vert \\vert p_\\mu )\\ge 0$ it follows that $F( \\alpha )\\ge \\sum _{\\nu =1}^L \\gamma (\\nu )H(q_\\nu )$ .",
"The lower bound is saturated when $D(q_\\nu \\vert \\vert p_\\mu )=0$ , i.e.",
"when $q_\\nu (\\mathbf {x})=p(\\mathbf {x}\\vert \\theta _{\\!\\mu })$ for all $(\\mu ,\\nu )$ , and the mapping between the sets $[L]$ and $[K]$ labelling these distributions is bijective.",
"This can only happen when $L=K$ and $\\alpha (\\nu , \\mu )=\\gamma (\\nu ){1}\\!\\left[D(q_\\nu \\vert \\vert p_\\mu )=0 \\right]$ , i.e.",
"when the `true' partitioning of the data is recovered.",
"Secondly, from $D(q_\\nu \\vert \\vert p_\\mu )\\!\\ge \\!",
"\\min _{\\tilde{\\mu }}D(q_\\nu \\vert \\vert p_{\\tilde{\\mu }})$ we deduce $F( \\alpha )\\!",
"\\ge \\!",
"\\sum _{\\nu =1}^L \\gamma (\\nu ) \\min _{\\tilde{\\mu }}D(q_\\nu \\vert \\vert p_{\\tilde{\\mu }})+\\sum _{\\nu =1}^L \\gamma (\\nu )H(q_\\nu ).$ This lower bound is saturated when $\\alpha (\\nu , \\mu )\\!=\\!\\gamma (\\nu ){1}\\!\\left[\\mu \\!=\\!\\mathrm {argmin}_{\\tilde{\\mu }}D\\left(q_\\nu \\vert \\vert p_{\\tilde{\\mu }}\\right)\\right]$ for all $(\\mu ,\\nu )$ .",
"For $K\\!\\le \\!",
"L$ , this state can be seen as the result of the following `macroscopic' clustering protocol: for all $\\nu \\in \\lbrace 1,\\ldots , L\\rbrace $ , find the distribution $p(\\mathbf {x}\\vert \\theta _{\\!\\mu })$ with the smallest distance $D(q_\\nu \\vert \\vert p_\\mu )$ to $q_\\nu (\\mathbf {x})$ , and assign all members of $\\nu $ to cluster $\\mu $ .",
"If $K\\!<\\!L$ this recipe will occasionally result in the data from more than one distribution being assigned to the same clusters, see Figure REF (a), but for $K\\!=\\!L$ , each cluster would hold only one distribution.",
"Hence, the protocol is able to recover the true partitioning even when the distributions $q_\\nu (\\mathbf {x})$ and $p(\\mathbf {x}\\vert \\theta _{\\!\\mu })$ are non-identical.",
"Figure: (Color online) Bayesian clustering: data (red rectangles) from LL different distributions q ν (𝐱)q_{\\nu }(\\mathbf {x}) are allocated to KK clusters (blue rectangles).",
"a) For K≤LK\\le L, data from q ν (𝐱)q_{\\nu }(\\mathbf {x}) occupy at most one cluster μ\\mu .",
"b) For K>LK>L, data from q ν (𝐱)q_{\\nu }(\\mathbf {x}) occupy at least one cluster.",
"c) Minimum F≡min α F(α)F\\equiv \\min _{ \\alpha } F( \\alpha ) of the mean-field entropy (blue line), shown as a function of KK and compared with the ground state entropy F ^ N ≡min 𝐜 F ^ N (𝐜,𝐗)\\hat{F}_N\\equiv \\min _{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X}) (red crosses), computed for the data of Figure .",
"The horizontal line corresponds to the lower bound ∑ ν≤L γ(ν)H(q ν )=4.853905\\sum _{\\nu \\le L}\\gamma (\\nu )H(q_\\nu )=4.853905.",
"Inset: the sum of F ^ N \\hat{F}_N and log(K)\\log (K) shown as a function of KK.",
"The minimum of this sum is obtained when K=LK=L.The inequality $D(q_\\nu \\vert \\vert p_\\mu )\\!\\ge \\min _{\\tilde{\\nu }}D(q_{\\tilde{\\nu }} \\vert \\vert p_{\\mu })$ gives the lower bound $F( \\alpha )\\!\\ge \\!\\sum _{\\mu =1}^K \\alpha (\\mu ) \\min _{\\tilde{\\nu }}D(q_{\\tilde{\\nu }} \\vert \\vert p_{\\mu })+ \\sum _{\\nu =1}^L \\gamma (\\nu )H(q_\\nu ),$ which is saturated when $\\alpha (\\nu , \\mu )=\\alpha (\\mu ){1}[\\nu \\!=\\!\\mathrm {argmin}_{\\tilde{\\nu }}D(q_{\\tilde{\\nu }} \\vert \\vert p_{\\mu })]$ for all $(\\mu ,\\nu )$ .",
"This state would result from to the following protocol: for all $\\nu \\in \\lbrace 1,\\ldots ,L\\rbrace $ , find the distribution $p(\\mathbf {x}\\vert \\theta _{\\!\\mu })$ with the smallest distance $D(q_\\nu \\vert \\vert p_\\mu )$ to $q_\\nu (\\mathbf {x})$ , and assign all members of $\\mu $ to cluster $\\nu $ .",
"For $K\\!>\\!L$ , this algorithm could allocate more than one distribution to the same cluster, see Figure REF (b).",
"Furthermore, since $\\sum _{\\nu =1}^L \\alpha (\\mu ){1}[\\nu \\!=\\!\\mathrm {argmin}_{\\tilde{\\nu }}D(q_{\\tilde{\\nu }} \\vert \\vert p_{\\mu })]=\\alpha (\\mu )$ , the properties of $\\alpha (\\nu , \\mu )$ imply validity of the set of $L$ linear equations $\\sum _{\\mu =1}^K \\alpha (\\mu ){1}[\\nu \\!=\\!\\mathrm {argmin}_{\\tilde{\\nu }}D(q_{\\tilde{\\nu }} \\vert \\vert p_{\\mu })]=\\gamma (\\nu )$ , which is underdetermined and hence has either infinitely many solutions, or no solutions at all.",
"We now consider the case where the cluster parameters are unknown, and $p(\\theta _{\\!\\mu })>0$ for all $\\lbrace \\theta _{\\!\\mu }\\rbrace $ .",
"For $N\\!\\rightarrow \\!\\infty $ , the entropy (REF ) is now strictly dominated via steepest descent by the following set of saddle point equations (see Appendix for details): $\\frac{\\partial }{\\partial \\theta _\\mu (\\ell )} \\frac{1}{N} \\sum _{i=1}^N\\!c_{i\\mu } \\log p\\left(\\mathbf {x}_{i}\\vert \\theta _{\\!\\mu }\\right)=0 .$ Solving (REF ) for Gaussian distributions $p(\\mathbf {x}_{i}\\vert \\theta _{\\!\\mu })\\equiv \\mathcal {N}\\big (\\mathbf {x}\\vert \\mathbf {m}_{\\mu },\\mathbf {\\Lambda }_{\\mu }^{-1}\\big )$ , with mean $\\mathbf {m}_{\\mu }$ and inverse covariance matrix $\\mathbf {\\Lambda }_{\\mu }$ , gives us (see Appendix ) : $\\hat{F}_N(\\mathbf {c},\\,\\mathbf {X})&=&\\sum _{\\mu =1}^K\\frac{ M_\\mu \\left(\\mathbf {c}\\right)}{2N} \\log \\Big ((2\\pi \\mathrm {e})^{d}\\big \\vert \\mathbf {\\Lambda }_{\\mu }^{-1}(\\mathbf {c},\\,\\mathbf {X})\\big \\vert \\Big ), ~$ where $\\mathbf {\\Lambda }_{\\mu }^{-1}(\\mathbf {c},\\,\\mathbf {X})$ is the empirical covariance matrix of the data in cluster $\\mu $ .",
"Since $ \\frac{1}{2} \\log \\left((2\\pi \\mathrm {e})^{d}\\left|\\mathbf {\\Lambda }_{\\mu }^{-1} \\left(\\mathbf {c},\\,\\mathbf {X}\\right)\\right|\\right) $ is the differential entropy [15] of a Gaussian distribution with covariance matrix $\\mathbf {\\Lambda }_{\\mu }^{-1}(\\mathbf {c},\\,\\mathbf {X})$ , (REF ) represents an average of $K$ entropies of Gaussian distributions, which for $N\\!\\rightarrow \\!\\infty $ will converge to the following mean-field entropy (see Appendix ): $F( \\alpha )&=& \\sum _{\\mu =1}^K\\alpha (\\mu ) \\frac{1}{2}\\log \\Big ((2\\pi \\mathrm {e})^{d}\\big \\vert \\mathbf {\\Lambda }_{\\mu }^{-1}( \\alpha )\\big \\vert \\Big ), $ in which $\\mathbf {\\Lambda }_{\\mu }^{-1}( \\alpha )$ denotes the covariance matrix $\\mathbf {\\Lambda }_{\\mu }^{-1}( \\alpha )&=&\\sum _{\\nu =1}^L\\!",
"\\alpha (\\nu \\vert \\mu )\\big \\langle \\!",
"(\\mathbf {x}\\!-\\!\\mathbf {m}_{\\mu }\\!",
"( \\alpha ))(\\mathbf {x}\\!-\\!\\mathbf {m}_{\\mu }\\!",
"( \\alpha ))^{\\!T}\\big \\rangle _\\nu ,~~~$ with $\\mathbf {m}_{\\mu }( \\alpha )=\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\langle \\mathbf {x}\\rangle _\\nu $ , and the short-hand $\\langle \\lbrace \\cdots \\rbrace \\rangle _\\nu =\\int \\!\\mathrm {d}\\mathbf {x}~ q_\\nu (\\mathbf {x}) \\lbrace \\cdots \\rbrace $ .",
"Note that (REF ) also equals $F( \\alpha )&=&\\!\\sum _{\\mu , \\nu }\\!\\alpha (\\nu , \\mu ) D(q_{\\nu } \\vert \\vert \\mathcal {N}_\\mu ( \\alpha ))+\\sum _{\\nu =1}^L\\!\\gamma (\\nu )H(q_\\nu ),~~ $ where $\\mathcal {N}_\\mu ( \\alpha )\\equiv \\mathcal {N}\\big (\\mathbf {x}\\vert \\mathbf {m}_{\\mu }( \\alpha ),\\mathbf {\\Lambda }_{\\mu }^{-1}( \\alpha )\\big )$ .",
"Moreover, as shown in Appendix , $F( \\alpha ) \\ge \\sum _{\\mu =1}^K\\alpha (\\mu )H(Q_\\mu )\\ge \\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu ).$ The second inequality in (REF ) has two consequences.",
"First, if $K\\le L$ then for any state $ \\alpha $ that corresponds to either of the scenarios depicted in Figures REF (a,b), we will have $F( \\alpha ) \\ge \\min _{K} \\min _{\\tilde{ \\alpha }} F(\\tilde{ \\alpha })=\\sum _{\\nu \\le L}\\gamma (\\nu )H \\left(q_\\nu \\right)$ .",
"The lower bound is satisfied when $L=K$ and $q_\\nu (\\mathbf {x})$ is Gaussian.",
"The `true' parameters $ \\alpha $ thus represent a locally stable state.",
"Second, when $K>L$ , the entropy $F( \\alpha )$ can only increase with $L$ .",
"This follows from (REF ) and $D(q_{\\nu } \\vert \\vert \\mathcal {N}_\\mu ( \\alpha ))\\ge 0$ .",
"If $q_\\nu (\\mathbf {x})$ is not Gaussian, then $F( \\alpha ) \\ge \\sum _{\\nu =1}^L\\gamma (\\nu )\\frac{1}{2} \\log \\left((2\\pi \\mathrm {e})^{d}\\left|\\mathbf {C}_\\nu \\right|\\right)$ , where $\\mathbf {C}_\\nu $ is the covariance matrix of $ q_\\nu (\\mathbf {x})$ (see Appendix ).",
"Equality corresponds to the state shown in the Figure REF (a) with $L=K$ , i.e.",
"here the `true' data partitioning is recovered.",
"The first inequality in (REF ) has an appealing geometric interpretation.",
"The entropy $H (Q_\\mu )$ of each cluster $\\mu $ can for large $N$ be estimated by $(d/M_\\mu (\\mathbf {c}))\\sum _{i=1}^N c_{i\\mu }\\log \\rho _{i\\mu }\\left(\\mathbf {c}\\right) +\\log \\big (M_\\mu (\\mathbf {c})\\!-\\!1\\big )+\\mbox{const.",
"}$ , where $\\rho _{i\\mu }(\\mathbf {c})=\\min _{i\\in S_\\mu \\left(\\mathbf {c}\\right)\\setminus i}\\vert \\vert \\mathbf {x}_i-\\mathbf {x}_j\\vert \\vert $ (i.e.",
"the Euclidean distance between particle $i$ and its nearest neighbour) [16].",
"The average entropy $\\sum _{\\mu =1}^K\\alpha (\\mu )H (Q_\\mu )$ is hence estimated by $(d/N)\\sum _{\\mu =1}^K\\sum _{i=1}^N c_{i\\mu }\\log \\rho _{i\\mu }(\\mathbf {c}) +\\sum _{\\mu =1}^K\\big ( M_\\mu (\\mathbf {c})/N\\big ) \\log \\big ( M_\\mu (\\mathbf {c})/N \\big ) +\\mbox{const.",
"}$ This is minimized by any state $\\mathbf {c}$ which simultaneously maximises the entropy $-\\sum _{\\mu =1}^K \\big (M_\\mu (\\mathbf {c})/N\\big ) \\log \\big ( M_\\mu (\\mathbf {c})/N \\big )$ , i.e.",
"`disperses' particles maximally over clusters, and minimizes the nearest neighbour distances $ \\lbrace \\rho _{i\\mu }(\\mathbf {c})\\rbrace $ , i.e.",
"favours high particle `densities' in each cluster.",
"The lower bound $\\sum _{\\nu =1}^L \\gamma (\\nu )H\\left(q_\\nu \\right)$ in (REF ) is saturated upon choosing any bijective map $\\alpha :\\nu \\rightarrow \\mu $ , since this immediately gives us $F( \\alpha )=\\sum _{\\nu =1}^L \\gamma (\\nu )H\\left(q_\\nu \\right)$ .",
"Such maps are special instances of the more general family $\\alpha (\\nu \\vert \\mu )&=&\\frac{{1}[\\nu \\in S_\\mu ]\\gamma (\\nu )}{\\sum _{\\tilde{\\nu } \\in S_\\mu } \\gamma (\\tilde{\\nu }) } ,$ where $\\Pi =\\lbrace S_1,\\ldots ,S_K\\rbrace $ is any partitioning of $[L]$ into $K$ subsets.",
"Finding $\\min _{ \\alpha } F( \\alpha )$ over all possible matrices of the form (REF ) by enumeration of all partitions of $[L]$ into $K$ subsets is feasible only for small $L$ , since the number of such partitions is given by the Stirling number of the second kind $\\mathcal {S}(L,K)$ which grows as $K^L$ for large $L$ [17].",
"One can also compute $\\min _{ \\alpha } F( \\alpha )$ via the following `greedy' algorithm.",
"Start with any partition $\\Pi $ and compute $F( \\alpha )$ .",
"For all $x\\in [L]$ : consider all possible moves which do not create empty clusters, and execute the one which gives the largest decrease in $F( \\alpha )$ , then update $\\Pi $ .",
"Continue the last two steps until convergence of $F( \\alpha )$ is observed.",
"This macroscopic algorithm can also be implemented `microscopically'.",
"At each step: for all $i\\in [N]$ , consider all possible moves of the particle $i$ from its current cluster $S_\\mu (\\mathbf {c})$ to a new cluster $S_\\nu (\\mathbf {c})$ and select the one which reduces $\\hat{F}_N(\\mathbf {c},\\,\\mathbf {X})$ most.",
"To evolve from a non-ordered state as in Figure REF (b) to an `ordered' state as in Figure REF (a), this microscopic algorithm has to move on average at least $N(K\\!-\\!1)/K$ particles (see Appendix ).",
"Each move was selected from among $N(K\\!-\\!1)$ possible moves, so the numerical complexity is at least of order $N^2(K\\!-\\!1)^2/K$ ."
],
[
"Results of numerical experiments",
"Our mean-field theory for was derived under the assumption that $\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ is self-averaging for $N\\rightarrow \\infty $ .",
"To investigate the correctness of its predictions for finite sample sizes $N$ , we studied low entropy states of (REF ) as obtained by the gradient descent algorithm on the data of the Figure REF .",
"For each $K\\in [17]$ we ran the algorithm from 100 different random initial states $\\mathbf {c}\\,(0)$ , and computed $\\hat{F}_N(\\mathbf {c}\\,(\\infty ),\\,\\mathbf {X})$ and the mean field entropy $F( \\alpha )$ (REF ) for each.",
"Figure: (Color online) Data used in our numerical experiments.",
"We generated L=8L=8 clusters with 1000 data-points each, of which 100 are shown here.",
"The data in each cluster (i,j,k)(i,j,k) are generated from a distinct Gaussian distribution, with mean (Δi,Δj,Δk)(\\Delta i, \\Delta j, \\Delta k), where i,j,k∈{0,1}i,j,k\\in \\lbrace 0,1\\rbrace (Δ=20\\Delta =20), and with covariance matrix sampled from the Wishart distribution with 4 degrees of freedom and precision matrix 1\\mathbf {1}.For $K\\!\\le \\!L$ , most final states $\\mathbf {c}\\,(\\infty )$ allocate data from the same distribution correctly to the same cluster, see Figure REF (a).",
"The values of $\\hat{F}_N(\\mathbf {c}\\,(\\infty ),\\,\\mathbf {X})$ are those predicted by $F( \\alpha )$ , and indeed correspond to local minima and saddle points of $F( \\alpha )$ (see Appendix ).",
"Also, according to Figure REF (c), the value $\\hat{F}_N\\!=\\!\\min _{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ as estimated from $\\mathbf {c}\\,(\\infty )$ is predicted accurately by $F\\!=\\!\\min _{ \\alpha } F( \\alpha )$ .",
"Residual differences between $\\hat{F}_N$ and $F$ reflect finite size effects.",
"These can be computed exactly when $K=L$ , and when $\\mathbf {c}\\,(\\infty )$ represents the true partitioning of the data: the average and variance of $\\hat{F}_N$ are in that case given by $\\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu )+Kd(d\\!+\\!1)/4N$ and $d/2N$ , respectively (see Appendix ).",
"Finally, we note that the number of particles `moved' by the algorithm in going from $\\mathbf {c}\\,(0)$ to $\\mathbf {c}(\\infty )$ is consistent with the lower bound $N(K\\!-\\!1)/K$ , so the algorithmic complexity is quadratic in $N$ , see Figure REF .",
"Figure: (Color online) Total (normalised) number of `moves' tt used by the gradient descent algorithm to travel from a random unbiased partition to a final partition, i.e.",
"the effective algorithmic runtime, shown as a function of the assumed number of clusters KK.",
"The minimum and maximum time (red crosses) obtained in 100 runs on the data of Figure are compared with the average lower bound (K-1)/K(K\\!-\\!1)/K (blue line).If $K\\!>\\!L$ , the states $\\mathbf {c}\\,(\\infty )$ will allocate data from the same distribution to multiple clusters, see Figure REF (b).",
"Such states are already present for small $K\\!\\le \\!",
"L$ , and proliferate as $K$ is increased (see Appendix ).",
"The lower bound $\\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu )$ is now violated, and the gap between this bound and the value of $\\hat{F}_N$ as obtained by gradient descent increases with $K$ , see Figure REF (c).",
"While some of the $\\hat{F}_N(\\mathbf {c}\\,(\\infty ),\\,\\mathbf {X})$ values are consistent with $F( \\alpha )$ (see Appendix ), the mean-field theory fails to predict $\\min _{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ in this regime, due to the non-commutation of the $N\\rightarrow \\infty $ limit and the $\\min $ operator.",
"Our estimate of $\\hat{F}_N=\\min _{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ can also be used to infer the true number of clusters $L$ .",
"Assuming uniform prior distributions of partitions $p(\\mathbf {c}\\vert K)=(K!",
"\\mathcal {S}(L,K))^{-1}$ and cluster sizes $p(K)=N^{-1}{1}[K\\!\\in \\!",
"[N]]$ in the Bayesian formulae (REF )-(REF ), the total entropy $\\hat{F}_N +\\frac{1}{N}\\log (K!",
"\\mathcal {S}(L,K))\\approx \\hat{F}_N+\\log (K)$ has its minimum at the correct value $K=L$ , see inset in Figure REF (c).",
"An interesting and important question, from a practical and a theoretical point view, is how Bayesian clustering is affected by the `separation' between different clusters.",
"The simplest non-trivial case is to consider the clustering of $d$ -dimensional data sampled from two isotropic Gaussian distributions $\\mathcal {N}(\\mathbf {m}_1,\\mathbf {1})$ and $\\mathcal {N}(\\mathbf {m}_2,\\mathbf {1})$ .",
"Here one can use the Euclidean distance $\\vert \\vert \\mathbf {m}_1-\\mathbf {m}_2\\vert \\vert =\\Delta $ , measured relative to the natural scale $\\sqrt{d}$ , as a measure of the degree of separation [18] between the `clusters' centred at $\\mathbf {m}_1$ and $\\mathbf {m}_2$ .",
"For large $d$ , most of the vectors $\\mathbf {x}$ sampled from $\\mathcal {N}\\left(\\mathbf {m},\\mathbf {1}\\right)$ will be found in the `sphere' of radius $\\sqrt{d}$ centred at $\\mathbf {m}$ , reflecting `concentration' phenomena observed for large $d$ .",
"In particular if we assume that $\\mathbf {x}$ is sampled from $\\mathcal {N}\\left(\\mathbf {m},\\mathbf {\\Lambda }\\right)$ , then $\\left\\langle \\vert \\vert \\mathbf {x}-\\mathbf {m}\\vert \\vert ^2\\right\\rangle =\\mathrm {Tr}\\,\\mathbf {\\Lambda }$ , and for $\\lambda ,\\epsilon >0$ : $&&{\\rm Prob}\\left(\\vert \\vert \\mathbf {x}-\\mathbf {m}\\vert \\vert ^2\\ge \\mathrm {Tr}\\,\\mathbf {\\Lambda }+d\\epsilon \\right)\\nonumber \\\\&&~~~~~~~~~~={\\rm Prob}\\left(\\mathrm {e}^{\\frac{\\lambda }{2} \\vert \\vert \\mathbf {x}-\\mathbf {m}\\vert \\vert ^2}\\ge \\mathrm {e}^{\\frac{\\lambda }{2}(\\mathrm {Tr}\\,\\mathbf {\\Lambda }+d\\epsilon ) }\\right)\\nonumber \\\\&&~~~~~~~~~~~~~\\le \\left\\langle \\mathrm {e}^{\\frac{\\lambda }{2} \\vert \\vert \\mathbf {x}-\\mathbf {m}\\vert \\vert ^2}\\right\\rangle \\mathrm {e}^{-\\frac{\\lambda }{2}(\\mathrm {Tr}\\,\\mathbf {\\Lambda }+d\\epsilon ) } \\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~= \\mathrm {e}^{-\\frac{1}{2}\\left(\\log \\vert \\mathbf {1}-\\lambda \\mathbf {\\Lambda }\\vert +\\lambda (\\mathrm {Tr}\\,\\mathbf {\\Lambda }+d\\epsilon ) \\right)} .$ The upper bound in the above expression was obtained using Markov's inequality and properties of Gaussian integrals.",
"For the choice $\\mathbf {\\Lambda }=\\mathbf {1}$ , the above inequality, after optimising the upper bound with respect to $\\lambda $ , gives us ${\\rm Prob}\\left(\\vert \\vert \\mathbf {x}\\!-\\!\\mathbf {m}\\vert \\vert ^2\\!\\ge \\!d(1+\\epsilon )\\right) \\le \\mathrm {e}^{-\\frac{d}{2}\\left(\\log \\frac{1}{1+\\epsilon }-\\epsilon \\right)}$ .",
"Let us now consider the MF entropy $\\min _{ \\alpha } F( \\alpha )$ for the distributions $\\mathcal {N}(\\mathbf {m}_1,\\mathbf {1})$ and $\\mathcal {N}(\\mathbf {m}_2,\\mathbf {1})$ , with separation $\\vert \\vert \\mathbf {m}_1-\\mathbf {m}_2\\vert \\vert =\\Delta $ .",
"For the assumed number of clusters $K=1$ this entropy is given by $F_1&=&\\frac{d}{2} \\log \\left(2\\pi \\mathrm {e}\\right)+\\!\\frac{1}{2}\\!\\log \\Big | \\mathbf {1} \\!+\\!",
"\\!",
"\\sum _{\\nu =1}^2\\!\\gamma (\\nu )\\!\\left( \\mathbf {m}_\\nu \\!-\\!\\mathbf {m}\\right)\\!\\left( \\mathbf {m}_\\nu \\!-\\!\\mathbf {m}\\right)^T\\!",
"\\Big | $ where $\\mathbf {m}=\\sum _{\\nu =1}^2\\gamma (\\nu )\\, \\mathbf {m}_\\nu $ , and $\\gamma (\\nu )$ is the fraction of data sampled from $\\mathcal {N}(\\mathbf {m}_\\nu ,\\mathbf {1})$ .",
"For $K=2$ we obtain $F_2&=& \\frac{d}{2} \\log \\left(2\\pi \\mathrm {e}\\right),$ which corresponds to the situation where the true clustering of data is recovered.",
"Furthermore, upon choosing $\\mathbf {m}_1\\!=\\!\\mathbf {0}$ and $\\gamma (\\nu )\\!=\\!\\frac{1}{2}$ we obtain $F_1=\\frac{d}{2} \\log (2\\pi \\mathrm {e})+\\frac{1}{2} \\log [ 1+(\\frac{\\Delta }{2})^2 ]$ .",
"Thus in this case $F_1\\ge F_2$ , as required.",
"However, if $\\log (2)\\ge \\frac{1}{2} \\log [ 1+(\\frac{\\Delta }{2})^2 ]$ then $F_2+\\log (K)\\ge F_1$ (note that we minimise $\\min _{ \\alpha } F( \\alpha ) +\\log (K)$ to infer true number of clusters), so that here we are unable to recover the correct number $K=2$ of clusters due to the cluster separation $\\Delta $ being too small.",
"This happens when $\\Delta \\le 2\\sqrt{3}\\approx 3.46$ .",
"We expect that a similar analysis can be also performed for more general scenarios.",
"Numerical experiments are in qualitative agreement with the predicted separation boundary $\\Delta =2\\sqrt{3}$ , as can be seen in Figure REF .",
"Figure: (Color online) Bayesian clustering of data 𝐱∈ℝ d \\mathbf {x}\\in \\mathbb {R}^d generated from the Gaussian distributions 𝒩(𝐦 1 ,1)\\mathcal {N}(\\mathbf {m}_1,\\mathbf {1}) and 𝒩(𝐦 2 ,1)\\mathcal {N}(\\mathbf {m}_2,\\mathbf {1}), with separation Δ=||𝐦 1 -𝐦 2 ||\\Delta =\\vert \\vert \\mathbf {m}_1-\\mathbf {m}_2\\vert \\vert .",
"The data sample, split equally between the constituent distributions, is of size N=2000N=2000 and has d=10d=10.",
"The data was generated for cluster separations Δ/d∈1 2,1,3 2,2,5 2\\Delta /\\sqrt{d}\\in \\left\\lbrace \\frac{1}{2},1,\\frac{3}{2},2,\\frac{5}{2} \\right\\rbrace .",
"Top: Data projected into two dimensions.",
"The separation Δ\\Delta of the clusters is increasing from the left to the right.",
"Bottom: the sum F ^ N +logK\\hat{F}_N+\\log K (red crosses connected by lines), where F ^ N ≡min 𝐜 F ^ N (𝐜,𝐗)\\hat{F}_N\\equiv \\min _{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X}), shown as a function of the assumed number of clusters KK, and compared with the mean-field prediction min α F(α)\\min _\\alpha F( \\alpha ) (blue crosses).",
"For K=2K=2, the mean-field prediction min α F(α)=d 2log(2πe)\\min _\\alpha F( \\alpha )=\\frac{d}{2}\\log (2\\pi \\mathrm {e}) is plotted with the finite size corrections (error bars indicate one standard deviation).In this Figure we also compare the mean-field theory results (REF , REF ) with the results of numerical simulations.",
"For $K=1$ the discrepancy between theory and simulations is a finite size effect.",
"In contrast, for $K=2$ it is a combination of finite size effects and the inability of the mean-field theory to account for correlations between the data in clusters for small separations $\\Delta $ .",
"Such correlations are also responsible for a breakdown of the mean-field theory when $K>L$ , see Figure REF .",
"For larger separations $\\Delta $ the theory is in good agreement with the simulations, see Figure REF , and discrepancies again reflect only finite size effects.",
"The magnitude of the finite size effects can be estimated when $K=L$ for any $d/N<1$ , by the following argument.",
"For the empirical covariance matrix $\\hat{\\mathbf {\\Lambda }}$ of a sample of $M$ $d$ -dimensional data vectors generated from the Gaussian distribution $\\mathcal {N}(\\mathbf {m},\\mathbf {\\Lambda })$ the random quantity $\\log \\vert \\hat{\\mathbf {\\Lambda }}\\vert $ will for large $M$ be described by the distribution $\\mathcal {N}(\\log |\\mathbf {\\Lambda }|+\\tau (M,d), \\sigma ^2(M,d) ),$ where $\\tau (M,d)=\\sum _{\\ell =1}^d \\psi ( \\frac{M-\\ell +1}{2})-d\\log (\\frac{M}{2})$ and $ \\sigma ^2(M,d)=\\sum _{\\ell =1}^d\\frac{2}{M-\\ell +1}$ [19].",
"Assuming that $K=L$ and that the clustering is perfect allows us to compute, by following steps similar to those followed in the Appendix , the average and variance of the entropy (REF ).",
"They are found to be given by $\\min _\\alpha F( \\alpha )+\\sum _{\\nu =1}^L \\gamma (\\nu )\\, \\tau \\left(\\gamma (\\nu )N,d\\right)$ and $\\frac{1}{4}\\sum _{\\nu =1}^L \\gamma ^2(\\nu )\\, \\sigma ^2\\left(\\gamma (\\nu )N,d\\right)$ , respectively.",
"When evaluated for real datasets, the entropy function (REF ) may also have value as an exploratory tool.",
"To show this, we consider the Wisconsin Diagnostic Breast Cancer (WDBC) dataset [20], which describes characteristics of cell nuclei in the images of cells extracted from tumours [21], and contains $N=569$ data-points of dimension $d=30$ .",
"This dataset has two (linearly separable) classes, which we assume to be the `true' clusters, one is `benign', represented by 357 data-points, and the other is `malignant', represented by 212 data-points [21].",
"A first simple unsupervised method which one might apply to this dataset is hierarchical clustering, which uses pairwise distances between the data-points to build a hierarchy of clusters, see e.g. [22].",
"The agglomerative version of this algorithm, with Euclidean distances, separates this data into clusters of sizes 549 and 20 at the $K=2$ clusters level of hierarchy, into clusters of sizes 549, 19 and 1 at the $K=3$ clusters level of hierarchy, into clusters of sizes 438, 111, 19 and 1 at the $K=4$ clusters level of hierarchy, etc.",
"Hence, upon assuming (correctly) that $K=2$ , one cannot recover the true clusters of the WDBC data with this algorithm.",
"Alternatively, the $K$ -Means clustering algorithm, see e.g.",
"[5], which minimises the squared Euclidean distance between the points in a cluster, `finds' in the WDBC dataset (again upon assuming $K=2$ ) clusters of sizes 438 and 131.",
"Upon comparing these with the true clusters, we observe that $K$ -Means `misclassifies' 83 data-points in total.",
"It is interesting that the clusters found by $K$ -Means were also present in the four clusters generated via hierarchical clustering.",
"Using instead the gradient descent minimisation of (REF ) as a clustering protocol suggests that there are more than $K=4$ clustersFor $K>4$ , this approach favours small clusters, i.e.",
"we are in non-asymptotic regime, which suggests that a full Bayesian framework is more appropriate for this data.",
"in the WDBC dataset (see Figure REF ).",
"For $K=2$ the algorithm outputs clusters of sizes 328 and 241, which is, compared with the hierarchical and $K$ -Means results, much closer to the true sizes 357 and 212 of the WDBC dataset.",
"Now 57 data-points were misclassified, which can be explained by the non-sphericity of clusters in this dataset.",
"In particular, for any data covariance matrix $\\hat{\\Sigma }$ the ratio $\\mathcal {S}(\\hat{\\Sigma })=\\mathrm {Tr}^2(\\hat{\\Sigma })/d\\mathrm {Tr}(\\hat{\\Sigma }^2)$ can be used as a measure of `sphericity' of data, see e.g. [23].",
"We note that $1/d\\le \\mathcal {S}(\\hat{\\Sigma }) \\le 1$ , and that the lower bound $1/d$ is saturated only when a few eigenvalues dominate all others for large $d$ , i.e.",
"when only a few `directions' in $\\mathbb {R}^d$ contribute to the variability in the data.",
"The upper bound is saturated when all eigenvalues are equal, i.e.",
"all directions in $\\mathbb {R}^d$ contribute equally to the variability.",
"The sphericity values of the `benign' and `malignant' clusters in the WDBC dataset are given by $0.034$ and $0.036$ , respectively, so the data in these clusters is highly non-spherical.",
"This indeed suggests that the entropy function (REF ), derived upon assuming arbitrary multivariate Gaussian distributions of a data in the clusters, is better equipped to deal with this scenario than hierarchical or $K$ -Means clustering.",
"Figure: (Color online) The sum F ^ N +logK\\hat{F}_N+\\log K, where F ^ N ≡min 𝐜 F ^ N (𝐜,𝐗)\\hat{F}_N\\equiv \\min _{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X}), as computed for the Wisconsin Diagnostic Breast Cancer data (red crosses connected by lines), shown as a function of the assumed number of clusters KK.",
"These results suggest that the true number of clusters in this dataset is at least K=4K=4."
],
[
"Summary",
"In conclusion, in this paper we have demonstrated that mapping Bayesian clustering of data to a statistical mechanical problem is not only possible, but in fact also quite intuitive and fruitful.",
"It enables us to identify objectively the most plausible number of clusters in a dataset, and to obtain transparent interpretations and explanations of why and how conventional clustering methods (which are quite often based on ad-hoc definitions) may or may not fail to detect clusters correctly, dependent on the quantitative features of the data.",
"One possible extension of this work, currently in progress, is a more general analytical treatment of this Bayesian clustering problem, in which the distribution $P(F)=\\int \\!\\mathrm {d} \\alpha ~ P( \\alpha )\\, \\delta (F \\!- \\!",
"F( \\alpha ))$ is no longer assumed to converge to a delta distribution for large $N$ .",
"This will allow us allow us to tackle also the nontrivial regime where $N,d\\rightarrow \\infty $ with $N/d$ finite, and to correct the present mean-field theory in the $K>L$ regime.",
"This work was supported by the Medical Research Council of the United Kingdom (grant MR/L01257X/1)."
],
[
"Model of data and Bayesian clustering",
"Let us assume that we observe the sample $\\mathbf {X}=\\lbrace \\mathbf {x}_1, \\ldots , \\mathbf {x}_N\\rbrace $ , where $\\mathbf {x}_i \\in \\mathbb {R}^d$ for all $i$ , drawn from the distribution $p\\left(\\mathbf {X}\\vert \\mathbf {\\Theta },\\Pi \\right)=\\prod _{\\mu =1}^{\\vert \\Pi \\vert }\\prod _{i_\\mu \\in S_\\mu } p(\\mathbf {x}_{i_\\mu }\\vert \\theta _{\\!\\mu }),$ generated by the partition $\\Pi =\\left\\lbrace S_1, S_2,\\ldots ,S_{\\vert \\Pi \\vert }\\right\\rbrace $ , where the index sets $S_\\mu \\ne \\emptyset $ obey $S_\\mu \\cap S_\\nu =\\emptyset $ for $\\mu \\ne \\nu $ , and $\\cup _{\\mu =1}^{\\vert \\Pi \\vert } S_\\mu =[N]$ , with the short-hand $[N]=\\lbrace 1,\\ldots ,N\\rbrace $ .",
"Furthermore, we assume that each parameter $\\theta _{\\!\\mu }$ is sampled randomly and independently from the distribution $p(\\theta _{\\!\\mu })$ , and that we are also given the prior distribution of $\\Pi $ , $P(\\Pi )$ .",
"This allows us to write down the joint distribution $p\\left(\\mathbf {X},\\mathbf {\\Theta },\\Pi \\right)&=&p\\left(\\mathbf {X}\\vert \\mathbf {\\Theta },\\Pi \\right)p\\left(\\Pi \\right)\\prod _{\\mu =1}^{\\vert \\Pi \\vert } p(\\theta _{\\!\\mu }),$ where $\\mathbf {\\Theta }=\\lbrace \\theta _1,\\ldots ,\\theta _{\\vert \\Pi \\vert }\\rbrace $ .",
"Upon integrating out the parameters $\\theta _{\\!\\mu }$ in the above we obtain the distribution $p\\left(\\mathbf {X}, \\Pi \\right)&=&\\left\\langle p\\left(\\mathbf {X}\\vert \\mathbf {\\Theta },\\Pi \\right) \\right\\rangle _{\\mathbf {\\Theta }\\vert \\Pi } p\\left(\\Pi \\right),$ where $\\left\\langle f\\left(\\mathbf {\\Theta }\\right) \\right\\rangle _{\\mathbf {\\Theta }\\vert \\Pi }=\\int f\\left(\\mathbf {\\Theta }\\right) \\!\\big \\lbrace \\prod _{\\mu =1}^{\\vert \\Pi \\vert } p(\\theta _{\\!\\mu })\\,\\mathrm {d}\\theta _{\\!\\mu }\\big \\rbrace $ .",
"From this follows the conditional distribution $p\\left( \\Pi \\vert \\mathbf {X}\\right)&=&\\frac{ p\\left(\\mathbf {X}\\vert \\Pi \\right) p\\left(\\Pi \\right)}{\\sum _{\\tilde{\\Pi }} p(\\mathbf {X}\\vert \\tilde{\\Pi }) p(\\tilde{\\Pi })}$ with $p\\left(\\mathbf {X}\\vert \\Pi \\right)&=&\\left\\langle p\\left(\\mathbf {X}\\vert \\mathbf {\\Theta },\\Pi \\right) \\right\\rangle _{\\mathbf {\\Theta }\\vert \\Pi } .$ Let us next consider the `partition function' $&&\\sum _{\\Pi } p\\left(\\mathbf {X}\\vert \\Pi \\right) p\\left(\\Pi \\right)\\nonumber \\\\[-1mm]&&~~~~~=\\sum _{K=1}^N\\sum _{\\Pi } p\\left(\\mathbf {X}\\vert \\Pi \\right) p\\left(\\Pi \\right){1}\\left[\\vert \\Pi \\vert =K\\right]\\\\&&~~~~~=\\sum _{K=1}^N\\sum _{\\Pi } p\\left(\\mathbf {X}\\vert \\Pi \\right) p\\left(\\Pi \\vert K\\right)p(K)\\nonumber ,$ where we have defined the two distributions $p\\left(\\Pi \\vert K\\right)&=&\\frac{p\\left(\\Pi \\right){1}\\left[\\vert \\Pi \\vert =K\\right]}{\\sum _{\\tilde{\\Pi }}p(\\tilde{\\Pi }){1}\\big [\\vert \\tilde{\\Pi }\\vert =K\\big ]}\\\\[1mm]p(K)&=&\\sum _{\\Pi }p\\left(\\Pi \\right){1}\\left[\\vert \\Pi \\vert =K\\right]\\nonumber .$ Furthermore, if we define $\\Pi _K$ to be a partition $\\Pi $ with $\\vert \\Pi \\vert =K$ , i.e.",
"$\\Pi _K=\\lbrace S_1,\\ldots ,S_K\\rbrace $ , then $&&\\sum _{\\Pi } p\\left(\\mathbf {X}\\vert \\Pi \\right) p\\left(\\Pi \\right)\\nonumber \\\\[-1mm]&&~~~~~=\\sum _{K=1}^N p(K)\\sum _{\\Pi _K} p\\left(\\mathbf {X}\\vert \\Pi _K\\right) p\\left(\\Pi _K\\vert K\\right)$ and the distribution of $\\Pi _K$ is given by $p\\left(\\Pi _K\\vert (\\mathbf {X}\\right)&=&\\frac{p\\left(\\mathbf {X}\\vert \\Pi _K\\right) p\\left(\\Pi _K\\vert K\\right) p(K)}{\\sum _{\\tilde{K}=1}^N p(\\tilde{K})\\sum _{\\tilde{\\Pi }_{\\tilde{K}}} p(\\mathbf {X}\\vert \\tilde{\\Pi }_{\\tilde{K}}) p(\\tilde{\\Pi }_{\\tilde{K}}\\vert \\tilde{K})}.$ The mode of this distribution is located at $\\hat{\\Pi }_K&=&\\mathrm {argmax}_{\\Pi _K} \\Big \\lbrace p\\left(\\mathbf {X}\\vert \\Pi _K\\right) p\\left(\\Pi _K\\vert K\\right)\\Big \\rbrace .",
"$ from which, in turn, it follows that the mode of the distribution (REF ) is located at $\\hat{\\Pi }&=&\\mathrm {argmax}_{\\hat{\\Pi }_K} \\Big \\lbrace p(\\mathbf {X}\\vert \\hat{\\Pi }_K) p(\\hat{\\Pi }_K\\vert K) p(K)\\Big \\rbrace .$ To see this one considers $\\hat{\\Pi }&=& \\mathrm {argmax}_{\\Pi } \\left\\lbrace p\\left(\\mathbf {X}\\vert \\Pi \\right) p\\left(\\Pi \\right)\\right\\rbrace \\nonumber \\\\&=&\\mathrm {argmax}_{\\Pi }\\Big \\lbrace \\left\\lbrace p\\left(\\mathbf {X}\\vert \\Pi _1\\right) p\\left(\\Pi _1\\right)\\right\\rbrace ,\\ldots \\nonumber \\\\&&~~~~~~~~~\\ldots ,\\left\\lbrace p\\left(\\mathbf {X}\\vert \\Pi _K\\right) p\\left(\\Pi _K\\right)\\right\\rbrace ,\\ldots \\nonumber \\\\&&~~~~~~~~~~~~~~~~\\ldots ,\\left\\lbrace p\\left(\\mathbf {X}\\vert \\Pi _N\\right) p\\left(\\Pi _N\\right)\\right\\rbrace \\Big \\rbrace \\nonumber ,$ where $\\left\\lbrace p\\left(\\mathbf {X}\\vert \\Pi _K\\right) p\\left(\\Pi _K\\right)\\right\\rbrace $ is a set generated by $\\lbrace \\Pi _K\\rbrace $ .",
"Clearly, $\\max _{\\Pi _K}\\left\\lbrace p\\left(\\mathbf {X}\\vert \\Pi _K\\right) p\\left(\\Pi _K\\right)\\right\\rbrace =p(\\mathbf {X}\\vert \\hat{\\Pi }_K) p(\\hat{\\Pi }_K)$ , in which $\\hat{\\Pi }_K=\\mathrm {argmax}_{\\Pi _K}\\left\\lbrace p(\\mathbf {X}\\vert \\Pi _K) p\\left(\\Pi _K\\right)\\right\\rbrace $ , from which follows that $\\hat{\\Pi }&=&\\mathrm {argmax}_{\\hat{\\Pi }_K}\\left\\lbrace p(\\mathbf {X}\\vert \\hat{\\Pi }_K) p(\\hat{\\Pi }_K)\\right\\rbrace \\nonumber \\\\&=&\\mathrm {argmax}_{\\hat{\\Pi }_K} \\Big \\lbrace p(\\mathbf {X}\\vert \\hat{\\Pi }_K) p(\\hat{\\Pi }_K\\vert K) p(K)\\Big \\rbrace .$ Any partition $\\Pi _K$ of the data into $K$ clusters can be specified by the binary `allocation' variables $c_{i\\mu }={1}\\left[i\\in S_\\mu \\right]$ , where $i\\in [N]$ and $\\mu \\in [K]$ , forming the matrix $\\mathbf {c}$ with $\\left[\\mathbf {c}\\right]_{i\\mu }=c_{i\\mu }$ .",
"Hence $\\Pi _K\\equiv \\Pi _K\\left(\\mathbf {c}\\right)=\\lbrace S_1\\left(\\mathbf {c}\\right),\\ldots ,S_K\\left(\\mathbf {c}\\right)\\rbrace $ .",
"Conversely, an $N\\!\\times \\!",
"K$ matrix $\\mathbf {c}$ with binary entries is a partition only if it satisfies the constraints $\\sum _{\\mu =1}^Kc_{i\\mu }=1\\mbox{ for all } i\\in [N]$ and $\\sum _{i=1}^N c_{i\\mu }\\ge 1 \\mbox{ for all } \\mu \\in [K]$ .",
"The simplest distribution implementing these constraints is the uniform distribution $p\\left(\\mathbf {c}\\vert K\\right)&=&\\frac{\\left\\lbrace \\prod _{i=1}^N {1}\\left[\\sum _{\\nu =1}^Kc_{i\\nu }=1\\right]\\right\\rbrace \\!\\!\\left\\lbrace \\prod _{\\mu =1}^K{1}\\left[\\sum _{j=1}^N c_{j\\mu }\\ge 1\\right]\\right\\rbrace }{\\sum _{ \\tilde{\\mathbf {c}}}\\left\\lbrace \\prod _{i=1}^N {1}\\left[\\sum _{\\nu =1}^K\\tilde{c}_{i\\nu }=1\\right]\\right\\rbrace \\!\\!\\left\\lbrace \\prod _{\\mu =1}^K{1}\\left[\\sum _{j=1}^N \\tilde{c}_{j\\mu }\\ge 1\\right]\\right\\rbrace }.$ The denominator in this expression gives the total number of partitions of the set $[N]$ into $K$ subsets $\\mathcal {S}(N,K)$ , i.e.",
"it equals the Stirling number of the second kind times the number $K!$ of subset permutations.",
"Thus the probability of each individual partition $\\mathbf {c}$ is given by $1/K!\\,\\mathcal {S}(N,K)$ .",
"We note that for $N\\rightarrow \\infty $ and $K\\in O(N^0)$ we have $N^{-1}\\log (K!\\mathcal {S}(N,K))\\rightarrow \\log (K)$ [17].",
"Using this new notation allows us to write the distribution $p(\\mathbf {X}\\vert \\Pi _K)$ as $p(\\mathbf {X}\\vert \\Pi _K)&\\equiv &p\\left(\\mathbf {X}\\vert \\mathbf {c},K\\right)\\\\[1mm]&=&\\left\\langle \\mathrm {e}^{\\sum _{\\mu =1}^K \\sum _{i=1}^N c_{i\\mu }\\log p\\left(\\mathbf {x}_{i}\\vert \\theta _{\\!\\mu }\\right)}\\right\\rangle _{\\mathbf {\\Theta }}\\nonumber \\\\&=&\\mathrm {e}^{-N\\hat{F}_N\\left(\\mathbf {c},\\, \\mathbf {X}\\right)}\\nonumber ,$ where $\\left\\langle f\\left(\\mathbf {\\Theta }\\right) \\right\\rangle _{\\mathbf {\\Theta }}=\\int f\\left(\\mathbf {\\Theta }\\right) \\!\\big \\lbrace \\prod _{\\mu =1}^K p(\\theta _{\\!\\mu })\\,\\mathrm {d}\\theta _{\\!\\mu }\\big \\rbrace $ , and we defined the log-likelihood $\\hat{F}_N(\\mathbf {c},\\, \\mathbf {X})&=& -\\frac{1}{N}\\log \\left\\langle \\mathrm {e}^{\\sum _{\\mu =1}^K \\sum _{i=1}^N c_{i\\mu }\\log p\\left(\\mathbf {x}_{i}\\vert \\theta _{\\!\\mu }\\right)}\\right\\rangle _{\\!\\mathbf {\\Theta }}\\nonumber \\\\&&.$ Furthermore, combining $p\\left(\\mathbf {c},K\\right)=p\\left(\\mathbf {c}\\vert K\\right)p(K)$ with (REF ) gives us the joint distribution $p\\left(\\mathbf {X}, \\mathbf {c}, K\\right)&=&\\mathrm {e}^{-N\\hat{F}_N\\left(\\mathbf {c},\\, \\mathbf {X}\\right)} p\\left(\\mathbf {c},K\\right)$ from which we can derive the conditional distribution $&&p\\left(\\mathbf {c},K\\vert \\mathbf {X}\\right)\\nonumber \\\\&&~~~~=\\frac{\\mathrm {e}^{-N\\hat{F}_N(\\mathbf {c},\\, \\mathbf {X})} p(\\mathbf {c}\\vert K) p(K) }{\\sum _{\\tilde{K}=1}^N\\!p(\\tilde{K})\\!\\sum _{ \\tilde{\\mathbf {c}} }\\mathrm {e}^{-N\\hat{F}_N\\left(\\tilde{\\mathbf {c}},\\, \\mathbf {X}\\right)} p( \\tilde{\\mathbf {c}} \\vert \\tilde{K})}.$ For $K\\in [N]$ the mode of this distribution is located at $\\hat{\\mathbf {c}}\\,\\vert K&=&\\mathrm {argmax}_{\\mathbf {c}}\\, p\\left(\\mathbf {c},K\\vert \\mathbf {X}\\right)\\nonumber \\\\&=&\\mathrm {argmax}_{\\mathbf {c}}\\big \\lbrace \\mathrm {e}^{-N\\hat{F}_N\\left(\\mathbf {c},\\, \\mathbf {X}\\right)} p\\left(\\mathbf {c}\\vert K\\right)\\big \\rbrace $ and hence the mode of (REF ) is given by $\\hat{\\Pi }&=&\\mathrm {argmax}_{ \\hat{\\mathbf {c}}\\, \\vert K}\\big \\lbrace \\mathrm {e}^{-N\\hat{F}_N\\left(\\mathbf {c},\\, \\mathbf {X}\\right)} p\\left( \\hat{\\mathbf {c}} \\vert K\\right) p(K)\\big \\rbrace $ which is our MAP estimator of the partition of data $\\Pi $ ."
],
[
"Laplace approximation",
"Let us consider the log-likelihood density (REF ).",
"We note that $\\hat{F}_N\\left(\\mathbf {c},\\, \\mathbf {X}\\right)=\\sum _{\\mu =1}^K \\hat{F}_\\mu ^N\\left(\\mathbf {c},\\, \\mathbf {X}\\right)$ , where $\\hat{F}_\\mu ^N\\left(\\mathbf {c},\\, \\mathbf {X}\\right)&=&-\\frac{1}{N}\\!\\log \\!\\!",
"\\int \\!\\!",
"\\mathrm {e}^{-N\\Phi _\\mu (\\theta _{\\!\\mu }\\vert \\mathbf {c},\\,\\mathbf {X})} p(\\theta _{\\!\\mu }) \\mathrm {d}\\theta _{\\!\\mu } \\\\\\Phi _\\mu (\\theta _{\\!\\mu }\\vert \\mathbf {c},\\,\\mathbf {X})&=&-\\frac{1}{N} \\sum _{i=1}^N\\!c_{i\\mu }\\!\\log p\\left(\\mathbf {x}_{i}\\vert \\theta _{\\!\\mu }\\right)\\nonumber $ $\\hat{F}_\\mu ^N(\\mathbf {c},\\, \\mathbf {X})$ is a log-likelihood density of cluster $\\mu $ .",
"For large $N$ it can be evaluated by the Laplace method [24]: $\\hat{F}_\\mu ^N\\left(\\mathbf {c},\\, \\mathbf {X}\\right)&=&-\\frac{1}{N}\\log \\Bigg (\\frac{\\int \\mathrm {e}^{-N\\Phi _\\mu (\\theta _{\\!\\mu }\\vert \\mathbf {c},\\,\\mathbf {X})} p(\\theta _{\\!\\mu }) \\mathrm {d}\\theta _{\\!\\mu }}{\\int \\mathrm {e}^{-N\\Phi _\\mu ( \\tilde{\\theta }_\\mu \\vert \\mathbf {c},\\,\\mathbf {X})} \\mathrm {d}\\tilde{\\theta }_\\mu } \\!\\int \\!",
"\\mathrm {e}^{-N\\Phi _\\mu ( \\tilde{\\theta }_\\mu \\vert \\mathbf {c},\\,\\mathbf {X})} \\mathrm {d}\\tilde{\\theta }_\\mu \\!",
"\\Bigg ) \\nonumber \\\\&&=\\Phi _\\mu (\\theta _{\\!\\mu }^*\\vert \\mathbf {c}),$ where $\\theta _{\\!\\mu }^*&=&\\mathrm {argmin}_{\\theta } \\Phi _\\mu (\\theta \\vert \\mathbf {c},\\,\\mathbf {X}).$ The stationarity condition $\\frac{\\partial }{\\partial \\theta _\\mu (\\ell )}\\Phi _\\mu (\\theta \\vert \\mathbf {c},\\,\\mathbf {X})=0$ for all $\\ell $ , from which to solve $\\theta _{\\!\\mu }^*$ , gives us the equations $\\frac{\\partial }{\\partial \\theta _\\mu (\\ell )} \\frac{1}{N} \\sum _{i=1}^N\\!c_{i\\mu } \\log p\\left(\\mathbf {x}_{i}\\vert \\theta _{\\!\\mu }\\right)=0 .$ Let us now evaluate (REF ) for the multivariate Gaussian distributions $\\mathcal {N}(\\mathbf {x}\\vert \\mathbf {m}_{\\mu },\\mathbf {\\Lambda }_{\\mu }^{-1})&=&\\frac{\\mathrm {e}^{-\\frac{1}{2} (\\mathbf {x}-\\mathbf {m}_{\\mu })^T\\mathbf {\\Lambda }_\\mu (\\mathbf {x}-\\mathbf {m}_{\\mu })}}{|2\\pi \\mathbf {\\Lambda }_{\\mu }^{-1}|^{\\frac{1}{2}}} ,$ with the means $\\mathbf {m}_{\\mu }$ and the inverse covariance matrices $\\mathbf {\\Lambda }_{\\mu }$ .",
"Upon assuming that $ p\\left(\\mathbf {x}_{i}\\vert \\theta _{\\!\\mu }\\right)\\equiv \\mathcal {N}(\\mathbf {x}\\vert \\mathbf {m}_{\\mu },\\mathbf {\\Lambda }_{\\mu }^{-1})$ , the desired log-likelihood density becomes $&&-\\frac{1}{N} \\sum _{i=1}^N\\!c_{i\\mu } \\log \\mathcal {N}\\left(\\mathbf {x}_i\\vert \\mathbf {m}_{\\mu },\\mathbf {\\Lambda }_{\\mu }^{-1}\\right)=\\nonumber \\\\[-1mm]&&~~~~~~~~~~~~~\\frac{1}{2N} \\sum _{i=1}^N\\!c_{i\\mu }(\\mathbf {x}_i-\\mathbf {m}_{\\mu })^T\\mathbf {\\Lambda }_\\mu (\\mathbf {x}_i-\\mathbf {m}_{\\mu })\\nonumber \\\\&&~~~~~~~~~~~~~-\\frac{ M_\\mu \\left(\\mathbf {c}\\right) }{2N}\\log \\left((2\\pi )^{-d}\\left|\\mathbf {\\Lambda }_\\mu \\right|\\right),$ Here $M_\\mu (\\mathbf {c})\\!=\\!\\sum _{i=1}^N c_{i\\mu }\\!=\\!\\vert S_\\mu \\left(\\mathbf {c}\\right)\\vert $ denotes the number of data points in cluster $\\mu $ .",
"Solving the equations $\\frac{\\partial }{\\partial m_{\\mu \\ell }} \\sum _{i=1}^N\\!c_{i\\mu } \\log \\mathcal {N}(\\mathbf {x}_i\\vert \\mathbf {m}_{\\mu },\\mathbf {\\Lambda }_{\\mu }^{-1})=0$ and $\\frac{\\partial }{\\partial \\left[\\mathbf {\\Lambda }_\\mu \\right]_{s\\ell } } \\sum _{i=1}^N\\!c_{i\\mu } \\log \\mathcal {N}(\\mathbf {x}_i\\vert \\mathbf {m}_{\\mu },\\mathbf {\\Lambda }_{\\mu }^{-1})=0$ gives us $\\mathbf {m}_{\\mu } &=&\\frac{1}{ M_\\mu \\left(\\mathbf {c}\\right) } \\sum _{i=1}^N\\!c_{i\\mu }\\mathbf {x}_i\\\\\\mathbf {\\Lambda }_{\\mu }^{-1}&=&\\frac{1}{ M_\\mu \\left(\\mathbf {c}\\right) } \\sum _{i=1}^N\\!c_{i\\mu }\\left(\\mathbf {x}_i-\\mathbf {m}_{\\mu }\\right)\\left(\\mathbf {x}_i-\\mathbf {m}_{\\mu }\\right)^T\\!,~$ i.e.",
"the empirical mean and covariance of the data in cluster $\\mu $ .",
"Using the above results in equation (REF ) we then obtain the log-likelihood density (REF )."
],
[
"Distribution of log-likelihood – A `field theory' approach ",
"Let us assume that the data $\\mathbf {X}=\\lbrace \\mathbf {x}_1,\\ldots ,\\mathbf {x}_N\\rbrace $ are sampled from the distribution $p(\\mathbf {X}\\vert L)&=&\\sum _{\\tilde{\\mathbf {c}}}q(\\tilde{\\mathbf {c}}\\vert L) \\left\\lbrace \\prod _{\\nu =1}^L \\prod _{i_\\nu \\in S_\\nu (\\tilde{\\mathbf {c}})} q_{\\nu }(\\mathbf {x}_{i_\\nu })\\right\\rbrace ,$ where $q(\\tilde{\\mathbf {c}}\\vert L)$ is the `true' distribution of the partitions $\\tilde{\\mathbf {c}}$ of size $L$ .",
"We are interested in computing the distribution of log-likelihoods $P_N(F)&=&\\sum _{\\mathbf {c}}p(\\mathbf {c}\\vert K) \\!",
"\\int \\!",
"\\mathrm {d}\\mathbf {X}~p(\\mathbf {X}\\vert L)\\delta \\big (F\\!-\\!\\hat{F}_N(\\mathbf {c},\\mathbf {X})\\big )\\nonumber \\\\[-1mm]&& \\\\\\hat{F}_N\\left(\\mathbf {c},\\,\\mathbf {X}\\right)&=&-\\frac{1}{N}\\sum _{\\mu =1}^K\\sum _{i=1}^N\\!c_{i\\mu }\\!\\log p\\left(\\mathbf {x}_{i}\\vert \\theta _{\\!\\mu }\\right).",
"$ Here $p(\\mathbf {c}\\vert K)$ is our `assumed' distribution of the partition $\\mathbf {c}$ of size $K$ .",
"Let us now evaluate $P_N(F)$ further: $P_N(F)&=&\\sum _{\\mathbf {c}, \\tilde{\\mathbf {c}}}p(\\mathbf {c}\\vert K) q(\\tilde{\\mathbf {c}}\\vert L) \\int \\left\\lbrace \\prod _{\\nu =1}^L \\prod _{i_\\nu \\in S_\\nu (\\tilde{\\mathbf {c}})} q_{\\nu }(\\mathbf {x}_{i_\\nu })\\right\\rbrace \\delta \\left(F-\\hat{F}_N(\\mathbf {c},\\mathbf {X})\\right) \\mathrm {d}\\mathbf {X}$ We note that the sum over $\\tilde{\\mathbf {c}}$ inside the function $\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ can be written in the following form $-\\hat{F}_N\\left(\\mathbf {c},\\mathbf {X}\\right)&=&\\sum _{\\mu =1}^K\\frac{\\vert S_\\mu (\\mathbf {c}) \\vert }{N}\\int \\frac{1}{\\vert S_\\mu (\\mathbf {c}) \\vert } \\sum _{i_\\mu \\in S_\\mu (\\mathbf {c})} \\delta \\left(\\mathbf {x}-\\mathbf {x}_{i_\\mu }\\right)\\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\mathrm {d}\\mathbf {x}\\nonumber \\\\&=&\\sum _{\\mu =1}^K\\!\\frac{\\vert S_\\mu (\\mathbf {c}) \\vert }{N}\\!\\!\\!\\int \\!",
"\\!\\!\\frac{1}{\\vert S_\\mu (\\mathbf {c}) \\vert }\\sum _{\\nu =1}^L\\sum _{i_{\\mu \\nu }\\in S_\\mu (\\mathbf {c}) \\cap S_\\nu (\\tilde{\\mathbf {c}})} \\delta \\!\\left(\\mathbf {x}-\\mathbf {x}_{i_{\\mu \\nu }}\\right)\\!\\log p\\!\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\!\\mathrm {d}\\mathbf {x}\\nonumber \\\\&=&\\sum _{\\mu =1}^K \\frac{\\vert S_\\mu (\\mathbf {c}) \\vert }{N} \\int Q_\\mu (\\mathbf {x}\\vert \\mathbf {c},\\tilde{\\mathbf {c}},\\mathbf {X}) \\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\mathrm {d}\\mathbf {x},$ where we have defined the density $Q_\\mu (\\mathbf {x}\\vert \\mathbf {c},\\tilde{\\mathbf {c}},\\mathbf {X})&=&\\frac{1}{\\vert S_\\mu (\\mathbf {c}) \\vert }\\!\\sum _{\\nu =1}^L \\sum _{i_{\\mu \\nu }\\in S_\\mu (\\mathbf {c}) \\cap S_\\nu (\\tilde{\\mathbf {c}})}\\!\\!",
"\\!\\!\\!\\delta \\!\\left(\\mathbf {x}-\\mathbf {x}_{i_{\\mu \\nu }}\\right)$ Using the above form in (REF ) we obtain $P_N(F)&=&\\sum _{\\mathbf {c}, \\tilde{\\mathbf {c}}}p(\\mathbf {c}\\vert K) \\,q(\\tilde{\\mathbf {c}}\\vert L)\\int \\!\\mathrm {d}\\mathbf {X}\\left\\lbrace \\prod _{\\nu =1}^L \\prod _{i_\\nu \\in S_\\nu (\\tilde{\\mathbf {c}})} q_{\\nu }(\\mathbf {x}_{i_\\nu })\\right\\rbrace \\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~~\\times \\delta \\left(F+ \\sum _{\\mu =1}^K \\frac{\\vert S_\\mu (\\mathbf {c}) \\vert }{N} \\int \\!",
"Q_\\mu (\\mathbf {x}\\vert \\mathbf {c},\\tilde{\\mathbf {c}},\\mathbf {X}) \\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\mathrm {d}\\mathbf {x}\\right) \\nonumber \\\\&=&\\sum _{\\mathbf {c}, \\tilde{\\mathbf {c}}}p(\\mathbf {c}\\vert K) \\,q(\\tilde{\\mathbf {c}}\\vert L) \\left\\lbrace \\prod _{\\mu =1}^K\\prod _{\\mathbf {x}}\\int \\!\\mathrm {d}Q_\\mu (\\mathbf {x}) \\right\\rbrace P_N\\left[\\lbrace Q_\\mu (\\mathbf {x})\\rbrace \\vert \\mathbf {c}, \\tilde{\\mathbf {c}}\\,\\right]\\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~~\\times \\delta \\left(\\!F+ \\!",
"\\sum _{\\mu =1}^K\\!",
"\\frac{\\vert S_\\mu (\\mathbf {c}) \\vert }{N}\\int \\!",
"\\!Q_\\mu (\\mathbf {x})\\!",
"\\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\!\\mathrm {d}\\mathbf {x}\\right),$ were we have defined the (functional) distribution $P_N\\left[\\lbrace Q_\\mu (\\mathbf {x})\\rbrace \\vert \\mathbf {c}, \\tilde{\\mathbf {c}}\\,\\right]&=&\\int \\left\\lbrace \\prod _{\\nu =1}^L \\prod _{i_\\nu \\in S_\\nu (\\tilde{\\mathbf {c}})} q_{\\nu }(\\mathbf {x}_{i_\\nu })\\right\\rbrace \\nonumber \\\\&&~~~~~~~~~~\\times \\left\\lbrace \\prod _{\\mu =1}^K\\prod _{\\mathbf {x}}\\delta \\left[ Q_\\mu (\\mathbf {x})-Q_\\mu (\\mathbf {x}\\vert \\mathbf {c},\\tilde{\\mathbf {c}},\\mathbf {X})\\right]\\right\\rbrace \\mathrm {d}\\mathbf {X}.$ Let us next consider $&&P_N\\left[\\lbrace Q_\\mu (\\mathbf {x})\\rbrace \\vert \\mathbf {c}, \\tilde{\\mathbf {c}}\\,\\right]\\nonumber \\\\&&~~~~~~~~~=\\int \\!\\mathrm {d}\\mathbf {X}\\left\\lbrace \\prod _{\\nu =1}^L \\prod _{i_\\nu \\in S_\\nu (\\tilde{\\mathbf {c}})} q_{\\nu }(\\mathbf {x}_{i_\\nu })\\right\\rbrace \\left\\lbrace \\prod _{\\mu =1}^K\\prod _{\\mathbf {x}} \\int \\frac{\\mathrm {d}\\hat{Q}_\\mu (\\mathbf {x})}{2\\pi /N}\\right\\rbrace \\nonumber \\\\&&~~~~~~~~~~~~~~~~~~\\times \\mathrm {e}^{\\mathrm {i}N\\sum _{\\mu =1}^K\\int \\hat{Q}_\\mu (\\mathbf {x})\\left[ Q_\\mu (\\mathbf {x})-Q_\\mu (\\mathbf {x}\\vert \\mathbf {c},\\tilde{\\mathbf {c}},\\mathbf {X})\\right]\\mathrm {d}\\mathbf {x}}\\nonumber \\\\&&~~~~~~~~~=\\left\\lbrace \\prod _{\\mu =1}^K\\prod _{\\mathbf {x}}\\int \\frac{\\mathrm {d}\\hat{Q}_\\mu (\\mathbf {x})}{2\\pi /N}\\right\\rbrace \\mathrm {e}^{\\mathrm {i}N\\sum _{\\mu =1}^K\\int \\hat{Q}_\\mu (\\mathbf {x})Q_\\mu (\\mathbf {x})\\mathrm {d}\\mathbf {x}} \\nonumber \\\\&&~~~~~~~~~~~~~~~~~~\\times \\int \\!\\mathrm {d}\\mathbf {X}\\left\\lbrace \\prod _{\\nu =1}^L \\prod _{i_\\nu \\in S_\\nu (\\tilde{\\mathbf {c}})} q_{\\nu }(\\mathbf {x}_{i_\\nu })\\right\\rbrace \\mathrm {e}^{\\sum _{\\nu =1}^L \\sum _{\\mu =1}^K \\frac{N}{\\vert S_\\mu (\\mathbf {c}) \\vert } \\sum _{i_{\\mu \\nu }\\in S_\\mu (\\mathbf {c}) \\cap S_\\nu (\\tilde{\\mathbf {c}})} -\\mathrm {i}\\hat{Q}_\\mu \\left(\\mathbf {x}_{i_{\\mu \\nu }}\\right) }\\nonumber \\\\&&~~~~~~~~~=\\left\\lbrace \\prod _{\\mu =1}^K\\prod _{\\mathbf {x}}\\int \\frac{\\mathrm {d}\\hat{Q}_\\mu (\\mathbf {x})}{2\\pi /N}\\right\\rbrace \\mathrm {e}^{\\mathrm {i}N\\sum _{\\mu =1}^K\\int \\hat{Q}_\\mu (\\mathbf {x})Q_\\mu (\\mathbf {x})\\mathrm {d}\\mathbf {x}} \\nonumber \\\\&&~~~~~~~~~~~~~~~~ \\times \\prod _{\\nu =1}^L \\prod _{\\mu =1}^K \\prod _{i_{\\mu \\nu }\\in S_\\mu (\\mathbf {c}) \\cap S_\\nu (\\tilde{\\mathbf {c}})} \\int \\!",
"q_{\\nu }\\left(\\mathbf {x}_{i_{\\mu \\nu }}\\right) \\mathrm {e}^{-\\mathrm {i}\\frac{ N}{\\vert S_\\mu (\\mathbf {c}) \\vert } \\hat{Q}_\\mu \\left(\\mathbf {x}_{i_{\\mu \\nu }}\\right) } \\mathrm {d}\\mathbf {x}_{i_{\\mu \\nu }} \\nonumber \\\\&&~~~~~~~~~=\\left\\lbrace \\prod _{\\mu =1}^K\\prod _{\\mathbf {x}}\\int \\!\\!\\frac{\\mathrm {d}\\hat{Q}_\\mu (\\mathbf {x})}{2\\pi /N}\\right\\rbrace \\nonumber \\\\&&~~~~~~~~~~~~~~~~ \\times \\mathrm {e}^{\\mathrm {i}N\\sum _{\\mu =1}^K\\!\\int \\!\\hat{Q}_\\mu (\\mathbf {x})Q_\\mu (\\mathbf {x})\\mathrm {d}\\mathbf {x}+N\\sum _{\\mu =1}^K \\sum _{\\nu =1}^L \\frac{ \\vert S_\\mu (\\mathbf {c}) \\cap S_\\nu (\\tilde{\\mathbf {c}})\\vert }{N} \\log \\!\\int \\!",
"\\mathrm {d}\\mathbf {x}~q_{\\nu }\\left(\\mathbf {x}\\right) \\mathrm {e}^{-\\mathrm {i}\\frac{N \\hat{Q}_\\mu (\\mathbf {x})}{\\vert S_\\mu (\\mathbf {c}) \\vert } }}.$ Thus for $P_N\\left[Q\\vert \\alpha (\\mathbf {c}, \\tilde{\\mathbf {c}})\\right]\\equiv P_N\\left[\\lbrace Q_\\mu (\\mathbf {x})\\rbrace \\vert \\mathbf {c}, \\tilde{\\mathbf {c}}\\,\\right]$ we have $P_N\\left[Q\\vert \\alpha (\\mathbf {c}, \\tilde{\\mathbf {c}})\\right]&=& \\int \\mathcal {D} \\hat{Q}\\, \\mathrm {e}^{N\\Psi \\left[Q, \\hat{Q}\\vert \\alpha (\\mathbf {c}, \\tilde{\\mathbf {c}})\\,\\right]},$ where $\\Psi \\big [Q, \\hat{Q}\\vert \\alpha (\\mathbf {c}, \\tilde{\\mathbf {c}})\\,\\big ]&=& \\mathrm {i}\\sum _{\\mu =1}^K \\int \\hat{Q}_\\mu (\\mathbf {x})Q_\\mu (\\mathbf {x})\\mathrm {d}\\mathbf {x}\\nonumber \\\\&&~~~~~~+ \\sum _{\\mu =1}^K\\sum _{\\nu =1}^L \\alpha (\\nu , \\mu \\vert \\mathbf {c}, \\tilde{\\mathbf {c}})\\log \\!\\!\\int \\!\\!",
"q_{\\nu }\\left(\\mathbf {x}\\right) \\,\\mathrm {e}^{\\frac{ -\\mathrm {i}}{ \\alpha (\\mu \\vert \\mathbf {c}) } \\hat{Q}_\\mu \\left(\\mathbf {x}\\right) } \\mathrm {d}\\mathbf {x},$ with the usual short-hand for the path integral measure, $\\int \\mathcal {D} \\hat{Q}\\equiv \\left\\lbrace \\prod _{\\mu =1}^K\\prod _{\\mathbf {x}}\\int [\\mathrm {d}\\hat{Q}_\\mu (\\mathbf {x})/(2\\pi /N)]\\right\\rbrace $ .",
"In the above formula we have also introduced the matrix $ \\alpha (\\mathbf {c}, \\tilde{\\mathbf {c}})$ , with entries $[ \\alpha (\\mathbf {c}, \\tilde{\\mathbf {c}})]_{\\nu \\mu }= \\alpha (\\nu , \\mu \\vert \\mathbf {c}, \\tilde{\\mathbf {c}})$ , where in turn $\\alpha (\\nu , \\mu \\vert \\mathbf {c}, \\tilde{\\mathbf {c}})=N^{-1}\\vert S_\\mu (\\mathbf {c}) \\!\\cap \\!",
"S_\\nu (\\tilde{\\mathbf {c}})\\vert $ .",
"We note that $\\cup _{\\mu =1}^K \\left(S_\\mu (\\mathbf {c}) \\cap S_\\nu (\\tilde{\\mathbf {c}})\\right)=S_\\nu (\\tilde{\\mathbf {c}})$ and that $\\cup _{\\nu =1}^L \\left(S_\\mu (\\mathbf {c}) \\cap S_\\nu (\\tilde{\\mathbf {c}})\\right)=S_\\mu (\\mathbf {c})$ .",
"From these properties it follows that the entries $\\alpha (\\nu , \\mu \\vert \\mathbf {c}, \\tilde{\\mathbf {c}})\\ge 0$ can be interpreted as representing a joint distribution, i.e.",
"$\\sum _{\\mu =1}^K\\sum _{\\nu =1}^L \\alpha (\\nu , \\mu \\vert \\mathbf {c}, \\tilde{\\mathbf {c}})=1$ , with the marginals $\\sum _{\\nu =1}^L \\alpha (\\nu , \\mu \\vert \\mathbf {c}, \\tilde{\\mathbf {c}})=\\alpha (\\mu \\vert \\mathbf {c})=\\vert S_\\mu (\\mathbf {c})\\vert /N$ and $\\sum _{\\mu =1}^K\\alpha (\\nu , \\mu \\vert \\mathbf {c}, \\tilde{\\mathbf {c}})=\\alpha (\\nu \\vert \\tilde{\\mathbf {c}})=\\vert S_\\nu (\\tilde{\\mathbf {c}})\\vert /N$ .",
"Using all these ingredients in equation (REF ) then leads us to $P_N(F)&=&\\sum _{\\mathbf {c}, \\tilde{\\mathbf {c}}}p(\\mathbf {c}\\vert K) \\,q(\\tilde{\\mathbf {c}}\\vert L) \\int \\mathcal {D} Q\\, P_N\\left[Q\\vert \\alpha (\\mathbf {c}, \\tilde{\\mathbf {c}})\\right] \\delta \\left(F+ \\sum _{\\mu =1}^K \\frac{\\vert S_\\mu (\\mathbf {c}) \\vert }{N} \\int Q_\\mu (\\mathbf {x}) \\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\mathrm {d}\\mathbf {x}\\right)\\nonumber \\\\&=&\\int \\mathrm {d} \\alpha \\, P_N( \\alpha ) \\int \\mathcal {D} Q\\, P_N\\left[Q\\vert \\alpha \\right] \\delta \\!\\left(\\!F\\!+\\!",
"\\sum _{\\mu =1}^K\\!\\!",
"\\alpha (\\mu )\\!",
"\\!\\int \\!",
"Q_\\mu (\\mathbf {x}) \\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\mathrm {d}\\mathbf {x}\\!\\right),$ where we have defined the integral measure $\\int \\mathcal {D} Q\\equiv \\big \\lbrace \\prod _{\\mu =1}^K\\prod _{\\mathbf {x}}\\int \\mathrm {d}Q_\\mu (\\mathbf {x}) \\big \\rbrace $ as well as the short-hand $\\int \\!\\mathrm {d} \\alpha \\equiv \\prod _{\\mu =1}^K\\prod _{\\nu =1}^L\\int \\mathrm {d}\\alpha (\\nu ,\\mu )$ .",
"The distribution of $ \\alpha $ is given by $P_N( \\alpha )&=&\\sum _{\\mathbf {c}, \\tilde{\\mathbf {c}}}\\,p(\\mathbf {c}\\vert K) \\,q(\\tilde{\\mathbf {c}}\\vert L) \\prod _{\\mu =1}^K\\!\\prod _{\\nu =1}^L\\!\\delta \\!\\left[\\alpha (\\nu , \\mu )\\!-\\!\\alpha (\\nu , \\mu \\vert \\mathbf {c},\\!",
"\\tilde{\\mathbf {c}})\\right].\\nonumber \\\\[-2mm]&&$ Now for any smooth function $g$ we can consider the following average: $\\int P_N(F)\\, g(F)\\,\\mathrm {d}F&=&\\int \\mathrm {d} \\alpha \\, P_N( \\alpha ) \\int \\mathcal {D} Q\\, P_N\\left[Q\\vert \\alpha \\right] ~g\\Big (\\!-\\sum _{\\mu =1}^K \\alpha (\\mu ) \\int Q_\\mu (\\mathbf {x}) \\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\mathrm {d}\\mathbf {x}\\Big )\\nonumber \\\\&=&\\int \\mathrm {d} \\alpha \\, P_N( \\alpha ) \\frac{\\int \\mathcal {D} Q\\, P_N\\left[Q\\vert \\alpha \\right]}{\\int \\mathcal {D} \\tilde{Q}\\, P_N\\big [\\tilde{Q}\\vert \\alpha \\big ]} ~g\\Big (\\!-\\sum _{\\mu =1}^K \\alpha (\\mu ) \\int Q_\\mu (\\mathbf {x}) \\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\mathrm {d}\\mathbf {x}\\Big )\\nonumber \\\\&=&\\int \\mathrm {d} \\alpha \\, P_N( \\alpha ) \\frac{ \\int \\mathcal {D} Q\\, \\int \\mathcal {D} \\hat{Q}\\, \\mathrm {e}^{N\\Psi \\left[Q, \\hat{Q}\\vert \\alpha \\right]} }{\\int \\mathcal {D} \\tilde{Q}\\, \\int \\mathcal {D} \\hat{Q}\\, \\mathrm {e}^{N\\Psi \\left[\\tilde{Q}, \\hat{Q}\\vert \\alpha \\right]}} \\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~~~~\\times g\\Big (\\!-\\sum _{\\mu =1}^K\\!",
"\\alpha (\\mu )\\!\\!",
"\\int \\!\\!Q_\\mu (\\mathbf {x}) \\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\mathrm {d}\\mathbf {x}\\Big ).$ Let us assume that $P_N( \\alpha )\\rightarrow P( \\alpha )$ as $N\\rightarrow \\infty $ .",
"Furthermore we expect that in this limit the functional integral in the above equation is dominated by the extremum of the functional $\\Psi $ and hence for the distribution $ P(F)=\\lim _{N\\rightarrow \\infty } P_N(F)$ we obtain $&&\\int P(F)\\, g(F)\\,\\mathrm {d}F=\\int P( \\alpha ) ~g\\Big (\\!-\\sum _{\\mu =1}^K \\alpha (\\mu )\\int \\!",
"\\!Q_\\mu (\\mathbf {x}\\vert \\alpha )\\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\mathrm {d}\\mathbf {x}\\Big )\\mathrm {d} \\alpha ,$ where $Q_\\mu (\\mathbf {x}\\vert \\alpha )$ is a solution of the saddle-point equations $\\delta \\Psi [Q, \\hat{Q}\\vert \\alpha ]/\\delta Q_\\mu (\\mathbf {x})=0$ and $\\delta \\Psi [Q, \\hat{Q}\\vert \\alpha ]/\\delta \\hat{Q}_\\mu (\\mathbf {x})=0$ .",
"Solving the latter gives us the following two equations: $\\mathrm {i}\\hat{Q}_\\mu (\\mathbf {x})&=&0\\\\Q_\\mu (\\mathbf {x})&=& \\sum _{\\nu =1}^L \\frac{\\alpha (\\nu , \\mu )}{\\alpha (\\mu )} \\frac{q_{\\nu }\\left(\\mathbf {x}\\right) \\mathrm {e}^{\\frac{ -\\mathrm {i}}{ \\alpha (\\mu ) } \\hat{Q}_\\mu \\left(\\mathbf {x}\\right) }}{\\int q_{\\nu }\\left(\\mathbf {x}^\\prime \\right) \\,\\mathrm {e}^{\\frac{ -\\mathrm {i}}{ \\alpha (\\mu ) } \\hat{Q}_\\mu \\left(\\mathbf {x}^\\prime \\right) } \\mathrm {d}\\mathbf {x}^\\prime }~~~$ from which follows the equation $Q_\\mu (\\mathbf {x}\\vert \\alpha )&=& \\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu ) q_{\\nu }\\left(\\mathbf {x}\\right),$ where $ \\alpha (\\nu \\vert \\mu )=\\alpha (\\nu , \\mu )/\\alpha (\\mu )$ is a conditional distribution.",
"From the above we conclude that $P(F)&=&\\int \\!\\mathrm {d} \\alpha ~ P( \\alpha )\\\\[-1mm]&&\\times ~\\delta \\Big (F+\\sum _{\\mu =1}^K\\!\\sum _{\\nu =1}^L \\alpha (\\nu ,\\mu )\\!",
"\\!",
"\\int \\!\\!",
"q_{\\nu }\\!\\left(\\mathbf {x}\\right) \\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right) \\mathrm {d}\\mathbf {x}\\Big ).\\nonumber $ If we assume that $P( \\alpha )$ is a delta function, this gives us the mean-field (MF) log-likelihood $F( \\alpha )&=&-\\!\\sum _{\\mu =1}^K\\!\\sum _{\\nu =1}^L\\!",
"\\alpha (\\nu ,\\mu )\\!\\!",
"\\int \\!\\!",
"q_{\\nu }\\!\\left(\\mathbf {x}\\right) \\log p\\left(\\mathbf {x}\\vert \\theta _{\\!\\mu }\\right)\\!",
"\\mathrm {d}\\mathbf {x}~~~~~$ which is seen to be equivalent to (REF ).",
"Let us next consider the distribution (REF ) of the log-likelihood density (REF ): $P_N(F)&=&\\sum _{\\mathbf {c}}p(\\mathbf {c}\\vert K) \\int \\!\\mathrm {d}\\mathbf {X}~p(\\mathbf {X}\\vert L)\\\\[-1mm]&&\\times \\delta \\Big (F- \\sum _{\\mu =1}^K\\frac{ \\vert S_\\mu \\left(\\mathbf {c}\\right)\\vert }{2N} \\log \\!\\left(\\!",
"(2\\pi \\mathrm {e})^{d}\\!\\left|\\mathbf {\\Lambda }_{\\mu }^{-1}(\\mathbf {c}, \\mathbf {X})\\right|\\right)\\!",
"\\!",
"\\Big )\\nonumber $ where $\\mathbf {\\Lambda }_{\\mu }^{-1}(\\mathbf {c}, \\mathbf {X})$ is the covariance matrix of the data in cluster $\\mu $ , which can be written in the form $\\mathbf {\\Lambda }_{\\mu }^{-1}(\\mathbf {c}, \\mathbf {X})&=&\\frac{1}{ \\vert S_\\mu \\left(\\mathbf {c}\\right)\\vert }\\!",
"\\sum _{i_\\mu \\in S_\\mu \\left(\\mathbf {c}\\right)}\\!\\!",
"\\!\\!\\left(\\mathbf {x}_{i_\\mu }\\!\\!-\\!\\mathbf {m}_{\\mu }\\left(\\mathbf {c}\\right)\\!\\right) \\left(\\mathbf {x}_{i_\\mu }\\!\\!-\\!\\mathbf {m}_{\\mu }\\left(\\mathbf {c}\\right)\\!\\right)^T,$ where $\\mathbf {m}_{\\mu }(\\mathbf {c}) =\\frac{1}{ \\vert S_\\mu \\left(\\mathbf {c}\\right)\\vert } \\sum _{i_\\mu \\in S_\\mu \\left(\\mathbf {c}\\right)} \\mathbf {x}_{i_\\mu }$ .",
"Further manipulation of $P_N(F)$ gives $P_N(F)&=&\\sum _{\\mathbf {c}, \\tilde{\\mathbf {c}}}p(\\mathbf {c}\\vert K)\\, q(\\tilde{\\mathbf {c}}\\vert L) \\int ~\\left\\lbrace \\prod _{\\nu =1}^L \\prod _{i_\\nu \\in S_\\nu (\\tilde{\\mathbf {c}})} q_{\\nu }(\\mathbf {x}_{i_\\nu })\\right\\rbrace \\nonumber \\\\&&~~~~~~~~~~~~~~\\times \\delta \\Big (F- \\sum _{\\mu =1}^K\\frac{ \\vert S_\\mu \\left(\\mathbf {c}\\right)\\vert }{2N} \\log \\left((2\\pi \\mathrm {e})^{d}\\left|\\mathbf {\\Lambda }_{\\mu }^{-1}(\\mathbf {c}, \\mathbf {X})\\right|\\right) \\Big )\\mathrm {d}\\mathbf {X}~~~$ and the covariance matrix can be written in the form $\\mathbf {\\Lambda }_{\\mu }^{-1}\\left(\\mathbf {c}, \\mathbf {X}\\right)&=&\\frac{1}{ \\vert S_\\mu \\left(\\mathbf {c}\\right)\\vert } \\sum _{\\nu =1}^L\\sum _{i_{\\nu \\mu }\\in S_\\mu (\\mathbf {c}) \\cap S_\\nu (\\tilde{\\mathbf {c}}) }\\left(\\mathbf {x}_{i_{\\nu \\mu }}-\\mathbf {m}_{\\mu }\\left(\\mathbf {c}\\right)\\right) \\!\\left(\\mathbf {x}_{i_{\\nu \\mu }}-\\mathbf {m}_{\\mu }\\left(\\mathbf {c}\\right)\\right)^T\\nonumber \\\\&=&\\int \\!\\mathrm {d}\\mathbf {x}~ Q_\\mu (\\mathbf {x}\\vert \\mathbf {c},\\tilde{\\mathbf {c}},\\mathbf {X}) \\left(\\mathbf {x}\\!-\\!",
"\\int \\!",
"Q_\\mu (\\mathbf {y}\\vert \\mathbf {c},\\tilde{\\mathbf {c}},\\mathbf {X})\\,\\mathbf {y}\\,\\mathrm {d}\\mathbf {y}\\right) \\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~~~~\\times \\left(\\mathbf {x}\\!-\\!",
"\\int \\!Q_\\mu (\\mathbf {z}\\vert \\mathbf {c},\\tilde{\\mathbf {c}},\\mathbf {X})\\,\\mathbf {z}\\,\\mathrm {d}\\mathbf {z}\\right)^T\\!\\!\\!\\!.",
"$ From the above it is clear that $\\hat{F}_N$ is a functional of the density $Q_\\mu (\\mathbf {x}\\vert \\mathbf {c},\\tilde{\\mathbf {c}},\\mathbf {X}) $ , defined in (REF ), and the matrix $ \\alpha (\\nu ,\\mu \\vert \\mathbf {c}\\tilde{\\mathbf {c}})$ .",
"Following the same steps as in deriving equations (REF )-(REF ) gives us $P_N(F)&=&\\int \\mathrm {d} \\alpha \\, P_N( \\alpha ) \\int \\mathcal {D} Q\\, P_N\\left[Q\\vert \\alpha \\right]\\\\&&\\times ~ \\delta \\Big (F- \\sum _{\\mu =1}^K \\alpha (\\mu ) \\frac{1}{2} \\log \\left((2\\pi \\mathrm {e})^{d}\\left|\\mathbf {\\Lambda }_{\\mu }^{-1} \\left[Q\\right] \\right|\\right) \\Big ),\\nonumber $ where $\\mathbf {\\Lambda }_{\\mu }^{-1}\\left[Q\\right]&=&\\int \\!\\mathrm {d}\\mathbf {x}~ Q_\\mu (\\mathbf {x}) \\Big (\\mathbf {x}\\!- \\!",
"\\int \\!",
"Q_\\mu (\\mathbf {y})\\,\\mathbf {y}\\,\\mathrm {d}\\mathbf {y}\\Big )\\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~~~~\\times \\Big (\\mathbf {x}\\!- \\!",
"\\int \\!",
"Q_\\mu (\\mathbf {z})\\,\\mathbf {z}\\,\\mathrm {d}\\mathbf {z}\\Big )^T .$ Furthermore, for $N\\rightarrow \\infty $ , using a similar argument as outlined in equations (REF )-(REF ), we obtain $P(F)&=&\\int \\!",
"\\mathrm {d} \\alpha ~ P( \\alpha )\\\\[-1mm]&&\\times ~\\delta \\Big (F- \\sum _{\\mu =1}^K \\alpha (\\mu )\\frac{1}{2} \\log \\left((2\\pi \\mathrm {e})^{d}\\left|\\mathbf {\\Lambda }_{\\mu }^{-1} \\left( \\alpha \\right) \\right|\\right) \\Big )\\nonumber $ where the covariance matrix $ \\mathbf {\\Lambda }_{\\mu }^{-1}( \\alpha ) $ is defined by $\\mathbf {\\Lambda }_{\\mu }^{-1}( \\alpha )&=&\\sum _{\\nu =1}^L\\!",
"\\alpha (\\nu \\vert \\mu )\\Big \\langle \\!\\left(\\mathbf {x}-\\mathbf {m}_{\\mu }\\!\\left( \\alpha \\right)\\right) \\left(\\mathbf {x}-\\mathbf {m}_{\\mu }\\!\\left( \\alpha \\right)\\right)^T\\Big \\rangle _\\nu ,$ where $\\mathbf {m}_{\\mu }\\left( \\alpha \\right)=\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\langle \\mathbf {x}\\rangle _\\nu $ is the mean, and we used the short-hand $\\langle \\lbrace \\cdots \\rbrace \\rangle _\\nu =\\int q_\\nu (\\mathbf {x}) \\lbrace \\cdots \\rbrace \\mathrm {d}\\mathbf {x}$ .",
"Assuming that $P( \\alpha )$ is a delta function subsequently gives us the MF log-likelihood expression (REF )."
],
[
"Proofs of information-theoretic inequalities",
"In this section we compute lower bounds for the MF entropy (REF ).",
"First, we show that $F( \\alpha ) $ satisfies the inequalities $F( \\alpha ) \\ge \\sum _{\\mu =1}^K\\alpha (\\mu )H \\left(Q_\\mu \\right)\\ge \\sum _{\\nu =1}^L\\gamma (\\nu )H \\left(q_\\nu \\right).$ Let us consider the Kullback-Leibler distance [15] $D\\left(Q_\\mu \\vert \\vert \\mathcal {N}_\\mu \\right)$ between the mixture $Q_\\mu (\\mathbf {x})=\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu ) q_\\nu (\\mathbf {x})$ and the Gaussian distribution $ \\mathcal {N}\\left(\\mathbf {x}\\vert \\mathbf {m}_{\\mu },\\mathbf {\\Lambda }_{\\mu }^{-1}\\right)$ : $D\\left(Q_\\mu \\vert \\vert \\mathcal {N}_\\mu \\right)&=&\\int \\!\\mathrm {d}\\mathbf {x}\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\, q_\\nu (\\mathbf {x})\\log \\left( \\frac{ \\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\, q_\\nu (\\mathbf {x})}{\\mathcal {N}\\left(\\mathbf {x}\\vert \\mathbf {m}_{\\mu },\\mathbf {\\Lambda }_{\\mu }^{-1}\\right)}\\right)\\nonumber \\\\&=& -H \\left(Q_\\mu \\right)-\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\int \\!\\mathrm {d}\\mathbf {x}~ q_\\nu (\\mathbf {x}) \\log \\mathcal {N}\\left(\\mathbf {x}\\vert \\mathbf {m}_{\\mu },\\mathbf {\\Lambda }_{\\mu }^{-1}\\right)\\nonumber \\\\&=& -H \\left(Q_\\mu \\right) -\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\int \\!",
"\\mathrm {d}\\mathbf {x}~ q_\\nu (\\mathbf {x}) \\log \\left( \\frac{\\mathrm {e}^{-\\frac{1}{2} (\\mathbf {x}-\\mathbf {m}_{\\mu })^T\\mathbf {\\Lambda }_\\mu (\\mathbf {x}-\\mathbf {m}_{\\mu })}}{\\left|2\\pi \\mathbf {\\Lambda }_{\\mu }^{-1}\\right|^{\\frac{1}{2}}}\\right)\\nonumber \\\\&=&-H \\left(Q_\\mu \\right) + \\frac{1}{2}\\log \\left( (2\\pi )^d \\left|\\mathbf {\\Lambda }_{\\mu }^{-1}\\right|\\right)\\nonumber \\\\&&~~~~~~~~~~~+\\frac{1}{2}\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\int \\!",
"\\mathrm {d}\\mathbf {x}~q_\\nu (\\mathbf {x})(\\mathbf {x}-\\mathbf {m}_{\\mu })^T\\mathbf {\\Lambda }_\\mu (\\mathbf {x}-\\mathbf {m}_{\\mu })\\nonumber \\\\&=&-H \\left(Q_\\mu \\right) + \\frac{1}{2}\\log \\left( (2\\pi )^d \\left|\\mathbf {\\Lambda }_{\\mu }^{-1}\\right|\\right)\\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~~~~+\\frac{1}{2}\\mathrm {Tr}\\Bigg \\lbrace \\mathbf {\\Lambda }_\\mu \\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu ) \\int q_\\nu (\\mathbf {x})(\\mathbf {x}-\\mathbf {m}_{\\mu }) (\\mathbf {x}-\\mathbf {m}_{\\mu })^T\\mathrm {d}\\mathbf {x}\\Bigg \\rbrace \\nonumber $ Let us define the mean and covariance of the distribution $Q_\\mu (\\mathbf {x})\\!=\\!\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu ) q_\\nu (\\mathbf {x})$ as $\\mathbf {m}_{\\mu }\\!=\\!\\int Q_\\mu (\\mathbf {x})\\,\\mathbf {x}\\, \\mathrm {d}\\mathbf {x}$ and $\\mathbf {\\Lambda }_{\\mu }^{-1}\\!=\\!\\int Q_\\mu (\\mathbf {x})(\\mathbf {x}\\!-\\!\\mathbf {m}_{\\mu }) (\\mathbf {x}\\!-\\!\\mathbf {m}_{\\mu })^T\\mathrm {d}\\mathbf {x}$ .",
"Then $D\\left(Q_\\mu \\vert \\vert \\mathcal {N}_\\mu \\right)=-H \\left(Q_\\mu \\right) +\\frac{1}{2}\\log \\left( (2\\pi \\mathrm {e})^d \\left|\\mathbf {\\Lambda }_{\\mu }^{-1}\\right|\\right)$ and from the simple property $D\\left(Q_\\mu \\vert \\vert \\mathcal {N}_\\mu \\right)\\ge 0$ we immediately deduce that $F( \\alpha )&\\ge & \\sum _{\\mu =1}^K\\alpha (\\mu )H \\left(Q_\\mu \\right).$ Furthermore, for the average entropy we find the following inequality $\\sum _{\\mu =1}^K\\alpha (\\mu )H \\left(Q_\\mu \\right)&=&\\sum _{\\mu =1}^K\\alpha (\\mu ) \\sum _{\\nu _1=1}^L\\alpha (\\nu _1\\vert \\mu )\\int q_{\\nu _1}(\\mathbf {x}) \\log \\Big (1/\\sum _{\\nu _2=1}^L\\alpha (\\nu _2\\vert \\mu ) \\,q_{\\nu _2}(\\mathbf {x})\\Big )\\mathrm {d}\\mathbf {x}\\nonumber \\\\&=&\\sum _{\\mu =1}^K\\sum _{\\nu _1=1}^L\\alpha (\\nu _1, \\mu )\\nonumber \\\\&&~~\\times \\int q_{\\nu _1}(\\mathbf {x}) \\Big \\lbrace \\log q_{\\nu _1}(\\mathbf {x})-\\log q_{\\nu _1}(\\mathbf {x})+\\log \\Big (1/\\sum _{\\nu _2=1}^L\\alpha (\\nu _2\\vert \\mu ) \\,q_{\\nu _2}(\\mathbf {x})\\Big ) \\Big \\rbrace \\mathrm {d}\\mathbf {x}\\nonumber \\\\&=&\\sum _{\\mu =1}^K\\sum _{\\nu =1}^L\\alpha (\\nu , \\mu ) D\\left(q_{\\nu } \\vert \\vert Q_\\mu \\right) + \\sum _{\\nu =1}^L\\gamma (\\nu )H \\left(q_\\nu \\right)\\nonumber \\\\&\\ge & \\sum _{\\nu =1}^L\\gamma (\\nu )H \\left(q_\\nu \\right).$ Secondly, for the average entropy $F_0&=&\\sum _{\\nu =1}^L\\gamma (\\nu )\\frac{1}{2} \\log \\left((2\\pi \\mathrm {e})^{d}\\left|\\mathbf {C}_\\nu \\right|\\right) ,$ where $\\mathbf {C}_\\nu = \\left\\langle \\mathbf {x}\\,\\mathbf {x}^T\\right\\rangle _\\nu - \\left\\langle \\mathbf {x}\\right\\rangle _\\nu \\, \\left\\langle \\mathbf {x}\\right\\rangle _\\nu ^T$ is the covariance matrix of $ q_\\nu (\\mathbf {x})$ , we can show that the following holds: $F( \\alpha )\\ge F_0$ for all $ \\alpha $ .",
"The above equality follows from properties of the covariance matrix $\\mathbf {\\Lambda }_{\\mu }^{-1}( \\alpha )&=&\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\, \\mathbf {C}_\\nu \\\\&& +\\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\left(\\langle \\mathbf {x}\\rangle _\\nu \\!-\\!\\mathbf {m}_{\\mu }\\left( \\alpha \\right)\\right)\\left(\\langle \\mathbf {x}\\rangle _\\nu \\!-\\!\\mathbf {m}_{\\mu }\\left( \\alpha \\right)\\right)^T.",
"\\nonumber $ To prove (REF ) we first derive the inequality $\\log \\left|\\sum _{\\nu =1}^L\\alpha (\\nu ) {\\mathbf {D}}_\\nu \\right|&\\ge &\\sum _{\\nu =1}^L\\alpha (\\nu ) \\log \\left|{\\mathbf {D}}_\\nu \\right|$ for symmetric positive definite matrices ${\\mathbf {D}}_\\nu $ and $\\sum _{\\nu =1}^L \\alpha (\\nu )=1$ , where $\\alpha (\\nu )\\ge 0$ .",
"This inequality can be derived by repeated application of Minkowski's inequality for determinants, viz.",
"$\\vert {\\mathbf {D}}+\\mathbf {B}\\vert ^{\\frac{1}{d}}\\ge \\vert {\\mathbf {D}}\\vert ^{\\frac{1}{d}} +\\vert \\mathbf {B}\\vert ^{\\frac{1}{d}}$ for symmetric positive definite matrices ${\\mathbf {D}}$ and $\\mathbf {B}$ : $\\left|\\sum _{\\nu =1}^L\\alpha (\\nu ) {\\mathbf {D}}_\\nu \\right|^{\\frac{1}{d}} &=& \\left|\\alpha (1) {\\mathbf {D}}_1+ \\sum _{\\nu =2}^L\\alpha (\\nu ) {\\mathbf {D}}_\\nu \\right|^{\\frac{1}{d}} \\nonumber \\\\&&~~~~~~~~~~~~~~ \\ge \\alpha (1) \\left|{\\mathbf {D}}_1\\right|^{\\frac{1}{d}}+ \\left|\\sum _{\\nu =2}^L\\alpha (\\nu ) {\\mathbf {D}}_\\nu \\right|^{\\frac{1}{d}} \\ge \\sum _{\\nu =1}^L\\alpha (\\nu ) \\left|{\\mathbf {D}}_\\nu \\right|^{\\frac{1}{d}} $ from which follows the result $\\log \\left|\\sum _{\\nu =1}^L\\alpha (\\nu ) {\\mathbf {D}}_\\nu \\right|&\\ge & d\\log \\left(\\sum _{\\nu =1}^L\\alpha (\\nu ) \\left|{\\mathbf {D}}_\\nu \\right|^{\\frac{1}{d}} \\right)\\ge \\sum _{\\nu =1}^L\\alpha (\\nu ) \\log \\left|{\\mathbf {D}}_\\nu \\right|.$ The last step in this argument relied on Jensen's inequality [15].",
"Let us now apply (REF ) to the difference of entropies $2\\left(F( \\alpha )-F_0\\right)&=&-\\sum _{\\nu =1}^L\\gamma (\\nu ) \\log \\left|\\mathbf {C}_\\nu \\right|+\\sum _{\\mu =1}^K\\alpha (\\mu ) \\log \\left|\\mathbf {\\Lambda }_{\\mu }^{-1}\\left( \\alpha \\right) \\right|\\\\&=&-\\sum _{\\nu =1}^L\\gamma (\\nu ) \\log \\left|\\mathbf {C}_\\nu \\right|+\\sum _{\\mu =1}^K\\alpha (\\mu )\\nonumber \\\\&&~~~~~~~~\\times \\log \\Bigg \\vert \\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu ) \\Big (\\mathbf {C}_\\nu +\\left(\\langle \\mathbf {x}\\rangle _\\nu -\\mathbf {m}_{\\mu }\\left( \\alpha \\right)\\right)\\!\\left(\\langle \\mathbf {x}\\rangle _\\nu -\\mathbf {m}_{\\mu }\\left( \\alpha \\right)\\right)^T\\Big )\\Bigg \\vert \\nonumber \\\\&&~~\\ge -\\sum _{\\nu =1}^L\\gamma (\\nu ) \\log \\left|\\mathbf {C}_\\nu \\right|+\\sum _{\\mu =1}^K\\alpha (\\mu ) \\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\nonumber \\\\&&~~~~~~~~~~~~\\times \\log \\left|\\mathbf {C}_\\nu +\\left(\\langle \\mathbf {x}\\rangle _\\nu -\\mathbf {m}_{\\mu }\\left( \\alpha \\right)\\right)\\!\\left(\\langle \\mathbf {x}\\rangle _\\nu -\\mathbf {m}_{\\mu }\\left( \\alpha \\right)\\right)^T\\right|\\nonumber \\\\&&~~\\ge -\\sum _{\\nu =1}^L\\gamma (\\nu ) \\log \\left|\\mathbf {C}_\\nu \\right|+d\\sum _{\\mu =1}^K\\alpha (\\mu ) \\sum _{\\nu =1}^L \\alpha (\\nu \\vert \\mu )\\nonumber \\\\&&~~~~~~~~~~~~~~\\times \\log \\!",
"\\left(\\!",
"\\left|\\mathbf {C}_\\nu \\right|^{\\frac{1}{d}}\\!+\\!\\left|\\!\\left(\\!\\langle \\mathbf {x}\\rangle _\\nu \\!-\\!\\mathbf {m}_{\\mu }\\left( \\alpha \\right)\\!\\right)\\!\\left(\\!\\langle \\mathbf {x}\\rangle _\\nu \\!-\\!",
"\\mathbf {m}_{\\mu }\\left( \\alpha \\right)\\!\\right)^T\\right|^{\\frac{1}{d}} \\!\\right).\\nonumber $ The last line in the above, obtained by Minkowski's inequality, is equal to zero, and hence $F( \\alpha )\\ge F_0$ for all $ \\alpha $ ."
],
[
"Algorithmic cost of ordering random unbiased partitions ",
"Let us assume that we have $N$ `particles' of $L$ different `colours' which are distributed into $K$ different reservoirs.",
"The probability that a particle has colour $\\nu \\in [L]$ is $\\gamma (\\nu )$ and that it is in the reservoir $\\mu $ is $1/K$ .",
"Assuming that colour and reservoir allocation are independent events, the probability of `configuration' $\\mathbf {A}=( \\mathbf {a} _1,\\ldots , \\mathbf {a} _N)$ , where $ \\mathbf {a} _i=(a_i(1) , a_i(2))$ with the colour $a_i(1)\\in [L]$ and reservoir number $a_i(2)\\in [K]$ of the particle $i$ , is given by $P(\\mathbf {A})&=&\\prod _{i=1}^N P( \\mathbf {a} _i)\\mbox{,}\\\\P( \\mathbf {a} _i)&\\equiv & P(a_i(1)=\\nu , a_i(2)=\\mu )=\\frac{\\gamma (\\nu )}{K}.$ The total number of particles in reservoir $\\mu $ is given by $N_{\\mu }(\\mathbf {A})=\\sum _{i=1}^N\\delta _{\\mu ; a_i(2)}$ .",
"Let us now consider the joint distribution of particle numbers in reservoirs $P(N_1,\\ldots ,N_K)&=&\\sum _{\\mathbf {A}}P(\\mathbf {A})\\prod _{\\mu =1}^K\\delta _{N_{\\mu }; N_{\\mu }(\\mathbf {A}) }\\\\~~~&=& K^{-N} \\sum _{a_1(2),\\ldots ,a_N(2)} \\prod _{\\mu =1}^K\\delta _{N_{\\mu }; \\sum _{i=1}^N\\delta _{\\mu ; a_i(2)} }\\nonumber \\\\&=& K^{-N} \\frac{N!",
"}{ \\prod _{\\mu =1}^K N_{\\mu }!",
"}\\nonumber ,$ where $\\sum _{\\mu =1}^K N_{\\mu }=N$ .",
"The probability of observing the event that at least one reservoir is empty is given by $&&\\hspace*{-28.45274pt}1-P(N_1>0,\\ldots ,N_K>0)\\nonumber \\\\~~~&=&1-\\sum _{N_1>0,\\ldots ,N_K>0} K^{-N} \\frac{N!",
"}{ \\prod _{\\mu =1}^K N_{\\mu }!",
"}\\nonumber \\\\&=& K^{-N} \\Big ( \\sum _{N_1\\ge 0,\\ldots ,N_K\\ge 0} \\frac{N!",
"}{ \\prod _{\\mu =1}^K N_{\\mu }!",
"}\\nonumber \\\\&&\\hspace*{85.35826pt}-\\sum _{N_1>0,\\ldots ,N_K>0} \\frac{N!",
"}{ \\prod _{\\mu =1}^K N_{\\mu }!}",
"\\Big ) \\nonumber \\\\&=& \\sum _{\\ell =1}^{K-1} {{K}\\atopwithdelims (){\\ell }} \\Big (1-\\frac{\\ell }{K}\\Big )^N.$ Thus the probability of this event decays exponentially with increasing $N$ and, as $N\\rightarrow \\infty $ , the sequence $a_1(2),\\ldots , a_N(2)$ , sampled from the distribution (REF ) is, with high probability, a partition of the set $[N]$ into $K$ subsets (or clusters).",
"Furthermore, the entropy density $N^{-1}\\log (K^N)=\\log K$ of such sequences approaches the entropy density $N^{-1}\\log \\left(K!\\,\\mathcal {S}(N,K)\\right)$ of the random partitions sampled uniformly from (REF ).",
"Let us assume that $K\\le L$ .",
"The total number of particles of colour $\\nu $ , and the number of particles of colour $\\nu $ in reservoir $\\mu $ are given, respectively, by $N_{\\nu }(\\mathbf {A})=\\sum _{i=1}^N\\delta _{\\nu ; a_i(1)}$ and $N_{\\nu \\mu }(\\mathbf {A})=\\sum _{i=1}^N\\delta _{\\nu ; a_i(1)}\\delta _{\\mu ; a_i(2)}$ .",
"The number of particles of colour $\\nu $ which are not in reservoir $\\mu $ is the difference $N_{\\nu }(\\mathbf {A})-N_{\\nu \\mu }(\\mathbf {A})$ .",
"Suppose that each reservoir has a preference for particles of a particular colour (or colours), i.e.",
"there is an onto mapping $\\nu \\rightarrow \\mu (\\nu )$ between colours and reservoirs, then the total number of particles which are not in `their' reservoirs, i.e.",
"the number of particles which are to be `moved' in order for all particles to be in reservoirs to which they belong, is given by the difference $\\sum _{\\nu =1}^L \\left(N_{\\nu }(\\mathbf {A})-N_{\\nu \\mu (\\nu )}(\\mathbf {A})\\right)=N-\\sum _{\\nu =1}^LN_{\\nu \\mu (\\nu )}(\\mathbf {A})$ .",
"We are interested in the average and variance of $N-\\sum _{\\nu =1}^LN_{\\nu \\mu (\\nu )}(\\mathbf {A})$ .",
"The average is given by $\\left\\langle N\\!-\\!\\sum _{\\nu =1}^LN_{\\nu \\mu (\\nu )}(\\mathbf {A})\\right\\rangle _{\\mathbf {A}}&=&N\\!-\\!",
"\\sum _{\\nu =1}^L \\sum _{i=1}^N \\left\\langle \\delta _{\\nu ; a_i(1)}\\delta _{\\mu (\\nu ); a_i(2)} \\right\\rangle _{\\!\\mathbf {A}} \\nonumber \\\\&=&N- \\sum _{\\nu =1}^L \\sum _{i=1}^N \\frac{\\gamma (\\nu )}{K}\\nonumber \\\\&=& N\\frac{K-1}{K}$ and the variance is given by $Var\\Big \\lbrace N\\!-\\!\\sum _{\\nu =1}^LN_{\\nu \\mu (\\nu )}(\\mathbf {A}) \\Big \\rbrace &=& Var\\Big \\lbrace \\sum _{\\nu =1}^LN_{\\nu \\mu (\\nu )}(\\mathbf {A}) \\Big \\rbrace \\nonumber \\\\&=& \\left\\langle \\Big ( \\sum _{\\nu =1}^LN_{\\nu \\mu (\\nu )}(\\mathbf {A}) \\!-\\!\\frac{N}{K}\\Big )^2 \\right\\rangle _{\\!\\mathbf {A}}\\nonumber \\\\&=&\\frac{N}{K} \\left(1- \\frac{1}{K} \\right).$ The average in the penultimate line of the above was computed as follows $&&\\hspace*{-19.91692pt}\\left\\langle \\Big ( \\sum _{\\nu =1}^LN_{\\nu \\mu (\\nu )}(\\mathbf {A}) \\Big )^2 \\right\\rangle _{\\!\\mathbf {A}}\\nonumber \\\\~~&=& \\sum _{\\nu } \\sum _{i_1, i_2}\\!\\left\\langle \\delta _{\\nu ; a_{i_1}\\!",
"(1)} \\delta _{\\mu (\\nu ); a_{i_1}\\!",
"(2)} \\delta _{\\nu ; a_{i_2}\\!",
"(1)} \\delta _{\\mu (\\nu ); a_{i_2}\\!",
"(2)} \\right\\rangle _{\\!\\mathbf {A}} + \\nonumber \\\\&& \\hspace*{-5.69054pt}\\sum _{\\nu _1\\ne \\nu _2} \\sum _{i_1, i_2=1}\\!",
"\\!\\Big \\langle \\!",
"\\delta _{\\nu _1; a_{i_1}(1)} \\delta _{\\mu (\\nu _1); a_{i_1}(2)} \\delta _{\\nu _2; a_{i_2}(1)} \\delta _{\\mu (\\nu _2); a_{i_2}(2)} \\!",
"\\Big \\rangle _{\\!\\mathbf {A}} \\nonumber \\\\&=& \\frac{ N }{K} \\!+\\!",
"\\frac{N(N\\!-\\!1)}{K^2} \\sum _{\\nu =1}^L \\gamma ^2(\\nu ) \\!+\\!",
"\\frac{N(N\\!-\\!1)}{K^2} \\sum _{\\nu _1\\ne \\nu _2} \\gamma (\\nu _1)\\gamma (\\nu _2) \\nonumber \\\\&=& \\frac{ N }{K} + \\frac{N(N-1)}{K^2}.$ From the above derivations it follows that for a random unbiased partition to be ordered, i.e.",
"for particles of the same colour to occupy at most one reservoir, a fraction of particles has to be moved that is on average $\\langle 1-\\frac{1}{N}\\sum _{\\nu =1}^LN_{\\nu \\mu (\\nu )}(\\mathbf {A})\\rangle _{\\mathbf {A}}=(K\\!-\\!1)/K$ , with variance $Var\\lbrace 1-\\frac{1}{N}\\sum _{\\nu =1}^LN_{\\nu \\mu (\\nu )}(\\mathbf {A}) \\rbrace =(1\\!-\\!",
"K^{-1})/NK$ ."
],
[
"Details of numerical experiments",
"In this section we study the performance of the simplest algorithm that minimises the log-likelihood function (REF ) via gradient descent, for the data described in Figure REF .",
"The algorithm is implemented as follows: Start with any initial partition $\\Pi (\\mathbf {c}\\,(0))=\\left\\lbrace S_1(\\mathbf {c}\\,(0)),\\ldots ,S_K(\\mathbf {c}\\,(0)) \\right\\rbrace $ , and compute the log-likelihood $\\hat{F}_N(\\mathbf {c}\\,(0),\\mathbf {X})$ .",
"For all $i\\in [N]$ , consider all possible moves of $i$ from its current cluster $S_\\mu (\\mathbf {c})$ to a new cluster $S_\\nu (\\mathbf {c})$ and compute the new value $\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ for each.",
"Select and execute a move which gives the largest decrease in $\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ , and update $\\Pi (\\mathbf {c})$ .",
"Continue the last two steps while the value of $\\hat{F}_N(\\mathbf {c}, \\mathbf {X})$ continues to change.",
"Output the partition $\\Pi (\\mathbf {c}\\,(\\infty ))$ and the value of $\\hat{F}_N\\left(\\mathbf {c}\\,(\\infty ),\\,\\mathbf {X}\\right)$ .",
"Using as initial states random partitions of data $\\mathbf {c}\\,(0)$ , where each $i\\in [N]$ has a probability $1/K$ of being allocated to one of the $K$ clustersIn section we proved that for $N\\rightarrow \\infty $ the matrix $\\mathbf {c}$ constructed in this way is, with high probability, a partition of the set $[N]$ into the $K$ subsets., we run the above algorithm for each value of $K\\in [17]$ for 100 different initalisations $\\mathbf {c}\\,(0)$ and select the final partition, $\\mathbf {c}\\,(\\infty )$ , with the smallest value of $\\hat{F}_N\\equiv \\hat{F}_N(\\mathbf {c}\\,(\\infty ),\\mathbf {X})$ .",
"The latter is our estimate of $\\min _{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ .",
"We also compute, with the same parameters used to generate our data, the mean-field log-likelihood $F( \\alpha )$ via equation (REF ).",
"When $K\\le L$ , the log-likelihood $\\hat{F}_N(\\mathbf {c}\\,(\\infty ),\\mathbf {X})$ is dominated by partitions $\\mathbf {c}\\,(\\infty )$ corresponding to local minima and saddlepoints of $F( \\alpha )$ .",
"The matrix $ \\alpha $ is defined by the entries $\\left[ \\alpha \\right]_{\\nu \\mu }\\!=\\!",
"{1}[\\nu \\!\\in \\!",
"S_\\mu ]\\gamma (\\nu )$ , generated by partitions $\\Pi =\\lbrace S_1,\\ldots ,S_K\\rbrace $ of the set $[L]$ into $K$ subsets.",
"The total number of partitions is given by $\\mathcal {S}(L,K)$ .",
"To enumerate all partitions we use the algorithm of [25].",
"We classify turning points of $F( \\alpha )$ as follows.",
"For a given $\\Pi $ and its associated matrix $ \\alpha $ we count the number $\\mathcal {N}_{+}( \\alpha )$ of elementary `moves' into the new partition $\\tilde{\\Pi }$ and $\\tilde{ \\alpha }$ (in a single elementary `move', a member of the set $S_\\mu $ , with $\\vert S_\\mu \\vert >1$ , is moved into the set $S_\\nu $ ) for which $F( \\alpha )>F(\\tilde{ \\alpha })$ , and the number $\\mathcal {N}_{-}( \\alpha )$ of moves for which $F( \\alpha )<F(\\tilde{ \\alpha })$ .",
"If $\\mathcal {N}_{+}( \\alpha )=0$ the state $ \\alpha $ is a (possibly local) minimum, and if $\\mathcal {N}_{-}( \\alpha )=0$ the state $ \\alpha $ is a (possibly local) maximum.",
"All other cases are saddle points.",
"In Figures REF , REF , REF and REF we compare $\\hat{F}_N\\left(\\mathbf {c}\\,(\\infty ),\\,\\mathbf {X}\\right)$ with $F( \\alpha )$ .",
"Figure: (Color online) Top left: F(α)F( \\alpha ) as a function of the number of FF-increasing directions 𝒩 - (α)\\mathcal {N}_{-}( \\alpha ).",
"Top: right: F(α)F( \\alpha ) as a function of the number of FF-decreasing directions 𝒩 + (α)\\mathcal {N}_{+}( \\alpha ).Bottom left: histogram of log-likelihood values F ^ N (𝐜(∞),𝐗)\\hat{F}_N(\\mathbf {c}\\,(\\infty ), \\mathbf {X}), obtained by running gradient descent from a 100 different random unbiased partitions, with the assumed number K=3K=3 of clusters.",
"Blue filled circles correspond to the MF log-likelihood, F(α)F( \\alpha ), computed for all possible values of α(ν,μ)=1[ν∈S μ ]γ(ν)\\alpha (\\nu , \\mu )= {1}[\\nu \\!\\in \\!",
"S_\\mu ]\\gamma (\\nu ).",
"Bottom right: 𝒩 - (α)\\mathcal {N}_{-}( \\alpha ) as a function of 𝒩 + (α).\\mathcal {N}_{+}( \\alpha ).Figure: (Color online) Evolution of the log-likelihood, F ^ N (t)≡F ^ N (𝐜(t),𝐗)\\hat{F}_N(t)\\equiv \\hat{F}_N(\\mathbf {c}(t),\\mathbf {X}), and the fraction of data in cluster μ\\mu , α(μ|t)≡α(μ|𝐜(t))\\alpha (\\mu \\vert t)\\equiv \\alpha (\\mu \\vert \\mathbf {c}(t)), where μ={1,2,3}\\mu =\\lbrace 1,2,3\\rbrace , shown as functions of time (normalised number of `moves') in the gradient descent algorithm evolving from a random unbiased initial partition.",
"The assumed number of clusters is K=3K=3.",
"Blue horizontal lines correspond to the levels 3/83/8, 4/84/8 and 5/85/8.Those turning points of $F( \\alpha )$ that are of the form $\\left[ \\alpha \\right]_{\\nu \\mu }={1}[\\nu \\in S_\\mu ]\\gamma (\\nu )$ also act as dynamic `attractors'.",
"This can be seen by comparing Figure REF to Figure REF , and Figure REF to Figure REF , etc.",
"Here $\\hat{F}_N(t)\\equiv \\hat{F}_N\\left(\\mathbf {c}\\,(t),\\,\\mathbf {X}\\right)$ , as computed during the simulated process, is seen to evolve from plateau to plateau by a succession of rapid relaxations, and the value of $\\hat{F}_N(t)$ at the beginning of each plateau can be (approximately) mapped to the value of $F( \\alpha )$ via the fractions $\\alpha (\\mu )=\\sum _{\\nu =1}^L{1}[\\nu \\in S_\\mu ]\\gamma (\\nu )$ of data in clusters $\\mu $ .",
"However as $K$ is increased, more and more attractors are not of the form ${1}[\\nu \\in S_\\mu ]\\gamma (\\nu )$ (see Figures REF , REF and REF ).",
"The predictions of the mean-field log-likelihood $F( \\alpha )$ for $\\min _{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ are incorrect when $K>L$ .",
"The log-likelihood $F( \\alpha )$ is bounded from below by the average entropy $\\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu )$ , but in this regime the gap between this lower bound and $\\min _{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ is widening as we increase the number of assumed clusters $K$ .",
"This effect can be clearly seen in Figure REF .",
"We also see in this Figure that $\\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu )$ separates the low entropy states obtained by gradient descent into two sets.",
"The first set, which includes $\\mathrm {argmin}_{\\mathbf {c}}\\hat{F}_N(\\mathbf {c},\\mathbf {X})$ , is given byThe equality in this definition can only be true when $K=L$ (see Figure REF ).",
"$\\lbrace \\mathbf {c}:~\\hat{F}_N(\\mathbf {c},\\mathbf {X})\\le \\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu )\\rbrace $ , and the second set is given by $\\lbrace \\mathbf {c}:~\\hat{F}_N(\\mathbf {c},\\mathbf {X})>\\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu )\\rbrace $ .",
"Since for $K>L$ we have $F( \\alpha )>\\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu )$ , we expect that $\\min _{ \\alpha }F( \\alpha )$ gives correct predictions for at least some of the low entropy states in the second set.",
"Figure: (Color online) Top left: F(α)F( \\alpha ) as a function of the number of FF-increasing directions 𝒩 - (α)\\mathcal {N}_{-}( \\alpha ).",
"Top: right: F(α)F( \\alpha ) as a function of the number of FF-decreasing directions 𝒩 + (α)\\mathcal {N}_{+}( \\alpha ).Bottom left: histogram of log-likelihood values F ^ N (𝐜(∞),𝐗)\\hat{F}_N(\\mathbf {c}\\,(\\infty ), \\mathbf {X}), obtained by running gradient descent from a 100 different random unbiased partitions, with the assumed number K=7K=7 of clusters.",
"Blue filled circles correspond to the MF log-likelihood, F(α)F( \\alpha ), computed for all possible values of α(ν,μ)=1[ν∈S μ ]γ(ν)\\alpha (\\nu , \\mu )= {1}[\\nu \\!\\in \\!",
"S_\\mu ]\\gamma (\\nu ).",
"Bottom right: 𝒩 - (α)\\mathcal {N}_{-}( \\alpha ) as a function of 𝒩 + (α).\\mathcal {N}_{+}( \\alpha ).Figure: (Color online) Evolution of the log-likelihood, F ^ N (t)≡F ^ N (𝐜(t),𝐗)\\hat{F}_N(t)\\equiv \\hat{F}_N(\\mathbf {c}(t),\\mathbf {X}), and the fraction of data in cluster μ\\mu , α(μ|t)≡α(μ|𝐜(t))\\alpha (\\mu \\vert t)\\equiv \\alpha (\\mu \\vert \\mathbf {c}(t)), where μ={1,2,...,7}\\mu =\\lbrace 1,2,\\ldots ,7\\rbrace , shown as functions of time (normalised number of `moves') in the gradient descent algorithm evolving from a random unbiased initial partition.",
"The assumed number of clusters is K=7K=7.",
"Blue horizontal lines correspond to the levels 3/83/8, 4/84/8 and 5/85/8.Figure: (Color online) Histogram of the log-likelihood values obtained by running the gradient descent algorithm from a 100 different random unbiased partitions, with the assumed number K=8K=8 of clusters.",
"The blue filled circle corresponds to the MF lower bound ∑ ν=1 L γ(ν)H(q ν )=4.853905\\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu )=4.853905.Figure: (Color online) Evolution of the log-likelihood, F ^ N (t)≡F ^ N (𝐜(t),𝐗)\\hat{F}_N(t)\\equiv \\hat{F}_N(\\mathbf {c}(t),\\mathbf {X}), and the fraction of data in cluster μ\\mu , α(μ|t)≡α(μ|𝐜(t))\\alpha (\\mu \\vert t)\\equiv \\alpha (\\mu \\vert \\mathbf {c}(t)), where μ={1,2,...,8}\\mu =\\lbrace 1,2,\\ldots ,8\\rbrace , shown as functions of time (normalised number of `moves') in the gradient descent algorithm evolving from a random unbiased initial partition.",
"The assumed number of clusters is K=8K=8.",
"Blue horizontal lines correspond to the levels 3/83/8, 4/84/8 and 5/85/8.Figure: (Color online)Histogram of the log-likelihood values obtained by running the gradient descent algorithm from a 100 different random unbiased partitions, with the assumed number K=9K=9 of clusters.",
"The blue filled circle corresponds to the MF lower bound ∑ ν=1 L γ(ν)H(q ν )=4.853905\\sum _{\\nu =1}^L\\gamma (\\nu )H(q_\\nu )=4.853905."
],
[
"Estimation of differential entropy",
"In this section we compute the finite sample-size corrections to the MF entropy (REF ).",
"In order to do this we first note that for a sample $\\lbrace \\mathbf {x}_1,\\ldots , \\mathbf {x}_N\\rbrace $ , where each $\\mathbf {x}_i \\in \\mathbb {R}^d$ is drawn from the multivariate Gaussian distribution $\\mathcal {N}(\\mathbf {x}\\vert \\mathbf {m},\\mathbf {\\Lambda })$ , the empirical covariance matrix $\\hat{\\mathbf {\\Lambda }}=N^{-1}\\sum _{i=1}^N(\\mathbf {x}_i\\!-\\!\\hat{\\mathbf {m}})(\\mathbf {x}_i\\!-\\!\\hat{\\mathbf {m}})^T$ , where $\\hat{\\mathbf {m}}=\\frac{1}{N}\\sum _{i=1}^N\\mathbf {x}_i$ is the empirical mean, obeys the following asymptotic law: $[\\log \\vert \\hat{\\mathbf {\\Lambda }}\\vert -\\log \\left|\\mathbf {\\Lambda }\\right|-d(d\\!+\\!1)/2N ]/\\sqrt{2d/N}\\rightarrow \\mathcal {N}(0,1)$ as $N\\rightarrow \\infty $ (see [19] and references therein).",
"This is equivalent to stating $\\log |\\hat{\\mathbf {\\Lambda }}|\\rightarrow \\log |\\mathbf {\\Lambda }|+d(d\\!+\\!1)/2N+z\\sqrt{2d/N}$ , where $z\\sim \\mathcal {N}(0,1)$ .",
"Let us assume that the above is true for the empirical covariance matrices that feature in the log-likelihood (REF ) and evaluate $\\hat{F}_N(\\mathbf {c})$ for large $N$ : $\\hat{F}_N(\\mathbf {c})&=&\\sum _{\\mu =1}^K\\frac{ M_\\mu \\left(\\mathbf {c}\\right)}{N} \\frac{1}{2} \\log \\left((2\\pi \\mathrm {e})^{d}\\left|\\mathbf {\\Lambda }_{\\mu }^{-1}\\left(\\mathbf {c}\\right)\\right|\\right) \\nonumber \\\\&=&\\sum _{\\mu =1}^K \\frac{ M_\\mu \\left(\\mathbf {c}\\right)}{2N} \\Bigg \\lbrace \\log \\left((2\\pi \\mathrm {e})^{d} \\left|\\mathbf {\\Lambda }_{\\mu }^{-1}( \\alpha )\\right|\\right)\\nonumber \\\\&&\\hspace*{28.45274pt} +~\\frac{d(d\\!+\\!1)}{2M_\\mu (\\mathbf {c})}+z_\\mu \\sqrt{\\frac{2d}{M_\\mu (\\mathbf {c})}} \\Bigg \\rbrace \\nonumber \\\\&=&F( \\alpha )+\\sum _{\\mu =1}^K \\frac{ M_\\mu (\\mathbf {c})}{2N} \\Bigg \\lbrace \\frac{d(d\\!+\\!1)}{2M_\\mu \\left(\\mathbf {c}\\right)}+z_\\mu \\sqrt{\\frac{2d}{M_\\mu (\\mathbf {c})}} \\Bigg \\rbrace \\nonumber \\\\&=&F( \\alpha )+\\frac{Kd(d\\!+\\!1)}{4N} + \\sum _{\\mu =1}^K z_\\mu \\sqrt{\\frac{d\\,\\alpha (\\mu )}{2N }}$ The average and variance of the above random variable are given by $F( \\alpha )+Kd(d\\!+\\!1)/4N$ and $d/2N$ , respectively.",
"We expect the above result to be exact when $F( \\alpha )=\\sum _{\\nu =1}^L\\gamma (\\nu )H \\left(q_\\nu \\right)$ , which can only happen when $K=L$ , and all $q_\\nu (\\mathbf {x})$ are Gaussian distributions."
]
] | 1709.01632 |
[
[
"Gamma rays from microquasars Cygnus X-1 and Cygnus X-3"
],
[
"Abstract Gamma-ray observations of microquasars at high and very-high energies can provide valuable information of the acceleration processes inside the jets, the jet-environment interaction and the disk-jet coupling.",
"Two high-mass microquasars have been deeply studied to shed light on these aspects: Cygnus X-1 and Cygnus X-3.",
"Both systems display the canonical hard and soft X-ray spectral states of black hole transients, where the radiation is dominated by non-thermal emission from the corona and jets and by thermal emission from the disk, respectively.",
"Here, we report on the detection of Cygnus X-1 above 60 MeV using 7.5 yr of Pass8 Fermi-LAT data, correlated with the hard X-ray state.",
"A hint of orbital flux modulation was also found, as the source is only detected in phases around the compact object superior conjunction.",
"We conclude that the high-energy gamma-ray emission from Cygnus X-1 is most likely associated with jets and its detection allow us to constrain the production site.",
"Moreover, we include in the discussion the final results of a MAGIC long-term campaign on Cygnus X-1 that reaches almost 100 hr of observations at different X-ray states.",
"On the other hand, during summer 2016, Cygnus X-3 underwent a flaring activity period in radio and high-energy gamma rays, similar to the one that led to its detection in the high-energy regime in 2009.",
"MAGIC performed comprehensive follow-up observations for a total of about 70 hr.",
"We discuss our results in a multi-wavelength context."
],
[
"Introduction",
"Cygnus X-1 is an X-ray binary comprised by a (19.2$\\pm $ 1.9) M$_{\\odot }$ O9.7Iab supergiant star and a (14.8$\\pm $ 1.0) M$_{\\odot }$ BH [1], classified as a microquasar after the detection of a one-sided relativistic radio-jet [2].",
"The jet seems to create a 5 pc ring-like structure detected in the radio that extends up to $10^{19}$ cm from the BH [3].",
"The system follows an almost circular orbit of $\\sim 5.6$ d period [4].",
"Flux modulation with the orbital period is detected in X-ray and radio [5], [6], [7], produced by the absorption/scattering of the radiation by the stellar wind.",
"Cygnus X-1 displays the two principal X-ray states of BH transients, the soft state (SS) and the hard state (HS).",
"Both are described by the sum of a blackbody-like emission from the accretion disk that peaks at $\\sim 1$ keV (dominant in the SS) and a power-law tail with a cutoff at hundred keV, expected to be originated by inverse Compton (IC) scattering on disk photons by thermal electrons in the so-called corona (dominant in the HS).",
"During HS the source displays persistent jets from which synchrotron radio emission is detected, whilst in the SS, these jets are disrupted.",
"Cygnus X-1 showed a $4\\sigma $ -hint above 100 MeV during HS reported by [9], using 3.8 yr of Fermi-LAT data.",
"Evidences of flaring activity were also reported by AGILE ($> 100$ MeV, [10], [11], [12]) and by MAGIC ($> 100$ GeV, [13]).",
"The microquasar Cygnus X-3 hosts a Wolf-Rayet (WR) star, although it follows a short 4.8 hr-orbit.",
"The compactness of the system produces an unusually high absorption, which complicates the identification of the compact object (1.4 M$_{\\odot }$ neutron star (NS) [14] or $< 10$ M$_{\\odot }$ BH [15]).",
"Despite this high absorption, its X-ray spectrum shows the two aforementioned states.",
"Cygnus X-3 is the strongest radio source among the X-ray binaries, whose flux can vary several orders of magnitude during its frequent radio outbursts.",
"These major flares happen only during SS (see [16]).",
"Cygnus X-3 was detected above 100 MeV, during SS by AGILE [17] and Fermi-LAT [18].",
"Its spectrum was described as a power law with photon indices 1.8$\\pm $ 0.2 and $2.70\\pm 0.25$ , respectively.",
"Here, we present the results for GeV and TeV searches on Cygnus X-1 using 7.5yr of Fermi-LAT data and $\\sim 100$ hr of MAGIC data.",
"We also show the latest results of Cygnus X-3 obtained with MAGIC during the August-September 2016 flare."
],
[
"Observations and Analysis",
"Fermi-LAT is the principal scientific instrument on the Fermi Gamma-ray Space Telescope spacecraft that studies the gamma-ray sky within an energy range of $\\sim 20$ MeV to $\\sim 500$ GeV (see [19]).",
"To study Cygnus X-1 in the high-energy (HE; $>60$ MeV) regime, we used 7.5 years of Pass8 Fermi-LAT data (from MJD 54682–57420).",
"The analysis was performed using Fermipyhttp://fermipy.readthedocs.io/en/latest/, a package of python tools to automatize the analysis with the FERMI SCIENCE TOOLS (v10r0p5 package).",
"We selected photon-like events between 60 MeV and 500 GeV, within a 30$^{\\circ }$ radius centered at the position of Cygnus X-1.",
"Find more details in [20].",
"MAGIC is a stereoscopic system of two 17 m diameter Cherenkov Telescopes located in La Palma (Spain).",
"Until 2009, MAGIC consisted in just one telescope [21].",
"After autumn 2009, MAGIC II started operation [22] and between 2011-2012, both telescopes underwent a major upgrade [23].",
"MAGIC observed Cygnus X-1 for $\\sim 100$ hours between 2007 and 2014 mostly during its HS (see [24]).",
"This analysis was carried out with standard MAGIC software (MARS, [25]).",
"Upper limits (ULs) at 95% confidence level (CL) were computed with the full likelihood analysis developed by [26], assuming 30% systematic uncertainty.",
"Between August and September 2016, Cygnus X-3 experienced strong flaring activity in radio and HE regimes during its SS [27], [28].",
"MAGIC observed the source $\\sim 70$ hours between MJD 57623 to 57653, under different moonlight conditions (moon analysis performed following [29]).",
"ULs at 95% CL were computed following Rolke method [30]."
],
[
"Cygnus X-1",
"Fermi-LAT skymap, between 60 MeV and 500 GeV, showed a point-like source at the position of Cygnus X-1 with a TS=53.",
"Moreover, detection only happens during HS (Figure REF ) with TS=49 above 60 MeV (division between HS and SS done following [31]).",
"Therefore, Cygnus X-1 is only detected while displaying persistent radio-jets, as claimed by [9] and confirmed afterwards by [32].",
"Making use of the HS sample, we searched for orbital modulation (assuming ephemeris $T_{0}=52872.788$ HJD, [33]).",
"Orbital phases ($\\phi $ ) were split into two bins, one centered at $\\phi =0$ , the superior conjunction of the compact object (0.75 $< \\phi <$ 0.25) and other at the inferior conjunction (0.25 < $\\phi $ < 0.75).",
"Detection only occurred during superior conjunction (TS=31).",
"Cygnus X-1 spectrum, from 60 MeV up to $\\sim 20$ GeV, is well defined by a power law with photon index $\\Gamma =2.3\\pm 0.1$ and normalization factor of $N_{0}=(5.8\\pm 0.9)\\times 10^{-13}$ MeV$^{-1}$ cm$^{-2}$ s$^{-1}$ , at an energy pivot of 1.3 GeV.",
"Daily basis analysis was also performed, but no short-term flux variability was observed.",
"The results between 0.1-20 GeV can be found in Figure REF .",
"Figure: TS maps above 1 GeV centered in Cygnus X-1, using HS (left) and SS subsamples (right).Figure: Multi-wavelength light curve for Cygnus X-1.",
"From top to bottom: Daily MAGIC ULs (>200> 200 GeV), HE gamma rays from the Fermi-LAT analysis (flux points are computed when TS>9TS>9), hard X-rays from Swift-BAT (15-50 keV, ), soft X-rays from MAXI (2–20 keV, ) and RXTE-ASM (3–5 keV range), and radio from AMI (15 GHz) and RATAN-600 (4.6 GHz).",
"In the HE pad, dashed lines correspond to AGILE transient events.",
"The horizontal green line in Swift-BAT pad shows the limit at 0.09 cts cm -2 ^{-2} s -1 ^{-1} given by to differentiate between X-ray states.",
"HS and SS periods are highlighted with grey and blue bands, respectively.With MAGIC, we searched for steady emission at energies above 200 GeV, making use of the total data set of $\\sim 100$ hr.",
"No significant excess was found, which led to an integral UL of $2.6\\times 10^{-12}$ photons cm$^{-2}$ s$^{-1}$ , assuming a power-law function with photon index $\\Gamma =3.2$ (following former MAGIC results, [13]).",
"We also looked for gamma-ray emission at each X-ray state separately.",
"In the HS, the source was observed for $\\sim 83$ hours between 2007-2011, which yielded no significant excess.",
"Differential ULs are included in the spectral energy distribution (SED) shown in Figure REF .",
"Orbital phase-folded and daily analysis were also carried out, with no evidence of emission.",
"Integral ULs in a night-by-night basis are depicted in Figure REF .",
"During SS, this microquasar was observed for $\\sim 14$ hours in 2014.",
"We searched for steady, orbital and short-term variability modulation, resulting in no detection.",
"Figure: SED of Cygnus X-1.",
"Soft X-rays from BeppoSAX are shown in green stars , while hard X-rays are taken from INTEGRAL-ISGRI (red diamonds,) and INTEGRAL-PICsIT (brown diamonds, ).",
"In the HE and VHE band, results presented in this proceeding obtained with Fermi-LAT (violet points) and MAGIC (black ULs) are depicted.",
"Sensitivity curves for CTA-North for 50 hours (https://www.cta-observatory.org/science/cta- performance/) and scaled to 200 hours of observations are shown in light blue and dark blue, respectively.",
"No statistical errors are drawn, apart from the Fermi-LAT butterfly."
],
[
"Cygnus X-3",
"We searched for steady emission with the MAGIC telescopes, making use of the available $\\sim 70$ hours.",
"No excess was found at energies above 300 GeV (accounting for the energy threshold of the sample with the highest moonlight) nor 100 GeV (using $\\sim 52$ hours of dark data, i.e.",
"under absence of Moon).",
"Differential ULs, assuming a power-law function with photon index $\\Gamma =2.6$ , are presented in Figure REF .",
"In this figure, Fermi-LAT spectrum from [18] is taken, nevertheless Fermi-LAT data for the August-September 2016 flare is currently being studied.",
"No orbital (assuming ephemeris $T_{0}=2440949.892\\pm 0.001$ JD, [39]) or daily modulation was detected either.",
"Figure: SED of Cygnus X-3.",
"Blue butterfly corresponds to Fermi-LAT spectrum during 2009 flare .",
"MAGIC ULs for the August-September flare are represented in light orange (∼52\\sim 52 hours, dark data) and dark orange (∼70\\sim 70 hours, dark+moon data).",
"Sensitivity curves for CTA-North for 50 hours (dot-dashed line) and 200 hours (dashed lines) observations are shown."
],
[
"Discussion and conclusions",
"HE and VHE gamma-ray emission were proposed in the literature from both leptonic and hadronic mechanisms (see e.g.",
"[40], [41]).",
"Among these mechanism, the most efficient process seems to be a leptonic one, the IC.",
"The target photons depend on the distance of the production site with respect to the compact object: close to it, thermal photons from the disk or synchrotron photons would dominate [42], [40]; at a binary scales ($\\sim R_{orb}$ , the size of the system), IC would take place on stellar photons; and finally, gamma-ray emission could also be produced in the interaction between the jet and the medium (as seen in radio for Cygnus X-1, [3]).",
"In the first two scenarios, gamma rays may suffer high absorption due to pair creation."
],
[
"Cygnus X-1",
"At the base of the jet, GeV photons would be absorbed by $\\sim 1$ keV X-rays.",
"Given the detection achieved with Fermi-LAT, and following [43] approach, we estimated the smallest region size for HE gamma-ray production at $2\\times 10^{9}$ cm.",
"The radius of the corona is $\\sim 20-50~R_{g}\\sim 5-10 \\times 10^{7}$ cm [44], which allows us to conclude that the observed GeV emission is not originated in the corona, but most likely inside the jets.",
"This scenario is reinforced by the fact that Fermi-LAT detection only happens during HS.",
"If the hint of orbital modulation here reported is finally confirmed, GeV emission must arrive from inside the jets and not from their interaction with the environment.",
"Assuming so, we can set an UL on the largest distance of the production site at $< 10^{13}$ cm (few times $R_{orb}$ for this source).",
"On the other hand, this flux variability is only expected if the radiative process that leads to GeV emission is anisotropic IC on stellar photons [45].",
"Given that the density of stellar photons is dominant over other photon fields at distances $>10^{11}$ cm, we place the GeV emitter at $10^{11}$ –$10^{13}$ cm from the BH.",
"On the other side, the MAGIC non-detection above 200 GeV allows us to discard jet-medium interaction as possible region for VHE emission above MAGIC sensitivity level, since these regions are not affected by photon-photon absorption.",
"At binary scales this non-detection is less conclusive because of the pair production.",
"Although VHE radiation is predicted in the models (see e.g.",
"[46], [47]), several factors can prevent detection: low flux below MAGIC sensitivity even under negligible absorption [32], no efficient acceleration on the jets or strong magnetic field.",
"Nevertheless, transient events by relaxation of attenuation at some distance from the BH or extended pair cascade [48], [49] cannot be discarded.",
"Transient emission related to discrete radio-emitting-blobs between HS and SS could also happen, as observed in the HE regime for Cygnus X-3.",
"Hint of transient event was indeed reported previously by MAGIC [13].",
"More sensitive instruments, like the future CTA (see Figure REF ), could provide interesting information on Cygnus X-1."
],
[
"Cygnus X-3",
"Despite observing the source during strong radio and HE outbursts, no significant excess was found by MAGIC.",
"One has to consider the extremely high absorption due to the WR, which may affect VHE gamma-ray emission.",
"At energies above 300 GeV, the maximum absorption is produced by near-infrared (NIR) photons ($E_{target}\\sim 1.7 $ eV).",
"Following [50], absorption can be estimated as $\\tau \\sim \\sigma _{\\gamma \\gamma }\\cdot n_{NIR} \\cdot R$ , where $\\sigma _{\\gamma \\gamma }\\sim 1\\times 10^{-25}$ cm$^{2}$ is the cross-section of the process, $n_{NIR}\\sim L_{NIR}/(4 \\pi R^{2} c E_{target})$ is the density of NIR photons and $R$ the size of the emitting region.",
"Assuming the $L_{NIR}$ to be the bolometric luminosity, $L_{NIR}=10^{38}$ erg s$^{-1}$ , the absorption is not negligible until a radius $R\\sim 10^{13}$ cm, i.e.",
"outside the binary scale ($R_{orb,CygX3}\\sim 2.5\\times 10^{11}$ cm).",
"Given the MAGIC non-detection, acceleration up to VHE could still happen inside the jets at a distance $\\lesssim 10^{13}$ cm, maybe related to the HE emission site (produced at $>10^{11}$ cm to avoid absorption by X-rays).",
"On the other hand, MAGIC observed the source simultaneously with the strongest radio flare (at 9.5 Jy on MJD 57651), being the MAGIC significance for this day compatible with background.",
"This could reinforce the idea that VHE gamma rays, if produced, are originated inside the binary scale and not at the radio-emitting regions of the jets far from the compact object.",
"Note, however, that the amount of time observed during strong radio flares is very limited.",
"Figure REF shows the Cygnus X-3 SED with the results at VHE during the 2016 flare, along with Fermi-LAT spectrum taken from the 2009 flare [18].",
"As mentioned above, dedicated Fermi-LAT analysis for the August-Sept 2016 flare is currently being performed.",
"Our constraining ULs are also put in context with the CTA-North sensitivity curve for 50 hours of observationsTaken from (https://www.cta-observatory.org/science/cta- performance/ and the scaled one for 200 hours."
]
] | 1709.01725 |
[
[
"Scalar Fields, Hierarchical UV/IR Mixing and The Weak Gravity Conjecture"
],
[
"Abstract The Weak Gravity Conjecture (WGC) bounds the mass of a particle by its charge.",
"It is expected that this bound can not be below the ultraviolet cut-off scale of the effective theory.",
"Recently, an extension of the WGC was proposed in the presence of scalar fields.",
"We show that this more general version can bound the mass of a particle to be arbitrarily far below the ultraviolet cut-off of the effective theory.",
"It therefore manifests a form of hierarchical UV/IR mixing.",
"This has possible implications for naturalness.",
"We also present new evidence for the proposed contribution of scalar fields to the WGC by showing that it matches the results of dimensional reduction.",
"In such a setup the UV/IR mixing is tied to the interaction between the WGC and non-local gauge operators."
],
[
"Introduction",
"The gauge hierarchy problem can be phrased as the question of how ultraviolet (UV) physics can lead to a scalar field mass which is far below the UV scale.",
"Separating an infrared (IR) scalar mass from the UV scale requires a very intricate cancellation between all the contributions to the mass.",
"The standard proposed solutions to this problem are typically based on introducing new physics near the scalar mass.",
"This can either be a new symmetry, like supersymmetry, or more drastically Quantum Gravity physics, for example large extra dimensions [1], or the more general formulation in terms of a large number of species [2].",
"However, the absence of any experimental signs of new physics at energy scales near the mass of the Higgs motivates thinking about the possibility that there could be some property of UV physics which is responsible for the cancellation between the contributions to the scalar mass even when the UV scale is far away.",
"Such a property would have to belong to UV physics and yet manifest as a restriction on the mass of a scalar at a much lower IR scale.",
"In this sense it should exhibit some form of UV/IR mixing.",
"It is expected that quantum gravity manifests UV/IR mixing.",
"The classic example being that increasingly massive black holes have increasingly large horizon areas.",
"One conjectured property of quantum gravity is the Weak Gravity Conjecture (WGC) [3].",
"It is therefore interesting to ask if the WGC could manifest some sort of UV/IR mixing such that it restricts the mass of a scalar to an IR scale far below the UV scale associated to quantum gravity physics.",
"In the original formulation of [3] this was not possible.",
"In this note we present evidence that for the more general formulation in the presence of scalar fields [4] it is, at least in principle, possible.",
"Specifically, we propose that quantum gravity physics can restrict the mass of a particle to an IR scale far below the UV cut-off scale of an effective theory if the particle couples to gauge fields and massless scalar fields with almost precisely equal strength.",
"The rest of the introduction is dedicated to expanding on this and forms an overview of some of the main points of this note.",
"Consider a theory with gravity, a $U(1)$ with gauge coupling $g$ , a charged scalar $h$ with charge $q=1$ and mass $m$ , and a neutral scalar $\\phi $ with mass $m_{\\phi }$ ${\\cal L} = \\frac{M_p^2}{2} R -\\frac{1}{4 g^2} F^2 - \\left|D h \\right|^2 - \\frac{1}{2}\\left(\\partial \\phi \\right)^2 - m^2 h^* h - \\frac{1}{2}m_{\\phi }^2 \\phi ^2 - 2 m \\mu \\phi h^* h + ... \\;.",
"$ Here $\\mu $ is a dimensionless parameter which parameterises the coupling of $\\phi $ to $h$ .",
"The $...$ denote arbitrary further terms in the Lagrangian.",
"All the quantities in the Lagrangian (REF ) are the quantum corrected expressions having integrated out all the UV physics.",
"The UV completion of the theory is taken to be as follows.",
"There is a scale $\\Lambda _{UV} \\sim g M_p$ above which the theory can not be completed by a quantum field theory, so it is the scale where quantum gravity related physics is reached.",
"Below this scale, but above $m$ and $m_{\\phi }$ , there may be other scales where new degrees of freedom appear but for each one the theory can be completed by a quantum field theory.",
"The new degrees of freedom are constrained such that the particle with the largest charge-to-mass ratio with respect to the $U(1)$ remains $h$ .",
"Then the claim is that in the limit $m_{\\phi } \\rightarrow 0$ , this theory must satisfy [4] $\\sqrt{g^2 - \\mu ^2} M_p \\equiv \\beta M_p \\ge m \\;.",
"$ We also impose $g \\ge \\mu $ .",
"For finite $m_{\\phi } \\ne 0$ the expression (REF ) will receive corrections suppressed by $\\frac{m_{\\phi }}{m}$ .",
"We estimate these to be of order $\\frac{\\mu ^2 m_{\\phi }}{\\beta m}M_p$ .",
"We therefore observe a new mass scale in the theory $\\beta M_p$ .",
"We will argue that, assuming sufficiently small $m_{\\phi }$ , this mass scale can be separated arbitrarily far from the quantum gravity UV scale of the theory $m \\le \\beta M_p \\ll \\Lambda _{UV}$ .",
"Since (REF ) is tied to quantum gravity physics, with an associated mass scale $\\Lambda _{UV}$ , but is a constraint on an arbitrarily low IR scale, it manifests a form of UV/IR mixing.",
"In other words, say the particle with the largest charge-to-mass ratio in the theory happens to couple almost precisely the same to gauge fields and massless scalar fields.",
"From the perspective of quantum field theory this would not have any implications, but from the perspective of quantum gravity we claim that this would imply that the mass of the particle would have to be at an IR scale far below the UV scale of the theory.",
"The reason is that if $m$ violates the bound (REF ) then the charged particle $h$ would form a tower of absolutely stable bound states of increasing charges.",
"While this will not lead to a direct violation of any physical principle, the bound states are similar to black hole remnants and would go against some expectations of quantum gravity.",
"There are two important and related points to state from the outset.",
"The first is that what we propose is UV/IR mixing is different from other known cases.",
"Specifically, one definition of UV/IR mixing is that it is a property of UV physics that naturally and inevitably one recovers an IR mass scale from it.",
"For example, T-duality in string theory implies that going to the UV automatically recovers IR physics.",
"This is not the case here.",
"Rather, we have to input an IR mass scale $m_{\\phi }$ into the theory, and further tune $\\beta \\ll g$ .",
"Nonetheless, (REF ) could only be deduced, to our knowledge, from quantum gravity physics.",
"This has an associated mass scale of the UV scale $\\Lambda _{UV}$ which as $\\frac{\\beta }{g} \\rightarrow 0$ becomes arbitrarily far away from the IR scale where this relation holds.",
"We therefore believe this should be called UV/IR mixing.",
"The second point, which is a consequence of the first, is that the UV/IR mixing does not form in itself a solution to the problem of the naturalness of a scalar mass.",
"This is because we have to put in an IR mass scale $m_{\\phi }$ by hand and tune $\\beta $ to obtain an IR bound on the particle mass.",
"A full solution to the naturalness problem from UV/IR mixing would be one where the UV physics automatically gives an IR scalar mass.",
"In other words, we propose that if a particle couples almost equally to gauge and light scalar fields then its mass is light.",
"There is nothing stopping the particle being heavy and coupling to gauge and scalar fields differently.",
"Our results are therefore only a reformulation of the question of the naturalness of a scalar mass.",
"Interestingly, this reformulation is one which we believe could only arise from quantum gravity and involves UV/IR mixing in the sense described above.",
"It therefore may have a role to play in a possible full solution to the naturalness problem that involves UV/IR mixing in some way.Indeed some interesting recent proposals that are potentially relevant for naturalness and which utilise the WGC [32], [34], [35] assume that it must have some UV/IR mixing properties.",
"Indeed, a striking property of (REF ) is that it is a bound on a dimensionful mass parameter by dimensionless parameters.",
"This is certainly encouraging in terms of technical naturalness.",
"Both $g$ and $\\mu $ are technically natural.",
"The parameter $\\beta $ is not technically natural in the sense of there being an enhanced symmetry when it vanishes, however, it is very well protected.",
"Consider the theory (REF ) but without any additional terms.",
"Then the 1-loop correction to the gauge coupling diverges logarithmically but goes as $g^3$ .",
"The correction to $\\mu $ is not divergent, and goes as $\\mu ^3$ and $g^2 \\mu $ .",
"Then as long as $g^2 \\sim \\mu ^2 \\lesssim \\beta $ , the hierarchy $\\beta \\ll g$ is not disturbed.",
"This tells us that if we recouple new physics then it does not need to disturb a hierarchy.",
"Of course, it is easy to couple in new physics that does spoil a hierarchy $\\beta \\ll g$ by loop corrections, but the hierarchy is stable with respect to the physics needed for the bound (REF ).",
"We have also introduced a light neutral scalar $\\phi $ by hand.",
"While keeping it light after coupling to new physics beyond (REF ) is generally difficult, within (REF ) the scalar only has to couple to $h$ , and so the contributions of loops of $h$ to its mass will go as $m_{\\phi } \\sim \\mu m$ .",
"Since $\\mu $ is small the scalar remains light with respect to $m$ .",
"Of course, if $\\phi $ couples directly to some new high mass scale then it is expected to gain a large mass, but no such coupling is required for (REF ).",
"Rather, it is a model-building challenge to UV complete (REF ) in a way that allows $\\phi $ to couple much more strongly to $h$ than to high scale UV physics.",
"The note is structured as follows.",
"In section we discuss generally UV/IR mixing and the Weak Gravity Conjecture.",
"In section we develop a toy model based on dimensional reduction of a 5-dimensional theory.",
"This serves the role of illustrating the relevant physics, but also leads to new insights on the WGC.",
"In section we discuss the UV cut-off scale associated to quantum gravity physics.",
"In section we discuss quantum corrections and the implications for naturalness.",
"We finish with a discussion in section ."
],
[
"The Weak Gravity Conjecture and UV/IR Mixing",
"The Weak Gravity Conjecture (WGC) [3] (see [7], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [6], [30], [31], [32], [33], [34], [35], [36], [16] for some recent work on this) states that given a $U(1)$ symmetry with gauge coupling $g$ , there must exist a particle of charge $q$ and mass $m$ which satisfies the inequality $g q M_p \\ge m \\;.",
"$ We will henceforth generally set $q=1$ .",
"This bound is interesting in the sense that even though it is based on quantum gravity physics, it can bound the mass of a particle to be parametrically light compared to the Planck scale.",
"It is therefore natural to consider if the bound on the WGC particle mass could be manifesting UV/IR mixing.",
"One property of the WGC that could be in support of this is that it appears to be in tension with Wilsonian naturalness.",
"The left hand side of (REF ) runs logarithmically with energy scales, but if the WGC particle is a scalar, then it is bounding a mass that runs linearly with the energy scale.",
"This means that the scalar mass $m$ is bounded to be only logarithmically sensitive to UV physics at energy scales above $g M_p$ .",
"However, it is expected that (REF ) does not manifest UV/IR mixing, and for the same reason is also consistent with the Wilsonian notion of naturalness [5].",
"The reason is that it is expected that the cut-off scale of the effective theory $\\Lambda $ is bounded by $g M_p \\ge \\Lambda \\;.",
"$ Indeed, (REF ) can be argued for by reasoning unrelated to naturalness and is termed the Magnetic WGC [3].",
"Therefore the WGC (REF ) only bounds the mass to be below the UV cut-off scale of the effective theory.",
"While it is very interesting that the cut-off scale $\\Lambda $ can be far below the expected quantum gravity physics scale $M_p$ , within the effective theory $g M_p$ is the UV scale.",
"Recently a generalisation of the WGC in the presence of massless scalar fields was proposed [4].",
"Restricted to a simplified case so as to be comparable with (REF ) it states that $\\left(g^2 - \\mu ^2 \\right)M^2_p \\ge m^2 \\;.",
"$ Here $\\mu $ is the coupling of the WGC particle to massless scalar fields.",
"For example, if we consider a single real canonically normalised massless scalar $\\phi $ , and expand it about its vacuum expectation value $\\phi =\\left<\\phi \\right>+\\delta \\phi $ , then the coupling to a complex scalar WGC particle $h$ arises from the Lagrangian term ${\\cal L} \\supset 2 m \\mu \\delta \\phi h^* h \\;.$ Similarly, for a fermion WGC particle $\\psi $ the coupling is ${\\cal L} \\supset \\mu \\delta \\phi \\overline{\\psi } \\psi \\;.$ Note that this can also be written as $\\mu =\\partial _{\\phi } m$ , which is how it was presented in [4].",
"The key point of interest for this note is that in the presence of massless scalar fields it is possible to bound the mass $m$ to an IR mass scale by sending $\\beta ^2 \\equiv g^2 - \\mu ^2 \\rightarrow 0 \\;, $ while keeping $g$ , and therefore $\\mu $ , finite.",
"If the UV scale associated to quantum gravity physics is still only constrained by (REF ), then for $\\beta \\ll g$ the WGC particle mass is bound far below the UV scale of the effective theory $m \\ll \\Lambda $ .",
"Since the WGC is associated to UV quantum gravity physics but is bounding a mass at an IR scale, this would be a manifestation of UV/IR mixing.",
"There are two crucial questions with regards to this proposal.",
"The first is whether (REF ) is indeed a property of quantum gravity.",
"The second is whether sending $\\beta \\rightarrow 0$ does not also modify the cut-off scale of the effective theory so that the quantum gravity UV scale would be $\\beta M_p$ rather than $g M_p$ .",
"If that happened then we would be back to the case where the bound on the mass is only the UV cut-off scale of the effective theory.",
"In this note we present new evidence that (REF ) is a property of quantum gravity and that sending $\\beta \\rightarrow 0$ does not lower the quantum gravity cut-off of the effective theory.",
"Note that it is possible that $\\beta \\rightarrow 0$ does lower the scale of some physics, like a finite number of particles, but we claim that the scale at which one must utilise quantum gravity physics can remain finite.",
"In [4] the bound (REF ) was proposed based on a number of reasons.",
"One is related to ${\\cal N}=2$ supersymmetry and is discussed below.",
"A more general argument is that if we consider the particle with the largest charge to mass ratio in the theory, then unless it satisfies (REF ) it would form a tower of stable gravitationally bound states.",
"Specifically, two such particles would feel a mutual repulsive force due to the gauge field with strength $g$ , an attractive force due to gravity with strength $m$ , and a further attractive force due to the massless scalar fields with strength $\\mu $ .",
"The inequality (REF ) is the requirement that the repulsive force beats the attractive forces so that the particles do not form a bound state.",
"Such a bound state would be completely stable by charge and energy conservation.",
"We could then keep building such stable states by adding more particles.",
"Now consider how many such bound states we could fit below a finite mass scale, say $M_p$ .",
"We know that the particle mass must satisfy $m \\ge \\beta M_p$ to form bound states.",
"Then we can fit at least $\\frac{1}{\\beta }$ such states below $M_p$ .",
"But now send $\\beta \\rightarrow 0$ and we find an infinite number of states.",
"By a species argument, as in [2] for example, we deduce that the cut-off scale of the theory must go to zero in this limit.",
"However, we have proposed, and will present evidence for this in section , that the cut-off scale of the theory stays finite in this limit.",
"This therefore implies that the stable states must not be present to start off with, and so (REF ) must be satisfied.",
"Note that one way to avoid this is to say that we are free to take $m \\gg \\beta M_p$ .",
"This is true, but nonetheless, we can imagine starting with $m < \\beta M_p$ and adiabatically increasing $m$ past the threshold $\\beta M_p$ turning a theory consistent with quantum gravity into a theory inconsistent with quantum gravity and therefore $\\beta M_p$ is clearly an intrinsic quantum gravity scale.",
"We can reach the same conclusion by thinking about black hole remnants [4].",
"If an extremal black hole sources scalar fields through its charge, then the relation between its charge and ADM mass is modified in such a way that, in principle, the ADM mass could be far smaller than the charge.",
"The relation between the ADM mass and the charge is the same as (REF ), at least for any black holes that can be described by a so-called `fake superpotential' [37], but with the scalar coupling $\\mu $ replaced by an appropriate coupling to the black hole.",
"Schematically, if the gauge coupling depends on some massless scalar field $g\\left( \\phi \\right)$ , and if we can write it as $g\\left( \\phi \\right)^2 = W\\left(\\phi \\right)^2 + \\left( \\partial _{\\phi } W \\right)^2 \\;, $ for some real function $W\\left(\\phi \\right)$ , then the black hole solution will be such that the ADM mass is $M_{ADM}=W\\left(\\phi _{\\infty }\\right)$ .",
"Here $\\phi _{\\infty }$ is the value of the field at spatial infinity.",
"We can therefore define $\\beta _{BH}^2 \\equiv g\\left( \\phi _{\\infty } \\right)^2 - \\left.",
"\\left( \\partial _{\\phi } W \\right)^2\\right|_{\\phi =\\phi _{\\infty }}\\;,$ and write $M_{ADM} = \\beta _{BH} M_p$ .",
"Now one can consider how many black hole remnants can fit below $M_p$ and again we find $\\frac{1}{\\beta _{BH}}$ such states.",
"By sending $\\beta _{BH} \\rightarrow 0$ we can reach an infinite number of remnants but with a finite cut-off.",
"This is inconsistent and therefore the remnants must be able to decay.",
"While this argument tells us that there should be a particle in the theory that must become arbitrarily super-extremal in order for black holes to decay and avoid the remnants, it does not quite imply (REF ) since we need some relation between $\\beta _{BH}$ and $\\beta $ for this.",
"There is a nice way to see such a relation, and also to connect the remnant picture with the bound states picture.",
"It is possible to think of the extremality bound on a black hole as the statement that the black hole mass should be at least the binding energy in bringing in all the charge in the black hole from infinity [38].",
"In the presence of scalar fields this binding energy decreases since the attractive force makes it easier to bring the charges in.",
"This is why the ADM mass of a black hole can be much less than its charge.",
"It also directly relates the bound states with the remnants, and justifies a relation between $\\beta _{BH}$ and $\\beta $ .",
"While it is important to find a more quantitative version of this relation, we believe that (REF ) is strongly motivated by this argument.Quantitatively, we are assuming that $q_{\\mathrm {particle}} \\beta _{BH} \\le q_{BH} \\beta $ , where $q$ denotes the charge.",
"The more general expression is ${\\cal Q}^2_{\\mathrm {particle}} \\beta ^2_{BH} \\le {\\cal Q}^2_{BH} \\beta ^2$ , with ${\\cal Q}^2$ defines as in [4].",
"Note that the remnants argument is in some sense stronger than the bound states one, but requires the additional assumptions about the dependence of $g$ (and thereby $\\mu $ ) on $\\phi $ .",
"We can therefore define two versions of the general WGC (REF ).",
"The first one is assuming the structure of $g\\left( \\phi \\right)$ (REF ) and is therefore motivated by both remnants and bound states.",
"The second one is a stronger statement which does not assume anything about the dependence of $g$ on $\\phi $ , and is only motivated by bound states.",
"We denote the former the Weak General WGC and the latter the Strong General WGC.",
"Note that if one adopts only the weak version, then the action (REF ) should be appropriately modified.",
"This can lead to an additional source of mass for the field $\\phi $ and should be accounted for in understanding $m_{\\phi }$ within a UV completion.",
"It is worth noting that these arguments are much stronger than those presented in [3] for the original WGC.",
"There the number of bound states, or remnants, below $M_p$ was $\\frac{1}{g}$ .",
"This is consistent with a species argument as long as the cut-off of the theory went to zero when $g \\rightarrow 0$ , which is indeed the case.",
"Therefore the presence of the bound states or remnants was not excluded by any known arguments.",
"At this point it is worth making a further clarifying remark.",
"In our analysis we will assume that the WGC particle has charge of order one $q \\sim 1$ .",
"This is a strong version of the WGC, and it avoids possible loopholes to do with increasing the charge and mass of the particle so that it ends up above the cut-off scale of the effective theory.",
"Such a possibility for avoiding a strong statement by the WGC was pointed out already in [3].",
"It was also noted within the context of axion alignment [7], [8], [11], and its dual version of higgsing a linear combination of $U(1)$ groups [26].",
"This is supported by evidence from string theory [14], [15], [23], [24].",
"The arguments presented above also rule out this possibility in the sense that if the WGC particle is kept at the cut-off scale of the theory, then we would still have an infinite number of bound states or remnants in the $\\beta \\rightarrow 0$ limit."
],
[
"Relation to ${\\cal N}=2$ Supersymmetry",
"The general version of the WGC (REF ) has close ties to ${\\cal N}=2$ supersymmetry.",
"This is because BPS states in ${\\cal N}=2$ saturate the bound (REF ).",
"Indeed, the simplest way to argue for (REF ) is that in ${\\cal N}=2$ supergravity supersymmetric black holes are themselves BPS states.",
"This means that they can only decay to other BPS states.",
"So the WGC particle, responsible for the decay of extremal black holes, must be a BPS particle and therefore satisfies (REF ).",
"The ${\\cal N}=2$ setting is also the simplest illustration of the UV/IR mixing property of (REF ).",
"Consider going to a point in moduli space where the mass of a BPS state is vanishingly small $m \\rightarrow 0$ but where $g$ remains finite.",
"Since a BPS state saturates (REF ) this is also a point in moduli space where $\\beta \\rightarrow 0$ for that state.",
"Now, consider just a pure gauge theory with no charged matter at this point in moduli space.",
"This would be perfectly fine from the perspective of quantum field theory.",
"However, the WGC would demand the existence of a charged particle.",
"The original version (REF ) would tell us that this charged particle must have a mass below $g M_p$ .",
"However, we know that the decay of black holes requires a much stronger condition which is that the charged particle must actually be BPS.",
"This implies that its mass is at an IR scale far below $g M_p$ and therefore, as we argue in section , far below the scale of quantum gravity physics.",
"The general WGC (REF ) is not quite as strong a statement as ${\\cal N}=2$ would imply, since it could be satisfied in principle by a non-BPS particle, but the UV/IR mixing aspect is the same.",
"There is a also a more practical relation between (REF ) and ${\\cal N}=2$ supersymmetry in that it is a setting that allows for protection against quantum corrections.",
"Of course with respect to the hierarchy problem this is not so interesting since supersymmetry also protects a scalar mass.",
"But from the perspective of trying to understand the microscopic physics behind (REF ) it is a useful starting point."
],
[
"The WGC with Scalars and Dimensional Reduction",
"In this section we present new evidence for (REF ) based on dimensional reduction.",
"The point is that the WGC (REF ) in 5 dimensions leads, upon a classical dimensional reduction, to the generalised version of the WGC (REF ) in 4 dimensions.See also [15], [16] for a dimensional reduction analysis of the WGC.",
"In this section we will consider a classical dimensional reduction.",
"In general, as discussed in the introduction, at the quantum level it is difficult to control $\\beta $ and the mass of the scalar mediators.",
"This example model is no exception to this, and quantum corrections can significantly modify the scenario.",
"However, we will be able to control them, at least to some extent, by utilising some supersymmetry and will argue that in that case the key physics insights of the classical results will remain.",
"We return to a discussion of this as part of the general discussion on quantum corrections in section REF .",
"We consider first the case where the WGC particle is a scalar and return to the fermion case later.",
"To simplify notation we henceforth work in units where $M_p=1$ .",
"The 5-dimensional theory of interest is $S_{5D} = \\int _{M_4 \\times S^1} d^5 X \\sqrt{-G} \\left[ \\frac{1}{2} R^{(5)} - \\frac{1}{4g_5^2} F_{MN} F^{MN} - D_M H \\left(D^M H \\right)^* - M_H^2 H H^* \\right] \\;.",
"$ Here, we have a higher dimensional gauge field $A^M$ , with gauge coupling $g_5$ , and a complex scalar $H$ .",
"The scalar $H$ is charged with charge $q$ and acts as the WGC particle to make this theory consistent.",
"The bound on its mass from the 5-dimensional WGC reads (see for example [15]) $g_5 q \\ge \\sqrt{\\frac{2}{3}} M_H \\;.",
"$ If we reduce this theory on a circle, then we get an effective 4-dimensional action for the zero modes $S^0_{4D} = \\int _{M_4} d^4 x \\sqrt{-g} \\left[ \\frac{1}{2} R + {\\cal I}_{IJ} F_{\\mu \\nu }^{I} F^{J,\\mu \\nu } - \\frac{1}{2} \\left(\\partial \\varphi \\right)^2 - \\frac{1}{2 r^2g_5^2} \\left(\\partial a \\right)^2 - \\left| D h \\right|^2 - m^2 h^* h\\right] \\;.",
"$ Here the 4-dimensional fields are the dilaton $\\varphi $ , an axion $a$ , a complex scalar $h$ , the graviphoton $A_0^{\\mu }$ , and the zero mode of the gauge field $A_1^{\\mu }$ .",
"The dilaton measures the circumference $r$ of the extra dimension $r = e^{-\\sqrt{\\frac{2}{3}}\\varphi } \\;.$ The covariant derivative of the scalar field is given by $D_{\\mu } h = \\left(\\partial _{\\mu } - i q A^1_{\\mu } \\right) h \\;,$ where $q$ is its charge, and the mass is given by $m^2 = \\frac{M_H^2}{r} +\\frac{q^2 a^2}{r^3} \\;.$ The index $I=0,1$ runs over the two 4-dimensional gauge fields $A_I^{\\mu }$ and reads ${\\cal I}_{IJ} = -\\frac{r}{4 g_5^2} \\left( \\begin{array}{cc} \\frac{r^2g_5^2}{2} + a^2 & -a \\\\ -a & 1 \\end{array} \\right) \\;,\\;\\; \\left({\\cal I}^{-1}\\right)^{IJ} = -\\frac{8}{r^3} \\left( \\begin{array}{cc} 1 & a \\\\ a & \\frac{r^2g_5^2}{2} + a^2 \\end{array} \\right)\\;.$ The axion $a$ originates from the extra-dimensional component of the gauge field $a=\\int _{S^1}A_4$ .",
"It can be thought of as a Wilson line in the extra dimension and it enjoys a gauge discrete shift symmetry.",
"There are no fields charged under the graviphoton $U(1)_0$ since we only write the action for the zero modes.",
"The WGC states for $U(1)_0$ are KK modes.",
"The lightest field charged under $U(1)_1$ is the scalar $h$ and so it plays the role of the WGC particle.",
"To utilise the WGC (REF ) we need the general formulation for it presented in [4].",
"For a Lagrangian (not including the WGC particle $h$ ) $\\frac{R}{2} - g_{ij}\\partial _{\\mu } \\phi ^i \\partial ^{\\mu } \\phi ^{j} + {\\cal I}_{IJ} F_{\\mu \\nu }^{I} F^{J,\\mu \\nu } + {\\cal R}_{IJ} F_{\\mu \\nu }^{I} \\left(\\star F\\right)^{J,\\mu \\nu } \\;, $ with scalar fields $\\phi ^i$ and gauge fields $A^I$ , the generalisation of (REF ) is ${\\cal Q}^2 \\ge m^2 + g^{ij}\\partial _{i} m \\partial _{j} m \\;.",
"$ For purely electric charges $q_I$ , we have ${\\cal Q}^2 = -\\frac{1}{2} q_I \\left({\\cal I}^{-1}\\right)^{IJ} q_J$ .",
"Applying the general expression (REF ) to (REF ) we have ${\\cal Q}^2 \\ge m^2 + \\mu _a^2 + \\mu _{\\varphi }^2 \\;, $ where we define $\\mu _a^2 = g^{aa} \\partial _a m \\partial _a m$ , and $\\mu _{\\varphi }^2 = g^{\\varphi \\varphi } \\partial _{\\varphi } m \\partial _{\\varphi } m$ .",
"These measure the strength of the force mediated by the axion and dilaton respectively.",
"The full expressions are ${\\cal Q}^2 &=& \\frac{2 q^2 g_5^2}{r} + \\frac{4a^2q^2}{r^3}\\;,\\;\\;m^2 = \\frac{M_H^2}{r} + \\frac{q^2 a^2}{r^3} \\;,\\nonumber \\\\\\mu _a^2 &=& \\frac{2 q^4 g_5^2 a^2 }{r^4m^2} \\;,\\;\\;\\mu _{\\varphi }^2 = \\frac{\\left(\\sqrt{\\frac{2}{3}} \\frac{M_H^2}{r} + \\sqrt{6} \\frac{q^2 a^2}{r^3} \\right)^2}{2m^2} \\;.",
"$ Using these, the 4-dimensional bound (REF ) reads $\\frac{2M_H^2 r}{3\\left(a^2 q^2 + M_H^2 r^2 \\right)}\\left(3 g_5^2 q^2 - 2 M_H^2 \\right) \\ge 0 \\;.",
"$ This precisely reproduces the 5-dimensional bound (REF ).",
"We therefore find that the general WGC with scalar fields can be deduced from a classical dimensional reduction of the WGC in the absence of scalar fields.",
"This lends some further weight to its validity.",
"It is also interesting in the sense that it forms an example where the bound (REF ) is not saturated.",
"For the case where the WGC particle is a fermion we can consider a 5-dimensional Dirac fermion $\\Psi $ , with 5-dimensional mass $M_{\\Psi }$ and 4-dimensional Dirac zero-mode $\\psi $ .",
"Dimensional reduction leads to exactly the same results as (REF ), with $M_H \\rightarrow M_{\\Psi }$ , and therefore again matches the 5-dimensional WGC.",
"Since, in this respect, it is no different from the scalar case we do not reproduce the calculations here.",
"It is, however, worth commenting about a point regarding the pseudo-scalar nature of the axion $a$ .",
"In both the scalar and fermion cases the expression for $\\mu _a$ , which is the non-relativistic force mediated by the axion, vanishes if $a$ has a vanishing expectation value (in which case the non-relativistic limit requires $M_{\\Psi } ,M_H\\ne 0$ ).",
"In the scalar case this is readily seen from the action (REF ) since there is no cubic coupling for $a=0$ , as is guaranteed by parity conservation.",
"In the fermion case there is a cubic coupling in the action $r^{-\\frac{3}{2}} qa \\overline{\\psi } \\gamma ^5 \\psi $ .",
"However, the straight exchange of the axion using this coupling does not lead to a long range force in the non-relativistic limit.",
"Indeed, for $a=0$ , the leading force is a dipole force which is spin-dependent and scales as $l^{-4}$ , with $l$ the separation distance (see for example [48]).",
"The expression for $\\mu _a$ is capturing an exchange of the axion with an insertion of its vacuum expectation value at each external leg.",
"As in the case of a scalar, it is a long range non-relativistic force that is only present due to the spontaneous breaking of parity."
],
[
"The UV Cut-off Scale",
"The results of [4], and those of section , lend support to the proposal that (REF ) is a property of quantum gravity.",
"The next question with regards to whether (REF ) can bound $m$ far below the UV cut-off scale is whether it is possible to send $\\beta \\rightarrow 0$ without also implying $\\Lambda \\rightarrow 0$ , where $\\Lambda $ is the cut-off scale of the effective theory.",
"In this section we present arguments that indeed this is possible.",
"Note that we take $\\Lambda $ as an enforced cut-off scale on any quantum field theory due to the appearance of quantum gravity physics.",
"Typically this would be the mass scale of an infinite number of states.",
"So we consider $\\Lambda $ as, for example, the string scale.",
"It could be that for a given quantum field theory there is a lower cut-off scale than $\\Lambda $ where new physics appears $\\Lambda _{NP}$ .",
"But this lower cut-off could in principle be completed by some other quantum field theory.",
"We do not make the claim that $\\Lambda _{NP}$ does not go to zero when $\\beta $ does.",
"It could well be that there must be new physics at the scale $\\beta M_p$ for a given quantum field theory.",
"For example, supersymmetry may have to appear.",
"But, although $\\beta M_p$ is a scale implied by quantum gravity, we claim that this new physics is not that of quantum gravity.",
"Sending $g \\rightarrow 0$ in (REF ) is clearly problematic in the sense that the gauge field decouples and we recover an exact $U(1)$ global symmetry.",
"Note that taking multiple $U(1)$ s, and therefore using the general expression (REF ) where $g^2 \\rightarrow {\\cal Q}^2$ does not help in this regard.",
"Since in ${\\cal Q}^2$ the charge vector $q$ is contracted with the gauge kinetic matrix ${\\cal I}$ , a vanishing ${\\cal Q}^2$ would imply a vanishing determinant for ${\\cal I}$ which would again decouple a gauge field leading to a global symmetry.",
"This is expected to not be possible in quantum gravity.",
"The bound (REF ), or (REF ), alone would only imply that a particle become massless in this limit.",
"This seems insufficient to block such a global symmetry, and it is therefore natural to expect that the stronger condition (REF ) should be imposed.",
"In contrast, the $\\beta \\rightarrow 0$ limit does not decouple the gauge field and therefore just a particle becoming massless appears consistent from this perspective.",
"In the presence of ${\\cal N}=2$ supersymmetry the WGC particle must be BPS, at least in the sense that supersymmetric extremal black holes can only decay to BPS particles.",
"BPS particles satisfy the equality limit of (REF ).",
"Therefore in this case $m \\rightarrow 0$ implies, rather than is just implied by, $\\beta \\rightarrow 0$ .",
"So if $\\beta \\rightarrow 0$ would imply $\\Lambda \\rightarrow 0$ then we would have a quantum gravity obstruction to massless BPS states.",
"This would be very strange.",
"Indeed, the classic example of BPS states in string theory are wrapped branes.",
"If we consider compactifications of type IIB string theory on Calabi-Yau manifolds, then D3 branes wrapped on 3-cycles lead to BPS particles in four dimensions.",
"If we go to the conifold locus in moduli space then one such BPS particle (hypermultiplet) becomes massless [45].",
"But only one BPS state is becoming massless which can be understood by looking at its effect on the gauge coupling.",
"While the conifold locus is a somewhat exotic point in complex-structure moduli space, there is no reason to expect that it is not possible to work with an effective theory with a finite cut-off in its vicinity.",
"This is in contrast to the $g \\rightarrow 0$ limit, where an infinite number of states become massless sending $\\Lambda \\rightarrow 0$ and matching general quantum gravity expectations.",
"In section we showed that (REF ) can be understood from classical dimensional reduction.",
"In such a setting there are a number of UV cut-off scales.",
"There is the scale where the 5-dimensional theory breaks down (see for example [47]), the Kaluza-Klein scale, and an expected 5-dimensional magnetic WGC scaleNote that in these estimates we have included an extra factor of $r^{-1}$ relative to the versions of these expressions in the literature, due to the change from the 5-dimensional to 4-dimensional Planck mass.",
"Either way, this extra factor is not important for the primary point.",
"$\\Lambda ^2_{5D} \\sim \\frac{1}{r}\\left(\\frac{4\\pi }{g_5}\\right)^2 \\;, \\;\\; \\Lambda ^2_{KK} \\sim \\frac{1}{r}\\left(\\frac{2\\pi }{r}\\right)^2 \\;, \\;\\; \\Lambda ^2_{WGC,5} \\sim \\frac{g^2_5}{r} \\;.$ We would like to consider how the parameter $\\beta ^2$ in (REF ), which bounds the mass $m^2$ , compares to these scales.",
"Again we will focus on the classical results, and return to quantum corrections in section REF .",
"The explicit form is ${\\cal Q}^2 - \\mu _a^2 -\\mu _{\\varphi }^2 \\equiv \\beta ^2 = \\frac{3 a^4 q^4+6 a^2 M_H^2 q^2 r^2+6 g_5^2 M_H^2 q^2 r^4-M_H^4 r^4}{3 r^3 \\left(a^2 q^2+M_H^2 r^2\\right)}\\;.$ There are two ways in which we could choose the parameters in this model to obtain a small $\\beta $ .",
"One of them violates the WGC while the other does not.",
"Therefore, the difference between them must arise due to interaction with UV physics and so is associated to UV/IR mixing.",
"Consider first the case which violates the WGC.",
"Let us set $a = 0$ so that we have $\\left.\\beta ^2\\right|_{a=0} = \\frac{6 g_5^2 q^2 - M_H^2}{3 r} \\;.$ Then if we forget about the 5-dimensional WGC (REF ), we can take $M_H^2 \\rightarrow 6 g_5^2 q^2$ .",
"In that case we have $\\beta \\rightarrow 0$ .",
"This limit is perfectly fine from a quantum field theory perspective and it leads to a large mass $m^2 \\rightarrow \\frac{6 g_5^2 q^2}{r}$ .",
"So from a quantum field theory perspective, there is nothing that stops us from considering a particle which couples equally to gauge and scalar fields, yet has a large mass.",
"This exemplifies the expectation that the generality of the connection between the coupling equality $\\beta =0$ and the mass of a particle is inherently quantum gravitational in nature.",
"Consider now the way to reach small $\\beta $ while respecting the WGC.",
"We take the parameter values $r^2 g_5^2 q\\gg a^2 q\\gg g_5 M_H r^2 \\;.",
"$ We have that $\\beta ^2$ and ${\\cal Q}^2$ read in this case $\\beta ^2 \\simeq \\frac{a^2 q^2}{r^3}\\;,\\;\\; {\\cal Q}^2 \\simeq \\frac{q^2 g^2_5}{r}\\;.$ Therefore, we find $\\beta ^2 \\ll {\\cal Q}^2$ in this regime.",
"Since (REF ) is compatible with the WGC, we indeed find that the mass of the WGC particle is bound far below any UV cut-off scale.",
"This is contrast to the field theory setting discussed above.",
"It is also in contrast to the scale ${\\cal Q}^2$ , which would bound the mass in the absence of scalar fields, and which can not be separated from $\\Lambda ^2_{WGC,5}$ .",
"We therefore conclude that the UV/IR mixing properties of the WGC impose that in a quantum theory of gravity having a small $\\beta $ implies a small mass for the WGC particle.",
"Let us return to the case $a=0$ .",
"If we do impose the 5-dimensional WGC (REF ) then we have $\\left.\\beta ^2\\right|_{a=0} \\ge \\frac{3 g_5^2 q^2}{2r}.$ This means that the WGC bound on the mass is tied to the UV scale and there is no UV/IR mixing.",
"This is interesting from the microscopic perspective since it implies that the UV/IR mixing is tied to $\\mu _a$ which in this case vanishes.",
"The contribution $\\mu _a$ due to the axion arises in the 5-dimensional theory as a Wilson line around the extra circle dimension.",
"A Wilson line is a non-local gauge operator.",
"Therefore we see that the UV/IR mixing aspect is manifesting due to the interaction of the WGC with non-local physics.",
"This matches the general expectation that UV/IR mixing should be tied to non-local physics.",
"If we have $a \\ne 0$ , then the UV/IR mixing is tied to the parameter range (REF ).",
"This can be written as $a q \\gg M_H r$ .",
"If we were to take $M_H r$ very large the WGC particle would have a localised Compton wavelength and would not sense the compactness of the circle direction or the associated Wilson line.",
"It would be sensitive to only local physics and accordingly we see that UV/IR mixing would not be possible.",
"Note that the WGC bound (REF ) can only be understood in terms of classical long range forces if the mass of the WGC particle is heavier than the mediator forces.",
"But the $\\beta \\rightarrow 0$ limit is the one where the WGC state becomes massless and therefore must be approached with care.",
"We should keep this in mind, but it does not form a fundamental obstruction for any realistic scenario where we are concerned with $\\beta \\ll g$ rather than $\\beta =0$ .",
"To summarise, in this section we presented evidence that the general formulation of the WGC (REF ) can exhibit a bound on the WGC particle mass parametrically far below the UV cut-off scale of the theory.",
"In other words, that the WGC particle possesses the property that if it couples almost precisely the same to gauge fields and scalar fields then its mass is far below the UV scale of the theory."
],
[
"Relation to a Scalar Weak Gravity Conjecture",
"In [4] it was shown that in an ${\\cal N}=2$ supersymmetric setting the scalar forces act stronger than gravity on the WGC particle.",
"More precisely, for each scalar field there is one particle on which the scalar force acts stronger than gravity.",
"This was proposed as a possible Scalar WGC.",
"If we apply this to the zero mode $h$ it reads $\\left(\\mu _a^2 + \\mu _{\\varphi }^2\\right) M_p^2 \\ge m^2 \\;.",
"$ Utilising (REF ) this implies $3 a^2 q^4 \\left(a^2 + g_5^2 r^2 \\right) \\ge \\left(M_Hr \\right)^4 \\;.",
"$ Therefore, for sufficiently large $M_H$ we find that the Scalar WGC is not satisfied by the WGC particle.",
"So gravity is acting stronger than the scalar forces on the WGC particle.",
"A possible interpretation of this result is that it is evidence against a Scalar WGC.",
"If that is the case it could have interesting implications for large field excursions in quantum gravity.",
"In [4] it shown that a scalar WGC could be a possible explanation for the exponentially decreasing mass of states for super-Planckian scalar field variations.See [39], [40], [41], [42], [43], [44] for recent work on this.",
"Here it appears that taking large $M_H$ could allow to violate this.",
"As we will discuss in section , in a quantum setting the mass $M_H$ should be tied to supersymmetry breaking.",
"In this sense it could be suggesting that looking at highly non-supersymmetric situations could allow for large field variations.",
"Having stated this, it is the case still that even in this model the Scalar WGC is satisfied in the sense that there is still a particle on which gravity acts weaker than the scalar forces.",
"This is because any Kaluza-Klein mode of the 5-dimensional field $H$ will satisfy (REF ).",
"This can be seen by noting that the replacement $q a \\rightarrow q a + 2 \\pi n$ , where $n$ is an integer (the Kaluza-Klein number), is a gauge symmetry of the theory.",
"For any non-zero $n$ the inequality (REF ) can not be violated within the effective theory limit $M_H r \\ll 2\\pi $ .",
"Yet another possibility is to require that the Scalar WGC applied to the zero-mode $h$ , which is the WGC particle in the sense that it has the largest charge-to-mass ratio, should be satisfied.",
"Then this would be interpreted as a bound on the parameters (REF ).",
"Or as a statement that the original 5-dimensional theory is not compatible with a quantum gravity UV completion as it is.",
"With respect to the point of interest in this note.",
"The Scalar WGC can also be used to restrict the mass of a particle since it imposes $\\mu M_p \\ge m$ .",
"This appears in many ways simpler than utilising (REF ).",
"However, it is natural to expect that sending $\\mu \\rightarrow 0$ for the Scalar WGC particle implies a vanishing cut-off $\\Lambda \\rightarrow 0$ [49].",
"Further, if we accept that there are no additional constraints of relevance on the 5-dimensional theory or the parameter $M_H$ , then the Scalar WGC particle must be a Kaluza-Klein mode rather than the particle with the largest charge-to-mass ratio.",
"This tells us that we can not utilise the Scalar WGC to constrain its mass to be below the UV cut-off scale of the theory.",
"It is interesting to note that both the Scalar WGC $\\mu M_p \\ge m$ and the original WGC $g M_p \\ge m$ can be unified in a sense if we modify (REF ) to $\\left|g^2 -\\mu ^2 \\right| M_p^2 \\ge m^2 \\;.$ This would be a natural generalisation of (REF ).",
"It would also explain a slight puzzle which is that if we send $M_p \\rightarrow \\infty $ in (REF ) we seem to still have information left about the sign of $g^2 - \\mu ^2$ ."
],
[
"Quantum Corrections and Naturalness",
"So far we have primarily discussed the evidence for UV/IR mixing in the WGC.",
"In this section we focus on quantum corrections and the implications of the WGC for naturalness.",
"There are two key questions in thinking about this.",
"The first is to do with the fact that (REF ) is taken to apply in the presence of massless scalar fields, but at the quantum level we expect these scalars to gain a mass.",
"This can lead to corrections to (REF ) which need to be quantified.",
"The second question is how to obtain a hierarchy between $\\beta $ and $g$ that is protected from quantum corrections.",
"Both of these are model-dependent questions, but there are some general results that we discuss in this section."
],
[
"Quantum Corrections to $m_{\\phi }$",
"Consider first the issue of the mass of the scalar force mediators.",
"For ease of notation let us consider a single scalar (or pseudo-scalar) mediator $\\phi $ with mass $m_{\\phi }$ .",
"The WGC bound (REF ) applies precisely only for $m_{\\phi }=0$ .",
"More generally, we can expect it to apply approximately as long as $m_{\\phi } \\ll m$ .",
"This is the regime where the scalar $\\phi $ acts as a long range classical force on the WGC particle.",
"One way to ensure a mass separation at the quantum level is to impose supersymmetry.",
"This protects the scalar mass $m_{\\phi }$ from perturbative quantum corrections, and as long as non-perturbative effects are small, this is sufficient.",
"Of course, the problem with this is that in protecting $m_{\\phi }$ we are also protecting $m$ .",
"This makes the WGC bound on $m$ less interesting, but it remains non-trivial.",
"For example, it could be used to explain the smallness of technically natural parameters, such as a supersymmetric mass term.",
"To gain insights into naturalness we must consider a non-supersymmetric setting.",
"In this case a massless, or at least arbitrarily light, scalar field at the quantum level is difficult to implement but not impossible.",
"For example, axions can have arbitrarily low masses.",
"More generally, if it couples only very weakly to other sectors its mass could be protected.",
"Also the scalar, or pseudo-scalar, does not have to be fundamental, it could be a type of pion.",
"With this in mind we should recall that, as discussed in section , the pseudo-scalar interaction required is a long range spin-independent one, which means that it relies on some breaking of parity.",
"The mass $m_{\\phi }$ is therefore model-dependent, and while it may or may not be difficult to protect it, this is a question of how it couples to other fields.",
"However, there is a universal minimum bound on its coupling because in order to apply (REF ) we require that the scalar field $\\phi $ couples to the WGC particle with coupling $\\mu $ that is not arbitrarily small.",
"Therefore, at the quantum level, we know that its mass at least receives corrections from the WGC particle so that we expect it to be driven to at least $m_{\\phi } \\sim \\mu m$ .",
"This is the generic expectation though and it may be possible to construct models where it is much lighter.",
"This is not reintroducing the full hierarchy problem because it is a question of protecting $m_{\\phi }$ from $m$ rather than from the full UV physics.",
"We can generate a hierarchy $m_{\\phi } \\ll m$ by taking $\\mu \\ll 1$ .",
"In general, we expect that we require to take $\\mu $ and $g$ very small so as to reduce sensitivity to loops.",
"However, we should not take $\\mu $ as small as $\\mu \\sim \\beta $ since then we are lowering the UV cut-off scale (REF ) all the way to $m$ .",
"In this sense there is a model-building challenge in coupling $\\phi $ much more strongly to $h$ than to UV physics.",
"An important question in a non-supersymmetric context therefore appears to be what are the quantitative corrections to the WGC (REF ) for a finite $m_{\\phi }$ but with $m_{\\phi } \\ll m$ .",
"We do not know, but one estimate could be as follows.",
"Since $m_{\\phi } \\ll m$ we can still approximate the effect of the scalar $\\phi $ as a long range classical Yukawa force.",
"The question of the presence of a bound state is now dependent on the separation scale $l$ between the WGC particles.",
"At separation scales $m_{\\phi } l \\lesssim 1$ the scalar force is still significant, and we can take $l$ as small as $m^{-1}$ .",
"At this scale the Yukawa exponential suppression gives a leading correction to the force $\\mu ^2 \\rightarrow \\mu ^2 - {\\cal O}\\left( \\frac{\\mu ^2 m_{\\phi }}{m}\\right) = \\mu ^2 - {\\cal O}\\left(\\mu ^3\\right) \\;.",
"$ In the last equality we used the expected contribution to the scalar mass from loops of the WGC particle as discussed above.",
"Such a correction would not modify the bound as long as $\\mu ^{\\frac{3}{2}} < \\beta \\ll \\mu $ .",
"This is a statement on the absence of a bound state at the smallest distance scale at which the force behaves classically.",
"This is a reasonable guess but, since it is not clear what exactly is the constraint from quantum gravity on the presence of bound states, the distance scale at which to apply it remains an even larger uncertainty.",
"However, this is a limitation on our calculation ability rather than a fundamental obstruction to a small $\\beta $ .",
"In principle, if we knew the corrected $\\mu $ we could tune this against $g$ to reach an arbitrarily small $\\beta $ unrelated to $\\mu $ .",
"There is a weaker bound on $\\beta $ relative to $\\mu $ coming from quantum corrections to which we now turn."
],
[
"Quantum Corrections to $\\beta $",
"In this subsection we will consider the quantum corrections to $\\beta $ .",
"We are particularly interested in whether a hierarchy $\\beta \\ll g$ could be protected against quantum corrections and so could be technically natural.",
"We will not focus on a possible mechanism to generate the hierarchy in the first place.",
"However, there are a some comments worth making regarding some universal features of the effects of UV physics on $\\beta $ .",
"The first is that since $g$ is the gauge coupling of a $U(1)$ , as long as this remains a $U(1)$ and does not complete into a non-Abelian symmetry, adding a heavy state and integrating it out can only decrease the coupling $g$ .",
"The second point is that introducing new scalars that couple to the WGC particle has a tree-level effect on $\\beta $ because the scalar force $\\mu $ is a sum over all contributions.",
"A scalar field can only lead to an attractive force, and so can only increase $\\mu $ .If the new particle is very heavy there is no sense it which it acts as a classical long range force.",
"Nonetheless, it is natural to expect that integrating it out leads to a positive tree-level contribution to $\\mu $ .",
"Therefore, there is a sense in which adding new physics only serves to decrease $\\beta $ .",
"This may be a possible way to reach a hierarchically small $\\beta $ to start off with.",
"The primary effect we are interested in are the quantum corrections to $\\beta $ , which depend on the UV completion.",
"We will not study in detail the possibility that $\\beta $ could be protected by a symmetry present in the UV theory that then ensures its technical naturalness.",
"It is not clear what type of symmetry could protect $\\beta $ .",
"It would have to relate gauge fields and scalar fields.",
"Of course, ${\\cal N}=2$ supersymmetry is sufficient in this respect.",
"The interesting thing is that we only requires its action on the bosonic fields.",
"Also the setting of a higher dimensional compactification, as in section , offers some protection against quantum corrections because the gauge and scalar fields are related by a higher-dimensional gauge symmetry.",
"For example, deep UV physics above the Kaluza-Klein scale, where the gauge symmetry is restored, does not correct the mass $m_a$ .",
"If the 5-dimensional mass $M_H$ or $M_{\\Psi }$ could be kept small, then deep UV physics would also not correct a small $\\beta $ .",
"In any case, however, threshold corrections from the Kaluza-Klein scale are too large.",
"A symmetry that relates gauge fields and scalars on branes in string theory is T-duality.",
"This is interesting in the sense that this duality also ties to UV/IR mixing.",
"There are two ways in which the WGC bound on a particle mass could restrict the mass of a scalar field.",
"One is directly, so if the WGC particle is itself a scalar field, then the WGC directly bounds its mass.",
"The other is indirectly.",
"Say the WGC particle is a fermion that gains a mass from Yukawa coupling to a scalar.",
"Then restricting the mass of the fermion WGC particle restricts the vacuum expectation value of the scalar.",
"Through minimising the scalar potential this indirectly bounds its mass.",
"We will consider each possibility in turn."
],
[
"A Scalar WGC Particle",
"The case of a scalar WGC particle was discussed briefly in the introduction.",
"Consider the model (REF ).",
"We will set $m_{\\phi }=0$ for simplicity, and consider no additional terms beyond those in (REF ) for now.",
"We are interested in the 1-loop corrections to the gauge coupling $g$ and the parameter $\\mu $ .",
"Consider starting at some UV scale $M$ , and running down to an IR scale $m$ , and evaluating the change in the IR values of the parameters.",
"For the gauge coupling this reads $\\delta g \\sim g^3 \\ln \\left(\\frac{M}{m}\\right)\\;.",
"$ Therefore, if we consider $g^2 \\lesssim \\beta $ , this variation will not disturb a hierarchy $\\beta \\ll g$ .",
"In fact, $g$ can only ever decrease as long as UV physics does not involves massive vectors coupling to it, which means that the UV corrections can only ever decrease $\\beta $ in any case.",
"Consider now the parameter $\\mu $ .",
"To understand its corrections we need to consider the 1PI diagrams correcting the cubic coupling.",
"There are two such diagrams, due to the exchange of $\\phi $ and the $U(1)$ photon, and neither is divergent due to the fact that they involve one loop momentum integral but three bosonic propagators.",
"Schematically, for $m \\ll M$ , the correction takes the form $\\delta \\mu \\sim \\mu ^3 + \\mu g^2 \\;.$ Again, these do not disturb a hierarchy a long as $\\mu ^2 \\lesssim \\beta $ .",
"These results tell us that coupling the theory (REF ) to new physics does not need to spoil a hierarchy $\\beta \\ll g$ .",
"In this sense $\\beta $ is a technically natural parameter.",
"It is of course possible to couple in new physics that does spoil the hierarchy at 1-loop.",
"For example, say we introduced a new scalar $S$ with mass $M_S$ that couples to the WGC particle as ${\\cal L} \\supset 2 m \\mu _S S h^* h \\;.$ Then we would have $\\delta \\mu \\sim \\mu \\mu _S^2 \\left(\\frac{m}{M_S}\\right)^2 \\;.$ We would therefore require $\\mu \\mu _S \\left(\\frac{m}{M_S}\\right) \\lesssim \\beta $ to maintain the hierarchy.",
"Since we need $\\mu \\gg \\beta $ this puts a limit on how strongly $S$ can couple to $h$ and on how light $S$ could be.",
"Overall, the question of whether a hierarchically small $\\beta \\ll g$ can be protected against UV physics is model dependent, but there is no serious obstruction to doing so as far as we can see."
],
[
"A Fermion WGC Particle",
"The case of a fermion WGC particle is similar.",
"Consider a Dirac WGC particle $\\psi $ with action ${\\cal L} = \\frac{M_p^2}{2} R -\\frac{1}{4 g^2} F^2 - \\overline{\\psi } D \\psi - \\frac{1}{2}\\left(\\partial \\phi \\right)^2 - m \\overline{\\psi } \\psi - \\frac{1}{2}m_{\\phi }^2 \\phi ^2 - \\mu \\phi \\overline{\\psi } \\psi + ... \\;.",
"$ Again we have $\\delta g \\sim g^3$ and $\\delta \\mu \\sim \\mu ^3 + \\mu g^2$ , and so $\\beta $ is protected.",
"The same analysis holds for the case of a pseudo-scalar coupling ${\\cal L} \\supset \\mu \\phi \\overline{\\psi } \\gamma ^5 \\psi $ .",
"In this case the bound (REF ) is on a fermion mass.",
"However, it can be turned into a bound on a scalar mass.",
"Consider setting $m=0$ in (REF ), which is technically natural, and instead adding a scalar $h$ with a potential ${\\cal L}_h \\supset - \\frac{1}{2}\\left(\\partial h \\right)^2 - \\mu _h h \\overline{\\psi } \\psi - \\left( -\\frac{1}{2}m_h^2 h^2 + \\frac{1}{4} \\lambda h^4 \\right) \\;.$ Then minimising the potential we obtain the new mass $m = m_h\\frac{\\mu _h }{\\sqrt{\\lambda }} \\;.$ We therefore obtain a bound on a scalar mass.",
"Adding the scalar $h$ modifies $\\mu $ due to a new 1-loop diagram which means that $\\delta \\mu \\sim \\mu \\mu _h^2 \\left( \\frac{m}{m_h} \\right)^2 \\;.$ We want to maintain $m \\sim m_h$ , which can be done by choosing $\\lambda \\sim \\mu _h^2$ , and so to control this correction to $\\beta $ we need to take $\\mu \\mu _h \\lesssim \\beta $ ."
],
[
"Quantum Corrections in the 5-dimensional Model",
"The analysis of the dimensional reduction of the 5-dimensional theory studied in sections and was performed at the classical level.",
"At the quantum level there are significant changes which exemplify the general problems discussed in this section.",
"The problems begin already with the 5-dimensional theory since it is non-renormalisable, the coupling $g_5$ should become strong and the mass $M_H$ is not protected from the UV scale.",
"Even, if we take the 5-dimensional theory with small $g_5$ and $M_H$ as a given starting point, then the 4-dimensional scalars $a$ and $\\varphi $ are expected to gain a mass near the Kaluza-Klein scale from loop effects.",
"Also they would have a potential, in the case of $a$ this would be periodic, which would fix dynamically their values, rather than having them as the free parameters utilised in section .",
"This would imply that reaching a small $\\beta $ dynamically is difficult.",
"Further, $\\beta $ would receive quantum corrections that would be on top of this dynamical problem.",
"The point of sections and was not to argue for a mechanism that can address naturalness questions, but rather to provide some evidence for UV/IR mixing in the WGC.",
"The question we are interesting in is therefore if the physical conclusions reached through the classical analysis could still hold at the quantum level.",
"One way to ensure this is to employ supersymmetry.",
"If we consider ${\\cal N}=1$ supersymmetry in the 5-dimensional theory then the quantum problems discussed above are all addressed.",
"Consider the 5-dimensional action (REF ).",
"In the case of $M_H=0$ it forms the bosonic sector of an ${\\cal N}=1$ supersymmetric theory.",
"Therefore we could complete it to a supersymmetric theory and the analysis of sections and would remain unchanged.",
"In such a supersymmetric setting the ${\\cal N}=1$ 5-dimensional supersymmetry reduces on a circle to ${\\cal N}=2$ 4-dimensional supersymmetry.",
"This explains the vanishing of (REF ) since $h$ becomes BPS.",
"It is still the case that the WGC bounds the mass of the particle below the UV scale of the theory, although this only amounts to the statement that it should be BPS.",
"Now consider still a supersymmetric 5-dimensional action apart from turning on $M_H \\ne 0$ .",
"This is a non-supersymmetric mass term and so breaks supersymmetry completely.",
"However, since it is the only source of supersymmetry breaking it forms a control parameter for its effects.",
"This setup therefore allows us to perturb away from ${\\cal N}=2$ , and we see this since (REF ) no longer vanishes.",
"It is reassuring that under this non-supersymmetric perturbation we recover the 5-dimensional WGC.",
"From the 5-dimensional perspective a small $M_H$ is fine since it is technically natural.",
"The 4-dimensional scalars will still obtain a potential, and $\\beta $ quantum corrections.",
"However, we are not after a mechanism that dynamically fixes a small $\\beta $ .",
"We therefore assume that as long as the masses $m_a$ and $m_{\\varphi }$ are sufficiently light we could imagine displacing the fields to the point of small $\\beta $ by hand.",
"So that it still makes sense to ask how the theory would behave at small $\\beta $ .",
"With this assumption, we are left with the question of the scalar masses.",
"We require a hierarchy $m_a \\ll m$ and $m_{\\varphi } \\ll m$ to be able to utilise the WGC.",
"Such a hierarchy can be induced by the general mechanism described in this section.",
"Since their masses are induced by $M_H$ as the source of supersymmetry breaking, they are of order $\\mu _a M_H$ and $\\mu _{\\varphi } M_H$ .",
"While $m$ is at least of order $M_H$ .",
"We therefore reach the situation described in (REF ).",
"If indeed the corrections to the WGC are as proposed in (REF ), then for sufficiently small couplings these are sufficiently small to trust the results.",
"We therefore conclude that, under some reasonable assumptions, the classical analysis of sections and is expected to capture the correct physics as long as $\\mu _a$ and $\\mu _{\\varphi }$ are small."
],
[
"Discussion",
"In this note we proposed that in the presence of scalar fields the general version of the WGC (REF ) can bound the mass of the WGC particle far below the UV cut-off scale of the effective theory.",
"We supported this by presenting new evidence for (REF ) and for the claim that taking small $\\beta $ does not necessarily lower the cut-off scale of the effective theory.",
"Such a bound on an IR mass from UV quantum gravity physics is interesting in that it manifests a form of UV/IR mixing.",
"Of course, our results depend on the validity of (REF ) and while we presented some new arguments for it, extending the work in [4], it is crucial to build up more evidence for it.",
"A primary motivation for the interest in such a bound is that it could have potential relevance to the question of the naturalness of a scalar mass.",
"At a general level this could be tied to UV/IR mixing in quantum gravity and so any insights into this are useful.",
"In this respect, it was interesting to see that in the toy model 5-dimensional reduction on a circle example, the UV/IR mixing was tied to the interaction of the WGC with a Wilson line in the extra dimensions which is a non-local operator.",
"A direct application of our current understanding of the bound from the WGC to the naturalness problem of a scalar mass requires addressing two key questions.",
"The first is how to protect the mass of the force mediator scalar $m_{\\phi }$ .",
"The second is how to protect a hierarchy $\\beta \\ll g$ .",
"We presented arguments that while this is a model-dependent question, there is no obstruction in principle to doing so in the sense that the ingredients that go into the bound, as in the toy model (REF ), do not in themselves obstruct a small $m_{\\phi }$ and $\\beta $ at the quantum level.",
"Having stated this, there is a model-building challenge in that $\\phi $ must couple much more strongly to $h$ than to UV physics.",
"Also, we only considered if a hierarchy $\\beta \\ll g$ is technically natural, and did not study in detail how such a hierarchy could be induced in the first place.",
"One of the largest uncertainties in our analysis is the magnitude of the correction to (REF ) for finite but small $m_{\\phi }$ .",
"We presented an estimate of this in (REF ), but further work on understanding this effect is crucial.",
"Our estimate, combined with a natural expectation for the minimal value of $m_{\\phi }$ , led to a bound on how large $g$ could be of $g \\lesssim \\beta ^{\\frac{2}{3}}$ .",
"This implies that the $U(1)$ and scalar mediators must be very weakly coupled.",
"It also implies a relatively low UV scale $\\Lambda _{UV} \\lesssim \\beta ^{\\frac{2}{3}} M_p$ .",
"However, these limitations were really due to our lack of knowledge of the precise effect of a finite $m_{\\phi }$ .",
"In principle, $\\beta $ could be tuned accounting for this correction arbitrarily far below $g$ .",
"Another relation between $g$ , $\\mu $ and $\\beta $ comes from controlling quantum corrections which implies $g^2\\sim \\mu ^2 \\lesssim \\beta $ .",
"This relation implies the less strong bound $\\Lambda _{UV} \\lesssim \\sqrt{\\beta } M_p$ .",
"We end with a few brief comments about the mechanism discussed in this note and the actual hierarchy problem as applies to the Higgs in the Standard Model.",
"Since the WGC bound relies on coupling to a $U(1)$ and light scalars, which should be in some hidden sector, it is easier to imagine an indirect bound on the Higgs mass.",
"So that the WGC particle obtains some of its mass from the Higgs vacuum expectation value and thereby restricts it.",
"An attractive feature of the bound on the mass from the WGC is that it involves only IR quantities that are potentially measurable by experiments.",
"So say that some particle had a light mass due to the WGC bound, it would imply a sharp prediction that its couplings to massless scalar fields and gauge fields would have to be equal up to an accuracy of the ratio of its mass to the UV scale.",
"If the scalar fields are not massless but still very light, then there would be small corrections to this equality.",
"One would expect a similar such striking relation between gauge and scalar couplings for some particle, not necessarily the Higgs, possibly dark matter, if this mechanism plays a role in the hierarchy problem of the Standard Model.",
"If the scalar force mediators have a significant mass then the required equality of the couplings can be significantly corrected.",
"More generally, the WGC argument is based on a classical long range force picture, but it should be better thought of as an argument for a more general microscopic property of quantum gravity that manifests UV/IR mixing and most likely is present even when the long range classical force picture breaks down.",
"In this more general sense, one could imagine even a very massive scalar mediator leading to the bound on the WGC particle mass.",
"In particular, it could be applied to the Standard Model matter fields with the Higgs as the scalar force mediator.",
"While the mass of the Higgs implies there is no classical long range force picture, the more general mechanism could still manifest as a more complicated relation between the gauge and scalar couplings.",
"A quantitative understanding of this would require understanding the effects of a large mass for the scalar force mediators in (REF ).",
"Such a formulation could then be applied directly to the Standard Model Yukawa and gauge couplings, with say the electron as the WGC particle.",
"If the hierarchy problem is then tied to the WGC, as applied to the Standard Model in this way, then the quantitative formulation would predict a specific relation to high precision between the gauge and Yukawa couplings that could be experimentally tested.",
"In this note we proposed a new way to think about separation of scales and UV/IR mixing.",
"While we focused on a possible application to the gauge hierarchy problem, it would be very interesting to establish if it also has implications for the cosmological constant problem.",
"We have discussed how quantum gravity may ensure the absence of towers of bound states that are protected from decay by their charge.",
"There is another way that a tower of bound states can be protected which is by the particle being the lightest in the theory.",
"In the Standard Model these are neutrinos and therefore neutrinos could form a tower of stable bound states.",
"At long distances the only force that neutrinos feel is gravity.",
"Therefore, in empty space they would indeed form bound states.",
"However, this is not the case in the presence of a cosmological constant.",
"In the weak gravity regime the cosmological constant can be modelled as a repulsive linear force (see for example [50]) so that the total gravitational force acting on two neutrinos is $F_{\\mathrm {Gravity}} = m_{\\nu }\\left(- \\frac{m_{\\nu }}{r^2} + \\frac{\\Lambda r}{3} \\right)\\;,$ where $m_{\\nu }$ denotes the lightest neutrino mass.",
"We can apply this approximately up to the scale of the neutrino mass $r \\sim m_{\\nu }^{-1}$ .",
"Therefore, for neutrinos to not form stable bound states we require $\\Lambda > m_{\\nu }^{4} \\;.$ This bound can be viewed either as a bound on how small the cosmological constant could be or as a bound on how heavy neutrinos could be.",
"In the latter case, this can be translated to a bound on the scale of electroweak symmetry breaking, and therefore the mass of the Higgs.",
"We therefore find the striking result that the mass of the Higgs could not be much higher else, for the given value of the cosmological constant, neutrinos would form a tower of stable bound states.",
"This could lead to a species problem and in that sense be inconsistent with quantum gravity.",
"It would be interesting to explore a possible connection to the results of [32], [34], [35].",
"Acknowledgements: We would like to thank Gia Dvali, Inaki Garcia-Etxebarria, Arthur Hebecker, Luis Ibanez, Daniel Junghans and Irene Valenzuela for useful discussions.",
"The work of DL is supported by the ERC Advanced Grant “Strings and Gravity\" (Grant No.",
"320040)."
]
] | 1709.01790 |
[
[
"Ohmic Dissipation in Mini-Neptunes"
],
[
"Abstract In the presence of a magnetic field and weakly ionizing winds, ohmic dissipation is expected to take place in the envelopes of Jovian and lower-mass planets alike.",
"While the process has been investigated on the former, there have been no studies done on mini-Neptunes so far.",
"From structure and thermal evolution models, we determine that the required energy deposition for halting the contraction of mini-Neptunes increases with planetary mass and envelope fraction.",
"Scaled to the insolation power, the ohmic heating needed is small $\\sim10^{-5}$ -- orders of magnitude lower than for exo-Jupiters $\\sim 10^{-2}$.",
"Conversely, from solving the magnetic induction equation, we find that ohmic energy is dissipated more readily for lower-mass planets and those with larger envelope fractions.",
"Combining these two trends, we find that ohmic dissipation in hot mini-Neptunes is strong enough to inflate their radii ($\\sim 10^{15}$ W for $T_{eq}=1400K$).",
"The implication is that the radii of hot mini-Neptunes may be attributed in part to ohmic heating.",
"Thus, there is a trade-off between ohmic dissipation and H/He content for hot mini-Neptunes, adding a new degeneracy for the interpretation of the composition of such planets.",
"In addition, ohmic dissipation would make mini-Neptunes more vulnerable to atmospheric evaporation."
],
[
"Introduction",
"As the research field of exoplanets has progressed with more discoveries uncovering trends in the data, there are several outstanding questions that need to be answered.",
"In terms of structure and composition there are arguably two main unresolved issues: For the mini-Neptunes it is important to determine what the composition of their small envelopes is, as this will carry information as to where and perhaps how they formed.",
"And for exo-Jupiters we need to explain why so many of them are inflated beyond a composition made out of pure H/He, the so called radius-anomaly.",
"One common thread to addressing in part both these questions, is to study how ohmic dissipation affects the structure and thermal evolution of a planet with an atmosphere.",
"While this mechanism has been heavily studied for the purposes of explaining the radius anomaly of exo-Jupiters [2], [30], [27], [3], [42], [18], [32], [33], and to constrain the wind structure on Jupiter and Saturn [25], here we focus on assessing how ohmic heating affects our inference of the composition of mini-Neptunes.",
"Mini-Neptunes are planets that have a low enough mass that they do not have substantial atmospheres (i.e.",
"$\\lesssim 15 M_\\oplus $ ) but are large enough that they cannot be made out of solid material entirely, a threshold that varies as a function of mass.",
"For a given mass and radius measurement, these planets are larger than the terrestrial threshold radius, so that we deduce they have an envelope.",
"However, it is not possible to determine if the composition of the small envelope is made mostly out of H/He or water, or both, on the basis of mass and radius measurements alone because of how inherently degenerate the problem is [40], [31], [37].",
"Here, we explore how ohmic dissipation in mini-Neptunes affects their structure and radius, and thus compositional inference.",
"In the context of exo-Jupiters, it is well established that a significant number of close-in transiting hot Jupiters exhibit radii that are larger than expected from standard models of giant planet evolution (see review [10]).",
"A variety of mechanisms have been invoked to explain this radius anomaly and can be divided into three categories (following [41]): incident flux-driven mechanisms (that include kinetic heating [35] and ohmic dissipation [2], [30], tidal mechanisms [4] and delayed contraction mechanisms (that include enhanced opacities [5] and supression of heat transport [6] such as layered convection [23].",
"For more details, see the review by [1].",
"Recently [36] has proposed yet a different mechanism by which atmospheric 2D circulation in tidally-locked planets leads to hotter interior than 1D models predict via the advection of potential temperature and thus a larger planetary radius.",
"While it is very possible that a combination of mechanisms is responsible for the inflation of exo-Jupiters, there is evidence that a dominant effect causing inflation is absent at warm and cold effective temperatures.",
"Using internal structure and evolution models on 14 warm exoplanets, [28] suggested that planets receiving less stellar irradiation than $\\sim 2\\times 10^{8}$ erg/sec/cm$^{2}$ (or $10^{5}$ W/m$^{2}$ ) equivalent to an equilibrium temperature of 1400 K, did not appear inflated (i.e.",
"planets were smaller than a pure H/He planet).",
"In addition, observational work on 115 Kepler candidates by [7] revealed a trend between planetary radius and incident radiation above $\\sim 2\\times 10^{8}$ erg/sec/m$^{2}$ only, showing consistency in the results.",
"This trend was also observed by [41] to only hold true for massive planets ($M \\ge 150 M_{\\oplus }$ ) and suggested a weak scaling of planetary radius and incident flux.",
"[22] analyzed the radius anomalies (difference between the observed and predicted radii) of hot exoJupiters and found a scaling with effective temperature that was too steep (coefficient in power law of $\\alpha \\simeq 1.4$ ) to be attributed to kinetic heating ($\\alpha _{kin} = 0.67$ ) but consistent with ohmic dissipation ($\\alpha _{ohm} \\simeq 2.4$ ).",
"On the other hand, [9] found a shallower radius anomaly scaling (of $\\alpha \\simeq 0.84$ ) when metallicity and semi-major axis of the planet were taken into account.",
"With this evidence, ohmic dissipation stands out as a viable mechanism to explain inflation in hot Jupiters.",
"Ohmic dissipation is both a mechanism that produces inflation (by increasing the interior entropy of the planet from dissipation within the convective interior), and delays contraction (by moving the convective-radiative boundary deeper in the interior from dissipation in the atmosphere).",
"Energy dissipation due to ohmic effects is expected to be present in planets that have weakly-ionized winds and a magnetic field, regardless of the planet's mass.",
"The ions in the atmospheric wind moving across the planetary magnetic field induce electrical currents that can dissipate ohmic energy deep in the interior by removing kinetic energy from the atmospheric wind [25], [2].",
"The primary source of electrons in the atmosphere comes from alkali metals (e.g.",
"Na, Al, and K) that are thermally ionized, leading to a strong dependence of ohmic dissipation with atmospheric temperature.",
"This is consistent with the dependency of hot-Jupiters' radius anomaly with temperature.",
"In addition, this process is self limiting as highly ionized winds start freezing with the planetary magnetic field, slowing down the wind through Lorentz drag [30], [27] imposing an upper limit in temperature for ohmic heating to take place.",
"Supported by evidence that ohmic dissipation may take place in hot-Jupiters, we aim our efforts in understanding the effect of ohmic dissipation in mini-Neptunes for two reasons.",
"To assess how it will affect our inference of envelope composition given that if large enough, ohmic dissipation may mascarade as more H/He content adding to the sources of compositional degeneracy for these planets.",
"And second, by looking at another region of the parameter space, we hope to shed light on the debate of how ohmic dissipation operates on planets.",
"We start with a brief history of how ohmic dissipation has been treated so far as to put into context our modeling and assumptions.",
"In 2010, [2], inspired by [25], proposed that ohmic dissipation (a new mechanism at that time) could inflate hot Jupiters.",
"They solved the magnetic induction equation in steady state in a weakly ionized atmosphere in the presence of a planetary dipole magnetic field provided a prescribed wind speed profile.",
"They used an exponential approximation to the conductivity profile that enabled them to solve the equations analytically in the atmosphere, and numerically in the interior for the induced current, and ohmic power as a function of radius up to a constant.",
"This constant is adjusted so that the amount of irradiation that gets converted into ohmic energy (i.e.",
"efficiency factor) stays fixed at the few percent level.",
"They ignored the effect of Lorentz drag and looked at static interior models for three exoJupiters as examples.",
"The same year, following up on the work by [29] on magnetic drag, [30] independently suggested ohmic dissipation to explain the radius anomaly solving for the wind profile via atmospheric circulation models of a tidally locked planet, calculating the ohmic power generated via the azimuthal component of the induced current envisioning a slightly different geometry.",
"While [2] had the radial currents looping throughout the interior depositing the ohmic energy deep within the planet, [30] solved for the meridional currents (ignoring radial component) depositing the energy at pressures of several tens of bars, still deep enough to halt contraction during the thermal evolution of a planet [35].",
"Another difference is that [30] include the Lorentz drag in the atmospheric wind (via a linear friction that relaxes the wind towards zero over a specified drag time), and show that it reduces ohmic heating by about a factor of 5 for HD209548b, not enough to preclude ohmic dissipation as a mechanism for exoJupiters inflation provided they have a strong magnetic field ($\\sim 10$ G).",
"The next year [3] coupled an internal structure and evolution planetary model to their ohmic dissipation results to track which exoJupiters would be inflated given their age.",
"They could prescribe ohmic dissipation as a function of radius given a pressure-temperature profile for a (coreless) planet and calculate consistently its thermal evolution.",
"They also included a linear (Rayleigh) prescription for Lorentz drag.",
"Their most notable result shows that around $1200-1800$ K, there is an increase in radius inflation due to ionization for exoJupiters.",
"Cool planets do not have enough ionization to generate ohmic dissipation, and very hot planets (above $\\sim 1800$ K) have somewhat saturated ionization levels and deeper convective regions that hinder deep energy deposition.",
"They were able to expain all the inflated planets at the time.",
"In fact, their nominal efficiency ($\\sim 1$ %) and coreless model over estimated the radii of the low-mass Jupiters $\\left(\\lesssim 0.7M_{J}\\right)$ .",
"By extension, the radii of these planets could be easily explained either by invoking a large core $\\left(20\\,M_{E}\\right)$ or a smaller efficiency.",
"[42] revisited the subject and with a different model for internal structure and conductivity prescription obtained similar results.",
"Ohmic dissipation can inflate exo-Jupiters provided an efficiency of a few percent ($\\sim 3\\%$ ) is prescribed.",
"To obtain the desired efficiency they adjust the wind velocity.",
"Furthermore, they changed the depth of the wind zone and noticed that deeper atmospheric winds can inflate planets more easily (the efficiency invoked for inflation may be lower).",
"The same year, [27] proposed simple scalings by doing an order of magnitude balance between the acceleration of the zonal flow, the presure gradients (related to thermal response of the atmosphere) and the magnetic drag in the atmosphere.",
"The study finds that magnetic drag can eventually slow down the winds to the point of imparing ohmic dissipation.",
"The behaviour of normalized ohmic dissipation to irradiation increases with temperature up to levels of $\\sim 1\\%$ and then decreases.",
"The location for the peak depends on the magnitude of the magnetic field and for nominal values of 3-30 G, it happens between 1300-1900K.",
"However, these values depend strongly on the atmospheric response to irradiation which are poorly known (e.g.",
"opacities, thermal inversions, etc).",
"Importantly, [27] found that the magnetic Reynolds number is larger than one for hot planets (above 1500K for magnetic fields in the order of 3-10 G), indicating a degree of coupling between the flow and the magnetic field, with the possibility of a poloidal induced field in the atmosphere in addition to that of the planet.",
"[18] implemented a different approach to previous studies by looking at the wind zone and interior of the planet separately.",
"They solved the induction equation in the interior and connected the solution to the wind zone by setting boundary conditions for the temperature and toroidal magnetic induced field (or radial induced current) disregarding initially that the two are connected.",
"Subsequently, they relate the temperature below the windzone and the toroidal magnetic field via the scalings from [27], and find that for the allowed combinations of temperature and toroidal field, ohmic dissipation is too low and cannot account for the radius inflation of most exoJupiters.",
"For all other quantities fixed, they find that the ohmic power decreases with mass $\\sim R^{4}/M$ , and once they include the effects of variable conductivity and feedback of ohmic dissipation on the structure, the relationship changes to $P_{ohm}\\sim R^{2.4}/M$ .",
"Later on, [33] implemented an MHD code to investigate the radius inflation of HD209458b.",
"They too find that the magnetic Reynolds number for realistic flows exceeds unity and as a consequence they find that the Lorentz force is very efficient in slowing down the winds leading to low values of ohmic dissipation.",
"However, even without ohmic drag, their wind profile is considerably slower than results coming from conventional GCM atmospheric studies.",
"They obtain values near $0.1$ km/s at the 1 bar level, one order of magnitude less than conventional studies [17], [34].",
"Given that ohmic dissipation scales as the velocity square, this may understimate the ohmic dissipation calculated by two orders of magnitude.",
"In addition, they mention two other assumptions that can influence their results.",
"First, their choice of boundary conditions that can affect the calculation of the radial current and thus, the heating rate profile, and second, the fact that conductivity is dependent on the reference state temperature (instead of the actual local temperature).",
"In addition, like all atmospheric models, the geometry is limited to the outer most shell that limits the study of deep deposition by radial currents.",
"Follow up work by [32] extended this study to include effects of different reference temperatures, day-night temperature differences, magnetic field strenghts, as well as viscous, thermal and magnetic diffusivities.",
"They find that adding a magnetic field to a hydrodynamic treatment produces more complicated effects in the flow (oscillatory, stationary and westward mean flows) than those captured by pure drag (that cause reduced easward flow), and that ohmic dissipation seems too low (about an order of magnitude) to explain the radius inflation of Hot Jupiters, but would affect more lower-mass planets.",
"The caveats they mention to their results are several including that their implementation of an anelastic treament precludes the resolution of fast wind speeds commonly seen in the atmospheres of Hot Jupiters (which may explain why the obtain such very slow winds).",
"Other assumptions include the fact that their model extends to 200 bars while ohmic dissipation is most efficient when it happens at depth where perhaps convective motions are important, the need to have a coupled evolutionary models with MHD, as well as their implementation of a constant magnetic diffusivity.",
"Most recently, [12], [13] developed an analytical model for the inflation of hot Jupiters due to deep energy deposition.",
"They assumed a power-law opacity and a simple equation of state including both thermal and an electron degeneracy pressure.",
"They found that heat deposition can generate an exterior convective region which serves to delay the cooling of the planet, provided that the heat source is sufficiently deep.",
"The critical quantity is the fraction of energy deposited compared to the equilibrium luminosity times the depth at which the energy is deposited.",
"They find that the estimated a critical efficiency of $\\sim 5\\%$ is required to reproduce the observed hot Jupiter radii; in comparison, they estimate the peak ohmic dissipation efficiency to be $\\sim 0.3$ , reached at a temperature of 1500 K. Such efficiencies can explain inflated radii up to $R \\approx 1.6 R_J$ , but fail to explain some of the more extremely inflated planets.",
"In sum, while it is well accepted that ohmic dissipation should be present in highly irradiated planets there is still no consensus on the magnitude of the magnitude of this effect and its efficiency in inflating hot Jupiters, while a few of these studies point to the fact that lower-mass planets would be more amenable to radius inflation."
],
[
"This work",
"In contrast to previous studies, which focused on planets with Jovian masses, in this study we direct our attention to mini-Neptune planets.",
"These planets differ from exo-Jupiters in two key aspects: Firstly, they feature lower surface gravities due to their lower masses, so that their envelopes are less gravitationally bound.",
"For example, the gravity of Corot-24b, a Neptune-like planet, is $g = 4.2$ m/s$^2$ [21] and six times lighter than Jupiter's gravity $g = 24.8$ m/s$^2$ or 2 times lighter than HD209458b with $g = 9.4$ m/s$^2$ .",
"Secondly, unlike Jupiters, which are entirely gaseous in their composition, mini-Neptunes are dominated by a rocky core, with the gaseous envelope accounting for less than $10 \\%$ of the total mass in most cases [37] .",
"As a result, the internal heat flux may be largely affected by the rocky core, and the geometry of the current flow is a thinner shell (assuming currents do not penetrate into the rocky interior, (see below), as opposed to a sphere for the case of Jupiter.",
"We assume an internal structure that is a rocky interior overlain by a small envelope (up to $10\\%$ by mass) which describes most of the mini-Neptunes found [37] and as a first step do not consider ices in the interior based on the fact that we are mostly interested in planets that are close to their star.",
"It is unclear if mini-Neptunes are formed in situ with an absence of water/ices [24], or formed beyond the snow line with subsequent migration [16].",
"Thus, we consider the in-situ scenario as a first step in understanding how ohmic dissipation would influence such type of planets.",
"To study ohmic dissipation in mini-Neptunes we follow a two step approach.",
"We first model the interior structure and thermal evolution of these planets with various degrees of energy deposition to obtain envelope profiles.",
"With these results, we construct the conductivity profile and solve the induction equation given a prescribed wind profile and obtain the current distribution and ohmic dissipation power.",
"For self-consistent solutions, we feed the calculated ohmic dissipation back into the original planet interior structure models, and iterate until the derived ohmic dissipation strength matches the heat deposited in the model.",
"The ohmic dissipation feedback on the planet's internal structure profile is parametrized as a single point heat source deep in the planet's interior, and taken to be constant in time.",
"This approximation was found to be consistent with time evolution models of mini-Neptunes, as models with significant ohmic dissipation will reach radiative equilibrium at fixed final constant dissipation value.",
"As a first step, in this study we do not account for the effect of Lorentz drag on the atmospheric flow.",
"We find that ohmic dissipation can indeed play a role in inflating the radii of hot mini-Neptunes, analogous to the case of hot Jupiters.",
"This manuscript is organized as follows: in section 2 we discuss the details of our modelling procedures, in section 3, we present the results of our calculations and the implications on inferring the composition of mini-Neptunes, in section 4 we discuss implications, as well as the validity and limitations of our assumptions, and in section 5 we summarize our results.",
"For this study, we consider planets that have masses ranging from $2 M_{\\oplus }$ to $16 M_{\\oplus }$ , and with gas envelope fractions $f \\equiv M_{env}/M_{tot}$ ranging from $1 \\%$ to $10 \\%$ .",
"The composition of the gaseous envelope is assumed to be $X = 0.70$ , $Y = 0.28$ and $Z = 0.02$ , where $X$ , $Y$ and $Z$ are the mass fractions of hydrogen, helium and metals respectively.",
"Solar abundances are used for the relative abundances of metals within $Z$ .",
"For the purposes of this study we ignore the possibility of the envelopes of mini-Neptunes having substantial water.",
"We focus on planets with an equilibrium temperature $T_{eq}$ between 1300 K and 1700 K. We find that, for temperatures below 1300 K, ohmic dissipation is insignificant due to lack of ionization in the atmosphere, whilst at temperatures above 1700 K, for smaller planets ($M \\le 9M_{\\oplus }$ ) ohmic dissipation leads to planetary radius expansion ($dR/dt > 0$ ) in most cases when Lorentz drag is not taken into account.",
"Therefore, such temperatures are at the upper limits of the regime of validity of this study."
],
[
"Interior structure modelling",
"To calculate the evolution of the interior structure of mini-Neptunes, we use CEPAM, a code originally derived from the stellar evolution code CESAM but with additional physics relevant to planetary bodies incorporated [15].",
"Through combining the standard atmospheric structure models with the realistic model for rocky objects described in [38], this model is capable of simulating differentiated planets with an Earth-like rocky core surrounded by a gaseous envelope composed of hydrogen, helium and/or water.",
"The algorithm divides the rocky interior of the planet into radial shells corresponding to the iron core, mantle, and the gaseous envelope, and solves their structural evolution in each region, and joined at the boundaries by mass and pressure continuity [37].",
"It includes internal heat sources as radioactive heating in the rocky mantle in chondritic proportions.",
"For the gaseous portion of the planet, the structure of the atmosphere is determined based on a semi-grey model described in [14], and extends to an optical depth of $\\tau = 10^6$ .",
"Below such depths, the temperature structure of the gaseous envelope is determined using the lesser of the radiative and adiabatic gradients.",
"We assume that the planetary magnetic field is generated in the convective liquid iron core [8].",
"In this work, we define the envelope as the entire portion of the planet that is comprised of gaseous H/He.",
"The uppermost layers of the envelope, from the planet's exterior down to an optical depth of $\\tau = 10^6$ is referred to as the 'atmosphere' and we call the interior rocky portion of the planet, containing both the mantle and the iron core, as the 'core'.",
"The value of maximum optical depth of the atmosphere is rather arbitrary but not important as long as it is high enough to map into the interior envelope.",
"The overall planet radius $R$ is taken to be the chord radius of the planet's envelope, defined in [14].",
"In our interior models, initial entropies of planets were set to be sufficiently large as to ensure a 'hot start'.",
"Planets that form at high temperatures go through an initial rapid cooling phase, during which the photosphere temperature cools down to $T_{ph} \\sim T_{eq}$ .",
"[12] estimated this timescale to be $t \\sim 1$ Myr at $T_{eq} = 1500$ K (with numerical values adjusted to suit mini-Neptunes), after which planet temperatures converge to the same state regardless of their initial starting temperatures, provided that the initial temperature $T_0 $ >$$ Teq$.",
"In this work, we define a ^{\\prime }hot start^{\\prime } loosely as any initial temperature sufficiently high to reach the convergent state (typical values of $ S0 1 109$ erg/K/g at age zero).",
"We ran the structure and thermal evolution models for up to $ 10$ Gyr.$ We consider cases with no additional internal heating (only radioactive), and models with ohmic dissipation.",
"A typical radius evolution for a mini-Neptune contracting only under the irradiation of the host star is shown in Figure REF (black curve), the other curves have dissipation added.",
"The planet's envelope decreases in size as the planet undergoes Helmholtz contraction, until an equilibrium is reached; the exact time and the planet's final radius and temperature depends on the planet's properties and the presence of any other heat sources.",
"For the cases without additional heating, most of the radius contraction happens before 1Gy.",
"Figure REF (top) shows a typical pressure-temperature profile for a H/He envelope of a mini-Neptune.",
"The structure is defined by two nearly isothermal structures below a pressure of $\\sim $ mbar, corresponding to a temperature of $T_{atm} \\sim 0.8 T_{eq}$ ; and between pressures of $\\sim 1$ to $\\sim 10^{3}$ bars, with $T \\simeq T_{rcb} \\sim 1.2 T_{eq}$ corresponding temperatures, where $RCB$ stands for the radiative-convective boundary, with a steady increase within these two isothermal layers.",
"Above $\\sim 10^{3}$ bar, the convective region of the envelope begins and becomes adiabatic.",
"Generally, at the very early stages of the evolution the $RCB$ is very shallow ($< 100$ bars or even $<10$ bars).",
"By 10 Myr the boundary is typically at 200 bars, and by 100 Myr, unless the planet's contraction is significantly halted, the $RCB$ is between 1000 and 2000 bars (typically).",
"If the planet cools to 10 Gyr without additional energy sources, the $RCB$ eventually reaches very deep pressures - $2 \\times 10^4$ bars.",
"The trend is that planets with thinner atmospheres (e.e.",
"less H/He %) have deeper $RCB$ s, and more massive planets have shallower $RCB$ s. The rate of $RCB$ becoming deeper is related to the rate of cooling.",
"The $RCB$ no longer moves if the planet reaches equilibrium, and it also moves slower for planets at high equilibrium temperatures.",
"The reason we care about the behaviour of the $RCB$ is because its size is connected to how much ohmic dissipation is available deep in the interior for inflation.",
"Figure: Typical planet radius evolution for different amounts of ohmic dissipation in the planet's interior.",
"This particular figure shows the cooling track of a 9M ⊕ 9 M_\\oplus - planet with 5% H/He and T eq =1500T_{eq} = 1500K.Figure: Typical atmospheric and conductivity profile of a mini-Neptune.",
"Calculations correspond to a M=9M ⊕ M = 9 M_{\\oplus } planet with f=5%f = 5\\% for various equilibrium temperatures, taken as a snapshot after 1 Gyr of evolution.",
"Top: The temperature-pressure profile in the envelope is characterized by two isothermal layers, the outer one with a temperature T∼0.8T eq T \\sim 0.8 T_{eq} and the inner one with a temperature of T∼1.2T eq T \\sim 1.2 T_{eq}.",
"Bottom: The conductivity profile is characterized by a minimum and maximum values around the non-isothermal portion of the atmosphere.",
"The radiative-convective boundary occurs at ∼1000\\sim 1000 bars and moves gradually inward during evolution as the planet cools."
],
[
"Conductivity profile",
"As stated before, we assume a geometry where the currents do not penetrate the rocky interior and are confined to the thin envelope.",
"It is unclear what the conductivity values for rocks would be under the interior conditions of mini-Neptunes ($T>5000$ K and $P = 10^5$ bars or 10 GPa).",
"The conductivity of rock minerals depends not only on temperature, but also sensitively on the presence of impurities controlled by factors such as the amount of water (hydrogen) and oxygen fugacity [19].",
"While there is experimental data on upper mantle materials, lower mantle materials such as perovskite and postperovskite, which would constitute the bulk of the interior of mini-Neptunes [39] is absent.",
"Because of these uncertainties, as a first step we regard the rocky outer shell of the mini-Neptunes as an insulator, and confine the induced currents to the envelope.",
"Having calculated the envelope structure ($P,T$ ), we compute the conductivity profile.",
"Two sources could potentially contribute to electrical conductivity in the atmospheres of mini-Neptunes.",
"At low pressures, solar irradiation causes partial thermal ionization of alkali metals, with elements K, Na, Li, Cs, and Fe being the dominant source of ions.",
"Deeper in the gaseous envelope, the conductivity profile is dominated by the pressure-ionization of hydrogen, which dominates over ionization at pressures greater than $P \\sim 10^5$ bar.",
"The thermal ionization is described by the Saha equation: $\\frac{n_{j}^{+}n_{e}}{n_{j} – n_{j}^{+}} = \\left(\\frac{m_e k_b T}{2 \\pi \\hbar ^2}\\right)^{3/2} \\exp {(-I_j / k_b T)},$ where $n_j$ and $n_j^+$ are the total and positively ionized number densities of constituent $j$ respectively, $n_e$ is the number density of electrons, $m_e$ is the electron mass, $k_b$ is the Boltzmann constant, $T$ is temperature, $\\hbar $ is reduced Planck’s constant, and $I_j$ is the ionization potential of constituent $j$ .",
"For this study, we calculated the relevant contributions of the first 28 elements of the period table, from He through Ni, with hydrogen being treated separately.",
"The electrical conductivity of such a gas $\\sigma _Z$ is given by: $\\sigma _{Z} = \\frac{n_{e}}{n} \\frac{e^2}{m_e \\nu },$ where $n_e$ is the electron number density, $n$ is the total number density of particles, and $\\nu $ is the collision frequency of electron-neutral collisions given by [18] $\\nu = 10^{-15} n \\left(\\frac{128k_BT}{9\\pi m_e}\\right)^{1/2} cm^3 s^{-1}.$ As the temperatures in question are usually much lower than the first ionization energy of even the most easily ionized element, K (with an ionization energy of 4.35 eV), it turns out appropriate to only consider the element K, which gives a conductivity [18] $\\sigma _Z \\approx 1.74 \\times 10^3 \\left(\\frac{T}{1600K}\\right)^{3/4} \\left(\\frac{p}{bar}\\right)^{-1/2} e^{-4.35eV / k_B T}.$ Below pressures of $\\sim 100$ bars, the above expression is accurate to within a few percent compared to our full implementation of the Saha equation.",
"At higher temperatures and pressures, the above approximation breaks down as ionizations of other metals become significant.",
"Deeper in the planet's interior, pressures are high enough to induce pressure ionization of hydrogen, and at such depths hydrogen becomes a semi-conductor.",
"For the conductivity, we use the prescription from appendix A4 in [18]): $\\sigma _{X} = \\sigma _0 \\exp {\\left(\\frac{-E_g(\\rho )}{k_B T}\\right)},$ where $E_g = 20.3 - 64.7 \\rho $ eV, $\\rho $ is the density in mol cm$^{-3}$ , and $\\sigma _0 = 3.4 \\times 10^{10}$ in SI units.",
"The total conductivity is then calculated as $\\sigma = \\sigma _X + \\sigma _Z.$ An example of the conductivity profiles for a $6 M_{\\oplus }$ planet with $5 \\%$ H/He content and different equilibrium temperatures is shown in Fig.",
"REF (bottom)."
],
[
"Wind Profile",
"To solve the induction equation, we need the conductivity structure and wind profile.",
"For planets on very short orbits (periods of days), aside from being subject to intense irradiation from their host stars, these tight orbits also cause the planets to be tidally locked, such that the same side of the planet always faces the host star.",
"This tidal locking mechanism creates strong winds flowing from the day-side to the night-side in the upper atmosphere of these planets.",
"In our models, it is this wind that produces the currents responsible for ohmic dissipation in the outer atmosphere.",
"Studies of global circulations on hot Jupiters have shown that winds can reach 1 to 2 km/s at the 1 bar level on these planets.",
"Similar studies by [26], [20] on the 6.5 $M_{\\oplus }$ planet GJ 1214b showed that much less massive planets can attain very similar wind speeds as hot Jupiters.",
"It turns out, in the absence of strong magnetic drag, the zonal wind speed due to temperature gradients only on tidally locked planets can be approximated by [27]: $v_{\\phi } &\\sim \\sqrt{k_B \\Delta T \\Delta \\log {P} / \\mu } \\\\&\\sim 1.4 \\text{km/s} \\times \\left(\\frac{\\Delta T}{500 K}\\right)^{1/2} \\left(\\frac{\\mu }{2 \\text{amu}}\\right)^{1/2} (\\ln {\\Delta P})^{1/2},$ where $v_{\\phi }$ is the wind velocity, $k_B$ is the Boltzmann constant, $\\mu $ is the mean molecular mass, $\\Delta T$ is the day-night temperature difference, and $\\Delta \\log {P}$ is the depth of the weather layer.",
"For values typical of hot-Neptunes and inflated planets in general, the temperature difference is a substantial fraction of $T_{eq}$ , and this results in winds of $\\sim 1 $ km/s.",
"In this study, we assume the same maximal wind speed of 1 km/s throughout our planet models.",
"Following the typical envelope structure of the two nearly isothermal layers we find a conductivity maxima and minima (see Fig.",
"REF ).",
"The minimum conductivity value happens at the base of the shallower isothermal layer ($\\sim 10$ mbar) and the maximum at the top of the deeper isothermal layer and $\\sim 1 $ bar.",
"In our set-up, we assume that the base of the wind zone starts at $p = 10$ bars, and that the winds grow in intensity from this depth up until reaching the top of the lower, hotter isothermal sphere, and then stays constant throughout the atmosphere (see Fig.",
"REF ).",
"Above the base of the wind zone at the pressure of 10 bars, we assume the winds to take the expression [2] : $v_\\phi (r) = v_{max} \\left(\\frac{r - r_{10}}{r_{iso} - r_{10}}\\right)^2 ,$ where $r_{iso}$ is defined to be the location of the conductivity maxima in the atmosphere, $r_{10}$ is the radius at $p = 10$ bars, and $r$ is the radius.",
"Above $r > r_{iso}$ , we assumed a constant wind speed of $v_{\\phi }(r) = 1$ km/s.",
"Note that the quantity $(r - r_{iso})$ is the same as $\\delta $ used by [2]."
],
[
"Ohmic Dissipation",
"Ohmic theory has been worked out in detail by different groups ([25], [30], [2], [3], [18], [42], among others).",
"We summarize their key findings below.",
"For a current density $\\mathbf {J}$ and conductivity $\\sigma $ , the volume power density of ohmic dissipation is given by: $dP_{Ohm} = \\frac{J^2}{\\sigma } dV \\sim 4 \\pi r^2 \\frac{J^2}{\\sigma } dr .$ The current density is related to the planet's magnetic field $\\mathbf {B}$ and the atmospheric wind velocity $\\mathbf {v}$ through Ohm's law: $\\mathbf {J} = \\sigma \\left(- \\nabla \\Phi + \\frac{\\mathbf {v}}{c} \\times \\mathbf {B} \\right) ,$ with $\\Phi $ being the electric potential.",
"The motion of the atmospheric wind under a fixed magnetic dipole field $\\mathbf {B_{dip}}$ generates an induced magnetic toroidal field given by the induction equation: $\\frac{\\partial \\mathbf {B}}{{\\partial t}} = \\nabla \\times (\\mathbf {v} \\times \\mathbf {B}) - \\nabla \\times \\lambda (\\nabla \\times \\mathbf {B}).$ Under assumptions of steady-state and continuity, the solutions to the electric potential are then given by: $\\nabla \\cdot \\sigma \\nabla \\Phi = \\nabla \\cdot \\sigma (\\mathbf {v} \\times \\mathbf {B_{dip}}).$ To calculate the distribution of currents inside the planet in detail, we consider a simple geometry with a zonal flow given by $\\mathbf {v} = v(r) \\sin {\\theta } \\hat{\\phi }$ .",
"To solve the above equation in spherical coordinates, we can write the induced toroidal field as: $\\mathbf {B_{ind}} = \\frac{g(r)}{r} \\sin {\\theta }\\cos {\\theta }\\hat{\\phi },$ whose radial dependence satisfies $\\frac{-4\\sigma M}{r} \\frac{d}{dr}\\left(\\frac{v(r)}{r}\\right) = \\frac{d^2 g(r)}{dr^2} - l(l+1) \\frac{g(r)}{r^2} - \\frac{d \\ln {\\sigma }}{dr}\\frac{dg(r)}{dr},$ where $M$ is the magnetic dipole moment of the planet and $l = 2$ is the index of the associated Legendre polynomial $P_l^1$ .",
"The current $\\mathbf {J}$ is determined by Ampere's law $\\mathbf {J} = (c / 4 \\pi ) \\nabla \\times \\mathbf {B}$ .",
"In this study, we solve equation (REF ) numerically to obtain the current distribution and induced magnetic fields, which we then used to evaluate the strength of ohmic dissipation.",
"Before the second-order equation (REF ) can be solved, we must first pick two appropriate boundary conditions.",
"[2] made the physical argument that the conductivity drops several orders of magnitude over the transition between the two isothermal layers, and therefore the conductivity minima there could be treated as an insulating shell; they assumed that all currents would vanish at this point and integrated inward from the conductivity minimum.",
"In a later study, [3] fixed the boundary conditions to match a certain prescribed dissipation efficiency $\\epsilon $ .",
"[18] used a different approach and solved the induction equation in the interior and connected the solution to the wind zone by setting boundary conditions for the temperature and toroidal magnetic induced field (or radial induced current) disregarding initially that the two are connected.",
"Subsequently they relate the temperature below the windzone and the toroidal magnetic field via the scalings from[27].",
"For this work we pick our boundary conditions by letting the radial current $j_r$ vanish at a pressure of $p = 10^{-6}$ bar (e.g.",
"insulating shell).",
"We also run models using the prescription from [2].",
"We find that the two choices for boundary conditions agree well below temperatures of $T_{eq} \\lesssim 1400$ K. Beyond this temperature, solar insolation heats up the outer isothermal layer enough that the conductivity drop across the layers is only a factor of $\\sim 50$ less, so that the approximation of treating the conductivity minima as an insulating shell is dubious.",
"For the lower boundary condition in the envelope, we set the radial current to vanish, consistent with the assumption of an insulating outer rocky layer.",
"We assume that all planets have a magnetic dipole field strength of $B = 1$ G arising from fluid motions within the iron-core, consistent with numerical simulations which find surface magnetic field strengths of super-Earths of $\\sim 0.6 - 2$ G, a strength that varies little with mass (for $1 M_{\\oplus } \\le M \\le 10 M_{\\oplus }$ ) but increases with increasing iron-core mass fractions [8].",
"While it is possible that for close-in planets the stellar magnetic field can induce ohmic heating on the planet , as a first step, we ignore these effects.",
"We solve the induction equation following the prescription by [25] and a numerical code based on a relaxation method to solve for the differential equations.",
"We obtain the current in the radial and azimuthal directions, and the ohmic power dissipated at each radius and cummulatively.",
"See Fig.",
"REF for the results of a typical calculation."
],
[
"Results",
"We calculate more than 11,000 interior structure models spanning equilibrium temperatures between 1300 and 1700 K in intervals of 50 K as well as 1325 K and 1375 K, masses of [2, 3, 4.5, 6, 9, 12, 15] $M_{\\oplus }$ and envelope fractions of $1, 2, 3... 10\\%$ , and depositing ohmic heating rates of $5\\times 10^{13}$ to $5\\times 10^{18}$ W in logarithmic intervals of 0.1 until consistency is achieved."
],
[
"Planet models without ohmic dissipation",
"We first obtain the thermal evolution and structure of planets without any additional heat sources.",
"One of the differences between mini-Neptunes and Jovian planets, is that the rocky interior, which is expected to contract negligibly in comparison to the envelope, makes up most of the planet's mass and hence greatly influences the overall size evolution of the planet.",
"A fit to the envelope's size evolution is shown in Fig.",
"REF and can be summarized as: $R_{env} \\equiv R - R_c \\sim 1.4 R_{\\oplus } \\left(\\frac{T_{eq}}{1600 K}\\right)^{0.5} \\left(\\frac{M_{core}}{10 M_{\\oplus }}\\right)^{-0.8} \\\\\\times \\left(\\frac{M_{env}}{0.1 M_{\\oplus }}\\right)^{0.5} \\left(\\frac{t}{\\text{Gyr}}\\right)^{-0.08}.$ The fit yields a mean error of 8% and a maximum error of 64%.",
"This cooling law has a weaker dependence with time compared to the analytical prescription by [12] who suggested a relationship for Jupiter-like planets of $R(t) \\propto t^{-0.25}$ .",
"We attribute the differences mainly to the effects of the rocky core, which in the case of mini-Neptunes, dominates the structure.",
"We also find that the radius at the convection zone scales as $R_c \\propto R^{0.7} M^{0.3} t^{-0.08}$ .",
"Figure: The radius of the envelope for planets with various rocky core and envelope masses and no ohmic heating with T eq =1500T_{eq}=1500 K."
],
[
"Planet models with ohmic dissipation",
"We then move onto calculating the ohmic dissipation in mini-Neptunes and the effect on their evolution.",
"With the results on the internal structure, we calculate the conductivity profile and solve for the induction equation.",
"We take the cummulative ohmic power up to the radiative-convection zone as the available dissipation for inflation and feed this back into the interior structure and thermal evolution model as a constant heat source at the deepest point in the envelope (i.e.",
"at $r = R_c$ ).",
"We iterate until the calculated cumulative ohmic dissipation at the $RBC$ matches the input at the 20 $\\%$ level.",
"In addition, for a subset of our models (with masses $M = [4.5, 9, 15] M_{\\oplus }$ , equilibrium temperatures $T_{eq} = [1300, 1500, 1700]$ K, and envelope fractions of $[2\\%, 5\\%, 10\\%] $ ) instead of a constant heat source, we re-calculate the ohmic dissipation at the $RCB$ every 1 Myr by solving the induction equation, and feed this value back into the model.",
"We find this subset of models incorporating the time-varying feedback to yield virtually identical final planet radii and atmospheric profiles as the models with constant coupling.",
"In fact, we find the final equilibrium radius and atmospheric profile of a planet to be determined entirely by the amount of heat deposition regardless of history, consistent with the analytic models derived by [12].",
"As a result, we adopt the constant ohmic dissipation prescription for the bulk of our models.",
"A typical solution to the induction equation is shown in Fig.",
"REF .",
"This specific example is for a planet with $M=9M_\\oplus $ , 5% H-He and $T_{eq}=1500$ K. The wind profile (purple) is confined to the upper layers of the atmosphere and has a quadratic shape with a maxima of 1 km/s.",
"The radial current (blue) varies mostly in the upper layers (above $\\sim 1$ bar), and much less below it.",
"The cumulative ohmic power (green olive with axis on the right) follows closely the shape of the inverse of the conductivity (cyan) given that conductivity varies many orders of magnitude more than the radial or azimuthal currents (green).",
"We also compute the ohmic power per unit mass (red).",
"This specific ohmic power can be compared to the results [3] showed in Fig.",
"4 for Jupiter-like planets.",
"Similar to [3], [18], [42], we find that the largest ohmic values are obtained in the upper atmosphere where they are not efficient at inflating the planet.",
"While the amount of ohmic dissipation within the convective zone is more than an order of magnitude smaller than in the upper atmosphere, it is still enough to halt the contraction of this planet during its evolution.",
"Figure: Typical results to the induction equation.",
"The induction equation is solved in a spherical 1D geometry given a wind profile (purple), and conductivity profile (cyan).",
"The results are the radial current (blue), the azimuthal current (the absolute value is shown as green), and ohmic dissipation as a snapshot at 1Gy.",
"We compute the cumulative ohmic dissipation (olive green and right axis), and the ohmic dissipation per volume (red).",
"The units of each parameter are shown in the figure and all but the cumulative ohmic dissipation correspond to the left (black) axis.",
"This particular planet corresponds to a 9 M ⊕ M_\\oplus - planet with 5% H-He at 1500K.",
"The radiative-convective boundary is shown as a dashed black line.In a planet with exponential profiles for the conductivity, and a constant current $j$ across the interior of the planet, the ohmic power deposited below the radiative-convective boundary is equal to the power density at the boundary multiplied by the volume of a shell with thickness the conductivity scale height $H$ : $P_{Ohm} \\sim 4 \\pi R_{RCB}^2 H_{RCB} j^2 \\sigma _{RCB}^{-1}.$ Here, quantities with subscript $RCB$ indicate the parameter is evaluated at the radiative-convective boundary.",
"In our numerical models, we find the best fit expression for the ohmic power to be: $P_{ohm} \\sim 2.6 \\times 10^{16} W \\left(\\frac{R}{3R_{\\oplus }}\\right)^{2.2} \\left(\\frac{M}{10 M_{\\oplus }}\\right)^{-2.2} \\\\ \\times \\left(\\frac{T_{eq}}{1500 K}\\right)^{3.9} \\left(\\frac{j}{10^{-4} A}\\right)^{1.88} \\left(\\frac{\\sigma }{10^{-3} \\text{S/m}}\\right)^{-1.2} .$ However, this fit has a RMS error or 41% and a maximum error of 216% showing that a simple scaling fails to capture the complexity of the system.",
"In fact, the fit does worse for larger planets with more envelope mass fractions, perhaps indicating a transition in the interplay of factors determining ohmic dissipation between mini-Neptunes and Jovian-like planets.",
"A useful result is to show the dependence of ohmic power on the equilibrium temperature of the planet for a variety of planet masses and compositions, which can be seen in Fig.",
"REF .",
"Given that the degree of partial ionization of metals increases exponentially with temperature, the relationship to ohmic power is very steep ( $P_{ohm} \\propto T^{23}$ ).",
"We see that planets with smaller masses and larger gaseous mass fractions tend to experience more ohmic heating, since these planets have larger envelope radii and therefore a larger $R_{RCB}$ .",
"This means that more ohmic power is generated in a hot small planet with a large gaseous envelope.",
"More importantly the amount of ohmic power generated in mini-Neptunes ranges between $10^{14}-10^{17}$ , and is two to four orders of magnitude smaller than in exoJupiters ([2], [30] obtained ($P_{ohm}\\sim 10^{19}$ for HD209458b).",
"We now turn to estimating if this amount of ohmic power is enough to halt the contraction of mini-Neptunes.",
"Figure: Ohmic power generated within the convection zone as a function of equilibrium temperature for mini-Neptunes.",
"We show the amount of ohmic dissipation below the radiative-convective boundary for planets of 3, 6 and 16 M ⊕ M_\\oplus and different envelope fractions spanning 2% (thin lines), 5% (medium lines) and 10% (thick lines) H-He.",
"Small, hotter planets generate more ohmic dissipation available for inflation."
],
[
"Time evolution of planets with ohmic dissipation",
"When ohmic dissipation is introduced to a planet, we find that the main effect of such an energy source is to stop its contraction.",
"Figure REF shows an example of the radius evolution of a $M = 16 M_{\\oplus }$ planet with various levels of ohmic heating.",
"We find empirically that, the amount of ohmic dissipation $P_{crit}$ required to halt the planet's cooling at $t = 1 $ Gyr is given by (see Fig.",
"REF and REF ): $ P_{crit} \\sim 1.9 \\times 10^{16} W \\left(\\frac{M_{env}}{10 M_{\\oplus }}\\right)^{1.32}\\left(\\frac{f}{\\%}\\right)^{0.6}.$ Here, $M_{env} = M \\times f$ is the total mass of the gaseous envelope.",
"This fit has a mean error of 11% and a maximum error of 48%.",
"Note that the choice of $t = 1 $ Gyr is a stringent one, since it takes more energy to halt contraction at an earlier rather than at a later time.",
"Compared to this nominal value of $10^{16}$ W, the energy needed to halt HD209458b we calculate to be $3\\times 10^{19}$ W or 3 orders of magnitude larger.",
"Notably, the amount of energy needed to stop the contraction of mini-Neptunes is of comparable magnitude or less than that which is generated through ohmic dissipation.",
"One can compare Eqs.",
"REF and REF , but preferably Figs.",
"REF and REF to arrive at this inference.",
"Thus, while the ohmic dissipation generated in mini-Neptunes is small (about 2 orders of magnitude lower than typical values obtained for exo-Jupiter), it is enough to inflate mini-Neptunes (about 3 orders of magnitude lower than for exo-Jupiters).",
"In short, mini-Neptunes can more easily be inflated with ohmic dissipation than their Jovian counterparts.",
"We can also translate this critical ohmic power needed for halting contraction as an efficiency $\\epsilon _{crit} \\equiv P_{crit} / 4\\pi R^2 \\sigma T_{eq}^4$ , defined as the total ohmic power divided by the amount of incident solar insolation (with zero albedo).",
"We find empirically that $\\epsilon _{crit} \\sim 3 \\times 10^{-5} \\left(\\frac{M}{10 M_{\\oplus }}\\right)^{1.03} \\left(\\frac{T}{1500\\text{K}}\\right)^{-4}.$ This fit has a mean error of 14% and a maximum error of 51%.",
"Interestingly, we observe that approximately, the critical ohmic efficiency does not depend on the gaseous fraction of the atmosphere.",
"Note that this relationship is valid for mini-Neptunes planets and should not be used for Jovian planets that obey a different mass-radius relationship.",
"We see that the ohmic efficiency required to inflate the atmospheres of mini-Neptunes is $\\sim 10^{-5}$ , orders of magnitude lower than the case of hot Jupiters, where values of $\\sim 1 \\%$ [35] are necessary to inflate their radii to account for observations.",
"Figure: Critical ohmic power needed to stop contraction at 1 Gy as a function of planet mass and envelope fraction.",
"Planets with low mass and low volatile content require less energy for their contraction to halt.Figure: Efficiency needed to stop the contraction at 1 Gy as a function of envelope fraction for different planet masses.",
"Efficiency calculated as critical ohmic power over insolation energy (σT 4 \\sigma T^4).Another useful way to see the effect of ohmic dissipation is to show the evolved radius of mini-Neptunes as a function of different equilibrium temperatures for different masses and envelope fractions.",
"Figure REF shows that the radius dramatically increases at equilibrium temperatures that depend on the envelope mass fraction and planet mass and range between $1400 - 1600$ K. For a given planetary mass at low temperatures (where ohmic dissipation is unimportant) the planets with more envelope fractions are larger, as expected.",
"However, as the temperature and hence, ohmic dissipation increases, the trend gets overturned: the planets with less envelope fraction puff up to the point that they become larger than those with higher envelope mass fractions (compare lines with same color in Fig.",
"REF ).",
"Counterintuitively, the $T_{eq}$ at which the radius dramatically increases happens at lower temperatures for more massive planets with a given envelope mass fraction (compare the lines with same thickness in Fig.",
"REF ).",
"A corollary to these effects is that a hot mini-Neptune would be considerably puffier and susceptible to atmospheric evaporation than a cooler one, and could experience a runaway evaporation as it loses envelope mass in the early stages of evolution.",
"This fits nicely with recent work by [11] that suggests there may be a deficit in occurrence rate distribution of planets with a radii between $1.5-2\\, R_\\oplus $ and that this gap seems to be at least partially shapped by stellar irradiation.",
"We note that some planets with little H/He ($\\le 2 \\%$ ) can undergo ohmic heating so extreme that the planetary radius starts expanding ($dR/dt > 0$ ) instead of contracting during its thermal evolution (shown as circles in Fig.",
"REF ).",
"We note that we do not explore this regime further.",
"For comparison to previous work, we obtain a simple scaling for the radius inflation with efficiency $\\Delta R \\propto \\epsilon ^{0.12}$ , and see that the dependence is shallower than for the case of hot Jupiters $\\Delta R \\propto \\epsilon ^{0.3}$ [12].",
"Figure: The final evolved radius as a function of T eq T_{eq}, when the effects of ohmic dissipation are taken into account.",
"The planet mass is indicated in colour and the thickness of the line represents the amount of volatiles from 2, 5 and 10% H-He.The fact that the radius of a mini-Neptune can be inflated due to ohmic dissipation means that when inferring the composition from mass-radius data, one can wrongly estimate the H/He content.",
"It is clear from Fig.",
"REF that there is a pervasive degeneracy in composition of hot mini-Neptunes.",
"For example, a 16 $M_\\oplus $ at $\\sim 1360$ K and radius of $\\sim 3.45 R_\\oplus $ can be composed of 2% or 10% H-He (crossing of lines of same color).",
"This degeneracy arises due to the fact that smaller H/He envelopes are more susceptible than the larger ones to inflation due to ohmic dissipation.",
"Another way to look at this is with a mass-radius diagram.",
"Figure REF shows the effect of ohmic dissipation (solid lines) on the MR relationships compared to mini-Neptunes without it (dashed lines).",
"It is clear that the most susceptible planets to experience radius inflation from ohmic dissipation are those with masses $\\lesssim 8 M_{\\oplus }$ and high envelope fractions $\\gtrsim 5\\%$ and $T_{eq} \\ge 1400$ K. Figure: Mass-radius relations for planets with various ratios of rocky material to gas.",
"The dashed lines shows the mass-radius relations if no ohmic dissipation is present, while the solid lines incorporates ohmic dissipation for planets with an equilibrium temperature of 1400 K for four different envelope fractions: 1%, 2%, 5% and 10% H-He (ranging from dark to light grey).",
"Curves for the 5 and 10% envelope fractions have been slightly smoothen out.",
"Exoplanets with known masses and radii with error bars below 50% are shown and color coded by equilibrium temperature (calculated with an albedo of 0.3).",
"Earth, Uranus and Neptune are shown for reference.",
"The mass-radius relation for an earth-like composition is shown in green."
],
[
"Discussion",
"Our results show that, in the case of mini-Neptunes with close orbits around their host stars, ohmic dissipation in many cases can alter significantly the expected mass-radius relationship of such planets.",
"The effect is more pronounced for less massive planets and planets with a smaller fraction of their mass in their gaseous envelope.",
"For a $6 M_{\\oplus }$ planet at $T_{eq} = 1500$ K with $10 \\%$ gaseous content, the effects of ohmic dissipation can double the planet's radius, whereas the increase in radius is only $\\sim 10\\%$ for a $16 M_{\\oplus }$ planet with the same composition.",
"Like in previous studies of ohmic dissipation, we find that the amount of ohmic heating depends sensitively on temperature, which plays two roles in ohmic dissipation.",
"First, the amount of partial ionization in the wind zone depends sensitively on the temperature there, and as a result the induced magnetic field has a sharp exponential dependence on the temperature.",
"At the same time, the ohmic heating is inversely proportional to the conductivity in the convection zone, and increasing the temperature there leads to a decrease in ohmic heating.",
"The net effect of varying $T_{eq}$ is dominated by the first factor, and as a result we see a steep dependence of $P_{ohm}$ on $T_{eq}$ with ohmic dissipation becoming increasingly relevant for $T_{eq} $ >$$ 1400$ K, similar to what \\cite {BS11} observed for hot Jupiters.$ This scaling result agrees qualitatively with [18], who found that the ohmic power increases for low-mass planets with extended radii.",
"For a fixed temperature of the isothermal atmosphere $T_{iso}$ and the induced magnetic field $B_{\\phi 0}$ , [18] propose a scaling of $P_{ohm} \\propto R^{2.4} /M$ .",
"In comparison (and fixing the same quantities) we obtain $P_{ohm} \\propto R^{3.3} M^{-0.8}$ .",
"It is easier to puff up lower-mass planets for two reasons: for one, the lower gravity leads to a greater scale height, increasing the volume in which ohmic dissipation takes place and thereby the ohmic dissipation; moreover, the amount of dissipation necessary to halt a planet's contraction $P_{crit}$ increases with mass, and therefore less massive planets require less power to remain puffed-up.",
"The differences in the exponents of the scalings may arise from several sources.",
"First, [18] proposed their scalings based on Jovian planets and not planets with a large core fraction and incipient envelopes.",
"Second, our choice of placing all the ohmic dissipation at the bottom of the envelope as a constant source is an upper bound to how much inflation can happen.",
"And lastly, we do not take into account the Lorentz drag and have chosen a different set of boundary conditions.",
"We now turn to discussing these assumptions.",
"For simplicity, we account for ohmic dissipation by placing it as a shell source of luminosity at the base of the planet's gaseous envelope, ignoring its evolution with time and its radial profile.",
"By depositing all ohmic heating deep in the planet's envelope our results constitute an upper limit to the amount of dissipation and inflation.",
"For example, [18] found that ohmic heating tends to push the convection zone inwards, decreasing the amount of ohmic dissipation inside the convection zone.",
"A precise determination of the location of the radiative-convective boundary is crucial for accurate measurement of the ohmic flux.",
"On the other hand, the second approximation turns out to be an excellent one.",
"This is due to two reasons.",
"First, the eventual configuration of the planet appears to depend solely on the final heat deposition at equilibrium; taking into account the time evolution of ohmic heating only serves to delay the planet's contraction as ohmic dissipation generally grows weaker with time, without implications on the final planet radius.",
"Secondly, the delay itself tends to be small, as the ohmic efficiency tends to evolve slowly.",
"Across all our models, we found that after the initial $\\sim $ Myr rapid cooling phase, the amount of ohmic dissipation only decreases by $56 \\%$ on average from $t = 10$ Myr to $t = 10$ Gyr.",
"Comparisons between our model with evolving feedback and the constant dissipation case show negligible differences in both time evolution and final planet radius.",
"Another approach we take that is different in comparison to previous studies is our choice of boundary conditions for solving the induction equation.",
"Our nominal choices are to assume that the radial current $j_r$ vanishes at an inner and outer boundary, forming a closed current loop.",
"We take the transition from rocky interior to gaseous envelope to be our inner boundary.",
"This boundary condition assumes that the rocky outer shell is an insulator.",
"For the outer boundary condition, we take the radial current $j_r$ to vanish at a set pressure of $P = 10^{-6}$ bars.",
"However, we explore two different boundary conditions and assess the effects on ohmic dissipation.",
"For the bottom boundary condition, we also explore the condition proposed by [25] where $j_\\theta - j_r/r = 0$ for Jupiter at $r=0$ .",
"This choice of boundary condition had a minor effect, with lower values of cumulative ohmic dissipation by about $20\\%$ .",
"For the top boundary condition, we explore the one suggested by [2].",
"They found that the atmospheres of hot Jupiters feature a local temperature minimum, which motivated them to set the radial current to vanish at this shell.",
"This boundary condition turned out to match ours well for $T_{eq} $ <$$ 1400$ K; at higher equilibrium temperatures, the conductivity at the temperature minima is not sufficiently insulating to enable \\cite {BS10} boundary condition.",
"In comparing both treatments, our choice of boundary condition (radial current vanishing at one millibar) yields an ohmic dissipation that is $ 50 - 100$ times higher than if we set the insulation at the conductivity minima.",
"\\leavevmode {\\color {black}This points to the importance of the boundary conditions in affecting the results.",
"We admit that our boundary conditions are simple and based on the assumption that the currents loop in the interior so that at some point in the atmosphere and the bottom of the envelope the radial currents have to vanish.",
"However, we have not investigated other more complicated boundary conditions such as open currents and complicated geometries.",
"}$ Our assumption that the rocky outer shell is insulating enough so that the currents originating in the winds do not penetrate the rocky interior stems from our goal of invoking a simple treatment for mini-Neptunes in light of the many uncertainties about their interior.",
"We know little about the possible phase structure of super-Earths and mini-Neptunes.",
"Perhaps they do not have a well segreated iron core from an overlying silicate mantle, which would put into question how and where would a magnetic field may be generated.",
"Or even under this mixed structure, there could be regions of high conductivity (semi-conductive state of mixture, presence of melts, favorable spin states, etc) that can influence the type of magnetic field being generated.",
"Or, perhaps the electrical currents generated in the wind can penetrate the planet throughout because at the relevant temperatures and pressures conduction is high enough and phase transitions do not hinder the induced currents.",
"We leave these scenarios for future work and focus on the simple treatment where the currents are confined to the envelope of the mini-Neptunes.",
"As previously mentioned, we do not include Lorentz drag that is expected to reduce the ohmic power at high enough temperatures at this point.",
"Because we focus on planets with equilibrium temperatures of up to 1700 K, we expect Lorentz drag to yet have become dominant to affect the flow and hence, our results."
],
[
"Summary",
"In summary, we investigate ohmic heating in planets with a rocky interior surrounded by an envelope of hydrogen and helium gas, based on a simplified model in which the evolution of the atmospheric structure and the ohmic dissipation are calculated separately and coupled via iteration.",
"We find that ohmic dissipation increases with shrinking planetary mass and increasing envelope mass fraction.",
"The overall normalization depends on the exact modelling details.",
"In addition, the energy needed to halt contraction during the thermal evolution increases with planetary mass and envelope fraction.",
"Thus, in combination, the ohmic dissipation available is enough to inflate the radius of low-mass mini-Neptunes.",
"The implications are two-fold.",
"First, it means that there is an added degeneracy to the problem of inferring the composition of mini-Neptunes.",
"There is a trade-off between H/He content and ohmic dissipation.",
"In fact, even while correcting for ohmic dissipation, planets with the same mass, radius and equilibrium temperature can have different H/He contents due to the fact that incipient envelopes are much more susceptible to inflation than more massive envelopes.",
"For planets with equilibrium temperatures in excess of 1400 K, one would first have to correct the radius for inflation caused by ohmic dissipation and calculate the possible ensembles that yield the radius for a given mass.",
"Without this treatment, the H/He inferred could be wrong.",
"Second, ohmic dissipation may be a very important effect when considering evaporating atmospheres of highly irradiated planets.",
"Already they are hot to drive evaporation, so that ohmic dissipation may be expected to be present, making the planets puffier than otherwise and susceptible to mass loss.",
"This might explain why for the planets with measured masses and radii, there is a lack of hot mini-Neptunes compared to warm ones.",
"We would like to thank Mathieu Havel for multiple discussions that helped benchmark our results, Konstantin Batygin for comments on the first draft of this paper, and the anonymous reviewer for her/his insights.",
"This work was funded under grant NSERC RGPIN-2014-06567."
]
] | 1709.01642 |
[
[
"Scattering theory without injectivity radius assumptions and spectral\n stability for the Ricci flow"
],
[
"Abstract We prove a completely new integral criterion for the existence and completeness of the wave operators $W_{\\pm}(-\\Delta_h,-\\Delta_g, I_{g,h})$ corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators $-\\Delta_j$, $j=1,2$, that are induced by two quasi-isometric complete Riemannian metrics $g$ and $h$ on an open manifold $M$.",
"In particular, this result provides a criterion for the absolutely continuous spectra of $-\\Delta_g$ and $-\\Delta_h$ to coincide.",
"Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup.",
"Unlike all previous results, our integral criterion only requires some lower control on the Ricci curvatures and some upper control on the heat kernels, but no control at all on the injectivity radii.",
"As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow."
],
[
"Introduction",
"As the (unique self-adjoint realization in $L^2(M,g)$ of the) Laplace-Beltrami operator $-\\Delta _g\\ge 0$ on a noncompact geodesically complete Riemannian manifold $(M,g)$ typically contains some continuous spectrum, a natural question that arises is to what extent one can control at least certain parts of the continuous spectrum.",
"A particular decomposition of the spectrum $\\mathrm {spec}(-\\Delta _g)$ is given (cf.",
"[9], [17] and the appendix of this paper) by $\\mathrm {spec}(-\\Delta _g)=\\mathrm {spec}_{\\mathrm {ac}}(-\\Delta _g)\\bigcup \\mathrm {spec}_{\\mathrm {sc}}(-\\Delta _g)\\bigcup \\mathrm {spec}_{\\mathrm {pp}}(-\\Delta _g),$ where $\\mathrm {spec}_{\\mathrm {ac}}(-\\Delta _g)$ denotes the absolutely continuous spectrum (cf. Appendix ).",
"$\\mathrm {spec}_{\\mathrm {sc}}(-\\Delta _g)$ the singular continuous spectrum, $\\mathrm {spec}_{\\mathrm {pp}}(-\\Delta _g)$ the pure point spectrum, so that $\\mathrm {spec}_{c}(-\\Delta _g)= \\mathrm {spec}_{\\mathrm {ac}}(-\\Delta _g)\\bigcup \\mathrm {spec}_{\\mathrm {sc}}(-\\Delta _g)$ is the whole continuous spectrum.",
"The absolutely continuous spectrum corresponds to the quantum dynamics in the following sense: by definition, $\\mathrm {spec}_{\\mathrm {ac}}(-\\Delta _g)$ is the spectrum of the restriction of $-\\Delta _g$ to the closed subspace of $L^2(M,g)$ given by absolutely continuous states $\\psi $ corresponding to $-\\Delta _g$ .",
"But for those $\\psi $ 's the RAGE theorem [9], [17] shows $\\lim _{|t|\\rightarrow \\infty } 1_{K}\\exp (it\\Delta _g)\\psi =0\\quad \\text{ in $L^2(M,g)$ for every compact $K\\subset M$.",
"}$ As by Schrödinger's equation $\\exp (it\\Delta _g)\\psi $ is the state at the time $t$ given that the initial state was $\\psi $ , property (REF ) shows that the quantum particle eventually leaves every compact set, if the initial state was an absolutely continuous one.",
"A perturbative approach to control $\\mathrm {spec}(-\\Delta _g)$ is provided by the machinery of 2-Hilbert-space scattering theory: namely, assume that $h$ is another Riemannian metric on $M$ which is quasi-isometric to $g$ and whose absolutely continuous spectrum is known.",
"Then, with the trivial identification map $I_{g,h}: L^2(M,g)\\longrightarrow L^2(M,h),\\:\\:f\\longmapsto f,$ one can ask whether the 2-Hilbert-space wave operators $W_{\\pm }(-\\Delta _{h},-\\Delta _{g}, I_{g,h})=\\mathop {\\rm {st}\\rule [0.5ex]{1ex}{0.1ex}\\mathrm {lim}}_{t\\rightarrow \\pm \\infty }\\exp (-it\\Delta _{h})I_{g,h}\\exp (it\\Delta _{g})P_{\\mathrm {ac}}(-\\Delta _g)$ exist and are complete (cf.",
"section for the precise definitions), the point being that the latter property implies $\\mathrm {spec}_{\\mathrm {ac}}(-\\Delta _g)=\\mathrm {spec}_{\\mathrm {ac}}(-\\Delta _h),$ and one has managed to transfer the known spectral information from $-\\Delta _h$ to $-\\Delta _g$ .",
"The current state of the art concerning criteria for the existence and completeness of the $W_{\\pm }(-\\Delta _{h},-\\Delta _{g}, I_{g,h})$ is the main result from [7].",
"There, Hempel, Post and Weder prove the following result: Assume that for both $j\\in \\lbrace g,h\\rbrace $ one has $\\int \\gamma _j(x) \\delta _{g,h}(x)d\\mu _j(x)<\\infty ,$ where $\\mu _j$ is the Riemannian volume measure, $\\gamma _j:M\\rightarrow [0,\\infty )$ is a certain explicitly function which depends in a monotonically decreasing way on a local lower bound of the injectivity radius and in a monotonically increasing way on a local lower bound on the Ricci curvature $\\mathrm {Ric}_j$ , $\\delta _{g,h}:M\\rightarrow [0,\\infty )$ denotes a certain zeroth order deviation of the metrics from each other (cf.",
"section below for the definition).",
"Then the $W_{\\pm }(-\\Delta _{h},-\\Delta _{g}, I_{g,h})$ exist and are complete.",
"An important feature of this result is that no assumptions on the geometry of $(M,g)$ and $(M,h)$ are imposed, only the deviation of $g$ from $h$ matters, as it should be in scattering theory.",
"The above result has considerably improved an earlier result by Müller and Salomonsen [8], where instead of the zeroth order deviation, the authors had to weight their integral condition with a much stronger second order deviation, in addition to assuming both metrics to have a $C^{\\infty }$ -bounded geometry.",
"Nevertheless, a certain drawback of the result from [7] is that injectivity radii are very hard to calculate or even to control in general.",
"In any case, one needs a very detailed control on the sectional curvatures, to get some control on the injectivity radius [3].",
"On the other hand, volumes of balls are much more handy and equivalent under quasi-isometry, and in fact any lower bound on the injectivity radius implies a lower bound on the volume function by Bishop-Günther's inequality.",
"In view of these remarks, our main result Theorem REF , which reads as follows, provides a remarkable improvement: Assume that $g$ and $h$ are geodesically complete and quasi-isometric Riemannian metrics on $M$ such that for both $j\\in \\lbrace g,h\\rbrace $ and some $s>0$ one has $\\int \\alpha _{j}(x,s) \\beta _{j}(x) \\delta _{g,h}(x) d\\mu _j(x)<\\infty ,$ where $\\alpha _{j}(\\cdot ,s):M\\rightarrow [0,\\infty )$ is a local upper bound on the heat kernel on $(M,j)$ at the time $s>0$ and $\\beta _{j}:M\\rightarrow [0,\\infty )$ is a certain explicitly given local lower bound on $\\mathrm {Ric}_j$ .",
"Then the wave operators $W_{\\pm }(-\\Delta _{h},-\\Delta _{g}, I_{g,h})$ exist and are complete.",
"Again, no assumptions on the geometry of $(M,g)$ are $(M,h)$ are imposed.",
"While Theorem REF can be expected to be disjoint from that of [7] in general, under global lower Ricci bounds it can be brought into a form which indeed is much more general and handy then the induced result from [7] in the sense of the above remarks.",
"In fact, assuming that both Ricci curvatures are bounded from below by constants, one can use Li-Yau type heat kernel estimates and Theorem REF boils down to give the following criterion (cf.",
"Corollary REF below): Assume that $g$ and $h$ are geodesically complete and quasi-isometric Riemannian metrics on $M$ with $\\mathrm {Ric}_j$ bounded from below by a constant for both $j\\in \\lbrace g,h\\rbrace $ and $\\int \\mu _j(x,1)^{-1} \\delta _{g,h}(x) d\\mu _j(x)<\\infty \\quad \\text{ for some $j\\in \\lbrace g,h\\rbrace $,}$ where $\\mu _j(x,1)$ denotes the volume of the geodesic ball with radius 1 centered at $x$ with respect to $(M,j)$ .",
"Then the wave operators $W_{\\pm }(-\\Delta _{h},-\\Delta _{g}, I_{g,h})$ exist and are complete.",
"Note that if $g$ and $h$ are geodesically complete and quasi-isometric Riemannian metrics on $M$ with $\\mathrm {Ric}_j$ bounded from below by a constant for both $j\\in \\lbrace g,h\\rbrace $ , then (REF ) requires control on some lower bounds of the injectivity radii, while in our Corollary REF this condition is replaced by a more general and much more handy lower control on the volume function.",
"The essential difference between our machinery and the one from [7] is that we rely on parabolic techniques, while in [7] use elliptic estimates.",
"In fact, our main tool is an $L^2\\rightarrow L^{\\infty }_{\\mathrm {loc}} $ estimate for the gradient of the heat semigroup that should be of an independent interest, which is valid on every geodesically complete Riemannian manifold, and which relies on an explicit Bismut type probabilistic formula (cf.",
"[1], [13] and the proof of Theorem REF below).",
"Finally, it is remarkable that the assumptions in Corollary REF are explicit enough to deduce the following stability of absolutely continuous spectra under a Ricci flow, which seems to be first result of its kind (cf.",
"Corollary REF ): Let $S>0$ , $\\kappa \\in \\mathbb {R}$ and assume that the family $(g_s)_{s\\in [0,S]}$ of Riemannian metric on $M$ evolves under a Ricci type flow $\\partial _s g_s=\\kappa \\mathrm {Ric}_{g_s}, \\quad s\\in [0,S],$ the initial metric $g_0$ is geodesically complete setting $A(x):=\\sup \\big \\lbrace \\left|\\mathrm {Ric}_{g_s}(v,v)\\right|_{g_s}:s\\in [0,S], v\\in T_x M, |v|_{g_s}\\le 1\\big \\rbrace ,\\quad x\\in M,$ one has $&\\sup _{ x\\in M}A(x)<\\infty ,\\\\&\\int \\mu _{g_0}(x,1)^{-1} \\sinh \\big ( (m/4)|\\kappa | A(x)S\\big ) d\\mu _{g_0}(x)<\\infty .$ Then one has $\\mathrm {spec}_{\\mathrm {ac}}(H_{g_s})=\\mathrm {spec}_{\\mathrm {ac}}(H_{g_0})$ for all $s\\in [0,S]$ .",
"Note that assumption (REF ) is natural in this context: for example, a typical short time existence result for Ricci flow by [11] Shi requires that $g_0 $ is geodesically complete with bounded sectional curvatures, yielding a solution $(g_s)_{s\\in [0,S^*)}$ of the Ricci flow equation $\\partial _s g_s=-2 \\mathrm {Ric}_{g_s}, \\quad s\\in [0,S^*),$ which exists up to a time $S^*>0$ and satisfies $\\sup _{s\\in [0,S^*),x\\in M}\\left|\\mathrm {Sec}_{g_r}(x)\\right| <\\infty .$ The latter finiteness clearly implies (REF ) for every $S<S^*$ ."
],
[
"Main results",
"Let $M$ be a smooth connected manifold of dimension $m\\ge 2$ .",
"We stress the fact that we understand all our spaces of functions on $M$ (or more generally, all our spaces of sections in vector bundles over $M$ ) to be complex-valued.",
"For example, $\\Omega ^1_{C^{\\infty }}(M)$ stands for the smooth complex-valued 1-forms on $M$ , that is, the smooth sections of $T^*M\\otimes \\mathbb {C}\\rightarrow M$ , and then ${d}:C^{\\infty }(M)\\longrightarrow \\Omega ^1_{C^{\\infty }}(M)$ stands for the complexification of the usual exterior derivative.",
"It will be convenient to set $&{M}(M):=\\big \\lbrace \\text{smooth Riemannian metrics on $M$}\\big \\rbrace ,\\\\&\\widetilde{{M}}(M):=\\big \\lbrace g\\in {M}(M): \\text{$g$ is geodesically complete}\\big \\rbrace .$ Given $g\\in {M}(M)$ we denote by $\\mathrm {Ric}_g$ its Ricci curvature, and by $\\mu _g$ the Riemannian volume measure, by $B_g(x,r)$ the open geodesic balls, and by $\\mu _g(x,r):=\\mu _g(B_g(x,r))$ the volume function.",
"The induced metric on $T^*M$ will be denoted by $g^*$ .",
"Complexifications of these data will be denoted by $g_{\\mathbb {C}}$ etc.",
"The complex Hilbert space $L^2(M,g)$ is given by $\\mu _g$ -equivalence classes of Borel functions $f:M\\rightarrow \\mathbb {C}$ with $\\int |f|^2 d\\mu _g$ finite, and $\\left\\langle \\psi _1,\\psi _2\\right\\rangle _{L^2(M,g)}=\\int \\overline{\\psi _1}\\psi _2d\\mu _g.$ The complex Sobolev space $W^{1,2}_0(M,g)\\subset L^2(M,g)$ is defined to be the closure of $C^{\\infty }_c(M)$ with respect to the scalar product $\\left\\langle \\psi _1,\\psi _2 \\right\\rangle _{W^{1,2}_0(M,g)}=\\big (\\int \\overline{\\psi _1}\\psi _2 {d}\\mu _g\\big )^{1/2}+ \\big (\\int g^*_{\\mathbb {C}} ({d}\\psi _1,d\\psi _2) {d}\\mu _g\\big )^{1/2},\\quad \\psi \\in C^{\\infty }_c(M).$ Let $H_{g}\\ge 0$ denote the Friedrichs realization of the Laplace-Beltrami operator $- \\Delta _{g}\\ge 0$ in $L^2(M,g)$ .",
"We will also need the operator ${d}_g$ , which denotes the minimal extension of the exterior differential ${d}$ with respect to $g$ .",
"In other words, ${d}_g$ is the closed unbounded operator from $L^2(M,g)$ to $\\Omega ^{1}_{L^2}(M,g)$ which is defined by $\\mathrm {Dom}({d}_g)=W^{1,2}_0(M,g)$ , and ${d}_gf:={d}f$ , in the distributional sense.",
"In fact, one has $H_g= {d}_g^{*}{d}_g$ .",
"If $g$ is geodesically complete, then $H_g$ is essentially self-adjoint on $C^{\\infty }_{ \\mathrm {c} }(M)$ .",
"Given $g,h\\in {M}(M)$ , we can define a smooth vector bundle morphism ${A}_{g,h}:T^* M\\otimes \\mathbb {C}\\longrightarrow T^* M\\otimes \\mathbb {C},\\:\\:g^*_{\\mathbb {C}}({A}_{g,h}(x)\\alpha ,\\beta ) :=h^*_{\\mathbb {C}}(\\alpha ,\\beta )\\:,\\:\\:\\alpha ,\\beta \\in T^*_xM\\otimes \\mathbb {C}, \\:x\\in M.$ The endomorphism ${A}_{g,h}$ is fiberwise self-adjoint with respect to $g^*_{\\mathbb {C}}$ , in view of $g^*_{\\mathbb {C}}({A}_{g,h}(x)\\alpha ,\\alpha )\\in \\mathbb {R}\\quad \\text{ for all $x\\in M$, $\\alpha \\in T^*_xM\\otimes \\mathbb {C}$}.$ In addition, ${A}_{g,h}$ has fiberwise strictly positive eigenvalues.",
"We further define $\\delta _{g,h}:M\\longrightarrow [0,\\infty ),\\quad \\delta _{g,h}(x):=2\\sinh \\big ((m/4)\\max _{\\lambda \\in \\mathrm {spec}({A}_{g,h}(x))}|\\log (\\lambda ) |\\big ).$ The function $\\delta _{g,h}$ measures a 0-th order deviation of the metrics when we consider them as multiplicative perturbations of each other.",
"We have ${d}\\mu _{h}=\\rho _{g,h}{d}\\mu _{g} \\quad \\text{ with a Radon-Nikodym density $0<\\rho _{g,h}\\in C^{\\infty }(M)$},$ where we record the following simple facts: $\\rho _{h,g}=1/\\rho _{g,h},\\quad {A}_{h,g}={A}_{g,h}^{-1},\\quad \\rho _{g,h}=\\mathrm {det}({A}_{g,h})^{-{2}},\\quad \\delta _{g,h}=\\delta _{h,g}.$ We write $g\\sim h$ , if $h$ is quasi-isometric to $g$ , that is, if there exists a constant $C\\ge 1$ such that $(1/C)g \\le h \\le Cg \\quad \\text{ pointwise, as bilinear forms.",
"}$ Let us see how these definitions works in the case of conformal perturbations: Example 2.1 Assume $h= \\exp (-(4/m)\\phi )g$ for some smooth function $\\phi :M\\rightarrow \\mathbb {R}$ , that is, $h$ is a conformal perturbation of $g$ .",
"Then one has $h^*= \\exp ((4/m)\\phi )g^*$ and $g\\sim h$ holds if and only if $\\phi $ is bounded, and then one has $\\delta _{g,h}=2\\sinh (|\\phi |)$ .",
"The scattering theory of conformal perturbations has been studied in detail in [2].",
"So assume $g\\sim h$ for the moment.",
"Then there exists the trivial bounded linear and bijective identification operator $I_{g,h}: L^2(M,g)\\longrightarrow L^2(M,h),\\:\\:f\\longmapsto f,$ and one has $ 0<\\inf \\rho _{g,h}\\le \\sup \\rho _{g,h}<\\infty ,\\quad \\sup \\delta _{g,h}<\\infty .$ Furthermore, the operator $I^*_{g,h}$ is given by the bounded multiplication operator $I^*_{g,h}: L^2(M,g)\\longrightarrow L^2(M,h),\\quad I^*_{g,h}f(x)=\\rho _{g,h}(x)f(x).$ For every $g\\in \\widetilde{{M}}(M)$ , with $(P^g_t)_{t>0}:=(\\exp (-tH_g))_{t>0}\\subset {L}(L^2(M,g))$ the heat semigroup defined by the spectral calculusIn the sequel, whenever a Borel equivalence class of $L^2$ -functions on $(M,g)$ has a smooth representative, we implicitly take the latter., and $(x,s)\\in M\\times (0,\\infty )$ , we define the finite quantities $&\\Psi _{1,g}(x):=\\max \\Big (0,\\inf \\big \\lbrace C\\in \\mathbb {R}:\\mathrm {Ric}_g(v,v) \\ge C |v|_g^2 \\text{ \\rm for all $y\\in B_g(x,1/2)$, $v\\in T_yM$} \\big \\rbrace \\Big ),\\\\&\\Psi _{2,g}(x):= \\pi ^2(m+3) +\\pi \\sqrt{\\Psi _{1,g}(x)(m-1)}+4\\Psi _{1,g}(x),\\\\&\\Psi _{3,g}(x,s):= \\Psi _{2,g}(x)\\big (1-\\exp (-\\Psi _{2,g}(x)s)\\big )^{-1},\\\\&\\Psi _{4,g}(x,s):=\\sup _{y\\in M}P^g_s(x,y).$ While it is well-known [5] that $\\Psi _{4,g}(x,s)<\\infty $ for all $(x,s)\\in M\\times (0,\\infty )$ , one can even prove [6] $\\sup _{x\\in K}\\Psi _{4,g}(x,s)<\\infty \\quad \\text{ for all $s\\in (0,\\infty )$, $K\\subset M$ compact.",
"}$ Here comes our main result: Theorem A Assume that $g,h\\in \\widetilde{{M}}(M)$ satisfy $g\\sim h$ and that for some $s\\in (0,\\infty )$ and both $j\\in \\lbrace g,h\\rbrace $ one has $\\int \\Psi _{3,j}(x,s) \\Psi _{4,j}(x,s) \\delta _{g,h}(x) d\\mu _j(x)<\\infty .$ Then the wave operators $W_{\\pm }(H_{h},H_{g}, I_{g,h})=\\mathop {\\rm {st}\\rule [0.5ex]{1ex}{0.1ex}\\mathrm {lim}}_{t\\rightarrow \\pm \\infty }\\exp (itH_{h})I_{g,h}\\exp (-itH_{g})P_{\\mathrm {ac}}(H_g)$ exist and are complete (cf.",
"Theorem REF for the definition of completeness).",
"Moreover, $W_{\\pm }\\big (H_{h},H_g, I_{g,h}\\big )$ are partial isometries with inital space $\\mathrm {Ran} \\: P_{\\mathrm {ac}}(H_g)$ and final space $\\mathrm {Ran} \\: P_{\\mathrm {ac}}(H_h)$ , and one has $\\mathrm {spec}_{\\mathrm {ac}}(H_g)=\\mathrm {spec}_{\\mathrm {ac}}(H_h)$ .",
"Note that the assumptions and the conclusions of Theorem REF are symmetric in $(g,h)$ .",
"The ultimate definition of the functions $\\Psi _{j,g/h}$ is dictated by the bound from Theorem REF below.",
"In case the Ricci curvatures are bounded from below by constants, Theorem REF can be brought into the following convenient form: Corollary A Assume that $g,h\\in \\widetilde{{M}}(M)$ satisfy the following assumptions, $g\\sim h$ , $\\mathrm {Ric}_j$ is bounded from below by a constant for both $j\\in \\lbrace g,h\\rbrace $ , there exists $j\\in \\lbrace g,h\\rbrace $ with $\\int \\mu _j(x,1)^{-1} \\delta _{g,h}(x) d\\mu _j(x)<\\infty .$ Then the wave operators $W_{\\pm }(H_{h},H_{g}, I_{g,h})=\\mathop {\\rm {st}\\rule [0.5ex]{1ex}{0.1ex}\\mathrm {lim}}_{t\\rightarrow \\pm \\infty }\\exp (itH_{h})I_{g,h}\\exp (-itH_{g})P_{\\mathrm {ac}}(H_g)$ exist and are complete.",
"Moreover, $W_{\\pm }\\big (H_{h},H_g, I_{g,h}\\big )$ are partial isometries with inital space $\\mathrm {Ran} \\: P_{\\mathrm {ac}}(H_g)$ and final space $\\mathrm {Ran} \\: P_{\\mathrm {ac}}(H_h)$ , and one has $\\mathrm {spec}_{\\mathrm {ac}}(H_g)=\\mathrm {spec}_{\\mathrm {ac}}(H_h)$ .",
"Firstly note that if one has $\\int \\mu _j(x,1)^{-1} \\delta _{g,h}(x) d\\mu _j(x)<\\infty $ for some $j\\in \\lbrace g,h\\rbrace $ , then by quasi-isometry the same is true for both $j\\in \\lbrace g,h\\rbrace $ .",
"One trivially has $\\sup _{x\\in M}\\Psi _{3,j}(x,1)<\\infty $ and by a Li-Yau type heat kernel estimate [12] there exist $a,b\\in (0,\\infty )$ , which only depend on the lower Ricci bounds and $m$ , such that for all $x\\in M$ one has $\\Psi _{4,j}(x,1)\\le a\\mu _j(x,1)^{-1}\\exp (b),$ so that the claim follows from Theorem REF .",
"Corollary REF implies the following result concerning the stability of the absolutely continuous spectrum of a family of metrics that evolve under a Ricci flow, as long as the initial metric has a bounded Ricci curvature: Corollary B Let $S>0$ , $\\kappa \\in \\mathbb {R}$ and assume that the family $(g_s)_{s\\in [0,S]}\\subset {M}(M)$ evolves under a Ricci type flow $\\partial _s g_s=\\kappa \\mathrm {Ric}_{g_s}, \\quad s\\in [0,S],$ the initial metric $g_0$ is geodesically complete setting $A(x):=\\sup \\big \\lbrace \\left|\\mathrm {Ric}_{g_s}(v,v)\\right|_{g_s}:s\\in [0,S], v\\in T_x M, |v|_{g_s}\\le 1\\big \\rbrace ,\\quad x\\in M,$ one has $& \\sup _{ x\\in M}A(x)<\\infty ,\\\\&\\int \\mu _{g_0}(x,1)^{-1} \\sinh \\big ( (m/4)\\,S\\,|\\kappa | A(x)\\big ) d\\mu _{g_0}(x)<\\infty .$ Then one has $\\mathrm {spec}_{\\mathrm {ac}}(H_{g_s})=\\mathrm {spec}_{\\mathrm {ac}}(H_{g_0})$ for all $s\\in [0,S]$ .",
"Let $s\\in [0,S]$ .",
"It is well-known that the Ricci flow equation in combination with (REF ) imply $g_s\\sim g_0$ , so that in particular all $g_s$ are geodesically complete [16].",
"We give the simple proof for the convenience of the reader: Set $A:=\\sup _{ x\\in M}A(x)$ and assume $x\\in M, v\\in T_xM$ .",
"If $\\kappa >0$ we have $&\\partial _s g_s(v,v)=\\kappa \\mathrm {Ric}_{g_s}(v,v)\\le \\kappa A g_s(v,v),\\\\&\\partial _s g_s(v,v)=\\kappa \\mathrm {Ric}_{g_s}(v,v)\\ge -\\kappa A g_s(v,v),$ so that from Gronwall's Lemma we get $&g_s(v,v)\\le \\exp (sA\\kappa )g_0(v,v),\\\\&g_s(v,v)\\ge \\exp (-sA\\kappa )g_0(v,v).$ In case $\\kappa <0$ we have $\\partial _s g_s(v,v)=\\kappa \\mathrm {Ric}_{g_s}(v,v)\\le -\\kappa A g_s(v,v),\\\\\\partial _s g_s(v,v)=\\kappa \\mathrm {Ric}_{g_s}(v,v)\\ge \\kappa A g_s(v,v),$ again Gronwall gives the asserted quasi-isometries.",
"It remains to prove that for all $s$ the integrability () implies $\\int \\mu _{g_0}(x,1)^{-1} \\delta _{g_s,g_0}(x) d\\mu _{g_0}(x)<\\infty .$ To this end, assume again $\\kappa >0$ first.",
"Given $x\\in M$ , $\\alpha \\in T^*_xM$ we have $&\\partial _s g_0^*({A}_{g_0,g_s}(x)\\alpha ,\\alpha )= \\partial _s g^*_s(\\alpha ,\\alpha )=\\partial _s g_s(\\alpha ^{\\sharp _s},\\alpha ^{\\sharp _s})\\\\&\\le A(x) \\kappa g_s(\\alpha ^{\\sharp _s},\\alpha ^{\\sharp _s})=A(x) \\kappa g^*_s(\\alpha ,\\alpha )=A(x) \\kappa g_0^*({A}_{g_0,g_s}(x)\\alpha ,\\alpha )$ and likewise $\\partial _s g^*_0({A}_{g_0,g_s}(x)\\alpha ,\\alpha )\\ge - A(x)\\kappa g_0^*({A}_{g_0,g_s}(x)\\alpha ,\\alpha ),$ so that by Gronwall one has $|\\mathrm {log}(\\lambda )| \\le A(x)\\kappa s\\le A(x)\\kappa S\\quad \\text{ for all $\\lambda \\in \\mathrm {spec}({A}_{g_0,g_s}(x))=\\mathrm {spec}({A}_{g_0,g_s}(x)|_{T^*_xM}).$}$ The same proof gives in case $\\kappa <0$ the inequality $|\\mathrm {log}(\\lambda )| \\le -A(x)\\kappa s\\le -A(x)\\kappa S\\quad \\text{ for all $\\lambda \\in \\mathrm {spec}({A}_{g_0,g_s}(x))=\\mathrm {spec}({A}_{g_0,g_s}(x)|_{T^*_xM})$},$ showing altogether that $\\delta _{g_s,g_0}(x)\\le 2\\sinh \\big ( (m/4)\\,S\\,|\\kappa |\\, A(x)\\big ),$ and completing the proof.",
"The operators $(\\widehat{P}^g_s)_{s>0}:=(d_g P^g_s)_{s>0}\\subset {L}\\big (L^2(M,g), \\Omega ^1_{L^2}(M,g)\\big )$ will play a crucial role in proof of Theorem REF .",
"In fact, the main ingredient of the proof is the parabolic gradient bound for the jointly smooth integral kernel $(0,\\infty )\\times M\\times M \\ni (s,x,y)\\longmapsto \\widehat{P}^g_s(x,y)\\in T^*_x M$ of $\\widehat{P}^g_s$ from Theorem REF below, which is certainly of an independent interest.",
"Note that $(s,x,y)\\mapsto \\widehat{P}^g_s(x,y)$ is the uniquely determined smooth map such that for all $(s,x)\\in (0,\\infty )\\times M$ , $f\\in L^2(M,g)$ one has $\\widehat{P}^g_sf(x)=\\int \\widehat{P}^g_s(x,y)f(y) d\\mu _g(y).$ Theorem B For every $g\\in \\widetilde{{M}}(M)$ , $(s,x)\\in (0,\\infty )\\times M$ one has $\\int \\Big |\\widehat{P}^g_s(x,y)\\Big |^2_{g^*}d\\mu _g(y)\\le \\Psi _{3,g}(x,s) \\Psi _{4,g}(x,s).$ Remark 2.2 Note that by Riesz-Fischer's duality theorem, the estimate from Theorem REF is equivalent to the following statement: For every $g\\in \\widetilde{{M}}(M)$ , $(s,x)\\in (0,\\infty )\\times M$ , $f\\in L^2(M,g)$ one has $\\Big |\\widehat{P}^g_sf(x)\\Big |_{g^*}\\le \\sqrt{\\Psi _{3,g}(x,s) \\Psi _{4,g}(x,s)} \\left\\Vert f\\right\\Vert _{L^2(M,g)}.$"
],
[
"Proof of Theorem ",
"Here we give the We can omit $g$ in the notation.",
"Let $X(x)$ be a Brownian motion on $M$ starting from $x$ and $\\zeta (x)$ its maximal lifetime.",
"Let us first assume $f$ is real-valued.",
"Then, for $v\\in T_xM$ one has the Bismut type formula (cf.",
"Theorem 6.2 in [4], Formula (6.2) in [15], [13], [1]) $\\left(dP_{s}f\\right)_{x}v=- \\mathbb {E}\\left[ f(X_s(x))1_{\\lbrace s<\\zeta (x)\\rbrace }\\int _{0}^{\\tau (x)\\wedge s}\\big ( \\Theta _{r}(x)\\dot{\\ell }_{r}(v),dW_{r}(x)\\big ) \\right],$ where $\\Theta _r(x)$ , $r<\\zeta (x)$ , is the $\\mathrm {Aut}(T_xM)$ -valued process defined by $\\frac{d}{d r}\\Theta _r(x)=- {\\operatorname{Ric}}_{/\\!/_{\\!r}^{\\phantom{.",
"}}(x)}({\\bf .",
"}$ ,r)♯, 0(x)=idTxM, $with $ /r.",
"(x):TxMTXr(x)M$, $ r<(x)$, the stochastic parallel transport along the paths of $ X(x)$;$ $\\tau (x)<\\zeta (x)$ is the first exit time of $X(x)$ from $B(x,1/2)$ ; $W(x)$ is a Brownian motion in $T_xM$ ; $\\ell (v)$ is any adapted process in $T_xM$ with absolutely continuous paths such that (for some $\\varepsilon >0$ ) $\\ell _0(v)=v,\\quad \\ell _{\\tau _{x}}(v)=0\\quad \\text{and}\\quad \\mathbb {E}\\left[\\Bigl (\\int _0^{\\tau (x)\\wedge s}\\vert \\dot{\\ell }_t(v)\\vert ^2\\,dt\\Bigr ){}^{1/2+1/\\epsilon }\\right]<\\infty .$ In fact, the smooth representative of $P_sf(x)$ is given by $P_sf(x)=\\mathbb {E}[f(X_s(x) 1_{\\lbrace s<\\zeta (x)\\rbrace }]=\\int P_s(x,y) f(y) d\\mu (y).$ By Cauchy-Schwarz we obtain $\\left|\\left(dP_{s}f\\right)_{x}v\\right|\\le \\left(\\mathbb {E}\\left[ |f|^2(X_s(x)) 1_{\\lbrace s<\\zeta (x)\\rbrace }\\right]\\right)^{1/2}\\left(\\mathbb {E}\\left[\\left(\\int _{0}^{\\tau (x)\\wedge s}\\big (\\Theta _{r}(x)\\dot{\\ell }_{r}(v),dW_{r}(x)\\big ) \\right)^2\\right]\\right)^{1/2}.$ It is well-known how to estimate the second factor on the right by choosing $\\ell (v)$ appropriately, e.g. [14].",
"Namely, for $|v|\\le 1$ one can achieve $\\mathbb {E}\\left[\\left(\\int _{0}^{\\tau (x)\\wedge s}\\big ( \\Theta _{r}(x)\\dot{\\ell }_{r}(v),dW_{r}(x)\\big ) \\right)^2\\right] \\le \\Psi _3(x,s) .$ Thus $\\left|\\left(dP_{s}f\\right)_{x}v\\right|&\\le \\left(\\mathbb {E}\\left[ |f|^2(X_s(x)) 1_{\\lbrace s<\\zeta (x)\\rbrace }\\right]\\right)^{1/2}\\sqrt{\\Psi _3(x,s)}\\\\&= \\left(\\int |f(y)|^2 P_s(x,y)d\\mu (y)\\right)^{1/2} \\sqrt{\\Psi _3(x,s)}\\\\&\\le \\sqrt{\\Psi _4(x,s)}\\sqrt{\\Psi _3(x,s)}\\left\\Vert f\\right\\Vert _{L^2(M)}.$ Using that complexifications are norm preserving and using Remark REF , the latter bound completes the proof."
],
[
"Proof of Theorem ",
"We start by noting that given a diagonizable linear operator $A$ on a finite dimensional linear space and a real-valued function $f$ on the spectrum of $A$ , the linear operator $f(A)$ can be defined using the projectors onto the eigenspaces of $A$ .",
"In particular, this procedure does not depend on a scalar product, but if $A$ is self-adjoint w.r.t.",
"some scalar product, then the above definition of $f(A)$ is consistent with the spectral calculus.",
"For example, if we are given $g,h\\in {M}(M)$ and a point $x\\in M$ , then $\\varrho _{g,h}(x){A}_{g,h}(x): T^*_xM\\otimes \\mathbb {C}\\longrightarrow T^*_xM\\otimes \\mathbb {C}$ is diagonizable.",
"We define a function and a Borel section in $ T^*M\\otimes \\mathbb {C}\\rightarrow M$ by setting $&S_{g,h}:M\\longrightarrow \\mathbb {R}, \\quad S_{g,h}(x):=\\varrho _{g,h}(x)^{1/2}-\\varrho _{g,h}(x)^{-1/2},\\\\&\\widehat{S}_{g,h}:M\\longrightarrow \\mathrm {End}(T^* M\\otimes \\mathbb {C}),\\quad \\widehat{S}_{g,h}(x):=\\big (\\varrho _{g,h}(x){A}_{g,h}(x)\\big )^{1/2}-\\big (\\varrho _{g,h}(x){A}_{g,h}(x)\\big )^{-1/2}.$ One has the elementary bounds (cf.",
"Lemma 3.3 in [7]) $\\max \\big (\\left|S_{g,h}(x)\\right| , \\big |\\widehat{S_{g,h}}(x)\\big |_{g} ,\\big |\\widehat{S_{g,h}}(x)\\big |_{h} \\big )\\le \\delta _{g,h}(x)\\quad \\text{for all $x\\in M$},$ in particular, the assumption $\\sup \\delta _{g,h}<\\infty $ (which is equivalent to $g\\sim h$ ) implies the boundedness of $S_{g,h}$ , $\\widehat{S}_{g,h}$ .",
"We will need the maximally defined multiplication operators $&S_{g,h;j}:L^2(M,j)\\longrightarrow L^2(M,j),\\quad S_{g,h;j}f(x)=|S_{g,h}(x)|^{{2}}f(x),\\quad j=g,h,\\\\&\\widehat{S_{g,h;j}}:\\Omega ^1_{L^2}(M,j )\\longrightarrow \\Omega ^1_{L^2}(M,j ),\\quad \\widehat{S_{g,h;j}}f(x)=|\\widehat{S_{g,h}}(x)|^{{2}}f(x),\\quad j=g,h,\\\\&U_{g,h}:L^2(M,g)\\longrightarrow L^2(M,h),\\quad U_{g,h}f(x)=\\big [\\mathrm {sgn}(S_{g,h})\\varrho _{g,h}^{-{2}}\\big ](x)f(x),\\\\&\\widehat{U_{g,h}}:\\Omega ^1_{L^2}(M,g )\\longrightarrow \\Omega ^1_{L^2}(M,h),\\quad \\widehat{U_{g,h}}\\alpha (x)=\\big [\\mathrm {sgn}(\\widehat{S_{g,h}})(\\varrho _{g,h}{A}_{g,h})^{-{2}}\\big ](x)\\alpha (x).$ The operators $U_{g,h}$ , $\\widehat{U_{g,h}}$ are always unitary and self-adjoint, and the operators $S_{g,h;j}$ , $\\widehat{S_{g,h;j}}$ ($j=g,h$ ) are always self-adjoint and additionally bounded in case $g\\sim h$ .",
"The following technical result provides the link between Theorem REF and the proof of Theorem REF .",
"It is a variant of a decomposition formula by Hempel-Post-Weder (cf.",
"Lemma 3.4 from in [7]): Lemma 4.1 (HPW formula) Let $g,h\\in {M}(M)$ be given with $g\\sim h$ .",
"Then defining the bounded operator $T_{g,h,s}:L^2(M,g)\\rightarrow L^2(M,h)$ by $T_{g,h,s}:=\\big (\\widehat{S_{g,h;h}}\\widehat{P}^h_s\\big )^{*}\\widehat{U_{g,h}}\\widehat{S_{g,h;g}}\\widehat{P}^g_s-\\big (S_{g,h;h} P^h_s\\big )^{*}U_{g,h}S_{g,h;g} P^g_{s/2}H_g P^g_{s/2},$ the following formula holds for all $s>0$ , $f_h\\in \\mathrm {Dom}(H_h)$ , $f_g\\in \\mathrm {Dom}(H_g)$ , $\\left\\langle f_h ,T_{g,h,s}f_g\\right\\rangle _{L^2(M,h)} =\\left\\langle H_hf_h, P^h_s I_{g,h} P^g_sf_g\\right\\rangle _{L^2(M,h)} -\\left\\langle f_h,P^h_s I_{g,h}P^g_s H_gf_g\\right\\rangle _{L^2(M,h)}.$ Adding and subtracting the term $\\left\\langle I^{-1}_{g,h} P^h_s f_h, H_g P^g_sf_g\\right\\rangle _{L^2(M,g)}$ we get $&\\left\\langle H_hf_h, P^h_s I_{g,h} P^g_sf_g\\right\\rangle _{L^2(M,h)} -\\left\\langle f_h, P^h_s I_{g,h} P^g_s H_gf_g\\right\\rangle _{L^2(M,h)}\\\\&=\\left\\langle H_hP^h_sf_h, I_{g,h} P^g_sf_g\\right\\rangle _{L^2(M,h)} -\\left\\langle I^{-1}_{g,h}P^h_sf_h, H_g P^g_sf_g\\right\\rangle _{L^2(M,g)}\\\\&\\quad - \\left\\langle P^h_s f_h, \\big (I_{g,h}-(I^{-1}_{g,h})^*\\big ) H_g P^g_sf_g\\right\\rangle _{L^2(M,g)}\\\\&=\\left\\langle {d}_hP^h_sf_h, {d}_h I_{g,h} P^g_sf_g\\right\\rangle _{\\Omega _{L^2}(M,h)} -\\left\\langle {d}_g I^{-1}_{g,h}P^h_sf_h, {d}_g P^g_sf_g\\right\\rangle _{L^2(M,g)}\\\\&\\quad - \\left\\langle P^h_s f_h, \\big (I_{g,h}-(I^{-1}_{g,h})^*\\big ) H_g P^g_sf_g\\right\\rangle _{L^2(M,g)}.$ Using (REF ) and (REF ), the latter is $&=\\int h^*_{\\mathbb {C}}\\Big ((1-\\varrho _{g,h}^{-1}{A}_{g,h}^{-1}){d}_h P^h_sf_h,{d}_g P^g_s f_g \\Big ) {d}\\mu _h-\\int \\overline{ P^h_s f_h}(1-\\varrho _{g,h}^{-1})H_gP^g_s f_g {d}\\mu _h\\\\&=\\int h^*_{\\mathbb {C}}\\Big (\\mathrm {sgn}(\\widehat{S_{g,h}})(\\varrho _{g,h}{A}_{g,h})^{-{2}}|\\widehat{S_{g,h}}|^{{2}}|\\widehat{S_{g,h}}|^{{2}}{d}_h P^h_s f_h \\ , \\ {d}_g P^g_s f_g \\Big ) {d}\\mu _h\\\\&\\quad -\\int \\overline{ P^h_s f_h} \\cdot \\mathrm {sgn}(S_{g,h}) \\cdot \\varrho _{g,h}^{-{2}} \\cdot | S_{g,h}|^{{2}} \\cdot |S_{g,h}|^{{2}} \\cdot H_gP^g_s f_g {d}\\mu _h\\\\&=\\int h^*_{\\mathbb {C}}\\Big ({d}_h P^h_s f_h \\ , \\ \\Big [\\mathrm {sgn}(\\widehat{S_{g,h}})(\\varrho _{g,h}{A}_{g,h})^{-{2}}|\\widehat{S_{g,h}}|^{{2}}|\\widehat{S_{g,h}}|^{{2}}\\Big ]^{\\dagger _h}{d}_g P^g_s f_g \\Big ) {d}\\mu _h\\\\&\\quad -\\int \\overline{ P^h_s f_h} \\mathrm {sgn}(S_{g,h}) \\varrho _{g,h}^{-{2}} | S_{g,h}|^{{2}} |S_{g,h}|^{{2}} H_gP^g_s f_g {d}\\mu _h\\\\&=\\int h^*_{\\mathbb {C}} \\Big ({d}_h P^h_s f_h \\ , \\ |\\widehat{S_{g,h}}|^{{2}}\\mathrm {sgn}(\\widehat{S_{g,h}})(\\varrho _{g,h}{A}_{g,h})^{-{2}}|\\widehat{S_{g,h}}|^{{2}}{d}_g P^g_s f_g \\Big ) {d}\\mu _h\\\\&\\quad -\\int \\overline{ P^h_s f_h} | S_{g,h}|^{{2}} \\mathrm {sgn}(S_{g,h}) \\varrho _{g,h}^{-{2}} |S_{g,h}|^{{2}} H_gP^g_s f_g {d}\\mu _h\\\\&= \\left\\langle {d}_h P^h_s f_h \\ , \\ \\widehat{S_{g,h;h}} \\widehat{U_{g,h}} \\widehat{S_{g,h;g}}{d}_g P^g_s f_g \\right\\rangle _{\\Omega ^1_{L^2}(M,h)}\\\\&\\quad - \\left\\langle P^h_s f_h, S_{g,h;h} U_{g,h}S_{g,h;g} H_gP^g_s f_g \\right\\rangle _{L^2(M,h)}\\\\&= \\left\\langle f_h \\ , \\ ({d}_h P^h_s)^* \\widehat{S_{g,h;h}} \\widehat{U_{g,h}} \\widehat{S_{g,h;g}}{d}_g P^g_s f_g \\right\\rangle _{ L^2 (M,h)}\\\\&\\quad - \\left\\langle f_h, P^h_s S_{g,h;h} U_{g,h}S_{g,h;g} P^g_{s/2} H_g P^g_{s/2} f_g \\right\\rangle _{L^2(M,h)},$ proving the claimed formula.",
"We can now give the actual proof of Theorem REF : We are going to check the assumptions of Belopol'skii-Birman's Theorem (cf.",
"Theorem REF ): Firstly, by $g\\sim h$ , the operator $I:=I_{g,h}$ is well-defined and bounded, with a bounded inverse $I^{-1}= I_{h, g}$ .",
"The sesquilinear form corresponding to $H_j$ has domain $W^{1,2}_0(M,j)$ for $j=g,h$ , so that in view of (REF ), (REF ), (REF ), the assumption $g\\sim h$ also implies $IW^{1,2}(M,h)=W^{1,2}(M,h)$ .",
"Next, we claim that $( I^*I-1)P^g_s$ is Hilbert-Schmidt (and thus compact) for some $s>0$ .",
"Indeed, by (REF ) the operator $( I^*I-1)P^g_s$ has the integral kernel $\\left[(I^*I-1)P^g_s \\right](x,y)&=(\\varrho _{g,h}(x)-1)P^g_s (x,y)\\\\&=\\varrho _{g,h}^{-1/2}\\mathrm {sgn}(S_{g,h})|S_{g,h}|^{1/2}|S_{g,h}|^{1/2}P^g_s (x,y),$ so that using $\\int P^g_s (x,y)d\\mu _g(y)\\le 1$ we get the bounds $&\\int \\Big |\\left[(I^*I-1)P^g_s \\right](x,y)\\Big |^2{d}\\mu _g(y)\\\\&\\le \\left\\Vert \\varrho _{g,h}^{-1}\\, S_{g,h}\\right\\Vert _{\\infty } |S_{g,h}(x)| \\int P^g_s (x,y)^2d\\mu _g(y)\\\\&\\le \\left\\Vert \\varrho _{g,h}^{-1}\\, S_{g,h}\\right\\Vert _{\\infty } |S_{g,h}(x)| \\Psi _{4,g}(x,s)\\int P^g_s (x,y)d\\mu _g(y)$ for some $s>0$ .",
"Using (REF ) we arrive at the following Hilbert-Schmidt estimate, $&\\int \\int \\Big |\\left[(I^*I-1)P^g_s \\right](x,y)\\Big |^2{d}\\mu _g(y){d}\\mu _g(x)\\\\&\\lesssim \\int \\delta _{g,h}(x) \\Psi _{4,g}(x,s)d\\mu _{g}(x)\\\\&\\lesssim \\int \\delta _{g,h}(x) \\Psi _{4,g}(x,s) \\Psi _{3,g}d\\mu _{g}(x)<\\infty ,$ as $\\Psi _{4,j}(x,s)\\ge 1$ .",
"Next, as the product of Hilbert-Schmidt operators is compact, the HPW formula shows that it remains to show that the operators $\\widehat{S_{g,h;j}}\\widehat{P}^j_s$ and $S_{g,h;j} P^j_s$ are Hilbert-Schmidt, for $j=g,h$ .",
"To see this, as $S_{g,h;j} P^j_s$ has the integral kernel $\\left[S_{g,h;j} P^j_s\\right](x,y)=S_{g,h}(x) P^j_s(x,y),$ it follows as above that $\\int \\int \\Big |\\left[S_{g,h;j} P^j_s\\right](x,y)\\Big |^2{d}\\mu _g(y){d}\\mu _g(x)&=\\int \\int \\Big | S_{g,h}(x) P^j_s(x,y)\\Big |^2{d}\\mu _g(y){d}\\mu _g(x)\\\\&\\lesssim \\int \\delta _{g,h}(x) \\Psi _{4,j}(x,s)d\\mu _{j}(x)\\\\&\\lesssim \\int \\delta _{g,h}(x) \\Psi _{4,j}(x,s)\\Psi _{3,j}(x,s)d\\mu _{j}(x)<\\infty .$ Likewise, in order to prove the Hilbert-Schmidt property of $\\widehat{S_{g,h;j}}\\widehat{P}^j_s$ , we use Theorem REF : $\\int \\Big |\\widehat{P}^j_s(x,y)\\Big |^2_{j^*} d\\mu _j(y)\\le \\Psi _{3,j}(x,s) \\Psi _{4,j}(x,s) ,$ so that in view of (REF ) one has $\\int \\int \\Big |[\\widehat{S_{g,h;j}}\\widehat{P}^j_s](x,y)\\Big |^2_{j^*}d\\mu _j(y)d\\mu _j(x)&=\\int \\int \\Big |\\widehat{S_{g,h}}(x)\\widehat{P}^j_s(x,y)\\Big |^2_{j^*}d\\mu _j(y)d\\mu _j(x)\\\\&\\lesssim \\int \\delta _{g,h}(x)\\int \\Big |\\widehat{P}^j_s(x,y)\\Big |^2_{j^*}d\\mu _j(y)d\\mu _j(x)\\\\&\\le \\int \\delta _{g,h}(x) \\Psi _{3,j}(x,s) \\Psi _{4,j}(x,s) d\\mu _{_j}(x).$ This completes the proof."
],
[
"Belopol'skii-Birman theorem",
"We recall [9], [17] that given a self-adjoint operator $H$ in a Hilbert space ${H}$ with its operator valued spectral measure $E_H$ , one defines the $H$ -absolutely continuous subspace ${H}_{\\mathrm {ac}}(H)$ of ${H}$ to be the space of all $f\\in {H}$ such that the Borel measure $\\left\\Vert E_H(\\cdot )f\\right\\Vert ^2$ on $\\mathbb {R}$ is absolutely continuous with respect to the Lebesgue measure.",
"Then ${H}_{\\mathrm {ac}}(H)$ becomes a closed subspace of ${H}$ and the restriction $H_{\\mathrm {ac}}$ of $H$ to ${H}_{\\mathrm {ac}}(H)$ is a well-defined self-adjoint operator.",
"The absolutely continuous spectrum $\\mathrm {spec}_{\\mathrm {ac}}(H)$ of $H$ is defined to be the spectrum of $H_{\\mathrm {ac}}$ .",
"We record a version of the abstract Belopol'skii-Birman theorem for two Hilbert space scattering theory, which is well-suited for our purpose: Theorem 1.1 (Belopol'skii-Birman) For $k=1,2$ , let $H_k\\ge 0$ be a self-adjoint operator in a Hilbert space ${H}_k$ , where $E_{k}$ denotes the operator valued spectral measure, and and $P_{\\mathrm {ac}}(H_k)$ the projection onto the absolutely continuous subspace of ${H}_k$ corresponding to $H_k$ .",
"Assume that $I\\in {L}({H}_1, {H}_2)$ is a bounded operator such that the following assumptions hold: $I$ has a two-sided bounded inverse One has $(I^*I-1)P^g_sH_1) &\\in {J}^{\\infty }({H}_1)\\:\\:\\text{ (compact) for some $s>0$}.$ There exists an operator $T\\in {J}^1({H}_1, {H}_2)$ (trace class) and a number $s>0$ such that for all $f_2\\in \\mathrm {Dom}(H_2)$ , $f_1\\in \\mathrm {Dom}(H_1)$ one has $\\left\\langle f_2 ,Tf_1\\right\\rangle _{{H}_2}\\:=\\:&\\left\\langle H_2f_2, \\exp (-sH_{2}) I \\exp (-sH_{1})f_1\\right\\rangle _{{H}_2} \\\\&-\\left\\langle f_2, \\exp (-sH_{2}) I \\exp (-sH_{1}) H_1f_1\\right\\rangle _{{H}_2}.$ One has either $I\\mathrm {Dom}(\\widehat{P}_1)=\\mathrm {Dom}(\\widehat{P}_2)$ or $I\\mathrm {Dom}(H_1)=\\mathrm {Dom}(H_2)$ .",
"Then the wave operators $W_{\\pm }(H_{2},H_1, I)=\\mathop {\\rm {st}\\rule [0.5ex]{1ex}{0.1ex}\\mathrm {lim}}_{t\\rightarrow \\pm \\infty }\\exp (itH_{2})I\\exp (-itH_{1})P_{\\mathrm {ac}}(H_1)$ exist$\\mathop {\\rm {st}\\rule [0.5ex]{1ex}{0.1ex}\\mathrm {lim}}_{t\\rightarrow \\pm \\infty }$ stands for the strong limit.",
"and are complete, where completeness means that $\\left(\\mathrm {Ker} \\: W_{\\pm }(H_{2},H_1, I)\\right)^{\\perp }=\\mathrm {Ran}\\: P_{\\mathrm {ac}}(H_1), \\quad \\overline{\\mathrm {Ran} \\: W_{\\pm }(H_{2},H_1, I)}=\\mathrm {Ran}\\: P_{\\mathrm {ac}}(H_2).$ Moreover, $W_{\\pm }\\big (H_{2},H_1, I\\big )$ are partial isometries with inital space $\\mathrm {Ran} \\: P_{\\mathrm {ac}}(H_1)$ and final space $\\mathrm {Ran} \\: P_{\\mathrm {ac}}(H_2)$ , and one has $\\mathrm {spec}_{\\mathrm {ac}}(H_1)=\\mathrm {spec}_{\\mathrm {ac}}(H_2)$ .",
"In view of Theorem XI.13 from [9] and its proof, it remains to show that for every bounded interval $\\mathbb {I}$ the operator $(I^*I-1)E_1(\\mathbb {I})$ is compact, and that there exists a trace class operator $D\\in {J}^1({H}_1, {H}_2)$ such that for every bounded interval $\\mathbb {I}$ and all $f_1,f_2$ as above one has $\\left\\langle \\varphi ,Df_1\\right\\rangle _{{H}_2}=\\left\\langle H_2f_2, E_2(\\mathbb {I})IE_1(\\mathbb {I})f_1\\right\\rangle _{{H}_2}-\\left\\langle f_2, E_2(\\mathbb {I})IE_1(\\mathbb {I})H_1f_1\\right\\rangle _{{H}_2}.$ However, using that for all self-adjoint operators $A$ and all Borel functions $\\phi ,\\phi ^{\\prime }{}: \\mathbb {R}\\rightarrow \\mathbb {C}$ one has $\\phi (A)\\phi ^{\\prime }{}(A)\\subset (\\phi \\cdot \\phi ^{\\prime }{}) (A),\\quad \\mathrm {Dom}(\\phi (A)\\phi ^{\\prime }{}(A))=\\mathrm {Dom}(\\phi (A)\\phi ^{\\prime }{}(A))\\cap \\mathrm {Dom}(\\phi ^{\\prime }{}(A)),$ the required compactness becomes obvious, and furthermore it is easily justified that $D:=\\exp (sH_2)E_2(\\mathbb {I})T\\exp (sH_1)E_1(\\mathbb {I})$ has the required trace class property.",
"The second author has been supported by the Fonds National de la Recherche Luxembourg (FNR) under the OPEN scheme (project GEOMREV O14/7628746)."
]
] | 1709.01612 |
[
[
"Metric in the moduli of SU(2) monopoles from spectral curves and\n Gauss-Manin connection in disguise"
],
[
"Abstract We show here that from the metric of the manifold $M^0_2$ , i.e., the reduced moduli of SU(2) 2-monopoles in Yang-Mills-Higgs theory, one can recover the respective moduli of spectral curves using the method Gauss-Manin connection in disguise.",
"This work is a step towards creating a inverse process of finding the metric of any $M^0_k$ , from spectral curves.",
"This is a thirty years old problem that we hope to shed some light in it."
],
[
"Introduction",
"The study of instantons and monopoles in three and four dimension are among the most comprehensive research areas at the interface of physics and mathematics.",
"From the physics point of view, they relate to solitons, dualities and non-perturbative Yang-Mills theories.",
"From the mathematical point of view, it involves knowledge of Analysis, Differential Geometry, Algebraic Geometry and Twistor theory.",
"In this work, we add elements of Hodge theory to relate the metric of the moduli space of charge k monopoles $M^0_k$ and its spectral curves in SU(2) Yang-Mills-Higgs (YMH) theory in three spacial dimensions (static monopoles).",
"YMH monopoles in three dimensions are equivalent to instanton solutions of Yang-Mills theory in Euclidian four dimensions constrained by the fact that gauge fields do not depend on the fourth direction.",
"In this equivalence, the Higgs field is the fourth component of the gauge field in four dimensions.",
"In this way, the twistor methods applied in instantons were also adapted to YMH monopoles [1].",
"In a recent work [2], while revising the metric of $M^0_2$ reduced moduli of 2-monopoles, the present author noticed that the moduli of enhanced elliptic curves obtained from the self-dual metric should correspond to the moduli of spectral curves of 2-monopoles in its enhanced version.",
"Notice[1] that the spectral curve of a $k$ -monopole is an algebraic curve of genus $(k-1)^2$ and for $k$ =2, the spectral curve is an elliptic curve [3].",
"The moduli of enhanced elliptic curves appears from the metric of $M^0_2$ using the method Gauss-Manin connection in disguise developed by Movasati[4], [5], [6] which shows that the set of Darboux-Halphen differential equations obtained from self-duality of the metric of $M^0_2$ are a vector field in the moduli of an enhanced elliptic curve.",
"In here, the guess made in our previous collaboration[2] is proved.",
"In sections and we review basic elements of SU(2) monopoles and spectral curves following closely some original articles [7], [1], [3] and reference books [8], [9].",
"In section we quickly review the program Gauss-Manin connection in disguise for elliptic curves[6] and in section we show how the moduli of spectral curves of 2-monopoles emerge from the metric of $M^0_2$ .",
"In we summarize this article and comment about the cases $k>2$ and the issue of different parametrization of universal families of curves and the weights of the respective set of modular-type functions attached to such curves."
],
[
"A $k$ -monopole or BPS-monopole of charge $k$ in Yang-Mills-Higgs theory is a static soliton in $\\mathbb {R}^3$ that is a solution of the Bogomolny equation [10]: $F=\\star D \\phi , \\qquad \\textnormal {with}\\\\F:=\\mathrm {d}{A} + {A}\\wedge {A}\\quad \\textrm {and}\\quad D:= \\mathrm {d}+{A},\\nonumber $ where ${A}$ is the gauge field or connection form on a principal SU(2)-bundle over $\\mathbb {R}^3$ , $F$ is its curvature 2-form or field strength, $D$ is the covariant exterior derivative or connection and the Higgs field $\\phi $ is a section of the associated $\\mathfrak {su}(2)$ -bundle.",
"$\\star $ is the Hodge dual operation and the Bogomolny equation (REF ) is part of the self-duality equations for the related instantons in four dimensions.",
"The monopole solution has also to satisfy the finite action condition $\\int |F|^2<\\infty $ and the boundary condition $|\\phi |=1-\\frac{k}{2r}+ O(r^{-2})\\quad \\text{as}\\;\\;r\\rightarrow \\infty ,$ where the charge $k$ is an integer number.",
"A clever treatment of monopoles was given by Hitchin [1] where he applied the twistor methods in the space $\\mathbf {\\tilde{T}}$ of oriented straight lines (geodesics) in $\\mathbb {R}^3$ .",
"$\\mathbf {\\tilde{T}}$ has a holomorphic structure given by cross product and $\\mathbf {\\tilde{T}}\\equiv \\mathbf {T}\\mathbb {P}_1(\\mathbb {C})$ the holomorphic tangent bundle of the projective line.",
"Then the solutions of Bogomolny equations were restated in terms of complex geometry of $\\mathbf {\\tilde{T}}$ where the spectral curves were introduced [1], [11].",
"First consider $E$ the rank 2 complex vector bundle on $\\mathbb {R}^3$ associated to the principal SU(2) bundle.",
"Now one defines a rank 2 complex vector bundle $\\tilde{E}$ by defining at each point $z\\in \\mathbf {\\tilde{T}}$ a fiber $E_z$ .",
"To each point $z\\in \\mathbf {\\tilde{T}}$ there is a corresponding oriented line $l_z \\in \\mathbb {R}^3$ .",
"$E_z$ is given by the the space of sections $s$ of E with support on $l_z$ such that $(u^jD_j-i\\phi )s=0,$ $u$ is the unit tangent vector pointi ng (in the positive direction) along the oriented line $l_z$ .",
"It follows from Bogomolny equations that $\\tilde{E}$ has a natural holomorphic structure [1].",
"Conversely, from a holomorphic vector bundle $\\tilde{E}$ on $\\mathbf {\\tilde{T}}$ one reconstructs the solution $(A,\\phi )$ to Bogomolny equations.",
"But not all section $s$ of $\\tilde{E}_l$ satisfy the boundary conditions, which is the vanishing of $s$ at both ends of $l$ ."
],
[
"Spectral curve of a k-monopole",
"For each oriented line $l\\in \\mathbb {R}^3$ , the space of solutions (REF ) which decay at $+\\infty $ is one-dimensional.",
"This space is a holomorphic line bundle and a subbundle of $\\tilde{E}_l$ and it belongs to a class of ansätze $\\mathcal {A}_k$ according to the charge $k$ of the monopole [7].",
"Furthermore, the set of lines for which equation (REF ) has a solution decaying to zero at both ends forms a compact algebraic curve $S$ in $\\mathbf {T}\\mathbb {P}_1(\\mathbb {C})$ .",
"$S$ is called the spectral curve and it has genus $(k-1)^2$ .",
"Inhomogeneous coordinates $(\\eta ,\\zeta ) $ on $\\mathbb {P}_1(\\mathbb {C})$ gives local coordinates on $\\mathbf {\\tilde{T}}: (\\eta ,\\zeta ) \\rightarrow \\eta \\partial /\\partial \\zeta $ .",
"$S$ in terms of such local coordinates is given by $p(\\eta ,\\zeta )=\\eta ^k+a_1(\\zeta )\\eta ^{k-1}+\\dots a_k(\\zeta )=0,$ where $a_i(\\zeta )$ is a polynomial of degree 2i.",
"The polynomial $p(\\eta ,\\zeta )$ is preserved by an antiholomorphic involution $\\tau (\\eta ,\\zeta )= (-\\overline{\\eta }/\\overline{\\zeta }^2,-\\overline{\\zeta }^{-1})$ , a real structure on $\\mathbf {T}\\mathbb {P}_1(\\mathbb {C})$ .",
"Therefore $p(\\eta ,\\zeta )$ depends on $(k+1)^2-1$ real parameters.",
"Since $S$ is constrained by its genus (transcendental or ES constraint[12]), the parameter space has dimension $(k+1)^2-1-(k-1)^2= 4k-1.$ This is the dimension of the moduli space of k-monopoles $M_k$ .",
"Out of these parameters, the center of mass position of a k-monopole in $\\mathbb {R}^3$ can be translated to the origin and the remaining parameter space corresponds to the reduced moduli space $M^0_k$ with dimension $4k-4$ ."
],
[
"Spectral curve for k= 2",
"This case was extensively studied by Hurtubise [3].",
"The spectral curve $S$ is an elliptic curve of genus 1.",
"The real structure of $S$ imposes via Weierstrass $\\mathfrak {p}$ -function the complex structure $\\tau $ of the corresponding torus $\\mathbb {C}/\\Lambda $ to be purely imaginary and its corresponding lattice $\\Lambda $ to be rectangular.",
"Factoring out six parameters from translation action and $SO(3)$ action from the polynomial (REF ) for the spectral curve of 2-monopoles, two real parameters remain: $\\eta ^2= r_1\\zeta ^3- r_2\\zeta ^2-r_1\\zeta , \\;r_i\\in \\mathbb {R}, r_1\\ge 0 $ The genus constraint is enforced by matching the above equation to a normal form of the elliptic curve.",
"First notice that when $r_1=0$ , the spectral curve degenerates to two $k=1$ spectral curves.",
"$\\eta =i\\frac{\\pi }{2}\\zeta \\quad \\textnormal {and} \\quad \\eta =-i\\frac{\\pi }{2}\\zeta .$ In this case $r_2=\\pi ^2/4$ , as we show below (REF ), and there is no free parameter.",
"This is the case[13], [14] where a 2-monopole is simply a superposition of two 1-monopoles both centered at the origin of $\\mathbb {R}^3$ and it agrees with the fact that the dimension of the reduced moduli $M^0_1$ is zero.",
"The spectral curves (REF ) are two complex lines tangent to two different points in $\\mathbb {P}_1\\equiv S^2$ .",
"The symmetries of the spectral curve determines a symmetry of the monopole.",
"In this case, the isotropy group $S^1\\times \\mathbb {Z}_2$ of the two $k=1$ spectral curves corresponds to the axial symmetry of the two 1-monopoles solution and the exchanging the two 1-monopoles.",
"For $r_1>0$ , the spectral curve can be reparametrized to: $\\tilde{\\eta }^2= 4\\tilde{\\zeta }^3-g_2(\\Lambda )\\tilde{\\zeta }-g_3(\\Lambda ), \\;\\textrm {where},\\\\ \\nonumber \\tilde{\\eta }=\\eta (4/r_1)^{1/2},\\quad \\tilde{\\zeta }=\\zeta -\\tfrac{r_2}{3r_1}, \\\\\\nonumber g_2(\\Lambda )=60 G_4(\\Lambda )= 12(r_2/3r_1)^2+4\\quad \\textrm {and},\\\\g_3(\\Lambda )=140 G_6(\\Lambda )=8(r_2/3r_1)^3+4(r_2/3r_1).$ and $G_4$ and $G_6$ are Eisenstein series of weight 4 and 6, respectively, functions of the retangular lattice $\\Lambda $ with real generator $l_r=\\sqrt{4r_1}$ and imaginary generator $l_i$ .",
"A homothetic scaling of the lattice transform $g_2$ and $g_3$ : $g_i(m\\Lambda )=m^{-2i}g_i(\\Lambda ),\\, i=2,3,\\quad m\\in \\mathbb {R}^*$ and the polynomial (REF ) is preserved if we reparametrize $(\\tilde{\\eta },\\tilde{\\zeta })$ to absorb such scaling $\\tilde{\\eta }\\rightarrow m^{-3}\\tilde{\\eta }\\quad \\textnormal {and}\\quad \\tilde{\\zeta }\\rightarrow m^{-2}\\tilde{\\zeta }.$ Therefore we should consider modular functions such as $I= 27g_3^2/g_2^3$ .",
"This function will be invariant to scaling of the lattice $\\Lambda $ and it will only depend on the ratio of the generators $\\tau =l_i/l_r$ , a purely imaginary number.",
"To make this dependency explicit we can reparametrize the variables as above with $m= l_r^{-1}$ and obtain $g_2(\\tau )= 16r_1^2g_2(\\Lambda )=60G_4(\\tau )$ and $g_3(\\tau )= (4r_1)^3g_3(\\Lambda )=140G_6(\\tau )$ .",
"In terms of the normalized Eisenstein series $E_{2i}$ and Riemann zeta functions, $G_{2i}(\\tau )=2\\zeta (2i)E_{2i}(\\tau )$ with, $E_{2i}(q):=1+b_i\\sum _{n=1}^\\infty \\left(\\sum _{d\\mid n}d^{2i-1}\\right)q^{n},\\ \\ i=1,2,3,$ with $(b_1,b_2,b_3)=(-24, 240, -504)$ and $q= e^{i2\\pi \\tau },\\;\\tau \\in \\mathbb {H}=\\lbrace \\tau \\in \\mathbb {C}|\\Im (\\tau )\\ge 0)$ for convergency of the series.",
"When $\\tau $ is purely imaginary, $E_{2i}(\\tau )$ take values in $\\mathbb {R}$ .",
"From (REF ), $I$ depends on the ratio $(r_1/r_2)^2$ .",
"Notice that in the limit $r_1\\rightarrow 0$ , the discriminant of the elliptic curve $\\Delta = g_2^3-27g_3^2=0$ and $I(\\tau )=1$ .",
"This corresponds to the limit $\\tau \\rightarrow i \\infty $ .",
"In order to proceed showing that $r_2=\\pi ^2/4$ in this limit, we explore $I$ near 1.",
"$\\frac{1-I(\\tau )}{27}= \\frac{64}{j(\\tau )}= 2^{12}3^3(q-744q^2+\\dots )\\quad \\textnormal {where,} \\\\\\nonumber j(\\tau )=\\frac{1728g_2^3}{\\Delta }\\;\\textnormal {is the Klein modular function }.$ From (REF ), $\\frac{1-I}{27}=\\frac{r^4(\\frac{1}{4}+r^2)}{(1+3r^2)^3},\\quad \\textnormal {with } r=\\frac{r_1}{r_2}.$ We see that the limit $r_1,r\\rightarrow 0$ coincides with $\\tau \\rightarrow i\\infty $ or $ q\\rightarrow 0$ .",
"Near this limit, we keep only the first term of the Eisenstein series $G_4(\\tau )$ .",
"From (REF ): $g_2(\\Lambda )= \\frac{1}{16 r_1^2}\\frac{64}{3}r_2^2(1+3r^2)=\\frac{1}{16 r_1^2}60G_4(\\tau )\\xrightarrow{}\\frac{1}{16 r_1^2} \\frac{(2\\pi )^4}{12}.\\\\\\textnormal {Therefore,}\\quad r_2\\xrightarrow{} \\pi ^2/4.$ Hence, the point $(r_1,r_2)=(0,\\pi ^2/4)$ corresponds to the singular point $(\\Delta =0)$ of the real elliptic curve $S$ factored out by $SO(3)$ action and $\\mathbb {R}^3$ translations, corresponding to $\\tau =i\\infty $ .",
"The total space of parameters has one real dimension and it corresponds to purely imaginary $\\tau , \\, -i\\tau \\in \\mathbb {R}_{\\ge 0}\\cup \\infty $ ."
],
[
"Gauss Manin connection in disguise",
"In [15], [16] Movasati realized that the Ramanujan relations between Eisenstein series can be computed using the Gauss-Manin connection of families of elliptic curves.",
"Later in a private communication, Pierre Deligne called \"Gauss-Manin connection in disguise\" the vector field that best express the property of Griffths transversality[17], [18] of a Gauss-Manin connection.",
"Since then the method Gauss-Manin connection in disguise has been applied in many families of algebraic curves and relating them to differential equations and automorphic forms or modular-type functions [4], [19], [20], [21].",
"Our interest are in finding differential equations in the universal families of spectral curves of $k$ -monopoles.",
"The method developed for the elliptic curve [16], [6] still need to be thought through for spectral curves because of the reality condition on the spectral curves, but our general argument is that the reality condition is lifted for the sake of finding the Gauss-Manin connection and the respective vector field and later the reality condition is imposed on the domain of solutions of the vector field equations.",
"We present here the two known cases of families of enhanced elliptic curves which correspond to geometric expressions of Ramanujan and Darboux-Halphen differential equations.",
"A good review is in Movasati's lectures[6].",
"In both cases, the idea is to define the moduli of enhanced elliptic curve by including information about its Hodge structure.",
"Then, one calculates its Gauss-Manin connection and finds the appropriate vector field."
],
[
"Ramanujan differential equations",
"We extend the one-parameter family of elliptic curves (REF ).",
"Recall that the first de Rham cohomology $H^1_{\\rm dR}(E)$ of an elliptic curve $E$ is a two-dimensional vector space.",
"The moduli $\\mathtt {T_R}$ of pairs $(E,[\\alpha , \\omega ])$ , where $\\alpha , \\omega \\in H^1_{\\rm dR}(E)$ are a basis of the cohomology classes of 1-forms in $E$ with $\\alpha $ a regular differential 1-form on $E$ and $\\omega $ such that $\\langle \\alpha ,\\omega \\rangle =1$ .",
"In a equivalent way, $\\mathtt {T_R}$ can be defined as the moduli of pairs$(E,[\\omega ])$ , $\\omega \\in H^1_{\\rm dR}(E)\\backslash F^1$ and there is a unique regular 1-form $\\alpha $ in the Hodge filtration $ F^1\\subset H^1_{{\\rm dR}}(E)$ such that $\\langle \\alpha ,\\omega \\rangle =1$ .",
"Therefore $\\mathtt {T_R}$ is a three-dimensional space and it has a corresponding universal family of elliptic curves $E_t: y^2=4(x-t_1)^3-t_2(x-t_1)-t_3,$ with $\\alpha = [\\frac{dx}{y}],\\; \\omega =[\\frac{xdx}{y}]$ and the moduli $\\mathtt {T_R}$ can be expressed as $\\mathtt {T_R}:=\\lbrace (t_1,t_2,t_3)\\in \\mathbb {C}^3| \\Delta =t_2^3-27t_3^2\\ne 0\\rbrace .$ The Gauss-Manin connection of the above family $E_t$ , written in the basis $(\\alpha ,\\omega )$ is given by $\\nabla \\begin{pmatrix}\\alpha \\\\ \\omega \\end{pmatrix}=A\\begin{pmatrix}\\alpha \\\\ \\omega \\end{pmatrix},$ where $A=\\frac{1}{\\Delta }\\begin{pmatrix}-\\tfrac{3}{2}t_1\\beta -\\tfrac{1}{12}d\\Delta &\\tfrac{3}{2}\\beta \\\\\\Delta dt_1-\\tfrac{1}{6}t_1d-(\\tfrac{3}{2}t_1^2+\\tfrac{1}{8}t_2)\\beta &\\tfrac{3}{2}t_1\\beta \\Delta +\\tfrac{1}{12}d\\Delta \\end{pmatrix},\\\\ \\nonumber \\beta = 3t_3dt_2-2t_2dt_3.$ In $\\mathtt {T_R}$ there is a unique vector field $R$ such that[6] $\\nabla _R(\\alpha )=-\\omega ,\\quad \\nabla _R(\\omega )= 0.$ The vector field $R$ is given by the Ramanujan differential equations[22] $\\left\\lbrace \\begin{aligned}\\frac{\\partial t_1}{\\partial \\tau }&=t_1^2-\\tfrac{1}{12}t_2,\\\\\\frac{\\partial t_2}{\\partial \\tau }&=4t_1t_2-6t_3,\\\\\\frac{\\partial t_3}{\\partial \\tau }&=6t_1t_3-\\tfrac{1}{3}t_2^2,\\end{aligned}\\right.$ where $\\tau $ is a direction in the moduli $\\mathtt {T_R}$ chosen by $R$ .",
"$R$ has been called Ramanujan vector field, modular vector field and lately, Gauss-Manin connection in disguise.",
"It may seem that in this process the moduli was not only enhanced but also enlarged since the moduli of elliptic curves has 1 complex dimension while $\\dim (\\mathtt {T_R})=3$ .",
"But if we look to the solution of (REF ), $(t_1(\\tau ),t_2(\\tau ),t_3(\\tau )):=(\\frac{2\\pi i}{12}E_2(\\tau ),\\ 12(\\frac{2\\pi i}{12})^2E_4(\\tau ), 8(\\frac{2\\pi i}{12})^3E_6(\\tau )),$ $E_4$ and $E_6$ are modular forms of weight $k=4, 6$ , respectively, $E_k(\\dfrac{a\\tau +b}{c\\tau +d})=({c\\tau +d})^kE_k(\\tau ),\\; \\begin{pmatrix} a&b\\\\c&d\\end{pmatrix}\\in SL(2,\\mathbb {Z}),$ and $E_2$ is a quasi-modular form of weight 2: $E_2(\\dfrac{a\\tau +b}{c\\tau +d})=({c\\tau +d})^2E_2(\\tau )+ \\frac{12}{2\\pi \\mathrm {i}}c(c\\tau +d) .$ Therefore, $R$ vector field corresponds to a map of $\\tau \\in \\mathbb {H}$ to $ (t_1,t_2,t_3)\\in \\mathtt {T_R}$ .",
"The transformation of $E_2, E_4,$ and $E_6$ , under the action of the modular group $SL(2,\\mathbb {Z})$ on $\\tau $ , preserving the lattice that defines the elliptic curve $E_t\\equiv \\mathbb {C}^2/\\Lambda _{\\tau }$ , reveals that there is a group of isomorphisms ${G}$ that acts on $\\mathtt {T_R}$ .",
"The quotient moduli $\\mathtt {T_R}/{G}$ has one complex dimension.",
"For $g\\in {G}$ , $ [\\alpha ,\\omega ]&\\xrightarrow{}&[\\alpha ,\\omega ]g=[c\\alpha ,c^{\\prime }\\alpha +\\omega /c]\\\\t=(t_1,t_2,t_3)&\\xrightarrow{}&t^{\\prime }=(c^{-2}t_1+c^{\\prime }/c,c^{-4}t_2,c^{-6}t_3)\\\\(x,y)&\\xrightarrow{}&(c^{-2}x+c^{\\prime }/c,c^{-3}y)\\\\{E_t}&\\equiv &{E_{t^{\\prime }}} $ $g=\\begin{pmatrix}c&c^{\\prime }\\\\0&c^{-1}\\end{pmatrix}=\\begin{pmatrix}c&0\\\\0&c^{-1}\\end{pmatrix}\\begin{pmatrix}1&c^{\\prime }/c\\\\0&1\\end{pmatrix},\\quad c\\in \\mathbb {C}^*, c^{\\prime }\\in \\mathbb {C}$ Notice that this group action preserves the intersection form $\\langle \\alpha ,\\omega \\rangle =\\langle c\\alpha ,c^{\\prime }\\alpha +\\omega /c\\rangle =1$ ."
],
[
"Darboux-Halphen differential equations",
"In this case, the enhanced elliptic curve is given by a triple $(E,(P,Q),\\omega )$ , where $E$ is an elliptic curve, $\\omega \\in H^1_{\\rm dR}(E)\\backslash F^1$ , and $P$ and $Q$ are a pair of points of $E$ that generates the 2-torsion subgroup with the Weil pairing $e(P,Q)=-1$ .",
"The points $P$ and $Q$ are given by $(T_1,0)$ and $(T_2,0)$ .",
"In here, the torsion data is necessary because the modular group , or group of lattice equivalence, of this enhanced curve is the congruence subgroup $\\Gamma (2)\\subset SL_2(\\mathbb {Z})$ , which has index $\\left[SL_2(\\mathbb {Z}):\\Gamma (2)\\right]=6$ .",
"The torsion data choose one out of six enhanced elliptic curves with same $(E,\\omega )$ pairs.",
"For each choice of $\\omega $ , there is a unique regular differential 1-form in the Hodge filtration $ \\omega _1\\in F^1$ , such that $\\langle \\omega ,\\omega _1\\rangle =1$ and $\\omega ,\\, \\omega _1$ together form a basis of $H^1_{\\rm dR}(E)$ .",
"The corresponding universal family of elliptic curves is given by $E_T:\\ \\ y^2-4(x-T_1)(x-T_2)(x-T_3)=0,\\\\\\nonumber \\textnormal {and moduli } \\mathtt {T_H}=\\lbrace (T_1,T_2,T_3)\\in \\mathbb {C}^3|\\, T_1\\ne T_2\\ne T_3\\rbrace .$ In fact, this universal family patches together all six enhanced elliptic curves, separated by singularity borders $T_i=T_j$ , due its symmetry under permutation of $T_1, T_2$ and $T_3$ .",
"Hence, it is a six-fold cover of the enhanced elliptic curve $(E,\\omega )$ that is isomorphic to the enhanced elliptic curve $(E,[\\alpha ,\\omega ])$ for the full modular group $SL_2(\\mathbb {Z})$ .",
"The Gauss-Manin connection of the family of elliptic curves $E_T$ written in the basis $\\frac{dx}{y},\\ \\frac{xdx}{y}$ is given as bellow: $\\nabla \\begin{pmatrix}\\frac{dx}{y}\\\\ \\frac{xdx}{y}\\end{pmatrix}=A\\begin{pmatrix}\\frac{dx}{y}\\\\ \\frac{xdx}{y}\\end{pmatrix}$ where $A &=&\\frac{dT_1}{2(T_1-T_2)(T_1-T_3)}\\begin{pmatrix}-T_1 & 1 \\\\T_2T_3-T_1(T_2+T_3) & T_1\\end{pmatrix}+\\\\& &\\frac{dT_2}{2(T_2-T_1)(T_2-T_3)}\\begin{pmatrix}-T_2 & 1 \\\\T_1T_3-T_2(T_1+T_3) & T_2\\end{pmatrix}+\\\\& & \\frac{dT_3}{2(T_3-T_1)(T_3-T_2)}\\begin{pmatrix}-T_3 & 1 \\\\T_1T_2-T_3(T_1+T_2) & T_3\\end{pmatrix}.$ In the parameter space of the family of elliptic curves $E_T$ there is a unique vector field $H$ , such that $\\nabla _{H}(\\frac{dx}{y})= -\\frac{xdx}{y},\\ \\nabla _{H}(\\frac{xdx}{y})= 0.$ The vector field $H$ is given by the Darboux-Halphen differential equation $\\rm \\left\\lbrace \\begin{aligned}\\frac{\\partial T_1}{\\partial \\tau }=T_1(T_2+T_3)-T_2T_3,\\\\\\frac{\\partial T_2}{\\partial \\tau }= T_2(T_1+T_3)-T_1T_3, \\\\\\frac{\\partial T_3}{\\partial \\tau }= T_3(T_1+T_2)-T_1T_2.\\end{aligned}\\right.$ where $\\tau $ is a direction in $\\mathtt {T_H}$ chosen by $H$ .",
"This vector field $H$ has been called Darboux-Halphen vector field, and lately, Gauss-Manin connection in disguise.",
"Similarly to the previous subsection, there is a group of isomorphism $G^{\\prime }$ in $\\mathtt {T_H}$ with two generators ( addictive and multiplicative) and the quotient $\\mathtt {T_H}/G^{\\prime }$ has one complex dimension.",
"For $g\\in {G^{\\prime }}$ , $T=(T_1,T_2,T_3)&\\xrightarrow{}&T^{\\prime }=(c^{-2}T_1+c^{\\prime }/c,c^{-2}T_2+c^{\\prime }/c,c^{-2}T_3+c^{\\prime }/c),\\\\(x,y)&\\xrightarrow{}&(c^{-2}x+c^{\\prime }/c,c^{-3}y),\\\\{E_T}&\\equiv &{E_{T^{\\prime }}}, \\qquad c\\in \\mathbb {C}^*,\\; c^{\\prime }\\in \\mathbb {C}$"
],
[
"$\\mathtt {T_R}$ and {{formula:da7b096d-9d39-4b1e-a6af-d6a4e8a0a094}}",
"There is a algebraic morphism between the moduli $f: \\mathtt {T_H}\\longrightarrow \\mathtt {T_R}$ given by a match between the elliptic curves (REF ) and (REF ): $(T_1,T_2,T_3)\\rightarrow (T, -4\\sum _{1\\le i< j\\le 3}(T-T_i)(T-T_j),4(T-T_1)(T-T_2)(T-T_3)),\\\\\\nonumber \\textnormal {where }\\quad T=\\frac{T_1+T_2+T_3}{3}$ Since the permutations of $T_1, T_2$ and $T_3$ in $\\mathtt {T_H}$ are mapped to the same point in $\\mathtt {T_R}$ , $f$ is a six to one map, but if we restrict to the region $|T_1|<|T_2|<|T_3|$ in $\\mathtt {T_H}$ , $f$ is an isomorphism."
],
[
"From the metric of the moduli space of 2-monopoles to the spectral curve",
"In their book[8], Atiyah and Hitchin showed that the reduced moduli $M^0_2$ of 2-monopoles is a four dimensional hyperkähler manifold and an anti-self-dual Einstein manifold.",
"Since $M^0_2$ admits $SO(3)$ isometry, the metric is a Bianch IX [23].",
"This is consequence of the hyperkähler structure of $M^0_2$ which has an $S^2$ -parameter family of complex structures: if I, J, K are covariant constant complex structures in $M^0_2$ then $aI+bJ+cK$ is also a covariant constant complex structure in $M^0_2$ given that $a^2+b^2+c^2=1,\\, (a,b,c)\\in \\mathbb {R}^3$ .",
"The SO(3) isometry rotates this $S^2$ in a standard way.",
"Following Atiyah and Hitchin[8] and our review[2], the 4-dimensional Bianchi IX metric is cast in the form $ ds^2 = (abc)^2 d\\rho ^2 + a^2 (\\sigma _1)^2 + b^2(\\sigma _2)^2 + c^2 (\\sigma _3)^2,$ where $a, b, c$ are real functions of $\\rho $ which parametrizes each SO(3) orbit in $M^0_2$ or, in other words, it parametrizes trajectories orthogonal to these orbits in $M^0_2$ .",
"The $SO(3)$ invariant 1-forms $\\sigma _i$ are dual to the standard basis $X_1, X_2, X_3$ of its Lie algebra.",
"They obey the structure equation $d\\sigma _i=-\\sigma _j\\wedge \\sigma _k,$ for all cyclic permutations $(i, j, k)$ of $(1, 2, 3)$ .",
"The self-duality equations lead to the following equation $\\frac{2}{a}\\frac{da}{d\\rho }= b^2+c^2-a^2- 2bc,$ and two other equations obtained by cyclic permutations of (a,b,c).",
"Upon reparametrization $a^2=\\frac{\\Omega _2\\Omega _3}{\\Omega _1},\\quad b^2=\\frac{\\Omega _3\\Omega _1}{\\Omega _2},\\quad c^2=\\frac{\\Omega _1\\Omega _2}{\\Omega _3}.$ we obtain from (REF ) the three Darboux-Halphen differential equations: $\\dot{\\Omega }_i= \\Omega _i(\\Omega _j+\\Omega _k)-\\Omega _j\\Omega _k,$ where (i,j,k) run over cyclic permutations of (1,2,3) and the derivative (denoted by dot) is with respect to $\\rho $ , a real parameter.",
"When we put together the fact that the $k=2$ spectral curve corresponds to an elliptic curve with purely imaginary $\\tau $ and consequently, real valued Eisenstein series $E_{2i}(\\tau )$ , we conclude that the solution (REF ) of Ramanujan equations (REF ) $(t_1(\\tau ),t_2(\\tau ),t_3(\\tau ))$ with purely imaginary $t_1$ and $t_3$ will only match $(\\Omega _1,\\Omega _2,\\Omega _3)\\in \\mathbb {R}^3$ via $f$ morphism (REF ) if, $\\tau =i\\rho \\quad \\textnormal {and}\\quad \\Omega _j=iT_j$ From the discussion in REF , the space $\\mathtt {T_\\Omega }:=\\lbrace (\\Omega _1,\\Omega _2,\\Omega _3)\\in \\mathbb {R}^3| \\Omega _1\\ne \\Omega _2\\ne \\Omega _3\\rbrace ,$ corresponds to a section of purely imaginary coordinates $(T_1,T_2,T_3)$ of the moduli space $\\mathtt {T_H}$ of the enhanced elliptic curve $E_T$ .",
"As shown below, this sectioning corresponds to the reality condition of the spectral curve.",
"In $\\mathbf {\\tilde{T}}$ , the reality condition is given by a real structure $\\varsigma $ that acts by reverting the orientation of the corresponding lines in $\\mathbb {R}^3$ [1].",
"In terms of $\\mathbf {\\tilde{T}}$ coordinates, it corresponds to $\\varsigma (\\eta ,\\zeta )= (-\\overline{\\eta }/\\overline{\\zeta }^2,-1/\\overline{\\zeta })$ .",
"Hence, a real curve in $\\mathbf {\\tilde{T}}$ is invariant under $\\varsigma $ as in (REF ).",
"For the reparameterized spectral curve (REF ) this means that the coefficients $g_2, g_3$ of the defining polynomial are real.",
"Looking at their values in terms of Eisenstein series and using the group $G$ of isomorphisms (REF ) of the moduli $\\mathtt {T_R}$ with $c= (-2\\pi i)^{1/2}$ , we see that these real coordinates correspond to $g_2(\\tau ) = (-2\\pi i)^2 t_2(\\tau ), \\qquad g_3(\\tau )=(-2\\pi i)^3t_3(\\tau ),$ where $t_i(\\tau )$ are the solution (REF ) of the Ramanujan equations (REF ).",
"By $G$ isomorphism $(0,g_2(\\tau ),g_3(\\tau ))\\equiv (t_1(\\tau ),t_2(\\tau ),t_3(\\tau ))$ in $\\mathtt {T_R}$ , where an additive transformation of $G$ sets the first coordinate to zero.",
"The values of $t_i(\\tau )$ in (REF ), for imaginary $\\tau $ implies by the algebraic morphism $f$ in REF that the solution $(T_1(\\tau ),T_2(\\tau ),T_3(\\tau ))$ of the Darboux-Halphen equations (REF ) is restricted to pure imaginary values when $\\tau $ is imaginary, as mentioned before.",
"Also by the morphism $f$ in and group isomorphisms in $\\mathtt {T_R}$ and $\\mathtt {T_H}$ , $\\mathtt {T_\\Omega }$ maps to the section $ \\lbrace (t_1,t_2,t_3)\\in \\mathtt {T_R}| (-2\\pi i)^{i}t_i\\in \\mathbb {R}\\rbrace $ of the moduli $\\mathtt {T_R}$ of the enhanced elliptic curve $E_t$ (REF ).",
"This section of $\\mathtt {T_R}$ satisfies the reality condition of the spectral curve, and we call it the real section of $\\mathtt {T_R}$ .",
"Therefore we recognize that $\\mathtt {T_\\Omega }$ is a six-fold cover of the moduli of the enhanced spectral curve, which we define below: Definition 1 The moduli of an enhanced spectral curve $\\tilde{S}_k$ of a k-monopole is the real section of the moduli of the enhanced algebraic curve $(S_k^{\\mathbb {C}}, \\lbrace [\\alpha _i]\\rbrace )$ where $S_k^{\\mathbb {C}}$ is the family of algebraic curves in $\\mathbf {\\tilde{T}}$ given by (REF ) without the real structure constraints and $\\lbrace [\\alpha _i]\\rbrace $ is a basis of classes of algebraic de Rham cohomology $H^1_{\\rm dR}(S_k^{\\mathbb {C}})$ of differential 1-forms on $S_k^{\\mathbb {C}}$ with fixed intersection matrix $\\Phi _{ij}=\\langle \\alpha _i,\\alpha _j\\rangle $ .",
"Below we summarize our findings in a form of a theorem: Theorem 1 The moduli of enhanced spectral curves of $SU(2)$ monopoles of charge 2 quotient by $SO(3)$ action and translations in $\\mathbb {R}^3$ corresponds to $\\mathtt {T_\\Omega }$ , a real section of $\\mathtt {T_H}$ , quotient by permutations of $\\Omega _1, \\Omega _2$ and $\\Omega _3$ .",
"Furthermore, the self-dual curvature equation (REF ) corresponds to the Ramanujan vector field in $\\mathtt {T_R}$ upon reparametrization (REF ,REF ) and $f$ isomorphism (REF ).",
"This theorem is proved by $f$ isomorphism under restriction $|T_1|< |T_2| <|T_3|$ in $\\mathtt {T_H}$ .",
"The SO(3) isometry in $M^0_2$ means that this 4-manifold can be expressed by 3-dimensional SO(3) orbits and a orthogonal trajectory parametrized by $\\rho $ .",
"Hence, $M^0_2$ quotient by SO(3) action is a space of one real dimension parametrized by $\\rho $ .",
"Accordingly, the spectral curve $S_2$ , quotient by SO(3) action and translations, depends on a single parameter, after genus or ES[12] constraint is imposed.",
"When we work with the moduli of enhanced spectral curves, two extra parameters related to Hodge structure of the curve are added, but the vector field equation Gauss-Manin connection in disguise shows that the three parameters of the moduli $\\mathtt {T_{H}}$ depend on a single real parameter.",
"The moduli of $\\tilde{S}_2$ does not include the point $r_1=0$ and $r_2=\\pi ^2/4$ of zero discriminant where the curve $S_2$ degenerates to two $S_1$ .",
"This point can be mapped to a point of zero discriminant in $E_T$ (REF ) and it is given by the $\\tau =i \\infty $ limit in the solution of the system (REF ), with appropriate lattice scaling to match $g_2$ in (REF ,REF ): $\\Omega _i(\\rho )=\\frac{\\pi }{r_1}\\tfrac{\\partial }{\\partial \\rho }\\left(\\log \\theta _{i+1}(i\\rho )\\right),\\quad \\textnormal {where}\\quad \\rho = -i \\tau \\quad \\textnormal {and}\\\\\\left\\lbrace \\begin{array}{l} \\theta _2(\\tau ):=\\sum _{n=-\\infty }^\\infty q^{\\frac{1}{2}(n+\\frac{1}{2})^2}\\\\\\theta _3(\\tau ):=\\sum _{n=-\\infty }^\\infty q^{\\frac{1}{2}n^2}\\\\\\theta _4(\\tau ):=\\sum _{n=-\\infty }^\\infty (-1)^nq^{\\frac{1}{2}n^2}\\end{array} \\right.,\\ q=e^{2\\pi i \\tau },\\ \\tau \\in {\\mathbb {H}}.$ Notice that $r_1/r_2$ is function of $\\tau =i\\rho $ and in the limit $\\rho \\rightarrow \\infty $ we can write a explicit relation $r_2\\rightarrow \\pi ^2/4,\\quad \\textnormal {and} \\quad r_1\\rightarrow \\pi ^2q^{1/4}$ Therefore, near the limit $\\rho \\rightarrow \\infty \\; (q<< 1)$ we have $\\Omega _1\\approx -\\frac{q^{1/4}}{4},\\quad \\Omega _2\\approx -2q^{1/4},\\quad \\Omega _3\\approx 2q^{1/4}$ and the metric of $M^0_2$ becomes $ds^2\\approx 4q^{1/4}d\\rho ^2+q^{3/4}(\\sigma _1)^2+\\dfrac{q^{-1/4}}{4}\\left((\\sigma _3)^2+(\\sigma _2)^2\\right)$ The metric is singular at $q=0$ , but asymptotically, two of the coefficients of the metric (REF ) become equal.",
"In this case, the isometry grows to $SO(3)\\times SO(2)$ [24], where $SO(2)$ action corresponds to the axial symmetry of two 1-monopole solution and it corresponds to the $S^1$ isotropy subgroup of the spectral curve [3].",
"In other words, the asymptotic behavior of the metric confirms the behavior of the spectral curve at $r_1=0$ .",
"Furthermore, at infinite $\\rho $ distance, the $SO(3)\\times SO(2)$ orbit is a 2-torus Hopf fibration of the 3-sphere $S^3$ , which confirms the fact that the manifold $M^0_2$ is an asymptotically locally Euclidean (ALE) space.",
"This fact together with self-duality equations, characterizes it as a gravitational instanton configuration[25].",
"These are elements to take in consideration for finding metrics of $ M^0_k, k>2$ ."
],
[
"Conclusion",
"We hope this article pave the way for future contributions in understanding the moduli $M^0_k$ of SU(2) monopoles in YMH theory.",
"Among the obstacles, there are the growing computational challenge of Gauss-Manin Connection in Disguise for larger $k$ and the need to understand the homomorphism between vector fields in the enhanced spectral curves and curvature equations in the moduli $M^0_k$ .",
"The later obstacle is related to the fact that the universal families of curves can be written using different choices of parametrization, which yield different set of differential equations with different algebraic group of transformations of the moduli (where lattice scaling is one of the operations) [6].",
"In the well known case of elliptic curves, the different choices of parametrization of the universal families for the enhanced elliptic curves takes place according to the choices of congruence subgroups $\\Gamma $ of the modular group $SL_2(\\mathbb {Z})$[6].",
"The moduli parametrization of the enhanced curves are lifted to modular-type functions under algebraic group action in the moduli with distinct weights.",
"We notice that the canonical form (REF ) of spectral curves of $k$ -monopoles leads to Ramanujan type of parametrization with parameters with distinct scaling weights, while we expect that the curvature equations from $M^0_k$ leads to Darboux-Halphen type of parametrization with parameters with same (scaling) weight.",
"Therefore, another future step in this projects is to find new modular-type functions attached to $S_k$ curves that will play a role on defining the metric of $M^0_k$ .",
"Such role will depend on the symmetries of the moduli that may define the behavior of the metric of $M^0_k$ under algebraic group of tranformations of the moduli $\\mathtt {T}$ of the enhanced curve.In summary, the results of $M^0_2$ suggest that the terms of the metric of $M^0_k$ will be (quasi-)homogeneous polynomials or rational functions of modular-type functions that will correspond to coordinates of the moduli $\\mathtt {T}$ of enhanced spectral curves $\\tilde{S}_k$ satisfying a unique set of vector field equations in $\\mathtt {T}$ [20].",
"During the period of preparation of the manuscript MACT was fully sponsored by CNpQ-Brasil.",
"The author profited by the rich academic environment at IMPA and by many interactions with Hossein Movasati, whose work is the basis of this project.",
"My sincere thanks go to him and my colleagues from the project on DH equations R. Roychowdhury, Y. Nikdelan and J.",
"A. Cruz Morales, with whom my first thoughts on this project were shared."
]
] | 1709.01545 |
[
[
"Electron-Capture Isotopes could Constrain Cosmic-Ray Propagation Models"
],
[
"Abstract Electron capture (EC) isotopes are known to provide constraints on the low energy behavior of cosmic rays (CRs), such as re-acceleration.",
"Here we study the EC isotopes within the framework of the dynamic spiral-arms CR propagation model in which most of the CR sources reside in the galactic spiral arms.",
"The model was previously used to explain the B/C and sub-Fe/Fe ratios \\citep{BoverC,Iron}.",
"We show that the known inconsistency between the $^{49}$Ti/$^{49}$V and $^{51}$V/$^{51}$Cr ratios remains also in the spiral-arms model.",
"On the other hand, unlike the general wisdom in which the isotope ratios depend primarily on reacceleration, we find here that the ratio also depends on the halo size ($z_\\mathrm{h}$) and in spiral-arms models also on the time since the last spiral arm passage ($\\tau_\\mathrm{arm}$).",
"Namely, EC isotopes can in principle provide interesting constraints on the diffusion geometry.",
"However, with the present uncertainties in the lab measurements of both the electron attachment rate and the fragmentation cross-sections, no meaningful constraint can be placed."
],
[
"introduction",
"Observations of the CR composition can teach about the origin of CRs, their initial composition, path length distribution (PLD) and interaction they undergo as they propagate in the interstellar medium.",
"Such observations include the ratio between secondary to primary cosmic rays, including the Boron to Carbon (B/C) ratio and the sub-Iron (Scandium through Manganese) to Iron (sub-Fe/Fe) ratio, the positron fraction ($e^{+}/(e^{-}+e^{+})$ ), and the ratios between the EC's daughter and parent isotopes which we study here.",
"The latter are known to constrain the process of re-acceleration, but as we show below, they can also be used, at least in principle, to constrain diffusion models.",
"If one calculates the secondary to primary ratio under the simplest leaky box model or a “disk-like\" model (with an azimuthally symmetric CRs source distribution), one finds that the ratio drops with energy [6].",
"Indeed, the positron fraction below 10 GeV and the nuclei ratios above 1 GeV/nuc.",
"exhibit this behavior.",
"However, the positron fraction above 10 GeV and nuclei spectra below 1 GeV/nuc.",
"appear to be behave differently [1], [22], [25].",
"A disk-like model must include galactic winds, re-acceleration or ad-hoc assumptions on the diffusivity in order to explain the observed behavior of the nuclei ratios [26], [11], while the behavior of the positrons require either astrophysics solutions such as pulsars [14], [7], [2], [15], [23], or more exotic physics such as dark matter decay [5], [16] Unlike a disk-like model, a spiral-arms model, in which a significant fraction of CR sources are located at the galactic spiral arms—a place where star formation is enhanced and hence young SNRs are abundant, can explain these anomalies at the outset.",
"[24] showed that by considering the CR sources to be at a finite distance from earth, as expected from the spiral structure, one recovers the positron fraction spectrum.",
"Moreover, [3] recovered the B/C ratio also at low energies by taking the spiral arms to be dynamic.",
"The astrophysical motivation of the spiral arms model and its success in explaining these phenomena has motivated us to explore other predictions of this model.",
"Another interesting problem arises when the grammage required to explain the B/C ratio is compared to the sub-Fe/Fe ratio.",
"When doing so, it reveals that the latter ratio requires about 20% more grammage than the former ratio in a disk-like model [13], [10].",
"[13] proposed a solution to this problem, by cutting the short path lengths from the CR PLD.",
"Obviously, taking the source to be primarily in the spiral arms naturally causes a paucity in short path lengths.",
"In a later study the Ulysses-HET group [12] also tested the option of truncating the short path lengths.",
"They found that the this is not necessary.",
"Namely, a simple exponential power-law (as is the case in a disk-like and leaky-box models) is sufficient to recover their own observations.",
"However, the Ulysses-HET sub-Iron/Iron measurement is clearly well below other measurement (while the B/C is consistently the same), which explains why a simple PLD is enough when only their measurement is considered.",
"In [4], we have shown that a spiral-arms model resolves this anomaly by finding the optimal model parameters required to separately recover the B/C and sub-Iron/Iron.",
"It was shown that while the disk-like model does not recover the two ratios with consistently the same model parameters, the spiral arm model does.",
"Here we continue our investigation of the Iron group CR nuclei (Scandium through Nickel)The iron group includes the isotopes of Scanadium through Nickel, with Iron and Nickel being the primaries and the rest secondaries.",
"the sub-Iron group is a sub group of the iron group that includes Scandium through Manganese.",
"within the context of the spiral arm model, and focus on isotopes that decay through EC.",
"At low energies, electrons are bound to the nuclei and as a consequence, these isotopes rapidly decay through EC.",
"However, at higher energies, typically above 1 GeV/nuc., these isotopes are stripped of their electrons and this inhibits their decay.",
"The probability for having bound electrons depends on two processes, the stripping and the attachment of electrons.",
"Both are strongly dependent on energy [18].",
"However, because the EC decay time scale is generally much shorter than the stripping time scale, the isotope ratios basically depends on the attachment rate.",
"[19] reported the first measurements of $^{44}$ Ti, $^{49}$ V, $^{51}$ Cr, $^{55}$ Fe and $^{57}$ Co from CRIS – the Cosmic Ray Isotope Spectrometer, that is located on the Advanced Composition Explorer (ACE).",
"In particular, they measured the $^{49}$ Ti/$^{49}$ V and $^{51}$ V/$^{51}$ Cr ratios.",
"Evident from the observations is a very strong dependance on energy—the ratios decrease with energy.",
"Since the enumerator in these ratios is the stable daughter product of the EC of the parent isotope in the denominator, the ratios reflect the EC reaction rate, and therefore the probability for the parent isotope to be stripped.",
"[17] modeled the isotopic ratios using the weighted slab model, while assuming different assumptions on the retainment of electrons and reacceleration.",
"They have shown that complete stripping results in almost energy independent ratios, and therefore cannot explain the decrease with energy.",
"In other words, there must be a transition from unstripped to stripped isotopes.",
"The decrease with energy of the two ratios $^{49}$ Ti/$^{49}$ V and $^{51}$ V/$^{51}$ Cr is consistent with this interpretation as well.",
"[17] then tried to explain the measurements by assuming that the nuclei retain their bound electrons at low energies, and then reaccelerate to higher energies, on their way to the solar system.",
"The time spent at low energies will cause the EC isotopes to produce more daughter isotopes compared to CRs that did not spend time at low energies.",
"In a subsequent study, [20] showed that the cross-section to bind electrons from the ISM to stripped nuclei is increasing for progressively smaller energies.",
"For energies lower than a few hundred MeV/nuc., the time scale is shorter than the escape (and therefore typical age) of the cosmic rays.",
"However, even at a few MeV/nuc.",
"the attachment rate time scale is still much longer than the EC decay.",
"This means that at energies of up to a few 100 MeV/nuc., the attachment process is the dominant one determining the EC isotope ratios.",
"They also considered reaccelaration as [17], but due to the large electron attachment cross-section which they include, [20] require a more feasible higher initial energy to accelerate these isotopes from than [17] require.",
"However, both [17], [20] obtained inconclusive results—some of the observations were more consistent with models that include reacceleration (in particular, the $^{51}$ V/$^{51}$ Cr isotopes ratios) while other observations indicate the opposite (the $^{49}$ Ti/$^{49}$ V isotopes ratio).",
"Both [17] and [20] point out that the main problem in reaching any firm conclusions was the uncertainty in the fragmentation cross sections.",
"Namely, the above inconsistency cannot be resolved with just reacceleration.",
"This conclusion about the fragmentation cross-sections was reaffirmed by [21], who showed that the typical 10-20% uncertainty in the fragmentation cross-sections [28], can explain away the discrepancy between the above two isotope datasets.",
"For example, reducing the $^{49}$ Ti fragmentation cross-section by 15% will resolve the discrepancy.",
"We elaborate on this correction in the discussion.",
"We note that the CRIS/ACE results are not the first to have reached these conclusions.",
"They are consistent with the previous measurements by the Ulysses HET team [8] of a single data point at 300 MeV/nuc.",
"(but with a similar error bar).",
"These authors also concluded that some isotopes are consistent with reacceleration and while others are consistent with no acceleration.",
"All these studies were done within the standard disk model.",
"One could have hoped that like other inconsistencies, this one will be resolved when we consider a dynamical spiral arms model instead.",
"Here we show that the inconsistency between atomic mass 49 and 51 isotopes remains also in this model.",
"This points out to the same conclusion that there might be a problem with the fragmentation cross-sections.",
"We also show that while the power law index of the cross-sections' energy dependence required to fit the observations agree with the lab experiments [29], [9], the attachment normalization needed to fit the data varies depending on the halo size, $z_\\mathrm {h}$ , and the time since last spiral arm passage, $\\tau _\\mathrm {arm}$ .",
"We begin in § by briefly describing the spiral arms model and the nominal model parameters.",
"We review the data used in §.",
"In § we carry out an extensive analysis of the model, including a parameter study used to find a fitting formula for the attachment rate which recovers the $^{49}$ Ti/$^{49}$ V and $^{51}$ V/$^{51}$ Cr ratios.",
"The implications of these results are discussed in §."
],
[
"The model",
"SNRs are generally believed to be the sources of the galactic CRs.",
"The spiral-arms model assumes that, since SNR are more abundant in galactic spiral arms, these arms are the main source of CRs.",
"At low energies the CRs diffuse slowly and the dynamical motion of the spiral arms cannot be neglected.",
"In [3] we describe a fully three dimensional numerical code for CRs diffusion in the Milky Way under these assumptions.",
"The code enables us to explore dynamic spiral arms as the main source of the CR.",
"Using this model, [3] recovered the B/C ratio and demonstrated that the dynamics of the spiral arms has a notable effect on the ratio between secondary and primary CRs, which below 1 GeV/nuc.",
"increase with the energy.",
"In [4] we have shown that a spiral-arms model, unlike a disk-like model, can explain the discrepancy between the grammage implied by the B/C ratio and by the sub-Fe/Fe ratio.",
"Naturally, the spiral arms model require different diffusion parameters than those commonly used in the galactic disk model.",
"The optimal parameters required to fit the B/C, sub-Iron/Iron and $^{10}$ Be/$^{9}$ Be ratios within the dynamic arms and homogeneous disk models are summarized in table REF [4], [3].",
"Table: Nominal Model ParametersOur code is different from present day simulations (such as galprop, [25], and dragon, [11]) which solve the partial differential equations (PDE) describing diffusion in that we use a Monte Carlo methodology.",
"It allows for more flexibility in adding various physical aspects to the code (such as the spiral arm advection), though at the price of reduced speed.",
"Here we will only discuss the changes we recently made to explore the EC reactions.",
"The full details of the code and of the the model are found in [3], [4]."
],
[
"Attachment Rate Formula",
"[18] studied the EC reaction in CRs using experimental data collected by [29] and [9].",
"In figs.",
"1 and 2 of [18], one can see that for $21<Z<28$ and for energies of a few 100 MeV/nuc.",
"the mean free path for attachment of an electron is roughly $\\lambda _\\mathrm {attachment} \\approx 1~$ gr/cm$^2$ , while for the stripping of an electron it is roughly $\\lambda _\\mathrm {stripping} \\approx 10^{-3} $ gr/cm$^2$ , which correspond to time scales of $\\tau _\\mathrm {attachment} \\approx 5~$ Myr and $\\tau _\\mathrm {stripping} \\approx 5 \\times 10^{-3}$ Myr respectively.",
"For the isotopes $^{44}$ Ti, $^{49}$ V, $^{51}$ Cr, $^{55}$ Fe and $^{57}$ Co the decay time scale is between several days to a few years, much smaller than $\\tau _\\mathrm {stripping}$ , implying that we can neglect the stripping process for those isotopes and assume that they decay immediately after they attach an electron from the ISM.",
"However for $^{53}$ Mn and $^{59}$ Ni, the half life time for the EC decay is 3.7 Myr and 0.076 Myr respectively, which is much longer than $\\tau _\\mathrm {stripping}$ .",
"This allows one to neglect the decay process and assume that these isotopes will be striped off their electrons before they could decay, and therefore remain stable.We note that $^{54}$ Mn is also an EC isotope.",
"In our calculations it decay immediately since its $\\beta $ decay mode have half life time that is significantly shorter than the typical propagation time.",
"When interpolating the data of [18]'s fit for the electron attachment mean free path, one can see that the energy dependance of the attachment cross-section is a power-law of the form $\\sigma _{a}(E,Z)=N Z^{\\nu } {(E/\\mathrm {500\\,MeV})^{-\\mu }} \\ ,$ with indices of $\\mu =1.8 \\pm 0.1$ and $\\nu =4.5 \\pm 0.1$ , and a normalization $N=(1.2 \\pm 0.2) \\times 10^{-4}~$ mb for $20<Z<28$ .",
"Here we allow for a generalized power-law attachment rate and add it to the description of EC isotopes in the numerical code, which includes $^{44}$ Ti, $^{49}$ V, $^{51}$ Cr, $^{55}$ Fe and $^{57}$ Co. Namely, we use the same power-law with the above two parameters, the normalization factor, $N$ , and the power index, $\\mu $ , but keep them as free parameters (we choose to keep $\\nu $ fixed because the difference between the Z's of the two observational datasets is less than $5\\%$ , see §).",
"Each time step we check whether the CR isotope attached an electron from the ISM (and let it decay immediately) with the same methodology as we do for the spallation process (see more details in [3], §3.10).",
"In this work we study the sensitivity of isotope ratio outcome to the parameter space describing the attachment.",
"The observed ratios which we use are $^{49}$ Ti/$^{49}$ V and $^{51}$ V/$^{51}$ Cr described below in §."
],
[
"Observational datasets",
"We compare the model predictions for the $^{49}$ Ti/$^{49}$ V and $^{51}$ V/$^{51}$ Cr ratios with the two CRIS datasets [21], one collected during the solar minimum years, 1997-1999, and one for the solar maximum years, 2000-2003, with average solar modulation of 510 MV and 920 MV respectively [27].",
"For each observation, there are 14 data points between 100 MeV/nuc.",
"and 1 GeV/nuc.",
"To account for the solar wind modulation, the energy of each specie outside the solar system obtained in the simulation is mapped to the modulated energy inside the solar system through $E_{obs}=E-(Z/A) \\times \\phi $ , where $\\phi $ is the modulation potential.",
"The modeled specie ratios can then be calculated and compared with the observations at a given observed energy from which a $\\chi ^2$ can be calculated.",
"The data (and the model fits) are depicted in figs.",
"REF .",
"Figure: The 49 ^{49}Ti/ 49 ^{49}V (left figure) and 51 ^{51}V/ 51 ^{51}Cr (right figure) ratios we obtain in our nominal model (black lines) with the set of parameters described in table and the attachment parameters obtained from the fit described in §.",
"The shaded regions correspond to the spectrum once solar wind modulation is added, with the red lines describing the minimum solar modulation while the blue describe the maximum solar modulation.",
"The green lines are the respective lines obtained when the EC isotopes are assumed to be entirely stable.",
"Data taken from: CRIS , Ulysses-HET ."
],
[
"A disk-like model",
"We begin with the analysis of the disk-like model.",
"We consider a nominal halo size of $z_h=1$ kpc and a diffusion coefficient normalization of D$_0=5 \\times 10^{27}$ cm$^{2}/$ sec that recovers the observed sub-Fe/Fe ratio, and its comparison with [18].",
"The rational of using the values that fit the sub-Fe/Fe ratio and not the B/C ratio is because the EC isotopes are much closer to Iron and the other isotopes that we consider here.",
"Fig.",
"REF provides a contour plot of the $\\chi ^2$ fit between model and observations, for the two parameters in the attachment process formula, $N$ and $\\mu $ .",
"Figure: A contour plot of χ 2 \\chi ^2 fit for a disk-like model with halo size of z h =1z_h=1 kpc and diffusion coefficient of D 0 =4×10 27 _0=4 \\times 10^{27} cm 2 /^{2}/sec.",
"The red contours correspond to the 51 ^{51}V/ 51 ^{51}Cr fit, the blue contours correspond to the 49 ^{49}Ti/ 49 ^{49}V fit and the dashed purple lines correspond to the combined χ 2 \\chi ^2 calculation.",
"Note the discrepancy between the 51 ^{51}V/ 51 ^{51}Cr and the 49 ^{49}Ti/ 49 ^{49}V fits—while the observation for 51 ^{51}V/ 51 ^{51}Cr require high normalization and a low power-law index, the observation for 49 ^{49}Ti/ 49 ^{49}V require a smaller normalization and a higher power-law index.",
"The green point denotes the electron attachment cross-section derived by , based on the lab measurements of and , that is required to explain experimental data.Fig.",
"REF depicts the $\\chi ^2$ fit to the datasets, when separately fitting the $^{51}$ V/$^{51}$ Cr data, the $^{49}$ Ti/$^{49}$ V, and fitting them together.",
"One can easily see that there is an inconsistency.",
"While the observations for $^{51}$ V/$^{51}$ Cr require a high normalization factor and a low power-law index, the observations for $^{49}$ Ti/$^{49}$ V require a lower normalization factor and a higher power-law index.",
"This inconsistency between the two observations was already demonstrated in all previous works [20], [21], [17], [28].",
"Despite this inconsistency, the optimal power-law index for the combined $\\chi ^2$ for both data sets, $\\mu =2.1 \\pm 0.7$ , is in agreement with [18]'s results, $\\mu =1.8 \\pm 0.1$ .",
"Even for each separate set of isotope ratio measurements, $\\mu =1.8$ is inside the respective $2\\sigma $ region.",
"We note, however, that the significance contours denote only the statistical uncertainties, but not the unknown systematic errors that should exist given the uncertainty concerning the cross-section .",
"In addition to the the fact that the normalization cross-section required to explain the two observations are inconsistent with each other, they are also inconsistent with [18] who require a somewhat larger attachment cross-section.",
"For the disk-like model, the $^{49}$ Ti/$^{49}$ V data require a normalization factor of $N=(5.6 \\pm 0.8) \\times 10^{-5}$ mb, the $^{51}$ V/$^{51}$ Cr data require $N=(8.2 \\pm 0.7) \\times 10^{-5}$ mb, while [18] finds $N=(1.2 \\pm 0.2) \\times 10^{-4}$ mb, which is about a factor of 1.5 higher than our result.",
"Next, we proceed to check whether the inconsistency between the two sets of observations and the inconsistency in the normalization of the cross-section between our results and those of [18] are an outcome of the model parameters or whether these inconsistencies remains for all disk-like models.",
"Fig.",
"REF depicts the $\\chi ^2$ fit for a disk-like model with halo size of $z_h=3$ kpc and diffusion coefficient of D$_0=1.5 \\times 10^{28}$ cm$^{2}/$ sec.",
"Figure: A contour plot of χ 2 \\chi ^2 for a disk-like model with halo size of z h =3z_h=3 kpc and diffusion coefficient of D 0 =1.5×10 28 _0=1.5 \\times 10^{28} cm 2 /^{2}/sec.",
"Similar to fig.",
", the red contours correspond to the 51 ^{51}V/ 51 ^{51}Cr fit, the blue contours correspond to the 49 ^{49}Ti/ 49 ^{49}V fit and the dashed purple lines correspond to the combined χ 2 \\chi ^2 calculation.",
"Note the discrepancy between the two observations remains the same as in fig.",
".Evidently, the discrepancy between the two sets of observations remains for different halo sizes, however, a larger halo can remove the discrepancy between the required attachment cross-section to fit the lab measurements and the average of the cross-section's normalization factor determined from the two EC datasets."
],
[
"The normalization dependence on $z_\\mathrm {h}$ in a disk-like model",
"When varying the Galactic halo size, one has to take into account other observational constraints on the secondary to primary ratios, such as B/C or sub-Iron/Iron.",
"In fact, imposing the sub-Iron to Iron ratio measurements imposes the linear relation $D_0/z_\\mathrm {h}=(5 \\pm 1) \\times 10^{27} ~$ (cm$^{2}/$ sec) kpc$^{-1}$ .",
"Namely, for each $z_\\mathrm {h}$ there is a corresponding diffusion coefficient normalization, $D_0$ .",
"We note again that under the disk-like models, the B/C ratio requires a different normalization for the diffusion coefficient than the sub-Iron/Iron ratio [4], [10], [13].",
"We choose here the diffusion normalization factor corresponding to the sub-Iron/Iron data because the EC isotopes belong to the Iron group as well.",
"Next, we fix now the attachment cross-sections power-law index, $\\mu =1.8$ .",
"This value is consistent with [18]'s results and our results.",
"With these constraints on $\\mu $ and $D_0$ , we can now proceed to obtain the fitted attachment cross-section normalization, $N_{disk}$ , as a function of $z_\\mathrm {h}$ .",
"$N_{disk}=(8.0 \\pm 0.7) \\times 10^{-5} \\times (z_\\mathrm {h}/1~\\mathrm {kpc})^{0.27 \\pm 0.04}~\\mathrm {mb}.$ One can easily see that in a disk-like model a halo size of about $z_\\mathrm {h} \\approx 3$ to 5 kpc is required in order to match [18]'s results.",
"Figure: A contour plot of χ 2 \\chi ^2 for the spiral-arms model.",
"Similar to figs.",
"and , the red contours correspond to the 51 ^{51}V/ 51 ^{51}Cr fit, the blue contours correspond to the 49 ^{49}Ti/ 49 ^{49}V fit and the dashed purple lines correspond to the combined χ 2 \\chi ^2 calculation.",
"Note the discrepancy between the two observations is somewhat larger but still similar in size to the disk-like model."
],
[
"A spiral-arms model",
"We now proceed to study our nominal spiral-arm model from [4].",
"In particular we are interested in finding the optimal parameters (of the attachment process formula) that recover the observations.",
"Fig.",
"REF depicts a contour map of $\\chi ^2$ , similar to figs.",
"REF and REF , but for the spiral-arms model.",
"As is the case in the disk-like model, the optimal power-law index for the combined $\\chi ^2$ calculation in the spiral-arms model is $\\mu =1.9 \\pm 0.4$ .",
"This value agrees with [18]'s result as well.",
"Nevertheless, the inconsistency between the required cross-section normalization of the two observations remains the same as in the disk-like model and all other previous works.",
"This suggests that by the changing the diffusion parameters or geometry one cannot resolve the discrepancy, which probably arises due to uncertainties in the cross-sections.",
"We find that the $^{49}$ Ti/$^{49}$ V data require a normalization of $N=(4.3 \\pm 0.7) \\times 10^{-5}$ mb and the $^{51}$ V/$^{51}$ Cr data require $N=(8.7 \\pm 0.7) \\times 10^{-5}$ mb.",
"For a comparison again, [18] finds $N=(1.2 \\pm 0.2) \\times 10^{-4}$ mb.",
"Fig.",
"REF depicts the two observations, $^{49}$ Ti/$^{49}$ V and $^{51}$ V/$^{51}$ Cr, with the model prediction for the optimal parameters derived above.",
"The shaded regions correspond to the spectrum once solar wind modulation is addedMore details on the solar modulations can be found in [3] §3.6.",
"Here we use $\\phi _{max}=920$ MV and $\\phi _{min}=510$ MV which are the solar modulation values that correspond to the years of the CRIS measurements [27].."
],
[
"The normalization dependence on $z_\\mathrm {h}$ and {{formula:a2cb6766-46c1-46f1-a7b5-e47e25bcd09c}} in a spiral-arms model",
"In a similar way to disk-like models, observational constraints on the secondary to primary ratios, such as B/C or sub-Iron/Iron ratios, imply that the normalization of the diffusion coefficient, $D_0$ , varies when changing the geometry of the arms and/or the galaxy.",
"Namely, for each pair of $z_\\mathrm {h}$ and $\\tau _\\mathrm {arm}$ , there is a corresponding value of $D_0$ .",
"While in the disk-like models we had to chose this normalization that will fit either the B/C ratio or the sub-Iron/Iron ratio, here in the spiral-arms model the same $D_0$ is consistent with both the B/C and sub-Iron/Iron data [4].",
"Fig.",
"REF shows a contour map of $D_0$ as a function of $z_\\mathrm {h}$ and $\\tau _\\mathrm {arm}$ .",
"As expected, $D_0$ increases with both $z_\\mathrm {h}$ and $\\tau _\\mathrm {arm}$ .",
"Figure: A contour map of the optimal D 0 D_0 required to fit the sub-Iron to Iron measurements, as a function of z h z_\\mathrm {h} and τ arm \\tau _\\mathrm {arm}.",
"Evidently, D 0 D_0 increases with both z h z_\\mathrm {h} and τ arm \\tau _\\mathrm {arm}.The next step is to fix the attachment cross-sections power-law index, $\\mu =1.8$ .",
"This value is consistent with [18]'s results.",
"With the above values of $\\mu $ and $D_0$ , we can now proceed to obtain the optimal normalization of the attachment cross-section, $N$ , as a function of $z_\\mathrm {h}$ and $\\tau _\\mathrm {arm}$ .",
"Fig.",
"REF depicts contour maps of $N$ for the combined $\\chi ^2$ fit of the two datasets as a function of $z_\\mathrm {h}$ and $\\tau _\\mathrm {arm}$ .",
"One can readily see that $N$ increases with $z_\\mathrm {h}$ but it decreases with $\\tau _\\mathrm {arm}$ .",
"Figure: A contour map of the optimal attachment cross-section normalization, NN, required to fit the combined χ 2 \\chi ^2 calculation of the two datasets, as a function of z h z_\\mathrm {h} and τ arm \\tau _\\mathrm {arm}.",
"It is readily seen that the normalization, NN, increases when z h z_\\mathrm {h} increases, while it decreases with τ arm \\tau _\\mathrm {arm}.",
"Note that the rugged behavior arises from the raw data having “Monte Carlo\" noise.We can quantify better the required normalization by using the form: $N_{SA}&=&(7.98 \\pm 0.02) \\times 10^{-5} \\nonumber \\\\&& \\times (\\tau _\\mathrm {arm}/10~\\mathrm {Myr})^{-0.278 \\pm 0.008} \\nonumber \\\\&& \\times (z_\\mathrm {h}/1~\\mathrm {kpc})^{0.236 \\pm 0.007}~\\mathrm {mb}.$ Figure: Contour plots of the χ 2 \\chi ^{2} fit for the 49 ^{49}Ti/ 49 ^{49}V and 51 ^{51}V/ 51 ^{51}Cr ratios and the combined χ 2 \\chi ^2 calculation (the same colours as in fig.",
"and fig. )",
"for the disk-like model (left panel) and the spiral-arms model (right panel) after reducing the fragmentation cross-sections for 49 ^{49}Ti by 15% and 20% respectively."
],
[
"Discussion & Summary",
"It is generally accepted that CR EC isotopes can be used to assess the importance of re-acceleration in the ISM [26].",
"Nonetheless, a comparison between model predictions and measurements of $^{51}$ V/$^{51}$ Cr and $^{49}$ Ti/$^{49}$ V gave inconsistent results, generally interpreted as arising from uncertainties in the nuclear spallation cross-sections.",
"[21] have shown that the typical 10-20% uncertainty in the fragmentation cross-sections [28] can explain away the discrepancy between the observations of the two isotopes.",
"Specifically, they found that a reduction of the fragmentation cross-sections of $^{49}$ Ti by 15% was sufficient to resolve the discrepancy.",
"Previous analyses, however, considered axisymmetric models in which the CR source distribution is relatively smooth.",
"More recently, we developed a fully 3D CR diffusion model which not only considers that most CR acceleration takes place in the vicinity of spiral arms, but also that these arms are dynamic [3].",
"One very important aspect of this model is that the path length distribution (PLD) is different from the one found in standard disk-like models.",
"In the latter, the PLD is typically close to being exponential.",
"However, if most CRs arrive from a distance, such as from a spiral-arm, then the PLD will exhibit a paucity of small path lengths (compare fig.",
"4 to fig.",
"6 in [3]).",
"It was therefore our goal to see whether a more realistic distribution of CR sources could alleviate the discrepancy between the model predictions and the measurements of $^{51}$ V/$^{51}$ Cr and $^{49}$ Ti/$^{49}$ V ratios.",
"In this work, we studied the EC isotopes using the observations of $^{51}$ V/$^{51}$ Cr and $^{49}$ Ti/$^{49}$ V ratios, as well as an empirical fit to the electron attachment cross-section of propagating nuclei.",
"This fit is based on the results of [18], who derived the attachment and stripping cross-section using experimental data from [29] and [9].",
"They measured the time scales of both processes which are much longer than the EC decay timescale, thus, we can neglect the stripping process and assume that when an EC isotope attaches electron, it will decay immediately.",
"[18] also showed that the attachment cross-section has an approximate power-law dependance on the energy and on $Z$ , with respective power indices of $\\mu =1.8 \\pm 0.1$ and $\\nu =4.5 \\pm 0.1$ , and a normalization of $N=(1.2 \\pm 0.2) \\times 10^{-4}~$ mb (for $E=500~$ MeV and $Z=1$ ).",
"We first found that the EC ratios in standard disk-like models are not only sensitive to the EC rates but also modestly sensitive to the halo size.",
"Specifically, the required cross-section in disk-like models is $\\sigma _{a}(E,Z)=N(z_\\mathrm {h}) \\times Z^{4.5} \\times {(E/\\mathrm {500\\,MeV})^{-1.8}}$ , with the normalization roughly given by $N_{disk}(z_\\mathrm {h})=8 \\times 10^{-5} ~$ mb$ \\times (z_\\mathrm {h}/1~$ kpc$)^{0.27}$ .",
"We then found that EC in spiral-arms models can also constrain the geometry of the galactic arms in addition to the halo size.",
"The required cross-section also depends on the time since last spiral arm passage.",
"Its normalization should satisfy $N_{SA}(z_\\mathrm {h},\\tau _\\mathrm {arm})=7.98 \\times 10^{-5} ~$ mb$ \\times (\\tau _\\mathrm {arm}/10~$ Myr$)^{-0.278} \\times (z_\\mathrm {h}/1~$ kpc$)^{0.236}$ .",
"However, even with the added spiral arms one cannot alleviate the discrepancy between the $^{51}$ V/$^{51}$ Cr and $^{49}$ Ti/$^{49}$ V measurements.",
"This strengthens the claim that this discrepancy is due to the uncertainty in the spallation cross-sections.",
"Thus, improved spallation cross-sections are required in order to use the EC CRs to constrain geometric properties of the diffusion models."
],
[
"Acknowledgements",
"This work is supported by an Advanced ERC grant (TP), the Israel Science Foundation (grant no.",
"1423/15, NS) and by the I-CORE Program of the Planning and Budgeting Committee and The Israel Science Foundation (1829/12)."
]
] | 1709.01585 |
[
[
"Detecting animals in African Savanna with UAVs and the crowds"
],
[
"Abstract Unmanned aerial vehicles (UAVs) offer new opportunities for wildlife monitoring, with several advantages over traditional field-based methods.",
"They have readily been used to count birds, marine mammals and large herbivores in different environments, tasks which are routinely performed through manual counting in large collections of images.",
"In this paper, we propose a semi-automatic system able to detect large mammals in semi-arid Savanna.",
"It relies on an animal-detection system based on machine learning, trained with crowd-sourced annotations provided by volunteers who manually interpreted sub-decimeter resolution color images.",
"The system achieves a high recall rate and a human operator can then eliminate false detections with limited effort.",
"Our system provides good perspectives for the development of data-driven management practices in wildlife conservation.",
"It shows that the detection of large mammals in semi-arid Savanna can be approached by processing data provided by standard RGB cameras mounted on affordable fixed wings UAVs."
],
[
"Introduction",
"In the fragile ecosystems of semi-arid Savanna, any change in the equilibrium between precipitation, grazing pressure and bush fires can lead to long-term land degradation, such as the reduction in grass cover and bush encroachment [20].",
"To avoid overgrazing, the populations of grazers must be kept in adequacy with the grass availability, which is subject to meteorological conditions.",
"For this purpose, land managers need to regularly estimate the amount of wildlife present on their territory.",
"Thus, monitoring wildlife populations is crucial towards conservation in wildlife farms and parks.",
"To carry out wildlife censuses, traditional methods include transect counts on land or from a helicopter, and camera traps.",
"While a total count is usually not possible over large areas, these methods estimate the population density based on observations localized along a predefined path (see [4], [1] and references therein).",
"These methods are expensive (e.g.",
"in the case of the Kuzikus reserve considered in this paper, helicopter costs for a single survey are between 1000$ and 2500$), require trained human experts to screen large amounts of data and are consequently not suitable for regular censuses over large areas.",
"In recent years, unmanned aerial vehicles (UAVs) have been used to detect and count wildlife such as birds, marine mammals, and large herbivores [12].",
"Compared to traditional methods, UAVs offer several advantages: they cover large areas in a short amount of time and can be used in inaccessible and remote areas, yet they are cheaper and easier to deploy than helicopters.",
"Moreover, they are safer for the pilot, who can stay on the ground and avoiding retaliations from poachers.",
"However, UAVs collect large amounts of color images with sub-meter to sub-decimeter spatial resolution, of which only few contain animals.",
"Furthermore, the animals cover only an infinitesimal area of the images and their color might blend in smoothly with background vegetation and soil.",
"Therefore, identifying and counting single animals across large collections of images is extremely complex and time-consuming, preventing land managers from using UAVs on a regular basis.",
"Despite these challenges, recent developments in object detection pipelines in both computer vision [14], [7] and remote sensing [21], [15], [2], provide promising techniques to semi-automatically localize and count animals.",
"We refer to these methods as semi-automated and not as fully automated since they rely on supervised learning paradigms, thus requiring annotated ground truth to be trained.",
"Still, as the human effort required to make sense of the aerial images is reduced, the overall benefits of using UAVs are significantly increased.",
"The use of UAVs in wildlife monitoring and conservation is well documented (e.g.",
"[12]), but only few studies have implemented semi-automatic detection pipelines.",
"[8] proposes to detect seagulls by combining supervised classification of RGB images with geometric rules.",
"[11] present a pipeline to count salmons in aerial images using simple color thresholding after contrast adjustment.",
"Such approaches are only possible if the animals are visually very similar and exhibit distinctive colors that contrast with the background.",
"[5] detect geese by manual counting single animals in UAV images.",
"[13] adopt more advanced machine learning tools for the detection of dugongs.",
"They obtain promising results by training a deep neural network and address the problem of scarcity of training samples by replicating them through random rotations and scalings applied to confident missclassifications (a technique related to hard negative mining [14]).",
"In this paper, we present a data-driven machine learning system for the semi-automatic detection of large mammals in the Savanna ecosystem characterized by complex land-cover.",
"We perform animal detection on a set of sub-decimeter resolution images acquired in the Namibian Kalahari desert and train our system using animals annotated by digital volunteers using the Micromappers crowdsourcing platform [17].",
"We focused on large mammals for two main reasons: first, they stood out compared to the background, while smaller animals such as meerkats are not clearly visible and could be too easily confused with rocks or bushes by the volunteers.",
"Secondly, larger animals also mean more pixels available to learn the appearance of the animals' furs, which leads to less signal mixing, to more discriminative features and to a more accurate system overall.",
"We show that the system achieves high recall rate, and high overall accuracy can be obtained if a human operator can verify the detections, reduce the false positives and verify true negatives, and retrain the detector.",
"This last technique, known as active learning [23], aims at focusing the operator's effort on instances with low detection confidence and its benefits are shown by our experimental results, where only 1h was required to correct the crowd-sourced dataset of several errors (mainly animals missed by the volunteers).",
"The main contributions of the paper are: - A pipeline for semi-automatic animal detection in semi-arid Savanna that uses affordable UAV platforms with off-the-shelf RGB cameras; - A complete study of the model's parameters to provide intuitions about the trade-offs between acquisition settings, image resolution and the complexity of the appearance descriptors involved; - A discussion of the promising performances of the system in a real deployment scenario in the Kuzikus reserve in Namibia, including the quasi real time improvement of the model.",
"Kuzikus is a private wildlife reserve that covers 103 km$^2$ (10'300 hectares), located on the edge of the Kalahari in Namibia.",
"The Kalahari is a semi-arid sandy Savanna that extends across Botswana, South Africa and Namibia.",
"It is home of an important variety of animals, including many large mammal species [18].",
"About 3'000 individuals from more than 20 species populate the reserve, including Common Elands (Taurotragus oryx), Greater Kudus (Tragelaphus strepsiceros), Gemsboks (Oryx gazella), Hartebeests (Alcelaphus buselaphus), Gnus (Connochaetes gnou and Connochaetes taurinus), Blesboks (Damaliscus albifrons), Springboks (Antidorcas marsupialis), Steenboks (Raphicerus campestris), Common Duickers (Sylvicapra grimmia), Impalas (Aepyceros melampus), Burchell's Zebras (Equus quagga burchellii), Ostriches (Struthio camelus australis) and Giraffes (Giraffa camelopardalis giraffa)."
],
[
"The SAVMAP 2014 UAV campaign",
"The SAVMAP Consortium (http://lasig.epfl.ch/savmap) acquired a large aerial image dataset during a two-week campaign in May 2014.",
"It is composed of five flights, between May 12 and May 15, 2014.",
"The images were acquired with a Canon PowerShot S110 compact camera mounted on an eBee, a light UAV commercialized by SenseFly (https://www.sensefly.com).",
"Each image is $3000 \\times 4000$ pixels in size and comprises three bands in the Red Green and Blue (RGB) domains, with a radiometric resolution of 24 bits.",
"The ground sampling distance is approximately 4 cm per pixel.",
"The extent of the reserve mapped by the 2014 SAVMAP campaign is illustrated in Fig.",
"REF .",
"Figure: Map of the Kuzikus Wildlife Conservation Park and areas covered by the 2014 RGB dataset."
],
[
"Animals annotation via crowd sourcing",
"In order to obtain a ground truth of the position of all large animals, a crowd-sourcing campaign was set by MicroMappers (https://micromappers.wordpress.com/).",
"A total of 232 digital volunteers participated in the operation.",
"The volunteers were asked to draw a polygon around each animal they detected in the images, without distinction between species.",
"They did not have to report signs of animal presence, such as Aardwolves' holes or termite mounds.",
"Each image is inspected by at least three volunteers, with a maximum of ten.",
"On average, the images were seen by five volunteers [17].",
"The volunteers visually analyzed 6'500 images and drew 7'474 polygons in 654 images containing animals.",
"After merging the overlapping polygons and removing objects tagged only by a single volunteer (as the bottom right annotation in Fig.",
"REF ), 976 annotations were kept.",
"Since the number of volunteers per image varied between three and ten, we used as ground truth the surface that was tagged as “animal” by at least half of the annotators who considered it (areas in green-to-yellow colors in the right panel Fig.",
"REF ).",
"To avoid false annotations, we visually inspected them to confirm or infirm animals presence.",
"It took 30 minutes to verify the 976 annotations, leading to the removal of 21 spurious ones.",
"More details on the acquisition of annotations can be found in [17].",
"Note that the same animals could be observed in several consecutive, overlapping images.",
"This effect is beneficial when training appearance models, since the different angles and poses characterizing animals better cover the appearance variability of the class of interest.",
"However, note that the current system has no tracking component (nor ambition to track animals), so it cannot detect if a same animal has been detected multiple times during the same flight.",
"This means that when detecting animals in new images, there is a potential risk of counting a same animal multiple times.",
"However, due to the scarcity of animals in images overall and since the task can only by definition lead to an approximation of the real animal number, there is still a clear advantage for using the proposed system as compared with traditional techniques.",
"Figure: Example of the crowdsourced image annotations.",
"Left: annotations of the volunteers, represented as red polygons (one polygon per annotation and user).",
"Right: Annotation confidence map.",
"All the areas in green-to-yellow are retained as the ground truth annotations, with the exception of the one in the bottom right, since it was annotated only by one volunteer.In the following, we present the main components of our machine learning pipeline, as well as the iterative refinement with active learning."
],
[
"Animals detection system",
"The proposed system is composed of three steps, as illustrated in Fig.",
"REF : Definition of object proposals [3], [25], i.e.",
"regions of interest, which are likely to contain an animal (Section REF ); Extraction of a set of mid-level appearance descriptors, or features, defining animals meaningful visual characteristics (Section REF ); A classification model, or detector, learning from the training set proposals and their features to detect animals in new regions (Section REF ).",
"Figure: Pipeline for the detection of animals by object proposalsNaïve approaches to object detection require a classifier to scan all possible windows centered on each pixel, at every possible scale.",
"Although exhaustive, this strategy is computationally heavy, since windows containing animals correspond to a very small fraction of the image data and most computations are wasted.",
"One could discard regions of the image where the object is unlikely to occur, for instance by modeling class co-occurrences and discarding unlikely backgrounds, as [16] did for the detection of cars.",
"However, this strategy is hardly applicable to a semi-arid Savanna where animals can stand everywhere and co-occurrence of animals to background is uniform.",
"The alternative solution proposed is to quickly find an overcomplete set of regions likely to contain objects of interest and to consider them as candidates to be screened.",
"In computer vision this concepts is known as object proposals generation.",
"Object proposals have long been used in object detection pipelines in natural images [26], [3], [24].",
"The aim of object proposals is to provide meaningful, context-dependent and adaptive spatial support from which it is possible to extract meaningful and informative appearance descriptors needed to train an accurate object detector, while reducing the prediction space of the latter (see Section REF ).",
"This is because we process only a much smaller candidate subset of all the possible image windows, which are likely to contain an animal.",
"This step mainly discards regions that are very likely to not contain any positive instance.",
"Such a subset can be defined, for example, as the ensemble of windows containing high density of closed contours [26], as a set of windows containing object-like color and edge distribution [3] or as windows containing sets of similar regions dissimilar from those not contained in the bounding box [24].",
"A good generator of object proposals must lead to high recall rates, i.e.",
"it must cover with proposals all the positive objects of interest.",
"It has usually low precision, because many ambiguous proposal windows that do not contain any object of interest are also included (overcomplete set).",
"Such a trade-off is acknowledged, since the detection of the animals is left to the detector.",
"Our object proposal system relies on two observations: - Standing animals cast a shadow.",
"We define proposal based on a thresholding of the value channel issued by the HSV transform.",
"Connected regions with an area smaller than three pixels are discarded and the centroids of the remaining regions define the proposals.",
"- Laying animals, as well as animals located in the shade of a tree do not cast a distinctive shadow.",
"To cope with this, we group responses of a Sobel edge detector applied on the blue channel.",
"The filter produces a map of edge scores which is binarized by a threshold.",
"The centroids of the connected regions larger than three pixels define the edge-based proposals.",
"This second approach proved to be very informative, because of the high contrast and sharp edges of animal furs.",
"Either method defines a set of proposals, but the highest recall is obtained by combining the two.",
"For most animals the two methods produce proposals in agreement, i.e.",
"very similar to each other, both in location and size, leading to duplicate proposals.",
"To avoid this problem, we merged proposals closer than 15 pixels (i.e.",
"closer than 60 cm).",
"This threshold is in principle smaller than the average distance between close-by animals, and also corresponds to the average width of an animal in our dataset.",
"Finally, all images are downgraded to 8 cm resolution (i.e.",
"by a factor 2), since we observed that the results did not change significantly, but computational effort was greatly reduced: after such downgrading, the number of pixels is reduced by 4, and consequently the computation of all the features is reduced by a proportional amount.",
"The efficiency of the rest of the proposed system is not affected by the change in resolution, as it scales quadratically to the number of training examples and only linearly with dimensionality, which depends on the type of appearance descriptor.",
"Note that the dimensionality of the appearance features does not depends on the spatial resolution, but only their computation is affected.",
"We will detail this observation on the resolution in the experimental section."
],
[
"Features",
"To train the detector, each proposal must be represented by features describing its appearance (e.g.",
"colors and textures).",
"Since our detector is based on a linear classifier (as detailed in Section REF ), we want these features to translate complex and nonlinear appearance variations into linearly separable characteristics.",
"Furthermore, we want to employ features that are invariant to the rotation of the window containing the animal, since absolute orientation must not affect the detection scores.",
"In this work, we considered two types of features: - Histogram of colors (HOC): This descriptor summarizes the probability distribution of colors in a given image patch, by computing their histogram.",
"It is computed over a square region of $25 \\times 25$ pixels centered on the proposal.",
"The values are summarized in 10 bins and, for each proposal, the histograms of the red, blue and green bands are concatenated, yielding a 30-dimensional feature vector.",
"- Bag-of-visual-words (BoVW, [19]): like color histograms, BoVW relies on a quantization of the image data.",
"However, rather than binning color channels independently, BoVW accounts for dependencies across the whole RGB space.",
"Our BoVW extraction pipeline is as follows (Fig.",
"REF ): Figure: Extraction of the BoVW descriptor in four steps.",
"We extract 20'000 $25 \\times 25$ pixels patches from 100 different images.",
"To ensure that animals are well represented, we enforce that 5'000 samples are located on animals annotations, while the rest is sampled randomly across background regions.",
"For each window, we then concatenate the RGB pixel values in a single 1'875-dimensional vector (i.e.",
"$25 \\times 25 \\times 3$ ).",
"We apply the $k$ -means clustering algorithm to group the feature vectors into $k$ clusters.",
"The centers of the clusters are used as the representative patterns in the images, or visual words.",
"All the possible $25 \\times 25 \\times 3$ patches in all the images are then assigned to the closest among the $k$ visual words, to generate a dense map of visual words.",
"Like the HOC features above, a BoVW feature vector is a $k$ -dimensional histogram of visual words occurring in the $25 \\times 25$ window surrounding the candidate pixel considered.",
"The BoVW procedure offers a series of beneficial aspects over pure color-based descriptors.",
"First, the binning of the image is more semantic, as the presence of a given visual word corresponds to the occurrence of a specific pattern in the window.",
"Secondly, mapping the images into a space extrapolated from overlapping windows ensures that descriptive features are spatially smooth, which is a prior belief on image data, while locally keeping signals variance."
],
[
"Detector: ensemble of Exemplar SVM",
"Once the proposals have been defined and features extracted, we train the animal detector on these inputs.",
"The detection task is formulated as a binary classification problem, involving a positive (“animal”) and negative (“background”) class.",
"The problem is challenging for two main reasons: - Both classes are very heterogeneous, as shown in Fig.",
"REF .",
"On the one hand, most animals have a light fur, but one can also find darker, brown and gray/black (Ostriches) individuals.",
"Shapes vary strongly and a projected shadow is a frequent, but not persistent characteristic.",
"On the other hand, the background class contains diverse land-cover types such as bare soil, sand, roads, annual and perennial grasses, with sparse shrubs and trees and the corresponding shadows.",
"Aardwolves holes are frequent and appear as black spots that are visually very similar to animals' shadows.",
"Beside the complexity of the classification task, many background objects can be confused with animals.",
"- Animals are very rare in terms of total number of instances and occupy only a tiny fraction of the images in terms of spatial coverage.",
"Depending on the local animal density and on the land cover type, which influences the amount of object candidates, the ratio of positive to negative proposals varies between 1:200 and 1:500, while the total area occupied by an animal and its shadow is around 5'000 pixels (roughly 8 m$^2$ ), which is only $0.04$ % of an image.",
"Figure: Visual heterogeneity within the positive class (animals).To tackle these issues we adopt the Ensemble of Exemplar Support Vector Machines (EESVM) detector [14].",
"The EESVM is composed by an ensemble, where each member learns a separate model for each positive instance in the training set, rather than learning a global model at once.",
"Each model is a binary SVM trained to discriminate between a single positive and many negative instances, and is known as an Exemplar SVM, see Fig.",
"REF a.",
"This strategy offers flexibility to encode very diverse positive examples, while keeping the overall robustness of a single detector given by the ensemble learning component.",
"Once all the ESVMs in the ensemble have been trained, they all evaluate the candidate object proposals in new images: each ESVM produces a score, which can be interpreted as a similarity of the sample under evaluation to the positive proposal on which the ESVM has been trained on.",
"Since it has been evaluated by every ESVM, the new sample receives as many scores as there are positive proposals in the training set.",
"We assign the proposal to the positive class “animals” if at least one ESVM has predicted this label (Fig.",
"REF b).",
"Figure: a) An exemplar SVM separating a single positive example (framed in red) against a negatives set made of background and other positive examples.",
"b) The final decision function is an ensemble of linear models, one per positive example.",
"Each color corresponds to a single ESVM; the orange area indicates the background class prediction.In general, the scores produced by the different ESVMs cannot be compared directly, because the ESVMs have been trained independently to score the largest possible value on every single positive instance.",
"As a consequence, each ESVM can score the same positive example with very different values, depending on the similarity of the candidate region to the example used for training.",
"To ensure that the score provided by an ESVM on a test sample is comparable to those of the other ESVMs, we normalize each score so that the distance between the margin (red line in Fig.",
"REF a) and the positive proposal is equal to one."
],
[
"Model improvement by active learning",
"It is known that ground truths crowdsourced by querying non-experts are prone to inevitable errors introducing label noise, resulting in models that are harder to train and ultimately to lower accuracy [9], [6].",
"To improve the quality of the ground truth and consequently the accuracy of the system specifically deal with two possible errors: [noitemsep,nolistsep] - False positives: ground truth objects wrongly labeled as animals, while their correct class is “background”; - False negatives: ground truth objects wrongly labeled as background, while their correct class is “animal”.",
"On the one hand, false positives can be removed by a visual inspection of the proposals in the training set, since their number is limited (typically, we use 300 to 700 proposals in this study).",
"On the other hand, false negatives cannot be found by systematic user inspection, since this set can easily contain tens of thousands of object proposals, and most of those will be of the actual background class.",
"To tackle this task and lead the selection of a few negative examples to be screened by a user, we propose to use an iterative technique known as active learning [23].",
"Active learning is based on a user-machine interaction: the machine asks the user to provide labels of the objects, for which the current prediction is not confident.",
"Given the answer of the user, newly labeled examples are added to the training set and the model becomes more robust in areas of low confidence.",
"Here, differently from standard active learning pipelines, we aim at finding wrongly labeled proposals in the training set, i.e.",
"background proposals that contain actual animals, thus possibly wrongly annotated by annotators.",
"By definition, these proposals have a visual appearance that is very similar to the “animal” class, and consequently they lie close to the current EESVM decision boundary.",
"We formulate our active learning routine as follows: during training, an ESVM is trained on a single positive proposal and all the negative proposals.",
"Then, this model assigns a detection score to all the negative objects in the training set.",
"If a false negative similar to the exemplar is present, it will receive a high detection score.",
"The top scoring objects are then shown to the user (Fig.",
"REF a), who is invited to inspect them via a graphic user interface.",
"Following the user's response, three actions can be undertaken: - The proposal correctly belongs to the class “background”: in this case, nothing happens and the proposal continues to be treated as a negative example in the training set.",
"The next ESVM is trained normally.",
"- The proposal is a false negative: in this case the proposal is removed from the negative training set and the ESVM retrained without the confusing example (Fig.",
"REF b).",
"The user can also choose to add the newly found animal to the positive training set, thus increasing the number of ESVMs by one; - The user cannot decide: in case of extremely ambiguous proposals, we simply remove the proposal from the training set to ensure that no conflicting information is harming the learning process.",
"Figure: Efficient search of false negative proposals with active learning.",
"(a) The negative proposals with the highest score, i.e.",
"the most similar to the positive proposal (framed in red in this example), are shown to the user, who recognizes that one of them (framed in blue) is a false negative.",
"(b) The false negative is removed from the negative training set, thus modifying the decision function (red line) of the positive ESVM being trained.",
"In addition, the false negative can be used as an additional ESVM in the model."
],
[
"Experiments and results",
"In this section we present results on the Kuzikus dataset.",
"First, we perform an evaluation on the impact of our hyperparametrization (Section REF ).",
"Then, we report the results obtained by our system after model selection, including the contribution of the active learning step (Section REF )."
],
[
"Ablation study: features used, their parameters and image resolution",
"The ablation study evaluates the contribution of the different components of the proposed pipeline to the full model.",
"To this end, we convert the animal detection problem into a balanced two-class classification problem and employ a linear SVM as base classifier.",
"The reason behind this choice is that the analysis of the global factors of variation of the problem are much more robust when training models considering the whole class-conditional distribution, rather than depending from single positive examples in extremely unbalanced settings (EESVM).",
"Furthermore, note that we employ linear SVM classifiers as main building blocks of our proposed EESVM.",
"The training and test sets comprise 1'324 and 568 proposals containing animals, respectively.",
"To compare to [17], we use the same number of positive and negative examples.",
"To assess the random variability of our results, each experiment is repeated five times.",
"Each run uses the same set of positive proposals, and a randomly drawn negative set sampled over non-animal ground truth regions.",
"The hyperparameter trading off training errors and margin width of the linear SVM $C$ is selected via a 5-fold cross-validation.",
"ROC curves averaged over the 5 runs report accuracy.",
"Table REF summarizes the parameters considered in each experiment.",
"Table: Summary of the parameters considered in the experiments in Section .Figure REF presents the detection scores obtained with the HOC and BoVW features independently and with their concatenation.",
"To balance the relative importance of each feature type, the features are first normalized to $z$ -scores.",
"Then, they are further normalized to a unit $\\ell _2$ -norm, as suggested by [10].",
"Figure: Balanced classification scores per feature: HOC, BoVW and their concatenation.When used alone, the HOC features perform very well in comparison to the more elaborated and complex BoVW (with 100 words).",
"This suggests that colors hold a large part of the relevant information, while the shapes and structures represented by the visual words seem to be less important.",
"Both features perform similarly when requiring a small false positives rate.",
"However, the combination of both features clearly improves over the single sets along the whole ROC curve.",
"For instance, if a false positive rate of $0.03$ is retained, the recall for the combined features is $0.75$ , while being only $0.50$ for HOC and $0.45$ for BOVW.",
"In the next experiments, we will use the concatenation of the feature types to build the base appearance models."
],
[
"Number of visual words – $k$",
"Curves in Fig.",
"REF illustrate how the number of visual words influences the performance of the models when trained on differently clustered BoVW features.",
"Here, we show the effects for $k = 100$ and $k = 300$ words, when .",
"Smaller $k$ values produced significantly worse results, while larger did not produce better accuracies.",
"As one could expect for problems involving complex appearance variations of positive and background classes, using more words improves the classification.",
"The benefit is maximal for recall rates between $0.35$ and $0.60$ .",
"In this range, using 300 words improves the recall rate up to 15% (Fig.",
"REF ).",
"The number of words required to properly describe the images content reveals the diversity and complexity of the patterns found in the dataset.",
"While several thousands of visual words are often used with natural images, here a few hundred words can already retain most of the information.",
"Figure: Detection scores with 100 and 300 words."
],
[
"Image resolution",
"The last ablation experiment concerns the spatial resolution of the images.",
"Results in Fig.",
"REF show the effect of reducing the original ground sampling distance (GSD), which is of approximatively 4 cm per pixel, to 8, 12 and 16 cm per pixel.",
"Using the original image GSD did not improve the results significantly if comparing to a half resolution degradation, while it increased the computational time needed to extract features significantly.",
"Remember that the relation between the resolution and the computation for the BoVW descriptor is linear with a factor proportional to the increase in number of pixels per spatial unit.",
"The curves suggest that a GSD of 16 cm is too coarse to detect animals.",
"Interestingly, the benefits of using a GSD of 8 cm over a GSD of 12 cm only appear for recall rates of $0.65$ .",
"This indicates that two thirds of the animals do not require a GSD higher than 12cm, but the last third of the animals becomes more distinguishable when the resolution is at least of 8 cm.",
"Figure: Detection ROC curves with four different spatial resolutions."
],
[
"Animals detection with ESVMs and unbalanced class-ratio",
"This section deals with the original task of detecting animals in the full dataset, characterized by a strongly unbalanced class-ratio.",
"We employ models optimized thanks to the ablation study.",
"From now on, all the background objects are included in the training and test sets.",
"Each of the original 654 annotated images (see Section ) was assigned entirely to either the training or the test set, meaning that all the animals annotated on one image are included in the same set.",
"This way, we avoid any spatial correlation between training and test sets.",
"Both sets are supposed to contain a similar number of large, medium and isolated animals and to show similar probability distributions.",
"The training set comprises 574 positive objects (animals) and 403'859 negatives, giving a ratio of 1:703.",
"The test set has 284 positives and 160'384 negatives, yielding a ratio of 1:564."
],
[
"Results of the proposed system",
"Figure REF shows the precision-recall curves obtained with the proposed system (blue curve).",
"Because animals are very rare in the dataset, achieving a high precision is difficult, but in our context it is more important to ensure a high recall.",
"The false detections can be manually eliminated by the user in a further step, while it is much more difficult to recover animals missed by the detector.",
"Our results show that indeed we can obtain high recall: for example we can achieve 75% of correct detections for a precision of 10%.",
"An interesting property of the EESVM is that a detection is always associated with the positive training example most similar to it.",
"If additional information about the training examples is available (e.g.",
"the species), it can be easily transferred to the detection directly at test time.",
"Unfortunately, we could not quantitatively test such an idea for our dataset, because the ground truth did not include species information (in the crowdsourcing campaign only the presence/absence of animals was recorded).",
"Nevertheless, we observe that many detections are visually very similar to their closest proposal, as illustrated in Fig.",
"REF .",
"The appearance of the detection is not simply a direct matching of the proposal itself, but each ESVM learns the color statistics independently from the spatial orientation of the tiles: for example the detection depicted in Fig.",
"REF e shows animals in very different positions, while in Fig.",
"REF f the shadowing is reversed.",
"Figure: Detected animals (columns 1 and 3) and associated proposals (columns 2 and 4).",
"The images are centred on the detections / proposals.",
"Note that they are presented in full resolution and offer a larger view than the 25×2525 \\times 25 pixels regions used for feature extraction."
],
[
"Effect of the time of the day",
"Here, we study the robustness of the detection pipeline with respect to the acquisition time of the day.",
"The time of the day strongly influences the presence of shadows and the spectral response of the camera: models trained on morning images could be suboptimal for the detection of animals on images acquired in afternoon, since image statistics are different.",
"This problem is known as domain adaptation [22].",
"To determine whether the time of the day influences the detection rate, we define two sets: one made of images taken in the morning (from 09h13 to 09h28) and another of images taken around midday (from 13h08 to 13h30), respectively.",
"Each set is then subdivided in a training and a test subset, like in the experiments above.",
"Table REF indicates the number of images and animals in each subset.",
"To ensure a fair comparison, we used 176 positive examples for both time steps, corresponding to the total number of animals in the midday acquisitions.",
"The detection results are reported in Fig.",
"REF : Table: Composition of the subsets “Morning” and “Midday”.Figure: Detection results obtained by models applied on morning acquisition (red lines) and at noon (blue lines), respectively.",
"Dashed lines refer to models trained with proposals from the morning image set, dotted lines of the midday image set, and solid lines of both sets combined.",
"(a) precision-recall curve; (b) recall and (c) precision curves as function of the detector threshold.- From the precision-recall curves (Fig.",
"REF a), we observe that the morning test set is easier to classify than the midday test set, regardless of the proposals involved in training.",
"We hypothesize that the more discriminative shadows play a role in this difference.",
"- The recall curves (Fig.",
"REF b) consider the question `Given a detector threshold and all the animals in the test set, how many were correctly identified as positive detections?'.",
"These curves indicate that models trained on morning data allow training more accurate models, even when such models are used to classify the midday test set.",
"However, this result is always outperformed by the situation where training sets from both times have been jointly used, since the training set is larger and covering more variations in appearance.",
"- The precision rates (Fig.",
"REF c) consider the question `Given a detector threshold and all the positive detections, how many were real animals?'.",
"In this case, we observe another behaviour: the best results are obtained when the training and the test subsets are from the same time period.",
"Similar results are observed for both the morning and midday datasets.",
"On the contrary, mixing the acquisition times in training leads to a degradation of the results, mostly due to the many false positives.",
"These curves underline that in order to minimize the number of false positives, the images used to train the models must be acquired at a time as close as possible as the one when one wants to detect animals.",
"We conclude that flying in the morning and always at the same hour of the day can lead to better results.",
"However, this analysis may be biased by the fact that the morning and midday image sets were not taken over the exact same locations.",
"Even thought the land cover is very similar, it is possible that one of the image sets contains more confusing objects or hiding places for the animals."
],
[
"Active learning",
"In this section, we study the possibility to refine the training data using active learning.",
"In the previous section, we considered an ensemble of 574 ESVMs, each one corresponding to a positive proposal, trained against 403'859 negative proposals.",
"We now aim at highlighting proposals of the negative set that can potentially contain an animal (false negatives) and have them screened by a human user.",
"We proceed sequentially one ESVM at a time: a real user is asked to provide feedback on the eight most uncertain negative examples provided by the given model (Fig.",
"REF ).",
"In one hour, the user screened 120 models.",
"Among all negative proposals, 55 were marked as animals and added to the positive set, thus raising the number of ESVM to 629.",
"In parallel, the user also marked 52 originally negative proposals as unclear: these were simply removed from the set of negative proposals.",
"After one hour, hardly any false negative could be found by the user.",
"The detection results obtained with the 629 ESVM is compared to the original one (obtained with 574) in Fig.",
"REF .",
"Note that these results cannot be compared directly with those of Fig.",
"REF , as in this case we use a complete test set including images acquired both in the morning and at noon.",
"The precision-recall curves reveal that for this dataset, active learning enhances the predictive ability of the EESVM for recall rates below 55%.",
"It does not help to find animals that are either difficult to detect (e.g.",
"species that have never been seen by the system at training time), or observed in drastically different conditions.",
"This form of sampling is an effective way to improve an existing ground truth: 55 additional animals were found in the training images and we were able to remove proposals, for which a human expert was unable to decide whether an animal was present or not.",
"This way, a single user was able to screen the entire negative set (with more than 400'000 proposals) within an hour.",
"The user prioritized low-confidence one, thus showing the interest of using active learning instead of a random sampling strategy."
],
[
"Conclusions",
"In this paper, we proposed a semi-automated data-driven system to detect large mammals in the Semi-arid African Savanna.",
"Such approaches are crucial to make the difference in near real-time conservation and war against wildlife poaching.",
"Our system first processes many sub-decimeter UAV images to highlight possible candidate regions likely to contain animals (or proposals), and then infer the presence of animals among them by means of an object recognition model, the ensemble of exemplar SVM (EESVM).",
"We study and discuss the impact of every system components by performing a complete ablation study, and highlight differences in the data representation (i.e.",
"features) and other crucial aspects such as image resolution and acquisition time.",
"For the purpose of detecting mammals, a resolution of 8 cm proved to be sufficient when combined with the powerful histogram of colors and bag-of-visual-words descriptors.",
"When applied to the full problem, the proposed system achieves promising results and demonstrates that the detection of animals in aerial images in the semi-arid Savanna is feasible when employing simple RGB camera mounted on a UAV.",
"Even if a high recall rate can be obtained, a human operator is required to verify the false negatives and to improve the available ground truth, a step that relies on active learning.",
"Since our model is based on object proposals, it is also computationally advantageous over naïve techniques, as we only probe windows candidates likely to contain an animal.",
"Furthermore, using the object proposal strategy jointly with the EESVM model opens for fine grained classification applications, such as the identification of animal species.",
"Since it relies on static images acquired over a pre-defined flight-path, the current system is not able to provide exact counts for two main reasons.",
"First, the same animal can be observed in more than one image with no way of disambiguating the detections.",
"One option could be to plan image acquisition when animals are less active, but then also less visible.",
"Instead, a promising idea would be to use UAV videos for making the detector aware of the temporal dimension: considering temporal trajectories makes it possible to re-identify animals based on their paths and more realistic counts can be provided.",
"The second reason is that the current system might detect two individuals that are very close as a single instance: in this case, one could use prior knowledge about the animal size to post-process the detections and possibly disambiguate animal clusters, for example by estimating size-constrained bounding boxes, of a size equal to the one of the animal being detected.",
"However, although the approach presented here is able to only approximate the actual number of animals, it provides much more realistic numbers compared to traditional techniques.",
"Finally, UAV data allow to process large geographical areas and the system presented in this paper is likely to represent a significant saving in money and time for wildlife land managers.",
"It also represents a safe way to carry out animal surveys."
],
[
"Acknowledgements",
"This work has been supported by the Swiss National Science Foundation (grant PZ00P2-136827 (DT, http://p3.snf.ch/project-136827).",
"The authors would like to acknowledge the SAVMAP consortium (in particular Dr. Friedrich Reinhard of Kuzikus Wildlife Reserve, Namibia) and the QCRI and Micromappers (in particular Dr. Ferda Ofli and Ji Kim Lucas) for the support in the collection of ground truth data."
],
[
"References",
"authoryear"
]
] | 1709.01722 |
[
[
"Decentralized and Recursive Identification for Cooperative Manipulation\n of Unknown Rigid Body with Local Measurements"
],
[
"Abstract This paper proposes a fully decentralized and recursive approach to online identification of unknown kinematic and dynamic parameters for cooperative manipulation of a rigid body based on commonly used local measurements.",
"To the best of our knowledge, this is the first paper addressing the identification problem for 3D rigid body cooperative manipulation, though the approach proposed here applies to the 2D case as well.",
"In this work, we derive truly linear observation models for kinematic and dynamic unknowns whose state-dependent uncertainties can be exactly evaluated.",
"Dynamic consensus in different coordinates and a filter for dual quaternion are developed with which the identification problem can be solved in a distributed way.",
"It can be seen that in our approach all unknowns to be identified are time-invariant constants.",
"Finally, we provide numerical simulation results to illustrate the efficacy of our approach indicating that it can be used for online identification and adaptive control of rigid body cooperative manipulation."
],
[
"Introduction",
"Multi-robot cooperative manipulation has seen great progress in the last several decades.",
"In the 2D planar scenario, multi-robotic system demonstrates its ability to transport times larger and heavier loads under various conditions [1], [2], [3].",
"Recently further attention has been paid to the more general 3D cooperative manipulation of a rigid body, such as multi-finger grasping [4], motion planing to transport a large object [5], impedance control for cooperating manipulators based on internal force [6], multiple quadrotors with a suspended rigid-body load [7] and robust cooperative manipulation without force and torque information [8].",
"In most of these works, even though some assert that they are adaptive and can deal with uncertainties, it is usually assumed that either kinematic parameters (relative position and orientation) or dynamic parameters (mass, mass center, inertia tensor) are at least partially known, or robots implicitly communicate with a central processing unit to help decision making.",
"These assumptions may be problematic in practice — it is impossible to always have prior knowledge of the unknown load while implicit communication with a central unit is only available under limited circumstances and essentially reduces the overall distributeness of the system.",
"As a result, a suitable identification approach to estimating unknown kinematic and dynamic parameters is needed, which benefits multi-robot cooperative manipulation.",
"Though decentralized parameter identification for planar cooperative manipulation has been studied in [9] using velocities and forces measured in inertia frame, it is difficult to implement these methods for systems involving a rigid body due to the complexity of dynamics and inconvenience of processing inertia frame measurements.",
"Moreover, forces and torques are practically measured or estimated in a local reference frame and thus estimation of the relative orientation and position from the local frame to inertia frame – which is typically time-varying – is additionally required to transform local forces and torques to the inertia frame.",
"In this paper, we propose a fully decentralized and recursive approach to identifying the kinematic and dynamic unknowns for cooperative manipulation of a 3D rigid body.",
"To the best of our knowledge, similar problems have not been addressed before.",
"The approach proposed can be used for planar cooperative manipulation as well.",
"An advantage of our approach is that the identification only relies on local measurements,We refer to measurements in a local reference frame as “local measurements” and those in an inertial frame as “global measurements”.",
"the benefits of which are three-fold: i) it is consistent with robotic manipulation where control laws and forces are applied in local reference frame; ii) the rigid body dynamics in a local reference frame are more concise and thus the identification is simplified; iii) it can be shown that all the kinematic and dynamic unknowns to be estimated are constant and no estimation on time-varying parameters/states is needed.",
"The other contributions of this paper include the derivation of linear observation models with evaluable state-dependent uncertainties, dynamic consensus in different coordinates and appropriate filtering of dual quaternions for our specific problem.",
"The rest of this paper is organized as follows: section::note defines the most frequently used notations in the paper.",
"section::preliminary briefly reviews quaternions and dual quaternions that are used to develop linear observation model for pose estimation.",
"section::problem formulates the identification problem with common assumptions in cooperative manipulation and in section::measure linear observation models for kinematic and dynamic unknowns are derived.",
"section::filter discusses state-dependent uncertainties evaluation, dynamic consensus in different coordinates and filtering on dual quaternions so that the identification problem can be properly solved in a distributed way.",
"Numerical results are given in section::num and conclusions are made in section::conclusion."
],
[
"Nomenclature",
" Table: NO_CAPTION"
],
[
"Preliminaries",
"In this section, we give a brief review of quaternions and dual quaternions that are often used to represent $SO(3)$ and $SE(3)$ .",
"A more detailed introduction to quaternions and dual quaternions can be found in [10], [11].",
"In this paper, $Q$ and $\\widehat{Q}$ are respectively used to denote quaternion and dual quaternion."
],
[
"Quaternion",
"A quaternion $\\widetilde{\\mathbf {q}}=(q_0,\\,\\mathbf {q})\\in Q$ is a 4-tuple where $q_0\\in \\mathbb {R}$ is the scalar part and $\\mathbf {q}\\in \\mathbb {R}^3$ the vector part.",
"The multiplication $\\odot $ of two quaternions $\\widetilde{\\mathbf {p}}$ and $\\widetilde{\\mathbf {q}}$ is defined as $\\begin{aligned}\\widetilde{\\mathbf {p}}\\odot \\widetilde{\\mathbf {q}} &= (p_0 q_0 - \\mathbf {p} \\cdot \\mathbf {q}, p_0 \\mathbf {q} + q_0 \\mathbf {p} + \\mathbf {p} \\times \\mathbf {q}).\\end{aligned}$ Furthermore, linear operators $(\\cdot )^+:Q\\rightarrow \\mathbb {R}^{4\\times 4}$ and $(\\cdot )^-:Q\\rightarrow \\mathbb {R}^{4\\times 4}$ associated with eq::quatmult are defined as $\\widetilde{\\mathbf {q}}^+ =\\begin{bmatrix}q_0 & -\\mathbf {q}^T\\\\\\mathbf {q} & {\\mathbf {q}}^\\times + q_0 \\text{I}\\end{bmatrix}, \\,\\widetilde{\\mathbf {q}}^- =\\begin{bmatrix}q_0 & -\\mathbf {q}^T\\\\\\mathbf {q} & -{\\mathbf {q}}^\\times + q_0 \\text{I}\\end{bmatrix}$ where $(\\cdot )^\\times : \\mathbb {R}^3\\rightarrow \\mathbb {R}^{3\\times 3}$ is a linear operator such that $\\mathbf {a}^\\times \\mathbf {b}=\\mathbf {a}\\times \\mathbf {b} $ and $\\text{I}\\in \\mathbb {R}^{3\\times 3}$ is the identity matrix, then $\\widetilde{\\mathbf {p}}\\odot \\widetilde{\\mathbf {q}} = \\widetilde{\\mathbf {p}}^+ \\cdot \\widetilde{\\mathbf {q}} = \\widetilde{\\mathbf {q}}^- \\cdot \\widetilde{\\mathbf {p}}.$ The conjugate $\\widetilde{\\mathbf {q}}^*$ of a quaternion $\\widetilde{\\mathbf {q}}$ is $\\widetilde{\\mathbf {q}}^* =(q_0,\\,-\\mathbf {q}) $ and $\\widetilde{q}\\widetilde{q}^*=\\widetilde{q}^*\\widetilde{q}=(\\Vert \\widetilde{q}\\Vert ^2,\\,0)$ .",
"Unit quaternions $\\widetilde{\\mathbf {q}}$ are quaternions with $\\Vert \\widetilde{\\mathbf {q}}\\Vert =1$ and can be used to represent $SO(3)$ such that $\\nonumber \\widetilde{\\mathbf {q}} = (\\cos \\dfrac{\\theta }{2},\\sin \\dfrac{\\theta }{2}\\mathbf {\\omega })$ where $\\theta \\in [-\\pi ,\\,\\pi ]$ is the angle and the unit vector $\\mathbf {\\omega }\\in \\mathbb {R}^3$ is the rotational axis.",
"Besides for unit quaternions we have $\\widetilde{\\mathbf {q}}\\odot \\widetilde{\\mathbf {q}}^* = \\widetilde{\\mathbf {q}}^*\\odot \\widetilde{\\mathbf {q}} =(1,\\,\\mathbf {0}).$ Let $b^{\\prime }\\in \\mathbb {R}^3$ be obtained by rotating $b\\in \\mathbb {R}^3$ with a unit quaternion $\\widetilde{\\mathbf {q}}$ and then we have $\\widetilde{b}^{\\prime } = \\widetilde{\\mathbf {q}}\\odot \\widetilde{b}\\odot \\widetilde{\\mathbf {q}}^*$ where $\\widetilde{b}=(0,\\,\\mathbf {b})$ and by the conjugate property of unit quaternion it is equivalent to $\\widetilde{\\mathbf {q}}\\odot \\widetilde{b} = \\widetilde{b}^{\\prime }\\odot \\widetilde{\\mathbf {q}}.$"
],
[
"Dual Quaternion",
"A dual quaternion $\\widehat{\\mathbf {x}}=\\widetilde{\\mathbf {p}}+\\epsilon \\widetilde{\\mathbf {q}}\\in \\widehat{Q}$ where $\\widetilde{\\mathbf {p}}, \\widetilde{\\mathbf {q}}\\in Q$ are quaternions and $\\epsilon $ is defined to be $\\epsilon \\ne 0$ and $\\epsilon ^2=0$ .",
"The multiplication $\\otimes $ of two dual quaternions $\\widehat{\\mathbf {x}}_1 = \\widetilde{\\mathbf {p}}_1 +\\epsilon \\widetilde{\\mathbf {q}}_1$ and $\\widehat{\\mathbf {x}}_2 = \\widetilde{\\mathbf {p}}_2+\\epsilon \\widetilde{\\mathbf {q}}_2$ is given by $\\widehat{\\mathbf {x}}_1\\otimes \\widehat{\\mathbf {x}}_2 = \\widetilde{\\mathbf {p}}_1\\odot \\widetilde{\\mathbf {p}}_2+\\epsilon (\\widetilde{\\mathbf {p}}_1\\odot \\widetilde{\\mathbf {q}}_2+\\widetilde{\\mathbf {q}}_1\\odot \\widetilde{\\mathbf {p}}_2).$ Similarly, linear operators $(\\cdot )^+$ and $(\\cdot )^-:\\widehat{Q} \\rightarrow \\mathbb {R}^{8\\times 8}$ for dual quaternions are defined by $\\widehat{\\mathbf {x}}^+=\\begin{bmatrix}\\widetilde{\\mathbf {p}}^+ & \\mathbf {O}\\\\\\widetilde{\\mathbf {q}}^+ & \\widetilde{\\mathbf {p}}^+\\end{bmatrix}, \\hspace{15.0pt}\\widehat{\\mathbf {x}}^-=\\begin{bmatrix}\\widetilde{\\mathbf {p}}^- & \\mathbf {O}\\\\\\widetilde{\\mathbf {q}}^- & \\widetilde{\\mathbf {p}}^-\\end{bmatrix}$ so that $\\widehat{\\mathbf {x}}_1\\otimes \\widehat{\\mathbf {x}}_2 = \\widehat{\\mathbf {x}}_1^+\\cdot \\widehat{\\mathbf {x}}_2 = \\widehat{\\mathbf {x}}_2^-\\cdot \\widehat{\\mathbf {x}}_1.$ Dual quaternions have three conjugates which are respectively $\\widehat{\\mathbf {x}}^{1*}=\\widetilde{\\mathbf {p}}-\\epsilon \\widetilde{\\mathbf {q}}$ , $\\widehat{\\mathbf {x}}^{2*}=\\widetilde{\\mathbf {p}}^*+\\epsilon \\widetilde{\\mathbf {q}}^*$ and $\\widehat{\\mathbf {x}}^{3*}=\\widetilde{\\mathbf {p}}^*-\\epsilon \\widetilde{\\mathbf {q}}^*$ .",
"A dual quaternion $\\widehat{\\mathbf {x}}$ is unit if $\\widehat{\\mathbf {x}}\\otimes \\widehat{\\mathbf {x}}^{2*}=1$ which can be used to represent $SE(3)$ .",
"Given $g=(R,\\,\\mathbf {t})\\in SE(3)$ where $R\\in SO(3)$ is rotation and $\\mathbf {t}\\in \\mathbb {R}^3$ the translation, then we may use unit dual quaternion $\\widehat{\\mathbf {x}}=\\widetilde{q}_r+\\epsilon \\widetilde{q}_d$ to represent $g$ where $\\widetilde{\\mathbf {q}}_r$ is a unit quaternion corresponding to the rotation $R\\in SO(3)$ and $\\widetilde{q}_d$ is a quaternion corresponding to the translation $\\mathbf {t}\\in \\mathbb {R}^3$ that is given by $\\widetilde{q}_d = \\dfrac{\\widetilde{\\mathbf {t}}\\odot \\widetilde{\\mathbf {q}}_r}{2}.$ The rigid body transformation of a point $b\\in \\mathbb {R}^3$ given by a unit dual quaternion $\\widehat{\\mathbf {x}}$ is $\\widehat{b}^{\\prime }=\\widehat{\\mathbf {x}}\\otimes \\widehat{b}\\otimes \\widehat{\\mathbf {x}}^{3*}$ where $\\widehat{b}=1+\\epsilon \\widetilde{b}$ and $\\widetilde{b}=(0,\\,b)$ .",
"It is known for unit dual quaternions that $\\widehat{\\mathbf {x}}^{1*}\\otimes \\widehat{\\mathbf {x}}^{3*}=\\widehat{\\mathbf {x}}^{3*}\\otimes \\widehat{\\mathbf {x}}^{1*}=1$ so that eq::transform can be rewritten as $\\widehat{\\mathbf {x}}\\otimes \\widehat{b} = \\widehat{b}^{\\prime }\\otimes \\widehat{\\mathbf {x}}^{1*}.$ eq::quatlin,eq::dquatlin are often used to derive linear observation models to estimate orientation and pose [11], [12].",
"Figure: Cooperative Manipulation of a 3D rigid body."
],
[
"Problem Statement",
"We consider the problem that a network of $n$ robots manipulate a rigid body load as shown in fig::demo.",
"The network is defined as an undirected graph $G=(V,\\,E)$ where $V=\\lbrace 1,\\,2,\\,3,\\,\\cdots ,\\, n\\rbrace $ is the node set of robots and $E\\subset V\\times V$ the edge set of communication links.",
"In this paper we make the following assumptions for our identification problem.",
"Assumption 1 The network $G$ is connected and each robot $i$ can only communicate with its one-hop neighbours $N_i=\\lbrace j\\in V|\\,(i,\\,j)\\in E\\rbrace $ .",
"Assumption 2 The end-effector of each robot is fixed with the rigid body.",
"Assumption 3 Each sensor frame $\\mathcal {S}_i$ is fixed with the end-effector of robot $i$ as well as the rigid body whose origin is the contact point of robot $i$ with the rigid body.",
"Remark: The rigid body transformation $g_{ji}$ between different sensor frame $\\mathcal {S}_i$ and $\\mathcal {S}_j$ is not priorly known and needs to be identified so that each robot does not know the contact point, position and orientation for the other robots.",
"Assumption 4 The body-fixed linear velocity $v_i$ , body-fixed angular velocity $\\mathbf {\\omega }_i$ , the body-fixed proper acceleration $\\overline{a}_i$ at $i^{\\mathrm {th}}$ contact point and force $\\mathbf {f_i}$ and torque $\\mathbf {\\tau }_i$ applied by robot $i$ to the rigid body are measurable.",
"All of $v_i$ , $\\mathbf {\\omega }_i$ , $\\overline{a}_i$ , $\\mathbf {f_i}$ and $\\mathbf {\\tau }_i$ are measured in sensor frame $\\mathcal {S}_i$ .",
"Remark: The proper acceleration $\\overline{a}_i$ is the acceleration relative to a free-fall which is measurable by a accelerometer.",
"The relationship between proper acceleration $\\overline{a}_i$ and the usual coordinate acceleration $a_i$ is $\\overline{a}_i = a_i - R_i^T\\mathbf {g}$ where $R_i\\in SO(3)$ is the rotation matrix from $\\mathcal {S}_i$ to $\\mathcal {W}$ .",
"The overall identification problem addressed in this paper is formulated as follows.",
"Problem Suppose a group of robots manipulate a rigid body load with assump::1-4 hold, we define a distributed and recursive algorithm so that each robot $i$ can identify The rigid body transformation $g_{ji}$ where $j\\in N_i$ The mass $m$ , mass center $p_c^i$ and inertia tensor $\\mathcal {I}_i$ .",
"Remark: The rigid body transformation $g_{ji}$ , mass $m$ , mass center $p_c^i$ and inertia tensor $\\mathcal {I}_i$ are all time-invariant constants in the identification problem.",
"One of the things that we are particularly interested in is adaptively estimating time-varying mass $m$ , mass center $p_c^i$ and inertia tensor $\\mathcal {I}_i$ with $g_{ji}$ given since these parameters may vary in cooperative manipulation and transportation, e.g., by removing and adding some loads, and play a significant role for the control law design and system stability analysis.",
"Generally speaking, assump::1-4 are common and reasonable for multi-robot coordination and robotic manipulation and some of them may be even further relaxed.",
"The feasibility of measurements in assump::4 is analyzed in subsection::sensor."
],
[
"Feasibility of Sensor Measurements",
"Body-fixed angular velocity $\\mathbf {\\omega }_i$ and proper acceleration $\\overline{a}_i$ are respectively measurable with gyroscope and accelerometer.",
"The major concern is the body-fixed linear velocity $v_i$ , force $\\mathbf {f}_i$ and torque $\\mathbf {\\tau }_i$ which may need a case by case discussion."
],
[
"body-fixed linear velocity $v_i$",
"The body-fixed linear velocity can be calculated by the angular velocity of each joint and the twist of mobile base provided the manipulator Jacobian is given, all of which are either measurable with common sensors or explicitly computable.",
"If the rigid body's orientation and spatial linear velocity is known, body-fixed linear velocity may be determined as well.",
"As for planar cooperative manipulation, body-fixed linear velocity can be measured directly by a optical laser sensor [13]."
],
[
"force $\\mathbf {f}_i$ and torque {{formula:9d7fa55a-06e2-4300-8998-c4ae00745cca}}",
"If the Jacobian matrix is full row rank and the torques at each joint are known, then $\\mathbf {f}_i$ and $\\mathbf {\\tau }_i$ are determined.",
"Sometimes we may need some further assumptions, e.g., if all contacts are point contacts so that robots can only apply forces, then only force sensors should be enough."
],
[
"Observation Modelling",
"In this section, we derive truly linear observation models for the identification problem formulated in section::problem.",
"It can be further shown that the state-dependent uncertainties of these observations can be exactly evaluated."
],
[
"Observation Model for $g_{ji}$",
"It is known that sensor frames are fixed w.r.t.",
"each other, then body-fixed angular and linear velocities $\\mathbf {\\omega }_i$ , $\\mathbf {v}_i$ and $\\mathbf {\\omega }_j$ , $\\mathbf {v}_j$ where $(i,\\,j)\\in E$ are related as $\\mathbf {\\xi }_j = \\mathrm {Ad}_{g_{ji}}\\mathbf {\\xi }_i$ where $\\mathbf {\\xi }_i=\\begin{bmatrix}\\mathbf {\\omega }_i^T & \\mathbf {v}_i^T\\end{bmatrix}^T$ , $\\mathbf {\\xi }_j=\\begin{bmatrix}\\mathbf {\\omega }_j^T & \\mathbf {v}_j^T\\end{bmatrix}^T$ and $\\mathrm {Ad}_{g_{ji}}$ is the adjoint matrix defined by $\\nonumber \\mathrm {Ad}_{g_{ji}}=\\begin{bmatrix}R_{ji} & \\mathbf {O}\\\\\\mathbf {t}_{ji}^\\times R_{ji} & R_{ji}\\end{bmatrix}.$ eq::Ad gives an observation with $\\mathbf {\\omega }_i$ , $\\mathbf {v}_i$ , $\\mathbf {\\omega }_j$ , $\\mathbf {v}_j$ for the relative pose $g_{ji}$ .",
"It is difficult and even intractable to estimate $R_{ji}$ and $p_{ji}$ directly on $SE(3)$ with eq::Ad which is nonlinear and complicated.",
"Even though numbers of papers [10], [11] have used dual quaternions to estimate elements of $SE(3)$ , all of them rely on position measurements in $\\mathbb {R}^3$ while for our problem only twist measurements in $\\mathbb {R}^6$ are provided.",
"The following proposition demonstrates how to do adjoint transformation with dual quaternions so that a linear observation model is developed to estimate $g_{ji}$ .",
"Proposition 1 Given twist $\\mathbf {\\xi }=\\begin{bmatrix}\\mathbf {\\omega }\\\\v\\end{bmatrix}\\in \\mathbb {R}^6$ and unit dual quaternion $\\widehat{\\mathbf {x}}$ for rigid body transformation $g\\in SE(3)$ , then the resulting twist $\\mathbf {\\xi }^{\\prime }=\\begin{bmatrix}\\mathbf {\\omega }^{\\prime }\\\\v^{\\prime }\\end{bmatrix}\\in \\mathbb {R}^6$ by adjoint transformation of $g$ can be calculated by $\\widehat{\\mathbf {x}}$ as $\\widehat{\\mathbf {\\xi }}^{\\prime } = \\widehat{\\mathbf {x}}\\otimes \\widehat{\\mathbf {\\xi }}\\otimes \\widehat{\\mathbf {x}}^{2*}.$ where $\\widehat{\\mathbf {\\xi }}=\\widetilde{\\mathbf {\\omega }}+\\epsilon \\widetilde{\\mathbf {v}}$ .",
"It can be shown merely by calculation that $\\widehat{\\mathbf {x}}\\otimes \\widehat{\\mathbf {\\xi }}\\otimes \\widehat{\\mathbf {x}}^{2*}=\\widetilde{q}_r\\odot \\widetilde{\\mathbf {\\omega }}\\odot \\widetilde{q}_r^*+\\epsilon \\big (\\widetilde{q}_r\\odot \\widetilde{\\mathbf {v}}\\odot \\widetilde{q}_r^*+\\\\ \\widetilde{q}_d\\odot \\widetilde{\\mathbf {\\omega }}\\odot \\widetilde{q}_r^*+\\widetilde{q}_r\\odot \\widetilde{\\mathbf {\\omega }}\\odot \\widetilde{q}_d^*\\big ).$ Note that $\\widetilde{R\\mathbf {\\omega }}=\\widetilde{q}_r\\odot \\widetilde{\\mathbf {\\omega }}\\odot \\widetilde{q}_r^*$ and $\\widetilde{R\\mathbf {v}}=\\widetilde{q}_r\\odot \\widetilde{\\mathbf {v}}\\odot \\widetilde{q}_r^*$ , then for $\\widetilde{q}_d\\odot \\widetilde{\\mathbf {v}}\\odot \\widetilde{q}_r^*+\\widetilde{q}_r\\odot \\widetilde{\\mathbf {v}}\\odot \\widetilde{q}_d^*$ , a pure algebraic manipulation indicates $\\begin{aligned}&\\widetilde{q}_d\\odot \\widetilde{\\mathbf {\\omega }}\\odot \\widetilde{q}_r^*+\\widetilde{q}_r\\odot \\widetilde{\\mathbf {\\omega }}\\odot \\widetilde{q}_d^*\\\\=&\\dfrac{1}{2}\\,\\widetilde{\\mathbf {t}}\\odot \\widetilde{q}_r\\odot \\widetilde{\\mathbf {\\omega }}\\odot \\widetilde{q}_r^* + \\dfrac{1}{2}\\,\\widetilde{q}_r\\odot \\widetilde{\\mathbf {\\omega }}\\odot \\widetilde{q}_r^*\\odot \\widetilde{\\mathbf {t}}^*\\\\= & \\dfrac{1}{2}\\widetilde{\\mathbf {t}\\times R\\mathbf {\\omega }} - \\dfrac{1}{2}\\widetilde{ R\\mathbf {\\omega }\\times \\mathbf {t}} \\\\= & \\widetilde{\\mathbf {t}\\times R\\mathbf {\\omega }}.\\end{aligned}$ Substitute eq::qrlin back to eq::qrall, it can be shown that eq::dquatAd holds, which completes the proof.",
"As a result of prop::1, eq::Ad is equivalent to $\\widehat{\\mathbf {\\xi }}_j =\\widehat{\\mathbf {x}}_{ji}\\otimes \\widehat{\\mathbf {\\xi }}_i\\otimes \\widehat{\\mathbf {x}}_{ji}^{2*}.$ Since $\\widehat{\\mathbf {x}}_{ji}^{2*}\\otimes \\widehat{\\mathbf {x}}_{ji}=1+\\epsilon \\widetilde{0}$ , eq::mea1 can be written as $\\widehat{\\mathbf {\\xi }}_j\\otimes \\widehat{\\mathbf {x}}_{ji} =\\widehat{\\mathbf {x}}_{ji}\\otimes \\widehat{\\mathbf {\\xi }}_i$ , and according to eq::duatLR2, we further have $\\widehat{\\mathbf {\\xi }}_{j}^+\\cdot \\widehat{\\mathbf {x}}_{ji}= \\widehat{\\mathbf {\\xi }}_{i}^-\\cdot \\widehat{\\mathbf {x}}_{ji}.$ Next, simplify the equation above with eq::quatLR,eq::duatLR, the final result is $\\begin{bmatrix}H_q(\\mathbf {\\omega }_i,\\,\\mathbf {\\omega }_j) & \\mathbf {O}\\\\H_q(\\mathbf {v}_i,\\,\\mathbf {v}_j) & H_q(\\mathbf {\\omega }_i,\\,\\mathbf {\\omega }_j)\\end{bmatrix}\\cdot \\begin{bmatrix}\\widetilde{q}^r_{ji}\\\\\\widetilde{q}^d_{ji}\\end{bmatrix}=0$ where $H_q:\\mathbb {R}^3\\times \\mathbb {R}^3 \\longrightarrow \\mathbb {R}^{4\\times 4}$ is defined as $\\nonumber H_q(a,\\,b) =\\begin{bmatrix}0 &(a-b)^T\\\\b-a& (a+b)^\\times \\end{bmatrix}.$ In practice, $\\mathbf {\\omega }_i$ , $\\mathbf {v}_i$ , $\\mathbf {\\omega }_j$ , $\\mathbf {v}_j$ are noisy measurements and we can construct a pseudo-observation model from eq::qrob $\\mathbf {y}=\\begin{bmatrix}H_q(\\mathbf {\\omega }_i,\\,\\mathbf {\\omega }_j) & \\mathbf {O}\\\\H_q(\\mathbf {v}_i,\\,\\mathbf {v}_j) & H_q(\\mathbf {\\omega }_i,\\,\\mathbf {\\omega }_j)\\end{bmatrix}\\cdot \\begin{bmatrix}\\widetilde{q}^r_{ji}\\\\\\widetilde{q}^d_{ji}\\end{bmatrix}$ and enforce the pseudo-observation $\\mathbf {y}=0$ just like [11], [12] with position measurements.",
"Thus a linear observation model $y= H\\cdot \\widehat{\\mathbf {x}}$ is developed to estimate relative rigid body transformation matrix $g_{ji}$ with dual quaternions.",
"A specific filter is developed to solve the dual-quaternion-based pose estimation problem in subsection::poseest."
],
[
"Observation Model for $\\mathbf {\\alpha }_i$",
"The body-fixed angular acceleration $\\mathbf {\\alpha }(t)$ can be got by differentiating $\\mathbf {\\omega }(t)$ as $\\mathbf {\\alpha }(t)=\\frac{\\mathbf {\\omega }(t+\\Delta t)-\\mathbf {\\omega }(t-\\Delta t)}{2\\Delta t}.$ However, the angular acceleration $\\mathbf {\\alpha }(t)$ evaluated by eq::diffw is noisy and an observation model may be needed.",
"Note that the relative pose $g_{ji}$ is time-invariant, then for angular and proper linear accelerations $\\mathbf {\\alpha }_i$ , $\\overline{a}_i$ and $\\mathbf {\\alpha }_j$ , $\\overline{a}_j$ we have $\\begin{bmatrix}\\mathbf {\\alpha }_j\\\\\\overline{a}_j\\end{bmatrix} = \\text{Ad}_{g_{ji}} \\begin{bmatrix}\\mathbf {\\alpha }_i\\\\\\overline{a}_i\\end{bmatrix}.$ Provided $g_{ji}=(R_{ji},\\,\\mathbf {t}_{ji})\\in SE(3)$ , $\\overline{a}_i$ and $\\overline{a}_j$ are given, a pseudo-observation model for $\\mathbf {\\alpha }_i$ can be derived $y= H_{\\mathbf {\\alpha }}\\mathbf {\\alpha }_i$ where $H_{\\mathbf {\\alpha }}=\\mathbf {t}_{ji}^\\times R_{ji}$ and the enforced pseudo-observation is $y=\\overline{a}_j-R_{ji}\\overline{a}_i$ .",
"In addition, from $\\mathbf {\\alpha }_j = R_{ji}\\mathbf {\\alpha }_i,$ the estimation can be further improved through making consensus over $\\mathbf {\\alpha }_i$ for all $i\\in V$ ."
],
[
"Observation Model for $p_c^i$",
"It is known that the dynamics on $SE(3)$ is $\\mathcal {I}\\dot{\\mathbf {\\omega }} = \\mathbf {T}- \\mathbf {\\omega }\\times \\mathcal {I}\\mathbf {\\omega },$ $\\dot{v}= \\dfrac{\\mathbf {F}}{m}+ R^T\\mathbf {g}-\\mathbf {\\omega }\\times v$ for which the body-fixed frame origin is at the mass center of the rigid body.",
"By eq::propacc we may rewrite eq::se3dyn as $\\mathcal {I}\\dot{\\mathbf {\\omega }} = \\mathbf {T}- \\mathbf {\\omega }\\times \\mathcal {I}\\mathbf {\\omega },$ $\\overline{a}= \\dfrac{\\mathbf {F}}{m}-\\mathbf {\\omega }\\times v$ where $\\overline{a}=\\dot{v}-R^T\\mathbf {g}$ is the proper acceleration of the mass center.",
"Let $p_c^i$ be the observation of the mass center in $\\mathcal {S}_i$ .",
"Then for each robot $i$ a local frame $\\mathcal {O}_i$ is assigned at the mass center so that the rigid body transformation matrix $g_i$ from $\\mathcal {S}_i$ to $\\mathcal {O}_i$ is $g_i=\\begin{bmatrix}\\mathrm {I} & -p_c^i\\\\0 & 1\\end{bmatrix}.",
"$ Given body-fixed angular and linear velocities $\\mathbf {\\alpha }_i$ and $v_i$ , angular and linear proper accelerations $\\mathbf {\\alpha }_i$ and $\\overline{a}_i$ , eq::dyn indicates that $\\mathcal {I}_i\\mathbf {\\alpha }_i = \\mathbf {F}_i\\times p_c^i +\\mathbf {T}_i- \\mathbf {\\omega }_i\\times \\mathcal {I}_i\\mathbf {\\omega }_i,$ $\\overline{a}_i+\\mathbf {\\alpha }_i\\times p_{c}^i= \\dfrac{\\mathbf {F}_i}{m}-\\mathbf {\\omega }_i\\times (\\mathbf {\\omega }_i\\times p_c^i+v_i)$ where $\\overline{a}_i$ is measured by a accelerometer and the total force $\\mathbf {F}_i$ and torque $\\mathbf {T}_i$ applied by all robots are calculated by $\\begin{bmatrix}\\mathbf {T}_i\\\\\\mathbf {F}_i\\end{bmatrix}=\\sum \\limits _{j\\in V} \\text{Ad}_{g_{ji}}^T\\begin{bmatrix}\\mathbf {\\tau }_j\\\\\\mathbf {f}_j\\end{bmatrix}.$ which is shown to be computable by making consensus in different coordinates in subsection::consensus.",
"eq::dynob2 is equivalent to $(\\mathbf {\\alpha }_i^\\times +{\\mathbf {\\omega }_i^\\times }^2)\\cdot p_c^i+\\mathbf {\\omega }_i\\times v_i+\\overline{a}_i = \\frac{\\mathbf {F}_i}{m}.$ Note that the mass $m$ remains unknown, however, if $\\mathbf {F}_i\\ne 0$ , we may let $\\mathbf {F}_i^\\perp \\in \\mathbb {R}^{2\\times 3}$ be the row-orthogonal matrix such that $\\mathbf {F}_i^\\perp \\cdot \\mathbf {F}_i =0, $ i.e., columns of ${\\mathbf {F}_i^\\perp }^T$ spans the null space of $\\mathbf {F}_i$ .",
"Multiply $\\mathbf {F}_i^\\perp $ on both sides of eq::obm, the resulting equation is $\\mathbf {F}_i^\\perp (\\mathbf {\\alpha }_i^\\times +{\\mathbf {\\omega }_i^\\times }^2)\\cdot p_c^i+\\mathbf {F}_i^\\perp \\cdot (\\mathbf {\\omega }_i\\times v_i+\\overline{a}_i)=0.$ Thus an observation model for $p_c^i$ is obtained by eq::obmi2 $y_{p_c}=H_{p_c}\\cdot p_c^i$ where $H_{p_c}=\\mathbf {F}_i^\\perp (\\mathbf {\\alpha }_i^\\times +{\\mathbf {\\omega }_i^\\times }^2)$ and $y_{p_c}=-\\mathbf {F}_i^\\perp \\cdot (\\mathbf {\\omega }_i\\times v_i+\\overline{a}_i)$ ."
],
[
"Observation Model for $\\mathcal {I}_i$",
"Suppose the mass center $p_c^i$ is known, then an observation model for $\\mathcal {I}_i$ can be derived by eq::dynob1 so that $y_{\\mathcal {I}}=H_{\\mathcal {I}}\\cdot \\mathcal {I}_i^S$ where $\\mathcal {I}_i^S=\\begin{bmatrix}\\mathcal {I}_{xx}^i & \\mathcal {I}_{yy}^i & \\mathcal {I}_{zz}^i & \\mathcal {I}_{xy}^i & \\mathcal {I}_{xz}^i & \\mathcal {I}_{yz}^i\\end{bmatrix}^T$ , $\\begin{aligned}\\!\\!&H_{\\mathcal {I}}=\\\\&\\!\\!\\begin{bmatrix}\\!\\!\\alpha _1 \\!\\!\\!& \\!\\!\\!",
"-\\omega _2 \\omega _3 \\!\\!\\!& \\!\\!\\!",
"\\omega _2 \\omega _3 \\!\\!& \\!\\!",
"\\alpha _2-\\omega _1 \\omega _3 \\!\\!& \\!\\!",
"\\alpha _3+\\omega _1 \\omega _2 \\!\\!& \\!\\!",
"\\omega _2^2-\\omega _3^2\\\\\\!\\!\\omega _1 \\omega _3 \\!\\!\\!& \\!\\!\\!",
"\\alpha _2 \\!\\!\\!& \\!\\!\\!",
"-\\omega _1 \\omega _3 \\!\\!& \\!\\!",
"\\alpha _1+\\omega _2 \\omega _3 \\!\\!& \\!\\!",
"\\omega _3^2-\\omega _1^2 \\!\\!& \\!\\!",
"\\alpha _3-\\omega _1 \\omega _2\\\\\\!\\!-\\omega _1 \\omega _2 \\!\\!\\!& \\!\\!\\!",
"\\omega _1 \\omega _2 \\!\\!& \\!\\!",
"\\alpha _3 \\!\\!& \\!\\!\\omega _1^2-\\omega _2^2 \\!\\!& \\!\\!",
"\\alpha _1-\\omega _2\\omega _3 \\!\\!& \\!\\!",
"\\alpha _2+\\omega _1\\omega _3\\!\\!\\end{bmatrix}\\!\\!.\\end{aligned}$ and $y_{\\mathcal {I}}=\\mathbf {F}_i\\times p_c^i +\\mathbf {T}_i$ ."
],
[
"Observation Model for $m$",
"Given $p_c^i$ and these measurements in assump::4, the pseudo-observation model for mass $m$ is trivial from eq::dynob2 $y_m = H_m \\!\\cdot \\!m$ where $H_m=\\overline{a}_i+\\mathbf {\\alpha }_i\\times p_{c}^i+\\mathbf {\\omega }_i\\times (\\mathbf {\\omega }_i\\times p_c^i+v_i)$ and $y_m=\\mathbf {F}_i.$"
],
[
"Observation Model for $p_c^i$ , {{formula:b2e4de68-0f09-45de-9e32-41ef7fc44385}} and {{formula:be37f7fa-93bb-4c1f-bda0-07e8060ae5af}}",
"We have developed individual observation models for mass center $p_c^i$ , mass $m$ and inertia tensor $\\mathcal {I}_i$ among which the estimations of $m$ and $\\mathcal {I}_i$ depend on that of $p_c^i$ .",
"In general, we may prefer to estimate $p_c^i$ , $m$ and $\\mathcal {I}_i$ at the same time rather than separately since the former should be more robust and more accurate.",
"Note that $p_c^i$ , $m$ and $\\mathcal {I}_i$ are all constants, we may observe all of them just in one model.",
"For $p_c^i$ and $\\mathcal {I}_i$ , by eq::dynob1,eq::obmi1 we have $\\begin{bmatrix}-\\mathbf {F}_i^\\times & H_{\\mathcal {I}}\\end{bmatrix}\\begin{bmatrix}p_c^i\\\\\\mathcal {I}^S\\end{bmatrix}=\\mathbf {T}_i.$ Besides if $m$ is replaced by $\\frac{1}{m}$ as the estimated unknown, an observation model for $p_c^i$ and $\\frac{1}{m}$ from eq::dynob2 is $\\begin{bmatrix}-\\mathbf {\\alpha }_i^\\times -{\\mathbf {\\omega }_i^\\times }^2 & \\mathbf {F}_i\\end{bmatrix}\\begin{bmatrix}p_{c}^i\\\\\\frac{1}{m}\\end{bmatrix}= \\mathbf {\\omega }_i\\times v_i+\\overline{a}_i.$ According to eq::oba1,eq::oba2,eq::obmi2, we may derive a model to simultaneously observe $p_c^i$ , $m$ and $\\mathcal {I}$ as $y_D = H_D \\begin{bmatrix}p_c^i\\\\\\mathcal {I}^S\\\\\\frac{1}{m}\\end{bmatrix}$ where $\\nonumber H_D = \\begin{bmatrix}\\mathbf {F}_i^\\perp (\\mathbf {\\alpha }_i^\\times +{\\mathbf {\\omega }_i^\\times }^2) & \\mathbf {O} & 0\\\\-\\mathbf {F}_i^\\times & H_{\\mathcal {I}} & 0\\\\-\\mathbf {\\alpha }_i^\\times -{\\mathbf {\\omega }_i^\\times }^2 &\\mathbf {O} & \\mathbf {F}_i\\end{bmatrix}$ and $\\nonumber y_D=\\begin{bmatrix}-\\mathbf {F}_i^\\perp \\cdot (\\mathbf {\\omega }_i\\times v_i+\\overline{a}_i)\\\\\\mathbf {T}_i\\\\\\mathbf {\\omega }_i\\times v_i+\\overline{a}_i\\end{bmatrix}.$ In this paper, eq::oball is used in section::num for numerical simulation.",
"Observation models of eq::obpci,eq::obmm,eq::obmi1 can be used if some of $p_c^i$ , $m$ and $\\mathcal {I}_i$ are assumed to be known.",
"The consensus over each inertia tensor $\\mathcal {I}_i$ and mass center $p_c^i$ estimated in $\\mathcal {S}_i$ can be made as $\\mathcal {I}_j=R_{ji}^T\\cdot \\mathcal {I}_i \\cdot R_{ji}$ and $\\begin{bmatrix}p_c^j\\\\1\\end{bmatrix}=\\begin{bmatrix}R_{ji} & \\mathbf {t}_{ji}\\\\0 & 1\\end{bmatrix}\\begin{bmatrix}p_c^i\\\\1\\end{bmatrix}$ by dynamic consensus in different coordinates to increase identification belief.",
"In section::measure, we construct truly linear models in forms of $y=Hx$ for each sub-problem and all unknowns to be estimated are time-invariant constants.",
"A general recursive-least-square-like filter may be formulated as eq::filter for estimation with linear measurements $x_{k+1} = \\arg \\min _{x}\\Big \\lbrace \\dfrac{1}{2}(x-x_k)^T P_k^{-1} (x-x_k)+\\\\ \\dfrac{1}{2}(y_k-H_kx_k)^TR_k^{-1}(y_k-H_kx_k)\\Big \\rbrace $ and $P_{k+1} = \\lambda \\cdot (P_k^{-1} +H_k^T R_{k}^{-1} H_k)^{-1}$ where $P_k$ and $R_k$ are the covariance matrices of $x_k$ and $H_kx_k-y_k$ while $\\lambda \\ge 1$ is the forgetting factor.",
"Besides if $\\lambda =1$ and $x_k$ is not constrained, then eq::filter is just the correction step in Kalman filtering[14].",
"In this section, we will discuss how to exactly solve the formulated identification problem for cooperative manipulation with appropriate distributed filtering techniques."
],
[
"State-dependent Uncertainties Evaluation",
"All the observation models constructed in section::measure are pseudo whose either observation matrix $H_k$ or observation $y_k$ or both depend on the unknown $x_k$ and the noisy measurements.",
"Though it remains numerically feasible by just assuming the covariance matrix $R_k$ is constant which is simplified to a basic least square estimation, a explicit evaluation of the covariance $R_k$ for $Hx_k-y_k$ is still preferable.",
"This is possible with suitable independence assumptions as the following proposition indicates [12], [11].",
"Proposition 2 Let us consider $\\mathbf {b}\\in \\mathbb {R}^m$ and $\\mathbf {c}\\in \\mathbb {R}^m$ which are sequences with zero mean.",
"Let $\\mathbf {h}\\in \\mathbb {R}^n$ , $x\\in \\mathbb {R}^n$ and a linear matrix function $G:\\mathbb {R}^l\\longrightarrow \\mathbb {R}^{n\\times m}$ , such that $\\mathbf {y} = G(x)\\mathbf {b}+\\mathbf {c}$ .",
"Assume that $x$ , $\\mathbf {b}$ and $\\mathbf {c}$ are independent.",
"Then $\\Sigma ^{\\mathbf {y}}$ $\\Sigma ^{\\mathbf {y}} = G(x)\\Sigma ^{b}G^T(x)+\\mathbf {N}(\\Sigma ^{b}\\circledast \\Sigma ^{x})\\mathbf {N}^T+\\Sigma ^{\\mathbf {c}}$ where $\\circledast $ is the Kronecker product, $\\Sigma ^{(\\cdot )}$ is the uncertainty associated with $\\lbrace \\cdot \\rbrace $ and $\\mathbf {N}\\in \\mathbb {R}^{n\\times lm}$ is defined as follows $\\mathbf {N} \\triangleq \\begin{bmatrix}\\mathbf {G}_1 & \\mathbf {G}_2 &\\cdots & \\mathbf {G}_m,\\end{bmatrix} $ $\\mathbf {G}\\in \\mathbb {R}^{n\\times l}$ is obtained from the following identity $ \\mathbf {G}_ix=\\mathbf {G}(x)\\mathbf {e}_i$ where $\\mathbf {e}_i$ is column $i$ of the identity matrix of $\\mathbb {R}^{m\\times m}$ .",
"prop::uncertain enables us to evaluate the covariance $R_k$ for $Hx_k-y_k$ only with some independence assumptions of the unknowns and measurements.",
"Due to space limitation, we may not demonstrate this in detail.",
"Readers may refer to [12], [11] for some examples to implement this proposition."
],
[
"Dynamic Consensus in Different Coordinates",
"For this identification problem, dynamic consensus is needed to compute total wrench and improve the belief over different estimations of unknowns that are essentially the same.",
"Even though each individual robot makes estimations and apply forces and torques in its local reference frame, it is still likely to make consensus in different coordinates by communicating with its neighbours as long as the network is connected.",
"Proposition 3 Suppose $G=(V,\\,E)$ is an undirected $n$ -node graph and each node $i$ has initial value $x_i(0)$ .",
"Let $\\lbrace A_{ji}|i,\\,j\\in V\\rbrace $ be a set of time-invariant linear transformations such that $A_{ii}=\\text{I}$ and $A_{ij}=A_{ik}\\cdot A_{kj}$ for any $i,\\,j,\\,k\\in V$ , then for the following dynamical system $\\dot{x}_i(t)= \\sum \\limits _{j\\in N_i}\\Big [ A_{ij}x_j(t)-x_i(t)\\Big ]$ we have $x_i(t)\\longrightarrow \\dfrac{1}{n}\\sum \\limits _{j\\in V}A_{ij}x_j(0)$ and $A_{ji}x_i(t)\\longrightarrow x_{j}(t) $ as $t\\rightarrow \\infty $ if $G$ is connected.",
"Let $y_i(t) = A_{ki}x_i(t)$ .",
"As $t\\rightarrow \\infty $ , we have $y_i(t) \\rightarrow \\dfrac{1}{n}\\sum \\limits _{j\\in V} y_j(0)$ with the following dynamics $\\dot{y}_i(t) =\\sum \\limits _{j\\in N_i}\\Big [y_{j}(t)-y_i(t)\\Big ].$ Note $A_{ij}=A_{ik}A_{kj}$ and then multiply $A_{ik}$ on both sides of eq::result22,eq::condyn2 for each $i$ , the resulting equations are eq::result11,eq::condyn11, which completes the proof.",
"prop3 can be further generalized to other cases as these in [15] so that distributed filtering techniques may be used [16].",
"As for consensus in our problem, suitable linear transformations $\\lbrace A_{ij}\\rbrace $ can be $g_{ji}$ , $R_{ji}$ , $\\text{Ad}_{g_{ji}}$ etc.",
"as those shown in eq::conalpha,eq::wrench,eq::conI,eq::conpc."
],
[
"Filtering on Dual Quaternions",
"Filtering on quaternions and dual quaternions have been studied in [17], [12], [11], [18].",
"One of critical problems for quaternion and dual quaternion filtering is how to fulfil the unit requirements.",
"Popular methods include regarding the constraint as an extra pseudo-observation [19], substituting the constraint to observation [18], or normalizing the filtering result at the end of each step [11], which often either result in nonlinear observation models or converge to a local minima.",
"In our specific problem, however, it can be shown that we may update the dual quaternion estimation $\\widetilde{q}_{ji}^r$ and $\\widetilde{q}_{ji}^d$ while simultaneously preserving the linearity of observation and satisfying the unit requirement without normalization.",
"This method used for dual quaternion estimation is based on the fact that the relative pose $g_{ji}$ to be estimated is a constant.",
"For brevity we have suppressed subscript ${ji}$ in Eqs (REF )-(REF ).",
"For the rotational part $\\widetilde{q}^r_k$ , since the pseudo-observation is $H_k\\widetilde{q}_k^r=0$ , eq::filter may be reformulated as $P_k^{-1}=P_{k-1}^{-1}+H_k^T R_k^{-1}H_k,$ $\\widetilde{q}^r_k = \\arg \\min _{\\begin{array}{c}\\Vert \\widetilde{q}\\Vert =1\\\\ q_0\\ge 0\\end{array}}\\frac{1}{2}\\widetilde{q}^TP_k^{-1}\\widetilde{q}.$ in which eq::quatfilter1 is equivalent to determining the minimal eigenvalue $\\lambda _{\\min }$ of $P_k^{-1}$ and can be exactly solved through eigenvalue decomposition.",
"In cases that the multiplicity of $\\lambda _{\\min }$ is greater than 1, which may sometimes happen if there are not enough observations, we may determine the estimation by minimizing $\\Vert \\widetilde{q}_k^r-\\widetilde{q}_{k-1}^r\\Vert _{P^{-1}_{k-1}}$ among possible choices of $\\widetilde{q}_k^r$ .",
"If $\\widetilde{q}^r_k$ is given, the estimation of $\\widetilde{q}_k^d$ is determined by $\\widetilde{q}_k^d = \\arg \\min _{\\widetilde{q}^T\\cdot \\widetilde{q}_k^r=0}\\frac{1}{2}\\sum \\limits _{l=1}^k \\left\\Vert H_l^{\\mathbf {\\omega }}\\widetilde{q}+H_l^{v}\\widetilde{q}_k^r\\right\\Vert _{R_k^{-1}}^2$ where $H_l^{\\mathbf {\\omega }}$ and $H_l^{v}$ are defined by eq::qrob and the solution to which is $\\begin{bmatrix}\\widetilde{q}_k^d\\\\\\mu \\end{bmatrix}=\\begin{bmatrix}P_k^{-1} & \\widetilde{q}_k^r\\\\{{\\widetilde{q}_k^r}}{}^T & 0\\end{bmatrix}^{-1}\\begin{bmatrix}S_k\\cdot \\widetilde{q}_k^r\\\\0\\end{bmatrix}$ in which $\\mu $ is the Lagrangian multiplier and $P_k^{-1}$ and $S_k$ are recursively updated by $P_k^{-1}=P_{k-1}^{-1}+{H_k^{\\mathbf {\\omega }}}^T R_k^{-1}{H_k^{\\mathbf {\\omega }}}=\\sum \\limits _{l=1}^k{H_l^{\\mathbf {\\omega }}}^T R_l^{-1}{H_l^{\\mathbf {\\omega }}},$ $S_k = S_{k-1}-{H_l^{\\mathbf {\\omega }}}^T R_l^{-1}H_k^{v}=-\\sum \\limits _{l=1}^k{H_l^{\\mathbf {\\omega }}}^T R_l^{-1}H_k^{v}.$ In this way, a recursive filter to estimate the dual quaternion $\\widehat{x}_{ji}=(\\widetilde{q}_{ji}^r,\\,\\widetilde{q}_{ji}^d)$ is developed for estimating relative $g_{ji}\\in SE(3)$ .",
"Note that with dual quaternion and eigenvalue decomposition, the resulting filter is linear and gives exact optimal solution to the generally nonlinear and nonconvex pose estimation problem on $SE(3)$ .",
"Figure: Identification results for relative pose g ji g_{ji} in which (a) is the estimation error for rotation and (b) for translation.",
"At t=8t=8s, the mean error for rotation is 0.036 rad 0.036\\,\\mathrm {rad} and mean error for translation is 0.032m0.032\\;\\mathrm {m}.Figure: Estimation inertia tensor ℐ\\mathcal {I}, mass center p c {p_c} and mass mm.",
"(a) is ℐ\\mathcal {I}, (b) is p c {p_c} and (c) is mm.",
"At t=20t=20s, the mean estimation error for ℐ\\mathcal {I} is 0.075 kg ·m 2 0.075\\,\\mathrm {kg}\\cdot \\mathrm {m}^2 and for p c p_c is 0.015m0.015\\,\\mathrm {m} and the estimated m ^=2.03 kg \\hat{m}=2.03\\,\\mathrm {kg} while the true m 0 =2 kg m_0=2\\,\\mathrm {kg}.Figure: Adaptive estimation of inertia tensor ℐ\\mathcal {I}, mass center p c {p_c} and mass mm after a sudden change of the load at t=35t=35s.",
"(a) is ℐ\\mathcal {I}, (b) is p c {p_c} and (c) is mm.",
"At t=50t=50s, the mean estimation error is 0.058 kg ·m 2 0.058\\,\\mathrm {kg}\\cdot \\mathrm {m}^2 for inertia tensor ℐ\\mathcal {I}, and 0.017m0.017\\,\\mathrm {m} for p c p_c, and the estimated mass m ^=1.196 kg \\hat{m}=1.196\\,\\mathrm {kg} while the true m 0 =1.2 kg m_0=1.2\\,\\mathrm {kg}."
],
[
"Numerical Results",
"In numerical simulations, there are two procedures for identification: first, each robot $i$ estimates the relative pose $g_{ji}$ with its neighbours $j\\in N_i$ ; when the estimation of $g_{ji}$ converges, each robot $i$ starts estimating $\\mathcal {I}_i$ , $p_c^i$ and $m$ in its local reference frame while communicating with its neighbours to make consensus.",
"We assume that there can be a sudden change of inertia parameters for the load during manipulation.",
"The numerical simulation results indicate that the approach proposed is able to identify kinematic and dynamic unknown parameters with satisfactory efficiency and accuracy.",
"According to our simulation results, the convergence speed mainly depends on measurement noise.",
"In simulation, there are $n=5$ robots manipulating a 3D rigid body and the robots are connected as a ring network.",
"The relative pose $g$ between each robot, inertia tensor $\\mathcal {I}$ , mass center $p_c$ and mass $m$ are priorly unknown.",
"The measurement noise for $\\mathbf {\\omega }_i$ , $\\mathbf {v}_i$ , $\\overline{\\mathbf {a}}_i$ , $\\mathbf {f}_i$ and $\\mathbf {\\tau }_i$ are zero-mean Gaussians with covariance $\\Sigma =\\delta ^2 \\cdot \\text{I}$ where $\\delta =0.4$ ."
],
[
"Estimation of $g_{ji}$",
"Each individual robot $i$ estimates the relative pose $g_{ji}$ with its neighbour $j\\in N_i$ using linear dual quaternion observation models.",
"In fact, for each pair $(i,\\,j)\\in E$ , only one of $g_{ij}$ or $g_{ji}$ needs to be estimated and the other can be got by $g_{ij}g_{ji}=\\text{I}$ .",
"All the initial guesses for $g_{ji}$ are zero rotation and zero translation.",
"The identification results are fig::errg and it can be seen that all robots accurately identify the relative pose with neighbours in several seconds."
],
[
"Estimation of $\\mathcal {I}_i$ , {{formula:a119ce5d-4775-4b39-809f-9292c13584c3}} and {{formula:7120f5c1-b4b4-4cc2-b0f9-f440d6fa62fb}}",
"The estimated relative pose $g_{ji}$ at $t=8$ s are used for inertia parameters identification.",
"Each robot only knows forces and torques applied by itself and can only communicate with its neighbours.",
"The total wrench $\\mathbf {F}_i$ and $\\mathbf {T}_i$ is computed through consensus in different coordinates.",
"We may also make consensus on $\\mathcal {I}_i$ , $p_c^i$ and $m$ , the resulting estimation errors of which may vary slightly since $g_{ji}$ are not perfectly known.",
"As for the initial guesses $\\mathcal {I}_i=\\mathrm {diag}\\lbrace 1,\\,1,\\,1\\rbrace \\mathrm {kg}\\cdot \\mathrm {m}^2$ , $p_c^i$ is the geometric center of contact points and $m=1$ kg.",
"The initial covariance is $P_0 = 100\\text{I}$ and the forgetting factor is $\\lambda =1.005$ .",
"The results are shown in fig::inertia and it can be seen that it takes less than 15s for the estimation to converge.",
"We also test the robustness of the approach to the sudden change of inertia, mass center and mass.",
"In simulation, at $t=35$ s after the estimation converges, there is a change of the load and the results of re-identification are fig::adaptive which indicate that our approach may be used for adaptive cooperative manipulation."
],
[
"Conclusion",
"In this paper, we present a distributed and recursive approach to online identification for rigid body cooperative manipulation.",
"Linear observation models with local measurements are derived whose uncertainties can be explicitly evaluated under independence assumptions.",
"We also develop dynamic consensus in different coordinates and an appropriate filter for pose estimation with dual quaternions."
],
[
"Acknowldgement",
"This material is based upon work supported by the National Science Foundation under Grant CNS 1329891.",
"Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation."
]
] | 1709.01555 |
[
[
"The rank function of a positroid and non-crossing partitions"
],
[
"Abstract A positroid is a special case of a realizable matroid, that arose from the study of totally nonnegative part of the Grassmannian by Postnikov.",
"Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations.",
"The bases of a positroid can be described directly in terms of the Grassmann necklace and decorated permutation.",
"In this paper, we show that the rank of an arbitrary set in a positroid can be computed directly from the associated decorated permutation using non-crossing partitions."
],
[
"Introduction",
"A matrix is totally positive (respectively totally nonnegative) if all its minors are positive (respectively nonnegative) real numbers.",
"These matrices have a number of remarkable properties: for example, an $n \\times n$ totally positive matrix has $n$ distinct positive eigenvalues.",
"The space of these matrices can be grouped up into topological cells, with each cell completely parametrized by a certain planar network [4].",
"The idea of total positivity found numerous applications and was studied from many different angles, including oscillations in mechanical systems, stochastic processes and approximation theory, and planar resistor networks [4].",
"Now, instead of considering $n \\times n$ matrices with nonnegative minors, consider a full-rank $k \\times n$ matrix with all maximal minors nonnegative.",
"This arose from the study of the totally nonnegative part of the Grassmannian by Postnikov [13].",
"The set of nonzero maximal minors of such matrices forms a positroid, which is a matroid used to encode the topological cells inside the nonnegative part of the Grassmannian.",
"Positroids have a number of nice combinatorial properties.",
"In particular, Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations.",
"Recently, positroids have seen increased applications in physics, with use in the study of scattering amplitudes [2] and the study of shallow water waves [7].",
"The set of bases of a positroid can be described nicely from the Grassmann necklace [10], and the polytope coming from the bases can be described using the cyclic intervals [9],[1].",
"Non-crossing partitions were used to construct positroids from its connected components in [1].",
"They were also used in [8] as an analogue of the bases for electroids.",
"In this paper, we provide yet another usage of cyclic intervals and non-crossing partition for positroids.",
"Given an arbitrary set, the rank (the size of the biggest intersection with a basis) can be obtained by going through all the bases.",
"In this paper, we show a method of obtaining the rank of an arbitrary set directly from the associated decorated permutation without having to go through the bases.",
"In particular, we get a collection of upper bounds of the rank coming from non-crossing partitions, and one of them will be shown to be tight.",
"The structure of the paper is as follows.",
"In section 2, we go over the background materials needed for this paper, including the basics of matroids, positroids, Grassmann necklaces and decorated permutations.",
"In section 3 we show a basis exchange like property for cyclic intervals that works for positroids.",
"In section 4, we show our main result: that the rank of an arbitrary set in a positroid can be obtained directly from the decorated permutation by using non-crossing partitions.",
"In section 5, we provide an example of how to use our main result to compute the rank of a set."
],
[
"Acknowledgement",
"The authors would also like to thank Lillian Bu, Wini Taylor-Williams and David Xiang for useful discussions."
],
[
"Matroids",
"In this section we review the basics of matroids that we will need.",
"We refer the reader to [12] for a more in-depth introduction to matroid theory.",
"Definition 1 A matroid is a pair $(E,\\mathcal {B})$ consisting of a finite set $E$ , called the ground set of the matroid, and a nonempty collection of subsets $\\mathcal {B}= \\mathcal {B}(\\mathcal {M})$ of $E$ , called the bases of $\\mathcal {M}$ , which satisfy the basis exchange axiom: If $B_1,B_2 \\in \\mathcal {B}$ and $b_1 \\in B_1 \\setminus B_2$ , then there exists $b_2 \\in B_2 \\setminus B_1$ such that $B_1 \\setminus \\lbrace b_1\\rbrace \\cup \\lbrace b_2\\rbrace \\in \\mathcal {B}$ .",
"A subset $F \\subseteq E$ is called independent if it is contained in some basis.",
"All maximal independent sets contained in a given set $A \\subseteq E$ have the same size, called the rank $\\operatorname{rk} *{A}$ of $A$ .",
"The rank of the matroid $\\mathcal {M}$ , denoted as $\\operatorname{rk} *{\\mathcal {M}}$ , is given by $\\operatorname{rk} *{E}$ .",
"An element $e \\in E$ is a loop if it is not contained in any basis.",
"An element $e \\in E$ is a coloop if it is contained in all bases.",
"A matroid $\\mathcal {M}$ is loopless if it does not contain any loops.",
"The dual of $\\mathcal {M}$ is a matroid $\\mathcal {M}^{*} = (E,\\mathcal {B}^{\\prime })$ where $\\mathcal {B}^{\\prime } = \\lbrace E \\setminus B | B \\in \\mathcal {B}(\\mathcal {M})\\rbrace $ .",
"By using the basis exchange axiom on the dual matroid, we get the following dual basis exchange axiom: If $B_1,B_2 \\in \\mathcal {B}$ and $b_2 \\in B_2 \\setminus B_1$ , then there exists $b_1 \\in B_1 \\setminus B_2$ such that $B_1 \\setminus \\lbrace b_1\\rbrace \\cup \\lbrace b_2\\rbrace \\in \\mathcal {B}$ .",
"Remark 1 In this paper, we will always use $[n] := \\lbrace 1,\\ldots ,n\\rbrace $ as our ground set, reserving the usage of $E$ for subsets of the ground set we analyze.",
"A matroid of rank $d$ will have bases in the set ${[n] \\atopwithdelims ()d}$ which stands for all cardinality $d$ -subsets of $[n]$ .",
"Let $E$ be an arbitrary subset of the ground set $[n]$ .",
"For a basis $J$ , if $|J \\cap E|$ is maximal among $|B \\cap E|$ for all bases $B$ of the matroid $\\mathcal {M}$ , we say that $J$ maximizes $E$ , or $J$ is maximal in $E$ .",
"Similarly, if $|J \\cap E|$ is minimal among $|B \\cap E|$ for all bases $B$ of $\\mathcal {M}$ , we say that $J$ minimizes $E$ , or $J$ is minimal in $E$ .",
"The following property of the rank function will be crucial: Theorem 1 [12] The rank function is semimodular, meaning that $\\operatorname{rk} *{A\\cup B}+\\operatorname{rk} *{A\\cap B}\\le \\operatorname{rk} *{A}+\\operatorname{rk} *{B}$ for any subset $A$ and $B$ of the ground set.",
"Consider a matrix with entries in $\\mathbb {R}$ that has $n$ columns and $r$ rows, with $r \\le n$ .",
"Column sets that forms a $r$ -by-$r$ submatrix with nonzero determinant forms (the set of bases of) a matroid.",
"Such matroids are called realizable matroids.",
"For example, consider the following matrix: $ A = \\left( \\begin{array}{cccc}1 & 0 & -3 & -1 \\\\0 & 1 & 4 & 0\\\\\\end{array} \\right)$ The column sets $\\lbrace 1,2\\rbrace , \\lbrace 1,3\\rbrace ,\\lbrace 2,3\\rbrace ,\\lbrace 2,4\\rbrace ,\\lbrace 3,4\\rbrace $ give two-by-two submatrices that has nonzero determinant.",
"So the collection $\\lbrace \\lbrace 1,2\\rbrace , \\lbrace 1,3\\rbrace ,\\lbrace 2,3\\rbrace ,\\lbrace 2,4\\rbrace ,\\lbrace 3,4\\rbrace \\rbrace $ is a realizable matroid.",
"Proposition 1 Let $\\mathcal {M}$ be a realizable matroid over the ground set $[n]$ , and let $B$ be a basis of $\\mathcal {M}$ .",
"Pick an arbitrary subset $E$ of $[n]$ such that $B$ maximizes $E$ and some $J \\subseteq E$ such that $|J|= \\operatorname{rk} *{J} = \\operatorname{rk} *{E}$ .",
"Then $B \\setminus (B \\cap E) \\cup J$ is another basis of $\\mathcal {M}$ .",
"From the condition $|J| = \\operatorname{rk} *{J} = \\operatorname{rk} *{E}$ , the span of the vectors indexed by the set $J$ is exactly same as the span of the vectors indexed by the set $E$ .",
"Since $B$ maximizes $E$ , the span of the vectors indexed by the set $B \\cap E$ is the same vector space.",
"Hence starting from a set of basis vectors indexed by the set $B$ , if we replace the set of vectors indexed by $B \\cap E$ with the set of vectors indexed by $J$ , we still get a set of basis vectors."
],
[
"Positroids",
"In this section we go over the basics of positroids.",
"Positroids were originally defined in [13] as the column sets coming from nonzero maximal minors in a matrix such that all maximal minors are nonnegative.",
"For example, the matrix we saw in the previous section has nonnegative maximal minors: $ A = \\left( \\begin{array}{cccc}1 & 0 & -3 & -1 \\\\0 & 1 & 4 & 0\\\\\\end{array} \\right)$ The nonzero maximal minors come from column sets $\\lbrace 1,2\\rbrace , \\lbrace 1,3\\rbrace ,\\lbrace 2,3\\rbrace ,\\lbrace 2,4\\rbrace ,\\lbrace 3,4\\rbrace $ .",
"This collection forms a positroid.",
"However in this paper, we will use an equivalent definition using Grassmann necklace and Gale orderings.",
"Definition 2 Let $d \\le n$ be positive integers.",
"A Grassmann necklace of type $(d,n)$ is a sequence $(I_1,\\ldots ,I_n)$ of $d$ -subsets $I_k \\in {[n] \\atopwithdelims ()d}$ such that for any $i \\in [n]$ , if $i \\in I_i$ then $I_{i+1} = I_i \\setminus \\lbrace i\\rbrace \\cup \\lbrace j\\rbrace $ for some $j \\in [n]$ , if $i \\notin I_i$ then $I_{i+1} = I_i$ , where $I_{n+1} = I_1$ .",
"The cyclically shifted order $<_i$ on the set $[n]$ is the total order $i <_i i+1 <_i \\cdots <_i n <_i 1 <_i \\cdots <_i i-1.",
"$ For any rank $d$ matroid $\\mathcal {M}$ with ground set $[n]$ , let $I_k$ be the lexicographically minimal basis of $\\mathcal {M}$ with respect to $<_k$ , and denote $I(\\mathcal {M}) := (I_1,\\ldots ,I_n),$ which forms a Grassmann necklace [13].",
"The Gale order on ${[n] \\atopwithdelims ()d}$ (with respect to $<_i$ ) is the partial order $<_i$ defined as follows: for any two $d$ -subsets $S = \\lbrace s_1 <_i \\cdots <_i s_d\\rbrace $ and $T = \\lbrace t_1 <_i \\cdots <_i t_d\\rbrace $ of $[n]$ , we have $S \\le _i T$ if and only if $s_j \\le _i t_j$ for all $j \\in [d]$ [5].",
"Theorem 2 ([13],[10]) Let $I= (I_1,\\ldots ,I_n)$ be a Grassmann necklace of type $(d,n)$ .",
"Then the collection $\\mathcal {B}(I) := \\lbrace B \\in {[n] \\atopwithdelims ()d} | B \\ge _j I_j, \\text{ for all } j \\in [n] \\rbrace $ is the collection of bases of a rank $d$ positroid $\\mathcal {M}(I) := ([n],\\mathcal {B}(I))$ .",
"Moreover, for any positroid $\\mathcal {M}$ , we have $\\mathcal {M}(I(\\mathcal {M})) = \\mathcal {M}$ .",
"In order to check if a set is a basis of a positroid or not, we do not have to check for all the cyclic orderings.",
"Corollary 1 Let $\\mathcal {M}\\subseteq {[n] \\atopwithdelims ()d}$ be a positroid and $I$ the associated Grassmann necklace.",
"A set $B \\in {[n] \\atopwithdelims ()d}$ is a basis of $\\mathcal {M}$ if and only if $B \\ge _b I_b$ for all $b \\in B$ .",
"For arbitrary $q \\in [n]$ , denote the elements of $B$ as $b_1 <_q b_2 <_q \\cdots <_q b_d$ .",
"If we had $B \\ge _{b_1} I_{b_1}$ , we would also have $B \\ge _q I_{b_1} \\ge _q I_q$ .",
"Definition 3 A decorated permutation of the set $[n]$ is a bijection $\\pi $ of $[n]$ whose fixed points are colored either white or black.",
"A weak $i$ -exceedance of a decorated permutation $\\pi $ is an element $j \\in [n]$ such that either $j <_i \\pi ^{-1}(j)$ or $j$ is a fixed point colored black.",
"Given a decorated permutation $\\pi $ of $[n]$ we can construct a Grassmann necklace $I= (I_1,\\ldots ,I_n)$ by letting $I_k$ be the set of weak $k$ -exceedances of $\\pi $ .",
"A graphical way to see this is to cut the circle off between $k-1$ and $k$ to get a horizontal straight line with leftmost endpoint being $k$ and rightmost endpoint being $k-1$ .",
"Redraw the arrows of the permutation accordingly so that it stays within the line.",
"Endpoints of the leftward arrows are exactly the weak $k$ -exceedances of $\\pi $ , hence the elements of $I_k$ .",
"There is a bijection between Grassmann necklaces and decorated permutations [13].",
"Figure: A decorated permutation.For example, take a look at the decorated permutation (since it has no fixed points, it is the usual permutation) in Figure REF .",
"It is the permutation $[2,8,6,7,9,4,5,14,13,3,10,11,1,12]$ under the usual bracket notation.",
"The weak 1-exceedances of the permutation is given by the set $\\lbrace 1,3,4,5,10,11,12\\rbrace $ , and this is $I_1$ of the associated Grassmann necklace.",
"Remark 2 When we are dealing with positroids, we will always envision the ground set $[n]$ to be drawn on a circle.",
"We will say that $a_1,\\ldots ,a_t \\in [n]$ are cyclically ordered if there exists some $i \\in [n]$ such that $a_1 <_i \\cdots <_i a_t$ .",
"Given $a,b \\in [n]$ , we define the cyclic interval $[a,b]$ to be the set $\\lbrace x| x \\le _a b\\rbrace $ .",
"These cyclic intervals play an important role in the structure of a positroid [6],[9],[1].",
"All intervals mentioned in this paper will actually be referring to cyclic intervals.",
"Remark 3 If a positroid $\\mathcal {M}$ has loops or coloops, it is enough to study the positroid $\\mathcal {M}^{\\prime }$ obtained by deleting the loops and the coloops to study the structural properties of $\\mathcal {M}$ .",
"So throughout this paper, we will assume that our positroid has neither loops nor coloops.",
"This means that the associated decorated permutation has no fixed points."
],
[
"Interval exchange and Morphing",
"In this section we develop a stronger basis exchange technique for positroids.",
"Throughout the paper, unless otherwise stated, we will always be working with a positroid $\\mathcal {M}$ on a ground set $[n]$ , with rank $d$ , having Grassmann necklace $I= (I_1,\\ldots ,I_n)$ , and an associated decorated permutation $\\pi $ that does not have any fixed points (see Remark REF ).",
"The example positroid that we will be using, again unless otherwise stated, will be the positroid associated to the decorated permutation of Figure REF .",
"The following property follows from the definition of Grassmann necklaces and the proof will be omitted.",
"Lemma 1 (Sharing property) Let $a$ and $b$ be arbitrary elements of $[n]$ .",
"Then we have $I_a \\cap [b,a) \\subseteq I_b \\cap [b,a)$ .",
"To illustrate using our running example, notice that since $I_3=\\lbrace 3,4,5,8,10,11,12\\rbrace $ , the set $I_3\\cap [9,3)=\\lbrace 10,11,12\\rbrace $ is contained in $I_9 = \\lbrace 9,10,11,12,14,4,5\\rbrace $ .",
"We begin our analysis of the cyclic intervals of a positroid.",
"The following lemma follows directly from Theorem REF .",
"Lemma 2 For any interval $[a,b] \\subseteq [n]$ , the interval is maximized by $I_a$ .",
"Any interval $(b,a)$ is minimized by $I_a$ .",
"This can easily be seen by taking some Grassmann necklace element and any arbitrary basis; say, $I_6=\\lbrace 6,7,8,9,10,11,12\\rbrace $ and $B=\\lbrace 6,7,10,11,12,1,4\\rbrace $ .",
"Examine how the lemma holds on the intervals $[6,10]$ and $(10,6)$ in $[14]$ : $I_6\\cap [6,10]=\\lbrace 6,7,8,9,10\\rbrace $ contains more elements than $B\\cap [6,10]=\\lbrace 6,7,10\\rbrace $ , while $I_6\\cap (10,6)=\\lbrace 11,12\\rbrace $ contains fewer elements than $B\\cap (10,6)=\\lbrace 11,12,1,4\\rbrace $ .",
"The above lemma also suggests that given a cyclic interval $[a,b]$ , the set $I_a \\cap [a,b]$ plays a crucial role in studying that interval.",
"The following claim follows directly from Proposition REF and Lemma REF .",
"Corollary 2 (Interval exchange property of positroids) If $J \\in \\mathcal {M}$ maximizes $[a,b] \\subseteq [n]$ , then $J \\setminus (J \\cap [a,b]) \\cup (I_a \\cap [a,b]) \\in \\mathcal {M}$ .",
"Similarly, if $J$ minimizes $(b,a) \\subseteq [n]$ , then $J \\setminus (J \\cap (b,a)) \\cup (I_a \\cap (b,a)) \\in \\mathcal {M}$ .",
"Here is an example of how the interval exchange property works.",
"In the positroid coming from Figure REF , we have $I_{13} = \\lbrace 13,14,3,4,5,10,11\\rbrace $ .",
"The set $B = \\lbrace 1,4,7,8,10,11,13\\rbrace $ is a basis of the positroid.",
"Now if we exchange $B \\cap [13,2] = \\lbrace 13,1\\rbrace $ with $I_{13} \\cap [13,2] = \\lbrace 13,14\\rbrace $ , the resulting set $\\lbrace 4,7,8,10,11,13,14\\rbrace $ is again a basis.",
"Our goal of the paper is to express the rank of an arbitrary set $E \\subseteq [n]$ using non-crossing partitions.",
"To do so, we need to construct the bases that maximize $E$ and analyze them.",
"Remark 4 When $E$ is a subset of the ground set $[n]$ and we are trying to write $E$ as a disjoint union of cyclic intervals so that $E = [a_1,b_1] \\cup \\cdots \\cup [a_s,b_s]$ , we will arrange the $a_i$ 's such that $a_1 < a_2 < \\cdots < a_s$ unless otherwise stated.",
"The symbol $s$ will always be reserved for the number of disjoint intervals that $E$ has.",
"Here the indices of $[s]$ are considered cyclically, so $a_{s+1}=a_1$ .",
"Our goal is to show that it is possible to find a basis that maximizes $E$ starting from some Grassmann necklace element and then applying a series of transformations to it.",
"Lemma 3 Let $E$ be an arbitrary subset of $[n]$ .",
"Write $E$ as in Remark REF .",
"Let $i$ be any element of $[s]$ .",
"Then there exists a basis $B$ that maximizes $E$ and satisfies $B \\cap (b_{i-1},b_{i}] = I_{a_i} \\cap (b_{i-1},b_i]$ .",
"Let $B$ be a basis which maximizes $E$ .",
"Pick any $e \\in B\\cap (b_{i-1},a_i)\\setminus I_{a_i}$ .",
"By the basis exchange axiom, there is an $e^{\\prime } \\in I_{a_i}\\setminus B$ such that $(B\\setminus \\lbrace e\\rbrace )\\cup \\lbrace e^{\\prime }\\rbrace $ is a basis; furthermore, this maximizes $E$ .",
"Set this as new $B$ , and repeat the process until we run of elements in $B\\cap (b_{i-1},a_i)\\setminus I_{a_i}$ .",
"Now, let $e^{\\prime } \\in I_{a_i}\\cap [a_i,b_i]\\setminus B$ .",
"By the dual basis exchange axiom, there is an $e \\in B\\setminus I_{a_i}$ such that $(B\\setminus \\lbrace e\\rbrace )\\cup \\lbrace e^{\\prime }\\rbrace $ is a basis; furthermore, this maximizes $E$ .",
"Set this as new $B$ , and repeat the process until we run of elements in $I_{a_i}\\cap [a_i,b_i]\\setminus B$ .",
"In particular, $B$ as above will minimize $(b_{i-1},a_i)$ and maximize $[a_i,b_i]$ .",
"To illustrate the above lemma with our running example, let $E=[1,4]\\cup [6,7]$ .",
"The set $B^0=\\lbrace 1,3,6,7,10,11,14\\rbrace $ happens to be a basis which maximizes $E$ .",
"Recall that $I_1=\\lbrace 1,3,4,5,10,11,12\\rbrace $ .",
"By exchanging to get $B^1:=(B^0\\setminus \\lbrace 14\\rbrace )\\cup \\lbrace 12\\rbrace $ , we have another basis which maximizes $E$ and satisfies the condition that $B^1\\cap (7,1)=I_1\\cap (7,1)$ .",
"By exchanging again to get $B^2=(B^1\\setminus \\lbrace 6\\rbrace )\\cup \\lbrace 4\\rbrace $ , we arrive at a final basis $B^2$ satisfying the condition that $B^2\\cap [1,4]=I_1\\cap [1,4]$ as well.",
"Now we develop a method of constructing a basis that maximizes $E$ , starting from some element of the Grassmann necklace.",
"Fix some cyclically ordered elements $b,c,d \\in [n]$ .",
"Recall that the number of elements a basis can have in the interval $(b,c)$ is bounded below by $|I_c \\cap (b,c)|$ and the number of elements a basis can have in the interval $[c,d]$ is bounded above by $|I_c \\cap [c,d]|$ .",
"We will say that a set $J$ is compatible with $I_c$ in $(b,d]$ if $J \\cap (b,c) \\supseteq I_c \\cap (b,c)$ and $J \\cap [c,d] \\subseteq I_c \\cap [c,d]$ .",
"When we are comparing such $J$ with $I_c$ , we will call the elements of $(J \\setminus I_c) \\cap (b,c)$ as the excessive elements and the elements of $(I_c \\setminus J) \\cap [c,d]$ as the gaps.",
"A set $J$ mimics $I_c$ in $(b,d]$ if it is compatible with $I_c$ in $(b,d]$ and at least one of the above containments is an equality.",
"For a set $J$ that mimics $I_c$ in $(b,d]$ , if we have $J \\cap [c,d] = I_c \\cap [c,d]$ , we say that $J$ is gap-free (with respect to $I_c$ in $(b,d]$ ).",
"Otherwise we say that $J$ has gaps (with respect to $I_c$ in $(b,d]$ ).",
"Remark 5 For any cyclically ordered $a,b,c,d$ , we have that $I_a$ is compatible with $I_c$ in $(b,d]$ from the sharing property, but it doesn't necessarily mimic $I_c$ in the same interval.",
"Let $J$ be a basis of $\\mathcal {M}$ that is compatible to $I_c$ in $(b,d]$ , where $b,c,d$ are cyclically ordered elements of $[n]$ .",
"Our goal is to transform $J$ into a basis that mimics $I_c$ in $(b,d]$ .",
"The idea is to replace the elements of $(J \\setminus I_c) \\cap (b,c)$ with $(I_c \\setminus J) \\cap [c,d]$ .",
"Let $\\alpha $ be $\\operatorname{min} *{|(J \\setminus I_c) \\cap (b,c)|,|(I_c \\setminus J) \\cap [c,d]|}$ .",
"Define $J^{\\prime }$ to be the set obtained from $J$ by replacing biggest (with respect to $<_b$ ) $\\alpha $ elements of $(J \\setminus I_c) \\cap (b,c)$ with the smallest (again with respect to $<_b$ ) $\\alpha $ elements of $(I_c \\setminus J) \\cap [c,d]$ .",
"We will say that $J^{\\prime }$ is obtained from $J$ by mimicking $I_c$ in $(b,d]$.",
"We will describe the process as excessive elements of $(J \\setminus I_c) \\cap (b,c)$ being moved to fill the gaps of $(I_c \\setminus J) \\cap [c,d]$ .",
"The newly created $J^{\\prime }$ mimics $J$ .",
"We say that this mimicking process has gaps or is gap-free depending on whether $J^{\\prime }$ has gaps or is gap-free (with respect to $I_c$ in $(b,d]$ ).",
"Now we will use the above process multiple times starting from a Grassmann necklace element and produce multiple sets, that will potentially be a basis that maximizes $E$ (again using Remark REF ).",
"We dedicate $J^0$ to stand for $I_{a_1}$ .",
"Recursively, $J^{t}$ is going to be obtained from $J^{t-1}$ by mimicking $I_{a_{t+1}}$ in $(b_t,b_s]$ for $t \\in \\lbrace 1,\\ldots ,s-1\\rbrace $ (this is possible since $J^{t-1}$ is compatible with $I_{a_{t+1}}$ in $(b_t,b_s]$ ).",
"We call this process the $t$ -th morph of $J^0 = I_{a_1}$ .",
"So we will say that the set $J^t$ is obtained from $J^0$ by morphing $t$ times.",
"Similarly, we will use $J_i^t$ to denote the set obtained from $I_{a_i}$ by imposing $a_1 <_i \\cdots <_i a_s$ when labeling the starting points of the intervals of $E$ , then morphing $t$ times.",
"For example, consider the set $E=[2,4]\\cup [7,10]$ in our example from Figure REF .",
"The set $J_1^0$ is defined as $I_2 = \\lbrace 2,3,4,5,10,11,12\\rbrace $ .",
"Since we are dealing with $J_1$ , we label $a_1 = 2, b_1 = 4, a_2 = 7, b_2 = 10$ .",
"The first morph of $J_1$ will be mimicking $I_7$ in $(4,10]$ .",
"From $I_7 = \\lbrace 7,8,9,10,11,12,4\\rbrace $ , we move the excessive elements $(J_1^0 \\setminus I_7) \\cap (4,7) = \\lbrace 5\\rbrace $ to fill the gaps of $(I_7 \\setminus J_1^0) \\cap [7,10] = \\lbrace 7,8,9\\rbrace $ .",
"This gives us $J_1^1 = \\lbrace 2,3,4,7,10,11,12\\rbrace $ , which has gaps (with respect to $I_7$ in $[7,10]$ ).",
"The reader should beaware that we do not know if $J_1^1$ is actually a basis of $\\mathcal {M}$ yet.",
"Similarly, the set $J_2^0$ is defined as $I_7 = \\lbrace 7,8,9,10,11,12,4\\rbrace $ .",
"When we are dealing with $J_2$ , we label $a_1 = 7, b_1 = 10, a_2 = 2, b_2=4$ .",
"The first morph of $J_2$ will be mimicking $I_2$ in $(10,4]$ .",
"Since there are no excessive elements in $(10,2)$ , we have $J_2^1 = J_2^0$ in this case.",
"We have an analogue of the sharing property for $J_i^t$ 's, which is straightforward from the sharing property: Lemma 4 (Sharing property for the morphs) We have $J_i^t \\cap [a_{i+1},a_i) \\subseteq J_{i+1}^{t-1} \\cap [a_{i+1},a_i)$ .",
"Our ultimate goal is to show that one of the $J_i^t$ 's will maximize $E$ .",
"Lemma 5 Fix a subset $E$ of the ground set as in Remark REF .",
"Fix some $1 \\le t \\le s-1$ , then consider the set $J^t$ .",
"For each $1 \\le p \\le s$ , there exists some nonnegative number $q$ and a sequence $i_1 < \\cdots < i_q < i_{q+1}=p$ such that $J^t$ maximizes $[a_1,b_{i_1}], [a_{i_1+1},b_{i_2}], \\ldots , [a_{i_q+1},b_p]$ .",
"Recall that the $t$ -th morph removes the excessive elements in $(b_t,a_{t+1})$ and fills the gaps of $[a_{t+1},b_s]$ from left to right.",
"Consider the intervals $[a_1,b_1], [a_2,b_2], \\ldots , [a_p,b_p]$ which are some of the components of $E$ .",
"We will associate a number on each interval in the following way : for each $[a_x,b_x]$ , let $\\gamma (x)$ denote the biggest number within $\\lbrace 0,\\ldots ,\\operatorname{max} *{t,x-1}\\rbrace $ such that all gaps of $[a_{\\gamma (x)+1},b_x]$ gets filled in the $\\gamma (x)$ -th morph (that is, when $J^{\\gamma (x)} \\cap [a_{\\gamma (x)+1},b_x] = I_{a_{\\gamma (x)+1}} \\cap [a_{\\gamma (x)+1},b_x]$ ).",
"Such number is guaranteed to exist, since $J^0 = I_{a_1}$ .",
"Now $J^t$ maximizes $[a_{\\gamma (x)+1},b_x]$ in $\\mathcal {M}$ , since the morphs after the $\\gamma (x)$ -th morph does not change the number of elements in that interval.",
"Starting from $p$ , take $\\gamma (p),\\gamma (\\gamma (p)),\\ldots $ until you get 0.",
"Delete 0 from this collection, and relabel them as $i_1 < i_2 < \\cdots $ to get the desired result.",
"From the above lemma, we are guaranteed that each $J^t$ maximizes some set in $[a_1,b_t]$ (setting $p$ as $t$ ) which is obtained from $E \\cap [a_1,b_t]$ by merging some nearby intervals and replacing them with a bigger interval (for example merging $[1,3] \\cup [6,9]$ to get $[1,9]$ ).",
"Now if $J^t$ was gap free (that is the morph to get $J^t$ from $J^{t-1}$ is gap-free) then $[a_{t+1},b_s]$ is also maximized.",
"In other words, $J^t$ that is gap free will maximize some set that is obtained from $E$ by merging some nearby intervals.",
"The remainder of this section will be dedicated to showing that there is some $i$ and $t$ such that $J_i^t$ is gap free and is a basis of the positroid.",
"The next section will use that result to obtain our main result.",
"Lemma 6 Let $b,c,d,e$ be cyclically ordered elements of the ground set $[n]$ and let $J$ be a basis of $\\mathcal {M}$ that is compatible to $I_c$ in $(b,d]$ .",
"Define $J^{\\prime }$ to be obtained from $J$ by mimicking $I_c$ in $(b,d]$ .",
"The following holds: If $J^{\\prime } \\in \\mathcal {M}$ and $J^{\\prime }$ has gaps (with respect to $I_c$ in $(b,d]$ ), then we have $|J \\cap (z,b]| \\ge |I_c \\cap (z,b]|$ for all $z \\in (d,b]$ .",
"If $|J \\cap (z,b]| \\ge |I_c \\cap (z,b]|$ for all $z \\in (d,b]$ , then $J^{\\prime } \\in \\mathcal {M}$ .",
"Moreover if we have $J^{\\prime } \\cap [c,e) \\subseteq I_c \\cap [c,e)$ , then the above inequality holding for all $z \\in [e,b]$ is enough to get $J^{\\prime } \\in \\mathcal {M}$ .",
"From the fact that $J^{\\prime } \\setminus J >_z J \\setminus J^{\\prime }$ for any $z \\notin (b,d]$ and using Corollary REF , it is enough to show $J^{\\prime } \\ge _c I_c$ in order to achieve $J^{\\prime } \\in \\mathcal {M}$ .",
"Therefore $J^{\\prime } \\in \\mathcal {M}$ is equivalent to $|J^{\\prime } \\cap [c,z]| \\le |I_c \\cap [c,z]|$ for all $z \\in [n]$ .",
"Since $J$ and $J^{\\prime }$ are compatible with $I_c$ in $(b,d]$ the inequality automatically holds for any $z \\in (b,d]$ .",
"Hence we only need to show $|J^{\\prime } \\cap (z,c)| \\ge |I_c \\cap (z,c)|$ for all $z \\in (d,b]$ .",
"Observe that $|J^{\\prime } \\cap (z,c]| = |J \\cap (z,c]| - \\alpha $ for $z \\in (d,b]$ , where $\\alpha $ is the minimum of $|(I_c \\setminus J) \\cap [c,d]|$ (process is gap-free) and $|(J \\setminus I_c) \\cap (b,c)|$ (has gaps).",
"Cleaning up the inequalities in the latter case gives us the desired results.",
"Using the above lemma, we will finish off the section with the following result.",
"Proposition 2 Let $\\mathcal {M}$ be a positroid over the ground set $[n]$ and $E = [a_1,b_1] \\cup \\cdots [a_s,b_s]$ be a subset of the ground set.",
"Again consider the sets of form $J_i^t$ for $1 \\le t \\le s-1$ , obtained from $I_{a_i}$ by morphing $t$ times with respect to $E$ .",
"Fix some $1 \\le h \\le s-1$ .",
"If $J_i^t \\in \\mathcal {M}$ and have gaps for all $i \\in [n]$ and $1 \\le t < h$ , then $J_i^h \\in \\mathcal {M}$ for all $i \\in [n]$ .",
"First of all, $J^1 \\in \\mathcal {M}$ follows from Lemma REF and the sharing property.",
"Hence we only need to consider the case when $h>1$ .",
"We will show that if $J^1,\\ldots ,J^{h-1},J_2^1,\\ldots ,J_2^{h-1}$ are bases of $\\mathcal {M}$ and have gaps, then $J^h \\in \\mathcal {M}$ .",
"Also notice that $J^h \\cap [a_{i_{h+1}},a_1) \\subseteq I_{a_{i_{h+1}}} \\cap [a_{i_{h+1}},a_1)$ .",
"Therefore in order to show $J^h \\in \\mathcal {M}$ , we need $|J^{h-1} \\cap (z,b_h]| \\ge |I_{a_{h+1}} \\cap (z,b_h]|$ for all $z \\in [a_1,b_h]$ from Lemma REF .",
"From lemma REF , there exists a sequence $i_1 < \\cdots < i_q < i_{q+1}=h$ (setting $t=h-1$ and $p=h$ in the lemma) such that $J^{h-1}$ maximizes $E^{\\prime } = [a_1,b_{i_1}], \\ldots , [a_{i_q+1},b_{h}]$ in $\\mathcal {M}$ .",
"Since $I_{a_1}$ and $J^{h-1}$ have the same number of elements in $[a_1,b_{i_1}]$ and $I_{a_1} \\le J^{h-1}$ , we have $|J^{h-1} \\cap (z,b_{i_1}]| \\ge |I_{a_1} \\cap (z,b_{i_1}]|$ for all $z \\in [a_1,b_{i_1}]$ .",
"From $J^1,\\ldots ,J^{h-1}$ having gaps, $J^{h-1}$ minimizes each $(b_i,a_{i+1})$ within $(b_s,b_h]$ .",
"From $J_2^1, \\ldots , J_2^{h-1}$ having gaps, $J_2^{h-2}$ minimizes each $(b_i,a_{i+1})$ within $(b_1,b_h]$ .",
"This implies that within intervals of form $(b_i,a_{i+1})$ contained in $(b_1,b_h]$ , the sets $J^{h-1}$ and $J_2^{h-2}$ are exactly the same.",
"From the sharing property of morphs, $J^{h-1}$ maximizing $[a_1,b_{i_1}] , \\ldots , [a_{i_q+1},b_{h}]$ implies $J_2^{h-2}$ also does too except potentially at $[a_1,b_{i_1}]$ .",
"Combining these facts, we get that $J^{h-1} \\cap (b_{i_1},b_h] = J_2^{h-2} \\cap (b_{i_1},b_h]$ .",
"We now have all the ingredients to show that $|J^{h-1} \\cap (z,b_h]| \\ge |I_{a_{h+1}} \\cap (z,b_h]|$ for each $z \\in [a_1,b_h]$ .",
"From $J_2^{h-1} \\in \\mathcal {M}$ and having gaps, Lemma REF tells us that the same inequality replacing $J^{h-1}$ with $J_2^{h-2}$ is true for $z \\in (b_1,b_h]$ .",
"Therefore for any $z \\in (b_{i_1},b_h]$ , we have $|J^{h-1} \\cap (z,b_h]| = |J_2^{h-2} \\cap (z,b_h]| \\ge |I_{a_{h+1}} \\cap (z,b_h]|$ .",
"For any $z \\in [a_1,b_{i_1}]$ , we have $|J^{h-1} \\cap (z,b_h]| = |J^{h-1} \\cap (z,b_{i_1}]| + |J^{h-1} \\cap (b_{i_1},b_h]| \\ge |I_{a_1} \\cap (z,b_{i_1}]| + |J_2^{h-2} \\cap (b_{i_1},b_h]| \\ge |I_{a_{h+1}} \\cap (z,b_h]|$ , since we have $I_{a_1} \\cap (z,b_{i_1}) \\supseteq I_{a_{h+1}} \\cap (z,b_{i_1}]$ from the sharing property.",
"The above proposition will be used as a key idea during the proof of the main result in the next section."
],
[
"Rank of arbitrary sets",
"Let $E$ be a subset of the ground set as in Remark REF .",
"We use $E_i$ to denote $[a_i,b_i]$ .",
"The rank of $E$ is bounded above by $\\operatorname{rk} *{\\mathcal {M}}$ minus the sum of the minimal number of elements that a basis of $\\mathcal {M}$ can possibly have in each cyclic interval of the complement of $E$ .",
"So we get $\\operatorname{rk} *{E} \\le \\operatorname{rk} *{\\mathcal {M}} - \\sum _i \\operatorname{minelts} *{b_i,a_{i+1}}$ , where $\\operatorname{minelts} *{b,a}$ stands for the minimal number of elements that a basis of $\\mathcal {M}$ can have in the interval $(b,a)$ .",
"We call this bound the natural rank bound of $E$: $\\operatorname{nbd} *{E} := \\operatorname{rk} *{\\mathcal {M}} - \\sum _i (\\operatorname{minelts} *{b_i,a_{i+1}})$ .",
"Notice that $\\operatorname{minelts} *{b,a} = \\operatorname{rk} *{\\mathcal {M}} - \\operatorname{rk} *{[a,b]}$ .",
"Definition 4 Let $\\Pi $ be a partition $T_1 \\sqcup \\cdots \\sqcup T_p$ of $[s]$ into pairwise disjoint non-empty subsets.",
"We say that $\\Pi $ is a non-crossing partition if there are no cyclically ordered $a,b,c,d$ such that $a,c \\in T_i$ and $b,d \\in T_j$ for some $i \\ne j$ .",
"We will call the $T_i$ 's as the blocks of the partition.",
"To illustrate with a simple example, $\\lbrace 1,3\\rbrace \\sqcup \\lbrace 2\\rbrace \\sqcup \\lbrace 4\\rbrace $ is a non-crossing partition of $[4]$ , but $\\lbrace 1,3\\rbrace \\sqcup \\lbrace 2,4\\rbrace $ is not.",
"This can be easily verified by drawing the points 1 to 4 on a circle and trying to cut the circle into distinct regions corresponding to the partitions; this can only be done in the case of non-crossing partitions.",
"Let $\\Pi $ be an arbitrary non-crossing partition of $[s]$ with $T_1,\\ldots ,T_p$ as its parts.",
"We define $E|_{T_i}$ as the subset of $E$ obtained by taking only the intervals indexed by elements of $T_i$ .",
"For example, $E|_{\\lbrace 1,3\\rbrace }$ would stand for $E_1 \\cup E_3$ .",
"By submodularity of the rank function, we get another upper bound on the rank of $E$ : $\\operatorname{rk} *{E} \\le \\operatorname{rk} *{E|_{T_1}} + \\cdots + \\operatorname{rk} *{E|_{T_p}} \\le \\operatorname{nbd} *{E, \\Pi } := \\operatorname{nbd} *{E|_{T_1}} + \\cdots + \\operatorname{nbd} *{E|_{T_p}}$ .",
"So for each non-crossing partition of $[s]$ , we get an upper bound on the rank of $E$ .",
"We show that one of those bounds has to be tight in the theorem below.",
"Theorem 3 Let $E = [a_1,b_1] \\cup \\cdots \\cup [a_s,b_s]$ be a disjoint union of $s$ cyclic intervals, where $a_1,b_1,a_2,b_2,\\ldots ,a_s,b_s$ are cyclically ordered.",
"We have $\\operatorname{rk} *{E} = \\operatorname{nbd} *{E,\\Pi }$ for some non-crossing partition $\\Pi $ of $[s]$ .",
"We use induction on $s$ , the number of disjoint cyclic intervals of $E$ .",
"In case $s=1$ , we have $\\operatorname{rk} *{E} = \\operatorname{rk} *{\\mathcal {M}} - \\operatorname{minelts} *{E^c} = \\operatorname{nbd} *{E} = \\operatorname{nbd} *{E, \\lbrace \\lbrace 1\\rbrace \\rbrace }$ .",
"Assume for the sake of induction that the claim is true for $1,\\ldots ,s-1$ intervals.",
"We define $J_i^t$ recursively as in the previous section.",
"From Proposition REF , we either have some $J_i^t \\in \\mathcal {M}$ that is gap-free or we have $J^{s-1} \\in \\mathcal {M}$ that isn't gap-free.",
"In the latter case, since $J^{s-1}$ minimizes every interval of form $(b_i,a_{i+1})$ , we have $|J^{s-1} \\cap E| = \\operatorname{nbd} *{E} = \\operatorname{nbd} *{E,\\lbrace \\lbrace 1,\\ldots ,s\\rbrace \\rbrace }$ .",
"Therefore we only have to take care of the case when we have some $J_i^t \\in \\mathcal {M}$ that is gap-free.",
"Without loss of generality, we will assume $i=1$ .",
"From Lemma REF , we have some sequence $i_1 < \\cdots < i_q < i_{q+1}=t$ such that $J_i^t$ maximizes $[a_1,b_{i_1}], \\ldots , [a_{i_q+1},b_{t}],[a_{t+1},b_s]$ (the last interval is maximized due to $J_i^t$ being gap-free).",
"We will use $F_1,\\ldots ,F_{q+2}$ to denote these intervals.",
"For each $1 \\le j \\le q+2$ , let $K_j$ be a basis that maximizes $F_j \\cap E$ .",
"Modify $K_j$ using Lemma REF so that it minimizes the complement of $F_j$ in $[n]$ .",
"Since $|K_j \\cap F_j| = |J^t \\cap F_j|$ , using Proposition REF we can replace $J^t \\cap F_j$ with $K_j \\cap F_j$ in $J^t$ for each $j$ to obtain a new basis $B$ .",
"By induction hypothesis, for each $j$ , we have $|B \\cap F_j \\cap E| = \\operatorname{rk} *{F_j \\cap E} = \\operatorname{nbd} *{F_j \\cap E, \\Pi _j}$ for some non-crossing partition $\\Pi _j$ .",
"Letting $\\Pi $ be a non-crossing partition obtained by collecting all blocks of $\\Pi _j$ 's, we get $|B \\cap E| = \\operatorname{rk} *{E} = \\operatorname{nbd} *{E,\\Pi }$ .",
"Figure: Information needed to compute the rank of [1,3]∪[8,10][1,3] \\cup [8,10].For example, take a look at Figure REF (the positroid is the one associated to Figure REF ).",
"The rank of $E = [1,3] \\cup [8,10]$ is bounded above by $\\operatorname{nbd} *{E, \\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace \\rbrace }$ and $\\operatorname{nbd} *{E,\\lbrace \\lbrace 1,2\\rbrace \\rbrace }$ .",
"We get $\\operatorname{nbd} *{E,\\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace \\rbrace } = \\operatorname{rk} *{[1,3]} + \\operatorname{rk} *{[8,10]} = 2 + 3 = 5$ , since rank of an interval $[a,b]$ is given by $|[a,b]|$ minus the number of intervals of form $[\\pi ^{-1}(x),x]$ contained in $[a,b]$ (from $I_a$ being given by $a$ -exceedances, and $\\operatorname{rk} *{[a,b]} = |I_a \\cap [a,b]|)$ .",
"We also have $\\operatorname{nbd} *{E,\\lbrace \\lbrace 1,2\\rbrace \\rbrace } = \\operatorname{rk} *{\\mathcal {M}} - \\operatorname{minelts} *{(3,8)} - \\operatorname{minelts} *{(10,1)} = 7 - 2 - 2 = 3$ , since $\\operatorname{minelts} *{(b,a)}$ is given by the number of intervals of form $[x,\\pi ^{-1}(x)]$ contained in $(b,a)$ .",
"Hence the above theorem tells us that $\\operatorname{rk} *{E} = 3$ ."
],
[
"Application",
"Let $\\mathcal {M}$ be a positroid and let $E$ be an arbitrary subset of the ground set $[n]$ .",
"In this section, we will show how to use Theorem REF to obtain the rank of $E$ .",
"We will call an interval of form $[x,\\pi (x)]$ a CW-arrow, and an interval of form $[x,\\pi ^{-1}(x)]$ a CCW-arrow (each standing for clockwise and counterclockwise).",
"Given a cyclic interval $T$ , we use $\\operatorname{cw} *{T}$ to denote the number of CW-arrows contained in $T$ .",
"Similarly, we will use $\\operatorname{ccw} *{T}$ for the number of CCW-arrows contained in $T$ .",
"These numbers can easily be read from the associated decorated permutation of $\\mathcal {M}$ .",
"Recall that $\\operatorname{nbd} *{[a_1,b_1] \\cup \\cdots \\cup [a_s,b_s]} = \\operatorname{rk} *{\\mathcal {M}} - \\sum _i (\\operatorname{minelts} *{b_i,a_{i+1}})$ .",
"And $\\operatorname{minelts} *{b_i,a_{i+1}}$ stands for the minimal possible number of elements a basis can have in the interval $(b_i,a_{i+1})$ , which equals the number $|I_{a+1} \\cap (b_i,a_{i+1})|$ .",
"Hence $\\operatorname{minelts} *{b_i,a_{i+1}} = \\operatorname{ccw} *{(b_i,a_{i+1})}$ .",
"This gives us another way to interpret $\\operatorname{nbd} *{E}$ : it is $\\operatorname{rk} *{\\mathcal {M}}$ minus the total number of CCW-arrows contained in the complement of $E$ .",
"In the special case when $E$ is a cyclic interval, $\\operatorname{nbd} *{E}$ is given by $|E|$ minus the number of CW-arrows contained in $E$ .",
"Therefore for any $E$ , we can obtain $\\operatorname{nbd} *{E,\\Pi }$ by counting CW-arrows and CCW-arrows.",
"If $E$ is the disjoint union of $s$ cyclic intervals, we first write all possible non-crossing partitions of $[s]$ .",
"Each one of them gives a sum of $\\operatorname{nbd} *{E^{\\prime }}$ 's where $E^{\\prime }$ obtained from $E$ by taking some of the $s$ cyclic intervals of $E$ , and we compute them by counting the CCW-arrows (or CW-arrows for intervals) of the decorated permutation.",
"Consider the positroid associated with Figure REF .",
"Let us try to compute the rank for $E = [1,2] \\cup [7,10] \\cup [13,13]$ .",
"We have 3 disjoint intervals, so the upper bounds of $\\operatorname{rk} *{E}$ will be coming from the non-crossing partitions of $\\lbrace 1,2,3\\rbrace $ .",
"The following are the upper bounds for $\\operatorname{rk} *{E}$ we get: $\\operatorname{nbd} *{E,\\lbrace \\lbrace 1,2,3\\rbrace \\rbrace } = \\operatorname{nbd} *{E} = \\operatorname{rk} *{\\mathcal {M}} - \\operatorname{ccw} *{(2,7)} - \\operatorname{ccw} *{(10,13)} - \\operatorname{ccw} *{(13,13)} = 7 - 1 - 1 - 0 = 5.$ $\\operatorname{nbd} *{E,\\lbrace \\lbrace 1\\rbrace ,\\lbrace 2,3\\rbrace \\rbrace } = \\operatorname{nbd} *{E_1} + \\operatorname{nbd} *{E_2 \\cup E_3} = |[1,2]| - \\operatorname{cw} *{[1,2]} + \\operatorname{rk} *{\\mathcal {M}} - \\operatorname{ccw} *{(10,13)} - \\operatorname{ccw} *{(13,7)} = 1 + 7 - 1 - 1 = 6.$ $\\operatorname{nbd} *{E,\\lbrace \\lbrace 1,2\\rbrace ,\\lbrace 3\\rbrace \\rbrace } = \\operatorname{nbd} *{E_1 \\cup E_2} + \\operatorname{nbd} *{E_3} = \\operatorname{rk} *{\\mathcal {M}} - \\operatorname{ccw} *{(2,7)} - \\operatorname{ccw} *{(10,1)} + |[13,13]| - \\operatorname{cw} *{(13,13)} = 7 - 1 - 2 + 1 - 0 = 5.$ $\\operatorname{nbd} *{E,\\lbrace \\lbrace 1,3\\rbrace ,\\lbrace 2\\rbrace \\rbrace } = \\operatorname{nbd} *{E_1 \\cup E_3} + \\operatorname{nbd} *{E_2} = \\operatorname{rk} *{\\mathcal {M}} - \\operatorname{ccw} *{(2,13)} - \\operatorname{ccw} *{(13,1)} + |[7,10]| - \\operatorname{cw} *{[7,10]} = 7 - 5 - 0 + 4 - 0 = 6.$ $\\operatorname{nbd} *{E,\\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace ,\\lbrace 3\\rbrace \\rbrace } = \\operatorname{nbd} *{E_1} + \\operatorname{nbd} *{E_2} + \\operatorname{nbd} *{E_3} = |[1,2]| - \\operatorname{cw} *{[1,2]} + |[7,10]| - \\operatorname{cw} *{[7,10]} + |[13,13]| - \\operatorname{cw} *{[13,13]} = 2 - 1 + 4 - 0 + 1 - 0 = 6.$ Theorem REF tells us that $\\operatorname{rk} *{E} = 5$ .",
"Using Theorem REF , in [11] it is shown that the facets of the matroid polytope are given by cyclic intervals whose complement is covered by CCW-arrows.",
"It is also shown that the facets of the independendent set polytope of a positroid are given by sets whose complement is again covered by CCW-arrows.",
"In [3], the condition for an arbitrary subset of the ground set being a flat of the positroid will be given in terms of the decorated permutation, again using Theorem REF ."
]
] | 1709.01580 |
[
[
"Optimizing for Measure of Performance in Max-Margin Parsing"
],
[
"Abstract Many statistical learning problems in the area of natural language processing including sequence tagging, sequence segmentation and syntactic parsing has been successfully approached by means of structured prediction methods.",
"An appealing property of the corresponding discriminative learning algorithms is their ability to integrate the loss function of interest directly into the optimization process, which potentially can increase the resulting performance accuracy.",
"Here, we demonstrate on the example of constituency parsing how to optimize for F1-score in the max-margin framework of structural SVM.",
"In particular, the optimization is with respect to the original (not binarized) trees."
],
[
"Introduction",
"Many statistical learning problems in the area of natural language processing (NLP) including sequence tagging, sequence segmentation and various kinds of syntactic parsing have been successfully approached by means of structured prediction methods, which correspond to a machine learning paradigm that considers learning with complex outputs like sequences, trees or even general graphs.",
"Popular examples of the corresponding methods include maximum margin Markov networks (M$^3$ N) [13], structural support vector machine (SSVM) [16], and on-line algorithms like MIRA [10].",
"Apart from maximizing the margin between the true and false outputs, another appealing property of these discriminative learning algorithms is their ability to incorporate the loss function of interest directly in the training procedure, which potentially can improve the resulting prediction accuracy.",
"However, the existing training approaches including cutting-plane algorithm [7], bundle methods [12] and Frank-Wolfe optimization [9] assume that an efficient inference algorithm is given during the training in order to compute a subgradient of the objective function or the most violating output with respect to a given loss function.",
"This usually results in a combinatorial problem which often can be solved by means of dynamic programming.",
"The success of the latter crucially depends on the form of the underlying model and the chosen loss function.",
"Usually, if both decompose over small sets of variables we can apply efficient inference algorithms e.g.",
"Viterbi algorithm [6] for sequence tagging, CKY algorithm [17] for syntactic parsing or sum-product belief propagation [5] for probabilistic inference.",
"Still, some popular performance measure like precision in information retrieval or $F_1$ -score in segmentation and parsing tasks do not decompose in this way and, therefore, are often referred to as high-order measures.",
"Nevertheless, for non-decomposable loss functions which are build by a composition of locally decomposable statistics and some non-decomposable wrapper function, we can perform inference efficiently in polynomial time as has been shown in [1] and [3].",
"In this paper we consider only the task of syntactic parsing, an important preprocessing step for many NLP applications which aim at processing the meaning of a natural text.",
"In particular, we focus on the constituency parsing [14], [4] where the goal is, for a given input sentence, to predict the most probable parse tree according to some context-free grammar.",
"A lot of progress has been done previously in order to train a corresponding model based on finite-state techniques like probabilistic context free grammars (PCFGs) [8] or more general weighted context free grammars (WCFGs) [16].",
"Here we build on the discriminative max-margin approach of SSVM aiming at optimizing $F_1$ -score with respect to the constituents of parse trees.",
"In order to achieve a cubic running time for prediction it is a conventional approach to binarize the grammar before training.",
"Unfortunately, this introduces a bias during the training procedure as the corresponding loss function is evaluated on the binary representation while the resulting performance is measured on the original not binarezd trees.",
"In this paper we extend the inference procedure presented in [3] to account for this difference.",
"The result corresponds to the inference on not binarized trees leading to a better prediction accuracy while keeping the computational advantage of binarized representation."
],
[
"Optimizing for $F_1$ -score",
"A common structured prediction approach is to learn a functional relationship $f \\colon \\rightarrow $ between an input space $$ and an arbitrary discrete output space $$ of the form $f(x) = \\underset{y \\in }{} \\hspace*{5.0pt}w^(x, y).$ Here, $\\Psi \\colon \\times \\rightarrow \\mathbb {R}^d$ is the joint feature map describing the compatibility between an input $x$ and a corresponding output $y$ , and $w$ is the vector of model weights to be learned from a training sample of input-output pairs $(x_1, y_1), ..., (x_n, y_n) \\in \\times $ .",
"Training the models weights $w$ according to the maximum-margin criterion of an SSVM corresponds to solving the following optimization problem $\\begin{aligned}\\underset{w,\\hspace*{2.0pt} \\xi \\geqslant 0}{\\text{min.}}",
"& \\hspace*{5.0pt} \\frac{1}{2}\\Vert w\\Vert ^2 + \\frac{C}{n}\\sum _{i = 1}^n \\xi _i \\hspace*{10.0pt} \\text{subject to}\\\\& w^(\\Psi (x_i, y_i^*) - \\Psi (x_i, y) \\geqslant 1 - \\frac{\\xi _i}{\\Delta (y^*_i, y)}\\\\& \\forall i \\in \\lbrace 1, ..., n\\rbrace , \\hspace*{1.0pt}\\forall y \\in \\backslash \\lbrace y_i^*\\rbrace \\end{aligned}$ where $C$ is a regularization constant and $\\Delta \\colon \\times \\rightarrow \\mathbb {R}$ denotes a corresponding loss function quantifying the discrepancy between a prediction $y$ and the ground truth $y^*$ .",
"As already mentioned in the introduction the most popular training approaches rely on the assumption that during training process we are given an additional inference algorithm to compute the subgradient (in case of bundle methods) or the most violating configuration (in case of cutting-plane approach).",
"In the literature, this problem is referred to as the loss augmented inference [15], [2].",
"In the most general form it corresponds to maximizing the following objective $\\underset{y \\in }{\\max }\\hspace*{5.0pt}\\Delta (y^*, y) \\cdot (\\text{const.}",
"+ w^(x,y))$ where the constant term is $1 - w^(x,y^*)$ .",
"The main computational difficulty here arises from the fact that the size of $$ may grow exponentially in the size of the input $x$ as it is the case in constituency parsing.",
"Therefore, in order to perform inference efficiently we have to restrict the range of possible models and loss functions.",
"An important observation here is that $F_1$ -score can be parameterized by the number of true (TP) and false (FP) positives according to $F_1(TP, FP) = \\frac{2TP}{|y^*|+TP+FP}$ where $|y^*|$ denotes the number of nodes in the true parse tree.",
"In particular, these counts (TP and FP) decompose over the individual nodes of parse trees.",
"The main idea for solving (REF ) is then to stratify the maximisation over all configurations of (TP, FP) and picking the best."
],
[
"Model Description",
"For the task of constituency parsing the set $$ in (REF ) corresponds to all valid parse trees with respect to the given grammar $G$ and an input sentence $x$ .",
"A popular approach for representing a parse tree $y \\in $ is based on a joint feature vector $\\Psi (x,y)$ where the individual dimensions correspond to the grammar rules and the entries are the counts how often a production rule from $G$ occurs in a tree $y$ .",
"The dimension of the resulting feature vector is therefore equal to the number of different production rules in the grammar.",
"Furthermore, due to such representation the score $w^(x,y)$ for each pair $x,y$ decomposes over individual productions in the tree $y$ enabling efficient dynamic programming algorithms (e.g., CKY algorithm)."
],
[
"Grammar Binarization",
"In order to achieve a cubic running time (in the length of the sentence) it is a common approach to binarize a given grammar (or equivalently the trees) before training by the left or right factorization introducing new artificial constituents as illustrated in Figure 1.We can vary the amount of annotation (e.g., number of missing nodes to the right in case of right-factorization) contained in these artificial constituents, which is referred to as the horizontal annotation.",
"Another useful annotation technique in order to increase the expressivity of the grammar is the parent annotation where the labels of individual nodes in a parse tree are extended by the label of their parent nodes introducing more contextual information into the labels (see Figure 1).",
"Figure: Grammar binarization due to binarization of trees.In the notation of artificial constituents A|C-DA|C-D, AA before ||denotes the parent in the original tree and C-DC-D the children nodesof AAwhich are spanned by the current artificial constituent.",
"\"?\"\"?",
"\"denotes the parent label of AA."
],
[
"Loss Augmented Inference",
"We now show how the problem in (REF ) can be solved via dynamic programming for constituency parsing with $\\Delta _{F_1}(y^*,y) = 1 - F_1(y^*,y)$ .",
"Let an input sentence $x = (x_1, ..., x_{|x|})$ with $x_i$ denoting the token on position $i$ and a corresponding true parse tree $y^*$ be given.",
"Similar to the conventional CKY algorithm the idea here is to iteratively compute the values for the subproblems $\\Pi _{i,j,A}^{tp,fp} := \\underset{y \\in \\mathcal {T}_{i,j,A}^{tp,fp}}{\\max }\\hspace*{5.0pt}\\sum _{p \\in \\text{Productions}(y)} q(p),$ where $\\mathcal {T}_{i,j,A}^{tp,fp}$ denotes the set of all valid subtrees spanning the tokens $(x_i, ..., x_j)$ and having the label $A$ at the root.",
"The parameters $tp$ , $fp$ encode the number of true and false positives with respect to $y^*$ .",
"$q(p)$ denotes the weight of a production $p$ times its frequency in the parse tree $y$ .",
"That is, the quantity $\\Pi _{i,j,A}^{tp,fp}$ denotes the value of an optimal label configuration over the parse trees $y \\in \\mathcal {T}_{i,j,A}$ which additionally result in a fixed value for true and false positives.",
"The values of these subproblems can be computed in a bottom-up manner according to the following equation $\\Pi _{i,j,A}^{tp,fp} = &\\underset{A \\rightarrow B \\hspace*{1.0pt}C,\\hspace*{2.0pt} s,\\hspace*{2.0pt} \\hat{tp},\\hat{fp}}{\\max }\\hspace*{5.0pt}& q(A \\rightarrow B C) + \\nonumber \\\\&&\\Pi _{i,s,B}^{\\bar{tp}, \\hspace*{2.0pt}\\bar{fp}} + \\nonumber \\\\&&\\Pi _{s+1, j, C}^{\\hat{tp}, \\hat{fp}}$ where we maximize over all possible grammar productions $A \\rightarrow B \\hspace*{1.0pt}C$ with fixed $A$ , over all split points of subtrees $i \\leqslant s < j$ , and over possible distributions of the loss parameters $\\hat{tp}$ and $\\hat{fp}$ with $\\bar{tp} := tp - \\mathbf {1}([A]_{ij} \\hspace*{1.0pt} \\in \\hspace*{1.0pt}y^*) - \\hat{tp}$ and $\\bar{fp} := fp - \\mathbf {1}([A]_{ij} \\hspace*{1.0pt} \\notin \\hspace*{1.0pt}y^*) - \\hat{fp}$ .",
"The term $[A]_{ij}$ denotes a node in a subtree that spans tokens $x_i, ..., x_j$ and has the label $A$ , and $\\mathbf {1}(\\cdot )$ is an indicator function yielding 1 if the expression inside the brackets is true and 0 otherwise.",
"With a slight abuse of notation we write $[A]_{ij} \\in y^*$ to check if a node with the corresponding label is in a tree $y^*$ .",
"After computing the values for all the subproblems, we can obtain the optimal value $p^*$ of the problem in (REF ) by maximizing over all possible values $tp, fp$ according to $p^* = \\underset{tp, fp}{\\max }\\hspace*{2.0pt} (1 - F_1(tp, fp)) \\cdot (const.",
"+ \\Pi _{1,|x|,\\mathcal {S}}^{tp, fp}).$ where $\\text{const.",
"}$ corresponds to the constant term $1 - w^(x,y^*)$ .",
"$|x|$ denotes the number of tokens in the input sentence and $$ is the start (or root) symbol of each parse derivation.",
"The corresponding maximizing argument can be found by backtracking the optimal decisions in each computation step as usually done in dynamic programming.",
"Note, that the counts of true and false positives in the above computation scheme is with respect to the binarized tree representation.",
"The resulting performance, however, is evaluated on the original tree representation after reversing the binarization.",
"It turns out that we can easily adjust the above computation scheme to keep track of the corresponding counts with respect to unbinarized trees.",
"First note that in order to transform a binarized tree in the original form we need to remove all the artificial constituents, that is the counts of true and false positives are not affected by their presence.",
"Furthermore, after removing an artificial constituents we need to attach its children in a tree to its parent.",
"In particular, the boundary indices of the corresponding spans of the children nodes do not change during this procedure.",
"Finally we have to remove the additional annotation from the labels of the remaining nodes.",
"To summarize, we can compute the counts of true and false positives with respect to unbinarized grammar from binarized trees if we completely ignore artificial nodes and the additional annotation (e.g.",
"parent annotation).",
"More precisely, we only need to replace the indicator function $\\mathbf {1}$ for computing $\\bar{tp}, \\bar{fp}$ in (REF ) by $\\bar{\\mathbf {1}}([A^?",
"]_{ij} \\in y^*)={\\left\\lbrace \\begin{array}{ll}0, & A \\text{ is artificial,}\\\\\\mathbf {1}([A]_{ij} \\in y^*), & \\text{else}\\end{array}\\right.",
"}$ where $?$ denotes the parent annotation of $[A]_{ij}$ and $y^*$ corresponds to the (unbinarized) ground truth.",
"Similarly, we define $\\bar{\\mathbf {1}}([A^?",
"]_{ij} \\notin y^*)={\\left\\lbrace \\begin{array}{ll}0, & A \\text{ is artificial,}\\\\\\mathbf {1}([A]_{ij} \\notin y^*), & \\text{else}\\end{array}\\right.",
"}$ This way we ensure that the corresponding counts of true and false positives are with respect to the unbinarized trees.",
"The overall computation scheme is provable correct, that is it computes the optimal value of the problem in ($\\ref {E_LAI}$ )."
],
[
"Experiments",
"In this section we present our preliminary experimental results for the task of constituency parsing by training an SSVM via cutting plane algorithm and optimizing for $F_1$ -score.",
"In particular, we compare the performance when optimizing on trees in the binarized representation (marked by \"(bin.)\")",
"versus the original non binarized trees.",
"Additionally we report results when optimizing for 0/1 accuracy, and the number of false positives (#FP).",
"The resulting perforce is evaluated in terms of precision (P), recall (R), $F_1$ -score ($F_1$ ), and 0/1 prediction accuracy (all with respect to the unbinarized trees).",
"As training data we used a subset of the Wall Street Journal (WSJ) from the Penn English Treebank-3 restricted to sentences of the length $\\leqslant $ 20.",
"We used the standard data split: sections 2-21 of WSJ for training (16667 sentences), section 22 for validation (725 sentences) and section 23 for testing (1034 sentences).",
"The parse trees were preprocessed in the standard way by removing functional tags and null elements.",
"The regularization hyperparameter $C$ was chosen by cross-validation over a grid of values $\\lbrace 10^i \\colon i = 0,1, ...5\\rbrace $ .",
"We report the corresponding results on the test data in Table REF .",
"The first column describes the measure we optimized during the training procedure.",
"Here we can make two observations.",
"First, we see that there is a little difference in performance between #FP (bin.)",
"and $F_1$ -score (bin.)",
"on binarized trees supporting the claims in [4] (see Proposition 2).",
"Second, we see that adjusting training with $F_1$ -score for unbinarized trees improves the resulting performance upon training with binarized representation.",
"According to the Wilcoxon signed-rank test [11] this result is statically significant.",
"Figure REF illustrates the difference in loss $\\Delta _{F_1}$ for models optimized according to original versus binarized representation.",
"A corresponding null-hypothesis is that the measurement difference for a pair of methods follows a symmetric distribution around zero.",
"Here, we can see a clear shift of the corresponding distribution to the leftNote that we do not count examples with zero difference in the loss value.. Table: Experimental results on the test set.Figure: Illustration of the loss difference (in Δ F 1 \\Delta _{F_1}) on the test set when optimizing for F 1 F_1 score on the original treesversus binary representation."
],
[
"Conclusion",
"We demonstrated on the example of constituency parsing how to optimize the weights of the model with respect to $F_1$ -score in the maximum-margin framework of SSVMs.",
"In particular, we showed how the optimization during the training procedure can be performed with respect to the original non binarized trees.",
"More precisely, the proposed modification allows to perform loss augmented inference on non binarized trees, which results in a better prediction accuracy, while keeping the computational advantage of binarized representation.",
"Our preliminary experimental results suggest an improvement in the prediction performance by applying this new technique.",
"According to the Wilcoxon signed-rank test the presented performance difference is statically significant."
],
[
"Acknowledgments",
"This work was supported by the Federal Ministry of Education and Research under the Berlin Big Data Center Project under Grant FKZ 01IS14013A.",
"The work of K.-R. Müller was supported in part by the BK21 Program of NRF Korea, BMBF, under Grant 01IS14013A and by Institute for Information and Communications Technology Promotion (IITP) grant funded by the Korea government (No.",
"2017-0-00451)"
]
] | 1709.01562 |
[
[
"A multiplicative coalescent with asynchronous multiple mergers"
],
[
"Abstract We define a Markov process on the partitions of $[n]=\\{1,\\ldots,n\\}$ by drawing a sample in $[n]$ at each time of a Poisson process, by merging blocks that contain one of these points and by leaving all other blocks unchanged.",
"This coalescent process appears in the study of the connected components of random graph processes in which connected subgraphs are added over time with probabilities that depend only on their size.",
"First, we determine the asymptotic distribution of the coalescent time.",
"Then, we define a Bienayme-Galton-Watson (BGW) process such that its total population size dominates the block size of an element.",
"We compute a bound for the distance between the total population size distribution and the block size distribution at a time proportional to $n$.",
"As a first application of this result, we establish the coagulation equations associated with this coalescent process.",
"As a second application, we estimate the size of the largest block in the subcritical and supercritical regimes as well as in the critical window."
],
[
"Introduction",
"The paper is devoted to studying a family of multiplicative coalescent processes on a finite set $S$ defined by a simple algorithm.",
"To present this algorithm, let us fix a probability distribution $p$ on $\\operatorname{\\mathbb {N}}^*$ .",
"We construct a coalescent process denoted $(\\Pi _{S,p}(t))_{t\\ge 0}$ by the following algorithm: $\\Pi _{S,p}(0)$ is the partition defined by the singletons of $S$ ; At each event $\\tau $ of a Poisson process $(Z_t)_t$ with intensity one, we choose a positive integer $k$ according to $p$ and we draw $k$ elements $x_1,\\ldots ,x_k$ in $S$ by a simple random sampling with replacement.",
"The partition at time $\\tau $ is defined by merging blocks of $\\Pi _{S,p}(\\tau ^{-})$ that contain $x_1,\\ldots ,x_k$ into one block and by leaving all other blocks unchanged.",
"By construction, only one merger can occur at a given time but it may involve more than two blocks.",
"The probability that blocks coalesce depends only on the product of their sizes.",
"Such a coalescent process naturally appears when considering a random hypergraph process on the set of vertices $S$ of size $|S|=n$ .",
"A random hypergraph process can be defined as a Markov process $(\\mathcal {G}(t))_{t\\ge 0}$ whose states are hypergraphs on $S$ : it starts with the empty graph and hyperedges (i.e.",
"subsets of $S$ ) are added over time according to a given rule.",
"There are several possible definitions of hypergraph components.",
"One way is to identify a hyperedge $A$ to a connected subgraph and then a hypergraph to a multigraph; the component of a vertex can be defined as usual in a graph.",
"The process defined by the connected components of $\\mathcal {G}(t)$ for $t\\ge 0$ is a coalescent process.",
"Here are two examples of classical random hypergraph processes.",
"Erdös-Rényi random graph.",
"If a pair, chosen with a uniform distribution on $S^2$ , is added at each time of a Poisson process with intensity one, we obtain a variant of the Erdös-Rényi random graph process denoted $(\\mathcal {H}(n,t))_{t\\ge 0}$ : the probability that ${e=(i,j)\\in S^2}$ is an edge of $\\mathcal {H}(n,t)$ is equal to $1-\\exp (-\\frac{2t}{n^2})$ and the coalescent process associated with $(\\mathcal {H}(n,t))_{t\\ge 0}$ has the same distribution as $(\\Pi _{S,p}(t))_{t\\ge 0}$ where $p$ is the Dirac measure on 2 and $|S|=n$ .",
"Uniform random graph process.",
"For a fixed $d>2$ , if a subset of size $d$ chosen with a uniform distribution on $S^d$ , is added at each time of a Poisson process with intensity one, it defines a random hypergraph process whose components have similar properties as a $d$ -uniform random graph process.",
"The partition defined by the connected components of this random hypergraph process has the same law as $(\\Pi _{S,p}(t))_{t\\ge 0}$ where $p$ is the Dirac measure on $d$ .",
"More generally, if each new hyperedge $A$ added is chosen with a distribution $\\nu $ that depends only on the number of vertices in $|A|$ , then the associated coalescent process has the same distribution as $(\\Pi _{S,p}(t))_t$ , where $p(|A|)=\\nu (A)$ for every $A\\subset S$ .",
"Let us present the properties of $\\mathcal {H}(n,\\frac{nt}{2})$ which are related to our study.",
"For each property, we shall also review works done on random hypergraphs to introduce our contribution.",
"Precise statements of our results will be described in Section .",
"Connectivity threshold Erdös and Rényi in [12] and independently Gilbert in [17] have studied the probability that the random graph models they introduced are connected.",
"Erdös and Rényi results can be formulated for the random graph process $(\\mathcal {H}(n,\\frac{nt}{2}))_t$ as follows: Theorem For every $c\\in \\operatorname{\\mathbb {R}}$ and every $k\\in \\operatorname{\\mathbb {N}}$ , the probability that $\\mathcal {H}(n,\\frac{n}{2}(\\log (n)+c))$ contains a connected component of size $n-k$ and $k$ isolated points converges to $\\displaystyle {\\exp (-e^{-c})\\frac{e^{-ck}}{k!",
"}}$ as $n\\rightarrow $ tends to $+\\infty $ .",
"This shows that $\\frac{n}{2}\\log (n)$ is a sharp threshold function for the connectivity property.",
"Poole in his thesis [35], has extended this result for uniform random hypergraphs: the threshold for connectivity of a $d$ -uniform random hypergraph is $\\frac{n}{d}\\log (n)$ for every $d\\ge 2$ .",
"Kordecki in [23] has given a general formula for the probability that a random hypergraph is connected for non-uniform random hypergraph with bounded hyperedges.",
"Poisson point processes of Markov loops on a finite graph give examples of random graph processes for which connected subgraphs (close walks here) are added over time (see [25] and [24] for a survey of their properties).",
"Some general properties of the coalescent process induced by them have been presented by Le Jan and the author in [26].",
"In particular, it has been shown that when loops are constructed by a random walk killed at a constant rate on the complete graph $K_n$ , the coalescent process associated with the Poissonian ensembles of loops can be constructed as $\\Pi _{S,p}$ , where $p$ is a logarithmic distribution with a parameter depending on the killing rate; the connectivity threshold function have been established.",
"By a similar study, we extend the statement of the previous theorem for a large class of distributions for $p$ that contains probability distributions having a finite moment of order two showing in particular that the connectivity threshold for a random hypergraph whose components are described by $\\Pi _{S,p}$ is $\\displaystyle {\\frac{|S|\\log (|S|)}{\\sum _{k \\ge 2}kp(k)}}$ (Theorem REF ).",
"Phase transition.",
"The largest block size of $\\mathcal {H}(n,\\frac{nt}{2})$ undergoes a phase transition.",
"It was first proved by Erdös and Rényi in [13].",
"The statement we present is taken from [42], where the proof is based on the use of Bienaymé-Galton-Watson (BGW) processes.",
"Theorem ([42]) Let $c^{(n)}_t(x)$ denote the component size of a vertex $x$ of $\\mathcal {H}(n,\\frac{nt}{2})$ and let $c^{(n)}_{1,t}\\ge c^{(n)}_{2,t}$ denote the two largest component sizes.",
"Assume that $t<1$ .",
"For every vertex $x$ , $c^{(n)}_t(x)$ converges in distribution to the total population size of a BGW process with one progenitor and Poisson($t$ ) offspring distribution.",
"Let $I_{t}$ be the value at 1 of the Cramér function of the Poisson$(t)$ -distribution: $I_{t}=t-1-\\log (t)$ .",
"The sequence $\\displaystyle {\\left(\\frac{c^{(n)}_{1,t}}{\\log (n)}\\right)_n}$ converges in probability to $1/I_{t}$ .",
"Assume that $t>1$ and denote by $q_t$ the extinction probability of a BGW process with one progenitor and Poisson($t$ ) offspring distribution.",
"For every $a\\in ]1/2,1[$ , there exist $b>0$ and $c>0$ such that $\\operatorname{\\mathbb {P}}(|c^{(n)}_{1,t}-(1-q_t)n|\\ge n^a)+\\operatorname{\\mathbb {P}}(c^{(n)}_{2,t}\\ge c\\log (n))=O(n^{-b}).$ Assume that $t=1+\\theta n^{-1/3}$ for some $\\theta \\in \\operatorname{\\mathbb {R}}$ .",
"There exists a constant $b(\\theta )>0$ such that for every $w>1$ and every $n\\in \\operatorname{\\mathbb {N}}^*$ , $\\operatorname{\\mathbb {P}}(c^{(n)}_{1,1+\\theta n^{-1/3}} > w n^{2/3})\\le \\frac{w}{b(\\theta )}\\;\\text{and}\\;\\operatorname{\\mathbb {P}}(c^{(n)}_{1,1+\\theta n^{-1/3}} < w^{-1}n^{2/3})\\le \\frac{w}{b(\\theta )}.$ In [40], Schmidt-Pruzan and Shamir studied the size of the largest component for non-uniform random hypergraphs: in their model, the size of hyperedges is bounded and the probability that the hypergraph has a fixed hyperedge depends only on the size of the hyperedge.",
"They established similar statements for the largest component when the average degree of a vertex in the hypergraph is less than 1, equal to 1 and greater than 1.",
"More precise results on the phase transition have been established later in the case of uniform random hypergraphs (see [21]).",
"Bollobás, Janson and Riordan in [5] have studied the size of the connected components for a general model of random hypergraph: in their model a type is associated with each vertex and the probability to add a hyperedge $A$ depends on the types of the elements in $A$ .",
"From their study we can deduce that the size of the largest block of $\\Pi _{S,p}(|S|t)$ is $o_p(|S|)$ if $t\\sum _{k\\ge 2} k(k-1)p(k)<1$ and $\\rho |S|+o_p(|S|)$ if $t\\sum _{k\\ge 2} k(k-1)p(k)>1$ , where $1-\\rho $ is the smallest positive solution of the following equation: $x=\\exp \\left(-t\\sum _{k\\ge 2}kp(k)(1-x^{k-1})\\right).$ ($\\rho $ can be seen as the survival probability of a BGW process with a compound Poisson offspring distribution).",
"Janson in [20] proved a conjecture proposed by Durrett in [9] saying that for a random graph with a power law degree distribution with exponent $\\gamma >3$ , the largest component in the subcritical phase is of order $n^{\\frac{1}{\\gamma -1}}$ .",
"This result suggests that the size of the largest block of $\\Pi _{S,p}(|S|t)$ in the subcritical phase would be also order $|S|^{\\alpha }$ for some $0<\\alpha $ if $p$ does not have all its power moments finite.",
"Under the assumption that $p$ has a finite third moment, we give a bound for the distance between the cumulative distributions of the block size of an element and of the total population size of a BGW process with compound Poisson offspring distribution (Theorem REF ).",
"We deduce from this the asymptotic distribution of two block size as $|S|$ tends to $+\\infty $ (Corollary REF ).",
"We also study the largest block size in three different regimes (Theorems REF and REF ): in the subcritical phase, we show that the size of the largest block is $\\displaystyle {o_p(|S|^{\\frac{1+\\varepsilon }{u-1}})}$ for every $\\varepsilon >0$ , if $p$ has a finite moment of order $u \\ge 3$ and is $O_p(\\log (|S|))$ , if $p$ is a light-tailed distribution.",
"When $p$ is a regularly varying distribution with index smaller than $-3$ , we also establish that the size of the largest block grows faster than a positive power of $|S|$ as $|S|$ tends to $+\\infty $ .",
"In the critical window, we show that the size of the largest block is $O_p(n^{2/3})$ .",
"Although the supercritical regime is studied in [5], to complete the analysis of the largest block we present a simple proof of the property stated in (b) for our model.",
"Hydrodynamic behavior Let us now consider the average number of components of size $x$ in $\\mathcal {H}(n,\\frac{nt}{2})$ .",
"For any $t>0$ and $x\\in \\operatorname{\\mathbb {N}}^*$ , the average number of components of size $x$ in $\\mathcal {H}(n,\\frac{nt}{2})$ converges in $L^2$ to $v(x,t)= \\frac{(tx)^{x-1}e^{-tx}}{x.x!",
"}.$ The value $xv(x,t)$ is equal to the probability that $x$ is the total population size of a BGW process with one progenitor and Poisson($t$ ) offspring distributionFor $t\\le 1$ , $\\lbrace xv(x,t), x\\in \\operatorname{\\mathbb {N}}^*\\rbrace $ is a probability distribution called Borel-Tanner distribution with parameter $t$ .. $\\lbrace v(x,\\cdot ),\\ x\\in \\operatorname{\\mathbb {N}}^*\\rbrace $ is the solution on $\\operatorname{\\mathbb {R}}_+$ of the Flory's coagulation equations with multiplicative kernel: $\\frac{d}{dt}v(x,t)=\\frac{1}{2}\\sum _{y=1}^{x-1}y(x-y)v(y,t)v(x-y,t)\\\\-\\sum _{y=1}^{+\\infty }xyv(x,t)v(y,t)- xv(x,t)\\sum _{y=1}^{+\\infty }y\\big (v(y,0)-v(y,t)\\big )$ Up to time 1, this solution coincides with the solution of the Smoluchowski's coagulation equations with multiplicative kernel starting from the monodisperse state: $\\frac{d}{dt}v(x,t)=\\frac{1}{2}\\sum _{y=1}^{x-1}y(x-y)v(y,t)v(x-y,t)-xv(x,t)\\sum _{y=1}^{+\\infty }yv(y,t).$ Equations (REF ) introduced by Smoluchowski in [41] are used for example to describe aggregations of polymers in an homogeneous medium where diffusion effects are ignored.",
"The first term in the right-hand side describes the formation of a particle of mass $x$ by aggregation of two particles, the second sum describes the ways a particle of mass $x$ can be aggregated with another particle.",
"If the total mass of particles decreases after a finite time, the system is said to exhibit a `phase transition' called `gelation': the loss of mass is interpreted as the formation of infinite mass particles called gel.",
"Smoluchowski's equations do not take into account interactions between gel and finite mass particles.",
"Equations (REF ) introduced by Flory in [14] are a modified version of the Smoluchowski's equations with an extra term describing the loss of a particle of mass $x$ by `absorption' in the gel.",
"Let $T_{gel}$ denote the largest time such that the Smoluchowski's coagulation equations with monodisperse initial condition have a solution which has the mass-conserving propertyDifferent definitions of the `gelation time' $T_{gel}$ are used in the literature: the gelation time is sometimes defined as the smallest time when the second moment diverges (see [1]).",
"Then, $T_{gel}=1$ and $T_{gel}$ coincides with the smallest time when the second moment $\\sum _{x=1}^{+\\infty }x^2v(x,t)$ diverges (see [30]).",
"Let us note that the random graph process $(\\mathcal {H}(n,\\frac{nt}{2}))_{t\\ge 0}$ is equivalent to the microscopic model introduced by Marcus [28] and further studied by Lushnikov [27] (see [7] for a first study of the relationship between these two models and [1] for a review, [33], [32] and [16] for convergence results of Marcus-Lushnikov's model to (REF )).",
"Recently, Riordan and Warnke in [38] gave sufficient conditions under which the average number of blocks of size $x$ converges for a class of random graph processes in which a bounded number of edges can be added at each step according to a fixed rule.",
"This class includes uniform random hypergraph processes.",
"As far as we know such a result has not been established for more general random hypergraph processes.",
"Under the assumption that $p$ has a finite third moment, we show that the average number of blocks of size $x$ in the coalescent process $\\Pi _{S,p}$ converges in $L^2$ to the solution of coagulation equations in which more than two particles can collide at the same time at a rate that depends on the product of their masses (Theorem REF ).",
"Remark.",
"Let us note that Darling, Levin and Norris have introduced in [8] a random hypergraph model called Poisson$(\\rho )$ random hypergraph process and denoted $(\\Lambda _t)_{t\\ge 0}$.",
"The process $(\\Lambda _t)_{t\\ge 0}$ is defined as follows: Start with the set of vertices $S$ ; At each event $\\tau $ of a Poisson process with intensity 1, choose a positive integer $k\\le |S|$ with probability $\\rho (k)$ and a subset $A$ uniformly at random from the subsets of $S$ of size $k$ .",
"Then, add $A$ in the hyperedges subset of $\\Lambda _{\\tau ^{-}}$ .",
"One can choose $\\rho $ so that the coalescent process defined by the connected components of $(\\Lambda _t)_{t\\ge 0}$ is described by $\\Pi _{S,p}$ .",
"Indeed, $p(k)$ in the definition of $\\Pi _{S,p}$ describes the probability to add a subset defined by $k$ elements of $S$ chosen by a simple random sampling with replacement.",
"Hence, if we set $\\rho (j)=\\frac{\\binom{|S|}{j}}{|S|}\\sum _{k=j}^{+\\infty }\\frac{p(k)}{|S|^k}\\sum _{\\begin{array}{c}(k_1,\\ldots , k_{j})\\in (\\operatorname{\\mathbb {N}}^*)^{j},\\\\ k_1+\\cdots +k_{j} =k\\end{array} }\\binom{k}{k_1,\\ldots ,k_{|A|}}\\;\\text{for every}\\;1\\le j\\le |S|,$ then $\\Pi _{S,p}$ is the coalescent process defined by the connected components of $(\\Lambda _t)_{t\\ge 0}$ .",
"In [8], the object of study is not the connected components of $(\\Lambda _t)_{t\\ge 0}$ as we have defined them in our study but identifiable vertices.",
"Organization of the paper.",
"Section is devoted to a presentation of general properties of the coalescent process we study.",
"The main results are stated in Section .",
"In Section , we first study the distribution of the number of singletons in the coalescent process and the first time $\\tau ^{(singl)}_n$ the coalescent $\\Pi _{\\llbracket {n} \\rrbracket ,p}$ does not have singleton.",
"Next we show that the distribution of the coalescent time $\\tau ^{(coal)}_n$ coincides with the asymptotic distribution of $\\tau ^{(singl)}_n$ as $n$ tends to $+\\infty $ which proves Theorem REF .",
"In Section , we describe the exploration process used to compute the block size of an element and to construct the associated BGW process.",
"The asymptotic distribution of the block size of an element is studied in Section : proofs of Theorem REF and its corollaries are presented.",
"Section is devoted to the proof of Theorem REF that describes the hydrodynamic behaviour of the coalescent process.",
"In Section , we prove Theorems REF and REF which present some properties of the largest block size in the subcritical, critical and supercritical regimes.",
"Appendix contains some properties of BGW processes with a compound Poisson offspring distribution."
],
[
"Description of the model and general properties",
"To study the properties of $(\\Pi _{S,p}(t))_{t>0}$ , it is useful to construct it by the mean of a Poisson point process instead of the algorithm presented in the introduction.",
"Let us first introduce some notations associated with a finite set $S$ : The number of elements of $S$ is denoted by $|S|$ .",
"$\\mathcal {W}(S):=\\cup _{k\\in \\operatorname{\\mathbb {N}}^*}S^k$ denotes the set of nonempty tuples over $S$ and $\\mathfrak {P}(S)$ is the set of nonempty subsets of $S$ .",
"A tuple is called nontrivial if it contains at least two different elements of $S$ .",
"We write $\\mathcal {W}^{*}(S)$ for the set on nontrivial tuples over $S$ .",
"The length of a tuple $w\\in \\mathcal {W}(S)$ is denoted by $\\ell (w)$ ."
],
[
"The Poisson sample sets",
"Let $p=\\sum _{i=1}^{+\\infty }p(i)\\delta _{i}$ be a probability measure on $\\operatorname{\\mathbb {N}}^*$ such that $p(1)<1$ .",
"We denote by $G_p$ its probability generating function: $G_p(s)=\\sum _{k=1}^{+\\infty }p(k)s^k$ for $|s|\\le 1$ .",
"The following algorithm `Choose an integer $K$ with probability distribution $p$ and sample with replacement $K$ elements of $S$' defines a probability measure on $\\mathcal {W}(S)$ denoted by $\\mu _{S,p}$ : $\\mu _{S,p}(\\lbrace x\\rbrace )=\\frac{p(\\ell (x))}{|S|^{\\ell (x)}}\\text{ for every }x\\in \\mathcal {W}(S).",
"$ We consider a Poisson point process $\\mathcal {P}_{S,p}$ with intensity $\\text{Leb}\\times \\mu _{S,p}$ on $\\operatorname{\\mathbb {R}}_+\\times \\mathcal {W}(S)$ and for $t\\ge 0$ , we define $\\mathcal {P}_{S,p}(t)$ as the projection of the set $\\mathcal {P}_{S,p} \\cap ([0,t]\\times \\mathcal {W}(S))$ on $\\mathcal {W}(S)$ : $\\mathcal {P}_{S,p}(t)$ corresponds to the set of samples chosen before time $t$ .",
"Remark 1.1 Let $S^{\\prime }$ be a subset of $S$ .",
"The conditional probability $\\mu _{S,p}(\\cdot \\mid \\mathcal {W}(S^{\\prime }))$ seen as a probability on $\\mathcal {W}(S^{\\prime })$ is equal to $\\mu _{S^{\\prime },p_{S|S^{\\prime }}}$ where $p_{S|S^{\\prime }}$ is the probability on $\\operatorname{\\mathbb {N}}^*$ defined by: $p_{S|S^{\\prime }}(k)=\\Big (\\frac{|S^{\\prime }|}{|S|}\\Big )^k\\frac{p(k)}{G_{p}\\big (\\frac{|S^{\\prime }|}{|S|}\\big )}\\text{ for every }k\\in \\operatorname{\\mathbb {N}}^*.$ In particular, the restriction of $\\mathcal {P}_{S,p}(t)$ to tuples in $\\mathcal {W}(S^{\\prime })$ before time $t$ has the same distribution as $\\mathcal {P}_{S^{\\prime },p_{S|S^{\\prime }}}\\Big (G_{p}\\big (\\frac{|S^{\\prime }|}{|S|}\\big )t\\Big )$ .",
"Let us also note that the pushforward measure of $\\mu _{S,p}$ by the projection $\\pi _{S,S^{\\prime }}$ from $\\mathcal {W}(S)$ to $\\mathcal {W}(S^{\\prime })$ is equal to $\\mu _{S^{\\prime },p^{\\lbrace S^{\\prime }\\rbrace }}$ where $p^{\\lbrace S^{\\prime }\\rbrace }$ is the probability on $\\operatorname{\\mathbb {N}}^*$ defined by: $p^{\\lbrace S^{\\prime }\\rbrace }(k)=\\Big (\\frac{|S^{\\prime }|}{|S|}\\Big )^k\\sum _{\\ell =0}^{+\\infty }p(k+\\ell )\\binom{k+\\ell }{k}\\Big (1-\\frac{|S^{\\prime }|}{|S|}\\Big )^\\ell \\text{ for every }k\\in \\operatorname{\\mathbb {N}}^*.$ Remark 1.2 The order of elements in a tuple $w$ will play no role in the definition of the coalescent process, the main object is the subset of $S$ formed by the elements of $w$ .",
"The pushforward measure of $\\mu _{S,p}$ on $\\mathfrak {P}(S)$ is the probability measure $\\bar{\\mu }_{S,p}$ defined by $\\bar{\\mu }_{S,p}(\\lbrace A\\rbrace )=\\sum _{k=|A|}^{+\\infty }\\frac{p(k)}{|S|^k}\\sum _{\\begin{array}{c}(k_1,\\ldots , k_{|A|})\\in (\\operatorname{\\mathbb {N}}^*)^{|A|},\\\\ k_1+\\cdots +k_{|A|} =k\\end{array} }\\binom{k}{k_1,\\ldots ,k_{|A|}}\\;\\text{for every}\\; A \\in \\mathfrak {P}(S).",
"$ We choose to work with the Poisson point process on $\\operatorname{\\mathbb {R}}_+\\times \\mathcal {W}(S)$ instead of the associated Poisson point process on $\\operatorname{\\mathbb {R}}_+\\times \\mathfrak {P}(S)$ because some proofs are simpler to write.",
"To shorten the description we use sometimes a tuple $w\\in \\mathcal {W}(S)$ as the subset formed by its elements and write $x\\in w$ for $x\\in S$ to mean that $x$ is an element of the tuple $w$ and $w\\cap A\\ne \\emptyset $ for $A\\subset S$ to mean that $w$ contains some elements of the subset $A$ ."
],
[
"The coalescent process",
"If $A$ is a subset of $S$ , we define the $\\mathcal {P}_{S,p}(t)$ -neighborhood of $A$ as follows: $\\mathcal {V}_{A}(t)=A\\cup \\lbrace i\\in S,\\; \\exists w\\in \\mathcal {P}_{S,p}(t) \\text{ such that } i\\in w\\text{ and } w\\cap A\\ne \\emptyset \\rbrace .$ We can iterate this definition by setting: $\\mathcal {V}^k_{A}(t)=\\mathcal {V}^{k-1}_{\\mathcal {V}_{A}(t)}(t)$ for $k\\in \\operatorname{\\mathbb {N}}^*$ .",
"Given any $(i,j)\\in S^2$ , set $i\\underset{t}{\\sim } j \\text{ if and only if }\\exists k\\in \\operatorname{\\mathbb {N}}^*\\; \\text{ such that } j\\in \\mathcal {V}^{k}_{\\lbrace i\\rbrace }(t)$ .",
"This defines an equivalence relation on $S$ .",
"We denote by $\\Pi _{S,p}(t)$ the partition of $S$ defined by $\\underset{t}{\\sim }$ .",
"In other words, two elements $i$ and $j$ are in a same block of the partition $\\Pi _{S,p}(t)$ if and only if there exists a finite number of tuples $w_1,w_2,\\ldots , w_k\\in \\mathcal {P}_{S,p}(t)$ such that $i\\in w_1$ , $j\\in w_k$ and $w_i\\cap w_{i+1}\\ne \\emptyset $ for every $1\\le i\\le k-1$ .",
"The evolution in $t$ of $\\Pi _{S,p}(t)$ defines a coalescent process on $S$ .",
"Let us note that this coalescent process depends only on the restriction of $\\mathcal {P}_{S,p}$ to $\\operatorname{\\mathbb {R}}_+\\times \\mathcal {W}^{*}(S)$ ."
],
[
"Transition rates and semigroup of the coalescent process",
"Let us describe the transition rates and the semigroup of $\\Pi _{S,p}$ .",
"Proposition 1.3 Let $\\pi $ be a partition of $S$ into non-empty blocks $\\lbrace B_i,\\; i\\in I\\rbrace $ .",
"From state $\\pi $ , the only possible transitions of $(\\Pi _{S,p}(t))_ {t\\ge 0}$ are to partitions $\\pi ^{\\oplus J}$ obtained by merging blocks, indexed by some subset $J$ of $I$ of size greater than or equal to two, to form one block $B_J=\\cup _{j\\in J}B_j$ and leaving all other blocks unchanged.",
"Its transition rate from $\\pi $ to $\\pi ^{\\oplus J}$ is equal to: $\\tau _{\\pi ,\\pi ^{\\oplus J}}&=\\sum _{k\\ge |J|}\\frac{p(k)}{|S|^k}\\sum _{\\begin{array}{c}(k_1,\\ldots ,k_{|J|})\\in (\\operatorname{\\mathbb {N}}^*)^{|J|},\\\\ k_1+\\cdots +k_{|J|}=k\\end{array}}\\binom{k}{k_1,\\ldots ,k_{|J|}} \\prod _{j\\in J}|B_{j}|^{k_j}\\\\&=\\sum _{H\\subset J}(-1)^{|H|}G_{p}\\Big (\\frac{|B_{J\\setminus H}|}{|S|}\\Big ).$ For every partition $\\pi _0$ of $S$ , $\\operatorname{\\mathbb {P}}(\\Pi _{S,p}(t) \\text{ is finer than } \\pi \\mid \\Pi _{S,p}(0)=\\pi _0)\\\\=\\exp \\left(-t\\left(1-\\sum _{i\\in I}G_{p}\\left(\\frac{|B_{i}|}{|S|}\\right)\\right)\\right)\\operatorname{{1}}_{\\lbrace \\pi _0\\;\\text{is finer than}\\;\\pi \\rbrace }$ The transition rate $\\tau _{\\pi ,\\pi ^{\\oplus J}}$ is equal to the $\\mu _{S,p}$ -measure of tuples $w\\in \\mathcal {W}(B_J)$ that contain elements of each block $B_j$ for $j\\in J$ .",
"The first formula is obtained by enumerating such tuples ordered by their length.",
"The inclusion-exclusion formula yields the second formula since $\\tau _{\\pi ,\\pi ^{\\oplus J}}=\\mu _{S,p}(\\mathcal {W}(B_J))-\\mu _{S,p}\\left(\\underset{i\\in J}{\\cup }\\mathcal {W}(B_{J\\setminus \\lbrace i\\rbrace })\\right).$ $\\Pi _{S,p}(t)$ is finer than $\\pi $ if and only if every tuple chosen before time $t$ is included in a block of the partition $\\pi $ .",
"Therefore, if $\\pi _0$ is finer than $\\pi $ , $\\operatorname{\\mathbb {P}}(\\Pi _{S,p}(t) \\text{ is finer than } \\pi \\mid \\Pi _{S,p}(0)=\\pi _0)=\\exp \\Big (-t\\big (\\mu _{S,p}(\\mathcal {W}(S))-\\sum _{i\\in I}\\mu _{S,p}(\\mathcal {W}(B_{i}))\\big )\\Big ).$ Example 1.4 If $p$ is the Dirac measure $\\delta _{\\lbrace 2\\rbrace }$ , then the only possible transitions of $(\\Pi _{S,p}(t))_t$ are from a partition $\\pi =(B_i,\\; i\\in I)$ to partitions obtained by merging two blocks $B_i$ and $B_j$ ; the transition rate for such a transition is: $\\displaystyle {\\tau _{\\pi ,\\pi ^{\\oplus \\lbrace i,j\\rbrace }}=2\\frac{|B_i||B_j|}{|S|^2}}$ .",
"Therefore, for a partition $\\pi $ of $S$ into non-empty blocks $\\lbrace B_i,\\; i\\in I\\rbrace $ coarser than a partition $\\pi _0$ of $S$ , $\\operatorname{\\mathbb {P}}(\\Pi _{S,\\delta _{\\lbrace 2\\rbrace }}(t) \\text{ is finer than } \\pi \\mid \\Pi _{S,\\delta _{\\lbrace 2\\rbrace }}(0)=\\pi _0)=\\exp \\left(-2t\\sum _{i,j\\in I \\text{ s.t. }",
"i<j}\\frac{|B_i||B_j|}{|S|^2}\\right).$ Example 1.5 Let $p$ be the logarithmic distribution with parameter $a\\in ]0,1[$ : $p(k)=c\\frac{a^k}{k}$ with $\\frac{1}{c}=-\\log (1-a)$ for every $k\\in \\operatorname{\\mathbb {N}}^*$ .",
"For a partition $\\pi $ of $S$ into non-empty blocks $\\lbrace B_i,\\; i\\in I\\rbrace $ coarser than a partition $\\pi _0$ of $S$ , $\\operatorname{\\mathbb {P}}(\\Pi _{S,p}(t) \\text{ is finer than } \\pi \\mid \\Pi _{S,p}(0)=\\pi _0)=(1-a)^{ct}\\prod _{i\\in I}\\left(1-a\\frac{|B_i|}{|S|}\\right)^{-ct}.$ This shows that $\\Pi _{S,p}(t)$ has the same distribution as a coalescent process describing the evolution of the clusters of Poissonian loop sets on a complete graph defined in [26].",
"Let us briefly present how these Poissonian loop sets are defined.",
"Let $S$ stand for the vertices of a finite graph $\\mathcal {G}$ with $n$ vertices and let consider a simple random walk on $\\mathcal {G}$ killed at each step with probability $1-a$ .",
"In other words, $\\mathcal {G}$ is endowed with unit conductances and a uniform killing measure with intensity $\\kappa _n=n(\\frac{1}{a}-1)$ .",
"A discrete based loop $\\ell $ of length $k\\in \\operatorname{\\mathbb {N}}^*$ on $\\mathcal {G}$ is defined as an element of $\\mathcal {G}^k$ .",
"To each element $\\ell =(x_1,\\ldots ,x_k)$ of $\\mathcal {G}^k$ of length $k\\ge 2$ is assigned the weight $\\dot{\\mu }(\\ell )=\\frac{1}{k}P_{x_1,x_2}\\ldots P_{x_{k},x_{1}}$ where $P$ denotes the transition matrix of the random walk.",
"When $\\mathcal {G}$ is the complete graph $K_n$ then $\\dot{\\mu }(\\ell )=\\frac{a^k}{k n^k}$ for every $\\ell \\in K_{n}^{k}$ .",
"A based loop $\\ell =(x_1,\\ldots ,x_k)$ is said to be equivalent to the based loop $(x_i,\\ldots ,x_{k},x_1,\\ldots ,x_{i-1})$ for every $i\\in \\lbrace 2,\\ldots ,k\\rbrace $ .",
"An equivalent class of based loops is called a loop.",
"Let $\\mathcal {DL}(\\mathcal {G})$ denote the set of loops on $\\mathcal {G}$ .",
"The measure $\\dot{\\mu }$ on the set of based loops of length at least two induces a measure on loops denoted by $\\mu $ .",
"The Poisson loop sets on $\\mathcal {G}$ is defined as a Poisson point process $\\mathcal {DP}$ with intensity $\\text{Leb}\\times \\mu $ on $\\operatorname{\\mathbb {R}}_+\\otimes \\mathcal {DL}(\\mathcal {G})$ .",
"For $t>0$ , let $\\mathcal {DL}^{(n)}_{t}$ be the projection of the set $\\mathcal {DP}\\cap ([0,t]\\times \\mathcal {DL}(\\mathcal {G}))$ on $\\mathcal {DL}(\\mathcal {G})$ .",
"The loop set $\\mathcal {DL}^{(n)}_{t}$ defines a subgraph of $\\mathcal {G}$ .",
"The connected components of this subgraph form a partition of $S$ denoted by $\\mathcal {C}_{t}$ .",
"the distribution of which is computed in [26].",
"It follows that if the graph $\\mathcal {G}$ is the complete graph $K_n$ then $\\mathcal {C}_{-t\\log (1-a)}$ has the same distribution as $\\Pi _{S,p}(t)$ .",
"Figure: A loop set on the complete graph K 9 K_9"
],
[
"Restriction of the coalescent process to a subset ",
"In our model: each element of $S$ plays the same role, for every subset $A$ of $S$ , the Poisson tuple set inside $A$ at time $t$ , $\\mathcal {P}_{S}(t,A)$ has the same distribution as $\\mathcal {P}_{A,p_{S|A}}\\Big (G_{p}\\big (\\frac{|A|}{|S|}\\big )t\\Big )$ where $p_{S|A}(k)=\\Big (\\frac{|A|}{|S|}\\Big )^k\\frac{p(k)}{G_{p}\\big (\\frac{|A|}{|S|}\\big )}\\text{ for every }k\\in \\operatorname{\\mathbb {N}}^*$ and is independent of $\\mathcal {P}_{S}(t)\\setminus \\mathcal {P}_{S}(t,A)$ .",
"We can deduce from these properties a formula for the block size distribution of the coalescent process associated with $\\mathcal {P}_{S}(t,A)$ for every subset $A$ of $S$ : Proposition 1.6 For $x\\in S$ , let $\\Pi _{S,p}^{(x)}(t)$ denote the block of the partition $\\Pi _{S,p}(t)$ that contains $x$ .",
"Let $A$ be a subset of $S$ that contains $x$ .",
"For $k\\in \\lbrace 1,\\ldots ,|S|\\rbrace $ , $\\operatorname{\\mathbb {P}}\\left(\\Big \\vert \\Pi _{A,p_{S|A}}^{(x)}\\Big (tG_{p}\\big (\\frac{|A|}{ |S|}\\big )\\Big )\\Big \\vert =k\\right)=H_p(t,|S|,|A|,k)\\operatorname{\\mathbb {P}}(|\\Pi _{S,p}^{(x)}(t))|=k)$ where $H_p(t,n,m,k)=\\left(\\prod _{i=1}^{k-1}\\frac{m-i}{n-i}\\right)e^{t\\big (1-G_p(1-\\frac{k}{n})-G_p(\\frac{m}{n})+G_{p}(\\frac{m-k}{n})\\big )}$ with the convention $\\prod _{i=1}^{0}=1$ .",
"In particular, $\\operatorname{\\mathbb {E}}\\big ( H_p(t, |S|, j, |\\Pi _{S,p}^{(x)}(t)| ) \\big )=1\\quad \\forall j \\in \\lbrace 1,\\ldots ,|S|\\rbrace .$ Remark 1.7 The system of equations (REF ) characterizes the distribution of $|\\Pi _{S,p}^{(x)}(t))|$ since it can be written as a lower triangular linear system with positive coefficients and with $\\operatorname{\\mathbb {P}}(|\\Pi _{S,p}^{(x)}(t))|=k)$ for $k\\in \\lbrace 1,\\ldots ,|S|\\rbrace $ as unknowns.",
"When $p=\\delta _{\\lbrace 2\\rbrace }$ , we recover a formula presented by Ràth in a recent preprint (formula (1.1) of [37]): as applications of this formula, Ràth proposes in [37] new proofs of some properties of the component sizes of the Erdös-Rényi random graph in the subcritical and supercritical phases.",
"[Proof of Proposition REF ] Let $\\Pi ^{(x)}_{S,p}(t,A)$ denote the block of $x$ in the partition generated by $\\mathcal {P}_{S}(t,A)$ .",
"Let $B$ be a subset of $A$ containing $x$ : $\\Pi ^{(x)}_{S,p}(t)=B\\iff &\\Pi ^{(x)}_{S,p}(t,A)=B \\operatorname{ and }\\\\&\\text{no tuple in}\\;\\mathcal {P}_{S}(t)\\;\\text{contains both }\\;B \\;\\text{elements and}\\;S\\setminus A\\;\\text{elements}.$ By property REF , $&\\operatorname{\\mathbb {P}}(\\Pi ^{(x)}_{S,p}(t,A)=B)=\\operatorname{\\mathbb {P}}\\left(\\Pi ^{(x)}_{A,p_{S|A}}\\Big (tG_{p}\\big (\\frac{|A|}{|S|}\\big )\\Big )=B\\right) \\operatorname{ and }\\\\& \\operatorname{\\mathbb {P}}(\\Pi _{S,p}^{(x)}(t)=B)=\\operatorname{\\mathbb {P}}(\\Pi ^{(x)}_{S,p}(t,A)=B)e^{-tI_{S,A}(B)}$ where $I_{S,A}(B)&=&\\mu _S(\\lbrace \\omega \\in \\mathcal {W}(S),\\ \\omega \\cap B\\ne \\emptyset \\operatorname{ and }\\omega \\cap (S\\setminus A)\\ne \\emptyset \\rbrace )\\\\&=&\\mu _S(\\mathcal {W}(S))-\\mu _S(\\mathcal {W}(S\\setminus B)) - \\mu _S(\\mathcal {W}(A))+\\mu _S(\\mathcal {W}(A\\setminus B))\\\\&=& 1-G_p\\left(1-\\frac{|B|}{|S|}\\right)-G_p\\left(\\frac{|A|}{|S|}\\right)+G_p\\left(\\frac{|A|-|B[}{|S|}\\right).$ Then, formula (REF ) follows from property REF .",
"Indeed, $\\operatorname{\\mathbb {P}}(|\\Pi ^{(x)}_{S,p}(t,A)|=|B|)&=&\\binom{|A|-1}{|B|-1}\\operatorname{\\mathbb {P}}(\\Pi ^{(x)}_{S,p}(t,A)=B)\\\\&=&\\binom{|A|-1}{|B|-1}\\binom{|S|-1}{|B|-1}^{-1}\\operatorname{\\mathbb {P}}(\\Pi _{S,p}^{(x)}(t)=|B|)e^{tI_{S,A}(B)}.$ Let us note that $H_p(t,n,m,k)=0$ if $m$ and $k$ are two integers such that $k\\ge m+1$ .",
"Therefore, equality (REF ) holds for every $k\\in \\lbrace 1,\\ldots ,|S|\\rbrace $ .",
"The sum of over $k\\in \\lbrace 1,\\ldots ,|S|\\rbrace $ of (REF ) yields equation (REF )."
],
[
"Main results",
"Let us recall that $p$ is a probability distribution on $\\operatorname{\\mathbb {N}}^*$ such that $p(1)<1$ .",
"To shorten the notations, we assume now that $S=\\llbracket {n} \\rrbracket $ and omit the reference to the probability $p$ in the notation: the shorten notations $\\mu _n$ , $\\mathcal {P}_n$ , $\\mathcal {P}_n(t)$ and $\\Pi _{n}(t)$ are used instead of $\\mu _{S,p}$ , $\\mathcal {P}_{S,p}$ , $\\mathcal {P}_{S,p}(t)$ and $\\Pi _{S,p}(t)$ .",
"Before stating the main results, let us introduce other notations.",
"For $t>0$ , $\\mathcal {P}^{*}_n(t)$ denotes the projection of the set $\\mathcal {P}_n([0,t]\\times \\mathcal {W}^{*}(S))$ on $\\mathcal {W}^{*}(S)$ .",
"For $x\\in \\llbracket {n} \\rrbracket $ , $\\Pi _{n}^{(x)}(t)$ designates the block of the partition $\\Pi _{n}(t)$ that contains $x$ .",
"The $i$ -th factorial moment of $p$ is denoted by $m_{p,i}=\\sum _{k=i}^{+\\infty }k(k-1)\\ldots (k-i+1)p(k)$ (let us recall that its probability generating function is denoted by $G_p$ ).",
"Let $\\tilde{p}$ denote the size-biased probability measure defined on $\\operatorname{\\mathbb {N}}^*$ by $\\tilde{p}(k)=\\frac{(k+1)p(k+1)}{m^{*}_{p,1}}$ for every $k\\in \\operatorname{\\mathbb {N}}^*$ , where $m^{*}_{p,1}=m_{p,1}-p(1)$ .",
"For a positive real $\\lambda $ and a probability distribution $\\nu $ on $\\operatorname{\\mathbb {R}}$ , let $\\text{CPois}(\\lambda ,\\nu )$ denote the compound Poisson distribution with parameters $\\lambda $ and $\\nu $ : $\\text{CPois}(\\lambda ,\\nu )$ is the probability distribution of $\\sum _{i=1}^{N}X_i$ , where $N$ is a Poisson distributed random variable with expected value $\\lambda $ and $(X_i)_i$ is a sequence of independent random variables with law $\\nu $ , which is independent of $N$ .",
"For an integer $u\\in \\operatorname{\\mathbb {N}}^*$ , a positive real $a$ and a probability measure $\\eta $ on $\\operatorname{\\mathbb {N}}$ , we write $\\text{BGW}(u, a,\\eta )$ for a BGW process with family size distribution $\\text{CPois}(a,\\eta )$ and $u$ ancestors.",
"Finally for $t>0$ and $u\\in \\operatorname{\\mathbb {N}}^*$ , we use $T^{(u)}_{p}(t)$ to denote the total number of descendants of a $\\text{BGW}(u, tm^{*}_{p,1},\\tilde{p})$ process."
],
[
"Time to coalescence",
"The first result shows that the properties of having no singleton and of having only one block have the same sharp threshold function $\\frac{n\\log (n)}{m^{*}_{p,1}}$ .",
"Theorem 2.1 Assume that $p$ is a probability distribution on $\\operatorname{\\mathbb {N}}^*$ such that $p(1)<1$ , $m_{p,1}$ is finite and $1-G_{p}(1-h)=hm_{p,1}+o(\\frac{h}{\\log (h)})$ as $h$ tends to $0^+$ .",
"Let $\\tau ^{(singl)}_n$ and $\\tau ^{(coal)}_n$ denote the first time $t$ for which the partition $\\Pi _{n}(t)$ has no singleton and consists of a single block respectively.",
"For every $n\\in \\operatorname{\\mathbb {N}}^*$ , set $t_n=\\frac{n}{m^{*}_{p,1}}(\\log (n)+a+o(1))$ , where $a$ is a fixed real.",
"For every $k\\in \\operatorname{\\mathbb {N}}$ , the probability that $\\Pi _{n}(t_n)$ has $k$ singletons converges to $\\displaystyle {\\frac{e^{-ak}}{k!",
"}}e^{-e^{-a}}$ as $n$ tends to $+\\infty $ .",
"For every $k\\in \\operatorname{\\mathbb {N}}$ , the probability that $\\Pi _{n}(t_n)$ consists of a block of size $n-k$ and $k$ singletons converges to $\\displaystyle {\\frac{e^{-ak}}{k!",
"}e^{-e^{-a}}}$ as $n$ tends to $+\\infty $ .",
"In particular, $\\Big (m^{*}_{p,1}\\dfrac{\\tau ^{(singl)}_n}{n}-\\log (n)\\Big )_n$ and $\\Big (m^{*}_{p,1}\\dfrac{\\tau ^{(coal)}_n}{n}-\\log (n)\\Big )_n$ converge in distribution to the Gumbel distributionThe cumulative distribution function of the Gumbel distribution is $x\\mapsto e^{-e^{-x}}$ ..",
"Remark 2.2 Assumptions on $p$ in Theorem REF are satisfied by probability distributions on $\\operatorname{\\mathbb {N}}^*$ having a finite second moment but not only: the distribution $p$ on $\\operatorname{\\mathbb {N}}^*$ defined by $p(k)=\\frac{4}{k(k+1)(k+2)}\\; \\text{for}\\; k\\in \\operatorname{\\mathbb {N}}^*$ , satisfies the assumptions of Theorem REF and has an infinite variance.",
"Its generating function is $G_p(z)=1+2(z-1)+2(z-1)^2\\frac{1}{z^2}(-\\log (1-z)-z)\\; \\forall z\\in [0,1].$ When $p=\\delta _{\\lbrace d\\rbrace }$ with $d\\ge 2$ , $\\Pi _{n}$ corresponds to the partition made by the components of a random hypergraph process $\\mathcal {G}_n$ that have similar properties as the $d$ -uniform random hypergraph process.",
"It is not surprising to recover the threshold function $\\frac{n\\log (n)}{d}$ for connectivity of a $d$ -uniform random hypergraph (see [35]).",
"When $p$ is a logarithmic distribution with parameter $a$ (example REF ), $m^{*}_{p,1}=\\frac{-a^2}{(1-a)\\log (1-a)}.$"
],
[
"Block sizes",
"Let us turn to the study of the block size of an element at a time proportional to $n$ : Theorem 2.3 Let $t$ be a positive real.",
"Assume that $p$ has a finite third moment and that $p(1)<1$ .",
"Then there exists $C(t)>0$ such that for all $k,n\\in \\operatorname{\\mathbb {N}}^*$ and $x\\in \\llbracket {n} \\rrbracket $ , $|\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|\\le k)-\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)\\le k)|\\le C(t)\\frac{k^{2}}{n}.$ Remark 2.4 Let us present some properties of the distribution of $T^{(1)}_{p}(t)$ for $t>0$ .",
"A BGW process with family size distribution $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ is subcritical if and only if $t<\\frac{1}{m_{p,2}}$ .",
"Let $q_{p,t}$ denote the extinction probability of such a BGW process starting with one ancestor.",
"It is a decreasing function of $t$ .",
"Moreover, $\\left\\lbrace \\begin{array}{l}\\operatorname{\\mathbb {P}}(T^{(u)}_{p}(t) = u) =\\displaystyle {e^{-tum^{*}_{p,1}}}\\\\\\operatorname{\\mathbb {P}}(T^{(u)}_{p}(t) = k) = \\displaystyle {\\frac{u}{k}e^{-ktm^{*}_{p,1}}\\sum _{j=1}^{k-u}\\frac{(tkm^{*}_{p,1})^j}{j!",
"}(\\tilde{p})^{\\star j}(k-u)}\\quad \\forall k\\ge u+1.\\end{array}\\right.$ For $t\\le \\frac{1}{m_{p,2}}$ , $T^{(u)}_{p}(t)$ is almost surely finite and for $t>\\frac{1}{m_{p,2}}$ , $\\operatorname{\\mathbb {P}}(T^{(u)}_{p}(t)<\\infty )=(q_{p,t})^{u}<1$ .",
"For $t<\\frac{1}{m_{p,2}}$ , the distribution of $T^{(u)}_{p}(t)$ has a light tail (that is there exists $s_0>0$ such that $\\operatorname{\\mathbb {E}}(e^{sT^{(u)}_{p}(t)})$ is finite for every $s\\le s_0$ ) if and only if $p$ is a light-tailed distribution (application of Theorem 1 in [19]).",
"The statement of Theorem REF still holds if $|\\Pi _{\\llbracket {n} \\rrbracket ,p}^{(x)}(nt)|$ is replaced by $|\\Pi _{\\llbracket {n} \\rrbracket ,p_n}^{(x)}(nt_n)|$ where $(t_n)_n$ and $(p_n)_n$ converge rapidly to $t$ and $p$ respectively: Corollary 2.5 Let $(t_n)_n$ be a sequence of positive reals that converges to a real $t$ and let $(p_n)_n$ be a sequence of probability measures on $\\operatorname{\\mathbb {N}}^*$ that converges weakly to a probability measure $p$ on $\\operatorname{\\mathbb {N}}^*$ such that $p(1)<1$ .",
"If $\\sup _{n\\in \\operatorname{\\mathbb {N}}^*}\\sum _k k^3p_n(k)$ is finite, $t_n m^{*}_{p_n,1}-tm^{*}_{p,1}=O(\\frac{1}{n})$ and $\\operatorname{d_{\\text{TV}}}(\\tilde{p}_n,\\tilde{p})=O(\\frac{1}{n})$ then there exists $C(t)>0$ such that $\\forall n,k\\in \\operatorname{\\mathbb {N}}^*$ and $\\forall x\\in \\llbracket {n} \\rrbracket $ , $|\\operatorname{\\mathbb {P}}(|\\Pi _{\\llbracket {n} \\rrbracket ,p_n}^{(x)}(nt_n)|\\le k)-\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)\\le k)|\\le C(t) \\frac{k^{2}}{n}.$ As a first application of Corollary REF , let us consider the block size distribution for the partition defined by the Poisson tuple set inside a macroscopic subset of $\\llbracket {n} \\rrbracket $ at time $t$ : Corollary 2.6 Assume that $p$ is a probability distribution on $\\operatorname{\\mathbb {N}}^*$ such that $p(1)<1$ .",
"Let $a\\in ]0,1[$ .",
"Set $a_n=\\lfloor an\\rfloor $ and $p_n=p_{\\llbracket {n} \\rrbracket | \\llbracket {a_n} \\rrbracket }$ for $n\\in \\operatorname{\\mathbb {N}}^*$ .",
"Let $\\hat{p}_a$ denote the probability distribution on $\\operatorname{\\mathbb {N}}^*$ defined by $\\hat{p}_a(k)=\\frac{a^k p(k)}{G_{p}(a)}$ for every $k\\in \\operatorname{\\mathbb {N}}^*$ .",
"There exists $C_a(t)>0$ such that for every $k,n\\in \\operatorname{\\mathbb {N}}^*$ , $\\Big \\vert \\operatorname{\\mathbb {P}}\\left(|\\Pi _{\\llbracket a_n \\rrbracket , p_n}^{(1)}\\big (nG_{p}(\\frac{a_n}{n})t\\big )|\\le k\\right)-\\operatorname{\\mathbb {P}}\\left(T^{(1)}_{\\hat{p}_a}\\big (\\frac{G_{p}(a)}{a}t\\big )\\le k\\right)\\Big \\vert \\le C_a(t) \\frac{k^{2}}{n}.$ For every $u,k\\in \\operatorname{\\mathbb {N}}^*$ such that $k\\ge u$ , $\\operatorname{\\mathbb {P}}\\left(T^{(u)}_{\\hat{p}_a}\\big (\\frac{G_{p}(a)}{a}t\\big ) = k\\right)= a^{k-u}e^{tk(m_{p,1}-G_{p}^{\\prime }(a))}\\operatorname{\\mathbb {P}}(T^{(u)}_{p}(t)=k).$ Remark 2.7 It is not necessary to assume that the first moments of $p$ are finite since $\\hat{p}_a$ has finite moments of all order for every $a\\in ]0,1[$ .",
"Formula (REF ) for $u=1$ corresponds to the limit as $n$ tends to $+\\infty $ of the identity (REF ) satisfied by $|\\Pi _{\\llbracket a_n \\rrbracket , p_n}^{(1)}(nG_{p}(\\frac{a_n}{n})t)|$ .",
"If $t\\ge 1/m_{p,2}$ and $a$ is equal to the probability extinction $q_{p,t}$ of the BGW$(1,tm^{*}_{p,1},\\tilde{p})$ process, then $T^{(1)}_{\\hat{p}_a}(\\frac{G_{p}(a)}{a}t)$ has the same distribution as the total population size of a BGW$(1,tm^{*}_{p,1},\\tilde{p})$ process conditioned to become extinct.",
"Properties REF and REF stated in Subsection REF and Corollary REF allow to prove a joint limit theorem for the block sizes of two elements: Corollary 2.8 Let $x$ and $y$ be two distinct elements of $\\llbracket {n} \\rrbracket $ .",
"For every $t>0$ , $j,k\\in \\operatorname{\\mathbb {N}}^*$ , $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|=j \\operatorname{ and }|\\Pi _{n}^{(y)}(nt)|=k)$ converges to $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)=j)\\operatorname{\\mathbb {P}}( T^{(1)}_{p}(t)=k)$ as $n$ tends to $+\\infty $ ."
],
[
"Coagulation equations",
"Let us consider now the hydrodynamic behavior of the coalescent process $(\\Pi _{n}(t))_{t\\ge 0}$ .",
"A block of size $k$ can be seen as a cluster of $k$ particles of unit mass; at the same time, several clusters of masses $k_1,\\ldots , k_j$ can merge into a single cluster of mass ${k_1+\\ldots +k_j}$ at a rate proportional to the product $k_1\\ldots k_j$ .",
"The initial state corresponds to the monodisperse configuration ($n$ particles of unit mass).",
"Corollary REF is used to establish the convergence of the average number of blocks of size $k$ at time $nt$ as the number of particles $n$ tends to $+\\infty $ .",
"The limit seen as a function of $k$ is a solution to coagulation equations: Theorem 2.9 Let $p$ be a probability measure on $\\operatorname{\\mathbb {N}}^*$ such that $p(1)<1$ and $m_{p,3}$ is finite.",
"For $k\\in \\operatorname{\\mathbb {N}}^*$ , $n\\in \\operatorname{\\mathbb {N}}$ and $t>0$ , let $\\rho _{n,k}(t) =\\frac{1}{nk}\\sum _{x=1}^{n}\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(x)}(nt)| = k\\rbrace }$ be the average number of blocks of size $k$ and let $\\rho _{k}(t) =\\frac{1}{k}P(T^{(1)}_{p}(t) = k)$ .",
"$(\\rho _{n,k}(t))_n$ converges to $\\rho _{k}(t)$ in $L^2$ for every $t>0$ .",
"$(\\rho _{k}(t),\\ k\\in \\operatorname{\\mathbb {N}}^* \\text{ and } t\\ge 0)$ is a solution to the following coagulation equations: $\\frac{d}{dt}\\rho _k(t) =\\sum _{j=2}^{+\\infty }p(j)\\mathcal {K}_{j}(\\rho (t),k)$ where $\\mathcal {K}_{j}(\\rho (t),k)=\\Big (\\sum _{\\begin{array}{c}(i_1,\\ldots ,i_{j})\\in (\\operatorname{\\mathbb {N}}^*)^{j}\\\\ i_1+\\cdots +i_j=k\\end{array}}\\prod _{u=1}^{j}i_u\\rho _{i_u}(t)\\Big )\\operatorname{{1}}_{\\lbrace j\\le k\\rbrace }-jk\\rho _{k}(t).$ Remark 2.10 Consider a medium with integer mass particles and let $\\rho _{k}(t)$ denote the density of mass $k$ particles at time $t$ .",
"Equation (REF ) describes the evolution of $\\rho _{k}(t)$ if for every $j\\ge 2$ the number of aggregations of $j$ particles of mass $i_1,\\ldots ,i_j$ in time interval $[t,t+dt]$ is assumed to be $p(j)\\rho _{i_1}(t)\\ldots \\rho _{i_j}(t)\\kappa _j(i_1,\\ldots ,i_j) dt$ , where $\\kappa _j(i_1,\\ldots ,i_j)=i_1\\cdots i_j$ is the multiplicative kernel.",
"The first term in $\\mathcal {K}_{j}$ describes the formation of a particle of mass $k$ by aggregation of $j$ particles, the second term $jk\\rho _{k}(t)$ can be decomposed into the sum of the following two terms: ${\\displaystyle jk\\rho _{k}(t)\\Big (\\sum _{i=1}^{+\\infty }i\\rho _i(t)\\Big )^{j-1}}$ that describes the ways a particle of mass $k$ can be aggregated with $j-1$ other particles.",
"${\\displaystyle jk\\rho _{k}(t)\\sum _{h=1}^{j-1}\\binom{j-1}{h}\\Big (\\sum _{i=1}^{+\\infty }i(\\rho _{i}(0)-\\rho _i(t))\\Big )^h \\Big (\\sum _{u=1}^{+\\infty }u\\rho _u(t)\\Big )^{j-1-h}}$ .",
"This term is null if the total mass is preserved.",
"Otherwise, the decrease of the total mass can be interpreted as the appearance of a `gel' and this term describes the different ways a particle of mass $k$ can be aggregated with the gel and other particles.",
"The system of equations $\\frac{d}{dt}\\rho _{k}(t) = \\mathcal {K}_{2}(\\rho (t),k), \\quad \\forall k\\in \\operatorname{\\mathbb {N}}^*$ corresponds to the Flory's coagulation equations with the multiplicative kernel (see equation (REF )).",
"An application of Theorem REF with $p=\\delta _{\\lbrace j\\rbrace }$ for $j\\ge 2$ , shows that an approximation of the solution of the system of equations $\\frac{d}{dt}\\rho _k(t) = \\mathcal {K}_{j}(\\rho (t),k), \\quad \\forall k\\in \\operatorname{\\mathbb {N}}^*$ can be constructed by drawing tuples of fixed size $j$ .",
"Corollary 2.11 Let $j$ be an integer greater than or equal to 2.",
"For $k\\in \\operatorname{\\mathbb {N}}^*$ , $n\\in \\operatorname{\\mathbb {N}}$ and $t>0$ , let $\\rho ^{(j)}_{n,k}(t)$ be the average number of blocks of size $k$ in the partition $\\Pi _{\\llbracket {n} \\rrbracket ,\\delta _{\\lbrace j\\rbrace }}(\\frac{nt}{j})$ .",
"$(\\rho ^{(j)}_{n,k}(t))_n$ converges to $\\rho ^{(j)}_{k}(t)=e^{-tk}\\dfrac{(tk)^{\\frac{k-1}{j-1}}}{k^2(\\frac{k-1}{j-1})!",
"}\\operatorname{{1}}_{\\lbrace k-1\\in (j-1)\\operatorname{\\mathbb {N}}\\rbrace }$ in $L^2$ for every $t>0$ .",
"$(\\rho ^{(j)}_k(t),\\ k\\in \\operatorname{\\mathbb {N}}^* \\text{ and } t\\ge 0)$ is a solution to the following coagulation equations: $\\frac{d}{dt}\\rho _k(t) =\\mathcal {K}_{j}(\\rho (t),k)$ where $\\mathcal {K}_{j}$ is defined by equation (REF ).",
"The function $\\rho (t)$ defined by $\\rho _{k}(t)=\\frac{1}{k}\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)=k)$ for every $k\\in \\operatorname{\\mathbb {N}}^*$ gives an explicit solution of (REF ) with mass-conserving property on the interval $[0;\\frac{1}{m_{p,2}}]$ .",
"Its second moment $\\displaystyle {\\sum _{k=1}^{+\\infty }k^2\\rho _{k}(t)=(1-tm_{p,2})^{-1}}$ diverges as $t$ tends to $m_{p,2}$ ."
],
[
"Phase transition",
"As a last application of Theorem REF , we show that the block sizes of $(\\Pi _{n}(nt))_{t\\ge 0}$ undergo a phase transition at $t=\\frac{1}{m_{p,2}}$ similar to the phase transition of the Erdös-Rényi random graph process and present some bounds for the sizes of the two largest blocks in the three phases: Theorem 2.12 Let $p$ be a probability measure on $\\operatorname{\\mathbb {N}}^*$ such that $p(1)<1$ .",
"Let $B_{n,1}(nt)$ and $B_{n,2}(nt)$ denote the first and second largest blocks of $\\Pi _{n}(nt)$ .",
"Subcritical regime.",
"Let $0<t<\\frac{1}{m_{p,2}}$ .",
"Assume that $p$ has a finite moment of order $u$ for some $u\\ge 3$ .",
"If $(a_n)_n$ is a sequence of reals that tends to $+\\infty $ , then $\\operatorname{\\mathbb {P}}(|B_{n,1}(nt)|> a_n n^{\\frac{1}{u-1}})$ converges to 0 as $n$ tends to $+\\infty $ .",
"Assume that $G_{p}$ is finite on $[0,r]$ for some $r>1$ .",
"Let $L_{t}$ denote the moment-generating function of the $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distribution.",
"Set$h(t)$ is the value of the Cramér function at 1 of the $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distribution.",
"$h(t)=\\sup _{\\theta >0}(\\theta -\\log (L_{t}(\\theta ))).$ Then $h(t)>0$ and for every $a>(h(t))^{-1}$ , $\\operatorname{\\mathbb {P}}(|B_{n,1}(nt)|> a\\log (n))$ converges to 0 as $n$ tends to $+\\infty $ .",
"Supercritical regime.",
"Assume that $p$ has a finite moment of order three and that $t>\\frac{1}{m_{p,2}}$ .",
"Let $q_{t}$ denote the extinction probability of a BGW process with one progenitor and $\\text{CPois}(tm_{p,1},\\tilde{p})$ offspring distribution.",
"For every $a\\in ]1/2,1[$ , there exist $b>0$ and $c>0$ such that $\\operatorname{\\mathbb {P}}\\left(\\big \\vert |B_{n,1}(nt)|-(1-q_{t})n\\big \\vert \\ge n^a\\right)+\\operatorname{\\mathbb {P}}\\left(|B_{n,2}(nt)|\\ge c\\log (n)\\right) = O(n^{-b}).$ Critical window.",
"Assume that $p$ has a finite moment of order three.",
"For every $\\theta \\ge 0$ , there exists a constant $b>0$ such that for every $c>1$ and $n\\in \\operatorname{\\mathbb {N}}^*$ $\\operatorname{\\mathbb {P}}\\left(|B_{n,1}(\\frac{n}{m_{p,2}}(1+\\theta n^{-1/3}))| > cn^{2/3}\\right)\\le \\frac{c}{b}.$ Remark 2.13 Let us provide further information on the subcritical regime (${0<t<\\frac{1}{m_{p,2}}}$ ).",
"The upper bound for $|B_{n,1}(nt)|$ given in assertion 1.",
"(b) is reached when ${p=\\delta _2}$ ; Indeed, it is known since the Erdös and Rényi's paper [13] that $\\frac{1}{\\log (n)}|B_{n,1}(\\frac{ns}{2})|$ converges in probability to $(s-1-\\log (s))^{-1}$ as $n$ tends to $+\\infty $ , when $0<s<1$ .",
"Let us assume now that $p$ is regularly varying with index $-\\alpha <-3$ : there exists a slowly varying function $\\ell $ such that $\\sum _{j>k}p(j)=k^{-\\alpha }\\ell (k)$ $\\forall k\\in \\operatorname{\\mathbb {N}}$ .",
"Assertion 1.",
"(a) implies that for every $\\varepsilon >0$ , $\\operatorname{\\mathbb {P}}(|B_{n,1}(nt)|> n^{\\frac{1}{\\alpha -1}+\\varepsilon })$ tends to 0 as $n$ tends to $+\\infty $ .",
"Let us note that $n^{\\frac{1}{\\alpha -1}}$ corresponds to the order of the largest size for the total progeny of $n$ independent BGW$(1,tm^{*}_{p,1},\\tilde{p})$ processes.",
"Indeed, one can show that: If $T_{1},\\ldots ,T_n$ are the total progeny of $n$ independent BGW$(1,tm^{*}_{p,1},\\tilde{p})$ processes, then for every $1<\\alpha _1<\\alpha <\\alpha _2$ , $\\operatorname{\\mathbb {P}}(\\max _{i=1,\\ldots ,n}T_{i}> n^{\\frac{1}{\\alpha _1-1}})+\\operatorname{\\mathbb {P}}(\\max _{i=1,\\ldots ,n}T_{i}< n^{\\frac{1}{\\alpha _2-1}})\\underset{n\\rightarrow +\\infty }{\\rightarrow 0}.$ An application of the second moment method allows to prove that the largest block size actually grows faster than a positive power of $n$ in the subcritical regime, but gives an exponent smaller than expected: Theorem 2.14 Assume that $p$ is regularly varying with index $-\\alpha <-3$ .",
"If $t< \\frac{1}{m_{p,2}}$ then for every $\\alpha ^{\\prime }>\\alpha $ , $\\operatorname{\\mathbb {P}}(\\max _{x\\in \\llbracket {n} \\rrbracket }|\\Pi _{n}^{(x)}(nt)|\\le n^{\\frac{1}{1+\\alpha ^{\\prime }}})$ converges to 0 as $n$ tends to $+\\infty $ ."
],
[
"The number of singletons and the coalesence time",
"In a first part, we investigate the distribution of the number of singletons in the partition at time $t$ and the asymptotic distribution of the first time $t$ at which $\\Pi _{n}(t)$ does not have singleton.",
"In a second part, we show that the asymptotic distribution of the coalescence time as $n$ tends to $+ \\infty $ (that is the first time $t$ at which $\\Pi _{n}(t)$ consists of a single block) coincides with the asymptotic distribution of the first time $\\Pi _{n}$ does not have singleton."
],
[
"Number of singletons",
"Let us observe that the block of an element $x$ in the partition $\\Pi _{n}(t)$ is a singleton if and only if tuples in $\\mathcal {P}^{*}_n(t)$ do not contain $x$ .",
"The model is thus a variant of a coupon collector's problem with group drawings.",
"The exclusion-inclusion lemma provides an exact formula for the number of singletons in $\\Pi _{n}(t)$ .",
"Proposition 3.1 Let $Y_{n,p}(t)$ denote the number of singletons in $\\Pi _{n}(t)$ .",
"For every ${k\\in \\lbrace 0,\\ldots ,n\\rbrace }$ , $\\operatorname{\\mathbb {P}}(Y_{n,p}(t)=k)=\\sum _{j=0}^{n-k}(-1)^{j}\\frac{n!}{k!j!(n-k-j)!",
"}\\exp \\Big (-t\\big (1-G_{p}(1-\\frac{k+j}{n})-(k+j)G_{p}(\\frac{1}{n})\\big )\\Big ).$ Let $N_{n}^{(x)}(t)$ denote the number of tuples in $\\mathcal {P}^{*}_n(t)$ that contain the element $x$ .",
"$\\operatorname{\\mathbb {P}}(Y_{n,p}(t)=k)=\\sum _{F\\subset \\llbracket {n} \\rrbracket ,\\ |F|=k}\\operatorname{\\mathbb {P}}\\left(N_{n}^{(x)}(t)>0\\;\\forall x\\notin F\\; \\operatorname{ and }\\;\\sum _{x\\in F}N_{n}^{(x)}(t)=0\\right).$ By the exclusion-inclusion lemma, $\\operatorname{\\mathbb {P}}\\left(N_{n}^{(x)}(t)>0\\;\\forall x\\notin F\\; \\operatorname{ and }\\;\\sum _{x\\in F}N_{n}^{(x)}(t)=0\\right)= \\sum _{K\\subset \\llbracket {n} \\rrbracket \\setminus F}(-1)^{|K|}\\operatorname{\\mathbb {P}}\\left(\\sum _{x\\in F\\cup K}N_{n}^{(x)}(t)=0\\right).$ We conclude by noting that for any subset $A\\subset \\llbracket {n} \\rrbracket $ , $\\operatorname{\\mathbb {P}}\\left(\\sum _{x\\in A}N_{n}^{(x)}(t)=0\\right)=\\exp \\Big (-t\\mu (w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\ w\\cap A\\ne \\emptyset )\\Big )$ with $\\mu \\left(w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\ w\\cap A\\ne \\emptyset \\right)=1-\\mu \\left(\\mathcal {W}(\\llbracket {n} \\rrbracket \\setminus A)\\right)-\\mu \\left(\\bigcup _{a\\in A}\\mathcal {W}(\\lbrace a\\rbrace )\\right)\\\\=1-G_{p}\\left(1-\\frac{|A|}{n}\\right)-|A|G_{p}\\left(\\frac{1}{n}\\right).$ An analogy to the classical coupon collector's problem provides an idea of the average time until $\\Pi _{n}$ has no singleton: the number of tuples in $\\mathcal {P}^{*}_n(t)$ is in average $t\\mu _n(\\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ))$ and the length of nontrivial tuples is in average $(\\mu _n(\\mathcal {W}^{*}(\\llbracket {n} \\rrbracket )))^{-1}\\sum _{k=2}^{+\\infty }kp(k)(1-\\frac{1}{n^{k-1}}).$ Therefore, the total number of elements drawing before time $t$ and belonging to nontrivial tuples is in average $t(m^{*}_{p,1}+ O(\\frac{1}{n}))$ .",
"If the elements are drawn one by one and not by groups of random sizes, then the solution of the classical coupon collector's problem, suggests that the time until $\\Pi _{n}$ has no singleton would be around $\\frac{n\\log (n)}{m^{*}_{p,1}}$ .",
"The following result shows that this analogy holds in particular when $p$ has a finite second moment.",
"Theorem (REF .",
"(i)) Assume that $m_{p,1}$ is finite and $1-G_{p}(1-h)=hm_{p,1}+o(\\frac{h}{\\log (h)})$ as $h$ tends to $0^+$ .",
"For every $a\\in \\operatorname{\\mathbb {R}}$ , the number of singletons in $\\Pi _{n}$ at time $\\frac{n}{m^{*}_{p,1}}(\\log (n)+a+o(1))$ converges in distribution to the Poisson distribution with parameter $e^{-a}$ as $n$ tends to $+\\infty $ .",
"Let $\\tau ^{(singl)}_n$ denote the first time $t$ when $\\Pi _{n}(t)$ has no singleton.",
"The sequence $\\Big (m^{*}_{p,1}\\frac{\\tau ^{(singl)}_n}{n}-\\log (n)\\Big )_n$ converges in distribution to the Gumbel distribution.",
"Set $t_n=\\frac{n}{m^{*}_{p,1}}(\\log (n)+a+o(1))$ .",
"Using the notation introduced in proof of Proposition REF , the number of singletons in $\\Pi _{n}(t_n)$ is $Y_{n,p}(t_n)=\\sum _{x\\in \\llbracket {n} \\rrbracket }\\operatorname{{1}}_{\\lbrace N_{x}(t_n)=0\\rbrace }.$ By the theory of moments, it suffices to show that the factorial moments of any order of $Y_{n,p}(t_n)$ converge to those of the Poisson distribution with parameter $e^{-a}$ to prove the convergence in distribution.",
"Let $k\\in \\operatorname{\\mathbb {N}}^*$ .",
"The $k$ -th factorial moment of $Y_{n,p}(t_n)$ is $\\operatorname{\\mathbb {E}}(Y_{n,p}(t_n))_k:=\\operatorname{\\mathbb {E}}(Y_{n,p}(t_n)(Y_{n,p}(t_n)-1)\\ldots (Y_{n,p}(t_n)-k+1))\\\\=\\sum _{F\\subset \\llbracket {n} \\rrbracket ,\\ |F|=k}k!\\operatorname{\\mathbb {P}}(\\sum _{x\\in F}N_{n}^{(x)}(t_n)=0).$ Therefore, $\\operatorname{\\mathbb {E}}(Y_{n,p}(t_n))_k=n^k\\prod _{i=1}^{k-1}(1-\\frac{i}{n})\\exp \\Big (-t_n(1-G_{p}(1-\\frac{k}{n})-kG_{p}(\\frac{1}{n}))\\Big ).$ Set $I_{n,k}=-t_n\\Big (1-G_{p}(1-\\frac{k}{n})-kG_{p}(\\frac{1}{n})\\Big )+k\\log (n)$ .",
"It can be rewritten $I_{n,k} = -t_n\\Big (1-G_{p}(1-\\frac{k}{n})-\\frac{k}{n}m_{p,1}-k(G_{p}(\\frac{1}{n})-\\frac{p(1)}{n})\\Big )-ak+o(1).$ Therefore, $I_{n,k}$ converges to $-ak$ as $n$ tends to $+\\infty $ since $G_{p}(\\frac{1}{n})=\\frac{1}{n}p(1) +O(\\frac{1}{n^2})$ and $1-G_{p}(1-\\frac{k}{n})-\\frac{k}{n}m_{p,1}=o((n\\log (n))^{-1})$ by assumption.",
"This shows that $\\operatorname{\\mathbb {E}}(Y_{n,p}(t_n))_k$ converges to $\\exp (-ka)$ for every $k\\in \\operatorname{\\mathbb {N}}^*$ .",
"To deduce the assertion for $\\tau ^{(singl)}_n$ , it suffices to note that for every $x\\in \\operatorname{\\mathbb {R}}$ , $m^{*}_{p,1}\\frac{\\tau ^{(singl)}_n}{n}-\\log (n)\\le x\\; \\iff \\; Y_{n,p}(t_{n,x})=0.$ where $t_{n,x}=\\frac{n(\\log (n)+x)}{m^{*}_{p,1}}$ ."
],
[
"Time to coalescence",
"Let $\\tau ^{(coal)}_n$ denote the first time $t$ for which the partition $\\Pi _{n}(t)$ consists of a single block.",
"Theorem (REF .",
"(ii)) Assume that $m_{p,1}$ is finite and $1-G_{p}(1-h)=hm_{p,1}+o(\\frac{h}{\\log (h)})$ as $h$ tends to $0^+$ .",
"For every $n\\in \\operatorname{\\mathbb {N}}^*$ , set $t_n=\\frac{n}{m^{*}_{p,1}}(\\log (n)+a+o(1))$ where $a$ is a fixed real.",
"For every $k\\in \\operatorname{\\mathbb {N}}$ , the probability that $\\Pi _{n}(t_n)$ consists of a block of size $n-k$ and $k$ singletons converges to $\\displaystyle {\\exp (-e^{-a})\\frac{e^{-ak}}{k!",
"}}$ as $n$ tends to $+\\infty $ .",
"In particular, $\\Big (m^{*}_{p,1}\\dfrac{\\tau ^{(coal)}_n}{n}-\\log (n)\\Big )_n$ converges in distribution to the Gumbel distribution.",
"We adapt the proof of Theorem 5.6 given in [26] in the context of Markov loops in the complete graph (that is when $p$ is a logarithmic distribution).",
"For $k\\in \\operatorname{\\mathbb {N}}$ , let $H_{n,k}$ denote the event `$\\Pi _{n}(t_n)$ consists only of a block of size $n-k$ and $k$ singletons' and let $J_n$ be the event `$\\Pi _{n}(t_n)$ has at least two blocks of size greater or equal to 2'.",
"We have to prove that $\\operatorname{\\mathbb {P}}(H_{n,k})$ converges to $e^{-e^{-a}}\\frac{e^{-ka}}{k!",
"}$ .",
"As $\\operatorname{\\mathbb {P}}(Y_{n,p}=k)$ converges to $e^{-e^{-a}}\\frac{e^{-ka}}{k!",
"}$ and is equal to ${\\operatorname{\\mathbb {P}}(H_{n,k})+\\operatorname{\\mathbb {P}}(\\lbrace Y_{n,p}=k\\rbrace \\cap J_n)}$ for $n\\ge k+2$ , it suffices to prove that $\\operatorname{\\mathbb {P}}(J_n)$ converges to 0.",
"For a subset $F$ of $\\llbracket {n} \\rrbracket $ , let $b_{n}(F)$ denote the probability that $F$ is a block of $\\Pi _{n}(t_n)$ and set $S_{n,r}=\\sum _{F\\subset \\llbracket {n} \\rrbracket , |F|=r} b_{n}(F)$ for $r\\in \\llbracket {n} \\rrbracket $ .",
"The proof consists in showing that $\\sum _{r=2}^{\\lfloor n/2\\rfloor }S_{n,r}$ , which is an upper bound of $\\operatorname{\\mathbb {P}}(J_n)$ , converges to 0.",
"For every subset $A$ of $\\llbracket {n} \\rrbracket $ , let $\\mathcal {P}_n(t,A)$ denote the set of tuples $w\\in \\mathcal {P}_n(t)$ the elements of which are in $A$ .",
"Similarly, let $\\mathcal {P}^{*}_n(t,A)$ denote the subset of nontrivial tuples of $\\mathcal {P}^{*}_n(t,A)$ .",
"As $\\mathcal {P}_n(t,F)$ is independent of $\\mathcal {P}_n(t)\\setminus \\mathcal {P}_n(t,F)$ , $b_{n}(F)=b^{(1)}_{n}(F)b^{(2)}_{n}(F)$ where: $b^{(1)}_{n}(F)$ is the probability that the partition associated with $\\mathcal {P}_n(t_n,F)$ consists of the block $\\lbrace F\\rbrace $ , $b^{(2)}_{n}(F)$ is the probability that there is no tuple $w\\in \\mathcal {P}_n(t_n)$ containing both elements of $F$ and $F^c$ .",
"Let $\\delta \\in ]0,1[$ .",
"For $|F|\\ge n^{1-\\delta }$ , it is sufficient to replace $b^{(1)}_{n}(F)$ by 1 as we show that $(b^{(2)}_{n}(F))_n$ converges to 0 rapidly.",
"For $2\\le |F|< n^{1-\\delta }$ , we use that $b^{(1)}_{n}(F)$ is bounded by the probability that the total number of elements in nontrivial tuples of $\\mathcal {P}_n(t_n,F)$ are greater or equal to $|F|$ .",
"The value of this upper bound depends only on $|F|$ and $n$ .",
"Let denote it $\\bar{b}^{(1)}_n(|F|)$ .",
"$S_{n,r}\\le \\left\\lbrace \\begin{array}{ll}\\sum _{\\stackrel{F\\subset \\llbracket {n} \\rrbracket }{|F|=r}} b^{(2)}_{n}(F)&\\text{if}\\;r\\ge n^{1-\\delta }\\\\\\bar{b}^{(1)}_n(r)\\sum _{\\stackrel{F\\subset \\llbracket {n} \\rrbracket }{|F|=r}} b^{(2)}_{n}(F)&\\text{if}\\; 2\\le r< n^{1-\\delta }\\end{array}\\right.\\text{where}\\; \\bar{b}^{(1)}_n(r)=\\operatorname{\\mathbb {P}}\\Big (\\sum _{w\\in \\mathcal {P}^{*}_n(t_n,\\llbracket {r}\\rrbracket )}\\ell (w)\\ge r\\Big ).$ The expression of $b^{(2)}_{n}(F)$ is $\\exp \\Big (-t_n(1-G_{p}(\\frac{|F|}{n})-G_{p}(1-\\frac{|F|}{n}))\\Big )$ .",
"Using that $\\binom{n}{r}\\le \\frac{1}{\\sqrt{2\\pi r}\\sqrt{1-\\frac{r}{n}}}(\\frac{n}{r})^r(1-\\frac{r}{n})^{-(n-r)}$ (see for example [4], formula 1.5 page 4), we obtain: $\\sum _{F\\subset \\llbracket {n} \\rrbracket , |F|=r} b^{(2)}_{n}(F)\\le \\frac{1}{\\sqrt{r}}\\exp (-nf_n(\\frac{|F|}{n})),$ where $f_n$ is the function defined by: $f_n(x)=x\\log (x)+(1-x)\\log (1-x)+\\frac{t_n}{n}(1-G_{p}(x)-G_{p}(1-x))\\;\\text{for}\\;x\\in ]0,1[.$ To conclude, we need the following two lemmas: Lemma 3.2 Let $\\delta $ and $\\bar{\\delta }$ be two positive reals such that $0<\\bar{\\delta }<\\delta <1$ .",
"Let $a\\in \\operatorname{\\mathbb {R}}$ .",
"Set $t_n=\\frac{n}{m^{*}_{p,1}}(\\log (n)+a+o(1))$ for every $n\\in \\operatorname{\\mathbb {N}}$ .",
"There exists $n_{\\delta ,\\bar{\\delta }}>0$ such that for every $n\\ge n_{\\delta ,\\bar{\\delta }}$ , and $F\\subset \\llbracket {n} \\rrbracket $ with $2\\le |F|\\le n^{1-\\delta }$ , $\\operatorname{\\mathbb {P}}\\Big (\\sum _{w\\in \\mathcal {P}^{*}_n(t_n,F)}\\ell (w)\\ge |F|\\Big )\\le n^{-\\frac{\\bar{\\delta }}{2}|F|}.$ Lemma 3.3 Let $f_n$ be the function defined by: $f_n(x)=x\\log (x)+(1-x)\\log (1-x)+\\frac{t_n}{n}(1-G_{p}(x)-G_{p}(1-x))\\;\\forall x\\in ]0,1[.$ Let $(u_n)$ be a positive sequence such that $\\liminf _n\\frac{u_n}{\\log (n)}>0$ .",
"For every $\\delta \\in ]0,1[$ , there is an integer $n_{\\delta }>0$ such that for $n\\ge n_{\\delta }$ , $f_n(x)\\ge \\frac{1-\\delta }{2n^{\\delta }}\\log (n)$ for every $x\\in [n^{-\\delta },1/2]$ , $f_n(x)+xu_n \\ge \\frac{u_n}{n}$ for every $x\\in [\\frac{2}{n},1/2]$ .",
"Before presenting the proofs of the two lemmas, let us apply them to complete the proof of Theorem REF .",
"By Lemma REF , for every $0<\\bar{\\delta }<\\delta <1$ , there exists $n_{\\delta ,\\bar{\\delta }}\\in \\operatorname{\\mathbb {N}}$ , such that for every $n>n_{\\delta ,\\bar{\\delta }}$ , $S_{n,r}\\le \\left\\lbrace \\begin{array}{ll}\\frac{1}{\\sqrt{r}}\\exp (-nf_n(\\frac{r}{n}))& \\text{ if }r\\in [n^{1-\\delta }, \\lfloor n/2\\rfloor ]\\\\\\frac{1}{\\sqrt{r}}\\exp \\Big (-n\\big (f_n(\\frac{r}{n})+\\frac{r}{n}\\frac{\\bar{\\delta }}{2}\\log (n)\\big )\\Big )&\\text{ if } r\\in [2, n^{1-\\delta }].\\\\\\end{array}\\right.$ We deduce from Lemma REF that for sufficiently large values of $n$ , $S_{n,r}\\le \\left\\lbrace \\begin{array}{ll} \\frac{1}{\\sqrt{r}}\\exp (-\\frac{1-\\delta }{2}n^{1-\\delta }\\log (n))&\\text{ if } r\\in [n^{1-\\delta }, \\lfloor n/2\\rfloor ]\\\\\\frac{1}{\\sqrt{r}}n^{-\\frac{\\bar{\\delta }}{2}}&\\text{ if } r\\in [2, n^{1-\\delta }].\\end{array}\\right.$ Thus for sufficiently large values of $n$ , $\\operatorname{\\mathbb {P}}(J_n)\\le n^{1-\\delta -\\frac{\\bar{\\delta }}{2}}+n\\exp (-\\frac{1-\\delta }{2}n^{1-\\delta }\\log (n))$ .",
"If we take $\\delta =\\frac{3}{4}$ and $\\bar{\\delta }=\\frac{2}{3}$ , we obtain that for sufficiently large values of $n$ , ${\\operatorname{\\mathbb {P}}(J_n)\\le n^{-1/12}+ ne^{-\\frac{1}{8}n^{1/4}\\log (n)}}.$ It remains to prove Lemma REF and Lemma REF ."
],
[
"Proof of Lemma ",
"The random variable $N_n(F):=\\displaystyle {\\sum _{w\\in \\mathcal {P}^{*}_n(t_n,F)}\\ell (w)}$ has a compound Poisson distribution $\\text{CPois}(t_n\\beta _{n,F},\\nu _{n,F})$ , where $\\beta _{n,F}=G_{p}(\\frac{|F|}{n})-|F|G_{p}(\\frac{1}{n})$ , $\\nu _{n,F}(j)=\\frac{1}{\\beta _{n,F}}\\mu (w\\in \\mathcal {W}^{*}(F),\\ \\ell (w)=j)=\\frac{p(j)}{\\beta _{n,F}}\\Big ((\\frac{|F|}{n})^j-\\frac{|F|}{n^j}\\Big )\\; \\forall j\\in \\operatorname{\\mathbb {N}}^*$ .",
"Its probability generating function at $0\\le s\\le \\frac{n}{|F|}$ is: $G_{N_n(F)}(s)=\\exp \\Big (-t_n\\beta _{n,F}(1-G_{\\nu _{n,F}}(s))\\Big )\\\\ =\\exp \\Big (-t_n\\big (G_{p}(\\frac{|F|}{n})-G_{p}(s\\frac{|F|}{n})-|F|(G_{p}(\\frac{1}{n})-G_{p}(\\frac{s}{n}))\\big )\\Big ).$ For $2\\le r\\le n$ and $0<\\theta \\le \\log (\\frac{n}{r})$ , set $\\psi _{n,r}(\\theta )= \\theta r +t_n\\Big (G_{p}(\\frac{r}{n})-rG_{p}(\\frac{1}{n})-G_{p}(e^{\\theta }\\frac{r}{n})+rG_{p}(\\frac{e^{\\theta }}{n})\\Big ).$ By Markov's inequality $\\operatorname{\\mathbb {P}}(N_n(F)\\ge |F|)\\le \\exp (-\\psi _{n,|F|}(\\theta ))$ for every ${0<\\theta \\le \\log (\\frac{n}{|F|})}$ .",
"As $G_{p}^{\\prime }$ and $G_{p}^{\\prime \\prime }$ are increasing functions on $[0,1[$ , for $s\\in [1,\\frac{n}{2r}]$ , $G_{p}(s\\frac{r}{n})-G_{p}(\\frac{r}{n})-r(G_{p}(\\frac{s}{n})-G_{p}(\\frac{1}{n}))\\\\\\le \\frac{r}{n}(s-1)(G_{p}^{\\prime }(s\\frac{r}{n})-G_{p}^{\\prime }(\\frac{1}{n}))\\le \\frac{r}{n^2}(s-1)(rs-1)G_{p}^{\\prime \\prime }(1/2).$ Thus for every $0<\\theta \\le \\log (\\frac{n}{2r})$ , $\\psi _{n,r}(\\theta )\\ge rh_{n,r}(\\theta )$ with $h_{n,r}(\\theta )=\\theta -t_n\\frac{r}{n^2}G_{p}^{\\prime \\prime }(1/2)e^{2\\theta }$ .",
"The function $h_{n,r}$ has a maximum point at $\\theta _{n,r}=\\frac{1}{2}\\log (\\frac{n^2}{2t_nrG_{p}^{\\prime \\prime }(1/2)})$ , which is less than $\\log (\\frac{n}{2r})$ for every $r\\le n$ when $n$ is large enough.",
"Its value at $\\theta _{n,r}$ is $h_{n,r}(\\theta _{n,r})=\\frac{1}{2}(\\log (n)-\\log (r)-\\log (\\frac{t_n}{n}))+O(1).$ Therefore, for every $0<\\bar{\\delta }<\\delta <1$ , there exists $n_{\\delta ,\\bar{\\delta }}\\in \\operatorname{\\mathbb {N}}$ such that for every $n\\ge n_{\\delta ,\\bar{\\delta }}\\in \\operatorname{\\mathbb {N}}$ and $2\\le r\\le n^{1-\\delta }$ , $h_{n,r}(\\theta _{n,r})\\ge \\frac{\\bar{\\delta }}{2}\\log (n)$ and thus $\\operatorname{\\mathbb {P}}(N_n(F)\\ge |F|)\\le \\exp (-|F|\\frac{\\bar{\\delta }}{2}\\log (n))$ for $2\\le |F|\\le n^{1-\\delta }$ ."
],
[
"Proof of Lemma ",
"The proof consists in showing that for sufficiently large $n$ , $f_n$ and $\\bar{f}_n:x \\mapsto f_n(x)+xu_n $ are increasing functions in $]n^{-\\delta },\\frac{1}{2}[$ and $]\\frac{2}{n},\\frac{1}{2}[$ respectively and to compute their values at $n^{-\\delta }$ and $\\frac{2}{n}$ respectively.",
"Let us prove the result for the function $f_n$ .",
"By computations, we obtain that for every $x\\in ]0,1[$ , $f_n^{\\prime }(x)=\\log (x)-\\log (1-x)+\\frac{t_n}{n}(G_{p}^{\\prime }(1-x)-G_{p}(x))\\;\\operatorname{ and }$ $f_n^{\\prime \\prime }(x)=\\frac{1}{x(1-x)}\\Big (1-\\frac{t_n}{n}\\sum _{k=2}^{+\\infty }k(k-1)g_k(x)\\Big )\\;\\text{where}\\;g_{k}(x)=x(1-x)(x^{k-2}+(1-x)^{k-2}).$ The first derivative of $g_k$ is positive on $]0,\\frac{1}{2}[$ .",
"As the value of $1-\\frac{t_n}{n}\\sum _{k=2}^{+\\infty }k(k-1)g_k$ at 0 is 1 and at $\\frac{1}{2}$ is negative for sufficiently large $n$ , we deduce that for sufficiently large $n$ , there exists $a_n\\in ]0,\\frac{1}{2}[$ such that $f_n^{\\prime }$ is increasing in $]0,a_n[$ and decreasing in $]a_n,\\frac{1}{2}[$ .",
"As $f_n^{\\prime }(\\frac{1}{2})=0$ and $f_n^{\\prime }(n^{-\\delta })>0$ for sufficiently large $n$ , $f_n$ is an increasing function in $]n^{-\\delta },\\frac{1}{2}[$ for sufficiently large $n$ .",
"Finally, using that $1-G_{p}(1-s)-G_{p}(s)=sm^{*}_{p,1}+ o(\\frac{s}{\\log (s)})$ as $s$ tends to 0, we obtain $f_n(\\frac{1}{n^{\\delta }})=\\dfrac{1-\\delta }{n^\\delta }\\log (n)+O(n^{-\\delta })$ .",
"We deduce that for sufficiently large $n$ , $f_n(x)\\ge \\dfrac{1-\\delta }{2n^\\delta }\\log (n)$ $\\forall x\\in [\\dfrac{1}{n^{\\delta }},\\frac{1}{2}]$ .",
"As $\\bar{f}^{\\prime }_n=f^{\\prime }_n+u_n$ , $f^{\\prime }_n(\\frac{2}{n})=o(1)\\log (n)$ and $\\bar{f}_n(\\frac{2}{n})=\\frac{2}{n}(O(1)+u_n)$ , we obtain that $\\bar{f}_n(x)\\ge \\frac{u_n}{n}$ $\\forall x\\in [\\frac{2}{n},\\frac{1}{2}]$ for sufficiently large $n$ ."
],
[
"Block exploration procedure and associated BGW process ",
"In this section, we describe an exploration procedure modeled on the Karp [22] and Martin-Löf [29] exploration algorithm.",
"The aim of this procedure is to find the block of an element $x$ in the partition $\\Pi _{n}(t)$ (this block is denoted by $\\Pi _{n}^{(x)}(t)$ ), and to construct a BGW process such that its total population size is an upper bound of $|\\Pi _{n}^{(x)}(t)|$ ."
],
[
"Block exploration procedure",
"For every subset $A$ of $\\llbracket {n} \\rrbracket $ , and $x\\in A$ , let $\\mathcal {P}_{n,x}(t,A)$ denote the set of tuples $w\\in \\mathcal {P}_n(t,A)$ that contain $x$ and let $\\mathcal {P}^{*}_{n,x}(t,A)$ denote those that are nontrivial.",
"Let define the set of `neighbours' of $x$ in $A$ as $\\mathcal {N}_{x}(t,A)=\\lbrace y\\in A\\setminus \\lbrace x\\rbrace ,\\; \\exists w \\in \\mathcal {P}_{n,x}(t,A)\\;\\text{that contains}\\;y\\rbrace .$ In each step of the algorithm, an element of $\\llbracket {n} \\rrbracket $ is either active, explored or neutral.",
"Let $A_k$ and $H_k$ be the sets of active elements and explored vertices in step $k$ respectively in the exploration procedure of the block of $x$ .",
"In step 0, $x_1=x$ is said to be active ($A_0=\\lbrace x_1\\rbrace $ ) and other elements are neutral.",
"In step 1, every neighbour of $x_1$ is declared active and $x_1$ is said to be an explored element: $A_{1}=\\mathcal {N}_{x_1}(t,\\llbracket {n} \\rrbracket )$ and $H_1=\\lbrace x_1\\rbrace $ .",
"In step $k\\ge 1$ , let us assume that $A_{k-1}$ is not empty.",
"Let $x_k$ denote the smallest active element in $A_{k-1}$ .",
"Neutral elements that are neighbours of $x_k$ are added to $A_{k-1}$ and the status of $x_k$ is changed: $A_{k}=A_{k-1}\\cup \\mathcal {N}_{x_k}(t,\\llbracket {n} \\rrbracket \\setminus H_{k-1})\\setminus \\lbrace x_k\\rbrace $ and ${H_{k}=H_{k-1}\\cup \\lbrace x_k\\rbrace }$ .",
"In particular, $|A_k| = |A_{k-1}| + \\xi _{n,k}(t)-1$ with ${\\xi _{n,k}(t) = |\\mathcal {N}_{x_k}(t,\\llbracket {n} \\rrbracket \\setminus H_{k-1})\\setminus A_{k-1}|}$ .",
"The process stops in step $T_n(t)=\\min (k,\\; A_k=\\emptyset )$ .",
"By construction, $T_n(t)=\\min (k,\\; \\sum _{i=1}^{k}\\xi _{n,i}(t)\\le k-1).$ The block of $x$ is $\\Pi _{n}^{(x)}(t)=H_{T_n(t)}$ and its size is $T_n(t)$ .",
"Example 4.1 Let $n\\ge 10$ .",
"Assume that $\\mathcal {P}_n(t)$ is formed by five tuples $(1,2,3,4)$ , $(2,5,2,3)$ , $(3,6,4)$ , $(6,7)$ and $(8,10)$ .",
"The steps of the exploration procedure starting from 1 are Step 1: $x_{1}=1$ and $A_{1}=\\lbrace 2,3,4\\rbrace $ so that $\\xi _{n,1}(t)=3$ .",
"Step 2: $x_{2}=2$ and $A_{2}=\\lbrace 3,4,5\\rbrace $ so that $\\xi _{n,2}(t)=1$ .",
"Step 3: $x_{3}=3$ and $A_{3}=\\lbrace 4,5,6\\rbrace $ so that $\\xi _{n,3}(t)=1$ .",
"Step 4: $x_{4}=4$ and $A_{4}=\\lbrace 5,6\\rbrace $ so that $\\xi _{n,4}(t)=0$ .",
"Step 5: $x_{5}=5$ and $A_{5}=\\lbrace 6\\rbrace $ so that $\\xi _{n,5}(t)=0$ .",
"Step 6: $x_{6}=6$ and $A_{6}=\\lbrace 7\\rbrace $ so that $\\xi _{n,6}(t)=1$ .",
"Step 7: $x_{7}=7$ and $A_{7}=\\emptyset $ so that $\\xi _{n,7}(t)=0$ , $T_n(t)=7$ and $\\Pi ^{(1)}_n(t)=\\lbrace 1,2,3,4,5,6,7\\rbrace $ ."
],
[
"The BGW process associated with a block",
"The random variable $\\xi _{n,k}(t)$ is bounded above by $\\zeta ^{(1)}_{n,k}(t)=\\sum _{w \\in \\mathcal {P}^{*}_{n,x_k}(t,\\llbracket {n} \\rrbracket \\setminus H_{k-1})}\\!\\!\\!",
"(\\ell (w)-1)$ in which a same element is counted as many times as it appears in ${w \\in \\mathcal {P}^{*}_{n,x_k}(t,\\llbracket {n} \\rrbracket \\setminus H_{k-1})}$ .",
"To obtain identically distributed random variables in each step, we have to consider also in step $k$ , tuples that contain $x_k$ and elements of $H_{k-1}$ before time $t$ .",
"Let denote this set of tuples $\\mathcal {P}_{n,x_k,H_{k-1}}(t)$ and set $\\displaystyle {\\zeta ^{(2)}_{n,k}(t)=\\sum _{w\\in \\mathcal {P}_{n,x_k,H_{k-1}}(t)}(\\ell (w)-1)}$ and $\\zeta _{n,k}(t)=\\zeta ^{(1)}_{n,k}(t)+\\zeta ^{(2)}_{n,k}(t)=\\sum _{w \\in \\mathcal {P}^{*}_{n,x_k}(t,\\llbracket {n} \\rrbracket )}(\\ell (w)-1).$ The distribution of $\\zeta _{n,k}(t)$ is the $\\text{CPois}(t\\beta _{n},\\nu _{n})$ -distribution with $\\beta _{n}=\\mu (\\lbrace w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ), x\\in w\\rbrace )=\\mu (\\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ))-\\mu (\\mathcal {W}^{*}(\\llbracket {n} \\rrbracket \\setminus \\lbrace x\\rbrace ))=1-G_{p}(1-\\frac{1}{n})-G_{p}(\\frac{1}{n}).$ and $\\forall j\\in \\operatorname{\\mathbb {N}}$ , $\\nu _{n}(j)=\\frac{1}{\\beta _{n}}\\mu (\\lbrace w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\ x\\in w \\;\\operatorname{ and }\\; \\ell (w)=j+1\\rbrace )=\\frac{p(j+1)}{\\beta _n}\\Big (1-(1-\\frac{1}{n})^{j+1}-(\\frac{1}{n})^{j+1}\\Big ).$ Example 4.2 In example REF , the random variables associated with the first three steps of the exploration procedure of the block of 1 are $\\zeta ^{(1)}_{n,1}(t)=3$ , $\\zeta ^{(2)}_{n,1}(t)=0$ , $\\zeta ^{(1)}_{n,2}(t)=3$ , $\\zeta ^{(2)}_{n,2}(t)=3$ , $\\zeta ^{(1)}_{n,3}(t)=2$ and $\\zeta ^{(2)}_{n,3}(t)=6$ .",
"Let $\\mathcal {F}_k=\\sigma (H_j, A_j,\\; j\\le k)$ .",
"Let us note that the random variables $\\zeta _{n,j}(t)$ and $\\zeta _{n,k}(t)$ for $j<k$ are not independent since a same tuple can belong to $\\mathcal {P}_{n,x_k,H_{k-1}}(t)$ and $\\mathcal {P}_{n,x_j,H_{j-1}}(t)$ .",
"Nevertheless, since disjoint subsets of tuples in $\\mathcal {P}_n(t)$ are independent, the random variables $\\zeta ^{(1)}_{n,j}(t)$ for $j\\le k$ are independent conditionally on $\\mathcal {F}_{k-1}$ , and the random variable $\\zeta ^{(1)}_{n,k}(t)$ is independent of $\\zeta ^{(2)}_{n,k}(t)$ conditionally on $\\mathcal {F}_{k-1}$ .",
"Therefore, by using independent copies of the Poisson point process $\\mathcal {P}_n$ , we can construct a sequence of nonnegative random variables $(\\bar{\\zeta }^{(2)}_{n,k}(t))_k$ such that: $\\bar{\\zeta }^{(2)}_{n,k}(t)$ has the same distribution as $\\zeta ^{(2)}_{n,k}(t)$ and is independent of $\\zeta ^{(1)}_{n,k}(t)$ conditionally on $\\mathcal {F}_{k-1}$ for every $k\\ge 2$ ; $\\bar{\\zeta }_{n,k}(t)=\\zeta ^{(1)}_{n,k}(t)+\\bar{\\zeta }^{(2)}_{n,k}(t)$ are independent with distribution $\\text{CPois}(\\beta _{n} t,\\nu _{n})$ for every $k\\in \\operatorname{\\mathbb {N}}^*$ .",
"Set $\\bar{T}_{n}(t)=\\min (k,\\ \\bar{\\zeta }_{n,1}(t)+\\ldots +\\bar{\\zeta }_{n,k}(t)=k-1)$ .",
"By construction, $\\bar{T}_{n}(t)\\ge |\\Pi _{n}^{(x)}(t)|$ .",
"If $\\bar{\\zeta }_{n,1}(t)$ is seen as the number of offspring of an individual $I$ and $\\bar{\\zeta }_{n,k}(t)$ for $k\\ge 2$ as the number of offspring of the $k$ -th individual explored by a breadth-first algorithm of the family tree of $I$ , then $\\bar{T}_{n}(t)$ is the total number of individuals in the family tree of $I$ .",
"We call $(\\bar{\\zeta }_{n,k}(t))_k$ the associated BGW process (a bijection between BGW trees and lattice walks was described by T. E. Harris [18] in Section 6, see also Section 6.2 in [34] for a review)."
],
[
"Approximation of block sizes",
"The number of neighbours of an element is used to approximate the number of active elements added in each step of the exploration process of a block.",
"We begin this section by studying its asymptotic distribution.",
"Next, we prove Theorem REF and Corollary REF .",
"Its proof is divided into two steps: we give an upper bound of the deviation between the cumulative distribution function of $|\\Pi _{n}^{(x)}(t)|$ and of the total population size of the associated BGW process and then we study the asymptotic distribution of the BGW process associated with $|\\Pi _{n}^{(x)}(nt)|$ .",
"We end this section by a proof of Corollary REF .",
"In this section, the third moment of the distribution $p$ is assumed to be finite."
],
[
"Neighbours of an element",
"Let $V_n$ be a subset of $\\llbracket {n} \\rrbracket $ and let $x\\in \\llbracket {n} \\rrbracket \\setminus V_n$ .",
"The aim of this section is to show that the number of neighbours of $x$ in $\\llbracket {n} \\rrbracket \\setminus V_n$ at time $nt$ (denoted by $|\\mathcal {N}_{x}(nt,\\llbracket {n} \\rrbracket \\setminus V_n)|$ ) converges in law to the $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distribution if $\\frac{V_n}{n}$ tends to 0.",
"The number of neighbours of $x$ in $\\llbracket {n} \\rrbracket \\setminus V_n$ at time $t$ is equal to $\\sum _{w\\in \\mathcal {P}^{*}_{n,x}(t,\\llbracket {n} \\rrbracket \\setminus V_n)}(\\ell (w)-1)$ except if there exists a tuple in $\\mathcal {P}^{*}_{n,x}(t,\\llbracket {n} \\rrbracket \\setminus V_n)$ which has several copies of a same element or if there is an element $y\\ne x$ which appears in several tuples of ${\\mathcal {P}^{*}_{x}(t,\\llbracket {n} \\rrbracket \\setminus V_n)}$ .",
"The following lemma yields an upper bound for the probability that such an event occurs: Lemma 5.1 Let $x\\in \\llbracket {n} \\rrbracket $ .",
"Set $F_{n,t}$ be the event `some tuples in $\\mathcal {P}^{*}_{x}(t,\\llbracket {n} \\rrbracket )$ contain several copies of a same element or have in common other elements than $x$ .'",
"$\\operatorname{\\mathbb {P}}(F_{n,t})\\le \\frac{t}{2n^2}\\left(m_{p,2}+m_{p,3}+\\frac{t}{n}(m_{p,2})^2\\right).$ We study separately the following two events: $F^{(1)}_{n,t}$ :`there exists $y\\ne x$ which is in several tuples of $\\mathcal {P}^{*}_{n,x}(t)$ or several times in one tuple of $\\mathcal {P}^{*}_{n,x}(t)$' $F^{(2)}_{n,t}$ : `some tuples of $\\mathcal {P}^{*}_{n,x}(t)$ contain several copies of $x$'.",
"To compute $\\operatorname{\\mathbb {P}}(F^{(1)}_{n,t})$ , we introduce the random variable $S_{t,x}$ as the total length of tuples in $\\mathcal {P}^{*}_{n,x}(t)$ minus the number of copies of $x$ in tuples of $\\mathcal {P}^{*}_{n,x}(t)$ : $S_{t,x}=\\sum _{w\\in \\mathcal {P}^{*}_{n,x}}\\ell _x(w)$ where $\\ell _x(w)$ denotes the number of elements different from $x$ in the tuple $w$ .",
"Since elements that form a tuple are chosen independently with the uniform distribution on $\\llbracket {n} \\rrbracket $ , $\\operatorname{\\mathbb {P}}(F^{(1)}_{n,t})=1-\\operatorname{\\mathbb {E}}\\left(\\prod _{i=0}^{S_{t,x}-1}(1-\\frac{i}{n-1})\\right)\\le \\frac{1}{2(n-1)}\\operatorname{\\mathbb {E}}(S_{t,x}(S_{t,x}-1)).$ By Campbell's formula, the probability-generating function of $S_{t,x}$ is $\\operatorname{\\mathbb {E}}( u^{S_{t,x}})=\\exp \\Big (\\sum _{w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\; x\\in w} (u^{\\ell _x(w)}-1)t\\mu _n(w)\\Big ).$ By decomposing $\\displaystyle {f_n(u)=\\sum _{w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\; x\\in w} (u^{\\ell _x(w)}-1)\\mu _n(w)}$ according to the size of a tuple and the number of copies of $x$ in it and then applying the binomial formula, we obtain: $f_n(u)=&\\sum _{j=1}^{+\\infty }p(j)\\sum _{i=1}^{j-1}(u^{j-i}-1)\\binom{j}{i}\\left(\\frac{1}{n}\\right)^{i}\\left(1-\\frac{1}{n}\\right)^{j-i}\\\\=&\\sum _{j=1}^{+\\infty }\\frac{p(j)}{n^j}\\Big ((u(n-1)+1)^j-u^j(n-1)^j-n^j+(n-1)^j\\Big )\\\\=&G_{p}\\left(\\frac{1}{n}+u(1-\\frac{1}{n})\\right)-G_{p}\\left(u(1-\\frac{1}{n})\\right)-1+G_{p}\\left(1-\\frac{1}{n}\\right).$ We deduce the following formula of $\\operatorname{\\mathbb {E}}(S_{t,x}(S_{t,x}-1))$ by computing the first two derivatives of $\\operatorname{\\mathbb {E}}( u^{S_{t,x}})$ : $\\operatorname{\\mathbb {E}}(S_{t,x}(S_{t,x}-1))=(1-\\frac{1}{n})^2\\left(t\\big (G_{p}^{(2)}(1)-G_{p}^{(2)}(1-\\frac{1}{n})\\big )+t^2\\big (G_{p}^{(1)}(1)-G_{p}^{(1)}(1-\\frac{1}{n})\\big )^2\\right).$ As the third moment of $p$ is finite, $G_{p}^{(1)}(1)-G_{p}^{(1)}(1-\\frac{1}{n})\\le \\frac{m_{p,2}}{n} \\operatorname{ and }G_{p}^{(2)}(1)-G_{p}^{(2)}(1-\\frac{1}{n})\\le \\frac{m_{p,3}}{n}.$ Thus we obtain: $\\operatorname{\\mathbb {P}}(F^{(1)}_{n,t,k})\\le \\frac{t}{2n^2}(m_{p,3}+\\frac{t}{n}m_{p,2}^2).",
"$ To study $F^{(2)}_{n,t}$ , let $N_x(w)$ denote the number of copies of $x$ in a tuple $w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket )$ : $\\operatorname{\\mathbb {P}}(F^{(2)}_{n,t})=1-\\exp \\big (-t\\mu _n(w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\ N_x(w)\\ge 2)\\big ).$ We have already seen in Proposition REF that $\\mu _n(w\\in \\mathcal {W}(\\llbracket {n} \\rrbracket ),\\ N_x(w)\\ge 1)=1-G_{p}(1-\\frac{1}{n})-G_{p}(\\frac{1}{n}).$ Finally, $\\mu _n(w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\ N_x(w)= 1)=\\sum _{k=2}^{+\\infty }p(k)\\frac{k}{n}(1-\\frac{1}{n})^{k-1}=\\frac{1}{n}\\Big (G_{p}^{(1)}(1-\\frac{1}{n})-p(1)\\Big ).$ Therefore, $\\mu _n(w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\ N_x(w)\\ge 2)=1-G_{p}(1-\\frac{1}{n})-\\frac{1}{n}G_{p}^{(1)}(1-\\frac{1}{n})-G_{p}^{(1)}(\\frac{1}{n})+\\frac{1}{n} p(1)\\le \\frac{1}{2n^2} m_{p,2}.",
"$ In summary, $\\operatorname{\\mathbb {P}}(F^{(2)}_{n,t})\\le t\\mu _n(w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\ N_x(w)\\ge 2)\\le \\frac{t}{2n^2} m_{p,2}.$ Let us now describe the distribution of the upper bound we have obtained for the number of neighbours of $x$ in $\\llbracket {n} \\rrbracket \\setminus V_n$ at time $nt$ and the total variation distance (denoted by $\\operatorname{d_{\\text{TV}}}$ ) between it and the compound Poisson distribution $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ : Proposition 5.2 For a subset $V$ of $\\llbracket {n} \\rrbracket $ and $x\\in \\llbracket {n} \\rrbracket \\setminus V$ , set $S_{t,V,x}=\\sum _{w\\in \\mathcal {P}^{*}_{x}(t,\\llbracket {n} \\rrbracket \\setminus V)}(\\ell (w)-1).$ (i) The random variable $S_{nt,V,x}$ has the compound Poisson distribution $\\text{CPois}(nt\\beta _{V},\\nu _{n,V})$ where: $&\\beta _{n,V}=G_{p}\\left(1-\\frac{|V|}{n}\\right)-G_{p}\\left(1-\\frac{|V|+1}{n}\\right)-G_{p}\\left(\\frac{1}{n}\\right)\\\\&\\nu _{n,V}(j)=\\frac{p(j+1)}{\\beta _{n,V}}\\left(\\left(1-\\frac{|V|}{n}\\right)^{j+1}-\\left(1-\\frac{|V|+1}{n}\\right)^{j+1}-\\left(\\frac{1}{n}\\right)^{j+1}\\right)\\;\\forall j\\in \\operatorname{\\mathbb {N}}.$ (ii) $\\operatorname{d_{\\text{TV}}}\\left(\\text{CPois}(nt\\beta _{n,V},\\nu _{n,V}),\\text{CPois}(tm^{*}_{p,1},\\tilde{p})\\right)\\le 2tm_{p,2}\\left(\\frac{|V|}{n}+\\frac{1}{2n}\\right) + \\frac{t}{2n}.$ (i) By definition of the Poisson tuple set, $S_{nt,V,x}$ has the compound Poisson distribution $\\text{CPois}(nt\\beta _{V},\\nu _{n,V})$ where $\\beta _{n,V}=\\mu _n(w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket \\setminus V),\\ x\\in w)$ and for every $j\\in \\operatorname{\\mathbb {N}}^*$ , $\\nu _{n,V}(j)=\\frac{1}{\\beta _{n,V}}\\mu _n(w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket \\setminus V),\\ x\\in w \\operatorname{ and }\\ell (w)=j+1).$ (ii) The total variation distance between two compound Poisson distributions can be bounded as follows using coupling arguments: Lemma 5.3 Let $p_1$ and $p_2$ be two probability measures on $\\operatorname{\\mathbb {N}}$ and let $\\lambda _1$ and $\\lambda _2$ be two positive reals such that $\\lambda _1\\le \\lambda _2$ .",
"Then $\\operatorname{d_{\\text{TV}}}(\\text{CPois}(\\lambda _1,p_1),\\text{CPois}(\\lambda _2,p_2))\\le 1-e^{-(\\lambda _2-\\lambda _1)}+\\lambda _1\\operatorname{d_{\\text{TV}}}(p_1,p_2).$ [Proof of Lemma REF ] By Strassen's theorem, there exist two independent sequences $(X_i)_{i\\in \\operatorname{\\mathbb {N}}^*}$ and $(Y_i)_{i\\in \\operatorname{\\mathbb {N}}^*}$ of i.i.d.",
"random variables with distributions $p_1$ and $p_2$ respectively such that ${\\operatorname{d_{\\text{TV}}}(p_1,p_2)=\\operatorname{\\mathbb {P}}(X_i\\ne Y_i)}$ for every $i\\in \\operatorname{\\mathbb {N}}\\*$ .",
"Let $Z_1$ and $Z_2$ be two independent Poisson-distributed random variables with parameters $\\lambda _1$ and $\\lambda _2-\\lambda _1$ respectively, which are independent of the two sequences $(X_i)_i$ and $(Y_i)_i$ (we take $Z_2=0$ if $\\lambda _2=\\lambda _1$ ).",
"Set $Z=Z_1+Z_2$ .",
"Then $\\operatorname{\\mathbb {P}}\\left(\\sum _{i=1}^{Z_1}X_i\\ne \\sum _{i=1}^{Z}Y_i\\right)\\le \\operatorname{\\mathbb {P}}(Z_2>0)+\\operatorname{\\mathbb {P}}\\left(\\sum _{i=1}^{Z_1}X_i\\ne \\sum _{i=1}^{Z_1}Y_i\\right)$ and $\\operatorname{\\mathbb {P}}\\left(\\sum _{i=1}^{Z_1}X_i\\ne \\sum _{i=1}^{Z_1}Y_i\\right)\\le \\sum _{k=0}^{+\\infty }\\operatorname{\\mathbb {P}}(Z_1=k)\\sum _{i=1}^{k}\\operatorname{\\mathbb {P}}(X_i\\ne Y_i)=\\operatorname{\\mathbb {E}}(Z_1)\\operatorname{d_{\\text{TV}}}(p_1,p_2).$ We apply Lemma REF with $\\lambda _1=tn\\beta _{n,V}$ , $\\lambda _2=tm^{*}_{p,1}$ , $p_1=\\nu _{n,V}$ and $p_2=\\tilde{p}$ and use the following inequalities with $u=\\frac{1}{n}$ : $\\forall j\\in \\operatorname{\\mathbb {N}}^*$ , $\\forall x,u\\ge 0$ such that $x+u\\le 1$ , $ju-j(j-1)(xu+\\frac{u^2}{2})\\le (1-x)^j-(1-x-u)^j\\le ju.$ We obtain $0\\le m^{*}_{p,1}-n\\beta _{n,V}\\le m_{p,2}\\Big (\\frac{|V|}{n}+\\frac{1}{2n}\\Big )$ and for every $j\\in \\operatorname{\\mathbb {N}}^*$ , $ n\\beta _{n,V}|\\nu _{n,V}(j)-\\tilde{p}(j)|\\le \\\\ p(j+1)\\Big ((j+1)(1-\\frac{n\\beta _{n,V}}{m^{*}_{p,1}})+j(j+1)\\big (\\frac{|V|}{n}+\\frac{1}{2n}\\big )+\\frac{1}{n^{j}}\\Big ).$ Therefore $ n\\beta _{n,V}\\operatorname{d_{\\text{TV}}}(\\nu _{n,V},\\tilde{p}) \\le \\frac{1}{2}m_{p,2}\\big (\\frac{|V|}{n}+\\frac{1}{2n}\\big )+\\frac{1}{2}(m_{p,1}-n\\beta _{n,V})+\\frac{1}{2n}\\\\\\le \\frac{1}{2n}\\big (m_{p,2}(2|V|+1)+1\\big )$ and $\\operatorname{d_{\\text{TV}}}(\\text{CPois}(\\lambda _1,p_1),\\text{CPois}(\\lambda _2,p_2))\\le &1-e^{-t(m_{p,1}-n\\beta _{n,V})}+tn\\beta _{n,V}\\operatorname{d_{\\text{TV}}}(\\nu _{n,V},\\tilde{p})\\\\\\le & 2tm_{p,2}\\Big (\\frac{|V|}{n}+\\frac{1}{2n}\\Big )+\\frac{t}{2n}.$ In summary, Lemma REF and Proposition REF yield the following result for the number of neighbours of an element: Proposition 5.4 For every $x\\in \\llbracket {n} \\rrbracket $ and $V\\subset \\llbracket {n} \\rrbracket \\setminus \\lbrace x\\rbrace $ , the total variation distance between the distribution of $|\\mathcal {N}_{x}(nt,\\llbracket {n} \\rrbracket \\setminus V)|$ and the $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ distribution is smaller than $2tm_{p,2}\\Big (\\frac{|V|}{n}+\\frac{1}{2n}\\Big )+\\frac{t}{2n}\\big (1+m_{p,2}+m_{p,3}+t(m_{p,2})^2\\big ).$"
],
[
"Comparison between a block size and the associated BGW\nprocess",
"The aim of this section is to prove that small block sizes at time $nt$ are well approximated by $\\bar{T}_{n}(nt)$ which has the same distribution as the total population size of a BGW$(1,nt\\beta _{n},\\nu _{n})$ process (first step of the proof of Theorem REF ): Proposition 5.5 Let $x\\in \\llbracket {n} \\rrbracket $ .",
"For every $k,n\\in \\operatorname{\\mathbb {N}}$ and $t\\ge 0$ , $|\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|\\le k)-\\operatorname{\\mathbb {P}}(\\bar{T}_{n}(nt)\\le k)|\\le \\frac{kt}{2n}\\big (m_{p,2}^2(k-1+t)+m_{p,2}(k+2)+m_{p,3}\\big ).$ Let us recall that the number of new active elements added in the $j$ -th step of the exploration procedure at time $t$ is $\\xi _{n,j}(t)=|\\mathcal {N}_{x_j}(t,\\llbracket {n} \\rrbracket \\setminus H_{j-1})\\setminus A_{j-1}|$ where $A_{j-1}$ and $H_{j}=\\lbrace x_1,\\ldots ,x_{j-1}\\rbrace $ are respectively the set of active elements and explored elements in step $j-1$ .",
"We have already seen one source of difference between $\\xi _{n,j}(t)$ and ${\\zeta _{n,j}(t)=\\sum _{w\\in \\mathcal {P}^{*}_{n,x_j}(t)}(\\ell (w)-1)}$ .",
"It is described by the event $F_{n,t,j}$ : `some tuples in $\\mathcal {P}^{*}_{n,x_j}(t,\\llbracket {n} \\rrbracket \\setminus H_{j-1})$ contain several copies of a same element or have in common other elements than $x_j$ '.",
"By Lemma REF , the probability of this event is bounded by: $\\frac{t}{2n^2}(m_{p,2}+m_{p,3}+\\frac{t}{n}m_{p,2}^2)$ .",
"There are two other sources of difference described by the following events: $\\lbrace \\bar{\\zeta }^{(2)}_{n,j}(t)>0$ }: `there exists a tuple containing $x_j$ and already explored elements (that is elements of $H_{j-1}$ )', $K_{n,t,j}$ : `there exists a tuple in $\\mathcal {P}_{n,x_j}(t,\\llbracket {n} \\rrbracket \\setminus H_{j-1})$ (i.e.",
"containing $x_j$ but no element of $H_{j-1}$ ) which contains active elements (i.e.",
"elements of $A_{j-1}$ )', The probability of these two events can be bounded by using the following lemma: Lemma 5.6 Let $V$ be a subset of $\\llbracket {n} \\rrbracket $ and let $x\\in \\llbracket {n} \\rrbracket \\setminus V$ .",
"For every $t>0$ , $\\operatorname{\\mathbb {P}}(\\exists w \\in \\mathcal {P}^{*}_{n,x}(t),\\ w\\cap V\\ne \\emptyset )\\le 1-\\exp (-\\frac{t|V|}{n^2}m_{p,2})$ Let $K_{x,V}$ be the subset of tuples $w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket )$ which contain $x$ and some elements of $V$ .",
"$\\operatorname{\\mathbb {P}}(\\exists w\\in \\mathcal {P}_{n,x}(t), w\\cap V\\ne \\emptyset ) = 1-\\exp (-t\\mu (K_{x,V}))$ and $\\mu (K_{x,V})&=\\mu (\\mathcal {P}_{n,x})-\\mu (\\mathcal {P}_{n,x}(\\llbracket {n} \\rrbracket \\setminus V))\\\\&=1-G_{p}(1-\\frac{1}{n})-\\big (G_{p}(1-\\frac{|V|}{n})-G_{p}(1-\\frac{|V|+1}{n})\\big )\\\\&\\le \\frac{|V|}{n^2}m_{p,2},$ where the last upper bound is a consequence of the following inequality: $1-(1-au)^j-(1-bu)^j+(1-(a+b)u)^j\\le j(j-1)abu^2\\;\\forall j\\in \\operatorname{\\mathbb {N}}^*,\\;\\forall a,b\\in \\operatorname{\\mathbb {R}}_+ \\;\\operatorname{ and }\\;\\forall u\\in [0,\\frac{1}{a+b}[.$ With the help of these estimates, we prove Proposition REF .",
"[Proof of Proposition REF] Set $\\Delta _k=|\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|\\le k)-\\operatorname{\\mathbb {P}}(\\bar{T}_{n}(nt)\\le k)|$ .",
"Since ${|\\Pi _{n}^{(x)}(nt)|\\le \\bar{T}_{n}(nt)}$ , $\\Delta _k=\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|\\le k\\; \\text{and}\\;\\bar{T}^{(n)}_{nt}> k).$ It is bounded above by $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|\\le k\\;\\text{and}\\; \\exists j\\le |\\Pi _{n}^{(x)}(nt)|,\\ \\xi _{n,j}(nt)<\\bar{\\zeta }_{n,j}(nt))\\\\ \\le \\sum _{j=1}^{k}\\operatorname{\\mathbb {E}}(\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(x)}(nt)|\\ge j\\rbrace }\\operatorname{\\mathbb {P}}(\\xi _{n,j}(nt)<\\bar{\\zeta }_{n,j}(nt)|\\mathcal {F}_{j-1})).$ We have seen that $\\operatorname{\\mathbb {P}}(\\xi _{n,j}(nt)<\\bar{\\zeta }_{n,j}(nt)|\\mathcal {F}_{j-1})\\le \\operatorname{\\mathbb {P}}(\\zeta ^{(2)}_{n,j}(nt)>0|\\mathcal {F}_{j-1})+\\operatorname{\\mathbb {P}}(K_{n,tn,j}|\\mathcal {F}_{j-1})+\\operatorname{\\mathbb {P}}(F_{n,tn,j}|\\mathcal {F}_{j-1})$ with the notations introduced page REF .",
"By Lemma REF $\\operatorname{\\mathbb {P}}(K_{n,tn,j}|\\mathcal {F}_{j-1})\\le \\frac{t|A_{j-1}|}{n}m_{p,2}\\;\\text{and}\\;\\operatorname{\\mathbb {P}}(\\zeta ^{(n,2)}_{nt,j}>0|\\mathcal {F}_{j-1})\\le \\frac{t(j-1)}{n}m_{p,2}$ and by Lemma REF $\\operatorname{\\mathbb {P}}(F_{n,tn,j}|\\mathcal {F}_{j-1})\\le \\frac{t}{2n}(m_{p,2}+m_{p,3}+tm_{p,2}^2).$ Therefore, $\\Delta _k \\le \\frac{t}{n}m_{p,2}\\sum _{j=1}^{k}\\operatorname{\\mathbb {E}}(|A_{j-1}|\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(x)}(nt)|\\ge j\\rbrace })\\\\ +\\frac{t}{2n}\\sum _{j=1}^{k}\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|\\ge j)(m_{p,3}+(2j-1)m_{p,2}+tm_{p,2}^2).$ By construction $|A_{j-1}|-1=\\sum _{i=1}^{j-1}(\\xi _{n,i}(nt)-1)$ .",
"Let us recall that $\\xi _{n,i}(nt)$ has nonnegative integer values, it is bounded above by $\\bar{\\zeta }_{n,i}(nt)$ and the conditional law of $\\bar{\\zeta }_{n,i}(nt)$ given $\\mathcal {F}_{i-1}$ is equal to the law of $\\zeta _{n,1}(nt)$ .",
"Thus, $\\operatorname{\\mathbb {E}}(\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(x)}(nt)|\\ge j\\rbrace }(|A_{j-1}|-1))\\le \\sum _{i=1}^{j-1}\\operatorname{\\mathbb {E}}(\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(x)}(nt)|\\ge i\\rbrace }\\operatorname{\\mathbb {E}}(\\bar{\\zeta }_{n,i}(nt)|\\mathcal {F}_{i-1}))\\le (j-1)\\operatorname{\\mathbb {E}}(\\zeta _{n,1}(nt)).$ with $\\operatorname{\\mathbb {E}}(\\zeta _{n,1}(nt))=tn\\sum _{j=1}^{+\\infty }jp(j+1)\\big (1-(1-\\frac{1}{n})^{j+1}\\big )\\le t m_{p,2}$ .",
"Therefore, $\\Delta _k \\le \\frac{t}{n}m_{p,2}^2\\sum _{j=1}^{k}(j-1)+\\frac{t}{2n}\\sum _{j=1}^{k}(m_{p,3}+(2j-1)m_{p,2}+tm_{p,2}^2)\\\\ =\\frac{kt}{2n}(m_{p,2}^2(k-1+t)+m_{p,2}(k+2)+m_{p,3}).$"
],
[
"The total progeny of the BGW process associated with a block",
"Recall that the offspring distribution of the BGW process associated with a block at time $nt$ is the $\\text{CPois}(tn\\beta _{n},\\nu _{n})$ -distribution with: $\\beta _{n}=\\mu (\\lbrace w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ), x\\in w\\rbrace )=1-G_{p}(1-\\frac{1}{n})-G_{p}(\\frac{1}{n})\\; \\operatorname{ and }$ $\\nu _{n}(j)=\\frac{1}{\\beta _{n}}\\mu (\\lbrace w\\in \\mathcal {W}^{*}(\\llbracket {n} \\rrbracket ),\\ x\\in w \\;\\operatorname{ and }\\; \\ell (w)=j+1\\rbrace )\\\\=\\frac{p(j+1)}{\\beta _n}\\Big (1-(1-\\frac{1}{n})^{j+1}-(\\frac{1}{n})^{j+1}\\Big ) \\quad \\forall j\\in \\operatorname{\\mathbb {N}}^*.$ We have shown (Proposition REF ) that the $\\text{CPois}(tn\\beta _{n},\\nu _{n})$ -distribution is close to the $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distribution for large $n$ .",
"We now consider the distribution of the total number of individuals in a BGW process with one ancestor and offspring distribution $\\text{CPois}(tn\\beta _{n}, \\nu _{n})$ .",
"Let us state a general result for the comparison of the total number of individuals in two BGW processes: Lemma 5.7 Let $\\nu _1$ and $\\nu _2$ be two probability distributions on $\\operatorname{\\mathbb {N}}$ .",
"Let $\\operatorname{d_{\\text{TV}}}$ denote the total variation distance between probability measures.",
"Let $T_1$ and $T_2$ be the total population sizes of the BGW processes with one ancestor and offspring distributions $\\nu _1$ and $\\nu _2$ respectively.",
"For every $k\\in \\operatorname{\\mathbb {N}}^*$ , $|\\operatorname{\\mathbb {P}}(T_1\\ge k)-\\operatorname{\\mathbb {P}}(T_2\\ge k)|\\le \\operatorname{d_{\\text{TV}}}(\\nu _1,\\nu _2)\\sum _{i=1}^{k-1}\\operatorname{\\mathbb {P}}(T_2\\ge i)$ .",
"We follow the proof of Theorem 3.20 in [42] which states an analogous result between binomial and Poisson BGW processes.",
"The proof is based on the description of the total population size by means of the hitting time of a random walk and coupling arguments.",
"By Strassen's theorem, there exist two independent sequences $(X_i)_{i\\in \\operatorname{\\mathbb {N}}^*}$ and $(Y_i)_{i\\in \\operatorname{\\mathbb {N}}^*}$ of i.i.d.",
"random variables with distributions $\\nu _1$ and $\\nu _2$ respectively such that ${\\operatorname{d_{\\text{TV}}}(\\nu _1,\\nu _2)=P(X_i\\ne Y_i)}$ for every $i\\in \\operatorname{\\mathbb {N}}\\*$ .",
"Let $\\tau _1=\\min (n,\\ X_1+ \\ldots +X_n=n-1)$ and $\\tau _2=\\min (n,\\ Y_1+ \\ldots +Y_n=n-1)$ .",
"$\\tau _1$ and $\\tau _2$ have the same laws as $T_1$ and $T_2$ respectively.",
"Let $k\\in \\operatorname{\\mathbb {N}}^*$ .",
"$|\\operatorname{\\mathbb {P}}(T_1\\ge k)-\\operatorname{\\mathbb {P}}(T_2\\ge k)|\\le \\max \\big (\\operatorname{\\mathbb {P}}(\\tau _1\\ge k\\; \\text{and}\\; \\tau _2<k),\\operatorname{\\mathbb {P}}(\\tau _1< k\\; \\text{and}\\; \\tau _2\\ge k)\\big ).$ First, let us note that $\\lbrace \\tau _1\\ge k\\; \\text{and}\\; \\tau _2<k\\rbrace \\subset \\bigcup _{i=1}^{k-1}\\lbrace X_j=Y_j\\;\\forall j\\le i-1, \\; X_i\\ne Y_i \\; \\text{and}\\;\\tau _1\\ge k\\rbrace $ As $\\lbrace X_j=Y_j\\;\\forall j\\le i-1\\; \\text{and}\\; \\tau _1\\ge k\\rbrace \\subset \\lbrace \\tau _2\\ge i\\rbrace $ for $i\\le k-1$ and $\\lbrace \\tau _2\\ge i\\rbrace $ depends only on $Y_1,\\ldots ,Y_{i-1}$ , we obtain: $\\operatorname{\\mathbb {P}}(\\tau _1\\ge k\\; \\text{and}\\; \\tau _2<k)\\le \\sum _{i=1}^{k-1}\\operatorname{\\mathbb {P}}(\\tau _2\\ge i)\\operatorname{\\mathbb {P}}(X_i\\ne Y_i)=\\operatorname{d_{\\text{TV}}}(\\nu _1,\\nu _2)\\sum _{i=1}^{k-1}\\operatorname{\\mathbb {P}}(\\tau _2\\ge i).",
"$ The same upper bound holds for $\\operatorname{\\mathbb {P}}(\\tau _1<k\\;\\text{and}\\;\\tau _2\\ge k)$ since $\\lbrace \\tau _1<k\\;\\text{and}\\;\\tau _2\\ge k)\\rbrace \\subset \\bigcup _{i=1}^{k-1}\\lbrace X_j=Y_j\\;\\forall j\\le i-1, \\; X_i\\ne Y_i \\; \\text{and}\\; \\tau _2\\ge k\\rbrace .$ and $\\lbrace \\tau _2\\ge k\\rbrace \\subset \\lbrace \\tau _2\\ge i\\rbrace $ for $i\\le k$ .",
"From Lemma REF and Proposition REF , we obtain: Proposition 5.8 Let $t>0$ and $n\\in \\operatorname{\\mathbb {N}}^*$ .",
"Let $\\bar{T}_{n}(t)$ and $T^{(1)}_{p}(t)$ denote the total number of individuals in a BGW$(1,t\\beta _{n}, \\nu _{n})$ and BGW$(1,tm^{*}_{p,1},\\tilde{p})$ processes respectively.",
"$|\\operatorname{\\mathbb {P}}(\\bar{T}_{n}(nt)\\ge k)-\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)\\ge k)|\\le \\frac{t}{2n}(2m_{p,2}+1)\\sum _{i=1}^{k-1}\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)\\ge i)\\quad \\text{for every}\\; k\\in \\operatorname{\\mathbb {N}}^*.$"
],
[
"Proof of Theorem ",
"Theorem REF follows from Propositions REF and REF : $|\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|\\le k)-\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)\\le k)|&\\le &|\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|\\le k)-\\operatorname{\\mathbb {P}}(\\bar{T}_{n}(nt)\\le k)|\\\\&& + |\\operatorname{\\mathbb {P}}(\\bar{T}_{n}(nt)\\le k)-\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)\\le k)|\\\\&\\le & \\frac{kt}{2n}\\big (m_{p,2}^2(k-1+t)+m_{p,2}(k+4)+m_{p,3}+1\\big ).$ $\\Box $"
],
[
"Proof of Corollary ",
"To deduce Corollary REF , we apply the above inequality to $|\\operatorname{\\mathbb {P}}(|\\Pi _{\\llbracket {n} \\rrbracket ,p_n}^{(x)}(nt_n)|\\le k)-\\operatorname{\\mathbb {P}}(T^{(1)}_{p_n}(t_n)\\le k)|$ .",
"By Lemma REF and Lemma REF : $|\\operatorname{\\mathbb {P}}(T^{(1)}_{p_n}(t_n)\\le k)-\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)\\le k)|\\le k\\Big (|t_n m^{*}_{p_n,1}-tm^{*}_{p,1}|+\\max (t_n m^{*}_{p_n,1},tm^{*}_{p,1})\\operatorname{d_{\\text{TV}}}(\\tilde{p}_n,\\tilde{p}) \\Big ).$ Under the hypotheses of Corollary REF , $(m_{p_n,i})_n$ for $i\\in \\lbrace 1,2,3\\rbrace $ are bounded, ${|t_n m^{*}_{p_n,1}-tm^{*}_{p,1}|=O(\\frac{1}{n})}$ and $\\operatorname{d_{\\text{TV}}}(\\tilde{p}_n,\\tilde{p})=O(\\frac{1}{n})$ .",
"Therefore, there exists $C(t)>0$ such that for every $k\\in \\operatorname{\\mathbb {N}}^*$ , $|\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt_n)|\\le k)-\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)\\le k)|\\le \\frac{C(t)k^2}{n}.$ $\\Box $"
],
[
"Proof of Corollary ",
"Let us now consider $\\Pi _{\\llbracket a_n \\rrbracket , p_n}^{(1)}(ntG_{p}(\\frac{a_n}{n}))$ , where $a_n=\\lfloor an\\rfloor $ and $p_n$ is the probability distribution on $\\operatorname{\\mathbb {N}}^*$ defined by: $p_n(k)= \\left(\\frac{a_n}{n}\\right)^k p(k)\\frac{1}{G_{p}(\\frac{a_n}{n})}\\; \\forall k\\in \\operatorname{\\mathbb {N}}^*.$ Set $t_n=t\\frac{n}{a_n}G_{p}(\\frac{a_n}{n})$ for $n\\in \\operatorname{\\mathbb {N}}^*$ .",
"To prove that there exists $C_a(t)>0$ such that for every $k,n\\in \\operatorname{\\mathbb {N}}^*$ , $\\left|\\operatorname{\\mathbb {P}}\\left(|\\Pi _{\\llbracket a_n \\rrbracket , p_n}^{(1)}(ntG_{p}(\\frac{a_n}{n}))|\\le k\\right)-\\operatorname{\\mathbb {P}}\\left(T^{(1)}_{\\hat{p}_a}(t\\frac{G_{p}(a)}{a}))\\le k\\right)\\right|\\le C_a(t) \\frac{k^{2}}{n},$ it suffices to verify that Corollary REF applies to the sequences $(t_n)_n$ and $(p_n)_n$ : Since $|\\frac{a_n}{n}-a|\\le \\frac{1}{n}$ and $G_{p}^{\\prime }$ is bounded on $[0,a]$ , $t_n-t\\frac{G_{p}(a)}{a}=O(\\frac{1}{n})$ .",
"The third moment of $p_n$ is bounded since $\\sum _{k=1}^{+\\infty }k^3 p_n(k)\\le \\frac{1}{G_{p}(a_n)}\\sum _{k=1}^{+\\infty }k^3a^kp(k)$ for every $n\\in \\operatorname{\\mathbb {N}}^*$ .",
"The difference $\\Delta _n:=t_n m_{p_n,1}^{*}-t\\frac{G_{p}(a)}{a}m_{\\hat{p}_a,1}^{*}$ can be split into the sum of two terms: $\\Delta _{n,1}&=&t\\left(\\frac{n}{a_n}-\\frac{1}{n}\\right)\\sum _{k\\ge 2}k(\\frac{a_n}{n})^kp(k)=O(\\frac{1}{n}),\\\\\\Delta _{n,2}&=&\\frac{1}{a}\\sum _{k\\ge 2}((\\frac{a_n}{n})^k-a^k)kp(k).$ By applying the following inequality $|x^k-y^k|\\le k|x-y|\\max (|x|,|y|)^{k-1}\\; \\forall x,y\\in \\operatorname{\\mathbb {R}},$ we obtain $|\\Delta _{n,2}|\\le \\frac{1}{an}\\sum _{k\\ge 2}k^2a^{k-1}p(k)=O(\\frac{1}{n})$ .",
"The last assumption of Corollary REF concerns the total variation distance between the probability distributions $\\tilde{p}_n$ and $\\widetilde{(\\hat{p}_a)}$ defined by: $\\tilde{p}_n(k)=\\frac{1}{S(\\frac{a_n}{n})}(k+1)(\\frac{a_n}{n})^k p(k+1) \\operatorname{ and }\\widetilde{(\\hat{p}_a)}(k)=\\frac{1}{S(a)}(k+1)a^kp(k+1)\\; \\forall k\\in \\operatorname{\\mathbb {N}}^*,$ where $S(x)=\\sum _{j\\ge 2}jx^j p(j)$ for $x\\in [0,1[$ .",
"Let us note that $d_{\\operatorname{d_{\\text{TV}}}}(\\tilde{p}_n,\\widetilde{(\\hat{p}_a)})\\le \\frac{1}{S(a)}\\sum _{k\\ge 2}kp(k)|(\\frac{a_n}{n})^k-a^k|$ .",
"Therefore, by inequality (REF ), $\\operatorname{d_{\\text{TV}}}(\\tilde{p}_n,\\widetilde{(\\hat{p}_a)})=O(\\frac{1}{n})$ .",
"In conclusion, the four assumptions of Corollary REF are satisfied.",
"Relation (REF ) between the probability mass functions of $T^{(u)}_{\\hat{p}_a}(\\frac{G_{p}(a)}{a}t)$ and $T^{(u)}_{p}(t)$ can be easily proven by applying formula (REF ) for the probability mass function of $T^{(u)}_{p}(t)$ (see Appendix REF for a proof of (REF )) and by expressing the probability mass function of $\\text{CPois}(\\lambda G_{\\tilde{p}}(a),\\widehat{(\\tilde{p})}_a)$ in terms of the probability mass function of $\\text{CPois}(\\lambda ,\\tilde{p})$ (see Lemma REF ).",
"$\\Box $"
],
[
"Asymptotic distribution of two block sizes",
"Let us prove Corollary REF stating that under the assumptions of Theorem REF , the block sizes of two elements converge in law to the total population sizes of two independent BGW$(1,tm^{*}_{p,1},\\tilde{p})$ processes."
],
[
"Proof of Corollary ",
"The proof is similar to the proof presented in [3] in order to study the joint limit of the component sizes of two vertices in the Erdös-Rényi random graph process.",
"It is based on the properties REF and REF stated in Subsection REF .",
"Let $x$ and $y$ be two distinct vertices and let $j,k$ be two nonnegative integers.",
"We have to study the convergence of $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|=j \\operatorname{ and }|\\Pi _{n}^{(y)}(nt)|=k)$ .",
"First, let us note that by REF , for every $n\\ge j$ , $\\operatorname{\\mathbb {P}}(y\\in \\Pi _{n}^{(x)}(nt) \\mid |\\Pi _{n}^{(x)}(nt)|=j)= \\frac{j-1}{n-1}$ .",
"Therefore, ${\\operatorname{\\mathbb {P}}(y\\in \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|=j)}$ converges to 0 as $n$ tends to $+\\infty $ .",
"It remains to study $\\operatorname{\\mathbb {P}}(y\\notin \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|=j \\operatorname{ and }|\\Pi _{n}^{(y)}(nt)|=k)$ which can be written: $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(y)}(nt)|=k \\mid y\\notin \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|=j) \\operatorname{\\mathbb {P}}(y\\notin \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|=j).$ Since $\\operatorname{\\mathbb {P}}(y\\notin \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|=j)=(1-\\frac{j-1}{n-1})\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|=j)$ , it converges to $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)=j)$ by Theorem REF .",
"By REF , $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(y)}(nt)|=k \\mid y\\notin \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|=j)=\\operatorname{\\mathbb {P}}(|\\Pi _{\\llbracket {n-j} \\rrbracket ,p_{n,j}}^{(y)}((n-j)t_{n,j})|=k )$ where $t_{n,j}=\\frac{tn}{n-j}G_{p}(1-\\frac{j}{n})$ and $p_{n,j}=p_{\\llbracket {n} \\rrbracket |\\llbracket {n-j} \\rrbracket }$ (i.e.",
"$p_{n,j}(k)=(1-\\frac{j}{n})^k\\frac{p(k)}{G_{p}(1-\\frac{j}{n})}$ for every $k\\in \\operatorname{\\mathbb {N}}^*$ ).",
"Let us verify that Corollary REF can be applied to the sequences $(t_{n,j})_n$ and $(p_{n,j})_n$ .",
"First, $(t_{n,j})_n$ converges to $t$ , $(p_{n,j})_n$ converges weakly to $p$ , and $(m_{p_{n,j},3})_n$ converges to $m_{p,3}$ .",
"By inequality (REF ), $0\\le tm^{*}_{p,1}-t_{n,j}m^{*}_{p_{n,j},1}\\le \\frac{t j}{n}m_{p,2}$ .",
"Finally, let us show that $\\operatorname{d_{\\text{TV}}}(\\tilde{p}_{n,j},\\tilde{p})=O(\\frac{1}{n})$ .",
"For $k\\in \\operatorname{\\mathbb {N}}$ , $\\tilde{p}(k)-\\tilde{p}_{n,j}(k)=(k+1)p(k+1)\\frac{V_n(k)}{m_{p,1}S_n}$ with $V_n(k)=\\sum _{\\ell \\ge 1}(\\ell +1)p(\\ell +1)\\Big ((1-\\frac{j}{n})^{\\ell +1} - (1-\\frac{j}{n})^{k+1}\\Big )\\;\\text{and}\\;S_n=\\sum _{\\ell \\ge 1}(\\ell +1)p(\\ell +1)(1-\\frac{j}{n})^{\\ell +1}.$ Using that the first $k$ terms in $V_n(k)$ are positive and the others are nonpositive, we obtain $|V_n(k)|\\le \\frac{j}{n}\\max (km^{*}_{p,1},m_{p,2})$ for every $k\\in \\operatorname{\\mathbb {N}}$ .",
"As $(1-x)^{\\ell }\\ge 1-\\ell x$ for every $x\\ge 0$ and $\\ell \\in \\operatorname{\\mathbb {N}}^*$ , $S_n\\ge m^{*}_{p,1}-\\frac{j}{n}(m^{*}_{p,1}+m_{p,2})$ .",
"Therefore, $\\operatorname{d_{\\text{TV}}}(\\tilde{p}_{n,j},\\tilde{p})\\le \\frac{j}{n}m_{p,2}\\big (m^{*}_{p,1}-\\frac{j}{n}(m^{*}_{p,1}+m_{p,2})\\big )^{-1}=O(\\frac{1}{n})$ .",
"Consequently, $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(y)}(nt)|=k \\mid y\\notin \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|=j)$ converges to $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}=k)$ , which completes the proof.",
"$\\Box $"
],
[
"Hydrodynamic behavior of the coalescent process",
"This section is devoted to the proof of Theorem REF describing the asymptotic limit of the average number of blocks having the same size.",
"Let $t>0$ and $k\\in \\operatorname{\\mathbb {N}}^*$ .",
"First, we prove that $\\rho _{n,k}(t) =\\frac{1}{nk}\\sum _{x=1}^{n}\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(x)}(nt)| = k\\rbrace }$ converges in $L^2$ to $\\rho _{k}(t) =\\frac{1}{k}\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t) = k)$ .",
"Theorem REF and Corollary REF imply the convergence of the first two moments of $\\rho _{n,k}(t)$ to $\\rho _{k}(t)$ and $(\\rho _{k}(t))^2$ respectively and thus the $L^2$ convergence of $(\\rho _{n,k}(t))_n$ .",
"Indeed, $\\operatorname{\\mathbb {E}}(\\rho _{n,k}(t))=\\frac{1}{k}\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)| = k)$ converges to $\\rho _{k}(t)$ .",
"The second moment is $\\operatorname{\\mathbb {E}}((\\rho _{n,k}(t))^2)=\\frac{1}{nk^2}\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)|=k)+(1-\\frac{1}{n})\\frac{1}{k^2}\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)|=k\\; \\operatorname{ and }\\;|\\Pi _{n}^{(2)}(nt)|=k).$ The first term converges to 0 and the second term converges to $(\\rho _{k}(t))^2$ .",
"It remains to show that $\\lbrace \\rho (t),\\; t\\in \\operatorname{\\mathbb {R}}_+\\rbrace $ is solution of the coagulation equations (REF ): $\\frac{d}{dt}\\rho _k(t) =\\sum _{j=2}^{+\\infty }p(j)\\mathcal {K}_{j}(\\rho (t),k)$ where $\\mathcal {K}_{j}(\\rho (t),k)=\\Big (\\sum _{\\begin{array}{c}(i_1,\\ldots ,i_{j})\\in (\\operatorname{\\mathbb {N}}^*)^{j}\\\\ i_1+\\cdots +i_j=k\\end{array}}\\prod _{u=1}^{j}i_u\\rho _{i_u}(t)\\Big )\\operatorname{{1}}_{\\lbrace j\\le k\\rbrace }-kj\\rho _k(t).$ By definition of $\\rho (t)$ , for $j\\in \\operatorname{\\mathbb {N}}\\setminus \\lbrace 0,1\\rbrace $ , $\\mathcal {K}_{j}(\\rho (t),k)=\\operatorname{\\mathbb {P}}(T^{(j)}_{p}(t)=k)\\operatorname{{1}}_{\\lbrace j\\le k\\rbrace } - j\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)=k),$ where $T^{(\\ell )}_{p}(t)$ denotes the total progeny of a BGW$(\\ell ,tm^{*}_{p,1},\\tilde{p})$ process for every $\\ell \\in \\operatorname{\\mathbb {N}}^*$ .",
"The probability distribution of $T^{(\\ell )}_{p}(t)$ is computed in the appendix (Lemma REF ): $\\left\\lbrace \\begin{array}{l}\\operatorname{\\mathbb {P}}(T^{(\\ell )}_{p}(t) = \\ell ) =\\displaystyle {e^{-t\\ell m^{*}_{p,1}}}\\\\\\operatorname{\\mathbb {P}}(T^{(\\ell )}_{p}(t) = k) = \\displaystyle {\\frac{\\ell }{k}e^{-ktm^{*}_{p,1}}\\sum _{h=1}^{k-\\ell }\\frac{(tm^{*}_{p,1}k)^h}{h!",
"}(\\tilde{p})^{\\star h}(k-\\ell )}\\quad \\forall k\\ge \\ell +1.\\end{array}\\right.$ For $k=1$ , $\\rho _1$ is solution of the equation $\\frac{d}{dt}\\rho _1(t)=-m^{*}_{p,1}\\rho _1(t)$ and the right hand side term is equal to $\\sum _{j=2}^{+\\infty }p(j)\\mathcal {K}_{j}(\\rho (t),1)$ .",
"Let us assume now that $k\\ge 2$ .",
"$\\sum _{j=2}^{+\\infty }p(j)\\mathcal {K}_{j}(\\rho (t),k)=\\frac{e^{-tkm^{*}_{p,1}}}{k}\\left(kp(k)+\\sum _{j=2}^{k-1}\\sum _{h=1}^{k-j}\\frac{(tm^{*}_{p,1}k)^h}{h!",
"}jp(j)(\\tilde{p})^{\\star h}(k-j)\\right)\\\\-m^{*}_{p,1}\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)=k).$ By using that $jp(j)=m^{*}_{p,1}\\tilde{p}(j-1)$ for every $j\\ge 2$ and by inverting the two sums we obtain $\\sum _{j=2}^{+\\infty }p(j)\\mathcal {K}_{j}(\\rho (t),k)=\\frac{m^{*}_{p,1}}{k}e^{-tkm^{*}_{p,1}}\\sum _{h=1}^{k-1}\\frac{(tm^{*}_{p,1}k)^{h-1}}{(h-1)!",
"}(\\tilde{p})^{\\star h}(k-1)-m^{*}_{p,1}\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)=k).$ Since the right-hand side of the last formula is equal to $\\frac{1}{k}\\frac{d}{dt}\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t) = k)$ , $\\frac{d}{dt}\\rho _k(t) =\\sum _{j=2}^{+\\infty }p(j)\\mathcal {K}_{j}(\\rho (t),k)$ This completes the proof of Theorem REF ."
],
[
"Phase transition",
"The expectation of the compound Poisson distribution $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ is $tm_{p,2}$ .",
"Thus the limiting BGW process associated with a block is subcritical, critical or supercritical depending on whether $t$ is smaller, equal or larger than $\\frac{1}{m_{p,2}}$ .",
"This section is devoted to the proofs of Theorems REF and REF , which provide some results on the size of the largest block at time $nt$ in these three cases."
],
[
"The subcritical regime",
"Let us assume that $t< \\frac{1}{m_{p,2}}$ .",
"An application of the block exploration procedure and Fuk-Nagaev inequality allows to prove that, if the moment of $p$ of order $u$ is finite for some $u\\ge 3$ , then the largest block size at time $nt$ is not greater than $n^{1/(u-1)+\\varepsilon }$ for any $\\varepsilon >0$ with probability that converges to 1.",
"If the probability generating function of $p$ is assumed to be finite for some real greater than 1, then it can be shown using a Chernoff bound that the largest block size at time $nt$ is at most of order $\\log (n)$ with probability that converges to 1.",
"Theorem (REF .",
"(1)) Let $0<t<\\frac{1}{m_{p,2}}$ .",
"Assume that $p$ has a finite moment of order $u$ for some $u\\ge 3$ .",
"If $(a_n)_n$ is a sequence of reals that tends to $+\\infty $ , then $\\operatorname{\\mathbb {P}}(\\max _{x\\in \\llbracket {n} \\rrbracket }|\\Pi _{n}^{(x)}(nt)|> a_n n^{\\frac{1}{u-1}})$ converges to 0 as $n$ tends to $+\\infty $ .",
"Assume that $G_{p}$ is finite on $[0,r]$ for some $r>1$ .",
"Set $h(t)=\\sup _{\\theta >0}(\\theta -\\log (L_{t}(\\theta )))$ where $L_{t}$ is the moment-generating function of the compound Poisson distribution $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ .$h(t)$ is the value of the Cramér function at 1 of $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ .",
"Then $h(t)>0$ and for every $a\\ge (h(t))^{-1}$ , $\\operatorname{\\mathbb {P}}(\\max _{x\\in \\llbracket {n} \\rrbracket }|\\Pi _{n}^{(x)}(nt)|> a\\log (n))$ converges to 0 as $n$ tends to $+\\infty $ .",
"For $k\\in \\operatorname{\\mathbb {N}}^*$ , let $Z_k(t)$ denote the number of blocks of size greater than $k$ at time $t$ .",
"Since each element of $\\llbracket {n} \\rrbracket $ plays the same role, $\\operatorname{\\mathbb {P}}(\\max _{x\\in \\llbracket {n} \\rrbracket }|\\Pi _{n}^{(x)}(t)|>k)\\le \\operatorname{\\mathbb {P}}(Z_k(t)>k)\\le \\frac{\\operatorname{\\mathbb {E}}(Z_k(t))}{k}=\\frac{n}{k} \\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(t)|>k).$ By construction of the random variables $\\xi _{n,j}(t)$ and $\\bar{\\zeta }_{n,j}(t)$ , $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(t)|> k)\\le \\operatorname{\\mathbb {P}}(\\sum _{i=1}^{k}\\xi _{n,i}(t)\\ge k)\\le \\operatorname{\\mathbb {P}}(\\sum _{i=1}^{k}\\bar{\\zeta }_{n,i}(t)\\ge k).$ First, let us assume that $p$ has a finite moment of order $u\\ge 3$ .",
"Set $e_n(t)=\\operatorname{\\mathbb {E}}(\\bar{\\zeta }_{n,1}(t))$ and $X_{i,n}= \\bar{\\zeta }_{n,i}(nt)-e_n(nt)$ for $i\\in \\llbracket {n} \\rrbracket $ .",
"Let us recall the Fuk-Nagaev inegality, we shall apply to the sequence $(X_{i,n})_{i=1\\ldots ,k}$ : Theorem (Corollary 1.8 of [31]) Let $s\\ge 2$ and let $Y_1,\\ldots , Y_k$ be independent random variables such that $\\operatorname{\\mathbb {E}}(\\max (Y_i,0)^s)<+\\infty $ and $\\operatorname{\\mathbb {E}}(Y_i)=0$ for every ${i\\in \\lbrace 1,\\ldots ,k\\rbrace }$ .",
"Set $ A_{s,k}^+=\\sum _{i=1}^{k}\\operatorname{\\mathbb {E}}(\\max (Y_i,0)^s)$ and $B_k=\\sum _{i=1}^{k}\\operatorname{Var}(Y_i)$ .",
"For every $x>0$ , $P(\\sum _{i=1}^{k}Y_i\\ge x)\\le x^{-s}c^{(1)}_s \\sum _{i=1}^{k}\\operatorname{\\mathbb {E}}(\\max (Y_i,0)^s) +\\exp (-c^{(2)}_s \\frac{x^2}{B_k}),$ where $c^{(1)}_s= (1+2/s)^s$ and $c^{(2)}_s= 2(s+2)^{-2}e^{-s}$ .",
"We begin by proving that $\\operatorname{\\mathbb {E}}|X_{1,n}|^{u-1}$ is uniformly bounded in $n$ .",
"Let us recall that the law of $\\bar{\\zeta }_{n,j}(nt)$ is $\\text{CPois}(nt\\beta _n,\\nu _n)$ with $\\nu _n(j)=\\frac{1}{\\beta _n}\\left(1-(1-\\frac{1}{n})^{j+1}-(\\frac{1}{n})^{j+1}\\right)p(j+1)\\le \\frac{m^{*}_{p,1}}{n\\beta _n}\\tilde{p}(j)\\quad \\forall j\\in \\operatorname{\\mathbb {N}}.$ Thus we can apply the following property of compound Poisson distributions, the proof of which is straightforward: Lemma 7.1 Let $p_1$ and $p_2$ be two probability measures on $\\operatorname{\\mathbb {N}}^*$ and let $\\lambda _1, \\lambda _2$ be two positive reals such that $p_1(j)\\le \\frac{\\lambda _2}{\\lambda _1}p_2(j)$ $\\forall j\\in \\operatorname{\\mathbb {N}}^*$ .",
"Let $X_1$ and $X_2$ be two random variables with compound Poisson distribution $\\text{CPois}(\\lambda _1,p_1)$ and $\\text{CPois}(\\lambda _2, p_2)$ respectively.",
"For every positive function $f$ , $\\operatorname{\\mathbb {E}}(f(X_1))\\le \\operatorname{\\mathbb {E}}(f(X_2))\\exp (\\lambda _2-\\lambda _1)$ .",
"This shows that $\\operatorname{\\mathbb {E}}\\big (|X_{1,n}|^{u-1}\\big )\\le e^{t(m^{*}_{p,1}-n\\beta _n)}\\operatorname{\\mathbb {E}}(|Y-e_n(nt)|^{u-1}),$ where $Y$ is a ${\\text{CPois}(tm^{*}_{p,1},\\tilde{p})}$ -distributed random variable.",
"Since $p$ has a finite moment of order $u$ , $\\tilde{p}$ has a finite moment of order $u-1$ .",
"Consequently, $\\operatorname{\\mathbb {E}}(|Y|^{u-1})$ is finite.",
"As $e_n(nt)$ converges to $tm_{p,2}$ and $n\\beta _n$ converges to $m^{*}_{p,1}$ , we deduce that $\\operatorname{\\mathbb {E}}|X_{1,n}|^{u-1}$ is uniformly bounded.",
"Therefore, by the Fuk-Nagaev inequality, for every $k,n\\in \\operatorname{\\mathbb {N}}^*$ , $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)|> k)\\le k^{2-u}M^{(1)}_{u,t}+\\exp (-k M^{(2)}_{u,t})$ where $M^{(1)}_{u,t}=c^{(1)}_{u-1}(1-tm_{p,2})^{1-u}\\sup _{n}\\operatorname{\\mathbb {E}}|X_{1,n}|^{u-1}$ and $M^{(2)}_{u,t}=\\frac{c^{(2)}_{u-1}(1-tm_{p,2})^{2}}{t(m_{p,3}+m_{p,2})}$ .",
"In conclusion, there exists a constant $C_{t,u}>0$ such that if $(b_n)_n$ is a positive sequence that converges to $+\\infty $ , $\\operatorname{\\mathbb {P}}(\\max _{x\\in \\llbracket {n} \\rrbracket } |\\Pi _{n}^{(x)}(t)|> b_n)\\le C_{t,u}\\frac{n}{b_{n}^{u-1}}$ for every $n\\in \\operatorname{\\mathbb {N}}$ , which completes the proof of assertion (a).",
"Let us assume now that $G_{p}$ is finite on $[0,r]$ for some $r>1$ .",
"The moment-generating function of $\\bar{\\zeta }_{n,i}(nt)$ is finite on $[0,\\log (r)[$ and is equal to $\\operatorname{\\mathbb {E}}(e^{\\theta \\bar{\\zeta }_{n,i}(nt)})=\\exp \\Big (nt \\sum _{j=1}^{+\\infty }(e^{\\theta j}-1)p(j+1)(1-(1-\\frac{1}{n})^{j+1}-(\\frac{1}{n})^{j+1}) \\Big ).$ It is smaller than $L_t(\\theta )=\\exp \\Big (t \\sum _{j=1}^{+\\infty }(e^{\\theta j}-1)(j+1)p(j+1)\\Big )$ .",
"By Markov's inequality: $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)|> k)\\le \\operatorname{\\mathbb {E}}(e^{\\theta \\bar{\\zeta }_{n,1}(nt)})^ke^{-k\\theta }\\le \\exp \\Big (-k\\big (\\theta -\\log (L_{t}(\\theta ))\\big )\\Big )\\quad \\forall 0<\\theta <\\log (r).$ Since the expectation of the $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distribution is assumed to be smaller than 1, $h(t)=\\sup _{\\theta >0}(\\theta -\\log (L_t(\\theta )))$ is positive.",
"We deduce that for every $ k\\in \\operatorname{\\mathbb {N}}^*$ , $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)|> k)\\le \\exp (-kh(t))$ .",
"In particular, for every $a>0$ , $\\operatorname{\\mathbb {P}}\\left(\\max _{x\\in [n]} |\\Pi _{n}^{(x)}(nt)|>a\\log (n)\\right)\\le \\frac{n^{1-ah(t)}}{\\lfloor a\\log (n)\\rfloor }\\exp (h(t)),$ which completes the proof of assertion (b).",
"Let us now prove the lower bound for the largest block stated in Theorem REF : Theorem (Theorem REF ) Set $0<t<\\frac{1}{m_{p,2}}$ .",
"Assume that $p$ is regularly varying with index $-\\alpha <-3$ .",
"For every $\\alpha ^{\\prime }>\\alpha $ , $\\operatorname{\\mathbb {P}}(\\max _{x\\in \\llbracket {n} \\rrbracket }|\\Pi _{n}^{(x)}(nt)|\\le n^{\\frac{1}{1+\\alpha ^{\\prime }}})$ converges to 0 as $n$ tends to $+\\infty $ .",
"[Proof of Theorem REF ] To prove this lower bound, we use a second moment method with the random variable $Z_k(nt)$ (which is the number of elements that belong to a block of size greater than $k$ at time $nt$ ).",
"$\\operatorname{\\mathbb {P}}(\\max _{x\\in \\llbracket {n} \\rrbracket }|\\Pi _{n}^{(x)}(nt)|\\le k)=\\operatorname{\\mathbb {P}}(Z_k(nt)=0)\\le \\frac{\\operatorname{Var}(Z_k(nt))}{\\operatorname{\\mathbb {E}}(Z_k(nt))^2}.",
"$ Let us first give an upper bound for the variance $\\operatorname{Var}(Z_k(nt))$ .",
"Using properties REF and REF of $\\mathcal {P}_{n}(t)$ stated in Subsection REF , one can proceed as in ([42], Proposition 4.7) to obtain the following inequality: $\\operatorname{Var}(Z_k(nt))\\le n\\operatorname{\\mathbb {E}}(|\\Pi _{n}^{(1)}(nt)|\\operatorname{{1}}_{|\\Pi _{n}^{(1)}(nt)|>k}).$ Let us continue the proof of Theorem REF before showing (REF ).",
"The right-hand side of (REF ) can be expressed by means of the tail distribution of $|\\Pi _{n}^{(1)}(nt)|$ : $\\operatorname{\\mathbb {E}}(|\\Pi _{n}^{(1)}(nt)|\\operatorname{{1}}_{|\\Pi _{n}^{(1)}(nt)|>k})=k\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)|>k)+\\int _{[k,+\\infty [}\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)|>s)ds$ As $p$ has a finite moment of order $\\alpha _1$ for every $0<\\alpha _1<\\alpha $ , an application of inequality (REF ) deduced from the Fuk-Nagaev inequality yieds the following upper bound: for every $\\alpha _1\\in ]3,\\alpha [$ .",
"there exists $A_{t,\\alpha _1}>0$ such that $\\operatorname{\\mathbb {E}}(|\\Pi _{n}^{(1)}(nt)|\\operatorname{{1}}_{|\\Pi _{n}^{(1)}(nt)|>k})\\le A_{t,\\alpha _1}k^{3-\\alpha _1}.$ Let us now establish a lower bound for $\\operatorname{\\mathbb {E}}(Z_k(nt))=n\\operatorname{\\mathbb {P}}(\\Pi _{n}^{(1)}(nt)>k)$ .",
"By Theorem REF , there exists $C(t)>0$ such that $\\operatorname{\\mathbb {E}}(Z_k(nt))\\ge n(\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)>k)-C(t)\\frac{k^2}{n})\\; \\text{for every }k,n\\in \\operatorname{\\mathbb {N}}^*.$ To obtain a lower bound for $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t)>k)$ , we shall apply several results on regularly varying distributions.",
"Let us first introduce a notation: for a nonnegative random variable $X$ with probability distribution $\\nu $ , let $\\bar{F}_{X}(t)$ or $\\bar{F}_{\\nu }$ denote its tail distribution: $\\bar{F}_{\\nu }(t)=P(X>t)$ $\\forall t\\in \\operatorname{\\mathbb {R}}$ .",
"The following Lemma is an application of a more general result on the solution of a fixed-point problem proven in [39]: Lemma 7.2 Let $\\nu $ be a probability distribution on $\\operatorname{\\mathbb {N}}$ such that its expectation $m$ is smaller than 1.",
"Let $T$ be the total population size of a BGW process with offspring distribution $\\nu $ and one ancestor.",
"If $\\nu $ is a regular varying distribution then $T$ has also a regular varying distribution and $\\bar{F}_T(x)\\underset{x\\rightarrow +\\infty }{\\sim }\\frac{1}{1-m}\\bar{F}_{\\nu }((x-1)(1-m))$ .",
"To apply this Lemma when the offspring distribution is a compound Poisson distribution, we can use the following result proven in ([11], Theorem 3): Lemma 7.3 Let $\\nu $ be a regularly varying distribution on $\\operatorname{\\mathbb {R}}_+$ and $\\lambda >0$ .",
"Then, $\\text{CPois}(\\lambda ,\\nu )$ is a regularly varying distribution on $\\operatorname{\\mathbb {R}}_+$ with the same index as $\\nu $ and $\\bar{F}_{\\text{CPois}(\\lambda ,\\nu )}(x)\\underset{x\\rightarrow +\\infty }{\\sim } \\lambda \\bar{F}_{\\nu }(x)$ .",
"As $T^{(1)}_{p}(t)$ is the total population of a BGWprocess with $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -offspring distribution, it remains to show that $\\bar{F}_{\\tilde{p}}$ is a regularly varying function with index $-\\alpha +1$ .",
"Let us note that for $k\\in \\operatorname{\\mathbb {N}}$ , $m^{*}_{p,1}\\bar{F}_{\\tilde{p}}(k)=(k +1)\\bar{F}_{p}(k+1)+\\int _{[k+1,+\\infty [}\\bar{F}_p(u)du.$ The following result known as `Karamata Theorem for distributions' yields an asymptotic result for the last term in (REF ): Lemma (see Theorem 2.45 in [15] for instance).",
"Let $F$ be a cumulative distribution function on $\\operatorname{\\mathbb {R}}_+$ .",
"If $F$ is a regularly varying function with index $-\\alpha <-1$ then the integrated tail distribution $F_I:x \\mapsto \\int _{x}^{+\\infty }(1-F(u))du$ is a regularly varying function with index $-\\alpha +1$ and $F_I(x)\\underset{x\\rightarrow +\\infty }{\\sim } (\\alpha -1)^{-1}x(1-F(x))$ .",
"By this lemma we obtain $m^{*}_{p,1}\\bar{F}_{\\tilde{p}}(x)\\underset{x\\rightarrow +\\infty }{\\sim }\\frac{\\alpha }{\\alpha -1} \\frac{\\ell (\\lfloor x \\rfloor +1)}{(\\lfloor x \\rfloor +1)^{\\alpha -1}}.$ We deduce from Lemma REF and Lemma REF that $\\bar{F}_{T^{(1)}_{p}(t)}(x)\\underset{x\\rightarrow +\\infty }{\\sim }\\frac{t\\alpha }{(1-tm_{p,2})^{\\alpha }(\\alpha -1)}\\frac{\\ell \\left(\\lfloor x \\rfloor (1-tm_{p,2})\\right)}{\\lfloor x \\rfloor ^{\\alpha -1}}.$ In summary, we have shown that there exists a slowly varying function $\\tilde{\\ell }$ such that for every ${k,n\\in \\operatorname{\\mathbb {N}}^*}$ , $\\operatorname{\\mathbb {E}}(Z_k(nt))\\ge n k^{-\\alpha +1}(A^{(2)}_{\\alpha ,t}\\tilde{\\ell }(k)-C(t)\\frac{k^{1+\\alpha }}{n})$ where $A^{(2)}_{\\alpha ,t}=\\frac{t\\alpha }{\\alpha -1}(1-tm_{p,2})^{-\\alpha }$ and $C(t)$ is the constant defined in Theorem REF .",
"Set $k_n=n^{\\frac{1}{1+\\alpha ^{\\prime }}}$ with $\\alpha ^{\\prime }>\\alpha $ .",
"For $n$ large enough, the lower bound (REF ) for $\\operatorname{\\mathbb {E}}(Z_{k_n}(nt))$ is positive.",
"Using (REF ) and the upper bound (REF ) for $\\operatorname{Var}(Z_{k_n}(nt))$ , we obtain that for every $3<\\alpha _1<\\alpha <\\alpha ^{\\prime }$ and $n$ large enough, $\\operatorname{\\mathbb {P}}(\\max _{x\\in \\llbracket {n} \\rrbracket }|\\Pi _{n}^{(x)}(nt)|\\le n^{\\frac{1}{1+\\alpha ^{\\prime }}})\\le n^{2\\alpha -\\alpha _1-\\alpha ^{\\prime }}\\frac{A^{(1)}_{t,\\alpha _1}}{(A^{(2)}_{\\alpha ,t}\\tilde{\\ell }(n^{\\frac{1}{1+\\alpha ^{\\prime }}})-C(t)n^{\\frac{\\alpha -\\alpha ^{\\prime }}{1+\\alpha }})^2}$ If we take $\\alpha _1\\in ]\\max (3,\\alpha -(\\alpha ^{\\prime }-\\alpha )), \\alpha [$ , the upper bound converges to 0 as $n$ tends to $+\\infty $ .",
"This ends the proof of Theorem REF .",
"It remains to show the upper bound for $\\operatorname{Var}(Z_k(nt))$ given by (REF ).",
"We expand the value of $\\operatorname{Var}(Z_k(nt))$ by using that $Z_k(nt)$ is a sum of $n$ indicator functions and by splitting $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|> k\\;\\text{and}\\;|\\Pi _{n}^{(y)}(nt)|> k)$ into two terms depending on whether $x$ and $y$ belong to a same block or not: $\\operatorname{Var}(Z_k(nt))=S^{(1)}_{n}(k)+S^{(2)}_{n}(k)$ , where $S^{(1)}_{n}(k)=\\sum _{x,y\\in \\llbracket {n} \\rrbracket }&\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)|> k\\;\\text{and}\\; y\\in \\Pi _{n}^{(x)}(nt)\\big )\\\\S^{(2)}_{n}(k)=\\sum _{x,y\\in \\llbracket {n} \\rrbracket }&\\Big (\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)|> k,\\ |\\Pi _{n}^{(y)}(nt)|> k \\operatorname{ and }y\\notin \\Pi _{n}^{(x)}(nt)\\big )\\\\&-\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)|> k\\big )\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(y)}(nt)|> k\\big )\\Big ).$ First, $S^{(1)}_{n}(k)=n\\operatorname{\\mathbb {E}}(|\\Pi _{n}^{(1)}(nt)|\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(1)}(nt)|>k\\rbrace }).$ We consider now the following term in $S^{(2)}_{n}(k)$ : $\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)|> k,\\ |\\Pi _{n}^{(y)}(nt)|> k \\operatorname{ and }y\\notin \\Pi _{n}^{(x)}(nt)\\big )\\\\ =\\sum _{h=k+1}^{n-k}\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(y)}(nt)|> k\\mid |\\Pi _{n}^{(x)}(nt)|=h \\operatorname{ and }y\\notin \\Pi _{n}^{(x)}(nt)\\big )\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)|=h \\operatorname{ and }y\\notin \\Pi _{n}^{(x)}(nt)\\big ).$ Let $\\Pi _{n,h}(nt)$ denote the partition generated by tuples the elements of which are in $\\llbracket {n-h} \\rrbracket $ at time $nt$ and let $\\Pi _{n,h}^{(1)}(nt)$ denote the block of $\\Pi _{n,h}(nt)$ that contains 1.",
"By the properties of the Poisson tuple set, for $h\\in \\lbrace k+1,\\ldots ,n\\rbrace $ $\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(y)}(nt)|> k \\mid y\\notin \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|= h\\big )=\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n,h}^{(1)}(nt)|> k \\big )\\le \\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(1)}(nt)|> k \\big ).$ Thus $S^{(2)}_{n}(k)\\le 0$ which ends the proof of (REF )."
],
[
"The supercritical regime",
"When $t>\\frac{1}{m_{p,2}}$ , BGW processes with family size distribution $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ are supercritical.",
"We show that there is a constant $c>0$ such that with high probability there is only one block with more than $c\\log (n)$ elements and the size of this block is of order $n$ .",
"Let us recall the precise statement: Theorem (REF .",
"(ii)) Let $B_{n,1}(nt)$ and $B_{n,2}(nt)$ denote the first and second largest blocks of $\\Pi _{n}(nt)$ .",
"Assume that $p$ has a finite moment of order three, $p(1)<1$ and $t>\\frac{1}{m_{p,2}}$ .",
"Let $q_{t}$ denote the extinction probability of the BGW$(1,tm^{*}_{p,1},\\tilde{p})$ process.",
"For every $a\\in ]1/2,1[$ , there exist $b>0$ and $c>0$ such that $\\operatorname{\\mathbb {P}}[||B_{n,1}(nt)|-(1-q_{t})n|\\ge n^a]+\\operatorname{\\mathbb {P}}[|B_{n,2}(nt)|\\ge c\\log (n)]=O(n^{-b}).$ In this section, we always assume the following hypothesis: $(\\text{Hyp}_{p,t})$ : $p$ has a finite moment of order three, $p(1)<1$ and $t>\\frac{1}{m_{p,2}}$ .",
"Let $h(t)$ be the Cramér function of the $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distribution at point 1: ${h(t)=\\sup _{\\theta \\le 0}(\\theta -\\log \\operatorname{\\mathbb {E}}(e^{\\theta X}))}$ , if $X$ denotes a $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distributed random variable.",
"The proof of Theorem REF .",
"(ii) consists of four steps: In the first step, we show that the block of an element has a size greater than $c\\log (n)$ with a probability equivalent to the BGW process survival probability $1-q_{t}$ .",
"Proposition 7.4 Under assumption $(\\text{Hyp}_{p,t})$ , $h(t)>0 \\operatorname{ and }\\forall a>h(t)^{-1},\\; \\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|\\ge a\\log (n))=1-q_{t}+O(\\frac{\\log ^2(n)}{n}).$ For $k\\in \\operatorname{\\mathbb {N}}$ , let $Z_k(nt)$ denote the number of elements that belong to a block of size greater than $k$ at time $nt$ .",
"In the second step, we study the first two moments of $Z_k(nt)$ in order to prove: Proposition 7.5 Under assumption $(\\text{Hyp}_{p,t})$ , for every $b\\in ]1/2;1[$ , there exists $\\delta >0$ such that if $a>h(t)^{-1}$ then $\\operatorname{\\mathbb {P}}(|Z_{a\\log (n)}(nt)-n(1-q_{t})| > n^{b})=O(n^{-\\delta })$ .",
"The aim of the third step is to prove that with high probability, there is no block of size between $c_1\\log (n)$ and $c_2n^\\beta $ for any constant $\\beta \\in ]1/2,1[$ .",
"More precisely, we show the following result on the set of active elements in step $k$ , denoted $A_{k}(x)$ : Proposition 7.6 Let $\\beta \\in ]1/2,1[$ .",
"Assume that $(\\text{Hyp}_{p,t})$ holds.",
"For every ${0<c_2<\\min (1,tm_{p,2}-1)}$ , there exists $\\delta (c_2)>0$ such that for ${c_1>\\delta ^{-1}(c_2)}$ , $\\operatorname{\\mathbb {P}}\\big (\\exists x\\in \\llbracket {n} \\rrbracket ,\\ A_{c_1\\log (n)}(x)\\ne \\emptyset \\text{ and }\\exists k\\in [c_1\\log (n),n^{\\beta }],\\ |A_k(x)|\\le c_2k\\big )=O(n^{1-c_1\\delta (c_2)}).$ In the fourth step, we deduce from Proposition REF that with high probability there exists at most one block of size greater than $a\\log (n)$ : Proposition 7.7 Assume that $(\\text{Hyp}_{p,t})$ holds.",
"For every $0<c_2<\\min (1,tm_{p,2}-1)$ , there exists $\\delta (c_2)>0$ such that for $c_1>\\delta ^{-1}(c_2)$ , $\\operatorname{\\mathbb {P}}\\big (\\text{there exist two distinct blocks of size greater than}\\;c_1\\log (n)\\big )=O(n^{1-c_1\\delta (c_2)}).$ Assertion (ii) of Theorem REF is then a direct consequence of Proposition REF and Proposition REF , since $Z_{c_1\\log (n)}(nt)$ is equal to the size of the largest block on the event: $\\lbrace |Z_{c_1\\log (n)}(nt)-n(1-q_{t})|\\le n^{b}\\rbrace \\cap \\lbrace \\text{there is at most one block of size greater than}\\;c_1\\log (n)\\rbrace .$ The first two steps of the proof of assertion (ii) of Theorem REF are similar to the first two steps detailed in [42] for the Erdös-Rényi random graph.",
"The last two steps follow the proof described in [6] for the Erdös-Rényi random graph.",
"[Proof of Proposition REF ] Let $x\\in \\llbracket {n} \\rrbracket $ .",
"By Theorem REF , for every $c>0$ , $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|> c\\log (n))=\\operatorname{\\mathbb {P}}(T^{(1)}_{p}> c\\log (n))+O(\\frac{\\log ^2(n)}{n}).$ Moreover, $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}=+\\infty )=1-q_{t}$ .",
"To complete the proof, we use the following result on the total progeny of a supercritical BGW process stated in [42]: Theorem (3.8 in [42]) Let $T$ denote the total progeny of a BGW process with offspring distribution $\\nu $ .",
"Assume that $\\sum _{k\\in \\operatorname{\\mathbb {N}}}k\\nu (k)>1$ .",
"Then, $I=\\sup _{\\theta \\le 0}\\left(\\theta -\\log \\left(\\sum _{k=0}^{+\\infty } e^{\\theta x}\\nu (k)\\right)\\right)>0\\;\\operatorname{ and }\\;\\operatorname{\\mathbb {P}}(k\\le T<+\\infty )\\le \\frac{e^{-kI}}{1-e^{-I}}.$ This theorem shows that for every $c>h(t)^{-1}$ , $\\operatorname{\\mathbb {P}}(c\\log (n)< T^{(1)}_{p}<+\\infty )=O(n^{-1})$ and $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|> c\\log (n))=1-q_{t}+O(\\frac{\\log ^2(n)}{n}).$ [Proof of Proposition REF ] In order to apply Bienaymé-Chebyshev inequality to obtain an upper bound for $\\operatorname{\\mathbb {P}}(|Z_{a\\log (n)}(nt)-n(1-q_{t})| > n^{b})$ , we compute the expectation and the variance of $Z_k(nt)=\\sum _{x\\in \\llbracket {n} \\rrbracket }\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(x)}(nt)|> k\\rbrace }$ .",
"First, we deduce from Proposition REF that if $a>h(t)^{-1}$ then $\\operatorname{\\mathbb {E}}(Z_{a\\log (n)}(nt))=n(1-q_{t})+O(\\log ^2(n)).$ We proceed as in ([42], Proposition 4.10) to prove the following upper bound for the variance of $Z_k(nt)$ : $\\operatorname{Var}(Z_k(nt))\\le n(1+ktm_{p,2})\\operatorname{\\mathbb {E}}(|\\Pi _{n}^{(1)}(nt)|\\operatorname{{1}}_{|\\Pi _{n}^{(1)}(nt)|\\le k})$ The beginning of the calculation is similar to the one used to prove inequality (REF ): the variance of $Z_k(nt)$ which is equal to the variance of $\\sum _{x\\in \\llbracket {n} \\rrbracket }\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(x)}(nt)|\\le k\\rbrace }$ can be written as the sum of the following two terms: $\\tilde{S}^{(1)}_{n}(k)=\\sum _{x,y\\in \\llbracket {n} \\rrbracket }&\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)| \\le k\\;\\text{and}\\; y\\in \\Pi _{n}^{(x)}(nt)\\big )=n\\operatorname{\\mathbb {E}}(|\\Pi _{n}^{(1)}(nt)|\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(1)}(nt)|\\le k\\rbrace })\\\\\\tilde{S}^{(2)}_{n}(k)=\\sum _{x,y\\in \\llbracket {n} \\rrbracket }&\\Big (\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)| \\le k,\\ |\\Pi _{n}^{(y)}(nt)|\\le k \\operatorname{ and }y\\notin \\Pi _{n}^{(x)}(nt)\\big )\\\\&-\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)| \\le k\\big )\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(y)}(nt)|\\le k\\big )\\Big ).$ We consider the following term in $\\tilde{S}^{(2)}_{n}(k)$ : $\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)|\\le k,& |\\Pi _{n}^{(y)}(nt)|\\le k \\operatorname{ and }y\\notin \\Pi _{n}^{(x)}(nt)\\big )\\\\ &=\\sum _{h=1}^{k}\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)|=h,\\ |\\Pi _{n}^{(y)}(nt)| \\le k \\operatorname{ and }y\\notin \\Pi _{n}^{(x)}(nt)\\big ).\\\\ &\\le \\sum _{h=1}^{k} \\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(x)}(nt)|= h\\big )\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(y)}(nt)|\\le k \\mid y\\notin \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|= h\\big ).$ By the properties of the Poisson tuple set, $\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n}^{(y)}(nt)|\\le k \\mid y\\notin \\Pi _{n}^{(x)}(nt) \\operatorname{ and }|\\Pi _{n}^{(x)}(nt)|= h\\big )=\\operatorname{\\mathbb {P}}\\big (|\\Pi _{n,h}^{(1)}(nt)|\\le k \\big )$ where $\\Pi _{n,h}(nt)$ denotes the partition generated by tuples the elements of which are in $\\llbracket {n-h} \\rrbracket $ at time $nt$ and $\\Pi _{n,h}^{(1)}(nt)$ denotes the block of $\\Pi _{n,h}(nt)$ that contains 1.",
"We can couple $\\mathcal {P}_n(nt,\\llbracket {n-h} \\rrbracket )$ and $\\mathcal {P}_n(nt)$ by adding to $\\mathcal {P}_n(nt,\\llbracket {n-h} \\rrbracket )$ tuples of an independent Poisson point process on $\\operatorname{\\mathbb {R}}^{+}\\otimes \\mathcal {W}(\\llbracket {n} \\rrbracket )$ at time $nt$ that are not included in $\\llbracket {n-h} \\rrbracket $ .",
"Therefore, $\\operatorname{\\mathbb {P}}(|\\Pi _{n,h}^{(1)}(nt)| \\le k )-\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)| \\le k )$ is equal to the probability that $|\\Pi _{n,h}^{(1)}(nt)|$ is smaller than or equal to $k$ and that $|\\Pi _{n}^{(1)}(nt)|$ is greater than $k$ .",
"This probability is bounded above by the probability that there exists $w\\in \\mathcal {P}_{n}(nt)$ that contains both elements of $\\lbrace 1,\\ldots ,k\\rbrace $ and elements of $\\lbrace n-h+1,\\ldots ,n\\rbrace $ .",
"Therefore, $\\operatorname{\\mathbb {P}}(|\\Pi _{n,h}^{(1)}(nt)|\\le k )-\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt)|\\le k )\\\\\\le 1-e^{-nt I_{n}(k,h)}$ with $I_{n}(k,h)&=&\\mu _n\\big (w\\in \\mathcal {W}(\\llbracket {n} \\rrbracket ),\\ w\\cap \\llbracket {k} \\rrbracket \\ne \\emptyset \\operatorname{ and }w\\cap \\lbrace k+1,\\ldots ,k+h\\rbrace \\ne \\emptyset \\big )\\\\&=&\\mu _n(\\mathcal {W}(\\llbracket {n} \\rrbracket ))-\\mu _n(\\mathcal {W}(\\lbrace k+1,\\ldots ,n\\rbrace ))\\\\&&-\\mu _n(\\mathcal {W}(\\llbracket {n} \\rrbracket \\setminus \\lbrace k+1,\\ldots ,k+h\\rbrace ))+\\mu _n(\\mathcal {W}(\\lbrace k+1+h,\\ldots ,n\\rbrace ))\\\\&=&1-G_{p}(1-\\frac{k}{n})-G_{p}(1-\\frac{h}{n})+G_{p}(1-\\frac{k+h}{n})\\\\&\\le & \\frac{k h}{n^2}m_{p,2}.$ We deduce that $\\tilde{S}^{(2)}_{n}(k)\\le \\sum _{x,y\\in \\llbracket {n} \\rrbracket }\\Big (\\sum _{h=1}^{k}\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(x)}(nt)|=h)\\frac{tkh}{n}m_{p,2}\\Big )= ntkm_{p,2}\\operatorname{\\mathbb {E}}(|\\Pi _{n}^{(1)}(nt)|\\operatorname{{1}}_{\\lbrace |\\Pi _{n}^{(1)}(nt)|\\le k\\rbrace })$ which yields (REF ).",
"Let us note that for every $\\delta >0$ , $\\dfrac{\\operatorname{Var}(Z_{a\\log (n)}(nt))}{n^{1+\\delta }}$ converges to 0 as $n$ tends to $+\\infty $ .",
"Therefore, Bienaymé-Chebyshev inequality is sufficient to complete the proof.",
"[Proof of Proposition REF ] Let $\\alpha \\in ]1/2,1[$ .",
"The idea of the proof is to lower bound the number of new active elements at the first steps of the block exploration procedure by considering only tuples inside a subset of $m_n=n-\\lceil 2n^\\alpha \\rceil $ elements.",
"For large $n$ , the BGW process associated with this block exploration procedure is still supercritical.",
"Let $\\tau =T^{(n)}_t\\wedge \\min (k\\in \\operatorname{\\mathbb {N}}^*,\\ \\sum _{i=1}^{k}\\xi _{n,i}(t)\\ge 2n^{\\alpha })$ .",
"On the event $\\lbrace k\\le \\tau \\rbrace $ , the number of neutral elements at step $k$ is greater than $m_n$ .",
"Let $U_k$ denote the set of the $m_n$ first neutral elements at step $k$ and let $Y_{n,k+1}(t)$ denote the number of $y\\in U_k$ which are contained in a tuple $w\\in \\mathcal {P}_{n,x_k}(t,U_k\\cup \\lbrace x_k\\rbrace )$ .",
"On the event $\\lbrace k\\le \\tau \\rbrace $ , $Y^{(n)}_{t,k+1}\\le \\xi _{n,k+1}(t)$ .",
"Therefore, $\\sum _{i=1}^{k\\wedge \\tau }Y_{n,i}(t)\\le \\sum _{i=1}^{k\\wedge \\tau }\\xi _{n,i}(t)$ .",
"For $x\\in \\llbracket {n} \\rrbracket $ , set $\\Omega ^{(n)}_{c_1,c_2}(x)=\\lbrace A_{c_1\\log (n)}(x)\\ne \\emptyset \\text{ and }\\exists k\\in [c_1\\log (n),n^{\\alpha }],\\ |A_k(x)|\\le c_2k\\rbrace .",
"$ On the event $\\lbrace k\\le \\tau \\operatorname{ and }|A_k(x)|\\le c_2k\\rbrace $ , $\\sum _{i=1}^{k}Y_{n,i}(t)$ is bounded above by $(c_2+1)k-1$ .",
"Thus, $\\operatorname{\\mathbb {P}}(\\Omega ^{(n)}_{c_1,c_2}(x))&\\le &\\sum _{k=c_1\\log (n)}^{n^{\\alpha }}\\operatorname{\\mathbb {E}}\\left(\\operatorname{\\mathbb {P}}( A_{c_1\\log (n)}(x)\\ne \\emptyset \\text{ and }|A_k(x)|\\le c_2k \\mid \\mathcal {F}_{k-1})\\operatorname{{1}}_{\\lbrace k\\le \\tau \\rbrace }\\right)\\\\&\\le & \\sum _{k=c_1\\log (n)}^{n^{\\alpha }}\\operatorname{\\mathbb {P}}\\left(\\sum _{i=1}^{k}\\tilde{Y}_{n,i}(t)\\le (c_2+1)k-1\\right).$ where $(\\tilde{Y}_{n,i}(t))_i$ denotes a sequence of independent random variables distributed as $|\\mathcal {N}_{1}(t,\\llbracket {m_n+1} \\rrbracket )|$ .",
"The last step consists in establishing an exponential bound for $p_{n,k}:=\\operatorname{\\mathbb {P}}\\Big (\\sum _{i=1}^{k}\\tilde{Y}_{n,i}(t)\\le (c_2+1)k-1\\Big )$ uniformly on $n$ .",
"A such exponential bound is an easy consequence of the following two facts: (i) $c_2+1$ is smaller than the expectation of the $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distribution.",
"(ii) $(\\tilde{Y}_{n,1}(t))_n$ converges in law to the $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distribution by Proposition REF .",
"For every $\\theta >0$ , $p_{n,k}\\le \\exp (k\\Lambda _n(-\\theta ))$ where $\\Lambda _n(\\theta )=\\log \\Big (\\operatorname{\\mathbb {E}}(e^{\\theta (\\tilde{Y}_{n,1}(t)-(c_2+1))})\\Big )$ .",
"Let $Y$ be a $\\text{CPois}(tm^{*}_{p,1},\\tilde{p})$ -distributed random variable.",
"Set $\\Lambda (\\theta )=\\log \\big (\\operatorname{\\mathbb {E}}(e^{\\theta ( Y-(c_2+1))})\\big )$ for $\\theta \\le 0$ .",
"Since $\\operatorname{\\mathbb {E}}(Y)=tm_{p,2}$ is finite, $\\Lambda ^{\\prime }(-\\theta )$ converges to $-\\operatorname{\\mathbb {E}}(Y)+c_2+1$ which is negative as $\\theta $ converges to 0.",
"Therefore, there exists $u^*<0$ such that $\\Lambda (u^*)<0$ .",
"Set $\\delta =-\\frac{1}{2}\\Lambda (u^*)$ .",
"By assertion (ii), $\\Lambda _n(u^*)$ converges to $\\Lambda (u^*)$ , hence there exists $n^*$ such that for every $n\\ge n^*$ and $k\\in \\operatorname{\\mathbb {N}}^*$ , $p_{n,k}\\le \\exp (-k\\delta )$ .",
"We deduce that for $n\\ge n^*$ , $\\operatorname{\\mathbb {P}}\\Big (\\underset{x\\in \\llbracket {n} \\rrbracket }{\\cup }\\Omega ^{(n)}_{c_1,c_2}(x)\\Big )\\le n\\operatorname{\\mathbb {P}}(\\Omega ^{(n)}_{c_1,c_2}(1))\\le n^{1-c_1\\delta }(1-e^{-\\delta })^{-1}$ which converges to 0 if $c_1>\\delta ^{-1}$ .",
"[Proof of Proposition REF ] For $0<c_1<1$ and $c_2>0$ , let $\\Omega ^{(n)}_{c_1,c_2}$ denote the event $\\lbrace \\exists x\\in \\llbracket {n} \\rrbracket \\;\\text{such that}\\;A_{c_1\\log (n)}(x)\\ne \\emptyset \\operatorname{ and }\\exists k\\in [c_1\\log (n), n^{\\alpha }]\\;\\text{such that}\\;|A_k(x)|\\le c_2k\\rbrace .$ It occurs with probability $O(n^{1-c_1\\delta (c_2)})$ by Proposition REF .",
"Assume that $\\Omega ^{(n)}_{c_1,c_2}$ does not hold and that there exist two elements $x_1$ and $x_2$ in $\\llbracket {n} \\rrbracket $ contained in two different blocks both of size greater than $c_1\\log (n)$ .",
"The subsets of active elements in step $n^{\\alpha }$ , $A_{n^\\alpha }(x_1)$ and $A_{n^\\alpha }(x_2)$ , are disjoint and both of size greater than $c_2n^\\alpha $ .",
"It means that no tuple $w\\in \\mathcal {P}_{n}(nt)$ contains both elements of $A_{n^\\alpha }(x_1)$ and $A_{n^\\alpha }(x_2)$ .",
"Let us note that if $F_1$ and $F_2$ are two disjoint subsets of $\\llbracket {n} \\rrbracket $ then $\\operatorname{\\mathbb {P}}(\\nexists w \\in \\mathcal {P}_{n}(nt),\\;& w \\cap F_1\\ne \\emptyset \\operatorname{ and }w \\cap F_2\\ne \\emptyset )\\\\&=\\exp \\Big (-nt\\mu (w \\in \\mathcal {W}(\\llbracket {n} \\rrbracket ),\\; w \\cap F_1\\ne \\emptyset \\operatorname{ and }w \\cap F_2\\ne \\emptyset )\\Big )\\\\&=\\exp \\Big (-nt\\big (1-G_{p}(1-\\frac{|F_1|}{n})-G_{p}(1-\\frac{|F_2|}{n})+G_{p}(1-\\frac{|F_1|+|F_2|}{n})\\big )\\Big ).$ Therefore if $F_1$ and $F_2$ are two disjoint subsets of $\\llbracket {n} \\rrbracket $ of size greater than $c_2n^\\alpha $ with $n$ large enough, $\\operatorname{\\mathbb {P}}(\\nexists w \\in \\mathcal {P}_{n}(nt),\\; w \\cap F_1\\ne \\emptyset \\operatorname{ and }w \\cap F_2\\ne \\emptyset ) \\\\\\le \\exp \\Big (-nt\\big (1-2G_{p}(1-c_2n^{\\alpha -1})+G_{p}(1-2c_2n^{\\alpha -1})\\big )\\Big )\\\\\\le \\exp (-\\frac{1}{2}c^2_2tm_{p,2}n^{2\\alpha -1}),$ since $x\\mapsto 1-G_{p}(1-x)-G_{p}(1-x-y)$ is an increasing function on $]0,1-y[$ for every $y\\in ]0,1[$ and for $x>0$ small enough, $1-2G_{p}(1-x)+G_{p}(1-2x)\\ge \\frac{x^2}{2}m_{p,2}$ .",
"Set $J_{n,\\alpha }=\\lbrace (x_1,x_2)\\in \\llbracket {n} \\rrbracket ^2,\\ A_{n^{\\alpha }}(x_1)\\cap A_{n^{\\alpha }}(x_2)= \\emptyset ,|A_{n^{\\alpha }}(x_1)| > c_2 n^{\\alpha },\\ |A_{n^{\\alpha }}(x_2)| > c_2 n^{\\alpha }\\rbrace $ .",
"It follows from the last inequality that there exists two different blocks of size greater than $c_1\\log (n)$ with a probability smaller than the sum of $\\operatorname{\\mathbb {P}}(\\Omega ^{(n)}_{c_1,c_2})$ and $\\operatorname{\\mathbb {E}}\\left(\\sum _{(x_1,x_2)\\in J_{n,\\alpha }}\\operatorname{\\mathbb {P}}(\\nexists w \\in \\mathcal {P}_{n}(nt),\\; w \\cap A_{n^{\\alpha }}(x_1)\\ne \\emptyset \\operatorname{ and }w \\cap A_{n^{\\alpha }}(x_2)\\ne \\emptyset \\mid \\mathcal {F}_{n^\\alpha })\\right)\\\\\\le n^2\\exp \\left(-\\frac{1}{2}tc_{2}^{2}m_{p,2}n^{2\\alpha -1}\\right).$ As $\\alpha \\in ]\\frac{1}{2},1[$ , this probability is of order $O(n^{1-c_1\\delta (c_2)})$ ."
],
[
"The critical regime",
"Let us now study block sizes at time $t_n=\\frac{1}{m_{p,2}}(1+\\theta \\varepsilon _n)$ where $\\theta >0$ and $(\\varepsilon _n)_n$ is a sequence of positive reals that converges to 0.",
"The aim of this section is to prove the third statement of Theorem REF : Theorem (REF .",
"(iii)) Assume that $p$ is a probability measure on $\\operatorname{\\mathbb {N}}^*$ with $p(1)<1$ and a finite third moment.",
"For every $\\theta >0$ , there exists a constant $b>0$ such that for every $c>1$ and $n\\in \\operatorname{\\mathbb {N}}^*$ $\\operatorname{\\mathbb {P}}\\left(\\max _{x\\in \\llbracket {n} \\rrbracket }\\Big \\vert \\Pi _{n}^{(x)}\\left(\\frac{n}{m_{p,2}}(1+\\theta n^{-1/3})\\right)\\Big \\vert >cn^{2/3}\\right)\\le \\frac{c}{b}.\\qquad \\mathrm {(\\ref {critupperbound})}$ Let us recall that the size of a block is smaller than the total population size of a BGW$(1,n\\beta _n,\\nu _n)$ process, which is itself close to the total population size of a BGW$(1,t_nm^{*}_{p,1},\\tilde{p})$ process.",
"Therefore the strategy of proof used to establish the same result for the Erdös-Rényi random graph in ([42], Theorem 5.1) can be followed if we are able to show the following two properties for the total population size $T^{(1)}_p(t_n)$ of the BGW$(1,t_nm^{*}_{p,1},\\tilde{p})$ process: the survival probability of a BGW$(1,t_nm^{*}_{p,1},\\tilde{p})$ process is of order $O(\\varepsilon _n)$ .",
"There exists a constant $c>0$ such that $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)=k)\\le ck^{-3/2}$ for every $n\\in \\operatorname{\\mathbb {N}}$ and $k\\in \\operatorname{\\mathbb {N}}^*.$ Let us now detail the proof of (REF ).",
"As in the study of the subcritical phase, we reduce the proof to the study of $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt_n)|\\ge k)$ , by noting that for every $k\\in \\llbracket {n} \\rrbracket $ , $\\operatorname{\\mathbb {P}}(\\max _{x\\in \\llbracket {n} \\rrbracket }|\\Pi _{n}^{(x)}(nt_n)|\\ge k)\\le \\frac{n}{k}\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt_n)|\\ge k)$ (see (REF )).",
"Since $|\\Pi _{n}^{(1)}(nt_n)|$ is smaller than the total population size of a BGW$(1,nt_n\\beta _n,\\nu _n)$ process, by Proposition REF , for every $k\\ge 1$ , $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt_n)|\\ge k)\\le \\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)\\ge k)+\\frac{t_n}{2n}(2m_{p,2}+1)\\sum _{i=1}^{k-1}\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)\\ge i)$ and $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)\\ge k)=\\sum _{i=k}^{\\infty }\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)= k)+1-q_{t_n}$ , where $q_{t_n}$ is the extinction probability of the BGW$(1,t_nm^{*}_{p,1},\\tilde{p})$ process.",
"To estimate the survival probability $1-q_{t_n}$ , we use the following inequalities: Lemma 7.8 Let $\\nu $ be a probability measure on $\\operatorname{\\mathbb {N}}$ .",
"Assume that $\\nu $ has a finite second moment, $\\nu (0)+\\nu (1)<1$ and the first moment $m_{\\nu ,1}$ is greater than 1.",
"Let $m_{\\nu ,2}$ denote the second factorial moment of $\\nu $ .",
"The survival probability $\\alpha $ of a BGW process with offspring distribution $\\nu $ and one ancestor satisfies: $2\\frac{m_{\\nu ,1}-1}{m_{\\nu ,2}}\\le \\alpha \\le \\frac{m_{\\nu ,1}-1}{m_{\\nu ,1}-1+\\nu (0)}.$ The lower bound was proved by Quine in [36].",
"A simple proof of this lemma is given in Appendix REF .",
"By Lemma REF , $\\frac{2\\theta \\varepsilon _n}{(t_nm_{p,2})^2+t_nm_{p,3}}\\le 1-q_{t_n} \\le \\frac{\\theta \\varepsilon _n}{\\theta \\varepsilon _n+\\exp (-t_nm^{*}_{p,1})}.$ To estimate $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)= k)$ , we first rewrite it by the mean of Dwass identity: $\\forall k\\in \\operatorname{\\mathbb {N}}^*,\\; \\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)= k)=\\frac{1}{k}\\operatorname{\\mathbb {P}}(\\sum _{i=1}^{k}\\xi _{i}= k-1),$ where $(\\xi _{i})_i$ denotes a sequence of independent random variables with $\\text{CPois}(t_nm^{*}_{p,1},\\tilde{p})$ -distribution (the statement of Dwass's theorem is recalled in Appendix REF ).",
"The local central limit theorem applied to the sequence $(\\xi _{i})_i$ yields $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)=k)=O(k^{2/3})$ at a fixed time $t_n$ .",
"But we need a bound uniform in $n$ .",
"A careful study of the local central limit theorem proof shows that the convergence is uniform if it is applied to well-chosen families of probability distributions.",
"In particular, in our setting: Lemma 7.9 Let $\\nu $ be a probability distribution on $\\operatorname{\\mathbb {N}}$ with a finite second moment.",
"Let $m_{\\nu ,1}$ and $m_{\\nu ,2}$ denote the first two factorial moments of $\\nu $ .",
"For $\\lambda >0$ , let $(X_{\\lambda ,n})_n$ be a sequence of independent $\\text{CPois}(\\lambda ,\\nu )$ -distributed random variables.",
"Let $r$ be the largest positive integer such that the support of the $\\text{CPois}(1,\\nu )$ -distribution is included in $r\\operatorname{\\mathbb {N}}$ .",
"For every $0<a<b$ , $\\sup _{\\lambda \\in [a,b],k\\in \\operatorname{\\mathbb {N}}}\\!\\!\\!\\sqrt{n}\\left|\\operatorname{\\mathbb {P}}(\\sum _{i=1}^{n}X_{\\lambda ,i}=kr)-\\frac{r}{\\sqrt{2\\pi n\\lambda (m_{\\nu ,2}+m_{\\nu ,1})}}\\exp \\left(-\\frac{(kr-n\\lambda m_{\\nu ,1})^2}{2n\\lambda (m_{\\nu ,2}+m_{\\nu ,1})}\\right)\\right|\\underset{n\\rightarrow +\\infty }{\\longrightarrow }0.$ (See Appendix REF for a proof of Lemma REF ) Lemma REF implies that there exists a sequence $(\\delta _k)_k$ (depending on $\\theta $ ) that converges to 0 such that for every $n\\in \\operatorname{\\mathbb {N}}^*$ and $k\\in \\operatorname{\\mathbb {N}}^*$ , $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)=k)\\le \\frac{c+\\delta _k}{k^{3/2}}$ , where ${c=\\frac{r}{\\sqrt{2\\pi a (m_{p,2}+m_{p,3})}}}$ and $r$ is the largest integer such that the support of $\\text{CPois}(1,\\tilde{p})$ is included in $r\\operatorname{\\mathbb {N}}$ .",
"From this and the upper bound of (REF ), we deduce that there exists $c_1(\\theta )>0$ such that for every $n,k\\in \\operatorname{\\mathbb {N}}^*$ , $\\operatorname{\\mathbb {P}}(T^{(1)}_{p}(t_n)\\ge k)\\le c_1(\\theta )(\\frac{1}{\\sqrt{k}}+\\theta \\varepsilon _n)$ .",
"Thus by (REF ), there exists $c_1(\\theta )>0$ such that for every $n\\in \\operatorname{\\mathbb {N}}^*$ and $k\\in \\llbracket {n} \\rrbracket $ , $\\operatorname{\\mathbb {P}}(|\\Pi _{n}^{(1)}(nt_n)|\\ge k)\\le c_1(\\theta )(\\frac{1}{\\sqrt{k}}+\\theta \\varepsilon _n)+c_2(\\theta )(\\frac{\\sqrt{k}}{n}+\\frac{k}{n}\\theta \\varepsilon _n)\\le (c_1(\\theta )+c_2(\\theta ))(\\frac{1}{\\sqrt{k}}+\\theta \\varepsilon _n).$ To complete the proof of the statement (iii) of Theorem REF , it suffices to apply inequality (REF )."
],
[
"Probability generating function ",
"First, let us establish inequalities for the probability generating function of a distribution having a finite second moment; This provides a simple proof for Lemma REF : Lemma A.1 Let $\\nu $ be a probability measure on $\\operatorname{\\mathbb {N}}$ .",
"Let us assume that $\\nu $ has a finite second moment and that $\\nu (0)+\\nu (1)<1$ .",
"Let $m_{\\nu ,1}$ and $m_{\\nu ,2}$ denote the first two factorial moments of $\\nu $ , The probability generating function of $\\nu $ denoted by $G_{\\nu }$ satisfies: $(m_{\\nu ,1}-1+\\nu (0))(s-1)^2\\le G_{\\nu }(s)-1-(s-1)m_{\\nu ,1}\\le \\frac{1}{2} m_{\\nu ,2}(s-1)^2\\; \\forall s\\in [0,1].$ Assume that $m_{\\nu ,1}$ is greater than 1.",
"The survival probability $\\alpha $ of a BGW process with offspring distribution $\\nu $ and one ancestor satisfies: $2\\frac{m_{\\nu ,1}-1}{m_{\\nu ,2}}\\le \\alpha \\le \\frac{m_{\\nu ,1}-1}{m_{\\nu ,1}-1+\\nu (0)} \\qquad \\mathrm {(\\ref {survineq})}$ Let us note that $1-G_{\\nu }(s)=(1-s)m_{\\nu ,1}H(s)$ where $H$ is the generating function of the probability $\\eta $ defined by $\\eta (k)=\\frac{1}{m_{\\nu ,1}}\\sum _{\\ell \\ge k+1}\\nu (\\ell )\\; \\forall k\\in \\operatorname{\\mathbb {N}}.$ By writing a similar formula for $1-H$ (it is possible since $H$ has a finite expectation), we obtain: $G_{\\nu }(s)=1+(s-1)m_{\\nu ,1}+\\frac{1}{2} m_{\\nu ,2}(s-1)^2K(s)$ where $K$ is the generating function of the probability $\\rho $ defined by $\\rho (k)=\\frac{2}{m_{\\nu ,2}}\\sum _{\\ell \\ge k+2}(\\ell -1-k)\\nu (\\ell )\\;\\forall \\ell \\in \\operatorname{\\mathbb {N}}.",
"$ In particular, for every $s\\in [0,1]$ , $\\frac{2}{m_{\\nu ,2}}(m_{\\nu ,1}-1+\\nu (0))\\le K(s)\\le 1$ .",
"The extinction probability $q=1-\\alpha $ is smaller than 1 and satisfies $G_{\\nu }(q)=q$ .",
"The second assertion is obtained by taking $s=q$ in (REF )."
],
[
"Total progeny distribution ",
"Let us turn to the total population size of a BGW process.",
"A useful tool to study its distribution is the following formula known as Dwass identity: Theorem ([10]) Consider a BGW process with offspring distribution $\\nu $ and $u\\ge 1$ ancestors.",
"Let $T$ denote its total progeny and let $(X_n)_n$ be a sequence of independent random variables with distribution $\\nu $ .",
"$\\forall k\\ge u,\\ \\operatorname{\\mathbb {P}}(T=k)=\\frac{u}{k}P(X_1+\\ldots +X_k=k-u).$ Recall that in the supercritical case (i.e.",
"$\\sum _{k}\\nu (k)>1$ ), $P(T<+\\infty )=q^u<1$ if $q$ denotes the extinction probability of the BGW process starting from one ancestor.",
"Using Dwass identity, we obtain: Lemma A.2 Let $T^{(u)}$ denote the total progeny of a BGW process with $u$ ancestors and with offspring distribution $\\text{CPois}(\\lambda ,\\nu )$ .",
"Then, $\\left\\lbrace \\begin{array}{l}P(T^{(u)}= u) = \\displaystyle {e^{-u \\lambda }}\\\\P(T^{(u)} = k) = \\displaystyle {\\frac{u}{k}e^{-k\\lambda }\\sum _{j=1}^{k-u}\\frac{(\\lambda k)^j}{j!",
"}\\nu ^{\\star j}(k-u)}\\quad \\forall k\\ge u+1.",
"\\\\\\end{array}\\right.$ where $\\nu ^{\\star j}$ denotes the $j$ -th convolution power of $\\nu $ .",
"It suffices to observe that the sum $X_1+\\ldots +X_k$ appearing in Dwass's theorem has the $\\text{CPois}(k\\lambda ,\\nu )$ -distribution."
],
[
"Local central limit theorem for a family of compound Poisson distributions\n",
"Let $\\nu $ be a probability distribution on $\\operatorname{\\mathbb {N}}$ with a finite second moment.",
"For $\\lambda >0$ , let $(X_{\\lambda ,n})_n$ denote a sequence of independent random variables with $\\text{CPois}(\\lambda ,\\nu )$ distribution.",
"The aim of this paragraph is to prove Lemma REF , which states that for every $0<a<b$ , the speed of convergence in the local limit theorem for $(X_{\\lambda ,n})_n$ can be bounded uniformly for $\\lambda \\in [a,b]$ by $o(\\frac{1}{\\sqrt{n}})$ .",
"Without loss of generality, we can assume that there is no $r>1$ such that $\\operatorname{\\mathbb {P}}(X_{1,1}\\in a+r\\operatorname{\\mathbb {Z}})=1$ for some $a\\in \\operatorname{\\mathbb {Z}}$ (otherwise, it suffices to consider $\\frac{1}{r_*}X_{\\lambda ,n}$ instead of $X_{\\lambda ,n}$ , where $r_*$ is the largest $r\\in \\operatorname{\\mathbb {N}}^*$ such that $\\operatorname{\\mathbb {P}}(X_{\\lambda ,1}\\in a+r\\operatorname{\\mathbb {Z}})=1$ for some $a\\in \\operatorname{\\mathbb {Z}}$ ; and to note that $r_{*}$ does not depend on $\\lambda $ ).",
"We follow the presentation of the local limit theorem proof proposed in ([24], Theorem 2.3.9).",
"Let $m_{\\lambda ,\\nu }$ and $\\sigma ^{2}_{\\lambda ,\\nu }$ denote the expectation and variance of $X_{\\lambda ,n}$ respectively.",
"We have to prove that for every $k\\in \\operatorname{\\mathbb {N}}$ , $R_{\\lambda ,n}(k)=2\\pi \\left(\\sqrt{n}\\operatorname{\\mathbb {P}}(\\sum _{i=1}^{n}X_{\\lambda ,i}=k)-\\frac{1}{\\sqrt{2\\pi \\sigma ^{2}_{\\lambda ,\\nu }}}\\exp \\left(-\\frac{(k-n m_{\\lambda ,\\nu })^2}{2\\sigma ^{2}_{\\lambda ,\\nu }}\\right)\\right)$ converges to 0 uniformly for $\\lambda \\in [a,b]$ as $n$ tends to $+\\infty $ .",
"Let $\\varphi _{\\lambda }$ denote the characteristic function of $X_{\\lambda ,1}-\\operatorname{\\mathbb {E}}(X_{\\lambda ,1})$ .",
"The first term of $R_{\\lambda ,n}(k)$ can be rewritten: $2\\pi \\sqrt{n}\\operatorname{\\mathbb {P}}(\\sum _{i=1}^{n}X_{\\lambda ,i}=k) =\\int _{-\\pi \\sqrt{n}}^{\\pi \\sqrt{n}}\\varphi _{\\lambda }(\\frac{x}{\\sqrt{n}})^n h_{\\lambda ,n,k}(x) dx,$ where $h_{\\lambda ,n,k}(x)=e^{ix(m_{\\lambda ,\\nu }\\sqrt{n}-\\frac{k}{\\sqrt{n}})}$ .",
"For the second term, the Fourier inversion theorem yields: $\\frac{\\sqrt{2\\pi }}{\\sigma _{\\lambda ,\\nu }}\\exp \\left(-\\frac{(k-nm_{\\lambda ,\\nu })^2}{2\\sigma ^{2}_{\\lambda ,\\nu }}\\right)=\\int _{\\operatorname{\\mathbb {R}}}e^{-\\frac{x^2}{2}\\sigma ^{2}_{\\lambda ,\\nu }}h_{\\lambda ,n,k}(x)dx.$ Thus, $|R_{\\lambda ,n}(k)|$ is bounded by the sum of three terms: $I_{1,\\lambda ,\\varepsilon }(n) & = & \\int _{|x|< \\varepsilon \\sqrt{n}}|\\varphi _{\\lambda }(\\frac{x}{\\sqrt{n}})^n-e^{-\\frac{x^2}{2}\\sigma ^{2}_{\\lambda ,\\nu }}|dx\\\\I_{2,\\lambda ,\\varepsilon }(n) & = & \\int _{\\varepsilon \\sqrt{n}\\le |x|\\le \\pi \\sqrt{n}}|\\varphi _{\\lambda }(\\frac{x}{\\sqrt{n}})^n|dx\\\\I_{3,\\lambda ,\\varepsilon }(n) & = & \\int _{|x|\\ge \\varepsilon \\sqrt{n}}e^{-\\frac{x^2}{2}\\sigma ^{2}_{\\lambda ,\\nu }}dx$ where $\\varepsilon \\in ]0,\\pi [$ .",
"Let us now use that $X_{\\lambda ,1}$ has the $\\text{CPois}(\\lambda ,\\nu )$ -distribution: $\\varphi _{\\lambda }(x)=\\exp (-i\\lambda x+ \\lambda (\\phi _{\\nu }(x)-1))$ , where $\\phi _{\\nu }$ denotes the characteristic function of $\\nu $ ; the expectation of $X_{\\lambda ,1}$ is $m_{\\lambda ,\\nu }=\\lambda m_{\\nu ,1}$ and its variance is $\\sigma ^{2}_{\\lambda ,\\nu }=\\lambda (m_{\\nu ,2}+m_{\\nu ,1})$ .",
"Therefore, $\\varphi _{\\lambda }(\\frac{x}{\\sqrt{n}})^n=e^{\\psi _{n,\\lambda }(x)}e^{-\\frac{x^2}{2}\\sigma ^{2}_{\\lambda ,\\nu }}\\;\\text{where}\\;\\psi _{n,\\lambda }(x)=n\\lambda (\\phi _{\\nu }(\\frac{x}{\\sqrt{n}})-\\phi _{\\nu }(0)-\\phi ^{\\prime }_{\\nu }(0)\\frac{x}{\\sqrt{n}}-\\phi ^{\\prime \\prime }_{\\nu }(0)\\frac{x^2}{2n}).$ The study of the remainder in the Taylor expansion of $\\phi _\\nu $ yields: $|\\psi _{n,\\lambda }(x)|\\le \\frac{1}{2}\\lambda x^2\\sup _{u\\le \\frac{|x|}{\\sqrt{n}}}|\\phi ^{\\prime \\prime }_{\\nu }(u)-\\phi ^{\\prime \\prime }_{\\nu }(0)|.$ Accordingly, there exists $\\varepsilon _0>0$ such that for $|x|\\le \\varepsilon _{0} \\sqrt{n}$ , $|e^{\\psi _{n,\\lambda }(x)}-1|\\le e^{\\frac{x^2}{4}\\sigma ^2_{\\lambda ,\\nu }}+1$ .",
"Let us split $I_{1,\\lambda ,\\varepsilon _0}(n) $ into the integral on $[-B,B]$ denoted by $J_{1,\\lambda ,B}(n)$ and the integral on $]-\\varepsilon _0 \\sqrt{n},-B[\\cup ]B, \\varepsilon _0 \\sqrt{n}[$ denoted by $J_{2,\\lambda ,B,\\varepsilon _0}(n)$ .",
"For every $B>0$ and $\\lambda \\in [a,b]$ , $J_{2,\\lambda ,B,\\varepsilon _0}(n)$ is bounded by $\\int _{B<|x|<\\varepsilon _0 \\sqrt{n}}2\\exp \\left(-\\frac{1}{4}x^2\\sigma ^{2}_{\\lambda ,\\nu }\\right)dx\\le \\frac{2}{Ba(m_{\\nu ,1}+m_{\\nu ,2})}\\exp \\left(-\\frac{B^2}{4}a(m_{\\nu ,1}+m_{\\nu ,2})\\right).$ Since $\\sup _{\\lambda \\in [a,b],|x|\\le B}|\\psi _{n,\\lambda }(x)|$ converges to 0 as $n$ tends to $+\\infty $ (by (REF )), $J_{1,\\lambda ,B}(n)$ converges to 0 uniformly for $\\lambda \\in [a,b]$ .",
"Therefore, $I_{1,\\lambda ,\\varepsilon _0}(n) $ converges to 0 uniformly for $\\lambda \\in [a,b]$ .",
"Let us now consider $I_{2,\\lambda ,\\varepsilon _0}(n)$ .",
"By assumption, $|\\varphi _{1}(x)|<1$ for every ${x\\in [-\\pi ,\\pi ]\\setminus \\lbrace 0\\rbrace }$ .",
"Thus $\\beta =\\sup _{\\varepsilon _0\\le |x|\\le \\pi }|\\varphi _{1}(x)|<1$ .",
"Since $\\varphi _{\\lambda }=(\\varphi _{1})^{\\lambda }$ for every $\\lambda >0$ , $I_{2,\\lambda ,\\varepsilon _0}(n)\\le 2\\pi \\sqrt{n}(\\beta )^{\\lambda n}\\;\\text{for every}\\;\\lambda >0 \\operatorname{ and }n\\in \\operatorname{\\mathbb {N}}^*.$ This shows that $(I_{2,\\lambda ,\\varepsilon _0}(n))_n$ converges to 0 uniformly for $\\lambda \\in [a,b]$ .",
"Finally, $I_{3,\\lambda ,\\varepsilon _0}(n)\\le \\frac{2}{\\varepsilon _0\\lambda (m_{\\nu ,1}+m_{\\nu ,2})\\sqrt{n}}e^{-\\frac{n}{2}\\varepsilon ^2_0\\lambda (m_{\\nu ,1}+m_{\\nu ,2})}$ .",
"It converges also to 0 uniformly for $\\lambda \\in [a,b]$ , which completes the proof."
],
[
"Dual BGW process ",
"A supercritical BGW process conditioned to become extinct is a subcritical BGW process: Theorem ([2], Theorem 3, p. 52) Let $(Z_n)_n$ be a supercritical BGW process with one ancestor.",
"Let $\\phi $ denote the generating function of its offspring distribution and let $q$ denote its extinction probability.",
"Assume that $\\phi (0)>0$ .",
"Then, $(Z_n)_n$ conditioned to become extinct has the same law as a subcritical BGW process with one ancestor and offspring generating function $s\\mapsto \\frac{1}{q}\\phi (qs)$ .",
"As a consequence of this theorem, if the offspring distribution is a compound Poisson distribution, the offspring distribution of the dual BGW process is also a compound Poisson distribution: Lemma A.3 Let $\\lambda $ be a positve real and let $\\nu $ be a probability measure on $\\operatorname{\\mathbb {N}}$ with a finite expectation $m$ and generating function $G_{\\nu }$ .",
"Let $Z$ be a BGW process with offspring distribution $\\text{CPois}(\\lambda ,\\nu )$ .",
"Assume that $\\lambda m > 1$ and let $q$ denote the extinction probability of $Z$ (that is the smallest positive solution of the equation $\\exp (\\lambda (G_{\\nu }(x)-1))=x$ ).",
"Then $Z$ conditioned to become extinct has the same law as the subcritical BGW process with offspring distribution $\\text{CPois}(\\lambda G_{\\nu }(q) ,\\hat{\\nu }_q)$ where $\\hat{\\nu }_q(k)=\\frac{q^k}{G_{\\nu }(q)}\\nu (k)$ for every $k\\in \\operatorname{\\mathbb {N}}$ .",
"Remark A.4 Let us note that if $\\nu $ is an heavy-tailed distribution (that is the couvergence radius of $G_{\\nu }$ is equal to 1) then it is not the case for $\\hat{\\nu }_{q}$ .",
"More generally, let us write out some properties of a BGW$(u,\\lambda G_{\\nu }(a) ,\\hat{\\nu }_a)$ process for any $a\\in ]0,1[$ (such a process appears in Corollary REF dealing with the restriction of the Poisson point process $\\mathcal {P}_n$ to tuples in $\\mathcal {W}(\\lbrace 1,\\ldots ,\\lfloor a n\\rfloor \\rbrace )$ ).",
"Lemma A.5 Let $\\lambda >0$ , let $\\nu $ be a probability measure on $\\operatorname{\\mathbb {N}}$ and let $G_{\\nu }$ denote its generating function.",
"For $a\\in ]0,1[$ , set $\\hat{\\nu }_a(k)=\\frac{a^k}{G_{\\nu }(a)}\\nu (k)$ for every $k\\in \\operatorname{\\mathbb {N}}$ .",
"Let $G_{\\lambda ,\\nu ,a}$ denote the generating function of the $\\text{CPois}(\\lambda ,\\hat{\\nu }_a)$ -distribution.",
"Then, $G_{\\lambda G_{\\nu }(a),\\nu ,a}(s)= G_{\\lambda ,\\nu ,1}(as)\\exp (\\lambda (1-G_{\\nu }(a)) )\\text{ for every }s\\text{ in the domain of }G_{\\lambda ,\\nu ,1}.$ Mass-function distribution: for every $k\\in \\operatorname{\\mathbb {N}}$ , $\\text{CPois}(\\lambda G_{\\nu }(a),\\hat{\\nu }_a)(\\lbrace k\\rbrace )=a^ke^{\\lambda (1-G_{\\nu }(a))}\\text{CPois}(\\lambda ,\\nu )(\\lbrace k\\rbrace )\\quad \\forall k\\in \\operatorname{\\mathbb {N}}.$ The expectation of the $\\text{CPois}(\\lambda ,\\hat{\\nu }_a)$ -distribution is an analytic and increasing function of $a\\in ]0,1[$ .",
"In particular, the maximal value of this function is greater than 1 on $]0,1[$ if and only if the expectation of the $\\text{CPois}(\\lambda ,\\nu )$ -distribution is greater than 1.",
"Let $\\hat{T}^{(u)}_a$ denote the total population size of a BGW$(u,\\lambda G_{\\nu }(a) ,\\hat{\\nu }_a)$ process.",
"For every $k\\in \\operatorname{\\mathbb {N}}^*$ greater than or equal to $u$ , $\\operatorname{\\mathbb {P}}(\\hat{T}^{(u)}_a=k)=a^{k-u}e^{k\\lambda (1-G_{\\nu }(a))}\\operatorname{\\mathbb {P}}(\\hat{T}^{(u)}_1=k).$ Equality (REF ) can be established by applying formulae (REF ) and (REF ).",
"Example A.6 Set $a\\in ]0,1]$ .",
"Let us present the BGW$(1,tm_{1,\\hat{p}_{a}},\\hat{p}_a)$ process used to approximated the distribution of $|\\Pi ^{(1)}_{\\llbracket {n} \\rrbracket {\\lfloor an\\rfloor }}(ant)|$ for two examples of distributions $p$ .",
"If $p$ is the Dirac mass on $d\\in \\operatorname{\\mathbb {N}}\\setminus \\lbrace 0,1\\rbrace $ , the offspring distribution of the BGW process is $(d-1)\\text{Poisson}(\\frac{td}{a})$ and the total population size distribution is: $\\sum _{k\\in 1+(d-1)\\operatorname{\\mathbb {N}}}e^{-tk}\\frac{(tk)^{\\frac{k-1}{d-1}}}{k(\\frac{k-1}{d-1})!",
"}\\delta _k.$ If $p$ is the logarithmic$(u)$ distribution for $u\\in ]0,1[$ , then $\\hat{p}_a$ is the geometric distribution on $\\operatorname{\\mathbb {N}}^*$ with parameter $1-au$ : $\\hat{p}_a=\\sum _{k=1}^{+\\infty }(1-au)(au)^{k-1}\\delta _{k}$ .",
"The offspring distribution of the BGW process is the geometric Poisson distribution CPois$(tc(a,u), \\hat{p}_a)$ where $c(a,u)=\\frac{(au)^2}{-(1-au)\\log (1-au)}$ .",
"This distribution is also known as Pólya-Aeppli $(tc(a,u),au)$ distribution and is defined by: $e^{-tc(a,u) }\\delta _{0}+\\sum _{k=1}^{+\\infty }\\left(e^{-tc(a,u)} (1-au)^k\\sum _{{j=1}}^{k}\\frac{1}{j!",
"}\\binom{k-1}{j-1} \\left(\\frac{tc(a,u)au}{1-au}\\right)^{j}\\right)\\delta _{k}.$ The total population size distribution of the BGW process with $i$ ancestors has the following distribution: $e^{-tc(a,u) }\\delta _{i}+\\sum _{k=i+1}^{+\\infty }\\left(\\frac{i}{k}e^{-tc(a,u) }(au)^{k-i}\\sum _{j=1}^{k-i} \\frac{k^j}{j!",
"}{\\binom{k-i-1}{j-1}}\\left(tc(a,u)(\\frac{1}{au}-1)\\right)^{j}\\right)\\delta _{k}.$"
]
] | 1709.01896 |
[
[
"Characterization of compactness of commutators of bilinear singular\n integral operators"
],
[
"Abstract The commutators of bilinear Calder\\'on-Zygmund operators and point-wise multiplication with a symbol in $cmo$ are bilinear compact operators on product of Lebesgue spaces.",
"This work shows that, for certain non-degenerate Calder\\'on-Zygmund operators, the symbol being in $cmo$ is not only sufficient but actually necessary for the compactness of the commutators."
],
[
"Introduction",
"In this note we resolve a problem that has been open for a while in the multilinear Calderón–Zygmund theory.",
"Namely, whether the compactness of the commutators of the bilinear Riesz transforms (see the next section for technical definitions) with point-wise multiplication can be used to characterize the space ${\\rm CMO}(\\mathbb {R}^n)$ .",
"For the purpose of this article, ${\\rm CMO}(\\mathbb {R}^n)$ is the closure in the John–Nirenberg ${\\rm BMO}(\\mathbb {R}^n)$ , with its usual topology, of the space of infinitely differentiable functions with compact support.",
"This problem has been motivated by the analogous situation in the classical (linear) Calderón–Zygmund theory and several preliminary existing results in the multilinear setting, which we summarize in what follows.",
"As is well-known, the first to study the commutator $[b,\\mathcal {R}^k](f):=\\mathcal {R}^k (bf)-b\\mathcal {R}^k(f)$ of the classical Riesz transforms $\\mathcal {R}^k$ with point-wise multiplication by a function $b$ were Coifman, Rochberg and Weiss [5].",
"They showed that $[b,\\mathcal {R}^k]$ is bounded on $L^p$ for some $p$ with $1< p<\\infty $ if and only if the symbol $b$ is in ${\\rm BMO}(\\mathbb {R}^n)$ .",
"Their result was then extended to other non-degenerate Calderón–Zygmund operators by Janson [7] and Uchiyama [14].",
"Moreover, Uchiyama showed that $[b,\\mathcal {R}^k]$ is compact on $L^p$ for some (then for all) $1< p<\\infty $ if and only if the function $b$ is not just in ${\\rm BMO}(\\mathbb {R}^n)$ but actually in ${\\rm CMO}(\\mathbb {R}^n)$ .",
"In the multilinear setting, an interesting situation arises: multilinear Calderón–Zygmund operators, their commutators, and other related operators tend to be bounded also into $L^p$ spaces outside the Banach space situation.",
"For example, in the bilinear case a Calderón–Zygmund operator $T$ in the sense of Grafakos and Torres [6] (see also the references therein) satisfies $ T:L^{p_1} \\times L^{p_2} \\rightarrow L^p,$ for all $1<p_1<\\infty $ , $1<p_2<\\infty $ and $1/p_1+1/p_2=1/p<2$ .",
"This creates complications when studying the case of $p<1$ in the target space, as some analytic tools (often depending on duality) fail in this situation.",
"For this reason the case $p>1$ and $p<1$ have been occasionally treated separately in the literatures and by different arguments.",
"For example, the boundedness of the commutators $[b, T]_1(f,g):=\\, & T(bf,g)-b T(f,g),\\\\[b,T]_2(f,g):=\\, & T(f,bg)-bT(f,g),$ of a bilinear Calderón–Zygmund operator $T$ with a ${\\rm BMO}$ function $b$ was first obtained by Pérez and Torres in [10] when $p>1$ , while the case of $p\\le 1$ was latter studied independently by Tang [12] and Lerner et al. [8].",
"The compactness of the same commutators when $b\\in {\\rm CMO}(\\mathbb {R}^n)$ was obtained by Bényi and Torres in [1] but only for $p\\ge 1$ .",
"Nonetheless, it was recently observed by Torres and Xue [13] that the result also holds for $1/2<p<1$ .",
"The partial converse fact that the boundedness of $[b, T]_1$ or $[b, T]_2$ for certain bilinear Calderón–Zygmund operators forces $b$ to be in ${\\rm BMO}(\\mathbb {R}^n)$ was first proved by Chaffee [2] and was then also revisited by Li and Wick [9] using different techniques.",
"In both cases the results are also under the assumption $p>1$ .",
"Finally, in a very recent manuscript posted in arXiv by Wang, Zhou and Teng [15], the result of Chaffee [2] was extended to $1/2<p\\le 1$ .",
"We will show in Theorem REF below that at least for the bilinear Riesz transforms, the compactness of the commutators forces the symbol $b$ to be in ${\\rm CMO}(\\mathbb {R}^n)$ .",
"Our work follows ideas of Uchiyama [14] and Chen, Ding and Wang [4] in the linear case, as well as modification done in [3] for the bilinear operators.",
"We note however that the main difference with respect to the work in [3], and a difficulty we overcome here, is that the operators in [3] are bilinear fractional integral operators which are hence positively defined, which is a property heavily used in [3] but certainly completely failing for Calderón–Zygmund operators.",
"We refer the reader to [3] and the references therein for more on commutators of fractional singular operators in both linear and multilinear settings.",
"Acknowledgement.",
"Part of the work leading to this article took place while the last two named authors were visiting the Mathematical Sciences Research Institute (MSRI) at Berkeley in February 2017, during the Harmonic Analysis program.",
"This stay at MSRI gave them a chance to combine different previous efforts by all the colleagues involved.",
"The authors would like to thank the Institute and the organizers of the program for providing the resources for such a fruitful opportunity to carry out this research."
],
[
"Definitions",
"As mentioned in the introduction, the space ${\\rm CMO}(\\mathbb {R}^n)$ is the closure in the ${\\rm BMO}(\\mathbb {R}^n)$ topology of the space of infinitely differentiable functions with compact support, denoted here by $C^\\infty _c(\\mathbb {R}^n)$ .",
"For brevity, throughout the paper we denote $L^p(\\mathbb {R}^n)$ by $L^p$ , and similarly for ${\\rm BMO}$ , ${\\rm CMO}$ and $C_c^\\infty $ .",
"Also, for convenience, we will use the ${\\rm BMO}$ norm (modulo constants) defined for a locally integrable function $b$ by $\\Vert b\\Vert _{{\\rm BMO}}:=\\sup _{Q}\\mathchoice{{\\displaystyle {\\textstyle -}{\\int }}\\hbox{$\\textstyle -$}}{\\hspace{0.0pt}}{-}{.",
"}5$ Q |b(x)-bQ| dx < , $with the supremum taken over all cubes $ Rn$ with edgesparallel to the coordinate axes, and where for any locallyintegrable function $ f$ we use the standard notation $ fQ = Q f:=1|Q|Qf(x) dx$ for the average of$ f$ over $ Q$.",
"In addition, we recall (see \\cite {U}) that $ b BMO$ is in $ CMO$ if and only if{\\begin{@align}{1}{-1}& \\displaystyle \\lim _{a\\rightarrow 0}\\sup _{|Q|=a}\\frac{1}{|Q|}\\int _Q|b(x)-b_Q|\\,dx=0, \\\\& \\displaystyle \\lim _{a\\rightarrow \\infty }\\sup _{|Q|=a}\\frac{1}{|Q|}\\int _Q|b(x)-b_Q|\\,dx=0, \\quad \\quad {\\text{and}}\\\\& \\displaystyle \\lim _{|y|\\rightarrow \\infty }\\frac{1}{|Q|}\\int _Q|b(x+y)-b_Q|\\,dx=0, \\mbox{ for each } Q.\\end{@align}}$ For $x\\in \\mathbb {R}^n$ we will use the notation $x=(x^1,\\dots ,x^n)$ and consider the $2n$ bilinear Riesz transform operators defined for $k=1,\\dots ,n$ by $\\mathcal {R}^k_1(f,g)(x):=\\, & \\text{p.v.",
"}\\iint _{\\mathbb {R}^{2n}}\\frac{x^k-y^k}{(|x-y|^2+|x-z|^2)^{n+1/2}}\\,f(y)g(z)\\,dydz,\\\\\\mathcal {R}^k_2(f,g)(x):=\\, & \\text{p.v.",
"}\\iint _{\\mathbb {R}^{2n}}\\frac{x^k-z^k}{(|x-y|^2+|x-z|^2)^{n+1/2}}\\,f(y)g(z)\\,dydz.$ The name of these operators is justified by the fact that they can be “obtained\" by considering the linear Riesz transforms in $\\mathbb {R}^{2n}$ defined by $\\mathcal {R}^k(F)(u):=\\, & \\text{p.v.",
"}\\int _{\\mathbb {R}^{2n}}\\frac{u^k-v^k}{|u-v|^{2n+1}}\\,F(v)\\,dv,$ where $u=(u^1, \\dots , u^{2n})$ and $v=(v^1, \\dots , v^{2n})$ , $k=1,\\dots ,2n$ .",
"Note that setting $u=(x,x)$ , $v=(y,z)$ with $x,y,z \\in \\mathbb {R}^n$ , and $F(y,z)=f(y)g(z)$ leads, formally, to the bilinear operators $\\mathcal {R}^k_j$ , $j=1,2$ .",
"For $k=1,\\dots , n$ , $\\mathcal {R}^k_1(f,g)(x)= \\mathcal {R}^k(fg)(x,x)$ , while $\\mathcal {R}^k_2(f,g)(x)= \\mathcal {R}^{k+n}(fg)(x,x)$ .",
"The boundedness of the $\\mathcal {R}^k_j$ operators from $L^{p_1}\\times L^{p_2}$ to $L^p$ , for $1<p_1<\\infty $ , $1<p_2<\\infty $ and $1/p_1+1/p_2=1/p<2$ , is by now well-known.",
"See for example [6] and the references therein.",
"For $j=1,2$ , and $k=1,\\dots ,n$ , the (first–order) commutators of the Riesz transform operators with a symbol $b$ are given by $\\begin{split}[b, \\mathcal {R}^k_j]_1(f,g):=\\, & \\mathcal {R}^k_j(bf,g)-b \\mathcal {R}^k_j(f,g),\\\\[b,\\mathcal {R}^k_j]_2(f,g):=\\, & \\mathcal {R}^k_j(f,bg)-b\\mathcal {R}^k_j(f,g).\\end{split}$ Notice that $b \\in {\\rm BMO}$ is consistent with the fact that, by linearity, for any complex number $C$ , $\\begin{split}[b-C, \\mathcal {R}^k_j]_1(f,g)= [b, \\mathcal {R}^k_j]_1(f,g),\\\\[b-C,\\mathcal {R}^k_j]_2(f,g)= [b, \\mathcal {R}^k_j]_2(f,g),\\end{split}$ a fact that we will later use.",
"By the results mentioned in the introduction the boundedness of any of these commutators from $L^{p_1} \\times L^{p_2}$ to $L^p$ , for the full range of exponents $1<p_1<\\infty $ , $1<p_2<\\infty $ and $1/p_1+1/p_2=1/p<2$ is equivalent to $b$ being in ${\\rm BMO}$ .",
"It is also known that they are compact for the same range of exponents if in addition $b\\in {\\rm CMO}$ .",
"The new result we shall present is the converse of this last statement."
],
[
"Characterization of compactness",
"Theorem 3.1 Let $1<p_1<\\infty $ , $1<p_2<\\infty $ and $\\frac{1}{p}=\\frac{1}{p_1}+\\frac{1}{p_2}<2$ .We note that in a first draft of this article we had stated Theorem REF only for $p>1$ .",
"Although the computations in the proof (the same presented here) work for all $1/2<p<\\infty $ , it was not known at the time whether the boundedness of the commutators when $1/2<p\\le 1$ implies $b \\in {\\rm BMO}$ , which is a condition needed to jump start our arguments in the proof.",
"Nothing else in the proof depends on the value of $p>1/2$ .",
"The recent result in [15] allows us now to state Theorem REF for the full range of exponents without altering its proof.",
"Then each of the commutators in (REF ) is a compact bilinear operator from $L^{p_1} \\times L^{p_2} \\rightarrow L^p$ , if and only if $b\\in {\\rm CMO}$ .",
"We only need to establish the necessity of $b\\in {\\rm CMO}$ since another direction was proved in [1] and [13] as noted in Introduction.",
"Moreover, by symmetry and a change of variables it is enough to consider, for example, $\\mathcal {R}^1_1$ and $ [b, \\mathcal {R}_1^1]_1$ .",
"To simplify notation we denote $\\mathcal {R}^1_1$ by $\\mathcal {R}$ .",
"Fix exponents $p_1,p_2, p$ as in the statement of the theorem.",
"Since bilinear compact operators are bounded, if we assume $\\mathcal {R}$ to be compact from $L^{p_1} \\times L^{p_2} \\rightarrow L^p$ we must have that $b \\in {\\rm BMO}$ ; see [2] for $p>1$ and [15] for $1/2<p\\le 1$ .",
"So for convenience, by linearity, we may assume that $b$ is real valued and with $\\Vert b\\Vert _{{\\rm BMO}}=1.$ We will follow very closely some arguments in [14], [4] and [3] to show that if $b$ fails to satisfy one of the conditions ()–(), then one arrives at a contradiction with the compactness of the operator.",
"So $b$ must be in ${\\rm CMO}$ .",
"We notice, however, that a main difference in the arguments below, in particular with respect to [4] and [3], is the fact alluded to in the introduction that the fractional integral operators considered in those works are actually positive operators, while the singular integrals studied here are not.",
"This requires a modification in the lower estimate (REF ) proved below.",
"Assume that $\\lbrace Q_j\\rbrace _j$ is a sequence of cubes such that $\\frac{1}{|Q_j|}\\int _{Q_j}|b(x)-b_{Q_j}|\\,dx\\ge \\varepsilon ,$ for some $\\varepsilon >0$ and all $j\\in \\mathbb {N}$ .",
"As in [4] and [3], define two sequences of functions $\\lbrace f_j\\rbrace $ and $\\lbrace g_j\\rbrace $ associated with the cubes $Q_j$ in the following way.",
"Let $c_0:=|Q_j|^{-1}\\int _{Q_j}\\text{sgn}(b(y)-b_{Q_j})\\,dy$ and define $f_j(y):=|Q_j|^{-\\frac{1}{p_1}}\\left({\\rm sgn}(b(y)-b_{Q_j})-c_0\\right)\\chi _{Q_j}(y).$ Here $\\text{sgn}$ denotes the usual signum function.",
"Define also $g_j(y):=|Q_j|^{-\\frac{1}{p_2}}\\chi _{Q_j}(y).$ These functions satisfy the following properties $\\text{supp}\\,f_j\\subset Q_j$ and $\\text{supp}\\,g_j\\subset Q_j,$ $f_j(y)(b(y)-b_{Q_j})\\ge 0,$ $\\int f_j(y)\\,dy=0,$ $\\int (b(y)-b_{Q_j})f_j(y)\\,dy=|Q_j|^{-\\frac{1}{p_1}}\\int _{Q_j}|b(y)-b_{Q_j}|\\,dy,$ $|f_j(y)|\\le 2|Q_j|^{-\\frac{1}{p_1}}$ and $|g_j(y)|\\le |Q_j|^{-\\frac{1}{p_2}},$ $\\Vert f_j\\Vert _{L^{p_1}}\\le 2$ , $\\Vert g_j\\Vert _{L^{p_2}}=1.$ Let $\\lbrace y_j\\rbrace $ be the collection of centers of the cubes $\\lbrace Q_j\\rbrace $ .",
"Then for all $x\\in \\left(2\\sqrt{n} Q_j\\right)^{c}$ the following standard pointwise estimates hold: $|\\mathcal {R}((b-b_{Q_j})f_j,g_j)(x)|&\\lesssim |Q_j|^{\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2}}|x-y_j|^{-2n}, \\\\|\\mathcal {R}(f_j,g_j)(x)|&\\lesssim |Q_j|^{\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2}+\\frac{1}{n}}|x-y_j|^{-2n-1},$ where the constants involved are independent of $j, b, f_j,g_j$ and $\\varepsilon $ .",
"Indeed, for all such $x$ and all $y\\in Q_j$ we have $|x-y| \\approx |x-y_j| >0$ , and hence by (a) and (e), $|\\mathcal {R}((b-b_{Q_j})&f_j,g_j)(x)|=\\left|\\ \\iint _{\\mathbb {R}^{2n}} \\frac{ (x^1-y^1)(b(y)-b_{Q_j})f_j(y)g_j(z)}{\\left(|x-y|^2+|x-z|^2\\right)^{n+1/2}}\\,dydz\\right|\\\\&\\lesssim \\frac{1}{|Q_j|^{\\frac{1}{p_1}+\\frac{1}{p_2}}}|x-y_j|^{-2n}\\int _{Q_j}\\int _{Q_j}|b(y)-b_{Q_j}|\\,dydz\\\\&\\lesssim |Q_j|^{\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2}}|x-y_j|^{-2n}\\Vert b\\Vert _{{\\rm BMO}}\\\\&\\lesssim |Q_j|^{\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2}}|x-y_j|^{-2n}.$ On the other hand, using (a), (e), the cancellation property (c) of $f_j$ and the regularity of the kernel of the operator $\\mathcal {R}$ , $|&\\mathcal {R}(f_j,g_j)(x)|=\\left|\\iint _{\\mathbb {R}^{2n}}\\frac{ (x^1-y^1)f_j(y)g_j(z)}{\\left(|x-y|^2+|x-z|^2\\right)^{n+1/2}}\\,dydz\\right|\\\\&=\\left| \\int _{\\mathbb {R}^n}\\!\\!",
"\\left( \\int _{\\mathbb {R}^n} \\left( \\frac{ (x^1-y^1)f_j(y)g_j(z)}{\\left(|x-y|^2+|x-z|^2\\right)^{n+1/2}} \\right.",
"\\right.",
"\\right.\\\\&\\lesssim \\hspace{113.81102pt} - \\left.",
"\\left.",
"\\left.",
"\\frac{(x^1-y_j^1)f_j(y)g_j(z)}{\\left(|x-y_j|^2+|x-z|^2\\right)^{n+1/2}} \\right)\\,dy \\right)\\, dz\\right|\\\\&\\lesssim \\int _{Q_j}\\int _{Q_j} \\frac{|y-y_j||f_j(y)|g_j(z)}{\\left(|x-y_j|^2+|x-z|^2\\right)^{n+1}}\\,dy dz\\\\&\\lesssim \\frac{|Q_j|^{\\frac{1}{n}}}{|x-y_j|^{2n+1}}\\int _{Q_j} \\int _{Q_j} |f_j(y)|g_j(z)\\,dydz\\\\&\\lesssim |Q_j|^{\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2}+\\frac{1}{n}}|x-y_j|^{-2n-1}.$ Next, we note that if $d_j$ is the side-length of $Q_j$ then for all positive numbers $\\widetilde{\\gamma }_1, \\widetilde{\\gamma }_2$ , with $\\widetilde{\\gamma }_2 = 8 \\widetilde{\\gamma }_1\\gg 1 $ there always exists a cube $\\widetilde{Q}_j$ of side-length $\\frac{\\widetilde{\\gamma }_2}{4\\sqrt{n}} d_j$ contained in the annulus $A=\\lbrace x\\in \\mathbb {R}^n: \\widetilde{\\gamma }_1 d_j < |x-y_j| < \\widetilde{\\gamma }_2 d_j\\rbrace ,$ and such that $|x-y|\\approx |x-y_j| \\approx x^1- y_j^1 \\approx x^1-y^1>0$ for all $x\\in \\widetilde{Q}_j$ and all $y\\in Q_j$ .",
"We claim that for all such $x$ , $|\\mathcal {R}((b-b_{Q_j})f_j,g_j)(x)| \\gtrsim \\varepsilon |Q_j|^{\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2}}|x-y_j|^{-2n},$ where again the constant involved is independent of $j, b, f_j,g_j$ and $\\varepsilon $ .",
"To see (REF ), we use properties (b) and (d) of $f_j$ to estimate $|\\mathcal {R}((b-b_{Q_j})&f_j,g_j)(x)|=\\left|\\iint _{\\mathbb {R}^{2n}} \\frac{(x^1-y^1)(b(y)-b_{Q_j})f_j(y)g_j(z)}{\\left(|x-y|^2+|x-z|^2\\right)^{n+1/2}}\\,dydz\\right|\\\\&\\gtrsim |Q_j|^{1-\\frac{1}{p_2}}|x-y_j|^{-2n} \\int _{Q_j}(b(y)-b_{Q_j})f_j(y)\\,dy\\\\&= C_1 |Q_j|^{1-\\frac{1}{p_2}}|x-y_j|^{-2n}|Q_j|^{1-\\frac{1}{p_1}}\\frac{1}{|Q_j|}\\int _{Q_j}|(b(y)-b_{Q_j})|\\,dy\\\\&\\ge C_1 |Q_j|^{\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2}}|x-y_j|^{-2n}\\varepsilon .$ We continue to follow the computations in [14], [4] and [3] and want to establish now that there exist constants $\\gamma _1,\\gamma _2$ with $\\gamma _2>\\gamma _1>0$ and $\\gamma _3>0$ , depending only on $p_1,\\ p_2,\\ n$ and $\\varepsilon $ , such that the following estimates hold: $\\left(\\int _{\\gamma _1d_j<|x-y_j|<\\gamma _2d_j}|[b,\\mathcal {R}]_1(f_j,g_j)(x)|^p\\,dx\\right)^{\\frac{1}{p}}&\\ge \\gamma _3,\\\\\\left(\\int _{|x-y_j|>\\gamma _2d_j}|[b,\\mathcal {R}]_1(f_j,g_j)(x)|^p\\,dx\\right)^{\\frac{1}{p}}&\\le \\frac{\\gamma _3}{4}.$ In order to prove (REF ) and (), we first observe that for every large enough number $\\widetilde{\\gamma }_1>(\\frac{1}{\\ln \\sqrt{2}})^2$ , by properties (a) and (e) and the John–Nirenberg inequality, $& \\int _{|x-y_j|>\\widetilde{\\gamma }_1d_j}\\left|(b(x)-b_{Q_j})\\mathcal {R}(f_j,g_j)(x)\\right|^p\\,dx\\\\&\\lesssim |Q_j|^{\\left(\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2}+\\frac{1}{n}\\right)p}\\sum _{s=\\lfloor \\log _2(\\widetilde{\\gamma }_1)\\rfloor }^\\infty \\int _{2^sd_j<|x-y_j|<2^{s+1}d_j}\\frac{|b(x)-b_{Q_j}|^p}{|x-y_j|^{p(2n+1)}} \\,dx\\\\&\\lesssim |Q_j|^{\\left(\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2}+\\frac{1}{n}\\right)p}\\times \\\\&\\,\\,\\,\\,\\,\\,\\sum _{s=\\lfloor \\log _2(\\widetilde{\\gamma }_1)\\rfloor }^\\infty 2^{-s(2n+1)p}|Q_j|^{-\\left(2+\\frac{1}{n}\\right)p}\\int _{2^sd_j<|x-y_j|<2^{s+1}d_j}|b(x)-b_{Q_j}|^p\\,dx\\\\&\\lesssim |Q_j|^{\\left(\\frac{1}{p^{\\prime }_1}+\\frac{1}{p^{\\prime }_2} -2\\right)p} \\sum _{s=\\lfloor \\log _2(\\widetilde{\\gamma }_1)\\rfloor }^\\infty 2^{-s(2n+1)p} s^p2^{sn}|Q_j| \\\\&\\lesssim \\sum _{s=\\lfloor \\log _2(\\widetilde{\\gamma }_1)\\rfloor }^\\infty 2^{-s\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)p},$ and hence by $1/p<2$ , $\\left(\\int _{|x-y_j|>\\widetilde{\\gamma }_1d_j}\\left|(b(x)-b_{Q_j})\\mathcal {R}(f_j,g_j)(x)\\right|^p\\,dx\\right)^{\\frac{1}{p}}\\le C_2 \\widetilde{\\gamma }_1^{-\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)}.$ Next, for $\\widetilde{\\gamma }_2=8\\widetilde{\\gamma }_1$ , using (REF ) and (REF ), we obtain the following estimates: for $p\\ge 1$ , $&\\left(\\int _{\\widetilde{\\gamma }_1d_j<|x-y_j|<\\widetilde{\\gamma }_2d_j}|[b,\\mathcal {R}]_1(f_j,g_j)(x)|^p\\,dx\\right)^{\\frac{1}{p}} \\nonumber \\\\&\\quad \\ge \\left(\\int _{\\widetilde{\\gamma }_1d_j<|x-y_j|<\\widetilde{\\gamma }_2d_j}|\\mathcal {R}\\left((b-b_Q)f_j,g_j\\right)(x)|^p\\,dx\\right)^{\\frac{1}{p}} \\nonumber \\\\&\\quad \\ \\ \\ \\ -\\left(\\int _{\\widetilde{\\gamma }_1d_j<|x-y_j|}|(b(x)-b_Q)\\mathcal {R}(f_j,g_j)(x)|^p\\,dx\\right)^{\\frac{1}{p}} \\nonumber \\\\\\ &\\quad \\ge C_1 \\varepsilon |Q_j|^{\\frac{1}{p_1^{\\prime }}+\\frac{1}{p_2^{\\prime }}}\\left(\\int _{\\widetilde{Q}_j}|x-y_j|^{-2np}\\,dx\\right)^{\\frac{1}{p}} -C_2\\widetilde{\\gamma }_1^{-\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)}\\nonumber \\\\&\\quad \\ge C_1 \\varepsilon |Q_j|^{\\frac{1}{p_1^{\\prime }}+\\frac{1}{p_2^{\\prime }}} |\\widetilde{Q}_j|^{\\frac{1}{p}} \\widetilde{\\gamma }_2^{-2n} |Q_j|^{-2} -C_2\\widetilde{\\gamma }_1^{-\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)}\\nonumber \\\\&\\quad \\ge C_1 \\varepsilon (4 \\sqrt{n})^{-\\frac{n}{p}} \\widetilde{\\gamma }_2^{-2n+\\frac{n}{p}} -C_2 8^{\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)} \\widetilde{\\gamma }_2^{-\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)}, $ and for $1/2<p<1$ , $& \\int _{\\widetilde{\\gamma }_1d_j<|x-y_j|<\\widetilde{\\gamma }_2d_j}|[b,\\mathcal {R}]_1(f_j,g_j)(x)|^p\\,dx \\nonumber \\\\&\\quad \\ge \\int _{\\widetilde{\\gamma }_1d_j<|x-y_j|<\\widetilde{\\gamma }_2d_j}|\\mathcal {R}\\left((b-b_Q)f_j,g_j\\right)(x)|^p\\,dx \\nonumber \\\\&\\quad \\ \\ \\ \\ -\\int _{\\widetilde{\\gamma }_1d_j<|x-y_j|}|(b(x)-b_Q)\\mathcal {R}(f_j,g_j)(x)|^p\\,dx \\nonumber \\\\\\ &\\quad \\ge C_1 \\varepsilon ^p|Q_j|^{\\left(\\frac{1}{p_1^{\\prime }}+\\frac{1}{p_2^{\\prime }}\\right)p}\\int _{\\widetilde{Q}_j}|x-y_j|^{-2np}\\,dx -C_2\\widetilde{\\gamma }_1^{-\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)p}\\nonumber \\\\&\\quad \\ge C_1 \\varepsilon ^p|Q_j|^{\\left(\\frac{1}{p_1^{\\prime }}+\\frac{1}{p_2^{\\prime }}\\right)p} |\\widetilde{Q}_j| \\widetilde{\\gamma }_2^{-2np} |Q_j|^{-2p} -C_2\\widetilde{\\gamma }_1^{-\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)p}\\nonumber \\\\&\\quad \\ge C_1 \\varepsilon ^p (4 \\sqrt{n})^{-n} \\widetilde{\\gamma }_2^{-2np+n} -C_2 8^{\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)p} \\widetilde{\\gamma }_2^{-\\left(2n-\\frac{n}{p}+\\frac{1}{2}\\right)p}.",
"$ We can now use (REF ) and (REF ) or (REF ) to replace $\\widetilde{\\gamma }_1,\\widetilde{\\gamma }_2$ with $\\gamma _1$ sufficiently large and $\\gamma _2=8 \\gamma _1$ , so that (REF ) and () are verified for some $\\gamma _3>0$ .",
"From here the arguments used in [3], which in turn followed the ones in [4], can be repeated without any changes.",
"Namely, it is possible to construct sequences of cubes $\\lbrace Q_j\\rbrace $ and functions $\\lbrace f_j\\rbrace $ , $\\lbrace g_j\\rbrace $ in exactly the same way as in [3] so that if any one of the conditions ()–() were to be violated by $b$ , then we would arrive at a contradiction with the compactness of $[b,\\mathcal {R}]_1$ .",
"The reader can easily follow the argument in [3], simply replacing $[b,I_\\alpha ]_1$ therein by $[b,\\mathcal {R}]_1$ .",
"To make our paper more self-contained, we now sketch an outline of the argument.",
"Using (REF ) and () it can be shown that given $\\gamma _1$ , $\\gamma _2$ , and $\\gamma _3$ from (REF ) and (), there exists a $\\beta $ with $0<\\beta \\ll \\gamma _2$ , depending on $p_1,\\ p_2,\\ n,$ and $\\varepsilon $ , such that for each measurable set $E\\subset \\lbrace x:\\gamma _1d_j<|x-y_j|<\\gamma _2d_j\\rbrace $ with $|E|/|Q_j|<\\beta ^n$ , we get $\\left\\Vert [b,\\mathcal {R}]_1(f_j,g_j)\\right\\Vert _{L^p(E)}\\le \\frac{\\gamma _3}{4}.$ This estimate relies on the fact that the result of Lemma 3.17 (1) of [11], which is stated there for $p=1$ , also holds for all $p>0$ , and hence also applies in our case, where $p>1/2$ .",
"In [4], the estimate corresponding to our (REF ) was obtained using the case $p\\ge 1$ of this lemma.",
"With this in hand, if we suppose that any one of the conditions ()–() on $b$ fails, we can construct a sequence of functions that will lead us to a contradiction with the compactness of $[b,\\mathcal {R}]_1$ .",
"For instance, if $b$ does not satisfy (), then there exist some $\\varepsilon >0$ and a sequence $\\lbrace Q_j\\rbrace $ of cubes with $|Q_j|\\rightarrow 0$ as $j\\rightarrow \\infty $ such that $\\frac{1}{|Q_j|}\\int _{Q_j}|b(y)-b_{Q_j}|\\,dy\\ge \\varepsilon ,$ for every $j$ .",
"First, select a subsequence, denoted by $\\lbrace Q_j^{(i)}\\rbrace $ , so that the side-lengths satisfy $\\frac{d_{j+1}^{(i)}}{d_{j}^{(i)}}&<\\frac{\\beta }{2\\gamma _2}.$ Next, let $f_j^{(i)}$ and $g_j^{(i)}$ , as defined before, be the functions associated to the selected cubes $Q_j^{(i)}$ .",
"Finally, for each $k$ , $m\\in \\mathbb {N}$ , consider the sets: $G&:=\\lbrace x:\\gamma _1d^{(i)}_k<|x-y_k^{(i)}|<\\gamma _2d_k^{(i)}\\rbrace ,\\\\G_1&:=G\\setminus \\lbrace x:|x-y_{k+m}^{(i)}|\\le \\gamma _2d_{k+m}^{(i)}\\rbrace ,\\\\G_2&:=\\lbrace x:|x-y_{k+m}^{(i)}|>\\gamma _2d_{k+m}^{(i)}\\rbrace .$ The choice of the $Q_j^{(i)}$ s implies that $\\frac{|G_2^c\\cap G|}{|Q_k^{(i)}|}\\le \\beta ^n; $ see again [4].",
"For $p\\ge 1$ , we can then estimate $\\Vert [b,\\mathcal {R}]_1&(f_k^{(i)},g_k^{(i)})-[b,\\mathcal {R}]_1(f_{k+m}^{(i)},g_{k+m}^{(i)})\\Vert _{L^p}\\nonumber \\\\&\\ge \\left(\\int _{G}\\left|[b,\\mathcal {R}]_1(f_k^{(i)},g_k^{(i)})\\right|^p - \\int _{G_2^c\\cap G}\\left|[b,\\mathcal {R}]_1(f_k^{(i)},g_k^{(i)})\\right|^p\\right)^{\\frac{1}{p}} \\\\&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-\\left(\\int _{G_2}\\left|[b,\\mathcal {R}]_1(f_{k+m}^{(i)},g_{k+m}^{(i)})\\right|^p\\right)^{\\frac{1}{p}}.\\nonumber $ Applying (REF ), (REF ), and () respectively to the three terms on the right-hand side of (REF ), we conclude $\\Vert [b,\\mathcal {R}]_1(f_k^{(i)},g_k^{(i)})-[b,\\mathcal {R}]_1(f_{k+m}^{(i)},g_{k+m}^{(i)})\\Vert _{L^p}&\\ge \\left(\\gamma _3^p-\\frac{\\gamma _{3}^p}{4^p}\\right)^{\\frac{1}{p}}-\\frac{\\gamma _3}{4}\\\\&\\ge \\frac{\\gamma _3}{2},$ at least for $p\\ge 1$ .",
"In the case of $1/2 < p < 1$ , a similar argument using the reverse triangle inequality applied to the $p^{\\text{th}}$ power of the left-hand side of (REF ) leads to the lower bound $\\Vert [b,\\mathcal {R}]_1(f_k^{(i)},g_k^{(i)})-[b,\\mathcal {R}]_1(f_{k+m}^{(i)},g_{k+m}^{(i)})\\Vert ^p_{L^p}\\ge \\left(1-\\frac{2}{4^p}\\right)\\gamma _3^p.$ This means that the image of the bounded set $\\lbrace (f_j,g_j)\\rbrace _j$ is not precompact, which contradicts our assumption on $[b,\\mathcal {R}]_1$ .",
"The cases where $b$ does not satisfy condition () or condition () are handled similarly, and we conclude our proof here.",
"Remark 3.2 We observe that the arguments used for the Riesz transforms $\\mathcal {R}^k_j$ in Theorem REF also go through in more generality.",
"In order to get the lower bound (as in formulas (REF ) and (REF ) above), one usually uses the assumption that the kernel of the operator is positive, if not in the whole space, then at least in a substantial portion of the space.",
"For the Riesz transforms $\\mathcal {R}^k_j$ , although the kernel is not positive, for each cube $Q_j$ we can find another cube $\\widetilde{Q}_j$ such that $\\widetilde{Q}_j$ lies in some large annulus centered at the centre $y_j$ of $Q_j$ , and for all $x\\in \\widetilde{Q}_j$ and $y$ , $z\\in Q_j$ , $K(x-y, x-z)>0 \\, \\quad \\quad {\\rm and}\\quad \\quad \\,|x-y| \\approx |x-z| \\approx |x-y_j|.$ This condition together with the Calderón–Zygmund conditions on the size and regularity of the kernel suffice to obtain the lower bound.",
"This idea applies to certain other bounded convolution-type singular operators, as we now discuss.",
"In the linear case, as is shown in Uchiyama's paper [14], the Riesz transform can be replaced by convolution-type singular integral operators with kernel of the form $K(x)= \\frac{ \\Omega \\left( x\\right)}{|x|^n},$ where $\\Omega $ is a homogeneous function of degree zero defined on the unit sphere in $\\mathbb {R}^n$ and is sufficiently smooth.",
"Such a kernel is locally positive in the sense that there is some spherical cap $A$ in the unit sphere $S^{n-1}$ such that $\\Omega \\left( x\\right) > c_0 > 0$ for all $x \\in A$ .",
"Turning to the bilinear case, the arguments used for the bilinear Riesz transforms $\\mathcal {R}^k_j$ in Theorem REF can be repeated for bounded convolution bilinear operators with kernel of the form $K(y,z)= \\frac{ \\Omega \\left( \\frac{(y,z)}{|(y,z)|}\\right)}{(|y|^2+|z|^2)^n},$ where $\\Omega $ is a homogeneous function of degree zero defined on the unit sphere in $\\mathbb {R}^n\\times \\mathbb {R}^n$ and is sufficiently smooth.",
"We need more assumptions on this kernel than in the linear case.",
"First, we assume that $1/K$ has an absolutely convergent Fourier series in some ball in $\\mathbb {R}^{2n}$ .",
"This assumption guarantees that the boundedness of the commutator operator with a function $b$ implies that $b \\in {\\rm BMO}$ , by the main result of [2].",
"Second, we assume that there is some spherical cap $A$ on the unit sphere $S^{n-1}$ such that $ \\Omega \\left( \\frac{(y,z)}{|(y,z)|}\\right) > c_0 > 0$ for all $y$ , $z \\in A$ .",
"This assumption enables us to get the lower bound estimate (REF ).",
"Indeed, given a cube $Q_j$ centred at $y_j$ , we can find another cube $\\widetilde{Q}_j$ such that $\\widetilde{Q}_j$ lies in some large annulus centered at $y_j$ , and for all $x\\in \\widetilde{Q}_j$ and all $y$ , $z\\in Q_j$ , $x-y$ and $x - z$ lie in an infinite cone in $\\mathbb {R}^n$ whose vertex is at the origin and which passes through the cap $A$ .",
"From our assumption, it follows that $K(x-y, x-z)>0 \\, \\quad \\quad {\\rm and}\\quad \\quad \\,|x-y| \\approx |x-z| \\approx |x-y_j|$ for all $x \\in \\widetilde{Q}_j$ and $y$ , $z\\in Q_j$ .",
"The computations in the proof of Theorem REF can now be repeated.",
"We leave the details to the interested reader."
]
] | 1709.01701 |
[
[
"Room-temperature superparamagnetism due to giant magnetic anisotropy in\n Mo$_{S}$ defected single-layer MoS$_{2}$"
],
[
"Abstract Room-temperature superparamagnetism due to a large magnetic anisotropy energy (MAE) of a single atom magnet has always been a prerequisite for nanoscale magnetic devices.",
"Realization of two dimensional (2D) materials such as single-layer (SL) MoS$_{2}$, has provided new platforms for exploring magnetic effects, which is important for both fundamental research and for industrial applications.",
"Here, we use density functional theory (DFT) to show that the antisite defect (Mo$_{S}$) in SL MoS$_{2}$ is magnetic in nature with a magnetic moment of $\\mu$ of $\\sim$ 2$\\mu_{B}$ and, remarkably, exhibits an exceptionally large atomic scale MAE$=\\varepsilon_{\\parallel}-\\varepsilon_{\\perp}$ of $\\sim$500 meV.",
"Our calculations reveal that this giant anisotropy is the joint effect of strong crystal field and significant spin-orbit coupling (SOC).",
"In addition, the magnetic moment $\\mu$ can be tuned between 1$\\mu_{B}$ and 3$\\mu_{B}$ by varying the Fermi energy $\\varepsilon_{F}$, which can be achieved either by changing the gate voltage or by chemical doping.",
"We also show that MAE can be raised to $\\sim$1 eV with n-type doping of the MoS$_{2}$:Mo$_{S}$ sample.",
"Our systematic investigations deepen our understanding of spin-related phenomena in SL MoS$_{2}$ and could provide a route to nanoscale spintronic devices."
],
[
"Appendix",
"We derive here the crystal field Hamiltonian for trigonal symmetry (Fig.",
"REF ).",
"The contribution of the surroundings point charges (Mo atoms, Fig.",
"REF ) to the electron potential energy at Mo$_{S}$ site can be expressed as $V_{CF}=\\displaystyle \\sum _{i=1}^{3} \\frac{Ze^2}{|\\vec{r}-\\vec{R}_{i}|}$ where $\\vec{r}$ is the electron corrdinate and $\\vec{R}_{i}$ are the position vectors of the neighboring point charges.",
"With the help of Mathematica [39] we can write down the expression for the crystal field Hamiltonian $\\begin{split}V_{CF}=C_{0}+C_{1}\\rho Y_{1}^0+C_{2}\\rho ^2Y_{2}^0+\\rho ^3[C_{3}Y_{3}^0\\\\+C_{3}^{\\prime }(Y_{3}^{-3}+Y_{3}^{3})]+\\rho ^4[C_{4}Y_{4}^0+C_{4}^{\\prime }((Y_{4}^{-3}+Y_{4}^{3}))]....,\\end{split}$ Figure: Trigonal symmetry seen by the Mo S _S atom.",
"The origin is set at the Mo S _{S} atom.",
"One of the Mo atom is set at x-axis and the coordinates of the 2 and 3 atoms are obtained through rotation of coordintes.where $\\rho =r/\\sqrt{R^2+P^2}$ and $Y_l^m$ are the spherical harmonics with orbital angular momentum quantum numbers $l$ and $m$ .",
"The expansion coefficients $C_j$ , $j=0,1,2,...$ can be adjusted to fit the DFT results.",
"Here we use $d$ -orbitals of the Mo$_{S}$ atom, i.e.",
"$d_{x^2-y^2}=(Y_{2}^{-2}+Y_{2}^{2})/\\sqrt{2}, d_{xy}=i(Y_{2}^{-2}-Y_{2}^{2})/\\sqrt{2},d_{z^2}=Y_{2}^0,d_{xz}=(Y_{2}^{-1}+Y_{2}^{1})/\\sqrt{2}$ and $d_{yz}=i(Y_{2}^{-1}-Y_{2}^{1})/\\sqrt{2}$ .",
"Spherical harmonics with odd magnetic quantum numbers do not contribute, thus $V_{CF}\\sim \\rho ^{2}Y_{2}^{0}$ in lowest order.",
"The matrix elements of the $V_{CF}$ between different $d$ -orbitals may be written as $\\hat{H}_{mm^{\\prime }}^{cry}\\sim \\int \\psi _{nl}^{*}(r)\\rho ^{2}\\psi _{nl}(r)r^{2}dr\\iint d_{m}(\\theta ,\\phi )Y_{2}^{0}d_{m^\\prime }(\\theta ,\\phi )d\\theta d\\phi $ where $\\psi _{nl}(r)$ is the radial function for Mo$_{S}$ atom ($n=4$ , $l=2$ ) and subscripts $m$ and $m^{\\prime }$ stand for different $d$ -orbitals of the Mo$_{S}$ atom.",
"In this work we are able to omit the radial parts by fitting the appearing integrals, this spatial distribution may be omitted, wich allows to simplify the treatment with any loss of accuracy.",
"The diagonal matrix elements are given by $\\begin{split}\\mathinner {\\langle {d_{z^2}}|}Y_{2}^{0}\\mathinner {|{d_{z^2}}\\rangle }=145\\sqrt{5\\pi }/512=E_0, \\\\\\mathinner {\\langle {d_{x^2-y^2}}|}Y_{2}^{0}\\mathinner {|{d_{x^2-y^2}}\\rangle }=-45\\sqrt{5\\pi }/1024=E_2, \\\\\\mathinner {\\langle {d_{xy}}|}Y_{2}^{0}\\mathinner {|{d_{xy}}\\rangle }=-45\\sqrt{5\\pi }/1024=E_2, \\\\\\mathinner {\\langle {d_{xz}}|}Y_{2}^{0}\\mathinner {|{d_{xz}}\\rangle }=-15\\sqrt{5\\pi }/256=E_1, \\\\\\mathinner {\\langle {d_{yz}}|}Y_{2}^{0}\\mathinner {|{d_{yz}}\\rangle }=-15\\sqrt{5\\pi }/256=E_1.",
"\\\\\\end{split}$ It should be noted that all the of diagonal terms are zero with in the lowest approximation ($V_{CF}\\sim Y_{2}^0$ ).",
"Eq.",
"(REF ) correctly reproduces the numerical results, i.e.",
"two doublets $d_{x^2-y^2}/d_{xy}$ , $d_{xz}/d_{yz}$ and a singlet $d_{z^2}$ with the correct energy sequence $E_{0}>E_{2}>E_{1}$ .",
"Considering the fact that crystal field theory preserves the level splittings with respect to the degenerate d-orbitals of the isolated Mo atoms, i.e.",
"$E_{0}+2E_{1}+2E_{2}=0$ , the eigenenergies of the crystal field Hamiltonian can be written in the form of energy differences $\\Delta _{1}=E_2-E_1$ and $\\Delta _{2}=E_0-E_2$ with $E_0>0$ , $E_1<0$ , $E_2>0$ , as shown in Fig.",
"REF .",
"The crystal field Hamiltonian may be written as $\\hat{H}^{cry} = \\left(\\begin{array}{ccccc}\\frac{1}{5}(2\\Delta _{1}-\\Delta _{2})&0&0&0&0\\\\0&\\frac{1}{5}(2\\Delta _{1}-\\Delta _{2})&0&0&0\\\\0&0&\\frac{2}{5}(\\Delta _{1}+2\\Delta _{2})&0&0\\\\0&0&0&-\\frac{1}{5}(3\\Delta _{1}+\\Delta _{2})&0\\\\0&0&0&0&-\\frac{1}{5}(3\\Delta _{1}+\\Delta _{2})\\\\\\end{array}\\right),$ SOC is considered as the onsite interaction $\\hat{H}^{SOC}=\\xi L\\cdot S$ .",
"Using the d-orbital bases $\\mathinner {|{d_{x^2-y^2},\\uparrow }\\rangle }, \\mathinner {|{d_{xy},\\uparrow }\\rangle },\\mathinner {|{d_{z^2},\\uparrow }\\rangle },\\mathinner {|{d_{xz},\\uparrow }\\rangle },\\mathinner {|{d_{yz},\\uparrow }\\rangle }$ and $\\mathinner {|{d_{x^2-y^2},\\downarrow }\\rangle }, \\mathinner {|{d_{xy},\\downarrow }\\rangle },\\mathinner {|{d_{z^2},\\downarrow }\\rangle },\\mathinner {|{d_{xz},\\downarrow }\\rangle },\\mathinner {|{d_{yz},\\downarrow }\\rangle }$ , we get the SOC contribution to the Hamiltonians $\\hat{H}^{SOC}(\\vec{M}\\parallel \\hat{z})$ and $\\hat{H}^{SOC}(\\vec{M}\\parallel \\hat{x})$ as $\\hat{H}^{SOC}(\\vec{M}\\parallel \\hat{z}) =\\left(\\begin{array}{cccccccccc}0 & -2i\\xi & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{\\xi }{2} & i\\frac{\\xi }{2} \\\\2i\\xi & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - i\\frac{\\xi }{2} & \\frac{\\xi }{2} \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{\\sqrt{3}}{2}\\xi & -i\\frac{\\sqrt{3}}{2}\\xi \\\\0 & 0 & 0 & 0 & -i\\xi & \\frac{\\xi }{2} & -i\\frac{\\xi }{2} &\\frac{\\sqrt{3}}{2}\\xi & 0 & 0 \\\\0 & 0 & 0 & i\\xi & 0 & i\\frac{\\xi }{2} & \\frac{\\xi }{2} & -i\\frac{\\sqrt{3}}{2}\\xi & 0 & 0 \\\\0 & 0 & 0 & \\frac{\\xi }{2} & -i\\frac{\\xi }{2} & 0 & -2i\\xi & 0 & 0 & 0 \\\\0 & 0 & 0 & i\\frac{\\xi }{2} & \\frac{\\xi }{2} & 2i\\xi & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & \\frac{\\sqrt{3}}{2}\\xi & i\\frac{\\sqrt{3}}{2}\\xi & 0 & 0 & 0 & 0 & 0 \\\\\\frac{\\xi }{2} & i\\frac{\\xi }{2} & \\frac{\\sqrt{3}}{2}\\xi & 0 & 0 & 0 & 0 & 0 & 0 & -i\\xi \\\\-i\\frac{\\xi }{2} & \\frac{\\xi }{2} & i\\frac{\\sqrt{3}}{2}\\xi & 0 & 0 & 0 & 0 & 0 & i\\xi & 0 \\\\\\end{array}\\right)$ and $\\hat{H}^{SOC}(\\vec{M}\\parallel \\hat{x}) = \\left(\\begin{array}{cccccccccc}0 & 0 & 0 & \\xi & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & \\xi & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & \\sqrt{3}\\xi & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\xi & 0 & \\sqrt{3}\\xi & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & \\xi & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\xi & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\xi \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\sqrt{3}\\xi & 0 \\\\0 & 0 & 0 & 0 & 0 & \\xi & 0 & \\sqrt{3}\\xi & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & \\xi & 0 & 0 & 0 \\\\\\end{array}\\right),$ respectively."
]
] | 1709.01653 |
[
[
"Crank-Nicolson scheme for stochastic differential equations driven by\n fractional Brownian motions"
],
[
"Abstract We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by multidimensional fractional Brownian motion $(B^{1}, \\dots, B^{m})$ with Hurst parameter $H \\in (\\frac 12,1)$.",
"It is well-known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of $n^{-2}$, regardless of the dimension.",
"In this paper we show that, due to the interactions between the driving processes $ B^{1}, \\dots, B^{m} $, the corresponding Crank-Nicolson scheme for $m$-dimensional SDEs has a slower rate than for the one-dimensional SDEs.",
"Precisely, we shall prove that when $m=1$ and when the drift term is zero, the Crank-Nicolson scheme achieves the exact convergence rate $n^{-2H}$, while in the case $m=1$ and the drift term is non-zero, the exact rate turns out to be $n^{-\\frac12 -H}$.",
"In the general case when $m>1$, the exact rate equals $n^{\\frac12 -2H}$.",
"In all these cases the limiting distribution of the leading error is proved to satisfy some linear SDE driven by Brownian motions independent of the given fractional Brownian motions."
],
[
"Introduction",
"This paper is concerned with the following stochastic differential equation (SDE) driven by fractional Brownian motion on $ \\mathbb {R}^d$ $X_t &= &x + \\int ^t_0 V (X_s)dB_s \\,, \\quad t \\in [0, T]\\,, $ where $B = (B^{0},B^{1},\\dots , B^{m}) $ , and $ ( B^{1},\\dots , B^{m}) $ is an $m$ -dimensional fractional Brownian motion (fBm) with Hurst parameter $H>\\frac{1}{2}$ .",
"For notational convenience we denote $B^{0}_{t}=t$ for $t \\in [0,T]$ in order to include the drift term in (REF ).",
"The integral on the right-hand side of (REF ) is of Riemann-Stieltjes type.",
"It is well-known that if the vector field $ V = (V_0,V_1, \\dots ,V_m): \\mathbb {R}^{d} \\rightarrow \\mathcal {L} (\\mathbb {R}^{m+1}, \\mathbb {R}^{d}) $ has bounded partial derivatives which are Hölder continuous of order $ {\\alpha }> \\frac{1}{H} - 1 $ , then there exists a unique solution for equation (REF ), which has bounded $\\frac{1}{{\\gamma }}$ -variation on $[0,T]$ for any ${\\gamma }<H$ ; see e.g.",
"[13], [22].",
"As in the Brownian motion case, the explicit solution of SDEs driven by fractional Brownian motions are rarely known.",
"Thus one has to rely on numerical methods for simulations of these equations.",
"The simplest time-discrete numerical approximation scheme is the Euler scheme $X^{n}_{t_{k+1}} &=& X^{n}_{t_{k}} + V(X^{n}_{t_{k}}) (B_{t_{k+1}} - B_{t_{k}}),\\nonumber \\\\X^{n }_0 &=& x,$ where $k=0,1,\\dots , n-1$ and $t_{k} = kT/n$ .",
"This scheme has first been considered in [16], [17] for SDEs in the one-dimensional case, and generalized in [10], [14] to the multidimensional case.",
"The solution of (REF ) has the exact strong convergence rate of $n^{1-2H}$ when $H>\\frac{1}{2}$ .",
"When $H=\\frac{1}{2}$ the scheme converges to the corresponding Itô SDE $X_{t} &=&x+ \\int _{0}^{t}V(X_{s}) \\delta B_{s} \\,, \\quad t \\in [0, T],$ where $\\delta $ denotes the Itô stochastic integral.",
"Note also that the Euler scheme is not convergent when $H< \\frac{1}{2} $ ; see e.g.",
"[5].",
"A modified Euler scheme, introduced in [10], generalizes the classical Euler scheme to the fBm case $X^{n}_{t_{k+1}} &=& X^{n}_{t_{k}} + V(X^{n}_{t_{k}}) (B_{t_{k+1}} - B_{t_{k}}) + \\frac{1}{2} \\sum _{ j=1}^{m} \\partial V_{j}V_{j} (X^{n}_{t_{k}}) \\Big ( \\frac{T}{n} \\Big )^{2H} ,\\nonumber \\\\X^{n }_0 &=& x.$ The modified Euler scheme has been shown to have a better convergence rate than (REF ).",
"More precisely, the rate is $n^{\\frac{1}{2} -2H}$ when $\\frac{1}{2} <H<\\frac{3}{4}$ and $ n^{-1} {\\sqrt{\\ln n}} $ when $H = \\frac{3}{4}$ , and in the case $\\frac{3}{4} <H<1$ the rate is $n^{-1}$ .",
"The weak convergence rates and the asymptotic error distributions were also obtained for this modified Euler scheme.",
"In [9], the authors considered Taylor schemes derived from the Taylor expansion in the one-dimensional case.",
"In [11], the Taylor schemes and their modifications were introduced for SDEs driven by fBms $B^{1},\\dots , B^{m}$ , with Hurst parameters $H_{1},\\dots , H_{m}$ , where $ H_{1},\\dots , H_{m} \\in (\\frac{1}{2}, 1] $ are not necessarily equal.",
"In [4], the Milstein scheme (or 2nd-order Taylor scheme) has been considered for the rough case $ H<\\frac{1}{2}$ and it is convergent as long as $H>\\frac{1}{3}$ .",
"An extension of the result to $m$ -th order Taylor schemes is contained in [7].",
"In [5], [6], some 2nd and 3rd order implementable schemes are studied via the Wong-Zakai approximation of (REF ).",
"The Crank-Nicolson (or Trapezoidal) scheme has been studied only recently.",
"Recall that the Crank-Nicolson scheme for $\\hbox{(\\ref {e.1.1})}$ is defined as follows: $X^{n }_{t_{k+1}} &=& X^{n }_{t_k}+\\frac{1}{2} \\big [ V (X^n_{ t_{k+1}} ) + V (X^n_{t_k } )\\big ] (B_{t_{k+1}}-B_{t_k}),\\nonumber \\\\X^{n }_0 &=& x ,$ where again $t_k=kT/n$ for $k=0,\\dots , n-1$ .",
"In [15], [17], the Crank-Nicolson scheme is considered for SDEs with Hurst parameter $H \\in (1/3, 1/2)$ .",
"It has been shown in [17] that if $V \\in C^{\\infty }_{b}$ the convergence rate of the Crank-Nicolson scheme is $n^{\\frac{1}{2}-3H}$ .",
"This rate is exact in the sense that the renormalized error process $n^{3H-\\frac{1}{2}} (X-X^{n})$ converges weakly to a non-zero limit (see e.g.",
"[15]).",
"However, due to the use of the Doss-Sussmann representation, these results are applicable only to the scalar SDE setting, which corresponds, with our notation, to the case $m=d=1$ and $ V_{0} \\equiv 0$ .",
"On the other hand, it has been conjectured in [18] that the Crank-Nicolson scheme has exact root mean square convergence rate $n^{ \\frac{1}{2}-2H}$ .",
"In view of these results, our first goal is to answer the following question: Question 1: Is the Crank-Nicolson scheme still convergent in the multidimensional setting, and is the convergence rate the same as that of the scalar SDE?",
"Let us recall that in the case of deterministic ordinary differential equations (ODEs), either in the one-dimensional or multidimensional settings, and with proper regularity assumptions on $V$ , the convergence rate of the Crank-Nicolson scheme is always $ n^{-2} $ .",
"Surprisingly, as we will show in this paper, the Crank-Nicolson scheme (REF ) for equation (REF ) has very different features comparing to the ODE cases.",
"It turns out that, while the Crank-Nicolson scheme in the multidimensional case still converges to the solution $X$ of equation (REF ), the convergence rate is largely “throttled” due to the interactions between the driving processes $B^{0}$ , $B^{1}$ , ..., $B^{m}$ .",
"More precisely, we will prove the following result.",
"Let $X^n_t$ denote the continuous time interpolation of the Crank-Nicolson scheme defined by $ X^{n }_t &=& X^{n }_{t_k}+\\frac{1}{2} \\left[ V (X^n_{t_k } ) + V (X^n_{ t_{k+1} } ) \\right] (B_t-B_{t_k}),$ for $t \\in [t_{k},t_{k+1})$ , $k=0, \\dots , n-1$ .",
"Theorem 1.1 Let $X$ be the solution of equation (REF ) and let $X^n$ be the continuous time interpolation of the Crank-Nicolson scheme $\\left\\lbrace X^{n }_{t_0}, X^{n }_{t_1}, \\cdots , X^{n }_{t_n}\\right\\rbrace $ defined by (REF ).",
"Assume that $V \\in C^{3}_{b}$ .",
"Then for any $p \\ge 1$ there exists a constant $K=K_p$ independent of $n$ such that the following strong convergence result holds true for all $n\\in \\mathbb {N}$ : $\\sup _{t\\in [0,T]} \\left( \\mathbb {E}|X_{t}-X^{n}_{t} |^{p } \\right)^{1/p} &\\le &K/\\vartheta _{n},$ where $\\vartheta _{n}$ is defined as $\\vartheta _{n}&= &{\\left\\lbrace \\begin{array}{ll}n^{2H-\\frac{1}{2}} & \\quad \\text{when}\\quad m>1,\\\\n^{H+\\frac{1}{2}} & \\quad \\text{when}\\quad m=1 \\text{ and } V_{0} \\lnot \\equiv 0,\\\\n^{2H} &\\quad \\text{when}\\quad m=1 \\text{ and } V_{0} \\equiv 0.\\end{array}\\right.",
"}$ Note from Theorem REF that if $m=1$ and $V_0\\equiv 0$ , the convergence rate of the Crank-Nicolson scheme (REF ) is $n^{-2H}$ .",
"This result coincides with the case of deterministic ODEs if we formally set $H=1$ , and also the case of SDEs driven by a one-dimensional Brownian motion, that is, $H=\\frac{1}{2}$ (see, e.g.",
"[15], [17]).",
"If $m=1$ and $V_0 \\lnot \\equiv 0$ , then the rate turns out to be $n^{-H-\\frac{1}{2}}$ .",
"In the general case when $m>1$ , the converges rate becomes $n^{\\frac{1}{2} -2H}$ , which coincides with the modified Euler scheme defined in (REF ) when $\\frac{1}{2} <H<\\frac{3}{4}$ .",
"Note also that this gives a positive answer to the conjecture raised in [18] under this general assumption.",
"The slowing down of convergence rate from one-dimensional case to multidimensional cases is due to the nonvanishing Lévy area term, while in the one-dimensional case, these Lévy area type processes disappear and the convergence rate of $X-X^{n}$ is then dictated by the higher order terms.",
"The second part of the paper is motivated by the following question: Question 2: Are the convergence rates obtained in Theorem REF exact?",
"If yes, what is the limiting distributions of the leading term for both the one-dimensional and multidimensional cases?",
"Note that the different features observed between the one-dimensional and the multidimensional cases are true only if the rates are exact.",
"To this aim, we consider the piecewise constant interpolations.",
"Namely, we consider the processes $\\tilde{X}^{n}$ and $\\tilde{X}$ , where $&&\\tilde{X}_t :=X_{t_k} \\quad \\text{and} \\quad \\tilde{X}^n_t:= {X}^n_{t_k} ,$ for $t\\in [t_k, t_{k+1})$ , $k=0,1,\\dots , n$ , and as a consequence we have $ \\tilde{X}_T:= X_T \\quad \\text{and} \\quad \\tilde{X}^n_T:= X^n_T$ .",
"Recall that here $X$ is the solution of equation (REF ) and $X^{n}$ is the solution of (REF ).",
"The following theorem provides a complete picture of the asymptotic behaviors of the Crank-Nicolson scheme.",
"Theorem 1.2 Let $\\tilde{X}$ and $\\tilde{X}^n$ be the processes defined in (REF ).",
"Suppose that $V\\in C^3_b$ .",
"Denote $ \\phi _{jj^{\\prime }} = \\partial V_j V_{j^{\\prime }}-\\partial V_{j^{\\prime }} V_{j }$ for $j,j^{\\prime }=0,1,\\dots ,m$ .",
"(i) Assume that $m>1$ or $m=1$ but $V_{0} \\lnot \\equiv 0$ .",
"Then we have the convergence $( \\vartheta _{n} ( \\tilde{X} - \\tilde{X}^{n} ), B) &\\rightarrow & (U,B) $ in the Skorohod space $D([0,T]; \\mathbb {R}^{d+m+1})$ as $n$ tends to infinity.",
"In the case $m>1$ , the process $U$ is the solution of the following linear SDE on $[0,T]$ $ dU_{t}&= & \\sum _{j=0}^m\\partial V_j ( X_t )U_{t} dB^{j }_t+T^{2H-\\frac{1}{2}} \\sqrt{ \\frac{ \\kappa }{2} } \\sum _{1 \\le j^{\\prime } < j \\le m } \\phi _{jj^{\\prime }} (X_{t} ) dW^{j^{\\prime }j}_{t}, \\quad U_{0}= 0,$ where $W=(W^{j^{\\prime }j})_{1\\le j^{\\prime }<j \\le m}$ is a standard $\\frac{m(m-1)}{2}$ -dimensional Brownian motion independent of $B$ and $\\kappa $ is the constant defined in (REF ) in Section 3.",
"In the case $m=1$ and $V_{0} \\lnot \\equiv 0$ , $U$ is the solution of the following linear SDE on $[0,T]$ $dU_{t}&= & \\sum _{j=0,1}\\partial V_j ( X_t )U_{t} dB^{j }_t+T^{H+\\frac{1}{2}} \\sqrt{ \\frac{ \\varrho }{2} } \\phi _{10} (X_{t} ) dW_{t}, \\quad U_{0}=0, $ where $W$ is a one-dimensional standard Brownian motion independent of $B$ and $\\rho $ is the constant defined in (REF ) in Section 3.",
"(ii) Assume that $m=1$ and $V_{0} \\equiv 0$ .",
"Then, we have the following convergence in $L^p(\\Omega )$ for all $p\\ge 1$ and $t\\in [0,T]$ : $ n^{2H}( \\tilde{X}_{t} - \\tilde{X}^{n}_{t} ) &\\rightarrow &U_{t}\\,,$ where the process $U$ satisfies the following linear SDE on $[0,T]$ $ dU_{t}&=&\\partial V ( X_t )U_{t} dB_t-\\frac{T^{2H}}{4} \\sum _{i,i^{\\prime }=1}^{d} ( V^{i}V^{i^{\\prime }}\\partial _{i}\\partial _{i^{\\prime }}V ) (X_{t} ) dB_{t}, \\quad U_{0}=0.$ Theorem REF shows that in the cases $m>1$ or $V_{0} \\lnot \\equiv 0$ , one obtains the central limit theorem for the renormalized error process $\\vartheta _{n} (X-X^{n})$ , while in the case $m=0$ and $V_{0} \\equiv 0$ , one gets the convergence in $L^{p}$ .",
"It is interesting to point out that the cutoff of the convergence rates observed in [10], [18] is not present in either of these cases here.",
"Our approach to prove Theorem REF and Theorem REF is based on the explicit expression of $X-X^{n}$ we have mentioned previously, similar to that established in [10].",
"A significant difficulty is the integrability of the Malliavin derivatives of the approximation $X^{n}$ .",
"This is due to the fact that the Crank-Nicolson scheme (REF ) is determined by an implicit equation.",
"This difficulty will be handled thanks to some fractional calculus techniques, see e.g.",
"[3], [11], [26].",
"A special attention has to be paid also to the Lévy area type processes mentioned above.",
"Our approach to handle these processes relies on a combination of fractional calculus and Malliavin calculus tools.",
"The paper is structured as follows.",
"In Section , we recall some basic results on the fBms as well as some upper bound estimate results and limit theorem results on fractional integrals.",
"In Section , we consider the moment estimates and the weak convergence of some Lévy area type processes.",
"In Section , we prove Theorem REF , and then in Section , we prove Theorem REF .",
"Some auxiliary results are stated and proved in the appendix."
],
[
"Fractional Brownian motions",
"We briefly review some basic facts about the stochastic calculus with respect to a fBm.",
"The reader is referred to [19], [20] for further details.",
"Let $B= \\lbrace B_{t}\\,, \\,t\\in [0,T]\\rbrace $ be a one-dimensional fBm with Hurst parameter $H \\in (\\frac{1}{2}, 1)$ , defined on some complete probability space $(\\Omega , {F}, P)$ .",
"Namely, $B$ is a mean zero Gaussian process with covariance $\\mathbb {E}(B_{s}B_{t}) &=& \\frac{1}{2} (t^{2H} +s^{2H}-|t-s|^{2H})$ for $s,t \\in [0,T]$ .",
"Let $\\mathcal {H}$ be the Hilbert space defined as the closure of the set of step functions on $[0,T]$ with respect to the scalar product $\\langle \\mathbf {1}_{[0,t]}, \\mathbf {1}_{[0,s]} \\rangle _{\\mathcal {H}} &=& \\frac{1}{2} (t^{2H} +s^{2H}-|t-s|^{2H}).$ It is easy to verify that $\\langle \\phi , \\psi \\rangle _{\\mathcal {H}} &=&H(2H-1) \\int _{0}^{T} \\int _{0}^{T} \\phi _{u} \\psi _{v} |u-v|^{2H-2} dudv$ for every pair of step functions $ \\phi , \\psi \\in \\mathcal {H} $ .",
"The mapping $ \\mathbf {1}_{[0,t]} \\mapsto B_{t} $ can be extended to a linear isometry between $\\mathcal {H}$ and the Gaussian space spanned by $B$ .",
"We denote this isometry by $h \\mapsto B(h)$ .",
"In this way, $\\lbrace B(h), \\, h \\in \\mathcal {H} \\rbrace $ is an isonormal Gaussian process indexed by the Hilbert space $\\mathcal {H}$ .",
"Let $\\mathcal {S}$ be the set of smooth and cylindrical random variable of the form $F&=& f(B_{t_{1}},\\dots , B_{t_{N}}),$ where $N\\ge 1$ , $t_1,\\dots ,t_{N} \\in [0,T]$ and $f \\in C^{\\infty }_{b} (\\mathbb {R}^{N})$ , namely, $f$ and all its partial derivatives are bounded.",
"The derivative operator $D $ on $F$ is defined as the $\\mathcal {H}$ -valued random variable $D_{t}F &=& \\sum _{i=1}^{N} \\frac{\\partial f}{\\partial x_{i}} ( B_{t_{1}},\\dots , B_{t_{N}} ) \\mathbf {1}_{[0,t_{i}]} (t), \\quad t \\in [0,T].$ For $p\\ge 1$ we define the Sobolev space $\\mathbb {D}^{1,p}_{B}$ (or simply $\\mathbb {D}^{1,p}$ ) as the closure of $\\mathcal {S}$ with respect to the norm $\\Vert F\\Vert _{\\mathbb {D}^{1,p}} &=& \\Big ( \\mathbb {E}[ |F|^{p}] +\\mathbb {E}\\left[ \\Vert DF\\Vert _{\\mathcal {H}}^{p} \\right] \\Big )^{1/p}.$ The above definition of the Sobolev space $\\mathbb {D}^{1,p}$ can be extended to $\\mathcal {H}$ -valued random variables (see Section 1.2 in [20]).",
"We denote by $\\mathbb {D}^{1,p}_{B} (\\mathcal {H})$ (or simply $\\mathbb {D}^{1,p} (\\mathcal {H})$ ) the corresponding Sobolev space.",
"We denote by $\\delta $ the adjoint of the derivative operator $D$ .",
"We say $u \\in \\text{Dom}\\, \\delta $ if there is a $\\delta (u) \\in L^{2}(\\Omega )$ such that for any $F \\in \\mathbb {D}^{1,2}$ the following duality relationship holds $ \\mathbb {E}(\\langle u, DF \\rangle _{\\mathcal {H}}) & =& \\mathbb {E}(F \\delta (u)).$ The random variable $\\delta (u)$ is also called the Skorohod integral of $u$ with respect to the fBm $B$ , and we use the notation $\\delta (u) = \\int _{0}^{T} u_{t} \\delta B_{t}$ .",
"The following result is an example of application of the duality relationship that will be used later in the paper.",
"Lemma 2.1 Let $B$ and $\\widetilde{B}$ be independent one-dimensional fBms with Hurst parameter $H \\in (\\frac{1}{2}, 1)$ .",
"Take $h \\in \\mathcal {H} \\otimes \\mathcal {H}$ , then the integral $\\int _{0}^{T} \\int _{0}^{T} h_{s,t} \\delta {B}_{s} \\delta \\widetilde{B}_{t}$ is well defined.",
"Denote by $D$ and $\\widetilde{D}$ the derivative operators associated with $B$ and $\\widetilde{B} $ , respectively.",
"Take $F\\in \\mathbb {D}_{\\widetilde{B}}^{1,2}$ and assume that ${\\widetilde{D}F \\in {\\mathbb {D}}_{B}^{1,2} } (\\mathcal {H}) $ .",
"Then, applying the integration by parts twice, we obtain $\\mathbb {E}(\\langle h, D\\widetilde{D} F \\rangle _{\\mathcal {H} \\otimes \\mathcal {H} }) &=& \\mathbb {E}\\Big (F \\int _{0}^{T}\\int _{0}^{T} h_{s,t} \\delta {B}_{s} \\delta \\widetilde{B}_{t} \\Big ).$"
],
[
"Weighted random sums",
"In this subsection, we recall some estimates and limit results for Riemann-Stieltjes integrals of stochastic processes.",
"Our main references are [3], [10], [11], [26].",
"Let us start with the definition of Hölder continuous functions in $L^p:=L^p(\\Omega )$ .",
"In the following $\\Vert \\cdot \\Vert _p $ denotes the $L^p$ -norm in the space $L^p$ , where $p\\ge 1$ .",
"Definition 2.2 Let $f=\\lbrace f(t), t \\in [a,b]\\rbrace $ be a continuous process such that $f(t) \\in L^p$ for all $t \\in [a,b]$ .",
"Then $f$ is called a ${\\beta }$ -Hölder continuous function in $L^p$ if the following relation holds true for all $s,t \\in [ a, b ]$ : $\\Vert f(t)- f(s) \\Vert _{p} &\\le & K |t-s|^{ \\beta } .$ We denote by $ \\Vert f \\Vert _{\\beta , p} $ the Hölder semi-norm $\\Vert f\\Vert _{ \\beta , p} &=& \\sup \\left\\lbrace \\frac{ \\Vert f(t) - f(s)\\Vert _p }{|t-s|^{\\beta } }: t, s \\in [a,b], t \\ne s\\right\\rbrace .$ Our first result provides an upper-bound estimate for the $L^p$ -norm of a Riemann-Stieltjes integral.",
"Lemma 2.3 Take $p\\ge 1$ , $p^{\\prime },q^{\\prime } >1: \\frac{1}{p^{\\prime }} +\\frac{1}{q^{\\prime }} =1$ and $\\beta ,\\beta ^{\\prime } \\in (0,1): \\beta +\\beta ^{\\prime }>1$ .",
"Let $f(t)$ , $g(t)$ , $t \\in [a, b]$ be Hölder continuous functions of order $\\beta $ and $\\beta ^{\\prime }$ in $L^{pp^{\\prime }} $ and $L^{pq^{\\prime }} $ , respectively.",
"Then the Riemann-Stieltjes integral $ \\int ^b_a f dg $ is well defined in $L^p$ , and we have the estimate $\\Big \\Vert \\int ^b_a f dg\\Big \\Vert _p&\\le &\\left( K \\Vert f\\Vert _{ \\beta , pp^{\\prime }}+ \\Vert f(a)\\Vert _{pp^{\\prime }} \\right) \\Vert g\\Vert _{ \\beta ^{\\prime }, pq^{\\prime }} (b-a)^{\\beta ^{\\prime }},$ where $K$ is a constant depending only on the parameters $p,p^{\\prime },q^{\\prime },\\beta ,\\beta ^{\\prime }$ .",
"Proof: The proof is based on the fractional integration by parts formula (see [26]), following the arguments used in the proof of Lemma 11.1 in [10].",
"$\\Box $ Given a double sequence of random variables $\\zeta = \\lbrace \\zeta _{k, n} , n \\in \\mathbb {N}, k=0, 1, \\dots , n \\rbrace $ , for each $t \\in [0,T]$ we set $g_n(t): = \\sum _{k=0}^{ \\left\\lfloor \\frac{nt}{T} \\right\\rfloor } \\zeta _{k , n} \\,,$ where $\\left\\lfloor \\frac{nt}{T} \\right\\rfloor $ denotes the integer part of $\\frac{nt}{T}$ .",
"We recall the following result from [11], which provides an upper-bound estimate for weighted random sums (or the so-called discrete integrals) of the process $g_{n}$ .",
"Lemma 2.4 Let $p$ , $p^{\\prime }$ , $q^{\\prime }$ , $\\beta $ , $\\beta ^{\\prime }$ be as in Lemma REF .",
"Let $f$ be a Hölder continuous function of order $\\beta $ in $L^{pp^{\\prime }}$ .",
"Let $g_n $ be as in (REF ) such that for any $j,k =0, 1, \\dots , n $ we have $\\mathbb {E}(| g_n( kT/n ) - g_n ( jT/n) |^{pq^{\\prime }} ) &\\le &K ( |k-j|/n)^{ \\beta ^{\\prime } pq^{\\prime }}.$ Then the following estimate holds true for $i,j=0, 1,\\dots , n $ , $ i>j$ : $\\Big \\Vert \\sum _{k= j +1 }^{ i } f(t_k) \\zeta _{k, n} \\Big \\Vert _p &\\le & K \\left( \\Vert f\\Vert _{\\beta , pp^{\\prime }} + \\Vert f(t_{j} )\\Vert _{pp^{\\prime }} \\right)\\Big (\\frac{i-j}{n}\\Big )^{\\beta ^{\\prime }} .$ Let us now recall some limit theorems for weighted random sums.",
"The first result says that if the “weight-free” random sum (REF ) converges weakly and if the weight process satisfies certain regularity assumption, then the weighted random sum also converges weakly.",
"The reader is referred to [3] for further details.",
"Proposition 2.5 Let $g_n$ be defined in (REF ).",
"Assume that $g_{n}$ satisfies the inequality $\\mathbb {E}(| g_n( kT/n ) - g_n ( jT/n) |^{4} ) &\\le &K ( |k-j|/n)^{ 2}$ for $j,k =0, 1, \\dots , n $ .",
"Suppose further that the finite dimensional distributions of $g_n$ converge stably to those of $W=\\lbrace W_{t}, t\\in [0,T]\\rbrace $ , where $W$ is a standard Brownian motion independent of $g_{n}$ .",
"Let $f=\\lbrace f(t), t\\in [0,T]\\rbrace $ be a $\\beta $ -Hölder continuous process for $\\beta >1/2$ .",
"Then the finite dimensional distributions of $\\sum _{k=0}^{\\left\\lfloor \\frac{nt}{T} \\right\\rfloor } f(t_{k}) \\zeta _{k, n}$ converge stably to those of $\\int _{0}^{t} f(s) dW_{s}$ , where recall that $\\left\\lfloor \\frac{nt}{T} \\right\\rfloor $ denotes the integer part of $\\frac{nt}{T}$ .",
"Recall that a sequence of random vectors $F_{n}$ converges stably to a random vector $F$ , where $F$ is defined on an extension $(\\Omega ^{\\prime }, {F}^{\\prime }, \\mathbb {P}^{\\prime })$ of the original probability $(\\Omega , {F}, \\mathbb {P})$ , if $(F_{n}, Z) \\rightarrow (F,Z)$ weakly for any ${F}$ -measurable random variable $Z$ .",
"The reader is referred to [1], [12], [24] for further details on stable convergence.",
"The following result can be viewed as the $L^p$ -convergence version of Proposition REF ; see [10].",
"Proposition 2.6 Take $\\beta , \\lambda \\in (0,1): \\beta +\\lambda >1 $ .",
"Let $p\\ge 1$ and $p^{\\prime }, q^{\\prime } > 1$ such that $\\frac{1}{p^{\\prime }} +\\frac{1}{q^{\\prime }} =1$ and $pp^{\\prime }>\\frac{1}{ \\beta }$ , $pq^{\\prime } > \\frac{1}{\\lambda }$ .",
"Let $g_n $ be defined in (REF ).",
"Suppose that the following two conditions hold true: (i) For $t \\in [0, T]$ , we have the convergence $g_n(t ) \\rightarrow z(t)$ in $L^{pq^{\\prime }}$ ; (ii) For $j,k =0, 1, \\dots , n $ we have the relation: $\\mathbb {E}(| g_n( kT/n ) - g_n ( jT/n) |^{pq^{\\prime }} ) & \\le & K ( |k-j|/n)^{\\lambda pq^{\\prime } }.$ Let $f=\\lbrace f(t) , t \\in [0, T]\\rbrace $ be a continuous process such that $\\mathbb {E}(\\Vert f\\Vert _{\\beta }^{pp^{\\prime }} ) \\le K$ and $\\mathbb {E}(|f( 0 )|^{pp^{\\prime }}) \\le K$ .",
"Then for each $t\\in [0, T]$ we have the convergence: $\\lim _{n\\rightarrow \\infty } \\sum _{k=0}^{ \\left\\lfloor \\frac{nt}{T} \\right\\rfloor }f(t_k) \\zeta _{k , n} = \\int _0^t f(s) dz(s)\\,,$ where the limit is understand as the limit in $L^p$ ."
],
[
"Lévy area type processes",
"Let $B= \\lbrace B_{t}, t \\ge 0 \\rbrace $ be a one-dimensional fBm with Hurst parameter $H \\in (\\frac{1}{2}, 1)$ , and let $\\widetilde{B} = \\lbrace \\widetilde{B}_{t} , t \\ge 0 \\rbrace $ be a Hölder continuous process of order $\\beta >\\frac{1}{2}$ .",
"Let $\\Pi = \\lbrace 0=t_{0}<t_{1}<\\cdots <t_{n}=T \\rbrace $ be the uniform partition on $[0,T]$ and take $t_{n+1} = \\frac{n+1}{n}T$ .",
"We consider the following Lévy area type process on $[0,T]$ ${Z}_n(t)&=&\\sum _{k=0}^{ l } \\left(\\int _{t_k}^{t_{k+1} } \\int _{t_k}^{s } d\\widetilde{B}_u dB_s -\\int _{t_k}^{t_{k+1} } \\int ^{t_{k+1} }_{s } d\\widetilde{B}_u dB_s \\right)\\,$ for $t\\in [t_{l}, t_{l+1})\\cap [0,T]$ , $l=0,\\dots , n $ .",
"In this section, we study the convergence rate and the asymptotic distribution of the sequence $\\lbrace Z_{n}, n \\in \\mathbb {N}\\rbrace $ .",
"We will mainly focus on two cases: (i) $\\widetilde{B}$ is an independent copy of $B$ ; and (ii) $\\widetilde{B}$ is the identity function, that is, $\\widetilde{B}_t = t $ for $t \\ge 0$ ."
],
[
"Case (i)",
"For simplicity, we denote by $\\mu $ the measure on the plane $\\mathbb {R}^2 $ given by $\\mu (dsdt) = H(2H-1)|s-t|^{2H-2}dsdt.$ For each $p\\in \\mathbb {Z}$ we set $Q( p)= \\int _{ {0}}^{ { 1}} \\int _{ { p} }^{ {{ p} +1}}\\int _{ {0}}^{ t } \\int _{ { p} }^{s}\\mu (dvdu) \\mu (ds dt),\\quad R(p) =\\int _{ {0}}^{ { 1}} \\int _{ p}^{ { p+1}}\\int _{ {t}}^{ 1 } \\int _{ p}^{s}\\mu (dv du) \\mu (ds dt).$ The following result provides some properties of the process $Z_{n}$ .",
"Proposition 3.1 Let $Z_n$ be the process defined in (REF ) and let $\\widetilde{B}$ be an independent copy of $B$ .",
"Then, there exists a constant $K$ depending on $H$ and $T$ such that for $t,s \\in \\Pi $ we have $n^{4H-1} \\mathbb {E}( [ {Z}_n(t)- {Z}_n(s) ]^2 ) &\\le & K |t-s|.$ Furthermore, the finite dimensional distributions of $\\left( n^{2H-\\frac{1}{2}} Z_n(t) , B_{t} , \\widetilde{B}_{t} , \\, t\\in [0,T] \\right)$ converge weakly to those of $\\left( T^{2H-\\frac{1}{2}} \\sqrt{ {2\\kappa } } W_{t} , B_{t} , \\widetilde{B}_{t}, \\, t\\in [0,T] \\right)$ as $n$ tends to infinity, where $W= \\lbrace W_t, t \\in [0, T]\\rbrace $ is a standard Brownian motion independent of $(B,\\widetilde{B})$ , and $\\kappa &=& \\sum _{p \\in \\,\\mathbb {Z} } ( Q( p)- R(p) ) .$ Remark 3.2 The following figure provides the graph of the parameter $\\kappa $ as a function of $H$ on $(\\frac{1}{2} , 1)$ .",
"We observe that $\\kappa $ converges to $\\frac{1}{2}$ as $H$ tends to $\\frac{1}{2}$ which corresponds to the Brownian motion, and $\\kappa $ approaches 0 when $H$ tends to 1.",
"Figure: NO_CAPTION Proof of Proposition REF : The proof is divided into four steps.",
"Step 1.",
"In this step, we show the convergence of $n^{ 4H - 1 } \\mathbb {E}({ {Z}}_n(t)^2 )$ and derive its limit as $n\\rightarrow \\infty $ .",
"We first calculate the second moment of $Z_{n}(t)$ .",
"Note that when $\\widetilde{B}$ is an independent copy of $B$ we have $Z_{n}(t) &=&\\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\left(\\int _{t_k}^{t_{k+1} } \\int _{t_k}^{s } \\delta \\widetilde{B}_u \\delta B_s -\\int _{t_k}^{t_{k+1} } \\int ^{t_{k+1} }_{s } \\delta \\widetilde{B}_u \\delta B_s \\right)\\nonumber \\\\&=& \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{0}^{T} \\int _{0}^{T} \\beta _{\\frac{k}{n}} (s) \\gamma _{t_k, s} (u) \\delta \\widetilde{B}_{u} \\delta {B}_{s}\\,,$ where $\\delta $ denotes the Skorohod integral and $\\beta _{\\frac{k}{n}} (s) = \\mathbf {1}_{[t_k, t_{k+1}] } (s), \\quad \\quad \\quad \\quad \\gamma _{t_k, s} (u) =\\mathbf {1}_{[t_k, s] }(u) - \\mathbf {1}_{[s, t_{k+1}] }(u).$ By the integration by parts formula (REF ) and taking into account the expression of $Z_{n}(t)$ in (REF ) we obtain $\\mathbb {E}[{ {Z}}_n(t)^2 ]&=& \\sum _{k =0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{0}^{T} \\int _{0}^{T} \\int _{0}^{T}\\int _{0}^{T} \\widetilde{D}_{u^{\\prime }}D_{s^{\\prime }} Z_{n}(t) \\beta _{\\frac{k}{n}} (s) \\gamma _{t_k, s} (u) \\mu (dudu^{\\prime }) \\mu (dsds^{\\prime }),$ where $D$ and $\\widetilde{D}$ are the derivative operators associated with $B$ and $\\widetilde{B}$ , respectively.",
"It is clear that $\\widetilde{D}_{u^{\\prime }}D_{s^{\\prime }} Z_{n}(t) &=&\\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\beta _{\\frac{k}{n}} (s^{\\prime }) \\gamma _{t_k, s^{\\prime }} (u^{\\prime }) .$ Therefore, we obtain the expression $\\mathbb {E}[{ {Z}}_n(t)^2 ]&=&\\sum _{k,k^{\\prime }=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{0}^{T} \\int _{0}^{T} \\int _{0}^{T}\\int _{0}^{T} \\beta _{\\frac{k^{\\prime }}{n}} (s^{\\prime }) \\beta _{\\frac{k}{n}} (s) \\gamma _{t_{k^{\\prime }}, s^{\\prime }} (u^{\\prime }) \\gamma _{t_k, s} (u) \\mu (dudu^{\\prime }) \\mu (dsds^{\\prime }).$ By changing the variables from $(u,u^{\\prime },s,s^{\\prime })$ to $\\frac{T}{n} (u,u^{\\prime },s,s^{\\prime }) $ we obtain $\\mathbb {E}[{ {Z}}_n(t)^2 ]&=& \\left( \\frac{T}{n} \\right)^{4H} \\sum _{k,k^{\\prime }=0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\int _{ {k^{\\prime }}}^{ {k^{\\prime }+1}} \\int _{ {k}}^{ {k+1}} \\int _{0}^{n}\\int _{0}^{n} \\varphi _{ {k^{\\prime }}, s^{\\prime }} (u^{\\prime }) \\varphi _{ k, s} (u) \\mu (dudu^{\\prime }) \\mu (dsds^{\\prime }),$ where $\\varphi _{ {k},s}(u) = \\mathbf {1}_{[k,s]}(u) - \\mathbf {1}_{[s,k+1]}(u).$ Denote $\\varphi _{ {k},s}^{0}(u) = \\mathbf {1}_{[k,s]}(u) $ , $ \\varphi _{ {k},s}^{1}(u) = \\mathbf {1}_{[s,k+1]}(u) $ , and set $e_{ij}&=&\\int _{ {k^{\\prime }}}^{ {k^{\\prime }+1}} \\int _{ {k}}^{ {k+1}} \\int _{0}^{n}\\int _{0}^{n} \\varphi _{ {k^{\\prime }}, s^{\\prime }}^{i} (u^{\\prime }) \\varphi _{ k, s}^{j} (u) \\mu (dudu^{\\prime }) \\mu (dsds^{\\prime }).$ Then we can write $\\mathbb {E}[{ {Z}}_n(t)^2 ]&=& \\left( \\frac{T}{n} \\right)^{4H} \\sum _{k,k^{\\prime }=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\sum _{i,j=0,1} (-1)^{i+j}e_{ij}.$ It is easy to see that $e_{00}=e_{11}=Q(k-k^{\\prime })$ and $e_{01}=e_{10}=R(k-k^{\\prime })$ .",
"Therefore, $\\mathbb {E}[{ {Z}}_n(t)^2 ]&=&2 \\left( \\frac{T}{n}\\right)^{4H} \\sum _{k,k^{\\prime } =0}^{ \\lfloor \\frac{nt}{T} \\rfloor } [Q(k-k^{\\prime })-R(k-k^{\\prime })]\\,.$ Taking $p =k-k^{\\prime }$ on the right-hand side of (REF ), we obtain $\\mathbb {E}[{ {Z}}_n(t)^2 ]&= &2 \\left( \\frac{T}{n}\\right)^{4H}\\left(\\sum _{p =0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\sum _{ k =p}^{\\lfloor \\frac{nt}{T} \\rfloor }[Q(p)-R(p)]+\\sum ^{-1}_{p= - \\lfloor \\frac{nt}{T} \\rfloor }\\sum _{ k =0}^{\\lfloor \\frac{nt}{T} \\rfloor +p}[Q(p)-R(p)]\\right)\\nonumber \\\\&:= &q_1+q_2\\,.$ We decompose $q_{1} $ as follows, $q_1 &= &2 \\left( \\frac{T}{n}\\right)^{4H}\\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }(\\lfloor \\frac{nt}{T} \\rfloor -p+1 )(Q(p) - R(p) )\\\\&= &2 \\left( \\frac{T}{n}\\right)^{4H}\\left(\\lfloor \\frac{nt}{T} \\rfloor \\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }(Q(p) - R(p) )-\\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }( p-1)(Q(p) - R(p) )\\right)\\\\&:= &q_{11}+q_{12}\\,.$ By mean value theorem, it is easy to show that $|Q(p) - R(p) | \\le Kp^{4H-5} $ for $p>0$ , so the sum $ \\sum _{p=0}^{\\infty } ( Q(p)- R(p) ) $ is convergent, and that $\\left|\\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }(p-1)(Q(p) - R(p) ) \\right|& \\le & K(n^{4H-3} \\vee 1) .$ Here $a\\vee b$ denotes the maximum of $a$ and $b$ .",
"Therefore, $\\lim _{n \\rightarrow \\infty } n^{4H-1} q_{11}&=&\\lim _{n \\rightarrow \\infty } 2n^{4H-1} \\left( \\frac{T}{n}\\right)^{4H} \\lfloor \\frac{nt}{T} \\rfloor \\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }(Q(p) - R(p) )\\nonumber \\\\&=&2 t T^{4H-1} \\sum _{p=0 }^{\\infty }(Q(p) - R(p) )$ and $\\lim _{n \\rightarrow \\infty } n^{4H-1} q_{12}&= &0 .$ In summary, from (REF ) and (REF ), we obtain $\\lim _{n \\rightarrow \\infty } n^{4H-1} q_1& = &2 t T^{4H-1} \\sum _{p=0 }^{\\infty }(Q(p) - R(p) ) .$ In a similar way, we can prove the following convergence for $q_{2}$ $\\lim _{n\\rightarrow \\infty } n^{ 4H - 1} q_2 &=& 2 t T^{4H-1} \\sum ^{-1}_{p= -\\infty } ( Q( p)- R(p) ) .$ Substituting (REF ) and (REF ) into (REF ) yields $\\lim _{ n \\rightarrow \\infty } n^{ 4H - 1 } \\mathbb {E}({ {Z}}_n(t)^2 ) &=& 2 T^{4H-1} \\kappa t,$ where recall that $\\kappa $ is a constant defined in (REF ).",
"Step 2.",
"In this step, we show the inequality (REF ).",
"This inequality is obvious when $s=t$ .",
"In the following, we consider the case when $t>s$ .",
"Take $t\\in \\Pi $ .",
"By the definition of $q_{1}$ we have $q_1 &\\le &2 \\left( \\frac{T}{n}\\right)^{4H}\\sum _{p=0 }^{ \\frac{nt }{T} }( \\frac{nt}{T} -p+1)\\,\\, | Q(p) - R(p) |\\\\&\\le &2 \\left( \\frac{T}{n}\\right)^{4H}( \\frac{nt}{T} +1)\\sum _{p=0 }^{ \\infty }| Q(p) - R(p) |\\,.$ In the same way, we can show that $q_2 & \\le &2 \\left( \\frac{T}{n}\\right)^{4H} ( \\frac{nt}{T} +1)\\sum _{p= - \\infty }^{ -1 }| Q(p) - R(p) |.$ Applying these two inequalities to (REF ) we obtain $n^{4H-1} \\mathbb {E}({ {Z}}_n(t)^2 ) &\\le & K (t+\\frac{T}{n})$ for $t\\in \\Pi $ , where $K$ is a constant depending on $H$ , $T$ .",
"Take $s, t \\in \\Pi : s<t$ .",
"The inequality (REF ) then follows by replacing $t$ in (REF ) by $t-s-\\frac{T}{n}$ and noticing that $Z_{n} (t) -Z_{n}(s)$ and $Z_{n} (t-s-\\frac{T}{n})$ are equal in distribution and thus have the same second moments.",
"Step 3.",
"Take $s,t \\in [0,T]: s<t$ .",
"In this step, we derive the limit of the quantity $n^{4H-1} \\mathbb {E}(Z_{n}(t) Z_{n}(s))$ .",
"Denote $\\eta (t) =t_{k}$ for $t \\in [t_{k},t_{k+1})$ , $k=0,1,\\dots , n$ .",
"Then we have $Z_{n}(t) = Z_{n}(\\eta (t))$ .",
"Since $Z_{n} (\\eta (t)) - Z_{n}(\\eta (s))$ and $Z_{n} (\\eta (t)-\\eta (s)-\\frac{T}{n}) $ have the same distribution, we have $\\mathbb {E}(|Z_{n} (t) - Z_{n}(s) |^{2}) &=& \\mathbb {E}(|Z_{n} (\\eta (t)) - Z_{n}(\\eta (s)) |^{2})\\nonumber \\\\&=& \\mathbb {E}(|Z_{n} (\\eta (t)-\\eta (s) -\\frac{T}{n}) |^{2}).$ Note that $0< (t-s) - (\\eta (t)-\\eta (s) - \\frac{T}{n}) < \\, 2\\frac{T}{n}$ , so either $Z_{n} (\\eta (t)-\\eta (s) - \\frac{T}{n}) = Z_{n}(t-s)$ or $Z_{n} (\\eta (t)-\\eta (s) - \\frac{T}{n}) = Z_{n}(t-s - \\frac{T}{n})$ .",
"In both cases we have $\\lim _{n\\rightarrow \\infty } n^{4H-1 } \\mathbb {E}(|Z_{n} (\\eta (t) -\\eta (s) - \\frac{T}{n} ) |^{2})&=& \\lim _{n\\rightarrow \\infty } n^{4H-1 } \\mathbb {E}(|Z_{n} (t -s ) |^{2}).$ Indeed, the identity is clear in the first case.",
"In the second case, it can be shown with the help of (REF ) that $\\lim _{n\\rightarrow \\infty } n^{4H-1 } \\left( \\mathbb {E}(|Z_{n} (t -s -\\frac{T}{n} ) |^{2}) - \\mathbb {E}(|Z_{n} (t -s ) |^{2}) \\right) &=&0.$ The identity (REF ) then follows.",
"Substituting (REF ) into (REF ) and with the help of (REF ) we obtain $\\lim _{n\\rightarrow \\infty } n^{4H-1 } \\mathbb {E}(|Z_{n} (t) - Z_{n} (s) |^{2})&=& 2 {T^{4H-1}} \\kappa (t-s).$ By expanding the left-hand side of (REF ) and using (REF ), we obtain $\\lim _{ n \\rightarrow \\infty } n^{4H - 1} \\mathbb {E}[ {Z}_n(t) {Z}_n(s) ]&= 2 {T^{4H-1}} \\kappa {(t\\wedge s)} , \\quad \\, s,t \\in [0,T].$ Step 4.",
"In this step, we prove the weak convergence for the finite dimensional distributions of $ (n^{2H-\\frac{1}{2}}Z_{n}, B, \\widetilde{B}) $ .",
"Given $r_1, \\dots , r_L \\in [0, T]$ , $L \\in \\mathbb {N}$ , we need to show that the random vector $\\Theta ^n_L:=\\left( n^{2H-\\frac{1}{2}} (Z_n(r_1), \\dots , Z_n(r_L)), B_{r_1}, \\dots , B_{r_L}, \\widetilde{B}_{r_1}, \\dots , \\widetilde{B}_{r_L} \\right)$ converges in law to $\\Theta _L:= \\left( T^{2H-\\frac{1}{2}}\\sqrt{ {2\\kappa } } \\left( W(r_1), \\dots , W(r_L) \\right), B_{r_1}, \\dots , B_{r_L}, \\widetilde{B}_{r_1}, \\dots , \\widetilde{B}_{r_L} \\right)$ as $n $ tends to infinity, where recall that $W= \\lbrace W_t, t \\in [0, T]\\rbrace $ is a standard Brownian motion independent of $(B,\\widetilde{B})$ .",
"According to [23] (see also Theorem 6.2.3 in [19]), this is true if we can show the weak convergence of each component of $\\Theta ^n_L$ to the corresponding component of $\\Theta _L $ and the convergence of its covariance matrix to that of $\\Theta _L $ .",
"The convergence of the covariance of $n^{2H-\\frac{1}{2} }Z_n (r_i) $ and $ n^{2H-\\frac{1}{2} }Z_n (r_j) $ to that of $ T^{2H-\\frac{1}{2}}\\sqrt{ {2\\kappa } } W(r_i ) $ and $ T^{2H-\\frac{1}{2}}\\sqrt{ {2\\kappa } } W(r_j ) $ follows from (REF ).",
"The covariance of $n^{2H-\\frac{1}{2} }Z_{n}(r_{i}) $ and $(B_{r_{j}}, \\widetilde{B}_{r_{j}})$ is zero since they are in different chaos, so the limit of the covariance is zero, which equals the covariance of $T^{2H-\\frac{1}{2}}\\sqrt{ {2\\kappa } } W(r_{i})$ and $(B_{r_{j}}, \\widetilde{B}_{r_{j}})$ since $W$ and $B$ are independent.",
"By the fourth moment theorem (see [21] and also Theorem 5.2.7 in [19]) and taking into account (REF ), to show the weak convergence of the components of $\\Theta ^n_L$ it remains to show that the limits of their fourth moments exist, and $\\lim _{n\\rightarrow \\infty } n^{8H-2} \\mathbb {E}\\left[ Z_{n} (t)^{4} \\right]&=&3 \\lim _{n\\rightarrow \\infty } n^{8H-2} \\left( \\mathbb {E}[ Z_{n} (t)^{2} ] \\right)^{2}$ for $t\\in [0,T]$ .",
"Applying the integration by parts formula (REF ) to $\\mathbb {E}[{ {Z}}_n(t)^4 ]$ and taking into account the expression of $Z_{n}(t)$ in (REF ), we obtain $\\mathbb {E}[{ {Z}}_n(t)^4 ]&=& \\mathbb {E}[{ {Z}}_n(t)^{3} \\cdot { {Z}}_n(t) ]\\nonumber \\\\&=&\\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\mathbb {E}\\int _0^T\\int _0^T\\int _0^T\\int _0^T\\widetilde{D}_{u^{\\prime }}D_{s^{\\prime }}\\left[ { {Z}}_n(t)^3 \\right]\\beta _{\\frac{k}{n} }(s) \\gamma _{t_k, s} (u)\\mu (dudu^{\\prime }) \\mu (dsds^{\\prime }),$ where $D$ and $\\widetilde{D}$ are the differential operators associated with $B$ and $\\widetilde{B}$ , respectively.",
"We expand the second derivative $ \\widetilde{D}_{u^{\\prime }}D_{s^{\\prime }}\\left[ { {Z}}_n(t)^3 \\right] $ as follows $\\widetilde{D}_{u^{\\prime }}D_{s^{\\prime }}\\left[ { {Z}}_n(t)^3 \\right] &= &3 Z_{n}(t)^{2} \\widetilde{D}_{u^{\\prime }}D_{s^{\\prime }} Z_{n}(t)+ 6 Z_{n}(t) \\widetilde{D}_{u^{\\prime }}Z_{n}(t) D_{s^{\\prime }} Z_{n}(t).$ Substituting the above identity into (REF ), we obtain $\\mathbb {E}[{ {Z}}_n(t)^4 ] &= & d_{1} +d_{2}\\,,$ where $d_2&=& 6 \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\mathbb {E}\\int _0^T\\int _{0}^{T}\\int _0^T\\int _0^T{ {Z}}_n(t)\\widetilde{D}_{u^{\\prime }}{ {Z}}_n(t)D_{s^{\\prime }}{ {Z}}_n(t)\\beta _{\\frac{k}{n} }(s)\\gamma _{t_k, s} (u)\\mu (dudu^{\\prime }) \\mu (dsds^{\\prime })$ and $ d_1&=&3 \\mathbb {E}[ { {Z}}_n(t) ^2 ] \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\int _0^T \\int _{0}^{T} \\int _0^T\\int _0^T\\widetilde{D}_{u^{\\prime }}D_{s^{\\prime }}{ {Z}}_n(t)\\beta _{\\frac{k}{n} }(s) \\gamma _{t_k, s} (u)\\mu (dudu^{\\prime }) \\mu (dsds^{\\prime }).$ Substituting (REF ) into $d_{1}$ , we obtain $d_{1} &=& 3\\mathbb {E}[ { {Z}}_n(t) ^2 ]\\mathbb {E}[ { {Z}}_n(t) ^2 ].$ The term $d_2$ is more sophisticated.",
"We shall prove in Section REF the following fact: $\\lim _{n\\rightarrow \\infty } n^{8H-2}d_2 &=& 0.$ The identity (REF ) and the convergence (REF ) together imply the identity (REF ).",
"This completes the proof.",
"$\\Box $"
],
[
"Case (ii)",
"In this subsection, we consider the process $Z_n$ in (REF ) under the assumption that $\\widetilde{B}_t = t$ , $t\\in [0, T]$ .",
"We denote $z_n:=Z_n$ in this subsection to distinguish it from the $Z_{n}$ in the previous subsection.",
"For each $p \\in \\mathbb {Z} $ , we denote $\\widetilde{Q}( p)=\\int _{ {0}}^{ { 1}} \\int _{ { p} }^{ {{ p} +1}}\\int _{ {0}}^{ t } \\int _{ { p} }^{s}dvdu \\mu (ds dt),~~~~\\widetilde{R}(p) =\\int _{ {0}}^{ { 1}} \\int _{ p}^{ { p+1}}\\int _{ {t}}^{ 1 } \\int _{ p}^{s}dv du \\mu (ds dt),$ where recall that $\\mu (dsdt)=H(2H-1)|s-t|^{2H-2}dsdt$ is a measure on $\\mathbb {R}^{2}$ .",
"Proposition 3.3 Let $z_n$ be the process defined in (REF ) with $\\widetilde{B}_{t} =t$ , $t\\in [0,T]$ .",
"Then, there exists a constant $K$ depending on $H$ and $T$ such that for $t,s \\in \\Pi = \\lbrace \\frac{T}{n}i: i=0,\\dots , n\\rbrace $ we have $n^{2H+1} \\mathbb {E}( [ {z}_n(t)- {z}_n(s) ]^2 ) & \\le & K |t-s|.$ Moreover, the finite dimensional distributions of the process $ ( n^{ H + \\frac{1}{2}} z_n \\,, \\,B )$ converge weakly to those of $( { \\sqrt{2 \\varrho }{ } }T^{H+\\frac{1}{2}} W \\,, \\,B )$ as $n \\rightarrow \\infty $ , where $W= \\lbrace W_t, t \\in [0, T]\\rbrace $ is a standard Brownian motion independent of $B$ , and $\\varrho &:= & \\sum _{p \\in \\mathbb {Z} } ( \\widetilde{Q}(p) - \\widetilde{R}(p) ) .",
"$ Remark 3.4 The following figure provides the graph of the parameter $\\rho $ as a function of $H$ on $(\\frac{1}{2} , 1)$ .",
"We see that $\\rho $ converges to $\\frac{1}{6}$ as $H$ tends to $\\frac{1}{2}$ , and $\\rho $ approaches 0 when $H$ tends to 1.",
"Figure: NO_CAPTION Proof of Proposition REF : The proof is done in three steps.",
"Step 1.",
"In this step, we calculate the second moment of $z_{n}(t)$ .",
"Write $z_{n}(t)$ as $z_{n}(t)&=&\\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{0}^{T} \\int _{0}^{T} \\beta _{\\frac{k}{n}} (s) \\gamma _{t_k, s} (u) d{u} \\delta {B}_{s}\\,,$ where $ \\beta _{\\frac{k}{n}} (s) $ and $ \\gamma _{t_k, s} (u) $ are defined in (REF ).",
"Then, applying the covariance formula (REF ), we end up with $\\mathbb {E}[{ {z}}_n(t)^2 ]&=& \\sum _{k,k^{\\prime } =0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{0}^{T} \\int _{0}^{T} \\int _{0}^{T} \\int _{0}^{T} \\beta _{\\frac{k^{\\prime }}{n}} (s^{\\prime })\\beta _{\\frac{k}{n}}(s) \\gamma _{t_{k^{\\prime }}, s^{\\prime }} (u^{\\prime })\\gamma _{t_k, s} (u) dudu^{\\prime } \\mu (dsds^{\\prime }).$ Now by a change of variables from $(u,u^{\\prime }, s,s^{\\prime })$ to $ \\frac{T}{n}(u,u^{\\prime }, s,s^{\\prime }) $ , we obtain $\\mathbb {E}[{ {z}}_n(t)^2 ]&=& \\left(\\frac{T}{n}\\right)^{2H+2} \\sum _{k,k^{\\prime } =0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{k^{\\prime }}^{k^{\\prime }+1} \\int _{k}^{k+1} \\int _{0}^{n} \\int _{0}^{n} \\varphi _{ {k^{\\prime }}, s^{\\prime }} (u^{\\prime })\\varphi _{ k, s} (u) dudu^{\\prime } \\mu (dsds^{\\prime })\\\\&=& \\left(\\frac{T}{n}\\right)^{2H+2} \\sum _{k,k^{\\prime } =0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\sum _{i,j=0,1} (-1)^{i+j}\\tilde{e}_{ij},$ where $\\tilde{e}_{ij} &=& \\int _{k^{\\prime }}^{k^{\\prime }+1} \\int _{k}^{k+1} \\int _{0}^{n} \\int _{0}^{n} \\varphi _{ {k^{\\prime }}, s^{\\prime }}^{i} (u^{\\prime })\\varphi _{ k, s}^{j} (u) dudu^{\\prime } \\mu (dsds^{\\prime }),$ and $\\varphi _{k,s}^{0} =\\mathbf {1}_{[k,s]}$ , $\\varphi _{k,s}^{1} =\\mathbf {1}_{[s, k+1]}$ , $\\varphi _{k,s} =\\varphi _{k,s}^{0} -\\varphi _{k,s}^{1} $ , as defined as in the previous subsection.",
"It is easy to see that $\\tilde{e}_{00} = \\widetilde{Q}(k-k^{\\prime }), \\quad \\quad \\text{and}\\quad \\tilde{e}_{10} = \\widetilde{R}(k-k^{\\prime }).$ By a change of variables from $(s,s^{\\prime })$ to $(k+1-s, k+1-s^{\\prime })$ , we obtain $\\tilde{e}_{11}&=&\\int _{ k-k^{\\prime }}^{ {k-k^{\\prime }+ 1} }\\int ^{ {1 } }_{ {0} } (s^{\\prime }-(k-k^{\\prime })) s \\mu ( ds ds^{\\prime } )= \\widetilde{Q}(k-k^{\\prime }),$ where the second equation follows by exchanging the orders of the two integrals.",
"By changing the variables from $(s,s^{\\prime })$ to $(k^{\\prime }+1-s, k^{\\prime }+1-s^{\\prime })$ for $\\tilde{e}_{11}$ , we obtain $\\tilde{e}_{01} &=&\\int _{0}^{1} \\int _{k^{\\prime }-k}^{k^{\\prime }-k+1} (1-s^{\\prime }) ( s-(k^{\\prime }-k)) \\mu (dsds^{\\prime })= \\widetilde{R}(k^{\\prime }-k),$ and so $\\sum _{k ,k^{\\prime }=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\tilde{e}_{01} = \\sum _{k ,k^{\\prime }=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\widetilde{R}(k^{\\prime }-k)= \\sum _{k ,k^{\\prime }=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\widetilde{R}(k-k^{\\prime }).$ In summary from (REF ), (REF ) and (REF ), we obtain $\\mathbb {E}[{ {z}}_n(t)^2 ]&= &2 \\left(\\frac{T}{n}\\right)^{2H+2}\\sum _{k,k^{\\prime } =0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\left(\\widetilde{Q }(k-k^{\\prime }) -\\widetilde{R}(k-k^{\\prime })\\right)\\nonumber \\\\& = &2 \\left(\\frac{T}{n}\\right)^{2H+2}\\left(\\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }\\sum _{k^{\\prime } = 0}^{ \\lfloor \\frac{nt}{T} \\rfloor -p}( \\widetilde{Q}(p) - \\widetilde{R}(p) )+\\sum _{p=- \\lfloor \\frac{nt}{T} \\rfloor }^{ -1 }\\sum _{k^{\\prime } = -p }^{\\lfloor \\frac{nt}{T} \\rfloor }( \\widetilde{Q}(p) - \\widetilde{R}(p) )\\right)\\nonumber \\\\&:= &{\\widetilde{q}}_1 +{\\widetilde{q}}_2 .$ Step 2.",
"In this step, we show the inequality (REF ).",
"Since $|\\widetilde{Q} (p) - \\widetilde{R}(p)|\\sim p^{2H-3}$ for sufficiently large $p$ , it is easy to see that the series $ \\sum _{p \\in \\mathbb {Z}} | \\widetilde{Q}(p) - \\widetilde{R}(p) | $ is convergent.",
"So we have the estimates ${\\tilde{q}}_1 &\\le & 2 \\left(\\frac{T}{n}\\right)^{2H+2} (\\frac{nt}{T}+1) \\sum _{p=0}^{\\infty } |\\widetilde{Q}(p) - \\widetilde{R}(p)|$ and ${\\tilde{q}}_2 &\\le &2 \\left(\\frac{T}{n}\\right)^{2H+2} ( \\frac{nt}{T} +1) \\sum _{p=-\\infty }^{ -1 } |\\widetilde{Q}(p) - \\widetilde{R}(p)|.$ Applying (REF ) and (REF ) to (REF ) yields $n^{2H+1} \\mathbb {E}( {z}_n(t)^2 ) &\\le & K( t+\\frac{T}{n}).$ Take $s,t \\in \\Pi $ .",
"By replacing $t $ in (REF ) by $t-s -\\frac{T}{n} $ and noticing that $z_{n}(t)-z_{n}(s)$ and $z_{n}(t-s-\\frac{T}{n})$ have the same distribution, we obtain $n^{2H+1} \\mathbb {E}(| {z}_n(t) - {z}_n(s)|^2 )&=& n^{2H+1} \\mathbb {E}(| {z}_n(t -s-\\frac{T}{n})|^2 )\\nonumber \\\\&\\le & K(t-s).$ This completes the proof of (REF ).",
"Step 3.",
"In this step, we show the convergence of the process $(n^{H+\\frac{1}{2}} z_{n}, B)$ .",
"Note that the finite dimensional distributions of $(n^{H+\\frac{1}{2}}z_{n}, B)$ are Gaussian, so to show their convergences it suffices to show the convergences of their covariances.",
"We first consider the convergence of $n^{2H+1}\\mathbb {E}(|z_{n}(t)|^{2})$ .",
"To this aim, we write ${\\tilde{q}}_1&=&2 \\left(\\frac{T}{n}\\right)^{2H+2}\\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }(\\lfloor \\frac{nt}{T} \\rfloor -p+1)( \\widetilde{Q}(p) - \\widetilde{R}(p) )\\nonumber \\\\&=&2 \\left(\\frac{T}{n}\\right)^{2H+2}\\left(\\lfloor \\frac{nt}{T} \\rfloor \\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }( \\widetilde{Q}(p) - \\widetilde{R}(p) )-\\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }(p-1)( \\widetilde{Q}(p) - \\widetilde{R}(p) )\\right)\\nonumber \\\\&:=& \\widetilde{q}_{11} + \\widetilde{q}_{12}.$ First, it is easy to verify the following convergence: $\\lim _{n \\rightarrow \\infty } n^{2H+1} {\\widetilde{q}}_{11}& = &\\lim _{n \\rightarrow \\infty } 2 n^{2H+1} \\left(\\frac{T}{n}\\right)^{2H+2}\\lfloor \\frac{nt}{T} \\rfloor \\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }( \\widetilde{Q}(p) - \\widetilde{R}(p) )\\nonumber \\\\& =&2 T^{2H+1} {t} \\sum _{p=0 }^{\\infty }( \\widetilde{Q}(p) - \\widetilde{R}(p) ) .$ On the other hand, since $ | \\sum _{p=0 }^{\\lfloor \\frac{nt}{T} \\rfloor }(p-1)( \\widetilde{Q}(p) - \\widetilde{R}(p) ) | \\le K n^{2H-1} $ , we have the convergence: $\\lim _{n \\rightarrow \\infty } n^{2H+1} {\\widetilde{q}}_{12} &=& 0.$ Putting together (REF ) and (REF ), and taking into account (REF ), we obtain: $\\lim _{n \\rightarrow \\infty } n^{2H+1} {\\widetilde{q}}_1 &=& 2 {T}^{2H+1}{t} \\sum _{p=0 }^{\\infty }( \\widetilde{Q}(p) - \\widetilde{R}(p) ) .$ The quantity $\\tilde{q}_{2}$ can be considered in a similar way.",
"We can show that $\\lim _{n\\rightarrow \\infty } n^{2H+1} {\\widetilde{q}}_2 &=& 2 {T}^{2H+1}{t} \\sum ^{ -1}_{p= -\\infty } ( \\widetilde{Q}(p) - \\widetilde{R}(p) ) .$ Applying (REF ) and (REF ) to (REF ) we obtain $\\lim _{ n \\rightarrow \\infty } n^{2H+1} \\mathbb {E}({ {z}}_n(t)^2 ) = 2 {T}^{2H+1} \\varrho {t}.$ Take $s,t \\in [0,T]$ .",
"By the same argument as in (REF ) and with the help of (REF ) and (REF ), we can show that $\\lim _{ n \\rightarrow \\infty } n^{2H+1} \\mathbb {E}( {z}_n(t) {z}_n(s) )&=&2T^{2H+1} \\varrho { ( t\\wedge s)}{ } .$ On the other hand, by some elementary computation (see Section REF ), one can show that $\\lim _{n\\rightarrow \\infty } \\mathbb {E}(z_{n}(t) B_{r}) &=& 0.$ Therefore, combining (REF ) and (REF ), we conclude that the covariances of the finite dimensional distributions of $ (n^{H+\\frac{1}{2}} {z}_n , B )$ converge to those of $(\\sqrt{ {2 \\varrho }{ } }T^{H+\\frac{1}{2}} W , B )$ .",
"The proof is now complete.",
"$\\Box $"
],
[
"The strong convergence",
"We recall that $X$ is the solution of equation (REF ) and $X^n$ is the continuous time interpolation of the Crank-Nicolson scheme defined in (REF ).",
"In this section we prove Theorem REF and some auxiliary results.",
"Proof of Theorem REF : The proof is divided into six steps.",
"Step 1: Decomposition of the error process.",
"In this step, we derive a decomposition for the error process $Y_t : = X_t - X^{n}_t $ , $t\\in [0, T]$ .",
"For convenience we set $\\eta (t) =t_{k}$ for $t \\in [t_{k}, t_{k+1})$ and $\\epsilon (t) = t_{k+1}$ for $t \\in (t_{k}, t_{k+1}]$ .",
"Putting together equations (REF ) and (REF ), it is easily seen that $Y_t &= & \\int _0^t \\left[ V (X_s) - V (X^n_s) \\right] dB_s +\\frac{1}{2} \\int _0^t \\left[ V (X^n_s) - V (X^n_{\\eta (s)})\\right] dB_s\\nonumber \\\\&&+\\frac{1}{2} \\int _0^t \\left[ V (X^n_s) - V (X^n_{ \\epsilon (s) }) \\right] dB_s\\nonumber \\\\&= &\\sum _{j=0}^m \\sum _{i =1}^d \\int _0^t V_{ji}(s) Y^{i}_s dB^j_s+\\frac{1}{2} J_1 (t)+\\frac{1}{2} J_2 (t) \\,,$ where we have set for $t \\in [0,T]$ : $V_{ji}(s) &= &\\int _0^1 \\partial _{i} V_j (\\theta X_s +(1-\\theta ) X^n_s ) d\\theta ,$ $J_{1}(t) = \\int _0^t \\left[ V (X^n_s) - V (X^n_{\\eta (s)})\\right] dB_s\\,,\\quad \\quad J_{2}(t) = \\int _0^t \\left[ V (X^n_s) - V (X^n_{ \\epsilon (s) }) \\right] dB_s \\,,$ and we denote by $\\partial _{i}$ the partial differential operator with respect to the $i$ th variable, that is, $\\partial _{i}f(x)=\\frac{\\partial f}{\\partial x_{i}}(x)$ for $f \\in C^{1}$ .",
"In addition, the chain rule for the Young integral enable us to write $V (X^n_s) - V (X^n_{\\eta (s)})&=&\\sum _{i=1}^{d} \\partial _{i} V (X^{n}_{\\eta (s)}) (X^{n,i}_{ s} - X^{n,i}_{\\eta (s)} ) \\\\&&+ \\sum _{i,i^{\\prime }=1}^{d}\\int _{\\eta (s)}^{s} \\int _{\\eta (s)}^{u} \\partial _{i^{\\prime }} \\partial _{i} V (X^{n}_{v}) dX^{n,i^{\\prime }}_{v} dX^{n,i}_{u}\\,.$ Substituting the above expression into $J_1 (t)$ , we obtain the following decomposition for $J_{1}(t)$ $J_1 (t)&= &R_0 (t) + R_1 (t), \\quad t\\in [0,T],$ where we define $R_1 (t) &= & \\int _0^t\\left[\\sum _{i,i^{\\prime }=1}^d \\int ^s_{ \\eta (s) } \\int ^{u}_{\\eta (s) } \\partial _{i^{\\prime }}\\partial _{i} V (X^n_{v}) dX^{n,i^{\\prime }}_{v} dX^{n,i}_{u} \\right]dB_s\\,$ and $R_0 (t)&= & \\int _0^t\\left[\\sum _{i=1}^{d} \\partial _{i} V (X^{n}_{\\eta (s)}) (X^{n,i}_{ s} - X^{n,i}_{\\eta (s)} )\\right]dB_s\\nonumber \\\\&= &\\frac{1}{2} \\sum _{i=1}^{d} \\sum _{j,j^{\\prime }=0}^{m} \\int _0^t\\partial _{i} V_{j} (X^n_{ \\eta (s) })\\left[ V_{j^{\\prime }}^{i} (X^n_{ \\epsilon (s) }) + V_{j^{\\prime }}^{i} (X^n_{\\eta (s )}) \\right] \\int _{\\eta (s)}^{s} dB^{j^{\\prime }}_{u}dB^{j}_s\\,,$ and in the second equation of (REF ) we have used relation (REF ).",
"We can proceed similarly as in (REF ) to derive the corresponding decomposition for $J_{2}(t)$ $J_2 (t)&= & - \\widetilde{R}_0 (t) + \\widetilde{R}_1 (t),\\quad t\\in [0,T],$ where $\\widetilde{R}_1 (t) &= & \\int _0^t\\left[\\sum _{i,i^{\\prime }=1}^d \\int _s^{ \\epsilon (s) } \\int _{u}^{ \\epsilon (s) } \\partial _{i^{\\prime }}\\partial _{i} V (X^n_{v}) dX^{n,i^{\\prime }}_{v} dX^{n,i}_{u} \\right]dB_s\\,,\\\\\\widetilde{R}_0 (t)&= &\\frac{1}{2} \\sum _{i=1}^{d} \\sum _{j,j^{\\prime }=0}^{m} \\int _0^t\\partial _{i} V_{j} (X^n_{ \\epsilon (s) })\\left[ V_{j^{\\prime }}^{i} (X^n_{ \\epsilon (s) }) + V_{j^{\\prime }}^{i} (X^n_{\\eta (s )}) \\right] \\int _{s}^{\\epsilon (s)} dB^{j^{\\prime }}_{u}dB^{j}_s \\,.\\nonumber $ To further decompose the process $J_{1}$ and $J_{2}$ , we introduce the processes $I_{1} $ and $I_{2} $ defined on $ \\Pi $ .",
"Namely, for $t \\in \\Pi \\setminus \\lbrace 0 \\rbrace $ we define $I_{ 1} (t) &= & \\sum _{ j,j^{\\prime }=0}^m\\sum _{k=0}^{ {nt}/{T} -1}( \\partial V_{j} V_{j^{\\prime }}) (X^n_{t_k})\\int _{t_k}^{t_{k+1} }\\int _{t_{k}}^{s} d B^{j^{\\prime }}_{u}dB^{j}_s\\,,\\\\I_{ 2} (t) &= & \\sum _{ j,j^{\\prime }=0}^m\\sum _{k=0}^{ {nt}/{T} -1}( \\partial V_{j} V_{j^{\\prime }} ) (X^n_{t_k})\\int _{t_k}^{t_{k+1} } \\int _{s}^{t_{k+1}} d B^{j^{\\prime }}_{u} dB^{j}_s\\,,$ and for $t=0$ we set $I_{1}(0) = I_{2}(0) = 0$ , where we used the notation $\\partial =(\\partial _{1},\\dots , \\partial _{d})$ and $\\partial V_jV_{j^{\\prime }} = \\sum _{i=1}^d \\partial _i V_jV_{j^{\\prime }}$ .",
"Subtracting () from (REF ) we obtain the following “Lévy area term” $I_{1}(t) - I_{2}(t) =E_{1}(t) := \\sum _{ j,j^{\\prime }=0}^m\\sum _{k=0}^{ {nt}/{T} -1}( \\partial V_{j} V_{j^{\\prime }} ) (X^n_{t_k}) \\zeta ^{j^{\\prime }j}_{t_{k},t_{k+1}},$ where we have denoted $\\zeta _{s t}^{ij} &=&\\int _{s}^{ t }\\int _{s}^{u} dB^{i}_{v}dB^j_u - \\int _{s}^{ t } \\int _{u}^{t} dB^{i}_{v} dB^j_u, \\quad 0\\le s\\le t\\le T.$ Note that a simple application of Fubini's theorem to $ \\zeta _{s, t}^{ij} $ yields the identity $\\zeta _{s, t}^{ij} = - \\zeta _{s, t}^{ji} $ .",
"So expression (REF ) can be reduced to $E_{1} (t) &= & \\sum _{j^{\\prime } < j }\\sum _{k=0}^{ {nt}/{T} -1 }\\phi _{jj^{\\prime }} (X^n_{t_k})\\zeta _{t_k, t_{k+1} }^{j^{\\prime }j}\\,, \\quad \\quad t \\in \\Pi \\, ,$ where $\\phi _{jj^{\\prime }}$ is defined as $\\phi _{jj^{\\prime }}=\\partial V_j V_{j^{\\prime }}-\\partial V_{j^{\\prime }} V_{j }$ .",
"In particular, when the driving process $B$ has dimension one we are left with $E_{1}\\equiv 0$ .",
"With these calculations in hand, we can now decompose $J_{1}(t)+J_{2}(t)$ for $t \\in \\Pi $ as follows: $J_{1}(t) + J_{2}(t) & = &\\left( I_1 (t) - I_2 (t) \\right)+\\left( R_0 (t) - I_1 (t) \\right)+\\left( I_2 (t)-\\widetilde{R}_{0} (t) \\right) +R_1(t) +\\widetilde{R}_1(t)\\nonumber \\\\&:= &E_1 (t) +E_2 (t) +E_3 (t) +E_4 (t) +E_5 (t) \\,.$ Step 2: Upper-bound for the Crank-Nicolson scheme.",
"It follows from Lemma 8.4 in [11] that there exists a constant $K$ such that $\\Vert X^n\\Vert _{\\infty }\\vee \\Vert X^n\\Vert _{\\beta } & \\le & K +{K \\Vert B\\Vert _{\\beta }^{1/\\beta } } .$ Furthermore, there exist constants $K_0 $ and $K^{\\prime }_0$ independent of $n$ such that for $0 \\le s<t\\le T$ and $(t-s)^\\beta \\Vert B\\Vert _{\\beta } \\le K_0 $ , we have $\\Vert X^n \\Vert _{s, t , \\beta } &\\le & K^{\\prime }_0 \\Vert B \\Vert _{\\beta } \\, .$ Step 3: Estimates of $E_{e}, 1\\le e\\le 5$ .",
"Take $ s,t \\in \\Pi $ such that $ s\\le t$ .",
"In this step, we derive a $L^{p}$ -estimate of $ E_{e} (t) - E_{e}(s)$ for $e=1,\\dots , 5$ .",
"We first show that for $e=2,3,4,5$ we have $\\Vert E_{e}(t) - E_{e}(s) \\Vert _{p}& \\le & K n^{-2H}(t-s)^{\\frac{1}{2}}, \\quad s,t \\in \\Pi ,$ where recall that $\\Vert \\cdot \\Vert _{p}$ denotes the $L^p$ -norm.",
"Let us start by bounding the term $E_2(t) -E_{2}(s) $ , $s,t \\in \\Pi $ .",
"Subtracting (REF ) from (REF ) we obtain $E_{2}(t) &= &\\sum _{i=1}^{d} \\sum _{j,j^{\\prime }=0}^{m} \\sum _{k=0}^{\\frac{nt}{T}-1} \\frac{1}{2} \\partial _{i}V_{j} (X^{n}_{t_{k}}) \\left[ V^{i}_{j^{\\prime }}(X^{n}_{t_{k+1}} )- V^{i}_{j^{\\prime }}(X^{n}_{t_{k }} ) \\right] \\int _{t_{k}}^{t_{k+1}} \\int _{t_{k}}^{s} dB^{j^{\\prime }}_{u} dB^{j}_{s}\\nonumber \\\\&=&\\frac{1}{4} \\sum _{i=1}^{d} \\sum _{j,j^{\\prime },j^{\\prime \\prime }=0}^m \\sum _{k=0}^{ \\frac{nt}{T} -1}\\partial _{i} V_j (X^n_{ t_k })\\int _{t_{k}}^{t_{k+1}} \\partial V^{i}_{j^{\\prime }} (X^n_{v }) \\left[ V_{j^{\\prime \\prime }}(X^{n}_{t_{k+1} } ) + V_{j^{\\prime \\prime }}(X^{n}_{t_{k}}) \\right] d B^{j^{\\prime \\prime }}_{v}\\nonumber \\\\&&\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\times \\int _{t_k}^{t_{k+1} } \\int ^s_{t_k} dB^{j^{\\prime }}_{u}dB^j_s\\,,$ where the second equation follows by applying the chain rule to $V^{i}_{j^{\\prime }}(X^{n}_{t_{k+1}} )- V^{i}_{j^{\\prime }}(X^{n}_{t_{k }} )$ and taking into account equation (REF ) for $X^{n}$ .",
"Take $h^{n}_{t_{k}} = \\partial _{i} V_{j} (X^{n}_{t_{k}}) \\left[ V_{j^{\\prime \\prime }}(X^{n}_{t_{k+1} } ) + V_{j^{\\prime \\prime }}(X^{n}_{t_{k}}) \\right] $ and $f_{v} = \\partial V^{i}_{j^{\\prime }} (X^n_{v }) $ .",
"The above expression becomes $E_{2}(t) &= & \\frac{1}{4} \\sum _{i=1}^{d} \\sum _{j,j^{\\prime },j^{\\prime \\prime }=0}^{m} \\sum _{k=0}^{\\frac{nt}{T}-1} \\int _{t_{k}}^{t_{k+1}}\\int _{t_k}^{t_{k+1} } \\int ^s_{t_k} f_{v} h_{t_{k}}^{n} dB^{j^{\\prime }}_{u}dB^j_s d B^{j^{\\prime \\prime }}_{v}.$ It is easy to verify that the triple integral on the right-hand side of (REF ) is equal to $&&\\int _{t_{k}}^{t_{k+1}}\\int _{t_k}^{v } \\int ^s_{t_k} f_{v} h_{t_{k}}^{n} dB^{j^{\\prime }}_{u}dB^j_s d B^{j^{\\prime \\prime }}_{v}+\\int _{t_{k}}^{t_{k+1}}\\int _{t_{k}}^{s } \\int ^v_{t_k} f_{v} h_{t_{k}}^{n} dB^{j^{\\prime }}_{u} d B^{j^{\\prime \\prime }}_{v}dB^j_s\\\\&&+ \\int _{t_{k}}^{t_{k+1}}\\int _{t_{k}}^{s } \\int ^{u}_{t_{k}} f_{v} h_{t_{k}}^{n} d B^{j^{\\prime \\prime }}_{v} dB^{j^{\\prime }}_{u}dB^j_s\\, .$ Substituting the above expression into (REF ), we obtain an expression of $E_{2}(t)$ of the form of (REF ).",
"One can show, with the help of the estimate of $X^{n}$ in (REF ), that $f$ and $h^{n}$ satisfy the conditions in Lemma REF .",
"So applying Lemma REF to $E_{2}(t)$ we obtain the estimate (REF ) for $e=2$ .",
"Estimate (REF ) still holds true for the cases when $e=3,4,5$ .",
"The proof is based on Lemma REF and is similar to the case $e=2$ .",
"We omit the details.",
"This completes the proof of (REF ).",
"Now we consider the process $E_1(t)$ , $t \\in \\Pi $ .",
"To this aim, we consider the decomposition $E_{1} (t) &= & \\sum _{0 \\ne j^{\\prime } < j }\\sum _{k=0}^{ {nt}/{T} -1 }\\phi _{jj^{\\prime }} (X^n_{t_k})\\zeta _{t_k, t_{k+1} }^{j^{\\prime }j} + \\sum _{0= j^{\\prime } < j }\\sum _{k=0}^{ {nt}/{T}-1 }\\phi _{jj^{\\prime }} (X^n_{t_k})\\zeta _{t_k, t_{k+1} }^{j^{\\prime }j}\\nonumber \\\\& := & E_{11}(t)+ E_{12} (t).$ Expression (REF ) and Lemma REF together suggest to consider the following “weight-free” random sum corresponding to $E_{11}$ ${g}_{n} (t) &= &n^{2H-\\frac{1}{2} } \\sum _{0 \\ne j^{\\prime } < j }\\sum _{k=0}^{\\lfloor \\frac{nt}{T} \\rfloor }\\zeta _{t_k, t_{k+1} }^{j^{\\prime }j} .$ It follows from relation (REF ) in Proposition REF that $ {g}_{n}$ satisfies the assumptions in Lemma REF .",
"Indeed, by Proposition REF the following estimate holds true for all $s,t\\in \\Pi $ $\\mathbb {E}( [ {g}_n(t)- {g}_n(s) ]^2 )^{\\frac{1}{2}} &\\le & K |t-s|^{\\frac{1}{2}}.$ Furthermore, since $g_{n}(t)-g_{n}(s)$ is a random variable in the second chaos of $B$ , by an hyper-contractivity argument we can show that estimate (REF ) holds in the $L^p$ -norm for all $p\\ge 1$ .",
"Take $f = \\phi _{jj^{\\prime }} (X^{n}_{\\cdot })$ , $\\beta ^{\\prime }=\\frac{1}{2}$ , $ \\frac{1}{2}< \\beta <H$ , $p=p^{\\prime }=q^{\\prime }=2$ .",
"Then applying Lemma REF to $E_{11} $ we obtain the estimate $\\Vert E_{11}(t)-E_{11} (s) \\Vert _p & \\le & Kn^{-2H+\\frac{1}{2}} (t-s)^{\\frac{1}{2}}, \\quad s,t \\in \\Pi .$ We proceed similarly to show the estimate for $E_{12}$ .",
"We first define the “weight-free” random sum corresponding to $E_{12} (t)$ $\\tilde{g}_{n} (t) &= & n^{1/2+H} \\sum _{0 = j^{\\prime } < j }\\sum _{k=0}^{\\lfloor \\frac{nt}{T} \\rfloor }\\zeta _{t_k, t_{k+1} }^{j^{\\prime }j}.$ Then as in (REF ), estimate (REF ) in Proposition REF together with some hyper-contractivity arguments yields that $ \\tilde{g}_{n} $ satisfies the conditions in Lemma REF for $\\beta ^{\\prime }=\\frac{1}{2}$ and $p=q^{\\prime }=2$ .",
"Taking $\\frac{1}{2}<\\beta <H$ , $q^{\\prime }=2$ and $f = \\phi _{jj^{\\prime }} (X^{n}_{\\cdot })$ as before and applying Lemma REF to $E_{12}$ , we obtain the estimate $\\Vert E_{12}(t)-E_{12} (s) \\Vert _p & \\le & Kn^{-H-\\frac{1}{2}} (t-s)^{\\frac{1}{2}}, \\quad s,t \\in \\Pi .$ In summary of relations (REF ), (REF ) and (REF ), and taking into account the fact that $E_{11}=0$ when $m=1$ and $E_{11}=E_{12}=0$ when $m=1$ and $V_{0} \\equiv 0$ , we obtain $\\sum _{e=1}^{5} \\Vert E_{e}(t) - E_{e}(s) \\Vert _{p} & \\le &K (t-s)^{\\frac{1}{2}} /\\vartheta _{n} , \\quad s,t \\in \\Pi .$ Step 4: Upper-bounds for the Jacobian.",
"In this step, we consider some linear equations associated with $X^{n}$ and $X$ .",
"Let $\\Lambda ^{n } = \\left( \\Lambda ^{n,i}_{i^{\\prime }} \\right)_{1\\le i,i^{\\prime }\\le d}$ be the solution of the linear equation $\\Lambda ^{n,i}_{{i^{\\prime }} }(t) &= & \\delta ^i_{i^{\\prime }}+ \\sum _{j=0}^m \\sum _{i^{\\prime \\prime } =1}^d\\int _0^t V^i_{j i^{\\prime \\prime }} (s) \\Lambda ^{n,i^{\\prime \\prime }}_{{i^{\\prime }} } (s) dB^{j }_s, ~~~~~~~i,i^{\\prime }=1, \\dots , d,\\quad t \\in [0, T].$ Here $\\delta ^i_{i^{\\prime }} $ is the Kronecker function, that is, $\\delta ^i_{i^{\\prime }} =1$ when $i= i^{\\prime }$ and $\\delta ^i_{i^{\\prime }} = 0 $ otherwise.",
"The $d\\times d$ matrix $\\Lambda ^n (t) $ is invertible.",
"We denote its inverse by $\\Gamma ^n(t) $ .",
"It is easy to verify that $\\Gamma ^n$ satisfies the equation $\\Gamma ^{n,i}_{{i^{\\prime }} }(t) &=& \\delta ^i_{i^{\\prime }} - \\sum _{j=0 }^m \\sum _{i^{\\prime \\prime } =1}^d\\int _0^t \\Gamma ^{n,i }_{{i^{\\prime \\prime } } } (s) V^{i^{\\prime \\prime }}_{j i^{\\prime }} (s) dB^{j }_s, ~~~~~~~i,i^{\\prime }=1, \\dots , d,\\quad t \\in [0 , T].$ By the product rule of Young integrals, and taking into account (REF ), it is easy to verify that $Y_t &= &\\frac{1}{2} \\Lambda ^{n }_{ t} \\sum _{i=1}^{2} \\int _{0}^{ t }\\Gamma ^{n }_{s}dJ_{i}(s), \\quad t\\in [0, T]\\,.$ Applying Lemma 3.2 (ii) in [10] and taking into account (REF ), we obtain the estimate $\\Vert \\Lambda ^n\\Vert _{\\infty } \\vee \\Vert \\Lambda ^n\\Vert _{\\beta } \\vee \\Vert \\Gamma ^n \\Vert _{\\infty } \\vee \\Vert \\Gamma ^n \\Vert _{\\beta } &\\le & K e^{K \\Vert B\\Vert _{\\beta }^{1/\\beta } } .$ It follows from Fernique's lemma that for $p\\ge 1$ we have $\\left\\Vert \\Vert \\Lambda ^n\\Vert _{\\infty }\\right\\Vert _{p} \\vee \\Vert \\Vert \\Lambda ^n\\Vert _{\\beta } \\Vert _{p}\\vee \\Vert \\Vert \\Gamma ^n \\Vert _{\\infty }\\Vert _{p} \\vee \\Vert \\Vert \\Gamma ^n \\Vert _{\\beta }\\Vert _{p} &\\le & K .$ Let $\\Lambda = \\left( \\Lambda ^i_{i^{\\prime }} \\right)_{1\\le i,i^{\\prime }\\le d}$ be the solution of the following equation, $\\Lambda ^{ i}_{{i^{\\prime }} }(t) &= &\\delta ^i_{i^{\\prime }}+ \\sum _{j=0}^m \\sum _{i^{\\prime \\prime } =1}^d\\int _0^t\\partial _{i^{\\prime \\prime }} V^i_j ( X_s )\\Lambda ^{ i^{\\prime \\prime }}_{{i^{\\prime }} } (s) dB^{j }_s,$ for $t \\in [0, T]$ , $ i,i^{\\prime }=1, \\dots , d$ , and denote by $\\Gamma (t)$ the inverse of $\\Lambda (t)$ .",
"As before, we can show that $\\Gamma $ satisfies the equation $\\Gamma ^{ i}_{{i^{\\prime }} }(t) &=& \\delta ^i_{i^{\\prime }} - \\sum _{j=0 }^m \\sum _{i^{\\prime \\prime } =1}^d\\int _0^t \\Gamma ^{n,i }_{{i^{\\prime \\prime } } } (s) \\partial _{i^{\\prime }}V_{j}^{i^{\\prime \\prime }}(X_{s}) dB^{j }_s$ for $t \\in [0 , T]$ , $ i,i^{\\prime }=1, \\dots , d$ .",
"It follows from Lemma 3.1 in [10] that the estimate (REF ) still holds true if we replace $\\Lambda ^{n}$ and $\\Gamma ^{n} $ in (REF ) by $\\Lambda $ and $\\Gamma $ .",
"Step 5: Estimates of $ \\Gamma ^{n} \\, Y $ .",
"In this step, we consider the process $\\Gamma ^{n} Y $ .",
"Multiplying both sides of (REF ) by $\\Gamma ^{n}_{t}$ , we obtain the expression $\\Gamma ^{n}_{t}\\, Y_{t} &=& \\frac{1}{2} \\sum _{i=1}^{2}\\int _{0}^{t } \\Gamma ^{n}_{u} d J_{i}(u).$ By writing $ \\Gamma ^{n}_{u} = \\Gamma ^{n}_{\\eta (u)} + ( \\Gamma ^{n}_{u} - \\Gamma ^{n}_{\\eta (u)} ) $ we obtain the following decomposition for $s,t \\in \\Pi $ , $s\\le t$ $\\sum _{i=1}^{2}\\int _{s}^{t } \\Gamma ^{n}_{u} d J_{i}(u) &= & \\sum _{i=1}^{2}\\int _{s}^{t } \\Gamma ^{n}_{\\eta (u)} d J_{i}(u) + \\sum _{i=1}^{2} \\int _{s}^{t } \\int _{\\eta (u)}^{u} d \\Gamma ^{n}_{v} d J_{i}(u).$ Revoking the decomposition (REF ) we get $\\sum _{i=1}^{2}\\int _{s}^{t } \\Gamma ^{n}_{\\eta (u)} d J_{i}(u)&=&\\sum _{e=1}^{5} \\sum _{t_{k}=s}^{t-\\frac{T}{n}}\\Gamma ^{n}_{t_{k}} (E_{e}(t_{k+1}) - E_{e} (t_{k})).$ For simplicity, we will denote the right-hand side of (REF ) as $\\sum _{t_{k}=s}^{t-\\frac{T}{n}}\\Gamma ^{n}_{t_{k}} (E_{e}(t_{k+1}) - E_{e} (t_{k}))& :=& \\int _{s}^{t } \\Gamma ^{n}_{\\eta (u)} d E_{e}(u).$ Note, however, that equation (REF ) is only valid for $s,t \\in \\Pi $ since $E_{e}$ , $e=1,\\dots , 5$ are only defined on $\\Pi $ .",
"Now substituting (REF ) into (REF ) and taking into account (REF ) we get $\\sum _{i=1}^{2}\\int _{s}^{t } \\Gamma ^{n}_{u} d J_{i}(u) &= & \\sum _{e=1}^{5} \\int _{s}^{t } \\Gamma ^{n}_{\\eta (u)} d E_{e}(u) + \\sum _{i=1}^{2} \\int _{s}^{t } \\int _{\\eta (u)}^{u} d \\Gamma ^{n}_{v} d J_{i}(u).$ As in (REF ), we handle the term $\\int _{s}^{t } \\Gamma ^{n}_{\\eta (u)} d E_{e}(u) $ on the right-hand side of (REF ) by Lemma REF .",
"Take $ \\hat{g}_{n}(t) = \\vartheta _{n} E_{e}(t) $ , $t \\in \\Pi $ and $f = \\Gamma ^{n} $ , and let $\\beta , \\beta ^{\\prime }, p, p^{\\prime }, q^{\\prime }$ be as before.",
"Then estimate (REF ) shows that $\\hat{g}_{n}$ satisfies the conditions in Lemma REF .",
"Applying Lemma REF to $\\int _{s}^{t } \\Gamma ^{n}_{\\eta (u)} d E_{e}(u) $ and invoking expression (REF ) we obtain $\\Big \\Vert \\sum _{e=1}^{5} \\int _{s}^{t } \\Gamma ^{n}_{\\eta (u)} d E_{e}(u) \\Big \\Vert _{p} & \\le &K (t-s)^{\\frac{1}{2}} / \\vartheta _{n}\\,, \\quad s,t \\in \\Pi .$ We turn to the second term in (REF ).",
"By the definition of $\\Gamma ^{n}$ and $J_{1}$ we have $\\int _{t_{k}}^{t_{k+1} } \\int _{t_{k}}^{u} d \\Gamma ^{n}_{v} d J_{1}(u) & = &\\frac{1}{2} \\sum _{jj^{\\prime }=0 }^m \\sum _{i,i^{\\prime } =1}^d \\int _{t_{k}}^{t_{k+1} } \\int _{t_{k}}^{u}\\left( - \\Gamma ^{n }_{{i^{\\prime } } } (v) V^{i^{\\prime }}_{j , i} (v) \\right) dB^{j }_v\\int _{t_{k}}^{u} \\partial V_{j^{\\prime }}^{i} (X^n_{r}) d X^{n}_{r} dB^{j^{\\prime }}_u .$ One can show that $\\int _{s}^{t } \\int _{\\eta (u)}^{u} d \\Gamma ^{n}_{v} d J_{1}(u)&=&\\sum _{k=\\frac{ns}{T}}^{\\frac{nt}{T}-1} \\int _{t_{k}}^{t_{k+1}} \\int _{t_{k}}^{u} d \\Gamma ^{n}_{v} d J_{1}(u)$ has the form of (REF ).",
"Applying Lemma REF we obtain $\\left\\Vert \\int _{s}^{t } \\int _{\\eta (u)}^{u} d \\Gamma ^{n}_{v} d J_{i}(u) \\right\\Vert _{p}&\\le & K n^{-2H}(t-s)^{1/2}$ for $i=1$ .",
"This estimate still holds true in the case $i=2$ , and the proof is similar.",
"Substituting (REF ) and (REF ) into (REF ) we obtain the estimate $\\left\\Vert \\sum _{i=1}^{2 } \\int _{s}^{t } \\Gamma ^{n}_{u} d J_{i}(u) \\right\\Vert _{p} & \\le &K (t-s)^{\\frac{1}{2}} /\\vartheta _{n}$ for $s,t \\in \\Pi $ .",
"It is easy to see that $\\left\\Vert \\int _{t_{k}}^{t } \\Gamma ^{n}_{u} d J_{e}(u) \\right\\Vert _{p} \\le \\, & Kn^{-2H}, \\quad t\\in [t_{k}, t_{k+1}].$ Combining this estimate with (REF ) we obtain the inequality $\\sup _{t\\in [0,T]}\\left\\Vert \\sum _{i=1}^{2}\\int _{0}^{t } \\Gamma ^{n}_{u} d J_{i}(u) \\right\\Vert _{p} & \\le &K / \\vartheta _{n} .$ Step 6: Conclusion.",
"The inequality (REF ) follows by applying the Hölder inequality to (REF ) and using the estimate (REF ) and the estimate (REF ) for $ \\Lambda ^{n} $ .",
"$\\Box $ In the last part of this section we will show some technical estimates that will be used in the proof of the convergence in law of the error.",
"Lemma 4.1 Under the assumptions and notation of Theorem REF , the error process $Y=X-X^n$ satisfies the following relation for all $s,t\\in \\Pi $ $\\mathbb {E}\\left(\\left| Y_{t} - Y_{s} \\right|^{p} \\right)^{1/p} &\\le &K |t-s|^{\\frac{1}{2}}/\\vartheta _n.$ Proof: Invoking the expression (REF ) of $Y$ , we can write $Y_t-Y_{s} &= &\\frac{1}{2} ( \\Lambda ^{n }_{ t} -\\Lambda ^{n }_{ s} ) \\sum _{i=1}^{2} \\int _{0}^{ t }\\Gamma ^{n }_{u}dJ_{i}(u) +\\frac{1}{2} \\Lambda ^{n }_{ s} \\sum _{i=1}^{2} \\int _{s}^{ t }\\Gamma ^{n }_{u}dJ_{i}(u).$ The inequality (REF ) then follows by applying the Hölder inequality to (REF ) and by taking into account the estimates (REF ) and (REF ).",
"This completes the proof.",
"$\\Box $ The following lemma is a convergence result for the processes $\\Lambda ^{n}$ and $\\Gamma ^{n}$ .",
"Lemma 4.2 Take ${\\beta }: \\frac{1}{2} < \\beta <H$ .",
"Let $\\Lambda ^{n}$ and $\\Lambda $ be the solutions of equations (REF ) and (REF ), respectively, and let $\\Gamma ^{n}$ and $\\Gamma $ be their inverses.",
"Then we have $\\Vert \\Lambda ^{n} -\\Lambda \\Vert _{\\beta , p } +\\Vert \\Gamma ^{n} -\\Gamma \\Vert _{\\beta , p }&\\le & Kn^{1-2\\beta }.$ Proof: See Section REF .",
"$\\Box $ We end this section with the following auxiliary results.",
"The reason we put these results here is because they are concerned with $\\Gamma $ .",
"As in (REF ), for the sake of conciseness we will denote $\\sum _{t_{k}=s}^{t-\\frac{T}{n}}\\Gamma ^{n}_{t_{k}} (E_{11}(t_{k+1}) - E_{1} (t_{k}))& :=& \\int _{s}^{t } \\Gamma ^{n}_{\\eta (u)} d E_{11}(u) \\quad \\text{ for } s,t \\in \\Pi .$ The integral $\\int _{s}^{t} \\Gamma ^{n}_{\\eta (u)} dE_{12} (u)$ is defined similarly.",
"Lemma 4.3 We continue to use the notation of in Theorem REF .",
"Let $s,t \\in \\Pi $ , $s\\le t$ .",
"If $m>1$ , we have the estimate $\\sup _{s,t\\in \\Pi } \\left\\Vert \\sum _{i=1}^{2 } \\int _{s}^{t } \\Gamma ^{n}_{u} d J_{i}(u) - \\int _{s}^{t} \\Gamma ^{n}_{\\eta (u)} d E_{11}(u) \\right\\Vert _{p} & \\le &K n^{-\\frac{1}{2}-H}.$ In the case when $m=1$ , we have the estimate $\\sup _{s,t\\in \\Pi } \\left\\Vert \\sum _{i=1}^{2 } \\int _{s}^{t } \\Gamma ^{n}_{u} d J_{i}(u) - \\int _{s}^{t} \\Gamma ^{n}_{\\eta (u)} d E_{12}(u) \\right\\Vert _{p} &\\le &K n^{-2H}.$ Take ${\\beta }: \\frac{1}{2} < \\beta <H$ .",
"Assume that $m=1$ and $V_{0}\\equiv 0$ .",
"Then we have the estimate $\\sup _{ t\\in [0,T]} \\left\\Vert \\sum _{i=1}^{2 } \\int _{0}^{t } \\Gamma ^{n}_{u} d J_{i}(u) - \\sum _{e=2}^{5 }\\int _{0}^{\\eta (t)} \\Gamma ^{n}_{\\eta (u)} d E_{e}(u) \\right\\Vert _{p} & \\le & K_{\\beta }n^{1-4\\beta },$ where $K_{{\\beta }} $ is a constant depending on ${\\beta }$ .",
"Proof: By subtracting $\\int _{s}^{t} \\Gamma ^{n}_{\\eta (u)} d E_{11}(u) $ from both sides of (REF ) we obtain $&&\\sum _{i=1}^{2}\\int _{s}^{t } \\Gamma ^{n}_{u} d J_{i}(u) -\\int _{s}^{t} \\Gamma ^{n}_{\\eta (u)} d E_{11}(u)\\nonumber \\\\&&=\\, \\int _{s}^{t} \\Gamma ^{n}_{\\eta (u)} d E_{12}(u) + \\sum _{e=2}^{5}\\int _{s}^{t } \\Gamma ^{n}_{\\eta (u)} d E_{e}(u) + \\sum _{i=1}^{2} \\int _{s}^{t } \\int _{\\eta (u)}^{u} d \\Gamma ^{n}_{v} d J_{i}(u).$ Similar to the proof of the estimate (REF ), we can show that the first and second terms on the right-hand side of (REF ) are bounded by $Kn^{\\frac{1}{2}+H}$ and $K n^{-2H}$ , respectively.",
"On the other hand, we have shown in (REF ) that the third term is bounded by $Kn^{-2H}$ .",
"In summary, we obtain the estimate (REF ).",
"The estimate (REF ) can be shown in a similar way.",
"The proof of estimate (REF ) is included in Section REF .",
"$\\Box $"
],
[
"Asymptotic error distribution",
"In this section, we prove Theorem REF .",
"Proof of Theorem REF : The proof will be done in four steps.",
"Step 1.",
"We first assume that $m>1$ or $V_{0}\\lnot \\equiv 0$ .",
"By Theorem 13.5 in [2] and taking into account inequality (REF ), to prove the weak convergence of $( \\vartheta _{n}( \\tilde{X} - \\tilde{X}^{n}) , B)$ it suffices to show the convergence of its finite dimensional distributions (f.d.d.).",
"By (REF ) we have $\\tilde{X}_{t} - \\tilde{X}^{n}_{t} =X_{t_{k}} - X^{n}_{t_{k}}&= & \\frac{1}{2} \\Lambda ^{n}_{ t_{k}} \\sum _{i=1}^{2}\\int _{0}^{t_{k} } \\Gamma ^{n}_{u} d J_{i}(u)$ for $t\\in [t_{k},t_{k+1})$ .",
"Step 2.",
"Assume that $m>1$ .",
"Set $S^{n}(t) &=&\\frac{1}{2} \\Lambda ^{n}_{t_{k}} \\int _{0}^{t_{k}} \\Gamma ^{n}_{\\eta (s)} d E_{11} (s)$ for $t \\in [t_{k},t_{k+1})$ .",
"It follows from the estimate (REF ) in Lemma REF that the difference $\\vartheta _{n}\\Vert S^{n}(t) - ( \\tilde{X}_{t} - \\tilde{X}^{n}_{t} ) \\Vert _{p}$ is uniformly bounded by $\\vartheta _{n} n^{-\\frac{1}{2} - H}$ and thus converges to zero as $n\\rightarrow \\infty $ .",
"This implies that the limit of the finite dimensional distributions of $ (\\vartheta _{n}( \\tilde{X} - \\tilde{X}^{n}) , B)$ is equal to that of $\\left( \\vartheta _{n} S^{n} , ~B \\right)$ .",
"Set $S(t) &=& \\frac{1}{2} \\Lambda _{ t_{k}} \\int _{0}^{t_{k+1}} \\Gamma _{\\eta (s)} d E_{11} (s)$ for $t\\in [t_{k} , t_{k+1})$ .",
"Then we have $S^{n}(t_{k}) - S (t_{k})&=&\\frac{1}{2} \\sum _{0 \\ne j^{\\prime } < j } \\int _{0}^{t_{k}} \\left[ \\Lambda ^{n}_{t_{k}} \\Gamma ^{n}_{ \\eta (s) } \\phi _{jj^{\\prime }} (X^{n}_{\\eta (s)}) - \\Lambda _{t_{k}} \\Gamma _{ \\eta (s) } \\phi _{jj^{\\prime }} (X_{\\eta (s)})\\right] d \\zeta ^{j^{\\prime }j}_{\\eta (s), s }\\nonumber \\\\&& - \\frac{1}{2} \\Lambda _{t_{k}} \\int _{t_{k}}^{t_{k+1}} \\Gamma _{\\eta (s)} d E_{11}(s) .$ It is easy to see that the $L^p$ -norm of the second term in the right-hand side of (REF ) is bounded by $Kn^{-2H}$ .",
"On the other hand, with the help of Lemma REF , one can show that $\\Vert \\Lambda ^{n}_{t} \\Gamma ^{n}_{\\cdot } \\phi _{jj^{\\prime }} (X^{n}_{\\cdot }) - \\Lambda _{t} \\Gamma _{\\cdot } \\phi _{jj^{\\prime }} (X_{\\cdot }) \\Vert _{\\beta , p } & \\le & K n^{1-2\\beta }.$ So by taking $f = \\Lambda ^{n}_{t} \\Gamma ^{n}_{\\cdot } \\phi _{jj^{\\prime }} (X^{n}_{\\cdot }) - \\Lambda _{t} \\Gamma _{\\cdot } \\phi _{jj^{\\prime }} (X_{\\cdot })$ and $\\zeta _{k,n} = \\zeta ^{j^{\\prime }j}_{t_{k},t_{k+1}}$ in Lemma REF , we see that the first term in the right-hand side of (REF ) is bounded by $K n^{1-2\\beta +1/2 -2H}$ .",
"In summary of these two estimates, we obtain $\\left\\Vert S^{n}(t) - S(t) \\right\\Vert _{p} & \\le & K n^{1-2\\beta +1/2 -2H} \\vee n^{-2H}$ for $t \\in \\Pi $ , and thus for $t \\in [0,T]$ .",
"Therefore, the f.d.d.",
"convergence of $\\left( \\vartheta _{n} S^{n} , B \\right)$ is the same as that of $\\left( \\vartheta _{n} S , B \\right)$ .",
"Applying Proposition REF to the process $\\left( \\vartheta _{n} S , B \\right)$ and taking into account the weak convergence result in Proposition REF , we obtain that the f.f.d.",
"of $\\left( \\vartheta _{n} S , B \\right)$ converge to that of $\\left( U , B \\right)$ , where $U_{t} &=& T^{2H-\\frac{1}{2}} \\sqrt{ \\frac{ \\kappa }{ 2 } } \\Lambda _{t} \\sum _{1 \\le j^{\\prime } < j \\le m } \\int _{0}^{t} \\Gamma _{s} \\phi _{jj^{\\prime }}(X_{s} ) dW^{j^{\\prime }j}_{s} .$ The convergence (REF ) follows from the fact that $\\lbrace U_{t}, \\, t\\in [0,T]\\rbrace $ solves the SDE (REF ).",
"Step 3.",
"We assume $m=1$ and $V_{0} \\lnot \\equiv 0$ .",
"The estimate (REF ) implies that the f.d.d.",
"convergence of $ (\\vartheta _{n}(\\tilde{X} - \\tilde{X}^{n}) , B)$ is equal to that of $( \\vartheta _{n}\\widetilde{S}^{n}, B )$ , where $\\widetilde{S}^{n}_{t} &=& \\frac{1}{2} \\Lambda ^{n}_{ \\eta (t)} \\int _{0}^{\\eta (t) } \\Gamma ^{n}_{\\eta (s)} d E_{12} (s) .$ As in the case $m>1$ , with the help of Lemma REF we can show that the convergence of the f.d.d.",
"of $( \\vartheta _{n}\\widetilde{S}^{n}, B )$ is the same as that of $( \\vartheta _{n}\\widetilde{S} , B )$ , where $\\widetilde{S}_{t} &=& \\frac{1}{2} \\Lambda _{ \\eta (t)} \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\Gamma _{ t_{k} }\\phi _{10} (X_{t_k})\\zeta _{t_k, t_{k+1} }^{01} .$ Applying Proposition REF to the above process and taking into account the weak convergence result in Proposition REF , we obtain that its f.d.d.",
"converges to those of $(\\widetilde{U}, B)$ , where $\\widetilde{U}_{t} &=& T^{H+\\frac{1}{2}} \\sqrt{\\frac{\\varrho }{2}} \\Lambda _{t} \\int _{0}^{t} \\Gamma _{s} \\phi _{10} (X_s) d W_{s}$ as $n\\rightarrow \\infty $ .",
"The convergence (REF ) follows from the fact that $\\lbrace \\widetilde{U}_{t}, \\, t\\in [0,T]\\rbrace $ solves equation (REF ).",
"Step 4.",
"We consider the case when $m=1$ and $V_{0}\\equiv 0$ .",
"The convergence (REF ) is clear for $t=0$ .",
"In the following, we consider $t>0$ .",
"The estimate (REF ) implies that the $L^p$ -convergence of $n^{2H}(\\tilde{X}_{t} - \\tilde{X}^{n}_{t})$ is the same as that of $\\frac{1}{2} n^{2H} \\Lambda _{ \\eta (t)}^{n} \\sum _{e=2}^{5} \\int _{0}^{ \\eta (t) }\\Gamma _{\\eta (s)}^{n}dE_{e}(s).$ As in the case $m>1$ , with the help of Lemma REF we can show that the quantity (REF ) has the same $L^p$ -limit as $\\frac{1}{2} n^{2H} \\Lambda _{ \\eta (t) } \\sum _{e=2}^{5} \\int _{0}^{ \\eta (t) }\\Gamma _{\\eta (s)}dE_{e}(s)&=&\\frac{1}{2} n^{2H} \\Lambda _{ \\eta (t)} \\sum _{e=2}^{5} \\sum _{k=0}^{\\lfloor \\frac{nt}{T} \\rfloor -1}\\Gamma _{t_{k}}\\left(E_{e}(t_{k+1}) - E_{e}(t_{k }) \\right).$ In the following, we show that the quantity in (REF ) converges to the solution of equation (REF ).",
"Take $t\\in \\Pi $ .",
"By (REF ) we have $\\sum _{k=0}^{ \\frac{nt}{T} -1}\\Gamma _{t_{k}}\\left(E_{2}(t_{k+1}) - E_{2}(t_{k }) \\right)&=&\\frac{1}{4} \\sum _{i =1}^d \\sum _{k=0}^{ \\frac{nt}{T} -1} \\Gamma _{t_{k}}\\partial _{i} V (X^n_{ t_k })\\int _{t_{k}}^{t_{k+1}} \\partial V^{i} (X^n_{v }) \\left[ V(X^{n}_{t_{k+1} } ) + V(X^{n}_{t_{k}}) \\right] d B_{v}\\\\&&\\quad \\quad \\quad \\quad \\times \\int _{t_k}^{t_{k+1} } \\int ^s_{t_k} dB_{u}dB_s.$ Take $\\widetilde{E}_{2}(t) &= &\\frac{1}{2} \\sum _{i =1}^d \\sum _{k=0}^{ \\frac{nt}{T} -1} \\Gamma _{t_{k}}(\\partial _{i} V \\partial V^{i} V) (X_{ t_k })\\int _{t_{k}}^{t_{k+1}} d B_{v}\\int _{t_k}^{t_{k+1} } \\int ^s_{t_k} dB_{u}dB_s\\\\&= &\\frac{1}{4} \\sum _{i =1}^d \\sum _{k=0}^{ \\frac{nt}{T} -1} \\Gamma _{t_{k}}(\\partial _{i} V \\partial V^{i} V) (X_{ t_k })( B_{t_{k},t_{k+1}})^{3} .$ It is easy to show that $n^{2H} \\left( \\sum _{k=0}^{ \\frac{nt}{T} -1}\\Gamma _{t_{k}}\\left(E_{2}(t_{k+1}) - E_{2}(t_{k }) \\right) - \\widetilde{E}_{2}(t) \\right) \\rightarrow 0 \\quad \\text{ in } L^{p} \\quad \\text{ as } n \\rightarrow \\infty $ for $t\\in \\Pi $ .",
"Similarly, we take $\\widetilde{E}_{3}(t) &= &- \\frac{1}{4} \\sum _{i=1}^d\\sum _{k=0}^{ {nt}/{T}-1 }\\Gamma _{t_{k}} \\left(\\partial (\\partial _{i} V V^{i}) V+ V^{i} \\partial ( \\partial _{i} V ) V\\right) ( X_{ t_{k}})( B_{t_{k}, t_{k+1}})^{3}\\,,\\\\\\widetilde{E}_{4} (t) &= & \\widetilde{E}_{5} (t)=\\frac{1}{6} \\sum _{i^{\\prime },i =1}^d \\sum _{k=0}^{ nt/T -1 } \\Gamma _{t_{k}}( V^{i^{\\prime }} V^{i } \\partial _{i }\\partial _{i^{\\prime }} V ) (X_{t_{k}})( B_{t_{k},t_{k+1}})^{3}\\,,$ then one can show that $n^{2H} \\left( \\sum _{k=0}^{ \\frac{nt}{T} -1}\\Gamma _{t_{k}}\\left( E_{e}(t_{k+1}) - E_{e}(t_{k }) \\right) - \\widetilde{E}_{e}(t) \\right) &\\rightarrow & 0$ in $L^{p}$ for $e=3,\\,4,\\,5$ .",
"In summary from (REF ) and (REF ), we obtain $n^{2H} \\sum _{e=2}^{5}\\sum _{k=0}^{ \\frac{nt}{T} -1}\\Gamma _{t_{k}}\\left( E_{e}(t_{k+1}) - E_{e}(t_{k }) \\right) - n^{2H} \\sum _{e=2}^{5} \\widetilde{E}_{e}(t) & \\rightarrow & 0$ in $L^{p}$ for $t\\in \\Pi $ .",
"It is easy to see that $\\sum _{e=2}^{5} \\widetilde{E}_{e}(t) &= & -\\frac{1}{6}\\sum _{i^{\\prime },i =1}^d \\sum _{k=0}^{ nt/T -1 } \\Gamma _{t_{k}} ( V^{i^{\\prime }} V^{i } \\partial _{i }\\partial _{i^{\\prime }} V ) (X_{t_{k}})( B_{t_{k},t_{k+1}})^{3}$ for $t\\in \\Pi $ .",
"Take $ f_{t} = \\Gamma _{t} ( V^{i^{\\prime }} V^{i } \\partial _{i }\\partial _{i^{\\prime }} V ) (X_{t })$ and $\\zeta _{k,n} = (B_{t_{k+1}} - B_{t_{k}})^{3} $ , then by applying Proposition REF to (REF ) and taking into account Lemma REF (ii) we obtain that $\\frac{1}{2} \\Lambda _{\\eta (t)} \\left( n^{2H} \\sum _{e=2}^{5} \\widetilde{E}_{e}(\\eta (t)) \\right) \\rightarrow \\bar{U}_{t}$ in $L^p$ for $t\\in [0,T]$ , where $\\bar{U}_{t} &= &- \\frac{T^{2H}}{4} \\sum _{i^{\\prime },i =1}^d \\Lambda _{t} \\int _{0}^{t} \\Gamma _{s}( V^{i^{\\prime }} V^{i } \\partial _{i }\\partial _{i^{\\prime }} V ) (X_{t }) dB_{s}\\,.$ Thanks to (REF ), this convergence implies that $\\frac{1}{2} n^{2H} \\Lambda _{ \\eta (t)} \\sum _{e=2}^{5} \\sum _{k=0}^{\\lfloor \\frac{nt}{T} \\rfloor -1}\\Gamma _{t_{k}}\\left(E_{e}(t_{k+1}) - E_{e}(t_{k }) \\right) &\\rightarrow &\\bar{U}_{t}$ for $t\\in [0,T]$ .",
"The convergence (REF ) follows from the fact that the process $\\bar{U}$ solves equation (REF ).",
"$\\Box $"
],
[
"Proof of (",
"The proof will be done in seven steps.",
"Step 1.",
"In this step, we derive a decomposition for $d_{2}$ .",
"First, applying the integration by parts formula (REF ), we obtain $\\mathbb {E}\\left[{{}{Z}}_n(t)\\widetilde{D}_{u^{\\prime }}{{}{Z}}_n(t)D_{s^{\\prime }}{{}{Z}}_n(t)\\right]& = &\\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\int _0^T\\int _{0}^{T}\\int _0^T\\int _0^T\\left[D_{r^{\\prime }} \\widetilde{D}_{u^{\\prime }}{{}{Z}}_n(t)\\right]\\left[\\widetilde{D}_{v^{\\prime }} D_{s^{\\prime }}{{}{Z}}_n(t)\\right]\\nonumber \\\\& &\\quad \\quad \\quad \\times \\beta _{\\frac{k}{n} }(r) \\gamma _{t_k, r } (v )\\mu (dvdv^{\\prime }) \\mu (drdr^{\\prime })\\nonumber \\\\&= &\\sum _{k, k_3,k_4=0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\int _0^T\\int _{0}^{T}\\int _0^T\\int _0^T\\beta _{\\frac{k_3}{n}} (r^{\\prime } ) \\gamma _{t_{k_3}, r^{\\prime } } (u^{\\prime } )\\beta _{\\frac{k_4}{n}} ( {s}^{\\prime } ) \\gamma _{t_{k_4} , {s}^{\\prime } } (v^{\\prime } )\\nonumber \\\\&&\\quad \\quad \\quad \\times \\beta _{\\frac{k}{n} }(r) \\gamma _{t_k, r } (v )\\mu (dvdv^{\\prime }) \\mu (drdr^{\\prime }),$ where the second equation follows from the fact that $\\widetilde{D}_v D_r {Z}_n(t) &=& \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\beta _{\\frac{k}{n}} (r) \\gamma _{t_k, r} (v), \\quad t\\in [0,T].$ Substituting the expression (REF ) into (REF ) we obtain $d_{2}&=&6 \\sum _{k_1, k_2, k_3,k_4 =0}^{ \\lfloor \\frac{nt}{T} \\rfloor }\\int _{t_{k_4} }^{t_{k_4+1}}\\int _{t_{k_1} }^{t_{k_1+1}}\\int _0^T\\int _0^T\\int _{t_{k_3}}^{t_{k_3+1 } }\\int _{t_{k_2} }^{t_{{k_2} +1}}\\int _0^T\\int _0^T\\gamma _{t_{k_3}, r^{\\prime } } (u^{\\prime } )\\gamma _{t_{k_4} , {s}^{\\prime } } (v^{\\prime } )\\nonumber \\\\&&\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\times \\gamma _{t_{k_2}, r } (v )\\gamma _{t_{k_1}, s} (u)\\mu (dvdv^{\\prime }) \\mu (drdr^{\\prime })\\mu (dudu^{\\prime }) \\mu (dsds^{\\prime }).$ By changing the variables from $(v,v^{\\prime },r,r^{\\prime },u,u^{\\prime },s,s^{\\prime })$ to $\\frac{T}{n}(v,v^{\\prime },r,r^{\\prime },u,u^{\\prime },s,s^{\\prime })$ and exchanging of the orders of integrals associated with $\\mu (dudu^{\\prime })$ and $\\mu (drdr^{\\prime })$ we obtain $d_{2} &=&6 \\left( \\frac{T}{n} \\right)^{8H}\\sum _{k_1, k_2, k_3,k_4 =0}^{ \\lfloor \\frac{nt}{T} \\rfloor }c(k_{1},k_{2},k_{3},k_{4}) ,$ where $c(k_{1},k_{2},k_{3},k_{4}) &=&\\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}}\\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}}\\int _0^n\\int _0^n \\int _0^n\\int _0^n\\varphi _{ {k_3}, r^{\\prime } } (u^{\\prime } )\\varphi _{ {k_4} , {s}^{\\prime } } (v^{\\prime } )\\nonumber \\\\& &\\quad \\quad \\quad \\times \\varphi _{ {k_2}, r } (v )\\varphi _{ {k_1}, s} (u)\\mu (dvdv^{\\prime }) \\mu (dudu^{\\prime }) \\mu (drdr^{\\prime })\\mu (dsds^{\\prime }),$ and recall that $\\varphi _{ k, s} (u)= \\varphi ^{0}_{ k, s} (u) - \\varphi ^{1}_{ k, s} (u), \\quad \\varphi ^{0}_{ k, s} (u) = \\mathbf {1}_{ [k, s] }(u) \\quad \\text{ and} \\quad \\varphi ^{1}_{ k, s} (u)= \\mathbf {1}_{[s, {k+1}] }(u) \\,,$ where $\\mathbf {1}_{[a,b]}$ denotes the indicator function of the interval $[a,b]$ .",
"Now we denote $I&:=&\\left\\lbrace k_{1},k_{2},k_{3},k_{4} =0,1,\\dots , \\lfloor \\frac{nt}{T} \\rfloor \\right\\rbrace .$ Take $i,j = 1,2,3,4 $ , and denote by $I_{ij}$ the set of $(k_{1},k_{2},k_{3},k_{4})$ in $I$ such that $ |k_{i} - k_{j}|>2 $ , that is, $I_{ij} = \\lbrace (k_{1},k_{2},k_{3},k_{4}) \\in I: |k_{i} - k_{j}|>2 \\rbrace $ .",
"Denote by $I_{ij}^{c}$ the complement of $I_{ij}$ .",
"We can decompose $I$ as follows.",
"$I&=& \\bigcup _{l=1}^{8} M_{l} \\,,$ where $M_{1} &=& I_{42} \\cap I_{41} \\cap I_{31} \\cap I_{32} ;\\\\[3mm]M_{2} &=& \\left( I_{42}^{c} \\cap I_{41} \\cap I_{31} \\cap I_{32}\\right) \\bigcup \\left( I_{42} \\cap I_{41} \\cap I_{31}^{c} \\cap I_{32}\\right)\\\\[1mm]&:=& M_{21}+M_{22};\\\\[3mm]M_{3} &=& \\left( I_{42} \\cap I_{41}^{c} \\cap I_{31} \\cap I_{32}\\right) \\bigcup \\left( I_{42} \\cap I_{41} \\cap I_{31} \\cap I_{32}^{c} \\right);\\\\[3mm]M_{4} &=& \\left( I_{42}^{c} \\cap I_{41}^{c} \\cap I_{31} \\cap I_{32}\\right) \\bigcup \\left( I_{42} \\cap I_{41}^{c} \\cap I_{31}^{c} \\cap I_{32}\\right)\\\\&& \\bigcup \\left( I_{42}^{c} \\cap I_{41} \\cap I_{31} \\cap I_{32}^{c} \\right) \\bigcup \\left( I_{42} \\cap I_{41} \\cap I_{31}^{c} \\cap I_{32}^{c } \\right)\\\\[1mm]&:=& M_{41} \\cup M_{42} \\cup M_{43} \\cup M_{44} ;\\\\[3mm]M_{5} &=& I_{42} \\cap I_{41}^{c} \\cap I_{31} \\cap I_{32}^{c} ;\\\\[3mm]M_{6} &=& I_{42}^{c} \\cap I_{41} \\cap I_{31}^{c} \\cap I_{32} ;\\\\[3mm]M_{7} &=& \\left( I_{42}^{c} \\cap I_{41}^{c} \\cap I_{31}^{c} \\cap I_{32}\\right) \\bigcup \\left( I_{42}^{c} \\cap I_{41}^{c} \\cap I_{31} \\cap I_{32}^{c} \\right)\\\\&& \\bigcup \\left( I_{42}^{c} \\cap I_{41} \\cap I_{31}^{c} \\cap I_{32}^{c}\\right) \\bigcup \\left( I_{42} \\cap I_{41}^{c} \\cap I_{31}^{c} \\cap I_{32}^{c} \\right) ;\\\\[3mm]M_{8} &=& I_{42}^{c} \\cap I_{41}^{c} \\cap I_{31}^{c} \\cap I_{32}^{c} .$ For any subset $M$ of $I$ , we denote $d_{2 } ( M)&:=&6 \\left( \\frac{T}{n} \\right)^{8H}\\sum _{(k_1, k_2, k_3,k_4) \\in M }c(k_{1},k_{2},k_{3},k_{4}) .$ It is clear that $d_{2} &=& \\sum _{l=1}^{8} d_{2}(M_{l}).$ Thus to show (REF ) it suffices to show that $n^{8H-2} d_{2 } (M_{l}) \\rightarrow 0$ as $n\\rightarrow \\infty $ for each $l=1,\\dots , 8$ .",
"Step 2.",
"In this step, we show the convergence of $n^{8H-2} d_{2}(M_{7})$ and $n^{8H-2} d_{2}(M_{8})$ .",
"Since $| \\varphi _{k,s} (u)| \\le \\mathbf {1}_{[k,k+1]}(u),$ we have $|c ( k_{1},k_{2},k_{3},k_{4} )| &\\le & 1.$ Applying this inequality to $d_{2}(M_{7})$ we obtain $|d_{2}(M_{7})| &\\le & 6 \\left( \\frac{T}{n} \\right)^{8H} \\sum _{( k_{1},k_{2},k_{3},k_{4}) \\in M_{7}} 1.$ Note that $M_{7} \\subset \\lbrace |k_{i}-k_{j}|\\le 6 \\text{ for }i,j=1,2,3,4 \\rbrace ,$ so the number of elements in $M_{7}$ is less than $6n$ .",
"This implies that $|d_{2}(M_{7})|&\\le & 36 n \\left( \\frac{T}{n} \\right)^{8H}.$ It follows from this estimate that $n^{8H-2} d_{2}(M_{7}) \\rightarrow 0$ as $n\\rightarrow 0$ .",
"Note that $M_{8} \\subset \\lbrace |k_{i}-k_{j}|\\le 4 \\text{ for }i,j=1,2,3,4 \\rbrace $ .",
"So in the same way, we can show that $n^{8H-2} d_{2}(M_{8}) \\rightarrow 0$ .",
"Step 3.",
"In this step, we consider $d_{2}(M_{5})$ and $d_{2}(M_{6})$ .",
"Take $(k_{1},k_{2},k_{3},k_{4} ) \\in M_{5}$ , we have $|k_{2}-k_{4}|>2$ and $|k_{1}-k_{3}|>2$ .",
"By the mean value theorem and with the help of (REF ), it is easy to see that $|c( k_{1},k_{2},k_{3},k_{4})| &\\le & K |k_{2}-k_{4}|^{2H-2}|k_{1}-k_{3}|^{2H-2}.$ Applying (REF ) to $d_{2}(M_{5})$ we obtain $|d_{2}(M_{5})|&\\le & K \\left( \\frac{T}{n} \\right)^{8H}\\sum _{k_1, k_2, k_3,k_4 \\in M_{5}} |k_{2}-k_{4}|^{2H-2}|k_{1}-k_{3}|^{2H-2}.$ Note that for $(k_1, k_2, k_3,k_4) \\in M_{5}$ we have $ |k_{1} - k_{4}|\\le 2 $ and $ |k_{2} - k_{3}|\\le 2 $ , so $|k_{2} -k_{4}| &\\le & |k_{2}-k_{3}| + |k_{3} - k_{1}| + |k_{1} - k_{4}|\\\\&\\le & 3|k_{3} - k_{1}|.$ Applying this inequality to the right-hand side of (REF ) yields $| d_{2}(M_{5})|&\\le & K \\left( \\frac{T}{n} \\right)^{8H}\\sum _{(k_1, k_2, k_3,k_4) \\in M_{5}} |k_{2}-k_{4}|^{2H-2}|k_{4}-k_{2}|^{2H-2}\\\\&\\le & K \\left( \\frac{T}{n} \\right)^{8H}\\sum _{ k_2, k_4 :|k_{2}-k_{4}|>2} |k_{2}-k_{4}|^{4H-4}.$ By taking $p=k_{2}-k_{4}$ , we obtain $|d_{2}(M_{5})| &\\le & K \\left( \\frac{T}{n} \\right)^{8H} \\sum _{k_{2} =0}^{n} \\sum _{n\\ge |p|>2} |p|^{4H-4}\\\\&\\le & Kn \\left( \\frac{T}{n} \\right)^{8H} (n^{4H-3}\\vee 1 ).$ It follows from the above estimate that $n^{8H-2}d_{2}(M_{5}) $ converges to zero as $n$ tends to infinity.",
"The proof for the convergence $n^{8H-2}d_{2}(M_{6}) \\rightarrow 0$ is similar.",
"Instead of (REF ), we have the estimate $|c( k_{1},k_{2},k_{3},k_{4})| &\\le & K |k_{1}-k_{4}|^{2H-2}|k_{2}-k_{3}|^{2H-2}$ for $(k_{1},k_{2},k_{3},k_{4}) \\in M_{6}$ .",
"Step 4.",
"In this step, we derive a new expression for $c( k_{1},k_{2},k_{3},k_{4})$ .",
"Recall that $ \\varphi _{ k_{4}, s^{\\prime }} (v^{\\prime })= \\varphi ^{0}_{ k_{4}, s^{\\prime }} (v^{\\prime }) - \\varphi ^{1}_{ k_{4}, s^{\\prime }} (v^{\\prime })$ .",
"Substituting this identity into (REF ) we obtain $c(k_{1},k_{2},k_{3},k_{4})&= & c_{0}-c_{1},$ where $c_{i} &=& \\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}}\\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}}\\int _0^n\\int _0^n \\int _0^n\\int _0^n\\varphi _{ {k_3}, r^{\\prime } } (u^{\\prime } )\\varphi ^{i}_{ {k_4} , {s}^{\\prime } } (v^{\\prime } )\\nonumber \\\\&&\\quad \\quad \\quad \\quad \\quad \\quad \\times \\varphi _{ {k_2}, r } (v )\\varphi _{ {k_1}, s} (u)\\mu (dvdv^{\\prime }) \\mu (dudu^{\\prime }) \\mu (drdr^{\\prime })\\mu (dsds^{\\prime })\\,.$ By exchanging the orders of the integrals associated with $v^{\\prime }$ and $ s^{\\prime } $ in $c_{1}$ , we obtain $c_{1} &= &\\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}}\\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}}\\int _{0}^{n} \\int _{0}^{n} \\int _{ {k_4 } }^{ v^{\\prime }}\\int _{ 0 }^{n }|v - v^{\\prime }|^{2H-2} | {s} - {s}^{\\prime }|^{2H-2}\\varphi _{ {k_3}, r^{\\prime } } (u^{\\prime } )\\\\&&\\quad \\quad \\quad \\quad \\quad \\quad \\times \\varphi _{ {k_2}, r } (v ) \\varphi _{ {k_1}, s} (u)dv ds^{\\prime } \\mu (dudu^{\\prime })\\mu ( drdr^{\\prime } )ds dv^{\\prime } ,$ which, by switching the notations $s^{\\prime }$ and $v^{\\prime }$ , is equal to $&\\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}}\\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}}\\int _{0}^{n} \\int _{0}^{n}\\int _{ {k_4 } }^{ s^{\\prime }}\\int _{ 0 }^{ n }|v - s^{\\prime } |^{2H-2} | {s} - v^{\\prime }|^{2H-2}\\varphi _{ {k_3}, r^{\\prime } } (u^{\\prime } )\\\\& \\quad \\quad \\quad \\quad \\quad \\times \\varphi _{ {k_2}, r } (v ) \\varphi _{ {k_1}, s} (u)dv dv^{\\prime } \\mu (dudu^{\\prime })\\mu ( drdr^{\\prime } )ds ds^{\\prime } .$ Substituting the above expression of $c_{1}$ into (REF ) we obtain $c(k_{1},k_{2},k_{3},k_{4})&=&\\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}}\\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}}\\int _{0}^{n} \\int _{0}^{n} \\int _{ {k_4 } }^{ s^{\\prime }}\\int _{ 0 }^{ n }\\phi (s,s^{\\prime },v , v^{\\prime } ) \\varphi _{ {k_3}, r^{\\prime } } (u^{\\prime } ) \\nonumber \\\\& & \\quad \\quad \\quad \\quad \\times \\varphi _{ {k_2}, r } (v )\\varphi _{ {k_1}, s} (u)dv dv^{\\prime } \\mu (dudu^{\\prime })\\mu ( drdr^{\\prime } ) ds ds^{\\prime },$ where we denote $\\phi (s,s^{\\prime },v , v^{\\prime } ) &=&| v - v^{\\prime }|^{2H-2} |s - s^{\\prime } |^{2H-2}-|v - s^{\\prime } |^{2H-2} | {s} - v^{\\prime }|^{2H-2} .$ Step 5.",
"We turn to $d_{2}(M_{4})$ .",
"It is easy to show that $d_{2} \\left(M_{4i}\\right) &=& d_{2} \\left( M_{4j}\\right), \\quad i,j =1,2,3,4.$ As an example, we show that $ d_{2}(M_{41}) =d_{2}(M_{44}) $ .",
"The other identities in (REF ) can be shown similarly.",
"First, by exchanging the orders of integrals associated with $\\mu (drdr^{\\prime })$ and $\\mu (dsds^{\\prime })$ and integrals associated with $\\mu (dvdv^{\\prime })$ and $\\mu (dudu^{\\prime })$ , we obtain $c(k_{1},k_{2},k_{3},k_{4}) &=&\\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}} \\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}}\\int _0^n\\int _0^n \\int _0^n\\int _0^n\\varphi _{ {k_3}, r^{\\prime } } (u^{\\prime } )\\varphi _{ {k_4} , {s}^{\\prime } } (v^{\\prime } )\\nonumber \\\\& &\\quad \\quad \\quad \\times \\varphi _{ {k_2}, r } (v )\\varphi _{ {k_1}, s} (u)\\mu (dudu^{\\prime }) \\mu (dvdv^{\\prime })\\mu (dsds^{\\prime }) \\mu (drdr^{\\prime }).$ Replacing $(v,v^{\\prime },u,u^{\\prime },r,r^{\\prime },s,s^{\\prime }) $ by $(u,u^{\\prime },v,v^{\\prime },s,s^{\\prime },r,r^{\\prime })$ in the above expression, we obtain $c(k_{1},k_{2},k_{3},k_{4}) &=&\\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}} \\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}}\\int _0^n\\int _0^n \\int _0^n\\int _0^n\\varphi _{ {k_3}, s^{\\prime } } (v^{\\prime } )\\varphi _{ {k_4} , {r}^{\\prime } } (u^{\\prime } )\\nonumber \\\\& &\\quad \\quad \\quad \\times \\varphi _{ {k_2}, s } (u )\\varphi _{ {k_1}, r} (v)\\mu (dvdv^{\\prime }) \\mu (dudu^{\\prime })\\mu (drdr^{\\prime }) \\mu (dsds^{\\prime })\\\\&=& c(k_{2},k_{1},k_{4},k_{3}).$ So we have $d_{2}(M_{41}) &=& 6 \\left( \\frac{T}{n} \\right)^{8H}\\sum _{(k_1, k_2, k_3,k_4 ) \\in M_{41}}c(k_{2},k_{1},k_{4},k_{3})= d_{2}(M_{44}),$ where the second identity follows by replacing $ (k_1, k_2, k_3,k_4 ) $ by $(k_{2},k_{1},k_{4},k_{3})$ .",
"The identities in (REF ) imply that to show the convergence $n^{8H-2}d_{2}(M_{4})\\rightarrow 0$ , it suffices to show that $n^{8H-2}d_{2}(M_{44})\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"Take $(k_{1},k_{2},k_{3},k_{4}) \\in M_{44}$ .",
"Then we have $|k_{1}-k_{4}|>2$ and $ |k_{2}-k_{4}|>2 $ .",
"This allows us to apply the mean value theorem to $\\phi $ to obtain the estimate $\\left|\\phi (s,s^{\\prime }, v, v^{\\prime })\\right|& \\le &K (|k_4-k_1|^{2H-3}|k_4-k_2|^{2H-2} + |k_4-k_1|^{2H-2 } |k_4-k_2|^{2H-3 } ).$ Applying (REF ) to (REF ) and taking into account (REF ) we obtain $\\left|c(k_1, k_2, k_3,k_4 )\\right|& \\le &K (|k_4-k_1|^{2H-3}|k_4-k_2|^{2H-2} + |k_4-k_1|^{2H-2 } |k_4-k_2|^{2H-3 } ).$ Since $|k_{1}-k_{2}| \\le |k_{1}-k_{3}| + |k_{3}-k_{2}| \\le 4 $ , we have $|k_{4}-k_{1}| \\le 3 |k_{4}-k_{2}| $ .",
"This applied to (REF ) yields $\\left|c(k_1, k_2, k_3,k_4 )\\right|& \\le &K|k_{4}-k_{1}|^{4H-5},$ and thus $|d_{2}(M_{44}) |&\\le & 6 \\left( \\frac{T}{n} \\right)^{8H} \\sum _{( k_{1},k_{2},k_{3},k_{4}) \\in M_{44}} | c(k_1, k_2, k_3,k_4 )|\\\\&\\le & 6 \\left( \\frac{T}{n} \\right)^{8H} \\sum _{k_{1},k_{4}: |k_{1}-k_{4}|>2} K |k_4-k_1|^{4H-5} .$ By taking $p=k_{1}-k_{4}$ , we obtain $|d_{2}(M_{44}) |&\\le & 6 \\left( \\frac{T}{n} \\right)^{8H} \\sum _{k_{1} =0}^{n} \\sum _{n\\ge |p|>2} p^{4H-5}\\le 12n \\left( \\frac{T}{n} \\right)^{8H} \\sum _{p=3}^{\\infty } p^{4H-5} ,$ which implies that $n^{8H-2} d_{2}(M_{44}) \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"Step 6.",
"In this step, we consider $ d_{2}(M_{2}) $ and $d_{2}(M_{3}) $ .",
"As in Step 4, it is easy to show that $d_{2}(M_{21}) = d_{2}(M_{22})$ .",
"So to show that $n^{8H-2}d_{2}(M_{2}) \\rightarrow 0$ it suffices to show that $n^{8H-2}d_{2}(M_{22}) \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"Take $(k_{1},k_{2},k_{3},k_{4}) \\in M_{22}$ , we have $|k_{1}-k_{4}|>2$ , $ |k_{2}-k_{4}|>2 $ , and so the inequality (REF ) holds.",
"Applying (REF ) to $d_{2}(M_{22})$ and taking $p_{1}=k_{1}-k_{4}$ and $p_{2}=k_{4}-k_{2}$ , we obtain $|d_{2}(M_{22})| &\\le & K \\left( \\frac{T}{n} \\right)^{8H}\\sum _{(k_1, k_2, k_{3}, k_4) \\in M_{22}} (|k_4-k_1|^{2H-3}|k_4-k_2|^{2H-2} + |k_4-k_1|^{2H-2 } |k_4-k_2|^{2H-3 } )\\\\&\\le & K \\left( \\frac{T}{n} \\right)^{8H}\\sum _{k_{4}=1}^{n}\\,\\,\\sum _{n \\ge |p_{1}|,\\, |p_{2}| \\ge 2 } ( |p_{1}|^{2H-3} |p_{2}|^{2H-2}+ |p_{1}|^{2H-2} |p_{2}|^{2H-3} ) .$ It is easy to see from the above estimate that $n^{8H-2}|d_{2}(M_{22})| \\le Kn^{2H-2}$ , which converges to zero as $n$ tends to infinity.",
"The proof for the convergence $n^{8H-2}d_{2}(M_{3}) \\rightarrow 0$ follows the same lines.",
"Step 7.",
"It remains to show that $n^{8H-2}d_{2}(M_{1}) \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"To do this, we first derive a new expression for $c(k_{1},k_{2},k_{3},k_{4})$ .",
"Recall that $\\varphi _{ {k_3}, r^{\\prime }} (u^{\\prime })&=&\\varphi _{ {k_3}, r^{\\prime }}^{0} (u^{\\prime }) - \\varphi _{ {k_3}, r^{\\prime }}^{1} (u^{\\prime }).$ Substituting this identity into (REF ), we obtain $c(k_{1},k_{2},k_{3},k_{4}) &=& \\tilde{c}_{0}-\\tilde{c}_{1} ,$ where $\\tilde{c}_{i} &=&\\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}}\\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}}\\int _{0}^{n} \\int _{0}^{n} \\int _{ {k_4 } }^{ s^{\\prime }}\\int _{ 0 }^{ n }\\phi (s,s^{\\prime },v , v^{\\prime } ) \\varphi _{ {k_3}, r^{\\prime } }^{i} (u^{\\prime } ) \\nonumber \\\\& & \\quad \\quad \\quad \\quad \\times \\varphi _{ {k_2}, r } (v )\\varphi _{ {k_1}, s} (u)dv dv^{\\prime } \\mu (dudu^{\\prime })\\mu ( drdr^{\\prime } ) ds ds^{\\prime }\\,.$ As in Step 3, by exchanging the order of the integrals associated with the variables $r^{\\prime }$ and $u^{\\prime }$ , and then switching the notations $r^{\\prime }$ and $u^{\\prime }$ , we obtain $\\tilde{c}_{1} &=&\\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}} \\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}}\\int _{ k_{3} }^{r^{\\prime } } \\int _{ 0 }^{n }\\int _{ {k_4 } }^{ s^{\\prime }}\\int _{ 0 }^{n }\\phi (s ,s^{\\prime },v , v^{\\prime } )|u-r^{\\prime }|^{2H-2} |r-u^{\\prime }|^{2H-2}\\\\& &\\quad \\quad \\quad \\quad \\times \\varphi _{ {k_2}, r } (v ) \\varphi _{ {k_1}, s } (u) dv dv^{\\prime }d ud {u}^{\\prime }drdr^{\\prime } ds ds^{\\prime }\\,.$ Substituting the above expression of $\\tilde{c}_{1}$ into (REF ), we obtain $c (k_{1},k_{2},k_{3},k_{4})&= &\\int _{ {k_4} }^{ {k_4+1}}\\int _{ {k_1} }^{ {k_1+1}}\\int _{ {k_3}}^{ {k_3+1 } }\\int _{ {k_2} }^{ {{k_2} +1}} \\int _{ k_{3} }^{r^{\\prime } } \\int _{ 0 }^{n }\\int _{ {k_4 } }^{ s^{\\prime }}\\int _{ 0 }^{n }\\phi (s,s^{\\prime }, v, v^{\\prime }) \\phi (r,r^{\\prime },u,u^{\\prime })\\nonumber \\\\&&\\quad \\quad \\quad \\quad \\quad \\times \\varphi _{ {k_2}, r } (v ) \\varphi _{ {k_1}, s } (u ) dv dv^{\\prime } d {u}d u^{\\prime }drdr^{\\prime } ds ds^{\\prime }.$ Take $(k_{1},k_{2},k_{3},k_{4}) \\in M_{1}$ , then it is clear that the inequality (REF ) holds true, and in the same way, we can show that $| \\phi (r,r^{\\prime },u,u^{\\prime }) |&\\le & K( |k_{1}-k_{3}|^{2H-3}|k_{2}-k_{3}|^{2H-2}+ |k_{1}-k_{3}|^{2H-2}|k_{1}-k_{3}|^{2H-3}).$ Applying inequalities (REF ) and (REF ) to (REF ) and taking $p_{1}=k_{3}-k_{1}$ , $p_{2} = k_{2}-k_{3}$ , $p_{3} = k_{4} - k_{2}$ we obtain $| d_{2}(M_{1}) | &\\le & 6 \\left( \\frac{T}{n} \\right)^{8H}\\sum _{(k_1, k_2, k_3,k_4 ) \\in M_{1}}|c(k_1, k_2, k_3,k_4)|\\nonumber \\\\&\\le & Kn^{-8H} \\sum _{k_{1}=0}^{n} \\sum _{(p_{1},p_{2},p_{3}) \\in J} ( |p_{1}|^{2H-3}|p_{2}|^{2H-2}+ |p_{1}|^{2H-2}|p_{2}|^{2H-3})\\tilde{c} (p_{1},p_{2},p_{3})\\nonumber \\\\&=& Kn^{-8H} \\sum _{k_{1}=0}^{n} \\sum _{p_{1},p_{2}: n \\ge |p_{1}|,|p_{2}| >2} ( |p_{1}|^{2H-3}|p_{2}|^{2H-2}+ |p_{1}|^{2H-2}|p_{2}|^{2H-3})\\nonumber \\\\&& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\times \\sum _{ p_{3} \\in J(p_{1},p_{2})} \\tilde{c} (p_{1},p_{2},p_{3}),$ where $\\tilde{c} (p_{1},p_{2},p_{3}) &=& |p_{1}+p_{2}+p_{3}|^{2H-3}|p_{3}|^{2H-2} +|p_{1}+p_{2}+p_{3}|^{2H-2}|p_{3}|^{2H-3} ,$ $J = \\lbrace (p_{1},p_{2},p_{3}): n \\ge |p_{1}|,|p_{2}| , |p_{3}|, |p_{1}+p_{2}+p_{3}|>2\\rbrace ,$ and $J(p_{1},p_{2})= \\lbrace p_{3}: n \\ge |p_{3}|, |p_{1}+p_{2}+p_{3}|> 2 \\rbrace .$ We claim that $\\sum _{p_{3} \\in J(p_{1},p_{2})} \\tilde{c} (p_{1},p_{2},p_{3}) $ is uniformly bounded in $(p_{1},p_{2})$ by a constant.",
"Take $p_{1},p_{2}$ such that $n \\ge |p_{1}|,|p_{2}|>2$ .",
"Since when $|p_{1}+p_{2} +p_{3}|< |p_{3}|$ we have $|p_{1}+p_{2} +p_{3}|^{{\\alpha }}> |p_{3}|^{{\\alpha }}$ for ${\\alpha }=2H-2$ and ${\\alpha }=2H-3$ .",
"So $\\sum _{p_{3}\\in J(p_{1},p_{2}): |p_{1}+p_{2} +p_{3}| < |p_{3}|}\\tilde{c} (p_{1},p_{2},p_{3})&\\le &\\sum _{p_{3}\\in J(p_{1},p_{2}) } |p_{1}+p_{2}+p_{3}|^{4H-5}\\nonumber \\\\&\\le &2 \\sum _{p =3}^{\\infty }p^{4H-5}.$ Similarly, we have $\\sum _{p_{3}\\in J(p_{1},p_{2}): |p_{1}+p_{2} +p_{3}|\\ge |p_{3}|} \\tilde{c} (p_{1},p_{2},p_{3})&\\le &\\sum _{p_{3}:n \\ge |p_{3}| >2} |p_{3}|^{4H-5}\\le 2\\sum _{p=3}^{\\infty } p^{4H-5}.$ In summary of (REF ) and (REF ), we have shown that $\\sum _{p_{3}\\in J(p_{1},p_{2}) } \\tilde{c} (p_{1},p_{2},p_{3})&\\le & 4\\sum _{p=2}^{\\infty } p^{4H-5}.$ Applying inequality (REF ) to (REF ), we obtain the estimate $| d_{2}(M_{1}) | &\\le &K n^{-6H},$ which implies that $n^{8H-2}d_{2}(M_{1}) \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"$\\Box $"
],
[
"Proof of (",
"By the integration by parts formula (REF ), we obtain $\\mathbb {E}(z_{n}(t) B_{r}) &=& \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{0}^{r} \\int _{0}^{T} \\int _{0}^{T} \\beta _{\\frac{k}{n}} (s) \\gamma _{t_k, s} (u) d{u} \\mu (d{s} ds^{\\prime }).$ By changing the variables from $(u,s,s^{\\prime })$ to $\\frac{T}{n} (u,s,s^{\\prime })$ in the above expression, we obtain $\\mathbb {E}(z_{n}(t) B_{r}) &=&\\left(\\frac{T}{n}\\right)^{2H+1} \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{0}^{\\frac{nr}{T}} \\int _{k}^{k+1} \\int _{0}^{n} \\varphi _{ k, s} (u) |s-s^{\\prime }|^{2H-2} d{u} d{s} ds^{\\prime }\\,,$ where $\\varphi _{k,s}(u)$ is defined in (REF ).",
"Let us denote $I_{1}(k) = [k-2, k+2] \\cap [0, \\frac{nr}{T}]$ and $I_{2}(k) = [0, \\frac{nr}{T}] \\setminus I_{1}(k) $ , and set $A_{i} &=&\\left(\\frac{T}{n}\\right)^{2H+1} \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{I_{i}(k)} \\int _{k}^{k+1} \\int _{0}^{n} \\varphi _{ k, s} (u) |s-s^{\\prime }|^{2H-2} d{u} d{s} ds^{\\prime }\\,.$ Then it is easy to show that $\\mathbb {E}(z_{n}(t) B_{r}) &=& A_{1}+A_{2}.$ So to show that $n^{H+\\frac{1}{2}} \\mathbb {E}(z_{n}(t) B_{r}) \\rightarrow 0$ it suffices to show that $n^{H+\\frac{1}{2}} A_{i} \\rightarrow 0$ as $n\\rightarrow \\infty $ for $i=1,2$ .",
"We can write $A_{1}&\\le & \\left(\\frac{T}{n}\\right)^{2H+1} \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{k-2}^{k+2} \\int _{k}^{k+1} \\int _{k}^{k+1} |s-s^{\\prime }|^{2H-2} d{u} d{s} ds^{\\prime }\\le K n \\left(\\frac{T}{n}\\right)^{2H+1} ,$ so we have $n^{H+\\frac{1}{2}} A_{1} \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"Now we turn to $A_{2}$ .",
"By exchanging the orders of the integrals with respect to $u$ and $s$ , we have $\\int _{k}^{k+1} \\int _{0}^{n} \\mathbf {1}_{ [k, s]} (u) |s-s^{\\prime }|^{2H-2} d{u} d{s} &=& \\int _{k}^{k+1} \\int _{k}^{s} |u-s^{\\prime }|^{2H-2} d{u} d{s} .$ Substituting (REF ) into $A_{2}$ we obtain $A_{2} &=&\\left(\\frac{T}{n}\\right)^{2H+1} \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{I_{2}(k)} \\int _{k}^{k+1} \\int _{k}^{s} ( |s-s^{\\prime }|^{2H-2} - |u-s^{\\prime }|^{2H-2} ) d{u} d{s} ds^{\\prime }\\,.$ Note that for $s^{\\prime } \\in I_{2}(k)$ we have $\\left| \\int _{k}^{k+1} \\int _{k}^{s} ( |s-s^{\\prime }|^{2H-2} - |u-s^{\\prime }|^{2H-2} ) d{u} d{s} \\right| &\\le & K |k-s^{\\prime }|^{2H-3},$ so $|A_{2}| &\\le & \\left(\\frac{T}{n}\\right)^{2H+1} \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } \\int _{I_{2}(k)} K |k-s^{\\prime }|^{2H-3} ds^{\\prime }\\\\&=& K \\left(\\frac{T}{n}\\right)^{2H+1} \\sum _{k=0}^{ \\lfloor \\frac{nt}{T} \\rfloor } ( 2^{2H-2}-( n-k )^{2H-2} + 2^{2H-2}- k^{2H-2} )\\\\&\\le & K \\left(\\frac{T}{n}\\right)^{2H+1} \\sum _{k=0}^{n} 2^{2H-2} = Kn \\left(\\frac{T}{n}\\right)^{2H+1} ,$ which implies that $n^{H+\\frac{1}{2}} A_{2} \\rightarrow 0$ as $n \\rightarrow \\infty $ .",
"This completes the proof.",
"$\\Box $"
],
[
"Estimates of some triple integrals",
"In this subsection, we provide estimates for some triple integrals which have been used the main body of the paper.",
"Lemma 6.1 (i) For $t\\in \\Pi $ , we define $G_t & = & \\sum _{k=0}^{ nt/T -1 } \\int _{t_{k}}^{t_{k+1}} \\int _{t_{k}}^{s } \\int _{t_{k}}^{u} dB^1_{v} dB^2_{u} dB^3_{s },$ where $B^{1}$ , $B^{2}$ , $B^{3}$ are either a fractional Brownian motion with Hurst parameter $H>\\frac{1}{2}$ or equal to the identity function.",
"Take $p \\ge 1$ .",
"Then we have $\\Vert {G}_t - G_s \\Vert _p \\le K n^{-2 H} |t-s|^{\\frac{1}{2}}$ for $s,t \\in \\Pi $ .",
"(ii) Let $B$ a one-dimensional fBm with Hurst parameter $H>\\frac{1}{2}$ .",
"Take $p\\ge 1$ and $t \\in [0, T]$ .",
"We have the following convergence in $L^p$ : $n^{2H} \\sum _{k=0}^{\\lfloor nt/T \\rfloor -1} (B_{ t_{k+1} } - B_{t_{k} })^{3} &\\rightarrow & 3 {T^{2H} } B_t\\,.$ Proof: The result in (i) follows from Proposition 5.10 in [11].",
"The convergence in (ii) follows immediately from results in [8] or [25].",
"$\\Box $ We need the following technical lemma.",
"Lemma 6.2 Let $f$ and $g$ be $\\beta $ -Hölder continuous stochastic processes on $[0,T]$ and $ \\mathbb {E}[ \\Vert f\\Vert _{\\beta }^{p}]+ \\mathbb {E}[ \\Vert g\\Vert _{\\beta }^{p}] \\le K$ for all $\\frac{1}{2} < \\beta < H$ and $p\\ge 1$ , and let $h^{n}$ , $n\\in \\mathbb {N}$ be processes on $[0,T]$ such that $\\Vert h^{n}_{t } - h^{n}_{s}\\Vert _{p}& \\le & K (t-s)^{\\beta }, \\quad s,t \\in \\Pi : s\\le t.$ Let $B^{1}$ , $B^{2}$ and $B^{3}$ be as in Lemma REF .",
"Define the process $\\widetilde{G}_{t} &= & \\sum _{k=0}^{ nt/T -1 } h^{n}_{t_{k}} \\int _{t_{k}}^{t_{k+1}} \\int _{t_{k}}^{s_{3}} \\int _{t_{k}}^{s_{2}} f_{s_{i}} g_{s_{j}} dB^1_{s_{1}} dB^2_{s_{2}} dB^3_{s_{3}}\\,,\\quad t \\in \\Pi \\,.$ Then the following estimate holds true for all $s,t \\in \\Pi $ : $\\Vert \\widetilde{G}_t - \\widetilde{G}_s \\Vert _p & \\le &K n^{-2 H} |t-s|^{\\frac{1}{2}}.$ Proof: We decompose $\\widetilde{G}$ as follows: $\\widetilde{G}_{t} & = & \\sum _{k=0}^{ nt/T -1} f_{t_{k}} g_{t_{k}} h^{n}_{t_{k}} \\int _{t_{k}}^{t_{k+1}} \\int _{t_{k}}^{s_{3}} \\int _{t_{k}}^{s_{2}} dB^1_{s_{1}} dB^2_{s_{2}} dB^3_{s_{3}}\\nonumber \\\\&&+ \\sum _{k=0}^{ nt/T -1} f_{t_{k}} h^{n}_{t_{k}} \\int _{t_{k}}^{t_{k+1}} \\int _{t_{k}}^{s_{3}} \\int _{t_{k}}^{s_{2}} \\int _{t_{k}}^{s_{j}} dg_{s_{4}} dB^1_{s_{1}} dB^2_{s_{2}} dB^3_{s_{3}}\\nonumber \\\\&& + \\sum _{k=0}^{ nt/T -1} h^{n}_{t_{k}} \\int _{t_{k}}^{t_{k+1}} \\int _{t_{k}}^{s_{3}} \\int _{t_{k}}^{s_{2}} \\int _{t_{k}}^{s_{i}} g_{s_{j}} d f_{s_{4}} dB^1_{s_{1}} dB^2_{s_{2}} dB^3_{s_{3}}\\,.$ Applying Proposition 5.10 in [11] to the second and third terms on the right-hand side of (REF ), and applying Lemma REF to the first term and taking into account the estimate in Lemma REF (i), we obtain the inequality (REF ).",
"$\\Box $"
],
[
"Proof of Lemma ",
"By the definition of $J_{1}$ , we have $\\int _{s}^{ t }\\Gamma ^{n }_{u}dJ_{1}(u)&= &\\sum _{i=1}^{d} \\int _{s}^{ t }\\Gamma ^{n }_{u}\\int _{\\eta (u)}^{u} \\partial _{i} V (X^{n}_{v}) d X^{n,i}_{v} dB_u\\\\&= &\\sum _{i=1}^{d} \\sum _{k=\\lfloor \\frac{ns}{T} \\rfloor }^{\\lfloor \\frac{nt}{T} \\rfloor } \\int _{t_{k}\\vee s}^{ t_{k+1}\\wedge t }\\Gamma ^{n }_{u}\\int _{\\eta (u)}^{u} \\partial _{i} V (X^{n}_{v}) d X^{n,i}_{v} dB_u$ for $ s,t \\in [0,T]$ .",
"Applying the Minkovski inequality to the right-hand side of the above equation and taking into account Lemma 8.2 in [11] we obtain the estimate $\\left\\Vert \\int _{s}^{ t }\\Gamma ^{n }_{u}dJ_{1}(u)\\right\\Vert _{p} & \\le &\\sum _{i=1}^{d} \\sum _{k=\\lfloor \\frac{ns}{T} \\rfloor }^{\\lfloor \\frac{nt}{T} \\rfloor } n^{-\\beta } ( t_{k+1}\\wedge t - t_{k}\\vee s )^{\\beta }\\nonumber \\\\& \\le & K |t-s|^{\\beta } n^{1-2\\beta },$ for $i=1$ .",
"In the same way we can show that estimate (REF ) holds while $J_{1}$ is replaced by $J_{2}$ .",
"Applying these two estimates to $Y_t &= &\\frac{1}{2} \\Lambda ^{n }_{ t} \\sum _{i=1}^{2} \\int _{0}^{ t }\\Gamma ^{n }_{s}dJ_{i}(s)\\,$ we obtain $\\Vert Y\\Vert _{\\beta , p} &\\le & K n^{1-2\\beta }.$ We denote $\\Phi := \\Lambda - \\Lambda ^{n } $ .",
"Subtracting (REF ) from (REF ) we obtain $\\Phi ^{i}_{i^{\\prime }} (t)&=&\\sum _{j=0}^m \\sum _{i^{\\prime \\prime } =1}^d\\int _0^t\\left[\\partial _{i^{\\prime \\prime }} V^i_j ( X_s )\\Lambda ^{ i^{\\prime \\prime }}_{i^{\\prime }}({s})-V^i_{j i^{\\prime \\prime }} (s) \\Lambda ^{n,i^{\\prime \\prime }}_{i^{\\prime }}( {s})\\right]dB^{j }_s\\\\&=&\\sum _{j=0}^m \\sum _{i^{\\prime \\prime } =1}^d\\int _0^t\\partial _{i^{\\prime \\prime }} V^i_j ( X_s )\\Phi ^{i^{\\prime \\prime }}_{i^{\\prime }}({s})dB^{j }_s+\\sum _{j=0}^m \\sum _{i^{\\prime \\prime } =1}^d\\int _0^t\\left[\\partial _{i^{\\prime \\prime }} V^i_j ( X_s )-V^i_{j i^{\\prime \\prime }}({s})\\right]\\Lambda ^{n,i^{\\prime \\prime }}_{i^{\\prime }} ({ s})dB^{j }_s \\,.$ By the product rule it is easy to verify the following identity, $\\Lambda (t) - \\Lambda ^{n} (t)&=&\\sum _{ i,i^{\\prime }=1}^d\\sum _{j=0}^m \\Lambda (t)\\int _0^t\\Gamma _{i^{\\prime }}(s)\\left[\\partial _{i} V^{i^{\\prime }}_j ( X_s )-V^{i^{\\prime }}_{j i}({s})\\right]\\Lambda ^{n,i} (s)dB^{j }_s \\,.$ Denote $\\widetilde{V}(X_{s}, X_{s}^{n}) &= &\\int _{0}^{1} \\int _{0}^{1} \\partial \\partial _{i^{\\prime }} V^{i^{\\prime \\prime }}_{j} \\Big (\\lambda X_{s} + (1-\\lambda ) (\\theta X_{s} +(1-\\theta ) X^{n}_{s} ) \\Big ) (1-\\theta ) d\\lambda d\\theta \\,.$ It is easy to verify that $\\partial _{i^{\\prime }} V^{i^{\\prime \\prime }}_j ( X_s )-V^{i^{\\prime \\prime }}_{j i^{\\prime }}({s}) &= & \\widetilde{V}(X_{s}, X^{n}_{s} ) Y_{s} \\, .$ Then (REF ) becomes $\\Lambda _{i}({t}) -\\Lambda ^{n}_{i}({t}) &= &\\Lambda _{t} \\int _0^tdg_s \\cdot Y_{s} \\,,$ where $g_{t} &= &\\sum _{ i^{\\prime },i^{\\prime \\prime }=1}^d\\sum _{j=0}^m \\int _0^t\\Gamma _{i^{\\prime \\prime }}(s)\\widetilde{V}(X_{s}, X^{n}_{s} )\\Lambda ^{n,i^{\\prime }}_{ i}(s)dB^{j }_s \\,.$ Applying Lemma REF and taking into account the estimate (REF ), we obtain $\\left\\Vert \\Lambda _{t} \\int _s^tdg_s \\cdot Y_{s}\\right\\Vert _{p} &\\le & K n^{1-2\\beta } (t-s)^{\\beta },$ which implies the estimate for $\\Vert \\Lambda -\\Lambda \\Vert _{\\beta , p}$ .",
"The estimate for the quantity $\\Gamma -\\Gamma ^{n}$ can be shown similarly.",
"$\\Box $"
],
[
"Proof of (",
"It is clear that $&&\\sum _{i=1}^{2 } \\int _{0}^{t } \\Gamma ^{n}_{u} d J_{i}(u) - \\sum _{e=2}^{5 }\\int _{0}^{\\eta (t)} \\Gamma ^{n}_{\\eta (u)} d E_{e}(u)\\nonumber \\\\&&=\\sum _{i=1}^{2 } \\int _{0}^{t } \\Gamma ^{n}_{u} d J_{i}(u) - \\sum _{i=1}^{2 }\\int _{0}^{\\eta (t)} \\Gamma ^{n}_{\\eta (u)} d J_{i}(u)\\nonumber \\\\&&=\\sum _{i=1}^{2 } \\int _{0}^{t } \\int _{\\eta (s)}^{s} d\\Gamma ^{n}_{u} d J_{i}(s) + \\sum _{i=1}^{2 }\\int _{\\eta (t)}^{t} \\Gamma ^{n}_{\\eta (u)} d J_{i}(u).$ In the following, we estimate the $L^p$ -norms of the two terms on the right-hand side of (REF ).",
"For $t \\in [0,T]$ , we define $I (t) &= & \\int _{0}^{t} (\\partial V V ) (X^n_{\\eta (s)}) (B_{s} - B_{\\eta (s)}) dB_{s} ,$ It is clear that $I(t_{k})=I_{1}(t_{k})=I_{2}(t_{k})$ for $k=0,1,\\dots , n$ .",
"As in (REF ), we take the decomposition $J_{1}(t) + J_{2}(t) &= &\\left( R_0 (t) - I (t) \\right)+\\left( I (t)-\\widetilde{R}_{0} (t) \\right) +R_1(t) +\\widetilde{R}_1(t)\\nonumber \\\\& := &E_2 (t) +E_3 (t) +E_4 (t) +E_5 (t)$ for $t \\in [0,T]$ , where $R_{0}, R_{1}, \\widetilde{R}_{0}$ and $\\widetilde{R}_{1}$ are defined as before.",
"Note that the $E_{2}$ and $E_{3}$ defined here are extensions of those in (REF ) from $\\Pi $ to $[0,T]$ .",
"By applying Lemma 8.2 in [11] to (REF ) and (REF ) we obtain $\\Vert E_{e}\\Vert _{[t_{k}, t_{k+1}], \\beta } &\\le & Ke^{K\\Vert B\\Vert _{\\beta }^{1/\\beta }} n^{ -2\\beta }, \\quad e=4,5.$ Similarly, we can show that inequality (REF ) also holds for $e=2,3$ .",
"Therefore, we obtain $\\Vert J_{1}+J_{2}\\Vert _{[t_{k}, t_{k+1}], \\beta } &= &\\left\\Vert \\sum _{e=2}^{5}E_{e} \\right\\Vert _{[t_{k}, t_{k+1}], \\beta }\\nonumber \\\\&\\le & Ke^{K\\Vert B\\Vert _{\\beta }^{1/\\beta }} n^{-2\\beta }.$ By applying Lemma 8.2 in [11] and with the help of the estimate (REF ) we obtain $\\left\\Vert \\sum _{i=1}^{2} \\int _{t^{\\prime }}^{t^{\\prime \\prime }} \\int _{\\eta (s)}^{s} d \\Gamma ^{n}_{u} dJ_{i}(s)\\right\\Vert _{p} &\\le & Kn^{ -4\\beta }$ for $t^{\\prime }, t^{\\prime \\prime } \\in [t_{k},t_{k+1}]$ .",
"Therefore, we have $\\left\\Vert \\sum _{i=1}^{2} \\int _{0}^{t} \\int _{\\eta (s)}^{s} d \\Gamma ^{n}_{u} dJ_{i}(s)\\right\\Vert _{p}&\\le &\\sum _{k=0}^{\\lfloor nt/T \\rfloor }\\left\\Vert \\sum _{i=1}^{2} \\int _{t_{k}}^{t_{k+1}\\wedge t} \\int _{\\eta (s)}^{s} d \\Gamma ^{n}_{u} dJ_{i}(s)\\right\\Vert _{p}\\\\& \\le & Kn^{ 1-4\\beta }.$ On the other hand, applying (REF ) to $\\sum _{i=1}^{2} \\int _{\\eta (t)}^{t} \\Gamma ^{n}_{\\eta (s)} dJ_{i}(s)&=&\\sum _{i=1}^{2} \\Gamma ^{n}_{\\eta (t)} ( J_{i}(t) - J_{i}(\\eta (t)) \\,,$ we obtain $\\left\\Vert \\sum _{i=1}^{2} \\int _{\\eta (t)}^{t} \\Gamma ^{n}_{\\eta (s)} dJ_{i}(s) \\right\\Vert _{p}\\le Kn^{-3\\beta }\\le \\, Kn^{1-4\\beta }.$ This completes the proof.",
"$\\Box $"
]
] | 1709.01614 |
[
[
"The MeerKAT International GHz Tiered Extragalactic Exploration (MIGHTEE)\n Survey"
],
[
"Abstract The MIGHTEE large survey project will survey four of the most well-studied extragalactic deep fields, totalling 20 square degrees to $\\mu$Jy sensitivity at Giga-Hertz frequencies, as well as an ultra-deep image of a single ~1 square degree MeerKAT pointing.",
"The observations will provide radio continuum, spectral line and polarisation information.",
"As such, MIGHTEE, along with the excellent multi-wavelength data already available in these deep fields, will allow a range of science to be achieved.",
"Specifically, MIGHTEE is designed to significantly enhance our understanding of, (i) the evolution of AGN and star-formation activity over cosmic time, as a function of stellar mass and environment, free of dust obscuration; (ii) the evolution of neutral hydrogen in the Universe and how this neutral gas eventually turns into stars after moving through the molecular phase, and how efficiently this can fuel AGN activity; (iii) the properties of cosmic magnetic fields and how they evolve in clusters, filaments and galaxies.",
"MIGHTEE will reach similar depth to the planned SKA all-sky survey, and thus will provide a pilot to the cosmology experiments that will be carried out by the SKA over a much larger survey volume."
],
[
"Introduction",
"The MeerKAT International GHz Tiered Extragalactic Exploration (MIGHTEE) survey is a project being undertaken by an international collaboration of researchers to explore cosmic evolution by creating deep images of the GHz radio emission in continuum, spectral line and polarisation.",
"The survey will be conducted over 20 deg$^2$ of the best studied regions of the extragalactic sky observable from the southern hemisphere, namely COSMOS, XMM-LSS, ECDFS and ELAIS-S1.",
"The nominal sensitivity will be $\\sim $ 1 $\\mu $ Jy over the full bandwidth of 900-1670 MHz, at a resolution of $\\sim $ 6 arcsec, with additional observations made over a smaller area with the S-band receiver.",
"In the following we outline the key science aims of MIGHTEE.",
"The MIGHTEE Hi survey component and the LADUMA Hi survey (Baker et al.",
"these proceedings) can be thought of as two tiers of a survey `wedding cake' at L-band.",
"The $\\sim $ 20 square degree MIGHTEE survey will form the wide, shallow tier out to intermediate redshifts (z$\\sim $ 0.5) and LADUMA the narrow, deep tier (z$<$ 1.4).",
"LADUMA will include a deep L-band component plus a deep UHF-band component.",
"The two surveys are therefore highly complementary in that MIGHTEE will observe a larger volume at low redshift and will gather a larger sample of low redshift galaxies, while for the overlapping L-band component, LADUMA's extreme depth will probe lower Hi masses, and LADUMA's UHF-band component will cover redshifts inaccessible to MIGHTEE.",
"There is substantial overlap in the science goals of the two surveys, notably in probing the Hi mass function (HIMF) and cosmic neutral gas density over a range of cosmic time and different environments.",
"At low redshifts MIGHTEE will observe more galaxies at the high-mass end of the HIMF due to the larger local cosmological volume probed and LADUMA, due to its $\\sim $ 4.5x higher sensitivity in the L-band, will detect more low Hi mass galaxies.",
"By combining Hi detections from the two surveys we will be able to measure both the low- and high-mass ends of the HIMF, significantly reducing the associated statistical errors due to low source counts, out to intermediate redshift [48].",
"Figure: (left) The expected coverage of theHi mass versus redshift plane by combining the LADUMA and MIGHTEE surveys.",
"The grey shaded regions show the MIGHTEE coverage and the red shaded regions show the LADUMA coverage.",
"(right) The expected constraints on the Hi mass function.",
"It is clear that MIGHTEE is required to measure the high-mass end whereas LADUMA pins down the low-mass end at z<0.5z<0.5.",
"Taken from ."
],
[
"From the gas to the stars: the onset of star formation\nover cosmic time",
"Galaxies follow known scaling relations, such as the so-called star-formation (SF) main sequence [39] relating SF with stellar mass.",
"Underlying these relations is a complex cycle of acquisition, storage, consumption, expulsion and re-acquisition of gas acting to regulate a galaxy's ability to form stars [17].",
"The vastly different evolution of cosmic SF density and Hi density from $z\\sim 1$ to the present day implies a complex, non-linear relation between the two processes.",
"State-of-the-art simulations which attempt to incorporate a neutral gas component into galaxies are unable to reproduce the detailed distribution of Hi content in galaxies.",
"In particular, they do not exhibit the mass-dependent relation between Hi mass and stellar mass, found to be at least partly due to dark matter halo angular momentum [47].",
"This mismatch between observations and simulations indicates missing physics in our galaxy formation prescriptions that can only be brought to light with large samples of Hi measurements spanning orders of magnitude in stellar mass.",
"Furthermore, while the stellar properties of galaxies spanning the full range of masses over a large range of redshifts have been well studied, our knowledge of the neutral gas (Hi) content of these same galaxies is restricted to the local universe.",
"Surveys such as ALFALFA [23] cover large areas of sky and include a variety of cosmic environments, but do not have the frequency coverage required to explore a range of redshifts.",
"If we hope to understand the build-up of stellar mass, we must observe the neutral gas reservoir of fuel from which the molecular gas forms, eventually turning into stars, along with the interface between this gas and the galaxies and environment in which it is located.",
"MIGHTEE will open up this new parameter space in the study of the neutral gas component of galaxies.",
"MIGHTEE will have the sensitivity and frequency coverage required to detect Hi in a statistically significant number of M$_{\\rm HI}^{\\star }$ galaxies at $z=0.2$ , and will detect the rare, most Hi-massive galaxies to $z\\sim 0.5$ , due to the large volume probed.",
"Thus, together with LADUMA, MIGHTEE will revolutionise our understanding of the Hi content of galaxies over a cosmologically significant redshift range, as a function of stellar mass and environment.",
"MIGHTEE not only provides us with the ability to trace the origins of the gas that eventually turns into stars, but with the exquisite sensitivity to radio-continuum emission, we will also measure the end-point, namely the star-formation rate (SFR).",
"Radio-continuum observations closely trace the far-infrared emission of galaxies at all redshifts, the so-called Far-infrared-radio correlation [35], [64], as such they offer a unique method of measuring the evolution of the SFR density over cosmic time, free of dust-obscuration, a key issue in sensitive UV-based surveys [46].",
"Furthermore, deep far-IR and sub-mm surveys are generally confusion limited at SFR $\\sim 100$ M$_{\\odot }$ yr$^{-1}$ [54], [21], and although it provides high angular resolution, ALMA cannot survey large areas efficiently [18].",
"At the depth of the MIGHTEE survey, we will be able to detect galaxies with SFR$\\sim 15$ M$_{\\odot }$ yr$^{-1}$ to $z = 1$ and SFR$\\sim 150$ M$_{\\odot }$ yr$^{-1}$ to $z\\sim 3$ .",
"By using the excellent existing optical and near-IR data we are able to trace the evolution of SFR in galaxies as a function of mass, colour, environment etc, thus gaining critical insight of which galaxies form the bulk of the stars, in what environments and when."
],
[
"The galaxy merger history from OH megamasers",
"The global SFR and the assembly of massive elliptical galaxies are inextricably connected to how galaxies merge over cosmic time.",
"Recently merged (luminous and ultra-luminous infrared) galaxies provide the perfect conditions for detecting Hydroxyl (OH) megamasers, which are often found within 1 kpc of heavily dust-obscured AGN [14].",
"OH megamasers are therefore ideal luminous radio beacons ($L_{\\rm OH} \\sim 10^{4}\\,L_{\\odot }$ ) for tracing the merger history of the Universe [10].",
"The Arecibo OH Megamaser survey [14], which detected 52 masers out to $z =0.23$ , represents the current state-of-the-art in our understanding of the nearby megamaser population and the luminosity function [15].",
"The MIGHTEE survey will provide a unique opportunity to carry out a deep blind search for OH megamaser emission (and OH absorption) between $z = 0$ and 0.85.",
"The low-redshift luminosity function of [15], would imply a detection yield of $\\sim $ 10 OH megamasers.",
"However, this number is highly dependent on the evolution of the galaxy merger rate as a function of redshift, often parameterised as $(1+z)^{m}$ with $m\\approx 2 - 8$ , which could easily lead to an order of magnitude more in MIGHTEE.",
"With MIGHTEE we will measure the merger rate traced by OH megamasers directly."
],
[
"Quenching: the role of environment",
"One of the key unknowns in models of galaxy evolution is how SF in galaxies becomes quenched.",
"Environmentally-related processes such as ram-pressure stripping and strangulation, where the cold and/or hot-gaseous haloes are removed due to the density of the medium through which a galaxy is moving, can explain some of the required quenching.",
"However, internal processes such as SF and AGN-driven winds may also play an important, if not critical, role [56].",
"Furthermore, it is clear from both semi-analytic models and hydro-dynamical simulations [20] and observations [27], [30] that these internal feedback processes are also dependent on the mass of the galaxy or halo in which they reside.",
"Recent work [28], [16], [29] using optical and near-infrared data has shown that the quenching of low-mass galaxies in the environments of large massive galaxies, which trace the largest dark matter haloes, does not appear to be strongly related to their position in the halo.",
"This suggests that quenching is an internal process, with the environment playing a lesser role.",
"However, when the massive tracer galaxy is strongly star forming, then the satellite galaxies are less quenched.",
"Thus, the process is not simple and requires an understanding of the evolution of haloes over time, and also an indication of the time at which the halo itself formed.",
"Furthermore, it is now clear that the so-called “Galactic Conformity\" [42], [31], where neighbouring haloes appear to be related, continues to high ($z> 2$ ) redshift [29].",
"The depth of MIGHTEE allows us to directly observe where and when the cold gas is being stripped from galaxies through the sensitivity to Hi, and the areal coverage provides enough cosmic volume to study such physical processes in the densest and sparsest environments.",
"With the same set of observations, and following the prescription outlined in [29], we will relate the prevalence of quenching to the halo mass, and the proximity to the centre of the halo, and to the galaxies which may accelerate quenching such as those exhibiting AGN activity."
],
[
"AGN fueling and feedback",
"It is widely thought that AGN activity may be responsible for switching off SF in massive galaxies, or at least maintaining a “quenched\" state (Section REF ) once the SF has terminated.",
"However, a direct observational link between AGN activity and SF at high redshifts remains elusive.",
"Recent studies from both a theoretical [63] and observational [40] perspective have shown that powerful radio-loud AGN may actually provide a positive form of feedback.",
"On the other hand, there is little evidence for any type of feedback from radio-quiet objects based on studies using Herschel [9], with recent studies suggesting that the host galaxies of radio-quiet AGN are similar to the general galaxy population [58]."
],
[
"Tracing the mechanical feedback from AGN jets",
"To understand AGN feedback and the interplay between SF and AGN activity, both in the AGN host and the wider environment, a survey is required that spans enough cosmological volume to include the rare powerful AGN at $z\\sim 1$ , but with a depth that detects SFGs at similar redshifts to the AGN.",
"Radio is arguably the best line of attack due to the sensitivity to both SF activity and AGN activity.",
"Given that different forms of AGN feedback are invoked in current semi-analytic and hydrodynamic models of galaxy formation [33], it is essential that we understand such processes if we are to understand the evolution of galaxies in general.",
"Observational evidence [6] suggests that many or most low-power (P $<$ 10$^{25}$ W Hz$^{-1}$ ) radio galaxies in the local universe (the numerically dominant population) correspond to a distinct type of AGN.",
"These sources accrete through a radiatively inefficient mode (the so-called “radio mode\"), rather than the radiatively efficient accretion mode typical of radio-quiet AGN selected at optical or X-ray wavelengths (sometimes called `quasar mode'; see [32] for a recent review covering these feedback processes).",
"The role of these two accretion modes appears to be strongly influenced by the environment [69] while the level of radio-jet activity appears to be a strong function of the stellar mass of the host galaxy [34], [74].",
"Therefore, deeper radio surveys that cover enough area of sky with the best multi-wavelength data are required to probe the evolution of these relationships and the accretion mode dichotomy over cosmic time; this is key information for any attempt to incorporate radiative and mechanical feedback from radio-loud AGN in models of galaxy, group and cluster formation and evolution.",
"The depth and breadth of MIGHTEE will enable unique studies of the entire AGN population from $z\\sim 0-6$ , providing a complete view of nuclear activity in galaxies and its evolution, unbiased by gas/dust selection effects.",
"If current simulations and measurements are correct, we will detect a factor of $\\sim 20$ more low-accretion rate radio sources at $z>1$ than the current VLA-COSMOS data [65], [66], [67], due to the greater depth and increased area.",
"This dramatic increase means that we will accurately measure the evolution of such sources to $z\\sim 2$ , testing the key ingredient of galaxy evolution simulations.",
"Furthermore, this will allow the amount of energy deposited into the intergalactic medium by such objects to be measured over the era when such sources are active, thus providing the key input to the evolution of mechanical feedback from AGN jets."
],
[
"MIGHTEE will provide a homogeneous survey of 21-cm H i absorption and emission from AGN across the radio luminosity function, enabling a direct investigation into the symbiotic relationship between AGN activity and neutral gas in galaxies.",
"The H i gas content of AGN has received increasing attention in recent years, both through emission and absorption studies.",
"H i emission probes the global properties of the neutral gas, such as mass, velocity structure and neutral gas fraction.",
"This offers a direct measurement of the large-scale fuel source for the AGN, and a probe of triggering mechanisms.",
"Complementary to this, H i absorption seen against the radio continuum source adds vital information on the state and kinematic behaviour of neutral gas within the central regions of radio-loud AGN [22].",
"Existing samples for both types of study are limited in size, with previous work based on relatively shallow observations of pre-selected (and possibly biased) galaxy samples.",
"MIGHTEE will provide a unique opportunity to revolutionise this field through a deep, wide-area dataset, with an unbiased sample of radio-selected AGN and a rich suite of ancillary data not normally available in wider 21-cm surveys.",
"Using both direct H i emission detections, and stacking analyses, the H i gas content of samples of galaxies selected to display AGN activity will be compared against samples of non-AGN matched in mass, environment, and other properties.",
"This will allow the investigation of the origin of the AGN activity and the influence of the AGN on its host galaxy.",
"At the sensitivity of MIGHTEE, it will be possible to detect H i masses down to around $2 \\times 10^{9}\\,\\mathrm {M}_{\\odot }$ out to $z=0.2$ at the $5\\sigma $ level, assuming typical line widths of 250 km s$^{-1}$ .",
"This is a substantial advance in sensitivity over previous studies, and should allow detection of the H i gas in the AGN hosts, given typical gas fractions of 10% [19].",
"There will be around 1000 AGN of all classes hosted by galaxies more massive than $10^{10}\\,\\mathrm {M}_{\\odot }$ in the MIGHTEE area.",
"For absorption studies, assuming a typical line width of 100 km s$^{-1}$ , we expect to obtain a sample of approximately 240 radio galaxies brighter than 1 mJy within the comoving volume bounded by $z_{\\rm HI} \\approx 0.58$ .",
"Approximately 50 of these sources will be brighter than 10 mJy and we will be sensitive to cold gas clouds with peak opacities greater than 1% and typical column densities greater than $2 \\times 10^{20}\\,$ cm$^{-2}$ .",
"This sample spans the radio luminosity function between $10^{23} < L_{1.4} / {\\rm W\\,Hz}^{-1} < 10^{25}$ , allowing us to directly observe and test evolutionary models of neutral gas accretion in low-excitation radio galaxies [70].",
"The brightest radio galaxies in our absorption sample will also enable a study of jet-mode feedback in neutral gas [53] out to intermediate cosmological redshifts, and is a natural complement to the MeerKAT Absorption Line Survey [24]."
],
[
"AGN polarisation as a probe of environment",
"Radio-loud AGN are intrinsically highly polarised; the fractional polarisation in high-resolution, high-frequency images can approach the theoretical maximum of $\\sim 70$ % [57].",
"MIGHTEE will provide extremely sensitive, broad-band L-band observations, supplemented in some cases with S-band data as described in Section REF , and so will give measurements of integrated or moderately resolved polarisation as a function of frequency for many thousands of radio-loud AGN, allowing samples to be subdivided by e.g.",
"luminosity and redshift to measure the evolution of intrinsic and environmental properties.",
"Among other projects, MIGHTEE's polarisation surveys, coupled with total intensity information and optical identifications/redshifts for the target sources, will allow us to (i) statistically relate observed (de)polarisation to source environment, by combining with optical and (where possible) X-ray data available in the target fields; (ii) investigate the dependence of observed (de)polarisation on physical size, testing the prediction of models that sources observed at low resolution become more polarised as they become physically larger, so that the average Faraday depth to/dispersion in front of the lobes becomes smaller [25]; it does indeed appear to be the cases that sources of larger angular size are more polarised in deep surveys [59] but MIGHTEE will give much larger samples and constrain the physical size scale on which this takes place, and hence the physical size of the hot-gas halo; (iii) carry out large-scale statistical studies of the Laing-Garrington effect for a large number of moderately resolved AGN, probing physical conditions in the centre of the host environment and giving a prior for every source on the angle to the line of sight; and (iv) carry out a statistical test of the [13] depolarisation law and hence probe the fine structure and possibly the power spectrum of the magnetic field in the intra-cluster medium (ICM)."
],
[
"Large scale structure & cosmology",
"Covering 20 deg$^2$ , MIGHTEE will provide the ideal data set to pave the way for large-scale cosmology experiments with the SKA [1], [37], [61]."
],
[
"An accurate measurement of the evolution of bias",
"The depth and breadth of MIGHTEE will allow us to measure the bias, i.e.",
"how radio sources of different types (e.g.",
"star-forming galaxies, Fanaroff-Riley Class I and II, etc), trace the underlying dark-matter distribution [44], [4], [49].",
"Carrying this out over the best multi-wavelength fields allows us to accurately disentangle the different types of sources, whilst obtaining the required number density (apart from very rare FRIIs).",
"Furthermore, cross-correlations with the optical/near-infrared has been shown to be a powerful technique to measure the bias of rarer sources [45].",
"Such measurements of the bias can then be used in the wider area surveys where it is much more difficult to measure, due to a lack of multi-wavelength and redshift information.",
"Indeed, these measurements will be needed in order to carry out large cosmological tests with the proposed all-sky SKA-MID survey, as it would be at a similar depth to MIGHTEE.",
"Figure REF shows the current state of the art in the measurement of the clustering of radio sources to high-redshift using the most recent JVLA-COSMOS data, and the expected constraints with a single 7.5 deg$^2$ MIGHTEE field.",
"The error ellipse on the amplitude and the slope is improved by around an order of magnitude.",
"Figure: (left) The measured two-point correlation function of radio sources in the recent ∼1.5\\sim 1.5 deg 2 ^2 JVLA-COSMOS 3GHz survey (PI: Smolčić), with an equivalent 5σ\\sigma depth of S 1.4 ∼20μS_{1.4} \\sim 20\\mu Jy (Hale et al.",
"submitted).",
"(center) The expected constraints on the two-point correlation function for MIGHTEE with S 1.4 =5μS_{1.4} = 5\\mu Jy, for a single 7.5 deg 2 ^{2} field.",
"(right) Forecasts for a measurement of the cosmic shear weak lensing power spectrum using MIGHTEE polarisation observations combined with galaxy shape measurements from overlapping optical surveys.",
"For comparison we show forecasts for the same sample of galaxies without the polarisation information.",
"We assume we can obtain useful polarisation measurements for ∼5%\\sim 5\\% of those galaxies detected in total intensity, and we assume a scatter of ∼10\\sim 10 degrees in the relation between the polarisation orientation and the disk structure of the source galaxies."
],
[
"Weak lensing",
"The coherent distortion of distant galaxy shapes due to gravitational light deflection by large scale structure (“weak lensing\" or “cosmic shear\") is recognised as one of the most powerful probes of dark matter and dark energy, and is a major cosmology science goal for the SKA [11], [26], [8].",
"MIGHTEE does not have the resolution to measure shapes of galaxies and thus cannot directly measure the shear for weak lensing measurements.",
"However, by conducting the survey over the fields with the best optical data (which have been and continue to be used for weak lensing measurements), we are able to use the integrated polarisation vector from MIGHTEE data, to inform on the intrinsic alignment of the galaxies before they are lensed.",
"This removes one of the most significant systematics in weak lensing surveys, and if proven would allow MIGHTEE to be the path finder for weak lensing with polarisation for the SKA.",
"Our forecasts suggest that MIGHTEE polarisation data combined with shape information from best overlapping optical data, will provide a $\\sim 3.5\\sigma $ detection of the cosmic shear effect (Fig.",
"REF ; right).",
"This would represent the first detection of weak lensing using the polarisation technique."
],
[
"Cosmic magnetism and large-scale structure: The magnetic cosmic web",
"MIGHTEE will provide a supremely-dense Rotation Measure (RM) grid - an effect caused by the interaction of a magneto-ionic medium, with the linearly polarised synchrotron emission from background or embedded cluster galaxies - and permit measurement of the properties of magnetic fields embedded in the large-scale structure of the universe.",
"While clusters contain hot plasma of $T>10^7$ K, filaments are expected to be filled with plasma of $10^7$ K $>$ T $>10^5$ K, referred to as the Warm Hot Intergalactic Medium (WHIM).",
"The plasmas may be magnetised; diverse processes for seed magnetic fields have been suggested, and the seed fields can be further amplified through compression and turbulent dynamos as well as leakage of galactic media during the hierarchical structure formation in the universe [60].",
"Simulations predict that the inter-galactic magnetic field (IGMF) in filaments would induce excess Faraday Rotation Measures with a flat second-order structure function of $\\sim $ 100 rad$^2$ m$^{-4}$ for angular separation of $r \\gtrsim 0.1^{\\circ }$ .",
"The power from this contribution to the RM structure function will be distinguishable from the foreground of the Milky Way galaxy on angular scales smaller than a few degrees.",
"An RM data set with a sky density of several 100 to 1000 polarised sources per square degree, and with RM precision of $\\sim $ 1 rad m$^{-2}$ is required to accurately reconstruct the structure function of RM variance due to the IGMF in the cosmic web [3].",
"MIGHTEE will probe this structure function with the required number density and RM precision, and over angular scales of a few arcminutes to several degrees, offering our best opportunity before the SKA to use this technique to detect the magnetic cosmic web."
],
[
"Resolving Massive Galaxies out to Cosmological Distances: Dark Matter in Galaxies",
"MIGHTEE's combination of high sensitivity and angular resolution for Hi produces a column density sensitivity ($\\sim $ 1 M$_{\\odot }$ /pc$^2$ over $\\delta _V$ = 20 km/s at S/N=3) that is sufficient to resolve the Hi distributions of Hi-rich galaxies out to cosmological distances.",
"MIGHTEE will therefore be the first survey to probe the distribution of dark and luminous matter in a statistical sample of massive galaxies beyond the local volume.",
"The tight correlation between Hi mass and Hi diameter for local galaxies [73], coupled with the expected number of Hi detections using the local HIMF [51], implies that MIGHTEE will resolve $>150$ galaxies with M$_{\\rm HI}$ $>$ 10$^{10}$ M$_{\\odot }$ at redshifts $0.07 < z < 0.12$ .",
"This is sufficient to estimate the Hi distribution and rotation curve shape by modelling the 3D Hi cubes and optical images [41].",
"MIGHTEE is the only planned Hi survey capable of resolving a significant sample of massive galaxies out to cosmological distances before the advent of SKA1, and highly complementary to samples at lower redshifts that will be produced by wide-field Hi surveys with the Australian Square Kilometre Array Pathfinder [38] and [72]."
],
[
"Low-mass, nearby galaxies and the satellite problem",
"Predictions based on $\\Lambda $ CDM excel at matching observations of large scale structure.",
"However, on small scales, the effects of baryonic physics are important and it can be difficult to match simulations to observations.",
"Observations of low-mass galaxies offer constraints on the implementation of baryonic physics, and are especially valuable as they include spatially resolved kinematic information.",
"Systems with Hi masses below $\\sim $ 10$^8$ M$_{\\odot }$ are especially important for testing our understanding of baryonic physics and $\\Lambda $ CDM.",
"The Hi kinematics of galaxies in this mass range place them in dark matter halos incompatible with expectations from abundance matching results [55].",
"The lowest mass systems (below $\\sim 2 \\times 10^7$ M$_{\\odot }$ ) are extremely rare; in the full ALFALFA survey there are only $\\sim $ 70.",
"Finding and studying these lowest mass galaxies is critical for understanding at what mass range and due to which processes dark matter halos stop hosting observable galaxies.",
"With MIGHTEE we expect to detect $\\sim $ 270 sources with M$_{\\rm HI} < 10^8$ M$_{\\odot }$ , with $\\sim $ 15 of those below 10$^7$ M$_{\\odot }$ .",
"Using the HI mass-diameter relation [73], we estimate that $\\sim $ 7 of the sources below 10$^8$ M$_{\\odot }$ will be resolved with at least three MeerKAT beams, and one to two sources may be resolved with five beams."
],
[
"Cluster radio halos and relics",
"Clusters of galaxies show radio emission on a very broad range of scales, from discrete sources associated with AGN to diffuse emission on Mpc scales.",
"The latter points to the existence of a non-thermal component (cosmic rays and magnetic fields) of the ICM.",
"Diffuse radio sources at the cluster center are generally known as “radio halos”, while elongated polarised Mpc-scale radio structures at the cluster periphery are known as “radio relics”.",
"Current models indicate that Mpc-scale diffuse radio emission in the form of radio halos traces the turbulent re-acceleration of particles due to merger events, whereas relics are the result of electron acceleration and magnetic field compression resulting from ICM shocks [12].",
"The angular resolution and excellent brightness sensitivity provided by MIGHTEE will probe radio emission on all scales relevant for cluster radio emission to high redshift.",
"MIGHTEE's multi-wavelength coverage will be crucial in shedding light on the origin of the non-thermal cluster components, as detailed multi-wavelength analyses of clusters are essential to properly characterise the physical differences between “radio-loud\" and “radio-quiet\" systems.",
"This kind of study is crucial to understand, for instance, why not all merging clusters host diffuse radio sources, or vice-versa [7], and the role played by gravitational and non-gravitational effects on the evolution of the ICM."
],
[
"Galaxy clusters and their magnetic fields",
"Galaxy clusters are the largest known magnetised structures in the universe and therefore are unique laboratories to investigate the origin of large-scale magnetic (B) fields.",
"In spite of the wealth of evidence for the existence of ICM magnetic fields, measurements of both the field strengths and morphologies of clusters are still scattered within the literature.",
"Information on the cluster B-fields can be derived from detailed images of extended radio emission in clusters [71] and through Faraday Rotation measure synthesis.",
"With MIGHTEE we will be able to measure the magnetic field properties of hundreds of galaxy clusters in the MIGHTEE cosmic volume.",
"The Faraday rotation provides a clean measure of the magnetic field strength along the line of sight for uniform fields, and the magnetic field strength directly for turbulent fields [43].",
"Constraining the behaviour of magnetic fields in galaxy clusters is not only important for studies of the intra-cluster medium itself, but can also be used to constrain the origin of cosmic magnetic fields more widely by examining the evolution of cluster fields as a function of redshift.",
"Early-type seed fields are expected to produce a characteristic evolution in cluster magnetic field properties when combined with expectations of turbulent amplification.",
"The high density and precision of the MIGHTEE RM grid will offer the best opportunity before SKA to constrain the redshift evolution of cluster magnetic fields, and provide the first test of the early-type model for the origin of cosmic magnetic fields."
],
[
"The emergence and evolution of magnetic fields in galaxies",
"Through its ability to detect polarisation of sources at high redshift, MIGHTEE will be a cornerstone for investigating the evolution of magnetic fields in galaxies.",
"For disk galaxies, this range is $z \\lesssim $ 2.5, and for AGN and starbursts it reaches out to $z \\lesssim $ 7, into the epoch of reionisation.",
"Over this period, galaxies formed and evolved, converting most of their gas into stars.",
"The evolution of plasma and magnetic fields in galaxies is expected to be closely related to the evolution of the cosmic SFR, and tied to the intergalactic medium through accretion, galactic winds, tidal and ram stripping, and AGN activity.",
"While $\\mu $ G strength magnetic fields on kpc scales may have formed in galaxy disks by $z \\sim 3$ , ordering on the scale of a galaxy may have taken until $z \\sim 0.5$ , depending on galaxy mass [5], [50].",
"Competing with this, galaxy interactions and continuous feedback by supernovae and stellar winds can enhance the turbulent component of the magnetic field, and drive outflows that transport plasma and magnetic field from the disk into the halo.",
"Since these processes scale with the global SFR, significant evolution is expected between $z \\sim 2$ and the present.",
"Also, the density of Faraday rotating plasma will gradually decrease over time as galaxies transform a significant fraction of their gaseous mass into stars, implying a gradual evolution in Faraday depth.",
"Based on models by [68] for spiral galaxies at $z =0$ , applied to normal SF galaxies in the SKADS Simulated Skies simulation [75], [76], at the sensitivity of the MIGHTEE deep commensal polarisation image of the LADUMA field we expect $\\sim $ 5000 galaxies per square degree above 10$\\sigma $ detected to $z\\sim 3$ .",
"For the wider area survey we will detect several hundred galaxies per square degree in polarisation out to $z \\sim 1$ .",
"The analysis of [68] has shown that the integrated polarisation properties of these distant galaxies can be used to detect and characterise both the large-scale and turbulent magnetic fields in galaxy disks.",
"As shown in Figure REF , the combined L- and S-band data set is critical to this study as the structure of the depolarisation of the signal as a function of frequency over this broad band is required to reveal the magnetic properties.",
"The MIGHTEE data set will provide probes of the emergence and evolution of magnetic fields in $\\sim $ 10,000 galaxies out to $z > 3$ .",
"Figure: Models of the polarised intensity ofintegrated emission from galaxy disks as a functionof frequency.",
"The frequency coverages of the MeerKATL-band and S-band are shown.Together these bands span the frequency range over which depolarisation becomes important, and both are critical to measure the depolarisation signature.The change in polarised emission is due to internalFaraday effects from the magnetic field and thermal plasma in the galaxy.Broad-band integrated spectro-polarisation data thus probes the evolution of both magnetic and thermal gas (ionisation) properties.",
"Depolarization models arefrom ."
],
[
"Digging into the noise with MIGHTEE",
"Classical analyses of radio survey data relies on the direct detection of the Hi, continuum or polarised emission from galaxies.",
"However, there is significant additional information in the images for the vast number of objects that produce radio signals too faint to individually reach the direct detection flux density threshold.",
"We will be able to take advantage of this information and investigate the statistical radio properties of classes of objects to flux densities substantially below the noise floor of the images.",
"Following techniques developed in [52] and [77] we plan to derive deep Hi mass functions, and total intensity and polarised luminosity functions through a Bayesian stacking method.",
"Information from the rich multi-wavelength data set for the MIGHTEE fields can be used as prior information in this process.",
"For example, $\\sim 200,000$ galaxies per square degree, based on the VIDEO survey data [36] to $K_s \\sim 23.7$ , would result in a total of $4\\times 10^6$ galaxies to stack on over all redshifts and stellar masses.",
"While ultimately limited by confusion in total intensity, MIGHTEE will provide an unique opportunity: due to its high sensitivity, a large number of the MIGHTEE galaxies detected in continuum will be SFGs containing Hi, and for which we can also measure their magnetic fields through the stacked polarised emission that is unaffected by confusion."
],
[
"Observing strategy",
"The key science outlined above, coupled with the current and anticipated availability of multi-wavelength data, lead to the mosaic pointing setups shown in Figure REF .",
"The number of pointings and the areas (calculated without including the area beyond the half power point of the perimeter pointings) are provided in the caption.",
"Not shown is the COSMOS field which will be observed with a single (or tightly dithered) pointing.",
"Using the current measurements of the MeerKAT system temperature, with 16 h per pointing a depth of 2 $\\mu $ Jy beam$^{-1}$ (thermal + confusion noise) will be achieved in the full-band mosaics, and a typical depth of 90 $\\mu $ Jy beam$^{-1}$ will be reached in the 26 kHz channels for the spectral line component of the survey.",
"We note that one of the E-CDFS pointings will be that of the MIGHTEE-DEEP tier commensal with LADUMA.",
"The S-band component will cover 4 deg$^{2}$ in E-CDFS and 1.5 deg$^{2}$ in COSMOS (we do not plan to survey XMM-LSS in S-band due to satellite RFI observed in test JVLA observations).",
"The S-band survey over a limited area of MIGHTEE will enable us to obtain a much larger bandwidth for RM synthesis, while also allowing the possibility of multi-frequency synthesis in combination with the L-band, which will help to deconfuse the L-band survey.",
"We plan to reach a depth of 1 $\\mu $ Jy beam$^{-1}$ in the S-band mosaics (matched to the L-band depth for a typical $\\alpha =-0.7$ source), which requires 12.7 h per pointing (ignoring the effects of confusion, but including the overlapping pointings, and assuming a 20% sensitivity loss due to the weighting required for reliable deconvolution).",
"Calibration overheads of $\\sim $ 20% means that the total time required is $\\sim $ 1920 h. Figure: Current plausible pointing strategies for (left-to-right) XMM-LSS (20 pointings, 6.7 deg 2 ^{2}), E-CDFS (24 pointings, 8.3 deg 2 ^{2}) and ELAIS-S1 (7 pointings, 1.6 deg 2 ^{2}).",
"Not shown here is the fourth COSMOS field, which will consist of a single deep pointing.",
"In practice the grid of E-CDFS pointings will be snapped to the LADUMA pointing centre, requiring only 23 additional pointings from MIGHTEE."
],
[
"MIGHTEE scientific data products",
"We plan to provide multiple intermediate MIGHTEE data releases of the reduced data following qualification of the data products for scientific use.",
"Data will be released along with ancillary data where appropriate.",
"Visibility data will be processed into images and catalogues using the data-centric processing facility at the South African Inter-University Institute for Data Intensive Astronomy (IDIA).",
"The data processing pipeline is under development by a coordinated development team using the IDIA cloud-based development framework.",
"We envisage that the MIGHTEE team will provide a data release 18 months after observations of a field are completed in either L- or S-band.",
"The release will include data cubes for spectral line work, band-averaged multi-frequency synthesis total intensity images, full-Stokes intensity and RM synthesis cubes.",
"Continuum products will have multiple resolutions optimised for different science goals, e.g.",
"high resolution for galaxy studies, and lower resolution for cluster halo/relic work.",
"We plan to release these data through the IDIA systems and are also investigating plans for a mirrored archive in Europe.",
"Final data products will be released to the MeerKAT legacy data archive."
],
[
"VLBI enhancement of MIGHTEE science",
"MeerKAT's data alone will be limited to an angular resolution of a few arcseconds.",
"The $\\mu $ Jy radio sources will therefore be unresolved in the MIGHTEE images.",
"This faint population is known to consist of emission from SFG and RQ AGN.",
"Although multi-frequency analysis can aid in distinguishing the two populations, higher resolution radio imaging would allow unambiguous distinction through morphology as well as precise measures of the relative contribution of SF and AGN activity in individual objects.",
"Very Long Baseline observations would thus offer a tremendous enhancement to the MIGHTEE project.",
"VLBI can also play an important role in Hi absorption components of MIGHTEE, contributing unique morphological insights to the interpretation of gas inflows and outflows in particular [53].",
"VLBI polarimetry would not only provide detailed information on sub-kpc magnetic fields and jet physics, but also enable comparison with (and separation from) the larger scale polarisation properties to be probed by MeerKAT [2].",
"Over the course of the MIGHTEE survey the project team will explore the possibility of including MeerKAT along with elements of the emerging Africa VLBI Network and European antennas in complementary wide-field VLBI experiments to provide high-resolution radio images of MIGHTEE sources."
],
[
"Author affiliations",
"$^{1}$ Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK $^{2}$ Department of Physics and Astronomy, University of the Western Cape, Robert Sobukwe Road, Bellville 7535, South Africa $^{3}$ Department of Astronomy, University of Cape Town, Rondebosch 7701, South Africa $^{4}$ Inter-University Institute for Data Intensive Astronomy, University of Cape Town, South Africa $^{5}$ Instituto de Astrofísica de Andalucía (CSIC), Apartado 3004, E-18080 Granada, Spain $^{6}$ CSIRO Astronomy and Space Science, PO Box 76, Epping NSW 1710, Australia $^{7}$ Department of Physics & Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa $^{8}$ Inter-University Centre for Astronomy and Astrophysics, India $^{9}$ ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, 7990AA, Dwingeloo, The Netherlands $^{10}$ Square Kilometre Array South Africa, Pinelands 7405, Cape Town, South Africa $^{11}$ Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL, UK $^{12}$ INAF - Istituto di Radioastronomia, via Gobetti 101, 40129 Bologna, Italy $^{13}$ Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth PO1 3FX, UK $^{14}$ Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA $^{15}$ African Institute for Mathematical Sciences, 6-8 Melrose Road, Muizenberg 7945, South Africa $^{16}$ Department of Maths and Applied Maths, University of Cape Town, Cape Town, South Africa $^{17}$ South African Astronomical Observatory, Observatory, Cape Town, 7925, South Africa $^{18}$ SUPA, Institute for Astronomy, Royal Observatory, Edinburgh EH9 3HJ, UK $^{19}$ Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, D-21029 Hamburg, Germany $^{20}$ School of Physics, University of the Witwatersrand, Private Bag 3, 2050 Johannesburg, South Africa $^{21}$ CHPC, CSIR, 15 Lower Hope Rd, Rosebank, 7700, South Africa $^{22}$ Laboratoire Lagrange, Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Blvd de l'Observatoire, CS 34229, F-06304 Nice cedex 4, France $^{23}$ Centre for Astrophysics Research, School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK $^{24}$ Max-Planck-Institut für Radioastronomie Auf dem Hügel 69, 53121 Bonn, Germany $^{25}$ European Southern Observatory, Karl-Schwarzschild-Str.",
"2, 85748, Garching, Germany $^{26}$ Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa $^{27}$ Western Sydney University, Locked Bag 1797, Penrith South, NSW 1797, Australia $^{28}$ Astronomy Centre, Department of Physics and Astronomy, University of Sussex, Falmer, Brighton, BN1 9QH, UK $^{29}$ Centre for Space Research, North-West University, Potchefstroom 2520, South Africa $^{30}$ Leiden Observatory, Leiden University, P.O.",
"Box 9513, NL-2300 RA Leiden, The Netherlands $^{31}$ International Centre for Radio Astronomy Research, Curtin University, Perth, Australia $^{32}$ Gemini Observatory, Northern Operations Center, 670 North A‘ohoku Place, Hilo, HI 96720-2700, USA $^{33}$ Department of Physics, Royal Military College of Canada, PO Box 17000, Station Forces, Kingston, ON K7K 7B4, Canada $^{34}$ Department of Physics and Astronomy, University of Calgary, Canada 0000-0003-2623-2064 $^{35}$ GEPI, Observatoire de Paris, CNRS, Universite Paris Diderot, 5 place Jules Janssen, F-92190 Meudon, France"
]
] | 1709.01901 |
[
[
"Using Cross-Model EgoSupervision to Learn Cooperative Basketball\n Intention"
],
[
"Abstract We present a first-person method for cooperative basketball intention prediction: we predict with whom the camera wearer will cooperate in the near future from unlabeled first-person images.",
"This is a challenging task that requires inferring the camera wearer's visual attention, and decoding the social cues of other players.",
"Our key observation is that a first-person view provides strong cues to infer the camera wearer's momentary visual attention, and his/her intentions.",
"We exploit this observation by proposing a new cross-model EgoSupervision learning scheme that allows us to predict with whom the camera wearer will cooperate in the near future, without using manually labeled intention labels.",
"Our cross-model EgoSupervision operates by transforming the outputs of a pretrained pose-estimation network, into pseudo ground truth labels, which are then used as a supervisory signal to train a new network for a cooperative intention task.",
"We evaluate our method, and show that it achieves similar or even better accuracy than the fully supervised methods do."
],
[
"Introduction",
"Consider a dynamic scene such as Figure REF , where you, as the camera wearer, are playing basketball.",
"You need to make a decision with whom you will cooperate to maximize the overall benefit for your team.",
"Looking ahead at your teammates, you make a conscious decision and then 2-3 seconds afterwards you perform a cooperative action such as passing the ball.",
"In a team sport such as basketball, an effective cooperation among teammates is essential.",
"Thus, in this paper, we aim to investigate whether we can use a single first-person image to infer with whom the camera wearer will cooperate 2-3 seconds from now?",
"This is a challenging task because predicting camera wearer's cooperative intention requires 1) inferring his/her momentary visual attention, 2) decoding dominant social signals expressed by other players who want to cooperate, and 3) knowing who your teammates are when the players are not wearing any team-specific uniforms.",
"Figure: With whom will I cooperate after 2-3 seconds?",
"Given an unlabeled set of first-person basketball images, we predict with whom the camera wearer will cooperate 2 seconds from now.",
"We refer to this problem as a cooperative basketball intention prediction.Figure: The illustration of our cross-model EgoSupervision training scheme.",
"As our base model we use a multi-person pose estimation network from , which predicts 1) pose estimates of all people in a given first-person image and 2) the bounding boxes around each person.",
"Next, we feed these outputs to an EgoTransformer, which transforms them such that the transformed output would approximately capture the camera wearer's attention and intentions.",
"Then, we use such transformed output as a supervisory signal to train the network for our cooperative basketball intention task.To make this problem even more challenging we ask a question: “Can we infer cooperative basketball intention without manually labeled first-person data?”.",
"Building an unsupervised learning framework is important because manually collecting basketball intention labels is a costly and a time consuming process.",
"In the context of a cooperative basketball intention task, an annotator needs to have highly specific basketball domain knowledge.",
"Such a requirement limits the scalability of the annotation process because such annotators are difficult to find and costly to employ.",
"However, we conjecture that we can learn cooperative basketball intention in an unsupervised fashion by exploiting the signal provided by the first-person camera.",
"What people see reflects how they are going to act.",
"A first-person camera placed on a basketball player's head allows us to indirectly tap into that person's mind and reason about his/her internal state based on what the camera wearer sees.",
"To do so we propose a novel cross-model EgoSupervision learning scheme, which allows us to learn the camera wearer's intention without the manually labeled intention data.",
"Our cross-model EgoSupervision scheme works as follows.",
"First we transform the output of a pretrained pose-estimation network such that it would approximately reflect the camera wearer's internal state such as his/her visual attention and intentions.",
"Then, we use such transformed output as a supervisory signal to train another network for our cooperative basketball intention task.",
"We show that such a learning scheme allows us to train our model without manually annotated intention labels, and achieve similar or even better results as the fully supervised methods do."
],
[
"Related Work",
"First-Person Vision.",
"In the past, most first-person methods have focused on first-person object detection [29], [10], [40], [15], [2], or activity recognition [44], [43], [38], [31], [35], [13].",
"Several methods have employed first-person videos to summarize videos [29], [34] while recently the work in [46] proposed to predict the camera wearer's engagement detection from first-person videos.",
"The work in [14] used a group of people wearing first-person cameras to infer their social interactions such as monologues, dialogues, or discussions.",
"The method in [37] predicted physical forces experienced by the camera wearer, while the work in [26] recognized the activities performed in various extreme sports.",
"Several recent methods [36], [45] also predicted the camera wearer's movement trajectories.",
"Finally, first-person cameras have also been used for various robotics applications [41], [18] In comparison to these prior methods, we propose a novel cooperative basketball intention prediction task, that allows us to study cooperative behaviors of the basketball players.",
"Furthermore, we note that these prior first-person methods (except [26]) rely on manually annotated labels for their respective tasks whether it would be an object-detection, activity recognition, intention prediction or some other task.",
"Instead, in this work, we demonstrate that we can solve a challenging cooperative basketball intention prediction task without using annotated first-person intention labels, which are time consuming and costly to obtain.",
"Knowledge Transfer across Models.",
"With the introduction of supervised CNN models [27], there has been a lot of interest in adapting generic set of features [11] for different tasks at hand [22], [3], [16], [47], [39], [42].",
"Recently, generic image classification features were successfully used for the tasks such as edge detection [3], [47], object detection [16], [39], [42], and semantic segmentation [4], [32], [33], [7].",
"More related to our work, a recent line of research investigated how to transfer knowledge across different models by a combination of parameter updates [1], [12], [24], transformation learning [28], [17], network distillation [21] or cross-model supervision [23], [19].",
"The most similar to our work are the methods in [23], [19] that use cross-model supervision to transfer knowledge from one model to another.",
"All of the above methods focus on the third-person data.",
"In contrast, we show how to exploit a first-person view to solve a novel camera wearer's cooperative intention prediction task without using manually labeled first-person data."
],
[
"Learning Cooperative Basketball Intention",
"The goal of our cooperative basketball intention task is to predict with whom the camera wearer will cooperate in the near future.",
"Formally, we aim to learn a function $g(I_i)$ that takes a single first-person image $I_i$ as an input and outputs a per-pixel likelihood map, where each pixel indicates the cooperation probability.",
"Ideally, we would want such function to produce high probability values at pixels around the person with whom the camera wearer will cooperate, and low probability values around all the other pixels.",
"We implement $g(I_i)$ via a fully convolutional neural network based on the architecture of a multi-person pose estimation network in [6].",
"Let $\\hat{y}$ denote a per-pixel mask that is given to our network as a target label.",
"We refer to $\\hat{y}$ as a pseudo ground truth because we obtain it automatically instead of relying on the manually annotated intention labels.",
"Then, we learn our cooperative basketball intention model by optimizing the following cross-entropy loss objective: $\\begin{split}\\mathcal {L}^{(i)}= -\\sum _{j=1}^{N} \\hat{y}^{(i)}_j \\log g_j(I_i) +(1-\\hat{y}^{(i)}_j) \\log \\left(1-g_j(I_i)\\right), \\end{split}$ where $\\hat{y}^{(i)}_j$ is the pseudo ground truth value of image $I_i$ at pixel $j$ , $g_j(I_i)$ refers to our network's output at pixel $j$ , and $N$ denotes the number of pixels in an image.",
"We now explain how we obtain the pseudo ground truth data $\\hat{y}$ ."
],
[
"EgoTransformer",
"To construct a pseudo ground truth supervisory signal $\\hat{y}$ , we transform the output of a pretrained multi-person pose estimation network [6], such that it would approximately capture the camera wearer's internal state such as his/her visual attention, and intentions.",
"We do so using our proposed EgoTransformer scheme.",
"Let $f(I_i)$ denote a pretrained fully convolutional network from [6] that takes a first-person image as an input, and outputs the 1) pose part estimates of every person in an image, and 2) their bounding-box detections.",
"We note that the pretrained network $f$ was never trained on any first-person images.",
"Then, formally, let $B \\in \\mathbb {R}^{n \\times 5}$ denote the bounding box of people detected by $f$ .",
"Each of $n$ detected bounding boxes is parameterized by 5 numbers $(x,y,h,w,c)$ denoting the top-left bounding-box coordinates $(x,y)$ , the height $h$ , and width $w$ of the bounding box, and its confidence value $c$ .",
"Additionally, let $P \\in \\mathbb {R}^{n \\times 18 \\times 2}$ denote the predicted $(x,y)$ locations of 18 pose parts (see [6]) for each of $n$ detected people.",
"Then our goal is to come up with a transformation function $T(B^{(i)},P^{(i)})$ that takes these two outputs and transforms them into a per-pixel pseudo ground truth mask $\\hat{y}^{(i)}$ for our cooperative basketball intention prediction task.",
"We do so by exploiting three different characteristics encoded in a first-person view: 1) egocentric location prior, 2) egocentric size prior, and 3) egocentric pose prior.",
"All of these characteristics can be used to reason about the camera wearer's internal state.",
"labelformat=empty [figure]skip=5pt Figure: Qualitative comparison of the pseudo ground truth labels obtained via an EgoTransformer versus the actual ground truth.",
"Note that while the pseudo ground truth is not always correct (see the third row), in most cases, it successfully assigns high values around the player with whom the camera wearer will cooperate (see the first two rows).labelformat=default [figure]skip=10pt For instance, the location where another person is detected in a first-person image can be used to assess how likely the camera wearer is looking at that person [31], [2].",
"The size of another person in a first-person image can be used to infer how far the camera wearer is from that person, and hence, how likely will the camera wearer interact with that person (the nearer the more likely).",
"Finally, most person-to-person interactions involve people looking at each other, which imposes a certain pose prior.",
"We can then use such a pose prior to predict whether two people will cooperate with each other in the near future based on whether another person is looking at the camera wearer at present.",
"labelformat=empty [figure]skip=5pt Figure: The qualitative cooperative basketball intention prediction results.",
"Despite not using any manually annotated first-person labels during training, in most cases, our cross-model EgoSupervision method correctly predicts with whom the camera wearer will cooperate (the first two rows).",
"In the third row, we also illustrate two cases where our method fails to produce correct predictions.labelformat=default [figure]skip=10pt We express our pseudo ground truth data $\\hat{y}$ using these three characteristics using what we refer to as an EgoTransformer scheme: $\\begin{split}\\hat{y} = & \\Big [ \\sum _{j=1}^n V(B_j, \\phi _{size}(B_j)) \\cdot V(B_j,\\phi _{pose}(B_j))\\Big ] \\cdot \\phi _{loc} (B)\\end{split}$ where $n$ denotes the number of detected bounding boxes in a given image, $B_j$ depicts a $j^{th}$ bounding box, $V$ is a function that takes two inputs: 1) a bounding box $B_j$ , and 2) a scalar value $v$ , and outputs a $H \\times W$ dimensional mask by assigning every pixel inside this bounding box $B_j$ to $v$ , and zeros to all the pixels outside $B_j$ .",
"Here, $H$ and $W$ depict the height and the width of the original input image.",
"Finally, $\\phi _{size}(B_j) \\in \\mathbb {R}^{1 \\times 1}$ and $\\phi _{pose}(B_j) \\in \\mathbb {R}^{1 \\times 1}$ are scalars that capture the size and pose priors associated with a bounding box $B_j$ , while $\\phi _{loc} \\in \\mathbb {R}^{H \\times W}$ is a first-person location prior of the same dimensions as the original input image.",
"Intuitively, the formulation above operates by first assigning a specific value to each of the detected bounding boxes.",
"This yields a $H \\times W$ dimensional prediction map where every pixel that does not belong to any bounding boxes is assigned a zero value.",
"Then, this prediction map is multiplied with the location prior $\\phi _{loc} \\in \\mathbb {R}^{H \\times W}$ (using elementwise multiplication).",
"Finally, all the values are normalized to be in range $[0,1]$ , which produces our final pseudo ground truth labels.",
"We now explain each of the components in more detail.",
"Egocentric Location Prior.",
"The location of the camera wearer's visual attention is essential for inferring his/her cooperative intentions.",
"We know that a first-person camera is aligned with the person's head direction, and thus, it captures exactly what the camera wearer sees.",
"As a result, the way the camera wearer positions himself with respect to other players affects the location where these players will be mapped in a first-person image.",
"Instead of assuming any specific location a-priori (e.g.",
"a center prior), as is done in [31], [29], we find the egocentric location prior directly from the data.",
"As before, Let $B \\in \\mathbb {R}^{n \\times 5}$ denote the bounding boxes detected by a pretrained network.",
"Then we can compute $\\phi _{loc} \\in \\mathbb {R}^{H \\times W}$ as follows: $\\begin{split}\\phi _{loc}(B)=\\sum _{j=1}^n V(B^{(i)}_j,c^{(i)}_j) \\cdot \\frac{1}{N} \\sum _{i=1}^N \\sum _{j=1}^n V(B^{(i)}_j,c^{(i)}_j))\\nonumber \\end{split}$ where $c^{(i)}_j$ is the predicted confidence of the $j^{th}$ bounding box in the $i^{th}$ image.",
"Intuitively, the first term $\\sum _{j=1}^n V(B_j,c^{(i)}_j)$ depicts a $H \\times W$ dimensional mask that is obtained by assigning confidence values to all pixels in their respective bounding boxes in the current image, and zero values to the pixels outside the bounding boxes.",
"The second term $\\frac{1}{N} \\sum _{i=1}^N \\sum _{j=1}^n V(B_j,c^{(i)}_j))$ also depicts a $H \\times W$ dimensional mask that is obtained using this same procedure but across the entire training training dataset rather than a single image.",
"In other words, the second term captures the locations in a first-person image where the bounding box predictions are usually most dense.",
"We conjecture, that $\\phi _{loc}(I_i)$ can then be used to approximate the camera wearer's visual attention location, which is essential for inferring the camera wearer's cooperative intentions.",
"labelformat=empty [figure]skip=5pt Figure: Several qualitative examples from the top 4 performing subjects in our conducted human study.",
"Each subject specified their prediction by clicking on the person, with whom he/she thought the camera wearer was going to cooperate.",
"We then placed a fixed size Gaussian around the location of the click.",
"Note that based on these results, we can conclude that some instances of this task are quite difficult even for humans, i.e.",
"in these examples, there is no general consensus among the subjects' responses.labelformat=default [figure]skip=10pt Egocentric Size Prior.",
"Spatial $3D$ cues provides important information to infer the camera wearer's intentions [36], [45].",
"For instance, the camera wearer is more likely to cooperate with a player who is near him/her.",
"We propose to capture this intuition, by exploiting an egocentric size prior.",
"We know that the size of a bounding box in a first-person image is inversely related to the distance between the camera wearer and the person in the bounding box.",
"Thus, let $h_j$ be the height of the bounding box $B_j$ .",
"Then we express the egocentric size prior $\\phi _{size}(B_j) \\in \\mathbb {R}^{1 \\times 1}$ for a given bounding box as: $\\begin{split}\\phi _{size}(B_j)= \\exp {(-\\frac{\\sigma }{h_j})}\\nonumber \\end{split}$ where $\\sigma $ denotes a hyperparameter controlling how much to penalize small bounding boxes.",
"Such a formulation allows us to capture the intuition that the camera wearer is more likely to cooperate with players who are physically closer to him/her.",
"Egocentric Pose Prior.",
"In basketball, people tend to look at each other to express their intentions before actually performing cooperative actions.",
"Detecting whether a particular person is facing the camera wearer can be easily done by examining the $x$ coordinates of the paired body parts such as eyes, arms, legs, etc of a person detected in a first-person image.",
"For instance, if a particular person is facing the camera wearer then, we will observe that for most of his/her paired parts visible in a first-person image the following will be true: $x(right\\_part)<x(left\\_part)$ .",
"In other words, the right parts of that person's body will have smaller $x$ coordinate value in a first-person image, than the left parts.",
"We use this intuition to encode the egocentric pose prior $\\phi _{pose}(B_j) \\in \\mathbb {R}^{1 \\times 1}$ for a given bounding box $B_j$ as follows: $\\begin{split}\\phi _{pose}(B_j)=\\frac{1}{|\\mathcal {P}|}\\sum _{p \\in \\mathcal {P}} 1 \\lbrace x(right\\_part)<x(left\\_part) \\rbrace \\nonumber \\end{split}$ where $\\mathcal {P}$ is the set of all paired parts, and $1 \\lbrace x(right\\_part)<x(left\\_part) \\rbrace $ is an indicator function that returns 1 if the $x$ coordinate of the right part in a first-person image is smaller than the $x$ coordinate of the left part.",
"The computed value $\\phi _{pose}(B_j)$ can then be viewed as a confidence that a person in the bounding box $B_j$ is facing the camera wearer, which is an important cue for inferring the camera wearer's cooperative intentions.",
"Pseudo Ground Truth.",
"We then combine all the above discussed components into a unified framework using the Equation REF .",
"Such a formulation allows us to automatically construct pseudo ground truth labels from the outputs of a pretrained multi-person pose estimation network.",
"We illustrate several examples of our obtained pseudo ground truth labels in Figure REF .",
"Notice that while our computed pseudo ground truth is not always correct, in many cases it correctly captures the player with whom the camera wearer will cooperate in the near future.",
"In our experimental section, we will demonstrate that despite the imperfections of our pseudo ground truth labels, we can use them to obtain a model that is almost as good as the model trained in a fully supervised fashion using manually annotated cooperation labels."
],
[
"Cross-Model EgoSupervision",
"After obtaining the pseudo ground truth data $\\hat{y}$ , we train our cooperative basketball intention FCN using the cross-model EgoSupervision scheme as shown in Figure REF .",
"We employ a multi-person pose estimation network from [6] as our base model, which is used to predict the 1) pose estimates of all people in a given image and 2) their bounding boxes.",
"The parameters inside the base network are fixed throughout the entire training procedure.",
"At each iteration, the outputs from the base network are fed to the EgoTransformer, which transforms them into the pseudo ground truth cooperate intention labels.",
"These pseudo ground truth labels are then used as a supervisory signal to train our cooperative basketball intention FCN using a sigmoid cross entropy per-pixel loss as illustrated in Equation REF ."
],
[
"Implementation Details",
"For all of our experiments, we used a Caffe deep learning library [25].",
"As our base FCN model we used a multi-person pose estimation network from [6].",
"Inspired by the success of this method, we also used the same architecture for our cooperative basketball intention FCN.",
"During training, we optimized the network for 5000 iterations with a learning rate of $10^{-7}$ , the momentum equal to $0.9$ , the weight decay of $0.0005$ , and the batch size of 15.",
"The weights inside the base FCN network were fixed throughout the entire training procedure.",
"To compute the egocentric size prior mask we used $\\sigma = 10$ ."
],
[
"Cooperative Basketball Intention Dataset",
"We build upon the dataset from [5], which captures first-person basketball videos of 48 distinct college-level players in an unscripted basketball game.",
"The work in [5] studies a basketball performance assessment problem, and provides 401 training and 343 testing examples of basketball cooperations among players from $10.3$ hours of videos.",
"To obtain ground truth labels corresponding to the specific players, with whom the camera wearer cooperated, we look at the video segments corresponding to all such cooperation.",
"We then identify the player with whom the camera wearer cooperated, go back to the frame about 2 seconds before the cooperation happens, and label that player with a bounding box.",
"The ground truth data is then generated by placing a Gaussian inside the bounding box, according to the height and width of the bounding box.",
"Once again we note that these labels are only used for the evaluation purposes, and also to train other baseline models.",
"In comparison, our method learns to detect the players with whom the camera wearer will cooperate, without relying on manually annotated intention labels."
],
[
"Experimental Results",
"In this section, we present quantitative and qualitative results for our cooperative basketball intention prediction task.",
"To compute the accuracy of each method, we select the player in the image with the maximum predicted probability as the the final prediction and then compute the fraction of all the correct predictions across the entire testing dataset.",
"Table: Quantitative human study results on our cooperative basketball intention task.",
"We ask 5 subjects to predict a player in the first-person image, with whom they think the camera wearer will cooperate after 2 seconds.",
"We then compute the accuracy as the fraction of correct responses.",
"The results indicate that most subjects achieve the accuracy of about 90%90\\%.",
"We conjecture that Subject-4 may be less familiar with the basketball game thus, the lower accuracy."
],
[
"Human Study",
"First, to see how well humans can predict cooperative basketball intention from first-person images, we conduct a human study consisting of 5 human subjects.",
"Each subject is shown 343 testing images one at a time, and asked to click on the player in an image, with whom he/she thinks the camera wearer will cooperate 2 seconds from now.",
"Then the accuracy of each subject is evaluated as the fraction of correct responses.",
"We present these results in Table REF , and demonstrate that this task is not trivial even for humans: most of the subjects achieve about $90\\%$ accuracy on our task, which is solid but not perfect.",
"We also point out that we did not collect information on how familiar each subject was with basketball.",
"However, based on the results, we conjecture that Subject-4 who achieved almost $10\\%$ lower accuracy than the other subjects was probably not very familiar with basketball, which contributed to his lower performance.",
"In Figure REF , we also visualize the qualitative examples that human subjects found the most difficult, i.e.",
"in these instances, the predictions among the subjects differed substantially.",
"Table: The quantitative cooperative basketball intention results evaluated as the fraction of correct predictions.",
"We compare our Cross-Model EgoSupervision (CMES) scheme with a variety of supervised methods (marked by ‡\\ddagger ).",
"These results indicate that even without using manually annotated intention labels, our method outperforms most supervised methods, and produces almost identical performance as our main baseline “MPP-finetuned”."
],
[
"Quantitative Results",
"In Table REF , we present quantitative cooperative basketball intention results of our method and several other baselines.",
"As our baselines, we use a collection of methods that were successfully used for other computer vision tasks such as image classification, semantic segmentation or saliency detection.",
"These include a 1) Deep Contrast Saliency (DCL) method [30], 2-3) several variations of highly successful DeepLab semantic segmentation systems [9], [8] adapted to our task, 4-5) image classification ResNets [20] adapted to our task, 6) one of the top performing semantic segmentation systems PSPNet [48], 7-8) a pretrained and finetuned multi-person pose estimation (MPP) network [6], and 9) a pseudo ground truth obtained from our EgoTransformer.",
"Note that our Cross-Model EgoSupervision (CMES) method is based on an MPP network architecture [6], and thus, as our main baseline we use the “MPP-finetuned” method, which uses the manually labeled bounding box intention labels to infer with whom the camera wearer will interact.",
"In contrast to this baseline, our CMES method is only trained on the automatically generated pseudo ground truth labels.",
"We note that the supervised methods employing manually labeled data are marked with $^{\\ddagger }$ .",
"We now discuss several interesting observations based on these results.",
"Table: The quantitative ablation studies documenting the importance of each component in our EgoTransformer scheme.",
"We separately remove each of φ loc \\phi _{loc}, φ size \\phi _{size}, φ pose \\phi _{pose} and investigate how the accuracy changes.",
"The second column in the table denotes the accuracy of a pseudo ground truth, while the third column depicts the accuracy of our trained model.",
"Based on these results, we can conclude that each component of our EgoTransformer is essential for an accurate cooperative basketball intention prediction.Comparison with the Supervised Methods.",
"Based on the results, we observe that despite not using manually annotated bounding box intention labels, our method outperforms a number of supervised baselines and achieves almost equivalent results to our main baseline “MPP-finetuned”, which was trained using manually annotated cooperative intention labels.",
"Thus, these results indicatee the effectiveness of our cross-model EgoSupervision scheme.",
"Comparison with the Pseudo Ground Truth.",
"One interesting and a bit surprising observation from Table REF , is that our cross-model EgoSupervision model achieves substantially better accuracy than the pseudo ground truth, which was used to optimize our model.",
"We conjecture that this happens due to the following reasons.",
"The pseudo ground truth labels are constructed using three different signals: 1) an egocentric location prior, 2) an egocentric size prior, and 3) an egocentric pose prior.",
"Note, that our constructed pseudo ground truth does not incorporate any visual appearance information, i.e.",
"it does not consider how the players look like.",
"In contrast, during training, our network, learns what are the visual appearance cues indicative of the players with high pseudo ground truth values.",
"Arguably, such visual cues provide a stronger signal for a cooperative intention recognition, which then leads to a substantially better performance than the pseudo ground truth labels."
],
[
"Qualitative Results",
"In Figure REF , we present our qualitative results, where we show that in most cases, our model successfully learns to predict with whom the camera wearer will cooperate.",
"Furthermore, to gain a better understanding of what the network learned, in Figure REF , we visualize the activations inside the second to last FCN's layer.",
"Note that our network has high activation values around the faces of people with whom the camera wearer intends to cooperate.",
"This makes intuitive sense, as face is probably the most useful cue to recognize the camera wearer's intention to cooperate."
],
[
"Ablation Experiments",
"In Table REF , we present the results analyzing the behavior of our EgoTransformer scheme.",
"Earlier we discussed that to implement our EgoTransformer scheme we exploit three characteristics: 1) egocentric location prior $\\phi _{loc}$ , 2) egocentric size prior $\\phi _{size}$ , and 3) egocentric pose prior $\\phi _{pose}$ .",
"We want to investigate how much each of these priors affect 1) the quality of our generated pseudo ground truth data, and 2) the quality of our model trained using such pseudo ground truth.",
"To do this, we run experiments with three baselines where for each baseline we remove one of $\\phi _{loc}, \\phi _{size},$ or $\\phi _{pose}$ components.",
"We denote these three baselines as “no $\\phi _{loc}$ ”, “no $\\phi _{size}$ ” and “no $\\phi _{pose}$ ” respectively.",
"Finally, we include the results of our model using the full EgoTransformer scheme.",
"labelformat=empty [figure]skip=5pt Figure: The visualization of the activation values inside the second to last layer in our trained network.",
"Note that the network produces high activation values around the faces of the players in the camera wearer's field of view.",
"This makes intuitive sense, as facial expressions provide the most informative cues for a cooperative basketball intention task.labelformat=default [figure]skip=10pt Based on the results in Table REF , we first observe that each of these components have a significant impact to the quality of pseudo ground truth that we obtain.",
"Specifically, using our full model yields $9.4\\%$ better pseudo ground truth results than the second best baseline.",
"Additionally, note that the network trained to the pseudo ground truth of our full model achieves $4.4\\%$ higher accuracy than the second best baseline.",
"These results indicate that each component in our EgoTransformer scheme is crucial for learning a high quality cooperative intention model."
],
[
"Conclusions",
"In this work, we present a new task of predicting cooperative basketball intention from a single first-person image.",
"We demonstrate that a first-person image provides strong cues to infer the camera wearer's intentions based on what he/she sees.",
"We use this observation to design a new cross-model EgoSupervision learning scheme that allows us to predict with whom the camera wearer will cooperate, without using manually labeled intention labels.",
"We demonstrate that despite not using such labels, our method achieves similar or even better results than fully supervised methods.",
"We believe that our proposed cross-model EgoSupervision scheme could be applied on various other first-person vision tasks without the need to manually collect labels for each of such tasks.",
"In the long run, a learning scheme such as ours could effectively replace the supervised methods, which require costly and time consuming annotation process."
]
] | 1709.01630 |
[
[
"Spin precession and spin Hall effect in monolayer graphene/Pt\n nanostructures"
],
[
"Abstract Spin Hall effects have surged as promising phenomena for spin logics operations without ferromagnets.",
"However, the magnitude of the detected electric signals at room temperature in metallic systems has been so far underwhelming.",
"Here, we demonstrate a two-order of magnitude enhancement of the signal in monolayer graphene/Pt devices when compared to their fully metallic counterparts.",
"The enhancement stems in part from efficient spin injection and the large resistivity of graphene but we also observe 100% spin absorption in Pt and find an unusually large effective spin Hall angle of up to 0.15.",
"The large spin-to-charge conversion allows us to characterise spin precession in graphene under the presence of a magnetic field.",
"Furthermore, by developing an analytical model based on the 1D diffusive spin-transport, we demonstrate that the effective spin-relaxation time in graphene can be accurately determined using the (inverse) spin Hall effect as a means of detection.",
"This is a necessary step to gather full understanding of the consequences of spin absorption in spin Hall devices, which is known to suppress effective spin lifetimes in both metallic and graphene systems."
],
[
"Spin precession and the spin Hall effect in graphene/Pt nanostructures Supplementary Information W. Savero Torres Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology (BIST), Campus UAB, Bellaterra, 08193 Barcelona, Spain.",
"J.F.",
"Sierra Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology (BIST), Campus UAB, Bellaterra, 08193 Barcelona, Spain.",
"L.A. Benítez Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology (BIST), Campus UAB, Bellaterra, 08193 Barcelona, Spain.",
"Universidad Autonoma de Barcelona, Bellaterra, 08010 Barcelona, Spain F. Bonell Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology (BIST), Campus UAB, Bellaterra, 08193 Barcelona, Spain.",
"M.V.",
"Costache Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology (BIST), Campus UAB, Bellaterra, 08193 Barcelona, Spain.",
"S.O.",
"Valenzuela Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology (BIST), Campus UAB, Bellaterra, 08193 Barcelona, Spain.",
"Institucion Catalana de Recerca i Estudis Avançats (ICREA), 08070 Barcelona, Spain.",
"In the following, the analytical expressions for the spin Hall voltage $V_{SHE}$ used in the main text are obtained.",
"The standard procedure used in the literature is described and adapted to introduce spin injection with tunnel barriers.",
"This procedure focuses on the non-precessing spin component and the rotation of the ferromagnetic injector magnetization [1], [2] combined with spin absorption in Pt [3].",
"Afterwards, the model is extended to introduce spin precession.",
"In all cases, we use a one-dimensional approximation.",
"Figure 1 shows the probe configuration and the geometry used to describe the relevant parameters for the calculation.",
"The coordinate origin is defined at the position of the injector electrode.",
"$L_{SHE}$ is the distance from the injector to the spin-Hall cross, while $W_{Pt}$ , $w_{G}$ , $t_{Pt}$ are the platinum width, graphene width and platinum thickness, respectively.",
"$S=W_{Pt}*w_{G}$ is the area of the Pt/graphene contact, where the current is absorbed.",
"In the standard measurement, the magnetic field $B$ is applied along $x$ direction, while in the spin precession measurements, $B$ is out of the sample plane, in the $z$ direction.",
"Figure: Schematic representation of the probe measurementThe charge current $J^{T}_{q}$ in the Pt electrode at $B=0$ can be written as [4]: $ \\textbf {J}^{T}_{q} = \\alpha ^{Pt}_{SHE}[\\hat{\\textbf {\\i }} \\times \\textbf {J}_{s}] + \\sigma _{Pt}\\textbf {E}$ where $\\alpha ^{Pt}_{SHE}$ and $\\sigma _{Pt}$ are the Hall angle and the electrical conductivity of Pt respectively and $\\textbf {J}_{s}$ is the spin current reaching the detector.",
"The first term describes the spin to charge conversion due to the ISHE while the second term contains the electrical field that counteracts the charge current in the steady state, in which both components compensate, therefore: $\\textbf {0} = \\alpha ^{Pt}_{SHE}[\\hat{\\textbf {\\i }} \\times \\textbf {J}_{s}] + \\sigma _{Pt}\\textbf {E}$ At $x=L_{SHE}$ the average spin current over the Pt wire is $\\textbf {J}_{s}= \\bar{J}_{s}\\hat{\\textbf {k}}$ and the resulting electric field along the $y$ direction is $\\textbf {E}= E_{y}\\hat{\\textbf {\\j }}$ .",
"The electric field $E_{y}$ is then : $E_{y}= \\alpha ^{Pt}_{SHE}\\rho _{Pt}\\bar{J}_{s}$ The integration over the graphene width provides the Hall voltage $V_{SHE}$ : $V_{SHE}= \\alpha ^{Pt}_{SHE}\\rho _{Pt}w_{G}\\bar{J}_{s}$ $\\bar{J}_{s}$ is derived from the spin accumulation $\\mu _{s}$ generated in the Pt wire, defined by $\\mu ^{Pt}_{s}=(\\mu _{+}-\\mu _{-})/2$ , where $\\mu _{\\pm }$ are the electrochemical potentials for each spin population.",
"Considering a 1D diffusive spin transport, $\\mu ^{Pt}_{s}$ is: $\\mu ^{Pt}_{s}(z)= A \\exp (z/\\lambda _{Pt})+B \\exp (-z/\\lambda _{Pt})$ where $z$ indicates the dependence on the Pt thickness.",
"Since $J_{s}=-\\sigma /e\\nabla \\mu _{s}$ , the spin current density $J^{Pt}_{s}$ can be expressed as: $J^{Pt}_{s}(z)= \\frac{-\\sigma _{Pt}}{e\\lambda _{Pt}}[A \\exp (z/\\lambda _{Pt})-B \\exp (-z/\\lambda _{Pt})]$ To determine the constants A and B, we consider the following boundary conditions: 1) At $z=0$ , $J^{Pt}_{s}= J_{z0}$ , where $J_{z0}$ is the current density reaching the detector after diffusing from the injection point and 2) At $z=t_{Pt}$ , $J^{Pt}_{s}= 0$ .",
"From these conditions, we obtain: $B=\\frac{-J_{z0}e\\lambda _{Pt}}{\\sigma _{Pt}}\\frac{1}{\\exp (-2t_{Pt}/\\lambda _{Pt})-1}$ $A=\\frac{-J_{z0}e\\lambda _{Pt}}{\\sigma _{Pt}}\\frac{\\exp (-2t_{Pt}/\\lambda _{Pt})}{\\exp (-2t_{Pt}/\\lambda _{Pt})-1}$ The spin current density along the thickness $J_{s}(z)$ is therefore given by: $J^{Pt}_{s}(z)=J_{z0}[\\frac{\\exp (-2t_{Pt}/\\lambda _{Pt})}{\\exp (-2t_{Pt}/\\lambda _{Pt})-1}]\\exp (z/\\lambda _{Pt})-J_{z0}[\\frac{1}{\\exp (-2t_{Pt}/\\lambda _{Pt})-1}]\\exp (-z/\\lambda _{Pt})$ The average of $J^{Pt}_{s}(z)$ , $\\bar{J}_{s}$ , is calculated from [5], [6], [7], [8]: $\\bar{J}_{s}= \\frac{1}{t_{Pt}}\\int ^{t_{Pt}}_{0}J^{Pt}_{s}(z)dz=\\frac{J_{z0}\\lambda _{Pt}}{t_{Pt}} \\frac{[1-\\exp (-t_{Pt}/\\lambda _{Pt})]}{[1+\\exp (-t_{Pt}/\\lambda _{Pt})]}$ The current $J_{z0}$ at the interface between graphene and Pt is determined from the boundary conditions with the graphene region.",
"It is straightforward to obtain the full solution analytically, however, a much simpler expression for $J_{z0}$ can be found by noting that $\\mu \\sim 0$ at the graphene/Pt interface due to the fast relaxation in Pt.",
"Such a solution is an excellent approximation to the full solution for the relevant magnitudes of the involved parameters, and can be easily deduced as follows.",
"We start by considering the solution of the diffusion equation for the graphene wire without Pt.",
"Because the spin current in graphene $J_{s}\\rightarrow 0$ when $x\\rightarrow \\infty $ and we use tunnel barriers for spin injection, $J_{z0}$ can be approximated by [4], $J_{z0}\\approx \\frac{pI}{2S}\\exp (-L_{SHE}/\\lambda _{G})$ where $\\lambda _{G}$ is the spin relaxation length of graphene, $p$ is the effective spin polarization and $S$ the contact surface shown in Fig.",
"1.",
"Note that in Eq.",
"3, it is assumed that the magnetization of the injector is saturated along the the magnetic field.",
"If the rotation is not complete, the additional factor $\\sin \\gamma $ should be added to account for the magnetization tilting angle $\\gamma $ .",
"[2].",
"Because the spin current is fully absorbed by the Pt wire, no spin current flows away from Pt in the graphene opposite to the side of the injector, where $\\mu \\sim 0$ , as verified experimentally.",
"In addition, this implies that the contact resistance is very small and does not play a significant role in determining $J_{z0}$ .",
"Considering that $\\lambda _{G}\\gg \\lambda _{Pt}$ and that the conductivity of Pt is much larger than that of graphene, it can be easily verified that $J_{z0}$ is weakly dependent on the exact value of $\\lambda _{Pt}$ .",
"Indeed, the region of graphene on the other side of the injector can safely be ignored.",
"The full solution of the spin diffusion in our device is thus nearly identical to a problem involving two spin sources of opposite polarities separated by $2L_{SHE}$ without the Pt wire.",
"In this equivalent problem, the condition $\\mu = 0$ in the equidistant point to the sources (Pt wire position)is rigorously satisfied.",
"The current $J_{z0}$ in the middle of the two sources, which is absorbed by Pt, is therefore twice the current in Eq.",
"3, $J_{z0}\\approx \\frac{pI}{S}\\exp (-L_{SHE}/\\lambda _{G})$ By replacing Eq.",
"2 and 4 in Eq.",
"1, the inverse spin Hall voltage for this case is: $V_{SHE}= \\frac{pI\\alpha ^{Pt}_{SHE}\\rho _{Pt}\\lambda _{Pt}}{w_{Pt}t_{Pt}} \\frac{[1-\\exp (-t_{Pt}/\\lambda _{Pt})]}{[1+\\exp (-t_{Pt}/\\lambda _{Pt})]}\\exp (-L_{SHE}/\\lambda _{G})$ In this case, the magnetic field, applied out of the sample plane, induces spin current precession therefore changing the expression for $J_{z0}$ .",
"To determine $J_{z0}$ , we first calculate the contribution from the precessing non-equilibrium spin density $n_{+}-n_{-}$ reaching the detector location, which is related to the electrochemical potentials $\\mu _{\\pm }$ as $\\mu _{\\pm }=\\frac{n_{\\pm }}{N_{\\pm }(E_{F})}$ , where $n_{\\pm }$ and $N_{\\pm }(E_{F})$ are the spin density and the density of states at the Fermi level for each spin orientation [6].",
"An argument analogous to that in the previous case can be made.",
"For the spins freely diffusing in graphene, without the Pt wire, the non equilibrium density of spins perpendicular to the spin Hall bar is: $n_{+}-n_{-}=\\frac{Ip}{eS} \\int ^{\\infty }_{0} \\frac{\\exp (-x^{2}/4Dt)}{\\sqrt{4\\pi Dt}}\\sin (w_{L}t)\\exp (-t/\\tau _{s})dt$ where $Ip/eS$ is the spin injection rate and $\\frac{\\exp (-x^{2}/4Dt)}{\\sqrt{4\\pi Dt}}\\exp (-t/\\tau _{s})$ the probability to reach the detector at time $t$ , $w_{L}t$ is the precession angle, and $w_{L}$ the Larmor frequency.",
"Since in the spin channel no charge current flows, $N_{+}(E_{F})+ N_{-}(E_{F}) = N(E_{F})$ and $N_{+}(E_{F})= N_{-}(E_{F}) = N(E_{F})/2$ .",
"The corresponding spin accumulation $\\mu ^{N}_{s}$ is then given by: $\\mu ^{N}_{s}=\\frac{Ip}{eSN(E_{F})} \\int ^{\\infty }_{0}\\frac{ \\exp (-x^{2}/4Dt)}{\\sqrt{4\\pi Dt}}\\sin (w_{L}t)\\exp (-t/\\tau _{s})dt$ As $J^{N}_{s}=-\\sigma _{N}/e\\nabla \\mu ^{N}_{s}$ , with $\\sigma _{N}$ the conductivity of the channel, the spin current is: $J^{N}_{s}=\\frac{Ipx\\sigma }{2De^{2}N(E_{F})S} \\int ^{\\infty }_{0}\\frac{\\exp (-x^{2}/4Dt)}{t\\sqrt{4\\pi Dt}}\\sin (w_{L}t)\\exp (-t/\\tau _{s})dt$ Using the Einstein relation: $\\sigma =e^{2}N(E_{F})D$ , then: $J^{N}_{s}=\\frac{Ipx}{2S} \\int ^{\\infty }_{0}\\frac{ \\exp (-x^{2}/4Dt)}{t\\sqrt{4\\pi Dt}}\\sin (w_{L}t)\\exp (-t/\\tau _{s})dt$ At $x=L_{SHE}$ the spin current $J^{N}_{s}$ = $J_{z0}$ is given by: $J_{z0}=\\frac{IpL_{SHE}}{2S} \\int ^{\\infty }_{0}\\frac{ P(t)}{t}\\sin (w_{L}t)\\exp (-t/\\tau _{s})dt$ By multiplying this expression by a factor 2, resulting from the condition $\\mu =0$ , and substituting in Eq.",
"1, we find the expression for the Hall voltage $V_{SHE}$ shown in Eq.",
"2 of the main text, $V_{SHE}= \\frac{I\\alpha ^{Pt}_{SHE}\\rho _{Pt}\\lambda _{Pt}pL_{SHE}}{W_{Pt}t_{Pt}} \\frac{[1-\\exp (-t_{Pt}/\\lambda _{Pt})]}{[1+\\exp (-t_{Pt}/\\lambda _{Pt})]} \\int ^{\\infty }_{0}\\frac{ P(t)}{t}\\sin (w_{L}t)\\exp (-t/\\tau _{s})dt$ where the precession of $J_{z0}$ inside the Pt wire is neglected because of the very small dephasing times in Pt($\\tau _{s}\\approx 3\\times 10^{-13}$ s [9]).",
"The equivalence between the standard (Eq.",
"5) and precessional approach (Eq.",
"7) in the limit of no spin precession can be verified by substituting $\\sin (w_{L}t)=1$ in the integral of Eq.",
"7 and noting that $\\lambda _{G}=\\sqrt{Dt}$ ."
]
] | 1709.01854 |
[
[
"Non-commutative peaking phenomena and a local version of the\n hyperrigidity conjecture"
],
[
"Abstract We investigate various notions of peaking behaviour for states on a $\\mathrm{C}^*$-algebra, where the peaking occurs within an operator system.",
"We pay particularly close attention to the existence of sequences of elements forming an approximation of the characteristic function of a point in the state space.",
"We exploit such characteristic sequences to localize the $\\mathrm{C}^*$-algebra at a given state, and use this localization procedure to verify a variation of Arveson's hyperrigidity conjecture for arbitrary operator systems."
],
[
"Introduction",
"The study of unital closed subalgebras of continuous functions on some compact metric space has a rich history, involving questions and techniques ranging from complex function theory to measure theory and functional analysis.",
"The resulting theory of uniform algebras is quite extensive , and we briefly recall one of its features that is relevant for our present purpose.",
"Given a uniform algebra ${\\mathcal {A}}\\subset C(X)$ and a point $\\xi \\in X$ , the linear functional on ${\\mathcal {A}}$ of evaluation at $\\xi $ can be represented as integration against some Borel probability measure on $X$ .",
"Obviously, the point mass at $\\xi $ is always a representing measure, and when it is the unique such, then the point $\\xi $ is said to belong to the Choquet boundary of ${\\mathcal {A}}$ .",
"Equivalently, we see that $\\xi $ lies in the Choquet boundary of ${\\mathcal {A}}$ if the corresponding evaluation functional admits a unique unital completely positive extension to $C(X)$ .",
"The Choquet boundary is a meaningful object to associate to a uniform algebra, for at least two reasons.",
"First, it is known that the Choquet boundary is dense in the Shilov boundary of ${\\mathcal {A}}$ , that is the smallest closed subset of $X$ on which every function in ${\\mathcal {A}}$ attains its maximum modulus.",
"In other words, the Choquet boundary provides a mechanism to identify the “minimal\" representation of the elements of ${\\mathcal {A}}$ as functions on some compact metric space.",
"This philosophy has been transplanted in the non-commutative context of unital operator algebras and operator systems, and has been exploited with great success to obtain a corresponding minimal representation of these objects inside a $\\mathrm {C}^*$ -algebra.",
"The visionary initial push to make this possible was furnished by Arveson , and it came to fruition several years later in the form of the $\\mathrm {C}^*$ -envelope ,,,,.",
"Central to the construction of this envelope are linear maps admitting a unique unital completely positive extension to the ambient $\\mathrm {C}^*$ -algebra, very much in the spirit of the defining condition of the classical Choquet boundary.",
"The theory surrounding the $\\mathrm {C}^*$ -envelope has since flourished into a widely used invariant for the basic objects of non-commutative functional analysis ,,,.",
"The second meaningful aspect of the classical Choquet boundary that we wish to emphasize here, is that it can be characterized in a drastically different fashion from the one alluded to above, thus allowing for a wealth of additional interpretations.",
"While the definition we gave above using completely positive extensions is somehow extrinsic, there is an equivalent definition that is intrinsic.",
"Indeed, it is known that a point $\\xi \\in X$ lies in the Choquet boundary of ${\\mathcal {A}}$ if and only if it is a peak point for ${\\mathcal {A}}$ , in the sense that there is a function $\\varphi \\in {\\mathcal {A}}$ with the property that $|\\varphi (x)|<1=\\varphi (\\xi )$ for every $x\\in X$ such that $x\\ne \\xi $ .",
"Non-commutative extensions of this phenomenon have attracted significant interest in recent years ,,,,, and beautiful generalizations of function theoretic results were obtained therein.",
"Nevertheless, it has been noted (see the discussion following , or ) that the classical symbiosis between peak points and the Choquet boundary appears to collapse dramatically in this non-commutative interpretation.",
"A robust connection between the non-commutative concepts is still lacking despite some subsequent efforts ,,.",
"Exploring a different foundation for such a connection is one of the purposes of the paper.",
"The other purpose of this work is to address the so-called hyperrigidity conjecture of Arveson .",
"To motivate this conjecture, we recall a classical result of Korovkin .",
"For each $n\\in {\\mathbb {N}}$ , let $\\Phi _n:C[0,1]\\rightarrow C[0,1]$ be a positive linear map and assume that $\\lim _{n\\rightarrow \\infty }\\Vert \\Phi _n(f)-f\\Vert =0$ for every $f\\in \\lbrace 1,x,x^2\\rbrace $ .",
"Then, it must be the case that $\\lim _{n\\rightarrow \\infty }\\Vert \\Phi _n(f)-f\\Vert =0$ for every $f\\in C[0,1]$ .",
"This striking rigidity phenomenon was elucidated by Šaškin in , who showed that the crucial property at play here is that the interval $[0,1]$ consists entirely of peak points for the subspace generated by $\\lbrace 1,x,x^2\\rbrace $ .",
"One then wonders whether this is a manifestation of a general phenomenon which is valid in the non-commutative context as well.",
"More precisely, let ${\\mathfrak {S}}$ be an operator system.",
"Arveson conjectured that if the non-commutative Choquet boundary of ${\\mathfrak {S}}$ is “maximal\" then ${\\mathfrak {S}}$ is hyperrigid, in the sense that for any injective $*$ -representation $\\pi :\\mathrm {C}^*({\\mathfrak {S}})\\rightarrow B({\\mathcal {H}}_\\pi )$ and for any sequence of unital completely positive linear maps $\\Phi _n:B({\\mathcal {H}}_\\pi )\\rightarrow B({\\mathcal {H}}_\\pi ), \\quad n\\in {\\mathbb {N}}$ such that $\\lim _{n\\rightarrow \\infty }\\Vert \\Phi _n(\\pi (s))-\\pi (s)\\Vert =0, \\quad s\\in {\\mathfrak {S}}$ we must have $\\lim _{n\\rightarrow \\infty }\\Vert \\Phi _n(\\pi (a))-\\pi (a)\\Vert =0, \\quad a\\in \\mathrm {C}^*({\\mathfrak {S}}).$ The notion of hyperrigidity is deeply rooted in various parts of operator algebras and operator theory; for instance, in the appropriate context it has been shown to be equivalent to the Arveson-Douglas essential normality conjecture involving quotient modules of the Drury-Arveson space .",
"This conjecture has generated a flurry of activity and has witnessed spectacular progress recently (see and and the references therein).",
"Although we will not discuss the essential normality conjecture in any more detail here, the aforementioned equivalence makes the hyperrigidity conjecture all the more tantalizing.",
"At present the hyperrigidity conjecture is still unresolved in full generality.",
"It has been verified in the case where $\\mathrm {C}^*({\\mathfrak {S}})$ has countable spectrum and in the case where $\\mathrm {C}^*({\\mathfrak {S}})$ is commutative .",
"Some partial results in other contexts have also appeared in ,.",
"The techniques used therein are geared towards establishing the unique extension property of unital completely positive maps.",
"In this paper, we choose a different route.",
"We wish to exhibit a link between hyperrigidity and a seemingly neglected facet of the non-commutative Choquet boundary, namely peaking phenomena; such a link would constitute a first step in a faithful adaptation of the aforementioned approximation results of Korovkin and Šaškin.",
"More precisely, we will witness hyperrigidity in some “local\" sense, where the precise localization procedure is accomplished via peaking states.",
"It should be noted here that Arveson himself had early results concerning local hyperrigidity in the commutative setting.",
"Our contribution is to show that Arveson's commutative local hyperrigidity, once appropriately interpreted with the help of peaking states, holds in full generality.",
"We now describe the organization of the paper more precisely.",
"In Section , we gather the necessary background material and prove some preliminary results that are required throughout.",
"In Section , we introduce a notion of peaking states, which goes as follows.",
"Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"We say that a state $\\psi $ on ${\\mathfrak {A}}$ is ${\\mathfrak {S}}$ -peaking if there is a self-adjoint element $s\\in {\\mathfrak {S}}$ such that $\\Vert s\\Vert =1$ and with the property that $\\psi (s)=1>|\\varphi (s)|$ for every state $\\varphi $ on ${\\mathfrak {A}}$ such that $\\varphi \\ne \\psi $ .",
"In Theorem REF we obtain the following characterization of these objects; it is noteworthy in view of the classical result of Klee which guarantees the existence of a large supply of weak-$*$ exposed points.",
"Theorem 1.1 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $\\psi $ be a state on ${\\mathfrak {A}}$ .",
"Then, the following statements are equivalent.",
"The state $\\psi $ is ${\\mathfrak {S}}$ -peaking.",
"The state $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ , and the restriction $\\psi |_{{\\mathfrak {S}}}$ is a weak-$*$ exposed point of the set of self-adjoint functionals on ${\\mathfrak {S}}$ with norm at most 1.",
"We consider another flavour of peaking behaviour, heavily inspired by the work of Hay .",
"A projection $p\\in {\\mathfrak {A}}^{**}$ is said to be ${\\mathfrak {S}}$ -peaking if there is a self-adjoint element $s\\in {\\mathfrak {S}}$ with $\\Vert s\\Vert =1$ such that $ sp=p$ and $| \\varphi (s)|<1$ whenever $\\varphi $ is a state on ${\\mathfrak {A}}$ with support projection ${\\mathfrak {s}}_\\varphi \\in {\\mathfrak {A}}^{**}$ orthogonal to $p$ .",
"We elucidate the relationship between ${\\mathfrak {S}}$ -peaking states and ${\\mathfrak {S}}$ -peaking projections in the following (Theorem REF ).",
"Theorem 1.2 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Then, a pure state $\\omega $ on ${\\mathfrak {A}}$ is ${\\mathfrak {S}}$ -peaking if and only if its support projection ${\\mathfrak {s}}_\\omega \\in {\\mathfrak {A}}^{**}$ is ${\\mathfrak {S}}$ -peaking.",
"In Section , in preparation for studying a local version of hyperrigidity and motivated by concrete examples, we introduce a more global type of peaking phenomenon for states.",
"Roughly speaking, we wish to find, within a unital $\\mathrm {C}^*$ -algebra ${\\mathfrak {A}}$ , an approximate version of the characteristic function corresponding to a given state $\\psi $ on ${\\mathfrak {A}}$ .",
"More precisely, given a sequence $(\\Delta _n)_n$ in ${\\mathfrak {A}}$ such that $\\Vert \\Delta _n\\Vert =1$ for every $n\\in {\\mathbb {N}}$ , we say that $(\\Delta _n)_n$ is a characteristic sequence for $\\psi $ if $\\lim _{n\\rightarrow \\infty }\\psi (\\Delta _n)=1$ and $\\limsup _{n\\rightarrow \\infty }\\Vert \\Delta _n^* a \\Delta _n\\Vert \\le |\\psi (a)|, \\quad a\\in {\\mathfrak {A}}.$ We show that states that admit a characteristic sequence share several properties with ${\\mathfrak {S}}$ -peaking states and can be thought of as being “asymptotically ${\\mathfrak {S}}$ -peaking\" (Theorem REF ).",
"Theorem 1.3 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let $\\psi $ a state on ${\\mathfrak {A}}$ .",
"Let $(\\Delta _n)_n$ be a characteristic sequence for $\\psi $ .",
"The following statements hold.",
"We have that $\\lim _{n\\rightarrow \\infty }\\Vert \\Delta _n a \\Delta _n^*\\Vert =|\\psi (a)|$ for every $a\\in {\\mathfrak {A}}$ .",
"We have that $\\limsup _{n\\rightarrow \\infty }|\\varphi (\\Delta _n)|<1$ for every state $\\varphi $ on ${\\mathfrak {A}}$ such that $\\varphi \\ne \\psi $ .",
"The state $\\psi $ is pure.",
"Let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system and assume that $\\Delta _n\\in {\\mathfrak {S}}$ for each $n\\in {\\mathbb {N}}$ .",
"Then, $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ .",
"The asymptotic notion of characteristic sequence is flexible enough to occur in natural and important examples such as the higher-dimensional Toeplitz algebra, which is a ubiquitous object in multivariate operator theory, dilation theory and function theory (Example REF ).",
"Finally, in Section we exploit characteristic sequences to localize a $\\mathrm {C}^*$ -algebra at a given state, and we establish the following (Theorem REF ).",
"Theorem 1.4 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $\\pi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ be a unital $*$ -representation and let $\\Pi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ be a unital completely positive extension of $\\pi |_{{\\mathfrak {S}}}$ .",
"Let $\\psi $ be a state on ${\\mathfrak {A}}$ which admits a characteristic sequence $(\\Delta _n)_n$ in ${\\mathfrak {S}}$ .",
"Then, we have $\\lim _{n\\rightarrow \\infty } \\Vert \\pi (\\Delta _n)^*(\\Pi (a)-\\pi (a))\\pi (\\Delta _n)\\Vert =0$ for every $a\\in {\\mathfrak {A}}$ .",
"In view of part (3) of Theorem REF , this result can be viewed as supporting evidence for Arveson's conjecture, as it establishes a local form of the desired statement.",
"Note that although the conclusion is merely local, so is the hypothesis.",
"Thus, the theorem appears to be relevant even in cases where the conjecture has been verified.",
"Moreover, it generalizes Arveson's local hyperrigidity theorem that applies to commutative $\\mathrm {C}^*$ -algebras, as we show in detail at the end of the paper.",
"Acknowledgements.",
"The author wishes to thank the referee for several insightful suggestions."
],
[
" Operator systems and completely positive maps",
"Throughout the paper, ${\\mathcal {H}}$ will denote a Hilbert space and $B({\\mathcal {H}})$ will denote the bounded linear operators on it.",
"We now briefly review the basics of operator systems and completely positive maps.",
"The reader may wish to consult for greater detail.",
"Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra.",
"A unital self-adjoint subspace ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ is called an operator system.",
"As is well-known, it is possible to define operator systems in a more abstract fashion that does not rely on an embedding inside of a unital $\\mathrm {C}^*$ -algebra , but the previous concrete definition is sufficient for our purposes.",
"For each $n\\in {\\mathbb {N}}$ , we may view the space ${\\mathbb {M}}_n({\\mathfrak {S}})$ of $n\\times n$ matrices with entries from ${\\mathfrak {S}}$ as a self-adjoint subspace of ${\\mathbb {M}}_n({\\mathfrak {A}})$ , and in particular there is an associated notion of positivity for elements of ${\\mathbb {M}}_n({\\mathfrak {S}})$ .",
"A linear map $\\varphi :{\\mathfrak {S}}\\rightarrow B({\\mathcal {H}})$ is said to be completely positive if $\\varphi ^{(n)}$ is positive for every $n\\in {\\mathbb {N}}$ .",
"Here, given $n\\in {\\mathbb {N}}$ , we denote by $\\varphi ^{(n)}:{\\mathbb {M}}_n({\\mathfrak {S}})\\rightarrow B({\\mathcal {H}}^{(n)})$ the induced linear map defined as $\\varphi ^{(n)}([s_{ij}]_{i,j})=[\\varphi (s_{ij})_{i,j}], \\quad [s_{ij}]_{i,j}\\in {\\mathbb {M}}_n({\\mathfrak {S}}).$ If $\\varphi $ is a unital completely positive map on ${\\mathfrak {A}}$ , then it satisfies the Schwarz inequality: $\\varphi ^{(n)}(A)^*\\varphi ^{(n)}(A)\\le \\varphi ^{(n)}(A^*A)$ for each $A\\in {\\mathbb {M}}_n({\\mathfrak {A}})$ and $n\\in {\\mathbb {N}}$ .",
"We will use this inequality repeatedly throughout the paper."
],
[
"Extensions of completely positive maps and hyperrigidity",
"Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Given a unital completely positive map $\\varphi :{\\mathfrak {S}}\\rightarrow B({\\mathcal {H}})$ , by virtue of Arveson's extension theorem there is a unital completely positive map $\\Phi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ which extends $\\varphi $ .",
"In general, the extension is not unique.",
"Accordingly, we say that a unital completely positive map $\\psi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ has the unique extension property with respect to ${\\mathfrak {S}}$ if it is the unique completely positive extension of $\\psi |_{{\\mathfrak {S}}}$ to ${\\mathfrak {A}}$ .",
"We advise the reader to exercise some care: in other works (such as ) the use of the terminology “unique extension property\" is reserved for $*$ -representations of $\\mathrm {C}^*({\\mathfrak {S}})$ .",
"In our context, we found our more lenient definition to be more convenient, but the reader should keep this discrepancy in mind throughout.",
"The unique extension property is closely related to the phenomenon of hyperrigidity for operator systems that was defined in the introduction.",
"Indeed, it follows from that an operator system ${\\mathfrak {S}}$ is hyperrigid if and only if every unital $*$ -representation of $\\mathrm {C}^*({\\mathfrak {S}})$ has the unique extension property with respect to ${\\mathfrak {S}}$ .",
"In the special case where we have an irreducible $*$ -representation $\\pi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ which has the unique extension property with respect to ${\\mathfrak {S}}$ , then we say that $\\pi $ is a boundary representation for ${\\mathfrak {S}}$.",
"This is motivated by the definition of the Choquet boundary of a uniform algebra, as described in the introduction.",
"A basic problem is to determine to which extent the boundary representations of $\\mathrm {C}^*({\\mathfrak {S}})$ determine the hyperrigidity of ${\\mathfrak {S}}$ , in the spirit of the theorems of Korovkin and Šaškin mentioned earlier.",
"This is the content of the following conjecture formulated in .",
"Arveson's hyperrigidity conjecture.",
"An operator system ${\\mathfrak {S}}$ is hyperrigid if and only if every irreducible $*$ -representation of $\\mathrm {C}^*({\\mathfrak {S}})$ is a boundary representation for ${\\mathfrak {S}}$ .",
"In the case where ${\\mathfrak {S}}$ is separable and $\\mathrm {C}^*({\\mathfrak {S}})$ is commutative, Arveson managed to establish a local version of this conjecture .",
"Since this is one of the main motivations for our work, we wish to state this particular result precisely.",
"First, we set up some notation.",
"Let $(X,\\rho )$ be a compact metric space and let ${\\mathfrak {S}}\\subset C(X)$ be an operator system (a function system) such that $\\mathrm {C}^*({\\mathfrak {S}})=C(X)$ .",
"We know that the irreducible $*$ -representations of $C(X)$ are precisely the characters of evaluation at points of $X$ .",
"In particular, the character of evaluation at $x\\in X$ is a boundary representation for ${\\mathfrak {S}}$ if and only if $x$ lies in the Choquet boundary of ${\\mathfrak {S}}$ (the Choquet boundary of a function system is defined in a manner completely analogous to that of a uniform algebra).",
"Let ${\\mathcal {H}}$ be a separable Hilbert space and let $\\pi :\\mathrm {C}^*({\\mathfrak {S}})\\rightarrow B({\\mathcal {H}})$ be a unital $*$ -representation.",
"Put ${\\mathfrak {M}}_\\pi =\\pi (\\mathrm {C}^*({\\mathfrak {S}}))^{\\prime \\prime }$ .",
"Then, there is a weak-$*$ homeomorphic $*$ -isomorphism $\\Theta :{\\mathfrak {M}}_\\pi \\rightarrow L^\\infty (X,\\mu )$ for some Borel probability measure $\\mu $ with support equal to $X$ .",
"For each $\\Omega \\subset X$ , we let $\\chi _\\Omega \\in L^\\infty (X,\\mu )$ denote the characteristic function of $\\Omega $ .",
"Moreover, for $x\\in X$ and $\\delta >0$ , we let $B(x,\\delta )=\\lbrace y\\in X:\\rho (x,y)<\\delta \\rbrace .$ Then, for $x\\in X$ and $\\delta >0$ we put $E_\\pi (x,\\delta )=\\Theta ^{-1}(\\chi _{B(x,\\delta )})\\in {\\mathfrak {M}}_\\pi .$ The following is a reformulation of , which can be easily extracted from its proof.",
"Theorem 2.1 Let $X$ be a compact metric space and let ${\\mathfrak {S}}\\subset C(X)$ be an operator system such that $\\mathrm {C}^*({\\mathfrak {S}})=C(X)$ .",
"Let ${\\mathcal {H}}$ be a separable Hilbert space, let $\\pi :\\mathrm {C}^*({\\mathfrak {S}})\\rightarrow B({\\mathcal {H}})$ be a unital $*$ -representation and let $\\Pi :\\mathrm {C}^*({\\mathfrak {S}})\\rightarrow B({\\mathcal {H}})$ be a unital completely positive extension of $\\pi |_{{\\mathfrak {S}}}$ .",
"Let $x\\in X$ be a point in the Choquet boundary of ${\\mathfrak {S}}$ .",
"Then, $\\lim _{\\delta \\rightarrow 0}\\Vert (\\pi (a)-\\Pi (a))E_\\pi (x,\\delta ) \\Vert =0, \\quad a\\in C(X).$ We should point out that even though the hyperrigidity conjecture has been verified recently in the commutative case , it is not entirely clear to us how to obtain Theorem REF as a consequence of that seemingly stronger result."
],
[
"States and support projections",
"Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"We will frequently be dealing with scalar valued unital completely positive maps on ${\\mathfrak {S}}$ , which are called states.",
"We will denote by ${\\mathcal {S}}({\\mathfrak {S}})$ the state space of ${\\mathfrak {S}}$ .",
"A state on ${\\mathfrak {S}}$ is pure if it is an extreme point of ${\\mathcal {S}}({\\mathfrak {S}})$ .",
"Assume now that we have a state $\\varphi $ defined on the unital $\\mathrm {C}^*$ -algebra ${\\mathfrak {A}}$ .",
"The associated GNS representation is the essentially unique triple $(\\sigma _\\varphi , {\\mathfrak {H}}_\\varphi , \\xi _\\varphi )$ consisting of a Hilbert space ${\\mathfrak {H}}_\\varphi $ , a unital $*$ -representation $\\sigma _\\varphi :{\\mathfrak {A}}\\rightarrow B({\\mathfrak {H}}_\\varphi )$ and a cyclic unit vector $\\xi _\\varphi \\in {\\mathfrak {H}}_\\varphi $ with the property that $\\varphi (a)=\\langle \\sigma _\\varphi (a) \\xi _\\varphi , \\xi _\\varphi \\rangle , \\quad a\\in {\\mathfrak {A}}.$ It is well-known that $\\sigma _\\varphi $ is irreducible precisely when $\\varphi $ is a pure state on ${\\mathfrak {A}}$ .",
"One important consequence of the Schwarz inequality applied to the state $\\varphi $ is that if an element $a\\in {\\mathfrak {A}}$ is such that $\\Vert a\\Vert =1$ and $|\\varphi (a)|=1$ , then $|\\varphi (a)|^2=\\varphi (a^*a)$ and consequently $a$ belongs to the multiplicative domain of $\\varphi $, in the sense that $\\varphi (ba)=\\varphi (b)\\varphi (a), \\quad b\\in {\\mathfrak {A}}.$ This classical observation is due to Choi and will be used repeatedly throughout.",
"Another consequence of the Schwarz inequality is that the set $\\lbrace a\\in {\\mathfrak {A}}:\\varphi (a^*a)=0\\rbrace $ coincides with $\\lbrace a\\in {\\mathfrak {A}}: \\varphi (ba)=0, b\\in {\\mathfrak {A}}\\rbrace $ and in particular it is a norm closed left ideal of ${\\mathfrak {A}}$ .",
"Recall that the bidual ${\\mathfrak {A}}^{**}$ can be identified with the universal enveloping von Neumann algebra of ${\\mathfrak {A}}$ , and it contains ${\\mathfrak {A}}$ as a $\\mathrm {C}^*$ -subalgebra .",
"There is a unique weak-$*$ continuous state $\\widehat{\\varphi }$ on ${\\mathfrak {A}}^{**}$ which extends $\\varphi $ .",
"Then, the set ${\\mathfrak {N}}_\\varphi =\\lbrace x \\in {\\mathfrak {A}}^{**}:\\widehat{\\varphi }(x^* x)=0\\rbrace =\\lbrace x\\in {\\mathfrak {A}}^{**}:\\widehat{\\varphi }(y x)=0, y\\in {\\mathfrak {A}}^{**}\\rbrace $ is a weak-$*$ closed left ideal of the von Neumann algebra ${\\mathfrak {A}}^{**}$ .",
"We infer that there is a projection ${\\mathfrak {s}}_\\varphi \\in {\\mathfrak {A}}^{**}$ such that ${\\mathfrak {N}}_\\varphi ={\\mathfrak {A}}^{**}(I-{\\mathfrak {s}}_\\varphi ).$ This projection ${\\mathfrak {s}}_\\varphi $ is called the support projection of $\\varphi $ .",
"We note that $\\widehat{\\varphi }({\\mathfrak {s}}_\\varphi )=1$ and hence ${\\mathfrak {s}}_\\varphi $ lies in the multiplicative domain of $\\widehat{\\varphi }$ .",
"Thus, $\\widehat{\\varphi }({\\mathfrak {s}}_\\varphi x {\\mathfrak {s}}_\\varphi )=\\widehat{\\varphi }(x), \\quad x\\in {\\mathfrak {A}}^{**}.$ We note also that the restriction of $\\widehat{\\varphi }$ to ${\\mathfrak {s}}_\\varphi {\\mathfrak {A}}^{**} {\\mathfrak {s}}_\\varphi $ is faithful, that is $\\widehat{\\varphi }(x^* x)>0$ for every non-zero element $x\\in {\\mathfrak {A}}^{**}{\\mathfrak {s}}_\\varphi $ .",
"The next result elucidates some properties of support projections of pure states.",
"Lemma 2.2 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let $\\omega $ be a pure state on ${\\mathfrak {A}}$ .",
"Then, ${\\mathfrak {s}}_\\omega {\\mathfrak {A}}^{**} {\\mathfrak {s}}_\\omega ={\\mathbb {C}}{\\mathfrak {s}}_\\omega $ .",
"In particular, if $\\varphi $ is a state on ${\\mathfrak {A}}$ with $\\widehat{\\varphi }({\\mathfrak {s}}_\\omega )=1$ , then $\\varphi =\\omega $ .",
"It is shown in the proof of that ${\\mathfrak {s}}_\\omega {\\mathfrak {A}}^{**} {\\mathfrak {s}}_\\omega ={\\mathbb {C}}{\\mathfrak {s}}_\\omega $ .",
"In particular, we see that ${\\mathfrak {s}}_\\omega x {\\mathfrak {s}}_\\omega =\\widehat{\\omega }(x) {\\mathfrak {s}}_\\omega , \\quad x\\in {\\mathfrak {A}}^{**}.$ If $\\widehat{\\varphi }({\\mathfrak {s}}_\\omega )=1$ , then ${\\mathfrak {s}}_\\omega $ belongs to the multiplicative domain of $\\widehat{\\varphi }$ and we find $\\widehat{\\omega }(x)=\\widehat{\\omega }(x)\\widehat{\\varphi }({\\mathfrak {s}}_\\omega )=\\widehat{\\varphi }({\\mathfrak {s}}_\\omega x {\\mathfrak {s}}_\\omega )=\\widehat{\\varphi }(x)$ for every $x\\in {\\mathfrak {A}}^{**}$ .",
"We conclude that $\\varphi =\\omega $ .",
"The next fact we will need concerning support projections is likely well-known to experts, but we could not locate a convenient reference.",
"Accordingly, we provide a detailed proof, as support projections will play a key role in our analysis of peaking states in the next section.",
"Before we state the result, recall that two states $\\varphi ,\\psi $ on ${\\mathfrak {A}}$ are said to be orthogonal if $\\Vert \\psi -\\varphi \\Vert =2$ .",
"This is equivalent to their support projections ${\\mathfrak {s}}_\\varphi ,{\\mathfrak {s}}_\\psi \\in {\\mathfrak {A}}^{**}$ being orthogonal .",
"Proposition 2.3 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra.",
"The following statements are equivalent.",
"${\\mathfrak {A}}$ is commutative.",
"Any two distinct pure states are orthogonal.",
"If ${\\mathfrak {A}}$ is commutative, then the pure states on ${\\mathfrak {A}}$ are characters.",
"Since the character space of ${\\mathfrak {A}}$ is compact and Hausdorff, Urysohn's lemma implies that distinct characters are orthogonal.",
"Conversely, recall that the direct sum of all irreducible $*$ -representations of ${\\mathfrak {A}}$ is injective .",
"Thus, if ${\\mathfrak {A}}$ is not commutative, we can find an irreducible $*$ -representation $\\pi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ where ${\\mathcal {H}}$ has dimension at least two.",
"Let $\\xi ,\\eta \\in {\\mathcal {H}}$ be two unit vectors which are not orthogonal.",
"Define two states $\\omega _\\xi $ and $\\omega _\\eta $ on ${\\mathfrak {A}}$ as $\\omega _\\xi (a)=\\langle \\pi (a)\\xi ,\\xi \\rangle \\quad \\text{and}\\quad \\omega _\\eta (a)=\\langle \\pi (a)\\eta ,\\eta \\rangle $ for every $a\\in {\\mathfrak {A}}$ .",
"Since $\\pi $ is irreducible, both $\\xi $ and $\\eta $ are cyclic vectors and it is easily verified that $(\\pi ,{\\mathcal {H}},\\xi )$ and $(\\pi ,{\\mathcal {H}},\\eta )$ are unitarily equivalent to the GNS representations of $\\omega _\\xi $ and $\\omega _\\eta $ respectively.",
"We conclude that $\\omega _\\xi $ and $\\omega _\\eta $ are pure.",
"As shown in the proof of , we may represent ${\\mathfrak {A}}^{**}$ faithfully on some Hilbert space ${\\mathfrak {H}}$ with the property that there are non-orthogonal unit vectors $\\xi ^{\\prime },\\eta ^{\\prime }\\in {\\mathfrak {H}}$ such that $\\omega _\\xi (a)=\\langle a\\xi ^{\\prime },\\xi ^{\\prime } \\rangle \\quad \\text{and}\\quad \\omega _\\eta (a)=\\langle a\\eta ^{\\prime },\\eta ^{\\prime } \\rangle $ for every $a\\in {\\mathfrak {A}}$ .",
"In particular, we have that $\\widehat{\\omega _\\xi }(x)=\\langle x\\xi ^{\\prime },\\xi ^{\\prime } \\rangle \\quad \\text{and}\\quad \\widehat{\\omega _\\eta }(x)=\\langle x\\eta ^{\\prime },\\eta ^{\\prime } \\rangle $ for every $x\\in {\\mathfrak {A}}^{**}$ .",
"Using that $\\widehat{\\omega _\\xi }({\\mathfrak {s}}_{\\omega _\\xi })=1$ , we find that $\\langle {\\mathfrak {s}}_{\\omega _\\xi }\\xi ^{\\prime },\\xi ^{\\prime } \\rangle =1$ and thus $\\xi ^{\\prime }\\in {\\mathfrak {s}}_{\\omega _\\xi }{\\mathfrak {H}}$ .",
"Likewise, we see that $\\eta ^{\\prime }\\in {\\mathfrak {s}}_{\\omega _\\eta }{\\mathfrak {H}}$ .",
"Since $\\xi ^{\\prime }$ and $\\eta ^{\\prime }$ are not orthogonal, we infer that the support projections ${\\mathfrak {s}}_{\\omega _\\xi }$ and ${\\mathfrak {s}}_{\\omega _\\eta }$ are not orthogonal, and thus neither are the states $\\omega _\\xi $ and $\\omega _\\eta $ ."
],
[
"Peaking states",
"In this section, we consider peaking phenomena in non-commutative contexts.",
"There has been some previous work done on this topic; we mention for instance the notions of peaking $*$ -representations , or of peaking projections .",
"We opt here to focus our attention on states.",
"After having characterized peaking states, we will clarify their relationship to peaking representations and peaking projections.",
"Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"We say that a state $\\psi $ on ${\\mathfrak {A}}$ is ${\\mathfrak {S}}$ -peaking if there is a self-adjoint element $s\\in {\\mathfrak {S}}$ such that $\\Vert s\\Vert =1$ and with the property that $\\psi (s)=1>|\\varphi (s)|$ for every state $\\varphi $ on ${\\mathfrak {A}}$ such that $\\varphi \\ne \\psi $ .",
"We then say that $\\psi $ peaks on $s$ .",
"This simple property imposes strong conditions on the state $\\psi $ , as we now show.",
"Lemma 3.1 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Then, every state on ${\\mathfrak {A}}$ which is ${\\mathfrak {S}}$ -peaking must be pure and must have the unique extension property with respect to ${\\mathfrak {S}}$ .",
"Let $\\psi $ be a state on ${\\mathfrak {A}}$ which is ${\\mathfrak {S}}$ -peaking, so that we can find a self-adjoint element $s\\in {\\mathfrak {S}}$ with $\\Vert s\\Vert =1$ and such that $\\psi (s)=1>| \\varphi (s)|$ for every state $\\varphi $ on ${\\mathfrak {A}}$ such that $\\varphi \\ne \\psi $ .",
"Fix such a state $\\varphi $ .",
"In particular, we observe that $\\psi |_{{\\mathfrak {S}}}\\ne \\varphi |_{{\\mathfrak {S}}}$ .",
"This shows that $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ .",
"Moreover, if we write $\\psi =\\frac{1}{2}(\\varphi _1+\\varphi _2)$ for some states $\\varphi _1,\\varphi _2$ on ${\\mathfrak {A}}$ , then we must have $\\varphi _1(s)=\\varphi _2(s)=1$ .",
"By choice of the element $s$ , this forces $\\psi =\\varphi _1=\\varphi _2$ and thus $\\psi $ is pure.",
"We see that a (necessarily pure) state on ${\\mathfrak {A}}$ which is ${\\mathfrak {S}}$ -peaking must admit a so-called determining element in ${\\mathfrak {S}}$ .",
"This notion apparently has a quantum mechanical interpretation as discussed in , and it is reminiscent of the following familiar convexity concept.",
"Let $X$ be a normed space and let $Q$ be a weak-$*$ closed convex subset of the closed unit ball of the dual space $X^*$ .",
"Assume that $0\\in Q$ .",
"Recall that a non-zero element $\\Lambda _0\\in Q$ is a weak-$*$ exposed point if there is $x_0\\in X$ with the property that $\\operatorname{Re}\\Lambda _0(x_0)=1>\\operatorname{Re}\\Lambda (x_0)$ whenever $\\Lambda \\in Q$ and $\\Lambda \\ne \\Lambda _0$ .",
"Arguing as in the proof of Lemma REF , it is straightforward to verify that weak-$*$ exposed points are necessarily extreme points.",
"Using this concept, we can characterize ${\\mathfrak {S}}$ -peaking states.",
"Theorem 3.2 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $\\psi $ be a state on ${\\mathfrak {A}}$ .",
"Then, the following statements are equivalent.",
"The state $\\psi $ is ${\\mathfrak {S}}$ -peaking.",
"The state $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ , and the restriction $\\psi |_{{\\mathfrak {S}}}$ is a weak-$*$ exposed point of the set of self-adjoint functionals on ${\\mathfrak {S}}$ with norm at most 1.",
"Throughout the proof, we let $Q$ denote the set of self-adjoint functionals on ${\\mathfrak {S}}$ with norm at most 1.",
"Assume that $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ and that $\\psi |_{{\\mathfrak {S}}}$ is a weak-$*$ exposed point of $Q$ .",
"Then, there is $x_0\\in {\\mathfrak {S}}$ with the property that $\\operatorname{Re}\\psi (x_0)=1>\\operatorname{Re}\\Lambda (x_0)$ for every $\\Lambda \\in Q$ such that $\\Lambda \\ne \\psi |_{{\\mathfrak {S}}}$ .",
"Upon replacing $x_0$ by $(x_0+x_0^*)/2$ if necessary, we may assume that $x_0$ is self-adjoint.",
"Hence, we see that $\\psi (x_0)=1>\\Lambda (x_0)$ for every $\\Lambda \\in Q$ such that $\\Lambda \\ne \\psi |_{{\\mathfrak {S}}}$ .",
"Since $-\\Lambda \\in Q$ whenever $\\Lambda \\in Q$ , we see that $\\psi (x_0)=1>|\\Lambda (x_0)|$ for every $\\Lambda \\in Q$ such that $\\Lambda \\ne \\pm \\psi |_{{\\mathfrak {S}}}$ .",
"Thus $|\\varphi (x_0)|\\le 1$ if $\\varphi $ is a state on ${\\mathfrak {S}}$ , which implies that $\\Vert x_0\\Vert =1$ as $x_0$ is assumed to be self-adjoint.",
"Finally, the unique extension property of $\\psi $ with respect to ${\\mathfrak {S}}$ along with Equation (REF ) shows that $\\psi $ is ${\\mathfrak {S}}$ -peaking.",
"Conversely, assume that $\\psi $ is ${\\mathfrak {S}}$ -peaking, so that there is a self-adjoint element $x_0\\in {\\mathfrak {S}}$ with $\\Vert x_0\\Vert = 1$ such that $\\psi (x_0)=1>|\\varphi (x_0)|$ whenever $\\varphi $ is a state on ${\\mathfrak {A}}$ with $\\varphi \\ne \\psi $ .",
"By Lemma REF , we see that $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ .",
"It remains only to show that the restriction $\\psi |_{{\\mathfrak {S}}}$ is a weak-$*$ exposed point of $Q$ .",
"For that purpose, we will show that $|\\Lambda (x_0)|<1$ whenever $\\Lambda \\in Q$ with $\\Lambda \\ne \\pm \\psi |_{{\\mathfrak {S}}}$ .",
"Clearly, it suffices to deal with the case where $\\Vert \\Lambda \\Vert =1$ and $\\Lambda $ is not of the form $\\pm \\varphi $ for some state $\\varphi $ on ${\\mathfrak {S}}$ .",
"Let $\\Theta $ be a functional on ${\\mathfrak {A}}$ with $\\Vert \\Theta \\Vert =1$ which agrees with $\\Lambda $ on ${\\mathfrak {S}}$ .",
"Upon replacing $\\Theta $ by its real part if necessary, we may assume that $\\Theta $ is self-adjoint.",
"By means of the Jordan decomposition , we may write $\\Theta =\\Theta _1-\\Theta _2$ for some positive linear functionals $\\Theta _1$ and $\\Theta _2$ on ${\\mathfrak {A}}$ with the property that $1=\\Vert \\Theta \\Vert =\\Vert \\Theta _1\\Vert +\\Vert \\Theta _2\\Vert .$ Since $\\Lambda $ is not of the form $\\pm \\varphi $ for some state $\\varphi $ on ${\\mathfrak {S}}$ , we see that $\\pm \\Theta _k|_{{\\mathfrak {S}}}\\ne \\Lambda $ and thus $\\Vert \\Theta _k\\Vert \\ne 0$ for each $k=1,2$ .",
"Note now that $\\Theta _k/\\Vert \\Theta _k\\Vert $ is a state on ${\\mathfrak {A}}$ for each $k=1,2$ , and $\\Theta =\\Vert \\Theta _1\\Vert \\frac{\\Theta _1}{\\Vert \\Theta _1\\Vert }+\\Vert \\Theta _2\\Vert \\frac{(-\\Theta _2)}{\\Vert \\Theta _2\\Vert }$ whence $\\Lambda =\\Vert \\Theta _1\\Vert \\frac{\\Theta _1|_{{\\mathfrak {S}}}}{\\Vert \\Theta _1\\Vert }+\\Vert \\Theta _2\\Vert \\frac{(-\\Theta _2|_{{\\mathfrak {S}}})}{\\Vert \\Theta _2\\Vert }.$ Using that $\\Vert \\Lambda \\Vert =1$ , a straightforward verification yields that either $\\frac{\\Theta _1|_{{\\mathfrak {S}}}}{\\Vert \\Theta _1\\Vert }\\ne \\psi |_{{\\mathfrak {S}}} \\quad \\text{or}\\quad \\frac{\\Theta _2|_{{\\mathfrak {S}}}}{\\Vert \\Theta _2\\Vert }\\ne \\psi |_{{\\mathfrak {S}}}.$ By choice of $x_0$ , we conclude that $|\\Lambda (x_0)|\\le \\Vert \\Theta _1\\Vert \\frac{|\\Theta _1(x_0)|}{\\Vert \\Theta _1\\Vert }+\\Vert \\Theta _2\\Vert \\frac{|\\Theta _2(x_0)|}{\\Vert \\Theta _2\\Vert }<\\Vert \\Theta _1\\Vert +\\Vert \\Theta _2\\Vert =1.$ It is natural to wonder just how common states having the unique extension property happen to be.",
"Irreducible $*$ -representations with the unique extension property with respect to an operator system are abundant enough to generate the $\\mathrm {C}^*$ -envelope ,,, but a corresponding statement for states is unknown to us.",
"In fact, show that similar questions are closely related to the hyperrigidity conjecture itself.",
"If a pure state $\\psi $ on ${\\mathfrak {A}}$ has the unique extension property with respect to ${\\mathfrak {S}}$ , then the restriction $\\psi |_{{\\mathfrak {S}}}$ is pure as well .",
"By the previous theorem, determining whether $\\psi $ is ${\\mathfrak {S}}$ -peaking thus reduces to distinguishing weak-$*$ exposed points among the extreme points in the set of self-adjoint functionals on ${\\mathfrak {S}}$ with norm at most 1.",
"The following additional remarks appear relevant in view of Theorem REF .",
"First, a classical result of Klee (see also for a detailed proof) guarantees that weak-$*$ exposed points are plentiful.",
"Indeed, the set of weak-$*$ exposed points of a weak-$*$ closed convex subset is weak-$*$ dense in the set extreme points.",
"Second, we mention that in the classical case of a uniform algebra, it is known () that the Choquet boundary coincide with the set of peak points.",
"The multiplicative structure of the algebra is crucial here, as the equality of these two sets fails for general function systems .",
"Nevertheless, the peak points are still abundant; they are in fact dense in the Choquet boundary .",
"We exhibit some non-commutative examples related to these ideas.",
"Example 3.3 Let ${\\mathcal {H}}$ be a Hilbert space and let ${\\mathfrak {A}}\\subset B({\\mathcal {H}})$ be a unital $\\mathrm {C}^*$ -algebra which contains the ideal ${\\mathfrak {K}}$ of compact operators on ${\\mathcal {H}}$ .",
"Assume that ${\\mathfrak {A}}/{\\mathfrak {K}}$ is not one-dimensional and put ${\\mathfrak {S}}={\\mathfrak {K}}+{\\mathbb {C}}I$ .",
"Repeating the argument of , we see that the only pure states on ${\\mathfrak {A}}$ that have the unique extension property with respect to ${\\mathfrak {S}}$ are the vector states on ${\\mathfrak {A}}$ .",
"We proceed to verify that these vector states are always ${\\mathfrak {S}}$ -peaking, regardless of whether ${\\mathfrak {A}}/{\\mathfrak {K}}$ is one-dimensional or not.",
"Let $\\xi \\in {\\mathcal {H}}$ be a unit vector and let $\\omega $ be the pure vector state on ${\\mathfrak {A}}$ defined as $\\omega (a)=\\langle a\\xi ,\\xi \\rangle , \\quad a\\in {\\mathfrak {A}}.$ Let $s\\in {\\mathfrak {K}}$ be the rank-one projection onto ${\\mathbb {C}}\\xi $ .",
"Clearly, we have that $\\omega (s)=1$ .",
"Let now $\\varphi $ be a state on ${\\mathfrak {A}}$ .",
"Standard facts about the representation theory of $\\mathrm {C}^*$ -algebras (see the discussion preceding ) then imply that the GNS representation $\\sigma _\\varphi $ is unitarily equivalent to $\\operatorname{id}^{(\\gamma )}\\oplus \\pi _Q$ where $\\gamma $ is some cardinal number and $\\pi _Q$ is a unital $*$ -representation of ${\\mathfrak {A}}$ which annihilates ${\\mathfrak {K}}$ .",
"This shows that there is a positive trace class operator $T\\in B({\\mathcal {H}})$ and a positive linear functional $\\theta $ on ${\\mathfrak {A}}$ satisfying ${\\mathfrak {K}}\\subset \\ker \\theta $ with the property that $\\varphi (a)=\\operatorname{tr}(aT)+\\theta (a), \\quad a\\in {\\mathfrak {A}}.$ In particular, we see that $\\varphi (s)=\\operatorname{tr}(sT)$ , whence $|\\varphi (s)|<1$ unless $\\theta =0$ and $T=s$ , which is equivalent to $\\varphi =\\omega $ .",
"Hence $\\omega $ is ${\\mathfrak {S}}$ -peaking.",
"Example 3.4 Let ${\\mathbb {M}}_2$ denote the $2\\times 2$ complex matrices and consider the standard matrix unit $E_{12}\\in {\\mathbb {M}}_2$ .",
"Let ${\\mathfrak {S}}\\subset {\\mathbb {M}}_2$ be the operator system generated by $I$ and $E_{12}$ .",
"Note that ${\\mathbb {M}}_2=\\mathrm {C}^*({\\mathfrak {S}})$ .",
"It is well-known that up to unitary equivalence, the only irreducible $*$ -representation of ${\\mathbb {M}}_2$ is the identity representation, so that the pure states on ${\\mathbb {M}}_2$ are the vector states.",
"For each unit vector $\\xi \\in {\\mathbb {C}}^2$ , we let $\\omega _\\xi $ be the state on ${\\mathbb {M}}_2$ defined as $\\omega _\\xi (a)=\\langle a\\xi ,\\xi \\rangle , \\quad a\\in {\\mathbb {M}}_2.$ Observe that $\\omega _\\xi =\\omega _{\\xi ^{\\prime }}$ if and only if $\\xi =\\alpha \\xi ^{\\prime }$ for some $\\alpha \\in {\\mathbb {C}}$ with $|\\alpha |=1$ .",
"Let $\\lbrace e_1,e_2\\rbrace $ be the canonical orthonormal basis of ${\\mathbb {C}}^2$ and assume that $\\xi =\\xi _1 e_1+\\xi _2 e_2$ is a unit vector, where $\\xi _1,\\xi _2\\in {\\mathbb {C}}$ .",
"We wish to determine when $\\omega _\\xi $ has the unique extension property with respect to ${\\mathfrak {S}}$ and when it is ${\\mathfrak {S}}$ -peaking.",
"We distinguish two cases.",
"First, suppose that $\\xi _1$ is not of the form $\\alpha \\overline{\\xi _2}$ for some $\\alpha \\in {\\mathbb {C}}$ with $|\\alpha |=1$ .",
"Consider the unit vector $\\xi ^{\\prime }=\\overline{\\xi _2}e_1+\\overline{\\xi _1}e_2.$ Then, $\\omega _{\\xi }\\ne \\omega _{\\xi ^{\\prime }}$ .",
"However, we note that $\\langle E_{12}\\xi ,\\xi \\rangle =\\xi _2 \\overline{\\xi _1}=\\langle E_{12}\\xi ^{\\prime },\\xi ^{\\prime }\\rangle $ so that $\\omega _\\xi (E_{12})=\\omega _{\\xi ^{\\prime }}(E_{12})$ .",
"This implies that $\\omega _{\\xi }|_{{\\mathfrak {S}}}=\\omega _{\\xi ^{\\prime }}|_{{\\mathfrak {S}}}$ , and we infer that $\\omega _\\xi $ does not have the unique extension property with respect to ${\\mathfrak {S}}$ , and in particular it is not ${\\mathfrak {S}}$ -peaking by Lemma REF .",
"Second, suppose that $\\xi _1=\\alpha \\overline{\\xi _2}$ for some $\\alpha \\in {\\mathbb {C}}$ with $|\\alpha |=1$ .",
"Then $\\xi _2=\\alpha \\overline{\\xi _1}$ and $|\\xi _1|=|\\xi _2|=1/\\sqrt{2}.$ We infer $\\xi _1=\\beta /\\sqrt{2}, \\quad \\xi _2=\\alpha \\overline{\\beta }/\\sqrt{2}$ for some $\\beta \\in {\\mathbb {C}}$ with $|\\beta |=1$ .",
"We now claim that $\\omega _\\xi $ is ${\\mathfrak {S}}$ -peaking.",
"Indeed, consider $s=\\frac{1}{2}\\left(I+\\overline{\\alpha }\\beta ^2E_{12}+\\alpha \\overline{\\beta }^2E_{21} \\right)\\in {\\mathfrak {S}}.$ An easy calculation shows that $s$ is the rank-one projection onto ${\\mathbb {C}}\\xi $ and thus that $\\omega _\\xi (a)=\\operatorname{tr}(as), \\quad a\\in {\\mathbb {M}}_2.$ It is clear that $\\omega _\\xi (s)=1$ .",
"Let $\\varphi $ be a state on ${\\mathbb {M}}_2$ .",
"Then, there is a positive element $T\\in {\\mathbb {M}}_2$ such that $\\operatorname{tr}T=1$ and $\\varphi (a)=\\operatorname{tr}(aT), \\quad a\\in {\\mathbb {M}}_2.$ If $|\\varphi (sT)|=1$ , then a moment's thought reveals that we must have $s=T$ which implies that $\\varphi =\\omega _\\xi $ .",
"Thus, $\\omega _\\xi $ is ${\\mathfrak {S}}$ -peaking and in particular it has the unique extension property with respect to ${\\mathfrak {S}}$ by Lemma REF .",
"We have shown that every pure state on ${\\mathbb {M}}_2$ which has the unique extension property with respect to ${\\mathfrak {S}}$ must be ${\\mathfrak {S}}$ -peaking.",
"Inspired by , we now make the following definition.",
"Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"A projection $p\\in {\\mathfrak {A}}^{**}$ is said to be ${\\mathfrak {S}}$ -peaking if there is a self-adjoint element $s\\in {\\mathfrak {S}}$ with $\\Vert s\\Vert =1$ such that $ sp=p$ and $| \\varphi (s)|<1$ whenever $\\varphi $ is a state on ${\\mathfrak {A}}$ with support projection ${\\mathfrak {s}}_\\varphi \\in {\\mathfrak {A}}^{**}$ orthogonal to $p$ .",
"In this case, we also have $ps=p$ and we say that $p$ peaks on $s$ .",
"We warn the reader that our notion of ${\\mathfrak {S}}$ -peaking projection is slightly different from the corresponding one given in .",
"We feel our version is more natural in the context of operator systems.",
"This deviation from the convention of is perhaps partly justified by the fact, which we will show next, that with our definition in its current form we have the very natural equivalence between a pure state $\\omega $ and its support projection ${\\mathfrak {s}}_\\omega $ being ${\\mathfrak {S}}$ -peaking.",
"At first glance, this is not completely obvious, as an ${\\mathfrak {S}}$ -peaking projection only offers control over a restricted subset of states.",
"Indeed, assuming that the projection ${\\mathfrak {s}}_\\omega $ is ${\\mathfrak {S}}$ -peaking on $s$ , we know that $|\\varphi (s)|<1$ whenever $\\varphi $ is a state on ${\\mathfrak {A}}$ with ${\\mathfrak {s}}_\\varphi {\\mathfrak {s}}_\\omega =0$ .",
"However, to guarantee that the state $\\omega $ is ${\\mathfrak {S}}$ -peaking, we need to know that $|\\varphi (s)|<1$ for every state $\\varphi $ on ${\\mathfrak {A}}$ which is merely distinct from $\\omega $ .",
"In light of Proposition REF , it may thus appear as though the equivalence we seek could only hold in the case where ${\\mathfrak {A}}$ is commutative.",
"Perhaps surprisingly, this is not the case.",
"To show this fact we need the following, which is an adaptation of to our alternative definition of peaking projection.",
"The proof is almost identical.",
"Lemma 3.5 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $p\\in {\\mathfrak {A}}^{**}$ be an ${\\mathfrak {S}}$ -peaking projection which peaks on the self-adjoint element $s\\in {\\mathfrak {S}}$ .",
"Then, the sequence $(s^n)_n$ converges to $p$ in the weak-$*$ topology of ${\\mathfrak {A}}^{**}$ .",
"We may assume that ${\\mathfrak {A}}^{**}\\subset B({\\mathcal {H}})$ for some Hilbert space ${\\mathcal {H}}$ , and that the weak-$*$ topology of ${\\mathfrak {A}}^{**}$ coincides with that of $B({\\mathcal {H}})$ .",
"Since $\\Vert s\\Vert =1$ , it suffices to show that the sequence $(s^n)_n$ converges to $p$ in the weak operator topology of $B({\\mathcal {H}})$ .",
"To see this, we use the fact that $p=s p=p s$ to infer $s=p+(I-p)s (I-p).$ Put $t=(I-p)s(I-p)$ and observe that $\\Vert t\\Vert \\le 1$ .",
"We claim that $t$ is completely non-unitary, in the sense that it has no reducing subspace on which it restricts to be unitary.",
"Assume on the contrary that there is a closed subspace ${\\mathcal {M}}\\subset (I-p){\\mathcal {H}}$ which is reducing for $t$ and such that the operator $u=t|_{\\mathcal {M}}$ is unitary.",
"Since $s$ is assumed to be self-adjoint, so is $u$ and we see that $u^2=I$ .",
"Thus, there is a vector state $\\varphi $ on $B({\\mathcal {M}})$ such that $|\\varphi (u)|=1$ .",
"Let $\\Psi $ be the normal state on ${\\mathfrak {A}}^{**}$ defined as $\\Psi (x)=\\varphi (P_{{\\mathcal {M}}}x|_{{\\mathcal {M}}}),\\quad x\\in {\\mathfrak {A}}^{**}.$ If we put $\\psi =\\Psi |_{{\\mathfrak {A}}}$ then it is easily verified that $\\psi $ is a state on ${\\mathfrak {A}}$ such that $\\widehat{\\psi }=\\Psi $ .",
"We see that $|\\psi (s)|=|\\varphi (u)|=1.$ On the other hand, since ${\\mathcal {M}}$ is orthogonal to the range of $p$ , we see that $ \\widehat{\\psi }(p)=0$ whence $p\\le I-{\\mathfrak {s}}_\\psi $ and ${\\mathfrak {s}}_\\psi p=0$ .",
"This contradicts the fact that $p$ peaks on $s$ .",
"Hence, we conclude that indeed $t$ is completely non-unitary.",
"In particular, it is an absolutely continuous contraction , so that $\\lim _{n\\rightarrow \\infty }\\langle t^n \\xi ,\\eta \\rangle =0, \\quad \\xi ,\\eta \\in {\\mathcal {H}}.$ (Alternatively, since $t$ is self-adjoint one can apply the sharper in this context).",
"Since $s^n=p+t^n$ for each $n\\in {\\mathbb {N}}$ , we conclude that the sequence $(s^n)_n$ converges to $p$ in the weak operator topology of $B({\\mathcal {H}})$ .",
"We can now establish a very natural link between ${\\mathfrak {S}}$ -peaking states and ${\\mathfrak {S}}$ -peaking projections.",
"Theorem 3.6 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $s\\in {\\mathfrak {S}}$ be a self-adjoint element with $\\Vert s\\Vert =1$ .",
"Then, a pure state $\\omega $ on ${\\mathfrak {A}}$ is ${\\mathfrak {S}}$ -peaking on $s$ if and only if its support projection ${\\mathfrak {s}}_\\omega \\in {\\mathfrak {A}}^{**}$ is ${\\mathfrak {S}}$ -peaking on $s$ .",
"Fix a pure state $\\omega $ on ${\\mathfrak {A}}$ .",
"Assume first that $\\omega $ is ${\\mathfrak {S}}$ -peaking on $s$ .",
"To see that ${\\mathfrak {s}}_\\omega $ is ${\\mathfrak {S}}$ -peaking on $s$ as well, it suffices to verify that $s {\\mathfrak {s}}_\\omega ={\\mathfrak {s}}_\\omega $ .",
"Now, by assumption we see that $\\omega (s)=1$ and $\\Vert s\\Vert =1$ , so that $s$ belongs to the multiplicative domain of $\\widehat{\\omega }$ and thus $\\widehat{\\omega }(xs)=\\widehat{\\omega }(x), \\quad x\\in {\\mathfrak {A}}^{**}.$ This shows that $I-s\\in {\\mathfrak {N}}_\\omega $ , whence $(I-s){\\mathfrak {s}}_\\omega =0$ or $s{\\mathfrak {s}}_\\omega ={\\mathfrak {s}}_\\omega $ .",
"We conclude that ${\\mathfrak {s}}_\\omega $ is ${\\mathfrak {S}}$ -peaking.",
"Conversely, assume that ${\\mathfrak {s}}_\\omega $ is ${\\mathfrak {S}}$ -peaking on $s$ .",
"We obtain $\\omega (s)=\\widehat{\\omega }(s{\\mathfrak {s}}_\\omega )=\\widehat{\\omega }({\\mathfrak {s}}_\\omega )=1.$ Next, by Lemma REF we see that the sequence $(s ^n)_n$ converges to ${\\mathfrak {s}}_\\omega $ in the weak-$*$ topology of ${\\mathfrak {A}}^{**}$ .",
"Let $\\varphi $ be a state on ${\\mathfrak {A}}$ such that $|\\varphi (s)|=1$ .",
"Since $\\Vert s\\Vert = 1$ , this forces $s$ to lie in the multiplicative domain of $\\varphi $ .",
"In particular, we have $|\\varphi (s^n)|=|\\varphi (s)|^n=1$ and thus $|\\widehat{\\varphi }({\\mathfrak {s}}_\\omega )|=\\lim _{n\\rightarrow \\infty } |\\varphi (s^n)|=1$ which implies $\\varphi =\\omega $ by Lemma REF .",
"We conclude that $\\omega $ is ${\\mathfrak {S}}$ -peaking on $s$ .",
"It is worthy of note that in the case where ${\\mathfrak {S}}$ is a unital subalgebra of ${\\mathfrak {A}}$ , some faithful analogues of classical characterizations were obtained for Hay's peaking projections (see and references therein).",
"We close this section by discussing another notion of non-commutative peaking behaviour, guided by .",
"Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $\\pi $ be a unital $*$ -representation of ${\\mathfrak {A}}$ .",
"Given another unital $*$ -representation $\\rho $ of ${\\mathfrak {A}}$ , we write $\\pi \\prec \\rho $ to signify that $\\Vert \\pi (a)\\Vert \\le \\Vert \\rho (a)\\Vert , \\quad a\\in {\\mathfrak {A}}.$ Equivalently, we see that $\\pi \\prec \\rho $ if and only if there is a unital $*$ -representation $\\theta :\\rho ({\\mathfrak {A}})\\rightarrow \\pi ({\\mathfrak {A}})$ such that $\\theta \\circ \\rho =\\pi $ .",
"We say that the $*$ -representation $\\pi $ of ${\\mathfrak {A}}$ is ${\\mathfrak {S}}$ -peaking if there is an element $s\\in {\\mathfrak {S}}$ with the property that $\\Vert \\pi (s)\\Vert >\\Vert \\rho (s)\\Vert $ for every unital $*$ -representation $\\rho $ of ${\\mathfrak {A}}$ such that $\\pi \\nprec \\rho $ .",
"We then say that $\\pi $ peaks on $s$ .",
"The reader will note that this definition apparently differs from that of .",
"However, in the setting where it is used therein, the definitions coincide as the irreducible $*$ -representations of $\\mathrm {C}^*$ -algebras of compact operators are particularly transparent.",
"The GNS construction allows one to construct $*$ -representations starting from states.",
"We show how the condition of being ${\\mathfrak {S}}$ -peaking is preserved by this procedure.",
"Proposition 3.7 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $\\psi $ be a state on ${\\mathfrak {A}}$ which is ${\\mathfrak {S}}$ -peaking.",
"Then, its GNS representation $\\sigma _\\psi $ is ${\\mathfrak {S}}$ -peaking as well.",
"Since $\\psi $ is ${\\mathfrak {S}}$ -peaking, there is a self-adjoint element $s\\in {\\mathfrak {S}}$ such that $\\Vert s\\Vert =1$ and with the property that $1=\\psi (s)>|\\varphi (s)|$ for every state $\\varphi $ on ${\\mathfrak {A}}$ such that $\\varphi \\ne \\psi $ .",
"Now, let $\\rho $ be a unital $*$ -representation such that $\\sigma _\\psi \\lnot \\prec \\rho $ .",
"Seeing as $s$ is self-adjoint, we may choose a state $\\chi $ on $\\rho ({\\mathfrak {A}})$ with the property that $|\\chi (\\rho (s))|=\\Vert \\rho (s)\\Vert .$ Note now that if $\\chi \\circ \\rho =\\psi $ then $\\sigma _\\chi \\circ \\rho $ is unitarily equivalent to $\\sigma _\\psi $ , and thus $\\sigma _\\psi \\prec \\rho $ which is absurd.",
"Hence, $\\chi \\circ \\rho \\ne \\psi $ .",
"We infer that $\\Vert \\rho (s)\\Vert =|\\chi (\\rho (s))|<\\psi (s)\\le \\Vert \\sigma _\\psi (s)\\Vert $ and therefore $\\sigma _\\psi $ is ${\\mathfrak {S}}$ -peaking.",
"One consequence of the previous proposition is that the condition that every pure state on ${\\mathfrak {A}}$ be ${\\mathfrak {S}}$ -peaking implies that every irreducible $*$ -representation of ${\\mathfrak {A}}$ is ${\\mathfrak {S}}$ -peaking.",
"The converse does not hold, as we now illustrate.",
"Example 3.8 We return to the setting of Example REF .",
"Let ${\\mathbb {M}}_2$ denote the $2\\times 2$ complex matrices and consider the standard matrix unit $E_{12}\\in {\\mathbb {M}}_2$ .",
"Let ${\\mathfrak {S}}\\subset {\\mathbb {M}}_2$ be the operator system generated by $I$ and $E_{12}$ .",
"We have ${\\mathbb {M}}_2=\\mathrm {C}^*({\\mathfrak {S}})$ .",
"Up to unitary equivalence, the only irreducible $*$ -representation of ${\\mathbb {M}}_2$ is the identity representation.",
"Tautologically, we see that every irreducible $*$ -representation of ${\\mathbb {M}}_2$ is ${\\mathfrak {S}}$ -peaking.",
"However, we saw in Example REF that there are plenty of pure states on ${\\mathbb {M}}_2$ which are not ${\\mathfrak {S}}$ -peaking."
],
[
"Characteristic sequences",
"In the previous section, we introduced and studied the notion of ${\\mathfrak {S}}$ -peaking states, a certain non-commutative analogue of classical peak points for uniform algebras.",
"For the remainder of the paper, we aim to exploit these ideas to obtain a local version of the hyperrigidity conjecture.",
"To do so, we will need to consider states that have a global type of peaking behaviour, such as in the following example.",
"Example 4.1 Let ${\\mathcal {H}}$ be a Hilbert space and let ${\\mathfrak {A}}\\subset B({\\mathcal {H}})$ be a unital $\\mathrm {C}^*$ -algebra which contains the ideal ${\\mathfrak {K}}$ of compact operators on ${\\mathcal {H}}$ .",
"Put ${\\mathfrak {S}}={\\mathfrak {K}}+{\\mathbb {C}}I$ .",
"Let $\\xi \\in {\\mathcal {H}}$ be a unit vector and let $\\omega $ be the pure vector state on ${\\mathfrak {A}}$ defined as $\\omega (a)=\\langle a\\xi ,\\xi \\rangle , \\quad a\\in {\\mathfrak {A}}.$ We saw in Example REF that $\\omega $ is ${\\mathfrak {S}}$ -peaking.",
"Furthermore, if we let $s\\in {\\mathfrak {S}}$ be the rank-one projection onto ${\\mathbb {C}}\\xi $ , then a simple calculation reveals that $s^* a s=\\langle a\\xi ,\\xi \\rangle s=\\omega (a) s$ whence $\\Vert s^* a s\\Vert =|\\omega (a)|$ for each $a\\in {\\mathfrak {A}}$ .",
"This last equality says that the ${\\mathfrak {S}}$ -peaking state $\\omega $ also possesses a certain global peaking property on ${\\mathfrak {A}}$ .",
"It will be sufficient for our purposes to have such a global peaking condition be satisfied in an asymptotic sense.",
"Before proceeding with the definition, we take a closer look at the classical setting so as to provide motivation for what is to come.",
"Example 4.2 Let $X$ be a compact Hausdorff space and ${\\mathcal {A}}\\subset C(X)$ be a unital subalgebra.",
"Let $x_0\\in X$ be a point such that there is a function $\\varphi _0\\in {\\mathcal {A}}$ with the property that $|\\varphi _0(y)|<\\varphi _0(x_0)=1$ for each $y\\in X, y\\ne x_0$ .",
"Then, the peak point $x_0$ has a certain asymptotic global peaking property.",
"More precisely, we have that $\\lim _{n\\rightarrow \\infty } \\Vert \\varphi _0^{n}f\\Vert = |f(x_0)|, \\quad f\\in C(X).$ To see this, fix a non-zero function $f\\in C(X)$ and a number $\\varepsilon >0$ .",
"By continuity, there is an open subset $U\\subset X$ containing $x_0$ with the property that $|f(y)|< |f(x_0)|+\\varepsilon $ if $y\\in U$ .",
"Note also that $X\\setminus U$ is compact, and thus there is a positive integer $N$ large enough so that $\\sup \\lbrace |\\varphi _0(y)|^{n}: y\\in X\\setminus U\\rbrace \\le \\varepsilon \\Vert f\\Vert ^{-1}, \\quad n\\ge N.$ Consequently, $\\limsup _{n\\rightarrow \\infty }\\Vert \\varphi _0^n f \\Vert \\le |f(x_0)|+\\varepsilon .$ Since $\\varepsilon >0$ is arbitrary, we infer that $\\limsup _{n\\rightarrow \\infty }\\Vert \\varphi _0^n f \\Vert \\le |f(x_0)|.$ On the other hand, for each positive integer $n$ we have $\\Vert \\varphi _0^n f \\Vert \\ge |f(x_0)|$ and thus $\\lim _{n\\rightarrow \\infty } \\Vert \\varphi _0^{n}f\\Vert = |f(x_0)|.$ Roughly speaking, the previous example shows that a peak point $x_0\\in X$ for a uniform algebra ${\\mathcal {A}}$ gives rise to a sequence lying in ${\\mathcal {A}}$ which behaves asymptotically like the characteristic function of the singleton $\\lbrace x_0\\rbrace $ .",
"With the help of this device, the uniform algebra can be “localized\" at the peak point.",
"This is the phenomenon we want to reproduce in the general non-commutative setting, and accordingly we make the following definition, motivated by Examples REF and REF .",
"Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let $\\psi $ be a state on ${\\mathfrak {A}}$ .",
"A sequence $(\\Delta _n)_n$ in ${\\mathfrak {A}}$ is said to be a characteristic sequence for $\\psi $ if the following conditions are satisfied: $\\Vert \\Delta _n\\Vert =1$ for every $n\\in {\\mathbb {N}}$ , $\\lim _{n\\rightarrow \\infty }\\psi (\\Delta _n)=1,$ and $\\limsup _{n\\rightarrow \\infty }\\Vert \\Delta _n^* a \\Delta _n\\Vert \\le |\\psi (a)|$ for every $a\\in {\\mathfrak {A}}$ .",
"To uncover some of the properties of characteristic sequences, we require the following fact.",
"Lemma 4.3 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let $\\varphi , \\psi $ be states on ${\\mathfrak {A}}$ .",
"Assume that $(\\Delta _n)_n$ is a characteristic sequence for $\\psi $ and that $\\lim _{n\\rightarrow \\infty }|\\varphi (\\Delta _n)|=1.$ Then, we have that $\\varphi (a)=\\lim _{n\\rightarrow \\infty }\\varphi (\\Delta _n^* a\\Delta _n), \\quad a\\in {\\mathfrak {A}}$ and $\\varphi =\\psi $ .",
"By the Schwarz inequality we have $|\\varphi (\\Delta _n)|^2=\\varphi (\\Delta _n)^*\\varphi (\\Delta _n)\\le \\varphi (\\Delta _n^*\\Delta _n)\\le 1$ and thus $\\lim _{n\\rightarrow \\infty }(\\varphi (\\Delta _n^*\\Delta _n) -\\varphi (\\Delta _n)^*\\varphi (\\Delta _n)) =0.$ Let $a\\in {\\mathfrak {A}}$ .",
"For each $n\\in {\\mathbb {N}}$ consider the element $A_n=\\begin{bmatrix}\\Delta _n & a^*\\\\0 & 0\\end{bmatrix}\\in {\\mathbb {M}}_2({\\mathfrak {A}}).$ By the Schwarz inequality, we find that $\\varphi ^{(2)}(A_n)^* \\varphi ^{(2)}(A_n)\\le \\varphi ^{(2)}(A_n^*A_n).$ Equivalently $\\begin{bmatrix}\\varphi (\\Delta _n)^*\\varphi (\\Delta _n) &\\varphi (\\Delta _n)^*\\varphi (a)^*\\\\\\varphi (a)\\varphi (\\Delta _n) & \\varphi (a)\\varphi (a)^*\\end{bmatrix}\\le \\begin{bmatrix}\\varphi (\\Delta _n^*\\Delta _n) & \\varphi (\\Delta _n^* a^*)\\\\\\varphi (a\\Delta _n) & \\varphi (aa^*)\\end{bmatrix},$ or $\\begin{bmatrix}\\varphi (\\Delta _n^*\\Delta _n) -\\varphi (\\Delta _n)^*\\varphi (\\Delta _n) & \\varphi (\\Delta _n^* a^*)-\\varphi (\\Delta _n)^*\\varphi (a)^*\\\\\\varphi (a\\Delta _n)-\\varphi (a)\\varphi (\\Delta _n) & \\varphi (aa^*)- \\varphi (a)\\varphi (a)^*\\end{bmatrix}\\ge 0.$ In particular, we find that $|\\varphi (a\\Delta _n)-\\varphi (a)\\varphi (\\Delta _n) |^2&\\le ( \\varphi (aa^*)- \\varphi (a)\\varphi (a)^*)(\\varphi (\\Delta _n^*\\Delta _n) -\\varphi (\\Delta _n)^*\\varphi (\\Delta _n))\\\\&\\le \\Vert a\\Vert ^2(\\varphi (\\Delta _n^*\\Delta _n) -\\varphi (\\Delta _n)^*\\varphi (\\Delta _n)).$ Repeating the argument with $a^*$ in place of $a$ yields $|\\varphi (\\Delta _n^*a )-\\varphi (a)\\varphi (\\Delta ^*_n) |^2&=|\\varphi (a^*\\Delta _n)-\\varphi (a^*)\\varphi (\\Delta _n) |^2\\\\&\\le \\Vert a\\Vert ^2(\\varphi (\\Delta _n^*\\Delta _n) -\\varphi (\\Delta _n)^*\\varphi (\\Delta _n)).$ Therefore $&|\\varphi (\\Delta _n^*a\\Delta _n)-\\varphi (\\Delta _n^*)\\varphi (a)\\varphi (\\Delta _n) |\\\\&\\le |\\varphi (\\Delta _n^*a\\Delta _n)-\\varphi (\\Delta _n^*a)\\varphi (\\Delta _n)|+|\\varphi (\\Delta _n^*a)\\varphi (\\Delta _n)-\\varphi (\\Delta _n^*)\\varphi (a)\\varphi (\\Delta _n)|\\\\&\\le (\\Vert \\Delta _n^* a\\Vert +|\\varphi (\\Delta _n)|\\Vert a\\Vert )(\\varphi (\\Delta _n^*\\Delta _n) -\\varphi (\\Delta _n)^*\\varphi (\\Delta _n))^{1/2}\\\\&\\le 2\\Vert a\\Vert (\\varphi (\\Delta _n^*\\Delta _n) -\\varphi (\\Delta _n)^*\\varphi (\\Delta _n))^{1/2}$ which then forces $\\lim _{n\\rightarrow \\infty }(\\varphi (\\Delta _n^* a\\Delta _n)-\\varphi (\\Delta _n^*)\\varphi (a)\\varphi (\\Delta _n) )=0.$ Since $\\lim _{n\\rightarrow \\infty }|\\varphi (\\Delta _n)|=1$ we find $\\varphi (a)=\\lim _{n\\rightarrow \\infty }\\varphi (\\Delta _n^* a\\Delta _n)$ and thus $|\\varphi (a)|\\le |\\psi (a)|$ since $(\\Delta _n)_n$ is a characteristic sequence for $\\psi $ .",
"Since this holds for every $a\\in {\\mathfrak {A}}$ , we infer that $\\ker \\psi \\subset \\ker \\varphi $ , so that $\\varphi $ is a scalar multiple of $\\psi $ .",
"Since they are both states, this forces $\\varphi =\\psi $ .",
"We can now establish some useful properties of characteristic sequences.",
"Theorem 4.4 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let $\\psi $ a state on ${\\mathfrak {A}}$ .",
"Let $(\\Delta _n)_n$ be a characteristic sequence for $\\psi $ .",
"The following statements hold.",
"We have that $\\lim _{n\\rightarrow \\infty }\\Vert \\Delta _n a \\Delta _n^*\\Vert =|\\psi (a)|$ for every $a\\in {\\mathfrak {A}}$ .",
"We have that $\\limsup _{n\\rightarrow \\infty }|\\varphi (\\Delta _n)|<1$ for every state $\\varphi $ on ${\\mathfrak {A}}$ such that $\\varphi \\ne \\psi $ .",
"The state $\\psi $ is pure.",
"Let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system and assume that $\\Delta _n\\in {\\mathfrak {S}}$ for each $n\\in {\\mathbb {N}}$ .",
"Then, $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ .",
"To show (1), fix $a\\in {\\mathfrak {A}}$ .",
"By definition of a characteristic sequence, we have that $\\lim _{n\\rightarrow \\infty }\\psi (\\Delta _n)=1$ and $\\limsup _{n\\rightarrow \\infty }\\Vert \\Delta _n^* a \\Delta _n\\Vert \\le |\\psi (a)|.$ On the other hand, invoking Lemma REF , we obtain $|\\psi (a)|=\\lim _{n\\rightarrow \\infty }|\\psi (\\Delta _n^* a\\Delta _n)|\\le \\liminf _{n\\rightarrow \\infty }\\Vert \\Delta _n^* a \\Delta _n\\Vert $ whence $\\lim _{n\\rightarrow \\infty }\\Vert \\Delta _n a \\Delta _n^*\\Vert =|\\psi (a)|.$ Next, let $\\varphi $ be a state on ${\\mathfrak {A}}$ with $\\varphi \\ne \\psi $ and let $(\\Delta _{n_m})_m$ be a subsequence.",
"It is readily seen that $(\\Delta _{n_m})_m$ is still a characteristic sequence for $\\psi $ .",
"Thus, by virtue of Lemma REF , we see that $\\lim _{m\\rightarrow \\infty }|\\varphi (\\Delta _{n_m})|<1.$ Consequently, we see that $\\limsup _{n\\rightarrow \\infty }|\\varphi (\\Delta _n)|<1$ and (2) holds.",
"Assume now that $\\psi =\\frac{1}{2}(\\varphi _1+\\varphi _2)$ for some states $\\varphi _1,\\varphi _2$ on ${\\mathfrak {A}}$ .",
"Since $\\lim _{n\\rightarrow \\infty } \\psi (\\Delta _n)=1$ a routine verification yields that $\\lim _{n\\rightarrow \\infty } |\\varphi _1(\\Delta _n)|=1=\\lim _{n\\rightarrow \\infty } |\\varphi _2(\\Delta _n)|.$ Therefore $\\varphi _1=\\varphi _2=\\psi $ by (2), and thus $\\psi $ is pure.",
"We have established (3).",
"For (4), we assume that $\\Delta _n\\in {\\mathfrak {S}}$ for each $n\\in {\\mathbb {N}}$ and we let $\\varphi $ be a state on ${\\mathfrak {A}}$ which agrees with $\\psi $ on ${\\mathfrak {S}}$ .",
"Then $\\lim _{n\\rightarrow \\infty }|\\varphi (\\Delta _n)|=\\lim _{n\\rightarrow \\infty }|\\psi (\\Delta _n)|=1$ so that $\\varphi =\\psi $ by part (2).",
"We conclude that $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ .",
"The previous result shows that if a state $\\psi $ on ${\\mathfrak {A}}$ admits a characteristic sequence in ${\\mathfrak {S}}$ , then $\\psi $ shares several properties with ${\\mathfrak {S}}$ -peaking states.",
"Indeed, (3) and (4) correspond to Lemma REF , while (2) says that $\\psi $ can be considered to be “asymptotically ${\\mathfrak {S}}$ -peaking\" on the sequence $(\\Delta _n)_n$ .",
"Note however that typically a characteristic sequence does not consist of self-adjoint elements (see Example REF below).",
"We mention an outstanding problem that we have not been able to settle.",
"Question 1 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $\\psi $ be a pure state on ${\\mathfrak {A}}$ that has the unique extension property with respect to ${\\mathfrak {S}}$ .",
"Does there always exist a characteristic sequence in ${\\mathfrak {S}}$ for $\\psi $ ?",
"A positive answer (even in the commutative case) would constitute a nice analogue of the classical fact that for uniform algebras, the Choquet boundary coincides with the set of peak points .",
"As mentioned before, this is known to fail for general function systems .",
"It is unclear to us at the moment whether the asymptotic peaking behaviour of characteristic sequences is flexible enough to remedy this defect.",
"We close this section by answering the previous question in the affirmative in some special cases.",
"For that purpose, recall that an operator $T\\in B({\\mathcal {H}})$ is said to be a pure contraction if $\\Vert T\\Vert \\le 1$ and $\\lim _{n\\rightarrow \\infty }\\Vert T^{*n}\\xi \\Vert =0, \\quad \\xi \\in {\\mathcal {H}}.$ Theorem 4.5 Let ${\\mathcal {H}}$ be a Hilbert space.",
"Let ${\\mathfrak {A}}\\subset B({\\mathcal {H}})$ be a unital $\\mathrm {C}^*$ -algebra which contains the ideal of compact operators ${\\mathfrak {K}}$ and such that the quotient ${\\mathfrak {A}}/{\\mathfrak {K}}$ is commutative.",
"Let ${\\mathcal {A}}\\subset {\\mathfrak {A}}$ be a unital closed subalgebra such that for every character $\\omega $ on ${\\mathfrak {A}}$ , there is a pure contraction $t_\\omega \\in {\\mathcal {A}}$ with the property that $|\\chi (t_\\omega )|<1=\\omega (t_\\omega )$ whenever $\\chi $ is a character on ${\\mathfrak {A}}$ such that $\\chi \\ne \\omega $ .",
"Then, every pure state on ${\\mathfrak {A}}$ admits a characteristic sequence in ${\\mathfrak {S}}+{\\mathfrak {K}}$ , where ${\\mathfrak {S}}$ is the operator system generated by ${\\mathcal {A}}$ .",
"Let $\\omega $ be a pure state on ${\\mathfrak {A}}$ .",
"Based on the discussion from Example REF , we see that $\\omega $ is either a vector state or it annihilates ${\\mathfrak {K}}$ .",
"It also follows from Example REF that every vector state on ${\\mathfrak {A}}$ admits a (constant) characteristic sequence in ${\\mathfrak {K}}$ .",
"Next, assume $\\omega $ annihilates ${\\mathfrak {K}}$ .",
"There is a pure state $\\widehat{\\omega }$ on ${\\mathfrak {A}}/{\\mathfrak {K}}$ with the property that if $q:{\\mathfrak {A}}\\rightarrow {\\mathfrak {A}}/{\\mathfrak {K}}$ denotes the quotient map, then $\\omega =\\widehat{\\omega }\\circ q$ .",
"Since ${\\mathfrak {A}}/{\\mathfrak {K}}$ is a commutative $\\mathrm {C}^*$ -algebra, $\\widehat{\\omega }$ must be a character and thus so is $\\omega $ .",
"By assumption, there is a pure contraction $t_\\omega \\in {\\mathcal {A}}$ with the property that $|\\chi (t_\\omega )|<1=\\omega (t_\\omega )$ for every character $\\chi $ on ${\\mathfrak {A}}$ such that $\\chi \\ne \\omega $ .",
"In particular, this means that $|\\theta (q(t_\\omega ))|<1=\\widehat{\\omega }(q(t_\\omega ))$ for every character $\\theta $ on ${\\mathfrak {A}}/{\\mathfrak {K}}$ such that $\\theta \\ne \\widehat{\\omega }$ .",
"We now verify that $\\Delta _n=t_\\omega ^{n}, \\quad n\\in {\\mathbb {N}}$ is a characteristic sequence for $\\omega $ in ${\\mathfrak {S}}$ .",
"Clearly, we have that $\\omega (\\Delta _n)=1$ for every $n\\in {\\mathbb {N}}$ .",
"Moreover, Example REF along with Equation (REF ) implies that $\\lim _{n\\rightarrow \\infty }\\Vert q(\\Delta ^*_n a \\Delta _n)\\Vert = |\\omega (a)|, \\quad a\\in {\\mathfrak {A}}.$ Fix $a\\in {\\mathfrak {A}}$ and let $\\varepsilon >0$ .",
"Choose $N\\in {\\mathbb {N}}$ with the property that $\\Vert q(\\Delta ^*_N a \\Delta _N)\\Vert < |\\omega (a)|+\\varepsilon .$ We can find a compact operator $K$ such that $\\Vert \\Delta ^*_N a \\Delta _N+K\\Vert < |\\omega (a)|+\\varepsilon .$ Since $\\Vert \\Delta _m\\Vert =1$ and $\\Delta _N \\Delta _m=\\Delta _{N+m}$ for every $m\\in {\\mathbb {N}}$ , we conclude that $\\Vert \\Delta ^*_{N+m} a \\Delta _{N+m}+\\Delta ^*_m K\\Delta _m\\Vert < |\\omega (a)|+\\varepsilon , \\quad m\\in {\\mathbb {N}}.$ Now, the sequence $(\\Delta ^*_m)_m$ converges to 0 in the strong operator topology as $t_\\omega $ is assumed to be a pure contraction.",
"Since $K$ is compact, the sequence $(\\Delta ^*_m K\\Delta _m)_m$ must converge to 0 in norm and we infer $\\limsup _{m\\rightarrow \\infty }\\Vert \\Delta ^*_{N+m} a \\Delta _{N+m}\\Vert \\le |\\omega (a)|+\\varepsilon .$ Since $\\varepsilon >0$ is arbitrary, we conclude that $(\\Delta _n)_n$ is a characteristic sequence for $\\omega $ .",
"We now exhibit a natural example satisfying the conditions of the previous theorem.",
"Example 4.6 Fix a positive integer $d\\ge 1$ .",
"Let ${\\mathbb {B}}_d\\subset {\\mathbb {C}}^d$ denote the open unit ball and let ${\\mathbb {S}}_d$ denotes its boundary, the unit sphere.",
"The Drury-Arveson space $H^2_d$ is the reproducing kernel Hilbert space on ${\\mathbb {B}}_d$ with reproducing kernel given by the formula $k(z,w)=\\frac{1}{1-\\langle z,w\\rangle _{{\\mathbb {C}}^d}}, \\quad z,w\\in {\\mathbb {B}}_d.$ This is a Hilbert space of holomorphic functions on ${\\mathbb {B}}_d$ .",
"If for each $\\lambda \\in {\\mathbb {B}}_d$ we put $k_\\lambda (z)=\\frac{1}{1-\\langle z,\\lambda \\rangle _{{\\mathbb {C}}^d}}, \\quad z\\in {\\mathbb {B}}_d$ then $f(\\lambda )=\\langle f,k_\\lambda \\rangle _{H^2_d}, \\quad f\\in H^2_d.$ A function $\\varphi :{\\mathbb {B}}_d\\rightarrow {\\mathbb {C}}$ is a multiplier for $H^2_d$ if $\\varphi f\\in H^2_d$ for every $f\\in H^2_d$ .",
"Examples of such functions include the holomorphic polynomials in $d$ variables.",
"Every multiplier $\\varphi $ gives rise to a multiplication operator $M_\\varphi \\in B(H^2_d)$ , and the identification $\\varphi \\mapsto M_\\varphi $ allows us to view the algebra of multipliers as an operator algebra on $H^2_d$ .",
"It is easily verified that if $\\varphi $ is a multiplier, then $M_\\varphi ^* k_\\lambda =\\overline{\\varphi (\\lambda )}k_\\lambda , \\quad \\lambda \\in {\\mathbb {B}}_d.$ We refer the reader to the book for an excellent reference on these topics.",
"Let ${\\mathcal {A}}_d\\subset B(H^2_d)$ denote the norm closure of the polynomial multipliers and define the Toeplitz algebra as ${\\mathfrak {T}}_d=\\mathrm {C}^*({\\mathcal {A}}_d)$ .",
"Recall that ${\\mathfrak {T}}_d$ contains the ideal of compact operators ${\\mathfrak {K}}$ on $H^2_d$ and that ${\\mathfrak {T}}_d/{\\mathfrak {K}}$ is $*$ -isomorphic to $C({\\mathbb {S}}_d)$ .",
"In particular, any character on ${\\mathfrak {T}}_d/{\\mathfrak {K}}$ can be identified with the character of evaluation at some point in ${\\mathbb {S}}_d$ .",
"Let $\\omega $ be a character on ${\\mathfrak {T}}_d$ , which necessarily annihilates ${\\mathfrak {K}}$ .",
"There is a character $\\widehat{\\omega }$ on ${\\mathfrak {T}}_d/{\\mathfrak {K}}$ with the property that if $q:{\\mathfrak {T}}_d\\rightarrow {\\mathfrak {T}}_d/{\\mathfrak {K}}$ denotes the quotient map, then $\\omega =\\widehat{\\omega }\\circ q$ .",
"Assume that $\\widehat{\\omega }$ is identified with the character of evaluation at $\\zeta \\in {\\mathbb {S}}_d$ .",
"Consider the function defined as $\\varphi _\\zeta (z)=\\frac{1}{2}(1+\\langle z,\\zeta \\rangle _{{\\mathbb {C}}^d}), \\quad z\\in \\overline{{\\mathbb {B}}_d}.$ The row operator $(M_{z_1},M_{z_2},\\ldots ,M_{z_d})$ is contractive, so that $M_{\\varphi _\\zeta }\\in {\\mathcal {A}}_d$ and $\\Vert M_{\\varphi _\\zeta }\\Vert =1$ .",
"Moreover, $\\varphi _\\zeta (\\zeta )=1$ and $|\\varphi _\\zeta (z)|<1$ for every $z\\in \\overline{{\\mathbb {B}}_d}$ with $z\\ne \\zeta $ .",
"In particular, we see that $\\lim _{n\\rightarrow \\infty }M_{\\varphi _\\zeta }^{*n}k_\\lambda =0, \\quad \\lambda \\in {\\mathbb {B}}_d.$ Since the subset $\\lbrace k_\\lambda :\\lambda \\in {\\mathbb {B}}_d\\rbrace $ spans a dense subspace of $H^2_d$ , we conclude that $M_{\\varphi _\\zeta }$ is a pure contraction.",
"It follows from Theorem REF that every pure state on ${\\mathfrak {T}}_d$ admits a characteristic sequence in ${\\mathfrak {K}}+{\\mathfrak {S}}_d$ , where ${\\mathfrak {S}}_d$ is the operator system generated by ${\\mathcal {A}}_d$ .",
"We note in passing that the Toeplitz algebra in the previous example is of prime importance in multivariate operator theory, dilation theory and function theory, as exposed in and (see also the references therein).",
"Furthermore, we mention that by virtue of , the argument above can easily adapted to the setting of quotient modules of the Drury-Arveson space for which the essential normality conjecture is known to hold , ."
],
[
"A local version of the hyperrigidity conjecture",
"Recall that Theorem REF exhibits, for commutative $\\mathrm {C}^*$ -algebras, a local version of hyperrigidity using the spectral measure associated to a separable $*$ -representation.",
"Our aim in this final section is to generalize this result to arbitrary $\\mathrm {C}^*$ -algebras.",
"In the absence of a spectral measure, a key step in this endeavour is to give an appropriate interpretation of what local hyperrigidity should be.",
"To do this, we capitalize on the foundations that were laid in Section and perform a localization procedure with the help of characteristic sequences.",
"We explain at the end of the section how this is equivalent to Theorem REF when we specialize our result to uniform algebras.",
"The main technical observation is the following.",
"Lemma 5.1 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $\\pi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ be a unital $*$ -representation and let $\\Pi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ be a unital completely positive extension of $\\pi |_{{\\mathfrak {S}}}$ .",
"Let $\\psi $ be a state on ${\\mathfrak {A}}$ which admits a characteristic sequence $(\\Delta _n)_n$ in ${\\mathfrak {S}}$ .",
"Then, we have $\\limsup _{n\\rightarrow \\infty }\\Vert \\Pi (a)\\pi (\\Delta _n)\\Vert \\le \\psi (a^*a)^{1/2}$ for every $a\\in {\\mathfrak {A}}$ .",
"Furthermore, we have $\\lim _{n\\rightarrow \\infty }\\Vert \\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)\\Vert =0$ for every self-adjoint element $a\\in {\\mathfrak {A}}$ such that $\\psi (a)=0$ .",
"Fix $a\\in {\\mathfrak {A}}$ .",
"If $s\\in {\\mathfrak {S}}$ is a self-adjoint element that satisfies $s\\ge a^*a$ , then since $\\Pi $ and $\\pi $ coincide on ${\\mathfrak {S}}$ we find $\\pi (\\Delta _n)^*\\Pi (a^*a)\\pi (\\Delta _n)\\le \\pi (\\Delta _n)^*\\pi (s)\\pi (\\Delta _n)=\\pi (\\Delta _n^* s\\Delta _n)$ whence $0\\le \\pi (\\Delta _n)^*\\Pi (a)^*\\Pi (a)\\pi (\\Delta _n)\\le \\pi (\\Delta _n^* s\\Delta _n)$ for every $n\\in {\\mathbb {N}}$ , by the Schwarz inequality.",
"In particular, we find $\\Vert \\Pi (a)\\pi (\\Delta _n)\\Vert ^2\\le \\Vert \\Delta _n^* s\\Delta _n\\Vert , \\quad n\\in {\\mathbb {N}}$ whenever $s\\in {\\mathfrak {S}}$ satisfies $s\\ge a^*a$ .",
"Using that $(\\Delta _n)_n$ is a characteristic sequence for $\\psi $ , we infer that $\\limsup _{n\\rightarrow \\infty }\\Vert \\Pi (a)\\pi (\\Delta _n)\\Vert \\le \\psi (s)^{1/2}$ for every $s\\in {\\mathfrak {S}}$ such that $s\\ge a^*a$ .",
"Consequently, $\\limsup _{n\\rightarrow \\infty }\\Vert \\Pi (a)\\pi (\\Delta _n)\\Vert \\le \\inf \\lbrace \\psi (s):s\\in {\\mathfrak {S}}, s\\ge a^* a\\rbrace ^{1/2}.$ Invoke now Theorem REF to see that $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ .",
"It follows from a straightforward adaptation of that $\\inf \\lbrace \\psi (s):s\\in {\\mathfrak {S}}, s\\ge a^* a\\rbrace =\\psi (a^*a)$ and thus $\\limsup _{n\\rightarrow \\infty }\\Vert \\Pi (a)\\pi (\\Delta _n)\\Vert \\le \\psi (a^*a)^{1/2}.$ We have established the first statement.",
"To establish the second, fix a self-adjoint element $a\\in {\\mathfrak {A}}$ such that $\\psi (a)=0$ and a number $\\varepsilon >0$ .",
"Then, we can find self-adjoint elements $b_\\varepsilon ,c_\\varepsilon \\in {\\mathfrak {S}}$ such that $c_\\varepsilon \\le a\\le b_\\varepsilon $ and $\\sup \\lbrace \\psi (c):c\\in {\\mathfrak {S}}, c\\le a\\rbrace \\le \\psi (c_\\varepsilon )+\\varepsilon ,$ $\\inf \\lbrace \\psi (b):b\\in {\\mathfrak {S}}, b\\ge a\\rbrace \\ge \\psi (b_\\varepsilon )-\\varepsilon .$ Next, note that since $(\\Delta _n)_n$ is a characteristic sequence for $\\psi $ , we can find $N\\in {\\mathbb {N}}$ such that $\\Vert \\Delta _n^* c_\\varepsilon \\Delta _n\\Vert \\le |\\psi (c_\\varepsilon )|+\\varepsilon , \\quad \\Vert \\Delta _n^* b_\\varepsilon \\Delta _n\\Vert \\le |\\psi (b_\\varepsilon )|+\\varepsilon $ if $n\\ge N$ .",
"In particular, we see that $-(|\\psi (c_\\varepsilon )|+\\varepsilon )I\\le \\pi (\\Delta _n^* c_\\varepsilon \\Delta _n), \\quad \\pi ( \\Delta _n^* b_\\varepsilon \\Delta _n)\\le (|\\psi (b_\\varepsilon )|+\\varepsilon )I$ if $n\\ge N$ .",
"On the other hand, using that $\\Pi $ and $\\pi $ coincide on ${\\mathfrak {S}}$ , we note that $\\pi (\\Delta _n^*)\\pi ( c_\\varepsilon ) \\pi (\\Delta _n)\\le \\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)\\le \\pi (\\Delta _n^*)\\pi ( b_\\varepsilon ) \\pi (\\Delta _n), \\quad n\\in {\\mathbb {N}}$ or $\\pi (\\Delta _n^* c_\\varepsilon \\Delta _n)\\le \\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)\\le \\pi (\\Delta _n^* b_\\varepsilon \\Delta _n), \\quad n\\in {\\mathbb {N}}.$ Hence, we find $-(|\\psi (c_\\varepsilon )|+\\varepsilon )I\\le \\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)\\le (|\\psi (b_\\varepsilon )|+\\varepsilon )I, \\quad n\\ge N.$ Recall now that $\\psi (a)=0$ so that $\\psi (b_\\varepsilon )\\ge 0$ and $\\psi (c_\\varepsilon )\\le 0$ , and therefore $(\\psi (c_\\varepsilon )-\\varepsilon )I\\le \\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)\\le (\\psi (b_\\varepsilon )+\\varepsilon )I, \\quad n\\ge N,$ which in turn implies $(\\sup \\lbrace \\psi (c):c\\in {\\mathfrak {S}}, c\\le a\\rbrace -2\\varepsilon )I \\le \\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)$ and $\\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)\\le (\\inf \\lbrace \\psi (b):b\\in {\\mathfrak {S}}, b\\ge a\\rbrace +2\\varepsilon )I$ if $n\\ge N$ , because of the choice of $b_\\varepsilon $ and $c_\\varepsilon $ .",
"As above, by Theorem REF we see that $\\psi $ has the unique extension property with respect to ${\\mathfrak {S}}$ , and it follows from that $\\sup \\lbrace \\psi (c):c\\in {\\mathfrak {S}}, c\\le a\\rbrace =0=\\inf \\lbrace \\psi (b):b\\in {\\mathfrak {S}}, b\\ge a\\rbrace .$ Therefore, $-2\\varepsilon \\le \\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)\\le 2\\varepsilon I, \\quad n\\ge N.$ Since $\\varepsilon >0$ was arbitrary, we conclude that $\\lim _{n\\rightarrow \\infty }\\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)=0.$ We now arrive at one of the main results of the paper, which yields a local form of the hyperrigidity conjecture for general operator systems.",
"Recall that given a state $\\psi $ , we denote by $\\sigma _\\psi $ the associated GNS representation.",
"Theorem 5.2 Let ${\\mathfrak {A}}$ be a unital $\\mathrm {C}^*$ -algebra and let ${\\mathfrak {S}}\\subset {\\mathfrak {A}}$ be an operator system.",
"Let $\\pi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ be a unital $*$ -representation and let $\\Pi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ be a unital completely positive extension of $\\pi |_{{\\mathfrak {S}}}$ .",
"Let $\\psi $ be a state on ${\\mathfrak {A}}$ which admits a characteristic sequence $(\\Delta _n)_n$ in ${\\mathfrak {S}}$ .",
"Then, we have $\\limsup _{n\\rightarrow \\infty } \\Vert (\\Pi (a)-\\pi (a))\\pi (\\Delta _n)\\Vert \\le 2\\operatorname{dist}(\\sigma _\\psi (a),\\sigma _\\psi ({\\mathfrak {S}}))$ and $\\lim _{n\\rightarrow \\infty } \\Vert \\pi (\\Delta _n)^*(\\Pi (a)-\\pi (a))\\pi (\\Delta _n)\\Vert =0$ for every $a\\in {\\mathfrak {A}}$ .",
"Let $a\\in {\\mathfrak {A}}$ .",
"Since $\\Pi $ and $\\pi $ coincide on ${\\mathfrak {S}}$ , we note that for every $s\\in {\\mathfrak {S}}$ we have $\\Pi (a)-\\pi (a)=\\Pi (a-s)-\\pi (a-s).$ On the other hand, by virtue of Lemma REF we see for each $s\\in {\\mathfrak {S}}$ that $\\limsup _{n\\rightarrow \\infty }\\Vert \\Pi (a-s)\\pi (\\Delta _n)\\Vert \\le \\psi ((a-s)^*(a-s))^{1/2}\\le \\Vert \\sigma _\\psi (a-s)\\Vert $ and $\\limsup _{n\\rightarrow \\infty }\\Vert \\pi (a-s)\\pi (\\Delta _n)\\Vert \\le \\psi ((a-s)^*(a-s))^{1/2}\\le \\Vert \\sigma _\\psi (a-s)\\Vert ,$ so that $\\limsup _{n\\rightarrow \\infty } \\Vert (\\Pi (a)-\\pi (a))\\pi (\\Delta _n)\\Vert \\le 2\\Vert \\sigma _\\psi (a-s)\\Vert .$ Hence $\\limsup _{n\\rightarrow \\infty } \\Vert (\\Pi (a)-\\pi (a))\\pi (\\Delta _n)\\Vert \\le 2 \\operatorname{dist}(\\sigma _\\psi (a),\\sigma _\\psi ({\\mathfrak {S}})).$ We have established the first statement.",
"Since ${\\mathfrak {A}}$ is spanned algebraically by its self-adjoint elements, to establish the second statement it is no loss of generality to assume that $a\\in {\\mathfrak {A}}$ is self-adjoint.",
"Furthermore, because $\\Pi (a)-\\pi (a)=\\Pi (a-\\psi (a)I)-\\pi (a-\\psi (a)I)$ we may also assume without loss of generality that $\\psi (a)=0$ .",
"By Lemma REF , we conclude that $\\lim _{n\\rightarrow \\infty } \\Vert \\pi (\\Delta _n)^*\\Pi (a)\\pi (\\Delta _n)\\Vert =0=\\lim _{n\\rightarrow \\infty } \\Vert \\pi (\\Delta _n)^*\\pi (a)\\pi (\\Delta _n)\\Vert $ which implies $\\lim _{n\\rightarrow \\infty } \\Vert \\pi (\\Delta _n)^*(\\Pi (a)-\\pi (a))\\pi (\\Delta _n)\\Vert =0.$ In view of Theorem REF and Example REF , one can now easily write down concrete consequences of Theorem REF .",
"We leave the details to the reader, which should compare them with those in .",
"To conclude the paper, we wish to examine Theorem REF carefully in the case where ${\\mathfrak {A}}$ is commutative.",
"First, we note that in this case, a pure state $\\omega $ on ${\\mathfrak {A}}$ must be a character, and so $\\omega $ must coincide with its GNS representation $\\sigma _\\omega $ .",
"Therefore, $\\sigma _\\omega ({\\mathfrak {A}})=\\sigma _\\omega ({\\mathfrak {S}})={\\mathbb {C}}$ and $\\operatorname{dist}(\\sigma _\\omega (a),\\sigma _\\omega ({\\mathfrak {S}}))=0, \\quad a\\in {\\mathfrak {A}}$ which simplifies the first conclusion of Theorem REF to read $\\lim _{n\\rightarrow \\infty } \\Vert (\\Pi (a)-\\pi (a))\\pi (\\Delta _n)\\Vert = 0, \\quad a\\in {\\mathfrak {A}}.$ We now explain how this is equivalent to Theorem REF for operator systems generated by uniform algebras.",
"Let $(X,\\rho )$ be a compact metric space, let ${\\mathfrak {A}}=C(X)$ and let ${\\mathcal {A}}\\subset {\\mathfrak {A}}$ be a closed unital subalgebra.",
"Put ${\\mathfrak {S}}={\\mathcal {A}}+{\\mathcal {A}}^*$ .",
"Let $x_0\\in X$ be a point in the Choquet boundary of ${\\mathcal {A}}$ , so that $x_0$ is a peak point for ${\\mathcal {A}}$ .",
"There is a function $\\varphi _0\\in {\\mathcal {A}}$ with the property that $|\\varphi _0(y)|<1=\\varphi _0(x_0)$ for every $y\\in X$ such that $y\\ne x_0$ .",
"Let $\\omega $ be the pure state on ${\\mathfrak {A}}$ of evaluation at $x_0$ .",
"If we let $\\Delta _n=\\varphi _0^n\\in {\\mathcal {A}}$ for each $n\\in {\\mathbb {N}}$ , then we saw in Example REF that $(\\Delta _n)_n$ is a characteristic sequence in ${\\mathfrak {S}}$ for $\\omega $ .",
"Now, let ${\\mathcal {H}}$ be a separable Hilbert space and let $\\pi :{\\mathfrak {A}}\\rightarrow B({\\mathcal {H}})$ be a unital $*$ -representation.",
"The reader should refer to Subsection REF for the notation that is used below.",
"Fix $\\delta >0$ .",
"Given $\\varepsilon >0$ , by compactness of $X$ we may find $N\\in {\\mathbb {N}}$ such that $|\\Delta _n(y)|^2<\\varepsilon $ if $n\\ge N$ and $y\\in X$ satisfies $\\rho (y,x_0)\\ge \\delta $ .",
"In particular this implies $\\pi (\\Delta _n) \\pi (\\Delta _n) ^* \\le (E_\\pi (x_0,\\delta )+\\varepsilon I)^2, \\quad n\\ge N.$ Hence, for every $T\\in B({\\mathcal {H}})$ we have $T\\pi (\\Delta _n) \\pi (\\Delta _n) ^*T^*\\le T (E_\\pi (x_0,\\delta )+\\varepsilon I)^2T^*, \\quad n\\ge N$ and therefore $\\Vert T\\pi (\\Delta _n) \\Vert \\le \\Vert TE_\\pi (x_0,\\delta )\\Vert +\\varepsilon \\Vert T\\Vert , \\quad n\\ge N.$ Since $\\varepsilon >0$ was arbitrary we obtain $\\limsup _{n\\rightarrow \\infty }\\Vert T\\pi (\\Delta _n) \\Vert \\le \\inf _{\\delta >0}\\Vert TE_\\pi (x_0,\\delta )\\Vert , \\quad T\\in B({\\mathcal {H}}).$ Conversely, fix $n\\in {\\mathbb {N}}$ .",
"Given $0<\\varepsilon <1$ we may find $\\delta _0$ with the property that $|\\Delta _n(y)|\\ge 1-\\varepsilon $ if $\\rho (y,x_0)<\\delta _0$ .",
"Hence, if $\\delta <\\delta _0$ then we see that $(1-\\varepsilon )^2E_\\pi (x_0,\\delta )\\le \\pi (\\Delta _n)\\pi (\\Delta _n)^*.$ Hence, for every $T\\in B({\\mathcal {H}})$ we have $(1-\\varepsilon )^2TE_\\pi (x_0,\\delta ) T^*\\le T\\pi (\\Delta _n) \\pi (\\Delta _n) ^*T^*, \\quad \\delta <\\delta _0$ and therefore $\\Vert TE_\\pi (x_0,\\delta )\\Vert \\le (1-\\varepsilon )^{-1} \\Vert T\\pi (\\Delta _n) \\Vert , \\quad \\delta <\\delta _0.$ Since $0<\\varepsilon <1$ was arbitrary we obtain $\\limsup _{\\delta \\rightarrow 0}\\Vert TE_\\pi (x_0,\\delta )\\Vert \\le \\inf _{n\\in {\\mathbb {N}}} \\Vert T\\pi (\\Delta _n) \\Vert , \\quad T\\in B({\\mathcal {H}}).$ Using Equations (REF ) and (REF ) we see that Theorems REF and REF are equivalent in the case of uniform algebras, so that the main result of this section is a genuine generalization of Arveson's local hyperrigidity theorem.",
"AM2000article author=Agler, Jim, author=McCarthy, John E., title=Complete Nevanlinna-Pick kernels, date=2000, ISSN=0022-1236, journal=J.",
"Funct.",
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[
[
"Power Consumption-based Detection of Sabotage Attacks in Additive\n Manufacturing"
],
[
"Abstract Additive Manufacturing (AM), a.k.a.",
"3D Printing, is increasingly used to manufacture functional parts of safety-critical systems.",
"AM's dependence on computerization raises the concern that the AM process can be tampered with, and a part's mechanical properties sabotaged.",
"This can lead to the destruction of a system employing the sabotaged part, causing loss of life, financial damage, and reputation loss.",
"To address this threat, we propose a novel approach for detecting sabotage attacks.",
"Our approach is based on continuous monitoring of the current delivered to all actuators during the manufacturing process and detection of deviations from a provable benign process.",
"The proposed approach has numerous advantages: (i) it is non-invasive in a time-critical process, (ii) it can be retrofitted in legacy systems, and (iii) it is airgapped from the computerized components of the AM process, preventing simultaneous compromise.",
"Evaluation on a desktop 3D Printer detects all attacks involving a modification of X or Y motor movement, with false positives at 0%."
],
[
"Introduction",
"Additive Manufacturing technology (AM), a.k.a.",
"3D Printing, is receiving immense attention due to the potential for improvements in product performance, decreased development times, and reduced costs.",
"As compared to either traditional, “subtractive” manufacturing methods in which material is removed from a part via machining or the use of pre-fabricated dies (e.g.",
"investment casting, die casting, injection molding) which necessitate substantial investments in capital equipment, longer lead times, and associated labor costs, AM allows for the production of components with minimum material waste, shorter design-to-production times, and economical, on-demand production of niche parts.",
"These advantages enable a broad range of applications, ranging from models and prototypes up to functional parts of safety-critical systems.",
"An example of the latter is the FAA-approved fuel nozzle for General Electric's LEAP jet engine [7].",
"Further examples of components produced using AM techniques include medical implants [13], [9], air ducts [8], and tooling [12].",
"According to the Wohlers Report [17], a renowned annual survey of advances in AM, in 2015 the AM industry accounted for $5.165 billion of revenue, with 32.5% of all AM-manufactured objects used as functional parts.",
"A study conducted by Ernst & Young [6] shows rapidly growing adoption of this technology worldwide.",
"In the U.S. alone, 16% of surveyed companies have experience with AM and another 16% are considering adopting this technology in the future.",
"The current world leader of AM adoption is Germany with 37% of surveyed companies already using and a further 12% considering AM.",
"Because of AM's dependence on computerization, there is a growing concern that the AM process can be tampered with, in order to sabotage a part's mechanical properties.",
"While several studies have sabotaged a part's mechanical properties in a lab [15], [20], [22], a recent dr0wned study [2] has proven experimentally that a complete sabotage attack is possible.",
"To address the emerging security threat, in this paper we propose and evaluate a novel solution for detecting sabotage attacks in AM.",
"The proposed solution is based on a monitoring of current supplied to individual actuators and the detection of anomalies in this data.",
"The proposed approach has numerous advantages: (i) it is non-invasive in a time-critical process (ii) it can be retrofitted in legacy systems, and (iii) it can be easily air-gapped from the computerized components involved in the AM process, increasing the difficulty of simultaneous compromise.",
"The remainder of this paper is structured as follows.",
"After discussing related work in Section and describing an attack model in Section , we present details of the proposed solution in Section .",
"We present an experimental evaluation of the proposed solution in Section .",
"After discussing the applicability of the proposed solution to industrial-grade metal AM systems in Section , we conclude this paper with a short review and an outline of planned future work."
],
[
"Related Work",
"While researchers have already discussed various threats in the context of AM, such as Intellectual Property (IP) and manufacturing legally prohibited objects, only sabotage attacks and proposed detection methods are of relevance to this paper.",
"Sturm et al., 2014 [15] demonstrated that a part's tensile strength can be degraded by introducing defects such as voids (internal cavities).",
"Zeltmann et al., 2016 [22] showed that similar results can be achieved by printing part of the structure with the contaminated material.",
"Belikovetsky et al, 2017 [2] proposed degrading a part's fatigue life; the authors argue that the defect's size, geometry, and location are factors in the degradation.",
"Yampolskiy et al, 2015 [20] argued that the anisotropy intrinsic to 3D printed parts can be misused to degrade a part's quality, if an object is printed in the wrong orientation.",
"Zeltmann et al., 2016 [22] have experimentally shown the impact of this attack on a part's tensile strength, using 90 and 45 degree rotations of the printed model.",
"Chhetri et al., 2016 [3] introduced a skew along one of the build axes as an attack.",
"Moore et al., 2016 [10] modified the amount of extruded source material to compromise the printed object's geometry.",
"Pope et al., 2016 [11] identified that indirect manipulations like the modification of network command timing and energy supply interruptions can be potential means of sabotaging a part.",
"Yampolskiy et al., 2015 [20] discussed various metal AM process parameters whose manipulation can sabotage a part's quality; for the powder bed fusion (PBF) process, the identified parameters include heat source energy, scanning strategy, layer thickness, source material properties like powder size and form, etc.",
"Slaughter et al., 2017[14] has shown that an indirect sabotage attack is possible via a compromised in-situ infrared thermography; authors evaluate identified attacks on a metal 3D printer that employs the PBF process.",
"Yampolskiy et al., 2016 [21] argued that in the case of metal AM, manipulations of manufacturing parameters can not only sabotage a part's quality, but also damage the AM machine, or lead to the contamination of its environment.",
"Several publications present methods for detecting sabotage attacks.",
"Chhetri et al., 2016 [3] used the acoustic side-channel inherent to the FDM process to detect tampering with a 3D printed object; the authors report that the detection rate of object modifications is 77.45%.",
"Strum et al., 2017 [16] proposed an impedance-based monitoring method.",
"The authors physically coupled a piezoceramic (PZT) sensor to the part being fabricated and measure the electrical impedance of the PZT.",
"These impedance measurements can be directly linked to the mechanical impedance of the part, assisting in detecting in-situ defects of part mass and stiffness.",
"Two further papers built upon the cross-domain attack notion introduced in Yampolskiy et al., 2013 [19], and propose a notion of cross-domain attack detection.",
"Chhetri et al., 2017 [4] demonstrated the flow of information between the cyber and physical domains and how this information can be used for performing cross-domain security analysis.",
"By estimating this relationship, the model can be used for the detection of new cross-domain attack models and attack detection techniques.",
"Wu et al., 2017 [18] leverage machine learning methods to detect cyber-attacks in the manufacturing process.",
"The authors have used vision and acoustics as the data sources for machine learning algorithms and were able to detect anomalies with high accuracy (96.1% and 91.1% respectively)."
],
[
"Threat Model",
"3D printed objects generally begin as digital 3D models, stored on a computer.",
"The most common file format remains .STL, with emerging file formats like .AMF or .3MF offering better accuracy and additional features like color.",
"For a 3D printing job, the 3D model is first “sliced” by a dedicated software (like Slic3r) into layers; for each layer a toolpath is generated that defines exactly what actuators (motors and similar 3D printer components) should act and in what sequence.",
"Afterwards, the PC communicates to the 3D printer either the whole toolpath or individual commands (for desktop 3D printers, usually in G-code).",
"For the communication either network protocols or a USB connection is utilized.",
"All commands are then interpreted and executed by the firmware installed on the 3D Printer.",
"If a command requests motion from an analog actuator, the firmware “translates” it to the actuator's input power supply characteristics, such as frequency and voltage.",
"For sabotage attacks, researchers have compromised various elements of the outlined 3D printing workflow.",
"Moore et al., 2016 identified numerous vulnerabilities in software, firmware, and communication protocols often employed in desktop 3D printing niche.",
"Belikovetsky et al., 2016 [2] used a phishing attack to enable remote access to the controller PC.",
"Sturm et al., 2014 [15] used malware running on the controller PC to automatically modify STL files.",
"Do et al., 2016 [5] exploited weaknesses in the communication protocol between the controller PC and a 3D printer and were able to cancel a print or submit a new job.",
"Moore et al., 2017 [10] presented a wide range of attacks possible through 3D Printer firmware compromise.",
"Figure: Considered Threat ModelWe consider the following threat model (summarized in Figure REF ): As a lesson learned from the demonstrated attacks, we assume that any computerized element in the AM workflow (controller PC, 3D printer, computer network) can be compromised.",
"Analog actuators (e.g., stepper motors) cannot be compromised via cyber means.",
"We assume that actuators behaving according to characteristics of analog input (such as Variable Frequency Drives, VFD) are not compromised.",
"Destructive testing can provide high-confidence data about mechanical characteristics and geometry of a manufactured 3D object.",
"This information will be used to validate that the 3D printing process has not been tampered with.",
"This approach is similar to the one used by Agrawal et al., 2007 [1] for detecting hardware Trojans using IC fingerprinting.",
"Our detection system (that includes induction probes, oscilloscope, and a monitoring PC for data analysis) is air-gapped from the manufacturing environment and is not compromised.",
"Results from destructive testing can be used to (manually) confirm the benign nature of a recorded manufacturing process.",
"Electrical connections (of power monitoring system and of 3D Printing environment) are not physically tampered with, so that they are identical during both unaltered and maliciously modified manufacturing."
],
[
"Proposed Solution",
"In this section, we outline the proposed approach, and provide details on how—according to our proposal—power supply signatures should be generated and compared."
],
[
"Considered Approach",
"Our approach hinges on the direct, causative relation between the amplitude and frequency of current delivered to a motor and the motor's rotation.",
"Any G-code move command specifies an (X,Y,Z) position to move the extruder head to, a speed to move at, and a path to get there.",
"When these commands are received by the printer, the firmware translates them into a series of motor activations.",
"The firmware communicates with the on-board motor controllers, which deliver current at a given frequency and amplitude to actuate the motors.",
"This translation from G-code, to motor activations, to current, is fully deterministic.",
"Moreover, a set of delivered current over time for each motor will always result in the same printed object.",
"The current in each motor is therefore the penultimate representation of the object, which began as a 3D model file.",
"While various factors, including the mechanical arrangement of the motors, the filament, and the temperature of the extruder and bed, may influence the translation from current to physical object, this analog representation is not alterable by cyber means.",
"The current passing through a wire may be measured, nonintrusively, using a current transformer.",
"It can be sampled by an oscilloscope or digitizer; the sequence of samples for each motor circuit across the duration of a print is a measurement of the motor-current representation of the printed object.",
"Each trace in this representation is a series of (time, amplitude) data.",
"If we align traces at some consistent starting point, e.g., the beginning of the first print layer, we should see the following: for prints with identical G-code, the traces will be equivalent, with periods of activity occuring at the same time and having the same frequency, with some level of desync due to the limited precision of the equipment.",
"For prints with wholly different G-code, any alignment of the traces will be accidental and temporary; the difference in amplitude at any given time will vary from 0 to the full amplitude of the signal.",
"It follows from the above that prints with identical G-code, apart from some number of malicious modifications, will behave like this: up until the first modified command, the traces will display the same average deviation from each other as a comparison of two unmodified prints.",
"When the first altered command is reached, the deviation will begin to vary across the full amplitude range.",
"Depending on the duration of the command, the traces could either re-sync, or continue to produce the full range of deviations.",
"We propose an attack detection method based on the comparison of motor current traces.",
"In brief, the method requires generating traces from several known-good prints, collecting the current traces of subsequent prints of the same object on that printer, and comparing the captured traces against the known-good traces.",
"Modifications are identified by deviations between the captured and known-good traces being substantially different from the standard deviation.",
"In the remainder of this section, we describe the individual stages in more detail."
],
[
"Trace Generation",
"For a single combination of G-code and printer, a set of traces must be captured for each motor (or other actuator) involved in the printing process.",
"The sample rate of the trace must be above the Nyquist rate, i.e., at least double the rate of the highest non-noise frequency component in the signal.",
"The size of the trace set is determined by the statistical power of the method.",
"In general, it will depend on the measurement error, standard deviation between “normal” traces, expected deviation of a malicious trace, and the acceptable false positive and negative rates.",
"An assumption of the method, as in the attack model, is that the printed objects can be physically verified.",
"If it is impossible to verify every required property from a single object, it is sufficient to verify them from the same pool of objects created while capturing the known-good trace data.",
"When printing the potentially compromised object, a trace for each actuator must be captured, meeting the same standards as the known-good traces."
],
[
"Comparison of Traces",
"Before comparing the traces, they must be time-aligned and preprocessed.",
"Time-alignment can be done by pattern matching software, but is more easily achieved by a consistent hardware signal from the printer.",
"Preprocessing should involve smoothing the traces to minimize the impact of sampling noise and error on the comparison.",
"The smoothing method and strength must preserve the meaningful components of the signal, i.e., those that strongly impact the operation of the motor or actuator.",
"In the case of multiphase AC motors, these would be the primary frequency components and their harmonics.",
"For each captured trace, compute the deviation across equivalent sample points on a known-good trace from the same motor.",
"If the captured trace does not correspond to unmodified G-code, it will produce a deviation over time similar to the standard deviation between known-good traces from that motor.",
"If it does correspond to modified G-code, the deviation will fluctuate across a much larger range starting at the first modified command.",
"The detection threshold can be set in several ways; a simple version places a boundary above the highest peak in the known-good deviation, and detects a print as malicious if a certain number of samples have a larger deviation.",
"It may be useful to normalize the deviation of the captured trace to the standard deviation, or to produce a measure of accumulated deviation over time.",
"These and other analysis methods are discussed in Future Work."
],
[
"Experimental Evaluation",
"In this section, we first present our experimental setup and describe the experiments performed.",
"We then present and analyze our results."
],
[
"Experimental Setup",
"Figure REF presents our experimental setup.",
"We are measuring the current delivered by a single phase of the X, Y, and Z axis motors, along with the extruder motor.",
"To avoid reducing the delivered current or introducing phase shifts, we use a noninvasive device: a Tektronix A622 AC/DC current probe.",
"It is connected to a Teledyne LeCroy Waverunner 610 Zi oscilloscope, running at a 25 KS/s sampling rate.",
"According to our preliminary evaluation, this was vastly in excess of the major frequency components, which were expected in the 0-200 Hz range.",
"We are therefore we above the Nyquist rate for accurately sampling these signals, which is double the highest frequency component.",
"As there were no difficulties collecting at the higher sampling rate, we did not reduce it.",
"To better synchronize our collected traces, we implemented a trigger signal.",
"We selected the 3D Printer's extruder fan, as it was not one of the actuator manipulations under test.",
"The fan control line was pulled to an external resistor, and set to generate a falling edge immediately before printing the first layer.",
"This triggered the oscilloscope to begin sampling.",
"The experiments were run using a Printrbot Plus 1404 desktop 3D printer, employing Fused Deposition Modeling (FDM) technology.",
"The printer uses Nema 17 stepper motors, rated for 4.2V and 1.5 A per phase.",
"A desktop PC using the Repetier-Host software controlled the print jobs.",
"Riscure Inspector performed the data preprocessing and analytics."
],
[
"Experiments Performed",
"The goal of the experiments is to test the proposed solution and to evaluate its sensitivity threshold to manipulations.",
"While maliciously inserted print defects might be small, as discussed in the literature, creating them can introduce significant changes to the G-code.",
"This is because a slicer will have to “work around” gaps or other changes.",
"To produce smaller and more controllable manipulations, we have altered individual G-code commands within a single print job.",
"We first designed a benign object, a 10 layer cube with a honeycomb fill.",
"For each of the X, Y, Z, and extruder motors, we collected a minimum of 10 traces of the object being printed, representing 40 total prints.",
"These measurements established a baseline or “golden” measurement for the motors.",
"We next created manipulated copies of this object, with the following modifications: Insertion of a new G-code command, a G0 move, in layer 7.",
"Deletion of a G-code command present in the original STL file, a G1 move in layer 7.",
"A reordering of two G-code commands present in the original STL file, two G1 moves in layer 7 and two more in layer 8.",
"Replacement of a G1 command for simultaneous movement and filament extrusion with the G0 movement only.",
"We collected at least 3 traces of each malicious print per motor.",
"A comparison against a known good trace produced the deviation over time.",
"The detection threshold for attacks is 0.1 amps above the peak amplitude of the normal traces' standard deviation."
],
[
"Experimental Results",
"The experimental results are summarized in Table REF and described below in detail.",
"All trace captures have been smoothed by a moving average filter spanning 20 samples.",
"Table: Results of the detection method on the considered attacks."
],
[
"Normal Operation Traces and Standard Deviations",
"The X and Y motors' normal operation produced a signal similar to our expectations.",
"These motors are active throughout most of the print.",
"In Figure REF , a smaller timescale shows that the active sections are strongly periodic, separated by constant-level sections with high frequency noise.",
"The standard deviation of the X and Y traces (Figure REF ) varies over time, but remains below 0.5 A until after the print is completed at approximately 75 seconds.",
"Figure: Normal X motor operation.Figure: Standard deviation of X traces over time.The Z motor traces show a different usage pattern.",
"After the beginning of the print, the Z motor is active infrequently (Figure REF ); this matches the common-sense observation that the Z motor is used only at layer transitions.",
"This effect also shows in the standard deviation plot (Figure REF ).",
"The deviation remains constant for long periods, and is higher than the standard deviation of the X and Y motor traces while inactive.",
"This indicates the constant current levels held between active periods are less consistent for the Z motor.",
"As observed in the other motors (Figure REF ), the level holds where the previous periodic section ended.",
"Figure: Standard deviation of Z traces over time.Traces from the extruder motor show lower-frequency operation with very few interruptions, but a consistently higher standard deviation (Figure REF ).",
"Figure: Standard deviation of extruder traces over time."
],
[
"Insertion Attack",
"The results of our method applied to an Insertion modified object are in Figure REF .",
"The execution of the extra command occurs between 52 s and 53 s in the second trace; traces 1 and 2 are desynchronized by the duration of the inserted command.",
"This appears in the difference plot as an increase in amplitude from below 0.5 A to nearly 2 A.",
"After reducing the difference plot by subtracting the standard deviation of normal X traces, the malicious section is still clearly visible.",
"This attack was detected on the X and Y motors by a safe margin, as seen in Figure REF .",
"It was not detected on the Z or Extruder motors, as seen in Figure REF .",
"The attacks were visible in the Z and Extruder traces, but did not result in larger deviations during or after the attack period.",
"The detection failure on the Z trace is due to the very brief active period of the Z motor.",
"The time delay of a single command insertion is enough to misalign the motor activity, producing the duplicated spikes seen in the final plot of Figure REF .",
"Detection failure on the extruder motor is more likely due to the already high standard deviation of the normal traces (Figure REF ).",
"The Extruder signal shows much less synchronization than other motors.",
"As is seen more clearly in the Void attack (Figure REF ), periods of inactivity or misalignment in the extruder do not show substantially higher deviation.",
"Figure: Z motor behavior during an Insertion attack."
],
[
"Deletion Attack",
"A Deletion attack produces similar deviations in all motors to the Insertion attack; the amount of desynchronization introduced is nearly identical.",
"The detectability is the same in each case."
],
[
"Reorder Attack",
"A Reorder attack produces the expected reversal of two adjacent sections of samples, but also results in a greater than average deviation after the G-code commands return to normal (Figure REF ).",
"This may be because the reordered commands were both move commands, with a starting and ending point specified.",
"Swapping them creates a greater distance between the endpoint of one move and the start point of the other, causing a delay.",
"Detectability of the attack is identical to the Insertion and Deletion attacks.",
"The high-activity X and Y motors show significant deviations due to the misaligned active sections.",
"The Z motor also shows misalignment, but the short activity periods result in separate, smaller deviation spikes.",
"The extruder motor deviation is even smaller than in the Insertion or Deletion attacks, but the inactive time is still visible."
],
[
"Void Attack",
"During the Void attack, the inactive time due to the modification, while similar in length to the X and Y motors in other attacks, is only a fraction of a single cycle (Figure REF ).",
"The deviation across these sections is not substantially higher than the maximum deviation throughout the print.",
"This attack is not detected in the X, Y, and Z traces; the deviation is below the threshold set by the standard deviation for each.",
"It was also not detected in the extruder trace, but was clearly visible.",
"The X, Y, and Z detectability is expected for this attack; the modified code did not alter the movement of the print head or bed z-level at all.",
"The inability of the method to detect changes in the extruder trace is due to the already high deviation of these traces, and that the attacks do not significantly misalign the traces.",
"In fact, the extruder activity was almost 180$^{\\circ }$ out of phase before the attack section.",
"The break in periodicity for the duration of the attack is clear, however (Figure REF , final plot).",
"Figure: Extruder motor operation during a Void print."
],
[
"Discussion",
"While the proposed approach has shown impressive results in the tested experimental setup, two questions should be discussed: (i) limitations of the approach, and (ii) its applicability to metal AM systems."
],
[
"Approach Limitations",
"Detection of effects on the Z-axis and extruder motors was not successful with an absolute-deviation-based threshold, although the effects were visible in the current traces.",
"As the deviation is a result of comparing out of phase periodic signals, a frequency-based measure may better detect these effects while still detecting the effects on the X and Y motors.",
"It is possible that alterations to the G-code can produce a deviation in the trace that is not detected by our method, which still results in a malicious effect.",
"For example, an accumulation of slight modifications to the movement speed of the X, Y, and Z motors may eventually deposit significantly more filament; the method would need to also detect small but prolonged deviations in this case.",
"Testing a wider range of possible G-code modifications would verify this.",
"Our methodology assumes an unmediated relationship between the parameters of the G-code commands and the resulting actuation signal.",
"This is not the case in closed-loop control systems, where the actuator signal is a function of both the input commands and sensor feedback.",
"In such cases, signals in the feedback loop would also need to be captured."
],
[
"Applicability to Metal AM Systems",
"While the proposed approach has shown good results for FDM technology, this technology is predominantly used with plastics, a source material that has little relevance for safety-critical applications.",
"Two other AM processes are dominating the field in metal additive manufacturing: Powder Bed Fusion (PBF) and Directed Energy Deposition (DED).",
"In PBF, a thin layer of powder is deposited in a bed and the next layer's profile is melted with a laser or electron beam.",
"Applying the proposed approach to a PBF system will likely require additional instrumentation of heat sources, in order to detect changes in scanning strategies: Selective Laser Sintering/Melting (SLS/SLM) might require a camera and an infrared image recognition system; monitoring Electron Beam Melting (EBM) might utilize EM emanations instead.",
"DED systems employ a multi-axis arm, through a nozzle mounted on its end melted material is deposited onto a surface, where it solidifies.",
"The source material is in either wire or powder form; it is melted using a laser, electron beam, or plasma arc.",
"The proposed approach may be directly applicable for sabotage attack detection in DED systems if the motors controlling the arm can be measured using the same techniques."
],
[
"Conclusion and Future Work",
"Additive Manufacturing (AM), a.k.a.",
"3D Printing, is increasingly adopted around the world and used to produce functional parts of safety-critical systems.",
"Because of AM's dependence on computerization, there is a growing concern that the AM process can be tampered with, in order to sabotage a part's mechanical properties.",
"To address this threat, we proposed a novel approach for detecting sabotage attacks in manufacturing systems.",
"Our approach is based on the continuous monitoring of current supplied to all actuators during the manufacturing process and detecting anomalies compared to a provable benign process.",
"The proposed approach has numerous advantages: (i) it is non-invasive in a time-critical process (ii) it can be retrofitted in legacy systems, and (iii) it is air-gapped from the computerized components involved in AM process, increasing the difficulty of a simultaneous compromise.",
"We have evaluated the proposed approach on a desktop 3D Printer employing Fused Deposition Modeling (FDM) technology.",
"We monitored power supply to four motors: the X/Y/Z-axis motors, and the filament extrusion motor.",
"Our results show that the insertion, deletion, and reordering of individual G-code movement commands can be detected with 100% precision through the X and Y motors.",
"Modifications to extrusion rate are visible, but not detectable with the current method.",
"This method can detect attacks depending on these modifications, which includes many of the void-insertion attacks discussed in the literature and any attacks relying on modifications to a 3D model file.",
"In our future work, we plan to overcome the identified limitations, including restriction to open-loop AM systems, switching to a frequency-based deviation measure, and accounting for the gradual accumulation of deviations.",
"We will also test the method against other FDM printers, and adapt it for other printing technologies, such as Powder Bed Fusion and Directed Energy Deposition.",
"The demonstrated anomaly detection performance and the potential applicability to metal AM systems makes the proposed approach an important milestone to ensure AM security in safety-critical systems."
]
] | 1709.01822 |
[
[
"A First Step Towards Effectively Nonperturbative Scattering Amplitudes\n in the Perturbative Regime"
],
[
"Abstract We propose an effectively nonperturbative approach to calculating scattering amplitudes in the perturbative regime.",
"We do this in a discretized momentum space by using the QSE method to calculate all the contributions (to all orders in perturbation theory) to the scattering eigenstates that are above a precision cutoff.",
"We then calculate the scattering amplitude by directly taking the inner product between these eigenstates.",
"In the current work we have analyzed this procedure for a $\\lambda\\phi^4$ theory in one spatial dimension and compared our results with perturbation theory obtaining favorable results suggestive that further research in this direction might be worthwhile.",
"In particular, we show that the efficiency of our method scales much better than second- and higher-order perturbation theory as the momentum lattice spacing decreases and as the eigenstate energy increases."
],
[
"Results and Comparison with First-Order Perturbation Theory",
"In order to make this concrete, we have focused on three eigenstates of the Hamiltonian [Eq.",
"(REF )].",
"The first is the vacuum, which is the state of lowest energy.",
"We call this eigenstate $\\Psi _v$ .",
"It can be seen in the first row of Fig.",
"REF .",
"(We will describe this figure in detail throughout this section.)",
"The second is the two-particle eigenstate where the two particles have equal but opposite momentum.",
"We chose the magnitude of each momentum to be $24m$ , where $m$ is the mass parameter.",
"We label this eigenstate $\\Psi _{24,-24}$ .",
"For our numerical calculations, we have taken $m=1$ .",
"The reason we chose this state was that it was a scattering-type eigenstate far above the vacuum where the Hilbert space is too large to directly diagonalize on a personal computer, yet it was still low enough that second- (and even third-) order perturbation theory could still be achieved for comparison.",
"This two-particle state is shown in the second row of Fig.",
"REF .",
"The third state we chose is the parity-even four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}=(\\Psi _{24,-8,-8,-8}+\\Psi _{8,8,8,-24})/\\sqrt{2}$ .",
"It is a linear combination of two eigenstates related by a parity transformation.",
"The first has one particle of momentum $24m$ and three particles of momentum $-8m$ and the second has the same momenta but with the opposite sign.",
"We chose this eigenstate because it was close in energy to our two-particle eigenstate, and was thus a potential final state of a scattering event.",
"In other words, we were interested in calculating an S-Matrix element between a two-particle eigenstate and a four-particle eigenstate and this seemed like a good candidate for that calculation.",
"We display this four-particle eigenstate in the third row of Fig.",
"REF .",
"The left column of Fig.",
"REF is the result of a first-order perturbation-theory calculation (see App.",
"for a brief review) while the right column is the result of ten iterations of our cyclic QSE code (described in App. ).",
"Although they are very similar, there are small, but significant, differences.",
"Our code fills in basis states missed by first-order perturbation theory.",
"We will show (see Sec. )",
"that these points are included in second- and third-order perturbation theory, but first, we would like to describe these plots for each eigenstate and the differences between the left and right columns (the results of first-order perturbation theory and our QSE code, respectively.)",
"The vertical axis of these plots is the absolute value of the coefficient [e.g.",
"see Eq.",
"(REF )] of the basis states while the horizontal axis gives the free-particle energy of the basis states normalized by the mass.",
"The numerical calculations done to produce these plots used a momentum spacing of $\\Delta p=2m$ .",
"This is a rather large momentum spacing and requires explanation since smaller values of $\\Delta p\\ll m$ give better physical results as described in [10].",
"Although this is true, perturbation theory becomes impossible for these scattering states at such small momentum spacings and a critical objective of the current paper is to compare this method with perturbation theory for these scattering states.",
"The final parameter in our calculation is the coupling constant $\\lambda $ .",
"We took a value of $\\lambda =0.1m^2$ for the plots in Fig.",
"REF .",
"We will consider other values of $\\lambda $ in Sec. .",
"We begin by describing the vacuum, which is the top row of Fig.",
"REF .",
"This is the state of lowest energy.",
"We found its energy to be 0 at first order in perturbation theory and -0.000119m in our cyclic QSE code.",
"This is inline with higher orders in perturbation theory as we will describe in the next section.",
"The color coding of the points is as follows: black, blue, red, green and yellow dots represent 0-, 2-, 4-, 6- and 8-particle basis states.",
"We see that the vacuum is dominated by a 0-particle basis state, with a coefficient of very nearly 1 (all of the eigenstates described in this paper are normalized so that the sum of the squares equals 1).",
"If the coupling constant $\\lambda $ were equal to zero, the vacuum would be identically equal to the 0-particle basis state and the contribution from all other basis states would vanish.",
"As the coupling constant is turned on and grows in size, the contribution from the other basis states grows along with it.",
"The next most important basis states to the vacuum are 4-particle basis state seen in red.",
"The left-most red point is the basis state with 4 free particles at rest.",
"It contributes to the vacuum with a coefficient of approximately -0.00255 at first order in perturbation theory (in the left plot) and -0.00244 in our QSE code (in the right plot).",
"(We have normalized the overall phase so that the coefficient of the 0-particle basis state is positive).",
"The basis state directly to its right has 2 free particles at rest and 2 free particles with momenta of $\\pm 2m$ .",
"It's coefficient is also approximately -0.00244 at first-order in perturbation theory (in the left plot) and -0.00237 in our QSE code (in the right plot).",
"We will describe the differences between these coefficients from perturbation theory and our code in greater detail in Sec. .",
"Other 4-particle basis states of greater complexity can be seen exponentially falling off in importance to the right in these plots.",
"Another feature of these plots is that the points come in clusters with a horizontal spacing between the groups.",
"The reason for this is the rather large value of $\\Delta p$ that we chose.",
"In Sec.",
", we will show plots with smaller values of $\\Delta p$ , where this feature will be less pronounced or disappear altogether.",
"The largest difference between the left plots and the right plots is the points that are present in the cyclic QSE code (on the right) but not in first-order perturbation theory (on the left).",
"For the vacuum, there are no 2-particle basis states (blue points) at first order in perturbation theory.",
"This may seem strange but can be easily understood.",
"At first order in perturbation theory, the coefficient of these 2-particle basis states is proportional to $\\langle b_{2p}|V|b_{0p}\\rangle $ where $|b_{2p}\\rangle $ is the 2-particle basis state we are interested in and $|b_{0p}\\rangle $ is the 0-particle basis state [see Eq.",
"(REF )].",
"$V$ is the potential of our Hamiltonian and is the second term of Eq.",
"(REF ).",
"In order to give a nonzero result, the potential $V$ would have to add 2 free particles to the 0-particle basis state.",
"The only term in the Hamiltonian that could do this is the fourth term of the potential, the term with $a^\\dagger _{p_1}a^\\dagger _{p_2}a^\\dagger _{p_3}a_{-p_4}$ .",
"However, we see that this term has an annihilation operator at the right-most position and, therefore, annihilates the 0-particle basis state on the right, giving zero for this coefficient.",
"As a result, the 2-particle basis states do not contribute to the vacuum until at least second order in perturbation theory.",
"Our QSE code has no problem discovering these points and found two above the cutoff.",
"The left-most blue point is the basis state with two free particles at rest and has a coefficient of 0.000156 while the one directly below and to the right of it has two free particles with momenta $\\pm 2m$ and has a coefficient of 0.000031.",
"Other 2-particle basis states contribute below the cutoff.",
"Finally, we also see contributions from 6-particle basis states (green points) in the plot on the right.",
"These were also missed by first-order perturbation theory for a similar reason.",
"Their coefficients are proportional to $\\langle b_{6p}|V|b_{0p}\\rangle $ and $\\langle b_{8p}|V|b_{0p}\\rangle $ , respectively, where $|b_{6p}\\rangle $ is a 6-particle basis state and $|b_{8p}\\rangle $ is an 8-particle basis state.",
"For these to be nonzero, the potential would have to contain operators that created 6 and 8 particles, respectively.",
"However, the potential in Eq.",
"(REF ) does not have any operators of this form.",
"All of these basis states (missing in first-order perturbation theory but present in our QSE code) receive contributions at second order in perturbation theory.",
"Their coefficients at second order, for example, are proportional to $\\langle b_j|V|\\psi ^1\\rangle $ [see Eq.",
"(REF )] where $|b_j\\rangle $ is any of the basis states that contributes only at second order and $\\psi ^1$ is the first-order perturbative eigenstate.",
"For the vacuum, the left-most green point is the basis state with six free particles at rest.",
"It's coefficient is 0.000068.",
"The green point just to the right has four free particles at rest and two free particles with momenta $\\pm 2m$ with a coefficient of 0.000059.",
"Other green points have exponentially smaller contributions as the free energy of the basis states increases to the right.",
"The second row of Fig.",
"REF shows the eigenstate with two particles of momenta $\\pm 24m$ .",
"We found its energy to be 48.0417m at first order in perturbation theory and 48.0416m in our cyclic QSE code.",
"As expected, it is dominated by the basis state with two free particles of momenta $\\pm 24m$ , which has a coefficient of 0.99994 at first order in perturbation theory (on the left) and the same 0.99994 in our cyclic QSE code (on the right).",
"If the coupling constant $\\lambda $ were set equal to zero, this coefficients would become 1 and all others would vanish.",
"The 0-particle basis state does not contribute to this state at first order in perturbation theory.",
"The reason is that it's contribution is proportional to $\\langle |V|24m,-24m\\rangle $ .",
"This would only be nonzero if $V$ contained an operator that removed two free particles.",
"The only operator in Eq.",
"(REF ) that does this is $a^\\dagger _{p_1}a_{-p_2}a_{-p_3}a_{-p_4}$ which completely annihilates $|24m,-24m\\rangle $ , and so gives zero.",
"(This is the hermitian conjugate of the reason the 2-particle basis state did not appear in the vacuum at first order.)",
"It does contribute at higher order, however, and our cyclic QSE code discovers it but finds its contribution to be below the cutoff.",
"Therefore, it does not appear in either plot.",
"The most important 4-free-particle basis states, that are above the cutoff, are found by first-order perturbation theory and appear in both plots.",
"The most important one is $|24m,-6m,-8m,-10m\\rangle _+$ [where throughout this paper $|p_1,p_2,\\cdots \\rangle _+=\\left(|p_1,p_2,\\cdots \\rangle +|-p_1,-p_2,\\cdots \\rangle \\right)/\\sqrt{2}$ ], whose coefficient is -0.00373 at first order in perturbation theory and -0.00351 in our cyclic QSE code.",
"After this is the basis state $|24m,-4m,-6m,-14m\\rangle _+$ with a coefficient of -0.00348 at first order in perturbation theory and -0.00328 in our cyclic QSE code.",
"Other 4-particle basis states have coefficients that fall off exponentially as their free energy increases or are below the cutoff entirely.",
"This is true for the other eigenstates as well.",
"The 6-particle basis states begin with the basis state $|24m,-24m,0,0,0,0\\rangle $ which contributes with a coefficient of -0.00255 at first order in perturbation theory and -0.00241 in our cyclic QSE code.",
"Following this is the basis state $|24m,-24m,1m,0,0,-1m\\rangle $ with a coefficient of -0.00244 at first order in perturbation theory and -0.00236 in our cyclic QSE code.",
"Other 6-particle basis states contribute at exponentially smaller levels as their free energy increases as can be seen in the figure.",
"The third row of Fig.",
"REF shows the parity-even 4-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ .",
"We found its energy to be 48.2095m at first order in perturbation theory and 48.2090m in our cyclic QSE code.",
"As expected, its most significant contribution is from the parity-even basis state $|24m,-8m,-8m,-8m\\rangle _+$ with a coefficient of 0.99239 at first order in perturbation theory and 0.99508 in our QSE code.",
"The next two most important basis states are also 4-particle basis states.",
"The first is the basis state $|24m,-6m,-8m,-10m\\rangle _+$ with a coefficient of 0.11869 at first order in perturbation theory and 0.09477 in our QSE code.",
"After this is the basis state $|24m,-4m,-8m,-12m\\rangle _+$ with a coefficient of 0.02662 at first order in perturbation theory and 0.02217 in our QSE code.",
"There are several more 4-particle basis states contributing at lower values.",
"Some appear only in the right plot.",
"An example of this is the basis state $|24m,-6m,-6m,-12m\\rangle _+$ .",
"The reason this does not appear at first order in perturbation theory is because its coefficient is proportional to $\\langle 24m,-6m,-6m,-12m|V|24m,-8m,-8m,-8m\\rangle $ .",
"For this to be nonzero, $V$ would have to have an operator that annihilates 3 free particles and creates 3 new free particles, but there is no such operator in $V$ .",
"The 0-particle basis state does not contribute above the cutoff.",
"There is one 2-particle basis state above the cutoff.",
"It is $|24m,-24m\\rangle $ .",
"It has a coefficient of 0.00154 at first order in perturbation theory and 0.00106 in our QSE code.",
"We will see in the next section, however, that these coefficients are inline with higher orders of perturbation theory.",
"This basis state is the main basis state for the state $\\Psi _{24,-24}$ .",
"Its large coefficient is the reason we chose this state for our study.",
"Other 2-particle basis states fall below the cutoff.",
"The largest 6-particle basis state is $|8m,8m,4m,2m,2m,-24m\\rangle _+$ with a coefficient of 0.00884 at first order in perturbation theory and 0.00802 in our QSE code.",
"The next largest is $|8m,8m,8m,0,0,-24m\\rangle _+$ with a coefficient of 0.00726 at first order in perturbation theory and 0.00676 in our QSE code.",
"Other 6-particle basis states give smaller contributions.",
"The most important 8-particle basis state is $|8m,8m,8m,2m,0,0,-2m,-24m\\rangle _+$ with a coefficient of 0.00242 at first order in perturbation theory and 0.00226 in our QSE code.",
"After this comes the basis state $|8m,8m,8m,0,0,0,0,-24m\\rangle _+$ which contributes with a coefficient of 0.00253 at first order in perturbation theory and 0.00222 in our QSE code.",
"A few other 8-particle basis states contribute to the eigenstate with smaller coefficients.",
"Our code did not find any basis states with 10 or more free particles above the cutoff.",
"As we will discuss in the next section, neither did second- or third-order perturbation theory.",
"We also calculated the inner product between $\\Psi _{24,-24}$ and $\\Psi ^+_{24,-8,-8,-8}$ .",
"We remind the reader that the eigenvalues ($48.0416m$ and $48.2909m$ respectively) are not exactly the same and so we expect the result to be zero.",
"This is a symptom of the very large $\\Delta p$ that we used for this calculation.",
"If we magnify the plot of $\\Psi _{24,-24}$ around the basis state $|24m,-24m\\rangle $ , as in Fig.",
"REF , Figure: A magnification of the right middle plot of Fig.",
".we see that there is a gap in basis states around $|24m,-24m\\rangle $ .",
"In fact, the nearest basis state in this plot is $|24m,-8m,-8m,-8m\\rangle _+$ at the left-most edge of the red points.",
"There are other 4-particle basis states that are closer in free energy to $|24m,-24m\\rangle $ .",
"For example, the basis state $|12m,12m,-12m,-12m\\rangle $ has a free energy of 48.17m.",
"However, its contribution to $\\Psi _{24,-24}$ is below the cutoff, and therefore does not appear in this plot.",
"Because of this, the eigenstate $\\Psi _{12,12,-12,-12}$ which is dominated by this basis state has an inner product with $\\Psi _{24,-24}$ of $7\\times 10^{-7}$ at first order in perturbation theory and $5\\times 10^{-7}$ in our cyclic QSE code, both below the precision of this calculation.",
"In other words, their inner product is consistent with zero.",
"We chose, instead, to focus on the eigenstate $\\Psi ^+_{24,-8,-8,-8}$ because its main basis state $|24m,-8m,-8m,-8m\\rangle _+$ had a larger contribution at leading order in perturbation theory to $\\Psi _{24,-24}$ and would be a better test of the eigenstates.",
"In fact, we found that the inner product between $\\Psi ^+_{24,-8,-8,-8}$ and $\\Psi _{24,-24}$ was 0.00565 at first order in perturbation theory and $5.7\\times 10^{-6}$ in our QSE code.",
"At first order in perturbation theory, we find that the result is above the precision of the calculation, which is approximately $10^{-4}$ .",
"(The reason this is the approximate precision of the calculation is because we did not include basis states below this cutoff.)",
"The reason first-order perturbation theory gave a nonzero result for this inner product is that it missed several important basis states, that are above the cutoff as can be seen in the comparison between the left and right plots of Fig.",
"REF and discussed in this section.",
"On the other hand, our cyclic QSE code is able to find all the basis states contributing above this cutoff, and therefore, the calculation of the inner product using our QSE code is in agreement with zero at the precision of the calculation.",
"These important basis states are filled in by higher orders in perturbation theory and the inner product is in agreement with zero at the precision of the calculation at higher order, as we will describe in the next section."
],
[
"Comparison with Higher Orders of Perturbation Theory",
"We also compared the results of our cyclic QSE code with second- and third-order perturbation theory.",
"We review perturbation theory in App.",
"including a subtlety regarding nearly degenerate basis states.",
"However, because of its importance for our calculation, we will also briefly discuss it here.",
"The textbook formulas for time-independent perturbation theory (e.g.",
"[34]) are valid when no basis states are degenerate or nearly degenerate with the main basis state.",
"At first order, we have no problem with this issue and that is why we have not brought it up before this point.",
"However, beginning at second order, this is an issue for the two- and four-particle eigenstates, $\\Psi _{24,-24}$ and $\\Psi ^+_{24,-8,-8,-8}$ .",
"We get very good agreement between naive perturbation theory and our QSE code for basis states whose free energies are not nearly degenerate with the main basis state, but very bad agreement for the basis states that are nearly degenerate.",
"The reason for this is a breakdown of perturbation theory for the degenerate and nearly degenerate basis states.",
"The textbook way to handle this at first order is to explicitly diagonalize the degenerate and nearly degenerate basis states before doing perturbation theory.",
"This gives a slightly better than naive perturbation theory for the degenerate and nearly degenerate basis states but the perturbative result for the other basis states.",
"This works well at first order, but our problem does not appear until second order, where the diagonalization step would require basis states that connect the nearly degenerate basis states (via the Hamiltonian), and these intermediate basis states are not necessarily nearby in free energy.",
"Our approach is to find all the basis states given by the naive first- and second- (and third-) order perturbation theory, that are above the cutoff with the naive formulas.",
"This already includes both the nearly degenerate basis states and the basis states that connect them (that are above the cutoff).",
"We then simply diagonalize the Hamiltonian with respect to this entire set.",
"We have done this at both second and third order and find that, for our eigenstates, the results are identical since the basis states above the cutoff are the same for second and third order, and therefore, their diagonalization is the same.",
"Since the results of diagonalized third-order perturbation theory are identical to those of diagonalized second-order perturbation theory, we only show plots for second order.",
"For the rest of this paper, we will assume that whenever we talk about perturbation theory beyond first order, we are discussing the diagonalized versions of perturbation theory.",
"The columns of Fig.",
"REF are as follows.",
"The left column shows the absolute value of the difference between our QSE-code results and first-order perturbation theory.",
"The right column shows the absolute difference with second-order perturbation theory.",
"We used the same parameters as in the previous section, $\\lambda =0.1m^2$ and $\\Delta p=2m$ .",
"Once again, the reason we chose such a large value of $\\Delta p$ in these first two sections was to enable comparison with perturbation theory all the way to the third order.",
"We will discuss timing further in Sec.",
", but for now we note that with these parameters, third-order perturbation theory took approximately seven and a half hours and grows exponentially as $\\Delta p$ becomes smaller.",
"The three rows represent the same eigenstates as in Fig.",
"REF and are the vacuum $\\Psi _v$ at the top, the two-particle eigenstate $\\Psi _{24,-24}$ in the middle and the four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ at the bottom.",
"The dashed blue lines give the cutoff below which we did not keep basis states and forms an approximate precision for our calculation.",
"This line is the same as in Fig.",
"REF .",
"We can not expect to do any better than this line and the distance below the line is not significant.",
"What is more important are the points above the line.",
"These are the basis states which are present in either perturbation theory or our QSE code but missing in the other or, if the basis state is present, both in perturbation theory and our QSE code, it is the difference between the coefficients.",
"We begin by focusing on the vacuum, the top row of Fig.",
"REF .",
"We obtained an energy of -0.000124m at second-order which is in agreement with the result from our cyclic code.",
"In the left plot, there are six points above the cutoff line.",
"Two (in blue) are 2-particle basis states, two (red) are 4-particle basis states and two (green) are 6-particle basis states.",
"The 2-(blue) and 6-(green) particle basis states are missed by first-order perturbation theory as described in Sec. .",
"Because of this, their difference with our code is the full size of their coefficient, which is above the cutoff.",
"The two red points are found by first-order perturbation theory but are given coefficients that are slightly higher than the value achieved by our cyclic QSE code.",
"If all we had were first-order perturbation theory, it would be impossible to tell whether our QSE code was more or less correct than perturbation theory.",
"This is the reason we also implemented second- and third-order perturbation theory.",
"In the right column of the first row, we see the difference with second-order perturbation theory.",
"We immediately see that the points that were above the cutoff line are now at or below it.",
"The 2- (blue) and 6- (green) particle basis states that were missed by first-order perturbation theory were filled in by second-order perturbation theory.",
"Moreover, the coefficients are in agreement with those of our QSE code within the precision of our calculation.",
"The coefficients of the four-particle basis states (in red) are also now in agreement with our QSE code.",
"We find the same agreement with third-order perturbation theory.",
"We note that where our cyclic QSE code disagrees with first-order perturbation theory, it agrees with higher orders of perturbation theory.",
"This suggests that our QSE code is correctly constructing the vacuum.",
"We now move on to the two-particle eigenstate $\\Psi _{24,-24}$ , which is shown in the second row of Fig.",
"REF .",
"The left column shows the difference with first-order perturbation theory.",
"We see that there are seven 4-particle basis states (in red) and one 6-particle basis state (in green) above the cut off.",
"All of these basis states were included in first-order perturbation theory but their coefficients are slightly higher than in our cyclic QSE code.",
"In the right column, on the other hand, we see that second-order perturbation theory is in agreement with our QSE code results.",
"We also find the same agreement with third-order perturbation theory.",
"Again, the agreement with higher orders of perturbation theory suggests that our QSE code is working properly.",
"Figure: Plots of the zeroth, first and second iteration of our cyclic QSE code.",
"The left column is the result of the first order of perturbation theory and is exactly the same as the left column of Fig. .",
"The middle column is the result of the first iteration of our QSE code and the right column is the result of the second iteration of our QSE code.",
"The rows, the color coding, the dashed line and the axes are the same as in Fig.",
".The four-particle eigenstate $\\Psi ^+_{12,-4,-4,-4}$ , shown in the final row of Fig.",
"REF , is similar to the previous two cases, but with a new twist.",
"At first order of perturbation theory, shown in the left column, there are one 2- (in blue), ten 4- (in red), thirty-seven 6- (in green) and seven 8- (in yellow) particle basis states above the cutoff.",
"All of these, with the exception of thirteen 6- and two 8-particle basis states, are missed by first-order perturbation theory but found by our cyclic QSE code and therefore their difference is above the cutoff.",
"The other fifteen, the thirteen 6- and two 8-particle basis states not missed, were found by both first-order perturbation theory and our QSE code but had slightly different coefficients at first order in perturbation theory.",
"All of the basis states missed by first-order perturbation theory are found at second and third order and all their coefficients are in agreement with our QSE code within the precision of our calculation.",
"The coefficients of the thirteen 6- and two 8-particle basis states are also brought into agreement with our QSE code at second and third order.",
"However, there are four basis states that are missed by both our QSE code and first-order perturbation theory but are found at second and third order.",
"Two of these are 6-particle basis states (green points), one is a 4-particle basis state (red point) and one is a 10-particle basis state (orange point).",
"They can be seen above the blue dashed line in the right plot.",
"The highest of these is the basis state $|24,-4,-4,-4,-4,-8\\rangle _+$ .",
"To understand the reason that our cyclic QSE code missed these points, we remind the reader of two points described in App. .",
"The first is that our code uses a random procedure to discover the important basis states and, therefore, it is always possible that basis states are missed.",
"However, it is also important to note that the way our random procedure chooses new points emphasizes basis states with higher coefficients more strongly than basis states with lower coefficients.",
"Therefore, it is much more likely that a basis state near the precision cutoff is missed than a basis state high above the cutoff.",
"Indeed, the coefficient of this basis state is $1.8\\times 10^{-4}$ (as determined by second- and third-order perturbation theory), which is just barely above the cutoff, which is approximately $1\\times 10^{-4}$ .",
"Whether these points are discovered by our cyclic QSE code or not is very sensitive to the size of the coupling constant $\\lambda $ and the number of new basis states randomly added to the reduced Hilbert space each cycle.",
"We will describe these effects further in Secs.",
"and , respectively.",
"In the previous paragraphs, we have described the differences between perturbation theory and the tenth and final iteration of our cyclic QSE code.",
"We would now like to explore how our cyclic QSE code builds up its final solution as the iterations progress.",
"In Fig.",
"REF , we show the first two iterations of our cyclic QSE code for each of the three states under investigation.",
"In the first column, we display the results of first-order perturbation theory for comparison.",
"This is the initial configuration that our QSE code works with.",
"It is the same as the first column of Fig.",
"REF .",
"We include it for the convenience of the reader.",
"In the second and third columns, we show, respectively, the result of the first and second iterations of our cyclic QSE code.",
"As the iterations progress, our results quickly approach second- and third-order perturbation theory, often filling in most of the missing points within the first few iterations.",
"Focusing on the vacuum (the first row), we remind the reader that first-order perturbation theory misses two 2- (blue) and two 6- (green) particle basis states.",
"We can see these points in the top left plot of Fig.",
"REF .",
"Figure: Plots of the largest differences between our cyclic QSE code and second-order perturbation theory for each iteration.",
"The blue dashed line is the cutoff used in our calculations.As we can see in the middle plot of the first row of Fig.",
"REF , the first iteration of our cyclic QSE code has already found both 2-particle basis states (in blue).",
"By the second iteration, shown in the right plot, one of the 6-particle basis states (in green) has also been found.",
"The other missing 6-particle basis state is found in the third iteration.",
"In the top row of Fig.",
"REF , we show the largest difference between our cyclic QSE code and second-order perturbation theory as a function of the iteration.",
"We see that after the third iteration the largest difference becomes equal with the precision of our calculation and thus our cyclic QSE code is in agreement with second- and third-order perturbation theory.",
"From this point on, all of the important basis states have been found and the cyclic QSE code is randomly generating less important basis states below the cutoff.",
"The two-particle eigenstate $\\Psi _{24,-24}$ (shown in the second row of Fig.",
"REF ) has the unique property that it initially begins with all of the important basis states.",
"Consequently, our cyclic QSE code does not add any new basis states above the cutoff.",
"However, the coefficients coming from first-order perturbation theory are not yet correct.",
"At the first iteration of our cyclic QSE code, although no new basis states are found above the cutoff, our code directly diagonalizes the Hamiltonian with respect to these basis states (as well as some random basis states below the cutoff).",
"This is already sufficient to bring this state into agreement with second- and third-order perturbation theory, as we can see in the second row of Fig.",
"REF .",
"After the first iteration, our cyclic QSE code continues to randomly add basis states below the cutoff.",
"Because no new important basis states are added, and the change to the coefficients is so small, the three plots in the second row of Fig.",
"REF are indistinguishable by eye.",
"Turning now to the four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ , the initial iterations are shown in the third row of Fig.",
"REF .",
"As described earlier in this section, one 2- (in blue), ten 4- (in red), thirty-seven 6- (in green) and seven 8- (in yellow) particle basis states are missing at first order in perturbation theory, and consequently, in the left-most plot.",
"We can see in the middle plot that three of the missing 4-particle basis states, five of the missing 6-particle basis states, and two of the missing 8-particle basis states have been filled in by the first iteration of our cyclic QSE code.",
"In the second iteration, it further adds four 4-particle basis states, six 6-particle basis states, and one 8-particle basis state, as can be seen in the right-most plot.",
"The rest of the missing basis states that are found by our cyclic QSE code appear in the following order: six basis states are found in the third iteration, eight basis states are found in the fourth iteration, zero basis states are found in the fifth through seventh iteration, three basis states are found in the eighth iteration and no basis states are found in the ninth and tenth iterations.",
"In addition to finding missing basis states, the diagonalization of the Hamiltonian improves the coefficients of the basis states as in the previous cases.",
"Since our cyclic QSE code does not find the final four basis state (two green, one red and one orange point above the dashed line in the bottom right plot of Fig REF ), the largest difference between our cyclic QSE code and second- and third-order perturbation theory never goes below the cutoff line, although it gets very close, as seen in the bottom plot of Fig.",
"REF .",
"The height of the black line above the dashed blue line is precisely due to the cyclic QSE code missing these basis states.",
"In Sec., we will describe how this basis state can be found by our cyclic QSE code by increasing the number of new basis states randomly chosen each cycle."
],
[
"Dependence on $\\mathbf {\\lambda }$",
"In this section, we explore the dependence of the eigenstates on $\\lambda $ , the coupling constant.",
"When $\\lambda \\rightarrow 0$ , the theory becomes free.",
"In this limit, the eigenstates become free states with only one basis state (which we call the main basis state) whose coefficient is 1.",
"When $\\lambda $ is turned on, other basis states begin to contribute to the eigenstate and grow with $\\lambda $ while the coefficient of the main basis state very slowly diminishes, but remains close to 1 while the theory remains perturbative.",
"If a basis state is connected to the main basis state by one application of the Hamiltonian, its coefficient grows linearly with $\\lambda $ .",
"This is because it appears at first order in perturbation theory.",
"Similarly, if a basis state is connected to the main basis state by two applications of the Hamiltonian, its coefficient grows quadratically with $\\lambda $ , as expected since it appears at second order in perturbation theory [see Eqs.",
"(REF )-(REF )].",
"We have calculated our eigenstates for a range of values of $\\lambda $ from $0.01m^2$ to $1m^2$ , beginning in the deeply perturbative region and ending in the nearly non-perturbative regime.",
"We show these eigenstates for $\\lambda =0.01m^2, 0.05m^2$ and $0.5m^2$ in Fig.",
"REF while the eigenstate for $\\lambda =0.1m^2$ can be seen in Fig.",
"REF .",
"We can see that the coefficients grow as expected in these plots, as we will describe in greater detail below.",
"Moreover, because the importance of the basis states grow with increasing $\\lambda $ , a greater number of basis states contribute above a fixed precision cutoff.",
"In order to focus on the same region of the eigenstate, we adjust the cutoff for each value of $\\lambda $ .",
"The basis states contributing at first order in perturbation theory give us a natural pattern to follow.",
"We scale the precision of the cutoff linearly with $\\lambda $ and present a table of our cutoffs in Table REF .",
"As in previous plots, these cutoffs are displayed as dashed blue lines in the figure.",
"The same $\\Delta p=2m$ was used in this section as in the previous sections.",
"Table: A table of the coefficient cutoffs used in this section.The vacuum is shown in the first row of Fig.",
"REF .",
"As described in Sec.",
", the four-particle basis states (in red) enter at first order in perturbation theory.",
"As just described, they grow proportional to $\\lambda $ .",
"For example, the two most important 4-particle basis states are $|0,0,0,0\\rangle $ and $|-2, 0, 0, 2\\rangle $ .",
"They have coefficients -0.000254 and -0.000244 (for $\\lambda =0.01m^2$ ), -0.00125 and -0.00120 (for $\\lambda =0.05m^2$ ), -0.00244 and -0.00237 (for $\\lambda =0.1m^2$ ), -0.0107 and -0.0109 (for $\\lambda =0.5m^2$ ), and -0.0194 and -0.0202 (for $\\lambda =1m^2$ ), respectively.",
"As we can see, the scaling behavior is very nearly linear, especially for smaller $\\lambda $ .",
"The small deviation from linearity for larger $\\lambda $ comes from two sources.",
"The first is that as the coefficients grow, the normalization changes.",
"The second is that although the leading contribution to these coefficients is at first order in perturbation theory, there is also a subleading contribution at second order.",
"This is most strongly manifested for larger values of $\\lambda $ as expected.",
"On the other hand, the 2-particle basis state $|0,0\\rangle $ has its coefficient below the cutoff at $\\lambda =0.01m^2$ .",
"Its coefficient is 0.00004 at $\\lambda =0.05m^2$ , 0.00016 at $\\lambda =0.1m^2$ , 0.0032 at $\\lambda =0.5m^2$ , and 0.0106 at $\\lambda =1m^2$ .",
"Again, the growth of the coefficient is very nearly quadratic in $\\lambda $ deviating mostly towards larger $\\lambda $ , where subleading corrections at third order in perturbation theory become important.",
"We can use this information to estimate when perturbation theory breaks down by evaluating when the contribution from second order is as great as that from first order.",
"If we fit the growth of the coefficient of $|0,0,0,0\\rangle $ to a straight line and the coefficient of $|0,0\\rangle $ to a parabola, we find a crossing point of $\\lambda \\sim 1.8m^2$ as shown in Fig.",
"REF .",
"Figure: For the vacuum, a plot of the coefficients of |0,0,0,0〉|0,0,0,0\\rangle (green dots) along with their best-fit straight line (green) and the coefficients of |0,0〉|0,0\\rangle (blue dots) along with their best-fit parabola (blue).",
"Their crossing point is a crude estimate of the breakdown of perturbation theory.However, we think this estimate is rather crude and that it needs further investigation.",
"The two most important 6-particle basis states just barely appear above the cutoff at $\\lambda =0.05m^2$ , both with coefficient 0.00002.",
"Their coefficients also grow quadratically as $\\lambda $ increases.",
"The largest is $|0,0,0,0,0,0\\rangle $ and its coefficient takes the values 0.00007 at $\\lambda =0.1m^2$ , 0.0010 at $\\lambda =0.5m^2$ and 0.0028 at $\\lambda =1m^2$ .",
"The eight-particle basis states do not appear above the cutoff until $\\lambda =0.5m^2$ .",
"The largest of these is $|0,0,0,0,0,0,0,0\\rangle $ .",
"Its coefficient takes the values 0.0002 at $\\lambda =0.5m^2$ and 0.0004 at $\\lambda =1m^2$ .",
"We expect the deviation from quadratic growth is due to subleading corrections from third order at these large values of $\\lambda $ since they are connected to the vacuum by two powers of the Hamiltonian.",
"Basis states with 10 or more free particles contribute to the vacuum at third- or higher order perturbation theory.",
"They are all below the cutoff for the values of $\\lambda $ that we tested.",
"The two-particle eigenstate $\\Psi _{24,-24}$ is shown in the second row of Fig.",
"REF .",
"The 4-particle (red) and 6-particle (green) basis states shown in the left plot contribute at first order.",
"They grow linearly with $\\lambda $ until higher-order corrections become important.",
"For example, the largest 4-particle basis state is $|-24,6,8,10\\rangle _+$ .",
"Its coefficient is 0.000371 at $\\lambda =0.01m^2$ , 0.00181 at $\\lambda =0.05m^2$ , 0.00351 at $\\lambda =0.1m^2$ , 0.0144 at $\\lambda =0.5m^2$ and 0.0237 at $\\lambda =1m^2$ .",
"The largest 6-particle basis state is $|-24,0,0,0,0,24\\rangle $ .",
"Its coefficient is -0.000254 at $\\lambda =0.01m^2$ , -0.00124 at $\\lambda =0.05m^2$ , -0.00241 at $\\lambda =0.1m^2$ , -0.0102 at $\\lambda =0.5m^2$ and -0.0178 at $\\lambda =1m^2$ .",
"Other 2-particle basis states contribute at first order but are below the cutoff.",
"The 0-particle basis state (the free vacuum) contributes at second order in perturbation theory and does not appear above the cutoff for the range of coupling constants that we tested.",
"Some further 6-particle basis states appear above the cutoff at $\\lambda =0.5m^2$ (green points near the blue dashed line with a free energy of approximately 50$m$ .)",
"One of these is $|-24,2,2,4,6,10\\rangle _+$ .",
"It has a coefficient of 0.0007 at $\\lambda =0.5m^2$ and 0.0017 at $\\lambda =1m^2$ , getting its dominant contribution at second order.",
"Since we only have the coefficient at large values of $\\lambda $ , subleading corrections are already important.",
"We also see three 8-particle basis states (yellow points) appear above the cutoff when $\\lambda =0.5m^2$ .",
"The largest is $|-24,0,0,0,0,0,0,24\\rangle $ and has a coefficient of 0.0013 at $\\lambda =0.5m^2$ and 0.0033 at $\\lambda =1m^2$ , which appears to be slightly less than quadratic growth due to subleading effects.",
"We do not see 10-particle basis states above the cutoff for any of our tested values of $\\lambda $ .",
"The same is true for basis states with 12 or more free particles, which contribute at higher than second order.",
"Because the basis states contributing at second order do not appear above the cutoff until large values of $\\lambda $ , where subleading contributions are important, we do not use them to estimate the breakdown of perturbation theory.",
"The four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ is shown in the third row of Fig.",
"REF .",
"Most of the other 4-particle basis states (red points) contribute at first order in perturbation theory.",
"For example, the most important 4-particle basis state (after the main basis state) is $|-10,-8,-6,24\\rangle _+$ .",
"Its coefficients are 0.011661, 0.05291, 0.09476, 0.2545 and 0.3064 at $\\lambda =0.01m^2, 0.05m^2, 0.1m^2, 0.5m^2$ and $1m^2$ , respectively.",
"However, a few of the 4-particle basis states contribute only at second order.",
"For example, the basis state $|-10,-10,-4,24\\rangle _+$ has coefficients 0.000053, 0.00114, 0.00383, 0.0337 and 0.0429 at $\\lambda =0.01m^2, 0.05m^2, 0.1m^2, 0.5m^2$ and $1m^2$ , respectively.",
"The reason that this basis state require the action of the Hamiltonian twice is that at least three of the momenta are different.",
"So, it has to remove at least three momenta and add at least three new momenta.",
"This requires the application of at least three annihilation operators and at least three creation operators which can only be achieved in two applications of the Hamiltonian.",
"[See Eq.",
"(REF ) where the maximum number of creation and annihilation operators is four.]",
"Using these coefficients, we might attempt to estimate the breakdown of perturbation theory again and see how it correlates with what we found using the vacuum.",
"However, there are two challenges.",
"The first is that the coefficients are already quite large and would naively reach unity levels at $\\lambda \\sim 1.1m^2$ and $\\lambda \\sim 1.8m^2$ before they even cross each other.",
"Also, as we will shortly see, our results for the four-particle eigenstate at large $\\lambda $ appear to be much less trust worthy than for the vacuum.",
"For this reason, in this paragraph, we give the coefficients coming from diagonalized second-order perturbation theory.",
"Moving on to other contributing basis states.",
"The 0-particle basis state also contributes at first order but is below the cutoff for all the values of $\\lambda $ that we tested.",
"There is a single 2-particle basis state above the cutoff.",
"It is the basis state $|24,-24\\rangle $ .",
"Its coefficients are 0.000149, 0.00063, 0.00105, 0.0019 and 0.0012 at $\\lambda =0.01m^2, 0.05m^2, 0.1m^2, 0.5m^2$ and $1m^2$ , respectively.",
"We can see that it grows nearly linearly for small $\\lambda $ .",
"This makes sense because it can be reached from the main basis state by one application of the Hamiltonian.",
"Some 6-particle basis states (green points), contribute at first order.",
"For example, the highest such point is $|-24,2,2,4,8,8\\rangle _+$ with coefficients 0.000881, 0.00422, 0.00803, 0.0300 and 0.0479 at $\\lambda =0.01m^2, 0.05m^2, 0.1m^2, 0.5m^2$ and $1m^2$ .",
"Others contribute at second order.",
"For example, the basis state $|-10,-8,-6,0,0,24\\rangle _+$ appears above the cutoff at $\\lambda =0.05m^2$ and has coefficients 0.00020, 0.00070, 0.0073 and 0.0142 at $\\lambda =0.05m^2, 0.1m^2, 0.5m^2$ and $1m^2$ .",
"We can see that it grows nearly quadratically for small $\\lambda $ .",
"The same story applies to the 8-particle basis states (yellow points).",
"Some contribute at first order while others at second order.",
"The basis state $|-24,0,0,0,0,8,8,8\\rangle _+$ has coefficients 0.000252, 0.00119, 0.00223, 0.0073 and 0.0101 at $\\lambda =0.01m^2, 0.05m^2, 0.1m^2, 0.5m^2$ and $1m^2$ and contributes at first order while the basis state $|-10,-8,-6,0,0,0,0,24\\rangle _+$ has coefficients 0.000063, 0.00021, 0.0017 and 0.0023 at $\\lambda =0.05m^2, 0.1m^2, 0.5m^2$ and $1m^2$ and contributes at second order.",
"Finally, the 10-particle basis state can only begin contributing at second order since they require the addition of six new particles and therefore need at least two applications of the Hamiltonian.",
"They first appear above the cutoff at $\\lambda =0.1m^2$ with a few more appearing at larger $\\lambda $ .",
"The first one is the basis state $|-24,0,0,0,0,0,0,8,8,8\\rangle _+$ which has coefficients 0.00001, 0.0016 and 0.0036 at $\\lambda =0.1m^2, 0.5m^2$ and $1m^2$ .",
"Figure: Plots of the largest differences between our cyclic QSE code and second-order perturbation for a range of the coupling constant λ\\lambda .",
"The blue dashed line is the cutoff used in our calculations.",
"The green lines give the maximum and minimum largest differences after 100 independent runs of the code.",
"The red dotted line gives the mean largest difference.We have also compared the results of our code with perturbation theory for the range of $\\lambda $ that we have tested here.",
"We show plots of the largest absolute difference with second-order perturbation theory in Fig.",
"REF .",
"Because of the random nature of our cyclic QSE code, each time it is run, we will potentially obtain different results.",
"It is important to analyze this randomness and develop distributions giving the statistics of how frequently it is successful.",
"We have chosen to do that here as it depends on the value of the coupling constant $\\lambda $ .",
"For each value of $\\lambda $ and each eigenstate, we ran our cyclic QSE code 100 independent times.",
"We then compared each run with second-order perturbation theory and found the largest difference.",
"We used the largest difference for each run to determine the statistics.",
"In Fig.",
"REF , we show the cutoff we used in dashed blue.",
"The solid green curves are for the maximum largest difference and minimum largest difference encountered in the 100 independent runs while the red dotted line is the average largest difference.",
"The top and middle plots are for the vacuum and the two-particle eigenstate, respectively.",
"We can see that our cyclic QSE code is extremely successful at accurately constructing these eigenstates.",
"In fact, if we are able to calculate these two-particle eigenstates in two spatial dimensions with as much success, we should be able to successfully calculate elastic $2\\rightarrow 2$ scattering amplitudes with this method.",
"On the other hand, we can see that the four-particle eigenstate, shown at the bottom of Fig.",
"REF , is much more difficult.",
"Our cyclic QSE code seems to do an acceptable job for $\\lambda \\lesssim 0.1m^2$ .",
"But, it does not appear to be trustworthy above this value.",
"In fact, what is happening is that our cyclic QSE code is sometimes not finding important basis states.",
"In particular, at the very peak at $\\lambda =0.2m^2$ , the maximum largest difference is due to the QSE code not finding the basis state $|18,6,-8,-16\\rangle _+$ which has a coefficient 0.45123.",
"At $\\lambda =0.3m^2$ , on the other hand, the maximum largest difference is caused by missing the basis state $|24,-6,-8,-10\\rangle _+$ which has a coefficient 0.01065.",
"At $\\lambda =0.5m^2$ , it is the basis state $|18,6,-10,-14\\rangle _+$ with a coefficient of 0.1570.",
"Whether or not it finds these basis states is very sensitive to what other basis states are present and their coefficients, as well as the size of the reduced Hilbert space (see Sec. ).",
"It may be possible to fix this deficiency by a more clever random choice of new basis states at each cycle.",
"We spent considerable time tuning this process but believe there is further room for improvement.",
"Clearly, if this method is to be used to calculate $2\\rightarrow 4$ scattering amplitudes, this will have to be improved if these larger values of $\\lambda $ are important.",
"Figure: A plot of the inner product of the two-particle eigenstate Ψ 24,-24 \\Psi _{24,-24} and the four-particle eigenstate Ψ 24,-8,-8,-8 + \\Psi ^+_{24,-8,-8,-8} for a range of coupling constants λ\\lambda .",
"The blue dashed line represents the cutoff we used for these states, the dotted green line gives the inner product at first order, the dot-dashed orange line gives the inner product at second order, the solid green lines give the largest and smallest inner products given by our code after 100 trials while the dotted red line gives the mean value given by our code.To end this section, we calculate the inner product between the eigenstates $\\Psi _{24,-24}$ and $\\Psi ^+_{24,-8,-8,-8}$ for each value of $\\lambda $ and show the results in Fig.",
"REF , where the dashed blue line gives the approximate precision of this calculation, which is the cutoff we used for the basis states.",
"As we discussed in Sec.",
", we expect this inner product to be zero because the energies of these eigenstates are slightly different so a nonzero result would indicate non-conservation of energy.",
"Our result from first-order perturbation theory is shown as the dotted green line.",
"We can see that it is above the precision of the calculation for most of the range of $\\lambda $ , indicating a nonzero result.",
"We are not claiming, by this result, that perturbation theory itself violates energy conservation.",
"The problem with our first-order result is that it is missing important basis states and the coefficients are not yet sufficiently accurate.",
"That combination leads to a violation of energy conservation.",
"On the other hand, the second-order result is shown in dot-dashed purple and is below the precision of the calculation for the entire range of $\\lambda $ .",
"The result of our cyclic QSE code is also at or below the precision of the calculation for the entire range of $\\lambda $ showing that it is also in agreement with a zero result and successfully conserves energy.",
"In fact, we calculated the inner product for the entire set of 100 trials for each value of $\\lambda $ .",
"We show the smallest and largest values of the inner product we got from our cyclic QSE code as the two solid green lines, while the average inner product is displayed as a dotted red line.",
"The entire range of our results is at or below the precision of the calculation.",
"It might, naturally, be wondered how our inner product did so well at large $\\lambda $ when the disagreement between our QSE code and second-order perturbation theory for the eigenstate $\\Psi ^+_{24,-8,-8,-8}$ was so large (see the bottom plot of Fig.",
"REF ).",
"The reason is that the basis states that were missed by our cyclic QSE code when constructing $\\Psi ^+_{24,-8,-8,-8}$ were below the cutoff for the eigenstate $\\Psi _{24,-24}$ , so the contribution of these basis states was also below the cutoff.",
"That is to say, these basis states don't contribute significantly to this inner product."
],
[
"Dependence on $\\mathbf {\\Delta p}$",
"In this section, we turn to the dependence of our results on the size of the momentum spacing $\\Delta p$ , while keeping $\\lambda =0.1m^2$ .",
"A major long-term objective is to obtain useful information about the continuum limit $\\Delta p\\rightarrow 0$ .",
"In order to do this, we need to do the calculations in this paper at much smaller $\\Delta p$ than we have already done.",
"In fact, we would like to do it for a range of small $\\Delta p$ so that we can meaningfully extrapolate to $\\Delta p\\rightarrow 0$ .",
"This section will be an initial step in that direction.",
"We have calculated our eigenstates at a few values of $\\Delta p$ and show plots for $\\Delta p=0.25m, 0.5m$ and $1m$ in Fig.",
"REF while the results for $\\Delta p=2m$ can be seen in the right column of Fig.",
"REF .",
"For this section, we have kept $\\lambda =0.1m^2$ and use the same cutoff on each state as we decreased $\\Delta p$ .",
"Because the cutoff has remained the same as we decrease $\\Delta p$ , the number of basis states above the cutoff has increased.",
"In particular, we have found that the vacuum $\\Psi _v$ has 109, 360, 1211 and 4168 basis states above the cutoff, the two-particle eigenstate $\\Psi _{24,-24}$ has 59, 173, 556 and 1796 basis states above the cutoff, and the four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ has 137, 393, 894 and 2486 basis states above the cutoff for $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ , respectively.",
"In order to keep the ratio of the reduced Hilbert space size to the number of basis states above the cutoff roughly the same as we decreased $\\Delta p$ (in order to make a fair comparison with perturbation theory and ensure the cyclic QSE code is equally effective at each $\\Delta p$ ), we have tripled the reduced Hilbert space size each time we halfed $\\Delta p$ .",
"So, we used a reduced Hilbert space size of 700, 2100, 6300 and 18900 for $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ , respectively.",
"This kept our code running at roughly the same effectiveness for each value of $\\Delta p$ .",
"As we look at the plots in Fig.",
"REF , we notice a few general features as we decrease $\\Delta p$ .",
"The first is that the general structure (the shape of the points on the plot) remains largely the same.",
"This is a good thing as it gives us some confidence that the results at larger $\\Delta p$ are approximating those at small $\\Delta p$ .",
"The second is that the density of basis states (the density of points on the plot) increases as we decrease $\\Delta p$ .",
"This also makes sense as the free energy of the basis states (the horizontal axis of these plots) is a (nearly linear) function of the momentum spacing.",
"In particular, it takes the form $E_f=\\sum _i\\sqrt{n_i^2\\Delta p^2+m^2}$ , where $n_i$ is an integer unique to each free particle in the basis state and determines its momentum $p_i=n_i\\Delta p$ .",
"So, as $\\Delta p$ decreases, a greater number of basis states fit into the same free energy region.",
"Ideally, we would prefer to plot not the bare coefficient of each basis state (as we have done in Fig.",
"REF and throughout this paper), but rather the coefficient divided by the free-energy space between the basis states.",
"This would normalize the contribution of the basis states by their density and would allow for a more stable eigenstate as $\\Delta p\\rightarrow 0$ .",
"This was done in [10] where only 0- and 2-particle basis states were included and the free-energy spacing between basis states was constant and, consequently, so was their density.",
"In the present work, however, we keep higher-multiplicity basis states and the free-energy spacing between basis states is not constant.",
"In fact, it is quite complicated.",
"We do not, presently, know the correct free-energy spacing to use for the normalization when higher-multiplicity basis states are included.",
"We feel that this is a very important topic for future research if this method is to succeed.",
"In the meantime, we will simply analyze the bare coefficient for the basis states and make a few comments about how normalizing by the density might affect our results.",
"All the plots in this section, and throughout this paper, use the bare coefficient.",
"Figure: Plots of the right-most point of each color from the plots in Fig.",
"as a function of Δp\\Delta p along with their best-fit curves.",
"The horizontal axis is the momentum spacing divided by the mass while the vertical axis is the free energy of the right-most basis state divided by m.The third feature that stands out about these plots is that, although the general shape remains the same, the fall off of the coefficients is more rapid as $\\Delta p$ decreases.",
"For example, if we focus on the vacuum, we see that the 4-particle basis states extend out to approximately $57m$ before falling below the cutoff when $\\Delta p=2m$ , but only out to approximately $46m$ when $\\Delta p=1m$ , $35m$ when $\\Delta p=0.5m$ and $26m$ when $\\Delta p=0.25m$ .",
"In fact, this can be well fit by a straight line on a log-log plot as seen in the top plot of Fig.",
"REF , where the best-fit line is given by $\\mbox{ln}\\left(E_f\\right) = 3.80+0.39\\ \\mbox{ln}\\left(\\Delta p\\right)\\ .$ The right-most 2-particle basis state (blue point) can also be fit by a straight line and is presented as the dashed blue line in the same plot.",
"Moving to the two-particle eigenstate $\\Psi _{24,-24}$ , the right-most 6-particle basis state (green point) can be well fit by the straight line $\\mbox{ln}\\left(E_f\\right) = 4.27+0.11\\mbox{ln}\\left(\\Delta p\\right)\\ ,$ and is shown as the dashed green line in the middle plot of Fig.",
"REF where we also present the right-most red point and its best-fit straight (dashed red) line.",
"We note that a straight-line best fit cannot continue to infinitesimal $\\Delta p$ because that would imply that the right-most point in the plots of Fig.",
"REF goes to zero.",
"We believe that the reason for this is the lack of a proper density normalization as mentioned earlier in this section.",
"If we properly normalize these coefficients to the density of the basis states, we believe that the free energy of the right-most point would approach a constant nonzero value as $\\Delta p\\rightarrow 0$ .",
"That is, we believe with the proper normalization, they would not be well fit by a straight line, but rather something like an exponential curve (on a log-log plot).",
"The four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ , on the other hand, did have right-most points that appear to be gradually approaching a nonzero limit.",
"We have plotted the right most 8-particle basis state (yellow point) for this eigenstate in the bottom plot of Fig.",
"REF along with its fit to an exponential curve (on a log-log plot).",
"We did not include the right-most 8-particle basis state at $\\Delta p=0.25m$ for technical reasons, which we will explain in the next paragraph.",
"We also plotted the right-most 6-particle basis state (green point) in the same plot and also fit it with an exponential, given by $\\mbox{ln}\\left(E_f\\right) = 3.98 + 0.13\\Delta p\\ .$ If we extrapolate this exponential curve all the way to $\\Delta p\\rightarrow 0$ , we estimate that the 6-particle basis states (green points) will end at approximately $E_f=53.5m$ , in the continuum limit for the same cutoff.",
"Of course, properly normalizing by the density of basis states will probably make these curves shallower and affect this extrapolation to $\\Delta p\\rightarrow 0$ .",
"Figure: Plots of the top ridge and for an individual point from the plots in Fig.",
"as a function of Δp\\Delta p along with their best-fit curves.",
"The horizontal axis is the momentum spacing divided by the mass while the vertical axis is the absolute value of the coefficient for each point.",
"See the text for greater detail.Although the right-most point is moving towards lower free energy as $\\Delta p$ decreases, the basis states themselves do not move towards lower free energy.",
"Their free energy is fixed by their free-particle momenta.",
"Instead, what is happening is that their coefficients are diminishing as $\\Delta p$ decreases.",
"The basis states that were just above the cutoff at $\\Delta p=2m$ at the far right of the right plot in Fig.",
"REF are being reduced to coefficient values below the cutoff so that they no longer appear in the plots for $\\Delta p=1m$ , $\\Delta p=0.5m$ or $\\Delta p=0.25m$ in Fig.",
"REF .",
"In particular, focusing on the four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ shown in the bottom row of Figs.",
"REF and REF , we can see a small hump in the 8-particle basis states (yellow points) at $E_f\\sim 66m$ .",
"As $\\Delta p$ decreases and the density of basis states increases, we see this hump resolved in greater detail, however, it remains at the same free-energy position.",
"It's contribution to the eigenstate, however, does decrease and we see this as the hump sinks lower and lower as $\\Delta p\\rightarrow 0$ .",
"In particular, the highest point of the hump has coefficients of 0.00053, 0.00031 and 0.00017 at $\\Delta p=2m, 1m$ and $0.5m$ , respectively.",
"By the time, $\\Delta p=0.25m$ , the hump is completely below the cutoff.",
"This is why we did not include a point at $\\Delta p=0.25m$ in the extrapolation.",
"This can be seen as the bottom dotted line in the bottom plot of Fig.",
"REF passes below the dashed blue line.",
"In order to explore this behavior further, we have plotted a series of points and curves in Fig.",
"REF , which we will now explore.",
"The dashed blue line is the cutoff as usual.",
"The color coding of these points is the same as in Fig.",
"REF .",
"For each plot, we have begun with two points at $\\Delta p=2m$ as can be seen at the right edge of these plots and then followed what happens with these points as $\\Delta p$ decreases.",
"To understand these plots in detail, we will begin by focussing on the top plot, which is for the vacuum.",
"For this plot, we have focused on 4-particle basis states.",
"The top point at $\\Delta p=2m$ (in the top plot of Fig.",
"REF ) is the basis state $|0,0,0,0\\rangle $ which has a free energy of 4m and a coefficient 0.00255.",
"It is the highest red point in the top right plot of Fig.",
"REF .",
"As we move towards smaller $\\Delta p$ , two things happen.",
"The first is that the coefficient of the highest red point decreases.",
"In order to see this, we have plotted the coefficient of the highest red point for the vacuum at each value of $\\Delta p$ as the top four red points of the top plot of Fig.",
"REF and have fit a red dotted straight line to them.",
"This is the highest red line in the plot.",
"We can see that it has a very gentle slope downward as $\\Delta p$ decreases.",
"However, the highest point is not the same basis state for all $\\Delta p$ .",
"In fact, at $\\Delta p=1m$ , it is the basis state $|-1m,0,0,1m\\rangle $ with a coefficient 0.002499, at $\\Delta p=0.5m$ , it is the basis state $|-1m,0,0.5m,0.5m\\rangle _+$ with a coefficient 0.00195, and at $\\Delta p=0.25m$ , it is the basis state $|-0.75m,0,0.25m,0.5m\\rangle _+$ with a coefficient 0.00161.",
"The original basis state actually moves down at a faster rate, moving inside of the dense region of red points at the same energy.",
"In particular, the basis state $|0,0,0,0\\rangle $ that was at the top at $\\Delta p=2m$ , has coefficients 0.00255, 0.00122, 0.00061 and 0.00031 at $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ , respectively.",
"We plot this as the top red-dashed best-fit line and the points that it passes through.",
"This behavior is not unique to the highest point of a color.",
"Other points along the ridge of the color also do this.",
"The bottom red point (in the top plot of Fig.",
"12) at $\\Delta p=2m$ is the basis state $|-8m,0,2m,6m\\rangle _+$ and has free energy of 17.4m and coefficient 0.00039.",
"If we follow the top ridge of the red points at the same free energy, we find that it is the basis state $|-7m,-1m,0,8m\\rangle _+$ at $\\Delta p=1m$ with a coefficient 0.00022, it is the basis state $|-8m,0,0.5m,7.5m\\rangle _+$ at $\\Delta p=0.5m$ with a coefficient 0.00012, and it is the basis state $|-7.5,-0.25,0.5,7.25\\rangle _+$ at $\\Delta p=0.25m$ with a coefficient 0.00006.",
"We also plot these points along with their best fit line (bottom dotted red line) in the top plot of Fig.",
"REF .",
"However, we also plot the same basis state $|-8m,0,2m,6m\\rangle _+$ and follow its coefficient as the best-fit dashed red line below it.",
"It has coefficients 0.00014, 0.00007 and 0.00003 at $\\Delta p=1m, 0.5m$ and $0.25m$ .",
"We again see that the coefficient of the basis state is reduced faster than the height of the ridge showing that the basis state sinks into the red points at the free energy 17.4m.",
"All of these points are well fit by a straight line on a log-log plot.",
"However, as discussed earlier, we believe that this is due to using the bare coefficients of the basis states.",
"If, as we suspect we should, normalized the coefficients by the density of basis states, we believe these curves would flatten out as we approach smaller $\\Delta p$ so that the density normalized coefficients would approach a constant nonzero value.",
"Focusing on the two-particle eigenstate $\\Psi _{24,-24}$ , we direct our attention to the middle plot of Fig.",
"REF .",
"Similar to the previous case, we follow the highest green point of this eigenstate, which is at a free energy of 52.0m.",
"This is the basis state $|-24m,0,0,0,0,24m\\rangle $ , $|-24m,-1m,0,0,1m,24m\\rangle $ , $|-24m,-0.5m,-0.5m,0,1m,24m\\rangle _+$ and $|-24m,-0.75m,0,0.25m,0.5m,24m\\rangle _+$ with coefficient 0.00240, 0.00246, 0.00195 and 0.00161 at $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ , respectively.",
"We have plotted this as the four green points along with their best-fit line at the top of the plot.",
"As before, the basis state at the top changes.",
"Each basis state sinks down faster than the height of the top green point.",
"In fact, the basis state $|-24m,0,0,0,0,24m\\rangle $ has coefficients 0.00240, 0.00122, 0.00061 and 0.00031 at $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ , respectively.",
"We have plotted these as the green points and top dashed green line in the plot.",
"Again, to show that this behavior is not special to the highest point, we consider a point at the higher free energy of 58.3m.",
"We find that the highest basis state at this free energy is $|-24m,-4m,0,0,4m,24m\\rangle $ , $|-24m,-5m,0,1m,4m,24m\\rangle _+$ , $|-24m,-4.5m,0,0.5m,4m,24m\\rangle _+$ and $|-24m,-4.25m,0,0.25m,4m,24m\\rangle _+$ with coefficients 0.00082, 0.00055, 0.00035 and 0.00020 at $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ , respectively.",
"We have plotted this as the lower of the green points and best-fit straight dotted green line.",
"But, as before, the basis state itself falls off more quickly.",
"The basis state $|-24m,-4m,0,0,4m,24m\\rangle $ has coefficients 0.00082, 0.00041, 0.00021 and 0.00010 at $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ , respectively, and is plotted as the lower four green points and their associated dashed green best-fit line.",
"As before, we believe all of these curves would flatten out and approach a nonzero constant value if we normalized the coefficients with the density of basis states.",
"Turning to the four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ , we focus on the bottom plot of Fig.",
"REF .",
"We again choose two points at $\\Delta p=2m$ to begin with.",
"The first is the highest yellow point from this eigenstate in Fig.",
"REF .",
"It is the basis state $|-24m,0,0,0,0,8m,8m,8m\\rangle _+$ at free energy 52.2m with a coefficient of 0.00222.",
"The top yellow point changes as $\\Delta p$ decreases.",
"At $\\Delta p=1m$ , it is $|-8m,-8m,-8m,-1m,0,0,1m,24m\\rangle _+$ with coefficient 0.00239, at $\\Delta p=0.5m$ , it is $|-8m,-8m,-8m,-0.5m,0,0,0.5m,24m\\rangle _+$ with coefficient 0.00173, and at $\\Delta p=0.25m$ , it is $|-8m,-8m,-8m,-0.5m,-0.25m,0,0.75m,24m\\rangle _+$ with coefficient 0.00119.",
"These four points are the top four yellow points of the bottom plot of Fig.",
"REF along with their best-fit straight line in dotted yellow.",
"As in the previous eigenstates, the basis state $|-24m,0,0,0,0,8m,8m,8m\\rangle _+$ itself, sinks down into the middle of the yellow region at a free energy of 52.2m.",
"It takes coefficients 0.00082, 0.00042 and 0.00021 at $\\Delta p=1m, 0.5m$ and $0.25m$ .",
"These points are plotted along with their best-fit dashed yellow line (the higher of the two).",
"We also show the behavior of the top of the hump located at a free energy of 65.6m.",
"At $\\Delta p=2m$ , this peak is held by the basis state $|-8m,-8m,-8m,-8m,0,2m,6m,24m\\rangle _+$ with a coefficient of 0.00053.",
"As $\\Delta p$ decreases, the coefficient of both this peak and this basis state decreases, that of the basis state is faster as before.",
"The peak is given by the basis states $|-8m,-8m,-8m,-8m,0,1m,7m,24m\\rangle $ (coefficient 0.00031) at $\\Delta p=1m$ and $|-8m,-8m,-8m,-8m,0,0.5m,7.5m,24m\\rangle _+$ (coefficient 0.00017) at $\\Delta p=0.5m$ .",
"On the other hand, the basis state $|-8m,-8m,-8m,-8m,0,2m,6m,24m\\rangle _+$ has coefficients 0.00019 and 0.00009 at $\\Delta p=1m$ and $0.5m$ , respectively.",
"They are plotted along with their best fit dotted and dashed lines, respectively at the bottom of the plot.",
"As we can see, both the peak of the hump as well as the basis state itself, sink below the cutoff by the time $\\Delta p=0.25m$ .",
"As in the previous paragraphs, we expect that normalizing the coefficients according to the density of the basis states would reduce the slope of these curves as $\\Delta p\\rightarrow 0$ so that they approach constant nonzero values appropriate to the continuum limit.",
"Figure: Plot of the inner product between the eigenstates Ψ 24,-24 \\Psi _{24,-24} and Ψ 24,-8,-8,-8 + \\Psi ^+_{24,-8,-8,-8} as a function of Δp\\Delta p. The color coding is the same as in Fig.",
".We have calculated the inner product between the eigenstates $\\Psi _{24,-24}$ and $\\Psi ^+_{24,-8,-8,-8}$ and plot it in Fig.",
"REF .",
"The dashed blue line is the cutoff used, which is the approximate precision of this calculation.",
"The dotted green line at the top gives the first-order perturbation result which is above the cutoff and not consistent with zero.",
"As described in Sec.",
", this is due to not yet having sufficiently accurate coefficients and missing important basis states.",
"We can see that this problem persists at lower $\\Delta p$ .",
"The dot-dashed purple line gives the second-order perturbative result which, as we can see, is below the cutoff and therefore in agreement with a zero result.",
"In dotted red, we see the results of our cyclic QSE code which are also below the precision of the calculation and, therefore, also in agreement with zero.",
"We further see that the results of our QSE code are in perfect agreement with second-order perturbation theory where it was possible to calculate it.",
"The shape of the red dotted curve below the cutoff is unimportant as its shape depends on contributions below the precision of the calculation that were not included.",
"Therefore, we do not think that any trends, other than being in agreement with zero, can be extracted from this result.",
"Figure: Plot of the time the code requires for these calculations in seconds as a function of Δp\\Delta p. The solid point and lines are measured while the dashed, dot-dashed and dotted lines are extrapolated based on the measured values at the smallest Δp\\Delta p. The blue and yellow curves are for first- and second-order perturbation theory, respectively, while the red point is for third-order perturbation theory.",
"The green curve is the time it takes our cyclic QSE code per cycle.As we have discussed, our goal is eventually to do this calculation at smaller $\\Delta p$ and extrapolate to $\\Delta p\\rightarrow 0$ .",
"However, each decrease in $\\Delta p$ takes an increase in computational time.",
"We have analyzed this time for the calculations we have done and present it in Fig.",
"REF where the vertical axis is the time in seconds and the horizontal axis is the momentum spacing $\\Delta p$ we used.",
"We base this on the time it takes to calculate the 2-particle eigenstate $\\Psi _{24,-24}$ .",
"We emphasize that all these calculations have been done on a single cpu on small servers and that improvements in time could certainly be achieved by parallelizing these calculations and potentially using supercomputer resources.",
"Moreover, since these times depend on the machine we used, we do not think the absolute times are the most meaningful aspect of this plot.",
"Rather, we are interested in the trends and how our cyclic QSE code compares to perturbation theory.",
"The solid curves come from the measured values using our calculations while the dashed, dot-dashed and dotted lines are extrapolations down to $\\Delta p=0.1m$ .",
"We have plotted first-order perturbation theory in blue.",
"It took 0.015s, 0.13s, 2.2s and 66s to complete first-order perturbation theory at $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ , respectively.",
"We have plotted these points and joined them with a solid blue line that is very nearly linear on a log-log scale.",
"We have also plotted an extrapolation based on only the times at $\\Delta p=0.25m$ and $0.5m$ to estimate how the time would grow as we further decreased $\\Delta p$ .",
"We estimate that it would take approximately $10^4s$ or approximately 3 hours at $\\Delta p=0.1m$ .",
"We were only able to achieve second-order perturbation theory at $\\Delta p=2m$ and $1m$ .",
"It took 170s and 84000s, respectively.",
"We plotted this data in orange and joined it with a solid orange line.",
"We also extrapolated this data with a dotted orange line to $\\Delta p=0.1m$ .",
"Looking at the extrapolation, we see that it would have taken approximately 3 years to complete second-order perturbation theory for $\\Delta p=0.5m$ and much more for smaller $\\Delta p$ .",
"Furthermore, we see that the slope of the line for second order is greater than that of first order, so that as $\\Delta p$ decreases, the time grows faster for second order than it does for first order.",
"Finally, we were only able to achieve third-order perturbation theory at $\\Delta p=2m$ , where it took 26000s.",
"Since we did not achieve third order at $\\Delta p=1m$ , we were not able to do an extrapolation.",
"However, based on first and second order, we expect that the slope would be even greater than that of second order.",
"The reason that perturbation theory takes longer as $\\Delta p$ decreases is that it has to calculate the contribution of every basis state connected by the Hamiltonian to the main basis state.",
"The number of these basis states grows exponentially with decreasing $\\Delta p$ .",
"At first order, it has to do this for every basis state connected by one application of the Hamiltonian.",
"However, at higher orders, the burden is greater because it has to calculate the contribution for every basis state connected to the main basis state by two applications of the Hamiltonian (at second order).",
"This requires a doubling of the number of sums performed in the code, because to determine if a test basis state is connected to the main basis state by two applications of the Hamiltonian, it has to check all intermediate basis state as in $\\sum _i\\langle b_m|V|b_i\\rangle \\langle b_i|V|b_t\\rangle $ , where $|b_m\\rangle $ and $|b_t\\rangle $ represent the main and test basis states, respectively.",
"At third order, the calculation has to be done for every basis state connected by three applications of the Hamiltonian, thus requiring a tripling of the sums in the code, as in $\\sum _{i j}\\langle b_m|V|b_i\\rangle \\langle b_i|V|b_j\\rangle \\langle b_j|V|b_t\\rangle $ .",
"This is the reason that second order has a greater slope than first order and third order is expected to have a greater slope than second order in Fig.",
"REF .",
"Our cyclic QSE code, on the other hand, took 52s, 170s, 670s and 80000s per cycle for $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ .",
"These times are plotted in green and are joined by a solid green line in Fig.",
"REF .",
"Since we ran our code for ten cycles, the total time is one order of magnitude greater for each of these $\\Delta p$ .",
"Our code does significantly better than second-order perturbation theory.",
"Not only does it obtain just-as-good results (even better compared to naive second-order perturbation theory before the diagonalization step we added) but it is orders of magnitude more efficient as $\\Delta p$ decreases.",
"We see that the time increase is nearly linear on a log-log plot between $\\Delta p=2m$ and $0.5m$ and has a very shallow slope, shallower even than first-order perturbation theory.",
"But, between $\\Delta p=0.5m$ and $0.25m$ , the slope becomes steeper, greater than first-order, but still less steep than second-order perturbation theory.",
"The reason for this change in slope at $\\Delta p=0.5m$ is that this is where the matrix size (that must be constructed and diagonalized) passes from a small memory footprint to a large one.",
"In particular, our matrix sizes were $700\\times 700$ , $2100\\times 2100$ , $6300\\times 6300$ and $18900\\times 18900$ for $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ , respectively.",
"(The matrix size is equal to the reduced Hilbert space size squared.)",
"These matrix sizes are not significant compared to our memory resources until the last step at $\\Delta p=0.25m$ where a computational bottleneck is encountered.",
"This bottleneck can likely be pushed down to lower $\\Delta p$ with further clever computational techniques, however, it will eventually become impassable and is a critical and very relevant challenge for this technique.",
"On the other hand, we used a relatively large reduced Hilbert space size throughout this paper in order to have very high confidence in our results.",
"We will show in the next section that a much smaller reduced Hilbert space (and therefore a much smaller matrix size) is adequate for the vacuum and two-particle eigenstate, although not for the four-particle eigenstate.",
"However, as mentioned in previous sections, it is likely that the efficiency of the code at finding the important basis states for the four-particle eigenstate can be further improved.",
"Further research is necessary on this point.",
"In any case, it currently looks like there should not be any fundamental challenge to calculating two-particle eignestates using this method."
],
[
"Dependence on the Size of the Reduced Hilbert Space and the Energy of the Eigenstate",
"In this section, we explore the dependence on the size of the reduced Hilbert space and the energy of the eigenstate while keeping $\\lambda =0.1m^2$ and $\\Delta p=2m$ .",
"Figure: Plots of the distribution of largest difference between our cyclic QSE code and second-order perturbation theory for a range of sizes of the reduced Hilbert space.",
"The color coding is the same as in Fig.",
".We begin with the size of the reduced Hilbert space which is only a factor for our cyclic QSE code, not for perturbation theory.",
"During each iteration, after removing basis states below the cutoff (see App.",
"), our cyclic QSE code randomly adds new basis states to the reduced Hilbert space until it reaches some predetermined size, that we call the reduced Hilbert space size.",
"The results are very sensitive to this size as we show in Fig.",
"REF , where the horizontal axis is the reduced Hilbert space size.",
"The plots in this figure show the largest absolute difference with second-order perturbation theory for different reduced Hilbert space sizes, with a range going from 200 to 1000 in increments of 100.",
"Since our cyclic QSE code is random, we ran it one-hundred independent times at each reduced Hilbert space size to build up a distribution of results.",
"For each plot in this figure, we show the cutoff we used in dashed blue, as usual.",
"For each value of the reduced Hilbert space size, we plot the maximum largest differences encountered and join them with a solid green line.",
"Similarly, we plot the minimum largest differences with a solid green line.",
"We further plot the average largest difference with a dotted red curve.",
"For the vacuum $\\Psi _v$ (top plot of Fig.",
"REF ), we see that once the reduced Hilbert space size is equal or greater than 600, our cyclic QSE code never misses any of the basis states above the cutoff.",
"Below this size, it does occasionally miss a basis state.",
"The solid green line begins at a value of approximately $7\\times 10^{-5}$ and remains there until a reduced size of 500.",
"This means that at least once in the one-hundred independent runs, a basis state near the cutoff was missed.",
"On the other hand, we can see by the red dotted line that this is not typical once the reduced size is 400 or greater.",
"Even at 300, our cyclic QSE code typically only misses a basis state which is even closer to the cutoff, and therefore a less severe mistake.",
"However, even when the reduced Hilbert space size is on the smaller size, the difference with second-order perturbation theory is still not very severe and only slightly above the cutoff.",
"The two particle eigenstate $\\Psi _{24,-24}$ , turns out to do even better.",
"The reason for this is, as we discussed in Secs.",
"and , that first-order perturbation theory finds all the basis states above the cutoff for this value of the coupling constant $\\lambda $ .",
"Their coefficients are not correct yet, but they are all there.",
"Therefore, all our cyclic QSE code needs to do is diagonalize the basis states found by first-order perturbation theory.",
"We can see this in the middle plot of Fig.",
"REF where all three lines are right at the precision of the calculation.",
"This is because all one-hundred independent trials gave exactly the same result, since only the diagonalization step was necessary.",
"The cyclic part of the QSE code and the generation of random basis states to fill the reduced Hilbert space are both irrelevant, even overkill, for the two-particle eigenstate with our cutoff and value of $\\lambda $ .",
"If a much lower cutoff had been used or a larger $\\lambda $ , then basis states contributing at second and higher order would begin to contribute above the cutoff and our cyclic QSE code would find them cyclically as it does for the vacuum and four-particle eigenstate.",
"However, at this precision and $\\lambda $ , in one spatial dimension at least, it is unnecessary.",
"This suggests that if this precision is sufficient for a calculation of an elastic $2\\rightarrow 2$ scattering amplitude in two or three spatial dimensions, it may be sufficient to find the eigenstates at first order, diagonalize them and take the inner product.",
"Of course, we hasten to state that this suggestion must be tested.",
"The results for the four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ are plotted in the bottom of Fig.",
"REF where we see that the size of the reduced Hilbert space is extremely significant for this eigenstate.",
"This is due to the large number of important basis states missed at first order as well as their complex structures.",
"The good news, however, is that these basis states can be found by our cyclic QSE code if we increase the size of the reduced Hilbert space.",
"As we see in the plot, the general trend is towards smaller largest differences as the reduced space size increases.",
"By the time the reduced Hilbert space size is 1000, the maximum largest difference out of one-hundred independent trials is only $2\\times 10^{-4}$ , where the cutoff was $1\\times 10^{-4}$ and the average was half of that.",
"On the other hand, when the reduced Hilbert space size is small even the minimum largest difference is significantly above the cutoff, even by an order of magnitude when the size is 200.",
"Figure: Plot of the time it takes our cyclic QSE code per cycle as a function of the reduced Hilbert space size.",
"The dots are data points and the dashed line is a best fit.Although larger reduced Hilbert space sizes can always be used to increase confidence in the results, this also results in an increase in the time it takes to do the calculation.",
"The reason is that the reduced Hilbert space size directly determines the matrix size, as discussed in Sec. .",
"In Fig.",
"REF , we show a plot of the time it takes our cyclic QSE code per cycle as a function of the size of the reduced Hilbert space.",
"We can see that it is very well fit by a straight line on a log-log scale.",
"This plot is complementary to the plot in Fig.",
"REF .",
"As a final note, we would like to explore how the efficiency of our cyclic QSE method compares with perturbation theory when the eigenvalue energy increases and the eigenstate rises higher above the vacuum.",
"We will see that our cyclic QSE code is not very strongly affected by an increase in energy while perturbation theory is.",
"However, to make this comparison more striking, we will use hindsight to tune perturbation theory to be as efficient as possible.",
"To clarify, we will first describe two sets of parameters that have a large influence on the efficiency of perturbation theory but have very little effect on our cyclic QSE code.",
"Strictly speaking, in perturbation theory, we should calculate the contribution of all basis states connected to the main basis state by some number of applications of the Hamiltonian (1 at first order, 2 at second order and so on).",
"So, this means that we should consider all basis states with free energy all the way down to zero and all the way up to infinity.",
"Of course, this is impossible.",
"We have to cut this off for perturbation theory to finish in finite time.",
"Furthermore, only a tiny number of these basis states are important, as we have discussed.",
"So, we have a $E_{min}$ and $E_{max}$ in our perturbation theory code that restricts the range of basis states included in the perturbative calculations.",
"We also have these parameters for the cyclic QSE code, but we have found very little sensitivity to them (as long as they are low and high enough, respectively).",
"The reason is clear; the QSE code only deals with basis states connected to the basis states in the reduced Hilbert space by an application of the Hamiltonian.",
"So, if the basis states in the reduced Hilbert space are not very high in free energy, then it is not likely to choose a basis state far above them.",
"For the previous sections of this paper, we used an $E_{min}=0$ and an $E_{max}=80m$ .",
"We could have raised $E_{min}$ for $\\Psi _{24,-24}$ and $\\Psi ^+_{24,-8,-8,-8}$ , but we wanted to allow the possibility that some basis states below the main basis state were above the cutoff.",
"Indeed, there were several just below the cutoff.",
"There is another parameter important in the efficiency of the perturbative code.",
"It is $N_{max}$ , the maximum number of free particles in a basis state.",
"Strictly speaking, perturbation theory should consider all basis states connected by the Hamiltonian once or twice (for first or second order).",
"This means that for $\\Psi _{24,-24}$ , we should keep up to 6 free particles in our basis states at first order and up to 10 free particles at second order.",
"For $\\Psi ^+_{24,-8,-8,-8}$ , we should keep up to 8 and 12 free particles at first and second order, respectively.",
"In our calculations, up until this point, we have used an $N_{max}=10$ which includes all possible basis states for $\\Psi _{24,-24}$ but falls slightly short for $\\Psi ^+_{24,-8,-8,-8}$ .",
"We did this because choosing $N_{max}=12$ just took longer than we wanted to wait, especially since we knew with hindsight that 12-particle basis states were not important for these eigenstates.",
"However, we note that the timing plots in Figs.",
"REF and REF and the times discussed in the text were for $\\Psi _{24,-24}$ and included the full perturbative calculation.",
"On the other hand, with hindsight, we will reduce $N_{max}$ to 6 for $\\Psi _{24,-24}$ .",
"As we can see in the plots of these eigenstates, higher multiplicity basis states are never above the cutoff for $\\lambda =0.1m^2$ and a cutoff of $1\\times 10^{-4}$ .",
"This will reduce the time for perturbation theory significantly.",
"Although, it should be remembered that a slight increase in $\\lambda $ brings 8-particle basis states above the cutoff.",
"So, the times we show for second-order perturbation theory are extremely generous to the perturbative time.",
"On the other hand, we have found our cyclic QSE code to be largely insensitive to this parameter.",
"Again, the reason is clear.",
"Our QSE code only randomly chooses basis states connected to the basis states already in the reduced Hilbert space.",
"And, it focuses more effort on those that are within $\\pm 2$ free particles of those already in the reduced Hilbert space.",
"So, as long as $N_{max}$ is not too low, our cyclic QSE code is unaffected by this parameter.",
"Figure: Plots of the time it takes to calculate the eigenstate for different eigen-energies.",
"In blue and orange, we show the time it takes first- and second-order, respectively.",
"The time it takes our cyclic QSE code is shown in green.In this comparison, then, we will push perturbation theory to near its efficiency limit.",
"We have recalculated $\\Psi _{24,-24}$ with an $E_{min}=40m$ , $E_{max}=80m$ and $N_{max}=6$ .",
"We then calculated the analogous eigenstate at twice the energy, namely $\\Psi _{48,-48}$ .",
"We did this calculation with $E_{min}=90m$ , $E_{max}=130m$ and $N_{max}=6$ .",
"We also calculated the analogous eigenstate with four times the energy, namely $\\Psi _{96,-96}$ with $E_{min}=190m$ and $E_{max}=230m$ and $N_{max}=6$ .",
"Before we did these calculations, we first did the calculations with more generous parameters to make sure that we wouldn't miss any important basis states when we restricted to these more efficient parameters.",
"On the other hand, we tried our cyclic QSE code both with the original parameters and with these more restricted ones and saw no significant difference in time.",
"We have plotted the times for perturbation theory and our QSE code in Fig.",
"REF .",
"We have plotted the time it takes first- and second-order perturbation theory in blue and orange, respectively.",
"We see that first-order perturbation theory is much faster than our cyclic QSE code and second-order perturbation theory as before, but it increases in time as the eigenstate grows higher in energy.",
"With our very restrictive settings, the slope is not too steep and we do not expect it to cross the time it takes for our cyclic QSE code until quite high energies.",
"Second-order perturbation theory continues to take longer than our QSE code and also increases in time as the eigenstate increases in energy.",
"It also has a fairly shallow slope for these extreme parameter choices.",
"However, on the other hand, we see a remarkable feature of our QSE code plotted in green.",
"It is essentially flat!",
"Our cyclic QSE code is largely insensitive to the height of the eigenstate above the vacuum.",
"The reason for this is that the QSE code doesn't try to calculate everything.",
"It just searches the Hilbert space near the basis states it already has that are above the cutoff."
],
[
"Summary and Conclusions",
"In Sec.",
", we presented the results of our cyclic QSE code calculations of three eigenstates along with the first-order pertubative result.",
"One of the eigenstates was the vacuum $\\Psi _v$ , while two were “high\" above the vacuum.",
"The first was the eigenstate $\\Psi _{24,-24}$ with two particles of momentum $24m$ and $-24m$ , where $m$ is the mass parameter in the theory.",
"The second eigenstate was $\\Psi ^+_{24,-8,-8,-8}$ , a parity-symmetric four-particle eigenstate with one particle of momentum $24m$ and the other three of momentum $-8m$ .",
"We chose these two because they were potentially the type of eigenstates that could represent in and out states of a scattering S-Matrix element.",
"We plotted these eigenstates in Fig.",
"REF along with the first-order perturbative approximation to them.",
"In the text, we described and compared and contrasted them.",
"We pointed out that although first-order perturbation theory failed to find many of the important basis states to the eigenstates (because they did not contribute until second order), our cyclic QSE code did find them.",
"We also pointed out that our cyclic QSE code obtained different coefficients for the basis states that were found by both it and first-order perturbation theory and claimed that the difference in coefficients was due to higher-order perturbative corrections.",
"We ended this section with a calculation of the inner product between $\\Psi _{24,-24}$ and $\\Psi ^+_{24,-8,-8,-8}$ (the S-Matrix between these eigenstates) and found that it was nonzero using the first-order perturbative results and was consistent with zero using the results of our cyclic QSE code.",
"Furthermore, we pointed out that the inner product should be zero because the energies of these eignestates (48.042m and 48.209m) were not the same and therefore, by energy conservation, it must be zero.",
"We noted that the reason first-order perturbation theory did not give a zero result is that we truncated the first-order perturbative state at a cutoff on the coefficients.",
"So, the first-order perturbative calculation of the inner product was not complete in this sense.",
"On the other hand, although our cyclic QSE code also only kept basis states above the cutoff, unlike first-order perturbation theory, it had an essentially complete set of basis states above the cutoff, including basis states that contribute at second order.",
"Moreover, it had more correct coefficients for those basis states.",
"Together, these properties allow it to achieve a more accurate result for the inner product.",
"Indeed, the inner product using the results of the cyclic QSE code was in agreement with what we expect, which is zero.",
"In Sec.",
", we compare the results of our cyclic QSE code with second- and third-order perturbation theory.",
"In Fig.",
"REF , we show the difference in coefficients for each basis state at first order on the left and with second order on the right.",
"We show that although the results of our QSE code disagrees with first-order perturbation theory, it is in full agreement with second and third order for the vacuum $\\Psi _v$ and the two-particle eigenstate $\\Psi _{24,-24}$ .",
"For the four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ , we show that the agreement with second-order perturbation theory is much better than with first order, but that there are four basis states that are still not in full agreement.",
"We point out that this is due to the random nature of our cyclic QSE code but that these missed basis states are very near the cutoff while basis states high above the cutoff are much more likely to be discovered by our QSE code.",
"We point out that two things are happening as our QSE code works.",
"The first is that it is discovering new basis states missed by first-order perturbation theory while the second is that it is diagonalizing the Hamiltonian with respect to whichever basis states it has found.",
"Although both of these are typically important, we find that for the two-particle eigenstate $\\Psi _{24,-24}$ , for these parameter values and cutoff, all the important basis states are found by first-order perturbation theory.",
"All that is necessary for this eigenstate is the diagonalization step.",
"For this reason, our QSE code shoots directly to agreement with second-order perturbation theory on the first iteration, as seen in the middle plot of Fig.",
"REF .",
"On the other hand, for the vacuum and the four-particle eigenstate, our QSE code fills in most of the missing basis states in the first few iterations as seen in Fig.",
"REF and Fig.",
"REF .",
"Since the agreement with second-order perturbation theory is better than that of frist order, we take this as a sign that our cyclic QSE code is working properly.",
"In Sec.",
", we scanned over the value of $\\lambda $ and plotted the eigenstates for several values in Fig.",
"REF .",
"We described the dependence of several basis states for each eigenstate on $\\lambda $ and how its growth related to first- or second-order perturbation theory.",
"In particular, we showed that basis states connected to the main basis state by one application of the Hamiltonian grew linearly with $\\lambda $ , when $\\lambda $ was small and perturbative.",
"On the other hand, we showed that basis states connected to the main basis state by two applications of the Hamiltonian grew quadratically with $\\lambda $ , again when $\\lambda $ was small and perturbative.",
"We also noted that the deviation from linear or quadratic growth at higher $\\lambda $ was due to the increasing importance of higher-order corrections.",
"We also compared the results of our cyclic QSE code with second-order perturbation theory over the range of $\\lambda $ .",
"Since our code is random in nature, we ran our code 100 independent times for each value of $\\lambda $ and compared each independent run with second-order perturbation theory.",
"We used these independent runs to build up a distribution of results.",
"In Fig.",
"REF , we plotted the range of largest differences between our QSE code and second-order perturbation theory.",
"We showed that the vacuum and two-particle eigenstates, $\\Psi _v$ and $\\Psi _{24,-24}$ , are nearly always in agreement within the precision of the calculation.",
"On the other hand, we showed that the comparison for the four-particle eigenstate gets worse as $\\lambda $ increases and is in bad agreement for $\\lambda \\gtrsim 0.1m^2$ .",
"We noted that this was due to increasingly missing important basis states as $\\lambda $ increases and was a severe challenge for this code properly constructing four-particle eigenstates such as this one.",
"We suggested that it might be possible to further improve the QSE code for four-particle eigenstates and that that would be an important area of future research.",
"Finally, we also calculated the inner product of $\\Psi _{24,-24}$ and $\\Psi ^+_{24,-8,-8,-8}$ for this range of $\\lambda $ and plotted it in Fig.",
"REF .",
"Since this result potentially depends on the random nature of our cyclic QSE code, we calculated it for the 100 independent trials of our QSE code and show the range of values in this plot.",
"We found the inner product to be in agreement with zero for the entire range of $\\lambda $ and for all 100 trials at each value of $\\lambda $ .",
"We noted that this was true even though we missed important basis states in $\\Psi ^+_{24,-8,-8,-8}$ at large $\\lambda $ because those missed basis states were below the cutoff in $\\Psi _{24,-24}$ , and thus, did not contribute above the cutoff.",
"In Sec.",
", we analyzed the dependence of our results on the momentum spacing $\\Delta p$ .",
"We calculated the eigenstates at $\\Delta p=2m, 1m, 0.5m$ and $0.25m$ and show plots for three of these $\\Delta p$ in Fig.",
"REF .",
"We noted several important features of these eigenstates as $\\Delta p$ decreased.",
"The first is that the shapes of the eigenstates are largely the same as $\\Delta p$ decreases towards smaller values.",
"This gives us some confidence that the results with large $\\Delta p$ approximate the results when $\\Delta p$ is small.",
"Secondly, the density of basis states increases as $\\Delta p$ decreases, filling in the gaps in the plots and resolving the structure with greater detail.",
"We accounted for this by understanding the dependence of the free energy on $\\Delta p$ through $E_f=\\sum _i\\sqrt{n_i^2\\Delta p^2+m^2}$ .",
"Third, we found that the fall off of the basis states as their free energy increased was greater as $\\Delta p$ became smaller.",
"We picked this apart and found that this was due to the coefficients of the basis states decreasing as $\\Delta p$ diminished.",
"We claimed that this was mainly due to plotting the coefficients of the basis states directly rather than the coefficients normalized by the density of basis states.",
"If we had instead plotted the coefficients normalized by the density of basis states, we believe the fall of the basis states would stabilize and approach a constant value as $\\Delta p\\rightarrow 0$ .",
"But, we noted that we do not yet know the correct density of basis states to use for this normalization.",
"We believe this is an extremely important question to answer if we are to accurately calculate the limit of these eigenstates and the S-Matrix inner product between eigenstates as $\\Delta p\\rightarrow 0$ and plan to research it further in the future.",
"We ended this section with a calculation of the inner product as a function of $\\Delta p$ and plotted it in Fig.",
"REF .",
"We found that the first-order perturbative result continued to be greater than zero for the entire range that we analyzed while both second order and our cyclic QSE code were both in agreement with zero at the precision of our calculation.",
"Again, this was encouraging that our QSE-code result was in agreement with the higher orders of perturbation theory as well as with expectation based on physical arguments of energy conservation as $\\Delta p\\rightarrow 0$ .",
"In Sec.",
", we explored the dependence of our code on the size of the reduced Hilbert space.",
"As we did with the dependence on $\\lambda $ , we calculated the eigenstates 100 independent times with our cyclic QSE code for each value of the reduced Hilbert space size.",
"We plotted the distribution of largest differences between the results of our QSE code and second-order perturbation theory in Fig.",
"REF .",
"We found that for the vacuum and two-particle eigenstate, $\\Psi _v$ and $\\Psi _{24,-24}$ , our cyclic QSE code was in very good agreement with second-order perturbation theory for a large range of sizes of reduced Hilbert spaces from 200 up to 1000.",
"We even suggested this might be used to our advantage if we calculate elastic $2\\rightarrow 2$ scattering in two spatial dimensions.",
"On the other hand, we found that the four-particle eigenstate $\\Psi ^+_{24,-8,-8,-8}$ was extremely sensitive to the reduced Hilbert space size and that we only obtained good results for larger values on the order, or larger than, 1000.",
"On the one hand, increasing the size of the reduced Hilbert space is a great way to improve the accuracy of the results coming from our QSE code.",
"However, we note that, on the other hand, the time our QSE code takes to do the calculation is exponentially sensitive to the reduced space size.",
"We plotted the time it took versus the size of the reduced Hilbert space in Fig.",
"REF .",
"Also in Sec.",
", we studied how the efficiency of our cyclic QSE code was affected as the eigenstate increased in energy above the vacuum.",
"We compared this with perturbation theory which took exponentially longer the higher the eigenstate was above the vacuum.",
"This was because the density of basis states that must be checked increases exponentially as the free energy increases.",
"On the other hand, we noted that the QSE code was essentially unaffected by an increase in the energy of the eigenstate.",
"We claimed that this was because the QSE code does not try to calculate the contribution of every basis state in the vicinity.",
"It simply searches nearby the basis states it already has and this algorithm seems to work well at a broad range of energies.",
"There are still several open questions that we feel are very important.",
"We have already mentioned that the search algorithm for new basis states to add to the reduced Hilbert space needs to be improved for four-particle eigenstates if this method is to be useful for their calculation.",
"We also mentioned that, in order to get the limit of these eigenstates and their inner products as $\\Delta p\\rightarrow 0$ , it is imperative that we understand the density of basis states better and use it to normalize the coefficients of the basis states.",
"However, there are others.",
"Although this QSE method appears to work well at the values of $\\Delta p$ that we have calculated and it appears that it scales better than perturbation theory as $\\Delta p\\rightarrow 0$ and as the eigenstate energy increases, it still grows too large for the very small values of $\\Delta p$ that we are interested in.",
"We obtained results as low as $\\Delta p=0.25m$ in this project but we believe we eventually need to get $\\Delta p$ down to perhaps around $0.01m$ in order to get physically meaningful results in the continuum limit.",
"We think this may be possible with this method on a supercomputer, but we also think there may be significant ways to yet improve on the algorithm itself.",
"This will form a major line of our future research.",
"Beyond these points, it is our long-term goal to calculate non-trivial, non-zero S-matrix elements between scattering eigenstates.",
"We were unable to do this in the present project because no two eigenstates were degenerate in energy where we could achieve both our QSE-code calculation as well as second- and even, at one value of $\\Delta p$ , third-order perturbation theory for comparison.",
"In order to calculate non-zero inner products, we will either need to reduce $\\Delta p$ to much smaller values where the eigenstates begin to overlap at the precision of the calculation or increase the spatial dimensions to two.",
"In the latter case, there would always be multiple two-particle eigenstates with the same magnitude of momenta, and therefore (due to the rotational symmetry) the same energy.",
"They would be degenerate.",
"On the other hand, the added complexity of increasing the spatial dimension will be a challenge.",
"However, we think that since we can control the granularity of the angle, we might make progress in this direction, and perhaps achieve a nonzero scattering matrix element.",
"In any case, achieving a nonzero scattering amplitude is a major objective of our future research.",
"Although it is not the purpose of this paper to give the technical details of how to extract the scattering amplitude from the non-zero inner product, perhaps it would be good to give some insight into how we hope to do this in the future.",
"Although our results will be effectively non-perturbative, our intended technique is based on the ideas of “old-fashioned\" perturbation theory.",
"Our discussion follows Weinberg [30].",
"The S matrix can be split into a non-interacting piece and a scattering amplitude.",
"In Weinberg's very compact notation, this relation reads $S_{\\beta \\alpha }=\\delta (\\beta -\\alpha )-2i\\pi \\delta (E_\\beta -E_\\alpha )T_{\\beta \\alpha }^+\\ ,$ where the greek symbols $\\alpha $ and $\\beta $ represent the full list of quantum numbers for all the particles of the in and out states, respectively.",
"This includes the momenta, the spins, the charges and masses.",
"When $\\alpha =\\beta $ , the in and out states are exactly the same with no interaction.",
"This is removed in the $\\delta (\\beta -\\alpha )$ term.",
"$T_{\\beta \\alpha }^+$ is the scattering amplitude, up to a total-momentum preserving delta function.",
"It is given by the inner product $T_{\\beta \\alpha }^+=\\left(\\Phi _\\beta ,V\\Psi _\\alpha ^+\\right)\\ ,$ where $\\Phi _\\beta $ is a free-particle state, analogous to our free-particle states $\\langle p_1,\\cdots |$ , $\\Psi _\\alpha ^+$ is a scattering eigenstate, and $V$ is the potential, the interacting part of the Hamiltonian.",
"In old-fashioned perturbation theory, the scattering eigenstate $\\Psi _\\alpha ^+$ is expanded in a power-series in the coupling constant.",
"At leading order, $\\Psi _\\alpha ^+=\\Phi _\\alpha $ , and the leading order contribution is simply $\\left(\\Phi _\\beta ,V\\Phi _\\alpha \\right)$ , or using our discrete basis states, the leading order contribution to the scattering amplitude is given by $T_{\\beta \\alpha }^{+(0)} = \\langle \\beta |V|\\alpha \\rangle \\ .$ Consider, for example, the elastic scattering of two particles in our $\\lambda \\phi ^4$ theory.",
"The scattering eigenstate is dominated by the free two-particle basis state, say $|p_1,p_2\\rangle $ , therefore, the leading-order contribution to the elastic scattering amplitude is given by $T_{p_1,p_2;p_1,p_2}^{+(0)} = \\langle p_1,p_2|V|p_1,p_2\\rangle \\ .$ If we look at the potential of our Hamiltonian given in Eq.",
"(REF ), we find a term that is $\\lambda a_{p_1}^\\dagger a_{p_2}^\\dagger a_{p_1} a_{p_2}$ , up to normalization factors.",
"This term will annihilate the two particles in the free-particle state on the right, then re-create them, leaving us with $\\lambda \\langle p_1,p_2|p_1,p_2\\rangle =\\lambda $ , again up to normalization factors.",
"But, this is the well-known elastic $2\\rightarrow 2$ scattering amplitude at tree level in $\\lambda \\phi ^4$ theory!",
"The normalization factors will have to be accounted for properly, of course, but they are the discrete-momentum analogs of the factors in the continuum theory, which has already been worked out in old-fashioned perturbation theory.",
"Naturally, we would like to go beyond tree level.",
"In old-fashioned perturbation theory, we find $\\Psi _\\alpha ^+$ to higher order, as we have done in Appendix , showing the perturbative expansion in terms of the free-particle basis states in Eqs.",
"(REF ) through (REF ).",
"We then plug these higher-order contributions into Eq.",
"(REF ).",
"As a result of these higher-order corrections to the scattering eigenstate, the scattering amplitude will pick up contributions from other basis states, and moreover, the contribution from each basis state will change slightly as higher orders in perturbation theory are included.",
"Additionally, because the contributions of the basis states at higher orders are energy dependent, it will lead to energy dependence in the scattering amplitude.",
"So far, we are not stating anything new.",
"All of this existed long ago, before Feynman diagrams were even introduced [30].",
"Our contribution is to suggest that these scattering eigenstates can be found more efficiently using the QSE algorithm and that the result will be effectively correct to all orders in perturbation theory, at least to the precision of the calculation, without doing the much more difficult perturbative calculations.",
"In this paper, we have shown that we can reproduce the scattering eigenstates with the QSE method and we have shown that it is more efficient and scales much better than perturbation theory as the momentum spacing $\\Delta p$ decreases.",
"Once we have constructed the scattering eigenstate, we can simply plug it into Eq.",
"(REF ) to determine the scattering amplitude.",
"We do this in the same way we would do it for perturbation theory, however, we simply do it all at once rather than order by order in the coupling constant.",
"Of course, there will be challenges as we approach this result, both expected and unexpected.",
"Among the most important near-term expected challenges are that we will have to reduce the momentum spacing $\\Delta p$ to be very small and determine the density of states so that our eigenstates asymptotically approach a stable continuum eigenstate.",
"We then hope to extrapolate our results to the continuum limit based on the small $\\Delta p$ results.",
"Our approach has some aspects in common with calculations of the S matrix on the lattice.",
"Our discretization of momentum spacing can be seen to come from a finite space with a periodic boundary condition.",
"However, we do not latticize space, we do not Euclideanize space and we do not deal directly with the fields.",
"In fact, it is partly our purpose to move away from the field formulation of particle physics.",
"Nevertheless, there is important research going into the lattice calculation of the S matrix, which complements and influences our own work.",
"An important breakthrough in this field was a method for calculating the elastic scattering amplitude on the lattice [31].",
"As part of that work, the authors state the importance of using finite spaces that are very large (the dual of our very small $\\Delta p$ ) so that the “wave function ... is accurately given by the free wave\" near the boundaries and to suppress the effects of virtual particles going “around the world\".",
"Of course, we must achieve a small $\\Delta p$ partly, at least, for the same reasons.",
"Lattice calculations of non-elastic scattering, on the other hand, have not been fully worked out yet, but a couple gateway references for the progress in this field are [32], [33]."
],
[
"Truncated Hilbert Space Size",
"In [10], after discretizing momentum space, the Hilbert space was truncated by setting an upper limit on the free-particle energy of the basis states.",
"A limit on the number of particles to a maximum of two was also imposed.",
"The reason this was done is that, when all the basis states are kept, even with a cutoff on the free energy of the basis states, the Hilbert space grows too rapidly and quickly overcomes the ability of computers to diagonalize.",
"Indeed, it quickly overcomes the ability of computers to even store the Hamiltonian or the Hilbert space itself.",
"For illustration, we have plotted the size of the truncated Hilbert space as a function of the cutoff on the free-particle energies of the basis states in the top plot of Figure REF .",
"Figure: The number of basis states in a Hilbert space truncated by an energy cutoff on the basis states in the top plot.",
"The dots are calculated and the lines are fit to the dots.",
"The slopes on a log-log plot are determined and plotted in the lower plot.",
"They are fit to a straight line, determining the slope as a function of Δp\\Delta p.We have done this for several momentum step sizes.",
"For example, if we set the momentum step to $\\Delta p=0.05m$ , the Hilbert space already contains over 100 million state with a cutoff of only 12m.",
"We have also fit straight lines to the results on a log-log plot.",
"As expected, we find that the slope of these lines increases as $\\Delta p$ becomes smaller.",
"We have further plotted the slopes of these lines on the bottom plot of the same figure and fit a straight line to them on a log-log plot.",
"We find that the slope grows roughly as $1/\\sqrt{\\Delta p}$ $N(\\Delta p)\\sim E_{cut}^{\\Delta p^{-1/2}}\\ ,$ which makes sense, since as $\\Delta p\\rightarrow 0$ , the slope should become infinite.",
"On the other hand, in [10], we showed that for reasonable results, $\\Delta p$ should be much smaller than $m$ while $E_{cut}$ should be much larger.",
"This means that, in order to achieve good results with a reasonably small $\\Delta p$ and a reasonably high cutoff energy, while including multiple free particles in the basis states, we must find an alternative to diagonalizing the Hamiltonian while keeping all basis states below the cutoff.",
"Such an alternative was outlined in [9] and is called the Quasi-Sparse Eigenvector (QSE) method."
],
[
"The Cyclic QSE Method",
"Since the QSE method has been described in detail in [9], we will review it here in the context of our own calculations and invite the reader to refer to [9] for more details on the method itself.",
"Since we are interested in perturbative coupling, each scattering eigenstate is dominated by one basis state (see Appendix ).",
"For example, the vacuum approaches the 0-free-particle basis state in the limit $\\lambda \\rightarrow 0$ .",
"As the coupling constant turns on, other basis states begin to contribute.",
"Schematically, a perturbative eigenstate looks like $\\Psi _m = c^{(0)}|m\\rangle + \\lambda \\sum _{i=1}^\\infty c^{(1)}_i|b^{(1)}_i\\rangle + \\lambda ^2\\sum _{i=1}^\\infty c^{(2)}_i|b^{(2)}_i\\rangle + \\cdots \\ ,$ where $|m\\rangle $ is the main basis state, $|b^{(1)}_i\\rangle $ is a basis state contributing at first order, $|b^{(2)}_i\\rangle $ is a basis state contributing at second order, and so on.",
"Although these sums are infinite, the contribution of the vast majority of the basis states is negligible, either because of the smallness of $\\lambda ^n$ or because of the smallness of $c^{(n)}_i$ which depends on the Hamiltonian matrix elements between basis states and inversely on their energy differences.",
"(See App.",
"for details.)",
"Since calculations using these eigenstates will have working precisions, it does not make sense to keep or use basis states whose contributions are below that working precision.",
"So, if possible, we would like to truncate the full eigenstate by keeping only the basis states whose contributions to the eigenstate is greater than the precision of the calculation.",
"We will call this the cutoff on the basis states.",
"After throwing away any basis states whose contribution is smaller than the cutoff, we have something like $\\Psi _m &=& c^{(0)}|m\\rangle + \\lambda \\sum _{i=1}^{N_1} c^{(1)}_i|b^{(1)}_i\\rangle + \\lambda ^2\\sum _{i=1}^{N_2} c^{(2)}_i|b^{(2)}_i\\rangle \\nonumber \\\\&&+ \\lambda ^3\\sum _{i=1}^{N_3} c^{(3)}_i|b^{(3)}_i\\rangle \\ ,$ where we are assuming, for the sake of illuminating this technique, that all order-4 contributions are below the cutoff and only the first $N_i$ basis states at order $i$ are above the cutoff.",
"For some given precision and momentum spacing $\\Delta p$ , we might have $N_1+N_2+N_3$ is on the order of one-hundred to one-thousand, out of the infinite number of possible basis states in the full Hilbert space.",
"If possible, we would like to calculate only the contributions from these basis states in perturbation theory, however, unfortunately, we do not know, a priori, which basis states are above the cutoff.",
"So, we must calculate them all, or at least all of the basis states below some reasonable energy cutoff and throw away any below the precision cutoff.",
"Unfortunately, above first order, this is a very inefficient process and we would like to find something better.",
"What we would like is a search algorithm that explores the Hilbert space and extracts the most important basis states.",
"Once we have these, we can simply construct the Hamiltonian matrix in this reduced Hilbert space and directly diagonalize it.",
"This is what the QSE method does.",
"It accomplishes this cyclically, with each cycle getting closer to the complete set of most important basis states.",
"This algorithm works in the following five steps: Begin with a seed for the scattering state.",
"This seed could be as simple as the main basis state, but since first-order perturbation theory is so efficient, we begin with the first-order perturbative result.",
"Of course, this is not yet sufficiently accurate for our purposes.",
"Remove from the current scattering state all basis states whose contributions are below the desired precision cutoff.",
"Call the remaining basis states the reduced Hilbert space.",
"Randomly add new basis states to the reduced Hilbert space.",
"Do this by choosing random basis states that are already in the reduced Hilbert space and act on them with a randomly chosen operator from the Hamiltonian.",
"This results in a new basis state chosen randomly from the Hilbert space which is, however, “close” to the basis states already in the reduced Hilbert space.",
"Construct the Hamiltonian matrix with the new reduced Hilbert space and diagonalize.",
"Repeat steps 2-4 until the scattering state is achieved.",
"The original authors of [9] were not considering scattering states or scattering amplitudes.",
"In fact, their interest was in non-perturbative results.",
"Since every cycle of the QSE method potentially reaches a higher order of perturbation theory and it potentially constructs all the basis states above the precision cutoff, we say that it is “effectively\" non-perturbative.",
"However, if the coupling constant were truly above the perturbative range where each higher order in the perturbative series was more important, not less, it is not clear to us how this method can work.",
"Therefore, we do not believe it is truly non-perturbative in a strict sense.",
"It is only effectively non-perturbative in the perturbative regime.",
"As far as we can tell, the coupling can be large, but must still be perturbative for this method to work.",
"In the rest of this appendix, we'll give details of how we choose a random basis state from the reduced Hilbert space and how we choose a random operator from the Hamiltonian.",
"Perhaps a reader will have a clever idea for improving it.",
"We will begin with our method for choosing a random basis state.",
"Each time we choose a basis state from the reduced Hilbert space, we begin by randomly choosing an integer.",
"If it is odd, then we choose a random basis state from the reduced Hilbert space with a flat distribution where they are all equally likely.",
"If the random integer is even, then we choose a basis state from the reduced Hilbert space weighted according to the log of their absolute coefficients.",
"We do this using a simple Monte-Carlo technique.",
"We first randomly choose a test basis state with a flat distribution.",
"We then choose a random number between the log of the precision cutoff (because the cutoff is less than one, this is a negative number) and zero.",
"If this random number is less than the log of the absolute value of the coefficient of the test basis state then we keep it.",
"If it is above it, we discard it and randomly choose a new test basis state.",
"We do this until we keep one.",
"Once we have a basis state chosen randomly from the reduced Hilbert space, we need to randomly choose an operator from the Hamiltonian in Eq.",
"(REF ) to generate a new basis state that we add to the reduced Hilbert space.",
"Our first step is to choose a random integer between 0 and 13.",
"If it is 0, we use the operator $a_{-p_1}a_{-p_2}a_{-p_3}a_{-p_4}$ which annihilates four momenta.",
"We will come back to how we choose the momenta shortly.",
"First, we complete how we choose which operator we use.",
"If the integer is equal to 1, 2 or 3, we use the operator $a^\\dagger _{p_1}a_{-p_2}a_{-p_3}a_{-p_4}$ which annihilates three momenta and creates one new momentum.",
"This reduces the number of free particles in the basis state by two and we have found that our code is more successful focusing on basis states with only two more, two fewer particles, or the same number of particles.",
"This is the reason we choose this operator three times more often than the one that annihilates four momenta.",
"In a similar vein, if the integer is equal to 4, 5 or 6, we use the operator $a^\\dagger _{p_1}a^\\dagger _{p_2}a_{-p_3}a_{-p_4}$ which annihilates two momenta and creates two new momenta.",
"This operator does not change the number of free particles in the basis state.",
"It only changes one or two of the momenta.",
"If the integer is equal to 7, 8 or 9, we use the operator $a^\\dagger _{p_1}a^\\dagger _{p_2}a^\\dagger _{p_3}a_{-p_4}$ which annihilates one momentum and creates three new momenta.",
"This operator increases the number of free particles in the basis state by two.",
"If the integer is equal to 10, we use the operator $a^\\dagger _{p_1}a^\\dagger _{p_2}a^\\dagger _{p_3}a^\\dagger _{p_4}$ which creates four new momenta and, as a result, increases the number of free particles in the basis state by four.",
"Again, since we find this is less important, we choose this operator less frequently.",
"Finally, if the integer is equal to 11, 12 or 13, we choose a special form of the operator $a^\\dagger _{p_1}a^\\dagger _{p_2}a_{-p_3}a_{-p_4}$ which simply shifts exactly two momenta by $\\pm \\Delta p$ .",
"We will describe it in further detail below, but we find this to be one of the most important operators to fill in important basis states directly adjacent to the current basis states in the eigenstate.",
"If any of these operations are unsuccessful, we simply return to the beginning and choose a random new basis state from the reduced Hilbert space and a random new operator from the Hamiltonian.",
"We continue doing this until we fill the requested number of basis states.",
"When the operator $a_{p_1}a_{p_2}a_{p_3}a_{p_4}$ is chosen, we first make a list of all possible combinations of four momenta whose sum is zero that does not completely annihilate the basis state.",
"If no such combination can be found (for example if it is a 2-particle basis state), the function returns failure and the code chooses a new basis state and a new operator as discussed in the previous paragraph.",
"If it does find some momentum combinations that do not annihilate the basis state, then it randomly chooses one of them and annihilates the four corresponding free particles from the basis state.",
"It then returns this as a new basis state that is added to the reduced Hilbert space.",
"If the operator $a^\\dagger _{p_1}a_{p_2}a_{p_3}a_{p_4}$ is chosen, it make a list of all combinations of three momenta from the basis state.",
"It then randomly chooses one of these combinations and creates a free particle with a momentum equal to the sum of the three momenta.",
"This is followed by annihilations of the three momenta.",
"If this is successful, the new basis state is returned, else failure and the codes starts its search for a new basis state over.",
"If the operator $a^\\dagger _{p_1}a^\\dagger _{p_2}a_{p_3}a_{p_4}$ is chosen, first a list of combinations of two momenta is created from the basis state.",
"These two basis states will be annihilated.",
"We then randomly choose a momentum between $-E_{max}$ and $E_{max}$ , which is a parameter that we can adjust.",
"We always set it to be higher than the highest basis state contributing above the precision cutoff.",
"For example, for the eigenstates shown in Fig.",
"REF , we set $E_{max}$ to be $80m$ .",
"This should be more than sufficient since the basis state energy is the sum of the energy of all the momenta in it.",
"One individual particle always has a much smaller momentum than this.",
"However, we keep this relatively high $E_{max}$ to be on the safe side.",
"This same $E_{max}$ is used for the rest of the operators we discuss in this appendix.",
"Once this random new momentum is chosen, the final momentum is given the value $p_1=p_3+p_4-p_2$ , where $p_3$ and $p_4$ are the momenta chosen from those already existing and $p_2$ is the momentum randomly chosen between $\\pm E_{max}$ .",
"We then create two new free particles with momenta $p_1$ and $p_2$ followed by annihilation of free particles with momenta $p_3$ and $p_4$ .",
"If this is successful, we return the new basis state, if not, we return failure.",
"When the operator $a^\\dagger _{p_1}a^\\dagger _{p_2}a^\\dagger _{p_3}a_{p_4}$ is chosen, we begin by randomly choosing one of the momenta in the basis state.",
"We then choose two new momenta, each randomly between $-E_{max}$ and $E_{max}$ .",
"The final momentum is taken as $p_1=p_4-p_2-p_3$ .",
"Three new free particles are created with momenta $p_1, p_2$ and $p_3$ followed by one annihilation of a free particle with momentum $p_4$ .",
"If success is found, the new basis state is returned, else failure is returned.",
"If the operator $a^\\dagger _{p_1}a^\\dagger _{p_2}a^\\dagger _{p_3}a^\\dagger _{p_4}$ is chosen, three new momenta are chosen, each randomly between $-E_{max}$ and $E_{max}$ .",
"The final momentum is chosen as $p_1=-p_2-p_3-p_4$ .",
"Four new free particles are created in the basis state with momenta $p_1, p_2, p_3$ and $p_4$ .",
"If this is a success, the new basis state is returned, otherwise, failure.",
"Finally, if the special operator is chosen that simply shifts exactly two momenta by $\\pm \\Delta p$ , then two of the momenta from the basis state are randomly chosen, say $p_1$ and $p_2$ .",
"Free particles with momenta $p_1$ and $p_2$ are annihilated and two new free particles with momenta $p_1+\\Delta p$ and $p_2-\\Delta p$ are created.",
"If this is successful, the new basis state is returned.",
"If not, failure is returned.",
"After randomly creating a new basis state as described, we add either the P-even or the P-odd version of that basis state to the reduced Hilbert space.",
"For example, suppose the new basis state randomly generated is $|b\\rangle $ .",
"We then find the P reversed state $P|b\\rangle $ by reversing all the momenta.",
"If these two states are the same ($P|b\\rangle =|b\\rangle $ ), we simply add the basis state $|b\\rangle $ .",
"If they are different, we choose a random integer.",
"If the random integer is even, we add the P-even basis state $(|b\\rangle +P|b\\rangle )/\\sqrt{2}$ .",
"If the random integer is odd, we add the P-odd basis state $(|b\\rangle -P|b\\rangle )/\\sqrt{2}$ .",
"We could have simply always added the P-even basis states as those were what we expected for the eignestates we studied, but we wanted to keep our algorithm more general and we wanted to be sure our algorithm was working correctly, so we included both P-even and P-odd basis states in our reduced Hilbert space at each cycle.",
"As expected, we found that the P-odd basis states were all found to be below the precision cutoff and removed from the reduced Hilbert space by our algorithm while all the basis states remaining above the cutoff were P even."
],
[
"The $\\mathbf {\\lambda \\phi ^4}$ Theory",
"We already worked out the discrete Hamiltonian of our theory in two spacetime dimensions in Section I of [10].",
"However, we make one modification in the present paper.",
"We renormalize the mass in order to cancel the second term of Eq.",
"(10) in that paper.",
"We review the derivation and explain the modification here.",
"We begin with the Lagrangian of our theory $\\mathcal {L} = \\frac{1}{2}\\partial _\\mu \\phi \\partial ^\\mu \\phi - \\frac{1}{2}m^2\\phi ^2 - \\frac{\\lambda }{4!",
"}\\phi ^4\\ ,$ which, after Legendre transformation, gives the Hamiltonian $H = \\int dx\\left[\\frac{1}{2}\\left(\\frac{\\partial \\phi }{\\partial t}\\right)^2+\\frac{1}{2}\\left(\\frac{\\partial \\phi }{\\partial x}\\right)^2+\\frac{1}{2}m^2\\phi ^2+\\frac{\\lambda }{24}\\phi ^4\\right]\\ .$ As before we replace the field $\\phi $ with a linear combination of creation and annihilation operators $\\phi (x) = \\int \\frac{dp}{2\\pi }\\frac{1}{\\sqrt{2\\omega }}\\left[a(p)e^{i \\left(\\omega t-p x\\right)}+a^\\dagger (p)e^{-i \\left(\\omega t-p x\\right)}\\right]\\ .$ However, unlike before, we define the free-particle energy as $\\omega = \\sqrt{p^2+\\tilde{m}^2}\\ ,$ where $\\tilde{m}\\ne m$ .",
"After inserting this definition of the fields, expanding and normal ordering, we get the following contribution from the bare-mass term $\\int dx\\frac{1}{2}m^2\\phi ^2 = \\frac{1}{2}\\int \\frac{dp}{\\left(2\\pi \\right)\\left(2\\omega \\right)}m^2\\Big [&a_pa_{-p}e^{i2\\omega t}+2a_pa^\\dagger _p&\\nonumber \\\\&+a^\\dagger _pa^\\dagger _{-p}e^{-i2\\omega t}\\Big ]\\ ,&\\nonumber \\\\$ where we have dropped a non-dynamical constant term.",
"On the other hand, the $\\lambda \\phi ^4$ term, after normal ordering, also gives terms quadratic in the creation and annihilation operators, namely $\\int dx\\frac{\\lambda }{24}\\phi ^4 = \\frac{1}{2}\\int \\frac{dp}{\\left(2\\pi \\right)\\left(2\\omega \\right)}\\Delta m^2\\Big [&a_pa_{-p}e^{i2\\omega t}+2a_pa^\\dagger _p&\\nonumber \\\\&+a^\\dagger _pa^\\dagger _{-p}e^{-i2\\omega t}\\Big ]&\\nonumber \\\\&+\\cdots \\ ,$ where the dots represent the terms with four creation and annihilation operators that will not contribute to the renormalization of mass.",
"We have again dropped a non-dynamical constant term and defined $\\Delta m^2 = \\frac{\\lambda }{4}\\int \\frac{dp^{\\prime }}{\\left(2\\pi \\right)\\omega ^{\\prime }}\\ .$ We see that the contribution to the Hamiltonian in Eq.",
"(REF ) has exactly the same form as the contribution from the bare-mass term in Eq.",
"(REF ), therefore, it has the effect of renormalizing the mass.",
"We now take $\\tilde{m}^2 = m^2+\\Delta m^2\\ .$ We continue here with the rest of the textbook derivation of the free part of the Hamiltonian for the convenience of the reader.",
"The contribution from the spatial derivative is given by $\\int dx\\frac{1}{2}\\left(\\frac{\\partial \\phi }{\\partial x}\\right)^2 = \\frac{1}{2}\\int \\frac{dp}{\\left(2\\pi \\right)\\left(2\\omega \\right)}p^2\\Big [&a_pa_{-p}e^{i2\\omega t}+2a_pa^\\dagger _p&\\nonumber \\\\&+a^\\dagger _pa^\\dagger _{-p}e^{-i2\\omega t}\\Big ]\\ ,&\\nonumber \\\\$ where an overall non-dynamical constant has been dropped.",
"When this is combined with the $m^2$ contribution from Eq.",
"(REF ) and the $\\phi ^4$ contribution from Eq.",
"(REF ), we can use $\\omega ^2=p^2+\\tilde{m}^2$ to obtain $\\int dx\\left[\\frac{1}{2}\\left(\\frac{\\partial \\phi }{\\partial x}\\right)^2+\\frac{1}{2}m^2\\phi ^2+\\frac{\\lambda }{24}\\phi ^4\\right] =\\hspace{72.26999pt}$ $\\frac{1}{2}\\int \\frac{dp}{\\left(2\\pi \\right)\\left(2\\omega \\right)}\\omega ^2\\Big [a_pa_{-p}e^{i2\\omega t}+2a_pa^\\dagger _p+a^\\dagger _pa^\\dagger _{-p}e^{-i2\\omega t}\\Big ]+\\cdots \\ .$ Finally, the time derivative term gives $\\int dx\\frac{1}{2}\\left(\\frac{\\partial \\phi }{\\partial t}\\right)^2 = \\frac{1}{2}\\int &\\frac{dp}{\\left(2\\pi \\right)\\left(2\\omega \\right)}\\omega ^2\\Big [-a_pa_{-p}e^{i2\\omega t}&\\nonumber \\\\&+2a_pa^\\dagger _p-a^\\dagger _pa^\\dagger _{-p}e^{-i2\\omega t}\\Big ]\\ ,&\\nonumber \\\\$ where, as usual, we drop a non-dynamical constant.",
"We see that the first and third terms cancel between the time derivative contribution and the other contributions while the second term adds.",
"Our final result is $H = \\int \\frac{dp}{\\left(2\\pi \\right)}\\omega a_p a^\\dagger _p + \\cdots \\ ,$ where the dots, again, represent the terms with four creation and annihilation operators coming from the interaction.",
"The final step, for our purposes, is to discretize momentum space.",
"We do this by taking $\\int dp/\\left(2\\pi \\right)\\rightarrow \\sum _p\\Delta p$ and $2\\pi \\delta (p)\\rightarrow \\delta _p/\\Delta p$ , where $\\Delta p$ is the momentum step size and $\\delta _p$ is the Kronecker delta (equal to 1 if $p=0$ and 0, otherwise).",
"Consequently, we take $a(p)\\rightarrow a_p/\\sqrt{\\Delta p}$ and $a^\\dagger (p)\\rightarrow a^\\dagger _p/\\sqrt{\\Delta p}$ for the discrete creation and annihilation operators and find $\\left[a_p,a^\\dagger _{p^{\\prime }}\\right] = \\delta _{p,p^{\\prime }}$ .",
"With this, the final discrete Hamiltonian is $H &=&\\sum _p \\omega a^\\dagger _p a_p+\\frac{\\lambda \\Delta p}{96}\\sum _{p_1+p_2+p_3+p_4=0}\\frac{1}{\\sqrt{\\omega _1\\omega _2\\omega _3\\omega _4}}\\Big [\\nonumber \\\\&&a_{-p_1}a_{-p_2}a_{-p_3}a_{-p_4} e^{i\\left(\\omega _1+\\omega _2+\\omega _3+\\omega _4\\right)t}\\nonumber \\\\&&+4a^\\dagger _{p_1}a_{-p_2}a_{-p_3}a_{-p_4} e^{i\\left(-\\omega _1+\\omega _2+\\omega _3+\\omega _4\\right)t}\\nonumber \\\\&&+6a^\\dagger _{p_1}a^\\dagger _{p_2}a_{-p_3}a_{-p_4} e^{i\\left(-\\omega _1-\\omega _2+\\omega _3+\\omega _4\\right)t}\\nonumber \\\\&&+4a^\\dagger _{p_1}a^\\dagger _{p_2}a^\\dagger _{p_3}a_{-p_4} e^{i\\left(-\\omega _1-\\omega _2-\\omega _3+\\omega _4\\right)t}\\nonumber \\\\&&+a^\\dagger _{p_1}a^\\dagger _{p_2}a^\\dagger _{p_3}a^\\dagger _{p_4} e^{i\\left(-\\omega _1-\\omega _2-\\omega _3-\\omega _4\\right)t}\\Big ]\\ ,\\nonumber \\\\$ which is the same as Eq.",
"(10) of [10], except that the second term (quadratic in creation and annihilation operators) has been dropped due to the present mass renormalization.",
"The reason we did not do this renormalization in [10] is that we only kept basis states with up to two free particles in [10].",
"A truncated Hilbert space with only zero-particle and two-particle basis states completely decouple after this mass renormalization.",
"We kept the extra term in [10] in order to achieve more interesting results with this highly truncated space.",
"However, now that we are including a large set of basis states, including basis states with greater numbers of free particles, it makes sense to use the renormalized mass from the beginning.",
"Before ending this appendix, we briefly describe our basis states.",
"They are the same as in [10], however, in this paper, we will write our basis states as $|p_1,p_2,\\cdots ,p_n\\rangle $ where $p_1, p_2,\\cdots ,p_n$ is the list of the momenta of the free particles, where we have dropped the vector symbols for convenience since we are working in 1 spatial dimension.",
"Moreover, because we are dealing with bosons, the order does not matter.",
"In the $|p_1,p_2,\\cdots ,p_n\\rangle $ example, there are $n$ free particles.",
"Some of the momenta could be the same.",
"If this is the case, that momentum is listed multiple times.",
"If there are no particles in the basis state, it is written $|\\rangle $ and called the free vacuum.",
"These basis states are eigenstates of the free part of the Hamiltonian, namely $\\sum _p\\omega _pa^\\dagger _p a_p$ .",
"We call their free-Hamiltonian eigenvalue the free energy of the basis state and write it as $\\tilde{\\omega }$ .",
"It is simply the sum of the free energy $\\omega _p$ for each particle.",
"The creation and annihilation operators acting on these states give $a^\\dagger _p|\\cdots ,p,\\cdots \\rangle &=& \\sqrt{n_p+1}|p\\cdots ,p,\\cdots \\rangle \\ ,\\\\a_p|\\cdots ,p,\\cdots \\rangle &=& \\sqrt{n_p}|\\cdots ,\\cdots \\rangle \\ .$ where $n_p$ is the number of times the momentum $p$ appears in the state.",
"(We have only showed it explicitly once, but there may be others in the $\\cdots $ .)",
"If it does not appear at all, then $n_p=0$ .",
"The Hamiltonian matrix is then constructed by sandwiching the Hamiltonian operator in Eq.",
"(REF ) between all pairs of basis states in the (truncated or reduced) Hilbert space.",
"The inner product between two states is equal to 1 if they have exactly the same list of momenta (although, again, the order does not matter) and 0 otherwise."
],
[
"Perturbative Solution",
"In this appendix, for completeness, we review time-independent perturbation theory, which can be found in many textbooks (e.g.",
"[34]).",
"The Schrodinger equation can be written $H|\\psi \\rangle = E|\\psi \\rangle \\ ,$ where $H$ is the Hamiltonian, $|\\psi \\rangle $ is an eigenstate and $E$ is the associated eigenenergy.",
"We assume that the Hamiltonian can be written $H = H_o+\\lambda V\\ ,$ where $H_o$ is the free Hamiltonian and is assumed exactly solvable, $\\lambda $ is a small coupling constant and $\\lambda V$ is the interaction part of the Hamiltonian.",
"Because $\\lambda $ is small, we can expand the eigenstate and the energy as a power series in $\\lambda $ as in $|\\psi \\rangle &=& |\\psi ^0\\rangle + \\lambda |\\psi ^1\\rangle + \\lambda ^2|\\psi ^2\\rangle + \\cdots \\\\E &=& E^0 + \\lambda E^1 + \\lambda ^2 E^2 + \\cdots \\ ,$ where the superscript on $\\psi $ and $E$ are labels and not powers.",
"Plugging these expansions into the Schrodinger Equation, we obtain $\\left(H_o+\\lambda V\\right)\\left(|\\psi ^0\\rangle + \\lambda |\\psi ^1\\rangle + \\lambda ^2|\\psi ^2\\rangle + \\cdots \\right) =\\hspace{72.26999pt}$ $\\left(E^0 + \\lambda E^1 + \\lambda ^2 E^2 + \\cdots \\right)\\left(|\\psi ^0\\rangle + \\lambda |\\psi ^1\\rangle + \\lambda ^2|\\psi ^2\\rangle + \\cdots \\right).$ Since this equation must be satisfied for any value of the coupling constant $\\lambda $ , it must be satisfied order by order in $\\lambda $ .",
"Therefore, we get the system of equations $H_o|\\psi ^0\\rangle &=& E^0|\\psi ^0\\rangle \\nonumber \\\\H_o|\\psi ^1\\rangle + V|\\psi ^0\\rangle &=& E^0|\\psi ^1\\rangle + E^1|\\psi ^0\\rangle \\nonumber \\\\H_o|\\psi ^2\\rangle + V|\\psi ^1\\rangle &=& E^0|\\psi ^2\\rangle + E^1|\\psi ^1\\rangle + E^2|\\psi ^0\\rangle \\nonumber \\\\&\\vdots &\\ .$ It turns out that this system of equations can be solved recursively.",
"We do this by introducing the eigenstates of the free Hamiltonian as a complete set of basis states.",
"We will call these basis states $|b_0\\rangle , |b_1\\rangle , |b_2\\rangle ,\\cdots $ and their associated free Hamiltonian eigenvalues (their free energies) $\\omega _0, \\omega _1, \\omega _2,\\cdots $ so that $H_o|b_j\\rangle = \\omega _j|b_j\\rangle $ .",
"We have seen above in Eq.",
"(REF ) that $|\\psi ^0\\rangle $ is an eigenstate of the free Hamiltonian, therefore, for notational convenience, we take it to be $|b_0\\rangle $ .",
"This tells us that $E^0 = \\omega _0\\ .$ We next expand $|\\psi ^1\\rangle $ in terms of the basis states so that $|\\psi ^1\\rangle = \\sum _j c_j^1 |b_j\\rangle \\ ,$ where $c_j^1$ are the coefficients of the basis states.",
"We plug this into the second line of Eq.",
"(REF ) to obtain, after moving $|\\psi ^1\\rangle $ to the left and $|\\psi ^0\\rangle $ to the right, $\\sum _j c_j^1\\left(\\omega _j-\\omega _0\\right)|b_j\\rangle = \\left(E^1-V\\right)|b_0\\rangle \\ .$ We can see that this equation does not determine the coefficient of $|b_o\\rangle $ , $c_0^1$ , because that term drops out of the left side ($\\omega _0-\\omega _0=0$ ).",
"In fact, if we take the inner product of this equation with $\\langle b_0|$ , we obtain the equation $E^1 = \\langle b_0|V|b_0\\rangle \\ ,$ which determines the first-order contribution to the energy.",
"However, we would get this even if the sum over $j$ in Eq.",
"(REF ) did not include $j=0$ .",
"In fact, there is no way to determine $c_0^1$ from the perturbation series.",
"It is not unique.",
"This is a result of the perturbation series itself not being unique.",
"In fact, we can multiply the eigenstate $|\\psi \\rangle $ by any constant function (that passes through the Hamiltonian) and it will still be an eigenstate.",
"For example, we consider $|\\tilde{\\psi }\\rangle = \\left(a_0+\\lambda a_1 + \\lambda ^2 a_2 + \\cdots \\right)|\\psi \\rangle \\ .$ We can see that this is also an eigenstate of $H$ with the same eigenvalue $H|\\tilde{\\psi }\\rangle = E|\\tilde{\\psi }\\rangle \\ .$ We can use this freedom to completely remove the basis state $|b_0\\rangle $ from all but the leading order solution.",
"To see this, suppose $|\\psi ^1\\rangle $ has a nonzero $c_0^1$ .",
"Now, consider the expansion of $|\\tilde{\\psi }\\rangle $ in a power series in $\\lambda $ .",
"It is given by $|\\tilde{\\psi }\\rangle = a_0|\\psi ^0\\rangle + \\lambda \\left(a_1|\\psi ^0\\rangle + a_0|\\psi ^1\\rangle \\right) + \\cdots \\ ,$ or, after plugging in the expansion in the basis states and focusing on $|\\tilde{\\psi }^1\\rangle $ , we have $|\\tilde{\\psi }^1\\rangle = a_1|b_0\\rangle + a_0\\sum _j c_j^1|b_j\\rangle \\ .$ Therefore, we can completely remove $|b_0\\rangle $ from $|\\tilde{\\psi }^1\\rangle $ by taking $a_1 = -a_0 c_0^1\\ .$ We can do this, order by order, so that the only place $|b_0\\rangle $ appears is in $|\\tilde{\\psi }^0\\rangle $ .",
"For the rest of this section, we will assume that this has been done and drop the tilde.",
"Returning to Eq.",
"(REF ), we now write $|\\psi ^1\\rangle = \\sum _{j\\ne 0} c_j^1 |b_j\\rangle \\ .$ We again plug this into the second line of Eq.",
"(REF ) to obtain, $\\sum _{j\\ne 0} c_j^1\\left(\\omega _j-\\omega _0\\right)|b_j\\rangle = \\left(E^1-V\\right)|b_0\\rangle \\ .$ If we take the inner product of this equation with $\\langle b_0|$ , we obtain the first-order contribution to the energy, $E^1 = \\langle b_0|V|\\psi ^0\\rangle \\ ,$ where we have replaced $|b_0\\rangle $ with $|\\psi ^0\\rangle $ , on the right.",
"If we, on the other hand, take the inner product of Eq.",
"(REF ) with $\\langle b_j|$ , where $j\\ne 0$ , we obtain $c_j^1\\left(\\omega _j-\\omega _0\\right) = -\\langle b_j|V|b_0\\rangle \\ .$ Solving for $c_j^1$ gives us, $c_j^1 = \\frac{\\langle b_j|V|\\psi ^0\\rangle }{\\omega _0-\\omega _j}\\ ,$ where we have again replaced $|b_0\\rangle $ with $|\\psi ^0\\rangle $ (and we are assuming that $|b_j\\rangle $ is not degenerate with $|b_0\\rangle $ ).",
"We will also do second order explicitly since an important new term appears in the coefficient.",
"Taking $|\\psi ^2\\rangle = \\sum _{j\\ne 0} c_j^2 |b_j\\rangle \\ .$ We plug this into the third line of Eq.",
"(REF ) to obtain, $\\sum _{j\\ne 0} c_j^2\\left(\\omega _j-\\omega _0\\right)|b_j\\rangle = \\left(E^1-V\\right)|\\psi ^1\\rangle + E^2|\\psi ^0\\rangle \\ .$ If we take the inner product of this equation with $\\langle b_0|$ , we obtain the second-order contribution to the energy, $E^2 = \\langle b_0|V|\\psi ^1\\rangle \\ .$ If we take the inner product with $\\langle b_j|$ , where $j\\ne 0$ , on the other hand, we obtain $c_j^2\\left(\\omega _j-\\omega _0\\right) = -\\langle b_j|V|\\psi ^1\\rangle + c_j^1 E^1\\ ,$ so that $c_j^2 = \\frac{1}{\\omega _0-\\omega _j}\\left(\\langle b_j|V|\\psi ^1\\rangle - c_j^1E^1\\right)\\ .$ Continuing in this way, we find the general nth-order energy $E^n = \\langle b_0|V|\\psi ^{n-1}\\rangle \\ ,$ where we remind the reader that $E = \\sum _n \\lambda ^n E^n\\ .$ We also find the general nth-order eigenstate $c_j^n = \\frac{1}{\\omega _0-\\omega _j}\\left(\\langle b_j|V|\\psi ^{n-1}\\rangle - \\sum _{k=1}^{n-1}c_j^{k}E^{n-k}\\right)\\ ,$ where $|\\psi ^n \\rangle = \\sum _{j\\ne 0}c_j^n|b_j\\rangle $ and $|\\psi \\rangle = \\sum _n \\lambda ^n |\\psi ^n\\rangle \\ .$ So far, we have only described non-degenerate perturbation theory, which is appropriate when none of the basis states are degenerate, or even nearly degenerate.",
"However, this technique, as so far described, breaks down when a basis state is encountered whose free energy is close to that of the main basis state.",
"The reason is that the difference in free energy $(\\omega _0-\\omega _j)$ appears in the denominator of Eq.",
"(REF ) causing the coefficient to be inappropriately large.",
"This signals the breakdown of non-degenerate perturbation theory.",
"The textbook method for dealing with this (e.g.",
"[34]) is to directly diagonalize the degenerate (or nearly degenerate) sector first and then apply the perturbative formulas to the eigenstates of that initial diagonalization along with the other non-degenerate basis states.",
"However, this becomes complicated at higher orders of perturbation theory because the basis states may not be connected by just one factor of the potential.",
"As a result, the diagonalization step may involve many other basis states that may not be degenerate or nearly degenerate.",
"So, in fact, we must diagonalize a much larger sector.",
"However, it is not clear, a priori, what basis states should be included in this diagonalization since it is no longer just the nearby basis states.",
"Our approach is to simply calculate the coefficients using the naive formulas described in this section, realizing that some of them will be incorrectly large due to the breakdown of perturbation theory.",
"However, after doing this naive perturbative calculation, we fortunately have a clear set of basis states that can be directly diagonalized.",
"It may be larger than necessary, but it is certainly sufficient as it includes all the basis states connecting the main basis state and the degenerate and nearly degenerate basis states at the current order.",
"So, after doing the naive perturbative calculation, we simply construct the Hamiltonian using the set of basis states coming from perturbation theory and directly diagonalize it.",
"We do this for second- and third-order perturbation theory in this article.",
"We do not diagonalize the Hamiltonian for the first-order results because we do not run into a problem with degenerate basis states at first order.",
"We use the perturbative formulas as is at first order."
]
] | 1709.01556 |
[
[
"Exploring the correlation between the folding rates of proteins and the\n entanglement of their native states"
],
[
"Abstract The folding of a protein towards its native state is a rather complicated process.",
"However there are empirical evidences that the folding time correlates with the contact order, a simple measure of the spatial organisation of the native state of the protein.",
"Contact order is related to the average length of the main chain loops formed by amino acids which are in contact.",
"Here we argue that folding kinetics can be influenced also by the entanglement that loops may undergo within the overall three dimensional protein structure.",
"In order to explore such possibility, we introduce a novel descriptor, which we call \"maximum intrachain contact entanglement\".",
"Specifically, we measure the maximum Gaussian entanglement between any looped portion of a protein and any other non-overlapping subchain of the same protein, which is easily computed by discretized line integrals on the coordinates of the $C_{\\alpha}$ atoms.",
"By analyzing experimental data sets of two-state and multistate folders, we show that also the new index is a good predictor of the folding rate.",
"Moreover, being only partially correlated with previous methods, it can be integrated with them to yield more accurate predictions."
],
[
"Introduction",
"Simple paradigms very often play an invaluable role to help understanding complex systems.",
"A well known example is given by protein folding.",
"Protein folding is the physical process by which a protein chain aquires its final three dimensional structure, the native state, that is usually biologically functional, in a reproducible manner.",
"The characteristic time of this process is named folding time.",
"Protein folding is complex because of the sheer size of protein molecules, the twenty types of constituent amino acids with distinct side chains, and the essential role played by the environment.",
"Nevertheless, it is by now widely accepted that several aspects of the process driving a sequence of amino-acids to the corresponding native structure can be inferred by simple descriptors of the native-state geometry , .",
"For instance, it has been shown that the folding nucleus of a protein, including the residues whose interactions are essential for the folding to the native state, can be predicted through simulations of homopolymer models based on the mere knowledge of the native contact map, that is of the whole list of residue pairs in contact with each other , , , .",
"Other evidences of this simplicity are the ability of effective energy scores, derived by a statistical analysis of folded protein structures, to discriminate real native states among set of competing decoys , , , , and the finding that the universe of possible proteins folds can be derived by simple coarse-grained models of polymers, which capture few universal properties typical of all amino-acids , , .",
"A further evidence of this emerging simplicity is the empirical result of Plaxco and coworkers , , who found a significant correlation between experimental folding rates of proteins, e.g.",
"the inverse folding times, and a simple descriptor of the native state organisation, such as the contact order, that is, the average chemical length (in terms of the number of amino-acids) of the loop formed between residues which are in contact.",
"This results is somewhat surprising because folding inevitably involves states other than the native one and these conformations might affect the kinetic process.",
"Despite some evidence that this correlation is weak for proteins belonging to the all-$\\beta $ structural class , later studies confirmed correlations between folding rates and other descriptors of the native state organisation.",
"These descriptors are long range order , the number of native contacts , , the total contact distance , the cliquishness , the local secondary structure content and the chain crosslinks contact order .",
"The contact order and all its possible variants are descriptors that focus on the network of pairs of residues that are nearby in space regardless of the full spatial arrangement of the protein conformation.",
"More realistically, one can however think of non-local descriptors that capture the degree of self-entanglement of the whole protein backbone, seen as a curve in a three dimensional space.",
"An example is the writhing number, a measure of how a curve winds around itself in space .",
"In protein physics, the writhing number was first used in to quantify the amount of self-threading in native state and it was later extended to perform a systematic classification of existing protein folds , .",
"After the seminal observation that the backbone of a protein may self entangle into physical knots , , , , , , , , a growing attention has been devoted to find new, topologically inspired, descriptors for quantifying accurately the winding of a protein with itself or with other molecules.",
"Specific descriptors have been proposed to measure the amount and location of mutual entanglement between protein complexes , , , or to detect specific topological knots, links and lassos within a single chain , , .",
"The aim of this study is to explore the correlation between protein folding rates and a novel topological descriptor of proteins three dimensional entanglement, which we name maximum intrachain contact entanglement.",
"This indicator is the maximum value of the mutual entanglement measured between any looped portion of a protein and any other non-overlapping subchain extracted from the same protein.",
"As a measure of the mutual entanglement, we consider the Gaussian double integral of two oriented curves.",
"For closed curves this measure reduces to the Gauss linking number, a topological invariant that quantifies how pairs of loops are (homologically) linked .",
"Being quite easy to compute, the Gauss linking number has been extensively used in the past to characterise the mutual entanglement of diluted and concentrated solutions of linear polymers , , , , to estimate the linking probability and link complexity of pair of loops under geometrical constraints , , , as well as to identify threadings in dense solutions of unlinked loops diffusing in a gel .",
"By exploring a data set of 48 proteins , , for which the folding time is known experimentally, we compute the linear correlation coefficient between the values of our descriptor and the experimental folding rates.",
"We show that the maximum intrachain contact entanglement captures aspects that are different from those highlighted by the contact order, and we describe how the two descriptors can be combined to improve the predictions of folding rates.",
"It is well known that the Gauss double integral ${G} \\equiv \\frac{1}{4 \\pi } \\oint _{\\gamma _1}\\oint _{\\gamma _2}\\frac{r^{(1)} -r^{(2)}}{\\left| r^{(1)} - r^{(2)}\\right|^3}\\cdot (d r^{(1)} \\times d r^{(2)})$ between two closed curves $\\gamma _1$ and $\\gamma _2$ in $\\mathbb {R}^3$ gives an integer number, known as the linking number, whose value is a topological invariant.",
"A nice feature of this measure, however, is that it provides a meaningful assessment of the mutual entanglement also if either one or both curves are open , , .",
"Figure: Following the color code of figure , examples of protein native structures in which we identified the subchains γ i \\gamma _i (i 1 →i 2 i_1 \\rightarrow i_2, yellow-red-yellow) and γ j \\gamma _j (j 1 →j 2 j_1 \\rightarrow j_2, blue) yielding the maximum intrachain contact entanglement.Our strategy here is to consider as $\\gamma _1$ and $\\gamma _2$ any pair $(\\gamma _i,\\gamma _j)$ of non-overlapping subchains extracted from the same protein backbone and to compute their Gaussian entanglement ${G}_{ij}$ .",
"Since the backbone of a native protein structure with $N$ residues can be described as a discrete chain of monomers ($i=1,\\ldots , N$ ) placed at the positions $r_i$ of the C$_\\alpha $ atoms, it is natural to define the average positions $R_i \\equiv \\frac{1}{2} ( r_i + r_{i+1} ) \\,,$ and the bond vectors $d R_i = r_{i+1} - r_{i}.$ Hence, for a Given subchain $\\gamma _i$ with monomers from index $i_1$ to $i_2$ , and a subchain $\\gamma _j$ with monomers from index $j_1$ to $j_2$ such that $\\gamma _j \\cap \\gamma _i =\\emptyset $ , their Gaussian entanglement is given by ${G^{\\prime }}_{i j} \\equiv \\frac{1}{4 \\pi }\\sum _{i=i_1}^{i_2-1} \\sum _{j=j_1}^{j_2-1}\\frac{R_i - R_j}{\\left|R_i - R_j\\right|^3} \\cdot ( d R_i \\times d R_j)$ where the prime in $G^{\\prime }$ highlights the fact that the measure is for open chains.",
"Note that, unlike in our previous study where the entanglement was estimated between two different protein backbones, here the pairs of subchains $(\\gamma _i,\\gamma _j)$ are extracted from the same protein backbone.",
"The definition (REF ) is rather generic and can be applied to any pair $(\\gamma _i,\\gamma _j)$ of not overlapping portions of the protein backbone.",
"Here we specialize the analysis to the subset of $(\\gamma _i,\\gamma _j)$ where the subchain $\\gamma _i$ has its first ($i_1$ ) and last ($i_2$ ) residues forming a contact ($i_1\\div i_2$ ), i.e.",
"when $|r_{i_1} - r_{i_2}| < d$ , with $d=9Å$ .",
"With this restriction $\\gamma _i$ is essentially a loop.",
"The same restriction is not applied to $\\gamma _j$ , which can either precede or follow $\\gamma _i$ along the protein backbone.",
"This way of detecting entangled configurations is sketched in figure REF .",
"It is similar to that leading to the definition of “lassos” , although here we do not restrict the contacts to chemically strong bonds as in cystein pairs.",
"A preliminary analysis of our data set showed that such entangled configurations are not rare.",
"Given their topological complexity, it is reasonable to think that these native states host proteins which might fold slowly, especially when the mutual entanglement between $\\gamma _i$ and $\\gamma _j$ is considerable.",
"Motivated by the considerations above, we perform a statistical analysis to test the existence of a negative correlation between folding rates and a quantitative measure of the intrachain entanglement present in a protein native structure.",
"We define this measure to be the largest absolute value of the mutual entanglement found for all possible pairs $(\\gamma _i,\\gamma _j)$ having the lasso structure discussed above: $|G^{\\prime }|_c = \\max _{[i_1,i_2],[j_1,j_2]}|G^{\\prime }_{ij}|.$ The index “c” indicates that this maximum intrachain contact entanglement is subject to the loop constraint of having the ends of $\\gamma _i$ in contact with each other, $i_1 \\div i_2$ .",
"With the same definition of contact between non consecutive residues we can introduce the absolute contact order (ACO).",
"This is the average chemical distance $|j-i|$ between monomers in contact.",
"Supposing that there are $n_c$ of these contacts in the native state of a protein, we have $\\textrm {ACO} \\equiv \\frac{1}{n_c} \\sum _{i\\div j}|j-i|$ The relative contact order (RCO) is simply the ACO divided by the chain length $N$ , which is the average of normalized chemical distances $|j-i| / N$ of residues in contact ."
],
[
"Data sets",
"We use two separate data sets.",
"A first data set for two-state folders includes single-domain, non-disulfide-bonded proteins that have been reported to fold via two-state kinetics under at least some conditions .",
"We use folding rates as reported previously , , see table REF .",
"Table: Discussion"
]
] | 1709.01815 |
[
[
"The Hunt for Red Quasars: Luminous Obscured Black Hole Growth Unveiled\n in the Stripe 82 X-ray Survey"
],
[
"Abstract We present results of a ground-based near-infrared campaign with Palomar TripleSpec, Keck NIRSPEC, and Gemini GNIRS to target two samples of reddened active galactic nucleus (AGN) candidates from the 31 deg$^2$ Stripe 82 X-ray survey.",
"One sample, which is $\\sim$89\\% complete to $K<16$ (Vega), consists of eight confirmed AGNs, four of which were identified with our follow-up program, and is selected to have red $R-K$ colors ($>4$, Vega).",
"The fainter sample ($K>17$, Vega) represents a pilot program to follow-up four sources from a parent sample of 34 that are not detected in the single-epoch SDSS catalog and have {\\it WISE} quasar colors.",
"All twelve sources are broad-line AGNs (at least one permitted emission line has a FWHM exceeding 1300 km s$^{-1}$) and span a redshift range $0.59 < z < 2.5$.",
"Half the ($R-K$)-selected AGNs have features in their spectra suggestive of outflows.",
"When comparing these sources to a matched sample of blue Type 1 AGNs, we find the reddened AGNs are more distant ($z > 0.5$) and a greater percentage have high X-ray luminosities ($L_{\\rm X,full} > 10^{44}$ erg s$^{-1}$).",
"Such outflows and high luminosities may be consistent with the paradigm that reddened broad-line AGNs represent a transitory phase in AGN evolution as described by the major merger model for black hole growth.",
"Results from our pilot program demonstrate proof-of-concept that our selection technique is successful in discovering reddened quasars at $z > 1$ missed by optical surveys."
],
[
"Introduction",
"To understand the growth of supermassive black holes over cosmic time, it is crucial to identify and study samples of active galactic nuclei (AGNs) with diverse properties.",
"Powerful emission from the accreting black hole imprints signatures on its surroundings which enables these sources to be detected across the electromagnetic spectrum.",
"Optical emission from the accretion disk and gas photoionized by the AGN can be prominent.",
"Indeed, hundreds of thousands of AGNs have been detected by large ground-based optical surveys, like the Sloan Digital Sky Survey [149], with almost 300,000 Type 1 quasars in the most recent release of the SDSS Quasar Catalog [108].",
"These Type 1 quasars are sources where we have a direct view of the growing black holes, allowing them to be easily identified at optical wavelengths due to their blue color, which imparts a power law slope to the optical spectra, and broad emission lines, from gas rapidly orbiting near the black hole.",
"However, obscured AGNs, where direct view of the central engine is blocked by the circumnuclear torus of the AGN unification scheme [4], [134] and/or large amounts of dust from the host galaxy, are less efficiently detected based on their optical emission alone.",
"In apparent defiance of the canonical AGN unification scheme, red quasars are typically broad-line AGNs, yet are enshrouded by large amounts of dust that reddens the spectrum and attenuates optical emission.",
"Studies of this extreme segment of the obscured AGN population indicate that their reddening is due to a stage of AGN evolution in the major merger model of black hole growth, rather than orientation of the putative torus with respect to the line-of-sight [51], [56].",
"According to this model [121], [69], when galaxies of comparable mass collide and coalesce, gas is funneled to the central supermassive black hole, which ignites AGN activity and circumnuclear star formation.",
"During this phase, the AGN is cocooned within large amounts of dust and gas associated with on-going star formation, potentially reaching Compton-thick levels [76], [115], causing the AGN to appear heavily reddened while it is intrinsically luminous.",
"According to this major merger evolution model, powerful winds from the AGN eventually clear out the obscuring material, revealing a blue Type 1 quasar, and potentially regulating galaxy growth by shutting down star formation and/or evacuating gas from the host [70].",
"While the Compton-thick phase lasts 10$^{7}$ -10$^{8}$ years [71], [131], the reddened AGN phase, when the quasar begins to clear out its surroundings, is shorter-lived, $\\sim 5\\times 10^{6}$ years [71], [56], making these sources rare.",
"This pathway for black hole growth only pertains to a portion of the AGN population [130], [68], with secular processes apparently responsible for triggering moderate luminosity AGNs [123], [77], [140] and some high luminosity AGNs [139].",
"Identifying reddened AGNs that may be in this evolutionary phase provides us an opportunity to test whether there is a causal or coincidental connection between mergers and black hole growth [101], [44] and learn about how black holes can shape their environment.",
"Many previous red quasar samples were identified by their radio, optical-to-near-infrared, near-infrared, and/or mid-infrared emission [57], [56], [55], [8], [7], [6], [36], [5], [118], [60].",
"The traits of some of these sources are consistent with being in the transitory reddened phase in the AGN evolution model: they are intrinsically luminous, after correcting for extinction [57], [56], [55], [6], [5]; they host outflows that may impart feedback onto the host galaxy [45], [132], [151]; and their host galaxies have morphologies consistent with having recently undergone a merger [133], [54].",
"As X-ray emission provides a direct probe of black hole fueling and can pierce through optically obscuring dust, honing in on this emission provides a complementary method for identifying this population.",
"Indeed, reddened AGNs selected from the $\\sim $ 2 deg$^2$ XMM-COSMOS survey [62], [24], [20] are launching powerful outflows, suggesting that they may be in the “clear-out” phase in the AGN evolution paradigm [18], [19], [17], [111], [110].",
"Before the launch of focusing X-ray instruments with sensitivity beyond a few keV, there had not been wide-area X-ray surveys with sufficient depth to identify heavily obscured AGN beyond the local Universe, limiting our census of these rare, reddened sources.",
"Stripe 82X is a $\\sim $ 31 deg$^2$ X-ray survey with XMM-Newton and Chandra [87], [86], [84], [3] that overlaps the legacy SDSS Stripe 82 field [48].",
"About 20 deg$^2$ of the Stripe 82X survey is from a dedicated XMM-Newton observing program from AO10 and AO13 (PI: Urry), reaching depths of 5-7.5 ks, while the rest of the survey is composed of archival XMM-Newton and Chandra observations in the field.",
"The flux limit at half the survey area is $\\sim 5.4 \\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ in the soft band (0.5-2 keV).",
"Due to the relatively wide area covered in X-rays plus rich multiwavelenth data, this survey provides an ideal dataset to identify reddened AGNs, building on the census from smaller-area X-ray surveys and complementing samples selected at other wavelengths.",
"We used a combination of optical and infrared clues to identify signatures of such obscured black hole growth.",
"The full Stripe 82 region is covered at relatively homogeneous depths in the optical by SDSS and in the near-infrared by the UKIRT Infrared Deep Sky Survey [93] and the VISTA Hemisphere Survey [100].",
"Though, in general, reddened extragalactic sources could be dusty starbursts that may not necessarily host an accreting black hole (e.g., some (ultra) luminous infrared galaxies), X-ray emission from Stripe 82X sources is a clear indicator of supermassive black hole accretion: at the relatively bright X-ray flux limits of Stripe 82X, faint X-ray emission from star formation processes in galaxies beyond the local Universe are not detectable.",
"Optical faintness in tandem with relatively brighter infrared emission is thus consistent with heavily reddened AGNs.",
"We present two samples of objects that we targeted for follow-up with ground-based near-infrared spectroscopy using Palomar TripleSpec [66], Keck NIRSPEC [99], and Gemini GNIRS [37], [38].",
"Our “bright NIR” sample ($K < 16$ , Vega) consists of objects selected on the basis of red $R-K$ colors, with an X-ray to optical flux ($X/O$ ) cut to mitigate contamination from stars; these sources were followed up with Palomar TripleSpec.",
"We used Keck NIRSPEC and Gemini GNIRS to follow-up our “faint” sample ($K > 17$ , Vega), which are sources that are not detected in the single-epoch imaging of the SDSS survey, yet have WISE colors consistent with quasars.",
"From our follow-up campaign, the bright NIR sample is nearly complete (i.e., 8 of the 9 sources from the parent sample have secure spectroscopic redshifts) while the faint NIR sample represents a pilot program of a larger sample.",
"In Sections and , we describe the target selection and follow-up observations.",
"We discuss the results of our spectroscopic campaigns, multiwavelength properties of the samples, and insight from spectral energy distribution (SED) analysis in Section .",
"In Section , we compare the properties of the $R-K$ sample with a matched sample of blue Type 1 AGNs and compare both samples with reddened quasars from the literature selected at other wavelengths.",
"We assume a cosmology of H$_{0}$ =67.8 km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _{M}$ =0.31, $\\Omega _{\\Lambda }$ =0.69 [113]"
],
[
"Target Selection",
"We focused on photometric signatures of reddening for both samples we targeted for near-infrared spectroscopy.",
"Such red colors can be induced by large amounts of dust (in the host galaxy and/or circumnuclear region) or by radio synchrotron emission [124].",
"Two of our 12 sources are detected in the radio by the 1.5 GHz FIRST survey [65] and are discussed in more detail below.",
"Thus the red colors for most of these sources are likely due to obscuration.",
"The sources that we followed up are reported in Tables REF and REF , where we list the full Stripe 82X name based on the X-ray coordinates.",
"For clarity, we use an abbreviated version of the source name in the main text and subsequent tables."
],
[
"Bright NIR Stripe 82X Sample: $R-K$ versus {{formula:ecb52735-4bc3-48e0-8dbb-5b654b37ba56}} Selection",
"To unveil the brighter end of the reddened AGN population in Stripe 82X, we focus on the 551 sources (9% of the 6181 unique X-ray sources in Stripe 82X) that are detected in the X-ray full bandThe full band is defined from 0.5-10 keV for XMM-Newton and from 0.5-7 keV for Chandra.",
"and have UKIDSS $K$ -band magnitudes brighter than 16 (Vega).",
"We retain the sources where the SDSS $r$ band and $i$ band magnitudes are well-measured (i.e., error is below 0.5) to avoid artificially reddened colors from poor photometric measurements, leaving us with 373 sources.We note that only one source had $r$ and $i$ band magnitude errors exceeding 0.5 and otherwise met our selection criteria.",
"This source, selected as a counterpart from the SDSS coadded catalog of [73], is confused with a nearby source and a stellar spike, and hence has unreliable photometry.",
"For a straightforward comparison to reddened populations from other studies that use $R-K$ (Vega) to identify obscured AGNs [8], [19], we convert the SDSS $r$ magnitude from the AB system to the Bessell $R$ bandpass [11] in the Vega system using the formulae in [12], which was calibrated on galaxies ranging in redshifts $0 < z < 1.5$ from SDSS, GALEX [98], DEEP2 [32], [43] and GOODS [53] surveys: $R_{\\rm AB} = r - 0.0576 -0.3718((r - i) - 0.2589)$ $R_{\\rm Vega} = R_{\\rm AB} - 0.21,$ where $r$ and $i$ are the SDSS pipeline “modelMag” magnitudes, which is a PSF model for point sources, and the better of a de Vauculeurs or exponential profile fit for extended sources.",
"To calculate $X/O$ , the ratio of X-ray to optical flux, we use the following [16]: $X/O = {\\rm Log}(f_{\\rm x}/f_{\\rm opt}) = {\\rm log}(f_{\\rm x}) + C + 0.4 \\times m_{r},$ where $C$ , a constant that depends on the optical filter, is 5.67 for the SDSS $r$ band [58], and $m_{r}$ is the “modelMag” reported by the SDSS pipeline.",
"Here, the X-ray flux is in the full band.",
"We applied a modified version of the color cuts presented in [20] to select our sample: $R-K > 4$ (Vega) and $X/O > 0$ .",
"We find 17 sources that meet these cuts (boxed region in Figure REF ), of which seven are spectroscopically confirmed as stars and four are Type 1 AGN with existing SDSS spectra (see Section REF ).",
"Of the sources lacking spectra, we removed the one object that lies along the stellar locus of $R-K$ versus $R-W1$ color space presented in [83], i.e., $R-W1 = 0.998(\\pm 0.02) \\times (R - K) + 0.18$ , leaving us with five reddened AGN candidates lacking spectra (red filled squares).",
"We note that though the $X/O > 0$ cut is designed to mitigate contamination from stars, this restriction in principle can also omit sources where the galaxy light dominates [25], including low luminosity AGNs or heavily obscured to Compton-thick AGNs where the observed X-ray emission appears weak [63], [90], [88].",
"The X-ray to optical flux cut selects AGN where the X-ray emission dominates over the host galaxy, implicitly favoring AGN with high X-ray luminosities.",
"However, as Figure REF shows, the sources at $R-K > 4$ with $X/O < 0$ are either spectroscopically confirmed as stars or are likely stars based on their $R-K$ and $R-W1$ colors.",
"We targeted the five reddened AGN candidates lacking spectra with Palomar TripleSpec, as discussed below, and summarize their properties in Table REF , where the magnitudes are given in the native units from their parent catalogs, while the $R-K$ color is in the Vega system, following the derivation above.",
"The corresponding X-ray identification numbers refer to those published in [42], for Chandra sources identified in the Chandra Source Catalog, and [87], [86], [84], for Stripe 82X sources detected by XMM-Newton.",
"We discuss the four extragalactic sources that have SDSS spectra in Section REF and include them in our subsequent analysis.",
"Figure: R-KR-K versus X/OX/O colors of Stripe 82X sources brighter than KK=16 (Vega) with significant detection in the full X-ray band (0.5-10 keV) and well measured rr and ii band magnitudes (magnitude errors less than 0.5).",
"The boxed region (R/K>4R/K > 4 and X/O>0X/O > 0) indicates the locus for reddened AGN candidates where we defined a sample for follow-up.",
"This sample excludes sources lacking spectra that lie along the R-KR-K versus R-W1R-W1 stellar locus presented in since they are likely not AGNs (blue Xs).",
"The sources with pre-existing spectroscopic redshifts (from SDSS or our optical follow-up programs) that are extragalactic and stellar are shown by the grey circles and purple stars, respectively, while the sources lacking identifications via spectroscopic redshifts are indicated by the black diamonds.",
"The reddened AGN candidates we targeted with TripleSpec on Palomar are shown by the red squares.",
"For reference, we show where the faint NIR sample (K>17K > 17) lie in this parameter space with the maroon triangles and lower limits.lrllll 0pt Bright NIR Stripe 82X Targets: $R-K$ versus $X/O$ Selection Stripe 82X Name X-ray ID1 $r$ (AB) $K$ (Vega) $R-K$ (Vega) $X/O$ S82X 013245.41$-$ 000835.5 4150 22.39 15.85 5.92 0.88 S82X 024219.20+000511.9 618 (108774C2) 22.32 15.94 5.78 0.80 S82X 030215.39$-$ 000335.5 783 21.37 15.49 5.25 0.74 S82X 030324.58$-$ 011508.3 855 21.43 15.89 4.97 0.36 S82X 232801.91$-$ 002822.9 1859 21.22 16.00 4.71 1.07 1XMM-Newton record number introduced in the Stripe 82X survey [86].",
"2Source also detected by archival Chandra observations in Stripe 82 [87].",
"The Chandra Source Catalog MSID identifying number [42] for this object is noted in parentheses."
],
[
"Faint NIR Stripe 82X Sample: Optical Drop-outs Recovered by ",
"An interesting population are X-ray sources that lack an optical counterpart in the single-epoch SDSS imaging, but which are detected at infrared wavelengths.",
"The depths of the single-epoch SDSS imaging,$r \\le 22.2$ and $i \\le 21.3$ (AB) for 95% completeness of point sources NIR imaging,$K < 18.1$ (Vega) for 5$\\sigma $ point source detection and MIR imaging5$\\sigma $ limit at $W1 < 17.30$ and $W2 < 15.84$ (Vega) for 95% sky coverage in Stripe 82 are comparable for an AGN SED: $R - K \\sim 3.6$ at the flux limits of these surveys, indicating that the infrared coverage is not systematically deeper than the optical.",
"Hence a non-detection in SDSS in conjunction with an NIR detection selects for reddening.",
"We creates a target list of 47 such optical drop-outs that have mid-infrared colors in the quasar locus of the WISE color-color diagram [143], and have $K$ -band detections in VHS, which is deeper than UKIDSS.",
"These optical drop-outs have no SDSS source within the nominal search radius we used to identify counterparts to the X-ray source (5$^{\\prime \\prime }$ for Chandra and 7$^{\\prime \\prime }$ for XMM-Newton).",
"We vet each potential target by eye to remove any sources that fall out of the SDSS footprintThough the X-ray observations are designed to overlap the Stripe 82 region, some of the archival X-ray observations partially overlap the Stripe while the rest of the field-of-view is outside of the SDSS footprint.",
"or where visual inspection shows a clear source that failed to be detected by the SDSS pipeline, leaving us 37 objects.",
"Thirty-two of these sources are detected in the deeper coadded SDSS catalogs, while five remain undetected [46], [73].",
"We further vet our target list to preserve the reddening criterion of our selection using the information from the coadded SDSS catalogs.",
"If the source is detected in the $r$ and $i$ bands in the coadded SDSS catalog such that we can calculate $R$ , we only retain the sources where $R - K > 4$ .",
"We also retain all sources that remain undetected in at least the $r$ or $i$ band in even this deeper imaging, which implies colors of $R - K > 5.9$ .",
"In total, we have 34 such reddened AGN candidates in our faint sample, including the five sources not detected in any SDSS band in the coadded catalogs.",
"We targeted six with NIRSPEC on Keck (two in 2014 September and four in 2015 October; Section 3.2), and three with GNIRS on Gemini (Section 3.3).",
"Though all sources were detected in the NIRSPEC $K$ -band spectra, we were only able to identify an emission line in one object.",
"It is likely that the remaining sources were unidentified because of the limited wavelength range in the $K$ -band order.",
"Indeed, when using the cross-dispersed mode on Gemini GNIRS, which yields simultaneous $J$ , $H$ , and $K$ -band coverage, we detected emission lines in all three targets.",
"Figure: WISE color-color plot for our parent sample of reddened AGN candidates which are optical drop-outs in single-epoch SDSS imaging and have NIR detections in at the least the VHS KK band and WISE colors in the QSO and Seyfert locus (cyan circle).",
"The red triangles indicate the objects from our pilot program where we identified emission lines and were thus able to determine redshifts and confirm these sources as quasars; the black circles represent sources from the parent sample currently lacking identifications.",
"We show the R-KR-K versus X/OX/O selected sources from the bright NIR sample as maroon squares in this color space for reference.For the remainder of this work, we focus on the four objects for which we were able to identify emission lines and thus derive redshifts.",
"We list this sample in Table REF , where we note whether a reliable optical counterpart was found in the deeper coadded SDSS catalog, and provide the $R-K$ colors or lower limit (if undetected in the coadded SDSS catalogs).",
"llllllcll 0pt Faint NIR Stripe 82X Targets: WISE-Selected Optical Drop-outs Stripe 82X Name X-ray ID1 $W1$ (Vega) $W2$ (Vega) $W3$ (Vega) $K$ (Vega) SDSS Coadd?2 $r$ (AB) $R-K$ (Vega) 7cKeck NIRSPEC S82X 022723.51+004253.3 129832C 16.86 15.84 12.36 18.75 Y 23.44 4.68 7cGemini GNIRS S82X 010019.25+000844.8 2589X 16.79 15.28 12.32 18.36 Y 24.37 5.45 S82X 011840.06+001806.0 3692X 16.27 15.10 12.42 18.04 Y 24.40 5.85 S82X 014152.06$-$ 001749.5 4583X 16.14 14.46 10.87 17.98 N $>$ 24.7 $>$ 6.40 1If the X-ray ID number is followed by a “C”, this indicates the Chandra MSID number from the Chandra Source Catalog [42].",
"If the X-ray ID number is appended by an “X”, this denotes the XMM-Newton record number introduced in the Stripe 82X survey [86], [84].",
"2Flag to indicate if optical dropout was recovered in the coadded SDSS catalog."
],
[
"Palomar TripleSpec",
"TripleSpec simultaneously covers wavelengths 1 - 2.4 $\\mu $ m in 4 orders, with an approximate resolution of 120 km/s (based on the instrument specificationshttp://www.astro.caltech.edu/palomar/observer/200inchResources/ tspecspecs.html).",
"All five sources from the bright NIR Stripe 82X sample were observed using the standard $ABBA$ nodding sequence, where we integrated for 300s per exposure.",
"Four of the sources were observed on 2015 October 27 (S82X 0242+0005, S82X 0302-0003, S82X 0303-0015, S82X 2328-0028) with 3 $ABBA$ sequences.",
"The fifth source, S82X 0132-0008, was observed on 12 December 2016 with 4 $ABBA$ sequences.",
"At the end of the science exposures for each target, we observed a standard A0V or A1V star for telluric correction.",
"All sources were observed at an airmass below 1.5.",
"Data were reduced with the IDL program Spextool [30] which creates normalized flat field images, performs a wavelength calibration based on sky lines, and extracts the spectra from each order.",
"We note that for S82X 2338-0028, we only extract the spectrum from the $B$ position due to a bad column affecting the emission feature in the $A$ nod.",
"Telluric correction is performed on each order with xtellcor [135], after which the spectra are merged into a continuous spectrum with the Spextool xmergerorders routine.",
"Finally, the spectrum is smoothed with task xcleanspec using the Savitzky-Golay routine, which preserves the average resolving power using a smoothing window that is two times the slit width."
],
[
"Keck NIRSPEC",
"Keck NIRSPEC is a near-infrared spectrograph on Keck II, with different filter wheels limiting the wavelength range of a given spectrum to a single waveband.",
"The approximate resolution, found from measuring resolved sky lines, is 200 km/s.",
"We observed S82X 0227+0042 at an airmass of $\\sim $ 1.1 with the K$^{\\prime }$ filter (1.950 - 2.295 $\\mu $ m) on 2014 Sep 7 with the 42 $\\times $ 0$$ 79 slit.",
"We acquired 4 $ABBA$ exposures at 600s per exposure.",
"Due to an apparent error with target acquisition or the dithering script, the source was only in the slit in the $A$ nod.",
"We observed A0V standard star HD 18571.",
"We reduced the data with IRAF routine WMKONSPEChttp://www2.keck.hawaii.edu/inst/nirspec/wmkonspec.html which corrects the distortion in the $x$ and $y$ directions before spectral extraction.",
"Wavelength calibration was performed using sky lines.",
"We corrected the source spectrum for telluric features using xtellcor_general [135], part of the Spextool package [30]."
],
[
"Gemini GNIRS",
"Our Gemini program, GN-2015B-Q-80 (PI: LaMassa), made use of GNIRS in cross-dispersed mode, with simultaneous coverage from 0.85 - 2.5 $\\mu $ m, on Gemini North.",
"We used a slit width of 1.0$^{\\prime \\prime }$ to maximize signal throughput from our sources, with a 32 l/mm grating and short blue camera.",
"With this instrumental set-up, the approximate resolution is 550 km/s.https://www.gemini.edu/node/1046?q=node/10543 In total, we were awarded 12 hours of queue time in Band 3.",
"Each target was observed for three hours, including acquisition from offset stars and standard star observations, with 24-26 $ABBA$ exposures at 300s and 4 $ABBA$ exposures at 270s.",
"Due to varying sky conditions from changes in cloud cover, not all science exposures were included in the analysis.",
"Thus, we discarded observations that added more noise than signal, with the resulting net exposure times listed in Table REF for each source.",
"lcc 0pt Gemini GNIRS Observing Log Stripe 82X Name Observation Dates Net Exposure1 (year-month-date) (s) S82X 0100+0008 2016-01-07 3600 2016-01-08 3540 S82X 0111+0018 2015-12-10 2400 2016-01-02 6480 S82X 0141-0017 2016-01-05 5280 1Net exposure time after discarding observations with poor sky conditions.",
"We reduced the spectra with the XDGNIRS pipeline which calls Gemini GNIRS IRAF routines to clean pattern noise, flat field the data, remove spikes from the data, correct the $S$ -distortion, perform the wavelength calibration based on arc lamps, and extract a spectrum from the combined $A$ and $B$ exposures [29].",
"We used xtellcor_general [135] to perform the telluric correction.",
"Spectra from separate nights were averaged, weighted by the number of exposures contributing to each spectrum."
],
[
"Near Infrared Spectroscopy",
"For all objects, we used photometry to flux calibrate the spectra to obtain estimates of the emission line fluxes: we interpolated the $K$ -band filter response onto the wavelength grid of our spectra, using the filter curve from UKIDSS [67] for the bright NIR sample (Section 2.1) and from VHS for the faint NIR sample (Section 2.2).",
"The integrated flux is then measured from this folded spectrum.",
"The ratio of the $K$ -band flux, derived from the observed $K$ -band catalog photometry, and this pseudo-flux gives us the scale factor by which we adjust the spectrum for an absolute flux calibration.",
"We note that variability between the photometric and spectroscopic observations induce uncertainty into this calibration beyond the statistical errors we report on the emission line fluxes.",
"The spectra for the sources from the bright and faint samples are shown in Figures REF and REF , respectively.",
"In the Palomar TripleSpec spectrum of source S82X 0132-0008, we clearly detect continuum, yet we find no emission lines, precluding us from including this source in the analysis and discussion below.",
"We note that the photometric redshift for this source is $z_{\\rm phot} \\sim 1.74$ [3], such that H$\\alpha $ would fall between the TripleSpec spectral orders, consistent with our lack of emission line detections.",
"For the remaining eight sources, we detected H$\\alpha $ emission and in two objects (S82X 0242+0005 and S82X 0141-0017), we also detect [O3] emission.",
"We indicate these emission features in the extracted one-dimensional spectra in Figure REF .",
"We verified that these emission lines are visible in the two-dimensional spectral images.",
"Figure: Palomar TripleSpec spectra of our bright NIR R-KR-K versus X/OX/O selected sample.",
"(Top left): Spectrum of S82X 0132-0008; though continuum is detected, no emission lines are present.",
"(Top right): Spectrum of S82X 0242+0005 with Hα\\alpha and the [O3] doublet marked.",
"(Middle left): Spectrum of S82X 0302-0003 with Hα\\alpha marked.",
"(Middle right): Extracted spectrum of S82X 0303-0115 with Hα\\alpha marked.",
"Due to a bad column that overlaps the Hα\\alpha emission feature at the AA position, the spectrum was extracted from the BB position only.",
"(Bottom): Extracted spectrum of S82X 2328-0028 with Hα\\alpha marked.",
"Marked transitions indicate the emission lines visible in the two-dimensional spectral images.Figure: Spectra of our faint NIR, SDSS dropout sample.",
"The spectrum in the top left is from Keck NIRSPEC while the others are from Gemini GNIRS.",
"Marked transitions indicate the emission lines visible in the two-dimensional spectral images.",
"We detected Hα\\alpha emission in each source and [O3] in S82X 0141-0017.To obtain precise redshift measurements of these sources, as well as to calculate emission line fluxes and full-width half maxima (FWHMs) of the emission lines, we analyzed the spectra in IDL.",
"To start, we interactively fit a first-order polynomial to the regions of the continuum free of emission and sky lines which was subsequently subtracted from the spectrum.",
"We then used the IDL tool mpfitfun to fit a Gaussian model to the emission lines [97].",
"In two cases (S82X 0242+0005 and S82X 0302-0003), two broad Gaussian components were required to adequately fit the H$\\alpha $ emission.",
"When fitting the [O3] doublet, the amplitude of the 4959Å line was fixed to 1/3 the amplitude of the 5007Å line, and the width of the lines were tied together.",
"We note that in S82X 0242+0005, the [O3] doublet has a blue wing to the narrow profile, which we accommodated with additional Gaussian components; we comment more on this feature in Section REF .",
"While the redshifts were not tied when fitting the H$\\alpha $ and [O3] lines, we obtained consistent redshifts when fitting these features independently, indicating no systematic [O3] blueshift.",
"We corrected the emission line FWHMs for the instrumental resolution using the relation FWHM$_{\\rm corrected} = \\sqrt{{\\rm FWHM}_{\\rm observed}^2 - {\\rm FWHM}_{\\rm instrument}^2}$ and the instrumental resolutions listed in Section .",
"The emission line fits are shown in Figures REF and REF and the derived redshifts and emission line properties are listed in Tables REF and REF , respectively.",
"The quoted errors represent the propagation of the returned uncertainties associated with the fitted parameters.",
"All the sources we observed with this program have an H$\\alpha $ FWHM exceeding 1300 km s$^{-1}$ .",
"As pointed out by [150], the FWHM dividing line between Type 1 and Type 2 AGNs is not firmly established.",
"Some studies use a FWHM value of 2000 km s$^{-1}$ to differentiate between Type 1 and Type 2 AGNs [150], [2] while others set the limit at 1000 km s$^{-1}$ [141] or 1100 km s$^{-1}$ [114].",
"[61] demonstrated that the distribution of H$\\alpha $ FWHMs for emission line galaxies in SDSS is bimodal: all broad line sources have a minimum FWHM of 1200 km s$^{-1}$ .",
"They thus define any source with a FWHM above this value as a Type 1 AGN.",
"Following this convention, and for consistency with previous red quasar studies which require a FWHM exceeding 1000 km s$^{-1}$ to define a source as a quasar [57], we classify all our Stripe 82X sources as Type 1 AGNs.",
"We note, however, that the classification of the two sources with H$\\alpha $ FWHMs below 2000 km s$^{-1}$ , S82X 0303-0115 and S82X 2328-0028, may be ambiguous.",
"As discussed below, all have X-ray luminosities consistent with accretion onto a supermassive black hole.",
"Figure: Close-up of Hα\\alpha emission line region, fit with a Gaussian model (dashed red line).",
"For 0242+0005 and 0302-0003, two Gaussian models, with a broad (blue dot-dash line) and narrow (blue dot-dot-dash line) component, were needed to adequately fit the emission feature.",
"Spectra are in the observed frame.Figure: Close-up of [O3] doublet fitted with a two-component Gaussian model, with the width of the lines tied together and the amplitude of the 4959Å line frozen to a third of the 5007Å line.",
"We included an additional Gaussian component to fit the blue wing to the [O3] doublet in S82X 0242+0005 (left), which is blueshifted with respect to the narrow component by Δv=-400\\Delta v = -400 km s -1 ^{-1}, suggestive of outflowing gas.",
"Spectra are in the observed frame.",
"The red dashed line indicates the sum of the emission lines and the blue dot-dashed line represents the individual narrow and broad components.lllll[h] 0pt Redshifts and X-ray Characteristics of Reddened AGNs Source $z$ $L_{\\rm X,full}$ 1 $HR$ 2 $N_{\\rm H}$ 3 (erg s$^{-1}$ ) (10$^{22}$ cm$^{-2}$ ) 4cPalomar TripleSpec S82X 0242+0005 2.476$\\pm $ 0.001 5.57$\\times 10^{44}$ -0.24$^{+0.25}_{-0.19}$ 4$^{+16}_{-4}$ S82X 0302$-$ 0003 1.2574$\\pm $ 0.0001 2.51$\\times 10^{44}$ 0.37$^{+0.30}_{-0.26}$ 10$^{+10}_{-4}$ S82X 0303$-$ 0115 0.5909$\\pm $ 0.0003 1.71$\\times 10^{43}$ 0.21$^{+0.36}_{-0.34}$ 3$^{+4}_{-2}$ S82X 2328$-$ 0028 0.5859$\\pm $ 0.0001 1.08$\\times 10^{44}$ -0.96$^{+0.04}_{-0.04}$ 0 4cKeck NIRSPEC S82X 0227+0042 2.16$\\pm $ 0.01 8.17$\\times 10^{44}$ -0.90$^{+0.10}_{-0.10}$ 0 4cGemini GNIRS S82X 0100+0008 1.49$\\pm $ 0.01 2.25$\\times 10^{44}$ -0.75$^{+0.12}_{-0.25}$ 0 S82X 0118+0018 1.103$\\pm $ 0.003 1.46$\\times 10^{44}$ -0.49$^{+0.17}_{-0.51}$ $<$ 0.6 S82X 0141$-$ 0017 1.792$\\pm $ 0.004 2.68$\\times 10^{44}$ -0.43$^{+0.25}_{-0.22}$ $<$ 3 1$k$ -corrected (i.e., rest-frame), non-absorption corrected luminosities.",
"2$HR = (H-S)/(H+S)$ , where $H$ ($S$ ) are net counts in the hard (soft) X-ray bands and were calculated with BEHR [109].",
"3Gas column density ($N_{\\rm H}$ ) implied by the $HR$ ranges.",
"lllllll[h] 0pt Emission Line Properties of Targeted Reddened AGNs1 Source $f_{\\rm {H\\alpha ,1}}$ FWHM$_{\\rm {H\\alpha ,1}}$ $f_{\\rm {H\\alpha ,2}}$ FWHM$_{\\rm {H\\alpha ,2}}$ $f_{{\\rm [OIII] 5007Å\\ }}$ FWHM$_{{\\rm [OIII] 5007Å\\ }}$ (10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ ) (km s$^{-1}$ ) (10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ ) (km s$^{-1}$ ) (10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ ) (km s$^{-1}$ ) 7cPalomar TripleSpec S82X 0242+0005 820$\\pm $ 70 4900$\\pm $ 200 190$\\pm $ 30 900$\\pm $ 90 300$\\pm $ 20 750$\\pm $ 20 S82X 0302$-$ 0003 1370$\\pm $ 60 4690$\\pm $ 60 760$\\pm $ 30 1720$\\pm $ 20 S82X 0303$-$ 0115 180$\\pm $ 10 1430$\\pm $ 60 S82X 2328$-$ 0028 290$\\pm $ 10 1350$\\pm $ 20 7cKeck NIRSPEC S82X 0227+0042 70$\\pm $ 60 3400$\\pm $ 1300 7cGemini GNIRS S82X 0100+0008 60$\\pm $ 50 3500$\\pm $ 1200 S82X 0118+0018 90$\\pm $ 40 2300$\\pm $ 300 S82X 0141$-$ 0017 150$\\pm $ 60 3200$\\pm $ 600 $<$ 392 400$\\pm $ 200 1Two Gaussian components were needed to fit the H$\\alpha $ emission profile for two sources (S82X 0242+0005 and S82X 0302-0003).",
"If a second H$\\alpha $ component was needed, the flux and FWHM of this feature is reported as $f_{\\rm {H\\alpha ,2}}$ and FWHM$_{\\rm {H\\alpha ,2}}$ , respectively.",
"$f_{\\rm {H\\alpha ,1}}$ and FWHM$_{\\rm {H\\alpha ,1}}$ represent the flux and FWHM of the broader H$\\alpha $ component, if two Gaussian profiles are needed to fit the spectrum, or the single Gaussian component for the remaining sources.",
"The narrow component of the [O3] 5007 Å line is reported for S82X 0242+0005; the broad wing to the [O3] doublet is recorded in Table REF and discussed in Section REF .",
"23$\\sigma $ upper limit."
],
[
"X-ray Properties of AGNs Targeted with Infrared Spectroscopy",
"From the redshifts measured above, we calculated the $k$ -corrected, observed (non-absorption corrected) full band X-ray luminosities ($L_{\\rm X,full}$ , where $f_{\\rm k-corr} = f_{\\rm observed} \\times (1+z)^{\\Gamma - 2}$ and $\\Gamma $ , the slope of the AGN continuum power law, is 1.7; [87], [86], [84]).",
"To estimate the approximate X-ray absorption, we calculated the hardness ratio: $HR = (H-S)/(H+S)$ , where $H$ is the net number of counts in the hard band and $S$ is the net number of counts in the soft band.",
"We note that while the soft range is 0.5-2 keV for both Chandra and XMM-Newton, the hard band is 2-10 keV for XMM-Newton and 2-7 keV for Chandra.",
"For this calculation, we used the Bayesian Estimation of Hardness Ratios [109] which provides robust estimates of $HR$ in the low-count regime and in the case of non-detections in either band.",
"BEHR takes as input the total counts in the soft and hard bands within user-defined source and background regions and the ratio of areas between the source and background regions.",
"While XMM-Newton has three detectors, we extracted the net counts from only the MOS1 detector for a straightforward comparison with model $HR$ values.",
"Gas column density estimates ($N_{\\rm H}$ ) derived from hardness ratios are redshift dependent: at higher redshifts, the higher energy photons ($>$ 7-10 keV), which are less attenuated by absorption, are shifted into the observed bandpass.",
"Assuming an absorbed powerlaw, we calculated a grid of hardness ratios for various $N_{\\rm H}$ values over a range of redshifts in bins of 0.05 for both Chandra and the MOS1 detector on XMM-Newton.",
"Using the redshift of the source and range of hardness ratios returned by BEHR, we determined the implied $N_{\\rm H}$ and report these values in Table REF .",
"Of the eight sources, the hardness ratios for three are consistent with an unabsorbed X-ray source and three have lower limits on $N_{\\rm H}$ of 0; we do not correct for Galactic absorption since such low column densities [34] have little impact on the X-ray spectrum.",
"Since S82X 0242+0005 is detected in hard X-rays (2-10 keV; $F_{\\rm 2-10 keV} = (1.56 \\pm 0.34) \\times 10^{-14}$ erg s$^{-1}$ ) and the [O3] 5007 Å line is measured, we can independently assess the X-ray obscuration by $F_{\\rm 2-10 keV}$ /$F_{\\rm [OIII]}$ : because [O3] forms in the AGN narrow line region, it is unaffected by circumnuclear obscuration that attenuates the X-ray emission and thus serves as a proxy of the intrinsic AGN luminosity [10], [64], [89].",
"The ratio of the hard X-ray to [O3] flux can then indicate whether the X-ray emission is heavily absorbed [10], [107], [92], [91].",
"We find log($F_{\\rm 2-10 keV}$ /$F_{\\rm [OIII]}$ ) = 0.72$\\pm $ 0.09 dex, which is significantly less than the mean value for unabsorbed Type 1 AGNs [63], but higher than the most heavily obscured, Compton-thick systems [90], [88].",
"Both $F_{\\rm 2-10 keV}$ /$F_{\\rm [OIII]}$ and the implied $N_{\\rm H}$ from the hardness ratio are consistent with a moderately obscured AGN.",
"However, the narrow line region can suffer extinction, which would translate into more luminous intrinsic [O3] emission, causing the true $F_{\\rm 2-10 keV}$ /$F_{\\rm [OIII]}$ to decline.",
"The implied X-ray column density can thus be higher.",
"We caution that the theoretical $HR$ -$N_{\\rm H}$ conversion assumes a simple absorbed power law while observed X-ray spectra of obscured AGN are generally more complex, with scattered emission, or leakage through a patchy obscuring medium, that can boost the observed soft X-ray flux compared with the model assumed here.",
"Furthermore, hardness ratios provide no information about the global distribution, or global column density, of obscuration around the AGN.",
"Indeed, AGNs can have significantly different line-of-sight and global column densities [85], [147], [82].",
"Though the implied column densities for some of these AGNs are consistent with column densities of FIRST-2MASS selected reddened quasars [56] derived via X-ray spectral fitting [82], the reported $N_{\\rm H}$ ranges should be considered approximate."
],
[
" Bright Sources with SDSS Spectra",
"Four Stripe 82X sources with $R-K >4$ and $X/O > 0$ have existing SDSS spectra and obey the quality control cuts and magnitude limits applied to our target list for the bright NIR sample.",
"The optical, infrared, and X-ray properties of these sources are presented in Table REF .",
"As can be seen by their SDSS spectra in Figure REF , they are all Type 1 AGN.",
"We also calculated BEHR-derived hardness ratios for these sources to estimate their column densities.",
"While one source has only an upper limit on the implied column density, the other three objects have hardness ratios consistent with non-zero absorption.",
"Two of these sources, S82X 0022+0020 and S82X 0040+0058, are detected in hard X-rays and are at sufficiently low redshift that [O3] 5007 Å is observed in the SDSS spectrum.",
"Based on our fits to the optical spectra (see Section REF ), we find rest-frame [O3] 5007 Å flux values of (2.06 $\\pm $ 0.17) $\\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ and (3.2 $\\pm $ 0.2) $\\times 10^{-16}$ erg s$^{-1}$ cm$^{-2}$ for S82X 0022+0020 and S82X 0040+0058, respectively.",
"With observed hard X-ray fluxes of (1.7 $\\pm $ 0.2) $\\times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$ and (1.5 $\\pm 0.2$ ) $\\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$ , we obtain log($F_{\\rm 2-10keV}$ /$F_{\\rm [OIII]}$ ) values of 0.92 $\\pm $ 0.06 dex and 2.68 $\\pm $ 0.07 dex for S82X 0022+0020 and S82X 0040+0058.",
"These values are consistent with the hardness ratios: S82X 0022+0020 is moderately X-ray obscured while S82X 0040+0058 is X-ray unobscured.",
"While this SDSS sample is optically brighter than the sources we identified with our spectroscopic campaign, their X-ray luminosities span a similar range.",
"Below, we consider these four sources alongside the four we targeted with Palomar TripleSpec when we discuss the bright NIR $R-K$ versus $X/O$ Stripe 82X sample.",
"lrllllllll 0pt Bright NIR $R-K$ versus $X/O$ Sample from SDSS Stripe 82X Name X-ray ID1 $r$ $K$ $R-K$ $X/O$ $z$ $L_{\\rm X,full}$ 2 $HR$ 3 $N_{\\rm H}$ 4 (AB) (Vega) (Vega) (erg s$^{-1}$ ) (10$^{22}$ cm$^{-2}$ ) S82X 001130.21+005751.5 111X 20.65 15.56 4.83 1.34 1.491 2.87$\\times 10^{45}$ -0.30$^{+0.06}_{-0.06}$ 1$^{+1}_{-0.6}$ S82X 002255.06+002055.7 34598C 20.51 15.84 4.25 0.03 0.799 3.79$\\times 10^{43}$ 0.72$^{+0.12}_{-0.11}$ 20$^{+0}_{-10}$ S82X 004003.87+005853.9 287X 20.62 15.92 4.24 1.37 0.811 7.66$\\times 10^{44}$ -0.47$^{+0.54}_{-0.53}$ $<$ 3 S82X 004341.18+005253.2 367X 19.45 14.34 4.65 0.33 0.828 2.17$\\times 10^{44}$ -0.09$^{+0.14}_{-0.17}$ 2$^{+1}_{-1.4}$ 1If the X-ray ID number is followed by a “C”, this indicates the Chandra MSID number from the Chandra Source Catalog [42].",
"If the X-ray ID number is appended by an “X”, this denotes the XMM-Newton record number introduced in the Stripe 82X survey [86], [84].",
"2$k$ -corrected (i.e., rest-frame), non-absorption corrected luminosities.",
"3$HR = (H-S)/(H+S)$ , where $H$ ($S$ ) are net counts in the hard (soft) X-ray bands and were calculated with BEHR.",
"4Gas column density ($N_{\\rm H}$ ) implied by the $HR$ ranges.",
"Figure: SDSS spectra of extragalactic Stripe 82X sources that meet the selection criteria of our bright NIR Stripe 82X sample, i.e., R-K>4R-K > 4, X/O>0X/O > 0, and K≤16K \\le 16."
],
[
"Radio Properties",
"As noted above, red colors can be induced by synchrotron emission from jets along the line-of-sight that boost the $K$ -band flux [124].",
"The FIRST survey covers the full Stripe 82 region [65], and only two of our 12 sources are detected by FIRST: S82X 0011+0057 and S82X 0302-0003.",
"Following the prescription to calculate radio loudness from [72], we first define an AB radio magnitude based on the integrated FIRST flux density at 20 cm: $t = -2.5 {\\rm log}\\left(\\frac{F_{\\rm int}}{3631 {\\rm Jy}}\\right).$ The FIRST flux densities are 156 mJy and 0.46 mJy for S82X 0011+0057 and S82X 0302-0003, respectively, corresponding to $t$ = 10.9 and 17.2.",
"We then calculate the radio loudness by taking the ratio of the radio and optical flux: $R = {\\rm log}\\left(\\frac{F_{\\rm radio}}{F_{\\rm optical}}\\right) = 0.4\\ (m_r - t),$ where $m_r$ is the SDSS $r$ -band magnitude.",
"We note that the $r$ -band magnitude is not corrected for extinction, making our $R$ values upper limits.",
"We find $R$ = 3.9 and 1.7 for S82X 0011+0057 and S82X 0302-0003.",
"While S82X 0302-0003 can be classified as radio-intermediate [104], S82X 0011+0057 is radio loud.",
"The prominent radio jet in this source might contribute to the $K$ -band flux, which may enhance the red $R-K$ color."
],
[
"Spectral Energy Distribution Analysis",
"Using any available photometric data from ultraviolet to mid-infrared wavelengths, we constructed Spectral Energy Distributions (SEDs) of these sources [3].",
"The ultraviolet data are from the GALEX Medium Imaging Survey [105].",
"Due to the optical faintness of our sources, only one object, S82X 0011+0057, is detected by GALEX, and only in the near-ultraviolet band.",
"The optical data were culled from the coadded Stripe 82 catalog from [46], if available, otherwise from [73]; one source, S82X 0141+0017, was not detected at any optical wavelength.",
"For the NIR data, we used VHS magnitudes where available, or UKIDSS for filters that did not have a detection in VHS.We only used both VHS and UKIDSS magnitudes if the magnitudes in filters in common between both observatories were consistent.",
"For S82X 0303-0015, the lack of common filters between VHS and UKIDSS is due to non-coverage in the VHS $K$ -band and UKIDSS $J$ -band; the source was at the edge of the detector in the $J$ -band in UKIDSS, precluding the UKIDSS pipeline from measuring a $J$ -band magnitude.",
"Here, we combined the UKIDSS and VHS magnitudes.",
"The optical and NIR magnitudes were corrected for Galactic extinction.",
"In the case of non-detections, we use the upper limits reported in the various multiwavelength catalogs: $m_{\\rm FUV,AB}$ = 22.6, $m_{\\rm NUV,AB}$ = 22.7 [105]; $m_{\\rm u, AB}$ = 24.2, $m_{\\rm g, AB}$ = 25.2, $m_{\\rm r, AB}$ = 24.7, $m_{\\rm i, AB}$ = 24.3, $m_{\\rm z, AB}$ = 23.0 [46]; $m_{\\rm J, AB}$ = 21.5, $m_{\\rm H, AB}$ = 21.2, $m_{\\rm K, AB}$ = 20.4 [100]; and $F_{\\rm \\nu , W1}$ = 0.08 mJy, $F_{\\rm \\nu , W2}$ = 0.11 mJy, $F_{\\rm \\nu , W3}$ = 1 mJy, and $F_{\\rm \\nu , W4}$ = 6 mJy [143].",
"We fit the SEDs with AGNFitter [22] to estimate the bolometric AGN luminosities ($L_{\\rm bol}$ ) and reddening (E(B-V)).",
"This algorithm employs a Bayesian Markov Chain Monte Carlo (MCMC) method, assuming a flat prior on the parameters listed in Table 1 of [22], and fits the following templates to the SED: accretion disk emission, which is a modified version of the [116] template, extended to wavelengths redward of 1$\\mu $ m assuming $F_\\nu \\propto \\nu ^{-2}$ ; hot dust emission from the putative torus using models from [127]; host galaxy emission using the stellar population models from [21]; and cold dust emission associated with star formation using templates from [27] and [31].",
"AGNFitter accounts for upper limits by creating a fictitious data point at half the value of the upper limit ($F_{\\rm UL}$ ), with an error bar of $\\pm $ 0.5 $F_{\\rm UL}$ , such that the upper limit data point spans the range from 0 to $F_{\\rm UL}$ .",
"Inclusion of upper limits allows the MCMC sampling to accept models that lie within the bounds defined by the upper limits.",
"The fitted parameter space is 10-dimensional [22] which is on the order of or larger than the number of photometric detections used in the fitting, so we caution that our results from this exercise are approximate.",
"However, it is the best we are able to do with our data and does provide a sense of the bolometric AGN luminosity and reddening.",
"We ran AGNFitter with two burn-in sets, with 4000 steps per set and 100 chains per set: after each burn-in, the starting point in the parameter space of the subsequent MCMC chains is that of the highest likelihood of the previous chains.",
"After the burn-in sets, the MCMC chain is run with 10,000 steps, where all sampled areas of the parameter space are used in the calculation of the posterior probability distribution functions (PDFs).",
"We show the fitted SEDs in Figures REF - REF , where the black circles represent our photometric data points.",
"Ten random realizations from the MCMC chain are plotted.",
"The sum of the individual templates from these realizations are shown by the solid red line, with the individual templates denoted by the other colored lines as indicated in the caption of Figure REF .",
"In Table REF , we list results from the SED fitting: the AGN and host galaxy reddening values, E(B-V)$_{\\rm AGN}$ and E(B-V)$_{\\rm Galaxy}$ , respectively; and $L_{\\rm bol}$ , found by integrating the luminosity from the de-reddened accretion disk template from 0.03 $\\mu $ m - 1 $\\mu $ m. We note that since AGNFitter does not include the X-ray emission when modeling the SEDs, the AGN bolometric luminosity derived from the accretion disk template is underestimated [79].",
"To estimate black hole masses (see below), we calculate monochromatic luminosities at 5100 Å ($\\lambda L_{\\rm 5100}$ ) and 3000 Å ($\\lambda L_{\\rm 3000}$ ) from $L_{\\rm bol}$ assuming a bolometric correction of $8.1\\pm 0.4$ and $5.2\\pm 0.2$ , respectively [120], and list these values, where appropriate, in Table REF .",
"The reported values derived from the SED fitting represent the median of the PDFs, with the lower and higher error bars indicating the bounds for the 16th and 84th percentiles of the PDFs.",
"llllllll 0pt AGN Parameters Derived from SED & Spectral Fitting Stripe 82X Name E(B-V)$_{\\rm AGN}$ E(B-V)$_{\\rm Galaxy}$ Log($L_{\\rm bol}$ )1 Log ($\\lambda L_{\\rm 5100}$ )2 Log ($\\lambda L_{\\rm 3000}$ )3 Log($M_{\\rm BH}$ ) $\\lambda _{\\rm Edd}$ (erg s$^{-1}$ ) (erg s$^{-1}$ ) (erg s$^{-1}$ ) (M$_{}$ ) 8cBright NIR $R-K$ versus $X/O$ Selected Sample S82X 0242+0005 0.45$\\pm $ 0.02 -0.03$\\pm $ 0.05 47.24$\\pm $ 0.01 46.34$\\pm $ 0.02 9.62$^{+0.05}_{-0.06}$ 0.32$\\pm $ 0.04 S82X 0302$-$ 0003 0.70$\\pm $ 0.02 1.52$^{+0.33}_{-0.40}$ 46.85$\\pm $ 0.01 45.94$\\pm $ 0.02 9.38$\\pm $ 0.04 0.23$\\pm $ 0.02 S82X 0303$-$ 0115 0.69$^{+0.03}_{-0.04}$ 1.48$\\pm 0.38$ 45.17$^{+0.05}_{-0.08}$ 44.26$^{+0.06}_{-0.07}$ 7.44$^{+0.06}_{-0.07}$ 0.41$\\pm $ 0.08 S82X 2328$-$ 0028 0.71$^{+0.20}_{-0.12}$ 0.40$^{+0.11}_{-0.06}$ 45.08$^{+0.23}_{-0.42}$ 44.17$^{+0.22}_{-0.48}$ 7.34$^{+0.13}_{-0.19}$ 0.42$\\pm $ 0.32 8cFaint NIR WISE-Selected Optical Dropout Sample S82X 0100+0008 0.54$\\pm $ 0.09 1.50$^{+0.38}_{-0.44}$ 45.52$\\pm $ 0.22 44.61$^{+0.18}_{-0.32}$ 8.43$^{+0.25}_{-0.62}$ 0.10$\\pm $ 0.09 S82X 0118+0018 1.18$^{+0.54}_{-0.41}$ 1.20$^{+0.55}_{-0.49}$ 45.41$^{+0.24}_{-0.99}$ 44.50$^{+0.26}_{-0.76}$ 7.99$^{+0.18}_{-0.31}$ S82X 0141$-$ 00174 1.21$^{+0.51}_{-0.65}$ 1.38$^{+0.41}_{-0.31}$ 44.59$^{+0.67}_{-0.44}$ S82X 0227+00424 -0.03$\\pm $ 0.05 0.97$^{+0.70}_{-0.52}$ 44.70$^{+0.06}_{-0.09}$ 8cBright NIR $R-K$ versus $X/O$ Selected Sample from SDSS S82X 0011+00575 0.10$\\pm 0.02$ 1.43$^{+0.39}_{-0.37}$ 46.02$\\pm $ 0.01 45.31$\\pm $ 0.02 8.85$^{+0.13}_{-0.18}$ 0.11$\\pm $ 0.04 S82X 0022+0020 0.85$^{+0.47}_{-0.07}$ 0.03$^{+0.04}_{-0.08}$ 45.85$^{+0.15}_{-0.34}$ 44.94$^{+0.17}_{-0.29}$ 8.75$^{+0.13}_{-0.18}$ 0.10$\\pm $ 0.06 S82X 0040+00586 0.63$^{+0.09}_{-0.13}$ -0.04$\\pm $ 0.04 45.12$^{+0.37}_{-0.43}$ 44.41$^{+0.30}_{-1.72}$ $<$ 7.3 S82X 0043+00525 0.55$\\pm $ 0.02 1.42$^{+0.39}_{-0.32}$ 45.93$^{+0.04}_{-0.03}$ 45.21$\\pm $ 0.04 8.84$^{+0.10}_{-0.13}$ 0.09$\\pm $ 0.03 1$L_{\\rm bol}$ is the AGN bolometric luminosity found by decomposing the SED in AGNFitter and integrating the de-reddened accretion disk luminosity from 0.03$\\mu $ m - 1.0$\\mu $ m. 2$\\lambda L_{\\rm 5100}$ is the monochromatic continuum luminosity at 5100 Å, calculated from $L_{\\rm bol}$ and assuming a bolometric correction of 8.1 $\\pm $ 0.4 [120], that we use to estimate $M_{\\rm BH}$ for sources where H$\\alpha $ or H$\\beta $ is detected (and in the case of H$\\beta $ , not blended with the [O3] doublet).",
"3$\\lambda L_{\\rm 3000}$ is the monochromatic continuum luminosity at 3000 Å, calculated from $L_{\\rm bol}$ and assuming a bolometric correction of 5.2 $\\pm $ 0.2 [120], that we use to estimate $M_{\\rm BH}$ along with the Mg2 FWHM for sources where H$\\alpha $ and H$\\beta $ are not detected or where H$\\beta $ is blended.",
"4$L_{\\rm bol}$ is on the order of or lower than the observed X-ray luminosity, indicating potential errors in the AGN and galaxy decomposition in the SED fitting.",
"We therefore refrain from calculating $M_{\\rm BH}$ and $\\lambda _{\\rm Edd}$ , and caution that the E(B-V) values may be unreliable.",
"5$M_{\\rm BH}$ is estimated using the Mg2 FWHM reported in [126] and $\\lambda L_{\\rm 3000}$ calculated from our SED decomposition (i.e., $\\lambda L_{\\rm 3000} = L_{\\rm bol}/(5.2 \\pm 0.2)$ ).",
"6Due to uncertainties in $\\lambda L_{\\rm 3000}$ and the Mg2 FWHM, we report the 3$\\sigma $ upper limit on $M_{\\rm BH}$ .",
"Figure: SEDs of our bright NIR R-KR-K versus X/OX/O selected Stripe 82X sample fitted with AGNFitter .",
"The black data points are the ultraviolet to mid-infrared photometric detections (circles) and upper limits (arrows) .",
"Overplotted on the SEDs are ten random realizations from the MCMC fit of the individual templates: accretion disk emission (dark blue lines), host galaxy emission (orange lines), AGN-heated dust emission (cyan lines), and cold dust emission associated with star formation (green lines).",
"The sum of the individual emission components, averaged over all MCMC runs, is shown by the red lines.",
"The AGN bolometric luminosity is based on the integrated luminosity from the de-reddened accretion disk template (i.e., dark blue lines) integrated from 0.03μ\\mu m to 1μ\\mu m.Figure: SEDs of our faint NIR WISE-selected optical dropout Stripe 82X sample fitted with AGNFitter .",
"Colors are the same as in Figure .Figure: SEDs of our Stripe 82X NIR supplementary sample fitted with AGNFitter .",
"Colors are the same as in Figure ."
],
[
"Black Hole Masses and Eddington Ratios",
"We emphasize that due to the limited number of photometric detections used in the SED decomposition, the monochromatic and bolometric luminosities are uncertain, which propagate to uncertainties in the estimated black hole masses ($M_{\\rm BH}$ ) and implied accretion rate as measured by the Eddington parameter ($\\lambda _{\\rm Edd} = L_{\\rm bol}/L_{\\rm Edd}$ ).",
"We estimate $M_{\\rm BH}$ and $\\lambda _{\\rm Edd}$ using the best available data, but caution that these values should be considered approximate."
],
[
"Targeted Sources",
"For the sources that we targeted with Palomar, Keck, and Gemini, we use the estimated de-reddened $\\lambda L_{\\rm 5100}$ and measured H$\\alpha $ FWHM values to derive $M_{\\rm BH}$ using: $\\begin{split}M_{\\rm BH} = (9.7 \\pm 0.5) \\times 10^6 \\left[\\frac{{\\rm FWHM (H\\alpha )}}{1000\\ {\\rm km\\ s^{-1}}}\\right]^{2.06 \\pm 0.06}\\times \\\\\\left[\\frac{\\lambda L_{\\rm 5100}}{10^{44}\\ {\\rm erg\\ s^{-1}}} \\right]^{0.519 \\pm 0.07} M_{},\\end{split}$ from [59].",
"For the two sources where two Gaussian components were needed to fit the H$\\alpha $ profile (S82X 0242+0005 and S82X 0302-0003), we use the broader H$\\alpha $ FWHM to estimate $M_{\\rm BH}$ since this component arises from gas closer to the black hole.",
"From these $M_{\\rm BH}$ values, we estimate the Eddington luminosity [47] and $\\lambda _{\\rm Edd}$ .",
"These values are listed in Table REF .",
"The errors represent the propagation of the statistical measurement errors of the individual parameters and the errors associated with the bolometric corrections and virial $M_{\\rm BH}$ relations.",
"We note that comparisons of black hole masses derived via single epoch measurements, as calculated here, with those determined from reverberation mapping studies show a sample dispersion of $\\sim $ 0.5 dex [138], which is an additional uncertainty to our $M_{\\rm BH}$ values beyond the formal errors that we report.",
"We caution that the view to the broad line region is likely obscured, such that we are not getting an unbiased view of the gas kinematics near the black hole.",
"If we are indeed viewing just the outer photosphere of the broad line region, the FWHM of H$\\alpha $ , and other broad emission lines we use below to calculate $M_{\\rm BH}$ , are systematically lower than those observed in unobscured AGN that were used to derive virial relations to calculate $M_{\\rm BH}$ .",
"Our estimated black hole masses may thus be lower limits to the true value."
],
[
" SDSS Sample",
"Since H$\\alpha $ is not covered in the optical spectra of the sources in the bright NIR $R-K$ SDSS sample, we estimate black hole masses, and associated Eddington ratios, using the H$\\beta $ or Mg2 emission lines.",
"Two of these sources have published black hole masses in the [126] catalog (S82X 0011+0057 and S82X 0043+0052) which were calculated using the FWHM of the Mg2 line and $\\lambda L_{\\rm 3000}$ .",
"For consistency with our targeted sample, we use the $\\lambda L_{\\rm 3000}$ we calculated from our SED fitting ($\\lambda L_{\\rm 3000}$ = $ L_{\\rm Bol}$ /(5.2$\\pm $ 0.2)) along with the reported Mg2 FWHM in [126] to estimate $M_{\\rm BH}$ given the relation published in [129]: $\\begin{split}M_{\\rm BH} = 5.6 \\times 10^{6} \\left[\\frac{{\\rm FWHM (MgII)}}{1000\\ {\\rm km\\ s^{-1}}}\\right]^{2}\\times \\\\\\left[\\frac{\\lambda L_{\\rm 3000}}{10^{44}\\ {\\rm erg\\ s^{-1}}} \\right]^{0.62} {\\rm M_{}}.\\end{split}$ The black hole masses and Eddington ratios are reported in Table REF .",
"For the remaining two sources, we fitted the SDSS spectra using the IRAF package specfit [80] to obtain emission line FWHMs.",
"This routine uses a $\\chi ^2$ minimization technique to find the best fit to the input model parameters, which consists of: 1) AGN powerlaw continuum, 2) star formation templates that span an age range from 56 Myr to 10 Gyr (S. Charlot & G. Bruzual, private communication), 3) a [26] dust extinction that attenuates the AGN power law continuum with $R$ =3.1, and 4) Gaussian components to fit emission lines.",
"The spectra were corrected for Galactic reddening, shifted to the rest-frame, and the flux was multiplied by $(1+z)$ to preserve the observed integrated line flux.",
"During the fitting procedure, all narrow emission lines are forced to have the same FWHM, and the [O3] 4959 Å intensity was fixed to a third of the [O3] 5007 Å flux.",
"Since H$\\beta $ is blended with the [O3] doublet and Fe2 emission in S82X 0040+0058, we use the Mg2 FWHM (1500$\\pm $ 500 km s$^{-1}$ ; see Figure REF ) and the virial relation above to estimate $M_{\\rm BH}$ .",
"However, due to the large errors in $\\lambda L_{\\rm 3000}$ and Mg2 FWHM, we are only able to estimate a 3$\\sigma $ upper limit on the black hole mass.",
"Figure: Top: Rest-frame SDSS spectrum of S82X 0040+0058 with our best fit model from IRAF specfit overplotted (red dashed line).",
"Fitted emission lines are marked.",
"Bottom: Continuum-subtracted spectra around (left) the Mg2 emission line, which we use to derive M BH M_{\\rm BH}.",
"The Hβ\\beta -[O3] complex is shown on the right, where we model the emission lines with an Fe2 optical template and Gaussian profiles for the narrow and broad, redshifted (Δv=1400\\Delta v = 1400 km s 1 ^{1}) Hβ\\beta lines (green dot-dash lines), and narrow and broad, blueshifted (Δv=-500\\Delta v = -500 km s -1 ^{-1}) [O3] lines (dot-dot-dot dash blue line).",
"The red dashed line shows the sum of these emission features.We use the H$\\beta $ FWHM to derive a black hole mass for S82X 0022+0020.",
"With a fitted FWHM of 5100$\\pm $ 300 km s$^{-1}$ (see Figure REF , top), and using: $\\begin{split}M_{\\rm BH} = 1.05\\times 10^{8} \\left[\\frac{{\\rm FWHM (H\\beta )}}{1000\\ {\\rm km\\ s^{-1}}}\\right]^{2}\\times \\\\\\left[\\frac{\\lambda L_{\\rm 5100}}{10^{46}\\ {\\rm erg\\ s^{-1}}} \\right]^{0.65} {\\rm M_{}}\\end{split}$ from [129], we find $M_{\\rm BH}$ = 8.75$^{+0.13}_{-0.18}$ M$_{}$ , with an associated $\\lambda _{\\rm Edd}$ of 0.10$\\pm $ 0.06.",
"[129] do not provide formal errors on the $M_{\\rm BH}$ -Mg2 or $M_{\\rm BH}$ -H$\\beta $ virial relations.",
"Typical standard deviations in the samples used for their calibrations range from $\\sim $ 0.13 - 0.15 dex.",
"When estimating black hole masses using the Mg2 and H$\\beta $ emission FWHMs, we only propagated formal errors on the fit parameters and bolometric corrections, and note that there is likely an additional uncertainty of up to $\\sim $ 0.15 dex as well as an $\\sim $ 0.5 dex uncertainty associated with single-epoch black hole mass measurments [138].",
"Figure: Top: Rest-frame SDSS spectrum of S82X 0022+0020 with our best fit model from IRAF specfit overplotted (red dashed line).",
"Fitted emission lines are marked.",
"Bottom: Close-up of the Hβ\\beta and [O3] complex for S82X 0022+0020.",
"Here, the continuum (from AGN and host galaxy) has been subtracted off.",
"Overplotted are the sum of the fitted emission lines (red dashed line): broad and narrow Hβ\\beta emission lines (dot-dash purple line) and [O3] emission (dot-dot-dot-dash blue line), including the narrow and broad broad (FWHM = 1100±\\pm 100 km s -1 ^{-1}), blueshifted (Δv=-740±50\\Delta v = -740\\pm 50 km s -1 ^{-1}) [O3] components.",
"This blue wing to the [O3] doublet is likely a signature of an AGN outflow."
],
[
"Absorption Line System & Asymmetric Line Profiles: Indications of Outflows?",
"Half of the sources from the bright NIR $R-K$ versus $X/O$ selected sample have spectroscopic signatures of narrow line region kinematics, with either absorption line troughs or broadend [O3] emission: S82X 0043+0052 (Figure REF , bottom), S82X 0022+0020 (Figure REF ), S82X 0040+0058 (Figure REF ), and S82X 0242+0005 (Figure REF ).",
"None of the spectra from the faint NIR WISE-selected optical dropout sample show any sign of outflowing gas, though this sample is only $\\sim 12$ % complete.",
"S82X 0043+0052 was identified as a Mg2 quasar narrow absorption line (FWHM $\\le $ 500 km s$^{-1}$ ) system in [94].",
"Such absorption line systems can be associated with quasar outflows or from gas within the quasar environment [142], [136].",
"For the remaining three sources, we found asymmetries in the [O3] 5007 Å line from our own fits to the spectra, as detailed below and summarized in Table REF .",
"In all cases, the widths of the [O3] doublet lines were tied together, the flux of the [O3] 4959 Å line was fixed to 1/3 of the [O3] 5007 Å line, and the central wavelength of the [O3] 4959 Å line was fixed to 0.99 of the [O3] 5007 Å line.",
"In S82X 0040+0058, H$\\beta $ , Fe2 emission, and the [O3] doublet are blended (Figure REF ).",
"To fit the spectrum, we include an optical Fe2 emission template [15] as well as broad components to the [O3] doublet.",
"The broad H$\\beta $ emission (FWHM = 6100$\\pm $ 600 km s$^{-1}$ ) is redshifted with respect to the narrow component ($\\Delta v = 1400$ km s$^{-1}$ ), while the broad [O3] component (FWHM = 2400$\\pm $ 200 km s$^{-1}$ ) is blueshifted compared to the fitted wavelength of the narrow component ($\\Delta v = -500$ km s$^{-1}$ ).",
"Shifted broad H$\\beta $ emission is sometimes observed in double-peaked emitters with asymmetric line profiles [41], [40], [9].",
"This feature is typically explained by asymmetries in a Keplarian accretion disk.",
"We note that similar signatures, i.e., high velocity shifts in the broad H$\\beta $ line, can also be produced by supermassive black hole binaries [39], [119] and rapidly recoiling black holes [14], [78].",
"An apparent blue wing to the [O3] doublet is present in S82X 0022+0020, which we are able to fit with broad Gaussian components in addition to narrow Gaussians to fit the narrow line doublet (Figure REF ).",
"The broad component of the [O3] line has a FWHM of 1200$\\pm $ 200 km s$^{-1}$ , and is blueshifted with respect to the narrow component (FWHM = 560$\\pm $ 20 km s$^{-1}$ ) by $\\Delta v = -700$ km s$^{-1}$ .",
"The Palomar spectrum of S82X 0242+0005 shows a blue wing to the [O3] doublet.",
"As shown in Figure REF , additional Gaussian components, with FWHM = $2300\\pm 200$ km s$^{-1}$ , accommodates this additional emission.",
"It is blueshifted by $\\Delta v = -400$ km s$^{-1}$ compared with the narrow component of the line.",
"We note that these [O3] FHWM values and velocities are on the order of those observed in XMM-COSMOS reddened quasars [19], but less extreme than the SDSS-selected luminous reddened ($r_{\\rm AB} - W4_{\\rm Vega} > 14$ ) quasars [118] presented in [151].",
"llll 0pt Asymmetric [O3] Line Profiles Stripe 82X Name FWHM $\\Delta v$ 1 Spectrum (km s$^{-1}$ ) (km s$^{-1}$ ) S82X 0022+0020 $1200\\pm 200$ $-700$ SDSS S82X 0040+0058 $2400\\pm 200$ $-500$ SDSS S82X 0242+0005 $2300\\pm 200$ $-400$ Palomar TSpec 1$\\Delta v$ is measured between the fitted wavelengths of the broad and narrow components of the [O3] 5007 Å line."
],
[
"AGN Properties Derived from Spectral Analysis and SED Fitting",
"Two of the sources from the NIR faint optical drop-out sample, S82X 0141-0017 and S82X 0227+0042, have estimated bolometric luminosities on the order of, or lower than, the observed full-band X-ray luminosity.",
"This apparent inconsistency points to limitations in the SED decomposition due to the relatively few photometric detections for these sources.",
"We therefore refrain from estimating their black hole masses and Eddington parameters, and note that their E(B-V)$_{\\rm AGN}$ and E(B-V)$_{\\rm Galaxy}$ values may also be unreliable.",
"We therefore discard these objects when considering the AGN properties derived from SED fitting below.",
"Based on the fitted E(B-V) values from the SED decomposition, nine out of the remaining ten sources are “reddened” Type 1 AGNs, with E(B-V) $\\ge $ 0.45.",
"We note that the blue source is S82X 0011+0057, which we showed to be radio loud (Section REF ).",
"The SDSS spectrum for this source (Figure REF ) also shows a blue powerlaw slope, consistent with an unobscured quasar.",
"Thus we conclude that the red $R-K$ color for S82X 0011+0057 is due to synchrotron emission boosting the $K$ -band flux and the low E(B-V) value is to be expected.",
"We point out, however, that S82X 0302-0003, which is radio-intermediate appears to be truly reddened based on the E(B-V) values derived from SED decomposition.",
"For the reddened sources, the extinction is along the line-of-sight to the AGN.",
"The black hole masses and Eddington ratios span a range of values, with similar $M_{\\rm BH}$ - $\\lambda _{\\rm Edd}$ relationships as unobscured quasars from SDSS [129].",
"However, we reiterate that the bolometric and monochromatic luminosities from which we derive these values are approximate.",
"Furthermore, obscuration in the broad line region can skew the emission line FWHM which results in systematically lower $M_{\\rm BH}$ estimates compared with unobscured quasars, such that a comparison between both populations is not straightforward."
],
[
"Bright NIR Reddened AGN Are Less Numerous and More Luminous Than Blue Type 1 AGNs",
"[57], [56], [55] analyzed properties of radio-selected reddened quasars, finding that their observed surface space density was $\\sim $ 17-21% lower than a matched sample of radio-selected blue quasars.",
"They also reported that after correcting the $K$ -band magnitude for reddening, these red quasars were more luminous than their unobscured counterparts.",
"Similarly, [6] and [5] found that reddened quasars selected on the basis of red near-infrared colors ($J-K > 2.5$ , $K < 16.5$ , Vega) and red WISE colors, respectively, have higher bolometric luminosities than blue Type 1 AGNs culled from SDSS.",
"Since our bright NIR $R-K$ sample is $\\sim $ 89% complete (only one source that fits our selection criteria lacks a spectroscopic redshift; Figure REF ), we compare the properties of these reddened AGNs with a matched sample of X-ray selected blue ($R-K < 3$ ) Type 1 AGNs, also drawn from Stripe 82X.",
"This comparison sample obeys the same infrared and optical magnitude cuts as the $R-K$ versus $X/O$ sample: $X/O > 0$ , $K < 16$ (Vega).",
"We discard all sources spectroscopically identified as stars or galaxies (i.e., they lack broad lines in their optical spectra).",
"For sources that lacks a redshift, we discarded objects that lie along the $R-K$ versus $R-W1$ stellar locus [83], so that the comparison sample is made up of likely extragalactic sources.",
"There are 62 such blue sources for comparison, 56 of which have spectra and are confirmed Type 1 AGNs.",
"First, we calculated the observed surface density for both the reddened and comparison blue AGN samples in X-ray flux bins with a width of 0.3 dex.",
"To account for spectroscopic incompleteness, we multiplied the observed space density (i.e., $N$ /31.3 deg$^2$ ) within each bin by the fraction of sources spectroscopically identified in that bin.",
"Figure REF shows the observed space density for the reddened and blue AGN, where the errors are Poissonian ($\\sqrt{N}$ ) (if there are 10 or more sources in the bin) or are derived from [50].",
"From this exercise, we find that blue Type 1 AGNs have a higher space density than the reddened AGNs, and that they have higher X-ray fluxes than the reddened population.",
"There are no blue Type 1 AGNs at X-ray fluxes below 10$^{-13}$ erg s$^{-1}$ cm$^{-2}$ , while the reddened AGNs have a roughly constant space density ($\\sim $ 0.06 deg$^{-2}$ ) as a function of observed X-ray flux.",
"Figure: Observed surface space density in bins of 0.3 dex of full-band X-ray flux (F X, full F_{\\rm X,full}) of reddened AGNs from the nearly complete bright NIR R-KR-K versus X/OX/O selected sample (red circles) compared with a matched sample of X-ray selected blue (R-K<3R - K < 3) Type 1 AGNs (blue stars).",
"While the space density of the reddened AGN show is relatively constant with X-ray flux, no blue Type 1 AGNs are found at fluxes under 10 -13 ^{-13} erg s -1 ^{-1} cm -2 ^{-2}.The blue AGNs extend to brighter X-ray fluxes because they are predominantly nearby compared with the reddened population.",
"As shown in Figure REF (left), most of the blue AGNs (66%) reside at $z < 0.5$ while all the reddened AGNs are more distant.",
"Furthermore, the X-ray luminosities of the reddened population are drawn from the higher end of that observed in the blue AGN population, as illustrated in Figure REF (right), where we also show the estimated instrinsic X-ray luminosity for the reddened AGNs for reference.",
"The mean X-ray luminosities of both the reddened (log($L_{\\rm X,full}$ /erg s$^{-1}$ ) = 44.7 $\\pm $ 0.4 (observed); 45.0$\\pm $ 0.4 (intrinsic)) and blue (log($L_{\\rm X,full}$ /erg s$^{-1}$ ) = 44.8 $\\pm $ 0.5) AGNs are consistent.",
"However, under half of the blue AGNs have observed X-ray luminosities exceeding 10$^{44}$ erg s$^{-1}$ while 67% of the reddened AGNs are at these high X-ray luminosites.",
"Focusing on reddened AGNs in a flux-limited X-ray sample favors detection of AGNs that are more distant, and more luminous, than their unreddened counterparts.",
"This bias is induced by the red $R-K$ criterion and the $K$ -band flux limit: sources that are more reddened are those where the AGN dominates over the host galaxy, which are preferentially high-luminosity AGN since lower luminosity AGN would fall below the $K$ -band flux limit.",
"Furthermore, wide-area coverage is required to identify this luminous population at a relatively bright near-infrared flux limit.",
"We find no X-ray AGNs (i.e., $L_{\\rm X,full} > 10^{42}$ erg s$^{-1}$ ) from the smaller 2.2 deg$^2$ Chandra COSMOS Legacy survey [28], [96] with the same colors ($R-K > 4$ , Vega; $X/O > 0$ ) at the same magnitude limit (i.e., $K <$ 16, Vega).To match the magnitude system used in this study, we converted the COSMOS $r$ -band magnitude from Subaru SuprimeCam reported in the PSF-homogeneized photometric catalog of [81] to the SDSS $r$ -band filter using the formula in [23].",
"We then transformed to the Bessel $R$ bandpass and converted to the Vega magnitude system using Eqs.",
"1 & 2 above.",
"Figure: Left: Redshift distribution of our bright NIR R-KR-K versus X/OX/O selected reddened AGNs compared with blue (R-K<3R-K < 3, Vega) Type 1 X-ray selected AGNs from the Stripe 82X survey at similar magnitude limits (i.e., K<16K < 16, Vega).",
"The blue AGNs are predominantly at lower redshift (z<0.5z < 0.5) compared with the reddened AGNs.",
"Right: Luminosity distribution for the reddened and blue AGNs samples, where the estimated intrinsic luminosities, as implied by the hardness ratios, for the reddened AGNs are shown for reference.",
"Though the average luminosities are similar between the reddened and blue populations, a higher fraction of reddened AGNs have X-ray luminosities exceeding 10 44 10^{44} erg s -1 ^{-1} than the blue AGNs."
],
[
"Stripe 82X Reddened AGNs Compared with Those Previously Known",
"We compare our Stripe 82X reddened quasars with samples from the literature selected based on radio emission and red optical-infrared colors [57], [56], [55], near-infrared colors [8], [7], [6], mid-infrared colors [36], [5], and reddened ($R - K > 4-4.5$ , Vega) X-ray selected AGNs presented in [13].",
"Though these previous samples of reddened AGNs have been selected via independent methods, there are several traits that many of these sources have in common: the extinction ranges from moderate [57], [56], [55], [8], [7], [6] to extreme [5], the black holes are massive [6], [13], [144], the AGNs are generally distributed beyond $z > 1$ [13] and $z > 2$ [8], [7], [6], [5], and they tend to be more luminous than blue Type 1 quasars at comparable redshifts [57], [56], [55], [6], [5].",
"Most of the AGNs have broad H$\\alpha $ emission, and are not narrow-line only Type 2 AGNs, with the exception of the $W1W2$ drop-outs detected by WISE which are a mixture of Type 1 and Type 2 AGNs [36], [5].",
"[56] presented a sample of 120 reddened quasar candidates selected from the FIRST radio and 2MASS near-infrared surveys that have red colors.",
"An analogous sample of radio-selected quasar candidates was presented in [55], with similar color selection, but pushed down to lower near-infrared flux limits, using sources detected in the deeper UKIDSS survey.",
"We find that our Stripe 82X sources span similar redshift ranges ($0.6 < z < 2.5$ ) as those in [57], [56], [55], $0.13 < z < 3.1$ .",
"We also obtain similar AGN reddening values, where ours have a range of $0.45 < E$ ($B-V$ ) $< 1.18$ (after excluding the radio loud AGN and the two optical dropout AGN with inconsistent bolometric and X-ray luminosities) compared with $0.1 < E$ ($B-V$ )$< 1.55$ ; our $E$ ($B-V$ ) values are derived from SED fitting while those from [57], [56], [55] are measured from fitting a reddened quasar template to the optical and/or near-infrared spectra.",
"Only two of our Stripe 82X sources are detected in the radio by FIRST, indicating that the orthogonal axis of X-ray selection aids in recovering reddened AGNs not detected by radio surveys.",
"Similar to [6], we estimated $M_{\\rm BH}$ using the FWHM of the broad H$\\alpha $ emission line, where the continuum and bolometric luminosities were calculated via SED fitting.",
"Both samples are subjected to similar broad line region obscuration biases that could potentially affect emission line FWHMs.",
"Compared with the 38 $z > 2$ red quasars presented in [8], [7], [6], where log($M_{\\rm BH}/M_{}$ )=9.7$\\pm $ 0.46 and log($L_{\\rm bol}$ /erg s$^{-1}$ ) = 47.1$\\pm $ 0.4 [6], our X-ray selected reddened AGNs have lower black hole masses (log($M_{\\rm BH}/M_{}$ ) = 9.0$\\pm $ 0.8) and bolometric luminosities (log($L_{\\rm bol}$ /erg s$^{-1}$ )= 46.5$\\pm $ 0.8), though there is a wide spread on these values for the Stripe 82X AGN.",
"Additionally, as the faint NIR optical dropout sample is only $\\sim $ 12% complete, and we are unable to derive estimates of $M_{\\rm BH}$ for half of the sample we have observed, more observations are needed to test whether the pilot sample observed thus far is representative of the parent sample.",
"The measured AGN reddening in the [6] sample ($0.5 < E(B-V) < 1.5$ ) spans a similar range to the values calculated in our Stripe 82X sample.",
"The most luminous, reddened quasars yet identified were selected based on their mid-infrared colors in WISE: these $W1W2$ dropouts are weak or undetected in WISE bands $W1$ and $W2$ but are bright in bands $W3$ and $W4$ [36].",
"[5] analyzed the SEDs of 52 $W1W2$ drop-outs at $z > 1$ and $W4 < 7.2$ (Vega) that have Spitzer IRAC data.",
"The reddening in these objects are much more extreme (i.e., $\\langle E$ ($B-V$ )$\\rangle $ = 6.8) than what we observe in the Stripe 82X reddened AGNs presented here and seen in other reddened AGN samples.",
"The typical bolometric luminosities of the $W1W2$ drop-outs, 10$^{47}$ - 10$^{48}$ erg s$^{-1}$ , are also much higher than the Stripe 82X sample.",
"Three of these sources have been followed up with X-ray observations, with XMM-Newton and NuSTAR, and were found to be X-ray faint, consistent with Compton-thick levels of obscuration [128].",
"With our X-ray-optical-infrared selection of reddened quasars, we appear to be selecting an AGN population that is less extreme than the WISE $W1W2$ drop-outs, which at a space density of $\\sim $ 1/30 deg$^2$ , are more rare than reddened AGN selected via other diagnostics.",
"Finally, we compare our Stripe 82X reddened AGNs with the 21 reddened ($R - K > 4.5$ , Vega) AGNs from [13] that were selected from small-to-moderate area X-ray surveys: the original Chandra Deep Field South [52], XMM-COSMOS [62], [24], [20], and the literature [1], [122], [102], [33].",
"In this sample, the X-ray emission was mildly absorbed ($N_{\\rm H} > 10^{21} - 10^{22}$ cm$^{-2}$ , as implied by X-ray spectral analysis or hardness ratios), similar to the implied obscuration of our bright $R-K$ sample.",
"Our faint NIR optical dropout sample, which largely has hardness ratios consistent with no X-ray absorption, spans a similar redshift range as the [13] sample ($1.2 < z < 2.6$ ).",
"The [13] sample has a similar average black hole mass (log($M_{\\rm BH}/M_{}$ )=9.3$\\pm $ 0.5) as the Stripe 82X reddened AGNs.",
"They used the FWHM of the H$\\alpha $ line in conjunction with the intrinsic hard X-ray (2-10 keV) luminosity as a proxy for the AGN continuum luminosity [95] to estimate $M_{\\rm BH}$ : $\\begin{split}M_{\\rm BH} = 10^{7.11} \\left[\\frac{{\\rm FWHM (H\\alpha )}}{1000\\ {\\rm km\\ s^{-1}}}\\right]^{2.06}\\times \\\\\\left[\\frac{\\lambda L_{\\rm 2-10keV, intrinsic}}{10^{44}\\ {\\rm erg\\ s^{-1}}} \\right]^{0.693} {\\rm M_{}}.\\end{split}$ [13] do not provide errors on the parameters in this virial relation, but note that there is a scatter of about 0.1 dex in the normalization.",
"For reference, in Table REF we list the black hole masses we obtain using this scaling relation for the four sources that have hard band X-ray detections and H$\\alpha $ coverage.",
"We propagate the errors on the H$\\alpha $ FWHM and $L_{\\rm 2-10keV, intrinsic}$ values and note that there is likely an additional $\\sim $ 0.1 dex uncertainty in $M_{\\rm BH}$ that is associated with this virial relation as well as a general $\\sim $ 0.5 dex uncertainty that is found for single-epoch measurements, as discussed above [138].",
"We obtain similar black hole masses compared with what we calculated using $\\lambda L_{\\rm 51000}$ as the continuum luminosity, though the intrinsic hard X-ray luminosities are based on column densities derived from hardness ratios which are a very crude measure of absorption.",
"lll 0pt Black Hole Masses Estimated from $L_{\\rm 2-10keV, intrinsic}$ and H$\\alpha $ FWHM Stripe 82X Name Log ($L_{\\rm 2-10keV, intrinsic}$ ) $M_{\\rm BH}$ (erg s$^{-1}$ cm$^{-2}$ ) (M$_{}$ ) S82X 0242+0005 45.05$^{+0.27}_{-0.31}$ 9.26$^{+0.17}_{-0.29}$ S82X 0302$-$ 0003 44.84$^{+0.14}_{-0.08}$ 9.07$^{+0.08}_{-0.09}$ S82X 0303$-$ 0115 43.77$^{+0.11}_{-0.10}$ 7.29$^{+0.07}_{-0.09}$ S82X 0118+00181 44.12$^{+0.14}_{-0.00}$ 7.98$^{+0.16}_{-0.26}$ 1Since we only have an upper limit on the estimated $N_{\\rm H}$ , the lower limit on the intrinsic luminosity is the observed luminosity."
],
[
"Conclusions",
"We presented the results of a ground-based, near-infrared spectroscopic campaign to follow up reddened AGN candidates in the wide-area (31 deg$^2$ ) Stripe 82 X-ray survey [87], [86], [84].",
"Our bright NIR sample selected on the basis of red $R-K$ colors ($>$ 4, Vega) and $X/O > 0$ [20] consists of 9 sources, four of which had existing spectroscopy in SDSS and five of which we targeted with Palomar TripleSpec (Figure REF ); four of the targeted sources were identified via spectroscopic redshifts (Figure REF ).",
"This sample is 89% complete to a magnitude limit of $K=16$ (Vega).",
"We also presented a pilot program to follow-up sources that are not detected in the single-epoch SDSS imaging, yet have WISE colors consistent with quasars [143].",
"The spectra of these four sources were obtained with Keck NIRSPEC and Gemini GNIRS since 8-10m class telescopes are required to spectroscopically identify sources at these faint NIR magnitudes (i.e., $K > 17$ , Vega; Figure REF ).",
"All sources have at least one permitted emission line with FWHM exceeding 1300 km s$^{-1}$ in their optical or infrared spectra, and can thus be classified as Type 1 AGNs [61], [57].",
"The bright NIR sample spans a range of redshifts, $0.59 < z < 2.5$ , while faint optical dropout AGNs all lie beyond a redshift of 1.",
"We used AGNFitter [22] to fit the SEDs and decompose AGN and galaxy emission (Figures REF - REF ), obtaining estimates of the reddening and AGN bolometric luminosity for each source.",
"Two sources from the optical dropout sample, S82X 0141-0017 and S82X 0027+0042, have estimated $L_{\\rm bol}$ values on the order of or less than the observed X-ray luminosity, suggesting limitations in the SED decomposition for these sources.",
"All but one of the remaining AGNs are reddened, with 0.45 $<$ E(B-V)$_{\\rm AGN}$ $<$ 1.18.",
"The blue source is radio loud, such that its red $R-K$ color is likely due to jet-dominated synchrotron emission.",
"Half the sources in the bright NIR sample have features in their optical spectra indicative of outflows (Figures REF left, REF bottom, REF bottom right, and REF bottom).",
"Since many quasars host outflows [49], follow-up high-resolution imaging of the host galaxies would be necessary to search for morphological signatures of mergers to test whether these features signify feedback predicted by the major merger AGN evolution paradigm [121], [69].",
"Because the bright NIR sample is nearly complete, we compared the characteristics of these AGNs with blue ($R-K < 3$ , Vega) Type 1 AGNs selected from the Stripe 82X survey.",
"While the blue Type 1 AGNs have systematically higher X-ray fluxes (Figure REF ), they are predominantly at low redshift ($z<0.5$ ), with a greater percentage at lower X-ray luminosities compared with the reddened AGNs (Figure REF ).",
"Hence, focusing on reddened populations in shallow X-ray surveys, like Stripe 82X, for follow-up will likely unveil more distant and more luminous AGNs than blue AGN at similar X-ray, optical, and infrared flux limits.",
"Compared with reddened AGNs selected on the basis of their radio, optical, near-infrared, and/or mid-infrared emission, the Stripe 82X red quasars have similar reddening [5], and a range of estimated black hole masses and Eddington parameters.",
"Our pilot sample of WISE-selected optical dropouts is only $\\sim $ 12% complete, precluding us from drawing any firm conclusions about this population as a whole.",
"Our program does demonstrate proof-of-concept for using this selection technique to recover reddened quasars at $z > 1$ that are missed by optical surveys like SDSS.",
"We highlight that Stripe 82X complements other X-ray surveys by discovering reddened AGNs at relatively bright NIR magnitudes (i.e., $K < 16$ , Vega) that are missed entirely by smaller-area X-ray surveys, like the 2.2 deg$^2$ Chandra COSMOS Legacy [28], [96].",
"As these sources are rare, potentially due to the reddened stage being a short-lived AGN evolutionary phase, wide-area X-ray surveys like Stripe 82X, XMM-XXL [112], XBoötes [74], [106], the upcoming eROSITA mission [103], and serendipitous surveys/catalogs like ChaMP [75], the Chandra Source Catalog [42], and the XMM Serendipitous catalog [117], are necessary to reveal this missing tier of luminous, obscured black hole growth at the brightest fluxes.",
"We thank the anonymous referee for a careful reading of this manuscript and providing helpful comments.",
"Most of this work was completed while S.M.L was supported by an appointment to the NASA Postdoctoral Program at the NASA Goddard Space Flight Center, administered by Universities Space Research Association under contract with NASA.",
"S.M.L.",
"thanks A.-N. Chene for support when running the Gemini GNIRS reduction pipeline and G. Calistro Rivera for guidance in using AGFNitter.",
"Palomar and Keck observations were obtained through guaranteed Yale time on these facilities.",
"E.G acknowledges the generous support of the Cottrell College Award through the Research Corporation for Science Advancement.",
"The work of D.S.",
"was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA.",
"Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.",
"The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.",
"The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community.",
"We are most fortunate to have the opportunity to conduct observations from this mountain.",
"XMM, CXC, Sloan, Hale (TSPEC), Gemini:Gillet (GNIRS), Keck:II (NIRSPEC)"
]
] | 1709.01578 |
[
[
"On the Triangle Clique Cover and $K_t$ Clique Cover Problems"
],
[
"Abstract An edge clique cover of a graph is a set of cliques that covers all edges of the graph.",
"We generalize this concept to \"$K_t$ clique cover\", i.e.",
"a set of cliques that covers all complete subgraphs on $t$ vertices of the graph, for every $t \\geq 1$.",
"In particular, we extend a classical result of Erd\\\"os, Goodman, and P\\'osa (1966) on the edge clique cover number ($t = 2$), also known as the intersection number, to the case $t = 3$.",
"The upper bound is tight, with equality holding only for the Tur\\'an graph $T(n,3)$.",
"We also extend an algorithm of Scheinerman and Trenk (1999) to solve a weighted version of the $K_t$ clique cover problem on a superclass of chordal graphs.",
"We also prove that the $K_t$ clique cover problem is NP-hard."
],
[
"Introduction",
"A clique in a graph $G$ is a set of vertices that induces a complete subgraph; all graphs considered in this paper are simple and undirected.",
"A vertex clique cover of a graph $G$ is a set of cliques in $G$ that collectively cover all of its vertices.",
"The vertex clique cover number of $G$ , denoted ${\\theta _{\\text{v}}}(G)$ , is the minimum number of cliques in a vertex clique cover of $G$ .",
"An edge clique cover of a graph $G$ is a set of cliques of $G$ that collectively cover all of its edges.",
"The edge clique cover number of $G$ , denoted ${\\theta _{\\text{e}}}(G)$ , is the minimum number of cliques in an edge clique cover.",
"The vertex clique cover number, which is the same as the chromatic number of the complement graph, and the edge clique cover number, also referred to as the intersection number of a graph, have been extensively studied in the literature (see, for instance [9], [12], [22]).",
"Figure: An illustration of a minimum vertex clique cover (left-most), a minimum edge clique cover (middle), and a minimum triangle clique cover (right-most) of the same graph on eight vertices.We generalize the notions of vertex and edge clique covers by the following definition.",
"Note that we use the word clique to refer to any vertex set inducing a complete subgraph.",
"Definition 1 Let $t$ be a nonnegative integer.",
"A $t$ -clique of $G$ is a clique containing exactly $t$ vertices.",
"A set $\\mathcal {C}{}$ of cliques is a $K_t$ clique cover of $G$ if for every $t$ -clique $S \\subset V(G)$ there is a clique $Q \\in \\mathcal {C}$ that covers $S$ (i.e., $S$ is a subgraph of $Q$ ).",
"A $K_1$ clique cover is simply a vertex clique cover, while a $K_2$ clique cover is an edge clique cover.",
"We refer to a $K_3$ clique cover as a triangle clique cover.",
"The $K_t$ clique cover number, denoted ${\\theta _{K_t}}$ , and the triangle clique cover number, denoted ${\\theta _{\\triangle }}$ , are also defined accordingly.",
"We illustrate these concepts in Fig.",
"REF .",
"For positive integers $n$ and $k$ , the Turán graph $T(n,k)$ is defined to be the complete $k$ -partite graph on $n$ vertices whose part sizes differ by at most 1.",
"Turán graphs frequently arise as extremal graphs for various graph parameters.",
"We pose the following conjecture, which states that Turán graphs are the unique extremal graphs for the $K_t$ clique cover problem.",
"Conjecture 1 If $n$ and $t$ are positive integers, then for every $n$ -vertex graph $G$ , $ {\\theta _{K_t}(G)}\\le {\\theta _{K_t}}(T(n,t)).",
"$ Equality holds if and only if $G \\cong T(n,t)$ .",
"As motivation for Conjecture REF , we now briefly consider the cases $t=1$ and $t=2$ .",
"When $t = 1$ , it is obvious that $\\theta _{K_1}(G) = {\\theta _{\\text{v}}(G)}\\le n$ for every graph $G$ on $n$ vertices.",
"Equality clearly holds if and only if $G$ has no edges; a graph with no edges is the trivial Turán graph $T(n,1)$ .",
"Erdös, Goodman, and Pósa [9] proved that $\\theta _{K_2}(G) = {\\theta _{\\text{e}}(G)}\\le \\lfloor \\frac{n^2}{4} \\rfloor $ for every graph $G$ , with equality holding if and only if $G$ is the Turán graph $T(n,2) = K_{\\left\\lfloor n/2 \\right\\rfloor , \\lceil n/2 \\rceil }$ .",
"Table: Status of Conjecture .In Sections and , we prove the $t=3$ case of Conjecture REF .",
"At the end of Section , we also discuss the connections between Conjecture REF and a theorem of Lehel [14] about covering edges in hypergraphs.",
"The status of Conjecture REF is summarized in Table REF .",
"We also consider a natural weighted version of the $K_t$ clique cover problem in Section REF , in which each $t$ -clique $S$ is assigned a nonnegative integer weight $w_S$ ; we seek a smallest multiset of cliques such that each $t$ -clique $S$ is covered at least $w_S$ times.",
"We give a polynomial-time algorithm to solve this problem on a superclass of chordal graphs, extending a result of Scheinerman and Trenk [24].",
"The specific class of graphs for which our algorithm works, which we call semichordal graphs, is defined and discussed in Section REF .",
"It was shown by Orlin [20] and by Kou, Stockmeyer, and Wong [13] that determining ${\\theta _{\\text{e}}(G)}$ is an NP-complete problem.",
"Their idea is to reduce the problem of determining ${\\theta _{\\text{v}}(G)}$ , which was known to be NP-complete, to the problem of determining ${\\theta _{\\text{e}}(G)}$ .",
"In Section REF , we generalize the reduction used in [13] to show, by induction, that determining ${\\theta _{K_t}(G)}$ is an NP-complete problem, for any constant $t \\ge 2$ , by reducing the problem of determining $\\theta _{K_{t-1}}(G)$ to the problem of determining $\\theta _{K_t}(G)$ .",
"Our study of triangle clique covers and $K_t$ clique covers is motivated by the recent developments in the literature of community detection (a.k.a network clustering).",
"Complex networks such as social networks and biological networks often exhibit strong community structures in which nodes are clustered into different communities based on dense intra-community connections and sparse inter-community connections.",
"Cliques or clique-like substructures in the network are often used to model such communities which usually represent groups of people with common affiliations or interests in social networks, or disciplines in the citation networks, or functional modules in the protein-protein interaction networks [21], [3], [15].",
"Most community detection algorithms are based on pairwise connections among nodes in the network.",
"Nevertheless, recent work [2], [4], [16], [17] has revealed the importance of motifs, that is, subgraphs of the graph that appear with a frequency exceeding the one predicted through certain random models.",
"Examples of motifs include triangles and small cliques and near-cliques, the former arising due to the principle of triadic closure, which assumes that two people having a common friend will be more likely to be connected [11], [5].",
"The problem of covering triangles with cliques corresponds to the problem of network clustering with triangle motifs in the ideal setting where each community is a clique."
],
[
"A Minimum-Degree Version of a Theorem of Lovász",
"In this section, we obtain results similar to the following theorem of Lovász [18].",
"In the next section, we will apply these results to obtain an upper bound on ${\\theta _{\\triangle }(G)}$ .",
"Theorem 1 (Lovász [18]) Let $G$ be an $n$ -vertex graph, and let $k = {n \\atopwithdelims ()2} - \\left|{E(G)}\\right|$ .",
"If $t$ is the greatest integer such that $t^2-t \\le k$ , then ${\\theta _{\\text{e}}}(G) \\le k+t$ .",
"In this section, we obtain a variant of Theorem REF by strengthening the hypothesis to include a lower bound on $\\delta (G)$ , the minimum degree of $G$ , rather than just a lower bound on $\\left|{E(G)}\\right|$ .",
"We start with a basic result and then strengthen its weakest nontrivial case.",
"Lemma 1 For any graph $G$ , ${\\theta _{\\text{e}}}(G) \\le (n - \\delta (G)) + \\frac{n(n-\\delta (G)-1)}{2}$ .",
"We mimic the proof of Theorem REF .",
"Let $A_1$ be a maximum clique in $G$ , and for $i > 1$ , let $A_i$ be a maximum clique in $G-(A_1 \\cup \\cdots \\cup A_{i-1})$ .",
"Set $a_i = \\left|{A_i}\\right|$ .",
"Let $p$ be the largest index for which $a_i > 0$ .",
"As each vertex of $A_p$ has at least one non-neighbor in $A_j$ for $j < p$ , we have $p \\le n - \\delta (G)$ .",
"For each $v \\in A_i$ and each $j < i$ , let $S_{v,j} = \\lbrace v\\rbrace \\cup (N(v) \\cap A_j)$ .",
"Each set $S_{v,j}$ is clearly a clique.",
"Let $\\mathcal {F}$ be the set consisting of all cliques $A_i$ together with all the cliques $S_{v,j}$ where $v \\in A_i$ and $j < i$ .",
"The set $\\mathcal {F}$ covers all edges of $G$ , and we have $ {\\theta _{\\text{e}}}(G) \\le \\left|{\\mathcal {F}}\\right| \\le p + \\sum _{i=1}^{p}(i-1)a_i.",
"$ Subject to the constraints $a_1 + \\ldots + a_p = n$ and $a_1 \\ge \\cdots \\ge a_p$ , and allowing $a_i$ to take fractional values, the sum $\\sum _{i=1}^p (i-1)a_i$ is clearly maximized when $a_1 = \\cdots = a_p = n/p$ .",
"Hence, $ {\\theta _{\\text{e}}}(G) \\le p + \\frac{n}{p}\\sum _{i=1}^p(i-1) = p + \\frac{n(p-1)}{2}.",
"$ As $p \\le n-\\delta (G)$ , the conclusion follows.",
"Note that a lower bound on $\\delta (G)$ indirectly gives an upper bound on ${\\theta _{\\text{e}}}(G)$ via Theorem REF , since we have $k \\ge \\frac{n(n-\\delta (G)-1)}{2}$ .",
"However, the upper bound in Lemma REF is often sharper than the bound guaranteed this way in Theorem REF ; for example, when $n=12$ and $\\delta (G) = 9$ , Theorem REF gives an upper bound of ${\\theta _{\\text{e}}}(G) \\le 16$ , while Lemma REF gives an upper bound of 15.",
"When $\\delta (G) = n/2$ , Lemma REF gives ${\\theta _{\\text{e}}}(G) \\le n^2/4$ , which is sharp when $G = K_{n/2, n/2}$ .",
"We wish to obtain a sharper bound when $\\delta (G)$ is slightly larger than $n/2$ .",
"Lemma 2 The following upper bounds hold for the edge clique cover number of a graph $G$ .",
"${\\theta _{\\text{e}}}(G) \\le {\\left\\lbrace \\begin{array}{ll}\\frac{n^2}{4} - \\frac{n}{2} + \\frac{1}{4},&\\text{ if } \\delta (G) = (n+1)/2,\\\\\\frac{n^2}{4} - n + 2, &\\text{ if } \\delta (G) = n/2+1.\\end{array}\\right.",
"}$ Define $A_1, \\ldots , A_p$ and $\\mathcal {F}$ as in the proof of Lemma REF .",
"With $a_i = \\left|{A_i}\\right|$ , we have the bound $ {\\theta _{\\text{e}}}(G) \\le \\left|{\\mathcal {F}}\\right| \\le p + \\sum _{i=1}^{p}(i-1)a_i.",
"$ Case 1: $p < n-\\delta (G)$ .",
"If $\\delta (G) = (n+1)/2$ , then repeating the argument in Lemma REF yields $ {\\theta _{\\text{e}}}(G) \\le p + \\frac{n(p-1)}{2} \\le \\frac{n^2}{4} - \\frac{3n}{4} - \\frac{3}{2} < \\frac{n^2}{4} - \\frac{n}{2} + \\frac{1}{4}.",
"$ If $\\delta (G) = n/2 + 1$ , then similarly, ${\\theta _{\\text{e}}}(G) \\le p + \\frac{n(p-1)}{2} \\le \\frac{n^2}{4} - n - 2< \\frac{n^2}{4} - n + 2.",
"$ Case 2: $p = n-\\delta (G)$ .",
"We now exploit the integrality of $a_i$ , which we ignored in the proof of Lemma REF .",
"If $\\delta (G) = (n+1)/2$ then $p = (n-1)/2$ and the sum $\\sum _{i=1}^p (i-1)a_i$ is maximized, subject to $a_1 + \\ldots + a_p = n$ and $a_1 \\ge \\cdots \\ge a_p$ , by the sequence with $a_1 = 3$ and $a_i = 2$ for $i \\ge 2$ .",
"Hence, we obtain the upper bound ${\\theta _{\\text{e}}}(G) \\le p + 2\\sum _{i=2}^p (i-1) = p + p(p-1) = \\frac{n^2}{4} - \\frac{n}{2} + \\frac{1}{4}.$ Similarly, when $\\delta (G) = n/2 + 1$ and $p = n/2-1$ , the sum $\\sum _{i=1}^p (i-1)a_i$ is maximized by the sequence with $a_1 = a_2 = 3$ and $a_i = 2$ for $i \\ge 3$ .",
"We obtain the upper bound ${\\theta _{\\text{e}}}(G) \\le p + 3 + 2\\sum _{i=3}^p(i-1) = p+3 + (p(p-1) - 2) = p+1+p(p-1) = \\frac{n^2}{4} - n + 2.$ Thus, the claimed upper bounds on ${\\theta _{\\text{e}}(G)}$ hold in both cases."
],
[
"An Upper Bound on ${\\theta _{\\triangle }(G)}$",
"In this section, our goal is to generalize the following result of Erdös, Goodman, and Pósa.",
"Theorem 2 (Erdös–Goodman–Posa [9]) If $G$ is an $n$ -vertex graph, then ${\\theta _{\\text{e}}}(G) \\le \\left\\lfloor n^2/4 \\right\\rfloor $ .",
"Equality holds if and only if $G \\cong T(n,2)$ .",
"Definition 2 When $G$ is a graph and $r$ is a nonnegative integer, $k_t(G)$ is the number of copies of $K_t$ in $G$ .",
"Observation 1 For any nonnegative integer $n$ , $ k_3(T(n,3)) = {\\left\\lbrace \\begin{array}{ll}\\dfrac{n^3}{27}, &\\text{ if } n \\equiv 0 \\pmod {3},\\vspace{5.0pt}\\\\\\dfrac{(n-1)^3}{27} + \\dfrac{(n-1)^2}{9}, &\\text{ if } n \\equiv 1 \\pmod {3},\\vspace{5.0pt}\\\\\\dfrac{(n+1)^3}{27} - \\dfrac{(n+1)^2}{9}, &\\text{ if } n \\equiv 2 \\pmod {3}.\\end{array}\\right.}",
"$ Observation 2 For all $n \\ge 3$ , $k_3(T(n,3)) - k_3(T(n-1,3)) &= \\left\\lfloor \\frac{\\left\\lfloor 2n/3 \\right\\rfloor ^2}{4} \\right\\rfloor \\\\&={\\left\\lbrace \\begin{array}{ll}\\dfrac{n^2}{9}, &\\text{ if $n \\equiv 0 \\pmod {3}$}\\\\\\dfrac{(n-1)^2}{9}, &\\text{ if $n \\equiv 1 \\pmod {3}$}\\\\\\dfrac{n^2-n-2}{9}, &\\text{ if $n \\equiv 2 \\pmod {3}$}\\end{array}\\right.",
"}\\\\&\\ge \\frac{(n-1)^2}{9}.$ Theorem 3 For any graph $G$ , ${\\theta _{\\triangle }(G)}\\le k_3(T(n,3))$ .",
"If equality holds, then $G \\cong T(n,3)$ .",
"We use induction on $n$ , with trivial base case when $n \\le 3$ .",
"Assume that $n > 3$ and the claim holds for smaller graphs.",
"Let $v$ be a vertex of minimum degree in $G$ , let $G^{\\prime } = G-v$ , and let $\\mathcal {C}^{\\prime }$ be a smallest $K_3$ clique cover of $G^{\\prime }$ .",
"By the induction hypothesis, $\\left|{\\mathcal {C}^{\\prime }}\\right| \\le k_3(T(n-1,3))$ .",
"We will extend $\\mathcal {C}^{\\prime }$ to a $K_3$ clique cover $\\mathcal {C}$ of $G$ such that $\\left|{\\mathcal {C}}\\right| \\le k_3(T(n,3))$ .",
"The only triangles of $G$ not yet covered by $\\mathcal {C}^{\\prime }$ are the triangles that contain $v$ .",
"Let $H$ be the subgraph of $G$ induced by $N(v)$ , and let $\\mathcal {F}$ be a smallest edge clique cover of $H$ .",
"By adding $v$ to each clique in $\\mathcal {F}$ , we obtain a set of cliques $\\mathcal {F}_1$ covering every triangle that contains $v$ .",
"Thus, $\\mathcal {F}_1 \\cup \\mathcal {C}^{\\prime }$ is a triangle edge cover in $G$ .",
"It therefore suffices to show that ${\\theta _{\\text{e}}(H)}\\le k_3(T(n,3)) - k_3(T(n-1,3))$ , and this is what we show next.",
"We split the proof into cases according to $d(v)$ .",
"Case 1: $d(v) \\le \\left\\lfloor 2n/3 \\right\\rfloor $ .",
"In this case, $\\left|{V(H)}\\right| \\le 2n/3$ , so by Theorem REF and Observation REF , we have $ {\\theta _{\\text{e}}}(H) \\le \\left\\lfloor \\left|{V(H)}\\right|^2/4 \\right\\rfloor \\le \\left\\lfloor \\left\\lfloor 2n/3 \\right\\rfloor ^2/4 \\right\\rfloor = k_3(T(n,3)) - k_3(T(n-1,3)), $ as desired.",
"If ${\\theta _{\\triangle }(G)}= k_3(T(n,3))$ , then equality must hold throughout the above inequality, and in particular we must have ${\\theta _{\\text{e}}}(H) = \\left|{V(H)}\\right|^2/4 = \\left\\lfloor \\left\\lfloor 2n/3 \\right\\rfloor ^2/4 \\right\\rfloor $ .",
"By Theorem REF , this implies that $H \\cong T(\\left\\lfloor 2n/3 \\right\\rfloor , 2)$ .",
"Furthermore, ${\\theta _{\\triangle }(G)}= k_3(T(n,3))$ requires that $\\left|{\\mathcal {C}^{\\prime }}\\right| = \\theta _{\\triangle }(G-v) = k_3(T(n-1,3))$ , so by the induction hypothesis, we have $G^{\\prime } \\cong T(n-1,3)$ .",
"Thus, $G$ is obtained from $T(n-1,3)$ by adding a new vertex adjacent to $\\left\\lfloor 2n/3 \\right\\rfloor $ vertices inducing a complete bipartite graph.",
"This implies that $G \\cong T(n,3)$ .",
"Since $v$ was a vertex of minimum degree, every $w \\in N(v)$ satisfies $d(w) \\ge d(v)$ .",
"At most $n-d(v)$ of those neighbors lie outside $N(v)$ , so for all $w \\in V(H)$ , we have $ d_H(w) \\ge d(v) - (n-d(v)) = 2d(v)-n. $ Thus, $\\delta (H) \\ge 2d(v)-n$ .",
"This inequality will be used in the subsequent cases.",
"Case 2: $d(v) \\ge 2n/3 + 1$ .",
"Lemma REF yields ${\\theta _{\\text{e}}}(H) \\le (d(v) - \\delta (H)) + \\frac{d(v)(d(v) -\\delta (H)-1)}{2} \\le \\frac{n(d(v)+2) - d(v)(d(v)+3)}{2}.$ If $d(v) \\ge (2n+4)/3$ , then this implies that ${\\theta _{\\text{e}}}(H) \\le \\frac{n\\Big (\\frac{2n+4}{3}+2\\Big ) - \\frac{2n+4}{3}\\Big (\\frac{2n+4}{3}+3\\Big )}{2} = \\frac{(n-1)^2-27}{9}< \\frac{(n-1)^2}{9} \\le k_3(T(n,3)) - k_3(T(n-1,3)),$ and we are done.",
"Similarly, if $d(v) = (2n+3)/3$ , then $n \\equiv 0 \\pmod {3}$ , so we again have ${\\theta _{\\text{e}}}(H) \\le \\frac{n\\Big (\\frac{2n+3}{3}+2\\Big ) - \\frac{2n+3}{3}\\Big (\\frac{2n+3}{3}+3\\Big )}{2} = \\frac{2n^2-3n-36}{18}< \\frac{n^2}{9} = k_3(T(n,3)) - k_3(T(n-1,3)).$ Case 3: $d(v) = (2n+1)/3$ or $d(v) = (2n+2)/3$ .",
"If $\\delta (H) \\ge 2d(v) - n + 1$ , then Lemma REF yields $ {\\theta _{\\text{e}}}(H) \\le (d(v) - \\delta (H)) + \\frac{d(v)(d(v) -\\delta (H)-1)}{2} = n-1 + \\frac{d(v)(n - d(v) - 4)}{2} < \\frac{(n-1)^2}{9}, $ where the last inequality follows from the assumption that $d(v) \\ge (2n+1)/3$ .",
"Hence we may assume that $\\delta (H) = 2d(v)-n$ .",
"We consider two subcases: either $d(v) = (2n+1)/3$ or $d(v) = (2n+2)/3$ .",
"Case 3a: $d(v) = (2n+1)/3$ .",
"Here $\\delta (H) = 2d(v) - n$ gives $\\delta (H) = (n+2)/3 = (d(v)+1)/2$ .",
"Hence, Lemma REF yields $ {\\theta _{\\text{e}}}(H) \\le \\frac{d(v)^2}{4} - \\frac{d(v)}{2} + \\frac{1}{4} = \\frac{(n-1)^2}{9}, $ and so ${\\theta _{\\text{e}}}(H) \\le k_3(T(n,3)) - k_3(T(n-1,3))$ .",
"This yields ${\\theta _{\\triangle }(G)}\\le k_3(T(n,3))$ for the case $d(v) = (2n+1)/3$ .",
"To obtain the strict inequality ${\\theta _{\\triangle }(G)}< k_3(T(n,3))$ , suppose to the contrary that ${\\theta _{\\triangle }(G)}= k_3(T(n,3))$ .",
"Since ${\\theta _{\\text{e}}}(H) \\le k_3(T(n,3)) - k_3(T(n-1,3))$ , we must also have $\\theta _{\\triangle }(G-v) = k_3(T(n-1,3))$ .",
"By the induction hypothesis, $G-v \\cong T(n-1,3)$ .",
"In particular, $G-v$ is $K_4$ -free.",
"Let $\\mathcal {C}^{\\prime }$ be a smallest $K_3$ clique cover in $G-v$ .",
"Since $G-v$ is $K_4$ -free, every clique in $\\mathcal {C}^{\\prime }$ is a triangle.",
"Since $\\delta (H) = \\frac{d(v)+1}{2} > \\frac{d(v)}{2} = \\left|{V(H)}\\right|/2$ , every edge in $H$ is contained in some triangle of $H$ .",
"Since $G-v$ is $K_4$ -free, every triangle of $H$ is contained in $\\mathcal {C}^{\\prime }$ .",
"Let $\\mathcal {C}$ be the collection of cliques obtained by replacing every triangle $T$ of $H$ contained in $\\mathcal {C}^{\\prime }$ with the clique $T \\cup \\lbrace v\\rbrace $ .",
"Now $\\mathcal {C}$ covers all triangles in $G$ , and $ \\left|{\\mathcal {C}}\\right| = \\left|{\\mathcal {C}^{\\prime }}\\right| \\le k_3(T(n-1,3)) < k_3(T(n,3)).",
"$ This contradicts the hypothesis that ${\\theta _{\\triangle }(G)}= k_3(T(n,3))$ .",
"We conclude that ${\\theta _{\\triangle }(G)}< k_3(T(n,3))$ when $d(v) = (2n+1)/3$ .",
"Case 3b: $d(v) = (2n+2)/3$ .",
"Here $\\delta (H) = 2d(v) - n$ gives $\\delta (H) = \\frac{n+4}{3} = \\frac{d(v)}{2} + 1$ .",
"In this case, $n \\equiv 2 \\pmod {3}$ .",
"Since we have assumed also that $n > 3$ , we have $n \\ge 5$ , so Lemma REF yields ${\\theta _{\\text{e}}}(H) \\le \\frac{d(v)^2}{4} - d(v) + 2 = \\frac{n^2}{9} - \\frac{4n}{9} + \\frac{13}{9} \\le \\frac{n^2 - n - 2}{9},$ where the inequality is strict for $n > 5$ .",
"Thus, ${\\theta _{\\text{e}}}(H) \\le k_3(T(n,3)) - k_3(T(n-1,3))$ , and this inequality is strict for $n > 5$ .",
"On the other hand, when $n=5$ , we have $\\delta (G) = d(v) = 4$ , so that $G$ is a complete graph, which forces ${\\theta _{\\triangle }(G)}= 1 < k_3(T(5;3))$ .",
"Thus, when $d(v) = (2n+2)/3$ , we have ${\\theta _{\\triangle }(G)}< k_3(T(n,3))$ .",
"Thus, in all cases, ${\\theta _{\\triangle }(G)}\\le k_3(T(n,3))$ , and equality holds only in Case 1 when $G \\cong T(n,3)$ .",
"Considering the way that Lemma REF is used in the proof of Theorem REF , one might hope to prove an analogous “minimum-degree version” of Theorem REF and then use it to prove the $t=4$ case of Conjecture REF .",
"Unfortunately, one runs into difficulties with this approach very quickly.",
"The main difficulty is that the proof of Lemma REF relies very strongly on there only being two types of edges that must be covered: edges within a single $A_i$ and edges with one endpoint in $A_i$ and the other in $A_j$ .",
"When $t=3$ , on the other hand, there are (at least) three possible types of $K_3$ : those contained within a single $A_i$ , those with two endpoints in one $A_i$ and the other in $A_j$ , and those with endpoints in three different sets $A_i, A_j, A_k$ .",
"This makes it considerably more difficult to find a way to cover all copies of $K_3$ and efficiently count the number of cliques used in the process.",
"As $t$ grows larger, even more configurations are possible, making this approach more difficult than anticipated.",
"The results stated in Theorem REF have a very close connection with their counterparts established for hypergraphs [6], [14].",
"In the following we discuss the similarity and the differences between our results and those known in the hypergraph literature.",
"A $t$ -uniform hypergraph $H = (V(H),E(H))$ consists of a vertex set $V(H)$ and a hyperedge set $E(H)$ , where each hyperedge is a set of some $t$ vertices.",
"A 2-uniform hypergraph is simply a graph.",
"For $p \\ge t \\ge 2$ , let $K_p^{(t)}$ denote the hyperclique on $p$ vertices, i.e., a set of $p$ vertices of a hypergraph where every subset of $t$ vertices forms a hyperedge.",
"Note that the usual clique $K_t$ is the same as $K_t^{(2)}$ .",
"Let $h_t(n,p)$ denote the maximum number of hyperedges that a $K_p^{(t)}$ -free $t$ -uniform hypergraph on $n$ vertices can have.",
"Let $k_t(n,p)$ denote the maximum number of $K_t$ in a $K_p$ -free graph.",
"It was proved by Moon and Moser [19], and by Sauer [23] that $k_t(n,p)$ is precisely the number of $K_t$ in the Turán graph $T(n,p-1)$ .",
"In other words, $k_t(n,p) = k_t(T(n,p-1))$ .",
"The following result was conjectured by Bollobás [6] and proved by Lehel [14].",
"Theorem 4 (Lehel [14]) The edges of every $t$ -uniform hypergraph of order $n$ can be covered by at most $h_t(n,p)$ edges and copies of $K_p^{(t)}$ .",
"When $t = 2$ and $p = 3$ , we have $h_2(n,3) = k_2(n,3) = k_2(T(n,2)) = \\left\\lfloor \\frac{n^2}{4}\\right\\rfloor $ (by Mantel's theorem and also by Turán [26]), and hence Theorem REF reduces to the classical result on edge clique cover by Erdös, Goodman, and Pósa [9] mentioned in the introduction, which states that the edges of every graph on $n$ vertices can be covered by using at most $\\left\\lfloor \\frac{n^2}{4}\\right\\rfloor $ edges and triangles.",
"When $t = 3$ , Theorem REF states that one can use at most $h_3(n,4)$ hyperedges and $K_4^{(3)}$ 's to cover all hyperedges in a 3-uniform hypergraph.",
"However, Theorem REF does not follow from this statement.",
"Indeed, our theorem establishes that one can use at most $k_3(n,4)$ cliques to cover all triangles in any graph on $n$ vertices.",
"We emphasize that $h_3(n,4)$ is strictly larger than $k_3(n,4)$ .",
"Recall that $k_3(n,4)$ , which is the number of triangles in the Turán graph $T(n,3)$ , can be computed explicitly as $\\lfloor n/3 \\rfloor \\lfloor (n+1)/3 \\rfloor \\lfloor (n+2)/3 \\rfloor \\approx n^3/27$ .",
"By contrast, the determination of $h_3(n,4)$ , even asymptotically, has remained open since the original work of Turán [26].",
"Moreover, Turán established that (see also [7]) $h_3(n,4) \\ge {\\left\\lbrace \\begin{array}{ll}m^2(5m-3)/2, &\\text{ if } n = 3m,\\vspace{7.22743pt} \\\\m(5m^2+2m-1)/2, &\\text{ if } n = 3m+1,\\vspace{7.22743pt} \\\\m(m+1)(5m_2)/2, &\\text{ if } n = 3m + 2.\\end{array}\\right.",
"}$ The hypergraph that gives rise to this lower bound is a modification of the usual Turán graph $T(n, 3)$ to the hypergraph setting, which can be constructed as follows.",
"The vertex set is partitioned into three (almost) equal sets $V_1$ , $V_2$ , and $V_3$ , where $|V_1| = \\lfloor n/3 \\rfloor $ , $|V_2| = \\lfloor (n+1)/3 \\rfloor $ , and $|V_3| = \\lfloor (n+2)/3 \\rfloor $ .",
"The hyperedge set consists of the 3-tuples $e = \\lbrace u,v,w\\rbrace $ , where either $u \\in V_1, v \\in V_2, w \\in V_3$ , or $u, v \\in V_i$ and $w \\in V_{(i+1) \\pmod {3}}$ .",
"Notice that the first type of hyperedges corresponds to the triangles of the Turán graph $T(n,3)$ , while the second type of hyperedges corresponds to new triangles that are unique to the hypergraph context.",
"From (REF ), $h_3(n,4)$ is in order of $\\frac{5}{2}\\frac{n^3}{27}$ , which is strictly larger than $k_3(n,4) \\approx \\frac{n^3}{27}$ .",
"In fact, Turán conjectured that (REF ) is actually an equality, which was known to be true for all $n \\le 13$ [25].",
"The key point that leads to the difference between $h_3(n,4)$ and $k_3(n,4)$ is that while a $K_4$ -free graph (removing all edges that do not belong to any triangles) can be considered as a $K_4^{(3)}$ -free 3-uniform hypergraph, the converse is not true.",
"For example, a hypergraph with the vertex set $\\lbrace u,v,w,x\\rbrace $ and the hyperedge set $\\lbrace \\lbrace u,x,w\\rbrace , \\lbrace v,x,w\\rbrace $ , $\\lbrace u,v,w\\rbrace \\rbrace $ is a $K_4^{(3)}$ -free hypergraph, but it corresponds exactly to a $K_4$ as a graph.",
"Therefore, counting edges in a $K_4^{(3)}$ -free 3-uniform hypergraph is not the same as counting triangles in a $K_4$ -free graph.",
"In fact, the maximum number of hyperedges in such $K_4^{(3)}$ -free hypergraphs is larger than the maximum number of triangles in $K_4$ -free graphs.",
"That is the reason Theorem REF produces a strictly worse upper bound than the tight upper bound we obtain in Theorem REF .",
"This is also evident from the fact that while according to Theorem REF , the hyperedges of every 3-uniform hypergraph can be covered by using at most $h_3(n,4)$ $K_4^{(3)}$ 's and hyperedges, Remark REF states that we cannot cover all triangles in $K_n$ $(n \\ge 18)$ by $k_3(n,4) = k_3(T(n,3))$ $K_4$ 's and triangles.",
"Remark 1 It is impossible to cover all triangles in $K_n$ $(n \\ge 18)$ by only $k_3(n,4)$ triangles and copies of $K_4$ .",
"Indeed, since $k_3(n,4) \\approx \\frac{n^3}{27}$ and each $K_4$ contains four triangles, $k_3(n,4)$ triangles and copies of $K_4$ can cover at most $\\approx \\frac{4}{27}n^3 < \\frac{n^3}{6} \\approx \\binom{n}{3}$ triangles, which is the number of triangles in $K_n$ , for $n$ sufficiently large.",
"It can be easily verified that this conclusion holds for $n \\ge 18$ .",
"More generally, as $k_t(n,t+1)\\approx \\frac{n^t}{t^t}$ and $(t+1)\\frac{n^t}{t^t} < \\frac{n^t}{t!}",
"\\approx \\binom{n}{t}$ for any fixed $t \\ge 3$ and sufficiently large $n$ , we cannot cover all $K_t$ 's in $K_n$ by only $k_t(n,t+1)$ $K_t$ 's and copies of $K_{t+1}$ .",
"Note also that the key lemma (Lemma 4.3) in the proof of Lehel [14] fails if we try to adapt it to the setting of graphs and triangles.",
"The lemma states that for any $2 \\le t < p$ , every $t$ -uniform hypergraph containing $m$ edges has a $K_p$ -free edge subset of cardinality at least $m/2$ .",
"However, its graph-and-triangle version, which would state that every graph containing $m$ triangles has a $K_4$ -free subset of triangles of cardinality at least $m/2$ , is no longer correct.",
"A counterexample is the graph $K_5$ , which contains exactly ten triangles, where we cannot find any subset of five triangles that does not include a $K_4$ .",
"Indeed, as established by Moon and Moser [19] and Sauer [23], the Turán graph $T(5,3)$ is the $K_4$ -free graph that contains the largest number of triangles, which is only four.",
"Thus, a direct adaptation of Lehel's arguments to our setting does not imply our result."
],
[
"Algorithmic Considerations",
"Scheinerman and Trenk [24] developed an algorithm which computes the edge clique cover number of a chordal graph $G$ .",
"Our primary goal in this section is to generalize their algorithm to the context of $K_t$ clique covers; however, we will also generalize the algorithm in two other respects.",
"These generalizations may be of interest even for the original edge clique cover problem.",
"Our first generalization is to consider a weighted version of the edge clique cover problem in which each $t$ -clique $k$ has an integer weight $w(k)$ specifying the number of times the clique needs to be covered.",
"While we are primarily concerned with the unweighted version of the problem (equivalently, the case where all $t$ -cliques have weight 1), the most natural recursive formulation of even the unweighted version of the algorithm involves passing to subproblems in which some $t$ -cliques no longer need to be covered, which is equivalent to giving those cliques weight 0.",
"(In fact, this distinction already appears in the original formulation of Scheinerman and Trenk [24], in which edges are labeled as “covered” as the algorithm proceeds.)",
"Since weighted subproblems arise naturally even when solving the unweighted problem, we formulate the algorithm for the weighted problem from the outset.",
"Our second generalization is to observe that the Scheinerman–Trenk algorithm works on a slightly more general class of graphs than the chordal graphs, which we dub semichordal graphs, and so in the interest of generality we state the $K_t$ clique cover version of the algorithm in terms of semichordal graphs rather than chordal graphs.",
"The remainder of this section consists of three subsections.",
"In the first subsection, we define the class of semichordal graphs and discuss some of their properties.",
"In the second subsection, we give our generalization of the Scheinerman–Trenk algorithm and prove its correctness.",
"In the third subsection, we prove that the $K_t$ clique cover problem on general graphs is NP-hard, justifying the development of algorithms on specialized graph classes."
],
[
"A Superclass of Chordal Graphs",
"A graph is said to be chordal if it has no induced cycle of length greater than 3.",
"A well-known characterization of chordal graphs, due to Dirac [8], is that they are the graphs which admit a simplicial elimination ordering: an ordering $v_1,\\ldots , v_n$ of the vertices of $G$ such that $N(v_i) \\cap \\lbrace v_{i+1},\\ldots , v_n\\rbrace $ is a clique for each $i$ .",
"The notion of a perfect elimination order admits a natural generalization, as follows.",
"Definition 3 (Aboulker–Charbit–Trotignon–Vušković [1]) Let $\\mathcal {F}$ be a set of graphs.",
"An $\\mathcal {F}$ -elimination ordering of a graph $G$ is an ordering $v_1, \\ldots , v_n$ of the vertices of $G$ such that for each $i$ , the induced subgraph $G[N(v_i) \\cap \\lbrace v_{i+1}, \\ldots , v_n\\rbrace ]$ has no induced subgraph isomorphic to a graph in $\\mathcal {F}$ .",
"Thus, Dirac's result states that the chordal graphs are precisely the graphs that admit a $\\lbrace \\overline{K_2}\\rbrace $ -elimination ordering.",
"In Section REF , we give an algorithm for computing weighted $K_t$ clique covers on a superclass of the chordal graphs, defined as follows.",
"Definition 4 A graph $G$ is semichordal if it admits a $\\lbrace P_3\\rbrace $ -elimination ordering, where $P_3$ is the path on three vertices.",
"Equivalently, a graph is semichordal if it admits a vertex ordering $v_1, \\ldots , v_n$ such that for each $i$ , the subgraph induced by $N(v_i) \\cap \\lbrace v_{i+1}, \\ldots , v_n\\rbrace $ is a disjoint union of complete graphs.",
"Since $\\overline{K_2}$ is an induced subgraph of $P_3$ , Dirac's characterization immediately implies that every chordal graph is semichordal.",
"On the other hand, any cycle $C_n$ for $n > 3$ is a semichordal graph that is not chordal.",
"As semichordal graphs are defined in terms of the existence of a certain elimination ordering, it would be desirable to have a characterization of these graphs in terms of their forbidden induced subgraphs, analogous to the definition of chordal graphs.",
"Unfortunately, we are aware of no such characterization.",
"The following sufficient (but not necessary) condition for a graph to be semichordal was discovered by Aboulker, Charbit, Trotignon, and Vušković [1].",
"Definition 5 ([1]) A graph $G$ is a wheel if there is a vertex $v$ of degree at least 3 such that $G-v$ is isomorphic to a cycle.",
"The vertex $v$ is the center of the wheel and the subgraph $G-v$ is the rim of the wheel.",
"A wheel is a 3-wheel if there are three consecutive vertices $x,y,z$ on the rim such that the center is adjacent to $x$ , $y$ , and $z$ .",
"Theorem 5 ([1]) If $G$ has no induced subgraph isomorphic to a 3-wheel, then $G$ is semichordal.",
"In fact, [1] proves that if $G$ has no induced subgraph isomorphic to a 3-wheel, then $G$ satisfies a stronger property guaranteeing that a $\\lbrace P_3\\rbrace $ -elimination ordering can be easily found.",
"We refer the reader to [1] for more details."
],
[
"Weighted Edges",
"In this section, we consider a weighted variant of the $K_t$ clique cover problem.",
"Given a graph $G$ , we assume that each $t$ -clique $k\\subset V(G)$ is assigned a weight $w(k)$ representing the number of times that $S$ must be covered.",
"Our goal is to find a multiset $\\mathcal {C}$ of cliques in $G$ such that each $t$ -clique $k$ is covered at least $w(k)$ times.",
"We formalize these notions as follows.",
"Definition 6 Given a graph $G$ and an integer $t \\ge 0$ , let $\\mathcal {K}(G)$ be the family of all cliques in $G$ , and let $\\mathcal {K}_t(G)$ be the family of all $t$ -cliques in $G$ .",
"Let $w : \\mathcal {K}_t \\rightarrow \\mathbb {Z}_{\\ge 0}$ be a weight function on the $t$ -cliques of $G$ .",
"A $(w,K_t)$ -cover of $G$ is a function $f : \\mathcal {K}(G) \\rightarrow \\mathbb {Z}_{\\ge 0}$ such that $\\sum _{k \\subset K}f(K) \\ge w(k)$ for all $k \\in \\mathcal {K}_t(G)$ , where the sum ranges over all $K \\in \\mathcal {K}(G)$ with $k \\subset K$ .",
"When $f$ is a $(w,K_t)$ -cover, we write $\\operatorname{cost}(f)$ for the sum $\\sum _{K \\in \\mathcal {K}(G)}f(K)$ .",
"The $(w,K_t)$ -cover number of $G$ , written $i_{w,t}(G)$ , is the minimum value of $\\operatorname{cost}(f)$ over all $(w,K_t)$ -covers of $G$ .",
"Observe that when $w(S) = 1$ for all $S \\in \\mathcal {K}_t$ , the $(w,K_t)$ -cover number of $G$ is just the $K_t$ clique cover number of $G$ .",
"We also define a corresponding dual problem.",
"Definition 7 The $(w,K_t)$ -clique packing number of $G$ , written $p_{w,t}(G)$ , is the optimum value of the following integer program: $\\text{maximize}\\sum _{k \\in \\mathcal {K}_t(G)}w(k)y(k),& \\text{ subject to} \\\\\\sum _{k \\subset K}y(k) &\\le 1, \\text{ for all $K \\in \\mathcal {K}(G)$}, \\\\y(k) &\\ge 0, \\text{for all $k \\in \\mathcal {K}_t(G)$} \\\\y(k) &\\in \\mathbb {Z}.$ A feasible solution to this integer program is called a $(w,K_t)$ -packing.",
"When $y$ is a $(w,K_t)$ -packing write $\\operatorname{val}(y)$ for $\\sum _{k \\in \\mathcal {K}_t(G)}w(k)y(k)$ .",
"(In some circumstances it may be ambiguous which weight function is used to calculate $\\operatorname{val}(y)$ , in which case we write $\\operatorname{val}_w(y)$ to specify the weight function being used.)",
"Let $i_{w,t}^*(G)$ and $p_{w,t}^*(G)$ denote the fractional relaxations of $i_{w,t}(G)$ and $p_{w,t}(G)$ , respectively.",
"Standard LP duality gives $ p_{w,t}(G) \\le p_{w,t}^*(G) = i_{w,t}^*(G) \\le i_{w,t}(G).",
"$ We wish to show that when $G$ is semichordal, equality holds throughout.",
"Recursive algorithm $\\textsf {optpair}$ to produce a pair $(f,y)$ , where $f$ is an optimal $(w,K_t)$ -cover and $y$ is an optimal $(w,K_t)$ -packing.",
"$G$ has no edges Return $(e, e)$ , where $e$ is the empty function.",
"Let $v_1, \\ldots , v_n$ be a $\\lbrace P_3\\rbrace $ -elimination ordering of $G$ .",
"Let $Q_1, \\ldots , Q_h$ be the components of $G[N(v_1)]$ of size at least $t-1$ .",
"Each $Q_i$ is a clique.",
"Let $G^{\\prime } = G-v_1$ .",
"$i \\in \\lbrace 1, \\ldots h\\rbrace $ Let $Q^*_i = \\lbrace v_1\\rbrace \\cup Q_i$ .",
"Pick $Z_i \\in \\mathcal {K}_{t-1}(Q_i)$ to maximize $w(\\lbrace v_1\\rbrace \\cup Z_i)$ and let $t_i = w(\\lbrace v_1\\rbrace \\cup Z_i)$ .",
"Let $\\mathcal {L}= \\mathcal {K}_t(Q_1) \\cup \\cdots \\cup \\mathcal {K}_t(Q_h)$ .",
"Let $w^{\\prime }(k) = w(k)$ for $k \\in \\mathcal {K}_t(G^{\\prime }) - \\mathcal {L}$ and let $w^{\\prime }(k) = \\max \\lbrace 0, w(k)-t_i\\rbrace $ for $k \\in \\mathcal {K}_t(Q_i)$ .",
"Let $(f^{\\prime }, y^{\\prime }) = \\textsf {optpair}(G^{\\prime }, w^{\\prime })$ .",
"Let $f(K) = f^{\\prime }(K)$ for $K \\in \\mathcal {K}(G^{\\prime })$ .",
"Let $y(k) = y^{\\prime }(k)$ for $k \\in \\mathcal {K}_t(G^{\\prime })$ .",
"$f$ and $y$ are only partially defined so far $i \\in \\lbrace 1, \\ldots , h\\rbrace $ Let $f(Q^*_i) = t_i$ .",
"Let $y(\\lbrace v_1\\rbrace \\cup Z) = 0$ for all $Z \\in \\mathcal {K}_{t-1}(Q_i) - Z_i$ .",
"$t_i=0$ or $y^{\\prime }(k) > 0$ for some $k \\in \\mathcal {K}_t(Q_i)$ Let $y(\\lbrace v_1\\rbrace \\cup Z_i) = 0$ .",
"Let $y(\\lbrace v_1\\rbrace \\cup Z_i) = 1$ .",
"Let $f(Q) = 0$ for all cliques $Q$ on which $f$ is not yet defined.",
"Return $(f,y)$ .",
"Theorem 6 If $G$ is semichordal, then for all $t \\ge 1$ and all $w : \\mathcal {K}_t(G) \\rightarrow \\mathbb {Z}_{\\ge 0}$ , $ i_{w,t}(G) = p_{w,t}(G).",
"$ We adapt the argument of Scheinerman and Trenk [24].",
"We claim that Algorithm REF produces a $(w,K_t)$ -cover $f$ and a $(w,K_t)$ -packing $y$ such that $\\operatorname{cost}(f) = \\operatorname{val}(y)$ , and such that $y(k) > 0$ only if $w(k) > 0$ .",
"Our proof proceeds by induction on $\\left|{E(G)}\\right|$ .",
"When $\\left|{E(G)}\\right| = 0$ it is clear that the pair of empty functions $(e,e)$ returned by Algorithm REF has the desired properties.",
"Now suppose that $\\left|{E(G)}\\right| > 0$ , let $v_1, \\ldots , v_n$ be the $\\lbrace P_3\\rbrace $ -elimination ordering used in Algorithm REF , and let $(f^{\\prime }, y^{\\prime }) = \\textsf {optpair}(G^{\\prime }, w^{\\prime })$ .",
"By the induction hypothesis, $f^{\\prime }$ and $y^{\\prime }$ are feasible for their respective integer programs, and $\\operatorname{cost}(f^{\\prime }) = \\operatorname{val}(y^{\\prime })$ .",
"First we argue that $f$ is feasible.",
"First observe that $\\mathcal {K}_t(G) = (\\mathcal {K}_t(G^{\\prime }) - \\mathcal {L}) \\cup (\\mathcal {K}_t(Q_1^*) \\cup \\cdots \\cup \\mathcal {K}_t(Q_h^*))$ , and that $f(K) = f^{\\prime }(K)$ for all $k \\in \\mathcal {K}(G^{\\prime })$ .",
"For all $k \\in \\mathcal {K}_t(G^{\\prime }) - \\mathcal {L}$ , we have $w^{\\prime }(k) = w(k)$ , and the feasibility of $f^{\\prime }$ implies that $ \\sum _{\\begin{array}{c}k \\subset K\\\\ K \\in \\mathcal {K}(G)\\end{array}}f(K) \\ge \\sum _{\\begin{array}{c}k \\subset K \\\\ K \\in \\mathcal {K}(G^{\\prime })\\end{array}}f^{\\prime }(K) \\ge w^{\\prime }(k) =w(k) $ for all $k \\in \\mathcal {K}_t(G^{\\prime })- \\mathcal {L}$ .",
"On the other hand, for $k \\in \\mathcal {K}_t(Q_i)$ , we have $w(k) \\le w^{\\prime }(k) + t_i$ .",
"Since $f^{\\prime }$ is feasible and $f(Q^*_i) = t_i$ , for these $t$ -cliques we have $ \\sum _{\\begin{array}{c}k \\subset K\\\\ K \\in \\mathcal {K}(G)\\end{array}}f(K) = f(Q^*_i) + \\sum _{\\begin{array}{c}k \\subset K\\\\ K \\in \\mathcal {K}(G^{\\prime })\\end{array}}f^{\\prime }(K) \\ge t + w^{\\prime }(k) \\ge w(k).",
"$ The remaining $t$ -cliques to consider are those in $\\mathcal {K}_t(Q^*_i) \\setminus \\mathcal {K}_t(Q_i)$ for some $i$ , that is, the $t$ -cliques containing $v_1$ .",
"Any such $t$ -clique $k$ is of the form $k = \\lbrace v_1\\rbrace \\cup Z$ for some $Z \\in \\mathcal {K}_{t-1}(Q_i)$ , and is contained in the clique $Q^*_i$ .",
"Since $f(Q^*_i) = t_i = \\max _{Z}(w(\\lbrace v_1\\rbrace \\cup Z))$ , where the maximum is taken over all $Z \\in \\mathcal {K}_{t-1}(Q_i)$ , we see that for such $k$ , $ \\sum _{\\begin{array}{c}k \\subset K\\\\ K \\in \\mathcal {K}(G)\\end{array}} f(K) \\ge f(Q^*_i) \\ge w(k), $ and so $f$ is feasible.",
"Now we argue that $y$ is feasible.",
"Consider any clique $K \\in \\mathcal {K}(G)$ .",
"If $v_1 \\notin K$ , then $K \\in \\mathcal {K}(G^{\\prime })$ and so $y(k) = y^{\\prime }(k)$ for all $t$ -cliques $k \\subset K$ , so by the induction hypothesis, we have $ \\sum _{k \\subset K} y(k) = \\sum _{k \\subset K} y^{\\prime }(k) \\le 1.",
"$ Thus, we may assume that $v_1 \\in K$ .",
"This implies that $K \\subset Q^*_i$ for some $i$ .",
"Observe that $\\sum _{k \\subset K} y(k) &\\le \\sum _{Z \\in \\mathcal {K}_{t-1}(Q_i)}y(\\lbrace v_1\\rbrace \\cup Z) + \\sum _{k \\in \\mathcal {K}_t(Q_i)}y(k) \\\\&= y(\\lbrace v\\rbrace \\cup Z_i) + \\sum _{k \\in \\mathcal {K}_t(Q_i)}y^{\\prime }(k),$ where $\\sum _{k \\in \\mathcal {K}_t(Q_i)}y^{\\prime }(k) \\le 1$ by the feasibility of $y^{\\prime }$ .",
"Thus, if $y(\\lbrace v_1\\rbrace \\cup Z_i) = 0$ , then the constraint for $K$ is satisfied.",
"The only way the algorithm allows $y(\\lbrace v_1\\rbrace \\cup Z_i) > 0$ is when $y^{\\prime }(k) = 0$ for all $k \\in \\mathcal {K}_t(Q_i)$ , in which case the constraint is again satisfied.",
"Next we argue that $y(k) > 0$ only if $w(k) > 0$ .",
"Consider any $k \\in \\mathcal {K}_t(G)$ with $y(k) > 0$ .",
"If $v_1 \\notin k$ , then $k \\in \\mathcal {K}_t(G^{\\prime })$ , so the induction hypothesis implies that $w^{\\prime }(k) > 0$ .",
"Since $w^{\\prime }(k) \\le w(k)$ , this implies that $w(k) > 0$ as well.",
"On the other hand, if $v_1 \\in k$ , then $y(k) > 0$ is only possible if $k = \\lbrace v_1\\rbrace \\cup Z_i$ for some $i$ with $t_i >0$ .",
"Since $t_i = w(\\lbrace v_1\\rbrace \\cup Z_i) = w(k)$ , we again see that $y(k) > 0$ implies $w(k) > 0$ .",
"Finally we argue that $\\operatorname{cost}(f) = \\operatorname{val}(y)$ .",
"Let $R$ be the set the of indices $i$ such that $y(\\lbrace v_1\\rbrace \\cup Z_i) = 1$ .",
"Observe that $\\operatorname{cost}(f) = \\operatorname{cost}(f^{\\prime }) + \\sum _{i \\in R}f(Q_i^*) = \\operatorname{cost}(f^{\\prime }) + \\sum _{i=1}^ht_i.$ By the induction hypothesis, $\\operatorname{cost}(f^{\\prime }) = \\operatorname{val}_{w^{\\prime }}(y^{\\prime })$ .",
"We wish to determine $\\operatorname{val}_{w}(y^{\\prime })$ .",
"Observe that $ \\operatorname{val}_{w}(y^{\\prime }) - \\operatorname{val}_{w^{\\prime }}(y^{\\prime }) = \\sum _{y^{\\prime }(k) = 1}[ w(k) - w^{\\prime }(k) ], $ and by the induction hypothesis, $y^{\\prime }(k) = 1$ implies that $w^{\\prime }(k) > 0$ , so that $w^{\\prime }(k) = w(k) - t_i$ .",
"Hence, $ \\operatorname{val}_{w}(y^{\\prime }) - \\operatorname{val}_{w^{\\prime }}(y^{\\prime }) = \\sum _{y^{\\prime }(k) = 1}t_i = \\sum _{i \\notin R} t_i, $ where the last equality holds because $i \\notin R$ implies that either $t_i = 0$ or that $y^{\\prime }(k) > 0$ for some $k \\in \\mathcal {K}_t(Q_i)$ , in which case feasibility of $y^{\\prime }$ implies that there is exactly one $k$ for which this is true.",
"The remaining cliques to count in $\\operatorname{val}(y)$ are the cliques $\\lbrace v_1\\rbrace \\cup Z_i$ where $i \\in R$ .",
"Thus, $ \\operatorname{val}(y) = \\operatorname{val}_{w}(y^{\\prime }) + \\sum _{i \\in R} t_i = \\operatorname{val}_{w^{\\prime }}(y^{\\prime }) + \\sum _{i=1}^h t_i= \\operatorname{cost}(f^{\\prime }) + \\sum _{i=1}^h t_i = \\operatorname{cost}(f), $ as desired."
],
[
"NP-hardness of $K_t$ clique cover problem on general graphs",
"In this section, we will prove that for any fixed $t \\ge 1$ , the $K_t$ clique cover problem is NP-hard.",
"This justifies developing algorithms to solve this problem on specialized graph classes such as the semichordal graphs, since (unless $P=NP$ ) there can be no polynomial-time algorithm to solve the problem on general graphs.",
"Kou, Stockmeyer, and Wong [13] proved that the edge clique cover problem, i.e., the $K_2$ clique cover problem, is NP-hard, via a reduction from the $K_1$ clique cover problem, which is equivalent to vertex coloring in the complementary graph and thus NP-hard.",
"We generalize their approach, reducing the $K_{t-1}$ clique cover problem to the $K_t$ clique cover problem for each $t \\ge 2$ , which implies that each of these problems is NP-hard.",
"We formalize that $K_t$ clique cover problem as a decision problem as follows: Table: NO_CAPTIONProposition 1 The decision problem $\\operatorname{KCC}(t)$ is NP-complete for any constant $t \\ge 1$ .",
"We adapt the proof of Kou, Stockmeyer, and Wong.",
"It is obvious that $\\operatorname{KCC}(t)$ is in NP.",
"We prove the NP-completeness of this problem by induction on $t$ .",
"It is known that $\\operatorname{KCC}(1)$ is NP-complete [12].",
"Now suppose that $t \\ge 2$ and that $\\operatorname{KCC}(t-1)$ is NP-complete.",
"We aim to show that $\\operatorname{KCC}(t)$ is also NP-complete.",
"Let $G$ be an arbitrary graph of order $n$ and $k \\ge 1$ , and (mirroring the notation of Kou–Stockmeyer–Wong for the case $t=2$ ) let $e = \\left|{\\mathcal {K}_t(G)}\\right|$ .",
"Let $G^{\\prime }$ be the graph obtained from $G$ by introducing $s = 1 + e$ new vertices $\\lbrace u_1,\\ldots , u_s\\rbrace $ , and $sn$ new edges that connect the new vertices to all existing vertices of $G$ .",
"Since there are at most ${n \\atopwithdelims ()t}$ possible $t$ -cliques in $\\mathcal {K}_t(G)$ , it is clear that this construction can be carried out in polynomial time.",
"(Since $t$ is fixed in the decision problem $\\operatorname{KCC}(t)$ , it does not matter that the degree of the polynomial depends on $t$ .)",
"Let $k^{\\prime } = sk + e$ .",
"We demonstrate that $\\theta _{K_{t-1}}(G) \\le k$ if and only if ${\\theta _{K_t}}(G^{\\prime }) \\le k^{\\prime }$ .",
"We first claim that if $\\theta _{K_{t-1}}(G) \\le k$ , then ${\\theta _{K_t}}(G^{\\prime }) \\le k^{\\prime }$ .",
"Suppose that ${\\mathcal {A}}$ is a $K_{t-1}$ clique cover in $G$ , with $\\left|{{\\mathcal {A}}}\\right| \\le k$ .",
"For each $i \\in \\lbrace 1, \\ldots , s\\rbrace $ , let ${\\mathcal {B}}_i = \\lbrace u_i \\cup S \\colon \\,S \\in {\\mathcal {A}}\\rbrace $ , and let ${\\mathcal {B}}= {\\mathcal {B}}_1 \\cup \\cdots \\cup {\\mathcal {B}}_s$ .",
"Now ${\\mathcal {B}}$ covers every $t$ -clique in $G^{\\prime }$ except perhaps for some $t$ -cliques totally contained in $G$ .",
"Adding each such clique to ${\\mathcal {B}}$ separately yields a $K_t$ clique cover in $G^{\\prime }$ having at most $sk + e$ cliques, so that ${\\theta _{K_t}}(G^{\\prime }) \\le sk+e = k^{\\prime }$ .",
"It remains to show that if ${\\theta _{K_t}}(G^{\\prime }) \\le k^{\\prime }$ , then $\\theta _{K_{t-1}}(G) \\le k$ .",
"Suppose that $ is a $ Kt$ cliquecover in $ G'$ with $ k'$.",
"For each$ i {1, ..., s}$, let $ i$ be the subset of cliques in $ that contain the vertex $u_i$ .",
"Observe that for $i \\ne j$ , the vertices $u_i$ and $u_j$ are not adjacent, so that $i$ and $j$ are disjoint.",
"Hence, $\\sum _{i = 1}^s |i| \\le | \\le k^{\\prime }.$ Therefore, if $i_{\\min }$ is an index such that $|{i_{\\min }}| = \\min _{1 \\le i \\le s} |i|$ , then $|{i_{\\min }}| \\le \\left\\lfloor \\dfrac{\\sum _{i = 1}^s |i|}{s} \\right\\rfloor \\le \\left\\lfloor \\dfrac{k^{\\prime }}{s}\\right\\rfloor =\\left\\lfloor \\dfrac{sk + e}{s} \\right\\rfloor = k,$ where the last equality holds because $s = 1 + e$ .",
"Then, by removing $u_{i_{\\min }}$ from all cliques in ${i_{\\min }}$ , we obtain a $K_{t-1}$ clique cover of $G$ of size at most $k$ .",
"The proof follows.",
"We note briefly that Kou, Stockmeyer, and Wong [13] actually proved a stronger property than NP-completeness of the edge clique cover problem: they used a result of Garey and Johnson [10] on the inapproximability of the $K_1$ clique cover problem to prove that if $P \\ne NP$ , then there is no $c$ -approximation algorithm for the edge clique cover problem for any $c < 2$ .",
"We believe that this inapproximability proof extends to the $K_t$ clique cover problem by making the same modifications we made to the NP-hardness proof, but in the interest of simplicity, we have chosen only to present the NP-hardness version of the proof."
],
[
"Acknowledgment",
"This work was supported by the NSF grants 1527636 and 1527636.",
"We thank the anonymous referees for their careful reading and their helpful comments which improved the presentation of the paper."
]
] | 1709.01590 |
[
[
"Hidden edge Dirac point and robust quantum edge transport in InAs/GaSb\n quantum wells"
],
[
"Abstract The robustness of quantum edge transport in InAs/GaSb quantum wells in the presence of magnetic fields raises an issue on the fate of topological phases of matter under time-reversal symmetry breaking.",
"A peculiar band structure evolution in InAs/GaSb quantum wells is revealed: the electron subbands cross the heavy hole subbands but anticross the light hole subbands.",
"The topologically protected band crossing point (Dirac point) of the helical edge states is pulled to be close to and even buried in the bulk valence bands when the system is in a deeply inverted regime, which is attributed to the existence of the light hole subbands.",
"A sizable Zeeman energy gap verified by the effective g-factors of edge states opens at the Dirac point by an in-plane or perpendicular magnetic field, however it can also be hidden in the bulk valance bands.",
"This provides a plausible explanation for the recent observation on the robustness of quantum edge transport in InAs/GaSb quantum wells subjected to strong magnetic fields."
],
[
"Introduction",
"The quantum spin Hall (QSH) insulator is a quantum state of matter with topologically protected helical edge states in the bulk insulating gap [1], [2], [3].",
"The helical edge states will give rise to the QSH effect which is featured by a quantized conductance (i.e., $2e^{2}/h$ ) in the two-terminal measurement at low temperatures [4].",
"Theoretically the band crossing point (Dirac point) of the helical edge states is topologically protected by time-reversal symmetry, and it opens a minigap once the symmetry is broken (if there is no other extra symmetry protection).",
"The QSH insulator has been predicted theoretically [5] and confirmed experimentally in HgTe/CdTe quantum wells [6], [7].",
"Another promising candidate for QSH insulator is the InAs/GaSb double quantum well [8], [9].",
"The InAs/GaSb quantum wells possess a particular electronic phase with inverted band structure, in which the hybridization of electrons and holes opens a minigap at finite $k$ -vectors, leading to the QSH phase.",
"Due to the mature technology of material fabrications and potential device applications, there have been growing efforts exploring the QSH phase in InAs/GaSb quantum wells [10], [9], [11], [12], [13], [14], [15], [16], [17].",
"Recently, it was observed that the conductance in InAs/GaSb quantum wells can keep quantized in an in-plane magnetic field up to 12 T and is insensitive to temperatures ranging from 250 mK to several Kelvins [18].",
"Similar feature was also observed in HgTe/CdTe quantum wells [19].",
"This raises a question about the fate of the QSH effect under time-reversal symmetry breaking, which has become a fundamental issue to understand the physics of topological matter.",
"A number of theoretical efforts have been simulated on this puzzle [20], [21], [22].",
"However the robustness of the quantized conductance remains poorly understood.",
"In InAs/GaSb quantum wells, the lowest conduction bands of InAs are about 150 meV lower than the highest valence bands of GaSb [23], [24], which forms a broken-gap band alignment and leads to the coexistence of electrons and holes near the charge neutrality point.",
"The application of gate voltages can shift the band alignment and drive the system to different electronic phases [25], [8], [14].",
"When the (lowest) electron subbands of InAs lie above the (highest) heavy hole (HH) subbands of GaSb, the system is in a normal insulator phase.",
"Whereas the electron subbands lie below the HH subbands, the system is in an inverted phase and the QSH effect is expected in the hybridization gap opened by coupling between electron and hole states.",
"Around the topological phase transition point, the system can be well described by Bernevig-Hughes-Zhang (BHZ) model which considers four bands in the lowest energy [5], [8].",
"The BHZ model, however, fails to explain the robust quantum edge transport in InAs/GaSb quantum wells in the presence of in-plane magnetic fields, in which the Dirac point of the helical edge states opens an mini-gap, leading to the breakdown of quantized conductance.",
"InAs/GaSb quantum wells could possibly be in a deeply inverted regime where the lower energy subbands, e.g., the light hole (LH) subbands, will reside above the electron subbands and may have important influence on the system.",
"The consideration of the LH subbands may be a resolution to the puzzle.",
"To this end, re-examination of the band structure of InAs/GaSb quantum wells and a more comprehensive effective model are needed.",
"In the present work, a peculiar band structure evolution in InAs/GaSb quantum wells is revealed when varying the gate voltages.",
"The electron subbands of InAs can cross the HH subbands of GaSb, and correspondingly the system transits between a trivial insulator phase and a topological insulator phase as described by the BHZ model.",
"In contrast, the electron subbands cannot touch but anticross the LH subbands of GaSb.",
"This anticrossing behavior does not alter the topology of the system as no gap closing occurs, however, it may modify the properties of system near the hybridization gap significantly.",
"We present a six-band effective model to capture the essential low-energy properties of InAs/GaSb quantum wells, including the topological phase transition and anticrossing behavior.",
"One of the key features is that the Dirac point of the edge states will be pulled to be close to the bulk valence bands when the electron subbands are lowered to anticross the LH subbands.",
"The application of a magnetic field, in-plane or perpendicular, opens a sizable Zeeman energy gap at the Dirac point of the helical edge states, which indicates the breaking down of the QSH effect.",
"Nevertheless, the energy gap of edge states could also be hidden in the bulk valence bands up to a large magnetic field, which may account for recent experimental observation on the robustness of quantum edge transport under in-plane magnetic fields [18].",
"We anticipate our results can shed some light on experimental observations on the InAs/GaSb quantum wells and explore novel topological phases of matter in the future.",
"The rest of this paper is organized as follows.",
"In Sec.",
"II the band structure evolution of InAs/GaSb quantum wells is studied, and in Sec.",
"III a six-band effective model is derived for low-energy physics of the quantum wells.",
"With the effective model, the properties of edge states are investigated in Sec.",
"IV.",
"To characterize the response of the helical edge states to magnetic fields, the effective g-factors of edge states are calculated in Sec.",
"V. In Sec.",
"VI the robustness of quantum edge transport under in-plane magnetic fields is addressed by the numerical calculation of conductance.",
"Finally, Sec.",
"VII contains the discussions and conclusions."
],
[
"Band structure evolution of $\\mathrm {InAs/GaSb}$ quantum wells ",
"Both InAs and GaSb have zinc-blende crystal structure and direct gaps near the $\\Gamma $ point, and their low-energy physics can be well described by the Kane model [26], [27].",
"Considering the broken-gap band alignment in InAs/GaSb quantum wells and focusing on the case where the $\\Gamma ^{6}$ bands of InAs and the $\\Gamma ^{8}$ bands of GaSb are very close while the $\\Gamma ^{7}$ bands are far away in energy and thus can be neglected here.",
"In the basis $\\lbrace |\\Gamma ^{6},1/2\\rangle ,|\\Gamma ^{6},-1/2\\rangle ,|\\Gamma ^{8},3/2\\rangle ,|\\Gamma ^{8},1/2\\rangle ,|\\Gamma ^{8},-1/2\\rangle ,$ $|\\Gamma ^{8},-3/2\\rangle \\rbrace $ (Here we use the standard notation that $|\\Gamma ^{6},\\pm 1/2\\rangle ,$ $|\\Gamma ^{8},\\pm 1/2\\rangle $ and $|\\Gamma ^{8},\\pm 3/2\\rangle $ represent the s-like conduction bands, the p-like LH bands, and the p-like HH bands, respectively), the Kane Hamiltonian for the [001] growth direction is given by [27], [28] $H=\\left(\\begin{array}{cccccc}T & 0 & -\\frac{1}{\\sqrt{2}}Pk_{+} & \\sqrt{\\frac{2}{3}}Pk_{z} & \\frac{1}{\\sqrt{6}}Pk_{-} & 0\\\\0 & T & 0 & -\\frac{1}{\\sqrt{6}}Pk_{+} & \\sqrt{\\frac{2}{3}}Pk_{z} & \\frac{1}{\\sqrt{2}}Pk_{-}\\\\-\\frac{1}{\\sqrt{2}}k_{-}P & 0 & U+V & -\\bar{S}_{-} & R & 0\\\\\\sqrt{\\frac{2}{3}}k_{z}P & -\\frac{1}{\\sqrt{6}}k_{-}P & -\\bar{S}_{-}^{\\dagger } & U-V & C & R\\\\\\frac{1}{\\sqrt{6}}k_{+}P & \\sqrt{\\frac{2}{3}}k_{z}P & R^{\\dagger } & C^{\\dagger } & U-V & \\bar{S}_{+}^{\\dagger }\\\\0 & \\frac{1}{\\sqrt{2}}k_{+}P & 0 & R^{\\dagger } & \\bar{S}_{+} & U+V\\end{array}\\right),$ where $T & =E_{c}+h^{\\prime }(\\gamma _{0}k_{||}^{2}+k_{z}\\gamma _{0}k_{z}),\\nonumber \\\\U & =E_{v}-h^{\\prime }(\\gamma _{1}k_{||}^{2}+k_{z}\\gamma _{1}k_{z}),\\nonumber \\\\V & =-h^{\\prime }(\\gamma _{2}k_{||}^{2}-2k_{z}\\gamma _{2}k_{z}),\\nonumber \\\\R & =\\sqrt{3}h^{\\prime }\\gamma _{2}(k_{x}^{2}-k_{y}^{2})-2\\sqrt{3}ih^{\\prime }\\gamma _{3}k_{x}k_{y},\\nonumber \\\\\\bar{S}_{\\pm } & =-\\sqrt{3}h^{\\prime }k_{\\pm }\\left(\\lbrace \\gamma _{3},k_{z}\\rbrace +[\\kappa ,k_{z}]\\right),\\nonumber \\\\C & =2h^{\\prime }k_{-}[\\kappa ,k_{z}],$ in which ${\\bf k}_{\\parallel }=(k_{x},k_{y})$ , $k_{||}^{2}=k_{x}^{2}+k_{y}^{2}$ , $k_{\\pm }=k_{x}\\pm ik_{y}$ , and $h^{\\prime }=\\hbar ^{2}/(2m_{0})$ .",
"$m_{0}$ is the free electron mass, and $P$ is the Kane momentum matrix element.",
"$E_{c}$ and $E_{v}$ are the conduction and valence band edges, respectively.",
"$\\gamma _{0,1,2,3}$ and $\\kappa $ are the band parameters in the Kane model.",
"The parameters for InAs, GaSb and AlSb are given in Table REF .",
"We consider the quantum well configuration with InAs and GaSb layers sandwiched by two AlSb layers at each side along the growth direction (the $z$ -direction).",
"Hence the parameters of the Kane model are spatial dependent, corresponding to different layers of the quantum wells.",
"To simulate the experimental setup and for illustration, we will take 12.5 nm InAs/10 nm GaSb with barriers made of 50 nm AlSb at each side in the quantum well system [18].",
"Table: Parameters in the Kane model for InAs, GaSb and AlSb , , .Table: Parameters in the six-band effective model for V 0 V_{0}=-100meV, L InAs =12.5L_{\\text{InAs}}=12.5nm, L GaSb =10L_{\\text{GaSb}}=10 nm and L AlSb =50L_{\\text{AlSb}}=50 nm.We assume the confinement effect in the $z$ -direction and replace the operator $k_{z}$ with $-i\\partial _{z}$ in the Hamiltonian.",
"The full Hamiltonian of the quantum wells takes the form $H_{\\text{full}}=H_{K}(k_{x},k_{y},-i\\partial _{z})+V(z).$ Here $V(z)$ is the confinement potential and is also spatial dependent.",
"The subbands dispersions and corresponding eigenstates are obtained by solving the Schrödinger equation: $H_{\\text{full}}|\\Psi ^{\\xi }(k_{x},k_{y},z)\\rangle =E^{\\xi }|\\Psi ^{\\xi }(k_{x},k_{y},z)\\rangle ,$ where $\\xi $ is the subband index, and $|\\Psi ^{\\xi }(k_{x},k_{y},z)\\rangle =\\exp [ik_{x}x+ik_{y}y]F^{\\xi }(z)$ with $F^{\\xi }(z)$ an envelope function.",
"The envelope function approximation can be employed to solve the eigen problem of the quantum wells [31].",
"$F^{\\xi }(z)$ can be expanded in terms of plane waves $F^{\\xi }(z)=\\sum _{\\lambda =1}^{6}\\sum _{n=-N}^{N}\\frac{1}{\\sqrt{L}}a_{n,\\lambda }^{\\xi }e^{ik_{n}z}|\\lambda \\rangle ,$ where $k_{n}=2\\pi n/L$ with $n=0,\\pm 1,\\pm 2,\\cdots ,\\pm N$ ($N$ is a positive integer), and $L=L_{\\mathrm {InAs}}+L_{\\mathrm {GaSb}}+2L_{\\mathrm {AlSb}}$ is the total width of InAs/GaSb quantum wells.",
"$a_{n,\\lambda }^{\\xi }$ are the corresponding expansion coefficients.",
"Here we use $|\\lambda \\rangle $ ($\\lambda $ = 1, 2, $\\cdots $ , 6) to denote the basis set of wave functions where $|1\\rangle $ and $|2\\rangle $ are for $|\\Gamma ^{6},\\pm 1/2\\rangle $ , $|3\\rangle $ and $|6\\rangle $ are for $|\\Gamma ^{8},\\pm 3/2\\rangle $ , and $|4\\rangle $ and $|5\\rangle $ are for $|\\Gamma ^{8},\\pm 1/2\\rangle $ .",
"For the numerical calculations, we take $N=30$ which is accurate enough for the low-energy physics.",
"Figure: The energy spectrum of InAs/GaSb quantum wells in different phases.",
"(a) The energies of the lowest-energy subbands at the Γ\\Gamma pointas functions of the broken gap V 0 V_{0}.",
"The band structures at k y =0k_{y}=0with (b) V 0 =-70V_{0}=-70 meV; (c) V 0 =-90V_{0}=-90 meV; (d) V 0 =-120V_{0}=-120meV.Different electronic phases can be realized by varying the broken gap $V_{0}$ , the energy difference of band edges between the $\\Gamma ^{6}$ bands of InAs and the $\\Gamma ^{8}$ bands of GaSb, which is supposed to be tunable by gate voltages [8], [14].",
"Figure REF (a) shows the energies of the lowest energy subbands at the $\\Gamma $ point as functions of $V_{0}$ .",
"One can see that when decreasing $V_{0}$ , the lowest electron ($E1$ ) subbands cross the highest HH ($HH1$ ) subbands, showing a topological phase transition.",
"For a large $V_{0}(>-80$ meV), the system is a trivial insulator as shown in Fig.",
"REF (b) and should not possess robust edge states, which is labeled as case (i).",
"For a smaller $V_{0}(<-80$ meV), the system transfers from the trivial insulating phase to a shallowly inverted phase labeled by case (ii).",
"A hybridization gap will open at the crossing point, as shown in Fig.",
"REF (c), and the QSH effect is expected [8].",
"The low-energy properties of the system near the phase transition point $V_{0}(\\sim -80$ meV) can be well described by the BHZ model [5].",
"Decreasing $V_{0}$ further, the $E1$ subbands does not touch but anticross the highest LH ($LH1$ ) subbands.",
"We label the deeply inverted phase after the anticrossing as case (iii).",
"The transition from cases (ii) to (iii) is topologically trivial since there is no gap closing, however, some important properties (e.g., the property of edge states) near the system gap are changed, as will be shown below.",
"The corresponding band structure for case (iii) is presented in Fig.",
"REF (d), which exhibits giant spin-orbit splitting close to the hybridization gap.",
"The spin-orbit splitting due to the structure inversion asymmetry may lead to fully spin polarized states [17]."
],
[
"Six-band effective model",
"The topologically non-trivial band structure indicates the existence of helical edge states across bulk insulating gap with the open boundaries according to the bulk-edge correspondence [32], [33], [34].",
"To find the helical edge states and investigate the low-energy properties of InAs/GaSb quantum wells, it is helpful to derive an effective model, just as the BHZ model [5].",
"Noting that without gate voltage the InAs/GaSb quantum wells tend to stay in the deeply inverted phase of case (iii), the $LH1$ subbands may have significant influence on the system and thus should also be considered.",
"A six-band effective model which involves the $E1$ , $HH1$ and $LH1$ subbands can be constructed, following a similar procedure of Refs.",
"[5], [35].",
"Generally the full bulk Hamiltonian can be split into two parts $H_{\\text{full}}=H_{0}({\\bf k}_{\\parallel }=0,-i\\partial _{z},z)+H^{\\prime }({\\bf k}_{\\parallel },-i\\partial _{z},z),$ where $H_{0}$ describes the system at the $\\Gamma $ point (i.e., ${\\bf k}_{\\parallel }=0)$ and $H^{\\prime }$ can be treated as a perturbation around the $\\Gamma $ point.",
"First, we can numerically solve the Schrödinger equation $H_{0}|\\Psi _{0}^{\\xi }\\rangle =E_{0}^{\\xi }|\\Psi _{0}^{\\xi }\\rangle $ and obtain the eigenenergies $E_{0}^{\\xi }$ and the corresponding eigenstates $|\\Psi _{0}^{\\xi }\\rangle $ .",
"The Hamiltonian $H_{0}$ is effectively decoupled to four blocks: the electron subbands couple only with the LH subbands, while the HH subbands decouple from them.",
"We can treat these decoupled blocks separately.",
"Three eigen wave functions with components of the $E1$ and $HH1$ bands, or of the $LH1$ subbands can be written as $\\langle z|E1,+\\rangle & =\\left(\\psi _{e1}(z),0,0,\\psi _{e4}(z),0,0\\right)^{T},\\\\\\langle z|HH1,+\\rangle & =\\left(0,0,\\psi _{h3}(z),0,0,0\\right)^{T},\\\\\\langle z|LH1,+\\rangle & =\\left(\\psi _{l1}(z),0,0,\\psi _{l4}(z),0,0\\right)^{T},$ where $T$ means transpose.",
"The envelope function components $\\psi _{e(h,l)}(z)$ can be found by expanding the eigenstates in terms of plane waves, as introduced previously.",
"Carrying out the time-reversal operation on the above wave functions, we have other three eigen wave functions $\\langle z|E1,-\\rangle & =\\left(0,\\psi _{e1}^{*}(z),0,0,-\\psi _{e4}^{*}(z),0\\right)^{T},\\\\\\langle z|HH1,-\\rangle & =\\left(0,0,0,0,0,\\psi _{h3}^{*}(z)\\right)^{T},\\\\\\langle z|LH1,-\\rangle & =\\left(0,-\\psi _{l1}^{*}(z),0,0,\\psi _{l4}^{*}(z),0\\right)^{T}.$ Next, with the six lowest energy states at the $\\Gamma $ point as a basis set, we can project the Hamiltonian (REF ) and obtain a two-dimensional six-band effective model.",
"In the ordered basis $\\left\\lbrace |E1,+\\rangle ,|E1,-\\rangle ,\\right.$$|HH1,+\\rangle ,$ $|LH1,+\\rangle ,\\left.|LH1,-\\rangle ,|HH1,-\\rangle \\right\\rbrace $ , the effective Hamiltonian reads $H & ({\\bf k}_{\\parallel })=H_{0}({\\bf k}_{\\parallel })+\\delta H,$ $H_{0}({\\bf k}_{\\parallel }) & =\\left(\\begin{array}{cccccc}T_{e} & -\\frac{P_{e}k_{-}}{\\sqrt{6}} & -\\frac{P_{eh}k_{+}}{\\sqrt{2}} & Dk^{2} & \\frac{P_{el}k_{-}}{\\sqrt{6}} & R_{eh}\\\\-\\frac{P_{e}k_{+}}{\\sqrt{6}} & T_{e} & -R_{eh}^{\\dagger } & -\\frac{P_{el}k_{+}}{\\sqrt{6}} & -Dk^{2} & \\frac{P_{eh}k_{-}}{\\sqrt{2}}\\\\-\\frac{P_{eh}k_{-}}{\\sqrt{2}} & -R_{eh} & T_{h} & -\\frac{P_{lh}k_{-}}{\\sqrt{2}} & R_{lh} & 0\\\\Dk^{2} & -\\frac{P_{el}k_{-}}{\\sqrt{6}} & -\\frac{P_{lh}k_{+}}{\\sqrt{2}} & T_{l} & \\frac{P_{l}k_{-}}{\\sqrt{6}} & R_{lh}\\\\\\frac{P_{el}k_{+}}{\\sqrt{6}} & -Dk^{2} & R_{lh}^{\\dagger } & \\frac{P_{l}k_{+}}{\\sqrt{6}} & T_{l} & -\\frac{P_{lh}k_{-}}{\\sqrt{2}}\\\\R_{eh}^{\\dagger } & \\frac{P_{eh}k_{+}}{\\sqrt{2}} & 0 & R_{lh}^{\\dagger } & -\\frac{P_{lh}k_{+}}{\\sqrt{2}} & T_{h}\\end{array}\\right),\\\\\\delta H & =\\Delta V\\left(\\begin{array}{cccccc}Q_{e} & 0 & 0 & Q_{el} & 0 & 0\\\\0 & Q_{e} & 0 & 0 & -Q_{el} & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\Q_{el} & 0 & 0 & Q_{l} & 0 & 0\\\\0 & -Q_{el} & 0 & 0 & Q_{l} & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right)$ where $k_{\\pm }=k_{x}\\pm ik_{y}$ , $T_{e(l,h)}=E_{e(l,h)}+B_{e(l,h)}k_{\\parallel }^{2}$ , and $R_{e(l)h}=\\sqrt{3}h^{\\prime }\\gamma _{2e(l)h}(k_{x}^{2}-k_{y}^{2})-i2\\sqrt{3}h^{\\prime }\\gamma {}_{3e(l)h}k_{x}k_{y}$ .",
"Here and after, we choose a fixed broken gap $V_{0}$ as reference and take $\\Delta V$ as a variation from $V_{0}$ to tune the band structure evolution for convenience.",
"$H_{0}({\\bf k}_{\\parallel })$ describes the system with the broken gap $V_{0}$ .",
"$\\delta H$ is the modification by $\\Delta V$ , the small change of the broken gap, it shows clearly how the whole band structure varies as tuning gate voltages.",
"The diagonal terms $Q_{e}$ and $Q_{l}$ in $\\Delta H$ will shift the position of $E1$ and $LH1$ subbands, as shown in Fig.",
"REF (a).",
"There is no diagonal term for the $HH1$ subbands in $\\Delta H$ , which is consistent with Fig.",
"REF (a) in which the $HH1$ subbands nearly do not shift.",
"The off-diagonal term $Q_{el}$ is crucial for the anticrossing behavior.",
"It couples the $E1$ and $LH1$ subbands even at the $\\Gamma (k_{x}=k_{y}=0)$ point, preventing them from touching with each other.",
"In this way, the effective model not only covers the physics of the BHZ model but also captures the anticrossing behavior of the energy bands.",
"The parameters in this effective Hamiltonian can be found straightforwardly in the projection, and they depend on the details of the quantum wells (i.e., the thickness of the quantum wells and the broken gap reference $V_{0}$ , etc.).",
"For the considered quantum well configuration (i.e., the thickness of 50/12.5/10/50 nm for AlSb/InAs/GaSb/AlSb), the parameters in the effect model are provided in Table REF .",
"Figure: Energy spectrum of bulk and edge states of the system with periodicand open boundaries in the xx and yy direction, respectively.",
"(a) for ΔV=30\\Delta V=30 meV , (b) for ΔV=10\\Delta V=10 meV, (c) for ΔV=-20\\Delta V=-20meV, and (d) the energy position of Dirac point (E D E_{D}) and maximumpoint of valence bands (E M E_{M}) as function of ΔV\\Delta V. V 0 =-100V_{0}=-100meV is taken for all figures."
],
[
"Hidden Dirac point of the helical edge states",
"With the six-band effective model, we are in a position to investigate the energy dispersions of the edge states for the topologically non-trivial cases (ii) and (iii).",
"This can be accomplished numerically by means of tight-binding calculations.",
"The tight-binding model can be obtained by discretizing the effective Hamiltonian Eq.",
"(REF ) on a square lattice.",
"In the long wavelength limit, we use the approximation $k_{i}\\approx \\sin (k_{i}a)/a$ and $k_{i}^{2}\\approx 2[1-\\cos (k_{i}a)]/a^{2}$ with $i=x,\\ y$ and $a$ the lattice constant.",
"We take $a=20\\mathring{\\mathrm {A}}$ which is a good approximation to the continuum limit.",
"To find the edge states solution, we apply the open boundary condition along the $y$ direction while the periodic boundary condition along the $x$ direction.",
"Thus $k_{x}$ remains a good quantum number and the system is diagonal in $k_{x}$ .",
"Figures REF (a)-(c) plot the energy spectrum of the effective model in the absence of external fields, corresponding to the cases (i)-(iii) as mentioned above.",
"For the trivial insulator case (i), there is a direct system gap and no edge dispersion as shown in Fig.",
"REF (a).",
"In both cases (ii) and (iii) as shown in Figs.",
"REF (b) and (c), there are two pairs of gapless and doubly degenerate helical edge bands across the bulk insulating gap, as expected for the QSH effect.",
"Nevertheless, for case (iii) the Dirac point of the helical edge states is close to and even “buried” by the bulk valence states, which is in contrast to case (ii) where the Dirac point is well exposed in the middle of the bulk gap [see Fig.",
"REF (b)].",
"As reducing $\\Delta V$ further, the Dirac point $E_{D}$ approaches the maximum point of the bulk valence bands $E_{M}$ , and eventually it is hidden by the bulk valence bands, as shown in Fig.",
"REF (d).",
"The hidden Dirac point of edge states in case (iii) can be attributed to the anticrossing between $E1$ and $LH1$ subbands by comparing with Fig.",
"REF (a).",
"We find that the Dirac point can be hidden only around the value of $\\Delta V$ where the anticrossing behavior occurs.",
"The Dirac point will not be buried in the bulk states but well exposed in the bulk gap if the $LH1$ subbands are not taken into account.",
"The hidden Dirac point is also related to the strong anisotropy in the system, which inherits from the bulk Kane model.",
"The finding that the Dirac point of edge states can be hidden in the bulk bands serves as the basis for the robust quantum edge transport in InAs/GaSb quantum wells under time-reversal breaking as will be discussed in the following, and it is one of our main results."
],
[
"Effective $g$ -factors of edge states",
"A magnetic field ${\\bf B}$ breaks time-reversal symmetry, and consequently the Dirac point of the edge states will no longer be topologically protected if there is no other hidden symmetry.",
"The time-reversal symmetry breaking can be evidenced by a gap opening in the helical edge states, which originates from the Zeeman and the orbital coupling effects of the bulk electrons in an external magnetic field.",
"In the six-band effective model, the Zeeman term can be written as $H_{Z}=H_{c}^{Z}\\oplus H_{v}^{Z},$ with $H_{c}^{Z}=(1/2)g_{e}\\mu _{B}{\\bf s}\\cdot {\\bf B},$ for electrons in the s-like $E1$ bands, and $H_{v}^{Z}=g_{h}\\mu _{B}{\\bf J}\\cdot {\\bf B}.$ for the p-like $HH1$ and $LH1$ bands [27], [36].",
"Here ${\\bf s}=\\left\\lbrace s_{x},s_{y},s_{z}\\right\\rbrace $ are the Pauli matrices for spin 1/2, ${\\bf J}$ are the $4\\times 4$ angular momentum matrices for $j=3/2$ , and $\\mu _{B}$ is the Bohr magneton.",
"$g_{e}$ and $g_{h}$ are the g-factors for bulk electrons and holes, respectively, and are taken to be $g_{e}$$=-10.0$ and $g_{h}$$=0.3$ [37], [38] in the following.",
"Figure: Effective g-factors of edge states and Zeeman energy gaps for edgestates spectrum.",
"The effective g-factor tensor elements of the edgestates as function of ΔV\\Delta V for magnetic field along (a) xx,(b) yy, and (c) zz direction, respectively.",
"(d) The energy gapsΔ Z x,y,z \\Delta _{Z}^{x,y,z} of edge states opened by three principal magneticfield of 0.5 T as functions of ΔV\\Delta V. V 0 =-100V_{0}=-100 meV is takenfor all figures.The response of the helical edge states to the magnetic fields can be examined by projecting the Zeeman term in the space spanned by the two helical edge states $|\\psi _{0+}\\rangle $ and $|\\psi _{0-}\\rangle $ at the $\\Gamma $ point.",
"Note that $|\\psi _{0+}\\rangle $ and $|\\psi _{0-}\\rangle $ are time-reversal to each other.",
"The corresponding effective Zeeman coupling can be summarized as $\\mathcal {H}_{\\text{edge}}^{Z}=\\dfrac{\\mu _{B}}{2}\\sum _{i,j=x,y,z}g_{ij}\\sigma _{i}B_{j},$ where the $g_{ij}$ is the effective g-factor tensor and $\\sigma _{x,y,z}$ are the Pauli matrices for the edge states space.",
"Reminding that the effective model for the helical edge states takes the form $\\mathcal {H}_{\\text{edge}}^{0}=\\hbar v_{F}k_{x}\\sigma _{z}$ where $v_{F}$ is the effective velocity.",
"The g-factor tensor is attributed to the the fact that the two helical edge states at the $\\Gamma $ point are not the eigenstates of electron spin.",
"Figures REF (a,b,c) plot the values of the g-factor elements $g_{ij}$ for different $\\Delta V$ , from which several points are worthy addressing.",
"For a perpendicular magnetic field $B_{z}$ , considering the contribution from the orbital angular momentum coupling to $B_{z}$ , a large value of $g_{zz}$ is obtained.",
"This large $g_{zz}$ just shifts the position of the degeneracy (Dirac) point of the helical edge states in the $k_{x}$ direction, whereas it does not open an energy gap (so we do not show it here).",
"However, a non-zero $g_{xz}$ does open an energy gap.",
"Here the Peierls substitution is performed as $t^{ij}\\rightarrow t^{ij}\\exp [\\frac{2\\pi i}{\\phi _{0}}\\int _{i}^{j}d{\\bf \\ell }\\cdot {\\bf A}]$ where $\\phi _{0}=h/e$ is the magnetic flux quantum, and $t^{ij}$ is the hopping integral between sites $i$ and $j$ .",
"For an in-plane field, the orbital contribution to g-factors is ignorable as electrons are confined in the quantum wells.",
"$g_{xx}$ and $g_{yy}$ always take non-zero values, which indicates that an in-plane magnetic field also opens a gap in the edge states.",
"These values of Zeeman gap calculated from the effective g-factor tensor of edge states match well with those obtained directly from the spectrum [see Fig.",
"REF (d)].",
"Therefore the non-zero g-factors indicate an opened gap at the Dirac point of helical edge states under time-reversal symmetry breaking [39], and the QSH effect is broken down.",
"It is also interesting to find that the effective g-factors of edge states show an evident anisotropy.",
"Especially for the in-plane magnetic fields, though both edge Zeeman gaps $\\Delta _{\\text{Z }}^{x,y}$ decay as decreasing $\\Delta V$ , $\\Delta _{\\text{Z }}^{x}$ decays much faster, which indicates that the anisotropy is enhanced for a small $\\Delta V$ .",
"Finally, we note that $\\Delta _{\\text{Z }}^{x,y}$ can reach the order of 1 meV for a magnetic field of 10 T, which are experimentally measurable at low temperatures.",
"However, these Zeeman gap $\\Delta _{Z}^{x,y}$ could be hidden since the Dirac point would be hidden by the bulk valence bands after the anticrossing behavior at a small $\\Delta V$ ."
],
[
"Robustness of the quantum edge transport",
"Now let us address the robustness of the edge transport in the InAs/GaSb quantum wells in the inverted regime.",
"It is known that the quantized two-terminal conductance $2e^{2}/h$ of a QSH insulator is a consequence of the helical edge states, which has been measured experimentally in the InAs/GaSb quantum wells.",
"Unexpectedly under in-plane magnetic fields either along or normal to the boundary the quantized conductance value remains quantized for mesoscopic samples and persists up to 12 T [18].",
"To understand the robustness of quantized conductance plateau, the evolution of the band structure subjected to an in-plane external magnetic field has been explored.",
"The in-plane magnetic field effect can be included by considering that the InAs and GaSb layers are spatially separated [24], [14], [22].",
"An in-plane magnetic field applied along the open boundary $B_{y}$ will not only open an energy gap at the Dirac point of the edge states, but also tilt the bulk energy spectra and reduce the bulk gap [14], [22].",
"Henceforth, there is no direct gap between the edge states and the valence bands if the Dirac point is buried in the bulk.",
"Similar effect happens for the in-plane magnetic field $B_{x}$ normal to the open boundary.",
"Figure: The two-terminal conductance GG of InAs/GaSb quantum wells in magneticfields.",
"(a) GG as a function of the Fermi energy E F E_{F} in thepresence of different magnetic field B x B_{x} along the xx direction.",
"(b) the same as (a) but with the magnetic fields applied along theyy direction.",
"ΔV=-20\\Delta V=-20 meV and V 0 =-100V_{0}=-100 meV are takenfor both figures.Consider a ribbon geometry of the InAs/GaSb quantum wells.",
"The two-terminal conductance is calculated as a function of the Fermi energy $E_{F}$ under different in-plane magnetic fields by means of the Landauer-BÃŒttiker formalism in a clean sample.",
"The sample geometry considered consists of a rectangular central region (size $L_{x}\\times L_{y}=200a\\times 150a$ ) and two semi-infinite leads are connected to it as source and drain leads.",
"With the help of recursive Green's function technique [40], [41], the conductance from the left terminal to the right terminal can be evaluated as $G=\\frac{e^{2}}{h}\\mathrm {Tr}\\left[\\Gamma _{L}G^{r}\\Gamma _{R}G^{a}\\right],$ where $\\Gamma _{L,R}$ are the line-width functions coupling to the left lead and the right lead respectively, and $G^{r}(G^{a})$ is the retarded (advanced) Green's function of the central region [42].",
"In the absence of a magnetic field, the value of two-terminal conductance is exactly quantized at $G=2e^{2}/h$ as predicted theoretically for the QSH effect.",
"The conductance remains nearly unchanged for different magnetic fields either along the boundary as shown in Fig.",
"REF (a) or normal to the boundary as shown in Fig.",
"REF (b), which can be attributed to the fact that the energy gap of edge states is buried in the bulk valence bands.",
"This support that the picture of hidden Dirac point may account for the experimental observations on robust quantum edge transport in InAs/GaSb quantum wells [18].",
"We also notice that a much stronger magnetic field makes the width of the conductance plateau narrower, which indicates that the system will be a semimetal under strong magnetic fields."
],
[
"Discussions and conclusions",
"The gap opened in the edge states under an in-plane magnetic field can be measured explicitly by means of reciprocal spin Hall effect in a multi-terminal measurement [43].",
"The edge state transport could survive even if the edge states and the bulk electrons of valence bands co-exist, and can be checked in the non-local measurement.",
"This provides a possible way to verify the existence of the edge states buried by the HH bands.",
"However, the non-local transport will disappear if the Fermi level sweeps over the energy gap of the edge states in the presence of magnetic field if the bulk electrons in the HH bands are presented.",
"In short, we re-examine the band structure and construct a six-band effective model for InAs/GaSb quantum wells from the bulk Kane model.",
"An energy gap for helical edge states opens under a magnetic field, which is well described by the effective g-factors of edge states.",
"The edge transport remains robust even though the magnetic field has already broken time-reversal symmetry and opened an energy gap for the helical edge states.",
"This robustness is attributed to the peculiar topological band structure that the Dirac point of the helical edge states is buried in the bulk valence band after the anticrossing behavior."
],
[
"Acknowledgments",
"C.L.",
"and S.Z.",
"thank Jia-Bin You, Jian Li and Lun-Hui Hu for helpful discussions.",
"This work was supported by the Research Grants Council, University Grants Committee, Hong Kong under Grant No.",
"17304414 and C6026-16W.",
"HKU ITS computing facilities supported by the Hong Kong UGC Special Equipment Grant (SEG HKU09)."
]
] | 1709.01645 |
[
[
"Robust Semi-Cooperative Multi-Agent Coordination in the Presence of\n Stochastic Disturbances"
],
[
"Abstract This paper presents a robust distributed coordination protocol that achieves generation of collision-free trajectories for multiple unicycle agents in the presence of stochastic uncertainties.",
"We build upon our earlier work on semi-cooperative coordination and we redesign the coordination controllers so that the agents counteract a class of state (wind) disturbances and measurement noise.",
"Safety and convergence is proved analytically, while simulation results demonstrate the efficacy of the proposed solution."
],
[
"Introduction",
"Coordination in multi-agent systems has attracted much attention over the last decade with a plethora of theoretical and practical problems that this paper can not cite in their entirety; for recent overviews the reader is referred to [1], [2], [3], [4].",
"A fundamental problem of interest in the area of distributed coordination and control is the decentralized multi-agent motion planning, which mainly focuses on generating collision-free trajectories for multiple agents (e.g., unmanned vehicles, robots) so that they reach preassigned goal locations under limited sensing, communication, and interaction capabilities.",
"Numerous elegant methodologies on planning the motion for a single agent (robot) have appeared in recent years, with the most popular being (i) sampling-based methods, including probabilistic roadmaps [5], and rapidly-exploring random trees [6], [7], (ii) Lyapunov-based methods, including either the definition of closed-form feedback motion plans via potential functions or vector fields, or computation of Lyapunov-based feedback motion plans via sum-of-squares programming [8], [9], and (iii) graph search and decision-theoretic methods, see also [10], [11] for a detailed presentation.",
"Although each method has its own merits and caveats, arguably Lyapunov-based methods (often termed reactive) are particularly popular for multi-agent motion planning problems, as they offer scalability with the number of agents, and the merits of Lyapunov-based control design and analysis.",
"In addition, robustness against modeling and/or measurement uncertainties is of primary importance for real-world systems and applications.",
"Hence the problems of modeling, quantifying, and treating uncertainty are of particular interest when it comes to multi-agent coordination.",
"A straightforward way to model uncertainty in multi-agent systems is by considering them as bounded disturbances.",
"In [12], a stable uncertainty is assumed to be bounded in $\\mathcal {H}_{\\infty }$ -norm by some prior given desired tolerance, and a state space observer along with a robust controller are designed by using the algebraic Riccati equations.",
"In [13], an additive $l_{2}$ -norm bounded disturbance is considered, and a robust controller is proposed for the distributed cooperative tracking problem in a leader-follower network by using Lyapunov stability theorems.",
"In [14], both bounded disturbances and unmodeled dynamics are assumed in the dynamics of agents; an identifier for each agent is designed to estimate the unknown disturbances and unmodeled dynamics.",
"Related work considering bounded deterministic disturbances can be found in the design of finite-time consensus algorithms with mismatched disturbances [15], and the rotating consensus control with mixed model uncertainties and external disturbances [16].",
"Another way of modeling uncertainty is by Gaussian random processes.",
"In [17], communication noises are described by a standard Brownian motion, and the mean square consensus in the multi-agent system is achieved by proposing a stochastic approximation-type gain vector, which attenuates the effect of noises.",
"This work is extended to networks with Markovian switching topologies [18], and leader-follower networks [19] with a similar noise-attenuation controller.",
"In [20], it is assumed that the measurements for each agent are disturbed by white noises.",
"Robust consensus can be achieved by applying stochastic Lyapunov analysis.",
"Based on this idea, in [21], the average-consensus problem of first-order multi-agent systems is considered, and a necessary and sufficient condition is proposed for robust consensus by using probability limit theory.",
"The aforementioned methods are efficient in solving the coordination problems with stochastic uncertainties in measurement or system dynamics; however, safety (i.e., the generation of collision-free trajectories) is not considered.",
"In contrast to the aforementioned results, in this paper we consider the problem of generating collision-free trajectories for multiple agents in the presence of uncertainty.",
"We propose a robust, Lyapunov-based coordination protocol that achieves collision-free motion for multiple agents in a distributed fashion, in the presence of state and measurement uncertainties.",
"The method builds upon our earlier work in [22], in which the nominal (uncertainty-free) case was considered.",
"More specifically, we redesign the semi-cooperative coordination protocol in [22] so that it accommodates the case of state and measurement uncertainties.",
"Our approach yields a method on the safe and robust motion planning of multiple agents that is based on analytic vector fields, hence offers scalability with the number of agents along with provable guarantees.",
"In summary, the contributions of this paper are: (i) a robust, semi-cooperative coordination protocol that accommodates for a class of stochastic disturbances in the agents' dynamics and measurements, and (ii) the derivation of analytical bounds on the navigation (estimation) and final state errors of the agents in terms of the considered uncertainties.",
"The paper is organized as follows: Section includes an overview of the modeling of the system under the effect of disturbances.",
"Section presents the robust coordination protocol along with the safety and convergence analysis.",
"Section evaluates the performance of the proposed method via two simulation scenarios.",
"Our conclusions and thoughts on future work are summarized in Section ."
],
[
"Modeling and Problem Statement",
"Let us consider $N$ identical agents $i\\in \\lbrace 1,\\dots ,N\\rbrace $ , which are assigned to move to goal locations of position coordinates $\\mathbf {r}_{gi}=\\begin{bmatrix}x_{gi}&y_{gi}\\end{bmatrix}^T$ relative to some global frame $\\mathcal {G}$ , while avoiding collisions.",
"The motion of each agent $i$ is modeled under unicycle kinematics with additive disturbances that stand for state and output uncertainty, for instance due to wind effects and sensor imperfections, respectively, as: $\\mathbf {\\dot{q}}_i&=\\mathbf {f}(\\mathbf {q}_i, \\mathbf {u}_i)+\\mathbf {\\Gamma }\\mathbf {w} \\Rightarrow \\begin{bmatrix}\\dot{x}_i \\\\ \\dot{y}_i \\\\ \\dot{\\theta }_i \\end{bmatrix}=\\begin{bmatrix}u_i c\\theta _i \\\\u_i s\\theta _i \\\\ \\omega _i \\end{bmatrix}+ \\begin{bmatrix}w_{x}\\\\w_y\\\\0\\end{bmatrix},\\\\\\mathbf {y}_i &= \\mathbf {h}_i(\\mathbf {q}_i)+ \\mathbf {v}_i,$ where $\\mathbf {q}_i=\\begin{bmatrix}\\mathbf {r}_i^T&\\theta _i\\end{bmatrix}^T$ is the state vector of agent $i$ , comprising the position vector $\\mathbf {r}_i=\\begin{bmatrix}x_i&y_i\\end{bmatrix}^T$ and the orientation $\\theta _i$ of the agent wrt the global frame $\\mathcal {G}$ , $\\mathbf {u}_i=\\begin{bmatrix}u_i&\\omega _i\\end{bmatrix}^T$ is the control input vector comprising the linear velocity $u_i$ and the angular velocity $\\omega _i$ of agent $i$ , $\\mathbf {f}(\\cdot , \\cdot ):\\mathbb {R}^3 \\times \\mathbb {R}^2 \\rightarrow \\mathbb {R}^3$ is the vector valued function of the agent dynamics, and $c(\\cdot )\\triangleq \\cos (\\cdot )$ , $s(\\cdot )\\triangleq \\sin (\\cdot )$ and $\\mathbf {\\Gamma } = \\begin{bmatrix}1& 0 \\\\0 & 1 \\\\0 &0\\end{bmatrix}.$ The random process $\\mathbf {w}=\\begin{bmatrix}w_x&w_y\\end{bmatrix}^T$ is assumed to be Gaussian, white, of known mean $\\mathbf {\\bar{w}}=\\begin{bmatrix}\\bar{w}_x&\\bar{w}_y\\end{bmatrix}^T$ and known covariance $\\mathbf {P}_w\\in \\mathbb {R}^{2\\times 2}$ .",
"To maintain the Lipschitz continuity of the proposed control law, we assume that the mean value of state disturbance is continuous in time and is bounded.",
"Furthermore, $\\mathbf {y}_i\\in \\mathbb {R}^m$ is the output vector comprising the available measurements, $\\mathbf {h}_i(\\cdot ):\\mathbb {R}^3\\rightarrow \\mathbb {R}^m$ is the output function, and $\\mathbf {v}_i\\in \\mathbb {R}^m$ is the measurement noise modeled as a Gaussian, white process of zero mean $\\bar{\\mathbf {v}}_i=\\mathbf {0}$ and known covariance $\\mathbf {P}_{v_i}\\in \\mathbb {R}^{m\\times m}$ .",
"For simplicity in the sequel we assume that the output function is the identity map so that the measurement model reduces to $\\mathbf {y}_i=\\mathbf {q}_i + \\mathbf {v}_i$ , and that the measurements are uncorrelated, so that the covariance matrix of $\\mathbf {v}_i$ reads $\\mathbf {P}_{v_i}=\\mathrm {diag}(\\sigma _{v_{i,1}},\\sigma _{v_{i,2}},\\sigma _{v_{i,3}})$ .",
"Each agent $i$ is modeled as a closed circular disk of radius $\\varrho _i$ , and has a circular communication/sensing region $\\mathcal {C}_i$ of radius $R_c$ centered at $\\mathbf {r}_i=\\begin{bmatrix}x_i&y_i\\end{bmatrix}^T$ , denoted as $\\mathcal {C}_i : \\lbrace \\mathbf {r}\\in \\operatorname{\\mathbb {R}}^2 \\; | \\; \\Vert \\mathbf {r}_i - \\mathbf {r}\\Vert \\le R_c\\rbrace .$ We denote $\\mathcal {N}_{i}$ the set of neighboring agents $k\\in \\mathcal {C}_i$ of agent $i$ .",
"We assume that each agent $i$ can measure the position $\\mathbf {r}_k$ , orientation $\\theta _k$ and receive the linear velocity $u_k$ of any agent $k$ lying in $\\mathcal {C}_i$ .",
"In our earlier work [22] we considered the nominal case of (), i.e., the case for $\\mathbf {w}=\\mathbf {0}$ , $\\mathbf {v}=\\mathbf {0}$ , and designed the following semi-cooperative distributed coordination protocol: Coordination of linear velocities: The linear velocity $u_i$ of each agent $i$ is governed by the control law: $u_i &= \\left\\lbrace \\begin{array}{rc}\\max \\left\\lbrace 0,\\min \\limits _{k\\in \\mathcal {N}_{i} | J_k<0} u_{i|k}\\right\\rbrace , & \\hbox{$d_m\\le d_{ik}\\le d_\\epsilon $,}\\\\u_{i\\epsilon }, & \\hbox{$d_\\epsilon < d_{ik} < d_c$,}\\\\u_{ic}, & \\hbox{$d_c\\le d_{ik}$;} \\\\\\end{array}\\right.$ where: $d_{ij}$ is the Euclidean distance between agents $i$ and $j$ , $d_m\\ge 2(2\\varrho +\\varrho _\\epsilon )$ is the minimum allowable pairwise distance, $d_c$ is a positive constant such that $d_c\\le R_c$ , and $d_r$ is a positive constant such that $d_m<d_r<d_c$ , $u_{ic}=k_{ui} \\tanh (\\Vert \\mathbf {r}_i - \\mathbf {r}_{gi}\\Vert )$ , $k_{ui}>0$ , $u_{i\\epsilon }$ is the value of the linear velocity $u_i$ of the agent $i$ when $d_{ij}=d_c$ , that is, $u_{i\\epsilon }=u_{ic}|{_{d_{ij}=d_c}}$ , the distance $d_\\epsilon $ is set equal to $d_\\epsilon =d_r-\\epsilon $ , $u_{i|k}$ is the safe velocity of agent $i$ wrt a neighbor agent $k\\in \\mathcal {N}_{i}$ , given as: $u_{i|k}&=u_{i\\epsilon }\\;\\frac{d_{ik}-d_m}{d_\\epsilon -d_m}+\\varepsilon _i \\; u_{is|k} \\; \\frac{d_\\epsilon -d_{ik}}{d_\\epsilon -d_m},$ with the terms in (REF ) defined as: $u_{is|k}&= u_k \\;\\frac{{\\mathbf {r}_{ki}}^T\\mathbf {\\eta }_k}{{\\mathbf {r}_{ki}}^T\\mathbf {\\eta }_i},\\;\\; \\mathbf {\\eta }_i=\\left[\\begin{matrix}\\cos \\theta _i\\\\\\sin \\theta _i\\end{matrix}\\right],\\;\\; J_k={\\mathbf {r}_{ki}}^T\\mathbf {\\eta }_i, \\\\\\mathbf {r}_{ki}&=\\mathbf {r}_i-\\mathbf {r}_k, \\quad \\mbox{and} \\quad 0<\\varepsilon _i<1.$ Coordination of angular velocities: The angular velocity $\\omega _i$ of each agent $i$ is governed by the control law: $\\omega _i &= -k_{\\omega i}\\left(\\theta _i-\\varphi _i\\right)+\\dot{\\varphi }_i,$ where $k_{\\omega i}>0$ , and $\\varphi _i\\triangleq \\arctan \\left(\\frac{\\operatorname{\\mathrm {F}}_{iy}}{\\operatorname{\\mathrm {F}}_{ix}}\\right)$ is the orientation of a reference vector field $\\mathbf {F}_i$ for agent $i$ , defined as: $\\mathbf {F}_i = \\prod _{j\\in \\mathcal {N}_i} (1-\\sigma _{ij}) \\mathbf {F}_{gi} + \\sum _{j\\in \\mathcal {N}_i} \\sigma _{ij} \\mathbf {F}^i_{oj},$ where details about the attractive and repulsive vector fields can be found in [22].",
"Under this protocol we were able to establish collision-free and almost globally convergent motion of the agents towards to their goal locations: Theorem 1 Consider $N$ agents $i\\in \\lbrace 1,\\dots ,N\\rbrace $ assigned to move to goal locations $\\mathbf {r}_{gi}$ .",
"Then, under the coordination protocol (REF ), (REF ), each agent safely converges to its goal configuration almost globally, except for a set of initial conditions of measure zero.",
"The design and analysis of the coordination controller is given in [22].",
"In this paper we seek to design a robust coordination protocol so that each agent $i$ can safely accommodate the effects of state and measurement uncertainties $\\mathbf {w}(t)$ , $\\mathbf {v}_i(t)$ , $t\\in [0,\\infty )$ , respectively.",
"Since we are only concerned about radial convergence of the agents to their respective goal locations, we re-define the radially attractive vector field $\\textbf {F}^r_{gi}$ for $\\mathbf {r}_i\\ne \\mathbf {r}_{gi}$ as: $\\operatorname{\\mathrm {F}}^r_{gix} = \\frac{-(x_i-x_{gi})}{(x_i-x_{gi})^2+(y_i-y_{gi})^2},\\\\\\operatorname{\\mathrm {F}}^r_{giy} = \\frac{-(y_i-y_{gi})}{(x_i-x_{gi})^2+(y_i-y_{gi})^2}.$ With this globally attractive field, the new reference field is given by $\\mathbf {F}_i = \\prod _{j\\in \\mathcal {N}_i} (1-\\sigma _{ij}) \\mathbf {F}^r_{gi} + \\sum _{j\\in \\mathcal {N}_i} \\sigma _{ij} \\mathbf {F}^i_{oj}.$"
],
[
"Robust Coordination: Design and Analysis",
"The problem of safe trajectory generation in the presence of disturbances is tackled in two steps.",
"The first step is to modify the nominal controller in order to accommodate for known state disturbances, that in our case can be thought of as wind effects.",
"This task is achieved by feed-forwarding the mean wind speed to each agent's control law as per (REF ) and (REF ).",
"Once we have a controller that handles this class of known disturbances, the next step is to make it robust wrt unknown, zero-mean state disturbances and sensor noises.",
"To accomplish this, an Extended Kalman Filter (EKF) based observer is used to estimate the state vector for the feedback coordination law of each agent.",
"In order to incorporate the estimation error, the crucial safety parameters involved in the nominal control law (i.e., the minimum allowed separation $d_m$ and the set $J_k$ of critical neighbors $k$ to agent $i$ ) in (REF ) are modified.",
"Then, with a controller in hand for nominal (or disturbance-free case), feed-forwarded with the known mean wind speed, and by using estimated states as feedback with properly modified safety parameters, we get a robust coordination protocol that steers the agents within a neighborhood of their goal locations, while maintaining safety at all times.",
"Recall that the system is safe if the inter-agent distance $d_{ij}$ between any pair of agents $i$ , $j$ is always greater than a minimum allowed separation $d_m$ ."
],
[
"Control design under bounded disturbances",
"We first consider the case where each agent $i$ is subject to known state disturbances, without any measurement uncertainty, i.e., we consider the agent dynamics: $\\mathbf {\\dot{q}}_i&=\\mathbf {f}(\\mathbf {q}_i, \\mathbf {u}^p_i)+\\mathbf {\\Gamma }\\mathbf {\\bar{w}}, \\Rightarrow \\begin{bmatrix}\\dot{x}_i \\\\ \\dot{y}_i \\\\ \\dot{\\theta }_i \\end{bmatrix}=\\begin{bmatrix}u_i^p\\cos \\theta _i \\\\u_i^p\\sin \\theta _i \\\\ \\omega _i^p\\end{bmatrix}+ \\begin{bmatrix}\\bar{w}_{x}\\\\\\bar{w}_y\\\\0\\end{bmatrix},\\\\\\mathbf {y}_i &= \\mathbf {h}_i(\\mathbf {q}_i) = \\mathbf {q}_i,$ where $\\mathbf {\\bar{w}}$ is the known mean of the state disturbance and $\\mathbf {u}^p_i = \\begin{bmatrix} u^p_i & \\omega ^p_i \\end{bmatrix}^T$ is the control input.",
"We propose the following coordination protocol yielding the feedback control law $\\mathbf {u}^p_i(\\mathbf {q},\\mathbf {\\bar{w}},d_m)$ for the perturbed dynamics (REF ) of agent $i$ : Coordination of linear velocities: The linear velocity of each agent $i$ is governed by the control law: $u_i^p &= \\left\\lbrace \\begin{array}{rc}-\\frac{1}{\\mu } \\log \\Big (\\sum \\limits _{k\\in \\mathcal {N}_i|J_k<0} e^{-\\mu u_{i|k}}\\Big ), & \\hbox{$d_m\\le d_{ik}\\le d_{\\epsilon }$,}\\\\u_{ic}, & \\hbox{$d_{\\epsilon }\\le d_{ik}$;} \\\\\\end{array}\\right.,$ where: the linear velocity of agent $i$ when free of conflicts is: $u_{ic} &= \\left\\Vert u_i\\frac{\\mathbf {F}_i}{\\Vert \\mathbf {F}_i\\Vert } - \\mathbf {\\bar{w}}\\right\\Vert , \\\\u_i &= k_{ui}\\tanh \\left(\\Vert \\mathbf {r}_i-\\mathbf {r}_{gi}\\Vert \\right),$ where $k_{ui}>0$ and $\\textbf {F}_i$ is given by the nominal vector field (REF ), for $\\mathbf {a} = \\begin{bmatrix}a_1,\\dots ,a_n\\end{bmatrix}^T$ , the function $g(\\mathbf {a}) = -\\frac{1}{\\mu } \\log \\Big (\\sum \\limits _{i=1}^N e^{-\\mu a_i}\\Big )$ is a smooth approximation of the minimum function $\\min \\lbrace a_1,\\dots ,a_n\\rbrace $ as $\\mu \\rightarrow \\infty $ , $u_{i|k}$ is the safe velocity of agent $i$ wrt a neighbor agent $k\\in \\mathcal {N}_{i}$ , given as: $u_{i|k}&=u_{ic}\\;\\frac{d_{ik}-d_m}{d_\\epsilon -d_m}+\\varepsilon _i \\; u_{is|k} \\; \\frac{d_\\epsilon -d_{ik}}{d_\\epsilon -d_m},$ with the terms in (REF ) defined as: $u_{is|k}&= u^p_k \\;\\frac{{\\mathbf {r}_{ki}}^T\\mathbf {\\eta }_k}{{\\mathbf {r}_{ki}}^T\\mathbf {\\eta }_i},\\;\\; \\mathbf {\\eta }_i=\\left[\\begin{matrix}\\cos \\theta _i\\\\\\sin \\theta _i\\end{matrix}\\right],\\;\\; J_k={\\mathbf {r}_{ki}}^T\\mathbf {\\eta }_i, \\\\\\mathbf {r}_{ki}&=\\mathbf {r}_i-\\mathbf {r}_k, \\quad \\mbox{and} \\quad 0<\\varepsilon _i<1.$ Figure: If J j ≜𝐫 ji T η i <0J_{j}\\triangleq {\\mathbf {r}_{ji}}^T\\mathbf {\\eta }_i < 0, i.e., if agent ii moves towards agent jj, then agent ii adjusts its linear velocity according to the velocity profile shown here, given analytically by ().Coordination of angular velocities: The angular velocity of each agent $i$ is governed by the control law: $\\omega ^p_i &=-k_{\\omega i}(\\theta _i-\\varphi ^p_i)+\\dot{\\varphi }^p_i, $ where $k_{\\omega i}>0$ , $\\varphi ^p_i\\triangleq \\arctan \\left(\\frac{\\operatorname{\\mathrm {F}}^p_{niy}}{\\operatorname{\\mathrm {F}}^p_{nix}}\\right)$ is the orientation of the normalized vector field $\\mathbf {F}^p_{ni} = \\frac{\\mathbf {F}^p_i}{\\Vert \\mathbf {F}^p_i\\Vert }$ for the perturbed system (REF ) at a point $(x,y)$ , with the vector field $\\mathbf {F}^p_i$ for the perturbed system (REF ) given out of: $\\mathbf {F}^p_i &= u_i\\frac{\\mathbf {F}_i}{\\Vert \\mathbf {F}_i\\Vert } - \\mathbf {\\bar{w}}, $ where $\\mathbf {F}_i$ is the nominal vector field given out of (REF ), and $u_i$ is given out of ().",
"The time derivative of $\\varphi ^p_i$ reads $\\dot{\\varphi }^p_i &=\\Big (\\left(\\begin{matrix}\\frac{\\partial \\operatorname{\\mathrm {F}}^p_{niy}}{\\partial x}\\;c\\theta ^p_i+\\frac{\\partial \\operatorname{\\mathrm {F}}^p_{niy}}{\\partial y}\\;s\\theta ^p_i\\end{matrix}\\right)\\operatorname{\\mathrm {F}}^p_{nix}\\\\&- \\left(\\begin{matrix}\\frac{\\partial \\operatorname{\\mathrm {F}}^p_{nix}}{\\partial x}\\;c\\theta ^p_i+\\frac{\\partial \\operatorname{\\mathrm {F}}^p_{nix}}{\\partial y}\\;s\\theta ^p_i\\end{matrix}\\right)\\operatorname{\\mathrm {F}}^p_{niy}\\Big )u_i.$ It can be readily seen that steady-state solution of $\\dot{\\theta }_i = \\omega ^p_i =-k_\\omega (\\theta _i-\\varphi ^p_i)+\\dot{\\varphi }^p_i$ is $\\theta _i = \\varphi ^p_i$ .",
"Figure: Construction of new vector Field 𝐅 i p \\textbf {F}^p_i in the presence of wind 𝐰 ¯\\mathbf {\\bar{w}} to follow the desired vector field 𝐅 i \\textbf {F}_i.",
"Here, blue line is the desired trajectory, green arrow shows the direction of the mean wind speed, dotted black arrow shows direction of nominal vector field 𝐅 i \\textbf {F}_i while solid orange arrow represents direction of constructed vector field 𝐅 i p \\textbf {F}^p_i.The following theorem proves that safety for the multi-agent can be guaranteed under the control law given by (REF ) and (REF ): Theorem 2 If each agent $i\\in \\lbrace 1,\\dots ,N\\rbrace $ subject to the system dynamics (REF ) follows the control law given by (REF ) and (REF ), then $\\forall i,j\\in \\lbrace 1,\\dots ,N\\rbrace , \\; i\\ne j$ , it holds that: $\\Vert \\mathbf {r}_i(t)-\\mathbf {r}_j(t)\\Vert &\\ge d_m, \\; \\forall t \\ge 0, $ and each agent converges to its goal location almost globally, i.e., $\\Vert \\mathbf {r}_{i\\infty }-\\mathbf {r}_{gi}\\Vert & \\triangleq \\lim _{t\\rightarrow \\infty } \\Vert \\mathbf {r}_i(t)-\\mathbf {r}_{gi}\\Vert = 0, $ except for a set of initial conditions of measure zero.",
"To prove (REF ), we have that under the control law (REF ), agent $i$ adjusts its linear velocity $u_i$ according to the velocity profile shown in Fig.",
"REF , given analytically out of (REF ), so that the distance $d_{ij}$ wrt the worst case neighbor $j$ remains greater than $d_m$ .",
"Mathematically, it is required that $\\frac{d}{dt}d_{ij} \\big \\vert _{d_{ij} = d_m} \\ge 0,$ so that the inter-agent distance does not decrease further once the agents $i$ , $j$ are at the minimum allowed separation $d_m$ .",
"The derivative of inter-agent distance is given in (REF ).",
"Figure: NO_CAPTIONHence, we have: $\\frac{d}{dt} \\;d_{ij} &=\\frac{u^p_i \\; {\\mathbf {r}_{ji}}^T\\mathbf {\\eta }_i - u^p_j \\; {\\mathbf {r}_{ji}}^T\\mathbf {\\eta }_j}{d_{ij}}.$ The worst case neighbor for agent $i$ is defined as the agent $j\\in \\lbrace \\mathcal {N}_{i} \\; | \\; J_j \\triangleq {\\mathbf {r}_{ji}}^T\\mathbf {\\eta }_i <0\\rbrace $ towards whom the rate of change of inter-agent distance $d_{ij}$ given by (REF ), due to the motion of agent $i$ , is maximum.",
"More specifically, the decision making process works as follows: The term $J_j<0$ describes the set of neighbor agents $j$ of agent $i$ towards whom agent $i$ is moving [23].",
"Thus agent $i$ computes safe velocities $u_{i|j}$ wrt each neighbor $j\\in \\lbrace \\mathcal {N}_{i} \\; | \\; J_j<0\\rbrace $ , and picks the minimum $u_{i|j}$ of the safe velocities so that the first term in (REF ) is as less negative as possible.",
"Now, the value of the safe velocity $u_{i|j}$ (REF ) when $d_{ij}=d_m$ is by construction equal to $\\varepsilon _i u_{is|k}=\\varepsilon _i u_j \\frac{{\\mathbf {r}_{ji}}^T\\mathbf {\\eta }_j}{{\\mathbf {r}_{ji}}^T\\mathbf {\\eta }_i}$ .",
"Plugging this value into (REF ) reads: $\\frac{d}{dt}\\;d_{ij}=\\frac{(\\varepsilon _i-1) u_j \\; {\\mathbf {r}_{ji}}^T\\mathbf {\\eta }_j}{d_{ij}}\\ge 0.$ To see why this condition is true, recall that $\\varepsilon _i-1<0$ , $u_j\\ge 0$ , and ${\\mathbf {r}_{ji}}^T\\mathbf {\\eta }_j\\le 0$ : this is because agent $j$ is either following a vector field $\\mathbf {F}_j$ that points away from agent $i$ , or happens to move away from agent $i$ in the first place.",
"This implies that the inter-agent distance $d_{ij}$ can not become less than $d_m$ [22].",
"To verify this argument, it is sufficient to show that in the presence of known disturbances $\\mathbf {\\bar{w}}$ , the motion of any agent $i,j\\in \\lbrace 1,2,\\dots ,N\\rbrace $ under the control law (REF ) and (REF ) is along their nominal vector fields given by (REF ), as follows: Let $\\theta ^d_i$ be the nominal direction that agent $i$ is supposed to follow under the nominal vector field $\\mathbf {F}_i$ , i.e., $\\theta ^d_i \\triangleq \\arctan \\left( \\frac{\\operatorname{\\mathrm {F}}_{ix}}{\\operatorname{\\mathrm {F}}_{iy}}\\right)$ .",
"From the steady-state solution of $\\dot{\\theta _i} = \\omega _i$ under the control law (REF ), we have $\\theta _i = \\varphi ^p_i$ , where $\\theta _i$ is the orientation of the agent $i$ .",
"Let $\\angle {(\\cdot )}$ be signed angle, defined to be positive in the clockwise direction and negative in the counter-clockwise direction.",
"Now, from (REF ) we have: $\\angle {(\\dot{\\mathbf {r}}_i-\\mathbf {\\bar{w}})} = \\theta _i$ .",
"Also, by construction of the new vector field (REF ), $\\angle \\left(u_i\\frac{\\textbf {F}_i}{\\Vert \\textbf {F}_i\\Vert }-\\mathbf {\\bar{w}}\\right) = \\varphi ^p_i \\overset{(\\ref {omega-p})}{=} \\theta _i$ out of the steady-state solution of (REF ), see also Figure REF .",
"Thus we have that $\\angle {(\\dot{\\mathbf {r}}_i-\\mathbf {\\bar{w}})} = \\angle \\left(u_i\\frac{\\textbf {F}_i}{\\Vert \\textbf {F}_i\\Vert }-\\mathbf {\\bar{w}}\\right)$ , which makes $\\angle {\\dot{\\mathbf {r}_i}} = \\angle {\\textbf {F}_i} = \\theta ^d_i$ .",
"Hence, the motion of agent $i$ is along the desired nominal vector field $\\mathbf {F}_i$ .",
"Analysis of convergence of the agents to their goal locations is given in [22].",
"Note that the analysis given in [22] is carried out for dipole type attractive vector field, which will still hold for the radially attractive vector field.",
"Since, except for a set of initial conditions of measure zero, the nominal vector field drives the agents to their goal location (Theorem REF ), it follows that the modified coordination protocol also drives the agents to their goal locations almost globally.",
"It is to be noted that while the direction of motion of agent $i$ is along the nominal vector field $\\textbf {F}_i$ , its orientation (heading angle) $\\theta _i$ remains along the modified vector field $\\textbf {F}_i^p$ ."
],
[
"Extension of the controller for stochastic disturbances",
"In order to estimate the states of the system in the presence of stochastic disturbances, we design a system observer based on the Extended Kalman Filter: $\\dot{\\mathbf {\\hat{q}}}_i &= \\mathbf {f}(\\mathbf {\\hat{q}}_i,\\mathbf {u}_i)+\\mathbf {\\Gamma }\\mathbf {\\bar{w}} + \\mathbf {K}_i(\\mathbf {y}_i-\\hat{\\mathbf {y}}_i),\\\\\\dot{\\mathbf {P}}_i &= \\mathbf {A}_i \\mathbf {P}_i+ \\mathbf {P}_i \\mathbf {A}_i^T -\\mathbf {K}_i\\mathbf {H}_i\\mathbf {P}_i + \\mathbf {\\Gamma }\\mathbf {P}_w\\mathbf {\\Gamma }^T, \\\\\\hat{\\mathbf {y}}_i &= \\mathbf {h}(\\mathbf {\\hat{q}}_i) = \\mathbf {\\hat{q}}_i, \\\\\\mathbf {K}_i &= -\\mathbf {P}_i\\mathbf {H}_i^T\\mathbf {P}_{v_i}^{-1}, \\\\\\mathbf {u}_i&= \\mathbf {u}_i(\\mathbf {\\hat{q}}, \\mathbf {\\bar{w}}, d_m^{\\prime },\\epsilon _J), $ where $\\mathbf {A}_i = \\left.\\frac{\\partial \\mathbf {f}}{\\partial \\mathbf {q}_i } \\right|_{\\hat{\\mathbf {q}}_i}$ is the state matrix of the linearized dynamics evaluated at $\\hat{\\mathbf {q}}_i$ , $\\mathbf {H}_i = \\left.",
"\\frac{\\partial \\mathbf {h}}{\\partial \\mathbf {q_i} } \\right|_{\\hat{\\mathbf {q}}_i}$ is the linearized output matrix evaluated at $\\hat{\\mathbf {q}}_i$ , $\\mathbf {P}_i$ is the covariance matrix of the estimation error $\\tilde{\\mathbf {q}_i}=\\mathbf {q}_i - \\hat{\\mathbf {q}}_i$ , and $\\mathbf {K}_i$ is the Kalman gain.",
"Note that the control law () has the same form as (REF ) but it uses the estimated states $\\hat{\\mathbf {q}}$ for feedback, the mean wind speed $\\mathbf {\\bar{w}}$ for feed-forward command, and involves $d_m^{\\prime }$ as the new safety parameter.",
"The last parameter $\\epsilon _J$ is introduced to re-define the worst-case neighbor, as per (REF ).",
"Stability of EKF-based estimator: In [24], authors have shown that the EKF is stable if there exist positive constants $\\bar{c}, \\underline{p}, \\underline{q}, \\delta , k$ such that: $\\Vert \\mathbf {H}(t)\\Vert \\le \\bar{c}$ System is uniformly observable $\\underline{p} \\le \\mathbf {P}_{v} \\le \\delta \\mathbf {I}$ $\\underline{q} \\le \\mathbf {P}_{w} \\le \\delta \\mathbf {I}$ $\\Vert H.O.T\\Vert \\le k\\Vert \\tilde{\\mathbf {q}}_i\\Vert ^2$ where $\\tilde{\\mathbf {q}}_i = \\mathbf {q}_i - \\hat{\\mathbf {q}}_i$ and $H.O.T$ stands for higher-order terms of the linearization .",
"Furthermore, if these conditions hold, then the estimation error is exponentially mean-square bounded.",
"Using this result, we now show that EKF based estimator will remain bounded for system () whose control law is given by ().",
"In our system, $\\mathbf {H}$ is just an identity mapping, which allows us to choose $\\bar{c} = 1$ .",
"Since $\\mathbf {H}$ is identity, we have that the system is always observable.",
"By the assumption on the type of disturbance we consider in (), the covariance matrices $\\mathbf {P}_{v_i}$ and $\\mathbf {P}_w$ are non-zero constant matrices and hence are bounded.",
"We assume that the initial estimation error is bounded.",
"Now, we show that our closed-loop system also satisfies the condition on boundedness of $H.O.T$ .",
"The function $\\mathbf {f}$ of the system dynamics given by () is continuously differentiable in $\\mathbf {q}_i$ and $\\mathbf {u}_i$ .",
"From (REF ) and (REF ) we have that the control law () is a bounded, continuously differentiable function of $\\mathbf {q}$ , since the vector field $\\textbf {F}^p_i$ , its partial derivatives wrt $x_i$ and $y_i$ , and the linear control input $u^p_i$ , are continuously differentiable and bounded.",
"Hence for all agents the closed loop function $\\mathbf {f}(\\mathbf {q}_i)$ is a continuously differentiable, bounded function of the states of the agent $i$ .",
"Hence all its derivatives $\\frac{d^k\\mathbf {f}}{d\\mathbf {q}^k_i}$ are bounded in $\\mathbb {R}^n$ .",
"Using the expression for Lagrange Remainder for Taylor series expansion [25], since we are using first order expansion, we have that the norm of the remainder of Taylor series expansion or the higher order terms $\\Vert H.O.T\\Vert \\le \\frac{1}{2}\\Vert \\frac{d^2 \\mathbf {f}}{d \\mathbf {q}^2_i}\\Vert \\Vert \\tilde{\\mathbf {q}}_i\\Vert ^2$ and $\\Vert \\frac{d^2 \\mathbf {f}}{d \\mathbf {q}^2_i}\\Vert \\le L$ .",
"Hence $\\Vert H.O.T\\Vert \\le k\\Vert \\tilde{\\mathbf {q}}_i\\Vert ^2$ with $k = L/2$ .",
"Thus, the EKF-based estimator (REF ) is stable, which implies that the estimation error $\\tilde{\\mathbf {q}}$ will remain bounded for sufficiently small disturbances, i.e., for small covariance matrices $\\mathbf {P}_w$ and $\\mathbf {P}_{v_i}$ .",
"We can therefore bound the norms of the estimation errors in individual states and position of agent $i$ by small, positive numbers.",
"Define the maximum of the errors in the estimation of the states for any agent $i$ as $\\epsilon _x \\triangleq 3\\max \\limits _i\\sqrt{\\mathbf {P}_{i11}} $ , $ \\epsilon _y \\triangleq 3\\max \\limits _i\\sqrt{\\mathbf {P}_{i22}} $ and $ \\epsilon _\\theta \\triangleq 3\\max \\limits _i\\sqrt{\\mathbf {P}_{i33}}$ where $\\mathbf {P}_{ill}, l\\in \\lbrace 1,\\dots ,n\\rbrace $ , are the diagonal terms of error covariance matrix $\\mathbf {P}_i$ .",
"Also, we define the maximum error in the estimation of the position as $\\epsilon _d \\triangleq \\sqrt{\\epsilon _x^2+\\epsilon _y^2}$ .",
"Using the 3-$\\sigma $ bound for the Gaussian noise, we have that $\\Vert x_i-\\hat{x}_i\\Vert \\le \\epsilon _x $ , $ \\Vert y_i-\\hat{y}_i\\Vert \\le \\epsilon _y $ , $ \\Vert \\theta _i-\\hat{\\theta _i}\\Vert \\le \\epsilon _{\\theta }$ and $\\Vert \\mathbf {r}_i-\\hat{\\mathbf {r}}_i\\Vert \\le \\epsilon _d$ .",
"Referring to Definition 4.1 of [24], we can choose a function $\\Lambda (\\mathbf {q}) = -1$ , so that $\\Vert \\Lambda \\Vert = 1$ .",
"Then, using Theorem 7 in [26], we have that $\\Vert \\mathbf {P}_i(t)\\Vert \\le \\Vert \\mathbf {P}_i(0)\\Vert + 1$ .",
"We choose $\\mathbf {P}_i(0) = \\mathbf {P}_{v_i}$ , so that $\\max \\Vert \\mathbf {P}(t)\\Vert = \\max \\limits _i\\Vert \\mathbf {P}_{v_i}\\Vert +1$ .",
"With this bound on the covariance matrix, we can express $\\epsilon _x = \\epsilon _y = \\epsilon _{\\theta } = \\sqrt{\\max \\limits _i \\Vert \\mathbf {P}_{v_i}\\Vert +1}$ and $\\epsilon _d = \\sqrt{2 \\left(\\max \\limits _i \\Vert \\mathbf {P}_{v_i}\\Vert +1\\right)}$ ."
],
[
"Safety Analysis",
"The linear velocity coordination law (REF ) depends upon the safety parameter $d_m$ .",
"As only estimates of the inter-agent distances are available to each agent, one can set the minimum allowed estimated distance $\\hat{d}_{ij_{min}}$ equal to $d_m^{\\prime }$ (given by (REF )), in order to ensure that actual inter-agent distance $d_{ij}$ remains greater than the minimum allowed distance $d_m$ .",
"We re-define the worst case neighbor (defined earlier) in terms of the estimated states as: $j &\\in \\lbrace \\mathcal {N}_{i}\\;|\\; {\\mathbf {\\hat{}}{\\mathbf {r}}_{ji}}^T\\mathbf {\\hat{}}{\\mathbf {\\eta }}_i \\le -\\epsilon _J\\rbrace .$ To proceed with the analysis, we first prove the following Lemma: Lemma 1 If $\\mathbf {\\hat{\\mathbf {r}}}^T_{ji}\\mathbf {\\hat{}}{\\mathbf {\\eta }}_i \\le -\\epsilon _J$ , where $\\epsilon _J = \\frac{2\\epsilon _d+s(\\epsilon _{\\theta })(d_m+2\\epsilon _d)}{c(\\epsilon _{\\theta })}$ , then with probability 0.997 ${\\mathbf {r}^T_{ji}}\\mathbf {\\eta }_i &\\le 0.$ Since we are concerned about safety of the system, let's assume that inter-agent distance $d_{ij} = d_m$ .",
"We use the fact that $\\Vert x_i-\\hat{x}_i\\Vert \\le \\epsilon _x $ , $ \\Vert y_i-\\hat{y}_i\\Vert \\le \\epsilon _y $ , $ \\Vert \\theta _i-\\hat{\\theta _i}\\Vert \\le \\epsilon _{\\theta }$ and $\\Vert \\mathbf {r}_i-\\hat{\\mathbf {r}}_i\\Vert \\le \\epsilon _d$ .",
"Therefore we have that $d_m-2\\epsilon _d\\le \\Vert \\hat{\\mathbf {r}}_i-\\hat{\\mathbf {r}}_j\\Vert \\le d_m+2\\epsilon _d.$ Now, ${\\mathbf {r}_{ji}}^T\\mathbf {\\eta }_i &= {\\mathbf {r}_{ji}}^T\\left[\\begin{matrix}\\cos \\theta _i\\\\\\sin \\theta _i\\end{matrix}\\right]= (x_i-x_j)c\\theta _i + (y_i-y_j)s\\theta _i,$ which further reads as in (REF ).",
"Figure: NO_CAPTIONLet $\\epsilon _J = \\frac{2\\epsilon _d+s(\\epsilon _{\\theta })(d_m+2\\epsilon _d)}{c(\\epsilon _{\\theta })}$ .",
"Since $\\epsilon _{\\theta }$ and $\\epsilon _d$ are small positive numbers, we have $\\cos (\\epsilon _{\\theta })>0$ and $\\epsilon _J>0$ .",
"Hence, if $((\\hat{x}_i-\\hat{x}_j)c\\hat{\\theta }_i+ (\\hat{y}_i-\\hat{y}_j)s\\hat{\\theta }_i)\\le -\\epsilon _J$ , then we have that $\\mathbf {r}_{ji}^T\\mathbf {\\eta }_i \\le 0$ .",
"Since the bounds on the estimation error are probabilistic, we only guarantee that this result holds with probability 0.997Note that this can be increased to a value arbitrarily close to 1 by using k-$\\sigma $ rule to bound the estimation error, where k is arbitrarily large.. Now we use this result to show the safety of the system.",
"In order to maintain the safety of the agents in the presence of disturbances, the safety parameter $d_m^{\\prime }$ in the control law () is given by: $d_{m}^{\\prime } &= d_{m}+2\\epsilon _{d}.$ Theorem 3 If each agent $i$ under the system dynamics () follows the control law given by (), and if $d_m^{\\prime }$ is given by (REF ), then with probability 0.997, $\\forall t \\ge 0, \\forall \\; i, j\\in \\lbrace 1,\\dots ,N\\rbrace , \\; j\\ne i$ : $\\Vert \\mathbf {r}_i(t)-\\mathbf {r}_j(t)\\Vert \\ge d_m.$ Using triangle inequality, $\\Vert \\hat{\\mathbf {r}}_i-\\hat{\\mathbf {r}}_j\\Vert &\\le \\Vert \\hat{\\mathbf {r}}_i-\\mathbf {r}_i\\Vert + \\Vert \\hat{\\mathbf {r}}_j-\\mathbf {r}_j\\Vert + \\Vert \\mathbf {r}_i-\\mathbf {r}_j\\Vert \\Rightarrow \\\\\\Vert \\hat{\\mathbf {r}}_i-\\hat{\\mathbf {r}}_j\\Vert &\\le 2\\epsilon _d + \\Vert \\mathbf {r}_i-\\mathbf {r}_j\\Vert \\Rightarrow d_m^{\\prime } \\le 2\\epsilon _d+\\Vert \\mathbf {r}_i-\\mathbf {r}_j\\Vert .$ Choosing $d_m^{\\prime }$ as per (REF ), we get $\\Vert \\mathbf {r}_i-\\mathbf {r}_j\\Vert \\ge d_m$ .",
"To complete the proof, it is sufficient to show that when $\\hat{d}_{ij} = d_m^{\\prime }$ , the time derivative of inter-agent distance is positive, i.e.",
"$\\dot{d}_{ij}>0$ .",
"In the presence of uncertainties, to guarantee that the system is safe, we modify the definition of the worst case neighbor as $j \\in \\lbrace \\mathcal {N}_{i} \\;|\\; \\hat{J}_j \\triangleq \\mathbf {\\hat{\\mathbf {r}}}^T_{ji}\\mathbf {\\hat{}}{\\mathbf {\\eta }}_j <-\\epsilon _J\\rbrace $ so that, from Lemma REF we have that ${\\mathbf {r}^T_{ji}}\\mathbf {\\eta }_j \\le 0$ , which implies $\\dot{d}_{ij} \\ge 0$ at $\\hat{d}_{ij} \\triangleq \\Vert \\hat{\\mathbf {r}}_i-\\hat{\\mathbf {r}}_j\\Vert = d_m^{\\prime }$ .",
"Therefore, we have that: (i) $\\hat{d}_{ij} = d_m^{\\prime } \\Rightarrow d_{ij} \\ge d_m$ , and (ii) $\\frac{d}{dt}d_{ij} \\big \\vert _{\\hat{d}_{ij} = d_m^{\\prime }} \\ge 0$ , i.e.",
"the inter-agent distance does not decrease beyond this point.",
"Hence the multi-agent system always remains safe."
],
[
"Analysis of convergence to goal location",
"Theorem 4 If agent $i$ under the system dynamics () follows the control law given by (), then with probability 0.997: $\\lim _{t\\rightarrow \\infty } \\Vert \\mathbf {r}_i(t)-\\mathbf {r}_{gi}\\Vert &= \\Vert \\mathbf {r}_{i\\infty }-\\mathbf {r}_{gi}\\Vert \\le \\epsilon _f,$ where $\\epsilon _f = \\epsilon _d + \\epsilon $ , where $\\epsilon $ is an arbitrary small positive constant.",
"Using triangle inequality, $\\Vert \\mathbf {r}_{i\\infty }-\\mathbf {r}_{gi}\\Vert &\\le \\Vert \\mathbf {r}_{i\\infty }-\\hat{\\mathbf {r}}_{i\\infty }\\Vert + \\Vert \\hat{\\mathbf {r}}_{i\\infty }-\\mathbf {r}_{gi}\\Vert .$ System (REF ) can be seen as a perturbed form of (REF ) with $\\mathbf {y}_i -\\hat{\\mathbf {y}}_i$ being the constantly acting, bounded perturbation.",
"From Theorem REF , we have that $\\mathbf {r}_{gi}$ is an (almost globally) asymptotically stable equilibrium of (REF ).",
"Furthermore, (assuming no interactions among agents in the vicinity of the goal locations) one has that $\\mathbf {r}_{gi}$ is a stable equilibrium of (REF ).",
"To verify this argument, use the candidate Lyapunov function $V= \\Vert \\hat{\\mathbf {r}}_i - \\mathbf {r}_{gi}\\Vert ^2$ .",
"Define $\\mathbf {r}_e \\triangleq \\hat{\\mathbf {r}}_i - \\mathbf {r}_{gi}$ .",
"Now, taking the time derivative of candidate Lyapunov function along the system (REF ), we get $\\dot{V} &= \\mathbf {r}_e^T\\dot{\\hat{\\mathbf {r}}}_i = \\mathbf {r}_e^T\\left[\\begin{matrix}u^p_ic\\varphi ^p_i + \\bar{w}_x \\\\ u^p_is\\varphi ^p_i +\\bar{w}_y\\end{matrix}\\right]+ \\mathbf {r}_e^T\\mathbf {\\Gamma }^T\\mathbf {K}_i(\\mathbf {y}_i-\\hat{\\mathbf {y}}_i).$ Note that $\\left[\\begin{matrix}u^p_ic\\varphi ^p_i + \\bar{w}_x \\\\u^p_is\\varphi ^p_i +\\bar{w}_y\\end{matrix}\\right]$ is the $\\dot{\\mathbf {r}}^p_i$ of agent $i$ in the absence of unknown disturbance with magnitude $|\\dot{\\mathbf {r}}^p_i| = u_i$ and is along the attractive vector field $\\textbf {F}^r_{gi}$ which points in the direction $-(\\hat{\\mathbf {r}}_i-\\mathbf {r}_{gi})$ .",
"Hence, we have that $\\left[\\begin{matrix}u^p_ic\\varphi ^p_i + \\bar{w}_x \\\\u^p_is\\varphi ^p_i +\\bar{w}_y\\end{matrix}\\right] = -u_i\\frac{\\mathbf {r}_e}{\\Vert \\mathbf {r}_e\\Vert }$ .",
"Finally, since the EKF based estimator is stable, the perturbation term $\\mathbf {K}_i(\\mathbf {y}_i-\\hat{\\mathbf {y}}_i)$ can be bounded by $\\Vert \\mathbf {K}_i\\Vert \\epsilon _d \\le \\delta $ .",
"Hence, the time derivative $\\dot{V}$ reads $\\dot{V} &= -\\mathbf {r}_e^Tu_i\\frac{\\mathbf {r}_e}{\\Vert \\mathbf {r}_e\\Vert } + \\mathbf {r}_e^T\\mathbf {\\Gamma }^T\\mathbf {K}_i(\\mathbf {y}_i-\\hat{\\mathbf {y}}_i) \\\\&\\le -k_{ui}\\Vert \\mathbf {r}_e\\Vert \\tanh (\\Vert \\mathbf {r}_e\\Vert ) + \\delta \\Vert \\mathbf {r}_e\\Vert .$ Hence, we have $\\dot{V}\\le 0$ for $\\Vert \\mathbf {r}_e\\Vert = \\Vert \\hat{\\mathbf {r}}_i - \\mathbf {r}_{gi}\\Vert \\ge \\tanh ^{-1}\\frac{\\delta }{k_{ui}}$ where $k_{ui}$ is chosen to be greater than $\\delta $ .",
"Using the stability of perturbed systems under the effect of constantly acting (non-vanishing) perturbations [27], we have that $\\mathbf {r}_{gi}$ is a stable equilibrium of (REF ), which ensures that $\\Vert \\hat{\\mathbf {r}}_{i\\infty }-\\mathbf {r}_{gi}\\Vert \\le \\epsilon $ for some small positive number $\\epsilon >0$ .",
"Therefore, $\\Vert \\mathbf {r}_{i\\infty }-\\mathbf {r}_{gi}\\Vert \\le \\Vert \\mathbf {r}_{i\\infty }-\\hat{\\mathbf {r}}_{i\\infty }\\Vert + \\epsilon \\Rightarrow \\Vert \\mathbf {r}_{i\\infty }-\\mathbf {r}_{gi}\\Vert \\le \\epsilon _d +\\epsilon .$ Choosing $\\epsilon _f = \\epsilon _d +\\epsilon $ completes the proof.",
"It is important to note that in the presence of stochastic state-disturbance it cannot guaranteed that an agent can achieve any given goal orientation.",
"In fact, the steady-state orientation of any agent would be opposite to the wind direction with small deviations because of uncertain state disturbance.",
"This is true because at goal location, $\\textbf {F}_i = \\mathbf {0}$ and hence, $\\textbf {F}^p_i$ is along the opposite direction of wind.",
"We consider two scenarios involving $\\operatorname{\\textrm {N}}=20$ agents which are assigned to move towards goal locations while avoiding collisions.",
"The goal locations are selected sufficiently far apart so that the agents' communication regions do not overlap when agents lie on their goal locations.",
"The covariance matrix of the state disturbance is taken as $\\mathbf {P}_w = \\mathrm {diag}(0.01, 0.01)$ and measurement noise as $\\mathbf {P}_{vi} = \\mathrm {diag}(0.01, 0.01, 0.01)$ .",
"In the first case, we assume the mean wind speed to be constant with time, with $\\mathbf {\\bar{w}} = \\begin{bmatrix}-0.2& 0.7\\end{bmatrix}^T (m/sec)$ .",
"With these uncertainties, we have $\\epsilon _d = 1.42\\;m$ and $\\epsilon _f = 1.52 \\;m$ .",
"The radii of the agents are $\\varrho =0.4\\;m$ , the minimum separation is set equal to $d_m=2 \\varrho =0.8\\;m$ , and the communication radius is set equal to $R_c= 2d_m^{\\prime } = 7.28\\;m$ .",
"Figure: The smallest pairwise distance at each time instant for constant 𝐰 ¯\\bar{\\mathbf {w}}.Figure: Final distance from the goal location for constant 𝐰 ¯\\bar{\\mathbf {w}}.Figure: Final orientations θ i∞ \\theta _{i\\infty } and opposite direction of the wind.Figures REF and REF respectively show the minimum distance between any pair of agents and the final distances of agents from their respective goal locations.",
"It can be seen that the agents reach the $\\epsilon _f$ ball around their respective goals, while maintaining the $d_m$ distance between each other.",
"Figure REF shows that agents align themselves opposite to the wind direction in the equilibrium.",
"Figure REF shows the initial, goal and the final positions reached by agents after 15,000 iterations with time-step $0.01\\;sec$ .",
"Figure: Initial, goal locations and the positions reached by the agents after 150 seconds (15,000 iterations).In the second case, we assume the mean wind speed to be time varying with $\\bar{w}_x(t),\\bar{w}_y(t)\\in [-1, 1]\\;(m/sec)$ .",
"Similar to the first case, plots are given to show the performance of the coordination protocol in presence of time-varying wind disturbance.",
"Figure REF and REF depict the minimum pairwise distances between the agents and their final distance from their goal locations, respectively.",
"Figure: The smallest pairwise distance at each time instant for time varying 𝐰 ¯\\bar{\\mathbf {w}}.Figure: Final distance from the goal location for time varying 𝐰 ¯\\bar{\\mathbf {w}}."
],
[
"Conclusions and Future Work",
"We presented a safe semi-cooperative multi-agent coordination protocol under state and measurement uncertainty.",
"The nominal case of our earlier work is redesigned by feed-forward control, vector-field-based feedback control, and nonlinear estimation techniques, so that safety and convergence of the agents up to some bound around the desired destination is guaranteed.",
"In the future, we would like to study the case when the mean state disturbance has a spatial distribution.",
"Ongoing work focuses on treating the case of input- and state- constrained agents under uncertainties, with application to fixed-wing aircraft operating in obstacle environments."
]
] | 1709.01586 |
[
[
"Towards Neural Machine Translation with Latent Tree Attention"
],
[
"Abstract Building models that take advantage of the hierarchical structure of language without a priori annotation is a longstanding goal in natural language processing.",
"We introduce such a model for the task of machine translation, pairing a recurrent neural network grammar encoder with a novel attentional RNNG decoder and applying policy gradient reinforcement learning to induce unsupervised tree structures on both the source and target.",
"When trained on character-level datasets with no explicit segmentation or parse annotation, the model learns a plausible segmentation and shallow parse, obtaining performance close to an attentional baseline."
],
[
"Introduction",
"Many efforts to exploit linguistic hierarchy in NLP tasks make use of the output of a self-contained parser system trained from a human-annotated treebank [10].",
"An alternative approach aims to jointly learn the task at hand and relevant aspects of linguistic hierarchy, inducing from an unannotated training dataset parse trees that may or may not correspond to treebank annotation practices [23], [2].",
"Most deep learning models for NLP that aim to make use of linguistic hierarchy integrate an external parser, either to prescribe the recursive structure of the neural network [17], [8], [21] or to provide a supervision signal or training data for a network that predicts its own structure [20], [1], [5].",
"But some recently described neural network models take the second approach and treat hierarchical structure as a latent variable, applying inference over graph-based conditional random fields [11], the straight-through estimator [3], or policy gradient reinforcement learning [24] to work around the inapplicability of gradient-based learning to problems with discrete latent states.",
"For the task of machine translation, syntactically-informed models have shown promise both inside and outside the deep learning context, with hierarchical phrase-based models frequently outperforming traditional ones [2] and neural MT models augmented with morphosyntactic input features [19], [16], a tree-structured encoder [6], [9], and a jointly trained parser [7] each outperforming purely-sequential baselines.",
"Drawing on many of these precedents, we introduce an attentional neural machine translation model whose encoder and decoder components are both tree-structured neural networks that predict their own constituency structure as they consume or emit text.",
"The encoder and decoder networks are variants of the RNNG model introduced by [5], allowing tree structures of unconstrained arity, while text is ingested at the character level, allowing the model to discover and make use of structure within words.",
"The parsing decisions of the encoder and decoder RNNGs are parameterized by a stochastic policy trained using a weighted sum of two objectives: a language model loss term that rewards predicting the next character with high likelihood, and a tree attention term that rewards one-to-one attentional correspondence between constituents in the encoder and decoder.",
"We evaluate this model on the German-English language pair of the flickr30k dataset, where it obtains similar performance to a strong character-level baseline.",
"Analysis of the latent trees produced by the encoder and decoder shows that the model learns a reasonable segmentation and shallow parse, and most phrase-level constituents constructed while ingesting the German input sentence correspond meaningfully to constituents built while generating the English output."
],
[
"Encoder/Decoder Architecture",
"The model consists of a coupled encoder and decoder, where the encoder is a modified stack-only recurrent neural network grammar [14] and the decoder is a stack-only RNNG augmented with constituent-level attention.",
"An RNNG is a top-down transition-based model that jointly builds a sentence representation and parse tree, representing the parser state with a StackLSTM and using a bidirectional LSTM as a constituent composition function.",
"Figure: At a given timestep during either encoding or decodingthere are three possible transitions (although one or more may be forbidden): begin a new nonterminal constituent (nt), predict and ingest a terminal (gen), or end the current nonterminal (reduce).",
"If the chosen transition is nt, the RNNG adds a new-nonterminal token 𝐱 i+1 \\mathbf {x}_{i+1} to the active constituent and begins a new nonterminal constituent (1a).",
"If the transition is gen, the RNNG predicts the next token (Section 2.3) and adds the ground-truth next token 𝐞\\mathbf {e} from the context buffer at the cursor location (1b).",
"If the transition is reduce, the contents of the active nonterminal are passed to the composition function, the new-nonterminal token 𝐱 i \\mathbf {x}_i is replaced with the result of the composition 𝐡 i \\mathbf {h}_i, and the StackLSTM rolls back to the previously active constituent (1c).",
"In all three cases, the StackLSTM then advances one step with the newly added token as input (𝐱 i+1 \\mathbf {x}_{i+1}, 𝐞\\mathbf {e}, or 𝐡 i \\mathbf {h}_i).Our implementation is detailed in Figure 1, and differs from [5] in that it lacks separate new-nonterminal tokens for different phrase types, and thus does not include the phrase type as an input to the composition function.",
"Instead, the values of $\\mathbf {x}_i$ for the encoder are fixed to a constant $\\mathbf {x}^{\\rm enc}$ while the values of $\\mathbf {x}_j$ for the decoder are determined through an attention procedure (Section 2.2).",
"As originally described, the RNNG predicts parser transitions using a one-layer $\\tanh $ perceptron with three concatenated inputs: the last state of a unidirectional LSTM over the stack contents ($\\mathbf {s}$ ), the last state of a unidirectional LSTM over the reversed buffer of unparsed tokens ($\\mathbf {b}$ ), and the result of an LSTM over the past transitions ($\\mathbf {a}$ ).",
"All three of these states can be computed with at most one LSTM step per parser transition using the StackLSTM algorithm [4].",
"But such a baseline RNNG is actually outperformed by one which conditions the parser transitions only on the stack representation [14].",
"Restricting our model to this stack-only case allows both the encoder and decoder to be supervised using a language model loss, while allowing the model access to $\\mathbf {b}$ would give it a trivial way to predict the next character and obtain zero loss."
],
[
"Attention",
"With the exception of the attention mechanism, the encoder and decoder are identical.",
"While the encoder uses a single token to represent a new nonterminal, the decoder represents a new nonterminal on the stack as a sum weighted by structural attention of the phrase representations of all nonterminal tree nodes produced by the encoder.",
"In particular, we use the normalized dot products between the decoder stack representation $\\mathbf {s}_j^{\\rm dec}$ and the stack representation at each encoder node $\\mathbf {s}^i_{\\rm enc}$ (that is, the hidden state of the StackLSTM up to and including $\\mathbf {x}_j^{\\rm enc}$ but not $\\mathbf {h}_j^{\\rm enc}$ ) as coefficients in a weighted sum of the phrase embeddings $\\mathbf {h}^i_{\\rm enc}$ corresponding to the encoder nodes: $\\begin{split}\\alpha _{ij}&=\\operatornamewithlimits{softmax}_{\\text{all }i}(\\mathbf {s}_i^{\\rm enc}\\cdot \\mathbf {s}_j^{\\rm dec})\\\\\\mathbf {x}_j^{\\rm dec}&=\\sum _i\\alpha _{ij}\\mathbf {h}_i^{\\rm enc}.\\\\\\end{split}$ Since the dot products between encoder and decoder stack representations are a measure of structural similarity between the (left context of) the current decoder state and the encoder state.",
"Within a particular decoder nonterminal, the model reduces to ordinary sequence-to-sequence transduction.",
"Starting from the encoder's representation of the corresponding nonterminal or a weighted combination of such representations, the decoder will emit a translated sequence of child constituents (both nonterminal and terminal) one by one—applying attention only when emitting nonterminal children."
],
[
"Training",
"We formulate our model as a stochastic computation graph [18], leading to a training paradigm that combines backpropagation (which provides the exact gradient through deterministic nodes) and vanilla policy gradient (which provides a Monte Carlo estimator for the gradient through stochastic nodes).",
"There are several kinds of training signals in our model.",
"First, when the encoder or decoder chooses the gen action it passes the current stack state $\\mathbf {s}$ through a one-layer softmax perceptron, giving the probability that the next token is each of the characters in the vocabulary.",
"The language model loss $\\mathcal {L}_k$ for each generated token is the negative log probability assigned to the ground-truth next token.",
"The other differentiable training signal is the coverage loss $\\mathcal {L}_c$ , which is a measure of how much the attention weights diverge from the ideal of a one-to-one mapping.",
"This penalty is computed as a sum of three MSE terms: $\\begin{split}\\mathcal {L}_c&=\\operatornamewithlimits{mean}_{\\text{all }i}(1 - \\sum _{\\text{all }j}\\alpha _{ij})^2\\\\&+\\operatornamewithlimits{mean}_{\\text{all }i}(1 - \\max _{\\text{all }j}\\alpha _{ij})^2\\\\&+\\operatornamewithlimits{mean}_{\\text{all }j}(1 - \\max _{\\text{all }i}\\alpha _{ij})^2\\\\\\end{split}$ Backpropagation using the differentiable losses affects only the weights of the output softmax perceptron.",
"The overall loss function for these weights is a weighted sum of all $\\mathcal {L}_k$ terms and $\\mathcal {L}_c$ : $\\begin{split}\\mathcal {L}&=100\\mathcal {L}_c+10\\sum _{\\text{all }k}\\mathcal {L}_k\\end{split}$ There are additionally nondifferentiable rewards $r$ that bias the model towards or away from certain kinds of tree structures.",
"Here, negative numbers correspond to penalties.",
"We assign a tree reward of $-1$ when the model predicts a reduce with only one child constituent (reduce with zero child constituents is forbidden) or predicts two reduce or nt transitions in a row.",
"This biases the model against unary branching and reduces its likelihood of producing an exclusively left- or right-branching tree structure.",
"In addition, for all constituents except the root, we assign a tree reward based on the size and type of its children.",
"If $n$ and $t$ are the number of nonterminal and terminal children, this reward is $4t$ if all children are terminal and $9\\sqrt{n}$ otherwise.",
"A reward structure like this biases the model against freely mixing terminals and nonterminals within the same constituent and provides incentive to build substantial tree structures early on in training so the model doesn't get stuck in trivial local minima.",
"Within both the encoder and decoder, each stochastic action node has a corresponding tree reward $r_k$ if the action was reduce (otherwise zero) and a corresponding language model loss $\\mathcal {L}_k$ if the action was gen (otherwise zero).",
"We subtract an exponential moving average baseline from each tree reward and additional exponential moving average baselines—computed independently for each character $z$ in the vocabulary, because we want to reduce the effect of character frequency—from the language model losses.",
"If $\\textsc {gen}(k)$ is the number of gen transitions among actions one through $k$ , and $\\gamma $ is a decay constant, the final reward $\\mathcal {R}_k^m$ for action $k$ with $m\\in \\lbrace \\text{enc}, \\text{dec}\\rbrace $ is: $\\begin{split}\\hat{r}_k&=r_k-r_{\\rm baseline}\\\\\\hat{\\mathcal {L}}_k&=\\mathcal {L}_k-\\mathcal {L}_{\\rm baseline}(z_k)\\\\\\hat{\\mathcal {R}}_k&=\\sum _{\\kappa =k}^{K_m}\\gamma ^{\\textsc {gen}(\\kappa )-\\textsc {gen}(k)}(\\hat{r}_\\kappa -\\hat{\\mathcal {L}}_\\kappa ^m)\\\\\\mathcal {R}_k^{\\rm m}&=\\hat{\\mathcal {R}}_k-\\mathcal {L}_c-(m=\\text{enc})\\sum _{\\kappa =1}^{K_{\\rm dec}}\\mathcal {L}_k^{\\rm dec}.\\end{split}$ These rewards define the gradient that each stochastic node (with normalized action probabilities $p_k^a$ and chosen action $a_k$ ) produces during backpropagation according to the standard multinomial score function estimator (REINFORCE): $\\begin{split}\\nabla _{\\theta } p_k^a &=\\operatornamewithlimits{mean}_{\\rm a_k=a}\\mathcal {R}_k \\nabla _{\\theta } \\log p_k^{a_k}=\\operatornamewithlimits{mean}_{\\rm a_k=a}\\frac{-\\mathcal {R}_k}{p_k^{a_k}}\\end{split}$"
],
[
"Results",
"We evaluated our model on the German-English language pair of the flickr30k data, the textual component of the WMT Multimodal Translation shared task [22].",
"An attentional sequence-to-sequence model with two layers and 384 hidden units from the OpenNMT project [13] was run at the character level as a baseline, obtaining 32.0 test BLEU with greedy inference.",
"Our model with the same hidden size and greedy inference achieves test BLEU of 28.5 after removing repeated bigrams.",
"We implemented the model in PyTorch, benefiting from its strong support for dynamic and stochastic computation graphs, and trained with batch size 10 and the Adam optimizer [12] with early stopping after 12 epochs.",
"Character embeddings and the encoder's $\\mathbf {x}^{\\rm enc}$ embedding were initialized to random 384-dimensional vectors.",
"The value of $\\gamma $ and the decay constant for the baselines' exponential moving average were both set to 0.95.",
"Figure: Attention visualizations for two sentences from the development set.",
"Attention between two constituents is represented by a shaded rectangle whose projections on the xx and yy axes cover the encoder and decoder constituents respectively.A random selection of translations is included in the supplemental material, while two attention plots are shown in Figure 2.",
"Figure 2b demonstrates a common pathology of the model, where a phrasal encoder constituent would be attended to during decoding of the head word of the corresponding decoder constituent, while the head word of the encoder constituent would be attended to during decoding of the decoder constituent corresponding to the whole phrase.",
"Another common pathology is repeated sentence fragments in the translation, which are likely generated because the model cannot condition future attention directly on past attention weights (the “input feeding” approach introduced by [15]).",
"Translation quality also suffers because of our use of a stack-only RNNG, which we chose because an RNNG with both stack and buffer inputs is incompatible with a language model loss.",
"During encoding, the model must decide at the very beginning of the sentence how deeply to embed the first character.",
"But with a stack-only RNNG, it must make this decision randomly, since it isn't able to use the buffer representation—which contains the entire sentence."
],
[
"Conclusion",
"We introduce a new approach to leveraging unsupervised tree structures in NLP tasks like machine translation.",
"Our experiments demonstrate that a small-scale MT dataset contains sufficient training signal to infer latent linguistic structure, and we are excited to learn what models like the one presented here can discover in full-size translation corpora.",
"One particularly promising avenue of research is to leverage the inherently compositional phrase representations $\\mathbf {h}_i^{\\rm enc}$ produced by the encoder for other NLP tasks.",
"There are also many possible directions for improving the model itself and the training process.",
"Value function baselines can replace exponential moving averages, pure reinforcement learning can replace teacher forcing, and beam search can be used in place of greedy inference.",
"Solutions to the translation pathologies presented in Section 3 are likely more complex, although one possible approach would leverage variational inference using a teacher model that can see the buffer and helps train a stack-only student model.",
"Figure: NO_CAPTION"
]
] | 1709.01915 |
[
[
"Concurrence Topology of Some Cancer Genomics Data"
],
[
"Abstract The topological data analysis method \"concurrence topology\" is applied to mutation frequencies in 69 genes in glioblastoma data.",
"In dimension 1 some apparent \"mutual exclusivity\" is found.",
"By simulation of data having approximately the same second order dependence structure as that found in the data, it appears that one triple of mutations, PTEN, RB1, TP53, exhibits mutual exclusivity that depends on special features of the third order dependence and may reflect global dependence among a larger group of genes.",
"A bootstrap analysis suggests that this form of mutual exclusivity is not uncommon in the population from which the data were drawn."
],
[
"Introduction",
"This is a report of some work I have done under the auspices of the the Rabadan Lab in the department of Systems Biology at Columbia University (https://rabadan.c2b2.columbia.edu/).",
"Drs.",
"Rabadan and Camara kindly provided me with some genomic data on glioblastoma (GBM).",
"(See section .)",
"I dichotomized those data and applied the “concurrence topology (CT)” (Ellis and Klein [3]) method to them.",
"Concurrence topology (CT) is a method of “topological data analysis” that uses persistent homology to describe aspects of the statistical dependence among binary variables.",
"I detected some one-dimensional homology with apparently long lifespan.",
"The first question I investigated was, is that lifespan statistically significantly long?",
"“Long” compared to what?",
"Statistical significance is always based on a “null” hypothesis.",
"I took the null hypothesis to be that the observed persistent homology among mutations can be explained simply by the first and second order dependence among the mutations.",
"Second order dependence is that which can be described fully by looking at genes just two at a time.",
"A $p$ -value can be computed by simulating data whose distribution is completely determined by the first and second order statistics of the GBM data.",
"Specifically, I endeavored to simulate binary data sets of the size of the GBM data in such a way that all such data sets whose first and second order statistics approximate those of the GBM data are equally likely.",
"What I mean by “approximate” is specified in section .",
"Simulating such data is itself rather challenging (section ) and I am not sure that my efforts were completely successful."
],
[
"Data and CT analysis",
"The GBM data set consists of data on 290 tumors.",
"Dr. Camara recommended 75 genes of which I was able to locate 69 in the data set.",
"Each entry in the $290 \\times 69$ matrix is a numerical score ranging from 0 to 4, inclusive.",
"I dichotomized the data by converting every positive value to 1.",
"So “1” indicates the presence of a mutation.",
"The following table lists for every gene the number of tumors in which it was mutated.",
"$\\begin{matrix}PTEN & TP53 & EGFR & PIK3R1 & NF1 \\\\90 & 84 & 77 & 33 & 32 \\\\PIK3CA & RB1 & MUC17 & HMCN12 & ATRX\\\\32 & 25 & 23 & 19 & 17 \\\\IDH1 & KEL & COL6A3 & STAG25 & GABRA6 \\\\15 & 15 & 14 & 12 & 11 \\\\LZTR1 & PIK3C2G & SEMG1 & F5 & RPL5 \\\\10 & 9 & 9 & 9 & 8 \\\\TPTE22 & NUP210L & IL4R & BCOR0 & BRAF\\\\8 & 8 & 8 & 7 & 6 \\\\TP63 & TRPA1 & TLR6 & QKI & PTPN11 \\\\6 & 5 & 5 & 5 & 5 \\\\PLCG1 & SETD22 & FAM126B & ZDHHC46 & TCF12 \\\\5 & 5 & 4 & 4 & 4 \\\\DDX5 & SLC6A3 & CLCN7 & RNF16818 & GLT8D2 \\\\4 & 4 & 4 & 4 & 4 \\\\TGFA & EEF1A1 & AOX1& ACAN& NIPBL \\\\4 & 4 & 4 & 4 & 3 \\\\ZNF292 & KRT13 & RBBP6 & EPHA3 & CLIP1 \\\\3 & 3 & 3 & 3 & 3 \\\\KRT15 & CREBZF & MAX & ST3GAL62 & ARID1A \\\\2 & 2 & 2 & 2 & 2 \\\\KRAS & C15orf48 & TYRP1 & ARID228 & PPM1J2\\\\2 & 2 & 2 & 2 & 2 \\\\ZBTB20 & NRAS & IL1RL1 & C10orf76 & EIF1AX \\\\1 & 1 & 1 & 1 & 1 \\\\CIC & SMARCA4 & ABCD1& EDAR\\end{matrix}$ I ran my CT code on this binary data set for dimension 1 and with $\\mathbb {Z}/2 \\mathbb {Z}$ coefficients.",
"Figure REF shows the persistence diagram.",
"Figure: CT persistence plot in dimension 1 for GBM data.",
"(Since CT represents data as a descending filtration, persistence is indicated by points below the main diagonal.)",
"In order to represent multiplicity, I employ a “sunflower plot”.",
"The large disk represents 43 classes born at frequency level 1.",
"A simple dot represents one persistent class.",
"Each ray coming out of a point represents one class with birth and death given by the coordinates at that point.",
"For example, six classes were born at frequency level 2 and died at frequency level 1.",
"Two classes were born at frequency level 15 and died at frequency level 3.The two persistence classes corresponding to the dot in the figure lying furthest below the diagonal line are the ones with the longest lifespan.",
"They are born in “frequency level” 15 and have lifespan 13.",
"Since at least three subjects (tumors) are needed to form a 1-cycle, each of these persistent classes involves at least 3*15 tumors, or about 15.5% of the sample.",
"In CT it is often possible to “localize” classes in the sense of looking for representative “short cycles”.",
"A “short cycle” in dimension 1 means a cycle consisting of three 1-simplices, line segments.",
"A short cycle is uniquely determined by its vertices.",
"In this context, vertices correspond to genes.",
"We find that each of the two classes with lifespan 13 has a short representative in frequency levels 15 and lower.",
"They are: EGFR, TP53, PTEN and PTEN, RB1, TP53.",
"Each of these triplets exhibit “mutual exclusivity” (Ciriello et al [1], Szczurek1 and Niko Beerenwinkel [6], and Melamed et al [5]).",
"Note that the mutations involved in these short cycles are the three most common in the sample, EGFR, TP53, PTEN, and the seventh most common, RB1.",
"Later we will find more evidence that there is something special about RB1.",
"These triples of genes reflect more than just mutual exclusivity.",
"The corresponding 1-cycles represent homology and homology is a global property involving all 69 genes.",
"So we have found an apparent global pattern of mutation in GBM.",
"However, I do not have a biological interpretation of this kind of structure."
],
[
"Simulation",
"My goal was to generate random data sets that share the same first and second order statistics as the real data, at least approximately.",
"For the cancer data this seems a little tricky.",
"Here we discuss the algorithm I used.",
"Let $D$ be the $N \\times d$ data matrix.",
"So $N = 290$ is the number of samples and $d = 69$ is the number of genes.",
"We have $N > d$ .",
"$D$ is a binary (0 – 1) matrix and its $(i,j)$ entry is 1 if and only if the $j^{th}$ gene for the $i^{th}$ tumor is mutated.",
"The first and second order statistics are captured by the $d \\times d$ matrix $C := D^{T} D$ , where “${}^{T}$ ” indicates matrix transposition.",
"For $i \\ne j$ , the $(i,j)$ entry of $C$ is the number of times mutations of both genes $i$ and $j$ are present in the same sample, a second order statistic.",
"The $i^{th}$ diagonal element of $C$ is the number of samples in which the $i^{th}$ gene is mutated, a first order statistic.",
"For $n = 1, 2, \\ldots $ , let $1_{n}$ be the column vector all of whose entries are 1.",
"Then $s := 1_{N}^{T} D$ is the $d$ -dimensional row vector column sums of $D$ .",
"Thus, $s$ is the same as the diagonal of $C$ .",
"“Center” $D$ by subtracting out the column means: $D_{0} := D - M$ where $M^{N \\times d} := N^{-1} 1_{N} \\, s$ .",
"(I use superscripts to indicate matrix dimensions.)",
"Thus, the column sums of $D_{0}$ are all 0: $1_{N}^{T} D_{0} = 0$ .",
"Let $D_{0} = U \\Lambda V^{T}$ be the singular value decomposition of $D_{0}$ (Wikipedia).",
"Thus, $U$ is an $N \\times d$ matrix with orthonormal columns, $\\Lambda $ is $d \\times d$ diagonal, and $V$ is orthogonal.",
"For our data sets no entry of $\\Lambda $ is 0 and all the entries are distinct.",
"Since $0 = 1_{N}^{T} D_{0} = 1_{N}^{T} U \\Lambda V^{T}$ it follows that $ 1_{N}^{T} U = 0.$ Moreover, $D_{0}^{T} D_{0} = V \\Lambda ^{2} V^{T}$ .",
"Hence, the diagonal of $\\Lambda ^{2}$ is the vector of eigenvalues of $D_{0}^{T} D_{0}$ and the columns of $V$ are the corresponding unit eigenvectors.",
"Therefore, since the eigenvalues are distinct, the columns of $V$ are unique up to sign.",
"Observe that $N^{-1} D_{0}^{T} D_{0}$ is just the (variance-)covariance matrix of $D$ .",
"Hence, columns of $V$ are the unit eigenvectors of the covariance matrix and the diagonal elements of $N^{-1} \\Lambda ^{2}$ are the eigenvalues.",
"We have $ D = D_{0} + M = U \\Lambda V^{T} + M$ where, you recall, $D$ is the original data matrix.",
"Let $W^{N \\times 1}$ be any matrix with orthogonal columns s.t.",
"(such that) $1_{N}^{T} W = 0$ .",
"For example, by (REF ), we can take $W = U$ .",
"Let $Y := W \\Lambda V^{T} + M$ .",
"Since $1_{N}^{T} W = 0$ we have $W^{T} M = N^{-1} W^{T} 1_{N} s = 0$ .",
"Similarly, $M^{T} M = N^{-2} s^{T} 1_{N}^{T} 1_{N} s = N^{-1} s^{T} s$ , since $1_{N}^{T} 1_{N} = N$ .",
"Thus, since the columns of $W$ are orthonormal, $Y^{T} Y = ( V \\Lambda W^{T} + M^{T} ) (W \\Lambda V^{T} + M) = V \\Lambda ^{2} V^{T} + N^{-1} s^{T} s.$ So $Y^{T} Y$ does not depend on $W$ .",
"In particular, $Y^{T} Y = D^{T} D = C$ .",
"We can use this fact to sample uniformly from the set of all $N \\times d$ matrices, $Y$ , s.t.",
"$Y^{T} Y = C$ , i.e.",
"to sample uniformly from the set of all $N \\times d$ matrices having the same first and second order statistics that $D$ has.",
"One merely has to sample uniformly from the space of all matrices $W^{N \\times 1}$ with orthogonal columns s.t.",
"(such that) $1_{N}^{T} W = 0$ .",
"Such sampling can be done easily as follows.",
"Let $w^{N \\times 1}$ be a random Gaussian column vector with statistically independent population mean 0 components.",
"Let $\\bar{w} = N^{-1} 1_{N}^{T} w$ be the sample mean of the components of $w$ .",
"($\\bar{w}$ is just a random number.)",
"Center $w$ , i.e., replace it by $w_{1} := w - \\bar{w} 1_{N}$ .",
"Thus, $w_{1}$ is a random $N$ -column vector and $1_{N}^{T} w_{1} = 0$ .",
"Repeat this operation independently $d$ times producing column vectors $w_{1}, \\ldots , w_{d}$ .",
"Apply the Gram-Schmidt orthogonalization process to these vectors to produce orthonormal column $N$ -vectors $w_{1}^{\\prime }, \\ldots , w_{d}^{\\prime }$ .",
"Since $w_{1}, \\ldots , w_{d}$ all have sample mean 0, so do $w_{1}^{\\prime }, \\ldots , w_{d}^{\\prime }$ .",
"Let $W^{N \\times d}$ be the matrix whose columns are $w_{1}^{\\prime }, \\ldots , w_{d}^{\\prime }$ .",
"Finally, take $Y := W \\Lambda V^{T} + M$ .",
"Then we know $Y^{T} Y = C$ .",
"I mentioned above that the columns of $V$ are unique up to sign.",
"One might think that one can make the distribution of $Y$ more uniform by randomly changing the signs of the columns of $V$ .",
"Let $E^{d \\times d}$ be a diagonal matrix with diagonal entries, $\\epsilon _{i} = \\pm 1$ , ($i=1, \\ldots , d$ ).",
"Since diagonal matrices commute, we have $W \\Lambda (V E)^{T} = W \\Lambda E V^{T} = (W E) \\Lambda V^{T}$ .",
"Thus, multiplying the columns of $V$ by $\\epsilon _{i}$ , ($i=1, \\ldots , d$ ) amounts to replacing $w_{1}^{\\prime }, \\ldots , w_{d}^{\\prime }$ by $\\epsilon _{1} w_{1}^{\\prime }, \\ldots , \\epsilon _{d} w_{d}^{\\prime }$ .",
"Now, examination of the Gram-Schmidt process shows that $\\epsilon _{1} w_{1}^{\\prime }, \\ldots , \\epsilon _{d} w_{d}^{\\prime }$ is exactly what one gets when one applies Gram-Schmidt to the original $\\epsilon _{1} w_{1}, \\ldots , \\epsilon _{d} w_{d}$ , i.e.",
"changing the signs of the independent Gaussian random vectors we started with.",
"But theses random vectors are independent Gaussian with mean 0, therefore changing their signs does not change their distribution.",
"This shows that changing the signs of the columns of $V$ does not change the distribution of $Y$ and, thus, is unnecessary.",
"$Y$ constructed as above is uniformly distributed over the space of matrices $X^{N \\times d}$ with $X^{T} X = C$ .",
"There is only one problem.",
"We want binary matrices and the probability that $Y$ generated as above is binary is 0.",
"The obvious remedy is to threshold: Write $Y = ( y_{ij} )$ .",
"For some number $t$ replace each entry $y_{ij}$ by 0 if $y_{ij} < t$ and by 1 otherwise.",
"Call the resulting binary matrix $B_{t}^{N \\times d}$ .",
"But what threshold $t$ should we use?",
"Alas, we have to accept the fact that no matter what $t$ we use we will not have $B_{t}^{T} B_{t} = C$ .",
"So we have to settle for an approximation $B_{t}^{T} B_{t} \\approx C$ .",
"We pick $t$ to get a “best” approximation.",
"In order to define what “best” means we need a definition of distance between $B_{t}^{T} B_{t}$ and $C$ .",
"One possibility is to use the (squared) Euclidean or Frobenius matrix distance: $trace \\, (B_{t}^{T} B_{t} - C)^{T} (B_{t}^{T} B_{t} - C)$ .",
"This is just the sum of the squared entries of $B_{t}^{T} B_{t} - C$ .",
"But remember that $D$ , and hence, $C$ are random and a more stable distance would use weights approximately equal to the variances of the entries of $C$ .",
"There are numerous ways might estimate these variances.",
"I employed a simple one.",
"Remember that the entries of $C$ are counts, non-negative integer-values.",
"Perhaps the simplest distribution of a non-negative integer-valued random variable is the Poisson.",
"So a crude estimate of the variance of an entry is just the entry itself.",
"However, there are hundreds of unique values in $C$ so the values themselves will be very noisy estimates.",
"Now, in statistics it is well known that when simultaneously estimating a large number of quantities one improves estimates by shrinking toward a constant.",
"I informally employed that technique.",
"Let $c$ , a number, be the sample mean of all the entries in $C$ and let $\\bar{C}$ be the $d \\times d$ matrix all of whose entries are $c$ .",
"Then for the purpose of weighting we replace $C$ by $\\hat{C} := (1/2) C + (1/2) \\bar{C}$ .",
"Now define the “distance”, $\\delta (B_{t}^{T} B_{t}, C)$ , between $B_{t}^{T} B_{t}$ and $C$ as follows.",
"Form the matrix $\\Delta _{2}^{d \\times d} = (\\delta _{ij})$ whose $ij^{th}$ entry is the squared difference between the $ij^{th}$ entry of $B_{t}^{T} B_{t}$ and the $ij^{th}$ entry of $C$ .",
"Now divide $\\delta _{ij}$ by the $ij^{th}$ entry of $\\hat{C}$ .",
"Add up all those quotients.",
"That's the “distance”, $\\delta (B_{t}^{T} B_{t}, C)$ .",
"However, $B_{t}^{T} B_{t}$ is symmetric, so in this procedure the unique off-diagonal elements are counted twice.",
"To make up for this, we modify $\\hat{C}$ by dividing its diagonal by 2.",
"This doubles the contribution of the diagonals of $B_{t}^{T} B_{t}$ and $C$ .",
"The binary matrix we want will not be all 0 or 1.",
"Therefore, the only thresholds we need try are the distinct numeric values in $Y$ .",
"We try all those values as thresholds and pick the one that minimizes $\\delta (B_{t}^{T} B_{t}, C)$ .",
"How do we know that the minimum distance we achieve is small enough?",
"Again, the data matrix $D$ itself is random.",
"Even if we gathered another data set $D_{2}$ using the same method used to gather $D$ we would not have $D_{2}^{T} D_{2} = C$ .",
"Therefore, it is unreasonable to insist that $\\delta (B_{t}^{T} B_{t}, C)$ be tiny.",
"But how small a value for $\\delta (B_{t}^{T} B_{t}, C)$ is acceptable?",
"Looking at the distribution of $\\delta (D_{2}^{T} D_{2}, C)$ , where $D_{2}$ is drawn at random from the same population that $D$ came from gives us a yardstick to use for judging sizes of $\\delta (B_{t}^{T} B_{t}, C)$ .",
"Now, we cannot draw new samples $D_{2}$ , but we can approximate that process by drawing samples, with replacement, from the rows of $D$ .",
"This is the non-parametric bootstrap (Efron and Tibshirani [4]).",
"One draws many samples, with replacement, from the rows of $D$ .",
"(These are called “resamples”.",
"I drew 2,000.)",
"Each time one obtains a matrix $D_{2}$ .",
"One then records the value of $\\delta (D_{2}^{T} D_{2}, C)$ for each resample.",
"The distribution of all these numbers approximates the distribution one would get by taking many samples $D_{2}$ from the population.",
"I chose the median $m_{2}$ of these distances as the cutoff for distinguishing matrices that are close to $D$ from those that are not.",
"Unfortunately, even the closest $B_{t}$ , call it $B_{t_{opt}}$ , generated by thresholding $Y$ practically always fails this test.",
"So more work needs to be done to $B_{t_{opt}}$ to make it acceptable.",
"To do this I used an informal “Markov Chain Monte Carlo (MCMC)” algorithm (Wikipedia).",
"Intialize $Z_{1} := B_{t_{opt}}$ and $b_{1} := \\delta (Z_{1}^{T} Z_{1}, C)$ .",
"At each step pick a random entry in $Z_{1}$ and flip it so 0 gets replaced by 1 or vice versa.",
"(Call that a “flip attempt”.)",
"Call the resulting matrix $Z$ .",
"Then compute $b := \\delta (Z^{T} Z, C)$ .",
"If $b < b_{1}$ (a “successful flip”) then set $Z_{1} := Z$ and $b_{1} := b$ .",
"Otherwise, $Z_{1}$ and $b_{1}$ are not changed.",
"That is the iteration.",
"What is the stopping rule?",
"I stopped the iteration as soon as $b_{1} < m_{2}$ .",
"One might be concerned that when the algorithm halts the distance $b_{1} := \\delta (Z_{1}^{T} Z_{1}, C)$ would be only slightly smaller than $m_{2}$ .",
"Values much smaller than $m_{2}$ would never be achieved.",
"However, it is well known that the volume of a high dimensional ball is almost entirely found near the boundary.",
"So if one did sample matrices from the ball centered at $D$ and having $\\delta $ -radius $m_{2}$ , one would rarely get a matrix whose $\\delta $ -distance from $D$ is much smaller than $m_{2}$ .",
"So that is not a legitimate objection to the “MCMC” algorithm.",
"A more justified concern is the following.",
"Above we argued that $Y$ constructed as above is uniformly distributed over the space of matrices $X^{N \\times d}$ with $X^{T} X = C$ .",
"Even the matrix $B_{t_{opt}}$ , though not close to $D$ , is unbiased in terms of the direction $B_{t_{opt}} - D$ .",
"I.e., I conjecture that it would not be hard to prove that the expected value of $B_{t_{opt}} - D$ is 0.",
"However, I fear that the informal MCMC step in the construction might introduce some bias.",
"Still, why not skip the SVD step and just perform the “MCMC” step?",
"As an experiment I generated a 0 matrix of the same dimensions as the data and randomly replaced 712 of the entries by 1's, where 712 is the number of 1's in the original data matrix..",
"It took more than 21,000 flip attempts with over 1,000 successful flips to bring this matrix close to the data matrix.",
"A second attempt produced similar results.",
"As a benchmark, note that there are 20010 positions in data matrix.",
"But it is not just to save flips that the svd-based starting matrix is helpful.",
"That approach also serves, I believe, to help generate binary matrices (nearly?)",
"uniformly distributed among matrices with first and second order statistics similar to the real population."
],
[
"Simulation results and bootstrap",
"I generated 500 synthetic data sets using the algorithm described in section .",
"In contrast to the experiments described in the last paragraph of the last section, in these 500 MCMC calculations, the range in the number of flip attempts needed to bring the second order statistics acceptably close to those of the data was 625 to 2076 with a median of 1061.",
"The number of successful flips ranged from 193 to 329 with a median of 255.",
"For each synthetic data set I found the persistent 1-D homology classes with the longest lifespan.",
"(There were practically always just one such class.)",
"Here are summary statistics for those lifespans: $\\begin{matrix}Min.",
"& 1st Qu.",
"& Median & Mean & 3rd Qu.",
"& Max.",
"\\\\5.00 & 12.00 & 14.00 & 14.36 & 17.00 & 25.00\\end{matrix}$ We observe that the maximum lifespan in the real data, viz., 13, is not remarkable in the simulated data.",
"In fact, 56.4% of the time the maximum lifespan obtained in simulation was larger than in the data.",
"For those classes with maximum lifespan I also recorded their frequency level of birth.",
"Here are summaries.",
"$\\begin{matrix}Min.",
"& 1st Qu.",
"& Median & Mean & 3rd Qu.",
"& Max.",
"\\\\6.00 & 19.00 & 22.00 & 20.63 & 24.00 & 32.00\\end{matrix}$ We observe that the births in the real data, viz., 15, is actually rather small by comparison.",
"In fact 84.4% of the time the birth of the class with maximum lifespan obtained in simulation was larger than in the data.",
"The vertices that appear in one or both of the short cycles I found in the longest lived classes in the GBM data (section ) are EGFR, PTEN, RB1, and TP53.",
"For each of them I recorded if it appeared in a short cycle in the simulated data.",
"(The persistent class with the longest lifespan was represented by one or more short cycles in all but ten of the simulations.)",
"Here are the proportions of the simulations in which that happened.",
"$\\begin{matrix}EGFR & PTEN & RB1 & TP53 \\\\0.914 & 0.968 & 0.080 & 0.970\\end{matrix}$ We see, then, that the homology class represented by the short cycle with vertices PTEN, RB1, TP53 appears rather uncommonly in the simulated data (because RB1 appears uncommonly).",
"But perhaps the cycle with vertices PTEN, RB1, TP53 also appears uncommonly in the population from which the GBM data is derived.",
"To answer this question, in the spirit of Chazal et al [2], I applied the bootstrap method mentioned in section with 500 resamples.",
"Thus, I resampled the tumors (rows) of the GBM matrix and computed the same summaries I just described for the SVD simulations.",
"I found that in over 25% of the resamples the longest lived classes had short cycles including RB1 as a vertex.",
"This is (circumstantial) evidence that in the population of tumors from which the data are drawn the homology class represented by the short cycle with vertices PTEN, RB1, TP53 is not uncommon."
],
[
"Discussion",
"Applying the Concurrence Topology method to the GBM data we found two cycles with apparently long lifespan.",
"But “long” compared to what?",
"Compared to results one would get just “by luck”.",
"But what kind of luck?",
"Complete independence among mutations is biologically unrealistic.",
"It is already known that mutations are not independent.",
"(See, for example, the afore-cited articles on mutual exclusivity.)",
"Instead, I focussed on the luck one would observe if the distribution of mutations reflected, approximately, the first and second order statistics in the data.",
"I used an algorithm that is intended to generate samples from such a distribution.",
"It is not hard to generate floating point-valued matrices whose first and second order statistics exactly match those of the data.",
"The difficulty is in satisfying the requirement that the algorithm produce binary matrices having the desired distribution.",
"Specifically, one would like all binary matrices having approximately the right statistics (where “approximately” is defined in section ) to be equally likely.",
"The method I used takes the floating point algorithm as starting point then discretizes and randomly flips entries to achieve an approximate match.",
"Perhaps better algorithms already exist somewhere, otherwise more work in this area is needed.",
"We found that neither the frequency level of birth nor lifespan of the classes in the real data is remarkable in that second order context.",
"However, one of the classes with the longest lifespan in the data appears rather infrequently in the second order distribution but not infrequently in bootstrapped resamples.",
"This suggest the involvement of third and higher order dependence among the mutations.",
"I find it surprising that first and second order dependence can give rise to persistent homology in dimension 1 with long lifespans.",
"I think this is partly due to the fact that some of the mutations, viz., EGFR, TP53, PTEN, are so common.",
"In section I pointed out that persistent homology reflects “mutual exclusivity”.",
"But mutual exclusivity is a “local phenomenon”: mutual exclusivity among a group of mutations, e.g.",
"EGFR, TP53, PTEN, is a property of the joint distribution of just those mutations.",
"But persistent homology is a global property.",
"In general, it reflects the joint distribution of all, in this case 69, genes.",
"In the case of the short cycle EGFR, TP53, PTEN the fact that it represents persistent homology with a long lifespan may just be local because those mutations are far more numerous than the others.",
"However, the short cycle PTEN, RB1, TP53 seems to reflect something more global because RB1 is not such a common mutation.",
"The simulations seem to confirm this."
]
] | 1709.01753 |
[
[
"Dirichlet-to-Robin Operators via Composition Semigroups"
],
[
"Abstract We show well-posedness for an evolution problem associated with the Dirichlet-to-Robin operator for certain Robin boundary data.",
"Moreover, it turns out that the semigroup generated by the Dirichlet-to-Robin operator is closely related to a weighted semigroup of composition operators on an appropriate Banach space of analytic functions."
],
[
"Introduction",
"In recent years, the Dirichlet-to-Neumann operator has been studied intensively.",
"In the beginning of the 20th century, these operators were dealt with theoretically, while in the 1980s and 1990s they were used to analyze inverse problems to determine coefficients of a differential operator.",
"These problems apply, e.g., to image techniques in medicine and also to find defects in materials.",
"According to Arendt and ter Elst, the Dirichlet-to-Neumann operator can be obtained as an example of an operator associated with $m$ -sectorial forms, see [3].",
"Using methods from function theory, our purpose is to give an alternative approach to Poincaré-Steklov operators and the related semigroups on boundary spaces of Banach spaces of analytic functions.",
"It turns out, as pointed out by Lax [14], that there is a surprising connection between semigroups of composition operators on spaces of harmonic functions on the unit disk referring to a specific semiflow and the Dirichlet-to-Neumann operator.",
"In fact, we can extend this observation to the Laplace equation with Robin boundary conditions on Jordan domains in $.More precisely, we study the evolution problem$ ${\\left\\lbrace \\begin{array}{ll}\\partial _{t}u-g\\cdot u-G\\cdot \\partial _{z}u=0 &\\textrm {on } (0,\\infty )\\times \\partial \\Omega ,\\\\-\\Delta u=0 & \\textrm {on } (0,\\infty )\\times \\partial \\Omega ,\\\\u(0,\\cdot )=u_{0} & \\textrm {on }\\partial \\Omega ,\\end{array}\\right.",
"}$ where $\\Omega \\subsetneq is a Jordan domain and $ G$ and $ g$ are boundary values of appropriate holomorphic functions on $$.",
"We prove well-posedness of (\\ref {eq:DTR}) in various spaces of distributions on $$ including the scale of $ Lp$-spaces.As mentioned above, our approach does not use form methods but the theory of (weighted) composition operators on spaces of holomorphic and harmonic functions (for the moment only) on planar domains.",
"Our method appears to be restricted to problems involving the Laplace operator, while the variational approach to Dirichlet-to-Neumann and Dirichlet-to-Robin operators using the theory of forms is quite flexible with respect the choice of elliptic operators in the domain $$.",
"However, there it seems difficult to handle coefficients in front of the associated Neumann derivative (at least, we do not see how to handle them).",
"Here, we can allow a large class of coefficient functions $ G$ and $ g$.",
"In particular, it may happen that $ G$ degenerates at one point on the boundary.",
"Moreover, using our method, we can define Dirichlet-to-Neumann and Dirichlet-to-Robin operators on several spaces of distributions.$ This article is organized as follows.",
"In Section 2 we introduce the notion of admissible spaces which is eventually our tool to solve the above posed evolution problem.",
"We discuss some examples of admissible spaces, and we investigate corresponding boundary spaces.",
"Then, in Section 3, we examine the connection between certain Poincaré-Stecklov operators, namely Dirichlet-to-Neumann and Dirichlet-to-Robin operators, and weighted semigroups of composition operators, and prove our main theorem."
],
[
"Admissible spaces",
"Initiated by the famous paper by Berkson and Porta [6], semigroups of composition operators were studied intensively by many authors on various spaces of holomorphic functions defined on the unit disk, see, for example, [2], [5], [13], [18], [17].",
"In our approach, we consider (weighted) semigroups of composition operators on spaces of harmonic and holomorphic functions which are defined on a simply connected domain $\\Omega \\subsetneq bounded by a Jordan curve.To give the definition of such a semigroup, we need the notion ofa semiflow of holomorphic functions.$"
],
[
"Semiflows of holomorphic functions",
"Definition 2.1 Let $\\Omega \\subsetneq be simply connected.",
"Let$ :$ be holomorphic (we write $ H()$)such that for every $ t>0$ the fractional iterates $ t$are holomorphic selfmaps in $$.",
"A family $ (t)t$is called a semiflow of holomorphic functions if it satisfies thefollowing properties:\\begin{enumerate}\\item \\varphi _{0}(z)=z for all z\\in \\Omega ,\\item \\varphi _{s+t}(z)=\\varphi _{s}(\\varphi _{t}(z)) for all s,t>0 andz\\in \\Omega ,\\item \\varphi _{t}(z)\\rightarrow z as t\\rightarrow 0^{+} for all z\\in \\Omega .\\end{enumerate}$ Given a semiflow $(\\varphi _{t})_{t}$ we define its generator by $G(z)\\colon =\\lim _{t\\rightarrow 0^{+}}\\frac{\\varphi _{t}(z)-z}{t}$ for every $z\\in \\Omega $ .",
"Since $\\Omega $ is simply connected, by the Riemann mapping theorem there exists a conformal map $k:\\Omega \\rightarrow , and thus every semiflowon $$ can be written in terms of a semiflow on the unit disk.Let $ (t)t$ be a semiflow on $ .",
"As a consequence of the chain rule, the generator of $(\\psi _{t})_{t}:=(k^{-1}\\circ \\varphi _{t}\\circ k)_{t}$ can be written in terms of the generator of $(\\varphi _{t})_{t}$ .",
"For all holomorphic selfmaps $\\varphi $ in the unit disk which are not automorphisms, the embeddability into a semiflow can be characterized in terms of the Denjoy-Wolff point of $\\varphi $ , see for instance [10].",
"The Denjoy-Wolff point is defined as the unique fixed point of a holomorphic selfmap in the unit disk which is not an automorphism in the unit disk.",
"Such a point can be found in the interior of the unit disk as well as on the boundary.",
"Thus we can use appropriate Möbius transforms to shift an interior Denjoy-Wolff point to zero and a Denjoy-Wolff point on the boundary to 1.",
"In our case, the representation of $(\\psi _{t})_{t}$ on $\\Omega $ in terms of a semiflow on the unit disk gives also the unique fixed point of every $\\psi _{t}$ as $k^{-1}(b)$ where $b$ is the Denjoy-Wolff point of $(\\varphi _{t})_{t}$ .",
"From the theory of differential equations, we obtain that $\\varphi _{t}$ is univalent for every $t>0$ , hence the same is true for $\\psi _{t}$ .",
"Let $b\\in \\bar{ be the Denjoy-Wolff point of a semiflow (\\varphi _{t})_{t}in \\mathcal {H}(.",
"Then, by \\cite {BP78}, the generator of (\\varphi _{t})_{t}is given by the Berkson and Porta formula\\begin{equation}G(z)=F(z)(\\bar{b}z-1)(z-b),\\end{equation}where F\\colon \\mathbb {D}\\rightarrow \\mathbb {C} is holomorphic and\\operatorname{Re}(F(z))\\ge 0\\,(z\\in .",
"It is also well known that G is holomorphicin and that \\frac{d}{dt}\\varphi _{t}=G(\\varphi _{t}).In fact, if a holomorphic function G\\colon extendscontinuously to \\bar{ and \\operatorname{Re}(G(z)\\bar{z})\\le 0 for everyz\\in , then G is the generator of a semiflow in , see\\cite [Thm 1]{ARS96}.",
"Conversely, a generator of a semiflow need notextend continuously to the closure of On the other hand, notethat, by Fatou^{\\prime }s theorem, a generator G has radial limits almosteverywhere since the function F is the composition of a boundedholomorphic function and a Möbius transform.",
"The angle condition atthe boundary still holds.\\begin{lemma}Let (\\varphi _{t})_{t} be a semiflow in the unit disk and G itsgenerator.",
"Then \\operatorname{Re}(G(z)\\bar{z})\\le 0,\\,\\text{for a.e.",
"}z\\in \\partial .\\end{lemma}}\\begin{proof}Let b\\in \\bar{ be the Denjoy-Wolff point of (\\varphi _{t})_{t}.Then, by \\cite {BP78}, the generator is given by (\\ref {eq:BP}) andradial limits exist almost everywhere.",
"For z\\in \\partial we have}\\begin{align*}\\operatorname{Re}(F(z)(\\bar{b}z-1)(z-b)\\bar{z}) &= \\operatorname{Re}(F(z)(\\bar{b}-\\bar{z})(z-b))\\\\&= \\operatorname{Re}(-F(z)|z-b|^{2})\\\\&\\le 0.\\end{align*}\\end{proof}The same result holds true for generators of semiflowson Jordan domains.\\begin{lemma}Let \\Omega \\subsetneq be a Jordan domain.Let (\\varphi _{t})_{t} be a semiflow in \\Omega and G its generator.Then \\operatorname{Re}(G(x)\\overline{\\nu (x)})\\le 0,\\text{ for a.e.",
"}\\,x\\in \\partial \\Omega ,where \\nu (x) is the normal vector at x.\\end{lemma}}\\begin{proof}Let k\\colon \\Omega \\rightarrow be conformal.",
"Therefore (\\psi _{t})_{t}=(k\\circ \\varphi _{t}\\circ k^{-1})_{t}is a semiflow in the unit disk.",
"Let \\tilde{G} be the generatorof (\\psi _{t})_{t}.",
"Then \\operatorname{Re}(\\tilde{G}(z)\\bar{z})\\le 0,\\,\\text{for a.e.",
"}z\\in \\partial .\\\\For z=k(x)\\in , we have\\begin{align}\\tilde{G}(k(x))=\\lim _{t\\rightarrow 0}\\frac{d}{dt}\\psi _{t}(k(x)) &= \\lim _{t\\rightarrow 0}\\frac{d}{dt}k\\circ \\varphi _{t}(x)\\nonumber \\\\&= \\lim _{t\\rightarrow 0}k^{\\prime }(\\varphi _{t}(x))\\frac{d}{dt}\\varphi _{t}(x)\\nonumber \\\\&= k^{\\prime }(x)G(x).\\end{align}The function k extends continuously to \\bar{ (see \\cite [Thm.",
"2.6]{Po92})and has non-vanishing angular derivative a.e.",
"(see \\cite [Thm.",
"6.8]{Po92}).",
"Furthermore, for x\\in \\partial \\Omega , we have \\nu (x)=\\frac{k(x)}{k^{\\prime }(x)}.",
"For every x\\in \\partial \\Omega there exists a unique z\\in \\partial such that k(x)=z, so}\\begin{align*}\\operatorname{Re}(G(x)\\overline{\\nu (x)}) &= \\operatorname{Re}\\left(\\frac{\\tilde{G}(z)}{k^{\\prime }(x)}\\overline{\\left(\\frac{z}{k^{\\prime }(x)}\\right)}\\right)\\\\&= \\frac{1}{|k^{\\prime }(x)|^{2}}\\operatorname{Re}(\\tilde{G}(z)\\overline{z})\\\\&\\le 0.\\end{align*}\\end{proof}$ Next, we transfer the characterization of generators of semiflows in the unit disk given above to Jordan domains.",
"Proposition 2.2 Let $G:\\Omega \\rightarrow be holomorphic, where $ is simply connected.",
"(I) If $\\partial \\Omega $ is Dini-smooth and $G$ extends continuously to $\\bar{\\Omega }$ and $\\operatorname{Re}(G(x)\\overline{\\nu (x)}\\le 0$ for a.e.",
"$x\\in \\partial \\Omega $ , then $G$ is the generator of a semiflow in $\\Omega $ .",
"(II) If for every conformal map $k:\\Omega \\rightarrow there exists $$and a holomorphic function $ F: with positive real part such that $G(x)=\\frac{F\\circ k(x)(\\overline{k(\\tau )}k(x)-1)(k(x)-k(\\tau ))}{k^{\\prime }(x)}\\quad (x\\in \\Omega ),$ $G$ is the generator of a semiflow in $\\Omega $ .",
"In this case, we say that $G$ admits a conformal Berkson and Porta representation.",
"(I) Let $k:\\Omega \\rightarrow conformal.",
"Define $ G(z)=k'(k-1(z))G(k-1(z))$for $ z.",
"Then $\\tilde{G}$ is a holomorphic function which admits a uniformly continuous extension to $\\bar{, by \\cite [Thm 3.5]{Po92}.Moreover, for z\\in \\partial ,\\begin{align*}\\operatorname{Re}(\\tilde{G}(z)\\bar{z}) &= \\operatorname{Re}\\left(G(k^{-1}(z))\\frac{k^{\\prime }(k^{-1}(z))}{k^{\\prime }(k^{-1}(z))}\\overline{\\left(\\frac{k(k^{-1}(z))}{k^{\\prime }(k^{-1}(z))}\\right)}\\right)\\\\&= \\operatorname{Re}(G(k^{-1}(z))\\overline{\\nu (k^{-1}(x)})\\frac{1}{\\left|k^{\\prime }(k^{-1}(z))\\right|}\\\\&\\le 0.\\end{align*}So we can apply \\cite [Thm.",
"1]{ARS96} which shows that \\tilde{G}is the generator of a semiflow \\psi _{t} in , and by (\\ref {eq:konfGen})G is the generator of the semiflow \\left(\\varphi _{t}\\right)_{t}=\\left(k^{-1}\\circ \\psi _{t}\\circ k\\right)_{t}.\\\\(II) The function (k^{\\prime }\\cdot G)\\circ k^{-1}: is given by theBerkson and Porta formula, hence it is the generator of a semiflowin with Denjoy-Wolff point b=k(\\tau ).",
"The assertion followsagain by (\\ref {eq:konfGen}).",
"}$ Semiflows of holomorphic mappings lead to semigroups of composition operators on spaces of holomorphic functions.",
"Let $\\Omega \\subset be simply connected, and consider the Frechét space$ H(,$ equipped with the topology of uniform convergenceon compact subset of $$.",
"Let $ (Kn)n$ be an increasingsequence of compact subsets of $$ such that $ nKn=$.We define a sequence of seminorms on $ H(,$ as follows$$p_{n}(f)\\colon =\\sup _{z\\in K_{n}}|f(z)|\\quad (f\\in \\mathcal {H}(\\Omega ,),$$and a metric induced by these seminorms by$$d(f,g)\\colon =\\sum _{n=1}^{\\infty }2^{-n}\\frac{p_{n}(f-g)}{p_{n}(f-g)+1}\\quad (f,g\\in \\mathcal {H}(\\Omega ,).$$For a given semiflow $ (t)t$, we define a family of compositionoperators $ (Tt)t0$ acting on $ H(,$as follows\\begin{align} T_{t}\\colon \\mathcal {H}(\\Omega , &\\rightarrow \\mathcal {H}(\\Omega ,\\\\f &\\mapsto f\\circ \\varphi _{t}.\\nonumber \\end{align}By the definiton of semiflows, this family is an operator semigroupwhich is, in particular, strongly continuous since for all $ nN$,we have$$\\sup _{z\\in K_{n}}|f(\\varphi _{t}(z))-f(z)|\\overset{t\\rightarrow 0^{+}}{\\rightarrow }0.$$This defintion makes also sense when the space $ h(,$ ofharmonic functions is under consideration.",
"Since, by the Cauchy-Riemannequations, for every function $ uh(,$, we have $ uth(,$.\\begin{definition}Let X\\subset \\mathbb {H}(\\Omega ,be a Banach space and (\\varphi _{t})_{t} a semiflow of holomorphicfunctions in \\mathcal {H}(\\Omega ) generated by G. The space X is called(G)-admissible if the family of operators (T_{t})_{t\\ge 0} definedby (\\ref {eq:CO}) satisfies the following two conditions:\\begin{itemize}\\item [(i)] X is invariant underT_{t}, i.e., T_{t}X\\subset X for all t\\ge 0.\\item [(ii)] (T_{t})_{t\\ge 0}is strongly continuous on X.\\end{itemize}\\end{definition}$ Given a semigroup of composition operators $(T_{t})_{t\\ge 0}$ on a $(G)$ -admissible Banach space $X$ , the generator $\\Gamma $ admits a special form: $\\Gamma f=\\lim _{t\\rightarrow 0^{+}}\\frac{T_{t}f-f}{t}=G\\cdot f^{\\prime }\\quad (f\\in \\text{dom}\\Gamma ).$ Note that $G\\cdot f^{\\prime }$ is a directional derivative.",
"This is true for holomorphic functions and harmonic functions as well, but for convenience we write $\\nabla f$ instead of $f^{\\prime }$ for harmonic functions to distinguish products of complex numbers from inner products.",
"Typical choices for the space $X$ are the Bergman spaces $\\mathcal {A}^{p}(\\colon =\\mathcal {H}(\\cap L^{p}(dA)\\,(p\\ge 1),$ where $dA$ denotes the normalized Lebesgue measure on $, andthe Hardy spaces$$\\mathcal {H}^{p}(\\colon =\\left\\lbrace f\\colon f\\in \\mathcal {H}(;\\,\\sup _{0<r<1}\\left(\\frac{1}{2\\pi }_{0}^{2\\pi }|f(re^{it})|^{p}\\mathrm {d}t\\right)^{\\frac{1}{p}}<\\infty \\right\\rbrace \\,(p\\ge 1).$$The invariance is a consequence of Littlewood^{\\prime }s subordination principle,and the strong continuity follows from the density of the polynomialsand the dominated convergence theorem, see \\cite {Sis98}, which isalso a comprehensive survey on semigroups of composition operators.$ Indeed, this result carries over to Bergman and Hardy spaces on simply connected domains.",
"The Bergman spaces can be defined analogously to the Bergman spaces for functions in the unit disk.",
"For the Hardy space, we can give at least two definitions for simply connected domains, see [9], either using harmonic majorants or via approximating the boundary of $\\Omega $ by rectifiable curves.",
"Both definitions are equivalent when analytic Jordan domains are considered.",
"We use the definition in terms of harmonic majorants.",
"Definition 2.3 Let $\\Omega \\subsetneq be simply connected.",
"For $ p[1,)$,the Hardy space $ Hp()$ consists of those functions $ fH(,$such that the subharmonic functions $ |f|p$ is dominated by a harmonicfunction $ u:R$.$ Equipped with the norm $\\left\\Vert f\\right\\Vert _{\\mathcal {H}^{p}(\\Omega )}:=(u_{0}(z_{0}))^{\\frac{1}{p}}\\,(f\\in \\mathcal {H}(\\Omega ))$ where $z_{0}\\in \\Omega $ is some fixed point and $u_{0}$ is the least harmonic majorant for $f$ , the Hardy space over $\\Omega $ is a Banach space.",
"As in the unit disk, functions in $\\mathcal {H}^{p}(\\Omega )$ admit non-tangential limits a.e.",
"on $\\partial \\Omega $ and the boundary function is in $L^{p}(\\partial \\Omega )$ .",
"For more details about Hardy spaces over general domains, we refer to [9].",
"Proposition 2.4 Let $\\Omega \\subsetneq be simply connected.Let $ (t)t$ be a semiflow of holomorphic functions in$ H()$ generated by $ G$.",
"The Hardy space $ Hp()$($ p[1,)$) is $ (G)$-admissible.$ Let $k\\colon \\Omega \\rightarrow be conformal.Then there exists a semiflow $ (t)t$ in $ H($ suchthat $ (t)t=(k-1tk)t$.",
"By \\cite [Cor.",
"to Thm.",
"10.1]{Dur70},$ fHp()$ if and only if$ fk-1Hp().",
"This and Littlewood's subordination principle gives invariance since $|f\\circ \\varphi _{t}|^{p}=|\\underset{\\in \\mathcal {H}^{p}(\\Omega )}{\\underbrace{\\underset{\\in \\mathcal {H}^{p}(}{\\underbrace{\\underset{\\in \\mathcal {H}^{p}(}{\\underbrace{f\\circ k^{-1}}\\circ \\psi _{t}}}}\\circ k}|^{p}.",
"}$ Without loss of generality, we assume that $k^{-1}(0)=z_{0}$ .",
"Then, by [8], we have $\\left\\Vert f\\circ \\varphi _{t}-f\\right\\Vert _{\\mathcal {H}^{p}(\\Omega )}=\\left\\Vert f\\circ \\varphi _{t}\\circ k^{-1}-f\\circ k^{-1}\\right\\Vert _{\\mathcal {H}^{p}(}\\overset{t\\rightarrow 0^{+}}{\\rightarrow }0.$ Remark 2.5 If we were using the definition of Hardy spaces by approximating level curves (sometimes called Hardy-Smirnov spaces), the last proof would involve boundary values of conformal maps.",
"This would have forced us to prescribe conditions concerning the boundary of $\\Omega $ .",
"Therefore it seems more appropriate to define Hardy spaces via harmonic majorants.",
"Proposition 2.6 Let $\\Omega \\subsetneq be a Jordan domain.Let $ (t)t$ be a semiflow of holomorphic functions in$ H()$ generated by $ G$.",
"The Bergman space$ Ap()$ ($ p[1,)$) is $ (G)$-admissible.$ Let $k\\colon \\Omega \\rightarrow be conformal.Then there exists a semiflow $ (t)t$ in $ H($ suchthat $ (t)t=(k-1tk)t$.",
"Thus,for $ fAp()$,\\begin{align*}_{\\Omega }|f\\circ \\varphi _{t}|^{p} &= _{|f\\circ \\varphi _{t}\\circ k^{-1}|^{p}|\\frac{1}{k^{\\prime }}|^{2}\\\\&\\le C_{|f\\circ k^{-1}\\circ \\psi _{t}|^{p}.",
"}The derivative of k is non-vanishing in \\bar{\\Omega }, see \\cite [Thm.",
"6.8]{Po92}.",
"}For invariance, we only need to show that f\\circ k^{-1}\\in \\mathcal {A}^{p}(.Indeed,_{|f\\circ k^{-1}|^{p}\\mathrm {d}A=_{\\Omega }|f|^{p}|k^{\\prime }|^{2}\\mathrm {d}A\\le C\\Vert f\\Vert _{\\mathcal {A}^{p}(\\Omega )}^{p}<\\infty .Now Littlewood^{\\prime }s subordination principle yields invariance.",
"}By the same calculation, we obtain strong continuityof (T_{t})_t on \\mathcal {A}^{p}(\\Omega ) from strong continuity on \\mathcal {A}^{p}(.\\end{align*}Further examples of holomorphic function spaceson the unit disk which appear in the literature concerning semigroupsof composition operators are the Bloch space $ B$ and thespace BMOA as well as their subspaces $ B0$ and VMOA.On these spaces the question of strong continuity is much more delicate,and in fact there is no nontrivial strongly continuous semigroup on$ B$ and BMOA.",
"So in these cases, one is studying so-calledmaximal subspaces of strong continuity denoted by $ [t,B]$and $ [t,BMOA]$ such that a given semiflow$ (t)t$ defines a strongly continuous semigroup of compositionoperators on $ [t,B]$ resp.",
"$ [t,BMOA]$.In \\cite {Bl13} it has been shown that $ B0[t,B]B$, and in the recent paper \\cite {AMW17} the analogous result for BMOAhas been obtained, that is, $ VMOA[t,BMOA]BMOA.$$ It is also natural to consider weighted semigroups of composition operators.",
"Let $\\Omega \\subsetneq be simply connected.",
"Let $ : be holomorphic.",
"For $t\\in \\mathbb {R}_{+}$ we define a weight as follows $m_{t}=\\frac{\\omega (\\varphi _{t})}{\\omega }.$ For a family of composition operators $(T_{t})_{t\\ge 0}$ on $\\mathcal {H}(\\Omega ,$ with semiflow $\\varphi _{t}\\in \\mathcal {H}(\\Omega )$ , we define a family of weighted composition operators as follows StH(, H(, f mtTtf.",
"This is again an operator semigroup on $\\mathcal {H}(\\Omega ,$ and also on $h(\\Omega ,$ but the question of strong continuity is more difficult since it depends heavily on the choice of $\\omega $ .",
"Special weights we are interested in are so-called cocycles.",
"Definition 2.7 Let $(\\varphi _{t})_{t}$ be a semiflow in $\\mathcal {H}(\\Omega ,\\Omega )$ .",
"A family of holomorphic functions $m_{t}\\colon \\Omega \\rightarrow t\\ge 0,$ is called cocycle if $m_{0}(z)=1,\\,z\\in \\Omega $ , $m_{s+t}(z)=(m_{s}\\cdot m_{t})(\\varphi _{t}(z))$ for all $t,s\\ge 0$ and $z\\in \\Omega $ , $t\\mapsto m_{t}(z)$ is continuous for every $z\\in \\Omega $ .",
"If there exists a holomorphic function $w\\colon \\Omega \\rightarrow such that $ mt(z)=w(t(z))w(z), z,$then the family $ (mt)t$ is called a coboundary of $ (t)t$.$ It is easy to see that a family of cocycle weighted composition operators is also an operator semigroup on $\\mathbb {H}(\\Omega ,$ .",
"Moreover, given an arbitrary holomorphic function $g:\\Omega \\rightarrow , we can easilyconstruct a cocycle to a semiflow $ (t)t$: for $ t0$,$ $m_{t}(z)=\\exp \\left(_{0}^{t}g(\\varphi _{s}(z))ds\\right)\\quad (z\\in \\Omega )$ is a cocycle.",
"Definition 2.8 Let $(S_{t})_{t\\ge 0}$ be a weighted semigroup of composition operators on $\\mathbb {H}(\\Omega ,$ , cf.",
"(REF ), with semiflow generated by the holomorphic function $G:\\Omega \\rightarrow and cocycle weight in terms of a holomorphic function $ g:, see (REF ) .",
"A Banach space $X\\subset \\mathbb {H}(\\Omega ,$ is called $(g,G)$ -admissible if it satisfies the following two conditions: $X$ is invariant under $S_{t},$ i.e., $S_{t}X\\subset X$ for all $t\\ge 0$ .",
"$(S_{t})_{t}$ is strongly continuous on $X$ .",
"Let $X\\subset \\mathcal {H}(\\Omega ,$ be $(g,G)$ -admissible.",
"Then the generator $\\Gamma $ of $(S_{t})_{t\\ge 0}$ is given by $\\Gamma f=g\\cdot f+G\\cdot f^{\\prime }\\quad (f\\in \\text{dom}\\Gamma ).$ In [13] it has been shown that for certain holomorphic functions $g:\\Omega \\rightarrow and their associated cocycles $ mt$as in (\\ref {eq:cocy}), and a semiflow $ (t)t$ generated by$ G:,the Hardy space $\\mathcal {H}^{p}(\\,(p\\in [1,\\infty ))$ is $(g,G$ -admissible in the sense of Definition REF .",
"By a slight adjustment of the arguments in Proposition REF , we obtain the result for Hardy spaces over simply connected sets.",
"Lemma 2.9 Let $\\Omega \\subsetneq be simply connected.",
"Let $ g be a holomorphic function such that $\\underset{z\\in \\Omega }{\\sup }\\operatorname{Re}g(z)<\\infty $ , and let $(\\varphi _{t})_{t}$ be a semiflow in $\\mathcal {H}(\\Omega )$ with generator $G$ .",
"Then $\\mathcal {\\mathcal {H}}^{p}(\\Omega )\\,(p\\in [1,\\infty ))$ is $(g,G)-$ admissible.",
"Invariance follows by boundedness of $m_{t}$ and Proposition REF .",
"To show strong continuity, we use the same technique as in Proposition REF , too.",
"Since the real part of $g\\circ k^{-1}$ is bounded as well, we obtain the assertion from [18].",
"Indeed, the proof of [18] works as well for a family of $m_{t}$ -weighted composition operators on the Bergman space $\\mathcal {A}^{p}(\\Omega )$ , where $\\Omega $ is a Jordan domain.",
"Lemma 2.10 Let $\\Omega \\subsetneq be a Jordan domain.",
"Let$ g be a holomorphic function such that $\\underset{z\\in \\Omega }{\\sup }\\operatorname{Re}g(z)<\\infty $ , and let $(\\varphi _{t})_{t}$ be a semiflow in $\\mathcal {H}(\\Omega )$ with generator $G$ .",
"Then $\\mathcal {A}^{p}(\\Omega )\\,(p\\in [1,\\infty ))$ is $(g,G)$ -admissible.",
"It suffices to prove the statement for $\\mathcal {A}^{p}(=\\mathcal {A}^{p}$ and then apply the same technique as in Proposition REF .",
"Due to Siskakis [18], strong continuity for a weighted SGCO on $\\mathcal {H}^{p}($ is achieved if $\\limsup _{t\\rightarrow 0}\\Vert m_{t}\\Vert _{\\infty }\\le 1$ which is satisfied by our assumptions on $g$ , see [13].",
"To prove the assertion, we can simply follow the steps in the proof of [18].",
"For all $t\\ge 0$ we have $m_{t}\\in \\mathcal {H}^{\\infty }($ .",
"This and the cocycle properties yield that $(S_{t})_{t}$ defines a family of bounded operators on $\\mathcal {A}^{p}$ .",
"Let $f\\in \\mathcal {A}^{p}$ .",
"By Littlewood's subordination principle we get that $\\Vert S_{t}f\\Vert _{\\mathcal {A}^{p}}^{p}\\le \\Vert m_{t}\\Vert _{\\infty }\\left(\\frac{1+|\\varphi _{t}(0)|}{1-|\\varphi _{t}(0)|}\\right)^{p}\\Vert f\\Vert _{\\mathcal {A}^{p}}^{p},$ thus $\\Vert S_{t}\\Vert _{\\mathcal {L}(\\mathcal {A}^{p},\\mathcal {A}^{p})}<\\infty $ for all $t\\ge 0$ .",
"First, we prove strong continuity if $p>1$ .",
"Let $(t_{n})_{n\\in \\mathbb {N}}$ be a sequence such that $t_{n}\\overset{n\\rightarrow \\infty }{\\rightarrow }0$ .",
"Then we have $\\limsup _{n\\rightarrow \\infty }\\Vert S_{t_{n}}f\\Vert _{2}\\le \\Vert f\\Vert _{2}$ .",
"Since $\\mathcal {A}^{p}$ is reflexive and by (REF ), after passing to a subsequence again denoted by $(t_{n})_{n\\text{$\\in \\mathbb {N}$}}$ , the sequence $(S_{t_{n}}f)_{t_{n}}$ is weakly convergent.",
"The weak limit is $f$ because $(S_{t_{n}}f(z))_{t_{n}}\\rightarrow f$ for all $z\\in .By lower-semicontinuity of the $ Ap$ norm, $ fA2nStnfAp$,and thus $ StnfApfAp$.",
"This yields the desired strong continuity.\\\\To show strong continuity in the case $ p=1,$ weuse that $ Aq$ ($ q>1)$ is dense in $ A1$.",
"Let $ >0$.For every $ fA1$ there exists $ gAq$ such that $ f-gA1<(St+1)2$.",
"Moreover,\\begin{align*}\\left\\Vert S_{t}f-f\\right\\Vert _{\\mathcal {A}^{1}} &\\le \\left\\Vert S_{t}f-S_{t}g\\right\\Vert _{A^{1}}+\\left\\Vert S_{t}g-g\\right\\Vert _{A^{1}}+\\left\\Vert f-g\\right\\Vert _{A^{1}}\\\\&\\le \\left\\Vert S_{t}f-S_{t}g\\right\\Vert _{A^{1}}+\\left\\Vert S_{t}g-g\\right\\Vert _{A^{q}}+\\left\\Vert f-g\\right\\Vert _{A^{1}}\\\\&\\le (\\left\\Vert S_{t}\\right\\Vert +1)\\left\\Vert f-g\\right\\Vert _{A^{1}}+\\left\\Vert S_{t}g-g\\right\\Vert _{A^{q}}.\\end{align*}Since $ q>1$, for all $ >0$ there exists a sufficiently small$ t>0$ such that $ Stg-gAq<2.$Thus $ Stf-fA10$ as $ t0+$.\\\\$ Remark 2.11 Several authors are especially interested in semigroups of composition operators weighted by the derivative of the semiflow $(\\varphi _{t})_{t}$ with respect to the complex varibale, i.e., $S_{t}f:=\\varphi ^{\\prime }_{t}\\cdot f\\circ \\varphi _{t}\\quad (f\\in X).$ See for example the recent paper [4].",
"Indeed, this weight is a cocycle given by $m_{t}(z)\\colon =\\varphi ^{\\prime }_{t}(z)=\\exp \\left(_{0}^{t}G^{^{\\prime }}(\\varphi _{s}(z))\\mathrm {d}s\\right).$ Finding boundary values of holomorphic functions is a fundamental problem in function theory.",
"Strong results concerning the boundary values of functions in Hardy spaces are Fatou's theorem and the theorem by F. and M. Riesz.",
"But, in many spaces of holomorphic functions, convergence to boundary values in a nontangential sense is a rather strong condition.",
"Therefore we consider boundary values in a weaker sense, namly in the sense of distributions.",
"Let $\\Omega \\subsetneq be a Jordan domain.",
"This restriction guaranteesexistence and nonvanishing of boundary values of derivatives of conformalmaps defined on $$.",
"Up to now, we are not sure if the establishedtheory works for rectifiable boundaries as well.$ In what follows, we are exploring boundary distributions of functions in Banach spaces $X\\subset \\mathbb {H}(\\Omega ,$ .",
"Our first aim is to define the boundary space of $X$ consisting of appropriately defined distributional boundary values of elements of $X$ .",
"Definition 2.12 Let $\\Omega \\subsetneq be a Jordan domain.",
"Let $ XH(,$be a Banach space.",
"If for every $ fX$ there exists a uniquelydefined boundary distribution $ f* in the following sense $\\lim _{r\\rightarrow 1^{-}}_{\\partial \\Omega }f_{r}\\cdot \\phi (x)\\mathrm {d}x=\\left\\langle f^{*},\\phi \\right\\rangle $ for every $\\phi \\in C^{\\infty }(\\partial \\Omega )$ , where $f_{r}(z):=f(k^{-1}(rk(z))$ , and $k:\\Omega \\rightarrow is any conformal map, then we denote the setconsisting of all such boundary values by $ X$.",
"If thereexists an isomorphism $ Tr:XX$, then $ X$is called the boundary space corresponding to $ X$.Moreover, we define a norm on $ X$ by $ f*X=fX$for every $ f*X$.$ A first (though artificial) example is the space $X=\\mathcal {A}$ where $\\mathcal {A}$ denotes the disk algebra.",
"The restriction to the boundary is an isometric homomorphism from $\\mathcal {A}$ into $C(\\partial $ .",
"So $\\mathcal {A}$ is a Banach subalgebra of $C(\\partial $ which is even maximal due to Wermer's maximality theorem.",
"Thus the boundary space $\\partial X$ can be defined as the space of continuous functions on $\\partial which are holomorphically extendable to$ .",
"Let $p\\in [1,\\infty )$ and define $X$ as the Hardy space $\\mathcal {H}^{p}($ .",
"Then it is well known that every function in $\\mathcal {H}^{p}($ has nontangential limits a.e.",
"and the boundary function is in $L^{p}(\\partial $ .",
"For a comprehensive overview, we refer especially to [9].",
"These boundary functions form a closed subspace of $L^{p}(\\partial )which consists of those function in $ Lp($ with vanishingnegative Fourier coefficients.",
"Note that this theory is almost applicablewhen the analogously defined Hardy space $ hp$ of harmonic functionsis considered.",
"However, the case $ p=1$ appears to be different.",
"Theboundary space on $ h1$ consists of finite Borel measures on theunit circle.$ In both examples, the boundary space inherits some properties of the underlying space of holomorphic functions.",
"Moreover, by the Luzin-Privalov theorem, a holomorphic function is in either case identically zero if the boundary function vanishes on a set of positive measure.",
"Given a function in one of the two boundary spaces from the examples above, we can recover the holomorphic function in $X$ via Cauchy's integral formula and the Poisson integral as well which acts as an isometric isomorphism between $X$ and $\\partial X$ .",
"The theory of boundary values for functions in Hardy spaces on the unit disc is well established.",
"The question of boundary functions is much more complicated if one wishes to work on Bergman spaces.",
"In fact, the Bergman spaces contain functions which do not admit nontangential or radial limits almost everywhere, such as the Lacunary series.",
"So it seems more appropriate to define boundary values in the sense of distributions.",
"To establish such distributional boundary values, we emphasize a connection between Hardy and Bergman spaces.",
"For simplicity we use the notation $\\mathcal {A}^{p}:=\\mathcal {A}^{p}($ and $\\mathcal {H}^{p}:=\\mathcal {H}^{p}(,p\\ge 1$ .",
"The following theorem can be found in [8].",
"Theorem 2.13 If $f\\in \\mathcal {A}^{1}$ and $F$ is an antiderivative of $f$ , then $F\\in \\mathcal {H}^{1}$ .",
"For $p\\in [1,\\infty )$ , Theorem REF can be generalized to $f\\in \\mathcal {A}^{p}$ in the following way.",
"Theorem 2.14 Let $f\\in \\mathcal {A}^{p}\\,(p\\ge 1)$ and $F$ an antiderivative of $f$ .",
"Then $F\\in \\mathcal {H}^{p}$ .",
"Let $\\varepsilon \\in (0,1)$ .",
"Then we have F(z) = zzf(w)dw-F(z)(w=tz)=1f(tz)zdt-F(z) To estimate $M_{p}(r,F)$ , we examine the following two integrals Mp(r,F) = (1202|F(reit)|pdt)1p = (1202|1f(sreit)reitds+F(reit)|pdt)1p =I1(1202|1f(sreit)reitds|pdt)1p+=I2Mp(r,F).",
"For the first term we have I1p = 1202|1f(sreit)reitds|pdt rp1202(1|f(sreit)|ds)pdt rp1202(1|f(sreit)|pds)dt rp+11202(rr|f(ueit)|puudu)dt rp1202(01|f(ueit)|pudu)dt rpfApp.",
"Without loss of generality, we assume $f(0)=0$ .",
"Thus we obtain for the second integral I2p = 1202|F(reit)|pdt = 1202|01f(sreit)reitds|pdt (r)p1202(01|f(sreit)|pds)dt = (r)p1202(0r|f(ueit)|pdu)dt (r)pfApp.",
"Combining these results, we have $M_{p}(r,F)\\le \\left(\\frac{r^{p}}{\\varepsilon }+(\\varepsilon r)^{p}\\right)\\Vert f\\Vert _{\\mathcal {A}^{p}}.$ Letting $r\\rightarrow 1^{-}$ , the right-hand side is still finite since $\\varepsilon $ can be chosen arbitrarily in $(0,1)$ .",
"This theorem remains true if we replace $ by a Jordan domain $ .",
"Corollary 2.15 Theorem REF remains true if $ is replacedby a Jordan domain $ .",
"By [9], it is enough to show that $F\\circ k^{-1}\\in \\mathcal {H}^{p}() for some conformalmapping $ k. Therefore, one can mostly copy the proof of Theorem REF , noting that for $f\\in \\mathcal {A}^{p}(\\Omega )$ one has $f\\circ k^{-1}\\cdot \\frac{1}{k^{\\prime }}\\in \\mathcal {A}^{p}($ .",
"The derivative of $k$ does not vanish in $\\bar{\\Omega }$ , and so we have $_{|f\\circ k^{-1}|^{p}\\left|\\frac{1}{k^{\\prime }}\\right|^{\\frac{1}{p}}\\mathrm {d}A\\le C\\Vert f\\circ k^{-1}\\Vert _{\\mathcal {A}^{p}(}^{p}.It remains to show that f\\circ k^{-1}\\in \\mathcal {A}^{p}(:_{|f\\circ k^{-1}|^{p}\\mathrm {d}A=_{\\Omega }|f|^{p}|k^{\\prime }|^{2}\\mathrm {d}A\\le C\\Vert f\\Vert _{\\mathcal {A}^{p}(\\Omega )}^{p}<\\infty .",
"}}Now we can define distributional boundary values for Bergman functions.\\begin{theorem}Let p\\in [1,\\infty ).",
"Every function f\\in \\mathcal {A}^{p}(admits a distributional boundary value in W^{-1,q}(\\partial :=(W^{1,p}(\\partial )^{\\prime },the dual space of W^{1,p}(\\partial ), where q is the usualconjugate exponent of p.\\end{theorem}$ Let $\\varphi \\in W^{1,p}(\\partial $ .",
"We denote by $F$ the antiderivative of $f$ , so we obtain 1202f(reit)(eit)dt = =012F(reit)(eit)|02-1202F(reit)'(eit)dt r1- -Tf,.",
"This limit exists by using Theorem REF , Hölder's inequality, and the dominated convergence theorem.",
"Corollary 2.16 Let $\\Omega \\subset be Jordan domain.",
"Then everyfunction $ fAp() (p[1,))$ admits adistributional boundary value in $ W-1,q()$.$ Let $\\varphi \\in W^{1,p}(\\partial \\Omega )$ , and let $k:\\Omega \\rightarrow be conformal.",
"For $ r(0,1)$ we define as usual$ fr:xf(k-1(rk(x))).$ Thus $ frf$as $ r1-$.",
"Then\\begin{align}_{\\partial \\Omega }f_{r}(x)\\varphi (x)\\mathrm {d}x &= _{\\partial f_{r}(k^{-1}(x))\\varphi (k^{-1}(x))\\left|\\frac{1}{k^{\\prime }(x)}\\right|^{2}\\mathrm {d}x\\nonumber \\\\&= _{0}^{2\\pi }f_{r}(k^{-1}(e^{it}))\\varphi (k^{-1}(e^{it}))\\left|\\frac{1}{k^{\\prime }(x)}\\right|^{2}ie^{it}\\mathrm {d}t.}It is easy to show that f_{r}\\circ k^{-1}\\in \\mathcal {A}^{p}( and \\varphi \\circ k^{-1}\\in L^{p}(\\partial .Since k is a conformal, we also have \\varphi \\circ k^{-1}\\in W^{1,p}(\\partial .So, by \\cite [Thm 6.8]{Po92}, we obtain convergence of the integral(\\ref {eq:dbvo}) as r\\rightarrow 1^{-}.\\end{align}Distributional boundary values of harmonic and holomorphic functionsdefined on a simply connected domain with smooth boundary have beenstudied in \\cite {St84}.",
"There it has been shown that a holomorphicfunction admits a distributional boundary value if and only if itlies in the Sobolev space $ H-k():=W-k,2()$ forsome $ kN$, see \\cite [Thm.",
"1.3]{St84}.",
"Moreover, by \\cite [Cor.",
"1.7]{St84},for all $ kN$ the map $ P$ defined by\\begin{align*}P:W^{-k-\\frac{1}{2},2}(\\partial \\Omega ) &\\rightarrow H^{-k}(\\Omega )\\cap \\mathcal {H}(\\Omega ,\\\\T_{f} &\\mapsto \\langle P_{z},T_{f}\\rangle =f(z),\\end{align*}where $ Pz$ is the Poisson kernel for $$, is an isomorphism.The inverse is given by assigning the distributional boundary valueto a given function.",
"Thus, functions in $ H-k()H(,$are uniquely determined by their boundary distributions.",
"Therefore,restricting the map $ P$ to the boundary space $ Ap()$for some $ p[1,)$, we can recover each function in $ Ap()$using the Poisson operator.$"
],
[
"Dirichlet-to-Robin via composition semigroups",
"In this section we work out our main result, the connection between partial differential equations on the boundary associated with Poincaré-Steklov operators and semigroups of composition operators on Banach spaces of holomorphic functions."
],
[
"The Lax semigroup",
"Let $h\\colon \\partial be a ^{\\prime }nice^{\\prime } function and considerthe following elliptic equation$ ${\\left\\lbrace \\begin{array}{ll}-\\Delta u=0 & \\textrm {in }\\\\u=h & \\textrm {on }\\partial \\end{array}\\right.",
"}$ The Dirichlet-to-Neumann operator $\\mathfrak {D}_{\\mathcal {N}}$ maps the function $h$ to the Neumann derivative of the solution of (REF ) provided that a solution exists and is sufficiently regular.",
"As it is shown by Lax [14], if $g\\in C(\\partial $ or in $L^{2}(\\partial $ , the Dirichlet-to-Neumann operator generates the following semigroup $T_{t}h(z)\\colon =u(ze^{-t})\\quad (z\\in \\partial .$ This semigroup solves the first order evolution equation associated with the Dirichlet-to-Neumann operator ${\\left\\lbrace \\begin{array}{ll}\\partial _{t}u+\\partial _{\\nu }u=\\ 0 & \\textrm {on }(0,\\infty )\\times \\partial \\\\-\\Delta u=0 & \\textrm {on }(0,\\infty )\\times u(0,\\cdot )=h& \\textrm {on }\\partial \\end{array}\\right.",
"}$ In fact, the semigroup (REF ) is an unweighted semigroup of composition operators on $h^{p}($ if $h\\in \\partial h^{p}(\\subseteq L^{p}(\\partial $ ($p\\in (1,\\infty )$ ) with associated semiflow $(\\varphi _{t})_t$ given by $\\varphi _t(z)=ze^{-t}\\,(z\\in $ .",
"The generator is given by $\\ G(z)=-z=-\\nu (z)\\,\\,(z\\in $ , and therefore the generator of the semigroup (REF ) is $\\Gamma u=-\\nu \\cdot \\nabla u\\,\\,(u\\in \\text{dom}(\\Gamma )\\subset h^{p}()$ .",
"So, for $h\\in \\text{dom}(\\mathfrak {D}_{\\mathcal {N}})\\subset L^{p}(\\partial $ and $u\\in h^{p}(\\Omega )$ the solution to (REF ), -DNh = -u = -u =Tr( u).",
"Replacing $ by a simply connected domain $$ with Dini-smoothboundary in (\\ref {eq:DP.D}) and (\\ref {eq:DtN}), we obtain a similarcorrespondence.",
"Let $ k: be conformal, then $\\nu (z)=\\frac{k(z)}{k^{\\prime }(z)}|k^{\\prime }(z)|$ is the unit normal vector at $z\\in \\partial \\Omega $ .",
"Since $\\partial \\Omega $ is Dini-smooth, $k\\in C^{1}(\\bar{\\Omega })$ by [16].",
"Thus $G(z)=-\\frac{k(z)}{k(z)^{\\prime }}\\,\\,(z\\in \\Omega )$ is holomorphic in $\\Omega $ and uniformly continuous on $\\bar{\\Omega }$ , and moreover, $\\operatorname{Re}(-G\\bar{\\nu })\\le 0$ on $\\partial \\Omega $ .",
"So, by Proposition REF (I), $G$ generates a semiflow in $\\mathcal {H}(\\Omega )$ .",
"Therefore, we obtain the following relation between the Dirichlet-to-Neumann operator on $\\partial h^{p}(\\Omega )\\subset L^{p}(\\partial \\Omega )$ and the unweighted semigroup of composition operators on $h^{p}(\\Omega )$ .",
"Let $u\\in h^{p}(\\Omega )$ be the solution to ${\\left\\lbrace \\begin{array}{ll}-\\Delta u=0 & \\textrm {in }\\Omega ,\\\\u=h & \\textrm {on }\\partial \\Omega ,\\end{array}\\right.",
"}$ where $h\\in L^{p}(\\partial \\Omega )$ .",
"Then, for $h\\in \\text{dom}(\\mathfrak {D}_{\\mathcal {N}})$ , -DNh = -u = Tr((Gu)|k'|), and $\\Gamma u:=G\\cdot \\nabla u$ is the generator of an unweighted semigroup of composition operators on $h^{p}(\\Omega )$ with semiflow generated by $G$ .",
"So the Dirichlet-to-Neumann operator is a multiplicative perturbation of the generator of the semigroup of composition operators.",
"Indeed, in [11], it has been shown that the Dirichlet-to-Neumann semigroup is the trace of a semigroup of composition operators only if $\\Omega $ is a disk.",
"This result relies on the the fact that the normal unit vector (viewed as a complex valued map on $\\partial \\Omega $ ) can only extended analytically to $\\Omega $ if $\\partial \\Omega $ is a circle [11].",
"From our previous investigations, it is now clear how to state well-posedness of the evolution problem (REF ) associated with the Dirichlet-to-Robin operator.",
"This is the main theorem of this article.",
"Theorem 3.1 (Main Theorem) Let $\\Omega \\subsetneq be a Jordan domain, and let $ G: be the generator of a semiflow of holomorphic functions in $\\mathcal {H}(\\Omega )$ and $g:\\Omega \\rightarrow holomorphic such that $ XH(,$is $ (g,G)$-admissible space.",
"Then the evolution problem associatedwith the Dirichlet-to-Robin operator\\begin{equation}{\\left\\lbrace \\begin{array}{ll}\\partial _{t}u-g\\cdot u-G\\cdot \\partial _{z}u=0 &\\textrm {on } (0,\\infty )\\times \\partial \\Omega ,\\\\-\\Delta u=0 & \\textrm {on } (0,\\infty )\\times \\partial \\Omega ,\\\\u(0,\\cdot )=u_{0} & \\textrm {on }\\partial \\Omega ,\\end{array}\\right.",
"}\\end{equation}is well-posed in $ X$, and the solution is given by the traceof a weighted semigroup of composition operators.$ Let $(S_{t})_{t}$ be the semigroup of weighted composition operators with semiflow $(\\varphi _{t})_{t}$ in $\\mathcal {H}(\\Omega )$ generated by $G$ and weight $m_{t}(z)=\\exp \\left(_{0}^{t}g(\\varphi _{s}(z))ds\\right)\\quad (z\\in \\Omega ).$ We denote by $\\Gamma $ the generator of $(S_{t})_{t}.$ Then the Dirichlet-to-Robin operator $\\mathfrak {D}_{\\mathcal {R}}:\\text{dom}(\\mathfrak {D}_{\\mathcal {R}})\\subset \\partial X\\rightarrow \\partial X,u_{0}\\mapsto (g\\cdot u+G\\cdot u^{\\prime })|_{\\partial \\Omega }$ is given by DRu0 = Tr(gu+Gu') = Tr(u).",
"So we obtain the Dirichlet-to-Robin semigroup as $e^{-t\\mathfrak {D}_{\\mathcal {R}}}u_{0}=\\text{Tr}(m_{t}\\cdot u\\circ \\varphi _{t})\\quad (u_{0}\\in \\partial X).$ Remark 3.2 We would like to emphasize that a boundary space in the sense of distributions is not necessary since we can always define boundary values using hyperfunctions.",
"In this case our initial value would be very general.",
"On the other hand, if $\\varphi =\\varphi _{1}$ has an interior Denjoy-Wolff point and is not an inner function, then for $z\\in \\partial \\Omega $ and $t$ sufficiently large, $\\varphi _{t}(z)$ lies strictly inside $\\Omega $ , see [15].",
"Thus there is actually no need to restrict to distributions in problem (REF ).",
"It is worth noting that the function $G$ may degenerate at some point $a\\in \\partial \\Omega $ .",
"This is even possible if $a$ is not a fixed point of the generated semiflow $(\\varphi _{t})_{t}$ ; on the other hand, if $a$ is a non-superrepulsive fixed point of $\\varphi $ (i.e., $\\varphi ^{\\prime }(a)\\ne \\infty $ ), then the angular limit $\\lim _{z\\rightarrow a}G(z)=0$ , see [7].",
"We repeat from the Introduction that we do not see how to include such a $G$ in the variational approach.",
"I am grateful to my supervisor Ralph Chill, who brought this topic to my attention, for support and valuable suggestions which improved the presentation of the paper."
],
[
"Dirichlet-to-Robin via composition semigroups",
"In this section we work out our main result, the connection between partial differential equations on the boundary associated with Poincaré-Steklov operators and semigroups of composition operators on Banach spaces of holomorphic functions."
],
[
"The Lax semigroup",
"Let $h\\colon \\partial be a ^{\\prime }nice^{\\prime } function and considerthe following elliptic equation$ ${\\left\\lbrace \\begin{array}{ll}-\\Delta u=0 & \\textrm {in }\\\\u=h & \\textrm {on }\\partial \\end{array}\\right.",
"}$ The Dirichlet-to-Neumann operator $\\mathfrak {D}_{\\mathcal {N}}$ maps the function $h$ to the Neumann derivative of the solution of (REF ) provided that a solution exists and is sufficiently regular.",
"As it is shown by Lax [14], if $g\\in C(\\partial $ or in $L^{2}(\\partial $ , the Dirichlet-to-Neumann operator generates the following semigroup $T_{t}h(z)\\colon =u(ze^{-t})\\quad (z\\in \\partial .$ This semigroup solves the first order evolution equation associated with the Dirichlet-to-Neumann operator ${\\left\\lbrace \\begin{array}{ll}\\partial _{t}u+\\partial _{\\nu }u=\\ 0 & \\textrm {on }(0,\\infty )\\times \\partial \\\\-\\Delta u=0 & \\textrm {on }(0,\\infty )\\times u(0,\\cdot )=h& \\textrm {on }\\partial \\end{array}\\right.",
"}$ In fact, the semigroup (REF ) is an unweighted semigroup of composition operators on $h^{p}($ if $h\\in \\partial h^{p}(\\subseteq L^{p}(\\partial $ ($p\\in (1,\\infty )$ ) with associated semiflow $(\\varphi _{t})_t$ given by $\\varphi _t(z)=ze^{-t}\\,(z\\in $ .",
"The generator is given by $\\ G(z)=-z=-\\nu (z)\\,\\,(z\\in $ , and therefore the generator of the semigroup (REF ) is $\\Gamma u=-\\nu \\cdot \\nabla u\\,\\,(u\\in \\text{dom}(\\Gamma )\\subset h^{p}()$ .",
"So, for $h\\in \\text{dom}(\\mathfrak {D}_{\\mathcal {N}})\\subset L^{p}(\\partial $ and $u\\in h^{p}(\\Omega )$ the solution to (REF ), -DNh = -u = -u =Tr( u).",
"Replacing $ by a simply connected domain $$ with Dini-smoothboundary in (\\ref {eq:DP.D}) and (\\ref {eq:DtN}), we obtain a similarcorrespondence.",
"Let $ k: be conformal, then $\\nu (z)=\\frac{k(z)}{k^{\\prime }(z)}|k^{\\prime }(z)|$ is the unit normal vector at $z\\in \\partial \\Omega $ .",
"Since $\\partial \\Omega $ is Dini-smooth, $k\\in C^{1}(\\bar{\\Omega })$ by [16].",
"Thus $G(z)=-\\frac{k(z)}{k(z)^{\\prime }}\\,\\,(z\\in \\Omega )$ is holomorphic in $\\Omega $ and uniformly continuous on $\\bar{\\Omega }$ , and moreover, $\\operatorname{Re}(-G\\bar{\\nu })\\le 0$ on $\\partial \\Omega $ .",
"So, by Proposition REF (I), $G$ generates a semiflow in $\\mathcal {H}(\\Omega )$ .",
"Therefore, we obtain the following relation between the Dirichlet-to-Neumann operator on $\\partial h^{p}(\\Omega )\\subset L^{p}(\\partial \\Omega )$ and the unweighted semigroup of composition operators on $h^{p}(\\Omega )$ .",
"Let $u\\in h^{p}(\\Omega )$ be the solution to ${\\left\\lbrace \\begin{array}{ll}-\\Delta u=0 & \\textrm {in }\\Omega ,\\\\u=h & \\textrm {on }\\partial \\Omega ,\\end{array}\\right.",
"}$ where $h\\in L^{p}(\\partial \\Omega )$ .",
"Then, for $h\\in \\text{dom}(\\mathfrak {D}_{\\mathcal {N}})$ , -DNh = -u = Tr((Gu)|k'|), and $\\Gamma u:=G\\cdot \\nabla u$ is the generator of an unweighted semigroup of composition operators on $h^{p}(\\Omega )$ with semiflow generated by $G$ .",
"So the Dirichlet-to-Neumann operator is a multiplicative perturbation of the generator of the semigroup of composition operators.",
"Indeed, in [11], it has been shown that the Dirichlet-to-Neumann semigroup is the trace of a semigroup of composition operators only if $\\Omega $ is a disk.",
"This result relies on the the fact that the normal unit vector (viewed as a complex valued map on $\\partial \\Omega $ ) can only extended analytically to $\\Omega $ if $\\partial \\Omega $ is a circle [11].",
"From our previous investigations, it is now clear how to state well-posedness of the evolution problem (REF ) associated with the Dirichlet-to-Robin operator.",
"This is the main theorem of this article.",
"Theorem 3.1 (Main Theorem) Let $\\Omega \\subsetneq be a Jordan domain, and let $ G: be the generator of a semiflow of holomorphic functions in $\\mathcal {H}(\\Omega )$ and $g:\\Omega \\rightarrow holomorphic such that $ XH(,$is $ (g,G)$-admissible space.",
"Then the evolution problem associatedwith the Dirichlet-to-Robin operator\\begin{equation}{\\left\\lbrace \\begin{array}{ll}\\partial _{t}u-g\\cdot u-G\\cdot \\partial _{z}u=0 &\\textrm {on } (0,\\infty )\\times \\partial \\Omega ,\\\\-\\Delta u=0 & \\textrm {on } (0,\\infty )\\times \\partial \\Omega ,\\\\u(0,\\cdot )=u_{0} & \\textrm {on }\\partial \\Omega ,\\end{array}\\right.",
"}\\end{equation}is well-posed in $ X$, and the solution is given by the traceof a weighted semigroup of composition operators.$ Let $(S_{t})_{t}$ be the semigroup of weighted composition operators with semiflow $(\\varphi _{t})_{t}$ in $\\mathcal {H}(\\Omega )$ generated by $G$ and weight $m_{t}(z)=\\exp \\left(_{0}^{t}g(\\varphi _{s}(z))ds\\right)\\quad (z\\in \\Omega ).$ We denote by $\\Gamma $ the generator of $(S_{t})_{t}.$ Then the Dirichlet-to-Robin operator $\\mathfrak {D}_{\\mathcal {R}}:\\text{dom}(\\mathfrak {D}_{\\mathcal {R}})\\subset \\partial X\\rightarrow \\partial X,u_{0}\\mapsto (g\\cdot u+G\\cdot u^{\\prime })|_{\\partial \\Omega }$ is given by DRu0 = Tr(gu+Gu') = Tr(u).",
"So we obtain the Dirichlet-to-Robin semigroup as $e^{-t\\mathfrak {D}_{\\mathcal {R}}}u_{0}=\\text{Tr}(m_{t}\\cdot u\\circ \\varphi _{t})\\quad (u_{0}\\in \\partial X).$ Remark 3.2 We would like to emphasize that a boundary space in the sense of distributions is not necessary since we can always define boundary values using hyperfunctions.",
"In this case our initial value would be very general.",
"On the other hand, if $\\varphi =\\varphi _{1}$ has an interior Denjoy-Wolff point and is not an inner function, then for $z\\in \\partial \\Omega $ and $t$ sufficiently large, $\\varphi _{t}(z)$ lies strictly inside $\\Omega $ , see [15].",
"Thus there is actually no need to restrict to distributions in problem (REF ).",
"It is worth noting that the function $G$ may degenerate at some point $a\\in \\partial \\Omega $ .",
"This is even possible if $a$ is not a fixed point of the generated semiflow $(\\varphi _{t})_{t}$ ; on the other hand, if $a$ is a non-superrepulsive fixed point of $\\varphi $ (i.e., $\\varphi ^{\\prime }(a)\\ne \\infty $ ), then the angular limit $\\lim _{z\\rightarrow a}G(z)=0$ , see [7].",
"We repeat from the Introduction that we do not see how to include such a $G$ in the variational approach.",
"I am grateful to my supervisor Ralph Chill, who brought this topic to my attention, for support and valuable suggestions which improved the presentation of the paper."
]
] | 1709.01711 |
[
[
"Boosting Deep Learning Risk Prediction with Generative Adversarial\n Networks for Electronic Health Records"
],
[
"Abstract The rapid growth of Electronic Health Records (EHRs), as well as the accompanied opportunities in Data-Driven Healthcare (DDH), has been attracting widespread interests and attentions.",
"Recent progress in the design and applications of deep learning methods has shown promising results and is forcing massive changes in healthcare academia and industry, but most of these methods rely on massive labeled data.",
"In this work, we propose a general deep learning framework which is able to boost risk prediction performance with limited EHR data.",
"Our model takes a modified generative adversarial network namely ehrGAN, which can provide plausible labeled EHR data by mimicking real patient records, to augment the training dataset in a semi-supervised learning manner.",
"We use this generative model together with a convolutional neural network (CNN) based prediction model to improve the onset prediction performance.",
"Experiments on two real healthcare datasets demonstrate that our proposed framework produces realistic data samples and achieves significant improvements on classification tasks with the generated data over several stat-of-the-art baselines."
],
[
"Introduction",
"The worldwide exponential surge in volume, detail, and availability of Electronic Health Records (EHRs) promises to usher in the era of personalized medicine, enhancing each stage of the healthcare chain from providers to patients.",
"This field of research and applications, commonly mentioned as Data-Driven Healthcare (DDH) [1], has been under rapid development and attracted many researchers and institutions to utilize state-of-the-art machine learning and statistical models on a broad set of clinical tasks which are difficult or even impossible to solve with traditional methods [2], [3], [4].",
"Among these frontier models, deep learning tends to be the most exciting and promising solution to those difficult and important tasks.",
"Recent success and development in deep learning is revolutionizing many domains such as computer vision [5], [6], natural language processing [7], [8], and healthcare, with notable innovations and applicable solutions.",
"A series of excellent work have been conducted in seek of novel deep learning solutions in different healthcare applications including but not limited to computational phenotyping [9], [10], risk prediction [11], [12], medical imaging analysis [13], [14], and clinical natural language processing [15].",
"These works have made us closest ever towards the ultimate goal of improving health quality, reducing cost, and most importantly saving lives.",
"While existing achievements on deep learning models for healthcare are encouraging, the peculiar properties of EHRs, such as heterogeneity, longitudinal irregularity, inherent noises, and incomplete nature, make it extremely difficult to apply most existing mature models to healthcare compared with other well-developed domains with clean data.",
"Properly-designed deep neural networks have the prospect of handling these issues if equipped with massive data, but the amount of clinical data, especially with accurate labels and for rare diseases and conditions, is somewhat limited and far from most models' requirements [16].",
"This comes from the following reasons: The diagnosis and patient labeling process highly relies on experienced human experts and is usually very time-consuming; Getting detailed results of lab tests and other medical features, though has become more feasible with modern facilities than ever, are still quite costly; Not to mention the privacy issues and regulations which makes it even harder to collect and keep enough medical data with desired details.",
"These unique challenges lying in healthcare domain prevent existing deep learning models from exerting their strength with enough available and high-quality labeled data.",
"One way to overcome the challenges arising from limited data in machine learning field is semi-supervised learning (SSL) [17].",
"Semi-supervised learning is a class of techniques that makes use of unlabeled or augmented data together with a relatively small set of labeled data to get better performance.",
"Though some previous work utilized semi-supervised learning methods on EHR data [18], most of them focus on clinical text data [19], [20], and only limited work attempt to perform semi-supervised learning method on structured quantitative EHR data [21].",
"Generative model is also considered as a promising solution.",
"As one type of semi-supervised learning algorithms, it aims at learning the joint probability distribution over observations and labels from the training data, and can be further used for downstream algorithms and applications such as data modeling [22], classifier and predictor training [23], and data augmentations [24].",
"Though generative model approaches have been well explored for years, deep generative models haven't caught enough attentions due to its complexity and computation issues until the recent development of generative adversarial network (GAN) [25].",
"GAN simultaneously trains a deep generative model and a deep discriminative model, which captures the data distribution and distinguishes generated data from original data respectively, as a mini-max game.",
"GANs have been mainly used on image, video and text data to learn useful features with better understandings [26], [27] or sample data for specific demand [28], [29].",
"However, few GANs have been applied for generating sequential or time series EHR data, where large amount of reliable data, either from real dataset or augmentations, are in great demand for powerful predictive models.",
"In this paper, we investigate and propose general deep learning solutions to the challenges on high dimensional temporal EHR data with limited labels.",
"We propose a generative model, ehrGAN, for EHR data via adversarial training, which is able to generate convincing and useful samples similar to realistic patient records in existing data.",
"We further propose a semi-supervised learning framework which achieves boosted risk prediction performance by utilizing the augmented data and representations from the proposed generative models.",
"We conduct experiments on two real clinical tasks and demonstrate the efficacy of both the generative model and prediction framework."
],
[
"Related Work",
"In this section we briefly review existing works which are closely related to the our proposed method in this paper from two areas.",
"The first one is recent works on exploiting deep learning methods to applications on Electronic Health Records.",
"The other is on the recent research of adversarial training and generative adversarial networks."
],
[
"Deep Learning for Healthcare Applications",
"As deep learning has achieved great success recently, researchers have begun attempting to apply neural network based methods to EHR to utilize the ability of deep networks to learn complex patterns from data.",
"Previous studies, such as phenotype learning [10] and representation learning [12], formed a multi-layer perception (MLP) architecture and applied it to EHRs.",
"Considering the natural temporarily in EHR data, Recurrent neural networks (RNNs) are used for sequence prediction with regular times series of real-valued variables collected from intensive care unit patients and interpretable prediction on the diagnosis code given medication, lab test and disease information [30].",
"Deep state space model is also designed for phenotype learning and handled the data irregularity issue [31].",
"Several other works [32], [11], [33] exploited convolutional neural networks (CNNs) as well, to capture local temporal dependency of data in risk prediction or other related tasks.",
"All of these models are purely or mainly in the supervised learning manner and use fully labeled data.",
"However, supervised information is limited for many EHR applications, since it requires expense human efforts in labeling or scoring the patient records so as to analyze them in a model.",
"Thus unsupervised and semi-supervised learning schema is in great demand to be designed.",
"There are some other works [30], [33] trying to learn the medical feature and concept embedding representations with unsupervised method and the learned representations incorporate both co-occurrence information and visit sequence information of the EHR data.",
"However, to our best of knowledge, none of existing work attempts to build a semi-supervised deep learning model for applications with time series EHR data, and our work is the first of its kind.",
"In health informatics domain, there are some works targeting semi-supervised problems in both deep and non-deep settings, but most of them focus on clinical natural language processing problems, including learning from structured quantitative EHR data [21], building graph-based model for clinical text classification [20], and handling question-answering task in healthcare forum data [19].",
"They are not directly related to our work since we consider different data types or tasks."
],
[
"Adversarial Learning and Generative Adversarial Networks",
"The idea of adversarial learning is to include a set of machines which learn together by pursuing competing goals.",
"In generative adversarial networks (GANs) [25], a generator function (usually formed as a deep neural network) learns to synthesize samples that best resemble some dataset, while a discriminator function (also usually a deep neural network) learns to distinguish between samples drawn from the dataset and samples synthesized by the generator.",
"There are lots of deep learning works out on GANs recently, and some of them have emerged as promising frameworks for unsupervised learning.",
"For instance, the generators are able to produce images of unprecedented visual quality [34], while the discriminators learn features with rich semantics that can benefit other learning paradigms such as semi-supervised learning and transfer learning [35], [36].",
"The semi-supervised learning frameworks with GAN are used to solve classification tasks and learn a generative model simultaneously.",
"The representations learned by the discriminator, the classifications from the semi-supervised classifier, and the sampled data from the generator improve each other.",
"There are several works to build theoretical semi-supervised learning frameworks with GANs and apply them to the classification task.",
"Generally speaking, existing methods include feature augmentation [37], virtual adversarial training [38], and joint training [36], [39].",
"Our proposed semi-supervised paradigm belongs to data augmentation methods.",
"It is noting that all these related models are only applied to and designed for vision or natural language processing (NLP) domains.",
"To extend existing GAN framework to EHR data is not straightforward.",
"Moreover, to facilitate GANs with semi-supervised learning for onset prediction is also difficult.",
"These two unsolved challenges are well addressed in our proposed framework."
],
[
"The Proposed Method",
"In this section, we first introduce the basic deep learning risk prediction model used as a strong baseline as well as a component in the proposed framework.",
"Then we describe ehrGAN, a modified generative adversarial network which is specifically designed to be applied on EHR data.",
"Finally, we present the data augmented semi-supervised learning schema which is able to perform boosted onset predictions."
],
[
"Basic Deep Prediction Model",
"The basic model used in the paper is a convolutional neural network (CNN) model with 1D convolutional layer over the temporal dimension and max over-time pooling layer, which was used in previous work [32], [11], [33].",
"The input to the model is the EHR records of patient $p$ , which is represented as a temporal embedding matrix $\\mathbf {x}^p \\in \\mathbb {R}^{T_p \\times M}$ , where $T_p$ , which can be different among patients, is the number of medical events in patient $p$ 's record, and $M$ is the dimension of the learned embedding.",
"The rows of $\\mathbf {x}^p$ are the embedding vectors for the medical events, arranged in the order of time of occurrence.",
"The embedding for medical events is trained by Word2vec model [40] on the same corpus of EHR data.",
"We apply convolutional operation only over the temporal dimension but not over embedding dimension.",
"We us a combination of filters with different lengths to capture temporal dependencies in multiple levels, and our preliminary experiments validated the performance improvement from such strategy.",
"After the convolutional step, we apply a max-pooling operation along the temporal dimension to keep the most important features across the time.",
"This temporal pooling converts the inputs with different temporal lengths into a fixed length output vector.",
"Finally a fully connected soft-max layer is used to produce prediction probabilities.",
"This CNN-based deep prediction model described above is shown to be the most competitive baseline among all other tested baselines in our experiments and serves as the basic prediction component in our proposed work."
],
[
"The original GAN [25] is trained by solving the following mini-max game in the form of $\\min _G \\max _D\\operatornamewithlimits{\\mathbb {E}}_{\\mathbf {x} \\sim p_{data}(\\mathbf {x})}\\left[\\log D(\\mathbf {x})\\right] +\\operatornamewithlimits{\\mathbb {E}}_{\\mathbf {z} \\sim p_{\\mathbf {z}}(\\mathbf {z})}\\left[\\log \\left(1 - D(G(\\mathbf {z}))\\right)\\right]$ where $p_{data}(\\mathbf {x})$ is the true data distribution; $D(\\mathbf {x})$ is the discriminator that takes a sample as the input and outputs a scalar between $[0,1]$ as the probability of the sample drawing from real dataset; $G(\\mathbf {z})$ is the generator that maps a noise variable $\\mathbf {z} \\in \\mathbb {R}^d$ drawn from a given distribution $p_{\\mathbf {z}}(\\mathbf {z})$ back to the input space.",
"The training procedure consists of two loops optimizing $G$ and $D$ iteratively.",
"After the mini-max game reaches its Nash equilibrium [41], $G$ defines an implicit distribution $p_g(\\mathbf {x})$ that recovers the data distribution, i.e., $p_g(\\mathbf {x}) = p_{data}(\\mathbf {x})$ .",
"Generally, both $D$ and $G$ are parameterized as deep neural networks.",
"In the context of EHR data, similar to the basic prediction models, our choice of $G$ and $D$ falls into the family of 1D convolutional neural networks (CNNs) and 1D deconvolutional neural networks (DCNNs).",
"The overview of the model is shown in Figure REF .",
"In the following parts, we will discuss model details in terms of the design of discriminator and generator, and some specific training techniques.",
"Figure: The structure of the generator in ehrGAN.",
"𝐳\\mathbf {z} and 𝐦\\mathbf {m} are drawn randomly.",
"𝐱 ˜\\mathbf {\\tilde{x}} is the generated synthetic sample based on the real sample 𝐱.\\mathbf {x}."
],
[
"Discriminator",
"We adopt the structure of the basic prediction model to the discriminator, due to its simplicity and excellent classification performance.",
"We replace the top prediction layer by a single sigmoid unit to output the probability of the input data being drawn from the real dataset."
],
[
"Generator",
"The goal of the generator in GAN is to translate a latent vector $\\mathbf {z}$ into the synthetic sample $\\mathbf {\\tilde{x}}$ .",
"The generator is encoded by a de-convolutional neural network with two consecutive fully connected layers, the latter of which is reshaped and followed by two de-convolutional layers to perform upsampling convolution.",
"Empirically, this generator is able to generate good samples.",
"However, this version of generator can not be directly used in semi-supervised learning setting as the model is trained only to differentiate real or synthetic data instead of the classes.",
"To solve this problem, we introduce a variational version of the generator, which also provides some new understandings of GANs."
],
[
"Generator with variational contrastive divergence",
"The design of the variational generator is based on the recently proposed variational contrastive divergence (VCD) [42].",
"Instead of directly learning a generator distribution defined by $G(\\mathbf {z})$ , we learn a transition distribution of the form $p(\\mathbf {\\tilde{x}}|\\mathbf {x})$ for the generated sample $\\mathbf {\\tilde{x}}$ , with $\\mathbf {x} \\sim p_{data}(\\mathbf {x})$ .",
"The marginal distribution of of the generator is then given by $p_g(\\mathbf {\\tilde{x}}) = \\operatornamewithlimits{\\mathbb {E}}_{\\mathbf {x} \\sim p_{data}(\\mathbf {x})}p(\\mathbf {\\tilde{x}}|\\mathbf {x})$ .",
"Intuitively, the transition distribution $p(\\mathbf {\\tilde{x}}|\\mathbf {x})$ encodes a generating process.",
"In this process, based on an example drawn from the training data distribution, a neighboring sample $\\mathbf {\\tilde{x}}$ is generated.",
"To be more specific, the generator is equipped with encoder-decoder CNN networks.",
"For each real sample $\\mathbf {x}$ , we can get the representation $\\mathbf {h}$ from encoder, and the reconstruction $\\mathbf {\\bar{x}}$ by feeding $\\mathbf {h}$ into decoder.",
"$\\mathbf {h}$ can be mixed with a random noise vector $\\mathbf {z}$ of the same dimensionality by a random binary mask vector $\\mathbf {m}$ to obtain $\\mathbf {\\tilde{h}} = \\mathbf {m} * \\mathbf {z} + \\left(\\mathbf {1} - \\mathbf {m}\\right) * \\mathbf {h}$ , where $*$ represents element-wise multiplication.",
"The synthetic sample $\\mathbf {\\tilde{x}}$ can be obtained by feeding $\\mathbf {\\tilde{h}}$ to the same decoder.",
"An illustration of this VCD-based generator is shown in Figure REF .",
"The generator attempts to minimize the objective as $\\operatornamewithlimits{\\mathbb {E}}_{\\mathbf {x} \\sim p_{data}(\\mathbf {x})} \\left[\\rho \\cdot \\operatornamewithlimits{\\mathbb {E}}_{\\mathbf {\\tilde{x}} \\sim p_g(\\mathbf {\\tilde{x}}|\\mathbf {x})} \\left[ - \\log D(\\mathbf {\\tilde{x}})\\right]+ (1 - \\rho ) \\cdot {\\Vert \\mathbf {\\bar{x}} - \\mathbf {x}\\Vert }_2^2\\right]$ where $D$ is the discriminator function and the hyperparameter $\\rho $ controls how close the synthetic sample should be to the corresponding real sample.",
"The usage of VCD-based ehrGAN brings two benefits.",
"First, while original GANs are known to have mode collapsing issues, i.e., $G$ is encouraged to generate only a few modes, ehrGAN eliminates mode collapsing issue by its design, as the diversity of the generated samples inherently approximates that of the training data.",
"Second and more importantly, the learned transition distribution $p(\\mathbf {\\tilde{x}}|\\mathbf {x})$ contains rich structures of the data manifold around training examples $\\mathbf {x}$ , which can be quite useful when incorporating with our semi-supervised learning framework to obtain effective classification models."
],
[
"Training techniques",
"We train the proposed ehrGAN by optimizing the generator and discriminator iteratively with stochastic gradient descent (SGD).",
"The training procedure (shown in Algorithm REF ) is similar to that of standard GANs.",
"We take several techniques to stabilize the training of GANs similar to those in [27], [36], and relieve the training instability and sensitivity to hyper-parameters.",
"Firstly, we switch the order of discriminator and generator training, and perform $k=5$ optimization steps for the generator for every one steps for the discriminators.",
"Secondly, we add an $l_2$ -norm regularizer in the cost function of discriminator.",
"Finally, batch normalization and label smoothing techniques are used.",
"[h] The optimization procedure of ehrGAN [1] enough iterations until convergence $k$ inner steps sample $N$ noise variables $\\lbrace \\mathbf {z}^1, \\dots , \\mathbf {z}^N\\rbrace $ , and $N$ binary mask vectors $\\lbrace \\mathbf {m}^1, \\dots , \\mathbf {m}^N\\rbrace $ ; update generator $G$ by one step gradient ascent of $ \\qquad \\frac{1}{N}\\sum _{i=1}^N \\log D({G(\\mathbf {z}^i, \\mathbf {m}^i)})$ sample $N$ training data $\\lbrace \\mathbf {x}^1, \\dots , \\mathbf {x}^N\\rbrace $ , $N$ noise variables $\\lbrace \\mathbf {z}^1, \\dots , \\mathbf {z}^N\\rbrace $ , and $N$ binary mask vectors $\\lbrace \\mathbf {m}^1, \\dots , \\mathbf {m}^N\\rbrace $ ; update discriminator $D$ with one step gradient descent of $-\\frac{1}{N}\\sum _{i=1}^N \\log D({\\mathbf {x}^i}) - \\frac{1}{N}\\sum _{i=1}^N \\log \\left(1 - D({G(\\mathbf {z}^i, \\mathbf {m}^i)})\\right)$ We next introduce our method of conducting semi-supervised learning (SSL) with a learned ehrGAN, in a way which is similar to our previous model for images [42].",
"The basic idea is to use the learned transition distribution to perform data augmentation.",
"To be concrete, within the SSL setting we minimize the follow loss function: $\\frac{1}{N}\\sum _{i=1}^N\\mathcal {L}(\\mathbf {x}^i, \\mathbf {y}^i) + \\mu \\cdot \\frac{1}{N}\\sum _{i=1}^N \\operatornamewithlimits{\\mathbb {E}}_{\\mathbf {\\tilde{x}}^i \\sim p(\\mathbf {\\tilde{x}} | \\mathbf {x}^i)}\\mathcal {L}(\\mathbf {\\tilde{x}}^i, \\mathbf {y}^i)$ where $\\mathcal {L}$ refers to the binary crossentropy loss on each data sample, and $\\mu $ leverages the ratio of the numbers of training data and augmented data from GANs.",
"In other words, this model assumes that a well trained generator with distribution $p(\\mathbf {\\tilde{x}}|\\mathbf {x})$ should be able to generate samples that are likely to align within the same class of $\\mathbf {x}$ , which can in turn provide valuable information to the classifier as additional training data.",
"This method is called SSL-GAN (Semi-supervised learning with a learned ehrGAN) in this paper."
],
[
"Experimental Results",
"In this section we apply our models to two real clinical datasets extracted from heart failure and diabetes cohorts.",
"It is a particularly interesting to investigate how well GANs can generate EHRs samples as the real ones.",
"Also, understanding how the proposed method can boost the performance of onset prediction is crucial for many healthcare applications.",
"We start this section by introducing the datasets and experimental settings, and provide the evaluation analysis, followed by the discussions on the selections of parameters."
],
[
"Datasets and Settings",
"The datasets came from a real-world longitudinal Electronic Health Record database of $218,680$ patients and $14,969,489$ observations of $14,690$ unique medical events, between the year 2011 to 2015 from a health insurance company.",
"In these datasets, a set of diseases related ICD-9 codes were recorded to indicate medical conditions as well as drug prescriptions.",
"We identify two following cohorts and predict whether a patient is from case or control group as a binary classification task.",
"The labels of both case and control groups are identified by domain experts according to ICD-9 codes.",
"Congestive heart failure (Heart Failure), which contains $3,357$ confirmed patients in case group and $6,714$ patients in control group; Diabetes (Diabetes), with $2,248$ patients in case group and $4,496$ patients in control group.",
"We import ICD-9 diagnosis and medications as the input features, eliminate those which show less than 5 times in this dataset, and get $8,627$ unique medical features.",
"We segmented the time dimension into disjoint 90-day windows and combined all the observations within each window.",
"We split datasets into training, validation and test with ratio 7:1:2, and limit the length of each record sequence between 50 and 250 and form it to the embedding matrix.",
"All sequences are 0-padded to match the longest sequence.",
"The embedding is trained by Word2vec [40] on the entire dataset with dimension of 200.",
"The ehrGAN is trained on only the training subset.",
"For the CNN discriminator, we employ filters of sizes $\\lbrace 3,4,5\\rbrace $ with 100 feature maps.",
"For the generator, the dimension of the latent variable $\\mathbf {z}$ is 100.",
"It is first projected and reshaped by the generator and up-sampled by two one-dimensional CNN layers with filers size 100 and 3.",
"The output of the generator is an embedding matrix with size $200\\times 150$ .",
"These hyperparameters are selected based on preliminary experimental results.",
"To generate samples with different length, we paddle a special embedding mark at the end of each training record.",
"The masks $\\mathbf {m}$ in the VCD-based generator is uniformly sampled with probability $0.5$ .",
"The Adam algorithm [43] with learning rate $0.001$ for both discriminator and generator is utilized for optimization.",
"Gradients are clipped if the norm of the parameter vector exceeds 5 [8].",
"After we get the generated data, we can map it into EHR record by finding the nearest-neighbor with cosine distance for each feature.",
"The selection of optimal values for hyper-parameters $\\mu $ and $\\rho $ will be discussed later in Section REF .",
"Table: Prediction performance comparison."
],
[
"Risk Prediction Comparison on Basic Models",
"First, we show the performance of our basic predict model (CNN), which explores the CNN model with pre-trained medical feature embedding and is a strong baseline even before boosted.",
"We compare it with logistic regression (LR), linear support vector machine (SVM), random forest (RF) and two other deep models, recurrent neural network models using gated recurrent units (GRU [44]) and long short-term memory (LSTM [45]).",
"For LR, SVM and RF, we use the same setting as mentioned in previous work [33].",
"We follow the similar settings from existing work [45] for GRU and LSTM.",
"Table REF shows the classification accuracy and AUROC (area under receiver operating characteristic curve) of all basic baseline models on the two prediction tasks.",
"CNN is among the best methods in Heart Failure task and significantly outperforms baselines in Diabetes task.",
"The performance improvement mainly comes from the learned embeddings in heart failure task, and from CNN model structures in diabetes task.",
"The other two deep models GRU and LSTM work well but can not beat CNN."
],
[
"Analysis of Generated Data",
"Before testing our semi-supervised prediction models with augmented data, we need to inspect whether the generated data from ehrGAN are able to simulate original data well enough, especially for the patient records in the case cohorts.",
"Having the generated data similar to original one is an important precondition to improve our model performance instead of hurting it.",
"We compared the length and features of the original data ($\\mathcal {D}_o$ ) and generated data $\\mathcal {D}_g$ for the two case groups.",
"As shown in Figure REF , the generated datasets have similar length distributions to original datasets.",
"Then we check the most frequency input features from the two datasets.",
"As shown in Figure REF , The generated data keeps similar frequencies for the 100 most frequent features.",
"Comorbidities (cooccurrences) in patient records are quite useful in clinical prediction tasks.",
"We select 20 most frequent diagnosis features from these two case cohorts, and show the comorbidity heatmaps in Figure REF .",
"We can find that both the feature frequencies and the comorbidity clusters are well simulated in our generated datasets.",
"The list of top 10 diagnosis features for the two cohorts are listed in Table REF and REF .",
"Most of them are common diagnoses in patient records, but with slightly different occurrence frequencies for different cohorts.",
"Our generated models are able to capture the occurrence patterns from different case cohorts and keep those patterns very similar to those in the corresponding original datasets.",
"These analyses not only verify the quality of our generated data, but also help us get better understandings on patterns in cohorts for different tasks.",
"Table: Top 10 most frequent ICD-9 diagnosis codes of heart failure cohort group in the generated data.Table: Top 10 most frequent ICD-9 diagnosis codes of diabetes cohort group in the generated data.Table: Performance compassion of different CNN and SSL prediction models on four sub-datasets."
],
[
"Evaluation of the Boosted Model",
"To evaluate the performance of the boosted model with semi-supervised learning setting, we conduct extensive experiments on the following six approaches.",
"CNN-BASIC: The basic model described in Section REF , trained only on the training subset; CNN-FULL: The basic model trained with the same amount of labeled data as SSL-GAN; CNN-RAND: The basic model trained with the same amount of data as SSL-GAN with random labels for additional data and true labels for training subset; SSL-SMIR: Squared-loss mutual information regularization [46]; LGC: Semi-supervised learning approach with local and global consistency [47]; SSL-GAN: The proposed method with ehrGAN based data augmentation.",
"It is notable that SSL-SMIR and SSL-LGC are strong and robust SSL baselines.",
"CNN-FULL, SSL-SMIR, and SSL-LGC are trained with additional samples from a held-off subset.",
"The parameters setting of SSL-SMIR and SSL-LGC follows those in the original papers [47], [46] and bag of words feature are used.",
"We choose the values of $\\rho $ and $mu$ in SSL-GAN with best performance by Section REF .",
"We summarize the classification performance in Table REF in different settings with different amounts of labeled data.",
"For example, HF50 means $50\\%$ of the training set of Heart Failure is used, and Dia67 means $2/3$ of the training set of Diabetes is used.",
"First of all, our model consistently beats CNN-BASIC and CNN-RAND.",
"On HF50, SSL-GAN achieves $0.8574$ on accuracy and $0.9075$ on AUROC score, compared with $0.8096$ and $0.8784$ for CNN-BASIC.",
"On Dia50 and Dia67, SSL-GAN also improves $3\\%-4\\%$ over the baseline in both measurements.",
"Due to the messed up label information, the performance of CNN-RAND is even worse than CNN-BASIC.",
"Second, compared with the CNN-FULL method, our model can also achieve comparable results.",
"Measured in AUROC score, SSL-GAN is about $2\\%$ lower than CNN-FULL on both Dia50 and Dia67 sets.",
"On HF50 and HF67, the margins are even smaller.",
"The two standard SSL methods SSL-SMIR and SSL-LGC do not perform very well, only achieving similar performances as CNN-BASIC, and our method easily beats them.",
"Overall, these evaluations show the strong boosting power of the proposed SSL-GAN model.",
"Table: AUROC score comparison with different ρ\\rho ."
],
[
"Selections of Parameters",
"We use HF50 and Dia50 to show the effects of values of the two hyper-parameters $\\rho $ and $\\mu $ ."
],
[
"The effectiveness of $\\rho $",
"In this part, we discuss how the selection of $\\rho $ in Equation REF affects the performance.",
"We fix other parameters and vary $\\rho $ from 0 to 1, and report the AUROC score with different settings in Table REF .",
"We see that on both datasets, with a properly chosen $\\rho $ the generator is able to provide good generations to improve learning.",
"$\\rho =0.1$ is an optimal selection for the model (results of $\\rho > 0.2$ are no better than $\\rho =0.2$ and thus omitted here).",
"On the other hand, with $\\rho =0$ , which corresponds to sample from an autoencoder, hurts performance.",
"$\\rho =1$ completely messes up training as the generated samples are not guaranteed to have the same label as the samples conditioned on.",
"This shows that the transition distribution is able to generate samples that are sufficiently different from training samples to boost the performance.",
"Figure: Diabetes"
],
[
"The effectiveness of $\\mu $",
"How to optimally utilize the augmented data from GANs to support the supervised learning is an important problem.",
"In our task, this is controlled by the parameter $\\mu $ in Equation REF , which leverages the ratio of labeled data and augmented data.",
"Generally, including more augmented data will help while too many augmented data may even hurt the performance.",
"With a fixed value of $\\rho $ , we vary $\\mu $ from $0.2$ to $1.4$ and test the prediction performance on two datasets.",
"We also include the setting with fully labeled data (FULL), and $\\mu $ represents the number of real labeled data used instead of from GANs in this setting.",
"The prediction AUROC scores of different methods are shown in Figure REF .",
"It is obvious for the method with fully labeled data that the prediction performance continues improving with $\\mu $ increased.",
"For SSL-GAN, when $\\rho =0.1$ (the optimal setting), we can see it achieved the best performance when $\\mu =0.6$ .",
"After that point, the performance decreased a little, which indicates that more augmented data can not help further.",
"For the setting with $\\rho =1$ and $\\rho =0$ , the prediction power continues falling as including more augmented data is harmful for both cases.",
"Similar trend are also observed under the measure of accuracy."
],
[
"Conclusion",
"In this paper, we focus on exploiting deep learning technique and its applications in healthcare.",
"We first present ehrGAN, a generation model via adversarial training, and discuss several techniques for learning such a model for EHR data.",
"We demonstrate that the proposed model can produce realistic data samples by mimicking the input real data, and the learned latent representation space can continuously encode plausible samples.",
"To boost risk prediction performance, we utilize the learned model to perform data augmentation by semi-supervised learning.",
"Experimental results on two datasets show that the proposed model improves the generalization power and the prediction performance compared with strong baselines.",
"In future work, we would like to perform more comprehensive quantitative comparisons from a clinical view with the help of domain experts.",
"This includes analyzing the important clinical patterns conditioning on disease, the concurrence of several diagnoses, and the correlations among other key medical features.",
"The proposed generation model can also be improved with better clinical interpretation or structure information and achieve more compelling results.",
"We may also try to incorporate state-of-art GANs [48] into our framework.",
"Furthermore, other boosted learning techniques with GANs, such as training a joint learning model, and sharing latent representation space between the networks, will be evaluated on EHR data.",
"Finally, the proposed framework can be naturally extended to other healthcare applications, such as readmission prediction and representation learning."
]
] | 1709.01648 |
[
[
"Energy dependent forward B $\\rightarrow$ $J/\\psi$ measurements in p+p\n collisions at PHENIX"
],
[
"Abstract The heavy flavor studies at RHIC help improve the knowledge of the bottom/charm quark production and can test Quantum Chromodynamics (QCD).",
"Compared to the LHC/Tevatron, the RHIC heavy flavor production originates from different partonic sub-processes and has a complementary kinematic coverage.",
"The PHENIX forward rapidity silicon vertex detector (FVTX) provides precise determination of the event vertex, tracking and the Distance of Closest Approach (DCA) of charged tracks.",
"This detector allows direct access to the $B$ meson production via measurements of non-prompt $J/\\psi \\to \\mu^{+} + \\mu^{-}$ within $1.2<|y|<2.2$ rapidity in $p$+$p$ collisions at $\\sqrt{s} = $ 510 and 200 GeV.",
"Comparison among PHENIX measurements of the $B \\rightarrow J/\\psi$ fraction with integrated $J/\\psi$ $p_{T}$ up to 5 GeV$/c$ and higher energy results at the Tevatron and the LHC presents a smooth center of mass energy dependence from 0.2 to 13 TeV in $p$+$p$ ($p$+$\\bar{p}$) collisions.",
"The Next-To-Leading order Perturbative QCD (NLO pQCD) calculations are in reasonable agreement with the extracted total $b\\bar{b}$ cross section based on the $B \\rightarrow J/\\psi$ fraction measurements at PHENIX."
],
[
"Introduction",
"Heavy quarks are produced in the early stage of high energy hadronic collisions due to their high mass ($m_{\\rm {c,b}} \\gg \\Lambda _{\\rm {QCD}}$ ).",
"Since the heavy flavor quarks do not vanish or change into other flavors during the hard scattering and the sequential fragmentation processes, they can be treated as hard probes to study the initial state partons.",
"At RHIC energies, the bottom production is dominated by the gluon-gluon fusion (pair creation) process in contrast to the Tevatron and the LHC measurements which are dominated by the flavor excitation process [1].",
"In additional to this, the kinematic range of the nucleon (nuclear) gluon distribution function (PDF) accessed by the RHIC heavy flavor measurements is different from what has been measured at the LHC.",
"Therefore, measuring the heavy flavor production at RHIC provides a unique test of Quantum Chromodynamics (QCD).",
"Yields of non-prompt $J/\\psi $ from $B$ meson decay are proportional to the total $B$ meson production.",
"Extracting $B$ meson production via this channel provides a better signal to background ratio compared to the bottom semi-leptonic decay measurements as backgrounds are suppressed by the $J/\\psi $ identification.",
"Decayed particles with a non-zero displaced vertex can be identified with the help of the Forward Silicon Vertex Detector (FVTX) at PHENIX, which can precisely measure the Distance of Closet Approach (DCA) of final state particles.",
"Using this technique, $B$ mesons in the forward/backward rapidities have been studied at PHENIX in $p$ +$p$ collisions.",
"The $B \\rightarrow J/\\psi $ fractions have been measured and the $b\\bar{b}$ cross sections are studied in 510 and 200 GeV $p$ +$p$ collisions.",
"The NLO pQCD calculations are in good agreements with these results.",
"Figure: The B→J/ψB \\rightarrow J/\\psi fraction fit to muon DCA R \\textrm {DCA}_{\\rm R} of J/ψJ/\\psi sample in the (a) 1.2<y<2.21.2<y<2.2 and (b) -2.2<y<-1.2-2.2<y<-1.2 regions.",
"The red solid curve stands for the total fit which includes the prompt J/ψJ/\\psi (solid blue), the BB-meson →J/ψ\\rightarrow J/\\psi (green filled region), the combinatorial background (magenta dashed curve), the cc ¯+bb ¯c\\bar{c}+b\\bar{b} background (brown long-dashed curve) and the detector mismatching background (purple short-dashed curve).",
"Figures from ."
],
[
"The fraction of $J/\\psi $ from {{formula:fae6e541-3212-48fc-8e64-a88fd55e6233}} meson decay in {{formula:f5ff71c7-6272-4d70-9019-999e83fc9fe3}} = 510 and 200 GeV {{formula:0ee3be45-0b23-4579-b796-3fd064724128}} +{{formula:cdf80c2b-9095-4844-b2e5-71a7d5c96277}} collisions",
"The fraction of $J/\\psi $ from $B$ meson decay at forward/backward rapidities ($1.2<|y|<2.2$ ) has been first studied at PHENIX in the 510 GeV $p$ +$p$ collisions during the 2012 RHIC run.",
"Identification of $J/\\psi $ from $B$ meson decay is based on measurements of the DCA radial projection ($\\textrm {DCA}_{\\rm R}$ ) of muons from $J/\\psi $ decay as the FVTX has a better spatial resolution along the radial direction compared to its azimuthal angle component.",
"The $\\textrm {DCA}_{\\rm R}$ of muons from prompt $J/\\psi $ decay follows a symmetric distribution.",
"Because of the decay kinematics, muons from $B$ meson decayed $J/\\psi $ produce an asymmetric tail in their $\\textrm {DCA}_{\\rm R}$ distribution.",
"Figure: The B→J/ψB \\rightarrow J/\\psi fraction with integrated J/ψJ/\\psi p T p_{T} and 1.2<|y|<2.21.2<|y|<2.2 rapidity in pp+pp collisions at s=\\sqrt{s} = 510 (left) GeV and s=\\sqrt{s} = 200 (right) GeV.",
"In the left panel, the red shaded band stands for the FONLL+CEM calculated p T p_{T} dependent B→J/ψB \\rightarrow J/\\psi fraction within 1.2<|y|<2.21.2<|y|<2.2 in 500 GeV pp+pp collisions and the blue shaded curve represents the p T p_{T} integrated (p T <p_{T}< 5 GeV/c/c)B→J/ψ B \\rightarrow J/\\psi fraction within the same kinematic region.",
"The yellow shaded curve in the right panel stands for the FONLL+CEM predicted rapidity dependent B→J/ψB \\rightarrow J/\\psi fractions in 200 GeV pp+pp collisions.This analysis requires muon tracks pass through quality cuts and have good matching between the FVTX and the Muon Tracker (MuTr).",
"$J/\\psi $ candidates are selected from unlike-sign dimuon pairs within the mass region of 2.7-3.5 GeV/$c^{2}$ .",
"After verifications of consistent $\\textrm {DCA}_{\\rm R}$ resolutions in both data and the full simulation of PYTHIA+GEANT+reconstruction with realistic vertex and dead maps etc, the $\\textrm {DCA}_{\\rm R}$ distribution shapes of muons from prompt $J/\\psi $ and $B$ meson decayed $J/\\psi $ are determined in the full simulation.",
"The detector mis-alignments have been corrected before and after the data production.",
"Any remaining $\\textrm {DCA}_{\\rm R}$ offset is determined by identified prompt hadrons in data.",
"Besides the prompt $J/\\psi $ and $B \\rightarrow J/\\psi $ components, the dimuon sample in data also contains the combinatorial background, the detector mis-matching background and the heavy flavor ($c\\bar{c}+b\\bar{b}$ ) continuum background.",
"The combinatorial background which represents the mis-identified muon and/or hadron pairs is determined by unlike-sign track pairs in normalized mixed event samples.",
"Since a nearly 1 $m$ long hadron absorber is located between the FVTX and the MuTr, there is a certain probability of matching an incorrect FVTX track to a MuTr track after the multiple scattering within the absorber.",
"The FVTX-MuTr mis-matching background is determined in mixed events with topologically rotated FVTX.",
"The heavy flavor continuum background is determined using a full simulation.",
"Details of the analysis procedure have been discussed in [2].",
"A maximum log-likelihood fit of the muon $\\textrm {DCA}_{\\rm R}$ distribution is applied to data to simultaneously determine the raw yields of prompt $J/\\psi $ and non-prompt $J/\\psi $ from $B$ meson decay.",
"Figure REF shows the fit results within $-2.2<y<-1.2$ and $1.2<y<2.2$ rapidity regions in 510 GeV $p$ +$p$ collisions.",
"Figure: Comparison of PHENIX BB →\\rightarrow J/ψJ/\\psi fraction measured in 510 GeV and 200 GeV pp+pp collisions with the global data from CDF, ALICE, CMS and LHCb experiments as a function of center of mass energy integrated in the J/ψJ/\\psi 0<p T <50<p_{T}<5 GeV/cc interval.",
"The uncertainty is statistical and systematic combined.Figure: Comparison of the bb ¯b\\bar{b} cross section extracted from the PHENIX BB →\\rightarrow J/ψJ/\\psi fraction measured in 510 GeV and 200 GeV pp+pp collisions with previous measurements at Fermilab, RHIC, LHC and NLO pQCD calculations.",
"The uncertainty of the NLO pQCD calculations is shown in green dashed line, and the systematic uncertainty of experimental results is shown in filled boxes.The detector acceptance$\\times $ efficiency corrected fraction of $J/\\psi $ from $B$ meson decay with integrated $J/\\psi $ $p_{T}$ ($p_{T}<5$ GeV/c) and $1.2<|y|<2.2$ rapidity coverage measured in 510 GeV $p$ +$p$ collisions is shown in the left panel of Figure REF .",
"Reasonable agreement between this result and the Fixed-Order Next-To-Leading Logarithm and Color-Evaporation-Model (FONLL+CEM) [4], [5], [6] calculations has been achieved.",
"Following the same analysis procedure, the $B \\rightarrow J/\\psi $ fraction is measured in 200 GeV $p$ +$p$ collisions with the 2015 RHIC data.",
"The right panel of Figure REF shows the measured results in 200 GeV $p$ +$p$ collisions with integrated $J/\\psi $ $p_{T}$ in the forward and backward rapidity regions, which are also in good agreement with the derived rapidity dependent $B \\rightarrow J/\\psi $ fractions based on the FONLL [7] and the CEM [8] calculations."
],
[
"Energy dependent $B \\rightarrow J/\\psi $ fraction and extrapolated {{formula:0e52754e-658d-4b33-96a8-b33fe122ad25}} cross section in {{formula:26b688cf-56b0-4b36-8cd3-4904674290f4}} +{{formula:f02c17c3-f669-46bc-9cd5-1be76879d971}} collisions",
"In order to understand the energy dependent $B$ hadron production, the PHENIX measured $B \\rightarrow J/\\psi $ fractions in 510 and 200 GeV $p$ +$p$ collisions are compared with other Tevatron and LHC measurements [9], [10], [11], [12], [13] with integrated $J/\\psi $ $p_{T}$ up to 5 GeV/c at different rapidities and higher center of mass energies.",
"As shown in Figure REF , the center of mass energy dependent $B \\rightarrow J/\\psi $ fraction results indicate a smooth transition of $B$ production from the RHIC energies which are dominated by the gluon-gluon fusion process to the LHC energies which are dominated by the gluon splitting process.",
"To further confirm this, the total $b\\bar{b}$ cross sections in 200 and 510 GeV $p$ +$p$ collisions are extracted from the forward $B \\rightarrow J/\\psi $ fraction results at PHENIX with the measured inclusive $J/\\psi $ cross section [14], the branching ratio of $B$ meson decay to $J/\\psi $ and the phase space scaling factor calculated by the FONLL.",
"The preliminary results of the extracted $b\\bar{b}$ cross sections in $p$ +$p$ collisions, as shown in Figure REF , are in reasonable agreement with the NLO pQCD calculations [15].",
"The PHENIX extracted $b\\bar{b}$ cross sections and other global measurements follow the NLO pQCD predicted energy dependence of the bottom production."
],
[
"Summary and Outlook",
"First measurements of $J/\\psi $ from $B$ meson decay in forward/backward rapidities with integrated $p_{T}$ starting from zero are achieved in both 510 and 200 GeV $p$ +$p$ collisions at PHENIX.",
"A smooth transition is found for the $B \\rightarrow J/\\psi $ fractions with integrated $J/\\psi $ $p_{T}$ up to 5 GeV$/c$ measured in $p$ +$p$ or $p$ +$\\bar{p}$ collisions from 0.2 to 13 TeV.",
"The NLO pQCD calculated $b\\bar{b}$ cross sections are consistent with the extracted values based on the $B \\rightarrow J/\\psi $ fraction measurements at PHENIX in 200 and 510 GeV $p$ +$p$ collisions.",
"Global data on the bottom cross section follow the energy dependence predicted by the NLO pQCD calculations.",
"Large data sets in $p$ +$p$ and $p$ +Au collisions collected by PHENIX provide opportunities to further study the open heavy flavor production in forward/backward rapidities.",
"This allows us to study the $p_{T}$ dependent $B \\rightarrow J/\\psi $ yields and explore the cold nuclear matter effects on the bottom production.",
"The method used for the $B$ meson decay to $J/\\psi $ fraction analysis through analyzing the muon $DCA_{R}$ can be extended to the study of $B$ and $D$ meson semi-leptonic decays to muons in the forward/backward rapidities to study the charm and bottom production in the low $p_{T}$ region."
]
] | 1709.01565 |
[
[
"Automorphism groups of finite topological rank"
],
[
"Abstract We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class.",
"We then show how finite topological rank of the automorphism group of an $\\omega$-categorical structure can go down to reducts.",
"Together, those results prove that a large number of $\\omega$-categorical structures that appear in the literature have an automorphism group of finite topological rank.",
"In fact, we are not aware of any $\\omega$-categorical structure to which they do not apply (assuming the automorphism group has no compact quotients).",
"We end with a few questions and conjectures."
],
[
"Introduction",
"Many automorphism groups of Fraïssé structures are known to admit a 2-generated dense subgroup.",
"This is the case for example for dense linear orders and the random graph [25], [7].",
"It is however not true that all automorphism groups of say $\\omega $ -categorical structures have even a finitely generated dense subgroup.",
"For instance a construction of Cherlin and Hrushovski yields an $\\omega $ -categorical structure whose automorphism groups admits $(\\mathbb {Z}/2\\mathbb {Z})^{\\omega }$ as a quotient, which implies that it cannot have a finitely generated dense subgroup (see Remark REF ).",
"However, it seems that the existence of such a large compact quotient is the only known obstruction.",
"We speculate that this might indeed be the case and ask: Let $G$ is the automorphism group of an $\\omega $ -categorical structure; assume $G$ has no compact quotient, then does it have a finitely generated dense subgroup?",
"This paper is our attempt at answering this question.",
"We fall short of providing a definitive answer, but we succeed in finding sufficient conditions for such a $G$ to admit a finitely generated dense subgroup which seem to apply to all known examples.",
"It is even plausible that those conditions are actually satisfied by all $\\omega $ -categorical structures with trivial $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , see Conjecture REF .",
"We now describe our main results.",
"We first define a notion of a canonical independence relation, or CIR.",
"It is a ternary independence relation $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ which satisfies in particular stationarity over $$and transitivity on both sides.",
"Importantly, we do not assume symmetry.We show that if an ultrahomogeneous structure $ M$ admits such a CIR,then it has a 2-generated dense subgroup, and even a cyclically denseconjugacy class (that is, for some $ f,gG$, the set $ { f-ngfn | nZ} $is dense).",
"Many, but not all, classical $$-categorical structureshave a CIR.",
"Examples that do not include the dense circular order(Corollary \\ref {cor:Circular order has no CIR but degree <=00003D3})and the dense infinitely-branching tree (Corollary \\ref {cor:Dense trees don^{\\prime }t have a groovy ind relation}).However, an expansion of any of those structures obtained by naminga point does have a CIR (Example \\ref {exa:trees with points}, Remark\\ref {rem:expanding a circular order by a point has a CIR}).",
"Thisleads to our second main theorem which completely solves a relativeversion of our initial question: we show that if $ Aut(M)$ has nocompact quotients and $ N$ is an $$-categorical expansion of$ M$, then if $ Aut(N)$ has a finitely generated densesubgroup, then so does $ Aut(M)$.",
"More precisely, we showthat adding two elements from $ Aut(M)$ to $ Aut(N)$yields a dense subgroup of $ Aut(M)$: see Theorem \\ref {thm:If no CQ then finitely generated}.$ In Section , we give a dynamical consequence of having a CIR.",
"Our main motivation is to relate it to the Ramsey property.",
"We observe in Proposition REF that Ramsey structures admit a weaker form of independence relation.",
"We know by [20] that the Ramsey property is equivalent to extreme amenability of the automorphism group.",
"It is then natural to look for a dynamical interpretation of having a CIR, one goal being to understand to what extent the Ramsey property is not sufficient to imply it.",
"We give a necessary condition for having a CIR which sheds some light on this notion and can be used to prove negative results.",
"The theorems presented in this paper lead to a number of open questions.",
"In particular, it would be interesting to understand the obstructions to having a CIR and to prove more results about the automorphism groups of structures with a CIR.",
"It is also our hope that some ideas introduced here could be used to develop a general theory of $\\omega $ -categorical structures.",
"One strategy we have in mind is to show that $\\omega $ -categorical structures admit nice expansions and then to prove relative statements which pull down properties from an expansion to the structure itself.",
"See Section for some precise conjectures.",
"We end this introduction by mentioning some previous work done on this question: The existence of a cyclically dense conjugacy class was shown for the random graph by Macpherson [25], for the Urysohn space by Solecki [36], for dense linear orders by Darji and Mitchell [7] and recently for generic posets by Glab, Gordinowicz and Strobin [11].",
"Kechris and Rosendal [21] study the property of having a dense conjugacy class for Polish groups.",
"They show that a number of Polish groups admit cyclically dense conjugacy classes (see Theorem 2.10).",
"Those include the group of homeomorphisms of the Cantor space and the automorphism group of a standard Borel space.",
"Those groups do not fit in our context, although it should be possible to generalize our results so as to include them.",
"In fact, the proof of Theorem 2.10 is very much in the same spirit as the proofs in this paper.",
"Kwiatkowska and Malicki [19] give sufficient conditions for an automorphism group $G$ to have a cyclically dense conjugacy class, which gives new examples such as structures with the free amalgamation property and tournaments.",
"Their conditions do not seem to formally imply ours, but all the examples that they give (and in particular structures with free amalgamation) are covered by our theorems.",
"They also show that under the same hypothesis $L_{0}(G)$ has a cyclically dense conjugacy class.",
"We did not study this."
],
[
"$\\omega $ -categoricity,\nUltrahomogeneous structures, Fraïssé limits and model companions",
"Here we recall the basic facts we need for this paper.",
"Let $L$ be some countable first order language (vocabulary).",
"An $L$ -theory is called $\\omega $-categorical if it has a unique infinite countable model up to isomorphism.",
"A countable model $M$ is $\\omega $ -categorical if its complete theory $Th\\left(M\\right)$ is.",
"By a theorem of Engeler, Ryll-Nardzewski and Svenonius, see e.g., [15], this is equivalent to saying that $G=\\operatorname{Aut}\\left(M\\right)$ is oligomorphic: for every $n<\\omega $ , there are only finitely many orbits of the action of $G$ on $M^{n}$ .",
"It is also equivalent to the property that every set $X\\subseteq M^{n}$ which is invariant under $G$ is $\\emptyset $ -definable in $M$ .",
"Also, if $M$ is $\\omega $ -categorical and $A$ is a finite subset of $M$ then $M_{A}$ is also $\\omega $ -categorical, where $M_{A}$ is the expansion of $M$ for the language $L_{A}$ which adds a constant for every element in $A$ .",
"A countable $L$ -structure $M$ is called ultrahomogeneous if whenever $f:A\\rightarrow B$ is an isomorphism between two finitely generated substructures $A,B$ of $M$ , there is $\\sigma \\in \\operatorname{Aut}\\left(M\\right)$ extending $f$ .",
"The age of an $L$ -structure $M$ , $\\operatorname{Age}\\left(M\\right)$ , is the class of all finitely generated substructures which can be embedded into $M$ .",
"Recall that a class of finitely generated $L$ -structures $K$ closed under isomorphisms has the hereditary property (HP) if whenever $A\\in K$ and $B\\subseteq A$ ($B$ is a substructure of $A$ ), $B\\in K$ .",
"The class $K$ has the joint embedding property (JEP) if whenever $A,B\\in K$ there is some $C$ such that both $A,B$ embed into $C$ .",
"It has the amalgamation property (AP) if whenever $A,B,C\\in K$ and $f_{B}:A\\rightarrow B$ , $f_{C}:A\\rightarrow C$ are embeddings, then there is some $D\\in K$ and embeddings $g_{B}:B\\rightarrow D$ , $g_{C}:C\\rightarrow D$ such that $g_{B}\\circ f_{B}=g_{C}\\circ f_{C}$ .",
"We say that $K$ is uniformly locally finite if for some function $f:\\omega \\rightarrow \\omega $ , for every $A\\in K$ and $X$ a subset of $A$ of size $n$ , the structure generated by $X$ has size $\\le f\\left(\\left|X\\right|\\right)$ .",
"In the following, “essentially countable” means that $K$ contains at most countably many isomorphism types of structures.",
"We also recall the notions of model companions and model completions.",
"A theory $T$ is called model complete if whenever $M\\subseteq N$ are models of $T$ , $M\\prec N$ .",
"Suppose that $T_{\\forall }$ is a universal theory.",
"A theory $T^{\\prime }$ is the model companion of $T_{\\forall }$ if $T^{\\prime }$ is model complete and $T^{\\prime }_{\\forall }=T_{\\forall }$ (they have the same universal consequences).",
"In other words, every model of $T_{\\forall }$ can be embedded in a model of $T^{\\prime }$ .",
"The theory $T^{\\prime }$ is a model completion of $T_{\\forall }$ if in addition it has elimination of quantifiers.",
"Models companions are unique, if they exist.",
"For more, see [37] and [15].",
"Fact 2.1 Suppose that that $K$ is an essentially countable class of finite $L$ -structures which has HP, JEP and AP.",
"[15] The class $K$ has a Fraïssé limit: a unique countable ultrahomogeneous model $M$ with the same age.",
"[15] If $K$ is uniformly locally finite in a finite language (or without assuming that $L$ is finite but instead that for each $n<\\omega $ there are finitely many isomorphism types of structures generated by $n$ elements), then $M$ is $\\omega $ -categorical and has quantifier elimination.",
"(See remark below.)",
"If $T$ is a countable universal $L$ -theory, $L$ is finite, $K$ is the class of finitely generated models of $T$ and $K$ is uniformly locally finite (or without assuming that $L$ is finite but instead that $K$ has at most finitely many isomorphism types of structures generated by $n$ elements for all $n<\\omega $ ), then the theory $Th\\left(M\\right)$ is $\\omega $ -categorical and is the model completion of $T$ .",
"(See remark below.)",
"[15]The converse to (1) also holds: if $M$ is an ultrahomogeneous then the age of $M$ satisfies HP, JEP and AP.",
"Remark 2.2 The assumptions in parenthesis in (2) is not stated in [15], but a straightforward modification of that proof gives this.",
"We could not find an explicit reference for (3) (which is well-known), so here is a short argument.",
"Since $Th\\left(M\\right)$ eliminates quantifiers by (2), it is enough to show that $Th\\left(M\\right)_{\\forall }=T_{\\forall }=T$ (since $T$ is universal).",
"If $\\psi $ is universal and $T\\models \\psi $ , then since $\\operatorname{Age}\\left(M\\right)\\subseteq K$ , $M\\models \\psi $ .",
"On the other hand, if $M\\models \\psi $ where $\\psi $ is universal, and $T\\lnot \\models \\psi $ , then there is a model $A^{\\prime }\\models T$ such that $A^{\\prime }\\models \\lnot \\psi $ , and since $\\lnot \\psi $ is existential, the same is true for some finitely generated model $A\\subseteq A^{\\prime }$ , so $A\\in K$ and since $A\\in \\operatorname{Age}\\left(M\\right)$ , we get a contradiction.",
"Classes $K$ as in Fact REF are called Fraïssé classes or amalgamation classes.",
"Some examples of Fraïssé limits include DLO (dense linear order), i.e., $Th\\left(\\mathbb {Q},<\\right)$ (here we identify the Fraïssé limit and its theory), the random graph, the random poset (partially ordered set), the random tournament, and more.",
"One example that we will be interested in is that of dense trees.",
"Example 2.3 Let $L_{dt}=\\left\\lbrace <,\\wedge \\right\\rbrace $ , and let $T_{dt,\\forall }$ be the universal theory of trees with a meet function $\\wedge $ .",
"Then $T_{dt,\\forall }$ has an $\\omega $ -categorical model completion by Fact REF (note that the tree generated by a finite set $B$ is just $B\\cup \\left\\lbrace x\\wedge y\\,|\\,x,y\\in B\\right\\rbrace $ ).",
"We denote the model companion by $T_{dt}$ and call the unique countable model the dense tree.",
"See also [35].",
"Recall that a structure $M$ is homogeneous if whenever $a,b$ are finite tuples of the same length, and $a\\equiv b$ (which means $\\operatorname{tp}\\left(a/\\emptyset \\right)=\\operatorname{tp}\\left(b/\\emptyset \\right)$ , i.e., the tuples $a,b$ have the same type), then there is an automorphism taking $a$ to $b$ .",
"Note that ultrahomogeneous structures are homogeneous and the same is true for $\\omega $ -categorical ones.",
"When $M$ is homogeneous, an elementary map $f:A\\rightarrow B$ for $A,B\\subseteq M$ is just a restriction of an automorphism of $M$ .",
"Finally, we use $\\mathfrak {C}$ to represent a monster model of the appropriate theory.",
"This is a big saturated (so also homogeneous) model that contains all the models and sets we will need.",
"This is standard in model theory.",
"For more, see [37]."
],
[
"A mix of two Fraïsé limits",
"Suppose that $K_{1},K_{2}$ are two amalgamation classes of finite structures in the languages $L_{1},L_{2}$ respectively.",
"Assume the following properties: The symmetric difference $L_{1}\\mathrel {\\triangle }L_{2}$ is relational.",
"The class $K$ of finite $L_{1}\\cup L_{2}$ -structures $A$ such that $A\\upharpoonright L_{1}\\in K_{1}$ and $A\\upharpoonright L_{2}\\in K_{2}$ is an amalgamation class.",
"Let $M$ be its Fraïsé limit.",
"For every $A\\in K_{1}$ , there is some expansion $A^{\\prime }$ of $A$ to an $L_{1}\\cup L_{2}$ -structure such that $A^{\\prime }\\upharpoonright L_{2}\\in K_{2}$ , and similarly that for every $B\\in K_{2}$ there is some expansion $B^{\\prime }$ to $L_{1}\\cup L_{2}$ whose restriction to $L_{1}$ is in $K_{1}$ .",
"If $A\\in K$ and $A\\upharpoonright L_{1}\\subseteq B\\in K_{1}$ then there is an expansion $B^{\\prime }$ of $B$ to $L_{1}\\cup L_{2}$ such that $A\\subseteq B^{\\prime }$ and $B^{\\prime }\\upharpoonright L_{2}\\in K_{2}$ , and similarly for $L_{2}$ .",
"Under all these conditions we have the following.",
"Proposition 2.4 The structure $M\\upharpoonright L_{1}=M_{1}$ is the Fraïsé limit of $K_{1}$ and $M\\upharpoonright L_{2}=M_{2}$ is the Fraïsé limit of $K_{2}$ .",
"Start with $M_{1}$ (for $M_{2}$ , the proof is the same).",
"It is enough to show that $M_{1}$ is ultrahomogeneous and that $\\operatorname{Age}\\left(M_{1}\\right)=K_{1}$ .",
"The second statement follows from (2), (3) above.",
"For the first, by [15] it is enough to show that if $A\\subseteq B$ are from $K_{1}$ and $f:A\\rightarrow M_{1}$ then there is some $g:B\\rightarrow M_{1}$ extending $f$ .",
"Using $f$ we can expand $A$ to an $L_{1}\\cup L_{2}$ -structure $A^{\\prime }$ in such a way that $f$ is an embedding to $M$ (this uses the fact that $L_{2}\\backslash L_{1}$ is relational).",
"By (4) we can expand $B$ to an $L_{1}\\cup L_{2}$ -structure $B^{\\prime }$ such that $A^{\\prime }\\subseteq B^{\\prime }$ and $B^{\\prime }\\in K$ .",
"Since $M$ is the Fraïssé limit of $K$ , it follows by [15] again that $f$ can be extended to $g:B^{\\prime }\\rightarrow M$ , and in particular, $g\\upharpoonright L_{1}$ is the embedding we seek."
],
[
"The maximal compact quotient of\nthe automorphism group of a countable $\\omega $ -categorical structure",
"Assume in this section that $M$ is $\\omega $ -categorical and countable, and let $G=\\operatorname{Aut}\\left(M\\right)$ , considered as a topological group in the product topology.",
"The contents of this section are folklore but we give the details for the sake of readability.",
"Recall that for a structure $M$ and $A\\subseteq M$ , $\\operatorname{acl}\\left(A\\right)$ is the set of all algebraic elements over $A$ (elements satisfying an algebraic formula over $A$ : one with finitely many solutions).",
"Similarly, $\\operatorname{dcl}\\left(A\\right)$ is the set of all elements definable over $A$ .",
"In the context of $\\omega $ -categorical structures, $\\operatorname{acl}\\left(A\\right)$ and $\\operatorname{dcl}\\left(A\\right)$ are defined in terms of the size of the orbit of the action of $G$ fixing $A$ being finite or a singleton respectively.",
"In the next proposition, we describe the maximal compact quotient of $\\operatorname{Aut}\\left(M\\right)$ in model theoretic terms.",
"This uses the notion of $M^{\\operatorname{eq}}$ : the expansion of $M$ obtained by adding a new sort for every $\\emptyset $ -definable quotient of some $\\emptyset $ -definable set.",
"See [37] for more.",
"For $A\\subseteq M$ , $\\operatorname{Aut}\\left(M/A\\right)$ is the group of automorphisms of $M$ fixing $A$ .",
"This is a closed subgroup of $G$ which is normal when $A$ is invariant under $\\operatorname{Aut}\\left(M\\right)$ , thus the quotient $G/\\operatorname{Aut}\\left(M/A\\right)$ is a Hausdorff topological group (with the quotient topology).",
"We identify $\\operatorname{Aut}\\left(M\\right)$ and $\\operatorname{Aut}\\left(M^{\\operatorname{eq}}\\right)$ so that we can put $A=\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"In this case, it is also compact as the next proposition says.",
"Proposition 2.5 The group $G/\\operatorname{Aut}\\left(M/\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is a compact Hausdorff (in fact — profinite) group.",
"Let $H$ be the group of all elementary maps from $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ to $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , also denoted by $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ .",
"The group $H$ is naturally profinite as an inverse system of the family $\\left\\lbrace H_{X}\\,|\\,X\\subseteq M^{\\operatorname{eq}},\\text{ a finite }\\emptyset \\text{-definable set}\\right\\rbrace ,$ where $H_{X}$ is the group of elementary permutations of $X$ .",
"Let $\\operatorname{res}:G\\rightarrow H$ be the restriction map $\\operatorname{res}\\left(\\sigma \\right)=\\sigma \\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"We will show that $\\operatorname{res}$ is onto.",
"Using back-and-forth, it is enough to show that given any complete type $p\\left(x\\right)$ for $x$ in the home sort over $A\\cup \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ where $A\\subseteq M$ is finite, $p$ can be realized in $M$ .",
"We work in the monster model $\\mathfrak {C}$ .",
"Let $E=\\mathord {\\equiv _{A\\cup \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)}}$ be the equivalence relation of having the same type over $A\\cup \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"Then $E$ is $A$ -invariant and has boundedly many classes in $\\mathfrak {C}$ .",
"By $\\omega $ -categoricity, as $A$ is finite $E$ is definable over $A$ , and by compactness, $E$ has finitely many classes.",
"But then for every $E$ -class there must be a representative in $M$ .",
"Since a realization of $p$ must have an $E$ -equivalent element in $M$ , $p$ is realized in $M$ .",
"Note that by compactness we get that every such $p$ is isolated by its restriction to $A\\cup X$ where $X$ is some finite $\\emptyset $ -definable set in $M^{\\operatorname{eq}}$ .",
"The kernel of $\\operatorname{res}$ is precisely $G^{0}=\\operatorname{Aut}\\left(M/\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ , so $\\operatorname{res}$ induces an isomorphism of groups $G/G^{0}\\rightarrow H$ .",
"The group $G/G^{0}$ is also a topological group when equipped with the quotient topology.",
"This map is easily seen to be continuous.",
"To see that it is open, it is enough to show that the image of an open neighborhood $V$ of $\\operatorname{id}\\cdot G^{0}$ in $G/G^{0}$ contains an open neighborhood of $\\operatorname{id}$ in $H$ .",
"The preimage of $V$ in $G$ is some open set $U\\subseteq G$ containing $\\operatorname{id}$ .",
"Suppose $\\operatorname{id}\\in U_{b}=\\left\\lbrace \\sigma \\in G\\,|\\,\\sigma \\left(b\\right)=b\\right\\rbrace \\subseteq U$ is some basic open set.",
"As we noted above, there is some finite $\\emptyset $ -definable set $X\\subseteq M^{\\operatorname{eq}}$ such that $\\operatorname{tp}\\left(b/\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is isolated by $\\operatorname{tp}\\left(b/X\\right)$ .",
"Then if $\\tau \\in G$ is such that $\\tau \\upharpoonright X=\\operatorname{id}_{X}$ , then there is some $\\sigma \\in \\operatorname{Aut}\\left(M/\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ such that $\\sigma \\tau \\left(b\\right)=b$ , so $\\sigma \\tau \\in U_{b}$ , but then $\\tau \\in U$ (because $U$ is a union of cosets of $\\operatorname{Aut}\\left(M/\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ ).",
"Hence, the image of $V$ contains the open set $\\left\\lbrace \\tau \\in H\\,|\\,\\tau \\upharpoonright X=\\operatorname{id}_{X}\\right\\rbrace $ .",
"Together these two groups are isomorphic as topological groups, so are profinite.",
"Definition 2.6 We let $G^{0}=\\operatorname{Aut}\\left(M/\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ .",
"Proposition 2.7 If $H\\trianglelefteq G$ is normal and closed and $G/H$ is compact then $G^{0}\\le H$ .",
"Let $E_{n}$ be the equivalence relation on $M^{n}$ of having the same $H$ -orbit.",
"Then $E_{n}$ refines $\\equiv $ (having the same type over $\\emptyset $ ) and is definable in $M$ .",
"Indeed, suppose that $\\sigma \\in G$ .",
"Then for every $a\\in M^{n}$ , $\\sigma O_{H}\\left(a\\right)=O_{H}\\left(\\sigma \\left(a\\right)\\right)$ (because $H$ is normal), where $O_{H}\\left(a\\right)$ denotes the orbit of $a$ under $H$ .",
"Hence $E_{n}$ is invariant under $G$ so $\\emptyset $ -definable.",
"In addition, $G$ acts transitively on the $H$ -orbits within each $\\equiv $ -class by $\\sigma \\cdot O_{H}\\left(a\\right)=O_{H}\\left(\\sigma \\left(a\\right)\\right)$ .",
"The stabilizer of $O_{H}\\left(a\\right)$ is $\\left\\lbrace \\sigma \\in G\\,|\\,\\sigma \\left(O_{H}\\left(a\\right)\\right)=O_{H}\\left(a\\right)\\right\\rbrace $ so open (if $\\sigma $ is there, then to make sure that $\\sigma ^{\\prime }$ is there, it is enough that $\\sigma ^{\\prime }\\left(a\\right)=\\sigma \\left(a\\right)$ ).",
"Note that this action factors through $H$ (i.e., the action $\\sigma H\\cdot O_{H}\\left(a\\right)=O_{H}\\left(\\sigma \\left(a\\right)\\right)$ is well-defined).",
"Hence, the stabilizer is open in $G/H$ and hence has finite index in $G/H$ , so the number of orbits of $H$ under this action in every $\\equiv $ -class is finite.",
"By $\\omega $ -categoricity, the number of $\\equiv $ -classes (of $n$ -tuples) is finite, so the number of orbits of $H$ is finite.",
"In summary, $E_{n}$ is definable and has finitely many classes.",
"Hence these classes belong to $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"Given $\\sigma \\in G^{0}$ , $\\sigma $ fixes the orbits of $H$ under its action on $M^{n}$ .",
"As $H$ is closed, this means that $\\sigma \\in H$ .",
"Corollary 2.8 The group $G/G^{0}$ is the maximal compact Hausdorff quotient of $G$ .",
"In light of Corollary REF , $G$ has no compact quotients (by which we mean that there is no nontrivial compact Hausdorff group which is an image of $G$ under a continuous group homomorphism) iff $G^{0}=G$ .",
"If $N$ is a normal closed subgroup of $G$ , then we would like to say that $\\left(G,N\\right)$ has no compact quotients iff $G/N$ has no compact quotients.",
"Let us generalize this to any closed subgroup.",
"Definition 2.9 Suppose that $H\\le G$ is closed.",
"We will say that the pair $\\left(G,H\\right)$ has no compact quotients if for all $g\\in G$ , there is some $h\\in H$ such that $g\\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=h\\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"Proposition 2.10 Suppose that $H\\le G$ is closed.",
"Then $\\left(G,H\\right)$ has no compact quotients iff for every closed and normal $N\\trianglelefteq G$ such that $G/N$ is compact, $NH=G$ .",
"First note that $\\left(G,H\\right)$ has no compact quotient iff $\\left\\lbrace gH\\,|\\,g\\in G\\right\\rbrace =\\left\\lbrace gH\\,|\\,g\\in G^{0}\\right\\rbrace $ .",
"This happens iff $G=G^{0}H$ .",
"Hence the direction from right to left follows by taking $N=G^{0}$ .",
"The direction from left to right is immediate by Proposition REF .",
"Corollary 2.11 If $H\\le G$ is closed and normal then $\\left(G,H\\right)$ has no compact quotients iff $G/H$ has no compact quotients as a topological group (i.e., there is no nontrivial compact Hausdorff quotient).",
"Left to right: suppose that $G/H$ has a compact quotient.",
"Then there is a normal closed subgroup $N\\trianglelefteq G$ such that $H\\le N$ and $G/N$ is compact.",
"By Proposition REF , $N$ contains $G^{0}$ .",
"Thus, $G^{0}H\\le N$ and hence $G=N$ (by Proposition REF ).",
"Right to left: for a finite $\\emptyset $ -definable subset $X\\subseteq \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , let $G_{X}^{0}=\\operatorname{Aut}\\left(M/X\\right)\\le G$ (where we identify $G$ with $\\operatorname{Aut}\\left(M^{\\operatorname{eq}}\\right)$ ).",
"Note that as $X$ is definable, $G_{X}^{0}$ is normal, hence so is the product with $H$ .",
"Since $\\left[G:G_{X}^{0}\\right]$ is finite, $HG_{X}^{0}$ is closed as a finite union of translates of $G_{X}^{0}$ so that $G/HG_{X}^{0}$ is a compact (even finite) Hausdorff quotient of $G/H$ , so it must be trivial and hence that $HG_{X}^{0}=G$ .",
"Take $g\\in G$ , we need to find $h\\in H$ such that $g\\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=h\\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"By what we just said, we have that (*) for every finite definable $X\\subseteq \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ there is some $h_{X}\\in H$ such that $g\\upharpoonright X=h_{X}\\upharpoonright X$ .",
"But since every finite $X\\subseteq \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ is contained in a finite $\\emptyset $ -definable set $X^{\\prime }\\subseteq \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ ($X^{\\prime }$ is just the union of all conjugates of $X$ ), (*) is true for all finite subset $X\\subseteq \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"For every finite tuple $a$ from $M$ , $O_{H}\\left(a\\right)$ (the orbit of $a$ under $H$ ) is $a$ -definable (because $H$ is normal, $g\\cdot O_{H}\\left(a\\right)=O_{H}\\left(g\\left(a\\right)\\right)$ for any $g\\in G$ , so that $O_{H}\\left(a\\right)$ is $a$ -invariant thus $a$ -definable by $\\omega $ -categoricity).",
"In $M^{\\operatorname{eq}}$ , every definable set $X$ has a code $X\\in M^{\\operatorname{eq}}$ (such that the automorphisms fixing $X$ setwise in $M$ are precisely the automorphisms fixing $X$ ).",
"(This notation is a bit misleading since there could be many possible codes for $X$ .)",
"Let $D\\subseteq M^{\\operatorname{eq}}$ be the collection of all possible codes $O_{H}\\left(a\\right)$ for all finite tuples $a$ from $M$ .",
"Then $D$ is invariant under $G$ since $g\\left(O_{H}\\left(a\\right)\\right)=O_{H}\\left(g\\left(a\\right)\\right)$ for all $g\\in G$ (i.e., $g\\left(O_{H}\\left(a\\right)\\right)$ is a code for $O_{H}\\left(g\\left(a\\right)\\right)$ ).",
"Let $\\bar{c}$ be a tuple enumerating $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"Since every $h\\in H$ fixes $D$ pointwise, (*) gives us that $\\bar{c}\\equiv _{D}g\\left(\\bar{c}\\right)$ (because to check this equation it is enough to consider finite subtuples).",
"Thus, the map $f$ taking $\\bar{c}$ to $g\\left(\\bar{c}\\right)$ fixing $D$ is an elementary map.",
"By a back-and-forth argument almost identical to the one given in the proof of Proposition REF , there is some automorphism $h\\in G$ extending $f$ (the point is that the relation $\\equiv _{\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\cup D\\cup A}$ is bounded and $A$ -invariant for any finite set $A$ , hence definable and hence has finitely many classes, all of them realized in $M$ ).",
"Since $H$ is closed, and $h$ fixes all $H$ -orbits setwise (as it fixes $D$ ), $h\\in H$ .",
"Finally, $h\\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=g\\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ as requested.",
"Example 2.12 If $M^{\\prime }$ is an expansion of $M$ and $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ in $M$ then $\\left(G,\\operatorname{Aut}\\left(M^{\\prime }\\right)\\right)$ has no compact quotients.",
"This is because in that case, $G^{0}=G$ ."
],
[
"Expansions and reducts of $\\omega $ -categorical\nstructures",
"A group $H$ acts oligomorphically on a set $X$ if for all $n<\\omega $ , the number of orbits of $X^{n}$ under the action of $H$ is finite for every $n<\\omega $ .",
"If $M$ is countable and $\\omega $ -categorical, and $H\\le G$ is closed, then $H=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ for some expansion of $M$ .",
"In addition, if $H$ acts oligomorphically on $M$ , then $M^{\\prime }$ is $\\omega $ -categorical by Ryll-Nardzewski.",
"On the other hand, if $G\\le H$ where $H$ is a closed subgroup of the group of permutations of $M$ , then $H=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ for some ($\\omega $ -categorical) reduct $M^{\\prime }$ of $M$ .",
"Two such reducts $M^{\\prime }$ , $M^{\\prime \\prime }$ are the same up to bi-definability if they have the same definable sets, which is equivalent to $\\operatorname{Aut}\\left(M^{\\prime }\\right)=\\operatorname{Aut}\\left(M^{\\prime \\prime }\\right)$ .",
"Proposition 2.13 Let $M$ be $\\omega $ -categorical and $G=\\operatorname{Aut}\\left(M\\right)$ .",
"Then $G^{0}\\le G$ acts oligomorphically on $M$ and $G^{0}=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ for an $\\omega $ -categorical expansion $M^{\\prime }$ of $M$ with no compact quotients.",
"Let $M^{\\prime }_{0}$ be the expansion of $M^{\\operatorname{eq}}$ obtained by naming (i.e., adding constants for) every element in $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"Then $M_{0}^{\\prime }$ is still $\\omega $ -categorical (as a many sorted structure) since for any given sort (or finite collection of sorts) $S$ , the equivalence relation $\\equiv _{\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)}$ on $S$ -tuples, of having the same type over $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , is bounded, definable and hence finite, as in the arguments above.",
"Let $M^{\\prime }$ be the reduct to the home sort (so it is also $\\omega $ -categorical).",
"By definition, $G^{0}=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ .",
"Moreover, letting $H=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ , we have that $H^{0}=H$ .",
"This is because $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{acl}^{\\operatorname{eq}}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ ."
],
[
"Ramsey Classes",
"Let us start with the definition.",
"Definition 2.14 For two $L$ -structures $A,B$ , we let ${B \\atopwithdelims ()A}$ be the set substructures of $B$ isomorphic to $A$ .",
"Suppose that $K$ is a class of finite $L$ -structures.",
"We say that $K$ is a Ramsey class if for every $A,B\\in K$ and $k<\\omega $ there is some $C\\in K$ such that $C\\rightarrow \\left(B\\right)_{k}^{A}$ : for every function $f:{C \\atopwithdelims ()A}\\rightarrow k$ there is some $B^{\\prime }\\in {C \\atopwithdelims ()B}$ such that $f$ is constant on ${B^{\\prime } \\atopwithdelims ()A}$ .",
"Say that an ultrahomogeneous $L$ -structure $M$ with a quantifier-free definable linear order is a Ramsey structure if $\\operatorname{Age}\\left(M\\right)$ is a Ramsey class.",
"Ramsey classes are extremely important classes of finite structure.",
"There are many examples of Ramsey classes, in particular the class of finite linear orders (this is just Ramsey's theorem) and furthermore, by a theorem of Nešetřil and Rödl [30], proved independently by Harrington and Abramson [1], the class of all finite linearly ordered graphs, or more generally the class of all finite linearly ordered structures in a fixed finite relational language is Ramsey.",
"In fact, [13] generalizes this to allow function symbols as well.",
"There is another definition of Ramsey structures that colors embeddings instead of copies, see [3].",
"This is equivalent to our definition since we asked for a quantifier-free definable linear order (so that finite substructures are rigid).",
"If we drop the requirement that there is a definable linear order then these definitions do not agree in general.",
"In fact, using the alternative definition there must be a definable order in the $\\omega $ -categorical case, so these are equivalent in this case.",
"Fact 2.15 [3] If $M$ is an $\\omega $ -categorical ultrahomogeneous Ramsey structure according to [3], then there is a definable linear order on $M$ .",
"What about dense trees?",
"Adding a generic linear order to a dense tree will not result in a Ramsey structure.",
"By this we mean the model completion of the theory $T_{dt,<,\\forall }$ in the language $\\left\\lbrace <,\\wedge ,<^{\\prime }\\right\\rbrace $ which says that the $\\left\\lbrace <,\\wedge \\right\\rbrace $ -part is a meet tree, and $<^{\\prime }$ is a linear order.",
"The class of finite structures of $T_{dt,<,\\forall }$ easily has HP, JEP and HP, thus this model completion exists (see Fact REF ).",
"Call its Fraïssé limit the generically linearly ordered tree.",
"It turns out that this structure is not Ramsey, see Claim REF .",
"In any case, we can add a linear order to the tree structure and make it Ramsey.",
"Example 2.16 ([33], and see there for more references) Let $L=\\left\\lbrace <,<_{\\operatorname{lex}},\\wedge \\right\\rbrace $ and let $M$ be the $L$ -structure whose universe is the tree $\\omega ^{<\\omega }$ with the natural interpretations of $<$ as the tree order, $\\wedge $ as the meet function ($s\\wedge t=s\\upharpoonright \\operatorname{len}\\left(s\\wedge t\\right)$ where $\\operatorname{len}\\left(s\\wedge t\\right)=\\max \\left\\lbrace k\\,|\\,s\\upharpoonright k=t\\upharpoonright k\\right\\rbrace $ ), $<_{\\operatorname{lex}}$ as the lexicographical order ($s<_{\\operatorname{lex}}t$ iff $s<t$ or $s\\left(\\operatorname{len}\\left(s\\wedge t\\right)\\right)<t\\left(\\operatorname{len}\\left(s\\wedge t\\right)\\right)$ ).",
"Then $M$ is a Ramsey structure.",
"Fact 2.17 [28] A Ramsey class that has HP and JEP has AP (see Section REF for the definitions).",
"It is easy to see that $K=\\operatorname{Age}\\left(\\omega ^{<\\omega }\\right)$ in the language $\\left\\lbrace <,<_{\\operatorname{lex}},\\wedge \\right\\rbrace $ has JEP, and thus we conclude that it has AP.",
"Let $M$ be its Fraïssé limit and let $T_{dt,<_{lex}}=Th\\left(M\\right)$ .",
"Since $K$ is uniformly locally finite it follows that $M$ is $\\omega $ -categorical and $T_{dt,<_{lex}}$ has quantifier elimination (see Fact REF ).",
"By Proposition REF we have that the restriction of $M$ to $\\left\\lbrace <,\\wedge \\right\\rbrace $ is a dense tree, i.e., a model of $T_{dt}$ (see Example REF ).",
"Here, $L_{1}=\\left\\lbrace <,\\wedge \\right\\rbrace $ , $L_{2}=\\left\\lbrace <,<_{\\operatorname{lex}},\\wedge \\right\\rbrace $ , $K_{1}$ the class of finite meet trees and $K_{2}=\\operatorname{Age}\\left(\\omega ^{<\\omega }\\right)$ .",
"Similarly, the restriction of the generically linearly ordered tree is also a dense tree.",
"Claim 2.18 The generically linearly ordered tree is not a Ramsey structure.",
"Let $K=\\operatorname{Age}\\left(M\\right)$ where $M$ is the countable generically linearly ordered tree.",
"As $N=M\\upharpoonright \\left\\lbrace <,\\wedge \\right\\rbrace \\models T_{dt}$ , by $\\omega $ -categoricity, there is some linear order $<_{lex}$ such that $\\left(N,<_{lex}\\right)\\models T_{dt,<_{lex}}$ .",
"Fix some $A\\in K$ whose universe contains 3 elements $a,b,a\\wedge b$ such that $a\\wedge b<a,b$ and $a<^{\\prime }b<^{\\prime }a\\wedge b$ .",
"Define a coloring $f:{M \\atopwithdelims ()A}\\rightarrow 2$ by $f\\left(A^{\\prime }\\right)=0$ iff [$a<^{\\prime }b$ iff $a<_{lex}b$ (in $A^{\\prime }$ )].",
"If $M$ were Ramsey, there would be some homogeneous $B^{\\prime }\\in {M \\atopwithdelims ()B}$ where $B\\in K$ is such that $B$ contains 5 elements $a,b,a\\wedge b,c,a\\wedge c$ where $a\\wedge c<a,c$ and $a\\wedge c<a\\wedge b<a,b$ and $a<^{\\prime }c<^{\\prime }b<^{\\prime }a\\wedge b<^{\\prime }b\\wedge c$ .",
"It follows that for any copy of $B$ in $M$ , $a<_{lex}c$ iff $b<_{lex}c$ .",
"However in $B^{\\prime }$ we have that $a<_{lex}c$ iff $c<_{lex}b$ — contradiction.",
"We end this discussion with the following fact.",
"Fact 2.19 [3] If $M$ is an ultrahomogeneous Ramsey structure, and $c\\in M$ , then the structure $M^{\\prime }=\\left(M,c\\right)$ where $c$ is a named constant is still Ramsey (and ultrahomogeneous)."
],
[
"Topological dynamics and extremely amenable groups",
"Let us first recall some basic notions from topological dynamics.",
"Suppose that $G$ is a topological group.",
"A $G$ -flow is a compact Hausdorff space $X$ with a continuous action of $G$ .",
"A subflow of $X$ is a compact subspace $Y\\subseteq X$ that is preserved by the action of $G$ , i.e., $gY=Y$ for all $g\\in G$ .",
"A $G$ -ambit is pair $\\left(X,x_{0}\\right)$ where $X$ is a $G$ -flow and $x_{0}$ has a dense orbit.",
"A universal $G$ -ambit is a $G$ -ambit $\\left(X,x_{0}\\right)$ such that for any ambit $\\left(Y,y_{0}\\right)$ there is a map $f:X\\rightarrow Y$ taking $x_{0}$ to $y_{0}$ that commutes with the action: $gf\\left(x\\right)=f\\left(gx\\right)$ for all $x\\in X$ (it follows that $f$ is onto).",
"A universal $G$ -ambit exists and is unique (see [2]).",
"Finally, $G$ is called extremely amenable if for every $G$ -flow $X$ , there is some fixed point $x\\in X$ (i.e., $gx=x$ for all $g\\in G$ ).",
"Kechris, Pestov, and Todorcevic [20] found a striking link between Ramsey classes and topological dynamics, described in the following theorem.",
"Fact 2.20 [20] Suppose that $M$ is a countable ultrahomogeneous linearly ordered structure in a countable language.",
"Then $\\operatorname{Aut}\\left(M\\right)$ is extremely amenable iff $M$ is a Ramsey structure."
],
[
"Having a cyclically dense conjugacy class",
"Definition 3.1 Suppose that $G$ is a topological group.",
"The group $G$ has finite topological rank if it has a finitely generated dense subgroup.",
"Similarly, $G$ has topological rank $n$ (or topologically $n$ -generated) if there are $\\left\\lbrace f_{i}\\,|\\,i<n\\right\\rbrace \\subseteq G$ which generate a dense subgroup.",
"The group $G$ has a cyclically dense conjugacy class if there are $f_{1},f_{2}\\in G$ such that $\\left\\lbrace f_{1}^{-n}f_{2}f_{1}^{n}\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is dense in $G$ .",
"Remark 3.2 If $f:G_{1}\\rightarrow G_{2}$ is a surjective continuous homomorphism, and $G_{1}$ has a dense conjugacy class, then so does $G_{2}$ .",
"Also, if $H$ is a finite nontrivial group (with the discrete topology), then $H$ cannot have a dense conjugacy class.",
"Therefore, the same is true for nontrivial profinite groups.",
"Hence if $G$ is a topological group with a nontrivial profinite quotient, it does not contain a dense conjugacy class.",
"Let $M$ be countable and $\\omega $ -categorical and $G=\\operatorname{Aut}\\left(M\\right)$ .",
"By Proposition REF , $G/G^{0}$ is profinite.",
"It follows that one constraint against having a dense conjugacy class is having a nontrivial compact quotient.",
"In model theoretic terms, it means that if $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\ne \\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ (equivalently, $G^{0}\\ne G$ ), then $G$ cannot have a dense conjugacy class.",
"Moreover, in general, we can have that $\\left(\\left(\\mathbb {Z}/2\\mathbb {Z}\\right)^{\\omega },+\\right)$ is a quotient of $\\operatorname{Aut}\\left(M\\right)$ , which is locally finite so certainly not topologically finitely generated, in which case $G=\\operatorname{Aut}\\left(M\\right)$ cannot be topologically finitely generated.",
"For example, let $L=\\left\\lbrace E_{n}\\,|\\,n<\\omega \\right\\rbrace $ where each $E_{n}$ is a $2n$ -ary relation.",
"Let $T_{\\forall }$ say that $E_{n}$ is an equivalence relation with two classes, and that $\\left(x_{1},\\ldots ,x_{n}\\right)\\mathrel {E_{n}}\\left(y_{1},\\ldots ,y_{n}\\right)\\rightarrow \\bigwedge _{i\\ne j}x_{i}\\ne x_{j}\\wedge y_{i}\\ne y_{j}$ (there is no relation between different $E_{n}$ 's).",
"The class of finite $T_{\\forall }$ models has AP and JEP (and it is essentially countable), and by Fact REF (which we can use since for any $n<\\omega $ there are finitely many isomorphism types of $n$ -element structures), there is a model completion $T$ which is $\\omega $ -categorical.",
"Let $M$ be the countable model.",
"Then $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ contains $M^{n}/E_{n}$ for all $n<\\omega $ , and for any $\\eta \\in \\mathbb {Z}/2\\mathbb {Z}$ , there is an automorphism $\\sigma \\in G=\\operatorname{Aut}\\left(M\\right)$ such that $\\sigma \\upharpoonright M^{n}/E_{n}$ is the identity iff $\\eta \\left(n\\right)=0$ .",
"In fact one can show that $G/G^{0}=\\left(\\left(\\mathbb {Z}/2\\mathbb {Z}\\right)^{\\omega },+\\right)$ and that $\\sigma $ fixes $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ iff it fixes all $E_{n}$ -classes.",
"This construction is due to Cherlin and Hrushovski (see [14] and [22]).",
"Using a similar technique, in [8] it is shown that any profinite group $H$ which has a countable basis of open subgroups can be realized as $G/G^{0}$ for some automorphism group $G$ of an $\\omega $ -categorical structure $M$ .",
"Definition 3.3 Suppose that $M$ is some structure and $a,b\\in M$ are some tuples.",
"We write $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsb$ to say that $ tp(a/b)$ does notsplit over $$.",
"When $ M$ is homogeneous, this means, letting$ B$ be the set $ b$ enumerates: if $ g:B'B”$ is a partial automorphismof $ B$ (i.e., $ B',B”B$ and $ g$ extends to an automorphismof $ M$) then $ g$ extends to an automorphism of $ M$ which fixes$ a$ pointwise.$ In the next definition, our convention is that for sets $A,B$ , we write $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ if this is true for tuples enumerating $ A,B$.\\begin{defn}An automorphism \\sigma \\in \\operatorname{Aut}\\left(M\\right) is \\emph {repulsive}if for every finite set A\\subseteq M there is some n such thatA\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{defn}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsn(A)$ and $ n(A)$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsA$.Say that $$ is \\emph {strongly repulsive} if this is true forall $ mn$ as well.$ Suppose that $M$ is some structure.",
"For $k<\\omega $ , add predicates $P_{1},\\ldots ,P_{k}$ to the language, and let $\\bigsqcup _{k}M$ be the disjoint union of $k$ copies of $M$ , one for each predicate, where each copy has the same structure as $M$ .",
"Then $\\operatorname{Aut}\\left(\\bigsqcup _{k}M\\right)=\\operatorname{Aut}\\left(M\\right)^{k}$ .",
"Proposition 3.4 If $\\sigma $ is a (strongly) repulsive automorphism of an $L$ -structure $M$ , then $\\sigma ^{\\times k}\\in \\operatorname{Aut}\\left(M\\right)^{k}$ is a (strongly) repulsive automorphism of the structure $\\bigsqcup _{k}M$ for all $k<\\omega $ .",
"Suppose that $A\\subseteq \\bigsqcup _{k}M$ is finite.",
"Then we may assume, enlarging $A$ , that $A=\\bigsqcup _{k}A_{0}$ for some finite $A_{0}\\subseteq M$ (i.e., the disjoint union of the same set in the different predicates).",
"Thus the proposition follows from the fact that if $A_{0}\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsB0$in $ M$, then $ kA0$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nskB0$ in$ kM$, which is clear.$ A repulsive automorphism is a special case of a topologically transitive map: Definition 3.5 Suppose that $X$ is a topological space.",
"A map $f:X\\rightarrow X$ is called topologically transitive if for every two nonempty open sets $U,V\\subseteq X$ , there is some $n<\\omega $ such that $f^{n}\\left(U\\right)\\cap V\\ne \\emptyset $ .",
"Lemma 3.6 Suppose that $M$ is a countable structure and that $\\sigma \\in G=\\operatorname{Aut}\\left(M\\right)$ is repulsive.",
"Then conjugation by $\\sigma $ in $G$ is topologically transitive.",
"Denote by $f:G\\rightarrow G$ the conjugation by $\\sigma $ .",
"Suppose that $U,V$ are two nonempty basic open subsets of $G$ , i.e., $U=U_{a,b}=\\left\\lbrace \\tau \\in G\\,|\\,\\tau \\left(a\\right)=b\\right\\rbrace $ (where $a,b$ are finite tuples) and $V=U_{c,d}$ .",
"Note that $f\\left(U_{a,b}\\right)=U_{\\sigma \\left(a\\right),\\sigma \\left(b\\right)}$ .",
"So we need to find some $n<\\omega $ such that $U_{\\sigma ^{n}\\left(a\\right),\\sigma ^{n}\\left(b\\right)}\\cap U_{c,d}$ is nonempty.",
"This means that we need to show that for some $n<\\omega $ , $\\sigma ^{n}\\left(a\\right)c\\equiv \\sigma ^{n}\\left(b\\right)d$ .",
"As $\\sigma $ is repulsive, there is some $n<\\omega $ such that $cd\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsn(ab)$and $ n(ab)$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nscd$.",
"Since both $ U,V$,$ ab$ and $ cd$, and thus $ n(a)cn(b)cn(b)d$.$ Fact 3.7 [34] If $X$ is second countable (i.e., have a countable basis) of second category (i.e., not meager) and separable, and $f:X\\rightarrow X$ is topologically transitive, then for some $x\\in X$ , $\\left\\lbrace f^{n}\\left(x\\right)\\,|\\,n<\\omega \\right\\rbrace $ is dense.",
"In fact, it follows from the proof there that the set of such $x$ 's is comeager.",
"Since it is not written explicitly in [34], we provide a proof of the last statement (based on the proof from there).",
"Consider the set $F$ of $x\\in X$ such that $\\left\\lbrace f^{n}\\left(x\\right)\\,|\\,n<\\omega \\right\\rbrace $ is not dense.",
"Fix some countable basis $\\mathcal {V}$ of open sets.",
"For each $x\\in F$ , there is some $U_{x}\\in \\mathcal {V}$ such that $f^{n}\\left(x\\right)\\notin U_{x}$ for all $n$ .",
"Now, $\\bigcup \\left\\lbrace f^{-n}\\left(U_{x}\\right)\\,|\\,n<\\omega \\right\\rbrace $ is open and dense since $f$ is topologically transitive.",
"Hence, its complement $A_{U_{x}}$ is closed, nowhere dense and contains $x$ .",
"The union $\\bigcup \\left\\lbrace A_{U_{x}}\\,|\\,x\\in F\\right\\rbrace $ is a countable union which contains $F$ , hence $F$ is meager.",
"In our case, $G=\\operatorname{Aut}\\left(M\\right)$ for a countable model $M$ is of second category, since it is even Polish.",
"Hence we immediately get the following corollary.",
"Corollary 3.8 Suppose that $M$ is a countable structure.",
"Suppose that $G=\\operatorname{Aut}\\left(M\\right)$ contains a repulsive automorphism $\\sigma $ .",
"Then $G$ is topologically 2-generated and moreover has a cyclically dense conjugacy class, and even: the set of $\\tau $ for which $\\left\\lbrace \\sigma ^{n}\\tau \\sigma ^{-n}\\,|\\,n\\in \\mathbb {N}\\right\\rbrace $ is dense is comeager.",
"An alternative, more direct proof is as follows.",
"Assume that $\\sigma $ is a repulsive automorphism, and construct $\\tau \\in \\operatorname{Aut}\\left(M\\right)$ by back-and-forth, so that $\\left\\lbrace \\sigma ^{-n}\\tau \\sigma ^{n}\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is dense in $G$ .",
"We leave the details as an exercise.",
"We also point out that according to [19], the fact that this set of $\\tau $ is comeager actually follows immediately from the fact that there is one such $\\tau $ .",
"Now we turn to the question of finding a repulsive automorphism.",
"Definition 3.9 Suppose that $M$ is a countable structure.",
"A ternary relation $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ on finite subsets of $ M$, invariant under $ Aut(M)$is a \\emph {canonical} independence relation (CIR) if it satisfiesthe following properties for all finite sets $ A,B,C,D$:\\begin{itemize}\\item (Stationarity over \\emptyset ) If A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{itemize}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$, $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B'$, $ a,b,a',b'$are tuples enumerating $ A,B,A',B'$, and $ aa'$, $ bb'$then $ aba'b'$.",
"Note that this (together with monotonicity,see below) implies non-splitting: if $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ then $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsB$and $ B$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsA$ (where $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ means $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$).\\item (Extension (on the right)) If $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ then for all finite tuples$ d$ there is some $ d'BCd$ such that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CBd'$.\\item (Transitivity on both sides) If $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DCB$ then if $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$then $ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and if $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CD$ then $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CBD$ ($ DC$means $ DC$).\\item (Monotonicity) If $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and $ A'A$, $ B'B$then $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB'$.\\item (Existence) $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CC$ and $ C$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CA$.$ For finite tuples $a,b,c$ enumerating sets $A,B,C$ respectively, write $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $cb$ for $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.$ Note that we do not ask for symmetry nor for base monotonicity (if $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CBD$ then $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CDB$).$ We say that a CIR is defined on finitely generated substructures if for all finite $A,B,C,A^{\\prime },B^{\\prime },C^{\\prime }\\subseteq M$ , if $\\left\\langle A\\right\\rangle =\\left\\langle A^{\\prime }\\right\\rangle $ , $\\left\\langle B\\right\\rangle =\\left\\langle B^{\\prime }\\right\\rangle $ and $\\left\\langle C\\right\\rangle =\\left\\langle C^{\\prime }\\right\\rangle $ then $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C'B'$.",
"($ A$is the substructure generated by $ A$).$ Remark 3.10 If a CIR is defined on finitely generated substructures, then it naturally induces a relation $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*$whose domain is finitely generated substructures by setting $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C*B$iff $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"The relation $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*$ satisfies the naturalvariants of Definition \\ref {def:groovy}.",
"For example, transitivityto the left becomes: for all finitely generated substructures $ A,B,C,D$,if $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DC*B$ and $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C*B$then $ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C*B$.",
"Similarly, extensionbecomes: if $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C*B$ then for all finite tuples $ d$ thereis some $ d'BCd$ such that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C*Bd'$.$ On the other hand, if we have a relation $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*$ satisfying thesenatural properties on finitely generated substructures of $ M$, thenthere is also a CIR defined on finitely generated substructures: define$ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C*B$.$ Theorem 3.11 Assume that $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR defined on finitely generated substructuresof an ultrahomogeneous structure $ M$.",
"Then there is a strongly repulsiveautomorphism in $ Aut(M)$.$ Let $S$ be the set of all closed nonempty intervals of integers, i.e., sets of the form $\\left[i,j\\right]$ for $i\\le j$ from $\\mathbb {Z}$ .",
"For every finite set $s\\in S$ we attach a countable tuple of variables $\\bar{x}_{s}=\\left\\langle x_{s,i}\\,|\\,i<\\omega \\right\\rangle $ in such a way that if $t\\ne s\\in S$ then $\\bar{x}_{s}\\cap \\bar{x}_{t}=\\emptyset $ .",
"For $s\\in S$ , let $\\bar{y}_{s}=\\bigcup \\left\\lbrace \\bar{x}_{t}\\,|\\,t\\in S,t\\subseteq s\\right\\rbrace $ , and $\\bar{y}=\\bigcup \\left\\lbrace \\bar{x}_{s}\\,|\\,s\\in S\\right\\rbrace =\\bigcup \\left\\lbrace \\bar{y}_{s}\\,|\\,s\\in S\\right\\rbrace $ .",
"For a $\\bar{y}$ -tuple ($\\bar{y}_{s}$ -tuple) $\\bar{a}$ and $t\\in S$ (contained in $s$ ), we write $\\bar{a}\\upharpoonright t$ for $\\bar{a}\\upharpoonright \\bar{y}_{t}$ and similarly for a type in $\\bar{y}$ ($\\bar{y}_{s}$ ).",
"By Fact REF , the age of $M$ , denoted by $K$ , has HP, JEP and AP.",
"Fix an enumeration $\\left\\langle \\left(A_{l},B_{l}\\right)\\,|\\,l<\\omega \\right\\rangle $ of all pairs $A,B\\in K$ such that $A\\subseteq B\\subseteq M$ , including the case $A=\\emptyset $ .",
"For every $1\\le n<\\omega $ , we construct a complete quantifier free type $r_{n}\\left(\\bar{y}_{\\left[0,n-1\\right]}\\right)\\in S^{\\operatorname{qf}}\\left(\\emptyset \\right)$ and a sequence $\\left\\langle f_{n,l,i}\\,|\\,l,i<\\omega \\right\\rangle $ such that: If $\\bar{a}\\models r_{n}$ then $\\bar{a}$ is enumerates a finitely generated substructure $A\\in K$ and for a fixed $l<\\omega $ , $\\left\\langle f_{n,l,i}\\,|\\,i<\\omega \\right\\rangle $ enumerates a countable set of functions that contains all embeddings of $A_{l}$ into $A$ .",
"(Formally, $f_{n,l,i}$ is a function from $A_{l}$ into the variables $\\bar{y}_{\\left[0,n-1\\right]}$ which the type $r_{n}$ “thinks” is an embedding.)",
"If $0<m<n$ then for every interval $s\\in S$ with $s\\subseteq n$ such that $\\left|s\\right|=m$ , $r_{n}\\upharpoonright s=r_{m}\\left(\\bar{y}_{s}\\right)$ .",
"If $\\bar{a}\\models r_{n}$ then for every $0<m<n$ , every pair from $\\left\\lbrace \\left(A_{l},B_{l}\\right)\\,|\\,l<n-1\\right\\rbrace $ and every $f\\in \\left\\lbrace f_{m,l,i}\\,|\\,l,i<n-1\\right\\rbrace $ (so $f$ is an embedding of $A_{l}$ into the structure enumerated by $\\bar{a}\\upharpoonright \\left[0,m-1\\right]$ ), there is an embedding $g$ of $B_{l}$ into the structure enumerated by $\\bar{a}$ such that $g$ extends $f$ .",
"If $s,t$ are two intervals contained in $n$ such that $\\min s\\le \\min t$ and $\\bar{a}\\models r_{n}$ then $\\bar{a}\\upharpoonright s\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $ast*at$(see Remark \\ref {rem:CIR on f.g. substructures}).$ How?",
"For $n=1$ , let $r_{1}\\left(\\bar{y}_{\\left\\lbrace 0\\right\\rbrace }\\right)$ be a complete quantifier free type of a tuple enumerating some $D_{0}\\in K$ .",
"Note that it trivially satisfies all the assumptions.",
"Suppose we found $r_{n}$ satisfying all the properties and we construct $r_{n+1}$ .",
"Let $\\bar{a}\\models r_{n}$ from $M$ .",
"Let $f:\\bar{a}\\upharpoonright \\left[0,n-2\\right]\\rightarrow \\bar{a}\\upharpoonright \\left[1,n-1\\right]$ be defined by setting $f\\left(a_{s,i}\\right)=a_{s+1,i}$ for all $s\\in S$ contained in $\\left[0,n-2\\right]$ (in general, $s+n$ is the translation of $s$ by $n$ ).",
"It is an isomorphism since both $\\bar{a}\\upharpoonright \\left[0,n-2\\right]$ and $\\bar{a}\\upharpoonright \\left[1,n-1\\right]$ realize $r_{n-1}$ .",
"By the homogeneity of $M$ , we can extend $\\bar{a}\\upharpoonright \\left[1,n-1\\right]$ to some tuple $\\bar{a}^{\\prime }$ enumerated by $\\bar{y}_{\\left[1,n\\right]}$ , and extend $f$ to an isomorphism $f^{\\prime }:\\bar{a}\\rightarrow \\bar{a}^{\\prime }$ such that $f^{\\prime }\\left(a_{s,i}\\right)=a_{s+1,i}$ for all $s\\in S$ contained in $\\left[0,n-1\\right]$ .",
"(When $n=1$ , $f=\\emptyset $ and $\\bar{a}^{\\prime }$ is just an isomorphic copy of $\\bar{a}$ .)",
"By existence, we have that $\\bar{a}\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a[1,n-1]*a[1,n-1]$.By extension, we may assume that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a[1,n-1]*a'$.Let $ a”$ be a $ yn+1$-tuple containing $ aa'$that contains witnesses to all the relevant pairs and embeddings from(\\ref {enu:AP}), ordered in such a way that $ a$ enumeratesthe $ yn$-part and $ a'$ enumerates the $ y[1,n]$-part(so the remaining parts of $ a”$ are enumerated by $ x[0,n]$).This can be done since $ K$ has AP and JEP.",
"Let $ rn+1=tp(a”)$.$ Now we have to check that (1)–(4) hold.",
"We prove this by induction on $n$ .",
"(REF ) and (REF ) are clear by construction and the induction hypothesis.",
"(REF ) follows by the choice of $f$ and $f^{\\prime }$ .",
"Let us prove (REF ), so fix $\\bar{a}\\models r_{n+1}$ .",
"By monotonicity it is enough to prove that $\\bar{a}\\upharpoonright \\left[0,k-1\\right]\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a[m,k-1]*a[m,n]$for any $ mn,1kn+1$.",
"We may assume that $ 1m$and $ kn$ (otherwise this is true by existence).$ Note that $\\bar{a}\\upharpoonright \\left[0,k-1\\right]\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a[1,n-1]*a[m,n]$by construction (and monotonicity).",
"By induction, $ a[1,n-1]$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a[m,n-1]*a[m,n]$.Hence by transitivity and monotonicity we have that $ a[0,k-1]$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a[m,n-1]*a[m,n]$.By induction we have that $ a[0,k-1]$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a[m,k-1]*a[m,n-1]$.By applying transitivity (and monotonicity) again, we get that $ a[0,k-1]$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a[m,k-1]*a[m,n]$.This finishes the proof of (4).$ By compactness, we can find a $\\bar{y}$ -tuple $\\bar{b}$ , such that for every $s\\in S$ , $\\bar{b}\\upharpoonright s\\models r_{\\left|s\\right|}$ .",
"Note that $\\bar{b}$ enumerates a Fraïssé limit $N$ by (REF ).",
"This is as in the proof of [15].",
"More precisely, by JEP, the age of $N$ is $K$ (given $A\\in K$ , by JEP there is some $B$ such that $\\left(D_{0},B\\right)$ is one of $\\left(A_{l},B_{l}\\right)$ and $A\\subseteq B$ , so this is taken care of in the construction).",
"In addition, by [15], $N$ is ultrahomogeneous.",
"Hence by uniqueness of the Fraïssé limit (see Fact REF ), we may assume that $\\bar{b}$ enumerates $M$ .",
"Let $\\sigma :M\\rightarrow M$ be defined by $\\sigma \\left(b_{s,i}\\right)=b_{s+1,i}$ .",
"By construction, $\\sigma $ is an automorphism.",
"Now, any finite subset of $M$ is contained in $\\bar{b}\\upharpoonright s$ for some $s\\in S$ .",
"Then for some $m<\\omega $ , $s+n\\cap s=\\emptyset $ for all $n\\ge m$ , so $\\sigma ^{n}\\left(\\bar{b}\\upharpoonright s\\right)=\\bar{b}\\upharpoonright \\left(s+n\\right)$ satisfies that $\\sigma ^{n}\\left(\\bar{b}\\upharpoonright s\\right)\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*bs$and so as $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*$ implies non-splitting, $$ is indeed stronglyrepulsive.$ Corollary 3.12 Suppose that $M$ is a countable $\\omega $ -categorical $L$ -structure, and that $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$is a CIR on $ M$.",
"Then there is a strongly repulsive automorphism$ Aut(M)$.$ For all $n<\\omega $ and $a\\in M^{n}$ , let $R_{a}\\subseteq M^{n}$ be the orbit of $a$ under $G=\\operatorname{Aut}\\left(M\\right)$ .",
"Let $L^{\\prime }=\\left\\lbrace R_{a}\\,|\\,a\\in M^{n},n<\\omega \\right\\rbrace $ and let $M^{\\prime }$ be the $L^{\\prime }$ -structure induced by $M$ .",
"Then $M^{\\prime }$ has the same definable sets as $M$ , has quantifier elimination and is ultrahomogeneous.",
"Now, $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is still a CIR on finite subsets of$ M'$.",
"Moreover, substructures of $ M'$ are subsets since $ L'$ isrelational, so $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is defined on finitely generated substructures.Thus we may apply Theorem \\ref {thm:existence of repulsive automorphism-ultrahomogeneous}to get a strongly repulsive automorphism $$ of $ M'$.",
"However,$ Aut(M)=Aut(M')$ and $$ is stronglyrepulsive as an automorphism of $ M'$ as well.$ From Corollary REF and Corollary REF we get: Corollary 3.13 If $M$ is a countable model of an $\\omega $ -categorical theory which has a canonical independence relation then $G=\\operatorname{Aut}\\left(M\\right)$ has a cyclically dense conjugacy class.",
"In fact, there is some $f\\in G$ such that the set of $g\\in G$ for which [$\\left\\lbrace f^{n}gf^{-n}\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is dense] is comeager.",
"Furthermore, by Proposition REF , the same is immediately true for $G^{n}$ for any $n<\\omega $ .",
"Ramsey structures and a weakening of having a CIR Lemma 3.14 Suppose that $M$ is a countable ultrahomogeneous structure.",
"Then (1) implies (2) implies (3) where: $M$ has a CIR defined on finitely generated substructures.",
"There are two models $M_{0},M_{1}$ isomorphic to $M$ and contained in $M$ , such that $M_{0}\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM1$ and $ M1$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM0$.\\item There is a binary relation $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ on finite subsets of $ M$ thatsatisfies all the properties of Definition \\ref {def:groovy} but onlyover $$.",
"Namely it satisfies stationarity, extension tothe right and the left (over $$), monotonicity (over $$)and existence (over $$).$ (1) implies (2).",
"Suppose that $M$ has a CIR $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ defined on finitelygenerated substructures.",
"Consider the tuple $ b$ constructedin the proof of Theorem \\ref {thm:existence of repulsive automorphism-ultrahomogeneous}.By condition (\\ref {enu:AP}) in that proof, both $ b[0,)$and $ b(-,0]$ are ultrahomogeneousand with the same age as $ M$, thus isomorphic to $ M$ by Fact \\ref {fact:Fraisse limits}.By stationarity and monotonicity, $ b[0,)$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsb(-,0]$and $ b(-,0]$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsb[0,)$.$ (2) implies (3).",
"For two finite sets $A,B\\subseteq M$ , let $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$iff there is some automorphism $ Aut(M)$ suchthat $ (A)M0$ and $ (B)M1$.Now, $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is stationary: suppose that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$, $ a'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$,$ ab$ and $ a'b'$.",
"We want to show that $ aba'b'$.Let $ ,Aut(M)$ be such that $ (a),(a')M0$and $ (b),(b')M1$.",
"Then$ (a)(a')$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $ns(b)(b')$and $ (b)(b')$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $ns(a)(a')$,and hence $ ab(ab)(a)(b')(a'b')a'b'$.$ Next, $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ satisfies extension (on the right): suppose that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$and $ d$ is a finite tuple of $ M$, and we want to find some $ d'Bd$such that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $Bd'$.",
"By definition, there is some $$with $ (A)M0$ and $ (B)M1$.As $ M1$ is ultrahomogeneous and $ B$ is finite, we can extend$ B$ to some $ f:BdM1$.As $ M$ is ultrahomogeneous, $ d'=-1(f(d))Bd$and $$ witnesses that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $Bd'$.",
"Extension on theleft is shown in the same way.$ Existence follows from the fact that $\\operatorname{Age}\\left(M_{1}\\right)=\\operatorname{Age}\\left(M_{0}\\right)=\\operatorname{Age}\\left(M\\right)$ .",
"Monotonicity is clear.",
"Remark 3.15 If in Lemma REF , if we had assumed that $M$ was $\\omega $ -categorical, in (3) we could define $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$on arbitrary subsets of $ M$, even infinite.",
"We would define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$iff for every finite subsets $ A'A$ and $ B'B$,$ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B'$.",
"One can then use König^{\\prime }s Lemma and Ryll-Nardzewskito show that this has the extension property.$ In addition, (3) would be equivalent to (2): enumerate $M$ as $\\bar{m}=\\left\\langle m\\,|\\,m\\in M\\right\\rangle $ (i.e., the identity function).",
"Let $\\bar{x}=\\left\\langle x_{m}\\,|\\,m\\in M\\right\\rangle $ and $\\bar{y}=\\left\\langle y_{m}\\,|\\,m\\in M\\right\\rangle $ be two disjoint sequences of variables.",
"For a finite tuple $a=\\left\\langle m_{i}\\,|\\,i<n\\right\\rangle $ in $M$ , write $x_{a}=\\left\\langle x_{m_{i}}\\,|\\,i<n\\right\\rangle $ , and similarly define $y_{a}$ .",
"Let $\\Gamma \\left(\\bar{x},\\bar{y}\\right)$ be the union of the sets $\\Gamma _{a,b,c}^{0}\\left(x_{a},x_{b},y_{c}\\right)$ and $\\Gamma _{a,b,c}^{1}\\left(x_{c},y_{a},y_{b}\\right)$ for all finite tuples $a,b,c$ from $M$ such that $a\\equiv b$ , where $\\Gamma _{a,b,c}^{0}\\left(x_{a},x_{b},y_{c}\\right)$ says that $x_{a}$ and $x_{b}$ have the same type over $y_{c}$ and similarly, $\\Gamma _{a,b,c}^{1}\\left(x_{c},y_{a},y_{b}\\right)$ says that $y_{a}$ and $y_{b}$ have the same type over $x_{c}$ .",
"Let $\\Sigma \\left(\\bar{x},\\bar{y}\\right)$ be $\\Gamma \\left(\\bar{x},\\bar{y}\\right)$ and the assertions that both $\\bar{x}$ and $\\bar{y}$ satisfy the type $\\operatorname{tp}\\left(\\bar{m}/\\emptyset \\right)$ .",
"Then by (3), $\\Sigma $ is consistent, so by $\\omega $ -categoricity, we can realize it in $M$ .",
"Proposition 3.16 Suppose that $M$ is a countable $\\omega $ -categorical Ramsey structure.",
"Then (2) from Lemma REF holds.",
"Let $G=\\operatorname{Aut}\\left(M\\right)$ and let $\\bar{m}=\\left\\langle m\\,|\\,m\\in M\\right\\rangle $ (i.e., the identity function).",
"Let $\\bar{x}=\\left\\langle x_{m}\\,|\\,m\\in M\\right\\rangle $ and $S_{\\bar{m}}\\left(M\\right)=\\left\\lbrace p\\left(\\bar{x}\\right)\\in S\\left(M\\right)\\,|\\,p\\upharpoonright \\emptyset =\\operatorname{tp}\\left(\\bar{m}/\\emptyset \\right)\\right\\rbrace $ , a compact Hausdorff space with the logic topology.",
"Then $G$ acts on $S_{\\bar{m}}\\left(M\\right)$ by setting $\\sigma *p=\\left\\lbrace \\sigma *\\varphi \\,|\\,\\varphi \\in p\\right\\rbrace $ where $\\sigma *\\varphi \\left(\\bar{x},m\\right)=\\varphi \\left(\\bar{x},\\sigma \\left(m\\right)\\right)$ .",
"A fixed point of this action is just an invariant type over $M$ .",
"As $G$ is extremely amenable (Fact REF ), there is an invariant type which enumerates a model $N$ such that that $N\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$.",
"Now consider the space $ P$ of invariant types:$ P={ qSm(M) | G*q=q} $.",
"A type $ q$ is invariantiff for every formula $ (x,y)$ over $$,if $ mm'$ are from $ M$ then $ (x,m)q$iff $ (x,m')q$.",
"This is easily a closedcondition, so $ P$ is compact, and we already know that it is nonempty.Now let $ G$ act on $ P$ by setting $ q={ | q} $where $ (x,m)=(x,m)$,where $ x=x(m) | mM$.Note that for all $ pP$ and $ G$, $ p=tp(-1(m) | mM/)=tp(m/)$,and that $ p$ remains invariant.",
"By extreme amenability,there is some $ qP$ such that $ q=q$ for all $ G$.Let $ N'$ be a model enumerated by a realization of $ q$, then $ N'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$and $ M$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsN'$.$ By $\\omega $ -categoricity, we may assume that these two models are contained in $M$ .",
"Remark 3.17 By Fact REF , expanding an ultrahomogeneous Ramsey structure by finitely many constants gives an independence relation as in (3) from Lemma REF over any finite set.",
"However there is no reason that it would satisfy transitivity.",
"Indeed, the lexicographically ordered dense tree (which is Ramsey, see Example REF ) does not have a CIR.",
"See Corollary REF below.",
"Examples of theories with a canonical independence relation There are many examples of countable ultrahomogeneous structures with a CIR.",
"Here we will give some of them.",
"We will define the relation $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$, but sometimes leave most of the details of checking thatit satisfies the axioms to the reader.",
"All the CIRs we define aredefined on finitely generated substructures.\\begin{example}The most trivial ultrahomogeneous structureis of course the structure with universe \\omega and no relationsbut equality.",
"Its automorphism group is S_{\\infty }.",
"For finitesets A,B,C define A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{example}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ by $ ABC$.",
"Thisis a CIR.$ Example 4.1 If $T$ is stable and $\\emptyset $ is a base (i.e., $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ so that every type over $\\emptyset $ has a unique non-forking extension), then $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $f$ (i.e., non-forking independence) is canonical.$ Example 4.2 Let $\\left(\\mathcal {B},<,\\wedge ,\\vee ,0,1,\\square ^{c}\\right)$ be the atomless Boolean algebra.",
"For finite sets $A,B,C$ , define $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff$ C=ACBC$and for every atom $ aAC$ and everyatom $ bBC$, if there is an atom $ cC$such that $ a,bc$ then $ ab0$.",
"Let us show that $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$is a (symmetric) CIR.$ Stationarity over $\\emptyset $ : $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ says that the atoms in$ AB$ are in bijection with (atoms of $ A$)$$ (atoms of $ B$).",
"Thus, if $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B'$ and $ f:AA'$,$ g:BB'$are isomorphisms, then $ h:ABA'B'$taking an atom $ ab$ to $ f(a)g(b)$is an isomorphism.",
"This easily implies stationarity.$ Transitivity: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CDB$ and $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, and wewant to show that $ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"Suppose that $ aACD$,$ bBC$ are atoms, and $ cC$is an atom such that $ a,bc$.",
"Let $ dDC$be an atom such that $ adc$.",
"Then $ db0$.",
"Let$ b'db$ be an atom of $ BCD$,so that $ a,b'd$.",
"Hence $ ab'0$ and thus $ ab0$.Transitivity to the right follows by symmetry.$ Extension: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and $ d$ is given.",
"We want tofind $ d'BCd$ such that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CBd$.",
"We may assume that$ dBC$.",
"An atom in $ BCd$has the form $ db$ or $ dcb$ for some atom $ b$ of$ BC$.",
"The type $ tp(d/BC)$is determined by knowing which of these terms $ db,dcb$is nonzero (for $ bBC$ an atom).",
"As$ B$ is atomless and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, we can find some $ d'$ suchthat for every $ a,b,c$ atoms in $ AC,BC,C$respectively such that $ a,bc$, if $ db0$ then $ ad'b0$(else $ d'b=0$) and if $ dcb0$ then $ a(d')cb0$(else $ (d')cb=0$).",
"In addition, we ask thatif $ db,dcb0$ then both $ ad'b$and $ a(d')cb$ are not in $ ABC$.It now follows that $ ACBCd'=C$:suppose that $ eACBCd'$.Then as $ eBCd'$, it can be writtenas $ b0(b1d')(b2(d')c)$where $ b0,b1,b2BC$ are pairwisedisjoint and for every atom $ b'b1b2$ from $ BC$,$ d'b',(d')cb'0$.",
"If both $ b1,b2=0$,then $ eBCAC$so $ eC$.",
"If $ b10$, let $ b1'b1$be an atom of $ BC$.",
"So $ eb1'ABC$(because $ eAC$) and has the form $ b1'd'$.Let $ aAC$ and $ cC$be atoms such that $ a,b1'c$.",
"Then $ eb1'aABC$and has the form $ ab1'd'$ which is not in $ ABC$by construction, contradiction.",
"Similarly $ b2=0$ and we are done.$ Existence and monotonicity are clear.",
"Example 4.3 Let $\\left(M,R\\right)$ be the random tournament (a tournament is a complete directed graph such that for all $x,y$ , it cannot be that both $R\\left(x,y\\right)$ and $R\\left(y,x\\right)$ , and the random tournament is the Fraïssé limit of the class of finite tournaments).",
"Given finite sets $A,B,C$ , write $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ AB=C$ andif $ aAC$, $ bBC$ then $ R(a,b)$.This easily satisfies all the requirements.$ Remark 4.4 The following definition is from [19].",
"Let $K$ be a class of finite $L$ -structures where $L$ is a relational language.",
"Then $K$ has the strong$^{+}$ amalgamation property if for all $A,B,C\\in K$ with $A\\subseteq B,C$ there is $D\\in K$ such that $D=C^{\\prime }\\cup B^{\\prime }$ with $C^{\\prime }\\cong _{A}C$ , $B^{\\prime }\\cong _{A}B$ and for every $n$ -ary relation $R$ and every $x_{1},\\ldots ,x_{n}$ and $y_{1},\\ldots ,y_{n}$ from $D\\backslash A$ which intersect both $C^{\\prime }$ and $B^{\\prime }$ , if [$x_{i}\\in B^{\\prime }$ iff $y_{i}\\in B^{\\prime }$ for all $1\\le i\\le n$ ], then $R^{D}\\left(x_{1},\\ldots ,x_{n}\\right)$ iff $R^{D}\\left(y_{1},\\ldots ,y_{n}\\right)$ (where $R^{D}$ is the interpretation of $R$ in $D$ ).",
"For example, the random tournament satisfies this property.",
"In [19] it is proved that if $M$ is ultrahomogeneous and $\\operatorname{Age}\\left(M\\right)$ has the strong$^{+}$ amalgamation property, then $\\operatorname{Aut}\\left(M\\right)$ has a cyclically dense conjugacy class.",
"They prove it using a condition they denote by $(\\Delta _{n})$ , see there, Theorem 5.12.",
"We do not know if this condition implies the existence of a CIR.",
"Free amalgamation classes In [38], [27], [4] there is an axiomatic framework for defining an abstract ternary relation close to our CIR.",
"More precisely, in [27] and [38], the notion of a stationary independence relation (SIR) is introduced (in [27] for finitely generated structures and in [38] for sets in general).",
"A similar notion is defined in [4], with more axioms.",
"In any case, all these notions imply ours, except perhaps that stationarity over $\\emptyset $ becomes stationarity over $\\operatorname{acl}\\left(\\emptyset \\right)$ , so that this becomes a CIR in the expansion $\\left(M,\\operatorname{acl}\\left(\\emptyset \\right)\\right)$ (note that our extension follows from full stationarity and their version of existence).",
"Thus, we can apply our results to the examples studied there.",
"In particular, we get the following examples.",
"Example 4.5 The rational Urysohn space $\\mathbb {Q}\\mathbb {U}$ is the Fraïssé limit of the class of finite metric spaces with rational distances.",
"Pick a point $q\\in \\mathbb {Q}\\mathbb {U}$ , and consider the structure $\\left(\\mathbb {Q}\\mathbb {U},q\\right)$ where we add a constant for $q$ .",
"In [37], it is proved that the relation $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ which holds for finite $ A,B,C$ ifffor every $ aAC$, $ bBC$, $ d(a,b)={ d(a,c)+d(c,b) | cC} $is a CIR in $ (QU,q)$.$ In all examples given by Conant [4] which we list now, $\\operatorname{acl}\\left(\\emptyset \\right)=\\emptyset $ , so we actually get a CIR in the structure (i.e., no need to take an expansion) by [4].",
"Example 4.6 Fraïssé limits with free amalgamation: suppose that $L$ is a relational language and $K$ is an essentially countable (see above Fact REF ) class of finite $L$ -structures, such that if $A,B,C\\in K$ and $A\\subseteq C,B$ , then the free amalgam of $A,B,C$ is in $K$ (i.e., a structure $D=C^{\\prime }\\cup B^{\\prime }$ with $C^{\\prime }\\cong _{A}C$ , $B^{\\prime }\\cong _{A}B$ , $B^{\\prime }\\cap C^{\\prime }\\subseteq A$ and for every tuple $a\\in D$ in the length of some relation $R\\in L$ , if $R\\left(a\\right)$ then $a\\in C^{\\prime }$ or $a\\in B^{\\prime }$ ) (here we also include the case $A=\\emptyset $ ).",
"Let $M$ be the Fraïssé limit of $K$ , and define $B\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $AC$ iff $ ABC$ is the free amalgam of$ A,AB,AC$.",
"If the language is finite or more generally in the contextof Fact \\ref {fact:Fraisse limits} (2), it is easy to see that inthis case $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR (this is also proved, for finite languages,in \\cite [Proposition 3.4]{MR3663421}).$ This class of examples contain e.g., the random graph, the universal $K_{n}$ -free graph (the Henson graph), and their hypergraph analogs.",
"Example 4.7 Let $L=\\left\\lbrace P_{n}\\,|\\,n<\\omega \\right\\rbrace $ and let $K$ be the class of finite $L$ -structures in which $P_{n}\\left(x_{0},\\ldots ,x_{n-1}\\right)$ implies that $x_{i}\\ne x_{j}$ for $i\\ne j$ .",
"Then $K$ is essentially countable and is a free amalgamation class, and moreover for each $n<\\omega $ there are finitely many isomorphism types of structures of size $n$ (so we can use Fact REF ).",
"Let $M$ be the limit.",
"Now recall the example $N$ described in Remark REF , with infinitely many independent equivalence relations with two classes.",
"Then $M$ is $N$ expanded by naming the classes.",
"In other words, $\\operatorname{Aut}\\left(M\\right)=\\operatorname{Aut}\\left(N\\right)^{0}$ .",
"Example 4.8 In [5] the authors describe a generic $K_{n}+K_{3}$ -free graph, where $K_{n}+K_{3}$ is the free amalgam of the complete graph on $n$ vertices and a triangle over a single vertex.",
"This structure is $\\aleph _{0}$ -categorical, with $\\operatorname{acl}\\left(\\emptyset \\right)=\\emptyset $ .",
"By [4] there is a CIR on this graph.",
"Example 4.9 $\\omega $ -categorical Hrushovski constructions.",
"Let $L$ be a finite relational language, and let $f:\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}_{\\ge 0}$ be a “control function”.",
"According to [10], there is an $\\omega $ -categorical generic Hrushovski construction $M_{f}$ for a “free amalgamation class” $K_{f}$ if $f$ satisfies certain conditions.",
"It follows from [4] that given extra conditions on the algebraic closure, $M_{f}$ admits a CIR.",
"See more details in [4], [10].",
"Ultrahomogenous partial orders In [11] the authors prove that the automorphism group of every ultrahomogeneous poset (partially order set) is topologically 2-generated.",
"They also characterize when they have a cyclically dense conjugacy class.",
"We can find such a conjugacy class by finding a CIR whenever possible.",
"We should remark that they prove more on the automorphism groups of those structures.",
"By [32] there are four types of ultrahomogeneous posets.",
"Fact 4.10 [32] Suppose that $\\left(H,<\\right)$ is an ultrahomogeneous poset.",
"Then $H$ is isomorphic to one of the following: The random poset: the Fraïssé limit of the class of finite partial orders.",
"The orders $\\mathcal {A}_{n}$ for $1\\le n\\le \\omega $ : $\\left(n,<\\right)$ where $<$ is trivial i.e., empty.",
"The orders $\\mathcal {B}_{n}$ for $1\\le n\\le \\omega $ : $\\left(n\\times \\mathbb {Q},<\\right)$ where $\\left(k,q\\right)<\\left(m,p\\right)$ iff $k=m$ and $p<q$ .",
"The orders $\\mathcal {C}_{n}$ for $1\\le n\\le \\omega $ : $\\left(n\\times \\mathbb {Q},<\\right)$ where $\\left(k,q\\right)<\\left(m,p\\right)$ iff $q<p$ .",
"Note that the orders $\\mathcal {A}_{n}$ have $S_{n}$ as their automorphism group, and thus for $n$ finite cannot have a dense conjugacy class.",
"For $n=\\omega $ , this is Example .",
"Also, the orders $\\mathcal {B}_{n}$ for $1<n<\\omega $ cannot have a dense conjugacy class by Remark REF : $S_{n}$ is a quotient of the automorphism group (define $a\\mathrel {E}b$ iff $a$ and $b$ are comparable, and note that there are $n$ equivalence classes, every permutation of which is induced by an automorphism).",
"The random poset Suppose that $\\left(\\mathcal {D},\\le \\right)$ is the random partial order.",
"For finite sets $A,B,C$ define $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ ABC$and if $ aA,bB$ then $ a$ is comparable with $ b$ iff forsome $ cC$, $ acb$ or $ bca$.",
"Then $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$is a (symmetric) CIR.",
"We will show only transitivity and extension,and leave the rest to the reader.",
"Suppose that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CDB$ and$ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"Given $ aA,bB$, such that $ ab$, theremust be some $ dCD$ such that $ adb$.",
"Hence there mustbe some $ cC$ with $ dcb$.",
"Together we are done.",
"Extension:suppose that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and we are given $ d$.",
"Assume $ dBC$(otherwise we are done).",
"Then let $ d'BCd$ be such that $ d'ABC$and for all $ aA$, $ ad'$ iff for some $ cC$, $ acd$,and similarly define when $ d'a$.",
"Now, $ (ABCd',)$is a poset since $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.$ The orders $\\protect \\mathcal {B}_{1}$ and $\\protect \\mathcal {B}_{\\omega }$ .",
"Example 4.11 For $\\left(\\mathbb {Q},<\\right)$ (which is $\\mathcal {B}_{1}$ ), for every finite $A,B,C\\subseteq \\mathbb {Q}$ , we let $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ if $ ABC$and for all $ aAC$ and $ bBC$ such that$ aCb$, $ a<b$.",
"Then $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR.",
"We prove only transitivityand leave the rest to the reader.$ Suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DCB$ and $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"We have to show that$ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, which amounts to showing that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"Fix $ aAC$and $ bBC$ such that $ aCb$.",
"We have to showthat $ a<b$.",
"Note that $ bD$.",
"If $ aD$ then this is trueby our assumption.",
"Otherwise, $ aCD$.",
"If $ aCDb$ thenwe are done.",
"Otherwise, $ a,b$ have different cuts over $ CD$.",
"Butsince they realize the same cut over $ C$, it follows that there issome $ dDC$ such that either $ a<d<b$ or $ b<d<a$.The former would imply what we want, so assume that $ b<d<a$.",
"Butthen $ bCd$ so $ d<b$ —{} a contradiction.",
"Theother direction of transitivity is proved similarly.$ Example 4.12 Consider $\\mathcal {B}_{\\omega }$ .",
"Then each equivalence class of the relation $E$ of being comparable is a DLO, and thus by Example REF , for each $n<\\omega $ , there is a CIR $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $F$ defined as in Example\\ref {exa:DLO has groovy} for each $ E$-class $ F$.",
"For $ (B,<)$and finite sets $ A,B,C$, define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ ABC$,$ A/EB/EC/E$(if $ aA$, $ bB$ and $ aEb$then there is some $ cC$ such that $ aEc$) and for every$ E$-class $ F$, $ AF$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CFFBF$.",
"This is easilyseen to be a CIR.",
"Note that we need infinitely many classes for extension.$ The orders $\\protect \\mathcal {C}_{n}$ for $1\\protect \\le n\\protect \\le \\omega $ .",
"In $\\mathcal {C}_{n}$ we have an equivalence relation $E$ , defined by $a\\mathrel {E}b$ iff $a$ and $b$ are incomparable (they have the same second coordinate).",
"Then $\\mathcal {C}_{n}/E\\models DLO$ , so we have a CIR $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $E$ defined on it by Example \\ref {exa:DLO has groovy}.For finite $ A,B,CCn$, define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ A/E$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C/EEB/E$.This trivially satisfies all the properties.$ Ultrahomogeneous graphs In [17], the authors prove that for every ultrahomogeneous graph $\\Gamma =\\left(V,E\\right)$ , $\\operatorname{Aut}\\left(\\Gamma \\right)$ is topologically 2-generated.",
"Similarly to the poset case, we can recover some results by finding a CIR whenever possible.",
"By [24] we have the following classification of ultrahomogeneous graphs.",
"Recall that for a graph $\\left(V,E\\right)$ , its dual is $\\left(V,E^{\\prime }\\right)$ where $E^{\\prime }=\\left[V\\right]^{2}\\backslash E$ .",
"Fact 4.13 [24] Any countable ultrahomogeneous graph $\\Gamma $ is isomorphic to one of the following graphs, or its dual.",
"The random graph.",
"For $n\\ge 3$ , the Henson graph, i.e., the $K_{n}$ -free universal graph (the Fraïssé limit of the class of $K_{n}$ -free finite graphs).",
"For any $1\\le n\\le \\omega $ , the graph $\\omega K_{n}$ consisting of a disjoint union of countably many copies of $K_{n}$ .",
"For any $2\\le n<\\omega $ , the graph $nK_{\\omega }$ consisting of a disjoint union of $n$ copies of $K_{\\omega }$ (the complete graph on $\\omega $ ).",
"Note that the dual of a graph has the same automorphism group, so we can ignore the duals.",
"We already saw in Example REF that both the random graph and the Henson graph have a CIR.",
"The graphs $nK_{\\omega }$ for $n<\\omega $ cannot have a a dense conjugacy class by Remark REF as in the case of the posets $\\mathcal {B}_{n}$ described above.",
"However, $\\omega K_{n}$ for $1\\le n\\le \\omega $ has a CIR, just like the cases $\\mathcal {C}_{n}$ above.",
"A mix of two Fraïsé limits with CIRs Suppose we are in the situation of Section REF : we have two amalgamation classes $K_{1},K_{2}$ with all the properties listed there.",
"Let $M_{1},M_{2}$ be the Fraïsé limits of $K_{1},K_{2}$ respectively, and let $M$ be the Fraïsé limit of $K$ , the class of finite $L_{1}\\cup L_{2}$ -structures $A$ such that $A\\upharpoonright L_{1}\\in K_{1}$ and $A\\upharpoonright L_{2}\\in K_{2}$ .",
"Add the extra assumption that $L_{1}\\cap L_{2}=\\emptyset $ .",
"Suppose that $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $1,$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $2$ areCIRs on $ M1,M2$ respectively.",
"By Proposition \\ref {prop:restriction of a mix of two Faisse},we may assume that $ M1=ML1$ and $ M2=ML2$.For finite subsets of $ M$, define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C1B$and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C2B$.\\begin{prop}The relation \\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{prop}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR.$ Stationarity follows from the fact that by quantifier elimination, for any finite tuples $a,a^{\\prime }$ from $M$ , if $a\\equiv a^{\\prime }$ in $L_{1}$ and in $L_{2}$ , then $a\\equiv a^{\\prime }$ in $L_{1}\\cup L_{2}$ .",
"Extension: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, and we are given $ dM$.Let $ d1M1$ be such that $ d1BCd$ in $ L1$and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C1Bd1$.",
"Similarly find $ d2$ for $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $2$and $ L2$.",
"Consider the finite structure $ D$ with universe $ ABCd$where the $ L1L2$-structure on $ ABC$ is as in $ M$, andsuch that its restriction to $ L1$, $ L2$ is $ ABCd1$, $ ABCd2$,respectively.",
"This structure exists since $ L1L2=$and by the assumptions of Section \\ref {subsec:A-mix-of}, both languagesare relational.",
"Thus, $ DK$, so it has an isomorphic copy $ D'M$containing copies $ A',B',C',d'$ of $ A,B,C,d$.",
"As $ M$ is ultrahomogeneous,we can apply an automorphism $$ mapping $ A'B'C'$ to $ ABC$,so that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB(d')$, and $ (d')BCd$.$ The other properties are easy to check.",
"Example 4.14 The ordered random graph $M=\\left(V,R,<\\right)$ .",
"It is the Fraïsé limit of the class of finite linearly ordered graphs in the language $\\left\\lbrace <,R\\right\\rbrace $ .",
"It easily satisfies all our assumptions with $L_{1}=\\left\\lbrace <\\right\\rbrace $ , $K_{1}$ the class of finite linear orders and $L_{2}=\\left\\lbrace R\\right\\rbrace $ , $K_{2}$ the class of finite graphs.",
"It has a CIR as both $\\left(M,<\\right)$ (which is a DLO) and $\\left(M,R\\right)$ (the random graph) have CIRs by the two previous subsections.",
"Similarly we may define the random ordered hypergraph, and it too has a CIR.",
"Trees The theory $T_{dt}$ (see Example REF ) does not admit a canonical independence relation.",
"We shall give a precise (and stronger) argument for this below in Corollary REF , but it is easy to see that a natural candidate fails.",
"Namely, one can try to define $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ in such a way that if $ C=$and $ a,b$ are singletons then $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$ iff $ ab<a,b$, andfor $ a,b,c$ such that $ c$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$, then $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $cb$ iff $ ac<cb$.But then $ ac$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $cb$, $ c$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$ but $ ac$\\displaystyle ㈶$$\\displaystyle \\mid $$\\displaystyle \\smile $$\\textstyle ㈶$$\\textstyle \\mid $$\\textstyle \\smile $$\\scriptstyle ㈶$$\\scriptstyle \\mid $$\\scriptstyle \\smile $$\\scriptscriptstyle ㈶$$\\scriptscriptstyle \\mid $$\\scriptscriptstyle \\smile $b$,so transitivity fails.$ However, we can expand it in such a way that it does.",
"We give two such expansions.",
"Example 4.15 Let $L_{dt}^{B}=\\left\\lbrace <,P,f,\\wedge \\right\\rbrace $ where $P$ is a unary predicate and $f$ is a unary function symbol, and let $T_{dt}^{B}$ be the model completion of the universal $L_{dt}^{B}$ -theory of trees where $P$ is a downwards closed linearly ordered subset and $f\\left(x\\right)$ is the maximal element in $P$ which is $\\le x$ .",
"In other words, $T_{dt}^{B}$ is the theory of the Fraïsé limit of the class of finite $L_{dt}^{B}$ -structures $M$ where $M\\upharpoonright \\left\\lbrace \\wedge ,<\\right\\rbrace $ is a tree with a meet function, $P^{M}$ is linearly ordered and downwards closed and $f\\left(x\\right)=\\max \\left\\lbrace y\\le x\\,|\\,y\\in P\\right\\rbrace $ (note that this class has JEP and AP).",
"Then $T_{dt}^{B}$ is the theory of dense trees with a predicate for a branch (a maximal chain), it has quantifier elimination and is $\\omega $ -categorical.",
"Let us see why $P$ is a maximal chain in every model $M\\models T_{dt}^{B}$ .",
"Of course it is downwards closed by definition, so if $a\\in M$ is comparable with $P$ but $a\\notin P$ , then $a>P$ .",
"As $T_{dt}^{B}$ is model-complete, $M$ is existentially closed (see Fact REF ) so there is some $b\\in P$ (from $M)$ such that $f\\left(a\\right)<b$ .",
"Thus, $a>b>f\\left(a\\right)$ which is a contradiction to the definition of $f$ .",
"For three sets $A,B,C$ , let $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ ACBCC$and for all $ aAC$ with $ f(a)C$and $ bBC$ with $ f(b)C$such that $ f(a)Cf(b)$ (which is thesame as $ f(a)f(C)f(b)$),$ f(a)<f(b)$.",
"Then $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is canonical.",
"Theonly nontrivial axioms to check are stationarity over $$,extension and transitivity.$ Suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$.",
"This just says that that $ B$ is placed above$ A$ with respect to the branch $ P$ (i.e., $ f(a)<f(b)$for all $ aA,bB$).",
"So $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is stationary by quantifierelimination.$ Extension: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and we are given a single element$ d$, which we may assume is not in $ BC$and even that $ f(d)BC$.First find some $ d”$ such that $ d”BCf(d)$and $ d”$ is greater than every $ f(a)$ such that $ aAC$and $ f(a)Cf(d)$.",
"Then find $ d'$ suchthat $ d'BCd$ and $ f(d')=d”$ (and $ ACBCd'=C$).$ Transitivity: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DCB$ and $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and wehave to show that $ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"Suppose that $ aADC,f(a)C$and $ bBC,f(b)C$are such that $ f(a)Cf(b)$ but $ f(b)f(a)$.Then there must be some $ dDC$ suchthat $ f(b)df(a)$, as otherwise $ f(a)CDf(b)$.But since $ d$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, $ f(b)<d$, which implies that$ f(b)$ and $ d$ do not have the same type over $ C$, sothere must be some $ cC$ between them, and in particular, it contradictsour assumption that $ f(a)Cf(b)$.",
"Theother direction of transitivity is proved similarly.$ Example 4.16 Let $L_{dt}^{p}=\\left\\lbrace <,p,\\wedge \\right\\rbrace $ where $p$ is a new constant.",
"Let $T_{dt}^{p}$ be the unique completion of $T_{dt}$ to $L_{dt}^{p}$ .",
"Let $M\\models T_{dt}^{p}$ be the unique countable model.",
"To simplify notation, we identify $p$ with $p^{M}$ .",
"For three sets $A,B,C$ , we let $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ ABBCC$and:\\begin{enumerate}\\item For all a\\in \\left\\langle AC\\right\\rangle with a\\wedge p\\notin \\left\\langle C\\right\\rangle and b\\in \\left\\langle BC\\right\\rangle with b\\wedge p\\notin \\left\\langle C\\right\\rangle such that a\\wedge p\\equiv _{C}b\\wedge p, a\\wedge p<b\\wedge p.\\item For all a\\in \\left\\langle AC\\right\\rangle such that a>p withno c\\in \\left\\langle C\\right\\rangle such that a\\wedge c>p, andall b\\in \\left\\langle BC\\right\\rangle with b>p and no c\\in \\left\\langle C\\right\\rangle such that b\\wedge c>p, a\\wedge b=p.\\end{enumerate}Then $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is canonical.",
"The only nontrivial axioms to check arestationarity over $$, extension and transitivity.$ It is stationary over $\\emptyset $ by elimination of quantifiers, since $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ iff $ A'={ aA | ap<p} $ is placed below$ B'={ bB | bp<p} $ with respect to the points below $ p$while $ A”={ aA | ap} $ and $ B”={ bB | bp} $are placed independently above $ p$.$ Extension: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and we are given $ d$ such that$ dBC$.",
"First assume that $ dp<p$.If $ dpBC$, similarly to Example\\ref {exa:trees with predicate}, first find some $ d”$ such that$ d”BCdp$ and $ d”>ap$ for all $ aAC$with $ apCdp$.",
"Then find $ d'$ such that $ d'BCd$with $ d'pd”$ (and $ BCd'AC=C$).Now assume that $ d>p$.",
"If there is some $ bBC$with $ bd>p$, any $ d'BCd$ such that $ BCd'AC=C$will work.",
"Otherwise find some $ d'BCd$ such that $ d'a=p$for all $ aAC$ with $ a>p$.$ Transitivity: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DCB$ and $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and wehave to show that $ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"If we are in case (1) of the definition,($ aADC$, $ apC$,etc.)",
"then we proceed exactly as in Example \\ref {exa:trees with predicate}.Otherwise, suppose that $ aADC$, $ bBC$are as in case (2).",
"If there is some $ dCD$with $ ad>p$, then for no $ cC$is it the case that $ cd>p$ (otherwise $ ac>p$).",
"Thus$ db=p$ because $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ hence $ ab=p$ as required.If there is no such $ d$ then $ ab=p$ because $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DCB$.$ Trees also satisfy the following interesting phenomenon.",
"Proposition 4.17 If $M$ is a dense tree as in Example REF (i.e., the model companion of the theory of trees in $\\left\\lbrace <,\\wedge \\right\\rbrace $ ) then for every $\\sigma \\in \\operatorname{Aut}\\left(M\\right)$ which does not have any fixed points, there is a branch $B\\subseteq M$ such that $\\sigma \\left(B\\right)=B$ .",
"Let $B$ be a maximal linearly ordered set such that $\\sigma \\left(B\\right)=B$ (which exists by Zorn's lemma).",
"We will show that $B$ is a branch.",
"Note that if $x\\in B$ and $y<x$ , then $B\\cup \\left\\lbrace \\sigma ^{n}\\left(y\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is still a chain: given any $z\\in B$ and any $n\\in \\mathbb {Z}$ , $\\sigma ^{n}\\left(x\\right),z$ are comparable and $\\sigma ^{n}\\left(y\\right)<\\sigma ^{n}\\left(x\\right)$ it follows that $\\sigma ^{n}\\left(y\\right)$ and $z$ are comparable (if $z\\le \\sigma ^{n}\\left(x\\right)$ then both $\\sigma ^{n}\\left(y\\right),z\\le \\sigma ^{n}\\left(x\\right)$ , so they are comparable by the tree axioms, and if $\\sigma ^{n}\\left(x\\right)<z$ , then $\\sigma ^{n}\\left(y\\right)<z$ ), and for any $n,m\\in \\mathbb {Z}$ , $\\sigma ^{n}\\left(y\\right),\\sigma ^{m}\\left(y\\right)$ are comparable since $\\sigma ^{n}\\left(x\\right)$ and $\\sigma ^{m}\\left(x\\right)$ are (if $\\sigma ^{n}\\left(x\\right)\\le \\sigma ^{m}\\left(x\\right)$ then both $\\sigma ^{n}\\left(y\\right),\\sigma ^{m}\\left(y\\right)\\le \\sigma ^{m}\\left(x\\right)$ so they are comparable by the tree axioms).",
"Hence $B$ is downwards closed.",
"Now, as $\\sigma $ has no fixed points, $B$ cannot have a maximum (which would have to be a fixed point).",
"Also, if $a\\ge B$ and $\\sigma \\left(a\\right)\\ge a$ or $\\sigma \\left(a\\right)\\le a$ then $B\\cup \\left\\lbrace \\sigma ^{n}\\left(a\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is still a chain (since $\\sigma ^{n}\\left(a\\right)\\ge \\sigma ^{n}\\left(B\\right)=B$ for all $n\\in \\mathbb {Z}$ ), so $a\\in B$ .",
"If $B$ is not a branch (in particular, if $B=\\emptyset $ , which we haven't ruled out yet), there is some $a\\in M$ such that $B<a$ .",
"Let $b=\\sigma \\left(a\\right)\\ne a$ (and by the above, $b,a$ are not comparable), so $B<b$ .",
"Hence $B\\le \\left(a\\wedge b\\right)<a,b$ .",
"Now, $\\sigma \\left(a\\wedge b\\right)<\\sigma \\left(a\\right)=b$ , so $a\\wedge b$ and $\\sigma \\left(a\\wedge b\\right)$ are comparable.",
"The previous paragraph implies that $a\\wedge b\\in B$ .",
"But then $B$ has a maximum — contradiction.",
"Having finite topological rank In this section we will find some criteria that ensure that $G$ has finite topological rank.",
"$\\omega $ -categorical stable theories Proposition 5.1 If $T$ is stable $\\omega $ -categorical, $M\\models T$ is countable and $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is finite, then $\\operatorname{Aut}\\left(M\\right)$ has finite topological rank.",
"Without loss of generality, $M=M^{\\operatorname{eq}}$ (if $S\\subseteq \\operatorname{Aut}\\left(M^{\\operatorname{eq}}\\right)$ generates a dense subgroup, then $S\\upharpoonright M=\\left\\lbrace f\\upharpoonright M\\,|\\,f\\in S\\right\\rbrace $ generates a dense subgroup of $\\operatorname{Aut}\\left(M\\right)$ ).",
"Let $N=M_{\\operatorname{acl}\\left(\\emptyset \\right)}$ (i.e., name the elements in $\\operatorname{acl}\\left(\\emptyset \\right)$ ).",
"Then $N$ is $\\omega $ -categorical by Propsition REF .",
"Then in $N$ , $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , so by Example REF , there is a canonical independence relation in $N$ , so $G^{0}=\\operatorname{Aut}\\left(N\\right)$ is topologically 2-generated by Corollary REF , say by $\\left\\lbrace f_{1},f_{2}\\right\\rbrace $ .",
"Now, $\\operatorname{Aut}\\left(M\\right)/G^{0}$ is finite by assumption, so let $S\\subseteq \\operatorname{Aut}\\left(M\\right)$ be a finite set of representatives.",
"Then $S\\cup \\left\\lbrace f_{1},f_{2}\\right\\rbrace $ generates a dense subgroup $\\operatorname{Aut}\\left(M\\right)$ : given two finite tuples $\\bar{a},\\bar{b}$ from $M$ such that $\\bar{a}\\equiv \\bar{b}$ , there is an automorphism $\\sigma \\in \\operatorname{Aut}\\left(M\\right)$ such that $\\sigma \\left(\\bar{a}\\right)=\\bar{b}$ .",
"Also, there is some $f\\in S$ such that $f^{-1}\\sigma \\in \\operatorname{Aut}\\left(N\\right)$ .",
"Hence for some $g$ in the group generated by $\\left\\lbrace f_{1},f_{2}\\right\\rbrace $ , $g\\left(\\bar{a}\\right)=f^{-1}\\sigma \\left(\\bar{a}\\right)=f^{-1}\\left(\\bar{b}\\right)$ , so $fg\\left(\\bar{a}\\right)=\\bar{b}$ .",
"The following fact implies immediately the next result.",
"Fact 5.2 [9] If $T$ is $\\omega $ -categorical and $\\omega $ -stable and $M\\models T$ is countable, then $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is finite.",
"Corollary 5.3 If $T$ is $\\omega $ -stable and $\\omega $ -categorical and $M\\models T$ is countable, then $\\operatorname{Aut}\\left(M\\right)$ has finite topological rank.",
"Reducing finite topological rank to expansions Suppose that $M$ is countable let $G=\\operatorname{Aut}\\left(M\\right)$ .",
"We now want to explore the idea that perhaps by expanding $M$ (i.e., moving to a subgroup), we can show that the topological rank of $G$ is small by showing that the rank of the automorphism group of the expansion is.",
"Suppose that $H\\le G$ .",
"If $\\left(G,H\\right)$ has a compact quotient (see Definition REF ), then we cannot hope to deduce anything.",
"For example, by Proposition REF we have that $G^{0}$ acts oligomorphically on $M$ and it can be that $G^{0}$ has a cyclically dense conjugacy class (so topological rank 2) while $G/G^{0}=\\left(\\mathbb {Z}/2\\mathbb {Z}\\right)^{\\omega }$ (so $G$ is not topologically finitely generated) — this happens in the example described in in Remark REF , see Example REF .",
"Indeed, we will see that $\\left(G,H\\right)$ having a compact quotient is the only obstruction.",
"$\\omega $ -categorical structures with finitely many reducts Theorem 5.4 Suppose that $H\\le G$ is closed and that $\\left(G,H\\right)$ has no compact quotients.",
"If there are only finitely many closed groups between $G$ and $H$ then there is some $g\\in G$ such that $H\\cup \\left\\lbrace g\\right\\rbrace $ topologically generate $G$ .",
"Remark 5.5 The condition of having finitely many closed groups in the theorem holds when for instance $M$ is a reduct of an $\\omega $ -categorical structure $M^{\\prime }$ where $H=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ , and $M^{\\prime }$ has only finitely many reducts up to bi-definability.",
"Let $\\left\\lbrace H_{i}\\,|\\,i<n\\right\\rbrace $ be the family of closed proper subgroups of $G$ containing $H$ (which is finite by assumption).",
"If $\\left[G:H_{i}\\right]<\\infty $ for some $i<n$ , then there would be a closed normal proper subgroup $N_{i}\\trianglelefteq G$ of finite index such that $N_{i}\\le H_{i}$ (in general, if $H^{\\prime }\\le G$ is closed of finite index, then there is a closed normal subgroup $N\\le H^{\\prime }$ , $N\\trianglelefteq G$ such that $\\left[G:N\\right]<\\infty $ .",
"In fact, $N=\\bigcap \\left\\lbrace gH^{\\prime }g^{-1}\\,|\\,g\\in G\\right\\rbrace $ and this intersection is finite as it is the orbit of $H^{\\prime }$ under the action of $G$ on conjugates of $H^{\\prime }$ and its stabilizer contains $H^{\\prime }$ ).",
"But then $N_{i}H=G$ by assumption and Proposition REF , so $G=N_{i}H\\subseteq H_{i}H=H_{i}$ contradicting the fact that $H_{i}$ was a proper subgroup.",
"By a theorem of Neumann [29], there is some $g\\in G\\backslash \\bigcup \\left\\lbrace H_{i}\\,|\\,i<n\\right\\rbrace $ .",
"If $G\\ne \\operatorname{cl}\\left(\\left\\langle H\\cup \\left\\lbrace g\\right\\rbrace \\right\\rangle \\right)$ (the topological closure of the group generated by $H\\cup \\left\\lbrace g\\right\\rbrace $ ), then $\\operatorname{cl}\\left(\\left\\langle H\\cup \\left\\lbrace g\\right\\rbrace \\right\\rangle \\right)$ is one of the groups $H_{i}$ , contradicting the choice of $g$ .",
"Corollary 5.6 If $G$ and $H$ are as in Theorem REF and $H$ has finite topological rank then so does $G$ .",
"By Example REF , in the $\\omega $ -categorical context we get that if $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ in $M$ and $M^{\\prime }$ is an expansion having finitely many reducts, then we can apply Corollary REF .",
"This is the case, for instance, when $M^{\\prime }$ is $\\left(\\mathbb {Q},<\\right)$ (see [18]).",
"By Lemma [18], an example of such a reduct of DLO is given by the countable dense circular order, which is the structure with universe $\\mathbb {Q}$ , and a ternary relation $C\\left(x,y,z\\right)$ given by $C\\left(x,y,z\\right)\\Leftrightarrow x<y<z\\vee y<z<x\\vee z<x<y$ .",
"Corollary 5.7 $\\operatorname{Aut}\\left(\\mathbb {Q},C\\right)$ has topological rank $\\le 3$ , but $\\left(\\mathbb {Q},C\\right)$ has no CIR.",
"We only have to show that it has no CIR.",
"By Lemma REF , if there was a CIR, then in particular there would be a type of a single element $q\\left(x\\right)$ over $\\mathbb {Q}$ which does not split over $\\emptyset $ .",
"But by quantifier elimination, every tuple of two distinct elements have the same type (i.e., $\\operatorname{Aut}\\left(\\mathbb {Q},C\\right)$ acts 2-transitively on $\\mathbb {Q}$ ).",
"Now, $q$ cannot be realized in $\\mathbb {Q}$ and must contain $C\\left(0,x,1\\right)$ or $C\\left(1,x,0\\right)$ , hence both, which is a contradiction.",
"Remark 5.8 For any point $a\\in \\mathbb {Q}$ , the expansion $\\left(\\mathbb {Q},C,a\\right)$ does have a CIR.",
"Indeed, in this case $C$ defines a dense linear order with no endpoints on $\\mathbb {Q}\\backslash \\left\\lbrace a\\right\\rbrace $ by $b<c\\iff C\\left(a,b,c\\right)$ .",
"Since $\\left(\\mathbb {Q},<\\right)$ has a CIR $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ (Example \\ref {exa:DLO has groovy}),we can define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C*B$ by $ A{ a} $\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C{ a} B{ a} $.Since for every finite tuples $ b,c$, $ bc$ in the expansioniff $ baca$ in the order, it followseasily that $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*$ is a CIR.$ An even closer look at the reducts of DLO, gives the following result.",
"Corollary 5.9 Every closed supergroup of $\\operatorname{Aut}\\left(\\mathbb {Q},<\\right)$ has topological rank $\\le 3$ .",
"The diagram in [18] of the lattice of closed groups between $\\operatorname{Aut}\\left(\\mathbb {Q},<\\right)$ and $\\operatorname{Aut}\\left(\\mathbb {Q},=\\right)$ shows that any such group contains at most two incomparable closed subgroups.",
"Since no group can be a union of two of its proper subgroups, we do not need to use Neumanns's lemma in the proof of Theorem REF above, allowing us to drop the assumption that $\\left(G,H\\right)$ has no compact quotients.",
"A general reduction theorem In the next theorem we drop the assumption of having finitely many reducts of the expansion (i.e., of having finitely many groups between $H$ and $G$ ), but we compensate for it by assuming that $H$ acts oligomorphically on $M$ and increasing the number of generators by 1.",
"Fact 5.10 [9] Suppose that $M$ is a countable $\\omega $ -saturated structure.",
"Then for any $A,B\\subseteq M$ , there is some $A^{\\prime }$ (in the monster model $\\mathfrak {C}$ , see just above Section REF ) such that $A^{\\prime }\\equiv A$ and $A^{\\prime }\\cap B\\subseteq \\operatorname{acl}\\left(\\emptyset \\right)$ .",
"Theorem 5.11 Suppose as usual that $M$ is countable and $\\omega $ -categorical and let $G=\\operatorname{Aut}\\left(M\\right)$ .",
"Suppose that $H\\le G$ is closed and acts oligomorphically on $M$ and that $\\left(G,H\\right)$ has no compact quotients.",
"Then there are $g_{1},g_{2}\\in G$ such that $H\\cup \\left\\lbrace g_{1},g_{2}\\right\\rbrace $ topologically generates $G$ .",
"Let $M^{\\prime }$ be an $\\omega $ -categorical expansion of $M$ to some language $L^{\\prime }$ containing $L$ (the language of $M$ ) such that $H=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ .",
"We use $^{\\prime }$ to indicate the expansion.",
"In particular, $\\mathfrak {C}^{\\prime }$ denotes the expansion of $\\mathfrak {C}$ to $L^{\\prime }$ .",
"By Fact REF , there is some $M_{0}$ such that $M_{0}\\equiv M$ and $M_{0}^{\\operatorname{eq}}\\cap M^{\\operatorname{eq}}=\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ (apply the fact in $\\mathfrak {C}^{\\operatorname{eq}}$ ).",
"There is some automorphism $\\sigma $ of $\\mathfrak {C}$ such that $\\sigma \\left(M_{0}\\right)=M$ .",
"Let $N_{0}^{\\prime }$ be a countable model containing $\\sigma ^{n}\\left(M_{0}\\right)$ for all $n\\in \\mathbb {Z}$ .",
"Let $N_{1}^{\\prime }$ be a countable model containing $\\sigma ^{n}\\left(N_{0}^{\\prime }\\right)$ for all $n\\in \\mathbb {Z}$ .",
"Continue like this and finally let $N_{\\omega }^{\\prime }=\\bigcup \\left\\lbrace N_{i}^{\\prime }\\,|\\,i<\\omega \\right\\rbrace $ .",
"So $M^{\\prime }\\prec N^{\\prime }_{\\omega }\\prec \\mathfrak {C}^{\\prime }$ is countable and $\\sigma \\upharpoonright N{}_{\\omega }\\in \\operatorname{Aut}\\left(N_{\\omega }\\right)$ .",
"By $\\omega $ -categoricity (of $M^{\\prime }$ ) we may assume that $N_{\\omega }^{\\prime }=M^{\\prime }$ : there is some $g_{1}\\in \\operatorname{Aut}\\left(M\\right)$ and $M_{0}^{\\prime }\\prec M^{\\prime }$ such that $g_{1}\\left(M_{0}^{\\operatorname{eq}}\\right)\\cap M_{0}^{\\operatorname{eq}}=\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"Then $H_{1}=\\operatorname{cl}\\left(\\left\\langle H,g_{1}\\right\\rangle \\right)$ is a closed group acting oligomorphically on $M$ .",
"Also, note that $\\left(G,H_{1}\\right)$ has no compact quotients.",
"Let $M^{\\prime \\prime }$ be the reduct of $M^{\\prime }$ , which is also an expansion of $M$ that corresponds to $H_{1}$ : $\\operatorname{Aut}\\left(M^{\\prime \\prime }\\right)=H_{1}$ .",
"As usual, we use $^{\\prime \\prime }$ to indicate that we work in this expansion.",
"Claim 5.12 If $X\\subseteq M^{n}$ is definable over $\\emptyset ^{\\prime \\prime }$ (i.e., definable in $L^{\\prime \\prime }$ over $\\emptyset $ ) and $M$ -definable (in $L$ ), then it is $\\emptyset $ -definable (in $L$ ).",
"First note that it is enough to show that $X$ is $\\operatorname{acl}_{L}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ -definable (the code $X$ of $X$ belongs to $\\operatorname{dcl}_{L^{\\prime \\prime }}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ and to $\\operatorname{acl}_{L}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , and if it were not in $\\operatorname{dcl}_{L}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ then there would be an automorphism of $M$ moving it, but then by the no-compact quotient assumption there would be an automorphism of $M^{\\prime \\prime }$ moving it as well — contradiction).",
"Now, since $X$ is $\\emptyset ^{\\prime \\prime }$ -definable and $M$ -definable, it is definable over $M_{0}$ (because $M_{0}^{\\prime }\\prec M^{\\prime }$ ), so its code $X\\in M_{0}^{\\operatorname{eq}}$ .",
"In addition, $g_{1}\\left(X\\right)=X$ , so $X$ is definable over $g_{1}\\left(M_{0}\\right)$ , hence $X\\in g_{1}\\left(M_{0}^{\\operatorname{eq}}\\right)$ .",
"Together it is in $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , which is what we wanted.",
"Now we construct $g_{2}$ by back-and-forth to ensure that $\\operatorname{cl}\\left(\\left\\langle H_{1},g_{2}\\right\\rangle \\right)=G$ .",
"Suppose that we have constructed $g_{2}\\upharpoonright A$ for some finite set $A$ .",
"Let $O$ be an orbit of the action of $G$ on $M^{m}$ , and we write it as $O=\\bigcup \\left\\lbrace O_{i}\\,|\\,i<n\\right\\rbrace $ where the $O_{i}$ 's are the orbits of the action of $H_{1}$ (recall that $H_{1}$ acts oligomorphically on $M$ , so there are only finitely many such orbits).",
"Claim 5.13 For any subset $s\\subsetneq n$ there are $a,b\\in O$ such that $a\\in O_{s}=\\bigcup \\left\\lbrace O_{i}\\,|\\,i\\in s\\right\\rbrace ,b\\in O_{n\\backslash s}$ , and $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle a,b\\right\\rangle \\right\\rbrace $ or $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle b,a\\right\\rangle \\right\\rbrace $ is an elementary map.",
"Note that $O_{s}$ is $\\emptyset ^{\\prime \\prime }$ -definable.",
"As it is not $\\emptyset $ -definable (because $s\\subsetneq n$ ), it is also not $M$ -definable by Claim REF .",
"In particular, it is not $A$ -definable.",
"Hence there are $a_{0}\\in O_{s},a_{1}\\in O_{n\\backslash s}$ such that $a_{0}\\equiv _{A}a_{1}$ .",
"There is some $b$ such that $a_{0}A\\equiv a_{1}A\\equiv bg_{2}\\left(A\\right)$ .",
"If $b\\in O_{s}$ , then $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle a_{1},b\\right\\rangle \\right\\rbrace $ is the required map.",
"Otherwise, pick $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle a_{0},b\\right\\rangle \\right\\rbrace $ .",
"In the back-and-forth construction of $g_{2}$ , we deal with all these orbits (for every $m<\\omega $ , there are only finitely many) and all these subsets $s$ and increase $g_{2}$ according to Claim REF .",
"We claim that $g_{2}$ is such that $\\operatorname{cl}\\left(\\left\\langle H_{1},g_{2}\\right\\rangle \\right)=G$ .",
"Indeed, it is enough to show that every orbit $O$ of $G$ is also an orbit of $\\left\\langle H_{1},g_{2}\\right\\rangle $ .",
"The orbit $O$ can be written as $\\bigcup \\left\\lbrace O_{i}\\,|\\,i<n\\right\\rbrace $ where the $O_{i}$ 's are the orbits of $H_{1}$ , and also as $\\bigcup \\left\\lbrace O^{\\prime }_{i}\\,|\\,i\\in I\\right\\rbrace $ where the $O^{\\prime }_{i}$ 's are orbits of $\\left\\langle H_{1},g_{2}\\right\\rangle $ .",
"Each such $O_{i}^{\\prime }$ is itself a union of $H_{1}$ -orbits, so has the form $O_{s}$ for some $s\\subseteq n$ .",
"But by construction, if $s\\ne n$ there are tuples $a\\in O_{s},b\\in O_{n\\backslash s}$ such that either $g_{2}$ or $g_{2}^{-1}$ maps $a$ to $b$ — contradiction.",
"So $s=n$ , and $O_{i}^{\\prime }=O$ .",
"A topological dynamics consequence of having a CIR Definition 6.1 Suppose that $M$ is a countable structure.",
"Call an automorphism $\\sigma \\in G$ shifty if there is some invariant binary relation on finite sets in $M$ , $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ (the base will always be $$) such that:\\begin{itemize}\\item (Monotonicity) If A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{itemize}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ and $ A'A$, $ B'B$then $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B'$.\\item (Right existence) For every finite tuple $ a$ there is some $ a'a$such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$ (by this we mean that sets enumerated by $ a$,$ a'$ are independent).\\item (Right shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$, then there exists some $ n<$such that $ b'An(b)$.$ Lemma 6.2 If $\\sigma $ is shifty then it also satisfies: (Left existence) For every finite tuple $a$ there is some $a^{\\prime }\\equiv a$ such that $a^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.\\item (Left shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ b'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$, then there exists some $ n<$such that $ b'A-n(b)$.$ Suppose that $\\sigma $ is shifty, as witnessed by $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.",
"Given $ a$,there is some $ a'a$ such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Applying an automorphismtaking $ a'$ to $ a$ we get some $ a”a$ such that $ a”$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$,which shows left existence.$ As for left shiftiness, suppose that $A$ is finite and enumerated by $a$ , $b,b^{\\prime }$ are finite tuples such that $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ and $ bb'$.Then applying an automorphism, we get some $ a'$ such that $ ab'a'b$,so $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Hence for some $ n<$, $ a'bn(a)$.From $ a'bn(a)b$ we get that $ ab'a'b-n(a')-n(b)a-n(b)$,i.e., $ b'A-n(b)$.$ Proposition 6.3 The automorphism $\\sigma $ is a shifty automorphism on $M$ iff for any type $p\\in S\\left(\\emptyset \\right)$ (with finitely many variables), letting $Y_{a}=\\bigcap \\left\\lbrace \\bigcup \\left\\lbrace \\operatorname{tp}\\left(a,\\sigma ^{n}\\left(a^{\\prime }\\right)\\right)\\,|\\,n<\\omega \\right\\rbrace \\,|\\,a^{\\prime }\\equiv a\\right\\rbrace $ for any $a\\models p$ , the intersection $Y_{p}=\\bigcap \\left\\lbrace Y_{a}\\,|\\,a\\models p\\right\\rbrace $ is nonempty.",
"Suppose that $\\sigma $ is shifty, and fix some type $p\\in S\\left(\\emptyset \\right)$ .",
"Let $a\\models p$ .",
"By existence, there is some $a^{\\prime }\\equiv a$ with $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Let $ q=tp(a,a')$ and fix some $ bp$.Let $ Aut(M)$ map $ a$ to $ b$ and let $ b'=(a')$.We have that $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$ and hence by right shiftiness, $ q=tp(b,b')Yb$.Since $ b$ was arbitrary, $ qYp$.$ Suppose that the right hand side holds.",
"Given a finite tuple $a$ and $a^{\\prime }\\equiv a$ , write $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ iff $ tp(a,a')Yp$where $ p=tp(a/)$.",
"For general finite sets $ A,B$,write $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ iff there is some $ C$ containing $ A$ and $ C'$containing $ B$ such that $ C$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*C'$.",
"Obviously, $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is invariantand monotone.",
"Right existence follows from the assumption that $ Yp$for all $ pS()$.",
"Right shiftiness: supposethat $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ and $ a”a'$.",
"Then $ tp(a,a')Yp$and in particular it belongs to $ Ya$.",
"By definition of $ Ya$,$ tp(a,a'){ tp(a,n(a”)) | n<} $,so for some $ n<$, $ aa'ain(a”)$.$ Proposition 6.4 If $M$ is an ultrahomogeneous structure and $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR on finite subsets of $ M$ which respects substructures,then there exists a shifty automorphism $$ on $ M$, as witnessedby $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.$ Monotonicity and right existence are parts of the properties of a CIR, so we only have to prove right shiftiness.",
"Suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$and $ b'b$.",
"By the proof of Theorem \\ref {thm:existence of repulsive automorphism-ultrahomogeneous},the repulsive automorphism $$ constructed there satisfies thatfor some $ n<$, $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $n(b')$.",
"By stationarity,$ bA(b')$.$ Recall the definitions of flow and subflow from Section REF .",
"Theorem 6.5 Let $M$ be a countable homogeneous structure and $G=\\operatorname{Aut}\\left(M\\right)$ .",
"Suppose that $\\sigma \\in G$ is a shifty automorphism and that $\\left(X,d\\right)$ is a compact metric $G$ -flow.",
"Then for every $x_{*}\\in X$ there is some conjugate $\\sigma _{*}\\in G$ of $\\sigma $ such that: (*) Both $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ and $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contain a subflow of $X$ .",
"Remark 6.6 Note that Theorem REF implies that both $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ and $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ contain a subflow of $X$ : if e.g., $Y_{0}$ is a flow contained in the left space, then $GY_{0}=Y_{0}$ , so $\\sigma _{*}^{-k}\\left(Y_{0}\\right)\\subseteq Y_{0}\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ , hence $Y_{0}\\subseteq \\sigma _{*}^{k}\\left(\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace \\right)=\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace $ .",
"Before the proof we note the following useful lemma.",
"Lemma 6.7 Suppose that $G$ is a topological group acting continuously on a compact metric space $\\left(X,d\\right)$ .",
"Then for every $0<\\varepsilon $ there is some open neighborhood $U$ of $\\operatorname{id}\\in G$ such that for every $g,h\\in G$ if $gh^{-1}\\in U$ then for all $x\\in X$ we have that $d\\left(gx,hx\\right)<\\varepsilon $ .",
"It is enough to show that there is some open neighborhood $U$ of $\\operatorname{id}$ such that if $g\\in U$ then for all $x\\in X$ , $d\\left(gx,x\\right)<\\varepsilon $ (since then if $gh^{-1}\\in U$ then $d\\left(gh^{-1}\\left(hx\\right),hx\\right)<\\varepsilon $ ).",
"For every $x\\in X$ , there is some neighborhood $V_{x}$ of $x$ in $X$ and some neighborhood $U_{x}$ of $\\operatorname{id}$ in $G$ such that for all $g\\in U_{x}$ , $x^{\\prime }\\in V_{x}$ , $d\\left(gx^{\\prime },x^{\\prime }\\right)<\\varepsilon $ .",
"By compactness, a finite union of $V_{x}$ 's covers $X$ .",
"Let $U$ be the intersection of the corresponding $U_{x}$ 's.",
"Suppose that $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ witnesses that $$ is shifty.",
"Let $ G0$be a countable dense subset of $ G$, enumerated as $ gi | i<$,such that $ g0=id$.$ We construct an automorphism $\\tau :M\\rightarrow M$ by back and forth such that eventually $\\sigma _{*}=\\tau ^{-1}\\sigma \\tau $ and such that at each finite stage, $\\tau $ will be an elementary map.",
"For the construction it is actually better to think of the domain and range of $\\tau $ as two different structures, so we have $M=M_{*}$ and suppose that $\\sigma :M\\rightarrow M$ , $\\sigma _{*}:M_{*}\\rightarrow M_{*}$ and $\\tau :M_{*}\\rightarrow M$ .",
"The subscript $*$ will denote tuples from $M_{*}$ throughout.",
"Suppose that we have constructed a partial elementary map $f:A_{*}\\rightarrow A$ (that will be part of $\\tau $ eventually) with $A_{*}\\subseteq M_{*},A\\subseteq M$ finite, enumerated by $a_{*},a$ .",
"Here is the main tool in the construction.",
"Claim 6.8 Suppose that $b_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a*$ and$ b*'ba$.",
"Let $ b*=f-1(b)$.",
"Thenthere is $ k<$ and an extension $ f'$ of $ f$ such that anyautomorphism $ '$ extending $ f'$ will satisfy that for $ *'='-1'$,$ *'k(b'*)=b*$.$ Similarly, if $a_{*}\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b*'$ then there is some $ k<$ andan extension $ f'$ of $ f$ such that any automorphism $ '$ extending$ f'$ will satisfy that for $ *'='-1'$,$ *'k(b*)=b'*$.$ First, find some tuple $b^{\\prime }$ in $M$ such that $b^{\\prime }a\\equiv b_{*}^{\\prime }a_{*}$ .",
"In particular, $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.",
"By left shiftiness, there is some $ k<$such that $ -k(b)ab'ab*'a*$.Extend $ f$ to $ f'$ which sends $ b*'$ to $ -k(b)$.Then, for any $ '$ extending $ f'$, $ '-1k'(b*')='-1(b)=b*$.$ The second statement is proved similarly, using right shiftiness.",
"We will make sure that for each $n<\\omega $ , the following condition holds.",
"$\\star $ There are $k_{n,0},\\ldots ,k_{n,n-1}<\\omega $ such that for all $i<n$ , $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)\\right)\\right)<1/n$ and $k_{n,0}^{\\prime },\\ldots ,k_{n,n-1}^{\\prime }<\\omega $ such that for all $i<n$ , $d\\left(\\sigma _{*}^{-k_{n,i}^{\\prime }}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{-k_{n,0}^{\\prime }}\\left(x_{*}\\right)\\right)\\right)<1/n$ .",
"Why is $\\star $ enough?",
"Let $y_{n}=\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)$ , and let $y$ be a limit of some subsequence $\\left\\langle y_{n_{j}}\\,|\\,j<\\omega \\right\\rangle $ (which exists by compactness), then $Gy\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ (so $\\operatorname{cl}\\left(Gy\\right)$ is a subflow): given $g\\in G$ and $0<\\varepsilon $ , first find an open neighborhood $U\\subseteq G$ of $g$ such that if $h\\in U$ then $d\\left(gx,hx\\right)<\\varepsilon /4$ for all $x\\in X$ (this $U$ is given to us by Lemma REF : it is $\\left(g^{-1}V\\right)^{-1}$ where $V$ is an open neighborhood of $\\operatorname{id}$ such that if $gh^{-1}\\in V$ , $d\\left(gx,hx\\right)<\\varepsilon /4$ ).",
"Take $n$ so large that $g_{i}\\in U$ for some $i<n$ and $1/n<\\varepsilon /4$ , and find $n_{j}$ even larger so that $d\\left(g_{i}y,g_{i}y_{n_{j}}\\right)<\\varepsilon /4$ .",
"Then $d\\left(gy,g_{i}y\\right)<\\varepsilon /4$ , $d\\left(g_{i}y,g_{i}y_{n_{j}}\\right)<\\varepsilon /4$ and $d\\left(g_{i}y_{n_{j}},\\sigma _{*}^{k_{n_{j},i}}\\left(x_{*}\\right)\\right)<\\varepsilon /4$ .",
"Together, $d\\left(gy,\\sigma _{*}^{k_{n_{j},i}}\\left(x_{0}\\right)\\right)<3\\varepsilon /4<\\varepsilon $ , which means that $gy$ is in the closure.",
"Similarly, if $y^{\\prime }$ is a limit of a subsequence of $\\sigma _{*}^{-k_{n,0}^{\\prime }}\\left(x_{*}\\right)$ , then $Gy^{\\prime }\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ .",
"So we consider $f:A_{*}\\rightarrow A$ a partial elementary map.",
"Our task now is to deal with $n<\\omega $ .",
"Let $\\varepsilon =1/n$ .",
"Let $A_{*}\\subseteq C_{*}\\subseteq M_{*}$ be finite such that if $g^{-1}\\upharpoonright C_{*}=h^{-1}\\upharpoonright C_{*}$ then $d\\left(gx,hx\\right)<\\varepsilon /4$ for all $x\\in X$ and any $g,h\\in G$ (this is by Lemma REF ).",
"Let $z_{0},\\ldots ,z_{l-1}$ be such that $\\bigcup \\left\\lbrace B\\left(z_{j},\\varepsilon /4\\right)\\,|\\,j<l\\right\\rbrace $ cover $X$ , and write $B_{j}=B\\left(z_{j},\\varepsilon /4\\right)$ .",
"Let $c_{*}$ be a finite tuple enumerating $C_{*}$ .",
"For every $c_{*}^{\\prime }\\equiv c_{*}$ , we say that $c_{*}^{\\prime }$ has color $j<l$ if $j$ is least such that there is $g\\in G$ such that $g\\left(c_{*}^{\\prime }\\right)=c_{*}$ and $gx_{*}\\in B_{j}$ .",
"Note that by the choice of $c_{*}$ , if $g^{\\prime }\\left(c_{*}^{\\prime }\\right)=c_{*}$ then $g^{\\prime }g^{-1}\\upharpoonright C_{*}=\\operatorname{id}$ , so $g^{\\prime }x_{0}\\in B\\left(z_{j},\\varepsilon /2\\right)$ .",
"Let $D_{*}=\\bigcup \\left\\lbrace g_{i}^{-1}\\left(C_{*}\\right)\\,|\\,i<n\\right\\rbrace $ .",
"Note that $C_{*}\\subseteq D_{*}$ because $g_{0}=\\operatorname{id}$ .",
"Let $d_{*}$ enumerate $D_{*}$ .",
"For any $d_{*}^{\\prime }\\equiv d_{*}$ and $s\\subseteq l$ , we say that $d_{*}^{\\prime }$ has color $s$ if $\\left\\lbrace j<l\\,|\\,c_{*}^{\\prime }\\equiv c_{*},c_{*}^{\\prime }\\subseteq d_{*}^{\\prime },c_{*}^{\\prime }\\text{ has color }j\\right\\rbrace =s$ .",
"By left existence, there is some $s_{0}\\subseteq l$ such that for every finite set $S\\subseteq M_{*}$ , there is some $d_{*}^{\\prime }\\equiv d_{*}$ with $d_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $S$ and $ d*'$ has color $ s0$.$ Let $d_{*}^{\\prime }\\equiv d_{*}$ be of color $s_{0}$ such that $d_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $d*$.For $ i<n$, let $ c*,i'd*'$ be the tuple correspondingto $ gi-1(c*)$, so in particular $ c*,i'c*$.Let $ ji<l$ be the color of $ c*,i'$.",
"By the choice of $ s0$,for every finite set $ SM*$, there is some $ c*'c*$such that $ c'*$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $S$ and $ c*'$ has color $ ji$.$ Since $M$ is homogeneous we can extend $f$ in such a way so that its domain equals $D_{*}$ .",
"By Claim REF (the first part), there is some $k_{n,0}<\\omega $ and an extension $f_{0}$ of $f$ that ensures that $\\sigma _{*}^{k_{n,0}}\\left(d_{*}^{\\prime }\\right)=d_{*}$ .",
"Starting with $f_{0}$ , we construct an increasing sequence $\\left\\langle f_{i}\\,|\\,i<n\\right\\rangle $ as follows.",
"Suppose we have $f_{i}$ whose domain is $D_{*,i}$ .",
"Find some $c^{\\prime \\prime }_{*,i+1}\\equiv c_{*}$ of color $j_{i+1}$ such that $c_{*,i+1}^{\\prime \\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $D*,i$.By Claim \\ref {claim:Using shiftiness}, we can find $ kn,i+1<$and extend $ fi$ to $ fi+1$ which ensures that $ *kn,i+1(c*,i+1”)=c*$.$ Now we have the first part of $\\star $ : we need to check that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)\\right)\\right)<\\varepsilon $ for all $i<n$ .",
"For $i=0$ this is clear since $g_{0}=\\operatorname{id}$ , so we may assume that $i>0$ .",
"As $\\sigma _{*}^{k_{n,i}}\\left(c_{*,i}^{\\prime \\prime }\\right)=c_{*}$ , it follows that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{0}\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .",
"Similarly, as $\\sigma _{*}^{k_{n,0}}\\left(c_{*,i}^{\\prime }\\right)=g_{i}^{-1}\\left(c_{*}\\right)$ , we have that $d\\left(g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{0}\\right)\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .",
"Together, we are done.",
"Now we have to take care of the other half of $\\star $ .",
"This is done similarly, using right existence and the second part of Claim REF .",
"The following proposition explains why we needed to take a conjugate of $\\sigma $ .",
"The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.",
"In Section REF , we mentioned that it is a Ramsey structure.",
"Note that the underlying order is dense (by Proposition REF ).",
"Proposition 6.9 Let $M=\\left(V,<,R\\right)$ be the countable ordered random graph.",
"Then there is no automorphism $\\sigma \\in G=\\operatorname{Aut}\\left(M\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\in X$ .",
"First we find $a\\ne b$ in $M$ such that $\\sigma ^{n}\\left(a\\right)\\ne \\sigma ^{m}\\left(b\\right)$ for all $m,n\\in \\mathbb {Z}$ .",
"To do that, take any $a\\in M$ .",
"Then $\\left\\lbrace \\sigma ^{n}\\left(a\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is discrete (in the order sense: it is either a $\\mathbb {Z}$ -chain or just $a$ ).",
"Since $\\left(V,<\\right)$ is dense, there is some $b\\ne \\sigma ^{n}\\left(a\\right)$ for all $n\\in \\mathbb {Z}$ .",
"It follows that $b$ is as required.",
"Let $X=S_{x}\\left(M\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).",
"Let $p\\in X$ be any completion of the partial type $\\left\\lbrace R\\left(x,\\sigma ^{n}\\left(a\\right)\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace \\cup \\left\\lbrace \\lnot R\\left(x,\\sigma ^{m}\\left(b\\right)\\right)\\,|\\,m\\in \\mathbb {Z}\\right\\rbrace $ .",
"Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\in \\operatorname{cl}\\left\\lbrace \\sigma ^{n}\\left(p\\right)\\,|\\,n<\\omega \\right\\rbrace $ which is a fixed point of $G$ .",
"In other words, $p_{0}$ is an invariant type over $M$ .",
"However $R\\left(x,a\\right)\\wedge \\lnot R\\left(x,b\\right)\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).",
"The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.",
"If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\left(X,x_{0}\\right)$ .",
"Thus, there would be a conjugate $\\sigma _{*}$ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow.",
"But then if $\\left(Y,y_{0}\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\pi :X\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .",
"Thus, $\\pi $ maps $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ to $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(y\\right)\\,|\\,n<\\omega \\right\\rbrace $ , and the latter contains a $G$ -subflow.",
"Thus we get that $\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .",
"Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\left\\lbrace <,\\wedge \\right\\rbrace $ , and let $M\\models T$ be countable.",
"Then $\\operatorname{Aut}\\left(M\\right)$ has no shifty automorphism.",
"In particular, $M$ has no CIR.",
"Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .",
"Suppose that $\\sigma $ was shifty.",
"Let $\\bar{m}=\\left\\langle m\\,|\\,m\\in M\\right\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\bar{x}=\\left\\langle x_{m}\\,|\\,m\\in M\\right\\rangle $ .",
"Let $X=S_{\\bar{m}}\\left(M\\right)$ be the space of $\\bar{x}$ -complete types $p$ over $M$ such that $p\\upharpoonright \\emptyset =\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .",
"Then $X$ is a compact metric space.",
"Let $x_{*}=\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .",
"By Theorem REF , there is some conjugate $\\tau $ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\tau ^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{+}\\subseteq X$ and similarly, $\\operatorname{cl}\\left\\lbrace \\tau ^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{-}$ .",
"By Proposition REF , $\\tau $ fixes a branch or a point.",
"Suppose that $\\tau \\left(m\\right)=m$ for some $m\\in M$ .",
"Then for every $p\\in Y^{+}$ , $p\\models x_{m}=m$ .",
"However $G=\\operatorname{Aut}\\left(M\\right)$ acts transitively on $M$ , so we have a contradiction.",
"Now suppose that $\\tau $ fixes a branch $B\\subseteq M$ , but does not fix any point.",
"Suppose that $\\tau \\left(m\\right)>m$ for some $m\\in B$ .",
"Then $\\tau ^{n}\\left(m\\right)>m$ for all $n<\\omega $ , so for any $p\\in Y^{+}$ , $p\\models x_{m}>m$ .",
"There is some $m^{\\prime }\\in M$ such that $m^{\\prime }>m$ and $m^{\\prime }\\notin B$ .",
"Since $m<\\tau ^{n}\\left(m\\right)\\in B$ for all $n<\\omega $ , it follows that $p\\models x_{m}\\wedge m^{\\prime }=m$ for all $p\\in Y^{+}$ .",
"Let $\\tau ^{\\prime }\\in G$ fix $m$ and map $m^{\\prime }$ to $B$ .",
"Then $\\tau ^{\\prime }\\left(p\\right)\\models \\left(x_{m}\\wedge \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\right)=m<x_{m}$ .",
"But $\\tau ^{\\prime }\\left(p\\right)\\in Y^{+}$ , so $\\tau ^{\\prime }\\left(p\\right)\\models x_{m}\\le \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\vee \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\le x_{m}$ , which is a contradiction.",
"If, on the other hand $\\tau \\left(m\\right)<m$ , then $\\tau ^{-1}\\left(m\\right)>m$ , so we can apply the same argument to $Y^{-}$ .",
"For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .",
"In addition, letting $H=\\operatorname{Aut}\\left(N\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).",
"In addition, if $B\\subseteq N$ is a branch, $m\\in B$ , there is always some $m^{\\prime }>m$ , $m^{\\prime }\\notin B$ and for any $n^{\\prime }>m$ in $B$ , $m^{\\prime }m\\equiv n^{\\prime }m$ .",
"Hence, we can apply Proposition REF and the same proof will work.",
"Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\omega $ -categorical structures.",
"We state here a few general conjectures and questions.",
"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.",
"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.",
"Conjecture 7.1 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion which admits a CIR.",
"Suppose that $M$ is a structure and $\\mathfrak {C}$ a monster model for $\\operatorname{Th}\\left(M\\right)$ .",
"The group of Lascar strong automorphisms of $M$, denoted by $\\operatorname{Aut}f\\left(M\\right)$ is the group of automorphisms of $M$ generated by the set $\\left\\lbrace \\sigma \\upharpoonright M\\,|\\,\\exists N\\prec \\mathfrak {C},\\left|N\\right|=\\left|T\\right|,\\sigma \\upharpoonright N=\\operatorname{id}\\right\\rbrace $ .",
"If $\\sigma $ is Lascar strong, then $\\sigma \\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{id}$ so $\\operatorname{Aut}f\\left(M\\right)$ is contained in $G^{0}$ .",
"However, there are examples (even $\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\operatorname{Aut}f\\left(M\\right)$ , see [16], [31].",
"The Lascar group of $M$ is the quotient $\\operatorname{Aut}\\left(M\\right)/\\operatorname{Aut}f\\left(M\\right)$ .",
"For more on the Lascar group, see [39].",
"In the $\\omega $ -categorical case, the quotient $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is also called the compact Lascar group.",
"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$.",
"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.",
"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.",
"Forinstance, by Lemma \\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\operatorname{Aut}\\left(M\\right)$ has no compact quotients then it has finite topological rank.",
"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.",
"Thus, potentially, the Lascar group — as a quotient of $\\operatorname{Aut}\\left(M\\right)$ — can be an obstruction to having finite topological rank.",
"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.",
"Conjecture REF (together with Theorem REF ) implies that it could not.",
"Conjecture 7.2 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion with trivial Lascar group.",
"By the above, this second conjecture is implied by Conjecture REF .",
"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.",
"It would be interesting to investigate other consequences of having a CIR.",
"For instance a CIR might have something to say about normal subgroups.",
"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ for some CIR $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\aleph _{0}}$ is open.",
"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.",
"We refer to [26] for a survey on this.",
"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.",
"In fact the compact quotients are hidden in the stabilizer of a finite set.",
"It seems that one can avoid this counterexample by restricting to dense subgroups.",
"This leads us to the following questions.",
"Question 7.3 Let $M$ be $\\omega $ -categorical such that $G=\\operatorname{Aut}\\left(M\\right)$ has no compact quotient.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open (and hence is equal to $G$ )?",
"Note that the assumption of having no compact quotient is necessary.",
"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\operatorname{Aut}\\left(M\\right)$ has a dense subgroup of index 2, see [23].",
"Question 7.4 Let $M$ be $\\omega $ -categorical and $N$ an $\\omega $ -categorical expansion of $M$ .",
"Set $G=\\operatorname{Aut}\\left(M\\right)$ and $H=\\operatorname{Aut}\\left(N\\right)\\le G$ .",
"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open?",
"Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.",
"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.",
"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].",
"We would also like to thank Daoud Siniora for his comments.",
"Finally, we would like to thank the anonymous referee for his comments."
],
[
"Examples of theories with a canonical independence relation",
"There are many examples of countable ultrahomogeneous structures with a CIR.",
"Here we will give some of them.",
"We will define the relation $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$, but sometimes leave most of the details of checking thatit satisfies the axioms to the reader.",
"All the CIRs we define aredefined on finitely generated substructures.\\begin{example}The most trivial ultrahomogeneous structureis of course the structure with universe \\omega and no relationsbut equality.",
"Its automorphism group is S_{\\infty }.",
"For finitesets A,B,C define A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{example}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ by $ ABC$.",
"Thisis a CIR.$ Example 4.1 If $T$ is stable and $\\emptyset $ is a base (i.e., $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ so that every type over $\\emptyset $ has a unique non-forking extension), then $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $f$ (i.e., non-forking independence) is canonical.$ Example 4.2 Let $\\left(\\mathcal {B},<,\\wedge ,\\vee ,0,1,\\square ^{c}\\right)$ be the atomless Boolean algebra.",
"For finite sets $A,B,C$ , define $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff$ C=ACBC$and for every atom $ aAC$ and everyatom $ bBC$, if there is an atom $ cC$such that $ a,bc$ then $ ab0$.",
"Let us show that $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$is a (symmetric) CIR.$ Stationarity over $\\emptyset $ : $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ says that the atoms in$ AB$ are in bijection with (atoms of $ A$)$$ (atoms of $ B$).",
"Thus, if $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B'$ and $ f:AA'$,$ g:BB'$are isomorphisms, then $ h:ABA'B'$taking an atom $ ab$ to $ f(a)g(b)$is an isomorphism.",
"This easily implies stationarity.$ Transitivity: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CDB$ and $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, and wewant to show that $ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"Suppose that $ aACD$,$ bBC$ are atoms, and $ cC$is an atom such that $ a,bc$.",
"Let $ dDC$be an atom such that $ adc$.",
"Then $ db0$.",
"Let$ b'db$ be an atom of $ BCD$,so that $ a,b'd$.",
"Hence $ ab'0$ and thus $ ab0$.Transitivity to the right follows by symmetry.$ Extension: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and $ d$ is given.",
"We want tofind $ d'BCd$ such that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CBd$.",
"We may assume that$ dBC$.",
"An atom in $ BCd$has the form $ db$ or $ dcb$ for some atom $ b$ of$ BC$.",
"The type $ tp(d/BC)$is determined by knowing which of these terms $ db,dcb$is nonzero (for $ bBC$ an atom).",
"As$ B$ is atomless and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, we can find some $ d'$ suchthat for every $ a,b,c$ atoms in $ AC,BC,C$respectively such that $ a,bc$, if $ db0$ then $ ad'b0$(else $ d'b=0$) and if $ dcb0$ then $ a(d')cb0$(else $ (d')cb=0$).",
"In addition, we ask thatif $ db,dcb0$ then both $ ad'b$and $ a(d')cb$ are not in $ ABC$.It now follows that $ ACBCd'=C$:suppose that $ eACBCd'$.Then as $ eBCd'$, it can be writtenas $ b0(b1d')(b2(d')c)$where $ b0,b1,b2BC$ are pairwisedisjoint and for every atom $ b'b1b2$ from $ BC$,$ d'b',(d')cb'0$.",
"If both $ b1,b2=0$,then $ eBCAC$so $ eC$.",
"If $ b10$, let $ b1'b1$be an atom of $ BC$.",
"So $ eb1'ABC$(because $ eAC$) and has the form $ b1'd'$.Let $ aAC$ and $ cC$be atoms such that $ a,b1'c$.",
"Then $ eb1'aABC$and has the form $ ab1'd'$ which is not in $ ABC$by construction, contradiction.",
"Similarly $ b2=0$ and we are done.$ Existence and monotonicity are clear.",
"Example 4.3 Let $\\left(M,R\\right)$ be the random tournament (a tournament is a complete directed graph such that for all $x,y$ , it cannot be that both $R\\left(x,y\\right)$ and $R\\left(y,x\\right)$ , and the random tournament is the Fraïssé limit of the class of finite tournaments).",
"Given finite sets $A,B,C$ , write $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ AB=C$ andif $ aAC$, $ bBC$ then $ R(a,b)$.This easily satisfies all the requirements.$ Remark 4.4 The following definition is from [19].",
"Let $K$ be a class of finite $L$ -structures where $L$ is a relational language.",
"Then $K$ has the strong$^{+}$ amalgamation property if for all $A,B,C\\in K$ with $A\\subseteq B,C$ there is $D\\in K$ such that $D=C^{\\prime }\\cup B^{\\prime }$ with $C^{\\prime }\\cong _{A}C$ , $B^{\\prime }\\cong _{A}B$ and for every $n$ -ary relation $R$ and every $x_{1},\\ldots ,x_{n}$ and $y_{1},\\ldots ,y_{n}$ from $D\\backslash A$ which intersect both $C^{\\prime }$ and $B^{\\prime }$ , if [$x_{i}\\in B^{\\prime }$ iff $y_{i}\\in B^{\\prime }$ for all $1\\le i\\le n$ ], then $R^{D}\\left(x_{1},\\ldots ,x_{n}\\right)$ iff $R^{D}\\left(y_{1},\\ldots ,y_{n}\\right)$ (where $R^{D}$ is the interpretation of $R$ in $D$ ).",
"For example, the random tournament satisfies this property.",
"In [19] it is proved that if $M$ is ultrahomogeneous and $\\operatorname{Age}\\left(M\\right)$ has the strong$^{+}$ amalgamation property, then $\\operatorname{Aut}\\left(M\\right)$ has a cyclically dense conjugacy class.",
"They prove it using a condition they denote by $(\\Delta _{n})$ , see there, Theorem 5.12.",
"We do not know if this condition implies the existence of a CIR."
],
[
"Free amalgamation classes",
"In [38], [27], [4] there is an axiomatic framework for defining an abstract ternary relation close to our CIR.",
"More precisely, in [27] and [38], the notion of a stationary independence relation (SIR) is introduced (in [27] for finitely generated structures and in [38] for sets in general).",
"A similar notion is defined in [4], with more axioms.",
"In any case, all these notions imply ours, except perhaps that stationarity over $\\emptyset $ becomes stationarity over $\\operatorname{acl}\\left(\\emptyset \\right)$ , so that this becomes a CIR in the expansion $\\left(M,\\operatorname{acl}\\left(\\emptyset \\right)\\right)$ (note that our extension follows from full stationarity and their version of existence).",
"Thus, we can apply our results to the examples studied there.",
"In particular, we get the following examples.",
"Example 4.5 The rational Urysohn space $\\mathbb {Q}\\mathbb {U}$ is the Fraïssé limit of the class of finite metric spaces with rational distances.",
"Pick a point $q\\in \\mathbb {Q}\\mathbb {U}$ , and consider the structure $\\left(\\mathbb {Q}\\mathbb {U},q\\right)$ where we add a constant for $q$ .",
"In [37], it is proved that the relation $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ which holds for finite $ A,B,C$ ifffor every $ aAC$, $ bBC$, $ d(a,b)={ d(a,c)+d(c,b) | cC} $is a CIR in $ (QU,q)$.$ In all examples given by Conant [4] which we list now, $\\operatorname{acl}\\left(\\emptyset \\right)=\\emptyset $ , so we actually get a CIR in the structure (i.e., no need to take an expansion) by [4].",
"Example 4.6 Fraïssé limits with free amalgamation: suppose that $L$ is a relational language and $K$ is an essentially countable (see above Fact REF ) class of finite $L$ -structures, such that if $A,B,C\\in K$ and $A\\subseteq C,B$ , then the free amalgam of $A,B,C$ is in $K$ (i.e., a structure $D=C^{\\prime }\\cup B^{\\prime }$ with $C^{\\prime }\\cong _{A}C$ , $B^{\\prime }\\cong _{A}B$ , $B^{\\prime }\\cap C^{\\prime }\\subseteq A$ and for every tuple $a\\in D$ in the length of some relation $R\\in L$ , if $R\\left(a\\right)$ then $a\\in C^{\\prime }$ or $a\\in B^{\\prime }$ ) (here we also include the case $A=\\emptyset $ ).",
"Let $M$ be the Fraïssé limit of $K$ , and define $B\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $AC$ iff $ ABC$ is the free amalgam of$ A,AB,AC$.",
"If the language is finite or more generally in the contextof Fact \\ref {fact:Fraisse limits} (2), it is easy to see that inthis case $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR (this is also proved, for finite languages,in \\cite [Proposition 3.4]{MR3663421}).$ This class of examples contain e.g., the random graph, the universal $K_{n}$ -free graph (the Henson graph), and their hypergraph analogs.",
"Example 4.7 Let $L=\\left\\lbrace P_{n}\\,|\\,n<\\omega \\right\\rbrace $ and let $K$ be the class of finite $L$ -structures in which $P_{n}\\left(x_{0},\\ldots ,x_{n-1}\\right)$ implies that $x_{i}\\ne x_{j}$ for $i\\ne j$ .",
"Then $K$ is essentially countable and is a free amalgamation class, and moreover for each $n<\\omega $ there are finitely many isomorphism types of structures of size $n$ (so we can use Fact REF ).",
"Let $M$ be the limit.",
"Now recall the example $N$ described in Remark REF , with infinitely many independent equivalence relations with two classes.",
"Then $M$ is $N$ expanded by naming the classes.",
"In other words, $\\operatorname{Aut}\\left(M\\right)=\\operatorname{Aut}\\left(N\\right)^{0}$ .",
"Example 4.8 In [5] the authors describe a generic $K_{n}+K_{3}$ -free graph, where $K_{n}+K_{3}$ is the free amalgam of the complete graph on $n$ vertices and a triangle over a single vertex.",
"This structure is $\\aleph _{0}$ -categorical, with $\\operatorname{acl}\\left(\\emptyset \\right)=\\emptyset $ .",
"By [4] there is a CIR on this graph.",
"Example 4.9 $\\omega $ -categorical Hrushovski constructions.",
"Let $L$ be a finite relational language, and let $f:\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}_{\\ge 0}$ be a “control function”.",
"According to [10], there is an $\\omega $ -categorical generic Hrushovski construction $M_{f}$ for a “free amalgamation class” $K_{f}$ if $f$ satisfies certain conditions.",
"It follows from [4] that given extra conditions on the algebraic closure, $M_{f}$ admits a CIR.",
"See more details in [4], [10]."
],
[
"Ultrahomogenous partial orders",
"In [11] the authors prove that the automorphism group of every ultrahomogeneous poset (partially order set) is topologically 2-generated.",
"They also characterize when they have a cyclically dense conjugacy class.",
"We can find such a conjugacy class by finding a CIR whenever possible.",
"We should remark that they prove more on the automorphism groups of those structures.",
"By [32] there are four types of ultrahomogeneous posets.",
"Fact 4.10 [32] Suppose that $\\left(H,<\\right)$ is an ultrahomogeneous poset.",
"Then $H$ is isomorphic to one of the following: The random poset: the Fraïssé limit of the class of finite partial orders.",
"The orders $\\mathcal {A}_{n}$ for $1\\le n\\le \\omega $ : $\\left(n,<\\right)$ where $<$ is trivial i.e., empty.",
"The orders $\\mathcal {B}_{n}$ for $1\\le n\\le \\omega $ : $\\left(n\\times \\mathbb {Q},<\\right)$ where $\\left(k,q\\right)<\\left(m,p\\right)$ iff $k=m$ and $p<q$ .",
"The orders $\\mathcal {C}_{n}$ for $1\\le n\\le \\omega $ : $\\left(n\\times \\mathbb {Q},<\\right)$ where $\\left(k,q\\right)<\\left(m,p\\right)$ iff $q<p$ .",
"Note that the orders $\\mathcal {A}_{n}$ have $S_{n}$ as their automorphism group, and thus for $n$ finite cannot have a dense conjugacy class.",
"For $n=\\omega $ , this is Example .",
"Also, the orders $\\mathcal {B}_{n}$ for $1<n<\\omega $ cannot have a dense conjugacy class by Remark REF : $S_{n}$ is a quotient of the automorphism group (define $a\\mathrel {E}b$ iff $a$ and $b$ are comparable, and note that there are $n$ equivalence classes, every permutation of which is induced by an automorphism)."
],
[
"The random poset",
"Suppose that $\\left(\\mathcal {D},\\le \\right)$ is the random partial order.",
"For finite sets $A,B,C$ define $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ ABC$and if $ aA,bB$ then $ a$ is comparable with $ b$ iff forsome $ cC$, $ acb$ or $ bca$.",
"Then $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$is a (symmetric) CIR.",
"We will show only transitivity and extension,and leave the rest to the reader.",
"Suppose that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CDB$ and$ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"Given $ aA,bB$, such that $ ab$, theremust be some $ dCD$ such that $ adb$.",
"Hence there mustbe some $ cC$ with $ dcb$.",
"Together we are done.",
"Extension:suppose that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and we are given $ d$.",
"Assume $ dBC$(otherwise we are done).",
"Then let $ d'BCd$ be such that $ d'ABC$and for all $ aA$, $ ad'$ iff for some $ cC$, $ acd$,and similarly define when $ d'a$.",
"Now, $ (ABCd',)$is a poset since $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.$"
],
[
"The orders $\\protect \\mathcal {B}_{1}$ and {{formula:06824f81-37b5-4b61-93af-3b2091afec21}} . ",
"Example 4.11 For $\\left(\\mathbb {Q},<\\right)$ (which is $\\mathcal {B}_{1}$ ), for every finite $A,B,C\\subseteq \\mathbb {Q}$ , we let $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ if $ ABC$and for all $ aAC$ and $ bBC$ such that$ aCb$, $ a<b$.",
"Then $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR.",
"We prove only transitivityand leave the rest to the reader.$ Suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DCB$ and $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"We have to show that$ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, which amounts to showing that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"Fix $ aAC$and $ bBC$ such that $ aCb$.",
"We have to showthat $ a<b$.",
"Note that $ bD$.",
"If $ aD$ then this is trueby our assumption.",
"Otherwise, $ aCD$.",
"If $ aCDb$ thenwe are done.",
"Otherwise, $ a,b$ have different cuts over $ CD$.",
"Butsince they realize the same cut over $ C$, it follows that there issome $ dDC$ such that either $ a<d<b$ or $ b<d<a$.The former would imply what we want, so assume that $ b<d<a$.",
"Butthen $ bCd$ so $ d<b$ —{} a contradiction.",
"Theother direction of transitivity is proved similarly.$ Example 4.12 Consider $\\mathcal {B}_{\\omega }$ .",
"Then each equivalence class of the relation $E$ of being comparable is a DLO, and thus by Example REF , for each $n<\\omega $ , there is a CIR $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $F$ defined as in Example\\ref {exa:DLO has groovy} for each $ E$-class $ F$.",
"For $ (B,<)$and finite sets $ A,B,C$, define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ ABC$,$ A/EB/EC/E$(if $ aA$, $ bB$ and $ aEb$then there is some $ cC$ such that $ aEc$) and for every$ E$-class $ F$, $ AF$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CFFBF$.",
"This is easilyseen to be a CIR.",
"Note that we need infinitely many classes for extension.$"
],
[
"The orders $\\protect \\mathcal {C}_{n}$ for {{formula:83c96725-a83c-4805-a7be-bdf9c4a5c04c}} . ",
"In $\\mathcal {C}_{n}$ we have an equivalence relation $E$ , defined by $a\\mathrel {E}b$ iff $a$ and $b$ are incomparable (they have the same second coordinate).",
"Then $\\mathcal {C}_{n}/E\\models DLO$ , so we have a CIR $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $E$ defined on it by Example \\ref {exa:DLO has groovy}.For finite $ A,B,CCn$, define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ A/E$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C/EEB/E$.This trivially satisfies all the properties.$"
],
[
"Ultrahomogeneous graphs",
"In [17], the authors prove that for every ultrahomogeneous graph $\\Gamma =\\left(V,E\\right)$ , $\\operatorname{Aut}\\left(\\Gamma \\right)$ is topologically 2-generated.",
"Similarly to the poset case, we can recover some results by finding a CIR whenever possible.",
"By [24] we have the following classification of ultrahomogeneous graphs.",
"Recall that for a graph $\\left(V,E\\right)$ , its dual is $\\left(V,E^{\\prime }\\right)$ where $E^{\\prime }=\\left[V\\right]^{2}\\backslash E$ .",
"Fact 4.13 [24] Any countable ultrahomogeneous graph $\\Gamma $ is isomorphic to one of the following graphs, or its dual.",
"The random graph.",
"For $n\\ge 3$ , the Henson graph, i.e., the $K_{n}$ -free universal graph (the Fraïssé limit of the class of $K_{n}$ -free finite graphs).",
"For any $1\\le n\\le \\omega $ , the graph $\\omega K_{n}$ consisting of a disjoint union of countably many copies of $K_{n}$ .",
"For any $2\\le n<\\omega $ , the graph $nK_{\\omega }$ consisting of a disjoint union of $n$ copies of $K_{\\omega }$ (the complete graph on $\\omega $ ).",
"Note that the dual of a graph has the same automorphism group, so we can ignore the duals.",
"We already saw in Example REF that both the random graph and the Henson graph have a CIR.",
"The graphs $nK_{\\omega }$ for $n<\\omega $ cannot have a a dense conjugacy class by Remark REF as in the case of the posets $\\mathcal {B}_{n}$ described above.",
"However, $\\omega K_{n}$ for $1\\le n\\le \\omega $ has a CIR, just like the cases $\\mathcal {C}_{n}$ above."
],
[
"A mix of two Fraïsé limits with\nCIRs",
"Suppose we are in the situation of Section REF : we have two amalgamation classes $K_{1},K_{2}$ with all the properties listed there.",
"Let $M_{1},M_{2}$ be the Fraïsé limits of $K_{1},K_{2}$ respectively, and let $M$ be the Fraïsé limit of $K$ , the class of finite $L_{1}\\cup L_{2}$ -structures $A$ such that $A\\upharpoonright L_{1}\\in K_{1}$ and $A\\upharpoonright L_{2}\\in K_{2}$ .",
"Add the extra assumption that $L_{1}\\cap L_{2}=\\emptyset $ .",
"Suppose that $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $1,$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $2$ areCIRs on $ M1,M2$ respectively.",
"By Proposition \\ref {prop:restriction of a mix of two Faisse},we may assume that $ M1=ML1$ and $ M2=ML2$.For finite subsets of $ M$, define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C1B$and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C2B$.\\begin{prop}The relation \\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{prop}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR.$ Stationarity follows from the fact that by quantifier elimination, for any finite tuples $a,a^{\\prime }$ from $M$ , if $a\\equiv a^{\\prime }$ in $L_{1}$ and in $L_{2}$ , then $a\\equiv a^{\\prime }$ in $L_{1}\\cup L_{2}$ .",
"Extension: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, and we are given $ dM$.Let $ d1M1$ be such that $ d1BCd$ in $ L1$and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C1Bd1$.",
"Similarly find $ d2$ for $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $2$and $ L2$.",
"Consider the finite structure $ D$ with universe $ ABCd$where the $ L1L2$-structure on $ ABC$ is as in $ M$, andsuch that its restriction to $ L1$, $ L2$ is $ ABCd1$, $ ABCd2$,respectively.",
"This structure exists since $ L1L2=$and by the assumptions of Section \\ref {subsec:A-mix-of}, both languagesare relational.",
"Thus, $ DK$, so it has an isomorphic copy $ D'M$containing copies $ A',B',C',d'$ of $ A,B,C,d$.",
"As $ M$ is ultrahomogeneous,we can apply an automorphism $$ mapping $ A'B'C'$ to $ ABC$,so that $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB(d')$, and $ (d')BCd$.$ The other properties are easy to check.",
"Example 4.14 The ordered random graph $M=\\left(V,R,<\\right)$ .",
"It is the Fraïsé limit of the class of finite linearly ordered graphs in the language $\\left\\lbrace <,R\\right\\rbrace $ .",
"It easily satisfies all our assumptions with $L_{1}=\\left\\lbrace <\\right\\rbrace $ , $K_{1}$ the class of finite linear orders and $L_{2}=\\left\\lbrace R\\right\\rbrace $ , $K_{2}$ the class of finite graphs.",
"It has a CIR as both $\\left(M,<\\right)$ (which is a DLO) and $\\left(M,R\\right)$ (the random graph) have CIRs by the two previous subsections.",
"Similarly we may define the random ordered hypergraph, and it too has a CIR."
],
[
"Trees",
"The theory $T_{dt}$ (see Example REF ) does not admit a canonical independence relation.",
"We shall give a precise (and stronger) argument for this below in Corollary REF , but it is easy to see that a natural candidate fails.",
"Namely, one can try to define $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ in such a way that if $ C=$and $ a,b$ are singletons then $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$ iff $ ab<a,b$, andfor $ a,b,c$ such that $ c$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$, then $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $cb$ iff $ ac<cb$.But then $ ac$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $cb$, $ c$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$ but $ ac$\\displaystyle ㈶$$\\displaystyle \\mid $$\\displaystyle \\smile $$\\textstyle ㈶$$\\textstyle \\mid $$\\textstyle \\smile $$\\scriptstyle ㈶$$\\scriptstyle \\mid $$\\scriptstyle \\smile $$\\scriptscriptstyle ㈶$$\\scriptscriptstyle \\mid $$\\scriptscriptstyle \\smile $b$,so transitivity fails.$ However, we can expand it in such a way that it does.",
"We give two such expansions.",
"Example 4.15 Let $L_{dt}^{B}=\\left\\lbrace <,P,f,\\wedge \\right\\rbrace $ where $P$ is a unary predicate and $f$ is a unary function symbol, and let $T_{dt}^{B}$ be the model completion of the universal $L_{dt}^{B}$ -theory of trees where $P$ is a downwards closed linearly ordered subset and $f\\left(x\\right)$ is the maximal element in $P$ which is $\\le x$ .",
"In other words, $T_{dt}^{B}$ is the theory of the Fraïsé limit of the class of finite $L_{dt}^{B}$ -structures $M$ where $M\\upharpoonright \\left\\lbrace \\wedge ,<\\right\\rbrace $ is a tree with a meet function, $P^{M}$ is linearly ordered and downwards closed and $f\\left(x\\right)=\\max \\left\\lbrace y\\le x\\,|\\,y\\in P\\right\\rbrace $ (note that this class has JEP and AP).",
"Then $T_{dt}^{B}$ is the theory of dense trees with a predicate for a branch (a maximal chain), it has quantifier elimination and is $\\omega $ -categorical.",
"Let us see why $P$ is a maximal chain in every model $M\\models T_{dt}^{B}$ .",
"Of course it is downwards closed by definition, so if $a\\in M$ is comparable with $P$ but $a\\notin P$ , then $a>P$ .",
"As $T_{dt}^{B}$ is model-complete, $M$ is existentially closed (see Fact REF ) so there is some $b\\in P$ (from $M)$ such that $f\\left(a\\right)<b$ .",
"Thus, $a>b>f\\left(a\\right)$ which is a contradiction to the definition of $f$ .",
"For three sets $A,B,C$ , let $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ ACBCC$and for all $ aAC$ with $ f(a)C$and $ bBC$ with $ f(b)C$such that $ f(a)Cf(b)$ (which is thesame as $ f(a)f(C)f(b)$),$ f(a)<f(b)$.",
"Then $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is canonical.",
"Theonly nontrivial axioms to check are stationarity over $$,extension and transitivity.$ Suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$.",
"This just says that that $ B$ is placed above$ A$ with respect to the branch $ P$ (i.e., $ f(a)<f(b)$for all $ aA,bB$).",
"So $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is stationary by quantifierelimination.$ Extension: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and we are given a single element$ d$, which we may assume is not in $ BC$and even that $ f(d)BC$.First find some $ d”$ such that $ d”BCf(d)$and $ d”$ is greater than every $ f(a)$ such that $ aAC$and $ f(a)Cf(d)$.",
"Then find $ d'$ suchthat $ d'BCd$ and $ f(d')=d”$ (and $ ACBCd'=C$).$ Transitivity: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DCB$ and $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and wehave to show that $ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"Suppose that $ aADC,f(a)C$and $ bBC,f(b)C$are such that $ f(a)Cf(b)$ but $ f(b)f(a)$.Then there must be some $ dDC$ suchthat $ f(b)df(a)$, as otherwise $ f(a)CDf(b)$.But since $ d$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$, $ f(b)<d$, which implies that$ f(b)$ and $ d$ do not have the same type over $ C$, sothere must be some $ cC$ between them, and in particular, it contradictsour assumption that $ f(a)Cf(b)$.",
"Theother direction of transitivity is proved similarly.$ Example 4.16 Let $L_{dt}^{p}=\\left\\lbrace <,p,\\wedge \\right\\rbrace $ where $p$ is a new constant.",
"Let $T_{dt}^{p}$ be the unique completion of $T_{dt}$ to $L_{dt}^{p}$ .",
"Let $M\\models T_{dt}^{p}$ be the unique countable model.",
"To simplify notation, we identify $p$ with $p^{M}$ .",
"For three sets $A,B,C$ , we let $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ iff $ ABBCC$and:\\begin{enumerate}\\item For all a\\in \\left\\langle AC\\right\\rangle with a\\wedge p\\notin \\left\\langle C\\right\\rangle and b\\in \\left\\langle BC\\right\\rangle with b\\wedge p\\notin \\left\\langle C\\right\\rangle such that a\\wedge p\\equiv _{C}b\\wedge p, a\\wedge p<b\\wedge p.\\item For all a\\in \\left\\langle AC\\right\\rangle such that a>p withno c\\in \\left\\langle C\\right\\rangle such that a\\wedge c>p, andall b\\in \\left\\langle BC\\right\\rangle with b>p and no c\\in \\left\\langle C\\right\\rangle such that b\\wedge c>p, a\\wedge b=p.\\end{enumerate}Then $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is canonical.",
"The only nontrivial axioms to check arestationarity over $$, extension and transitivity.$ It is stationary over $\\emptyset $ by elimination of quantifiers, since $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ iff $ A'={ aA | ap<p} $ is placed below$ B'={ bB | bp<p} $ with respect to the points below $ p$while $ A”={ aA | ap} $ and $ B”={ bB | bp} $are placed independently above $ p$.$ Extension: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and we are given $ d$ such that$ dBC$.",
"First assume that $ dp<p$.If $ dpBC$, similarly to Example\\ref {exa:trees with predicate}, first find some $ d”$ such that$ d”BCdp$ and $ d”>ap$ for all $ aAC$with $ apCdp$.",
"Then find $ d'$ such that $ d'BCd$with $ d'pd”$ (and $ BCd'AC=C$).Now assume that $ d>p$.",
"If there is some $ bBC$with $ bd>p$, any $ d'BCd$ such that $ BCd'AC=C$will work.",
"Otherwise find some $ d'BCd$ such that $ d'a=p$for all $ aAC$ with $ a>p$.$ Transitivity: suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DCB$ and $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ and wehave to show that $ AD$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$.",
"If we are in case (1) of the definition,($ aADC$, $ apC$,etc.)",
"then we proceed exactly as in Example \\ref {exa:trees with predicate}.Otherwise, suppose that $ aADC$, $ bBC$are as in case (2).",
"If there is some $ dCD$with $ ad>p$, then for no $ cC$is it the case that $ cd>p$ (otherwise $ ac>p$).",
"Thus$ db=p$ because $ D$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $CB$ hence $ ab=p$ as required.If there is no such $ d$ then $ ab=p$ because $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $DCB$.$ Trees also satisfy the following interesting phenomenon.",
"Proposition 4.17 If $M$ is a dense tree as in Example REF (i.e., the model companion of the theory of trees in $\\left\\lbrace <,\\wedge \\right\\rbrace $ ) then for every $\\sigma \\in \\operatorname{Aut}\\left(M\\right)$ which does not have any fixed points, there is a branch $B\\subseteq M$ such that $\\sigma \\left(B\\right)=B$ .",
"Let $B$ be a maximal linearly ordered set such that $\\sigma \\left(B\\right)=B$ (which exists by Zorn's lemma).",
"We will show that $B$ is a branch.",
"Note that if $x\\in B$ and $y<x$ , then $B\\cup \\left\\lbrace \\sigma ^{n}\\left(y\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is still a chain: given any $z\\in B$ and any $n\\in \\mathbb {Z}$ , $\\sigma ^{n}\\left(x\\right),z$ are comparable and $\\sigma ^{n}\\left(y\\right)<\\sigma ^{n}\\left(x\\right)$ it follows that $\\sigma ^{n}\\left(y\\right)$ and $z$ are comparable (if $z\\le \\sigma ^{n}\\left(x\\right)$ then both $\\sigma ^{n}\\left(y\\right),z\\le \\sigma ^{n}\\left(x\\right)$ , so they are comparable by the tree axioms, and if $\\sigma ^{n}\\left(x\\right)<z$ , then $\\sigma ^{n}\\left(y\\right)<z$ ), and for any $n,m\\in \\mathbb {Z}$ , $\\sigma ^{n}\\left(y\\right),\\sigma ^{m}\\left(y\\right)$ are comparable since $\\sigma ^{n}\\left(x\\right)$ and $\\sigma ^{m}\\left(x\\right)$ are (if $\\sigma ^{n}\\left(x\\right)\\le \\sigma ^{m}\\left(x\\right)$ then both $\\sigma ^{n}\\left(y\\right),\\sigma ^{m}\\left(y\\right)\\le \\sigma ^{m}\\left(x\\right)$ so they are comparable by the tree axioms).",
"Hence $B$ is downwards closed.",
"Now, as $\\sigma $ has no fixed points, $B$ cannot have a maximum (which would have to be a fixed point).",
"Also, if $a\\ge B$ and $\\sigma \\left(a\\right)\\ge a$ or $\\sigma \\left(a\\right)\\le a$ then $B\\cup \\left\\lbrace \\sigma ^{n}\\left(a\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is still a chain (since $\\sigma ^{n}\\left(a\\right)\\ge \\sigma ^{n}\\left(B\\right)=B$ for all $n\\in \\mathbb {Z}$ ), so $a\\in B$ .",
"If $B$ is not a branch (in particular, if $B=\\emptyset $ , which we haven't ruled out yet), there is some $a\\in M$ such that $B<a$ .",
"Let $b=\\sigma \\left(a\\right)\\ne a$ (and by the above, $b,a$ are not comparable), so $B<b$ .",
"Hence $B\\le \\left(a\\wedge b\\right)<a,b$ .",
"Now, $\\sigma \\left(a\\wedge b\\right)<\\sigma \\left(a\\right)=b$ , so $a\\wedge b$ and $\\sigma \\left(a\\wedge b\\right)$ are comparable.",
"The previous paragraph implies that $a\\wedge b\\in B$ .",
"But then $B$ has a maximum — contradiction."
],
[
"Having finite topological rank ",
"In this section we will find some criteria that ensure that $G$ has finite topological rank."
],
[
"$\\omega $ -categorical stable theories ",
"Proposition 5.1 If $T$ is stable $\\omega $ -categorical, $M\\models T$ is countable and $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is finite, then $\\operatorname{Aut}\\left(M\\right)$ has finite topological rank.",
"Without loss of generality, $M=M^{\\operatorname{eq}}$ (if $S\\subseteq \\operatorname{Aut}\\left(M^{\\operatorname{eq}}\\right)$ generates a dense subgroup, then $S\\upharpoonright M=\\left\\lbrace f\\upharpoonright M\\,|\\,f\\in S\\right\\rbrace $ generates a dense subgroup of $\\operatorname{Aut}\\left(M\\right)$ ).",
"Let $N=M_{\\operatorname{acl}\\left(\\emptyset \\right)}$ (i.e., name the elements in $\\operatorname{acl}\\left(\\emptyset \\right)$ ).",
"Then $N$ is $\\omega $ -categorical by Propsition REF .",
"Then in $N$ , $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , so by Example REF , there is a canonical independence relation in $N$ , so $G^{0}=\\operatorname{Aut}\\left(N\\right)$ is topologically 2-generated by Corollary REF , say by $\\left\\lbrace f_{1},f_{2}\\right\\rbrace $ .",
"Now, $\\operatorname{Aut}\\left(M\\right)/G^{0}$ is finite by assumption, so let $S\\subseteq \\operatorname{Aut}\\left(M\\right)$ be a finite set of representatives.",
"Then $S\\cup \\left\\lbrace f_{1},f_{2}\\right\\rbrace $ generates a dense subgroup $\\operatorname{Aut}\\left(M\\right)$ : given two finite tuples $\\bar{a},\\bar{b}$ from $M$ such that $\\bar{a}\\equiv \\bar{b}$ , there is an automorphism $\\sigma \\in \\operatorname{Aut}\\left(M\\right)$ such that $\\sigma \\left(\\bar{a}\\right)=\\bar{b}$ .",
"Also, there is some $f\\in S$ such that $f^{-1}\\sigma \\in \\operatorname{Aut}\\left(N\\right)$ .",
"Hence for some $g$ in the group generated by $\\left\\lbrace f_{1},f_{2}\\right\\rbrace $ , $g\\left(\\bar{a}\\right)=f^{-1}\\sigma \\left(\\bar{a}\\right)=f^{-1}\\left(\\bar{b}\\right)$ , so $fg\\left(\\bar{a}\\right)=\\bar{b}$ .",
"The following fact implies immediately the next result.",
"Fact 5.2 [9] If $T$ is $\\omega $ -categorical and $\\omega $ -stable and $M\\models T$ is countable, then $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is finite.",
"Corollary 5.3 If $T$ is $\\omega $ -stable and $\\omega $ -categorical and $M\\models T$ is countable, then $\\operatorname{Aut}\\left(M\\right)$ has finite topological rank."
],
[
"Reducing finite topological rank to expansions",
"Suppose that $M$ is countable let $G=\\operatorname{Aut}\\left(M\\right)$ .",
"We now want to explore the idea that perhaps by expanding $M$ (i.e., moving to a subgroup), we can show that the topological rank of $G$ is small by showing that the rank of the automorphism group of the expansion is.",
"Suppose that $H\\le G$ .",
"If $\\left(G,H\\right)$ has a compact quotient (see Definition REF ), then we cannot hope to deduce anything.",
"For example, by Proposition REF we have that $G^{0}$ acts oligomorphically on $M$ and it can be that $G^{0}$ has a cyclically dense conjugacy class (so topological rank 2) while $G/G^{0}=\\left(\\mathbb {Z}/2\\mathbb {Z}\\right)^{\\omega }$ (so $G$ is not topologically finitely generated) — this happens in the example described in in Remark REF , see Example REF .",
"Indeed, we will see that $\\left(G,H\\right)$ having a compact quotient is the only obstruction."
],
[
"$\\omega $ -categorical structures with finitely many reducts",
"Theorem 5.4 Suppose that $H\\le G$ is closed and that $\\left(G,H\\right)$ has no compact quotients.",
"If there are only finitely many closed groups between $G$ and $H$ then there is some $g\\in G$ such that $H\\cup \\left\\lbrace g\\right\\rbrace $ topologically generate $G$ .",
"Remark 5.5 The condition of having finitely many closed groups in the theorem holds when for instance $M$ is a reduct of an $\\omega $ -categorical structure $M^{\\prime }$ where $H=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ , and $M^{\\prime }$ has only finitely many reducts up to bi-definability.",
"Let $\\left\\lbrace H_{i}\\,|\\,i<n\\right\\rbrace $ be the family of closed proper subgroups of $G$ containing $H$ (which is finite by assumption).",
"If $\\left[G:H_{i}\\right]<\\infty $ for some $i<n$ , then there would be a closed normal proper subgroup $N_{i}\\trianglelefteq G$ of finite index such that $N_{i}\\le H_{i}$ (in general, if $H^{\\prime }\\le G$ is closed of finite index, then there is a closed normal subgroup $N\\le H^{\\prime }$ , $N\\trianglelefteq G$ such that $\\left[G:N\\right]<\\infty $ .",
"In fact, $N=\\bigcap \\left\\lbrace gH^{\\prime }g^{-1}\\,|\\,g\\in G\\right\\rbrace $ and this intersection is finite as it is the orbit of $H^{\\prime }$ under the action of $G$ on conjugates of $H^{\\prime }$ and its stabilizer contains $H^{\\prime }$ ).",
"But then $N_{i}H=G$ by assumption and Proposition REF , so $G=N_{i}H\\subseteq H_{i}H=H_{i}$ contradicting the fact that $H_{i}$ was a proper subgroup.",
"By a theorem of Neumann [29], there is some $g\\in G\\backslash \\bigcup \\left\\lbrace H_{i}\\,|\\,i<n\\right\\rbrace $ .",
"If $G\\ne \\operatorname{cl}\\left(\\left\\langle H\\cup \\left\\lbrace g\\right\\rbrace \\right\\rangle \\right)$ (the topological closure of the group generated by $H\\cup \\left\\lbrace g\\right\\rbrace $ ), then $\\operatorname{cl}\\left(\\left\\langle H\\cup \\left\\lbrace g\\right\\rbrace \\right\\rangle \\right)$ is one of the groups $H_{i}$ , contradicting the choice of $g$ .",
"Corollary 5.6 If $G$ and $H$ are as in Theorem REF and $H$ has finite topological rank then so does $G$ .",
"By Example REF , in the $\\omega $ -categorical context we get that if $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ in $M$ and $M^{\\prime }$ is an expansion having finitely many reducts, then we can apply Corollary REF .",
"This is the case, for instance, when $M^{\\prime }$ is $\\left(\\mathbb {Q},<\\right)$ (see [18]).",
"By Lemma [18], an example of such a reduct of DLO is given by the countable dense circular order, which is the structure with universe $\\mathbb {Q}$ , and a ternary relation $C\\left(x,y,z\\right)$ given by $C\\left(x,y,z\\right)\\Leftrightarrow x<y<z\\vee y<z<x\\vee z<x<y$ .",
"Corollary 5.7 $\\operatorname{Aut}\\left(\\mathbb {Q},C\\right)$ has topological rank $\\le 3$ , but $\\left(\\mathbb {Q},C\\right)$ has no CIR.",
"We only have to show that it has no CIR.",
"By Lemma REF , if there was a CIR, then in particular there would be a type of a single element $q\\left(x\\right)$ over $\\mathbb {Q}$ which does not split over $\\emptyset $ .",
"But by quantifier elimination, every tuple of two distinct elements have the same type (i.e., $\\operatorname{Aut}\\left(\\mathbb {Q},C\\right)$ acts 2-transitively on $\\mathbb {Q}$ ).",
"Now, $q$ cannot be realized in $\\mathbb {Q}$ and must contain $C\\left(0,x,1\\right)$ or $C\\left(1,x,0\\right)$ , hence both, which is a contradiction.",
"Remark 5.8 For any point $a\\in \\mathbb {Q}$ , the expansion $\\left(\\mathbb {Q},C,a\\right)$ does have a CIR.",
"Indeed, in this case $C$ defines a dense linear order with no endpoints on $\\mathbb {Q}\\backslash \\left\\lbrace a\\right\\rbrace $ by $b<c\\iff C\\left(a,b,c\\right)$ .",
"Since $\\left(\\mathbb {Q},<\\right)$ has a CIR $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ (Example \\ref {exa:DLO has groovy}),we can define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C*B$ by $ A{ a} $\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C{ a} B{ a} $.Since for every finite tuples $ b,c$, $ bc$ in the expansioniff $ baca$ in the order, it followseasily that $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*$ is a CIR.$ An even closer look at the reducts of DLO, gives the following result.",
"Corollary 5.9 Every closed supergroup of $\\operatorname{Aut}\\left(\\mathbb {Q},<\\right)$ has topological rank $\\le 3$ .",
"The diagram in [18] of the lattice of closed groups between $\\operatorname{Aut}\\left(\\mathbb {Q},<\\right)$ and $\\operatorname{Aut}\\left(\\mathbb {Q},=\\right)$ shows that any such group contains at most two incomparable closed subgroups.",
"Since no group can be a union of two of its proper subgroups, we do not need to use Neumanns's lemma in the proof of Theorem REF above, allowing us to drop the assumption that $\\left(G,H\\right)$ has no compact quotients."
],
[
"A general reduction theorem",
"In the next theorem we drop the assumption of having finitely many reducts of the expansion (i.e., of having finitely many groups between $H$ and $G$ ), but we compensate for it by assuming that $H$ acts oligomorphically on $M$ and increasing the number of generators by 1.",
"Fact 5.10 [9] Suppose that $M$ is a countable $\\omega $ -saturated structure.",
"Then for any $A,B\\subseteq M$ , there is some $A^{\\prime }$ (in the monster model $\\mathfrak {C}$ , see just above Section REF ) such that $A^{\\prime }\\equiv A$ and $A^{\\prime }\\cap B\\subseteq \\operatorname{acl}\\left(\\emptyset \\right)$ .",
"Theorem 5.11 Suppose as usual that $M$ is countable and $\\omega $ -categorical and let $G=\\operatorname{Aut}\\left(M\\right)$ .",
"Suppose that $H\\le G$ is closed and acts oligomorphically on $M$ and that $\\left(G,H\\right)$ has no compact quotients.",
"Then there are $g_{1},g_{2}\\in G$ such that $H\\cup \\left\\lbrace g_{1},g_{2}\\right\\rbrace $ topologically generates $G$ .",
"Let $M^{\\prime }$ be an $\\omega $ -categorical expansion of $M$ to some language $L^{\\prime }$ containing $L$ (the language of $M$ ) such that $H=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ .",
"We use $^{\\prime }$ to indicate the expansion.",
"In particular, $\\mathfrak {C}^{\\prime }$ denotes the expansion of $\\mathfrak {C}$ to $L^{\\prime }$ .",
"By Fact REF , there is some $M_{0}$ such that $M_{0}\\equiv M$ and $M_{0}^{\\operatorname{eq}}\\cap M^{\\operatorname{eq}}=\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ (apply the fact in $\\mathfrak {C}^{\\operatorname{eq}}$ ).",
"There is some automorphism $\\sigma $ of $\\mathfrak {C}$ such that $\\sigma \\left(M_{0}\\right)=M$ .",
"Let $N_{0}^{\\prime }$ be a countable model containing $\\sigma ^{n}\\left(M_{0}\\right)$ for all $n\\in \\mathbb {Z}$ .",
"Let $N_{1}^{\\prime }$ be a countable model containing $\\sigma ^{n}\\left(N_{0}^{\\prime }\\right)$ for all $n\\in \\mathbb {Z}$ .",
"Continue like this and finally let $N_{\\omega }^{\\prime }=\\bigcup \\left\\lbrace N_{i}^{\\prime }\\,|\\,i<\\omega \\right\\rbrace $ .",
"So $M^{\\prime }\\prec N^{\\prime }_{\\omega }\\prec \\mathfrak {C}^{\\prime }$ is countable and $\\sigma \\upharpoonright N{}_{\\omega }\\in \\operatorname{Aut}\\left(N_{\\omega }\\right)$ .",
"By $\\omega $ -categoricity (of $M^{\\prime }$ ) we may assume that $N_{\\omega }^{\\prime }=M^{\\prime }$ : there is some $g_{1}\\in \\operatorname{Aut}\\left(M\\right)$ and $M_{0}^{\\prime }\\prec M^{\\prime }$ such that $g_{1}\\left(M_{0}^{\\operatorname{eq}}\\right)\\cap M_{0}^{\\operatorname{eq}}=\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"Then $H_{1}=\\operatorname{cl}\\left(\\left\\langle H,g_{1}\\right\\rangle \\right)$ is a closed group acting oligomorphically on $M$ .",
"Also, note that $\\left(G,H_{1}\\right)$ has no compact quotients.",
"Let $M^{\\prime \\prime }$ be the reduct of $M^{\\prime }$ , which is also an expansion of $M$ that corresponds to $H_{1}$ : $\\operatorname{Aut}\\left(M^{\\prime \\prime }\\right)=H_{1}$ .",
"As usual, we use $^{\\prime \\prime }$ to indicate that we work in this expansion.",
"Claim 5.12 If $X\\subseteq M^{n}$ is definable over $\\emptyset ^{\\prime \\prime }$ (i.e., definable in $L^{\\prime \\prime }$ over $\\emptyset $ ) and $M$ -definable (in $L$ ), then it is $\\emptyset $ -definable (in $L$ ).",
"First note that it is enough to show that $X$ is $\\operatorname{acl}_{L}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ -definable (the code $X$ of $X$ belongs to $\\operatorname{dcl}_{L^{\\prime \\prime }}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ and to $\\operatorname{acl}_{L}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , and if it were not in $\\operatorname{dcl}_{L}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ then there would be an automorphism of $M$ moving it, but then by the no-compact quotient assumption there would be an automorphism of $M^{\\prime \\prime }$ moving it as well — contradiction).",
"Now, since $X$ is $\\emptyset ^{\\prime \\prime }$ -definable and $M$ -definable, it is definable over $M_{0}$ (because $M_{0}^{\\prime }\\prec M^{\\prime }$ ), so its code $X\\in M_{0}^{\\operatorname{eq}}$ .",
"In addition, $g_{1}\\left(X\\right)=X$ , so $X$ is definable over $g_{1}\\left(M_{0}\\right)$ , hence $X\\in g_{1}\\left(M_{0}^{\\operatorname{eq}}\\right)$ .",
"Together it is in $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , which is what we wanted.",
"Now we construct $g_{2}$ by back-and-forth to ensure that $\\operatorname{cl}\\left(\\left\\langle H_{1},g_{2}\\right\\rangle \\right)=G$ .",
"Suppose that we have constructed $g_{2}\\upharpoonright A$ for some finite set $A$ .",
"Let $O$ be an orbit of the action of $G$ on $M^{m}$ , and we write it as $O=\\bigcup \\left\\lbrace O_{i}\\,|\\,i<n\\right\\rbrace $ where the $O_{i}$ 's are the orbits of the action of $H_{1}$ (recall that $H_{1}$ acts oligomorphically on $M$ , so there are only finitely many such orbits).",
"Claim 5.13 For any subset $s\\subsetneq n$ there are $a,b\\in O$ such that $a\\in O_{s}=\\bigcup \\left\\lbrace O_{i}\\,|\\,i\\in s\\right\\rbrace ,b\\in O_{n\\backslash s}$ , and $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle a,b\\right\\rangle \\right\\rbrace $ or $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle b,a\\right\\rangle \\right\\rbrace $ is an elementary map.",
"Note that $O_{s}$ is $\\emptyset ^{\\prime \\prime }$ -definable.",
"As it is not $\\emptyset $ -definable (because $s\\subsetneq n$ ), it is also not $M$ -definable by Claim REF .",
"In particular, it is not $A$ -definable.",
"Hence there are $a_{0}\\in O_{s},a_{1}\\in O_{n\\backslash s}$ such that $a_{0}\\equiv _{A}a_{1}$ .",
"There is some $b$ such that $a_{0}A\\equiv a_{1}A\\equiv bg_{2}\\left(A\\right)$ .",
"If $b\\in O_{s}$ , then $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle a_{1},b\\right\\rangle \\right\\rbrace $ is the required map.",
"Otherwise, pick $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle a_{0},b\\right\\rangle \\right\\rbrace $ .",
"In the back-and-forth construction of $g_{2}$ , we deal with all these orbits (for every $m<\\omega $ , there are only finitely many) and all these subsets $s$ and increase $g_{2}$ according to Claim REF .",
"We claim that $g_{2}$ is such that $\\operatorname{cl}\\left(\\left\\langle H_{1},g_{2}\\right\\rangle \\right)=G$ .",
"Indeed, it is enough to show that every orbit $O$ of $G$ is also an orbit of $\\left\\langle H_{1},g_{2}\\right\\rangle $ .",
"The orbit $O$ can be written as $\\bigcup \\left\\lbrace O_{i}\\,|\\,i<n\\right\\rbrace $ where the $O_{i}$ 's are the orbits of $H_{1}$ , and also as $\\bigcup \\left\\lbrace O^{\\prime }_{i}\\,|\\,i\\in I\\right\\rbrace $ where the $O^{\\prime }_{i}$ 's are orbits of $\\left\\langle H_{1},g_{2}\\right\\rangle $ .",
"Each such $O_{i}^{\\prime }$ is itself a union of $H_{1}$ -orbits, so has the form $O_{s}$ for some $s\\subseteq n$ .",
"But by construction, if $s\\ne n$ there are tuples $a\\in O_{s},b\\in O_{n\\backslash s}$ such that either $g_{2}$ or $g_{2}^{-1}$ maps $a$ to $b$ — contradiction.",
"So $s=n$ , and $O_{i}^{\\prime }=O$ ."
],
[
"A topological dynamics consequence\nof having a CIR ",
"Definition 6.1 Suppose that $M$ is a countable structure.",
"Call an automorphism $\\sigma \\in G$ shifty if there is some invariant binary relation on finite sets in $M$ , $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ (the base will always be $$) such that:\\begin{itemize}\\item (Monotonicity) If A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{itemize}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ and $ A'A$, $ B'B$then $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B'$.\\item (Right existence) For every finite tuple $ a$ there is some $ a'a$such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$ (by this we mean that sets enumerated by $ a$,$ a'$ are independent).\\item (Right shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$, then there exists some $ n<$such that $ b'An(b)$.$ Lemma 6.2 If $\\sigma $ is shifty then it also satisfies: (Left existence) For every finite tuple $a$ there is some $a^{\\prime }\\equiv a$ such that $a^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.\\item (Left shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ b'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$, then there exists some $ n<$such that $ b'A-n(b)$.$ Suppose that $\\sigma $ is shifty, as witnessed by $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.",
"Given $ a$,there is some $ a'a$ such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Applying an automorphismtaking $ a'$ to $ a$ we get some $ a”a$ such that $ a”$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$,which shows left existence.$ As for left shiftiness, suppose that $A$ is finite and enumerated by $a$ , $b,b^{\\prime }$ are finite tuples such that $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ and $ bb'$.Then applying an automorphism, we get some $ a'$ such that $ ab'a'b$,so $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Hence for some $ n<$, $ a'bn(a)$.From $ a'bn(a)b$ we get that $ ab'a'b-n(a')-n(b)a-n(b)$,i.e., $ b'A-n(b)$.$ Proposition 6.3 The automorphism $\\sigma $ is a shifty automorphism on $M$ iff for any type $p\\in S\\left(\\emptyset \\right)$ (with finitely many variables), letting $Y_{a}=\\bigcap \\left\\lbrace \\bigcup \\left\\lbrace \\operatorname{tp}\\left(a,\\sigma ^{n}\\left(a^{\\prime }\\right)\\right)\\,|\\,n<\\omega \\right\\rbrace \\,|\\,a^{\\prime }\\equiv a\\right\\rbrace $ for any $a\\models p$ , the intersection $Y_{p}=\\bigcap \\left\\lbrace Y_{a}\\,|\\,a\\models p\\right\\rbrace $ is nonempty.",
"Suppose that $\\sigma $ is shifty, and fix some type $p\\in S\\left(\\emptyset \\right)$ .",
"Let $a\\models p$ .",
"By existence, there is some $a^{\\prime }\\equiv a$ with $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Let $ q=tp(a,a')$ and fix some $ bp$.Let $ Aut(M)$ map $ a$ to $ b$ and let $ b'=(a')$.We have that $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$ and hence by right shiftiness, $ q=tp(b,b')Yb$.Since $ b$ was arbitrary, $ qYp$.$ Suppose that the right hand side holds.",
"Given a finite tuple $a$ and $a^{\\prime }\\equiv a$ , write $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ iff $ tp(a,a')Yp$where $ p=tp(a/)$.",
"For general finite sets $ A,B$,write $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ iff there is some $ C$ containing $ A$ and $ C'$containing $ B$ such that $ C$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*C'$.",
"Obviously, $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is invariantand monotone.",
"Right existence follows from the assumption that $ Yp$for all $ pS()$.",
"Right shiftiness: supposethat $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ and $ a”a'$.",
"Then $ tp(a,a')Yp$and in particular it belongs to $ Ya$.",
"By definition of $ Ya$,$ tp(a,a'){ tp(a,n(a”)) | n<} $,so for some $ n<$, $ aa'ain(a”)$.$ Proposition 6.4 If $M$ is an ultrahomogeneous structure and $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR on finite subsets of $ M$ which respects substructures,then there exists a shifty automorphism $$ on $ M$, as witnessedby $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.$ Monotonicity and right existence are parts of the properties of a CIR, so we only have to prove right shiftiness.",
"Suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$and $ b'b$.",
"By the proof of Theorem \\ref {thm:existence of repulsive automorphism-ultrahomogeneous},the repulsive automorphism $$ constructed there satisfies thatfor some $ n<$, $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $n(b')$.",
"By stationarity,$ bA(b')$.$ Recall the definitions of flow and subflow from Section REF .",
"Theorem 6.5 Let $M$ be a countable homogeneous structure and $G=\\operatorname{Aut}\\left(M\\right)$ .",
"Suppose that $\\sigma \\in G$ is a shifty automorphism and that $\\left(X,d\\right)$ is a compact metric $G$ -flow.",
"Then for every $x_{*}\\in X$ there is some conjugate $\\sigma _{*}\\in G$ of $\\sigma $ such that: (*) Both $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ and $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contain a subflow of $X$ .",
"Remark 6.6 Note that Theorem REF implies that both $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ and $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ contain a subflow of $X$ : if e.g., $Y_{0}$ is a flow contained in the left space, then $GY_{0}=Y_{0}$ , so $\\sigma _{*}^{-k}\\left(Y_{0}\\right)\\subseteq Y_{0}\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ , hence $Y_{0}\\subseteq \\sigma _{*}^{k}\\left(\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace \\right)=\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace $ .",
"Before the proof we note the following useful lemma.",
"Lemma 6.7 Suppose that $G$ is a topological group acting continuously on a compact metric space $\\left(X,d\\right)$ .",
"Then for every $0<\\varepsilon $ there is some open neighborhood $U$ of $\\operatorname{id}\\in G$ such that for every $g,h\\in G$ if $gh^{-1}\\in U$ then for all $x\\in X$ we have that $d\\left(gx,hx\\right)<\\varepsilon $ .",
"It is enough to show that there is some open neighborhood $U$ of $\\operatorname{id}$ such that if $g\\in U$ then for all $x\\in X$ , $d\\left(gx,x\\right)<\\varepsilon $ (since then if $gh^{-1}\\in U$ then $d\\left(gh^{-1}\\left(hx\\right),hx\\right)<\\varepsilon $ ).",
"For every $x\\in X$ , there is some neighborhood $V_{x}$ of $x$ in $X$ and some neighborhood $U_{x}$ of $\\operatorname{id}$ in $G$ such that for all $g\\in U_{x}$ , $x^{\\prime }\\in V_{x}$ , $d\\left(gx^{\\prime },x^{\\prime }\\right)<\\varepsilon $ .",
"By compactness, a finite union of $V_{x}$ 's covers $X$ .",
"Let $U$ be the intersection of the corresponding $U_{x}$ 's.",
"Suppose that $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ witnesses that $$ is shifty.",
"Let $ G0$be a countable dense subset of $ G$, enumerated as $ gi | i<$,such that $ g0=id$.$ We construct an automorphism $\\tau :M\\rightarrow M$ by back and forth such that eventually $\\sigma _{*}=\\tau ^{-1}\\sigma \\tau $ and such that at each finite stage, $\\tau $ will be an elementary map.",
"For the construction it is actually better to think of the domain and range of $\\tau $ as two different structures, so we have $M=M_{*}$ and suppose that $\\sigma :M\\rightarrow M$ , $\\sigma _{*}:M_{*}\\rightarrow M_{*}$ and $\\tau :M_{*}\\rightarrow M$ .",
"The subscript $*$ will denote tuples from $M_{*}$ throughout.",
"Suppose that we have constructed a partial elementary map $f:A_{*}\\rightarrow A$ (that will be part of $\\tau $ eventually) with $A_{*}\\subseteq M_{*},A\\subseteq M$ finite, enumerated by $a_{*},a$ .",
"Here is the main tool in the construction.",
"Claim 6.8 Suppose that $b_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a*$ and$ b*'ba$.",
"Let $ b*=f-1(b)$.",
"Thenthere is $ k<$ and an extension $ f'$ of $ f$ such that anyautomorphism $ '$ extending $ f'$ will satisfy that for $ *'='-1'$,$ *'k(b'*)=b*$.$ Similarly, if $a_{*}\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b*'$ then there is some $ k<$ andan extension $ f'$ of $ f$ such that any automorphism $ '$ extending$ f'$ will satisfy that for $ *'='-1'$,$ *'k(b*)=b'*$.$ First, find some tuple $b^{\\prime }$ in $M$ such that $b^{\\prime }a\\equiv b_{*}^{\\prime }a_{*}$ .",
"In particular, $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.",
"By left shiftiness, there is some $ k<$such that $ -k(b)ab'ab*'a*$.Extend $ f$ to $ f'$ which sends $ b*'$ to $ -k(b)$.Then, for any $ '$ extending $ f'$, $ '-1k'(b*')='-1(b)=b*$.$ The second statement is proved similarly, using right shiftiness.",
"We will make sure that for each $n<\\omega $ , the following condition holds.",
"$\\star $ There are $k_{n,0},\\ldots ,k_{n,n-1}<\\omega $ such that for all $i<n$ , $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)\\right)\\right)<1/n$ and $k_{n,0}^{\\prime },\\ldots ,k_{n,n-1}^{\\prime }<\\omega $ such that for all $i<n$ , $d\\left(\\sigma _{*}^{-k_{n,i}^{\\prime }}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{-k_{n,0}^{\\prime }}\\left(x_{*}\\right)\\right)\\right)<1/n$ .",
"Why is $\\star $ enough?",
"Let $y_{n}=\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)$ , and let $y$ be a limit of some subsequence $\\left\\langle y_{n_{j}}\\,|\\,j<\\omega \\right\\rangle $ (which exists by compactness), then $Gy\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ (so $\\operatorname{cl}\\left(Gy\\right)$ is a subflow): given $g\\in G$ and $0<\\varepsilon $ , first find an open neighborhood $U\\subseteq G$ of $g$ such that if $h\\in U$ then $d\\left(gx,hx\\right)<\\varepsilon /4$ for all $x\\in X$ (this $U$ is given to us by Lemma REF : it is $\\left(g^{-1}V\\right)^{-1}$ where $V$ is an open neighborhood of $\\operatorname{id}$ such that if $gh^{-1}\\in V$ , $d\\left(gx,hx\\right)<\\varepsilon /4$ ).",
"Take $n$ so large that $g_{i}\\in U$ for some $i<n$ and $1/n<\\varepsilon /4$ , and find $n_{j}$ even larger so that $d\\left(g_{i}y,g_{i}y_{n_{j}}\\right)<\\varepsilon /4$ .",
"Then $d\\left(gy,g_{i}y\\right)<\\varepsilon /4$ , $d\\left(g_{i}y,g_{i}y_{n_{j}}\\right)<\\varepsilon /4$ and $d\\left(g_{i}y_{n_{j}},\\sigma _{*}^{k_{n_{j},i}}\\left(x_{*}\\right)\\right)<\\varepsilon /4$ .",
"Together, $d\\left(gy,\\sigma _{*}^{k_{n_{j},i}}\\left(x_{0}\\right)\\right)<3\\varepsilon /4<\\varepsilon $ , which means that $gy$ is in the closure.",
"Similarly, if $y^{\\prime }$ is a limit of a subsequence of $\\sigma _{*}^{-k_{n,0}^{\\prime }}\\left(x_{*}\\right)$ , then $Gy^{\\prime }\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ .",
"So we consider $f:A_{*}\\rightarrow A$ a partial elementary map.",
"Our task now is to deal with $n<\\omega $ .",
"Let $\\varepsilon =1/n$ .",
"Let $A_{*}\\subseteq C_{*}\\subseteq M_{*}$ be finite such that if $g^{-1}\\upharpoonright C_{*}=h^{-1}\\upharpoonright C_{*}$ then $d\\left(gx,hx\\right)<\\varepsilon /4$ for all $x\\in X$ and any $g,h\\in G$ (this is by Lemma REF ).",
"Let $z_{0},\\ldots ,z_{l-1}$ be such that $\\bigcup \\left\\lbrace B\\left(z_{j},\\varepsilon /4\\right)\\,|\\,j<l\\right\\rbrace $ cover $X$ , and write $B_{j}=B\\left(z_{j},\\varepsilon /4\\right)$ .",
"Let $c_{*}$ be a finite tuple enumerating $C_{*}$ .",
"For every $c_{*}^{\\prime }\\equiv c_{*}$ , we say that $c_{*}^{\\prime }$ has color $j<l$ if $j$ is least such that there is $g\\in G$ such that $g\\left(c_{*}^{\\prime }\\right)=c_{*}$ and $gx_{*}\\in B_{j}$ .",
"Note that by the choice of $c_{*}$ , if $g^{\\prime }\\left(c_{*}^{\\prime }\\right)=c_{*}$ then $g^{\\prime }g^{-1}\\upharpoonright C_{*}=\\operatorname{id}$ , so $g^{\\prime }x_{0}\\in B\\left(z_{j},\\varepsilon /2\\right)$ .",
"Let $D_{*}=\\bigcup \\left\\lbrace g_{i}^{-1}\\left(C_{*}\\right)\\,|\\,i<n\\right\\rbrace $ .",
"Note that $C_{*}\\subseteq D_{*}$ because $g_{0}=\\operatorname{id}$ .",
"Let $d_{*}$ enumerate $D_{*}$ .",
"For any $d_{*}^{\\prime }\\equiv d_{*}$ and $s\\subseteq l$ , we say that $d_{*}^{\\prime }$ has color $s$ if $\\left\\lbrace j<l\\,|\\,c_{*}^{\\prime }\\equiv c_{*},c_{*}^{\\prime }\\subseteq d_{*}^{\\prime },c_{*}^{\\prime }\\text{ has color }j\\right\\rbrace =s$ .",
"By left existence, there is some $s_{0}\\subseteq l$ such that for every finite set $S\\subseteq M_{*}$ , there is some $d_{*}^{\\prime }\\equiv d_{*}$ with $d_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $S$ and $ d*'$ has color $ s0$.$ Let $d_{*}^{\\prime }\\equiv d_{*}$ be of color $s_{0}$ such that $d_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $d*$.For $ i<n$, let $ c*,i'd*'$ be the tuple correspondingto $ gi-1(c*)$, so in particular $ c*,i'c*$.Let $ ji<l$ be the color of $ c*,i'$.",
"By the choice of $ s0$,for every finite set $ SM*$, there is some $ c*'c*$such that $ c'*$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $S$ and $ c*'$ has color $ ji$.$ Since $M$ is homogeneous we can extend $f$ in such a way so that its domain equals $D_{*}$ .",
"By Claim REF (the first part), there is some $k_{n,0}<\\omega $ and an extension $f_{0}$ of $f$ that ensures that $\\sigma _{*}^{k_{n,0}}\\left(d_{*}^{\\prime }\\right)=d_{*}$ .",
"Starting with $f_{0}$ , we construct an increasing sequence $\\left\\langle f_{i}\\,|\\,i<n\\right\\rangle $ as follows.",
"Suppose we have $f_{i}$ whose domain is $D_{*,i}$ .",
"Find some $c^{\\prime \\prime }_{*,i+1}\\equiv c_{*}$ of color $j_{i+1}$ such that $c_{*,i+1}^{\\prime \\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $D*,i$.By Claim \\ref {claim:Using shiftiness}, we can find $ kn,i+1<$and extend $ fi$ to $ fi+1$ which ensures that $ *kn,i+1(c*,i+1”)=c*$.$ Now we have the first part of $\\star $ : we need to check that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)\\right)\\right)<\\varepsilon $ for all $i<n$ .",
"For $i=0$ this is clear since $g_{0}=\\operatorname{id}$ , so we may assume that $i>0$ .",
"As $\\sigma _{*}^{k_{n,i}}\\left(c_{*,i}^{\\prime \\prime }\\right)=c_{*}$ , it follows that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{0}\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .",
"Similarly, as $\\sigma _{*}^{k_{n,0}}\\left(c_{*,i}^{\\prime }\\right)=g_{i}^{-1}\\left(c_{*}\\right)$ , we have that $d\\left(g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{0}\\right)\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .",
"Together, we are done.",
"Now we have to take care of the other half of $\\star $ .",
"This is done similarly, using right existence and the second part of Claim REF .",
"The following proposition explains why we needed to take a conjugate of $\\sigma $ .",
"The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.",
"In Section REF , we mentioned that it is a Ramsey structure.",
"Note that the underlying order is dense (by Proposition REF ).",
"Proposition 6.9 Let $M=\\left(V,<,R\\right)$ be the countable ordered random graph.",
"Then there is no automorphism $\\sigma \\in G=\\operatorname{Aut}\\left(M\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\in X$ .",
"First we find $a\\ne b$ in $M$ such that $\\sigma ^{n}\\left(a\\right)\\ne \\sigma ^{m}\\left(b\\right)$ for all $m,n\\in \\mathbb {Z}$ .",
"To do that, take any $a\\in M$ .",
"Then $\\left\\lbrace \\sigma ^{n}\\left(a\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is discrete (in the order sense: it is either a $\\mathbb {Z}$ -chain or just $a$ ).",
"Since $\\left(V,<\\right)$ is dense, there is some $b\\ne \\sigma ^{n}\\left(a\\right)$ for all $n\\in \\mathbb {Z}$ .",
"It follows that $b$ is as required.",
"Let $X=S_{x}\\left(M\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).",
"Let $p\\in X$ be any completion of the partial type $\\left\\lbrace R\\left(x,\\sigma ^{n}\\left(a\\right)\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace \\cup \\left\\lbrace \\lnot R\\left(x,\\sigma ^{m}\\left(b\\right)\\right)\\,|\\,m\\in \\mathbb {Z}\\right\\rbrace $ .",
"Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\in \\operatorname{cl}\\left\\lbrace \\sigma ^{n}\\left(p\\right)\\,|\\,n<\\omega \\right\\rbrace $ which is a fixed point of $G$ .",
"In other words, $p_{0}$ is an invariant type over $M$ .",
"However $R\\left(x,a\\right)\\wedge \\lnot R\\left(x,b\\right)\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).",
"The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.",
"If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\left(X,x_{0}\\right)$ .",
"Thus, there would be a conjugate $\\sigma _{*}$ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow.",
"But then if $\\left(Y,y_{0}\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\pi :X\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .",
"Thus, $\\pi $ maps $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ to $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(y\\right)\\,|\\,n<\\omega \\right\\rbrace $ , and the latter contains a $G$ -subflow.",
"Thus we get that $\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .",
"Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\left\\lbrace <,\\wedge \\right\\rbrace $ , and let $M\\models T$ be countable.",
"Then $\\operatorname{Aut}\\left(M\\right)$ has no shifty automorphism.",
"In particular, $M$ has no CIR.",
"Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .",
"Suppose that $\\sigma $ was shifty.",
"Let $\\bar{m}=\\left\\langle m\\,|\\,m\\in M\\right\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\bar{x}=\\left\\langle x_{m}\\,|\\,m\\in M\\right\\rangle $ .",
"Let $X=S_{\\bar{m}}\\left(M\\right)$ be the space of $\\bar{x}$ -complete types $p$ over $M$ such that $p\\upharpoonright \\emptyset =\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .",
"Then $X$ is a compact metric space.",
"Let $x_{*}=\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .",
"By Theorem REF , there is some conjugate $\\tau $ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\tau ^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{+}\\subseteq X$ and similarly, $\\operatorname{cl}\\left\\lbrace \\tau ^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{-}$ .",
"By Proposition REF , $\\tau $ fixes a branch or a point.",
"Suppose that $\\tau \\left(m\\right)=m$ for some $m\\in M$ .",
"Then for every $p\\in Y^{+}$ , $p\\models x_{m}=m$ .",
"However $G=\\operatorname{Aut}\\left(M\\right)$ acts transitively on $M$ , so we have a contradiction.",
"Now suppose that $\\tau $ fixes a branch $B\\subseteq M$ , but does not fix any point.",
"Suppose that $\\tau \\left(m\\right)>m$ for some $m\\in B$ .",
"Then $\\tau ^{n}\\left(m\\right)>m$ for all $n<\\omega $ , so for any $p\\in Y^{+}$ , $p\\models x_{m}>m$ .",
"There is some $m^{\\prime }\\in M$ such that $m^{\\prime }>m$ and $m^{\\prime }\\notin B$ .",
"Since $m<\\tau ^{n}\\left(m\\right)\\in B$ for all $n<\\omega $ , it follows that $p\\models x_{m}\\wedge m^{\\prime }=m$ for all $p\\in Y^{+}$ .",
"Let $\\tau ^{\\prime }\\in G$ fix $m$ and map $m^{\\prime }$ to $B$ .",
"Then $\\tau ^{\\prime }\\left(p\\right)\\models \\left(x_{m}\\wedge \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\right)=m<x_{m}$ .",
"But $\\tau ^{\\prime }\\left(p\\right)\\in Y^{+}$ , so $\\tau ^{\\prime }\\left(p\\right)\\models x_{m}\\le \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\vee \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\le x_{m}$ , which is a contradiction.",
"If, on the other hand $\\tau \\left(m\\right)<m$ , then $\\tau ^{-1}\\left(m\\right)>m$ , so we can apply the same argument to $Y^{-}$ .",
"For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .",
"In addition, letting $H=\\operatorname{Aut}\\left(N\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).",
"In addition, if $B\\subseteq N$ is a branch, $m\\in B$ , there is always some $m^{\\prime }>m$ , $m^{\\prime }\\notin B$ and for any $n^{\\prime }>m$ in $B$ , $m^{\\prime }m\\equiv n^{\\prime }m$ .",
"Hence, we can apply Proposition REF and the same proof will work.",
"Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\omega $ -categorical structures.",
"We state here a few general conjectures and questions.",
"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.",
"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.",
"Conjecture 7.1 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion which admits a CIR.",
"Suppose that $M$ is a structure and $\\mathfrak {C}$ a monster model for $\\operatorname{Th}\\left(M\\right)$ .",
"The group of Lascar strong automorphisms of $M$, denoted by $\\operatorname{Aut}f\\left(M\\right)$ is the group of automorphisms of $M$ generated by the set $\\left\\lbrace \\sigma \\upharpoonright M\\,|\\,\\exists N\\prec \\mathfrak {C},\\left|N\\right|=\\left|T\\right|,\\sigma \\upharpoonright N=\\operatorname{id}\\right\\rbrace $ .",
"If $\\sigma $ is Lascar strong, then $\\sigma \\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{id}$ so $\\operatorname{Aut}f\\left(M\\right)$ is contained in $G^{0}$ .",
"However, there are examples (even $\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\operatorname{Aut}f\\left(M\\right)$ , see [16], [31].",
"The Lascar group of $M$ is the quotient $\\operatorname{Aut}\\left(M\\right)/\\operatorname{Aut}f\\left(M\\right)$ .",
"For more on the Lascar group, see [39].",
"In the $\\omega $ -categorical case, the quotient $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is also called the compact Lascar group.",
"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$.",
"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.",
"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.",
"Forinstance, by Lemma \\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\operatorname{Aut}\\left(M\\right)$ has no compact quotients then it has finite topological rank.",
"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.",
"Thus, potentially, the Lascar group — as a quotient of $\\operatorname{Aut}\\left(M\\right)$ — can be an obstruction to having finite topological rank.",
"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.",
"Conjecture REF (together with Theorem REF ) implies that it could not.",
"Conjecture 7.2 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion with trivial Lascar group.",
"By the above, this second conjecture is implied by Conjecture REF .",
"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.",
"It would be interesting to investigate other consequences of having a CIR.",
"For instance a CIR might have something to say about normal subgroups.",
"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ for some CIR $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\aleph _{0}}$ is open.",
"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.",
"We refer to [26] for a survey on this.",
"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.",
"In fact the compact quotients are hidden in the stabilizer of a finite set.",
"It seems that one can avoid this counterexample by restricting to dense subgroups.",
"This leads us to the following questions.",
"Question 7.3 Let $M$ be $\\omega $ -categorical such that $G=\\operatorname{Aut}\\left(M\\right)$ has no compact quotient.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open (and hence is equal to $G$ )?",
"Note that the assumption of having no compact quotient is necessary.",
"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\operatorname{Aut}\\left(M\\right)$ has a dense subgroup of index 2, see [23].",
"Question 7.4 Let $M$ be $\\omega $ -categorical and $N$ an $\\omega $ -categorical expansion of $M$ .",
"Set $G=\\operatorname{Aut}\\left(M\\right)$ and $H=\\operatorname{Aut}\\left(N\\right)\\le G$ .",
"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open?",
"Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.",
"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.",
"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].",
"We would also like to thank Daoud Siniora for his comments.",
"Finally, we would like to thank the anonymous referee for his comments."
],
[
"Having finite topological rank ",
"In this section we will find some criteria that ensure that $G$ has finite topological rank."
],
[
"$\\omega $ -categorical stable theories ",
"Proposition 5.1 If $T$ is stable $\\omega $ -categorical, $M\\models T$ is countable and $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is finite, then $\\operatorname{Aut}\\left(M\\right)$ has finite topological rank.",
"Without loss of generality, $M=M^{\\operatorname{eq}}$ (if $S\\subseteq \\operatorname{Aut}\\left(M^{\\operatorname{eq}}\\right)$ generates a dense subgroup, then $S\\upharpoonright M=\\left\\lbrace f\\upharpoonright M\\,|\\,f\\in S\\right\\rbrace $ generates a dense subgroup of $\\operatorname{Aut}\\left(M\\right)$ ).",
"Let $N=M_{\\operatorname{acl}\\left(\\emptyset \\right)}$ (i.e., name the elements in $\\operatorname{acl}\\left(\\emptyset \\right)$ ).",
"Then $N$ is $\\omega $ -categorical by Propsition REF .",
"Then in $N$ , $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , so by Example REF , there is a canonical independence relation in $N$ , so $G^{0}=\\operatorname{Aut}\\left(N\\right)$ is topologically 2-generated by Corollary REF , say by $\\left\\lbrace f_{1},f_{2}\\right\\rbrace $ .",
"Now, $\\operatorname{Aut}\\left(M\\right)/G^{0}$ is finite by assumption, so let $S\\subseteq \\operatorname{Aut}\\left(M\\right)$ be a finite set of representatives.",
"Then $S\\cup \\left\\lbrace f_{1},f_{2}\\right\\rbrace $ generates a dense subgroup $\\operatorname{Aut}\\left(M\\right)$ : given two finite tuples $\\bar{a},\\bar{b}$ from $M$ such that $\\bar{a}\\equiv \\bar{b}$ , there is an automorphism $\\sigma \\in \\operatorname{Aut}\\left(M\\right)$ such that $\\sigma \\left(\\bar{a}\\right)=\\bar{b}$ .",
"Also, there is some $f\\in S$ such that $f^{-1}\\sigma \\in \\operatorname{Aut}\\left(N\\right)$ .",
"Hence for some $g$ in the group generated by $\\left\\lbrace f_{1},f_{2}\\right\\rbrace $ , $g\\left(\\bar{a}\\right)=f^{-1}\\sigma \\left(\\bar{a}\\right)=f^{-1}\\left(\\bar{b}\\right)$ , so $fg\\left(\\bar{a}\\right)=\\bar{b}$ .",
"The following fact implies immediately the next result.",
"Fact 5.2 [9] If $T$ is $\\omega $ -categorical and $\\omega $ -stable and $M\\models T$ is countable, then $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is finite.",
"Corollary 5.3 If $T$ is $\\omega $ -stable and $\\omega $ -categorical and $M\\models T$ is countable, then $\\operatorname{Aut}\\left(M\\right)$ has finite topological rank."
],
[
"Reducing finite topological rank to expansions",
"Suppose that $M$ is countable let $G=\\operatorname{Aut}\\left(M\\right)$ .",
"We now want to explore the idea that perhaps by expanding $M$ (i.e., moving to a subgroup), we can show that the topological rank of $G$ is small by showing that the rank of the automorphism group of the expansion is.",
"Suppose that $H\\le G$ .",
"If $\\left(G,H\\right)$ has a compact quotient (see Definition REF ), then we cannot hope to deduce anything.",
"For example, by Proposition REF we have that $G^{0}$ acts oligomorphically on $M$ and it can be that $G^{0}$ has a cyclically dense conjugacy class (so topological rank 2) while $G/G^{0}=\\left(\\mathbb {Z}/2\\mathbb {Z}\\right)^{\\omega }$ (so $G$ is not topologically finitely generated) — this happens in the example described in in Remark REF , see Example REF .",
"Indeed, we will see that $\\left(G,H\\right)$ having a compact quotient is the only obstruction."
],
[
"$\\omega $ -categorical structures with finitely many reducts",
"Theorem 5.4 Suppose that $H\\le G$ is closed and that $\\left(G,H\\right)$ has no compact quotients.",
"If there are only finitely many closed groups between $G$ and $H$ then there is some $g\\in G$ such that $H\\cup \\left\\lbrace g\\right\\rbrace $ topologically generate $G$ .",
"Remark 5.5 The condition of having finitely many closed groups in the theorem holds when for instance $M$ is a reduct of an $\\omega $ -categorical structure $M^{\\prime }$ where $H=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ , and $M^{\\prime }$ has only finitely many reducts up to bi-definability.",
"Let $\\left\\lbrace H_{i}\\,|\\,i<n\\right\\rbrace $ be the family of closed proper subgroups of $G$ containing $H$ (which is finite by assumption).",
"If $\\left[G:H_{i}\\right]<\\infty $ for some $i<n$ , then there would be a closed normal proper subgroup $N_{i}\\trianglelefteq G$ of finite index such that $N_{i}\\le H_{i}$ (in general, if $H^{\\prime }\\le G$ is closed of finite index, then there is a closed normal subgroup $N\\le H^{\\prime }$ , $N\\trianglelefteq G$ such that $\\left[G:N\\right]<\\infty $ .",
"In fact, $N=\\bigcap \\left\\lbrace gH^{\\prime }g^{-1}\\,|\\,g\\in G\\right\\rbrace $ and this intersection is finite as it is the orbit of $H^{\\prime }$ under the action of $G$ on conjugates of $H^{\\prime }$ and its stabilizer contains $H^{\\prime }$ ).",
"But then $N_{i}H=G$ by assumption and Proposition REF , so $G=N_{i}H\\subseteq H_{i}H=H_{i}$ contradicting the fact that $H_{i}$ was a proper subgroup.",
"By a theorem of Neumann [29], there is some $g\\in G\\backslash \\bigcup \\left\\lbrace H_{i}\\,|\\,i<n\\right\\rbrace $ .",
"If $G\\ne \\operatorname{cl}\\left(\\left\\langle H\\cup \\left\\lbrace g\\right\\rbrace \\right\\rangle \\right)$ (the topological closure of the group generated by $H\\cup \\left\\lbrace g\\right\\rbrace $ ), then $\\operatorname{cl}\\left(\\left\\langle H\\cup \\left\\lbrace g\\right\\rbrace \\right\\rangle \\right)$ is one of the groups $H_{i}$ , contradicting the choice of $g$ .",
"Corollary 5.6 If $G$ and $H$ are as in Theorem REF and $H$ has finite topological rank then so does $G$ .",
"By Example REF , in the $\\omega $ -categorical context we get that if $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{dcl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ in $M$ and $M^{\\prime }$ is an expansion having finitely many reducts, then we can apply Corollary REF .",
"This is the case, for instance, when $M^{\\prime }$ is $\\left(\\mathbb {Q},<\\right)$ (see [18]).",
"By Lemma [18], an example of such a reduct of DLO is given by the countable dense circular order, which is the structure with universe $\\mathbb {Q}$ , and a ternary relation $C\\left(x,y,z\\right)$ given by $C\\left(x,y,z\\right)\\Leftrightarrow x<y<z\\vee y<z<x\\vee z<x<y$ .",
"Corollary 5.7 $\\operatorname{Aut}\\left(\\mathbb {Q},C\\right)$ has topological rank $\\le 3$ , but $\\left(\\mathbb {Q},C\\right)$ has no CIR.",
"We only have to show that it has no CIR.",
"By Lemma REF , if there was a CIR, then in particular there would be a type of a single element $q\\left(x\\right)$ over $\\mathbb {Q}$ which does not split over $\\emptyset $ .",
"But by quantifier elimination, every tuple of two distinct elements have the same type (i.e., $\\operatorname{Aut}\\left(\\mathbb {Q},C\\right)$ acts 2-transitively on $\\mathbb {Q}$ ).",
"Now, $q$ cannot be realized in $\\mathbb {Q}$ and must contain $C\\left(0,x,1\\right)$ or $C\\left(1,x,0\\right)$ , hence both, which is a contradiction.",
"Remark 5.8 For any point $a\\in \\mathbb {Q}$ , the expansion $\\left(\\mathbb {Q},C,a\\right)$ does have a CIR.",
"Indeed, in this case $C$ defines a dense linear order with no endpoints on $\\mathbb {Q}\\backslash \\left\\lbrace a\\right\\rbrace $ by $b<c\\iff C\\left(a,b,c\\right)$ .",
"Since $\\left(\\mathbb {Q},<\\right)$ has a CIR $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ (Example \\ref {exa:DLO has groovy}),we can define $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C*B$ by $ A{ a} $\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $C{ a} B{ a} $.Since for every finite tuples $ b,c$, $ bc$ in the expansioniff $ baca$ in the order, it followseasily that $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*$ is a CIR.$ An even closer look at the reducts of DLO, gives the following result.",
"Corollary 5.9 Every closed supergroup of $\\operatorname{Aut}\\left(\\mathbb {Q},<\\right)$ has topological rank $\\le 3$ .",
"The diagram in [18] of the lattice of closed groups between $\\operatorname{Aut}\\left(\\mathbb {Q},<\\right)$ and $\\operatorname{Aut}\\left(\\mathbb {Q},=\\right)$ shows that any such group contains at most two incomparable closed subgroups.",
"Since no group can be a union of two of its proper subgroups, we do not need to use Neumanns's lemma in the proof of Theorem REF above, allowing us to drop the assumption that $\\left(G,H\\right)$ has no compact quotients."
],
[
"A general reduction theorem",
"In the next theorem we drop the assumption of having finitely many reducts of the expansion (i.e., of having finitely many groups between $H$ and $G$ ), but we compensate for it by assuming that $H$ acts oligomorphically on $M$ and increasing the number of generators by 1.",
"Fact 5.10 [9] Suppose that $M$ is a countable $\\omega $ -saturated structure.",
"Then for any $A,B\\subseteq M$ , there is some $A^{\\prime }$ (in the monster model $\\mathfrak {C}$ , see just above Section REF ) such that $A^{\\prime }\\equiv A$ and $A^{\\prime }\\cap B\\subseteq \\operatorname{acl}\\left(\\emptyset \\right)$ .",
"Theorem 5.11 Suppose as usual that $M$ is countable and $\\omega $ -categorical and let $G=\\operatorname{Aut}\\left(M\\right)$ .",
"Suppose that $H\\le G$ is closed and acts oligomorphically on $M$ and that $\\left(G,H\\right)$ has no compact quotients.",
"Then there are $g_{1},g_{2}\\in G$ such that $H\\cup \\left\\lbrace g_{1},g_{2}\\right\\rbrace $ topologically generates $G$ .",
"Let $M^{\\prime }$ be an $\\omega $ -categorical expansion of $M$ to some language $L^{\\prime }$ containing $L$ (the language of $M$ ) such that $H=\\operatorname{Aut}\\left(M^{\\prime }\\right)$ .",
"We use $^{\\prime }$ to indicate the expansion.",
"In particular, $\\mathfrak {C}^{\\prime }$ denotes the expansion of $\\mathfrak {C}$ to $L^{\\prime }$ .",
"By Fact REF , there is some $M_{0}$ such that $M_{0}\\equiv M$ and $M_{0}^{\\operatorname{eq}}\\cap M^{\\operatorname{eq}}=\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ (apply the fact in $\\mathfrak {C}^{\\operatorname{eq}}$ ).",
"There is some automorphism $\\sigma $ of $\\mathfrak {C}$ such that $\\sigma \\left(M_{0}\\right)=M$ .",
"Let $N_{0}^{\\prime }$ be a countable model containing $\\sigma ^{n}\\left(M_{0}\\right)$ for all $n\\in \\mathbb {Z}$ .",
"Let $N_{1}^{\\prime }$ be a countable model containing $\\sigma ^{n}\\left(N_{0}^{\\prime }\\right)$ for all $n\\in \\mathbb {Z}$ .",
"Continue like this and finally let $N_{\\omega }^{\\prime }=\\bigcup \\left\\lbrace N_{i}^{\\prime }\\,|\\,i<\\omega \\right\\rbrace $ .",
"So $M^{\\prime }\\prec N^{\\prime }_{\\omega }\\prec \\mathfrak {C}^{\\prime }$ is countable and $\\sigma \\upharpoonright N{}_{\\omega }\\in \\operatorname{Aut}\\left(N_{\\omega }\\right)$ .",
"By $\\omega $ -categoricity (of $M^{\\prime }$ ) we may assume that $N_{\\omega }^{\\prime }=M^{\\prime }$ : there is some $g_{1}\\in \\operatorname{Aut}\\left(M\\right)$ and $M_{0}^{\\prime }\\prec M^{\\prime }$ such that $g_{1}\\left(M_{0}^{\\operatorname{eq}}\\right)\\cap M_{0}^{\\operatorname{eq}}=\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ .",
"Then $H_{1}=\\operatorname{cl}\\left(\\left\\langle H,g_{1}\\right\\rangle \\right)$ is a closed group acting oligomorphically on $M$ .",
"Also, note that $\\left(G,H_{1}\\right)$ has no compact quotients.",
"Let $M^{\\prime \\prime }$ be the reduct of $M^{\\prime }$ , which is also an expansion of $M$ that corresponds to $H_{1}$ : $\\operatorname{Aut}\\left(M^{\\prime \\prime }\\right)=H_{1}$ .",
"As usual, we use $^{\\prime \\prime }$ to indicate that we work in this expansion.",
"Claim 5.12 If $X\\subseteq M^{n}$ is definable over $\\emptyset ^{\\prime \\prime }$ (i.e., definable in $L^{\\prime \\prime }$ over $\\emptyset $ ) and $M$ -definable (in $L$ ), then it is $\\emptyset $ -definable (in $L$ ).",
"First note that it is enough to show that $X$ is $\\operatorname{acl}_{L}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ -definable (the code $X$ of $X$ belongs to $\\operatorname{dcl}_{L^{\\prime \\prime }}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ and to $\\operatorname{acl}_{L}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , and if it were not in $\\operatorname{dcl}_{L}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ then there would be an automorphism of $M$ moving it, but then by the no-compact quotient assumption there would be an automorphism of $M^{\\prime \\prime }$ moving it as well — contradiction).",
"Now, since $X$ is $\\emptyset ^{\\prime \\prime }$ -definable and $M$ -definable, it is definable over $M_{0}$ (because $M_{0}^{\\prime }\\prec M^{\\prime }$ ), so its code $X\\in M_{0}^{\\operatorname{eq}}$ .",
"In addition, $g_{1}\\left(X\\right)=X$ , so $X$ is definable over $g_{1}\\left(M_{0}\\right)$ , hence $X\\in g_{1}\\left(M_{0}^{\\operatorname{eq}}\\right)$ .",
"Together it is in $\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)$ , which is what we wanted.",
"Now we construct $g_{2}$ by back-and-forth to ensure that $\\operatorname{cl}\\left(\\left\\langle H_{1},g_{2}\\right\\rangle \\right)=G$ .",
"Suppose that we have constructed $g_{2}\\upharpoonright A$ for some finite set $A$ .",
"Let $O$ be an orbit of the action of $G$ on $M^{m}$ , and we write it as $O=\\bigcup \\left\\lbrace O_{i}\\,|\\,i<n\\right\\rbrace $ where the $O_{i}$ 's are the orbits of the action of $H_{1}$ (recall that $H_{1}$ acts oligomorphically on $M$ , so there are only finitely many such orbits).",
"Claim 5.13 For any subset $s\\subsetneq n$ there are $a,b\\in O$ such that $a\\in O_{s}=\\bigcup \\left\\lbrace O_{i}\\,|\\,i\\in s\\right\\rbrace ,b\\in O_{n\\backslash s}$ , and $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle a,b\\right\\rangle \\right\\rbrace $ or $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle b,a\\right\\rangle \\right\\rbrace $ is an elementary map.",
"Note that $O_{s}$ is $\\emptyset ^{\\prime \\prime }$ -definable.",
"As it is not $\\emptyset $ -definable (because $s\\subsetneq n$ ), it is also not $M$ -definable by Claim REF .",
"In particular, it is not $A$ -definable.",
"Hence there are $a_{0}\\in O_{s},a_{1}\\in O_{n\\backslash s}$ such that $a_{0}\\equiv _{A}a_{1}$ .",
"There is some $b$ such that $a_{0}A\\equiv a_{1}A\\equiv bg_{2}\\left(A\\right)$ .",
"If $b\\in O_{s}$ , then $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle a_{1},b\\right\\rangle \\right\\rbrace $ is the required map.",
"Otherwise, pick $g_{2}\\upharpoonright A\\cup \\left\\lbrace \\left\\langle a_{0},b\\right\\rangle \\right\\rbrace $ .",
"In the back-and-forth construction of $g_{2}$ , we deal with all these orbits (for every $m<\\omega $ , there are only finitely many) and all these subsets $s$ and increase $g_{2}$ according to Claim REF .",
"We claim that $g_{2}$ is such that $\\operatorname{cl}\\left(\\left\\langle H_{1},g_{2}\\right\\rangle \\right)=G$ .",
"Indeed, it is enough to show that every orbit $O$ of $G$ is also an orbit of $\\left\\langle H_{1},g_{2}\\right\\rangle $ .",
"The orbit $O$ can be written as $\\bigcup \\left\\lbrace O_{i}\\,|\\,i<n\\right\\rbrace $ where the $O_{i}$ 's are the orbits of $H_{1}$ , and also as $\\bigcup \\left\\lbrace O^{\\prime }_{i}\\,|\\,i\\in I\\right\\rbrace $ where the $O^{\\prime }_{i}$ 's are orbits of $\\left\\langle H_{1},g_{2}\\right\\rangle $ .",
"Each such $O_{i}^{\\prime }$ is itself a union of $H_{1}$ -orbits, so has the form $O_{s}$ for some $s\\subseteq n$ .",
"But by construction, if $s\\ne n$ there are tuples $a\\in O_{s},b\\in O_{n\\backslash s}$ such that either $g_{2}$ or $g_{2}^{-1}$ maps $a$ to $b$ — contradiction.",
"So $s=n$ , and $O_{i}^{\\prime }=O$ ."
],
[
"A topological dynamics consequence\nof having a CIR ",
"Definition 6.1 Suppose that $M$ is a countable structure.",
"Call an automorphism $\\sigma \\in G$ shifty if there is some invariant binary relation on finite sets in $M$ , $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ (the base will always be $$) such that:\\begin{itemize}\\item (Monotonicity) If A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{itemize}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ and $ A'A$, $ B'B$then $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B'$.\\item (Right existence) For every finite tuple $ a$ there is some $ a'a$such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$ (by this we mean that sets enumerated by $ a$,$ a'$ are independent).\\item (Right shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$, then there exists some $ n<$such that $ b'An(b)$.$ Lemma 6.2 If $\\sigma $ is shifty then it also satisfies: (Left existence) For every finite tuple $a$ there is some $a^{\\prime }\\equiv a$ such that $a^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.\\item (Left shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ b'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$, then there exists some $ n<$such that $ b'A-n(b)$.$ Suppose that $\\sigma $ is shifty, as witnessed by $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.",
"Given $ a$,there is some $ a'a$ such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Applying an automorphismtaking $ a'$ to $ a$ we get some $ a”a$ such that $ a”$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$,which shows left existence.$ As for left shiftiness, suppose that $A$ is finite and enumerated by $a$ , $b,b^{\\prime }$ are finite tuples such that $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ and $ bb'$.Then applying an automorphism, we get some $ a'$ such that $ ab'a'b$,so $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Hence for some $ n<$, $ a'bn(a)$.From $ a'bn(a)b$ we get that $ ab'a'b-n(a')-n(b)a-n(b)$,i.e., $ b'A-n(b)$.$ Proposition 6.3 The automorphism $\\sigma $ is a shifty automorphism on $M$ iff for any type $p\\in S\\left(\\emptyset \\right)$ (with finitely many variables), letting $Y_{a}=\\bigcap \\left\\lbrace \\bigcup \\left\\lbrace \\operatorname{tp}\\left(a,\\sigma ^{n}\\left(a^{\\prime }\\right)\\right)\\,|\\,n<\\omega \\right\\rbrace \\,|\\,a^{\\prime }\\equiv a\\right\\rbrace $ for any $a\\models p$ , the intersection $Y_{p}=\\bigcap \\left\\lbrace Y_{a}\\,|\\,a\\models p\\right\\rbrace $ is nonempty.",
"Suppose that $\\sigma $ is shifty, and fix some type $p\\in S\\left(\\emptyset \\right)$ .",
"Let $a\\models p$ .",
"By existence, there is some $a^{\\prime }\\equiv a$ with $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Let $ q=tp(a,a')$ and fix some $ bp$.Let $ Aut(M)$ map $ a$ to $ b$ and let $ b'=(a')$.We have that $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$ and hence by right shiftiness, $ q=tp(b,b')Yb$.Since $ b$ was arbitrary, $ qYp$.$ Suppose that the right hand side holds.",
"Given a finite tuple $a$ and $a^{\\prime }\\equiv a$ , write $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ iff $ tp(a,a')Yp$where $ p=tp(a/)$.",
"For general finite sets $ A,B$,write $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ iff there is some $ C$ containing $ A$ and $ C'$containing $ B$ such that $ C$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*C'$.",
"Obviously, $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is invariantand monotone.",
"Right existence follows from the assumption that $ Yp$for all $ pS()$.",
"Right shiftiness: supposethat $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ and $ a”a'$.",
"Then $ tp(a,a')Yp$and in particular it belongs to $ Ya$.",
"By definition of $ Ya$,$ tp(a,a'){ tp(a,n(a”)) | n<} $,so for some $ n<$, $ aa'ain(a”)$.$ Proposition 6.4 If $M$ is an ultrahomogeneous structure and $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR on finite subsets of $ M$ which respects substructures,then there exists a shifty automorphism $$ on $ M$, as witnessedby $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.$ Monotonicity and right existence are parts of the properties of a CIR, so we only have to prove right shiftiness.",
"Suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$and $ b'b$.",
"By the proof of Theorem \\ref {thm:existence of repulsive automorphism-ultrahomogeneous},the repulsive automorphism $$ constructed there satisfies thatfor some $ n<$, $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $n(b')$.",
"By stationarity,$ bA(b')$.$ Recall the definitions of flow and subflow from Section REF .",
"Theorem 6.5 Let $M$ be a countable homogeneous structure and $G=\\operatorname{Aut}\\left(M\\right)$ .",
"Suppose that $\\sigma \\in G$ is a shifty automorphism and that $\\left(X,d\\right)$ is a compact metric $G$ -flow.",
"Then for every $x_{*}\\in X$ there is some conjugate $\\sigma _{*}\\in G$ of $\\sigma $ such that: (*) Both $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ and $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contain a subflow of $X$ .",
"Remark 6.6 Note that Theorem REF implies that both $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ and $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ contain a subflow of $X$ : if e.g., $Y_{0}$ is a flow contained in the left space, then $GY_{0}=Y_{0}$ , so $\\sigma _{*}^{-k}\\left(Y_{0}\\right)\\subseteq Y_{0}\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ , hence $Y_{0}\\subseteq \\sigma _{*}^{k}\\left(\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace \\right)=\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace $ .",
"Before the proof we note the following useful lemma.",
"Lemma 6.7 Suppose that $G$ is a topological group acting continuously on a compact metric space $\\left(X,d\\right)$ .",
"Then for every $0<\\varepsilon $ there is some open neighborhood $U$ of $\\operatorname{id}\\in G$ such that for every $g,h\\in G$ if $gh^{-1}\\in U$ then for all $x\\in X$ we have that $d\\left(gx,hx\\right)<\\varepsilon $ .",
"It is enough to show that there is some open neighborhood $U$ of $\\operatorname{id}$ such that if $g\\in U$ then for all $x\\in X$ , $d\\left(gx,x\\right)<\\varepsilon $ (since then if $gh^{-1}\\in U$ then $d\\left(gh^{-1}\\left(hx\\right),hx\\right)<\\varepsilon $ ).",
"For every $x\\in X$ , there is some neighborhood $V_{x}$ of $x$ in $X$ and some neighborhood $U_{x}$ of $\\operatorname{id}$ in $G$ such that for all $g\\in U_{x}$ , $x^{\\prime }\\in V_{x}$ , $d\\left(gx^{\\prime },x^{\\prime }\\right)<\\varepsilon $ .",
"By compactness, a finite union of $V_{x}$ 's covers $X$ .",
"Let $U$ be the intersection of the corresponding $U_{x}$ 's.",
"Suppose that $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ witnesses that $$ is shifty.",
"Let $ G0$be a countable dense subset of $ G$, enumerated as $ gi | i<$,such that $ g0=id$.$ We construct an automorphism $\\tau :M\\rightarrow M$ by back and forth such that eventually $\\sigma _{*}=\\tau ^{-1}\\sigma \\tau $ and such that at each finite stage, $\\tau $ will be an elementary map.",
"For the construction it is actually better to think of the domain and range of $\\tau $ as two different structures, so we have $M=M_{*}$ and suppose that $\\sigma :M\\rightarrow M$ , $\\sigma _{*}:M_{*}\\rightarrow M_{*}$ and $\\tau :M_{*}\\rightarrow M$ .",
"The subscript $*$ will denote tuples from $M_{*}$ throughout.",
"Suppose that we have constructed a partial elementary map $f:A_{*}\\rightarrow A$ (that will be part of $\\tau $ eventually) with $A_{*}\\subseteq M_{*},A\\subseteq M$ finite, enumerated by $a_{*},a$ .",
"Here is the main tool in the construction.",
"Claim 6.8 Suppose that $b_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a*$ and$ b*'ba$.",
"Let $ b*=f-1(b)$.",
"Thenthere is $ k<$ and an extension $ f'$ of $ f$ such that anyautomorphism $ '$ extending $ f'$ will satisfy that for $ *'='-1'$,$ *'k(b'*)=b*$.$ Similarly, if $a_{*}\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b*'$ then there is some $ k<$ andan extension $ f'$ of $ f$ such that any automorphism $ '$ extending$ f'$ will satisfy that for $ *'='-1'$,$ *'k(b*)=b'*$.$ First, find some tuple $b^{\\prime }$ in $M$ such that $b^{\\prime }a\\equiv b_{*}^{\\prime }a_{*}$ .",
"In particular, $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.",
"By left shiftiness, there is some $ k<$such that $ -k(b)ab'ab*'a*$.Extend $ f$ to $ f'$ which sends $ b*'$ to $ -k(b)$.Then, for any $ '$ extending $ f'$, $ '-1k'(b*')='-1(b)=b*$.$ The second statement is proved similarly, using right shiftiness.",
"We will make sure that for each $n<\\omega $ , the following condition holds.",
"$\\star $ There are $k_{n,0},\\ldots ,k_{n,n-1}<\\omega $ such that for all $i<n$ , $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)\\right)\\right)<1/n$ and $k_{n,0}^{\\prime },\\ldots ,k_{n,n-1}^{\\prime }<\\omega $ such that for all $i<n$ , $d\\left(\\sigma _{*}^{-k_{n,i}^{\\prime }}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{-k_{n,0}^{\\prime }}\\left(x_{*}\\right)\\right)\\right)<1/n$ .",
"Why is $\\star $ enough?",
"Let $y_{n}=\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)$ , and let $y$ be a limit of some subsequence $\\left\\langle y_{n_{j}}\\,|\\,j<\\omega \\right\\rangle $ (which exists by compactness), then $Gy\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ (so $\\operatorname{cl}\\left(Gy\\right)$ is a subflow): given $g\\in G$ and $0<\\varepsilon $ , first find an open neighborhood $U\\subseteq G$ of $g$ such that if $h\\in U$ then $d\\left(gx,hx\\right)<\\varepsilon /4$ for all $x\\in X$ (this $U$ is given to us by Lemma REF : it is $\\left(g^{-1}V\\right)^{-1}$ where $V$ is an open neighborhood of $\\operatorname{id}$ such that if $gh^{-1}\\in V$ , $d\\left(gx,hx\\right)<\\varepsilon /4$ ).",
"Take $n$ so large that $g_{i}\\in U$ for some $i<n$ and $1/n<\\varepsilon /4$ , and find $n_{j}$ even larger so that $d\\left(g_{i}y,g_{i}y_{n_{j}}\\right)<\\varepsilon /4$ .",
"Then $d\\left(gy,g_{i}y\\right)<\\varepsilon /4$ , $d\\left(g_{i}y,g_{i}y_{n_{j}}\\right)<\\varepsilon /4$ and $d\\left(g_{i}y_{n_{j}},\\sigma _{*}^{k_{n_{j},i}}\\left(x_{*}\\right)\\right)<\\varepsilon /4$ .",
"Together, $d\\left(gy,\\sigma _{*}^{k_{n_{j},i}}\\left(x_{0}\\right)\\right)<3\\varepsilon /4<\\varepsilon $ , which means that $gy$ is in the closure.",
"Similarly, if $y^{\\prime }$ is a limit of a subsequence of $\\sigma _{*}^{-k_{n,0}^{\\prime }}\\left(x_{*}\\right)$ , then $Gy^{\\prime }\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ .",
"So we consider $f:A_{*}\\rightarrow A$ a partial elementary map.",
"Our task now is to deal with $n<\\omega $ .",
"Let $\\varepsilon =1/n$ .",
"Let $A_{*}\\subseteq C_{*}\\subseteq M_{*}$ be finite such that if $g^{-1}\\upharpoonright C_{*}=h^{-1}\\upharpoonright C_{*}$ then $d\\left(gx,hx\\right)<\\varepsilon /4$ for all $x\\in X$ and any $g,h\\in G$ (this is by Lemma REF ).",
"Let $z_{0},\\ldots ,z_{l-1}$ be such that $\\bigcup \\left\\lbrace B\\left(z_{j},\\varepsilon /4\\right)\\,|\\,j<l\\right\\rbrace $ cover $X$ , and write $B_{j}=B\\left(z_{j},\\varepsilon /4\\right)$ .",
"Let $c_{*}$ be a finite tuple enumerating $C_{*}$ .",
"For every $c_{*}^{\\prime }\\equiv c_{*}$ , we say that $c_{*}^{\\prime }$ has color $j<l$ if $j$ is least such that there is $g\\in G$ such that $g\\left(c_{*}^{\\prime }\\right)=c_{*}$ and $gx_{*}\\in B_{j}$ .",
"Note that by the choice of $c_{*}$ , if $g^{\\prime }\\left(c_{*}^{\\prime }\\right)=c_{*}$ then $g^{\\prime }g^{-1}\\upharpoonright C_{*}=\\operatorname{id}$ , so $g^{\\prime }x_{0}\\in B\\left(z_{j},\\varepsilon /2\\right)$ .",
"Let $D_{*}=\\bigcup \\left\\lbrace g_{i}^{-1}\\left(C_{*}\\right)\\,|\\,i<n\\right\\rbrace $ .",
"Note that $C_{*}\\subseteq D_{*}$ because $g_{0}=\\operatorname{id}$ .",
"Let $d_{*}$ enumerate $D_{*}$ .",
"For any $d_{*}^{\\prime }\\equiv d_{*}$ and $s\\subseteq l$ , we say that $d_{*}^{\\prime }$ has color $s$ if $\\left\\lbrace j<l\\,|\\,c_{*}^{\\prime }\\equiv c_{*},c_{*}^{\\prime }\\subseteq d_{*}^{\\prime },c_{*}^{\\prime }\\text{ has color }j\\right\\rbrace =s$ .",
"By left existence, there is some $s_{0}\\subseteq l$ such that for every finite set $S\\subseteq M_{*}$ , there is some $d_{*}^{\\prime }\\equiv d_{*}$ with $d_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $S$ and $ d*'$ has color $ s0$.$ Let $d_{*}^{\\prime }\\equiv d_{*}$ be of color $s_{0}$ such that $d_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $d*$.For $ i<n$, let $ c*,i'd*'$ be the tuple correspondingto $ gi-1(c*)$, so in particular $ c*,i'c*$.Let $ ji<l$ be the color of $ c*,i'$.",
"By the choice of $ s0$,for every finite set $ SM*$, there is some $ c*'c*$such that $ c'*$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $S$ and $ c*'$ has color $ ji$.$ Since $M$ is homogeneous we can extend $f$ in such a way so that its domain equals $D_{*}$ .",
"By Claim REF (the first part), there is some $k_{n,0}<\\omega $ and an extension $f_{0}$ of $f$ that ensures that $\\sigma _{*}^{k_{n,0}}\\left(d_{*}^{\\prime }\\right)=d_{*}$ .",
"Starting with $f_{0}$ , we construct an increasing sequence $\\left\\langle f_{i}\\,|\\,i<n\\right\\rangle $ as follows.",
"Suppose we have $f_{i}$ whose domain is $D_{*,i}$ .",
"Find some $c^{\\prime \\prime }_{*,i+1}\\equiv c_{*}$ of color $j_{i+1}$ such that $c_{*,i+1}^{\\prime \\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $D*,i$.By Claim \\ref {claim:Using shiftiness}, we can find $ kn,i+1<$and extend $ fi$ to $ fi+1$ which ensures that $ *kn,i+1(c*,i+1”)=c*$.$ Now we have the first part of $\\star $ : we need to check that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)\\right)\\right)<\\varepsilon $ for all $i<n$ .",
"For $i=0$ this is clear since $g_{0}=\\operatorname{id}$ , so we may assume that $i>0$ .",
"As $\\sigma _{*}^{k_{n,i}}\\left(c_{*,i}^{\\prime \\prime }\\right)=c_{*}$ , it follows that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{0}\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .",
"Similarly, as $\\sigma _{*}^{k_{n,0}}\\left(c_{*,i}^{\\prime }\\right)=g_{i}^{-1}\\left(c_{*}\\right)$ , we have that $d\\left(g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{0}\\right)\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .",
"Together, we are done.",
"Now we have to take care of the other half of $\\star $ .",
"This is done similarly, using right existence and the second part of Claim REF .",
"The following proposition explains why we needed to take a conjugate of $\\sigma $ .",
"The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.",
"In Section REF , we mentioned that it is a Ramsey structure.",
"Note that the underlying order is dense (by Proposition REF ).",
"Proposition 6.9 Let $M=\\left(V,<,R\\right)$ be the countable ordered random graph.",
"Then there is no automorphism $\\sigma \\in G=\\operatorname{Aut}\\left(M\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\in X$ .",
"First we find $a\\ne b$ in $M$ such that $\\sigma ^{n}\\left(a\\right)\\ne \\sigma ^{m}\\left(b\\right)$ for all $m,n\\in \\mathbb {Z}$ .",
"To do that, take any $a\\in M$ .",
"Then $\\left\\lbrace \\sigma ^{n}\\left(a\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is discrete (in the order sense: it is either a $\\mathbb {Z}$ -chain or just $a$ ).",
"Since $\\left(V,<\\right)$ is dense, there is some $b\\ne \\sigma ^{n}\\left(a\\right)$ for all $n\\in \\mathbb {Z}$ .",
"It follows that $b$ is as required.",
"Let $X=S_{x}\\left(M\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).",
"Let $p\\in X$ be any completion of the partial type $\\left\\lbrace R\\left(x,\\sigma ^{n}\\left(a\\right)\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace \\cup \\left\\lbrace \\lnot R\\left(x,\\sigma ^{m}\\left(b\\right)\\right)\\,|\\,m\\in \\mathbb {Z}\\right\\rbrace $ .",
"Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\in \\operatorname{cl}\\left\\lbrace \\sigma ^{n}\\left(p\\right)\\,|\\,n<\\omega \\right\\rbrace $ which is a fixed point of $G$ .",
"In other words, $p_{0}$ is an invariant type over $M$ .",
"However $R\\left(x,a\\right)\\wedge \\lnot R\\left(x,b\\right)\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).",
"The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.",
"If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\left(X,x_{0}\\right)$ .",
"Thus, there would be a conjugate $\\sigma _{*}$ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow.",
"But then if $\\left(Y,y_{0}\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\pi :X\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .",
"Thus, $\\pi $ maps $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ to $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(y\\right)\\,|\\,n<\\omega \\right\\rbrace $ , and the latter contains a $G$ -subflow.",
"Thus we get that $\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .",
"Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\left\\lbrace <,\\wedge \\right\\rbrace $ , and let $M\\models T$ be countable.",
"Then $\\operatorname{Aut}\\left(M\\right)$ has no shifty automorphism.",
"In particular, $M$ has no CIR.",
"Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .",
"Suppose that $\\sigma $ was shifty.",
"Let $\\bar{m}=\\left\\langle m\\,|\\,m\\in M\\right\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\bar{x}=\\left\\langle x_{m}\\,|\\,m\\in M\\right\\rangle $ .",
"Let $X=S_{\\bar{m}}\\left(M\\right)$ be the space of $\\bar{x}$ -complete types $p$ over $M$ such that $p\\upharpoonright \\emptyset =\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .",
"Then $X$ is a compact metric space.",
"Let $x_{*}=\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .",
"By Theorem REF , there is some conjugate $\\tau $ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\tau ^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{+}\\subseteq X$ and similarly, $\\operatorname{cl}\\left\\lbrace \\tau ^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{-}$ .",
"By Proposition REF , $\\tau $ fixes a branch or a point.",
"Suppose that $\\tau \\left(m\\right)=m$ for some $m\\in M$ .",
"Then for every $p\\in Y^{+}$ , $p\\models x_{m}=m$ .",
"However $G=\\operatorname{Aut}\\left(M\\right)$ acts transitively on $M$ , so we have a contradiction.",
"Now suppose that $\\tau $ fixes a branch $B\\subseteq M$ , but does not fix any point.",
"Suppose that $\\tau \\left(m\\right)>m$ for some $m\\in B$ .",
"Then $\\tau ^{n}\\left(m\\right)>m$ for all $n<\\omega $ , so for any $p\\in Y^{+}$ , $p\\models x_{m}>m$ .",
"There is some $m^{\\prime }\\in M$ such that $m^{\\prime }>m$ and $m^{\\prime }\\notin B$ .",
"Since $m<\\tau ^{n}\\left(m\\right)\\in B$ for all $n<\\omega $ , it follows that $p\\models x_{m}\\wedge m^{\\prime }=m$ for all $p\\in Y^{+}$ .",
"Let $\\tau ^{\\prime }\\in G$ fix $m$ and map $m^{\\prime }$ to $B$ .",
"Then $\\tau ^{\\prime }\\left(p\\right)\\models \\left(x_{m}\\wedge \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\right)=m<x_{m}$ .",
"But $\\tau ^{\\prime }\\left(p\\right)\\in Y^{+}$ , so $\\tau ^{\\prime }\\left(p\\right)\\models x_{m}\\le \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\vee \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\le x_{m}$ , which is a contradiction.",
"If, on the other hand $\\tau \\left(m\\right)<m$ , then $\\tau ^{-1}\\left(m\\right)>m$ , so we can apply the same argument to $Y^{-}$ .",
"For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .",
"In addition, letting $H=\\operatorname{Aut}\\left(N\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).",
"In addition, if $B\\subseteq N$ is a branch, $m\\in B$ , there is always some $m^{\\prime }>m$ , $m^{\\prime }\\notin B$ and for any $n^{\\prime }>m$ in $B$ , $m^{\\prime }m\\equiv n^{\\prime }m$ .",
"Hence, we can apply Proposition REF and the same proof will work.",
"Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\omega $ -categorical structures.",
"We state here a few general conjectures and questions.",
"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.",
"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.",
"Conjecture 7.1 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion which admits a CIR.",
"Suppose that $M$ is a structure and $\\mathfrak {C}$ a monster model for $\\operatorname{Th}\\left(M\\right)$ .",
"The group of Lascar strong automorphisms of $M$, denoted by $\\operatorname{Aut}f\\left(M\\right)$ is the group of automorphisms of $M$ generated by the set $\\left\\lbrace \\sigma \\upharpoonright M\\,|\\,\\exists N\\prec \\mathfrak {C},\\left|N\\right|=\\left|T\\right|,\\sigma \\upharpoonright N=\\operatorname{id}\\right\\rbrace $ .",
"If $\\sigma $ is Lascar strong, then $\\sigma \\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{id}$ so $\\operatorname{Aut}f\\left(M\\right)$ is contained in $G^{0}$ .",
"However, there are examples (even $\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\operatorname{Aut}f\\left(M\\right)$ , see [16], [31].",
"The Lascar group of $M$ is the quotient $\\operatorname{Aut}\\left(M\\right)/\\operatorname{Aut}f\\left(M\\right)$ .",
"For more on the Lascar group, see [39].",
"In the $\\omega $ -categorical case, the quotient $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is also called the compact Lascar group.",
"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$.",
"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.",
"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.",
"Forinstance, by Lemma \\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\operatorname{Aut}\\left(M\\right)$ has no compact quotients then it has finite topological rank.",
"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.",
"Thus, potentially, the Lascar group — as a quotient of $\\operatorname{Aut}\\left(M\\right)$ — can be an obstruction to having finite topological rank.",
"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.",
"Conjecture REF (together with Theorem REF ) implies that it could not.",
"Conjecture 7.2 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion with trivial Lascar group.",
"By the above, this second conjecture is implied by Conjecture REF .",
"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.",
"It would be interesting to investigate other consequences of having a CIR.",
"For instance a CIR might have something to say about normal subgroups.",
"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ for some CIR $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\aleph _{0}}$ is open.",
"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.",
"We refer to [26] for a survey on this.",
"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.",
"In fact the compact quotients are hidden in the stabilizer of a finite set.",
"It seems that one can avoid this counterexample by restricting to dense subgroups.",
"This leads us to the following questions.",
"Question 7.3 Let $M$ be $\\omega $ -categorical such that $G=\\operatorname{Aut}\\left(M\\right)$ has no compact quotient.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open (and hence is equal to $G$ )?",
"Note that the assumption of having no compact quotient is necessary.",
"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\operatorname{Aut}\\left(M\\right)$ has a dense subgroup of index 2, see [23].",
"Question 7.4 Let $M$ be $\\omega $ -categorical and $N$ an $\\omega $ -categorical expansion of $M$ .",
"Set $G=\\operatorname{Aut}\\left(M\\right)$ and $H=\\operatorname{Aut}\\left(N\\right)\\le G$ .",
"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open?",
"Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.",
"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.",
"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].",
"We would also like to thank Daoud Siniora for his comments.",
"Finally, we would like to thank the anonymous referee for his comments.",
"Definition 6.1 Suppose that $M$ is a countable structure.",
"Call an automorphism $\\sigma \\in G$ shifty if there is some invariant binary relation on finite sets in $M$ , $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ (the base will always be $$) such that:\\begin{itemize}\\item (Monotonicity) If A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}\\end{itemize}{\\textstyle \\textstyle x}\\hspace{0.0pt}\\hbox{t}o 0pt{\\hss \\textstyle \\mid \\hss } \\hss $$\\hss $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ and $ A'A$, $ B'B$then $ A'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B'$.\\item (Right existence) For every finite tuple $ a$ there is some $ a'a$such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$ (by this we mean that sets enumerated by $ a$,$ a'$ are independent).\\item (Right shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$, then there exists some $ n<$such that $ b'An(b)$.$ Lemma 6.2 If $\\sigma $ is shifty then it also satisfies: (Left existence) For every finite tuple $a$ there is some $a^{\\prime }\\equiv a$ such that $a^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.\\item (Left shiftiness) If $ A$ is finite and $ b,b'$ are finite tuplessuch that $ b'b$ and $ b'$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$, then there exists some $ n<$such that $ b'A-n(b)$.$ Suppose that $\\sigma $ is shifty, as witnessed by $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.",
"Given $ a$,there is some $ a'a$ such that $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Applying an automorphismtaking $ a'$ to $ a$ we get some $ a”a$ such that $ a”$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$,which shows left existence.$ As for left shiftiness, suppose that $A$ is finite and enumerated by $a$ , $b,b^{\\prime }$ are finite tuples such that $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ and $ bb'$.Then applying an automorphism, we get some $ a'$ such that $ ab'a'b$,so $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Hence for some $ n<$, $ a'bn(a)$.From $ a'bn(a)b$ we get that $ ab'a'b-n(a')-n(b)a-n(b)$,i.e., $ b'A-n(b)$.$ Proposition 6.3 The automorphism $\\sigma $ is a shifty automorphism on $M$ iff for any type $p\\in S\\left(\\emptyset \\right)$ (with finitely many variables), letting $Y_{a}=\\bigcap \\left\\lbrace \\bigcup \\left\\lbrace \\operatorname{tp}\\left(a,\\sigma ^{n}\\left(a^{\\prime }\\right)\\right)\\,|\\,n<\\omega \\right\\rbrace \\,|\\,a^{\\prime }\\equiv a\\right\\rbrace $ for any $a\\models p$ , the intersection $Y_{p}=\\bigcap \\left\\lbrace Y_{a}\\,|\\,a\\models p\\right\\rbrace $ is nonempty.",
"Suppose that $\\sigma $ is shifty, and fix some type $p\\in S\\left(\\emptyset \\right)$ .",
"Let $a\\models p$ .",
"By existence, there is some $a^{\\prime }\\equiv a$ with $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a'$.",
"Let $ q=tp(a,a')$ and fix some $ bp$.Let $ Aut(M)$ map $ a$ to $ b$ and let $ b'=(a')$.We have that $ b$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b'$ and hence by right shiftiness, $ q=tp(b,b')Yb$.Since $ b$ was arbitrary, $ qYp$.$ Suppose that the right hand side holds.",
"Given a finite tuple $a$ and $a^{\\prime }\\equiv a$ , write $a\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ iff $ tp(a,a')Yp$where $ p=tp(a/)$.",
"For general finite sets $ A,B$,write $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $B$ iff there is some $ C$ containing $ A$ and $ C'$containing $ B$ such that $ C$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*C'$.",
"Obviously, $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is invariantand monotone.",
"Right existence follows from the assumption that $ Yp$for all $ pS()$.",
"Right shiftiness: supposethat $ a$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $*a'$ and $ a”a'$.",
"Then $ tp(a,a')Yp$and in particular it belongs to $ Ya$.",
"By definition of $ Ya$,$ tp(a,a'){ tp(a,n(a”)) | n<} $,so for some $ n<$, $ aa'ain(a”)$.$ Proposition 6.4 If $M$ is an ultrahomogeneous structure and $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ is a CIR on finite subsets of $ M$ which respects substructures,then there exists a shifty automorphism $$ on $ M$, as witnessedby $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$.$ Monotonicity and right existence are parts of the properties of a CIR, so we only have to prove right shiftiness.",
"Suppose that $A\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b$and $ b'b$.",
"By the proof of Theorem \\ref {thm:existence of repulsive automorphism-ultrahomogeneous},the repulsive automorphism $$ constructed there satisfies thatfor some $ n<$, $ A$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $n(b')$.",
"By stationarity,$ bA(b')$.$ Recall the definitions of flow and subflow from Section REF .",
"Theorem 6.5 Let $M$ be a countable homogeneous structure and $G=\\operatorname{Aut}\\left(M\\right)$ .",
"Suppose that $\\sigma \\in G$ is a shifty automorphism and that $\\left(X,d\\right)$ is a compact metric $G$ -flow.",
"Then for every $x_{*}\\in X$ there is some conjugate $\\sigma _{*}\\in G$ of $\\sigma $ such that: (*) Both $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ and $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contain a subflow of $X$ .",
"Remark 6.6 Note that Theorem REF implies that both $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ and $\\bigcap \\left\\lbrace \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{0}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace \\,|\\,k<\\omega \\right\\rbrace $ contain a subflow of $X$ : if e.g., $Y_{0}$ is a flow contained in the left space, then $GY_{0}=Y_{0}$ , so $\\sigma _{*}^{-k}\\left(Y_{0}\\right)\\subseteq Y_{0}\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ , hence $Y_{0}\\subseteq \\sigma _{*}^{k}\\left(\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace \\right)=\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,k\\le n<\\omega \\right\\rbrace $ .",
"Before the proof we note the following useful lemma.",
"Lemma 6.7 Suppose that $G$ is a topological group acting continuously on a compact metric space $\\left(X,d\\right)$ .",
"Then for every $0<\\varepsilon $ there is some open neighborhood $U$ of $\\operatorname{id}\\in G$ such that for every $g,h\\in G$ if $gh^{-1}\\in U$ then for all $x\\in X$ we have that $d\\left(gx,hx\\right)<\\varepsilon $ .",
"It is enough to show that there is some open neighborhood $U$ of $\\operatorname{id}$ such that if $g\\in U$ then for all $x\\in X$ , $d\\left(gx,x\\right)<\\varepsilon $ (since then if $gh^{-1}\\in U$ then $d\\left(gh^{-1}\\left(hx\\right),hx\\right)<\\varepsilon $ ).",
"For every $x\\in X$ , there is some neighborhood $V_{x}$ of $x$ in $X$ and some neighborhood $U_{x}$ of $\\operatorname{id}$ in $G$ such that for all $g\\in U_{x}$ , $x^{\\prime }\\in V_{x}$ , $d\\left(gx^{\\prime },x^{\\prime }\\right)<\\varepsilon $ .",
"By compactness, a finite union of $V_{x}$ 's covers $X$ .",
"Let $U$ be the intersection of the corresponding $U_{x}$ 's.",
"Suppose that $\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ witnesses that $$ is shifty.",
"Let $ G0$be a countable dense subset of $ G$, enumerated as $ gi | i<$,such that $ g0=id$.$ We construct an automorphism $\\tau :M\\rightarrow M$ by back and forth such that eventually $\\sigma _{*}=\\tau ^{-1}\\sigma \\tau $ and such that at each finite stage, $\\tau $ will be an elementary map.",
"For the construction it is actually better to think of the domain and range of $\\tau $ as two different structures, so we have $M=M_{*}$ and suppose that $\\sigma :M\\rightarrow M$ , $\\sigma _{*}:M_{*}\\rightarrow M_{*}$ and $\\tau :M_{*}\\rightarrow M$ .",
"The subscript $*$ will denote tuples from $M_{*}$ throughout.",
"Suppose that we have constructed a partial elementary map $f:A_{*}\\rightarrow A$ (that will be part of $\\tau $ eventually) with $A_{*}\\subseteq M_{*},A\\subseteq M$ finite, enumerated by $a_{*},a$ .",
"Here is the main tool in the construction.",
"Claim 6.8 Suppose that $b_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a*$ and$ b*'ba$.",
"Let $ b*=f-1(b)$.",
"Thenthere is $ k<$ and an extension $ f'$ of $ f$ such that anyautomorphism $ '$ extending $ f'$ will satisfy that for $ *'='-1'$,$ *'k(b'*)=b*$.$ Similarly, if $a_{*}\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $b*'$ then there is some $ k<$ andan extension $ f'$ of $ f$ such that any automorphism $ '$ extending$ f'$ will satisfy that for $ *'='-1'$,$ *'k(b*)=b'*$.$ First, find some tuple $b^{\\prime }$ in $M$ such that $b^{\\prime }a\\equiv b_{*}^{\\prime }a_{*}$ .",
"In particular, $b^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $a$.",
"By left shiftiness, there is some $ k<$such that $ -k(b)ab'ab*'a*$.Extend $ f$ to $ f'$ which sends $ b*'$ to $ -k(b)$.Then, for any $ '$ extending $ f'$, $ '-1k'(b*')='-1(b)=b*$.$ The second statement is proved similarly, using right shiftiness.",
"We will make sure that for each $n<\\omega $ , the following condition holds.",
"$\\star $ There are $k_{n,0},\\ldots ,k_{n,n-1}<\\omega $ such that for all $i<n$ , $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)\\right)\\right)<1/n$ and $k_{n,0}^{\\prime },\\ldots ,k_{n,n-1}^{\\prime }<\\omega $ such that for all $i<n$ , $d\\left(\\sigma _{*}^{-k_{n,i}^{\\prime }}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{-k_{n,0}^{\\prime }}\\left(x_{*}\\right)\\right)\\right)<1/n$ .",
"Why is $\\star $ enough?",
"Let $y_{n}=\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)$ , and let $y$ be a limit of some subsequence $\\left\\langle y_{n_{j}}\\,|\\,j<\\omega \\right\\rangle $ (which exists by compactness), then $Gy\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ (so $\\operatorname{cl}\\left(Gy\\right)$ is a subflow): given $g\\in G$ and $0<\\varepsilon $ , first find an open neighborhood $U\\subseteq G$ of $g$ such that if $h\\in U$ then $d\\left(gx,hx\\right)<\\varepsilon /4$ for all $x\\in X$ (this $U$ is given to us by Lemma REF : it is $\\left(g^{-1}V\\right)^{-1}$ where $V$ is an open neighborhood of $\\operatorname{id}$ such that if $gh^{-1}\\in V$ , $d\\left(gx,hx\\right)<\\varepsilon /4$ ).",
"Take $n$ so large that $g_{i}\\in U$ for some $i<n$ and $1/n<\\varepsilon /4$ , and find $n_{j}$ even larger so that $d\\left(g_{i}y,g_{i}y_{n_{j}}\\right)<\\varepsilon /4$ .",
"Then $d\\left(gy,g_{i}y\\right)<\\varepsilon /4$ , $d\\left(g_{i}y,g_{i}y_{n_{j}}\\right)<\\varepsilon /4$ and $d\\left(g_{i}y_{n_{j}},\\sigma _{*}^{k_{n_{j},i}}\\left(x_{*}\\right)\\right)<\\varepsilon /4$ .",
"Together, $d\\left(gy,\\sigma _{*}^{k_{n_{j},i}}\\left(x_{0}\\right)\\right)<3\\varepsilon /4<\\varepsilon $ , which means that $gy$ is in the closure.",
"Similarly, if $y^{\\prime }$ is a limit of a subsequence of $\\sigma _{*}^{-k_{n,0}^{\\prime }}\\left(x_{*}\\right)$ , then $Gy^{\\prime }\\subseteq \\operatorname{cl}\\left\\lbrace \\sigma _{*}^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ .",
"So we consider $f:A_{*}\\rightarrow A$ a partial elementary map.",
"Our task now is to deal with $n<\\omega $ .",
"Let $\\varepsilon =1/n$ .",
"Let $A_{*}\\subseteq C_{*}\\subseteq M_{*}$ be finite such that if $g^{-1}\\upharpoonright C_{*}=h^{-1}\\upharpoonright C_{*}$ then $d\\left(gx,hx\\right)<\\varepsilon /4$ for all $x\\in X$ and any $g,h\\in G$ (this is by Lemma REF ).",
"Let $z_{0},\\ldots ,z_{l-1}$ be such that $\\bigcup \\left\\lbrace B\\left(z_{j},\\varepsilon /4\\right)\\,|\\,j<l\\right\\rbrace $ cover $X$ , and write $B_{j}=B\\left(z_{j},\\varepsilon /4\\right)$ .",
"Let $c_{*}$ be a finite tuple enumerating $C_{*}$ .",
"For every $c_{*}^{\\prime }\\equiv c_{*}$ , we say that $c_{*}^{\\prime }$ has color $j<l$ if $j$ is least such that there is $g\\in G$ such that $g\\left(c_{*}^{\\prime }\\right)=c_{*}$ and $gx_{*}\\in B_{j}$ .",
"Note that by the choice of $c_{*}$ , if $g^{\\prime }\\left(c_{*}^{\\prime }\\right)=c_{*}$ then $g^{\\prime }g^{-1}\\upharpoonright C_{*}=\\operatorname{id}$ , so $g^{\\prime }x_{0}\\in B\\left(z_{j},\\varepsilon /2\\right)$ .",
"Let $D_{*}=\\bigcup \\left\\lbrace g_{i}^{-1}\\left(C_{*}\\right)\\,|\\,i<n\\right\\rbrace $ .",
"Note that $C_{*}\\subseteq D_{*}$ because $g_{0}=\\operatorname{id}$ .",
"Let $d_{*}$ enumerate $D_{*}$ .",
"For any $d_{*}^{\\prime }\\equiv d_{*}$ and $s\\subseteq l$ , we say that $d_{*}^{\\prime }$ has color $s$ if $\\left\\lbrace j<l\\,|\\,c_{*}^{\\prime }\\equiv c_{*},c_{*}^{\\prime }\\subseteq d_{*}^{\\prime },c_{*}^{\\prime }\\text{ has color }j\\right\\rbrace =s$ .",
"By left existence, there is some $s_{0}\\subseteq l$ such that for every finite set $S\\subseteq M_{*}$ , there is some $d_{*}^{\\prime }\\equiv d_{*}$ with $d_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $S$ and $ d*'$ has color $ s0$.$ Let $d_{*}^{\\prime }\\equiv d_{*}$ be of color $s_{0}$ such that $d_{*}^{\\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $d*$.For $ i<n$, let $ c*,i'd*'$ be the tuple correspondingto $ gi-1(c*)$, so in particular $ c*,i'c*$.Let $ ji<l$ be the color of $ c*,i'$.",
"By the choice of $ s0$,for every finite set $ SM*$, there is some $ c*'c*$such that $ c'*$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $S$ and $ c*'$ has color $ ji$.$ Since $M$ is homogeneous we can extend $f$ in such a way so that its domain equals $D_{*}$ .",
"By Claim REF (the first part), there is some $k_{n,0}<\\omega $ and an extension $f_{0}$ of $f$ that ensures that $\\sigma _{*}^{k_{n,0}}\\left(d_{*}^{\\prime }\\right)=d_{*}$ .",
"Starting with $f_{0}$ , we construct an increasing sequence $\\left\\langle f_{i}\\,|\\,i<n\\right\\rangle $ as follows.",
"Suppose we have $f_{i}$ whose domain is $D_{*,i}$ .",
"Find some $c^{\\prime \\prime }_{*,i+1}\\equiv c_{*}$ of color $j_{i+1}$ such that $c_{*,i+1}^{\\prime \\prime }\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $D*,i$.By Claim \\ref {claim:Using shiftiness}, we can find $ kn,i+1<$and extend $ fi$ to $ fi+1$ which ensures that $ *kn,i+1(c*,i+1”)=c*$.$ Now we have the first part of $\\star $ : we need to check that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{*}\\right),g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{*}\\right)\\right)\\right)<\\varepsilon $ for all $i<n$ .",
"For $i=0$ this is clear since $g_{0}=\\operatorname{id}$ , so we may assume that $i>0$ .",
"As $\\sigma _{*}^{k_{n,i}}\\left(c_{*,i}^{\\prime \\prime }\\right)=c_{*}$ , it follows that $d\\left(\\sigma _{*}^{k_{n,i}}\\left(x_{0}\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .",
"Similarly, as $\\sigma _{*}^{k_{n,0}}\\left(c_{*,i}^{\\prime }\\right)=g_{i}^{-1}\\left(c_{*}\\right)$ , we have that $d\\left(g_{i}\\left(\\sigma _{*}^{k_{n,0}}\\left(x_{0}\\right)\\right),z_{j_{i}}\\right)<\\varepsilon /2$ .",
"Together, we are done.",
"Now we have to take care of the other half of $\\star $ .",
"This is done similarly, using right existence and the second part of Claim REF .",
"The following proposition explains why we needed to take a conjugate of $\\sigma $ .",
"The countable ordered random graph has a CIR by Example REF , thus Theorem REF applies to it.",
"In Section REF , we mentioned that it is a Ramsey structure.",
"Note that the underlying order is dense (by Proposition REF ).",
"Proposition 6.9 Let $M=\\left(V,<,R\\right)$ be the countable ordered random graph.",
"Then there is no automorphism $\\sigma \\in G=\\operatorname{Aut}\\left(M\\right)$ which satisfies (*) for every continuous action on a compact metric space $X$ on which $G$ acts and every $x_{*}\\in X$ .",
"First we find $a\\ne b$ in $M$ such that $\\sigma ^{n}\\left(a\\right)\\ne \\sigma ^{m}\\left(b\\right)$ for all $m,n\\in \\mathbb {Z}$ .",
"To do that, take any $a\\in M$ .",
"Then $\\left\\lbrace \\sigma ^{n}\\left(a\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace $ is discrete (in the order sense: it is either a $\\mathbb {Z}$ -chain or just $a$ ).",
"Since $\\left(V,<\\right)$ is dense, there is some $b\\ne \\sigma ^{n}\\left(a\\right)$ for all $n\\in \\mathbb {Z}$ .",
"It follows that $b$ is as required.",
"Let $X=S_{x}\\left(M\\right)$ be the space of complete types over $M$ (in one variable $x$ ) (it is a compact metric space).",
"Let $p\\in X$ be any completion of the partial type $\\left\\lbrace R\\left(x,\\sigma ^{n}\\left(a\\right)\\right)\\,|\\,n\\in \\mathbb {Z}\\right\\rbrace \\cup \\left\\lbrace \\lnot R\\left(x,\\sigma ^{m}\\left(b\\right)\\right)\\,|\\,m\\in \\mathbb {Z}\\right\\rbrace $ .",
"Then if (*) holds for $p$ , then by Fact REF , there is some point $p_{0}\\in \\operatorname{cl}\\left\\lbrace \\sigma ^{n}\\left(p\\right)\\,|\\,n<\\omega \\right\\rbrace $ which is a fixed point of $G$ .",
"In other words, $p_{0}$ is an invariant type over $M$ .",
"However $R\\left(x,a\\right)\\wedge \\lnot R\\left(x,b\\right)\\in p_{0}$ (this is true for any type in the closure), so $p_{0}$ cannot be invariant (because $G$ is transitive).",
"The example of the ordered random graph also explains why we needed to restrict to compact metric spaces, and could not prove this for all compact spaces.",
"If Theorem REF had worked for all compact spaces, it would also work for the universal $G$ -ambit (see Section REF ), $\\left(X,x_{0}\\right)$ .",
"Thus, there would be a conjugate $\\sigma _{*}$ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow.",
"But then if $\\left(Y,y_{0}\\right)$ is any other $G$ -ambit, by universality, there is a continuous surjection $\\pi :X\\rightarrow Y$ mapping $x_{0}$ to $y_{0}$ and commuting with the action of $G$ .",
"Thus, $\\pi $ maps $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(x_{0}\\right)\\,|\\,n<\\omega \\right\\rbrace $ to $\\operatorname{cl}\\left\\lbrace \\sigma _{*}^{n}\\left(y\\right)\\,|\\,n<\\omega \\right\\rbrace $ , and the latter contains a $G$ -subflow.",
"Thus we get that $\\sigma _{*}$ satisfies (*) for every $G$ -ambit, which contradicts Proposition REF .",
"Corollary 6.10 Let $T=T_{dt}$ be the theory of dense trees in the language $\\left\\lbrace <,\\wedge \\right\\rbrace $ , and let $M\\models T$ be countable.",
"Then $\\operatorname{Aut}\\left(M\\right)$ has no shifty automorphism.",
"In particular, $M$ has no CIR.",
"Furthermore, the same is true for $T_{dt,<_{lex}}$ , the theory of the lexicographically ordered dense tree $N$ , see Example REF .",
"Suppose that $\\sigma $ was shifty.",
"Let $\\bar{m}=\\left\\langle m\\,|\\,m\\in M\\right\\rangle $ be an enumeration of $M$ (really the identity function), and let $\\bar{x}=\\left\\langle x_{m}\\,|\\,m\\in M\\right\\rangle $ .",
"Let $X=S_{\\bar{m}}\\left(M\\right)$ be the space of $\\bar{x}$ -complete types $p$ over $M$ such that $p\\upharpoonright \\emptyset =\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .",
"Then $X$ is a compact metric space.",
"Let $x_{*}=\\operatorname{tp}\\left(\\bar{m}/M\\right)$ .",
"By Theorem REF , there is some conjugate $\\tau $ of $\\sigma $ such that $\\operatorname{cl}\\left\\lbrace \\tau ^{n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{+}\\subseteq X$ and similarly, $\\operatorname{cl}\\left\\lbrace \\tau ^{-n}\\left(x_{*}\\right)\\,|\\,n<\\omega \\right\\rbrace $ contains a subflow $Y^{-}$ .",
"By Proposition REF , $\\tau $ fixes a branch or a point.",
"Suppose that $\\tau \\left(m\\right)=m$ for some $m\\in M$ .",
"Then for every $p\\in Y^{+}$ , $p\\models x_{m}=m$ .",
"However $G=\\operatorname{Aut}\\left(M\\right)$ acts transitively on $M$ , so we have a contradiction.",
"Now suppose that $\\tau $ fixes a branch $B\\subseteq M$ , but does not fix any point.",
"Suppose that $\\tau \\left(m\\right)>m$ for some $m\\in B$ .",
"Then $\\tau ^{n}\\left(m\\right)>m$ for all $n<\\omega $ , so for any $p\\in Y^{+}$ , $p\\models x_{m}>m$ .",
"There is some $m^{\\prime }\\in M$ such that $m^{\\prime }>m$ and $m^{\\prime }\\notin B$ .",
"Since $m<\\tau ^{n}\\left(m\\right)\\in B$ for all $n<\\omega $ , it follows that $p\\models x_{m}\\wedge m^{\\prime }=m$ for all $p\\in Y^{+}$ .",
"Let $\\tau ^{\\prime }\\in G$ fix $m$ and map $m^{\\prime }$ to $B$ .",
"Then $\\tau ^{\\prime }\\left(p\\right)\\models \\left(x_{m}\\wedge \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\right)=m<x_{m}$ .",
"But $\\tau ^{\\prime }\\left(p\\right)\\in Y^{+}$ , so $\\tau ^{\\prime }\\left(p\\right)\\models x_{m}\\le \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\vee \\tau ^{\\prime }\\left(m^{\\prime }\\right)\\le x_{m}$ , which is a contradiction.",
"If, on the other hand $\\tau \\left(m\\right)<m$ , then $\\tau ^{-1}\\left(m\\right)>m$ , so we can apply the same argument to $Y^{-}$ .",
"For the furthermore part, note that by Proposition REF , the reduct of $T_{dt,<_{lex}}$ to the tree language is $T_{dt}$ .",
"In addition, letting $H=\\operatorname{Aut}\\left(N\\right)$ , $H$ acts transitively on $N$ (by quantifier elimination, as $N$ is ultrahomogeneous).",
"In addition, if $B\\subseteq N$ is a branch, $m\\in B$ , there is always some $m^{\\prime }>m$ , $m^{\\prime }\\notin B$ and for any $n^{\\prime }>m$ in $B$ , $m^{\\prime }m\\equiv n^{\\prime }m$ .",
"Hence, we can apply Proposition REF and the same proof will work.",
"Further questions The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\omega $ -categorical structures.",
"We state here a few general conjectures and questions.",
"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.",
"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.",
"Conjecture 7.1 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion which admits a CIR.",
"Suppose that $M$ is a structure and $\\mathfrak {C}$ a monster model for $\\operatorname{Th}\\left(M\\right)$ .",
"The group of Lascar strong automorphisms of $M$, denoted by $\\operatorname{Aut}f\\left(M\\right)$ is the group of automorphisms of $M$ generated by the set $\\left\\lbrace \\sigma \\upharpoonright M\\,|\\,\\exists N\\prec \\mathfrak {C},\\left|N\\right|=\\left|T\\right|,\\sigma \\upharpoonright N=\\operatorname{id}\\right\\rbrace $ .",
"If $\\sigma $ is Lascar strong, then $\\sigma \\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{id}$ so $\\operatorname{Aut}f\\left(M\\right)$ is contained in $G^{0}$ .",
"However, there are examples (even $\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\operatorname{Aut}f\\left(M\\right)$ , see [16], [31].",
"The Lascar group of $M$ is the quotient $\\operatorname{Aut}\\left(M\\right)/\\operatorname{Aut}f\\left(M\\right)$ .",
"For more on the Lascar group, see [39].",
"In the $\\omega $ -categorical case, the quotient $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is also called the compact Lascar group.",
"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$.",
"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.",
"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.",
"Forinstance, by Lemma \\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\operatorname{Aut}\\left(M\\right)$ has no compact quotients then it has finite topological rank.",
"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.",
"Thus, potentially, the Lascar group — as a quotient of $\\operatorname{Aut}\\left(M\\right)$ — can be an obstruction to having finite topological rank.",
"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.",
"Conjecture REF (together with Theorem REF ) implies that it could not.",
"Conjecture 7.2 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion with trivial Lascar group.",
"By the above, this second conjecture is implied by Conjecture REF .",
"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.",
"It would be interesting to investigate other consequences of having a CIR.",
"For instance a CIR might have something to say about normal subgroups.",
"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ for some CIR $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\aleph _{0}}$ is open.",
"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.",
"We refer to [26] for a survey on this.",
"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.",
"In fact the compact quotients are hidden in the stabilizer of a finite set.",
"It seems that one can avoid this counterexample by restricting to dense subgroups.",
"This leads us to the following questions.",
"Question 7.3 Let $M$ be $\\omega $ -categorical such that $G=\\operatorname{Aut}\\left(M\\right)$ has no compact quotient.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open (and hence is equal to $G$ )?",
"Note that the assumption of having no compact quotient is necessary.",
"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\operatorname{Aut}\\left(M\\right)$ has a dense subgroup of index 2, see [23].",
"Question 7.4 Let $M$ be $\\omega $ -categorical and $N$ an $\\omega $ -categorical expansion of $M$ .",
"Set $G=\\operatorname{Aut}\\left(M\\right)$ and $H=\\operatorname{Aut}\\left(N\\right)\\le G$ .",
"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open?",
"Acknowlegements Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.",
"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.",
"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].",
"We would also like to thank Daoud Siniora for his comments.",
"Finally, we would like to thank the anonymous referee for his comments."
],
[
"Further questions",
"The results presented in the previous sections lead to a number of questions, both related to CIR and more generally on $\\omega $ -categorical structures.",
"We state here a few general conjectures and questions.",
"If they turn out to be false at this level of generality, they could be weakened by restricting to finitely homogeneous structures or other subclasses.",
"The following conjecture, along with Theorem REF (and Example REF ), would imply that indeed compact quotients are the only obstruction to having finite topological rank.",
"Conjecture 7.1 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion which admits a CIR.",
"Suppose that $M$ is a structure and $\\mathfrak {C}$ a monster model for $\\operatorname{Th}\\left(M\\right)$ .",
"The group of Lascar strong automorphisms of $M$, denoted by $\\operatorname{Aut}f\\left(M\\right)$ is the group of automorphisms of $M$ generated by the set $\\left\\lbrace \\sigma \\upharpoonright M\\,|\\,\\exists N\\prec \\mathfrak {C},\\left|N\\right|=\\left|T\\right|,\\sigma \\upharpoonright N=\\operatorname{id}\\right\\rbrace $ .",
"If $\\sigma $ is Lascar strong, then $\\sigma \\upharpoonright \\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)=\\operatorname{id}$ so $\\operatorname{Aut}f\\left(M\\right)$ is contained in $G^{0}$ .",
"However, there are examples (even $\\omega $ -categorical examples) where $G^{0}$ is strictly bigger than $\\operatorname{Aut}f\\left(M\\right)$ , see [16], [31].",
"The Lascar group of $M$ is the quotient $\\operatorname{Aut}\\left(M\\right)/\\operatorname{Aut}f\\left(M\\right)$ .",
"For more on the Lascar group, see [39].",
"In the $\\omega $ -categorical case, the quotient $\\operatorname{Aut}\\left(\\operatorname{acl}^{\\operatorname{eq}}\\left(\\emptyset \\right)\\right)$ is also called the compact Lascar group.",
"If $M$ is an ultrahomogeneous linearly ordered Ramsey structure, then by Proposition REF , there is some model $N$ such that $N\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $nsM$.",
"In particular,$ (M)NM$, for every $ Aut(M)$which implies that $$ is Lascar strong.",
"Thus in Ramsey structures,and in fact for any model $ M$ for which there is some such $ N$,the Lascar group is trivial, and there are no compact quotients.",
"Forinstance, by Lemma \\ref {lem:CIR implies just over 0} this happensalso when $ M$ is $$-categorical with a CIR.$ As we said above, we conjecture that if $\\operatorname{Aut}\\left(M\\right)$ has no compact quotients then it has finite topological rank.",
"However, as we pointed out, it could be that $G^{0}=G$ but the Lascar group is nontrivial.",
"Thus, potentially, the Lascar group — as a quotient of $\\operatorname{Aut}\\left(M\\right)$ — can be an obstruction to having finite topological rank.",
"During a talk given on this paper by the second author, Anand Pillay asked if this scenario could happen.",
"Conjecture REF (together with Theorem REF ) implies that it could not.",
"Conjecture 7.2 Any $\\omega $ -categorical structure has an $\\omega $ -categorical expansion with trivial Lascar group.",
"By the above, this second conjecture is implied by Conjecture REF .",
"Note also that by Proposition REF , the conjecture is true when we replace the Lascar group by the compact Lascar group.",
"It would be interesting to investigate other consequences of having a CIR.",
"For instance a CIR might have something to say about normal subgroups.",
"The analysis in [6] of automorphism groups of trees seems to suggest that there is a link: normal subgroups appear as groups fixing a set of points roughly corresponding to the set of $x$ such that $x\\mathop {\\mathchoice{\\displaystyle \\displaystyle x}{\\hspace{0.0pt}}{}{0}\\hbox{t}o 0pt{\\hss \\displaystyle \\mid \\hss } \\hss \\displaystyle \\smile \\hss }\\hspace{0.0pt}$$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $A$ for some CIR $$\\displaystyle \\mid $ $\\displaystyle \\smile $$\\textstyle \\mid $ $\\textstyle \\smile $$\\scriptstyle \\mid $ $\\scriptstyle \\smile $$\\scriptscriptstyle \\mid $ $\\scriptscriptstyle \\smile $$ and finite set $ A$.A similar phenomenon happens in DLO, where there are only three normalsubgroups (the group of automorphism fixing a cone to the left, tothe right, and the intersection of these two), see \\cite [Theorem 2.3.2]{MR645351}.$ In another direction, recall that an automorphism group $G$ (or more generally a Polish group) has the small index property (sip) if every subgroup of index less than $2^{\\aleph _{0}}$ is open.",
"Many groups are known to have this property, but there are at least two different types of techniques used to show it—the Hrushovski property (or extension property) and direct combinatorial methods—which have yet to be unified.",
"We refer to [26] for a survey on this.",
"As in the case of finite topological rank, large compact quotients seem to be only known obstruction to having sip, although the situation is more complicated: Lascar [23] gives an example of an automorphism group without the sip and with no compact quotients.",
"In fact the compact quotients are hidden in the stabilizer of a finite set.",
"It seems that one can avoid this counterexample by restricting to dense subgroups.",
"This leads us to the following questions.",
"Question 7.3 Let $M$ be $\\omega $ -categorical such that $G=\\operatorname{Aut}\\left(M\\right)$ has no compact quotient.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open (and hence is equal to $G$ )?",
"Note that the assumption of having no compact quotient is necessary.",
"Indeed, in the example suggested by Cherlin and Hrushovski (the one described in Remark REF ), we have that $G=\\operatorname{Aut}\\left(M\\right)$ has a dense subgroup of index 2, see [23].",
"Question 7.4 Let $M$ be $\\omega $ -categorical and $N$ an $\\omega $ -categorical expansion of $M$ .",
"Set $G=\\operatorname{Aut}\\left(M\\right)$ and $H=\\operatorname{Aut}\\left(N\\right)\\le G$ .",
"Assume that $(G,H)$ has no compact quotients and that $H$ has the sip.",
"Is it true that any dense subgroup of $G$ of index less than $2^{\\aleph _{0}}$ is open?"
],
[
"Acknowlegements",
"Thanks to Alejandra Garrido for bringing up some questions that lead to this work and to Dugald Macpherson for helping us get a grasp of the area through several interesting discussions.",
"We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.",
"Thanks to Katrin Tent for comments on a previous draft and for telling us about [19].",
"We would also like to thank Daoud Siniora for his comments.",
"Finally, we would like to thank the anonymous referee for his comments."
]
] | 1709.01918 |
[
[
"The ALMA Early Science View of FUor/EXor objects. IV. Misaligned\n Outflows in the Complex Star-forming Environment of V1647 Ori and McNeil's\n Nebula"
],
[
"Abstract We present Atacama Large Millimeter/sub-millimeter Array (ALMA) observations of the star-forming environment surrounding V1647 Ori, an outbursting FUor/EXor pre-MS star.",
"Dust continuum and the (J = 2 - 1) $^{12}$CO, $^{13}$CO, C$^{18}$O molecular emission lines were observed to characterize the V1647 Ori circumstellar disc and any large scale molecular features present.",
"We detect continuum emission from the circumstellar disc and determine a radius r = 40 au, inclination i = 17$^{\\circ}$$^{+6}_{-9}$ and total disc mass of M$_{\\mathrm{disk}}$ of ~0.1 M$_{\\odot}$.",
"We do not identify any disc structures associated with nearby companions, massive planets or fragmentation.",
"The molecular cloud environment surrounding V1647 Ori is both structured and complex.",
"We confirm the presence of an excavated cavity north of V1647 Ori and have identified dense material at the base of the optical reflection nebula (McNeil's Nebula) that is actively shaping its surrounding environment.",
"Two distinct outflows have been detected with dynamical ages of ~11,700 and 17,200 years.",
"These outflows are misaligned suggesting disc precession over ~5500 years as a result of anisotropic accretion events is responsible.",
"The collimated outflows exhibit velocities of ~2 km s$^{-1}$, similar in velocity to that of other FUor objects presented in this series but significantly slower than previous observations and model predictions.",
"The V1647 Ori system is seemingly connected by an \"arm\" of material to a large unresolved structure located ~20$\"$ to the west.",
"The complex environment surrounding V1647 Ori suggests it is in the early stages of star formation which may relate to its classification as both an FUor and EXor type object."
],
[
"Introduction",
" Stars form as a result of gravitational contraction of rotating molecular clouds.",
"As the gravitational collapse proceeds, a system emerges composed of an accreting star, disc and envelope.",
"Observations of this stage of pre-main sequence (pre-MS) stellar evolution display large scale molecular outflows that act to transport energy back into the surrounding environment and are likely the main dispersing mechanism of molecular cloud material, responsible for the transition from a deeply embedded Class 0 protostar to the more evolved Class I/II pre-MS stellar systems.",
"An empirical relationship has been identified between the opening angle of these outflows and the pre-MS stage of stellar evolution .",
"The earliest stages of evolution (Class 0) exhibit outflows that are highly collimated and have opening angles of 20-50$^{\\circ }$ whereas the more evolved Class I and II stages are associated with less collimated opening angles of 80-120$^{\\circ }$ and 100-160$^{\\circ }$ , respectively .",
"The origin of these outflows are debated due to the au spatial scales required to resolve them however several theories relate the outflows to intense mass accretion events from the disc and/or envelope onto the central pre-MS star .",
"Collimated and wide-angle outflows appear to represent two distinct structures in the star-forming environment of their host stars.",
"In particular, wide-angle outflows tend to be more massive and slower (velocities ranging from $\\sim $ 10-30 km s$^{-1}$ ) compared to highly collimated outflows which can reach velocities of $\\sim $ 100-1000 km s$^{-1}$ .",
"Their distinct velocities and masses arise from the mechanisms responsible for their production; wide angle outflows are likely produced from the ambient molecular material getting swept up when interacting with a jet bow shock whereas highly collimated jets likely arise from the interaction between a strongly magnetized central star and the inner edge of its accretion disc during intense mass accretion events .",
"Recent observations indicate that extended disc winds may also drive protostellar outflows .",
"Mass accretion events where material gets loaded from the disc onto the protostar are essential to star formation as these events are the primary means by which stars gain their mass.",
"The standard model of star formation indicates that a molecular cloud gravitationally contracts and forms a star-disc system in about $\\sim $ 5 $\\times $ 10$^{5}$ years .",
"The mass accretion rate necessary to produce a star-disc system after this timescale is inconsistent with the age and the implied accretion rates derived from the observed accretion luminosity of a large sample of stars in Taurus .",
"One solution to this \"luminosity problem\" is short-lived major accretion events of significant mass.",
"In the past several decades, a phenomenon has been observed in a small subset of pre-MS stars that is consistent with brief and intense mass accretion events identified as \"outbursts\" , , .",
"These outbursting pre-MS stars have been named FUor/EXor objects based on their namesakes FU Ori and EX Lup.",
"This phenomenon is identifiable when pre-MS stars suddenly increase in brightness by several magnitudes at optical/near-IR wavelengths as a result of intense mass accretion outbursts where accretion rates can reach as high as 10$^{-4}$ – 10$^{-5}$ M$_{\\odot }$ yr$^{-1}$ .",
"FU Ori type (FUor) and EX Lup type (EXor) are generally distinguished by the frequency and length of their protostellar outbursts where FUor type objects tend to have longer (years to decades) timescale and EXors are characterized by more frequent and shorter outbursts (months-years).",
"While it is evident these outbursts are caused by large mass-accretion events, it is not clear what underlying mechanism is responsible.",
"Several theories have been proposed which include binary companions and/or massive planets generating disc instabilities , , disc fragmentation , and a combination of magneto rotational instability (MRI) and gravitational instabilities .",
"A detailed summary of these mechanisms and their relevant references can be found in .",
"Regardless of the underlying mechanism, it is clear that sudden intense outbursts can affect the circumstellar disc of protostars and thus, may impact planet formation , .",
"Recently, an analysis of high resolution ALMA observations of the outbursting FUor type object V883 Ori revealed the first detection of a water snow line in a circumstellar disc .",
"Such a discovery was feasible because the water snow line, which typically resides < 5 au from a $\\sim $ 1 M$_{\\odot }$ central star, was temporarily extended to $\\sim $ 40 au in response to the increase in luminosity of the FUor object during outburst.",
"V1647 Ori, a protostar in the Lynds 1630 (L1630) region of the Orion Molecular Cloud , was first noted in 2003 when a large ($\\sim $ 1$^{\\prime }$ ) reflection nebula (McNeil's Nebula) appeared coincident with a sudden increase in optical stellar brightness , , .",
"After its identification in 2003, two more outbursts were identified: a 1966 outburst identified using archival photometric plates and a 2008 outburst which increased the flux values of V1647 Ori to similar values as in 2003.",
"During the 2008 outburst, carried out a 4.5 year photometric study of V1647 Ori and report that at the end of their campaign in 2012, V1647 Ori was still in outburst after almost half a decade.",
"The frequency and length of these outbursts, combined with several spectral features characteristic of both FUors and EXors emphasize the ambiguity between the FUor or EXor classification of V1647 Ori .",
"Spectral parameters of V1647 Ori were determined from data taken during quiescence and presented in .",
"They identify V1647 Ori as a young (< 0.5 Myr) pre-MS star of spectral type M0 $\\pm $ 2 with an approximate bolometric luminosity, mass and radius of $\\sim $ 5.2 L$_{\\odot }$ , 0.8 M$_{\\odot }$ , and 5 R$_{\\odot }$ , respectively.",
"During its 2003 outburst, its bolometric luminosity was measured to be L$_\\mathrm {bol}$ = 44 L$_{\\odot }$ .",
"Given the transient nature of V1647 Ori, several multiwavelength campaigns were setup to observe the star in its outbursting state.",
"The 2003 outburst was characterized by a $\\sim $ 3 magnitude increase in the near-IR , a factor of 25 increase in 12 $\\mu $ m flux and a factor of 50 increase in X-ray flux .",
"report no apparent changes in the (350 $\\mu $ m-1.3mm) submm dust continuum brightness during outburst compared with its quiescent state.",
"Plasma temperatures derived from the X-ray imaging spectrum presented in were too high to be produced via accretion alone suggesting extreme changes in magnetic field configuration (i.e., magnetic reconnection events) were be present.",
"Spectroscopic measurements during the 2003 outburst indicated accretion rates of a few 10$^{-6}$ to 10$^{-5}$ M$_{\\odot }$ year $^{-1}$ and H$\\alpha $ P Cygni profiles indicate wind velocities up to $\\sim $ 600 km s$^{-1}$ .",
"Mid-IR interferometric observations probing the circumstellar disc indicate a moderately flaring disc with no signatures of close companions at radii < 100 au [1].",
"All three outbursts re-illuminated the reflection nebula, where dust grains preferentially scatter blue light, just north of V1647 Ori and several studies have identified small variations in the local environment as probed by this extended nebular emission , .",
"The time delay between brightness variations of the star and of clumps in the nebulosity were used to estimate a stellar inclination angle of $\\sim $ 61$^{\\circ }$ .",
"An X-ray periodicity study of V1647 Ori found a similar stellar inclination angle of i $\\sim $ 68$^{\\circ }$ based on modeling the rotation of accretion hot spots on the stellar surface .",
"These inclinations are in disagreement with studies that suggest a more face-on inclination of $<$ 30$^{\\circ }$ based on the absence of a 9.7 $\\mu $ m amorphous silicate feature and a low column density of CO in absorption from the disc .",
"Here, we present ALMA 1.3 mm observations of the gas and dust surrounding V1647 Ori in order to characterize its circumstellar disc and local star-forming environment.",
"V1647 Ori is one of 8 FUor/EXor type objects observed as part of an ALMA Cycle 2 project (2013.1.00710.S ; PI L. Cieza) to investigate star-forming environments at this critical stage of pre-MS stellar evolution.",
"All other FUor/EXor objects observed as part of this program, including V883 Ori, HBC 494, V2775 Ori, NY Ori, V1143 Ori, V1118 Ori, ASASSN-13db, are presented in papers I, II, III, IV (this paper) and V , , , .",
"These recently published results indicate that the star-forming environments of these objects are structured and complex.",
"In particular, features such as rings/shells and both wide-angle and collimated outflows have all been identified.",
"Outflow velocities measured for V883 Ori, HBC 494, V2775 Ori, and V1647 Ori (Section 3.6) as part of this series of papers indicate that significantly slower velocities (v$\\sim $ 1 - 4 km $s^{-1}$ ) are present than those typically observed and associated with protostellar outflows at these wavelengths .",
"The resolving power and sensitivity of the observations presented in this series also provides strict constraints on circumstellar disc characteristics such as disc size, orientation, and mass."
],
[
"Observations",
"V1647 Ori was observed with ALMA band-6 at three epochs (December 12th, 2014, April 5th, 2015 and August 30th, 2015).",
"The first two observations were performed with an ALMA configuration of 45 antennas (12 meter diameter) and baselines ranging from 14.6 to 348.5 meters which achieved an angular resolution of $\\sim $ 1.0 $^{\\prime \\prime }$ .",
"The final epoch had a configuration of 35 antennas and longer baselines of 42-1574 meters resulting in a higher angular resolution of 0.2 $^{\\prime \\prime }$ .",
"The precipitable water vapor (PWV) levels were 0.7, 1.3 and 1 mm for the December 2014, April 2015, and August 2015 observations, respectively.",
"Each ALMA observation of V1647 Ori include two broad (2 GHz) continuum spectral windows centered at $\\sim $ 232 GHz and $\\sim $ 218 GHz and three narrow (59 MHz) windows centered near the rest frequencies of $^{12}$ CO (J = 2-1; 230.5380 GHz), $^{13}$ CO (J = 2-1; 220.3987 GHz) and C180 (J = 2-1; 219.5603 GHz).",
"Flux calibration was performed with Ganymede and J0423-013 while bandpass calibration was performed with quasars J0538-4405 and J0607-0834.",
"The time dependence variations of the complex gains were calibrated by alternating V1647 Ori with nearby phase calibrators J0541-0541, J0532-0307 and/or J0529-0519.",
"The ALMA pipeline-calibrated observations were further reduced with the Common Astronomical Software Application .",
"The two epochs of $\\sim $ 1.0$^{\\prime \\prime }$ resolution data were continuum subtracted in the visibility domain, concatenated, and cleaned to form a single dataset for the continuum and each spectral line.",
"The same procedure was performed for the single epoch of $\\sim $ 0.2$^{\\prime \\prime }$ resolution data.",
"While features of the V1647 Ori disc/outflows are clearly present in the $\\sim $ 1$^{\\prime \\prime }$ resolution datasets, none of these features are identifiable in the high resolution line imaging from the 0.2$^{\\prime \\prime }$ resolution dataset.",
"This is consistent with most of the flux from the system being fairly uniform on large scales.",
"Therefore we only present the detection of continuum emission from the 0.2$^{\\prime \\prime }$ dataset and the line and continuum emission from the combined two $\\sim $ 1$^{\\prime \\prime }$ datasets.",
"The beam size, position angle, and 3$\\sigma $ sensitivities for each spectral line channel map and the continuum are indicated in Table REF .",
"Integrated intensity images (moment 0) were creating using the CASA routine $immoments$ .",
"The moment 0 images were created using only pixels with a signal detection of 3$\\sigma $ or higher.",
"To better describe the morphology and dynamics of the V1647 Ori star-forming environment, we created moment 0 images for each spectral line with three distinct velocity ranges: $5.0-9.49$ km s$^{-1}$ , $9.5-10.5$ km s$^{-1}$ and $10.51-13.0$ km s$^{-1}$ .",
"These velocity bands will henceforth be referred to in this paper as blueshifted, systemic, and redshifted, respectively.",
"The images presented in this work are not primary-beam corrected.",
"However, primary-beam corrected channel maps were used when estimating physical parameters of the outflow from the line fluxes (Section 3.6).",
"Optical Gemini-GMOS imaging observations of V1647 Ori were performed on September 22, 2008 with an exposure time of 60 seconds in R band.",
"Optical imaging data was retrieved from the Gemini Science Archive and presented here to supplement the ALMA observations.",
"Optical analysis of these and similar data are presented in other work and the discussion in this paper is limited only to correlations between large-scale optical and mm emission morphology.",
"Table: V1647 Ori ALMA Continuum and Integrated Intensity Image Parameters"
],
[
"Continuum Emission",
"The integrated intensity image of the resolved 0.2$^{\\prime \\prime }$ V1647 Ori continuum is displayed in Figure REF .",
"Two-dimensional gaussian fitting of the $\\sim $ 225 GHz continuum emission performed in CASA indicate a flux, inclination, position angle and centroid of 82.62 $\\pm $ 8.26 mJy, ${\\it i} = 17^{\\circ }$ $^{+6}_{-9}$ , PA$ = 109^{\\circ } \\pm $ 19, and RA= 05:46:13.139, Dec= -00:06:04.90, respectively.",
"The inclination was calculated using the aspect ratio of an assumed circular disc.",
"All position angles reported in this work are east of north.",
"A semi major axis FWHM angular size of 189.0 $\\pm $ 3.6 milliarcseconds and a distance of 414 $\\pm {7}$ pc results in a disc dust radius of $\\sim $ 40 au.",
"Evidence that the continuum emission from the circumstellar disc is spatially resolved is identified by its amplitude as a function of UV distance and asymmetry in the continuum emission (Figure REF ) indicates a potentially non-gaussian distribution near the outer edges of the image.",
"A more in-depth analysis of continuum emission from V1647 Ori as well as the other FUor/EXor objects in this series will be presented in .",
"The mass of the V1647 Ori circumstellar disc dust can be estimated from the continuum emission (assuming continuum emission is thermal and optically thin) following where: $\\hspace{86.72377pt}M_{\\rm {disc}}= \\frac{F_\\nu d^2}{\\kappa _\\nu B_\\nu (T)},$ with $B_\\nu (T)$ representing the Planck function for dust at a temperature $T$ and $\\kappa _\\nu $ as the dust opacity, which is a power-law function of submm frequency, i.e., $\\kappa _\\nu =0.1(\\nu / 1000$ $\\mathrm {GHz})^\\beta $ cm$^{2}$ g$^{-1}$ .",
"The power law opacity index $\\beta $ is related to the size distribution and composition of the disc dust particles.",
"While $\\beta $ can range from $\\beta $ =0 for certain grain mineralogies or size distributions to $\\beta $ =2 for interstellar dust grains in the ISM , it is more likely to be in the range of 0.5-1.5 for circumstellar discs .",
"Without more sophisticated continuum modeling to estimate the disc dust temperature of V1647 Ori, a dust temperature and range of $T=20 \\pm 10$ K is chosen based on the median temperature of discs in Taurus-Auriga and the relatively small temperature range indicated in modeling of other circumstellar discs .",
"Assuming values of $\\beta $ =1, d= 414 pc, $\\kappa _\\nu $ = 0.023 cm$^{2}$ g$^{-1}$ , and a dust temperature of $T=20$ K, we estimate the disc dust mass of V1647 Ori to be $M_\\mathrm {dust}$ = 427 M$_\\mathrm {E}$ .",
"This corresponds to a total disc mass $M_\\mathrm {disc}$ = 0.13 $M_{\\odot }$ ($\\sim $ 136 M$_\\mathrm {Jup}$ ) assuming a gas to dust mass ratio of 100.",
"When converting from flux to disc mass using Equation 1, the largest sources of uncertainty are the ranges in $\\beta $ and dust temperature.",
"While the uncertainty in the continuum flux is dominated by the 10% absolute flux uncertainty of ALMA Band 6, the potential range in $\\beta $ ($\\beta $ = 1 $\\pm {0.5}$ ) and dust temperature (T = 20 $\\pm {10}$ ) introduce much higher uncertainties in dust mass that can result in masses in the extreme ranges of M$_{\\rm {dust}}$ = [110 - 2688] M$_\\mathrm {E}$ or $M_{\\rm {disc}}$ = [0.03 - 0.80] $M_{\\odot }$ assuming a gas to dust ratio of 100.",
"A summary of V1647 Ori disc parameters are displayed in Table REF .",
"Table: V1647 Ori Disc Parameters Derived From Continuum EmissionFigure: Resolved continuum emission of the V1467 Ori circumstellar disc with contours overlaid of values 1.75, 3, 7, 20, 30, 45 and 60 ×\\times 3σ3\\sigma .",
"A two-dimensional gaussian fit indicates a disc with inclination of 17 ∘ ^{\\circ } -9 +6 ^{+6}_{-9} and PA = 109±19\\pm 19 ∘ ^{\\circ }.",
"The beam size is displayed in the bottom left and north is up and east is left."
],
[
"Gas Emission Lines",
"Spectral flux density as a function of radial velocity for the $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O lines are displayed in 0.25 km s$^{-1}$ bins in Figure REF .",
"Spectral extraction was performed using two different apertures both centered at the position of V1647 Ori (J2000 RA= 05:46:13.135, Dec= -00:06:04.82).",
"An aperture diameter of 2.0 $^{\\prime \\prime }$ was chosen to extract spectral information at the precise location of the continuum emission (Fig REF top) and an aperture diameter of 20$^{\\prime \\prime }$ was chosen to encompass the nebulosity surrounding V1647 Ori (Fig REF bottom).",
"In both the small and large apertures, a double peaked $^{12}$ CO and single peaked $^{13}$ CO emission feature is observed.",
"A clear detection of C$^{18}$ O material is seen in the large 20$^{\\prime \\prime }$ aperture indicating an abundance of dense material traveling at a radial velocity equidistant between the two $^{12}$ CO peaks.",
"Due to the embedded nature of this object, the rest velocity of the system (i.e., systemic velocity) is not clear.",
"In the absence of a literature value of the systemic velocity of V1647 Ori, we estimate a systemic velocity of $\\sim $ 10.0 km s$^{-1}$ assuming the star forms in the densest part of the molecular cloud (i.e.",
"at the same velocity as the bulk of the dense C$^{18}$ O emission; Figure REF ).",
"This systemic velocity is also consistent with being equidistant between the peaks of the blue and red shifted $^{12}$ CO emission lines.",
"The blueshifted, systemic, and redshifted velocity ranges are indicated at the top of Figure REF .",
"Individual channel maps for each spectral line where emission associated with the V1647 Ori environment is observed is shown in Appendix Figures 1-3.",
"The integrated intensity images of $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O emission over their entire velocity range are displayed in Figure REF with white circles corresponding to the extraction apertures associated with the line profile displayed in Figure REF .",
"Several features are readily identifiable and will be discussed in detail in the following sections.",
"An illustration of the three-dimensional system geometry is displayed in Figure REF .",
"Figure: Line profiles from apertures centered on the dust continuum emission of V1647 Ori with diameters of 2 '' ^{\\prime \\prime } (top) and 20 '' ^{\\prime \\prime } (bottom) and velocity bins of 0.25 km s -1 ^{-1}.",
"A systemic velocity of 10.0 km s -1 ^{-1} is indicated with a vertical dashed line.",
"The blue, green and red lines above the plot represent the velocity ranges described in Section 2 for the blueshifted, systemic, and redshifted images presented throughout the rest of this paper.",
"Individual line channel maps are included in the Appendix.Figure: Integrated intensity (moment 0) images of 12 ^{12}CO, 13 ^{13}CO and C 18 ^{18}O integrated over each lines entire velocity range.",
"V1647 Ori is at the center of each panel (offset [0,0] corresponds to RA= 05:46:13.135, Dec= -00:06:04.82).",
"The small solid and large dashed white lines indicate the spectral extraction regions from Figure (aperture diameters of 2 '' ^{\\prime \\prime } and 20 '' ^{\\prime \\prime }, respectively) with north is up and east is left.",
"The beam is displayed in the lower left.Figure: A three-dimensional illustration of V1647 Ori and its surrounding misaligned outflows (labels D and E discussed in Sections 3.3 and 3.4).",
"The nearly face-on circumstellar disc is shown in green.",
"Features in the illustration are not to scale."
],
[
"Emission from $^{12}$ CO",
"The $^{12}$ CO emission line is typically optically thick in molecular cloud environments and is generally not a good probe of column density.",
"Instead, $^{12}$ CO emission traces molecular material of a particular temperature and is useful for probing excavated cavities as well as wide-angle outflows , .",
"Figure REF displays the blue, systemic and red shifted $^{12}$ CO emission summed over the velocity ranges indicated in the top of Figure REF and in Section 2.",
"The base of McNeil's Nebula, the familiar reflection nebula first identified from optical observations during its 2003 outburst is clearly seen in the blue shifted $^{12}$ CO emission line (label A).",
"This northern cavity has a well-defined wall and an opening angle to the north of $\\sim $ 100$^{\\circ }$ which extends $\\sim $ 10$^{\\prime \\prime }$ ($\\sim $ 4140 au).",
"A bright extended region spatially coincident with the star-disc system is identifiable at radii of $r \\lesssim 3^{\\prime \\prime }$ (1245 au) from V1647 Ori.",
"A distinct clump of material is seen $\\sim $ 2$^{\\prime \\prime }$ south-west from the central emission (label B) appearing to be cut off from the main nebula.",
"Meanwhile, the redshifted morphology of the $^{12}$ CO emission line is distinct from the blueshifted emission.",
"Emission coincident with the position of the circumstellar disc of V1647 Ori is detected in the arc-like shape bending from the north-west to the south-west.",
"Such a well defined structure is seemingly connected with an 'arm' to an atypically large column of $^{12}$ CO emission located 20$^{\\prime \\prime }$ (8300 au) west of V1647 Ori near the edge of the ALMA field of view (label C).",
"The northern bowl-like nebula may be faintly detected behind the arc-like feature in the red-shifted $^{12}$ CO emission.",
"More prominently displayed is an apparent southern feature extending in a straight column south of the V1647 Ori disc with an angular length of $\\sim $ 20$^{\\prime \\prime }$ ($\\sim $ 8300 au; label D ).",
"Parts of both the large southern and western columns are identified in the blueshifted $^{12}$ CO emission (Figure REF ; label D).",
"The systemic $^{12}$ CO emission of V1647 Ori appears to be resolved out or undetected and instead displays features that may originate from foreground cloud contamination.",
"Figure: Side by side comparison of blue, systemic, and red shifted moment 0 images of 12 ^{12}CO.",
"The central location of the V1647 Ori disc continuum (Fig ) is shown with a black cross and the beam is displayed in the lower left.",
"Figures are labeled according to the text in Section 3.",
"All 12 ^{12}CO channels associated with emission can be seen in Figure 1 of the Appendix."
],
[
"Emission from $^{13}$ CO",
"Unlike $^{12}$ CO, typical densities of $^{13}$ CO in molecular cloud environments are low enough such that the emission line is optically thin and is a better indicator of column density.",
"As such, the blue and redshifted emission features displayed in Figure REF likely trace material outflowing away from the central star whereas the systemic emission features likely represent material located near the central star-disc system, out of which the star has been forming.",
"Similar to the case of the $^{12}$ CO blueshifted emission, a clump of material is identifiable in the blueshifted $^{13}$ CO emission at position angle east of north of $\\sim $ 220$^{\\circ }$ seemingly disconnected or 'pinched' from the central region (label B).",
"The prominent blueshifted outflow feature (Figure REF left) identifiable at position angle $\\sim $ 330$^{\\circ }$ (label E) has an opening angle of $\\sim $ 50$^{\\circ }$ extending to distances of $\\sim $ 11$^{\\prime \\prime }$ ($\\sim $ 4500 au).",
"The peak intensity in the blueshifted image is located $\\sim $ 1$^{\\prime \\prime }$ north-west of V1647 Ori, in the direction of the outflow.",
"A second collimated outflow is detected in the redshifted line emission pointing almost directly south and extending a distance of 12.8$^{\\prime \\prime }$ ($\\sim $ 5300 au; label D).",
"Similar to the case of the blueshifted emission, the peak intensity of the redshifted image is located $\\sim $ 1$^{\\prime \\prime }$ south-east of V1647 Ori, in the direction of the southern outflow.",
"The systemic $^{13}$ CO emission feature (Figure REF center) displays an apparent \"hole\" coincident with the location of the collimated $^{13}$ CO blueshifted outflow (label F).",
"Figure: Side by side comparison of blue, systemic, and red shifted moment 0 images of 13 ^{13}CO.",
"The central location of the V1647 Ori disc continuum (Fig ) is shown with a black cross.",
"Figures are labeled according to the text in Section 3.",
"All 13 ^{13}CO channels associated with emission can be seen in Figure 1 of the Appendix."
],
[
"Emission from C$^{18}$ O",
"C$^{18}$ O emission, typically a tracer of higher column density gas in molecular cloud environments, is displayed in Figure REF .",
"Distinct features are seen in the blue, systemic and red shifted velocity channels.",
"The blue shifted C$^{18}$ O features trace the collimated outflowing material (label E) previously identified in the $^{13}$ CO integrated intensity images (Figure REF ).",
"This feature has roughly the same opening angle and angular size as its blueshifted $^{13}$ CO counterpart.",
"Similar to the case of the blueshifted $^{13}$ CO emission, the peak intensity location in the blueshifted C$^{18}$ O is located $\\sim $ 1$^{\\prime \\prime }$ northwest of V1647 Ori in the direction of the blueshifted outflow.",
"The same clump located at position angle $\\sim $ 220 from the $^{12}$ CO and $^{13}$ CO blueshifted integrated intensity images, apparently disconnected ('pinched') from the main emission features of V1647 Ori, is also observed to the south west in blueshifted C$^{18}$ O emission (label B).",
"While the redshifted C$^{18}$ O emission does not display the same collimated feature as is evident in the $^{13}$ CO redshifted emission, it does however display two bright peaks, one of which is coincident with the location of the peak intensity of the $^{13}$ CO redshifted emission south east of V1647 Ori (i.e., in the direction of the southern collimated outflow).",
"The second bright peak in the C$^{18}$ O redshifted image is very close to the location of the V1647 Ori disc and may be a detection of circumstellar disc gas.",
"Figure: Side by side comparison of blue, systemic, and red shifted moment 0 images of C 18 ^{18}O.",
"The central location of the V1647 Ori disc continuum (Fig ) is shown with a black cross.",
"Figures are labeled according to the text in Section 3.",
"All C 18 ^{18}O channels associated with emission can be seen in Figure 1 of the Appendix.A strong detection of C$^{18}$ O at systemic velocities likely represents the dense material from which the V1647 Ori protostar formed and may impact the structures identifiable in the other spectral lines.",
"To emphasize this role, the systemic C$^{18}$ O emission contours are overlaid on blueshifted $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O integrated intensity images in Figure REF .",
"The systemic C$^{18}$ O emission rests as the base of the bowl identified in blueshifted $^{12}$ CO (Figure REF left) and also is spatially coincident with an apparent discontinuity between the blueshifted $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O emission at position angle $\\sim $ 220$^{\\circ }$ (label B; i.e., these emission features appear to 'bend' around the material probed by systemic C$^{18}$ O emission).",
"Moreover, the large ($\\sim $ 10$^{\\prime \\prime }$ ) bright extended emission feature in the south east part of the systemic C$^{18}$ O emission appears to be in line with the axis of the $^{13}$ CO and C$^{18}$ O collimated outflow (label G in Figure REF ).",
"The redshifted C$^{18}$ O material may also play a role in shaping of the nebula as displayed in Figure REF .",
"Redshifted C$^{18}$ O emission material may be shaping the long extended arm observed in redshifted $^{12}$ CO that is apparently connecting the V1647 Ori system to an extended feature at a projected distance of $\\sim $ 8300 au.",
"Figure: C 18 ^{18}O systemic emission (white contours) is shown overlaid on blueshifted 12 ^{12}CO, 13 ^{13}CO and C 18 ^{18}O integrated intensity images.",
"The contour levels for the systemic C 18 ^{18}O emission are 4.0, 5.5, 6.0, 6.5, and 6.75 ×\\times 3σ\\sigma .",
"The beam is displayed in the lower left.Figure: Redshifted C 18 ^{18}O contours (0.6, 0.7, 0.85, 1.0, 1.1, and 1.2 ×\\times 3σ\\sigma ) overlaid on the redshifted 12 ^{12}CO integrated intensity image of V1647 Ori."
],
[
"Dynamics of the V1647 Ori Environment",
"Figure REF highlights the various outflows/features identified in the previous sections (labels D and E from Figures REF , REF , and REF ).",
"These features can be summarized in the following way: a $\\sim $ 50$^{\\circ }$ collimated outflow at position angle of $\\sim $ 330$^{\\circ }$ is detected in blueshifted $^{13}$ CO and C$^{18}$ O (Figure REF left), a $\\sim $ 100$^{\\circ }$ excavated cavity pointing directly north of V1647 Ori is detected in both blueshifted $^{12}$ CO and systemic C$^{18}$ O and misaligned with the aforementioned collimated outflow (Figure REF center), and a $\\sim $ 30$^{\\circ }$ collimated outflow detected in redshifted $^{12}$ CO and $^{13}$ CO pointing directly south of V1647 Ori (Figure REF right).",
"The dynamical age ($\\tau _{d}$ ) of these outflows can be estimated using their physical extent measured from the location of the V1647 Ori disc ($r_\\mathrm {outflow}$ ) and their maximum velocities relative to the velocity of the system (i.e., $\\tau _{d}$ = $\\frac{r_\\mathrm {outflow}}{v_\\mathrm {system}-v_\\mathrm {max}}$ ).",
"However, these dynamical ages are relatively uncertain given the low inclination of this system (i = 17$^{\\circ }$ ).",
"Since each of the two collimated outflows were detected with multiple emission lines, we can estimate two independent dynamical ages for each outflow.",
"These dynamical ages and the values used to determine them are displayed in Table REF and indicate the outflow identified in blueshifted $^{13}$ CO and C$^{18}$ O has a dynamical age of $\\sim $ 11,700 years whereas the outflow identified in redshifted $^{12}$ CO and $^{13}$ CO has a dynamical age of $\\sim $ 17,200 years.",
"Given an uncertainty of $\\pm $ 7 pc for a distance of 414 towards the Orion Nebula , these dynamical age estimates may be different by $\\sim $ 200 and 300 years, respectively.",
"However, we caution that the dynamical age of the redshifted $^{12}$ CO feature is highly uncertain given that it extends to radii larger than the ALMA primary beam.",
"We do not estimate the dynamical age of the $^{12}$ CO bowl-like feature pointed directly north because this feature is more likely a cavity rather than an outflow.",
"However, it should be noted that the northern bowl-like emission has a similar blueshifted $^{12}$ CO maximum velocity ($v_\\mathrm {max}$ = 3) as the collimated redshifted $^{12}$ CO feature pointed directly south ($v_\\mathrm {max}$ = 2.5).",
"Figure: Contours overplotted on blueshifted 13 ^{13}CO, C 18 ^{18}O and redshifted 12 ^{12}CO integrated intensity images of V1647 Ori to highlight outflow/cavity morphology.",
"Left: Blueshifted C 18 ^{18}O contours with levels of 1.0, 1.75, 4.0, 6.0, and 8.0 ×\\times 3σ\\sigma .",
"Middle: Blueshifted 12 ^{12}CO contours with levels 4.0, 10.0, 12.0, 25.0 and 35.0 ×\\times 3σ\\sigma .",
"Right: Redshifted 13 ^{13}CO contours with levels 1.0, 1.5, 2.0, 2.5, and 3.2 ×\\times 3σ\\sigma .Table: Dynamical Ages of the Outflows in V1647 OriPhysical characteristics (e.g., mass, momentum, and kinetic energy) of the molecular gas surrounding the V1647 Ori protostar can be estimated using primary beam corrected $^{12}$ CO and $^{13}$ CO line emission.",
"We follow the procedure outlined in and adopt a $^{12}$ CO to H$_{2}$ abundance ratio of 10$^{-4}$ and a $^{12}$ CO to $^{13}$ CO relative abundance of 62 .",
"This analysis assumes a constant excitation temperature of T$_\\mathrm {ex}$ = 50 K along the line of sight and a beam filling factor of 1.",
"We then calculate the outflow mass M =$\\sum _{v}$ M($x,y,v$ ), momentum P = $\\sum _{v}$ M($x,y,v$ ) $\\times $ $v$ , and kinetic energy KE = $\\sum _{v}$ M($x,y,v$ ) $\\times $ $\\frac{1}{2}$$v^{2}$ for each velocity channel within a radius of 10$^{\\prime \\prime }$ of V1646 Ori (e.g., see Figure REF ).",
"This calculation was performed on primary beam-corrected channel maps.",
"Parameters for the blue and redshifted regions of $^{12}$ CO and $^{13}$ CO are displayed in Table REF and correspond to the velocity ranges indicated at the top of Figure REF and displayed in Figures REF and REF .",
"Optical depth corrections, typically performed using the ratio of line fluxes between the optically thick $^{12}$ CO and optically thin $^{13}$ CO , are unable to be applied to this dataset because there are few channels where both $^{13}$ CO and $^{12}$ CO are detected.",
"Therefore, the $^{12}$ CO parameters estimated in this work (Table REF ) are lower limits, and not necessarily representative of the entire mass of the outflow in $^{12}$ CO. Table: Physical Properties of the V1647 Ori Outflows"
],
[
"Optical Reflection Nebula (McNeil's Nebula)",
"The $\\sim $ 1$^{\\prime }$ diameter optical r-band reflection nebula to the north of V1647 Ori is displayed in Figure REF .",
"These data were taken during the 2008 outburst and display a generally diffuse nebulosity of dust with a clump of material to the north west near RA and Dec offset [-4, 10].",
"Overlays of the $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O contours on the optical data indicate the millimeter gas emission is well aligned with the base of the reflection nebula.",
"Moreover, the collimated outflow observed in $^{13}$ CO and C$^{18}$ O appear to be traveling in the direction of the bright dust clump observable at optical wavelengths.",
"Figure: Optical Gemini GMOS r band observation of V1647 Ori and its reflection nebula taken during the 2008 outburst.",
"Overlaid are integrated intensity image contours of 12 ^{12}CO, 13 ^{13}CO, and C 18 ^{18}O as indicated in the lower left of each panel.",
"North is up and east is left"
],
[
"The Circumstellar Disc of V1647 and Potential Mechanism for Accretion Outbursts",
"The resolved continuum image of the V1647 Ori circumstellar disc (Figure REF ) does not display clear evidence of spirals or clumps associated with disc fragmentation or perturbations from a potential binary companion.",
"Therefore, it is unlikely there are any binary or massive substellar companions $\\gtrsim $ 40 au from the central star capable of producing the intense outbursts observed in V1647 Ori.",
"However, the potential non-gaussian radial brightness profile identifiable at large radii in Figure REF suggests some amount of asymmetry and warrants a more detailed investigation than that performed for the continuum data and presented here.",
"While the circumstellar disc is clearly detected in the dust continuum, it is less evident in the molecular line data.",
"Molecular emission spatially coincident with the location of the disc is identifiable in the redshifted $^{12}$ CO integrated intensity image (Figure REF ) and is potentially detected but unresolved from extended cloud/outflow emission at blueshifted $^{12}$ CO and $^{13}$ CO and redshifted $^{13}$ CO and C$^{18}$ O.",
"Such emission in these cases may not be from the disc but instead could be from a small region surrounding the disc.",
"Moreover, gaseous emission from the disc was not detected in any of the emission lines at the systemic velocities (e.g., see Figure REF top and Appendix).",
"Given the inconsistency of its detection, we were unable to search for signatures of Keplerian rotation.",
"The non-detection of the disc at systemic velocities is likely due to the deeply embedded nature of this system which is consistent with its high extinction at optical and X-ray wavelengths.",
"The radius of the disc derived from the continuum emission (Table REF ) is similar to that of the namesake of the FUor class as well as HBC 494 and V2775 Ori , .",
"However, compared with V883 Ori, the V1647 Ori disc is a factor of $\\sim $ 2 smaller and its star/disc bolometric luminosity is a factor of 10 fainter .",
"The V1647 Ori disc mass derived from the continuum is the least massive compared with the other FUor objects presented in this series.",
"This suggests objects that have been classified strictly as FUors may be more massive than EXor objects where V1647 Ori is in the middle of this classification as it displays outbursts indicative of both classifications."
],
[
"Excavated Cavities and Outflows",
"The molecular emission line data indicate that the star-forming environment surrounding V1647 Ori is both structured and complex.",
"Several features indicating the three-dimensional structure of this system are readily identifiable and show that the protostar and disc system are still deeply embedded in its parent molecular cloud.",
"The orientation of the system can be further constrained when considering the direction of the blueshifted outflow (label E) coincident with northern reflection nebula (Figure REF ).",
"The dust cavity appears in the north at optical wavelengths because it is preferentially forward scattering emission from its illuminating source (i.e., V1647 Ori during outburst).",
"This indicates that the cavity is located between V1647 Ori and our line of sight which is consistent with the cavity detection in blueshifted $^{12}$ CO emission (label A).",
"The co-alignment of this excavated northern cavity with the outflow at PA$\\sim $ 330$^{\\circ }$ is further substantiated by the location of the bright clumps visible during outburst in the optical reflection nebula since they are aligned with direction of the collimated outflow (Figure REF ).",
"This orientation also explains the absence of a reflection nebula at optical wavelengths to the south of V1647 Ori even though a southern cavity is present (Figure REF ; i.e., the southern cavity is pointed away from us and any reflected light will be backscattered and too faint to detect in shallow observations).",
"Given the appearance of its complex environment, the strongest constraints on the orientation of the V1647 Ori system is measured by the gaussian fit to the continuum data which indicates the disc is almost face-on (i = 17$^{\\circ }$ ) with a position angle of 109$^{\\circ }$ .",
"This disc inclination is in good agreement with previous estimates of i $<$ 30$^{\\circ }$ , but is not consistent with the stellar inclination modeled from periodic X-ray emission generated by rotating accretion hot spots .",
"The measured position angle of the disc (109$^{\\circ }$ $\\pm $ 19) appears to align more with the outflow at PA$\\sim $ 330$^{\\circ }$ (label E) than the southern collimated outflow at PA$\\sim $ 180$^{\\circ }$ (label D) assuming that outflows travel perpendicular to the plane of the disc and the disc is inclined such that the northern extended emission is blueshifted towards our line of sight.",
"Since the disc orientation is a strong constraint on the system as a whole, our interpretation of the large scale (r $\\gtrsim $ 1-3$^{\\prime \\prime }$ ) emission features (e.g., collimated outflows) assumes that these features were formed from material traveling roughly perpendicular to the circumstellar disc of V1647 Ori at the time of their respective outflow triggering events.",
"Such an assumption is reasonable given our current understanding of launching mechanism of material during protostellar evolution .",
"The blueshifted $^{12}$ CO emission whose bowl-like shape conforms well to the base of the optical reflection nebula (Figure REF ) is seemingly misaligned with the aforementioned blueshifted $^{13}$ CO and C$^{18}$ O collimated feature (Figure REF , center).",
"Moreover, the non-detection in $^{13}$ CO of this $^{12}$ CO blueshifted bowl-like shape suggests that there are no active wide-angle outflows at this location and that the $^{12}$ CO is probing the temperature of the walls of the excavated cavity.",
"The misalignment between the collimated southern outflow (Figures REF , REF , label D) and the collimated northern outflow at PA$\\sim $ 330$^{\\circ }$ (Figures REF , REF , label E) may be the result of a re-orientation or precession of the mass-launching area (i.e., the orientation of the circumstellar disc may have changed from one outburst to another resulting in two outflows orientated with different position angles).",
"The difference in position angle and the dynamical age between these collimated outflows (Table REF ) suggest each outflow was likely the result of a specific mass-accretion event with the southern outflow occurring $\\sim $ 5000 years before the north-west outflow.",
"Given that a radical change in disc position angle is unlikely, these collimated outflows were probably launched from opposite sides of the circumstellar disc during their respective mass accretion events.",
"While some asymmetric jets, like those detected from Herbig-Haro objects, appear to be shaped by their local star-forming environment , others systems exhibit clear evidence that the jet-launching region (i.e., the region perpendicular to the plane of the disc) has precessed over time resulting in outflows with asymmetric and misaligned features , , , .",
"While the amount of precession over the lifetime of the outflow varies from case to case, it appears that jet precession of more than a few degrees may be rare .",
"In cases where the degree of disc precession is high (like in the case of V1647 Ori), several mechanisms have been proposed capable of significantly re-orienting the outflow/jet: (a) radiative-induced disc warping where a jet, presumably oriented perpendicular to a disc, precesses in response to a disc warp which is generated in cases of high-incident radiation ; (b) tidal interaction from a companion in a non-coplanar orbit truncating and/or distorting the disc , ; or (c) anisotropic accretion events where an impact/merging of material onto the disc during anisotropic accretion can change the orientation of the disc angular momentum vector resulting in a net torque in the rotation of the disc , .",
"Estimating the critical radius at which the disc becomes unstable to warping following equation 5 in , the assumption that $\\eta $ =1 in , and the physical characteristics of V1647 Ori during outburst (M = 0.8 $M_{\\odot }$ , $\\overset{.",
"}{M}$$_\\mathrm {acc}$ = 4 $\\times $ $10^{-6}$ $M_{\\odot }$ yr$^{-1}$ , $L = 44 L_{\\odot }$ ), $R_\\mathrm {crit}$ = 2 $\\times $ 10$^{6}$ au.",
"Given that this radius is orders of magnitude larger than the radius of the V1647 Ori disc, it is unlikely a disc warp is the origin of its misaligned outflows.",
"However, it should be noted that the stellar parameters during the outburst event that produced these outflows ($\\sim $ 14,000 years ago; Table REF ) are not necessarily the same as those measured during the 2003 outburst.",
"As noted in Section 4.1, the non-detection of any companions to V1647 Ori at radii > 40 au likely rules out tidal interactions via a companion causing disc precession unless such a companion exists at smaller radii and has not disrupted the V1647 Ori disc in such a way as to be evident from the continuum observations.",
"Therefore, given the intrinsic connection between accretion and FUor/EXor outbursts, it appears that significant mass-loading (e.g., from a large clump of envelope material or a companion) during anisotropic accretion may be the most likely scenario to have caused the disc to precess between each outflow-generating outburst in V1647 Ori.",
"If this anisotropic accretion scenario is true in the case of V1647 Ori, one might expect more FUor/EXor type objects to display multiple misaligned outflows.",
"However, there appears to be little evidence for this, at least in the case of the other FUors studied as part of this series , , .",
"Given the complex geometry of the V1647 Ori star-forming environment, it may be that other FUor/EXor objects underwent smoother mass-accretion outbursts which did not result in disc precession or simply had fewer large-scale mass accretion events.",
"Z CMa, a binary FUor object that also exhibits properties of an EXor, displays two slightly misaligned outflows where a wiggling shape of one of the outflows may be generated via precession from an unseen additional companion , .",
"Given that Z CMa and V1647 Ori represent a rare class of objects that exhibit properties of both FUor and EXor objects, we speculate that disc precession may be related to this intermediate classification.",
"While the disc precession argument seems the most plausible, other scenarios could be invoked to describe the complicated geometry of these two collimated emission features.",
"The two outflows could originate from separate binary members ( e.g., each with their own outflow) if V1647 Ori has an undetected companion.",
"In this case, outflow misalignment may be a signature that the potential binary formation of V1647 Ori resulted from turbulent fragmentation .",
"However, this scenario generally considers binaries with separations much larger than what would likely be the case for V1647 Ori ($\\lesssim $ 40 au).",
"Another scenario alternative to changes in disc orientation is the case that the outflows instead are located at different position angles as a result of a dense material within the molecular cloud shaping the outflow.",
"V1647 Ori clearly resides in a complex and embedded environment and the outflow structure could simply be following a path of least resistance due to some unobserved molecular cloud component whose mass would be difficult to determine without knowledge of the pre-outflow geometry.",
"This scenario would also affect the dynamical age calculations if the outflowing material does not travel in a straight line.",
"Another explanation alternative to disc precession could be that the collimated features observed are actually inflows and not outflows.",
"This interpretation would require our understanding of the system geometry to be reversed (i.e., the blueshifted outflow situated between V1647 Ori and our line of sight would instead be behind V1647 Ori and traveling towards the system presumably in a large accretion stream).",
"However, this scenario is unlikely to be true given the spatial correlation between the blueshifted $^{13}$ CO and C$^{18}$ O emission with the northern cavity of the optical reflection nebula which independently indicates this northern region is between V1647 Ori and our line of sight and thus, the emission feature is more likely an outflow."
],
[
"The Shaping Potential of Dense Molecular Gas Surrounding V1647 Ori",
"Given the structure of the red and blueshifted molecular line emitting material compared to that of the systemic C$^{18}$ O emission (Figure REF ), it is evident the dense material in which the protostar is embedded influences the overall structure of the system.",
"While it is not clear whether this dense material can result in two misaligned outflows (Section 4.2), it is still evident that the blueshifted $^{12}$ CO bowl-like shape is surrounded by the dense C$^{18}$ O material of similar shape (Figure REF left).",
"This is in good agreement with the location of the wall of the excavated cavity as seen in $^{12}$ CO, suggesting the shaping mechanism that produced this cavity compressed the low-density ambient molecular material until it collided with the dense material as probed by C$^{18}$ O.",
"Moreover, the western arm of the systemic C$^{18}$ O material appears to have cut off or 'pinched' the material associated with blueshifted $^{12}$ CO and $^{13}$ CO material $\\sim $ 3$^{\\prime \\prime }$ southwest of the V1647 Ori disc (label B).",
"The outflowing blueshifted material appears to travel around the dense material at systemic velocities as it may be more energetically favorable.",
"If this interpretation is true then the location of the dense systemic C$^{18}$ O material must be between our line of sight and the V1647 Ori disc (i.e., in the foreground but still part of the V1647 Ori star-forming environment).",
"This interpretation suggests that the local dense gas plays a large role in the shaping of cavities and molecular outflows during this stage of pre-MS stellar evolution.",
"A similar shaping scenario may be taking place where the redshifted C$^{18}$ O emission appears along the border of the 'arm' seemingly connecting the V1647 Ori system to the large unresolved feature $\\sim $ 20$^{\\prime \\prime }$ (8300 au) to the west (Figure REF ).",
"This large $\\sim $ 40$^{\\prime \\prime }$ (17,000 au) column of material is also detected in a few channels of $^{13}$ CO emission traveling at the same velocity (Figure 2 in the appendix).",
"The source of this structure is unclear and it may just be leftover cloud material that either has yet to form a star or does not meet the physical requirements to do so.",
"However, we caution that this feature may be an artifact from emission outside the primary beam.",
"The closest pre-MS star (in projection) to this structure (2MASS J05461162-0006279) is located $\\sim $ 7$^{\\prime \\prime }$ (3150 au) from this extended structure's south-western edge.",
"It is identified as a Class II object in the L1630 star-forming region based on its infrared-excess and detection in X-rays and has a moderate absorption of N$_{H}$ = 5 $\\times $ 10$^{21}$ cm$^{-2}$ ."
],
[
"Outflow Dynamics and a Comparison with Other FUors",
"The complex structures identified in the V1647 Ori at spatial scales of $\\sim $ 1.0$^{\\prime \\prime }$ can be compared with the other similarly structured FUor/EXor type objects presented in this series , , .",
"Of the eight FUor/EXor objects observed, only V2775 Ori, V883 Ori, HBC 494 and V1647 Ori display significant amounts of extended $^{12}$ CO, $^{13}$ CO and/or C$^{18}$ O emission.",
"Given the high A$_V$ , N$_H$ , complex C$^{18}$ O environment, and ambiguous detection of its gaseous circumstellar disc, V1647 Ori appears to be in the earliest stage of stellar evolution compared with the other FUor objects presented.",
"In particular, the structures displayed in these ALMA observations of V1647 Ori could represent the progenitor stages of those of V883 Ori and V2775 Ori given their similar inclination (and thus similar viewing angle).",
"Similar to the case of V2775 Ori , V1647 Ori also exhibits a small outflow opening angle ($\\sim $ 30$^{\\circ }$ ) in $^{13}$ CO, C$^{18}$ O, further indicating its primordial nature if opening angle is a correlation to age.",
"In contrast, HBC 494 and V883 Ori , have much larger opening angles of $\\sim $ 150$^{\\circ }$ .",
"Almost counterintuitively, the outflows of V1647 Ori have the oldest dynamical ages (11,700 and 17,200 years) when compared to that of V2775 Ori , HBC 494 and V883 Ori .",
"These dynamical ages are very small given the timescale of pre-MS stellar evolution with respect to disc dissipation (e.g., $\\sim $ 1-3 Myr).",
"Therefore the older dynamical timescale of V1647 Ori's outflows do not necessarily mean it is inconsistent with the interpretation here that this system is less evolved than its other FUor counterparts.",
"It could simply mean that whatever mechanism initiates these outbursts may have started earlier in V1647 Ori.",
"Given the complex environment surrounding V1647 Ori, it may indicate that inhomogeneous patches of envelope material could have prompted more frequent or earlier accretion outbursts.",
"The outflow mass surrounding V1647 Ori (as indicated in Section 3.6) appears similar to those obtained for other Class 0-II pre-MS stars with outflows as well as the small sample of FUor type objects whose outflows have been measured .",
"However, when comparing the outflow velocities associated with these objects, it is clear that the outflows surrounding V1647 Ori are quite slow ($\\sim $ 1-2 km s$^{-1}$ ).",
"Given the similar sensitivity and resolution of the other FUors detected as part of this series, we compare the results of V1647 Ori directly with those of V2775 Ori, HBC 494, and V883 Ori.",
"Of these sources, V1647 Ori has the least massive circumstellar disc and is the only source considered both an FUor and EXor type object based on its unusual outburst frequency.",
"V1647 Ori is the most similar in disc mass and inclination to V2775 Ori , however their local environments are quite distinct.",
"While V2775 Ori displays an impressive ring-like structure in its blue and red shifted emission , there is little evidence of complex structure surrounding the large-scale environment of the protostar indicating that this system may be more evolved than V1647 Ori.",
"Moreover, V2775 Ori exhibits a redshifted $^{12}$ CO outflow velocity of 4.4 km s$^{-1}$ , a factor of $\\sim $ 2 faster than those of V1647 Ori, potentially indicative of a emptier environment surrounding the star where the outflow doesn't slow down from interactions with surrounding material.",
"A comparison of outflow mass and energy indicates the outflow of V1647 Ori as probed by $^{12}$ CO and $^{13}$ CO is more massive than those in V2775 Ori by a factor of 3 and 15, respectively.",
"A comparison of V1647 Ori with that of HBC 494 and V883 Ori, both of which have more massive and larger circumstellar discs, indicates the outflow mass of V1647 Ori is significantly smaller , .",
"Both these systems exhibit slow outflow speeds similar to or smaller than V1647 Ori possibly indicating they too still are interacting with or have been influenced by surrounding material."
],
[
"Summary and Conclusions",
"We have presented ALMA observations of the V1647 Ori continuum at a resolution of 0.2$^{\\prime \\prime }$ as well as molecular line emission of $^{12}$ CO,$^{13}$ CO and C$^{18}$ O at a resolution of $\\sim $ 1$^{\\prime \\prime }$ .",
"No emission line features were detected in the 0.2$^{\\prime \\prime }$ dataset indicating that structure surrounding V1647 Ori are fairly uniform on large scales.",
"These observations are presented as part of a series and we compare these results of V1647 Ori to that of other FUor type objects with extended emission (e.g., V2775 Ori, HBC 494, and V883 Ori).",
"We have identified a complex yet structured system surrounding the V1647 Ori protostar and circumstellar disc and are able to draw several conclusions: $\\bullet $ The circumstellar disc of V1647 Ori is resolved and has a nearly face-on inclination (i = 17$^{\\circ }$ ), a radius of 40 au and a total disc mass of $\\sim $ 0.1 M$_{\\odot }$ .",
"The continuum resolved disc does not display any disc structures (e.g., spirals and/or fragmentation) associated with nearby interacting companions (r $\\gtrsim $ 40 au) and therefore such a mechanism, under these constraints, is unlikely to be responsible for the various observed accretion outbursts.",
"The mass of the disc is less than the other FUor type objects presented in this series may be related to its classification as both an FUor and EXor type object.",
"$\\bullet $ Blueshifted $^{12}$ CO molecular line emission spatially coincident with the V1647 Ori optical reflection nebula (McNeil's Nebula) and a non-detection of this feature in $^{13}$ CO and C$^{18}$ O emission confirm that this structure is a cavity excavated during a previous accretion outburst.",
"A collimated $^{13}$ CO and C$^{18}$ O emission feature aligns with clumps in the nebula identified during the recent multiwavelength outburst.",
"$\\bullet $ Dense molecular material as probed via C$^{18}$ O emission traveling at a systemic velocity of $\\sim $ 10 km s$^{-1}$ is spatially coincident with the base of the northern excavated cavity and likely shaped the outflow material identified in blueshifted $^{12}$ CO, $^{13}$ CO, and C$^{18}$ O emission.",
"$\\bullet $ Two distinct collimated ($\\sim $ 30-50$^{\\circ }$ ) outflows with position angles of 330$^{\\circ }$ and 180$^{\\circ }$ have been detected with dynamical ages of $\\sim $ 11,700 and $\\sim $ 17,100 years, respectively.",
"These outflows are likely the result of two distinct mass accretion events where outflows were launched from opposite sides of a disc that has changed orientation over the last $\\sim $ 5000 years from previous anisotropic accretion events.",
"$\\bullet $ Compared to the other FUor objects included in our series, V1647 Ori displays the most complex large-scale structures likely the result of its comparatively early stage of stellar evolution.",
"A comparison of outflow parameters supports this notion and also indicates that the outflow velocities of these FUor and FUor/EXor type objects ($\\sim $ 0.5 - 4 km s$^{-1}$ ) are relatively small compared to those observed in typical pre-MS stars and predicted by simulations ($v$$\\sim $ 10-30 km s$^{-1}$ ).",
"The complex star-forming environment surrounding V1647 Ori may result in more frequent yet still extreme outburst events which justify its classification as both an FUor and EXor type object.",
"$\\bullet $ A large ($\\sim $ 17,000 au) unresolved structure is identified $\\sim $ 8300 au from V1647 Ori.",
"This structure is not associated with any obvious source but is seemly connected to V1647 Ori by an 'arm' of molecular material as probed with redshifted $^{12}$ CO emission."
],
[
"Acknowledgements",
"This research was supported by CONICYT-FONDECYT awards (3150550, 1171246) and support from the Millennium Science Initiative (Chilean Ministry of Economy; grant Nucleus RC 130007).",
"This paper makes use of the following ALMA data: ADS/JAO.ALMA #2013.1.00710.S .",
"ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.",
"The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.",
"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.. D. P. acknowledges funding by the National Aeronautics and Space Administration through Chandra Award Number GO6-17013A.",
"K.M acknowledges funding by the Joint Committee of ESO/Government of Chile, and funding by the Science and Technology Foundation of Portugal (FCT), grant No.",
"IF/00194/2015.",
"H.C. acknowledges support from the Spanish Ministerio de Economía y Competitividad under grant AYA 2014-55840-P. J.J.T acknowledges support from the University of Oklahoma, the Homer L. Dodge endowed chair, and grant 639.041.439 from the Netherlands Organisation for Scientific Research (NWO).",
"The authors would like to thank the referee whose comments and suggestions increased the quality of this work."
]
] | 1709.01924 |
[
[
"Linear gyrokinetic investigation of the geodesic acoustic modes in\n realistic tokamak configurations"
],
[
"Abstract Geodesic acoustic modes (GAMs) are studied by means of the gyrokinetic global particle-in-cell code ORB5.",
"Linear electromagnetic simulations in the low electron beta limit have been performed, in order to separate acoustic and Alfv\\'enic time scales and obtain more accurate measurements.",
"The dependence of the frequency and damping rate on several parameters such as the safety factor, the GAM radial wavenumber and the plasma elongation is studied.",
"All simulations have been performed with kinetic electrons with realistic electron/ion mass ratio.",
"Interpolating formulae for the GAM frequency and damping rate, based on the results of the gyrokinetic simulations, have been derived.",
"Using these expressions, the influence of the temperature gradient on the damping rate is also investigated.",
"Finally, the results are applied to the study of a real discharge of the ASDEX Upgrade tokamak."
],
[
"Introduction",
"The ion heat transport in the plasma core is governed by turbulence formed by a class of microinstabilities such as toroidal ion temperature gradient (ITG) driven modes [1].",
"ITG turbulence is known to self-organize to form macroscopic structures [2].",
"These structures take the form of a macroscopic radial electric field which depends only on the radial coordinate.",
"$E \\times B$ poloidal flows associated with this electric field are referred to as zonal flows (ZFs) [3], [4], [5], [6].",
"The action of the toroidal magnetic field curvature on the ZF gives rise to oscillations of the radial electric field.",
"These oscillations of the ZFs are called geodesic acoustic modes (GAMs) [7], [8].",
"The modes are observed predominantly in the edge region of the tokamak plasmas with characteristic frequency of the order of the sound frequency $\\sim c_s/R$ , where $c_s = \\sqrt{T_e/m_i}$ is the sound speed, $R$ is the major radius.",
"One of the main linear damping mechanisms for the stationary ZF are collisional processes and for the GAM it is a collisionless wave-particle interaction, namely the Landau damping, and collisional damping at the very edge of the plasma, where equilibrium temperatures drastically decrease [9].",
"A recent comparison of collisionless and collisional damping of GAMs, using existing analytical theories, for experimentally relevant plasmas was done in Ref.",
"[10].",
"The importance of the ZF is that they can regulate the drift-wave (DW) turbulence [11].",
"But it is still a question how the GAMs influence the ZF efficiency of the DW suppression [8], [12], [13].",
"On the other hand, the development of zonal structures can play a key role in the transition from the low to the high confinement regime (L-H transition) [14].",
"In Ref.",
"[15] interaction of the mean and oscillatory poloidal flows with the turbulence were experimentally observed.",
"The turbulence suppression by the ZFs was observed in experiments described in Ref.",
"[16].",
"On the other hand, in Ref.",
"[17] the role of the mean flow in the dynamic evolution towards the H-mode is emphasized.",
"In Ref.",
"[12] two predators - one prey system, including ZF, GAM and turbulence, was developed to study transitions between states with different combinations of the ZF and GAM.",
"In this paper, we investigate the GAM frequency and collisionless damping rate, carrying out linear collisionless simulations with kinetic electrons.",
"The electromagnetic global gyrokinetic particle-in-cell code ORB5 is used [18], [19].",
"As it has been reported previously [20], [21], models, numerical or analytical, derived with adiabatic electrons, result in considerably smaller GAM damping rate in comparison to simulations performed with kinetic electrons.",
"By adiabatic electron models, we mean here models treating the $m\\ne 0$ component of the electrons as adiabatic, and setting the zonal component of the electron density perturbation to zero.",
"In simulations considered in this paper, electrondocumentclasss are treated drift-kinetically, and a realistic ion-electron mass ratio is used.",
"Moreover, to study the influence of the plasma elongation on the GAM dynamics, magnetic equilibria with realistic plasma shapes are considered.",
"To summarize the results obtained in different plasma regimes, interpolating formulae for the GAM frequency and damping rate, based on the gyrokinetic simulations with ORB5, are derived.",
"Due to the so-called phase mixing effect, the GAM damping rate is increased in the presence of a temperature gradient or the safety factor profile [8], [22], [23].",
"This effect arises when the damping rate of the wave depends on its wavenumber.",
"In the case of the GAM the damping rate increases with the GAM radial wavenumber (more precisely, with the radial wavenumber of the radial electric field).",
"Since the GAM frequency depends on the temperature and safety factor, the GAM oscillates with different frequencies at different radial points in presence of the temperature gradient or magnetic shear.",
"Distorting the GAM radial structure and creating higher radial wavenumbers, this process can strongly increase the GAM damping rate [8], [24].",
"A section of our paper is dedicated to the extension of previous works [21], [23], which were done treating the electrons as adiabatic, and in circular flux surfaces, to the inclusion of kinetic electrons and realistic tokamak configurations.",
"Finally, the last section of this paper is dedicated to the investigation of a realistic discharge of ASDEX Upgrade, described in Ref. [25].",
"In Appendix we have shown a comparison between ORB5 and GENE for the case of non-flat temperature profile."
],
[
"Model",
"The gyrokinetic simulations presented in this work have been performed with the code ORB5 [18], [19].",
"ORB5 is a nonlinear gyrokinetic multi-species global particle-in-cell (PIC) code, which solves the Vlasov-Maxwell system in the electrostatic or electromagnetic limit, and has a capability of handling true MHD equilibrium for an axisymmetric toroidal plasma.",
"The particle-in-cell method consists of coupling a particle-based algorithm for the Vlasov equation with a grid-based method for the computation of the self-consistent electromagnetic fields.",
"Several physical models are available in ORB5, all of them derived from a systematic Hamiltonian theory [19], [26] to provide exact energy and momentum conservation.",
"In this work, only one ion species (deuterium) has been considered while the electrons are assumed to be drift-kinetic.",
"This corresponds to the following gyrokinetic total Lagrangian: $L =&& \\sum _{\\rm {sp}}\\int \\mathrm {d}V \\mathrm {d}W \\left(\\left(\\frac{q}{c}{A}+p_\\parallel {b}\\right)\\cdot \\dot{{R}} + \\frac{m c}{q}\\mu \\dot{\\theta } - H_0-H_1\\right)f \\\\&&+ \\int \\mathrm {d}V \\mathrm {d}W H_2 f_{M,ions}-\\int \\mathrm {d}V \\frac{B_\\perp ^2}{8\\pi }.\\nonumber $ The velocity variables are the magnetic moment $\\mu \\equiv (mv_\\perp ^2)/(2B)$ , the canonical parallel momentum $p_\\parallel $ and the gyroangle $\\theta $ .",
"The equilibrium magnetic field is ${B}=\\nabla \\times {A}$ , $m$ and $q$ are the mass and charge of the particle species $sp$ and $c$ is the speed of light.",
"The volume element of the velocity space is $\\mathrm {d}W\\equiv (2\\pi )/m^2 B^*_\\parallel \\mathrm {d}p_\\parallel \\mathrm {d}\\mu $ with $B^*_\\parallel ={B^*}\\cdot {{b}}$ , ${b}={B}/B$ and ${B^*}={B}+(c/q)p_\\parallel \\nabla \\times {b}$ ; $\\mathrm {d}V$ denotes the volume element in physical space.",
"Here $f$ is the distribution function for the species $sp$ , while $f_{M,ions}$ is the equilibrium time independent distribution function of the ions.",
"In this system, only long wavelength electrostatic perturbation and magnetic perturbations perpendicular to the equilibrium magnetic field are considered.",
"Note that no second order term in the fields is retained for the electrons, this is equivalent to neglect the electron polarization density in the Polarization equation (drift-kinetic approximation, see Ref.",
"[19] for details).",
"The first two terms in the total Lagrangian define the charged particles Lagrangian [27].",
"The GK Hamiltonian in general depends on the electrostatic potentials $\\Phi $ and on the parallel component of the fluctuation magnetic potential $A_\\parallel $ .",
"The third term in the total Lagrangian is the electromagnetic field Lagrangian, in which the electric field component has been neglected (quasi-neutrality approximation, see [19] for details).",
"In this work we used the following Hamiltonian: $H&=&H_0+H_1+H_2\\\\H_0&=&\\frac{p_\\parallel ^2}{2m}+\\mu B\\nonumber \\\\H_1&=& e(J_0\\Phi - \\frac{p_\\parallel }{mc} J_0A_\\parallel )\\nonumber \\\\H_2 &=& - \\frac{q^2}{2mc^2}(J_0A_\\parallel )^2+\\frac{mc^2}{2B^2}|\\nabla _\\perp \\Phi |^2\\nonumber $ the gyroaveraging (Hermitian) operator $J_0$ , applied to an arbitrary function $\\psi $ in configuration space, is defined by $(J_0\\psi ) ( \\mathbf {R},\\mu ) = \\frac{1}{2\\pi }\\int _0^{2\\pi } \\psi ( \\mathbf {R}+{\\rho }(\\alpha )) \\,d \\alpha ,$ where ${\\rho }$ is the vector going from the guiding center position to the particle position.",
"In this work we have assumed $J_0=1$ for the electrons (drift-kinetic approximation).",
"The gyrokinetic equations for the particle distribution function and the GK field equations can be derived from the GK Lagrangian using variational principles.",
"In summary, the GK model used in the following is: $\\bullet $ gyrokinetic full-f Vlasov equation for the ions $&&\\frac{\\partial {f_i}}{\\partial {t}}+\\dot{{R}_i}\\cdot \\nabla f_i+\\dot{p_{\\parallel ,i}}\\frac{\\partial {f_i}}{\\partial {p_{\\parallel ,i}}}=0,\\\\&&\\dot{{R}_i}=\\left(\\frac{p_{\\parallel ,i}}{m_i}-\\frac{Z_ie}{m_ic}J_0A_\\parallel \\right)\\frac{{B_i^*}}{B^*_{\\parallel ,i}}+\\frac{c}{Z_ieB^*_{\\parallel ,i}}{b}\\times \\left[\\mu _i\\nabla B + Z_ie \\nabla J_0 \\Psi _i \\right],\\\\&&\\dot{p_{\\parallel ,i}}=-\\frac{{B_i^*}}{B^*_{\\parallel ,i}}\\cdot \\left[\\mu _i \\nabla B + Z_ie\\nabla J_0\\Psi _i \\right],$ $\\bullet $ drift-kinetic full-f Vlasov equation for the electrons: $&&\\frac{\\partial {f_e}}{\\partial {t}}+\\dot{{R}_e}\\cdot \\nabla f_e+\\dot{p_{\\parallel ,e}}\\frac{\\partial {f_e}}{\\partial {p_{\\parallel ,e}}}=0,\\\\&&\\dot{{R}_e}=\\left(\\frac{p_{\\parallel ,e}}{m_e}+\\frac{e}{m_ec}A_\\parallel \\right)\\frac{{B_e^*}}{B^*_{\\parallel ,e}}-\\frac{c}{eB^*_{\\parallel ,e}}{b}\\times \\left[\\mu _e\\nabla B - e \\nabla \\Psi _e \\right],\\\\&&\\dot{p_{\\parallel ,e}}=-\\frac{{B_e^*}}{B^*_{\\parallel ,e}}\\cdot \\left[\\mu _e \\nabla B -e\\nabla \\Psi _e \\right],$ having introduced the generalized potential $\\Psi \\equiv \\Phi - \\frac{p_{\\parallel ,sp} }{m{_{sp}}c} A_\\parallel .$ $\\bullet $ Linear polarization equation in the long wave-length limit (and drift-kinetic electrons): $&&\\int \\mathrm {d}W_i Z_ieJ_0 f_i-\\int \\mathrm {d}W_e e f_e=-\\nabla \\cdot \\left(\\frac{n_0 m_ic^2}{B^2} \\nabla _\\perp \\Phi \\right)$ $\\bullet $ Linear Ampère's law: $\\int \\mathrm {d}W_i \\frac{4\\pi Z_ie }{m_i c} p_{\\parallel ,i} J_0 f_i &-& \\int \\mathrm {d}W_e \\frac{4\\pi e }{m_e c} p_{\\parallel ,e} f_e = \\\\&&\\frac{1}{d_e^2}A_\\parallel +\\frac{1}{d_i^2}A_\\parallel -\\nabla _\\perp ^2 A_\\parallel -\\nabla \\cdot \\frac{\\beta _{i}}{4}\\nabla _\\perp A_\\parallel $ where $n_0$ is the density associated with the equilibrium Maxwellian $f_M$ and $\\beta _{sp}=(4\\pi n_{sp} T_{sp})/B^2$ .",
"The skin depth is defined by $d^{-2}_{sp}=\\beta _{sp}/\\rho ^2_{sp}$ , where $\\rho ^2_{sp} = T_{sp}m_{sp}c^2/q^2_{sp}B^2$ , and it appears on the right-hand-side of the Ampére's law because of the choice of the velocity space variables $(p_\\parallel ,\\mu )$ instead of the usual $(v_\\parallel ,\\mu )$ .",
"The indexes $i$ and $e$ indicate ions and electrons respectively and $q_i=Z_ie$ is the ion charge while $q_e=-e$ is the electron charge.",
"Despite all the approximations made, this model is highly physically relevant and it can be used to describe not only the GAM and ZF dynamics, but also a large class of micro-instabilities excited by the density and temperature gradients, like ion temperature gradient (ITG) driven modes, trapped electron modes (TEM) or kinetic ballooning modes (KBM).",
"It also contained the reduced MHD model as a subset (see, among other, Ref.",
"[28]).",
"According to the PIC method the particle distribution function is discretized with macroparticles, known as markers.",
"The motion of the markers is calculated using the equations of motions of the gyrokinetic model while the electromagnetic fields are evolved on a spatial grid using the two field equations.",
"The charge and current density, that are necessary to solve the field equations, are calculated by projecting the marker weights on a spatial grid.",
"After that, the fields are calculated using a finite elements method.",
"The code is based on a straight-field-line coordinate system $(s, \\chi , \\phi )$ .",
"Here, radial coordinate is $s = \\sqrt{\\psi /\\psi _{edge}}$ (where $\\psi $ is the poloidal flux), $\\chi = \\displaystyle {\\frac{1}{q(s)}} \\int _0^\\Theta \\displaystyle {\\frac{{B}\\cdot \\nabla \\phi }{{B}\\cdot \\nabla \\Theta _1}} d\\Theta _1$ is the straight-field-line coordinate (where $\\Theta $ and $q$ are the poloidal angle and the safety factor respectively) and $\\phi $ is a toroidal angle.",
"Two different kinds of magnetic equilibria are implemented: analytical equilibria with circular concentric magnetic surfaces and ideal MHD realistic equilibria.",
"For the latter case, the ORB5 code is coupled with the CHEASE code [29], which solves the Grad-Shafranov equation with a fixed plasma boundary."
],
[
"Equilibrium and simulations parameters",
"Linear electromangetic gyrokinetic collisionless simulations with drift-kinetic electrons and a realistic electron - ion (deuterium) mass ratio $m_e / m_i = 2.5\\cdot 10^{-4}$ have been performed.",
"Electrostatic simulations with kinetic electrons are, in principle, faster than electromagnetic simulations, due to the smaller number of equations to be solved.",
"Nevertheless, a high frequency oscillation, called the $\\omega _H$ -mode [30], is observed to be often numerically unstable.",
"To decrease the level of the high-frequency oscillations, electromagnetic simulations in the small-$\\beta _e$ ($\\beta _e = 10^{-5}$ ) limit have been performed instead of the electrostatic ones.",
"MHD equilibria of the circular and elongated plasma have been calculated with an external code CHEASE [29].",
"Simulations have been carried out with a flat density profile, which have been shown to not impact the GAM frequency and damping rate in linear simulations (see Appendix ).",
"To focus on the Landau damping in the absence of the phase mixing effect, flat temperature profile has been considered in simulations used for the results of this section.",
"Since the safety factor profiles have been taken from the CHEASE, there is a magnetic shear, that also causes the phase mixing, but its influence on the GAM damping rate is much smaller in comparison to the temperature gradient effect.",
"Plasma parameters have been taken close to the ASDEX Upgrade parameters near the plasma edge [25]: the major radius $R = 1.65$ m, the minor radius $a = 0.5$ m (inverse aspect ratio is $\\epsilon = 0.303$ ), the magnetic field on the axis $B = 2$ T. Since in the CHEASE code the plasma elongation is defined at the edge and changes gradually to the plasma center, the GAM frequency and damping rate have been measured at the same radial position $s_0 = 0.90$ to perform more accurate scan on the elongation.",
"The temperature has been taken to be $T_i = T_e = 70$ eV.",
"It means that $c_s = \\sqrt{q_e T_e / 2 m_p} = 5.8 \\cdot 10^{3}$ m/s, $\\rho _s = c_s / \\omega _{ci} = 6.1\\cdot 10^{-4}$ m and $\\rho ^* = \\rho _s / a = 1.2\\cdot 10^{-3}$ .",
"Here, $\\omega _{ci} = q_e B/2m_p$ ($\\omega _{ci}/2\\pi = 15.2$ MHz) is the ion gyro-frequency, $q_e$ is an electron charge, and $m_p$ is a proton mass.",
"To weaken the constraint on the space step and to reduce the effect of the charge accumulation at the edge of the numerical work box, we have simulated only a ring from $s_1 = 0.85$ to $s_2 = 0.95$ in a poloidal cross section with the Dirichlet condition for the potential $\\phi $ on inner boundaries ($\\phi (s_1) = \\phi (s_2) = 0$ ).",
"A typical simulation has the following parameters.",
"Number of nodes in radial direction is taken to be $n_s = 256$ , in toroidal directions $n_{\\phi } = 4$ and along the straight-field-line coordinate $\\chi $ the number of nodes is $n_{\\chi } = 64$ .",
"Time step is $dt \\cdot \\omega _{ci} = 2$ .",
"The GAM damping rate and frequency have been calculated (see Appendix ) for different GAM radial wavenumbers $k = k_r\\rho _i \\in [0.054, 0.377]$ (where $\\rho _i = \\sqrt{2}v_{Ti}/\\omega _{ci} = 8.57 \\cdot 10^{-4}$ m is the ion Larmor radius of the deuterium and $v_{Ti} = \\sqrt{q_eT_i/2m_p}$ is the thermal speed), the safety factor $q \\in [3.5, 5.0]$ at $s_0 = 0.90$ and the plasma elongation $e \\in [1.0, 1.6]$ at the edge.",
"This is the regime where GAMs are typically observed in tokamak plasmas (see, for example, Ref.",
"[25]).",
"To simulate the GAM dynamics, the ORB5 simulations have been initialized by introducing an axisimmetric density perturbation designed to produce an initial electric potential field of the form $\\sim \\sin (ks)$ , where $s \\in [s_1, s_2]$ (as in the so-called Rosenbluth-Hinton test [4]).",
"All toroidal modes $n \\ne 0$ and poloidal modes $|m| > 10$ have been filtered out.",
"To study the GAM dynamics, the frequency and damping rate of the poloidally averaged radial electric field have been calculated."
],
[
"Results of gyrokinetic simulations",
"In Fig.",
"REF , a comparison of the GAM frequencies and damping rate obtained from numerical simulations with two analytical theories of Qiu 2009 [32] and Gao 2010 [31] is shown.",
"A good agreement between numerical results and analytical predictions of the GAM frequency has been found.",
"Nevertheless, the GAM damping rate, obtained from the theories, derived using adiabatic electrons, is smaller in comparison to numerical simulations with kinetic electrons, and the divergence increases for smaller values of the GAM radial wavenumber.",
"Moreover, since the frequency stops increasing in the domain of higher wavenumbers (subplot $a$ of Fig.",
"REF ), a divergence between numerical results and analytical theories is observed.",
"The same effect was observed in Ref.",
"[33], where the GAMs were studied using drift reduced Braginskii equations.",
"The Gao 2010 theory describes the GAM dependence on the plasma elongation and it is in a good agreement with numerical results for the frequency.",
"Although the Gao 2010 theory provides considerably smaller damping rate, it seems to give similar trend of the damping coefficient with the plasma elongation, i.e., the damping rate is weakened by the elongation.",
"The Gao 2010 theory was derived in the large orbit drift width limit, where the dominant damping mechanism is the resonance $\\omega \\sim \\omega _d$ (here, $\\omega _d = {k_r}\\cdot {v_d}$ is a magnetic drift frequency, ${k_r}$ is a wave vector of the zonal potential in the radial direction and ${v_d}$ is a magnetic drift velocity) [34].",
"As explained in Ref.",
"[31], the GAM frequency decreases with the elongation less rapidly than the drift frequency.",
"To satisfy the resonance $\\omega \\sim \\omega _d$ particles have to have higher drift velocities, which involves fewer particles in the wave-particle interaction and, as a result, the GAM damping rate decreases.",
"Figure: Comparison between numerically simulated values (dots, traingles and squares) of the GAM frequency and values obtained using the interpolating expression provided in Eq.",
"() (solid lines).",
"Dotted lines indicate 95% confidence bounds of the fitting."
],
[
"Interpolating formulae",
"To provide a scaling of the GAM frequency and damping rate, corresponding interpolating expressions have been fitted to the results of the gyrokinetic simulations described in Sec.",
"REF .",
"For consistency, the regime has been chosen for the GAM wavenumbers $k = k_r \\rho _i$ in the range $[0.054, 0.377]$ , safety factor $q \\in [3.5, 5.0]$ and plasma elongation $e \\in [1.0, 1.6]$ .",
"Figure: Comparison between numerically simulated values (dots, triangles or squares) of the GAM damping rate and values obtained by using the interpolating expression provided in Eq.",
"() (solid lines).",
"Dotted lines indicate 95% confidence bounds of the fitting.To derive an interpolating expression for the frequency several assumptions have been used.",
"The experimentally obtained dependence [25] on the plasma elongation $1 / (1 + e)$ has been slightly modified to $1 / (1 + g_6 e)$ , where $g_6$ is an adjustable coefficient.",
"The dependence on the safety factor has been taken in the form $\\exp (-g_5 q^2)$ .",
"In fact, the $q$ -dependence in a form of $\\sqrt{1 + g_5/q^2}$ , that is given in Ref.",
"[21], gives the same results.",
"To describe how the frequency changes with the radial wavenumber, a polynomial has been taken.",
"Moreover, to take into account the frequency saturation for higher wavenumbers[33] we have introduced a function of the form $1 / (1 + g_4 k)$ .",
"Here, $k = k_r \\rho _{i}$ , $v_{Ti} = \\sqrt{q_eT_i/2m_p}$ .",
"The resulting frequency interpolating formula is the following one: $f_{\\omega } \\left[\\displaystyle {\\frac{\\sqrt{2} v_{Ti}}{R}} \\right] = \\displaystyle {\\frac{g_1 + g_2 k^2 + g_3 k^4}{1 + g_4 k}} \\displaystyle {\\frac{\\exp \\left( -g_5 q^2 \\right)}{1 + g_6 e}}.$ Among different tested functions, this form gives the best approximation to numerically simulated values of the GAM frequency, it has one of the smallest 95% confidential bounds and is not overfitted.",
"The corresponding coefficients $g$ with their 95% confidential bounds (lower $g_{lc}$ and upper $g_{uc}$ bounds) are $g = &&[3.7733,\\ 6.3505,\\ -1.9741e1,\\ 1.3557e-1,\\ 1.4620e-3,\\ 1.1684],\\\\*g_{lc} = &&[3.6745,\\ 3.3168,\\ -2.8800e1,\\ -6.0078e-2,\\ 1.1373e-3,\\ 1.1234],\\\\*g_{uc} = &&[3.8720,\\ 9.3843,\\ -1.0682e1,\\ 3.3121e-1,\\ 1.7866e-3,\\ 1.2135].$ Results for the Eq.",
"(REF ) are depicted in Fig.",
"REF .",
"For the damping rate we have derived the following expression (here, the damping rate is normalized to $\\sqrt{2} v_{Ti}/R$ ): $f_{\\gamma } \\left[\\displaystyle {\\frac{\\sqrt{2} v_{Ti}}{R}} \\right] = \\displaystyle {\\frac{\\left(h_1 + h_2 k^2\\right)\\exp \\left[-h_3 q^2\\right]}{1 + h_4 e^2}} + \\displaystyle {\\frac{\\left(h_5 + h_6 k^2\\right)\\exp \\left[-h_7 q^2 \\right]}{1+h_8 e^4}}.$ with interpolating coefficients $h =&& [-1.2494e-2,\\ -8.9688e-1,\\ 4.5498e-2,\\ -1.9884e-1,\\\\*&&-1.1248e-2, -2.5481,\\ -5.3340e-3,\\ 7.7748e-1],\\\\h_{lc} =&& [-2.3115e-2,\\ -1.6490,\\ 2.5215e-2,\\ -3.3573e-1,\\\\*&&-2.5523e-2, -3.1909,\\ -1.9665e-2,\\ 5.1924e-2],\\\\h_{uc} =&& [-1.8723e-3,\\ -1.4471e-1,\\ 6.5781e-2,\\ -6.1955e-2,\\\\*&&3.0272e-3, -1.9053,\\ 8.9973e-3,\\ 1.5030].$ Comparison between the results from the gyrokinetic simulations and the interpolation expression for the GAM damping rate is shown in Fig.",
"REF for some specific values of parameters taken as examples."
],
[
"Phase mixing",
"To investigate the influence of the phase mixing on the GAM dynamics the same parameters as described in chapter REF has been used, but the temperature gradient (the same for both the electrons and ions to have $\\tau _e = T_e/T_i = 1$ , $T_i(s_0) = 70$ eV) at a radial position $s_0 = 0.90$ has been introduced.",
"The radial point $s_0 = 0.90$ has been chosen here to be in agreement with the section REF .",
"Initial radial wavenumber of the radial electric field is $k = 0.108$ .",
"The safety factor is $q(s_0) = 4.0$ .",
"We consider a temperature profile of the following form, similarly to Ref.",
"[23]: $\\displaystyle {\\frac{T_e(s)}{T_e(s_0)}} = \\exp \\left[-\\Delta \\cdot k_T \\cdot \\tanh \\left(\\displaystyle {\\frac{s - s_0}{\\Delta }}\\right)\\right],$ where $\\Delta = 0.04$ , $k_T = - \\left.",
"d[\\ln (T)]/ds \\right|_{s = s_0}$ .",
"The temperature profiles and the corresponding temperature gradient profiles for different $k_T$ in a radial interval $s = [0.85, 0.95]$ , are shown in Fig.",
"REF .",
"Dependence of the GAM half-decay time $t_{1/2}$ on the temperature gradient has been investigated in the domain $k_T \\in [1, 15]$ .",
"Figure: Temperature and temperature gradient radial profiles for different k T k_T: k T =[1,5,15]k_T = [1, 5, 15].A scan of gyrokinetic simulations with the temperature gradient $k_T$ has been performed, and the results are depicted in Fig.",
"REF .",
"In presence of a temperature gradient, the GAM is observed to oscillate with different frequencies at different radial points, that leads to the distortion of the initial GAM radial structure.",
"Producing higher radial wavenumbers, this distortion amplifies the GAM damping.",
"This combined effect, already investigated for a more simplified configuration in Ref.",
"[22], [23], has been observed even more pronounced in the simulations described here.",
"In fact, here the phase mixing effect is investigated using gyrokinetic simulations with kinetic electrons that significantly influences the GAM damping, and, as a consequence, the GAM half-decay time.",
"For example, using the Sugama-Watanabe model[35], which is derived with adiabatic electrons, for the Landau damping and combining with phase mixing, we have obtained $t_{1/2}[R/ \\sqrt{2} v_{Ti}] = 118$ for the $k_T = 1$ and $t_{1/2}[R/ \\sqrt{2} v_{Ti}] = 23.4$ for $k_T = 10$ , that predicts much longer half-decay time of the GAM in comparison to the calculations based on the simulations with the kinetic electrons (compare with a Fig.",
"REF ).",
"In order to verify the results of gyrokinetic simulations, we have used a theoretical simplified model of the phase mixing, proposed in Ref.",
"[8], [22], [23], where the linear growth in time of the radial wavenumber is considered.",
"In the phase mixing simulations a space point $s_0$ is considered with a certain temperature $T(s_0)$ and temperature gradient $k_T(s_0)$ .",
"Initial radial electric field has the following radial structure: $E(s) = E_0 \\cos (k_0 s)$ with an initial amplitude $E_0$ and initial normalized radial wavenumber $k_0$ .",
"The electric field is assumed to evolve in time at a point $s_0$ according to a simple rule $E(s_0, t) = E_{a}(s_0, t) \\cos (\\omega (s_0) t),$ where $E_{a}(s_0, t)$ is an amplitude of the electric field, that changes in time due to the damping, $E_{a}(0) = E_0$ .",
"The general form of the GAM frequency is $\\omega (s, t) = \\sqrt{\\frac{2 T_e(s)}{2m_p}} \\omega ^*(k, q, e),$ where $\\omega ^*(k, q, e)$ describes frequency dependence on the radial wavenumber, the safety factor and the elongation.",
"The safety factor profile is taken to be flat, and plasma with a circular cross-section is considered: $e = 1.00$ , $r \\approx a s$ .",
"The damping rate is defined as $\\gamma (s_0, t) = \\displaystyle {\\frac{1}{E(s_0,t)}}\\displaystyle {\\frac{ d E(s_0,t)}{d t}}.$ At the beginning of every time interval $[t_1, t_1 + \\Delta t]$ , new values of the damping rate $\\gamma (s_0, t_1)$ and frequency $\\omega (s_0, t_1)$ are found with the scaling formulae given in Eq.",
"(REF ) and (REF ), using a current value of the wavenumber $k(s_0, t_1)$ .",
"A new value of the electric field can be found, assuming that the damping rate is constant at the lapse of time $[t_1, t_1 + \\Delta t]$ : $E(s_0, t_1 + \\Delta t) = E(s_0, t_1)\\cdot (1 + \\gamma (s_0, t_1) \\Delta t).$ After that, a new value of the wavenumber $k(s_0, t_1+\\Delta t)$ is calculated using the radial derivative of the frequency $\\left.",
"\\frac{\\partial {\\omega (s, t_1)}}{\\partial {s}} \\right|_{s = s_0} = - \\displaystyle {\\frac{1}{2}} \\omega (s_0, t_1) k_T.$ With that, the wavenumber is assumed to change linearly in time as $k(s_0, t_1 + \\Delta t) = k(s_0, t_1) - \\sqrt{2} \\rho ^* \\left.",
"\\frac{\\partial {\\omega (s, t_1)}}{\\partial {s}} \\right|_{s = s_0} \\Delta t,$ where $\\rho ^* = \\rho _s / a$ , $\\rho _s = c_s/ \\omega _{ci}$ .",
"Another option, it is to estimate the time evolution of the radial wavenumber directly from numerical calculations in ORB5.",
"Substituting new value of the normalized wavenumber $k(s_0, t_1+\\Delta t)$ into Eq.",
"(REF ), we can find the damping rate $\\gamma (s_0, t_1+\\Delta t)$ at the next time point.",
"The results obtained with this reduced theoretical model are also shown in Fig.",
"REF .",
"As it can be seen in Fig.",
"REF , the qualitative dependence of the half-decay time on the temperature gradient finds a good match of gyrokinetic simulations of ORB5 and analytical theory.",
"The difference is due to the global dynamics of the ORB5 simulations, which is compared here with a theory where the phase mixing follows a local estimation given in Ref. [8].",
"Figure: Dependence of the GAM half-decay time on the temperature gradient obtained from the simulations in ORB5 (green squares), from the theory using a linear estimation Eq.",
"() (blue dots) and estimation from ORB5 (red triangles) of the radial wavenumber."
],
[
"Comparison with experimental data",
"The dispersion relations obtained in Sec.",
"REF as an interpolation of gyrokinetic simulations and given in Eqs.",
"(REF ), (REF ) can be used to compare numerical estimations of the GAM behaviour to measurements of the GAM frequency, performed on ASDEX Upgrade tokamak [25] using Doppler reflectometry.",
"More precisely, we consider the discharge AUG#20787 with the plasma elongation at the edge $e = 1.09$ (and we assume that it is constant at the considered radial region $\\rho = r/a = [0.8, 1.0]$ ).",
"The GAM radial wavenumber is considered to be constant and is estimated to be $k_ra = 40\\pi $ from the experimental radial profile of the GAM amplitude (see Fig.",
"5f in Ref.",
"[25]).",
"Experimental safety factor and ion temperature profiles have been taken to estimate the GAM frequency and damping rate using the scaling formulae (REF ), (REF ) at different radial points $\\rho $ .",
"In Fig.",
"REF the GAM frequency profiles with corresponding theoretical prediction are depicted.",
"A good general agreement is found in the central region of interest, where the GAM intensity, measured in the experiments, is peaked.",
"On the other hand, the linear dispersion relation REF can not explain neither the staircase nature (the plateaus) of the frequencies nor the GAM peak splitting that is observed experimentally at the radius positions $\\rho = 0.922$ or $\\rho = 0.932$ (although the presence of GAM eigenmodes has been suggested by simplified analytical models [36], [37], whose detailed analysis is out of the scope of this paper).",
"For this reason, we can conjecture that the coherent phenomena at the basis of the formation of GAM extended eigenmode or frequency splitting must have a nonlinear origin.",
"For reference we have given here estimation of the GAM collisional damping rate using formulae, derived by Gao in Ref.",
"[9].",
"Introducing normalized ion collision rate $\\hat{\\nu }_i = \\nu _i qR / v_{ti}$ , the collisional damping rate is calculated as[9] $\\displaystyle {\\frac{\\gamma ^{col}}{v_{Ti}/qR}} = - \\displaystyle {\\frac{3\\hat{\\nu }_i}{14 + 8\\tau _i}},$ if $\\hat{\\nu }_i \\ll 1$ , and as: $\\displaystyle {\\frac{\\gamma ^{col}}{v_{Ti}/qR}} = - \\displaystyle {\\frac{3}{8}} \\hat{\\nu }_i \\left(\\displaystyle {\\frac{7}{4}} + \\tau _i + \\displaystyle {\\frac{\\hat{\\nu }_i^2}{q^2}} \\right)^{-1},$ if $\\hat{\\nu }_i \\ge 1$ .",
"To find the ion collisional rate we have used classical expressions: $\\nu _i &=& 4.8\\cdot 10^{-8} Z^4 \\mu ^{-1/2} n_i[cm^{-3}] T_i[eV]^{-3/2}\\ln \\Lambda ,\\\\\\ln \\Lambda &=& 23 - \\ln \\left[ \\sqrt{2n_i[cm^{-3}]}\\displaystyle {\\frac{Z^3}{T_i[eV]^{3/2}}} \\right],$ where $\\mu \\equiv m_i / m_p = 2$ , $Z = 1$ .",
"Figure: Comparison of the experimental GAM frequencies to the numerical values, obtained with the formula given in Eq.",
"().",
"Numerical damping rate is depicted on the right plot.",
"The grey dotted line is an estimation of the collisional damping rate of the GAM found using expressions given by Gao in Ref.",
".",
"The red dotted lines are the 95% confidential bounds of the approximated damping rate.According to the Fig.",
"REF , the collisional damping is found to be negligible in the radial domain where GAMs are experimentally measured, except in a very narrow region close to the separatrix, where it can be of the same order of magnitude as the Landau damping."
],
[
"Conclusions",
"In tokamak plasmas, the drift-wave turbulence gives rise to the zonal flows that in their turn shear and distort convective and turbulent cells leading to the saturation of turbulence and, consequently, to a reduction of the radial heat transport.",
"Action of the magnetic curvature results in the oscillatory zonal flows, so-called geodesic acoustic modes.",
"The peculiarity of the GAM oscillations resides in the different shearing efficiency that the ZF have in relation to their oscillatory behavior.",
"The nonlinear interactions between the GAM and the DW turbulence is defined in a high degree by the GAM damping rate.",
"Lack of the experimental data of this characteristic of the GAM makes the results from linear gyrokinetic simulations particularly important for analytical and numerical investigation of the nonlinear GAM-DW systems.",
"In this work, linear gyrokinetic simulations have been performed with kinetic electrons to study the GAM dynamics.",
"Numerical results have been compared to analytical theories, derived with adiabatic electrons.",
"It has been shown that analytical theories, derived with adiabatic electrons, result in smaller values of the damping rate and for higher wavenumbers diverge from numerical calculations of the frequency.",
"That is why, investigating the GAM dependence on the plasma safety factor, elongation and radial wavenumber, we have found approximating analytic expressions for the frequency and damping rate to predict the GAM behaviour in different plasma regimes.",
"The derived expressions can be used to estimate the GAM linear characteristics used in analytical models of the nonlinear interactions between the GAM and the DW, such as different reduced models [38], [8].",
"Using these formulae, the phase mixing effect on the damping rate has also been calculated.",
"Based on the gyrokinetic simulations with kinetic electrons, the results have shown smaller half-decay times of the GAM in comparison with the Ref.",
"[22], [23].",
"The GAM is one of the special features of the I-mode and can be observed in the L-mode [15], [39], [40].",
"Comparison of the characteristic drive time of the GAM $t_{RD} \\sim 1/\\gamma _{RD}$ , which is given by the nonlinear coupling with the ion-temperature-mode (ITG)[8], [38], with the GAM half-decay time $t_{1/2}$ confirms the results of the Ref.",
"[22], [23].",
"Indeed, we estimate the GAM drive time to be $t_{RD} < t_s$ (where $t_s \\sim 2^{-1/2} R/v_{Ti}$ ) in the L-mode, $t_{RD} \\sim t_s$ in the I-mode and $t_{RD} \\sim 10t_s$ in the H-mode, according to Ref.",
"[23].",
"In this case, it can be seen from the Fig.",
"REF that the GAM half-decay time, which is defined by both the Landau damping and the phase mixing effect, is much higher than the drive-time in the L and I modes, $t_{1/2} > t^{L,I}_{RD}$ , for all considered values of $k_T$ .",
"This means that the energy transfer rate from the ITG turbulence to the GAM exceeds the Landau-phase mixing damping rate of the GAM.",
"As a result, the GAM can be observed in the L and I modes, but not in the H-mode, where $t_{1/2} < t^H_{RD}$ already for $k_T > 3$ .",
"This could be the explanation for the result that the GAM are not observed in the high-confinement mode, as proposed in Ref.",
"[22], [23].",
"The approximating expressions have showed quite good agreement with experimental data, but estimations of the GAM frequency, obtained from linear gyrokinetic simulations, does not explain the staircase radial profile of the frequency and the GAM peak splitting.",
"The frequency expression describes only the continuum or dispersive mode in contrast to eigenmode.",
"The latter is characterized by the GAM mode frequencies which are predicted to remain constant over a large radial extent, but a significant radial overlap in the frequency radial profile can be observed, that lead to the GAM frequency peak splitting[40]."
],
[
"Acknowledgments",
"Part of this work was done while two of the authors, I. Novikau and A. Biancalani, were visiting LPP-Palaiseau (France), whose team is kindly acknowledged for the hospitality.",
"The authors acknowledge discussions with F. Jenko, L. Villard, X. Garbet and T. Görler."
],
[
"Numerical convergence tests",
"To calculate the GAM damping rate and frequency, poloidally averaged radial electric field has been fitted, using the Levenberg–Marquardt algorithm [41], to a function of the form $\\exp (\\gamma t) \\cos (\\omega t)$ , where $\\gamma $ , $\\omega $ are sought-for damping rate and frequency.",
"Before the fitting it's necessary to filter the radial electric field to get ride of the high-frequency Alfvén oscillations.",
"In Fig.",
"REF the fitting is depicted for the case: $e = 1.30$ , $q = 4.0$ , $k = 0.108$ .",
"It is worth to mention that the choice of a time interval, where the fitting is performed, can influence the result damping rate, and it is not so crucial for the GAM frequency calculation.",
"This ambiguity in the choice of the time interval can be explained by the fact that at the beginning of the simulations there are some transient processes that must be excluded from the damping rate measurements.",
"Moreover, with the time, the global effects start to play a significant role, distorting initial radial structure of the radial electric field, that makes the damping rate to be variable in time.",
"The GAM dynamics (frequency and damping rate) doesn't depend on the plasma density in linear calculations (see Fig.",
"REF and REF ).",
"But the frequency of the Alfvén waves decreases with the increase of the density, and for high values of the plasma density it becomes difficult to separate acoustic and Alfvénic time scales (see Fig.",
"REF ).",
"Figure: Fourier spectrum of the radial electric field for different values of the electron beta: β e =5e-5\\beta _e = 5e-5 (blue line), β e =1e-3\\beta _e = 1e-3 (red line).",
"Only the Alfvén frequency changes with the electron beta.",
"The GAM frequency remains the same.",
"Here, the frequency is normalized to ω ci \\omega _{ci}.The transition from the simulations with the adiabatic electrons to the ones with the kinetic electrons applies additional restrictions on several numerical parameters such as the time step and the number of markers (see Fig.",
"REF ).",
"Figure: Convergence tests on the number of points in the radial space grid n s n_s (e=1.60e = 1.60, q=4.0q = 4.0, k r ρ i =0.108k_r \\rho _i = 0.108) (Fig.",
"), on the normalized time step dt norm dt_{norm} for different values of the plasma elongation e=1.00,1.60e = 1.00, 1.60 (q=3.5q = 3.5, k r ρ i =0.108k_r \\rho _i = 0.108) (Fig. )",
"and on the number of markers N markers N_{markers} (e=1.60e = 1.60, q=4.0q = 4.0, k r ρ i =0.108k_r \\rho _i = 0.108) (Fig.",
").In projects with adiabatic electrons the normalized time step $dt_{norm} = dt\\cdot \\omega _{ci}$ can be of the order of 20, but in case of the kinetic electrons it has to be significantly reduced till 2 because of the high parallel velocity of the passing electrons.",
"Also electrostatic simulations of the kinetic electrons reveal high-frequency oscillations[30].",
"These oscillations can lead to numerical instabilities in case of low number of markers.",
"To reduce their level we have passed to electromagnetic simulations with small values of electron beta that gave us an opportunity to keep the number of markers on the level of $10^7$ .",
"The radial space step (or number of the points in the radial space grid) is determined by, among other parameters, the GAM wavenumber.",
"To investigate the GAM dynamics with higher values of the radial wavenumber, we simulated a narrow poloidal ring near the edge instead of the full plasma cross-section to reduce the number of radial space points."
],
[
"Comparison with the code GENE",
"A complete cross-code verification between the gyrokinetic ORB5 and GENE [42], [43] codes has already been done in Ref.",
"[21] on the linear collisionless dynamics of the GAMs with adiabatic and kinetic electrons in the specific case of flat temperature profiles.",
"For completeness, in this paper a comparison between these different codes is shown including the additional phase mixing physical effect, which is driven by non-flat temperature profiles.",
"The motivation behind this study is that although the linear physical models between ORB5 and GENE are equivalent [44], the numerical schemes are different.",
"GENE is an Eulerian code, where the distribution function is not discretized with markers, but it is discretized on a 5D fixed grid in phase-space $({R}, v_{\\parallel }, \\mu )$ , where ${R}$ is the gyrocenter position, $v_{\\parallel }$ is the parallel velocity, and $\\mu $ is the magnetic momentum.",
"The simulation plasma parameters have been taken as in Sec.",
"for both GENE and ORB5.",
"A sinusoidal perturbation in the potential field is initialised, as defined in Sec.",
"REF and is let evolved in time.",
"In GENE, the radial box size is $60 \\rho _s$ .",
"We have used 128 grid points in radial direction in order to have at least two points per ion Larmor radius.",
"Along the field line 68 points have been used.",
"In velocity space, 68 points and 128 equidistant symmetric grid points have been used for resolving respectively the $\\mu $ and the $v_{\\parallel }$ space.",
"The velocity space domain has been fixed to 3 and 9 times the thermal velocity, respectively in the $v_{\\parallel }$ and $\\mu $ space.",
"In order to avoid any recurrence problem, an hyperdiffusivity scheme has been used in the $v_{\\parallel }$ direction.",
"In Fig.",
"REF peaks of the flux-surface averaged radial electric field, measured at the radial position $s_0 = 0.90$ for the ion-electron temperature gradient $k_T = 10$ , are shown for both GENE and ORB5.",
"Half-decay time, calculated in ORB5, is $t_{1/2}^{orb}[R/\\sqrt{2} v_{Ti}] = 4.9$ and for the case of GENE it is $t_{1/2}^{GENE}[R/\\sqrt{2} v_{Ti}] = 4.8$ ."
]
] | 1709.01818 |
[
[
"Interfacing MHD Single Fluid and Kinetic Exospheric Solar Wind Models\n and Comparing Their Energetics"
],
[
"Abstract An exospheric kinetic solar wind model is interfaced with an observation-driven single fluid magnetohydrodynamic (MHD) model.",
"Initially, a photospheric magnetogram serves as observational input in the fluid approach to extrapolate the heliospheric magnetic field.",
"Then semi-empirical coronal models are used for estimating the plasma characteristics up to a heliocentric distance of 0.1AU.",
"From there on a full MHD model which computes the three-dimensional time-dependent evolution of the solar wind macroscopic variables up to the orbit of the Earth is used.",
"After interfacing the density and velocity at the inner MHD boundary, we compare with the results of a kinetic exospheric solar wind model based on the assumption of Maxwell and Kappa velocity distribution functions for protons and electrons respectively, as well as with \\textit{in situ} observations at 1AU.",
"This provides insight on more physically detailed processes, such as coronal heating and solar wind acceleration, that naturally arise by inclusion of suprathermal electrons in the model.",
"We are interested in the profile of the solar wind speed and density at 1AU, in characterizing the slow and fast source regions of the wind and in comparing MHD with exospheric models in similar conditions.",
"We calculate the energetics of both models from low to high heliocentric distances."
],
[
"Introduction",
"Solar wind heating and acceleration mechanisms are still subjects of active research.",
"In computational models, physical quantities estimated or observationally inferred close to the Sun serve as boundary or initial conditions that will examine the solar wind evolution and its underlying physics as it propagates through interplanetary space.",
"The solar wind plasma can be studied macroscopically through the magnetohydrodynamic (MHD) approach or microscopically when using the kinetic approach.",
"In the following Subsections REF and REF , we briefly review the main aspects of both approaches used in this paper, which are here for the first time interfaced in a global model.",
"[32] presented an empirical relation between the expansion of magnetic flux tubes and the solar wind speed, showing that they evolve inversely proportional to each other.",
"This assumption was tested using more than two decades of observations, and can give predictions of the solar wind speed at Earth.",
"The model involves synoptic magnetograms of photospheric field, which allow a Potential Field Source Surface (PFSS, with the source surface typically at 2.5$\\mathrm {R}_\\odot $ ) extrapolation which quantifies the expansion factors.",
"It was found that at the Earth's orbit, greater expansion corresponded to magnetic field lines near the centre of coronal holes, which diverge more slowly than the ones coming from the hole boundaries.",
"This is consistent with the fact that lower densities are found in the fast wind regions.",
"[1] improved the Wang-Sheeley model by using daily updated magnetogram data from the Wilcox Solar Observatory (WSO) and relating the magnetic flux tube expansion factor with the solar wind speed at the source surface, while including effects of stream interactions from the source surface to the Earth.",
"A statistical study which covered three years and compared the Wang-Sheeley model predictions with data from the WindNASA spacecraft at the L$_1$ Lagrangian point of the Earth designed for long-term solar wind measurements and its effects on the terrestrial magnetosphere (https://wind.nasa.gov).",
"satellite was presented.",
"The interplanetary magnetic field (IMF) polarity was properly predicted 75% of the time, while solar wind speeds were within 10-15% of actual values, when a 6-month period with data gaps was removed.",
"In the computational work of [14], solar wind variations were examined in the corotating frame with a three-dimensional MHD model with a CME (Coronal Mass Ejection) injection scheme in the streamer belt.",
"Such MHD models take into account magnetic field variations due to the CME interaction with the solar wind during the CME's evolution.",
"The CME movement depends on the background solar wind density and velocity and the vector properties of the solar wind magnetic field and velocity are affected by the passing disturbance.",
"ENLIL [16], [17], [15] is a heliospheric MHD model that provides a three-dimensional description of the time-dependent solar wind evolution.",
"It can use the Wang-Sheeley-Arge (WSA) [32], [1] semi-empirical model as its boundary condition.",
"The WSA model was used by [13] to study the solar wind at low heliocentric distances including an empirical method to link the magnetic field information with the velocity at 21.5$\\mathrm {R}_\\odot $ .",
"The new method was cross-validated using the 3D MHD code ENLIL and by comparing the results with observations at 1AU and at further distances as provided by Ulysses.",
"The estimation of the solar wind speed at 21.5$\\mathrm {R}_\\odot $ was indeed better than previous models and it captured both fast and slow solar wind.",
"Similar to ENLIL, we use a fully 3D MHD code EUHFORIA (European heliospheric forecasting information asset) that from 0.1AU onwards models the evolution of the plasma environment in the inner heliosphere.",
"The code details are discussed in [26] and in this paper we adopt it to get the macroscopic description of the solar wind."
],
[
"Kinetic Exospheric Models",
"Exospheric kinetic models are simplified collisionless, stationary models, that are meant to explain the acceleration of the solar wind in a self-consistent way and they were first established by [7] and [10].",
"The model was one-dimensional and time-independent and provided the state of the solar wind plasma along a magnetic field line.",
"The acceleration of the solar wind was due to the induced electric field even without suprathermal electrons (Maxwellian distribution), but when accounting for the presence of suprathermal electrons, the terminal speed at 1AU increased.",
"The original exospheric model [7], [10] assumed Maxwellian velocity distribution functions (VDFs) for protons and electrons and supersonic winds of 300 $\\mathrm {km\\ s^{-1}}$ could be reached at 1AU with temperatures of the order of 1 MK for both species at the exobase (the distance beyond which collisions become negligible).",
"Nevertheless, it remained difficult for the model to achieve higher bulk velocities, such as the ones observed in the fast solar wind, without increasing the temperature to unrealistic high values (10MK) at the exobase, or by adding other sources of solar wind acceleration.",
"After the induced electric field is calculated, the solar wind acceleration spontaneously follows giving the solution of the solar wind from sub- to supersonic, without any extra energy terms assumed.",
"A Lorentzian (Kappa) velocity distribution function is used instead of the classic Maxwellian in search for better agreement with observations in [20].",
"Indeed, suprathermal electrons are generally observed in the velocity distribution functions measured in situ in the solar wind.",
"[20] have shown that the presence of such suprathermal electrons accelerates the wind to higher bulk velocities, so that no other source of energy needs to be considered to reach the values observed in the high speed solar wind.",
"[11] applied the kinetic model developed by [20] using Kappa VDFs for both electron and proton populations that escape from the Sun to describe the solar wind.",
"Since the first exospheric model, the semi-analytic kinetic model has been able to describe not only the fast but also the slow solar wind together with their sources, in the cold coronal hole and hot equatorial regions respectively, without unrealistic assumptions of too high temperatures and extra heating in the corona, as required by non-turbulence driven fluid models.",
"While previous exospheric models placed the exobase at a distance of about 5-10$\\mathrm {R}_\\odot $ , from where on the proton total potential energy was a monotonic function of the heliocentric distance, [8] calculated the exobase to be positioned at about 1.1-5$\\mathrm {R}_\\odot $ .",
"This deeper location of the exobase, lowered under the radial location of the maximum of the total potential energy of the protons, gives the solar wind the observed acceleration to high velocities.",
"A low exobase leads indeed to higher bulk velocities at 1AU in the case of suprathermal electrons.",
"Collisionless (exospheric) theoretical models and collisional simulations were compared in [35].",
"Including suprathermal tails in the velocity distribution function of the electrons and employing a self-consistently computed heat flux, the models were able to reproduce fast solar wind speeds.",
"Results of collisional kinetic simulations with non-Maxwellian velocity distribution functions and collisionless exospheric models are in good agreement.",
"Taking into account that the exospheric and collisional models provide comparable results, in this paper we will go a step further and try to interface exospheric models with MHD ones.",
"On the way to developing predictive tools and 3D solar wind models, a 2D observationally driven kinetic exospheric solar wind model was developed by [21], presenting solar wind variations on the ecliptic plane and how they compare to observations from close to the Sun up to 1AU.",
"For the ecliptic variational study OMNIMulti-source data set for the near Earth solar wind of combined and normalized observational data from ACE (Advanced Composition Explorer), Wind, IMP 8 (Interplanetary Monitoring Platform) and GOES (Geostationary Operational Environmental Satellite) satellite missions.",
"observations were used for the time period 26 September to 23 October 2008.",
"The $\\kappa $ parameter was chosen as 2.35 and 3.82 for fast and slow wind respectively, to match bulk speed observations close to the orbit of the Earth.",
"We will present a three dimensional generalization of this exospheric model, i.e.",
"we find the solar wind characteristics along a collection of magnetic field lines each passing through a point on the spherical shell at the exobase level in latitude and longitude $(\\theta ,\\phi )$ .",
"The basic principles, boundary conditions, physical assumptions and computational methods used by the MHD and the exospheric kinetic models are described in Section 2.",
"The specific criteria and the observational data that are chosen for this work are explained and presented in Section 3.",
"The interfacing method as well as explicit results of both approaches and their energetics are discussed in Section 4, while in Section 5 we compare the two approaches and we close by discussing the main conclusions of the study in Section 6."
],
[
"MHD Modeling: EUHFORIA",
"The inner heliosphere model EUHFORIA [26] is used for our MHD approach.",
"EUHFORIA is a three-dimensional observationally driven model providing an accurate description of the large-scale time-dependent solar wind including transient events such as CMEs.",
"As such, it allows to inject CMEs at the inner radial boundary at 0.1AU as a time-dependent boundary condition.",
"Apart from CMEs, the variables at the inner radial boundary are constructed in order to capture the large-scale variations in the solar wind for the particular time period under study.",
"This is accomplished using a model for the coronal magnetic field and employing empirical relations between the coronal magnetic topology and the state of the solar wind.",
"The magnetic field model consists of a potential field source surface (PFSS) model in the low corona coupled with a current sheet model higher in the corona.",
"The PFSS model requires a magnetogram to be provided as input.",
"To finally compute the super-sonic state of the solar wind at 0.1AU, empirical relations inspired by the success of the WSA (Wang-Sheeley-Arge) model [32], [1] are used.",
"The MHD model is able to provide density and speed profiles at the Earth's orbit, it allows for slow and fast solar wind source region tracing, and can serve as the MHD counterpart in a comparison project together with kinetic exospheric models that correspond to similar initial and boundary conditions at 0.1AU.",
"This will be our first goal in this paper, and the way the two approaches are coupled will be described next.",
"EUHFORIA uses a finite volume discretization scheme to solve the hyperbolic conservative MHD equations.",
"The equations solved are those of ideal MHD with gravity included as a source term in the equations of momentum and energy: $\\frac{\\partial \\rho }{\\partial t}+\\nabla \\cdot (\\rho {v})=0{,} \\\\\\frac{\\partial (\\rho {v})}{\\partial t}+\\nabla \\cdot \\left[ \\rho {v}{v}+\\left( p+\\frac{B^2}{2\\mu _0}\\right)\\mathcal {I}-\\frac{1}{\\mu _0}{B}{B}\\right]=\\rho {g}{,} \\\\\\frac{\\partial \\mathcal {E}}{\\partial t}+\\nabla \\cdot \\left[\\left(\\mathcal {E}+p-\\frac{B^2}{2\\mu _0}\\right){v}+\\frac{1}{\\mu _0}{B}\\times \\left( {v}\\times {B}\\right)\\right]=\\rho {v}\\cdot {g}{,} \\\\\\frac{\\partial {B}}{\\partial t}-\\nabla \\times \\left( {v}\\times {B}\\right)=0{,} \\\\\\nabla \\cdot {B}=0{,} \\\\\\mathcal {E}= \\frac{p}{\\gamma -1}+\\frac{\\rho v^2}{2}+\\frac{B^2}{2\\mu _0} {,} \\qquad \\gamma =1.5$ where $\\rho $ is the mass density, ${v}$ the velocity vector, ${g}$ the gravitational acceleration, ${B}$ the magnetic field vector, $p$ the thermal pressure, $\\gamma $ the polytropic index, $\\mathcal {E}$ the total energy density, $\\mu _0$ the magnetic permeability and $\\mathcal {I}$ the unit tensor.",
"Note that we are working in the inertial frame, so no Coriolis nor centrifugal forces need to be added in Equation .",
"The polytropic index is chosen to be slightly smaller than the expected $\\gamma =5/3$ value for a monatomic gas.",
"This causes a finite energy to be injected into the system in the form of heat [27].",
"The use of either a non-adiabatic polytropic index or explicit source terms in the momentum and energy equation to drive the solar wind and heat the corona have been used in several works.",
"In EUHFORIA, the reduced polytropic index is used in order to slightly accelerate the solar wind further out in the heliosphere, from the speed values at the boundary at 0.1AU.",
"The single fluid MHD description still leaves freedom to vary $\\gamma $ , which allows to account for expected deviations from the mono-atomic ideal gas value of 5/3.",
"For a discussion of more self-consistent models that attempt to capture and explain the physical mechanisms resulting in the observed coronal heating and acceleration, we refer to [2].",
"The employed numerical grid is uniform in spherical coordinates, with the number of cells in $r$ , $\\theta $ , $\\phi $ chosen to be 800, 60, 180, respectively.",
"The outer boundary is set at 2AU.",
"Further details of the numerical solution scheme are described in [26]."
],
[
"Boundary Conditions",
"The essential input to EUHFORIA is a synoptic magnetogram.",
"We select a magnetogram from the GONG (Global Oscillation Network Group) standard synoptic data product, which are available with one hour cadence from GONG.",
"The chosen magnetogram corresponds closely to Carrington Rotation (CR) 2059.",
"During this Carrington rotation, an equatorial coronal hole was visible near the central meridian to about $60^\\circ $ degrees west.",
"The solar wind plasma state at 0.1AU in EUHFORIA is determined using a semi-empirical approach similar to the Wang-Sheeley-Arge model.",
"The method consists of constructing a model of the coronal magnetic field consisting of a PFSS extrapolation in the low corona while the \"Schatten\" Current Sheet is used from $2.5\\mathrm {R}_\\odot $ to 0.1AU.",
"The solar wind speed is then given through an empirical relation which is a function of the magnetic flux tube expansion factor.",
"The formula used in this work is given by $V(f_s)=240.0+675.0(1+f_s)^{-0.22} \\mathrm {km\\ s^{-1}} {,}$ where $f_s$ is given by $f_s=\\left(\\frac{\\mathrm {R}_\\odot }{r}\\right)^2\\frac{B_r(\\mathrm {R}_\\odot ,\\theta _0,\\phi _0)}{B_r(r,\\theta ,\\phi )}$ and it quantifies the expansion factor of the flux tube from the photospheric footpoint $(\\mathrm {R}_\\odot ,\\theta _0,\\phi _0)$ of the specific field line to its position further outwards $(r,\\theta ,\\phi )$ at a heliocentric distance $r$ [33].",
"As explained in [33], the expansion factor takes values greater or equal to unity for flux divergence more rapid than or equal to $r^2$ , respectively.",
"Simple scaling laws that are functions of $V$ are used in order to determine the plasma density and temperature.",
"For further details of the empirical model, see [26].",
"For the kinetic component of our analysis, we are using an exospheric model, which is a way to simulate low density plasmas, where the importance of collisions is limited.",
"The solar atmosphere is considered to have a collision-dominated barosphere at low altitude [8] and a collisionless exosphere, which is the region modeled kinetically.",
"These regions are separated by a surface called the exobase $r_0$ , beyond which collisions become negligible.",
"This exobase level is defined as the altitude where the particle mean free path $l_f$ and the local density scale height $H$ become equal, i.e.",
"where the dimensionless Knudsen number $K_n=l_\\mathrm {f}/H$ is equal to unity.",
"The kinetic modelA 1D version of the kinetic exospheric model developed by the group in IASB-BIRA and collaborators can be found in CCMC (http://ccmc.gsfc.nasa.gov/models/exo.php) and it can run online for user-defined setups.",
"[10], [20], [11], [8] gives different temperatures for electrons and protons as indeed observed [9] and can include different characteristics of any other ion species.",
"Sources of the fast solar wind are considered to be coronal holes and in these regions the electron VDFs are assumed to correspond to a Lorentzian function with a small $\\kappa $ -value and thus have a large suprathermal tail [11].",
"The low speed solar wind usually comes from equatorial regions, with larger $\\kappa $ -values.",
"When $\\kappa \\rightarrow \\infty $ the VDF tends to a Maxwellian.",
"In this work, we consider two particle species, namely electrons and protons and therefore their respective exobase levels need to be defined.",
"The proton exobase is located where the Coulomb mean free path for the protons $l_{\\mathrm {f},p}$ according to [29] as estimated for coronal values by [11] is equal to the local density scale height $H$ , where $l_{\\mathrm {f},p}\\approx 7.2\\times 10^7\\frac{T^2_p}{n_e}{,} { \\ \\ \\ \\ }H=\\left(-\\frac{\\mathrm {d}\\ln n_e}{\\mathrm {d}r} \\right)^{-1} {,}$ where all the quantities are in SI.",
"The proton mean free path is shown in Figure REF as a function of the proton temperature and the electron number density.",
"For the electrons, a similar electron exobase height can be estimated from the Coulomb mean free path in a plasma consisting only of electrons and protons: $l_{\\mathrm {f},e}=0.416 \\left(\\frac{T_e}{T_p}\\right)^2l_{\\mathrm {f},p} {,}$ as in [11] for a hydrogen plasma, assuming that the electrons have the mean thermal velocity $(8k_\\mathrm {B}T_e/m_e\\pi )^{1/2}$ , with $l_{\\mathrm {f},e},l_{\\mathrm {f},p}$ in meters and $T_e,T_p$ in kelvin.",
"A crude estimate of the scale height can be obtained assuming hydrostatic equilibrium in a stratified atmosphere with isothermal plane parallel layers.",
"This is not the case for the solar wind, since expansion is taking place and rather hydrodynamic conditions apply, but it gives a good approximation of the order of magnitude of the scale height.",
"Figure: The Coulomb mean free path l f,p l_{\\mathrm {f},p} in solar radii as a function of proton temperature and electron density.",
"This figure quantifies the variations in Equation .When we adopt the same proton and electron temperature in the above formulae, the electron mean free path becomes smaller than the proton one $l_{\\mathrm {f},e}<l_{\\mathrm {f},p}$ , such that the electron collisions are more important for higher altitudes and thus the proton exobase is found at lower altitudes [11].",
"In this study, we make the assumption that both populations have the same exobase altitude, and we choose it to correspond to the source surface location $r_{0,p}=r_{0,e}=r_0=2.5\\mathrm {R}_\\odot $ where we by construction obtain purely radial magnetic fields.",
"The comparison between $l_\\mathrm {f}$ and $H$ shows anyway that the collisions become negligible already at very low radial distances in the solar corona.",
"Some indicative values for the different source regions on the Sun, namely coronal hole and equatorial regions, are estimated by [6] and [34].",
"Figure REF illustrates the Coulomb mean free path $l_{\\mathrm {f},p}$ as a function of temperature and number density to show the possible positions of the exobase.",
"According to [8], the exobase for equatorial regions is estimated to be at about 5 – 10$\\mathrm {R}_\\odot $ , whereas for coronal holes the exobase is estimated to be positioned at about 1.1 – 5$\\mathrm {R}_\\odot $ .",
"[28] have shown that suprathermal particles are already collisionless for $K_n>0.01$ , due to the velocity dependence of the mean free path of the particles.",
"This shows that it is not especially important that the exobase is chosen to correspond exactly to the level where $K_n=1$ , but it will appear where the density gradient is very sharp and thus where the plasma becomes collisionless to a good approximation."
],
[
"Velocity Distribution Functions",
"When collisions are ignored as in the exospheric theory developed by [10], the Fokker-Planck equation reduces to the Vlasov equation for the evolution of the velocity distribution function: $\\frac{\\partial f}{\\partial t}+{v}\\cdot \\frac{\\partial f}{\\partial {r}}+{a}\\cdot \\frac{\\partial f}{\\partial {v}}=0 {.",
"}$ Our kinetic exospheric model works by constructing stationary solution to the Vlasov equation, starting from an exact stationary solution for protons and electrons prescribed at the exobase.",
"Kinetic models based on this equation were developed and are discussed in [11] for radial magnetic field lines and in [23] taking into account the spiral interplanetary magnetic field topology.",
"It was shown in [11] that the specific moments in the solar wind, namely densities and temperatures, as well as the electrostatic potential characteristics from the corona to the interplanetary space are already well described, agreeing with observations at 1AU, when the collision term is neglected, since the collisions would rather modify the temperature anisotropies.",
"Apart from the Maxwellians the generalised Lorentzian or Kappa function is also a solution of the Vlasov equation and can be used as a boundary condition to study the effect of suprathermal particles on the kinetic moments.",
"Observations suggest that the velocity distribution functions of the electrons have strong suprathermal tails.",
"We therefore assume a Lorentzian VDF for the electrons and a Maxwellian VDF for the protons at the exobase: $& f^p_{\\mathrm {Maxwell}}(r_0,v)=n_p(r_0)\\left(\\frac{m_p}{2\\pi k_\\mathrm {B}T_p(r_0)} \\right)^{3/2}\\exp \\left( -\\frac{m_pv_p^2}{2k_\\mathrm {B}T_p(r_0)}\\right) {,} \\\\& f^e_{\\mathrm {kappa}}(r_0,v)=\\frac{n_e(r_0)}{2\\pi \\kappa ^{3/2}}\\left(\\frac{m_e}{2k_\\mathrm {B}T_e(r_0)} \\right)^{3/2}A(\\kappa )\\left(1+\\frac{m_ev_e^2}{2k_\\mathrm {B}T_e(r_0)\\kappa } \\right)^{-(\\kappa +1)} {,} \\qquad $ where $A(\\kappa )=\\frac{\\Gamma (\\kappa +1)}{\\Gamma (\\kappa -1/2)\\Gamma (3/2)} {.",
"}$ We note in passing that the moments of the Lorentzian VDF are not well defined for every $\\kappa $ value, but rather every $i$ th moment is defined for $\\kappa >(i+1)/2$ [20].",
"Suprathermal protons have almost no influence on the solar wind velocity, so for them a Maxwellian VDF can suffice [11].",
"Liouville's theorem [3] implies that any function that depends on the constants of motion of a collection of particles satisfies the Vlasov equation.",
"The relevant constants of motion in this study are the total energy and the magnetic moment.",
"Knowing the velocity distribution functions for our particle species at the exobase, the velocity distribution as a function of the radial distance can be deduced from energy conservation [20].",
"Thereby, the electron and proton VDFs can be computed as a function of radius and speed along a purely radial magnetic field line.",
"The analytic expressions for the kinetic moments of the exospheric models were derived for a Maxwellian VDF by [10] and for a Lorentzian VDF by [20].",
"As explained above the exospheric model used in this paper includes only radial velocities along open radial magnetic field lines.",
"Like in previous exospheric models, it is assumed that there are no particles coming from the interplanetary space to the Sun.",
"The anisotropy of the distribution leads to the solar wind flux.",
"The density, temperature and $\\kappa $ -index are determined at the exobase by either the MHD model or constrained via observations.",
"The model provides then the velocity distribution function at any other distance as well as the rest of the kinetic moments.",
"We use the code and the analytical expressions for the Maxwellian and the $\\kappa $ distributions by [20].",
"Provided the number density, the electron and proton temperatures and the $\\kappa $ index for the electron VDF at the exobase, the quasi-neutrality and zero-current conditions are solved iteratively at a fixed radial distance $r_m$ using a Newton-Raphson scheme.",
"The value of $r_m$ is iteratively modified using a dichotomy method until the electric field is found to be continuous within a predefined tolerance [8].",
"On the other hand, EUHFORIA solves the 3D MHD equations taking self-consistently stream interactions into account thereby providing a $v_\\phi $ for any point, whereas in the kinetic model we impose that this velocity is constant on each spherical shell.",
"The kinetic model thus proceeds without accounting for stream interactions instead keeping the same topology of fast and slow solar wind sources at every radial distance as at the exobase, reducing the computational time.",
"As was argued in [23], the effects due to rotation as compared to the purely radial case change only the estimated proton and electron temperatures and their anisotropies.",
"More specifically, the spiral structure predicts higher electron temperatures $T_e$ and lower proton temperatures $T_p$ than the radial case, but the number density, the electric potential and the bulk speed remain almost unaffected up to $300\\mathrm {R}_\\odot $ .",
"It's worth noting here that stream interactions are once again not taken into account.",
"Therefore, in this study we use the radial magnetic field topology and simply rotate by $v_\\phi $ each spherical shell to account for solar rotation.",
"The advantage of the kinetic model used in this paper is the direct quantification of species-specific temperature profiles, densities, speeds, energy fluxes etc.",
"once the electric potential is calculated.",
"Even if $T_e=T_p$ is chosen at the exobase, the kinetic model self-consistently calculates the species-specific heating with distance and the two temperatures depart from each other.",
"The $T_e$ is indeed observed to be different than $T_p$ at 1AU for slow and fast wind cases, e.g.",
"as reported in [9]."
],
[
"Observational Input: Cases and Selection Criteria",
"Several missions have observed, or continue to observe the Sun, as well as measure the physical parameters that characterize the solar wind, at different heliocentric distances as well as heliographic latitudes.",
"They provide high resolution data, not only in the ecliptic plane, but also at higher latitudes, requiring simulations that explain and predict the behavior of the solar wind not only in the ecliptic plane, but rather in three dimensions.",
"In this study we will be using OMNI data, and data from the Ulysses spacecraft due to its large latitudinal coverage.",
"We based our synoptic magnetogram and solar state selection on the following criteria, that allow us to perform a crosscheck on their prediction ability with available spacecraft observations: i) quiet Sun periods (since the exospheric models are particularly tailored to quiet Sun conditions); ii) the presence of equatorial coronal holes, such that significant differences between high and slow speed wind may be expected at the orbit of the Earth; iii) position of Ulysses for combination of simulations and different spacecraft observations from different angles/telescopes; and iv) very few CME events according to the available catalogue CACTUSMore information can be found at http://sidc.oma.be/cactus/..",
"In this study we focus on the year 2007, as it was a mostly quiet Sun year coinciding with the third orbit of Ulysses.",
"For the global 3D comparison we focus on the period July-August 2007, when Ulysses crossed the ecliptic plane.",
"We have confirmed the relative paucity of CME events in the selected time period using the CACTUS CME list."
],
[
"Constraining the Kinetic Model using EUHFORIA",
"First, we constrain the input to the kinetic model based on data-driven results at the inner radial boundary of EUHFORIA at $21.5 \\mathrm {R}_\\odot $ and examine both models' efficiency at capturing the solar wind bulk quantities by comparing with observations at 1AU and at 1.4AU (the Ulysses orbit).",
"We compare the results of the MHD and kinetic models after interfacing their velocities and densities at the inner MHD boundary at $21.5\\mathrm {R}_\\odot $ .",
"In Figure REF we show a slice at the equator (left panels) and a slice in latitude (right panels) that corresponds to the mass density scaled with the inverse square of the heliocenteric distance, and the radial speed also indicating the positions of Mercury, Venus, Earth, Mars and the STEREO (Solar Terrestrial Relations Observatory) A spacecraft.",
"The grey part of the colorscale used in the Figure corresponds to high velocities and densities that occur especially during CME events.",
"The velocity ranges ocurring at this time are roughly from 350 to 650 $\\mathrm {km\\ s^{-1}}$ with a clear separation between streams of different speeds, forming a clear Parker spiral-like configuration.",
"There is a configuration of several distinct high and slow speed streams.",
"We can see the different streams with high density associated to low speed and vice versa, while the highest speed captured is around $650\\mathrm {km\\ s^{-1}}$ and corresponds to a density similar to the one measured at 1AU.",
"The highest density contrast with respect to the one measured at 1AU is about 10.",
"The latitudinal panels suggest that there is compression and rarefaction as the plasma flows outwards with the corresponding speed and density.",
"We thus conclude that the plasma is not following exactly the ideal Parker spiral moving on perfect cones as it expands, but following a rather more complicated motion, as the simulation is observation-driven and the photospheric magnetogram shows a complex topology, as discussed previously.",
"Figure: EUHFORIA longitudinal and latitudinal variations of the three velocity and the three magnetic field components at 0.1AU (left panels) and at 1AU (right panels) for a standard magnetogram centered at 10 August 2007.In Figure REF , we show the components of the velocity and magnetic field in the spherical ($r,\\theta ,\\phi $ ) basis at two different distances, 0.1AU and 1AU, as functions of heliospheric longitude and latitude, as calculated by EUHFORIA.",
"We observe a clear and narrow undulating current sheet showing clear discrimination between low- and high-latitude solar wind.",
"This undulating sheet is characterized by lower velocity than other regions at 0.1AU.",
"It separates the outward (southern hemisphere) and inward (northern hemisphere) magnetic field topologies.",
"As the solar wind propagates outwards, the interactions between streams of different speeds become increasingly important with the sharp features at 0.1AU turn into more diffused, smoothed and extended ones at larger distances.",
"At the Earth's orbit the current sheet and thus the slow speed region is thicker covering about $10^\\circ $ in latitude.",
"The speed difference at 1AU with respect to the inner boundary is about $50\\mathrm {km\\ s^{-1}}$ .",
"The $\\theta $ , $\\phi $ -components of the velocity increase in magnitude from zero to about $\\approx 15 \\mathrm {km\\ s^{-1}}$ at the Earth's orbit, roughly following the pattern that the slow speeds appear at small latitudes contrary to the high solar wind speeds.",
"The radial component of the magnetic field decreases by about 2 orders of magnitude from the inner boundary to the orbit of the Earth showing a broader current sheet ($\\approx 5^\\circ $ ), where the magnetic field is close to zero.",
"The $\\theta $ -component of the magnetic field increases from zero to about $0.36$ nT in magnitude up to 1AU.",
"On the other hand, the $B_\\phi $ decreases by almost a factor of 8 up to 1AU and is at both distances opposite in polarity to the radial magnetic field, thus having positive polarity at the North Pole and negative at the South.",
"In all the depicted quantities the rotation of the plasma shows as the entire structure in the 2D maps moves to the left.",
"[htbp] Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONEUHFORIA longitudinal and latitudinal variations of the radial velocity (first row), the number density (second row) and the temperature (third row) at 0.1AU (left panels), at 1AU (middle panels) and at 1.4AU (right panels).",
"They can be contrasted with their kinetic counterparts in Figure REF .",
"In Figure REF , we show the velocity (first row), the density (second row) and the temperature (third row) variation at three different distances, 0.1AU, 1AU, and at the orbit of Ulysses at 1.4 AU, i.e.",
"$\\approx 300\\mathrm {R}_\\odot $ , corresponding to left, middle and right columns, respectively.",
"Most of the acceleration has already taken place at 1AU and only a small increase is happening above that distance due to the heating implemented by the reduced polytropic index ($\\gamma =1.5$ ) as the wind flows away from the Sun.",
"The density decreases by 2 orders of magnitude from the inner boundary of the MHD simulation to the Earth's orbit and only by 50% from there onwards up to 1.4AU.",
"The current sheet, seen as the high density structure that appears in the middle of the Figures in the second row seems to have expanded from $2^\\circ $ at the inner boundary to about $10^\\circ $ from there on, while it diffuses.",
"The temperature decreases by a factor 8 from the inner boundary up to 1AU and only by 30% up to Ulysses' orbit.",
"There is a temperature reversal captured, in the sense that while at 0.1AU the equatorial region appears cold and the poles hot, the opposite is happening from 1AU onwards, with the temperature shows a peak at a longitude of $-100^\\circ $ .",
"The current sheet region gets very diffused outwards and from the Earth's orbit outwards appears discontinuous in this temperature view of the expanding plasma."
],
[
"Interfacing",
"In order to interface the two models at 0.1AU, we run the kinetic model up to $21.5\\mathrm {R}_\\odot $ with $r_0=r_s=2.5\\mathrm {R}_\\odot $ , $T_e=1\\mathrm {MK}$ , $T_p=1\\mathrm {MK}$ , using 600 $\\kappa $ -indexes in the range [2,8] with step 0.01 and with $n_e=n_p=3\\times 10^{10}\\mathrm {m^{-3}}$ [8].",
"With the results we create a matrix with solar wind speeds at $21.5\\mathrm {R}_\\odot $ and by comparison with the EUHFORIA results for $v_r$ , we estimate the appropriate $\\kappa $ for every speed at the internal boundary of the MHD run.",
"We obtain a 2D map of $N_\\theta \\times N_\\phi $ values of $\\kappa (\\theta ,\\phi )$ each corresponding to a field line.",
"From the matrix, we also compare the number density given by EUHFORIA at the same distance (0.1AU) with the number density given by the kinetic model and get a scaling factor that we assume to be valid throughout all the considered radial distances [8].",
"Thereby, we can estimate the appropriate initial density at the exobase that would give us the same density with EUHFORIA at 0.1AU.",
"For the temperatures, the relation is more complicated, but [8] have demonstrated that the temperature does not affect the kinetic moments as much as the $\\kappa $ -indexes and the exobase altitude even for extreme changes (1 – 2MK) and thus for convenience we take $T_e=T_p=1$ MK for all the latitudes and longitudes at $r_0=2.5\\mathrm {R}_\\odot $ and focus instead on the $\\kappa $ and the density parameters.",
"Note that EUHFORIA is not solving the MHD equations below 0.1AU, while the kinetic model provides results at any distance above the exobase and especially in the crucial region close to the Sun where the solar wind is being accelerated."
],
[
"Kinetic Model",
"Similar to Figure REF , in Figure REF , we show the velocity (first row), the density (second row), the electron temperature (third row) and the proton temperature (fourth row) at three different distances, 0.1AU, 1AU, and at the orbit of Ulysses at 1.4AU, i.e.",
"$\\approx 300\\mathrm {R}_\\odot $ , corresponding to the left, middle and right columns, respectively.",
"Unlike the MHD results discussed previously, in the kinetic approach the sharp structures that appear at the interfacing boundary (0.1AU) remain unchanged as the plasma moves outwards, as we don't account for stream interactions.",
"In other words, neighboring streams with different speeds will not interact as they propagate and they will keep the same topological features up to large radial distances.",
"The $\\kappa $ indexes corresponding to the kinetic velocities of this simulation lie in the range $(2,4)$ .",
"The bulk speed accelerates more than in the MHD case, reaching terminal speeds about 100 $\\mathrm {km\\ s^{-1}}$ higher.",
"Thus in the kinetic approach, where the acceleration is self-consistent and due to the induced electric field, the acceleration is more efficient than in the MHD approach, where semi-empirical schemes are used to accelerate the wind.",
"Similarly to the MHD case, the terminal speed is reached at 1AU and from there on the acceleration is very slow up to 1.4AU.",
"The density decreases faster by 20% at 1AU, to reach the orbit of Ulysses 30% smaller than the MHD case.",
"The temperatures of the electron and proton populations are depicted in the last two rows.",
"We have started at the exobase by setting both temperatures equal to 1MK independent of longitude and latitude.",
"The electron temperature drops by a factor 5 at the orbit of the Earth and it doesn't change much up to 1.4AU, while the proton temperature decreases by a factor 2 and remains about the same up to the orbit of Ulysses.",
"The temperature of the MHD plasma is always in between the electron and proton temperatures, being one order of magnitude smaller than the electron and one order of magnitude larger than the proton temperature.",
"The MHD temperature is not the average between the two particle species temperatures, since the two species are not in thermodynamic equilibrium.",
"[htbp] Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONKinetic results in the form of color maps in heliographic longitude and latitude from top to bottom rows: of the bulk speed, the number density, the electron and proton temperatures of the solar wind respectively at 0.1AU (left panels), at 1AU (middle panels) and at Ulysses' orbit 1.4AU (right panels).",
"These can be contrasted directly with the MHD counterparts in Figure REF ."
],
[
"Models ",
"In Figure REF , the speed, number density and temperature of the solar wind are shown for i) the MHD model, ii) the kinetic model and iii) near-Earth observations using the OMNI dataset.",
"From the speed plot we conclude that both models reproduce the number of peaks, their position with respect to each other and have a similar width.",
"The most prominent difference between the models and the observations appears at the double peak of August 13th.",
"EUHFORIA reproduces peaks of similar amplitude as observed, but showing a higher global minimum at about 400 $\\mathrm {km\\ s^{-1}}$ .",
"The heights of the peaks and their relative ratio to one another are not reproduced exactly by any model.",
"The kinetic model systematically overestimates the speeds varying from 400 to 800 $\\mathrm {km\\ s^{-1}}$ .",
"This can be explained by the fact that the acceleration in that model is more efficient than the MHD case and continues at a significant rate at distances larger than 0.1AU to about 50$\\mathrm {R}_\\odot $ .",
"This indicates that the final velocity obtained with the kinetic model can be improved by adapting the boundary conditions, for instance by lowering the empirical solar wind speed close to the Sun.",
"The second panel shows the number density variations for the models and the observations.",
"We observe that both models and the OMNI observations of the number density agree roughly in order of magnitude and in number of peaks, with the peaks not coinciding perfectly.",
"The peak amplitudes are about 2.5 times larger in the observations than in both models, that are in agreement with one another.",
"For the Alfvénic Mach number (third panel), we conclude that the Alfvénic Mach number of the MHD simulation is about a factor 3 higher than the average measured value most of the time, with three lows that reach lower than the observed values.",
"The average observed proton temperature agrees with the EUHFORIA plasma temperature in order of magnitude, varying from the kinetic proton temperature $T_p$ at its minimum to the kinetic electron temperature $T_e$ at its maximum, while it is located between the two species' temperatures, about an order of magnitude larger than the kinetic proton temperature and an order of magnitude smaller than the kinetic electron temperature, at all times.",
"The variation profile of the observations doesn't match with any model.",
"Finally, the plasma $\\beta $ shown in the final panel for the MHD simulation lies most of the time at the lower limit of the observed one, with three peaks that reach values of about a factor 3 higher than the observed highest values, showing the opposite trend when compared to the Alfvénic Mach number shown in the third panel.",
"Thus we conclude that, in the MHD simulation the magnetic field is higher with respect to the plasma pressure and the number density than the one indicated by observations at 1AU.",
"The same case corresponding to CR 2059 was analyzed in a comparative study published in [4] for different observational inputs and MHD modeling schemes.",
"None of their models seemed to accurately represent observation.",
"The case directly comparable to ours was the one using GONG magnetograms and the WSA empirical model and ENLIL for the MHD modeling (dark red curve of lower panel in Figure 3).",
"Before 27 July their maxima are not synchronous nor do they reach the same values, but after that the model captures the times of the peaks, but underestimates their value on the panel in the figure showing the speed.",
"Figure: Time variations as measured by Ulysses (black) and calculated by the kinetic (blue) and the MHD (red) model, for i) top left panel: the position of Ulysses in heliographic coordinates, ii) top right panel: the speed, iii) bottom left panel: the number density and iv) bottom right panel: the Ulysses small (black) and large (green) proton, the kinetic proton (magenta), the kinetic electron (blue) and MHD (red) temperature.For a further comparison of the models with respect to observations, we overplot in Figure REF observations from Ulysses at 1.4AU and at latitudes between $-10^\\circ $ and $5^\\circ $ together with the results of the kinetic and MHD models.",
"In the first panel (top left), we show the trajectory of Ulysses during the period of interest.",
"At the same 3D heliographic spherical coordinates we extract the quantity of interest from the MHD and the kinetic model to accommodate further comparisons.",
"More specifically, for each $(r,\\theta ,\\phi )$ position of Ulysses at each hour of a specific date, we choose the closest available point for the specific resolution of each model.",
"In the second panel, we show the observed speed by Ulysses with black, together with the kinetic prediction in blue and the MHD results in red.",
"All three exhibit different time profiles for each case.",
"The kinetic model systematically overestimates the speeds reaching maximum values of about 800 $\\mathrm {km\\ s}^{-1}$ , whereas the MHD model ranges from 400 to 650 $\\mathrm {km\\ s}^{-1}$ , while at the same period the Ulysses measurements lie between 300 and 650 $\\mathrm {km\\ s}^{-1}$ .",
"The peaks for both models are not well synchronized with the measured temporal velocity profiles.",
"For the density the two models don't reproduce the observed number of peaks and they are not synchronized either.",
"The number densities of the kinetic and MHD models agree with each other varying from 1 – 5 $\\mathrm {cm}^{-3}$ whereas the observed number density profile is much more variable with a number of peaks and reaching densities of 20 $\\mathrm {cm}^{-3}$ .",
"The average observed proton temperatureAs shown in Figure REF , there are two different proton temperatures estimated, that in general bracket the real temperature at 1AU.",
"As $T$ -large we denote the integral in the 3D velocity space of the distribution over all measured angles and energy bandwidths.",
"The $T$ -small is calculated by the sum over all angles for a determined energy, then summing the moments of the estimated spectrum of the plasma and by taking the radial component of the temperature tensor (http://www.cosmos.esa.int/web/ulysses/swoops-ions-user-notes).",
"agrees with the plasma temperature predicted by the MHD model and lies in between the kinetic proton (magenta) and kinetic electron (blue) temperatures of the kinetic model, being one order of magnitude smaller than the electron and one order of magnitude larger than the proton temperatures.",
"No model reproduces in high accuracy the temporal variations of the observed temperature profile.",
"The models show some correspondence with the Ulysses observations and the correct orders of magnitude are roughly reproduced, but certainly there is need for improvement.",
"To reach better agreement with observations, we can modify the parameters in the semi-empirical model of EUHFORIA to better adjust and reproduce the measurements in each case individually."
],
[
"Heat Flux",
"[27] presented an analysis of the different ways that energy source considerations can be used in MHD solar wind models.",
"A generalized formulation of the energy Equation with an extra energy source term at the right hand side $S$ was examined for different cases of $S$ .",
"The relevant models studied therein include i) a model with a polytropic index with spatial dependance $\\Gamma (r)$ and ii) a model with a polytropic index fixed at $\\gamma =5/3$ constant in the entire coronal volume.",
"In particular, the authors showed that a steady-state solar wind solution accelerated by a given non-adiabatic polytropic wind can equivalently be re-written using an energy source term, given by: $S=\\nabla \\cdot \\Bigg [{v}P\\bigg ( \\frac{1}{\\Gamma -1}-\\frac{1}{\\gamma -1}\\bigg ) \\Bigg ] {,}$ with ${v}$ , $P$ and $\\Gamma $ being obtained by the model, as explained in [27], with the non-adiabatic index $\\Gamma (r)=1.5$ for our case.",
"Figure: Heat flux along magnetic field lines in the equatorial plane as given by EUHFORIA.In this section, we will discuss the heat flux of the MHD and the kinetic models.",
"According to the formulation presented above we have that the analytic expression of the energy flux responsible for accelerating the wind in the MHD model is given by ${v}P\\bigg (\\frac{1}{\\Gamma -1}-\\frac{1}{\\gamma -1}\\bigg )$ .",
"In Figure REF , we present the heat flux of the MHD model along selected magnetic field lines as a function of distance from 0.1AU to 2AU in the equatorial plane.",
"Figure: Electron (top panel) and proton (bottom panel) heat fluxes for CR2059 with red, orange, light green, green and blue curves corresponding to κ\\kappa values of 3, 4, 5, 6 and 7, respectively.In Figure REF , we illustrate the radial profiles of the electron (top panel) and proton (bottom panel) heat fluxes, as calculated by the kinetic model for an exobase at 2.5$\\mathrm {R}_\\odot $ , 1MK electron and proton temperatures at the exobase $T_e=T_p=1\\mathrm {MK}$ and the same densities as our default case for $\\kappa $ -indexes 3, 4, 5, 6 and 7 corresponding to red, orange, light green, green and blue respectively.",
"We observe that the more suprathermal particles are present, i.e.",
"lower $\\kappa $ , the higher the heat flux is.",
"The electron heat flux is higher by an order of magnitude than the proton heat flux close to the Sun and it decreases faster than the proton heat flux with the heliocentric distance to reach similar values at 1AU.",
"We conclude that the differences between the proton heat flux curves corresponding to different $\\kappa $ -indexes are smaller than the respective electron flux curves.",
"In general, the heat fluxes are overestimated by exospheric models, since the corresponding VDFs have the highest possible anisotropy due to lack of collisions.",
"As shown in [35], exospheric models and kinetic simulations that include collisions are in good agreement, making the exospheric model a convenient tool for the study of weakly collisional plasmas.",
"Both the exospheric model and the collisional simulations found heat fluxes several times larger than the classical value, suggesting that the classical formulation is not appropriate for weakly collisional plasmas, since it was based on the assumption of a collision-dominated medium.",
"If one really needs to prescribe a realistic heat flux, then collisions need to be taken into account and further improvements in the kinetic model need to be considered.",
"The inclusion of interactions, through e.g.",
"Alfvén or whistler waves, will decrease the anisotropies and it will improve the higher moments, i.e.",
"temperatures and heat fluxes for the considered species [22], [24], [31].",
"Using such sophisticated schemes to improve the heat flux agreement with observations is outside of the scope of this paper, due to i) the consequent increased computational expense, ii) the fact that the temperatures are not the most important geo-effective parameters, making these improvements rather impractical for future operational space weather applications.",
"Thus, the heat flux profiles for the electron and protons, that are quantified by the kinetic model (Figure REF ) can serve as upper limits and are more physics-based than the MHD heating prescriptions that are based on the empirical determination of the polytropic index value.",
"The heat flux profiles along the magnetic field lines of the MHD model are in agreement in order of magnitude with the profiles of the protons of the kinetic model closer to the Sun, but they drop faster further out reaching electron heat flux values at the orbit of the Earth.",
"Our results for the heat flux for the MHD model depicted in Figure REF are roughly in agreement with the literature [5], [30].",
"In accordance to the kinetic model results, the electron heat flux is higher than the proton heat flux according to [5] and [30], respectively, with our results roughly being closer to the electron energetics profile, but at 1AU approaching the heat flux values expected for the protons.",
"In this study, we are interested in making a first comparison between single fluid and kinetic models, that have the potential to be used in space weather applications.",
"The two models are very different in nature, making use of very different formulations for the plasma physics.",
"EUHFORIA is a single fluid code that uses the MHD equations to describe the plasma.",
"Whereas the kinetic model used in this study begins by making an observationally-inspired assumption for the VDFs of the electron and proton species, that are considered to constitute the solar wind plasma, and based on that all the physically interesting quantities are calculated as moments over the velocity space.",
"EUHFORIA is observationally-driven and provides three-dimensional information that accounts for stream interactions and complicated magnetic topologies.",
"The kinetic model is a semi-analytic model solving for the plasma characteristics along a magnetic field line and accounting for heating and acceleration in a self consistent way.",
"Note that the electric field (used in the kinetic model) is hidden in the pressure term .",
"Parker explained that the momentum equation of the electrons is given by $\\frac{\\mathrm {d}p_e}{\\mathrm {d}r}+neE_\\mathrm {LS}=0 {,}$ which is the hydrodynamic condition, with $E_\\mathrm {LS}$ the Lemaire-Scherer electric field [10].",
"In [18], Parker showed that the electric field of Lemaire-Scherer is in the fluid approach given by $E_\\mathrm {LS}=-\\frac{1}{ne}\\frac{\\mathrm {d}p_e}{\\mathrm {d}r}=\\frac{m_p}{e}\\left( v\\frac{\\mathrm {d}v}{\\mathrm {d}r}+\\frac{GM_\\odot }{r^2}\\right) {,}$ which finally, taking into account that the Pannekoek-Rosseland electric field is $E_\\mathrm {PR}=\\frac{m_pg}{e}$ , is written as $E_\\mathrm {LS}=E_\\mathrm {PR}+\\frac{m_pv}{e} \\frac{\\mathrm {d}v}{\\mathrm {d}r} {.",
"}$ The Lemaire-Scherer electric field (used in exospheric models) is several times larger than the Pannekoek-Rosseland one, corresponding to hydrodynamic equilibrium that is used to describe the solar wind expansion.",
"It is able to lift and accelerate to supersonic speeds the initially slow and heavy protons through trapping the fast and light electrons, while keeping the quasi-neutrality and almost zero-current condition.",
"Any difference in the bulk speeds of the two populations would lead to the generation of a current, which due to Ampère's law has to remain small.",
"From Figure REF , we deduce that at large radial distances from the Sun, the MHD current sheet gets expanded and becomes thicker, evolving from about $2^\\circ $ at the inner boundary to about $10^\\circ $ at 1AU, while the fine structures that appear in the 0.1AU longitudinal and latitudinal map get diffused and smoothed.",
"On the contrary, in Figure REF we see that the fine structures and the current sheet size don't change under the kinetic approach, as there are no stream interactions taken into account.",
"Both models give speeds of the same order of magnitude at every altitude and have most of their acceleration taking place already before 1AU.",
"The kinetic model shows a more efficient acceleration, reaching terminal speeds of about 100 $\\mathrm {km\\ s^{-1}}$ larger than the MHD one up to 1AU.",
"The acceleration in the MHD approach is taking place due to the reduced polytropic index, while in the kinetic approach the acceleration is related to the induced electric field that assures quasi-neutrality and equal outward electron and proton fluxes.",
"Regarding the number density, both models give densities of the same order of magnitude, with the kinetic model though showing a sharper profile that decreases faster outwards.",
"The number density of the kinetic model is 20% and 30% lower at 1AU and 1.4 AU respectively than the MHD number density.",
"Moreover, for the temperature in the MHD approach, as is shown in the bottom row of Figure REF , there is a reversal of the hot-cold regions from the inner boundary up to large distances.",
"A faster cooling takes place at higher latitudes, making the initially colder equatorial region appear hotter at large radial distances with respect to the rest of the latitudes.",
"Furthermore, the temperature ranges fall by a factor 4 from 0.1AU up to 1AU, and continue decreasing by 30% up to the orbit of Ulysses.",
"For the kinetic approach, the proton temperature only falls by factor two up to the Earth's orbit and seems constant up to 1.4 AU, while the electron temperature drops by a factor 5 up to 1AU and doesn't seem to vary much from there on.",
"Contrary to the MHD single fluid results, in the kinetic case there is no temperature profile change and the cooling seems to take place uniformly at all latitudes.",
"There is a structure close to the equator at a heliographic longitude of $-70^\\circ $ that appears like a spike in contact with the current sheet in both Figures REF and REF .",
"In Figure REF though, in the middle and right panels, corresponding to large distances, that spike appears to change inclination from pointing to the left of the figure at the panels corresponding to 0.1AU to pointing to the right from 1AU outwards, unlike Figure REF , where the spike's inclination remains unchanged.",
"This difference is then likely due to stream interactions that are captured in the 3D MHD model, while these are excluded in the essentially 1D, radial field kinetic approach."
],
[
"Conclusions",
"After the parallelization of the kinetic model and its use in a quasi-3D approach we linked it to more robust 3D MHD models and compared both to observations at 1AU, using similar boundary conditions at $21.5\\mathrm {R}_\\odot $ .",
"When the two models were compared, starting from the same boundary conditions, the kinetic model gives systematically higher speeds than the MHD model at large radial distances.",
"This is due to the fact that the acceleration of the solar wind continues at a higher rate in the kinetic model after $21.5\\mathrm {R}_\\odot $ .",
"The acceleration mechanism in the kinetic model is due to the induced electric field that assures quasi-neutrality and prevents charge separation and also \"bounds\" the two species to move with the same bulk speed.",
"There is no explicit heating term in the MHD equations used by EUHFORIA.",
"The MHD model accelerates further the solar wind due to the reduced polytropic index $\\gamma $ , but at a very slow rate accounting for an acceleration of about 50 $\\mathrm {km\\ s}^{-1}$ from 0.1AU to 1AU.",
"The exospheric models overestimate the heat flux, especially for the electrons that have a thermal speed comparable to their bulk speed, but such a heat flux can be improved by inclusion of interactions through waves, e.g.",
"Alfvén or whistler waves [22], [24], [31].",
"The heat flux calculated by the kinetic exospheric model can be used as an upper limit for more physically-driven heat flux prescriptions in a global MHD model.",
"The heat flux of the kinetic model is in qualitative agreement with other studies [5], [30] and the heat flux profile of the MHD models close to the Sun resembles the proton profile, only to decrease faster outwards resembling the electron profile at 1AU.",
"In the exospheric model the fast electrons are slowed down and the protons are accelerated by the Lemaire-Scherer electric field that eventually leads to the observed supersonic solar wind.",
"As shown by [18] the acceleration of the solar wind in collisionless plasmas to supersonic values lies in the hydrodynamic equation in combination to the mass ratio between electrons and protons.",
"According to Parker, the exospheric model describes a very efficient heat transport mechanism with an electron temperature that decreases very slowly at large distances and through the induced electric field it elevates the protons and causes the transonic solar wind.",
"There is some agreement in the high- and slow-speed streams in the velocity profiles for both OMNI and Ulysses observations.",
"The high speed positions of the models are better synchronized for OMNI rather than Ulysses observations, as we showed earlier.",
"The number densities of both models were approximately in the same order of magnitude with the observed ones at 1AU and at the orbit of Ulysses.",
"The MHD temperature is one order of magnitude smaller than the electron temperature $T_e$ and one order of magnitude larger than the proton temperature $T_p$ of the kinetic model.",
"The average observed proton temperatures agreed with the temperature predicted by the MHD model in order of magnitude and thus were also lying in the range between the electron and proton kinetic temperatures.",
"In this study we assumed that the $\\kappa $ -value is independent of the radial distance, but observations such as [12] indicate that the $\\kappa $ can actually change as the distance from the Sun increases.",
"However, the fitting model used in [12] was not with a Kappa distribution for the full range, but a sum of a Maxwellian for the core and a Kappa function for the halo, so that the parameter $\\kappa $ does not represent the same quantity as in the model used in the current study.",
"The density ratio between the core and the halo remains constant with the distance, as analyzed by [25].",
"A more realistic exobase profile with temperatures having a latitudinal and even longitudinal variation can be taken into account and will be the next step towards a fully 3D kinetic numerical code of the solar corona and the solar wind.",
"The kinetic model is a semi-analytic model ignoring stream interactions and thus conserves the slow and fast wind distributions for every radial distance.",
"Accounting for stream interactions as the solar wind propagates outwards will be one of the main points of interest for a more realistic 3D model.",
"Shocks associated to sharp velocity gradients can be included in an empirical way or by using more sophisticated kinetic models including collisions and wave-particle interactions [19].",
"Here we have ignored the spiral shape of the magnetic field in the calculation of the moments in the kinetic model, adopting purely radial magnetic fields, because as argued in [23] this aspect won't affect the main average quantities apart from the temperature anisotropies.",
"But in the future we are planing on including the spiral magnetic field effects in a similar study.",
"[23] quantified the effect that the spiral magnetic field topology has on the particle temperatures and their anisotropies.",
"More specifically, in the radial case the electron temperature is slightly underestimated, whereas the opposite is true for the proton species.",
"The temperature anisotropies are overestimated by the kinetic model in comparison to observations [9].",
"Another important aspect that can be improved and would provide a deeper comparison between operational MHD codes and kinetic exospheric ones, would be to change the formulation of the kinetic model, so that boundary conditions for the speed, the density and temperature directly from MHD models can be used at each grid point including the magnetic field information.",
"These aspects will allow us to reach more fundamental conclusions about the two different models and will upgrade the kinetic exospheric model into a computational equivalent to the robust 3D MHD code.",
"When the 3D magnetic field topology from the source surface, through the Schatten current sheet region, and throughout the region covered by the MHD model, is used directly within the kinetic description, we can use its predicted heat fluxes and higher order moment information to turn the model into a self-consistent hybrid kinetic-MHD modeling tool.",
"SPM acknowledges financial support by the FWO and NASA Living with a Star grant number NNX16AC11G.",
"This research was supported by projects GOA/2015 – 014 (KU Leuven, 2014 – 2018), and the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7/08 CHARM)."
],
[
"Conflict of Interest",
"The authors declare that they have no conflict of interest."
]
] | 1709.01605 |
[
[
"Did we learn from LLC Side Channel Attacks? A Cache Leakage Detection\n Tool for Crypto Libraries"
],
[
"Abstract This work presents a new tool to verify the correctness of cryptographic implementations with respect to cache attacks.",
"Our methodology discovers vulnerabilities that are hard to find with other techniques, observed as exploitable leakage.",
"The methodology works by identifying secret dependent memory and introducing forced evictions inside potentially vulnerable code to obtain cache traces that are analyzed using Mutual Information.",
"If dependence is observed, the cryptographic implementation is classified as to leak information.",
"We demonstrate the viability of our technique in the design of the three main cryptographic primitives, i.e., AES, RSA and ECC, in eight popular up to date cryptographic libraries, including OpenSSL, Libgcrypt, Intel IPP and NSS.",
"Our results show that cryptographic code designers are far away from incorporating the appropriate countermeasures to avoid cache leakages, as we found that 50% of the default implementations analyzed leaked information that lead to key extraction.",
"We responsibly notified the designers of all the leakages found and suggested patches to solve these vulnerabilities."
],
[
"Introduction",
"In the last decade we have witnessed the cloud revolution enabled by sandboxing mechanisms such as virtualization.",
"These technologies allow cloud service providers to place data and processes of various customers on the same hardware without jeopardizing their security.",
"IaaS clouds for instance rent expensive hardware resource to multiple customers by offering guest OS instances sandboxed inside virtualized machines (VMs).",
"PaaS cloud services go one step further and allow users to share the application space while sandboxed, e.g.",
"using containers, at the OS level.",
"Similar sandboxing techniques are used to isolate semi-trusted apps running on mobile devices.",
"Even browsers use sandboxing to execute untrusted code without the risk of harming the local host.",
"Resource sharing improves utilization and thereby helps reducing IT costs and saves power.",
"However, resource sharing also enables information leakages at the hardware level which sandboxing techniques cannot prevent and that can be exploited by malicious code.",
"One of the most prominent leakage sources, i.e.",
"the Last Level Cache (LLC), has been heavily targeted.",
"From the adversaries' point of view, the LLC leakage channel has a number of advantages, including high resolution information and cross core information leakage.LLC attacks have demonstrated to recover a wide range of information, ranging from private cryptographic keys to user's private shopping behavior .",
"Furthermore, LLC attacks have been successfully applied in a wide range of scenarios, e.g., malicious VMs in IaaS or PaaS public clouds, malicious Javascript execution in web browsers or even as malicious smartphone apps .",
"In short, cache attacks have demonstrated to pose a severe threat applicable in a wide range of use scenarios.",
"These attacks can be stopped at three different layers; at the hardware , the OS/hypervisor or at the application layer .",
"While the first two involve overheads that hardware/OS designers might not be willing to pay, preventing leakage in security critical applications can be achieved by aware code designers through proper implementation techniques.",
"Indeed, cache attacks usually exploit human coding mistakes that lead to private information leakage.",
"Although these leakages might sometimes be difficult to detect, they need to be carefully verified, especially when designing cryptographic code.",
"Cryptographic primitives make the core of the security infrastructure used to protect our digital communications.",
"If the used cryptographic code is poorly designed, any solution built on top is also susceptible to fail.",
"In summary, cache attacks diverge from traditional attacks in the following ways: Stealthy: Leakage attacks are extremely hard to detect.",
"The effects of an attack may only be felt through performance degradation for a short duration while leakage traces are being collected.",
"After the attack is performed no digital footprint is left.",
"Hence standard detection techniques such as traffic, access and privilege monitoring are completely blind to leakage attacks.",
"For instance, any application submitted to the app store is checked for access violations to devices, however, any attack code exploiting hardware leakage would only monitor legitimate memory access time variations.",
"Hard to Prevent: It is rather difficult to design leakage proof code, especially if good performance is also an objective.",
"Leakages are quite subtle and may not be detected after deployment for a long time.",
"Even if the code is not leaky on one platform, with gradual optimizations implemented at the microarchitectural level, new leakage channels may emerge with newer releases of a platform.",
"Difficult to Test: Verification of leakage resistant code is painfully slow and typically requires inspection by experts well versed in cache attacks.",
"In this paper, we introduce a tool that helps cryptographic code designers to analyze and expose any leakages in cryptographic implementations.",
"We demonstrate the viability of the technique by analyzing cache leakages in a number of popular cryptographic libraries with respect to three main cryptographic primitives we rely on every day to securely communicate over the internet: AES, RSA and ECC.",
"In order to achieve this goal, we first find secret dependent instructions/data with dynamic taint analysis, introduce cache evictions inside code routines, record cache traces and then verify the existing dependence with respect to the secret.",
"Our results show that, despite the efforts of cryptography library designers in reaction to the multitude of recently published LLC cache attacks several popular libraries are still vulnerable.",
"Our Contributions.",
"This paper presents a proactive tool to analyze leakage behavior of security critical code.",
"Unlike other approaches, our tool can be used to find real exploitable leakage in the design of cryptographic implementations.",
"The tool identifies secret dependent data, obtains cache traces obtained from the code execution on the microarchitecture and uses the generic Mutual Information Analysis (MIA) as a metric to determine the existence of cache leakage.",
"The detection technique is agnostic to the implementation, i.e.",
"the testing code can be run across all target platforms without having to redesign it, yet pinpoints parts of code that cause found leakages.",
"We perform the first big scale analysis of popular cryptographic libraries on three of the most commonly used primitives: AES, RSA and ECC.",
"We demonstrate that several cache leakages are still present in up-to-date cryptographic libraries (i.e.",
"50% still leak information) and need to be fixed."
],
[
"Preliminaries",
"The proposed tool employs techniques from modern cache attacks to monitor cache activity and measures information leakage using Mutual Information."
],
[
"LLC Attacks",
"LLC cache attacks are one of the most dangerous side channel attacks since they do not rely on physical proximity.",
"Furthermore, these attacks do not require root privileges and can be executed in user mode.",
"There are two main classes of LLC attacks: Flush and Reload Attack The Flush and Reload attack was first introduced in Gullasch et al.",
"but acquired its name in Yarom and Falkner , who were able to extract RSA cryptographic keys across VMs situated in different cores.",
"This attack was later shown to be successful also breaking symmetric key algorithms , , TLS sessions or the isolation in PaaS clouds and smartphone devices , .",
"In particular, the attack assumes the following; The attacker and the victim are executing processes in the same CPU but in different cores.",
"Attacker and victim share read-only memory pages.",
"The Flush and Reload attack can only succeed when attacking memory blocks shared with the potential victim.",
"These usually imply static code and global variables, but never dynamically allocated variables.",
"Prime and Probe Attack The Prime and Probe attack was first introduced for the L1 cache by Osvik et al.",
"and was later utilized by Ristenpart et al.",
"and Zhang et al.",
"to recover, in IaaS clouds, keystrokes and El Gamal decryption keys respectively.",
"Recently the attack was expanded to attack the LLC by Liu et al.",
"and Irazoqui et al.",
", and has been successfully applied in commercial IaaS clouds , as Javascript executions and as smartphone applications .",
"The attack presents advantages and disadvantages over Flush and Reload: Prime and Probe does not require any memory sharing between victim and attacker.",
"Prime and Probe can increase the attack vector to dynamically allocated variables.",
"Prime and Probe implies some reverse engineering to know which sets in the cache the attacker is using, while Flush and Reload does not.",
"Prime and Probe is noisier than Flush and Reload.",
"Taking these two attacks into account, we explore the cryptographic libraries looking for leakages exploitable by either attack.",
"Thus, we examine both statically and dynamically allocated memory in our analysis."
],
[
"Mutual Information Analysis",
"The metric that we use to quantify leakage is Mutual Information (MI), that is given by the following equation: $I(X;Y) = H(X) - H(X|Y) = H(X) + H(Y) - H(X, Y)$ Where $H(X)$ is the entropy of random variable $X$ and $H(X|Y)$ is the entropy of the random variable $X$ given the knowledge of the random variable $Y$ .",
"In a way $I(X; Y)$ gives us how related variables $X$ and $Y$ are.",
"Note that if $X$ and $Y$ are independent, $I(X;Y)=0$ since $H(X|Y)=H(X)$ .",
"In contrast, if $X$ and $Y$ are fully dependent we obtain maximum MI, i.e., $I(X;Y)=H(X)$ , since $H(X|Y)=0$ ($Y$ fully determines $X$ ).",
"MI has been used in prior work for side channel attacks and leakage quantification.",
"Gierlichs et al.",
"utilize MI as a side channel distinguisher to mount differential side channel attacks.",
"Standaert et al.",
"utilized MI as a leakage quantifier to evaluate side channel attack security.",
"Prouff et al.",
"further expanded on the limitations and strengths of MI as a side channel attack metric.",
"Recently Zhang and Lee used MI to measure the security level of several cache architectures which they modeled as finite state machines."
],
[
"Methodology",
"Our goal is to determine whether specific memory addresses being used inside a cryptographic routine reveal information based on their cache accesses."
],
[
"Main Idea",
"Our approach involves the monitorization of the cache usage for all the memory addresses considered susceptible of carrying information.",
"In order to achieve this goal we utilize common known techniques from cache attacks, like the usage of cache flushing or cycle counter instructions.",
"We illustrate our approach on a toy example shown in Figure REF .",
"The Hello code snippet simply returns different messages depending on whether the caller is a female or a male.",
"We assume that the designer of this simple code does not want a potential attacker to know the gender of the caller.",
"Assume that the code designers would like to know whether they did a good job, i.e., whether the cache traces do not reveal the gender of the user.",
"Here, it is easy to observe that there is a $gender$ dependent branch that could reveal whether the user is a male or female.",
"We call $B1$ and $B2$ the two possible outputs of the branch.",
"Figure: NO_CAPTION"
]
] | 1709.01552 |
[
[
"An active-learning algorithm that combines sparse polynomial chaos\n expansions and bootstrap for structural reliability analysis"
],
[
"Abstract Polynomial chaos expansions (PCE) have seen widespread use in the context of uncertainty quantification.",
"However, their application to structural reliability problems has been hindered by the limited performance of PCE in the tails of the model response and due to the lack of local metamodel error estimates.",
"We propose a new method to provide local metamodel error estimates based on bootstrap resampling and sparse PCE.",
"An initial experimental design is iteratively updated based on the current estimation of the limit-state surface in an active learning algorithm.",
"The greedy algorithm uses the bootstrap-based local error estimates for the polynomial chaos predictor to identify the best candidate set of points to enrich the experimental design.",
"We demonstrate the effectiveness of this approach on a well-known analytical benchmark representing a series system, on a truss structure and on a complex realistic frame structure problem."
],
[
"Introduction",
"Structural reliability analysis aims at computing the probability of failure of a system with respect to some performance criterion in the presence of uncertainty in its structural and operating parameters.",
"Such uncertainty can be modelled by a random vector $\\mathbf {X}\\in {\\mathbb {R}}^M$ with prescribed joint probability density function $f_{\\mathbf {X}}$ .",
"The limit-state function $g$ is defined over the support of $\\mathbf {X}$ such that $\\left\\lbrace \\mathbf {x}: g(\\mathbf {x})\\le 0\\right\\rbrace $ defines the failure domain, while $\\left\\lbrace \\mathbf {x}: g(\\mathbf {x})>0\\right\\rbrace $ defines the safe domain.",
"The limit state surface implicitly defined by $g(\\mathbf {x}) = 0$ lies at the boundary between the two domains.",
"The probability of failure of such a system can be defined as [30], [25]: $P_F = \\int _{\\left\\lbrace \\mathbf {x}: g(\\mathbf {x})\\le 0\\right\\rbrace } f_{\\mathbf {X}}(\\mathbf {x}) d\\mathbf {x}.$ A straightforward approach to compute the integral in Eq.",
"(REF ) is to use of Monte Carlo Simulation (MCS).",
"However, standard MCS approaches can often not be used in the presence of complex and computationally expensive engineering models, because of the large number of samples they require to estimate small probabilities (typically in the order of $\\sim 10^{k+2}$ for $P_F \\approx 10^{-k}$ ) with acceptable accuracy.",
"Well-known methods based on local approximation of the limit-state function close to the failure domain (such as FORM [20] and SORM [35]) can be more efficient, yet they are usually based on linearisation and tend to fail in real-case scenarios with highly non-linear structural models.",
"In contrast, methods based on surrogate modelling have gradually gained momentum in the last few years.",
"Due to the nature of the problem of estimating low probabilities, most recent methods combine active-learning-based greedy algorithms with Gaussian process surrogate models (Kriging).",
"Among the first works to propose this approach, the earliest applications in this context were the efficient global reliability analysis method (EGRA) by [2], [3], and the active-learning reliability (AK-MCS) method based on Kriging by [14].",
"More recently, Kriging has been employed to devise quasi-optimal importance densities in [12], [10].",
"Amongst other variations, polynomial-chaos-based Kriging has also been used as an alternative metamodelling technique [37] to overcome some of the limitations of pure Kriging-based methods.",
"Additional works on the topic of Kriging and structural reliability can be found, including extensions of the original AK-MCS algorithm to more advanced sampling techniques [15], [1], system reliability [18] and for the exploration of multiple-failure regions [7].",
"Polynomial chaos expansions (PCE) [19] are a well-established tool in the context of uncertainty quantification, with applications in uncertainty propagation [43], sensitivity analysis [23] and, to a lesser degree, structural reliability [38].",
"While often considered as an efficient surrogate modelling technique due to their global convergence behaviour, PCEs have been employed only seldom in reliability analysis (see, e.g.",
"[32]) due to their lack of accuracy in the tails of the model response distribution, which are essential in this field.",
"In addition, most active-learning approaches with surrogates require some form of local error estimate to adaptively enrich a small set of model evaluations close to the limit state surface.",
"Kriging-based methods can rely on the Kriging variance for this task, but PCEs do not provide a natural equivalent.",
"In this paper, we leverage on the properties of regression-based sparse-PCE [6] to derive a local error estimator based on bootstrap resampling.",
"We then use this estimator to construct an active-learning strategy that adaptively approximates the limit-state function with PCE by minimizing a misclassification probability-based learning function at every iteration.",
"The method is then showcased on a standard benchmark functions representing a series system and on a realistic structural frame engineering example."
],
[
"Polynomial Chaos Expansions",
"Consider a finite variance model $Y = {\\mathcal {M}}(\\mathbf {X})$ representing the response of some quantity of interest (QoI) $Y$ to the random input parameters $\\mathbf {X}\\in {\\mathbb {R}}^M$ , modelled by a joint probability distribution function (PDF) $f_{\\mathbf {X}}$ .",
"Also consider the functional inner product defined by: $\\left<g,h\\right> \\equiv \\int _{\\mathbf {x}\\in \\Omega _{\\mathbf {X}}}g(\\mathbf {x})h(\\mathbf {x})f_{\\mathbf {X}}(\\mathbf {x})d\\mathbf {x}= {\\mathbb {E}}\\left[ g(\\mathbf {X}) h(\\mathbf {X}) \\right]$ where $\\Omega _{\\mathbf {X}}$ represents the input domain.",
"Under the assumption of independence of the input variables, that is $f_{\\mathbf {X}}(\\mathbf {x}) =\\prod \\limits _{i=1}^M f_{X_i}(x_i)$ , one can represent ${\\mathcal {M}}(\\mathbf {X})$ as the following generalised polynomial chaos expansion (see, e.g.",
"[19], [43]): $Y = {\\mathcal {M}}(\\mathbf {X}) = \\sum \\limits _{\\mathbf {\\alpha }\\in {\\mathbb {N}}^M}y_{\\mathbf {\\alpha }}\\Psi _{\\mathbf {\\alpha }}(\\mathbf {X}),$ where the $y_{\\mathbf {\\alpha }}$ are real coefficients and $\\mathbf {\\alpha }$ is a multi-index that identifies the degree of the multivariate polynomial $\\Psi _{\\mathbf {\\alpha }}$ in each of the input variables $X_i$ : $\\Psi _{\\mathbf {\\alpha }}= \\prod \\limits _{i = 1}^M \\phi ^{(i)}_{\\alpha _i}(X_i).$ Here $\\phi ^{(i)}_{\\alpha _i}$ is a polynomial of degree $\\alpha _i$ that belongs to the family of orthogonal polynomials w.r.t.",
"the marginal PDF $f_{X_i}$ .",
"For more details on the construction of such polynomials for both standard and arbitrary distributions, the reader is referred to [43].",
"In the presence of a complex dependence structure between the input variables, it is always possible to construct isoprobabilistic transforms (e.g.",
"Rosenblatt or Nataf transforms, see e.g.",
"[24]) to decorrelate the input variables prior to the expansion, even in the case of complex dependence modelled by vine copulas [40].",
"For the sake of notational simplicity and without loss of generality, we will hereafter assume independent input variables.",
"In practical applications, the series expansion in Eq.",
"(REF ) is traditionally truncated based on the maximal degree $p$ of the expansion, thus yielding a set of basis elements identified by the multi-indices ${\\mathbf {\\alpha }\\in {\\mathcal {A}}: \\sum \\limits _{i = 1}^M \\alpha _i\\le p}$ , with $\\text{card}({\\mathcal {A}}) \\equiv P = \\binom{M+p}{p}$ , or using more advanced truncation schemes that favour sparsity, e.g.",
"hyperbolic truncation [4].",
"The corresponding expansion coefficients $\\mathbf {y_{\\mathbf {\\alpha }}}$ can then be calculated efficiently via least-square analysis based on an existing sample of the input random vector ${\\mathcal {X}}= \\left\\lbrace \\mathbf {x}^{(1)},\\cdots ,\\mathbf {x}^{(N)}\\right\\rbrace $ , known as the experimental design (ED), and the corresponding model responses ${\\mathcal {Y}}=\\left\\lbrace y^{(1)},\\cdots ,y^{(N)}\\right\\rbrace $ as follows: $\\mathbf {y_{\\mathbf {\\alpha }}} = \\text{argmin}\\frac{1}{N} \\sum \\limits _{i = 1}^N\\left[y^{(i)} - \\sum \\limits _{\\mathbf {\\alpha }\\in {\\mathcal {A}}} y_{\\mathbf {\\alpha }}\\Psi _{\\mathbf {\\alpha }}(\\mathbf {x}^{(i)})\\right]^2.$ When the number of unknown coefficients $P$ is high (e.g.",
"for high-dimensional inputs or high-degree expansions), regression strategies that favour sparsity are needed to avoid over-fitting in the presence of a limited-size experimental design and to make the analysis at all feasible with a reasonable sample size $N$ .",
"Amongst them, least angle regression (LARS, [17]), based on a regularized version of Eq.",
"(REF ), has proven to be very effective in tackling realistic engineering problems even in relatively high dimensions (i.e.",
"$M \\sim 100$ ).",
"In this paper, we adopt the full degree-adaptive, sparse PCE based on hybrid-LARS introduced in [6]), as implemented in the UQLab Matlab software ([28], [29])."
],
[
"Bootstrap in least-square regression",
"Adopting a least-square regression strategy to calculate the coefficients in Eq.",
"(REF ) allows one to use the bootstrap resampling method [16] to obtain information on the variability in the estimated coefficients due to the finite size of the experimental design.",
"Suppose that a set of estimators $\\mathbf {\\theta }$ is a function of a finite-size sample ${\\mathcal {X}}= \\left\\lbrace \\mathbf {x}^{(1)},\\cdots ,\\mathbf {x}^{(N)}\\right\\rbrace $ drawn from the random vector $\\mathbf {X}$ .",
"Then the bootstrap method consists in drawing $B$ new sample sets $\\left\\lbrace {\\mathcal {X}}^{(1)},\\cdots ,{\\mathcal {X}}^{(B)}\\right\\rbrace $ from the original ${\\mathcal {X}}$ by resampling with substitution.",
"This is achieved by randomly assembling $B-$ times $N$ realizations $\\mathbf {x}^{(i)}\\in {\\mathcal {X}}$ , possibly including repeatedly the same realization multiple times within each sample.",
"The set of estimated quantities can then be re-calculated from each of the $B$ samples, thus yielding a set of estimators $\\mathbf {\\Theta } =\\left\\lbrace \\mathbf {\\theta }^{(1)},\\cdots ,\\theta ^{(B)}\\right\\rbrace $ .",
"This set of estimators can then be used to directly assess the variability of $\\mathbf {\\theta }$ due to the finite size of the experimental design ${\\mathcal {X}}$ , at no additional costs, e.g.",
"by calculating statistics, or directly using each realization separately.",
"Application of the bootstrap method combined with PCE to provide confidence bounds in the estimated $P_F$ in structural reliability applications can be found in e.g.",
"[32], [33]."
],
[
"Bootstrap-PCE",
"We propose to use the bootstrap technique to provide local error estimates to the PCE predictions.",
"The rationale is the following: the PCE coefficients $\\mathbf {y_{\\mathbf {\\alpha }}}$ in Eq.",
"(REF ) are estimated from the experimental design ${\\mathcal {X}}$ , therefore they can be resampled through bootstrap.",
"This can be achieved by first generating a set of bootstrap-resampled experimental designs $\\left\\lbrace {\\mathcal {X}}^{(b)},{\\mathcal {Y}}^{(b)}, b = 1,\\cdots ,B\\right\\rbrace $ .",
"For each of the generated designs, one can calculate a corresponding set of coefficients $\\mathbf {y_{\\mathbf {\\alpha }}}^{(b)}$ , effectively resulting in a set of $B$ different PCEs.",
"Correspondingly, the response of each PCE can be evaluated at a point $\\mathbf {x}$ as follows: $Y_{PC}^{(b)}(\\mathbf {x}) = \\sum \\limits _{\\mathbf {\\alpha }\\in {\\mathcal {A}}}y^{(b)}_{\\mathbf {\\alpha }}\\,\\Psi _{\\mathbf {\\alpha }}(\\mathbf {x}),$ thus yielding a full response sample at each point $\\left\\lbrace Y_{PC}^{(b)}(\\mathbf {x}), b =1,\\cdots ,B\\right\\rbrace $ .",
"Therefore, empirical quantiles can be employed to provide local error bounds on the PCE prediction at each point, as well as to any derived quantity (e.g.",
"$P_F$ or sensitivity indices, see e.g.",
"[33], [13]).",
"This bootstrap-resampling strategy in Eq.",
"(REF ) yields in fact a family of $B$ surrogate models that can be interpreted as trajectories.",
"Figure showcases how such trajectories can be directly employed to assess confidence bounds on point-wise predictions on a simple 1D test function given by: $f(x) = x\\sin (x),\\qquad x\\in [0, 2\\pi ],$ where the single random variable is assumed to be uniformly distributed within the bounds $X\\sim {\\mathcal {U}}(0,2\\pi ) $ , and where $B = 100$ bootstrap samples have been used.",
"Figure: Bootstrap-resampled trajectories (B=100B = 100) of the simple 1Danalytical function in Eq.",
"().The black line represents the true model, sampled at the 8 experimental designpoints 𝐱 (i) \\mathbf {x}^{(i)} (green dots).The PCE surrogate is represented by the blue line, while the bootstraptrajectories are given by the gray lines.",
"The corresponding 95% empiricalinter quantile-range is given by the dashed blue lines.This process of bootstrap-based trajectory resampling to provide better estimates of point-wise confidence bounds has been recently explored in the Gaussian process modelling literature, see e.g., [8], [41].",
"We refer to this approach as to bootstrap-PCE, or bPCE in short."
],
[
"Fast bPCE",
"Because the training of a PCE model with sparse least-square analysis may be time consuming, especially in high dimension and/or when an already large experimental design is available (i.e.",
"$N\\sim 10^3$ ), and because in this particular application we do not need very accurate estimates on the bounds of the derived quantities, we adopt a fast bPCE approach.",
"In this approach, the sparse polynomial basis identified by the LARS algorithm during calibration is calculated only once from the available full experimental design ${\\mathcal {X}}$ , and bootstrapping is applied only to the final hybrid step, which consists in a classic ordinary least-square regression on the sparse basis [6].",
"In the presence of a very expensive model, however (i.e.",
"requiring several hours for a single model run), we recommended to adopt full bootstrapping, including the estimation of the sparse PCE basis for each of the $B$ bootstrapped experimental designs ${\\mathcal {X}}^{(1,\\cdots ,B)}$ ."
],
[
"Active bPCE-based reliability analysis",
"In this section we present an adaptation of the Adaptive PC-Kriging MCS algorithm in [37] (based in turn on the original AK-MCS algorithm by [14]), that makes use of the bPCE just introduced.",
"Consistently with [14], [37], in the following we will refer to this algorithm as active bootstrap-polynomial-chaos Monte-Carlo simulation (A-bPCE).",
"We follow the original idea of adaptively building a surrogate of the limit-state function starting from a small initial experimental design and subsequently refining it to optimize the surrogate performance for structural reliability.",
"The ultimate goal of the adaptation is to retrieve an estimate of $P_F$ that is comparable to that of a direct Monte Carlo simulation (MCS) using a large sample set with a much smaller experimental design.",
"The algorithm is summarized as follows: Initialization: Generate an initial experimental design (e.g.",
"through Latin hypercube sampling or uniform sampling of a ball [9]) and calculate the corresponding bPCE surrogate (see Section REF ).",
"Generate a large reference MCS sample ${\\mathcal {X}}_{MCS} =\\left\\lbrace \\mathbf {x}_{MCS}^{(i)},~i = 1,\\cdots ,N_{MCS}\\right\\rbrace $ of size $N_{MCS}$ (e.g.",
"$N_{MCS} = 10^6\\gg N$ ).",
"A discussion on the choice of a suitable MCS sample is given in Section ).",
"Calculate a set of MCS estimators of the probability of failure: $\\left\\lbrace \\widehat{P}_F^{(b)},~b=1,\\cdots ,B\\right\\rbrace $ with the current bPCE surrogate.",
"Evaluate one or more suitable convergence criteria on $\\widehat{P}_F^{(1,\\cdots ,B)}$ (see Section REF ).",
"If they are met, go to Step REF (terminate the algorithm).",
"Otherwise continue to the next step.",
"Evaluate a suitable learning function on the MCS sample ${\\mathcal {X}}_{MCS}$ (see Section REF ).",
"Choose one or more additional $\\mathbf {x}_{MCS}^{(i)}\\in {\\mathcal {X}}_{MCS}$ and add them to the ED (see Section REF ).",
"Update the bPCE surrogate on the new ED and return to Step REF Algorithm termination: return the $\\widehat{P}_F$ resulting from the PCE on the current ED, as well as the error bounds derived e.g.",
"from the extremes or the empirical quantiles of the current $\\widehat{P}_F^{(1,\\cdots ,B)}$ set.",
"A detailed description of each step of the algorithm is given in the following sections."
],
[
"Initial experimental design",
"The initial experimental design is usually generated by space-filling sampling techniques of the random vector $\\mathbf {X}$ , such as Latin hypercube sampling (LHS) or pseudo-random sequences (e.g.",
"Sobol' sequence).",
"Alternative sampling techniques, such as the uniform sampling of a ball, have also proven effective in the context of structural reliability when low probabilities of failure are expected [9].",
"Note that this initial set of model evaluations does not need to be a subset of the reference sample ${\\mathcal {X}}_{MCS}$ used later to evaluate the $P_F$ estimates during the iterations of the algorithm."
],
[
"Inner MCS-based estimate of $P_F$",
"While the estimation of the $P_F$ via MCS is trivial, as it simply entails counting the number of samples that belong to the failure domain, some discussion about the number of samples $N_{MCS}$ in this step is needed.",
"Throughout this paper, we opted to choose a single MCS sample ${\\mathcal {X}}_{MCS}$ large enough to ensure a relatively small CoV for the $P_F$ estimate at every iteration.",
"This is by no means a requirement of this algorithm, but it simplifies significantly the notation (because ${\\mathcal {X}}_{MCS}$ becomes independent on the current iteration) and in some cases (as noted in both [14] and [37]) it can result in stabler convergence, due to the lowered MCS noise in the estimation of $P_F$ during each iteration.",
"This technique is known as common random numbers in the context of repeated reliability analysis .e.g.",
"in reliability-based design optimization [39].",
"It is entirely possible to redraw the ${\\mathcal {X}}_{MCS}$ during every iteration, possibly each time with a different number of samples $N_{MCS}$ .",
"The choice of $N_{MCS} = 10^6$ ensures that the CoV estimated probabilities of failure in the order of $P_F\\ge 10^{-3} $ is always smaller than $5\\%$ , which we found suitable in our application examples.",
"The choice of a single MCS sample drawn during the algorithm initialization also allows us to use the application examples to focus more on the convergence of the active learning part of A-bPCE, which is the focus of this paper.",
"In more general applications, the order of magnitude of $P_F$ may be unknown.",
"In this case, it is recommended instead to set a target desired CoV for the estimation of $P_F$ at each iteration (as proposed in the original AK-MCS algorithm in [14]), and gradually add samples to ${\\mathcal {X}}_{MCS}$ until it is reached."
],
[
"Convergence criteria",
"The proposed convergence criterion of choice is directly inspired by [37], [32] and it depends on the stability of the $P_F$ estimate at the current iteration.",
"Let us define: $\\begin{split}\\widehat{P}_F^+ &= \\max \\limits _{b = 1,\\cdots ,B}\\left(\\widehat{P}_F^{(b)}\\right)\\\\\\widehat{P}_F^- &= \\min \\limits _{b = 1,\\cdots ,B}\\left(\\widehat{P}_F^{(b)}\\right).\\end{split}$ Convergence is achieved when the following condition is satisfied for at least two consecutive iterations of the algorithm: $\\frac{\\widehat{P}_F^+ - \\widehat{P}_F^-}{\\widehat{P}_F} \\le \\epsilon _{\\widehat{P}_F},$ with $0.05\\le \\epsilon _{\\widehat{P}_F}\\le 0.15$ in typical usage scenarios."
],
[
"Learning function",
"A learning function is a function that allows one to rank a set of candidate points based on some utility criterion that depends on the desired application.",
"In this case, we adopt the same heuristic approach proposed in [37], by focusing on the probability of misclassification of the bPCE model on the candidate set given by ${\\mathcal {X}}_{MCS}$ .",
"Due to the availability of the bootstrap response samples ${\\mathcal {Y}}_{MCS}^{(1,\\cdots ,B)}$ , it is straightforward to define a measure of the misclassification probability $U_{FBR}$ (where the subscript FBR stands for failed bootstrap replicates) at each point $\\mathbf {x}_{MCS}^{(i)}\\in {\\mathcal {X}}_{MCS}$ as follows: $U_{FBR}(\\mathbf {x}_{MCS}^{(i)}) =\\left|\\frac{B_S(\\mathbf {x}_{MCS}^{(i)})-B_F(\\mathbf {x}_{MCS}^{(i)})}{B}\\right|$ where $B_S(\\mathbf {x}_{MCS}^{(i)})$ and $B_F(\\mathbf {x}_{MCS}^{(i)})$ are the number of safe (resp.",
"failed) bPCE replicate predictions at point $\\mathbf {x}_{MCS}^{(i)}$ (with $B_S(\\mathbf {x}_{MCS}^{(i)}) + B_F(\\mathbf {x}_{MCS}^{(i)}) = B$ ).",
"When all the $B$ replicates consistently classify $\\mathbf {x}_{MCS}^{(i)}$ in the safe or in the failure domain, $U_{FBR} = 1$ (minimum misclassification probability).",
"In contrast, $U_{FBR} = 0$ corresponds to the case when the replicates are equally distributed between the two domains.",
"In the latter case, 50% of the $B$ bootstrap PCEs predict that $\\mathbf {x}_{MCS}^{(i)}$ is in the safe domain, while the other 50% predicts that $\\mathbf {x}_{MCS}^{(i)}$ belongs to the failure domain.",
"Therefore, maximum epistemic uncertainty on the classification of a point $\\mathbf {x}^{(i)}_{MCS}$ is attained when $U_{FBR}$ is minimum."
],
[
"Enrichment of the experimental design",
"The aim of the iterative algorithm described in Section REF is to obtain a surrogate model that minimizes the misclassification probability.",
"As a consequence, the learning function in Eq.",
"(REF ) can be directly used to obtain a single-point enrichment criterion.",
"The next best candidate point for the ED $\\mathbf {x}^*\\in {\\mathcal {X}}_{MCS}$ is given by: $\\mathbf {x}^* = \\operatornamewithlimits{argmin}\\limits _{\\mathbf {x}^{(i)} \\in {\\mathcal {X}}_{MCS}}\\left(U_{FBR}(\\mathbf {x}^{(i)})\\right).$ Due to the global character of regression-based PCE, it can be beneficial to add multiple points in each iteration to sample several interesting regions of the parameter space simultaneously.",
"The criterion in Eq.",
"(REF ) can be extended to include $K$ distinct points simultaneously by following the approach in [37].",
"A limit state margin region is first defined as the set of points such that $U_{FBR}<1$ (i.e.",
"those point with non-zero misclassification probability at the current iteration).",
"Subsequently, $k$ -means clustering techniques (see, e.g., [44]) can be used at each iteration to identify $K$ disjoint regions $\\left\\lbrace {\\mathcal {X}}^{(1,\\cdots ,K)}_{MCS}\\right\\rbrace $ in the limit-state margin.",
"Then, Eq.",
"(REF ) can be directly applied to each of the subregions to obtain $K$ different enrichment points: $\\mathbf {x}^{*}_k = \\operatornamewithlimits{argmin}\\limits _{\\mathbf {x}^{(i)} \\in {\\mathcal {X}}_{MCS}^{(k)}}\\left(U^{(k)}_{FBR}(\\mathbf {x}_k^{(i)})\\right), \\quad k =1,\\cdots ,K$ where $\\mathbf {x}^{*}_k\\in {\\mathcal {X}}^{(k)}_{MCS}$ is the $k-$ th enrichment sample and $U^{(k)}_{FBR}$ is the learning function evaluated on the $k-$ th region of the parameter space.",
"Note that this approach is also convenient when parallel computing facilities are available and in the presence of computationally expensive objective functions, as the evaluation of the $K$ enrichment points can be carried out simultaneously."
],
[
"Results on benchmark applications",
"All the algorithm development and the final calculations presented in this section were performed with the polynomial chaos expansions and reliability analysis modules of the UQLab software for uncertainty quantification [28], [29], [27]."
],
[
"Series system",
"A common benchmark for reliability analysis functions is given by the four-branch function, originally proposed in [42], that represents a series system comprising four components with different failure criteria.",
"Although it is a simple analytical function, it shows multiple failure regions and a composite limit-state surface.",
"Its two-dimensional limit state function reads: $g(\\mathbf {x}) = \\min \\left\\lbrace \\begin{matrix}3+0.1(x_1+x_2)^2 - \\frac{x_1 + x_2}{\\sqrt{2}}\\\\3+0.1(x_1+x_2)^2 + \\frac{x_1 + x_2}{\\sqrt{2}}\\\\(x_1-x_2) + \\frac{6}{\\sqrt{2}}\\\\(x_2-x_1) + \\frac{6}{\\sqrt{2}}\\\\\\end{matrix} \\right\\rbrace $ where the two random input variables $X_1\\sim {\\mathcal {N}}(0,1)$ and $X_2\\sim {\\mathcal {N}}(0,1)$ are modelled as independent standard normals.",
"Failure occurs when $g(\\mathbf {x})\\le 0$ .",
"Due to the multi-failure shape of the limit-state surface (represented as a solid black line in Figure REF ), classic methods like FORM/SORM and importance sampling tend to fail with this benchmark problem.",
"The reference failure probability of $P_F = 4.460\\cdot 10^{-3}$ is obtained through an extremely large MCS ($N_{MCS} = 10^8$ ).",
"Figure: Four branch function: limit state surface (black line) andexperimental design before (black circles) and after (red crosses)enrichmentThe initial experimental design for the A-bPCE algorithm was obtained with a space-filling LHS sample consisting of $N_{ini} = 20$ points drawn from the input distributions (black dots in Figure REF ).",
"Three points at a time were added to the experimental design during the enrichment phase of the algorithm.",
"The number of replications for the A-bPCE algorithm is set to $B = 100$ .",
"After extensive testing, the algorithm was found to be very weakly dependent on the number of bootstrap replications, provided a minimum of $B \\ge 20$ was provided.",
"Indeed, the boostrap samples are used to identify areas of relatively large prediction variability, but an accurate estimate of such variability is never really needed by the algorithm.",
"Degree adaptive sparse PCE (with maximum degree in the range $p\\in [2, 10]$ ) based on LARS [6] was used to calibrate the PCE metamodel at each iteration.",
"For validation and comparison purposes, a similar analysis was performed on the same initial ED with the AK-MCS module of UQLab, with an anisotropic Matérn 5/2 ellipsoidal multivariate Kriging correlation function [36], [27], [22].",
"The convergence criterion in Eq.",
"(REF ) was set to $\\epsilon _{\\widehat{P}_F} = 0.05$ for both the AK-MCS and A-bPCE algorithms.",
"Convergence was achieved after 49 iterations, resulting in a total cost (including the initial experimental design) of $N_{tot} = 185$ model evaluations.",
"The experimental design points added during the iterations are marked by red crosses on panel (b) of Figure REF .",
"As expected, the adaptive algorithm tends to enrich the experimental design close to the limit state surface as it is adaptively learned during the iterations.",
"A graphical representation of the convergence of the algorithm is shown in Figure REF , where the estimated $\\widehat{P}_F$ is plotted against the total number of model evaluations $N_{tot}$ .",
"The shaded area represents the $95\\%$ confidence bounds based on the empirical quantiles as estimated from the bootstrap sample.",
"Figure: Convergence curves of the four-branchlimit-state function.",
"The reference P F P_F and β\\beta are given by dotted lines.The final results of the analysis are summarized in Table , where the generalised reliability index $\\beta = -\\Phi ^{-1}(P_F)$ is also given for reference.",
"For comparison, the reference MCS probability as well as an estimate from AK-MCS are also given.",
"The latter converged to a comparably accurate estimate of $P_F$ , at the cost of a slightly higher number of model evaluations.",
"Note that for engineering purposes, the algorithm could have been stopped earlier, i.e.",
"when a $5\\%$ accuracy on the generalized reliability index is attained.",
"In this case, the algorithm would have converged to a comparable result ($\\widehat{\\beta } = 2.56$ ) with only 50 runs of the model.",
"The final sparse PCE model after enrichment contained a total of $P = 12$ basis elements of degree up to $p = 5$ .",
"Table: Comparison of different reliability analysis methods for the fourbranch function"
],
[
"Two-dimensional truss structure",
"To test the algorithm on a more realistic engineering benchmark, consider the two-dimensional truss structure sketched in Figure REF .",
"This structure has been previously analysed in several works, see e.g.",
"[6], [5], [37].",
"The truss comprises 23 bars and 13 nodes, with deterministic geometry yet with uncertain material properties and random loads.",
"The components of the input random vector $\\mathbf {X}=\\left[A_{1,2},E_{1,2},P_{1,...,6}\\right]^{\\textsf {T}}$ include the cross-section and the Young's modulus $\\left\\lbrace A_1, E_1\\right\\rbrace $ of the horizontal bars, the cross-section and the Young's modulus $\\left\\lbrace A_2, E_2\\right\\rbrace $ of the diagonal bars and the six random loads $\\left\\lbrace P_1,\\cdots ,P_6\\right\\rbrace $ .",
"They are considered mutually independent and their distributions are given in Table REF .",
"An in-house developed Matlab-based finite-element solver is used to calculate the displacement at midspan $u(\\mathbf {X})$ , counted positively downwards.",
"Figure: Two-dimensional truss structure with uncertain parameters.Probabilitydistributions of the geometrical parameters A i ,E i \\left\\lbrace A_i,E_i\\right\\rbrace and of theloadsP 1 ,⋯,P 6 \\left\\lbrace P_1,\\cdots ,P_6\\right\\rbrace are given in Table Table: Two-dimensional truss structure: definition of the probabilisticmodel of the input variables This structure operates in the nominal range as long as the midspan displacement is smaller than a critical threshold $\\tau = 12$ cm, which can be cast as the following limit-state function: $g(\\mathbf {x}) = \\tau - u(\\mathbf {x})$ where $g(\\mathbf {x}) \\le 0$ if the system is in a failure state.",
"Because the FEM computational model is relatively cheap to evaluate, we could run a direct MCS-analysis with $N = 10^6$ samples to provide the reference $P_F= 1.52\\cdot 10^{-3}$ for validation purposes.",
"Additionally, standard FORM and SORM analyses were run to estimate the non-linearity of the limit-state surface.",
"FORM underestimated the failure probability of a factor of almost 2 and a cost of $N_{FORM} = 160$ model runs, while SORM achieved a good accuracy at a cost of $N_{SORM} = 372$ model runs, which suggests that the underlying problem is non-linear.",
"Neither of the two methods, however, provides confidence interval on their estimates.",
"The A-bPCE algorithm was initialized with an experimental design consisting in a uniform sampling of a ball (for details, see e.g.",
"[9]) of size $N_{ini} = 30$ , while the sparse adaptive PCE was given a polynomial degree range $1 \\le p \\le 10$ , hyperbolic truncation with q-norm $q = 0.75$ [6] and maximum allowed interaction $r = 2$ [29].",
"The internal MCS sample size was $N_{MCS} = 10^6$ and the algorithm was set to add $K = 3$ new samples per iteration.",
"The stopping criterion in Eq.",
"(REF ) was set to $\\epsilon _{\\widehat{P}_F} = 0.10$ .",
"For comparison purposes, we also ran a standard AK-MCS analysis with the same initial experimental design and convergence criterion.",
"The covariance family of choice for the underlying Kriging model was chosen as Gaussian.",
"Table: Comparison of the estimation of P F P_F with several algorithms forthe truss structure example.Table REF presents a comparison of the estimated $\\widehat{P}_F$ with the aforementioned analyses.",
"Both AK-MCS and A-bPCE estimates of $P_F$ include the reference value within the confidence bounds set by the convergence criterion.",
"However, for this particular example and choice of convergence criterion, A-bPCE achieved convergence significantly faster than AK-MCS, with a total cost of 129 model evaluations, as compared to the 300 required by AK-MCS, resulting in a final PCE of degree $p=3$ with $P =43$ basis elements.",
"Overall, A-bPCE provides a stable estimate of the failure probability and confidence intervals at a cost that is lower than FORM for this example."
],
[
"Top-floor displacement of a structural frame",
"Figure REF shows a well known, high dimensional benchmark in structural reliability applications [26], [4].",
"It consists on a three-span, five story frame structure that is subject to horizontal loads.",
"Both the loads and the properties of the elements of the frame (see Table REF ) are uncertain.",
"Of interest is the top-floor horizontal displacement at the top right corner $u$ .",
"Figure: 20-dimensional structural frame inSection .The distributions of the input variables are reported inTable Table: Frame structure: properties of the elements shown inFigure The uncertainties on the applied loads $P_1,...,P_3$ , the Young's moduli $E_1$ and $E_2$ , the moments of inertia $I_6,...,I_{13}$ and the cross sections $A_{14},...,A_{21}$ are modelled by a 21-dimensional joint random vector $\\mathbf {Z} = \\left\\lbrace P_1,...,A_{21}\\right\\rbrace $ with marginal distributions given in Table REF .",
"Table: Frame structure: definition of the probabilisticmodel of the input variables Additionally, a Gaussian copula [24] is used to model dependence between the variables.",
"The elements of the Gaussian copula correlation matrix $\\mathbf {R}$ are given as: $R_{E_1,E_2} = 0.9$ – the two Young's moduli are highly correlated; $R_{A_i,I_i} = 0.95$ – each element's cross-sectional area is highly correlated to the corresponding moment of inertia; ${R_{A_i,I_j} = R_{I_i,I_j} = R_{A_i,A_j} = 0.13}$ – the correlation between the properties of different elements is much lower; All the remaining elements of $\\mathbf {R}$ are set to 0.",
"A critical displacement of $\\tau = 5$ cm is identified as the maximum admissible threshold for the displacement $u$ , hence resulting in the limit-state function: $g(\\mathbf {z}) = \\tau - u(\\mathbf {z})$ where $u(\\mathbf {z})$ is the displacement on the top right corner calculated with an in-house FEM code.",
"Due to the associated computational costs, the maximum available budget for the calculation of a reference solution is in this case limited to $N_{REF} = 4\\cdot 10^4$ .",
"Therefore, the reference solution is calculated with standard importance sampling (IS) [30] instead of direct MCS.",
"In addition to Importance sampling, we also ran FORM and SORM.",
"Due to the non-linearity of the problem, FORM significantly underestimated $P_F$ , while SORM provided an accurate estimate.",
"However, due to the high dimensionality of the input space, the associated cost in terms of model evaluation was relatively high, with $N_{SORM} = 1146$ model runs, since all the gradients of the limit-state function are computed using finite-differences.",
"The A-bPCE algorithm was initialized with an experimental design consisting of an LHS sampling of the input random vector of size $N_{ini} = 40$ .",
"Sparse PCE was carried out with a q-norm truncation with $q = 0.75$ and maximum allowed interaction $r = 2$ .",
"Note that the initialization is essentially the same as for the truss structure in the previous application.",
"The internal MCS sample size was $N_{MCS} = 10^6$ , with single point enrichment per iteration.",
"The stopping criterion in Eq.",
"(REF ) was in this case set to $\\epsilon _{\\widehat{P}_F} = 0.15 $ .",
"For comparison purposes, an AK-MCS analysis was also run on the same initial design, with similar settings and a Gaussian covariance family.",
"A comparison of the results is gathered in Table REF .",
"Due to the different estimation method between the reference probability (importance sampling) and the active learning-based methods (which rely on an inner MCS), no direct comparison of the results is possible as in the previous cases.",
"Indeed, even fixing the same random seeds would result in different estimates due to the different methodologies.",
"Therefore, confidence bounds are given for all the three methods: 95% confidence bounds for IS [30], and $P_F^{\\pm }$ for both AK-MCS and A-bPCE.",
"The three methods give comparable results, albeit with significant differences in the convergence behaviour.",
"In particular, both AK- and A-bPCE resulted in a slight underestimation of the probability of failure w.r.t.",
"the reference solution by IS, which in turn is slightly overestimated with respect to the reference result quoted in the literature [4].",
"However, AK-MCS did not converge in the allotted maximum number of model evaluations, and its confidence bounds remained remarkably large with respect to A-bPCE.",
"A-bPCE converged instead at a total cost of approximately 200 model evaluations to the target $\\epsilon _{\\widehat{P}_F}$ , with a final sparse PCE of degree 2, counting 30 non-zero coefficients.",
"For both active-learning-based methods, the reference solution lies within the given confidence bounds.",
"Moreover, the confidence bounds on the reliability index $\\widehat{\\beta }$ show that the results are stable to within $2\\%$ of the calculated values.",
"Finally, it is interesting to mention that for this example the costs of FORM and A-bPCE were comparable, but the latter provides a much less biased estimate, and includes confidence bounds.",
"Table: Comparison of the estimation of P F P_F with several algorithms forthe top-floor displacement of a structural frame example"
],
[
"Conclusions and outlook",
"A novel approach to solving reliability problems with polynomial chaos expansions has been proposed.",
"The combination of the bootstrap method and sparse regression enabled us to introduce local error estimation in the standard PCE predictor.",
"In turn, this allows one to construct active learning algorithms similar to AK-MCS to greedily enrich a relatively small initial experimental design so as to efficiently estimate the probability of failure of complex systems.",
"This approach has shown comparable performance w.r.t.",
"to the well established AK-MCS method on both a simple analytical benchmark function and in two high-dimensional engineering applications of increasing complexity.",
"Extensions of this approach can be envisioned in two main directions: the simulation-based reliability analysis method can be extended beyond simple MCS (e.g.",
"by using importance sampling [12], line sampling [34] or subset simulation [11]) to achieve better $\\widehat{P}_F$ estimates at each iteration, especially for very low probabilities of failure; remote parallel computing facilities may be used during the enrichment phase of the algorithms with expensive computational models when adding more than one point at a time; the use of bootstrap to enable local error estimation in an active learning context can be used also with different regression-based surrogate modelling techniques, including e.g.",
"low rank tensor approximations [21].",
"Additionally, the bPCE approach itself introduced in this work can be used also outside of a pure reliability analysis context, as it provides an effective local error estimate for PCE.",
"It has been used, e.g.",
"in the context of reliability-based design optimization in [31].",
"Indeed the lack of this feature (as opposed to Kriging) has somewhat hindered its usage in more advanced active-learning applications."
]
] | 1709.01589 |
[
[
"Deep and Confident Prediction for Time Series at Uber"
],
[
"Abstract Reliable uncertainty estimation for time series prediction is critical in many fields, including physics, biology, and manufacturing.",
"At Uber, probabilistic time series forecasting is used for robust prediction of number of trips during special events, driver incentive allocation, as well as real-time anomaly detection across millions of metrics.",
"Classical time series models are often used in conjunction with a probabilistic formulation for uncertainty estimation.",
"However, such models are hard to tune, scale, and add exogenous variables to.",
"Motivated by the recent resurgence of Long Short Term Memory networks, we propose a novel end-to-end Bayesian deep model that provides time series prediction along with uncertainty estimation.",
"We provide detailed experiments of the proposed solution on completed trips data, and successfully apply it to large-scale time series anomaly detection at Uber."
],
[
"Introduction",
"Accurate time series forecasting and reliable estimation of the prediction uncertainty are critical for anomaly detection, optimal resource allocation, budget planning, and other related tasks.",
"This problem is challenging, especially during high variance segments (e.g., holidays, sporting events), because extreme event prediction depends on numerous external factors that can include weather, city population growth, or marketing changes (e.g., driver incentives) [1] that all contribute to the uncertainty of the forecast.",
"These exogenous variables, however, are difficult to incorporate in many classical time series models, such as those found in the standard $R$ forecast[2] package.",
"In addition, these models usually require manual tuning to set model and uncertainty parameters.",
"Relatively recently, time series modeling based on the Long Short Term Memory (LSTM) model [3] has gained popularity due to its end-to-end modeling, ease of incorporating exogenous variables, and automatic feature extraction abilities [4].",
"By providing a large amount of data across numerous dimensions, it has been shown that an LSTM network can model complex nonlinear feature interactions [5], which is critical for modeling complex extreme events.",
"A recent paper [6] has shown that a neural network forecasting model is able to outperform classical time series methods in cases with long, interdependent time series.",
"However, the problem of estimating the uncertainty in time-series predictions using neural networks remains an open question.",
"The prediction uncertainty is important for assessing how much to trust the forecast produced by the model, and has profound impact in anomaly detection.",
"The previous model proposed in [6] had no information regarding the uncertainty.",
"Specifically, this resulted in a large false anomaly rates during holidays where the model prediction has large variance.",
"In this paper, we propose a novel end-to-end model architecture for time series prediction, and quantify the prediction uncertainty using Bayesian Neural Network, which is further used for large-scale anomaly detection.",
"Recently, Bayesian neural networks (BNNs) have garnered increasing attention as a principled framework to provide uncertainty estimation for deep models.",
"Under this framework, the prediction uncertainty can be decomposed into three types: model uncertainty, inherent noise, and model misspecification.",
"Model uncertainty, also referred to as epistemic uncertainty, captures our ignorance of the model parameters, and can be reduced as more samples being collected.",
"Inherent noise, on the other hand, captures the uncertainty in the data generation process and is irreducible.",
"These two sources have been previously recognized with successful application in computer visions [7].",
"The third uncertainty from model misspecification, however, has been long-overlooked.",
"This captures the scenario where the testing samples come from a different population than the training set, which is often the case in time series anomaly detection.",
"Similar ideas have gained attention in deep learning under the concept of adversarial examples in computer vision [8], but its implication in prediction uncertainty remains unexplored.",
"Here, we propose a principled solution to incorporate this uncertainty using an encoder-decoder framework.",
"To the best of our knowledge, this is the first time that misspecification uncertainty has been successfully applied to prediction and anomaly detection in a principled way.",
"In summary, this paper makes the following contributions: Provides a generic and scalable uncertainty estimation implementation for deep prediction models.",
"Quantifies the prediction uncertainty from three sources: (i) model uncertainty, (ii) inherent noise, and (iii) model misspecification.",
"The third uncertainty has been previously overlooked, and we propose a potential solution with an encoder-decoder.",
"Motivates a real-world anomaly detection use-case at Uber that uses Bayesian Neural Networks with uncertainty estimation to improve performance at scale.",
"The rest of this paper is organized as follows: Section gives an overview of previous work on time series prediction for both classical and deep learning models, as well as the various approaches for uncertainty estimation in neural networks.",
"The approach of Monte Carlo dropout (MC dropout) is used in this paper due to its simplicity, strong generalization ability, and scalability.",
"In Section , we present our uncertainty estimation algorithm that accounts for the three different sources of uncertainty.",
"Section provides detailed experiments to evaluate the model performance on Uber trip data, and lays out a successful application to large-scale anomaly detection for millions of metrics at Uber.",
"Finally, Section concludes the paper.",
"Classical time series models, such as those found in the standard $R$ forecast[2] package are popular methods to provide an univariate base-level forecast.",
"These models usually require manual tuning to set seasonality and other parameters.",
"Furthermore, while there are time series models that can incorporate exogenous variables [9], they suffer from the curse of dimensionality and require frequent retraining.",
"To more effectively deal with exogenous variables, a combination of univariate modeling and a machine learning model to handle residuals was introduced in [10].",
"The resulting two-stage model, however, is hard to tune, requires manual feature extraction and frequent retraining, which is prohibitive to millions of time series.",
"Relatively recently, time series modeling based on LSTM [3] technique gained popularity due to its end-to-end modeling, ease of incorporating exogenous variables, and automatic feature extraction abilities [4].",
"By providing a large amount of data across numerous dimensions, it has been shown that an LSTM approach can model complex extreme events by allowing nonlinear feature interactions [5], [6].",
"While uncertainty estimation for classical forecasting models has been widely studied [11], this is not the case for neural networks.",
"Approaches such as a modified loss function or using a collection of heterogenous networks [12] were proposed, however they require changes to the underlying model architecture.",
"A more detailed review is given in the next section.",
"In this work, we use a simple and scalable approach for deep model uncertainty estimation that builds on [13].",
"This framework provides a generic error estimator that runs in production at Uber-scale to mitigate against bad decisions (e.g., false anomaly alerts) resulting from poor forecasts due to high prediction variance."
],
[
"Bayesian Neural Networks",
"Bayesian Neural Networks (BNNs) introduce uncertainty to deep learning models from a Bayesian perspective.",
"By giving a prior to the network parameters $W$ , the network aims to find the posterior distribution of $W$ , instead of a point estimation.",
"This procedure is usually referred to as posterior inference in traditional Bayesian models.",
"Unfortunately, due to the complicated non-linearity and non-conjugacy in deep models, exact posterior inference is rarely available; in addition, most traditional algorithms for approximate Bayesian inference cannot scale to the large number of parameters in most neural networks.",
"Recently, several approximate inference methods are proposed for Bayesian Neural Networks.",
"Most approaches are based on variational inference that optimizes the variational lower bound, including stochastic search [14], variational Bayes [15], probabilistic backpropagation [16], Bayes by BackProp [17] and its extension [18].",
"Several algorithms further extend the approximation framework to $\\alpha $ -divergence optimization, including [19], [20].",
"We refer the readers to [21] for a more detailed and complete review of these methods.",
"All of the aforementioned algorithms require different training methods for the neural network.",
"Specifically, the loss function must be adjusted to different optimization problems, and the training algorithm has to be modified in a usually non-trivial sense.",
"In practice, however, an out-of-the-box solution is often preferred, without changing the neural network architecture and can be directly applied to the previously trained model.",
"In addition, most existing inference algorithms introduce extra model parameters, sometimes even double, which is difficult to scale given the large amount of parameters used in practice.",
"This paper is inspired by the Monte Carlo dropout (MC dropout) framework proposed in [13] and [22], which requires no change of the existing model architecture and provides uncertainty estimation almost for free.",
"Specifically, stochastic dropouts are applied after each hidden layer, and the model output can be approximately viewed as a random sample generated from the posterior predictive distribution [21].",
"As a result, the model uncertainty can be estimated by the sample variance of the model predictions in a few repetitions.",
"Details of this algorithm will be reviewed in the next section.",
"The MC dropout framework is particularly appealing to practitioners because it is generic, easy to implement, and directly applicable to any existing neural networks.",
"However, the exploration of its application to real-world problems remains extremely limited.",
"This paper takes an important step forward by successfully adapting this framework to conduct time series prediction and anomaly detection at large scale."
],
[
"Method",
"Given a trained neural network $f^{\\hat{W}}(\\cdot )$ where $\\hat{W}$ represents the fitted parameters, as well as a new sample $x^*$ , our goal is to evaluate the uncertainty of the model prediction, $\\hat{y}^* = f^{\\hat{W}}(x^*)$ .",
"Specifically, we would like to quantify the prediction standard error, $\\eta $ , so that an approximate $\\alpha $ -level prediction interval can be constructed by $[\\hat{y}^* - z_{\\alpha /2} \\eta , ~ \\hat{y}^* + z_{\\alpha /2} \\eta ]$ where $z_{\\alpha /2}$ is the upper $\\alpha /2$ quantile of a standard Normal.",
"This prediction interval is critical for various tasks.",
"For example, in anomaly detection, anomaly alerts will be fired when the observed value falls outside the constructed 95% interval.",
"As a result, underestimating $\\eta $ will lead to high false positive rates.",
"In the rest of this section, we will present our uncertainty estimation algorithm in Section REF , which accounts for three different sources of prediction uncertainties.",
"This framework can be generalized to any neural network architectures.",
"Then, in Section REF , we will present our neural network design for predicting time series at Uber."
],
[
"Prediction Uncertainty",
"We denote a neural network as function $f^W(\\cdot )$ , where $f$ captures the network architecture, and $W$ is the collection of model parameters.",
"In a Bayesian neural network, a prior is introduced for the weight parameters, and the model aims to fit the optimal posterior distribution.",
"For example, a Gaussian prior is commonly assumed: $W \\sim N(0, I)$ We further specify the data generating distribution $p(y \\,|\\, f^W(x))$ .",
"In regression, we often assume $ y\\,|\\, W \\sim N(f^W(x), \\sigma ^2) $ with some noise level $\\sigma $ .",
"In classification, the softmax likelihood is often used.",
"For time series prediction, we will focus on the regression setting in this paper.",
"Given a set of $N$ observations $X=\\lbrace x_1, ..., x_N\\rbrace $ and $Y=\\lbrace y_1, ..., y_N\\rbrace $ , Bayesian inference aims at finding the posterior distribution over model parameters $p(W \\,|\\, X, Y)$ .",
"With a new data point $x^*$ , the prediction distribution is obtained by marginalizing out the posterior distribution: $ p(y^* \\,|\\, x^*) = \\int _W p(y^* \\,|\\, f^W(x^*)) p(W \\,|\\, X, Y)\\, dW $ In particular, the variance of the prediction distribution quantifies the prediction uncertainty, which can be further decomposed using law of total variance: $\\begin{split}\\textrm {Var}(y^* \\,|\\, x^*) & =\\textrm {Var}\\left[ \\mathbb {E}(y^* \\,|\\, W, x^*) \\right] +\\mathbb {E}\\left[\\textrm {Var}(y^* \\,|\\, W, x^*) \\right] \\\\& = \\textrm {Var}(f^W(x^*)) + \\sigma ^2\\end{split}$ Immediately, we see that the variance is decomposed into two terms: (i) $\\textrm {Var}(f^W(x^*))$ , which reflects our ignorance over model parameter $W$ , referred to as the model uncertainty; and (ii) $\\sigma ^2$ which is the noise level during data generating process, referred to as the inherent noise.",
"An underlying assumption for (REF ) is that $y^*$ is generated by the same procedure.",
"However, this is not always the case in practice.",
"In anomaly detection, in particular, it is expected that certain time series will have unusual patterns, which can be very different from the trained model.",
"Therefore, we propose that a complete measurement of prediction uncertainty should be a combination from three sources: (i) model uncertainty, (ii) model misspecification, and (iii) inherent noise level.",
"The following sections provide details on how we handle these three terms.",
"The key to estimating model uncertainty is the posterior distribution $p(W\\,|\\, X, Y)$ , also referred to as Bayesian inference.",
"This is particularly challenging in neural networks because the non-conjugacy due to nonlinearities.",
"There have been various research efforts on approximate inference in deep learning (see Section REF for a review).",
"Here, we follow the idea in [13] and [22] to approximate model uncertainty using Monte Carlo dropout (MC dropout).",
"The algorithm proceeds as follows: given a new input $x^*$ , we compute the neural network output with stochastic dropouts at each layer.",
"That is, randomly dropout each hidden unit with certain probability $p$ .",
"This stochastic feedforward is repeated $B$ times, and we obtain $\\lbrace \\hat{y}^*_{(1)}, ..., \\hat{y}^*_{(B)}\\rbrace $ .",
"Then the model uncertainty can be approximated by the sample variance: $\\widehat{\\textrm {Var}}(f^W(x^*)) = \\frac{1}{B} \\sum _{b=1}^B \\left(\\hat{y}^*_{(b)} - \\overline{\\hat{y}}^* \\right)^2$ where $\\overline{\\hat{y}}^* = \\frac{1}{B} \\sum _{b=1}^B \\hat{y}^*_{(b)}$ [13].",
"There has been recent work done on choosing the optimal dropout probability $p$ adaptively by treating it as part of the model parameter, but this approach requires modifying the training phase [12].",
"In practice, we find that the uncertainty estimation is usually robust within a reasonable range of $p$ ."
],
[
"Model misspecification",
"Next, we address the problem of capturing potential model misspecification.",
"In particular, we would like to capture the uncertainty when predicting unseen samples with very different patterns from the training data set.",
"We propose to account for this source of uncertainty by introducing an encoder-decoder to the model framework.",
"The idea is to train an encoder that extracts the representative features from a time series, in the sense that a decoder can reconstruct the time series from the encoded space.",
"At test time, the quality of encoding of each sample will provide insight on how close it is to the training set.",
"Another way to think of this approach is that we first fit a latent embedding space for all training time series using an encoder-decoder framework.",
"Then, we measure the distance between test cases and training samples in the embedded space.",
"The next question is how to incorporate this uncertainty in the variance calculation.",
"Here, we take a principled approach by connecting the encoder, $g(\\cdot )$ , with a prediction network, $h(\\cdot )$ , and treat them as one large network $f = h(g(\\cdot ))$ during inference.",
"REF illustrates such an inference network, and Algorithm REF presents the MC dropout algorithm.",
"Specifically, given an input time series $x = \\lbrace x_1, ..., x_T\\rbrace $ , the encoder $g(\\cdot )$ constructs the learned embedding $e = g(x)$ , which is further concatenated with external features, and the final vector is fed to the final prediction network $h$ .",
"During this feedforward pass, MC dropout is applied to all layers in both the encoder $g$ and the prediction network $h$ .",
"As a result, the random dropout in the encoder perturbs the input intelligently in the embedding space, which accounts for potential model misspecification and gets further propagated through the prediction network.",
"Here, variational dropout for recurrent neural networks [22] is applied to the LSTM layers in the encoder, and regular dropout [13] is applied to the prediction network.",
"[H] [1] data $x^*$ , encoder $g(\\cdot )$ , prediction network $h(\\cdot )$ , dropout probability $p$ , number of iterations $B$ prediction $\\hat{y}^*_{mc}$ , uncertainty $\\eta _1$ $b = 1$ to $B$ $e^*_{(b)} \\leftarrow $ VariationalDropout$(g(x^*), p)$ $z^*_{(b)} \\leftarrow \\textrm {Concatenate}(e^*_{(b)}, \\textrm {extFeatures})$ $\\hat{y}^*_{(b)} \\leftarrow $ Dropout $(h(z^*_{(b)}), p)$ // prediction $\\hat{y}^*_{mc} \\leftarrow \\frac{1}{B} \\sum _{b=1}^B \\hat{y}^*_{(b)}$ // model uncertainty and misspecification $\\eta _1^2 \\leftarrow \\frac{1}{B} \\sum _{b=1}^B (\\hat{y}^*_{(b)} - \\hat{y}^* )^2$ $\\hat{y}^*_{mc}, \\, \\eta _1$ MCdropout"
],
[
"Inherent noise",
"Finally, we estimate the inherent noise level $\\sigma ^2$ .",
"In the original MC dropout algorithm [13], this parameter is implicitly determined by a prior over the smoothness of $W$ .",
"As a result, the model could end up with drastically different estimations of the uncertainty level depending on this pre-specified smoothness (see [21], chapter 4).",
"This dependency is undesirable in anomaly detection, because we want the uncertainty estimation to also have robust frequentist coverage, but it is rarely the case that we would know the correct noise level a priori.",
"Here, we propose a simple and adaptive approach that estimates the noise level via the residual sum of squares, evaluated on an independent held-out validation set.",
"Specifically, let $f^{\\hat{W}}(\\cdot )$ be the fitted model on training data, and $X^{\\prime }=\\lbrace x^{\\prime }_1, ..., x^{\\prime }_V\\rbrace , Y^{\\prime }=\\lbrace y^{\\prime }_1, ..., y^{\\prime }_V\\rbrace $ be an independent validation set, then we estimate $\\sigma ^2$ via $\\hat{\\sigma }^2 = \\frac{1}{V} \\sum _{v=1}^V \\left( y^{\\prime }_v - f^{\\hat{W}}(x^{\\prime }_v) \\right)^2 \\,.$ Note that $(X^{\\prime }, Y^{\\prime })$ are independent from $f^{\\hat{W}}(\\cdot )$ , and if we further assume that $f^{\\hat{W}}(x^{\\prime }_v)$ is an unbiased estimation of the true model, we have $\\begin{split}\\mathbb {E}(\\hat{\\sigma }^2) &=\\sigma ^2 + \\frac{1}{V} \\sum _{v=1}^V \\mathbb {E} \\left[ f^{\\hat{W}}(x^{\\prime }_v) - f^{W}(x^{\\prime }_v) \\right]^2 \\\\&= \\sigma ^2 + \\textrm {Var}_{\\rm TRN}(f^{\\hat{W}}(x^{\\prime }_v))\\end{split}$ where $\\textrm {Var}_{\\rm TRN}$ is w.r.t the training data, which decreases as the training sample size increases, and $\\rightarrow 0$ as the training sample size $N \\rightarrow \\infty $ .",
"Therefore, $\\hat{\\sigma }^2$ provides an asymptotically unbiased estimation on the inherent noise level.",
"In the finite sample scenario, it always overestimates the noise level and tends to be more conservative.",
"The final inference algorithm combines inherent noise estimation with MC dropout, and is presented in Algorithm REF .",
"[H] [1] data $x^*$ , encoder $g(\\cdot )$ , prediction network $h(\\cdot )$ , dropout probability $p$ , number of iterations $B$ prediction $\\hat{y}^*$ , predictive uncertainty $\\eta $ // prediction, model uncertainty and misspecification $\\hat{y}^*, \\, \\eta _1 \\leftarrow $ MCdropout $(x^*, g, h, p, B)$ // Inherent noise $x^{\\prime }_v$ in validation set $\\lbrace x^{\\prime }_1, ..., x^{\\prime }_V\\rbrace $ $\\hat{y^{\\prime }}_v \\leftarrow h(g(x^{\\prime }_v))$ $\\eta _2^2 \\leftarrow \\frac{1}{V} \\sum _{v=1}^V \\left( \\hat{y^{\\prime }}_v - y^{\\prime }_v \\right)^2$ // total prediction uncertainty $\\eta \\leftarrow \\sqrt{\\eta _1^2 + \\eta _2^2}$ $\\hat{y}^*, \\, \\eta $ Inference"
],
[
"Model Design",
"The complete architecture of the neural network is shown in REF .",
"The network contains two major components: (i) an encoder-decoder framework that captures the inherent pattern in the time series, which is learned during pre-training step, and (ii) a prediction network that takes input from both the learned embedding from encoder-decoder, as well as any potential external features to guide the prediction.",
"We discuss the two components in more details below.",
"Figure: Neural network architecture, with a pre-training phase using a LSTM encoder-decoder, followed by a prediction network, with input being the learned embedding concatenated with external features."
],
[
"Encoder-decoder",
"Prior to fitting the prediction model, we first conduct a pre-training step to fit an encoder that can extract useful and representative embeddings from a time series.",
"The goals are to ensure that (i) the learned embedding provides useful features for prediction and (ii) unusual input can be captured in the embedded space, which will get further propagated to the prediction network in the next step.",
"Here, we use an encoder-decoder framework with two-layer LSTM cells.",
"Specifically, given a univariate time series $\\lbrace x_t\\rbrace _t$ , the encoder reads in the first $T$ timestamps $\\lbrace x_1, ..., x_T\\rbrace $ , and constructs a fixed-dimensional embedding state.",
"After then, from this embedding state, the decoder constructs the following $F$ timestamps $\\lbrace x_{T+1}, ..., x_{T+F}\\rbrace $ with guidance from $\\lbrace x_{T-F+1}, ..., x_T\\rbrace $ ( REF , bottom panel).",
"The intuition is that in order to construct the next few timestamps, the embedding state must extract representative and meaningful features from the input time series.",
"This design is inspired from the success of video representation learning using a similar architecture [23]."
],
[
"Prediction network",
"After the encoder-decoder is pre-trained, it is treated as an intelligent feature-extraction blackbox.",
"Specifically, the last LSTM cell states of the encoder are extracted as learned embedding.",
"Then, a prediction network is trained to forecast the next one or more timestamps using the learned embedding as features.",
"In the scenario where external features are available, these can be concatenated to the embedding vector and passed together to the final prediction network.",
"Here, we use a multi-layer perceptron as the prediction network.",
"We will show in Section REF that the learned embedding from the encoder successfully captures interesting patterns from the input time series.",
"In addition, including external features significantly improves the prediction accuracy during holidays and special events (see Section )"
],
[
"Inference",
"After the full model is trained, the inference stage involves only the encoder and the prediction network ( REF , left panel).",
"The complete inference algorithm is presented in Algorithm REF , where the prediction uncertainty, $\\eta $ , contains two terms: (i) the model and misspecification uncertainty, estimated by applying MC dropout to both the encoder and the prediction network, as presented in Algorithm REF ; and (ii) the inherent noise level, estimated by the residuals on a held-out validation set.",
"Finally, an approximate $\\alpha $ -level prediction interval is constructed by $[\\hat{y}^* - z_{\\alpha /2} \\eta , ~ \\hat{y}^* + z_{\\alpha /2} \\eta ]$ , where $z_{\\alpha /2}$ is the upper $\\alpha /2$ quantile of a standard Normal.",
"Two hyper-parameters need to be specified in Algorithm REF : the dropout probability, $p$ , and the number of iterations, $B$ .",
"As for the dropout probability, we find in our experiments that the uncertainty estimation is relatively stable across a range of $p$ , and we choose the one that achieves the best performance on the validation set.",
"As for the number of iterations, the standard error of the estimated prediction uncertainty is proportional to $1/\\sqrt{B}$ .",
"We measure the standard error across different repetitions, and find that a few hundreds of iterations are usually suffice to achieve a stable estimation."
],
[
"Evaluation",
"This section contains two sets of results.",
"We first evaluate the model performance on a moderately sized data set of daily trips processed by the Uber platform.",
"We will evaluate the prediction accuracy and the quality of uncertain estimation during both holidays and non-holidays.",
"We will also present how the encoder recognizes the day of the week pattern in the embedding space.",
"Next, we will illustrate the application of this model to real-time large-scale anomaly detection for millions of metrics at Uber."
],
[
"Experimental settings",
"In this section, we illustrate the model performance using the daily completed trips over four years across eight representative large cities in U.S. and Canada, including Atlanta, Boston, Chicago, Los Angeles, New York City, San Francisco, Toronto, and Washington D.C. We use three years of data as the training set, the following four months as the validation set, and the final eight months as the testing set.",
"The encoder-decoder is constructed with two-layer LSTM cells, with 128 and 32 hidden states, respectively.",
"The prediction network has three fully connected layers with tanh activation, with 128, 64, and 16 hidden units, respectively.",
"Samples are constructed using a sliding window with step size one, where each sliding window contains the previous 28 days as input, and aims to forecast the upcoming day.",
"The raw data are log-transformed to alleviate exponential effects.",
"Next, within each sliding window, the first day is subtracted from all values, so that trends are removed and the neural network is trained for the incremental value.",
"At test time, it is straightforward to revert these transformations to obtain predictions at the original scale."
],
[
"Prediction performance",
"We compare the prediction accuracy among four different models: Last-Day: A naive model that uses the last day's completed trips as the prediction for the next day.",
"QRF: Based on the naive last-day prediction, a quantile random forest (QRF) is further trained to estimate the holiday lifts, i.e., the ratio to adjust the forecast during holidays.",
"The final prediction is calculated from the last-day forecast multiplied by the estimated ratio.",
"LSTM: A vanilla LSTM model with similar size as our model.",
"Specifically, a two-layer sacked LSTM is constructed, with 128 and 32 hidden states, respectively, followed by a fully connected layer for the final output.",
"This neural network also takes 28 days as input, and predicts the next day.",
"Our Model: Our model that combines an encoder-decoder and a prediction network, as described in REF .",
"Table REF reports the Symmetric Mean Absolute Percentage Error (SMAPE) of the four models, evaluated on the testing set.",
"We see that using a QRF to adjust for holiday lifts is only slightly better than the naive prediction.",
"On the other hand, a vanilla LSTM neural network provides an average of 26% improvement across the eight cities.",
"As we further incorporate the encoder-decoder framework and introduce external features for holidays to the prediction network ( REF ), our proposed model achieves another 36% improvement in prediction accuracy.",
"Note that when using LSTM and our model, only one generic model is trained, where the neural network is not tuned for any city-specific patterns; nevertheless, we still observe significant improvement on SMAPE across all cities when compared to traditional approaches.",
"Table: SMAPE of Four Different Prediction Models, Evaluated on the Test Data.Finally, REF visualizes the true values and our predictions during the testing period in San Francisco as an example.",
"We observe that accurate predictions are achieved not only in regular days, but also during holiday seasons.",
"Figure: Daily completed trips in San Francisco during eight months of the testing set.",
"True values are shown with the orange solid line, and predictions are shown with the blue dashed line, where the 95% prediction band is shown as the grey area.",
"Exact values are anonymized."
],
[
"Uncertainty estimation",
"Next, we evaluate the quality of our uncertainty estimation by calibrating the empirical coverage of the prediction intervals.",
"Here, the dropout probability is set to be 5% at each layer, and Table REF reports the empirical coverage of the 95% predictive intervals under three different scenarios: PredNet: Use only model uncertainty estimated from MC dropout in the prediction network, with no dropout layers in the encoder.",
"Enc+Pred: Use MC dropout in both the encoder and the prediction network, but without the inherent noise level.",
"This is the term $\\eta _1$ in Algorithm REF .",
"Enc+Pred+Noise: Use the full prediction uncertainty $\\eta $ as presented in Algorithm REF , including $\\eta _1$ as in 2), as well as the inherent noise level $\\eta _2$ .",
"Table: Empirical Coverage of 95% Predictive Intervals, Evaluated on the Test Data.By comparing PredNet with Enc+Pred, it is clear that introducing MC dropout to the encoder network is critical, which significantly improves the empirical coverage from 78% to 90% by capturing potential model misspecification.",
"In addition, by further accounting for the inherent noise level, the empirical coverage of the final uncertainty estimation, Enc+Pred+Noise, nicely centers around 95% as desired.",
"One important use-case of the uncertainty estimation is to provide insight for unusual patterns in the time series.",
"REF shows the estimated predictive uncertainty on six U.S. holidays in the testing data.",
"We see that New Year's Eve has significantly higher uncertainty than all other holidays.",
"This pattern is consistent with our previous experience, where New Year's Eve is usually the most difficult day to predict.",
"Figure: Estimated prediction standard deviations on six U.S. holidays during testing period for eight cities.",
"Exact values are anonymized."
],
[
"Embedding features",
"As illustrated previously, the encoder is critical for both improving prediction accuracy, as well as for estimating prediction uncertainty.",
"One natural follow-up question is whether we can interpret the embedding features extracted by the encoder.",
"This can also provide valuable insights for model selection and anomaly detection.",
"Here, we visualize our training data, each being a 28-day time series segment, in the embedding space.",
"We use the last LSTM cell in the encoder, and project its cell states to 2D for visualization using PCA ( REF ).",
"The strongest pattern we observe is day of the week, where weekdays and weekends form different clusters, with Fridays usually sitting in between.",
"We do not observe city-level clusters, which is probably due to the fact all cities in this data set are large cities in North America, where riders and drivers tend to have similar behaviors.",
"Figure: Training set of time series, visualized in the embedding space.",
"Each point represents a 28-day segment, colored by the day of the week of the last day.",
"We evaluate the cell states of the two LSTM layers, where the first layer with dimension 128 is plotted on the left, and second layer with dimension 32 is plotted on the right.",
"PCA is used to project into 2D space for visualization."
],
[
"Application to Anomaly Detection\nat Uber\n",
"At Uber, we track millions of metrics each day to monitor the status of various services across the company.",
"One important application of uncertainty estimation is to provide real-time anomaly detection and deploy alerts for potential outages and unusual behaviors.",
"A natural approach is to trigger an alarm when the observed value falls outside of the 95% predictive interval.",
"There are two main challenges we need to address in this application: Scalability: In order to provide real-time anomaly detection at the current scale, each predictive interval must be calculated within a few milliseconds during inference stage.",
"Performance: With highly imbalanced data, we aim to reduce the false positive rate as much as possible to avoid unnecessary on-call duties, while making sure the false negative rate is properly controlled so that real outages will be captured."
],
[
"Scalability",
"Our model inference is implemented in Go.",
"Our implementation involves efficient matrix manipulation operations, as well as stochastic dropout by randomly setting hidden units to zero with pre-specified probability.",
"A few hundred stochastic passes are executed to calculate the prediction uncertainty, which is updated every few minutes for each metric.",
"We find that the uncertainty estimation step adds only a small amount of computation overhead and can be conducted within ten milliseconds per metric."
],
[
"Performance",
"Here, we illustrate the precision and recall of this framework on an example data set containing 100 metrics with manual annotation available, where 17 of them are true anomalies.",
"Note that the neural network was previously trained on a separate and much larger data set.",
"By adding MC dropout layers in the neural network, the estimated predictive intervals achieved 100% recall rate and a 80.95% precision rate.",
"REF visualizes the neural network predictive intervals on four representative metrics, where alerts are correctly fired for two of them.",
"When applying this framework to all metrics, we observe a 4% improvement in precision compared to the previous ad-hoc solution, which is substantial at Uber's scale.",
"Figure: Four example metrics during a 12-hour span, and anomaly detection is performed for the following 30 minutes.",
"All metrics are evaluated by minutes.",
"The neural network constructs predictive intervals for the following 30 minutes, visualized by the shaded area in each plot.",
"(a) A normal metric with large fluctuation, where the observation falls within the predictive interval.",
"(b) A normal metric with small fluctuation, and an unusual inflation has just ended.",
"The predictive interval still captures the observation.",
"(c) An anomalous metric with a single spike that falls outside the predictive interval.",
"(d) An anomalous metric with two consecutive spikes, also captured by our model."
],
[
"Conclusion",
"We have presented an end-to-end neural network architecture for uncertainty estimation used at Uber.",
"Using the MC dropout technique and model misspecification distribution, we showed a simple way to provide uncertainty estimation for a neural network forecast at scale while providing a 95% uncertainty coverage.",
"A critical feature about our framework is its applicability to any neural network without modifying the underlying architecture.",
"We have used the proposed uncertainty estimate to measure special event (e.g., holiday) uncertainty and to improve anomaly detection accuracy.",
"For special event uncertainty estimation, we found New Year's Eve to be the most uncertain time.",
"Using the uncertainty information, we adjusted the confidence bands of an internal anomaly detection model to improve precision during high uncertainty events, resulting in a 4% accuracy improvement, which is large given the number of metrics we track at Uber.",
"Our future work will be focused on utilizing the uncertainty information for neural network debugging during high error periods."
]
] | 1709.01907 |
[
[
"Half-vicinity model and a phase diagram for quantum oscillations in\n confined and degenerate Fermi gases"
],
[
"Abstract We propose an analytical model for the accurate calculation of size and density dependent quantum oscillations in thermodynamic and transport properties of confined and degenerate non-interacting Fermi gases.",
"We provide a universal, material independent, recipe that explicitly separates oscillatory quantum regime from stationary classical regime.",
"Our model quite accurately estimates quantum oscillations depending on confinement and degeneracy.",
"We construct a phase diagram representing stationary and oscillatory regimes on degeneracy-confinement space.",
"Analytical expressions of phase transition interfaces are derived for different dimensions.",
"The critical point on the phase diagram, which separates entirely stationary and entirely oscillatory regions, is determined and their aspect ratio dependencies are examined.",
"Quantum oscillations as well as their periods are analytically expressed for one-dimensional case.",
"Accuracy of our model is verified through quantum oscillations in electronic specific heat capacity.",
"We also compare the predictions of our half-vicinity model, based on bounded sums, with those of infinite sums, for the oscillatory violation of entropy-heat capacity equivalence in degenerate limit to show the accuracy of our model.",
"Furthermore, similarities between functional behaviors of total occupancy variance and conventional density of states functions at Fermi level are discussed."
],
[
"Introduction",
"A great deal of attention has been given to the physics of low-dimensional nanostructures associated with the intensive developments of nanoscience and nanotechnology in recent years.",
"When the mean de Broglie wavelength of particles inside a domain becomes comparable to the size of the domain, quantum size effects start to reveal themselves and their presence has to be taken into account in the calculation of the physical properties of such nanoscale systems [1], [2], [3], [4].",
"In Fermi gases, instead of fully occupied states, only partially occupied ones around Fermi level contribute to some thermodynamic quantities like entropy and specific heat and transport properties like electrical and thermal conductivities.",
"Definitional expressions of such quantities contain the variance of distribution function instead of the distribution function itself.",
"Hence, variance function plays a crucial role in the behavior of these quantities.",
"In degenerate and strongly confined ideal Fermi gases, the quantities containing variance function exhibit oscillatory behaviors.",
"The basic reason of this behavior is the oscillatory behavior of total occupancy variance (TOV) function due to changes in size and density.",
"It has been shown experimentally in literature that, size and density dependent oscillations occur in some certain thermodynamic and transport properties of charge carriers in semiconductors/metals or Fermi gases confined at nano domains in general.",
"First experimental studies of size dependent oscillations due to quantum size effects has been reported in late 60's [5], [6], [7], [8].",
"Thermodynamic properties like entropy and specific heat capacity at constant volume, thermoelectric properties such as Seebeck coefficient and electronic transport properties such as charge carrier mobility, electrical and thermal conductivities are some examples of these oscillatory quantities [4], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35].",
"Examination of size dependent oscillations also has crucial importance in superconductors and topological insulators [33], [34], [36], [37], [38], [39].",
"Size and density dependent oscillations in aforementioned quantities are attributed to the quantization of energy spectrum and fluctuations in density of states around Fermi level at nanoscale [15], [16], [17], [18], [19], [20], [40].",
"For Fermi systems with linear energy dispersion relation (e.g.",
"a Fermi system under an applied magnetic field or harmonic trap potential), quantum oscillations and its implications are extensively studied especially for specific materials [41], [15], [16], [17], [18], [19], [20], [42], [43].",
"For the systems obeying quadratic energy-momentum dispersion relation, on the other hand, there is a need for a mathematical model predicting quantum oscillations and explaining their nature in a comprehensive and material/case independent way.",
"Moreover, a phase diagram separating oscillatory and stationary regimes on confinement-degeneracy space has never been proposed for any dispersion relation.",
"Constructing a mathematical model and a phase diagram may help to deeply understand and efficiently control the oscillations to design novel nanoscale devices [20], [27].",
"In this article, we propose a theoretical model, half-vicinity model (HVM), that accurately predicts the oscillations appearing in confined Fermi gases.",
"A rectangular confinement domain is chosen, because it is one of the most suitable geometries for common manufacturing methods of low dimensional nanostructures.",
"In the following section, by examining the nature of TOV of quantum states, we introduce the model by defining a half-vicinity shell around Fermi level.",
"$d$ -dimensional discrete and continuum expressions of TOV function are given and thickness of half-vicinity shell, which plays an important role in HVM, is obtained for various dimensions.",
"For 1D case, we obtain analytical expressions for TOV oscillations and their periods.",
"In Sec.",
"III, a phase diagram of quantum oscillations is established and phase transition interfaces between stationary (classical, continuous) and oscillatory (quantum, discrete) regimes are analytically given for 1D, 2D and 3D cases.",
"Critical confinement and degeneracy values, which separates entirely stationary and oscillatory regions, are also examined by considering their aspect ratio dependencies.",
"In Sec.",
"IV, results of exact (definitional expressions based on infinite sums) and HVMs are compared for size and density dependent oscillations in the electronic specific heat capacity of strongly degenerate and confined ideal Fermi gases.",
"Furthermore, broken equivalence of entropy-heat capacity in quantum degenerate limit of ideal Fermi gases is also well predicted by HVM.",
"Finally, we discuss (Sec.",
"V) the relationship between TOV and conventional density of states functions at Fermi level."
],
[
"Half-vicinity model for total occupancy variance",
"Oscillations in thermodynamic and transport quantities originate from the nature of occupancy variance function.",
"For this reason, in order to establish a model to understand the nature of oscillations, we need to examine TOV in detail."
],
[
"Total occupancy variance revisited",
"Occupancies of quantum states for Fermions are described by Fermi-Dirac distribution function, $f=g_s/\\left[\\exp (-\\Lambda +\\tilde{\\varepsilon })+1\\right]$ where $g_s$ is spin degree of freedom, $\\Lambda =\\mu /(k_BT)$ is dimensionless chemical potential (or degeneracy parameter) indicating the strength of degeneracy, in which $\\mu $ is chemical potential, $k_B$ is Boltzmann's constant, $T$ is temperature.",
"$\\tilde{\\varepsilon }$ represents dimensionless energy eigenvalues (normalized to thermal energy $k_B T$ ).",
"For quadratic dispersion relation of massive particles confined in a $d$ -dimensional rectangular domain, $\\tilde{\\varepsilon }=\\varepsilon /(k_BT)=(\\alpha _1 i_1)^2+\\cdots +(\\alpha _d i_d)^2$ where $i_n$ is quantum state variable for a particular direction $n=\\lbrace 1,2,\\cdots ,d\\rbrace $ .",
"Here $\\alpha _n$ is confinement parameter indicating the strength of quantum confinement in a particular direction $n$ and defined as $\\alpha _n=h/\\left(\\sqrt{8m k_B T}L_n\\right)$ where $h$ is Planck's constant, $m$ is mass of the particle and $L_n$ is domain size in direction $n$ .",
"Derivative of Fermi-Dirac distribution function with respect to $\\Lambda $ , a bell-shaped function peaked at Fermi level, is called occupancy variance function.",
"Occupancy variance of an energy state $\\tilde{\\varepsilon }=\\tilde{\\varepsilon }(i_1,\\ldots ,i_d)$ is $\\sigma ^2(i_1,\\ldots ,i_d)=\\frac{\\partial f}{\\partial \\Lambda }=-\\frac{\\partial f}{\\partial \\tilde{\\varepsilon }}=\\frac{g_s}{4}\\operatorname{sech}^2\\left(\\frac{\\tilde{\\varepsilon }-\\Lambda }{2}\\right).$ In literature, it's sometimes called also as thermal broadening function, since it represents thermal broadening of energy states around Fermi level [44].",
"Due to the fact that Fermi-Dirac distribution is a Bernoulli-type distribution (basically a sigmoid function), occupancy variance can also be represented by a Bernoulli-type variance form $\\sigma ^2=g_s\\tilde{f}(1-\\tilde{f})$ , where $\\tilde{f}=f/g_s$ is the spin normalized Fermi-Dirac distribution function.",
"Spin factor $g_s$ is taken simply two in all numerical calculations in this article.",
"In order to understand the behavior of oscillatory quantities and how occupancy variance behaves under the accumulation operators such as summation or integration, we need to investigate the nature of TOV.",
"TOV of Fermi particles inside a $d$ -dimensional box is written in its exact form based on infinite sums as $\\Sigma _D^2=\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }\\sigma ^2(i_1,\\ldots ,i_d),$ where subscript $D$ stands for discrete calculations.",
"When the summations in Eq.",
"(2) are calculated by using the first two terms of Poisson summation formula (PSF) [45], [30], the following expression can be obtained after some mathematical operations, $\\begin{split}\\Sigma _{W}^2=& g_s\\sum _{n=0}^d\\frac{(-1)^{d-n+1}\\hspace{2.22214pt}\\pi ^{n/2}}{2^d \\hspace{2.22214pt}d^{1-\\Theta (n)\\Theta (d-n)}}Li_{\\frac{n-2}{2}}\\left[-\\exp (\\Lambda )\\right] \\\\& \\times \\sum _{k=1}^{d}\\prod _{m=k}^{k+n-1}{\\alpha _{\\text{mod}(m,d)+1}^{-1}},\\end{split}$ where $Li$ is polylogarithm function [46] and $\\Theta $ is left-continuous Heaviside step function.",
"Since the expressions calculated by using the first two terms of PSF are equivalent to those obtained from Weyl conjecture [45], [3], [47], we denote this expression by subscript $W$ .",
"In strongly degenerate conditions ($\\Lambda >>1$ ), by using the asymptotic forms of polylogarithm functions [48], Eq.",
"(3) can be approximated by $\\begin{split}\\Sigma _{W\\hspace{-2.77771pt}A}^2=& g_s\\sum _{n=0}^d\\frac{(-1)^{d-n}\\hspace{2.22214pt}\\pi ^{n/2}}{2^d \\hspace{2.22214pt}d^{1-\\Theta (n)\\Theta (d-n)}}\\frac{\\Lambda ^{(n-2)/2}}{\\Gamma (n/2)} \\\\& \\times \\sum _{k=1}^{d}\\prod _{m=k}^{k+n-1}{\\alpha _{\\text{mod}(m,d)+1}^{-1}},\\end{split}$ where $\\Gamma $ is gamma function.",
"When confinement parameters are much smaller than unity, namely in macroscale, summations may be converted directly into integrals by using continuum approximation and continuous TOV is obtained $\\Sigma _C^2=-\\frac{g_s\\pi ^{d/2}}{2^d \\alpha _1 \\cdots \\alpha _d}Li_{\\frac{d-2}{2}}\\left[-\\exp (\\Lambda )\\right],$ where $C$ subscript denotes the continuous expressions obtained under continuum approximation.",
"In strongly degenerate case ($\\Lambda >>1$ ), Eq.",
"(5) can be approximated by using asymptotic expansions of polylogarithm functions $\\Sigma _{C\\hspace{-2.77771pt}A}^2=\\frac{g_s\\pi ^{d/2}}{2^d \\alpha _1 \\cdots \\alpha _d}\\frac{\\Lambda ^{(d-2)/2}}{\\Gamma (d/2)}.$ As it should be, Eqs.",
"(5) and (6) are actually the first (bulk) terms of Eqs.",
"(3) and (4) respectively.",
"Expressions of thermodynamic and transport properties exhibiting oscillatory behavior contain Eq.",
"(1) multiplied by some relevant functions of the quantity to be calculated.",
"Then, it's proper to construct our HVM on TOV, in order to predict oscillatory behaviors in thermodynamic and transport properties."
],
[
"Basic concept of half-vicinity model",
"Contributions to oscillatory physical quantities substantially come from partially occupied states around Fermi level.",
"Under strongly confined and degenerate conditions, sharpness of the occupancy variance function around Fermi level increases and only few states contribute to TOV.",
"In that case, it becomes possible to make a quantitative treatment of these states by constructing a shell which contains them inside.",
"Therefore, for the accurate calculation of oscillatory physical quantities, instead of summing over all possible quantum states from unity to infinity, considering only those several states around Fermi level (inside the half-vicinity shell) is enough to represent the oscillations.",
"Half-vicinity shell in state space is defined as the interval containing the states in the half-vicinities ($\\pm 1/2$ ) of Fermi level.",
"It can be extended to any dimension, by considering the $\\pm 1/2$ intervals of Fermi level in each direction at quantum state space.",
"Magnitude of the contribution of each state to TOV is determined by its proximity to the Fermi level.",
"Contribution of a state is maximum when it lies exactly on the Fermi level, and exponentially decays as the state is away from it.",
"The number of half-vicinity states and their proximities to Fermi level change with degeneracy and/or confinement.",
"When confinement becomes stronger, spacing between energy levels increases and causes a decrement in number of half-vicinity states.",
"Because of this decrement, addition (removal) of a state to (from) half-vicinity shell leads to a strong change in TOV.",
"This causes strong fluctuations both in number of half-vicinity states and their proximities to the Fermi level.",
"In other words, half-vicinity states reorganize their distribution and proximities to the Fermi level when for instance sizes of the domain or density of the gas change.",
"Each state gives distinct contribution to TOV.",
"For macro domains with highly populated Fermi level, effect of this reorganization becomes statistically insignificant.",
"Conversely, in confined and degenerate Fermi gases these reorganization of half-vicinity states manifest themselves as quantum oscillations in several thermodynamic and transport quantities due to changes in size and density.",
"For strongly confined and degenerate Fermi gases, TOV, Eq.",
"(2), can approximately be calculated by restricting the summations over infinite number of quantum states to only the states inside the constructed half-vicinity shell.",
"TOV based on HVM is then given by $\\Sigma _{HV}^2=\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }\\sigma ^2w_{HV}.$ Here, half-vicinity window function is define as $w_{HV}=\\Theta \\left(\\Lambda -\\tilde{\\varepsilon }_{-1/2}\\right)\\Theta \\left(\\tilde{\\varepsilon }_{1/2}-\\Lambda \\right),$ where $\\Theta $ is Heaviside step function and $\\tilde{\\varepsilon }_{\\pm 1/2}=\\alpha _1^2\\left(i_1\\pm 1/2\\right)^2+\\ldots +\\alpha _d^2(i_d\\pm 1/2)^2$ give upper and lower energy bounds of half-vicinity window.",
"Window function $w_{HV}$ filters the states which do not significantly contribute to the oscillatory quantities and serves as a half-vicinity states pass filter, that restricts the upper and lower limits of summations.",
"By definition, in state space, half-vicinity shell thickness is equal to unity in any particular direction, which is the minimum possible thickness in state space for each direction.",
"On the other hand, in energy space its thickness varies with the strength of confinement and degeneracy.",
"Half-vicinity window can equivalently be written as $w_{HV}=\\Theta (\\tilde{\\varepsilon }_H-\\tilde{\\varepsilon })\\Theta (\\tilde{\\varepsilon }-\\tilde{\\varepsilon }_L),$ where $\\tilde{\\varepsilon }_H=\\Lambda +\\delta _{\\tilde{\\varepsilon }_H}$ , $\\tilde{\\varepsilon }_L=\\Lambda -\\delta _{\\tilde{\\varepsilon }_L}$ , $\\delta _{\\tilde{\\varepsilon }_H}=\\delta _{\\tilde{\\varepsilon }}/2+\\tilde{\\varepsilon }_0/4$ , $\\delta _{\\tilde{\\varepsilon }_L}=\\delta _{\\tilde{\\varepsilon }}/2-\\tilde{\\varepsilon }_0/4$ .",
"Here, the thickness of half-vicinity shell, $\\delta _{\\tilde{\\varepsilon }}$ , in its discrete form and the ground state energy in normalized energy space, $\\tilde{\\varepsilon }_0$ , are given respectively as follows =2(12 i1+...+d2 id) 0=12+...+d2 Instead of discrete definition of $\\delta _{\\tilde{\\varepsilon }}$ , it's also possible to define it based on continuous expressions which simplifies mathematical operations and will be given in the following subsection.",
"Half-vicinity shell thickness is a crucial parameter since contributions to oscillatory properties mainly come from the states within this half-vicinity shell.",
"Figure: (Color online) (a) Spin normalized 1D Fermi distribution and occupancy variance functions in energy space for Λ=25\\Lambda =25 and α=0.5\\alpha =0.5.",
"Thickness of half-vicinity shell in energy space (δ ε ˜ =5\\delta _{\\tilde{\\varepsilon }}=5) denoted by green line.",
"(b) Spin normalized occupancy variance function in 1D state space for weakly confined (α 1 =0.25\\alpha _1=0.25) and confined (α 1 =1\\alpha _1=1) domains for a constant chemical potential (Λ=10\\Lambda =10).",
"Shaded region denotes the thickness of half-vicinity shell which is equal to unity in state space.",
"Red and green points represent the half-vicinity states (Fermi points in 1D) and off-half-vicinity states respectively.In Fig.",
"1(a), spin normalized 1D Fermi distribution and occupancy variance functions in energy space are shown by dashed-blue and solid-red curves respectively.",
"Thickness of half-vicinity shell is represented by the solid horizontal green line.",
"In Fig.",
"1(b), occupancy variance is plotted in 1D state space for two different confinement values for a constant chemical potential ($\\Lambda =10$ ).",
"As is seen from Fig.",
"1(b), when confinement increases (denoted by the leftward arrow), occupancy variance peak becomes sharper.",
"Red states are the closest states to the respective Fermi levels, so their contributions are the largest.",
"Light blue shaded region represents HV window.",
"These closest states to Fermi level in 1D state space are called Fermi points.",
"They constitute a Fermi line in 2D and Fermi surface in 3D cases.",
"Green points represent the states outside of half-vicinity shell, which we call off-half-vicinity (OHV) states.",
"Note that in strongly confined case contributions of OHV states are negligible.",
"Thus, HVM represents the true behavior of occupancy variance function accurately in strongly degenerate cases, where oscillations are strong.",
"On the contrary, for weakly confined or unconfined cases, where oscillations weaken or disappear, HVM representation of variance function becomes inaccurate because of non-negligible contributions of OHV states (Fig.",
"1(b) for $\\alpha =0.25$ ).",
"However, in this case, continuum expressions of variance function already represent the true behavior properly."
],
[
"Thickness of half-vicinity shell in energy space and renormalized HVM",
"Thickness of half-vicinity shell in energy space plays an important role in HVM.",
"To describe the thickness in a general way, we invoke number of states (NOS) and density of states (DOS) concepts.",
"NOS inside the half-vicinity shell is found by $NOS_{HV}=\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }w_{HV}$ , which is an exact equation but does not lead to analytical expressions, which we seek here.",
"Therefore, we use continuum approximation to convert summations into integrals and find the thickness of half-vicinity shell in energy space analytically as $\\delta _{\\tilde{\\varepsilon }}=CNOS_{HV}/CDOS(\\Lambda )$ , where $CNOS_{HV}=\\int _{0}^{\\infty }\\cdots \\int _0^{\\infty }w_{HV}di_1\\ldots di_d$ .",
"By calculating the integrals of CNOS and using the known expressions of CDOS [47], $\\delta _{\\tilde{\\varepsilon }}$ can be obtained for various dimensions in degenerate limit as given in Table I.",
"Note that when $\\sqrt{\\Lambda }/\\alpha _n$ value of a system becomes less than unity for a particular direction $n$ , momentum modes in that direction becomes empty except the ground state and the system can be considered as if it has one dimension less.",
"Table: CNOS, CDOS and thickness of half-vicinity shell in energy space for various dimensions.$\\delta _{\\tilde{\\varepsilon }}$ can be generalized into $d$ -dimensions as $\\delta _{\\tilde{\\varepsilon }}(d)=2\\sqrt{\\frac{\\Lambda }{\\pi }}\\frac{\\Gamma (d/2)}{\\Gamma [(d+1)/2]}\\sum _{n=1}^{d}\\alpha _n.$ In continuum case, $(\\alpha _1,\\ldots ,\\alpha _d\\rightarrow 0)$ , HVM gives the following expression for TOV, HVC2 =02 wHVdi1...did =02 wHVCDOS()d =LH2 CDOS()d. It should be noted that $\\tilde{\\varepsilon }_0\\rightarrow 0$ in continuous case and $\\tilde{\\varepsilon }_H$ and $\\tilde{\\varepsilon }_L$ become $\\Lambda +\\delta _{\\tilde{\\varepsilon }}/2$ and $\\Lambda -\\delta _{\\tilde{\\varepsilon }}/2$ respectively.",
"It's possible to obtain an analytical solution for the integral in Eq.",
"(12c) for any dimension by considering strongly degenerate conditions $(\\Lambda >>1)$ , $\\Sigma _{HVC}^2\\cong \\tanh \\left(\\frac{\\delta _{\\tilde{\\varepsilon }}}{4}\\right)\\Sigma _{CA}^2.$ Here, $\\tanh $ factor acts as an amplitude renormalization factor.",
"Although this factor is found by a continuous approach, the similar renormalization factor is expected also for the discrete case, because amplitude renormalization process is independent of the type of accumulation operators which is verified by the numerical results in this article.",
"Thus, we may do the following approximation $\\begin{aligned}\\Sigma _{HV}^2 &=\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }\\sigma ^2w_{HV} \\\\&\\cong \\tanh \\left(\\frac{\\delta _{\\tilde{\\varepsilon }}}{4}\\right)\\left[\\sum _{i_1=1}^{\\infty }\\cdots \\sum _{i_d=1}^{\\infty }\\sigma ^2\\right]=\\tanh \\left(\\frac{\\delta _{\\tilde{\\varepsilon }}}{4}\\right)\\Sigma _{D}^2.\\end{aligned}$ Then, we can express discrete TOV, $\\Sigma _{D}^2$ , in terms of $\\Sigma _{HV}^2$ as follows, $\\Sigma _{D}^2\\approx \\frac{\\Sigma _{HV}^2}{\\tanh \\left(\\delta _{\\tilde{\\varepsilon }}/4\\right)}=\\Sigma _{RHV}^2,$ where $RHV$ subscript denotes renormalized half-vicinity.",
"Factor of $1/\\tanh $ renormalizes the amplitude of variance function found by HVM (which only considers HV states) in order to represent also the contributions of OHV states.",
"In other words, $\\Sigma _{RHV}^2$ is the renormalized TOV based on HVM and it includes contributions of both HV and OHV states.",
"Accuracy of this expression depends on the thickness of half-vicinity shell inside the factor of $\\tanh $ , which is a function of confinement and degeneracy.",
"The remarkable accuracy of Eq.",
"(15) in comparison with the exact equation (Eq.",
"(2)) is shown in Fig.",
"2.",
"Although Weyl representation (Eq.",
"(3)) does not predict the oscillations but only their trend, HVM quite perfectly matches with the exact results especially for high degeneracy and confinement conditions where oscillations are significant.",
"Consequently, by considering Eqs.",
"(2), (7) and (15), HVM proposes the following transformation: $\\sigma ^2\\xrightarrow{}\\sigma ^2\\frac{w_{HV}}{\\tanh \\left(\\delta _{\\tilde{\\varepsilon }}/4\\right)}$ for the calculation of any thermodynamic and transport quantity containing $\\sigma ^2$ term.",
"By this transformation, infinite sums are converted into finite sums over the minimum number of states near to Fermi level.",
"As is clear from the comparisons in Fig.",
"2, HVM can be used as a reliable tool to predict and represent quantum oscillations.",
"Figure: (Color online) Comparison of TOV results based on exact model, HVM and Weyl expressions.",
"(a) 1D (b) 2D (c) 3D occupancy variance functions changing with degeneracy and (d) 1D (e) 2D (f) 3D occupancy variance functions changing with isometric confinement in all available directions.",
"Red, blue and green solid curves represent Eq.",
"(2) in 1D, 2D and 3D respectively.",
"Eq.",
"(15) (HVM estimation) is represented by dashed black curves.",
"Dot-dashed purple curves are Weyl representations of TOV functions from Eq.",
"(3)."
],
[
"Analytical expression for TOV in 1D case",
"In highly degenerate and strongly confined 1D case, summation of TOV in Eq.",
"(2) vanishes and HVM represents only the contribution of Fermi point inside $i_F\\pm 1/2$ interval in state space, where $i_F=\\sqrt{\\Lambda }/\\alpha _1$ .",
"Therefore, for this special case, there is no need to do summation over half-vicinity shell states (since there is only one) and a special value of that state variable can be found analytically as $i_{*}=\\left[i_F\\right]=i_F-\\frac{1}{\\pi }\\arctan {\\left[\\tan \\left(\\pi i_F\\right)\\right]},$ where $\\left[\\cdots \\right]$ denotes the round function which can be represented by elementary functions on the right hand side.",
"Then, renormalized TOV in 1D case is $\\Sigma _{RHV}^2=\\frac{\\sigma ^2(i_{*})}{\\tanh \\left(\\alpha _1\\sqrt{\\Lambda }/2\\right)}.$ A comparison for variations of exact and HVM's TOV functions with confinement is given in Fig.",
"3.",
"Solid red, solid blue and dashed black curves are results of Eqs.",
"(2), (5) and (18) respectively.",
"Accuracy of HVM in representing the prominent oscillations is quite good, even though it only considers contribution of Fermi points.",
"Since Eqs.",
"(3) and (5) are equal to each other for 1D case, there is no need to examine $\\Sigma _W^2$ additionally.",
"It's seen from the red curve in Fig.",
"3 that transitions between stationary and oscillatory regimes are smooth and $\\ln 3/\\sqrt{\\Lambda }$ line perfectly separates these regimes, which will be examined in detail in the following section.",
"Figure: (Color online) A comparison of exact and HVM results for the variation of TOV with confinement when Λ=30\\Lambda =30.",
"Green vertical line, α 1 =ln3/Λ\\alpha _1=\\ln 3/\\sqrt{\\Lambda }, separates the oscillatory regime from the stationary one.",
"Δα pp \\Delta \\alpha _{pp} denotes the period of TOV varying with confinement.Extremum points of oscillations in Fig.",
"3 correspond to the values where the total number of particles is integer.",
"We can determine the periodicity of TOV changing with $\\Lambda $ and $\\alpha $ , considering the fact that maxima and minima of oscillations correspond to odd and even integer particle numbers respectively, for $g_s=2$ .",
"Special to the 1D case, expression found by considering first two terms of PSF exactly represent number of particles for a given chemical potential in degenerate case [30].",
"We find the values of $\\Lambda $ and $\\alpha $ correspond to integer numbers of particles as $\\Lambda =[\\alpha _1(\\tilde{N}+1/2)]^2$ and $\\alpha _1=\\sqrt{\\Lambda }/(\\tilde{N}+1/2)$ respectively.",
"Periods of oscillations in $\\alpha $ and $\\Lambda $ spaces can then easily be found respectively as pp=4(2N+1)(2N+3)=212N+3, pp=212(N+1)=8N+1(2N+1)2, where $\\tilde{N}=N/g_s$ is spin normalized particle number.",
"Considering the fact that $\\Sigma _{RHV}^2$ mimics the oscillations in an almost perfect way, an intriguing question comes to the mind: Is it possible to determine analytically where quantum oscillations start as the confinement and/or degeneracy changes?"
],
[
"A phase diagram for quantum oscillations: Transition from classical to quantum behavior",
"HVM takes the discreteness of quantum states into account and allows us to accurately calculate the thermodynamic and transport properties exhibiting size and density dependent quantum oscillations without using infinite summations.",
"In addition to that, HVM can accurately estimate where the transition from classical behavior to quantum behavior starts, in the framework of quantum oscillations.",
"The states contributing to TOV consist of HV states and OHV states (i.e., $\\Sigma _{D}^2=\\Sigma _{HV}^2+\\Sigma _{OHV}^2$ ).",
"HV states are responsible from the oscillatory part of TOV, whereas OHV states represent the stationary part.",
"They constitute two competing parts of TOV.",
"Domination of contributions of HV states over OHV ones leads to oscillations and vice versa.",
"Hence, by considering $\\Sigma _{HV}^2=\\Sigma _{OHV}^2$ balance, we can compare contributions of HV states and OHV states to define a universal (material independent) recipe for the separation of stationary regime (SR) from oscillatory regime (OR).",
"According to the recipe, when $\\Sigma _{HV}^2<\\Sigma _{OHV}^2$ , oscillations disappear (SR) and on the opposite condition $\\Sigma _{HV}^2>\\Sigma _{OHV}^2$ , oscillations reveal (OR).",
"Since $\\Sigma _{OHV}^2=\\Sigma _{D}^2-\\Sigma _{HV}^2$ , SR-OR transition can be quantified as the balance of the contributions of HV and OHV states ($\\Sigma _{HV}=\\Sigma _{OHV}$ ) by $\\Sigma _{D}^2=2\\Sigma _{HV}^2$ .",
"Since the transition is not sharp but smooth, we can safely use analytical expressions of TOV, instead of their exact expressions, to find an analytical expression for SR-OR separation.",
"From the balance condition between contributions of HV and OHV states as well as Eq.",
"(13), we can determine the following analytical condition for the transition between SR and OR, $\\Sigma _{CA}^2=2\\Sigma _{HVC}^2 \\;\\Rightarrow \\; 1=2\\tanh \\left(\\delta _{\\tilde{\\varepsilon }}/4\\right) \\;\\Rightarrow \\; \\delta _{\\tilde{\\varepsilon }}=2\\ln 3.$ It's seen from red curve in Fig.",
"3, balance condition quantified in Eq.",
"(20) clearly gives the SR-OR separation.",
"By decreasing confinement or degeneracy, the number of states around Fermi level increases, contributions of OHV states become also appreciable and their contributions make oscillation amplitude smaller.",
"The more number of states around Fermi level, the smaller oscillation amplitude.",
"When contributions of OHV states exceeds that of HV states, oscillations disappear.",
"To complete the construction of the phase diagram, it is necessary to check also the number of particles, ($N$ ), inside the system, since we are dealing with extremely confined systems having relatively low number of particles.",
"For statistical representations there has to be sufficiently large number of particles inside the system.",
"Nevertheless, the physically meaningful region in a phase diagram can be stated by the condition $N\\ge 1$ .",
"Full list of recipes and conditions to establish the phase diagram is given in the Table 2.",
"Table: Recipes and conditions for the construction of the phase diagram of quantum oscillations.According to the recipes and conditions given in Table 2, phase diagrams of quantum oscillations in degeneracy-confinement space for various dimensions can be determined.",
"For isometric 3D domains, the phase diagram is constructed and given in Fig.",
"4.",
"Solid-black curves represent interfaces defined by the conditions in Table II.",
"$\\delta _{\\tilde{\\varepsilon }}=2\\ln 3$ condition, representing the balance of HV and OHV states, separates OR and SR.",
"When both confinement and degeneracy sufficiently increase, system enters into the quantum regime where certain thermodynamic and transport quantities oscillate with varying size or density.",
"Figure: (Color online) A phase diagram on degeneracy-confinement space for quantum oscillations.",
"δ ε ˜ =2ln3\\delta _{\\tilde{\\varepsilon }}=2\\ln 3 condition defines the boundary between stationary (classical) and oscillatory (quantum) regimes.",
"Blue dot represents the critical point in the phase diagram for isometric 3D domains.Intersection of SR-OR interface curve with $N=1$ curve denotes the critical point which is represented by blue dot in Fig.",
"4.",
"Critical value of the confinement parameter at this point is $\\alpha _{*}^{3D}=0.78$ and the corresponding degeneracy value is $\\Lambda _{*}^{3D}=0.88$ .",
"These values are universal for isometric rectangular confinement domains.",
"For confinement values below $\\alpha _{*}$ , existence of oscillations can be controlled and they can even be suppressed by decreasing degeneracy (through density or temperature).",
"Similarly, for degeneracy values higher than $\\Lambda _*$ , existence of oscillations can be controlled by changing confinement.",
"This region defined by $\\lbrace \\Lambda >\\Lambda _{*},\\alpha <\\alpha _*\\rbrace $ is the crossover region restricted by dashed lines in the phase diagram.",
"On the contrary, for higher confinement values than $\\alpha _{*}$ , quantum oscillations cannot be suppressed and the region is called entirely oscillatory region.",
"Below the critical degeneracy values, system does not exhibit oscillatory behaviors regardless of the values of confinements.",
"Hence, this critical point is used to define different regions (crossover region, entirely OR and entirely SR).",
"Regimes on phase diagram and corresponding conditions are summarized in Table III.",
"Table: Conditions and regions of the phase diagram.Although the phase diagram in Fig.",
"4 represents only isometric 3D domain, the form of the phase diagrams for 1D and isometric 2D domains are also very similar to the 3D one.",
"Only SR-OR interface slightly shifts upward while $N=1$ curve slightly rotates in clockwise direction around the critical point as the dimension decreases.",
"For 1D and isometric 2D domains, universal critical values can also be found as $\\alpha _{*}^{1D}=1.07$ , $\\alpha _{*}^{2D}=0.87$ and $\\Lambda _{*}^{1D}=1.05$ , $\\Lambda _{*}^{2D}=0.98$ .",
"It's also possible to investigate the anisometric domains by defining aspect ratios.",
"For 2D domain, $r_{12}=L_1/L_2=\\alpha _2/\\alpha _1$ denotes the aspect ratio.",
"For 3D, additionally we define $r_{13}=L_1/L_3=\\alpha _3/\\alpha _1$ .",
"Here, the most confined direction is chosen as direction 1.",
"In Fig.",
"5, we show how the critical confinement values are changing with respect to aspect ratios of 2D and 3D domains.",
"Figure: (Color online) Aspect ratio dependencies of critical confinement values for 2D and 3D cases.Although the values in Fig.",
"5 are calculated in an exact way by using Tables I and II, $\\alpha _{1*}^{2D}$ can be approximated by $\\alpha _{1*}^{2D}\\approx 0.87-0.40\\ln r_{12}$ with less than $1.5\\%$ mean absolute percentage error (MAPE) for $0.1\\le r_{12} \\le 1$ interval.",
"Similarly, $\\alpha _{1*}^{3D}$ is approximated by $\\alpha _{1*}^{3D}\\approx 0.78-0.27\\ln r_{12}-0.27\\ln r_{13}$ with less than $2.2\\%$ MAPE for $0.1\\le \\lbrace r_{12},r_{13}\\rbrace \\le 1$ intervals.",
"Critical degeneracy values for anisometric cases can easily be found for 3D, 2D and 1D cases respectively, *3D=[231*3D(1+r12+r13)]2, *2D=[321*2D(1+r12)]2, *1D=(31*1D)2.",
"It's important to note that thickness of half-vicinity shell indicates the quantumness of the system.",
"When volume $V\\rightarrow \\infty $ or temperature $T\\rightarrow \\infty $ or density $n\\rightarrow 0$ , thickness $\\delta _{\\tilde{\\varepsilon }}\\rightarrow 0 <2\\ln 3$ and system behaves classically.",
"Conversely, for sets of $(V,T,n)$ making $\\delta _{\\tilde{\\varepsilon }}>2\\ln 3$ , the system shows quantum oscillatory behaviors.",
"As a matter of fact, $\\delta _{\\tilde{\\varepsilon }}$ is a precise indicator of whether system is in quantum regime or not in terms of oscillations.",
"In order to see the full picture of quantum oscillations in 1D case, variation of TOV, Eq.",
"(2), with confinement and degeneracy is given in Fig.",
"6.",
"It is seen now even more clearly that $\\delta _{\\tilde{\\varepsilon }}=2\\ln 3$ plane perfectly separates oscillatory region from the stationary one regardless of the values of $\\alpha $ and $\\Lambda $ .",
"Increment in degeneracy and/or confinement make $\\delta _{\\tilde{\\varepsilon }}$ larger and cause strong oscillations.",
"Figure: (Color online) Variation of discrete TOV (Σ D 2 \\Sigma ^2_D) with confinement (α\\alpha ) and degeneracy (Λ\\Lambda ).",
"Gray 2ln32\\ln 3 plane separates the oscillatory (quantum) and stationary (classical) regimes.HVM is constructed to predict oscillations appearing in strongly confined and highly degenerate conditions.",
"Accuracy of its predictions increases with increasing confinement and degeneracy, while accuracy becomes weaker for the regions near to SR-OR interface in phase diagram, Fig.",
"4.",
"To be precise, HVM predicts oscillations almost perfectly, as long as confinement and degeneracy are larger than their critical values ($\\alpha _*$ and $\\Lambda _*$ )."
],
[
"Size and density dependent oscillations in electronic heat capacity",
"In previous sections we investigated the oscillatory behavior of TOV function and constructed a theoretical model (HVM) for its calculation.",
"In this subsection, we examine quantum oscillations of heat capacity of an electron gas confined in a nanowire by considering both exact and HVM.",
"Electronic heat capacity from the derivative of internal energy with respect to temperature is written in its exact (based on infinite sums) form as $C_V=g_s k_B\\left[\\sum _{i_n=1}^{\\infty }\\tilde{\\varepsilon }^2\\sigma ^2-\\frac{\\left(\\sum _{i_n=1}^{\\infty }\\tilde{\\varepsilon }\\sigma ^2\\right)^2}{\\sum _{i_n=1}^{\\infty }\\sigma ^2}\\right].$ Heat capacity expression, Eq.",
"(22), contains zeroth, first and second order energy moments of occupancy variance.",
"Each summation in Eq.",
"(22) has distinct oscillatory behavior, which leads to the characteristic oscillations in heat capacity.",
"In order to reveal quantum oscillations and make electronic contribution of heat capacity appreciable, we focus on very low temperature ($T=5$ K), high electron density $(\\sim 10^{25} \\text{m}^{-3})$ and nanoscale confinements in which quantum effects are observable.",
"To make our discussions quantity-independent, we examine electronic heat capacity per particle $(c_V=C_V/N)$ .",
"In the calculations, chemical potential is obtained numerically from particle number equation as functions of temperature, density and confinement.",
"In Fig.",
"7, accuracy of HVM is examined by considering size and density dependent oscillations of normalized electronic specific heat.",
"$c_V^0$ represents the specific heat expression under continuum approximation, where all summations in Eq.",
"(22) replaced by integrals.",
"In all cases in Fig.",
"7, an electron gas confined in a quantum wire with 200 nm long $(L_3)$ is considered at 5K temperature.",
"Red, blue and green curves in Fig.",
"7(a), 7(b) and 7(c) respectively represent the results of the exact model, while the black dashed curves are the results of HVM.",
"Although confinement domain is a nanowire, it cannot be considered as a pure 1D structure since there are still a few excited states in transverse directions.",
"Thus, calculations are done over triple sums.",
"Figure: (Color online) Accuracy of HVM on predicting (a) size dependent and (b), (c) density dependent oscillations in normalized heat capacity for a nanowire with 200 nm long at 5K temperature.",
"Solid curves are the results obtained by using exact expressions (based on infinite sums), while dashed black curves are obtained by using HVM.In Fig.",
"7(a), sizes of the domain in transverse directions ($L_1$ and $L_2$ ) are changed simultaneously from 10 nm to 40 nm for $n=10^{25}\\text{m}^{-3}$ where $\\Lambda $ is always larger than $\\Lambda _{*}$ .",
"Confinement in transverse directions spans the values larger and smaller than $\\alpha _*$ .",
"HVM predicts oscillatory size dependence of electronic heat capacity pretty well between 10 nm and 25 nm, where confinement is strong, oscillations are large and unavoidable ($\\alpha \\ge \\alpha _*$ ).",
"For 25 nm and 40 nm interval, however, confinement becomes weaker ($\\alpha \\le \\alpha _*$ ) and an appreciable difference appears in between exact and HVM results while oscillations are also weak.",
"This is an expected result, since the system approaches to SR-OR interface in phase diagram.",
"In Fig.",
"7(b), quantum oscillations in heat capacity varying with electron density are shown for a square wire having 20 nm size in transverse directions.",
"Note that density variation for a constant confinement corresponds to a vertical movement on the phase diagram.",
"HVM perfectly matches with the exact model almost everywhere in Fig.",
"7(b) because of strong confinement, $\\alpha _{1}=\\alpha _{2}=1.48>\\alpha _{1*}^{3D}=1.27$ .",
"System shows oscillatory behavior regardless of the value of degeneracy (or electron density).",
"In other words, oscillations are unavoidable for the system discussed in Fig.",
"7(b) as long as the confinement is sustained.",
"By increasing the domain size from 20 nm to 25 nm in confined directions (reducing the confinement), we can put the system into the crossover regime where $\\alpha _{1}=\\alpha _2=1.18<\\alpha _{1*}^{3D}=1.23$ as it's shown in Fig.",
"7(c).",
"For this case, HVM does not match well with the exact model for lower electron densities, where the system is getting closer to SR-OR interface.",
"Oscillations decay by decreasing confinement and degeneracy but the decay rate of oscillations with degeneracy depends on how much the confinement smaller than the critical confinement value.",
"Since the confinement is not much smaller than $\\alpha _*$ , there is no considerable decay in oscillations in Fig.",
"7(c).",
"All these behaviors confirm the predictions of the phase diagram presented in Fig.",
"4 as well as aspect ratio dependencies of critical confinement parameters given in Fig.",
"5.",
"In contrast to size and density, variation of temperature does not lead to oscillatory behaviors in thermodynamic and transport quantities.",
"The reason can clearly be seen from Eqs.",
"(17) and (18) where oscillations are originated from the variation of round of $i_F$ .",
"Since temperature dependence of chemical potential disappears in strongly degenerate case, $[i_F]=[\\sqrt{\\Lambda }/\\alpha ]$ becomes independent of temperature.",
"Hence, in strongly degenerate case, temperature variation does not affect the proximities of states to the Fermi level in state space, but just changes the occupation probabilities of the states.",
"In other words, thermal broadening happens only in energy space, not in state space.",
"Since the proximities do not variate, oscillations do not appear.",
"This explanation can be directly extended to higher dimensional cases also, due to the orthogonality of state space dimensions."
],
[
"Oscillatory violation of entropy-heat capacity equivalence in degenerate limit",
"Continuum expressions of entropy and heat capacity of an ideal Fermi gas are equal to each other in degenerate limit [3].",
"When we consider the discrete nature of quantum states, however, entropy-heat capacity equivalence is also broken and quantum behaviors lead to an oscillatory non-equivalence of entropy and heat capacity of an ideal degenerate Fermi gas at nanoscale.",
"The well-known form of entropy is written as $S=-g_s k_B\\left[\\sum _{i_n=1}^{\\infty }{\\tilde{f}\\ln (\\tilde{f})+(1-\\tilde{f})\\ln (1-\\tilde{f})}\\right].$ In degenerate limit, both Eq.",
"(22) and (23) give the same expression (for instance, $g_s\\pi ^2Nk_B/(2\\Lambda )$ for 3D case) under continuum approximation.",
"However, as it is seen from Fig.",
"8, this equivalence is broken.",
"Heat capacity/entropy ratio deviates from unity and variates with domain size and density in an oscillatory fashion.",
"A nanowire in the previous subsection is reconsidered with the same specifications here for the examination of heat capacity-entropy ratio.",
"Both electronic heat capacity and entropy oscillate in confined and degenerate conditions.",
"Although their scales are almost the same, due to the phase and magnitude differences of both function's characteristic oscillations, their ratio also oscillates around unity.",
"Increasing domain size or temperature decreases the amplitude of oscillations.",
"Similar to heat capacity results, the success of HVM is quite good for higher electron density and confinement conditions.",
"Figure: (Color online) Oscillatory violation of entropy-heat capacity equivalence in degenerate limit depending on (a) size and (b), (c) density for a nanowire with 200 nm long at 5K temperature.",
"Solid curves are the results obtained by using exact expressions, while dashed black curves are obtained by using HVM.Very similar to the behavior of TOV, fully occupied or completely unoccupied states do not contribute to the entropy of the gas and contributions come only from the states around Fermi level, which are identified as HV states in HVM.",
"Therefore, it is possible to use HVM also for the quantities which does not directly contain occupancy variance, as long as HV states are dominant over the quantity to be calculated like in entropy."
],
[
"Resemblance of total occupancy variance and density of states functions",
"There is a direct relationship between DOS at Fermi level and TOV.",
"It's well-known that, in strongly degenerate case, spin normalized distribution function nearly turns into a Heaviside step function $f(\\tilde{\\varepsilon })\\approx g_s\\Theta (\\Lambda -\\tilde{\\varepsilon })$ .",
"Since the number of states below Fermi level $\\Lambda $ is approximately equal to the number of Fermions per spin state, $N=\\sum f(\\tilde{\\varepsilon })\\approx g_s\\sum \\Theta (\\Lambda -\\tilde{\\varepsilon })=NOS(\\Lambda )$ , derivative of $NOS(\\Lambda )$ with respect to $\\Lambda =\\tilde{\\varepsilon }_F$ gives $DOS(\\Lambda )$ and then TOV is approximately equal to $DOS(\\Lambda )$ .",
"$\\Sigma _{D}^2=-\\sum _{i_n}{\\frac{\\partial f}{\\partial \\tilde{\\varepsilon }}}\\approx g_s\\sum _{i_n}{\\delta (\\Lambda -\\tilde{\\varepsilon })}=DOS(\\Lambda ).$ In Fig.",
"9, resemblance of TOV and DOS functions is shown for two different subbands with constant confinement parameters for various dimensions.",
"Note that the results of lower dimensions are obtained by strongly confining a 3D domain in different directions.",
"Even the lower dimensional domains are represented by triple sums where sums in strongly confined directions represent subbands.",
"Hence, it's possible to directly obtain Eq.",
"(6) by using continuous DOS functions at Fermi level $\\tilde{\\varepsilon }_F=\\Lambda $ given in Table I.",
"While HVM predicts oscillations quite good, DOS functions represents only the trend of the oscillations (Fig.",
"9) like Weyl representations of TOV functions in Fig.",
"2 and continuous TOV function in Fig.",
"3.",
"Figure: (Color online) Similarity of TOV functions (solid curves) and continuous density of states functions at Fermi level (dashed curves) for various dimensions.Although the structure of two functions are very similar, due to thermal broadening at Fermi level there are small differences around the beginning of each subband especially for low dimensional cases.",
"The origin of this resemblance is basically coming from the fact that occupancy variance function has a peakwise nature at Fermi level and vanishing behavior elsewhere, which is directly related with density of states at Fermi level."
],
[
"Conclusion",
"In this article, investigation of size and density dependent quantum oscillations is done by proposing the HVM for the calculation of oscillatory quantities.",
"Proposition of our model not only allows to calculate oscillatory thermodynamic or transport quantities in a more efficient way, but also offers a phase diagram which predict quantum oscillations clearly.",
"Moreover, HVM introduces analytical conditions separating oscillatory and stationary regimes on the phase diagram.",
"Occupancy variance function appearing in many oscillatory quantities is examined in detail and the model is constructed on the half-vicinity of states around Fermi level.",
"Two main factors determine the value of TOV: the number of states inside the half-vicinity shell and their proximities to the Fermi level.",
"Confinement and degeneracy directly affect the distribution of the states around Fermi level besides the thickness of half-vicinity shell and so the number of states inside the shell.",
"The results show that HVM can be used as a reliable tool to predict and calculate quantum oscillations of a degenerate and confined ideal Fermi gas.",
"The proposed phase diagram and analytical conditions for quantum oscillations evidently describe entirely stationary/oscillatory and crossover regions on degeneracy-confinement space.",
"Consequently, our model and the phase diagram not only clarify the theoretical understanding of the quantum oscillations but also may help to determine the proper material parameters for experimental studies."
]
] | 1709.01816 |
[
[
"Blind image deblurring using class-adapted image priors"
],
[
"Abstract Blind image deblurring (BID) is an ill-posed inverse problem, usually addressed by imposing prior knowledge on the (unknown) image and on the blurring filter.",
"Most of the work on BID has focused on natural images, using image priors based on statistical properties of generic natural images.",
"However, in many applications, it is known that the image being recovered belongs to some specific class (e.g., text, face, fingerprints), and exploiting this knowledge allows obtaining more accurate priors.",
"In this work, we propose a method where a Gaussian mixture model (GMM) is used to learn a class-adapted prior, by training on a dataset of clean images of that class.",
"Experiments show the competitiveness of the proposed method in terms of restoration quality when dealing with images containing text, faces, or fingerprints.",
"Additionally, experiments show that the proposed method is able to handle text images at high noise levels, outperforming state-of-the-art methods specifically designed for BID of text images."
],
[
"Introduction",
"Blind image deblurring (BID) is an inverse problem where the observed image is modeled as the convolution of an underlying (sharp) image and an unknown blurring filter, often followed by additive noise.",
"The goal of BID is usually to estimate both the underlying image and the blurring filter.",
"The problem is obviously severely ill-posed.",
"In addition, since the convolution operator itself is typically ill-conditioned, the inverse problem is highly sensitive to the presence of noise.",
"In recent years, researchers have investigated a variety approaches to single image BID, mostly considering generic natural images [1], [2], [3], [4], [5], [6].",
"To deal with the ill-posed nature of the BID problem, most methods use prior information on both the image and the blurring filter.",
"The most common choice for the image prior exploits the statistics of natural images [1], [3], [4], [7], [8], [5], [9] and is usually based on implicit or explicit restoration of salient edges.",
"Although that approach gives good results for natural images, the prior itself is not designed for images that belong to specific classes (e.g., text, face, medical structures, fingerprints) appearing in many important applications, like document analysis, surveillance, and forensics.",
"Methods that use priors that capture the properties of images belonging to specific classes are more likely to provide better results, when dealing with those images, e.g., text [10], [11], [12] or face images [13], [14], [15].",
"Furthermore, images that belong to different specific classes may have different characteristics that are hard to capture with a unique prior.",
"For example, face images do not contain much texture and text images have specific structure due to the contents of interest being mainly in two tones (commonly, black and white).",
"Here, we proposed a method that uses patch-based image priors learned from a set of clean images of the specific class of interest.",
"The method is based on the so-called plug-and-play approach, recently proposed in [16].",
"In contrast with [16], we do not use a fixed denoiser, but a denoiser based on a Gaussian mixture model (GMM) that is learned from patches of clean images belonging to a specific class.",
"A similar idea was recently proposed for non-blind image deblurring and compressive imaging [17].",
"Here, in addition to the GMM-based image prior, we also adopt a weak prior on the blurring filter.",
"Considering the blur, earlier methods typically impose hard constrains for the (arguably) most relevant case of a generic motion blur by encouraging sparsity of the blur filter estimate [4], [18], [1], [8], [3], [19], [7].",
"In this paper, we use a weaker prior on the blur (limited support), thus being able to recover a wide variety of filters then those methods."
],
[
"Observation model",
"Consider the linear observation model $\\textbf {y} = \\textbf {H} \\textbf {x} + \\textbf {n}$ , where $\\textbf {y} \\in \\mathbb {R}^{n}$ , $\\textbf {x} \\in \\mathbb {R}^{m}$ denote the vectorized (lexicographically ordered) observed data and the (unknown) original image, respectively, and $\\textbf {n}$ is noise, assumed to be Gaussian, with zero mean and known variance $\\sigma ^2$ .",
"For computational convenience, $\\textbf {H} \\in \\mathbb {R}^{n \\times m}$ is the matrix that represents the convolution with the blurring filter $\\textbf {h}$ with periodic boundary conditions, thus with $n = m$ .",
"As explained above, to deal with the blind image deblurring problem, prior information (a regularizer) is imposed on both the underlying image and the blurring filter.",
"The image $\\textbf {x}$ and the blurring operator $\\textbf {H}$ (equivalently, the filter $\\textbf {h}$ ) are estimated by minimizing the cost function $O_\\lambda (\\textbf {x},\\textbf {h}) = \\frac{1}{2} ||\\textbf {y} - \\textbf {H x}||_2^2 + \\lambda \\phi (\\textbf {x}) + \\Psi _\\textit {S}(\\textbf {h}).$ We assume a weak prior on the blurring filter, $\\Psi _{\\textit {S}}$ , the indicator function of the set $\\textit {S}$ (set of filters with positive entries on a given support).",
"The rationale behind using a weak prior is that it covers a wider variety of blurring filters.",
"$\\Psi _{\\textit {S}}(\\textbf {u}) ={\\left\\lbrace \\begin{array}{ll}0 & \\quad \\text{if } \\textbf {u} \\in \\textit {S} \\\\\\infty & \\quad \\text{if } \\textbf {u} \\notin \\textit {S}.\\\\\\end{array}\\right.",
"}$ The function $\\phi $ represents the prior on the image used to promote characteristics that the underlying sharp image is assumed to have, while parameter $\\lambda $ controls the trade-off between data-fidelity term and the regularizer.",
"As shown recently [20], [6], good results can be obtained by alternating estimation of the image and the blur kernel (Algorithm 1).",
"Both steps are performed by using the alternating direction method of multipliers (ADMM) [6].",
"[H] Blind Image Deblurring Algorithm [1] Blurred image $\\textbf {y}$ Estimated sharp image $\\hat{\\textbf {x}}$ and the blur kernel $\\hat{\\textbf {h}}$ Initialization: Initial estimate $ \\hat{\\textbf {x}} = \\textbf {y}$ , $\\hat{\\textbf {h}}$ set to the identity filter, $\\lambda > 0$ stopping criterion is not satisfied $\\hat{\\textbf {x}} \\leftarrow \\underset{\\textbf {x}}{\\text{argmin}} \\hspace{5.69054pt} O_\\lambda (\\textbf {x}, \\hat{\\textbf {h}})$ estimating $\\textbf {x}$ with $\\textbf {h}$ fixed $\\hat{\\textbf {h}} \\leftarrow \\underset{\\textbf {h}}{\\text{argmin}} \\hspace{5.69054pt} O_\\lambda (\\hat{\\textbf {x}},\\textbf {h})$ estimating $\\textbf {h}$ with $\\textbf {x}$ fixed In contrast with [6], we do not decrease the regularization parameter $\\lambda $ at every iteration.",
"We found that a fixed parameter yields better results arguably due to the more expressive prior herein used, when compared with the total-variation regularizer in [6]."
],
[
"ADMM for image inverse problems",
"As discussed above, we use the ADMM optimization algorithm to perform estimation of both the image and the blurring filter, and therefore, in this section, we will briefly explain the ADMM for image inverse problems.",
"Consider an unconstrained optimization problem in which the objective function is the sum of two functions $\\underset{\\textbf {z}}{\\text{min}} \\hspace{5.69054pt} f_1 ( \\textbf {z}) + f_2 ( \\textbf {z}).$ By using a variable splitting procedure, we introduce a new variable $\\textbf {v}$ as the argument of the function $f_2$ , under the constrain that $\\textbf {z}=\\textbf {v}$ .",
"This leads to rewriting the unconstrained problem from above as a constrained one: $\\underset{\\textbf {z}, \\textbf {v}}{\\text{min}} \\hspace{5.69054pt} f_1(\\textbf {z}) + f_2 (\\textbf {v})\\hspace{11.38109pt} \\text{subject to} \\hspace{11.38109pt} \\textbf {z} = \\textbf {v}.$ The rationale behind variable splitting methods, such as the method of multipliers or augmented Lagrangian method (ALM), is that it may be easier to solve the constrained problem (4) instead of the unconstrained one (3).",
"The main idea behind the ALM is to minimize alternatingly the so-called augmented Lagrangian function $\\hat{\\textbf {z}}, \\hat{\\textbf {v}} \\leftarrow \\underset{\\textbf {z}, \\textbf {v}}{\\text{min}} \\hspace{5.69054pt} f_1(\\textbf {z}) + f_2 (\\textbf {v})+ \\textbf {d}^T(\\textbf {z}-\\textbf {v}) + \\frac{\\mu }{2}||\\textbf {z} - \\textbf {v}||_2^2,$ and updating the vector of Lagrange multipliers $\\textbf {d}$ (Algorithm 2).",
"In Equation (5), $\\mu \\ge 0$ is called the penalty parameter.",
"If we recall that, by definition, the proximity operator (PO) of some convex function $g$ , computed at the point $\\textbf {u}$ is defined as $\\text{prox}_g(\\textbf {u}) = \\underset{\\textbf {x}}{\\text{argmin}} \\hspace{5.69054pt}\\frac{1}{2}||\\textbf {x} - \\textbf {u}||_2^2 + g(\\textbf {x}),$ it is clear that in Algorithm 2, lines 3 and 4 are the PO of $f_1$ and $f_2$ , computed at $\\textbf {v}^k + \\textbf {d}^k$ and $\\textbf {z}^{k+1} - \\textbf {d}^k$ , respectively.",
"Formulation (6) can be considered as the solution to a denoising problem, with $\\textbf {u}$ as the noisy observation and $g$ the regularizer.",
"ADMM [1] Initialization: Set $k = 0$ , $\\mu > 0$ , initialize $\\textbf {v}_0$ and $\\textbf {d}_0 $ stopping criterion is not satisfied $ \\textbf {z}^{k+1} \\leftarrow \\underset{\\textbf {z}}{\\text{min}} \\hspace{5.69054pt} f_1(\\textbf {z}) + \\frac{\\mu }{2}||\\textbf {z} - \\textbf {v}^k - \\textbf {d}^k||_2^2$ $ \\textbf {v}^{k+1} \\leftarrow \\underset{\\textbf {v}}{\\text{min}} \\hspace{5.69054pt} f_2(\\textbf {v}) + \\frac{\\mu }{2}||\\textbf {z}^{k+1} - \\textbf {v} - \\textbf {d}^k||_2^2$ $\\textbf {d}^{k+1} \\leftarrow \\textbf {d}^k - ( \\textbf {z}^{k+1} - \\textbf {v}^{k+1} )$ $k \\leftarrow k + 1$"
],
[
"GMM-based Denoiser",
"For the image estimate update (line 3 of Algorithm 1), instead of using the PO of a convex regularizer (line 4 of the Algorithm 2), we implemented a state-of-the-art denoiser considering the fact that the PO itself is a denoising function.",
"This approach, also known as plug-and-play, was recently exploited in [16], but instead of using a fixed denoiser, such as BM3D [21] or K-SVD [22], we consider a class-adapted GMM-based denoiser [23].",
"In [23], the authors show that clean image patches are well modeled by a GMM estimated from a collection of clean images using the expectation-maximization (EM) algorithm.",
"Furthermore, for a GMM-based prior for the clean patches, the corresponding minimum mean squared error (MMSE) estimate can be obtained in closed form [24].",
"We use these facts to obtain a GMM-based prior learned from the set of clean images that belong to the specific class.",
"The rationale behind this approach is that with the class-adapted image prior, we may achieve better performance than with a fixed, generic denoiser, when we process images that do belong to the same specific class."
],
[
"Proposed method",
"The proposed method uses ADMM for solving each of the inner minimization problems in Algorithm 1 (lines 3 and 4), with $O_\\lambda (\\textbf {x},\\textbf {h})$ as defined in (1) with the PO of $\\phi $ replaced by the MMSE estimator using a class-adapted GMM prior."
],
[
"Image Estimate",
"The image estimation problem (line 3 of Algorithm 1) can be formulated as: $\\hat{\\textbf {x}} = \\underset{\\textbf {x}}{\\text{argmin}} \\hspace{5.69054pt} \\frac{1}{2} ||\\textbf {y} - \\textbf {H x}||_2^2 + \\lambda \\phi (\\textbf {x}).$ This problem can be written in the form (3) by setting $f_1(\\textbf {x}) = \\frac{1}{2} ||\\textbf {y} - \\textbf {H x}||_2^2$ and $f_2(\\textbf {x}) = \\lambda \\phi (\\textbf {x})$ .",
"Applying ADMM to problem (7), yields the so-called SALSA algorithm [25].",
"Line 3 of Algorithm 2 becomes a quadratic optimization problem, which has a linear solution: $\\textbf {x}^{k+1} = (\\textbf {H}^T\\textbf {H}+\\mu \\textbf {I})^{-1} (\\textbf {H}^T \\textbf {y} + \\mu (\\textbf {v}^k + \\textbf {d}^k)).$ As shown in some previous work [25], [26], the matrix inversion in (8) can be efficiently computed in the discrete Fourier transform (DFT) domain (using the FFT) in the case of cyclic deblurring, which we consider in this paper.",
"Extension to other boundary conditions can be obtained via the technique proposed in [6]."
],
[
"Blur Estimate",
"The blur estimation problem (line 4 of the Algorithm 1) can be formulated as: $\\hat{\\textbf {h}} = \\underset{\\textbf {h}}{\\text{argmin}} \\hspace{5.69054pt} \\frac{1}{2} ||\\textbf {y} - \\textbf {Xh}||_2^2 + \\Psi _\\textit {S}(\\textbf {h}),$ where $\\textbf {h} \\in \\mathbb {R}^{n}$ is the vector containing the lexicographically ordered blurring filter elements and $\\textbf {X} \\in \\mathbb {R}^{n \\times n}$ is the square matrix representing the convolution of the image $\\textbf {x}$ and the filter $\\textbf {h}$ .",
"Considering formulation (3) we have $f_1(\\textbf {h}) = \\frac{1}{2} ||\\textbf {y} - \\textbf {X h}||_2^2$ and $f_2(\\textbf {h}) = \\Psi _\\textit {S}(\\textbf {h})$ .",
"The resulting instance of line 3 of Algorithm 2 has the same form as (8) and, as previously explained, the matrix inversion can be efficiently computed in the DFT domain, using the FFT.",
"Since the proximity operator of the indicator of a convex set is the orthogonal projection on that set [27], line 4 of the Algorithm 2 becomes $\\text{prox}_{\\Psi _{\\textit {S}}}(\\textbf {u}) = P_{\\textit {S}}(\\textbf {u}),$ which simply sets to zero all negative elements and any elements outside of the given support."
],
[
"Experiments",
"In all the experiments, we use the following setting for the two ADMM algorithms described in Section 5: the image estimate is computed with 20 iterations of the algorithm in Subsection 5.1, initialized with the image estimate from the previous iteration, $\\textbf {d}_0 = 0$ , and $\\mu $ hand-tuned for the best visual results or best ISNR (improvement in SNR [6]) in terms of synthetic data.",
"The blur estimate is computed with two iteration of the algorithm explained in Subsection 5.2, initialized with the blur estimate from the previous iteration, $\\textbf {d}_0 = 0$ , and $\\mu = 0.01$ .",
"Furthermore, the experiments were performed on three sets of images: (a) a dataset containing 10 text images that is available from the author of [28] (one for testing and nine for training the mixture), (b) a dataset containing 100 face images from the same author as the text dataset, and (c) a dataset containing 128 fingerprints from the publicly available the UPEK fingerprint database.",
"The GMM-based prior is obtained by using patches of size $6 \\times 6$ pixels and a 20-component mixture."
],
[
"Results",
"For the experiments with text images, we created five test images using the same clean image of text and $15 \\times 15$ synthetic kernels that represent Gaussian, linear motion, out-of-focus, uniform, and nonlinear motion blur, respectively, and noise level corresponding to BSNR = 30 dB (Table 1).",
"For the face images, we created four $11 \\times 11$ synthetic blur kernels that represent Gaussian, linear motion, out-of-focus, and uniform blurs, respectively, and for the fifth experiment, we used blur kernel number 5 from [2], with noise level corresponding to BSNR = 40 dB (Table 2).",
"Experiments on the image containg fingerprints are performed with the $15 \\times 15$ linear motion blur kernel and noise level corresponding to BSNR = 40 dB (Fig.",
"2).",
"Results of all experiments are compared with two state-of-the-art BID algorithms constructed for natural images ([6] and [5]), and additionally with the algorithm explained in Subsection 5.1 with the BM3D denoiser plugged into it (PlugBM3D), instead of the GMM-based denoiser (PlugGMM).",
"Note that the generic algorithm [6] is designed for a wide variety of blur filters, while [5], like the majority of blind deblurring algorithms, is designed mostly for motion blur.",
"Figure: Text image blurred with nonlinear motion blur number 2 from and high noise level (BSNR = 20 dB): (a) Original image and ground truth kernel; (b) Blurred image; (c) Results of , ISNR = -2.72; (d) PlugBM3D, ISNR = 9.97; (e) PlugGMM, ISNR = 11.16.Figure: Fingerprint image blurred with 9×99 \\times 9 linear motion blur and noise level (BSNR = 40 dB): (a) Original image and ground truth kernel; (b) Blurred image; (c) Results of , ISNR = 0.36; (d) Results of , ISNR = -0.64; (e) PlugBM3D, ISNR = 0.56; (f) PlugGMM, ISNR = 1.19.Table: Results in terms of ISNR of the generic methods and , our method using the BM3D denoiser, and our method with the class-adapted GMM prior, tested for text images (BSNR = 30 dB).Table: Results in terms of ISNR of the generic methods and , our method using the BM3D denoiser, and our method with the class-adapted GMM prior, tested for face images (BSNR = 40 dB).Moreover, we tested our method on text images blurred with the blurring filter number 2 from [2], followed by a higher noise level (BSNR = 20 dB) (Fig.",
"1).",
"Results are compared, as previously explained, with the BM3D denoiser plugged into the ADMM loop and the method from Pan et al.",
"[11], which was designed for BID of text images.",
"As the BM3D denoiser is based on exploiting non-local patch similarities, which is highly present in the images we tested, visual results of using PlugBM3D are very good, but in terms of ISNR, PlugGMM clearly outperforms it."
],
[
"Conclusion",
"In this paper, we have proposed a class-adapted blind image deblurring method, built upon the so-called plug-and-play approach.",
"The method uses Gaussian mixture model (GMM) based denoisers, adapted to specific image classes, plugged into the ADMM optimization algorithm, and a weak prior (positivity and limited support) on the blurring filter.",
"Experiments show that the proposed method yields state-of-the-art results, when applied to images that belong to a specific class (e.g., text, face, and fingerprints), outperforming several generic techniques for blind image deblurring [5], [6].",
"In addition, experiments show that the proposed method can be used for a variety of blurring filters and is able to handle strong noise in the case of images known to contain text, outperforming the state-of-the-art method for BID of text images [11].",
"The proposed method suffers from some potential limitations, such as setting of the regularization parameter and stopping criteria for the inner ADMM algorithms, as well as for the outer iterations, that we aim to improve in future work."
]
] | 1709.01710 |
[
[
"Measurement of the shape of the $\\Lambda_b^0\\to\\Lambda_c^+ \\mu^-\n \\overline{\\nu}$ differential decay rate"
],
[
"Abstract A measurement of the shape of the differential decay rate and the associated Isgur-Wise function for the decay $\\Lambda_b^0\\to\\Lambda_c^+\\mu^-\\overline{\\nu}$ is reported, using data corresponding to $3 fb^{-1}$ collected with the LHCb detector in proton-proton collisions.",
"The $\\Lambda_c^+\\mu^-\\overline{\\nu}$(+ anything) final states are reconstructed through the detection of a muon and a $\\Lambda_c^+$ baryon decaying into $pK^-\\pi^+$, and the decays $\\Lambda_b^0\\to\\Lambda_c^+\\pi^+\\pi^-\\mu^-\\overline{\\nu}$ are used to determine contributions from $\\Lambda_b^0\\to \\Lambda_c^{\\star+}\\mu ^-\\bar{\\nu}$ decays.",
"The measured dependence of the differential decay rate upon the squared four-momentum transfer between the heavy baryons, $q^2$, is compared with expectations from heavy-quark effective theory and from unquenched lattice QCD predictions."
],
[
"0.5em"
]
] | 1709.01920 |
[
[
"Logarithmic Correction to BMSFT Entanglement Entropy"
],
[
"Abstract Using Rindler method we derive the logarithmic correction to the entanglement entropy of a two dimensional BMS-invariant field theory (BMSFT).",
"In particular, we present a general formula for extraction of the logarithmic corrections to both the thermal and the entanglement entropies.",
"We also present a CFT formula related to the logarithmic correction of the BTZ inner horizon entropy which results in our formula after taking appropriate limit."
],
[
"Introduction",
"One of the lessons of the AdS/CFT correspondence [1] is that the asymptotic symmetry of the asymptotically AdS spacetimes in d+1 dimensions is the same as conformal symmetry in one dimension lower.",
"One can use this idea to generalize the gauge/gravity duality beyond the AdS/CFT correspondence.",
"Accordingly , in any non-AdS/non-CFT correspondence, the symmetry of the dual field theory should be the same as the asymptotic symmetry of the gravity solutions.",
"The asymptotic symmetry of asymptotically flat spacetimes were known long before Maldacena's conjecture.",
"Such symmetries are known as BMS, which were first found at null infinity of four-dimensional asymptotically flat spacetimes [2], and later were generalized to the three-dimensional case [3].",
"Recently, Barnich and his collaborates [4] have shown that imposing just locally well-definiteness condition is enough to enhance both the translation and the rotation symmetry of the Poincare group at null infinity to an infinite-dimensional symmetry group.",
"In this work BMS group refers to this infinite-dimensional group which consists of super-rotation and super-translation generators.",
"In one dimension lower than gravity theory, the BMS algebra is given by an ultra-relativistic contraction of the conformal algebra.",
"Thus BMS symmetry can be the symmetry of a lower dimensional field theory.",
"It is proposed that the holographic dual of d+1-dimensional asymptotically flat spacetimes are ultra-relativistic d-dimensional BMS-invariant field theories (BMSFTs) [5],[6].",
"In flat-space holography, BMSFT entanglement entropy and its holographic dual were first studied in [7] and the entanglement entropy of a two-dimensional BMSFT was computed by a method similar to the CFT [8].",
"A holographic interpretation of the BMSFT entanglement entropy in [7] (similar to the Ryu-Takayanagi proposal in the context of AdS/CFT correspondence [9]) is given using Chern-Simons formulation of three-dimensional flat-space gravity [10].",
"Other aspects of BMSFT entanglement entropy has been studied in [11],[12].",
"A remarkable progress in this subject has been achieved recently in [13] where Rindler method [14] is used to derive not only the BMSFT entanglement entropy formula but also the holographic description in terms of some curves length in the bulk theory.",
"The central idea of the Rindler method is to find local unitary transformations which map entangled states to the thermal states in the field theory.",
"Then one can use the thermal entropy formula and find the entanglement entropy.",
"Calculation of thermal entropies in BMSFTs is performed using Cardy-like formula first introduced in [15].",
"Using Rindler method, the entanglement entropy is given by Cardy-like formula.",
"Similar to the Cardy-formula in CFT , the saddle point approximation is used to derive this formula.",
"Thus, it is possible to improve approximation and find possible corrections of this formula.",
"In [16], employing a method first used for the Cardy formula in [17]We note however that the Cardy (high temperature) limit is not always reliable for extracting the black hole entropy [18].",
"On the other hand, in two dimensional CFTs the logarithmic corrections are only universal in the Cardy regime.",
"From the dual three dimensional gravity perspective, the universality of the correction is usually because heat kernels do not contain logarithmic terms [19].",
"Thus the logarithmic correction to the BMSFT thermal and entanglement entropies are universal in the limit that BMSFT Cardy-like formula is reliable., logarithmic correction to the Cardy-like formula has been derived.",
"In this paper, we use the logarithmic correction to Cardy-like formula along with Rindler method to find the logarithmic correction of BMSFT entanglement entropy.",
"Similar to the leading term, this correction depends on central charges of BMS algebra, the interval of the sub-system and the cut-off.",
"However, the interesting point is that we can rewrite the corrections in a universal form: $S=S_0-3\\log \\left(C_M^{\\frac{1}{3}}\\dfrac{\\partial S_0}{\\partial C_L}\\right)$ where $S_0$ is the leading term and $C_L$ and $C_M$ are the central charges of BMS algebra.",
"$S$ can be both of the thermal entropy (which is given by the Cardy-like formula) and the entanglement entropy.",
"This formula works for all BMSFTs on plane or cylinder at zero or finite temperature.",
"The entanglement entropy of the boundary theory can be used to reconstruct the bulk dynamics [20], [21].",
"In this view, the logarithmic correction of the entanglement entropy might be helpful to find quantum correction to the Einstein field equations.",
"This paper is the first step on this road.",
"The idea is to use the first law of the entanglement entropy as the analogue of the first law of (black hole) thermodynamics [20], [21], [22].",
"The better understanding of BMSFT modular hamiltonian might allow us to achieve not only the classical bulk dynamics but also the quantum correction to the Einstein equation without the cosmological constant.",
"One approach to study Flat/BMSFT is to take limit from the AdS/CFT calculations.",
"According to the proposal of [6], the flat space limit in the gravity side corresponds to taking an ultra-relativistic limit from the CFT calculations.",
"Hence one can find all BMSFT formulas by taking limit from a CFT counterpart.",
"It was shown in [23] and [24] that the Cardy-like formula of BMSFT is given by taking limit from a formula in the CFT which is related to the inner horizon entropy of the gravity solution.",
"If we assume the same relation for the logarithmic correction, the logarithmic term in (REF ) is related to a logarithmic correction in the entropy of the inner horizon.",
"We propose a suitable logarithmic correction in the inner horizon formula which corresponds to the logarithmic term of (REF ) after taking limit.",
"Using logarithmically corrected inner and outer horizon entropy formulas we can calculate their multiplication.",
"The observation is that the previously known result, the multiplication being mass independent for the Einstein gravity [25] (see also [26]) , is violated in the presence of the logarithmic terms.",
"The structure of this paper is as follows: In section two we review the results of [13] which uses the Rindler method to derive the BMSFT entanglement entropy and its holographic description.",
"In section three we review the derivation of BMSFT Cardy-like formula and its logarithmic correction.",
"In section four we put together the results of section two and three to derive the logarithmic correction of BMSFT entanglement entropy.",
"Section five is devoted to the derivation of the BMSFT entanglememnt entropy formula by taking flat-space limit."
],
[
"Entanglement Entropy of BMSFT using Rindler method",
"One of the advantages of the Rindler method is the convenience it provides for the calculation of entanglement entropy in the context of gauge/gravity duality.",
"In this view, the thermal entropy of the boundary theory is mapped to the horizon entropy of the black hole (object) in the gravity side.",
"Thus, one can use it to prove Ryu-Takayanagi (RT) formula for the holographic entanglement of CFTs.",
"Moreover, since many thermal properties of the gravitational systems are known for the non-AdS cases, we expect that the Rindler method gives a lot of insights to find the analogue of RT formula for the dualities beyond the AdS/CFT correspondence.",
"In Rindler method, the asymptotic symmetries play an essential role.",
"We are interested in BMSFTs which are proposed to be the holographic dual of asymptotically flat spacetimes.",
"In our case, a Rindler transformation is of the form of BMS transformation and final coordinates are invariant under specific thermal identifications.",
"These transformations act like unitary operators $U_{\\mathcal {R}}$ on the fields and map the reduced density matrix in the subregion $\\mathcal {A}$ (which we want to calculate the entanglement entropy for) to a thermal density matrix in the interval $\\mathcal {B}$ , $\\rho _{\\mathcal {A}} = U_{\\mathcal {R}}\\, \\rho _{\\mathcal {B}}\\, U_{\\mathcal {R}}^{-1}.$ Since the unitary transformations do not change the entropy, the thermal entropy of the subregion $\\mathcal {B}$ is the same as entanglement entropy of the subregion $\\mathcal {A}$ .",
"A two-dimensional BMSFT has the following symmetry [27]: $\\nonumber \\tilde{u} &= \\partial _{\\phi }{f(\\phi )} u +g(\\phi ),\\\\\\tilde{\\phi } &= f(\\phi ),$ where $f(\\phi )$ and $g(\\phi )$ are arbitrary functions of the original coordinate $(u,\\phi )$ .",
"The BMS algebra is given by using infinitesimal BMS transformation as $\\nonumber [L_n,L_m]&=(n-m)L_{n+m}+{C_L\\over 12}n(n^2-1)\\delta _{n+m,0},\\\\\\nonumber [L_n,M_m]&=(n-m)M_{n+m}+{C_M\\over 12}n(n^2-1)\\delta _{n+m,0},\\\\{[}M_n,M_m]&=0.$ where for a BMSFT on a plane with coordinates $(u,\\phi )$ the generators $L_n$ and $M_n$ are given by $\\nonumber L_{n} &= -u (n+1) \\phi ^n \\partial _{u} - \\phi ^{n+1} \\partial _{\\phi }\\\\M_{n} &= \\phi ^{n+1}\\partial _{u}.$ The global part of this algebra is identified with $n=0,\\pm 1$ .",
"A Rindler transformation is of the form $\\tilde{x}=T(x)$ , but it should be invariant under some imaginary identification (thermal identification) of the new coordinate $\\tilde{x}^i\\sim \\tilde{x}^i+i\\tilde{\\beta }^i$ .",
"Moreover, vectors $\\partial _{\\tilde{x}^i}$ annihilate the vacuum and hence should be written as the linear combination of the global part of the BMS algebra: $\\partial _{\\tilde{x}^i} = \\sum _{n=-1}^{1}{( b_{n} L_{n} +d_{n} M_{n} )}.$ Using (REF ) and (REF ) we conclude that $\\partial _{\\phi }{f(\\phi )} = \\frac{1}{Y}, \\qquad \\partial _{\\phi }{g(\\phi )} = -\\frac{T}{Y^2},$ where $Y &= - b_{-1} -b_{0} \\phi -b_{1} \\phi ^2,\\\\T &= d_{-1} + d_{0} \\phi + d_{1} \\phi ^2.$ Solutions to (REF ) determine vector $ \\partial _{\\tilde{x}^i}$ .",
"It is assumed that entangled region is given by $\\mathcal {A} = \\lbrace (\\frac{-l_{\\phi }}{2} , \\frac{-l_{u}}{2}) \\cup (\\frac{l_{\\phi }}{2} , \\frac{l_{u}}{2})\\rbrace $ .",
"We can use the following constraints to find vector $ \\partial _{\\tilde{x}^i}$ and the geometric (modular) flow $ k_t = -\\tilde{\\beta _{\\phi }} \\partial _{\\tilde{\\phi }} + \\tilde{\\beta _{u}} \\partial _{\\tilde{u}} $ : The finite interval on $\\mathcal {A}$ should be mapped to the infinite interval on $\\mathcal {B}$ , $( \\frac{-l_{\\phi }}{2} , \\frac{-l_u}{2} ) &\\rightarrow ( -\\infty , -\\infty ),\\\\( \\frac{l_{\\phi }}{2} , \\frac{l_u}{2} ) &\\rightarrow ( \\infty , \\infty ).$ The origin of the entangled interval is mapped to the origin of the thermal interval $(0, 0) \\rightarrow (0,0)$ The thermal interval (tilde coordinate) obeys a thermal identification of the following form $( \\tilde{\\phi } , \\tilde{u} ) \\sim ( \\tilde{\\phi } + i \\tilde{\\beta }_{\\phi } ,\\tilde{u} - i \\tilde{\\beta }_{u} ) .$ The modular flow $k_t$ vanishes on the boundary of the entangled region $k_t(\\partial {\\mathcal {A}} ) = 0 \\Rightarrow {\\left\\lbrace \\begin{array}{ll}k_t( \\frac{-l_\\phi }{2} ,\\frac{-l_u}{2} ) = 0,\\\\k_t( \\frac{l_\\phi }{2} ,\\frac{l_u}{2} ) =0\\end{array}\\right.",
"}$ Using these conditions, [13] completely determines the Rindler transformation and modular flow of a BMSFT on the plane as below: $\\tilde{\\phi } &= \\frac{ \\tilde{ \\beta }_{\\phi } }{ \\pi } \\tanh ^{-1}{ \\frac{2 \\phi }{ l_{\\phi }} } ,\\\\\\tilde{u} + \\frac{ \\tilde{\\beta }_{u} }{ \\tilde{\\beta }_{\\phi } } \\tilde{\\phi } &= \\frac{2 \\tilde{\\beta }_{\\phi } ( u l_{\\phi } - l_{u} \\phi )}{\\pi (l_{\\phi }^2 - 4 \\phi ^2)},$ $k_t = -\\tilde{\\beta }_{\\phi } \\partial _{\\phi } + \\tilde{\\beta }_{u} \\partial _{u} =\\frac{- \\pi }{2 l_{\\phi }} \\left( \\left(l_{\\phi }^2 - 4 \\phi ^2\\right)\\partial _{\\phi } + \\left(l_u l_{\\phi } +4 \\frac{lu}{l_{\\phi }} \\phi ^2- 8 u \\phi \\right)\\partial _{u} \\right)$ For a BMSFT with identification of coordinates as $(\\tilde{u} , \\tilde{\\phi } ) \\sim ( \\tilde{u} + i \\bar{a} , \\tilde{\\phi } - i a) \\sim ( \\tilde{u} + 2 \\pi \\bar{b} , \\tilde{\\phi } - 2 \\pi b),$ The degeneracy of states is given by a Cardy-like formula [15], [13], [28] (see next section).",
"$S_{\\bar{b}|b}(\\bar{a}|a) = \\frac{- \\pi ^2}{3} \\left(C_{L} \\frac{b}{a} +C_{M} \\frac{(\\bar{a} b - a \\bar{b} )}{a^2}\\right).$ Using (REF ), the entanglement entropy of a BMSFT on the plane for the interval $\\mathcal {A} = \\lbrace (\\frac{-l_{\\phi }}{2} + \\epsilon _{\\phi } , \\frac{-l_{u}}{2} + \\epsilon _{u} ) \\cup (\\frac{l_{\\phi }}{2} -\\epsilon _{\\phi } , \\frac{-l_{u}}{2} - \\epsilon _{\\phi } )\\rbrace $ becomes [13] $S_{EE} = \\frac{C_{L}}{6} \\log { \\frac{l_{\\phi }}{\\epsilon _{\\phi }} } + \\frac{C_{M} }{6} \\left( \\frac{l_{u}}{l_{\\phi } }- \\frac{\\epsilon _{u}}{\\epsilon _{\\phi }}\\right),$ where $\\epsilon _u$ and $\\epsilon _\\phi $ are ultraviolet cut-offs in $u$ and $\\phi $ coordinates.",
"Similarly the entanglement entropy for finite temperature BMSFT on the cylinder has been calculated in [13]."
],
[
"Logarithmic Correction to Cardy-like Formula of BMSFT",
"The entropy of CFT thermal states is calculated using Cardy formula.",
"Using the saddle point approximation, it is possible to find a similar formula for the degeneracy of thermal states in a BMSFT [15], [13], [28].",
"Due to the Rindler method, any correction to Cardy-like formula has influence on the entanglement entropy formula.",
"In this section, we review the logarithmic correction to Cardy-like formula [16] and then use it to find the logarithmic correction to the entanglement entropy.",
"Method of [16] is based on [17] that introduces the first order logarithmic correction to the Cardy formula (For a calculation of all order corrections see [29]) We start from the modular invariant partition function of BMSFT on a torus defined by $Z_{0}( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ) = \\operatorname{Tr}e^{-\\hat{\\beta }_{u} (M_{0} - \\frac{C_{M}}{24}) + \\hat{\\beta }_{\\phi } (L_{0} -\\frac{C_{L}}{24}) }= e^{\\hat{\\beta }_{u} \\frac{C_{M}}{24} - \\hat{\\beta }_{\\phi } \\frac{C_{L}}{24} } Z( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ),$ where $ Z( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } )$ is $Z( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ) =\\operatorname{Tr}e^{-\\hat{\\beta }_{u} M_{0} + \\hat{\\beta }_{\\phi } L_{0}}= \\sum _{h_M,h_L} e^{( -\\hat{\\beta }_{u} h_{M} + \\hat{\\beta }_{\\phi } h_{L})} d(h_{M},h_{L}),$ and identification of torus are $(\\hat{u},\\hat{\\phi })\\sim (\\hat{u}+i\\hat{\\beta }_u,\\hat{\\phi }-i\\hat{\\beta }_\\phi )\\sim (\\hat{u},\\hat{\\phi }-2\\pi ).$ Here, $h_L$ and $h_M$ are respectively the eigenvalues of $L_0$ and $M_0$ .",
"It is shown that the BMS modular invariant partition function satisfies [13] $Z_{0}( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ) =Z_{0}\\left(-4 \\pi ^2 \\frac{\\hat{\\beta }_{u}}{\\hat{\\beta }_{\\phi }^2} | \\frac{4 \\pi ^2}{\\hat{\\beta }_{\\phi }}\\right).$ Plugging equation (REF ) into (REF ) gives, $Z( \\hat{\\beta }_{u} | \\hat{\\beta }_{\\phi } ) = e^{ \\hat{\\beta }_{u} \\frac{C_M}{24} - \\hat{\\beta }_{\\phi } \\frac{C_L}{24} - \\frac{ \\pi ^2 \\hat{\\beta }_{u} C_{M}}{6 \\hat{\\beta }_{\\phi }^2} - \\frac{ \\pi ^2 C_{L}}{6 \\hat{\\beta }_{\\phi }} } Z\\left(-4 \\pi ^2 \\frac{\\hat{\\beta }_{u}}{\\hat{\\beta }_{\\phi }^2} | \\frac{4 \\pi ^2}{\\hat{\\beta }_{\\phi }}\\right).$ By using the inverse Laplace transformation in last term of (REF ) we find $d(h_{L} , h_{M} ) = \\int d\\hat{\\beta }_{u} d\\hat{\\beta }_{\\phi } e^{ \\hat{\\beta }_{u} \\frac{C_M}{24} - \\hat{\\beta }_{\\phi } \\frac{C_L}{24} - \\frac{ \\pi ^2 \\hat{\\beta }_{u} C_{M}}{6 \\hat{\\beta }_{\\phi }^2} - \\frac{ \\pi ^2 C_{L}}{6 \\hat{\\beta }_{\\phi }} + \\hat{\\beta }_{u} h_M - \\hat{\\beta }_{\\phi } h_L } Z\\left(-4 \\pi ^2 \\frac{\\hat{\\beta }_{u}}{\\hat{\\beta }_{\\phi }^2} | \\frac{4 \\pi ^2}{\\hat{\\beta }_{\\phi }} \\right).$ In order to simplify (REF ), we use two approximations.",
"First, we consider large charges which yields $d(h_{L} , h_{M} ) = \\int d\\hat{\\beta }_{u} d\\hat{\\beta }_{\\phi } e^{ - \\frac{ \\pi ^2 \\hat{\\beta }_{u} C_{M}}{6 \\hat{\\beta }_{\\phi }^2} - \\frac{ \\pi ^2 C_{L}}{6 \\hat{\\beta }_{\\phi }} + \\hat{\\beta }_{u} h_M - \\hat{\\beta }_{\\phi } h_L } Z\\left(-4 \\pi ^2 \\frac{\\hat{\\beta }_{u}}{\\hat{\\beta }_{\\phi }^2} | \\frac{4 \\pi ^2}{\\hat{\\beta }_{\\phi }}\\right).$ Then, we approximate (REF ) around the saddle point given by $(\\hat{\\beta }_{\\phi }^{s})^2 = \\frac{\\pi ^2 C_{M}}{6 h_{M}},\\qquad \\hat{\\beta }_{u}^{s} ={\\hat{\\beta }_{\\phi }^{s}\\over 2h_M\\,C_M} (C_{M} h_{L} - C_{L} h_{M} ) .$ Finally, the entropy or Cardy like formula for BMSFT reads as It is assumed that the partition function is slowly varying at the extremum.",
"$S^{0} = \\log {d(h_{L} , h_{M} )} = -{\\pi ^2\\over 3 (\\hat{\\beta }_{\\phi }^{s})^2 }\\left(C_M \\hat{\\beta }_{u}^{s}+C_L \\hat{\\beta }_{\\phi }^{s}\\right).$ To find logarithmic correction we expand the integral (REF ) around the saddle point (REF ) up to quadratic term: $d(h_{L} , h_{M} ) = e^{S^{0}} \\int d\\hat{\\beta }_{u} d\\hat{\\beta }_{\\phi } e^{\\frac{1}{2} (X^2 -Y^2)},$ where $X &= {\\pi C_M\\over 3 \\hat{\\beta }^{s}_{\\phi } A}(\\hat{\\beta }_{u} - \\hat{\\beta }^{s}_{u}) , \\qquad Y = {\\pi A\\over (\\hat{\\beta }^{s}_{\\phi })^2} \\left[ -(\\hat{\\beta }_{\\phi } - \\hat{\\beta }^{s}_{\\phi })+ {C_M \\hat{\\beta }^{s}_{\\phi }\\over 3 A^2} (\\hat{\\beta }_{u}-\\hat{\\beta }^{s}_{u})\\right],\\\\A &= \\sqrt{C_M \\hat{\\beta }^{s}_{u}+\\frac{1}{3} C_L\\hat{\\beta }^{s}_{\\phi }}.$ Using (REF ), Eq.",
"(REF ) takes the form $d(h_{L} , h_{M} ) = e^{S^{0}}\\left(-{\\pi ^2 C_M\\over 3(\\hat{\\beta }^s_\\phi )^3}\\right)^{-1} \\int dX dY e^{\\frac{X^2 -Y^2}{2} }.$ In the above equation the result of integration is just a number.",
"Thus we find correction to the entropy up to a constant as [16] $S = \\log { d(h_{L} , h_{M} ) } = -{\\pi ^2\\over 3 (\\hat{\\beta }_{\\phi }^{s})^2 }\\left(C_M \\hat{\\beta }_{u}^{s}+C_L \\hat{\\beta }_{\\phi }^{s}\\right)-3\\log \\left(-{C_M^\\frac{1}{3}\\over \\hat{\\beta }_{\\phi }^{s} }\\right)+\\text{constant}.$ The interesting point is that the logarithmically corrected term can be rewritten (up to a constant) as the derivative of the leading term with respect to $C_L$ : $S=S_0-3\\log \\left(C_M^{\\frac{1}{3}}\\dfrac{\\partial S_0}{\\partial C_L}\\right).$"
],
[
"Logarithmic Correction to Entanglement Entropy",
"In this section, we put together the results from sections two and three to find the logarithmic correction of BMSFT entanglement entropy.",
"As mentioned before, the idea is to map entangled states to thermal states and then calculate entropy.",
"The entropy is computed using BMSFT Cardy-like formula (REF ) which also has a logarithmic correction.",
"In the derivation of (REF ) the identification of coordinates (REF ) plays an essential role.",
"It is possible to use (REF ) for finding the degeneracy of thermal states with more generic identification of coordinates as $(\\tilde{u},\\tilde{\\phi })\\sim (\\tilde{u}+i\\bar{a},\\tilde{\\phi }-ia)\\sim (\\tilde{u}+2\\pi \\bar{b},\\tilde{\\phi }-2\\pi b)$ The coordinate change between $(\\tilde{u},\\tilde{\\phi })$ and $(\\hat{u},\\hat{\\phi })$ is a BMS transformation, $\\hat{\\phi }={\\tilde{\\phi }\\over b},\\qquad \\hat{u}={\\tilde{u} \\over b}+{\\bar{b}\\over b^2}\\tilde{\\phi },$ where $\\hat{\\beta }_\\phi ={a\\over b},\\qquad \\hat{\\beta }_u={\\bar{a}\\,b-a\\,\\bar{b}\\over b^2}.$ Thus Cardy-like formula (REF ) can be written as $S=-{\\pi ^2\\over 3}\\left(C_L{b\\over a}+C_M{\\bar{a}\\,b-a\\,\\bar{b}\\over a^2}\\right)-3\\log \\left(-C_M^\\frac{1}{3}{b\\over a}\\right)$ In order to find the logarithmic correction of BMSFT entanglement entropy, it is enough to map the entanglement entropy to a thermal entropy and then use (REF ).",
"As it was reviewed in section , the Rindler transformation which governs this map is determined in such a way that finally induces the thermal identification (REF ).",
"Comparing (REF ) to (REF ) shows that $a=\\tilde{\\beta }_\\phi ,\\qquad \\bar{a}=\\tilde{\\beta }_u.$ The values of $\\tilde{\\beta }_\\phi $ , $\\tilde{\\beta }_u$ and $b$ , $\\bar{b}$ depend on the details of the Rindler transformation.",
"These are given in terms of the cut-offs and the interval for which entanglement entropy is calculated.",
"Starting from a regulated interval in the BMSFT given by $(-{l_u\\over 2}+\\epsilon _u,-{l_\\phi \\over 2}+\\epsilon _\\phi )\\rightarrow ({l_u\\over 2}-\\epsilon _u,{l_\\phi \\over 2}-\\epsilon _\\phi ),$ the Rindler transformation yields the following results [13]: For the zero temperature BMSFT on the plane we have $a&=\\tilde{\\beta }_\\phi =-{2\\pi ^2\\over \\log {l_\\phi \\over \\epsilon _\\phi }},\\qquad \\bar{a}=\\tilde{\\beta }_u=-{\\tilde{\\beta }_\\phi ^2\\over 2\\pi ^2}\\left({l_u\\over l_\\phi }-{\\epsilon _u\\over \\epsilon _\\phi }\\right),\\\\b&=-{\\tilde{\\beta }_\\phi \\over 2\\pi ^2}\\log {l_\\phi \\over \\epsilon _\\phi },\\qquad \\bar{b}={1\\over 2\\pi ^2}\\left({\\tilde{\\beta }_\\phi l_u\\over l_\\phi }-{\\tilde{\\beta }_\\phi \\epsilon _u\\over \\epsilon _\\phi }-\\tilde{\\beta }_u\\log {l_\\phi \\over \\epsilon _\\phi }\\right),$ Then the logarithmically corrected Cardy-like formula (REF ) becomes $S_{EE}={C_L\\over 6}\\log {l_\\phi \\over \\epsilon _\\phi }+{C_M\\over 6}\\left({l_u\\over l_\\phi }-{\\epsilon _u\\over \\epsilon _\\phi }\\right)-3\\log \\left(C_M^\\frac{1}{3}\\log {l_\\phi \\over \\epsilon _\\phi }\\right)+\\text{constant}.$ The third term in the above formula is the calculated correction.",
"For the finite temperature BMSFT with identification $(u,\\phi )\\sim (u+i\\beta _u,\\phi -i\\beta _\\phi ),$ we can use the results of [13] and Eq.",
"(REF ) to write $\\nonumber S_{EE}={C_L\\over 6}\\log \\left({\\beta _\\phi \\over \\pi \\epsilon _\\phi }\\sinh {\\pi l_\\phi \\over \\beta _\\phi }\\right)&+{C_M\\over 6}{1\\over \\beta _\\phi }\\left[\\pi \\left(l_u+{\\beta _u\\over \\beta _\\phi }l_\\phi \\right)\\coth {\\pi l_\\phi \\over \\beta _\\phi }-\\beta _u\\right]-{C_M\\epsilon _u\\over 6\\,\\epsilon _\\phi }\\\\ &-3\\log \\left(C_M^{1\\over 3}\\log \\left({\\beta _\\phi \\over \\pi \\epsilon _\\phi }\\sinh {\\pi l_\\phi \\over \\beta _\\phi }\\right)\\right)+\\text{constant}.$ The forth term in the above equation is the obtained correction for finite temperature.",
"For the zero temperature BMSFT on the cylinder, the entanglement entropy with logarithmic correction reads as $S_{EE}={C_L\\over 6}\\log \\left({2\\over \\epsilon _\\phi }\\sin {l_\\phi \\over 2}\\right)+{C_M\\over 12}\\left(l_u\\cot {l_\\phi \\over 2}-{2\\epsilon _u\\over \\epsilon _\\phi }\\right)-3\\log \\left(C_M^{1\\over 3}\\log \\left({2\\over \\epsilon _\\phi }\\sin {l_\\phi \\over 2}\\right)\\right)+\\text{costant}.$ It is clear from all the above cases that up to a constant, the entanglement entropy formula including the logarithmic correction is given by $S_{EE}=S_0-3\\log \\left(C_M^{\\frac{1}{3}}\\dfrac{\\partial S_0}{\\partial C_L}\\right)$ This formula is also valid for the thermal entropy when $S_0$ is the Cardy-like formula.",
"Thus, we propose a universal form for the logarithmic correction of entanglement entropy and thermal entropy.",
"We expect the same common form for the CFT case.",
"In the next section, we propose a similar form of logarithmic correction to CFT thermal and entanglement entropy and try to find (REF ) by taking limit from CFT formula."
],
[
"Logarithmic correction of BMSFT entropy by taking limit from CFT counterpart",
"For a two dimensional CFT with central charges $c$ and $\\bar{c}$ and right and left temperatures $\\beta $ and $\\bar{\\beta }$ , the Cardy formula is $S_0={\\pi ^2\\over 3}\\left({c\\over \\beta }+{\\bar{c}\\over \\bar{\\beta }}\\right).$ Using AdS/CFT correspondence, this formula results in the same entropy as the outer horizon entropy of asymptotically AdS black holes in the gravity side.",
"The logarithmic correction to (REF ) was evaluated in [17]: $S_{log}=-{3\\over 2}\\log \\left({c^{1/3}\\over \\beta }\\right)-{3\\over 2}\\log \\left({\\bar{c}^{1/3}\\over \\bar{\\beta }}\\right)+\\text{constant}.$ Employing (REF ) and (REF ) we can write $S=S_0+S_{log}=S_0-{3\\over 2}\\log \\left(c^{1/3}\\dfrac{\\partial S_0}{\\partial c}\\right)-{3\\over 2}\\log \\left(\\bar{c}^{1/3}\\dfrac{\\partial S_0}{\\partial \\bar{c}}\\right)+\\text{constant}.$ Using Rindler method we can expect the same formula for the logarithmically corrected CFT entanglement entropy.",
"Thus we propose (REF ) as a universal formula that can be used for both the thermal entropy and the entanglement entropy.",
"It is known that taking flat space limit from the asymptotically AdS spacetimes (written in the appropriate coordinate) yields asymptotically flat spacetimes.",
"It is proposed in [6] that the flat space limit in the bulk corresponds to taking the ultra-relativistic limit in the boundary CFT.",
"In other words, BMSFT is given by taking ultra-relativistic limit from the CFT.",
"Starting from conformal algebra in two dimensions, one can introduce an ultra-relativistic contraction and generate BMS algebra [6].",
"The relation between central charges of conformal algebra and BMS algebra is given by $C_L=\\lim _{\\epsilon \\rightarrow 0}\\left(c-\\bar{c}\\right),\\qquad C_M=\\lim _{\\epsilon \\rightarrow 0}\\epsilon \\left(c+\\bar{c}\\right)$ where $\\epsilon $ is a dimensionless parameter which corresponds to $G/\\ell $ on the gravity side.",
"$G$ is the Newton constant and $\\ell $ is the AdS-radius.",
"Thus $\\epsilon \\rightarrow 0$ in the boundary corresponds to $\\ell \\rightarrow \\infty $ .",
"It is plausible to find all BMSFT quantities by taking limit from the CFT counterparts.",
"Thus one can look for the possible relation between (REF ) and (REF ).",
"However, assuming $S_0$ in (REF ) as the Cardy formula (REF ) and using (REF ) does not result in the Cardy-like formula (REF ).",
"It was shown in [23] and [24] that the appropriate formula which its limit yields the Cardy-like formula is $S_{inner}^0={\\pi ^2\\over 3}\\left({c\\over \\beta }-{\\bar{c}\\over \\bar{\\beta }}\\right).$ For taking limit we should scale the temperatures as below: $\\beta _u=\\lim _{\\epsilon \\rightarrow 0}\\left({\\beta -\\bar{\\beta }\\over 2 \\epsilon }\\right), \\qquad \\beta _\\phi =\\lim _{\\epsilon \\rightarrow 0}-\\left({\\beta +\\bar{\\beta }\\over 2 }\\right).$ Using AdS/CFT correspondence, (REF ) corresponds to the inner horizon entropy of asymptotically AdS black holes.",
"Thus, we expect that taking limit from (REF ) which is written for the outer horizon does not yield (REF ).",
"In the rest of this section, we introduce appropriate formula which taking limit from it result in (REF ).",
"Let us start from $S^0$ in (REF ).",
"When it is the thermal entropy given by the Cardy-like formula, the corresponding formula in the CFT will be (REF ).",
"The question is then what is the corresponding CFT formula when $S^0$ in (REF ) is not the leading order of entanglement entropy?",
"It is known that taking ultra-relativistic limit from the entanglement entropy of CFT is not well-defined.",
"For example, for a zero temperature CFT on a plane, the entanglement entropy of an interval $-{R\\over 2}(\\cosh \\kappa ,\\sinh \\kappa )\\rightarrow {R\\over 2}(\\cosh \\kappa ,\\sinh \\kappa ),$ is given by [30] $S_{EE}={c+\\bar{c}\\over 2}\\log {R\\over \\epsilon _R}-{c-\\bar{c}\\over 6}\\kappa $ where $\\epsilon _R$ is a cut-off.",
"Using (REF ) together with ultra-relativistic contraction $t\\rightarrow \\epsilon t$ shows that (REF ) is divergent in the $\\epsilon \\rightarrow 0$ limit.",
"It is proposed in [11] that BMSFT entanglement entropy can be found by taking the limit from the CFT counterpart.",
"In the prescription of [11] it is argued that the symmetries are enough to fix the form of the entanglement entropy.",
"Then they use a new contraction of conformal algebra which results in BMS algebra.",
"But the relation between central charges of conformal algebra and BMS algebra differs from (REF ).",
"Since the final algebra is the same as the ultra-relativistic contraction of [6], the author of [11] argued that their results are the entanglement entropy of BMSFT.",
"If we continue the logic which relates the Cardy-like formula to the limit of inner horizon entropy then we conclude that the entanglement entropy of BMSFT should be related to a formula in the CFT which is transformed to the inner horizon formula by using Rindler transformation [13].",
"It is not difficult to check that the formula $S_{inner}={c-\\bar{c}\\over 2}\\log {R\\over \\epsilon _R}-{c+\\bar{c}\\over 6}\\kappa $ results in the leading term of entanglement entropy of zero temperature BMSFT on the plane after taking the ultra-relativistic limit.",
"This formula is nothing but the inner horizon Cardy formula (REF ) if we use Rindler transformation [13].",
"Therefore we conclude that $S^0$ in (REF ) is given by taking limit from a formula which is related to the inner horizon of dual spacetime.",
"Similarly, the logarithmic correction in (REF ) is also given by taking limit from a formula which is the logarithmic correction to the inner horizon entropy.",
"It is not difficult to check that taking $\\epsilon \\rightarrow 0$ limit from $S_{log,inner}=-{3\\over 2}\\log \\left| c^{1/3}\\dfrac{\\partial S_0}{\\partial c}\\right|-{3\\over 2}\\log \\left| \\bar{c}^{1/3}\\dfrac{\\partial S_0}{\\partial \\bar{c}}\\right|-\\log \\epsilon ,$ results in the logarithmic term of (REF ) up to a constant.",
"The interpretation of (REF ) in the gravity is of importance.",
"As mentioned before $\\epsilon $ in the field theory corresponds to $G/\\ell $ on the bulk side.",
"For the BTZ black holes, we have $\\beta ={2\\pi \\ell \\over (r_++r_-)},\\qquad \\bar{\\beta }={2\\pi \\ell \\over (r_+-r_-)}.$ where $r_\\pm $ are the radii of inner and outer horizons.",
"Thus, using (REF ), (REF ) and (REF ) , we can find the logarithmic correction to the inner horizon entropy of BTZ black hole as $S^{BTZ}_{inner}={\\pi r_-\\over 2 G}-{3\\over 2}\\log {r_+^2-r_-^2\\over \\ell ^2}+\\text{constant},$ where we have substituted central charges as $c=\\bar{c}={3\\ell \\over 2 G}$ .",
"On the other hand, using (REF ), (REF ) and (REF ) we find that $S^{BTZ}_{outer}={\\pi r_+\\over 2 G}-{3\\over 2}\\log {r_+^2-r_-^2\\over \\ell ^2}-\\log {\\ell \\over G}+\\text{constant}.$ It is clear from (REF ) and (REF ) that due to the logarithmic corrections, multiplication of inner and outer horizon entropies depends not only on the angular momentum but also on the mass of BTZ.",
"This corrects the result of [25] which in the leading term the multiplication of inner and outer entropies is mass independent."
],
[
"Conclusion",
"In this paper we introduced a generic formula for the logarithmic corrections of the BMSFT thermal and the entanglement entropies.",
"Our derivation is based on the Rindler method which makes connections between the thermal and the entanglement entropies.",
"The most important application of the present work will reveal itself in the reconstruction of the bulk dynamics beyond the classical regime.",
"This reconstruction has been done recently in the bulk AdS case through the perturbation of the entanglement entropy [20], [21].",
"After generalizing the method of [20] for the flat-space holography, we will be able to derive the quantum corrections to the Einstein gravity without the cosmological constant field equations using the results of the current paper.",
"Most of the works in the flat-space holography can be performed by taking the flat space-limit from the AdS/CFT calculations.",
"One of the possible roads for finding the corrections of the BMSFT entanglement entropy formula is taking limit from the calculation of [31] which studies the one-loop bulk corrections to the Ryu-Takayanagi formula."
],
[
"Acknowledgements",
"The authors would like to thank S. M. Hosseini for his useful comments on the manuscript.",
"We are also grateful to Yousef Izadi for his comments on the revised version.",
"We would like to thank the referee for his/her useful comments.",
"This research is supported by research grant No.",
"600/1476 of Shahid Beheshti University, G.C.",
"."
]
] | 1709.01804 |
[
[
"The GTC exoplanet transit spectroscopy survey VIII. Flat transmission\n spectrum for the warm gas giant WASP-80"
],
[
"Abstract We set out to study the atmosphere of WASP-80b, a warm inflated gas giant with an equilibrium temperature of $\\sim$800~K, using ground-based transmission spectroscopy covering the spectral range from 520~to~910~nm.",
"The observations allow us to probe the existence and abundance of K and Na in WASP-80b's atmosphere, existence of high-altitude clouds, and Rayleigh-scattering in the blue end of the spectrum.",
"We observed two spectroscopic time series of WASP-80b transits with the OSIRIS spectrograph installed in the Gran Telescopio CANARIAS, and use the observations to estimate the planet's transmission spectrum between 520~nm and 910~nm in 20~nm-wide passbands, and around the K~I and Na~I resonance doublets in 6~nm-wide passbands.",
"We model three previously published broadband datasets consisting of 27 light curves jointly prior to the transmission spectroscopy analysis in order to obtain improved prior estimates for the planet's orbital parameters, average radius ratio, and stellar density.",
"We recover a flat transmission spectrum with no evidence of Rayleigh scattering or K~I or Na~I absorption, and obtain an improved system characterisation as a by-product of the broadband- and GTC-dataset modelling.",
"The transmission spectra estimated separately from the two observing runs are consistent with each other, as are the transmission spectra estimated using either a parametric or nonparametric systematics models.",
"The flat transmission spectrum favours an atmosphere model with high-altitude clouds over cloud-free models with stellar or sub-stellar metallicities."
],
[
"Introduction",
"Transmission spectroscopy allows us to probe the existence and abundance of atmospheric species in the atmospheres of transiting extrasolar planets [34], [5].",
"The method requires a high observing precision, which has made the space-based studies most successful in finding significant features in the transmission spectra [7], [36], [13], but the developments in instrumentation, observing techniques, and data analysis methods have also enabled transmission spectroscopy studies to be carried out successfully using ground-based telescopes.",
"The signal of interest – variations in the effective planetary radius as a function of wavelength – is minute, corresponding to changes of $\\sim 0.01\\%$ in the observed transit depth and $\\sim 0.1\\%$ in the effective planet-star radius ratio.",
"Further complications arise from possible high-altitude clouds, which can mask any atmospheric extinction features, leading to a flat transmission spectrum [21], [4], and from the fact that atmospheric extinction is not the only source of wavelength-dependent features in transmission spectra.",
"Both instrumental and astrophysical sources, such as host star's spots [2] and plages [26], flux contamination from a possible unresolved source, and incorrectly accounted for stellar limb darkening, can all imprint features that can be difficult to disentangle from the atmospheric signal.",
"Notwithstanding the complications, ground-based transmission spectroscopy has been used successfully to identify features attributed to absorption in planetary atmospheres.",
"Simultaneous measurements of the target star and several comparison stars – a process similar to relative photometry [3], [14] – the use of Gaussian processes have facilitated the robust modelling of systematics [15], [32], [31], and the use of Bayesian inference methods has allowed for realistic uncertainty estimation that is crucial when assessing the true significance of the identified transmission spectrum features.",
"We report a ground-based transmission spectroscopy study of WASP-80b [37].",
"We have observed spectroscopic time series of two WASP-80b transits with the OSIRIS spectrograph [33] installed in the 10.4 m Gran Telescopio CANARIAS (GTC) on La Palma, Spain.",
"The observations cover the spectral range from 520 to 910 nm, probing the planet atmosphere for a possible Rayleigh scattering signal in the blue end of the spectrum, and the visible-light extinction features of the K I and Na I resonance doublets at 767 nm and 589.4 nm, respectively.",
"WASP-80b [37], [22], [12], [38], a warm gas giant orbiting a bright (V=11.87) late-K / early-M dwarf on a 3.07 d orbit, was identified as a promising target for transmission spectroscopy from its discovery.",
"The planet has a low surface gravity ($M_\\mathrm {p} = 0.56\\;\\mathrm {M_{Jup}} $ , $R_\\mathrm {p} = 0.99 \\mathrm {R_{Jup}} $ , $g = 14.34$ ms$^{-2}$ , [22]), and its large radius ratio leads to $\\sim $ 3% deep transits, which, combined with the brightness of its host star, enhance our abilities to detect any possible transmission spectrum features.",
"WASP-80b is a warm gas giant with an equilibrium temperature $\\sim $ 800 K [37], [22].",
"This very likely places the planet into the pL class (no temperature inversion) in the classification by [11].",
"The main spectroscopic features in the visible passband for pL class planets are expected to be from Rayleigh scattering and K I and Na I resonance doublet absorption, of which K I absorption detection was recently claimed by [35] based on transmission spectroscopy analysis carried out with the FORS2 spectrograph installed in the VLT.",
"Table: Identifiers for WASP-80 with its coordinates and magnitudes (SIMBAD, retrieved 2017-07-04).Our study consists of two main analyses: joint analysis of 27 previously published broadband transit light curves observed in 7 passbands (broadband dataset), joint analysis of the two GTC-observed spectroscopic transit time series (transmission spectroscopy dataset), which both consist of a set of analyses with different prior assumptions and modelling approaches carried out to ensure the robustness of the final results.",
"The broadband dataset analysis is carried out to obtain improved estimates for the planet's broadband radius ratios, orbital parameters, and stellar density (system parameters).",
"The marginal parameter posteriors from the broadband dataset analysis are then used as priors in the GTC-data analysis.",
"The GTC transmission spectroscopy starts with a direct-modelling analysis with a flexible Gaussian-process-based systematics model to further constrain the system parameters, and the final transmission spectroscopy uses a divide-by-white approach with either a parametric or nonparametric residual systematics model.",
"This paper is divided roughly into three sections.",
"We outline the numerical methods and the generic equations for the calculation of posterior probability densities in §.",
"We continue in § by carrying out a detailed joint modelling of three priorly observed broadband datasets described in [37], [22], and [12].",
"The datasets cover 27 transit light curves observed in $g^{\\prime }$ , $r^{\\prime }$ , $i^{\\prime }$ , $I$ , $z^{\\prime }$ , $J$ , $H$ , and $K$ .",
"We describe the GTC observations and data reduction in § , detail the analysis in §, present the transmission spectroscopy results in §, and discuss the results in §.",
"Finally, we conclude the paper in §.",
"The raw data are publicly available from Zenodo and GTC data archive, and the whole analysis with reduced data is available from GitHub github.com/hpparvi/Parviainen-2017-WASP-80b as an easy-to-follow set of IPython notebooks and Python codes to help with the reproducibility of the study.",
"We use a fully Bayesian approach to transmission spectroscopy: our parameter estimates are based on the marginal posterior densities derived from a joint posterior density estimated with Markov Chain Monte Carlo (MCMC) sampling, and the marginal parameter posteriors from the broadband dataset analysis are used as priors in the GTC transmission spectroscopy analysis.",
"Our datasets consist of light curves observed either photometrically, or constructed from spectroscopic observations.",
"A light curve is modelled as a product of a baseline and transit model, where the combined model is parametrised with a parameter vector $\\theta $ .",
"When modelling multiple light curves jointly, the parameter vector is divided into parameters shared between all the light curves, passband-specific parameters shared between light curves observed in the same passband, observing-run-specific parameters, and light-curve-specific parameters.",
"Especially, the parameters defining the planetary orbit (zero epoch, orbital period, impact parameter, and stellar density) are shared between all the light curves included into the analysis, and thus the likelihoods from all the light curves contribute to the parameter posteriors.",
"The planet-star radius ratios and stellar limb darkening coefficients are considered passband-dependent, but observing-run-independent parameters (although this is not strictly true, since spots and plages, whether occulted or not, have an effect on the transit depths).",
"That is, the light curves observed in a given passband all contribute to the radius ratio and limb darkening posteriors for the passband.",
"Finally, the parameters defining the baseline and noise properties are considered either observing-run- or light-curve-specific, i.e., they depend both on the passband and observing run (this depends on the specific modelling approach, as detailed later).",
"Joint modelling leads to relatively high-dimensional models.",
"The number of free parameters in the analyses presented here varies from 10 to $\\sim $ 250.",
"However, the approach allows us to utilise the data fully.",
"Simultaneous modelling of different observing runs reduces our sensitivity on systematics, and simultaneous multiband analysis reduces the degeneracies between the estimated radius ratios, orbital impact parameter, and stellar limb darkening.",
"The parameter posteriors are estimated as a two-step process.",
"First, a population-based global optimisation method (Differential evolution implemented in PyDE) is used to obtain a parameter vector population that is clumped close to the global posterior maximum.",
"The parameter vector population is then used to initialise the emcee Markov Chain Monte Carlo (MCMC) sampler [10], [16], which is used to create a sample of parameter vectors drawn from the model posterior (see the analysis-specific sections for practical details).",
"The transit model uses the quadratic limb darkening formalism by [23], and is calculated using PyTransit [27].",
"PyTransit contains optimisations to compute a transit in multiple passbands with only a minor additional computational cost to the computation of a single passband transit, which reduces the computational burden due to the joint modelling approach.",
"The analyses have been carried out both with and without LDTk-based constraints on the stellar limb darkening.",
"Quadratic limb darkening is parametrised using the parametrisation presented in [20], which is aimed for efficient sampling of the physically allowed limb darkening coefficient space.",
"When the limb darkening is not constrained, we marginalise over the whole limb darkening parameter space allowed by the data.",
"When the limb darkening is constrained, it is done by fitting the observational data jointly with the limb darkening profiles created using the LDTk-package [28], and by marginalising over the limb darkening coefficients allowed by the stellar density profiles.",
"LDTk uses PHOENIX-calculated stellar atmosphere library by [18] to construct limb darkening profiles with the uncertainties in the stellar properties propagated into the uncertainties in the limb darkening profiles.",
"We also repeat the analyses for different systematics-modelling approaches, for separate subsets of data, and with synthethic mock data, and using the target star alone without dividing by the comparison star, to test the reliability of our approach.",
"Unless otherwise specified, the parameter point estimates correspond to posterior medians, and the uncertainties correspond to the central 68% posterior intervals.",
"We do not plot point parameter estimates (these are listed in tables), but prefer to show either the posterior distributions or limits based on central posterior intervals.",
"The analyses rely on Python- and Fortran-based code utilising SciPy, NumPy [39], IPython [29], Pandas [24], matplotlib [17], seaborn,$\\!$http://stanford.edu/~mwaskom/software/seaborn PyFITS,$\\!$PyFITS is a product of the Space Telescope Science Institute, which is operated by AURA for NASA and F2PY [30].",
"The transits were modelled with PyTransit Freely available from https://github.com/hpparvi/PyTransit [27], the limb darkening computations were carried out with LDTk $\\!$Available from https://github.com/hpparvi/ldtk [28], global optimisation was carried out with PyDE,$\\!$Available from https://github.com/hpparvi/PyDE the MCMC sampling was carried out with emcee [10], [16], and the Gaussian processes were computed using GeorgeAvailable from https://dan.iel.fm/george [1]."
],
[
"Posteriors and likelihoods",
"The unnormalised log posterior density for a dataset consisting of $n_\\mathrm {lc}$ light curves observed in $n_\\mathrm {pb}$ passbands is $\\ln P(\\theta |D) = \\ln P(\\theta ) + \\sum _i^{n_\\mathrm {lc}} \\ln P(\\vec{D_{\\mathrm {LC}}} _{,i}|\\theta ) +\\sum _i^{n_\\mathrm {pb}} \\ln P(\\vec{D_{\\mathrm {LD}}} _{,i}|\\theta ),$ where $\\theta $ is the parameter vector encapsulating all the model parameters, $\\ln P(\\theta )$ is the log prior, $\\vec{D_{\\mathrm {LC}}} $ are the light curves, $\\ln P(\\vec{D_{\\mathrm {LC}}} |\\theta )$ is the log likelihood for the photometry, $\\vec{D_{\\mathrm {LD}}} $ are the theoretical limb darkening profiles calculated by LDTk, and $\\ln P(\\vec{D_{\\mathrm {LD}}} |\\theta )$ is the log likelihood for the limb darkening profile.",
"Assuming that the uncertainties (noise) in the observations are normally distributed, we can write the log likelihood for the data $\\vec{D}$ in vector form as $\\ln P(\\vec{D}|\\theta ) = -\\frac{1}{2} \\left( n_\\mathrm {D} \\ln 2\\pi +\\ln |\\Sigma | +\\vec{r}^\\mathrm {T}\\Sigma ^{-1}\\vec{r}\\right),$ where $n_\\mathrm {D}$ is the number of datapoints, $\\vec{r}$ is the residual vector, and $\\Sigma $ is the covariance matrix.",
"If the noise is white (uncorrelated), the covariance matrix is diagonal, and the likelihood can be written out explicitly in scalar form as $\\ln P(\\vec{D}|\\theta ) = -\\frac{1}{2}\\left(n_\\mathrm {D}\\ln 2\\pi +\\sum _j^{n_\\mathrm {D}} \\ln \\sigma _{\\mathrm {j}}^2 + \\sum _{j=1}^{n_\\mathrm {D}}\\frac{\\vec{r}^2}{2\\sigma _{\\mathrm {j}}^2} \\right),$ where $\\sigma _{\\mathrm {j}}$ is the uncertainty of the $j$ th datapoint.",
"If the per-point uncertainty does not vary significantly, this equation can be simplified further into $\\ln P(\\vec{D}|\\theta ) = -\\frac{n_\\mathrm {D}}{2} \\ln 2\\pi \\sigma _{\\mathrm {i}}^2 -\\frac{1}{2} \\sum _{j=1}^{n_\\mathrm {D}}\\frac{\\vec{r}^2}{2\\sigma _{\\mathrm {j}}^2}.$ If the noise (here used to describe the leftover variation not explained by the parametric transit and baseline models) is not white, the covariance matrix will have off-diagonal elements, and the matrix needs to be inverted for the likelihood evaluation.",
"This is the case when the noise is presented as a Gaussian process [31], [15], [32].",
"The covariance matrix $\\Sigma $ in Eq.",
"(REF ) is now $\\Sigma = \\vec{K}(\\vec{x},\\vec{x}) + \\sigma ^2\\vec{I},$ where $\\vec{K}(\\vec{x},\\vec{x})$ is defined by a covariance function (also known as a covariance kernel).",
"We describe the likelihood and covariance functions separately for the broadband dataset analysis and GTC transmission spectroscopy in Sects.",
"and , respectively, since the two analyses use slightly different modelling approaches."
],
[
"Overview",
"We estimate the posterior densities for the WASP-80b orbital parameters based on the three datasets observed by [22], [37], and [12], abbreviated from herein as M14, T13, and F14, respectively.",
"The datasets contain 27 light curves (Fig.",
"REF ) observed in $g^{\\prime }$ , $r^{\\prime }$ , $i^{\\prime }$ , $I$ , $z^{\\prime }$ , $J$ , $H$ , and $K$ .",
"We model all the light curves jointly, and, in contrast to [22], include the M14 GROND J, H, and K light curves.",
"We also simplify the analysis slightly by merging the $I$ and $i^{\\prime }$ passbands.",
"The analysis is carried out assuming either a parametric or nonparametric systematics model (white or red noise), constant or wavelength-dependent radius ratio, and with and without LDTk to constrain the stellar limb darkening, which leads to eight separate analysis sets defined in Table REF .",
"The T13 and M14 datasets, as obtained from VizieR, were detrended by their original authors, and do not include other information than the observation time, flux, and error estimates.",
"The F14 dataset was kindly provided by the author without detrending, and included all the auxiliary information (such as the airmass, and x- and y-centroid shifts for each exposure) used in the original analysis presented in [12].",
"Table: Broadband analysis runs for the external datasets.",
"The white-noise runs include only the and datasets, while the red-noise runs also include the light curvesby .",
"The constant and varying radius ratios mark whether the radius ratiowas allowed to vary from passband to passband, or whether it was assumed to be wavelength-independent.Table: Marginal posterior medians from the external data broadband analysis (run ckrn_ldtkwith systematics modelled using GPs, wavelength-independent radius ratio, and limb darkeningconstrained with LDTk).",
"The uncertainties correspond to the central 68% posterior intervals.Table: Broadband radius ratio estimates and their uncertainties from the vkrn_ldtk run.We used the T13 and M14 datasets for an initial modelling with a parametric systematics model and white additive noise, and include the F14 dataset in the final runs using nonparametric systematics model.",
"The nonparametric systematics model represents the systematics (and white noise) as a Gaussian process (GP) with time as the only covariate for the T13 and M14 datasets, and with time, airmass, x-shift, and y-shift as covariates for the F14 dataset.",
"We do not marginalise over the GP hyperparameters in the broadband data analysis, but fix them to the values fitted from the white-noise run residuals.",
"As described in Sect.",
"REF , the parameter estimation starts with a parameter vector population that fills the prior space uniformly.",
"A differential evolution (DE) optimisation is used to clump the population close to the global posterior maximum (the number of DE iterations depending slightly on the run, but is usually close to 1000), after which MCMC sampling is carried out using emcee.",
"The sampler is run for 10$\\,$ 000 iterations, which yields 12000 independent posterior samples when using a population size of 150, thinning factor of 100, and burn-in period of 2000 iterations, where the thinning factor and burn-in period were chosen by studying the parameter chain populations and the average parameter autocorrelation lengths."
],
[
"Log posterior and likelihoods",
"The log posterior for the combined broadband dataset given a parameter vector $\\theta $ is $\\ln P(\\theta |D) &= \\ln P(\\theta ) \\nonumber \\\\&+ \\ln P(D_\\mathrm {T13}|\\theta ) + \\ln P(D_\\mathrm {M14}|\\theta ) +\\ln P(D_\\mathrm {F14}|\\theta ) \\\\&+ \\sum _i^{n_\\mathrm {pb}} \\ln P(\\vec{D_{\\mathrm {LD}}} _{,i}|\\theta ), \\nonumber $ where the first term is the log prior, followed by the per-dataset log likelihoods, and the last term is the sum of the LDTk-calculated log likelihoods for the limb darkening coefficients for each passband.",
"The exact form of the three likelihoods follows either Eq.",
"(REF ) or Eq.",
"(REF ), depending on the chosen systematics model.",
"The parametric (white noise) model assumes a constant average per-light-curve uncertainty (that is, the observation noise is the same for all the datapoints in a single light curve), and the kernels for the GP model are detailed below.",
"The light curve model consists of a product of a baseline function and a transit model with quadratic limb darkening calculated using PyTransit.",
"The residual vector for a single light curve in Eq.",
"(REF ) is $\\vec{r} = \\vec{f}_o - \\vec{f}_m(\\vec{X}, \\theta ) = \\vec{f}_o - \\mathcal {B}(\\vec{X}, \\theta ) \\,\\mathcal {T}(\\vec{t}, \\theta )$ where $\\vec{f}_o$ is the observed flux, $\\vec{f}_m$ the modelled flux, $\\mathcal {B}$ the baseline model, $\\mathcal {T}$ the transit model, $\\vec{X}$ is a matrix containing the the input parameters (covariates), $\\vec{t}$ are the mid-exposure times, and $\\theta $ is the model parameter vector."
],
[
"Gaussian process kernels",
"The T13 and M14 datasets do not include other auxiliary information than the mid-exposure times.",
"Thus, we model the noise as a Gaussian process with time as the only covariate, the kernel being a sum of an exponential kernel and a white noise term $k = A_t \\exp \\left[ - \\eta _t (t_i-t_j) \\right] + \\sigma ^2 \\delta _{ij},$ where $t$ are the time values, $A_t$ is the GP output amplitude, $\\eta _t$ the inverse time scale, $\\sigma $ the average white noise standard deviation, and $\\delta $ the Kronecker delta.",
"The F14 dataset includes also airmass, and per-observation x- and y-centroid estimates.",
"This allows us to use a slightly more complex kernel, where we have an exponential time component, squared exponential airmass component, squared exponential PSF-centroid component, and a white noise term.",
"The kernel becomes $k &= A_t \\exp \\left[-\\eta _t (t_i-t_j) \\right] \\nonumber \\\\& + A_a \\exp \\left[-\\eta _a (a_i-a_j)^2 \\right] \\nonumber \\\\& + A_{xy} \\exp \\left[-\\eta _x (x_i-x_j)^2 -\\eta _y (y_i-y_j)^2 \\right] \\nonumber \\\\& + \\sigma ^2\\delta _{ij},$ where $t$ , $a$ , $x$ , and $y$ are the time, airmass, x, and y estimates, respectively; $A_t$ , $A_a$ , $A_{xy}$ are the GP time, airmass and xy output amplitudes; and $\\eta _t$ , $\\eta _a$ , $\\eta _x$ , and $\\eta _y$ the time, airmass, x and y inverse time scales."
],
[
"Results",
"We adopt the ckrn_ldtk (passband-independent radius ratio, GP systematics, and limb darkening constrained using the LDTk) run as our final analysis, and report the stellar, orbital, and planetary parameter estimates in Tables REF and REF .",
"The posterior distributions are all close-to normal, as shown in Figs.",
"REF and REF .",
"The parameter estimates agree with the previous WASP-80b studies.",
"The broadband transmission spectrum from the vkrn_ldtk run, shown in Fig.",
"REF is consistent with a flat line.",
"The result agrees with a previous broadband transmission spectrum analysis by [38].",
"We used the ExoTransmit transmission spectrum modelling package to test different atmosphere scenarios, but the precision in radius ratios is not sufficient to meaningfully distinguish between any of the physically plausible scenarios.",
"Figure: Marginal- and joint-posteriors for the radius ratio, stellar density, and impactparameter corresponding to the final ckrn_ldtk run.",
"The 68% centralinterval is marked with a darker shade.Figure: Broadband radius ratio posteriors estimated by jointly modelling the three priordatasets.",
"The results correspond to the final vkrn_ldtk run with rednoise modelled using GPs and limb darkening constrained using LDTk.Figure: Broadband quadratic limb darkening coefficient posteriors.",
"The results correspond to thefinal ckrn runs with red noise modelled with GPs and limb darkening eitherunconstrained (left) or constrained using LDTk (right).The limb darkening coefficient posteriors are plotted in Fig.",
"REF , both with and without constraints from LDTk.",
"The two versions agree with each other within uncertainties.",
"The observed light curve, conditional model distribution (for the red noise model), and the residuals are shown in Fig.",
"REF .",
"Figure: The 27 broadband light curves observed by ,, and after the removal of the mean GP trend, theposterior model medians, and the residuals."
],
[
"Observations",
"We observed two WASP-80b transits simultaneously with one comparison star using the OSIRIS (Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy) spectrograph installed in the GTC (Gran Telescopio CANARIAS) on the nights starting 16 July 2014 and 25 August 2014 (observing runs R1 and R2 respectively).",
"The R1 observations carried from 23:30 UT till 3:41 UT and the R2 observations from 20:35 UT till 1:00 UT.",
"The observing conditions were good, with a seeing of $\\sim $ 0.8 in the beginning of each night, and all the observations were carried out with an airmass smaller than 1.4.",
"OSIRIS [6] contains two Marconi CCD42-82 2048$\\times $ 4096 pixel CCDs, which were used in the standard 2$\\times $ 2 binning mode yielding a plate scale of 0.254.",
"The observations were carried out using grism R1000R with a 40-wide custom-designed slit that aims to minimise the systematics related to variations in flux loss.",
"We chose TYC 5165-00235-1 as the comparison star, located at a distance of 6.9 from WASP-80.",
"The star has a similar V magnitude ($V=11.62$ ), but is slightly redder ($J=8.376$ ).",
"The two stars were positioned equidistantly from the optical axis close to the centre of each CCD.",
"The slit does not include other stars bright enough to be useful in the analysis.",
"The exposure time was 6 s for R1 and 5 s for R2.",
"While the exposure time was short, it was long enough for the comparison star to saturate during the final parts of the WASP-80b ingress during R1.",
"This was fixed by defocusing the instrument, but meant that the comparison star cannot be used in the reduction during this saturated period.",
"For R1, 81 flat fields were taken before the transit, and 66 bias frames after the transit.",
"For R2, 100 flat fields were taken after the transit, and 15 bias frames before the transit.",
"Three arc frames (Xe, HgAr, and Ne) for R1 were observed on 14 July and for R2 on the same night as the observations, after the transit."
],
[
"Data reduction and passband sets",
"The passband-integrated light curves are produced from the raw data using the pipeline described in [9], [8].",
"The pipeline carries out the basic CCD data reduction steps, calculates a 2D wavelength solution, removes the sky, and generates a reduced 1D spectrum for WASP-80 and the comparison star (Fig.",
"REF ).",
"The light curves are then generated by integrating the spectra multiplied by a transmission function defining the passband.",
"We created four sets of passbands (datasets) listed in Table REF .",
"The W dataset consists of a single light curve covering the whole usable spectrum (white light curve), the NB (narrow-band dataset) covers the whole usable spectrum in 20 nm-wide bins, and the Na I and K I datasets cover the Na and K lines, respectively, in 6 nm-wide bins.",
"The final white-light WASP-80 and reference star lightcurves with the relative light curve are shown in Fig.",
"REF .",
"The white-light white noise estimates for R1 and R2 are 400 and 520 ppm, respectively.",
"For the NB narrow-band dataset, the white noise level varies from 600 to 2100 ppm, with a median level of 720 and 870 ppm for R1 and R2, respectively.",
"The difference in the white noise levels is expected due to the different exposure times used in R1 and R2 (6 and 5 s, respectively).",
"Figure: Normalised, sky-subtracted, and wavelength-calibrated spectra for WASP-80b (dark blueline) and the comparison star (orange line).Figure: White-light light curves for the comparison star (top), WASP-80 (middle), and therelative light curve (bottom) for both observing runs.",
"A small number of clear outliers have beenomitted for clarity.Table: Passband sets extracted from the GTC spectroscopy."
],
[
"Systematics",
"The relative light curves in Fig.",
"REF feature systematics not corrected by division by the comparison star.",
"Specially, the baseline is affected by a smooth trend that cannot be modelled by a simple parametric model as a function of time (or any of the simultaneously measured auxiliary variables).",
"The trend causes a pre- and post-transit baseline difference of $\\sim 2\\%$ during run 1, and a $\\sim 1\\%$ difference during run 2.",
"A similar, smooth, nearly linear, variation as a function of the rotator angle accompanied by relatively smooth 'bumps', has been observed in previous OSIRIS transmission spectroscopy studies, and is likely caused by vignetting in the telescope pupil space [25].",
"Fortunately, the systematics are mainly common-mode, without significant variation across the spectrum.",
"Common-mode systematics can be removed by either fitting the white-light light curves with a flexible GP, which can be used to create a common-mode systematics model, or by using a divide-by-white approach.",
"After the common-mode correction, the residual wavelength-dependent systematics can be accounted for either with parametric or non-parametric approaches, both of which were used in our analyses."
],
[
"Overview",
"The GTC transmission spectroscopy covers the modelling of the white light curves and the three narrowband datasets described in Sec.",
"REF .",
"We used three modelling approaches: DIR direct modelling with flexible GP-based systematics DWW divide-by-white with parametric systematics DWR divide-by-white with GP-based systematics The first approach, direct modelling of the light curves with a flexible GP-based systematics model, was used to obtain a model for the common-mode systematics (from the white light curves), and to improve the system characterisation (from the full-spectrum 20 nm-wide dataset).",
"The divide-by-white (DW) approaches were then used for transmission spectroscopy, with the direct-modelling posteriors used as priors on the wavelength-independent system parameters (the motivation for this is discussed later).",
"The analyses were repeated with and without LDTk to assess how sensitive the results are to assumptions about limb darkening, and modelling the two nights separately and jointly, to test whether the results are consistent from night to night.",
"The parameter posteriors are estimated in similar fashion to the prior data analysis in Sec.",
"REF .",
"Differential evolution is used to create a parameter vector population clumped around the global posterior maximum, which used to initialise the emcee sampler.",
"The sampler is ran over $10\\times 15\\;000$ iterations (that is, ten sets of $15\\;000$ iterations where each set is initialised from the final state of the previous set) with 600 chains and a thinning factor of 100, which gives us a final set of 75$\\;$ 000 independent posterior samples.",
"The number of iterations, chains, and the thinning factor were chosen after studying the chain populations and per-parameter autocorrelation lengths.",
"Unlike in the broadband dataset analysis, we keep the GP hyperparameters free and marginalize over them.",
"This slows down the sampling process, but yields more reliable posteriors, and allows us to study how well the GP kernels represent the systematics.",
"The direct-modelling approach DIR reproduces the light curves as a product of a baseline and a transit model directly.",
"We model all the passbands for both nights in a dataset jointly, which leads to a slightly involved parameterisation, but aims to utilise the data fully.",
"As mentioned earlier, the model parameters can be divided into four categories: a) passband- and baseline-independent parameters, b) achromatic per-night baseline parameters, c) chromatic per-light-curve parameters, and d) chromatic parameters that should stay constant between observation runs.",
"The direct model parametrisation and the parameter priors are listed in Table REF .",
"Priors for the zero epoch, orbital period, impact parameter, and stellar density are based on the broadband dataset analysis posteriors (run ckrn_ldtk).",
"The $u$ and $v$ limb darkening coefficients correspond to the [20] quadratic limb darkening model parametrisation, where uniform priors from 0 to 1 lead to uninformative priors covering the physically viable values for the quadratic limb darkening priors.",
"The GP hyperparameters can be chosen to be night- or light-curve-dependent, but we choose to keep them independent for practical reasons.",
"Our approach adds three GP hyperparameters to the analysis, and requires two covariance matrix inversions per posterior evaluation (one per night).",
"Making the GP hyperparameters light-curve-dependent would add three free parameters to the model for each light curve, and require a covariance matrix inversion for each light curve, which would make marginalisation over the GP hyperparameters costly.",
"Making the GP hyperparameters night-dependent would be feasible, since the approach would add only three more parameters to the model, and would not require additional covariance matrix inversions.",
"However, our analyses for separate nights result with compatible GP hyperparameter posteriors, and we choose the simplest approach.",
"Table: Model parametrisations and priors.",
"𝒰(a,b)\\mathcal {U}(a,b) stands for a uniform prior from a to b, where a andb are omitted when the range is chosen to be wide enough not to affect the posteriors.",
"𝒩(μ,σ)\\mathcal {N}(\\mu ,\\sigma )stands for a normal prior with mean μ\\mu and standard deviation σ\\sigma .",
"The system parametershave normal priors with means and standard deviations corresponding to the values shown in Table .The direct model represents each light curve as a Gaussian process $f \\sim \\mathcal {N}(c_b\\;\\mathcal {T}(\\vec{t}, \\theta ), \\Sigma ),$ where $c_b$ is a baseline constant, $\\mathcal {T}$ is the transit model, and $\\Sigma $ is the covariance matrix.",
"We use airmass $x$ , and telescope rotator angle, $\\alpha $ , as GP input parameters (covariates), with a covariance matrix defined as a sum of a linear kernel and squared-exponential kernel, $\\Sigma (i,j) = \\left( \\frac{x_j \\cdot x_i}{\\gamma _x}\\right) + a_\\alpha ^2 \\exp \\left( \\frac{(\\alpha _j - \\alpha _i)^2}{\\gamma _\\alpha }\\right) + \\delta _{ij} \\sigma $ where $a_\\alpha $ is an output scale parameter, $\\gamma _x$ and $\\gamma _\\alpha $ are the input scale parameters, and $\\sigma $ is the average white noise.",
"Any residual airmass-dependent systematics should be approximately linear, and can be modelled with a linear kernel, while the possible chromatic rotator-angle dependencies are expected to be smooth, and well-modelled by a squared-exponential kernel.",
"We also tested a GP with time as a covariate (with a Matern kernel), but it did not affect the posteriors significantly.",
"Also, the power spectral density (PSD) of the residuals with the transit and the two-covariate GP mean removed is approximately constant, which suggests that the noise is white after the residual airmass and rotator-angle dependencies are accounted for.",
"The white noise level does not vary significantly from passband to passband in our datasets, and we choose to use a single average white noise estimate for an observing run (again, so that we do not need to invert the covariance matrix separately for each light curve).",
"The white noise estimate is an average of the estimates calculated separately for each light curve, which are calculated as $\\sigma = \\mathrm {std}(\\vec{f}_d) / \\sqrt{2}$ where $\\vec{f}_d = f_i - f_{i-1}$ .",
"That is, we estimate the white noise from the standard deviation of the light curve differentials.",
"The radius ratios and limb darkening coefficients yield three free parameters per passband, and the baseline constant a further free parameter per light curve.",
"In total, the number of free parameters for a direct model reaches 108 for the 20-nm run with 20 passbands.",
"The computations can still be carried out with a modern laptop in a matter of hoursPyTransit, which was used to compute the light curves, is optimised to compute a set of multiband light curves with only a minor additional cost to calculating a single passband.",
"for a single analysisHowever, while a single analysis can be carried out in a laptop, the Glamdring-cluster in Oxford University and the TeideHPC supercomputer in Spain were used to carry out the final analyses., but tests must be carried out to ensure that the final posterior sample gives a reliable representation of the true posterior distribution."
],
[
"Divide-by-white model with and without GP systematics",
"The divide-by-white (DW) approachOur approach would be more accurately named as divide-by-average, since we're dividing each dataset by the dataset mean.",
"allows us to remove any common-mode systematics from the dataset with the cost of increased per-passband white noise.",
"The residual vector $\\vec{r}$ in Eq.",
"(REF ) for the DW approach is $\\vec{r} = \\frac{1}{\\vec{b}(\\theta , \\mathbf {X})} \\left( \\frac{\\vec{f}}{1/n_\\mathrm {pb} \\sum \\vec{f}} - \\frac{\\mathcal {T} (\\vec{t}, \\theta )}{1/n_\\mathrm {pb} \\sum \\mathcal {T} (\\vec{t}, \\theta )} \\right), $ where $\\mathbf {f}$ is the observed flux vector, $\\mathcal {T}$ is the transit model, and $\\vec{b}$ is the residual baseline model.",
"That is, we divide both the observed and modelled fluxes by the values averaged over all the passbands in the dataset.",
"The DW approach does not only remove the common-mode systematics, but all colour-independent signals.",
"Effectively, the signals left in the data are due to colour-dependent systematics, changes in the radius radio, and changes in stellar limb darkening.",
"The system parameters, such as the orbital period and impact parameter, are poorly constrained, so we set informative priors based on the nb_ldtk direct-model analysis on them.",
"The average radius ratio is not well constrained either, so we set a prior on the median radius ratio based on the broadband dataset analysis.",
"We use median instead of mean to ensure that the prior does not affect the scaling of the transmission spectrum, as using mean would.",
"We repeat the DW analysis using first a parametric baseline model (DWW approach), and then a more flexible Gaussian process-based baseline model (DWR approach).",
"The parametric model represents the baseline for each light curve as a linear function of time and airmass $\\vec{b}_i(\\vec{t}, \\vec{x}, \\theta ) = \\theta _{i,a} + \\theta _{i,b} \\vec{t} + \\theta _{i,c} \\vec{x},$ where $\\theta _{i,a}$ , $\\theta _{i,b}$ , and $\\theta _{i,b}$ are the light curve specific baseline constant, linear time coefficient and linear airmass coefficient, respectively.",
"The residual noise is considered white, which makes the approach significantly faster than using GPs, but yields three free parameters per light curve to the model.",
"The DWR analysis uses the same GP kernel as the DIR approach, changing only the GP mean function to the one in Eq.",
"REF ."
],
[
"System characterisation",
"The direct-modelling DIR runs were used to improve the system characterisation from the broadband dataset analysis, and yield the final WASP-80b parameter estimates listed in Table REF .",
"The estimates correspond to the full-spectrum narrow-band dataset (NB) with LDTk.",
"The narrow-band run was preferred over the white-light run since the colour information allows us to mitigate the degeneracy between the average impact parameter, radius ratio, and limb darkening, leading to improved posterior estimates.",
"Table: Results from the final GTC characterisation run (nb_12_ldtkwith systematics modelled using GPs, wavelength-dependent radius ratio, and limb darkeningconstrained with LDTk).",
"The uncertainty is based on the central 68% posteriorinterval.",
"We do not list the radius ratio estimates, since the (common-mode)systematics are too strong for direct modelling to constrain them."
],
[
"Transmission spectroscopy",
"We show the transmission spectra for the NB, K I, and Na I datasets in Fig.",
"REF .",
"These results correspond to the joint DWR run with a GP systematics model and limb darkening coefficients constrainted by LDTk.",
"The figures show the central 68% and 99% radius ratio posterior intervals, and the point estimates are listed in Table REF .",
"Table: Passband centres, ranges, and radius ratio estimates for the three narrow-band datasetsfrom the final DWR approach.Comparison of the results from the two different divide-by-white approaches can be found below in Sect.",
"REF , and a comparison of the results obtained using only the light curves from one of the two can be found Sect.",
"REF .",
"We do not show the results for the analysis without LDTk.",
"Unconstrained limb darkening does not have a significant effect on the radius ratio posterior medians, but does increase the posterior width significantly due to the strong degeneracy caused by the DW approach.",
"Figure: The NB light curves for both observing runs and the DIR approach posterior median model(black line) as a function of the phase.",
"Passband centres are marked in nanometres.Figure: Transmission spectra for the NB dataset with 20 passbands spanning from 520 nm to 920 nm (top),the Na I set covering the Na I doublet (bottom left), and the K I dataset covering the K I double (bottom right).The inner boxes correspond to the central 68% posterior intervals, the outer boxes to the central 99% posteriorintervals, and Na I and K I lines are marked as black lines.",
"The light gray line shows an Exo-Transmit spectrumwith Na and K described in Sect.",
", the black dots show the Exo-Transmitspectrum binned to the dataset binning, and the slashed horizontal line shows the transmission spectrum mean."
],
[
"Flat transmission spectrum",
"We adopt the results from the more conservative approach, DWR, as the final results of the analysis.",
"All the spectra in Fig.",
"REF are flat within uncertainties, in agreement with the broadband analysis, and the broadband transmission spectroscopy carried out by [38].",
"The NB features a single strongly deviating passband centred at 807 nm, with a significantly smaller radius ratio than the average.",
"The passband is likely affected by instrumental observation-geometry-dependent systematics (the deviation is evident in both observing runs, albeit slightly different), and we discuss it in more detail in Sect.",
"REF .",
"The flat transmission spectrum allows us to rule out strong Rayleigh scattering or Na I or K I absorption in WASP-80b's atmosphere, but does not justify detailed atmospheric modelling.",
"However, basic modelling with Exo-transmit [19] strongly favours a flat spectrum over a spectrum from an atmosphere with 0.1 or 1 solar metallicity and K and Na: a likelihood ratio test between the Exo-Transmit spectra and a flat spectrum favours the flat model with likelihood ratios around 1000-5000.",
"(We fit the model spectra to the estimated spectrum with a free scaling factor, and the flat spectrum is the mean of the estimated spectrum.)",
"Figure REF includes an Exo-Transmit spectrum with K and Na calculated assuming Solar metallicity, gas-phase chemistry, $T_\\mathrm {eq}=800$ K, $log {\\it g}= 3.18$ , $R_\\star = 0.57 R_\\odot $ , and $k=0.17$ as an example.",
"The DWR approach was chosen over DWW because the DWW results show greater discrepancies in the spectra estimated from R1 and R2 separately, as discussed below.",
"It is likely that the parametric systematics model is not flexible enough, and the unaccounted-for systematics lead to biases in the radius ratio values.",
"The GP kernel in the DWR approach is flexible enough to marginalize over the systematics, and still allows for a reliable radius ratio estimation (tested with two mock datasets discussed in Sect.",
"REF )."
],
[
"Comparison with the previous K I detection",
"[35] reported of a detection of strong K I absorption in WASP-80b's atmosphere based on transmission spectroscopy with the FORS2 spectrograph.",
"Our results disagree with the reported K I detection, as shown in Fig.",
"REF , but agree with the redmost part of their spectrum.",
"Both of our divide-by-white approaches are consistent with each other, as are the transmission spectra estimated separately from the two observing runs (as discussed below in Sects.",
"REF and REF ), which leads us to believe that the strong K I signal reported by [35] is due to systematics, even though the analysis described in their paper seems rigorous in all standards.",
"Figure: Comparison between the results from the GTC analysis and the results presented by ."
],
[
"Comparison of modelling approaches",
"The two divide-by-white approaches differ in their way of modelling the systematics.",
"The DWW approach models the systematics with parametric model as a combination of linear functions of time and airmass, and assumes that the everything else is white normally distributed noise, while the DWR approach models the systematics as a Gaussian process.",
"In the DWW case all systematics not corrected by the division by the dataset average light curve or modelled by the parametric model may cause biases in the radius ratio estimates.",
"The GP kernel in the DWR case is relatively flexible, even when we impose a linear relation on airmass, and should yield less bias-prone radius ratio estimates.",
"This is true especially since we're integrating over the GP hyperparameters, and do not significantly constrain them with priorsThe GP hyperparameters are actually allowed to probe the parameter space where they model time correlation rather than rotator-angle correlation, since the rotator angle itself is a slowly and smoothly varying function of time.",
"We compare the transmission spectra obtained with DWW and DWR approaches in Figs.",
"REF , REF , and REF .",
"The differences are small for the Na and K analyses, which can be expected due to the small wavelength range covered by the datasets.",
"However, the NB set analysed with DWW approach features a Rayleigh-like signal in the bluemost passbands that is not visible in the DWR results.",
"When analysing the two nights separately (Sect.",
"REF ), we can see that the Rayleigh-like signal arises from the second night, and is not significant in the first night.",
"However, the first night shows an increasing trend towards the red end of the spectrum, not visible in the second night spectrum.",
"In theory, the varying Rayleigh-like slope could be interpreted as variations in stellar activity.",
"However, a more likely explanation is residual wavelength-dependent systematics not accounted for properly by the parametric model.",
"Figure: Radius ratio posterior distributions approximated as normal distributions for the NB DWW (left) and DWR (right) runs.Figure: Radius ratio posterior distributions approximated as normal distributions for the Na DWW (left) and DWR (right) runs.Figure: Radius ratio posterior distributions approximated as normal distributions for the K DWW (left) and DWR (right) runs."
],
[
"Comparison of individual nights",
"We show the transmission spectra estimated separately from run 1 or run 2 either with the DWW or DWR approach in Figs REF , REF , and REF .",
"The results from the different nights agree with some exceptions.",
"Both the blue and red ends of the DWW NB spectrum differ systematically, as mentioned earlier.",
"the Rayleigh-like signal visible in the DWW model originates clearly only from R2, while the corresponding passbands are close or below the mean level for R1.",
"However the three redmost passbands are well above the mean level for R1, while R2 shows a flat spectrum in the red.",
"These differences disappear in DWR analysis, where the both runs give compatible flat-within-uncertainties spectrum.",
"Figure: Radius ratio posterior distributions approximated as normal distributions for the NB DWW (up) and DWR (below) runsusing only the data from night 1 (left) or night 2 (right).Figure: Radius ratio posterior distributions approximated as normal distributions for the Na I DWW (up) and DWR (below) runsusing only the data from night 1 (left) or night 2 (right).Figure: Radius ratio posterior distributions approximated as normal distributions for the K I DWW (up) and DWR (below) runsusing only the data from night 1 (left) or night 2 (right)."
],
[
"Outlier passband at 807 nm",
"A single passband centred at 807 nm in the NB dataset transmission spectrum deviates from the mean.",
"This signal is visible on both nights, and a detailed analysis shows that the features is smooth, but somewhat different between the nights.",
"We could not trace the source of the feature, but can only speculate.",
"First, the feature is very unlikely of astrophysical origin.",
"Certain astrophysical phenomena, such as non-transited star spots and flares, can affect the radius ratio estimates, but the feature would require a bright narrow-band source.",
"Second, the feature is not caused by the comparison star.",
"We repeated the DW NB analyses using absolute photometry without dividing with the comparison star (practical since the divide-by-white approach removes the common-mode systematics), but this did not affect the feature."
],
[
"Mock dataset analysis",
"We also carry out a sensitivity (reality) checks by using two mock datasets based on the NB dataset.",
"We calculate synthethic light curves using the observed time stamps and the system parameter posterior medians from the direct modelling.",
"The radius ratios follow a saw-tooth-like pattern for the mock dataset 1, and a transmission spectrum calculated by Exo-Transmit for the mock dataset 2, and the limb darkening coefficients are based on the theoretical values calculated using LDTk.",
"The baselines are modelled as sums of a linear time trend and a linear airmass trend with normally distributed random coefficients, and we add normally distributed white noise with standard deviation corresponding to the true light curve standard deviation estimates.",
"The radius ratio posterior densities from the mock dataset analyses agree with the true radius ratios for both DW approaches."
],
[
"Conclusions",
"We have carried out a transmission spectroscopy analysis for WASP-80b using two OSIRIS-observed spectroscopic time series, and a joint analysis of 27 previously observed broadband light curves to provide reliable parameter priors for the transmission spectroscopy analysis.",
"The OSIRIS data feature strong common-mode systematics, but these can be removed using either a divide-by-white (DW) approach or initial white-light GP modelling.",
"We chose the divide-by-white approach, (or, more accurately, divide-by-dataset-average) which increases the per-passband white noise levels, but removes any systematics-model dependencies from the common-mode systematics removal.",
"The transmission spectroscopy analyses were repeated modelling both datasets jointly and separately, with or without LDTk-constructed stellar limb darkening priors, and using either a parametric or nonparametric (GP-based) systematics model.",
"We chose the most conservative approach as the final analysis detailed in this paper: a DW approach accompanied with a flexible Gaussian process systematics model where the GP hyperparameters are marginalized over in the posterior sampling process.",
"This was motivated by significant nightly variations in the transmission spectrum obtained using the parametric systematics model.",
"Despite the nightly variations in the DWW analysis, the joint analyses are consistent with each other: the transmission spectrum is flat within uncertainties.",
"Especially, we do not detect significant Na or K absorption, and, our results do not agree with the detection of potassium by [35].",
"The absence of significant features does not justify detailed atmospheric modelling.",
"However, basic Exo-Transmit modelling favours a truly flat spectrum over atmosphere models with stellar or sub-stellar metallicity and K and Na.",
"Ground-based observations are prone to complex systematics, and transmission spectroscopy is carried out at the limits of what the instruments are capable of, or were designed for.",
"The most robust approach for ground-based transmission spectroscopy should try to account for this by repeating the observations several times, preferably with different instruments covering the same wavelength range.",
"Repeated observations do not only improve the final precision that can be reached, but also our capability to decouple the systematics from the minute transit depth variations.",
"We thank the anonymous referee for their constructive and useful comments.",
"HP has received support from the Leverhulme Research Project grant RPG-2012-661.",
"FM acknowledges the support of the French Agence Nationale de la Recherche (ANR), under the program ANR-12-BS05-0012 Exo-atmos.",
"The work has been supported by the Spanish MINECO grants ESP2013-48391-C4-2-R and ESP2014-57495-C2-1-R. GC acknowledges the support by the National Natural Science Foundation of China (Grant No.",
"11503088) and the Natural Science Foundation of Jiangsu Province (Grant No.",
"BK20151051).",
"Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma.",
"The authors wish to acknowledge the contribution of Teide High-Performance Computing facilities to the results of this research.",
"TeideHPC facilities are provided by the Instituto Tecnológico y de Energías Renovables (ITER, SA).",
"The authors wish to acknowledge the contribution of Glamdring computing cluster in the Subdepartment of Astrophysics, Department of Physics, University of Oxford, to the results of this research."
],
[
"DW model fit and residuals",
"Figures REF and REF show the DWR analysis model fit and residuals, respectively.",
"Figure: The NB light curves for both observing runs and the DWR approach posterior median model(black line) as a function of the phase.",
"Passband centres are marked in nanometres.Figure: The NB residuals from the DWR analysis.",
"Passband centres are marked in nanometres."
]
] | 1709.01875 |
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