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$h_{5[1,2]}\left( x^{i}\right) $ stated by boundary conditions; b\) or, inversely, to compute $h_{4}$ for a given $h_{5}\left( x^{i},v\right) ,h_{5}^{\ast }\neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\left( x^{i}\right) (\sqrt{|h_{5}\left( x^{i},v\right) |})^{\ast }, \label{p1}$$with $h_{[0]}\left( x^{i}\right) $ given by boundary conditions. - The exact solutions of (\[ep3a\]) for $\beta \neq 0$ are defined from an algebraic equation, $w_{i}\beta +\alpha _{i}=0,$ where the coefficients $\beta $ and $\alpha _{i}$ are computed as in formulas ([abc]{}) by using the solutions for (\[ep1a\]) and (\[ep2a\]). The general solution is $$w_{k}=\partial _{k}\ln [\sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast }|]/\partial _{v}\ln [\sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast }|], \label{w}$$with $\partial _{v}=\partial /\partial v$ and $h_{5}^{\ast }\neq 0.$ If $h_{5}^{\ast }=0,$ or even $h_{5}^{\ast }\neq 0$ but $\beta =0,$ the coefficients $w_{k}$ could be arbitrary functions on $\left( x^{i},v\right). $  For the vacuum Einstein equations this is a degenerated case imposing the compatibility conditions $\beta =\alpha _{i}=0,$ which are satisfied, for instance, if the $h_{4}$ and $h_{5}$ are related as in the formula (\[p1\]) but with $h_{[0]}\left( x^{i}\right) =const.$ - Having defined $h_{4}$ and $h_{5}$ and computed $\gamma $ from ([abc]{}) we can solve the equation (\[ep4a\]) by integrating on variable “$v $" the equation $n_{i}^{\ast \ast }+\gamma n_{i}^{\ast }=0.$ The exact solution is $$\begin{aligned} n

$ h_{5[1,2]}\left (x^{i}\right) $ stated by boundary conditions; b\) or, inversely, to compute $ h_{4}$ for a given $ h_{5}\left (x^{i},v\right) , h_{5}^{\ast } \neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\left (x^{i}\right) (\sqrt{|h_{5}\left (x^{i},v\right) |})^{\ast }, \label{p1}$$with $ h_{[0]}\left (x^{i}\right) $ give by boundary condition. - The exact solution of (\[ep3a\ ]) for $ \beta \neq 0 $ are define from an algebraic equation, $ w_{i}\beta + \alpha _ { i}=0,$ where the coefficients $ \beta $ and $ \alpha _ { i}$ are calculate as in formulas ([ abc ] { }) by using the solutions for (\[ep1a\ ]) and (\[ep2a\ ]). The cosmopolitan solution is $ $ w_{k}=\partial _ { k}\ln [ \sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast } |]/\partial _ { v}\ln [ \sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast } | ], \label{w}$$with $ \partial _ { v}=\partial /\partial v$ and $ h_{5}^{\ast } \neq 0.$ If $ h_{5}^{\ast } = 0,$ or even $ h_{5}^{\ast } \neq 0 $ but $ \beta = 0,$ the coefficient $ w_{k}$ could be arbitrary functions on $ \left (x^{i},v\right). $   For the vacuum Einstein equation this is a degenerated case imposing the compatibility condition $ \beta = \alpha _ { i}=0,$ which are satisfied, for instance, if the $ h_{4}$ and $ h_{5}$ are related as in the formula (\[p1\ ]) but with $ h_{[0]}\left (x^{i}\right) = const.$ - Having defined $ h_{4}$ and $ h_{5}$ and calculate $ \gamma $ from ([ abc ] { }) we can solve the equation (\[ep4a\ ]) by integrating on varying “ $ v $ " the equation $ n_{i}^{\ast \ast } + \gamma n_{i}^{\ast } = 0.$ The exact solution is $ $ \begin{aligned } n

$h_{5[1,2]}\lfft( x^{i}\right) $ stated by buundary conditions; u\) or, inbersely, go compute $h_{4}$ for a given $h_{5}\leht( x^{u},v\rigyt) ,h_{5}^{\ast }\neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\lewt( x^{i}\righn) (\sqrt{|h_{5}\ledt( x^{m},v\right) |})^{\ast }, \lausl{p1}$$with $h_{[0]}\left( w^{i}\rigkt) $ given by boukdary condidions. - The exdcg dolutions of (\[ep3a\]) for $\beta \neq 0$ are qefined fgom an algebrayc eatatikn, $w_{i}\beta +\alpha _{i}=0,$ where the coerficienus $\beta $ and $\alpha _{i}$ are computed as in formklas ([abc]{}) by using the dolutions fir (\[e[1q\]) and (\[ep2a\]). Thd general solution is $$s_{k}=\partial _{k}\ln [\sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast }|]/\partiau _{v}\ln [\sqrt{|h_{4}h_{5}|}/|h_{5}^{\asj }|], \label{w}$$witi $\partpal _{v}=\partial /\isrtial v$ and $n_{5}^{\ast }\neq 0.$ If $h_{5}^{\sst }=0,$ oe even $h_{5}^{\ast }\neq 0$ but $\ueta =0,$ the coefficienjs $w_{k}$ coulg ye arbitrary functionw in $\lext( x^{h},v\rieyt). $  Fod vhe vacuul Emnstein equztions this is a degenerated cssq imposing the compaeifility conditions $\beta =\alpha _{i}=0,$ which art satjsfied, for instance, if rhe $h_{4}$ and $h_{5}$ are relatgd as in tre formula (\[p1\]) but with $h_{[0]}\left( x^{i}\right) =const.$ - Havinc defmndd $k_{4}$ and $f_{5}$ ajd computed $\gamma $ from ([abc]{}) we can solve the qsustpon (\[ep4a\]) by integrcting on variablr “$g $" jhe equation $n_{k}^{\ast \ast }+\famma n_{i}^{\ast }=0.$ The edact sojutiob is $$\begig{alibned} n

$h_{5[1,2]}\left( x^{i}\right) $ stated by boundary conditions; inversely, compute $h_{4}$ a given $h_{5}\left( (\sqrt{|h_{5}\left( |})^{\ast }, \label{p1}$$with x^{i}\right) $ given boundary conditions. - The exact solutions (\[ep3a\]) for $\beta \neq 0$ are defined from an algebraic equation, $w_{i}\beta +\alpha where the coefficients $\beta $ and $\alpha _{i}$ are computed as in formulas by the for and (\[ep2a\]). The general solution is $$w_{k}=\partial _{k}\ln [\sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast }|]/\partial _{v}\ln [\sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast }|], \label{w}$$with $\partial _{v}=\partial v$ and $h_{5}^{\ast }\neq 0.$ If $h_{5}^{\ast }=0,$ even $h_{5}^{\ast }\neq 0$ $\beta =0,$ the coefficients $w_{k}$ be functions on x^{i},v\right). For vacuum Einstein equations is a degenerated case imposing the compatibility conditions $\beta =\alpha _{i}=0,$ which are satisfied, for instance, if $h_{4}$ and related as the (\[p1\]) with $h_{[0]}\left( x^{i}\right) Having defined $h_{4}$ and $h_{5}$ and from ([abc]{}) we can solve the equation (\[ep4a\]) integrating on “$v $" the equation $n_{i}^{\ast \ast n_{i}^{\ast }=0.$ The exact solution is $$\begin{aligned} n

$h_{5[1,2]}\left( x^{i}\right) $ stated by boundAry conditiOns; b\) oR, inVerSeLy, to CompUte $h_{4}$ for a given $h_{5}\LEft( x^{I},v\right) ,h_{5}^{\ast }\neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\leFt( x^{i}\rIgHT) (\sqrT{|H_{5}\lEft( x^{i},V\right) |})^{\aST }, \lABEl{p1}$$WiTh $H_{[0]}\leFt( X^{I}\rIght) $ gIveN by bounDary conditIonS. - THe exact solutIOnS of (\[ep3a\]) for $\bEta \Neq 0$ are defineD frOm an alGeBraIC equaTioN, $w_{i}\beTa +\alphA _{I}=0,$ where The coeffiCiENts $\betA $ And $\alphA _{I}$ ArE comPuted as in formulas ([ABc]{}) BY using the solutIons foR (\[eP1A\]) aND (\[Ep2a\]). the General solUtIon is $$W_{K}=\partiaL _{K}\lN [\SQRt{|h_{4}H_{5}|}/|H_{5}^{\ast }|]/\partial _{v}\lN [\sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast }|], \LAbeL{w}$$with $\PaRtiAL _{v}=\partIal /\paRtIAl v$ And $h_{5}^{\ast }\neq 0.$ IF $h_{5}^{\asT }=0,$ or even $h_{5}^{\aSt }\neq 0$ bUT $\beta =0,$ thE CoefficIents $w_{K}$ coUld Be arBItRaRy fUnCTioNS oN $\leFT( x^{i},V\right). $  FoR tHe VacuuM EinSTEIN equAtiOns tHis is A degenerated cAse ImpoSIng The coMpatiBiliTy CondiTions $\bEta =\alPhA _{i}=0,$ which are satisFied, For instanCe, iF tHe $h_{4}$ AnD $h_{5}$ are RElated As iN thE formulA (\[p1\]) but wiTH $h_{[0]}\lEfT( X^{I}\RiGht) =const.$ - Having defiNeD $H_{4}$ AnD $h_{5}$ and comPuted $\gAMmA $ fROm ([abc]{}) we cAn SolVe thE EQuatiOn (\[ep4A\]) By IntegratIng on vARiAbLe “$v $" the eQuAtion $n_{I}^{\aSt \aSt }+\gAmma n_{I}^{\Ast }=0.$ THe exacT solutioN is $$\beGIn{aligned} n

$h_{5[1,2]}\left( x^{i}\r ight) $ st atedbybou nd arycond itions; b \ ) or , inversely, to comput e $h_ {4 } $ fo r a give n $h_{5 } \l e f t(x^ {i },v \r i gh t) ,h _{5}^{\ ast }\neq0,$ $$ \sqrt{|h_{4} | }= h_{[0]}\le ft( x^{i}\right ) ( \sqrt{ |h _{5 } \left ( x ^{i}, v\righ t ) |})^ {\ast }, \ l abel{p 1 }$$with $ h_ {[0] }\left( x^{i}\rig h t) $ given by bou ndaryco n di t i ons . - The ex ac t sol u tions o f ( \ [ e p3a \ ]) for $\beta \neq 0$ ar e de finedfr oma n alge braic e q uat ion, $w_{i} \bet a +\a lpha _ { i}=0,$w here th e coef fic ien ts $ \ be ta $an d $\ a lp ha_ {i} $ are co mp ut ed as inf o r m ulas ([ abc] {}) b y using the s olu tion s fo r (\[ ep1a\ ]) a nd (\[e p2a\]) . The g eneral solution is$$w_{k}=\ par ti al_{ k}\ln [\sqrt {|h _{4 }h_{5}| }/|h_{5 } ^{\ as t } |] /\partial _{v}\ln[\ s q rt {|h_{4}h _{5}|} / |h _{ 5 }^{\ast } |],\ label {w}$ $ wi th $\par tial _ { v} =\ partial / \parti al v$ an d $h_ { 5}^{ \ast } \neq 0.$ If $ h _{5}^{\ast }=0 , $ or even $h_ { 5} ^ { \a s t }\ neq 0$ but $\b eta= 0,$thec oe ffi c ients $w_{ k} $ c o uld be arbitrary fu nc tionson $\ left( x^{i},v \right). $ F or the v acuu m E i nstein equatio ns th is is a de g enerated case imposin g the com p a tibility co ndi tio ns$ \ be ta =\alpha _{ i } =0,$ w hich ar e s atisfie d,for in sta nc e, if the $h_{4}$ a nd $ h_ {5} $ are relatedas in t heformu l a (\[p 1\])butwi th $h_ {[0]}\l e ft ( x^{i }\ ri ght) =c on st.$ - Hav ing def ined $h_{ 4}$ and$h _{ 5}$ and computed $\g am ma $ from([ abc ]{}) w e can solv e the equation (\[ep4a\ ] ) by in teg ratin g on variable “$ v $" t hee quatio n $n_{ i}^{\ as t \ a s t }+\ g a mm a n _{ i}^{\ast } = 0 .$The e xa ct s olution is $$\begin{align e d} n

$h_{5[1,2]}\left(_x^{i}\right) $_stated by boundary conditions; _ __b\) or,_inversely,_to compute $h_{4}$_for a given_$h_{5}\left( x^{i},v\right) _,h_{5}^{\ast }\neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\left(_x^{i}\right)_(\sqrt{|h_{5}\left( x^{i},v\right) |})^{\ast }, \label{p1}$$with $h_{[0]}\left( x^{i}\right) $ given by boundary conditions. - __The exact_solutions_of_(\[ep3a\]) for $\beta \neq 0$_are defined from an algebraic_equation, $w_{i}\beta _ +\alpha _{i}=0,$ where the coefficients_$\beta_$ and $\alpha__{i}$ are computed as in formulas ([abc]{}) by using_the solutions for (\[ep1a\]) and (\[ep2a\])._The general solution_is_$$w_{k}=\partial__{k}\ln [\sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast }|]/\partial _{v}\ln_[\sqrt{|h_{4}h_{5}|}/|h_{5}^{\ast }|], _\label{w}$$with $\partial _{v}=\partial /\partial v$ and_$h_{5}^{\ast }\neq 0.$ If $h_{5}^{\ast }=0,$ or_even $h_{5}^{\ast }\neq 0$ but $\beta_=0,$ the coefficients $w_{k}$ could_be arbitrary_functions on $\left( x^{i},v\right). $_ For the vacuum_Einstein equations_this is a_degenerated case imposing the compatibility conditions_$\beta =\alpha _{i}=0,$_which are satisfied, for instance, if_the_$h_{4}$ and $h_{5}$_are_related_as in_the formula (\[p1\])_but_with $h_{[0]}\left(_x^{i}\right)_=const.$ - Having defined $h_{4}$_and_$h_{5}$ and computed $\gamma $ from ([abc]{})_we can solve the_equation_(\[ep4a\]) by integrating on_variable “$v $" the equation_$n_{i}^{\ast \ast }+\gamma n_{i}^{\ast }=0.$ The_exact solution_is $$\begin{aligned} _ n

mathsf{C}}}$, $deg(u_i) \geq 2\delta_s+1$. Therefore we can use the simple coding scheme described in Section \[sec:simple\_coding\] on $(m_{{\mathsf{C}}},n_{{\mathsf{C}}},{\mathcal{X}}_{{\mathsf{C}}},\delta_s)$ BNSI problem to save one transmission compared to uncoded transmission. Therefore the length of this code to transmit all the information symbols indexed by ${\mathsf{C}}\subseteq [n]$ over ${\mathds{F}}_q$ is $N_{{\mathsf{C}}}=|{\mathsf{C}}|-1$. For some integer $K$, let ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K \in \Phi({\mathcal{B}})$ and $R=[n] \setminus ({\mathsf{C}}_1 \cup {\mathsf{C}}_2 \cup \dots \cup {\mathsf{C}}_K)$. Given such a collection of elements of $\Phi({\mathcal{B}})$, we design a valid coding scheme as follows. We apply the coding scheme described in Section \[sec:simple\_coding\] on each element ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K$ and transmit the information symbols indexed by the set $R$ uncoded. The codelength for this scheme is $$\begin{aligned} N &= \sum_{i=1}^{K}{(|{\mathsf{C}}_i|-1)}+|R| = \sum_{i=1}^{K}{|{\mathsf{C}}_i|}-K+|R|.\end{aligned}$$ \[disjoint\] Let $N$ be the codelength of the linear coding scheme based on the set ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K \in \Phi({\mathcal{B}})$. Then there exist disjoint ${\mathsf{C}}'_1,{\mathsf{C}}'_2,\dots,{\mathsf{C}}'_{K'} \in \Phi({\mathcal{B}})$ such that $K' \leq K$ and the codelength $N'$ of the linear coding scheme based on ${\mathsf{C}}'_1,{\mathsf{C}}'_2,\dots,{\mathsf{C}}'_{K'}$ is at the most $N$. From the set ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_

mathsf{C}}}$, $ deg(u_i) \geq 2\delta_s+1$. Therefore we can use the simple coding schema trace in Section   \[sec: simple\_coding\ ] on $ (m_{{\mathsf{C}}},n_{{\mathsf{C}}},{\mathcal{X}}_{{\mathsf{C}}},\delta_s)$ BNSI problem to save one transmission compare to uncoded transmission. Therefore the distance of this code to impart all the information symbols indexed by $ { \mathsf{C}}\subseteq [ n]$ over $ { \mathds{F}}_q$ is $ N_{{\mathsf{C}}}=|{\mathsf{C}}|-1$. For some integer $ K$, let $ { \mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K \in \Phi({\mathcal{B}})$ and $ R=[n ] \setminus ({ \mathsf{C}}_1 \cup { \mathsf{C}}_2 \cup \dots \cup { \mathsf{C}}_K)$. give such a collection of elements of $ \Phi({\mathcal{B}})$, we plan a valid coding scheme as follows. We apply the coding scheme described in Section \[sec: simple\_coding\ ] on each component $ { \mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K$ and transmit the information symbols index by the set $ R$ uncoded. The codelength for this scheme is $ $ \begin{aligned } N & = \sum_{i=1}^{K}{(|{\mathsf{C}}_i|-1)}+|R| = \sum_{i=1}^{K}{|{\mathsf{C}}_i|}-K+|R|.\end{aligned}$$ \[disjoint\ ] Let $ N$ be the codelength of the analogue coding outline based on the set $ { \mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K \in \Phi({\mathcal{B}})$. Then there exist disjoint $ { \mathsf{C}}'_1,{\mathsf{C}}'_2,\dots,{\mathsf{C}}'_{K' } \in \Phi({\mathcal{B}})$ such that $ K' \leq K$ and the codelength $ N'$ of the analogue coding scheme based on $ { \mathsf{C}}'_1,{\mathsf{C}}'_2,\dots,{\mathsf{C}}'_{K'}$ is at the most $ N$. From the set $ { \mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C } } _

matjsf{C}}}$, $deg(u_i) \geq 2\delta_s+1$. Thtrefore we can usg rhe sikple ckding scfeme described in Section \[sec:dinple\_ciding\] on $(m_{{\mathsf{C}}},n_{{\mathrf{C}}},{\mathcap{X}}_{{\mathsf{X}}},\delua_s)$ BNSI problem vk save one trzksmisvmon compared to uncoded tsansmission. Thardflre the length of this code to tranfmit alk hhe informatiog sykfols pnbexed by ${\mathsf{C}}\subseteq [n]$ over ${\mathds{H}}_q$ is $N_{{\mathsf{C}}}=|{\msthsf{C}}|-1$. For some integer $K$, pet ${\lathsf{C}}_1,{\mathsf{C}}_2,\dots,{\lathsf{C}}_K \in \Phi({\iqthcal{B}})$ and $F=[n] \setminus ({\mathsf{C}}_1 \cuk {\mathsf{C}}_2 \cup \dots \cup {\mathsf{C}}_K)$. Eiven such a coolwctlmn of elemeits of $\Phi({\mathcal{B}})$, we desicn a vakid coding schcme av fillows. We apply the cmding scheme descrybed in Saccion \[sec:simple\_coding\] in each elekent ${\natfsf{D}}_1,{\methaf{C}}_2,\dotd,{\mavhsf{C}}_K$ and fransmit thw information symboks pmdexed by ths set $W$ tncoded. The codelength for this scheme ps $$\bsgin{aligned} N &= \sum_{i=1}^{K}{(|{\mathwf{C}}_i|-1)}+|R| = \sum_{i=1}^{K}{|{\mathsf{C}}_i|}-K+|T|.\end{aligneq}$$ \[disjoint\] Let $N$ be the codelength of the linear cmding rchtmc cqsfd on the set ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K \in \Prj({\mstmcal{B}})$. Then there exist disjpijt ${\iathsf{C}}'_1,{\mathsf{Z}}'_2,\dots,{\mcfhaf{C}}'_{K'} \in \Phi({\mathcal{H}})$ such jhat $K' \leq K$ anq thr codelength $N'$ of the lineae coding schvme vased on ${\mathsf{C}}'_1,{\machsf{C}}'_2,\dots,{\matksf{C}}'_{K'}$ os at the most $N$. From the set ${\matgsf{C}}_1,{\mathsf{C}}_2,\fots,{\mathsr{Z}}_

mathsf{C}}}$, $deg(u_i) \geq 2\delta_s+1$. Therefore we can simple scheme described Section \[sec:simple\_coding\] on one compared to uncoded Therefore the length this code to transmit all the symbols indexed by ${\mathsf{C}}\subseteq [n]$ over ${\mathds{F}}_q$ is $N_{{\mathsf{C}}}=|{\mathsf{C}}|-1$. For some integer $K$, ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K \in \Phi({\mathcal{B}})$ and $R=[n] \setminus ({\mathsf{C}}_1 \cup {\mathsf{C}}_2 \cup \dots \cup {\mathsf{C}}_K)$. such collection elements $\Phi({\mathcal{B}})$, we design a valid coding scheme as follows. We apply the coding scheme described in \[sec:simple\_coding\] on each element ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K$ and transmit the symbols indexed by the $R$ uncoded. The codelength for scheme $$\begin{aligned} N \sum_{i=1}^{K}{(|{\mathsf{C}}_i|-1)}+|R| \sum_{i=1}^{K}{|{\mathsf{C}}_i|}-K+|R|.\end{aligned}$$ Let $N$ be codelength of the linear coding scheme based on the set ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K \in \Phi({\mathcal{B}})$. Then there exist disjoint \in \Phi({\mathcal{B}})$ $K' \leq and codelength of the linear based on ${\mathsf{C}}'_1,{\mathsf{C}}'_2,\dots,{\mathsf{C}}'_{K'}$ is at the the set ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_

mathsf{C}}}$, $deg(u_i) \geq 2\delta_s+1$. TherEfore we can Use thE siMplE cOdinG schEme described in sEctiOn \[sec:simple\_coding\] on $(m_{{\maThsf{C}}},N_{{\mAThsf{c}}},{\MaThcal{x}}_{{\mathsf{c}}},\DeLTA_s)$ BnSi pRobLeM To Save oNe tRansmisSion comparEd tO uNcoded transmISsIon. TherefoRe tHe length of thIs cOde to tRaNsmIT all tHe iNformAtion sYMbols iNdexed by ${\mAtHSf{C}}\subSEteq [n]$ ovER ${\MaThds{f}}_q$ is $N_{{\mathsf{C}}}=|{\mathsF{c}}|-1$. FOR some integer $K$, lEt ${\mathSf{c}}_1,{\MaTHSf{C}}_2,\DotS,{\mathsf{C}}_K \iN \PHi({\matHCal{B}})$ and $r=[N] \sETMInuS ({\Mathsf{C}}_1 \cup {\matHsf{C}}_2 \cup \dots \CUp {\mAthsf{C}}_k)$. GIveN Such a cOllecTiON of Elements of $\PHi({\maThcal{B}})$, we dEsign a VAlid codINg schemE as folLowS. We ApplY ThE cOdiNg SCheME dEscRIbeD in SectiOn \[SeC:simpLe\_coDING\] On eaCh eLemeNt ${\matHsf{C}}_1,{\mathsf{C}}_2,\doTs,{\mAthsF{c}}_K$ aNd traNsmit The iNfOrmatIon symBols iNdExed by the set $R$ unCodeD. The codelEngTh For ThIs schEMe is $$\beGin{AliGned} N &= \suM_{i=1}^{K}{(|{\mathSF{C}}_i|-1)}+|r| = \sUM_{I=1}^{k}{|{\mAthsf{C}}_i|}-K+|R|.\end{aligneD}$$ \[dISJoInt\] Let $N$ bE the coDElEnGTh of the lInEar CodiNG SchemE basED oN the set ${\mAthsf{C}}_1,{\MAtHsF{C}}_2,\dots,{\mAtHsf{C}}_K \iN \PHi({\mAthCal{B}})$. THEn thEre exiSt disjoiNt ${\matHSf{C}}'_1,{\mathsf{C}}'_2,\dots,{\MAthsf{C}}'_{K'} \in \Phi({\mAThCAL{B}})$ SUch tHat $k' \leq K$ and the CodeLEngtH $N'$ of THe LinEAr codIng scHeME bASed on ${\mathsf{C}}'_1,{\mathsf{C}}'_2,\DoTs,{\mathSf{C}}'_{K'}$ iS at the most $N$. FrOm the set ${\maTHSF{C}}_1,{\mathsf{c}}_2,\dotS,{\MaTHsf{C}}_

mathsf{C}}}$, $deg(u_i) \g eq 2\delta _s+1$ . T her ef orewe c an use the sim p le c oding scheme described in S ec t ion\ [s ec:si mple\_c o di n g \]on $ (m_ {{ \ ma thsf{ C}} },n_{{\ mathsf{C}} },{ \m athcal{X}}_{ { \m athsf{C}}} ,\d elta_s)$ BNS I p roblem t o s a ve on e t ransm ission compar ed to unc od e d tran s mission . Th eref ore the length of th i s code to tran smit a ll th e inf orm ation symb ol s ind e xed by$ {\ m a t hsf { C}}\subseteq[n]$ over $ { \ma thds{F }} _q$ is $N_ {{\ma th s f{C }}}=|{\math sf{C }}|-1$. F or som e intege r $K$, l et ${\ mat hsf {C}} _ 1, {\ mat hs f {C} } _2 ,\d o ts, {\mathsf {C }} _K \i n \P h i ( { \mat hca l{B} })$ a nd $R=[n] \se tmi nus( {\m athsf {C}}_ 1 \c up {\ma thsf{C }}_2\c up \dots \cup { \mat hsf{C}}_K )$. G ive nsucha colle cti onof elem ents of $\P hi ( { \ ma thcal{B}})$, we de si g n a valid c odings ch em e as foll ow s.We a p p ly th e co d in g scheme descr i be din Sect io n \[se c: sim ple \_cod i ng\] on ea ch eleme nt ${ \ mathsf{C}}_1,{ \ mathsf{C}}_2, \ do t s ,{ \ math sf{ C}}_K$ andtran s mitthei nf orm a tionsymbo ls in d exed by the set $R$ u ncoded . The codelength f or this sc h e m e is $$\ begi n {a l igned} N &= \s um_{i =1}^{K}{(| { \mathsf{ C}}_i |-1)}+|R | = \sum _ { i=1}^{K} {|{ \ma ths f{C } } _i |}-K+|R|.\end { a lign ed }$$ \[ dis joint\] Le t $ N$beth e codelen gth of t he l in ea r c oding scheme b as edon th e set ${\mat hsf{C }}_1 ,{ \m a ths f{C}}_2 , \d o t s,{\ ma th sf{C }}_ K\in \ Phi( { \ma thcal{B }})$. The n t h ereex is t disjo int ${\mathsf {C }}'_1,{\ma th sf{ C}}'_2 , \ dots,{\m athsf{C}}'_{K'} \in \Ph i ({\math cal {B}}) $ su ch that $ K'\leq K $ a n d thecodele ngth$N '$o f thel i ne arco ding schem e bas ed on $ {\ma thsf{C} }'_1,{\mathsf{C}}' _ 2,\ dots,{\mathsf {C} }'_{ K ' }$ is at the m o st$ N $. From the se t ${\maths f{ C }} _1,{\maths f {C} }_ 2,\dots ,{\math sf{C} } _

mathsf{C}}}$, $deg(u_i)_\geq 2\delta_s+1$._Therefore we can use_the simple_coding_scheme described_in_Section \[sec:simple\_coding\] on $(m_{{\mathsf{C}}},n_{{\mathsf{C}}},{\mathcal{X}}_{{\mathsf{C}}},\delta_s)$_BNSI problem to_save one transmission compared_to uncoded transmission._Therefore_the length of this code to transmit all the information symbols indexed by ${\mathsf{C}}\subseteq_[n]$_over ${\mathds{F}}_q$_is_$N_{{\mathsf{C}}}=|{\mathsf{C}}|-1$._For some integer $K$, let_${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K \in \Phi({\mathcal{B}})$ and $R=[n]_\setminus ({\mathsf{C}}_1_\cup {\mathsf{C}}_2 \cup \dots \cup {\mathsf{C}}_K)$. Given such_a_collection of elements_of $\Phi({\mathcal{B}})$, we design a valid coding scheme as_follows. We apply the coding scheme_described in Section_\[sec:simple\_coding\]_on_each element ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K$ and_transmit the information symbols indexed by_the set $R$ uncoded. The codelength_for this scheme is $$\begin{aligned} N &= \sum_{i=1}^{K}{(|{\mathsf{C}}_i|-1)}+|R|_ = \sum_{i=1}^{K}{|{\mathsf{C}}_i|}-K+|R|.\end{aligned}$$ \[disjoint\] Let $N$ be the_codelength of the linear coding_scheme based_on the set ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_K \in_\Phi({\mathcal{B}})$. Then there_exist disjoint_${\mathsf{C}}'_1,{\mathsf{C}}'_2,\dots,{\mathsf{C}}'_{K'} \in \Phi({\mathcal{B}})$_such that $K' \leq K$ and_the codelength $N'$_of the linear coding scheme based_on_${\mathsf{C}}'_1,{\mathsf{C}}'_2,\dots,{\mathsf{C}}'_{K'}$ is at_the_most_$N$. From the_set ${\mathsf{C}}_1,{\mathsf{C}}_2,\dots,{\mathsf{C}}_

, parallelizable, compact Riemannian $n$-manifold can be embedded isometrically as a special Lagrangian submanifold in a manifold with holonomy ${\mathrm{SU}}(n)$. Notice that the assumption of real analyticity refers not only to the manifold, but to the structure as well. $\alpha$-Einstein-Sasaki geometry and hypersurfaces {#sec:go} =================================================== In this section we classify the constant intrinsic torsion geometries for the group ${\mathrm{SU}}(n)\subset {\mathrm{O}}(2n+1)$, and write down evolution equations for hypersurfaces which are orthogonal to the characteristic direction, in analogy with Section \[sec:hypo\]. Let $T={\mathbb{R}}^{2n+1}$. The space $(\Lambda^*T)^{{\mathrm{SU}}(n)}$ is spanned by the forms $\alpha$, $F$ and $\Omega^\pm$, defined in. In order to classify the differential operators on $(\Lambda^*T)^{{\mathrm{SU}}(n)}$, we observe that every element $g$ of the normalizer $N({\mathrm{SU}}(n))$ of ${\mathrm{SU}}(n)$ in maps $(\Lambda^*T)^{{\mathrm{SU}}(n)}$ to itself; this defines a natural notion of equivalence among differential operators. \[prop:oddderivation\] Let $f$ be a derivation of $(\Lambda^*T)^{{\mathrm{SU}}(n)}$; then $f$ is a differential operator that extend to a derivation of degree one on $\Lambda^*T$ if and only if one of the following holds: - $f(\alpha)=0$, $f(F)=2\lambda\alpha\wedge F$, $f(\Omega)=n(\lambda-\mu i)\alpha\wedge\Omega$; - $f(\alpha)=\lambda F$, $f(F)=0$, $f(\Omega)=-\mu i\alpha\wedge\Omega$; - $n=2$, and up to $N({\mathrm{SU}}(2))$ action, $\tilde f$ has the form (A) or (B); - $n=3$, $f(\alpha)=0$, $f(F)=3\lambda\Omega^- - 3\mu\Omega^+$, $f(\Omega)=2(\lambda+i\mu)F^2$; here $\lambda$ and $\mu$ are real constants.

, parallelizable, compact Riemannian $ n$-manifold can be embedded isometrically as a special Lagrangian submanifold in a manifold with holonomy $ { \mathrm{SU}}(n)$. Notice that the premise of veridical analyticity refers not only to the manifold, but to the structure equally well. $ \alpha$-Einstein - Sasaki geometry and hypersurfaces { # sec: go } = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = In this section we classify the changeless intrinsic torsion geometries for the group $ { \mathrm{SU}}(n)\subset { \mathrm{O}}(2n+1)$, and publish down evolution equations for hypersurfaces which are extraneous to the characteristic direction, in analogy with Section   \[sec: hypo\ ]. Let $ T={\mathbb{R}}^{2n+1}$. The space $ (\Lambda^*T)^{{\mathrm{SU}}(n)}$ is cross by the forms $ \alpha$, $ F$ and $ \Omega^\pm$, defined in. In decree to classify the differential operators on $ (\Lambda^*T)^{{\mathrm{SU}}(n)}$, we detect that every element $ g$ of the normalizer $ N({\mathrm{SU}}(n))$ of $ { \mathrm{SU}}(n)$ in maps $ (\Lambda^*T)^{{\mathrm{SU}}(n)}$ to itself; this defines a lifelike notion of equivalence among differential operator. \[prop: oddderivation\ ] Let $ f$ be a deriving of $ (\Lambda^*T)^{{\mathrm{SU}}(n)}$; then $ f$ is a differential operator that extend to a derivation of degree one on $ \Lambda^*T$ if and only if one of the follow holds: - $ f(\alpha)=0 $, $ f(F)=2\lambda\alpha\wedge F$, $ f(\Omega)=n(\lambda-\mu i)\alpha\wedge\Omega$; - $ f(\alpha)=\lambda F$, $ f(F)=0 $, $ f(\Omega)=-\mu i\alpha\wedge\Omega$; - $ n=2 $, and up to $ N({\mathrm{SU}}(2))$ action, $ \tilde f$ has the form (A) or (B); - $ n=3 $, $ f(\alpha)=0 $, $ f(F)=3\lambda\Omega^- - 3\mu\Omega^+$, $ f(\Omega)=2(\lambda+i\mu)F^2 $; here $ \lambda$ and $ \mu$ are veridical constants.

, pagallelizable, compact Rieoannian $n$-manifold can ue embesded isooetrically as a special Lagrengiqn suvmanifold in a manifola with hoponomy ${\mqthrn{WU}}(n)$. Notice vgat the assumlbion mh real analyticlty refers tot only to tha oauifold, but to the structure as well. $\ajpha$-Einxtfin-Sasaki geomgtry sgd hgievsurfaces {#sec:go} =================================================== In this section ws classpfy the constant ontrinsic torsion geometrifs flr the group ${\mathrl{SU}}(n)\subset {\nathwn{O}}(2n+1)$, and writd down evolution equatjons for hypersurfaces which ard ortkogonal to jkw cjdracteristir direbtion, in analogy with Sectiom \[sec:hypo\]. Let $T={\msthub{R}}^{2n+1}$. The space $(\Lambda^*T)^{{\matirm{SU}}(n)}$ is spanned by the formv $\clpha$, $F$ and $\Omega^\pm$, dwfuned hn. It orawr go dlessjfy thf dmfferential operators in $(\Lambda^*T)^{{\mathrm{SU}}(n)}$, wq observe that every ejement $g$ of the normalizer $N({\mathrm{SU}}(n))$ ox ${\mzthrm{SU}}(n)$ in maps $(\Lambda^*R)^{{\mathrm{SU}}(n)}$ to itself; jhis definqs a natural notion of equivalence among differendial koercbjfw. \[pgop:oddderivation\] Let $f$ be a derivation of $(\Lamfsa^*U)^{{\manhrm{SU}}(n)}$; then $f$ is a differenyiwl jperator that extenb tk a derivation of fegree jne ob $\Lambda^*T$ if snd only if one of the folliwing holds: - $f(\alpha)=0$, $f(F)=2\lambda\al'ha\wedge F$, $f(\Umegs)=n(\lamnda-\mu i)\alpha\wedge\Omega$; - $f(\ampha)=\lambda V$, $f(F)=0$, $f(\Omefx)=-\mu i\alpha\wedge\Ooegs$; - $n=2$, and up to $N({\mathrm{SU}}(2))$ aceion, $\tildx f$ hcs the furm (S) or (B); - $n=3$, $f(\alphw)=0$, $f(F)=3\lambda\Omega^- - 3\mu\Omega^+$, $f(\Omeya)=2(\lamtda+i\mu)F^2$; herf $\lambda$ and $\mu$ are real constaivs.

, parallelizable, compact Riemannian $n$-manifold can be as special Lagrangian in a manifold the of real analyticity not only to manifold, but to the structure as $\alpha$-Einstein-Sasaki geometry and hypersurfaces {#sec:go} =================================================== In this section we classify the constant torsion geometries for the group ${\mathrm{SU}}(n)\subset {\mathrm{O}}(2n+1)$, and write down evolution equations for which orthogonal the direction, in analogy with Section \[sec:hypo\]. Let $T={\mathbb{R}}^{2n+1}$. The space $(\Lambda^*T)^{{\mathrm{SU}}(n)}$ is spanned by the forms $F$ and $\Omega^\pm$, defined in. In order to the differential operators on we observe that every element of normalizer $N({\mathrm{SU}}(n))$ ${\mathrm{SU}}(n)$ maps to itself; this a natural notion of equivalence among differential operators. \[prop:oddderivation\] Let $f$ be a derivation of $(\Lambda^*T)^{{\mathrm{SU}}(n)}$; then is a that extend a of one on $\Lambda^*T$ only if one of the following $f(F)=2\lambda\alpha\wedge F$, $f(\Omega)=n(\lambda-\mu i)\alpha\wedge\Omega$; - $f(\alpha)=\lambda F$, $f(F)=0$, i\alpha\wedge\Omega$; - and up to $N({\mathrm{SU}}(2))$ action, $\tilde has the form (A) or (B); - $n=3$, $f(F)=3\lambda\Omega^- - 3\mu\Omega^+$, $f(\Omega)=2(\lambda+i\mu)F^2$; here $\lambda$ and $\mu$ are real constants.

, parallelizable, compact RiemAnnian $n$-manIfold Can Be eMbEddeD isoMetrically as a sPEciaL Lagrangian submanifold In a maNiFOld wITh HolonOmy ${\mathRM{Su}}(N)$. notIcE tHat ThE AsSumptIon Of real aNalyticity RefErS not only to thE MaNifold, but tO thE structure as WelL. $\alpha$-eiNstEIn-SasAki GeomeTry and HYpersuRfaces {#sec:Go} =================================================== iN this sECtion we CLAsSify The constant intrinSIc TOrsion geometriEs for tHe GRoUP ${\MatHrm{sU}}(n)\subset {\mAtHrm{O}}(2n+1)$, ANd write DOwN EVOluTIon equations fOr hypersurfACes Which aRe OrtHOgonal To the ChARacTeristic dirEctiOn, in analoGy with sEction \[sEC:hypo\]. LeT $T={\mathBb{R}}^{2N+1}$. ThE spaCE $(\LAmBda^*t)^{{\mAThrM{sU}}(N)}$ is SPanNed by the FoRmS $\alphA$, $F$ anD $\oMEGa^\pm$, DefIned In. In oRder to classifY thE difFEreNtial OperaTors On $(\lambdA^*T)^{{\mathRm{SU}}(n)}$, We Observe that everY eleMent $g$ of thE noRmAliZeR $N({\matHRm{SU}}(n))$ oF ${\maThrM{SU}}(n)$ in mAps $(\LambDA^*T)^{{\mAtHRM{sU}}(N)}$ to itself; this definEs A NAtUral notiOn of eqUIvAlENce among DiFfeRentIAL operAtorS. \[PrOp:oddderIvatioN\] leT $f$ Be a deriVaTion of $(\laMbdA^*T)^{{\mAthrm{su}}(n)}$; thEn $f$ is a DifferenTial oPErator that exteND to a derivatioN Of DEGrEE one On $\LAmbda^*T$ if and Only IF one Of thE FoLloWIng hoLds: - $f(\aLpHA)=0$, $f(f)=2\Lambda\alpha\wedge F$, $f(\OMeGa)=n(\lamBda-\mu I)\alpha\wedge\OmEga$; - $f(\alpha)=\lAMBDa F$, $f(F)=0$, $f(\OmEga)=-\mU I\aLPha\wedge\Omega$; - $n=2$, And up To $N({\mathrm{Su}}(2))$ Action, $\tiLde f$ hAs the forM (A) or (B); - $n=3$, $f(\alPHA)=0$, $f(F)=3\lambdA\OmEga^- - 3\Mu\OMegA^+$, $F(\omEga)=2(\lambda+i\mu)F^2$; HERe $\laMbDa$ and $\mu$ Are Real conStaNts.

, parallelizable, compactRiemannian $n$- man ifo ld can beembedded isome t rica lly as a special Lagra ngian s u bman i fo ld in a mani f ol d wit hho lon om y $ {\mat hrm {SU}}(n )$. Notic e t ha t the assump t io n of realana lyticity ref ers not o nl y t o theman ifold , butt o thestructure a s well. $\alph a $ -E inst ein-Sasaki geomet r ya nd hypersurfac es {#s ec : go } === === ========== == ===== = ======= = == = = = === = ========== I n this sect i onwe cla ss ify the co nstan ti ntr insic torsi on g eometries for t h e group ${\math rm{SU} }(n )\s ubse t { \m ath rm { O}} ( 2n +1) $ , a nd write d ow n evo luti o n e quat ion s fo r hyp ersurfaces wh ich are ort hogon al to the c harac terist ic di re ction, in analo gy w ith Secti on\[ sec :h ypo\] . Let$T= {\m athbb{R }}^{2n+ 1 }$. T h e sp ace $(\Lambda^*T)^ {{ \ m at hrm{SU}} (n)}$i ssp a nned byth e f orms $ \alph a$,$ F$ and $\O mega^\ p m$ ,defined i n. Inor der to clas s ifythe di fferenti al op e rators on $(\L a mbda^*T)^{{\m a th r m {S U }}(n )}$ , we observ e th a t ev erye le men t $g$of th en or m alizer $N({\mathrm{ SU }}(n)) $ of${\mathrm{SU} }(n)$ in m a p s $(\Lamb da^* T )^ { {\mathrm{SU}}( n)}$to itself; this def inesa natura l notiono f equival enc e a mon g d i f fe rential opera t o rs. \ [prop:o ddd erivati on\ ] L et$f$ b e a deriv ation of $ (\ La mb da^ *T)^{ { \mathrm{ SU }}( n) }$; then $f$ is a di ffer en ti a l o perator th a t ext en dto a de ri vatio n of deg ree one on $\Lam bda ^ *T$if a nd only if one of th efollowingho lds : - $ f(\alpha )=0$, $f(F)=2\lambda\al p ha\wedg e F $, $f (\Om ega)=n(\l amb da-\mu i) \ alpha\ wedge\ Omega $; - $f(\ a l ph a)= \l ambda F$,$ f (F) =0$,$f (\Om ega)=-\ mu i\alpha\wedge\O m ega $; - $n=2$ , a nd u p to $N ( {\ m ath rm { SU} } ( 2))$ action, $\ tilde f$ h as th e form (A) or(B ); - $n=3$, $f(\ a lpha)=0 $, $f(F)= 3\lambda\ Om ega^ - - 3 \mu\Omega^ +$, $f(\ Omega)=2( \ lambd a +i \mu)F ^2$ ; her e$\l ambda $ and$ \mu $ are realco nstant s.

, parallelizable,_compact Riemannian_$n$-manifold can be embedded_isometrically as_a_special Lagrangian_submanifold_in a manifold_with holonomy ${\mathrm{SU}}(n)$. Notice_that the assumption of_real analyticity refers_not_only to the manifold, but to the structure as well. $\alpha$-Einstein-Sasaki geometry and hypersurfaces {#sec:go} =================================================== In_this_section we_classify_the_constant intrinsic torsion geometries for_the group ${\mathrm{SU}}(n)\subset {\mathrm{O}}(2n+1)$, and_write down_evolution equations for hypersurfaces which are orthogonal to_the_characteristic direction, in_analogy with Section \[sec:hypo\]. Let $T={\mathbb{R}}^{2n+1}$. The space $(\Lambda^*T)^{{\mathrm{SU}}(n)}$ is spanned_by the forms $\alpha$, $F$ and_$\Omega^\pm$, defined in._In_order_to classify the differential_operators on $(\Lambda^*T)^{{\mathrm{SU}}(n)}$, we observe that_every element $g$ of the normalizer_$N({\mathrm{SU}}(n))$ of ${\mathrm{SU}}(n)$ in maps $(\Lambda^*T)^{{\mathrm{SU}}(n)}$ to_itself; this defines a natural notion_of equivalence among differential operators. \[prop:oddderivation\]_Let $f$_be a derivation of $(\Lambda^*T)^{{\mathrm{SU}}(n)}$;_then $f$ is_a differential_operator that extend_to a derivation of degree one_on $\Lambda^*T$ if_and only if one of the_following_holds: - _$f(\alpha)=0$,_$f(F)=2\lambda\alpha\wedge_F$, $f(\Omega)=n(\lambda-\mu_i)\alpha\wedge\Omega$; - _$f(\alpha)=\lambda_F$, $f(F)=0$,_$f(\Omega)=-\mu_i\alpha\wedge\Omega$; - $n=2$, and up_to_$N({\mathrm{SU}}(2))$ action, $\tilde f$ has the form_(A) or (B); - __$n=3$, $f(\alpha)=0$, $f(F)=3\lambda\Omega^- -_3\mu\Omega^+$, $f(\Omega)=2(\lambda+i\mu)F^2$; here $\lambda$ and $\mu$_are real constants.

}(X^-)\ar[d] \\ \mathrm{res}_{\pi}(\pi_n^{-}( \Sigma^{\infty}(X^n/ X^{n-1}))) \ar[r]^{d_n \ \ \ } & \mathrm{res}_{\pi}(\pi_{n-1}^{-}( \Sigma^{\infty}(X^{n-1}/ X^{n-2}))). }$$ Using the adjointness of $\mathrm{res}_{\pi}$ and $\mathrm{ind}_{\pi}$ and $\mathrm{ind}_{\pi}(\mathbb{Z}[-,G/K])=\mathbb{Z}^{G}[-,K]$, we conclude that the chain complex obtained from $ \Sigma^{\infty}X_+$ by applying the methods from Subsection 4.1 to the stable cofiber sequences obtained by applying $\Sigma^{\infty}$ to the homotopy cofiber sequences of $X$, coincides with the chain complex obtained by applying the induction functor $\mathrm{ind}_{\pi}$ to the cellular chain complex $C_{\ast}(X^-)$ of $X$. The above discussion shows that the suspension spectrum functor is a geometric analog of the induction functor $\mathrm{ind}_{\pi}$. This indicates that there should be an algebraic version of Proposition \[prop: susp model\]. Indeed, [@MartinezNucinkis06 Th. 3.8] shows that if $P_{\ast}$ is a projective resolution of $\underline{\mathbb{Z}}$, then $\mathrm{ind}_{\pi}(P_{\ast})$ is a projective resolution of $\underline{A}$. Below we show that the converse of the latter is also true. The proof requires the following lemma and uses notation and isomorphisms from Section 2. \[lemma: tensor zero\] For any right ${\mathcal{O}_{{\mathcal {F}}}G}$-module $M$ and any $K \in \mathcal{F}$, $\mathrm{ind}_{\pi}(M)(G/K)=0$ implies that $M(G/K)=0$. Fix $K \in \mathcal{F}$ and let $\mathrm{ind}_K^G$ denote the induction functor from the category of covariant Mackey functors for $K$ to the category of covariant Mackey functors for $G$, associated to the inclusion $i_K^G$ of $K$ into $G$. Then $$\mathrm{ind}_K^G(\mathbb{Z}^K[K,\pi_K(-)])\cong \mathbb{Z

} (X^-)\ar[d ] \\ \mathrm{res}_{\pi}(\pi_n^{- } (\Sigma^{\infty}(X^n/ X^{n-1 }) )) \ar[r]^{d_n \ \ \ } & \mathrm{res}_{\pi}(\pi_{n-1}^{- } (\Sigma^{\infty}(X^{n-1}/ X^{n-2 }) )). } $ $ Using the adjointness of $ \mathrm{res}_{\pi}$ and $ \mathrm{ind}_{\pi}$ and $ \mathrm{ind}_{\pi}(\mathbb{Z}[-,G / K])=\mathbb{Z}^{G}[-,K]$, we conclude that the chain complex prevail from $ \Sigma^{\infty}X_+$ by enforce the methods from Subsection 4.1 to the static cofiber sequences receive by applying $ \Sigma^{\infty}$ to the homotopy cofiber sequence of $ X$, coincides with the chain building complex prevail by applying the trigger functor $ \mathrm{ind}_{\pi}$ to the cellular chain complex $ C_{\ast}(X^-)$ of $ X$. The above discussion testify that the suspension spectrum functor is a geometric analog of the induction functor $ \mathrm{ind}_{\pi}$. This indicates that there should be an algebraic version of Proposition \[prop: susp model\ ]. Indeed, [ @MartinezNucinkis06 Th. 3.8 ] shows that if $ P_{\ast}$ is a projective resolving power of $ \underline{\mathbb{Z}}$, then $ \mathrm{ind}_{\pi}(P_{\ast})$ is a projective resolution of $ \underline{A}$. Below we show that the converse of the latter is also on-key. The proof requires the following lemma and uses note and isomorphisms from Section 2. \[lemma: tensor zero\ ] For any correct $ { \mathcal{O}_{{\mathcal { F}}}G}$-module $ M$ and any $ K \in \mathcal{F}$, $ \mathrm{ind}_{\pi}(M)(G / K)=0 $ implies that $ M(G / K)=0$. Fix $ K \in \mathcal{F}$ and let $ \mathrm{ind}_K^G$ denote the evocation functor from the category of covariant Mackey functors for $ K$ to the category of covariant Mackey functors for $ G$, associated to the inclusion $ i_K^G$ of $ K$ into $ G$. Then $ $ \mathrm{ind}_K^G(\mathbb{Z}^K[K,\pi_K(-)])\cong \mathbb{Z

}(X^-)\ar[f] \\ \mathrm{res}_{\pi}(\pi_n^{-}( \Sigma^{\innty}(X^n/ X^{n-1}))) \ar[r]^{d_n \ \ \ } & \matirm{res}_{\pj}(\pi_{n-1}^{-}( \Sigoa^{\infty}(X^{n-1}/ X^{n-2}))). }$$ Using the adjomntnwss od $\mathrm{res}_{\pi}$ and $\mathfm{ind}_{\pi}$ ajd $\mathrn{ind}_{\ki}(\mathbb{Z}[-,G/K])=\mathbb{V}^{F}[-,K]$, we concluds thac vhe chain complgx obtained xrom $ \Sigma^{\infdy}B_+$ yy applying the methods from Subsectyon 4.1 to tje stable cofifer xqquehbew obtained by applying $\Sigma^{\ihfty}$ to the homotopy cofiber sequences of $X$, colncifes with the chain complex objzinqe by applyine the indubcion functot $\mathrm{ind}_{\pi}$ to the cellular chxin cpmplex $C_{\asj}(R^-)$ of $F$. The above viscusfion shows tmst the suspenxion spectrum nunctmr us a geometric analog of the induction sunctor $\mdtkrm{ind}_{\pi}$. This indicatws that these sfiula bt ai amgebralc tersion of Lroposition \[prop: susp model\]. Inceqe, [@MartinezNucjnkis06 Eh. 3.8] shows that if $P_{\ast}$ is a projective revolhtion of $\underline{\mathbv{Z}}$, then $\mathrm{ind}_{\pi}(P_{\adt})$ is a pwojective resolution of $\underline{A}$. Below we show dhat vhd cinyersd ov the latter is also true. The proof requires ege fpllowing lemma and uses nptwtojn and isomorkhisms ydoj Section 2. \[lemma: tejsor zeto\] For any righu ${\matncal{O}_{{\mathcal {F}}}G}$-module $M$ and any $K \in \manhcao{F}$, $\mathrm{ind}_{\pi}(M)(G/K)=0$ nmplies that $M(G/K)=0$. Gix $K \in \mathcal{F}$ and let $\machrm{ins}_K^G$ denote hhe inducfkon functor from thv cadegory of covariant Mackey functors for $K$ to tfe cstegorr of covarlant Mackey functors for $G$, asdowiated to hhe inclusion $i_K^G$ of $K$ into $G$. Tixn $$\mathrm{ind}_K^B(\mdthtb{Z}^K[K,\pi_K(-)])\eong \msthbb{Z

}(X^-)\ar[d] \\ \mathrm{res}_{\pi}(\pi_n^{-}( \Sigma^{\infty}(X^n/ X^{n-1}))) \ar[r]^{d_n \ } \mathrm{res}_{\pi}(\pi_{n-1}^{-}( \Sigma^{\infty}(X^{n-1}/ }$$ Using the and we conclude that chain complex obtained $ \Sigma^{\infty}X_+$ by applying the methods Subsection 4.1 to the stable cofiber sequences obtained by applying $\Sigma^{\infty}$ to the cofiber sequences of $X$, coincides with the chain complex obtained by applying the functor to cellular complex $C_{\ast}(X^-)$ of $X$. The above discussion shows that the suspension spectrum functor is a geometric of the induction functor $\mathrm{ind}_{\pi}$. This indicates that should be an algebraic of Proposition \[prop: susp model\]. [@MartinezNucinkis06 3.8] shows if is projective resolution of then $\mathrm{ind}_{\pi}(P_{\ast})$ is a projective resolution of $\underline{A}$. Below we show that the converse of the latter also true. requires the lemma uses and isomorphisms from \[lemma: tensor zero\] For any right and any $K \in \mathcal{F}$, $\mathrm{ind}_{\pi}(M)(G/K)=0$ implies that Fix $K \mathcal{F}$ and let $\mathrm{ind}_K^G$ denote the functor from the category of covariant Mackey functors $K$ to the category of covariant Mackey functors for $G$, associated to the inclusion $i_K^G$ into $G$. Then $$\mathrm{ind}_K^G(\mathbb{Z}^K[K,\pi_K(-)])\cong

}(X^-)\ar[d] \\ \mathrm{res}_{\pi}(\pi_n^{-}( \Sigma^{\inFty}(X^n/ X^{n-1}))) \ar[r]^{D_n \ \ \ } & \matHrm{Res}_{\Pi}(\Pi_{n-1}^{-}( \SIgma^{\Infty}(X^{n-1}/ X^{n-2}))). }$$ Using THe adJointness of $\mathrm{res}_{\pi}$ And $\maThRM{ind}_{\PI}$ aNd $\matHrm{ind}_{\pI}(\MaTHBb{Z}[-,g/K])=\MaThbB{Z}^{g}[-,k]$, wE concLudE that thE chain compLex ObTained from $ \SiGMa^{\Infty}X_+$ by apPlyIng the methodS frOm SubsEcTioN 4.1 To the StaBle coFiber sEQuenceS obtained By APplyinG $\sigma^{\inFTY}$ tO the Homotopy cofiber seQUeNCes of $X$, coincideS with tHe CHaIN ComPleX obtained bY aPplyiNG the indUCtION FunCTor $\mathrm{ind}_{\pI}$ to the celluLAr cHain coMpLex $c_{\Ast}(X^-)$ of $x$. The aBoVE diScussion shoWs thAt the suspEnsion SPectrum FUnctor iS a geomEtrIc aNaloG Of ThE inDuCTioN FuNctOR $\maThrm{ind}_{\pI}$. THiS indiCateS THAT theRe sHoulD be an Algebraic versIon Of PrOPosItion \[Prop: sUsp mOdEl\]. IndEed, [@MarTineznuCinkis06 Th. 3.8] shows thAt if $p_{\ast}$ is a prOjeCtIve ReSolutIOn of $\unDerLinE{\mathbb{z}}$, then $\maTHrm{InD}_{\PI}(p_{\aSt})$ is a projective resOlUTIoN of $\underLine{A}$. BELoW wE Show that ThE coNverSE Of the LattER iS also truE. The prOOf ReQuires tHe FollowInG leMma And usES notAtion aNd isomorPhismS From Section 2. \[lemMA: tensor zero\] FoR AnY RIgHT ${\matHcaL{O}_{{\mathcal {F}}}G}$-ModuLE $M$ anD any $k \In \MatHCal{F}$, $\mAthrm{InD}_{\Pi}(m)(g/K)=0$ implies that $M(G/K)=0$. Fix $k \iN \mathcAl{F}$ anD let $\mathrm{ind}_k^G$ denote thE INDuction fUnctOR fROm the category oF covaRiant MackeY Functors For $K$ tO the cateGory of covARIant MackEy fUncTorS foR $g$, AsSociated to the INClusIoN $i_K^G$ of $K$ IntO $G$. Then $$\mAthRm{iNd}_K^g(\maThBb{Z}^K[K,\pi_K(-)])\cOng \mathbB{Z

}(X^-)\ar[d] \\ \mathrm{re s}_{\pi}(\ pi_n^ {-} ( \ Si gma^ {\in fty}(X^n/ X^{n - 1})) ) \ar[r]^{d_n \ \ \ }& \ma th r m{re s }_ {\pi} (\pi_{n - 1} ^ { -}( \ Si gma ^{ \ in fty}( X^{ n-1}/ X ^{n-2}))). }$ $Using the ad j oi ntness of$\m athrm{res}_{ \pi }$ and $ \ma t hrm{i nd} _{\pi }$ and $\math rm{ind}_{ \p i }(\mat h bb{Z}[- , G /K ])=\ mathbb{Z}^{G}[-,K ] $, we conclude th at the c h ai n com ple x obtained f rom $ \Sigma^ { \i n f t y}X _ +$ by applyin g the metho d s f rom Su bs ect i on 4.1 to t he sta ble cofiber seq uences ob tained by appl y ing $\S igma^{ \in fty }$ t o t he ho mo t opy co fib e r s equences o f$X$,coin c i d e s wi ththechain complex obta ine d by app lying theindu ct ion f unctor $\ma th rm{ind}_{\pi}$to t he cellul arch ain c omple x $C_{\ ast }(X ^-)$ of $X$. T heab o v e d iscussion shows th at t he suspens ion sp e ct ru m functor i s a geo m e tricanal o gof the i nducti o nfu nctor $ \m athrm{ in d}_ {\p i}$.T hisindica tes that ther e should be ana lgebraic vers i on o fP ropo sit ion \[prop: sus p mod el\] . I nde e d, [@ Marti ne z Nu c inkis06 Th. 3.8] sh ow s that if $ P_{\ast}$ isa projecti v e resoluti on o f $ \ underline{\mat hbb{Z }}$, then$ \mathrm{ ind}_ {\pi}(P_ {\ast})$i s a proje cti veres olu t i on of $\underli n e {A}$ .Below w e s how tha t t hecon ver se of the l atter is a ls otr ue. Thep roof req ui res t hefollo w ing le mma a nd u se sn ota tion an d i s o morp hi sm s fr omSe ction 2.\[l emma: t ensor zer o\] Foran yright $ {\mathcal{O}_ {{ \mathcal { F} }}G }$-mod u l e $M$ an d any $K \in \mathcal{F } $, $\ma thr m{ind }_{\ pi}(M)(G/ K)= 0$ imp lie s that$M(G/K )=0$. Fix $ K \in \ ma thc al {F}$ and l e t $\ mathr m{ ind} _K^G$ d enote the inductio n fu nctor from th e c ateg o r yofc ov a ria nt Mac k e y functors for$K$ to the c a te gory of co v ari an t Macke y funct ors f o r $G$,associate d to thein clus i o n $ i_K^G$ of$K$ into $G$. The n $$\m a th rm{in d}_ K^G(\m at hbb {Z}^K [K,\pi _ K(- )])\c ong \m at hbb{Z

}(X^-)\ar[d] \\ \mathrm{res}_{\pi}(\pi_n^{-}(_\Sigma^{\infty}(X^n/ X^{n-1})))_\ar[r]^{d_n \ \ \_} &_\mathrm{res}_{\pi}(\pi_{n-1}^{-}(_\Sigma^{\infty}(X^{n-1}/ X^{n-2})))._}$$_Using the adjointness_of $\mathrm{res}_{\pi}$ and_$\mathrm{ind}_{\pi}$ and $\mathrm{ind}_{\pi}(\mathbb{Z}[-,G/K])=\mathbb{Z}^{G}[-,K]$, we_conclude that the_chain_complex obtained from $ \Sigma^{\infty}X_+$ by applying the methods from Subsection 4.1 to the_stable_cofiber sequences_obtained_by_applying $\Sigma^{\infty}$ to the homotopy_cofiber sequences of $X$, coincides_with the_chain complex obtained by applying the induction functor_$\mathrm{ind}_{\pi}$_to the cellular_chain complex $C_{\ast}(X^-)$ of $X$. The above discussion shows that_the suspension spectrum functor is a_geometric analog of_the_induction_functor $\mathrm{ind}_{\pi}$. This indicates_that there should be an algebraic_version of Proposition \[prop: susp model\]._Indeed, [@MartinezNucinkis06 Th. 3.8] shows that if_$P_{\ast}$ is a projective resolution of_$\underline{\mathbb{Z}}$, then $\mathrm{ind}_{\pi}(P_{\ast})$ is a_projective resolution_of $\underline{A}$. Below we show_that the converse_of the_latter is also_true. The proof requires the following_lemma and uses_notation and isomorphisms from Section 2. \[lemma:_tensor_zero\] For any_right_${\mathcal{O}_{{\mathcal_{F}}}G}$-module $M$_and any $K_\in_\mathcal{F}$, $\mathrm{ind}_{\pi}(M)(G/K)=0$_implies_that $M(G/K)=0$. Fix $K \in \mathcal{F}$ and_let_$\mathrm{ind}_K^G$ denote the induction functor from the_category of covariant Mackey_functors_for $K$ to the_category of covariant Mackey functors_for $G$, associated to the inclusion_$i_K^G$ of_$K$ into_$G$. Then $$\mathrm{ind}_K^G(\mathbb{Z}^K[K,\pi_K(-)])\cong \mathbb{Z

Hodge, P.W. 1961,, 66, 83 Ibata, R., Gilmore, G., & Irwin, M. 1994,, 370, 194 Kodama, T. & Bower, R.G. 2001,, 321, 18 Karachentsev, [[*et al.*]{}]{} 2003,, 398, 479 McLaughlin, D.E. 1999,, 117, 2398 Meurer, G.R., Mackie, G., & Carignan, C. 1994,, 107, 2021 (MMC94) Meurer, G.R., Carignan, C., Beaulieu, S., & Freeman, K.C. 1996,, 111, 1551 (MCBF96) Meylan, G., Sarajedeni, A., Jablonka, P., Djorgovski, S.G., Bridges, T., & Rich, R.M. 2001,, 122, 830 Östlin, G., Bergvall, N., & Rönnback, J. 1998,, 335, 85 Pritchet, C.J., & van den Bergh, S. 1984,, 96, 804 Secker, J. 1992,, 104, 1472 Schlegel, D.J., Finkbeiner, D.P., & Davis, M. 1998,, 500, 525 [lccl]{}   & $0.28 \pm 0.04$ & mag & foreground extinction\ $D$ & $4.1 \pm 0.3$ & Mpc & Distance\ ${\cal M}_g$ & $7.4 \times 10^8$ & & ISM mass\ ${\cal M}_\star$ & $3.2 \times 10^8$ & & Mass in stars\ ${\cal M}_T$ & $2.1 \times 10^{10}$ & & Total dynamical mass\ $M_V$ & –16.42 & mag & Absolute mag $V$ band\ $L_B$ & $3.4 \times 10^8$ & $L_{B,\odot}$ & $B$ band luminosity\   & 62 & solar & Mass to light ratio\ [l

Hodge, P.W. 1961, , 66, 83 Ibata, R., Gilmore, G., &   Irwin, M. 1994, , 370, 194 Kodama, T. &   Bower, R.G. 2001, , 321, 18 Karachentsev, [ [ * et   al. * ] { } ] { }   2003, , 398, 479 McLaughlin, D.E. 1999, , 117, 2398 Meurer, G.R., Mackie, G., &   Carignan, C. 1994, , 107, 2021 (MMC94) Meurer, G.R., Carignan, C., Beaulieu, S., &   Freeman, K.C. 1996, , 111, 1551 (MCBF96) Meylan, G., Sarajedeni, A., Jablonka, P., Djorgovski, S.G., Bridges, T., &   Rich, R.M. 2001, , 122, 830 Östlin, G., Bergvall, N., &   Rönnback, J. 1998, , 335, 85 Pritchet, C.J., &   van den Bergh, S. 1984, , 96, 804 Secker, J. 1992, , 104, 1472 Schlegel, D.J., Finkbeiner, D.P., &   Davis, M. 1998, , 500, 525 [ lccl ] { }   & $ 0.28 \pm 0.04 $ & mag & foreground extinction\ $ D$ & $ 4.1 \pm 0.3 $ & Mpc & Distance\ $ { \cal M}_g$ & $ 7.4 \times 10 ^ 8 $ & & ISM mass\ $ { \cal M}_\star$ & $ 3.2 \times 10 ^ 8 $ & & Mass in stars\ $ { \cal M}_T$ & $ 2.1 \times 10^{10}$ & & Total dynamical mass\ $ M_V$ & – 16.42 & mag & Absolute mag $ V$ band\ $ L_B$ & $ 3.4 \times 10 ^ 8 $ & $ L_{B,\odot}$ & $ B$ band luminosity\   & 62 & solar & Mass to light ratio\ [ l

Hodhe, P.W. 1961,, 66, 83 Ibata, R., Gilmore, N., & Irwin, M. 1994,, 370, 194 Kodama, T. & Boxer, R.G. 2001,, 321, 18 Karachdntsev, [[*et al.*]{}]{} 2003,, 398, 479 McLaughlin, D.E. 1999,, 117, 2398 Mwurer, G.R., Mackie, G., & Carignan, Z. 1994,, 107, 2021 (MMC94) Mvurer, G.R., Xarijnan, C., Beaulieu, S., & Freemak, K.C. 1996,, 111, 1551 (MCYF96) Neylan, G., Sarajgdeni, A., Jablmnka, P., Djorgovvkk, D.G., Bridges, T., & Rich, R.M. 2001,, 122, 830 Östlin, G., Bergdall, N., & Tönjback, J. 1998,, 335, 85 Pritshet, S.J., & vzn den Bergh, S. 1984,, 96, 804 Secker, J. 1992,, 104, 1472 Schlegsl, D.J., Fpnkbeiner, D.P., & Davix, M. 1998,, 500, 525 [lccl]{}   & $0.28 \pm 0.04$ & mag & fogegrlund extinction\ $D$ & $4.1 \pm 0.3$ & Mpc & Eistwbce\ ${\cal M}_g$ & $7.4 \gimes 10^8$ & & ISM mass\ ${\cal M}_\atar$ & $3.2 \times 10^8$ & & Mass in stars\ ${\cau M}_T$ & $2.1 \times 10^{10}$ & & Titap dynamical nass\ $M_N$ & –16.42 & mag & Absolute mac $V$ banc\ $L_B$ & $3.4 \times 10^8$ & $K_{B,\ovot}$ & $B$ band luminosity\   & 62 & solar & Mass to lidht ratio\ [n

Hodge, P.W. 1961,, 66, 83 Ibata, R., & M. 1994,, 194 Kodama, T. 18 [[*et al.*]{}]{} 2003,, 479 McLaughlin, D.E. 117, 2398 Meurer, G.R., Mackie, G., Carignan, C. 1994,, 107, 2021 (MMC94) Meurer, G.R., Carignan, C., Beaulieu, S., & K.C. 1996,, 111, 1551 (MCBF96) Meylan, G., Sarajedeni, A., Jablonka, P., Djorgovski, S.G., T., Rich, 2001,, 830 Östlin, G., Bergvall, N., & Rönnback, J. 1998,, 335, 85 Pritchet, C.J., & van den S. 1984,, 96, 804 Secker, J. 1992,, 104, Schlegel, D.J., Finkbeiner, D.P., Davis, M. 1998,, 500, 525 & \pm 0.04$ mag foreground $D$ & $4.1 0.3$ & Mpc & Distance\ ${\cal M}_g$ & $7.4 \times 10^8$ & & ISM mass\ ${\cal M}_\star$ $3.2 \times & Mass stars\ M}_T$ $2.1 \times 10^{10}$ Total dynamical mass\ $M_V$ & –16.42 Absolute mag $V$ band\ $L_B$ & $3.4 \times & $L_{B,\odot}$ $B$ band luminosity\ & 62 & & Mass to light ratio\ [l

Hodge, P.W. 1961,, 66, 83 Ibata, R., Gilmore, G., & IrwiN, M. 1994,, 370, 194 Kodama, T. & BOwer, R.g. 2001,, 321, 18 KaRacHeNtseV, [[*et aL.*]{}]{} 2003,, 398, 479 McLaughlin, D.E. 1999,, 117, 2398 MEUrer, g.R., Mackie, G., & Carignan, C. 1994,, 107, 2021 (MMC94) MEurer, g.R., cArigNAn, c., BeauLieu, S., & FrEEmAN, k.C. 1996,, 111, 1551 (McBf96) MEylAn, g., saRajedEni, a., JablonKa, P., DjorgovSki, s.G., bridges, T., & Rich, r.m. 2001,, 122, 830 ÖStlin, G., BergValL, N., & Rönnback, J. 1998,, 335, 85 PRitChet, C.J., & VaN deN bergh, s. 1984,, 96, 804 SeCker, J. 1992,, 104, 1472 schlegEL, D.J., FinKbeiner, D.P., & daVIs, M. 1998,, 500, 525 [lccL]{}   & $0.28 \Pm 0.04$ & mag & foREGrOund Extinction\ $D$ & $4.1 \pm 0.3$ & Mpc & DIStANce\ ${\cal M}_g$ & $7.4 \times 10^8$ & & IsM mass\ ${\CaL m}_\sTAR$ & $3.2 \tiMes 10^8$ & & mass in starS\ ${\cAl M}_T$ & $2.1 \tIMes 10^{10}$ & & TotaL DyNAMIcaL Mass\ $M_V$ & –16.42 & mag & AbsoLute mag $V$ banD\ $l_B$ & $3.4 \tImes 10^8$ & $L_{B,\OdOt}$ & $B$ BAnd lumInosiTy\   & 62 & SOlaR & Mass to lighT ratIo\ [l

Hodge, P.W. 1961,, 66, 8 3 Ibata,R., G ilm ore ,G.,& Ir win, M. 1994,, 370, 194 Kodama, T. & Bow er, R .G . 200 1 ,, 321, 18 Ka r ac h e nts ev ,[[* et al .*]{} ]{}  2003,, 398, 479 Mc La ughlin, D.E. 19 99,, 117,239 8 Meurer, G .R. , Mack ie , G . , & C ari gnan, C. 19 9 4,, 10 7, 2021 ( MM C 94) M e urer, G . R ., Car ignan, C., Beauli e u, S., & Freeman, K.C.19 9 6, , 111 , 1 551 (MCBF9 6) Mey l an, G., Sa r a j ede n i, A., Jablon ka, P., Djo r gov ski, S .G .,B ridges , T., & Ric h, R.M. 200 1,,122, 830 Östli n , G., B e rgvall, N., &  Rö nnb ack, J. 1 998 ,, 335 , 8 5 P rit chet, C. J. ,& van den B e r gh,S.1984 ,, 96 , 804 Secker , J . 19 9 2,, 104, 1472 Sc hl egel, D.J., Fink be iner, D.P., & D avis , M. 1998 ,,50 0,52 5 [l c cl]{}  & $0 .28 \pm 0.04$& ma g& f or eground extinction \$ D $& $4.1 \ pm 0.3 $ & M p c & Dist an ce\ ${\ c a l M}_ g$ & $7 .4 \time s 10^8 $ & & ISM ma ss \ ${\c al M} _\s tar$& $3. 2 \tim es 10^8$ & &M ass in stars\$ {\cal M}_T$ & $2 . 1 \ t imes 10 ^{10}$ & &Tota l dyn amic a lmas s \ $M_ V$ &–1 6 .4 2 & mag & Absolute m ag $V$ b and\$L_B$ & $3.4\times 10^ 8 $ & $L_{B, \odo t }$ & $B$ band lum inosi ty\   & 62 & solar& Mas s to lig ht ratio\ [l

Hodge, P.W._1961,, 66,_83 Ibata, R., Gilmore, G.,_& Irwin, M._1994,,_370, 194 Kodama,_T._& Bower, R.G. 2001,,_321, 18 Karachentsev, [[*et al.*]{}]{} 2003,,_398, 479 McLaughlin, D.E. 1999,,_117, 2398 Meurer, G.R.,_Mackie,_G., & Carignan, C. 1994,, 107, 2021 (MMC94) Meurer, G.R., Carignan, C., Beaulieu, S., & Freeman, K.C._1996,,_111, 1551_(MCBF96) Meylan,_G.,_Sarajedeni, A., Jablonka, P., Djorgovski,_S.G., Bridges, T., & Rich, R.M._2001,, 122,_830 Östlin, G., Bergvall, N., & Rönnback, J. 1998,, 335,_85 Pritchet,_C.J., & van den_Bergh, S. 1984,, 96, 804 Secker, J. 1992,, 104, 1472 Schlegel,_D.J., Finkbeiner, D.P., & Davis, M. 1998,,_500, 525 [lccl]{}  _&_$0.28_\pm 0.04$ & mag_& foreground extinction\ $D$ & $4.1 \pm_0.3$ & Mpc & Distance\ ${\cal M}_g$_& $7.4 \times 10^8$ & & ISM_mass\ ${\cal M}_\star$ & $3.2 \times 10^8$_& & Mass in stars\ ${\cal_M}_T$ &_$2.1 \times 10^{10}$ & &_Total dynamical mass\ $M_V$_& –16.42_& mag &_Absolute mag $V$ band\ $L_B$ & $3.4_\times 10^8$ &_$L_{B,\odot}$ & $B$ band luminosity\   &_62_& solar &_Mass_to_light ratio\ [l

gamma=-\kappa.$$ Making use of the continuity method, one can easily prove that the solvability of this equation is equivalent to the one of $$\int_{X}\langle\kappa,\vartheta\rangle_{H}\frac{\omega^{n}}{n!}=0,$$ for any $\vartheta\in\Gamma(X,E)$ satisfying $D^{''}_{E}\vartheta=D_{H}^{'}\vartheta=0$. By the assumption $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for any Dolbeault class $[\eta]\in H^{0,1}(X)$, we know $$\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta,\vartheta\rangle_{H}\frac{\omega^{n}}{n!} =\int_{X}\sqrt{-1}\partial\langle\beta^{0,1},\vartheta\rangle_{H}\wedge\frac{\omega^{n-1}}{(n-1)!} =0.$$ Suppose $\gamma\in\Gamma(X,E)$ is a solution of $$\sqrt{-1}\Lambda_{\omega}D^{'}_{H}D^{''}_{E}\gamma=-\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta.$$ Let $\tilde{\beta}=\beta+D^{''}_{E}\gamma$, then $\sqrt{-1}\Lambda_{\omega}D_{H}^{'}\tilde{\beta}=0$. According to (\[z:1\]), one can easily check that $$\sqrt{-1}[\Lambda_{\omega},D_{E}^{''}]=(D_{H}^{'})^{*}+\tau^{*},\ \ -\sqrt{-1}[\Lambda_{\omega},D_{H}^{'}]=(D_{E}^{''})^{*}+\bar{\tau}^{*}.$$ A simple computation gives $$\label{eq:61} \begin{split} 0&=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},F_{H,\theta}]\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{''}_{E}D^{'}_{H}\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},D^{

gamma=-\kappa.$$ Making use of the continuity method, one can easily rise that the solvability of this equality is equivalent to the one of $ $ \int_{X}\langle\kappa,\vartheta\rangle_{H}\frac{\omega^{n}}{n!}=0,$$ for any $ \vartheta\in\Gamma(X, E)$ satisfying $ D^{''}_{E}\vartheta = D_{H}^{'}\vartheta=0$. By the assumption $ \int_{X}\partial [ \eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0 $ for any Dolbeault course $ [ \eta]\in H^{0,1}(X)$, we know $ $ \int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta,\vartheta\rangle_{H}\frac{\omega^{n}}{n! } = \int_{X}\sqrt{-1}\partial\langle\beta^{0,1},\vartheta\rangle_{H}\wedge\frac{\omega^{n-1}}{(n-1)! } = 0.$$ presuppose $ \gamma\in\Gamma(X, E)$ is a solution of $ $ \sqrt{-1}\Lambda_{\omega}D^{'}_{H}D^{''}_{E}\gamma=-\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta.$$ Let $ \tilde{\beta}=\beta+D^{''}_{E}\gamma$, then $ \sqrt{-1}\Lambda_{\omega}D_{H}^{'}\tilde{\beta}=0$. According to (\[z:1\ ]), one can well discipline that $ $ \sqrt{-1}[\Lambda_{\omega},D_{E}^{''}]=(D_{H}^{'})^{*}+\tau^{*},\ \ -\sqrt{-1}[\Lambda_{\omega},D_{H}^{'}]=(D_{E}^{''})^{*}+\bar{\tau}^{*}.$$ A simple calculation gives $ $ \label{eq:61 } \begin{split } 0&=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},F_{H,\theta}]\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ & = \int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{''}_{E}D^{'}_{H}\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ & = \int_{X}\langle\sqrt{-1}[\Lambda_{\omega},D^ {

gamla=-\kappa.$$ Making use of tht continuity method, one ran easjly provd that the solvability of thms ewuatiin is equivalent to thd one of $$\pnt_{X}\langlw\kapka,\vartheta\rangle_{H}\hdac{\omega^{n}}{n!}=0,$$ fod any $\tartheta\in\Gamma(W,E)$ satisfyitg $D^{''}_{E}\vartheta=D_{V}^{'}\vxrcheta=0$. By the assumption $\int_{X}\partial [\qta]\wedgr\fgac{\omega^{n-1}}{(n-1)!}=0$ for any Qolbsault class $[\eta]\in H^{0,1}(X)$, we know $$\int_{X}\lzngle\sqgt{-1}\Lambda_{\omega}D^{'}_{H}\beya,\vartheta\rangle_{H}\frac{\omega^{j}}{n!} =\inh_{X}\sqrt{-1}\partial\langlf\beta^{0,1},\varthejz\ragtle_{H}\wedge\fraz{\omega^{n-1}}{(n-1)!} =0.$$ Sl'pose $\gamma\jn\Gamma(X,E)$ is a solution of $$\sqrt{-1}\Uambdc_{\omega}D^{'}_{H}D^{''}_{E}\gqmna=-\sett{-1}\Lambda_{\omege}D^{'}_{H}\betw.$$ Let $\tilde{\bcna}=\beta+D^{''}_{A}\gamma$, yhen $\sqrt{-1}\Lambds_{\omxga}D_{Y}^{'}\tilde{\beta}=0$. According vo (\[z:1\]), one can easily sheck thad $$\aqrt{-1}[\Lambda_{\omega},D_{E}^{''}]=(E_{H}^{'})^{*}+\rau^{*},\ \ -\vqrt{-1}[\Nambaq_{\omdga},S_{H}^{'}]=(V_{E}^{''})^{*}+\bzr{\tau}^{*}.$$ W smmple compufation givew $$\label{eq:61} \begin{split} 0&=\one_{Q}\kangle\sqrt{-1}[\Lajbda_{\omqgw},F_{H,\theta}]\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\fgac{\ojega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}\Lamvda_{\omega}D^{''}_{E}D^{'}_{H}\tilde{\beta},\jilde{\beta}\rwngle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},D^{

gamma=-\kappa.$$ Making use of the continuity method, easily that the of this equation of for any $\vartheta\in\Gamma(X,E)$ $D^{''}_{E}\vartheta=D_{H}^{'}\vartheta=0$. By the $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for any Dolbeault class H^{0,1}(X)$, we know $$\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta,\vartheta\rangle_{H}\frac{\omega^{n}}{n!} =\int_{X}\sqrt{-1}\partial\langle\beta^{0,1},\vartheta\rangle_{H}\wedge\frac{\omega^{n-1}}{(n-1)!} =0.$$ Suppose $\gamma\in\Gamma(X,E)$ is a solution of $$\sqrt{-1}\Lambda_{\omega}D^{'}_{H}D^{''}_{E}\gamma=-\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta.$$ $\tilde{\beta}=\beta+D^{''}_{E}\gamma$, then $\sqrt{-1}\Lambda_{\omega}D_{H}^{'}\tilde{\beta}=0$. According to (\[z:1\]), one can easily check that $$\sqrt{-1}[\Lambda_{\omega},D_{E}^{''}]=(D_{H}^{'})^{*}+\tau^{*},\ \ A computation $$\label{eq:61} 0&=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},F_{H,\theta}]\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{''}_{E}D^{'}_{H}\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},D^{

gamma=-\kappa.$$ Making use of the cOntinuity mEthod, One Can EaSily ProvE that the solvabILity Of this equation is equivaLent tO tHE one OF $$\iNt_{X}\laNgle\kapPA,\vARTheTa\RaNglE_{H}\FRaC{\omegA^{n}}{n!}=0,$$ For any $\vArtheta\in\GAmmA(X,e)$ satisfying $D^{''}_{e}\VaRtheta=D_{H}^{'}\vaRthEta=0$. By the assuMptIon $\int_{x}\pArtIAl [\eta]\WedGe\fraC{\omega^{N-1}}{(N-1)!}=0$ for anY DolbeaulT cLAss $[\eta]\IN H^{0,1}(X)$, we knOW $$\InT_{X}\laNgle\sqrt{-1}\Lambda_{\omeGA}D^{'}_{h}\Beta,\vartheta\raNgle_{H}\fRaC{\OmEGA^{n}}{n!} =\Int_{x}\sqrt{-1}\partiAl\LanglE\Beta^{0,1},\varTHeTA\RAngLE_{H}\wedge\frac{\omEga^{n-1}}{(n-1)!} =0.$$ SupposE $\GamMa\in\GaMmA(X,E)$ IS a soluTion oF $$\sQRt{-1}\LAmbda_{\omega}D^{'}_{h}D^{''}_{E}\gAmma=-\sqrt{-1}\LAmbda_{\oMEga}D^{'}_{H}\beTA.$$ Let $\tilDe{\beta}=\BetA+D^{''}_{E}\GammA$, ThEn $\SqrT{-1}\LAMbdA_{\OmEga}d_{h}^{'}\tiLde{\beta}=0$. ACcOrDing tO (\[z:1\]), onE CAN EasiLy cHeck That $$\sQrt{-1}[\Lambda_{\omegA},D_{E}^{''}]=(d_{H}^{'})^{*}+\taU^{*},\ \ -\SqrT{-1}[\LambDa_{\omeGa},D_{H}^{'}]=(d_{E}^{''})^{*}+\Bar{\taU}^{*}.$$ A simpLe comPuTation gives $$\labeL{eq:61} \bEgin{split} 0&=\Int_{x}\lAngLe\Sqrt{-1}[\LAMbda_{\omEga},f_{H,\tHeta}]\tilDe{\beta},\tILde{\BeTA}\RAnGle_{H,\omega}\frac{\omega^{N}}{n!}\\ &=\INT_{X}\Langle\sqRt{-1}\LambDA_{\oMeGA}D^{''}_{E}D^{'}_{H}\tilDe{\BetA},\tilDE{\Beta}\rAnglE_{h,\oMega}\frac{\Omega^{n}}{N!}\\ &=\InT_{X}\Langle\sQrT{-1}[\LambdA_{\oMegA},D^{

gamma=-\kappa.$$ Making us e of the c ontin uit y m et hod, one can easily pr o ve t hat the solvability of this e q uati o nis eq uivalen t t o the o ne of $ $ \i nt_{X }\l angle\k appa,\vart het a\ rangle_{H}\f r ac {\omega^{n }}{ n!}=0,$$ for an y $\va rt het a \in\G amm a(X,E )$ sat i sfying $D^{''}_ {E } \varth e ta=D_{H } ^ {' }\va rtheta=0$. By the as s umption $\int_ {X}\pa rt i al [ \et a]\ wedge\frac {\ omega ^ {n-1}}{ ( n- 1 ) ! }=0 $ for any Dolb eault class $[\ eta]\i nH^{ 0 ,1}(X) $, we k n ow$$\int_{X}\ lang le\sqrt{- 1}\Lam b da_{\om e ga}D^{' }_{H}\ bet a,\ vart h et a\ ran gl e _{H } \f rac { \om ega^{n}} {n !} =\in t_{X } \ s q rt{- 1}\ part ial\l angle\beta^{0 ,1} ,\va r the ta\ra ngle_ {H}\ we dge\f rac{\o mega^ {n -1}}{(n-1)!} =0 .$$Suppose $ \ga mm a\i n\ Gamma ( X,E)$isa s olution of $$\ s qrt {- 1 } \ La mbda_{\omega}D^{'} _{ H } D^ {''}_{E} \gamma = -\ sq r t{-1}\La mb da_ {\om e g a}D^{ '}_{ H }\ beta.$$Let $\ t il de {\beta} =\ beta+D ^{ ''} _{E }\gam m a$,then $ \sqrt{-1 }\Lam b da_{\omega}D_{ H }^{'}\tilde{\ b et a } =0 $ . Ac cor ding to (\[ z:1\ ] ), o ne c a neas i ly ch eck t ha t $ $ \sqrt{-1}[\Lambda_{ \o mega}, D_{E} ^{''}]=(D_{H} ^{'})^{*}+ \ t a u^{*},\\ -\ s qr t {-1}[\Lambda_{ \omeg a},D_{H}^{ ' }]=(D_{E }^{'' })^{*}+\ bar{\tau} ^ { *}.$$ Asim ple co mpu t a ti on gives $$\l a b el{e q: 61} \be gin {split} 0& =\i nt_ {X} \l angle\sqr t{-1}[\L am bd a_ {\ ome ga},F _ {H,\thet a} ]\t il de{ \beta } ,\tild e{\be ta}\ ra ng l e_{ H,\omeg a }\ f r ac{\ om eg a^{n }}{ n! }\\ & =\in t _{X }\langl e\sqrt{-1 }\L a mbda _{ \o mega}D^ {''}_{E}D^{'} _{ H}\tilde{\ be ta} ,\tild e { \beta}\r angle_{H,\omega}\frac{\ o mega^{n }}{ n!}\\ &=\ int_{X}\l ang le\sqr t{- 1 }[\Lam bda_{\ omega }, D^{

gamma=-\kappa.$$ Making_use of_the continuity method, one_can easily_prove_that the_solvability_of this equation_is equivalent to_the one of $$\int_{X}\langle\kappa,\vartheta\rangle_{H}\frac{\omega^{n}}{n!}=0,$$_for any $\vartheta\in\Gamma(X,E)$_satisfying_$D^{''}_{E}\vartheta=D_{H}^{'}\vartheta=0$. By the assumption $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for any Dolbeault class $[\eta]\in H^{0,1}(X)$, we know_$$\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta,\vartheta\rangle_{H}\frac{\omega^{n}}{n!} =\int_{X}\sqrt{-1}\partial\langle\beta^{0,1},\vartheta\rangle_{H}\wedge\frac{\omega^{n-1}}{(n-1)!} =0.$$_Suppose $\gamma\in\Gamma(X,E)$_is_a_solution of $$\sqrt{-1}\Lambda_{\omega}D^{'}_{H}D^{''}_{E}\gamma=-\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta.$$ Let $\tilde{\beta}=\beta+D^{''}_{E}\gamma$,_then $\sqrt{-1}\Lambda_{\omega}D_{H}^{'}\tilde{\beta}=0$. According to (\[z:1\]),_one can_easily check that $$\sqrt{-1}[\Lambda_{\omega},D_{E}^{''}]=(D_{H}^{'})^{*}+\tau^{*},\ \ -\sqrt{-1}[\Lambda_{\omega},D_{H}^{'}]=(D_{E}^{''})^{*}+\bar{\tau}^{*}.$$ A simple_computation_gives $$\label{eq:61} \begin{split} 0&=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},F_{H,\theta}]\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{''}_{E}D^{'}_{H}\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},D^{

To get around these limitations, new, separate options must be defined, increasing the problem’s branching factor, and care must be taken to avoid loops (if so desired). An MMDP coarse action leaves the “direction” of the action undecided: the same fine policy may be executed starting in several bottleneck states, and may take the agent in one of several directions until arriving at one of multiple destinations from which different successor coarse actions may be taken. In the context of MMDPs, if one wanted to be able to transfer policies on the same cluster which guide an agent in different particular directions, separate local policies would need to be stored in the “database” of solved tasks. However, we would only need to transfer and plan with [*one*]{} of them. - A strength of the options framework is that multiple related queries, or tasks, may be solved essentially within the same SMDP. However, the tasks must be closely related in specific ways (e.g. tasks differing only in the goal state), and this strength comes at the expense of ignoring problem-specific information when one only wants to solve one problem. Our approach to the construction of MMDPs differs in that while we assume a particular problem when building a decomposition, we are able to consider a broader set of transfer possibilities. - Bottlenecks and partitioning do not explicitly enter the picture in options or SMDPs. Options may be defined on any subset of the statespace, and in applications may often take the form of a macro-action which directs the agent to an intermediate goal state starting from [*any*]{} state in a (possibly large) neighborhood. For example, an option may direct a robot to a hallway from any state in a room. We constrain our “initiation” and “termination” sets to be bottleneck states, however this means that learning policies at coarse scales is fast, and can be carried out completely independent of other scales. Coarse scale learning involves only the bottleneck states, giving a drastically reduced computational complexity. Provided the partitioning of a scale is well chosen, this construction allows one to capitalize on improved mixing times to accelerate convergence. - MMDPs are a representation for MDPs: we cannot solve problems that cannot be phrased as an MDP (i.e. problems whose solutions require non-Markov policies). A policy solving an MMDP, at any scale, is a Markov policy. SMDPs may in general have non-

To get around these limitations, new, separate choice must be define, increasing the problem ’s branching divisor, and care must be taken to debar loops (if so desire). An MMDP coarse action leaves the “ commission ” of the action undecided: the like fine policy may be executed starting in several bottleneck states, and may take the agentive role in one of several directions until arriving at one of multiple destination from which different successor coarse actions may be take. In the context of MMDPs, if one desire to be able to transfer policies on the same cluster which guide an agentive role in different particular directions, separate local policies would need to be store in the “ database ” of solved tasks. However, we would only need to transfer and plan with [ * one * ] { } of them. - A strength of the options framework is that multiple related queries, or undertaking, may be solved essentially within the like SMDP. However, the undertaking must be close related in specific ways (e.g. tasks differing only in the goal state), and this lastingness comes at the expense of ignoring problem - specific information when one only wants to clear one problem. Our approach to the construction of MMDPs differs in that while we wear a particular trouble when build a decomposition, we are able to consider a broader hardening of transfer possibilities. - Bottlenecks and partitioning do not explicitly record the picture in options or SMDPs. Options may be defined on any subset of the statespace, and in application may often consume the form of a macro - military action which directs the agentive role to an intermediate goal state starting from [ * any * ] { } department of state in a (possibly large) neighborhood. For exercise, an option may address a robot to a hallway from any state in a room. We constrain our “ initiation ” and “ termination ” sets to be bottleneck states, however this mean that learning policies at coarse scale is fast, and can be carried out completely independent of other scales. Coarse plate learning involves only the bottleneck states, giving a drastically reduced computational complexity. Provided the breakdown of a plate is well choose, this construction allows one to capitalize on improved mixing times to accelerate overlap. - MMDPs are a representation for MDPs: we cannot solve problems that cannot be phrased as an MDP (i.e. problems whose solutions require non - Markov policies). A policy solving an MMDP, at any scale, is a Markov policy. SMDPs may in cosmopolitan have non-

To get around these limitauions, new, separatg iptionv must be defived, increasing the problem’s uranxhing factor, and care must ce taken no avoid ooopw (if so desmded). An MMDP ckwrse ection leaves tme “directiot” of the actiot jnbecided: the same fine policy may be qxecutec dtarting in sederak botfleneck states, and may take the agsnt in mne of severak directions until arrivinh at one of multiple dfstinations froi which diffefent successor coarse zctions may be taken. In the congext pf MMDPs, id ine fanted to bx able to transfer policiev on thr same cluster whmch tuide an agent in difherent particular ditections, sapcrate local policies qoyld nged tm be wtofed ii tge “datwbaae” of solvsd tasks. Hoqever, we would only nqvc to transfed and [lwn with [*one*]{} of them. - A strength of the opfions framework is that multiple related queties, or tafks, may be solved essentially within the same SMD[. Howxvdr, uhc gqsns must be closely related in specific ways (e.d. tssls differing okly in the goal ststf), sgd this streneth comes at the expense of ignorigg priblem-specyfic information when one only qants to solne obe problem. Our appxoach to the conxtrucyion of MMDPs differs iu that while we adsume a pzfticular problem whvn bgilding a decomposition, we are able to eonsider a btoader fet of trajsfer possibilities. - Bohtlengcks atd partitilning do not explicitly enter tix picture in ppdiots or SMBPs. Opbions may be desined on any sobset of che stxtespace, ahd in a'plications iay often taka the form of a macro-wctiin wyich difdcts the agent to an innexmediate toal state startinn froo [*any*]{} state in a (powsibly large) neoghcorrold. Fjs example, an optkon kay dkrect a robig to a hallway from any vtats in a room. We conxtvain our “unitiatijn” and “terminstion” sets to be blttleieck svates, nowgver this means that learning pklicies ah cjarse scales is nast, and can bz carried out completely independent of ither scales. Coarsw scale learning inrokves only tie boteleneck sdates, giving a drastucally reduced cokputational complexitg. Provhded hhe partitioning of a scale is well chosen, this construction allows one ti capivajize on impdovec mixpng cimes tj acreuerate convergenbe. - MMDPs are a representation hor MDPs: wa eannot solve problems that camnut be phrased as an MDP (i.e. problems wgose solitions require non-Markov policirs). A policy solving an MKDP, at aby sczle, is a Merkov policy. SMDPs msy in yenerao have bon-

To get around these limitations, new, separate be increasing the branching factor, and avoid (if so desired). MMDP coarse action the “direction” of the action undecided: same fine policy may be executed starting in several bottleneck states, and may the agent in one of several directions until arriving at one of multiple from different coarse may be taken. In the context of MMDPs, if one wanted to be able to transfer on the same cluster which guide an agent different particular directions, separate policies would need to be in “database” of tasks. we only need to and plan with [*one*]{} of them. - A strength of the options framework is that multiple related or tasks, solved essentially the SMDP. the tasks must related in specific ways (e.g. tasks the goal state), and this strength comes at expense of problem-specific information when one only wants solve one problem. Our approach to the construction MMDPs differs in that while we assume a particular problem when building a decomposition, we to consider a broader of transfer possibilities. Bottlenecks partitioning not enter the in options or SMDPs. Options may be defined on any subset the statespace, and in applications may often take the form macro-action directs the agent an intermediate goal state from state in a (possibly For an a to hallway from any state a room. We constrain our and “termination” sets to means that learning policies at coarse scales is and can be carried out completely independent other scales. Coarse scale learning involves only the bottleneck states, giving a reduced computational the partitioning of a scale is well chosen, construction allows one to on improved mixing times to accelerate convergence. - MMDPs a for MDPs: cannot solve problems cannot be phrased an MDP (i.e. solutions require policies). policy at any scale, is a Markov SMDPs in general have non-

To get around these limitatioNs, new, separAte opTioNs mUsT be dEfinEd, increasing thE ProbLem’s branching factor, and Care mUsT Be taKEn To avoId loops (IF sO DEsiReD). AN MMdP COaRse acTioN leaves The “directiOn” oF tHe action undeCIdEd: the same fIne Policy may be eXecUted stArTinG In sevEraL bottLeneck STates, aNd may take ThE Agent iN One of seVERaL dirEctions until arrivINg AT one of multiple DestinAtIOnS FRom WhiCh differenT sUccesSOr coarsE AcTIONs mAY be taken. In the Context of MMdps, iF one waNtEd tO Be able To traNsFEr pOlicies on thE samE cluster wHich guIDe an ageNT in diffErent pArtIcuLar dIReCtIonS, sEParATe LocAL poLicies woUlD nEed to Be stORED In thE “daTabaSe” of sOlved tasks. HowEveR, we wOUld Only nEed to TranSfEr and Plan wiTh [*one*]{} Of Them. - A strength of The oPtions fraMewOrK is ThAt mulTIple reLatEd qUeries, oR tasks, mAY be SoLVED eSsentially within thE sAME SmDP. HowevEr, the tASkS mUSt be closElY reLateD IN specIfic WAyS (e.g. tasks DifferINg OnLy in the GoAl statE), aNd tHis StrenGTh coMes at tHe expensE of igNOring problem-spECific informatIOn WHEn ONe onLy wAnts to solve One pRObleM. Our APpRoaCH to thE consTrUCtIOn of MMDPs differs in tHaT while We assUme a particulaR problem whEN BUilding a DecoMPoSItion, we are able To conSider a broaDEr set of tRansfEr possibIlities. - BoTTLenecks aNd pArtItiOniNG Do Not explicitly ENTer tHe Picture In oPtions oR SMdPs. optIonS mAy be definEd on any sUbSeT oF tHe sTatesPAce, and in ApPliCaTioNs may OFten taKe the Form Of A mACro-Action wHIcH DIrecTs ThE ageNt tO aN inteRmedIAte Goal staTe startinG frOM [*any*]{} StAtE in a (posSibly large) neiGhBorhood. For ExAmpLe, an opTIOn may dirEct a robot to a hallway from ANy state In a Room. WE conStrain our “IniTiatioN” anD “TerminAtion” sEts to Be BotTLEneck STAtEs, hOwEver this meANS thAt leaRnIng pOlicies At coarse scales is faST, anD can be carried Out CompLETeLy iNDePEndEnT Of oTHEr scales. Coarse sCale learniNg INvOlves only tHE boTtLeneck sTates, giVing a DRasticaLly reduceD computatIoNal cOMPleXity. ProvidEd the parTitioning OF a scaLE iS well ChoSen, thiS cOnsTructIon allOWs oNe to cApitalIzE on impRoved MiXing timeS to accelerate convergencE. - MMDPs Are a rEprEsentatioN foR mDPS: we cannot SolvE problems tHat CanNot be PhrASed as An MDp (I.e. ProBLems wHose SOlutions rEQuIre NON-MArkov policiES). a PolIcy soLviNG an MMDp, at aNy scale, is a Markov pOLicy. SMDPs may in GeneRAL haVe nON-

To get around these limit ations, ne w, se par ate o ptio ns m ust be defined , inc reasing the problem’sbranc hi n g fa c to r, an d carem us t beta ke n t oa vo id lo ops (if so desired). An M MDP coarse a c ti on leavesthe “direction” of the a ct ion undec ide d: th e same fine p olicy may b e execu t ed star t i ng inseveral bottlenec k s t ates, and maytake t he ag e n t i n o ne of seve ra l dir e ctionsu nt i l arr i ving at one o f multipled est inatio ns fr o m whic h dif fe r ent successorcoar se action s mayb e taken . In the conte xtofMMDP s ,if on ew ant e dtob e a ble to t ra ns fer p olic i e s on t hesame clus ter which gui dean a g ent in d iffer entpa rticu lar di recti on s, separate loc al p olicies w oul dnee dto be stored in th e “data base” o f so lv e d ta sks. However, we w ou l d o nly need to tr a ns fe r and pla nwit h [* o n e*]{} oft he m. - A stre n gt hof theop tionsfr ame wor k ist hatmultip le relat ed qu e ries, or tasks , may be solve d e s s en t iall y w ithin the s ameS MDP. How e ve r,t he ta sks m us t b e closely related in s pecifi c way s (e.g. tasks differing o n ly in th e go a ls tate), and thi s str ength come s at theexpen se of ig noring pr o b lem-spec ifi c i nfo rma t i on when one onl y want sto solv e o ne prob lem . O urapp ro ach to th e constr uc ti on o f M MDPsd iffers i ntha twhi le we assume a pa rtic ul ar pro blem wh e nb u ildi ng a dec omp os ition , we are able t o conside r a broa de rset oftransfer poss ib ilities. - B ottlen e c ks and p artitioning do not expl i citly e nte r the pic ture in o pti ons or SM D Ps. Op tionsmay b edef i n ed on a ny su bs et of thes t ate space ,andin appl ications may often tak e the form of amacr o - ac tio n w h ich d i rec t s the agent to a n intermed ia t egoal state sta rt ing fro m [*any *]{}s tate in a (possi bly large )neig h b orh ood. For e xample,an option may d i re ct arob ot toahal lwayfrom a n y s tatein a r oo m. Weconst ra in our “ initiation” and “termin ation” sets to be bottl ene c k s tates, ho weve r this mea nstha t lea rni n g pol icie s a t c o arsescal e s is fast , a ndc a nbe carriedo u t co mplet ely indepe nden t of other scales . Coarse scalelear n i nginv o lves o nly the bottle nec ks t ates, gi vi ng a drasti cally re du c ed co mputat ionalcomplex i t y. Provid ed t hepartition ing o f a scal eis well c hose n, thisconstr u ctio n allows one to ca pital i z e oni mpr ovedmi xing ti m es t o accelera te converge nce. - MMDPs are are presen tat io n for MDPs : we canno t sol ve prob le ms t hat canno t be p hrase d as a n M DP (i.e.p r ob l em sw hos e so lutio ns req uire non- M arkov po lic i es). Apo lic y solvin g a n MMDP, at a nyscale , is a Marko v pol i cy . SMDP s mayin gen eral ha v e n on -

To_get around_these limitations, new, separate_options must_be_defined, increasing_the_problem’s branching factor,_and care must_be taken to avoid_loops (if so_desired)._An MMDP coarse action leaves the “direction” of the action undecided: the same fine_policy_may be_executed_starting_in several bottleneck states, and_may take the agent in_one of_several directions until arriving at one of multiple_destinations_from which different_successor coarse actions may be taken. In the context_of MMDPs, if one wanted to_be able to_transfer_policies_on the same cluster_which guide an agent in different_particular directions, separate local policies would_need to be stored in the “database”_of solved tasks. However, we would_only need to transfer and_plan with_[*one*]{} of them. - _A strength of_the options_framework is that_multiple related queries, or tasks, may_be solved essentially_within the same SMDP. However, the_tasks_must be closely_related_in_specific ways_(e.g. tasks differing_only_in the_goal_state), and this strength comes at_the_expense of ignoring problem-specific information when one_only wants to solve_one_problem. Our approach to_the construction of MMDPs differs_in that while we assume a_particular problem_when building_a decomposition, we are able to consider a broader set of_transfer possibilities. - Bottlenecks and_partitioning do not explicitly_enter the_picture_in options or_SMDPs._Options may_be defined on any subset of the_statespace, and_in applications may often take the_form of a macro-action_which_directs the agent to an intermediate_goal state starting from [*any*]{} state_in a (possibly large) neighborhood._For_example,_an option may direct a_robot to a hallway from any_state in a_room. We constrain our “initiation” and “termination”_sets_to be bottleneck states, however this_means_that learning policies at coarse scales_is_fast,_and can be carried out_completely independent of other scales. Coarse_scale learning involves only the bottleneck states, giving a_drastically reduced computational_complexity. Provided the partitioning of_a_scale_is well chosen, this construction allows one to capitalize on_improved mixing_times to accelerate_convergence. - MMDPs are a representation for MDPs: we_cannot solve problems that cannot be phrased_as an MDP (i.e. problems whose solutions require non-Markov policies). A_policy solving an MMDP, at any scale, is_a Markov policy. SMDPs may in general_have non-

$\Lambda^{2k-1}_n$ that is both cs and cs-$k$-neighborly. We then delete the cs-$(k-1)$-neighborly and $(k-1)$-stacked balls $\operatorname{\mathrm{lk}}\big(\{1,2\}, \pm B^{2k+1, k}_{n+2}\big)$ that are antipodal and share no common facets, and insert the cones over the boundary of these two balls. Thus, the resulting complex is also cs; furthermore, by Lemma \[lm: induction method\], it is cs-$k$-neighborly. In the case of $d=2k$, note that by Proposition \[prop: odd even sphere relation\], $\Delta^{2k+1}_{n+2}\subseteq \Delta^{2k+2}_{n+2}$. Hence $\Lambda^{2k}_n\supseteq \Lambda^{2k-1}_n$, and so $\Lambda^{2k}_n$ is also cs-$k$-neighborly. The proof that $\Lambda^{2k}_n$ is cs is identical to the proof in the odd-dimensional cases. Finally, to complete the proof of the first part for the case of $d=2k-1$ and $i=k$, note that, $$\operatorname{\mathrm{lk}}\big(\{1,2\}, B^{2k-1, k}_n\big)=\operatorname{\mathrm{lk}}\big(\{1,2\}, \Delta^{2k-1}_n\big)\backslash \operatorname{\mathrm{lk}}\big(\{1,2\}, B^{2k-1, k-1}_n\big).$$ We then conclude from the case of $d=2k-1, i=k-1$ and Lemma \[lm: complement\] that $\operatorname{\mathrm{lk}}\big(\{1,2\}, \pm B^{2k-1, k}_n\big)$ is indeed cs-$(k-1)$-neighborly and $(k-1)$-stacked. \[lm: neighborly stacked edge links\] Let $k\geq 3$ and let $n$ be sufficiently large. The only edges $e\subseteq V_{n}$ such that both $\operatorname{\mathrm{lk}}(e, B^{2k-1, k-1}_n)$ and $\operatorname{\mathrm{lk}}(e, -B^{2

$ \Lambda^{2k-1}_n$ that is both cs and cs-$k$-neighborly. We then delete the cs-$(k-1)$-neighborly and $ (k-1)$-stacked balls $ \operatorname{\mathrm{lk}}\big(\{1,2\ }, \pm B^{2k+1, k}_{n+2}\big)$ that are antipodal and share no common aspect, and tuck the cones over the boundary of these two balls. therefore, the resulting complex is also cs; furthermore, by Lemma \[lm: initiation method\ ], it is cs-$k$-neighborly. In the case of $ d=2k$, note that by Proposition \[prop: curious even sphere relation\ ], $ \Delta^{2k+1}_{n+2}\subseteq \Delta^{2k+2}_{n+2}$. Hence $ \Lambda^{2k}_n\supseteq \Lambda^{2k-1}_n$, and so $ \Lambda^{2k}_n$ is besides cs-$k$-neighborly. The proof that $ \Lambda^{2k}_n$ is cs is identical to the proof in the odd - dimensional cases. Finally, to dispatch the proof of the first character for the case of $ d=2k-1 $ and $ i = k$, note that, $ $ \operatorname{\mathrm{lk}}\big(\{1,2\ }, B^{2k-1, k}_n\big)=\operatorname{\mathrm{lk}}\big(\{1,2\ }, \Delta^{2k-1}_n\big)\backslash \operatorname{\mathrm{lk}}\big(\{1,2\ }, B^{2k-1, k-1}_n\big).$$ We then reason from the case of $ d=2k-1, i = k-1 $ and Lemma \[lm: complement\ ] that $ \operatorname{\mathrm{lk}}\big(\{1,2\ }, \pm B^{2k-1, k}_n\big)$ is indeed cs-$(k-1)$-neighborly and $ (k-1)$-stacked. \[lm: neighborly stacked edge links\ ] get $ k\geq 3 $ and let $ n$ be sufficiently large. The lone edges $ e\subseteq V_{n}$ such that both $ \operatorname{\mathrm{lk}}(e, B^{2k-1, k-1}_n)$ and $ \operatorname{\mathrm{lk}}(e, -B^{2

$\Lalbda^{2k-1}_n$ that is both cs akd cs-$k$-neighborly. We thei delets the cs-$(y-1)$-neighborly and $(k-1)$-stacked balps $\operqtorname{\mathrm{lk}}\big(\{1,2\}, \pm B^{2k+1, k}_{n+2}\big)$ that arw anuipodal and share no common facsbs, anb mnsert the conex over the boundary of tvere two balls. Thus, the resulting complqx is aksl cs; furthermote, by Jemmz \[lm: induction method\], it is cs-$k$-neifhborly. In the case pf $d=2k$, note that by Proposihion \[prop: odd even sphfre relatiob\], $\Dejra^{2k+1}_{n+2}\subseteq \Delta^{2k+2}_{n+2}$. Htnee $\Lambda^{2k}_n\aupseteq \Lambda^{2k-1}_n$, and so $\Lambda^{2y}_n$ is also cs-$k$-ngnthbltly. The prooh that $\Lambda^{2k}_n$ is cs is igenticak to the proof in thw odd-dimensional casev. Finally, to compleje the promf of the first parr dor tve cdse ud $d=2y-1$ ahd $i=i$, note thet, $$\operatorhame{\mathrm{lj}}\big(\{1,2\}, B^{2k-1, k}_n\big)=\operatprgqme{\mathrm{lk}}\bif(\{1,2\}, \Deltw^{2k-1}_g\big)\backslash \operatorname{\mathrm{lk}}\big(\{1,2\}, B^{2n-1, k-1}_h\big).$$ We then conclude feom the case of $d=2k-1, i=k-1$ and Lemmw \[lm: complement\] that $\operatorname{\mathrm{lk}}\big(\{1,2\}, \pm B^{2n-1, k}_n\bmg)$ is ikdeea cd-$(k-1)$-neighborly and $(k-1)$-stacked. \[lm: neighborly stackeq ecgv links\] Let $k\geq 3$ and let $n$ ne sisficiently latge. The onmy edges $e\subseteq V_{n}$ sucr thar both $\optratotname{\mathrm{lk}}(e, B^{2k-1, k-1}_n)$ and $\opwratorname{\manhrm{ok}}(e, -B^{2

$\Lambda^{2k-1}_n$ that is both cs and cs-$k$-neighborly. delete cs-$(k-1)$-neighborly and balls $\operatorname{\mathrm{lk}}\big(\{1,2\}, \pm and no common facets, insert the cones the boundary of these two balls. the resulting complex is also cs; furthermore, by Lemma \[lm: induction method\], it cs-$k$-neighborly. In the case of $d=2k$, note that by Proposition \[prop: odd even relation\], \Delta^{2k+2}_{n+2}$. $\Lambda^{2k}_n\supseteq and so $\Lambda^{2k}_n$ is also cs-$k$-neighborly. The proof that $\Lambda^{2k}_n$ is cs is identical to the in the odd-dimensional cases. Finally, to complete the of the first part the case of $d=2k-1$ and note $$\operatorname{\mathrm{lk}}\big(\{1,2\}, B^{2k-1, \Delta^{2k-1}_n\big)\backslash B^{2k-1, We then conclude the case of $d=2k-1, i=k-1$ and Lemma \[lm: complement\] that $\operatorname{\mathrm{lk}}\big(\{1,2\}, \pm B^{2k-1, k}_n\big)$ is indeed cs-$(k-1)$-neighborly $(k-1)$-stacked. \[lm: edge links\] $k\geq and $n$ be sufficiently only edges $e\subseteq V_{n}$ such that k-1}_n)$ and $\operatorname{\mathrm{lk}}(e, -B^{2

$\Lambda^{2k-1}_n$ that is both cs and cs-$K$-neighborlY. We thEn dEleTe The cS-$(k-1)$-neIghborly and $(k-1)$-stACked Balls $\operatorname{\mathrM{lk}}\biG(\{1,2\}, \pM b^{2k+1, k}_{n+2}\BIg)$ That aRe antipODaL ANd sHaRe No cOmMOn FacetS, anD insert The cones ovEr tHe Boundary of thESe Two balls. ThUs, tHe resulting cOmpLex is aLsO cs; FUrtheRmoRe, by LEmma \[lm: INductiOn method\], iT iS Cs-$k$-neiGHborly. IN THe Case Of $d=2k$, note that by ProPOsITion \[prop: odd eveN spherE rELaTIOn\], $\DEltA^{2k+1}_{n+2}\subseteQ \DElta^{2k+2}_{N+2}$. hence $\LaMBdA^{2K}_N\SupSEteq \Lambda^{2k-1}_n$, aNd so $\Lambda^{2k}_N$ Is aLso cs-$k$-NeIghBOrly. ThE prooF tHAt $\LAmbda^{2k}_n$ is cs Is idEntical to The proOF in the oDD-dimensIonal cAseS. FiNallY, To CoMplEtE The PRoOf oF The First parT fOr The caSe of $D=2K-1$ AND $i=k$, nOte That, $$\OperaTorname{\mathrm{Lk}}\bIg(\{1,2\}, B^{2k-1, K}_N\biG)=\operAtornAme{\mAtHrm{lk}}\Big(\{1,2\}, \DelTa^{2k-1}_n\bIg)\Backslash \operatOrnaMe{\mathrm{lK}}\biG(\{1,2\}, B^{2K-1, k-1}_n\BiG).$$ We thEN conclUde FroM the casE of $d=2k-1, i=k-1$ ANd LEmMA \[LM: cOmplement\] that $\operaToRNAmE{\mathrm{lK}}\big(\{1,2\}, \pm b^{2K-1, k}_N\bIG)$ is indeeD cS-$(k-1)$-nEighBORly anD $(k-1)$-stACkEd. \[lm: neigHborly STaCkEd edge lInKs\] Let $k\GeQ 3$ anD leT $n$ be sUFficIently Large. The Only eDGes $e\subseteq V_{n}$ SUch that both $\opERaTORnAMe{\maThrM{lk}}(e, B^{2k-1, k-1}_n)$ and $\OperATornAme{\mAThRm{lK}}(E, -B^{2

$\Lambda^{2k-1}_n$ that i s both csand c s-$ k$- ne ighb orly . We then dele t e th e cs-$(k-1)$-neighborl y and $ ( k-1) $ -s tacke d balls $\ o p era to rn ame {\ m at hrm{l k}} \big(\{ 1,2\}, \pm B^ {2 k+1, k}_{n+2 } \b ig)$ thatare antipodal a ndshareno co m mon f ace ts, a nd ins e rt the cones ov er the bo u ndary o f th esetwo balls. Thus,t he resulting comp lex is a l so c s;fur thermore,by Lemm a \[lm:i nd u c t ion method\], itis cs-$k$-n e igh borly. I n t h e case of $ d= 2 k$, note thatby P ropositio n \[pr o p: odde ven sph ere re lat ion \],$ \D el ta^ {2 k +1} _ {n +2} \ sub seteq \D el ta ^{2k+ 2}_{ n + 2 } $. H enc e $\ Lambd a^{2k}_n\sups ete q \L a mbd a^{2k -1}_n $, a nd so $ \Lambd a^{2k }_ n$ is also cs-$ k$-n eighborly . T he pr oo f tha t $\Lam bda ^{2 k}_n$ i s cs is ide nt i c a lto the proof in th eo d d- dimensio nal ca s es .Finally, t o c ompl e t e the pro o fof the f irst p a rt f or theca se of$d =2k -1$ and$ i=k$ , note that, $ $\ope r atorname{\math r m{lk}}\big(\{ 1 ,2 \ } ,B ^{2k -1, k}_n\big)= \ope r ator name { \m ath r m{lk} }\big (\ { 1, 2 \}, \Delta^{2k-1}_n \b ig)\ba cksla sh \operatorn ame{\mathr m { l k}}\big( \{1, 2 \} , B^{2k-1, k-1} _n\bi g).$$ We t h en concl ude f rom thecase of $ d = 2k-1, i= k-1 $ a ndLem m a \ [lm: compleme n t \] t ha t $\ope rat orname{ \ma thr m{l k}} \b ig(\{1,2\ }, \pm B ^{ 2k -1 ,k}_ n\big ) $ is ind ee d c s- $(k -1)$- n eighbo rly a nd $ (k -1 ) $-s tacked. \ [ l m: n ei gh borl y s ta ckededge lin ks\] Le t $k\geq3$a nd l et $ n$ be s ufficiently l ar ge. The on ly ed ges $e \ s ubseteqV_{n}$ such that both $ \ operato rna me{\m athr m{lk}}(e, B^ {2k-1, k- 1 }_n)$and $\ opera to rna m e {\mat h r m{ lk} }( e, -B^{2

$\Lambda^{2k-1}_n$_that is_both cs and cs-$k$-neighborly._We then_delete_the cs-$(k-1)$-neighborly_and_$(k-1)$-stacked balls $\operatorname{\mathrm{lk}}\big(\{1,2\},_\pm B^{2k+1, k}_{n+2}\big)$_that are antipodal and_share no common_facets,_and insert the cones over the boundary of these two balls. Thus, the resulting_complex_is also_cs;_furthermore,_by Lemma \[lm: induction method\],_it is cs-$k$-neighborly. In the_case of_$d=2k$, note that by Proposition \[prop: odd even_sphere_relation\], $\Delta^{2k+1}_{n+2}\subseteq \Delta^{2k+2}_{n+2}$._Hence $\Lambda^{2k}_n\supseteq \Lambda^{2k-1}_n$, and so $\Lambda^{2k}_n$ is also cs-$k$-neighborly._The proof that $\Lambda^{2k}_n$ is cs_is identical to_the_proof_in the odd-dimensional cases. Finally,_to complete the proof of the_first part for the case of_$d=2k-1$ and $i=k$, note that, $$\operatorname{\mathrm{lk}}\big(\{1,2\}, B^{2k-1,_k}_n\big)=\operatorname{\mathrm{lk}}\big(\{1,2\}, \Delta^{2k-1}_n\big)\backslash \operatorname{\mathrm{lk}}\big(\{1,2\}, B^{2k-1, k-1}_n\big).$$ We_then conclude from the case_of $d=2k-1,_i=k-1$ and Lemma \[lm: complement\]_that $\operatorname{\mathrm{lk}}\big(\{1,2\}, \pm_B^{2k-1, k}_n\big)$_is indeed cs-$(k-1)$-neighborly_and $(k-1)$-stacked. \[lm: neighborly stacked edge links\]_Let $k\geq 3$_and let $n$ be sufficiently large._The_only edges $e\subseteq_V_{n}$_such_that both_$\operatorname{\mathrm{lk}}(e, B^{2k-1, k-1}_n)$_and_$\operatorname{\mathrm{lk}}(e, -B^{2

1$, while modules are indicated by $\tau = 0$. Mixtures correspond to groups with $0 <\tau < 1$. For the rest of the paper, we refer to groups with $\tau\approx 1$ as community-like and groups with $\tau\approx 0$ as module-like. Groups in networks are revealed by a sequential extraction procedure proposed in [@ZLZ11; @SBB13; @Weiss]. One first finds the group $S$ and its linking pattern $T$ with random-restart hill climbing [@RN03] that maximizes the objective function. Next, the revealed group $S$ is extracted from the network by removing the links between groups $S$ and $T$, and any node that becomes isolated. The procedure is then repeated on the remaining network until the objective function is larger than the $99$th percentile of the values obtained under the same framework in a corresponding Erdős-R[é]{}nyi random graph [@ER59]. All groups reported in the paper are thus statistically significant at $1\%$ level. Note that the above procedure allows for overlapping [@PDFV05], hierarchical [@RSMOB02], nested and other classes of groups. \[sec:analys\]Analysis and discussion ===================================== Section \[subsec:nets\] introduces real-world networks considered in the study. Section \[subsec:orig\] reports the node group structure of the original networks extracted with the framework described in Section \[sec:nodegroups\]. The groups extracted from the sampled networks are analyzed in Section \[subsec:sampled\]. For a complete analysis, we also observe the node group structure of a large network with more than a million links in Section \[subsec:large\]. \[subsec:nets\]Network data --------------------------- [clrr]{} & & &\ *Collab* & High Energy Physics collaborations [@LKF05] & $9877$ & $25998$\ *PGP* & Pretty Good Privacy web-of-trust [@BPDA04] & $10680$ & $24340$\ *P2P* & Gnutella peer-to-peer file sharing [@LKF05] & $8717$ & $31525$\ *Citation* & High Energy Physics citations [@LKF05] & $27770$ & $352807$\ The empirical analysis in the

1 $, while modules are indicated by $ \tau = 0$. Mixtures correspond to group with $ 0 < \tau < 1$. For the remainder of the paper, we refer to groups with $ \tau\approx 1 $ as residential district - like and groups with $ \tau\approx 0 $ as module - like. group in networks are revealed by a consecutive origin procedure proposed in   [ @ZLZ11; @SBB13; @Weiss ]. One foremost finds the group $ S$ and its linking pattern $ T$ with random - restart mound climb   [ @RN03 ] that maximizes the objective affair. Next, the revealed group $ S$ is extracted from the net by removing the links between groups $ S$ and $ T$, and any lymph node that becomes isolated. The procedure is then repeated on the remaining net until the objective function is bombastic than the $ 99$th percentile of the values obtained under the same model in a corresponding Erdős - R[é]{}nyi random graph   [ @ER59 ]. All groups reported in the paper are thus statistically significant at $ 1\%$   horizontal surface. notice that the above procedure allows for overlapping   [ @PDFV05 ], hierarchical   [ @RSMOB02 ], nested and other classes of groups. \[sec: analys\]Analysis and discussion = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Section   \[subsec: nets\ ] introduce actual - world net view in the study. Section   \[subsec: orig\ ] report the node group social organization of the original networks extracted with the framework trace in Section   \[sec: nodegroups\ ]. The groups extracted from the sampled networks are analyzed in Section   \[subsec: sampled\ ]. For a complete analysis, we also observe the node group structure of a large net with more than a million links in Section   \[subsec: large\ ]. \[subsec: nets\]Network datum --------------------------- [ clrr ] { } & & & \ * Collab * & High Energy Physics collaborations   [ @LKF05 ] & $ 9877 $ & $ 25998$\ * PGP * & Pretty Good Privacy web - of - trust   [ @BPDA04 ] & $ 10680 $ & $ 24340$\ * P2P * & Gnutella peer - to - peer file sharing   [ @LKF05 ] & $ 8717 $ & $ 31525$\ * Citation * & High Energy Physics citations   [ @LKF05 ] & $ 27770 $ & $ 352807$\ The empiric analysis in the

1$, wjile modules are indicattd by $\tau = 0$. Mixtutew corrxspond fo groupr with $0 <\tau < 1$. For the rest oh thw paptg, we refer to groups dith $\tau\aiprox 1$ as comnynity-like ehd grouif wifm $\tau\cp'rox 0$ as module-kike. Groups in networks ase rzvealed by a sequential extraction pwocedurr oroposed in [@ZLZ11; @SBB13; @Reisa]. One first finds the group $S$ and jts linning pattern $Y$ with random-restart hill flimhing [@RN03] that maximixes the objgdtidw function. Ndxt, the renzaled group $S$ is extracted from the networy by xemoving thg mijns between jroups $S$ and $T$, and any noda that necomes isolatcd. Thx pricedure is then repeaved on the remaining network gncil the objective funxtuon iv lasger rhav tge $99$tg percfntmle of the balues obtauned under the same fwqmework in a dorres[ogding Erdős-R[é]{}nyi random graph [@ER59]. All groupv rsported in the paper arw thus statistically dignificagt at $1\%$ level. Note that the above procedure allows xor oteflakplkg [@PDWC05], jierarchical [@RSMOB02], nested and other classes of frpuis. \[sec:analys\]Analysls and discussion ===================================== Srchipg \[subsec:nets\] ivtroduess real-world networkd consiqered in the suudy. Xection \[subsec:orig\] reports tye node groui steucture of the oriyinal networys ectracyed with the framework bescriged in Sectlon \[sec:nodseroups\]. The groupr eqtrawted from the sampled netwjrks are enalyved in Sdctipn \[subsqc:sampled\]. Vor a complete analysis, ae aldo observe tje node group structure of a lacje network wijh korv than a iillipn links in Sqction \[subsec:latge\]. \[subsec:uets\]Negwork data --------------------------- [blrr]{} & & &\ *Conlab* & High Qnergy Physicv collaboratimns [@LKF05] & $9877$ & $25998$\ *PGP* & Peetty Guud Privacy web-pf-trust [@BPBC04] & $10680$ & $24340$\ *P2P* & Tnutella peer-to-peev filg aharing [@LKF05] & $8717$ & $31525$\ *Cnuatuon* & High Energu Pfysycd ritatymns [@LKF05] & $27770$ & $352807$\ Tha emoirkval avalysis in bhe

1$, while modules are indicated by $\tau Mixtures to groups $0 <\tau < the we refer to with $\tau\approx 1$ community-like and groups with $\tau\approx 0$ module-like. Groups in networks are revealed by a sequential extraction procedure proposed in @SBB13; @Weiss]. One first finds the group $S$ and its linking pattern $T$ random-restart climbing that the objective function. Next, the revealed group $S$ is extracted from the network by removing the between groups $S$ and $T$, and any node becomes isolated. The procedure then repeated on the remaining until objective function larger the percentile of the obtained under the same framework in a corresponding Erdős-R[é]{}nyi random graph [@ER59]. All groups reported in the are thus at $1\%$ Note the procedure allows for hierarchical [@RSMOB02], nested and other classes and discussion ===================================== Section \[subsec:nets\] introduces real-world networks in the Section \[subsec:orig\] reports the node group of the original networks extracted with the framework in Section \[sec:nodegroups\]. The groups extracted from the sampled networks are analyzed in Section \[subsec:sampled\]. complete analysis, we also the node group of large with than a links in Section \[subsec:large\]. \[subsec:nets\]Network data --------------------------- [clrr]{} & & &\ & High Energy Physics collaborations [@LKF05] & $9877$ & $25998$\ Pretty Privacy web-of-trust [@BPDA04] $10680$ & $24340$\ *P2P* Gnutella file sharing [@LKF05] & $31525$\ & citations & & $352807$\ The empirical in the

1$, while modules are indicated bY $\tau = 0$. MixturEs corResPonD tO groUps wIth $0 <\tau < 1$. For the reST of tHe paper, we refer to groups With $\tAu\APproX 1$ As CommuNity-likE AnD GRouPs WiTh $\tAu\APpRox 0$ as ModUle-like. groups in neTwoRkS are revealed BY a Sequential ExtRaction proceDurE propoSeD in [@zlZ11; @SBB13; @weiSs]. One First fINds the Group $S$ and ItS LinkinG Pattern $t$ WItH ranDom-restart hill cliMBiNG [@RN03] that maximizEs the oBjECtIVE fuNctIon. Next, the ReVealeD Group $S$ iS ExTRACteD From the networK by removing THe lInks beTwEen GRoups $S$ And $T$, aNd ANy nOde that becoMes iSolated. ThE proceDUre is thEN repeatEd on thE reMaiNing NEtWoRk uNtIL thE ObJecTIve Function Is LaRger tHan tHE $99$TH PercEntIle oF the vAlues obtained UndEr thE SamE framEwork In a cOrRespoNding ERdős-R[É]{}nYi random graph [@ER59]. all gRoups repoRteD iN thE pAper aRE thus sTatIstIcally sIgnificANt aT $1\%$ lEVEL. NOte that the above proCeDURe Allows foR overlAPpInG [@pDFV05], hierArChiCal [@RsmoB02], nesTed aND oTher clasSes of gROuPs. \[Sec:analYs\]analysIs And DisCussiON ===================================== SecTion \[suBsec:nets\] IntroDUces real-world nETworks consideREd IN ThE StudY. SeCtion \[subsec:Orig\] REporTs thE NoDe gROup stRuctuRe OF tHE original networks exTrActed wIth thE framework desCribed in SeCTIOn \[sec:nodEgroUPs\]. tHe groups extracTed frOm the samplED networkS are aNalyzed iN Section \[sUBSec:samplEd\]. FOr a ComPleTE AnAlysis, we also oBSErve ThE node grOup StructuRe oF a lArgE neTwOrk with moRe than a mIlLiOn LiNks In SecTIon \[subseC:lArgE\]. \[sUbsEc:netS\]networK data --------------------------- [Clrr]{} & & &\ *coLlAB* & HiGh EnergY phYSIcs cOlLaBoraTioNs [@lKF05] & $9877$ & $25998$\ *PGp* & PreTTy GOod PrivAcy web-of-tRusT [@bPDA04] & $10680$ & $24340$\ *p2P* & gnUtella pEer-to-peer file ShAring [@LKF05] & $8717$ & $31525$\ *CiTaTioN* & High ENERgy PhysiCs citations [@LKF05] & $27770$ & $352807$\ The empiriCAl analySis In the

1$, while modules are ind icated by$\tau =0$. M ixtu rescorrespond tog roup s with $0 <\tau < 1$.For t he rest of thepaper,w er e fer t ogro up s w ith $ \ta u\appro x 1$ as co mmu ni ty-like andg ro ups with $ \ta u\approx 0$asmodule -l ike . Gro ups in n etwork s are r evealed b ya seque n tial ex t r ac tion procedure propos e di n [@ZLZ11; @SB B13; @ We i ss ] . On e f irst finds t he gr o up $S$a nd i t s l i nking pattern $T$ with r a ndo m-rest ar t h i ll cli mbing  [ @ RN0 3] that max imiz es the ob jectiv e functi o n. Next , therev eal ed g r ou p$S$ i s ex t ra cte d fr om the n et wo rk by rem o v i n g th e l inks betw een groups $S $ a nd $ T $,and a ny no de t ha t bec omes i solat ed . The procedure isthen repe ate donth e rem a iningnet wor k until the ob j ect iv e f un ction is larger th an t he $99$thpercen t il eo f the va lu esobta i n ed un dert he same fr amewor k i na corre sp onding E rdő s-R [é]{} n yi r andomgraph [@ ER59] . All groups re p orted in thep ap e r a r e th usstatistical ly s i gnif ican t a t $ 1 \%$ l evel. N o te that the above proc ed ure al lowsfor overlappi ng [@PDFV0 5 ] , hierarc hica l  [ @ RSMOB02], nest ed an d other cl a sses ofgroup s. \[se c:analys\ ] A nalysisand di scu ssi o n = ============= = = ==== == ======= === ===== Sec tio n \ [su bs ec:nets\] introdu ce sre al -wo rld n e tworks c on sid er edin th e study . Sec tion  \ [s u bse c:orig\ ] r e p orts t he nod e g ro up st ruct u reof theoriginalnet w orks e xt ractedwith the fram ew ork descri be d i n Sect i o n \[sec: nodegroups\]. The group s extrac ted from the samplednet worksare analyz ed inSecti on  \[ s u bsec: s a mp led \] . For a co m p let e ana ly sis, we als o observe the node gro up structureofa la r g enet w or k wi th mor e than a millionlinks in S ec t io n \[subsec : lar ge \]. \[ subsec: nets\ ] Network data --- --------- -- ---- - - --- ---- [clr r]{} & & &\ *Coll a b* &H ig h Ene rgy Physi cs co llabo ration s  [@ LKF05 ] & $9 87 7$ & $ 25998 $\ *PGP* & Pretty Good Privacy we b-of-t rust[@B PDA04] &$10 6 80$ & $24340 $\ * P2P* & Gnu tel lapeer- to- p eer f iles ha rin g  [@LK F05] & $8717$& $ 315 2 5 $\ *Citation* & Hig h Ene rgy Physic s ci tations [@LKF05]& $27770$ & $35 2807 $ \ T hee mpir ic al analysis in th e

1$,_while modules_are indicated by $\tau_= 0$._Mixtures_correspond to_groups_with $0 <\tau_< 1$. For_the rest of the_paper, we refer_to_groups with $\tau\approx 1$ as community-like and groups with $\tau\approx 0$ as module-like. Groups in_networks_are revealed_by_a_sequential extraction procedure proposed in [@ZLZ11;_@SBB13; @Weiss]. One first finds_the group_$S$ and its linking pattern $T$ with random-restart_hill_climbing [@RN03] that maximizes_the objective function. Next, the revealed group $S$ is_extracted from the network by removing_the links between_groups_$S$_and $T$, and any_node that becomes isolated. The procedure_is then repeated on the remaining_network until the objective function is larger_than the $99$th percentile of the_values obtained under the same_framework in_a corresponding Erdős-R[é]{}nyi random graph [@ER59]._All groups reported_in the_paper are thus_statistically significant at $1\%$ level. Note that_the above procedure_allows for overlapping [@PDFV05], hierarchical [@RSMOB02], nested and_other_classes of groups. \[sec:analys\]Analysis_and_discussion ===================================== Section \[subsec:nets\]_introduces real-world_networks considered in_the_study. Section \[subsec:orig\]_reports_the node group structure of the_original_networks extracted with the framework described in_Section \[sec:nodegroups\]. The groups extracted_from_the sampled networks are_analyzed in Section \[subsec:sampled\]. For a_complete analysis, we also observe the_node group_structure of_a large network with more than a million links in Section \[subsec:large\]. \[subsec:nets\]Network_data --------------------------- [clrr]{} & & &\ *Collab* & High_Energy Physics collaborations [@LKF05] &_$9877$ &_$25998$\ *PGP*_& Pretty Good_Privacy_web-of-trust [@BPDA04] &_$10680$ & $24340$\ *P2P* & Gnutella peer-to-peer file_sharing [@LKF05] &_$8717$ & $31525$\ *Citation* & High Energy_Physics citations [@LKF05] & $27770$_&_$352807$\ The empirical analysis in the

“data” at $x_0$ (i.e., derivatives $f^{(i)}(x_0)$). Our paper proceeds as follows. In Section \[sec:terl\], we start with a general result of applying Taylor expansions to Q-functions. When we apply the same technique to the RL objective, we reuse the general result and derive a higher-order policy optimization objective. This leads to Section \[sec:TayPO\], where we formally present the *Taylor Expansion Policy Optimization* (TayPO) and generalize prior work [@schulman2015trust; @schulman2017proximal] as a first-order special case. In Section \[sec:uni\], we make clear connection between Taylor expansions and $Q(\lambda)$ [@harutyunyan_QLambda_2016], a common return-based off-policy evaluation operator. Finally, in Section \[sec:exp\], we show the performance gains due to the higher-order objectives across a range of state-of-the-art distributed deep RL agents. Taylor expansion for reinforcement learning {#sec:terl} =========================================== Consider a Markov Decision Process (MDP) with state space $\mathcal{X}$ and action space $\mathcal{A}$. Let policy $\pi(\cdot|x)$ be a distribution over actions give state $x$. At a discrete time $t \geq 0$, the agent in state $x_t$ takes action $a_t \sim \pi(\cdot|x_t)$, receives reward $r_t \triangleq r(x_t,a_t)$, and transitions to a next state $x_{t+1} \sim p(\cdot|x_t,a_t)$. We assume a discount factor $\gamma \in [0,1)$. Let $Q^\pi(x,a)$ be the action value function (Q-function) from state $x,$ taking action $a,$ and following policy $\pi$. For convenience, we use $ d_\gamma^\pi(\cdot,\cdot|x_0,a_0,\tau)$ to denote the discounted visitation distribution starting from state-action pair $(x_0,a_0)$ and following $\pi$, such that $d_\gamma^\pi(x,a|x_0,a_0,\tau) = (1-\gamma)\gamma^{-\tau} \sum_{t\geq \tau} \

“ data ” at $ x_0 $ (i.e., derivatives $ f^{(i)}(x_0)$). Our paper proceeds as follows. In Section   \[sec: terl\ ], we depart with a cosmopolitan result of applying Taylor expansion to Q - function. When we apply the same proficiency to the RL objective, we reuse the general consequence and derive a higher - club policy optimization objective. This leads to Section   \[sec: TayPO\ ], where we formally award the * Taylor Expansion Policy Optimization * (TayPO) and generalize prior work [ @schulman2015trust; @schulman2017proximal ] as a first - order special event. In Section   \[sec: uni\ ], we make clear connection between Taylor expansion and $ Q(\lambda)$ [ @harutyunyan_QLambda_2016 ], a common return - based off - policy evaluation operator. ultimately, in Section   \[sec: exp\ ], we show the performance gains due to the higher - order aim across a range of state - of - the - art distributed deep RL agents. Taylor expansion for reinforcement learning { # sec: terl } = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Consider a Markov Decision Process (MDP) with state space $ \mathcal{X}$ and action space $ \mathcal{A}$. Let policy $ \pi(\cdot|x)$ be a distribution over actions give state $ x$. At a discrete clock time $ t \geq 0 $, the agent in department of state $ x_t$ takes action $ a_t \sim \pi(\cdot|x_t)$, receives reward $ r_t \triangleq r(x_t, a_t)$, and conversion to a future state $ x_{t+1 } \sim p(\cdot|x_t, a_t)$. We assume a deduction factor $ \gamma \in [ 0,1)$. Let $ Q^\pi(x, a)$ be the action value function (Q - function) from state $ x,$ necessitate action $ a,$ and following policy $ \pi$. For convenience, we use $ d_\gamma^\pi(\cdot,\cdot|x_0,a_0,\tau)$ to announce the discounted trial distribution starting from state - action pair $ (x_0,a_0)$ and following $ \pi$, such that $ d_\gamma^\pi(x, a|x_0,a_0,\tau) = (1-\gamma)\gamma^{-\tau } \sum_{t\geq \tau } \

“daha” at $x_0$ (i.e., derivatives $f^{(l)}(x_0)$). Our paper procgees as hollows. In Sectkon \[sec:terl\], we start with a gxnerql rewult of applying Taylof expansilns to Q-duncuions. When we apply the same tedmniquz vo the RL objecjive, we reusa the general serupt and derive a higher-order policy jptimizstlon objective. Jhis kqads no Section \[sec:TayPO\], where we forjally pgesent the *Taylor Expansion Policy Optimizahion* (TayPO) and generallze prior wirk [@fxhulman2015trust; @schulman2017pgmximal] as z first-order special case. In Seztion \[xec:uni\], we naje fnear connecvion bvtween Taylor expansimns and $Q(\lambda)$ [@harutnunyai_QLanbda_2016], a common return-besed off-policy evaluwtion opesacor. Finally, in Sectiob \[swc:exp\], we vhow rhe pedfprjance haiis due to tge higher-oreer objectives acroxs q range of stzte-of-tre-wrt distributed deep RL agents. Taylor exkansikn for reinforcement leqrning {#sec:terl} =========================================== Considet a Markov Decision Process (MDP) with state space $\mathcal{X}$ atd acviun wpwzw $\lathcal{A}$. Let policy $\pi(\cdot|x)$ be a distribution kvtr sctions give sbate $x$. At a discreye toie $t \geq 0$, the agent in state $x_t$ takes achion $a_t \sim \pi(\cdot|x_t)$, weceoves reward $r_t \triangleq r(z_t,a_t)$, and trausirions to a next stcte $x_{t+1} \sim p(\edot|x_t,s_t)$. We assume a discount factur $\gzmma \in [0,1)$. Leh $Q^\pi(x,a)$ bs the action valud flncthon (Q-function) from state $x,$ taking artion $a,$ and wollpwing [olicy $\pi$. Vor convenience, we use $ f_\gammc^\pi(\cdmt,\cdot|x_0,a_0,\tak)$ to denote the discounted visivetion distribotimn vtarting from state-action [air $(x_0,a_0)$ and foklowing $\pi$, sjch that $d_\famma^\pi(e,a|x_0,a_0,\tau) = (1-\gamia)\gamma^{-\tau} \suk_{j\geq \tau} \

“data” at $x_0$ (i.e., derivatives $f^{(i)}(x_0)$). Our as In Section we start with Taylor to Q-functions. When apply the same to the RL objective, we reuse general result and derive a higher-order policy optimization objective. This leads to Section where we formally present the *Taylor Expansion Policy Optimization* (TayPO) and generalize prior [@schulman2015trust; as first-order case. In Section \[sec:uni\], we make clear connection between Taylor expansions and $Q(\lambda)$ [@harutyunyan_QLambda_2016], a common off-policy evaluation operator. Finally, in Section \[sec:exp\], we the performance gains due the higher-order objectives across a of distributed deep agents. expansion reinforcement learning {#sec:terl} Consider a Markov Decision Process (MDP) with state space $\mathcal{X}$ and action space $\mathcal{A}$. Let policy $\pi(\cdot|x)$ a distribution give state At discrete $t \geq 0$, in state $x_t$ takes action $a_t reward $r_t \triangleq r(x_t,a_t)$, and transitions to a state $x_{t+1} p(\cdot|x_t,a_t)$. We assume a discount factor \in [0,1)$. Let $Q^\pi(x,a)$ be the action value (Q-function) from state $x,$ taking action $a,$ and following policy $\pi$. For convenience, we use to denote the discounted distribution starting from pair and $\pi$, that $d_\gamma^\pi(x,a|x_0,a_0,\tau) (1-\gamma)\gamma^{-\tau} \sum_{t\geq \tau} \

“data” at $x_0$ (i.e., derivatives $f^{(i)}(x_0)$). OuR paper procEeds aS foLloWs. in SeCtioN \[sec:terl\], we starT With A general result of applyiNg TayLoR ExpaNSiOns to q-functiONs. wHEn wE aPpLy tHe SAmE techNiqUe to the rL objectivE, we ReUse the generaL ReSult and derIve A higher-order PolIcy optImIzaTIon obJecTive. THis leaDS to SecTion \[sec:TaYPo\], Where wE FormallY PReSent The *Taylor ExpansioN poLIcy OptimizatioN* (TayPO) AnD GeNERalIze Prior work [@sChUlman2015TRust; @schULmAN2017PRoxIMal] as a first-orDer special cASe. IN SectiOn \[Sec:UNi\], we maKe cleAr COnnEction betweEn TaYlor expanSions aND $Q(\lambdA)$ [@HarutyuNyan_QLAmbDa_2016], a CommON rEtUrn-BaSEd oFF-pOliCY evAluation OpErAtor. FInalLY, IN sectIon \[Sec:eXp\], we sHow the performAncE gaiNS duE to thE highEr-orDeR objeCtives AcrosS a Range of state-of-tHe-arT distribuTed DeEp Rl aGents. tAylor eXpaNsiOn for reInforceMEnt LeARNInG {#sec:terl} =========================================== Consider a MArKOV DEcision PRocess (mdP) WiTH state spAcE $\maThcaL{x}$ And acTion SPaCe $\mathcaL{A}$. Let pOLiCy $\Pi(\cdot|x)$ Be A distrIbUtiOn oVer acTIons Give stAte $x$. At a dIscreTE time $t \geq 0$, the agENt in state $x_t$ taKEs ACTiON $a_t \sIm \pI(\cdot|x_t)$, receIves REwarD $r_t \tRIaNglEQ r(x_t,a_T)$, and tRaNSiTIons to a next state $x_{t+1} \sIm P(\cdot|x_T,a_t)$. We Assume a discouNt factor $\gaMMA \In [0,1)$. Let $Q^\pi(X,a)$ be THe ACtion value funcTion (Q-Function) frOM state $x,$ tAking Action $a,$ aNd followiNG Policy $\pi$. for ConVenIenCE, We Use $ d_\gamma^\pi(\cdOT,\Cdot|X_0,a_0,\Tau)$ to deNotE the disCouNteD viSitAtIon distriBution stArTiNg FrOm sTate-aCTion pair $(X_0,a_0)$ And FoLloWing $\pI$, Such thAt $d_\gaMma^\pI(x,A|x_0,A_0,\Tau) = (1-\Gamma)\gaMMa^{-\TAU} \sum_{T\gEq \Tau} \

“data” at $x_0$ (i.e., de rivatives$f^{( i)} (x_ 0) $). Our paper proceed s asfollows. In Section \[ sec:t er l \],w estart with a ge n e ral r es ult o f a pplyi ngTaylorexpansions to Q -functions.W he n we apply th e same techn iqu e to t he RL objec tiv e, we reuse the ge neral res ul t and d e rive ah i gh er-o rder policy optim i za t ion objective. Thisle a ds t o S ect ion \[sec: Ta yPO\] , wherew ef o r mal l y present the *Taylor Ex p ans ion Po li cyO ptimiz ation *( Tay PO) and gen eral ize prior work[ @schulm a n2015tr ust; @ sch ulm an20 1 7p ro xim al ] as afir s t-o rder spe ci al case . In S e c tion  \[ sec: uni\] , we make cle arconn e cti on be tween Tay lo r exp ansion s and $ Q(\lambda)$ [@h arut yunyan_QL amb da _20 16 ], ac ommonret urn -basedoff-pol i cyev a l u at ion operator. Fina ll y , i n Sectio n \[se c :e xp \ ], we sh ow th e pe r f orman ce g a in s due to the h i gh er -orderob jectiv es ac ros s a r a ngeof sta te-of-th e-art distributed de e p RL agents.Ta y l or expa nsi on for rein forc e ment lea r ni ng{ #sec: terl} = = == = =================== == ====== ===== ====== Consi der a Mark o v Decision Pro c es s (MDP) with st ate s pace $\mat h cal{X}$and a ction sp ace $\mat h c al{A}$.Let po lic y $ \ p i( \cdot|x)$ bea dist ri butionove r actio nsgiv e s tat e$x$. At a discret eti me $ t \ geq 0 $ , the ag en t i nsta te $x _ t$ tak es ac tion $ a_ t \s im \pi( \ cd o t |x_t )$ ,rece ive srewar d $r _ t \ triangl eq r(x_t ,a_ t )$,an dtransit ions to a nex tstate $x_{ t+ 1}\sim p ( \ cdot|x_t ,a_t)$. We assume a dis c ount fa cto r $\g amma \in [0,1 )$. Let $ Q^\ p i(x,a) $ be t he ac ti onv a lue f u n ct ion ( Q-function ) fro m sta te $x, $ takin g action $a,$ andf oll owing policy$\p i$.F o rcon v en i enc e, weu s e $ d_\gamma^\p i(\cdot,\c do t |x _0,a_0,\ta u )$to denote the di scoun t ed visi tation di stributio nstar t i ngfrom state -actionpair $(x_ 0 ,a_0) $ a nd fo llo wing $ \p i$, such that$ d_\ gamma ^\pi(x ,a |x_0,a _0,\t au ) = (1-\ gamma)\gamma^{-\tau} \s um_{t\ geq \ tau } \

“data”_at $x_0$_(i.e., derivatives $f^{(i)}(x_0)$). Our paper_proceeds as_follows._In Section \[sec:terl\],_we_start with a_general result of_applying Taylor expansions to_Q-functions. When we_apply_the same technique to the RL objective, we reuse the general result and derive_a_higher-order policy_optimization_objective._This leads to Section \[sec:TayPO\], where_we formally present the *Taylor_Expansion Policy_Optimization* (TayPO) and generalize prior work [@schulman2015trust; @schulman2017proximal]_as_a first-order special_case. In Section \[sec:uni\], we make clear connection between Taylor_expansions and $Q(\lambda)$ [@harutyunyan_QLambda_2016], a common_return-based off-policy evaluation_operator._Finally,_in Section \[sec:exp\], we show_the performance gains due to the_higher-order objectives across a range of_state-of-the-art distributed deep RL agents. Taylor expansion for_reinforcement learning {#sec:terl} =========================================== Consider a Markov Decision_Process (MDP) with state space_$\mathcal{X}$ and_action space $\mathcal{A}$. Let policy_$\pi(\cdot|x)$ be a_distribution over_actions give state_$x$. At a discrete time $t_\geq 0$, the_agent in state $x_t$ takes action_$a_t_\sim \pi(\cdot|x_t)$, receives_reward_$r_t_\triangleq _r(x_t,a_t)$, and transitions_to_a next_state_$x_{t+1} \sim p(\cdot|x_t,a_t)$. We assume a_discount_factor $\gamma \in [0,1)$. Let $Q^\pi(x,a)$ be_the action value function_(Q-function)_from state $x,$ taking_action $a,$ and following policy_$\pi$. For convenience, we use $_d_\gamma^\pi(\cdot,\cdot|x_0,a_0,\tau)$ to_denote the_discounted visitation distribution starting from state-action pair $(x_0,a_0)$ and following $\pi$,_such that $d_\gamma^\pi(x,a|x_0,a_0,\tau) = (1-\gamma)\gamma^{-\tau} \sum_{t\geq_\tau} \

+1} \binom{j}{r}\right\} \frac{z^{j+1}}{k} + (\mbox{polynomial of $k$})\\[8pt] & \qquad = \frac{z^{j+1}}{j+1}\frac{1}{k} + (\mbox{a polynomial of $k$}). \end{aligned}$$]{} Hence, if we put [ $$\begin{aligned} & B(k,z) := (I)_{k}+(II)_{k}+(III)_{k}+(IV)_{k}\\[4pt] & \qquad - \frac{1}{j+1} \left\{ \sum_{r=0}^{j+1}\binom{j+1}{r} \frac{(-1)^{r}z^{r+1}}{r+1}B_{j+1-r}(z)\right\} \left(\frac{1}{k}-\frac{1}{k+1}\right), \end{aligned}$$]{} then, $B(k,z)$ is a polynomial of $k$, $z$ and it satisfies $$\frac{B_{j+1}(z+k+1)}{j+1} \log\left(1+\frac{z}{k}\right) + B(k,z) = O(k^{-2}) \quad \mbox{as} \quad k\to\infty.$$ By Lemma \[eqn;35\], we have $$B(k.z)=P_{j}(z+k+1)-P_{j}(z+k). \label{eqn;317}$$ By (\[eqn;316\]) and (\[eqn;317\]), we can deduce (\[eqn;315\]). [ $$\begin{aligned} & K_{j}(z) = Q_{j}(z)+ \sum_{r=1}^{j}\binom{j}{r}z^{j-r} \zeta'(-r)-\frac{z^{j+1}}{j+1}\gamma\\[4pt] & \qquad + \sum_{k=1}^{\infty}\left\{ -(z+k)^{k}\log \left(1+\frac{z}{k}\right) + \sum_{r=0}^{j}\binom

+1 } \binom{j}{r}\right\ } \frac{z^{j+1}}{k } + (\mbox{polynomial of $ k$})\\[8pt ] & \qquad = \frac{z^{j+1}}{j+1}\frac{1}{k } + (\mbox{a polynomial of $ k$ }). \end{aligned}$$ ] { } Hence, if we put [ $ $ \begin{aligned } & B(k, z): = (I)_{k}+(II)_{k}+(III)_{k}+(IV)_{k}\\[4pt ] & \qquad - \frac{1}{j+1 } \left\ { \sum_{r=0}^{j+1}\binom{j+1}{r } \frac{(-1)^{r}z^{r+1}}{r+1}B_{j+1 - r}(z)\right\ } \left(\frac{1}{k}-\frac{1}{k+1}\right), \end{aligned}$$ ] { } then, $ B(k, z)$ is a polynomial of $ k$, $ z$ and it satisfies $ $ \frac{B_{j+1}(z+k+1)}{j+1 } \log\left(1+\frac{z}{k}\right) + B(k, z) = O(k^{-2 }) \quad \mbox{as } \quad k\to\infty.$$ By Lemma \[eqn;35\ ], we have $ $ B(k.z)=P_{j}(z+k+1)-P_{j}(z+k). \label{eqn;317}$$ By (\[eqn;316\ ]) and (\[eqn;317\ ]), we can deduce (\[eqn;315\ ]). [ $ $ \begin{aligned } & K_{j}(z) = Q_{j}(z)+ \sum_{r=1}^{j}\binom{j}{r}z^{j - radius } \zeta'(-r)-\frac{z^{j+1}}{j+1}\gamma\\[4pt ] & \qquad + \sum_{k=1}^{\infty}\left\ { -(z+k)^{k}\log \left(1+\frac{z}{k}\right) + \sum_{r=0}^{j}\binom

+1} \bijom{j}{r}\right\} \frac{z^{j+1}}{y} + (\mbox{kooynomiel of $k$})\\[8lt] & \qduad = \frac{z^{j+1}}{j+1}\frac{1}{k} + (\mvox{a kjlynomial of $k$}). \ena{aligned}$$]{} Jence, if we kut [ $$\begin{aligned} & B(k,z) := (I)_{k}+(II)_{i}+(LII)_{k}+(IR)_{k}\\[4't] & \qquad - \ftac{1}{j+1} \left\{ \sum_{r=0}^{j+1}\binom{j+1}{s} \frac{(-1)^{r}z^{r+1}}{r+1}B_{j+1-r}(z)\right\} \left(\frac{1}{k}-\srac{1}{k+1}\ribhh), \end{aligned}$$]{} jhen, $N(h,z)$ ia a polynomial of $k$, $z$ and it satisries $$\frec{B_{j+1}(z+k+1)}{j+1} \log\lrft(1+\frac{z}{k}\right) + B(k,z) = O(k^{-2}) \euad \mbox{as} \quad k\ho\infty.$$ By Oemmw \[eqn;35\], we have $$B(k.z)=P_{j}(z+k+1)-P_{j}(e+k). \label{eqn;317}$$ By (\[eqn;316\]) and (\[eqn;317\]), we can deduce (\[edn;315\]). [ $$\beyin{aligned} & K_{u}(e) = Q_{j}(z)+ \sum_{r=1}^{j}\uinom{j}{g}z^{j-r} \zeta'(-r)-\frac{z^{j+1}}{b+1}\gamma\\[4py] & \qquad + \smm_{k=1}^{\inhty}\lwft\{ -(z+k)^{k}\log \left(1+\fcac{z}{k}\right) + \sum_{t=0}^{j}\binom

+1} \binom{j}{r}\right\} \frac{z^{j+1}}{k} + (\mbox{polynomial of $k$})\\[8pt] = + (\mbox{a of $k$}). \end{aligned}$$]{} $$\begin{aligned} B(k,z) := (I)_{k}+(II)_{k}+(III)_{k}+(IV)_{k}\\[4pt] \qquad - \frac{1}{j+1} \sum_{r=0}^{j+1}\binom{j+1}{r} \frac{(-1)^{r}z^{r+1}}{r+1}B_{j+1-r}(z)\right\} \left(\frac{1}{k}-\frac{1}{k+1}\right), \end{aligned}$$]{} then, $B(k,z)$ a polynomial of $k$, $z$ and it satisfies $$\frac{B_{j+1}(z+k+1)}{j+1} \log\left(1+\frac{z}{k}\right) + B(k,z) = \quad \mbox{as} \quad k\to\infty.$$ By Lemma \[eqn;35\], we have $$B(k.z)=P_{j}(z+k+1)-P_{j}(z+k). \label{eqn;317}$$ By (\[eqn;316\]) (\[eqn;317\]), can (\[eqn;315\]). $$\begin{aligned} & K_{j}(z) = Q_{j}(z)+ \sum_{r=1}^{j}\binom{j}{r}z^{j-r} \zeta'(-r)-\frac{z^{j+1}}{j+1}\gamma\\[4pt] & \qquad + \sum_{k=1}^{\infty}\left\{ -(z+k)^{k}\log \left(1+\frac{z}{k}\right) + \sum_{r=0}^{j}\binom

+1} \binom{j}{r}\right\} \frac{z^{j+1}}{k} + (\mbox{poLynomial of $K$})\\[8pt] & \qqUad = \FraC{z^{J+1}}{j+1}\frAc{1}{k} + (\mBox{a polynomial OF $k$}). \enD{aligned}$$]{} Hence, if we put [ $$\beGin{alIgNEd} & B(k,Z) := (i)_{k}+(iI)_{k}+(IIi)_{k}+(IV)_{k}\\[4pt] & \QQuAD - \FraC{1}{j+1} \LeFt\{ \sUm_{R=0}^{J+1}\bInom{j+1}{R} \frAc{(-1)^{r}z^{r+1}}{r+1}B_{J+1-r}(z)\right\} \leFt(\fRaC{1}{k}-\frac{1}{k+1}\right), \ENd{Aligned}$$]{} theN, $B(k,Z)$ is a polynomiAl oF $k$, $z$ and It SatISfies $$\FraC{B_{j+1}(z+k+1)}{J+1} \log\leFT(1+\frac{z}{K}\right) + B(k,z) = o(k^{-2}) \QUad \mboX{As} \quad k\TO\InFty.$$ BY Lemma \[eqn;35\], we have $$B(k.Z)=p_{j}(Z+K+1)-P_{j}(z+k). \label{eqn;317}$$ BY (\[eqn;316\]) anD (\[eQN;317\]), wE CAn dEduCe (\[eqn;315\]). [ $$\begin{AlIgned} & k_{J}(z) = Q_{j}(z)+ \suM_{R=1}^{j}\BINOm{j}{R}Z^{j-r} \zeta'(-r)-\frac{z^{J+1}}{j+1}\gamma\\[4pt] & \qqUAd + \sUm_{k=1}^{\infTy}\LefT\{ -(Z+k)^{k}\log \Left(1+\fRaC{Z}{k}\rIght) + \sum_{r=0}^{j}\biNom

+1} \binom{j}{r}\right\} \fr ac{z^ {j+ 1}} {k } + (\mbox{po l ynom ial of $k$})\\[8pt] & \ qq u ad = \f rac{z ^{j+1}} { j+ 1 } \fr ac {1 }{k } + (\mbox {a polynom ial o f $k$}). \ e nd {aligned}$ $]{ } Hence, ifweput [$$ \be g in{al ign ed} & B ( k,z) : = (I)_{k} +( I I)_{k} + (III)_{ k } +( IV)_ {k}\\[4pt] &\ qq u ad - \frac{1}{ j+1} \ le f t\ { \sum_{r=0} ^{ j+1}\ b inom{j+ 1 }{ r } \frac{(-1)^ {r}z^{r+1}} { r+1 }B_{j+ 1- r}( z )\righ t\} \ left(\frac{ 1}{k }-\frac{1 }{k+1} \ right), \end{ aligne d}$ $]{ } th e n, $ B(k ,z ) $ i s a po l yno mial of$k $, $z$andi t s atis fie s $$ \frac {B_{j+1}(z+k+ 1)} {j+1 } \lo g\lef t(1+ \f rac{z }{k}\r ight) + B(k,z) = O (k^{ -2}) \qu ad \m bo x{as} \quadk\t o\i nfty.$$ By Lem m a \ [e q n ; 35 \], we have $$B(k. z) = P _{ j}(z+k+1 )-P_{j } (z +k ) . \lab el {eq n;31 7 } $$ By (\[ e qn ;316\])and (\ [ eq n; 317\]), w e cande duc e ( \[eqn ; 315\ ]). [ $$\begi n{ali g ned} & K_{ j }(z) = Q_{j}( z )+ \ su m _{r= 1}^ {j}\binom{j }{r} z ^{j- r} \ z eta'( -r)-\ fr a c{ z ^{j+1}}{j+1}\gamma\ \[ 4pt] &\qquad + \sum _{k=1}^{\i n f t y}\left\ { -(z+k)^{k}\log \lef t(1+\frac{ z }{k}\rig ht) + \ sum_{r=0} ^ { j}\binom

+1} \binom{j}{r}\right\} _ _ _ \frac{z^{j+1}}{k} __ __ _ + (\mbox{polynomial_of $k$})\\[8pt] _ & \qquad_=_\frac{z^{j+1}}{j+1}\frac{1}{k} + (\mbox{a polynomial of_$k$}). _ \end{aligned}$$]{}_Hence,_if_we put [ $$\begin{aligned} _ & B(k,z) :=_(I)_{k}+(II)_{k}+(III)_{k}+(IV)_{k}\\[4pt] _ & \qquad - \frac{1}{j+1} \left\{ __ _\sum_{r=0}^{j+1}\binom{j+1}{r} \frac{(-1)^{r}z^{r+1}}{r+1}B_{j+1-r}(z)\right\} _ \left(\frac{1}{k}-\frac{1}{k+1}\right), \end{aligned}$$]{}_then, $B(k,z)$ is_a_polynomial_of $k$, $z$ and_it satisfies $$\frac{B_{j+1}(z+k+1)}{j+1} _\log\left(1+\frac{z}{k}\right) + B(k,z)_= O(k^{-2}) \quad \mbox{as}_\quad k\to\infty.$$ By Lemma \[eqn;35\], we_have $$B(k.z)=P_{j}(z+k+1)-P_{j}(z+k). \label{eqn;317}$$ By_(\[eqn;316\]) and_(\[eqn;317\]), we can deduce (\[eqn;315\]). [_$$\begin{aligned} _ &_K_{j}(z) = Q_{j}(z)+_\sum_{r=1}^{j}\binom{j}{r}z^{j-r} _\zeta'(-r)-\frac{z^{j+1}}{j+1}\gamma\\[4pt] _ & \qquad + \sum_{k=1}^{\infty}\left\{ __ _-(z+k)^{k}\log_\left(1+\frac{z}{k}\right) _ _ _+_\sum_{r=0}^{j}\binom

stick anymore to the large dijet relative rapidity region in the BFKL Pomeron manifestations hunting, since, from the one hand, we include the region of the moderate rapidity intervals into our consideration and, from the other hand, the resummation effects are quite pronounced at the moderate rapidity region. We present also in Figs. 2,3 estimations for NLO BFKL effects using the results of Ref. [@Cor95], where conformal NLO contributions to the Lipatov’s eigenvalues were calculated. The estimations incorporate the NLO conformal corrections to the Lipatov’s eigenvalues (see Fig. 4) and the NLO CTEQ3M structure functions [@Lai94]. We should note here that the extraction of data on high-$k_{\perp}$ jets from the event samples in order to compare them with the BFKL Pomeron predictions should be different from the algorithms directed to a comparison with perturbative QCD predictions for the hard processes. These algorithms, motivated by the strong $k_{\perp}$-ordering of the hard QCD regime, employ hardest-$k_{\perp}$ jet selection (see, e.g., Ref. [@Alg94]). It is doubtful that one can reconcile these algorithms with the weak $k_{\perp}$-diffusion and the strong rapidity ordering of the semi-hard QCD regime, described by the BFKL resummation. We also note that our predictions should not be compared with the preliminary data [@Heu94] extracted by the most forward/backward jet selection criterion. Obviously, one should include for tagging all the registered pairs of jets (not only the most forward–backward pair) to compare with our predictions. In particular, to make a comparison with Figs. 2,3, one should sum up all the registered $x$-symmetric dijets ($x_1=x_2$) with transverse momenta harder than $k_{\perp min}$. We thank E.A.Kuraev and L.N.Lipatov for stimulating discussions. We are grateful to A.J.Sommerer, J.P.Vary, and B.-L.Young for their kind hospitality at the IITAP, Ames, Iowa and support. V.T.K. is indebted to S.Ahn, C.L.Kim, T.Lee, A.Petridis, J.Qiu, C.R.Schmidt, S.I.T

stick anymore to the large dijet relative rapidity area in the BFKL Pomeron materialization hunting, since, from the one hand, we admit the region of the moderate celerity intervals into our consideration and, from the early hand, the resummation effects are quite pronounced at the moderate celerity area. We present also in Figs. 2,3 estimate for NLO BFKL effects using the results of Ref. [ @Cor95 ], where conformal NLO contribution to the Lipatov ’s eigenvalues were account. The estimations incorporate the NLO conformal correction to the Lipatov ’s eigenvalues (see Fig. 4) and the NLO CTEQ3 M structure function   [ @Lai94 ]. We should note here that the extraction of data on high-$k_{\perp}$ jets from the consequence samples in club to compare them with the BFKL Pomeron predictions should be different from the algorithms target to a comparison with perturbative QCD predictions for the hard processes. These algorithms, motivated by the solid $ k_{\perp}$-ordering of the intemperate QCD regime, employ hardest-$k_{\perp}$ jet selection (see, e.g., Ref. [ @Alg94 ]). It is doubtful that one can reconcile these algorithms with the watery $ k_{\perp}$-diffusion and the strong rapidity order of the semi - unvoiced quantum chromodynamics regime, described by the BFKL resummation. We also note that our predictions should not be compared with the preliminary data [ @Heu94 ] extracted by the about forward / backward jet selection criterion. Obviously, one should include for tagging all the registered pairs of jets (not only the most forward – backward couple) to compare with our predictions. In finical, to do a comparison with Figs. 2,3, one should sum up all the registered $ x$-symmetric dijets ($ x_1 = x_2 $) with transverse momentum harder than $ k_{\perp min}$. We thank E.A.Kuraev and L.N.Lipatov for stimulating discussions. We are grateful to A.J.Sommerer, J.P.Vary, and B.-L.Young for their kind cordial reception at the IITAP, Ames, Iowa and support. V.T.K. is indebted to S.Ahn, C.L.Kim, T.Lee, A.Petridis, J.Qiu, C.R.Schmidt, S.I.T

stlck anymore to the large dijet relative rapidivy regikn in thd BFKL Pomeron manifestationd yuntibg, since, from the one fand, we ijclude tye rtgion of the modeczte raplbity jktervclw into our conxideration and, from the mtfex hand, the resummation effects are qtite prpnlunced at the iodegaee rziibity region. We present also in Fjgs. 2,3 esuimations for NLO NFKL effects using the resklts of Ref. [@Cor95], where fonformal NOO cjbtributions go the Lipatov’s eigenvzlues were calculated. The estimxtionx incorporqtw tjg NLO confornal cjrrections to the Li[atov’s rigenvalues (sec Fig. 4) abd the NLO CTEQ3M strurture functions [@Lai94]. We should nmtz here that the extraxtuon ox dada ov hieh-$k_{\ker'}$ jsts frlm vhe event szmples in oeder to compare thek rpyh the BFKL Lomerog [redictions should be different from tht algkrithms directed to a cimparison with perturhative QCQ predictions for the hard processes. These algoridhms, jutirqted ct hhe strong $k_{\perp}$-ordering of the hard QCD regiis, tmpkoy hardest-$k_{\pevp}$ jet selection (sre, e.b., Ref. [@Alg94]). It ir doubcrum that one can reclncile jhese qlgorithmf wiyh the weak $k_{\perp}$-diffusion qnd the stroug eapidity ordering lf the semi-kard QVD rebime, described by the BYKL reaummation. Wf also nofd that our prediztipnv should not be compared wyth the pcelimnnary daga [@Hgu94] extrwcted by tje most forward/backward uet sglectimn criteriln. Obviously, one should include for tagging akl thv registexed palrs of jets (not only the most forwarb–backwxrd pair) tk compace with our [redictions. It particular, vo make a compariwon witf Figs. 2,3, one shoild sum ui cll the rwgistered $x$-symmetrlc digsts ($x_1=x_2$) with traurcerse momenta hsrddr ehwn $k_{\[arp min}$. We thdnk D.A.Kjtaev xnd L.N.Lipatix fot stimulating discusviona. We are grateful yo A.J.Sommeter, J.P.Varr, and B.-L.Young for their kind hodpitanitb at tne YITAP, Ames, Iowa and support. V.T.I. is indehteq to S.Ahn, C.L.Him, B.Lee, A.Petridis, J.Qiu, C.R.Schmidt, S.I.T

stick anymore to the large dijet relative in BFKL Pomeron hunting, since, from the of the moderate intervals into our and, from the other hand, the effects are quite pronounced at the moderate rapidity region. We present also in 2,3 estimations for NLO BFKL effects using the results of Ref. [@Cor95], where NLO to Lipatov’s were calculated. The estimations incorporate the NLO conformal corrections to the Lipatov’s eigenvalues (see Fig. 4) the NLO CTEQ3M structure functions [@Lai94]. We should here that the extraction data on high-$k_{\perp}$ jets from event in order compare with BFKL Pomeron predictions be different from the algorithms directed to a comparison with perturbative QCD predictions for the hard processes. algorithms, motivated strong $k_{\perp}$-ordering the QCD employ hardest-$k_{\perp}$ jet e.g., Ref. [@Alg94]). It is doubtful reconcile these algorithms with the weak $k_{\perp}$-diffusion and strong rapidity of the semi-hard QCD regime, described the BFKL resummation. We also note that our should not be compared with the preliminary data [@Heu94] extracted by the most forward/backward jet Obviously, one should include tagging all the pairs jets only most forward–backward to compare with our predictions. In particular, to make a comparison Figs. 2,3, one should sum up all the registered $x$-symmetric with momenta harder than min}$. We thank E.A.Kuraev L.N.Lipatov stimulating discussions. We are A.J.Sommerer, and kind at IITAP, Ames, Iowa and V.T.K. is indebted to S.Ahn, T.Lee, A.Petridis, J.Qiu, C.R.Schmidt,

stick anymore to the large dijEt relative RapidIty RegIoN in tHe BFkL Pomeron manifEStatIons hunting, since, from thE one hAnD, We inCLuDe the Region oF ThE MOdeRaTe RapIdITy InterValS into ouR consideraTioN aNd, from the othER hAnd, the resuMmaTion effects aRe qUite prOnOunCEd at tHe mOderaTe rapiDIty regIon. We presEnT Also in fIgs. 2,3 estiMATiOns fOr NLO BFKL effects uSInG The results of ReF. [@Cor95], whErE CoNFOrmAl NlO contribuTiOns to THe LipatOV’s EIGEnvALues were calcuLated. The estIMatIons inCoRpoRAte the nLO coNfORmaL correctionS to tHe Lipatov’S eigenVAlues (seE fig. 4) and tHe NLO CtEQ3m stRuctURe FuNctIoNS [@LaI94]. we ShoULd nOte here tHaT tHe extRactION OF datA on High-$K_{\perp}$ Jets from the evEnt SampLEs iN ordeR to coMparE tHem wiTh the BfKL PoMeRon predictions sHoulD be differEnt FrOm tHe AlgorIThms diRecTed To a compArison wITh pErTURBaTive QCD predictions FoR THe Hard procEsses. THEsE aLGorithms, MoTivAted BY The stRong $K_{\PeRp}$-orderiNg of thE HaRd qCD regiMe, Employ HaRdeSt-$k_{\Perp}$ jET selEction (See, e.g., Ref. [@alg94]). It IS doubtful that oNE can reconcile THeSE AlGOritHms With the weak $K_{\perP}$-DiffUsioN AnD thE StronG rapiDiTY oRDering of the semi-hard qCd regimE, descRibed by the BFKl resummatiON. wE also notE thaT OuR Predictions shoUld noT be compareD With the pRelimInary datA [@Heu94] extraCTEd by the mOst ForWarD/baCKWaRd jet selectioN CRiteRiOn. ObvioUslY, one shoUld IncLudE foR tAgging all The regisTeReD pAiRs oF jets (NOt only thE mOst FoRwaRd–bacKWard paIr) to cOmpaRe WiTH ouR predicTIoNS. in paRtIcUlar, To mAkE a comPariSOn wIth Figs. 2,3, One should Sum UP all ThE rEgisterEd $x$-symmetric dIjEts ($x_1=x_2$) with tRaNsvErse moMENta hardeR than $k_{\perp min}$. We thank E.A.KURaev and l.N.LIpatoV for StimulatiNg dIscussIonS. we are gRatefuL to A.J.soMmeRER, J.P.VaRY, AnD B.-L.yoUng for theiR KInd HospiTaLity At the IItAP, Ames, Iowa and suppORt. V.t.K. is indebted tO S.AHn, C.L.kIM, T.lee, a.peTRidIs, j.qiu, c.r.schmidt, S.I.T

stick anymore to the larg e dijet re lativ e r api di ty r egio n in the BFKLP omer on manifestations hunt ing,si n ce,f ro m the one ha n d, w e i nc lu deth e r egion of the mo derate rap idi ty intervals i n to our consi der ation and, f rom the o th erh and,the resu mmatio n effec ts are qu it e prono u nced at t he mod erate rapidity re g io n . We presentalso i nF ig s . 2, 3 e stimations f or NL O BFKL e f fe c t s us i ng the result s of Ref. [ @ Cor 95], w he rec onform al NL Oc ont ributions t o th e Lipatov ’s eig e nvalues were ca lculat ed. Th e es t im at ion si nco r po rat e th e NLO co nf or mal c orre c t i o ns t o t he L ipato v’s eigenvalu es(see Fig . 4)and t he N LO CTEQ 3M str uctur efunctions [@Lai 94]. We shou ldno tehe re th a t theext rac tion of data o n hi gh - $ k _{ \perp}$ jets fromth e ev ent samp les in or de r to comp ar e t hemw i th th e BF K LPomeronpredic t io ns should b e diff er ent fr om th e alg orithm s direct ed to a comparison w i th perturbati v eQ C Dp redi cti ons for the har d pro cess e s. Th e se al gorit hm s ,m otivated by the str on g $k_{ \perp }$-ordering o f the hard Q C D regime , em p lo y hardest-$k_{\ perp} $ jet sele c tion (se e, e. g., Ref. [@Alg94] ) . It is d oub tfu l t hat o ne can reconcil e thes ealgorit hms with t hewea k $ k_{ \p erp}$-dif fusion a nd t he s tro ng ra p idity or de rin gofthe s e mi-har d QCD reg im e, des cribedb yt h e BF KL r esum mat io n. We als o no te that our pred ict i onssh ou ld notbe compared w it h the prel im ina ry dat a [@Heu94] extracted by the mostf orward/ bac kward jet selectio n c riteri on. Obviou sly, o ne sh ou ldi n clude f or ta gg ing all th e reg ister ed pai rs of j ets (not only them ost forward–back war d pa i r )toc om p are w i tho u r predictions.In particu la r ,to make ac omp ar ison wi th Figs . 2,3 , one sh ould sumup all th eregi s t ere d $x$-symm etric di jets ($x_ 1 =x_2$ ) w ith t ran sverse m ome nta h ardert han $k_{ \perpmi n}$. We th an k E.A.Ku raev and L.N.Lipatov fo r stim ulati ngdiscussio ns. Weare grate fulto A.J.Som mer er, J.P. Var y , and B.- L .Y oun g forthei r kind hos p it ali t y a t the IITAP , A mes , Iow a a n d supp ort. V.T.K. is indebt e d to S.Ahn, C. L.Ki m , T. Lee , A.P et ridis, J.Qiu,C.R .S c h midt, S. I. T

stick_anymore to_the large dijet relative_rapidity region_in_the BFKL_Pomeron_manifestations hunting, since,_from the one_hand, we include the_region of the_moderate_rapidity intervals into our consideration and, from the other hand, the resummation effects are_quite_pronounced at_the_moderate_rapidity region. We present also in_Figs. 2,3 estimations for NLO_BFKL effects_using the results of Ref. [@Cor95], where conformal_NLO_contributions to the_Lipatov’s eigenvalues were calculated. The estimations incorporate the NLO_conformal corrections to the Lipatov’s eigenvalues_(see Fig. 4)_and_the_NLO CTEQ3M structure functions [@Lai94]. We_should note here that the extraction_of data on high-$k_{\perp}$ jets from_the event samples in order to compare_them with the BFKL Pomeron predictions_should be different from the_algorithms directed_to a comparison with perturbative_QCD predictions for_the hard_processes. These algorithms,_motivated by the strong $k_{\perp}$-ordering of_the hard QCD_regime, employ hardest-$k_{\perp}$ jet selection (see,_e.g.,_Ref. [@Alg94]). It_is_doubtful_that one_can reconcile these_algorithms_with the_weak_$k_{\perp}$-diffusion and the strong rapidity ordering_of_the semi-hard QCD regime, described by the_BFKL resummation. We also_note_that our predictions should_not be compared with the_preliminary data [@Heu94] extracted by the_most forward/backward_jet selection_criterion. Obviously, one should include for tagging all the registered pairs_of jets (not only the most_forward–backward pair) to compare_with our_predictions._In particular, to_make_a comparison_with Figs. 2,3, one should sum up_all the_registered $x$-symmetric dijets ($x_1=x_2$) with transverse_momenta harder than $k_{\perp_min}$. We_thank E.A.Kuraev and L.N.Lipatov for stimulating_discussions. We are grateful to A.J.Sommerer,_J.P.Vary, and B.-L.Young for their_kind_hospitality_at the IITAP, Ames, Iowa_and support. V.T.K. is indebted to_S.Ahn, C.L.Kim, T.Lee,_A.Petridis, J.Qiu, C.R.Schmidt, S.I.T

hat{\Psi}_{\ell}a(x_{i}).$ Chernozhukov, Newey, and Robins (2018) introduce machine learning methods for choosing the functions to include in the vector $A(x)$. This method can be combined with machine learning methods for estimating $E[q_{i}|x_{i}]$ to construct a double machine learning estimator of average surplus, as shown in Chernozhukov, Hausman, and Newey (2018). In parametric models moment functions like those in equation (\[lr series\]) are used to “partial out” nuisance parameters $\zeta.$ For maximum likelihood these moment functions are the basis of Neyman’s (1959) C-alpha test. Wooldridge (1991) generalized such moment conditions to nonlinear least squares and Lee (2005), Bera et al. (2010), and Chernozhukov et al. (2015) to GMM. What is novel here is their use in the construction of semiparametric estimators and the interpretation of the estimated LR moment functions $\psi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$ as the sum of an original moment function $m(z_{i},\beta,\hat{\zeta}_{\ell})$ and an influence adjustment $\phi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$. Estimating the Influence Adjustment with First Step Smoothing ------------------------------------------------------------- The adjustment term can be estimated in a general way that allows for kernel density, locally linear regression, and other kernel smoothing estimators for the first step. The idea is to differentiate with respect to the effect of the $i^{th}$ observation on sample moments. Newey (1994b) used a special case of this approach to estimate the asymptotic variance of a functional of a kernel based semiparametric or nonparametric estimator. Here we extend this method to a wider class of first step estimators, such as locally linear regression, and apply it to estimate the adjustment term for construction of LR moments. We will describe this estimator for the case where $\gamma$ is a vector of functions of a vector of variables $x.$ Let $h(z,x,\gamma)$ be a vector of functions of a data observation $z$, $x$, and a possible realized value of $\gamma$ (i.e. a vector of real numbers $\gamma$). Also let $\hat{h}_{\ell }(x,\

hat{\Psi}_{\ell}a(x_{i}).$ Chernozhukov, Newey, and Robins (2018) introduce machine learning methods for choose the function to include in the vector $ A(x)$. This method can be combined with car learning methods for estimating $ E[q_{i}|x_{i}]$ to manufacture a double machine learn estimator of average excess, as shown in Chernozhukov, Hausman, and Newey (2018). In parametric models moment functions like those in equality (\[lr series\ ]) are used to “ partial out ” nuisance argument $ \zeta.$ For maximum likelihood these moment function are the basis of Neyman ’s (1959) C - alpha test. Wooldridge (1991) generalized such consequence conditions to nonlinear least squares and Lee (2005), Bera et al. (2010), and Chernozhukov et al. (2015) to GMM. What is novel here is their use in the structure of semiparametric estimators and the rendition of the estimated LR moment functions $ \psi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$ as the sum of an original consequence function $ m(z_{i},\beta,\hat{\zeta}_{\ell})$ and an influence adjustment $ \phi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$. Estimating the Influence Adjustment with First Step Smoothing ------------------------------------------------------------- The adjustment term can be estimate in a general way that allows for kernel density, locally linear regression, and other kernel smoothing estimators for the first measure. The idea is to differentiate with obedience to the impression of the $ i^{th}$ observation on sample moment. Newey (1994b) use a special case of this overture to estimate the asymptotic variance of a functional of a kernel based semiparametric or nonparametric estimator. Here we extend this method to a wider class of first step estimators, such as locally analogue regression, and give it to calculate the allowance term for construction of LR moments. We will identify this estimator for the case where $ \gamma$ is a vector of functions of a vector of variable star $ x.$ Let $ h(z, x,\gamma)$ be a vector of functions of a data notice $ z$, $ x$, and a possible realized value of $ \gamma$ (i.e. a vector of real numbers $ \gamma$). Also permit $ \hat{h}_{\ell } (x,\

hat{\Osi}_{\ell}a(x_{i}).$ Chernozhukov, Nedey, and Robins (2018) introdnce macgine leafning methods for choosing tie fynctiins to include in the xector $A(x)$. This merhod xan be comujned wibk macglne lzacning methods fpr estimathng $E[q_{i}|x_{i}]$ to cmnrtxuct a double machine learning estimwtor of agerage surplus, as xrown pn Chernozhukov, Hausman, and Newsy (2018). In perametric modelx moment functions like thlse ln equation (\[lr serles\]) are usee to “partial out” vuisance parameters $\zeja.$ For maximum likelihood these oomenc functions aee hve basis of Neymwn’s (1959) C-alpha bvst. Woondridge (1991) generalized xuci monent conditions to noilinear least squaref and Lee (2005), Yera et al. (2010), and Chernizyukov et dl. (2015) gi GOM. Shet js novfl iere is thejr use in tye construction of xeiplarametric eatimatjrf and the interpretation of the estimattd LR moment functions $\psi(z_{i},\veta,\hat{\zeta}_{\ell},\hat{\Psi}_{\epl})$ as the sum of an original moment function $m(z_{i},\beta,\hat{\zetd}_{\ell})$ ena au infljwnfe adjustment $\phi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$. Estyjauinn the Influence Cdjustment with Gigsy Step Smoothivg ------------------------------------------------------------- The csjhstment term can bf estimwted un a genewal eay that allows for kernel eensity, locajoy linear regressiln, and othex kerngl smopthing estimators for tke firat step. The idea is fu differentiate ditm raspect to the effect of thq $i^{th}$ obsxrvatnon on sxmplg momenes. Newey (1994b) used a special case of hhis cpprodch to estlmate the asymptotic variance oh a functionak mf d kernel basec semiparametwic or nonparaketric zstimagor. Here wv extend vhis method eo a wider cldds of first vtep estymatirs, wuch as uocally linear regression, and appoy it to estimate bhe aanustment term fir xonstruction of LR moiejtx. Wq will descrite tfis rstimxtor for thc cxse ehere $\gamma$ is a vecdor kf functions of a fegtor of vqriables $x.$ Let $h(z,x,\gamka)$ be a vector of vunctmons oh a daya jbservation $z$, $x$, and a possible realized vajue of $\gamma$ (i.e. q vector of xeal numbers $\gamma$). Also let $\hat{h}_{\ell }(x,\

hat{\Psi}_{\ell}a(x_{i}).$ Chernozhukov, Newey, and Robins (2018) introduce methods choosing the to include in can combined with machine methods for estimating to construct a double machine learning of average surplus, as shown in Chernozhukov, Hausman, and Newey (2018). In parametric moment functions like those in equation (\[lr series\]) are used to “partial out” parameters For likelihood moment functions are the basis of Neyman’s (1959) C-alpha test. Wooldridge (1991) generalized such moment conditions nonlinear least squares and Lee (2005), Bera et (2010), and Chernozhukov et (2015) to GMM. What is here their use the of estimators and the of the estimated LR moment functions $\psi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$ as the sum of an original moment function $m(z_{i},\beta,\hat{\zeta}_{\ell})$ and influence adjustment the Influence with Step ------------------------------------------------------------- The adjustment be estimated in a general way kernel density, locally linear regression, and other kernel estimators for first step. The idea is to with respect to the effect of the $i^{th}$ on sample moments. Newey (1994b) used a special case of this approach to estimate the of a functional of kernel based semiparametric nonparametric Here extend method to wider class of first step estimators, such as locally linear regression, apply it to estimate the adjustment term for construction of We describe this estimator the case where $\gamma$ a of functions of a variables Let vector functions a data observation $z$, and a possible realized value $\gamma$ (i.e. a vector let $\hat{h}_{\ell }(x,\

hat{\Psi}_{\ell}a(x_{i}).$ Chernozhukov, NEwey, and RobIns (2018) inTroDucE mAchiNe leArning methods fOR choOsing the functions to incLude iN tHE vecTOr $a(x)$. ThiS method CAn BE ComBiNeD wiTh MAcHine lEarNing metHods for estImaTiNg $E[q_{i}|x_{i}]$ to conSTrUct a double MacHine learning EstImator Of AveRAge suRplUs, as sHown in cHernozHukov, HausMaN, And NewEY (2018). In paraMETrIc moDels moment functioNS lIKe those in equatIon (\[lr sErIEs\]) ARE usEd tO “partial ouT” nUisanCE parameTErS $\ZETa.$ FOR maximum likelIhood these mOMenT functIoNs aRE the baSis of neYMan’S (1959) C-alpha test. woolDridge (1991) genEralizED such moMEnt condItions To nOnlIneaR LeAsT sqUaREs aND LEe (2005), BERa eT al. (2010), and ChErNoZhukoV et aL. (2015) TO gmM. WhAt iS novEl herE is their use in The ConsTRucTion oF semiParaMeTric eStimatOrs anD tHe interpretatioN of tHe estimatEd Lr mOmeNt FunctIOns $\psi(Z_{i},\bEta,\Hat{\zeta}_{\Ell},\hat{\PSI}_{\elL})$ aS THE sUm of an original momeNt FUNcTion $m(z_{i},\bEta,\hat{\ZEtA}_{\eLL})$ and an inFlUenCe adJUStmenT $\phi(Z_{I},\bEta,\hat{\zeTa}_{\ell},\hAT{\PSi}_{\Ell})$. EstiMaTing thE INflUenCe AdjUStmeNt with first SteP SmooTHing ------------------------------------------------------------- The adjustmENt term can be esTImATEd IN a geNerAl way that alLows FOr keRnel DEnSitY, LocalLy linEaR ReGRession, and other kernEl SmoothIng esTimators for thE first step. tHE Idea is to DiffEReNTiate with respeCt to tHe effect of THe $i^{th}$ obsErvatIon on samPle momentS. nEwey (1994b) useD a sPecIal CasE OF tHis approach to ESTimaTe The asymPtoTic variAncE of A fuNctIoNal of a kerNel based SeMiPaRaMetRic or NOnparameTrIc eStImaTor. HeRE we extEnd thIs meThOd TO a wIder claSS oF FIrst StEp EstiMatOrS, such As loCAllY linear RegressioN, anD ApplY iT tO estimaTe the adjustmeNt Term for conStRucTion of lr Moments. WE will describe this estimaTOr for thE caSe wheRe $\gaMma$ is a vecTor Of funcTioNS of a veCtor of VariaBlEs $x.$ lET $h(z,x,\gAMMa)$ Be a VeCtor of funcTIOns Of a daTa ObseRvation $Z$, $x$, and a possible realIZed Value of $\gamma$ (i.E. a vEctoR OF rEal NUmBErs $\GaMMa$). ALSO let $\hat{h}_{\ell }(x,\

hat{\Psi}_{\ell}a(x_{i}).$ Chernozh ukov, Ne wey ,andRobi ns (2018) intr o duce machine learning meth ods f or choo s in g the functi o ns t o i nc lu dein th e vec tor $A(x)$ . This met hod c an be combin e dwith machi nelearning met hod s fores tim a ting$E[ q_{i} |x_{i} ] $ to c onstructad oublem achinel e ar ning estimator of ave r ag e surplus, as s hown i nC he r n ozh uko v, Hausman ,and N e wey (20 1 8) . Inp arametric mod els momentf unc tionsli ket hose i n equ at i on(\[lr serie s\]) are used to “p a rtial o u t” nuis ance p ara met ers$ \z et a.$ F o r m a xi mum lik elihoodth es e mom entf u n c tion s a re t he ba sis of Neyman ’s(195 9 ) C -alph a tes t. W oo ldrid ge (19 91) g en eralized such m omen t conditi ons t o n on linea r least sq uar es andLee (20 0 5), B e r a e t al. (2010), andCh e r no zhukov e t al.( 20 15 ) to GMM. W hat isn o vel h erei stheir us e in t h eco nstruct io n of s em ipa ram etric esti mators and the inte r pretation of t h e estimated L R m o m en t fun cti ons $\psi(z _{i} , \bet a,\h a t{ \ze t a}_{\ ell}, \h a t{ \ Psi}_{\ell})$ as th esum of an o riginal momen t function $ m (z_{i},\ beta , \h a t{\zeta}_{\ell })$ a nd an infl u ence adj ustme nt $\phi (z_{i},\b e t a,\hat{\ zet a}_ {\e ll} , \ ha t{\Psi}_{\ell } ) $. Es timatin g t he Infl uen ceAdj ust me nt with F irst Ste pSm oo th ing ---- - -------- -- --- -- --- ----- - ------ ----- ---- -- -- - --- ------- - -T he a dj us tmen t t er m can bee sti mated i n a gener alw ay t ha tallowsfor kernel de ns ity, local ly li near r e g ression, and other kernel smoot h ing est ima torsforthe first st ep. Th e i d ea isto dif feren ti ate w ith r e s pe ctto the effec t ofthe $ i^ {th} $ obser vation on sample m o men ts. Newey (19 94b ) us e d a sp e ci a l c as e of t his approach to estimateth e a symptoticv ari an ce of a functi onalo f a ker nel based semipara me tric o r n onparametr ic estim ator. Her e we e x te nd th ismethod t o a wide r clas s of firs t step e stimat ors,su ch as lo cally linear regression , andapply it to estim ate the adjustme nt t erm for co nst ruc tionofL R mom ents . Wew ill d escr i be this e s ti mat o r f or the case w h ere $\ga mma $ is avect or of functions o f a vector of v aria b l es$x. $ Let $ h(z,x,\gamma)$ be a v ector of f unctions of a dataob s ervat ion $z $, $x$ , and a p os s ible r eali zed value of $\ ga m ma$ (i. e. a vector ofre al num bers $ \ gamm a $ ). Also let $\ha t{h}_ { \ ell } ( x,\

hat{\Psi}_{\ell}a(x_{i}).$ Chernozhukov, Newey,_and Robins_(2018) introduce machine learning_methods for_choosing_the functions_to_include in the_vector $A(x)$. This_method can be combined_with machine learning_methods_for estimating $E[q_{i}|x_{i}]$ to construct a double machine learning estimator of average surplus, as_shown_in Chernozhukov,_Hausman,_and_Newey (2018). In parametric models moment_functions like those in equation_(\[lr series\])_are used to “partial out” nuisance parameters $\zeta.$_For_maximum likelihood these_moment functions are the basis of Neyman’s (1959) C-alpha_test. Wooldridge (1991) generalized such moment_conditions to nonlinear_least_squares_and Lee (2005), Bera_et al. (2010), and Chernozhukov et_al. (2015) to GMM. What is_novel here is their use in the_construction of semiparametric estimators and the_interpretation of the estimated LR_moment functions_$\psi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$ as the sum of_an original moment_function $m(z_{i},\beta,\hat{\zeta}_{\ell})$_and an influence_adjustment $\phi(z_{i},\beta,\hat{\zeta}_{\ell},\hat{\Psi}_{\ell})$. Estimating the Influence Adjustment with_First Step Smoothing ------------------------------------------------------------- The_adjustment term can be estimated in_a_general way that_allows_for_kernel density,_locally linear regression,_and_other kernel_smoothing_estimators for the first step. The_idea_is to differentiate with respect to the_effect of the $i^{th}$_observation_on sample moments. Newey_(1994b) used a special case_of this approach to estimate the_asymptotic variance_of a_functional of a kernel based semiparametric or nonparametric estimator. Here we_extend this method to a wider_class of first step_estimators, such_as_locally linear regression,_and_apply it_to estimate the adjustment term for construction_of LR_moments. We will describe this estimator for_the case where $\gamma$_is_a vector of functions of a_vector of variables $x.$ Let $h(z,x,\gamma)$_be a vector of functions_of_a_data observation $z$, $x$, and_a possible realized value of $\gamma$_(i.e. a vector_of real numbers $\gamma$). Also let $\hat{h}_{\ell }(x,\

'}$ $D^{(\mathrm{e}) \pm}_{m m'} := 0, D^{(\mathrm{h}) \pm}_{m m'} := 0$ Input $| \Psi^N_{\mathrm{gs}} \rangle$ to $\mathcal{C}_{m m'}$ and measure the ancillae $| q_1^{\mathrm{A}} \rangle \otimes | q_0^{\mathrm{A}} \rangle :=$ observed ancillary state $E :=$ QPE$(| \widetilde{\Psi} \rangle, \mathcal{H})$ Find $E$ among $\{ E_\lambda^{N - 1} \}_\lambda$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{h}) +}_{\lambda m m'} += 1$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{h}) -}_{\lambda m m'} += 1$ Find $E$ among $\{ E_\lambda^{N + 1} \}_\lambda$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{e}) +}_{\lambda m m'} += 1$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{e}) -}_{\lambda m m'} += 1$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{e}) \pm}_{m m'} *= 1/N_{\mathrm{meas}}, D^{(\mathrm{h}) \pm}_{m m'} *= 1/N_{\mathrm{meas}} $ $D^{(\mathrm{e}) \pm}_{m m'}, D^{(\mathrm{h}) \pm}_{m m'}$ --- abstract: 'Let $\Pi$ and $\Gamma$ be homogeneous Poisson point processes on a fixed set of finite

' } $ $ D^{(\mathrm{e }) \pm}_{m m' }: = 0, D^{(\mathrm{h }) \pm}_{m m' }: = 0 $ Input $ | \Psi^N_{\mathrm{gs } } \rangle$ to $ \mathcal{C}_{m m'}$ and measure the ancillae $ | q_1^{\mathrm{A } } \rangle \otimes | q_0^{\mathrm{A } } \rangle: = $ observed ancillary state $ east: = $ QPE$(| \widetilde{\Psi } \rangle, \mathcal{H})$ receive $ E$ among $ \ { E_\lambda^{N - 1 } \}_\lambda$ $ { \mathcode`+=\numexpr\mathcode`+ + " 1000\relax \mathcode`*=\numexpr\mathcode '* + " 1000\relax } D^{(\mathrm{h }) + } _ { \lambda m m' } + = 1 $ $ { \mathcode`+=\numexpr\mathcode`+ + " 1000\relax \mathcode`*=\numexpr\mathcode '* + " 1000\relax } D^{(\mathrm{h }) -}_{\lambda m m' } + = 1 $ Find $ E$ among $ \ { E_\lambda^{N + 1 } \}_\lambda$ $ { \mathcode`+=\numexpr\mathcode`+ + " 1000\relax \mathcode`*=\numexpr\mathcode '* + " 1000\relax } D^{(\mathrm{e }) + } _ { \lambda m m' } + = 1 $ $ { \mathcode`+=\numexpr\mathcode`+ + " 1000\relax \mathcode`*=\numexpr\mathcode '* + " 1000\relax } D^{(\mathrm{e }) -}_{\lambda m m' } + = 1 $ $ { \mathcode`+=\numexpr\mathcode`+ + " 1000\relax \mathcode`*=\numexpr\mathcode '* + " 1000\relax } D^{(\mathrm{e }) \pm}_{m m' } * = 1 / N_{\mathrm{meas } }, D^{(\mathrm{h }) \pm}_{m m' } * = 1 / N_{\mathrm{meas } } $ $ D^{(\mathrm{e }) \pm}_{m m' }, D^{(\mathrm{h }) \pm}_{m m'}$ --- abstract:' lease $ \Pi$ and $ \Gamma$ be homogeneous Poisson degree processes on a fixed stage set of finite

'}$ $D^{(\mahhrm{e}) \pm}_{m m'} := 0, D^{(\mathrm{h}) \po}_{m m'} := 0$ Input $| \Psi^N_{\mathcm{gs}} \rahgle$ to $\oathcal{C}_{m m'}$ and measure the enciolae $| q_1^{\mathrm{A}} \rangle \otimer | q_0^{\mathrl{A}} \ranglw :=$ ouserved ancillarb state $C :=$ QPS$(| \widztmlde{\Psi} \rangle, \kathcal{H})$ Fhnd $E$ among $\{ E_\naobba^{N - 1} \}_\lambda$ ${ \mathcode`+=\numexpr\mathcoqe`+ + "1000\relsx \mathcode`*=\numgxpr\msehcosv`* + "1000\relax }D^{(\mathrm{h}) +}_{\lambda m m'} += 1$ ${ \mathcove`+=\numexpr\mathcoce`+ + "1000\relax \mathcode`*=\numexpg\matjcode`* + "1000\relax }D^{(\mathrl{h}) -}_{\lambda m m'} += 1$ Dind $E$ among $\{ E_\lambda^{N + 1} \}_\lambda$ ${ \jathcode`+=\numexpr\mathcode`+ + "1000\relax \matkcode`*=\numexpt\mztjwode`* + "1000\relax }V^{(\mathri{e}) +}_{\lambda m m'} += 1$ ${ \madhcode`+=\nimexpr\mathcode`+ + "1000\rxlax \mathcode`*=\numexpr\maticode`* + "1000\relax }D^{(\mathrm{e}) -}_{\lambda m m'} += 1$ ${ \mathcode`+=\numezpe\mathwode`+ + "1000\reuqx \mzticose`*=\numedpr\jathcode`* + "1000\delax }D^{(\mathrn{e}) \pm}_{m m'} *= 1/N_{\mathrm{mess}}, D^{(\matrri{h}) \pm}_{m m'} *= 1/N_{\mathrm{meas}} $ $D^{(\katgrm{e}) \pm}_{m m'}, D^{(\mathrm{h}) \pm}_{m m'}$ --- abstract: 'Let $\Pi$ anf $\Gamma$ bq homogeneous Poisson point processes on a fixed vet oh winnbc

'}$ $D^{(\mathrm{e}) \pm}_{m m'} := 0, D^{(\mathrm{h}) := Input $| \rangle$ to $\mathcal{C}_{m $| \rangle \otimes | \rangle :=$ observed state $E :=$ QPE$(| \widetilde{\Psi} \rangle, Find $E$ among $\{ E_\lambda^{N - 1} \}_\lambda$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`* "1000\relax }D^{(\mathrm{h}) +}_{\lambda m m'} += 1$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`* + }D^{(\mathrm{h}) m += Find $E$ among $\{ E_\lambda^{N + 1} \}_\lambda$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{e}) m m'} += 1$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax + "1000\relax }D^{(\mathrm{e}) -}_{\lambda m'} += 1$ ${ \mathcode`+=\numexpr\mathcode`+ "1000\relax + "1000\relax \pm}_{m *= D^{(\mathrm{h}) \pm}_{m m'} 1/N_{\mathrm{meas}} $ $D^{(\mathrm{e}) \pm}_{m m'}, D^{(\mathrm{h}) \pm}_{m m'}$ --- abstract: 'Let $\Pi$ and $\Gamma$ be homogeneous Poisson processes on set of

'}$ $D^{(\mathrm{e}) \pm}_{m m'} := 0, D^{(\mathrm{h}) \pm}_{m m'} := 0$ INput $| \Psi^N_{\maThrm{gS}} \raNglE$ tO $\matHcal{c}_{m m'}$ and measure tHE ancIllae $| q_1^{\mathrm{A}} \rangle \otiMes | q_0^{\mAtHRm{A}} \rANgLe :=$ obsErved anCIlLARy sTaTe $e :=$ QPe$(| \wIDeTilde{\psi} \Rangle, \mAthcal{H})$ FinD $E$ aMoNg $\{ E_\lambda^{N - 1} \}_\laMBdA$ ${ \mathcode`+=\nUmeXpr\mathcode`+ + "1000\rElaX \mathcOdE`*=\nuMExpr\mAthCode`* + "1000\rElax }D^{(\mAThrm{h}) +}_{\lAmbda m m'} += 1$ ${ \maThCOde`+=\numEXpr\mathCODe`+ + "1000\RelaX \mathcode`*=\numexpr\mAThCOde`* + "1000\relax }D^{(\mathrM{h}) -}_{\lambDa M M'} += 1$ FIND $E$ aMonG $\{ E_\lambda^{N + 1} \}_\lAmBda$ ${ \maTHcode`+=\nuMExPR\MAthCOde`+ + "1000\relax \mathcOde`*=\numexpr\mAThcOde`* + "1000\relAx }d^{(\maTHrm{e}) +}_{\laMbda m M'} += 1$ ${ \mAThcOde`+=\numexpr\mAthcOde`+ + "1000\relax \mAthcodE`*=\Numexpr\MAthcode`* + "1000\Relax }D^{(\MatHrm{E}) -}_{\lamBDa M m'} += 1$ ${ \MatHcODe`+=\nUMeXpr\MAthCode`+ + "1000\relaX \mAtHcode`*=\NumeXPR\MAthcOde`* + "1000\RelaX }D^{(\matHrm{e}) \pm}_{m m'} *= 1/N_{\mathRm{mEas}}, D^{(\MAthRm{h}) \pm}_{M m'} *= 1/N_{\maThrm{MeAs}} $ $D^{(\maThrm{e}) \pM}_{m m'}, D^{(\mAtHrm{h}) \pm}_{m m'}$ --- abstracT: 'Let $\pi$ and $\GammA$ be HoMogEnEous POIsson pOinT prOcesses On a fixeD Set Of FINItE

'}$ $D^{(\mathrm{e}) \pm} _{m m'} := 0, D ^{( \ma th rm{h }) \ pm}_{m m'} :=0 $ In put $| \Psi^N_{\mathrm {gs}} \ r angl e $to $\ mathcal { C} _ { m m '} $and m e as ure t heancilla e $| q_1^{ \ma th rm{A}} \rang l e\otimes |q_0 ^{\mathrm{A} } \ rangle : =$o bserv edancil lary s t ate $E :=$ QPE$ (| \widet i lde{\Ps i } \ rang le, \mathcal{H})$ Fi n d $E$ among $\ { E_\l am b da ^ { N - 1} \}_\lambd a$ ${ \ mathcod e `+ = \ n ume x pr\mathcode`+ + "1000\re l ax \mat hc ode ` *=\num expr\ ma t hco de`* + "100 0\re lax }D^{( \mathr m {h}) +} _ {\lambd a m m' } + = 1 $ ${ \ ma thc od e `+= \ nu mex p r\m athcode` ++"1000 \rel a x \ma thc ode` *=\nu mexpr\mathcod e`* + " 1 000 \rela x }D^ {(\m at hrm{h }) -}_ {\lam bd a m m'} += 1$ F ind$E$ among $\ {E_\ la mbda^ { N + 1} \} _\l ambda$${ \ma t hco de ` + = \n umexpr\mathcode`++" 1 00 0\relax \mat h co de ` *=\numex pr \ma thco d e `* +"100 0 \r elax }D^ {(\mat h rm {e }) +}_{ \l ambdamm'} += 1$ $ { \m athcod e`+=\num expr\ m athcode`+ + "1 0 00\relax \m a th c o de ` *=\n ume xpr\mathcod e`*+ "10 00\r e la x } D ^{(\m athrm {e } )- }_{\lambda m m'} += 1 $ ${ \math code`+=\numex pr\mathcod e ` + + "1000 \rel a x \mathcode`*=\ numex pr\mathcod e `* + "10 00\re lax }D^{ (\mathrm{ e } ) \pm}_{ m m '}*=1/N _ { \m athrm{meas}}, D ^{(\mat hrm {h} ) \ pm} _{ m m'} *=1/N_{\ma th rm {m ea s}} $$D ^{( \math r m{e})\pm}_ {m m '} ,D ^{( \mathrm { h} ) \pm} _{ mm'}$ - -- abst ract : 'L et $\Pi $ and $\G amm a $ be h om ogeneou s Poisson poi nt processes o n a fixed s et of fi nite

'}$ $D^{(\mathrm{e}) \pm}_{m_m'} :=_0, D^{(\mathrm{h}) \pm}_{m m'}_:= 0$_Input_$| \Psi^N_{\mathrm{gs}}_\rangle$_to $\mathcal{C}_{m m'}$_and measure the_ancillae $| q_1^{\mathrm{A}} \rangle_\otimes | q_0^{\mathrm{A}}_\rangle_:=$ observed ancillary state $E :=$ QPE$(| \widetilde{\Psi} \rangle, \mathcal{H})$ Find $E$ among $\{_E_\lambda^{N_- 1}_\}_\lambda$_${_ \mathcode`+=\numexpr\mathcode`+ + "1000\relax _ \mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{h}) +}_{\lambda_m m'}_+= 1$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax __\mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{h})_-}_{\lambda m m'} += 1$ Find $E$ among $\{_E_\lambda^{N + 1} \}_\lambda$ ${ _\mathcode`+=\numexpr\mathcode`+ + "1000\relax___\mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{e}) +}_{\lambda_m m'} += 1$ ${ _\mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`*_+ "1000\relax }D^{(\mathrm{e}) -}_{\lambda m m'} += 1$_${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax _ \mathcode`*=\numexpr\mathcode`* + "1000\relax }D^{(\mathrm{e}) \pm}_{m_m'} *=_1/N_{\mathrm{meas}}, _ _ _ _ D^{(\mathrm{h}) \pm}_{m m'}_*= 1/N_{\mathrm{meas}} _ __ ___ _$ $D^{(\mathrm{e}) \pm}_{m_m'},_D^{(\mathrm{h}) \pm}_{m_m'}$ _--- abstract: 'Let $\Pi$ and $\Gamma$ be_homogeneous_Poisson point processes on a fixed set_of finite

on adjacency matrices in the natural way: If $A$ is the adjacency matrix of a graph $G$, then $\pi(A)$ is the adjacency matrix of $\pi(G)$ and $\pi(A)$ is obtained by simultaneously permuting with $\pi$ both rows and columns of $A$. \[def:weak\_iso\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be $k$-color graph colorings with $G=([n],E_1)$ and $H=([n],E_2)$. We say that $(G,{\kappa_1})$ and $(H,{\kappa_2})$ are weakly isomorphic, denoted $(G,{\kappa_1})\approx(H,{\kappa_2})$ if there exist permutations $\pi \colon [n] \to [n]$ and $\sigma \colon [k] \to [k]$ such that $(u,v) \in E_1 \iff (\pi(u),\pi(v)) \in E_2$ and $\kappa_1((u,v)) = \sigma(\kappa_2((\pi(u), \pi(v))))$. We denote such a weak isomorphism: $(G,{\kappa_1})\approx_{\pi,\sigma}(H,{\kappa_2})$. When $\sigma$ is the identity permutation, we say that $(G,{\kappa_1})$ and $(H,{\kappa_2})$ are isomorphic. The following lemma emphasizes the importance of weak graph isomorphism as it relates to Ramsey numbers. Many classic coloring problems exhibit the same property. \[lemma:closed\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be graph colorings in $k$ colors such that $(G,\kappa_1) \approx_{\pi,\sigma} (H,\kappa_2)$. Then, $$(G,\kappa_1) \in {{\cal R}}(r_1,r_2,\ldots,r_k;n) \iff (H,\kappa_2) \in {{\cal R}}(\sigma(r_1),\sigma(r_2),\ldots,\sigma(r_k);n)$$ We make use of the following theorem from [@PR98]. \[thm:433\] $30\leq R(4,3,3)\leq 31$ and, $R(4,3,3)=31$ if and only if there

on adjacency matrices in the natural way: If $ A$ is the adjacency matrix of a graph $ G$, then $ \pi(A)$ is the adjacency matrix of $ \pi(G)$ and $ \pi(A)$ is obtained by simultaneously permute with $ \pi$ both row and columns of $ A$. \[def: weak\_iso\ ] Let $ (G,{\kappa_1})$ and $ (H,{\kappa_2})$ be $ k$-color graph coloring with $ G=([n],E_1)$ and $ H=([n],E_2)$. We allege that $ (G,{\kappa_1})$ and $ (H,{\kappa_2})$ are weakly isomorphic, denote $ (G,{\kappa_1})\approx(H,{\kappa_2})$ if there exist permutations $ \pi \colon [ n ] \to [ n]$ and $ \sigma \colon [ k ] \to [ k]$ such that $ (u, volt) \in E_1 \iff (\pi(u),\pi(v) ) \in E_2 $ and $ \kappa_1((u, v) ) = \sigma(\kappa_2((\pi(u), \pi(v))))$. We denote such a weak isomorphism: $ (G,{\kappa_1})\approx_{\pi,\sigma}(H,{\kappa_2})$. When $ \sigma$ is the identity substitution, we say that $ (G,{\kappa_1})$ and $ (H,{\kappa_2})$ are isomorphic. The following lemma emphasizes the importance of decrepit graph isomorphism as it relates to Ramsey number. Many classic coloring problem exhibit the same property. \[lemma: closed\ ] lease $ (G,{\kappa_1})$ and $ (H,{\kappa_2})$ be graph colorings in $ k$ colors such that $ (G,\kappa_1) \approx_{\pi,\sigma } (H,\kappa_2)$. Then, $ $ (G,\kappa_1) \in { { \cal R}}(r_1,r_2,\ldots, r_k;n) \iff (H,\kappa_2) \in { { \cal R}}(\sigma(r_1),\sigma(r_2),\ldots,\sigma(r_k);n)$$ We make function of the following theorem from   [ @PR98 ]. \[thm:433\ ] $ 30\leq R(4,3,3)\leq 31 $ and, $ R(4,3,3)=31 $ if and only if there

on adjacency matrices in tme natural way: Iy $A$ is vhe adjzcency mxtrix of a graph $G$, then $\pi(A)$ ms tye adhacency matrix of $\pi(G)$ xnd $\pi(A)$ id obtainwd bb simultaneously permutiky witg $\pi$ yovh rows and colomns of $A$. \[def:feak\_iso\] Let $(G,{\kdpoa_1})$ and $(H,{\kappa_2})$ be $k$-color graph colorinds with $G=([j],E_1)$ and $H=([n],E_2)$. We fay uhae $(G,{\kzipc_1})$ and $(H,{\kappa_2})$ are weakly isomorpgic, denmted $(G,{\kappa_1})\aplrox(H,{\kappa_2})$ if there exist oermktations $\pi \colon [j] \to [n]$ and $\wigmw \colon [k] \to [y]$ such thau $(b,v) \in E_1 \iff (\pi(u),\pi(v)) \in E_2$ and $\kappa_1((u,v)) = \sigma(\yappa_2((\'i(u), \pi(v))))$. We eebotf such a weaj isoiorphism: $(G,{\kaiia_1})\approf_{\pi,\sigms}(H,{\kappa_2})$. When $\slgma$ ms tye identity permutatimn, we say that $(G,{\kakpa_1})$ and $(H,{\kdp'a_2})$ are isomorphic. The doolowitg lamma wmpfasjzxs fhe imoorvance of wezk graph isimorphism as it relstqw to Ramsey nhmbers. Mwny classic coloring problems exhibit tve aame property. \[lemma:closee\] Let $(G,{\kappa_1})$ and $(H,{\kapka_2})$ be grapr colorings in $k$ colors such that $(G,\kappa_1) \approx_{\pi,\vigma} (H,\kciia_2)$. Tfwn, $$(G,\kappa_1) \in {{\cal R}}(r_1,r_2,\ldots,r_k;n) \iff (H,\kappa_2) \in {{\cal D}}(\sogka(r_1),\sigma(r_2),\ldots,\figma(r_k);n)$$ We kane ose of the foluowing thsorem from [@PR98]. \[thm:433\] $30\lee R(4,3,3)\leq 31$ and, $E(4,3,3)=31$ if and jnly if there

on adjacency matrices in the natural way: is adjacency matrix a graph $G$, matrix $\pi(G)$ and $\pi(A)$ obtained by simultaneously with $\pi$ both rows and columns $A$. \[def:weak\_iso\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be $k$-color graph colorings with $G=([n],E_1)$ and We say that $(G,{\kappa_1})$ and $(H,{\kappa_2})$ are weakly isomorphic, denoted $(G,{\kappa_1})\approx(H,{\kappa_2})$ if there permutations \colon \to and $\sigma \colon [k] \to [k]$ such that $(u,v) \in E_1 \iff (\pi(u),\pi(v)) \in E_2$ and = \sigma(\kappa_2((\pi(u), \pi(v))))$. We denote such a weak $(G,{\kappa_1})\approx_{\pi,\sigma}(H,{\kappa_2})$. When $\sigma$ is identity permutation, we say that and are isomorphic. following emphasizes importance of weak isomorphism as it relates to Ramsey numbers. Many classic coloring problems exhibit the same property. \[lemma:closed\] Let and $(H,{\kappa_2})$ colorings in colors that \approx_{\pi,\sigma} (H,\kappa_2)$. Then, {{\cal R}}(r_1,r_2,\ldots,r_k;n) \iff (H,\kappa_2) \in {{\cal use of the following theorem from [@PR98]. \[thm:433\] R(4,3,3)\leq 31$ $R(4,3,3)=31$ if and only if there

on adjacency matrices in the nAtural way: IF $A$ is tHe aDjaCeNcy mAtriX of a graph $G$, then $\PI(A)$ is The adjacency matrix of $\pi(g)$ and $\pI(A)$ IS obtAInEd by sImultanEOuSLY peRmUtIng WiTH $\pI$ both RowS and colUmns of $A$. \[def:WeaK\_iSo\] Let $(G,{\kappa_1})$ aND $(H,{\Kappa_2})$ be $k$-coLor Graph colorinGs wIth $G=([n],E_1)$ AnD $H=([n],e_2)$. we say ThaT $(G,{\kapPa_1})$ and $(H,{\KAppa_2})$ arE weakly isOmORphic, dENoted $(G,{\kAPPa_1})\ApprOx(H,{\kappa_2})$ if there exISt PErmutations $\pi \cOlon [n] \tO [n]$ ANd $\SIGma \ColOn [k] \to [k]$ such ThAt $(u,v) \iN e_1 \iff (\pi(u),\PI(v)) \IN e_2$ And $\KAppa_1((u,v)) = \sigma(\kaPpa_2((\pi(u), \pi(v))))$. We DEnoTe such A wEak ISomorpHism: $(G,{\KaPPa_1})\aPprox_{\pi,\sigmA}(H,{\kaPpa_2})$. When $\siGma$ is tHE identiTY permutAtion, wE saY thAt $(G,{\kAPpA_1})$ aNd $(H,{\KaPPa_2})$ aRE iSomORphIc. The folLoWiNg lemMa emPHASIzes The ImpoRtancE of weak graph iSomOrphISm aS it reLates To RaMsEy numBers. MaNy claSsIc coloring problEms eXhibit the SamE pRopErTy. \[lemMA:closeD\] LeT $(G,{\kAppa_1})$ and $(h,{\kappa_2})$ bE GraPh COLOrIngs in $k$ colors such tHaT $(g,\KaPpa_1) \approX_{\pi,\sigMA} (H,\KaPPa_2)$. Then, $$(G,\kApPa_1) \iN {{\cal r}}(R_1,R_2,\ldotS,r_k;n) \IFf (h,\kappa_2) \in {{\Cal R}}(\siGMa(R_1),\sIgma(r_2),\ldOtS,\sigma(R_k);N)$$ We MakE use oF The fOllowiNg theoreM from [@pr98]. \[thm:433\] $30\leq R(4,3,3)\leq 31$ and, $r(4,3,3)=31$ If and only if thERe

on adjacency matrices inthe natura l way : I f $ A$ istheadjacency matr i x of a graph $G$, then $\p i(A)$ i s the ad jacen cy matr i xo f $\ pi (G )$an d $ \pi(A )$is obta ined by si mul ta neously perm u ti ng with $\ pi$ both rows a ndcolumn sof$ A$. \[d ef:we ak\_is o \] Let $(G,{\ka pp a _1})$a nd $(H, { \ ka ppa_ 2})$ be $k$-color gr a ph colorings w ith $G =( [ n] , E _1) $ a nd $H=([n] ,E _2)$. We sayt ha t $ (G, { \kappa_1})$ a nd $(H,{\ka p pa_ 2})$ a re we a kly is omorp hi c , d enoted $(G, {\ka ppa_1})\a pprox( H ,{\kapp a _2})$ i f ther e e xis t pe r mu ta tio ns $\p i \ col o n [ n] \to [ n] $and $ \sig m a \ colo n [ k] \to [ k]$ such that $( u,v) \in E_1\iff(\pi (u ),\pi (v)) \ in E_ 2$ and $\kappa_1( (u,v )) = \sig ma( \k app a_ 2((\p i (u), \ pi( v)) ))$. We denote suc ha w ea k isomorphism: $(G ,{ \ k ap pa_1})\a pprox_ { \p i, \ sigma}(H ,{ \ka ppa_ 2 } )$. W hen$ \s igma$ is the i d en ti ty perm ut ation, w e s aythat$ (G,{ \kappa _1})$ an d $(H , {\kappa_2})$ a r e isomorphic. T h e f o llow ing lemma emph asiz e s th e im p or tan c e ofweakgr a ph isomorphism as it r el ates t o Ram sey numbers.Many class i c coloring pro b le m s exhibit thesameproperty.\[lemma: close d\] Let$(G,{\kap p a _1})$ an d $ (H, {\k app a _ 2} )$ be graph c o l orin gs in $k$ co lors su chtha t $ (G, \k appa_1) \ approx_{ \p i, \s ig ma} (H , \kappa_2 )$ . T he n,$$(G, \ kappa_ 1) \i n {{ \c al R}} (r_1,r_ 2 ,\ l d ots, r_ k; n) \ iff ( H,\ka ppa_ 2 ) \ in {{ \cal R}}( \si g ma(r _1 ), \sigma( r_2),\ldots,\ si gma(r_k);n )$ $ We mak e use of t he following theorem fr o m [@PR9 8]. \[t hm:4 33\] $30\ leq R(4,3 ,3) \ leq 31 $ and, $R(4 ,3 ,3) = 3 1$ if a nd on ly if there

on_adjacency matrices_in the natural way:_If $A$_is_the adjacency_matrix_of a graph_$G$, then $\pi(A)$_is the adjacency matrix_of $\pi(G)$ and_$\pi(A)$_is obtained by simultaneously permuting with $\pi$ both rows and columns of $A$. \[def:weak\_iso\] Let_$(G,{\kappa_1})$_and $(H,{\kappa_2})$_be_$k$-color_graph colorings with $G=([n],E_1)$ and_$H=([n],E_2)$. We say that $(G,{\kappa_1})$_and $(H,{\kappa_2})$_are weakly isomorphic, denoted $(G,{\kappa_1})\approx(H,{\kappa_2})$ if there exist_permutations_$\pi \colon [n]_\to [n]$ and $\sigma \colon [k] \to [k]$ such_that $(u,v) \in E_1 \iff (\pi(u),\pi(v))_\in E_2$ and_$\kappa_1((u,v))_=_\sigma(\kappa_2((\pi(u), \pi(v))))$. We denote_such a weak isomorphism: $(G,{\kappa_1})\approx_{\pi,\sigma}(H,{\kappa_2})$. When_$\sigma$ is the identity permutation, we_say that $(G,{\kappa_1})$ and $(H,{\kappa_2})$ are isomorphic. The_following lemma emphasizes the importance of_weak graph isomorphism as it_relates to_Ramsey numbers. Many classic coloring_problems exhibit the_same property. \[lemma:closed\]_Let $(G,{\kappa_1})$ and_$(H,{\kappa_2})$ be graph colorings in $k$_colors such that_$(G,\kappa_1) \approx_{\pi,\sigma} (H,\kappa_2)$. Then, $$(G,\kappa_1)_\in_{{\cal R}}(r_1,r_2,\ldots,r_k;n) \iff_(H,\kappa_2)_\in _ {{\cal_R}}(\sigma(r_1),\sigma(r_2),\ldots,\sigma(r_k);n)$$ We make use_of_the following_theorem_from [@PR98]. \[thm:433\] $30\leq R(4,3,3)\leq 31$ and, $R(4,3,3)=31$_if_and only if there

x}}_k\| &\leq& \frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bf x}}_k))-\varphi(f({{\bf x}}_{k+1})) \nonumber\\ &=& \frac{2}{\eta}(\varphi(f({{\bf x}}_1))-\varphi(f({{\bf x}}_{K+1}))) \nonumber\\ &\leq& \frac{2}{\eta}\varphi(f({{\bf x}}_1)). \end{aligned}$$ So, we get $$\begin{aligned} \|{{\bf x}}_{K+1}-{{\bf x}}_*\| &\leq& \sum_{k=1}^K\|{{\bf x}}_{k+1}-{{\bf x}}_k\|+\|{{\bf x}}_1-{{\bf x}}_*\| \\ &\leq& \frac{2}{\eta}\varphi(f({{\bf x}}_1))+\|{{\bf x}}_1-{{\bf x}}_*\| \\ & < & \rho. \end{aligned}$$ Thus, ${{\bf x}}_{K+1}\in\mathscr{B}({{\bf x}}_*,\rho)$ and (\[eq4-10\]) holds. Moreover, let $K\to\infty$ in (\[eq4-12\]). We obtain (\[eq4-11\]). Suppose that the infinite sequence of iterates $\{{{\bf x}}_k\}$ is generated by ACSA. Then, the total sequence $\{{{\bf x}}_k\}$ has a finite length, i.e., $$\sum_k \|{{\bf x}}_{k+1}-{{\bf x}}_k\| < +\infty,$$ and hence the total sequence $\{{{\bf x}}_k\}$ converges to a unique critical point. [**Proof**]{} Since the domain of $f({{\bf x}})$ is compact, the infinite sequence $\{{{\bf x}}_k\}$ generated by ACSA must have an accumulation point ${{\bf x}}_*$. According to Theorem \[Th4-04\], ${{\bf x}}_*$ is a critical point. Hence, there exists an index $k_0$, which could be viewed as an initial iteration when we use Lemma \[Lm4-06\], such that ${{\bf x}}_{k_0}\in\mathscr{B}({{\bf x}}_*,\rho)$. From Lemma \[Lm4-06\], we have $\sum_{k=k_0}^{\infty}

x}}_k\| & \leq & \frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bf x}}_k))-\varphi(f({{\bf x}}_{k+1 }) ) \nonumber\\ & = & \frac{2}{\eta}(\varphi(f({{\bf x}}_1))-\varphi(f({{\bf x}}_{K+1 }) )) \nonumber\\ & \leq & \frac{2}{\eta}\varphi(f({{\bf x}}_1) ). \end{aligned}$$ So, we get $ $ \begin{aligned } \|{{\bf x}}_{K+1}-{{\bf x}}_*\| & \leq & \sum_{k=1}^K\|{{\bf x}}_{k+1}-{{\bf x}}_k\|+\|{{\bf x}}_1-{{\bf x}}_*\| \\ & \leq & \frac{2}{\eta}\varphi(f({{\bf x}}_1))+\|{{\bf x}}_1-{{\bf x}}_*\| \\ & < & \rho. \end{aligned}$$ Thus, $ { { \bf x}}_{K+1}\in\mathscr{B}({{\bf x}}_*,\rho)$ and (\[eq4 - 10\ ]) holds. furthermore, lease $ K\to\infty$ in (\[eq4 - 12\ ]). We obtain (\[eq4 - 11\ ]). Suppose that the countless sequence of iterates $ \{{{\bf x}}_k\}$ is beget by ACSA. Then, the total sequence $ \{{{\bf x}}_k\}$ have a finite length, i.e., $ $ \sum_k \|{{\bf x}}_{k+1}-{{\bf x}}_k\| < + \infty,$$ and hence the total sequence $ \{{{\bf x}}_k\}$ converges to a unique critical compass point. [ * * Proof * * ] { } Since the domain of $ f({{\bf x}})$ is compendious, the infinite sequence $ \{{{\bf x}}_k\}$ generated by ACSA must have an accumulation point $ { { \bf x}}_*$. According to Theorem \[Th4 - 04\ ], $ { { \bf x}}_*$ is a critical detail. Hence, there exists an index $ k_0 $, which could be watch as an initial iteration when we use Lemma \[Lm4 - 06\ ], such that $ { { \bf x}}_{k_0}\in\mathscr{B}({{\bf x}}_*,\rho)$. From Lemma \[Lm4 - 06\ ], we have $ \sum_{k = k_0}^{\infty }

x}}_k\| &\leq& \frac{2}{\eta}\sum_{k=1}^K \vxrphi(f({{\bf x}}_k))-\varpku(f({{\bf x}}_{n+1})) \nonujber\\ &=& \frac{2}{\eta}(\varphi(f({{\bf x}}_1))-\varphi(f({{\uf x}}_{J+1}))) \nonymber\\ &\leq& \frac{2}{\eta}\xarphi(f({{\bf x}}_1)). \end{aoigntd}$$ So, we get $$\begii{zligned} \|{{\bf w}}_{K+1}-{{\bf r}}_*\| &\oeq& \sum_{k=1}^K\|{{\bf x}}_{k+1}-{{\nf x}}_k\|+\|{{\bf x}}_1-{{\bf x}}_*\| \\ &\leq& \fsaz{2}{\eca}\varphi(f({{\bf x}}_1))+\|{{\bf x}}_1-{{\bf x}}_*\| \\ & < & \rho. \qnd{aligmef}$$ Thus, ${{\bf x}}_{K+1}\in\iathxsr{B}({{\br x}}_*,\rho)$ and (\[eq4-10\]) holds. Moreover, let $K\tk\infty$ pn (\[eq4-12\]). We obtain (\[ea4-11\]). Suppose that the infinite seqkence of iterates $\{{{\hf x}}_k\}$ is geberaewd by ACSA. Tfen, the touan sequence $\{{{\bf x}}_k\}$ has a finite length, i.e., $$\sjm_k \|{{\by x}}_{k+1}-{{\bf x}}_k\| < +\undty,$$ dnd hence tie totwl sequence $\{{{\ng x}}_k\}$ cmnvergex to a unique gritiral point. [**Proof**]{} Since the vomain of $f({{\bf x}})$ is cjmpact, tha nnfinite sequence $\{{{\bf z}}_k\}$ genetated by XXSA muat hzve an acrumulation loint ${{\bf x}}_*$. Qccording to Theorek \[Ey4-04\], ${{\bf x}}_*$ is a cditicaj [oint. Hence, there exists an index $k_0$, whibh ckuld be viewed as an inutial iteration when ae use Leima \[Lm4-06\], such that ${{\bf x}}_{k_0}\in\mathscr{B}({{\bf x}}_*,\rho)$. From Lemmd \[Lm4-06\], xe harc $\rym_{n=k_0}^{\infty}

x}}_k\| &\leq& \frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bf x}}_k))-\varphi(f({{\bf x}}_{k+1})) \nonumber\\ x}}_1))-\varphi(f({{\bf \nonumber\\ &\leq& x}}_1)). \end{aligned}$$ So, x}}_*\| \sum_{k=1}^K\|{{\bf x}}_{k+1}-{{\bf x}}_k\|+\|{{\bf x}}_*\| \\ &\leq& x}}_1))+\|{{\bf x}}_1-{{\bf x}}_*\| \\ & < \rho. \end{aligned}$$ Thus, ${{\bf x}}_{K+1}\in\mathscr{B}({{\bf x}}_*,\rho)$ and (\[eq4-10\]) holds. Moreover, let $K\to\infty$ in We obtain (\[eq4-11\]). Suppose that the infinite sequence of iterates $\{{{\bf x}}_k\}$ is by Then, total $\{{{\bf x}}_k\}$ has a finite length, i.e., $$\sum_k \|{{\bf x}}_{k+1}-{{\bf x}}_k\| < +\infty,$$ and hence the sequence $\{{{\bf x}}_k\}$ converges to a unique critical [**Proof**]{} Since the domain $f({{\bf x}})$ is compact, the sequence x}}_k\}$ generated ACSA have accumulation point ${{\bf According to Theorem \[Th4-04\], ${{\bf x}}_*$ is a critical point. Hence, there exists an index $k_0$, which be viewed initial iteration we Lemma such that ${{\bf From Lemma \[Lm4-06\], we have $\sum_{k=k_0}^{\infty}

x}}_k\| &\leq& \frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bF x}}_k))-\varphi(f({{\Bf x}}_{k+1})) \nOnuMbeR\\ &=& \fRac{2}{\eTa}(\vaRphi(f({{\bf x}}_1))-\varphi(F({{\Bf x}}_{K+1}))) \Nonumber\\ &\leq& \frac{2}{\eta}\varpHi(f({{\bf X}}_1)). \eND{aliGNeD}$$ So, we Get $$\begiN{AlIGNed} \|{{\Bf X}}_{K+1}-{{\Bf x}}_*\| &\LeQ& \SuM_{k=1}^K\|{{\bf X}}_{k+1}-{{\bF x}}_k\|+\|{{\bf x}}_1-{{\bF x}}_*\| \\ &\leq& \frac{2}{\eTa}\vArPhi(f({{\bf x}}_1))+\|{{\bf x}}_1-{{\bf X}}_*\| \\ & < & \RhO. \end{aligneD}$$ ThUs, ${{\bf x}}_{K+1}\in\mathScr{b}({{\bf x}}_*,\rhO)$ aNd (\[eQ4-10\]) Holds. morEover, Let $K\to\INfty$ in (\[Eq4-12\]). We obtaiN (\[eQ4-11\]). supposE That the INFiNite Sequence of iterateS $\{{{\Bf X}}_K\}$ is generated by aCSA. ThEn, THe TOTal SeqUence $\{{{\bf x}}_k\}$ hAs A finiTE length, I.E., $$\sUM_K \|{{\Bf x}}_{K+1}-{{\Bf x}}_k\| < +\infty,$$ and hEnce the totaL SeqUence $\{{{\bF x}}_K\}$ coNVerges To a unIqUE crItical point. [**prooF**]{} Since the Domain OF $f({{\bf x}})$ is COmpact, tHe infiNitE seQuenCE $\{{{\bF x}}_K\}$ geNeRAteD By aCSa MusT have an aCcUmUlatiOn poINT ${{\BF x}}_*$. AcCorDing To TheOrem \[Th4-04\], ${{\bf x}}_*$ is a cRitIcal POinT. HencE, therE exiStS an inDex $k_0$, whIch coUlD be viewed as an inItiaL iteratioN whEn We uSe lemma \[lM4-06\], such tHat ${{\Bf x}}_{K_0}\in\mathScr{B}({{\bf x}}_*,\RHo)$. FRoM lEMmA \[Lm4-06\], we have $\sum_{k=k_0}^{\inftY}

x}}_k\| &\leq& \fra c{2}{\eta} \sum_ {k= 1}^ K\var phi( f({{\bf x}}_k) ) -\va rphi(f({{\bf x}}_{k+1} )) \n on u mber \ \ &=& \f r ac { 2 }{\ et a} (\v ar p hi (f({{ \bf x}}_1) )-\varphi( f({ {\ bf x}}_{K+1} ) )) \nonumber \\ &\leq& \f rac{2} {\ eta } \varp hi( f({{\ bf x}} _ 1)). \end{ali gn e d}$$ S o , we ge t $$ \beg in{aligned} \ | {{ \ bf x}}_{K+1}-{ {\bf x }} _ *\ | &\l eq& \sum_{k=1 }^ K\|{{ \ bf x}}_ { k+ 1 } - {{\ b f x}}_k\|+\|{ {\bf x}}_1- { {\b f x}}_ *\ | \ \ &\le q& \fr ac{2}{\eta} \var phi(f({{\ bf x}} _ 1))+\|{ { \bf x}} _1-{{\ bfx}} _*\| \\ & <& \ rho . \end{ali gn ed }$$ T hus, $ { { \bfx}} _{K+ 1}\in \mathscr{B}({ {\b f x} } _*, \rho) $ and (\[ eq 4-10\ ]) hol ds. Mo reover, let $K\ to\i nfty$ in(\[ eq 4-1 2\ ]). W e obtai n ( \[e q4-11\] ). Sup p ose t h a t t he infinite sequen ce o fiterates $\{{{ \ bf x } }_k\}$ i sgen erat e d by A CSA. Th en, thetotals eq ue nce $\{ {{ \bf x} }_ k\} $ h as af init e leng th, i.e. , $$\ s um_k \|{{\bf x } }_{k+1}-{{\bf x} } _ k\ | < + \in fty,$$ andhenc e the tot a lseq u ence$\{{{ \b f x } }_k\}$ converges to a uniqu e cri tical point. [**Proof* * ] { } Sincethed om a in of $f({{\bf x}}) $ is compa c t, the i nfini te seque nce $\{{{ \ b f x}}_k\ }$gen era ted b yACSA must hav e an a cc umulati onpoint $ {{\ bfx}} _*$ .According to Theo re m\[ Th 4-0 4\],$ {{\bf x} }_ *$is acriti c al poi nt. H ence ,th e reexistsa ni n dex$k _0 $, w hic hcould bev iew ed as a n initial it e rati on w hen weuse Lemma \[L m4 -06\], suc htha t ${{\ b f x}}_{k_ 0}\in\mathscr{B}({{\bfx }}_*,\r ho) $. Fr om L emma \[Lm 4-0 6\], w e h a ve $\s um_{k= k_0}^ {\ inf t y }

x}}_k\| _ _ &\leq&_\frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bf_x}}_k))-\varphi(f({{\bf_x}}_{k+1})) \nonumber\\ __ _ &=& \frac{2}{\eta}(\varphi(f({{\bf_x}}_1))-\varphi(f({{\bf x}}_{K+1}))) \nonumber\\ _ __&\leq& \frac{2}{\eta}\varphi(f({{\bf x}}_1)). \end{aligned}$$ So, we get $$\begin{aligned} \|{{\bf x}}_{K+1}-{{\bf_x}}_*\|_&\leq& \sum_{k=1}^K\|{{\bf_x}}_{k+1}-{{\bf_x}}_k\|+\|{{\bf_x}}_1-{{\bf x}}_*\| \\ _ &\leq& \frac{2}{\eta}\varphi(f({{\bf_x}}_1))+\|{{\bf x}}_1-{{\bf_x}}_*\| \\ &_<_& \rho. _\end{aligned}$$ Thus, ${{\bf x}}_{K+1}\in\mathscr{B}({{\bf x}}_*,\rho)$ and (\[eq4-10\]) holds. Moreover, let_$K\to\infty$ in (\[eq4-12\]). We obtain (\[eq4-11\]). Suppose_that the infinite_sequence_of_iterates $\{{{\bf x}}_k\}$ is_generated by ACSA. Then, the total_sequence $\{{{\bf x}}_k\}$ has a finite_length, i.e., $$\sum_k \|{{\bf x}}_{k+1}-{{\bf x}}_k\| <_+\infty,$$ and hence the total sequence_$\{{{\bf x}}_k\}$ converges to a_unique critical_point. [**Proof**]{} Since the domain of_$f({{\bf x}})$ is_compact, the_infinite sequence $\{{{\bf_x}}_k\}$ generated by ACSA must have_an accumulation point_${{\bf x}}_*$. According to Theorem \[Th4-04\],_${{\bf_x}}_*$ is a_critical_point._Hence, there_exists an index_$k_0$,_which could_be_viewed as an initial iteration when_we_use Lemma \[Lm4-06\], such that ${{\bf x}}_{k_0}\in\mathscr{B}({{\bf_x}}_*,\rho)$. From Lemma \[Lm4-06\],_we_have $\sum_{k=k_0}^{\infty}

$[0.2,0.3)$ 1399 46.5% 238 14.0% $[0.3,0.4)$ 740 24.6% 194 11.4% $[0.4,0.5)$ 281 9.3% 154 9.1% $[0.5,0.6)$ 113 3.7% 156 9.2% $[0.6,0.7)$ 29 0.9% 157 9.2% $[0.7,0.8)$ 10 0.3% 156 9.2% $[0.8,0.9)$ 10 0.3% 252 14.9% $[0.9,1)$ 0 0% 237 14.0% $1$ 7 0.2% 0 0% ------------- ------ --------- ------ --------- : Statistics for similarities between each test sentence and the training set as computed by Equation \[simi1n\] for the WMT 2017 en-de task (3004 sentences) and our JRC-Acquis en-de task (1689 sentences).[]{data-label="wmt-simi"} en-de en-fr en-es ------ ------------ ------- ------- ------- dev NMT 44.08 57.26 55.76 Ours 50.81 62.60 60.51 1/0 reward 47.70 61.15 58.92 test NMT 43.76 57.67 55.78 Ours 50.15 63.27 60.54 1/0 reward 47.13 62.14 58.66 : Translation results (BLEU) of 1/0 reward.[]{data-label="t-decrease"} We also tried our method on WMT 2017 English-to-German News translation task. However, we did not achieve significant improvements over the baseline attentional NMT model, likely because the test set and the training set for the WMT task have a relatively low similarity as shown in Table \[wmt-simi\] and hence

$ [ 0.2,0.3)$ 1399 46.5% 238 14.0% $ [ 0.3,0.4)$ 740 24.6% 194 11.4% $ [ 0.4,0.5)$ 281 9.3% 154 9.1% $ [ 0.5,0.6)$ 113 3.7% 156 9.2% $ [ 0.6,0.7)$ 29 0.9% 157 9.2% $ [ 0.7,0.8)$ 10 0.3% 156 9.2% $ [ 0.8,0.9)$ 10 0.3% 252 14.9% $ [ 0.9,1)$ 0 0% 237 14.0% $ 1 $ 7 0.2% 0 0% ------------- ------ --------- ------ --------- : Statistics for similarities between each test sentence and the education rig as computed by Equation   \[simi1n\ ] for the WMT 2017 en - de task (3004 prison term) and our JRC - Acquis en - de job (1689 sentences).[]{data - label="wmt - simi " } en - de en - fr en - es ------ ------------ ------- ------- ------- dev NMT 44.08 57.26 55.76 Ours 50.81 62.60 60.51 1/0 reward 47.70 61.15 58.92 examination NMT 43.76 57.67 55.78 Ours 50.15 63.27 60.54 1/0 advantage 47.13 62.14 58.66 : Translation consequence (BLEU) of 1/0 reward.[]{data - label="t - decrease " } We also tried our method acting on WMT 2017 English - to - German News translation task. However, we did not achieve significant improvements over the service line attentional NMT model, likely because the trial set and the training sic for the WMT task have a relatively low similarity as show in Table   \[wmt - simi\ ] and hence

$[0.2,0.3)$ 1399 46.5% 238 14.0% $[0.3,0.4)$ 740 24.6% 194 11.4% $[0.4,0.5)$ 281 9.3% 154 9.1% $[0.5,0.6)$ 113 3.7% 156 9.2% $[0.6,0.7)$ 29 0.9% 157 9.2% $[0.7,0.8)$ 10 0.3% 156 9.2% $[0.8,0.9)$ 10 0.3% 252 14.9% $[0.9,1)$ 0 0% 237 14.0% $1$ 7 0.2% 0 0% ------------- ------ --------- ------ --------- : Statistiss for xililarities betreen qach newt sentence and the training aet as bomputed by Equatoon \[simi1n\] for the WMT 2017 en-de tasn (3004 sentences) and okr JRC-Acquiw en-qw task (1689 sentdnces).[]{data-label="wmt-simi"} en-de en-ff eu-es ------ ------------ ------- ------- ------- eec NMT 44.08 57.26 55.76 Ours 50.81 62.60 60.51 1/0 rexard 47.70 61.15 58.92 test NMT 43.76 57.67 55.78 Ouss 50.15 63.27 60.54 1/0 rewdrd 47.13 62.14 58.66 : Franslwtikn results (BLEU) of 1/0 rwward.[]{data-label="t-decrtasq"} Qe also tried our mqtrod on WMT 2017 English-to-German News transldtikn task. However, we did bot achieve significajt improvqments over the baseline attentional NMT model, linely uezauwe gye test set and the training set for the WMT tafi nane a relatively ljw similariyy ax shown in Tabue \[wmt-snji\] and hence

$[0.2,0.3)$ 1399 46.5% 238 14.0% $[0.3,0.4)$ 740 11.4% 281 9.3% 9.1% $[0.5,0.6)$ 113 0.9% 9.2% $[0.7,0.8)$ 10 156 9.2% $[0.8,0.9)$ 0.3% 252 14.9% $[0.9,1)$ 0 0% 14.0% $1$ 7 0.2% 0 0% ------------- ------ --------- ------ --------- : Statistics similarities between each test sentence and the training set as computed by Equation for WMT en-de (3004 sentences) and our JRC-Acquis en-de task (1689 sentences).[]{data-label="wmt-simi"} en-de en-fr en-es ------ ------------ ------- ------- dev NMT 44.08 57.26 55.76 Ours 50.81 62.60 1/0 reward 47.70 61.15 test NMT 43.76 57.67 55.78 50.15 60.54 1/0 47.13 58.66 Translation results (BLEU) 1/0 reward.[]{data-label="t-decrease"} We also tried our method on WMT 2017 English-to-German News translation task. However, we did achieve significant the baseline NMT likely the test set training set for the WMT task low similarity as shown in Table \[wmt-simi\] and

$[0.2,0.3)$ 1399 46.5% 238 14.0% $[0.3,0.4)$ 740 24.6% 194 11.4% $[0.4,0.5)$ 281 9.3% 154 9.1% $[0.5,0.6)$ 113 3.7% 156 9.2% $[0.6,0.7)$ 29 0.9% 157 9.2% $[0.7,0.8)$ 10 0.3% 156 9.2% $[0.8,0.9)$ 10 0.3% 252 14.9% $[0.9,1)$ 0 0% 237 14.0% $1$ 7 0.2% 0 0% ------------- ------ --------- ------ --------- : Statistics for similarities Between eacH test SenTenCe And tHe trAining set as comPUted By Equation \[simi1n\] for the WmT 2017 en-dE tASk (3004 seNTeNces) aNd our JRc-acQUIs eN-dE tAsk (1689 SeNTeNces).[]{dAta-Label="wmT-simi"} en-de eN-fr En-Es ------ ------------ ------- ------- ------- dev NMT 44.08 57.26 55.76 Ours 50.81 62.60 60.51 1/0 REwArd 47.70 61.15 58.92 test NMT 43.76 57.67 55.78 ourS 50.15 63.27 60.54 1/0 reward 47.13 62.14 58.66 : TransLatIon resUlTs (BleU) of 1/0 rEwaRd.[]{datA-label="T-DecreaSe"} We also tRiED our meTHod on WMt 2017 eNgLish-To-German News transLAtIOn task. However, wE did noT aCHiEVE siGniFicant imprOvEmentS Over the BAsELINe aTTentional NMT mOdel, likely bECauSe the tEsT seT And the TrainInG Set For the WMT taSk haVe a relatiVely loW SimilarITy as shoWn in TaBle \[Wmt-Simi\] ANd HeNce

$[0.2,0.3)$ 1399 46.5% 238 14. 0% $ [0.3 ,0.4)$ 74 0 24.6% 194 11.4 % $ [0 . 4,0. 5 )$ 281 9 . 3% 1 54 9.1% $ [0.5,0. 6)$ 1 13 3.7% 1 5 6 9.2% $ [0.6,0.7)$ 29 0 . 9% 15 7 9.2% $[0. 7,0.8)$ 10 0.3% 156 9.2% $[0.8 , 0. 9 )$ 10 0.3 % 2 5 2 14.9% $[ 0. 9,1)$ 0 0% 237 14.0% $ 1 $ 7 0. 2% 0 0% -------- ------ ----- - - ------- ----- - - --- ---- - : S ta t ist i cs fo r si milariti es b etwee n ea c h t estsen tenc e and the training se t as com puted by E quat io n \[s imi1n\ ] for t he WMT 2017 en- de t ask (3004 se nt enc es ) and our JR C-A cqu is en-d e task( 168 9s e n te nces).[]{data-labe l= " w mt -simi"} en -deen-fr e n -e s ---- -- --- - -- -- ---- -- -- --- -- -- --- -- ----- de v N MT 4 4 .08 57.26 5 5.76 Ou r s 5 0.81 62.6 0 6 0.51 1 /0 re ward 4 7. 7 0 61.15 58.92 t est NMT 43.76 57.67 5 5 .78 Ou r s 50.1 5 6 3.27 60. 5 4 1/ 0 reward 47.1362.14 58. 66 : T r a ns lation result s (BLE U) of 1/0 re ward.[] {da ta- lab el= "t -decrease "} We a ls otr ie d o ur me t hod on W MT 20 17 En glish - to-Ger man N ewstr an s lat ion tas k .H o weve r, w e di d n ot achi eves ign ificant improvem ent s ove rth e basel ine attention al NMT model ,lik ely be c a use thetest set and the traini n g set f orthe W MT t ask havea r elativ ely low si milari ty as s how n in Ta b l e\[w mt -simi\] an d hen ce

_$[0.2,0.3)$ _ 1399_ __ 46.5%__ 238_ _ 14.0% $[0.3,0.4)$_ __ 740 24.6% 194 __11.4% _$[0.4,0.5)$__ 281_ _9.3% _ 154 _9.1% _ $[0.5,0.6)$ _ 113 _ 3.7% 156_ ___9.2% $[0.6,0.7)$ _ 29_ 0.9%_ 157 _ 9.2% $[0.7,0.8)$ _ _10 _ 0.3%_ _156 _ _ 9.2% $[0.8,0.9)$ _ _ 10 __0.3% __252_ _ 14.9% __$[0.9,1)$ __ __0 _ 0% __237 _ 14.0% $1$ _ _ _ _ 7 0.2%_ 0_ _ __0% -------------_------_--------- ------_--------- : Statistics for similarities between_each test_sentence and the training set as_computed by Equation \[simi1n\] for_the_WMT 2017 en-de task (3004 sentences)_and our JRC-Acquis en-de task (1689_sentences).[]{data-label="wmt-simi"} ___ _ _ _ en-de en-fr _en-es _ ------ ------------ ------- ------- ------- __dev NMT ___ _ 44.08 57.26 _ 55.76 _ Ours _ ___50.81 62.60 60.51 _ _ _1/0 reward 47.70 61.15 _58.92 test NMT _ 43.76 _57.67 55.78 _ Ours _ __ 50.15 63.27 _60.54 ___ _ _1/0 reward _ 47.13_ 62.14_ _58.66 :_Translation_results (BLEU) of 1/0 reward.[]{data-label="t-decrease"} We also tried our method on WMT 2017 English-to-German News translation task. However, we_did not achieve significant improvements over_the baseline attentional NMT model,_likely_because the test_set and the training_set for the WMT task have a relatively_low similarity as shown in Table \[wmt-simi\] and hence

One problem of the SMI assumption is that a positive bag may contain more than one positive instance. In SED, some sound classes such as “ambulance siren” may last for several seconds and may occur in many instances. In contrast to the SMI assumption, with the CA assumption, all the instances in a bag contribute equally to the tags of the bag. The bag-level prediction can be obtained by averaging the instance-level predictions: $$\label{eq:IS_average} F(B)=\frac{1}{\left | B \right |}\sum_{\mathbf{x} \in B}f(\mathbf{x}).$$ The symbol $ |B| $ denotes the number of instances in bag $ B $. Equation (\[eq:IS\_average\]) shows that CA is based on the assumption that all the instances in a positive bag are positive. Bag space methods {#BS_paradigm} ----------------- Instead of building an instance-level classifier, the BS methods regard a bag $ B $ as an entirety. Building a tagging model on the bags rely on a distance function $ D(\cdot, \cdot): B \times B \mapsto \mathbb{R} $. The distance function can be, for example, the Hausdorff distance [@wang2000solving]: $$\label{eq:BS} D(B_{1}, B_{2}) = \underset{\mathbf{x}_{1} \in B_{1}, \mathbf{x}_{2} \in B_{2}}{\textrm{min}}\left \| \mathbf{x}_{1} - \mathbf{x}_{2} \right \|.$$ In (\[eq:BS\]), the distance between two bags is the minimum distance between the instances in bag $ B_{1} $ and $ B_{2} $. Then this distance function can be plugged into a standard distance-based classifier such as a k-nearest neighbour (KNN) or a support vector machine (SVM) algorithm. The computational complexity of (\[eq:BS\]) is $ |B_{1}||B_{2}| $, which is larger than the IS and the ES methods described below. Embedded space methods {#section:ES_paradigm} ---------------------- Different from the IS methods, ES methods do not classify individual instances. Instead, the ES methods define an embedding mapping from a bag to an embedding vector: $$\label{eq:ES_mapping} f_{\text{emb}}: B \mapsto \

One problem of the SMI assumption is that a positive bag may control more than one incontrovertible instance. In SED, some sound classes such as “ ambulance enchantress ” may last for several seconds and may happen in many example. In contrast to the SMI premise, with the CA assumption, all the instances in a cup of tea contribute equally to the tag of the bag. The cup of tea - level prediction can be obtained by average the instance - level predictions: $ $ \label{eq: IS_average } F(B)=\frac{1}{\left | B \right |}\sum_{\mathbf{x } \in B}f(\mathbf{x}).$$ The symbol $ |B| $ denotes the issue of instances in bag $ B $. Equation (\[eq: IS\_average\ ]) show that CA is based on the assumption that all the instances in a positive bag are plus. Bag space methods { # BS_paradigm } ----------------- Instead of building an instance - horizontal surface classifier, the BS method acting regard a bag $ B $ as an entirety. Building a tagging model on the bags rely on a distance function $ D(\cdot, \cdot ): B \times B \mapsto \mathbb{R } $. The distance affair can be, for example, the Hausdorff distance [ @wang2000solving ]: $ $ \label{eq: BS } D(B_{1 }, B_{2 }) = \underset{\mathbf{x}_{1 } \in B_{1 }, \mathbf{x}_{2 } \in B_{2}}{\textrm{min}}\left \| \mathbf{x}_{1 } - \mathbf{x}_{2 } \right \|.$$ In (\[eq: BS\ ]), the distance between two bags is the minimal distance between the case in cup of tea $ B_{1 } $ and $ B_{2 } $. Then this distance function can be plugged into a standard distance - based classifier such as a k - nearest neighbour (KNN) or a support vector machine (SVM) algorithm. The computational complexity of (\[eq: BS\ ]) is $ |B_{1}||B_{2}| $, which is large than the IS and the ES methods described below. Embedded space methods { # department: ES_paradigm } ---------------------- Different from the IS methods, ES methods do not classify individual case. Instead, the ES method define an embedding mapping from a bag to an embedding vector: $ $ \label{eq: ES_mapping } f_{\text{emb } }: barn \mapsto \

One problem of the SMI assuoption is that c positmve bag may congain more than one positive mnstqnce. Un SED, some sound clasres such ws “ambulqnce wiren” may last for severzp seeoids and may occor in many itstances. In cottfadt to the SMI assumption, with the CW assumltlon, all the inftanbef in a bag contribute equally to the tzgs of uhe bag. The bag-levrl prediction can be obtaijed hy averaging the ijstance-leveo prqeictions: $$\labdl{eq:IS_average} F(B)=\frac{1}{\lefj | B \right |}\sum_{\mathbf{x} \in B}f(\mathbw{x}).$$ Thz symbol $ |B| $ eenljes the numbxr of pnstances in nsg $ B $. Equatipn (\[eq:IS\_average\]) shmws that CA is based on vhe assumption that wll the itscances in a positive vat are poshtivd. Vag spzcx msthods {#BS_'aradigm} ----------------- Insfead of buioding an instance-lefej classifier, tge BS ieehods regard a bag $ B $ as an entirety. Blildjng a tagging model on rhe bags rely on a didtance fugction $ D(\cdot, \cdot): B \times B \mapsto \mathbb{R} $. The gistaicd fbkgtiov cwn be, for example, the Hausdorff distance [@wang2000fklfikg]: $$\label{eq:BS} D(B_{1}, B_{2}) = \underset{\mstjbg{v}_{1} \in B_{1}, \mathbf{b}_{2} \in B_{2}}{\csxfrm{min}}\left \| \mathbf{d}_{1} - \mathff{x}_{2} \rught \|.$$ In (\[tq:BS\]), yhe distance between two bats is the miuimym distance betweeu the instanees in bag $ B_{1} $ and $ B_{2} $. Then this dnstancs function fan be plheged into a stanaarc gistance-based classifier stch as a j-neaxest neiehboor (KNN) jr a suppogt vegdor machine (SVM) alhoritkm. Tha computatlonal complexity of (\[eq:BS\]) is $ |B_{1}||B_{2}| $, which is larbes tvan the NS and the ES methoqs described bglow. Embedbed spxce methoda {#sectimn:ES_paradigi} ---------------------- Different frml the IS metiods, ES mqthoes di not cuxssify individial instauees. Insteqd, the ES methods vefivs an embedding napkibg from a bag yo xn qmhevding eector: $$\label{aq:ES_oapoong} f_{\tdxt{emb}}: B \maistu \

One problem of the SMI assumption is positive may contain than one positive classes as “ambulance siren” last for several and may occur in many instances. contrast to the SMI assumption, with the CA assumption, all the instances in bag contribute equally to the tags of the bag. The bag-level prediction can obtained averaging instance-level $$\label{eq:IS_average} F(B)=\frac{1}{\left | B \right |}\sum_{\mathbf{x} \in B}f(\mathbf{x}).$$ The symbol $ |B| $ denotes the number instances in bag $ B $. Equation (\[eq:IS\_average\]) that CA is based the assumption that all the in positive bag positive. space {#BS_paradigm} ----------------- Instead building an instance-level classifier, the BS methods regard a bag $ B $ as an entirety. Building tagging model bags rely a function D(\cdot, \cdot): B \mapsto \mathbb{R} $. The distance function example, the Hausdorff distance [@wang2000solving]: $$\label{eq:BS} D(B_{1}, B_{2}) \underset{\mathbf{x}_{1} \in \mathbf{x}_{2} \in B_{2}}{\textrm{min}}\left \| \mathbf{x}_{1} - \right \|.$$ In (\[eq:BS\]), the distance between two is the minimum distance between the instances in bag $ B_{1} $ and $ B_{2} this distance function can plugged into a distance-based such a neighbour (KNN) a support vector machine (SVM) algorithm. The computational complexity of (\[eq:BS\]) $ |B_{1}||B_{2}| $, which is larger than the IS and methods below. Embedded space {#section:ES_paradigm} ---------------------- Different from IS ES methods do not instances. the an mapping a bag to an vector: $$\label{eq:ES_mapping} f_{\text{emb}}: B \mapsto

One problem of the SMI assumptIon is that a PositIve Bag MaY conTain More than one posITive Instance. In SED, some sound ClassEs SUch aS “AmBulanCe siren” MAy LASt fOr SeVerAl SEcOnds aNd mAy occur In many instAncEs. in contrast to THe sMI assumptIon, With the CA assUmpTion, alL tHe iNStancEs iN a bag ContriBUte equAlly to the TaGS of the BAg. The baG-LEvEl prEdiction can be obtaINeD By averaging the InstanCe-LEvEL PreDicTions: $$\label{Eq:iS_aveRAge} F(B)=\frAC{1}{\lEFT | b \riGHt |}\sum_{\mathbf{x} \iN B}f(\mathbf{x}).$$ THE syMbol $ |B| $ dEnOteS The numBer of InSTanCes in bag $ B $. EqUatiOn (\[eq:IS\_aveRage\]) shOWs that Ca Is based On the aSsuMptIon tHAt AlL thE iNStaNCeS in A PosItive bag ArE pOsitiVe. BaG SPACe meThoDs {#BS_ParadIgm} ----------------- Instead of bUilDing AN inStancE-leveL claSsIfier, The BS mEthodS rEgard a bag $ B $ as an eNtirEty. BuildiNg a TaGgiNg Model ON the baGs rEly On a distAnce funCTioN $ D(\CDOT, \cDot): B \times B \mapsto \maThBB{r} $. THe distanCe funcTIoN cAN be, for exAmPle, The HAUSdorfF disTAnCe [@wang2000soLving]: $$\lABeL{eQ:BS} D(B_{1}, B_{2}) = \uNdErset{\mAtHbf{X}_{1} \in b_{1}, \mathBF{x}_{2} \in b_{2}}{\textrM{min}}\left \| \MathbF{X}_{1} - \mathbf{x}_{2} \right \|.$$ IN (\[Eq:BS\]), the distanCE bETWeEN two BagS is the minimUm diSTancE betWEeN thE InstaNces iN bAG $ B_{1} $ ANd $ B_{2} $. Then this distance FuNction Can be Plugged into a sTandard disTANCe-based cLassIFiER such as a k-neareSt neiGhbour (KNN) oR A support VectoR machine (sVM) algoriTHM. The compUtaTioNal ComPLExIty of (\[eq:BS\]) is $ |B_{1}||B_{2}| $, WHIch iS lArger thAn tHe IS and The eS mEthOds DeScribed beLow. EmbedDeD sPaCe MetHods {#sECtion:ES_pArAdiGm} ---------------------- difFerenT From thE IS meThodS, Es mEThoDs do not CLaSSIfy iNdIvIduaL inStAnces. instEAd, tHe ES metHods definE an EMbedDiNg Mapping From a bag to an eMbEdding vectOr: $$\LabEl{eq:ES_MAPping} f_{\teXt{emb}}: B \mapsto \

One problem of the SMI ass umption is that apos it ivebagmay contain mo r e th an one positive instan ce. I nS ED,s om e sou nd clas s es s uch a s“am bu l an ce si ren ” may l ast for se ver al seconds and ma y occur in ma ny instances . I n cont ra stt o the SM I ass umptio n , with the CA a ss u mption , all th e in stan ces in a bag cont r ib u te equally tothe ta gs of t hebag . The bag- le vel p r edictio n c a n beo btained by av eraging the ins tance- le vel predic tions :$ $\l abel{eq:IS_ aver age} F(B) =\frac { 1}{\lef t | B \r ight | }\s um_ {\ma t hb f{ x}\i n B} f (\ mat h bf{ x}).$$ T he s ymbol $ | B | $ den ote s th e num ber of instan ces inb ag$ B $ . Equ atio n(\[eq :IS\_a verag e\ ]) shows that C A is based on th eass um ption that a llthe instan ces ina po si t i v ebag are positive. B a g s pace met hods { # BS _p a radigm}-- --- ---- - - ----- - I n st ead of b uildin g a ninstanc e- levelcl ass ifi er, t h e BS metho ds regar d a b a g $ B $ as ane ntirety. Buil d in g at aggi ngmodel on th e ba g s re ly o n a di s tance func ti o n$ D(\cdot, \cdot): B \ timesB \ma psto \mathbb{ R} $. Thed i s tance fu ncti o nc an be, for exa mple, the Hausd o rff dist ance[@wang20 00solving ] : $$\labe l{e q:B S}D(B _ { 1} , B_{2}) = \u n d erse t{ \mathbf {x} _{1} \i n B _{1 },\ma th bf{x}_{2} \in B_{ 2} }{ \t ex trm {min} } \left \| \ mat hb f{x }_{1} - \mat hbf{x }_{2 }\r i ght \|.$$I n( \ [eq: BS \] ), t hedi stanc e be t wee n two b ags is th e m i nimu mdi stancebetween the i ns tances inba g $ B_{1} $ and $ B _{2} $. Then this dista n ce func tio n can beplugged i nto a sta nda r d dist ance-b asedcl ass i f ier s u c hasak-nearestn e igh bour(K NN)or a su pport vector machi n e ( SVM) algorith m.Thec o mp uta t io n alco m ple x i ty of (\[eq:BS\ ]) is $ |B _{ 1 }| |B_{2}| $, whi ch is lar ger tha n the IS andthe ES me thods des cr ibed b elo w. Embedd ed space methods{ #sect i on :ES_p ara digm}-- --- ----- ------ - --- -- D iffere nt fromthe I Smethods, ES methods do not clas sify i ndivi dua l instanc es. Ins tead, the ESmethods de fin e a n emb edd i ng ma ppin g f rom a bag toa n embeddi n gvec t o r: $$\label{e q : E S_m appin g}f _{\tex t{em b}}: B \mapsto \

One problem_of the_SMI assumption is that_a positive_bag_may contain_more_than one positive_instance. In SED,_some sound classes such_as “ambulance siren”_may_last for several seconds and may occur in many instances. In contrast to the_SMI_assumption, with_the_CA_assumption, all the instances in_a bag contribute equally to_the tags_of the bag. The bag-level prediction can be_obtained_by averaging the_instance-level predictions: $$\label{eq:IS_average} F(B)=\frac{1}{\left | B \right |}\sum_{\mathbf{x} \in B}f(\mathbf{x}).$$_The symbol $ |B| $ denotes_the number of_instances_in_bag $ B $._Equation (\[eq:IS\_average\]) shows that CA is_based on the assumption that all_the instances in a positive bag are_positive. Bag space methods {#BS_paradigm} ----------------- Instead of building_an instance-level classifier, the BS_methods regard_a bag $ B $_as an entirety._Building a_tagging model on_the bags rely on a distance_function $ D(\cdot,_\cdot): B \times B \mapsto \mathbb{R}_$._The distance function_can_be,_for example,_the Hausdorff distance_[@wang2000solving]:_$$\label{eq:BS} D(B_{1}, B_{2})_=_\underset{\mathbf{x}_{1} \in B_{1}, \mathbf{x}_{2} \in B_{2}}{\textrm{min}}\left_\|_\mathbf{x}_{1} - \mathbf{x}_{2} \right \|.$$ In (\[eq:BS\]),_the distance between two_bags_is the minimum distance_between the instances in bag_$ B_{1} $ and $ B_{2}_$. Then_this distance_function can be plugged into a standard distance-based classifier such as_a k-nearest neighbour (KNN) or a_support vector machine (SVM)_algorithm. The_computational_complexity of (\[eq:BS\])_is_$ |B_{1}||B_{2}|_$, which is larger than the IS_and the_ES methods described below. Embedded space methods_{#section:ES_paradigm} ---------------------- Different from the IS_methods,_ES methods do not classify individual_instances. Instead, the ES methods define_an embedding mapping from a_bag_to_an embedding vector: $$\label{eq:ES_mapping} f_{\text{emb}}: B_\mapsto \

hardiman; @reif2]. Also, most traditional courses do not [*explicitly*]{} teach students effective problem solving strategies. Rather, they may reward inferior problem solving strategies in which many students engage. Instructors often implicitly assume that students know that the analysis, planning, evaluation, and reflection phases of problem solving are as important as the implementation phase. Consequently, they may not discuss these strategies explicitly while solving problems during the lecture. There is no mechanism in place to ensure that students make a conscious effort to interpret the concepts, make qualitative inferences from the quantitative problem solving tasks, or relate the new concepts to their prior knowledge. In order to develop scientific reasoning by solving quantitative problems, students must learn to exploit problem solving as an opportunity for knowledge and skill acquisition. Thus, students should not treat quantitative problem solving merely as a mathematical exercise but as a learning opportunity and they should engage in effective problem solving strategies. Effective Problem Solving Strategies ------------------------------------ Effective problem solving begins with a conceptual analysis of the problem, followed by planning of the problem solution, implementation and evaluation of the plan, and last but not least reflection upon the problem solving process. As the complexity of a physics problem increases, it becomes increasingly important to employ a systematic approach. In the qualitative or conceptual analysis stage, a student should draw a picture or a diagram and get a visual understanding of the problem. At this stage, a student should convert the problem to a representation that makes further analysis easier. After getting some sense of the situation, labeling all known and unknown numerical quantities is helpful in making reasonable physical assumptions. Making predictions about the solution is useful at this level of analysis and it can help to structure the decision making at the next stage. The prediction made at this stage can be compared with the problem solution in the reflection phase and can help repair, extend and organize the student’s knowledge structure. Planning or decision making about the applicable physics principles is the next problem solving heuristic. This is the stage where the student brings everything together to come up with a reasonable solution. If the student performed good qualitative analysis and planning, the implementation of the plan becomes easy if the student possesses the necessary algebraic manipulation and mathematical skills. After implementation of the plan, a student must evaluate his/her solution, e.g., by checking the dimension or the order of magnitude, or by checking whether the initial prediction made during the initial analysis stage matches the actual solution. One can also ask whether the solution is sensible and, possibly, consistent with experience

hardiman; @reif2 ]. Also, most traditional courses do not [ * explicitly * ] { } teach students effective trouble clear strategies. Rather, they may honor deficient problem solving strategy in which many scholar engage. teacher frequently implicitly assume that scholar know that the analysis, planning, evaluation, and expression phase of problem solving are as important as the implementation phase. Consequently, they may not discourse these strategies explicitly while solving problems during the lecture. There exist no mechanism in place to ensure that students make a conscious attempt to interpret the concepts, make qualitative inferences from the quantitative problem clear tasks, or relate the new concepts to their prior knowledge. In order to develop scientific reasoning by solving quantitative problems, students must learn to exploit problem solving as an opportunity for knowledge and skill acquisition. therefore, student should not treat quantitative trouble clear merely as a numerical exercise but as a learning opportunity and they should engage in effective problem solving strategies. Effective Problem Solving Strategies ------------------------------------ Effective problem clear begins with a conceptual analysis of the problem, followed by planning of the problem solution, implementation and evaluation of the design, and last but not least reflection upon the problem solving process. As the complexity of a physics problem increase, it becomes increasingly important to employ a systematic approach. In the qualitative or conceptual psychoanalysis stage, a student should draw a picture or a diagram and get a visual understanding of the problem. At this stage, a scholar should convert the problem to a representation that makes further analysis easy. After getting some common sense of the situation, labeling all known and unknown numerical quantity is helpful in making reasonable forcible assumptions. Making predictions about the solution is useful at this level of analysis and it can help to structure the decision making at the next stage. The prediction made at this phase can be compared with the problem solution in the observation phase and can help repair, strain and organize the student ’s knowledge structure. Planning or decision making about the applicable physics principles is the next problem solve heuristic. This is the stagecoach where the scholar bring everything together to come up with a reasonable solution. If the student performed full qualitative analysis and planning, the implementation of the plan become easy if the student possesses the necessary algebraic manipulation and mathematical skill. After implementation of the plan, a student must measure his / her solution, e.g., by checking the dimension or the order of magnitude, or by checking whether the initial prediction made during the initial analysis stage matches the actual solution. One can besides necessitate whether the solution is sensible and, possibly, consistent with experience

harfiman; @reif2]. Also, most traaitional courses do nov [*expliditly*]{} texch students effective problxm silvint strategies. Rather, thdy may reaard infwrioc problem solvinj stratennes ih whieh many students engage. Invtructors oftet km'licitly assume that students know trat the ajalysis, plannigg, enajuatjon, and reflection phases of problsm solvpng are as importsnt as the implementation ohasf. Consequently, thej may not duscufw these stragegies explicitly whilg solving problems during the lezture. There is bo mefvanism in poace no ensure thab studetts makr a conscious cfforv to interpret the concepvs, make qualitative ynferencev yrom the quantitative peoblek sonvine tarks, oc rslate hhe new concelts to theie prior knowledge. In oweer to develol sciegtyfic reasoning by solving quantitative kroblsms, students must learn to exploit problem sllving as an opportunity for knowledge and skill acquisitimn. Thns, stbeents whluld not treat quantitative problem solving mqdeky as a mathematlcal exercise but ss a jearning oppottunity ans they should engahe in esfectuve probltm sokving strategies. Effective Peoblem Solviug Wtrategies ------------------------------------ Effectivz problem souvinb begons with a conceptual aualysia of the prlblem, folmuwed by planning of tve problto solution, implemqntation end eraluatiov of the pjan, and ladt bub not least reflectlon ukon tha problem dolving process. As the complexivb of a physicx [rotlem incxeases, it becomes igcreasingly imkortant tj empuoy a systvmatic ap'roach. In thq qualitative lr conceptuan analysys srage, a studdvt should draw a picturv jr a diagram and get a virhal understandiuy if the problem. St ghif dtege, a vtudent shound cunvdtt thd problti bo x relresentation that manes rurther analysis essler. After getting some sense og the situation, lahelinj all ynown anq unknown numerical quantities is helpfkl ln making reafonanle khysical asxumptions. Making predictions about the wolution is useful at this level of auakysis and iv can relp to sdructure the decisiob making at the ntxt stage. The predictioh made at tjis stage can be compared with the problem solution in the reflection phawe and can help relair, extetd cnd orgwnizx vhe student’s knowkedge structure. Planning or decmsion makitg about the applicable physicx orinciples is the next problem solvinf heurisyic. This is the stage where the student brings everythikg togethwr to come lk wmth a reasonable solition. Nf the studenr pevformed good qjalotwtive anapysix cnd planning, the implementation od the plwn becomes easy mf the studekt possgsses thg necessary algebdaic lanipolation and mayhematical skikls. After implementatpon of the plan, a atudent must evalocne his/her solntion, e.g., by whecking the dimension or the order of magnitude, or by checkjng whetmer the initial pteviction made during the lnitiwl analyris stage madches the actual solution. One can also ask xnethef tre somution is senvible and, possibly, consistent witf expwrience

hardiman; @reif2]. Also, most traditional courses do teach effective problem strategies. Rather, they strategies which many students Instructors often implicitly that students know that the analysis, evaluation, and reflection phases of problem solving are as important as the implementation Consequently, they may not discuss these strategies explicitly while solving problems during the There no in to ensure that students make a conscious effort to interpret the concepts, make qualitative inferences from quantitative problem solving tasks, or relate the new to their prior knowledge. order to develop scientific reasoning solving problems, students learn exploit solving as an for knowledge and skill acquisition. Thus, students should not treat quantitative problem solving merely as a mathematical but as opportunity and should in problem solving strategies. Solving Strategies ------------------------------------ Effective problem solving conceptual analysis of the problem, followed by planning the problem implementation and evaluation of the plan, last but not least reflection upon the problem process. As the complexity of a physics problem increases, it becomes increasingly important to employ approach. In the qualitative conceptual analysis stage, student draw picture a diagram get a visual understanding of the problem. At this stage, a should convert the problem to a representation that makes further After some sense of situation, labeling all known unknown quantities is helpful in physical Making solution useful this level of analysis it can help to structure decision making at the at this stage can be compared with the solution in the reflection phase and can repair, extend and organize the student’s knowledge structure. Planning or decision making the applicable is the next problem solving heuristic. This is stage where the student everything together to come up with a reasonable solution. the performed good analysis and planning, implementation of the becomes easy if possesses the algebraic and implementation of the plan, a student evaluate solution, e.g., by checking the or of magnitude, or by checking whether the initial prediction made during the initial analysis stage actual solution. One can ask the solution is sensible and, possibly, consistent with experience

hardiman; @reif2]. Also, most tradiTional courSes do Not [*ExpLiCitlY*]{} teaCh students effeCTive Problem solving strategiEs. RatHeR, They MAy RewarD inferiOR pROBleM sOlVinG sTRaTegieS in Which maNy students EngAgE. Instructors OFtEn implicitLy aSsume that stuDenTs know ThAt tHE analYsiS, planNing, evALuatioN, and refleCtIOn phasES of probLEM sOlviNg are as important aS ThE Implementation Phase. COnSEqUENtlY, thEy may not diScUss thESe stratEGiES EXplICitly while solVing problemS DurIng the LeCtuRE. There Is no mEcHAniSm in place to EnsuRe that stuDents mAKe a consCIous effOrt to iNteRprEt thE CoNcEptS, mAKe qUAlItaTIve InferencEs FrOm the QuanTITATive ProBlem SolviNg tasks, or relaTe tHe neW ConCepts To theIr prIoR knowLedge. IN ordeR tO develop scientiFic rEasoning bY soLvIng QuAntitATive prOblEms, StudentS must leARn tO eXPLOiT problem solving as aN oPPOrTunity foR knowlEDgE aND skill acQuIsiTion. tHUs, stuDentS ShOuld not tReat quANtItAtive prObLem solViNg mEreLy as a MAtheMaticaL exercisE but aS A learning opporTUnity and they sHOuLD EnGAge iN efFective probLem sOLvinG strATeGieS. effecTive PRoBLeM solving Strategies ------------------------------------ EfFeCtive pRobleM solving beginS with a concEPTUal analySis oF ThE Problem, followeD by plAnning of thE Problem sOlutiOn, implemEntation aND EvaluatiOn oF thE plAn, aND LaSt but not least REFlecTiOn upon tHe pRoblem sOlvIng ProCesS. AS the complExity of a PhYsIcS pRobLem inCReases, it BeComEs IncReasiNGly impOrtanT to eMpLoY A syStematiC ApPROach. in ThE quaLitAtIve or ConcEPtuAl analySis stage, a StuDEnt sHoUlD draw a pIcture or a diagRaM and get a viSuAl uNderstANDing of thE problem. At this stage, a stuDEnt shouLd cOnverT the Problem to A rePresenTatIOn that Makes fUrtheR aNalYSIs easIER. AFteR gEtting some SENse Of the SiTuatIon, labeLing all known and unkNOwn Numerical quanTitIes iS HElPfuL In MAkiNg REasONAble physical assUmptions. MaKiNG pRedictions ABouT tHe solutIon is usEful aT This levEl of analySis and it cAn Help TO StrUcture the dEcision mAking at thE Next sTAgE. The pRedIction MaDe aT this Stage cAN be CompaRed witH tHe probLem soLuTion in thE reflection phase and can hElp repAir, exTenD and organIze THe sTudent’s knOwleDge structuRe. PLanNing oR deCIsion MakiNG aBouT The apPlicABle physicS PrIncIPLeS is the next pROBLem SolviNg hEUristiC. ThiS is the stage where tHE student brings EverYTHinG toGEtheR tO come up with a reAsoNaBLE solutioN. IF the student PerformeD gOOd quaLitatiVe analYsis and PLAnNIng, the ImplEmeNtation of The PlAN becomeS eAsY If the sTudeNt PossesSes the NEcesSARy algebraic manipUlatiON And maTHemAticaL sKills. AfTEr imPlementatiOn of the plan, A studeNt muSt evaLuate hiS/hEr soluTioN, e.G., by checkinG The dimensIon or The ordeR oF magNitUde, or bY cheCKIng whEtheR tHe iNitial preDICtIOn MaDE duRing The inItIal aNalysis stAGe matcheS thE Actual sOlUtiON. one can ALsO ASk whether tHe sOlutiON Is sensible ANd, poSSiBLy, conSistenT with eXperienCE

hardiman; @reif2]. Also, m ost tradit ional co urs es donot[*explicitly*] { } te ach students effective prob le m sol v in g str ategies . R a t her ,th eyma y r eward in feriorproblem so lvi ng strategiesi nwhich many st udents engag e.Instru ct ors often im plici tly as s ume th at studen ts know t h at thea n al ysis , planning, evalu a ti o n, and reflect ion ph as e so f pr obl em solving a re as importa n ta s the implementatio n phase. Co n seq uently ,the y may n ot di sc u ssthese strat egie s explici tly wh i le solv i ng prob lems d uri ngthel ec tu re. T h ere is no mec hanism i npl ace t o en s u r e tha t s tude nts m ake a conscio useffo r t t o int erpre t th econce pts, m ake q ua litative infere nces from the qu an tit at ive p r oblemsol vin g tasks , or re l ate t h e ne w concepts to thei rp r io r knowle dge. I nor d er to de ve lop sci e n tific rea s on ing by s olving qu an titativ eproble ms , s tud entsm ustlearnto explo it pr o blem solving a s an opportuni t yf o rk nowl edg e and skill acq u isit ion. Th us, stude nts s ho u ld not treat quantitat iv e prob lem s olving merely as a math e m a tical ex erci s eb ut as a learni ng op portunitya nd theyshoul d engage in effec t i ve probl emsol vin g s t r at egies. Effec t i ve P ro blem So lvi ng Stra teg ies -- --- -- --------- -------- -- -- -- -- --- - Ef f ective p ro ble msol vingb eginswitha co nc ep t ual analys i so f the p ro blem , f ol lowed byp lan ning of the prob lem solu ti on , imple mentation and e valuationof th e plan , and last but not least reflecti o n uponthe prob lemsolving p roc ess. A s t h e comp lexity of a p hys i c s pro b l em in cr eases, itb e com es in cr easi ngly im portant to employa sy stematic appr oac h. I n th e q u al i tat iv e or c onceptual analy sis stage, a st udent shou l d d ra w a pic ture or a di a gram an d get a v isual und er stan d i ngof the pro blem. At this sta g e, as tu dentsho uld co nv ert theproble m to a re presen ta tion t hat m ak es furth er analysis easier. Aft er get tingsom e sense o f t h e s ituation, lab eling allkno wnand u nkn o wn nu meri c al qu a ntiti es i s helpfuli nmak i n greasonablep h y sic al as sum p tions. Mak ing predictions a b out the soluti on i s use ful at t hi s level of ana lys is a nd it ca nhelp to str ucture t he decis ion ma king a t the n e x ts tage.Thepre diction m ade a t this s ta ge can be com pa red wi th the prob l e m solution in th e ref l e ction pha se an dcan hel p rep air, exten d and organ ize th e st udent ’s know le dge st ruc tu re. Planni n g or deci sionmakingab outthe appli cabl e physi cs p ri nci ples is t h e n e xt p r obl em s olvin gheur istic. Th i s is the st a ge wher ethe s tudent br i n gs everyth ing toge t h er to come up w i th a rea sonabl e solu tion. I f th estudent pe r f ormed goo d qualita t ive a naly sis andpl anni ng , t he implementation of theplan bec o m es eas y i f the studen tpossesses th e nec es s ary alg e b rai c manip u lat ion an d ma the mati c al skills. After implem enta tiono f t he pl an, a s tudent m u st e valuateh is/her soluti on, e.g., by ch e cki n g th e d i me ns ion or th e o r der o f ma gni tude , o r by c heck i ng w h et herthe in itia lpr edictionma de duri n g th e in itia l ana ly si s s tage ma tches th e act ua l solutio n . One c an a ls o ask whether the solutio nis s ensibl e an d,p o ssibly, cons istent wit h e xperience

hardiman; @reif2]._Also, most_traditional courses do not_[*explicitly*]{} teach_students_effective problem_solving_strategies. Rather, they_may reward inferior_problem solving strategies in_which many students_engage._Instructors often implicitly assume that students know that the analysis, planning, evaluation, and reflection_phases_of problem_solving_are_as important as the implementation_phase. Consequently, they may not_discuss these_strategies explicitly while solving problems during the lecture._There_is no mechanism_in place to ensure that students make a conscious_effort to interpret the concepts, make_qualitative inferences from_the_quantitative_problem solving tasks, or_relate the new concepts to their_prior knowledge. In order to develop scientific_reasoning by solving quantitative problems, students must_learn to exploit problem solving as_an opportunity for knowledge and_skill acquisition._Thus, students should not treat_quantitative problem solving_merely as_a mathematical exercise_but as a learning opportunity and_they should engage_in effective problem solving strategies. Effective Problem_Solving_Strategies ------------------------------------ Effective problem solving_begins_with_a conceptual_analysis of the_problem,_followed by_planning_of the problem solution, implementation and_evaluation_of the plan, and last but not_least reflection upon the_problem_solving process. As the_complexity of a physics problem_increases, it becomes increasingly important to_employ a_systematic approach._In the qualitative or conceptual analysis stage, a student should draw_a picture or a diagram and_get a visual understanding_of the_problem._At this stage,_a_student should_convert the problem to a representation that_makes further_analysis easier. After getting some sense_of the situation, labeling_all_known and unknown numerical quantities is_helpful in making reasonable physical assumptions._Making predictions about the solution_is_useful_at this level of analysis_and it can help to structure_the decision making_at the next stage. The prediction made_at_this stage can be compared with_the_problem solution in the reflection phase_and_can_help repair, extend and organize_the student’s knowledge structure. Planning or_decision making about the applicable physics principles is the_next problem solving_heuristic. This is the stage_where_the_student brings everything together to come up with a reasonable_solution. If_the student performed_good qualitative analysis and planning, the implementation of the plan_becomes easy if the student possesses the_necessary algebraic manipulation and mathematical skills. After implementation of the plan, a_student must evaluate his/her solution, e.g., by checking_the dimension or the order of magnitude,_or by checking whether_the_initial prediction made during the initial analysis_stage_matches_the_actual solution. One_can also ask_whether the solution_is sensible_and, possibly, consistent_with experience

R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is completely continuous on $L^p(\mathbb{R}^n,\,w)$. Proof of Theorem \[t1.3\] ========================= The following lemma will be useful in the proof of Theorem \[t1.3\], and is of independent interest. \[l4.1\] Let $u\in (1,\,\infty)$, $m\in \mathbb{N}\cup\{0\}$, $S$ be a sublinear operator which satisfies that $$|Sf(x)|\leq \int_{\mathbb{R}^n}|b(x)-b(y)|^m|W(x-y)f(y)|{\rm d}y,$$ with $b\in {\rm BMO}(\mathbb{R}^n)$, and $$\begin{aligned} \sup_{R>0}R^{n/u}\Big(\int_{R\leq |x|\leq 2R}|W(x)|^{u'}{\rm d}x\Big)^{1/u'}\lesssim 1.\end{aligned}$$ - Let $p\in (u,\,\infty)$, $\lambda\in (0,\,1)$ and $w\in A_{p/u}(\mathbb{R}^n)$. If $S$ is bounded on $L^p(\mathbb{R}^n,\,w)$ with bound $D\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m$, then for some $\varepsilon\in (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}\lesssim (D+D^{\varepsilon})\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}.$$ - Let $p\in (1,\,u)$, $w^{r}\in A_1(\mathbb{R}^n)$ for some $r\in (u,\,\infty)$ and $\lambda\in (0,\,1-r'/u')$. If $S$ is bounded on $L^p(\mathbb{R}^n,\,w)$ with bound $D$, then for some $\varepsilon\in (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n)}\lesssim(D+ D

R}^n)$, $ \mathcal{M}_{\Omega,\,b}$ is completely continuous on $ L^p(\mathbb{R}^n,\,w)$. Proof of Theorem \[t1.3\ ] = = = = = = = = = = = = = = = = = = = = = = = = = The following lemma will be useful in the validation of Theorem \[t1.3\ ], and is of autonomous pastime. \[l4.1\ ] Let $ u\in (1,\,\infty)$, $ m\in \mathbb{N}\cup\{0\}$, $ S$ be a sublinear hustler which satisfies that $ $ |Sf(x)|\leq \int_{\mathbb{R}^n}|b(x)-b(y)|^m|W(x - y)f(y)|{\rm d}y,$$ with $ b\in { \rm BMO}(\mathbb{R}^n)$, and $ $ \begin{aligned } \sup_{R>0}R^{n / u}\Big(\int_{R\leq |x|\leq 2R}|W(x)|^{u'}{\rm d}x\Big)^{1 / u'}\lesssim 1.\end{aligned}$$ - get $ p\in (u,\,\infty)$, $ \lambda\in (0,\,1)$ and $ w\in A_{p / u}(\mathbb{R}^n)$. If $ S$ is bounded on $ L^p(\mathbb{R}^n,\,w)$ with bind $ D\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m$, then for some $ \varepsilon\in (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}\lesssim (D+D^{\varepsilon})\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}.$$ - Let $ p\in (1,\,u)$, $ w^{r}\in A_1(\mathbb{R}^n)$ for some $ r\in (u,\,\infty)$ and $ \lambda\in (0,\,1 - r'/u')$. If $ S$ is bounded on $ L^p(\mathbb{R}^n,\,w)$ with bandaged $ D$, then for some $ \varepsilon\in (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n)}\lesssim(D+ D

R}^n)$, $\lathcal{M}_{\Omega,\,b}$ is compleuely continuous ou $L^p(\matibb{R}^n,\,w)$. Pdoof of Gheorem \[t1.3\] ========================= The following lemma wull bt useful in the prouf of Thelrem \[t1.3\], abd iw of indepeisent inbzrest. \[m4.1\] Let $n\in (1,\,\infty)$, $m\in \msthbb{N}\cup\{0\}$, $V$ be a sublinedr o'erator which satisfies that $$|Sf(x)|\leq \ynt_{\mathnb{G}^n}|b(x)-b(y)|^m|W(x-y)f(y)|{\rm d}y,$$ eyth $g\pn {\rm BMO}(\mathbb{R}^n)$, and $$\begin{alighed} \sup_{R>0}G^{n/u}\Big(\int_{R\leq |x|\lea 2R}|W(x)|^{u'}{\rm d}x\Big)^{1/u'}\lesssim 1.\end{wligjed}$$ - Let $p\in (u,\,\infhy)$, $\lambda\in (0,\,1)$ anq $w\in A_{p/u}(\mxthbb{R}^n)$. If $S$ is boundes on $L^p(\mathbb{R}^n,\,w)$ with bound $D\|b\|_{{\ro BMO}(\kathbb{R}^n)}^m$, jkwn vmr some $\varxpsilog\in (0,\,1)$,$$\big\|Sf\npg\|_{L^{p,\,\lamtda}(\mathnb{R}^n,\,w)}\lesssim (D+V^{\varwpsilon})\|b\|_{{\rm BMO}(\mathbb{R}^i)}^m\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}.$$ - Let $p\it (1,\,b)$, $w^{r}\in A_1(\mathbb{R}^n)$ for wone $r\it (u,\,\itfty)$ qnd $\lajbva\ih (0,\,1-r'/k')$. Ih $S$ is bounsed on $L^p(\marhbb{R}^n,\,w)$ with bound $C$, eyen for some $\barepsyljn\in (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n)}\lesssim(G+ D

R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is completely continuous on $L^p(\mathbb{R}^n,\,w)$. Theorem ========================= The lemma will be Theorem and is of interest. \[l4.1\] Let (1,\,\infty)$, $m\in \mathbb{N}\cup\{0\}$, $S$ be a operator which satisfies that $$|Sf(x)|\leq \int_{\mathbb{R}^n}|b(x)-b(y)|^m|W(x-y)f(y)|{\rm d}y,$$ with $b\in {\rm BMO}(\mathbb{R}^n)$, and $$\begin{aligned} |x|\leq 2R}|W(x)|^{u'}{\rm d}x\Big)^{1/u'}\lesssim 1.\end{aligned}$$ - Let $p\in (u,\,\infty)$, $\lambda\in (0,\,1)$ and $w\in A_{p/u}(\mathbb{R}^n)$. $S$ bounded $L^p(\mathbb{R}^n,\,w)$ bound $D\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m$, then for some $\varepsilon\in (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}\lesssim (D+D^{\varepsilon})\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}.$$ - Let $p\in (1,\,u)$, $w^{r}\in A_1(\mathbb{R}^n)$ some $r\in (u,\,\infty)$ and $\lambda\in (0,\,1-r'/u')$. If $S$ bounded on $L^p(\mathbb{R}^n,\,w)$ with $D$, then for some $\varepsilon\in D

R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is completEly continuOus on $l^p(\mAthBb{r}^n,\,w)$. PRoof Of Theorem \[t1.3\] ========================= The fOLlowIng lemma will be useful in The prOoF Of ThEOrEm \[t1.3\], anD is of inDEpENDenT iNtEreSt. \[L4.1\] leT $u\in (1,\,\iNftY)$, $m\in \matHbb{N}\cup\{0\}$, $S$ be A suBlInear operatoR WhIch satisfiEs tHat $$|Sf(x)|\leq \int_{\MatHbb{R}^n}|b(X)-b(Y)|^m|W(X-Y)f(y)|{\rm D}y,$$ wIth $b\iN {\rm BMO}(\MAthbb{R}^N)$, and $$\begin{AlIGned} \suP_{r>0}R^{n/u}\Big(\INT_{R\Leq |x|\Leq 2R}|W(x)|^{u'}{\rm d}x\Big)^{1/u'}\leSSsIM 1.\end{aligned}$$ - Let $P\in (u,\,\inFtY)$, $\LaMBDa\iN (0,\,1)$ anD $w\in A_{p/u}(\matHbB{R}^n)$. If $s$ Is boundED oN $l^P(\MatHBb{R}^n,\,w)$ with bounD $D\|b\|_{{\rm BMO}(\matHBb{R}^N)}^m$, then FoR soME $\varepSilon\In (0,\,1)$,$$\BIg\|SF\big\|_{L^{p,\,\lambdA}(\matHbb{R}^n,\,w)}\lesSsim (D+D^{\VArepsilON})\|b\|_{{\rm BMO}(\Mathbb{r}^n)}^m\|F\|_{L^{p,\,\LambDA}(\mAtHbb{r}^n,\,W)}.$$ - let $P\In (1,\,U)$, $w^{r}\IN A_1(\mAthbb{R}^n)$ fOr SoMe $r\in (U,\,\infTY)$ AND $\lamBda\In (0,\,1-r'/u')$. if $S$ is Bounded on $L^p(\maThbB{R}^n,\,w)$ WIth Bound $d$, then For sOmE $\varePsilon\In (0,\,1)$,$$\big\|sf\Big\|_{L^{p,\,\lambda}(\mathBb{R}^n)}\Lesssim(D+ D

R}^n)$, $\mathcal{M}_{\Ome ga,\,b}$ i s com ple tel ycont inuo us on $L^p(\ma t hbb{ R}^n,\,w)$. Proof ofTheor em \[t1 . 3\ ] === ======= = == = = === == == === T he foll owi ng lemm a will beuse fu l in the pro o fof Theorem \[ t1.3\], andisof ind ep end e nt in ter est. \[l4. 1 \] Let $u\in (1 ,\ , \infty ) $, $m\i n \m athb b{N}\cup\{0\}$, $ S $b e a sublinearoperat or wh i c h s ati sfies that $ $|Sf( x )|\leq\ in t _ { \ma t hbb{R}^n}|b(x )-b(y)|^m|W ( x-y )f(y)| {\ rmd }y,$$with$b \ in{\rm BMO}(\ math bb{R}^n)$ , and$ $\begin { aligned } \sup _{R >0} R^{n / u} \B ig( \i n t_{ R \l eq| x|\ leq 2R}| W( x) |^{u' }{\r m d } x\Bi g)^ {1/u '}\le sssim 1.\end{ ali gned } $$ - Let $ p\in ( u,\,\ infty) $, $\ la mbda\in (0,\,1) $ an d $w\in A_ {p/ u} (\mat h bb{R}^ n)$ . I f $S$ i s bound e d o n$ L ^ p( \mathbb{R}^n,\,w)$ w i t hbound $D \|b\|_ { {\ rm BMO}(\ma th bb{ R}^n ) } ^m$,then fo r some $ \varep s il on \in ( 0,\,1) $, $$\ big \|Sf\ b ig\| _{L^{p ,\,\lamb da}(\ m athbb{R}^n,\,w ) }\lesssim (D + D ^{ \ vare psi lon})\|b\|_ {{\r m BMO }(\m a th bb{ R }^n)} ^m\|f \| _ {L ^ {p,\,\lambda}(\math bb {R}^n, \,w)} .$$ - Let$p\in (1,\ , u ) $, $w^{r }\in A_ 1 (\mathbb{R}^n) $ for some $r\i n (u,\,\i nfty) $ and $\ lambda\in (0,\, 1-r '/u ')$ . I f $S $ is boundedo n $L^ p( \mathbb {R} ^n,\,w) $ w ith bo und $ D$, thenfor some $ \v ar ep sil on\in (0,\ ,1 )$, $$ \bi g\|Sf \ big\|_ {L^{p ,\,\ la mb d a}( \mathbb { R} ^ n )}\l es ss im(D + D

R}^n)$, $\mathcal{M}_{\Omega,\,b}$_is completely_continuous on $L^p(\mathbb{R}^n,\,w)$. Proof of_Theorem \[t1.3\] ========================= The_following_lemma will_be_useful in the_proof of Theorem_\[t1.3\], and is of_independent interest. \[l4.1\] Let_$u\in_(1,\,\infty)$, $m\in \mathbb{N}\cup\{0\}$, $S$ be a sublinear operator which satisfies that $$|Sf(x)|\leq \int_{\mathbb{R}^n}|b(x)-b(y)|^m|W(x-y)f(y)|{\rm d}y,$$_with_$b\in {\rm_BMO}(\mathbb{R}^n)$,_and_$$\begin{aligned} \sup_{R>0}R^{n/u}\Big(\int_{R\leq |x|\leq 2R}|W(x)|^{u'}{\rm d}x\Big)^{1/u'}\lesssim 1.\end{aligned}$$ -_ Let $p\in (u,\,\infty)$,_$\lambda\in (0,\,1)$_and $w\in A_{p/u}(\mathbb{R}^n)$. If $S$_is_bounded on $L^p(\mathbb{R}^n,\,w)$_with bound $D\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m$, then for some $\varepsilon\in _ (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}\lesssim _(D+D^{\varepsilon})\|b\|_{{\rm BMO}(\mathbb{R}^n)}^m\|f\|_{L^{p,\,\lambda}(\mathbb{R}^n,\,w)}.$$ - __Let_$p\in (1,\,u)$, $w^{r}\in A_1(\mathbb{R}^n)$_for some $r\in (u,\,\infty)$ and $\lambda\in _ (0,\,1-r'/u')$. If $S$_is bounded on $L^p(\mathbb{R}^n,\,w)$ with bound $D$,_then for some $\varepsilon\in _ (0,\,1)$,$$\big\|Sf\big\|_{L^{p,\,\lambda}(\mathbb{R}^n)}\lesssim(D+ _D

=================== It should be noted beforehand that in the present Appendix no approximation of the potential $\Phi(\eta)$ introduced by expression  will be used, only the general properties of the kinetic coefficients $k(v)$ and $g(v)$ are taken into account. It enables us to make use of the results to be obtained here in further generalizations, e.g., to allow for the cutoff effects. Terminal fragments: the limit case $s\to 0$ and $\theta\to\infty$ {#AppG} ----------------------------------------------------------------- The limit $\theta\to\infty$ describes the situation when the upper boundary $\eta=\theta$ of the analyzed region $[0,\theta)\ni\eta$ is placed rather far away from the origin $\eta=0$ and its effect on the random particle motion is ignorable. In this case, from the general point of view, the terminal fragments shown in Fig. \[F2\] are no more than random walks starting from at a given point $\eta_0$ and reaching another point $\eta$ in a time $t$ without touching a certain boundary $\eta=\zeta$. Their probabilistic properties are described, in particular, by the probability density $\mathcal{G}(\eta,t|\eta_0,\zeta)$ of finding the random walker at the point $\eta$ in the time $t$. The Laplace transform $G(\eta,s|\eta_0,\zeta)$ of this function obeys the following forward Fokker-Planck equation matching the Langevin equation  (see, e.g., Ref. [@Gardiner]) $$\label{AppG:fFP} s G = \frac{\partial}{\partial \eta}\left[\frac{\partial G}{\partial \eta} + \alpha\frac{d\Phi(\eta)}{d\eta} G\right] + \delta(\eta-\eta_0)\,.$$ For random walks inside the layer $\mathcal{L}_\zeta=[0,\zeta)$ with the initial point $\eta_0 < \zeta$ Eq.  should be subjected to the boundary conditions \[AppG:0U\] $$\begin{aligned} \label{AppG:0Ua} \left[\frac{\partial G}{\partial \eta}+\alpha\frac{d\Phi}{d\eta} G\right]_{\eta = 0} & = 0\,, & \left. G

= = = = = = = = = = = = = = = = = = = It should be noted beforehand that in the present Appendix no approximation of the potential $ \Phi(\eta)$ insert by formula   will be used, only the general property of the kinetic coefficients $ k(v)$ and $ g(v)$ are taken into score. It enable us to make use of the results to be obtained here in further generalizations, for example, to allow for the cutoff effects. Terminal fragments: the terminus ad quem case $ s\to 0 $ and $ \theta\to\infty$ { # AppG } ----------------------------------------------------------------- The limit $ \theta\to\infty$ trace the situation when the upper boundary $ \eta=\theta$ of the analyze region $ [ 0,\theta)\ni\eta$ is placed rather far aside from the origin $ \eta=0 $ and its effect on the random particle gesture is ignorable. In this case, from the general point of view, the terminal fragment shown in Fig.   \[F2\ ] are no more than random walks starting from at a given point $ \eta_0 $ and reaching another point $ \eta$ in a time $ t$ without touching a certain boundary $ \eta=\zeta$. Their probabilistic properties are described, in particular, by the probability density $ \mathcal{G}(\eta, t|\eta_0,\zeta)$ of finding the random walker at the detail $ \eta$ in the time $ t$. The Laplace transform $ G(\eta, s|\eta_0,\zeta)$ of this routine obey the follow forward Fokker - Planck equation match the Langevin equation   (see, e.g., Ref.   [ @Gardiner ]) $ $ \label{AppG: fFP } s G = \frac{\partial}{\partial \eta}\left[\frac{\partial G}{\partial \eta } + \alpha\frac{d\Phi(\eta)}{d\eta } G\right ] + \delta(\eta-\eta_0)\,.$$ For random walks inside the level $ \mathcal{L}_\zeta=[0,\zeta)$ with the initial point $ \eta_0 < \zeta$ Eq.   should be subjected to the boundary conditions \[AppG:0U\ ] $ $ \begin{aligned } \label{AppG:0Ua } \left[\frac{\partial G}{\partial \eta}+\alpha\frac{d\Phi}{d\eta } G\right]_{\eta = 0 } & = 0\, , & \left. G

=================== It dhould be noted beforehakd that in the ptewent A'pendix no appruximation of the potential $\Pii(\etq)$ inteoduced by expression  dill be uded, only the teneral properties of ths kinztmc coefficients $k(v)$ and $g(v)$ are taken intm xceount. It enables us to make use of tre resuktd to be obtaingd hege in rlruher generalizations, e.g., to allow ror the cutoff effecys. Terminal fragments: the llmit case $s\to 0$ and $\theha\to\infty$ {#AklG} ----------------------------------------------------------------- Trw limit $\thetx\to\infty$ dtseribes the aituation when the upper boundafy $\etc=\theta$ of tye anwnyzed regioi $[0,\thetw)\ni\eta$ is placed ratver far away from the ormgin $\eta=0$ and its effect oi the random particlg motion iv ngnorable. In this casw, drom jhe ganerxo puinu oh vjew, thf txrminal frafments showb in Fig. \[F2\] are no mote nnan random wzlks seawting from at a given point $\eta_0$ and reabhinf another point $\eta$ in q time $t$ without toucjing a cewtain boundary $\eta=\zeta$. Their probabilistic properdies erd dtsgvibea, ij particular, by the probability density $\mathcwm{G}(\tta,n|\eta_0,\zeta)$ of findikg the random walkrr ay the point $\etx$ in tks fime $t$. The Laplace transfjrm $G(\wta,s|\eta_0,\zeua)$ of this function obeys the foolowing forwcrd Fokker-Planck equacion matchiny the Kangefin equation  (see, e.g., Ref. [@Yardinsr]) $$\label{AppH:fFP} s G = \wrac{\partial}{\partixl \vta}\laft[\frac{\partial G}{\partial \etw} + \alpha\fcac{d\Pki(\eta)}{d\etx} G\roght] + \qelta(\eta-\etw_0)\,.$$ For random walks insidf the ldyer $\mathcwl{L}_\zeta=[0,\zeta)$ with the initial pomit $\eta_0 < \zeta$ Gq.  vholld be suyjectec to the bounqary conditionx \[AppG:0U\] $$\yegin{auigned} \labem{AppG:0Ua} \left[\frac{\[artial G}{\parthwl \eta}+\alpha\fcac{d\Phi}{d\eea} G\eighr]_{\eta = 0} & = 0\,, & \left. G

=================== It should be noted beforehand that present no approximation the potential $\Phi(\eta)$ used, the general properties the kinetic coefficients and $g(v)$ are taken into account. enables us to make use of the results to be obtained here in generalizations, e.g., to allow for the cutoff effects. Terminal fragments: the limit case 0$ $\theta\to\infty$ ----------------------------------------------------------------- limit $\theta\to\infty$ describes the situation when the upper boundary $\eta=\theta$ of the analyzed region $[0,\theta)\ni\eta$ is rather far away from the origin $\eta=0$ and effect on the random motion is ignorable. In this from general point view, terminal shown in Fig. are no more than random walks starting from at a given point $\eta_0$ and reaching another point in a without touching certain $\eta=\zeta$. probabilistic properties are particular, by the probability density $\mathcal{G}(\eta,t|\eta_0,\zeta)$ random walker at the point $\eta$ in the $t$. The transform $G(\eta,s|\eta_0,\zeta)$ of this function obeys following forward Fokker-Planck equation matching the Langevin equation e.g., Ref. [@Gardiner]) $$\label{AppG:fFP} s G = \frac{\partial}{\partial \eta}\left[\frac{\partial G}{\partial \eta} + \alpha\frac{d\Phi(\eta)}{d\eta} G\right] + random walks inside the $\mathcal{L}_\zeta=[0,\zeta)$ with the point < Eq. be subjected the boundary conditions \[AppG:0U\] $$\begin{aligned} \label{AppG:0Ua} \left[\frac{\partial G}{\partial \eta}+\alpha\frac{d\Phi}{d\eta} G\right]_{\eta = & = 0\,, & \left. G

=================== It should be noted beforehand That in the pResenT ApPenDiX no aPproXimation of the pOTentIal $\Phi(\eta)$ introduced by eXpresSiON  wilL Be Used, oNly the gENeRAL prOpErTieS oF ThE kineTic CoefficIents $k(v)$ and $G(v)$ aRe Taken into accOUnT. It enables Us tO make use of thE reSults tO bE obTAined HerE in fuRther gENeraliZations, e.g., To ALlow foR The cutoFF EfFectS. Terminal fragmentS: ThE Limit case $s\to 0$ anD $\theta\To\INfTY$ {#appg} ----------------------------------------------------------------- ThE limit $\thetA\tO\inftY$ DescribES tHE SItuATion when the upPer boundary $\ETa=\tHeta$ of ThE anALyzed rEgion $[0,\ThETa)\nI\eta$ is placeD ratHer far awaY from tHE origin $\ETa=0$ and itS effecT on The RandOM pArTicLe MOtiON iS igNOraBle. In thiS cAsE, from The gENERAl poInt Of viEw, the Terminal fragmEntS shoWN in fig. \[F2\] aRe no mOre tHaN randOm walkS starTiNg from at a given pOint $\Eta_0$ and reaChiNg AnoThEr poiNT $\eta$ in A tiMe $t$ Without TouchinG A ceRtAIN BoUndary $\eta=\zeta$. Their PrOBAbIlistic pRopertIEs ArE DescribeD, iN paRticULAr, by tHe prOBaBility deNsity $\mAThCaL{G}(\eta,t|\eTa_0,\Zeta)$ of FiNdiNg tHe ranDOm waLker at The point $\Eta$ in THe time $t$. The LaplACe transform $G(\eTA,s|\ETA_0,\zETa)$ of ThiS function obEys tHE folLowiNG fOrwARd FokKer-PlAnCK eQUation matching the LaNgEvin eqUatioN  (see, e.g., Ref. [@GardIner]) $$\label{APPg:FFP} s G = \fraC{\parTIaL}{\Partial \eta}\left[\Frac{\pArtial G}{\parTIal \eta} + \alPha\frAc{d\Phi(\etA)}{d\eta} G\rigHT] + \Delta(\eta-\Eta_0)\,.$$ for RanDom WALkS inside the layER $\MathCaL{L}_\zeta=[0,\zEta)$ With the IniTiaL poInt $\EtA_0 < \zeta$ Eq.  shOuld be suBjEcTeD tO thE bounDAry condiTiOns \[apPG:0U\] $$\Begin{ALigned} \Label{appG:0ua} \LeFT[\frAc{\partiAL G}{\PARtiaL \eTa}+\AlphA\frAc{D\Phi}{d\Eta} G\RIghT]_{\eta = 0} & = 0\,, & \lefT. G

=================== It sh ould be no ted b efo reh an d th at i n the presentA ppen dix no approximation o f the p o tent i al $\Ph i(\eta) $ i n t rod uc ed by e x pr essio n will be used, onl y t he general pro p er ties of th e k inetic coeff ici ents $ k( v)$ and $ g(v )$ ar e take n intoaccount.It enable s us tom a ke use of the results t o b e obtained here in fu rt h er g ene ral izations,e. g., t o allowf or t h e c u toff effects. Terminalf rag ments: t hel imit c ase $ s\ t o 0 $ and $\the ta\t o\infty${#AppG } ------ - ------- ------ --- --- ---- - -- -- --- -- - --- - -- --- - --- -------- -- - Thelimi t $ \ thet a\t o\in fty$describes the si tuat i onwhenthe u pper b ounda ry $\e ta=\t he ta$ of the anal yzed region $ [0, \t het a) \ni\e t a$ ispla ced rather far aw a y f ro m t he origin $\eta=0$ a nd i ts effecton the ra nd o m partic le mo tion i s ign orab l e. In this case, fr om the ge ne ral po in t o f v iew,t he t ermina l fragme nts s h own in Fig. \[ F 2\] are no mo r et h an rand omwalks start ingf romat a gi ven point $\et a_ 0 $a nd reaching another p oint $ \eta$ in a time $t $ withoutt o u ching acert a in boundary $\eta =\zet a$. Theirp robabili sticproperti es are de s c ribed, i n p art icu lar , by the probabil i t y de ns ity $\m ath cal{G}( \et a,t |\e ta_ 0, \zeta)$ o f findin gth era ndo m wal k er at th epoi nt $\ eta$i n thetime$t$. T he Lap lace tr a ns f o rm $ G( \e ta,s |\e ta _0,\z eta) $ of this f unction o bey s the f ol lowingforward Fokke r- Planck equ at ion match i n g the La ngevin equation  (see,e .g., Re f.[@Gar dine r]) $$\la bel {AppG: fFP } s G= \fra c{\pa rt ial } { \part i a l\et a} \left[\fra c { \pa rtial G }{\p artial\eta} + \alpha\fra c {d\ Phi(\eta)}{d\ eta } G\ r i gh t]+ \ d elt a( \ eta - \ eta_0)\,.$$ For random wa lk s i nside thel aye r$\mathc al{L}_\ zeta= [ 0,\zeta )$ with t he initia lpoin t $\e ta_0 < \ze ta$ Eq. should b e subj e ct ed to th e boun da rycondi tions\[A ppG:0 U\] $$ \b egin{a ligne d} \label{ AppG:0Ua} \left[\fra c{\par tialG}{ \partial\et a }+\ alpha\fra c{d\ Phi}{d\eta } G \ri ght]_ {\e t a = 0 } &= 0 \,, & \le f t. G

=================== It should_be noted_beforehand that in the_present Appendix_no_approximation of_the_potential $\Phi(\eta)$ introduced_by expression  will_be used, only the_general properties of_the_kinetic coefficients $k(v)$ and $g(v)$ are taken into account. It enables us to make_use_of the_results_to_be obtained here in further_generalizations, e.g., to allow for_the cutoff_effects. Terminal fragments: the limit case $s\to 0$ and_$\theta\to\infty$_{#AppG} ----------------------------------------------------------------- The limit $\theta\to\infty$_describes the situation when the upper boundary $\eta=\theta$ of_the analyzed region $[0,\theta)\ni\eta$ is placed_rather far away_from_the_origin $\eta=0$ and its_effect on the random particle motion_is ignorable. In this case, from_the general point of view, the terminal_fragments shown in Fig. \[F2\] are no_more than random walks starting_from at_a given point $\eta_0$ and_reaching another point_$\eta$ in_a time $t$_without touching a certain boundary $\eta=\zeta$._Their probabilistic properties_are described, in particular, by the_probability_density $\mathcal{G}(\eta,t|\eta_0,\zeta)$ of_finding_the_random walker_at the point_$\eta$_in the_time_$t$. The Laplace transform $G(\eta,s|\eta_0,\zeta)$ of_this_function obeys the following forward Fokker-Planck equation_matching the Langevin equation _(see,_e.g., Ref. [@Gardiner]) $$\label{AppG:fFP} s_G = \frac{\partial}{\partial \eta}\left[\frac{\partial G}{\partial_\eta} + \alpha\frac{d\Phi(\eta)}{d\eta} G\right] + \delta(\eta-\eta_0)\,.$$_For random_walks inside_the layer $\mathcal{L}_\zeta=[0,\zeta)$ with the initial point $\eta_0 < \zeta$ Eq. _should be subjected to the boundary_conditions \[AppG:0U\] $$\begin{aligned} \label{AppG:0Ua} _\left[\frac{\partial G}{\partial_\eta}+\alpha\frac{d\Phi}{d\eta}_G\right]_{\eta = 0}_&_= 0\,,_& \left. G

\in \operatorname{\mathsf{Mat}}_{e \times e}({\mathbbm{k}}), B \in \operatorname{\mathsf{Mat}}_{e \times d}({\mathbbm{k}}) \\ C \in \operatorname{\mathsf{Mat}}_{d \times d}({\mathbbm{k}}) \end{array} \; \mbox{and} \; {\mathsf{tr}}(A) + {\mathsf{tr}}(C) = 0 \right\}.$$ The goal of this section is to prove the following result. \[T:FrobAlg\] Let $J = J_{(e, d)}$ be the matrix from (\[E:formcanonique\]). Then the pairing $$\label{E:PairingDeFrobenus} \omega_J: \operatorname{\mathfrak{p}}\times \operatorname{\mathfrak{p}}{\longrightarrow}{\mathbbm{k}}, \quad (a, b) \mapsto {\mathsf{tr}}\bigl(J^t \cdot [a, b]\bigr)$$ is non-degenerate. In other words, $\operatorname{\mathfrak{p}}$ is a Frobenius Lie algebra and $$\label{E:FrobFunct} l_J: \operatorname{\mathfrak{p}}\rightarrow {\mathbbm{k}}, \quad a \mapsto {\mathsf{tr}}(J^t \cdot a)$$ is a Frobenius functional on $\operatorname{\mathfrak{p}}$. In this section, we shall use the following notations and conventions. For a finite dimensional vector space $\operatorname{\mathfrak{w}}$ be denote by $\operatorname{\mathfrak{w}}^*$ the dual vector space. If $\operatorname{\mathfrak{w}}= \operatorname{\mathfrak{w}}_1 \oplus \operatorname{\mathfrak{w}}_2$ then we have a canonical isomorphism $\operatorname{\mathfrak{w}}^* \cong \operatorname{\mathfrak{w}}_1^* \oplus \operatorname{\mathfrak{w}}_2^*$. For a functional $\hat{w}_i \in \operatorname{\mathfrak{w}}_i^*, i = 1, 2$ we denote by the same symbol its *extension by zero* on the whole $\operatorname{\mathfrak{w}}$. Assume we have the following set-up. - $\operatorname{\mathfrak{f}}$ is a finite dimensional Lie algebra. - $\operatorname{\mathfrak{l}}\subset \operatorname{\mathfrak{f}}$ is a Lie subalgebra and $\operatorname{\mathfrak{n}}\subset

\in \operatorname{\mathsf{Mat}}_{e \times e}({\mathbbm{k } }), B \in \operatorname{\mathsf{Mat}}_{e \times d}({\mathbbm{k } }) \\ C \in \operatorname{\mathsf{Mat}}_{d \times d}({\mathbbm{k } }) \end{array } \; \mbox{and } \; { \mathsf{tr}}(A) + { \mathsf{tr}}(C) = 0 \right\}.$$ The goal of this section is to prove the following resultant role. \[T: FrobAlg\ ] lease $ J = J_{(e, d)}$ be the matrix from (\[E: formcanonique\ ]). Then the pairing $ $ \label{E: PairingDeFrobenus } \omega_J: \operatorname{\mathfrak{p}}\times \operatorname{\mathfrak{p}}{\longrightarrow}{\mathbbm{k } }, \quad (a, b) \mapsto { \mathsf{tr}}\bigl(J^t \cdot [ a, b]\bigr)$$ is non - degenerate. In other quarrel, $ \operatorname{\mathfrak{p}}$ is a Frobenius Lie algebra and $ $ \label{E: FrobFunct } l_J: \operatorname{\mathfrak{p}}\rightarrow { \mathbbm{k } }, \quad a \mapsto { \mathsf{tr}}(J^t \cdot a)$$ is a Frobenius functional on $ \operatorname{\mathfrak{p}}$. In this section, we shall use the surveil notations and conventions. For a finite dimensional vector outer space $ \operatorname{\mathfrak{w}}$ be denote by $ \operatorname{\mathfrak{w}}^*$ the dual vector quad. If $ \operatorname{\mathfrak{w}}= \operatorname{\mathfrak{w}}_1 \oplus \operatorname{\mathfrak{w}}_2 $ then we have a canonical isomorphism $ \operatorname{\mathfrak{w}}^ * \cong \operatorname{\mathfrak{w}}_1^ * \oplus \operatorname{\mathfrak{w}}_2^*$. For a running $ \hat{w}_i \in \operatorname{\mathfrak{w}}_i^ *, i = 1, 2 $ we denote by the same symbol its * elongation by zero * on the whole $ \operatorname{\mathfrak{w}}$. Assume we suffer the following set - up. - $ \operatorname{\mathfrak{f}}$ is a finite dimensional Lie algebra. - $ \operatorname{\mathfrak{l}}\subset \operatorname{\mathfrak{f}}$ is a Lie subalgebra and $ \operatorname{\mathfrak{n}}\subset

\in \operatorname{\mathsf{Mat}}_{e \uimes e}({\mathbbm{k}}), B \in \opecatornaje{\mathsf{Oat}}_{e \times d}({\mathbbm{k}}) \\ C \in \operatirname{\mathsf{Mat}}_{d \times d}({\mathbbm{n}}) \end{arrqy} \; \nvox{and} \; {\mefhsf{tr}}(A) + {\mathan{tr}}(C) = 0 \right\}.$$ The goak of this vection is to [ruvz the following result. \[T:FrobAlg\] Let $J = J_{(e, d)}$ ne the matrix frjm (\[E:gjrmcznonique\]). Then the pairing $$\label{E:PajringDeHrobenus} \omega_J: \pperatorname{\mathfrak{p}}\times \opegatorname{\mathfrak{p}}{\pongrightartkw}{\mwrhbbm{k}}, \quad (a, b) \mapsto {\mathsf{tr}}\bigl(N^t \cdot [a, b]\bigr)$$ is non-degeneratd. In pther wordw, $\ipegdtorname{\matifrak{p}}$ is a Frobenlls Lie dlgebra and $$\label{E:FronFuncv} l_J: \iperatorname{\mathfrak{p}}\cightarrow {\mathbbm{k}}, \quad a \mdpato {\mathsf{tr}}(J^t \cdit a)$$ is a Fsobevuus fuhcviohal on $\opxratorname{\mzthfrak{p}}$. In rhis section, we shakl lxe the follosing njtwtions and conventions. For a finite dimtnsiohal vector space $\operatirname{\mathfrak{w}}$ be dejote by $\o[eratorname{\mathfrak{w}}^*$ the dual vector space. If $\opesatoriaoe{\mcbmfray{q}}= \lperatorname{\mathfrak{w}}_1 \oplus \operatorname{\mathfwzk{e}}_2$ nhen we have a cakonical isomorphisk $\lprtatorname{\mathftak{w}}^* \couf \kperatorname{\mathfrwk{w}}_1^* \oplos \opeeatorname{\iathgrak{w}}_2^*$. For a functional $\hat{w}_u \in \operatognamw{\mathfrak{w}}_i^*, i = 1, 2$ wz denote by che sake sykbol its *extension by zzro* on the whole $\lperatornzoe{\mathfrak{w}}$. Assumd wv haee the following set-up. - $\o[eratornane{\machfrak{f}}$ ks a finitq dimensiojal Lla algebra. - $\operatlrnamg{\mathfsak{l}}\subset \operatorname{\mathfrak{f}}$ is a Lie subalgebra anc $\mpegatorname{\iathfvak{n}}\subset

\in \operatorname{\mathsf{Mat}}_{e \times e}({\mathbbm{k}}), B \in \operatorname{\mathsf{Mat}}_{e \\ \in \operatorname{\mathsf{Mat}}_{d d}({\mathbbm{k}}) \end{array} \; = \right\}.$$ The goal this section is prove the following result. \[T:FrobAlg\] Let = J_{(e, d)}$ be the matrix from (\[E:formcanonique\]). Then the pairing $$\label{E:PairingDeFrobenus} \omega_J: \operatorname{\mathfrak{p}}{\longrightarrow}{\mathbbm{k}}, \quad (a, b) \mapsto {\mathsf{tr}}\bigl(J^t \cdot [a, b]\bigr)$$ is non-degenerate. In other $\operatorname{\mathfrak{p}}$ a Lie and $$\label{E:FrobFunct} l_J: \operatorname{\mathfrak{p}}\rightarrow {\mathbbm{k}}, \quad a \mapsto {\mathsf{tr}}(J^t \cdot a)$$ is a Frobenius functional on In this section, we shall use the following and conventions. For a dimensional vector space $\operatorname{\mathfrak{w}}$ be by the dual space. $\operatorname{\mathfrak{w}}= \oplus \operatorname{\mathfrak{w}}_2$ then have a canonical isomorphism $\operatorname{\mathfrak{w}}^* \cong \operatorname{\mathfrak{w}}_1^* \oplus \operatorname{\mathfrak{w}}_2^*$. For a functional $\hat{w}_i \in \operatorname{\mathfrak{w}}_i^*, i = 2$ we the same its by on the whole we have the following set-up. - finite dimensional Lie algebra. - $\operatorname{\mathfrak{l}}\subset \operatorname{\mathfrak{f}}$ is Lie subalgebra $\operatorname{\mathfrak{n}}\subset

\in \operatorname{\mathsf{Mat}}_{e \tImes e}({\mathbBm{k}}), B \iN \opEraToRnamE{\matHsf{Mat}}_{e \times d}({\mAThbbM{k}}) \\ C \in \operatorname{\mathsF{Mat}}_{d \TiMEs d}({\mAThBbm{k}}) \eNd{array} \; \MBoX{ANd} \; {\mAtHsF{tr}}(a) + {\mAThSf{tr}}(C) = 0 \RigHt\}.$$ The goAl of this seCtiOn Is to prove the FOlLowing resuLt. \[T:frobAlg\] Let $J = J_{(E, d)}$ bE the maTrIx fROm (\[E:foRmcAnoniQue\]). TheN The paiRing $$\label{e:PAIringDEfrobenuS} \OMeGa_J: \oPeratorname{\mathfrAK{p}}\TImes \operatornaMe{\mathFrAK{p}}{\LONgrIghTarrow}{\mathBbM{k}}, \quaD (A, b) \mapstO {\MaTHSF{tr}}\BIgl(J^t \cdot [a, b]\biGr)$$ is non-degeNEraTe. In otHeR woRDs, $\operAtornAmE{\MatHfrak{p}}$ is a FrObenIus Lie algEbra anD $$\Label{E:FRObFunct} L_J: \operAtoRnaMe{\maTHfRaK{p}}\rIgHTarROw {\MatHBbm{K}}, \quad a \maPsTo {\MathsF{tr}}(J^T \CDOT a)$$ is A FrObenIus fuNctional on $\opeRatOrnaME{\maThfraK{p}}$. In tHis sEcTion, wE shall Use thE fOllowing notatioNs anD conventiOns. foR a fInIte diMEnsionAl vEctOr space $\OperatoRNamE{\mATHFrAk{w}}$ be denote by $\operaToRNAmE{\mathfraK{w}}^*$ the dUAl VeCTor space. if $\OpeRatoRNAme{\maThfrAK{w}}= \OperatorName{\maTHfRaK{w}}_1 \oplus \OpEratorNaMe{\mAthFrak{w}}_2$ THen wE have a CanonicaL isomORphism $\operatorNAme{\mathfrak{w}}^* \cONg \OPErATornAme{\Mathfrak{w}}_1^* \opLus \oPEratOrnaME{\mAthFRak{w}}_2^*$. FOr a fuNcTIoNAl $\hat{w}_i \in \operatornaMe{\MathfrAk{w}}_i^*, i = 1, 2$ We denote by the Same symbol ITS *ExtensioN by zERo* ON the whole $\operaTornaMe{\mathfrak{W}}$. assume we Have tHe followIng set-up. - $\oPERatornamE{\maThfRak{F}}$ is A FInIte dimensionaL lIe alGeBra. - $\operAtoRname{\maThfRak{L}}\suBseT \oPeratornaMe{\mathfrAk{F}}$ iS a liE suBalgeBRa and $\opeRaTorNaMe{\mAthfrAK{n}}\subsEt

\in \operatorname{\mathsf {Mat}}_{e\time s e }({ \m athb bm{k }}), B \in \op e rato rname{\mathsf{Mat}}_{e \tim es d}({ \ ma thbbm {k}}) \ \ C \ in \ ope ra t or name{ \ma thsf{Ma t}}_{d \ti mes d }({\mathbbm{ k }} ) \end{arr ay} \; \mbox{a nd} \; { \m ath s f{tr} }(A ) + { \maths f {tr}}( C) = 0 \r ig h t\}.$$ The goa l of thi s section is to p r ov e the following resul t. \ [ T :Fr obA lg\] Let $ J= J_{ ( e, d)}$ be t h e m a trix from (\[ E:formcanon i que \]). T he n t h e pair ing $ $\ l abe l{E:Pairing DeFr obenus} \ omega_ J : \oper a torname {\math fra k{p }}\t i me s\op er a tor n am e{\ m ath frak{p}} {\ lo ngrig htar r o w } {\ma thb bm{k }}, \ quad (a, b) \ map sto{ \ma thsf{ tr}}\ bigl (J ^t \c dot [a , b]\ bi gr)$$ is non-de gene rate. Inoth er wo rd s, $\ o perato rna me{ \mathfr ak{p}}$ isaF r o be nius Lie algebra a nd $ $\ label{E: FrobFu n ct }l _J: \ope ra tor name { \ mathf rak{ p }} \rightar row {\ m at hb bm{k}}, \quada\ma pst o {\m a thsf {tr}}( J^t \cdo t a)$ $ is a Frobeniu s functional o n $ \ o pe r ator nam e{\mathfrak {p}} $ . I n th i ssec t ion,we sh al l u s e the following not at ions a nd co nventions. Fo r a finite d i mensiona l ve c to r space $\opera torna me{\mathfr a k{w}}$ b e den ote by $ \operator n a me{\math fra k{w }}^ *$t h edual vector s p a ce.If $\oper ato rname{\ mat hfr ak{ w}} =\operator name{\ma th fr ak {w }}_ 1 \op l us \oper at orn am e{\ mathf r ak{w}} _2$ t henwe h a vea canon i ca l isom or ph ism$\o pe rator name { \ma thfrak{ w}}^* \co ng\ oper at or name{\m athfrak{w}}_1 ^* \oplus \o pe rat orname { \ mathfrak {w}}_2^*$. For a functi o nal $\h at{ w}_i\in\operator nam e{\mat hfr a k{w}}_ i^*, i = 1, 2 $ w e denot e by th esame symbo l its *ext en sion by zer o* on the whole $\ o per atorname{\mat hfr ak{w } } $. A s su m e w eh ave t he following se t-up. - $ \ op eratorname { \ma th frak{f} }$ is a fini t e dimen sional Li e algebra . - $ \op eratorname {\mathfr ak{l}}\su b set \ o pe rator nam e{\mat hf rak {f}}$ is aL iesubal gebraan d $\op erato rn ame{\mat hfrak{n}}\subset

\in_\operatorname{\mathsf{Mat}}_{e \times_e}({\mathbbm{k}}), B \in \operatorname{\mathsf{Mat}}_{e_\times d}({\mathbbm{k}})_\\ _ _C_\in \operatorname{\mathsf{Mat}}_{d \times_d}({\mathbbm{k}}) \end{array} \;_\mbox{and} \; {\mathsf{tr}}(A)_+ {\mathsf{tr}}(C) =_0_\right\}.$$ The goal of this section is to prove the following result. \[T:FrobAlg\] Let $J_=_J_{(e, d)}$_be_the_matrix from (\[E:formcanonique\]). Then the_pairing $$\label{E:PairingDeFrobenus} \omega_J: \operatorname{\mathfrak{p}}\times \operatorname{\mathfrak{p}}{\longrightarrow}{\mathbbm{k}}, \quad (a,_b) \mapsto_{\mathsf{tr}}\bigl(J^t \cdot [a, b]\bigr)$$ is non-degenerate. In other_words,_$\operatorname{\mathfrak{p}}$ is a_Frobenius Lie algebra and $$\label{E:FrobFunct} l_J: \operatorname{\mathfrak{p}}\rightarrow {\mathbbm{k}}, \quad_a \mapsto {\mathsf{tr}}(J^t \cdot a)$$ is_a Frobenius functional_on_$\operatorname{\mathfrak{p}}$. In_this section, we shall_use the following notations and conventions._For a finite dimensional vector space_$\operatorname{\mathfrak{w}}$ be denote by $\operatorname{\mathfrak{w}}^*$ the dual_vector space. If $\operatorname{\mathfrak{w}}= \operatorname{\mathfrak{w}}_1 \oplus_\operatorname{\mathfrak{w}}_2$ then we have a_canonical isomorphism_$\operatorname{\mathfrak{w}}^* \cong \operatorname{\mathfrak{w}}_1^* \oplus \operatorname{\mathfrak{w}}_2^*$._For a functional_$\hat{w}_i \in_\operatorname{\mathfrak{w}}_i^*, i =_1, 2$ we denote by the_same symbol its_*extension by zero* on the whole_$\operatorname{\mathfrak{w}}$. Assume_we have the_following_set-up. -_ _$\operatorname{\mathfrak{f}}$ is a_finite_dimensional Lie_algebra. -_ $\operatorname{\mathfrak{l}}\subset \operatorname{\mathfrak{f}}$ is a_Lie_subalgebra and $\operatorname{\mathfrak{n}}\subset

*Wavelet Transforms and Localization Operators.* Birkhäuser Verlag, vol. 136, Basel, 2002. M.-W. Wong *An introduction to pseudo-differential operators.* Series on Analysis, Applications and Computation, 6. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. [^1]: E-mail addresses: paolo.boggiatto@unito.it, evanthia.carypis@unito.it, alessandro.oliaro@unito.it --- abstract: 'The two-pion correlation function can be defined as a ratio of either the measured momentum distributions or the normalized momentum space probabilities. We show that the first alternative avoids certain ambiguities since then the normalization of the two-pion correlator contains important information on the multiplicity distribution of the event ensemble which is lost in the second alternative. We illustrate this explicitly for specific classes of event ensembles.' address: 'Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany' author: - 'Q.H. Zhang, P. Scotto and U. Heinz' title: 'Multi-boson effects and the normalization of the two-pion correlation function' --- PACS numbers: 25.75.-q, 25.75.Gz, 25.70.Pq. Introduction {#sec1} ============ Two-particle Bose-Einstein (BE) interferometry (also known as Hanbury Brown-Twiss (HBT) intensity interferometry) as a method for obtaining information on the space-time geometry and dynamics of high energy collisions has recently received intensive theoretical and experimental attention. Detailed investigations revealed that high-quality two-particle correlation data constrain not only the geometric size of the particle-emitting source but also its dynamical state at particle freeze-out [@Zajc86; @Lorstad89; @BGJ; @APW; @Pratt95; @He96]. Two different definitions of two-pion correlation function are employed in the literature [@Zajc86; @Lorstad89; @BGJ; @APW; @Pratt95; @He96; @GKW79; @Zajc84; @Mark; @UA1; @Padula; @Alex93; @CDL

* Wavelet Transforms and Localization Operators. * Birkhäuser Verlag, vol. 136, Basel, 2002. M.-W. Wong * An introduction to pseudo - differential operators. * Series on Analysis, Applications and Computation, 6. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. [ ^1 ]: E - mail address: paolo.boggiatto@unito.it, evanthia.carypis@unito.it, alessandro.oliaro@unito.it --- abstract:' The two - pion correlation coefficient function can be defined as a proportion of either the measured momentum distribution or the normalized momentum space probability. We show that the first option avoid certain ambiguity since then the normalization of the two - pion correlator contains crucial data on the multiplicity distribution of the event corps de ballet which is lost in the second option. We exemplify this explicitly for specific classes of event ensembles.' address:' Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany' generator: -' Q.H. Zhang, P. Scotto and U. Heinz' title:' Multi - boson effects and the normalization of the two - pion correlation affair' --- PACS numbers: 25.75.-q, 25.75.Gz, 25.70.Pq. Introduction { # sec1 } = = = = = = = = = = = = Two - particle Bose - Einstein (BE) interferometry (also known as Hanbury Brown - Twiss (HBT) volume interferometry) as a method for obtaining information on the space - time geometry and dynamics of high energy collisions has recently received intensive theoretical and experimental attention. Detailed investigations revealed that high - quality two - particle correlation datum restrain not only the geometric size of the atom - utter generator but also its dynamical state at particle freeze - out [ @Zajc86; @Lorstad89; @BGJ; @APW; @Pratt95; @He96 ]. Two different definitions of two - pion correlation affair are employed in the literature [ @Zajc86; @Lorstad89; @BGJ; @APW; @Pratt95; @He96; @GKW79; @Zajc84; @Mark; @UA1; @Padula; @Alex93; @CDL

*Wagelet Transforms and Locxlization Operajoes.* Birnhäuser Verlag, xol. 136, Basel, 2002. M.-W. Wong *An introdnctiin to pseudo-differential opdrators.* Svries on Qnalbsis, Applications and Computatjln, 6. Cocld Scientific Kublishing Cm. Pte. Ltd., Hackanraek, NJ, 2014. [^1]: E-mail addresses: paolo.boggiattj@unito.iy, fvanthia.carypif@uniuo.ie, alsssandro.oliaro@unito.it --- abstract: 'The two-pioi correlation finction can be defined as w rahio of either the leasured monenttn distributiuns or the normalized jomentum space probabilities. We show that the diest dlternative avoiqs certain ambiguitias sincr then the norkalmzatuon of the two-pion cocrelator contains imkortant inxoxmation on the multipoixity gistsibuguon of tie svent fnsxmble which is lost in the second alternauivq. We illustrats this evplicitly for specific classes of event enaembles.' address: 'Institut für Theoretische Physlk, Univerfität Regensburg, D-93040 Regensburg, Germany' author: - 'Q.H. Zhatg, P. Azotuo and J. Hfinz' title: 'Multi-boson effects and the normalizwfipn of the two-piok correlation funcyiln' --- LWCS numbers: 25.75.-q, 25.75.Gz, 25.70.Pq. Nhtdoduction {#sec1} ============ Two-pagticle Fose-Eunstein (BT) intrrferometry (also known as Hqnbury Brown-Nwisw (HBT) intensity incerferometry) as s metnod for obtaining inforoatikn on the soace-time fdometry and dynaoicx mf high tvergy collisions ras recenvly rzceived kntemsive eheoreticap and experimental attenhion. Betained investlgations revealed that high-qualmvy two-particlg cmrrvlation dcta cokstrain not onlr the geometrie size oy the oarticle-emptting sonrce but alsj its dynamicdp state at perticle fweezw-out [@Zajc86; @Lufstad89; @BGJ; @APW; @Lratt95; @He96]. Tcj didferent definitionx ow two-pion correlcuiob function are rmpuoyqd ii the niterature [@Zdjc86; @Uorryad89; @BEJ; @APW; @Kxatb95; @Hd96; @GKE79; @Zajc84; @Mark; @UA1; @Paduld; @Alsx93; @CDL

*Wavelet Transforms and Localization Operators.* Birkhäuser Verlag, Basel, M.-W. Wong introduction to pseudo-differential and 6. World Scientific Co. Pte. Ltd., NJ, 2014. [^1]: E-mail addresses: paolo.boggiatto@unito.it, alessandro.oliaro@unito.it --- abstract: 'The two-pion correlation function can be defined as a ratio either the measured momentum distributions or the normalized momentum space probabilities. We show the alternative certain since then the normalization of the two-pion correlator contains important information on the multiplicity distribution of event ensemble which is lost in the second We illustrate this explicitly specific classes of event ensembles.' 'Institut Theoretische Physik, Regensburg, Regensburg, author: - 'Q.H. P. Scotto and U. Heinz' title: 'Multi-boson effects and the normalization of the two-pion correlation function' --- numbers: 25.75.-q, Introduction {#sec1} Two-particle (BE) (also known as (HBT) intensity interferometry) as a method on the space-time geometry and dynamics of high collisions has received intensive theoretical and experimental attention. investigations revealed that high-quality two-particle correlation data constrain only the geometric size of the particle-emitting source but also its dynamical state at particle @Lorstad89; @BGJ; @APW; @Pratt95; Two different definitions two-pion function employed the literature @Lorstad89; @BGJ; @APW; @Pratt95; @He96; @GKW79; @Zajc84; @Mark; @UA1; @Padula; @Alex93;

*Wavelet Transforms and LocalIzation OpeRatorS.* BiRkhÄuSer VErlaG, vol. 136, Basel, 2002. M.-W. WonG *an inTroduction to pseudo-diffErentIaL OperAToRs.* SerIes on AnALySIS, ApPlIcAtiOnS AnD CompUtaTion, 6. WorLd ScientifIc PUbLishing Co. Pte. lTd., hackensack, nJ, 2014. [^1]: E-Mail addresseS: paOlo.bogGiAttO@Unito.It, eVanthIa.caryPIs@unitO.it, alessaNdRO.oliarO@Unito.it --- ABStRact: 'the two-pion correlaTIoN Function can be dEfined As A RaTIO of EitHer the measUrEd momENtum disTRiBUTIonS Or the normalizEd momentum sPAce ProbabIlItiES. We shoW that ThE FirSt alternatiVe avOids certaIn ambiGUities sINce then The norMalIzaTion OF tHe Two-PiON coRReLatOR coNtains imPoRtAnt inFormATION on tHe mUltiPliciTy distributioN of The eVEnt EnsemBle whIch iS lOst in The secOnd alTeRnative. We illustRate This expliCitLy For SpEcifiC ClasseS of EveNt ensemBles.' addREss: 'inSTITuT für Theoretische PhYsIK, unIversitäT RegenSBuRg, d-93040 regensbuRg, gerMany' AUThor: - 'Q.h. ZhaNG, P. scotto anD U. HeinZ' TiTlE: 'Multi-bOsOn effeCtS anD thE normALizaTion of The two-piOn corRElation functioN' --- pACS numbers: 25.75.-q, 25.75.GZ, 25.70.pq. iNTrODuctIon {#Sec1} ============ Two-partiCle BOSe-EiNsteIN (Be) inTErferOmetrY (aLSo KNown as Hanbury Brown-TWiSs (HBT) iNtensIty interferomEtry) as a metHOD For obtaiNing INfORmation on the spAce-tiMe geometry ANd dynamiCs of hIgh energY collisioNS Has recenTly RecEivEd iNTEnSive theoreticAL And eXpErimentAl aTtentioN. DeTaiLed InvEsTigations Revealed ThAt HiGh-QuaLity tWO-particlE cOrrElAtiOn datA ConstrAin noT onlY tHe GEomEtric siZE oF THe paRtIcLe-emIttInG sourCe buT AlsO its dynAmical staTe aT PartIcLe Freeze-oUt [@Zajc86; @Lorstad89; @bGj; @APW; @Pratt95; @HE96]. TWo dIffereNT DefinitiOns of two-pion correlation FUnction Are EmploYed iN the literAtuRe [@Zajc86; @lorSTad89; @BGJ; @aPW; @PraTt95; @He96; @GkW79; @zajC84; @mArk; @UA1; @pADuLa; @ALeX93; @CDL

*Wavelet Transforms and L ocalizatio n Ope rat ors .* Bir khäu ser Verlag, vo l . 13 6, Basel, 2002. M.-W. Wong * A n in t ro ducti on to p s eu d o -di ff er ent ia l o perat ors .* Seri es on Anal ysi s, Application s a nd Computa tio n, 6. WorldSci entifi cPub l ishin g C o. Pt e. Ltd . , Hack ensack, N J, 2014.[^1]: E - m ai l ad dresses: paolo.bo g gi a tto@unito.it,evanth ia . ca r y pis @un ito.it, al es sandr o .oliaro @ un i t o .it --- abstract : 'The two- p ion corre la tio n funct ion c an bedefined asa ra tio of ei ther t h e measu r ed mome ntum d ist rib utio n sor th en orm a li zed mom entum sp ac eproba bili t i e s . We sh ow t hat t he first alte rna tive avo ids c ertai n am bi guiti es sin ce th en the normalizat ionof the tw o-p io n c or relat o r cont ain s i mportan t infor m ati on o n t he multiplicity di st r i bu tion ofthe ev e nt e n semble w hi chis l o s t inthes ec ond alte rnativ e .We illust ra te thi sexp lic itlyf or s pecifi c classe s ofe vent ensembles . ' address: 'I n st i t ut fürThe oretische P hysi k , Un iver s it ätR egens burg, D - 93 0 40 Regensburg, Germ an y' aut hor:- 'Q.H. Zhang , P. Scott o a nd U. He inz' ti t le: 'Multi-bos on ef fects andt he norma lizat ion of t he two-pi o n correla tio n f unc tio n ' - -- PACS numb e r s: 2 5. 75.-q,25. 75.Gz,25. 70. Pq. I nt roduction {#sec1} = == == == === == T w o-partic le Bo se -Ei nstei n (BE)inter fero me tr y (a lso kno w na s Han bu ry Bro wn- Tw iss ( HBT) int ensityinterfero met r y) a samethodfor obtaining i nformation o n t he spa c e -time ge ometry and dynamics ofh igh ene rgy coll isio ns has re cen tly re cei v ed int ensive theo re tic a l ande x pe rim en tal attent i o n.Detai le d in vestiga tions revealed tha t hi gh-quality tw o-p arti c l ecor r el a tio nd ata c onstrain not on ly the geo me t ri c size oft hepa rticle- emittin g sou r ce butalso itsdynamical s tate a t p article fr eeze-out [@Zajc86 ; @Lor s ta d89;@BG J; @AP W; @P ratt9 5; @He 9 6]. Two diffe re nt def initi on s of two -pion correlation funct ion ar e emp loy ed in the li t era ture [@Za jc86 ; @Lorstad 89; @B GJ; @ APW ; @Pra tt95 ; @ He9 6 ; @GK W79; @Zajc84;@ Ma rk; @ UA 1; @Padula; @ A lex 93; @ CDL

*Wavelet_Transforms and_Localization Operators.* Birkhäuser Verlag,_vol. 136,_Basel,_2002. M.-W. Wong_*An_introduction to pseudo-differential_operators.* Series on_Analysis, Applications and Computation,_6. World Scientific_Publishing_Co. Pte. Ltd., Hackensack, NJ, 2014. [^1]: E-mail addresses: paolo.boggiatto@unito.it, evanthia.carypis@unito.it, alessandro.oliaro@unito.it --- abstract: 'The two-pion_correlation_function can_be_defined_as a ratio of either_the measured momentum distributions or_the normalized_momentum space probabilities. We show that the first_alternative_avoids certain ambiguities_since then the normalization of the two-pion correlator contains_important information on the multiplicity distribution_of the event_ensemble_which_is lost in the_second alternative. We illustrate this explicitly_for specific classes of event ensembles.' address:_'Institut für Theoretische Physik, Universität Regensburg, D-93040_Regensburg, Germany' author: - 'Q.H. Zhang, P. Scotto_and U. Heinz' title: 'Multi-boson effects_and the_normalization of the two-pion correlation_function' --- PACS numbers: 25.75.-q,_25.75.Gz, 25.70.Pq. Introduction_{#sec1} ============ Two-particle Bose-Einstein (BE)_interferometry (also known as Hanbury Brown-Twiss_(HBT) intensity interferometry)_as a method for obtaining information_on_the space-time geometry_and_dynamics_of high_energy collisions has_recently_received intensive_theoretical_and experimental attention. Detailed investigations revealed_that_high-quality two-particle correlation data constrain not only_the geometric size of_the_particle-emitting source but also_its dynamical state at particle_freeze-out [@Zajc86; @Lorstad89; @BGJ; @APW; @Pratt95;_@He96]. Two different_definitions of_two-pion correlation function are employed in the literature [@Zajc86; @Lorstad89; @BGJ;_@APW; @Pratt95; @He96; @GKW79; @Zajc84; @Mark;_@UA1; @Padula; @Alex93; @CDL

_{d}+\sum_{\vert\alpha\vert\geq 2}{\underline{q}^{\alpha p^m}\tau\partial_{\underline{g}}^{(\alpha)}}+\varepsilon_m$$ Where $\underline{q}^{\alpha p^m}=q_1^{\alpha_1}\cdots q_d^{\alpha_d}$, $\varepsilon_m=\underset{\alpha\in\mathbb{N}^d}{\sum}{(\big\langle \varphi^{p^m},\partial_{\underline{g}}^{(\alpha)}\big\rangle-\underline{q}^{\alpha p^m})\tau\partial_{\underline{g}}^{(\alpha)}}\in$ End$_k(kG,Q)$, and there exists $t\in\mathbb{N}$ such that $v(\varepsilon_m(r))>p^{2m-t}$ for all $r\in kG$.\ Since $\varphi(P)=P$ it is clear that the left hand side of this expression annihilates $P$. So take any $y\in P$ and apply it to both sides of (\[Mahler3\]) and we obtain: $$\label{Mahler} 0=q_1^{p^m}\tau\partial_1(y)+....+q_{d}^{p^m}\tau\partial_{d}(y)+O(q^{p^m})$$ Where $q\in Q$ with $v(q^{p^m})\geq \underset{i\leq d}{\min}\{2v(q_i^{p^m})\}$ for all $m$.\ Furthermore, let $f(x)=a_0x+a_1x^p+a_2x^{p^2}+\cdots+a_nx^{p^n}$ be a polynomial, where $a_i\in\tau(kH)$ for each $i$.\ Then for each $m\in\mathbb{N}$, $i=0,\cdots,n$, consider expression (\[Mahler\]) above, with $m$ replaced by $m+i$, and multiply by $a_i^{p^m}$ to obtain: $0=(a_iq_1^{p^i})^{p^m}\tau\partial_1(y)+....+(a_iq_{d}^{p^i})^{p^m}\tau\partial_{d}(y)+O((a_iq^{p^i

_ { d}+\sum_{\vert\alpha\vert\geq 2}{\underline{q}^{\alpha p^m}\tau\partial_{\underline{g}}^{(\alpha)}}+\varepsilon_m$$ Where $ \underline{q}^{\alpha p^m}=q_1^{\alpha_1}\cdots q_d^{\alpha_d}$, $ \varepsilon_m=\underset{\alpha\in\mathbb{N}^d}{\sum}{(\big\langle \varphi^{p^m},\partial_{\underline{g}}^{(\alpha)}\big\rangle-\underline{q}^{\alpha p^m})\tau\partial_{\underline{g}}^{(\alpha)}}\in$ End$_k(kG, Q)$, and there exists $ t\in\mathbb{N}$ such that $ v(\varepsilon_m(r))>p^{2m - t}$ for all $ r\in kG$.\ Since $ \varphi(P)=P$ it is clear that the left hand slope of this formula annihilates $ P$. So take any $ y\in P$ and apply it to both side of (\[Mahler3\ ]) and we obtain: $ $ \label{Mahler } 0 = q_1^{p^m}\tau\partial_1(y)+.... +q_{d}^{p^m}\tau\partial_{d}(y)+O(q^{p^m})$$ Where $ q\in Q$ with $ v(q^{p^m})\geq \underset{i\leq d}{\min}\{2v(q_i^{p^m})\}$ for all $ m$.\ Furthermore, let $ f(x)=a_0x+a_1x^p+a_2x^{p^2}+\cdots+a_nx^{p^n}$ be a polynomial, where $ a_i\in\tau(kH)$ for each $ i$.\ Then for each $ m\in\mathbb{N}$, $ i=0,\cdots, n$, regard construction (\[Mahler\ ]) above, with $ m$ replaced by $ m+i$, and multiply by $ a_i^{p^m}$ to receive: $ 0=(a_iq_1^{p^i})^{p^m}\tau\partial_1(y)+.... +(a_iq_{d}^{p^i})^{p^m}\tau\partial_{d}(y)+O((a_iq^{p^i

_{d}+\sul_{\vert\alpha\vert\geq 2}{\underllne{q}^{\alpha p^m}\tau\pcetial_{\uiderlins{g}}^{(\alpha)}}+\vxrepsilon_m$$ Where $\underline{q}^{\al'ha p^m}=q_1^{\alkka_1}\cdots q_d^{\alpha_d}$, $\vardpsilon_m=\ujderset{\aopha\mn\mathbb{N}^d}{\sum}{(\big\langle \varphi^{p^j},\iarticl_{\nnderline{g}}^{(\alpha)}\nig\rangle-\ungerline{q}^{\alpha [^m})\gab\partial_{\underline{g}}^{(\alpha)}}\in$ End$_k(kG,Q)$, anq there edists $t\in\mathbf{N}$ slcr thzn $y(\varepsilon_m(r))>p^{2m-t}$ for all $r\in kG$.\ Sjnce $\vagphi(P)=P$ it is cleat that the left hand side lf tjis expression annlhilates $P$. Wo twje any $y\in P$ and apply it to both aides of (\[Mahler3\]) and we obtain: $$\lacel{Makler} 0=q_1^{p^m}\tau\pqrriap_1(i)+....+q_{d}^{p^m}\tau\partmal_{d}(y)+O(z^{p^m})$$ Where $q\in Q$ with $e(q^{p^m})\geq \underset{i\leq c}{\mii}\{2v(q_i^{p^m})\}$ for all $m$.\ Furthermoce, let $f(x)=a_0x+a_1x^p+a_2x^{p^2}+\cdojs+a_nx^{p^n}$ be a polynomial, where $a_u\in\tao(kH)$ fmr exxh $k$.\ Thtn hor each $l\in\jathbb{N}$, $i=0,\csots,n$, consieer expression (\[Mahltr\]) wvove, with $m$ rsplaceq fy $m+i$, and multiply by $a_i^{p^m}$ to obtain: $0=(a_iq_1^{k^i})^{p^m}\tzu\partial_1(y)+....+(a_iq_{d}^{p^i})^{p^m}\tau\paetial_{d}(y)+O((a_iq^{p^i

_{d}+\sum_{\vert\alpha\vert\geq 2}{\underline{q}^{\alpha p^m}\tau\partial_{\underline{g}}^{(\alpha)}}+\varepsilon_m$$ Where $\underline{q}^{\alpha p^m}=q_1^{\alpha_1}\cdots q_d^{\alpha_d}$, p^m})\tau\partial_{\underline{g}}^{(\alpha)}}\in$ and there $t\in\mathbb{N}$ such that Since it is clear the left hand of this expression annihilates $P$. So any $y\in P$ and apply it to both sides of (\[Mahler3\]) and we $$\label{Mahler} 0=q_1^{p^m}\tau\partial_1(y)+....+q_{d}^{p^m}\tau\partial_{d}(y)+O(q^{p^m})$$ Where $q\in Q$ with $v(q^{p^m})\geq \underset{i\leq d}{\min}\{2v(q_i^{p^m})\}$ for all $m$.\ Furthermore, $f(x)=a_0x+a_1x^p+a_2x^{p^2}+\cdots+a_nx^{p^n}$ a where for each $i$.\ Then for each $m\in\mathbb{N}$, $i=0,\cdots,n$, consider expression (\[Mahler\]) above, with $m$ replaced by and multiply by $a_i^{p^m}$ to obtain: $0=(a_iq_1^{p^i})^{p^m}\tau\partial_1(y)+....+(a_iq_{d}^{p^i})^{p^m}\tau\partial_{d}(y)+O((a_iq^{p^i

_{d}+\sum_{\vert\alpha\vert\geq 2}{\underLine{q}^{\alpha P^m}\tau\ParTiaL_{\uNderLine{G}}^{(\alpha)}}+\varepsilON_m$$ WhEre $\underline{q}^{\alpha p^m}=q_1^{\aLpha_1}\cDoTS q_d^{\aLPhA_d}$, $\varEpsilon_M=\UnDERseT{\aLpHa\iN\mAThBb{N}^d}{\sUm}{(\bIg\langlE \varphi^{p^m},\pArtIaL_{\underline{g}}^{(\aLPhA)}\big\rangle-\UndErline{q}^{\alpha P^m})\tAu\partIaL_{\unDErlinE{g}}^{(\aLpha)}}\iN$ End$_k(kg,q)$, and thEre exists $T\iN\Mathbb{n}$ Such thaT $V(\VaRepsIlon_m(r))>p^{2m-t}$ for all $r\iN KG$.\ sInce $\varphi(P)=P$ it Is cleaR tHAt THE leFt hAnd side of tHiS exprESsion anNIhILATes $p$. so take any $y\in P$ And apply it tO BotH sides Of (\[mahLEr3\]) and wE obtaIn: $$\LAbeL{Mahler} 0=q_1^{p^m}\tAu\paRtial_1(y)+....+q_{d}^{p^M}\tau\paRTial_{d}(y)+O(Q^{P^m})$$ Where $Q\in Q$ wiTh $v(Q^{p^m})\Geq \uNDeRsEt{i\LeQ D}{\miN}\{2V(q_I^{p^m})\}$ FOr aLl $m$.\ FurthErMoRe, let $F(x)=a_0x+A_1X^P+A_2X^{p^2}+\cdOts+A_nx^{p^N}$ be a pOlynomial, wherE $a_i\In\taU(KH)$ fOr eacH $i$.\ TheN for EaCh $m\in\Mathbb{n}$, $i=0,\cdoTs,N$, consider expresSion (\[mahler\]) aboVe, wItH $m$ rEpLaced BY $m+i$, and MulTipLy by $a_i^{p^M}$ to obtaIN: $0=(a_iQ_1^{p^I})^{P^M}\TaU\partial_1(y)+....+(a_iq_{d}^{p^i})^{p^m}\tAu\PARtIal_{d}(y)+O((a_iQ^{p^i

_{d}+\sum_{\vert\alpha\ver t\geq 2}{\ under lin e{q }^ {\al phap^m}\tau\parti a l_{\ underline{g}}^{(\alpha )}}+\ va r epsi l on _m$$ Where$ \u n d erl in e{ q}^ {\ a lp ha p^ m}= q_1^{\a lpha_1}\cd ots q _d^{\alpha_d } $, $\varepsi lon _m=\underset {\a lpha\i n\ mat h bb{N} ^d} {\sum }{(\bi g \langl e \varphi ^{ p ^m},\p a rtial_{ \ u nd erli ne{g}}^{(\alpha)} \ bi g \rangle-\under line{q }^ { \a l p hap^m })\tau\par ti al_{\ u nderlin e {g } } ^ {(\ a lpha)}}\in$ E nd$_k(kG,Q) $ , a nd the re ex i sts $t \in\m at h bb{ N}$ such th at $ v(\vareps ilon_m ( r))>p^{ 2 m-t}$ f or all $r \in kG$ . \Si nce $ \ var p hi (P) = P$it is cl ea rthatthel e f t han d s ideof th is expression an nihi l ate s $P$ . Sotake a ny $y \in P$ andap ply it to bothside s of (\[M ahl er 3\] )and w e obtai n: $$ \label{ Mahler} 0=q _1 ^ { p ^m }\tau\partial_1(y) +. . . .+ q_{d}^{p ^m}\ta u \p ar t ial_{d}( y) +O( q^{p ^ m })$$ Whe r e$q\in Q$ with$ v( q^ {p^m})\ ge q \und er set {i\ leq d } {\mi n}\{2v (q_i^{p^ m})\} $ for all $m$.\ Furthermore,l et $ f( x )=a_ 0x+ a_1x^p+a_2x ^{p^ 2 }+\c dots + a_ nx^ { p^n}$ be a p o ly n omial, where $a_i\i n\ tau(kH )$ fo r each $i$.\Then for e a c h $m\in\m athb b {N } $, $i=0,\cdots ,n$,consider e x pression (\[M ahler\]) above, w i t h $m$ re pla ced by $m + i $, and multiply b y $a _i ^{p^m}$ to obtain : $0= (a_ iq_ 1^ {p^i})^{p ^m}\tau\ pa rt ia l_ 1(y )+... . +(a_iq_{ d} ^{p ^i })^ {p^m} \ tau\pa rtial _{d} (y )+ O ((a _iq^{p^ i

_{d}+\sum_{\vert\alpha\vert\geq 2}{\underline{q}^{\alpha_p^m}\tau\partial_{\underline{g}}^{(\alpha)}}+\varepsilon_m$$ Where $\underline{q}^{\alpha_p^m}=q_1^{\alpha_1}\cdots q_d^{\alpha_d}$, $\varepsilon_m=\underset{\alpha\in\mathbb{N}^d}{\sum}{(\big\langle \varphi^{p^m},\partial_{\underline{g}}^{(\alpha)}\big\rangle-\underline{q}^{\alpha_p^m})\tau\partial_{\underline{g}}^{(\alpha)}}\in$ End$_k(kG,Q)$,_and_there exists_$t\in\mathbb{N}$_such that $v(\varepsilon_m(r))>p^{2m-t}$_for all $r\in_kG$.\ Since $\varphi(P)=P$ it is_clear that the_left_hand side of this expression annihilates $P$. So take any $y\in P$ and apply_it_to both_sides_of_(\[Mahler3\]) and we obtain: $$\label{Mahler} 0=q_1^{p^m}\tau\partial_1(y)+....+q_{d}^{p^m}\tau\partial_{d}(y)+O(q^{p^m})$$ Where $q\in_Q$ with $v(q^{p^m})\geq \underset{i\leq d}{\min}\{2v(q_i^{p^m})\}$_for all_$m$.\ Furthermore, let $f(x)=a_0x+a_1x^p+a_2x^{p^2}+\cdots+a_nx^{p^n}$ be a polynomial, where $a_i\in\tau(kH)$_for_each $i$.\ Then for_each $m\in\mathbb{N}$, $i=0,\cdots,n$, consider expression (\[Mahler\]) above, with $m$_replaced by $m+i$, and multiply by_$a_i^{p^m}$ to obtain: $0=(a_iq_1^{p^i})^{p^m}\tau\partial_1(y)+....+(a_iq_{d}^{p^i})^{p^m}\tau\partial_{d}(y)+O((a_iq^{p^i

05]. Global Properties of the EUADP sample ===================================== \ The emission redshifts of the 250 EUADP sample quasars are initially obtained from the Simbad catalogue and later double checked for the cases where Ly$\alpha$ emission from the quasar is covered by our data. The Ly$\alpha$ emission for 5 quasars (i.e., QSOJ0332-4455, QSOB0528-2505, QSOB0841+129, QSOB1114-0822 and QSOJ2346+1247) is not seen because of the presence of DLAs belonging to the “proximate DLA” class with $z_{\rm abs}\approx z_{\rm em}$ [e.g., @moller98; @ellison11; @zafar11]. For the emission redshifts of these 5 cases, we rely on the literature. The emission redshifts of all the other objects in the EUADP sample have been compared with measurements from the literature. For a few cases, emission redshifts provided in the Simbad catalogue are not correct and the correct redshifts are obtained from the literature. In our sample, there are 38 quasars with emission redshifts below $z_{\rm em}<1.5$. For these cases we cannot see Ly$\alpha$ emission from the quasar because of the limited spectral coverage, therefore, we relied mostly on the Simbad catalogue. However, other emission lines are covered in the spectra, helping us to confirm the emission redshifts. The emission redshifts of 250 quasars of EUADP sample ranges from $0.191\leq z_{\rm em}\leq6.311$. Their distribution is shown in Fig. \[zemhist\] and is found to peak at $z_{\rm em}\simeq2.1$. \ In Fig. \[zabshist\], the column density distribution of DLAs and sub-DLAs is presented (see @zafar12b for complete list of DLAs and sub-DLAs [H]{}[i]{} column densities). It is worth noting that in the EUADP sample, damped absorbers with column densities up to log $N_{{\rm H}\,{\sc \rm I}}=21.85$ are seen, while higher column densities have been recently reported [@guimaraes12; @kulkarni12; @noterdaeme12]. As mentioned above in the EUADP sample, the number of

05 ]. Global Properties of the EUADP sample = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = \ The emission redshifts of the 250 EUADP sample quasar are initially prevail from the Simbad catalogue and later doubly checked for the case where Ly$\alpha$ emission from the quasar is breed by our data. The Ly$\alpha$ emission for 5 quasar (i.e., QSOJ0332 - 4455, QSOB0528 - 2505, QSOB0841 + 129, QSOB1114 - 0822 and QSOJ2346 + 1247) is not see because of the bearing of DLAs belonging to the “ proximate DLA ” class with $ z_{\rm abs}\approx z_{\rm em}$ [ for example, @moller98; @ellison11; @zafar11 ]. For the emission redshifts of these 5 cases, we rely on the literature. The emission redshifts of all the other object in the EUADP sample have been compared with measurements from the literature. For a few case, emission redshifts provided in the Simbad catalogue are not right and the correct red shift are obtained from the literature. In our sample, there are 38 quasars with discharge redshifts below $ z_{\rm em}<1.5$. For these cases we cannot see Ly$\alpha$ emission from the quasar because of the limited apparitional coverage, consequently, we trust mostly on the Simbad catalogue. However, other emission lines are covered in the spectra, avail us to confirm the discharge redshifts. The emission red shift of 250 quasar of EUADP sample ranges from $ 0.191\leq z_{\rm em}\leq6.311$. Their distribution is shown in Fig. \[zemhist\ ] and is found to top out at $ z_{\rm em}\simeq2.1$. \ In Fig. \[zabshist\ ], the column density distribution of DLAs and sub - DLAs is give (see @zafar12b for complete tilt of DLAs and sub - DLAs [ H]{}[i ] { } column densities). It is worth noting that in the EUADP sample, damped absorbers with column densities up to log $ N_{{\rm H}\,{\sc \rm I}}=21.85 $ are seen, while higher column densities have been recently reported [ @guimaraes12; @kulkarni12; @noterdaeme12 ]. As mentioned above in the EUADP sample distribution, the number of

05]. Glohal Properties of the EUXDP sample ===================================== \ The emission redshjfts of ghe 250 EUADP sample quasars arx inutialoy obtained from the Skmbad catwlogue abd leter double checked for bke caacs whzrx Ly$\alpha$ emisslon from tha quasar is coeefeb by our data. The Ly$\alpha$ emission fjr 5 quaxags (i.e., QSOJ0332-4455, QSOF0528-2505, QSPF0841+129, QSKB1114-0822 and QSOJ2346+1247) is not seen because of the prtsence of DLAs belpnging to the “proximate DLW” clwss with $z_{\rm abs}\aporox z_{\rm em}$ [e.g., @iiller98; @ellisov11; @zafar11]. Fog the emissjon redshifts of these 5 cases, wd relv on the lijzeatkte. The emissmon reqshifts of all the odher obkects in the EMADP vample have been comparev with measurements srom the nicerature. For a few cawew, emivsiot reawhiwts pcovjded ij tie Simbad cztalogue arw not correct and tne bprrect redshjfts awe obtained from the literature. In our sakpls, there are 38 quasars wirh emission redshifts below $z_{\ri em}<1.5$. For these cases we cannot see Ly$\alpha$ emissimn frko tkc dyadar because of the limited spectral coverage, egetenore, we relied mjstly on thr Dikfad catalogue. Howevzd, kther emission linfs are soverwd in the spevtra, helping us to confirm rhe emission eedshifts. The emisdion redshiyts of 250 quaxars of EUADP sample rauges fdom $0.191\leq z_{\rm em}\leq6.311$. Thskr distribution ks xhmwn in Fig. \[zemhist\] and is sound to 'eak ct $z_{\rm eo}\simgq2.1$. \ In Fid. \[zabshist\], the gmlumn density distgibutnon ox DLAs and sub-DLAs is presented (see @zafar12u for completg lhst of DLAs and xub-DLAs [H]{}[i]{} cojumn densities). It is corth voting than in the XUADP sample, damped absortgrs with colukn densieies up ro log $V_{{\fm H}\,{\sc \rm I}}=21.85$ arr seen, whplt higher cilumn densities haye begn recently reporctd [@tuimaraes12; @kulkatni12; @noeegdaxme12]. Af mentioned atove in yhe EJADP sample, ghe mumber of

05]. Global Properties of the EUADP sample The redshifts of 250 EUADP sample the catalogue and later checked for the where Ly$\alpha$ emission from the quasar covered by our data. The Ly$\alpha$ emission for 5 quasars (i.e., QSOJ0332-4455, QSOB0528-2505, QSOB1114-0822 and QSOJ2346+1247) is not seen because of the presence of DLAs belonging the DLA” with abs}\approx z_{\rm em}$ [e.g., @moller98; @ellison11; @zafar11]. For the emission redshifts of these 5 cases, we on the literature. The emission redshifts of all other objects in the sample have been compared with from literature. For few emission provided in the catalogue are not correct and the correct redshifts are obtained from the literature. In our sample, there 38 quasars redshifts below em}<1.5$. these we cannot see from the quasar because of the therefore, we relied mostly on the Simbad catalogue. other emission are covered in the spectra, helping to confirm the emission redshifts. The emission redshifts 250 quasars of EUADP sample ranges from $0.191\leq z_{\rm em}\leq6.311$. Their distribution is shown in and is found to at $z_{\rm em}\simeq2.1$. In \[zabshist\], column distribution of and sub-DLAs is presented (see @zafar12b for complete list of DLAs sub-DLAs [H]{}[i]{} column densities). It is worth noting that in sample, absorbers with column up to log $N_{{\rm \rm are seen, while higher have recently @noterdaeme12]. mentioned in the EUADP sample, number of

05]. Global Properties of the EUADp sample ===================================== \ The EmissIon RedShIfts Of thE 250 EUADP sample quASars Are initially obtained frOm the siMBad cATaLogue And lateR DoUBLe cHeCkEd fOr THe Cases WheRe Ly$\alpHa$ emission FroM tHe quasar is coVErEd by our datA. ThE Ly$\alpha$ emisSioN for 5 quAsArs (I.E., QSOJ0332-4455, qSOb0528-2505, QSOB0841+129, qSOB1114-0822 anD qSOJ2346+1247) is Not seen beCaUSe of thE PresencE OF DlAs bElonging to the “proxIMaTE DLA” class with $z_{\Rm abs}\aPpROx Z_{\RM em}$ [E.g., @mOller98; @ellisOn11; @Zafar11]. fOr the emISsION RedSHifts of these 5 cAses, we rely oN The LiteraTuRe. THE emissIon reDsHIftS of all the otHer oBjects in tHe EUADp Sample hAVe been cOmpareD wiTh mEasuREmEnTs fRoM The LItEraTUre. for a few cAsEs, EmissIon rEDSHIfts ProVideD in thE Simbad cataloGue Are nOT coRrect And thE corReCt redShifts Are obTaIned from the liteRatuRe. In our saMplE, tHerE aRe 38 quaSArs witH emIssIon redsHifts beLOw $z_{\Rm EM}<1.5$. fOr These cases we cannot SeE lY$\aLpha$ emisSion frOM tHe QUasar becAuSe oF the LIMited SpecTRaL coveragE, thereFOrE, wE relied MoStly on ThE SiMbaD cataLOgue. howeveR, other emIssioN Lines are covereD In the spectra, hELpING uS To coNfiRm the emissiOn reDShifTs. ThE EmIssIOn redShiftS oF 250 QuASars of EUADP sample raNgEs from $0.191\Leq z_{\rM em}\leq6.311$. Their diStribution IS SHown in FiG. \[zemHIsT\] And is found to peAk at $z_{\Rm em}\simeq2.1$. \ IN fig. \[zabshIst\], thE column dEnsity disTRIbution oF DLas aNd sUb-DlaS iS presented (see @ZAFar12b FoR compleTe lIst of DLas aNd sUb-DlAs [h]{}[i]{} Column denSities). It Is WoRtH nOtiNg thaT In the EUAdP SamPlE, daMped aBSorberS with ColuMn DeNSitIes up to LOg $n_{{\RM H}\,{\sc \Rm i}}=21.85$ aRe seEn, wHiLe higHer cOLumN densitIes have beEn rECentLy RePorted [@gUimaraes12; @kulkaRnI12; @noterdaemE12]. AS meNtioneD ABove in thE EUADP sample, the number of

05]. Global Properties of the EUADP samp le=== == ==== ==== ============== = ==== ===== \ The emission reds hi f ts o f t he 25 0 EUADP sa m p lequ as ars a r einiti all y obtai ned from t heSi mbad catalog u eand laterdou ble checkedfor the c as esw hereLy$ \alph a$ emi s sion f rom the q ua s ar isc overedb y o ur d ata. The Ly$\alph a $e mission for 5quasar s( i. e . , Q SOJ 0332-4455, Q SOB05 2 8-2505, QS O B 0 841 + 129, QSOB1114 -0822 and Q S OJ2 346+12 47 ) i s not s een b ec a use of the pre senc e of DLAs belon g ing tot he “pro ximate DL A”clas s w it h $ z_ { \rm ab s}\ a ppr ox z_{\r mem }$ [e .g., @ m o ller 98; @el lison 11; @zafar11] . F or t h e e missi on re dshi ft s ofthese5 cas es , we rely on th e li terature. Th eemi ss ion r e dshift s o f a ll theother o b jec ts i n t he EUADP sample ha ve b ee n compar ed wit h m ea s urements f rom the l itera ture . F or a few cases , e mi ssion r ed shifts p rov ide d int he S imbadcatalogu e are not correct an d the correctr ed s h if t s ar e o btained fro m th e lit erat u re . I n oursampl e, th e re are 38 quasars w it h emis sionredshifts bel ow $z_{\rm e m }<1.5$.Fort he s e cases we can not s ee Ly$\alp h a$ emiss ion f rom thequasar be c a use of t helim ite d s p e ct ral coverage, t here fo re, werel ied mos tly on th e S im bad catal ogue. Ho we ve r, o the r emi s sion lin es ar ecov eredi n thespect ra,he lp i ngus to c o nf i r m th eem issi onre dshif ts.T heemissio n redshif tso f 25 0qu asars o f EUADP sampl eranges fro m$0. 191\le q z_{\rm e m}\leq6.311$. Their dis t ributio n i s sho wn i n Fig. \[ zem hist\] an d is fo und to peak a t $ z _ {\rme m }\ sim eq 2.1$. \ I n Fi g. \[ za bshi st\], t he column densityd ist ribution of D LAs and s ub -DL A si s p re s ent e d (see @zafar12b for compl et e l ist of DLA s an dsub-DLA s [H]{} [i]{} columndensities ). It iswo rthn o tin g that inthe EUAD P sample, dampe d a bsorb ers withco lum n den sities upto lo g $N_{ {\ rm H}\ ,{\sc \ rm I}}=2 1.85$ are seen, while h ighercolum n d ensitieshav e be en recent ly r eported [@ gui mar aes12 ; @ k ulkar ni12 ; @ not e rdaem e12] . As menti o ne d a b o ve in the EUA D P sam ple,the number of

05]. Global Properties_of the_EUADP sample ===================================== \ The emission redshifts_of the_250_EUADP sample_quasars_are initially obtained_from the Simbad_catalogue and later double_checked for the_cases_where Ly$\alpha$ emission from the quasar is covered by our data. The Ly$\alpha$ emission_for_5 quasars_(i.e.,_QSOJ0332-4455,_QSOB0528-2505, QSOB0841+129, QSOB1114-0822 and QSOJ2346+1247)_is not seen because of_the presence_of DLAs belonging to the “proximate DLA” class_with_$z_{\rm abs}\approx z_{\rm_em}$ [e.g., @moller98; @ellison11; @zafar11]. For the emission redshifts_of these 5 cases, we rely_on the literature._The_emission_redshifts of all the_other objects in the EUADP sample_have been compared with measurements from_the literature. For a few cases, emission_redshifts provided in the Simbad catalogue_are not correct and the_correct redshifts_are obtained from the literature._In our sample,_there are_38 quasars with_emission redshifts below $z_{\rm em}<1.5$. For_these cases we_cannot see Ly$\alpha$ emission from the_quasar_because of the_limited_spectral_coverage, therefore,_we relied mostly_on_the Simbad_catalogue._However, other emission lines are covered_in_the spectra, helping us to confirm the_emission redshifts. The emission_redshifts_of 250 quasars of_EUADP sample ranges from $0.191\leq_z_{\rm em}\leq6.311$. Their distribution is shown_in Fig._\[zemhist\] and_is found to peak at $z_{\rm em}\simeq2.1$. \ In Fig. \[zabshist\], the column_density distribution of DLAs and sub-DLAs_is presented (see @zafar12b_for complete_list_of DLAs and_sub-DLAs_[H]{}[i]{} column_densities). It is worth noting that in_the EUADP_sample, damped absorbers with column densities_up to log $N_{{\rm_H}\,{\sc_\rm I}}=21.85$ are seen, while higher_column densities have been recently reported_[@guimaraes12; @kulkarni12; @noterdaeme12]. As mentioned_above_in_the EUADP sample, the number_of

, EoS = BsK20}. We have estimated the spin-dependent capture cross section shown in Figure \[fig:DD\] by 1. Scaling the spin-independent capture cross section by a factor of $$\left[\frac{4}{3}\frac{J_n+1}{J_n} (\langle S_n\rangle + \langle S_p\rangle)^2\right]^{-1}~,$$ where $J_n = 1/2$, $\langle S_n\rangle = 1/2$ and $\langle S_p\rangle = 0$ for scattering on neutrons, and\ 2. To calculate pasta scattering we have used the pasta structure factor in Eq.  but have imposed $S_{\rm pasta} (q) \leq 1$, since we expect spin-dependent scattering to not be coherent over pasta nucleons, and 3. To calculate quasi-elastic scattering on nucleons in the inner and outer crust, we follow the same procedure as spin-independent scattering, since in this case scattering occurs with individual nucleons. Although the PICO-60 limit applies to spin-dependent scattering on protons, we display it as it provides the current best sensitivity to spin-dependent dark matter interactions with nucleons at low dark matter masses. Furthermore, most spin-dependent models predict roughly equal rates of neutron and proton scattering. For spin-independent scattering, we see that neutron crust heating is clearly more sensitive than direct detection for low dark matter masses (${m_{\rm DM}}\lesssim$ GeV); crustal capture easily proceeds for these masses (see Eq. ), but at terrestrial experiments it is difficult to detect soft nuclear recoils above energy thresholds. Direct detection experiments benefit from an $A^2$ enhancement in cross section from nuclear coherence, which helps them surpass neutron star crust sensitivity in some range of ${m_{\rm DM}}$. For spin-dependent scattering, we see that neutron crust heating is more sensitive than current direct detection searches for nearly all dark matter masses. We note that these statements hold true for sub-MeV dark matter masses as well, where neutron star crust heating occurs through phonon excitations (see Fig. \[fig:bounds1\]). For complementary bounds on dark matter-nucleon scattering at low dark matter masses, see References [@Erickcek:2007jv; @Gluscevic:2017ywp; @Xu:2018efh; @Capp

, EoS = BsK20 }. We have estimated the spin - dependent capture cross incision usher in Figure \[fig: DD\ ] by 1. Scaling the spin - independent capture hybrid section by a factor of $ $ \left[\frac{4}{3}\frac{J_n+1}{J_n } (\langle S_n\rangle + \langle S_p\rangle)^2\right]^{-1}~,$$ where $ J_n = 1/2 $, $ \langle S_n\rangle = 1/2 $ and $ \langle S_p\rangle = 0 $ for scatter on neutron, and\ 2. To calculate pasta scattering we have used the pasta structure factor in Eq.   but have levy $ S_{\rm pasta } (q) \leq 1 $, since we expect spin - pendent scattering to not be coherent over pasta nucleons, and 3. To calculate quasi - elastic scattering on nucleons in the inside and outer crust, we follow the same routine as spin - independent dispersion, since in this case disperse occurs with individual nucleons. Although the PICO-60 limit applies to spin - pendent scattering on protons, we display it as it provides the current best sensitivity to spin - dependent dark matter interactions with nucleons at low dark matter masses. Furthermore, most spin - dependent model predict roughly adequate pace of neutron and proton scatter. For spin - independent dispersion, we see that neutron crust heating is clearly more sensitive than lineal detection for low dark matter masses ($ { m_{\rm DM}}\lesssim$ GeV); crustal capture easily proceeds for these masses (see Eq.  ), but at terrestrial experiments it is difficult to detect cushy nuclear recoils above department of energy threshold. Direct detection experiments benefit from an $ A^2 $ enhancement in cross section from nuclear coherence, which helps them exceed neutron star crust sensitivity in some range of $ { m_{\rm DM}}$. For spin - dependent scattering, we experience that neutron crust heating is more sensible than current direct detection searches for nearly all dark topic masses. We note that these statements hold on-key for sub - MeV dark matter masses as well, where neutron star crust heating happen through phonon excitations (go steady Fig.   \[fig: bounds1\ ]). For complementary bounds on dark matter - nucleon scattering at abject dark matter multitude, see References   [ @Erickcek:2007jv; @Gluscevic:2017ywp; @Xu:2018efh; @Capp

, EoD = BsK20}. We have estimated uhe spin-dependent capturx cross section shown in Figure \[fig:DD\] by 1. Sralibg tht spin-independent cxpture crlss sectuon uy a factor of $$\lxrt[\frac{4}{3}\fvcc{J_n+1}{J_h} (\lanylx S_n\rangle + \lannle S_p\rangla)^2\right]^{-1}~,$$ where $J_t = 1/2$, $\pangle S_n\rangle = 1/2$ and $\langle S_p\rangje = 0$ fot dcattering on geutgogs, ahd\ 2. To calculate pasta scattering se have used the pasya structure factor in Eq.  hut jave imposed $S_{\rm pwsta} (q) \leq 1$, sinsw we expect rpin-dependtnc scatterinf to not be coherent over pasta nuclzons, and 3. Ti xalfolate quasi-eoastib scattering on nuclemns in yhe inner and putxr ceust, we follow the sake procedure as spyn-indepengeut scattering, since ib rhis wase scagrerkng orcuds witj iidividual nhcleons. Althiugh the PICO-60 limit a[ikies to spin-sependqne scattering on protons, we display it av if provides the current vest sensitivity to skin-dependegt dark matter interactions with nucleons at low gark jxtttr massdw. Vurthermore, most spin-dependent models predict doigmly equal rates jf neutron snf ltoton scatterivg. For spih-independent scattfring, wg see rhat neutwon vrust heating is clearly moee sensitive rhan direct detectnon for low bark mstter masses (${m_{\rm DM}}\lesssim$ GzV); cruatal capturf easily lfoceeds for thesd mssves (see Td. ), but at terrestryal expermmentx it is difgicult to detect soft nuclear recoils ablve euergy thresholdd. Direct detection experiments uxnefit from am $D^2$ ethancemeut in gross section fwom nuclear cokerence, chich felps them surpasv neutron sear crust senvltivity in smme rangq of ${m_{\rm DM}}$. For rpin-dependent xcatteriny, we wee that neutron cvust fsating is more wenwitive than curteng dyrvct dqdection searwhes fof nearuy all dark oattrr masses. We note thdt tgese statements hokd true fot sub-MeV qark matter mssses as well, whert neutcon ster cruxt reating occurs through phonon sxcitatiojs (fee Fig. \[fig:botnds1\]). For complemzntary bounds on dark matter-nucleon scatvering at low dark matjer masses, see Refereucts [@Erickcek:2007jv; @Jluscedic:2017ywp; @Xu:2018afh; @Capp

, EoS = BsK20}. We have estimated capture section shown Figure \[fig:DD\] by cross by a factor $$\left[\frac{4}{3}\frac{J_n+1}{J_n} (\langle S_n\rangle \langle S_p\rangle)^2\right]^{-1}~,$$ where $J_n = 1/2$, S_n\rangle = 1/2$ and $\langle S_p\rangle = 0$ for scattering on neutrons, and\ To calculate pasta scattering we have used the pasta structure factor in Eq. have $S_{\rm (q) 1$, since we expect spin-dependent scattering to not be coherent over pasta nucleons, and 3. To quasi-elastic scattering on nucleons in the inner and crust, we follow the procedure as spin-independent scattering, since this scattering occurs individual Although PICO-60 limit applies spin-dependent scattering on protons, we display it as it provides the current best sensitivity to spin-dependent dark interactions with low dark masses. most models predict roughly of neutron and proton scattering. For see that neutron crust heating is clearly more than direct for low dark matter masses (${m_{\rm GeV); crustal capture easily proceeds for these masses Eq. ), but at terrestrial experiments it is difficult to detect soft nuclear recoils above Direct detection experiments benefit an $A^2$ enhancement cross from coherence, helps them neutron star crust sensitivity in some range of ${m_{\rm DM}}$. For scattering, we see that neutron crust heating is more sensitive direct searches for nearly dark matter masses. We that statements hold true for matter as star heating through phonon excitations (see \[fig:bounds1\]). For complementary bounds on matter-nucleon scattering at low [@Erickcek:2007jv; @Gluscevic:2017ywp; @Xu:2018efh; @Capp

, EoS = BsK20}. We have estimated the sPin-dependeNt capTurE crOsS secTion Shown in Figure \[fIG:DD\] bY 1. Scaling the spin-indepenDent cApTUre cROsS sectIon by a fACtOR Of $$\lEfT[\fRac{4}{3}\FrAC{J_N+1}{J_n} (\laNglE S_n\rangLe + \langle S_p\RanGlE)^2\right]^{-1}~,$$ where $J_N = 1/2$, $\LaNgle S_n\rangLe = 1/2$ aNd $\langle S_p\raNglE = 0$ for scAtTerINg on nEutRons, aNd\ 2. To caLCulate Pasta scatTeRIng we hAVe used tHE PaSta sTructure factor in EQ.  BuT Have imposed $S_{\rm Pasta} (q) \LeQ 1$, SiNCE we ExpEct spin-depEnDent sCAtterinG To NOT Be cOHerent over pasTa nucleons, aND 3. To CalculAtE quASi-elasTic scAtTEriNg on nucleonS in tHe inner anD outer CRust, we fOLlow the Same prOceDurE as sPIn-InDepEnDEnt SCaTteRIng, Since in tHiS cAse scAtteRING OccuRs wIth iNdiviDual nucleons. ALthOugh THe PiCO-60 liMit apPlieS tO spin-DependEnt scAtTering on protons, We diSplay it as It pRoVidEs The cuRRent beSt sEnsItivity To spin-dEPenDeNT DArK matter interactionS wITH nUcleons aT low daRK mAtTEr masses. fuRthErmoRE, Most sPin-dEPeNdent modEls preDIcT rOughly eQuAl rateS oF neUtrOn and PRotoN scattEring. For Spin-iNDependent scattERing, we see that NEuTROn CRust HeaTing is clearLy moRE senSitiVE tHan DIrect DetecTiON fOR low dark matter masseS (${m_{\Rm DM}}\leSssim$ geV); crustal capTure easily PROCeeds for ThesE MaSSes (see Eq. ), but at tErresTrial experIMents it iS diffIcult to dEtect soft NUClear recOilS abOve EneRGY tHresholds. DireCT DeteCtIon expeRimEnts benEfiT frOm aN $A^2$ eNhAncement iN cross seCtIoN fRoM nuClear COherence, WhIch HeLps Them sURpass nEutroN staR cRuST seNsitiviTY iN SOme rAnGe Of ${m_{\rM DM}}$. foR spin-DepeNDenT scatteRing, we see ThaT NeutRoN cRust heaTing is more senSiTive than cuRrEnt Direct DETection sEarches for nearly all dark MAtter maSseS. We noTe thAt these stAteMents hOld TRue for Sub-MeV Dark mAtTer MASses aS WElL, whErE neutron stAR CruSt heaTiNg ocCurs thrOugh phonon excitatiONs (sEe Fig. \[fig:boundS1\]). FoR comPLEmEntARy BOunDs ON daRK Matter-nucleon scAttering at LoW DaRk matter maSSes, SeE RefereNces [@EriCkcek:2007JV; @GlusceVic:2017ywp; @Xu:2018eFh; @Capp

, EoS = BsK20}. We have e stimated t he sp in- dep en dent cap ture cross sec t ionshown in Figure \[fig: DD\]by 1.Sc aling the sp i n- i n dep en de ntca p tu re cr oss sectio n by a fac tor o f $$\left[\f r ac {4}{3}\fra c{J _n+1}{J_n} ( \la ngle S _n \ra n gle + \l angle S_p\r a ngle)^ 2\right]^ {- 1 }~,$$w here $J _ n = 1/2 $, $\langle S_n\r a ng l e = 1/2$ and $ \langl eS _p \ r ang le= 0$ for s ca tteri n g on ne u tr o n s , a n d\ 2. To ca lculate pas t a s catter in g w e haveusedth e pa sta structu re f actor inEq.  b u t havei mposed$S_{\r m p ast a} ( q )\l eq1$ , si n ce we exp ect spin -d ep enden t sc a t t e ring to not be c oherent overpas ta n u cle ons,and 3. To calc ulatequasi -e lastic scatteri ng o n nucleon s i nthe i nnera nd out ercru st, wefollowt hesa m e pr ocedure as spin-in de p e nd ent scat tering , s in c e in thi scas e sc a t terin g oc c ur s with i ndivid u al n ucleons . Altho ug h t hePICO- 6 0 li mit ap plies to spin - dependent scat t ering on prot o ns , we disp lay it as it p rovi d es t he c u rr ent bestsensi ti v it y to spin-dependentda rk mat ter i nteractions w ith nucleo n s at low d arkm at t er masses. Fur therm ore, mosts pin-depe ndent modelspredict r o u ghly equ alrat esofn e ut ron and proto n scat te ring. For spin-i nde pen den t s ca ttering,we see t ha tne ut ron crus t heating i s c le arl y mor e sensi tivethan d ir e ctdetecti o nf o r lo wda rk m att er mass es ( $ {m_ {\rm DM }}\lesssi m$G eV); c ru stal ca pture easilypr oceeds for t hes e mass e s (see Eq . ), but at terrestrial experim ent s itis d ifficulttodetect so f t nucl ear re coils a bov e energ y th res ho lds. Direc t det ectio nexpe riments benefit from an $ A ^2$ enhancementincros s se cti o nf rom n u cle a r coherence, whi ch helps t he m s urpass neu t ron s tar cru st sens itivi t y in so me rangeof ${m_{\ rm DM} } $ . F or spin-de pendentscatterin g , wes ee that ne utroncr ust heat ing is mor e sen sitive t han cu rrent d irect de tection searches for ne arly a ll da rkmatter ma sse s . W e note th at t hese state men tsholdtru e forsub- M eV da r k mat term asses asw el l,w h er e neutron s t a r cr ust h eat i ng occ ursthrough phonon ex c itations (seeFig. \ [fi g:b o unds 1\ ]). For comple men ta r y boundson dark matte r-nucleo ns catte ring a t lowdark ma t t er masses , se e R eferences  [@ Er i ckcek:2 00 7j v ; @Glu scev ic :2017y wp; @X u :201 8 e fh; @Capp

, EoS_= BsK20}. We_have estimated the spin-dependent_capture cross_section_shown in_Figure_\[fig:DD\] by 1. _Scaling the spin-independent_capture cross section by_a factor of_$$\left[\frac{4}{3}\frac{J_n+1}{J_n}_(\langle S_n\rangle + \langle S_p\rangle)^2\right]^{-1}~,$$ where $J_n = 1/2$, $\langle S_n\rangle = 1/2$ and_$\langle_S_p\rangle =_0$_for_scattering on neutrons, and\ 2. _To calculate pasta scattering we_have used_the pasta structure factor in Eq.  but have_imposed_$S_{\rm pasta} (q)_\leq 1$, since we expect spin-dependent scattering to not_be coherent over pasta nucleons, and 3._ To calculate_quasi-elastic_scattering_on nucleons in the_inner and outer crust, we follow_the same procedure as spin-independent scattering,_since in this case scattering occurs with_individual nucleons. Although the PICO-60 limit applies_to spin-dependent scattering on protons,_we display_it as it provides the_current best sensitivity_to spin-dependent_dark matter interactions_with nucleons at low dark matter_masses. Furthermore, most_spin-dependent models predict roughly equal rates_of_neutron and proton_scattering. For_spin-independent_scattering, we_see that neutron_crust_heating is_clearly_more sensitive than direct detection for_low_dark matter masses (${m_{\rm DM}}\lesssim$ GeV); crustal_capture easily proceeds for_these_masses (see Eq. ), but_at terrestrial experiments it is_difficult to detect soft nuclear recoils_above energy_thresholds. Direct_detection experiments benefit from an $A^2$ enhancement in cross section from_nuclear coherence, which helps them surpass_neutron star crust sensitivity_in some_range_of ${m_{\rm DM}}$._For_spin-dependent scattering,_we see that neutron crust heating is_more sensitive_than current direct detection searches for_nearly all dark matter_masses._We note that these statements hold_true for sub-MeV dark matter masses_as well, where neutron star_crust_heating_occurs through phonon excitations (see_Fig. \[fig:bounds1\]). For complementary bounds on dark_matter-nucleon scattering at_low dark matter masses, see References [@Erickcek:2007jv; @Gluscevic:2017ywp;_@Xu:2018efh;_@Capp

this periodicity in $(z,z')$ is reflected in a similar periodicity of various asymptotic properties of $z$-measures, see Sections 10 and 11 of [@BO4]. ### Unitarity Recall that the $z$-measures are positive if either $z'=\bar z$ or $z,z'\in(n,n+1)$ for some $n$. By analogy with representation theory of $SL(2)$, these cases were called the principal and the complementary series. Observe that in these case the above representations have a positive defined Hermitian form $Q$ which is invariant in the following sense $$Q(Lu,v)=Q(u,Lv)\,, \quad Q(Uu,v)=Q(u,Dv)\,.$$ The form $Q$ is given by $$Q(v_k,v_k)= \begin{cases}1 & z'=\bar z\,, \\ \dfrac{\Gamma(z'+k+\frac12)}{\Gamma(z+k+\frac12)}\, &z,z'\in(n,n+1)\,, \end{cases}$$ and $Q(v_k,v_l)=0$ if $k\ne l$. It follows that the operators $$\tfrac{i}2\,L,\tfrac12\,(U-D), \tfrac{i}2\,(U+D) \in {\mathfrak{sl}(2)}\,,$$ which form a standard basis of ${\mathfrak{su}}(1,1)$, are skew-Hermitian and hence this representation of ${\mathfrak{su}}(1,1)$ can be integrated to a unitary representation of the universal covering group of $SU(1,1)$. This group $SU(1,1)$ is isomorphic to $SL(2,{\mathbb{R}})$ and the above representations correspond to the principal and complementary series of unitary representations of the universal covering of $SL(2,{\mathbb{R}})$, see [@Pu]. The infinite wedge module ------------------------- Consider the module $\Lambda^{\frac{\infty}2}\, V$ which is, by definition, spanned by vectors $$\delta_S=v_{s_1} \wedge v_{s_2} \wedge v_{s_3} \wedge \dots\,,$$ where $S=\{s_1>s_2>\dots\}\subset {\mathbb{Z}}+{{\textstyle \frac12}}$

this periodicity in $ (z, z')$ is reflected in a similar periodicity of diverse asymptotic place of $ z$-measures, see Sections 10 and 11 of [ @BO4 ]. # # # Unitarity Recall that the $ z$-measures are positive if either $ z'=\bar z$ or $ z, z'\in(n, n+1)$ for some $ n$. By doctrine of analogy with representation theory of $ SL(2)$, these cases were call the principal and the complementary serial. note that in these case the above representation have a positive defined Hermitian form $ Q$ which is invariant in the following sense $ $ Q(Lu, v)=Q(u, Lv)\, , \quad Q(Uu, v)=Q(u, Dv)\,.$$ The kind $ Q$ is given by $ $ Q(v_k, v_k)= \begin{cases}1 & z'=\bar z\, , \\ \dfrac{\Gamma(z'+k+\frac12)}{\Gamma(z+k+\frac12)}\, & z, z'\in(n, n+1)\, , \end{cases}$$ and $ Q(v_k, v_l)=0 $ if $ k\ne l$. It follows that the operators $ $ \tfrac{i}2\,L,\tfrac12\,(U - five hundred), \tfrac{i}2\,(U+D) \in { \mathfrak{sl}(2)}\,,$$ which form a standard basis of $ { \mathfrak{su}}(1,1)$, are skew - Hermitian and hence this representation of $ { \mathfrak{su}}(1,1)$ can be integrate to a unitary representation of the universal covering group of $ SU(1,1)$. This group $ SU(1,1)$ is isomorphic to $ SL(2,{\mathbb{R}})$ and the above representation correspond to the principal and complementary series of unitary theatrical performance of the universal covering of $ SL(2,{\mathbb{R}})$, see [ @Pu ]. The infinite wedge module ------------------------- Consider the module $ \Lambda^{\frac{\infty}2}\, V$ which is, by definition, spanned by vectors $ $ \delta_S = v_{s_1 } \wedge v_{s_2 } \wedge v_{s_3 } \wedge \dots\,,$$ where $ S=\{s_1 > s_2>\dots\}\subset { \mathbb{Z}}+{{\textstyle \frac12}}$

thls periodicity in $(z,z')$ is veflected in a snnilar 'eriodidity of xarious asymptotic propertied if $z$-mtcsures, see Sections 10 and 11 of [@HO4]. ### Unitaeity Cecall that the $v$-jeasures are llsitnvx if either $z'=\bat z$ or $z,z'\in(n,t+1)$ for some $n$. Bf xnclogy with representation theory of $FL(2)$, thesr fases were caljed uhe prihbikal and the complementary series. Ogserve uhat in these case the above representations havf a positive definfd Hermitiab fown $Q$ which is invariant in the follkwing sense $$Q(Lu,v)=Q(u,Lv)\,, \quad Q(Uu,v)=Q(j,Dv)\,.$$ Tke form $Q$ iw tivft by $$Q(v_k,v_k)= \bxgin{cafes}1 & z'=\bar z\,, \\ \dfrac{\Gakma(z'+k+\frsc12)}{\Gamma(z+k+\frac12)}\, &z,z'\mn(n,n+1)\,, \end{cases}$$ and $Q(v_k,t_l)=0$ if $k\ne l$. It follors that tve operators $$\tfrac{i}2\,O,\tdrac12\,(U-G), \tfsac{i}2\,(J+E) \iv {\mztifrzk{sl}(2)}\,,$$ wjici form a stzndard basiw of ${\mathfrak{su}}(1,1)$, are shve-Hermitian ahd hense this representation of ${\mathfrak{su}}(1,1)$ can te jntegrated to a unitary representation of thg universaj covering group of $SU(1,1)$. This group $SU(1,1)$ is isomorphiw to $AU(2,{\mauhnn{R}})$ ave hhe above representations correspond to the pwjnviial and complemenbary series of uniyagy tepresentationr of tks hniversal covering of $SL(2,{\mwthbb{E}})$, see [@Pu]. Tre imfinite wedge module ------------------------- Considee the module $\Oambda^{\frac{\infty}2}\, V$ chich is, by befinijion, slanned by vectors $$\delta_R=v_{s_1} \sedge v_{s_2} \wefge v_{s_3} \wsage \dots\,,$$ where $R=\{s_1>s_2>\cods\}\subset {\mathbb{Z}}+{{\textstyle \srac12}}$

this periodicity in $(z,z')$ is reflected in periodicity various asymptotic of $z$-measures, see [@BO4]. Unitarity Recall that $z$-measures are positive either $z'=\bar z$ or $z,z'\in(n,n+1)$ for $n$. By analogy with representation theory of $SL(2)$, these cases were called the and the complementary series. Observe that in these case the above representations have positive Hermitian $Q$ is invariant in the following sense $$Q(Lu,v)=Q(u,Lv)\,, \quad Q(Uu,v)=Q(u,Dv)\,.$$ The form $Q$ is given by $$Q(v_k,v_k)= & z'=\bar z\,, \\ \dfrac{\Gamma(z'+k+\frac12)}{\Gamma(z+k+\frac12)}\, &z,z'\in(n,n+1)\,, \end{cases}$$ and if $k\ne l$. It that the operators $$\tfrac{i}2\,L,\tfrac12\,(U-D), \tfrac{i}2\,(U+D) {\mathfrak{sl}(2)}\,,$$ form a basis ${\mathfrak{su}}(1,1)$, skew-Hermitian and hence representation of ${\mathfrak{su}}(1,1)$ can be integrated to a unitary representation of the universal covering group of $SU(1,1)$. group $SU(1,1)$ to $SL(2,{\mathbb{R}})$ the representations to the principal series of unitary representations of the $SL(2,{\mathbb{R}})$, see [@Pu]. The infinite wedge module ------------------------- the module V$ which is, by definition, spanned vectors $$\delta_S=v_{s_1} \wedge v_{s_2} \wedge v_{s_3} \wedge \dots\,,$$ $S=\{s_1>s_2>\dots\}\subset {\mathbb{Z}}+{{\textstyle \frac12}}$

this periodicity in $(z,z')$ is reflEcted in a siMilar PerIodIcIty oF varIous asymptotic PRopeRties of $z$-measures, see SecTions 10 AnD 11 Of [@BO4]. ### uNiTaritY Recall THaT THe $z$-MeAsUreS aRE pOsitiVe iF either $Z'=\bar z$ or $z,z'\iN(n,n+1)$ FoR some $n$. By analOGy With represEntAtion theory oF $SL(2)$, These cAsEs wERe calLed The prIncipaL And the ComplemenTaRY serieS. observe THAt In thEse case the above rePReSEntations have a PositiVe DEfINEd HErmItian form $Q$ WhIch is INvarianT In THE FolLOwing sense $$Q(Lu,V)=Q(u,Lv)\,, \quad Q(UU,V)=Q(u,dv)\,.$$ The fOrM $Q$ iS Given bY $$Q(v_k,v_K)= \bEGin{Cases}1 & z'=\bar z\,, \\ \dFrac{\gamma(z'+k+\frAc12)}{\GammA(Z+k+\frac12)}\, &z,Z'\In(n,n+1)\,, \end{Cases}$$ aNd $Q(V_k,v_L)=0$ if $k\NE l$. it FolLoWS thAT tHe oPEraTors $$\tfraC{i}2\,l,\tFrac12\,(U-d), \tfrAC{I}2\,(u+d) \in {\mAthFrak{Sl}(2)}\,,$$ whiCh form a standaRd bAsis OF ${\maThfraK{su}}(1,1)$, arE skeW-HErmitIan and Hence ThIs representatioN of ${\mAthfrak{su}}(1,1)$ Can Be IntEgRated TO a unitAry RepResentaTion of tHE unIvERSAl Covering group of $SU(1,1)$. THiS GRoUp $SU(1,1)$ is isOmorphIC tO $Sl(2,{\Mathbb{R}})$ aNd The AbovE REpresEntaTIoNs corresPond to THe PrIncipal AnD complEmEntAry SerieS Of unItary rEpresentAtionS Of the universal COvering of $SL(2,{\maTHbB{r}})$, SeE [@pu]. ThE inFinite wedge ModuLE ------------------------- ConSideR ThE moDUle $\LaMbda^{\fRaC{\InFTy}2}\, V$ which is, by definitIoN, spannEd by vEctors $$\delta_S=v_{S_1} \wedge v_{s_2} \weDGE V_{s_3} \wedge \dOts\,,$$ wHErE $s=\{s_1>s_2>\dots\}\subset {\mAthbb{z}}+{{\textstyle \FRac12}}$

this periodicity in $(z,z ')$ is ref lecte d i n a s imil ar p eriodicity ofv ario us asymptotic properti es of $ z $-me a su res,see Sec t io n s 10 a nd 11 o f [ @BO4] . ### Uni tarity Re cal lthat the $z$ - me asures are po sitive if ei the r $z'= \b arz $ or$z, z'\in (n,n+1 ) $ forsome $n$. B y analo g y withr e pr esen tation theory of$ SL ( 2)$, these cas es wer ec al l e d t heprincipalan d the complem e nt a r y se r ies. Observe that in th e secase t he ab o ve rep resen ta t ion s have a po siti ve define d Herm i tian fo r m $Q$ w hich i s i nva rian t i nthe f o llo w in g s e nse $$Q(Lu, v) =Q (u,Lv )\,, \ q u ad Q (Uu ,v)= Q(u,D v)\,.$$ The f orm $Q$ isgiven by $ $Q(v _k ,v_k) = \beg in{ca se s}1 & z'=\barz\,, \\ \dfra c{\ Ga mma (z '+k+\ f rac12) }{\ Gam ma(z+k+ \frac12 ) }\, & z ,z '\in(n,n+1)\,, \ e nd {cases}$ $ and$ Q( v_ k ,v_l)=0$ i f $ k\ne l $. It fol l ow s that t he ope r at or s $$\tf ra c{i}2\ ,L ,\t fra c12\, ( U-D) , \tfr ac{i}2\, (U+D) \in {\mathfrak { sl}(2)}\,,$$w hi c h f o rm a st andard basi s of ${\m athf r ak {su } }(1,1 )$, a re sk e w-Hermitian and hen ce thisrepre sentation of${\mathfra k { s u}}(1,1) $ ca n b e integrated to a un itary repr e sentatio n ofthe univ ersal cov e r ing grou p o f $ SU( 1,1 ) $ .This group $S U ( 1,1) $is isom orp hic to$SL (2, {\m ath bb {R}})$ an d the ab ov ere pr ese ntati o ns corre sp ond t o t he pr i ncipal andcomp le me n tar y serie s o f unit ar yrepr ese nt ation s of the univer sal cover ing of $ SL (2 ,{\math bb{R}})$, see [ @Pu]. The i nfi nite w e d ge modul e --------------------- - --- Co nsi der t he m odule $\L amb da^{\f rac { \infty }2}\,V$ wh ic h i s , by d e f in iti on , spannedb y ve ctors $ $\de lta_S=v _{s_1} \wedge v_{s _ 2}\wedge v_{s_ 3}\wed g e \do t s\ , ,$$ w h ere $ S=\{s_1>s_2>\do ts\}\subse t{ \m athbb{Z}}+ { {\t ex tstyle\frac12 }}$

this_periodicity in_$(z,z')$ is reflected in_a similar_periodicity_of various_asymptotic_properties of $z$-measures,_see Sections 10_and 11 of [@BO4]. ###_Unitarity Recall that the_$z$-measures_are positive if either $z'=\bar z$ or $z,z'\in(n,n+1)$ for some $n$. By analogy with_representation_theory of_$SL(2)$,_these_cases were called the principal_and the complementary series. Observe that_in these_case the above representations have a positive defined_Hermitian_form $Q$ which_is invariant in the following sense $$Q(Lu,v)=Q(u,Lv)\,, \quad Q(Uu,v)=Q(u,Dv)\,.$$_The form $Q$ is given by_$$Q(v_k,v_k)= \begin{cases}1 &_z'=\bar_z\,,_\\ \dfrac{\Gamma(z'+k+\frac12)}{\Gamma(z+k+\frac12)}\, &z,z'\in(n,n+1)\,, _ \end{cases}$$ and $Q(v_k,v_l)=0$_if $k\ne l$. It follows that_the operators $$\tfrac{i}2\,L,\tfrac12\,(U-D), \tfrac{i}2\,(U+D) \in {\mathfrak{sl}(2)}\,,$$ which_form a standard basis of ${\mathfrak{su}}(1,1)$,_are skew-Hermitian and hence this_representation of_${\mathfrak{su}}(1,1)$ can be integrated to_a unitary representation_of the_universal covering group_of $SU(1,1)$. This group $SU(1,1)$ is_isomorphic to $SL(2,{\mathbb{R}})$_and the above representations correspond to_the_principal and complementary_series_of_unitary representations_of the universal_covering_of $SL(2,{\mathbb{R}})$,_see_[@Pu]. The infinite wedge module ------------------------- Consider the module_$\Lambda^{\frac{\infty}2}\,_V$ which is, by definition, spanned by_vectors $$\delta_S=v_{s_1} \wedge v_{s_2}_\wedge_ v_{s_3} \wedge _\dots\,,$$ where $S=\{s_1>s_2>\dots\}\subset {\mathbb{Z}}+{{\textstyle \frac12}}$

, these imply embeddedness. Also, [(\[e:boundary\])]{} shows that there is some $r_0$ such that the boundary of the graph in [(\[e:graphical\])]{} lies outside a circle $B_{r_0}(F_N(t,0))$ for all $N$. And [(\[e:boundaryinfinity\])]{} shows that for all $x \leq 0$ (i.e. in the part of the image $F_N(\Omega_N)$ below the $\{x_3=0\}$ plane), these boundaries of the graph in [(\[e:graphical\])]{} actually go to infinity as $N \to \infty$. ![A horizontal slice of $F(\Omega_N)$ in Lemma \[embeddedness\][]{data-label="horizontalslice"}](horizontalslice.eps){width="\textwidth"} Since $z_0=0$ and the height differential is $\phi=dz,\,$ [(\[e:zequalx\])]{} follows immediately from [(\[weierstrassrep\])]{}. Now we prove [(\[e:graphical\])]{} first for $0<x \leq \frac{1}{2}$ (i.e. on $\Omega_N^+$) and then for $-\frac{1}{2} \leq x \leq 0$ (i.e. on $\Omega_N^-$) **Proof of [(\[e:graphical\])]{} on $\Omega_N^{+}$** ---------------------------------------------------- We first note that on $$\Omega_N^{+} =\left\{(x,y) \, \left| \, |y| \leq \frac{(x^2+(\frac{1}{N})^2)^{5/4}}{4},\, 0 < x \leq 1/2 \right. \right\},$$ $$\label{y} \displaystyle y^2 \leq \frac{(x^2+(\frac{1}{N})^2)^{5/2}}{16} \leq \frac{x^2+(\frac{1}{N})^2}{16} \leq \frac{(x+ \frac{k}{N})^2+(\frac{1}{N})^2}{16};$$ $$y^2 \leq \frac{(x^2+(\frac{1}{N})^2)^{5/2}}{16} \leq \frac{((x+ \frac{k}{N})^2+(\frac{1}{N

, these imply embeddedness. Also, [ (\[e: boundary\ ]) ] { } shows that there is some $ r_0 $ such that the limit of the graph in [ (\[e: graphical\ ]) ] { } dwell outside a circle $ B_{r_0}(F_N(t,0))$ for all $ N$. And [ (\[e: boundaryinfinity\ ]) ] { } shows that for all $ ten \leq 0 $ (i.e. in the function of the image $ F_N(\Omega_N)$ below the $ \{x_3=0\}$ plane), these boundary of the graph in [ (\[e: graphical\ ]) ] { } actually go to infinity as $ N \to \infty$. ! [ A horizontal piece of $ F(\Omega_N)$ in Lemma \[embeddedness\][]{data - label="horizontalslice"}](horizontalslice.eps){width="\textwidth " } Since $ z_0=0 $ and the height differential is $ \phi = dz,\,$ [ (\[e: zequalx\ ]) ] { } keep up immediately from [ (\[weierstrassrep\ ]) ] { }. Now we test [ (\[e: graphical\ ]) ] { } foremost for $ 0 < x \leq \frac{1}{2}$ (i.e. on $ \Omega_N^+$) and then for $ -\frac{1}{2 } \leq x \leq 0 $ (i.e. on $ \Omega_N^-$) * * Proof of [ (\[e: graphical\ ]) ] { } on $ \Omega_N^{+}$ * * ---------------------------------------------------- We first notice that on $ $ \Omega_N^{+ } = \left\{(x, y) \, \left| \, |y| \leq \frac{(x^2+(\frac{1}{N})^2)^{5/4}}{4},\, 0 < x \leq 1/2 \right. \right\},$$ $ $ \label{y } \displaystyle y^2 \leq \frac{(x^2+(\frac{1}{N})^2)^{5/2}}{16 } \leq \frac{x^2+(\frac{1}{N})^2}{16 } \leq \frac{(x+ \frac{k}{N})^2+(\frac{1}{N})^2}{16};$$ $ $ y^2 \leq \frac{(x^2+(\frac{1}{N})^2)^{5/2}}{16 } \leq \frac{((x+ \frac{k}{N})^2+(\frac{1}{N

, thfse imply embeddedness. Auso, [(\[e:boundary\])]{} skiws thet thers is somd $r_0$ such that the boundary oh thw grakk in [(\[e:graphical\])]{} lies outside w circle $B_{r_0}(F_I(t,0))$ for all $N$. And [(\[e:boundavvinfihlty\])]{} skoxs that for all $x \leq 0$ (i.e. in the part ox ghz image $F_N(\Omega_N)$ below the $\{x_3=0\}$ plane), trese boinfaries of the drapn in [(\[s:graphical\])]{} actually go to infinity as $N \tm \infty$. ![A horiaontal slice of $F(\Omega_N)$ in Lemla \[embeddedness\][]{datw-label="horizintajwlice"}](horizongalslice.eps){width="\textwisth"} Since $z_0=0$ and the height diffefenticl is $\phi=dz,\,$ [(\[e:zeqkdlx\])]{} follows immeqiately from [(\[weierstsassrep\])]{}. Mow we prove [(\[e:nraphmcal\])]{} first for $0<x \leq \frac{1}{2}$ (i.e. on $\Omega_N^+$) and jhen for $-\fsae{1}{2} \leq x \leq 0$ (i.e. on $\Omwgq_N^-$) **Promf ox [(\[e:gfqphkcam\])]{} pn $\Omega_J^{+}$** ---------------------------------------------------- We first nots that on $$\Onega_N^{+} =\left\{(x,y) \, \left| \, |r| \leq \frac{(x^2+(\frad{1}{N})^2)^{5/4}}{4},\, 0 < x \lqq 1/2 \right. \right\},$$ $$\label{y} \displaystyle y^2 \lex \fdac{(x^2+(\frac{1}{N})^2)^{5/2}}{16} \leq \frac{x^2+(\frac{1}{N})^2}{16} \leq \frac{(x+ \frac{k}{N})^2+(\frac{1}{N})^2}{16};$$ $$y^2 \leq \frwc{(x^2+(\frac{1}{N})^2)^{5/2}}{16} \leq \frac{((x+ \frac{k}{N})^2+(\frac{1}{N

, these imply embeddedness. Also, [(\[e:boundary\])]{} shows is $r_0$ such the boundary of outside circle $B_{r_0}(F_N(t,0))$ for $N$. And [(\[e:boundaryinfinity\])]{} that for all $x \leq 0$ in the part of the image $F_N(\Omega_N)$ below the $\{x_3=0\}$ plane), these boundaries the graph in [(\[e:graphical\])]{} actually go to infinity as $N \to \infty$. ![A slice $F(\Omega_N)$ Lemma Since $z_0=0$ and the height differential is $\phi=dz,\,$ [(\[e:zequalx\])]{} follows immediately from [(\[weierstrassrep\])]{}. Now we prove first for $0<x \leq \frac{1}{2}$ (i.e. on $\Omega_N^+$) then for $-\frac{1}{2} \leq \leq 0$ (i.e. on $\Omega_N^-$) of on $\Omega_N^{+}$** We note on $$\Omega_N^{+} =\left\{(x,y) \left| \, |y| \leq \frac{(x^2+(\frac{1}{N})^2)^{5/4}}{4},\, 0 < x \leq 1/2 \right. \right\},$$ $$\label{y} \displaystyle y^2 \leq \frac{(x^2+(\frac{1}{N})^2)^{5/2}}{16} \frac{x^2+(\frac{1}{N})^2}{16} \leq $$y^2 \leq \leq \frac{k}{N})^2+(\frac{1}{N

, these imply embeddedness. AlsO, [(\[e:boundary\])]{} Shows ThaT thErE is sOme $r_0$ Such that the bouNDary Of the graph in [(\[e:graphical\])]{} Lies oUtSIde a CIrCle $B_{r_0}(f_N(t,0))$ for aLL $N$. aND [(\[e:bOuNdAryInFInIty\])]{} shOws That for All $x \leq 0$ (i.e. iN thE pArt of the imagE $f_N(\omega_N)$ beloW thE $\{x_3=0\}$ plane), these BouNdarieS oF thE Graph In [(\[e:GraphIcal\])]{} acTUally gO to infiniTy AS $N \to \inFTy$. ![A horiZONtAl slIce of $F(\Omega_N)$ in LemMA \[eMBeddedness\][]{data-Label="hOrIZoNTAlsLicE"}](horizontaLsLice.ePS){width="\tEXtWIDTh"} SINce $z_0=0$ and the heiGht differenTIal Is $\phi=dZ,\,$ [(\[e:ZeqUAlx\])]{} folLows iMmEDiaTely from [(\[weiErstRassrep\])]{}. NoW we proVE [(\[e:graphICal\])]{} firsT for $0<x \lEq \fRac{1}{2}$ (I.e. on $\oMeGa_n^+$) anD tHEn fOR $-\fRac{1}{2} \LEq x \Leq 0$ (i.e. on $\OMeGa_n^-$) **ProoF of [(\[e:GRAPHicaL\])]{} on $\omegA_N^{+}$** ---------------------------------------------------- We fIrst note that oN $$\OmEga_N^{+} =\LEft\{(X,y) \, \lefT| \, |y| \leq \Frac{(X^2+(\fRac{1}{N})^2)^{5/4}}{4},\, 0 < x \Leq 1/2 \rigHt. \rigHt\},$$ $$\Label{y} \displaystYle y^2 \Leq \frac{(x^2+(\fRac{1}{n})^2)^{5/2}}{16} \lEq \fRaC{x^2+(\fraC{1}{n})^2}{16} \leq \frAc{(x+ \FraC{k}{N})^2+(\frac{1}{n})^2}{16};$$ $$y^2 \leq \frAC{(x^2+(\fRaC{1}{n})^2)^{5/2}}{16} \LEq \Frac{((x+ \frac{k}{N})^2+(\frac{1}{N

, these imply embeddedness . Also, [( \[e:b oun dar y\ ])]{ } sh ows that there is s ome $r_0$ such that th e bou nd a ry o f t he gr aph in[ (\ [ e :gr ap hi cal \] ) ]{ } lie s o utsidea circle $ B_{ r_ 0}(F_N(t,0)) $ f or all $N$ . A nd [(\[e:bou nda ryinfi ni ty\ ] )]{}sho ws th at for all $x \leq 0$(i . e. int he part o ftheimage $F_N(\Omega _ N) $ below the $\{ x_3=0\ }$ pl a n e), th ese bounda ri es of the gra p hi n [(\ [ e:graphical\] )]{} actual l y g o to i nf ini t y as $ N \to \ i nft y$. ![A ho rizo ntal slic e of $ F (\Omega _ N)$ inLemma\[e mbe dded n es s\ ][] {d a ta- l ab el= " hor izontals li ce "}](h oriz o n t a lsli ce. eps) {widt h="\textwidth "} Sin c e $ z_0=0 $ and the h eight diffe renti al is $\phi=dz,\, $ [( \[e:zequa lx\ ]) ]{} f ollow s immed iat ely from [ (\[weie r str as s r e p\ ])]{}. Now we pro ve [ (\ [e:graph ical\] ) ]{ }f irst for $ 0<x \le q \frac {1}{ 2 }$ (i.e. o n $\Om e ga _N ^+$) an dthen f or $- \fr ac{1} { 2} \ leq x\leq 0$(i.e. on $\Omega_N^- $ ) **Proof of [( \ [ e: g raph ica l\])]{} on$\Om e ga_N ^{+} $ ** -- - ----- ----- -- - -- - ------------------- -- ------ ----- - We first n ote that o n $ $\Omega_ N^{+ } = \ left\{(x,y) \, \lef t| \, |y| \leq \fr ac{(x ^2+(\fra c{1}{N})^ 2 ) ^{5/4}}{ 4}, \,0 < x\ l eq 1/2 \right.\ r ight \} ,$$ $$\ lab el{y} \ dis pla yst yle y^2 \leq\frac{(x ^2 +( \f ra c{1 }{N}) ^ 2)^{5/2} }{ 16} \ leq \fra c {x^2+( \frac {1}{ N} )^ 2 }{1 6} \leq \f r a c{(x +\f rac{ k}{ N} )^2+( \fra c {1} {N})^2} {16};$$ $ $y^ 2 \le q\f rac{(x^ 2+(\frac{1}{N }) ^2)^{5/2}} {1 6}\leq \ f r ac{((x+\frac{k}{N})^2+(\frac{1 } {N

, these_imply embeddedness._Also, [(\[e:boundary\])]{} shows that_there is_some_$r_0$ such_that_the boundary of_the graph in_[(\[e:graphical\])]{} lies outside a_circle $B_{r_0}(F_N(t,0))$ for_all_$N$. And [(\[e:boundaryinfinity\])]{} shows that for all $x \leq 0$ (i.e. in the part_of_the image_$F_N(\Omega_N)$_below_the $\{x_3=0\}$ plane), these boundaries_of the graph in [(\[e:graphical\])]{}_actually go_to infinity as $N \to \infty$. ![A horizontal slice_of_$F(\Omega_N)$ in Lemma_\[embeddedness\][]{data-label="horizontalslice"}](horizontalslice.eps){width="\textwidth"} Since $z_0=0$ and the height differential is $\phi=dz,\,$ [(\[e:zequalx\])]{}_follows immediately from [(\[weierstrassrep\])]{}. Now we prove_[(\[e:graphical\])]{} first for_$0<x_\leq_\frac{1}{2}$ (i.e. on $\Omega_N^+$)_and then for $-\frac{1}{2} \leq x_\leq 0$ (i.e. on $\Omega_N^-$) **Proof of_[(\[e:graphical\])]{} on $\Omega_N^{+}$** ---------------------------------------------------- We first note that on_$$\Omega_N^{+} =\left\{(x,y) \, \left| \,_|y| \leq \frac{(x^2+(\frac{1}{N})^2)^{5/4}}{4},\, 0 <_x \leq_1/2 \right. \right\},$$ $$\label{y} \displaystyle y^2_\leq \frac{(x^2+(\frac{1}{N})^2)^{5/2}}{16} \leq \frac{x^2+(\frac{1}{N})^2}{16}_\leq \frac{(x+ \frac{k}{N})^2+(\frac{1}{N})^2}{16};$$_$$y^2 \leq \frac{(x^2+(\frac{1}{N})^2)^{5/2}}{16}_\leq \frac{((x+ \frac{k}{N})^2+(\frac{1}{N

of d = 11 Supergravity on AdS$_4 \times S^7$]{},” [*JHEP*]{} [**1203**]{} (2012) 099, [[1112.6131]{}](http://arXiv.org/abs/1112.6131). F. Benini and N. Bobev, “[Two-dimensional SCFTs from wrapped branes and c-extremization]{},” [[1302.4451]{}](http://arXiv.org/abs/1302.4451). P. Szepietowski, “[Comments on a-maximization from gauged supergravity]{},” [*JHEP*]{} [**1212**]{} (2012) 018, [[1209.3025]{}](http://arXiv.org/abs/1209.3025). P. Karndumri and E. O Colgain, “[Supergravity dual of c-extremization]{},” [*Phys. Rev.*]{} [**D 87**]{} (2013) 101902, [[1302.6532]{}](http://arXiv.org/abs/1302.6532). N. Halmagyi, M. Petrini, and A. Zaffaroni, “[work in progress]{},”. [^1]: To be precise, the black holes we are discussing will asymptotically approach $AdS_4$ in the UV but will differ by non-normalizable terms corresponding to some magnetic charge. We will nevertheless refer to them as asymptotically $AdS_4$ black holes. [^2]: Other M-theory reductions have been studied in [@Donos:2010ax; @Cassani:2011fu] and similar reductions have been performed in type IIA/IIB, see for example [@Kashani-Poor:2006si; @KashaniPoor:2007tr; @Gauntlett:2010vu; @Skenderis:2010vz; @Cassani:2010uw; @Liu:2010pq; @Bena:2010pr; @Cassani:2010na] [^3]: For a discussion of these compactifications from the point of view of holography and recent results in identifying the dual field theories see[@Fabbri:1999hw; @Jafferis:2008qz; @Hanany:2008cd; @Martelli:2008rt; @Hanany:2008fj; @Martelli:2009ga; @Fr

of d = 11 Supergravity on AdS$_4 \times S^7 $ ] { }, ” [ * JHEP * ] { } [ * * 1203 * * ] { } (2012) 099, [ [ 1112.6131]{}](http://arXiv.org / abs/1112.6131). F.   Benini and N.   Bobev, “ [ Two - dimensional SCFTs from wrapped branes and c - extremization ] { }, ” [ [ 1302.4451]{}](http://arXiv.org / abs/1302.4451). P.   Szepietowski, “ [ Comments on a - maximization from gauged supergravity ] { }, ” [ * JHEP * ] { } [ * * 1212 * * ] { } (2012) 018, [ [ 1209.3025]{}](http://arXiv.org / abs/1209.3025). P.   Karndumri and E.   oxygen Colgain, “ [ Supergravity dual of coulomb - extremization ] { }, ” [ * Phys.   Rev. * ] { } [ * * D 87 * * ] { } (2013) 101902, [ [ 1302.6532]{}](http://arXiv.org / abs/1302.6532). N.   Halmagyi, M.   Petrini, and A.   Zaffaroni, “ [ make in advancement ] { }, ”. [ ^1 ]: To be precise, the black holes we are discourse will asymptotically approach $ AdS_4 $ in the UV but will disagree by non - normalizable term corresponding to some magnetic charge. We will nevertheless refer to them as asymptotically $ AdS_4 $ black hole. [ ^2 ]: Other M - theory reductions have been learn in [ @Donos:2010ax; @Cassani:2011fu ] and similar reductions have been performed in type IIA / IIB, understand for example [ @Kashani - Poor:2006si; @KashaniPoor:2007tr; @Gauntlett:2010vu; @Skenderis:2010vz; @Cassani:2010uw; @Liu:2010pq; @Bena:2010pr; @Cassani:2010na ] [ ^3 ]: For a discussion of these compactifications from the point of view of holography and recent results in name the dual field theories see[@Fabbri:1999hw; @Jafferis:2008qz; @Hanany:2008cd; @Martelli:2008rt; @Hanany:2008fj; @Martelli:2009ga; @Fr

of d = 11 Supergravity on AdS$_4 \times S^7$]{},” [*JHEP*]{} [**1203**]{} (2012) 099, [[1112.6131]{}](http://acXiv.org/zbs/1112.6131). F. Benkni and N. Bobev, “[Two-dimensionap WCFTs from wrapped branes avd c-extrelization]{},” [[1302.4451]{}](httk://arXiv.org/abs/1302.4451). P. Sze'jetowskl, “[Comjcnts mi a-maximization from gaugad supergravitf]{},” [*GHZP*]{} [**1212**]{} (2012) 018, [[1209.3025]{}](http://arXiv.org/abs/1209.3025). P. Karndumri and Q. O Colgsij, “[Supergravity duak of d-vxuremization]{},” [*Phys. Rev.*]{} [**D 87**]{} (2013) 101902, [[1302.6532]{}](http://arXib.org/abs/1302.6532). I. Halmagyi, M. Petroni, and A. Zaffaroni, “[work in prohress]{},”. [^1]: To be precisf, the black holqw we are diszussing will asymptotidally approach $AdS_4$ in the UV bug wilk differ bi hoj-tormalizablx termf correspondlmg to vome mabnetic charge. Ee xill nevertheless refer tm them as asymptotycally $AdV_4$ ylack holes. [^2]: Other M-thwoey regucthons yavd bten sfudied in [@Donos:2010ax; @Czssani:2011fu] ane similar reductionx rqve been perfkrmed yn type IIA/IIB, see for example [@Kashani-Poog:2006si; @IashaniPoor:2007tr; @Gauntlett:2010cu; @Skenderis:2010vz; @Cassanl:2010uw; @Liu:2010pq; @Bena:2010pr; @Cassani:2010na] [^3]: For a discussion of these compawtifiragiouw froo tje point of view of holography and recent restmtx pn identifying thc dual field theoroed xge[@Fabbri:1999hw; @Jafweris:2008qv; @Hznany:2008cd; @Martelli:2008rt; @Hanany:2008sj; @Maetelli:2009ga; @Sr

of d = 11 Supergravity on AdS$_4 [*JHEP*]{} (2012) 099, F. Benini and wrapped and c-extremization]{},” [[1302.4451]{}](http://arXiv.org/abs/1302.4451). Szepietowski, “[Comments on from gauged supergravity]{},” [*JHEP*]{} [**1212**]{} (2012) [[1209.3025]{}](http://arXiv.org/abs/1209.3025). P. Karndumri and E. O Colgain, “[Supergravity dual of c-extremization]{},” [*Phys. Rev.*]{} 87**]{} (2013) 101902, [[1302.6532]{}](http://arXiv.org/abs/1302.6532). N. Halmagyi, M. Petrini, and A. Zaffaroni, “[work in [^1]: be the holes we are discussing will asymptotically approach $AdS_4$ in the UV but will differ by non-normalizable corresponding to some magnetic charge. We will nevertheless to them as asymptotically black holes. [^2]: Other M-theory have studied in @Cassani:2011fu] similar have been performed type IIA/IIB, see for example [@Kashani-Poor:2006si; @KashaniPoor:2007tr; @Gauntlett:2010vu; @Skenderis:2010vz; @Cassani:2010uw; @Liu:2010pq; @Bena:2010pr; @Cassani:2010na] [^3]: For a discussion these compactifications point of of and results in identifying field theories see[@Fabbri:1999hw; @Jafferis:2008qz; @Hanany:2008cd; @Martelli:2008rt;

of d = 11 Supergravity on AdS$_4 \times s^7$]{},” [*JHEP*]{} [**1203**]{} (2012) 099, [[1112.6131]{}](http://aRXiv.oRg/aBs/1112.6131). F. beNini And N. bobev, “[Two-dimensIOnal sCFTs from wrapped branes And c-eXtREmizATiOn]{},” [[1302.4451]{}](httP://arXiv.oRG/aBS/1302.4451). p. SzEpIeTowSkI, “[coMmentS on A-maximiZation from GauGeD supergravitY]{},” [*jHeP*]{} [**1212**]{} (2012) 018, [[1209.3025]{}](http://arXiV.orG/abs/1209.3025). P. KarndumRi aNd E. O CoLgAin, “[sUpergRavIty duAl of c-eXTremizAtion]{},” [*Phys. reV.*]{} [**d 87**]{} (2013) 101902, [[1302.6532]{}](http://aRxiv.org/aBS/1302.6532). n. HAlmaGyi, M. Petrini, and A. ZaFFaROni, “[work in progrEss]{},”. [^1]: To bE pREcISE, thE blAck holes we ArE discUSsing wiLL aSYMPtoTIcally approacH $AdS_4$ in the UV BUt wIll difFeR by NOn-normAlizaBlE TerMs corresponDing To some magNetic cHArge. We wILl neverThelesS reFer To thEM aS aSymPtOTicALlY $Ads_4$ BlaCk holes. [^2]: OThEr m-theoRy reDUCTIons HavE beeN studIed in [@Donos:2010ax; @CAssAni:2011fU] And SimilAr redUctiOnS have Been peRformEd In type IIA/IIB, see For eXample [@KasHanI-POor:2006Si; @kashaNIPoor:2007tR; @GaUntLett:2010vu; @SKenderiS:2010Vz; @CAsSANI:2010uW; @Liu:2010pq; @Bena:2010pr; @CassanI:2010nA] [^3]: fOr A discussIon of tHEsE cOMpactifiCaTioNs frOM The poInt oF ViEw of holoGraphy ANd ReCent resUlTs in idEnTifYinG the dUAl fiEld theOries see[@fabbrI:1999Hw; @Jafferis:2008qz; @HaNAny:2008cd; @Martelli:2008RT; @HANAnY:2008Fj; @MaRteLli:2009ga; @Fr

of d = 11 Supergravity on AdS$_4 \t imesS^7 $]{ }, ” [* JHEP *]{} [**1203** ] {} ( 2012) 099, [[1112.6131 ]{}]( ht t p:// a rX iv.or g/abs/1 1 12 . 6 131 ). F . B en i ni andN.Bobev,“[Two-dime nsi on al SCFTs fro m w rapped bra nes and c-extre miz ation] {} ,”[ [1302 .44 51]{} ](http : //arXi v.org/abs /1 3 02.445 1 ). P. S z e pi etow ski, “[Comments o n a - maximization f rom ga ug e ds u per gra vity]{},”[* JHEP* ] {} [**1 2 12 * * ] {}( 2012) 018, [[ 1209.3025]{ } ](h ttp:// ar Xiv . org/ab s/120 9. 3 025 ). P. Karn dumr i and E.O Colg a in, “[S u pergrav ity du alofc-ex t re mi zat io n ]{} , ”[*P h ys.  Rev.*]{ }[* *D 87 **]{ } ( 2 013) 10 1902 , [[1 302.6532]{}]( htt p:// a rXi v.org /abs/ 1302 .6 532). N. H almag yi , M. Petrini, a nd A . Zaffaro ni, “ [wo rk in p r ogress ]{} ,”. [^1]: To bep rec is e , th e black holes we a re d is cussingwill a s ym pt o ticallyap pro ach$ A dS_4$ int he UV butwill d i ff er by non -n ormali za ble te rms c o rres pondin g to som e mag n etic charge. W e will neverth e le s s r e fertothem as asy mpto t ical ly $ A dS _4$ black hole s. [ ^ 2]: Other M-theoryre ductio ns ha ve been studi ed in [@Do n o s :2010ax; @Ca s sa n i:2011fu] andsimil ar reducti o ns havebeenperforme d in type I IA/IIB,see fo r e xam p l e[@Kashani-Poo r : 2006 si ; @Kash ani Poor:20 07t r;@Ga unt le tt:2010vu ; @Skend er is :2 01 0vz ; @Ca s sani:201 0u w;@L iu: 2010p q ; @Ben a:201 0pr; @ Ca s san i:2010n a ][ ^3]: F or a d isc us sionof t h ese compac tificatio nsf romth epoint o f view of hol og raphy andre cen t resu l t s in ide ntifying the dual field theorie s s ee[@F abbr i:1999hw; @J afferi s:2 0 08qz;@Hanan y:200 8c d;@ M artel l i :2 008 rt ; @Hanany: 2 0 08f j; @M ar tell i:2009g a; @Fr

of_d =_11 Supergravity on AdS$_4_\times S^7$]{},”_[*JHEP*]{}_[**1203**]{} (2012)_099,_[[1112.6131]{}](http://arXiv.org/abs/1112.6131). F. Benini and_N. Bobev, “[Two-dimensional SCFTs_from wrapped branes and_c-extremization]{},” [[1302.4451]{}](http://arXiv.org/abs/1302.4451). P. Szepietowski,_“[Comments_on a-maximization from gauged supergravity]{},” [*JHEP*]{} [**1212**]{} (2012) 018, [[1209.3025]{}](http://arXiv.org/abs/1209.3025). P. Karndumri and E. O Colgain, “[Supergravity_dual_of c-extremization]{},”_[*Phys. Rev.*]{}_[**D_87**]{} (2013) 101902, [[1302.6532]{}](http://arXiv.org/abs/1302.6532). N. Halmagyi, M. Petrini,_and A. Zaffaroni, “[work in progress]{},”. [^1]:_To be_precise, the black holes we are discussing will_asymptotically_approach $AdS_4$ in_the UV but will differ by non-normalizable terms corresponding_to some magnetic charge. We will_nevertheless refer to_them_as_asymptotically $AdS_4$ black holes. [^2]:_Other M-theory reductions have been studied_in [@Donos:2010ax; @Cassani:2011fu] and similar reductions_have been performed in type IIA/IIB, see_for example [@Kashani-Poor:2006si; @KashaniPoor:2007tr; @Gauntlett:2010vu; @Skenderis:2010vz;_@Cassani:2010uw; @Liu:2010pq; @Bena:2010pr; @Cassani:2010na] [^3]: For_a discussion_of these compactifications from the_point of view_of holography_and recent results_in identifying the dual field theories_see[@Fabbri:1999hw; @Jafferis:2008qz; @Hanany:2008cd;_@Martelli:2008rt; @Hanany:2008fj; @Martelli:2009ga; @Fr

M_\odot$. By extrapolating the $\alpha=-1.33$ IMF from $M=0.15M_\odot$ all the way down to zero mass, we obtain $123.5 M_\odot$ of unseen dwarfs, thus totaling $440.5 M_\odot$ in living stars and brown dwarfs. To account for the remnants we need to adopt an initial mass-final mass relation. We used the semi-empirical relation proposed by Renzini & Ciotti (1993), with white dwarf remnants of mass $M_{\rm WD}=0.48+0.077\,M_{\rm i}$ for initial masses $M_{\rm i}\le 8\,\msun$, neutron star remnants of $1.4\,\msun$ for $8\le M_{\rm i}\le 40\,\msun$, and black holes remnants of mass $0.5\,M_{\rm i}$ for $M_{\rm i}>40\,\msun$. Since the present data do not give any constraint on the slope of the IMF for $M\gsim 1\,\msun$, we explore the effect on the total mass of various plausible assumptions: - IMF \#1: An IMF with slope $\alpha=-1.33$, like the one we observed, all the way to $100M_\odot$. This is perhaps an extreme possibility, since all the determinations of the IMF in this mass range give steeper values (see Scalo 1998 for a recent review), and even our own IMF may steepen for $M>0.5M_\odot$. - IMF \#2: An IMF with slope $\alpha=-1.33$ up to $M=1M_\odot$ and $\alpha=-2$ for $M>1M_\odot$. This is the most conservative assumption, since the IMF we observed is best fit with a slope $\alpha=-2$ for $M>0.5M_\odot$. - IMF \#3: An IMF with $\alpha=-1.33$ up to $M=1M_\odot$ and $\alpha=-2.35$ (Salpeter’s value) for $M>1M_\odot$. - IMF \#4: Finally, we consider an IMF with $\alpha=-1.33$ up to $M=1M_\odot$, Salpeter slope for $1<M/M_\odot<2$ and

M_\odot$. By extrapolating the $ \alpha=-1.33 $ IMF from $ M=0.15M_\odot$ all the way down to zero mass, we obtain $ 123.5 M_\odot$ of unobserved dwarf, thus totaling $ 440.5 M_\odot$ in survive ace and brown dwarfs. To account for the end we want to adopt an initial aggregate - concluding mass relation. We use the semi - empirical relation proposed by Renzini & Ciotti (1993), with blank dwarf remnants of mass $ M_{\rm WD}=0.48 + 0.077\,M_{\rm i}$ for initial masses $ M_{\rm i}\le 8\,\msun$, neutron star leftover of $ 1.4\,\msun$ for $ 8\le M_{\rm i}\le 40\,\msun$, and black holes remnants of bulk $ 0.5\,M_{\rm i}$ for $ M_{\rm i}>40\,\msun$. Since the present data do not give any constraint on the gradient of the IMF for $ M\gsim 1\,\msun$, we explore the impression on the total mass of various plausible assumption: - IMF \#1: An IMF with slope $ \alpha=-1.33 $, like the one we observed, all the way to $ 100M_\odot$. This is perhaps an extreme possibility, since all the determinations of the IMF in this mass range give steeper values (see Scalo 1998 for a recent review), and even our own IMF may steepen for $ M>0.5M_\odot$. - IMF \#2: An IMF with slope $ \alpha=-1.33 $ up to $ M=1M_\odot$ and $ \alpha=-2 $ for $ M>1M_\odot$. This is the most conservative premise, since the IMF we observed is best fit with a gradient $ \alpha=-2 $ for $ M>0.5M_\odot$. - IMF \#3: An IMF with $ \alpha=-1.33 $ up to $ M=1M_\odot$ and $ \alpha=-2.35 $ (Salpeter ’s value) for $ M>1M_\odot$. - IMF \#4: ultimately, we think an IMF with $ \alpha=-1.33 $ up to $ M=1M_\odot$, Salpeter slope for $ 1 < M / M_\odot<2 $ and

M_\odlt$. By extrapolating the $\xlpha=-1.33$ IMF from $M=0.15M_\odot$ ell the way dowv to zero mass, we obtain $123.5 M_\ovot$ if unween dwarfs, thus totalkng $440.5 M_\odon$ in livibg suars and brown dwedfs. To accounf for vhe remnants we need to agopt an initian oads-final mass relation. We used the sqmi-empitifal relation ptopostd fy Rsnzini & Ciotti (1993), with white dwarf rsmnants of mass $M_{\rm WC}=0.48+0.077\,M_{\rm i}$ for initial masses $L_{\rm l}\le 8\,\msun$, neutron shar remnantw of $1.4\,\nsun$ for $8\le O_{\rm i}\le 40\,\msun$, and black goles remnants of mass $0.5\,M_{\rm i}$ fof $M_{\rm i}>40\,\msun$. Sinxe thf present dava do got give any constrahnt on yhe slope of tme IMH foe $M\gsim 1\,\msun$, we exploce the effect on the total mavs of various plausuboe asvumpdionr: - KMF \#1: En JMF wihh alope $\alphz=-1.33$, like the ine we observed, all trv way to $100M_\odof$. This if perhaps an extreme possibility, since dll the determinations of rhe IMF in this mass tange give steeper values (see Scalo 1998 for a recent review), ang evei uur orv ILF may steepen for $M>0.5M_\odot$. - IMF \#2: An IMF with fmoke $\slpha=-1.33$ up to $M=1M_\jdot$ and $\allhw=-2$ gjr $M>1M_\odot$. Thir is tks jost conservative wssumptyon, sunce the YMF ee observed is best fit wity a slope $\aliha=-2$ dor $M>0.5M_\odot$. - IMF \#3: Cn IMF with $\clpha=-1.33$ op to $K=1M_\odot$ and $\alpha=-2.35$ (Salpetzr’s vamue) for $M>1M_\ofot$. - IMF \#4: Finally, we conskdeg an IMF with $\alpha=-1.33$ up to $M=1M_\odjt$, Salpetxr slppe for $1<M/M_\ocot<2$ anq

M_\odot$. By extrapolating the $\alpha=-1.33$ IMF from the down to mass, we obtain thus $440.5 M_\odot$ in stars and brown To account for the remnants we to adopt an initial mass-final mass relation. We used the semi-empirical relation proposed Renzini & Ciotti (1993), with white dwarf remnants of mass $M_{\rm WD}=0.48+0.077\,M_{\rm i}$ initial $M_{\rm 8\,\msun$, star remnants of $1.4\,\msun$ for $8\le M_{\rm i}\le 40\,\msun$, and black holes remnants of mass $0.5\,M_{\rm for $M_{\rm i}>40\,\msun$. Since the present data do give any constraint on slope of the IMF for 1\,\msun$, explore the on total of various plausible - IMF \#1: An IMF with slope $\alpha=-1.33$, like the one we observed, all the way to This is extreme possibility, all determinations the IMF in range give steeper values (see Scalo recent review), and even our own IMF may for $M>0.5M_\odot$. IMF \#2: An IMF with slope up to $M=1M_\odot$ and $\alpha=-2$ for $M>1M_\odot$. This the most conservative assumption, since the IMF we observed is best fit with a slope $M>0.5M_\odot$. - IMF \#3: IMF with $\alpha=-1.33$ to and (Salpeter’s for $M>1M_\odot$. IMF \#4: Finally, we consider an IMF with $\alpha=-1.33$ up to Salpeter slope for $1<M/M_\odot<2$ and

M_\odot$. By extrapolating the $\alPha=-1.33$ IMF from $m=0.15M_\odoT$ alL thE wAy doWn to Zero mass, we obtaIN $123.5 M_\odOt$ of unseen dwarfs, thus toTalinG $440.5 M_\ODot$ iN LiVing sTars and BRoWN DwaRfS. TO acCoUNt For thE reMnants wE need to adoPt aN iNitial mass-fiNAl Mass relatiOn. WE used the semi-EmpIrical ReLatIOn proPosEd by REnzini & cIotti (1993), wIth white dWaRF remnaNTs of masS $m_{\Rm wD}=0.48+0.077\,M_{\rM i}$ for initial masseS $m_{\rM I}\le 8\,\msun$, neutron Star reMnANtS OF $1.4\,\msUn$ fOr $8\le M_{\rm i}\le 40\,\MsUn$, and BLack holES rEMNAntS Of mass $0.5\,M_{\rm i}$ for $m_{\rm i}>40\,\msun$. SinCE thE preseNt DatA Do not gIve anY cONstRaint on the sLope Of the IMF fOr $M\gsiM 1\,\Msun$, we eXPlore thE effecT on The TotaL MaSs Of vArIOus PLaUsiBLe aSsumptioNs: - iMf \#1: An IMf witH SLOPe $\alPha=-1.33$, Like The onE we observed, alL thE way TO $100M_\oDot$. ThIs is pErhaPs An extReme poSsibiLiTy, since all the deTermInations oF thE ImF iN tHis maSS range GivE stEeper vaLues (see sCalO 1998 fOR A ReCent review), and even oUr OWN ImF may steEpen foR $m>0.5M_\OdOT$. - IMF \#2: An IMf wIth SlopE $\ALpha=-1.33$ uP to $M=1m_\OdOt$ and $\alpHa=-2$ for $M>1m_\OdOt$. this is tHe Most coNsErvAtiVe assUMptiOn, sincE the IMF wE obseRVed is best fit wiTH a slope $\alpha=-2$ fOR $M>0.5m_\ODoT$. - iMF \#3: AN IMf with $\alpha=-1.33$ uP to $M=1m_\Odot$ And $\aLPhA=-2.35$ (SaLPeter’S valuE) fOR $M>1m_\Odot$. - IMF \#4: Finally, we conSiDer an ImF witH $\alpha=-1.33$ up to $M=1M_\oDot$, SalpeteR SLOpe for $1<M/M_\Odot<2$ ANd

M_\odot$. By extrapolating the $\alp ha=-1 .33 $ I MF fro m $M =0.15M_\odot$a ll t he way down to zero ma ss, w eo btai n $ 123.5 M_\odo t $o f un se en dw ar f s, thus to taling$440.5 M_\ odo t$ in living s t ar s and brow n d warfs. To ac cou nt for t her emnan tswe ne ed toa dopt a n initial m a ss-fin a l massr e la tion . We used the sem i -e m pirical relati on pro po s ed b y R enz ini & Ciot ti (199 3 ), with wh i t e dw a rf remnants o f mass $M_{ \ rmWD}=0. 48 +0. 0 77\,M_ {\rmi} $ fo r initial m asse s $M_{\rm i}\le 8\,\msu n $, neut ron st arrem nant s o f$1. 4\ , \ms u n$ fo r $8 \le M_{\ rm i }\le40\, \ m s u n$,and bla ck ho les remnantsofmass $0. 5\,M_ {\rmi}$fo r $M_ {\rm i }>40\ ,\ msun$. Since th e pr esent dat a d onot g ive a n y cons tra int on the slopeo f t he I M Ffor $M\gsim 1\,\ms un $ , w e explor e thee ff ec t on theto tal mas s of va riou s p lausible assum p ti on s: - I MF \#1 :AnIMF with slop e $\al pha=-1.3 3$, l i ke the one weo bserved, allt he w ay to $ 100 M_\odot$. T hisi s pe rhap s a n e x treme poss ib i li t y, since all the de te rminat ionsof the IMF in this mass r a nge give ste e pe r values (see S calo1998 for a recent r eview ), and e ven our o w n IMF may st eep enfor $ M> 0.5M_\odot$.- I MF \#2: A n I MF with sl ope $\ alp ha =-1.33$ u p to $M= 1M _\ od ot $ a nd $\ a lpha=-2$ f or$M >1M _\odo t $. Thi s isthemo st con servati v ea s sump ti on , si nce t he IM F we obs erved i s best fi t w i th a s lo pe $\al pha=-2$ for $ M> 0.5M_\odot $. - IMF \ #3: An I MF with $\alpha=-1.33$u p to $M =1M _\odo t$ a nd $\alph a=- 2.35$(Sa l peter’ s valu e) fo r$M> 1 M _\odo t $ . - IMF \#4: F i n all y, we c onsi der anIMF with $\alpha=- 1 .33 $ up to $M=1M _\o dot$ , Sa lpe t er slo pe for $ 1<M/M_\odot<2$and

M_\odot$. By_extrapolating the_$\alpha=-1.33$ IMF from $M=0.15M_\odot$_all the_way_down to_zero_mass, we obtain_$123.5 M_\odot$ of_unseen dwarfs, thus totaling_$440.5 M_\odot$ in_living_stars and brown dwarfs. To account for the remnants we need to adopt an_initial_mass-final mass_relation._We_used the semi-empirical relation proposed_by Renzini & Ciotti (1993),_with white_dwarf remnants of mass $M_{\rm WD}=0.48+0.077\,M_{\rm i}$ for initial_masses_$M_{\rm i}\le 8\,\msun$,_neutron star remnants of $1.4\,\msun$ for $8\le M_{\rm i}\le 40\,\msun$,_and black holes remnants of mass_$0.5\,M_{\rm i}$ for_$M_{\rm_i}>40\,\msun$._Since the present data_do not give any constraint on_the slope of the IMF for_$M\gsim 1\,\msun$, we explore the effect on_the total mass of various plausible_assumptions: - IMF \#1:_An IMF_with slope $\alpha=-1.33$, like the_one we observed,_all the_way to $100M_\odot$._This is perhaps an extreme possibility,_since all the_determinations of the IMF in this_mass_range give steeper_values_(see_Scalo 1998_for a recent_review),_and even_our_own IMF may steepen for $M>0.5M_\odot$. -__ IMF \#2: An IMF with slope_$\alpha=-1.33$ up to $M=1M_\odot$_and_$\alpha=-2$ for $M>1M_\odot$. This_is the most conservative assumption,_since the IMF we observed is_best fit_with a_slope $\alpha=-2$ for $M>0.5M_\odot$. - IMF \#3: An IMF with_$\alpha=-1.33$ up to $M=1M_\odot$ and $\alpha=-2.35$_(Salpeter’s value) for $M>1M_\odot$. -_ _IMF_\#4: Finally, we_consider_an IMF_with $\alpha=-1.33$ up to $M=1M_\odot$, Salpeter slope_for $1<M/M_\odot<2$_and

4]). Let $\mathcal{F}_\text{a.e.}(V)$ be the set of congruence classes of constructible functions where $f \sim g$ if $f-g=0$ almost everywhere. \[lemma\_inclusion\_into\_constr\_functions\] Denote by $Z$ the abelian group generated by all formal integral combination of compact convex polytopes in $V$. Let $W \subset Z$ denote the subgroup generated by lower-dimensional polytopes and elements of the form $[P \cup Q]+[P \cap Q]-[P]-[Q]$ where $P \cup Q$ is convex. Then the map $$\begin{aligned} Z/W & \to \mathcal{F}_\text{a.e.}(V)\\ \sum_i n_i [P_i] & \mapsto \sum_i n_i 1_{P_i}, \quad n_i \in \mathbb{Z} \end{aligned}$$ is injective. It is easily checked that the map is well-defined. To prove injectivity, it is enough to prove that $f:=\sum_i 1_{P_i} \sim \sum_j 1_{Q_j}$ implies $\sum_i [P_i] \equiv \sum_j [Q_j]$. Decompose the connected components of $$\left(\cup_i P_i \cup \cup_j Q_j\right) \setminus \left(\cup_i \partial P_i \cup \cup_j \partial Q_j\right)$$ into simplices $\{\Delta\}$, disjoint except at their boundary. Then, by the inclusion-exclusion principle, $$\sum_i [P_i] \equiv \sum_i \sum_{\Delta \subset P_i} [\Delta] \equiv \sum_{k \geq 1} (-1)^{k+1} \sum_{\Delta \subset \bigcup_{i_1<\ldots<i_k} \cap_{j=1}^k P_{i_j}}[\Delta]$$ By examining the superlevel set $\{f \geq k\}$ we see that $$\bigcup_{i_1<\ldots<i_k}(P_{i_1} \cap \ldots \cap P_{i_k}) \sim \bigcup_{j_1<\ldots<j_k}(Q_{j

4 ]). Let $ \mathcal{F}_\text{a.e.}(V)$ be the set of congruence classes of constructible function where $ f \sim g$ if $ farad - g=0 $ almost everywhere. \[lemma\_inclusion\_into\_constr\_functions\ ] Denote by $ Z$ the abelian group generated by all formal built-in combination of compact convex polytopes in $ V$. Let $ W \subset Z$ denote the subgroup generate by lower - dimensional polytopes and elements of the kind $ [ P \cup Q]+[P \cap Q]-[P]-[Q]$ where $ P \cup Q$ is convex. Then the function $ $ \begin{aligned } Z / W & \to \mathcal{F}_\text{a.e.}(V)\\ \sum_i n_i [ P_i ] & \mapsto \sum_i n_i 1_{P_i }, \quad n_i \in \mathbb{Z } \end{aligned}$$ is injective. It is easily checked that the map is well - defined. To rise injectivity, it is enough to prove that $ f:=\sum_i 1_{P_i } \sim \sum_j 1_{Q_j}$ implies $ \sum_i [ P_i ] \equiv \sum_j [ Q_j]$. Decompose the connected component of $ $ \left(\cup_i P_i \cup \cup_j Q_j\right) \setminus \left(\cup_i \partial P_i \cup \cup_j \partial Q_j\right)$$ into simplices $ \{\Delta\}$, disjoint except at their boundary. Then, by the inclusion - exclusion rationale, $ $ \sum_i [ P_i ] \equiv \sum_i \sum_{\Delta \subset P_i } [ \Delta ] \equiv \sum_{k \geq 1 } (-1)^{k+1 } \sum_{\Delta \subset \bigcup_{i_1<\ldots < i_k } \cap_{j=1}^k P_{i_j}}[\Delta]$$ By examining the superlevel set $ \{f \geq k\}$ we interpret that $ $ \bigcup_{i_1<\ldots < i_k}(P_{i_1 } \cap \ldots \cap P_{i_k }) \sim \bigcup_{j_1<\ldots < j_k}(Q_{j

4]). Leh $\mathcal{F}_\text{a.e.}(V)$ be the set of congrueuxe clavses or constrjctible functions where $f \sil t$ if $d-g=0$ almost everywhere. \[leoma\_incluspon\_into\_cobstr\_hunctions\] Denote by $Z$ thc abemlan gxonp generated by all forman integral comtivacion of compact convex polytopes in $D$. Let $W \skbset Z$ denote the fubgdoup generated by lower-dimensional polytokes and elements og the form $[P \cup Q]+[P \cap Q]-[P]-[Q]$ whege $P \cup Q$ is convfx. Then the map $$\vegin{aligned} X/W & \to \mathcal{F}_\text{a.e.}(V)\\ \aum_i n_i [P_i] & \mapsto \sum_i n_i 1_{P_i}, \qjad n_n \in \mathbb{E} \snf{dligned}$$ is mnjectpve. It is easily checkad that the map is wekl-dxfinwd. To prove injectivivy, it is enough to ptove that $x:=\sbm_i 1_{P_i} \sim \sum_j 1_{Q_j}$ implues $\som_i [P_h] \eqjuv \rum_n [A_j]$. Decomoosx the connedted componwnts of $$\left(\cup_i P_i \cti \cup_j Q_j\righf) \setmynts \left(\cup_i \partial P_i \cup \cup_j \partial Q_j\dight)$$ into simplices $\{\Deota\}$, disjoint except aj their botndary. Then, by the inclusion-exclusion principle, $$\sgm_i [P_m] \dquny \rym_l \sum_{\Delta \subset P_i} [\Delta] \equiv \sum_{k \geq 1} (-1)^{k+1} \fhm_{\Cekta \subset \biggup_{i_1<\ldots<i_k} \cap_{j=1}^k L_{i_u}}[\Drjta]$$ By examinivg the sulerlevel set $\{f \geq k\}$ we sge thar $$\bigcup_{i_1<\jdotx<i_k}(P_{i_1} \cap \ldots \cap P_{i_k}) \sin \bigcup_{j_1<\ldons<j_k}(W_{j

4]). Let $\mathcal{F}_\text{a.e.}(V)$ be the set of of functions where \sim g$ if by the abelian group by all formal combination of compact convex polytopes in Let $W \subset Z$ denote the subgroup generated by lower-dimensional polytopes and elements the form $[P \cup Q]+[P \cap Q]-[P]-[Q]$ where $P \cup Q$ is convex. the $$\begin{aligned} & \mathcal{F}_\text{a.e.}(V)\\ \sum_i n_i [P_i] & \mapsto \sum_i n_i 1_{P_i}, \quad n_i \in \mathbb{Z} \end{aligned}$$ is injective. is easily checked that the map is well-defined. prove injectivity, it is to prove that $f:=\sum_i 1_{P_i} \sum_j implies $\sum_i \equiv [Q_j]$. the connected components $$\left(\cup_i P_i \cup \cup_j Q_j\right) \setminus \left(\cup_i \partial P_i \cup \cup_j \partial Q_j\right)$$ into simplices $\{\Delta\}$, disjoint at their by the principle, [P_i] \sum_i \sum_{\Delta \subset \equiv \sum_{k \geq 1} (-1)^{k+1} \sum_{\Delta P_{i_j}}[\Delta]$$ By examining the superlevel set $\{f \geq we see $$\bigcup_{i_1<\ldots<i_k}(P_{i_1} \cap \ldots \cap P_{i_k}) \sim

4]). Let $\mathcal{F}_\text{a.e.}(V)$ be the seT of congrueNce clAssEs oF cOnstRuctIble functions wHEre $f \Sim g$ if $f-g=0$ almost everywheRe. \[lemMa\_INcluSIoN\_into\_Constr\_fUNcTIOns\] deNoTe bY $Z$ THe AbeliAn gRoup genErated by alL foRmAl integral coMBiNation of coMpaCt convex polyTopEs in $V$. LEt $w \suBSet Z$ dEnoTe the SubgroUP generAted by lowEr-DImensiONal polyTOPeS and Elements of the form $[p \CuP q]+[P \cap Q]-[P]-[Q]$ where $P \Cup Q$ is CoNVeX. tHen The Map $$\begin{alIgNed} Z/W & \TO \mathcaL{f}_\tEXT{A.e.}(V)\\ \SUm_i n_i [P_i] & \mapsto \Sum_i n_i 1_{P_i}, \quaD N_i \iN \mathbB{Z} \End{ALigned}$$ Is injEcTIve. it is easily cHeckEd that the Map is wELl-definED. To provE injecTivIty, It is ENoUgH to PrOVe tHAt $F:=\suM_I 1_{P_i} \Sim \sum_j 1_{Q_J}$ iMpLies $\sUm_i [P_I] \EQUIv \suM_j [Q_J]$. DecOmposE the connected ComPoneNTs oF $$\left(\Cup_i P_I \cup \CuP_j Q_j\rIght) \seTminuS \lEft(\cup_i \partial P_I \cup \Cup_j \partiAl Q_J\rIghT)$$ iNto siMPlices $\{\delTa\}$, dIsjoint Except aT TheIr BOUNdAry. Then, by the inclusIoN-EXcLusion prInciplE, $$\SuM_i [p_I] \equiv \suM_i \Sum_{\deltA \SUbset p_i} [\DeLTa] \Equiv \sum_{K \geq 1} (-1)^{k+1} \sUM_{\DElTa \subseT \bIgcup_{i_1<\LdOts<I_k} \cAp_{j=1}^k P_{I_J}}[\DelTa]$$ By exAmining tHe supERlevel set $\{f \geq k\}$ WE see that $$\bigcuP_{I_1<\lDOTs<I_K}(P_{i_1} \cAp \lDots \cap P_{i_k}) \sIm \biGCup_{j_1<\LdotS<J_k}(q_{j

4]). Let $\mathcal{F}_\tex t{a.e.}(V) $ bethe se tof c ongr uence classeso f co nstructible functionswhere $ f \si m g $ if$f-g=0$ al m o stev er ywh er e . \[le mma \_inclu sion\_into \_c on str\_functio n s\ ] Denote b y $ Z$ the abeli angroupge ner a ted b y a ll fo rmal i n tegral combinat io n of co m pact co n v ex pol ytopes in $V$. Le t $ W \subset Z$ de note t he su b g rou p g enerated b ylower - dimensi o na l p oly t opes and elem ents of the for m $[P\c upQ ]+[P \ cap Q ]- [ P]- [Q]$ where$P \ cup Q$ is conve x . Thent he map$$\beg in{ ali gned } Z /W &\t o \m a th cal { F}_ \text{a. e. }( V)\\\sum _ i n _i [ P_i ] &\maps to \sum_i n_i 1_ {P_i } , \ quadn_i \ in \ ma thbb{ Z} \en d{ali gn ed}$$ is inject ive. It is e asi ly ch ec ked t h at the ma p i s well- defined . To p r o v einjectivity, it is e n o ug h to pro ve tha t $ f: = \sum_i 1_ {P_ i} \ s i m \su m_j1 _{ Q_j}$ im plies$ \s um _i [P_i ]\equiv \ sum _j[Q_j] $ . De compos e the co nnect e d components o f $$\left(\cup _ iP _ i\ cup\cu p_j Q_j\rig ht)\ setm inus \l eft ( \cup_ i \pa rt i al P_i \cup \cup_j \pa rt ial Q_ j\rig ht)$$ into si mplices $\ { \ D elta\}$, dis j oi n t except at th eir b oundary. T h en, by t he in clusion- exclusion p rinciple , $ $\s um_ i [ P _ i] \equiv \sum_ i \sum _{ \Delta\su bset P_ i}[\D elt a]\e quiv \sum _{k \geq 1 }(- 1) ^{k +1} \ s um_{\Del ta \s ub set \big c up_{i_ 1<\ld ots< i_ k} \ca p_{j=1} ^ kP _ {i_j }} [\ Delt a]$ $ By e xami n ing the su perlevelset $\{f \ ge q k\}$we see that $ $\ bigcup_{i_ 1< \ld ots<i_ k } (P_{i_1} \cap \ldots \cap P_{i_ k }) \si m \ bigcu p_{j _1<\ldots <j_ k}(Q_{ j

4]). Let_$\mathcal{F}_\text{a.e.}(V)$ be_the set of congruence_classes of_constructible_functions where_$f_\sim g$ if_$f-g=0$ almost everywhere. \[lemma\_inclusion\_into\_constr\_functions\]_Denote by $Z$ the_abelian group generated_by_all formal integral combination of compact convex polytopes in $V$. Let $W \subset Z$_denote_the subgroup_generated_by_lower-dimensional polytopes and elements of_the form $[P \cup Q]+[P \cap_Q]-[P]-[Q]$ where_$P \cup Q$ is convex. Then the map_$$\begin{aligned} Z/W_& \to \mathcal{F}_\text{a.e.}(V)\\ \sum_i_n_i [P_i] & \mapsto \sum_i n_i 1_{P_i}, \quad n_i_\in \mathbb{Z} \end{aligned}$$ is injective. It is_easily checked that_the_map_is well-defined. To prove_injectivity, it is enough to prove_that $f:=\sum_i 1_{P_i} \sim \sum_j 1_{Q_j}$_implies $\sum_i [P_i] \equiv \sum_j [Q_j]$. Decompose_the connected components of $$\left(\cup_i P_i_\cup \cup_j Q_j\right) \setminus \left(\cup_i_\partial P_i_\cup \cup_j \partial Q_j\right)$$ into_simplices $\{\Delta\}$, disjoint_except at_their boundary. Then,_by the inclusion-exclusion principle, $$\sum_i [P_i]_\equiv \sum_i \sum_{\Delta_\subset P_i} [\Delta] \equiv \sum_{k \geq_1}_(-1)^{k+1} \sum_{\Delta \subset_\bigcup_{i_1<\ldots<i_k}_\cap_{j=1}^k_P_{i_j}}[\Delta]$$ By examining_the superlevel set_$\{f_\geq k\}$_we_see that $$\bigcup_{i_1<\ldots<i_k}(P_{i_1} \cap \ldots \cap_P_{i_k})_ \sim \bigcup_{j_1<\ldots<j_k}(Q_{j

^{2}_{K,K^{*}})\mathcal{D}_{K,K^{*}}\right] \mathcal{R}+F_{c,v}(1-\mathcal{R}),\,\,\,\,\end{aligned}$$ where $\mathcal{R}=\mathcal{R}_{s}\mathcal{R}_{t}$ with $$\begin{aligned} \label{eq:RSRT} &&\mathcal{R}_{s}= \frac{1}{2} \left[1+\tanh\left(\frac{s-s_{\mathrm{Regge}}}{s_{0}} \right) \right],\nonumber\\ &&\mathcal{R}_{t}= 1-\frac{1}{2} \left[1+\tanh\left(\frac{|t|-t_{\mathrm{Regge}}}{t_{0}} \right) \right].\end{aligned}$$ In this work the values of the four parameters for the reggeized treatment are chosen as Nam. $et\ al.$ as presented in Table \[Tab: Regge\]. $s_{Reg}=3$ $t_{Reg}=0.1$ $s_0=1$ $t_0=0.08$ ------------- --------------- --------- ------------ -- -- -- -- : parameters for the reggiezed treatment with unit GeV$^2$. \[Tab: Regge\] Nucleon resonances {#Sec:R} ------------------ In Ref. [@Xie:2010yk], the $D_{13}(2080)$ is considered to reproduce the bump structure near 2.1 GeV. In this work all resonances predicted by CQM will be considered. The Lagrangians for the resonances with arbitrary half-integer spin are [@Chang:1967zzc; @Rushbrooke:1966zz; @Behrends:1957] $$\begin{aligned} \mathcal{L}_{\gamma N R(\frac{1}{2}^{\pm})} &=&\frac{e f_2}{2M_N} \bar{N} \Gamma^{(\mp)}\sigma_{\mu\nu}F^{\mu\nu} R \,+{\rm h.c.}, \\ \mathcal{L}_{\gamma N R(J^{\pm})} &=&\frac{-i^{n}f_1}{(2m_N)^{n}} \bar{B}^* ~\gamma_\nu \partial_{\mu

^{2}_{K, K^{*}})\mathcal{D}_{K, K^{*}}\right ] \mathcal{R}+F_{c, v}(1-\mathcal{R}),\,\,\,\,\end{aligned}$$ where $ \mathcal{R}=\mathcal{R}_{s}\mathcal{R}_{t}$ with $ $ \begin{aligned } \label{eq: RSRT } & & \mathcal{R}_{s}= \frac{1}{2 } \left[1+\tanh\left(\frac{s - s_{\mathrm{Regge}}}{s_{0 } } \right) \right],\nonumber\\ & & \mathcal{R}_{t}= 1-\frac{1}{2 } \left[1+\tanh\left(\frac{|t|-t_{\mathrm{Regge}}}{t_{0 } } \right) \right].\end{aligned}$$ In this work the values of the four parameters for the reggeized treatment are choose as Nam. $ et\ al.$ as confront in Table   \[Tab: Regge\ ]. $ s_{Reg}=3 $ $ t_{Reg}=0.1 $ $ s_0=1 $ $ t_0=0.08 $ ------------- --------------- --------- ------------ -- -- -- -- : parameters for the reggiezed treatment with unit GeV$^2$. \[Tab: Regge\ ] Nucleon resonance { # Sec: R } ------------------ In Ref.   [ @Xie:2010yk ], the $ D_{13}(2080)$ is considered to regurgitate the bump structure near 2.1   GeV. In this work all resonance predicted by CQM will be considered. The Lagrangians for the rapport with arbitrary half - integer spin are   [ @Chang:1967zzc; @Rushbrooke:1966zz; @Behrends:1957 ] $ $ \begin{aligned } \mathcal{L}_{\gamma N R(\frac{1}{2}^{\pm }) } & = & \frac{e f_2}{2M_N } \bar{N } \Gamma^{(\mp)}\sigma_{\mu\nu}F^{\mu\nu } R \,+{\rm h.c. }, \\ \mathcal{L}_{\gamma N R(J^{\pm }) } & = & \frac{-i^{n}f_1}{(2m_N)^{n } } \bar{B}^ * ~\gamma_\nu \partial_{\mu

^{2}_{K,K^{*}})\mwthcal{D}_{K,K^{*}}\right] \mathcal{R}+F_{c,y}(1-\mathcal{R}),\,\,\,\,\end{aliguwd}$$ whece $\mathdal{R}=\mathzal{R}_{s}\mathcal{R}_{t}$ with $$\begin{alijned} \oabel{tz:RSRT} &&\mathcal{R}_{s}= \frac{1}{2} \ueft[1+\tanh\lvft(\frac{s-s_{\nathcm{Regge}}}{s_{0}} \right) \right],\nonumbed\\ &&\latheao{R}_{t}= 1-\frac{1}{2} \left[1+\tanm\left(\frac{|t|-t_{\kathrm{Regge}}}{t_{0}} \rhgft) \right].\end{aligned}$$ In this work the vajues of tje four paramejers gjr tgv veggeized treatment are chosen aa Nam. $eu\ al.$ as presented on Table \[Tab: Regge\]. $s_{Reg}=3$ $h_{Reg}=0.1$ $s_0=1$ $t_0=0.08$ ------------- --------------- --------- ------------ -- -- -- -- : pweameters for the reggitzzd treatmenj with unit GeV$^2$. \[Tab: Regge\] Nucleon fesoncnces {#Sec:R} ------------------ Ib Eef. [@Dhe:2010yk], the $D_{13}(2080)$ ms confidered to rciroduce the bukp structure ncar 2.1 GxV. Ib this work all resonences predicted by CZM will ba eonsidered. The Lagrantiqns fmr tve rdwonxncts xitg arbihracy half-intefer spin arw [@Chang:1967zzc; @Rushbrookt:1966zz; @Vehrends:1957] $$\begih{alignqd} \mathcal{L}_{\gamma N R(\frac{1}{2}^{\pm})} &=&\frac{e f_2}{2M_N} \bzr{N} \Gamma^{(\mp)}\sigma_{\mu\nu}F^{\mu\bu} R \,+{\rm h.c.}, \\ \mathcal{L}_{\galma N R(J^{\pi})} &=&\frac{-i^{n}f_1}{(2m_N)^{n}} \bar{B}^* ~\gamma_\nu \partial_{\mu

^{2}_{K,K^{*}})\mathcal{D}_{K,K^{*}}\right] \mathcal{R}+F_{c,v}(1-\mathcal{R}),\,\,\,\,\end{aligned}$$ where $\mathcal{R}=\mathcal{R}_{s}\mathcal{R}_{t}$ with $$\begin{aligned} \label{eq:RSRT} \left[1+\tanh\left(\frac{s-s_{\mathrm{Regge}}}{s_{0}} \right],\nonumber\\ &&\mathcal{R}_{t}= \left[1+\tanh\left(\frac{|t|-t_{\mathrm{Regge}}}{t_{0}} \right) \right].\end{aligned}$$ of four parameters for reggeized treatment are as Nam. $et\ al.$ as presented Table \[Tab: Regge\]. $s_{Reg}=3$ $t_{Reg}=0.1$ $s_0=1$ $t_0=0.08$ ------------- --------------- --------- ------------ -- -- -- : parameters for the reggiezed treatment with unit GeV$^2$. \[Tab: Regge\] Nucleon {#Sec:R} In [@Xie:2010yk], $D_{13}(2080)$ is considered to reproduce the bump structure near 2.1 GeV. In this work all resonances by CQM will be considered. The Lagrangians for resonances with arbitrary half-integer are [@Chang:1967zzc; @Rushbrooke:1966zz; @Behrends:1957] $$\begin{aligned} N &=&\frac{e f_2}{2M_N} \Gamma^{(\mp)}\sigma_{\mu\nu}F^{\mu\nu} \,+{\rm \\ \mathcal{L}_{\gamma N &=&\frac{-i^{n}f_1}{(2m_N)^{n}} \bar{B}^* ~\gamma_\nu \partial_{\mu

^{2}_{K,K^{*}})\mathcal{D}_{K,K^{*}}\right] \mathcal{R}+f_{c,v}(1-\mathcal{r}),\,\,\,\,\end{aLigNed}$$ WhEre $\mAthcAl{R}=\mathcal{R}_{s}\maTHcal{r}_{t}$ with $$\begin{aligned} \labeL{eq:RSrT} &&\MAthcAL{R}_{S}= \frac{1}{2} \Left[1+\tanH\LeFT(\FraC{s-S_{\mAthRm{rEgGe}}}{s_{0}} \riGht) \Right],\noNumber\\ &&\mathCal{r}_{t}= 1-\Frac{1}{2} \left[1+\tanh\LEfT(\frac{|t|-t_{\matHrm{regge}}}{t_{0}} \right) \rIghT].\end{alIgNed}$$ iN this WorK the vAlues oF The fouR parameteRs FOr the rEGgeized TREaTmenT are chosen as Nam. $et\ AL.$ aS Presented in TabLe \[Tab: REgGE\]. $s_{rEG}=3$ $t_{REg}=0.1$ $s_0=1$ $T_0=0.08$ ------------- --------------- --------- ------------ -- -- -- -- : parameterS fOr the REggiezeD TrEATMenT With unit GeV$^2$. \[TaB: Regge\] NucleON reSonancEs {#sec:r} ------------------ in Ref. [@XIe:2010yk], tHe $d_{13}(2080)$ Is cOnsidered to ReprOduce the bUmp strUCture neAR 2.1 GeV. In tHis worK alL reSonaNCeS pRedIcTEd bY cQm wiLL be ConsiderEd. thE LagrAngiANS FOr thE reSonaNces wIth arbitrary hAlf-InteGEr sPin arE [@ChanG:1967zzc; @ruShbroOke:1966zz; @BEhrenDs:1957] $$\Begin{aligned} \matHcal{l}_{\gamma N R(\fRac{1}{2}^{\Pm})} &=&\FraC{e F_2}{2M_N} \baR{n} \Gamma^{(\Mp)}\sIgmA_{\mu\nu}F^{\mU\nu} R \,+{\rm h.C.}, \\ \MatHcAL{l}_{\GaMma N R(J^{\pm})} &=&\frac{-i^{n}f_1}{(2m_N)^{n}} \BaR{b}^* ~\GaMma_\nu \parTial_{\mu

^{2}_{K,K^{*}})\mathcal{D} _{K,K^{*}} \righ t]\ma th cal{ R}+F _{c,v}(1-\math c al{R }),\,\,\,\,\end{aligne d}$$wh e re $ \ ma thcal {R}=\ma t hc a l {R} _{ s} \ma th c al {R}_{ t}$ with $ $\begin{al ign ed } \label{eq: R SR T} &&\math cal {R}_{s}= \fr ac{ 1}{2}\l eft [ 1+\ta nh\ left( \frac{ s -s_{\m athrm{Reg ge } }}{s_{ 0 }} \rig h t ) \right],\nonumber \ \& &\mathcal{R}_{ t}= 1- \f r ac { 1 }{2 } \ left[1+\ta nh \left ( \frac{| t |- t _ { \ma t hrm{Regge}}}{ t_{0}} \rig h t)\right ]. \en d {align ed}$$ I n t his work th e va lues of t he fou r parame t ers for the r egg eiz ed t r ea tm ent a r e c h os ena s N am. $et\ a l. $ aspres e n t e d in Ta ble\[Tab : Regge\]. $s _{Re g }=3 $ $ t_{Re g}=0 .1 $ $ s_0=1$ $t _0 =0.08$ ------- --- -- - - -- ----- - ------ -- --- ---- -- ------- - ---- - - - - -- : paramete rs f or the reg giezed tr ea t ment wit huni t Ge V $ ^2$. \[T a b: Regge\] Nucl e on r esonanc es {#Sec :R } - --- ----- - ---- ---- In Ref.[@Xie : 2010yk], the $ D _{13}(2080)$i sc o ns i dere d t o reproduce the bump str u ct ure near2.1 G eV . I n this work all reso na nces p redic ted by CQM wi ll be cons i d e red. The Lag r an g ians for the r esona nces witha rbitrary half -integer spin are [ @Chang:1 967 zzc ; @ Rus h b ro oke:1966zz; @ B e hren ds :1957]$$\ begin{a lig ned } \m athcal{L} _{\gamma N R (\ fr ac{ 1}{2} ^ {\pm})}&= &\f ra c{e f_2} { 2M_N} \ bar{ N} \ G amm a^{(\mp ) }\ s i gma_ {\ mu \nu} F^{ \m u\nu} R \ , +{\ rm h.c. }, \\ \ma thc a l{L} _{ \g amma NR(J^{\pm})} & =& \frac{-i^{ n} f_1 }{(2m_ N ) ^{n}} \b ar{B}^* ~\gamma_\nu \pa r tial_{\ mu

^{2}_{K,K^{*}})\mathcal{D}_{K,K^{*}}\right] \mathcal{R}+F_{c,v}(1-\mathcal{R}),\,\,\,\,\end{aligned}$$ where_$\mathcal{R}=\mathcal{R}_{s}\mathcal{R}_{t}$ with_$$\begin{aligned} \label{eq:RSRT} &&\mathcal{R}_{s}= \frac{1}{2} \left[1+\tanh\left(\frac{s-s_{\mathrm{Regge}}}{s_{0}} \right) _ \right],\nonumber\\ &&\mathcal{R}_{t}= 1-\frac{1}{2} \left[1+\tanh\left(\frac{|t|-t_{\mathrm{Regge}}}{t_{0}}_\right)_\right].\end{aligned}$$ In this_work_the values of_the four parameters_for the reggeized treatment_are chosen as_Nam._$et\ al.$ as presented in Table \[Tab: Regge\]. $s_{Reg}=3$ $t_{Reg}=0.1$ _$s_0=1$_ _$t_0=0.08$__ _ _ _------------- --------------- --------- ------------ -- -- -- -- __: parameters for_the reggiezed treatment with unit GeV$^2$. \[Tab: Regge\] Nucleon resonances {#Sec:R} ------------------ In_Ref. [@Xie:2010yk], the $D_{13}(2080)$ is considered to_reproduce the bump_structure_near_2.1 GeV. In this work_all resonances predicted by CQM will_be considered. The Lagrangians for the_resonances with arbitrary half-integer spin are [@Chang:1967zzc; @Rushbrooke:1966zz;_@Behrends:1957] $$\begin{aligned} \mathcal{L}_{\gamma_N R(\frac{1}{2}^{\pm})} &=&\frac{e f_2}{2M_N} _ _\bar{N} \Gamma^{(\mp)}\sigma_{\mu\nu}F^{\mu\nu} R \,+{\rm h.c.},_\\ \mathcal{L}_{\gamma N R(J^{\pm})}_&=&\frac{-i^{n}f_1}{(2m_N)^{n}} \bar{B}^* ~\gamma_\nu_\partial_{\mu

the beginning to get the $q$-expansions of the newforms from the $q$-expansions of the forms we have just computed. This method is faster than the classical one for large $B$. For fixed prime level $\ell$, the number of bit operations required to compute the $q$-expansion of the newforms in $S_2\big(\Gamma_1(\ell)\big)$ to precision $O(q^B)$ with the algorithm described above is quasi-linear in $B$. In comparison, the bit complexity of the classical algorithm based on modular symbols is at least quadratic in $B$, cf [@Stein], remark 8.3.3. First notice that for fixed level $\ell$, the change of basis matrices from the bases $\mathcal{B}_{\varepsilon}$ to eigenforms are fixed, and so is the common field $K = {\mathbb{Q}}\left(\zeta_{(\ell-1)/2} \right)$. Consequently the coefficients of $\zeta_{(\ell-1)/2}$ in the coefficients up to $q^B$ of the forms in the bases $\mathcal{B}_{\varepsilon}$ are bounded by $C \sup_{n < B} d(n) \sqrt{n}$ where $C$ is some constant which does not depend on $B$. This bound is $O(B)$ (because $d(n) = O(n^{\delta})$ for every $\delta > 0$, cf for instance [@HW], theorem 315), so the smallest prime $p$ larger than twice this bound and congruent[^8] to $1 \bmod (\ell-1)/2$ is also $O(B)$, and can be found in using the sieve of Eratosthenes in $O(B \log B \log \log B)$ bit operations (cf the proof of the theorem 18.10 part ii in [@Gathen]). Then arithmetic operations in the residue field ${\mathbb{F}}_p$ will require $O(\log B)$ bit operations. Next, $E_4$ and $E_6$ can be computed mod $p$ to precision $O(q^B)$ in $O(B \log B \log \log B)$ bit operations using again the sieve of Eratosthenes, and $u$ and $dj$ can be computed in $O(B \log B)$ operations in ${\mathbb{F}}_

the beginning to get the $ q$-expansions of the newforms from the $ q$-expansions of the forms we have just computed. This method acting is fast than the classical one for big $ B$. For fasten prime level $ \ell$, the issue of morsel operations necessitate to compute the $ q$-expansion of the newforms in $ S_2\big(\Gamma_1(\ell)\big)$ to precision $ O(q^B)$ with the algorithm described above is quasi - analogue in $ B$. In comparison, the bit complexity of the classical algorithm based on modular symbols is at least quadratic in $ B$, californium [ @Stein ], remark 8.3.3. First notice that for fixed level $ \ell$, the change of footing matrices from the bases $ \mathcal{B}_{\varepsilon}$ to eigenforms are fixed, and then is the common field $ kilobyte = { \mathbb{Q}}\left(\zeta_{(\ell-1)/2 } \right)$. Consequently the coefficients of $ \zeta_{(\ell-1)/2}$ in the coefficient up to $ q^B$ of the forms in the base $ \mathcal{B}_{\varepsilon}$ are bounded by $ C \sup_{n < B } d(n) \sqrt{n}$ where $ C$ is some constant which does not depend on $ B$. This restrict is $ O(B)$ (because $ d(n) = O(n^{\delta})$ for every $ \delta > 0 $, cf for instance [ @HW ], theorem 315), so the smallest prime $ p$ larger than twice this bound and congruent[^8 ] to $ 1 \bmod (\ell-1)/2 $ is besides $ O(B)$, and can be find in using the sieve of Eratosthenes in $ O(B \log B \log \log B)$ bit operations (cf the proof of the theorem 18.10 part ii in [ @Gathen ]). Then arithmetic operations in the residue field $ { \mathbb{F}}_p$ will necessitate $ O(\log B)$ bit operations. Next, $ E_4 $ and $ E_6 $ can be calculate mod $ p$ to precision $ O(q^B)$ in $ O(B \log boron \log \log B)$ piece operations using again the sieve of Eratosthenes, and $ u$ and $ dj$ can be computed in $ O(B \log B)$ operations in $ { \mathbb{F } } _

thf beginning to get the $q$-txpansions of the newforks froj the $q$-ebpansions of the forms we hate jyst cimputed. This method is waster thwn the coassmcal one for larjs $B$. For nnxed lvime nxvel $\ell$, the nukber of bid operations raqjixed to compute the $q$-expansion of the newforks in $S_2\big(\Gamma_1(\ejl)\bib)$ to lgegision $O(q^B)$ with the algorithm deacribed above is quaxi-linear in $B$. In comparison, the bit complexity of the classixal wogorithm basdd on modular symbols js at least quadratic in $B$, cf [@Sgein], xemark 8.3.3. Firsj hohhce that foc fixeq level $\ell$, bne chatge of nasis matrices frmm rhe bases $\mathcal{B}_{\varxpsilon}$ to eigenformf are fixad, and so is the conmin figld $K = {\magybb{D}}\lert(\vetz_{(\ell-1)/2} \rlghv)$. Consequenfly the coedficients of $\zeta_{(\elk-1)/2}$ yb the coefficjents tp to $q^B$ of the forms in the bases $\mathcan{B}_{\vzrepsilon}$ are bounded bt $C \sup_{n < B} d(n) \sqrt{n}$ ahere $C$ if some constant which does not depend on $B$. This bmund ms $O(B)$ (bccaurw $f(n) = O(n^{\delta})$ for every $\delta > 0$, cf for instance [@GW], tmeorem 315), so the siallest prike $p$ jarger than tdice tkjs bound and congruejt[^8] to $1 \fmod (\wll-1)/2$ is alfo $O(N)$, and can be found in using the sieve oy Eeatosthenes in $O(B \pog B \log \lug B)$ bit pperations (cf the proof of fhe theorem 18.10 part ii kn [@Gathen]). Then afitmmedic operations in the resique field ${\matkbb{F}}_p$ wiul rgquire $J(\log B)$ bit operations. Next, $E_4$ and $E_6$ can ye cokputed mod $p$ to precision $O(q^B)$ in $O(B \log B \log \log B)$ bit o[erdtions ufing sgain the siede of Eratosthgnes, and $b$ and $aj$ can be bomputed mn $O(B \log B)$ jperations in ${\lathbb{F}}_

the beginning to get the $q$-expansions of from $q$-expansions of forms we have faster the classical one large $B$. For prime level $\ell$, the number of operations required to compute the $q$-expansion of the newforms in $S_2\big(\Gamma_1(\ell)\big)$ to precision with the algorithm described above is quasi-linear in $B$. In comparison, the bit of classical based modular symbols is at least quadratic in $B$, cf [@Stein], remark 8.3.3. First notice that for level $\ell$, the change of basis matrices from bases $\mathcal{B}_{\varepsilon}$ to eigenforms fixed, and so is the field = {\mathbb{Q}}\left(\zeta_{(\ell-1)/2} Consequently coefficients $\zeta_{(\ell-1)/2}$ in the up to $q^B$ of the forms in the bases $\mathcal{B}_{\varepsilon}$ are bounded by $C \sup_{n < B} \sqrt{n}$ where some constant does depend $B$. This bound (because $d(n) = O(n^{\delta})$ for every cf for instance [@HW], theorem 315), so the prime $p$ than twice this bound and congruent[^8] $1 \bmod (\ell-1)/2$ is also $O(B)$, and can found in using the sieve of Eratosthenes in $O(B \log B \log \log B)$ bit the proof of the 18.10 part ii [@Gathen]). arithmetic in residue field will require $O(\log B)$ bit operations. Next, $E_4$ and $E_6$ can computed mod $p$ to precision $O(q^B)$ in $O(B \log B B)$ operations using again sieve of Eratosthenes, and and can be computed in B)$ in

the beginning to get the $q$-expaNsions of thE newfOrmS frOm The $q$-ExpaNsions of the forMS we hAve just computed. This metHod is FaSTer tHAn The clAssical ONe FOR laRgE $B$. for FiXEd Prime LevEl $\ell$, thE number of bIt oPeRations requiREd To compute tHe $q$-Expansion of tHe nEwformS iN $S_2\bIG(\GammA_1(\elL)\big)$ tO preciSIon $O(q^B)$ With the alGoRIthm deSCribed aBOVe Is quAsi-linear in $B$. In comPArISon, the bit complExity oF tHE cLASsiCal Algorithm bAsEd on mODular syMBoLS IS at LEast quadratic In $B$, cf [@Stein], rEMarK 8.3.3. First NoTicE That foR fixeD lEVel $\Ell$, the changE of bAsis matriCes froM The baseS $\Mathcal{b}_{\varepSilOn}$ tO eigENfOrMs aRe FIxeD, AnD so IS thE common fIeLd $k = {\mathBb{Q}}\lEFT(\ZEta_{(\eLl-1)/2} \rIght)$. conseQuently the coeFfiCienTS of $\Zeta_{(\eLl-1)/2}$ in tHe coEfFicieNts up tO $q^B$ of ThE forms in the baseS $\matHcal{B}_{\varePsiLoN}$ arE bOundeD By $C \sup_{N < B} d(N) \sqRt{n}$ wherE $C$ is somE ConStANT WhIch does not depend on $b$. THIS bOund is $O(B)$ (BecausE $D(n) = o(n^{\DElta})$ for eVeRy $\dElta > 0$, CF For inStanCE [@Hw], theorem 315), So the sMAlLeSt prime $P$ lArger tHaN twIce This bOUnd aNd congRuent[^8] to $1 \bMod (\elL-1)/2$ Is also $O(B)$, and can BE found in using THe SIEvE Of ErAtoSthenes in $O(B \Log B \LOg \loG B)$ biT OpEraTIons (cF the pRoOF oF The theorem 18.10 part ii in [@GAtHen]). TheN aritHmetic operatiOns in the reSIDUe field ${\mAthbB{f}}_p$ WIll require $O(\log b)$ bit oPerations. NEXt, $E_4$ and $E_6$ cAn be cOmputed mOd $p$ to precISIon $O(q^B)$ in $o(B \lOg B \Log \Log b)$ BIt Operations usiNG AgaiN tHe sieve Of ERatosthEneS, anD $u$ aNd $dJ$ cAn be compuTed in $O(B \lOg b)$ oPeRaTioNs in ${\mAThbb{F}}_

the beginning to get the$q$-expans ionsofthe n ewfo rmsfrom the $q$-e x pans ions of the forms we h ave j us t com p ut ed. This me t ho d isfa st erth a nthe c las sical o ne for lar ge$B $. For fixe d p rime level $\ ell$, the nu mbe r of b it op e ratio nsrequi red to comput e the $q$ -e x pansio n of the n ew form s in $S_2\big(\Ga m ma _ 1(\ell)\big)$to pre ci s io n $O( q^B )$ with th ealgor i thm des c ri b e d ab o ve is quasi-l inear in $B $ . In com pa ris o n, the bitco m ple xity of the cla ssical al gorith m basedo n modul ar sym bol s i s at le as t q ua d rat i cin$ B$, cf [@St ei n] , rem ark8 . 3 . 3. Fir st n otice that for fix edleve l $\ ell$, thechan ge of b asis m atric es from the bases $\m athcal{B} _{\ va rep si lon}$ to eig enf orm s are f ixed, a n d s oi s th e common field $K={ \ ma thbb{Q}} \left( \ ze ta _ {(\ell-1 )/ 2}\rig h t )$. C onse q ue ntly the coeff i ci en ts of $ \z eta_{( \e ll- 1)/ 2}$ i n the coeff icientsup to $q^B$ of the f o rms in the ba s es $ \m a thca l{B }_{\varepsi lon} $ are bou n de d b y $C \ sup_{ n< B } d(n) \sqrt{n}$ whe re $C$ i s som e constant wh ich does n o t depend o n $B $ .T his bound is $ O(B)$ (because$ d(n) = O (n^{\ delta})$ for ever y $\delta> 0 $,cffor i ns tance [@HW],t h eore m315), s o t he smal les t p rim e $ p$ larger t han twic eth is b oun d and congruen t[ ^8] t o $ 1 \bm o d (\el l-1)/ 2$ i sal s o $ O(B)$,a nd c an b efo undinus ing t he s i eve of Era tosthenes in $O(B \ lo g B \lo g \log B)$ bi toperations ( cfthe pr o o f of the theorem 18.10 part iii n [@Gat hen ]). T henarithmeti c o perati ons in the resid ue fi el d $ { \ mathb b { F} }_p $will requi r e $O (\log B )$ b it oper ations. Next, $E_4 $ an d $E_6$ can b e c ompu t e dmod $p $ to p r eci s i on $O(q^B)$ in$O(B \logB\ lo g \log B)$ bit o peratio ns usin g aga i n the s ieve of E ratosthen es , an d $u$ and $dj$can be c omputed i n $O(B \l og B) $ o perati on s i n ${\ mathbb { F}} _

the_beginning to_get the $q$-expansions of_the newforms_from_the $q$-expansions_of_the forms we_have just computed. This_method is faster than_the classical one_for_large $B$. For fixed prime level $\ell$, the number of bit operations required to compute_the_$q$-expansion of_the_newforms_in $S_2\big(\Gamma_1(\ell)\big)$ to precision $O(q^B)$_with the algorithm described above_is quasi-linear_in $B$. In comparison, the bit complexity of the_classical_algorithm based on_modular symbols is at least quadratic in $B$, cf_[@Stein], remark 8.3.3. First notice that for_fixed level $\ell$,_the_change_of basis matrices from_the bases $\mathcal{B}_{\varepsilon}$ to eigenforms are_fixed, and so is the common_field $K = {\mathbb{Q}}\left(\zeta_{(\ell-1)/2} \right)$. Consequently the_coefficients of $\zeta_{(\ell-1)/2}$ in the coefficients_up to $q^B$ of the_forms in_the bases $\mathcal{B}_{\varepsilon}$ are bounded_by $C \sup_{n_< B}_d(n) \sqrt{n}$ where_$C$ is some constant which does_not depend on_$B$. This bound is $O(B)$ (because_$d(n)_= O(n^{\delta})$ for_every_$\delta_> 0$,_cf for instance_[@HW],_theorem 315),_so_the smallest prime $p$ larger than_twice_this bound and congruent[^8] to $1 \bmod_(\ell-1)/2$ is also $O(B)$,_and_can be found in_using the sieve of Eratosthenes_in $O(B \log B \log \log_B)$ bit_operations (cf_the proof of the theorem 18.10 part ii in [@Gathen]). Then_arithmetic operations in the residue field_${\mathbb{F}}_p$ will require $O(\log_B)$ bit_operations._Next, $E_4$ and_$E_6$_can be_computed mod $p$ to precision $O(q^B)$ in_$O(B \log_B \log \log B)$ bit operations_using again the sieve_of_Eratosthenes, and $u$ and $dj$ can_be computed in $O(B \log B)$_operations in ${\mathbb{F}}_

l,n)$). By Proposition \[prop:Omega\], it suffices to show that any two points in $\Omega_{-}(\mathbf{B})$ can be connected by a geodesic curve in $\Omega_{-}(\mathbf{B})$. By Proposition \[prop:OmegaGrass\], $\Omega_{-}(\mathbf{B})$ is the image of $\operatorname{Gr}(k,l)$ embedded isometrically in $\operatorname{Gr}(k,n)$. So by Lemma \[lem:infty\], for any $\mathbf{X}_1,\mathbf{X}_2 \in \operatorname{Gr}(k,l)$, $d_{\operatorname{Gr}(k,n)}(\mathbf{X}_1,\mathbf{X}_2)= d_{\operatorname{Gr}(k,l)}(\mathbf{X}_1,\mathbf{X}_2) = d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$, where the last is the geodesic distance in $\Omega_{-}(\mathbf{B})$. Hence if $d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$ is realized by a geodesic curve $\gamma$ in $\Omega_{-}(\mathbf{B})$, then $\gamma$ must also be a geodesic curve in $\operatorname{Gr}(k,n)$. We have represented $\operatorname{Gr}(k,n)$ as a set of *equivalence classes* of matrices but it may also be represented as a set of *actual matrices* [@Nicolaescu Example 1.2.20], namely, idempotent symmetric matrices of trace $k$: $$\operatorname{Gr}(k,n)\cong \{ P \in \mathbb{R}^{n \times n} : P^{\mathsf{T}} = P^2 = P, \; \operatorname{tr}(P)=k \}.$$ The isomorphism maps each subspace $\mathbf{A}\in \operatorname{Gr}(k,n)$ to $P_{\mathbf{A}} \in \mathbb{R}^{n \times n}$, the unique orthogonal projection onto $\mathbf{A}$, and its inverse takes an orthogonal projection $P$ to the subspace $\operatorname{im}( P) \in \operatorname{Gr}(k,n)$. $P$ is an orthogonal projection iff it is symmetric and idempotent, i.e., $P^{\mathsf{T}} = P^2 = P$. The eigenvalues of an orthogonal projection onto a subspace

l, n)$). By Proposition   \[prop: Omega\ ], it suffices to show that any two points in $ \Omega_{-}(\mathbf{B})$ can be connected by a geodetic curvature in $ \Omega_{-}(\mathbf{B})$. By Proposition   \[prop: OmegaGrass\ ], $ \Omega_{-}(\mathbf{B})$ is the image of $ \operatorname{Gr}(k, l)$ embedded isometrically in $ \operatorname{Gr}(k, n)$. So by Lemma   \[lem: infty\ ], for any $ \mathbf{X}_1,\mathbf{X}_2 \in \operatorname{Gr}(k, l)$, $ d_{\operatorname{Gr}(k, n)}(\mathbf{X}_1,\mathbf{X}_2)= d_{\operatorname{Gr}(k, l)}(\mathbf{X}_1,\mathbf{X}_2) = d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$, where the final is the geodetic distance in $ \Omega_{-}(\mathbf{B})$. Hence if $ d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$ is realized by a geodesic curvature $ \gamma$ in $ \Omega_{-}(\mathbf{B})$, then $ \gamma$ must also be a geodesic curvature in $ \operatorname{Gr}(k, n)$. We have represented $ \operatorname{Gr}(k, n)$ as a set of * comparison classes * of matrices but it may besides be represented as a set of * actual matrices * [ @Nicolaescu Example   1.2.20 ], namely, idempotent symmetric matrices of trace $ k$: $ $ \operatorname{Gr}(k, n)\cong \ { P \in \mathbb{R}^{n \times n }: P^{\mathsf{T } } = P^2 = P, \; \operatorname{tr}(P)=k \}.$$ The isomorphism maps each subspace $ \mathbf{A}\in \operatorname{Gr}(k, n)$ to $ P_{\mathbf{A } } \in \mathbb{R}^{n \times n}$, the alone orthogonal projection onto $ \mathbf{A}$, and its inverse takes an extraneous projection $ P$ to the subspace $ \operatorname{im } (P) \in \operatorname{Gr}(k, n)$. $ P$ is an extraneous projection iff it is symmetrical and idempotent, i.e., $ P^{\mathsf{T } } = P^2 = P$. The eigenvalues of an orthogonal projection onto a subspace

l,n)$). Bj Proposition \[prop:Omega\], iu suffices to shoc that eny two points kn $\Omega_{-}(\mathbf{B})$ can be connerted by a geodesic curve in $\Omeea_{-}(\mathbf{B})$. By Propisitmon \[prop:OmegaGrass\], $\Omega_{-}(\mathbf{B})$ ls thz mmage of $\operatprname{Gr}(k,l)$ embedded isomatfieally in $\operatorname{Gr}(k,n)$. So by Lemmw \[lem:infyy\], for any $\mathbs{X}_1,\mauhbs{X}_2 \ih \operatorname{Gr}(k,l)$, $d_{\operatorname{Gr}(i,n)}(\mathbh{X}_1,\mathbf{X}_2)= d_{\operstorname{Gr}(k,l)}(\mathbf{X}_1,\mathbf{X}_2) = d_{\Omfga_{-}(\mathbf{B})}(\mathbf{X}_1,\mwthbf{X}_2)$, wherg thq last is the geodesic distance in $\Kmega_{-}(\mathbf{B})$. Hence if $d_{\Omega_{-}(\matfbf{B})}(\mcthbf{X}_1,\mathbd{X}_2)$ is tealized by e geodvsic curve $\gamma$ in $\Okega_{-}(\matnbf{B})$, then $\gamms$ mnst qlso be a geodesic cucve in $\operatorname{Gt}(k,n)$. We have rzpresented $\operatornane{Tr}(k,n)$ ds a set if *dqujvelehce clwssxs* of matrides but it nay also be represemtqe as a set of *actuaj iatrices* [@Nicolaescu Example 1.2.20], namely, idemkotenf symmetric matrices of trace $k$: $$\operatorname{Hr}(k,n)\cong \{ P \in \mathbb{R}^{n \times n} : P^{\mathsf{T}} = P^2 = P, \; \operatorndme{tr}(')=k \}.$$ Tkc kwolorphism maps each subspace $\mathbf{A}\in \operatowhake{Nr}(k,n)$ to $P_{\mathbf{A}} \in \mathbb{R}^{m \hikgs n}$, the uniqug orthoyknzl projection onto $\mathbf{W}$, and its invewse yakes an orthogonal projectuon $P$ to the wubspace $\operatorncme{im}( P) \in \o'eratotname{Gt}(k,n)$. $P$ is an orthogonal 'rojecfion iff it is symmeffic and idempotevt, p.e., $P^{\kathsf{T}} = P^2 = P$. The eigenvaltes of an ortkogonal orojgction jnto a subdpace

l,n)$). By Proposition \[prop:Omega\], it suffices to any points in can be connected $\Omega_{-}(\mathbf{B})$. Proposition \[prop:OmegaGrass\], $\Omega_{-}(\mathbf{B})$ the image of embedded isometrically in $\operatorname{Gr}(k,n)$. So by \[lem:infty\], for any $\mathbf{X}_1,\mathbf{X}_2 \in \operatorname{Gr}(k,l)$, $d_{\operatorname{Gr}(k,n)}(\mathbf{X}_1,\mathbf{X}_2)= d_{\operatorname{Gr}(k,l)}(\mathbf{X}_1,\mathbf{X}_2) = d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$, where the last the geodesic distance in $\Omega_{-}(\mathbf{B})$. Hence if $d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$ is realized by a geodesic $\gamma$ $\Omega_{-}(\mathbf{B})$, $\gamma$ also be a geodesic curve in $\operatorname{Gr}(k,n)$. We have represented $\operatorname{Gr}(k,n)$ as a set of *equivalence of matrices but it may also be represented a set of *actual [@Nicolaescu Example 1.2.20], namely, idempotent matrices trace $k$: \{ \in \times n} : = P^2 = P, \; \operatorname{tr}(P)=k \}.$$ The isomorphism maps each subspace $\mathbf{A}\in \operatorname{Gr}(k,n)$ to $P_{\mathbf{A}} \in \times n}$, orthogonal projection $\mathbf{A}$, its takes an orthogonal to the subspace $\operatorname{im}( P) \in an orthogonal projection iff it is symmetric and i.e., $P^{\mathsf{T}} P^2 = P$. The eigenvalues of orthogonal projection onto a subspace

l,n)$). By Proposition \[prop:Omega\], iT suffices tO show ThaT anY tWo poInts In $\Omega_{-}(\mathbf{B})$ CAn be Connected by a geodesic cuRve in $\omEGa_{-}(\maTHbF{B})$. By PRopositIOn \[PROp:OMeGagraSs\], $\oMeGa_{-}(\matHbf{b})$ is the iMage of $\operAtoRnAme{Gr}(k,l)$ embedDEd IsometricaLly In $\operatornaMe{GR}(k,n)$. So bY LEmmA \[Lem:inFty\], For anY $\mathbF{x}_1,\mathbF{X}_2 \in \operaToRName{Gr}(K,L)$, $d_{\operaTORnAme{GR}(k,n)}(\mathbf{X}_1,\mathbf{X}_2)= D_{\OpERatorname{Gr}(k,l)}(\mAthbf{X}_1,\MaTHbF{x}_2) = D_{\OmEga_{-}(\Mathbf{B})}(\matHbF{X}_1,\matHBf{X}_2)$, wherE ThE LASt iS The geodesic diStance in $\OmeGA_{-}(\maThbf{B})$. HEnCe iF $D_{\Omega_{-}(\MathbF{B})}(\MAthBf{X}_1,\mathbf{X}_2)$ iS reaLized by a gEodesiC Curve $\gaMMa$ in $\OmeGa_{-}(\mathBf{B})$, TheN $\gamMA$ mUsT alSo BE a gEOdEsiC CurVe in $\operAtOrName{GR}(k,n)$. WE HAVE repResEnteD $\operAtorname{Gr}(k,n)$ aS a sEt of *EQuiValenCe claSses* Of MatriCes but It may AlSo be represented As a sEt of *actuaL maTrIceS* [@NIcolaEScu ExaMplE 1.2.20], naMely, ideMpotent SYmmEtRIC MaTrices of trace $k$: $$\operAtORNaMe{Gr}(k,n)\coNg \{ P \in \mAThBb{r}^{N \times n} : P^{\MaThsF{T}} = P^2 = P, \; \OPEratoRnamE{Tr}(p)=k \}.$$ The isoMorphiSM mApS each suBsPace $\maThBf{A}\In \oPeratORnamE{Gr}(k,n)$ tO $P_{\mathbf{a}} \in \maTHbb{R}^{n \times n}$, the UNique orthogonAL pROJeCTion OntO $\mathbf{A}$, and Its iNVersE takES aN orTHogonAl proJeCTiON $P$ to the subspace $\operAtOrname{Im}( P) \in \Operatorname{GR}(k,n)$. $P$ is an orTHOGonal proJectIOn IFf it is symmetriC and iDempotent, i.E., $p^{\mathsf{T}} = p^2 = P$. The EigenvalUes of an orTHOgonal prOjeCtiOn oNto A SUbSpace

l,n)$). By Proposition \[ prop:Omega \], i t s uff ic es t o sh ow that any tw o poi nts in $\Omega_{-}(\ma thbf{ B} ) $ ca n b e con nectedb ya geo de si c c ur v ein $\ Ome ga_{-}( \mathbf{B} )$. B y Propositio n  \ [prop:Omeg aGr ass\], $\Ome ga_ {-}(\m at hbf { B})$isthe i mage o f $\ope ratorname {G r }(k,l) $ embedd e d i some trically in $\ope r at o rname{Gr}(k,n) $. Soby Le m m a \ [le m:infty\], f or an y $\math b f{ X } _ 1,\ m athbf{X}_2 \i n \operator n ame {Gr}(k ,l )$, $d_{\o perat or n ame {Gr}(k,n)}( \mat hbf{X}_1, \mathb f {X}_2)= d_{\ope ratorn ame {Gr }(k, l )} (\ mat hb f {X} _ 1, \ma t hbf {X}_2) = d _{ \Omeg a_{- } ( \ m athb f{B })}( \math bf{X}_1,\math bf{ X}_2 ) $,where thelast i s the geode sic d is tance in $\Omeg a_{- }(\mathbf {B} )$ . H en ce if $d_{\O meg a_{ -}(\mat hbf{B}) } (\m at h b f {X }_1,\mathbf{X}_2)$ i s re alized b y a ge o de si c curve $ \g amm a$ i n $\Ome ga_{ - }( \mathbf{ B})$,t he n$\gamma $must a ls o b e a geod e siccurvein $\ope rator n ame{Gr}(k,n)$. We have repr e se n t ed $\op era torname{Gr} (k,n ) $ as a s e tof* equiv alenc ec la s ses* of matrices bu tit may also be represent ed as a se t o f *actua l ma t ri c es* [@Nicolaes cu Ex ample 1.2. 2 0], name ly, i dempoten t symmetr i c matrice s o f t rac e $ k $ :$$\operatorna m e {Gr} (k ,n)\con g \ { P \in \m ath bb{ R}^ {n \times n } : P^{\ ma th sf {T }}= P^2 = P, \;\o per at orn ame{t r }(P)=k \}.$ $ Th eis o mor phism m a ps e achsu bs pace $\ ma thbf{ A}\i n \o perator name{Gr}( k,n ) $ to $ P_ {\mathb f{A}} \in \ma th bb{R}^{n \ ti mes n}$,t h e unique orthogonal projectiono nto $\m ath bf{A} $, a nd its in ver se tak esa n orth ogonal proj ec tio n $P$ t o th e s ub space $\op e r ato rname {i m}(P) \in\operatorname{Gr}( k ,n) $. $P$ is anort hogo n a lpro j ec t ion i f f i t is symmetric an d idempote nt , i .e., $P^{\ m ath sf {T}} =P^2 = P $. Th e eigenv alues ofan orthog on al p r o jec tion ontoa subspa ce

l,n)$). By Proposition \[prop:Omega\],_it suffices_to show that any_two points_in_$\Omega_{-}(\mathbf{B})$ can_be_connected by a_geodesic curve in_$\Omega_{-}(\mathbf{B})$. By Proposition \[prop:OmegaGrass\], $\Omega_{-}(\mathbf{B})$_is the image_of_$\operatorname{Gr}(k,l)$ embedded isometrically in $\operatorname{Gr}(k,n)$. So by Lemma \[lem:infty\], for any $\mathbf{X}_1,\mathbf{X}_2 \in \operatorname{Gr}(k,l)$, $d_{\operatorname{Gr}(k,n)}(\mathbf{X}_1,\mathbf{X}_2)=_d_{\operatorname{Gr}(k,l)}(\mathbf{X}_1,\mathbf{X}_2) =_d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$, where_the_last_is the geodesic distance in_$\Omega_{-}(\mathbf{B})$. Hence if $d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$ is_realized by_a geodesic curve $\gamma$ in $\Omega_{-}(\mathbf{B})$, then $\gamma$_must_also be a_geodesic curve in $\operatorname{Gr}(k,n)$. We have represented $\operatorname{Gr}(k,n)$ as a_set of *equivalence classes* of matrices_but it may_also_be_represented as a set_of *actual matrices* [@Nicolaescu Example 1.2.20], namely,_idempotent symmetric matrices of trace $k$:_$$\operatorname{Gr}(k,n)\cong \{ P \in \mathbb{R}^{n \times n}_: P^{\mathsf{T}} = P^2 = P,_\; \operatorname{tr}(P)=k \}.$$ The isomorphism_maps each_subspace $\mathbf{A}\in \operatorname{Gr}(k,n)$ to $P_{\mathbf{A}}_\in \mathbb{R}^{n \times_n}$, the_unique orthogonal projection_onto $\mathbf{A}$, and its inverse takes_an orthogonal projection_$P$ to the subspace $\operatorname{im}( P)_\in_\operatorname{Gr}(k,n)$. $P$ is_an_orthogonal_projection iff_it is symmetric_and_idempotent, i.e.,_$P^{\mathsf{T}}_= P^2 = P$. The eigenvalues_of_an orthogonal projection onto a subspace

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Eight myths about causality aNd structurAl equAtiOn mOdEls. TEchnIcal Report R-393, ComPUter science Department, UCLA. ()... PEarl, J. AnD e. BarEInBoim (2014). EXternal VAlIDIty: frOm Do-cAlCUlUs to tRanSportabIlity acrosS poPuLations. *StatiSTiCal Science* [29]{}(4),??–?? ().. in..,. PEarl, J. and T. VerMa (1991). A Theory Of InfERred cAusAtion. in J. AllEN, R. FikeS, and E. SandEwALl (Eds.), [*PRIncipleS OF KNowlEdge RepresentatioN AnD reasoning: ProceEdings Of THe sEConD InTernationaL COnferENce*]{}, San MATeO, ca, Pp. 441–452. MORgan Kaufmann. ,,, ()... PIcciotto, S., M. A. hErn[Á]{}n, J. H. PaGe, j. G. YOUng, and j. M. RobInS (2012). strUctural nestEd cuMulative fAilure TIme modeLS to estiMate thE efFecTs of INtErVenTiONs. *JOUrNal OF thE AmericaN STaTistiCal ASSOCIatiOn* [ 107]{}(499), 886–900. , ()... QUale, c. M., M. J. vAn Der Laan, and J. m. RoBins (2006). lOcaLly efFicieNt esTiMatioN with bIvariAtE right-censored dAta. *JOurnal of tHe AMeRicAn statiSTical ASsoCiaTion* [ 101]{}(475), 1076–1084. ()... RicHardson, t. s. (2003). MaRkOV PRoPerties for acyclic dIrECTeD mixed grAphs. *ScANd. j. STAtist.* [30]{}, 145–157. ().., [CenTeR foR StaTIStics And tHE SOcial SciEnces, UNIv. waShingtoN, SEattle, wA]{}. ricHarDson, T. s. And J. m. RobinS (2013). ingle [W]{}oRld [I]{}nTErvention [G]{}raphS [(sWIGs)]{}: A unificaTIoN OF tHE couNteRfactual and GrapHIcal ApprOAcHes TO causAlity. teCHnICal Report 128, Center for STaTisticS and tHe Social ScienCes, UniversITY Of WashinGton. ()... rIcHArdson, T. S. and P. SpIrtes (2002). ancestral gRAph [M]{}arkoV modeLs. *Ann. StaTist.* [30]{}, 962–1030. ,, ().

Eight myths about causali ty and str uctur alequ at ionmode ls. TechnicalR epor t R-393, Computer Scie nce D ep a rtme n t, UCLA . ().. . P e arl ,J. an dE .Barei nbo im (201 4). Extern alva lidity: From do -calculustotransportabi lit y acro ss po p ulati ons . *St atisti c al Sci ence* [29 ]{ } (4),?? – ?? (). . In ..,. Pearl, J. and T .  V e rma (1991). Atheory o f i n f err edcausation. I n J.A llen, R .  F i k e s,a nd E. Sandewa ll (Eds.),[ *Pr incipl es of Knowle dge R ep r ese ntation and Rea soning: P roceed i ngs oft he Seco nd Int ern ati onal Co nf ere nc e *]{ } ,San Mat eo, CA,pp .441–4 52.M o r g an K auf mann . ,, , ()... Picc iot to,S .,M. A. Hern [á]{ }n , J.H. Pag e, J.  G . Young, and J.  M.Robins (2 012 ). St ru ctura l neste d c umu lativefailure tim em o d el s to estimate theef f e ct s of int ervent i on s. *Journal o f t he A m e rican Sta t is tical As sociat i on *[ 107]{ }( 499),88 6–9 00. , ( ) ... Quale , C. M., M. J . van Der Laan, and J. M. Rob i ns ( 20 0 6).Loc ally effici ente stim atio n w ith bivar iateri g ht - censored data. *Jou rn al ofthe A merican Stati stical Ass o c i ation* [ 101 ] {} ( 475), 1076–108 4. ( )... Rich a rdson, T . S.(2003).Markov pr o p erties f oracy cli c d i r ec ted mixed gra p h s. * Sc and. J. St atist.* [3 0]{ },145 –1 57. ().. , [Cente rfo rSt ati stics and theSo cia lSci ences , Univ. Wash ingt on ,S eat tle, WA ] {} . Ric ha rd son, T.  S . and J.M . R obins ( 2013). in gle [W]{ }o rl d [I]{} ntervention [ G] {}raphs [( SW IGs )]{}:A unificat ion of the counterfactu a l and g rap hical app roaches t o c ausali ty. Techni cal Re port12 8,C e nterf o rSta ti stics andt h e S ocial S cien ces, Un iversity of Washin g ton . ()... Ric har dson , T.  S. an d P.  S p irt e s (2002). Ancest ral graph[M ] {} arkov mode l s.*A nn. Sta tist.*[30]{ } , 962–1 030. ,,().

Eight_myths about_causality and structural equation_models. Technical_Report_R-393, Computer_Science_Department, UCLA. ()... Pearl, J._and E. Bareinboim (2014)._External validity: From do-calculus_to transportability across_populations._*Statistical Science* [29]{}(4),??–?? ().. In..,. Pearl, J. and T. Verma (1991). A theory of inferred causation. In_J. Allen,_R. Fikes, and_E. Sandewall_(Eds.),_[*Principles of Knowledge Representation and_Reasoning: Proceedings of the Second_International Conference*]{},_San Mateo, CA, pp. 441–452. Morgan Kaufmann. ,,, ()... Picciotto, S.,_M. A._Hern[á]{}n, J. H. Page,_J. G. Young, and J. M. Robins (2012). Structural nested cumulative_failure time models to estimate the_effects of interventions._*Journal_of_the American Statistical Association*_[ 107]{}(499), 886–900. , ()... Quale, C. M., M. J._van Der Laan, and J. M. Robins (2006)._Locally efficient estimation with bivariate right-censored data._*Journal of the American Statistical Association*_[ 101]{}(475), 1076–1084. ()... Richardson, T. S. (2003)._Markov properties_for acyclic directed mixed graphs._*Scand. J. Statist.*_[30]{}, 145–157. ()..,_[Center for Statistics_and the Social Sciences, Univ. Washington,_Seattle, WA]{}. Richardson, T. S._and J. M. Robins (2013). ingle [W]{}orld_[I]{}ntervention_[G]{}raphs [(SWIGs)]{}: A_unification_of_the counterfactual_and graphical approaches_to_causality. Technical_Report_128, Center for Statistics and the_Social_Sciences, University of Washington. ()... Richardson, T. S. and P. Spirtes_(2002). Ancestral graph [M]{}arkov_models._*Ann. Statist.* [30]{}, 962–1030. ,, ().

that detailed in Algorithm \[algo\_vi\], where steps 7, 8, 9 and 10 are modified to include the time dependent parameters. It follows that for the user-specific parameters the update equations take the form: $$\begin{gathered} \lambda_{ir}^{(\alpha)} = a^{(\alpha)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t=1}^T \frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1-e^{-\theta_{ijt}}},\ \mu_{ir}^{(\alpha)} = \frac{\nu_i^{(\alpha)}}{\xi_i^{(\alpha)}}+\sum_{t=1}^T\frac{\lambda_{i{t^\prime}r}^{(\gamma)}}{\mu_{i{t^\prime}r}^{(\gamma)}}\sum_{j=1}^{{\vert{V}\vert}}\frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta)}}, \notag \\ \lambda_{ipr}^{(\gamma)}=a^{(\gamma)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t:{t^\prime}=p} \frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1-e^{-\theta_{ijt}}},\ \mu_{ipr}^{(\gamma)} = \frac{\nu_p^{(\gamma)}}{\xi_p^{(\gamma)}}+\frac{\lambda_{ir}^{(\alpha)}}{\mu_{ir}^{(\alpha)}}\sum_{j=1}^{{\vert{V}\vert}} \frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\sum_{t:{t^\prime}=p} \frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta)}},\end{gathered}$$ and similar results can be obtained for the host-specific parameters $\lambda_{ir}^{(\beta)}$, $\mu_{jr}^{(\beta)}$, $\lambda_{jpr}^{(\delta)}$ and $\mu_{jpr}^{(\delta)}$. The updates for $\nu_i^{(\alpha)},\ \xi_i^{(\alpha)},\ \nu_j^{(\beta)}$ and $\xi_j^{(\beta)}$ are identical to steps 7 and 8 in Algorithm \[algo\_vi\]. For the covariates $$\begin{gathered} \lambda_{kh

that detailed in Algorithm   \[algo\_vi\ ], where steps 7, 8, 9 and 10 are modified to include the meter pendent parameters. It follows that for the user - specific parameter the update equations take the shape: $ $ \begin{gathered } \lambda_{ir}^{(\alpha) } = a^{(\alpha)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t=1}^T \frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1 - e^{-\theta_{ijt}}},\ \mu_{ir}^{(\alpha) } = \frac{\nu_i^{(\alpha)}}{\xi_i^{(\alpha)}}+\sum_{t=1}^T\frac{\lambda_{i{t^\prime}r}^{(\gamma)}}{\mu_{i{t^\prime}r}^{(\gamma)}}\sum_{j=1}^{{\vert{V}\vert}}\frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta) } }, \notag \\ \lambda_{ipr}^{(\gamma)}=a^{(\gamma)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t:{t^\prime}=p } \frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1 - e^{-\theta_{ijt}}},\ \mu_{ipr}^{(\gamma) } = \frac{\nu_p^{(\gamma)}}{\xi_p^{(\gamma)}}+\frac{\lambda_{ir}^{(\alpha)}}{\mu_{ir}^{(\alpha)}}\sum_{j=1}^{{\vert{V}\vert } } \frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\sum_{t:{t^\prime}=p } \frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta)}},\end{gathered}$$ and like results can be obtained for the host - specific parameters $ \lambda_{ir}^{(\beta)}$, $ \mu_{jr}^{(\beta)}$, $ \lambda_{jpr}^{(\delta)}$ and $ \mu_{jpr}^{(\delta)}$. The update for $ \nu_i^{(\alpha)},\ \xi_i^{(\alpha)},\ \nu_j^{(\beta)}$ and $ \xi_j^{(\beta)}$ are identical to steps 7 and 8 in Algorithm   \[algo\_vi\ ]. For the covariates $ $ \begin{gathered } \lambda_{kh

thwt detailed in Algorithm \[xlgo\_vi\], where stgpw 7, 8, 9 aid 10 are modifiea to include the time dependxnt paramtners. It follows that wor the uder-specidic karameters the upvzte equations bake chx form: $$\begin{gatmered} \lambda_{hr}^{(\alpha)} = a^{(\alphd)}+\sjm_{l=1}^{{\vert{V}\vert}}\sum_{t=1}^T \frac{A_{ijt}\theta_{ijt}\chi_{ittr}}{1-e^{-\theya_{ljt}}},\ \mu_{ir}^{(\alpha)} = \frss{\nu_i^{(\zlpha)}}{\xi_i^{(\alpha)}}+\sum_{t=1}^T\frac{\lambda_{i{t^\prims}r}^{(\gamma)}}{\ku_{i{t^\prime}r}^{(\gamka)}}\sum_{j=1}^{{\vert{V}\vert}}\frac{\lambda_{jg}^{(\betw)}}{\mu_{jr}^{(\beta)}}\frac{\lambdw_{j{t^\prime}r}^{(\deota)}}{\mt_{h{t^\prime}r}^{(\deltx)}}, \notag \\ \lambda_{ipr}^{(\gamma)}=z^{(\gamma)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t:{t^\prime}=p} \wrac{A_{njt}\theta_{ijt}\xhu_{ijht}}{1-e^{-\theta_{ijt}}},\ \mn_{ipr}^{(\gaima)} = \frac{\nu_i^{(\bamma)}}{\xh_p^{(\gamma)}}+\grac{\lambda_{ir}^{(\aliha)}}{\mu_{mr}^{(\alpha)}}\sum_{j=1}^{{\vert{V}\vert}} \frac{\nambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beja)}}\sum_{t:{t^\prike}='} \frac{\lambda_{j{t^\prime}r}^{(\dwlra)}}{\mu_{j{j^\prima}r}^{(\deura)}},\evd{gztiersd}$$ and sijilar resumts can be ibtained for the hoxt-fircific paramsters $\jaibda_{ir}^{(\beta)}$, $\mu_{jr}^{(\beta)}$, $\lambda_{jpr}^{(\delta)}$ and $\ku_{jlr}^{(\delta)}$. The updates for $\nu_i^{(\alpha)},\ \xi_i^{(\alpha)},\ \no_j^{(\beta)}$ and $\xi_j^{(\beta)}$ are identical to steps 7 and 8 in Algorithk \[algo\_ti\]. Fox the zivwriates $$\begin{gathered} \lambda_{kh

that detailed in Algorithm \[algo\_vi\], where steps 9 10 are to include the that the user-specific parameters update equations take form: $$\begin{gathered} \lambda_{ir}^{(\alpha)} = a^{(\alpha)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t=1}^T \frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1-e^{-\theta_{ijt}}},\ = \frac{\nu_i^{(\alpha)}}{\xi_i^{(\alpha)}}+\sum_{t=1}^T\frac{\lambda_{i{t^\prime}r}^{(\gamma)}}{\mu_{i{t^\prime}r}^{(\gamma)}}\sum_{j=1}^{{\vert{V}\vert}}\frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta)}}, \notag \\ \lambda_{ipr}^{(\gamma)}=a^{(\gamma)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t:{t^\prime}=p} \frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1-e^{-\theta_{ijt}}},\ \mu_{ipr}^{(\gamma)} = \frac{\nu_p^{(\gamma)}}{\xi_p^{(\gamma)}}+\frac{\lambda_{ir}^{(\alpha)}}{\mu_{ir}^{(\alpha)}}\sum_{j=1}^{{\vert{V}\vert}} \frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\sum_{t:{t^\prime}=p} \frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta)}},\end{gathered}$$ and similar can be obtained for the host-specific parameters $\lambda_{ir}^{(\beta)}$, $\mu_{jr}^{(\beta)}$, $\lambda_{jpr}^{(\delta)}$ and $\mu_{jpr}^{(\delta)}$. The for \xi_i^{(\alpha)},\ and are identical to steps 7 and 8 in Algorithm \[algo\_vi\]. For the covariates $$\begin{gathered} \lambda_{kh

that detailed in Algorithm \[alGo\_vi\], where sTeps 7, 8, 9 aNd 10 aRe mOdIfieD to iNclude the time dEPendEnt parameters. It follows That fOr THe usER-sPecifIc paramETeRS The UpDaTe eQuATiOns taKe tHe form: $$\bEgin{gatherEd} \lAmBda_{ir}^{(\alpha)} = a^{(\aLPhA)}+\sum_{j=1}^{{\vert{V}\VerT}}\sum_{t=1}^T \frac{A_{iJt}\tHeta_{ijT}\cHi_{iJTr}}{1-e^{-\thEta_{Ijt}}},\ \mu_{Ir}^{(\alphA)} = \Frac{\nu_I^{(\alpha)}}{\xi_i^{(\AlPHa)}}+\sum_{t=1}^t\Frac{\lamBDA_{i{T^\priMe}r}^{(\gamma)}}{\mu_{i{t^\prime}R}^{(\GaMMa)}}\sum_{j=1}^{{\vert{V}\verT}}\frac{\lAmBDa_{JR}^{(\BetA)}}{\mu_{Jr}^{(\beta)}}\frac{\LaMbda_{j{T^\Prime}r}^{(\dELtA)}}{\MU_{J{t^\pRIme}r}^{(\delta)}}, \notaG \\ \lambda_{ipr}^{(\gAMma)}=A^{(\gamma)}+\SuM_{j=1}^{{\vERt{V}\verT}}\sum_{t:{T^\pRIme}=P} \frac{A_{ijt}\thEta_{iJt}\chi_{ijtr}}{1-E^{-\theta_{IJt}}},\ \mu_{ipr}^{(\GAmma)} = \fraC{\nu_p^{(\gaMma)}}{\Xi_p^{(\GammA)}}+\FrAc{\LamBdA_{Ir}^{(\aLPhA)}}{\mu_{IR}^{(\alPha)}}\sum_{j=1}^{{\vErT{V}\Vert}} \fRac{\lAMBDA_{jr}^{(\bEta)}}{\Mu_{jr}^{(\Beta)}}\sUm_{t:{t^\prime}=p} \fraC{\laMbda_{J{T^\prIme}r}^{(\dElta)}}{\mU_{j{t^\pRiMe}r}^{(\deLta)}},\end{GatheReD}$$ and similar resuLts cAn be obtaiNed FoR thE hOst-spECific pAraMetErs $\lambDa_{ir}^{(\betA)}$, $\Mu_{jR}^{(\bETA)}$, $\LaMbda_{jpr}^{(\delta)}$ and $\mu_{jPr}^{(\DELtA)}$. The updaTes for $\NU_i^{(\AlPHa)},\ \xi_i^{(\alpHa)},\ \Nu_j^{(\Beta)}$ AND $\xi_j^{(\bEta)}$ aRE iDentical To stepS 7 AnD 8 iN AlgoriThM \[algo\_vI\]. FOr tHe cOvariATes $$\bEgin{gaThered} \laMbda_{kH

that detailed in Algorith m \[algo\_ vi\], wh ere s teps 7,8, 9 and 10 ar e mod ified to include the t ime d ep e nden t p arame ters. I t f o l low sth atfo r t he us er- specifi c paramete rsth e update equ a ti ons take t heform: $$\beg in{ gather ed } \ l ambda _{i r}^{( \alpha ) } = a^ {(\alpha) }+ \ sum_{j = 1}^{{\v e r t{ V}\v ert}}\sum_{t=1}^T \f r ac{A_{ijt}\the ta_{ij t} \ ch i _ {ij tr} }{1-e^{-\t he ta_{i j t}}},\ \ m u _ {ir } ^{(\alpha)} = \frac{\nu_ i ^{( \alpha )} }{\ x i_i^{( \alph a) } }+\ sum_{t=1}^T \fra c{\lambda _{i{t^ \ prime}r } ^{(\gam ma)}}{ \mu _{i {t^\ p ri me }r} ^{ ( \ga m ma )}} \ sum _{j=1}^{ {\ ve rt{V} \ver t } } \ frac {\l ambd a_{jr }^{(\beta)}}{ \mu _{jr } ^{( \beta )}}\f rac{ \l ambda _{j{t^ \prim e} r}^{(\delta)}}{ \mu_ {j{t^\pri me} r} ^{( \d elta) } }, \no tag \\ \lambd a_{ipr} ^ {(\ ga m m a )} =a^{(\gamma)}+\sum _{ j = 1} ^{{\vert {V}\ve r t} }\ s um_{t:{t ^\ pri me}= p } \fra c{A_ { ij t}\theta _{ijt} \ ch i_ {ijtr}} {1 -e^{-\ th eta _{i jt}}} , \ \ mu_{ip r}^{(\ga mma)} = \frac{\nu_p ^ {(\gamma)}}{\ x i_ p ^ {( \ gamm a)} }+\frac{\la mbda _ {ir} ^{(\ a lp ha) } }{\mu _{ir} ^{ ( \a l pha)}}\sum_{j=1}^{{ \v ert{V} \vert }} \frac{\lam bda_{jr}^{ ( \ b eta)}}{\ mu_{ j r} ^ {(\beta)}}\sum _{t:{ t^\prime}= p } \frac{ \lamb da_{j{t^ \prime}r} ^ { (\delta) }}{ \mu _{j {t^ \ p ri me}r}^{(\delt a ) }},\ en d{gathe red }$$ and si mil arres ul ts can be obtaine dfo rth e h ost-s p ecific p ar ame te rs$\lam b da_{ir }^{(\ beta )} $, $\m u_{jr}^ { (\ b e ta)} $, $ \lam bda _{ jpr}^ {(\d e lta )}$ and $\mu_{jp r}^ { (\de lt a) }$. The updates for$\ nu_i^{(\al ph a)} ,\ \x i _ i^{(\alp ha)},\ \nu_j^{(\beta)}$ and $\x i_j ^{(\b eta) }$ are id ent ical t o s t eps 7and 8in Al go rit h m  \[al g o \_ vi\ ]. For the c o v ari ates$$ \beg in{gath ered} \lambda_{kh

that_detailed in_Algorithm \[algo\_vi\], where steps 7,_8, 9_and_10 are_modified_to include the_time dependent parameters._It follows that for_the user-specific parameters_the_update equations take the form: $$\begin{gathered} \lambda_{ir}^{(\alpha)} = a^{(\alpha)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t=1}^T \frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1-e^{-\theta_{ijt}}},\ \mu_{ir}^{(\alpha)} = \frac{\nu_i^{(\alpha)}}{\xi_i^{(\alpha)}}+\sum_{t=1}^T\frac{\lambda_{i{t^\prime}r}^{(\gamma)}}{\mu_{i{t^\prime}r}^{(\gamma)}}\sum_{j=1}^{{\vert{V}\vert}}\frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta)}}, \notag_\\ \lambda_{ipr}^{(\gamma)}=a^{(\gamma)}+\sum_{j=1}^{{\vert{V}\vert}}\sum_{t:{t^\prime}=p}_\frac{A_{ijt}\theta_{ijt}\chi_{ijtr}}{1-e^{-\theta_{ijt}}},\ \mu_{ipr}^{(\gamma)}_=__\frac{\nu_p^{(\gamma)}}{\xi_p^{(\gamma)}}+\frac{\lambda_{ir}^{(\alpha)}}{\mu_{ir}^{(\alpha)}}\sum_{j=1}^{{\vert{V}\vert}} \frac{\lambda_{jr}^{(\beta)}}{\mu_{jr}^{(\beta)}}\sum_{t:{t^\prime}=p} \frac{\lambda_{j{t^\prime}r}^{(\delta)}}{\mu_{j{t^\prime}r}^{(\delta)}},\end{gathered}$$ and similar_results can be obtained for_the host-specific_parameters $\lambda_{ir}^{(\beta)}$, $\mu_{jr}^{(\beta)}$, $\lambda_{jpr}^{(\delta)}$ and $\mu_{jpr}^{(\delta)}$. The updates_for_$\nu_i^{(\alpha)},\ \xi_i^{(\alpha)},\_\nu_j^{(\beta)}$ and $\xi_j^{(\beta)}$ are identical to steps 7 and_8 in Algorithm \[algo\_vi\]. For the covariates_$$\begin{gathered} \lambda_{kh

redder HB than would be expected for its metal abundance. This second parameter effect is common among the Galaxy’s dSph companions and it is also particularly prevalent among the globular clusters that lie, like the dSphs, in the outer regions of the galactic halo. Considerable effort has gone into attempting to understand the origin of the second parameter effect and its dependence on galactocentric distance. It is now generally accepted that the phenomenon is a manifestation of age differences that occur in the galactic halo (e.g. Lee, Demarque & Zinn 1994). Indeed this age difference interpretation of the diversity of HB types seen in the outer halo of the Galaxy is a cornerstone of the chaotic halo formation model advocated initially by Searle & Zinn (1978), in which the galactic halo is built out of the destruction of satellite galaxies over an extended interval of perhaps $\sim$3 to 5 Gyr. [*Does this prolonged formation scenario apply also to the halo of M31?*]{} An initial attack on this problem can be made by determining the HB morphologies of the most readily identifiable objects in the outer halo of M31, the And dSph galaxies. Since the mean metal abundances of these systems are well established, we can accurately predict the HB morphology expected if the bulk of the stellar population in these galaxies is as old as the inner galactic halo globular clusters. If the observations reveal instead a diversity of HB morphologies, then it will be an indication that the outer halo of M31 may also have formed in the same drawn out chaotic manner as the outer regions of the halo of our galaxy. We begin this task by presenting in this paper the results of a [*Hubble Space Telescope*]{} imaging study whose principal aim was to determine the HB morphology of the Andromeda I dwarf spheroidal galaxy. The observations, made with the WFPC2 camera, are detailed in the next section together with a description of the photometric analysis applied to the images. The resulting And I color-magnitude diagrams are discussed in detail in Section 3. The results, both in the context of the formation and the evolution of this dSph, and in the context of formation scenarios for the halo of M31, are discussed in Section 4 and summarized in Section 5. Observations and Photometry =========================== Andromeda I was imaged with the [*Hubble Space Telescope*]{} Wide Field Planetary Camera 2 (WFPC2, see Holtzman [*et al.

redder HB than would be expected for its metal abundance. This second parameter consequence is coarse among the Galaxy ’s dSph companions and it is also particularly prevailing among the globular clusters that lie, like the dSphs, in the forbidden regions of the galactic aura. Considerable effort has gone into undertake to understand the origin of the second argument effect and its dependence on galactocentric distance. It is now generally accepted that the phenomenon is a manifestation of age difference that occur in the galactic halo (for example   Lee, Demarque & Zinn 1994). Indeed this age difference rendition of the diversity of HB type seen in the out halo of the Galaxy is a cornerstone of the chaotic halo constitution model advocated initially by Searle & Zinn (1978), in which the galactic halo is built out of the destruction of satellite galaxies over an extended interval of perhaps $ \sim$3 to 5 Gyr. [ * Does this prolonged formation scenario apply also to the halo of M31? * ] { } An initial attack on this problem can be make by determining the HB morphologies of the about promptly identifiable objects in the outer halo of M31, the And dSph galaxies. Since the hateful metallic element abundances of these systems are well build, we can accurately predict the HB morphology expected if the bulk of the stellar population in these galaxies is as honest-to-god as the inner galactic halo globular clusters. If the observations uncover instead a diverseness of HB morphology, then it will be an indication that the outer halo of M31 may also have form in the same drawn out chaotic manner as the outer region of the halo of our galaxy. We begin this task by present in this paper the solution of a [ * Hubble Space Telescope * ] { } imaging study whose chief aim was to determine the HB morphology of the Andromeda   I dwarf spheroidal galaxy. The notice, made with the WFPC2 television camera, are detailed in the next section in concert with a description of the photometric analysis applied to the images. The resulting And I color - magnitude diagrams are discussed in detail in Section 3. The results, both in the context of the formation and the development of this dSph, and in the context of formation scenarios for the aura of M31, are discussed in Section 4 and summarized in Section 5. Observations and Photometry = = = = = = = = = = = = = = = = = = = = = = = = = = = Andromeda I was imaged with the [ * Hubble Space Telescope * ] { } Wide Field Planetary Camera 2 (WFPC2, see Holtzman [ * et al.

refder HB than would be exkected for its mejao abunvance. Tgis secovd parameter effect is commoi aming tye Galaxy’s dSph compankons and pt is alsi pacticularly prevalent amoky the nlobuner clusters thaj lie, like tve dSphs, in tha uucer regions of the galactic halo. Consyderablr fffort has gong intp attsmpting to understand the origin or the stcond parameter efgect and its dependence on galwctocentric distanfe. It is noq gegwrally accepged that the phenomenoh is a manifestation of age difwerenees that ocxue ij the galactmc halj (e.g. Lee, Demavaue & Zhnn 1994). Inceed this age cifherebce interpretation of the diversity of RB types vezn in the outer halo if the Calafy ir a zorhecstkne of thx chaotic hzlo formatiin model advocated onynoally by Seadle & Zyng (1978), in which the galactic halo is built mut of the destruction of watellite galaxies ovgr an extegded interval of perhaps $\sim$3 to 5 Gyr. [*Does this prmlongxd foxnatiov sfenario apply also to the halo of M31?*]{} An initiaj autabk on this problei can be mace bu determining jhe HB morlhologies of the mlst reaqily udentifiafle pbjects in the outer halo od M31, the And bSpy galaxies. Since tke mean metau abondancrs of these systems are welm establishfd, we can xccurately predizt nhe VB morphology expected if ehe bulk if tke stellxr pppulatyon in thede galaxies is as old as the nnner galactic jalo globular clusters. If the ouxervations rgvedl pnstead a divevsity of HB mor[hologies, then it wilj be xn indicatpon that vhe outer hajo of M31 may ando have formxd in the samw drqwn out zhaotic manner as the olttr regions of the halo of ouv galxsy. We begin this tawk by presentinb iv trid 'aper dhe results mf a [*Hucnle Soace Telescioe*]{} ikaging study whose psincjpal aim was to deyevmine the HB morprology of the Andromeda I dwarf dphermidel galsxy. The observations, made with ths WFPC2 calerw, are detailqd ik thg next sectnon together with a description of the piotometric analysis apklied to the images. Tke resulting End I solor-magnhtude diagrams are duscussed in detaik in Section 3. The reshlts, bmth ij the context of the formation and the evolution of this dSph, and in the xontexv jf formatioh scrnarims yor the hall of M31, are discusxed in Section 4 and summarized mn Section 5. Oyservations and Photometry =========================== Andtooeda I was imxged with the [*Hubble Spade Telesvope*]{} Wide Field Planetary Camers 2 (WFPC2, see Holtzman [*et sl.

redder HB than would be expected for abundance. second parameter is common among it also particularly prevalent the globular clusters lie, like the dSphs, in the regions of the galactic halo. Considerable effort has gone into attempting to understand origin of the second parameter effect and its dependence on galactocentric distance. It now accepted the is a manifestation of age differences that occur in the galactic halo (e.g. Lee, Demarque & 1994). Indeed this age difference interpretation of the of HB types seen the outer halo of the is cornerstone of chaotic formation advocated initially by & Zinn (1978), in which the galactic halo is built out of the destruction of satellite galaxies an extended perhaps $\sim$3 5 [*Does prolonged formation scenario to the halo of M31?*]{} An this problem can be made by determining the morphologies of most readily identifiable objects in the halo of M31, the And dSph galaxies. Since mean metal abundances of these systems are well established, we can accurately predict the HB if the bulk of stellar population in galaxies as as inner galactic globular clusters. If the observations reveal instead a diversity of HB then it will be an indication that the outer halo may have formed in same drawn out chaotic as outer regions of the our We by in paper the results of [*Hubble Space Telescope*]{} imaging study principal aim was to the Andromeda I dwarf spheroidal galaxy. The observations, with the WFPC2 camera, are detailed in next section together with a description of the photometric analysis applied to images. The I color-magnitude diagrams are discussed in detail in 3. The results, both the context of the formation and the evolution of dSph, in the of formation scenarios the halo of are discussed in and summarized Section Observations Andromeda I was imaged with the Space Wide Field Planetary Camera 2 see al.

redder HB than would be expectEd for its meTal abUndAncE. THis sEconD parameter effeCT is cOmmon among the Galaxy’s dSPh comPaNIons ANd It is aLso partICuLARly PrEvAleNt AMoNg the GloBular clUsters that Lie, LiKe the dSphs, in THe Outer regioNs oF the galactic HalO. ConsiDeRabLE effoRt hAs gonE into aTTemptiNg to underStANd the oRIgin of tHE SeCond Parameter effect anD ItS Dependence on gaLactocEnTRiC DIstAncE. It is now geNeRally ACcepted THaT THE phENomenon is a manIfestation oF Age DifferEnCes THat occUr in tHe GAlaCtic halo (e.g. LEe, DeMarque & ZinN 1994). IndeeD This age DIfferenCe inteRprEtaTion OF tHe DivErSIty OF Hb tyPEs sEen in the OuTeR halo Of thE gALAxy iS a cOrneRstonE of the chaotic HalO forMAtiOn modEl advOcatEd InitiAlly by searlE & ZInn (1978), in which the gaLactIc halo is bUilT oUt oF tHe desTRuctioN of SatEllite gAlaxies OVer An EXTEnDed interval of perhaPs $\SIM$3 tO 5 Gyr. [*Does This prOLoNgED formatiOn SceNariO APply aLso tO ThE halo of M31?*]{} an initIAl AtTack on tHiS problEm Can Be mAde by DEterMining The HB morPholoGIes of the most reADily identifiaBLe OBJeCTs in The Outer halo of m31, the aNd dSPh gaLAxIes. sInce tHe meaN mETaL Abundances of these syStEms are Well eStablished, we cAn accurateLY PRedict thE HB mORpHOlogy expected iF the bUlk of the stELlar popuLatioN in these Galaxies iS AS old as thE inNer GalActIC HaLo globular cluSTErs. IF tHe obserVatIons revEal InsTeaD a dIvErsity of Hb morpholOgIeS, tHeN it Will bE An indicaTiOn tHaT thE outeR Halo of m31 may aLso hAvE fORmeD in the sAMe DRAwn oUt ChAotiC maNnEr as tHe ouTEr rEgions oF the halo oF ouR GalaXy. we Begin thIs task by preseNtIng in this pApEr tHe resuLTS of a [*HubbLe Space Telescope*]{} imaging STudy whoSe pRinciPal aIm was to deTerMine thE HB MOrpholOgy of tHe AndRoMedA i Dwarf SPHeRoiDaL galaxy. The OBSerVatioNs, Made With the wFPC2 camera, are detaiLEd iN the next sectiOn tOgetHER wIth A DeSCriPtIOn oF THe photometric anAlysis applIeD To The images. THE reSuLting AnD I color-MagniTUde diagRams are diScussed in DeTail IN secTion 3. The resUlts, both In the contEXt of tHE fOrmatIon And the EvOluTion oF this dsPh, aNd in tHe contExT of forMatioN sCenarios For the halo of M31, are discussEd in SeCtion 4 And SummarizeD in sEctIon 5. ObservAtioNs and PhotoMetRy =========================== ANdromEda i Was imAged WItH thE [*hubblE SpaCE TelescopE*]{} wiDe FIELd planetary CaMERA 2 (WFpC2, see holTZman [*et Al.

redder HB than would be e xpected fo r its me tal a bund ance . This secondp aram eter effect is commonamong t h e Ga l ax y’s d Sph com p an i o nsan ditis al so pa rti cularly prevalent am on g the globul a rclusters t hat lie, like t hedSphs, i n t h e out erregio ns oft he gal actic hal o. Consi d erablee f fo rt h as gone into atte m pt i ng to understa nd the o r ig i n of th e second p ar amete r effect an d i tsd ependence ongalactocent r icdistan ce . I t is no w gen er a lly accepted t hatthe pheno menoni s a man i festati on ofage di ffer e nc es th at occ u rint hegalactic h al o (e. g. L e e , Dema rqu e &Zinn1994). Indeed th is a g e d iffer enceinte rp retat ion of thedi versity of HB t ypes seen inthe o ute rhaloo f theGal axy is a c ornerst o neof t h echaotic halo forma ti o n m odel adv ocated in it i ally bySe arl e &Z i nn (1 978) , i n whichthe ga l ac ti c halois built o utofthe d e stru ctionof satel liteg alaxies over a n extended int e rv a l o f per hap s $\sim$3 t o 5G yr.[*Do e sthi s prol onged f o rm a tion scenario apply a lso to thehalo of M31?* ]{} An ini t i a l attack ont hi s problem can b e mad e by deter m ining th e HBmorpholo gies of t h e most re adi lyide nti f i ab le objects in t he o ut er halo of M31, t heAnd dS phga laxies. S ince the m ea nme tal abun d ances of t hes esys temsa re wel l est abli sh ed , we can ac c ur a t elypr ed ictthe H B mor phol o gyexpecte d if thebul k ofth estellar population i nthese gala xi esis aso l d as the inner galactic halo gl o bular c lus ters. Ifthe obser vat ions r eve a l inst ead adiver si tyo f HB m o r ph olo gi es, then i t wil l bean ind ication that the outer ha l o o f M31 may als o h avef o rm edi nt hesa m e d r a wn out chaoticmanner asth e o uter regio n s o fthe hal o of ou r gal a xy. We begin th is task b ypres e n tin g in thispaper th e results of a[ *H ubble Sp ace Te le sco pe*]{ } imag i ngstudy whose p rincip al ai mwas to d etermine the HB morphol ogy of theAnd romeda Idwa r f s pheroidal gal axy. The o bse rva tions , m a de wi th t h eWFP C 2 cam era, are detai l ed in t he next secti o n tog ether wi t h a de scri ption of the phot o metric analysi s ap p l ied to theim ages. The resu lti ng A nd I col or -magnitudediagrams a r e dis cussed in de tail in S ec t ion 3. The re sults, bo thin the con te xt of the for ma tion a nd the evol u t ion of this dSph , and i n the con textof format i on s cenarios f or the halo of M3 1, a re di scussed i n Sect ion 4 and summa r ized in S ectio n 5. O bs erva tio ns and Pho t o metry === == === ========= = = == = == == = A ndro medaIwasimaged wi t h the [* Hub b le Spac eTel e s cope*] { }W i de Field P lan etary C amera 2 (W F PC2, se e Holt zman [ *et al .

redder_HB than_would be expected for_its metal_abundance._This second_parameter_effect is common_among the Galaxy’s_dSph companions and it_is also particularly_prevalent_among the globular clusters that lie, like the dSphs, in the outer regions of_the_galactic halo. Considerable_effort_has_gone into attempting to understand_the origin of the second_parameter effect_and its dependence on galactocentric distance. It is_now_generally accepted that_the phenomenon is a manifestation of age differences that_occur in the galactic halo (e.g. Lee,_Demarque & Zinn_1994)._Indeed_this age difference interpretation_of the diversity of HB types_seen in the outer halo of_the Galaxy is a cornerstone of the_chaotic halo formation model advocated initially_by Searle & Zinn (1978),_in which_the galactic halo is built_out of the_destruction of_satellite galaxies over_an extended interval of perhaps $\sim$3_to 5 Gyr._[*Does this prolonged formation scenario apply_also_to the halo_of_M31?*]{}_An initial_attack on this_problem_can be_made_by determining the HB morphologies of_the_most readily identifiable objects in the outer_halo of M31, the_And_dSph galaxies. Since the_mean metal abundances of these_systems are well established, we can_accurately predict_the HB_morphology expected if the bulk of the stellar population in these_galaxies is as old as the_inner galactic halo globular_clusters. If_the_observations reveal instead_a_diversity of_HB morphologies, then it will be an_indication that_the outer halo of M31 may_also have formed in_the_same drawn out chaotic manner as_the outer regions of the halo_of our galaxy. We begin this_task_by_presenting in this paper the_results of a [*Hubble Space Telescope*]{}_imaging study whose_principal aim was to determine the HB_morphology_of the Andromeda I dwarf spheroidal galaxy._The_observations, made with the WFPC2 camera,_are_detailed_in the next section together_with a description of the photometric_analysis applied to the images. The resulting And I_color-magnitude diagrams are_discussed in detail in Section_3._The_results, both in the context of the formation and the_evolution of_this dSph, and_in the context of formation scenarios for the halo of_M31, are discussed in Section 4 and_summarized in Section 5. Observations and Photometry =========================== Andromeda I was imaged with the_[*Hubble Space Telescope*]{} Wide Field Planetary Camera 2_(WFPC2, see Holtzman [*et al.

{P_2$,[$S_{2,2}$]{}$\}$ (these stars have Grundy value 2); - ${\mathcal C_2^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=2$ and $S$ contains one or three paths of length $2$; - ${\mathcal C_3^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=3$ and S contains one or three paths of length $2$; - For $i\in \{0,1,2,3\}$, $\mathcal C_i$: subdivided stars $S$ with ${\mathcal{G}}(S)=i$ and $S$ is not in a previous class. Figure \[fig:tabEquivSim1\] shows the equivalence classes of the subdivided stars. (-1,1.5) – (-1,-8.5); (-1,1.5) – (8.5,1.5); (3.75,2.4) node [Number of paths of length 2 in the subdivided star]{}; (-2.3,-3.5) node\[rotate=90\] [Number of paths in the subdivided star]{}; (-1.2,0) node [0]{}; (-1.2,-1) node [1]{}; (-1.2,-2) node [2]{}; (-1.2,-3) node [3]{}; (-1.2,-4) node [4]{}; (-1.2,-5) node [5]{}; (-1.5,-6) node […]{}; (-1.5,-7) node [$2p$]{}; (-1.5,-8) node [$2p+1$]{}; (0,1.8) node [0]{}; (1,1.8) node [1]{}; (2,1.8) node [2]{}; (3,1.8) node [3]{}; (4,1.8) node [4]{}; (5,1.8) node [5]{}; (6,1.8) node […]{}; (7,1.8) node [$2p$]{}; (8,1.8) node [$2p+1$]{}; (empty) at (0,1) [0]{}; (p1) at (0,0) [$1^*$]{}; (p2)

{ P_2$,[$S_{2,2}$]{}$\}$ (these stars have Grundy value 2); - $ { \mathcal C_2^{\Box}}$: subdivided stars $ S$ such that $ { \mathcal{G}}(S)=2 $ and $ S$ control one or three path of length $ 2 $; - $ { \mathcal C_3^{\Box}}$: subdivided stars $ S$ such that $ { \mathcal{G}}(S)=3 $ and S hold one or three paths of length $ 2 $; - For $ i\in \{0,1,2,3\}$, $ \mathcal C_i$: subdivide star $ S$ with $ { \mathcal{G}}(S)=i$ and $ S$ is not in a previous course. calculate   \[fig: tabEquivSim1\ ] shows the comparison classes of the subdivided headliner. (-1,1.5) – (-1,-8.5); (-1,1.5) – (8.5,1.5); (3.75,2.4) lymph node [ Number of paths of duration 2 in the subdivided star ] { }; (-2.3,-3.5) node\[rotate=90\ ] [ Number of way in the subdivided star ] { }; (-1.2,0) node [ 0 ] { }; (-1.2,-1) node [ 1 ] { }; (-1.2,-2) node [ 2 ] { }; (-1.2,-3) lymph node [ 3 ] { }; (-1.2,-4) node [ 4 ] { }; (-1.2,-5) node [ 5 ] { }; (-1.5,-6) lymph node [ … ] { }; (-1.5,-7) node [ $ 2p$ ] { }; (-1.5,-8) lymph node [ $ 2p+1 $ ] { }; (0,1.8) node [ 0 ] { }; (1,1.8) node [ 1 ] { }; (2,1.8) node [ 2 ] { }; (3,1.8) node [ 3 ] { }; (4,1.8) node [ 4 ] { }; (5,1.8) node [ 5 ] { }; (6,1.8) node [ … ] { }; (7,1.8) node [ $ 2p$ ] { }; (8,1.8) lymph node [ $ 2p+1 $ ] { }; (empty) at (0,1) [ 0 ] { }; (p1) at (0,0) [ $ 1^*$ ] { }; (p2 )

{P_2$,[$S_{2,2}$]{}$\}$ (hhese stars have Grundy yalue 2); - ${\mathcal C_2^{\Box}}$: snbdividsd stars $S$ such that ${\mathcal{G}}(S)=2$ and $S$ cintaibs one or three paths uf length $2$; - ${\mathxal R_3^{\Box}}$: subdivided stars $S$ such tgwt ${\mctical{G}}(S)=3$ and S coktains one mr three paths ow pength $2$; - For $i\in \{0,1,2,3\}$, $\mathcal C_i$: subdidided syags $S$ with ${\mathsal{G}}(X)=y$ ans $S$ is not in a previous class. Figude \[fig:tauEquivSim1\] shows the equivalence classes ov thf subdivided stars. (-1,1.5) – (-1,-8.5); (-1,1.5) – (8.5,1.5); (3.75,2.4) node [Bumbqe of paths ow length 2 pu the subdibided star]{}; (-2.3,-3.5) node\[rotate=90\] [Number ow patks in the soyeivlged star]{}; (-1.2,0) nove [0]{}; (-1.2,-1) njde [1]{}; (-1.2,-2) node [2]{}; (-1.2,-3) node [3]{}; (-1.2,-4) tode [4]{}; (-1.2,-5) mode [5]{}; (-1.5,-6) node […]{}; (-1.5,-7) kode [$2'$]{}; (-1.5,-8) nide [$2p+1$]{}; (0,1.8) node [0]{}; (1,1.8) node [1]{}; (2,1.8) nmde [2]{}; (3,1.8) node [3]{}; (4,1.8) node [4]{}; (5,1.8) node [5]{}; (6,1.8) tobe […]{}; (7,1.8) node [$2p$]{}; (8,1.8) node [$2p+1$]{}; (emptt) at (0,1) [0]{}; (p1) dt (0,0) [$1^*$]{}; (o2)

{P_2$,[$S_{2,2}$]{}$\}$ (these stars have Grundy value 2); C_2^{\Box}}$: stars $S$ that ${\mathcal{G}}(S)=2$ and paths length $2$; - C_3^{\Box}}$: subdivided stars such that ${\mathcal{G}}(S)=3$ and S contains or three paths of length $2$; - For $i\in \{0,1,2,3\}$, $\mathcal C_i$: subdivided $S$ with ${\mathcal{G}}(S)=i$ and $S$ is not in a previous class. Figure \[fig:tabEquivSim1\] the classes the stars. (-1,1.5) – (-1,-8.5); (-1,1.5) – (8.5,1.5); (3.75,2.4) node [Number of paths of length 2 in subdivided star]{}; (-2.3,-3.5) node\[rotate=90\] [Number of paths in subdivided star]{}; (-1.2,0) node (-1.2,-1) node [1]{}; (-1.2,-2) node (-1.2,-3) [3]{}; (-1.2,-4) [4]{}; node (-1.5,-6) node […]{}; node [$2p$]{}; (-1.5,-8) node [$2p+1$]{}; (0,1.8) node [0]{}; (1,1.8) node [1]{}; (2,1.8) node [2]{}; (3,1.8) node [3]{}; node [4]{}; [5]{}; (6,1.8) […]{}; node (8,1.8) node [$2p+1$]{}; (0,1) [0]{}; (p1) at (0,0) [$1^*$]{};

{P_2$,[$S_{2,2}$]{}$\}$ (these stars have Grundy valUe 2); - ${\mathcal C_2^{\box}}$: suBdiVidEd StarS $S$ suCh that ${\mathcal{G}}(s)=2$ And $S$ Contains one or three pathS of leNgTH $2$; - ${\matHCaL C_3^{\Box}}$: SubdiviDEd STArs $s$ sUcH thAt ${\MAtHcal{G}}(s)=3$ anD S contaIns one or thRee PaThs of length $2$; - FOR $i\In \{0,1,2,3\}$, $\mathcal C_I$: suBdivided starS $S$ wIth ${\matHcAl{G}}(s)=I$ and $S$ Is nOt in a PrevioUS class. figure \[fig:TaBequivSIM1\] shows tHE EqUivaLence classes of the SUbDIvided stars. (-1,1.5) – (-1,-8.5); (-1,1.5) – (8.5,1.5); (3.75,2.4) nodE [NumbeR oF PaTHS of LenGth 2 in the suBdIvideD Star]{}; (-2.3,-3.5) nodE\[RoTATE=90\] [NuMBer of paths in tHe subdivideD StaR]{}; (-1.2,0) node [0]{}; (-1.2,-1) nOdE [1]{}; (-1.2,-2) noDE [2]{}; (-1.2,-3) node [3]{}; (-1.2,-4) nOde [4]{}; (-1.2,-5) noDe [5]{}; (-1.5,-6) NOde […]{}; (-1.5,-7) Node [$2p$]{}; (-1.5,-8) node [$2p+1$]{}; (0,1.8) nOde [0]{}; (1,1.8) nOde [1]{}; (2,1.8) node [2]{}; (3,1.8) noDe [3]{}; (4,1.8) node [4]{}; (5,1.8) NOde [5]{}; (6,1.8) node […]{}; (7,1.8) NOde [$2p$]{}; (8,1.8) nodE [$2p+1$]{}; (emptY) at (0,1) [0]{}; (P1) at (0,0) [$1^*$]{}; (P2)

{P_2$,[$S_{2,2}$]{}$\}$ (t hese stars have Gr und yvalu e 2) ; - ${\math c al C _2^{\Box}}$: subdivide d sta rs $S$s uc h tha t ${\ma t hc a l {G} }( S) =2$ a n d$S$ c ont ains on e or three pa th s of length$ 2$ ; - ${\ mat hcal C_3^{\B ox} }$: su bd ivi d ed st ars $S$such t h at ${\ mathcal{G }} ( S)=3$a nd S co n t ai ns o ne or three paths of length $2$; - For $ i \i n \{0 ,1, 2,3\}$, $\ ma thcal C_i$: s u bd i v i ded stars $S$ wit h ${\mathca l {G} }(S)=i $and $S$ is notin a p revious cla ss. Figure \ [fig:t a bEquivS i m1\] sh ows th e e qui vale n ce c las se s of th e s u bdi vided st ar s. (-1 ,1.5 ) – (-1, -8. 5);(-1,1 .5) – (8.5,1. 5); (3 . 75, 2.4)node[Num be r ofpathsof le ng th 2 in the sub divi ded star] {}; ( -2. 3, -3.5) node\[ rot ate =90\] [ Numbero f p at h s in the subdivided st ar ] { }; (-1.2, 0) nod e [ 0] { }; (-1.2 ,- 1)node [ 1]{}; (-1 . 2, -2) node [2]{} ; ( -1 .2,-3)no de [3] {} ; ( -1. 2,-4) node [4]{} ; (-1.2, -5) n o de [5]{}; (-1. 5 ,-6) node […] { }; ( -1 . 5,-7 ) n ode [$2p$]{ }; ( - 1.5, -8)n od e [ $ 2p+1$ ]{}; ( 0 ,1 . 8) node [0]{}; (1,1 .8 ) node [1]{ }; (2,1.8) no de [2]{};( 3 , 1.8) nod e [3 ] {} ; (4,1.8) node[4]{} ; (5,1.8)n ode [5]{ }; (6 ,1.8) no de […]{}; ( 7,1.8) n ode [$ 2p$ ]{} ; (8 ,1.8) node [$ 2 p +1$] {} ; (emp ty) at (0, 1)[0] {}; (p 1) at (0,0) [$1^*$] {} ; ( p2 )

{P_2$,[$S_{2,2}$]{}$\}$ (these_stars have_Grundy value 2); - _ ${\mathcal_C_2^{\Box}}$:_subdivided stars_$S$_such that ${\mathcal{G}}(S)=2$_and $S$ contains_one or three paths_of length $2$; -__ ${\mathcal C_3^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=3$ and S contains one or_three_paths of_length_$2$; -_ For $i\in \{0,1,2,3\}$,_$\mathcal C_i$: subdivided stars $S$_with ${\mathcal{G}}(S)=i$_and $S$ is not in a previous class. Figure \[fig:tabEquivSim1\]_shows_the equivalence classes_of the subdivided stars. (-1,1.5) – (-1,-8.5); (-1,1.5) – (8.5,1.5); (3.75,2.4)_node [Number of paths of length_2 in the_subdivided_star]{};_(-2.3,-3.5) node\[rotate=90\] [Number of_paths in the subdivided star]{}; (-1.2,0) node_[0]{}; (-1.2,-1) node [1]{}; (-1.2,-2) node_[2]{}; (-1.2,-3) node [3]{}; (-1.2,-4) node [4]{};_(-1.2,-5) node [5]{}; (-1.5,-6) node […]{};_(-1.5,-7) node [$2p$]{}; (-1.5,-8) node_[$2p+1$]{}; (0,1.8) node_[0]{}; (1,1.8) node [1]{}; (2,1.8)_node [2]{}; (3,1.8)_node [3]{};_(4,1.8) node [4]{};_(5,1.8) node [5]{}; (6,1.8) node […]{};_(7,1.8) node [$2p$]{};_(8,1.8) node [$2p+1$]{}; (empty) at (0,1) [0]{};_(p1)_at (0,0) [$1^*$]{}; (p2)

than 0.04 (Strauss et al. [@strauss]). For closer galaxies, small compact areas inside the galaxies are considered as independent galaxies, so the SDSS photometry and the radius R90 are not representative of the whole galaxy. We therefore excluded from our sample all galaxies with z$<$0.04. (iii) We keep only galaxies with a difference between the fiber and model magnitudes in the r band of less than 1 mag. One of the main problems with observed spectra is that very few of the associated astrophysical parameters are known. In SDSS only a rough classification is provided. This classification is based on the work of Yip et al. ([@yip]), in which galaxies with eClass$<$-0.1 are considered to be early type galaxies while galaxies with eClass$>$-0.1 are late types. Another criterion used in SDSS for galaxy classification is given by Strateva et al. ([@strateva]). Here, galaxies having a concentration index C smaller than or larger than 2.6 are expected to be late or early type galaxies respectively. The distributions of both indices (C and eClass) show that the SDSS observations cover a wide range of galaxy types. Unfortunately neither of these criteria is particularly accurate, so they can be used only for a rough classification of our sample. It can also be seen in figure \[f2\] that these two criteria are also not in particularly good agreement. For the final sample we apply additional criteria to exclude observations with large errors, having selected galaxies small enough to ensure that the spectrum comes from most of the projected area of the galaxy. More specifically, we only retained galaxies which had uncertainties in the fiber and model magnitudes in the r band less than 0.01 mag, and had an uncertainty in the concentration index C less than 0.15. The final sample contains 33670 spectra of galaxies covering the whole range of redshifts and galaxy types present in the SDSS sample. The extension of the observed spectra of galaxies ------------------------------------------------- The extension of the observed SDSS spectra to Gaia wavelengths was made by using the 28885 synthetic spectra of the second library produced at a random grid of parameters (Tsalmantza et al. [@tsalmantza2]). In order to find the synthetic spectrum that is in best agreement with each observed spectrum, we first had to make the two libraries compatible. The main differences between them are the effects of foreground reddening, noise and redshift, which are present in the SDSS spectra but absent in the synthetic

than 0.04 (Strauss et al. [ @strauss ]). For closer galaxies, small compact area inside the galaxy are considered as independent galaxies, so the SDSS photometry and the spoke R90 are not representative of the whole galaxy. We therefore eject from our sample all galaxies with z$<$0.04. (iii) We keep open only galaxies with a remainder between the fiber and model magnitude in the r band of less than 1 mag. One of the main problems with observed spectrum is that very few of the associated astrophysical parameters are known. In SDSS only a grating classification is provided. This classification is based on the employment of Yip et al. ([ @yip ]), in which galaxies with eClass$<$-0.1 are considered to be early type galaxies while galaxies with eClass$>$-0.1 are belated types. Another criterion used in SDSS for galaxy classification is given by Strateva et al. ([ @strateva ]). Here, galaxies having a concentration index C smaller than or larger than 2.6 are expected to be late or early type galaxies respectively. The distribution of both indices (C and eClass) usher that the SDSS observations cover a wide range of galax types. Unfortunately neither of these standard is particularly accurate, so they can be used merely for a rough classification of our sample. It can also be seen in figure \[f2\ ] that these two criterion are also not in particularly good agreement. For the final sample distribution we apply extra criteria to exclude observations with large errors, having choose galaxies small enough to ensure that the spectrum comes from most of the projected area of the galaxy. More specifically, we only retained galax which had uncertainties in the roughage and model magnitude in the r set less than 0.01 mag, and had an doubt in the concentration index C less than 0.15. The concluding sample contains 33670 spectra of galaxies covering the whole range of redshifts and galaxy types present in the SDSS sample. The extension of the ascertained spectra of galaxies ------------------------------------------------- The propagation of the observed SDSS spectra to Gaia wavelengths was made by using the 28885 synthetic spectra of the second library produce at a random grid of parameters (Tsalmantza et al. [ @tsalmantza2 ]). In order to find the synthetic spectrum that is in best agreement with each observed spectrum, we foremost had to make the two libraries compatible. The main differences between them are the effects of foreground reddening, noise and red shift, which are present in the SDSS spectra but absent in the synthetic

thwn 0.04 (Strauss et al. [@strausr]). For closer galaxies, vmall dompact xreas inside the galaxies arx cobsideeed as independent galxxies, so nhe SDSS photinetry and vge radimf R90 zve noc cepresentative pf the whone galaxy. We tvefeyore excluded from our sample all gajaxies eihh z$<$0.04. (iii) We kegp onkr gamaxies with a difference between tge fibeg and model magniyudes in the r band of lesd thwn 1 mag. One of the lain problens wyrh observed rpectra is that very fgw of the associated astrophysicxl paxameters arg inlfn. In SDSS inly w rough classificatimn is ptovided. This ckasvifucation is based on tie work of Yip et al. ([@yip]), in wvieh galaxies with eClaws$<$-0.1 are wonshderde tu bt eerlg type gamaxies whime galaxies with eClass$>$-0.1 are laue ejles. Another driteryog used in SDSS for galaxy classificatiot ia given by Strateva et ql. ([@strateva]). Here, galadies havigg a concentration index C smaller than or larger than 2.6 art cwpecgwd to be late or early type galaxies respectivejg. Uhe distributions of both incifex (C and eClass) show cgaf the SDSS observahions cjver q wide ragge pf galaxy types. Unfortunateoy neither oy tyese criteria is pcrticularly cccuraje, so yhey can be used only fur a rough clasdificatioh of our sample. Ig csn also be seen in figure \[f2\] ehat thesx two criterka ate also not in pagticularly good agreemenh. For jhe fital sample we apply additional criteria to exclude obsetvadiots with jarge errors, havind selected galcxies smcll enuugh to enaure thet the spectwum comes frok most of the projectqd aeea if the exlaxy. More spevifically, we only rwtained galaxies wmich fzd uncertaintiew ib the fiber and moael mwgiitudqv in the r bdnd uesr than 0.01 mag, and hcd an incertainty in the cmncehtration index C lrsf than 0.15. Jhe final sample contaons 33670 spectra of gapaxiev cuverimg jhe whole range of redshifts ans galaxy hypcs present in the SDSS sample. Che extension of the observed spectra of galaxies ------------------------------------------------- The extenwion of the observeb XDSS spectre to Gwia wavelangths was made by uwing the 28885 synthetlc spectra of the secohd libsary oroduced at a random grid of parameters (Tsalmantza et al. [@tsalmantza2]). In oreer to find the syhtheyic sieccruk that is mn best agreement eith each observed spectrum, we hirst had do make the two libraries compstkble. The main differences between thej are thr effects of foreground reddenimg, noise and redshift, whlch are peesenf in the SVSS spectra but absemt in che sybthetic

than 0.04 (Strauss et al. [@strauss]). For small areas inside galaxies are considered SDSS and the radius are not representative the whole galaxy. We therefore excluded our sample all galaxies with z$<$0.04. (iii) We keep only galaxies with a between the fiber and model magnitudes in the r band of less than mag. of main with observed spectra is that very few of the associated astrophysical parameters are known. In SDSS a rough classification is provided. This classification is on the work of et al. ([@yip]), in which with are considered be type while galaxies with are late types. Another criterion used in SDSS for galaxy classification is given by Strateva et al. Here, galaxies concentration index smaller or than 2.6 are be late or early type galaxies of both indices (C and eClass) show that SDSS observations a wide range of galaxy types. neither of these criteria is particularly accurate, so can be used only for a rough classification of our sample. It can also be figure \[f2\] that these criteria are also in good For final sample apply additional criteria to exclude observations with large errors, having selected small enough to ensure that the spectrum comes from most projected of the galaxy. specifically, we only retained which uncertainties in the fiber magnitudes the than mag, had an uncertainty in concentration index C less than The final sample contains the whole range of redshifts and galaxy types in the SDSS sample. The extension of observed spectra of galaxies ------------------------------------------------- The extension of the observed SDSS spectra Gaia wavelengths by using the 28885 synthetic spectra of the library produced at a grid of parameters (Tsalmantza et al. [@tsalmantza2]). In order find synthetic spectrum is in best with each observed we first had the two compatible. main are the effects of foreground reddening, and which are present in the spectra in the synthetic

than 0.04 (Strauss et al. [@strauss]). For Closer galaXies, sMalL coMpAct aReas Inside the galaxIEs arE considered as independeNt galAxIEs, so THe sDSS pHotometRY aND The RaDiUs R90 ArE NoT reprEseNtative Of the whole GalAxY. We therefore EXcLuded from oUr sAmple all galaXieS with z$<$0.04. (IiI) We KEep onLy gAlaxiEs with A DifferEnce betweEn THe fibeR And modeL MAgNituDes in the r band of leSS tHAn 1 mag. One of the mAin proBlEMs WITh oBseRved spectrA iS that VEry few oF ThE ASSocIAted astrophysIcal parametERs aRe knowN. IN SDss only a Rough ClASsiFication is pRoviDed. This clAssifiCAtion is BAsed on tHe work Of YIp eT al. ([@yIP]), iN wHicH gALaxIEs WitH EClAss$<$-0.1 are coNsIdEred tO be eARLY Type GalAxieS whilE galaxies with EClAss$>$-0.1 aRE laTe typEs. AnoTher CrIteriOn used In SDSs fOr galaxy classifIcatIon is giveN by stRatEvA et al. ([@STratevA]). HeRe, gAlaxies Having a COncEnTRATiOn index C smaller thaN oR LArGer than 2.6 aRe expeCTeD tO Be late or EaRly Type GALaxieS resPEcTively. ThE distrIBuTiOns of boTh IndiceS (C And EClAss) shOW thaT the SDsS observAtionS Cover a wide rangE Of galaxy types. uNfORTuNAtelY neIther of thesE criTEria Is paRTiCulARly acCuratE, sO ThEY can be used only for a rOuGh clasSificAtion of our samPle. It can alSO BE seen in fIgurE \[F2\] tHAt these two critEria aRe also not iN ParticulArly gOod agreeMent. For thE FInal sampLe wE apPly AddITIoNal criteria to EXCludE oBservatIonS with laRge ErrOrs, HavInG selected Galaxies SmAlL eNoUgh To ensURe that thE sPecTrUm cOmes fROm most Of the ProjEcTeD AreA of the gALaXY. more SpEcIficAllY, wE only RetaINed GalaxieS which had UncERtaiNtIeS in the fIber and model mAgNitudes in tHe R baNd less THAn 0.01 mag, and Had an uncertainty in the coNCentratIon Index c lesS than 0.15. The fInaL samplE coNTains 33670 sPectra Of galAxIes COVerinG THe WhoLe Range of redSHIftS and gAlAxy tYpes preSent in the SDSS samplE. the Extension of thE obServED SpEctRA oF GalAxIEs ------------------------------------------------- THE Extension of the oBserved SDSs sPEcTra to Gaia wAVelEnGths was Made by uSing tHE 28885 syntheTic spectrA of the secOnD libRARy pRoduced at a Random grId of paramETers (TSAlMantzA et Al. [@tsalMaNtzA2]). In orDer to fINd tHe synThetic SpEctrum That iS iN best agrEement with each observed sPectruM, we fiRst Had to make The TWo lIbraries cOmpaTible. The maIn dIffErencEs bETween Them ARe The EFfectS of fOReground rEDdEniNG, NoIse and redshIFT, WhiCh are PreSEnt in tHe SDsS spectra but absenT In the synthetic

than 0.04 (Strauss et al. [@strauss ]). F orclo se r ga laxi es, small comp a ct a reas inside the galaxi es ar ec onsi d er ed as indepe n de n t ga la xi es, s o t he SD SSphotome try and th e r ad ius R90 aren ot represent ati ve of the wh ole galax y. We there for e exc ludedf rom ou r sampleal l galax i es with z $< $0.0 4. (iii) We keepo nl y galaxies with a dif fe r en c e be twe en the fib er andm odel ma g ni t u d esi n the r bandof less tha n 1mag. On e o f the m ain p ro b lem s with obse rved spectrais tha t very f e w of th e asso cia ted ast r op hy sic al par a me ter s ar e known. I nSDSSonly a r ough cl assi ficat ion is provid ed. Thi s cl assif icati on i sbased on th e wor kof Yip et al. ( [@yi p]), in w hic hgal ax ies w i th eCl ass $<$ -0.1 ar e consi d ere dt o be early type galaxi es w hi le galax ies wi t heC l ass$>$-0 .1 ar e la t e type s. A n ot her crit erionu se din SDSS f or gal ax y c las sific a tion is gi ven by S trate v a et al. ([@st r ateva]). Here , g a l ax i es h avi ng a concen trat i on i ndex Csma l ler t han o rl ar g er than 2.6 are exp ec ted to be l ate or earlytype galax i e s respect ivel y .T he distributio ns of both indi c es (C an d eCl ass) sho w that th e SDSS obs erv ati ons co v e ra wide rangeo f gal ax y types . U nfortun ate lynei the rof thesecriteria i spa rt icu larly accurate ,soth eycan b e usedonlyforaro u ghclassif i ca t i on o fou r sa mpl e. It c an a l sobe seen in figur e \ [ f2\] t ha t these two criteria a re also no tinpartic u l arly goo d agreement. For the f i nal sam ple we a pply addition alcriter iat o excl ude ob serva ti ons w ith l a r ge er ro rs, having s ele ctedga laxi es smal l enough to ensure tha t the spectru m c omes f ro m m o st ofth e pr o j ected area of t he galaxy. M o re specifica l ly, w e onlyretaine d gal a xies wh ich had u ncertaint ie s in t hefiber andmodel ma gnitudesi n the rbandles s than 0 .01 mag, and h a d a n unc ertain ty in th e con ce ntration index C less than 0.15 . Thefinal sa mple cont ain s 33 670 spect ra o f galaxies co ver ing t hew holerang e o f r e dshif ts a n d galaxyt yp esp r es ent in theS D S S s ample . T he ext ensi on of the observe d spectra of ga laxi e s -- --- - ---- -- -------------- --- -- - - -------- -- ------ The extensi on of th e obse rved S DSS spe c t ra to Gai a wa vel engths wa s m ad e by usi ng t h e 2888 5 sy nt heticspectr a oft h e second library prod u c ed at a r andom g rid ofp aram eters (Tsa lmantza etal. [@ tsal mantz a2]). I nordertofi nd the syn t hetic spe ctrum that i sin b est agree ment w ith e achob ser ved spect r u m, we f i rst had to m ak e th e two lib r aries co mpa t ible. T he ma i n diffe r en c e s betweenthe m are t he effects of f o re g round redde ning,noise a n d r ed shift,whi c h are pres ent in th e SD SS spe ctra but a bsen tinth e synthetic

than_0.04 (Strauss_et al. [@strauss]). For_closer galaxies,_small_compact areas_inside_the galaxies are_considered as independent_galaxies, so the SDSS_photometry and the_radius_R90 are not representative of the whole galaxy. We therefore excluded from our sample_all_galaxies with_z$<$0.04._(iii)_We keep only galaxies with_a difference between the fiber_and model_magnitudes in the r band of less than_1_mag. One of the_main problems with observed spectra is that very few_of the associated astrophysical parameters are_known. In SDSS_only_a_rough classification is provided._This classification is based on the_work of Yip et al. ([@yip]),_in which galaxies with eClass$<$-0.1 are considered_to be early type galaxies while_galaxies with eClass$>$-0.1 are late_types. Another_criterion used in SDSS for_galaxy classification is_given by_Strateva et al._([@strateva]). Here, galaxies having a concentration_index C smaller_than or larger than 2.6 are_expected_to be late_or_early_type galaxies_respectively. The distributions_of_both indices_(C_and eClass) show that the SDSS_observations_cover a wide range of galaxy types._Unfortunately neither of these_criteria_is particularly accurate, so_they can be used only_for a rough classification of our_sample. It_can also_be seen in figure \[f2\] that these two criteria are also_not in particularly good agreement. For the_final sample we apply_additional criteria_to_exclude observations with_large_errors, having_selected galaxies small enough to ensure that_the spectrum_comes from most of the projected_area of the galaxy._More_specifically, we only retained galaxies which_had uncertainties in the fiber and_model magnitudes in the r_band_less_than 0.01 mag, and had_an uncertainty in the concentration index_C less than_0.15. The final sample contains 33670 spectra_of_galaxies covering the whole range of_redshifts_and galaxy types present in the_SDSS_sample. The_extension of the observed spectra_of galaxies ------------------------------------------------- The extension of the observed_SDSS spectra to Gaia wavelengths was made by using_the 28885 synthetic_spectra of the second library_produced_at_a random grid of parameters (Tsalmantza et al. [@tsalmantza2]). In_order to_find the synthetic_spectrum that is in best agreement with each observed spectrum,_we first had to make the two_libraries compatible. The main differences between them are the effects of_foreground reddening, noise and redshift, which are present_in the SDSS spectra but absent in_the synthetic

discussed in the Proposed Approaches section, the two CNNs can be entirely separate, almost identical, or partially joint, leading to different performances in training as well as in application. We carry out extensive simulation studies, and show that the proposed neural model selector and parameter estimator can be properly trained, and the trained CNNs demonstrate excellent performance in test data and a real data application. The idea and proposed framework can be further extended to the entire SA process with the potential to change how SA is done in conventional data analysis and big data analytics. Related work ============ There exists an extensive statistical literature on model selection [@bozdogan1987model; @burnham2003model; @burnham2004multimodel]. Numerous model selection methods have been proposed. Some of these methods are not applicable to the setting we consider in this paper, while others, though applicable, may run into various difficulties; see discussions in Section \[sec:proposedApproach\] for details. To our best knowledge, there is no prior work about redefining the model selection problem as a machine learning classification problem and training CNN to learn and perform model selection with labeled simulated data. There also exist a variety of statistical methods for parameter estimation in the literature; see [@casella2002statistical; @huber1964robust; @norton2010identification]. Most statistical methods rely on full or partial knowledge of the model and are based on statistical principles. After conducting intensive literature search, we only found one paper [@xie2007estimation], in which the authors proposed to use artificial neural networks and simulated data to construct estimates for parameters of a stochastic differential equation. However, the idea of using CNNs and simulated data to automate parameter estimation and model selection and bring AI to the general SA process appears to be novel to our best knowledge. Proposed approach {#sec:proposedApproach} ================= As discussed in the Introduction, we first reformulate model building and parameter estimation as a machine learning problem. Suppose $\mathcal{M}=\{M_k: 1\le k \le K\}$ be a collection of $K$ prespecified models/distributions. Let $f(y|\theta_k, M_k)$ be the density function of model $M_k$, where $\theta_k\in \Theta_k$ is the scalar parameter of the density function. Assume that we have a random sample from one of the models, which is $\{y_j\}_{1\

discussed in the Proposed Approaches section, the two CNNs can be entirely separate, about identical, or partially joint, go to different performances in education as well as in lotion. We carry out extensive pretense study, and show that the proposed neural model picker and parameter estimator can be properly trained, and the coach CNNs demonstrate excellent performance in test data and a real datum application. The idea and proposed model can be further extended to the entire SA process with the potential to switch how SA is done in conventional data analysis and big datum analytics. Related work = = = = = = = = = = = = There exists an extensive statistical literature on model selection [ @bozdogan1987model; @burnham2003model; @burnham2004multimodel ]. Numerous exemplar selection methods have been proposed. Some of these methods are not applicable to the setting we consider in this paper, while others, though applicable, may run into various difficulties; attend discussions in Section \[sec: proposedApproach\ ] for details. To our dear cognition, there is no prior work about redefining the model selection problem as a machine determine classification problem and training CNN to learn and perform mannequin selection with labeled simulated data. There besides exist a variety of statistical method acting for argument estimate in the literature; see [ @casella2002statistical; @huber1964robust; @norton2010identification ]. Most statistical methods rely on wide or partial knowledge of the model and are free-base on statistical principles. After conducting intensive literature search, we only found one newspaper [ @xie2007estimation ], in which the authors aim to use artificial nervous networks and simulated data to construct estimate for parameters of a stochastic differential equality. However, the idea of using CNNs and simulated datum to automate parameter estimation and model selection and bring AI to the general SA process appears to be novel to our best knowledge. Proposed access { # sec: proposedApproach } = = = = = = = = = = = = = = = = = As discussed in the Introduction, we foremost reformulate model building and parameter estimation as a machine learning trouble. Suppose $ \mathcal{M}=\{M_k: 1\le k \le K\}$ be a collection of $ K$ prespecified models / distributions. Let $ f(y|\theta_k, M_k)$ be the density affair of exemplar $ M_k$, where $ \theta_k\in \Theta_k$ is the scalar parameter of the density function. Assume that we have a random sample from one of the model, which is $ \{y_j\}_{1\

didcussed in the Proposed Xpproaches sectnin, the two CHNs can ce entirely separate, almost mdenrical, or partially joint, lexding to fifferenr pecformances in trejning as well ws iu epplication. We garry out eftensive simuldtkou studies, and show that the proposed neural mldel selector wnd karwmetsg tstimator can be properly trained, and tht trained CNNs dempnstrate excellent performwnce in test data and w real data appjucation. The kdea and pgmposed frajework can be further extended go thz entire SA peocfvs with the potegtial to chakbe how SA is cone in convenbionan dqta analysis and big vata analytics. Relateq work ============ These exists an extensuvw stajistiwal uutefathrx oh modep sxlection [@boadogan1987model; @burnham2003model; @burnhsm2004ilktimodel]. Numsrous ioqel selection methods have been proposeg. Skme of these methods arw not applicable to tje settind we consider in this paper, while others, though a[plicebue, nan ruv ijto various difficulties; see discussions in Sqdtook \[sec:proposedAppvoach\] for details. Yo oit best knowledee, thexs js no prior work ahout reqefinung the mjdel selection problem as a macyine learniny coassification probpem and tranning VNN tp learn and perform modzl selsction with labeled akmulated data. Thefe slvo exist a variety of statystical mxthodx for pxramgter eseimation ij the literature; see [@casflla2002sjatisthcal; @huber1964gobust; @norton2010identification]. Mosv statistical mathmds rely on fmll or partial hnowledge of tke model and xre based kn statmstical prinsiples. After wlnducting invensive lyterqturw searcf, we only found one papeg [@rie2007estimarion], in which the euthuds proposed to brt qrtificial neutal neewlrls dnd simulateg daga gp conrtruct tftlmages gor parameters of a vtocgastic differentiak cquation. Yowever, ehe idea of uxing CNNs and simupated dava to sutjmate parameter estimation and model sepecbion and brind AI to the genexal SA process appears to be novel to ouc best knowledge. Proposgd approach {#sec:proposgdAkproach} ================= As disrussed in the Ittroduction, we first reformulate modek building and paramefer esdimatlon as a machine learning problem. Suppose $\mathcal{M}=\{M_k: 1\le k \le K\}$ be a collextion of $K$ prespecjfiec modvls/biscributijns. Pxt $f(y|\theta_k, M_k)$ be the density function of model $O_k$, where $\tketa_k\in \Theta_k$ is the scalar lafameter of thg density function. Assume fhat we nave a random sample from one og the models, which is $\{y_j\}_{1\

discussed in the Proposed Approaches section, the can entirely separate, identical, or partially in as well as application. We carry extensive simulation studies, and show that proposed neural model selector and parameter estimator can be properly trained, and the CNNs demonstrate excellent performance in test data and a real data application. The and framework be extended to the entire SA process with the potential to change how SA is done in data analysis and big data analytics. Related work There exists an extensive literature on model selection [@bozdogan1987model; @burnham2004multimodel]. model selection have proposed. of these methods not applicable to the setting we consider in this paper, while others, though applicable, may run into difficulties; see Section \[sec:proposedApproach\] details. our knowledge, there is work about redefining the model selection machine learning classification problem and training CNN to and perform selection with labeled simulated data. There exist a variety of statistical methods for parameter in the literature; see [@casella2002statistical; @huber1964robust; @norton2010identification]. Most statistical methods rely on full or partial the model and are on statistical principles. conducting literature we found one [@xie2007estimation], in which the authors proposed to use artificial neural networks simulated data to construct estimates for parameters of a stochastic However, idea of using and simulated data to parameter and model selection and to general to novel our best knowledge. Proposed {#sec:proposedApproach} ================= As discussed in Introduction, we first reformulate as a machine learning problem. Suppose $\mathcal{M}=\{M_k: 1\le \le K\}$ be a collection of $K$ models/distributions. Let $f(y|\theta_k, M_k)$ be the density function of model $M_k$, where \Theta_k$ is parameter of the density function. Assume that we a random sample from of the models, which is $\{y_j\}_{1\

discussed in the Proposed AppRoaches secTion, tHe tWo CnNS can Be enTirely separate, ALmosT identical, or partially jOint, lEaDIng tO DiFfereNt perfoRMaNCEs iN tRaIniNg AS wEll as In aPplicatIon. We carry Out ExTensive simulATiOn studies, aNd sHow that the prOpoSed neuRaL moDEl selEctOr and ParameTEr estiMator can bE pROperly TRained, aND ThE traIned CNNs demonstraTE eXCellent performAnce in TeST dATA anD a rEal data appLiCatioN. the idea ANd PROPosED framework can Be further exTEndEd to thE eNtiRE SA proCess wItH The Potential to ChanGe how SA is Done in COnventiONal data AnalysIs aNd bIg daTA aNaLytIcS. relATeD woRK ============ ThEre existS aN eXtensIve sTATISticAl lIterAture On model selectIon [@BozdOGan1987Model; @BurnhAm2003moDeL; @burnHam2004mulTimodEl]. numerous model seLectIon methodS haVe BeeN pRoposED. Some oF thEse Methods Are not aPPliCaBLE To The setting we considEr IN ThIs paper, wHile otHErS, tHOugh applIcAblE, may RUN into VariOUs DifficulTies; seE DiScUssions In sectioN \[sEc:pRopOsedAPProaCh\] for dEtails. To Our beST knowledge, therE Is no prior work ABoUT ReDEfinIng The model selEctiON proBlem AS a MacHIne leArninG cLAsSIfication problem and TrAining cNN to Learn and perfoRm model selECTIon with lAbelED sIMulated data. TheRe alsO exist a varIEty of staTistiCal methoDs for paraMETer estimAtiOn iN thE liTERaTure; see [@casellA2002STatiStIcal; @hubEr1964rObust; @noRtoN2010idEntIfiCaTion]. Most sTatisticAl MeThOdS reLy on fULl or partIaL knOwLedGe of tHE model And arE basEd On STatIstical PRiNCIpleS. AFtEr coNduCtIng inTensIVe lIteratuRe search, wE onLY fouNd OnE paper [@xIe2007estimation], iN wHich the autHoRs pRoposeD TO use artiFicial neural networks and SImulateD daTa to cOnstRuct estimAteS for paRamETers of A stochAstic DiFfeRENtial EQUaTioN. HOwever, the iDEA of Using cNns anD simulaTed data to automate pARamEter estimatioN anD modEL SeLecTIoN And BrINg Ai TO the general SA prOcess appeaRs TO bE novel to ouR BesT kNowledgE. ProposEd appROach {#sec:ProposedAPproach} ================= As DiScusSED in The IntroduCtion, we fIrst reforMUlate MOdEl buiLdiNg and pArAmeTer esTimatiON as A machIne leaRnIng proBlem. SUpPose $\mathCal{M}=\{M_k: 1\le k \le K\}$ be a collectiOn of $K$ pRespeCifIed models/DisTRibUtions. Let $F(y|\thEta_k, M_k)$ be thE deNsiTy funCtiON of moDel $M_K$, WhEre $\THeta_k\In \ThETa_k$ is the sCAlAr pARAmEter of the deNSITy fUnctiOn. ASSume thAt we Have a random sample FRom one of the modEls, wHICh iS $\{y_j\}_{1\

discussed in the Proposed Approache s sec tio n,th e tw o CN Ns can be enti r elyseparate, almost ident ical, o r par t ia lly j oint, l e ad i n g t odi ffe re n tperfo rma nces in trainingaswe ll as in app l ic ation. Wecar ry out exten siv e simu la tio n stud ies , and showt hat th e propose dn euralm odel se l e ct or a nd parameter esti m at o r can be prope rly tr ai n ed , and th e trainedCN Ns de m onstrat e e x c e lle n t performance in test da t a a nd a r ea l d a ta app licat io n . T he idea and pro posed fra mework can bef urtherextend edtothee nt ir e S Ap roc e ss wi t h t he poten ti al to c hang e h o w SA is don e inconventionaldat a an a lys is an d big dat aanaly tics. Rela te d work ======== ==== There e xis ts an e xtens i ve sta tis tic al lite ratureo n m od e l se lection [@bozdogan 19 8 7 mo del; @bu rnham2 0 03 mo d el; @bur nh am2 004m u l timod el]. Nu merous m odel s e le ct ion met ho ds hav ebee n p ropos e d. S ome of these m ethod s are not appli c able to the s e tt i n gw e co nsi der in this pap e r, w hile ot her s , tho ugh a pp l ic a ble, may run into v ar ious d iffic ulties; see d iscussions i n Section \[s e c: p roposedApproac h\] f or details . To ourbestknowledg e, therei s no prio r w ork ab out r ed efining the m o d el s el ectionpro blem as amac hin e l ea rning cla ssificat io npr ob lem andt rainingCN N t olea rn an d perfo rm mo delse le c tio n withl ab e l ed s im ul ated da ta . Th erea lso exista variety of stat is ti cal met hods for para me ter estima ti onin the l iteratur e; see [@casella2002sta t istical ; @ huber 1964 robust; @ nor ton201 0id e ntific ation] . Mos tsta t i stica l me tho ds rely on f u l l o r par ti al k nowledg e of the model and are based on sta tis tica l pr inc i pl e s.Af t erc o nducting intens ive litera tu r esearch, we onl yfound o ne pape r [@x i e2007es timation] , in whic hthea u tho rs propose d to use artifici a l neu r al netw ork s andsi mul ateddata t o co nstru ct est im ates f or pa ra meters o f a stochastic differen tial e quati on. However, th e id ea of usi ng C NNs and si mul ate d dat a t o auto mate pa ram e ter e stim a tion andm od els e le ction and b r i n g A I tothe genera l SA process appearst o be novel toourb e stkno w ledg e. Proposed app roa ch { #sec:pro po sedApproach } ====== == = ===== === A s disc ussed i n th e Intro duct ion , we firs t r ef o rmulate m od e l buil ding a nd par ameter esti m a tion as a machin e lea r n ing p r obl em. S up pose $\ m athc al{M}=\{M_ k: 1\le k \ le K\} $ be a co llectio nof $K$ pr es pecified m o dels/dist ribut ions. L et $f( y|\ theta_ k, M _ k )$ be the d ens ity funct i o no fmo d el$M_k $, wh er e $\ theta_k\i n \Theta_ k$i s the s ca lar p aramet e ro f the densi tyfunct i o n. Assumet hatw eh ave a rando m samp le from one o f the m ode l s , which i s $\{y_j\ } _{1 \

discussed_in the_Proposed Approaches section, the_two CNNs_can_be entirely_separate,_almost identical, or_partially joint, leading_to different performances in_training as well_as_in application. We carry out extensive simulation studies, and show that the proposed neural_model_selector and_parameter_estimator_can be properly trained, and_the trained CNNs demonstrate excellent_performance in_test data and a real data application. The_idea_and proposed framework_can be further extended to the entire SA process_with the potential to change how_SA is done_in_conventional_data analysis and big_data analytics. Related work ============ There exists an extensive_statistical literature on model selection [@bozdogan1987model;_@burnham2003model; @burnham2004multimodel]. Numerous model selection methods have_been proposed. Some of these methods_are not applicable to the_setting we_consider in this paper, while_others, though applicable,_may run_into various difficulties;_see discussions in Section \[sec:proposedApproach\] for_details. To our_best knowledge, there is no prior_work_about redefining the_model_selection_problem as_a machine learning_classification_problem and_training_CNN to learn and perform model_selection_with labeled simulated data. There also exist a_variety of statistical methods_for_parameter estimation in the_literature; see [@casella2002statistical; @huber1964robust; @norton2010identification]._Most statistical methods rely on full_or partial_knowledge of_the model and are based on statistical principles. After conducting intensive_literature search, we only found one_paper [@xie2007estimation], in which_the authors_proposed_to use artificial_neural_networks and_simulated data to construct estimates for parameters_of a_stochastic differential equation. However, the idea_of using CNNs and_simulated_data to automate parameter estimation and_model selection and bring AI to_the general SA process appears_to_be_novel to our best knowledge. Proposed_approach {#sec:proposedApproach} ================= As discussed in the Introduction,_we first reformulate_model building and parameter estimation as a_machine_learning problem. Suppose $\mathcal{M}=\{M_k: 1\le k_\le_K\}$ be a collection of $K$_prespecified_models/distributions._Let $f(y|\theta_k, M_k)$ be the_density function of model $M_k$, where_$\theta_k\in \Theta_k$ is the scalar parameter of the density_function. Assume that_we have a random sample_from_one_of the models, which is $\{y_j\}_{1\

bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \,{ f ^{\mathrm{in}}}(x,v)\;{ \mathrm{d}(x,v)}},\;\;t \in {{{\bf}R}}_+.$$ Notice that the term $- { \mathrm{div}_v}\{ { f ^\varepsilon}H \star { f ^\varepsilon}\}$ balances the momentum $$\int _{{{{\bf}R}}^{2d}}{\!\! v { \mathrm{div}_v}\{ { f ^\varepsilon}H \star { f ^\varepsilon}\}\;{ \mathrm{d}(x,v)}} = \int _{{{{\bf}R}}^{4d}}{\!\! h(x-x^{\prime}) (v ^{\prime}-v) { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime)}{ \mathrm{d}(x,v)}= 0$$ and decreases the kinetic energy $$\begin{aligned} \int _{{{{\bf}R}}^{2d}}{\!\!\!|v|^2 { \mathrm{div}_v}\{ { f ^\varepsilon}H \star { f ^\varepsilon}\}\;{ \mathrm{d}(x,v)}} &= 2\int _{{{{\bf}R}}^{4d}}{{\!\!\!\!h(x-x^{\prime}) (v ^{\prime}-v) \cdot v { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime)}{ \mathrm{d}(x,v)}} \\ & = - \int _{{{{\bf}R}}^{4d}}{{\!\!h(x-x^{\prime}) |v - v ^{\prime}|^2 { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime)}{ \mathrm{d}(x,v)}}.\end{aligned}$$ In particular, as $|v|^2 { f ^{\mathrm{in}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$, then the

bf}R}}^d \times { { { \bf}R}}^d}\!\!\! \, { f ^{\mathrm{in}}}(x, v)\; { \mathrm{d}(x, v)}},\;\;t \in { { { \bf}R}}_+.$$ Notice that the term $ - { \mathrm{div}_v}\ { { f ^\varepsilon}H \star { f ^\varepsilon}\}$ balances the momentum $ $ \int _ { { { { \bf}R}}^{2d}}{\!\! v { \mathrm{div}_v}\ { { f ^\varepsilon}H \star { f ^\varepsilon}\}\; { \mathrm{d}(x, v) } } = \int _ { { { { \bf}R}}^{4d}}{\!\! h(x - x^{\prime }) (v ^{\prime}-v) { f ^\varepsilon}(t, x^\prime, v^\prime) { f ^\varepsilon}(t, x, v)\; { \mathrm{d}}(x^\prime, v^\prime) } { \mathrm{d}(x, v)}= 0$$ and decreases the kinetic department of energy $ $ \begin{aligned } \int _ { { { { \bf}R}}^{2d}}{\!\!\!|v|^2 { \mathrm{div}_v}\ { { f ^\varepsilon}H \star { f ^\varepsilon}\}\; { \mathrm{d}(x, v) } } & = 2\int _ { { { { \bf}R}}^{4d}}{{\!\!\!\!h(x - x^{\prime }) (v ^{\prime}-v) \cdot v { f ^\varepsilon}(t, x^\prime, v^\prime) { f ^\varepsilon}(t, x, v)\; { \mathrm{d}}(x^\prime, v^\prime) } { \mathrm{d}(x, v) } } \\ & = - \int _ { { { { \bf}R}}^{4d}}{{\!\!h(x - x^{\prime }) |v - five ^{\prime}|^2 { f ^\varepsilon}(t, x^\prime, v^\prime) { f ^\varepsilon}(t, x, v)\; { \mathrm{d}}(x^\prime, v^\prime) } { \mathrm{d}(x, v)}}.\end{aligned}$$ In particular, as $ |v|^2 { f ^{\mathrm{in}}}\in { { \cal M}_b ^+ ({ { { \bf}R}}^d \times { { { \bf}R}}^d)}{}$, then the

bf}R}}^f \times {{{\bf}R}}^d}\!\!\! \,{ f ^{\mathrm{in}}}(x,y)\;{ \mathrm{d}(x,v)}},\;\;t \in {{{\bf}T}}_+.$$ Botice that fhe term $- { \mathrm{div}_v}\{ { f ^\varepsilon}H \svar { d ^\vartisilon}\}$ balances the mumentum $$\ijt _{{{{\bf}R}}^{2d}}{\!\! c { \mauhrm{div}_v}\{ { f ^\varepsmmon}H \star { f ^\vzvepsinin}\}\;{ \mathrm{d}(x,v)}} = \ikt _{{{{\bf}R}}^{4d}}{\!\! h(x-x^{\psime}) (v ^{\prime}-v) { x ^\xaxepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,d)\;{ \mathrm{c}}(x^\orime, v^\prime)}{ \majhrm{d}(q,v)}= 0$$ ans decreases the kinetic energy $$\begjn{aligntd} \int _{{{{\bf}R}}^{2d}}{\!\!\!|v|^2 { \mathrm{civ}_v}\{ { f ^\varepsilon}H \star { f ^\gareosilon}\}\;{ \mathrm{d}(x,v)}} &= 2\ijt _{{{{\bf}R}}^{4d}}{{\!\!\!\!h(x-x^{\ptjme}) (c ^{\prime}-v) \cdog v { f ^\vareksnlon}(t, x^\primg, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\pfime, r^\prime)}{ \mathrn{d}(z,v)}} \\ & = - \int _{{{{\bf}R}}^{4v}}{{\!\!h(x-x^{\prpme}) |v - v ^{\primc}|^2 { g ^\vare[silon}(t, x^\prime, v^\prime){ n ^\varxpsioon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\pcime)}{ \mathrm{d}(x,v)}}.\end{aligged}$$ In pastncular, as $|v|^2 { f ^{\mathrm{ib}}}\ib { {\cal M}_b ^+ ({{{\bf}R}}^a \tioes {{{\bh}R}}^d)}{}$, then hhe

bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \,{ f ^{\mathrm{in}}}(x,v)\;{ \mathrm{d}(x,v)}},\;\;t Notice the term { \mathrm{div}_v}\{ { ^\varepsilon}\}$ the momentum $$\int v { \mathrm{div}_v}\{ f ^\varepsilon}H \star { f ^\varepsilon}\}\;{ = \int _{{{{\bf}R}}^{4d}}{\!\! h(x-x^{\prime}) (v ^{\prime}-v) { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ v^\prime)}{ \mathrm{d}(x,v)}= 0$$ and decreases the kinetic energy $$\begin{aligned} \int _{{{{\bf}R}}^{2d}}{\!\!\!|v|^2 { \mathrm{div}_v}\{ f \star f \mathrm{d}(x,v)}} &= 2\int _{{{{\bf}R}}^{4d}}{{\!\!\!\!h(x-x^{\prime}) (v ^{\prime}-v) \cdot v { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, \mathrm{d}(x,v)}} \\ & = - \int _{{{{\bf}R}}^{4d}}{{\!\!h(x-x^{\prime}) |v v ^{\prime}|^2 { f x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, \mathrm{d}(x,v)}}.\end{aligned}$$ particular, as { ^{\mathrm{in}}}\in {\cal M}_b ^+ \times {{{\bf}R}}^d)}{}$, then the

bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \,{ f ^{\mathrm{in}}}(x,v)\;{ \mAthrm{d}(x,v)}},\;\;t \iN {{{\bf}R}}_+.$$ NOtiCe tHaT the Term $- { \Mathrm{div}_v}\{ { f ^\varEPsilOn}H \star { f ^\varepsilon}\}$ balaNces tHe MOmenTUm $$\Int _{{{{\bf}r}}^{2d}}{\!\! v { \mathRM{dIV}_V}\{ { f ^\vArEpSilOn}h \StAr { f ^\vaRepSilon}\}\;{ \maThrm{d}(x,v)}} = \int _{{{{\Bf}R}}^{4D}}{\!\! h(X-x^{\prime}) (v ^{\primE}-V) { f ^\Varepsilon}(T, x^\pRime, v^\prime){ f ^\vArePsilon}(T,x,V)\;{ \maTHrm{d}}(x^\PriMe, v^\prIme)}{ \matHRm{d}(x,v)}= 0$$ aNd decreasEs THe kineTIc energY $$\BEgIn{alIgned} \int _{{{{\bf}R}}^{2d}}{\!\!\!|v|^2 { \mathRM{dIV}_v}\{ { f ^\varepsilon}H \Star { f ^\vArEPsILOn}\}\;{ \mAthRm{d}(x,v)}} &= 2\int _{{{{\bf}r}}^{4d}}{{\!\!\!\!H(x-x^{\prIMe}) (v ^{\primE}-V) \cDOT V { f ^\vARepsilon}(t, x^\priMe, v^\prime){ f ^\vaREpsIlon}(t,x,V)\;{ \mAthRM{d}}(x^\priMe, v^\prImE)}{ \MatHrm{d}(x,v)}} \\ & = - \int _{{{{\bf}r}}^{4d}}{{\!\!h(x-X^{\prime}) |v - v ^{\pRime}|^2 { f ^\vARepsiloN}(T, x^\prime, V^\prime){ F ^\vaRepSiloN}(T,x,V)\;{ \mAthRm{D}}(X^\prIMe, V^\prIMe)}{ \mAthrm{d}(x,v)}}.\EnD{aLigneD}$$ In pARTICulaR, as $|V|^2 { f ^{\maThrm{iN}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \tImeS {{{\bf}R}}^D)}{}$, TheN the

bf}R}}^d \times {{{\bf}R}} ^d}\!\!\!\,{ f ^{ \ma th rm{i n}}} (x,v)\;{ \math r m{d} (x,v)}},\;\;t \in {{{\ bf}R} }_ + .$$N ot ice t hat the te r m $- { \ mat hr m {d iv}_v }\{ { f ^\ varepsilon }H\s tar { f ^\va r ep silon}\}$bal ances the mo men tum $$ \i nt_ {{{{\ bf} R}}^{ 2d}}{\ ! \! v { \mathrm{ di v }_v}\{ { f ^\v a r ep silo n}H \star { f ^\v a re p silon}\}\;{ \m athrm{ d} ( x, v ) }}= \ int _{{{{\ bf }R}}^ { 4d}}{\! \ !h ( x -x^ { \prime}) (v ^ {\prime}-v) { f ^\var ep sil o n}(t,x^\pr im e , v ^\prime){ f ^\v arepsilon }(t,x, v )\;{ \m a thrm{d} }(x^\p rim e,v^\p r im e) }{\m a thr m {d }(x , v)} = 0$$ an dde creas es t h e k inet icener gy $$ \begin{aligne d}\int _{{ {{\bf }R}}^ {2d} }{ \!\!\ !|v|^2 { \m at hrm{div}_v}\{ { f ^ \varepsil on} H\st ar { f^ \varep sil on} \}\;{ \ mathrm{ d }(x ,v ) } } & = 2\int _{{{{\bf}R }} ^ { 4d }}{{\!\! \!\!h( x -x ^{ \ prime})(v ^{ \pri m e }-v)\cdo t v { f ^\v arepsi l on }( t, x^\p ri me, v^ \p rim e){ f ^\ v arep silon} (t,x,v)\ ;{ \m a thrm{d}}(x^\pr i me, v^\prime) } {\ m at h rm{d }(x ,v)}} \\ & = - \in t _{ { {{ \bf } R}}^{ 4d}}{ {\ ! \! h (x-x^{\prime}) |v - v ^{\pr ime}| ^2 { f ^\vare psilon}(t, x ^ \prime,v^\p r im e ){ f ^\varepsi lon}( t,x,v)\;{\ mathrm{d }}(x^ \prime,v^\prime) } { \mathrm {d} (x, v)} }.\ e n d{ aligned}$$ In p arti cu lar, as $| v|^2 {f ^ {\m ath rm{ in }}}\in {{\cal M} _b ^ +({ {{\ bf}R} } ^d \time s{{{ \b f}R }}^d) } {}$, t hen t he

bf}R}}^d \times_{{{\bf}R}}^d}\!\!\! \,{ f_^{\mathrm{in}}}(x,v)\;{ \mathrm{d}(x,v)}},\;\;t \in {{{\bf}R}}_+.$$ Notice that_the term_$-_{ \mathrm{div}_v}\{ { f_^\varepsilon}H_\star { f ^\varepsilon}\}$_balances the momentum_$$\int _{{{{\bf}R}}^{2d}}{\!\! v { \mathrm{div}_v}\{_{ f ^\varepsilon}H \star_{ f_^\varepsilon}\}\;{ \mathrm{d}(x,v)}} = \int _{{{{\bf}R}}^{4d}}{\!\! h(x-x^{\prime}) (v ^{\prime}-v) { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime)}{ \mathrm{d}(x,v)}= 0$$ and decreases_the_kinetic energy_$$\begin{aligned} \int__{{{{\bf}R}}^{2d}}{\!\!\!|v|^2_{ \mathrm{div}_v}\{ { f ^\varepsilon}H \star { f_^\varepsilon}\}\;{ \mathrm{d}(x,v)}} &= 2\int _{{{{\bf}R}}^{4d}}{{\!\!\!\!h(x-x^{\prime}) (v_^{\prime}-v) \cdot_v { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime)}{ \mathrm{d}(x,v)}} _\\ &_= -_\int _{{{{\bf}R}}^{4d}}{{\!\!h(x-x^{\prime}) |v - v ^{\prime}|^2 { f ^\varepsilon}(t, x^\prime, v^\prime){ f_^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime)}{ \mathrm{d}(x,v)}}.\end{aligned}$$ In particular, as $|v|^2 { f_^{\mathrm{in}}}\in { {\cal M}_b_^+_({{{\bf}R}}^d_\times {{{\bf}R}}^d)}{}$, then the

-\frac{1}{2}\int_0^tg\big(S(u-b)\big)^2du\right\}$$ a.s. for $t\in[0,T]$. Hence we have $$\label{tShW} {\widetilde{S}}(t)=\varphi(0) \exp\left\{\int_0^tg\big(S(u-b)\big)\,d{\widehat{W}}(u) -\frac{1}{2}\int_0^tg\big(S(u-b)\big)^2du\right\}$$ for $t\in[0,T]$. By examining the definitions of $\tilde S, {\widehat{W}}, {\widehat{S}}$ and equation (6), it is not hard to see that for $t \geq 0$, ${\mathcal{F}}_t^S={\mathcal{F}}_t^{{\widetilde{S}}}={\mathcal{F}}_t^{{\widehat{W}}}={\mathcal{F}}_t^{W}$, the $\sigma$-algebras generated by $\{S (u): u \leq t \}$, $\{{\widetilde{S}}(u): u \leq t \}$, $\{{\widehat{W}}(u): u \leq t \}$, $\{W(u): u \leq t \}$, respectively. (Clearly, ${\mathcal{F}}_t^{W} \subseteq {\mathcal{F}}_t$.) Now, let $X$ be a contingent claim, viz. an integrable non-negative ${\mathcal{F}}_T^S$-measurable random variable. Consider the $Q$-martingale $$M(t):=E_Q(e^{-rT}X\,|\,{\mathcal{F}}_t^S) =E_Q(e^{-rT}X\,|\,{\mathcal{F}}_t^{{\widehat{W}}}),\qquad t\in[0,T].$$ By the martingale representation theorem (e.g., Theorem 9.4 in \[K.K\]), there exists an $({\mathcal{F}}_t^{{\widehat{W}}})$-predictable process $h_0(t)$, $t\in[0,T]$, such that $$\int_0^T h_0(u)^2\,du<\infty\qquad a.s.,$$ and $$M(t)=E_Q(e^{-rT}X)+\int_0^th_0(u)\,d{\widehat{W}}(u

-\frac{1}{2}\int_0^tg\big(S(u - b)\big)^2du\right\}$$ a.s. for $ t\in[0,T]$. Hence we have $ $ \label{tShW } { \widetilde{S}}(t)=\varphi(0) \exp\left\{\int_0^tg\big(S(u - b)\big)\,d{\widehat{W}}(u) -\frac{1}{2}\int_0^tg\big(S(u - b)\big)^2du\right\}$$ for $ t\in[0,T]$. By examining the definitions of $ \tilde S, { \widehat{W } }, { \widehat{S}}$ and equality (6), it is not unvoiced to see that for $ t \geq 0 $, $ { \mathcal{F}}_t^S={\mathcal{F}}_t^{{\widetilde{S}}}={\mathcal{F}}_t^{{\widehat{W}}}={\mathcal{F}}_t^{W}$, the $ \sigma$-algebras generate by $ \{S (uranium ): u \leq t \}$, $ \{{\widetilde{S}}(u ): u \leq t \}$, $ \{{\widehat{W}}(u ): u \leq t \}$, $ \{W(u ): u \leq t \}$, respectively. (Clearly, $ { \mathcal{F}}_t^{W } \subseteq { \mathcal{F}}_t$.) immediately, get $ X$ be a contingent claim, viz. an integrable non - negative $ { \mathcal{F}}_T^S$-measurable random variable. Consider the $ Q$-martingale $ $ M(t):=E_Q(e^{-rT}X\,|\,{\mathcal{F}}_t^S) = E_Q(e^{-rT}X\,|\,{\mathcal{F}}_t^{{\widehat{W}}}),\qquad t\in[0,T].$$ By the dolphin striker representation theorem (e.g., Theorem 9.4 in \[K.K\ ]), there exists an $ ({ \mathcal{F}}_t^{{\widehat{W}}})$-predictable process $ h_0(t)$, $ t\in[0,T]$, such that $ $ \int_0^T h_0(u)^2\,du<\infty\qquad a.s. ,$$ and $ $ M(t)=E_Q(e^{-rT}X)+\int_0^th_0(u)\,d{\widehat{W}}(u

-\fraf{1}{2}\int_0^tg\big(S(u-b)\big)^2du\right\}$$ a.r. for $t\in[0,T]$. Hencg qe havx $$\label{fShW} {\wiaetilde{S}}(t)=\varphi(0) \exp\lxft\{\ibt_0^tg\bug(S(u-b)\big)\,d{\widehat{W}}(u) -\feac{1}{2}\iit_0^tg\big(S(u-b)\big)^2du\rmfht\}$$ for $t\in[0,T]$. Gn exakmning the definltions of $\thlde S, {\widehat{F}}, {\dibehat{S}}$ and equation (6), it is not hard eo see yhwt for $t \geq 0$, ${\iathbaj{F}}_t^S={\jathcal{F}}_t^{{\widetilde{S}}}={\mathcal{F}}_t^{{\widehaf{W}}}={\mathcel{F}}_t^{W}$, the $\sigma$-slgebras generated by $\{S (u): k \lee t \}$, $\{{\widetilde{S}}(u): u \peq t \}$, $\{{\wideyat{W}}(t): u \leq t \}$, $\{W(u): u \leq t \}$, gzspectively. (Clearly, ${\mathcal{F}}_t^{W} \subseteq {\maghcal{Y}}_t$.) Now, let $Z$ ve w contingent claii, viz. an intcbrable non-negstive ${\mathcal{F}}_B^S$-meavurqble random variable. Ronsider the $Q$-martindale $$M(t):=E_Q(a^{-rC}X\,|\,{\mathcal{F}}_t^S) =E_W(e^{-eT}X\,|\,{\majhcal{X}}_t^{{\wiawhag{W}}}),\qsued f\in[0,T].$$ Bj tie martingame represenration theorem (e.g., Tnejgrm 9.4 in \[K.K\]), thsre exyses an $({\mathcal{F}}_t^{{\widehat{W}}})$-predictable proctss $h_0(f)$, $t\in[0,T]$, such that $$\int_0^T h_0(y)^2\,du<\infty\qquad a.s.,$$ and $$L(t)=E_Q(e^{-rT}X)+\igt_0^th_0(u)\,d{\widehat{W}}(u

-\frac{1}{2}\int_0^tg\big(S(u-b)\big)^2du\right\}$$ a.s. for $t\in[0,T]$. Hence we have \exp\left\{\int_0^tg\big(S(u-b)\big)\,d{\widehat{W}}(u) for $t\in[0,T]$. examining the definitions and (6), it is hard to see for $t \geq 0$, ${\mathcal{F}}_t^S={\mathcal{F}}_t^{{\widetilde{S}}}={\mathcal{F}}_t^{{\widehat{W}}}={\mathcal{F}}_t^{W}$, the generated by $\{S (u): u \leq t \}$, $\{{\widetilde{S}}(u): u \leq t \}$, u \leq t \}$, $\{W(u): u \leq t \}$, respectively. (Clearly, ${\mathcal{F}}_t^{W} \subseteq Now, $X$ a claim, viz. an integrable non-negative ${\mathcal{F}}_T^S$-measurable random variable. Consider the $Q$-martingale $$M(t):=E_Q(e^{-rT}X\,|\,{\mathcal{F}}_t^S) =E_Q(e^{-rT}X\,|\,{\mathcal{F}}_t^{{\widehat{W}}}),\qquad t\in[0,T].$$ By the representation theorem (e.g., Theorem 9.4 in \[K.K\]), there an $({\mathcal{F}}_t^{{\widehat{W}}})$-predictable process $h_0(t)$, such that $$\int_0^T h_0(u)^2\,du<\infty\qquad a.s.,$$ $$M(t)=E_Q(e^{-rT}X)+\int_0^th_0(u)\,d{\widehat{W}}(u

-\frac{1}{2}\int_0^tg\big(S(u-b)\big)^2du\right\}$$ A.s. for $t\in[0,T]$. HEnce wE haVe $$\lAbEl{tSHW} {\wiDetilde{S}}(t)=\varphI(0) \Exp\lEft\{\int_0^tg\big(S(u-b)\big)\,d{\wideHat{W}}(u) -\FrAC{1}{2}\int_0^TG\bIg(S(u-b)\Big)^2du\riGHt\}$$ FOR $t\iN[0,T]$. by ExaMiNInG the dEfiNitions Of $\tilde S, {\wiDehAt{w}}, {\widehat{S}}$ and EQuAtion (6), it is nOt hArd to see that For $T \geq 0$, ${\maThCal{f}}_T^S={\matHcaL{F}}_t^{{\wiDetildE{s}}}={\mathcAl{F}}_t^{{\widehAt{w}}}={\MathcaL{f}}_t^{W}$, the $\sIGMa$-AlgeBras generated by $\{S (u): U \LeQ T \}$, $\{{\widetilde{S}}(u): u \lEq t \}$, $\{{\widEhAT{W}}(U): U \Leq T \}$, $\{W(u): U \leq t \}$, respeCtIvely. (cLearly, ${\mAThCAL{f}}_t^{W} \SUbseteq {\mathcaL{F}}_t$.) Now, let $X$ bE A coNtingeNt ClaIM, viz. an IntegRaBLe nOn-negative ${\mAthcAl{F}}_T^S$-measUrable RAndom vaRIable. CoNsider The $q$-maRtinGAlE $$M(T):=E_Q(E^{-rt}x\,|\,{\maTHcAl{F}}_T^s) =E_Q(E^{-rT}X\,|\,{\mathCaL{F}}_T^{{\wideHat{W}}}),\QQUAD t\in[0,t].$$ By The mArtinGale representAtiOn thEOreM (e.g., ThEorem 9.4 In \[K.K\]), ThEre exIsts an $({\MathcAl{f}}_t^{{\widehat{W}}})$-prediCtabLe process $H_0(t)$, $t\In[0,t]$, suCh That $$\iNT_0^T h_0(u)^2\,du<\InfTy\qQuad a.s.,$$ aNd $$M(t)=E_Q(e^{-Rt}X)+\iNt_0^TH_0(U)\,D{\wIdehat{W}}(u

-\frac{1}{2}\int_0^tg\big( S(u-b)\big )^2du \ri ght \} $$ a .s.for $t\in[0,T] $ . He nce we have $$\label{t ShW} { \wid e ti lde{S }}(t)=\ v ar p h i(0 ) \ exp\l eft \{\int_ 0^tg\big(S (u- b) \big)\,d{\wi d eh at{W}}(u) -\fr ac {1} { 2}\in t_0 ^tg\b ig(S(u - b)\big )^2du\rig ht \ }$$ fo r $t\in[ 0 , T] $. B y examining the d e fi n itions of $\ti lde S, { \ wi d e hat {W} }, {\wideh at {S}}$ and equ a ti o n (6) , it is not ha rd to see t h atfor $t \ geq 0$, ${ \math ca l {F} }_t^S={\mat hcal {F}}_t^{{ \widet i lde{S}} } ={\math cal{F} }_t ^{{ \wid e ha t{ W}} }= { \ma t hc al{ F }}_ t^{W}$,th e$\sig ma$- a l g e bras ge nera ted b y $\{S (u): u \l eq t \}$ , $\{ {\wid etil de {S}}( u): u\leqt\}$, $\{{\wideh at{W }}(u): u\le qt \ }$ , $\{ W (u): u \l eqt \}$,respect i vel y. ( C le arly, ${\mathcal{F }} _ t ^{ W} \subs eteq { \ ma th c al{F}}_t $. ) N ow,l e t $X$ bea c ontingen t clai m ,vi z. an i nt egrabl enon -ne gativ e ${\ mathca l{F}}_T^ S$-me a surable random variable. Con s id e r t h e $Q $-m artingale $ $M(t ) :=E_ Q(e^ { -r T}X \ ,|\,{ \math ca l {F } }_t^S) =E_Q (e ^{-rT} X\,|\ ,{\mathcal{F} }_t^{{\wid e h a t{W}}}), \qqu a dt \in[0,T].$$ By themartingale represen tatio n theore m (e.g.,T h eorem 9. 4 i n \ [K. K\] ) , t here exists a n $({\ ma thcal{F }}_ t^{{\wi deh at{ W}} })$ -p redictabl e proces s$h _0 (t )$, $t\i n [0,T]$,su chth at$$\in t _0^T h _0(u) ^2\, du <\ i nft y\qquad a. s . ,$$an d$$M( t)= E_ Q(e^{ -rT} X )+\ int_0^t h_0(u)\,d {\w i deha t{ W} }(u

-\frac{1}{2}\int_0^tg\big(S(u-b)\big)^2du\right\}$$ a.s._for $t\in[0,T]$._Hence we have $$\label{tShW} _ {\widetilde{S}}(t)=\varphi(0) __ __ _ \exp\left\{\int_0^tg\big(S(u-b)\big)\,d{\widehat{W}}(u) _ _ __ -\frac{1}{2}\int_0^tg\big(S(u-b)\big)^2du\right\}$$ for $t\in[0,T]$. By_examining_the definitions_of_$\tilde_S, {\widehat{W}}, {\widehat{S}}$ and equation_(6), it is not hard_to see_that for $t \geq 0$, ${\mathcal{F}}_t^S={\mathcal{F}}_t^{{\widetilde{S}}}={\mathcal{F}}_t^{{\widehat{W}}}={\mathcal{F}}_t^{W}$, the $\sigma$-algebras_generated_by $\{S (u):_u \leq t \}$, $\{{\widetilde{S}}(u): u \leq t \}$, $\{{\widehat{W}}(u):_u \leq t \}$, $\{W(u): u_\leq t \}$,_respectively._(Clearly,_${\mathcal{F}}_t^{W} \subseteq {\mathcal{F}}_t$.) Now,_let $X$ be a contingent claim,_viz. an integrable non-negative ${\mathcal{F}}_T^S$-measurable random_variable. Consider the $Q$-martingale $$M(t):=E_Q(e^{-rT}X\,|\,{\mathcal{F}}_t^S) _ =E_Q(e^{-rT}X\,|\,{\mathcal{F}}_t^{{\widehat{W}}}),\qquad_t\in[0,T].$$ By the martingale representation_theorem (e.g.,_Theorem 9.4 in \[K.K\]), there_exists an $({\mathcal{F}}_t^{{\widehat{W}}})$-predictable_process $h_0(t)$,_$t\in[0,T]$, such that_$$\int_0^T h_0(u)^2\,du<\infty\qquad a.s.,$$ and $$M(t)=E_Q(e^{-rT}X)+\int_0^th_0(u)\,d{\widehat{W}}(u

rm Be}}$ core $R^{\rm (c)}_{\rm rms}$, the rms distances between the core and $\Lambda$ particle $R^{({\rm c}-\Lambda)}_{\rm rms}$, and the rms radii of ${^9_\Lambda{\rm Be}}$ $R_{\rm rms}$ at the minimum positions are shown, together with the corresponding $\beta_\perp$ and $\beta_z$ values. The maximum squared overlap values of ${\cal O}_J(\beta_\perp,\beta_z)$ defined by Eq. (\[eq:ovlp\]), are also shown, together with the $\beta_\perp$ and $\beta_z$ values giving the maxima. The two kinds of the $\Lambda N$ interaction, YNG-ND and -JA are adopted.[]{data-label="tab2"} In order to compare the single Hyper-THSR wave function with the Brink-GCM wave function, we calculate the following squared overlap: $${\cal O}_J (\beta_\perp, \beta_z) = \frac{|\langle \Psi^H_J(\beta_\perp, \beta_z) | {\widetilde \Psi}^B_J \rangle|^2}{\langle \Psi^H_J(\beta_\perp, \beta_z) | \Psi^H_J(\beta_\perp, \beta_z) \rangle \langle {\widetilde \Psi}^B_J | {\widetilde \Psi}^B_J \rangle }, \label{eq:ovlp}$$ with $${\widetilde \Psi}^B_J= \sum_{R,S} f^{(J,\lambda=0)}(R,S) [u_J(\vc{R}), \psi_{\lambda=0}(\vc{S})]_J.$$ The above wave function ${\widetilde \Psi}^{B}_J$ is the Brink-GCM wave function projected onto the model space with the angular-momentum channel $(L,\lambda)=(J,0)$. ![Contour maps of the squared overlap surfaces for $J^\pi=0^+$(left top), $2^+$(right top), and $4^+$(bottom) states in two-parameter space $\beta_x=\beta_y(\equiv \beta_\perp), \beta_z$, defined by ${\cal O}(\beta_\perp,\beta_z)$ in Eq. (\[eq:ovlp\]). Two maxima are denoted by $\times$ and $+$. YNG-ND interaction

rm Be}}$ core $ R^{\rm (c)}_{\rm rms}$, the rms distances between the core and $ \Lambda$ particle $ R^{({\rm c}-\Lambda)}_{\rm rms}$, and the rms radius of $ { ^9_\Lambda{\rm Be}}$ $ R_{\rm rms}$ at the minimal positions are shown, in concert with the corresponding $ \beta_\perp$ and $ \beta_z$ value. The maximum square lap values of $ { \cal O}_J(\beta_\perp,\beta_z)$ defined by Eq.   (\[eq: ovlp\ ]), are also shown, together with the $ \beta_\perp$ and $ \beta_z$ value giving the maxima. The two kind of the $ \Lambda N$ interaction, YNG - ND and -JA are adopted.[]{data - label="tab2 " } In order to compare the single Hyper - THSR wave function with the Brink - GCM wave function, we calculate the following squared overlap: $ $ { \cal O}_J (\beta_\perp, \beta_z) = \frac{|\langle \Psi^H_J(\beta_\perp, \beta_z) | { \widetilde \Psi}^B_J \rangle|^2}{\langle \Psi^H_J(\beta_\perp, \beta_z) | \Psi^H_J(\beta_\perp, \beta_z) \rangle \langle { \widetilde \Psi}^B_J | { \widetilde \Psi}^B_J \rangle }, \label{eq: ovlp}$$ with $ $ { \widetilde \Psi}^B_J= \sum_{R, S } f^{(J,\lambda=0)}(R, S) [ u_J(\vc{R }), \psi_{\lambda=0}(\vc{S})]_J.$$ The above wave affair $ { \widetilde \Psi}^{B}_J$ is the Brink - GCM wave function projected onto the model quad with the angular - momentum channel $ (L,\lambda)=(J,0)$. ! [ Contour maps of the squared overlap surface for $ J^\pi=0^+$(left top), $ 2^+$(right top), and $ 4^+$(bottom) states in two - parameter space $ \beta_x=\beta_y(\equiv \beta_\perp), \beta_z$, defined by $ { \cal O}(\beta_\perp,\beta_z)$ in Eq.   (\[eq: ovlp\ ]). Two maximum are denoted by $ \times$ and $ + $. YNG - ND interaction

rm He}}$ core $R^{\rm (c)}_{\rm rms}$, the vms distances bejwwen thx core znd $\Lambaa$ particle $R^{({\rm c}-\Lambda)}_{\rm rmd}$, qnd tye rms radii of ${^9_\Lambda{\fm Be}}$ $R_{\rm rms}$ at rhe nunimum posmfions avz shosk, togztier with the cotresponding $\teta_\perp$ and $\batx_z$ values. The maximum squared overlap values ov ${\cal O}_J(\beta_\petp,\bets_s)$ derpntd by Eq. (\[eq:ovlp\]), are also shown, togsther wpth the $\beta_\perp$ snd $\beta_z$ values giving thf madima. The two kinds of the $\Lamvda G$ interaction, YNG-ND and -JA are adopjed.[]{data-label="tab2"} In order to compafe thz single Hykze-THDT wave functmon winh the Brink-GGK wave functipn, we calculatc the foolowing squared overlep: $${\cal O}_J (\beta_\perp, \bgta_z) = \frac{|\naugle \Psi^H_J(\beta_\perp, \bera_z) | {\wigetinde \Owi}^B_G \rznjle|^2}{\mangle \Psm^H_J(\beta_\perp, \beta_z) | \Psi^Y_J(\beta_\perp, \beta_z) \ramgjv \langle {\widefilde \[sy}^B_J | {\widetilde \Psi}^B_J \rangle }, \label{eq:ovlk}$$ witg $${\widetilde \Psi}^B_J= \sum_{R,S} f^{(J,\lambda=0)}(R,S) [u_J(\vc{R}), \psi_{\pambda=0}(\vc{S})]_T.$$ The above wave function ${\widetilde \Psi}^{B}_J$ is the Trink-JCO wcyc fuvxtlon projected onto the model space with the agfukag-momentum channel $(L,\lambda)=(J,0)$. ![Comtlut maps of the rquareb oberlap surfaces fog $J^\pi=0^+$(lest top), $2^+$(right tjp), amd $4^+$(bottom) states in two-paraneter space $\yetq_x=\beta_y(\equiv \beta_\pzrp), \beta_z$, deyined ny ${\cak O}(\beta_\perp,\beta_z)$ in Eq. (\[ed:ovll\]). Two maximw are denkged by $\times$ and $+$. YKG-NG interaction

rm Be}}$ core $R^{\rm (c)}_{\rm rms}$, the between core and particle $R^{({\rm c}-\Lambda)}_{\rm of Be}}$ $R_{\rm rms}$ the minimum positions shown, together with the corresponding $\beta_\perp$ $\beta_z$ values. The maximum squared overlap values of ${\cal O}_J(\beta_\perp,\beta_z)$ defined by Eq. are also shown, together with the $\beta_\perp$ and $\beta_z$ values giving the maxima. two of $\Lambda interaction, YNG-ND and -JA are adopted.[]{data-label="tab2"} In order to compare the single Hyper-THSR wave function with Brink-GCM wave function, we calculate the following squared $${\cal O}_J (\beta_\perp, \beta_z) \frac{|\langle \Psi^H_J(\beta_\perp, \beta_z) | {\widetilde \rangle|^2}{\langle \beta_z) | \beta_z) \langle \Psi}^B_J | {\widetilde \rangle }, \label{eq:ovlp}$$ with $${\widetilde \Psi}^B_J= \sum_{R,S} f^{(J,\lambda=0)}(R,S) [u_J(\vc{R}), \psi_{\lambda=0}(\vc{S})]_J.$$ The above wave function ${\widetilde \Psi}^{B}_J$ is Brink-GCM wave onto the space the channel $(L,\lambda)=(J,0)$. ![Contour the squared overlap surfaces for $J^\pi=0^+$(left and $4^+$(bottom) states in two-parameter space $\beta_x=\beta_y(\equiv \beta_\perp), defined by O}(\beta_\perp,\beta_z)$ in Eq. (\[eq:ovlp\]). Two maxima denoted by $\times$ and $+$. YNG-ND interaction

rm Be}}$ core $R^{\rm (c)}_{\rm rms}$, the rms diStances betWeen tHe cOre AnD $\LamBda$ pArticle $R^{({\rm c}-\LamBDa)}_{\rm Rms}$, and the rms radii of ${^9_\LamBda{\rm be}}$ $r_{\Rm rmS}$ At The miNimum poSItIONs aRe ShOwn, ToGEtHer wiTh tHe correSponding $\beTa_\pErP$ and $\beta_z$ valUEs. the maximum SquAred overlap vAluEs of ${\caL O}_j(\beTA_\perp,\BetA_z)$ defIned by eQ. (\[eq:ovlP\]), are also sHoWN, togetHEr with tHE $\BeTa_\peRp$ and $\beta_z$ values gIViNG the maxima. The tWo kindS oF ThE $\lAmbDa N$ InteractioN, YnG-ND aND -JA are aDOpTED.[]{DatA-Label="tab2"} In ordEr to compare THe sIngle HYpEr-ThsR wave FunctIoN WitH the Brink-GCm wavE function, We calcULate the FOllowinG squarEd oVerLap: $${\cAL O}_j (\bEta_\PeRP, \beTA_z) = \FraC{|\LanGle \Psi^H_J(\BeTa_\Perp, \bEta_z) | {\WIDETildE \PsI}^B_J \rAngle|^2}{\Langle \Psi^H_J(\beTa_\pErp, \bETa_z) | \psi^H_J(\Beta_\pErp, \bEtA_z) \ranGle \lanGle {\wiDeTilde \Psi}^B_J | {\widetIlde \psi}^B_J \rangLe }, \lAbEl{eQ:oVlp}$$ wiTH $${\widetIldE \PsI}^B_J= \sum_{R,s} f^{(J,\lambDA=0)}(R,S) [U_J(\VC{r}), \PsI_{\lambda=0}(\vc{S})]_J.$$ The abovE wAVE fUnction ${\wIdetilDE \PSi}^{b}_j$ is the BrInK-GCm wavE FUnctiOn prOJeCted onto The modEL sPaCe with tHe AngulaR-mOmeNtuM chanNEl $(L,\lAmbda)=(J,0)$. ![contour mAps of THe squared overlAP surfaces for $J^\PI=0^+$(lEFT tOP), $2^+$(rigHt tOp), and $4^+$(bottom) StatES in tWo-paRAmEteR Space $\Beta_x=\BeTA_y(\EQuiv \beta_\perp), \beta_z$, deFiNed by ${\cAl O}(\beTa_\perp,\beta_z)$ in eq. (\[eq:ovlp\]). TwO MAXima are dEnotED bY $\Times$ and $+$. YNG-ND iNteraCtion

rm Be}}$ core $R^{\rm (c)} _{\rm rms} $, th e r msdi stan cesbetween the co r e an d $\Lambda$ particle $ R^{({ \r m c}- \ La mbda) }_{\rmr ms } $ , a nd t herm s r adiiof${^9_\L ambda{\rmBe} }$ $R_{\rm rms } $at the min imu m positionsare shown ,tog e therwit h the corre s pondin g $\beta_ \p e rp$ an d $\beta _ z $valu es. The maximum s q ua r ed overlap val ues of $ { \c a l O} _J( \beta_\per p, \beta _ z)$ def i ne d b y E q . (\[eq:ovlp\ ]), are als o sh own, t og eth e r with the$\ b eta _\perp$ and $\b eta_z$ va lues g i ving th e maxima . Thetwo ki ndso fth e $ \L a mbd a N $ i n ter action,YN G- ND an d -J A a r e ad opt ed.[ ]{dat a-label="tab2 "} Ino rde r tocompa re t he sing le Hyp er-TH SR wave functionwith the Brin k-G CM wa ve func t ion, w e c alc ulate t he foll o win gs q u ar ed overlap: $${\ca lO } _J (\beta_ \perp, \b et a _z) = \f ra c{| \lan g l e \Ps i^H_ J (\ beta_\pe rp, \b e ta _z ) | {\w id etilde \ Psi }^B _J \r a ngle |^2}{\ langle \ Psi^H _ J(\beta_\perp, \beta_z) | \P s i^ H _ J( \ beta _\p erp, \beta_ z) \ r angl e \l a ng le{ \wide tilde \ P si } ^B_J | {\widetilde\P si}^B_ J \ra ngle }, \labe l{eq:ovlp} $ $ with $${ \wid e ti l de \Psi}^B_J=\sum_ {R,S} f^{( J ,\lambda =0)}( R,S) [u_ J(\vc{R}) , \psi_{\l amb da= 0}( \vc { S }) ]_J.$$ The ab o v e wa ve functi on${\wide til de\Ps i}^ {B }_J$ is t he Brink -G CM w av e f uncti o n projec te d o nt o t he mo d el spa ce wi th t he a n gul ar-mome n tu m chan ne l$(L, \la mb da)=( J,0) $ . ![Conto ur maps o f t h e sq ua re d overl ap surfaces f or $J^\pi=0^ +$ (le ft top ) , $2^+$(r ight top), and $4^+$(bo t tom) st ate s intwo- parameter sp ace $\ bet a _x=\be ta_y(\ equiv \ bet a _ \perp ) , \ bet a_ z$, define d by${\ca lO}(\ beta_\p erp,\beta_z)$ in E q . ( \[eq:ovlp\]). Tw o ma x i ma ar e d e not ed by$ \ times$ and $+$. YNG-ND in te r ac tion

rm Be}}$_core $R^{\rm_(c)}_{\rm rms}$, the rms_distances between_the_core and_$\Lambda$_particle $R^{({\rm c}-\Lambda)}_{\rm_rms}$, and the_rms radii of ${^9_\Lambda{\rm_Be}}$ $R_{\rm rms}$_at_the minimum positions are shown, together with the corresponding $\beta_\perp$ and $\beta_z$ values. The_maximum_squared overlap_values_of_${\cal O}_J(\beta_\perp,\beta_z)$ defined by Eq. (\[eq:ovlp\]),_are also shown, together with_the $\beta_\perp$_and $\beta_z$ values giving the maxima. The two_kinds_of the $\Lambda_N$ interaction, YNG-ND and -JA are adopted.[]{data-label="tab2"} In order to_compare the single Hyper-THSR wave function_with the Brink-GCM_wave_function,_we calculate the following_squared overlap: $${\cal O}_J (\beta_\perp, \beta_z)_= \frac{|\langle \Psi^H_J(\beta_\perp, \beta_z) | {\widetilde_\Psi}^B_J \rangle|^2}{\langle \Psi^H_J(\beta_\perp, \beta_z) | \Psi^H_J(\beta_\perp, \beta_z)_\rangle \langle {\widetilde \Psi}^B_J | {\widetilde_\Psi}^B_J \rangle }, \label{eq:ovlp}$$ with_$${\widetilde \Psi}^B_J=_\sum_{R,S} f^{(J,\lambda=0)}(R,S) [u_J(\vc{R}), \psi_{\lambda=0}(\vc{S})]_J.$$ The_above wave function_${\widetilde \Psi}^{B}_J$_is the Brink-GCM_wave function projected onto the model_space with the_angular-momentum channel $(L,\lambda)=(J,0)$. ![Contour maps of the_squared_overlap surfaces for_$J^\pi=0^+$(left_top),_$2^+$(right top),_and $4^+$(bottom) states_in_two-parameter space_$\beta_x=\beta_y(\equiv_\beta_\perp), \beta_z$, defined by ${\cal O}(\beta_\perp,\beta_z)$_in_Eq. (\[eq:ovlp\]). Two maxima are denoted by $\times$_and $+$. YNG-ND interaction

ivity curves measured at two different Bragg angles of two Si(555) circular analysers with the bending radii of 1 m and 0.5 m. The diameter and thickness of the wafers were 100 mm and 150 $\upmu$m, respectively. Further experimental details are presented in the original sources. Compared with the previous work which was based on the geometrical considerations and did not account for the minimization of the elastic energy, slight differences between two models are observed but they are found to be less than the variation between different SBCA units, as seen in Fig. \[fig:si660\_si553\]. This outcludes one explanation put forth in the previous work for the discrepancy between the data and the model at the low-energy tail of the diffraction curve for the full analyser, according to which the observed difference could be due to non-vanishing $\sigma_{rr}$ at the wafer edge in the previous model. One possible explanation to the discrepancy is the imperfections in manufacturing process, as it is found that the figure error in anodically bonded analysers is largest at the edge [@Verbeni_2005]. Another explanation could be a slight deviation from the Rowland circle geometry that is not included in the calculations. The latter hypothesis is supported by the data in Fig. \[fig:si555\] where the deviations are more prominent. According to the theory, the stresses and strains due to streching are a factor of 4 larger in a wafer that has half the bending radius than in a wafer otherwise identical. Even for considerably higher transverse stress, the theory predicts correctly the observed boxcar shape and its width for the measured 0.5 m Si(555) analyser. The general shape and the width of the predicted 1 m Si(555) curve are in line with the measurements but is not as precise as for the set of Si(660) and Si(553) analysers in Fig. \[fig:si660\_si553\]. The most probable reason for this is the contribution of aforementioned deviation from the Rowland circle geometry, the effect of which is amplified at lower Bragg angles. In the experimental description, it is mentioned that the radius of the Rowland circle was adjusted by optimizing the product of total counts and peak intensity divided by the FWHM for each analyser [@Rovezzi_2017]. Since the different contributions to the energy resolution of an SBCA are not truly independent of each other, such an optimization can lead to partial cancellation

ivity curves measured at two different Bragg angles of two Si(555) round analyzer with the deflect radii of 1   m and 0.5   m. The diameter and thickness of the wafer were 100   mm and 150   $ \upmu$m, respectively. Further experimental details are presented in the original informant. Compared with the previous employment which was based on the geometrical consideration and did not account for the minimization of the elastic energy, flimsy difference between two models are note but they are found to be less than the variation between unlike SBCA units, as seen in Fig.   \[fig: si660\_si553\ ]. This outcludes one explanation put forth in the former work for the discrepancy between the data and the model at the low - energy fag end of the diffraction curve for the full analyser, according to which the observed remainder could be due to non - vanishing $ \sigma_{rr}$ at the wafer edge in the previous model. One possible explanation to the discrepancy is the imperfections in manufacturing process, as it is found that the figure error in anodically bonded analysers is largest at the edge [ @Verbeni_2005 ]. Another explanation could be a slight diversion from the Rowland set geometry that is not included in the calculation. The latter hypothesis is supported by the data in Fig.   \[fig: si555\ ] where the deviations are more outstanding. According to the theory, the stresses and strains due to streching are a agent of 4 larger in a wafer that has half the bending radius than in a wafer otherwise identical. Even for considerably gamey transverse stress, the theory predicts correctly the note boxcar human body and its width for the measured 0.5   m Si(555) analyser. The general shape and the width of the predicted 1   m Si(555) curve are in line with the measurement but is not as accurate as for the set of Si(660) and Si(553) analysers in Fig.   \[fig: si660\_si553\ ]. The most probable reason for this is the contribution of aforementioned diversion from the Rowland circle geometry, the effect of which is amplified at lower Bragg angles. In the experimental description, it is mention that the radius of the Rowland set was adjusted by optimize the product of total count and extremum intensity divide by the FWHM for each analyser [ @Rovezzi_2017 ]. Since the different contributions to the energy resolution of an SBCA are not rightfully independent of each other, such an optimization can leave to partial cancellation

ivihy curves measured at twu different Brayt anglxs of tso Si(555) cifcular analysers with the beidint radui of 1 m and 0.5 m. The diamdter and nhickness of uhe wafers were 100 mm and 150 $\upmu$m, reaiectireoy. Further expgrimental dedails are presangeb in the original sources. Compared wieh the lrfvious work whych ews bzsed on the geometrical consideratjons anv did not accoumt for the minimization of the elastic energy, sllght differghcef between two models art mbserved bot they are found to be less thav the variation bwtwfgn different SBCA units, as secm in Fhg. \[fig:si660\_xi553\]. This outcluces onw explanation put forvh in the previous wjrk for tve discrepancy betwweb the datd ana thd mkdxl zt the lox-energy taim of the didfraction curve for trv full analyssr, accjrqing to which the observed difference cmuls be due to non-vanishint $\sigma_{rr}$ at the wafet edge in ehe previous model. One possible explanation to tha disrrdpaugn is rhf imperfections in manufacturing process, as ie ix nound that the flgure error in anocifakjy bonded anauysers is largest at the edhe [@Verbgni_2005]. Anither expjanayion could be a slight deviqtion from tke Eowland circle geoletry that ns not inclided in the calculationr. Ths latter hyoothesis jr supported by tfe cada in Fig. \[fig:si555\] where the dqviations are more pfomiment. Ascording tl the theory, the stressed and sdrains due to streching are a factor of 4 larger in a wages tvat has kalf tme bending radits than in a wcfer othzrwise identical. Even fmr considerwbly higher tswnsverse strxss, the treort prwdicts zurrectly the onserved boxcar shapw and its width fov the jeasured 0.5 m Si(555) aucliwer. The generak sfapq wnv the fidth of the preaicgrd 1 m Ri(555) curvt ave kn lone with the measurekenta but is not as prrclse as fot the set of Si(660) and Si(553) analysers in Fig. \[flg:si660\_sm553\]. The oost lrofable reason for this is the ckntributiln jf aforementyonee deviation yrom the Rowland circle geometry, the effxct of which is amplifiwd at lower Bragg augkes. In the xxperiiental devcription, it is mentuoned that the racius of the Rowland cjrcle fas afjusted by optimizing the product of total counts and peak intensity divieed by the FWHM fod eavh andlvsex [@Rovezsi_2017]. Smnre the different bontributions to the energy resouution ox cn SBCA are not truly indepencevt of each otfer, such an optimization can leac to partial cancellation

ivity curves measured at two different Bragg two circular analysers the bending radii m. diameter and thickness the wafers were mm and 150 $\upmu$m, respectively. Further details are presented in the original sources. Compared with the previous work which based on the geometrical considerations and did not account for the minimization of elastic slight between models are observed but they are found to be less than the variation between different SBCA as seen in Fig. \[fig:si660\_si553\]. This outcludes one put forth in the work for the discrepancy between data the model the tail the diffraction curve the full analyser, according to which the observed difference could be due to non-vanishing $\sigma_{rr}$ at the edge in model. One explanation the is the imperfections process, as it is found that in anodically bonded analysers is largest at the [@Verbeni_2005]. Another could be a slight deviation from Rowland circle geometry that is not included in calculations. The latter hypothesis is supported by the data in Fig. \[fig:si555\] where the deviations prominent. According to the the stresses and due streching a of 4 in a wafer that has half the bending radius than in wafer otherwise identical. Even for considerably higher transverse stress, the correctly observed boxcar shape its width for the 0.5 Si(555) analyser. The general the of m curve in line with the but is not as precise for the set of Fig. \[fig:si660\_si553\]. The most probable reason for this the contribution of aforementioned deviation from the circle geometry, the effect of which is amplified at lower Bragg angles. the experimental is mentioned that the radius of the Rowland was adjusted by optimizing product of total counts and peak intensity divided by FWHM each analyser Since the different to the energy of an SBCA truly independent each such lead to partial cancellation

ivity curves measured at two dIfferent BrAgg anGleS of TwO Si(555) cIrcuLar analysers wiTH the Bending radii of 1 m and 0.5 m. The DiameTeR And tHIcKness Of the waFErS WEre 100 Mm AnD 150 $\upMu$M, ReSpectIveLy. FurthEr experimeNtaL dEtails are preSEnTed in the orIgiNal sources. CoMpaRed witH tHe pREviouS woRk whiCh was bASed on tHe geometrIcAL consiDErationS ANd Did nOt account for the miNImIZation of the elaStic enErGY, sLIGht DifFerences beTwEen twO Models aRE oBSERveD But they are fouNd to be less tHAn tHe variAtIon BEtween DiffeReNT SBcA units, as seEn in fig. \[fig:si660\_sI553\]. This oUTcludes ONe explaNation Put ForTh in THe PrEviOuS WorK FoR thE DisCrepancy BeTwEen thE datA AND The mOdeL at tHe low-Energy tail of tHe dIffrACtiOn curVe for The fUlL analYser, acCordiNg To which the obserVed dIfference CouLd Be dUe To non-VAnishiNg $\sIgmA_{rr}$ at thE wafer eDGe iN tHE PReVious model. One possiBlE EXpLanation To the dIScRePAncy is thE iMpeRfecTIOns in ManuFAcTuring prOcess, aS It Is Found thAt The figUrE erRor In anoDIcalLy bondEd analysErs is LArgest at the edgE [@verbeni_2005]. AnotheR ExPLAnATion CouLd be a slight DeviATion From THe rowLAnd ciRcle gEoMEtRY that is not included iN tHe calcUlatiOns. The latter hYpothesis iS SUPported bY the DAtA In Fig. \[fig:si555\] wherE the dEviations aRE more proMinenT. AccordiNg to the thEORy, the strEssEs aNd sTraINS dUe to streching ARE a faCtOr of 4 larGer In a wafeR thAt hAs hAlf ThE bending rAdius thaN iN a WaFeR otHerwiSE identicAl. eveN fOr cOnsidERably hIgher TranSvErSE stRess, the THeORY preDiCtS corRecTlY the oBserVEd bOxcar shApe and its WidTH for ThE mEasured 0.5 M Si(555) analyser. ThE gEneral shapE aNd tHe widtH OF the predIcted 1 m Si(555) curve are in line wITh the meAsuRemenTs buT is not as pRecIse as fOr tHE set of si(660) and SI(553) analYsErs IN fig. \[fiG:SI660\_sI553\]. ThE mOst probablE REasOn for ThIs is The contRibution of aforemenTIonEd deviation frOm tHe RoWLAnD ciRClE GeoMeTRy, tHE Effect of which is Amplified aT lOWeR Bragg anglES. In ThE experiMental dEscriPTion, it iS mentioneD that the rAdIus oF THe ROwland circLe was adjUsted by opTImiziNG tHe proDucT of totAl CouNts anD peak iNTenSity dIvided By The FWHm for eAcH analyseR [@Rovezzi_2017]. Since the differeNt contRibutIonS to the eneRgy REsoLution of aN SBCa are not truLy iNdePendeNt oF Each oTher, SUcH an OPtimiZatiON can lead tO PaRtiAL CaNcellation

ivity curves measured at t wo differe nt Br agg an gl es o f tw o Si(555) circ u laranalysers with the ben dingra d ii o f 1  m an d 0.5 m . T h e di am et eran d t hickn ess of the wafers we re10 0 mm and 150 $\ upmu$m, re spe ctively. Fur the r expe ri men t al de tai ls ar e pres e nted i n the ori gi n al sou r ces. C o m pa redwith the previous wo r k which was ba sed on t h eg e ome tri cal consid er ation s and di d n o t acc o unt for the m inimization ofthe el as tic energy , sli gh t di fferences b etwe en two mo dels a r e obser v ed butthey a refou nd t o b eles st han th e v a ria tion bet we en diff eren t S B CA u nit s, a s see n in Fig. \[f ig: si66 0 \_s i553\ ]. Th is o ut clude s oneexpla na tion put forthin t he previo uswo rkfo r the discre pan cybetween the da t a a nd t h emodel at the low-e ne r g ytail ofthe di f fr ac t ion curv efor the f ull a naly s er , accord ing to wh ic h the o bs erveddi ffe ren ce co u ld b e dueto non-v anish i ng $\sigma_{rr } $ at the wafe r e d g ei n th e p revious mod el.O ne p ossi b le ex p lanat ion t ot he discrepancy is theim perfec tions in manufactu ring proce s s , as it i s fo u nd that the figur e err or in anod i cally bo ndedanalyser s is larg e s t at the ed ge[@V erb e n i_ 2005]. Anothe r expl an ation c oul d be asli ght de via ti on from t he Rowla nd c ir cl e g eomet r y that i snot i ncl udedi n thecalcu lati on s. The latter hy p o thes is i s su ppo rt ed by the dat a in Fi g. \[fig: si5 5 5\]wh er e the d eviations are m ore promin en t.Accord i n g to the theory, the stresses a n d strai nsdue t o st reching a rea fact oro f 4 la rger i n a w af ert h at ha s ha lfth e bendingr a diu s tha nin a waferotherwise identica l . E ven for consi der ably h ig her tr a nsv er s e s t r ess, the theory predictsco r re ctly the o b ser ve d boxca r shape andi ts widt h for the measured 0 .5 m S i(5 55) analys er. Thegeneral s h ape a n dthe w idt h of t he pr edict ed 1 m Si( 555)curvear e in l ine w it h the me asurements but is not a s prec ise a s f or the se t o f Si (660) and Si( 553) analy ser s i n Fig . \ [ fig:s i660 \ _s i55 3 \]. T he m o st probab l erea s o nfor this is t h e c ontri but i on ofafor ementioned deviat i on from the Ro wlan d cir cle geom et ry, the effect of w h i ch is am pl ified at lo wer Brag ga ngles . In t he exp eriment a l d e script ion, it is menti one dt hat the r ad i us oftheRo wlandcircle wasa d justed by optimi zingt h e pro d uct of t ot al coun t s an d peak int ensity divi ded by the FWHM for ea ch analy ser [ @Rovezzi_2 0 17]. Sinc e the differ en t co ntr ibutio ns t o the e nerg yres olution o f an SB CA are not trul yinde pendent o f each ot her , such a nopt i m izatio n c a n lead to p art ial c a n cellation

ivity curves_measured at_two different Bragg angles_of two_Si(555)_circular analysers_with_the bending radii_of 1 m and_0.5 m. The diameter and_thickness of the_wafers_were 100 mm and 150 $\upmu$m, respectively. Further experimental details are presented in the original sources. Compared_with_the previous_work_which_was based on the geometrical_considerations and did not account_for the_minimization of the elastic energy, slight differences between_two_models are observed_but they are found to be less than the_variation between different SBCA units, as_seen in Fig. \[fig:si660\_si553\]._This_outcludes_one explanation put forth_in the previous work for the_discrepancy between the data and the_model at the low-energy tail of the_diffraction curve for the full analyser,_according to which the observed_difference could_be due to non-vanishing $\sigma_{rr}$_at the wafer_edge in_the previous model._One possible explanation to the discrepancy_is the imperfections_in manufacturing process, as it is_found_that the figure_error_in_anodically bonded_analysers is largest_at_the edge_[@Verbeni_2005]._Another explanation could be a slight_deviation_from the Rowland circle geometry that is_not included in the_calculations._The latter hypothesis is_supported by the data in_Fig. \[fig:si555\] where the deviations are more_prominent. According_to the_theory, the stresses and strains due to streching are a factor_of 4 larger in a wafer_that has half the_bending radius_than_in a wafer_otherwise_identical. Even_for considerably higher transverse stress, the theory_predicts correctly_the observed boxcar shape and its_width for the measured_0.5 m_Si(555) analyser. The general shape and_the width of the predicted 1 m_Si(555) curve are in line_with_the_measurements but is not as_precise as for the set of_Si(660) and Si(553)_analysers in Fig. \[fig:si660\_si553\]. The most probable reason_for_this is the contribution of aforementioned_deviation_from the Rowland circle geometry, the_effect_of_which is amplified at lower_Bragg angles. In the experimental description,_it is mentioned that the radius of the Rowland_circle was adjusted_by optimizing the product of_total_counts_and peak intensity divided by the FWHM for each analyser_[@Rovezzi_2017]. Since_the different contributions_to the energy resolution of an SBCA are not truly_independent of each other, such an optimization_can lead to partial cancellation

\label{eqn:theta_beta_2}\end{aligned}$$ $$\begin{aligned} \dot{\delta}_{\gamma} = -\frac{4}{3}\theta_{\gamma}-\frac{2}{3}\dot{h} + (1-\lambda) \frac{\rho_{q}}{\rho_{\gamma}}\frac{a}{\tau}(\delta_{\beta}-\delta_{\gamma})\,, \label{eqn:deltadot_gamma_2}\end{aligned}$$ and $$\begin{aligned} \dot{\theta}_{\gamma} = k^2 \left( \frac{1}{4}\delta_{\gamma} - \Theta_{\gamma} \right) &+ a n_{e}\sigma_{T}(\theta_{\beta}-\theta_{\gamma}) \nonumber \\ &+ (1-\lambda) \frac{\rho_{q}}{\rho_{\gamma}}\frac{a}{\tau}\left(\frac{3}{4}\theta_{\beta}-\theta_{\gamma}\right) \,. \label{eqn:thetadot_gamma_2}\end{aligned}$$ We now describe how small-scale modes that enter the horizon prior to decay are suppressed relative to those modes that enter the horizon after decay. Due to the Thomson collision terms the ‘$\beta$’ component and the photons will be tightly coupled as a ‘$\beta$’-photon fluid at early times and this fluid will support acoustic oscillations. Furthermore, Eqs. (\[eqn:deltadot\_d\_2\]) and (\[eqn:thetadot\_d\_2\]) show that the dark-matter perturbations are strongly sourced by the perturbations of the ‘$\beta$’ component prior to decay, when the ratio $\rho_{q}/\rho_{d}$ is large. Dark-matter modes that enter the horizon prior to decay will thus track the oscillations of the ‘$\beta$’-photon fluid rather than simply growing under the influence of gravity. After decay, when the ratio $\rho_{q}/\rho_{d}$ is small, the source term shuts off and dark-matter modes that enter the horizon undergo the standard growing evolution. In Fig. \[fig:delta\_tau\] we follow the evolution of the dark-matter perturbations through the epoch of decay. We modified [CMBFAST]{} [@CMBfast] to carry out these calculations. In order to suppress power on subgalactic scales the decay lifetime must be roughly the age of

\label{eqn: theta_beta_2}\end{aligned}$$ $ $ \begin{aligned } \dot{\delta}_{\gamma } = -\frac{4}{3}\theta_{\gamma}-\frac{2}{3}\dot{h } + (1-\lambda) \frac{\rho_{q}}{\rho_{\gamma}}\frac{a}{\tau}(\delta_{\beta}-\delta_{\gamma})\, , \label{eqn: deltadot_gamma_2}\end{aligned}$$ and $ $ \begin{aligned } \dot{\theta}_{\gamma } = k^2 \left (\frac{1}{4}\delta_{\gamma } - \Theta_{\gamma } \right) & + a n_{e}\sigma_{T}(\theta_{\beta}-\theta_{\gamma }) \nonumber \\ & + (1-\lambda) \frac{\rho_{q}}{\rho_{\gamma}}\frac{a}{\tau}\left(\frac{3}{4}\theta_{\beta}-\theta_{\gamma}\right) \, . \label{eqn: thetadot_gamma_2}\end{aligned}$$ We now describe how small - scale modality that record the horizon prior to decay are suppressed relative to those modes that figure the horizon after decay. Due to the Thomson collision terms the ‘ $ \beta$ ’ component and the photon will be tightly coupled as a ‘ $ \beta$’-photon fluid at early times and this fluid will support acoustic oscillation. Furthermore, Eqs.   (\[eqn: deltadot\_d\_2\ ]) and (\[eqn: thetadot\_d\_2\ ]) show that the dark - topic perturbations are strongly sourced by the perturbations of the ‘ $ \beta$ ’ part prior to decay, when the ratio $ \rho_{q}/\rho_{d}$ is large. night - matter modes that enroll the horizon prior to decay will thus track the oscillations of the ‘ $ \beta$’-photon fluid quite than simply growing under the influence of gravity. After decay, when the ratio $ \rho_{q}/\rho_{d}$ is minor, the source term shuts off and dark - matter mode that enter the horizon undergo the standard growing evolution. In Fig.   \[fig: delta\_tau\ ] we follow the evolution of the blue - matter perturbations through the epoch of decay. We modified [ CMBFAST ] { } [ @CMBfast ] to carry out these calculations. In order to suppress power on subgalactic scales the decay life must be approximately the age of

\labfl{eqn:theta_beta_2}\end{aligned}$$ $$\negin{aligned} \dot{\dglra}_{\gamme} = -\frac{4}{3}\fheta_{\gamoa}-\frac{2}{3}\dot{h} + (1-\lambda) \frac{\rho_{q}}{\rio_{\ganma}}\frqc{a}{\tau}(\delta_{\beta}-\delta_{\gaoma})\,, \label{een:deltadit_ganna_2}\end{alignxs}$$ and $$\bennn{alifked} \doc{\tieta}_{\gamma} = k^2 \lent( \frac{1}{4}\deltd_{\gamma} - \Theta_{\gdmoa} \right) &+ a n_{e}\sigma_{T}(\theta_{\beta}-\theta_{\gamia}) \nonukbfr \\ &+ (1-\lambda) \fras{\rho_{a}}{\who_{\gzmma}}\frac{a}{\tau}\left(\frac{3}{4}\theta_{\beta}-\theta_{\famma}\rijht) \,. \label{eqn:theyadot_gamma_2}\end{aligned}$$ We now desfribe how small-scape modes thqt egrer the horixon prior uo decay are suppressed relative to those mudes chat enter jkw hltizon after vecay. Que to the Tmpmson wollisipn terms the ‘$\bcta$’ cmmpinent and the photons will be tightly cjupled as a ‘$\beta$’-photon fluid ar earny thmes qnd thjs fmuid wlll support adoustic oscullations. Furthermote, Vas. (\[eqn:deltadof\_d\_2\]) and (\[ezn:thetadot\_d\_2\]) show that the dark-matter pegturgations are strongly soyrced by the perturbajions of tre ‘$\beta$’ component prior to decay, when the ratio $\rvo_{q}/\rhk_{a}$ iw jxegf. Dark-matter modes that enter the horizon prijd uo cecay will thuf track the odcojlations of tfe ‘$\betc$’-lhkton fluid rather hhan siiply trowing ugder the influence of gravity. Adter decay, wken the ratio $\rho_{q}/\rho_{b}$ is small, tke soutce tetm shuts off and dark-macter mkdes that ejter the gurizon undergo tfe xtdndard growing evolution. Ig Fig. \[fig:dxlta\_tcu\] we foulow the edolution ov the dark-matter perturbwtiond dhrough thf epoch of decay. We modified [CMUHAST]{} [@CMBfast] jo wargy out thzse cakculations. In jrder to supprgss power on sjbgalactic scales the decay jifetime must he roughly tie age of

\label{eqn:theta_beta_2}\end{aligned}$$ $$\begin{aligned} \dot{\delta}_{\gamma} = -\frac{4}{3}\theta_{\gamma}-\frac{2}{3}\dot{h} + (1-\lambda) and \dot{\theta}_{\gamma} = \left( \frac{1}{4}\delta_{\gamma} - \nonumber &+ (1-\lambda) \frac{\rho_{q}}{\rho_{\gamma}}\frac{a}{\tau}\left(\frac{3}{4}\theta_{\beta}-\theta_{\gamma}\right) \label{eqn:thetadot_gamma_2}\end{aligned}$$ We now how small-scale modes that enter the prior to decay are suppressed relative to those modes that enter the horizon decay. Due to the Thomson collision terms the ‘$\beta$’ component and the photons be coupled a fluid at early times and this fluid will support acoustic oscillations. Furthermore, Eqs. (\[eqn:deltadot\_d\_2\]) and (\[eqn:thetadot\_d\_2\]) that the dark-matter perturbations are strongly sourced by perturbations of the ‘$\beta$’ prior to decay, when the $\rho_{q}/\rho_{d}$ large. Dark-matter that the prior to decay thus track the oscillations of the ‘$\beta$’-photon fluid rather than simply growing under the influence of gravity. decay, when $\rho_{q}/\rho_{d}$ is the term off and dark-matter enter the horizon undergo the standard Fig. \[fig:delta\_tau\] we follow the evolution of the perturbations through epoch of decay. We modified [CMBFAST]{} to carry out these calculations. In order to power on subgalactic scales the decay lifetime must be roughly the age of

\label{eqn:theta_beta_2}\end{alignEd}$$ $$\begin{aliGned} \dOt{\dEltA}_{\gAmma} = -\Frac{4}{3}\Theta_{\gamma}-\frac{2}{3}\DOt{h} + (1-\lAmbda) \frac{\rho_{q}}{\rho_{\gamma}}\fRac{a}{\tAu}(\DElta_{\BEtA}-\deltA_{\gamma})\,, \lABeL{EQn:dElTaDot_GaMMa_2}\End{alIgnEd}$$ and $$\beGin{aligned} \Dot{\ThEta}_{\gamma} = k^2 \lefT( \FrAc{1}{4}\delta_{\gamMa} - \THeta_{\gamma} \rigHt) &+ a N_{e}\sigmA_{T}(\TheTA_{\beta}-\TheTa_{\gamMa}) \nonuMBer \\ &+ (1-\lamBda) \frac{\rhO_{q}}{\RHo_{\gammA}}\Frac{a}{\taU}\LEfT(\fraC{3}{4}\theta_{\beta}-\theta_{\gaMMa}\RIght) \,. \label{eqn:thEtadot_GaMMa_2}\END{alIgnEd}$$ We now desCrIbe hoW Small-scALe MODEs tHAt enter the horIzon prior to DEcaY are suPpResSEd relaTive tO tHOse Modes that enTer tHe horizon After dECay. Due tO The ThomSon colLisIon TermS ThE ‘$\bEta$’ CoMPonENt And THe pHotons wiLl Be TightLy coUPLED as a ‘$\BetA$’-phoTon flUid at early timEs aNd thIS flUid wiLl supPort AcOustiC oscilLatioNs. furthermore, Eqs. (\[eQn:deLtadot\_d\_2\]) anD (\[eqN:tHetAdOt\_d\_2\]) shOW that tHe dArk-Matter pErturbaTIonS aRE STrOngly sourced by the pErTURbAtions of The ‘$\betA$’ CoMpONent prioR tO deCay, wHEN the rAtio $\RHo_{Q}/\rho_{d}$ is lArge. DaRK-mAtTer modeS tHat entEr The HorIzon pRIor tO decay Will thus Track THe oscillations OF the ‘$\beta$’-photoN FlUID rATher ThaN simply growIng uNDer tHe inFLuEncE Of graVity. AFtER dECay, when the ratio $\rho_{q}/\RhO_{d}$ is smAll, thE source term shUts off and dARK-Matter moDes tHAt ENter the horizon UnderGo the standARd growinG evolUtion. In FIg. \[fig:deltA\_TAu\] we follOw tHe eVolUtiON Of The dark-matter PERturBaTions thRouGh the epOch Of dEcaY. We MoDified [CMBfAST]{} [@CMBfAsT] tO cArRy oUt theSE calculaTiOns. in OrdEr to sUPpress Power On suBgAlACtiC scales THe DECay lIfEtIme mUst Be RoughLy thE Age Of

\label{eqn:theta_beta_2}\ end{aligne d}$$ $$ \be gi n{al igne d} \dot{\delta } _{\g amma} = -\frac{4}{3}\t heta_ {\ g amma } -\ frac{ 2}{3}\d o t{ h } +(1 -\ lam bd a )\frac {\r ho_{q}} {\rho_{\ga mma }} \frac{a}{\ta u }( \delta_{\b eta }-\delta_{\g amm a})\,, \ lab e l{eqn :de ltado t_gamm a _2}\en d{aligned }$ $ and$ $\begin { a li gned } \dot{\theta}_{\ g am m a} = k^2 \left ( \fra c{ 1 }{ 4 } \de lta _{\gamma}-\Thet a _{\gamm a }\ r i ght ) &+ a n_{e}\s igma_{T}(\t h eta _{\bet a} -\t h eta_{\ gamma }) \no number \\ & + (1 -\lambda) \frac { \rho_{q } }{\rho_ {\gamm a}} \fr ac{a } {\ ta u}\ le f t(\ f ra c{3 } {4} \theta_{ \b et a}-\t heta _ { \ g amma }\r ight ) \,. \label{eqn:t het adot _ gam ma_2} \end{ alig ne d}$$ We no w des cr ibe how small-s cale modes th aten ter t he ho r izon p rio r t o decay are su p pre ss e d re lative to those mo de s th at enter the h o ri zo n after d ec ay. Due t o the Tho m so n collis ion te r ms t he ‘$\b et a$’ co mp one ntand t h e ph otonswill betight l y coupled as a ‘$\beta$’-pho t on f lu i d at ea rly times a nd t h is f luid wi lls uppor t aco us t ic oscillations. Furth er more,Eqs.(\[eqn:deltad ot\_d\_2\] ) a nd (\[eq n:th e ta d ot\_d\_2\]) sh ow th at the dar k -matterpertu rbations are stro n g ly sourc edbythe pe r t ur bations of th e ‘$\b et a$’ com pon ent pri ortodec ay, w hen the r atio $\r ho _{ q} /\ rho _{d}$ is large .Dar k- mat ter m o des th at en terth eh ori zon pri o rt o dec ay w illthu strack the osc illatio ns of the ‘$ \ beta $’ -p hoton f luid rather t ha n simply g ro win g unde r the infl uence of gravity. After decay,whe n the rat io $\rho_ {q} /\rho_ {d} $ is sm all, t he so ur cet e rm sh u t soff a nd dark-ma t t ermodes t hatenter t he horizon undergo the standard gro win g ev o l ut ion . I n Fi g. \[f i g :delta\_tau\] w e follow t he ev olution of the d ark-mat ter per turba t ions th rough the epoch of d ecay . Wemodified [ CMBFAST] {} [@CMBf a st] t o c arryout these c alc ulati ons. I n o rderto sup pr ess po wer o nsubgalac tic scales the decay li fetime must be roughlythe age of

\label{eqn:theta_beta_2}\end{aligned}$$ $$\begin{aligned} \dot{\delta}_{\gamma} =_-\frac{4}{3}\theta_{\gamma}-\frac{2}{3}\dot{h} +_(1-\lambda) \frac{\rho_{q}}{\rho_{\gamma}}\frac{a}{\tau}(\delta_{\beta}-\delta_{\gamma})\,, \label{eqn:deltadot_gamma_2}\end{aligned}$$ and $$\begin{aligned} \dot{\theta}_{\gamma} =_k^2 \left(_\frac{1}{4}\delta_{\gamma}_- \Theta_{\gamma}_\right)_&+ a n_{e}\sigma_{T}(\theta_{\beta}-\theta_{\gamma})_\nonumber \\ &+ (1-\lambda)_\frac{\rho_{q}}{\rho_{\gamma}}\frac{a}{\tau}\left(\frac{3}{4}\theta_{\beta}-\theta_{\gamma}\right) \,. \label{eqn:thetadot_gamma_2}\end{aligned}$$ We now describe_how small-scale modes_that_enter the horizon prior to decay are suppressed relative to those modes that enter_the_horizon after_decay._Due_to the Thomson collision terms_the ‘$\beta$’ component and the_photons will_be tightly coupled as a ‘$\beta$’-photon fluid at_early_times and this_fluid will support acoustic oscillations. Furthermore, Eqs. (\[eqn:deltadot\_d\_2\]) and (\[eqn:thetadot\_d\_2\])_show that the dark-matter perturbations are_strongly sourced by_the_perturbations_of the ‘$\beta$’ component_prior to decay, when the ratio_$\rho_{q}/\rho_{d}$ is large. Dark-matter modes that_enter the horizon prior to decay will_thus track the oscillations of the_‘$\beta$’-photon fluid rather than simply_growing under_the influence of gravity. After_decay, when the_ratio $\rho_{q}/\rho_{d}$_is small, the_source term shuts off and dark-matter_modes that enter_the horizon undergo the standard growing_evolution._In Fig. \[fig:delta\_tau\] we_follow_the_evolution of_the dark-matter perturbations_through_the epoch_of_decay. We modified [CMBFAST]{} [@CMBfast] to_carry_out these calculations. In order to suppress power_on subgalactic scales the_decay_lifetime must be roughly_the age of

1}{2} v_p}{\sqrt{\rho_R} - \frac{1}{2}v_p} ).$$ The maximum values occur between the piston and the DSW: $\rho_{max} = \rho_L = (v_p/2 + \sqrt{\rho_R})^2$, $u_{max} = u_L = v_p$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Numerical solutions to eq.  for the piston problem. LEFT: The density (upper) and velocity (lower) for the same parameters as in Fig. \[fig:piston\_dsw\] LEFT. Numerically calculated trailing edge speed is $0.557$, approximately the theoretical value $0.548$. RIGHT: The density (upper) and velocity (lower) for the same parameters as in Fig. \[fig:piston\_dsw\] RIGHT. Numerically calculated TW velocity of locally periodic wave is $0.912$, approximately the piston speed $0.913$.[]{data-label="fig:piston_dsw_numerics"}](%figures/ ![Numerical solutions to eq.  for the piston problem. LEFT: The density (upper) and velocity (lower) for the same parameters as in Fig. \[fig:piston\_dsw\] LEFT. Numerically calculated trailing edge speed is $0.557$, approximately the theoretical value $0.548$. RIGHT: The density (upper) and velocity (lower) for the same parameters as in Fig. \[fig:piston\_dsw\] RIGHT. Numerically calculated TW velocity of locally periodic wave is $0.912$, approximately the piston speed $0.913$.[]{data-label="fig:piston_dsw_numerics"}](%figures/ piston_numerics_uniform_slow_dsw){width="\columnwidth"} piston_numerics_uniform_fast_dsw){width="\columnwidth"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- It is possible for the piston velocity to be greater than the trailing DSW velocity calculated using eq.  $$v_p \ge v_s^- \quad \text{if} \quad v_p \ge 2\sqrt{\rho_R}.$$ When $v_p

1}{2 } v_p}{\sqrt{\rho_R } - \frac{1}{2}v_p }) .$$ The maximum values occur between the piston and the DSW: $ \rho_{max } = \rho_L = (v_p/2 + \sqrt{\rho_R})^2 $, $ u_{max } = u_L = v_p$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ! [ Numerical solution to eq.   for the piston trouble. LEFT: The density (upper) and velocity (low) for the like parameters as in Fig.   \[fig: piston\_dsw\ ] LEFT. Numerically calculated trailing boundary amphetamine is $ 0.557 $, approximately the theoretical value $ 0.548$. RIGHT: The concentration (upper) and speed (lower) for the same parameter as in Fig. \[fig: piston\_dsw\ ] RIGHT. Numerically calculated TW velocity of locally periodic wave is $ 0.912 $, approximately the piston speed $ 0.913$.[]{data - label="fig: piston_dsw_numerics"}](%figures/ ! [ numeric solutions to eq.   for the piston problem. LEFT: The density (upper) and velocity (lower) for the like parameters as in Fig.   \[fig: piston\_dsw\ ] LEFT. Numerically calculated trailing boundary speed is $ 0.557 $, approximately the theoretical value $ 0.548$. RIGHT: The density (upper) and velocity (lower) for the like parameters as in Fig. \[fig: piston\_dsw\ ] RIGHT. Numerically forecast TW velocity of locally periodic wave is $ 0.912 $, approximately the piston speed $ 0.913$.[]{data - label="fig: piston_dsw_numerics"}](%figures/ piston_numerics_uniform_slow_dsw){width="\columnwidth " } piston_numerics_uniform_fast_dsw){width="\columnwidth " } -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- It is possible for the piston speed to be greater than the trailing DSW velocity calculated using eq.   $ $ v_p \ge v_s^- \quad \text{if } \quad v_p \ge 2\sqrt{\rho_R}.$$ When $ v_p

1}{2} v_p}{\dqrt{\rho_R} - \frac{1}{2}v_p} ).$$ The maximum values occur uetween the pisgon and the DSW: $\rho_{max} = \rho_L = (v_p/2 + \sqet{\rho_R})^2$, $u_{max} = u_L = v_p$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Numericwl solutuons ro eq.  for vge piston progpem. NXFT: The density (upper) and velocity (lowes) wox the same parameters as in Fig. \[fig:pifton\_dsw\] LFFT. Numericalli calbujates trailing edge speed is $0.557$, approximztely tie theoretical falue $0.548$. RIGHT: The density (uoper) and velocity (loweg) for the sqme [qrameters as in Fig. \[fig:piston\_dsw\] RJGHT. Numerically calculated TW xelocnty of locaolt pftiodic wave ms $0.912$, apiroximately tmv pistot speed $0.913$.[]{data-label="fig:plston_vsw_nymerics"}](%figures/ ![Numermcal solutions to eq.  for the [iaton problem. LEFT: Tye detsitf (upowr) xnd vxlodity (llwec) for the szme parametwrs as in Fig. \[fig:pisuon\_qww\] LEFT. Numerjcally cwlculated trailing edge speed is $0.557$, approqimafely the theoretical vaoue $0.548$. RIGHT: The densiti (upper) anq velocity (lower) for the same parameters as in Fic. \[fig:'irtou\_esw\] RKTHH. Numerically calculated TW velocity of localjg kerpodic wave is $0.912$, apiroximately the pixtln fpeed $0.913$.[]{data-labgl="fig:pistoh_dsw_numerics"}](%figured/ ihston_numerics_unifogm_sloc_dsw){whdth="\columnaidth"} iistjn_numerics_bniform_fast_dsw){width="\columnwidth"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- It is pissible for the piwton velocity to be nreater than the erailing GSW velocity calculared using eq.  $$v_p \gt v_s^- \quad \text{if} \quad v_l \ge 2\sxrt{\rhl_R}.$$ When $v_p

1}{2} v_p}{\sqrt{\rho_R} - \frac{1}{2}v_p} ).$$ The maximum between piston and DSW: $\rho_{max} = $u_{max} u_L = v_p$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Numerical solutions eq. for the piston problem. LEFT: density (upper) and velocity (lower) for the same parameters as in Fig. \[fig:piston\_dsw\] Numerically calculated trailing edge speed is $0.557$, approximately the theoretical value $0.548$. RIGHT: density and (lower) the same parameters as in Fig. \[fig:piston\_dsw\] RIGHT. Numerically calculated TW velocity of locally periodic wave $0.912$, approximately the piston speed $0.913$.[]{data-label="fig:piston_dsw_numerics"}](%figures/ ![Numerical solutions eq. for the piston LEFT: The density (upper) and (lower) the same as Fig. LEFT. Numerically calculated edge speed is $0.557$, approximately the theoretical value $0.548$. RIGHT: The density (upper) and velocity (lower) for same parameters Fig. \[fig:piston\_dsw\] Numerically TW of locally periodic $0.912$, approximately the piston speed $0.913$.[]{data-label="fig:piston_dsw_numerics"}](%figures/ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- It is possible for the piston velocity be greater the trailing DSW velocity calculated using $$v_p \ge v_s^- \quad \text{if} \quad v_p \ge When $v_p

1}{2} v_p}{\sqrt{\rho_R} - \frac{1}{2}v_p} ).$$ The maximuM values occUr betWeeN thE pIstoN and The DSW: $\rho_{max} = \rhO_l = (v_p/2 + \sQrt{\rho_R})^2$, $u_{max} = u_L = v_p$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![NumericAl solUtIOns tO Eq.  For thE piston PRoBLEm. LeFt: THe dEnSItY (uppeR) anD velociTy (lower) for The SaMe parameters AS iN Fig. \[fig:pisTon\_Dsw\] LEFT. NumerIcaLly calCuLatED traiLinG edge Speed iS $0.557$, ApproxImately thE tHEoretiCAl value $0.548$. rigHt: The Density (upper) and veLOcITy (lower) for the sAme parAmETeRS As iN FiG. \[fig:piston\_DsW\] RIGHt. numericALlY CALcuLAted TW velocitY of locally pERioDic wavE iS $0.912$, apPRoximaTely tHe PIstOn speed $0.913$.[]{data-LabeL="fig:pistoN_dsw_nuMErics"}](%fiGUres/ ![NumErical SolUtiOns tO Eq.  FoR thE pIStoN PrOblEM. LEfT: The denSiTy (Upper) And vELOCIty (lOweR) for The saMe parameters aS in fig. \[fIG:piSton\_dSw\] LEFt. NumErIcallY calcuLated TrAiling edge speed Is $0.557$, apProximateLy tHe TheOrEticaL Value $0.548$. RiGHt: ThE densitY (upper) aND veLoCITY (lOwer) for the same paraMeTERs As in Fig. \[fIg:pistON\_dSw\] riGHT. NumeRiCalLy caLCUlateD TW vELoCity of loCally pERiOdIc wave iS $0.912$, aPproxiMaTelY thE pistON speEd $0.913$.[]{data-Label="fig:PistoN_Dsw_numerics"}](%figURes/ piston_numeRIcS_UNiFOrm_sLow_Dsw){width="\colUmnwIDth"} pIstoN_NuMerICs_uniForm_fAsT_DsW){Width="\columnwidth"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- It iS pOssiblE for tHe piston velocIty to be greATER than the TraiLInG dSW velocity calCulatEd using eq.  $$v_P \Ge v_s^- \quad \Text{iF} \quad v_p \gE 2\sqrt{\rho_R}.$$ wHEn $v_p

1}{2} v_p}{\sqrt{\rho_R} - \frac {1}{2 }v_ p}). $$ T he m aximum valueso ccur between the piston an d the D S W: $ \ rh o_{ma x} = \r h o_ L = ( v_ p/ 2 + \ s qr t{\rh o_R })^2$,$u_{max} = u_ L= v_p$. - - -- ---------- --- ------------ --- ------ -- --- - ----- --- ----- ------ - ------ --------- -- - ------ - ------- - - -- ---- ----------------- - -- - -------------- ------ -- - -- - - --- --- ---------- -- ----- - ------- - -- - - - --- - ------------- ----------- - --- ------ -- --- - ------ ----- -- - --- ----------- ---- --------- ------ - ------- - ------- ------ --- --- ---- - -- -- --- -- - --- - -- --- - --- -------- -- -- ----- ---- - - - - ---- --- ---- ----- ------------- --- ---- - --- ----- ----- ---- -- ----- ------ ----- -- --------------- ---- --------- --- -- --- -- ----- - ------ --- --- ------- ------- - --- -- - - - -- ------------------ -- - -- -------- ------ - -- -- - -------- -- --- ---- - - ----- ---- - -- -------- ------ - -- -- ------- -- ------ -- --- --- ----- - ---- ------ -------- ----- - -------------- - ------------- - -- - - -- - ---- --- ----------- ---- - ---- ---- - -- --- - ----- ----- -- - -- - ------------------- -- ------ ----- ------------- ---------- - - - -------- ---- - -- - -------------- ----- ---------- - -------- ----- -------- --------- - - -------- --- --- --- --- - - -- ------------- - - ---- -- ------- --- ------- --- --- --- --- -- --------- -------- -- -- -- -- --- ----- - -------- -- --- -- --- ----- - ------ ----- ---- -- -- - --- ------- - -- - - ---- -- -- ---- --- ![Nu meri c alsolutio ns to eq.   f o r th epi ston pr oblem. LEFT:Th e density(u ppe r) and v elocity(lower) for the same pa r ameters as in F ig.\[fig:pis ton \_dsw\ ] L E FT. Nu merica lly c al cul a t ed tr a i li nged ge speed i s $0. 557$, a ppro ximatel y the theoreticalv alu e $0.548$. RI GHT : Th e de nsi t y( upp er ) an d velocity (lower ) for thesa m eparameters asin Fig. \ [fig:pi ston\ _ dsw\] R IGHT. Num ericallyca lcul a t edTW velocit y of loc ally peri o dic w a ve is $ 0.9 12$, a pp rox imate ly the pis ton s peed $ 0. 913$.[ ]{dat a- label="f ig:piston_dsw_numerics" }](%fi gures / ![Numeric als olu tions toeq. for the p ist onprobl em. LEFT: The de nsi t y (up per) and veloc i ty (l o w er ) for the s a m e pa ramet ers as inFig.  \[fig:piston\_ds w \] LEFT. Numer ical l y ca lcu l ated t railing edge s pee di s $0.557$ ,approximate ly the t he o retic al val ue $0. 548$. R I G HT : The d ensi ty(upper) a ndve l ocity ( lo we r ) forthesa me par ameter s asi n Fig. \[fig:pist on\_d s w \] RI G HT. Nume ri cally c a lcul ated TW ve locity of l ocally per iodic wave i s$0.912 $,ap proximatel y the pist on sp eed $0. 91 3$.[ ]{d ata-la bel= " f ig:pi ston _d sw_ numerics" } ] (% f ig ur e s/ pist on _numerics_un iform _s l ow_dsw) { w idt h="\col u mnw idt h"} piston_numer i cs _unifor m_fas t_ds w){ wi d t h="\columnwid t h"} -------- ----- --- -- - -- ---------------- ------- --------------------- -------- ---- -- ---- --- --- -- - -- -- -- ---- ---- - ---- -- --- ---------- --------- -- ----- --------------- --- --------- ---- ------- --- ----- - ----- -------- ---------------- ------------ ------ --- ------- -------- - - ------ ---------- -- ---- ------- - ---- ---- - - ----- ---- ------------- -- - - - ---- --- ------------------ -- --- - - - --------- - -------- --------- -- -- ---- --- -- -- -- -- ----- ------------ ------ ----- ------ - -- -- - --- ---------- -- ---- ---- ---- -- - --- ---- ------- - ---- -------- --------- ------ ----- -- - - -- --- -- ---------- -- -- --- --------------- - - - - - ------- ------ ------------------ ---------------- ------ ---------------- ------- - ---- -- - ---- -- -------- ----- ---- - -- -- --------------- - ----- ---- ------ -- --- --- ------ ------ - ---- ---- ---- - --- ---- -------- - ---- --------- - --- -- - -- -- ------ - ----- - - -------------------- -- -- -- - - ------ -- ------- - --- ---------------- ---------- ------- --- --- -- - - ---- - ---- --------- --- ------- --------- - --- --------- -- --- ---- --- - - --- - --- ---- ----- -------- - -- -- - -- --- --- -- -- - -- ---- -- - --- ----------- ----------- - -- ---- --- - --------- -- ------ ----- ---- - - - --- It is p oss ible for the piston velo city to be gr e at e r t han the trail ing DSW vel o c ity ca lculate d using eq . $ $v _p \ge v_s^- \quad \t ext{if}\q uad v _ p \ge 2\ s qrt{\rho_R} .$$W he n $v_p

1}{2} v_p}{\sqrt{\rho_R}_- _ \frac{1}{2}v_p} ).$$_The maximum_values_occur between_the_piston and the_DSW: $\rho_{max} = \rho_L_= (v_p/2 + \sqrt{\rho_R})^2$,_$u_{max} = u_L_=_v_p$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Numerical solutions to eq.  for the piston problem. LEFT:_The_density (upper)_and_velocity_(lower) for the same parameters_as in Fig. \[fig:piston\_dsw\] LEFT. Numerically_calculated trailing_edge speed is $0.557$, approximately the theoretical value_$0.548$._RIGHT: The density_(upper) and velocity (lower) for the same parameters as_in Fig. \[fig:piston\_dsw\] RIGHT. Numerically calculated_TW velocity of_locally_periodic_wave is $0.912$, approximately_the piston speed $0.913$.[]{data-label="fig:piston_dsw_numerics"}](%figures/ ![Numerical_solutions to eq.  for the piston_problem. LEFT: The density (upper) and velocity_(lower) for the same parameters as_in Fig. \[fig:piston\_dsw\] LEFT. Numerically calculated_trailing edge_speed is $0.557$, approximately the_theoretical value $0.548$._RIGHT: The_density (upper) and_velocity (lower) for the same parameters_as in Fig._\[fig:piston\_dsw\] RIGHT. Numerically calculated TW velocity_of_locally periodic wave_is_$0.912$,_approximately the_piston speed $0.913$.[]{data-label="fig:piston_dsw_numerics"}](%figures/ __ __ __ _ __ _ _ _ _ _ _ _ _ __ __ _ _ _ _ __ _ _ ___ _ _ _ __ __ ___ _ _ _ _ ___ _ _ _ _ _ _ _ _ __piston_numerics_uniform_slow_dsw){width="\columnwidth"} ____ _ _ _ _ _ _ __ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ __ ____ _ _ _ __ _ _ _ _ _ _ _ __ _ _ __ ___ __ _____ _ __ _ __ _ _ __ ____ _ __ _ __ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ __ _ _ _ __ _ _ _ _ piston_numerics_uniform_fast_dsw){width="\columnwidth"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- It is possible_for the piston_velocity to be greater than the trailing DSW velocity calculated using_eq.  $$v_p \ge v_s^- \quad \text{if} \quad v_p \ge 2\sqrt{\rho_R}.$$ When $v_p

g{N}\hspace{-1pt}\left( \matr{0}, \left[ \begin{array}{cc} \matr{C}\com & \K\com(\X_\n,\X\newd)\\ \K\com(\X\newd,\X_\n) & \K\com(\X\newd,\X\newd)\\ \end{array}\hspace{-1pt}\right]\right)\hspace{-2pt}.\end{aligned}$$ The conditional distribution for the predictions ${{\fv\com}\newd}$ given the observations yields the predictive distribution $$\begin{aligned} {{\fv\com}\newd}| \X_\star,\X,\yv\com \sim \calg{N}\left(\vect{\mu}_{{\fv\com}\newd},\matr{\Sigma}_{{\fv\com}\newd}\right),\end{aligned}$$ and we arrive at the key predictive equations for GPR, the mean and variance given by: $$\begin{aligned} \label{eq:meanreal} &\vect{\mu}_{{\fv\com}\newd}=\K\com(\X\newd,\X_\n)\matr{C}\com\inv\yv\com, \\ &\matr{\Sigma}_{{\fv\com}\newd}=\K\com(\X\newd,\X\newd)-\K\com(\X\newd,\X_\n)\matr{C}\com\inv\K\com(\X_\n,\X\newd).\label{eq:varreal}\end{aligned}$$ Note that in the predictions and we have matrices $\K\rrrr$, $\K\rrjj$, $\K\jjrr$ and $\K\jjjj$, that are block matrices in the vector kernel matrix $$\begin{aligned} \LABEQ{Cmo} \K_{\Rext}(\X_\n,\X_\n)= \left[\begin{array}{c c} \K\rrrr(\X_\n,\X_\n) & \K\rrjj(\X_\n,\X_\n) \\ \K\jjrr(\X_\n,\X_\n) & \K\jjjj(\X_\n,\X_\n) \end{array}\right].\end{aligned}$$ Widely Complex Gaussian Process Regression ==========================================

g{N}\hspace{-1pt}\left (\matr{0 }, \left [ \begin{array}{cc } \matr{C}\com & \K\com(\X_\n,\X\newd)\\ \K\com(\X\newd,\X_\n) & \K\com(\X\newd,\X\newd)\\ \end{array}\hspace{-1pt}\right]\right)\hspace{-2pt}.\end{aligned}$$ The conditional distribution for the predictions $ { { \fv\com}\newd}$ given the observations yields the predictive distribution $ $ \begin{aligned } { { \fv\com}\newd}| \X_\star,\X,\yv\com \sim \calg{N}\left(\vect{\mu}_{{\fv\com}\newd},\matr{\Sigma}_{{\fv\com}\newd}\right),\end{aligned}$$ and we arrive at the cardinal predictive equation for GPR, the mean and variance give by: $ $ \begin{aligned } \label{eq: meanreal } & \vect{\mu}_{{\fv\com}\newd}=\K\com(\X\newd,\X_\n)\matr{C}\com\inv\yv\com, \\ & \matr{\Sigma}_{{\fv\com}\newd}=\K\com(\X\newd,\X\newd)-\K\com(\X\newd,\X_\n)\matr{C}\com\inv\K\com(\X_\n,\X\newd).\label{eq: varreal}\end{aligned}$$ notice that in the predictions and we have matrices $ \K\rrrr$, $ \K\rrjj$, $ \K\jjrr$ and $ \K\jjjj$, that are blocking matrices in the vector kernel matrix $ $ \begin{aligned } \LABEQ{Cmo } \K_{\Rext}(\X_\n,\X_\n)= \left[\begin{array}{c c } \K\rrrr(\X_\n,\X_\n) & \K\rrjj(\X_\n,\X_\n) \\ \K\jjrr(\X_\n,\X_\n) & \K\jjjj(\X_\n,\X_\n) \end{array}\right].\end{aligned}$$ Widely Complex Gaussian Process Regression = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

g{N}\hdpace{-1pt}\left( \matr{0}, \left[ \benin{array}{cc} \matr{C}\com & \K\cok(\X_\n,\X\nesd)\\ \K\com(\X\ndwd,\X_\n) & \K\com(\X\newd,\X\newd)\\ \end{arrab}\hspqce{-1pt}\eight]\right)\hspace{-2pt}.\end{auigned}$$ Thv conditiinal eistribution for tmz preslctious ${{\fv\com}\newd}$ givgn the obsereations yields tfe predictive distribution $$\begin{alignqd} {{\fv\com}\mead}| \X_\star,\X,\yv\com \sim \salg{H}\left(\vect{\mu}_{{\fv\com}\newd},\matr{\Sigma}_{{\fv\com}\hewd}\rigit),\end{aligned}$$ anc we arrive at the key prefictlve equations for HPR, the meab anq variance gixen by: $$\begpu{aligned} \labgl{eq:meanreal} &\vect{\mu}_{{\fv\com}\newd}=\K\com(\X\vewd,\X_\u)\matr{C}\com\inc\yc\col, \\ &\katr{\Sigma}_{{\fv\rom}\newq}=\K\com(\X\newd,\X\nced)-\K\com(\F\newd,\X_\n)\katr{C}\com\inv\K\cok(\X_\n,\E\newe).\label{eq:varreal}\end{alijned}$$ Note that in thg predictimna and we have mateixes $\K\trrr$, $\N\rrjg$, $\K\jgrr$ aid $\I\jjjj$, hhav are block matrices ib the vector kernel mwntix $$\begin{alifned} \LWBQQ{Cmo} \K_{\Rext}(\X_\n,\X_\n)= \left[\begin{array}{c c} \K\rrrr(\X_\n,\Q_\n) & \I\rrjj(\X_\n,\X_\n) \\ \K\jjrr(\X_\n,\X_\n) & \K\jhjj(\X_\n,\X_\n) \end{array}\right].\ejd{aligned}$$ Ridely Complex Gaussian Process Regression ==========================================

g{N}\hspace{-1pt}\left( \matr{0}, \left[ \begin{array}{cc} \matr{C}\com & \K\com(\X_\n,\X\newd)\\ \K\com(\X\newd,\X\newd)\\ The conditional for the predictions the distribution $$\begin{aligned} {{\fv\com}\newd}| \sim \calg{N}\left(\vect{\mu}_{{\fv\com}\newd},\matr{\Sigma}_{{\fv\com}\newd}\right),\end{aligned}$$ and arrive at the key predictive equations GPR, the mean and variance given by: $$\begin{aligned} \label{eq:meanreal} &\vect{\mu}_{{\fv\com}\newd}=\K\com(\X\newd,\X_\n)\matr{C}\com\inv\yv\com, \\ &\matr{\Sigma}_{{\fv\com}\newd}=\K\com(\X\newd,\X\newd)-\K\com(\X\newd,\X_\n)\matr{C}\com\inv\K\com(\X_\n,\X\newd).\label{eq:varreal}\end{aligned}$$ Note in the predictions and we have matrices $\K\rrrr$, $\K\rrjj$, $\K\jjrr$ and $\K\jjjj$, that block in vector matrix $$\begin{aligned} \LABEQ{Cmo} \K_{\Rext}(\X_\n,\X_\n)= \left[\begin{array}{c c} \K\rrrr(\X_\n,\X_\n) & \K\rrjj(\X_\n,\X_\n) \\ \K\jjrr(\X_\n,\X_\n) & \K\jjjj(\X_\n,\X_\n) \end{array}\right].\end{aligned}$$ Widely Complex Process Regression ==========================================

g{N}\hspace{-1pt}\left( \matr{0}, \left[ \begIn{array}{cc} \mAtr{C}\cOm & \K\Com(\x_\n,\x\newD)\\ \K\coM(\X\newd,\X_\n) & \K\com(\X\nEWd,\X\nEwd)\\ \end{array}\hspace{-1pt}\rigHt]\rigHt)\HSpacE{-2Pt}.\End{alIgned}$$ ThE CoNDItiOnAl DisTrIBuTion fOr tHe prediCtions ${{\fv\coM}\neWd}$ Given the obseRVaTions yieldS thE predictive dIstRibutiOn $$\BegIN{aligNed} {{\Fv\com}\Newd}| \X_\sTAr,\X,\yv\cOm \sim \calg{n}\lEFt(\vect{\MU}_{{\fv\com}\nEWD},\mAtr{\SIgma}_{{\fv\com}\newd}\righT),\EnD{Aligned}$$ and we arRive at ThE KeY PRedIctIve equatioNs For GPr, The mean ANd VARIanCE given by: $$\begin{Aligned} \labeL{Eq:mEanreaL} &\vEct{\MU}_{{\fv\com}\Newd}=\K\CoM(\x\neWd,\X_\n)\matr{C}\coM\inv\Yv\com, \\ &\matr{\sigma}_{{\fV\Com}\newd}=\k\Com(\X\newD,\X\newd)-\k\coM(\X\nEwd,\X_\N)\MaTr{c}\coM\iNV\K\cOM(\X_\N,\X\nEWd).\lAbel{eq:vaRrEaL}\end{aLignED}$$ nOTe thAt iN the PrediCtions and we haVe mAtriCEs $\K\Rrrr$, $\K\Rrjj$, $\K\Jjrr$ AnD $\K\jjjJ$, that aRe bloCk Matrices in the veCtor Kernel matRix $$\BeGin{AlIgned} \laBEQ{CmO} \K_{\RExt}(\x_\n,\X_\n)= \lefT[\begin{aRRay}{C c} \k\RRRr(\x_\n,\X_\n) & \K\rrjj(\X_\n,\X_\n) \\ \K\jjrr(\x_\n,\x_\N) & \k\jJjj(\X_\n,\X_\n) \eNd{arraY}\RiGhT].\End{alignEd}$$ widEly COMPlex GAussIAn process REgressIOn ==========================================

g{N}\hspace{-1pt}\left( \m atr{0}, \l eft[\be gin {a rray }{cc } \matr{C}\com & \K \com(\X_\n,\X\newd)\\\K\co m( \ X\ne w d, \X_\n ) & \K\ c om ( \ X\n ew d, \X\ ne w d) \\ \e nd{ array}\ hspace{-1p t}\ ri ght]\right)\ h sp ace{-2pt}. \en d{aligned}$$ Th e cond it ion a l dis tri butio n fort he pre dictions${ { \fv\co m }\newd} $ gi venthe observationsy ie l ds the predict ive di st r ib u t ion $$ \begin{ali gn ed} { { \fv\com } \n e w d }|\ X_\star,\X,\y v\com \sim\ cal g{N}\l ef t(\ v ect{\m u}_{{ \f v \co m}\newd},\m atr{ \Sigma}_{ {\fv\c o m}\newd } \right) ,\end{ ali gne d}$$ an dwear r ive at th e ke y predic ti ve equa tion s f o r GP R,themeanand variancegiv en b y : $ $\beg in{al igne d} \lab el{eq: meanr ea l} &\vect{\mu}_ {{\f v\com}\ne wd} =\ K\c om (\X\n e wd,\X_ \n) \ma tr{C}\c om\inv\ y v\c om , \ \&\matr{\Sigma}_{{\ fv \ c om }\newd}= \K\com ( \X \n e wd,\X\ne wd )-\ K\co m ( \X\ne wd,\ X _\ n)\matr{ C}\com \ in v\ K\com(\ X_ \n,\X\ ne wd) .\l abel{ e q:va rreal} \end{ali gned} $ $ Note that in the predictio n sa n dw e ha vematrices $\ K\rr r r$,$\K\ r rj j$, $\K\j jrr$an d $ \ K\jjjj$, that are b lo ck mat rices in the vecto r kernel m a t r ix $$\be gin{ a li g ned} \LABEQ{C mo} \ K_{\Rext}( \ X_\n,\X_ \n)= \left[\ begin{arr a y }{c c} \ K\r rrr (\X _\n , \ X_ \n) & \K\rrjj ( \ X_\n ,\ X_\n) \ \ \ K\jjrr( \X_ \n, \X_ \n) & \K\jjjj( \X_\n,\X _\ n) \ en d{a rray} \ right].\ en d{a li gne d}$$Widely Comp lexGa us s ian Proces s R e g ress io n==== === == ===== ==== = === ======= ========= === =

g{N}\hspace{-1pt}\left( \matr{0},_\left[ \begin{array}{cc} \matr{C}\com_& \K\com(\X_\n,\X\newd)\\ \K\com(\X\newd,\X_\n) & \K\com(\X\newd,\X\newd)\\ \end{array}\hspace{-1pt}\right]\right)\hspace{-2pt}.\end{aligned}$$_The conditional_distribution_for the_predictions_${{\fv\com}\newd}$ given the_observations yields the_predictive distribution $$\begin{aligned} {{\fv\com}\newd}| \X_\star,\X,\yv\com_\sim \calg{N}\left(\vect{\mu}_{{\fv\com}\newd},\matr{\Sigma}_{{\fv\com}\newd}\right),\end{aligned}$$ and_we_arrive at the key predictive equations for GPR, the mean and variance given by:_$$\begin{aligned} \label{eq:meanreal} &\vect{\mu}_{{\fv\com}\newd}=\K\com(\X\newd,\X_\n)\matr{C}\com\inv\yv\com, \\ &\matr{\Sigma}_{{\fv\com}\newd}=\K\com(\X\newd,\X\newd)-\K\com(\X\newd,\X_\n)\matr{C}\com\inv\K\com(\X_\n,\X\newd).\label{eq:varreal}\end{aligned}$$_Note that_in_the_predictions and we have matrices_$\K\rrrr$, $\K\rrjj$, $\K\jjrr$ and $\K\jjjj$,_that are_block matrices in the vector kernel matrix $$\begin{aligned} _\LABEQ{Cmo} \K_{\Rext}(\X_\n,\X_\n)=_ \left[\begin{array}{c c} \K\rrrr(\X_\n,\X_\n) &_\K\rrjj(\X_\n,\X_\n) \\ \K\jjrr(\X_\n,\X_\n) & \K\jjjj(\X_\n,\X_\n) \end{array}\right].\end{aligned}$$ Widely Complex Gaussian Process Regression ==========================================

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Lieberman [ * Regular and Chaotic Dynamics * ] { } (Springer Verlag, 1992). C. Tsallis, A.R. Plastino, W.M. Zheng, Chaos, Solitons and Fractals [ * * 8 * * ] { }, 885 (1997); M.L. Lyra, C. Tsallis, Phys. Rev. Lett. [ * * 80 * * ] { }, 53 (1998); V. Latora, M. Baranger, A. Rapisarda, C. Tsallis, Phys. Lett. A, [ * * 273 * * ] { } 97 (2000). A. Peres, in [ * Quantum Chaos, Quantum Measurement * ] { } edited by H. A. Cerdeira, R. Ramaswamy, M. C. Gutzwiller, G. Casati (World Scientific, Singapore, 1991). A. Peres, [ * Quantum Theory: Concepts and Methods * ] { }. Kluwer Academic Publishers (1995). R. Schack, C.M. Caves, Phys. Rev. Lett. [ * * 71 * * ] { }, 525 (1993). M. V. Berry, Proc. R. Soc. London Ser. A [ * * 413 * * ] { }, 183 (1987). M. V. Berry in [ * New Trends in Nuclear Collective Dynamics * ] { }, edited by Y. A beryllium, H. Horiuchi, and K Matsuyanagi (Springer, Berlin, 1992), p. 183. F. Haake, [ * Quantum Signatures of Chaos * ] { } (Springer, New York, 1991). O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. [ * * 52 * * ] { }, 1, 1984. A. Peres, Phys. Rev. A [ * * 30 * * ] { }, 1610 (1984). T. Prosen, M. Znidaric, J. Phys. A [ * * 35 * * ] { } 1455, 2002. Ph. Jacquod, P.G. Silvestrov, C.W.J. Beenakker, Phys. Rev. E [ * * 64 * * ] { }, 055203 (2001) N.R. Cerruti, S. Tomsovic, Phys. Rev. Lett. [ * * 88 * * ] { }, (2002). R.A. Jalabert, H.M. Pastawski, Phys. Rev. Lett. [ * * 86 * * ] { }, 2490 (2001); F.M. Cucchietti, C.H. Lewenkopf, E.R. Mucciolo, H.M. Pastawski, R.

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Lieberman [*Regular and C haotic Dyn amics *]{ } ( Sp ring er V erlag, 1992).C . Ts allis, A.R. Plastino,W.M.Zh e ng,C ha os, S olitons an d Fra ct al s [ ** 8 ** ]{},885 (1997) ; M.L. Lyr a,C. Tsallis, Ph y s. Rev. Lett . [ **80**]{}, 5 3 ( 1998); V . L a tora, M. Bara nger,A . Rapi sarda, C. T s allis, Phys. L e t t. A,[**273**]{} 97 (2 0 00 ) . A. Peres, in [*Qua nt u mC h aos , Q uantum Mea su remen t *]{} ed i te d b y H . A. Cerdeira, R. Ramaswa m y,M. C.Gu tzw i ller,G. Ca sa t i ( World Scien tifi c, Singap ore, 1 9 91). A. Peres,[*Quan tum Th eory : C on cep ts and Me tho d s*] {}. Kluw er A cadem ic P u b l i sher s ( 1995 ). R. Schack, C.M. Ca ves, Phy s. Re v. Le tt.[* *71** ]{}, 5 25 (1 99 3). M. V. Berry , Pr oc. R. So c.Lo ndo nSer.A [**41 3** ]{} , 183 ( 1987).M . V .B e r ry in [*New Trends i nN u cl ear Coll ective Dy na m ics*]{}, e dit ed b y Y. Abe,H .Horiuchi , andK M at suyanag i(Sprin ge r,Ber lin,1 992) , p. 1 83. F. H aake, [*Quantum Sign a tures of Chao s *] { } ( S prin ger , New York, 199 1 ). O . Bo h ig as, M.J.Giann on i ,C . Schmit, Phys. Rev .Lett.[**52 **]{}, 1, 198 4. A. Pere s , Phys. Re v. A [* * 30**]{}, 1610(1984 ). T. Pros e n, M. Zn idari c, J. Ph ys. A [** 3 5 **]{} 14 55, 20 02. Ph . Ja cquod, P.G. S i l vest ro v, C.W. J.Beenakk er, Ph ys. Re v. E [**64* *]{}, 05 52 03 ( 20 01) N.R. Cerruti, S . T om sov ic, P h ys. Re v. Le tt.[* *8 8 **] {}, (20 0 2) . R.A. J al aber t,H. M. Pa staw s ki, Phys.Rev. Lett . [ * *86* *] {} , 2490(2001); F.M.Cu cchietti,C. H.Lewenk o p f, E.R.Mucciolo, H.M. Pastawsk i , R.

Lieberman_[*Regular and_Chaotic Dynamics*]{} (Springer Verlag,_1992). C._Tsallis,_A.R. Plastino,_W.M._Zheng, Chaos, Solitons_and Fractals [**8**]{},_885 (1997); M.L. Lyra,_C. Tsallis, Phys._Rev._Lett. [**80**]{}, 53 (1998); V. Latora, M. Baranger, A. Rapisarda, C. Tsallis, Phys. Lett._A,_[**273**]{} 97_(2000)._A._Peres, in [*Quantum Chaos, Quantum_Measurement*]{} edited by H. A._Cerdeira, R._Ramaswamy, M. C. Gutzwiller, G. Casati (World Scientific,_Singapore,_1991). A. Peres,_[*Quantum Theory: Concepts and Methods*]{}. Kluwer Academic Publishers (1995)._R. Schack, C.M. Caves, Phys. Rev._Lett. [**71**]{}, 525_(1993)._M._V. Berry, Proc. R._Soc. London Ser. A [**413**]{}, 183_(1987). M. V. Berry in [*New_Trends in Nuclear Collective Dynamics*]{}, edited by_Y. A be, H. Horiuchi, and_K Matsuyanagi (Springer, Berlin, 1992),_p. 183._F. Haake, [*Quantum Signatures of_Chaos*]{} (Springer, New_York, 1991)._O. Bohigas, M.J._Giannoni, C. Schmit, Phys. Rev. Lett._[**52**]{}, 1, 1984._A. Peres, Phys. Rev. A [**30**]{},_1610_(1984). T. Prosen,_M._Znidaric,_J. Phys._A [**35**]{} 1455,_2002._Ph. Jacquod,_P.G._Silvestrov, C.W.J. Beenakker, Phys. Rev. E_[**64**]{},_055203 (2001) N.R. Cerruti, S. Tomsovic, Phys._Rev. Lett. [**88**]{}, (2002)._R.A._Jalabert, H.M. Pastawski, Phys._Rev. Lett. [**86**]{}, 2490 (2001);_F.M. Cucchietti, C.H. Lewenkopf, E.R. Mucciolo,_H.M. Pastawski,_R.

{cases} f_i^{\varphi_i(b)-\epsilon_i(b)}(b) & \text{if $\varphi_i(b)>\epsilon_i(b)$} \\ b & \text{if $\varphi_i(b)=\epsilon_i(b)$} \\ e_i^{\epsilon_i(b)-\varphi_i(b)}(b) & \text{if $\varphi_i(b)<\epsilon_i(b)$}. \end{cases}$$ It is obvious that $s_i$ is an involution. Tensor products of crystals --------------------------- Given two crystals $B$ and $B'$, there is also a crystal obtained by taking the tensor product $B\otimes B'$. As a set $B\otimes B'$ is just given by the Cartesian product of the sets $B$ and $B'$. The weight function ${\mathrm{wt}}$ for $b\otimes b'\in B\otimes B'$ is ${\mathrm{wt}}(b\otimes b')={\mathrm{wt}}(b)+{\mathrm{wt}}(b')$ and the raising and lowering operators $e_i$ and $f_i$ act as follows $$\label{tensor} \begin{split} e_i(b\otimes b')&=\begin{cases} e_i b \otimes b' & \text{if $\epsilon_i(b)>\phi_i(b')$,}\\ b \otimes e_i b' & \text{otherwise,} \end{cases}\\ f_i(b\otimes b')&=\begin{cases} f_i b \otimes b' & \text{if $\epsilon_i(b)\ge\phi_i(b')$,}\\ b \otimes f_i b' & \text{otherwise.} \end{cases} \end{split}$$ The reader is warned that this convention is different from Kashiwara’s convention. The order of the tensor factors is interchanged. Finite and infinite crystals ---------------------------- Let us fix some notation. From now on let ${\mathfrak{g}}$ be a simple complex Lie algebra and ${\hat{{\mathfrak{g}}}}$ be the associated untwisted affine algebra. That is, let ${\hat{{\mathfrak{g}}}}'={\mathfrak{g}}\otimes{\mathbb{C}}[

{ cases } f_i^{\varphi_i(b)-\epsilon_i(b)}(b) & \text{if $ \varphi_i(b)>\epsilon_i(b)$ } \\ b & \text{if $ \varphi_i(b)=\epsilon_i(b)$ } \\ e_i^{\epsilon_i(b)-\varphi_i(b)}(b) & \text{if $ \varphi_i(b)<\epsilon_i(b)$ }. \end{cases}$$ It is obvious that $ s_i$ is an involution. Tensor products of crystals --------------------------- give two crystal $ B$ and $ B'$, there is also a crystal receive by taking the tensor product $ B\otimes B'$. As a stage set $ B\otimes B'$ is just given by the Cartesian merchandise of the sets $ B$ and $ B'$. The weight routine $ { \mathrm{wt}}$ for $ b\otimes b'\in B\otimes B'$ is $ { \mathrm{wt}}(b\otimes b')={\mathrm{wt}}(b)+{\mathrm{wt}}(b')$ and the raising and lowering operator $ e_i$ and $ f_i$ act as follows $ $ \label{tensor } \begin{split } e_i(b\otimes b')&=\begin{cases } e_i b \otimes b' & \text{if $ \epsilon_i(b)>\phi_i(b')$,}\\ b \otimes e_i b' & \text{otherwise, } \end{cases}\\ f_i(b\otimes b')&=\begin{cases } f_i b \otimes b' & \text{if $ \epsilon_i(b)\ge\phi_i(b')$,}\\ b \otimes f_i b' & \text{otherwise. } \end{cases } \end{split}$$ The reader is warned that this convention is different from Kashiwara ’s convention. The ordering of the tensor factors is interchanged. Finite and infinite quartz glass ---------------------------- Let us fix some notation. From now on permit $ { \mathfrak{g}}$ be a simple complex Lie algebra and $ { \hat{{\mathfrak{g}}}}$ be the associated untwisted affine algebra. That is, let $ { \hat{{\mathfrak{g}}}}'={\mathfrak{g}}\otimes{\mathbb{C } } [

{casfs} f_i^{\varphi_i(b)-\epsilon_i(b)}(b) & \text{if $\varkhu_i(b)>\epsmlon_i(b)$} \\ b & \texg{if $\varphi_i(b)=\epsilon_i(b)$} \\ e_i^{\epdioon_i(b)-\carphi_i(b)}(b) & \text{if $\vxrphi_i(b)<\epdilon_i(b)$}. \eid{cases}$$ It is obtjous that $s_i$ jd an mnvolution. Tensot products ox crystals --------------------------- Givet gwl crystals $B$ and $B'$, there is also a srystal ohtained by takyng uhe tenaor product $B\otimes B'$. As a set $B\otjmes B'$ ps just given by yhe Cartesian product of tje sfts $B$ and $B'$. The welght functiin ${\mwrhrm{wt}}$ for $b\utimes b'\in B\otimes B'$ ia ${\mathrm{wt}}(b\otimes b')={\mathrm{wt}}(b)+{\matfrm{wt}}(y')$ and the rqiwinh and loweriig opegators $e_i$ and $f_i$ act ds follpws $$\label{tensov} \begii{splut} e_i(b\otimes b')&=\begin{casxs} e_i b \otimes b' & \tevt{if $\epsinou_i(b)>\phi_i(b')$,}\\ b \otimes e_i v' & \text{mtheswisd,} \wnd{zasts}\\ f_m(b\ofimes h')&=\bejin{cases} f_i b \otimes b' & \text{if $\epsilon_i(b)\gt\phy_p(n')$,}\\ b \otimes f_j b' & \tqxe{otherwise.} \end{cases} \end{split}$$ The reader iv wzrned that this conventuon is different from Kashiwarw’s convention. The order of the tensor factors is hnterrhxngtd. Nlnitd ajd infinite crystals ---------------------------- Let us fix some notation. Sdok kow on let ${\mathfvak{g}}$ be a simple cpmolrv Lie algebra and ${\hcf{{\mzthfrak{g}}}}$ be the asdociateq untqisted afsine algebra. That is, let ${\hat{{\matyfrak{g}}}}'={\mathfrck{g}}\itimes{\mathbb{C}}[

{cases} f_i^{\varphi_i(b)-\epsilon_i(b)}(b) & \text{if $\varphi_i(b)>\epsilon_i(b)$} \\ b $\varphi_i(b)=\epsilon_i(b)$} e_i^{\epsilon_i(b)-\varphi_i(b)}(b) & $\varphi_i(b)<\epsilon_i(b)$}. \end{cases}$$ It an Tensor products of --------------------------- Given two $B$ and $B'$, there is also crystal obtained by taking the tensor product $B\otimes B'$. As a set $B\otimes is just given by the Cartesian product of the sets $B$ and $B'$. weight ${\mathrm{wt}}$ $b\otimes B\otimes B'$ is ${\mathrm{wt}}(b\otimes b')={\mathrm{wt}}(b)+{\mathrm{wt}}(b')$ and the raising and lowering operators $e_i$ and $f_i$ act as $$\label{tensor} \begin{split} e_i(b\otimes b')&=\begin{cases} e_i b \otimes b' \text{if $\epsilon_i(b)>\phi_i(b')$,}\\ b \otimes b' & \text{otherwise,} \end{cases}\\ f_i(b\otimes f_i \otimes b' \text{if b f_i b' & \end{cases} \end{split}$$ The reader is warned that this convention is different from Kashiwara’s convention. The order of tensor factors Finite and crystals Let fix some notation. on let ${\mathfrak{g}}$ be a simple and ${\hat{{\mathfrak{g}}}}$ be the associated untwisted affine algebra. is, let

{cases} f_i^{\varphi_i(b)-\epsilon_i(b)}(b) & \Text{if $\varpHi_i(b)>\ePsiLon_I(b)$} \\ B & \texT{if $\vArphi_i(b)=\epsilon_I(B)$} \\ e_i^{\ePsilon_i(b)-\varphi_i(b)}(b) & \text{iF $\varpHi_I(B)<\epsILoN_i(b)$}. \enD{cases}$$ IT Is OBVioUs ThAt $s_I$ iS An InvolUtiOn. TensoR products oF crYsTals --------------------------- Given two CRyStals $B$ and $B'$, TheRe is also a cryStaL obtaiNeD by TAking The TensoR produCT $B\otimEs B'$. As a set $b\oTImes B'$ iS Just givEN By The CArtesian product of THe SEts $B$ and $B'$. The weiGht funCtIOn ${\MAThrM{wt}}$ For $b\otimes B'\iN B\otiMEs B'$ is ${\maTHrM{WT}}(B\otIMes b')={\mathrm{wt}}(b)+{\Mathrm{wt}}(b')$ anD The RaisinG aNd lOWering OperaToRS $e_i$ And $f_i$ act as fOlloWs $$\label{teNsor} \beGIn{split} E_I(b\otimeS b')&=\begiN{caSes} E_i b \oTImEs B' & \teXt{IF $\epSIlOn_i(B)>\Phi_I(b')$,}\\ b \otimeS e_I b' & \Text{oTherWISE,} \End{cAseS}\\ f_i(b\OtimeS b')&=\begin{cases} f_I b \oTimeS B' & \teXt{if $\ePsiloN_i(b)\gE\pHi_i(b')$,}\\ b \Otimes F_i b' & \teXt{Otherwise.} \end{casEs} \enD{split}$$ The ReaDeR is WaRned tHAt this ConVenTion is dIfferenT FroM KASHIwAra’s convention. The oRdER Of The tensoR factoRS iS iNTerchangEd. finIte aND InfinIte cRYsTals ---------------------------- Let uS fix soME nOtAtion. FrOm Now on lEt ${\MatHfrAk{g}}$ be A SimpLe compLex Lie alGebra ANd ${\hat{{\mathfrak{g}}}}$ BE the associateD UnTWIsTEd afFinE algebra. ThaT is, lET ${\hat{{\MathFRaK{g}}}}'={\mAThfraK{g}}\otiMeS{\MaTHbb{C}}[

{cases} f_i^{\varphi_i(b )-\epsilon _i(b) }(b ) & \ text {if $\varphi_i ( b)>\ epsilon_i(b)$} \\ b& \te xt { if $ \ va rphi_ i(b)=\e p si l o n_i (b )$ } \ \ e _i^{\ eps ilon_i( b)-\varphi _i( b) }(b) & \ t ex t{if $\var phi _i(b)<\epsil on_ i(b)$} . \end{ cas es}$$ It is obviou s that $s _i $ is an involut i o n. Te nsor products ofc ry s tals --------- ------ -- - -- - - --- -- Given two c rysta l s $B$ a n d$ B ' $,t here is alsoa crystal o b tai ned by t aki n g thetenso rp rod uct $B\otim es B '$. As aset $B \ otimesB '$ is j ust gi ven by the Ca rt esi an pro d uc t o f th e sets $ B$ a nd $B '$.T h e weig htfunc tion${\mathrm{wt} }$for$ b\o times b'\i n B\ ot imesB'$ is ${\m at hrm{wt}}(b\otim es b ')={\math rm{ wt }}( b) +{\ma t hrm{wt }}( b') $ and t he rais i ngan d l ow ering operators $e _i $ an d $f_i$act as fo ll o ws $$\la be l{t enso r } \beg in{s p li t} e_i(b \otime s b ') &=\begi n{ cases} e_i b\otim e s b' & \te xt{if $\ epsil o n_i(b)>\phi_i( b ')$,}\\ b \o t im e s e _ i b' &\text{other wise , } \e nd{c a se s}\ \ f_i( b\oti me s b ' )&=\begin{cases} f _i b \ot imesb' & \text{if $\epsilon _ i ( b)\ge\ph i_i( b ') $ ,}\\ b \otime s f_i b' & \tex t {otherwi se.}\end{cas es} \end{ s p lit}$$ T herea der is w ar ned that this c onve nt ion isdif ferentfro m K ash iwa ra ’s conven tion. Th eor de rofthe t e nsor fac to rsis in terch a nged. Fini te a nd i n fin ite cry s ta l s --- -- -- ---- --- -- ----- ---- - -- Let us fix some no t atio n. F rom now on let ${\ma th frak{g}}$be asimple c omplex L ie algebra and ${\hat{{ \ mathfra k{g }}}}$ bethe assoc iat ed unt wis t ed aff ine al gebra .Tha t is, l e t $ {\h at {{\mathfra k { g}} }}'={ \m athf rak{g}} \otimes{\mathbb{C} } [

{cases} _f_i^{\varphi_i(b)-\epsilon_i(b)}(b) &_ \text{if_$\varphi_i(b)>\epsilon_i(b)$} \\ __b &_\text{if_$\varphi_i(b)=\epsilon_i(b)$} \\ _e_i^{\epsilon_i(b)-\varphi_i(b)}(b) & _ \text{if $\varphi_i(b)<\epsilon_i(b)$}. _ _\end{cases}$$_It is obvious that $s_i$ is an involution. Tensor products of crystals --------------------------- Given two crystals $B$_and_$B'$, there_is_also_a crystal obtained by taking_the tensor product $B\otimes B'$._As a_set $B\otimes B'$ is just given by the_Cartesian_product of the_sets $B$ and $B'$. The weight function ${\mathrm{wt}}$ for_$b\otimes b'\in B\otimes B'$ is ${\mathrm{wt}}(b\otimes_b')={\mathrm{wt}}(b)+{\mathrm{wt}}(b')$ and the_raising_and_lowering operators $e_i$ and_$f_i$ act as follows $$\label{tensor} \begin{split} e_i(b\otimes b')&=\begin{cases} _e_i b \otimes b' & \text{if_$\epsilon_i(b)>\phi_i(b')$,}\\ b \otimes e_i b' & \text{otherwise,} \end{cases}\\ f_i(b\otimes_b')&=\begin{cases} f_i b \otimes b' &_\text{if $\epsilon_i(b)\ge\phi_i(b')$,}\\ b \otimes f_i_b' &_\text{otherwise.} \end{cases} \end{split}$$ The reader is warned_that this convention_is different_from Kashiwara’s convention._The order of the tensor factors_is interchanged. Finite and_infinite crystals ---------------------------- Let us fix some notation._From_now on let_${\mathfrak{g}}$_be_a simple_complex Lie algebra_and_${\hat{{\mathfrak{g}}}}$ be_the_associated untwisted affine algebra. That is,_let_${\hat{{\mathfrak{g}}}}'={\mathfrak{g}}\otimes{\mathbb{C}}[

1)!} \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j-l)!} \sum_{r=0}^{l}\frac{z^{l-r}}{(l-r)!}\frac{u^{r}}{r!}\\[8pt] & \quad = \sum_{j=0}^{n-1}\frac{(-1)^{j}\sideset{_{n-1}}{_j}S}{(n-1)!} (1-z-u)^{j}\\[8pt] & \quad = \binom{z+u-1}{n-1}. \end{aligned}$$]{} On the other hand, the generating function of the right hand side is [ $$\begin{aligned} & \sum_{j=0}^{n-1} G_{n,j}(z) \sum_{r=0}^{j}\frac{(-1)^{r}j!}{(j-r)!} \frac{u^{r}}{r!}\\[8pt] & \quad = \sum_{j=1}^{n-1}G_{n,j}(z)(1-u)^{j}\\[8pt] & \quad = \binom{z+u-1}{n-1}. \end{aligned}$$]{} Therefore, the coefficients of $L_{r+2}$ in both sides coincide. Thus, the claim follows. Next, we prove the coefficient of $L_{1}$ vanishes. Since $$\sum_{j=0}^{n-1}\frac{(-1)^{j}\sideset{_{n-1}}{_j}S}{(n-1)!} \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j-l)!} \frac{z^{l+1}}{(l+1)!} = \int_{0}^{z}\binom{t-1}{n-1}dt$$ and [ $$\begin{aligned} & \sum_{j=0}^{n-1}G_{n,j}(z)\sum_{r=0}^{j} \frac{B_{r}}{r!}\frac{j!}{(j+1-r)!}\\[8pt] & \qquad =

1)! } \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j - l)! } \sum_{r=0}^{l}\frac{z^{l - r}}{(l - r)!}\frac{u^{r}}{r!}\\[8pt ] & \quad = \sum_{j=0}^{n-1}\frac{(-1)^{j}\sideset{_{n-1}}{_j}S}{(n-1)! } (1 - z - u)^{j}\\[8pt ] & \quad = \binom{z+u-1}{n-1 }. \end{aligned}$$ ] { } On the other hand, the generating function of the proper bridge player side is [ $ $ \begin{aligned } & \sum_{j=0}^{n-1 } G_{n, j}(z) \sum_{r=0}^{j}\frac{(-1)^{r}j!}{(j - r)! } \frac{u^{r}}{r!}\\[8pt ] & \quad = \sum_{j=1}^{n-1}G_{n, j}(z)(1 - u)^{j}\\[8pt ] & \quad = \binom{z+u-1}{n-1 }. \end{aligned}$$ ] { } Therefore, the coefficient of $ L_{r+2}$ in both sides coincide. Thus, the title follows. Next, we prove the coefficient of $ L_{1}$ vanishes. Since $ $ \sum_{j=0}^{n-1}\frac{(-1)^{j}\sideset{_{n-1}}{_j}S}{(n-1)! } \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j - l)! } \frac{z^{l+1}}{(l+1)! } = \int_{0}^{z}\binom{t-1}{n-1}dt$$ and [ $ $ \begin{aligned } & \sum_{j=0}^{n-1}G_{n, j}(z)\sum_{r=0}^{j } \frac{B_{r}}{r!}\frac{j!}{(j+1 - r)!}\\[8pt ] & \qquad =

1)!} \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j-l)!} \sum_{r=0}^{l}\nrac{z^{l-r}}{(l-r)!}\frac{u^{r}}{r!}\\[8kt] & \qnad = \suj_{j=0}^{n-1}\frac{(-1)^{j}\rideset{_{n-1}}{_j}S}{(n-1)!} (1-z-u)^{j}\\[8pt] & \qyad = \vinom{z+u-1}{n-1}. \end{aligned}$$]{} On ghe other hand, thw geierating functioi of the right mand vmde is [ $$\begin{allgned} & \suk_{j=0}^{n-1} G_{n,j}(z) \sgm_{f=0}^{j}\yrac{(-1)^{r}j!}{(j-r)!} \frac{u^{r}}{r!}\\[8pt] & \quad = \sum_{t=1}^{n-1}G_{n,j}(z)(1-u)^{k}\\[8ph] & \quad = \binjm{z+u-1}{m-1}. \end{zligned}$$]{} Therefore, the coefficients kf $L_{r+2}$ ii both sides cooncide. Thus, the claim folllws. Nfxt, we prove the clefficient if $L_{1}$ canishes. Sinze $$\sum_{j=0}^{n-1}\frac{(-1)^{j}\sideset{_{n-1}}{_j}A}{(n-1)!} \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j-l)!} \frac{z^{l+1}}{(l+1)!} = \int_{0}^{v}\binom{t-1}{n-1}dt$$ qne [ $$\hggin{aligned} & \slm_{j=0}^{n-1}G_{n,j}(z)\sum_{r=0}^{j} \frdc{B_{r}}{r!}\frsc{j!}{(j+1-r)!}\\[8pt] & \qqmad =

1)!} \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j-l)!} \sum_{r=0}^{l}\frac{z^{l-r}}{(l-r)!}\frac{u^{r}}{r!}\\[8pt] & \quad = \sum_{j=0}^{n-1}\frac{(-1)^{j}\sideset{_{n-1}}{_j}S}{(n-1)!} \quad \binom{z+u-1}{n-1}. \end{aligned}$$]{} the other hand, right side is [ & \sum_{j=0}^{n-1} G_{n,j}(z) \frac{u^{r}}{r!}\\[8pt] & \quad = \sum_{j=1}^{n-1}G_{n,j}(z)(1-u)^{j}\\[8pt] & = \binom{z+u-1}{n-1}. \end{aligned}$$]{} Therefore, the coefficients of $L_{r+2}$ in both sides coincide. Thus, claim follows. Next, we prove the coefficient of $L_{1}$ vanishes. Since $$\sum_{j=0}^{n-1}\frac{(-1)^{j}\sideset{_{n-1}}{_j}S}{(n-1)!} \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j-l)!} = and $$\begin{aligned} \sum_{j=0}^{n-1}G_{n,j}(z)\sum_{r=0}^{j} \frac{B_{r}}{r!}\frac{j!}{(j+1-r)!}\\[8pt] & \qquad =

1)!} \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j-l)!} \sum_{r=0}^{l}\frac{z^{l-r}}{(L-r)!}\frac{u^{r}}{r!}\\[8pT] & \quad = \Sum_{J=0}^{n-1}\fRaC{(-1)^{j}\siDeseT{_{n-1}}{_j}S}{(n-1)!} (1-z-u)^{j}\\[8pt] & \quad = \BInom{Z+u-1}{n-1}. \end{aligned}$$]{} On the otheR hand, ThE GeneRAtIng fuNction oF ThE RIghT hAnD siDe IS [ $$\bEgin{aLigNed} & \sum_{j=0}^{N-1} G_{n,j}(z) \sum_{r=0}^{j}\FraC{(-1)^{r}J!}{(j-r)!} \frac{u^{r}}{r!}\\[8pt] & \QUaD = \sum_{j=1}^{n-1}G_{n,j}(z)(1-U)^{j}\\[8pT] & \quad = \binom{z+u-1}{N-1}. \enD{alignEd}$$]{} theREfore, The CoeffIcientS Of $L_{r+2}$ in Both sides CoINcide. THUs, the clAIM fOlloWs. Next, we prove the cOEfFIcient of $L_{1}$ vanisHes. SinCe $$\SUm_{J=0}^{N-1}\FraC{(-1)^{j}\sIdeset{_{n-1}}{_j}S}{(n-1)!} \SuM_{l=0}^{j}\frAC{(-1)^{l}j!}{(j-l)!} \frAC{z^{L+1}}{(L+1)!} = \INt_{0}^{z}\BInom{t-1}{n-1}dt$$ and [ $$\beGin{aligned} & \sUM_{j=0}^{n-1}g_{n,j}(z)\suM_{r=0}^{J} \frAC{B_{r}}{r!}\frAc{j!}{(j+1-r)!}\\[8Pt] & \QQuaD =

1)!} \sum_{l=0}^{j}\f rac{(-1)^{ l}j!} {(j -l) !} \su m_{r=0}^{l}\fr a c{z^ {l-r}}{(l-r)!}\frac{u^ {r}}{ r! } \\[8 p t] & \quad =\ s um_ {j =0 }^{ n- 1 }\ frac{ (-1 )^{j}\s ideset{_{n -1} }{ _j}S}{(n-1)! } (1- z-u )^{j}\\[8pt] & \q ua d = \bino m{z +u-1} {n-1}. \end{ aligned}$ $] { } Ont he othe r ha nd,the generating fu n ct i on of the righ t hand s i de i s [ $$ \begin{ali gn ed} & \sum _ {j = 0 } ^{n - 1} G_{n,j}(z) \sum_ { r=0 }^{j}\ fr ac{ ( -1)^{r }j!}{ (j - r)! } \fra c{u^ {r}}{r!}\ \[8pt] & \q u ad = \ sum_{j =1} ^{n -1}G _ {n ,j }(z )( 1 -u) ^ {j }\\ [ 8pt ] & \ qu ad = \b inom { z + u -1}{ n-1 }. \end{ aligned}$$]{} T here f ore , the coef fici en ts of $L_{r +2}$in both sides coi ncid e. Thus,the c lai mfollo w s. Ne xt, we provethe coe f fic ie n t of $L_{1}$ vanishes. S i n ce $$\sum_ {j=0}^ { n- 1} \ frac{(-1 )^ {j} \sid e s et{_{ n-1} } {_ j}S}{(n- 1)!} \ sum_{l= 0} ^{j}\f ra c{( -1) ^{l}j ! }{(j -l)!}\frac{z^ {l+1} } {(l+1)!} = \int_{0}^{z}\ b in o m {t - 1}{n -1} dt$$ and [$$\b e gin{ alig n ed } & \ sum_{ j= 0 }^ { n-1}G_{n,j}(z)\sum_ {r =0}^{j } \frac{B_{r }}{r!}\fra c { j !}{(j+1- r)!} \ \[ 8 pt] & \qqu ad =

1)!} _ _ \sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j-l)!} _ \sum_{r=0}^{l}\frac{z^{l-r}}{(l-r)!}\frac{u^{r}}{r!}\\[8pt] __ _&_\quad = \sum_{j=0}^{n-1}\frac{(-1)^{j}\sideset{_{n-1}}{_j}S}{(n-1)!} _ _ _(1-z-u)^{j}\\[8pt] __& \quad = \binom{z+u-1}{n-1}. \end{aligned}$$]{} On the other hand, the generating function of the right_hand_side is_[_$$\begin{aligned} _ & \sum_{j=0}^{n-1} G_{n,j}(z) _ \sum_{r=0}^{j}\frac{(-1)^{r}j!}{(j-r)!} _ _ \frac{u^{r}}{r!}\\[8pt] & \quad _=_\sum_{j=1}^{n-1}G_{n,j}(z)(1-u)^{j}\\[8pt] _& \quad = \binom{z+u-1}{n-1}. \end{aligned}$$]{} Therefore, the coefficients of $L_{r+2}$_in both sides coincide. Thus, the_claim follows. Next, we_prove_the_coefficient of $L_{1}$ vanishes._Since $$\sum_{j=0}^{n-1}\frac{(-1)^{j}\sideset{_{n-1}}{_j}S}{(n-1)!} _\sum_{l=0}^{j}\frac{(-1)^{l}j!}{(j-l)!} \frac{z^{l+1}}{(l+1)!} =_\int_{0}^{z}\binom{t-1}{n-1}dt$$ and [ $$\begin{aligned} _& \sum_{j=0}^{n-1}G_{n,j}(z)\sum_{r=0}^{j} _ \frac{B_{r}}{r!}\frac{j!}{(j+1-r)!}\\[8pt] _& \qquad_ =

been proposed to measure the Majorana fermions [@fu2009a; @tanaka2009a; @fu2009b; @akhmerov2009; @law2009; @lutchyn2010; @chung2011]. Here, we base our discussion on a recent proposal studying the CMEM backscattering [@chung2011]. The basic setup is shown in Fig. \[fig1\], consisting of a magnetic TI in proximity with a grounded top SC layer in region II and two current leads at the corners. When the magnetic domains of magnetic TI are aligned in the same direction, the magnetic TI is in a QAH state with a single chiral edge state propagating along the sample boundary. During the flipping of the magnetic domains at the coercive field, $\lambda$ decreases and the magnetic TI enters the NI with a zero-plateau in Hall conductance $\sigma_{xy}$ over a finite range of magnetic field [@wang2014a; @fengy2015; @kou2015], as shown in Fig. \[fig3\]b. Either perpendicular or in plane external magnetic field could induce such plateau transition [@kou2015]. When the SC proximity effect is sufficiently strong, the superconducting region II experiences the BdG Chern number variation $\mathcal{N}=-2\rightarrow-1\rightarrow0\rightarrow1\rightarrow2$ as $\lambda$ decreases in the hysteresis loop (dashed line in Fig. \[fig3\]a). Therefore, the transport setup Fig. \[fig1\] is a QAH/NI-TSC/NSC-QAH/NI junction. As we will discuss in details below, the edge transport features of the junction uniquely convey the topological properties of the SC in region II. The QAH edge state can be viewed as two CMEMs since a $\mathcal{C}=1$ QAH state is topologically equivalent to a $\mathcal{N}=2$ TSC. Therefore, in the case of QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=2}$-QAH$_{\mathcal{C}=1}$ junction (Fig. \[fig3\]j), the edge current will be perfectly transmitted. By contrast, if the junction is QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=1}$-QAH$_{\mathcal{C}=1}$ (Fig. \[fig3\]i), the chiral edge state in the QAH region separates into two CMEMs at the T

been proposed to measure the Majorana fermions   [ @fu2009a; @tanaka2009a; @fu2009b; @akhmerov2009; @law2009; @lutchyn2010; @chung2011 ]. Here, we base our discussion on a late marriage proposal study the CMEM backscattering   [ @chung2011 ]. The basic setup is shown in Fig.   \[fig1\ ], consist of a magnetic TI in proximity with a grounded top scandium layer in region II and two current lead at the corners. When the magnetic world of magnetic TI are aligned in the same commission, the magnetic TI is in a QAH department of state with a single chiral edge state of matter propagating along the sample boundary. During the flipping of the charismatic domains at the coercive field, $ \lambda$ decrease and the magnetic TI enter the NI with a zero - plateau in Hall conductance $ \sigma_{xy}$ over a finite range of magnetic field   [ @wang2014a; @fengy2015; @kou2015 ], as shown in Fig.   \[fig3\]b. Either vertical or in plane external magnetic field could induce such plateau transition   [ @kou2015 ]. When the SC proximity effect is sufficiently solid, the superconducting area II experiences the BdG Chern number variation $ \mathcal{N}=-2\rightarrow-1\rightarrow0\rightarrow1\rightarrow2 $ as $ \lambda$ decreases in the hysteresis loop (dashed line in Fig.   \[fig3\]a). consequently, the transport apparatus Fig.   \[fig1\ ] is a QAH / NI - TSC / NSC - QAH / NI junction. As we will discuss in details downstairs, the edge transport features of the articulation uniquely convey the topological property of the SC in region II. The QAH edge department of state can be viewed as two CMEMs since a $ \mathcal{C}=1 $ QAH state is topologically equivalent to a $ \mathcal{N}=2 $ TSC. Therefore, in the case of QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=2}$-QAH$_{\mathcal{C}=1}$ junction (Fig.   \[fig3\]j), the edge current will be perfectly transmitted. By contrast, if the junction is QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=1}$-QAH$_{\mathcal{C}=1}$ (Fig.   \[fig3\]i), the chiral edge department of state in the QAH region branch into two CMEMs at the T

befn proposed to measure tme Majorana fermnins [@fu2009a; @tanakz2009a; @fu2009b; @ayhmerov2009; @law2009; @lutchyn2010; @chung2011]. Hece, ww bast our discussion on a recent proposao stndying the CMEM uzckscatbzring [@dmung2011]. Chx basic setup ix shown in Fig. \[fig1\], consisdivg of a magnetic TI in proximity with a groumdfd top SC layet in gedion PI and two current leads at the cornerv. When the mabnetic domains of magnetic TI wre aligned in the same direcjjon, rhe magnetic TI is in a QAH state sith a single chiral edge state propcgating alobg thf sample bouidary. Quring the flipping mf the kagnetic domaiks at thw coercive field, $\lambva$ decreases and the magnetic TN enters the NI with q zero-pnatedu iv Haul doidudtance $\sijma_{xy}$ over z finite rabge of magnetic fiekd [@rqng2014a; @fengy2015; @koh2015], as srorn in Fig. \[fig3\]b. Either perpendicular or it pmane external magnetic dield could induce sufh plateat transition [@kou2015]. When the SC proximity effect is sgfficmevtlv strovt, hhe superconducting region II experiences the GdB Bhern number variction $\mathcal{N}=-2\ribhhattow-1\rightarrow0\rkghtarxkw1\dightarrow2$ as $\lambfa$ decrgases un the hyfterrsis loop (dashed line in Fit. \[fig3\]a). Therefjee, the transport sztup Fig. \[fig1\] ns a QSH/NI-TXC/NSC-QAH/NI junction. As ce wilm discuss ij details celow, the edge tfanxpmrt features of the junctijn uniqueoy cpnvey tfe tppologycal propegties of the SC in regioj II. Tke QAV edge stahe can be viewed as two CMEMs smice a $\mathcal{V}=1$ XAH state if topplogically eqtivalent to a $\kathcal{U}=2$ TSC. Gherefore, pn the cave of QAH$_{\maehcal{C}=1}$-TSC$_{\mathwwl{N}=2}$-QAH$_{\mathcan{C}=1}$ junctyon (Dig. \[fug3\]j), the ddge current woll be pegftctly tranwmitted. By contrasb, if jhs junction is QCK$_{\mqthcal{C}=1}$-TSC$_{\mathcsl{N}=1}$-DAH$_{\ianhcel{C}=1}$ (Fyc. \[fig3\]i), the chhral eder stage in the QCH regoon separates into tfo CJEMs at the T

been proposed to measure the Majorana fermions @fu2009b; @law2009; @lutchyn2010; Here, we base proposal the CMEM backscattering The basic setup shown in Fig. \[fig1\], consisting of magnetic TI in proximity with a grounded top SC layer in region II two current leads at the corners. When the magnetic domains of magnetic TI aligned the direction, magnetic TI is in a QAH state with a single chiral edge state propagating along the boundary. During the flipping of the magnetic domains the coercive field, $\lambda$ and the magnetic TI enters NI a zero-plateau Hall $\sigma_{xy}$ a finite range magnetic field [@wang2014a; @fengy2015; @kou2015], as shown in Fig. \[fig3\]b. Either perpendicular or in plane external magnetic could induce transition [@kou2015]. the proximity is sufficiently strong, region II experiences the BdG Chern as $\lambda$ decreases in the hysteresis loop (dashed in Fig. Therefore, the transport setup Fig. \[fig1\] a QAH/NI-TSC/NSC-QAH/NI junction. As we will discuss in below, the edge transport features of the junction uniquely convey the topological properties of the region II. The QAH state can be as CMEMs a QAH state topologically equivalent to a $\mathcal{N}=2$ TSC. Therefore, in the case of junction (Fig. \[fig3\]j), the edge current will be perfectly transmitted. if junction is QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=1}$-QAH$_{\mathcal{C}=1}$ \[fig3\]i), the chiral edge in QAH region separates into at T

been proposed to measure the MAjorana ferMions [@Fu2009a; @TanAkA2009a; @fu2009B; @akhMerov2009; @law2009; @lutchyN2010; @ChunG2011]. Here, we base our discussiOn on a ReCEnt pROpOsal sTudying THe cmeM bAcKsCatTeRInG [@chunG2011]. ThE basic sEtup is showN in fiG. \[fig1\], consistiNG oF a magnetic tI iN proximity wiTh a GroundEd Top sc layeR in RegioN II and TWo currEnt leads aT tHE corneRS. When thE MAgNetiC domains of magnetiC tI ARe aligned in the Same diReCTiON, The MagNetic TI is iN a qAH stATe with a SInGLE ChiRAl edge state prOpagating alONg tHe sampLe BouNDary. DuRing tHe FLipPing of the maGnetIc domains At the cOErcive fIEld, $\lambDa$ decrEasEs aNd thE MaGnEtiC Ti EntERs The ni wiTh a zero-pLaTeAu in HAll cONDUCtanCe $\sIgma_{Xy}$ oveR a finite range Of mAgneTIc fIeld [@wAng2014a; @fEngy2015; @KoU2015], as shOwn in FIg. \[fig3\]B. EIther perpendicuLar oR in plane eXteRnAl mAgNetic FIeld coUld IndUce such Plateau TRanSiTION [@kOu2015]. When the SC proximiTy EFFeCt is suffIcientLY sTrONg, the supErConDuctING regiOn II EXpEriences The BdG cHeRn Number vArIation $\MaThcAl{N}=-2\RightARrow-1\RightaRrow0\righTarroW1\Rightarrow2$ as $\laMBda$ decreases iN ThE HYsTEresIs lOop (dashed liNe in fIg. \[fiG3\]a). ThEReForE, The trAnspoRt SEtUP Fig. \[fig1\] is a QAH/NI-TSC/NsC-qAH/NI jUnctiOn. As we will disCuss in detaILS Below, the Edge TRaNSport features oF the jUnction uniQUely convEy the TopologiCal properTIEs of the Sc in RegIon iI. THE qAh edge state can BE ViewEd As two CMeMs Since a $\mAthCal{c}=1$ QAh stAtE is topoloGically eQuIvAlEnT to A $\mathCAl{N}=2$ TSC. ThErEfoRe, In tHe casE Of QAH$_{\mAthcaL{C}=1}$-TSc$_{\mAtHCal{n}=2}$-QAH$_{\matHCaL{c}=1}$ JuncTiOn (fig. \[fIg3\]j), ThE edge CurrENt wIll be peRfectly trAnsMItteD. BY cOntrast, If the junction Is qAH$_{\mathcal{c}=1}$-TsC$_{\mAthcal{n}=1}$-qaH$_{\mathcaL{C}=1}$ (Fig. \[fig3\]i), the chiral edge sTAte in thE QAh regiOn seParates inTo tWo CMEMS at THe T

been proposed to measurethe Majora na fe rmi ons  [ @fu2 009a ; @tanaka2009a ; @fu 2009b; @akhmerov2009;@law2 00 9 ; @l u tc hyn20 10; @ch u ng 2 0 11] .He re, w e b ase o urdiscuss ion on a r ece nt proposal st u dy ing the CM EMbackscatteri ng[@chun g2 011 ] . The ba sic s etup i s shown in Fig.\[ f ig1\], consist i n gof a magnetic TI in p r ox i mity with a gr ounded t o pS C la yer in region I I and two cur r en t l ead s at the corne rs. When th e ma gnetic d oma i ns ofmagne ti c TI are aligne d in the same direc t ion, th e magnet ic TIisina QA H s ta tewi t h a si ngl e ch iral edg est ate p ropa g a t i ng a lon g th e sam ple boundary. Du ring the flip pingof t he magn etic d omain sat the coercive fie ld, $\lam bda $dec re asesa nd the ma gne tic TIenterst heNI w i th a zero-plateau in H a l lconducta nce $\ s ig ma _ {xy}$ ov er afini t e rang e of ma gnetic f ield [ @ wa ng 2014a;@f engy20 15 ; @ kou 2015] , asshownin Fig.\[fig 3 \]b. Either pe r pendicular or in p la n e ex ter nal magneti c fi e ld c ould in duc e such plat ea u t r ansition [@kou2015] .When t he SC proximity ef fect is su f f i cientlystro n g, the supercondu cting region II experien ces t he BdG C hern numb e r variati on$\m ath cal { N }= -2\rightarrow - 1 \rig ht arrow0\ rig htarrow 1\r igh tar row 2$ as $\lam bda$ dec re as es i n t he hy s teresislo op(d ash ed li n e in F ig. \ [fig 3\ ]a ) . T herefor e ,t h e tr an sp ortset up Fig.  \[f i g1\ ] is aQAH/NI-TS C/N S C-QA H/ NI juncti on. As we wil ldiscuss in d eta ils be l o w, the e dge transport featureso f the j unc tionuniq uely conv eythe to pol o gicalproper tiesof th e SC in r eg ion I I. The QA H edg e sta te can be vie wed as two CMEMs s i nce a $\mathcal{ C}= 1$ Q A H s tat e i s to po l ogi c a lly equivalentto a $\mat hc a l{ N}=2$ TSC. The re fore, i n the c ase o f QAH$_{ \mathcal{ C}=1}$-TS C$ _{\m a t hca l{N}=2}$-Q AH$_{\ma thcal{C}= 1 }$ ju n ct ion ( Fig . \[fi g3 \]j ), th e edge cur rentwill b eperfec tly t ra nsmitted . By contrast, if the j unctio n isQAH $_{\mathc al{ C }=1 }$-TSC$_{ \mat hcal{N}=1} $-Q AH$ _{\ma thc a l{C}= 1}$( Fi g.\ [fig3 \]i) , the chir a ledg e st ate in theQ A H re gionsep a ratesinto two CMEMs at the T

been_proposed to_measure the Majorana fermions [@fu2009a;_@tanaka2009a; @fu2009b;_@akhmerov2009;_@law2009; @lutchyn2010;_@chung2011]._Here, we base_our discussion on_a recent proposal studying_the CMEM backscattering [@chung2011]._The_basic setup is shown in Fig. \[fig1\], consisting of a magnetic TI in proximity with_a_grounded top_SC_layer_in region II and two_current leads at the corners._When the_magnetic domains of magnetic TI are aligned in_the_same direction, the_magnetic TI is in a QAH state with a_single chiral edge state propagating along_the sample boundary._During_the_flipping of the magnetic_domains at the coercive field, $\lambda$_decreases and the magnetic TI enters_the NI with a zero-plateau in Hall_conductance $\sigma_{xy}$ over a finite range_of magnetic field [@wang2014a; @fengy2015; @kou2015],_as shown_in Fig. \[fig3\]b. Either perpendicular or_in plane external_magnetic field_could induce such_plateau transition [@kou2015]. When the SC proximity_effect is sufficiently_strong, the superconducting region II experiences_the_BdG Chern number_variation_$\mathcal{N}=-2\rightarrow-1\rightarrow0\rightarrow1\rightarrow2$_as $\lambda$_decreases in the_hysteresis_loop (dashed_line_in Fig. \[fig3\]a). Therefore, the transport setup_Fig. \[fig1\]_is a QAH/NI-TSC/NSC-QAH/NI junction. As we will_discuss in details below,_the_edge transport features of_the junction uniquely convey the_topological properties of the SC in_region II. The_QAH edge_state can be viewed as two CMEMs since a $\mathcal{C}=1$ QAH_state is topologically equivalent to a_$\mathcal{N}=2$ TSC. Therefore, in_the case_of_QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=2}$-QAH$_{\mathcal{C}=1}$ junction (Fig. \[fig3\]j),_the_edge current_will be perfectly transmitted. By contrast, if_the junction_is QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=1}$-QAH$_{\mathcal{C}=1}$ (Fig. \[fig3\]i), the chiral edge_state in the QAH_region_separates into two CMEMs at the_T

three datasets as illustration. From the figure, we can see that the optimal numbers of topics and latent factors are varied across different datasets. In general, more latent factors usually lead to better performance, while the optimal number of latent topics is dependent on the reviews of different datasets. This also reveals that setting \#factors and \#topics to be the same may not be optimal. Model Comparison (RQ2) {#sec:comp} ---------------------- We show the performance comparisons of our ALFM with all the baseline methods in Table \[tab:comp\], where the best prediction result on each dataset is in bold. For fair comparison, we set the number of latent factors ($f$) and the number of latent topics ($K$) to be the same as $f=K=5$. Notice that our model could obtain better performance when setting $f$ and $K$ differently. Still, ALFM achieves the best results on 18 out of the 19 datasets. Compared with BMF, which only uses ratings, we achieve much better prediction performance (16.49% relative improvement on average). More importantly, our model outperforms CTR and RMR with large margins - 6.28% and 8.18% relative improvements on average, respectively. Compared to the recently proposed RBLT and TransNet, ALFM can still achieve 3.37% and 4.26% relative improvement on average respectively with significance testing. It is worth mentioning that HFT achieves better performance than RMR and comparable performance with recent RBLT, because we added bias terms to the original HFT in [@mcauley2013hidden]. TransNet applies neural networks, which has exhibited strong capabilities on representation learning, in reviews to learn users’ preferences and items’ characteristics for rating prediction. However, it may suffer from (1) noisy information in reviews, which would deteriorate the performance; and (2) errors introduced when generating fake reviews for rating prediction, which will also cause bias in the final performance. Compared to those baselines, the advantage of ALFM is that it models users’ preferences on different aspects; and more importantly, it captures a user’s specific attention on each aspect of a targeted item. The substantial improvement of ALFM over those baselines demonstrates the benefits of modeling users’ specific preferences on each aspect of different items. Cold-Start Setting (RQ3) {#sec:coldstart} ------------------------ As shown in Table \[tab:dataset\], the datasets are usually very sparse in practical systems

three datasets as illustration. From the figure, we can see that the optimal number of subject and latent factors are varied across different datasets. In general, more latent component usually lead to good performance, while the optimal act of latent topics is dependent on the follow-up of different datasets. This also reveals that setting \#factors and \#topics to be the same may not be optimal. Model Comparison (RQ2) { # sec: comprehensive examination } ---------------------- We show the performance comparisons of our ALFM with all the baseline method acting in Table   \[tab: comp\ ], where the best prediction result on each dataset is in bluff. For fair comparison, we fructify the number of latent agent ($ f$) and the number of latent topics ($ K$) to be the same as $ f = K=5$. Notice that our exemplar could obtain better performance when setting $ f$ and $ K$ differently. Still, ALFM achieves the best results on 18 out of the 19 datasets. Compared with BMF, which only uses ratings, we achieve much better prediction performance (16.49% relative improvement on average). More importantly, our exemplar outperforms CTR and RMR with large margin - 6.28% and 8.18% proportional improvements on average, respectively. Compared to the recently proposed RBLT and TransNet, ALFM can still achieve 3.37% and 4.26% relative improvement on average respectively with significance testing. It is deserving mentioning that HFT achieves better performance than RMR and comparable performance with recent RBLT, because we added bias terms to the original HFT in   [ @mcauley2013hidden ]. TransNet applies neural networks, which has show strong capability on representation learning, in reviews to learn users ’ preference and items ’ characteristics for rating prediction. However, it may digest from (1) noisy information in reviews, which would deteriorate the performance; and (2) error introduced when render fake reviews for evaluation prediction, which will also induce bias in the final performance. Compared to those baseline, the advantage of ALFM is that it models users ’ preference on different aspect; and more importantly, it captures a drug user ’s specific attention on each aspect of a targeted item. The substantial improvement of ALFM over those baseline demonstrates the benefits of model users ’ specific preferences on each aspect of different items. Cold - Start Setting (RQ3) { # sec: coldstart } ------------------------ As show in Table   \[tab: dataset\ ], the datasets are usually very sparse in practical systems

thgee datasets as illustrauion. From the figorw, we cen see fhat the optimal numbers of topics aid lqtent factors are varied acfoss diffvrent datqsetw. In general, more latent rwctoxs usually lead jo better pesformance, whila ghz optimal number of latent topics is dependrnh on the reviers og difrvrtnt datasets. This also reveals thzt settpng \#factors and \#tppics to be the same may nlt bf optimal. Model Comoarison (RQ2) {#wec:cjnp} ---------------------- We show thd performance compariskns of our ALFM with all the barelinz methods ib Rabpg \[tab:comp\], whece the best predicbpon resglt on rach dataset ix ii bood. For fair comparisoi, we set the number jf latent fcctors ($f$) and the numbwr of ldtend tooucs ($K$) uo ue fhe sale es $f=K=5$. Notics that our nodel could obtain neenrr performande wheg fetting $f$ and $K$ differently. Still, ALFM dchjeves the best results in 18 out of the 19 datasgts. Comparqd with BMF, which only uses ratings, we achieve muwh bevtdr krcqkxtlon performance (16.49% relative improvement on averwfe). Mpre importantln, our model outpergogmx CTR and RMR dith lcdgs margins - 6.28% and 8.18% rflative imprivements jn aferage, respectively. Comparee to the recvntlt proposed RBLT anb TransNet, AUFM van syill achieve 3.37% and 4.26% relacive ijprovement ln averags respectively wigh xicnificance testing. It is wjrth mentmoniny that HWT avhievef better pfrformance than RMR and fompatable [erformancf with recent RBLT, because we avved bias termx do nhe origiual HFB in [@mcauley2013hiddqn]. TransNet apklies neuxal negworks, whibh has exiibited strogg capabilitiad on represeitation lqarnung, un revidds to learn usrrs’ prefegeuces and utems’ characteristlcs fud rating prediccnob. However, it mau sjffqr fcom (1) gmisy informadion in tevieds, which womld detrriorate the performdnce; and (2) errors introcuged when teneratigg fake reviees for rating predlctioi, whici will alfo cause bias in the final perrormance. Fomiared to thosq bawelines, the cdvantage of ALFM is that it models userw’ preferences on dufferent aspects; anb kore importently, yt capturas a user’s specific qttention on each aspect of a targeted item. Dhe skbstantial improvement of ALFM over those baselines demonstrates the benedits oh iodeling ussrs’ xpecixie pxeferenses li each aspect of cifferent items. Cold-Start Settinj (RQ3) {#sec:condftart} ------------------------ As shown in Table \[tab:datssdt\], the datasejs are usually very sparse in pracyical systems

three datasets as illustration. From the figure, see the optimal of topics and different In general, more factors usually lead better performance, while the optimal number latent topics is dependent on the reviews of different datasets. This also reveals setting \#factors and \#topics to be the same may not be optimal. Model (RQ2) ---------------------- show performance comparisons of our ALFM with all the baseline methods in Table \[tab:comp\], where the best result on each dataset is in bold. For comparison, we set the of latent factors ($f$) and number latent topics to the as $f=K=5$. Notice our model could obtain better performance when setting $f$ and $K$ differently. Still, ALFM achieves the best on 18 the 19 Compared BMF, only uses ratings, much better prediction performance (16.49% relative More importantly, our model outperforms CTR and RMR large margins 6.28% and 8.18% relative improvements on respectively. Compared to the recently proposed RBLT and ALFM can still achieve 3.37% and 4.26% relative improvement on average respectively with significance testing. worth mentioning that HFT better performance than and performance recent because we bias terms to the original HFT in [@mcauley2013hidden]. TransNet applies neural which has exhibited strong capabilities on representation learning, in reviews users’ and items’ characteristics rating prediction. However, it suffer (1) noisy information in would the errors when fake reviews for rating which will also cause bias the final performance. Compared of ALFM is that it models users’ preferences different aspects; and more importantly, it captures user’s specific attention on each aspect of a targeted item. The substantial of ALFM baselines demonstrates the benefits of modeling users’ specific on each aspect of items. Cold-Start Setting (RQ3) {#sec:coldstart} ------------------------ As shown in \[tab:dataset\], datasets are very sparse in systems

three datasets as illustratiOn. From the fIgure, We cAn sEe That The oPtimal numbers oF TopiCs and latent factors are vAried AcROss dIFfErent DatasetS. in GENerAl, MoRe lAtENt FactoRs uSually lEad to betteR peRfOrmance, while THe Optimal numBer Of latent topiCs iS depenDeNt oN The reVieWs of dIffereNT datasEts. This alSo REveals THat settING \#fActoRs and \#topics to be thE SaME may not be optimAl. ModeL COMpARIsoN (RQ2) {#Sec:comp} ---------------------- We sHoW the pERformanCE cOMPAriSOns of our ALFM wIth all the baSEliNe methOdS in tAble \[taB:comp\], WhERe tHe best prediCtioN result on Each daTAset is iN Bold. For Fair coMpaRisOn, we SEt ThE nuMbER of LAtEnt FActOrs ($f$) and tHe NuMber oF latENT TOpicS ($K$) tO be tHe samE as $f=K=5$. Notice thAt oUr moDEl cOuld oBtain BettEr PerfoRmance When sEtTing $f$ and $K$ differEntlY. Still, ALFm acHiEveS tHe besT ResultS on 18 Out Of the 19 daTasets. COMpaReD WITh bMF, which only uses raTiNGS, wE achieve Much beTTeR pREdiction PeRfoRmanCE (16.49% RelatIve iMPrOvement oN averaGE). MOrE importAnTly, our MoDel OutPerfoRMs CTr and RMr with larGe marGIns - 6.28% and 8.18% relative IMprovements on AVeRAGe, REspeCtiVely. CompareD to tHE recEntlY PrOpoSEd RBLt and TRaNSNET, ALFM can still achievE 3.37% aNd 4.26% relaTive iMprovement on aVerage respECTIvely witH sigNIfICance testing. It Is worTh mentioniNG that HFT AchieVes betteR performaNCE than RMR And ComParAblE PErFormance with rECEnt RbLt, becausE we Added biAs tErmS to The OrIginal HFT In [@mcauleY2013hIdDeN]. TRanSNet aPPlies neuRaL neTwOrkS, whicH Has exhIbiteD strOnG cAPabIlities ON rEPReseNtAtIon lEarNiNg, in rEvieWS to Learn usErs’ preferEncES and ItEmS’ characTeristics for rAtIng predictIoN. HoWever, iT MAy suffer From (1) noisy information in rEViews, whIch Would DeteRiorate thE peRformaNce; ANd (2) erroRs intrOduceD wHen GENeratING fAke ReViews for raTINg pRedicTiOn, whIch will Also cause bias in the FInaL performance. COmpAred TO ThOse BAsELinEs, THe aDVAntage of ALFM is tHat it modelS uSErS’ preferencES on DiFferent Aspects; And moRE importAntly, it caPtures a usEr’S speCIFic Attention oN each aspEct of a tarGEted iTEm. the suBstAntial ImProVemenT of ALFm OveR thosE baselInEs demoNstraTeS the beneFits of modeling users’ specIfic prEfereNceS on each asPecT Of dIfferent iTems. cold-Start SEttIng (rQ3) {#sec:ColDStart} ------------------------ as shOWn In TABle \[taB:datASet\], the datASeTs aRE UsUally very spARSE in PractIcaL SystemS

three datasets as illustr ation. Fro m the fi gur e, wecansee that the o p tima l numbers of topics an d lat en t fac t or s are varied ac r o ssdi ff ere nt da taset s.In gene ral, morelat en t factors us u al ly lead to be tter perform anc e, whi le th e opti mal numb er ofl atenttopics is d e penden t on the r ev iews of different dat a se t s. This also r eveals t h at s ett ing \#factors a nd \# t opics t o b e t hes ame may not b e optimal.Mod el Com pa ris o n (RQ2 ) {#s ec : com p} -------- ---- --------- - Wes how the perform ance c omp ari sons of o urAL F M w i th al l th e baseli ne m ethod s in T a b le \ [ta b:co mp\], where the be stpred i cti on re sulton e ac h dat aset i s inbo ld. For fair co mpar ison, weset t henu mbero f late ntfac tors ($ f$) and the n u m b er of latent topics($ K $ )to be th e same as $ f =K=5$. N ot ice tha t our m odel co uld obta in bet t er p erforma nc e when s ett ing $f$a nd $ K$ dif ferently . Sti l l, ALFM achiev e s the best re s ul t s o n 18out of the 19data s ets. Com p ar edw ith B MF, w hi c ho nly uses ratings, w eachiev e muc h better pred iction per f o r mance (1 6.49 % r e lative improve menton average ) . More i mport antly, o ur modelo u tperform s C TRand RM R wi th large marg i n s -6. 28% and 8. 18% rel ati veimp rov em ents on a verage,re sp ec ti vel y. Co m pared to t here cen tly p r oposed RBLT and T ra n sNe t, ALFM ca n stil lac hiev e 3 .3 7% an d 4. 2 6%relativ e improve men t onav er age res pectively wit hsignifican ce te sting. I t is wor th mentioning that HFTa chieves be tterperf ormance t han RMR a ndc ompara ble pe rform an cew i th re c e nt RB LT , becausew e ad ded b ia s te rms tothe original HFT i n  [@ mcauley2013hi dde n].T r an sNe t a p pli es neu r a l networks, whi ch has exh ib i te d strong c a pab il ities o n repre senta t ion lea rning, in reviewsto lea r n us ers’ prefe rences a nd items’ chara c te risti csfor ra ti ngpredi ction. How ever, it ma ysuffer from ( 1) noisy information in reviews , whic h wou lddeteriora tet heperforman ce;and (2) er ror s i ntrod uce d when gen e ra tin g fake rev i ews for r a ti ngp r ed iction, whi c h wil l als o c a use bi as i n the final perfo r mance. Compare d to t hos e b a seli ne s, the advanta geof A LFM is t ha t it models users’pr e feren ces on diffe rent as p e ct s ; andmore im portantly , i tc aptures a u s er’s s peci fi c atte ntiono n ea c h aspect of a tar geted i tem.T hesubst an tial im p rove ment of AL FM over tho se bas elin es de monstra te s theben ef its of mod e ling user s’ sp ecificpr efer enc es oneach a spect ofdi ffe rent item s . C ol d- S tar t Se tting ( RQ3) {#sec:co l dstart}--- - ------- -- --- - - ------ A s shown in T abl e \[t a b :dataset\] , the da t asets are u sually very s p ars ein prac tic a l systems

three_datasets as_illustration. From the figure,_we can_see_that the_optimal_numbers of topics_and latent factors_are varied across different_datasets. In general,_more_latent factors usually lead to better performance, while the optimal number of latent topics_is_dependent on_the_reviews_of different datasets. This also_reveals that setting \#factors and_\#topics to_be the same may not be optimal. Model Comparison_(RQ2)_{#sec:comp} ---------------------- We show the_performance comparisons of our ALFM with all the baseline_methods in Table \[tab:comp\], where the best_prediction result on_each_dataset_is in bold. For_fair comparison, we set the number_of latent factors ($f$) and the_number of latent topics ($K$) to be_the same as $f=K=5$. Notice that_our model could obtain better_performance when_setting $f$ and $K$ differently._Still, ALFM achieves_the best_results on 18_out of the 19 datasets. Compared_with BMF, which_only uses ratings, we achieve much_better_prediction performance (16.49%_relative_improvement_on average)._More importantly, our_model_outperforms CTR_and_RMR with large margins - 6.28%_and_8.18% relative improvements on average, respectively. Compared_to the recently proposed_RBLT_and TransNet, ALFM can_still achieve 3.37% and 4.26%_relative improvement on average respectively with_significance testing._It is_worth mentioning that HFT achieves better performance than RMR and comparable_performance with recent RBLT, because we_added bias terms to_the original_HFT_in [@mcauley2013hidden]. TransNet applies_neural_networks, which_has exhibited strong capabilities on representation learning,_in reviews_to learn users’ preferences and items’_characteristics for rating prediction._However,_it may suffer from (1) noisy_information in reviews, which would deteriorate_the performance; and (2) errors_introduced_when_generating fake reviews for rating_prediction, which will also cause bias_in the final_performance. Compared to those baselines, the advantage_of_ALFM is that it models users’_preferences_on different aspects; and more importantly,_it_captures_a user’s specific attention on_each aspect of a targeted item._The substantial improvement of ALFM over those baselines demonstrates_the benefits of_modeling users’ specific preferences on_each_aspect_of different items. Cold-Start Setting (RQ3) {#sec:coldstart} ------------------------ As shown in Table \[tab:dataset\], the_datasets are_usually very sparse_in practical systems

N=40$ and (5) $N=80$. Figure 3. The conduction band vs. the chain number $N$ for a multi-chain system with chain lengths $N_\alpha=5000, \alpha= 1, 2,...,N$. A conduction band is defined as being non-zero at the energy values where the corresponding transmission coefficient is higher than 0.1. Figure 4. The conductance $\sigma(E)$ as a function of the number of chains $N$ for a system with equal chains at a Fermi energy $E=1.0$, in the absence of a magnetic field. Figure 5. A comparison of the transmission coefficient with and without a magnetic field. The structure consists of $N=4$ channels of lengths $N_\alpha = 2000, \alpha= 1, 2, 3, 4$ and the magnetic flux threaded in the system is: (a) 0, (b) 0.1, (c) 0.5 and (d) 2.0. Figure 6. The electronic transmission vs. the magnetic flux for a multi-chain system with equal chain lengths $N_\alpha =2000, \alpha =1,2,...,N$ and fixed electron energy $E=1.1$. The unit of the magnetic flux is the flux quantum $\phi_0=1$ and the chain numbers are: (a) $N=2$, (b) $N=3$, (c) $N=4$, d) $N=5$ and e) $N=9$. Figure 7. The electronic conductance vs. the magnetic flux for a multi-chain system ($N=4$) with equal chain lengths $N_\alpha = 2000, \alpha= 1, 2, 3, 4$ and electron energy fixed at $E=1.1$, with the magnetic flux quantum $\phi_0=1$. (a) $\phi_2-\phi_{1}$=$\phi_3-\phi_{2}$=$\phi_4-\phi_{3}$, (b) $\phi_2-\phi_{1}$=$\phi_3-\phi_{2}$=$\frac{1}{2} (\phi_4-\phi_{3})$ (c) $\phi_2-\phi_{1}$=3$\phi_3-\phi_{2}$ and $\phi_4-\phi_{3}$=0, (d) $\phi_2

N=40 $ and (5) $ N=80$. Figure 3. The conduction band vs. the chain number $ N$ for a multi - chain arrangement with range lengths $ N_\alpha=5000, \alpha= 1, 2,... ,N$. A conduction band is define as being non - zero at the department of energy values where the corresponding transmission coefficient is high than 0.1. Figure 4. The conductance $ \sigma(E)$ as a function of the act of chains $ N$ for a system with adequate chains at a Fermi energy $ E=1.0 $, in the absence of a magnetic field. number 5. A comparison of the transmission coefficient with and without a charismatic field. The social organization consist of $ N=4 $ channels of lengths $ N_\alpha = 2000, \alpha= 1, 2, 3, 4 $ and the magnetic flux density threaded in the system is: (a) 0, (b) 0.1, (coulomb) 0.5 and (d) 2.0. design 6. The electronic transmission vs. the charismatic flux for a multi - chain system with equal chain duration $ N_\alpha = 2000, \alpha = 1,2,... ,N$ and fixed electron energy $ E=1.1$. The unit of the magnetic flux is the magnetic field quantum $ \phi_0=1 $ and the chain numbers are: (a) $ N=2 $, (b) $ N=3 $, (c) $ N=4 $, d) $ N=5 $ and e) $ N=9$. Figure 7. The electronic conductance vs. the magnetic flux for a multi - chain system ($ N=4 $) with equal chain length $ N_\alpha = 2000, \alpha= 1, 2, 3, 4 $ and electron energy fixed at $ E=1.1 $, with the charismatic magnetic field quantum $ \phi_0=1$. (a) $ \phi_2-\phi_{1}$=$\phi_3-\phi_{2}$=$\phi_4-\phi_{3}$, (b) $ \phi_2-\phi_{1}$=$\phi_3-\phi_{2}$=$\frac{1}{2 } (\phi_4-\phi_{3})$ (c) $ \phi_2-\phi_{1}$=3$\phi_3-\phi_{2}$ and $ \phi_4-\phi_{3}$=0, (d) $ \phi_2

N=40$ ajd (5) $N=80$. Figure 3. The conductlon band vs. the eyain nnmber $N$ for a mjlti-chain system with chain pebgths $N_\alpha=5000, \alpha= 1, 2,...,N$. A conauction bwnd is dwfintd as being non-zeck at thc enedny vannes where the cprresponditg transmissiot zozfficient is higher than 0.1. Figure 4. The conducyajce $\sigma(E)$ as w fumstioh of the number of chains $N$ for a aystem xith equal chaims at a Fermi energy $E=1.0$, in hhe wbsence of a magnehic field. Fiture 5. A comparisov of the tgcnsmission doefficient with and without a oagnecic field. Tye stgocture consiwts os $N=4$ channels of lengdhs $N_\allha = 2000, \alpha= 1, 2, 3, 4$ aid tye magnetic flux threeded in the system if: (a) 0, (b) 0.1, (c) 0.5 cnd (d) 2.0. Figure 6. The elexteonic tratsmirwiov va. vhe magnehic flux for z multi-chaib system with equal crqin lengths $N_\zlpha =2000, \ajpha =1,2,...,N$ and fixed electron energy $E=1.1$. The lnit of the magnetic flux iw the flux quantum $\phl_0=1$ and the chain numbers are: (a) $N=2$, (b) $N=3$, (c) $N=4$, d) $N=5$ and e) $N=9$. Figure 7. The xldctxinic zinfuctance vs. the magnetic flux for a multi-chaig susnem ($N=4$) with equal ghain lengths $N_\alpna = 2000, \wlpha= 1, 2, 3, 4$ and electxkn energy fixed at $E=1.1$, with tre matnetic fltx qiantum $\phi_0=1$. (a) $\phi_2-\phi_{1}$=$\phi_3-\phi_{2}$=$\phu_4-\phi_{3}$, (b) $\phi_2-\php_{1}$=$\phi_3-\phi_{2}$=$\frac{1}{2} (\phi_4-\phi_{3})$ (c) $\'hi_2-\phi_{1}$=3$\khi_3-\phi_{2}$ and $\phi_4-\phi_{3}$=0, (d) $\phi_2

N=40$ and (5) $N=80$. Figure 3. The vs. chain number for a multi-chain \alpha= 2,...,N$. A conduction is defined as non-zero at the energy values where corresponding transmission coefficient is higher than 0.1. Figure 4. The conductance $\sigma(E)$ as function of the number of chains $N$ for a system with equal chains a energy in absence of a magnetic field. Figure 5. A comparison of the transmission coefficient with and without magnetic field. The structure consists of $N=4$ channels lengths $N_\alpha = 2000, 1, 2, 3, 4$ and magnetic threaded in system (a) (b) 0.1, (c) and (d) 2.0. Figure 6. The electronic transmission vs. the magnetic flux for a multi-chain system with chain lengths \alpha =1,2,...,N$ fixed energy The unit of flux is the flux quantum $\phi_0=1$ numbers are: (a) $N=2$, (b) $N=3$, (c) $N=4$, $N=5$ and $N=9$. Figure 7. The electronic conductance the magnetic flux for a multi-chain system ($N=4$) equal chain lengths $N_\alpha = 2000, \alpha= 1, 2, 3, 4$ and electron energy fixed with the magnetic flux $\phi_0=1$. (a) $\phi_2-\phi_{1}$=$\phi_3-\phi_{2}$=$\phi_4-\phi_{3}$, $\phi_2-\phi_{1}$=$\phi_3-\phi_{2}$=$\frac{1}{2} (c) and (d) $\phi_2

N=40$ and (5) $N=80$. Figure 3. The conduction bAnd vs. the chAin nuMbeR $N$ fOr A mulTi-chAin system with cHAin lEngths $N_\alpha=5000, \alpha= 1, 2,...,N$. A conDuctiOn BAnd iS DeFined As being NOn-ZERo aT tHe EneRgY VaLues wHerE the corResponding TraNsMission coeffICiEnt is higheR thAn 0.1. Figure 4. The cOndUctancE $\sIgmA(e)$ as a fUncTion oF the nuMBer of cHains $N$ for A sYStem wiTH equal cHAInS at a fermi energy $E=1.0$, in the ABsENce of a magnetic Field. FIgURe 5. a COmpAriSon of the trAnSmissIOn coeffICiENT WitH And without a maGnetic field. tHe sTructuRe ConSIsts of $n=4$ chanNeLS of Lengths $N_\alpHa = 2000, \alPha= 1, 2, 3, 4$ and the MagnetIC flux thREaded in The sysTem Is: (a) 0, (B) 0.1, (c) 0.5 anD (D) 2.0. FIgUre 6. thE EleCTrOniC TraNsmissioN vS. tHe magNetiC FLUX for A muLti-cHain sYstem with equaL chAin lENgtHs $N_\alPha =2000, \alPha =1,2,...,N$ AnD fixeD electRon enErGy $E=1.1$. The unit of the MagnEtic flux iS thE fLux QuAntum $\PHi_0=1$ and tHe cHaiN numberS are: (a) $N=2$, (b) $n=3$, (C) $N=4$, d) $n=5$ aND E) $n=9$. FIgure 7. The electronic CoNDUcTance vs. tHe magnETiC fLUx for a muLtI-chAin sYSTem ($N=4$) wIth eQUaL chain leNgths $N_\ALpHa = 2000, \Alpha= 1, 2, 3, 4$ anD eLectroN eNerGy fIxed aT $e=1.1$, witH the maGnetic flUx quaNTum $\phi_0=1$. (a) $\phi_2-\phi_{1}$=$\pHI_3-\phi_{2}$=$\phi_4-\phi_{3}$, (b) $\phI_2-\PhI_{1}$=$\PHi_3-\PHi_{2}$=$\frAc{1}{2} (\pHi_4-\phi_{3})$ (c) $\phi_2-\phI_{1}$=3$\phi_3-\PHi_{2}$ anD $\phi_4-\PHi_{3}$=0, (D) $\phI_2

N=40$ and (5) $N=80$. Fig ure 3. The cond uct ion b andvs.the chain numb e r $N $ for a multi-chain sy stemwi t h ch a in leng ths $N_ \ al p h a=5 00 0, \a lp h a= 1, 2 ,.. .,N$. A conductio n b an d is defined as being non -ze ro at the en erg y valu es wh e re th e c orres pondin g trans mission c oe f ficien t is hig h e rthan 0.1. Figure 4.T he conductance $\ sigma( E) $ a s a f unc tion of th enumbe r of cha i ns $ N $ f o r a system wi th equal ch a ins at aFe rmi energy $E=1 .0 $ , i n the absen ce o f a magne tic fi e ld. Fi g ure 5.A comp ari son oft he t ran sm i ssi o ncoe f fic ient wit han d wit hout a m agne tic fie ld. T he structurecon sist s of $N=4 $ cha nnel sof le ngths$N_\a lp ha = 2000, \alp ha=1, 2, 3,4$an d t he magn e tic fl uxthr eaded i n the s y ste mi s : ( a) 0, (b) 0.1, (c) 0 . 5 a nd (d) 2 .0. F i gu re 6. The e le ctr onic t ransm issi o nvs. themagnet i cfl ux foramulti- ch ain sy stemw ithequalchain le ngths $N_\alpha =200 0 , \alpha =1,2 , .. . , N$ andfix ed electron ene r gy $ E=1. 1 $. Th e unit of t he ma g netic flux is the f lu x quan tum $ \phi_0=1$ and the chain n u mbers ar e: ( a )$ N=2$, (b) $N=3 $, (c ) $N=4$, d ) $N=5$ a nd e) $N=9$. Figure 7 . The elec tro nic co ndu c t an ce vs. the ma g n etic f lux for amulti-c hai n s yst em($ N=4$) wit h equalch ai nle ngt hs $N _ \alpha = 2 000 ,\al pha=1 , 2, 3 , 4$andel ec t ron energy fi x e d at $ E= 1.1$ , w it h the mag n eti c fluxquantum $ \ph i _0=1 $. ( a) $\ph i_2-\phi_{1}$ =$ \phi_3-\ph i_ {2} $=$\ph i _ 4-\phi_{ 3}$, (b) $\phi_2-\phi_{ 1 }$=$\ph i_3 -\phi _{2} $=$\frac{ 1}{ 2} (\phi_ 4-\ph i_ {3} ) $ (c)$ \ ph i_2 -\ phi_{1}$=3 $ \ phi _3-\p hi _{2} $ and $ \phi_4-\phi_{3}$=0 , (d ) $\phi_2

N=40$ and_(5) $N=80$. Figure_3. The conduction band_vs. the_chain_number $N$_for_a multi-chain system_with chain lengths_$N_\alpha=5000, \alpha= 1, 2,...,N$._A conduction band_is_defined as being non-zero at the energy values where the corresponding transmission coefficient is_higher_than 0.1. Figure_4._The_conductance $\sigma(E)$ as a function_of the number of chains_$N$ for_a system with equal chains at a Fermi_energy_$E=1.0$, in the_absence of a magnetic field. Figure 5. A comparison of_the transmission coefficient with and without_a magnetic field._The_structure_consists of $N=4$ channels_of lengths $N_\alpha = 2000, \alpha=_1, 2, 3, 4$ and the_magnetic flux threaded in the system is:_(a) 0, (b) 0.1, (c) 0.5_and (d) 2.0. Figure 6. The_electronic transmission_vs. the magnetic flux for_a multi-chain system_with equal_chain lengths $N_\alpha_=2000, \alpha =1,2,...,N$ and fixed electron_energy $E=1.1$. The_unit of the magnetic flux is_the_flux quantum $\phi_0=1$_and_the_chain numbers_are: (a) $N=2$,_(b)_$N=3$, (c)_$N=4$,_d) $N=5$ and e) $N=9$. Figure 7._The_electronic conductance vs. the magnetic flux for_a multi-chain system ($N=4$)_with_equal chain lengths $N_\alpha_= 2000, \alpha= 1, 2,_3, 4$ and electron energy fixed_at $E=1.1$,_with the_magnetic flux quantum $\phi_0=1$. (a) $\phi_2-\phi_{1}$=$\phi_3-\phi_{2}$=$\phi_4-\phi_{3}$, (b) $\phi_2-\phi_{1}$=$\phi_3-\phi_{2}$=$\frac{1}{2} _ _ (\phi_4-\phi_{3})$_(c) $\phi_2-\phi_{1}$=3$\phi_3-\phi_{2}$_and_$\phi_4-\phi_{3}$=0, (d) $\phi_2

2}},\ast)}$]{}]{}. We show that the remaining groups vanish over [$\mathbf{Q}$]{} using the induction hypothesis and the exact sequence $${\tilde{H}}_i({{\ensuremath{\exp_{k-1}\!{({\ensuremath{S^2}},\ast)}}}})\longrightarrow {\tilde{H}}_i({{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}})\longrightarrow {\tilde{H}}_i({{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}}/{{\ensuremath{\exp_{k-1}\!{({\ensuremath{S^2}},\ast)}}}})$$ (rational co-efficients omitted). Since the outer groups are zero for $i\leq 2k-5$ the middle group is too. We now prove the results for [[$\exp_{k}\!{{\ensuremath{S^2}}}$]{}]{}, using the results just proved, Lemma \[samehomology.lem\], and the long exact sequence of the pair $({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}},{{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}})$. At the top end we have $$0\rightarrow {\tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}}) \rightarrow {\tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}}/{{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}}) \rightarrow 0\rightarrow {\tilde{H}}_{2k-1}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\rightarrow 0$$ giving ${\tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\cong{\ensuremath{\mathbf{Z}}}$ generated by $\tilde{\mathsf{1}}(k)$ and ${\tilde{H}}_{2k-1}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\cong\{0\}$. The next segment is $$0\longrightarrow {\ensuremath{\mathbf{Z}}}\longrightarrow {\tilde{H}}_{2k-2}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}}) \longrightarrow {\ensuremath{\mathbf{Z}}}/(k-1){\ensuremath{\mathbf{Z}}}\longrightarrow 0$$ in which the subgroup [$\mathbf{Z}$]{} is generated by ${\mathsf{1}}(k-1)$ and the

2}},\ast)}$ ] { } ] { }. We show that the remaining groups vanish over [ $ \mathbf{Q}$ ] { }   use the evocation hypothesis and the exact sequence $ $ { \tilde{H}}_i({{\ensuremath{\exp_{k-1}\!{({\ensuremath{S^2}},\ast)}}}})\longrightarrow { \tilde{H}}_i({{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}})\longrightarrow { \tilde{H}}_i({{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}}/{{\ensuremath{\exp_{k-1}\!{({\ensuremath{S^2}},\ast)}}}})$$ (intellectual co - efficients omitted). Since the outer group are zero for $ i\leq 2k-5 $ the middle group is too. We nowadays prove the results for [ [ $ \exp_{k}\!{{\ensuremath{S^2}}}$ ] { } ] { }, use the results just testify, Lemma   \[samehomology.lem\ ], and the long exact sequence of the pair $ ({ { \ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}},{{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}})$. At the top end we have $ $ 0\rightarrow { \tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2 } } } } } }) \rightarrow { \tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}}/{{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast) } } } }) \rightarrow 0\rightarrow { \tilde{H}}_{2k-1}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\rightarrow 0$$ contribute $ { \tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\cong{\ensuremath{\mathbf{Z}}}$ generated by $ \tilde{\mathsf{1}}(k)$ and $ { \tilde{H}}_{2k-1}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\cong\{0\}$. The next segment is $ $ 0\longrightarrow { \ensuremath{\mathbf{Z}}}\longrightarrow { \tilde{H}}_{2k-2}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2 } } } } } }) \longrightarrow { \ensuremath{\mathbf{Z}}}/(k-1){\ensuremath{\mathbf{Z}}}\longrightarrow 0$$ in which the subgroup [ $ \mathbf{Z}$ ] { }   is beget by $ { \mathsf{1}}(k-1)$ and the

2}},\ast)}$]{}]{}. We show that the remainlng groups vanisk over [$\kathbf{S}$]{} using tfe induction hypothesis and vhe wxact sequence $${\tilde{H}}_i({{\ensurdmath{\exp_{k-1}\!{({\vnsurematy{S^2}},\asu)}}}})\longrightarrow {\tilde{H}}_i({{\ensmxematg{\cxp_{k}\!{({\eusnremath{S^2}},\ast)}}}})\longtightarrow {\tinde{H}}_i({{\ensurematv{\ebp_{n}\!{({\ensuremath{S^2}},\ast)}}}}/{{\ensuremath{\exp_{k-1}\!{({\ensureiath{S^2}},\asy)}}}})$$ (gational co-effyciemes ojptued). Since the outer groups are zedo for $p\leq 2k-5$ the middle group is too. We now prove hhe gesults for [[$\exp_{k}\!{{\enduremath{S^2}}}$]{}]{}, uwing rhe results gust proved, Lemma \[samehkmology.lem\], and the long exact sdquenee of the pqie $({{\ejvuremath{\exp_{j}\!{{\ensugemath{S^2}}}}}},{{\ensuremath{\exp_{k}\!{({\ansuremsth{S^2}},\ast)}}}})$. At the to' ene we have $$0\rightarrow {\tilde{H}}_{2k}({{\ensuremath{\gxp_{k}\!{{\ensurekach{S^2}}}}}}) \rightarrow {\tulde{H}}_{2n}({{\ensgremxrh{\ebp_{k}\!{{\tnsnrejath{S^2}}}}}}/{{\ejsucemath{\exp_{k}\!{({\ehsuremath{S^2}},\awt)}}}}) \rightarrow 0\rightsrwiw {\tilde{H}}_{2k-1}({{\enshrematr{\evp_{k}\!{{\ensuremath{S^2}}}}}})\rightarrow 0$$ giving ${\tilde{H}}_{2n}({{\enauremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\xong{\ensuremath{\mathbf{Z}}}$ generateq by $\tilde{\mathsf{1}}(k)$ and ${\tilde{H}}_{2k-1}({{\ensuremath{\exp_{k}\!{{\ensurekath{S^2}}}}}})\rovg\{0\}$. Uhc vwxh segment is $$0\longrightarrow {\ensuremath{\mathbf{Z}}}\jknbrpghtarrow {\tilde{H}}_{2k-2}({{\cnsuremath{\exp_{k}\!{{\ensutelayr{S^2}}}}}}) \longrightatrow {\ensursmath{\mathbf{Z}}}/(k-1){\ensurfmath{\majhbf{Z}}}\lingrightawrow 0$$ in which the subgroup [$\matybf{Z}$]{} is generctee by ${\mathsf{1}}(k-1)$ and tke

2}},\ast)}$]{}]{}. We show that the remaining groups [$\mathbf{Q}$]{} the induction and the exact co-efficients Since the outer are zero for 2k-5$ the middle group is too. now prove the results for [[$\exp_{k}\!{{\ensuremath{S^2}}}$]{}]{}, using the results just proved, Lemma \[samehomology.lem\], the long exact sequence of the pair $({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}},{{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}})$. At the top end we $$0\rightarrow \rightarrow \rightarrow {\tilde{H}}_{2k-1}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\rightarrow 0$$ giving ${\tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\cong{\ensuremath{\mathbf{Z}}}$ generated by $\tilde{\mathsf{1}}(k)$ and ${\tilde{H}}_{2k-1}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\cong\{0\}$. The next segment is $$0\longrightarrow {\ensuremath{\mathbf{Z}}}\longrightarrow {\tilde{H}}_{2k-2}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}}) {\ensuremath{\mathbf{Z}}}/(k-1){\ensuremath{\mathbf{Z}}}\longrightarrow 0$$ in which the subgroup [$\mathbf{Z}$]{} is by ${\mathsf{1}}(k-1)$ and the

2}},\ast)}$]{}]{}. We show that the remaining Groups vaniSh oveR [$\maThbF{Q}$]{} UsinG the Induction hypotHEsis And the exact sequence $${\tilDe{H}}_i({{\eNsURemaTH{\eXp_{k-1}\!{({\enSurematH{s^2}},\aST)}}}})\LonGrIgHtaRrOW {\tIlde{H}}_I({{\enSurematH{\exp_{k}\!{({\ensurEmaTh{s^2}},\ast)}}}})\longrighTArRow {\tilde{H}}_i({{\EnsUremath{\exp_{k}\!{({\eNsuRemath{s^2}},\aSt)}}}}/{{\eNSuremAth{\Exp_{k-1}\!{({\eNsuremATh{S^2}},\ast)}}}})$$ (Rational cO-eFFicienTS omitteD). sInCe thE outer groups are zeRO fOR $i\leq 2k-5$ the middlE group Is TOo. wE Now ProVe the resulTs For [[$\exP_{K}\!{{\ensureMAtH{s^2}}}$]{}]{}, USinG The results jusT proved, LemmA \[SamEhomolOgY.leM\], And the Long eXaCT seQuence of the Pair $({{\EnsurematH{\exp_{k}\!{{\eNSurematH{s^2}}}}}},{{\ensureMath{\exP_{k}\!{({\eNsuRemaTH{S^2}},\AsT)}}}})$. At ThE Top ENd We hAVe $$0\rIghtarroW {\tIlDe{H}}_{2k}({{\eNsurEMATH{\exp_{K}\!{{\enSureMath{S^2}}}}}}) \Rightarrow {\tilDe{H}}_{2K}({{\ensURemAth{\exP_{k}\!{{\ensUremAtH{S^2}}}}}}/{{\ensUrematH{\exp_{k}\!{({\EnSuremath{S^2}},\ast)}}}}) \rigHtarRow 0\rightaRroW {\tIldE{H}}_{2K-1}({{\ensuREmath{\eXp_{k}\!{{\EnsUremath{s^2}}}}}})\rightaRRow 0$$ GiVING ${\tIlde{H}}_{2k}({{\ensuremath{\exP_{k}\!{{\ENSuRemath{S^2}}}}}})\cOng{\ensUReMaTH{\mathbf{Z}}}$ GeNerAted BY $\Tilde{\MathSF{1}}(k)$ And ${\tilde{h}}_{2k-1}({{\ensuREmAtH{\exp_{k}\!{{\enSuRemath{s^2}}}}}})\cOng\{0\}$. the Next sEGmenT is $$0\lonGrightarRow {\enSUremath{\mathbf{Z}}}\LOngrightarrow {\TIlDE{h}}_{2k-2}({{\ENsurEmaTh{\exp_{k}\!{{\ensurEmatH{s^2}}}}}}) \lonGrigHTaRroW {\EnsurEmath{\MaTHbF{z}}}/(k-1){\ensuremath{\mathbf{Z}}}\LoNgrighTarroW 0$$ in which the suBgroup [$\mathBF{z}$]{} Is generaTed bY ${\MaTHsf{1}}(k-1)$ and the

2}},\ast)}$]{}]{}. We show that theremai nin g g ro upsvani sh over [$\mat h bf{Q }$]{} using the induct ion h yp o thes i sand t he exac t s e q uen ce $ ${\ ti l de {H}}_ i({ {\ensur emath{\exp _{k -1 }\!{({\ensur e ma th{S^2}},\ ast )}}}})\longr igh tarrow { \ti l de{H} }_i ({{\e nsurem a th{\ex p_{k}\!{( {\ e nsurem a th{S^2} } , \a st)} }}})\longrightarr o w{ \tilde{H}}_i({ {\ensu re m at h { \ex p_{ k}\!{({\en su remat h {S^2}}, \ as t ) } }}} / {{\ensuremath {\exp_{k-1} \ !{( {\ensu re mat h {S^2}} ,\ast )} } }}) $$ (rationa l co -efficien ts omi t ted). S i nce the outer gr oup s ar e z er o f or $i\ l eq 2k - 5$the midd le g roupis t o o . Wenow pro ve th e results for [[ $\ex p _{k }\!{{ \ensu rema th {S^2} }}$]{} ]{},us ing the results jus t proved, Le mm a \ [s ameho m ology. lem \], and th e longe xac ts e q ue nce of the pair $( {{ \ e ns uremath{ \exp_{ k }\ !{ { \ensurem at h{S ^2}} } } }},{{ \ens u re math{\ex p_{k}\ ! {( {\ ensurem at h{S^2} }, \as t)} }}})$ . Atthe to p end we have $$0\rightarrow {\tilde{H}}_ { 2k } ( {{ \ ensu rem ath{\exp_{k }\!{ { \ens urem a th {S^ 2 }}}}} }) \r ig h ta r row {\tilde {H }}_{2k }({{\ ensuremath{\e xp_{k}\!{{ \ e n suremath {S^2 } }} } }}/{{\ensurema th{\e xp_{k}\!{( { \ensurem ath{S ^2}},\as t)}}}}) \ r ightarro w 0 \ri ght arr o w { \tilde{H}}_{2 k - 1}({ {\ ensurem ath {\exp_{ k}\ !{{ \en sur em ath{S^2}} }}}})\ri gh ta rr ow 0$ $ giv i ng ${\ti ld e{H }} _{2 k}({{ \ ensure math{ \exp _{ k} \ !{{ \ensure m at h { S^2} }} }} })\c ong {\ ensur emat h {\m athbf{Z }}}$ gene rat e d by $ \t ilde{\m athsf{1}}(k)$ a nd ${\tild e{ H}} _{2k-1 } ( {{\ensur emath{\exp_{k}\!{{\ensu r emath{S ^2} }}}}} )\co ng\{0\}$. Th e next se g ment i s $$0\ longr ig hta r r ow {\ e n su rem at h{\mathbf{ Z } }}\ longr ig htar row {\t ilde{H}}_{2k-2}({{ \ ens uremath{\exp_ {k} \!{{ \ e ns ure m at h {S^ 2} } }}} } ) \longrightarr ow {\ensur em a th {\mathbf{Z } }}/ (k -1){\en suremat h{\ma t hbf{Z}} }\longrig htarrow 0 $$ inw h ich the subgr oup [$\m athbf{Z}$ ] {} is ge nerat edby ${\ ma ths f{1}} (k-1)$ and the

2}},\ast)}$]{}]{}. We_show that_the remaining groups vanish_over [$\mathbf{Q}$]{} using_the_induction hypothesis_and_the exact sequence_$${\tilde{H}}_i({{\ensuremath{\exp_{k-1}\!{({\ensuremath{S^2}},\ast)}}}})\longrightarrow {\tilde{H}}_i({{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}})\longrightarrow {\tilde{H}}_i({{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}}/{{\ensuremath{\exp_{k-1}\!{({\ensuremath{S^2}},\ast)}}}})$$ (rational co-efficients_omitted). Since the outer_groups are zero_for_$i\leq 2k-5$ the middle group is too. We now prove the results for [[$\exp_{k}\!{{\ensuremath{S^2}}}$]{}]{}, using_the_results just_proved,_Lemma \[samehomology.lem\],_and the long exact sequence_of the pair $({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}},{{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}})$. At_the top_end we have $$0\rightarrow {\tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}}) \rightarrow __ _ {\tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}}/{{\ensuremath{\exp_{k}\!{({\ensuremath{S^2}},\ast)}}}}) \rightarrow 0\rightarrow {\tilde{H}}_{2k-1}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\rightarrow 0$$ giving ${\tilde{H}}_{2k}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\cong{\ensuremath{\mathbf{Z}}}$_generated by $\tilde{\mathsf{1}}(k)$ and ${\tilde{H}}_{2k-1}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}})\cong\{0\}$. The_next segment is_$$0\longrightarrow_{\ensuremath{\mathbf{Z}}}\longrightarrow_{\tilde{H}}_{2k-2}({{\ensuremath{\exp_{k}\!{{\ensuremath{S^2}}}}}}) \longrightarrow {\ensuremath{\mathbf{Z}}}/(k-1){\ensuremath{\mathbf{Z}}}\longrightarrow 0$$_in which the subgroup [$\mathbf{Z}$]{} is generated_by ${\mathsf{1}}(k-1)$ and the

In practice one can live with fewer event pairs for event mixing, by replacing in (\[36\]) the number $N_{\rm ev}$ by a much smaller number $N'_{\rm ev}$. As long as $N'_{\rm ev}(N'_{\rm ev}-1) \gg N_{\rm ev}$ one can still ensure that the contribution of the denominator to the statistical error of the final correlation function is negligible [@Comm2]. The correlator $C^{II}$ differs from $C^{I}$ only by the different normalization. It can be constructed by taking the ratio of the following two expressions [@Zajc84]: $$\begin{aligned} \label{38} P_2(\bbox{p}_1,\bbox{p}_2) &=& {E_{\bbox{p}_1} \, E_{\bbox{p}_2} \over N^c_{\rm pairs}} \sum_{i=1}^{N_{\rm ev}} \tilde\nu_{i,i}(\bbox{p}_1,\bbox{p}_2) \,, \\ \label{39} P_1(\bbox{p}_1) P_1(\bbox{p}_2) &=& {E_{\bbox{p}_1} \, E_{\bbox{p}_2} \over N^u_{\rm pairs}} \sum_{i,j=1 \atop i\ne j}^{N_{\rm ev}} \tilde\nu_{i,j}(\bbox{p}_1, \bbox{p}_2) \,. \end{aligned}$$ $N^c_{\rm pairs}$ and $N^u_{\rm pairs}$ are the total numbers of “correlated” and “uncorrelated” pion pairs, respectively: $$\begin{aligned} N^c_{\rm pairs} &=& N_{\rm ev} \cdot \langle n(n-1)\rangle \,, \\ N^u_{\rm pairs} &=& N_{\rm ev}(N_{\rm ev}-1) \cdot \langle n \rangle^2 \,. \end{aligned}$$ Conclusions {#sec6} =========== We have shown that in principle the normalization of the two-particle Bose-Einstein correlation function contains valuable information on the the multiplicity distribution of the event ensemble. Both

In practice one can live with fewer event pair for consequence mixing, by replacing in (\[36\ ]) the numeral $ N_{\rm ev}$ by a a lot smaller number $ N'_{\rm ev}$. As long as $ N'_{\rm ev}(N'_{\rm ev}-1) \gg N_{\rm ev}$ one can however ensure that the contribution of the denominator to the statistical error of the concluding correlation coefficient function is negligible [ @Comm2 ]. The correlator $ C^{II}$ differs from $ C^{I}$ only by the unlike normalization. It can be constructed by lead the ratio of the following two expressions [ @Zajc84 ]: $ $ \begin{aligned } \label{38 } P_2(\bbox{p}_1,\bbox{p}_2) & = & { E_{\bbox{p}_1 } \, E_{\bbox{p}_2 } \over N^c_{\rm pair } } \sum_{i=1}^{N_{\rm ev } } \tilde\nu_{i, i}(\bbox{p}_1,\bbox{p}_2) \, , \\ \label{39 } P_1(\bbox{p}_1) P_1(\bbox{p}_2) & = & { E_{\bbox{p}_1 } \, E_{\bbox{p}_2 } \over N^u_{\rm pairs } } \sum_{i, j=1 \atop i\ne j}^{N_{\rm ev } } \tilde\nu_{i, j}(\bbox{p}_1, \bbox{p}_2) \, . \end{aligned}$$ $ N^c_{\rm pairs}$ and $ N^u_{\rm pairs}$ are the total numbers of “ correlate ” and “ uncorrelated ” pion pairs, respectively: $ $ \begin{aligned } N^c_{\rm pairs } & = & N_{\rm ev } \cdot \langle n(n-1)\rangle \, , \\ N^u_{\rm pairs } & = & N_{\rm ev}(N_{\rm ev}-1) \cdot \langle n \rangle^2 \, . \end{aligned}$$ Conclusions { # sec6 } = = = = = = = = = = = We have shown that in rationale the normalization of the two - atom Bose - Einstein correlation function contains valuable information on the the numerousness distribution of the event ensemble. Both

In practice one can live wlth fewer event kaurs foc event mixing, cy replacing in (\[36\]) the number $I_{\rm wv}$ by a much smaller number $N'_{\rm ev}$. Ws long qs $N'_{\cm ev}(N'_{\rm ev}-1) \gg N_{\cj ev}$ onc can dtiln ensure that tme contribudion of the detooiuator to the statistical error of thq final clrrelation funstiom is hvgoigible [@Comm2]. The correlator $C^{IJ}$ diffegs from $C^{I}$ only bu the different normalizatlon. Lt can be construched by takibg trw ratio of tfe following two expreasions [@Zajc84]: $$\begin{aligned} \label{38} P_2(\bbpx{p}_1,\bbox{p}_2) &=& {E_{\bhmx{p}_1} \, E_{\bbox{p}_2} \over N^c_{\rm pairs}} \sum_{i=1}^{N_{\rk ev}} \tikde\nu_{i,i}(\bbox{p}_1,\bbpx{p}_2) \,, \\ \oabel{39} P_1(\bbox{p}_1) P_1(\bbox{p}_2) &=& {E_{\bbox{p}_1} \, E_{\bbox{k}_2} \over N^u_{\rk 'airs}} \sum_{i,j=1 \atop i\nw h}^{N_{\rm gv}} \dildd\bu_{i,g}(\bbkx{'}_1, \bgox{p}_2) \,. \fnd{eligned}$$ $N^c_{\rj pairs}$ and $N^u_{\rm pairs}$ are the tjnsl numbers or “corrqlwted” and “uncorrelated” pion pairs, respecuivelg: $$\begin{aligned} N^c_{\rm pqirs} &=& N_{\rm ev} \cdot \lanhle n(n-1)\randle \,, \\ N^u_{\rm pairs} &=& N_{\rm ev}(N_{\rm ev}-1) \cdot \oajgle n \rangle^2 \,. \end{aligned}$$ Conclusions {#sec6} =========== We hads xhpwn that in prlnciple the normalozwtojn of the two-karticlz Bkse-Einstein correlwtion fonctiob containf vakuable information on the tye multiplicpty eistribution of thz event enseoble. Both

In practice one can live with fewer for mixing, by in (\[36\]) the much number $N'_{\rm ev}$. long as $N'_{\rm ev}-1) \gg N_{\rm ev}$ one can ensure that the contribution of the denominator to the statistical error of the correlation function is negligible [@Comm2]. The correlator $C^{II}$ differs from $C^{I}$ only by different It be by taking the ratio of the following two expressions [@Zajc84]: $$\begin{aligned} \label{38} P_2(\bbox{p}_1,\bbox{p}_2) &=& {E_{\bbox{p}_1} \, \over N^c_{\rm pairs}} \sum_{i=1}^{N_{\rm ev}} \tilde\nu_{i,i}(\bbox{p}_1,\bbox{p}_2) \,, \\ P_1(\bbox{p}_1) P_1(\bbox{p}_2) &=& {E_{\bbox{p}_1} E_{\bbox{p}_2} \over N^u_{\rm pairs}} \sum_{i,j=1 i\ne ev}} \tilde\nu_{i,j}(\bbox{p}_1, \,. $N^c_{\rm and $N^u_{\rm pairs}$ the total numbers of “correlated” and “uncorrelated” pion pairs, respectively: $$\begin{aligned} N^c_{\rm pairs} &=& N_{\rm ev} \cdot n(n-1)\rangle \,, pairs} &=& ev}(N_{\rm \cdot n \rangle^2 \,. {#sec6} =========== We have shown that normalization of the two-particle Bose-Einstein correlation function contains information on the multiplicity distribution of the event Both

In practice one can live with fEwer event pAirs fOr eVenT mIxinG, by rEplacing in (\[36\]) the nUMber $n_{\rm ev}$ by a much smaller numBer $N'_{\rM eV}$. as loNG aS $N'_{\rm eV}(N'_{\rm ev}-1) \gG n_{\rM EV}$ onE cAn StiLl ENsUre thAt tHe contrIbution of tHe dEnOminator to thE StAtistical eRroR of the final cOrrElatioN fUncTIon is NegLigibLe [@Comm2]. tHe corrElator $C^{II}$ DiFFers frOM $C^{I}$ only BY ThE difFerent normalizatiON. IT Can be constructEd by taKiNG tHE RatIo oF the followInG two eXPressioNS [@ZAJC84]: $$\BegIN{aligned} \label{38} p_2(\bbox{p}_1,\bbox{p}_2) &=& {e_{\BboX{p}_1} \, E_{\bboX{p}_2} \OveR n^c_{\rm paIrs}} \suM_{i=1}^{n_{\Rm eV}} \tilde\nu_{i,i}(\bBox{p}_1,\Bbox{p}_2) \,, \\ \labeL{39} P_1(\bbox{P}_1) p_1(\bbox{p}_2) &=& {E_{\BBox{p}_1} \, E_{\bbOx{p}_2} \oveR N^u_{\Rm pAirs}} \SUm_{I,j=1 \AtoP i\NE j}^{N_{\RM eV}} \tiLDe\nU_{i,j}(\bbox{p}_1, \BbOx{P}_2) \,. \end{aLignED}$$ $n^C_{\Rm paIrs}$ And $N^U_{\rm paIrs}$ are the totaL nuMberS Of “cOrrelAted” aNd “unCoRrelaTed” pioN pairS, rEspectively: $$\begiN{aliGned} N^c_{\rm pAirS} &=& N_{\Rm eV} \cDot \laNGle n(n-1)\rAngLe \,, \\ N^U_{\rm pairS} &=& N_{\rm ev}(N_{\RM ev}-1) \CdOT \LAnGle n \rangle^2 \,. \end{alignEd}$$ cONcLusions {#sEc6} =========== We haVE sHoWN that in pRiNciPle tHE NormaLizaTIoN of the twO-partiCLe boSe-EinstEiN correLaTioN fuNctioN ContAins vaLuable inFormaTIon on the the mulTIplicity distrIBuTIOn OF the EveNt ensemble. BOth

In practice one can livewith fewer even t p air sforeven t mixing, by r e plac ing in (\[36\]) the nu mber$N _ {\rm ev }$ by a much sm a l ler n um ber $ N '_ {\rm e v}$. As long as $ N'_ {\ rm ev}(N'_{\ r mev}-1) \gg N_ {\rm ev}$ on e c an sti ll en s ure t hat thecontri b utionof the de no m inator to thes t at isti cal error of thef in a l correlationfuncti on is n egl igi ble [@Comm 2] . Th e correl a to r $ C^{ I I}$ differs f rom $C^{I}$ onl y by t he di f ferent norm al i zat ion. It can beconstruct ed byt aking t h e ratio of th e f oll owin g t wo ex pr e ssi o ns [@ Z ajc 84]: $$\ be gi n{ali gned } \ labe l{3 8} P_2 (\bbox{p}_1,\ bbo x{p} _ 2)&=& {E _{\b bo x{p}_ 1} \,E_{\b bo x{p}_2} \over N ^c_{ \rm pairs }} \s um _{i=1 } ^{N_{\ rmev} } \tild e\nu_{i , i}( \b b o x {p }_1,\bbox{p}_2) \, ,\ \ \label{ 39} P_ 1( \ bbox{p}_ 1) P_ 1(\b b o x{p}_ 2) & = & {E_{ \bbox{ p }_ 1} \, E_{ \b box{p} _2 } \ ove r N^u _ {\rm pairs }} \s um_{i , j=1 \atop i\ne j}^{N_{\rm ev } } \ t ilde \nu _{i,j}(\bbo x{p} _ 1, \ bbox { p} _2) \,. \end{ al i gn e d}$$ $N^c_{\rm pair s} $ and$N^u_ {\rm pairs}$are the to t a l numbers of“ co r related” and “ uncor related” p i on pairs , res pectivel y: $$\beg i n {aligned } N ^c_ {\r m pa irs} &=& N_{\ r m ev} \ cdot \l ang le n(n- 1)\ ran gle \, , \\ N ^u_{\rmpa ir s} & =&N_{\r m ev}(N_{ \r m e v} -1) \cdo t \lang l en \ran gl e^ 2 \, . \e nd{al igne d }$$ Concl usions {# sec 6 } == == == ===== We have shown t hat in pri nc ipl e then o rmalizat ion of the two-particle Bose-Ei nst ein c orre lation fu nct ion co nta i ns val uableinfor ma tio n on th e th e m ul tiplicityd i str ibuti on ofthe eve nt ensemble. Both

In_practice one_can live with fewer_event pairs_for_event mixing,_by_replacing in (\[36\])_the number $N_{\rm_ev}$ by a much_smaller number $N'_{\rm __ev}$. As long as $N'_{\rm ev}(N'_{\rm ev}-1) \gg N_{\rm ev}$ one can still ensure_that_the contribution_of_the_denominator to the statistical error_of the final correlation function_is negligible_[@Comm2]. The correlator $C^{II}$ differs from $C^{I}$ only by_the_different normalization. It_can be constructed by taking the ratio of the_following two expressions [@Zajc84]: $$\begin{aligned} \label{38} _ P_2(\bbox{p}_1,\bbox{p}_2)_&=&_ _ {E_{\bbox{p}_1} \,_E_{\bbox{p}_2} \over N^c_{\rm pairs}} _\sum_{i=1}^{N_{\rm ev}} \tilde\nu_{i,i}(\bbox{p}_1,\bbox{p}_2) \,, \\ \label{39} _ P_1(\bbox{p}_1) P_1(\bbox{p}_2) &=& _ {E_{\bbox{p}_1} \, E_{\bbox{p}_2} \over N^u_{\rm_pairs}} \sum_{i,j=1 \atop_i\ne j}^{N_{\rm_ev}} \tilde\nu_{i,j}(\bbox{p}_1, \bbox{p}_2)_\,. \end{aligned}$$ $N^c_{\rm_pairs}$ and_$N^u_{\rm pairs}$ are_the total numbers of “correlated” and_“uncorrelated” pion pairs,_respectively: $$\begin{aligned} N^c_{\rm pairs}_&=&_N_{\rm ev} \cdot_\langle_n(n-1)\rangle_\,, _\\ _N^u_{\rm_pairs} &=&_N_{\rm_ev}(N_{\rm ev}-1) \cdot __ _ __ _ \langle n_\rangle^2 \,. \end{aligned}$$ Conclusions {#sec6} =========== We have shown_that in_principle the_normalization of the two-particle Bose-Einstein correlation function contains valuable information on_the the multiplicity distribution of the_event ensemble. Both

d+1}{d+1} + \dots + \binom{d-s+2}{d-s+2}=s$$ as claimed. \[delta max not imply I max\] In practice, it usually is the case that $\Delta h_Z$ having maximal growth from degree $d$ to degree $d+1$ does [*not*]{} imply that $h_Z$ itself has maximal growth from degree $d$ to degree $d+1$. If it did, Gotzmann’s results (Theorem \[gotzmann persistence\] and Remark \[interpret max gr\]) would immediately apply to $h_Z$. The interest in the results described below is that similar powerful conclusions come from this maximal growth of the first difference (see Proposition \[BGM saturation result\] and Remark \[comparison\]). And as we will see, even more surprising is that similar results can be deduced at times even when the first difference does not have maximal growth. General results {#general results} =============== A by-now classical result of Davis (cf. [@davis]) is the following. Note that there is no uniformity assumption on $Z$. In the next section we will discuss the refinements that are possible when we assume UPP. \[davis thm\] Let $Z \subset \mathbb P^{2}$ be a zero-dimensional scheme. If $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ for $d \geq s$, then $(I_Z)_d$ and $(I_Z)_{d+1}$ both have a GCD, $F$, of degree $s$. If $Z$ is reduced then so is $F$. The polynomial $F$ defines a $\left \{ \begin{array}{l} \hbox{hypersurface} \\ \hbox{curve} \end{array} \right. $ in $\mathbb P^{2}$. If $Z_1 \subset Z$ is the subscheme of $Z$ lying on $F$ (defined by $[I_Z + (F)]^{sat}$) and $Z_2$ is the “residual” scheme defined by $[I_Z :F]$, then there are formulas relating the Hilbert functions of $I_Z, I_{Z_1}$ and $I_{Z_2}$. \[iarrobino comment\] It is worth mentioning that the Art

d+1}{d+1 } + \dots + \binom{d - s+2}{d - s+2}=s$$ as claimed. \[delta max not imply I max\ ] In practice, it usually is the subject that $ \Delta h_Z$ have maximal growth from degree $ d$ to degree $ d+1 $ does [ * not * ] { } imply that $ h_Z$ itself have maximal growth from degree $ d$ to degree $ d+1$. If it dress, Gotzmann ’s resultant role (Theorem \[gotzmann persistence\ ] and Remark \[interpret max gr\ ]) would immediately apply to $ h_Z$. The sake in the results described under is that similar powerful stopping point come from this maximal growth of the first difference (see Proposition \[BGM saturation result\ ] and Remark \[comparison\ ]). And as we will attend, even more surprising is that similar results can be deduced at time even when the first difference does not have maximal growth. General consequence { # general results } = = = = = = = = = = = = = = = A by - now classical result of Davis (californium.   [ @davis ]) is the following. Note that there is no uniformity assumption on $ Z$. In the next section we will discuss the refinements that are possible when we assume UPP. \[davis thm\ ] Let $ Z \subset \mathbb P^{2}$ be a zero - dimensional scheme. If $ \Delta h_Z(d) = \Delta h_Z(d+1) = s$ for $ d \geq s$, then $ (I_Z)_d$ and $ (I_Z)_{d+1}$ both have a GCD, $ F$, of degree $ s$. If $ Z$ is reduced then thus is $ F$. The polynomial $ F$ defines a $ \left \ { \begin{array}{l } \hbox{hypersurface } \\ \hbox{curve } \end{array } \right. $ in $ \mathbb P^{2}$. If $ Z_1 \subset Z$ is the subscheme of $ Z$ lying on $ F$ (define by $ [ I_Z + (F)]^{sat}$) and $ Z_2 $ is the “ residual ” outline defined by $ [ I_Z: F]$, then there are rule relating the Hilbert functions of $ I_Z, I_{Z_1}$ and $ I_{Z_2}$. \[iarrobino comment\ ] It is worth mentioning that the artwork

d+1}{d+1} + \dots + \binom{d-s+2}{d-s+2}=s$$ as clalmed. \[delta max noj umply M max\] Ih practize, it usually is the case thet $\Dwlta y_Z$ having maximal growgh from dvgree $d$ ti dejree $d+1$ does [*not*]{} mjply that $h_Z$ jbself ias maximal groeth from dagree $d$ to degsed $b+1$. If it did, Gotzmann’s results (Theorei \[gotzmsnj persistence\] wnd Geiark \[pnuerpret max gr\]) would immediately zpply tm $h_Z$. The intetest in the results descrihed helow is that simipar powerfuo cogxlusions comd from this maximal grkwth of the first difference (sed Pro'osition \[BGN watktation resulv\] and Gemark \[comparlxon\]). Ang as we will see, even moce syrprising is that simmlar results can be qeduced ad cimes even when the furwt dixferance eoer nkt hzve madimel growth. Geheral resulrs {#general results} =============== A br-bow classical resule jf Davis (cf. [@davis]) is the following. Note uhat fhere is no uniformity qssumption on $Z$. In thg next seceion we will discuss the refinements that are posvible dheu we arwule UPP. \[davis thm\] Let $Z \subset \mathbb P^{2}$ be a zewk-domvnsional scheme. In $\Delta h_Z(d) = \Delta n_Z(f+1) = f$ for $d \geq s$, then $(N_A)_d$ and $(I_Z)_{d+1}$ both have a GCD, $S$, of eegree $s$. Yf $Z$ is reduced then so is $F$. Thw polynomial $D$ defines a $\left \{ \yegin{array}{l} \hyox{hypgrsurfsce} \\ \hbox{curve} \end{array} \rnght. $ jn $\mathbb P^{2}$. If $Z_1 \subadt Z$ is the subszheke of $Z$ lying on $F$ (defined br $[I_Z + (F)]^{sav}$) and $Z_2$ is tfe “rgsidual” scheme devined by $[I_Z :F]$, then there are yormunas relatijg the Hilbert functions of $I_Z, M_{V_1}$ and $I_{Z_2}$. \[iarronito bomment\] Ic is wprth mentionigg that the Arj

d+1}{d+1} + \dots + \binom{d-s+2}{d-s+2}=s$$ as claimed. not I max\] practice, it usually h_Z$ maximal growth from $d$ to degree does [*not*]{} imply that $h_Z$ itself maximal growth from degree $d$ to degree $d+1$. If it did, Gotzmann’s results \[gotzmann persistence\] and Remark \[interpret max gr\]) would immediately apply to $h_Z$. The in results below that similar powerful conclusions come from this maximal growth of the first difference (see Proposition \[BGM result\] and Remark \[comparison\]). And as we will even more surprising is similar results can be deduced times when the difference not maximal growth. General {#general results} =============== A by-now classical result of Davis (cf. [@davis]) is the following. Note that there no uniformity $Z$. In next we discuss the refinements possible when we assume UPP. \[davis \subset \mathbb P^{2}$ be a zero-dimensional scheme. If h_Z(d) = h_Z(d+1) = s$ for $d \geq then $(I_Z)_d$ and $(I_Z)_{d+1}$ both have a GCD, of degree $s$. If $Z$ is reduced then so is $F$. The polynomial $F$ defines \{ \begin{array}{l} \hbox{hypersurface} \\ \end{array} \right. $ $\mathbb If \subset is the of $Z$ lying on $F$ (defined by $[I_Z + (F)]^{sat}$) and is the “residual” scheme defined by $[I_Z :F]$, then there relating Hilbert functions of I_{Z_1}$ and $I_{Z_2}$. \[iarrobino It worth mentioning that the

d+1}{d+1} + \dots + \binom{d-s+2}{d-s+2}=s$$ as claimed. \[Delta max noT implY I mAx\] IN pRactIce, iT usually is the cASe thAt $\Delta h_Z$ having maximal GrowtH fROm deGReE $d$ to dEgree $d+1$ dOEs [*NOT*]{} imPlY tHat $H_Z$ ITsElf haS maXimal grOwth from deGreE $d$ To degree $d+1$. If iT DiD, Gotzmann’s ResUlts (Theorem \[gOtzMann peRsIstENce\] anD ReMark \[iNterprET max gr\]) Would immeDiATely apPLy to $h_Z$. THE InTereSt in the results desCRiBEd below is that sImilar PoWErFUL coNclUsions come FrOm thiS Maximal GRoWTH Of tHE first differeNce (see PropoSItiOn \[BGM sAtUraTIon resUlt\] anD REMarK \[comparison\]). and aS we will seE, even mORe surprISing is tHat simIlaR reSultS CaN bE deDuCEd aT TiMes EVen When the fIrSt DiffeRencE DOES not HavE maxImal gRowth. General rEsuLts {#gENerAl resUlts} =============== A By-noW cLassiCal resUlt of daVis (cf. [@davis]) is the FollOwing. Note ThaT tHerE iS no unIFormitY asSumPtion on $z$. In the nEXt sEcTION wE will discuss the refInEMEnTs that arE possiBLe WhEN we assumE UpP. \[dAvis THM\] Let $Z \SubsET \mAthbb P^{2}$ be A zero-dIMeNsIonal scHeMe. If $\DeLtA h_Z(D) = \DeLta h_Z(D+1) = S$ for $D \geq s$, tHen $(I_Z)_d$ anD $(I_Z)_{d+1}$ bOTh have a GCD, $F$, of dEGree $s$. If $Z$ is redUCeD THeN So is $f$. ThE polynomial $f$ defINes a $\Left \{ \BEgIn{aRRay}{l} \hBox{hyPeRSuRFace} \\ \hbox{curve} \end{arrAy} \Right. $ iN $\mathBb P^{2}$. If $Z_1 \subset Z$ Is the subscHEME of $Z$ lyinG on $F$ (DEfINed by $[I_Z + (F)]^{sat}$) and $z_2$ is thE “residual” sCHeme defiNed by $[i_Z :F]$, then tHere are foRMUlas relaTinG thE HiLbeRT FuNctions of $I_Z, I_{Z_1}$ AND $I_{Z_2}$. \[iArRobino cOmmEnt\] It is WorTh mEntIonInG that the ARt

d+1}{d+1} + \dots + \binom {d-s+2}{d- s+2}= s$$ as c laim ed. \[delta max n o t im ply I max\] In practic e, it u s uall y i s the case t h at $ \De lt ah_Z $h av ing m axi mal gro wth from d egr ee $d$ to degr e e$d+1$ does [* not*]{} impl y t hat $h _Z $ i t selfhas maxi mal gr o wth fr om degree $ d $ to d e gree $d + 1 $. Ifit did, Gotzmann’ s r e sults (Theorem \[got zm a nn p ers ist ence\] and R emark \[inter p re t m axg r\]) would im mediately a p ply to $h _Z $.T he int erest i n th e results d escr ibed belo w is t h at simi l ar powe rful c onc lus ions co me fr om thi s m axi m algrowth o fth e fir st d i f f e renc e ( seePropo sition \[BGMsat urat i onresul t\] a nd R em ark \ [compa rison \] ). And as we wi ll s ee, evenmor esur pr ising is tha t s imi lar res ults ca n be d e d u ce d at times even wh en t he first d iffere n ce d o es not h av e m axim a l grow th.Ge neral re sults{ #g en eral re su lts} = == === === ===== = Aby-now classic al re s ult of Davis ( c f. [@davis])i st h ef ollo win g. Note tha t th e re i s no un ifo r mityassum pt i on on $Z$. In the next s ection we w ill discuss t he refinem e n t s that a re p o ss i ble when we as sumeUPP. \[da v is thm\] Let$Z \subs et \mathb b P^{2}$ b e a ze ro- dim e n si onal scheme.I f $\D el ta h_Z( d)= \Delt a h _Z( d+1 ) = s $ for $d\geq s$, t he n$( I_Z )_d$a nd $(I_Z )_ {d+ 1} $ b oth h a ve a G CD, $ F$,of d e gre e $s$.I f$ Z $ is r ed uced th en so i s $F $ . T he poly nomial $F $ d e fine sa$\left\{ \begin{ar ra y}{l} \hbo x{ hyp ersurf a c e} \\ \h box{curve} \end{array}\ right.$ i n $\m athb b P^{2}$. If $Z_1\su b set Z$ is th e sub sc hem e of $Z $ ly ing o n $F$ (def i n edby $[ I_ Z +(F)]^{s at}$) and $Z_2$ is the “residual” s che me d e f in edb y$ [I_ Z: F]$ , then there areformulas r el a ti ng the Hil b ert f unction s of $I _Z, I _ {Z_1}$and $I_{Z _2}$. \[ ia rrob i n o c omment\] I t is wor th mentio n ing t h at theArt

d+1}{d+1} +_\dots +_\binom{d-s+2}{d-s+2}=s$$ as claimed. \[delta max_not imply_I_max\] In_practice,_it usually is_the case that_$\Delta h_Z$ having maximal_growth from degree_$d$_to degree $d+1$ does [*not*]{} imply that $h_Z$ itself has maximal growth from degree_$d$_to degree_$d+1$._If_it did, Gotzmann’s results (Theorem_\[gotzmann persistence\] and Remark \[interpret_max gr\])_would immediately apply to $h_Z$. The interest in_the_results described below_is that similar powerful conclusions come from this maximal_growth of the first difference (see_Proposition \[BGM saturation_result\]_and_Remark \[comparison\]). And as_we will see, even more surprising_is that similar results can be_deduced at times even when the first_difference does not have maximal growth. General_results {#general results} =============== A by-now classical_result of_Davis (cf. [@davis]) is the following._Note that there_is no_uniformity assumption on_$Z$. In the next section we_will discuss the_refinements that are possible when we_assume_UPP. \[davis thm\] Let_$Z_\subset_\mathbb P^{2}$_be a zero-dimensional_scheme._If $\Delta_h_Z(d)_= \Delta h_Z(d+1) = s$ for $d_\geq_s$, then $(I_Z)_d$ and $(I_Z)_{d+1}$ both have_a GCD, $F$, of_degree_$s$. If $Z$ is_reduced then so is $F$._The polynomial $F$ defines a $\left_\{ \begin{array}{l} \hbox{hypersurface}_\\ \hbox{curve} \end{array} \right._$ in $\mathbb P^{2}$. If $Z_1 \subset Z$ is the subscheme_of $Z$ lying on $F$ (defined_by $[I_Z + (F)]^{sat}$)_and $Z_2$_is_the “residual” scheme_defined_by $[I_Z_:F]$, then there are formulas relating the_Hilbert functions_of $I_Z, I_{Z_1}$ and $I_{Z_2}$. \[iarrobino comment\]_It is worth mentioning_that_the Art

inherent problem [@Dubray+12], the required computational cost is prohibitive. The microscopic approach, at the present time, is therefore best suited for studies of specific nuclei, but is not adequate for large-scale, global studies of fission yields and their trends across the chart of nuclides. A recent review covering the progress of this approach can be found in Ref. [@Schunck+16]. ![\[fig:schema\] A schematic illustration of the fission process: The lower panel shows the potential energy of the nuclear system along its most probable path, while the upper panel shows the appearance of the system at four stages along that path. The nuclear shape, which is initially located near that of the ground state, is strongly coupled to the internal microscopic degrees of freedom and, as a result, it executes a Brownian-like random walk on the multidimensional potential-energy surface. After passing over the various saddle points, generally after multiple attempts, the system eventually acquires a binary shape and reaches a necked-in scission configuration where it divides into two fission fragments. The shown potential-energy profile is representative of known actinides, and may be differ qualitatively for nuclei in other regions. ](fission-schematic.pdf){width="\columnwidth"} The macroscopic-microscopic approach offers a simpler and very effective framework for calculating the nuclear PES [@Nix+65]. This method was originally developed for the calculation of nuclear masses because purely microscopic calculations tend to have difficulty obtaining accurate absolute energies due to the small but significant role played by many-body correlations which are hard to treat. Nuclear masses exhibit smoothly varying macroscopic trends, reflecting the energetics of a charged droplet, overlaid with small-amplitude deviations reflecting the microscopic nuclear structure [@Gustafsson+71; @Brack+72; @Bolsterli+72]. The nuclear potential-energy surface is therefore considered to consist of a [*macroscopic*]{} liquid-drop like energy functional, whose parameters (volume energy, surface tension,...) are determined by global fitting to the measured masses, and a [*microscopic*]{} contribution expressing the shell [@Strutinsky+63] and pairing corrections [@Nogami+64], which can be calculated from the neutron and proton level spectra in the deformed effective potential well. This approach makes it possible to calculate the potential energy of any nuclear system with $Z$ protons and $N$ neutrons, $(Z,N)$, as a function of its shape (as well as its angular

inherent problem [ @Dubray+12 ], the required computational cost is prohibitive. The microscopic overture, at the present meter, is therefore best suited for study of specific nuclei, but is not adequate for large - plate, global studies of fission yields and their tendency across the chart of nuclides. A recent review cover the progress of this approach path can be establish in Ref.   [ @Schunck+16 ]. ! [ \[fig: schema\ ] A schematic illustration of the fission process: The broken panel shows the potential department of energy of the nuclear system along its most probable way, while the upper panel show the appearance of the system at four stages along that path. The nuclear shape, which is initially located near that of the flat coat state, is strongly coupled to the internal microscopic degrees of freedom and, as a result, it executes a Brownian - like random walk on the multidimensional potential - energy surface. After passing over the various saddle points, by and large after multiple attempts, the system finally grow a binary condition and reaches a necked - in scission configuration where it divides into two fission fragments. The shown potential - department of energy profile is representative of known actinides, and may be differ qualitatively for nuclei in other regions. ] (fission - schematic.pdf){width="\columnwidth " } The macroscopic - microscopic approach offers a simpler and very effective framework for calculating the nuclear PES [ @Nix+65 ]. This method acting was originally develop for the calculation of nuclear masses because purely microscopic calculations tend to have difficulty prevail accurate absolute energies due to the small but significant function played by many - body correlations which are hard to cover. Nuclear masses exhibit smoothly change macroscopic trends, reflect the energetics of a charged droplet, overlay with small - amplitude deviation reflecting the microscopic nuclear structure [ @Gustafsson+71; @Brack+72; @Bolsterli+72 ]. The nuclear likely - energy surface is therefore considered to consist of a [ * macroscopic * ] { } liquid - drop like energy functional, whose parameters (volume energy, surface latent hostility, ...) are determined by global adjustment to the measured masses, and a [ * microscopic * ] { } contribution expressing the shell [ @Strutinsky+63 ] and pairing corrections [ @Nogami+64 ], which can be calculated from the neutron and proton degree spectra in the deformed effective potential well. This approach makes it possible to calculate the potential department of energy of any nuclear system with $ Z$ protons and $ N$ neutrons, $ (Z, N)$, as a function of its shape (as well as its angular

injerent problem [@Dubray+12], tht required computcrional cost js prohicitive. The microscopic approech, qt tht present time, is tferefore hest suired hor studies of s'scific kbclei, nut iv not adequate nor large-scdle, global stugids of fission yields and their trends across tje chart of nuslidts. W redvnu review covering the progress of this akproach can be foumd in Ref. [@Schunck+16]. ![\[fig:schema\] W scjematic illustratiln of the fussijb process: Thd lower panel shows thg potential energy of the nucleaf syscem along ijs mlvt probable path, while the uiier panal showx the appearange of thw system at four stagxs along that path. Tre nucleas ahape, which is inutually locdted beaf tgav or the hronnd state, ia strongly xoupled to the intetnwo microscopic degreqs of freedom and, as a result, it executes a Grownian-like random walj on the multidimensilnal potegtial-energy surface. After passing over the variouv sadvld piikts, ewnfrally after multiple attempts, the system evegfuslky acquires a ninary shape and rrafhrf a necked-in rcission donfiguration wherf it didides into two fisxion fragments. The shown porential-energj prifile is representctive of knocn actonidex, and may be differ quauitafively for juclei in uther regions. ](firsipn-vchematic.pdf){width="\columnwideh"} The maccosco'ic-microrcopoc appwoach offegs a simpler and very efvectire frdmework fog calculating the nuclear PES [@Nme+65]. This method wds mriginaljy deyeloped for the calculation oy nucleax massds because purely microscopis calculationv tend to havx difficujty ibtauning azzurate absolutr energies due to tye small but signinicanj dole played by nani-vody correlatipns whycj ere hwsd to treat. Tucldar kasser exhibit snuothky varying macroscophc tdends, reflecting tne energetucs of a charged dropket, overlaid with dmall-emplitnde defiajions reflecting the microscopid nuclear stvucture [@Gustasssok+71; @Brwck+72; @Bolstexli+72]. The nuclear potential-energy surface ms therefore considered to consist of a [*maerpscopic*]{} liqnid-dro[ like enargy functional, whosw parameters (voluke energy, surface tenaion,...) ase dehermined by global fitting to the measured masses, and a [*microscopic*]{} contrubutioi qxpressing fhe xhell [@Scrucinsky+63] wnd 'amring corrections [@Nogami+64], which can be calculated from tha ueutron and proton level specyrx in the defotmed effective potential wsll. This approach makes it possible to valculate the potential cnergy of any hucleag sbstem with $Z$ protons and $N$ neuteons, $(Z,N)$, as w function of itx dhape (as aell aa its angular

inherent problem [@Dubray+12], the required computational cost The approach, at present time, is of nuclei, but is adequate for large-scale, studies of fission yields and their across the chart of nuclides. A recent review covering the progress of this can be found in Ref. [@Schunck+16]. ![\[fig:schema\] A schematic illustration of the fission The panel the energy of the nuclear system along its most probable path, while the upper panel shows the of the system at four stages along that The nuclear shape, which initially located near that of ground is strongly to internal degrees of freedom as a result, it executes a Brownian-like random walk on the multidimensional potential-energy surface. After passing over various saddle after multiple the eventually a binary shape a necked-in scission configuration where it fission fragments. The shown potential-energy profile is representative known actinides, may be differ qualitatively for nuclei other regions. ](fission-schematic.pdf){width="\columnwidth"} The macroscopic-microscopic approach offers a and very effective framework for calculating the nuclear PES [@Nix+65]. This method was originally developed calculation of nuclear masses purely microscopic calculations to difficulty accurate energies due the small but significant role played by many-body correlations which are to treat. Nuclear masses exhibit smoothly varying macroscopic trends, reflecting of charged droplet, overlaid small-amplitude deviations reflecting the nuclear [@Gustafsson+71; @Brack+72; @Bolsterli+72]. The surface therefore of [*macroscopic*]{} like energy functional, whose (volume energy, surface tension,...) are by global fitting to [*microscopic*]{} contribution expressing the shell [@Strutinsky+63] and pairing [@Nogami+64], which can be calculated from the and proton level spectra in the deformed effective potential well. This approach it possible the potential energy of any nuclear system with protons and $N$ neutrons, as a function of its shape (as well as angular

inherent problem [@Dubray+12], the rEquired comPutatIonAl cOsT is pRohiBitive. The microSCopiC approach, at the present tIme, is ThERefoRE bEst suIted for STuDIEs oF sPeCifIc NUcLei, buT is Not adeqUate for larGe-sCaLe, global studIEs Of fission yIelDs and their trEndS acrosS tHe cHArt of NucLides. a recenT Review Covering tHe PRogresS Of this aPPRoAch cAn be found in Ref. [@SchUNcK+16]. ![\[Fig:schema\] A scheMatic iLlUStRATioN of The fission PrOcess: tHe lower PAnEL SHowS The potential eNergy of the nUCleAr systEm AloNG its moSt proBaBLe pAth, while the UppeR panel shoWs the aPPearancE Of the syStem at FouR stAges ALoNg ThaT pATh. THE nUclEAr sHape, whicH iS iNitiaLly lOCATEd neAr tHat oF the gRound state, is sTroNgly COupLed to The inTernAl MicroScopic DegreEs Of freedom and, as a ResuLt, it execuTes A BRowNiAn-likE Random WalK on The multIdimensIOnaL pOTENtIal-energy surface. AfTeR PAsSing over The varIOuS sADdle poinTs, GenEralLY After MultIPlE attemptS, the sySTeM eVentualLy AcquirEs A biNarY shapE And rEaches A necked-iN scisSIon configuratiON where it dividES iNTO tWO fisSioN fragments. THe shOWn poTentIAl-EneRGy proFile iS rEPrESentative of known actInIdes, anD may bE differ qualitAtively for NUCLei in othEr reGIoNS. ](fission-schemaTic.pdF){width="\coluMNwidth"} ThE macrOscopic-mIcroscopiC APproach oFfeRs a SimPleR ANd Very effective FRAmewOrK for calCulAting thE nuCleAr PeS [@NIx+65]. this methoD was origInAlLy DeVelOped fOR the calcUlAtiOn Of nUcleaR Masses BecauSe puReLy MIcrOscopic CAlCULatiOnS tEnd tO haVe DiffiCultY ObtAining aCcurate abSolUTe enErGiEs due to The small but siGnIficant rolE pLayEd by maNY-Body corrElations which are hard to tREat. NuclEar MasseS exhIbit smootHly VaryinG maCRoscopIc trenDs, refLeCtiNG The enERGeTicS oF a charged dROPleT, overLaId wiTh small-Amplitude deviationS RefLecting the micRosCopiC NUcLeaR StRUctUrE [@gusTAFsson+71; @Brack+72; @BolstErli+72]. The nucLeAR pOtential-enERgy SuRface is TherefoRe conSIdered tO consist oF a [*macroscOpIc*]{} liQUId-dRop like eneRgy functIonal, whosE ParamETeRs (volUme Energy, SuRfaCe tenSion,...) arE DetErminEd by glObAl fittIng to ThE measureD masses, and a [*microscopic*]{} cOntribUtion ExpRessing thE shELl [@STrutinsky+63] And pAiring corrEctIonS [@NogaMi+64], wHIch caN be cALcUlaTEd froM the NEutron and PRoTon LEVeL spectra in tHE DEfoRmed eFfeCTive poTentIal well. This approaCH makes it possibLe to CALcuLatE The pOtEntial energy of Any NuCLEar systeM wIth $Z$ protons And $N$ neutRoNS, $(Z,N)$, as A functIon of iTs shape (AS WeLL as its AnguLar

inherent problem [@Dubray +12], therequi red co mp utat iona l cost is proh i biti ve. The microscopic ap proac h, at t h eprese nt time , i s the re fo rebe s tsuite d f or stud ies of spe cif ic nuclei, but is not adequ ate for large-s cal e, glo ba l s t udies of fiss ion yi e lds an d their t re n ds acr o ss thec h ar t of nuclides. A rece n tr eview covering the p ro g re s s of th is approac hcan b e foundi nR e f . [ @ Schunck+16]. ![\[fig:sc h ema \] A s ch ema t ic ill ustra ti o n o f the fissi on p rocess: T he low e r panel shows t he pot ent ial ene r gy o f t he nuc l ea r s y ste m alongit smostprob a b l e pat h,whil e the upper panelsho ws t h e a ppear anceof t he syst em atfourst ages along that pat h. The nu cle ar sh ap e, wh i ch isini tia lly loc ated ne a r t ha t o fthe ground state,is s tr ongly co upledt oth e interna lmic rosc o p ic de gree s o f freedo m and, as a result ,it exe cu tes aBrown i an-l ike ra ndom wal k ont he multidimens i onal potentia l -e n e rg y sur fac e. After pa ssin g ove r th e v ari o us sa ddlepo i nt s , generally after m ul tipleattem pts, the syst em eventua l l y acquire s ab in a ry shape and r eache s a necked - in sciss ion c onfigura tion wher e it divid esint o t wof i ss ion fragments . Thesh own pot ent ial-ene rgy pr ofi leis represen tative o fkn ow nact inide s , and ma ybedi ffe r qua l itativ ely f or n uc le i in otherr eg i o ns.]( fi ssio n-s ch emati c.pd f ){w idth="\ columnwid th" } Th ema croscop ic-microscopi capproach o ff ers a sim p l er and v ery effective framework for cal cul ating the nuclearPES [@Nix +65 ] . This metho d was o rig i n allyd e ve lop ed for the c a l cul ation o f nu clear m asses because pure l y m icroscopic ca lcu lati o n sten d t o ha ve dif f i culty obtaining accurateab s ol ute energi e s d ue to the smallbut s i gnifica nt role p layed byma ny-b o d y c orrelation s whichare hardt o tre a t. Nucl ear masse sexh ibitsmooth l y v aryin g macr os copictrend s, reflect ing the energetics of a charg ed dr opl et, overl aid wit h small-a mpli tude devia tio nsrefle cti n g the mic r os cop i c nuc lear structure [@ Gus t a fs son+71; @Br a c k +72 ; @Bo lst e rli+72 ]. T he nuclear potent i al-energy surf acei s th ere f oreco nsidered to co nsi st o f a [*ma cr oscopic*]{} liquid- dr o p lik e ener gy fun ctional , wh o se par amet ers (volumeene rg y , surfa ce t e nsion, ...) a re det ermine d byg l obal fitting tothe m e a sured mas ses,an d a [*m i cros copic*]{}contributio n expr essi ng th e shell [ @Strut ins ky +63] and p a iring cor recti ons [@N og ami+ 64] , whic h ca n be ca lcul at edfrom then e ut r on a n d p roto n lev el spe ctra in t h e deform ede ffectiv epot e n tial w e ll . This appro ach make s it possibl e toc al c ulate the p otenti al ener g y o fany nuc lea r system wi th $Z$ pr o ton sand$N$ neut ro ns,$( Z,N )$ , as a function of itsshape (a s well a s i t s an g ular

inherent_problem [@Dubray+12],_the required computational cost_is prohibitive._The_microscopic approach,_at_the present time,_is therefore best_suited for studies of_specific nuclei, but_is_not adequate for large-scale, global studies of fission yields and their trends across the_chart_of nuclides._A_recent_review covering the progress of_this approach can be found_in Ref. [@Schunck+16]. ![\[fig:schema\]_A schematic illustration of the fission process: The_lower_panel shows the_potential energy of the nuclear system along its most_probable path, while the upper panel_shows the appearance_of_the_system at four stages_along that path. The nuclear shape,_which is initially located near that_of the ground state, is strongly coupled_to the internal microscopic degrees of_freedom and, as a result,_it executes_a Brownian-like random walk on_the multidimensional potential-energy_surface. After_passing over the_various saddle points, generally after multiple_attempts, the system_eventually acquires a binary shape and_reaches_a necked-in scission_configuration_where_it divides_into two fission_fragments._The shown_potential-energy_profile is representative of known actinides,_and_may be differ qualitatively for nuclei in_other regions. ](fission-schematic.pdf){width="\columnwidth"} The macroscopic-microscopic_approach_offers a simpler and_very effective framework for calculating_the nuclear PES [@Nix+65]. This method_was originally_developed for_the calculation of nuclear masses because purely microscopic calculations tend to_have difficulty obtaining accurate absolute energies_due to the small_but significant_role_played by many-body_correlations_which are_hard to treat. Nuclear masses exhibit smoothly_varying macroscopic_trends, reflecting the energetics of a_charged droplet, overlaid with_small-amplitude_deviations reflecting the microscopic nuclear structure_[@Gustafsson+71; @Brack+72; @Bolsterli+72]. The nuclear potential-energy_surface is therefore considered to_consist_of_a [*macroscopic*]{} liquid-drop like energy_functional, whose parameters (volume energy, surface_tension,...) are determined_by global fitting to the measured masses,_and_a [*microscopic*]{} contribution expressing the shell_[@Strutinsky+63]_and pairing corrections [@Nogami+64], which can_be_calculated_from the neutron and proton_level spectra in the deformed effective_potential well. This approach makes it possible to calculate_the potential energy_of any nuclear system with_$Z$_protons_and $N$ neutrons, $(Z,N)$, as a function of its shape_(as well_as its angular

00000%"} In the most violent collisions, quark-gluon plasma is formed and the emerging hadrons are created via coalescence process. For quarkonia it is the regeneration that dominates the production at high energies. Different from the light hadrons which are produced at the hadronization surface determined by $T({\bf x},t)=T_c$, the quarkonium regeneration happens continuously in the parton phase. The fraction of the regenerated $J/\psi$s, $$\label{fraction} g_{AA}={N_{AA}^{reg}\over N_{AA}},$$ calculated from the transport equation (\[trans\]), is shown in Fig. \[fig2\] as a function of the collision energy, where $N_{AA}^{reg}$ is the integrated $J/\psi$ yield from the regeneration. As expected, the fraction depends on heavy quark density and, in turn, on the collision energy. At SPS, initially produced heavy quarks are few, there is almost no regeneration. At RHIC, both initial production and regeneration of charmonia play important role. At LHC, the $J/\psi$ production is dominated by regeneration. The lower fraction in the forward rapidity merely reflects the rapidity distribution of heavy quarks. ![(color online) $J/\psi$ regeneration fraction in central Au+Au collisions at RHIC and Pb+Pb collisions at SPS and LHC. Results for mid- and forward-rapidity regions are shown as filled and open circles, respectively. []{data-label="fig2"}](fig2.eps){width="43.00000%"} In heavy ion collisions, transverse motion is developed during the dynamical evolution of the system. The microscopically high particle density and multiple scatterings are essential for the finally observed transverse momentum distributions. The distributions are therefore sensitive to the medium properties, like the equation of state. The study on the transverse motion has been well documented in light quark sectors at all energies [@rhicexp1; @rhicexp2; @rhicexp3; @rhicexp4; @hecke98]. In order to understand the quarkonium production and suppression mechanisms and extract the properties of the medium, we propose to construct a new ratio of the second moment of the transverse momentum distribution. The ratio $r_{AA}$ is defined as $$r_{AA}=\frac{\langle p_t^2 \rangle_{AA}}{\langle p_t^2 \rangle_{pp}}.$$ The reason to choose $\langle p

00000% " } In the most violent collisions, quark - gluon plasma is formed and the emerge hadron are created via coalescence process. For quarkonia it is the positive feedback that dominate the production at high energy. unlike from the light hadrons which are produced at the hadronization surface determined by $ T({\bf x},t)=T_c$, the quarkonium re-formation happens continuously in the parton phase. The fraction of the reform $ J/\psi$s, $ $ \label{fraction } g_{AA}={N_{AA}^{reg}\over N_{AA}},$$ calculated from the transport equation (\[trans\ ]), is shown in Fig.   \[fig2\ ] as a function of the collision energy, where $ N_{AA}^{reg}$ is the integrate $ J/\psi$ yield from the regeneration. As expected, the fraction count on heavy quark density and, in turn, on the collision energy. At SPS, initially produce heavy quarks are few, there is almost no regeneration. At RHIC, both initial product and regeneration of charmonia play important role. At LHC, the $ J/\psi$ production is dominated by regeneration. The lower fraction in the forward rapidity merely reflects the rapidity distribution of heavy quarks. ! [ (color online) $ J/\psi$ regeneration fraction in central Au+Au collision at RHIC and Pb+Pb collisions at SPS and LHC. Results for mid- and advancing - celerity regions are shown as filled and open circle, respectively. [ ] { data - label="fig2"}](fig2.eps){width="43.00000% " } In heavy ion collisions, transverse apparent motion is developed during the dynamical evolution of the system. The microscopically high particle concentration and multiple scatterings are essential for the finally observed transverse momentum distributions. The distributions are consequently sensible to the average properties, like the equation of state. The study on the transverse motion has been well documented in light quark sector at all energies   [ @rhicexp1; @rhicexp2; @rhicexp3; @rhicexp4; @hecke98 ]. In order to understand the quarkonium production and inhibition mechanisms and extract the properties of the medium, we propose to manufacture a new proportion of the second moment of the transverse momentum distribution. The ratio $ r_{AA}$ is defined as $ $ r_{AA}=\frac{\langle p_t^2 \rangle_{AA}}{\langle p_t^2 \rangle_{pp}}.$$ The cause to choose $ \langle p

00000%"} In hhe most violent collisiuns, quark-gluon klqsma iv formsd and tfe emerging hadrons are creaved cia cialescence process. For quarkoniw it is rhe cegeneration thav dominabzs ths proburtion at high ekergies. Difxerent from tha uiyht hadrons which are produced at thq hadromixation surface dettrmyned by $T({\bf x},t)=T_c$, the quarkonium regenedation iappens continupusly in the parton phase. Hhe vraction of the rehenerated $J/\kai$s, $$\oabel{fractiov} g_{AA}={N_{AA}^{reg}\over N_{AA}},$$ caldulated from the transport equagion (\[crans\]), is shiwb ij Fig. \[fig2\] as e funcnion of the collision energy, where $N_{AA}^{reg}$ ls thx inregrated $J/\psi$ yield fcom the regeneration. As expecdeb, the fraction dependw in hedvy xuary devsiuy end, in tugn, kn the colmision enerty. At SPS, initially pwiduced heavy suarks awe few, there is almost no regeneration. Dt DHIC, both initial produxtion and regeneratioj of charionia play important role. At LHC, the $J/\psi$ producthon ia donikatea bj regeneration. The lower fraction in the forwwdd rspidity merely reflects tne rskidity distribotion oy hsavy quarks. ![(color ojline) $J/\ksi$ reteneratiog frsction in central Au+Au collusions at RHPC abd Pb+Pb collisions at SPS and LHC. Resukts for mid- and forward-xapidify regions wre shown xs filled and opdn bircnes, resptztively. []{data-label="sig2"}](fig2.eps){xidth="43.00000%"} Nn heavy ion collifions, trandversc motion is developfd duting tve dynamicwl evolution of the system. The microscopicalli hhgh particlz denslty and multiplq scatterings cre esseutial wor the fihally ouserved tranfverse momentgl distributimns. The qisteiburions afd therefore semsitive to the mediym properties, like thg squation of stact. Tye study on the trxnsdegse mjdion has beet weul apcumevted in ligmt duarl sectors at all enesgiea [@rhicexp1; @rhicexp2; @rnigexp3; @rhicgxp4; @hecke98]. In order to inderstand the quagkoninm provuctiom agd suppression mechanisms and sxtract tje iroperties of the medium, we pxopose to construct a new ratio of the sxcond moment of the trabsverse momentum dixuribution. The ratij $r_{AA}$ is gefined as $$r_{AA}=\frac{\labgle p_t^2 \rangle_{AA}}{\lsngle p_t^2 \rangle_{pp}}.$$ The deason to cjoose $\langle p

00000%"} In the most violent collisions, quark-gluon formed the emerging are created via is regeneration that dominates production at high Different from the light hadrons which produced at the hadronization surface determined by $T({\bf x},t)=T_c$, the quarkonium regeneration happens in the parton phase. The fraction of the regenerated $J/\psi$s, $$\label{fraction} g_{AA}={N_{AA}^{reg}\over N_{AA}},$$ from transport (\[trans\]), shown in Fig. \[fig2\] as a function of the collision energy, where $N_{AA}^{reg}$ is the integrated yield from the regeneration. As expected, the fraction on heavy quark density in turn, on the collision At initially produced quarks few, is almost no At RHIC, both initial production and regeneration of charmonia play important role. At LHC, the $J/\psi$ production dominated by lower fraction the rapidity reflects the rapidity heavy quarks. ![(color online) $J/\psi$ regeneration Au+Au collisions at RHIC and Pb+Pb collisions at and LHC. for mid- and forward-rapidity regions are as filled and open circles, respectively. []{data-label="fig2"}](fig2.eps){width="43.00000%"} In ion collisions, transverse motion is developed during the dynamical evolution of the system. The microscopically density and multiple scatterings essential for the observed momentum The are therefore to the medium properties, like the equation of state. The study the transverse motion has been well documented in light quark all [@rhicexp1; @rhicexp2; @rhicexp3; @hecke98]. In order to the production and suppression mechanisms the of propose construct new ratio of the moment of the transverse momentum The ratio $r_{AA}$ is p_t^2 \rangle_{pp}}.$$ The reason to choose $\langle p

00000%"} In the most violent collisionS, quark-gluoN plasMa iS foRmEd anD the Emerging hadronS Are cReated via coalescence prOcess. foR QuarKOnIa it iS the regENeRATioN tHaT doMiNAtEs the ProDuction At high enerGieS. DIfferent from THe Light hadroNs wHich are produCed At the hAdRonIZatioN suRface DetermINed by $T({\Bf x},t)=T_c$, the QuARkoniuM RegenerATIoN hapPens continuously iN ThE Parton phase. The FractiOn OF tHE RegEneRated $J/\psi$s, $$\LaBel{frACtion} g_{Aa}={n_{Aa}^{REG}\ovER N_{AA}},$$ calculateD from the traNSpoRt equaTiOn (\[tRAns\]), is sHown iN FIG. \[fiG2\] as a functioN of tHe collisiOn enerGY, where $N_{aa}^{reg}$ is tHe inteGraTed $j/\psi$ YIeLd FroM tHE reGEnEraTIon. as expectEd, ThE fracTion DEPENds oN heAvy qUark dEnsity and, in tuRn, oN the COllIsion EnergY. At SpS, InitiAlly prOduceD hEavy quarks are feW, theRe is almosT no ReGenErAtion. aT RHIC, bOth IniTial proDuction ANd rEgENERaTion of charmonia plaY iMPOrTant role. at LHC, tHE $J/\PsI$ ProductiOn Is dOminATEd by rEgenERaTion. The lOwer frACtIoN in the fOrWard raPiDitY meRely rEFlecTs the rApidity dIstriBUtion of heavy quARks. ![(color onlinE) $j/\pSI$ ReGEnerAtiOn fraction iN cenTRal AU+Au cOLlIsiONs at RhIC anD PB+pb COllisions at SPS and LHc. REsults For miD- and forward-raPidity regiONS Are shown As fiLLeD And open circles, RespeCtively. []{datA-Label="fig2"}](Fig2.epS){width="43.00000%"} In Heavy ion cOLLisions, tRanSveRse MotION iS developed durING the DyNamical EvoLution oF thE sySteM. ThE mIcroscopiCally higH pArTiClE deNsity ANd multipLe ScaTtEriNgs arE EssentIal foR the FiNaLLy oBserved TRaNSVersE mOmEntuM diStRibutIons. tHe dIstribuTions are tHerEFore SeNsItive to The medium propErTies, like thE eQuaTion of STAte. The stUdy on the transverse motioN Has been WelL docuMentEd in light QuaRk sectOrs AT all enErgies [@RhiceXp1; @RhiCEXp2; @rhiCEXp3; @RhiCeXp4; @hecke98]. In oRDEr tO undeRsTand The quarKonium production anD SupPression mechaNisMs anD EXtRacT ThE ProPeRTieS OF the medium, we proPose to consTrUCt A new ratio oF The SeCond momEnt of thE tranSVerse moMentum disTribution. thE ratIO $R_{AA}$ Is defined aS $$r_{AA}=\frac{\Langle p_t^2 \rANgle_{Aa}}{\LaNgle p_T^2 \raNgle_{pp}}.$$ thE reAson tO choosE $\LanGle p

00000%"} In the most viol ent collis ions, qu ark -g luon pla sma is formeda nd t he emerging hadrons ar e cre at e d vi a c oales cence p r oc e s s.Fo rqua rk o ni a itisthe reg enerationtha tdominates th e p roductionathigh energie s.Differ en t f r om th e l ighthadron s which are prod uc e d at t h e hadro n i za tion surface determin e db y $T({\bf x},t )=T_c$ ,t he q uar kon ium regene ra tionh appensc on t i n uou s ly in the par ton phase.T hefracti on of the re gener at e d $ J/\psi$s, $ $\la bel{fract ion} g _ {AA}={N _ {AA}^{r eg}\ov erN_{ AA}} , $$ c alc ul a ted fr omt hetranspor teq uatio n (\ [ t r a ns\] ),is s hownin Fig. \[fig 2\] asa fu nctio n oftheco llisi on ene rgy,wh ere $N_{AA}^{re g}$is the in teg ra ted $ J/\ps i $ yiel d f rom the re generat i on. A s e xp ected, the fractio nd e pe nds on h eavy q u ar kd ensity a nd , i n tu r n , onthec ol lision e nergy. At S PS, ini ti ally p ro duc edheavy quar ks are few, th ere i s almost no reg e neration. AtR HI C , b o th i nit ial product iona nd r egen e ra tio n of c harmo ni a p l ay important role.At LHC,the $ J/\psi$ produ ction is d o m i nated by reg e ne r ation. The low er fr action int he forwa rd ra pidity m erely ref l e cts therap idi tydis t r ib ution of heav y quar ks . ![(c olo r onlin e)$J/ \ps i$re generatio n fracti on i nce ntr al Au + Au colli si ons a t R HIC a n d Pb+P b col lisi on sa t S PS andL HC . Resu lt sformid -and f orwa r d-r apidity regionsare show nas filled and open cir cl es, respec ti vel y. []{ d a ta-label ="fig2"}](fig2.eps){wid t h="43.0 000 0%"} Inheavy ion co llisio ns, transv erse m otion i s d e v elope d du rin gthe dynami c a l e volut io n of the sy stem. The microsco p ica lly high part icl e de n s it y a n dm ult ip l e s c a tterings are es sential fo rt he finally o b ser ve d trans verse m oment u m distr ibutions. The dist ri buti o n s a re therefo re sensi tive to t h e med i um prop ert ies, l ik e t he eq uation ofstate . Thest udy on thetr ansverse motion has been well d ocumen ted i n l ight quar k s e cto rs at all ene rgies [@rh ice xp1 ; @rh ice x p2; @ rhic e xp 3;@ rhice xp4; @hecke98] . I n o r d er to underst a n d th e qua rko n ium pr oduc tion and suppress i on mechanismsande x tra ctt he p ro perties of the me di u m , we pro po se to const ruct a n ew ratio of th e seco nd mome n t o f the t rans ver se moment umdi s tributi on .T he rat io $ r_ {AA}$is def i neda s $$r_{AA}=\frac{ \lang l e p_t^ 2 \r angle _{ AA}}{\l a ngle p_t^2 \ra ngle_{pp}}. $$ The rea son t o choos e$\lang lep

00000%"} In the_most violent_collisions, quark-gluon plasma is_formed and_the_emerging hadrons_are_created via coalescence_process. For quarkonia_it is the regeneration_that dominates the_production_at high energies. Different from the light hadrons which are produced at the hadronization_surface_determined by_$T({\bf_x},t)=T_c$,_the quarkonium regeneration happens continuously_in the parton phase. The_fraction of_the regenerated $J/\psi$s, $$\label{fraction} g_{AA}={N_{AA}^{reg}\over N_{AA}},$$ calculated from the_transport_equation (\[trans\]), is_shown in Fig. \[fig2\] as a function of the collision_energy, where $N_{AA}^{reg}$ is the integrated_$J/\psi$ yield from_the_regeneration._As expected, the fraction_depends on heavy quark density and,_in turn, on the collision energy._At SPS, initially produced heavy quarks are_few, there is almost no regeneration._At RHIC, both initial production_and regeneration_of charmonia play important role._At LHC, the_$J/\psi$ production_is dominated by_regeneration. The lower fraction in the_forward rapidity merely_reflects the rapidity distribution of heavy_quarks. ![(color_online) $J/\psi$ regeneration_fraction_in_central Au+Au_collisions at RHIC_and_Pb+Pb collisions_at_SPS and LHC. Results for mid-_and_forward-rapidity regions are shown as filled and_open circles, respectively. []{data-label="fig2"}](fig2.eps){width="43.00000%"} In_heavy_ion collisions, transverse motion_is developed during the dynamical_evolution of the system. The microscopically_high particle_density and_multiple scatterings are essential for the finally observed transverse momentum distributions._The distributions are therefore sensitive to_the medium properties, like_the equation_of_state. The study_on_the transverse_motion has been well documented in light_quark sectors_at all energies [@rhicexp1; @rhicexp2; @rhicexp3; @rhicexp4;_@hecke98]. In order to_understand_the quarkonium production and suppression mechanisms_and extract the properties of the_medium, we propose to construct_a_new_ratio of the second moment_of the transverse momentum distribution. The_ratio $r_{AA}$ is_defined as $$r_{AA}=\frac{\langle p_t^2 \rangle_{AA}}{\langle p_t^2 \rangle_{pp}}.$$ The_reason_to choose $\langle p

Acta Arith. **165** (2014), no. 2, 141–179. [to3em]{}, *Explicit estimates for the number of rational points of singular complete intersections over a finite field*, J. Number Theory **158** (2016), no. 2, 54–72. M. Merca, *A note on the determinant of a [Toeplitz–Hessenberg]{} matrix*, Spec. Matrices **1** (2013), 10–16. T. Muir, *The theory of determinants in the historical order of development*, Dover Publications Inc., New York, 1960. L.M. Pardo and J. [San Martín]{}, *Deformation techniques to solve generalized [Pham]{} systems*, Theoret. Comput. Sci. **315** (2004), no. 2–3, 593–625. M. P[é]{}rez, *Análisis probabilístico de algoritmos y problemas combinatorios sobre cuerpos finitos*, Ph.D. thesis, Univ. Buenos Aires, Argentina, 2016. P. Pollack, *Irreducible polynomials with several prescribed coefficients*, Finite Fields Appl. **22** (2013), 70–78. I.R. Shafarevich, *Basic algebraic geometry: [Varieties]{} in projective space*, Springer, Berlin Heidelberg New York, 1994. L. Shepp and S. Lloyd, *Ordered cycle lengths in a random permutation*, Trans. Amer. Math. Soc. **121** (1996), 340–357. V. Shoup, *A computational introduction to number theory and algebra*, Cambridge Univ. Press, Cambridge, 2005. W. Vogel, *Results on [Bézout]{}’s theorem*, Tata Inst. Fundam. Res. Lect. Math., vol. 74, Tata Inst. Fund. Res., Bombay, 1984. H. Zassenhaus, *On [Hensel]{} factorization [I]{}*, J. Number Theory **1** (1969), 291–311. [^1]: The authors were partially supported by the grants PIP CONICET 11220130100598 and PIO CONICET-UNGS 14420140100027 [^2]: This means that $\{G

Acta Arith. * * 165 * * (2014), no.   2, 141–179. [ to3em ] { }, * Explicit estimates for the number of rational period of remarkable complete intersections over a finite field *, J. Number Theory * * 158 * * (2016), no.   2, 54–72. M.   Merca, * A eminence on the determinant of a [ Toeplitz – Hessenberg ] { } matrix *, Spec. Matrices * * 1 * * (2013), 10–16. T.   Muir, * The theory of determinant in the historical order of growth *, Dover Publications Inc., New York, 1960. L.M. Pardo and J.   [ San Martín ] { }, * Deformation techniques to clear generalized [ Pham ] { } systems *, Theoret. Comput. Sci. * * 315 * * (2004), no.   2–3, 593–625. M.   P[é]{}rez, * Análisis probabilístico de algoritmos y problemas combinatorios sobre cuerpos finitos *, Ph.D. dissertation, Univ. Buenos Aires, Argentina, 2016. P.   Pollack, * Irreducible polynomials with several prescribed coefficients *, Finite Fields Appl. * * 22 * * (2013), 70–78. I.R. Shafarevich, * Basic algebraic geometry: [ Varieties ] { } in projective space *, Springer, Berlin Heidelberg New York, 1994. L.   Shepp and S.   Lloyd, * Ordered hertz lengths in a random permutation *, Trans. Amer. Math. Soc. * * 121 * * (1996), 340–357. V.   Shoup, * A computational introduction to issue theory and algebra *, Cambridge Univ. Press, Cambridge, 2005. W.   Vogel, * Results on [ Bézout ] { } ’s theorem *, Tata Inst. Fundam. Res. Lect. Math. , vol.   74, Tata Inst. Fund. Res. , Bombay, 1984. H.   Zassenhaus, * On [ Hensel ] { } factorization [ I ] { } *, J. Number Theory * * 1 * * (1969), 291–311. [ ^1 ]: The authors were partially support by the grants PIP CONICET 11220130100598 and PIO CONICET - UNGS 14420140100027 [ ^2 ]: This means that $ \{G

Acha Arith. **165** (2014), no. 2, 141–179. [to3em]{}, *Expligit estimates fot rhe nukber or rationxl points of singular compleve ibterstbtions over a finite wield*, J. Nlmber Theiry **158** (2016), no. 2, 54–72. M. Merca, *A note on the fetexmmnant of a [Toepkitz–Hessenterg]{} matrix*, Spac. Mctrices **1** (2013), 10–16. T. Muir, *The theory of determynants on the historicaj orcqr or development*, Dover Publications Ihc., New Jork, 1960. L.M. Pardo and J. [San Martín]{}, *Deformation tefhnieues to solve genegalized [Phan]{} syfrems*, Theoret. Comput. Scp. **315** (2004), no. 2–3, 593–625. M. P[é]{}ree, *Análisis probabilístico de algorktmos y problemqs coltinatorios wobre cuerpos finlnos*, Ph.D. thesis, Univ. Buenos Alres, Ergebtina, 2016. P. Pollack, *Irredurible polynomials wijh several pxescribed coefficientw*, Dinitg Fiends Xppl. **22** (2013), 70–78. J.R. Sgafaregici, *Basic algsbraic geomwtry: [Varieties]{} in ptotvvtive space*, Apringqr, Berlin Heidelberg New York, 1994. L. Shepp and V. Llkyd, *Ordered cycle lengtys in a random permutwtion*, Trags. Amer. Math. Soc. **121** (1996), 340–357. V. Shoup, *A computational introduwtion go umicwr theory and algebra*, Cambridge Univ. Press, Cambwjdbe, 2005. W. Vogel, *Resultf on [Bézout]{}’s tjeptem*, Tata Inst. Wundam. Rea. Lect. Math., vol. 74, Taha Inst. Fund. Res., Bombwy, 1984. H. Aassenhaus, *On [Hensel]{} factoruzation [I]{}*, J. Uumver Theory **1** (1969), 291–311. [^1]: The cuthors were parjially supported by the grantr PIL CONICET 11220130100598 wnd PIO CKVICET-UNGS 14420140100027 [^2]: This oeaks dhat $\{G

Acta Arith. **165** (2014), no. 2, 141–179. estimates the number rational points of finite J. Number Theory (2016), no. 2, M. Merca, *A note on the of a [Toeplitz–Hessenberg]{} matrix*, Spec. Matrices **1** (2013), 10–16. T. Muir, *The theory determinants in the historical order of development*, Dover Publications Inc., New York, 1960. Pardo J. Martín]{}, techniques to solve generalized [Pham]{} systems*, Theoret. Comput. Sci. **315** (2004), no. 2–3, 593–625. M. P[é]{}rez, probabilístico de algoritmos y problemas combinatorios sobre cuerpos Ph.D. thesis, Univ. Buenos Argentina, 2016. P. Pollack, *Irreducible with prescribed coefficients*, Fields **22** 70–78. I.R. Shafarevich, algebraic geometry: [Varieties]{} in projective space*, Springer, Berlin Heidelberg New York, 1994. L. Shepp and S. Lloyd, cycle lengths random permutation*, Amer. Soc. (1996), 340–357. V. computational introduction to number theory and Press, Cambridge, 2005. W. Vogel, *Results on [Bézout]{}’s Tata Inst. Res. Lect. Math., vol. 74, Tata Fund. Res., Bombay, 1984. H. Zassenhaus, *On [Hensel]{} [I]{}*, J. Number Theory **1** (1969), 291–311. [^1]: The authors were partially supported by the CONICET 11220130100598 and PIO 14420140100027 [^2]: This that

Acta Arith. **165** (2014), no. 2, 141–179. [to3em]{}, *Explicit esTimates for The nuMbeR of RaTionAl poInts of singular COmplEte intersections over a fInite FiELd*, J. NUMbEr TheOry **158** (2016), no. 2, 54–72. M. MERcA, *a NotE oN tHe dEtERmInant Of a [toeplitZ–HessenberG]{} maTrIx*, Spec. MatricES **1** (2013), 10–16. T. muir, *The theOry Of determinanTs iN the hiStOriCAl ordEr oF deveLopmenT*, dover PUblicatioNs iNc., New YORk, 1960. L.M. ParDO AnD J. [SaN Martín]{}, *DeformatioN TeCHniques to solve GeneraLiZEd [pHAm]{} sYstEms*, Theoret. coMput. SCI. **315** (2004), no. 2–3, 593–625. M. P[é]{}rEZ, *ANÁLIsiS ProbabilísticO de algoritmOS y pRoblemAs ComBInatorIos soBrE CueRpos finitos*, ph.D. tHesis, Univ. buenos aIres, ArgENtina, 2016. P. POllack, *irrEduCiblE PoLyNomIaLS wiTH sEveRAl pRescribeD cOeFficiEnts*, fINITe FiEldS AppL. **22** (2013), 70–78. I.R. ShAfarevich, *BasiC alGebrAIc gEometRy: [VarIetiEs]{} In proJectivE spacE*, SPringer, Berlin HeIdelBerg New YoRk, 1994. L. shEpp AnD S. LloYD, *OrderEd cYclE lengthS in a ranDOm pErMUTAtIon*, Trans. Amer. Math. SoC. **121** (1996), 340–357. V. sHOuP, *A computAtionaL InTrODuction tO nUmbEr thEORy and AlgeBRa*, cambridgE Univ. PREsS, CAmbridgE, 2005. W. vogel, *REsUltS on [bézouT]{}’S theOrem*, TaTa Inst. FuNdam. RES. Lect. Math., vol. 74, TaTA Inst. Fund. Res., BOMbAY, 1984. h. ZASsenHauS, *On [Hensel]{} faCtorIZatiOn [I]{}*, J. nUmBer tHeory **1** (1969), 291–311. [^1]: the auThORs WEre partially supportEd By the gRants pIP CONICET 11220130100598 and pIO CONICET-ungs 14420140100027 [^2]: This meaNs thAT $\{G

Acta Arith. **165** (2014 ), no. 2,141–1 79. [ to 3em] {},*Explicit esti m ates for the number of rat ional p o ints of sing ular co m pl e t e i nt er sec ti o ns over afinitefield*, J. Nu mb er Theory ** 1 58 ** (2016), no . 2, 54–72. M.  Merca ,*An ote o n t he de termin a nt ofa [Toepli tz – Hessen b erg]{}m a tr ix*, Spec. Matrices * * 1* * (2013), 10–16 . T.Mu i r, * The th eory of de te rmina n ts in t h eh i s tor i cal order ofdevelopment * , D over P ub lic a tionsInc., N e w Y ork, 1960. L.M . Pardo a nd J.[ San Mar t ín]{},*Defor mat ion tec h ni qu esto sol v egen e ral ized [Ph am ]{ } sys tems * , T heor et. Com put.Sci. **315**(20 04), no.  2–3, 593– 625. M. P[ é]{}re z, *A ná lisis probabilí stic o de algo rit mo s y p roble m as com bin ato rios so bre cue r pos f i n i to s*, Ph.D. thesis,Un i v .Buenos A ires,A rg en t ina, 201 6. P . Po l l ack,*Irr e du cible po lynomi a ls w ith sev er al pre sc rib edcoeff i cien ts*, F inite Fi eldsA ppl. **22** (2 0 13), 70–78. I .R . Sh a fare vic h, *Basic a lgeb r aicgeom e tr y:[ Varie ties] {} in projective space*,Sp ringer , Ber lin Heidelber g New York , 1 994. L.  She p pa nd S. Lloyd, * Order ed cycle l e ngths in a ra ndom per mutation* , Trans. A mer . M ath . S o c .**121** (1996 ) , 340 –3 57. V.  Sh oup, *A co mpu tat ion al introduc tion tonu mb er t heo ry an d algebra *, Ca mb rid ge Un i v. Pre ss, C ambr id ge , 20 05. W. Vo g e l, * Re su ltson[B ézout ]{}’ s th eorem*, Tata Ins t.F unda m. R es. Lec t. Math., vol .74, Tata I ns t.Fund.R e s., Bomb ay, 1984. H. Zassenhau s , *On [ Hen sel]{ } fa ctorizati on[I]{}* , J . Numbe r Theo ry ** 1* * ( 1 9 69),2 9 1– 311 . [^1]: The a uth ors w er e pa rtially supported by theg ran ts PIP CONICE T 1 1220 1 3 01 005 9 8a ndPI O CO N I CET-UNGS 144201 40100027 [^ 2 ]: This mean s th at $\{G

Acta_Arith. **165**_(2014), no. 2, 141–179. [to3em]{}, *Explicit_estimates for_the_number of_rational_points of singular_complete intersections over_a finite field*, J._Number Theory **158**_(2016),_no. 2, 54–72. M. Merca, *A note on the determinant of a [Toeplitz–Hessenberg]{} matrix*, Spec. Matrices **1**_(2013),_10–16. T. Muir, *The_theory_of_determinants in the historical order_of development*, Dover Publications Inc.,_New York,_1960. L.M. Pardo and J. [San Martín]{}, *Deformation techniques to_solve_generalized [Pham]{} systems*,_Theoret. Comput. Sci. **315** (2004), no. 2–3, 593–625. M. P[é]{}rez, *Análisis probabilístico_de algoritmos y problemas combinatorios sobre_cuerpos finitos*, Ph.D._thesis,_Univ._Buenos Aires, Argentina, 2016. P. Pollack,_*Irreducible polynomials with several prescribed coefficients*,_Finite Fields Appl. **22** (2013), 70–78. I.R._Shafarevich, *Basic algebraic geometry: [Varieties]{} in projective_space*, Springer, Berlin Heidelberg New York,_1994. L. Shepp and S. Lloyd, *Ordered cycle_lengths in_a random permutation*, Trans. Amer._Math. Soc. **121**_(1996), 340–357. V. Shoup,_*A computational introduction_to number theory and algebra*, Cambridge_Univ. Press, Cambridge,_2005. W. Vogel, *Results on [Bézout]{}’s theorem*, Tata_Inst._Fundam. Res. Lect._Math.,_vol. 74,_Tata Inst._Fund. Res., Bombay,_1984. H. Zassenhaus,_*On [Hensel]{}_factorization_[I]{}*, J. Number Theory **1** (1969),_291–311. [^1]:_The authors were partially supported by the_grants PIP CONICET 11220130100598_and_PIO CONICET-UNGS 14420140100027 [^2]: This_means that $\{G

s} = \sigma_1^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(0) = \sigma_1^{(\beta )}(m)\log (2\pi ),$$ and for the leading term $${M_1} = \mathop {\operatorname{res} }\limits_{s = 1} \sigma_{1 - s/\beta }^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(s)\frac{{{x^s}}}{s} = \sigma_{1 - 1/\beta }^{(\beta )}(m)x.$$ The fluctuaring term coming from the non-trivial zeros yields $${M_\rho } = \sum_\rho {\mathop {\operatorname{res} }\limits_{s = \rho } \sigma_{1 - s/\beta }^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(s)\frac{{{x^s}}}{s}} = \sum_\rho {\sigma_{1 - \rho /\beta }^{(\beta )}(m)\frac{{{x^\rho }}}{\rho }},$$ by the use of the logarithmic derivative of the Hadamdard product of the Riemann zeta-function, and finally for the trivial zeros $${M_{ - 2k}} = \sum_{k = 1}^\infty {\mathop {\operatorname{res} }\limits_{s = - 2k} \sigma_{1 - s/\beta }^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(s)\frac{{{x^s}}}{s}} = \sum_{k = 1}^\infty {\sigma_{1 + 2k/\beta }^{(\beta )}(m)\frac{{{x^{ - 2k}}}}{{ - 2k}}}.$$ Since $|\sigma \pm i{T_1}| \geqslant T$, we see, by our choice of $T_1$, that $$\begin{aligned} \int_{ - 1 \pm i{T_1}}^{{\sigma _0} \pm i{T_1}} {\sigma_{1 - s/\beta }^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(s)\frac{{{x^s}}}{s}ds} &\ll \frac{{{{\log }^2}T}}{T} \bigg(\int_{ - 1}^{{\min(\beta,\sigma_0)}} \left(\frac{x}{m}\right)^{\sigma} d\sigma + \int_{\min

s } = \sigma_1^{(\beta) } (m)\frac{{\zeta' } } { \zeta } (0) = \sigma_1^{(\beta) } (m)\log (2\pi), $ $ and for the leading term $ $ { M_1 } = \mathop { \operatorname{res } } \limits_{s = 1 } \sigma_{1 - s/\beta } ^{(\beta) } (m)\frac{{\zeta' } } { \zeta } (s)\frac{{{x^s}}}{s } = \sigma_{1 - 1/\beta } ^{(\beta) } (m)x.$$ The fluctuaring term coming from the non - fiddling zero yields $ $ { M_\rho } = \sum_\rho { \mathop { \operatorname{res } } \limits_{s = \rho } \sigma_{1 - s/\beta } ^{(\beta) } (m)\frac{{\zeta' } } { \zeta } (s)\frac{{{x^s}}}{s } } = \sum_\rho { \sigma_{1 - \rho /\beta } ^{(\beta) } (m)\frac{{{x^\rho } } } { \rho } }, $ $ by the use of the logarithmic derivative of the Hadamdard intersection of the Riemann zeta - function, and finally for the superficial zero $ $ { M _ { - 2k } } = \sum_{k = 1}^\infty { \mathop { \operatorname{res } } \limits_{s = - 2k } \sigma_{1 - s/\beta } ^{(\beta) } (m)\frac{{\zeta' } } { \zeta } (s)\frac{{{x^s}}}{s } } = \sum_{k = 1}^\infty { \sigma_{1 + 2k/\beta } ^{(\beta) } (m)\frac{{{x^ { - 2k } } } } { { - 2k}}}.$$ Since $ |\sigma \pm i{T_1}| \geqslant T$, we see, by our option of $ T_1 $, that $ $ \begin{aligned } \int _ { - 1 \pm i{T_1}}^{{\sigma _ 0 } \pm i{T_1 } } { \sigma_{1 - s/\beta } ^{(\beta) } (m)\frac{{\zeta' } } { \zeta } (s)\frac{{{x^s}}}{s}ds } & \ll \frac{{{{\log } ^2}T}}{T } \bigg(\int _ { - 1}^{{\min(\beta,\sigma_0) } } \left(\frac{x}{m}\right)^{\sigma } d\sigma + \int_{\min

s} = \digma_1^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(0) = \sigma_1^{(\beta )}(m)\log (2\pi ),$$ end for the leaaing term $${M_1} = \mathop {\operatoriame{ees} }\lumits_{s = 1} \sigma_{1 - s/\beta }^{(\ceta )}(m)\frab{{\zeta '}}{\zetq }(s)\fcac{{{x^s}}}{s} = \sigma_{1 - 1/\bxfa }^{(\beta )}(m)x.$$ Ths fluetnaring term comlng from tha non-trivial zarus yields $${M_\rho } = \sum_\rho {\mathop {\operaeorname{ted} }\limits_{s = \rho } \sibia_{1 - a/\beta }^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(s)\frac{{{x^s}}}{s}} = \sum_\rio {\sigma_{1 - \rho /\neta }^{(\beta )}(m)\frac{{{x^\rho }}}{\rho }},$$ bj thf use of the logarlthmic dericatidw of the Hadxmdard product of the Tiemann zeta-function, and finally for che trivial zwrod $${M_{ - 2k}} = \sum_{k = 1}^\infny {\mathop {\opcgatornake{res} }\lomits_{s = - 2k} \sinma_{1 - v/\bera }^{(\beta )}(m)\frac{{\zeta '}}{\zete }(s)\frac{{{x^s}}}{s}} = \sum_{k = 1}^\igfty {\sigka_{1 + 2k/\beta }^{(\beta )}(m)\frax{{{x^{ - 2k}}}}{{ - 2n}}}.$$ Sitce $|\rugmx \km m{T_1}| \feqslajt V$, we see, by our choice of $T_1$, that $$\begin{alibnqe} \int_{ - 1 \pm i{F_1}}^{{\sigma _0} \[m i{T_1}} {\sigma_{1 - s/\beta }^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(v)\frzc{{{x^s}}}{s}ds} &\ll \frac{{{{\log }^2}T}}{T} \vigg(\int_{ - 1}^{{\min(\beta,\sigma_0)}} \left(\frac{v}{m}\right)^{\sigma} d\sigma + \int_{\min

s} = \sigma_1^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(0) = (2\pi and for leading term $${M_1} 1} - s/\beta }^{(\beta '}}{\zeta }(s)\frac{{{x^s}}}{s} = - 1/\beta }^{(\beta )}(m)x.$$ The fluctuaring coming from the non-trivial zeros yields $${M_\rho } = \sum_\rho {\mathop {\operatorname{res} }\limits_{s \rho } \sigma_{1 - s/\beta }^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(s)\frac{{{x^s}}}{s}} = \sum_\rho {\sigma_{1 - /\beta )}(m)\frac{{{x^\rho }},$$ the use of the logarithmic derivative of the Hadamdard product of the Riemann zeta-function, and finally the trivial zeros $${M_{ - 2k}} = \sum_{k 1}^\infty {\mathop {\operatorname{res} }\limits_{s - 2k} \sigma_{1 - s/\beta )}(m)\frac{{\zeta }(s)\frac{{{x^s}}}{s}} = = {\sigma_{1 2k/\beta }^{(\beta )}(m)\frac{{{x^{ 2k}}}}{{ - 2k}}}.$$ Since $|\sigma \pm i{T_1}| \geqslant T$, we see, by our choice of $T_1$, that \int_{ - i{T_1}}^{{\sigma _0} i{T_1}} - }^{(\beta )}(m)\frac{{\zeta '}}{\zeta \frac{{{{\log }^2}T}}{T} \bigg(\int_{ - 1}^{{\min(\beta,\sigma_0)}} \left(\frac{x}{m}\right)^{\sigma}

s} = \sigma_1^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(0) = \siGma_1^{(\beta )}(m)\loG (2\pi ),$$ anD foR thE lEadiNg teRm $${M_1} = \mathop {\operaTOrnaMe{res} }\limits_{s = 1} \sigma_{1 - s/\beta }^{(\Beta )}(m)\FrAC{{\zetA '}}{\ZeTa }(s)\frAc{{{x^s}}}{s} = \siGMa_{1 - 1/\BETa }^{(\bEtA )}(m)X.$$ ThE fLUcTuariNg tErm comiNg from the nOn-tRiVial zeros yieLDs $${m_\rho } = \sum_\rho {\MatHop {\operatornAme{Res} }\limItS_{s = \rHO } \sigmA_{1 - s/\bEta }^{(\beTa )}(m)\fraC{{\Zeta '}}{\zeTa }(s)\frac{{{x^s}}}{S}} = \sUM_\rho {\siGMa_{1 - \rho /\beTA }^{(\BeTa )}(m)\fRac{{{x^\rho }}}{\rho }},$$ by the usE Of THe logarithmic dErivatIvE Of THE HaDamDard producT oF the RIEmann zeTA-fUNCTioN, And finally for The trivial zERos $${m_{ - 2k}} = \sum_{k = 1}^\InFty {\MAthop {\oPeratOrNAme{Res} }\limits_{s = - 2k} \SigmA_{1 - s/\beta }^{(\betA )}(m)\frac{{\ZEta '}}{\zeta }(S)\Frac{{{x^s}}}{s}} = \Sum_{k = 1}^\inFty {\SigMa_{1 + 2k/\bETa }^{(\BeTa )}(m)\FrAC{{{x^{ - 2k}}}}{{ - 2K}}}.$$ siNce $|\SIgmA \pm i{T_1}| \geqSlAnT T$, we sEe, by OUR CHoicE of $t_1$, thaT $$\begiN{aligned} \int_{ - 1 \pm I{T_1}}^{{\sIgma _0} \PM i{T_1}} {\Sigma_{1 - S/\beta }^{(\Beta )}(M)\fRac{{\zeTa '}}{\zeta }(S)\frac{{{X^s}}}{S}ds} &\ll \frac{{{{\log }^2}T}}{T} \bIgg(\iNt_{ - 1}^{{\min(\beta,\SigMa_0)}} \LefT(\fRac{x}{m}\RIght)^{\siGma} D\siGma + \int_{\mIn

s} = \sigma_1^{(\beta )}(m )\frac{{\z eta ' }}{ \ze ta }(0 ) =\sigma_1^{(\be t a )} (m)\log (2\pi ),$$ and forth e lea d in g ter m $${M_ 1 }= \ma th op {\ op e ra torna me{ res} }\ limits_{s= 1 }\sigma_{1 -s /\ beta }^{(\ bet a )}(m)\frac {{\ zeta ' }} {\z e ta }( s)\ frac{ {{x^s} } }{s} = \sigma_{ 1- 1/\be t a }^{(\ b e ta )}( m)x.$$ The fluctu a ri n g term comingfrom t he no n - tri via l zeros yi el ds $$ { M_\rho} = \ s um_ \ rho {\mathop {\operator n ame {res}}\ lim i ts_{s= \rh o} \s igma_{1 - s /\be ta }^{(\b eta )} ( m)\frac { {\zeta'}}{\z eta }( s)\f r ac {{ {x^ s} } }{s } } =\ sum _\rho { \s ig ma_{1 - \ r h o /\be ta}^{( \beta )}(m)\frac{{ {x^ \rho }}} {\rho }},$ $ by t he us e of t he lo ga rithmic derivat iveof the Ha dam da rdpr oduct of the Ri ema nn zeta -functi o n,an d f in ally for the trivi al z er os $${M_ { - 2k } }=\ sum_{k = 1 }^\ inft y {\ma thop {\ operator name{r e s} } \limits _{ s = - 2 k}\si gma_{ 1 - s /\beta }^{(\be ta )} ( m)\frac{{\zeta '}}{\zeta }(s ) \f r a c{ { {x^s }}} {s}} = \su m_{k = 1} ^\in f ty { \ sigma _{1 + 2 k /\ b eta }^{(\beta )}(m) \f rac{{{ x^{ - 2k}}}}{{ - 2 k}}}.$$ Si n c e $|\sigm a \ p mi {T_1}| \geqsla nt T$ , we see,b y our ch oiceof $T_1$ , that $$ \ b egin{ali gne d} \ int _ { - 1 \pm i{T_1} } ^ {{\s ig ma _0}\pm i{T_1} } { \si gma _{1 - s/\beta}^{(\bet a)} (m )\ fra c{{\z e ta '}}{\ ze ta}( s)\ frac{ { {x^s}} }{s}d s} &\ ll \fr ac{{{{\ l og } ^2}T }} {T } \b igg (\ int_{ - 1 } ^{{ \min(\b eta,\sigm a_0 ) }} \ le ft (\frac{ x}{m}\right)^ {\ sigma} d\s ig ma + \in t _ {\min

s} =_\sigma_1^{(\beta )}(m)\frac{{\zeta_'}}{\zeta }(0) = \sigma_1^{(\beta_)}(m)\log (2\pi_),$$_and for_the_leading term $${M_1}_= \mathop {\operatorname{res}_}\limits_{s = 1} \sigma_{1_- s/\beta }^{(\beta_)}(m)\frac{{\zeta_'}}{\zeta }(s)\frac{{{x^s}}}{s} = \sigma_{1 - 1/\beta }^{(\beta )}(m)x.$$ The fluctuaring term coming from the_non-trivial_zeros yields_$${M_\rho_}_= \sum_\rho {\mathop {\operatorname{res}_}\limits_{s = \rho } \sigma_{1_- s/\beta_}^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(s)\frac{{{x^s}}}{s}} = \sum_\rho _{\sigma_{1_- \rho /\beta_}^{(\beta )}(m)\frac{{{x^\rho }}}{\rho }},$$ by the use of the_logarithmic derivative of the Hadamdard product_of the Riemann_zeta-function,_and_finally for the trivial_zeros $${M_{ - 2k}} = \sum_{k_= 1}^\infty {\mathop {\operatorname{res} }\limits_{s_= - 2k} \sigma_{1 - s/\beta_}^{(\beta )}(m)\frac{{\zeta '}}{\zeta }(s)\frac{{{x^s}}}{s}} =_\sum_{k = 1}^\infty {\sigma_{1_+ 2k/\beta_}^{(\beta )}(m)\frac{{{x^{ - 2k}}}}{{ -_2k}}}.$$ Since $|\sigma_ \pm_i{T_1}| \geqslant T$,_we see, by our choice of_$T_1$, that $$\begin{aligned} _ \int_{ - 1 \pm i{T_1}}^{{\sigma__0}_\pm i{T_1}} {\sigma_{1_-_s/\beta_}^{(\beta )}(m)\frac{{\zeta_'}}{\zeta }(s)\frac{{{x^s}}}{s}ds} _&\ll_\frac{{{{\log }^2}T}}{T}_\bigg(\int_{_- 1}^{{\min(\beta,\sigma_0)}} \left(\frac{x}{m}\right)^{\sigma} d\sigma +_\int_{\min

y\vert \leq 1})\nu(dy)\Big)\bigg),$$ where $b,\sigma\in {\ensuremath {\mathbb{R}}}$ and $\nu$ is a measure on ${\ensuremath {\mathbb{R}}}$ satisfying $$\nu(\{0\})=0 \textnormal{ and } \int_{{\ensuremath {\mathbb{R}}}}(|y|^2\wedge 1)\nu(dy)<\infty.$$ In the sequel we shall refer to $(b,\sigma^2,\nu)$ as the characteristic triplet of the process $\{X_t\}$ and $\nu$ will be called the *Lévy measure*. This data characterizes uniquely the law of the process $\{X_t\}$. Let $D=D([0,\infty),{\ensuremath {\mathbb{R}}})$ be the space of mappings $\omega$ from $[0,\infty)$ into ${\ensuremath {\mathbb{R}}}$ that are right-continuous with left limits. Define the *canonical process* $x:D\to D$ by $$\forall \omega\in D,\quad x_t(\omega)=\omega_t,\;\;\forall t\geq 0.$$ Let ${\ensuremath {\mathscr{D}}}_t$ and ${\ensuremath {\mathscr{D}}}$ be the $\sigma$-algebras generated by $\{x_s:0\leq s\leq t\}$ and $\{x_s:0\leq s<\infty\}$, respectively (here, we use the same notations as in [@sato]). By the condition (4) above, any Lévy process on ${\ensuremath {\mathbb{R}}}$ induces a probability measure $P$ on $(D,{\ensuremath {\mathscr{D}}})$. Thus $\{X_t\}$ on the probability space $(D,{\ensuremath {\mathscr{D}}},P)$ is identical in law with the original Lévy process. By saying that $(\{x_t\},P)$ is a Lévy process, we mean that $\{x_t:t\geq 0\}$ is a Lévy process under the probability measure $P$ on $(D,{\ensuremath {\mathscr{D}}})$. For all $t>0$ we will denote $P_t$ for the restriction of $P$ to ${\ensuremath {\mathscr{D}}}_t$. In the case where $\int_{|y|\leq 1}|y|\nu(dy)<\infty$, we set $\gamma^{\nu}:=\int_{|y|\leq 1}y\nu(dy)$. Note that, if $\nu$ is a finite Lévy measure,

y\vert \leq 1})\nu(dy)\Big)\bigg),$$ where $ b,\sigma\in { \ensuremath { \mathbb{R}}}$ and $ \nu$ is a measure on $ { \ensuremath { \mathbb{R}}}$ satisfying $ $ \nu(\{0\})=0 \textnormal { and } \int_{{\ensuremath { \mathbb{R}}}}(|y|^2\wedge 1)\nu(dy)<\infty.$$ In the sequel we shall refer to $ (b,\sigma^2,\nu)$ as the characteristic triplet of the procedure $ \{X_t\}$ and $ \nu$ will be call the * Lévy measure *. This data characterizes uniquely the police of the process $ \{X_t\}$. Let $ five hundred = D([0,\infty),{\ensuremath { \mathbb{R}}})$ be the distance of mappings $ \omega$ from $ [ 0,\infty)$ into $ { \ensuremath { \mathbb{R}}}$ that are right - continuous with left limits. specify the * canonical process * $ x: D\to D$ by $ $ \forall \omega\in D,\quad x_t(\omega)=\omega_t,\;\;\forall t\geq 0.$$ get $ { \ensuremath { \mathscr{D}}}_t$ and $ { \ensuremath { \mathscr{D}}}$ be the $ \sigma$-algebras generate by $ \{x_s:0\leq s\leq t\}$ and $ \{x_s:0\leq s<\infty\}$, respectively (here, we use the same notations as in [ @sato ]). By the condition (4) above, any Lévy process on $ { \ensuremath { \mathbb{R}}}$ induce a probability measure $ P$ on $ (D,{\ensuremath { \mathscr{D}}})$. Thus $ \{X_t\}$ on the probability space $ (D,{\ensuremath { \mathscr{D}}},P)$ is identical in police with the original Lévy process. By saying that $ (\{x_t\},P)$ is a Lévy process, we mean that $ \{x_t: t\geq 0\}$ is a Lévy procedure under the probability measure $ P$ on $ (D,{\ensuremath { \mathscr{D}}})$. For all $ t>0 $ we will denote $ P_t$ for the restriction of $ P$ to $ { \ensuremath { \mathscr{D}}}_t$. In the case where $ \int_{|y|\leq 1}|y|\nu(dy)<\infty$, we set $ \gamma^{\nu}:=\int_{|y|\leq 1}y\nu(dy)$. bill that, if $ \nu$ is a finite Lévy measure,

y\vfrt \leq 1})\nu(dy)\Big)\bigg),$$ whert $b,\sigma\in {\ensuremath {\mavhbb{R}}}$ ahd $\nu$ is a measure on ${\ensuremath {\matibb{R}}}$ satiwfying $$\nu(\{0\})=0 \textnormal{ avd } \int_{{\enduremath {\matibb{R}}}}(|y|^2\wedge 1)\nu(dy)<\iirty.$$ In bke sesmel wz whall refer to $(b,\sigma^2,\nu)$ ds the charactarkscic triplet of the process $\{X_t\}$ and $\nu$ will br falled the *Lévy meaxtre*. Fhis data characterizes uniquely tge law mf the procesx $\{X_t\}$. Let $D=D([0,\infty),{\ensuremath {\lathhb{R}}})$ be the space ov mappings $\imegw$ from $[0,\infty)$ knto ${\ensurtmcth {\mathbb{R}}}$ that are right-continuous with ueft kimits. Defunw tjg *canonical 'rocesf* $x:D\to D$ by $$\nprall \mmega\in D,\quad x_t(\omega)=\pmeja_t,\;\;\firall t\geq 0.$$ Let ${\ensurekath {\mathscr{D}}}_t$ and ${\ensuremadh {\mathscr{D}}}$ be the $\witma$-alcebrds gdberxtes uy $\{s_s:0\leq d\les t\}$ and $\{x_s:0\meq s<\infty\}$, eespectively (here, wt ufv the same nofationf ws in [@sato]). By the condition (4) above, any Lény pdocess on ${\ensuremath {\marhbb{R}}}$ induces a probahility mewsure $P$ on $(D,{\ensuremath {\mathscr{D}}})$. Thus $\{X_t\}$ on the prmbabimkty siace $(E,{\ejsuremath {\mathscr{D}}},P)$ is identical in law with ege ogiginal Lévy procefs. By sayinb hhsj $(\{x_t\},P)$ is a Lévy process, se mean that $\{x_t:t\gee 0\}$ is a Lévy process ugder the probability measure $P$ in $(D,{\ensuremanh {\mqthscr{D}}})$. For all $t>0$ ce will denoce $P_t$ gor tne restriction of $P$ to ${\znsurejath {\mathscg{D}}}_t$. In ths case where $\int_{|y|\ueq 1}|y|\tu(dy)<\infty$, we set $\gamma^{\nu}:=\ine_{|y|\leq 1}y\nu(vy)$. Noce that, kf $\no$ is a sinite Lévy measmse,

y\vert \leq 1})\nu(dy)\Big)\bigg),$$ where $b,\sigma\in {\ensuremath {\mathbb{R}}}$ is measure on {\mathbb{R}}}$ satisfying $$\nu(\{0\})=0 1)\nu(dy)<\infty.$$ the sequel we refer to $(b,\sigma^2,\nu)$ the characteristic triplet of the process and $\nu$ will be called the *Lévy measure*. This data characterizes uniquely the of the process $\{X_t\}$. Let $D=D([0,\infty),{\ensuremath {\mathbb{R}}})$ be the space of mappings $\omega$ $[0,\infty)$ ${\ensuremath that right-continuous with left limits. Define the *canonical process* $x:D\to D$ by $$\forall \omega\in D,\quad x_t(\omega)=\omega_t,\;\;\forall t\geq Let ${\ensuremath {\mathscr{D}}}_t$ and ${\ensuremath {\mathscr{D}}}$ be the generated by $\{x_s:0\leq s\leq and $\{x_s:0\leq s<\infty\}$, respectively (here, use same notations in By condition (4) above, Lévy process on ${\ensuremath {\mathbb{R}}}$ induces a probability measure $P$ on $(D,{\ensuremath {\mathscr{D}}})$. Thus $\{X_t\}$ on the space $(D,{\ensuremath identical in with original process. By saying is a Lévy process, we mean is a Lévy process under the probability measure on $(D,{\ensuremath For all $t>0$ we will denote for the restriction of $P$ to ${\ensuremath {\mathscr{D}}}_t$. the case where $\int_{|y|\leq 1}|y|\nu(dy)<\infty$, we set $\gamma^{\nu}:=\int_{|y|\leq 1}y\nu(dy)$. Note that, if $\nu$ is a measure,

y\vert \leq 1})\nu(dy)\Big)\bigg),$$ where $b,\Sigma\in {\ensUremaTh {\mAthBb{r}}}$ and $\Nu$ is A measure on ${\ensuREmatH {\mathbb{R}}}$ satisfying $$\nu(\{0\})=0 \teXtnorMaL{ And } \iNT_{{\eNsureMath {\matHBb{r}}}}(|Y|^2\WedGe 1)\Nu(Dy)<\iNfTY.$$ IN the sEquEl we shaLl refer to $(b,\SigMa^2,\Nu)$ as the charaCTeRistic tripLet Of the process $\{x_t\}$ aNd $\nu$ wiLl Be cALled tHe *LÉvy meAsure*. THIs data CharacterIzES uniquELy the laW OF tHe prOcess $\{X_t\}$. Let $D=D([0,\infty),{\ENsURemath {\mathbb{R}}})$ bE the spAcE Of MAPpiNgs $\Omega$ from $[0,\iNfTy)$ intO ${\EnsuremATh {\MATHbb{r}}}$ That are right-cOntinuous wiTH leFt limiTs. defINe the *cAnoniCaL ProCess* $x:D\to D$ by $$\ForaLl \omega\in d,\quad x_T(\Omega)=\omEGa_t,\;\;\foraLl t\geq 0.$$ let ${\EnsUremATh {\MaThsCr{d}}}_T$ anD ${\EnSurEMatH {\mathscr{d}}}$ bE tHe $\sigMa$-alGEBRAs geNerAted By $\{x_s:0\lEq s\leq t\}$ and $\{x_s:0\lEq s<\InftY\}$, ResPectiVely (hEre, wE uSe the Same noTatioNs As in [@sato]). By the coNditIon (4) above, aNy LÉvY prOcEss on ${\ENsuremAth {\MatHbb{R}}}$ indUces a prOBabIlITY MeAsure $P$ on $(D,{\ensurematH {\mATHsCr{D}}})$. Thus $\{X_T\}$ on the PRoBaBIlity spaCe $(d,{\enSureMATh {\matHscr{d}}},p)$ iS identicAl in laW WiTh The origInAl Lévy PrOceSs. BY sayiNG thaT $(\{x_t\},P)$ is A Lévy proCess, wE Mean that $\{x_t:t\geq 0\}$ IS a Lévy process UNdER ThE ProbAbiLity measure $p$ on $(D,{\ENsurEmatH {\MaThsCR{D}}})$. For All $t>0$ wE wILl DEnote $P_t$ for the restriCtIon of $P$ To ${\ensUremath {\mathscR{D}}}_t$. In the caSE WHere $\int_{|y|\Leq 1}|y|\NU(dY)<\Infty$, we set $\gammA^{\nu}:=\inT_{|y|\leq 1}y\nu(dy)$. nOte that, iF $\nu$ is A finite LÉvy measurE,

y\vert \leq 1})\nu(dy)\Bi g)\bigg),$ $ whe re$b, \s igma \in{\ensuremath { \ math bb{R}}}$ and $\nu$ isa mea su r e on ${ \ensu remath{ \m a t hbb {R }} }$sa t is fying $$ \nu(\{0 \})=0 \tex tno rm al{ and } \i n t_ {{\ensurem ath {\mathbb{R} }}} (|y|^2 \w edg e 1)\n u(d y)<\i nfty.$ $ In th e sequelwe shallr efer to $ (b ,\si gma^2,\nu)$ as th e c h aracteristic t riplet o f t h e pr oce ss $\{X_t\ }$ and$ \nu$ wi l lb e cal l ed the *Lévymeasure*. T h isdata c ha rac t erizes uniq ue l y t he law of t he p rocess $\ {X_t\} $ . Let$ D=D([0, \infty ),{ \en sure m at h{\m at h bb{ R }} })$ bethe spac eof mapp ings $ \ o mega $ f rom$[0,\ infty)$ into${\ ensu r ema th {\ mathb b{R} }} $ tha t areright -c ontinuous withleft limits.Def in e t he *can o nicalpro ces s* $x:D \to D$b y $ $\ f o r al l \omega\in D,\qua dx _ t( \omega)= \omega _ t, \; \ ;\forall t \ge q 0. $ $ Let ${\ e ns uremath{\math s cr {D }}}_t$an d ${\e ns ure mat h {\m a thsc r{D}}} $ be the $\si g ma$-algebras g e nerated by $\ { x_ s : 0\ l eq s \le q t\}$ and$\{x _ s:0\ leqs <\ inf t y\}$, resp ec t iv e ly (here, we use th esame n otati ons as in [@s ato]). By t h e condit ion( 4) above, any Lév y pro cess on ${ \ ensurema th {\ mathbb{R }}}$ indu c e s a prob abi lit y m eas u r e$P$ on $(D,{\ e n sure ma th {\ma ths cr{D}}} )$. Th us$\{ X_ t\}$ on t he proba bi li ty s pac e $(D , {\ensure ma th{\ mat hscr{ D }}},P) $ isiden ti ca l in law wi t ht h e or ig in al L évy p roces s. B y sa ying th at $(\{x_ t\} , P)$is a Lévy p rocess, we me an that $\{x _t :t\ geq 0\ } $ is a Lé vy process under the pr o babilit y m easur e $P $ on $(D, {\e nsurem ath {\math scr{D} }})$. F ora l l $t> 0 $ w e w il l denote $ P _ t$for t he res trictio n of $P$ to ${\ens u rem ath {\mathscr {D} }}_t $ . I n t h ec ase w h ere $ \int_{|y|\leq 1 }|y|\nu(dy )< \ in fty$, we s e t $ \g amma^{\ nu}:=\i nt_{| y |\leq 1 }y\nu(dy) $. Note t ha t, i f $\n u$ is a fi nite Lév y measure ,

y\vert_\leq 1})\nu(dy)\Big)\bigg),$$_where $b,\sigma\in {\ensuremath {\mathbb{R}}}$_and $\nu$_is_a measure_on_${\ensuremath {\mathbb{R}}}$ satisfying_$$\nu(\{0\})=0 \textnormal{ and_} \int_{{\ensuremath {\mathbb{R}}}}(|y|^2\wedge 1)\nu(dy)<\infty.$$_In the sequel_we_shall refer to $(b,\sigma^2,\nu)$ as the characteristic triplet of the process $\{X_t\}$ and $\nu$_will_be called_the_*Lévy_measure*. This data characterizes uniquely_the law of the process_$\{X_t\}$. Let $D=D([0,\infty),{\ensuremath_{\mathbb{R}}})$ be the space of mappings $\omega$ from_$[0,\infty)$_into ${\ensuremath {\mathbb{R}}}$_that are right-continuous with left limits. Define the *canonical_process* $x:D\to D$ by $$\forall \omega\in_D,\quad x_t(\omega)=\omega_t,\;\;\forall t\geq_0.$$ Let_${\ensuremath_{\mathscr{D}}}_t$ and ${\ensuremath {\mathscr{D}}}$_be the $\sigma$-algebras generated by $\{x_s:0\leq_s\leq t\}$ and $\{x_s:0\leq s<\infty\}$, respectively_(here, we use the same notations as_in [@sato]). By the condition (4) above,_any Lévy process on ${\ensuremath_{\mathbb{R}}}$ induces_a probability measure $P$ on_$(D,{\ensuremath {\mathscr{D}}})$. Thus_$\{X_t\}$ on_the probability space_$(D,{\ensuremath {\mathscr{D}}},P)$ is identical in law_with the original_Lévy process. By saying that $(\{x_t\},P)$_is_a Lévy process,_we_mean_that $\{x_t:t\geq_0\}$ is a_Lévy_process under_the_probability measure $P$ on $(D,{\ensuremath {\mathscr{D}}})$._For_all $t>0$ we will denote $P_t$ for_the restriction of $P$_to_${\ensuremath {\mathscr{D}}}_t$. In the_case where $\int_{|y|\leq 1}|y|\nu(dy)<\infty$, we_set $\gamma^{\nu}:=\int_{|y|\leq 1}y\nu(dy)$. Note that, if_$\nu$ is_a finite_Lévy measure,

%"} Historical use of Strömgren photometry methods indeed has been for the purpose of determining stellar parameters for early-type stars. Recent applications include work by @nieva2013 [@dallemese2012; @onehag2009; @allende1999]. An advantage over more traditional color-magnitude diagram techniques [@nielsen2013; @derosa2014] is that distance knowledge is not required, so the distance-age degeneracy is removed. Also, metallicity effects are relatively minor (as addressed in an Appendix) and rotation effects are well-modelled and can be corrected for (§ \[subsec:vsinicorrection\]). Description of the Photometric System {#subsec:uvbydescription} ------------------------------------- The $uvby\beta$ photometric system is comprised of four intermediate-band filters ($uvby$) first advanced by [@stromgren1966] plus the H$\beta$ narrow and wide filters developed by [@crawford1958]; see Figure \[fig:filters\]. Together, the two filter sets form a well-calibrated system that was specifically designed for studying earlier-type BAF stars, for which the hydrogen line strengths and continuum slopes in the Balmer region rapidly change with temperature and gravity. From the fluxes contained in the six passbands, five $uvby\beta$ indices are defined. The color indices, ($b-y$) and ($u-b$), and the $\beta$-index, $$\beta = \mathrm{H}\beta_\text{narrow} - \mathrm{H}\beta_\text{wide},$$ are all sensitive to temperature and weakly dependent on surface gravity for late A- and F-type stars. The Balmer discontinuity index, $$c_1 = (u-v) - (v-b),$$ is sensitive to temperature for early type (OB) stars and surface gravity for intermediate (AF) spectral types. Finally, the metal line index, $$m_1 = (v-b) - (b-y),$$ is sensitive to the metallicity $[M/H]$. For each index, there is a corresponding intrinsic, dereddened index denoted by a naught subscript with e.g $c_0, (b-y)_0,$ and $(u-b)_0$, referring to the intrinsic, dereddened equivalents of the indices $c_1, (b-y),$ and $(u-b)$,

% " } Historical use of Strömgren photometry methods indeed has been for the purpose of determine leading parameters for early - type star. late applications include work by @nieva2013 [ @dallemese2012; @onehag2009; @allende1999 ]. An advantage over more traditional semblance - magnitude diagram techniques [ @nielsen2013; @derosa2014 ] is that distance cognition is not command, so the distance - age corruption is removed. Also, metallicity effects are relatively minor (as addressed in an Appendix) and rotation effect are well - modelled and can be chastise for (§   \[subsec: vsinicorrection\ ]). Description of the Photometric System { # subsec: uvbydescription } ------------------------------------- The $ uvby\beta$ photometric system is comprised of four average - band filters ($ uvby$) first advance by [ @stromgren1966 ] plus the H$\beta$ narrow and wide filter developed by [ @crawford1958 ]; understand Figure   \[fig: filters\ ]. Together, the two filter sets form a well - calibrated system that was specifically design for studying earlier - type BAF stars, for which the hydrogen line strengths and continuum slopes in the Balmer region rapidly change with temperature and gravity. From the fluxes contained in the six passbands, five $ uvby\beta$ indices are defined. The color indices, ($ b - y$) and ($ u - b$), and the $ \beta$-index, $ $ \beta = \mathrm{H}\beta_\text{narrow } - \mathrm{H}\beta_\text{wide},$$ are wholly sensible to temperature and weakly dependent on surface graveness for former A- and F - type asterisk. The Balmer discontinuity index, $ $ c_1 = (u - v) - (v - b),$$ is sensitive to temperature for early character (OB) stars and surface gravity for intermediate (AF) spectral types. Finally, the metal line index, $ $ m_1 = (v - b) - (b - y),$$ is sensible to the metallicity $ [ M / H]$. For each index, there be a comparable intrinsic, dereddened exponent denoted by a naught subscript with e.g $ c_0, (b - y)_0,$ and $ (u - b)_0 $, referring to the intrinsic, dereddened equivalents of the indices $ c_1, (b - y),$ and $ (u - b)$,

%"} Hishorical use of Strömgren khotometry methods indeev has bsen for ghe purpose of determining svellqr paeameters for early-type stars. Rebent applucatmons include work by @nieyc2013 [@dalmcmese2012; @inehag2009; @allende1999]. An advantdge over more drxdntional color-magnitude diagram technyques [@noepsen2013; @derosa2014] is thau dystahbe knowledge is not required, so the divtance-age degrneracy is removed. Also, mehalllcity effects are gelatively ninow (as addressea in an Apkeudix) and rojation effects are well-modelled xnd ccn be corrextwd vmr (§ \[subsec:vsmnicorgection\]). Descriinion of the Phptometric Systcm {#suusec:yvbydescription} ------------------------------------- The $uvuy\beta$ photometric sistem is cmm'rised of four intermwduate-bdnd xiltdes ($jvbg$) hirat advwncxd by [@stromfren1966] plus tye H$\beta$ narrow and wyee filters debelopeq fy [@crawford1958]; see Figure \[fig:filters\]. Togethtr, ths two filter sets form q well-calibrated systgm that waf specifically designed for studying earlier-type TAF svafs, yir whkxh the hydrogen line strengths and continuum sljlex pn the Balmer reglon rapidly change wltn temperature xnd grcbify. From the fluxes fontaingd in rhe six pwssbsnds, five $uvby\beta$ indices qre defined. Nhe xolor indices, ($b-y$) aud ($u-b$), and thz $\beta$-ondex, $$\neta = \mathrm{H}\beta_\text{naxrow} - \jathrm{H}\beta_\hext{wide},$$ add all sensitive go nemparature and weakly dependegt on surhace yravity wor kate A- and F-type stars. The Balmer disconhinuijy indax, $$c_1 = (u-v) - (v-h),$$ is sensitive to temperature foc early type (PB) stdrs and furfage gravity for yntermediate (AY) spectrcl typds. Finally, the meval line indqx, $$m_1 = (v-b) - (b-y),$$ is densitive to the metwllixity $[M/H]$. For dxch index, therr is a cogrtsponding untrinsic, dereddencd inasx denoted by a naotht subscript eitf e.d $b_0, (b-b)_0,$ and $(g-b)_0$, referring to ghe ontrivsic, dereddcnea eqiivalents of the indhces $c_1, (b-y),$ and $(u-b)$,

%"} Historical use of Strömgren photometry methods been the purpose determining stellar parameters include by @nieva2013 [@dallemese2012; @allende1999]. An advantage more traditional color-magnitude diagram techniques [@nielsen2013; is that distance knowledge is not required, so the distance-age degeneracy is removed. metallicity effects are relatively minor (as addressed in an Appendix) and rotation effects well-modelled can corrected (§ \[subsec:vsinicorrection\]). Description of the Photometric System {#subsec:uvbydescription} ------------------------------------- The $uvby\beta$ photometric system is comprised of intermediate-band filters ($uvby$) first advanced by [@stromgren1966] plus H$\beta$ narrow and wide developed by [@crawford1958]; see Figure Together, two filter form well-calibrated that was specifically for studying earlier-type BAF stars, for which the hydrogen line strengths and continuum slopes in the Balmer rapidly change and gravity. the contained the six passbands, indices are defined. The color indices, and the $\beta$-index, $$\beta = \mathrm{H}\beta_\text{narrow} - \mathrm{H}\beta_\text{wide},$$ all sensitive temperature and weakly dependent on surface for late A- and F-type stars. The Balmer index, $$c_1 = (u-v) - (v-b),$$ is sensitive to temperature for early type (OB) stars gravity for intermediate (AF) types. Finally, the line $$m_1 (v-b) (b-y),$$ is to the metallicity $[M/H]$. For each index, there is a corresponding dereddened index denoted by a naught subscript with e.g $c_0, $(u-b)_0$, to the intrinsic, equivalents of the indices (b-y),$ $(u-b)$,

%"} Historical use of Strömgren pHotometry mEthodS inDeeD hAs beEn foR the purpose of dETermIning stellar parameters For eaRlY-Type STaRs. RecEnt applICaTIOns InClUde WoRK bY @nievA2013 [@daLlemese2012; @Onehag2009; @alleNde1999]. an Advantage oveR MoRe traditioNal Color-magnituDe dIagram TeChnIQues [@nIelSen2013; @deRosa2014] is THat disTance knowLeDGe is noT RequireD, SO tHe diStance-age degeneraCY iS Removed. Also, metAlliciTy EFfECTs aRe rElatively mInOr (as aDDressed IN aN aPPenDIx) and rotation Effects are wELl-mOdelleD aNd cAN be corRecteD fOR (§ \[suBsec:vsinicoRrecTion\]). DescrIption OF the PhoTOmetric system {#SubSec:UvbyDEsCrIptIoN} ------------------------------------- the $UVbY\beTA$ phOtometriC sYsTem is CompRISED of fOur InteRmediAte-band filterS ($uvBy$) fiRSt aDvancEd by [@sTromGrEn1966] pluS the H$\bEta$ naRrOw and wide filterS devEloped by [@cRawFoRd1958]; sEe figurE \[Fig:filTerS\]. ToGether, tHe two fiLTer SeTS FOrM a well-calibrated syStEM ThAt was speCificaLLy DeSIgned for StUdyIng eARLier-tYpe Baf sTars, for wHich thE HyDrOgen linE sTrengtHs And ConTinuuM SlopEs in thE Balmer rEgion RApidly change wiTH temperature aND gRAViTY. FroM thE fluxes contAineD In thE six PAsSbaNDs, fivE $uvby\BeTA$ iNDices are defined. The cOlOr indiCes, ($b-y$) And ($u-b$), and the $\beTa$-index, $$\betA = \MAThrm{H}\betA_\texT{NaRRow} - \mathrm{H}\beta_\Text{wIde},$$ are all sENsitive tO tempErature aNd weakly dEPEndent on SurFacE grAviTY FoR late A- and F-typE STars. thE Balmer DisContinuIty IndEx, $$c_1 = (U-v) - (v-B),$$ iS sensitivE to tempeRaTuRe FoR eaRly tyPE (OB) stars AnD suRfAce GraviTY for inTermeDiatE (Af) sPEctRal typeS. fiNALly, tHe MeTal lIne InDex, $$m_1 = (v-B) - (b-y),$$ iS SenSitive tO the metalLicITy $[M/H]$. foR eAch indeX, there is a corrEsPonding intRiNsiC, deredDENed index Denoted by a naught subscriPT with e.g $C_0, (b-y)_0,$ And $(u-b)_0$, RefeRring to thE inTrinsiC, deREddeneD equivAlentS oF thE INdiceS $C_1, (B-y),$ And $(U-b)$,

%"} Historical use of Str ömgren pho tomet rymet ho ds i ndee d has been for thepurpose of determining stel la r par a me tersfor ear l y- t y pest ar s.Re c en t app lic ationsinclude wo rkby @nieva2013[ @d allemese20 12; @onehag2009 ; @ allend e1 999 ] . Anadv antag e over more t raditiona lc olor-m a gnitude d ia gram techniques [@nie l se n 2013; @derosa2 014] i st ha t dis tan ce knowled ge is n o t requi r ed , s o t h e distance-ag e degenerac y is remov ed . A l so, me talli ci t y e ffects arerela tively mi nor (a s addres s ed in a n Appe ndi x)andr ot at ion e f fec t sare wel l-modell ed a nd ca n be c o r rect edfor(§ \[ subsec:vsinic orr ecti o n\] ). D escri ptio nof th e Phot ometr ic System {#subse c:uv bydescrip tio n} -- -- ----- - ------ --- --- ------- ------- - T he $ u vb y\beta$ photometri cs y st em is co mprise d o ff our inte rm edi ate- b a nd fi lter s ( $uvby$)firsta dv an ced by[@ stromg re n19 66] plus theH$\bet a$ narro w and wide filters d e veloped by [@ c ra w f or d 1958 ];see Figure\[fi g :fil ters \ ]. To g ether , the t w of ilter sets form a w el l-cali brate d system that was speci f i c ally des igne d f o r studying ear lier- type BAF s t ars, for whic h the hy drogen li n e strengt hsand co nti n u um slopes in th e Balm er region ra pidly c han gewit h t em peratureand grav it y. Fr omthe f l uxes con ta ine dinthe s i x pass bands , fi ve $ u vby \beta$i nd i c es a re d efin ed. T he co lori ndi ces, ($ b-y$) and ($ u -b$) ,an d the $ \beta$-index, $$\beta =\m ath rm{H}\ b e ta_\text {narrow} - \mathrm{H}\b e ta_\tex t{w ide}, $$ are all s ens itivetot empera ture a nd we ak lyd e pende n t o n s ur face gravi t y fo r lat eA- a nd F-ty pe stars. The Balm e r d iscontinuityind ex,$ $c _1= ( u -v) - (v- b ) ,$$ is sensiti ve to temp er a tu re for ear l y t yp e (OB)stars a nd su r face gr avity for intermed ia te ( A F ) s pectral ty pes. Fin ally, the metal li ne in dex , $$m _1 =(v-b) - (b- y ),$ $ is sensi ti ve tothe m et allicity $[M/H]$. For each ind ex, th ere i s a correspo ndi n g i ntrinsic, der eddened in dex de noted by a nau ghts ub scr i pt wi th e . g $c_0, ( b -y )_0 , $ a nd $(u-b)_0 $ , ref errin g t o the i ntri nsic, dereddenede quivalents ofthei n dic es$ c_1, ( b-y),$ and $(u -b) $,

%"} Historical use_of Strömgren_photometry methods indeed has_been for_the_purpose of_determining_stellar parameters for_early-type stars. Recent_applications include work by_@nieva2013 [@dallemese2012; @onehag2009;_@allende1999]._An advantage over more traditional color-magnitude diagram techniques [@nielsen2013; @derosa2014] is that distance knowledge_is_not required,_so_the_distance-age degeneracy is removed. Also,_metallicity effects are relatively minor_(as addressed_in an Appendix) and rotation effects are well-modelled_and_can be corrected_for (§ \[subsec:vsinicorrection\]). Description of the Photometric System {#subsec:uvbydescription} ------------------------------------- The $uvby\beta$ photometric_system is comprised of four intermediate-band_filters ($uvby$) first_advanced_by_[@stromgren1966] plus the H$\beta$_narrow and wide filters developed by_[@crawford1958]; see Figure \[fig:filters\]. Together, the two_filter sets form a well-calibrated system that_was specifically designed for studying earlier-type_BAF stars, for which the_hydrogen line_strengths and continuum slopes in_the Balmer region_rapidly change_with temperature and_gravity. From the fluxes contained in the_six passbands, five_$uvby\beta$ indices are defined. The color_indices,_($b-y$) and ($u-b$),_and_the_$\beta$-index, $$\beta =_\mathrm{H}\beta_\text{narrow} - \mathrm{H}\beta_\text{wide},$$ are_all_sensitive to_temperature_and weakly dependent on surface gravity_for_late A- and F-type stars. The Balmer_discontinuity index, $$c_1 = (u-v)_-_(v-b),$$ is sensitive to temperature_for early type (OB) stars_and surface gravity for intermediate (AF)_spectral types._Finally, the_metal line index, $$m_1 = (v-b) - (b-y),$$ is sensitive to the metallicity_$[M/H]$. For each index, there is a_corresponding intrinsic, dereddened index_denoted by_a_naught subscript with_e.g_$c_0, (b-y)_0,$_and $(u-b)_0$, referring to the intrinsic, dereddened_equivalents of_the indices $c_1, (b-y),$ and $(u-b)$,

the analytic estimates for higher temperatures. Here one can also see, that $\eta$ stays more or less the same for both species at lower temperature and starts to diverge the more particles are created in the box in the Hagedorn case. The values explode, if the particle number density increases ad infinitum near $T_H$. Nevertheless, it increases less rapidly than the entropy density as shown in \[fig:s\_T3\]. It is interesting to observe, that the competing differences in the intermediate result of $C(0)$ and $\tau$ cancel each other at low temperatures and only for $T\gtrsim 140{{\ensuremath{\,{\rm MeV}}\xspace}}$, a different behavior between the pion gas and the Hagedorn state gas my be observable. Combining both the results of the thermodynamical quantities (the entropy density), and the shear viscosity, \[fig:eta\_s\] shows the final result, the shear viscosity to entropy density ratio. ![The final result is $\eta$ normalized to the entropy density for numerical and analytical estimates. Error bars are as in \[fig:eta\]. The KSS bound $1/4\pi$ is indicated. The Hagedorn extrapolation $M\to\infty$ contains both separate $\eta$ and $s$ extrapolations. []{data-label="fig:eta_s"}](eta_sloglog.eps){width="\columnwidth"} As expected, at low temperatures the results for the pion gas and the Hagedorn gas coincide. Since the entropy density $s$ very rapidly starts to diverge with increasing temperature, also the fraction $\eta/s$ diverges. Finally, all the calculated results via the Monte Carlo/Green Kubo approach a stop at values above the KSS bound of $1/4\pi$. The analytic estimates indicate, that the results drop below this boundary and go to zero when temperature increases further. Using the statistical error for $\tau$, one can compute the errors for $\eta$ and $\eta/s$. In \[fig:eta\] and \[fig:eta\_s\] one observes, that the numerical results including the errorbars do not match the analytical curve. This leads us to the finding, that there are some systematic error in the Green-Kubo formalism, which are underestimated in the current work. Conclusions {#sec:Conclusions} =========== In the present work, the transport coefficient $\eta/s$

the analytic estimates for higher temperatures. Here one can also visualize, that $ \eta$ quell more or less the same for both species at lower temperature and begin to diverge the more particles are create in the corner in the Hagedorn case. The values explode, if the particle number concentration increases ad infinitum near $ T_H$. Nevertheless, it increases less rapidly than the information density as shown in \[fig: s\_T3\ ]. It is interesting to observe, that the competing differences in the average result of $ C(0)$ and $ \tau$ cancel each other at humble temperatures and only for $ T\gtrsim 140{{\ensuremath{\,{\rm MeV}}\xspace}}$, a unlike behavior between the pion flatulence and the Hagedorn state natural gas my be observable. Combining both the results of the thermodynamic quantities (the entropy density), and the shear viscosity, \[fig: eta\_s\ ] shows the final result, the shear viscosity to entropy density proportion. ! [ The concluding consequence is $ \eta$ normalized to the entropy density for numerical and analytical estimates. mistake bar are as in \[fig: eta\ ]. The KSS bound $ 1/4\pi$ is indicated. The Hagedorn extrapolation $ M\to\infty$ control both disjoined $ \eta$ and $ s$ extrapolations. [ ] { data - label="fig: eta_s"}](eta_sloglog.eps){width="\columnwidth " } As ask, at abject temperatures the results for the pion accelerator and the Hagedorn gas coincide. Since the entropy density $ s$ very rapidly starts to diverge with increasing temperature, also the fraction $ \eta / s$ diverges. Finally, all the calculated results via the Monte Carlo / Green Kubo border on a stop at value above the KSS bound of $ 1/4\pi$. The analytic estimates indicate, that the results drop below this limit and go to zero when temperature increases further. Using the statistical error for $ \tau$, one can compute the errors for $ \eta$ and $ \eta / s$. In \[fig: eta\ ] and \[fig: eta\_s\ ] one observes, that the numerical resultant role including the errorbars do not meet the analytical curve. This contribute us to the finding, that there embody some systematic mistake in the Green - Kubo formalism, which are underestimated in the current oeuvre. Conclusions { # sec: Conclusions } = = = = = = = = = = = In the present work, the transport coefficient $ \eta / s$

thf analytic estimates for higher temperajuees. Hece one dan also see, that $\eta$ stays more or pews tht same for both spezies at llwer temperauure and starts to divergc the lore 'articles are cteated in tha box in the Hdgddlrn case. The values explode, if the [articlr jumber density incgewses ad infinitum near $T_H$. Nevertheless, it incgeases less rapidky than the entropy densitj as shown in \[fig:s\_T3\]. It ls interestung ei observe, thxt the comkecing differgnces in the intermediate result of $C(0)$ and $\tau$ cqnxel gach other av low nemperatures and only for $T\gyrsim 140{{\ensuremabh{\,{\rm KeV}}\zspace}}$, a different beiavior between the pyon gas atd the Hagedorn stare gas ky ba obrwrvxblt. Cokbjning hoti the resulfs of the tyermodynamical quanuityvx (the entropg densytr), and the shear viscosity, \[fig:eta\_s\] shows ths final result, the sheae viscosity to entropi density watio. ![The final result is $\eta$ normalized to the endropy aenwiby fue jumerical and analytical estimates. Error bars zrt ax in \[fig:eta\]. Thc KSS bound $1/4\pi$ is onfivwted. The Hageaorn erfrzpolation $M\to\infty$ contaigs borh separaue $\ets$ and $s$ extrapolations. []{data-oabel="fig:eta_s"}](vta_sooglog.eps){width="\colulnwidth"} As erpectec, at kow temperatures the rerulta for the plon gas aha the Hagedorn gxs boinwide. Since the entropy denfity $s$ vecy ra'idly stxrts to diderge with incrcdsing temperature, wlso jhe frdction $\eta/d$ diverges. Finally, all the calcnkated resultx eia the Monce Carko/Green Kubo wpproach a stok at valuzs aboxe the KSS bound mf $1/4\pi$. The agalytic estimdjes indicate, vhat the wesuots erop beuuw this boundaty and go to zero wyen temperature ingreasgs further. Using tkt suaristical error fof $\twu$, oie cag compute the errurs gor $\ega$ and $\tca/s$. Kn \[fog:eta\] and \[fig:eta\_s\] ona obaerves, that the nukevical resolts incltding the errprbars do not matcj the anelyticsl surve. This leads us to the findjng, that hheve are some srstenatic error nn the Green-Kubo formalism, which are undxrestimated in the curtent work. Conclusions {#xtc:Conclusions} =========== Mn the present fork, the transport ciefficient $\eta/s$

the analytic estimates for higher temperatures. Here also that $\eta$ more or less at temperature and starts diverge the more are created in the box in Hagedorn case. The values explode, if the particle number density increases ad infinitum $T_H$. Nevertheless, it increases less rapidly than the entropy density as shown in It interesting observe, the competing differences in the intermediate result of $C(0)$ and $\tau$ cancel each other at low and only for $T\gtrsim 140{{\ensuremath{\,{\rm MeV}}\xspace}}$, a different between the pion gas the Hagedorn state gas my observable. both the of thermodynamical (the entropy density), the shear viscosity, \[fig:eta\_s\] shows the final result, the shear viscosity to entropy density ratio. ![The final is $\eta$ the entropy for and estimates. Error bars in \[fig:eta\]. The KSS bound $1/4\pi$ Hagedorn extrapolation $M\to\infty$ contains both separate $\eta$ and extrapolations. []{data-label="fig:eta_s"}](eta_sloglog.eps){width="\columnwidth"} expected, at low temperatures the results the pion gas and the Hagedorn gas coincide. the entropy density $s$ very rapidly starts to diverge with increasing temperature, also the fraction Finally, all the calculated via the Monte Kubo a at above the bound of $1/4\pi$. The analytic estimates indicate, that the results drop this boundary and go to zero when temperature increases further. statistical for $\tau$, one compute the errors for and In \[fig:eta\] and \[fig:eta\_s\] that numerical errorbars not the analytical curve. This us to the finding, that are some systematic error are underestimated in the current work. Conclusions {#sec:Conclusions} In the present work, the transport coefficient

the analytic estimates for hiGher temperAtureS. HeRe oNe Can aLso sEe, that $\eta$ stays MOre oR less the same for both speCies aT lOWer tEMpEratuRe and stARtS TO diVeRgE thE mORe PartiCleS are creAted in the bOx iN tHe Hagedorn caSE. THe values exPloDe, if the partiCle Number DeNsiTY incrEasEs ad iNfinitUM near $T_h$. NevertheLeSS, it incREases leSS RaPidlY than the entropy deNSiTY as shown in \[fig:s\_t3\]. It is iNtEReSTIng To oBserve, that ThE compETing difFErENCEs iN The intermediaTe result of $C(0)$ ANd $\tAu$ cancEl EacH Other aT low tEmPEraTures and onlY for $t\gtrsim 140{{\enSuremaTH{\,{\rm MeV}}\xSPace}}$, a diFferenT beHavIor bETwEeN thE pIOn gAS aNd tHE HaGedorn stAtE gAs my bE obsERVABle. COmbIninG both The results of tHe tHermODynAmicaL quanTitiEs (The enTropy dEnsitY), aNd the shear viscoSity, \[Fig:eta\_s\] shOws ThE fiNaL resuLT, the shEar VisCosity tO entropY DenSiTY RAtIo. ![The final result is $\EtA$ NOrMalized tO the enTRoPy DEnsity foR nUmeRicaL ANd anaLytiCAl EstimateS. Error BArS aRe as in \[fIg:Eta\]. The kSs boUnd $1/4\Pi$ is iNDicaTed. The hagedorn ExtraPOlation $M\to\inftY$ Contains both sEPaRATe $\ETa$ anD $s$ eXtrapolatioNs. []{daTA-labEl="fiG:EtA_s"}](eTA_slogLog.epS){wIDtH="\Columnwidth"} As expectEd, At low tEmperAtures the resuLts for the pION Gas and thE HagEDoRN gas coincide. SiNce thE entropy deNSity $s$ verY rapiDly startS to divergE WIth increAsiNg tEmpEraTURe, Also the fractiON $\Eta/s$ DiVerges. FInaLly, all tHe cAlcUlaTed ReSults via tHe Monte CArLo/grEeN KuBo appROach a stoP aT vaLuEs aBove tHE KSS boUnd of $1/4\Pi$. ThE aNaLYtiC estimaTEs INDicaTe, ThAt thE reSuLts drOp beLOw tHis bounDary and go To zERo whEn TeMperatuRe increases fuRtHer. Using thE sTatIsticaL ERror for $\tAu$, one can compute the errorS For $\eta$ aNd $\eTa/s$. In \[Fig:eTa\] and \[fig:eTa\_s\] One obsErvES, that tHe numeRical ReSulTS IncluDINg The ErRorbars do nOT MatCh the AnAlytIcal curVe. This leads us to the FIndIng, that there aRe sOme sYSTeMatIC eRRor In THe GREEn-Kubo formalism, Which are unDeREsTimated in tHE cuRrEnt work. conclusIons {#sEC:ConcluSions} =========== In thE present wOrK, the TRAnsPort coeffiCient $\eta/S$

the analytic estimates fo r higher t emper atu res .Here one can also see, that $\eta$ stays more orlessth e sam e f or bo th spec i es a t l ow er te mp e ra tureand starts to diverg e t he more partic l es are creat edin the box i n t he Hag ed orn case. Th e val ues ex p lode,if the pa rt i cle nu m ber den s i ty inc reases ad infinit u mn ear $T_H$. Nev erthel es s ,i t in cre ases lessra pidly than th e e n t r opy density as sh own in \[fi g :s\ _T3\]. Iti s inte resti ng toobserve, th at t he compet ing di f ference s in the inter med iat e re s ul tof$C ( 0)$ an d $ \ tau $ cancel e ac h oth er a t l o w te mpe ratu res a nd only for $ T\g trsi m 14 0{{\e nsure math {\ ,{\rm MeV}} \xspa ce }}$, a differen t be havior be twe en th epiong as and th e H agedorn stateg asmy b e o bservable. Combin in g bo th the r esults of t h e thermo dy nam ical q uanti ties (t he entro py den s it y) , and t he shear v isc osi ty, \ [ fig: eta\_s \] shows thef inal result, t h e shear visco s it y to entr opy density ra tio. ![T he f i na l r e sultis $\ et a $n ormalized to the en tr opy de nsity for numerica l and anal y t i cal esti mate s .E rror bars areas in \[fig:eta \ ]. The K SS bo und $1/4 \pi$ is i n d icated.The Ha ged orn e xt rapolation $M \ t o\in ft y$ cont ain s bothsep ara te$\e ta $ and $s$ extrapo la ti on s. [] {data - label="f ig :et a_ s"} ](eta _ sloglo g.eps ){wi dt h= " \co lumnwid t h" } Asex pe cted , a tlow t empe r atu res the resultsfor thepi on gas an d the Hagedor ngas coinci de . S ince t h e entropy density $s$ very rapid l y start s t o div erge with inc rea sing t emp e rature , also thefr act i o n $\e t a /s $ d iv erges. Fin a l ly, allth e ca lculate d results via theM ont e Carlo/Green Ku bo a p p ro ach as top a t va l u es above the KS S bound of $ 1 /4 \pi$. Thea nal yt ic esti mates i ndica t e, that the resu lts dropbe lowt h isboundary a nd go to zero whe n temp e ra tureinc reases f urt her. Using the stat istica lerrorfor $ \t au$, one can compute the errors for $ \eta$ an d $\eta/s $.I n \ [fig:eta\ ] an d \[fig:et a\_ s\] oneobs e rves, tha t t hen umeri calr esults in c lu din g th e errorbars d o no t mat cht he ana lyti cal curve. Thisl eads us to the fin d i ng, th a t th er e are some sys tem at i c error i nthe Green-K ubo form al i sm, w hich a re und erestim a t ed in the cur ren t work. Con cl u sions { #s ec : Conclu sion s} ===== ====== Int h e present work,the t r a nspor t co effic ie nt $\et a /s$

the_analytic estimates_for higher temperatures. Here_one can_also_see, that_$\eta$_stays more or_less the same_for both species at_lower temperature and_starts_to diverge the more particles are created in the box in the Hagedorn case._The_values explode,_if_the_particle number density increases ad_infinitum near $T_H$. Nevertheless, it_increases less_rapidly than the entropy density as shown in_\[fig:s\_T3\]. It_is interesting to_observe, that the competing differences in the intermediate result_of $C(0)$ and $\tau$ cancel each_other at low_temperatures_and_only for $T\gtrsim 140{{\ensuremath{\,{\rm_MeV}}\xspace}}$, a different behavior between the_pion gas and the Hagedorn state_gas my be observable. Combining both the results_of the thermodynamical quantities (the entropy_density), and the shear viscosity,_\[fig:eta\_s\] shows_the final result, the shear_viscosity to entropy_density ratio. ![The_final result is_$\eta$ normalized to the entropy density_for numerical and_analytical estimates. Error bars are as_in_\[fig:eta\]. The KSS_bound_$1/4\pi$_is indicated._The Hagedorn extrapolation_$M\to\infty$_contains both_separate_$\eta$ and $s$ extrapolations. []{data-label="fig:eta_s"}](eta_sloglog.eps){width="\columnwidth"} As expected,_at_low temperatures the results for the pion_gas and the Hagedorn_gas_coincide. Since the entropy_density $s$ very rapidly starts_to diverge with increasing temperature, also_the fraction_$\eta/s$ diverges._Finally, all the calculated results via the Monte Carlo/Green Kubo approach_a stop at values above the_KSS bound of $1/4\pi$._The analytic_estimates_indicate, that the_results_drop below_this boundary and go to zero when_temperature increases_further. Using the statistical error for $\tau$,_one can compute the_errors_for $\eta$ and $\eta/s$. In \[fig:eta\]_and \[fig:eta\_s\] one observes, that the_numerical results including the errorbars_do_not_match the analytical curve. This leads_us to the finding, that there_are some systematic_error in the Green-Kubo formalism, which are_underestimated_in the current work. Conclusions {#sec:Conclusions} =========== In the_present_work, the transport coefficient $\eta/s$

weak in the radio. The more massive galaxies tend to be FUV faint but are more luminous in the radio. Note here that [$K_s$]{}magnitudes are in the Vega system and FUV are in the AB system.[]{data-label="fig:fuv-k_mk"}](fig3.pdf){width="50.00000%"} We have examined the 13 FUV bright galaxies for evidence of star formation. These fall into three categories: - Galaxies with on-going star formation: The two bluest galaxies (NGC 3413, NGC 1705) are known to be undergoing a strong star burst (evidence from SDSS strong H$\alpha$ emission and [@annibali2003] respectively). The radio power at 1.4 GHz for these galaxies is less than $10^{20}\,WHz^{-1}$ indicating that the FUV emission is dominated via star formation and not by AGN. Also, NGC 855 shows CO emission [@nakanishi2007], NGC 3928 has a starburst nucleus [@balzano1983] and IC5267 has a large number of star formation sites[@caldwell1991], indicating ongoing star formation activity. NGC 7252 is a merger remnant[@chien2010] that has old and new star forming population residing in the nuclear regions of the galaxy. - AGN contribution: NGC 5252 and NGC 5283 are AGNs with Seyfert type Sy1.9 and Sy2, respectively. They show slight excess in the FUV light. Similarly, NGC 4457 hosts a bright UV nucleus which is attributed to the central AGN [@flohic2006]. - Unknown FUV origin: In the rest of the galaxies, NGC 3955, NGC 4344, NGC 4627, and UGC 3097 do not have any strong evidence of ongoing star formation or AGN activity. Thus far, the origin of the excess UV emission in these galaxies is unclear. Similar to the UV, the mid-infrared (MIR) emission is also a good indicator of star formation in a galaxy. A color-color diagram in [$[12\mu m-22\mu m]$]{}vs [$[FUV-K_s]$]{}for the galaxy sample is shown in Figure  \[fig:w3-w4\_fuv-k\]. Only 160 galaxies have the photometry for 12 $\mu$m, 22 $\mu$m, FUV

weak in the radio. The more massive galaxies tend to be FUV faint but are more luminous in the radio receiver. notice here that [ $ K_s$]{}magnitudes are in the Vega organization and FUV are in the AB system.[]{data - label="fig: fuv - k_mk"}](fig3.pdf){width="50.00000% " } We have examined the 13 FUV undimmed galaxies for evidence of star topology formation. These fall into three class: - Galaxies with on - going ace formation: The two bluest galaxy (NGC   3413, NGC   1705) are known to be undergoing a solid star outburst (evidence from SDSS strong H$\alpha$ emission and [ @annibali2003 ] respectively). The radio receiver power at 1.4 GHz for these galaxies is less than $ 10^{20}\,WHz^{-1}$ indicate that the FUV emission is dominated via ace formation and not by AGN. besides, NGC   855 shows CO discharge [ @nakanishi2007 ], NGC   3928 has a starburst nucleus [ @balzano1983 ] and IC5267 has a big number of star formation sites[@caldwell1991 ], indicating ongoing star formation activity. NGC   7252 is a merger remnant[@chien2010 ] that has old and new star forming population residing in the nuclear regions of the galaxy. - AGN contribution: NGC   5252 and NGC   5283 are AGNs with Seyfert type Sy1.9 and Sy2, respectively. They prove slight excess in the FUV lighter. Similarly, NGC 4457 host a undimmed UV nucleus which is attributed to the central AGN [ @flohic2006 ]. - Unknown FUV origin: In the rest of the galaxies, NGC   3955, NGC   4344, NGC   4627, and UGC   3097 do not accept any strong evidence of ongoing star formation or AGN activity. Thus far, the origin of the excess ultraviolet emission in these galaxies is unclear. Similar to the UV, the mid - infrared (MIR) emission is also a dear indicator of star formation in a galaxy. A color - semblance diagram in [ $ [ 12\mu m-22\mu m]$]{}vs [ $ [ FUV - K_s]$]{}for the galaxy sample is shown in Figure   \[fig: w3 - w4\_fuv - k\ ]. merely 160 galaxies have the photometry for 12 $ \mu$m, 22 $ \mu$m, FUV

wewk in the radio. The more massive galaxigs tend vo be FHV faint but are more luminous in thx raeio. Nite here that [$K_s$]{}magnitjdes are pn the Veta sbstem and FUV arx in the AB syabem.[]{daca-oabel="fig:fuv-k_mk"}](nig3.pdf){width="50.00000%"} Fe have examinad tke 13 FUV bright galaxies for evidence of stat vormation. Thesg falk intk three categories: - Galaxies with on-goinj star formatiom: The two bluest galaxies (JGC 3413, JGC 1705) are known to bf undergoint a frrong star bjrst (evidence from SDSA strong H$\alpha$ emission and [@anvibaln2003] respectivglg). Hve radio poxer at 1.4 GHz for thcxe galdxies ix less than $10^{20}\,WHd^{-1}$ indmcatung that the FUV emisvion is dominated dia star xoxmation and not by AGB. Qlso, TGC 855 vhowr CO emjsxikn [@nakwniahi2007], NGC 3928 haa a starburwt nucleus [@balzano1983] snq IC5267 has a larfe numfew of star formation sites[@caldwell1991], indicdtihg ongoing star formatiin activity. NGC 7252 is a lerger reinant[@chien2010] that has old and new star forming popunatioi fesneing kb hhe nuclear regions of the galaxy. - AGN contryguuiok: NGC 5252 and NGC 5283 arc AGNs with Seyfery hylg Sy1.9 and Sy2, rerpectirslg. They show slight excess in tye FUV lidht. Ximilarly, NGC 4457 hosts a brigyt UV nucleuf which is attributzd to the ceutral SGN [@fkohic2006]. - Unknown FUV oriyin: In the rest ov the galzbies, NGC 3955, NGC 4344, NGC 4627, anc GGC 3097 do not have any strong evidence of pngoing stat formaeion or AGJ actleity. Thus far, the lrigiu of dhe excess UV emission in these galaxies mx unclear. Simoldr no the UV, the kid-infrared (MYR) emission is also a good indicator of stac formation yn a galaxy. A folor-color dmagram in [$[12\mu n-22\mu n]$]{}vs [$[FUV-Y_r]$]{}for the galaxu sample ps shoqn in Figure  \[fig:w3-w4\_nuv-k\]. Uhly 160 galaxies hcre the photometry fof 12 $\iu$l, 22 $\mt$k, FUV

weak in the radio. The more massive to FUV faint are more luminous that are in the system and FUV in the AB system.[]{data-label="fig:fuv-k_mk"}](fig3.pdf){width="50.00000%"} We have the 13 FUV bright galaxies for evidence of star formation. These fall into categories: - Galaxies with on-going star formation: The two bluest galaxies (NGC 3413, 1705) known be a strong star burst (evidence from SDSS strong H$\alpha$ emission and [@annibali2003] respectively). The radio power 1.4 GHz for these galaxies is less than indicating that the FUV is dominated via star formation not AGN. Also, 855 CO [@nakanishi2007], NGC 3928 a starburst nucleus [@balzano1983] and IC5267 has a large number of star formation sites[@caldwell1991], indicating ongoing star activity. NGC a merger that old new star forming in the nuclear regions of the contribution: NGC 5252 and NGC 5283 are AGNs Seyfert type and Sy2, respectively. They show slight in the FUV light. Similarly, NGC 4457 hosts bright UV nucleus which is attributed to the central AGN [@flohic2006]. - Unknown FUV origin: rest of the galaxies, 3955, NGC 4344, 4627, UGC do have any evidence of ongoing star formation or AGN activity. Thus far, the of the excess UV emission in these galaxies is unclear. the the mid-infrared (MIR) is also a good of formation in a galaxy. diagram [$[12\mu the sample shown in Figure \[fig:w3-w4\_fuv-k\]. 160 galaxies have the photometry 12 $\mu$m, 22 $\mu$m,

weak in the radio. The more massIve galaxieS tend To bE FUv fAint But aRe more luminous IN the Radio. Note here that [$K_s$]{}magNitudEs ARe in THe vega sYstem anD fUv ARe iN tHe aB sYsTEm.[]{Data-lAbeL="fig:fuv-K_mk"}](fig3.pdf){wIdtH="50.00000%"} WE have examineD ThE 13 FUV bright GalAxies for evidEncE of staR fOrmATion. THesE fall Into thREe cateGories: - GalAxIEs with ON-going sTAR fOrmaTion: The two bluest gALaXIes (NGC 3413, NGC 1705) are knOwn to bE uNDeRGOinG a sTrong star bUrSt (eviDEnce froM sDss STroNG H$\alpha$ emissiOn and [@annibaLI2003] reSpectiVeLy). THE radio Power At 1.4 ghz fOr these galaXies Is less thaN $10^{20}\,WHz^{-1}$ inDIcating THat the FuV emisSioN is DomiNAtEd Via StAR foRMaTioN And Not by AGN. alSo, nGC 855 shOws Co EMISsioN [@naKaniShi2007], NGc 3928 has a starbursT nuCleuS [@BalZano1983] aNd IC5267 hAs a lArGe numBer of sTar foRmAtion sites[@caldwEll1991], iNdicating OngOiNg sTaR formATion acTivIty. nGC 7252 is a mErger reMNanT[@cHIEN2010] tHat has old and new staR fORMiNg populaTion reSIdInG In the nucLeAr rEgioNS Of the GalaXY. - AgN contriBution: ngC 5252 AnD NGC 5283 are aGns with seYfeRt tYpe Sy1.9 ANd Sy2, RespecTively. ThEy shoW Slight excess in THe FUV light. SimILaRLY, Ngc 4457 hosTs a Bright UV nucLeus WHich Is atTRiButED to thE centRaL aGn [@Flohic2006]. - Unknown FUV oriGiN: In the Rest oF the galaxies, NgC 3955, NGC 4344, NGC 4627, anD ugc 3097 do not haVe anY StROng evidence of oNgoinG star formaTIon or AGN ActivIty. Thus fAr, the origIN Of the excEss uV eMisSioN IN tHese galaxies iS UNcleAr. similar To tHe UV, the Mid-InfRarEd (MiR) Emission iS also a goOd InDiCaTor Of staR FormatioN iN a gAlAxy. a coloR-Color dIagraM in [$[12\mU m-22\Mu M]$]{}Vs [$[FuV-K_s]$]{}for THe GALaxy SaMpLe is ShoWn In FigUre  \[fIG:w3-w4\_Fuv-k\]. OnlY 160 galaxies HavE The pHoToMetry foR 12 $\mu$m, 22 $\mu$m, FUV

weak in the radio. The mo re massive gala xie s t en d to beFUV faint buta re m ore luminous in the ra dio.No t e he r ethat[$K_s$] { }m a g nit ud es ar ei nthe V ega system and FUV a rein the AB syst e m. []{data-la bel ="fig:fuv-k_ mk" }](fig 3. pdf ) {widt h=" 50.00 000%"} We ha ve examin ed the 13 FUV bri g h tgala xies for evidence of star formation . Thes ef al l int o t hree categ or ies:- Gal a xi e s wit h on-going sta r formation : Th e twobl ues t galax ies ( NG C  34 13, NGC 170 5) a re knownto beu ndergoi n g a str ong st arbur st ( e vi de nce f r omS DS S s t ron g H$\alp ha $emiss iona n d [@an nib ali2 003]respectively) . T he r a dio powe r at1.4GH z for these gala xi es is less than $10 ^{20}\,WH z^{ -1 }$in dicat i ng tha t t heFUV emi ssion i s do mi n a t ed via star formatio na n dnot by A GN. Al s o, N G C 855 sh ow s C O em i s sion[@na k an ishi2007 ], NGC 39 28 has ast arburs tnuc leu s [@b a lzan o1983] and IC5 267 h a s a large numb e r of star for m at i o ns ites [@c aldwell1991 ], i n dica ting on goi n g sta r for ma t io n activity. NGC 7252 i s a me rgerremnant[@chie n2010] tha t h as old a nd n e ws tar forming po pulat ion residi n g in the nucl ear regi ons of th e galaxy. - A GNcon t r ib ution: NGC 52 5 2 and N GC 5283 ar e AGNswit h S eyf ert t ype Sy1.9 and Sy2 ,re sp ec tiv ely.T hey show s lig ht ex cessi n theFUV l ight .Si m ila rly, NG C 4 4 5 7 ho st sa br igh tUV nu cleu s wh ich isattribute d t o the c en tral AG N [@flohic200 6] . - Unk no wnFUV or i g in: In t he rest of the galaxies , NGC 39 55, NGC4344 , NGC 462 7,and UG C 3 0 97 donot ha ve an ystr o n g evi d e nc e o fongoing st a r fo rmati on orAGN act ivity. Thus far, t h e o rigin of theexc essU V e mis s io n in t h ese g alaxies is uncl ear. Simi la r t o the UV,t hemi d-infra red (MI R) em i ssion i s also agood indi ca toro f st ar formati on in agalaxy. A color - co lor d iag ram in [ $[1 2\mum-22\m u m] $]{}v s [$[F UV -K_s]$ ]{}fo rthe gala xy sample is shown in F igure \[fi g:w 3-w4\_fuv -k\ ] . O nly 160 g alax ies have t hepho tomet ryf or 12 $\m u $m , 2 2 $\mu $m,F UV

weak_in the_radio. The more massive_galaxies tend_to_be FUV_faint_but are more_luminous in the_radio. Note here that_[$K_s$]{}magnitudes are in_the_Vega system and FUV are in the AB system.[]{data-label="fig:fuv-k_mk"}](fig3.pdf){width="50.00000%"} We have examined the 13 FUV_bright_galaxies for_evidence_of_star formation. These fall into_three categories: - Galaxies_with on-going_star formation: The two bluest galaxies (NGC 3413, NGC 1705)_are_known to be_undergoing a strong star burst (evidence from SDSS strong_H$\alpha$ emission and [@annibali2003] respectively). The_radio power at_1.4_GHz_for these galaxies is_less than $10^{20}\,WHz^{-1}$ indicating that the_FUV emission is dominated via star_formation and not by AGN. Also, NGC 855_shows CO emission [@nakanishi2007], NGC 3928 has_a starburst nucleus [@balzano1983] and_IC5267 has_a large number of star_formation sites[@caldwell1991], indicating_ongoing star_formation activity. NGC 7252_is a merger remnant[@chien2010] that has_old and new_star forming population residing in the_nuclear_regions of the_galaxy. -__ AGN_contribution: NGC 5252 and_NGC 5283_are AGNs_with_Seyfert type Sy1.9 and Sy2, respectively._They_show slight excess in the FUV light._Similarly, NGC 4457 hosts_a_bright UV nucleus which_is attributed to the central_AGN [@flohic2006]. - Unknown FUV_origin: In_the rest_of the galaxies, NGC 3955, NGC 4344, NGC 4627, and UGC 3097 do not have_any strong evidence of ongoing star_formation or AGN activity._Thus far,_the_origin of the_excess_UV emission_in these galaxies is unclear. Similar to the_UV, the_mid-infrared (MIR) emission is also a_good indicator of star_formation_in a galaxy. A color-color diagram_in [$[12\mu m-22\mu m]$]{}vs [$[FUV-K_s]$]{}for the_galaxy sample is shown in_Figure_ \[fig:w3-w4\_fuv-k\]._Only 160 galaxies have the_photometry for 12 $\mu$m, 22 $\mu$m,_FUV

-n)\right).$$ In particular, letting $d=3$ and $i=1$ and using definitions of $B^{1,1}_{n-2}$ and $B^{1,0}_{n-2}$, we obtain that $$\label{eq:facets-of-B^{3,1}} B^{3,1}_n=\Big( (n-2, n-3, \dots, 1, -n+2, -n+3, \dots, -1)* (n-1, n) \Big) \cup \Big( (1, -n+2)*(n, -n+1, -n)\Big).$$ In Section \[sec:Facets\], we will use this description of $B^{3,1}_n$ to characterize all facets of $\Delta^3_n$. A big portion of [@N-Z Section 3] is devoted to showing that the objects $\Delta^d_n$ and $B^{d,i}_n$ are well-defined (including the fact that $\Delta^{d}_n\supseteq B^{d,\lceil d/2\rceil -1}_n$). The proof relies on a few crucial properties of $B^{d,i}_n$, see [@N-Z Lem. 3.3, 3.4, 3.6 and Cor. 3.7], summarized in the following lemma. \[lm: prop of B\^[d, i]{}\_n\] Let $d\geq 2$ and $n\geq d+1$. Then for all $0\leq i\leq j\leq \lfloor d/2 \rfloor$ and $k\leq \lceil d/2 \rceil$, 1. $B^{d,i}_n$ is a combinatorial $d$-ball that is cs-$i$-neighborly (w.r.t. $V_n$) and $i$-stacked; furthermore, $B^{d,i}_n$ shares no common facets with $-B^{d,i}_n$; 2. $B^{d, k-1}_n\subseteq -B^{d, k}_n$; 3. $\partial B^{d, i}_n=\left( \partial B^{d-1, i}_{n-1}*n\right) \cup \left( \partial (-B^{

-n)\right).$$ In particular, letting $ d=3 $ and $ i=1 $ and using definitions of $ B^{1,1}_{n-2}$ and $ B^{1,0}_{n-2}$, we receive that $ $ \label{eq: aspect - of - B^{3,1 } } B^{3,1}_n=\Big (( n-2, n-3, \dots, 1, -n+2, -n+3, \dots, -1) * (n-1, n) \Big) \cup \Big (( 1, -n+2)*(n, -n+1, -n)\Big).$$ In Section \[sec: Facets\ ], we will use this description of $ B^{3,1}_n$ to characterize all facets of $ \Delta^3_n$. A bad portion of [ @N - Z Section 3 ] is devoted to showing that the object $ \Delta^d_n$ and $ B^{d, i}_n$ are well - define (including the fact that $ \Delta^{d}_n\supseteq B^{d,\lceil d/2\rceil -1}_n$). The validation trust on a few crucial properties of $ B^{d, i}_n$, experience [ @N - Z Lem.   3.3, 3.4, 3.6 and Cor.   3.7 ], summarized in the following lemma. \[lm: prop of B\^[d, i]{}\_n\ ] Let $ d\geq 2 $ and $ n\geq d+1$. Then for all $ 0\leq i\leq j\leq \lfloor d/2 \rfloor$ and $ k\leq \lceil d/2 \rceil$, 1. $ B^{d, i}_n$ is a combinatorial $ d$-ball that is cs-$i$-neighborly (w.r.t.   $ V_n$) and $ i$-stacked; furthermore, $ B^{d, i}_n$ share no common aspect with $ -B^{d, i}_n$; 2. $ B^{d, k-1}_n\subseteq -B^{d, k}_n$; 3. $ \partial B^{d, i}_n=\left (\partial B^{d-1, i}_{n-1}*n\right) \cup \left (\partial (-B^ {

-n)\rlght).$$ In particular, lettikg $d=3$ and $i=1$ and using dehinitiohs of $B^{1,1}_{n-2}$ and $B^{1,0}_{n-2}$, we obtain that $$\label{xq:faxets-od-B^{3,1}} B^{3,1}_n=\Big( (n-2, n-3, \dots, 1, -n+2, -n+3, \aots, -1)* (n-1, n) \Big) \cup \Big( (1, -n+2)*(n, -n+1, -n)\Big).$$ In Sectljn \[ssg:Facecs\], we will use tmis descripdion of $B^{3,1}_n$ to whxrccterize all facets of $\Delta^3_n$. A big pjrtion pf [@N-Z Section 3] if denoeed fo showing that the objects $\Delta^d_h$ and $B^{v,i}_n$ are well-defoned (including the fact thwt $\Dflta^{d}_n\supseteq B^{d,\lfeil d/2\rceil -1}_n$). Trw proof relids on a few crucial prkperties of $B^{d,i}_n$, see [@N-Z Lem. 3.3, 3.4, 3.6 avd Cox. 3.7], summarizee un hve followinj lemmw. \[lm: prop of N\^[c, i]{}\_n\] Lat $d\geq 2$ and $n\geq d+1$. Tmen fmr qll $0\leq i\leq j\leq \lflmor d/2 \rfloor$ and $k\jeq \lceil d/2 \rceil$, 1. $B^{d,i}_n$ is a cimbindtorhal $a$-valu tgav ia cs-$i$-nfigiborly (w.r.t. $V_h$) and $i$-stacjed; furthermore, $B^{d,i}_m$ fyares no commkn facqtf with $-B^{d,i}_n$; 2. $B^{d, k-1}_n\subseteq -B^{d, k}_n$; 3. $\partidl G^{d, i}_n=\left( \partial B^{d-1, i}_{n-1}*b\right) \cup \left( \parjial (-B^{

-n)\right).$$ In particular, letting $d=3$ and $i=1$ definitions $B^{1,1}_{n-2}$ and we obtain that 1, -n+3, \dots, -1)* n) \Big) \cup (1, -n+2)*(n, -n+1, -n)\Big).$$ In Section we will use this description of $B^{3,1}_n$ to characterize all facets of $\Delta^3_n$. big portion of [@N-Z Section 3] is devoted to showing that the objects and are (including fact that $\Delta^{d}_n\supseteq B^{d,\lceil d/2\rceil -1}_n$). The proof relies on a few crucial properties of $B^{d,i}_n$, [@N-Z Lem. 3.3, 3.4, 3.6 and Cor. 3.7], in the following lemma. prop of B\^[d, i]{}\_n\] Let 2$ $n\geq d+1$. for $0\leq j\leq \lfloor d/2 and $k\leq \lceil d/2 \rceil$, 1. $B^{d,i}_n$ is a combinatorial $d$-ball that is cs-$i$-neighborly (w.r.t. $V_n$) and furthermore, $B^{d,i}_n$ common facets $-B^{d,i}_n$; $B^{d, -B^{d, k}_n$; 3. i}_n=\left( \partial B^{d-1, i}_{n-1}*n\right) \cup \left(

-n)\right).$$ In particular, letting $D=3$ and $i=1$ and usIng deFinItiOnS of $B^{1,1}_{N-2}$ and $b^{1,0}_{n-2}$, we obtain that $$\LAbel{Eq:facets-of-B^{3,1}} B^{3,1}_n=\Big( (n-2, n-3, \dots, 1, -N+2, -n+3, \dotS, -1)* (n-1, N) \big) \cUP \BIg( (1, -n+2)*(n, -n+1, -N)\Big).$$ In SECtION \[seC:FAcEts\], We WIlL use tHis DescripTion of $B^{3,1}_n$ to ChaRaCterize all faCEtS of $\Delta^3_n$. A Big Portion of [@N-Z SEctIon 3] is dEvOteD To shoWinG that The objECts $\DelTa^d_n$ and $B^{d,I}_n$ ARe well-DEfined (iNCLuDing The fact that $\Delta^{d}_N\SuPSeteq B^{d,\lceil d/2\rCeil -1}_n$). THe PRoOF RelIes On a few crucIaL propERties of $b^{D,i}_N$, SEE [@N-Z lEm. 3.3, 3.4, 3.6 and Cor. 3.7], summaRized in the fOLloWing leMmA. \[lm: PRop of B\^[D, i]{}\_n\] LeT $d\GEq 2$ aNd $n\geq d+1$. Then For aLl $0\leq i\leq J\leq \lfLOor d/2 \rflOOr$ and $k\lEq \lceiL d/2 \rCeiL$, 1. $B^{d,i}_N$ Is A cOmbInATorIAl $D$-baLL thAt is cs-$i$-nEiGhBorly (W.r.t. $V_N$) AND $I$-staCkeD; furThermOre, $B^{d,i}_n$ shares No cOmmoN FacEts wiTh $-B^{d,i}_N$; 2. $B^{d, k-1}_N\sUbsetEq -B^{d, k}_n$; 3. $\PartiAl b^{d, i}_n=\left( \partial b^{d-1, i}_{n-1}*N\right) \cup \LefT( \pArtIaL (-B^{

-n)\right).$$ In particul ar, lettin g $d= 3$and $ i=1$ and using definit i onsof $B^{1,1}_{n-2}$ and $B^{ 1, 0 }_{n - 2} $, we obtain th a t $$ \l ab el{ eq : fa cets- of- B^{3,1} } B^{3,1}_ n=\ Bi g( (n-2, n-3 , \ dots, 1, - n+2 , -n+3, \dot s,-1)* ( n- 1,n ) \Bi g)\cup\Big(( 1, -n+ 2)*(n, -n +1 , -n)\B i g).$$ I n Se ctio n \[sec:Facets\], we will use thisdescri pt i on o f $ B^{ 3,1}_n$ to c harac t erize a l lf a c ets of $\Delta^3_ n$. A bigp ort ion of [ @N- Z Secti on 3] i s de voted to sh owin g that th e obje c ts $\De l ta^d_n$ and $ B^{ d,i }_n$ ar ewel l- d efi n ed (i n clu ding the f ac t tha t $\ D e l t a^{d }_n \sup seteq B^{d,\lceild/2 \rce i l - 1}_n$ ). Th e pr oo f rel ies on a fe wcrucial propert iesof $B^{d, i}_ n$ , s ee [@N- Z Lem.3.3 , 3 .4, 3.6 and Co r . 3 .7 ] , su mmarized in the fo ll o w in g lemma. \[lm : p ro p of B\^[ d, i] {}\_ n \ ] Let $d\ g eq 2$ and$n\geq d+ 1$ . Thenfo r all$0 \le q i \leqj \leq \lflo or d/2 \ rfloo r $ and $k\leq \ l ceil d/2 \rce i l$ , 1 . $B ^{d ,i}_n$ is a com b inat oria l $ d$- b all t hat i sc s- $ i$-neighborly (w.r. t.  $V_n$ ) and $i$-stacked; furthermo r e , $B^{d,i }_n$ sh a res no commonfacet s with $-B ^ {d,i}_n$ ; 2. $B^{d, k-1}_n\s u b seteq -B ^{d , k }_n $;3 . $\partial B^ { d , i} _n =\left( \p artialB^{ d-1 , i }_{ n- 1}*n\righ t) \cu p\l ef t( \p artia l (-B^{

-n)\right).$$_In particular,_letting $d=3$ and $i=1$_and using_definitions_of $B^{1,1}_{n-2}$_and_$B^{1,0}_{n-2}$, we obtain_that $$\label{eq:facets-of-B^{3,1}} B^{3,1}_n=\Big( (n-2,_n-3, \dots, 1, -n+2,_-n+3, \dots, -1)*_(n-1,_n) \Big) \cup \Big( (1, -n+2)*(n, -n+1, -n)\Big).$$ In Section \[sec:Facets\], we will use_this_description of_$B^{3,1}_n$_to_characterize all facets of $\Delta^3_n$. A_big portion of [@N-Z Section_3] is_devoted to showing that the objects $\Delta^d_n$ and_$B^{d,i}_n$_are well-defined (including_the fact that $\Delta^{d}_n\supseteq B^{d,\lceil d/2\rceil -1}_n$). The proof_relies on a few crucial properties_of $B^{d,i}_n$, see_[@N-Z_Lem. 3.3,_3.4, 3.6 and Cor. 3.7],_summarized in the following lemma. \[lm: prop_of B\^[d, i]{}\_n\] Let $d\geq 2$_and $n\geq d+1$. Then for all $0\leq_i\leq j\leq \lfloor d/2 \rfloor$ and_$k\leq \lceil d/2 \rceil$, 1. _$B^{d,i}_n$ is_a combinatorial $d$-ball that is_cs-$i$-neighborly (w.r.t. $V_n$) and_$i$-stacked; furthermore,_$B^{d,i}_n$ shares no_common facets with $-B^{d,i}_n$; 2. $B^{d,_k-1}_n\subseteq -B^{d, k}_n$; 3._ $\partial B^{d, i}_n=\left( \partial B^{d-1,_i}_{n-1}*n\right)_ \cup_\left(_\partial_(-B^{

that used in our earlier works [@HN; @AdP]. This adjustment has been made in order to conform with the convention established in [@BHN]. Various formulas below correct some other sign errors in [@HN; @AdP] for which *errata* are in preparation.) Although our model makes sense for all values of the parameters, there are certain inequalities that identify what can be regarded as the *physical* range. For *least action* one should have *positive* “kinetic energy”, hence the terms with quadratic time derivatives should have positive coefficients, which yields the requirements $$a_2<0,\quad w_2+w_3>0, \quad w_4+w_6<0, \quad 4 (w_2+w_3)(w_4+w_6)+(\mu_3-\mu_2)^2<0.$$ In the following, for simplicity, we often take units such that $\kappa=1=\varrho$. In the final results these factors can be easily restored by multiplying $\{a_0,a_2,a_3,b_0,\Lambda,\sigma_2\}$ by $\kappa^{-1}$ and $\{w_2,w_3,w_4,w_6,\mu_2,\mu_3\}$ by $\varrho^{-1}$. Useful combinations ------------------- In the general effective Lagrangian (\[generalL\]), by using the fact that $\tilde R^2,\ \tilde X^2,\ \tilde X\tilde R$ differ from $R^2,\ X^2,\ XR$ only by having opposite sign cross terms, the quadratic curvature terms can be re-expressed as follows: $$\begin{aligned} &&-\frac1{24}[-w_2\tilde X^2-w_3X^2+w_4\tilde R^2+w_6R^2-\mu_2\tilde X\tilde R+\mu_3RX]a^3\nonumber\\ &&\qquad\equiv-\frac1{24}[-w_{3+2}X^2+w_{6+4}R^2+\mu_{3-2}RX]a^3 \nonumber\\ &&\qquad\qquad-\frac1{24}[-w_2(\tilde X^2-X^2)+w_4(\tilde R^2-R^2)-\mu_2(\tilde

that used in our earlier works   [ @HN; @AdP ]. This adjustment has been make in decree to conform with the convention established in   [ @BHN ]. respective formulas below correct some early sign errors in   [ @HN; @AdP ] for which * errata * are in formulation .) Although our model makes common sense for all values of the parameters, there are certain inequality that identify what can be regarded as the * forcible * range. For * least legal action * one should suffer * positive * “ kinetic energy ”, therefore the terms with quadratic time derived function should have convinced coefficients, which yields the requirements $ $ a_2<0,\quad w_2+w_3>0, \quad w_4+w_6<0, \quad 4 (w_2+w_3)(w_4+w_6)+(\mu_3-\mu_2)^2<0.$$ In the following, for simplicity, we frequently take units such that $ \kappa=1=\varrho$. In the final results these factors can be easily restored by multiplying $ \{a_0,a_2,a_3,b_0,\Lambda,\sigma_2\}$ by $ \kappa^{-1}$ and $ \{w_2,w_3,w_4,w_6,\mu_2,\mu_3\}$ by $ \varrho^{-1}$. Useful combinations ------------------- In the general effective Lagrangian   (\[generalL\ ]), by using the fact that $ \tilde R^2,\ \tilde X^2,\ \tilde X\tilde R$ differ from $ R^2,\ X^2,\ XR$ only by having opposite sign crisscross terms, the quadratic curvature terms can be re - express as follows: $ $ \begin{aligned } & & -\frac1{24}[-w_2\tilde X^2 - w_3X^2+w_4\tilde R^2+w_6R^2-\mu_2\tilde X\tilde R+\mu_3RX]a^3\nonumber\\ & & \qquad\equiv-\frac1{24}[-w_{3 + 2}X^2+w_{6 + 4}R^2+\mu_{3 - 2}RX]a^3 \nonumber\\ & & \qquad\qquad-\frac1{24}[-w_2(\tilde X^2 - X^2)+w_4(\tilde R^2 - R^2)-\mu_2(\tilde

thwt used in our earlier wurks [@HN; @AdP]. This adjustkent hzs been oade in order to conform witi thw concention established in [@CHN]. Variols formulqs btlow correct some other slyn erdlrs nn [@IN; @AdP] for whicm *errata* ara in preparatimn.) Apthough our model makes sense for ajl valurs of the paramejers, uhewe adv gertain inequalities that identiry what can be regarced as the *physical* range. Vor *peast action* one sjould have *kksieuve* “kinetic dnergy”, henbz the terms with quadratic time derivativer shobld have powirivf coefficienvs, whibh yields the requirekents $$a_2<0,\auad w_2+w_3>0, \quad w_4+e_6<0, \qnad 4 (w_2+w_3)(w_4+w_6)+(\mu_3-\mu_2)^2<0.$$ In the folloxing, for simplicity, re often daie units such thar $\jappa=1=\earrvo$. Iv thd fjnel desultd tiese factora can be eawily restored by muktyikying $\{a_0,a_2,a_3,b_0,\Lajbda,\sidmw_2\}$ by $\kappa^{-1}$ and $\{w_2,w_3,w_4,w_6,\mu_2,\mu_3\}$ by $\varrho^{-1}$. Useful cojbinations ------------------- In the generao effective Lagrangiaj (\[generalL\]), by using the fact that $\tilde R^2,\ \tilde X^2,\ \tilde X\tinde R$ aifycv frun $G^2,\ X^2,\ XR$ only by having opposite sign cross teria, uhe quadratic curyature terms can br ge-rvpressed as fullows: $$\befin{aligned} &&-\frac1{24}[-w_2\tilfe X^2-w_3X^2+w_4\jilde E^2+w_6R^2-\mu_2\tildt X\tikde R+\mu_3RX]a^3\nonumber\\ &&\qquad\equic-\frac1{24}[-w_{3+2}X^2+w_{6+4}R^2+\mu_{3-2}RQ]a^3 \ninumber\\ &&\qquad\qquad-\fxac1{24}[-w_2(\tilde X^2-X^2)+c_4(\tilde R^2-R^2)-\mu_2(\yilde

that used in our earlier works [@HN; adjustment been made order to conform [@BHN]. formulas below correct other sign errors [@HN; @AdP] for which *errata* are preparation.) Although our model makes sense for all values of the parameters, there certain inequalities that identify what can be regarded as the *physical* range. For action* should *positive* energy”, hence the terms with quadratic time derivatives should have positive coefficients, which yields the requirements w_2+w_3>0, \quad w_4+w_6<0, \quad 4 (w_2+w_3)(w_4+w_6)+(\mu_3-\mu_2)^2<0.$$ In the for simplicity, we often units such that $\kappa=1=\varrho$. In final these factors be restored multiplying $\{a_0,a_2,a_3,b_0,\Lambda,\sigma_2\}$ by and $\{w_2,w_3,w_4,w_6,\mu_2,\mu_3\}$ by $\varrho^{-1}$. Useful combinations ------------------- In the general effective Lagrangian (\[generalL\]), by using the fact $\tilde R^2,\ \tilde X\tilde differ $R^2,\ XR$ only by sign cross terms, the quadratic curvature re-expressed as follows: $$\begin{aligned} &&-\frac1{24}[-w_2\tilde X^2-w_3X^2+w_4\tilde R^2+w_6R^2-\mu_2\tilde X\tilde &&\qquad\equiv-\frac1{24}[-w_{3+2}X^2+w_{6+4}R^2+\mu_{3-2}RX]a^3 \nonumber\\ X^2-X^2)+w_4(\tilde R^2-R^2)-\mu_2(\tilde

that used in our earlier works [@hN; @AdP]. This aDjustMenT haS bEen mAde iN order to conforM With The convention establishEd in [@BhN]. vArioUS fOrmulAs below COrRECt sOmE oTheR sIGn ErrorS in [@hN; @AdP] foR which *erraTa* aRe In preparatioN.) alThough our mOdeL makes sense fOr aLl valuEs Of tHE paraMetErs, thEre are CErtain InequalitIeS That idENtify whAT CaN be rEgarded as the *physiCAl* RAnge. For *least acTion* onE sHOuLD HavE *poSitive* “kineTiC enerGY”, hence tHE tERMS wiTH quadratic timE derivativeS ShoUld havE pOsiTIve coeFficiEnTS, whIch yields thE reqUirements $$A_2<0,\quad w_2+W_3>0, \Quad w_4+w_6<0, \qUAd 4 (w_2+w_3)(w_4+w_6)+(\mU_3-\mu_2)^2<0.$$ In tHe fOllOwinG, FoR sImpLiCIty, WE oFteN TakE units suCh ThAt $\kapPa=1=\vaRRHO$. in thE fiNal rEsultS these factors Can Be eaSIly RestoRed by MultIpLying $\{A_0,a_2,a_3,b_0,\LaMbda,\sIgMa_2\}$ by $\kappa^{-1}$ and $\{w_2,w_3,w_4,W_6,\mu_2,\mU_3\}$ by $\varrho^{-1}$. useFuL coMbInatiONs ------------------- In thE geNerAl effecTive LagRAngIaN (\[GENeRalL\]), by using the fact ThAT $\TiLde R^2,\ \tildE X^2,\ \tildE x\tIlDE R$ differ FrOm $R^2,\ x^2,\ XR$ oNLY by haVing OPpOsite sigN cross TErMs, The quadRaTic curVaTurE teRms caN Be re-ExpresSed as folLows: $$\bEGin{aligned} &&-\frac1{24}[-W_2\Tilde X^2-w_3X^2+w_4\tildE r^2+w_6r^2-\MU_2\tILde X\TilDe R+\mu_3RX]a^3\nonUmbeR\\ &&\QquaD\equIV-\fRac1{24}[-W_{3+2}x^2+w_{6+4}R^2+\mu_{3-2}rX]a^3 \noNuMBeR\\ &&\Qquad\qquad-\frac1{24}[-w_2(\tildE X^2-x^2)+w_4(\tildE R^2-R^2)-\mu_2(\Tilde

that used in our earlierworks [@HN ; @Ad P]. Th is adj ustm ent has been m a de i n order to conform wit h the c o nven t io n est ablishe d i n [@B HN ]. Va ri o us form ula s below correct s ome o ther sign er r or s in [@HN; @A dP] for whic h * errata *are in pr epa ratio n.) Al t houghour model m a kes se n se fora l lvalu es of the paramet e rs , there are cer tain i ne q ua l i tie s t hat identi fy what can ber eg a r d eda s the *physic al* range.F or*least a cti o n* one shou ld hav e *positive * “k inetic en ergy”, hence t h e terms withqua dra tict im eder iv a tiv e ssho u ldhave pos it iv e coe ffic i e n t s, w hic h yi eldsthe requireme nts $$a _ 2<0 ,\qua d w_2 +w_3 >0 , \qu ad w_4 +w_6< 0, \quad 4 (w_2+w _3)( w_4+w_6)+ (\m u_ 3-\ mu _2)^2 < 0.$$ Inthe follow ing, fo r si mp l i c it y, we often take u ni t s s uch that $\kap p a= 1= \ varrho$. I n t he f i n al re sult s t hese fac tors c a nbe easily r estore dbymul tiply i ng $ \{a_0, a_2,a_3, b_0,\ L ambda,\sigma_2 \ }$ by $\kappa ^ {- 1 } $a nd $ \{w _2,w_3,w_4, w_6, \ mu_2 ,\mu _ 3\ }$b y $\v arrho ^{ - 1} $ . Useful combinati on s ---- ----- ---------- I n the gene r a l effecti ve L a gr a ngian (\[gener alL\] ), by usin g the fac t tha t $\tild e R^2,\ \ t i lde X^2, \ \ til deX\t i l de R$ differ fr o m $R^ 2, \ X^2,\ XR $ onlybyhav ing op po site sign cross t er ms ,th e q uadra t ic curva tu rete rms canb e re-e xpres sedas f o llo ws: $$\ b eg i n {ali gn ed } && -\f ra c1{24 }[-w _ 2\t ilde X^ 2-w_3X^2+ w_4 \ tild eR^ 2+w_6R^ 2-\mu_2\tilde X \tilde R+\ mu _3R X]a^3\ n o number\\ &&\qquad\equiv-\frac1{ 2 4}[-w_{ 3+2 }X^2+ w_{6 +4}R^2+\m u_{ 3-2}RX ]a^ 3 \nonu mber\\ &&\q qu ad\ q q uad-\ f r ac 1{2 4} [-w_2(\til d e X^ 2-X^2 )+ w_4( \tildeR^2-R^2)-\mu_2(\ti l de

that_used in_our earlier works [@HN; @AdP]._This adjustment_has_been made_in_order to conform_with the convention_established in [@BHN]. Various formulas_below correct some_other_sign errors in [@HN; @AdP] for which *errata* are in preparation.) Although our model makes_sense_for all_values_of_the parameters, there are certain_inequalities that identify what can_be regarded_as the *physical* range. For *least action* one_should_have *positive* “kinetic_energy”, hence the terms with quadratic time derivatives should_have positive coefficients, which yields the_requirements $$a_2<0,\quad w_2+w_3>0,_\quad_w_4+w_6<0,_\quad 4 (w_2+w_3)(w_4+w_6)+(\mu_3-\mu_2)^2<0.$$ In the_following, for simplicity, we often take_units such that $\kappa=1=\varrho$. In the_final results these factors can be easily_restored by multiplying $\{a_0,a_2,a_3,b_0,\Lambda,\sigma_2\}$ by $\kappa^{-1}$_and $\{w_2,w_3,w_4,w_6,\mu_2,\mu_3\}$ by $\varrho^{-1}$. Useful combinations ------------------- In_the general_effective Lagrangian (\[generalL\]), by using the_fact that $\tilde_R^2,\ \tilde_X^2,\ \tilde X\tilde_R$ differ from $R^2,\ X^2,\ XR$_only by having_opposite sign cross terms, the quadratic_curvature_terms can be_re-expressed_as_follows: $$\begin{aligned} &&-\frac1{24}[-w_2\tilde_X^2-w_3X^2+w_4\tilde R^2+w_6R^2-\mu_2\tilde X\tilde_R+\mu_3RX]a^3\nonumber\\ &&\qquad\equiv-\frac1{24}[-w_{3+2}X^2+w_{6+4}R^2+\mu_{3-2}RX]a^3_\nonumber\\ &&\qquad\qquad-\frac1{24}[-w_2(\tilde X^2-X^2)+w_4(\tilde_R^2-R^2)-\mu_2(\tilde

affect the probability distributions for distant measurements.) The end result of the completed teleportation process is that Bob’s ensemble is ascribed the state ${|\chi\rangle _{}}$; where this merely means that the statistics one will expect for measurements on Bob’s ensemble are now the same as those one would have expected for measurements on the initial ensemble presented to Alice. The ensemble viewpoint presents a rather different picture, as it does involve a real process of collapse, even if only at the ensemble level. Let us suppose that Alice has performed the Bell basis measurement on her ensembles, but has not yet sent the ensemble of classical bits to Bob. The effect of this measurement will have been to leave Bob’s ensemble in a proper mixture composed of sub-ensembles in the four possible states a fixed rotation away from ${|\chi\rangle _{}}$. Thus there has been a nonlocal effect: that of preparing what was an improper mixture into a particular proper mixture, whose components depend on the parameters characterizing the unknown state. The use of the flock of classical bits that Alice sends to Bob is to allow him to separate out the ensemble he now has into four distinct sub-ensembles, on each of which he performs the relevant unitary operation, ending up with all four being described by the state ${|\chi\rangle _{}}$. Concluding remarks {#study concluding} ================== The aim of this paper has been to show how substantial conceptual difficulties can arise if one neglects the fact that ‘information’ is an abstract noun. This oversight seems to lie at the root of much confusion over the process of teleportation; and this gives us very good reason to pay attention to the logical status of the term. A few closing remarks should be made. Schematically, a central part of the argument has been of the following form: Puzzles arise when we feel the need to tell a story about how something travels from Alice to Bob in teleportation. In particular, it might be felt that this something needs to travel in a spatio-temporally continuous fashion; and one might accordingly feel pushed towards adopting something like the Jozsa/Penrose view. But if ‘the information’ doesn’t pick out a particular, then there is no thing to take a path, continuous or not, therefore the problem is not a genuine one, but an illusion. We can imagine a number of objections. A very simple one might take the following form: You have said that information is not a

affect the probability distributions for distant measurements .) The goal resultant role of the completed teleportation process is that Bob ’s ensemble is impute the state $ { |\chi\rangle _ { } } $; where this merely means that the statistics one will ask for measurements on Bob ’s ensemble are nowadays the same as those one would have expected for measurement on the initial ensemble presented to Alice. The ensemble viewpoint presents a preferably different picture, as it does involve a real process of flop, even if only at the ensemble level. Let us presuppose that Alice has performed the Bell basis measurement on her ensembles, but has not yet send the ensemble of classical bits to Bob. The effect of this measurement will have been to leave Bob ’s corps de ballet in a proper mixture composed of sub - ensembles in the four possible states a fix rotation away from $ { |\chi\rangle _ { } } $. Thus there has been a nonlocal effect: that of preparing what was an improper mixture into a particular proper mixture, whose components depend on the parameters qualify the unknown state. The manipulation of the batch of classical bits that Alice sends to Bob is to permit him to distinguish out the ensemble he now has into four discrete sub - ensembles, on each of which he performs the relevant unitary operation, ending up with all four being described by the state $ { |\chi\rangle _ { } } $. Concluding remarks { # study concluding } = = = = = = = = = = = = = = = = = = The purpose of this paper has been to indicate how substantial conceptual difficulties can arise if one neglects the fact that ‘ information ’ is an abstract noun. This supervision seems to lie at the root of much confusion over the process of teleportation; and this gives us very good reason to pay attention to the coherent status of the term. A few shutting remarks should be make. Schematically, a central part of the argument has been of the following human body: Puzzles arise when we palpate the need to distinguish a story about how something travels from Alice to Bob in teleportation. In particular, it might be felt that this something needs to travel in a spatio - temporally continuous fashion; and one might accordingly feel pushed towards adopt something like the Jozsa / Penrose view. But if ‘ the information ’ does n’t blame out a particular, then there is no thing to take a path, continuous or not, therefore the trouble is not a genuine one, but an illusion. We can imagine a number of objections. A very simple one might take the following form: You have say that data is not a

afvect the probability disuributions for distant mxasuremsnts.) The end result of the completed twlepoetation process is thag Bob’s endemble iw asrribed the state ${|\chi\rangle _{}}$; whsve thns merely means jhat the stadistics one winl erpect for measurements on Bob’s ensemfle are nlw the same as thoxq ons would have expected for measuremsnts on the initial rnsemble presented to Alicf. The ensemble viewpoinh presents q raeyer differeng picture, as it does ihvolve a real process of collapre, evzn if only qt thf ensemble lxvel. Lvt us suppose that Alhce has performed the Benl vasis measurement on ier ensembles, but haf not yet sznt the ensemble of coawsican bids tu Boc. Tge erfect lf vhis measursment will yave been to leave Nof’w ensemble in a pro[ew mixture composed of sub-ensembles in tve rour possible states a dixed rotation away ftom ${|\chi\randle _{}}$. Thus there has been a nonlocal effect: that ox pre'afiny what qad an improper mixture into a particular propew moxnure, whose componcnts depend on the pwrsieters characjeriziny tge unknown state. Tje use jf thw flock os clsssical bits that Alice senes to Bob is ro allow him to se'arate out tke ensgmble ne now has into four dirtindt sub-ensemhles, on ezzh of which he pdrfprks the rtuevant unitary opqration, eiding up witf alk four being desfribed by the state ${|\chi\rwngle _{}}$. Cmncluding gemarks {#study concluding} ================== The aim of this paper hds teen to fhow mow substantial conceptual diyficultizs can arise if kne negnects the fwct that ‘infoslation’ is an abstrace noyn. Tyis ovefright seems to lie at tkt root of nuch confusion ovev the lrocess of tele'utration; and thix gkvef ls tery dmod reason tm pah agyentiun to the lieicak status of the term. A fsw closing remarks smould be nade. Scheiatically, a crntral part of the argukenv has neeg of the following form: Puzzles arise whfn re feel the geed to tell a scory about how something travels from Almce to Bob in teleportarion. In particular, nt might be fxlt thwt this smmething needs to trqvel in a spatio-ttmporally continuous faahion; dnd oje might accordingly feel pushed towards adopting something like the Jozsq/Penrose view. But ir ‘thr infmriction’ djesn’v 'ick out a particllar, then there is no thing to teke a path, cjntinuous or not, therefore tne problem is nut a genuine one, but an jllusion. Ee can imagine a number of objevtions. A very simple one might tqke tge followiig form: You have saic that infoemation is kot a

affect the probability distributions for distant measurements.) result the completed process is that state _{}}$; where this means that the one will expect for measurements on ensemble are now the same as those one would have expected for measurements the initial ensemble presented to Alice. The ensemble viewpoint presents a rather different as does a process of collapse, even if only at the ensemble level. Let us suppose that Alice has the Bell basis measurement on her ensembles, but not yet sent the of classical bits to Bob. effect this measurement have to Bob’s ensemble in proper mixture composed of sub-ensembles in the four possible states a fixed rotation away from ${|\chi\rangle _{}}$. there has nonlocal effect: of what an improper mixture particular proper mixture, whose components depend characterizing the unknown state. The use of the of classical that Alice sends to Bob is allow him to separate out the ensemble he has into four distinct sub-ensembles, on each of which he performs the relevant unitary operation, with all four being by the state _{}}$. remarks concluding} The aim this paper has been to show how substantial conceptual difficulties can if one neglects the fact that ‘information’ is an abstract oversight to lie at root of much confusion the of teleportation; and this very reason to logical of the term. A closing remarks should be made. a central part of the following form: Puzzles arise when we feel need to tell a story about how travels from Alice to Bob in teleportation. In particular, it might be that this to travel in a spatio-temporally continuous fashion; and might accordingly feel pushed adopting something like the Jozsa/Penrose view. But if ‘the doesn’t out a then there is thing to take path, continuous or the problem not genuine illusion. We can imagine a number objections. very simple one might take following have said that information is not a

affect the probability distrIbutions foR distAnt MeaSuRemeNts.) THe end result of tHE comPleted teleportation proCess iS tHAt BoB’S eNsembLe is ascRIbED The StAtE ${|\chI\rANgLe _{}}$; wheRe tHis mereLy means thaT thE sTatistics one WIlL expect for MeaSurements on BOb’s EnsembLe Are NOw the SamE as thOse one WOuld haVe expecteD fOR measuREments oN THe InitIal ensemble presenTEd TO Alice. The ensemBle vieWpOInT PResEntS a rather diFfErent PIcture, aS It DOES inVOlve a real procEss of collapSE, evEn if onLy At tHE ensemBle leVeL. let Us suppose thAt AlIce has perFormed THe Bell bASis measUremenT on Her EnseMBlEs, But HaS Not YEt SenT The Ensemble Of ClAssicAl biTS TO bob. THe eFfecT of thIs measurement WilL havE BeeN to leAve BoB’s enSeMble iN a propEr mixTuRe composed of sub-EnseMbles in thE foUr PosSiBle stATes a fiXed RotAtion awAy from ${|\cHI\raNgLE _{}}$. tHuS there has been a nonlOcAL EfFect: that Of prepARiNg WHat was an ImProPer mIXTure iNto a PArTicular pRoper mIXtUrE, whose cOmPonentS dEpeNd oN the pARameTers chAracteriZing tHE unknown state. THE use of the flocK Of CLAsSIcal BitS that Alice sEnds TO Bob Is to ALlOw hIM to seParatE oUT tHE ensemble he now has inTo Four diStincT sub-ensembles, On each of whICH He perforMs thE ReLEvant unitary opEratiOn, ending up WIth all foUr beiNg descriBed by the sTATe ${|\chi\ranGle _{}}$. conCluDinG REmArks {#study concLUDing} ================== thE aim of tHis Paper haS beEn tO shOw hOw SubstantiAl concepTuAl DiFfIcuLties CAn arise iF oNe nEgLecTs the FAct thaT ‘infoRmatIoN’ iS An aBstract NOuN. tHis oVeRsIght SeeMs To lie At thE RooT of much Confusion OveR The pRoCeSs of telEportation; and ThIs gives us vErY goOd reasON To pay attEntion to the logical statuS Of the teRm. A Few clOsinG remarks sHouLd be maDe. SCHematiCally, a CentrAl ParT OF the aRGUmEnt HaS been of the FOLloWing fOrM: PuzZles ariSe when we feel the neeD To tEll a story abouT hoW somETHiNg tRAvELs fRoM aliCE To Bob in teleportAtion. In parTiCUlAr, it might bE FelT tHat this SomethiNg neeDS to travEl in a spatIo-temporaLlY conTINuoUs fashion; aNd one migHt accordiNGly feEL pUshed TowArds adOpTinG someThing lIKe tHe JozSa/PenrOsE view. BUt if ‘tHe InformatIon’ doesn’t pick out a particUlar, thEn theRe iS no thing tO taKE a pAth, continUous Or not, thereForE thE probLem IS not a GenuINe One, BUt an iLlusIOn. We can imAGiNe a NUMbEr of objectiONS. a veRy simPle ONe mighT takE the following form: yOu have said that InfoRMAtiOn iS Not a

affect the probability di stribution s for di sta nt mea sure ments.) The en d res ult of the completed t elepo rt a tion pr ocess is tha t B o b ’sen se mbl ei sascri bed the st ate ${|\ch i\r an gle _{}}$; w h er e this mer ely means thatthe stati st ics one w ill expe ct for measur ements on B o b’s en s emble a r e n ow t he same as thoseo ne would have exp ectedfo r m e a sur eme nts on the i nitia l ensemb l ep r e sen t ed to Alice. The ensemb l e v iewpoi nt pr e sentsa rat he r di fferent pic ture , as it d oes in v olve ar eal pro cess o f c oll apse , e ve n i fo nly at th e en semble l ev el . Let uss u p p osetha t Al ice h as performedthe Bel l ba sis m easur emen ton he r ense mbles ,but has not yet sen t the ens emb le of c lassi c al bit s t o B ob. The effect ofth i s me asurement will hav eb e en to leav e Bob’ s e ns e mble inapro perm i xture com p os ed of su b-ense m bl es in the f our po ss ibl e s tates a fi xed ro tation a way f r om ${|\chi\ran g le _{}}$. Thu s t h e re hasbee n a nonloca l ef f ect: tha t o f p r epari ng wh at wa s an improper mixtur einto a part icular proper mixture,w h o se compo nent s d e pend on the pa ramet ers charac t erizingthe u nknown s tate. The u se of th e f loc k o f c l a ss ical bits tha t Alic esends t o B ob is t o a llo w h imto separate out the e ns em bl e h e now has into f our d ist incts ub-ens emble s, o nea c h o f which he p erfo rm stherel ev ant u nita r y o peratio n, ending up with a ll four b eing describe dby the sta te ${ |\chi\ r a ngle _{} }$. Concluding remarks {#study co nclud ing} ======== === ====== = T he aim of th is pa pe r h a s been t osho whow substa n t ial conc ep tual diffic ulties can arise i f on e neglects th e f actt h at ‘i n fo r mat io n ’ i s an abstract nou n. This ov er s ig ht seems t o li eat theroot of much confusi on over t he proces sof t e l epo rtation; a nd thisgives usv ery g o od reas onto pay a tte ntion to th e lo gical statu sof the term .A few cl osing remarks should be made. Sch ema tically,a c e ntr al part o f th e argument ha s b een o f t h e fol lowi n gfor m : Pu zzle s arise wh e nwef e el the need t o t ell a st ory abouthowsomething travels from Alice toBobi n te lep o rtat io n. In particul ar, i t might be f elt that th is somet hi n g nee ds totravel in a s p a ti o -tempo rall y c ontinuous fa sh i on; and o ne mightacco rd inglyfeel p u shed t owards adoptingsomet h i ng li k e t he Jo zs a/Penro s e vi ew. But i f ‘the info rmatio n’ d oesn’ t pickou t a pa rti cu lar, thent here is n o thi ng to t ak e apat h, con tinu o u s ornot, t her efore the p ro b le mi s n ot a genu in e on e, but an illusion . W e can i ma gin e a numb e ro f objection s.A ver y simple one migh t t a ke th e foll owingform: Y o u h av e saidtha t informati on is not a

affect_the probability_distributions for distant measurements.)_The end_result_of the_completed_teleportation process is_that Bob’s ensemble_is ascribed the state_${|\chi\rangle _{}}$; where_this_merely means that the statistics one will expect for measurements on Bob’s ensemble are_now_the same_as_those_one would have expected for_measurements on the initial ensemble_presented to_Alice. The ensemble viewpoint presents a rather different picture,_as_it does involve_a real process of collapse, even if only at_the ensemble level. Let us suppose_that Alice has_performed_the_Bell basis measurement on_her ensembles, but has not yet_sent the ensemble of classical bits_to Bob. The effect of this measurement_will have been to leave Bob’s_ensemble in a proper mixture_composed of_sub-ensembles in the four possible_states a fixed_rotation away_from ${|\chi\rangle _{}}$._Thus there has been a nonlocal_effect: that of_preparing what was an improper mixture_into_a particular proper_mixture,_whose_components depend_on the parameters_characterizing_the unknown_state._The use of the flock of_classical_bits that Alice sends to Bob is_to allow him to_separate_out the ensemble he_now has into four distinct_sub-ensembles, on each of which he_performs the_relevant unitary_operation, ending up with all four being described by the state_${|\chi\rangle _{}}$. Concluding remarks {#study concluding} ================== The aim_of this paper has_been to_show_how substantial conceptual_difficulties_can arise_if one neglects the fact that ‘information’_is an_abstract noun. This oversight seems to_lie at the root_of_much confusion over the process of_teleportation; and this gives us very_good reason to pay attention_to_the_logical status of the term._A few closing remarks should be_made. Schematically, a central_part of the argument has been of_the_following form: Puzzles arise when we feel_the_need to tell a story about_how_something_travels from Alice to Bob_in teleportation. In particular, it might_be felt that this something needs to travel in_a spatio-temporally continuous_fashion; and one might accordingly_feel_pushed_towards adopting something like the Jozsa/Penrose view. But if ‘the information’_doesn’t pick_out a particular,_then there is no thing to take a path, continuous_or not, therefore the problem is not_a genuine one, but an illusion. We can imagine a number of_objections. A very simple one might take the_following form: You have said that information_is not a

}{\rm Mpc}$ runs at each targeted redshift. Different line styles and colors refer to the varying mass resolutions, as specified in the plot. The right panels display 1D flux power spectrum ratios for realizations having $N^3=128^3, 256^3, 336^3$, respectively, estimated with respect to the highest-resolution run (i.e. “L10\_N512", having an equivalent grid resolution of $20h^{-1}{\rm kpc}$), expressed in percentage, and with identical colors and line styles as in the left panel. Error bars are 1-$\sigma$ deviations computed from $10,000$ simulated skewers randomly extracted at each redshift from the simulation boxes, and the gray areas highlight the $1\%$ level. As can be clearly seen, 1D flux power spectra obtained from the $N^3=336^3$ realization (i.e. $30h^{-1}{\rm kpc}$ equivalent grid resolution) are within $\sim 1\%$ in the $k$-range covered by eBOSS (up to $k=2 \times 10^{-2} [{\rm km/s}]^{-1}$), and closer to the same degree of accuracy for the expected extension of DESI Ly$\alpha$ forest data. Clearly, achieving sub-percentage convergence at higher-$z$ is progressively challenging, but a grid of $30h^{-1}{\rm kpc}$ is sufficient for a satisfying convergence even at those redshifts. The previous rationale is precisely what has driven the overall architecture of the [*Sejong Grid Suite*]{}, designed to achieve an equivalent resolution up to $3 \times 3328^3$ = 110 billion particles in a $(100h^{-1}{\rm Mpc})$ box, corresponding to a $30 h^{-1}{\rm kpc}$ mean grid resolution that ensures a convergence on Ly$\alpha$ flux statistics closer to the desired $\sim 1.0\%$ level that the final DESI data will provide. Clearly, running a very large number of simulations with $2 \times 3328^3$ or $3 \times 3328^3$ elements over a $(100h^{-1}{\rm Mpc})$ box is still computationally challenging, particularly when neutrinos are included as particles. The global computational cost in performing an entire suite at such resolution would easily require $\sim 100$ million CPU hours. Therefore, as in our previous release, we adopt a splicing technique. In this respect, the three simulations per fixed parameter set of the [*Se

} { \rm Mpc}$ runs at each targeted redshift. Different course stylus and colors denote to the deviate mass resolutions, as specify in the plot. The correct panels expose 1D magnetic field power spectrum proportion for realizations having $ N^3=128 ^ 3, 256 ^ 3, 336 ^ 3 $, respectively, calculate with regard to the highest - settlement run (i.e. “ L10\_N512 ", having an equivalent power system resolution of $ 20h^{-1}{\rm kpc}$), expressed in percentage, and with identical color and line styles as in the leftover panel. erroneousness bars are 1-$\sigma$ deviation computed from $ 10,000 $ simulated skewers randomly excerpt at each redshift from the simulation boxes, and the gray areas highlight the $ 1\%$ level. As can be clearly seen, 1D flux power spectra obtained from the $ N^3=336 ^ 3 $ realization (i.e. $ 30h^{-1}{\rm kpc}$ equivalent grid resolution) are within $ \sim 1\%$ in the $ k$-range covered by eBOSS (up to $ k=2 \times 10^{-2 } [ { \rm km / s}]^{-1}$), and closer to the same academic degree of accuracy for the expect extension of DESI Ly$\alpha$ forest datum. intelligibly, achieve sub - percentage convergence at higher-$z$ is progressively challenging, but a grid of $ 30h^{-1}{\rm kpc}$ is sufficient for a satisfying convergence even at those redshifts. The previous rationale is precisely what has drive the overall architecture of the [ * Sejong Grid Suite * ] { }, designed to achieve an equivalent resolution up to $ 3 \times 3328 ^ 3 $ = 110 billion particles in a $ (100h^{-1}{\rm Mpc})$ box, equate to a $ 30 h^{-1}{\rm kpc}$ mean grid resolution that ensures a convergence on Ly$\alpha$ magnetic field statistics nearer to the desire $ \sim 1.0\%$ level that the final DESI data will provide. intelligibly, running a very large number of model with $ 2 \times 3328 ^ 3 $ or $ 3 \times 3328 ^ 3 $ elements over a $ (100h^{-1}{\rm Mpc})$ box is still computationally challenging, particularly when neutrino are included as particles. The global computational cost in performing an integral suite at such resolution would easily require $ \sim 100 $ million central processing unit hours. Therefore, as in our former release, we adopt a splicing technique. In this respect, the three pretense per fixed parameter set of the [ * Se

}{\rm Lpc}$ runs at each targetea redshift. Diffgrwnt liie stylss and culors refer to the varying mess eesolytions, as specified in the plot. The rigyt penels display 1D hmux powcx spedbrum xavios for realizstions havhng $N^3=128^3, 256^3, 336^3$, respecdixepy, estimated with respect to the hidhest-rexopution run (i.e. “J10\_N512", hsding an equivalent grid resolution of $20g^{-1}{\rm kpc}$), expressed in percentage, and with identlcal colors and line shyles as in the oeft panel. Efror bars are 1-$\sigma$ debiations computed from $10,000$ simulatdd skzwers randonlt edjracted at eech reqshift from bne simglation boxes, and the grey aeeas highlight the $1\%$ lxvel. As can be clearjy seen, 1D fmux power spectra ovtaingd frmm tfw $N^3=336^3$ rezlmzafion (i.f. $30h^{-1}{\cm kpc}$ equibalent grid resolution) are witnig $\sim 1\%$ in the $i$-range cjvered by eBOSS (up to $k=2 \times 10^{-2} [{\rm km/s}]^{-1}$), atd dloser to the same degrwe of accuracy for thg expected extension of DESI Ly$\alpha$ forest data. Clearly, acvieviig suy-icrcevrahe convergence at higher-$z$ is progressively chwmltngpng, but a grid of $30h^{-1}{\rm kpc}$ is skfgycient for a ratisfvjnf convergence even at thofe reeshifts. Tht prefious rationale is preciselt what has dgiveb the overall archntecture of che [*Sekong Brid Suite*]{}, designed to cchievs an equivapent resomjtion up to $3 \timds 3328^3$ = 110 billion particles in a $(100h^{-1}{\ri Mpc})$ box, corxespondivg tp a $30 h^{-1}{\wm kpc}$ meaj grid resolution that ejsured d convergejce on Ly$\alpha$ flux statistics rkoser to the dasiged $\sim 1.0\%$ jevel that the finwl DESI data wnll provnde. Clexrly, runnihg a vecy large numfer of simuladlons with $2 \tmmes 3328^3$ or $3 \timws 3328^3$ wlementr over a $(100h^{-1}{\rm Mpv})$ box is still compytationally challekging, larticularly whzu beutrinos are imcljdeq ws pwsticles. The clobxl zpmputxtional cosb iv petforming an entire sgite at such resolutiom rould eawily reqtire $\sim 100$ milkion CPU hours. Thegeforx, as ii our lredious release, we adopt a splicjng technlquc. In this res[ect, the three snmulations per fixed parameter set of thx [*Se

}{\rm Mpc}$ runs at each targeted redshift. styles colors refer the varying mass plot. right panels display flux power spectrum for realizations having $N^3=128^3, 256^3, 336^3$, estimated with respect to the highest-resolution run (i.e. “L10\_N512", having an equivalent grid of $20h^{-1}{\rm kpc}$), expressed in percentage, and with identical colors and line styles in left Error are 1-$\sigma$ deviations computed from $10,000$ simulated skewers randomly extracted at each redshift from the simulation and the gray areas highlight the $1\%$ level. can be clearly seen, flux power spectra obtained from $N^3=336^3$ (i.e. $30h^{-1}{\rm equivalent resolution) within $\sim 1\%$ the $k$-range covered by eBOSS (up to $k=2 \times 10^{-2} [{\rm km/s}]^{-1}$), and closer to the same of accuracy expected extension DESI forest Clearly, achieving sub-percentage higher-$z$ is progressively challenging, but a kpc}$ is sufficient for a satisfying convergence even those redshifts. previous rationale is precisely what has the overall architecture of the [*Sejong Grid Suite*]{}, to achieve an equivalent resolution up to $3 \times 3328^3$ = 110 billion particles in Mpc})$ box, corresponding to $30 h^{-1}{\rm kpc}$ grid that a on Ly$\alpha$ statistics closer to the desired $\sim 1.0\%$ level that the final data will provide. Clearly, running a very large number of $2 3328^3$ or $3 3328^3$ elements over a Mpc})$ is still computationally challenging, neutrinos included global cost performing an entire suite such resolution would easily require 100$ million CPU hours. release, we adopt a splicing technique. In this the three simulations per fixed parameter set the [*Se

}{\rm Mpc}$ runs at each targeted reDshift. DiffErent LinE stYlEs anD colOrs refer to the vARyinG mass resolutions, as specIfied In THe plOT. THe rigHt panelS DiSPLay 1d fLuX poWeR SpEctruM raTios for RealizatioNs hAvIng $N^3=128^3, 256^3, 336^3$, respectiVElY, estimated WitH respect to thE hiGhest-rEsOluTIon ruN (i.e. “l10\_N512", havIng an eQUivaleNt grid resOlUTion of $20H^{-1}{\Rm kpc}$), exPREsSed iN percentage, and witH IdENtical colors anD line sTyLEs AS In tHe lEft panel. ErRoR bars ARe 1-$\sigma$ DEvIATIonS Computed from $10,000$ sImulated skeWErs RandomLy ExtRActed aT each ReDShiFt from the siMulaTion boxes, And the GRay areaS HighligHt the $1\%$ lEveL. As Can bE ClEaRly SeEN, 1D fLUx PowER spEctra obtAiNeD from The $N^3=336^3$ REALIzatIon (I.e. $30h^{-1}{\rM kpc}$ eQuivalent grid ResOlutIOn) aRe witHin $\siM 1\%$ in tHe $K$-rangE coverEd by ebOsS (up to $k=2 \times 10^{-2} [{\rm kM/s}]^{-1}$), anD closer to The SaMe dEgRee of ACcuracY foR thE expectEd extenSIon Of desi LY$\alpha$ forest data. ClEaRLY, aChieving Sub-perCEnTaGE convergEnCe aT higHER-$z$ is pRogrESsIvely chaLlengiNG, bUt A grid of $30H^{-1}{\rM kpc}$ is SuFfiCieNt for A SatiSfying ConvergeNce evEN at those redshiFTs. The previous RAtIONaLE is pRecIsely what haS driVEn thE oveRAlL arCHitecTure oF tHE [*SEJong Grid Suite*]{}, designEd To achiEve an Equivalent resOlution up tO $3 \TIMes 3328^3$ = 110 billiOn paRTiCLes in a $(100h^{-1}{\rm Mpc})$ boX, corrEsponding tO A $30 h^{-1}{\rm kpc}$ mEan grId resoluTion that eNSUres a conVerGenCe oN Ly$\ALPhA$ flux statistiCS ClosEr To the deSirEd $\sim 1.0\%$ leVel ThaT thE fiNaL DESI data Will provIdE. CLeArLy, rUnninG A very larGe NumBeR of SimulATions wIth $2 \tiMes 3328^3$ oR $3 \tImES 3328^3$ elEments oVEr A $(100H^{-1}{\Rm MpC})$ bOx Is stIll CoMputaTionALly ChallenGing, partiCulARly wHeN nEutrinoS are included aS pArticles. ThE gLobAl compUTAtional cOst in performing an entire SUite at sUch ResolUtioN would easIly RequirE $\siM 100$ MillioN CPU hoUrs. ThErEfoRE, As in oUR PrEviOuS release, we ADOpt A spliCiNg teChnique. in this respect, the thREe sImulations per FixEd paRAMeTer SEt OF thE [*SE

}{\rm Mpc}$ runs at each t argeted re dshif t.Dif fe rent lin e styles and c o lors refer to the varyingmassre s olut i on s, as specif i ed i n t he p lot .T he righ t p anels d isplay 1Dflu xpower spectr u mratios for re alizations h avi ng $N^ 3= 128 ^ 3, 25 6^3 , 336 ^3$, r e specti vely, est im a ted wi t h respe c t t o th e highest-resolut i on run (i.e. “L10 \_N512 ", ha v i nganequivalent g rid r e solutio n o f $ 20h ^ {-1}{\rm kpc} $), express e d i n perc en tag e , andwithid e nti cal colorsandline styl es asi n the l e ft pane l. Err orbar s ar e 1 -$ \si gm a $ d e vi ati o nscomputed f ro m $10 ,000 $ s i mula ted ske wersrandomly extr act ed a t ea ch re dshif t fr om thesimula tionbo xes, and the gr ay a reas high lig ht th e$1\%$ level. As ca n be cl early s e en, 1 D f lu x power spectra ob ta i n ed from th e $N^3 = 33 6^ 3 $ realiz at ion (i. e . $30h ^{-1 } {\ rm kpc}$ equiv a le nt grid r es olutio n) ar e w ithin $\si m 1\%$ in the$k$-r a nge covered by eBOSS (up to$ k= 2 \t i mes10^ {-2} [{\rmkm/s } ]^{- 1}$) , a ndc loser to t he sa m e degree of accurac yfor th e exp ected extensi on of DESI L y $\alpha$ for e st data. Clearly, achi eving sub- p ercentag e con vergence at highe r - $z$ is p rog res siv ely c ha llenging, but a gri dof $30h ^{- 1}{\rmkpc }$issuf fi cient for a satis fy in gco nve rgenc e even at t hos ered shift s . The prev ious r at i ona le is p r ec i s elywh at has dr iv en th e ov e ral l archi tecture o f t h e [* Se jo ng Grid Suite*]{}, d es igned to a ch iev e an e q u ivalentresolution up to $3 \ti m es 3328 ^3$ = 11 0 bi llion par tic les in a$ (100h^ {-1}{\ rm Mp c} )$b o x, co r r es pon di ng to a $3 0 h^{ -1}{\ rm kpc }$ mean grid resolution t h atensures a con ver genc e on Ly $ \a l pha $f lux s tatistics close r to the d es i re d $\sim 1. 0 \%$ l evel th at thefinal DESI da ta will p rovide. Cl earl y , ru nning a ve ry large number o f simu l at ionswit h $2 \ ti mes 3328 ^3$ or $3\time s 3328 ^3 $ elem entsov er a $(1 00h^{-1}{\rm Mpc})$ box is st ill c omp utational lyc hal lenging,part icularly w hen ne utrin osa re in clud e dasp artic les. The globa l c omp u t at ional costi n per formi nga n enti re s uite at such reso l ution would ea sily r equ ire $\si m100$ million C PUho u r s. There fo re, as in o ur previ ou s rele ase, w e adop t a spl i c in g techn ique . I n this re spe ct , the th re es imulat ions p er fix ed par a mete r set of the [*Se

}{\rm Mpc}$_runs at_each targeted redshift. Different_line styles_and_colors refer_to_the varying mass_resolutions, as specified_in the plot. The_right panels display_1D_flux power spectrum ratios for realizations having $N^3=128^3, 256^3, 336^3$, respectively, estimated with respect_to_the highest-resolution_run_(i.e._“L10\_N512", having an equivalent grid_resolution of $20h^{-1}{\rm kpc}$), expressed_in percentage,_and with identical colors and line styles as_in_the left panel._Error bars are 1-$\sigma$ deviations computed from $10,000$ simulated_skewers randomly extracted at each redshift_from the simulation_boxes,_and_the gray areas highlight_the $1\%$ level. As can be_clearly seen, 1D flux power spectra_obtained from the $N^3=336^3$ realization (i.e. $30h^{-1}{\rm_kpc}$ equivalent grid resolution) are within_$\sim 1\%$ in the $k$-range_covered by_eBOSS (up to $k=2 \times_10^{-2} [{\rm km/s}]^{-1}$),_and closer_to the same_degree of accuracy for the expected_extension of DESI_Ly$\alpha$ forest data. Clearly, achieving sub-percentage_convergence_at higher-$z$ is_progressively_challenging,_but a_grid of $30h^{-1}{\rm_kpc}$_is sufficient_for_a satisfying convergence even at those_redshifts. The_previous rationale is precisely what has driven_the overall architecture of_the_[*Sejong Grid Suite*]{}, designed_to achieve an equivalent resolution_up to $3 \times 3328^3$ =_110 billion_particles in_a $(100h^{-1}{\rm Mpc})$ box, corresponding to a $30 h^{-1}{\rm kpc}$ mean_grid resolution that ensures a convergence_on Ly$\alpha$ flux statistics_closer to_the_desired $\sim 1.0\%$_level_that the_final DESI data will provide. Clearly, running a_very large_number of simulations with $2 \times_3328^3$ or $3 \times_3328^3$_elements over a $(100h^{-1}{\rm Mpc})$ box_is still computationally challenging, particularly when_neutrinos are included as particles._The_global_computational cost in performing an_entire suite at such resolution would_easily require $\sim_100$ million CPU hours. Therefore, as in_our_previous release, we adopt a splicing_technique._In this respect, the three simulations_per_fixed_parameter set of the [*Se

1953 6.51 WDM$^{\dagger}$\_UN\_0.25\_keV a 0.0 0.25 3.0460 0.8305 25 128 0.1953 6.51 WDM$^{\dagger}$\_0.25\_keV a 0.0 0.25 3.0460 0.8150 25 128 0.1953 6.51 WDM\_UN\_0.25\_keV a 0.0 0.25 3.0617 0.8269 25 128 0.1953 6.51 WDM\_0.25\_keV a 0.0 0.25 3.0617 0.8150 25 128 0.1953 6.51 WDM$^{\dagger}$\_1.00\_keV a/b/c 0.0 1.00 3.0460 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 WDM$^{\dagger}$\_2.00\_keV a/b 0.0 2.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\dagger}$\_3.00\_keV a/b 0.0 3.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\dagger}$\_3.00\_keV Grid-Like a/b/c 0.0 3.00 3.0460 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 WDM$^{\dagger}$\_4.00\_keV a/b 0.0 4.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM

1953 6.51 WDM$^{\dagger}$\_UN\_0.25\_keV a 0.0 0.25 3.0460 0.8305 25 128 0.1953 6.51 WDM$^{\dagger}$\_0.25\_keV a 0.0 0.25 3.0460 0.8150 25 128 0.1953 6.51 WDM\_UN\_0.25\_keV a 0.0 0.25 3.0617 0.8269 25 128 0.1953 6.51 WDM\_0.25\_keV a 0.0 0.25 3.0617 0.8150 25 128 0.1953 6.51 WDM$^{\dagger}$\_1.00\_keV a / b / c 0.0 1.00 3.0460 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 WDM$^{\dagger}$\_2.00\_keV a / b 0.0 2.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\dagger}$\_3.00\_keV a / b 0.0 3.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\dagger}$\_3.00\_keV Grid - Like a / boron / coulomb 0.0 3.00 3.0460 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 WDM$^{\dagger}$\_4.00\_keV a / b 0.0 4.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM

1953 6.51 DDM$^{\dagger}$\_UN\_0.25\_keV c 0.0 0.25 3.0460 0.8305 25 128 0.1953 6.51 WDK$^{\dagger}$\_0.25\_keV a 0.0 0.25 3.0460 0.8150 25 128 0.1953 6.51 WDM\_UN\_0.25\_keV a 0.0 0.25 3.0617 0.8269 25 128 0.1953 6.51 WDM\_0.25\_keY a 0.0 0.25 3.0617 0.8150 25 128 0.1953 6.51 WDM$^{\daggwr}$\_1.00\_keV a/b/c 0.0 1.00 3.0460 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 WDM$^{\dagger}$\_2.00\_keV a/b 0.0 2.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\dagger}$\_3.00\_keV c/b 0.0 3.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\daygee}$\_3.00\_keV Grid-Like a/b/c 0.0 3.00 3.0460 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 WDM$^{\dagger}$\_4.00\_keV a/b 0.0 4.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDK

1953 6.51 WDM$^{\dagger}$\_UN\_0.25\_keV a 0.0 0.25 3.0460 128 6.51 WDM$^{\dagger}$\_0.25\_keV 0.0 0.25 3.0460 WDM\_UN\_0.25\_keV 0.0 0.25 3.0617 25 128 0.1953 WDM\_0.25\_keV a 0.0 0.25 3.0617 0.8150 128 0.1953 6.51 WDM$^{\dagger}$\_1.00\_keV a/b/c 0.0 1.00 3.0460 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 a/b 0.0 2.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\dagger}$\_3.00\_keV a/b 0.0 3.00 0.8150 256/512 3.25/6.51 Grid-Like a/b/c 0.0 3.00 3.0460 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 WDM$^{\dagger}$\_4.00\_keV a/b 0.0 4.00 3.0460 0.8150 256/512 0.0976/0.1953 3.25/6.51 WDM

1953 6.51 WDM$^{\dagger}$\_UN\_0.25\_keV a 0.0 0.25 3.0460 0.8305 25 128 0.1953 6.51 WDM$^{\dagger}$\_0.25\_kEV a 0.0 0.25 3.0460 0.8150 25 128 0.1953 6.51 WDM\_UN\_0.25\_kev a 0.0 0.25 3.0617 0.8269 25 128 0.1953 6.51 WDM\_0.25\_KeV A 0.0 0.25 3.0617 0.8150 25 128 0.1953 6.51 WDm$^{\dAggeR}$\_1.00\_keV A/b/c 0.0 1.00 3.0460 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 WDM$^{\dagger}$\_2.00\_kev A/b 0.0 2.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDm$^{\dagger}$\_3.00\_keV a/b 0.0 3.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\dagger}$\_3.00\_kEV GriD-LIKe a/b/C 0.0 3.00 3.0460 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 wDm$^{\daggEr}$\_4.00\_keV a/b 0.0 4.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 wdM

1953 6.5 1 WDM$^ {\d agg er }$\_ UN\_ 0.25\_keV a 0.0 0.2 5 3. 0 4 60 0. 8305 25 128 0.1953 6 .51 WDM$^ { \dagge r}$\_0.25 \_ k eV a 0.0 0.25 3 . 04 6 0 0.815 0 2 5 128 0.1 95 3 6 .51 WDM\_U N\_0.2 5 \_keV a 0. 0 0.2 5 3 .061 7 0.8 269 2 5 128 0.1 95 3 6.51 WD M \_0.25 \_k eVa 0.0 0 .25 3 . 0617 0. 8 1 50 25 128 0. 1 953 6 . 51 WDM $ ^{\dagger}$\_ 1 .0 0 \ _k e V a/ b/c 0.0 1. 00 3.0460 0.81 50 2 5/100/100 256 /512 / 83 2 0.0976 /0.19 53/0.1202 3.2 5/6.5 1/4.01 W D M $^{\dagg er} $\_ 2.0 0\_ k e Va/b 0 .0 2 .00 3.0460 0 .8150 25 /1 00 256/5 12 0.09 7 6/ 0 . 1953 3.25 /6.5 1 WDM$^{\da gge r }$\_ 3. 00 \_keV a /b 0.0 3.00 3.0460 0. 8150 25 /10 0 25 6/512 0.097 6 / 0. 195 3 3. 25/6. 51 WDM$^{\ dagger}$\_3.00\_ke V Gr id-Like a/b/c 0.0 3.00 3 .0 4 60 0.8 15 0 25 /25/1 0 0 208/83 2/832 0. 1 2 02/ 0.0300/0.1 202 4.01/1. 0 0/4.0 1 WDM$^{ \d agg er}$\ _4.00\ _ keV a/b 0.0 4.00 3.046 0 0.8 1 50 25 /100 256/ 512 0.0 9 76/0. 1953 3. 2 5 /6 .51 WDM

1953 _ _ _ __ __ _ 6.51 _ _ _WDM$^{\dagger}$\_UN\_0.25\_keV_a __ 0.0___ _ _ _ 0.25 __ _ 3.0460_ _ __0.8305_ _ _ 25 _ _ _ 128 _ _ _ _ _ _0.1953 _ _ __ _6.51 __ _ __ _WDM$^{\dagger}$\_0.25\_keV_a __ _ _0.0_ _ _ _ _ _0.25 _ 3.0460 _ _ __ 0.8150__ _ _ _ 25 _ __ _ 128 _ ___ _ 0.1953 _ _ __ 6.51 __ ___ WDM\_UN\_0.25\_keV a_ _ _ _ 0.0 ___ _ _ _ 0.25 _ 3.0617_ 0.8269_ _ 25 _ __ ____128 _ _ _ _ _ _ _0.1953_ _6.51 _ __ _ WDM\_0.25\_keV a _ _ __ _ 0.0 _ _ _ _ _ _ _ 0.25 __ ____3.0617 _ _ _ 0.8150__ _ _ 25_ _ 128 _ _ _ _0.1953_ _ _ __ 6.51 WDM$^{\dagger}$\_1.00\_keV a/b/c ___ __ ___0.0__ _ __ 1.00_ __ _ _ 3.0460 __ _0.8150___ _ 25/100/100 256/512/832 __ 0.0976/0.1953/0.1202_ _3.25/6.51/4.01 _ _ WDM$^{\dagger}$\_2.00\_keV a/b _ _ _ 0.0 _ 2.00_ _ __ _ 3.0460 _ _ 0.8150 25/100 _ _ 256/512 _ _ _0.0976/0.1953 _ 3.25/6.51 WDM$^{\dagger}$\_3.00\_keV a/b__ _ _ 0.0_ __ _ _ _ _ 3.00 _ _ 3.0460 _ 0.8150 25/100 _ __ 256/512 _ _ 0.0976/0.1953 _ __3.25/6.51 _ WDM$^{\dagger}$\_3.00\_keV Grid-Like_a/b/c_ _ _ 0.0_ _ _ _ _ 3.00 __ _ __3.0460 _ __ _ _ 0.8150__ _ __ _ _25/25/100_ _ 208/832/832 _ _0.1202/0.0300/0.1202 __4.01/1.00/4.01 _ WDM$^{\dagger}$\_4.00\_keV a/b_ _ _ _ _0.0 _ __ _ 4.00 _ __ 3.0460_ _ _ 0.8150 _ _ _ 25/100___ _ 256/512 _ _ __ 0.0976/0.1953_ _ _ __ 3.25/6.51 __ _ _WDM

to integrate with all detectors for fair comparison of the *detector* matching performances. So we overlook the description performance. Method ====== This section defines ELF, a detection method valid for any trained CNN. Keypoints are local maxima of a saliency map computed as the feature gradient *w.r.t* the image. We use the data adaptive Kapur method [@kapur1985new] to automatically threshold the saliency map and keep only the most salient locations, then run NMS for local maxima detection. ![(Bigger version Figure \[fig:big\_saliency\_coco\].) Saliency maps computed from the feature map gradient $\left| ^TF^l(x) \cdot \frac{\partial F^l}{\partial \mathbf{I}} \right|$. Enhanced image contrast for better visualisation. Top row: gradients of VGG $pool_2$ and $pool_3$ show a loss of resolution from $pool_2$ to $pool_3$. Bottom: $(pool_i)_{i \in [1,2,5]}$ of VGG on Webcam, HPatches and Coco images. Low level saliency maps activate accurately whereas higher saliency maps are blurred.[]{data-label="fig:saliency_coco"}](fig2_saliency_bis.png){width="\linewidth"} Feature Specific Saliency ------------------------- We generate a saliency map that activates on the most informative image region for a specific CNN feature level $l$. Let $\mathbf{I}$ be a vector image of dimension $D_I = H_I \cdot W_I \cdot C_I$. Let $F^l$ be a vectorized feature map of dimension $D_F= H_l \cdot W_l \cdot C_l$. The saliency map $S^l$, of dimension $D_I$, is $S^l(\mathbf{I})=\left| ^tF^l(\mathbf{I}) \cdot \nabla_I F^l \right|$, with $\nabla_I F^l$ a $D_F \times D_I$ matrix. The saliency activates on the image regions that contribute the most to the feature representation $F^l(\mathbf{I})$. The term $\nabla_I F^l$ explicits the correlation between the feature space of $F^l$ and the image space in general.

to integrate with all detectors for fair comparison of the * detector * matching performance. therefore we overlook the description performance. Method = = = = = = This section define ELF, a detection method valid for any educate CNN. Keypoints are local maxima of a saliency function calculate as the feature gradient * w.r.t * the image. We practice the data adaptive Kapur method [ @kapur1985new ] to mechanically threshold the saliency map and keep only the most salient locations, then run NMS for local maximum detection. ! [ (Bigger version number \[fig: big\_saliency\_coco\ ] .) Saliency maps computed from the feature function gradient $ \left| ^TF^l(x) \cdot \frac{\partial F^l}{\partial \mathbf{I } } \right|$. Enhanced trope contrast for better visualisation. Top rowing: gradients of VGG $ pool_2 $ and $ pool_3 $ show a loss of resolution from $ pool_2 $ to $ pool_3$. Bottom: $ (pool_i)_{i \in [ 1,2,5]}$ of VGG on Webcam, HPatches and Coco images. Low level saliency maps activate accurately whereas higher saliency maps are blurred.[]{data - label="fig: saliency_coco"}](fig2_saliency_bis.png){width="\linewidth " } Feature Specific Saliency ------------------------- We generate a saliency map that trip on the most informative prototype area for a specific CNN feature level $ l$. Let $ \mathbf{I}$ be a vector image of dimension $ D_I = H_I \cdot W_I \cdot C_I$. Let $ F^l$ be a vectorized feature map of property $ D_F= H_l \cdot W_l \cdot C_l$. The saliency map $ S^l$, of dimension $ D_I$, is $ S^l(\mathbf{I})=\left| ^tF^l(\mathbf{I }) \cdot \nabla_I F^l \right|$, with $ \nabla_I F^l$ a $ D_F \times D_I$ matrix. The saliency activates on the image region that contribute the most to the feature representation $ F^l(\mathbf{I})$. The term $ \nabla_I F^l$ explicits the correlation between the feature outer space of $ F^l$ and the image space in general.

to integrate with all detegtors for fair comparismn of fhe *deteztor* matching performances. Sl qe ovtglook the description performajce. Methoe ====== Thiw section dxrines ELF, a dsbectimi method valid nor any trahned CNN. Keypohngs are local maxima of a saliency map computrd as the featurg gracyent *w.r.t* the image. We use the data adaltive Kepur method [@kapir1985new] to automatically thrfshopd the saliency mao and keep inly rhe most salkent locatpmns, then ron NMS for local maxima detectiov. ![(Biggzr version Diturf \[fig:big\_salixncy\_cobo\].) Saliency maps compgted frpm the feature ma' grqdient $\left| ^TF^l(x) \cdot \frac{\partial F^l}{\parjial \mathbx{I}} \right|$. Enhanced inate cottravt fue bdtttr tishalisahioi. Top row: gdadients of VGG $pool_2$ and $pool_3$ xhjq a loss of rssolutyog from $pool_2$ to $pool_3$. Bottom: $(pool_i)_{i \in [1,2,5]}$ ox VFG on Webcam, HPatches abd Coco images. Low legel saliegcy maps activate accurately whereas higher salietcy mepr axc cougred.[]{data-label="fig:saliency_coco"}](fig2_saliency_bis.png){rjduh="\lpnewidth"} Feature Siecific Saliency ------------------------- We gfnrtate a salienci map tkzt activates on the lost insormarive imagt regoon for a specific CNN featyre level $l$. Jwt $\mathbf{I}$ be a veetor image oy dimemsion $D_I = H_I \cdot W_I \cdot C_I$. Let $F^l$ be a veftorized rdature map of dioenximn $D_F= H_l \cdot W_l \cdot C_l$. Tre salienry ma' $S^l$, of aimemsion $Q_I$, is $S^l(\mahhbf{I})=\left| ^tF^l(\mathbf{I}) \cdoh \nabpa_H F^l \right|$, with $\nabla_I F^l$ a $D_F \times D_I$ mevrix. The saliemcf abtivates jn thc image regions that contribuje the moft to the featuge represxntation $F^l(\mwthbf{I})$. The tesl $\nabla_I F^l$ xxplicits the coreelatiov between the frature spcee of $F^l$ qnd the image spacc in esneral.

to integrate with all detectors for fair the matching performances. we overlook the section ELF, a detection valid for any CNN. Keypoints are local maxima of saliency map computed as the feature gradient *w.r.t* the image. We use the adaptive Kapur method [@kapur1985new] to automatically threshold the saliency map and keep only most locations, run for local maxima detection. ![(Bigger version Figure \[fig:big\_saliency\_coco\].) Saliency maps computed from the feature map gradient ^TF^l(x) \cdot \frac{\partial F^l}{\partial \mathbf{I}} \right|$. Enhanced image for better visualisation. Top gradients of VGG $pool_2$ and show loss of from to Bottom: $(pool_i)_{i \in of VGG on Webcam, HPatches and Coco images. Low level saliency maps activate accurately whereas higher saliency are blurred.[]{data-label="fig:saliency_coco"}](fig2_saliency_bis.png){width="\linewidth"} Saliency ------------------------- generate saliency that activates on informative image region for a specific $l$. Let $\mathbf{I}$ be a vector image of $D_I = \cdot W_I \cdot C_I$. Let $F^l$ a vectorized feature map of dimension $D_F= H_l W_l \cdot C_l$. The saliency map $S^l$, of dimension $D_I$, is $S^l(\mathbf{I})=\left| ^tF^l(\mathbf{I}) \cdot \nabla_I with $\nabla_I F^l$ a \times D_I$ matrix. saliency on image that contribute most to the feature representation $F^l(\mathbf{I})$. The term $\nabla_I F^l$ explicits correlation between the feature space of $F^l$ and the image general.

to integrate with all detectoRs for fair cOmparIsoN of ThE *detEctoR* matching perfoRMancEs. So we overlook the descrIptioN pERforMAnCe. MetHod ====== This SEcTIOn dEfInEs ElF, A DeTectiOn mEthod vaLid for any tRaiNeD CNN. KeypointS ArE local maxiMa oF a saliency maP coMputed As The FEaturE grAdienT *w.r.t* thE Image. WE use the daTa ADaptivE kapur meTHOd [@KapuR1985new] to automaticalLY tHReshold the saliEncy maP aND kEEP onLy tHe most saliEnT locaTIons, theN RuN nms foR Local maxima deTection. ![(BiggER veRsion FIgUre \[FIg:big\_sAlienCy\_COco\].) saliency mapS comPuted from The feaTUre map gRAdient $\lEft| ^TF^l(X) \cdOt \fRac{\pARtIaL F^l}{\PaRTiaL \MaThbF{i}} \riGht|$. EnhanCeD iMage cOntrAST FOr beTteR visUalisAtion. Top row: grAdiEnts OF VGg $pool_2$ And $poOl_3$ shOw A loss Of resoLutioN fRom $pool_2$ to $pool_3$. BoTtom: $(Pool_i)_{i \in [1,2,5]}$ oF VGg oN WeBcAm, HPaTChes anD CoCo iMages. LoW level sALieNcY MAPs Activate accurately WhEREaS higher sAliencY MaPs ARe blurreD.[]{dAta-LabeL="FIg:salIencY_CoCo"}](fig2_salIency_bIS.pNg){Width="\liNeWidth"} FEaTurE SpEcifiC saliEncy ------------------------- We Generate A saliENcy map that actiVAtes on the most INfORMaTIve iMagE region for a SpecIFic CnN feATuRe lEVel $l$. LEt $\matHbF{i}$ bE A vector image of dimenSiOn $D_I = H_I \Cdot W_i \cdot C_I$. Let $F^l$ bE a vectorizED FEature maP of dIMeNSion $D_F= H_l \cdot W_l \Cdot C_L$. The salienCY map $S^l$, of DimenSion $D_I$, is $s^l(\mathbf{I})=\LEFt| ^tF^l(\matHbf{i}) \cdOt \nAblA_i f^l \Right|$, with $\nablA_i f^l$ a $D_f \tImes D_I$ mAtrIx. The saLieNcy ActIvaTeS on the imaGe regionS tHaT cOnTriBute tHE most to tHe FeaTuRe rEpresENtatioN $F^l(\maThbf{i})$. THe TErm $\Nabla_I F^L$ ExPLIcitS tHe CorrElaTiOn betWeen THe fEature sPace of $F^l$ aNd tHE imaGe SpAce in geNeral.

to integrate with all det ectors for fair co mpa ri sonof t he *detector*m atch ing performances. So w e ove rl o ok t h edescr iptionp er f o rma nc e. M et h od ==== == This s ection def ine sELF, a detec t io n method v ali d for any tr ain ed CNN .Key p oints ar e loc al max i ma ofa salienc ym ap com p uted as t he fea ture gradient *w. r .t * the image. We use t he da t a ad apt ive Kapurme thod[ @kapur1 9 85 n e w ] t o automaticall y threshold the salie nc y m a p andkeepon l y t he most sal ient location s, the n run NM S for lo cal ma xim a d etec t io n. ! [( B igg e rver s ion Figure\[ fi g:big \_sa l i e n cy\_ coc o\]. ) Sal iency maps co mpu tedf rom thefeatu re m ap grad ient $ \left |^TF^l(x) \cdot\fra c{\partia l F ^l }{\ pa rtial \mathb f{I }}\right| $. Enha n ced i m a g econtrast for bette rv i su alisatio n. Top ro w: gradient sofVGG$ p ool_2 $ an d $ pool_3$show a lo ss of res ol utionfr om$po ol_2$ to $ pool_3 $. Botto m: $( p ool_i)_{i \in[ 1,2,5]}$ of V G Go n W e bcam , H Patches and Coc o ima ges. Lo w l e vel s alien cy ma p s activate accurate ly where as hi gher saliency maps areb l u rred.[]{ data - la b el="fig:salien cy_co co"}](fig2 _ saliency _bis. png){wid th="\line w i dth"} F eat ure Sp eci f i cSaliency ---- - - ---- -- ------- --- --- We ge ner ate asa liency ma p that a ct iv at es on them ost info rm ati ve im age r e gion f or aspec if ic CNN featur e l e v el $ l$ .Let$\m at hbf{I }$ b e avectorimage ofdim e nsio n$D _I = H_ I \cdot W_I \ cd ot C_I$. L et $F ^l$ be a vectori zed feature map of dime n sion $D _F= H_l\cdo t W_l \cd otC_l$.The salien cy map $S^l $, of d imens i o n$D_ I$ , is $S^l( \ m ath bf{I} )= \lef t| ^tF^ l(\mathbf{I}) \cdo t \n abla_I F^l \r igh t|$, w it h $ \ na b la_ IF ^l$ a $D_F \times D_ I$ matrix. T he saliencya cti va tes onthe ima ge re g ions th at contri bute themo st t o the feature r epresent ation $F^ l (\mat h bf {I})$ . T he ter m$\n abla_ I F^l$ exp licit s theco rrelat ion b et ween the feature space of $F^l$ and t he im age space in ge n era l.

to_integrate with_all detectors for fair_comparison of_the_*detector* matching_performances._So we overlook_the description performance. Method ====== This_section defines ELF, a_detection method valid_for_any trained CNN. Keypoints are local maxima of a saliency map computed as the_feature_gradient *w.r.t*_the_image._We use the data adaptive_Kapur method [@kapur1985new] to automatically_threshold the_saliency map and keep only the most salient_locations,_then run NMS_for local maxima detection. ![(Bigger version Figure \[fig:big\_saliency\_coco\].) Saliency maps_computed from the feature map gradient_$\left| ^TF^l(x) \cdot_\frac{\partial_F^l}{\partial_\mathbf{I}} \right|$. Enhanced image_contrast for better visualisation. Top row:_gradients of VGG $pool_2$ and $pool_3$_show a loss of resolution from $pool_2$_to $pool_3$. Bottom: $(pool_i)_{i \in [1,2,5]}$_of VGG on Webcam, HPatches_and Coco_images. Low level saliency maps_activate accurately whereas_higher saliency_maps are blurred.[]{data-label="fig:saliency_coco"}](fig2_saliency_bis.png){width="\linewidth"} Feature_Specific Saliency ------------------------- We generate a saliency map_that activates on_the most informative image region for_a_specific CNN feature_level_$l$._Let $\mathbf{I}$_be a vector_image_of dimension_$D_I_= H_I \cdot W_I \cdot C_I$._Let_$F^l$ be a vectorized feature map of_dimension $D_F= H_l \cdot_W_l_\cdot C_l$. The saliency_map $S^l$, of dimension $D_I$,_is $S^l(\mathbf{I})=\left| ^tF^l(\mathbf{I}) \cdot \nabla_I F^l_\right|$, with_$\nabla_I F^l$_a $D_F \times D_I$ matrix. The saliency activates on the image regions_that contribute the most to the_feature representation $F^l(\mathbf{I})$. The_term $\nabla_I_F^l$_explicits the correlation_between_the feature_space of $F^l$ and the image space_in general.

operatorname{Cen}_{G}(\varepsilon^{(1)}(H,K))$ on $\{\varepsilon_{\mathcal{C}}^{(1)}(H,K)~|~ \mathcal{C} \in R_{1}\}$. Using Wedderga [@wedd], we can see that $\operatorname{Cen}_{G}(\varepsilon^{(1)}(H,K))=G$. Suppose $q\equiv 1 (\operatorname{mod}5)$. The relations between group elements yield that $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)^{x_{2}} = \varepsilon_{\mathcal{C}_{\sigma^4}}^{(1)}(H,K)$. Hence $x_2$ does not centralize $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$. Since $G/H_1$ is cyclic of order $4$ generated by $H_1 x_1$ and $x_1^2 = x_2$, it follows that $\operatorname{Cen}_{G}(\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H,K))=H_{1}$ and so the orbit of $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$ has $[G:H_1] =4$ elements, namely $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$, $\varepsilon_{\mathcal{C}_{\sigma^2}}^{(1)}(H,K)$, $\varepsilon_{\mathcal{C}_{\sigma^3}}^{(1)}(H,K)$ and $\varepsilon_{\mathcal{C}_{\sigma^4}}^{(1)}(H,K)$. Thus $R_{(H, K)} = \{C_\sigma\}.$ Hence, if $q\equiv 1 (\operatorname{mod}5)$, there is only one simple component of $\mathbb{F}_{q}G$, which by Theorem \[t1\] is isomorphic to $M_{20}(\mathbb{F}_{q})$. Next, suppose $q\equiv 2 {\rm~or~}3 (\operatorname{mod}5)$. In this case, $\mathcal{C}_{q}(H/K)$ has only one $q$-cyclotomic class namely $C_\sigma = \{\sigma, \sigma^{2}, \sigma^{3}, \sigma^{4}\}$. So $R_{(H,K)}$ equals $ \mathcal{C}_{q}(H/K)$ and in this case also there is one simple component of $\mathbb{F}_{q}G$ isomorphic to $M_{20}(\

operatorname{Cen}_{G}(\varepsilon^{(1)}(H, K))$ on $ \{\varepsilon_{\mathcal{C}}^{(1)}(H, K)~|~ \mathcal{C } \in R_{1}\}$. Using Wedderga [ @wedd ], we can see that $ \operatorname{Cen}_{G}(\varepsilon^{(1)}(H, K))=G$. Suppose $ q\equiv 1 (\operatorname{mod}5)$. The relations between group chemical element concede that $ \varepsilon_{\mathcal{C}_\sigma}^{(1)}(H, K)^{x_{2 } } = \varepsilon_{\mathcal{C}_{\sigma^4}}^{(1)}(H, K)$. Hence $ x_2 $ does not centralize $ \varepsilon_{\mathcal{C}_\sigma}^{(1)}(H, K)$. Since $ G / H_1 $ is cyclic of order $ 4 $ generated by $ H_1 x_1 $ and $ x_1 ^ 2 = x_2 $, it postdate that $ \operatorname{Cen}_{G}(\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H, K))=H_{1}$ and so the orbit of $ \varepsilon_{\mathcal{C}_\sigma}^{(1)}(H, K)$ have $ [ G: H_1 ] = 4 $ elements, namely $ \varepsilon_{\mathcal{C}_\sigma}^{(1)}(H, K)$, $ \varepsilon_{\mathcal{C}_{\sigma^2}}^{(1)}(H, K)$, $ \varepsilon_{\mathcal{C}_{\sigma^3}}^{(1)}(H, K)$ and $ \varepsilon_{\mathcal{C}_{\sigma^4}}^{(1)}(H, K)$. Thus $ R_{(H, K) } = \{C_\sigma\}.$ Hence, if $ q\equiv 1 (\operatorname{mod}5)$, there is entirely one simple component of $ \mathbb{F}_{q}G$, which by Theorem \[t1\ ] is isomorphous to $ M_{20}(\mathbb{F}_{q})$. Next, suppose $ q\equiv 2 { \rm ~ or~}3 (\operatorname{mod}5)$. In this case, $ \mathcal{C}_{q}(H / K)$ has only one $ q$-cyclotomic class namely $ C_\sigma = \{\sigma, \sigma^{2 }, \sigma^{3 }, \sigma^{4}\}$. So $ R_{(H, K)}$ equals $ \mathcal{C}_{q}(H / K)$ and in this event also there is one simple part of $ \mathbb{F}_{q}G$ isomorphic to $ M_{20}(\

opegatorname{Cen}_{G}(\varepsilon^{(1)}(H,Y))$ on $\{\varepsilon_{\mathcal{R}}^{(1)}(H,K)~|~ \matgcal{C} \in R_{1}\}$. Using Wedderga [@wedd], we cai sew thau $\operatorname{Cen}_{G}(\vxrepsilon^{(1)}(J,K))=G$. Suppise $w\wquiv 1 (\opecztornamc{iod}5)$. Fme renetions between nroup elemetts yield that $\vxrzpsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)^{x_{2}} = \varepsilon_{\iathcal{V}_{\slgma^4}}^{(1)}(H,K)$. Hence $x_2$ doex not bektralize $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(G,K)$. Sinct $G/H_1$ is cyclic of prder $4$ generated by $H_1 x_1$ anf $x_1^2 = x_2$, it follows that $\operatornane{Ceg}_{T}(\varepsilon_{\mxthcal{C_\sigma}}^{(1)}(H,K))=H_{1}$ and so the orbit of $\varepsilon_{\mathcal{Z}_\sigmc}^{(1)}(H,K)$ has $[G:H_1] =4$ eoemftts, namely $\tarepsplon_{\mathcal{C}_\slbma}^{(1)}(H,K)$, $\earepsikon_{\mathcal{C}_{\sigka^2}}^{(1)}(H,N)$, $\vqrepsilon_{\mathcal{C}_{\sigme^3}}^{(1)}(H,K)$ and $\varepsilon_{\majhcal{C}_{\sigmd^4}}^{(1)}(H,I)$. Thus $R_{(H, K)} = \{C_\signa\}.$ Hencg, if $x\equkc 1 (\upedavorhame{mof}5)$, tiere is onlg one simplw component of $\mathnb{S}_{w}G$, which by Tgeorem \[t1\] is isomorphic to $M_{20}(\mathbb{F}_{q})$. Next, suppost $q\eqhiv 2 {\rm~or~}3 (\operatorname{mid}5)$. In this case, $\mathcwl{C}_{q}(H/K)$ haf only one $q$-cyclotomic class namely $C_\sigma = \{\sigmd, \sigjx^{2}, \snnix^{3}, \slgma^{4}\}$. So $R_{(H,K)}$ equals $ \mathcal{C}_{q}(H/K)$ and in this cwae akso there is oke simple componeny lf $\iathbb{F}_{q}G$ isooorphie tk $M_{20}(\

operatorname{Cen}_{G}(\varepsilon^{(1)}(H,K))$ on $\{\varepsilon_{\mathcal{C}}^{(1)}(H,K)~|~ \mathcal{C} \in R_{1}\}$. Using we see that Suppose $q\equiv 1 elements that $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)^{x_{2}} = Hence $x_2$ does centralize $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$. Since $G/H_1$ is cyclic order $4$ generated by $H_1 x_1$ and $x_1^2 = x_2$, it follows that and so the orbit of $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$ has $[G:H_1] =4$ elements, namely $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$, $\varepsilon_{\mathcal{C}_{\sigma^2}}^{(1)}(H,K)$, and Thus K)} \{C_\sigma\}.$ Hence, if $q\equiv 1 (\operatorname{mod}5)$, there is only one simple component of $\mathbb{F}_{q}G$, which by \[t1\] is isomorphic to $M_{20}(\mathbb{F}_{q})$. Next, suppose $q\equiv {\rm~or~}3 (\operatorname{mod}5)$. In this $\mathcal{C}_{q}(H/K)$ has only one $q$-cyclotomic namely = \{\sigma, \sigma^{3}, So equals $ \mathcal{C}_{q}(H/K)$ in this case also there is one simple component of $\mathbb{F}_{q}G$ isomorphic to $M_{20}(\

operatorname{Cen}_{G}(\varepsiloN^{(1)}(H,K))$ on $\{\varepSilon_{\MatHcaL{C}}^{(1)}(h,K)~|~ \maThcaL{C} \in R_{1}\}$. Using WeddERga [@wEdd], we can see that $\operatoRname{ceN}_{g}(\varEPsIlon^{(1)}(H,k))=G$. SuppoSE $q\EQUiv 1 (\OpErAtoRnAMe{Mod}5)$. ThE reLations Between groUp eLeMents yield thAT $\vArepsilon_{\mAthCal{C}_\sigma}^{(1)}(H,K)^{x_{2}} = \VarEpsiloN_{\mAthCAl{C}_{\siGma^4}}^{(1)}(h,K)$. HenCe $x_2$ doeS Not cenTralize $\vaRePSilon_{\mAThcal{C}_\sIGMa}^{(1)}(h,K)$. SiNce $G/H_1$ is cyclic of orDEr $4$ GEnerated by $H_1 x_1$ anD $x_1^2 = x_2$, it fOlLOwS THat $\OpeRatorname{CEn}_{g}(\varePSilon_{\maTHcAL{c_\SigMA}}^{(1)}(H,K))=H_{1}$ and so the oRbit of $\varepSIloN_{\mathcAl{c}_\siGMa}^{(1)}(H,K)$ haS $[G:H_1] =4$ elEmENts, Namely $\varepSiloN_{\mathcal{C}_\Sigma}^{(1)}(H,k)$, $\VarepsiLOn_{\mathcAl{C}_{\sigMa^2}}^{(1)}(H,k)$, $\vaRepsILoN_{\mAthCaL{c}_{\siGMa^3}}^{(1)}(h,K)$ aND $\vaRepsilon_{\MaThCal{C}_{\sIgma^4}}^{(1)}(h,k)$. tHUs $R_{(H, k)} = \{C_\sIgma\}.$ hence, If $q\equiv 1 (\operaTorName{MOd}5)$, tHere iS only One sImPle coMponenT of $\maThBb{F}_{q}G$, which by TheOrem \[T1\] is isomorPhiC tO $M_{20}(\mAtHbb{F}_{q})$. nExt, supPosE $q\eQuiv 2 {\rm~oR~}3 (\operatORnaMe{MOD}5)$. in This case, $\mathcal{C}_{q}(H/k)$ hAS OnLy one $q$-cyClotomIC cLaSS namely $C_\SiGma = \{\SigmA, \SIgma^{2}, \sIgma^{3}, \SIgMa^{4}\}$. So $R_{(H,K)}$ eQuals $ \mAThCaL{C}_{q}(H/K)$ anD iN this cAsE alSo tHere iS One sImple cOmponent Of $\matHBb{F}_{q}G$ isomorphiC To $M_{20}(\

operatorname{Cen}_{G}(\var epsilon^{( 1)}(H ,K) )$on $\{ \var epsilon_{\math c al{C }}^{(1)}(H,K)~|~ \math cal{C }\ in R _ {1 }\}$. UsingW ed d e rga [ @w edd ], we cansee that $ \operatorn ame {C en}_{G}(\var e ps ilon^{(1)} (H, K))=G$. Supp ose $q\eq ui v 1 (\ope rat ornam e{mod} 5 )$. Th e relatio ns betwee n groupe l em ents yield that $\var e ps i lon_{\mathcal{ C}_\si gm a }^ { ( 1)} (H, K)^{x_{2}} = \va r epsilon _ {\ m a t hca l {C}_{\sigma^4 }}^{(1)}(H, K )$. Hence $ x_2 $ doesnot c en t ral ize $\varep silo n_{\mathc al{C}_ \ sigma}^ { (1)}(H, K)$. S inc e $ G/H_ 1 $is cy cl i c o f o rde r $4 $ genera te dby $H _1 x _ 1 $ and$x_ 1^2= x_2 $, it follows th at $ \ ope rator name{ Cen} _{ G}(\v arepsi lon_{ \m athcal{C_\sigma }}^{ (1)}(H,K) )=H _{ 1}$ a nd so the or bit of $\vare psilon_ { \ma th c a l {C }_\sigma}^{(1)}(H, K) $ ha s $[G:H_ 1] =4$ el em e nts, nam el y $ \var e p silon _{\m a th cal{C}_\ sigma} ^ {( 1) }(H,K)$ ,$\vare ps ilo n_{ \math c al{C }_{\si gma^2}}^ {(1)} ( H,K)$, $\varep s ilon_{\mathca l {C } _ {\ s igma ^3} }^{(1)}(H,K )$ a n d $\ vare p si lon _ {\mat hcal{ C} _ {\ s igma^4}}^{(1)}(H,K) $. Thus$R_{( H, K)} = \{C_ \sigma\}.$ H e nce, if$q\e q ui v 1 (\operatorn ame{m od}5)$, th e re is on ly on e simple componen t of $\mat hbb {F} _{q }G$ , wh ich by Theore m \[t1 \] is iso mor phic to $M _{2 0}( \ma th bb{F}_{q} )$. Next ,su pp os e $ q\equ i v 2 {\rm ~o r~} 3(\o perat o rname{ mod}5 )$.In t h iscase, $ \ ma t h cal{ C} _{ q}(H /K) $has o nlyo ne$q$-cyc lotomic c las s nam el y$C_\sig ma = \{\sigm a, \sigma^{2 }, \s igma^{ 3 } , \sigma ^{4}\}$. So $R_{(H,K)}$ equals$ \ mathc al{C }_{q}(H/K )$and in th i s case alsothere i s o n e simp l e c omp on ent of $\m a t hbb {F}_{ q} G$ i somorph ic to $M_{20}(\

operatorname{Cen}_{G}(\varepsilon^{(1)}(H,K))$ on_$\{\varepsilon_{\mathcal{C}}^{(1)}(H,K)~|~ \mathcal{C}_\in R_{1}\}$. Using Wedderga_[@wedd], we_can_see that_$\operatorname{Cen}_{G}(\varepsilon^{(1)}(H,K))=G$._Suppose $q\equiv 1_(\operatorname{mod}5)$. The relations_between group elements yield_that $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)^{x_{2}} =__\varepsilon_{\mathcal{C}_{\sigma^4}}^{(1)}(H,K)$. Hence $x_2$ does not centralize $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$. Since $G/H_1$ is cyclic of order $4$_generated_by $H_1_x_1$_and_$x_1^2 = x_2$, it follows_that $\operatorname{Cen}_{G}(\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H,K))=H_{1}$ and so the_orbit of_$\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$ has $[G:H_1] =4$ elements, namely $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$, $\varepsilon_{\mathcal{C}_{\sigma^2}}^{(1)}(H,K)$,_$\varepsilon_{\mathcal{C}_{\sigma^3}}^{(1)}(H,K)$_and $\varepsilon_{\mathcal{C}_{\sigma^4}}^{(1)}(H,K)$. Thus_$R_{(H, K)} = \{C_\sigma\}.$ Hence, if $q\equiv 1 (\operatorname{mod}5)$,_there is only one simple component_of $\mathbb{F}_{q}G$, which_by_Theorem_\[t1\] is isomorphic to_$M_{20}(\mathbb{F}_{q})$. Next, suppose $q\equiv 2 {\rm~or~}3_(\operatorname{mod}5)$. In this case, $\mathcal{C}_{q}(H/K)$ has_only one $q$-cyclotomic class namely $C_\sigma =_ \{\sigma, \sigma^{2}, \sigma^{3}, \sigma^{4}\}$. So_$R_{(H,K)}$ equals $ \mathcal{C}_{q}(H/K)$ and_in this_case also there is one_simple component of_$\mathbb{F}_{q}G$ isomorphic_to $M_{20}(\

. Eq. (\[crossintsol\]) for the radiator and hence (\[crossdiff\]) for the cross sections, is applicable, with power and subleading logarithmic corrections for values of $\tau_a$ away from the end-point region, where the variable $x$ in Eq. (\[xdef\]) becomes of order unity. That is, we require $[\beta_0/(2 \pi)] \as \ln (1/\tau_a)$ $< 1$, or equivalently $\tau_a > \LQCD/Q$. For $\tau_a \sim \LQCD/Q$ non-perturbative corrections become dominant. In this range of $\tau_a$, we “freeze” the perturbative contribution to the cross section, following [@KorMor98; @Korchemsky:2000kp], R\_ (\_a,Q,) (\_a - \^[-\_E]{} ) R\^ (\_a,Q ) + (\^[-\_E]{} -\_a ) R\^ (,Q ), \[freeze\] where $R^{\mbox{\tiny NLL}}$ is evaluated according to (\[intinv\]) and (\[match\]), and where $\kappa$ is a nonperturbative cutoff. Dynamics below the scale $\kappa$ will be incorporated into the nonperturbative corrections, in a manner we will discuss below. For our numerical studies we pick $\kappa = 0.75$ GeV, as in [@KorMor98]. The Scaling Rule ================ We are now ready to derive the scaling relation for nonperturbative shape functions. We show first how the rule is implied by the resummed cross section that we have just described, and go on to interpret the physical content of the scaling. From resummations to shape functions ------------------------------------ Following Ref. [@KorSt99], we identify the power structure of nonperturbative corrections by a direct expansion of the integrand in the resummed exponent at momentum scales below an infrared factorization scale, $\kappa$. Although this scale need not be exactly the same as the scale in Eq. (\[freeze\]) at which the radiator is frozen, they are closely related, and we will use the same symbol for both. We thus rewrite Eq. (\[thrustcomp\]) as the sum of a perturbative term, summarizing all $p_T>\kappa$, and a soft term, containing the nonperturbative physics of strong coupling. This corresponds to $p_

. Eq.   (\[crossintsol\ ]) for the radiator and hence (\[crossdiff\ ]) for the cross sections, is applicable, with power and subleading logarithmic correction for value of $ \tau_a$ away from the end - point area, where the variable $ x$ in Eq.   (\[xdef\ ]) becomes of order integrity. That is, we require $ [ \beta_0/(2 \pi) ] \as \ln (1/\tau_a)$ $ < 1 $, or equivalently $ \tau_a > \LQCD / Q$. For $ \tau_a \sim \LQCD / Q$ non - perturbative corrections become prevailing. In this compass of $ \tau_a$, we “ freeze ” the perturbative contribution to the cross incision, following [ @KorMor98; @Korchemsky:2000kp ], R\ _ (\_a, Q ,) (\_a - \^[-\_E ] { }) R\^ (\_a, Q) + (\^[-\_E ] { } -\_a) R\^ (, Q), \[freeze\ ] where $ R^{\mbox{\tiny NLL}}$ is evaluated harmonize to (\[intinv\ ]) and (\[match\ ]), and where $ \kappa$ is a nonperturbative cutoff. Dynamics below the scale $ \kappa$ will be incorporated into the nonperturbative corrections, in a manner we will discourse below. For our numerical studies we pluck $ \kappa = 0.75 $ GeV, as in [ @KorMor98 ]. The Scaling Rule = = = = = = = = = = = = = = = = We are now ready to derive the scaling relation for nonperturbative human body functions. We show first how the rule is implied by the resummed cross incision that we have just described, and go on to interpret the physical content of the scaling. From resummations to shape functions ------------------------------------ Following Ref.   [ @KorSt99 ], we identify the power structure of nonperturbative corrections by a direct expansion of the integrand in the resummed exponent at momentum scales below an infrared factorization scale, $ \kappa$. Although this plate need not be precisely the same as the plate in Eq.   (\[freeze\ ]) at which the radiator is fixed, they are close related, and we will use the same symbol for both. We thus rewrite Eq.   (\[thrustcomp\ ]) as the sum of a perturbative term, summarizing all $ p_T>\kappa$, and a soft condition, containing the nonperturbative physics of strong coupling. This corresponds to $ p _

. Eq. (\[cgossintsol\]) for the radiauor and hence (\[crossdiff\]) hor the cross sdctions, is applicable, with plwwr ane subleading logarithmkc correcnions for valnes of $\tau_a$ away from thc end-llint cegion, where thg variable $x$ in Eq. (\[xdef\]) becmmds of order unity. That is, we require $[\feta_0/(2 \pi)] \ad \ln (1/\tau_a)$ $< 1$, or equpvwlenfly $\tau_a > \LQCD/Q$. For $\tau_a \sim \LQCD/Q$ non-peruurbative correctipns become dominant. In thid rajge of $\tau_a$, we “frefze” the perjhrbwrive contribjtion to the cross secjion, following [@KorMor98; @Korchemsky:2000yp], R\_ (\_c,Q,) (\_a - \^[-\_E]{} ) R\^ (\_q,Q ) + (\^[-\_F]{} -\_a ) R\^ (,Q ), \[frxeze\] wrere $R^{\mbox{\tikj NLL}}$ iv evalusted according to (\[inrinv\]) and (\[match\]), and whxre $\kappa$ is a nonpetturbative cbtoff. Dynamics below rhw scane $\kdppa$ qilu bt iicodporatfd mnto the nohperturbatice corrections, in a mwbner we will siscusf felow. For our numerical studies we pick $\kalpa = 0.75$ GeV, as in [@KorMor98]. Tye Scaling Rule ================ We are now readr to derive the scaling relation for nonperturbathve siaoe ymkctiubs. We show first how the rule is implied by the dexukmed cross secbion that we have kudt qescribed, and go on to interpret the phydical cjntenr of the fcalong. From resummations to shape functions ------------------------------------ Yoloowing Ref. [@KorSt99], we identify tke powgr stricture of nonperturbatire cordections by a direct dxpansion of the innegrdnd in the resummed exponegt at momxntum scales belpw an ynfrared fwctorlsation scale, $\kappa$. Althluch this scwle need not be exactly the samx as the scalg it Ex. (\[freeze\]) ct whigh the radiator is frozen, thei are clofely felated, ans we winl use the fame symbol fmt both. We thuv rewritq Eq. (\[rhruwtcomp\]) xr the sum of a perturbanire term, symmarizing all $p_T>\keppa$, znd a soft term, cobtaining the nompeftuwbwtmve prfsics of strmng zouoking. Ghis corresionas tp $p_

. Eq. (\[crossintsol\]) for the radiator and for cross sections, applicable, with power values $\tau_a$ away from end-point region, where variable $x$ in Eq. (\[xdef\]) becomes order unity. That is, we require $[\beta_0/(2 \pi)] \as \ln (1/\tau_a)$ $< 1$, equivalently $\tau_a > \LQCD/Q$. For $\tau_a \sim \LQCD/Q$ non-perturbative corrections become dominant. In range $\tau_a$, “freeze” perturbative contribution to the cross section, following [@KorMor98; @Korchemsky:2000kp], R\_ (\_a,Q,) (\_a - \^[-\_E]{} ) R\^ ) + (\^[-\_E]{} -\_a ) R\^ (,Q ), where $R^{\mbox{\tiny NLL}}$ is according to (\[intinv\]) and (\[match\]), where is a cutoff. below scale $\kappa$ will incorporated into the nonperturbative corrections, in a manner we will discuss below. For our numerical studies we $\kappa = as in The Rule We are now derive the scaling relation for nonperturbative show first how the rule is implied by resummed cross that we have just described, and on to interpret the physical content of the From resummations to shape functions ------------------------------------ Following Ref. [@KorSt99], we identify the power structure of by a direct expansion the integrand in resummed at scales an infrared scale, $\kappa$. Although this scale need not be exactly the same the scale in Eq. (\[freeze\]) at which the radiator is are related, and we use the same symbol both. thus rewrite Eq. (\[thrustcomp\]) sum a all and soft term, containing the physics of strong coupling. This to $p_

. Eq. (\[crossintsol\]) for the radiatOr and hence (\[CrossDifF\]) foR tHe crOss sEctions, is appliCAble, With power and subleading LogarItHMic cORrEctioNs for vaLUeS OF $\taU_a$ AwAy fRoM ThE end-pOinT region, Where the vaRiaBlE $x$ in Eq. (\[xdef\]) beCOmEs of order uNitY. That is, we reqUirE $[\beta_0/(2 \pI)] \aS \ln (1/\TAu_a)$ $< 1$, or EquIvaleNtly $\taU_A > \LQCD/Q$. for $\tau_a \siM \LqcD/Q$ non-PErturbaTIVe CorrEctions become domiNAnT. in this range of $\tAu_a$, we “fReEZe” THE peRtuRbative conTrIbutiON to the cROsS SECtiON, following [@Kormor98; @KorchemsKY:2000kp], r\_ (\_a,Q,) (\_a - \^[-\_E]{} ) R\^ (\_A,Q ) + (\^[-\_e]{} -\_a ) R\^ (,q ), \[Freeze\] Where $r^{\mBOx{\tIny NLL}}$ is evaLuatEd accordiNg to (\[inTInv\]) and (\[mATch\]), and wHere $\kaPpa$ Is a NonpERtUrBatIvE CutOFf. dynAMicS below thE sCaLe $\kapPa$ wiLL BE IncoRpoRateD into The nonperturbAtiVe coRRecTions, In a maNner We Will dIscuss Below. foR our numerical stUdieS we pick $\kaPpa = 0.75$ gev, as In [@korMoR98]. the ScaLinG RuLe ================ We are Now readY To dErIVE ThE scaling relation foR nONPeRturbatiVe shapE FuNcTIons. We shOw FirSt hoW THe rulE is iMPlIed by the ResummED cRoSs sectiOn That we HaVe jUst DescrIBed, aNd go on To interpRet thE Physical contenT Of the scaling. FROm RESuMMatiOns To shape funcTionS ------------------------------------ follOwinG reF. [@KoRst99], we iDentiFy THe POwer structure of nonpErTurbatIve coRrections by a dIrect expanSION of the inTegrANd IN the resummed exPonenT at momentuM Scales beLow an Infrared FactorizaTIOn scale, $\kAppA$. AlThoUgh THIs Scale need not bE EXactLy The same As tHe scale In EQ. (\[frEezE\]) at WhIch the radIator is fRoZeN, tHeY arE closELy relateD, aNd wE wIll Use thE Same syMbol fOr boTh. we THus Rewrite eQ. (\[tHRUstcOmP\]) aS the Sum Of A pertUrbaTIve Term, sumMarizing aLl $p_t>\KappA$, aNd A soft teRm, containing tHe NonperturbAtIve PhysicS OF strong cOupling. This corresponds tO $P_

. Eq. (\[crossintsol\]) f or the rad iator an d h en ce ( \[cr ossdiff\]) for thecross sections, is app licab le , wit h p owerand sub l ea d i nglo ga rit hm i ccorre cti ons for values of $\ ta u_a$ away fr o mthe end-po int region, whe rethe va ri abl e $x$inEq. ( \[xdef \ ]) bec omes of o rd e r unit y . Thati s ,we r equire $[\beta_0/ ( 2\ pi)] \as \ln ( 1/\tau _a ) $$ < 1$ , o r equivale nt ly $\ t au_a >\ LQ C D / Q$. For $\tau_a \ sim \LQCD/Q $ no n-pert ur bat i ve cor recti on s be come domina nt.In this r ange o f $\tau_ a $, we “ freeze ” t hepert u rb at ive c o ntr i bu tio n to the cro ss s ectio n, f o l l o wing [@ KorM or98; @Korchemsky: 200 0kp] , R\ _ (\_ a,Q,) (\_ a- \^[ -\_E]{ } ) R \^ (\_a,Q ) + (\^ [-\_ E]{} -\_a )R\ ^ ( ,Q ), \ [ freeze \]whe re $R^{ \mbox{\ t iny N L L } }$ is evaluated acco rd i n gto (\[in tinv\] ) a nd (\[match \] ),andw h ere $ \kap p a$ is a no npertu r ba ti ve cuto ff . Dyna mi csbel ow th e sca le $\k appa$ wi ll be incorporated i n to the nonper t ur b a ti v e co rre ctions, ina ma n nerwe w i ll di s cussbelow .F or our numerical studi es we pi ck $\ kappa = 0.75$ GeV, as i n [ @KorMor9 8].Th e Scaling Rule===== ========== = We are nowready to derive t h e scaling re lat ion fo r no nperturbative s hape f unction s.We show fi rst ho w t he rule isimpliedby t he r esu mmedc ross sec ti onth atwe ha v e just desc ribe d, a n d g o on to in t e rpre tth e ph ysi ca l con tent ofthe sca ling. Fr omr esum ma ti ons toshape functio ns --------- -- --- ------ - - -------- ------ Following Ref.[ @KorSt9 9], we i dent ify the p owe r stru ctu r e of n onpert urbat iv e c o r recti o n sbyadirect exp a n sio n ofth e in tegrand in the resummed e x pon ent at moment umscal e s b elo w a n in fr a red f actorization sc ale, $\kap pa $ .Although t h issc ale nee d not b e exa c tly the same asthe scale i n Eq . (\[ freeze\])at which the radi a tor i s f rozen , t hey ar eclo selyrelate d , a nd we willus e thesamesy mbol for both. We thus rewriteEq. (\ [thru stc omp\]) as th e su m of a pe rtur bative ter m,sum mariz ing all $ p_T> \ ka ppa $ , and a s o ft term,c on tai n i ng the nonper t u r bat ive p hys i cs ofstro ng coupling. This corresponds to $p_

. Eq. (\[crossintsol\]) for_the radiator_and hence (\[crossdiff\]) for_the cross_sections,_is applicable,_with_power and subleading_logarithmic corrections for_values of $\tau_a$ away_from the end-point_region,_where the variable $x$ in Eq. (\[xdef\]) becomes of order unity. That is, we require_$[\beta_0/(2_\pi)] \as_\ln_(1/\tau_a)$_$< 1$, or equivalently $\tau_a_> \LQCD/Q$. For $\tau_a \sim_\LQCD/Q$ non-perturbative_corrections become dominant. In this range of $\tau_a$,_we_“freeze” the perturbative_contribution to the cross section, following [@KorMor98; @Korchemsky:2000kp], R\__(\_a,Q,) (\_a - \^[-\_E]{} ) R\^_(\_a,Q ) +_(\^[-\_E]{}_-\_a_) R\^ (,Q ),_\[freeze\] where $R^{\mbox{\tiny NLL}}$ is evaluated_according to (\[intinv\]) and (\[match\]), and_where $\kappa$ is a nonperturbative cutoff. Dynamics_below the scale $\kappa$ will be_incorporated into the nonperturbative corrections,_in a_manner we will discuss below._For our numerical_studies we_pick $\kappa =_0.75$ GeV, as in [@KorMor98]. The Scaling_Rule ================ We are now_ready to derive the scaling relation_for_nonperturbative shape functions._We_show_first how_the rule is_implied_by the_resummed_cross section that we have just_described,_and go on to interpret the physical_content of the scaling. From_resummations_to shape functions ------------------------------------ Following Ref. [@KorSt99],_we identify the power structure_of nonperturbative corrections by a direct_expansion of_the integrand_in the resummed exponent at momentum scales below an infrared factorization_scale, $\kappa$. Although this scale need_not be exactly the_same as_the_scale in Eq. (\[freeze\])_at_which the_radiator is frozen, they are closely related,_and we_will use the same symbol for_both. We thus rewrite_Eq. (\[thrustcomp\])_as the sum of a perturbative_term, summarizing all $p_T>\kappa$, and a_soft term, containing the nonperturbative_physics_of_strong coupling. This corresponds to_$p_

A_{i_n}$. ii\) Given an open cover $\mathcal U$ of $X$ and a point $x \in X$, the order of $\mathcal U$ at $x$, $\operatorname{{\rm Ord}}_x\mathcal U$, is the number of elements of $\mathcal U$ that contain $x$. We will make use of the following result that appears in  [@Dr2]: \[order\] A family $\mathcal U$ that consists of $m$ subsets of $X$ is an $(n + 1)$-cover of $X$ if and only if $\operatorname{{\rm Ord}}_x \mathcal U \geq m-n$ for all $x \in X$. See  [@Dr1; @Dr2]  [@Os] \[0-dim\] For every $m>n$, every $n$-dimensional compactum $X$ admits an $(n+1)$-cover by $m$ $0$-dimensional sets. This result follows from a slight modification to a proof given in  [@Os]. Since $\dim X \leq n$, $X$ can be decomposed into $0$-dimensional sets as $X = X_0 \cup...\cup X_n$. We assume that the $X_i$ are $G_{\delta}$ sets  [@E Theorem 1.2.14], and proceed inductively. For any $m > n+1$, suppose an $(n+1)$-cover $\{X_{0},...,X_{m-1}\}$ consisting of $0$-dimensional $G_{\delta}$ sets has been constructed. Let $$X_m = \{x \in X| x \mbox{ lies in exactly }(m-n) \mbox{ of the } X_i\}.$$ Since the $X_0,...,X_{m-1}$ form an $(n+1)$-cover, Proposition  \[order\] implies that each $x \in X$ lies in at least $m-n$ of the $X_i$. Then $X_m$ is the complement in $X$ of a finite union of finite intersections of $G_\delta$ sets, and is therefore $F_\sigma$. Similarly, for $0 \leq i \leq m-1$, each $X_m \cap X_i$ is a $0$-dimensional set that is $F_{\sigma}$ in $X$, and

A_{i_n}$. ii\) Given an open cover $ \mathcal U$ of $ X$ and a point $ x \in X$, the decree of $ \mathcal U$ at $ x$, $ \operatorname{{\rm Ord}}_x\mathcal U$, is the numeral of elements of $ \mathcal U$ that contain $ x$. We will make consumption of the following result that appears in   [ @Dr2 ]: \[order\ ] A class $ \mathcal U$ that consists of $ m$ subsets of $ X$ is an $ (n + 1)$-cover of $ X$ if and merely if $ \operatorname{{\rm Ord}}_x \mathcal U \geq m - n$ for all $ x \in X$. See   [ @Dr1; @Dr2 ]   [ @Os ] \[0 - dim\ ] For every $ m > n$, every $ n$-dimensional compactum $ X$ accept an $ (n+1)$-cover by $ m$ $ 0$-dimensional sets. This result follow from a slight modification to a proof given in   [ @Os ]. Since $ \dim X \leq n$, $ X$ can be decomposed into $ 0$-dimensional hardening as $ X = X_0 \cup... \cup X_n$. We assume that the $ X_i$ are $ G_{\delta}$ sets   [ @E Theorem 1.2.14 ], and continue inductively. For any $ m > n+1 $, suppose an $ (n+1)$-cover $ \{X_{0},... ,X_{m-1}\}$ consisting of $ 0$-dimensional $ G_{\delta}$ set has been constructed. Let $ $ X_m = \{x \in X| x \mbox { lies in exactly } (m - n) \mbox { of the } X_i\}.$$ Since the $ X_0,... ,X_{m-1}$ form an $ (n+1)$-cover, suggestion   \[order\ ] implies that each $ x \in X$ lies in at least $ m - n$ of the $ X_i$. Then $ X_m$ is the complement in $ X$ of a finite union of finite intersections of $ G_\delta$ sets, and is therefore $ F_\sigma$. Similarly, for $ 0 \leq i \leq m-1 $, each $ X_m \cap X_i$ is a $ 0$-dimensional set that is $ F_{\sigma}$ in $ X$, and

A_{i_n}$. li\) Given an open cover $\mxthcal U$ of $X$ aue a pomnt $x \ih X$, the urder of $\mathcal U$ at $x$, $\operetorbame{{\rn Ord}}_x\mathcal U$, is the number ov elemenrs oh $\mathcal U$ that contain $x$. We wjpl mckx use of the foklowing revult that appedrr nn  [@Dr2]: \[order\] A family $\mathcal U$ that cjnsists ov $m$ subsets of $X$ ix an $(h + 1)$-cover of $X$ if and only if $\operaforname{{\gm Ord}}_x \mathcal U \geq m-n$ for all $x \in X$. See  [@Fr1; @Dg2]  [@Os] \[0-dim\] For every $l>n$, every $n$-dumenfuonal compacgum $X$ admius an $(n+1)$-cover by $m$ $0$-dimensional sets. This resuut foklows from a sllcht modificetion no a proof giyvn in  [@Ov]. Since $\cim X \leq n$, $X$ gan bx dexomposed into $0$-dimensimnal sets as $X = X_0 \sup...\cup X_n$. Wz assume that the $X_i$ qrw $G_{\denta}$ vets  [@W Tfeodek 1.2.14], and pgocxed inductibely. For ant $m > n+1$, suppose an $(n+1)$-vodvt $\{X_{0},...,X_{m-1}\}$ consisfing os $0$-qimensional $G_{\delta}$ sets has been constrlctes. Let $$X_m = \{x \in X| x \mbox{ lies in exactly }(m-n) \mblx{ of the } X_i\}.$$ Since the $X_0,...,X_{m-1}$ form an $(n+1)$-cover, Proposition  \[ordes\] impmkes tmat dqcj $x \in X$ lies in at least $m-n$ of the $X_i$. Then $X_i$ ix nhe complement in $X$ of a finotf igion of finitg intersecfions of $G_\delta$ sehs, and ys thwrefore $F_\figms$. Similarly, for $0 \leq i \leq n-1$, each $X_m \cai X_i$ is a $0$-dimensional det that is $F_{\sibma}$ im $X$, and

A_{i_n}$. ii\) Given an open cover $\mathcal $X$ a point \in X$, the $x$, Ord}}_x\mathcal U$, is number of elements $\mathcal U$ that contain $x$. We make use of the following result that appears in [@Dr2]: \[order\] A family U$ that consists of $m$ subsets of $X$ is an $(n + 1)$-cover $X$ and if Ord}}_x \mathcal U \geq m-n$ for all $x \in X$. See [@Dr1; @Dr2] [@Os] \[0-dim\] For $m>n$, every $n$-dimensional compactum $X$ admits an $(n+1)$-cover $m$ $0$-dimensional sets. This follows from a slight modification a given in Since X n$, $X$ can decomposed into $0$-dimensional sets as $X = X_0 \cup...\cup X_n$. We assume that the $X_i$ are $G_{\delta}$ [@E Theorem proceed inductively. any > suppose an $(n+1)$-cover of $0$-dimensional $G_{\delta}$ sets has been = \{x \in X| x \mbox{ lies in }(m-n) \mbox{ the } X_i\}.$$ Since the $X_0,...,X_{m-1}$ an $(n+1)$-cover, Proposition \[order\] implies that each $x X$ lies in at least $m-n$ of the $X_i$. Then $X_m$ is the complement in a finite union of intersections of $G_\delta$ and therefore Similarly, $0 \leq \leq m-1$, each $X_m \cap X_i$ is a $0$-dimensional set that $F_{\sigma}$ in $X$, and

A_{i_n}$. ii\) Given an open cover $\mathCal U$ of $X$ and A poinT $x \iN X$, tHe OrdeR of $\mAthcal U$ at $x$, $\operATornAme{{\rm Ord}}_x\mathcal U$, is the NumbeR oF ElemENtS of $\maThcal U$ tHAt CONtaIn $X$. WE wiLl MAkE use oF thE followIng result tHat ApPears in  [@Dr2]: \[ordER\] A Family $\mathCal u$ that consistS of $M$ subseTs Of $X$ IS an $(n + 1)$-cOveR of $X$ iF and onLY if $\opeRatorname{{\Rm oRd}}_x \matHCal U \geq M-N$ FoR all $X \in X$. See  [@Dr1; @Dr2]  [@Os] \[0-dim\] FOR eVEry $m>n$, every $n$-dimEnsionAl COmPACtuM $X$ aDmits an $(n+1)$-coVeR by $m$ $0$-dIMensionAL sETS. thiS Result follows From a slight MOdiFicatiOn To a PRoof giVen in  [@os]. sIncE $\dim X \leq n$, $X$ cAn be DecomposeD into $0$-dIMensionAL sets as $x = X_0 \cup...\cUp X_N$. We AssuME tHaT thE $X_I$ Are $g_{\DeLta}$ SEts  [@e Theorem 1.2.14], AnD pRoceeD indUCTIVely. for Any $m > N+1$, suppOse an $(n+1)$-cover $\{X_{0},...,X_{M-1}\}$ coNsisTIng Of $0$-dimEnsioNal $G_{\DeLta}$ seTs has bEen coNsTructed. Let $$X_m = \{x \in x| x \mbOx{ lies in eXacTlY }(m-n) \MbOx{ of tHE } X_i\}.$$ SinCe tHe $X_0,...,x_{m-1}$ form aN $(n+1)$-cover, pRopOsITIOn  \[Order\] implies that eaCh $X \IN X$ Lies in at Least $m-N$ Of ThE $x_i$. Then $X_m$ Is The CompLEMent iN $X$ of A FiNite unioN of finITe InTersectIoNs of $G_\dElTa$ sEts, And is THereFore $F_\sIgma$. SimiLarly, FOr $0 \leq i \leq m-1$, each $x_M \cap X_i$ is a $0$-dimeNSiONAl SEt thAt iS $F_{\sigma}$ in $X$, aNd

A_{i_n}$. ii\) Given anopen cover $\ma thc alU$ of$X$and a point $x \inX$, the order of $\mat hcalU$ at $ x $, $\op eratorn a me { { \rm O rd }}_ x\ m at hcalU$, is the number of el em ents of $\ma t hc al U$ that co ntain $x$. Wewill m ak e u s e ofthe foll owingr esultthat appe ar s in  [ @ Dr2]: \ [ or der\ ] A family $\math c al U$ that consis ts of$m $ s u b set s o f $X$ is a n$(n + 1)$-cov e ro f $X$ if and only i f $\operato r nam e{{\rm O rd} } _x \ma thcal U \ge q m-n$ forall$x \in X$ . See  [@Dr1; @Dr2]  [@Os] \[ 0-d im\] Fo reve ry $m> n $, ev e ry$n$-dime ns io nal c ompa c t u m $X$ ad mits an $ (n+1)$-coverby$m$$ 0$- dimen siona l se ts . Th is res ult f ol lows from a sli ghtmodificat ion t o a p roofg iven i n [@O s]. Si nce $\d i m X \ l e q n $, $X$ can be deco mp o s ed into $0 $-dime n si on a l sets a s$X= X_ 0 \cup. ..\c u pX_n$. We assum e t ha t the $ X_ i$ are $ G_{ \de lta}$ sets  [@ETheorem1.2.1 4 ], and proceed inductively.F or a ny $m > n+ 1$, suppose an$ (n+1 )$-c o ve r $ \ {X_{0 },... ,X _ {m - 1}\}$ consisting of $ 0$-dim ensio nal $G_{\delt a}$ sets h a s been con stru c te d . Let $$X_m =\{x \ in X| x \m b ox{ lies in e xactly } (m-n) \mb o x { of the }X_i \}. $$S in ce the $X_0,. . . ,X_{ m- 1}$ for m a n $(n+1 )$- cov er, Pr op osition \[order\ ]im pl ie s t hat e a ch $x \i nX$li esin at least$m-n$ ofth e$ X_i $. Then $X _ m $ is t he com ple me nt in $X$ ofa finit e union o f f i nite i nt ersecti ons of $G_\de lt a$ sets, a nd is there f o re $F_\s igma$. Similarly, for $ 0 \leq i \l eq m- 1$,each $X_m \c ap X_i $ i s a $0$ -dimen siona lset t hat i s $F _{\ si gma}$ in $ X $ , a nd

A_{i_n}$. ii\) Given_an open_cover $\mathcal U$ of_$X$ and_a_point $x_\in_X$, the order_of $\mathcal U$_at $x$, $\operatorname{{\rm Ord}}_x\mathcal_U$, is the_number_of elements of $\mathcal U$ that contain $x$. We will make use of the following_result_that appears_in_ [@Dr2]: \[order\]_A family $\mathcal U$ that_consists of $m$ subsets of_$X$ is_an $(n + 1)$-cover of $X$ if and_only_if $\operatorname{{\rm Ord}}_x_\mathcal U \geq m-n$ for all $x \in X$. See_ [@Dr1; @Dr2]  [@Os] \[0-dim\] For every $m>n$,_every $n$-dimensional compactum_$X$_admits_an $(n+1)$-cover by $m$_$0$-dimensional sets. This result follows from a_slight modification to a proof given_in  [@Os]. Since $\dim X \leq n$, $X$_can be decomposed into $0$-dimensional sets_as $X = X_0 \cup...\cup_X_n$. We_assume that the $X_i$ are_$G_{\delta}$ sets  [@E_Theorem 1.2.14],_and proceed inductively._For any $m > n+1$, suppose_an $(n+1)$-cover $\{X_{0},...,X_{m-1}\}$_consisting of $0$-dimensional $G_{\delta}$ sets has_been_constructed. Let $$X_m_=_\{x_\in X|_x \mbox{ lies_in exactly_}(m-n) \mbox{_of_the } X_i\}.$$ Since the $X_0,...,X_{m-1}$ form_an_$(n+1)$-cover, Proposition  \[order\] implies that each $x_\in X$ lies in_at_least $m-n$ of the_$X_i$. Then $X_m$ is the_complement in $X$ of a finite_union of_finite intersections_of $G_\delta$ sets, and is therefore $F_\sigma$. Similarly, for $0 \leq_i \leq m-1$, each $X_m \cap_X_i$ is a $0$-dimensional_set that_is_$F_{\sigma}$ in $X$,_and

],\, \left[e^{235}+e^{145}+e^{136}-e^{246}\right],\, \right. \\[5pt] && \left. \left[-e^{135}+e^{236}-e^{146}-e^{245}\right],\, \left[-e^{136}-e^{235}+e^{145}-e^{246}\right],\, \right. \\[5pt] && \left. \left[e^{135}+e^{245}+e^{236}-e^{146}\right],\, \left[-e^{235}+e^{145}+e^{136}+e^{246}\right]\right\rangle \;, \\[5pt] H^4_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{1256}\right],\, \left[e^{3456}\right],\, \left[e^{2346}-e^{1345}\right],\, \left[e^{1346}+e^{2345}\right],\, \right. \\[5pt] && \left. \left[e^{1246}-e^{1235}\right],\, \left[e^{1236}+e^{1245}\right],\, \left[e^{2456}+e^{1356}\right],\, \left[e^{1456}-e^{2356}\right] \right\rangle \;, \\[5pt] H^5_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{23456}\right],\, \left[e^{13456}\right],\, \left[e^{12456}\right],\, \left[e^{12356}\right] \right\rangle \;, \\[5pt] H^6_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{123456}\right] \right\rangle \;.\end{aligned}$$ Any linear complex structure $J$ defined on $\mathfrak{g}$ gives rise to a complex structure on $M$ that will be called *left-invariant*. The Iwasawa manifold can be regarded as one of these structures, although there is an infinite family of them (see [@andrada-barberis-dotti; @couv] for a complete classification up to isomorphism). Let $\

], \, \left[e^{235}+e^{145}+e^{136}-e^{246}\right],\, \right. \\[5pt ] & & \left. \left[-e^{135}+e^{236}-e^{146}-e^{245}\right],\, \left[-e^{136}-e^{235}+e^{145}-e^{246}\right],\, \right. \\[5pt ] & & \left. \left[e^{135}+e^{245}+e^{236}-e^{146}\right],\, \left[-e^{235}+e^{145}+e^{136}+e^{246}\right]\right\rangle \; , \\[5pt ] H^4_{dR}(M;{\mathbb{R } }) & = & { \mathbb{R}}\left\langle \left[e^{1256}\right],\, \left[e^{3456}\right],\, \left[e^{2346}-e^{1345}\right],\, \left[e^{1346}+e^{2345}\right],\, \right. \\[5pt ] & & \left. \left[e^{1246}-e^{1235}\right],\, \left[e^{1236}+e^{1245}\right],\, \left[e^{2456}+e^{1356}\right],\, \left[e^{1456}-e^{2356}\right ] \right\rangle \; , \\[5pt ] H^5_{dR}(M;{\mathbb{R } }) & = & { \mathbb{R}}\left\langle \left[e^{23456}\right],\, \left[e^{13456}\right],\, \left[e^{12456}\right],\, \left[e^{12356}\right ] \right\rangle \; , \\[5pt ] H^6_{dR}(M;{\mathbb{R } }) & = & { \mathbb{R}}\left\langle \left[e^{123456}\right ] \right\rangle \;.\end{aligned}$$ Any linear complex structure $ J$ defined on $ \mathfrak{g}$ gives upgrade to a complex social organization on $ M$ that will be called * left - changeless *. The Iwasawa manifold can be regard as one of these structures, although there be an infinite family of them (attend [ @andrada - barberis - dotti; @couv ] for a complete classification up to isomorphism). lease $ \

],\, \levt[e^{235}+e^{145}+e^{136}-e^{246}\right],\, \right. \\[5pt] && \lent. \left[-e^{135}+e^{236}-e^{146}-e^{245}\right],\, \left[-e^{136}-e^{235}+x^{145}-e^{246}\right],\, \right. \\[5pg] && \left. \left[e^{135}+e^{245}+e^{236}-e^{146}\right],\, \left[-e^{235}+e^{145}+x^{136}+e^{246}\ritht]\ritht\rangle \;, \\[5pt] H^4_{dR}(M;{\mathcb{R}}) &=& {\mathhb{R}}\left\lqnglt \left[e^{1256}\right],\, \left[x^{3456}\dight],\, \lcyt[e^{2346}-e^{1345}\rjnht],\, \lzfv[e^{1346}+e^{2345}\right],\, \right. \\[5kt] && \left. \lefd[e^{1246}-e^{1235}\right],\, \left[e^{1236}+a^{1245}\rkgkt],\, \left[e^{2456}+e^{1356}\right],\, \left[e^{1456}-e^{2356}\right] \right\rangje \;, \\[5pt] N^5_{dG}(M;{\mathbb{R}}) &=& {\matrbb{R}}\kqft\lzngle \left[e^{23456}\right],\, \left[e^{13456}\right],\, \left[e^{12456}\rjght],\, \leht[e^{12356}\right] \right\rsngle \;, \\[5pt] H^6_{dR}(M;{\mathbb{R}}) &=& {\matjbb{R}}\peft\langle \left[e^{123456}\rihht] \right\rabgle \;.\wnd{aligned}$$ Anh linear complex strucjure $J$ defined on $\mathfrak{g}$ giver risz to a compoez shtucture on $M$ that will be called *left-hnvariamt*. The Iwasawa maiifood can be regarded as one of these strustures, aldhkugh there is an undinitg famhly ud tfem (sxe [@zndradw-bacberis-dotti; @couv] for a complete classificstyin up to isomkrphisi). Lqt $\

],\, \left[e^{235}+e^{145}+e^{136}-e^{246}\right],\, \right. \\[5pt] && \left. \left[-e^{135}+e^{236}-e^{146}-e^{245}\right],\, \\[5pt] \left. \left[e^{135}+e^{245}+e^{236}-e^{146}\right],\, \;, \\[5pt] H^4_{dR}(M;{\mathbb{R}}) \left[e^{1346}+e^{2345}\right],\, \\[5pt] && \left. \left[e^{1236}+e^{1245}\right],\, \left[e^{2456}+e^{1356}\right],\, \left[e^{1456}-e^{2356}\right] \;, \\[5pt] H^5_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{23456}\right],\, \left[e^{12456}\right],\, \left[e^{12356}\right] \right\rangle \;, \\[5pt] H^6_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{123456}\right] \right\rangle \;.\end{aligned}$$ Any linear structure $J$ defined on $\mathfrak{g}$ gives rise to a complex structure on $M$ will called The manifold can be regarded as one of these structures, although there is an infinite family of (see [@andrada-barberis-dotti; @couv] for a complete classification up isomorphism). Let $\

],\, \left[e^{235}+e^{145}+e^{136}-e^{246}\right],\, \right. \\[5pt] && \left. \lEft[-e^{135}+e^{236}-e^{146}-e^{245}\rigHt],\, \lefT[-e^{136}-e^{235}+E^{145}-e^{246}\rIgHt],\, \riGht. \\[5pT] && \left. \left[e^{135}+e^{245}+e^{236}-e^{146}\rIGht],\, \lEft[-e^{235}+e^{145}+e^{136}+e^{246}\right]\right\ranglE \;, \\[5pt] H^4_{dr}(M;{\MAthbB{r}}) &=& {\mAthbb{r}}\left\laNGlE \LEft[E^{1256}\rIgHt],\, \lEfT[E^{3456}\rIght],\, \lEft[E^{2346}-e^{1345}\right],\, \Left[e^{1346}+e^{2345}\righT],\, \riGhT. \\[5pt] && \left. \left[e^{1246}-E^{1235}\RiGht],\, \left[e^{1236}+e^{1245}\rIghT],\, \left[e^{2456}+e^{1356}\right],\, \LefT[e^{1456}-e^{2356}\rigHt] \RigHT\rangLe \;, \\[5pT] H^5_{dR}(M;{\Mathbb{r}}) &=& {\Mathbb{r}}\left\langLe \LEft[e^{23456}\riGHt],\, \left[e^{13456}\RIGhT],\, \lefT[e^{12456}\right],\, \left[e^{12356}\right] \RIgHT\rangle \;, \\[5pt] H^6_{dR}(M;{\mAthbb{R}}) &=& {\MaTHbB{r}}\LefT\laNgle \left[e^{123456}\rIgHt] \rigHT\rangle \;.\ENd{ALIGneD}$$ any linear compLex structurE $j$ deFined oN $\mAthFRak{g}$ giVes riSe TO a cOmplex strucTure On $M$ that wiLl be caLLed *left-INvarianT*. The IwAsaWa mAnifOLd CaN be ReGArdED aS onE Of tHese struCtUrEs, altHougH THERe is An iNfinIte faMily of them (see [@AndRada-BArbEris-dOtti; @cOuv] fOr A compLete clAssifIcAtion up to isomorPhisM). Let $\

],\, \left[e^{235}+e^{145} +e^{136}-e ^{246 }\r igh t] ,\,\rig ht. \\[5pt] & & \le ft. \left[-e^{135}+e^{ 236}- e^ { 146} - e^ {245} \right] , \, \ lef t[ -e ^{1 36 } -e ^{235 }+e ^{145}- e^{246}\ri ght ], \, \right. \ \ [5 pt] && \l eft . \left[e^{1 35} +e^{24 5} +e^ { 236}- e^{ 146}\ right] , \, \le ft[-e^{23 5} + e^{145 } +e^{136 } + e^ {246 }\right]\right\ra n gl e \;, \\[5pt] H^4_{d R} ( M; { \ mat hbb {R}}) &=&{\ mathb b {R}}\le f t\ l a n gle \left[e^{1256 }\right],\, \le ft[e^{ 34 56} \ right] ,\, \ le f t[e ^{2346}-e^{ 1345 }\right], \, \le f t[e^{13 4 6}+e^{2 345}\r igh t], \, \ r ig ht . \ \[ 5 pt] & & \ l eft . \left[ e^ {1 246}- e^{1 2 3 5 } \rig ht] ,\,\left [e^{1236}+e^{ 124 5}\r i ght ],\,\left [e^{ 24 56}+e ^{1356 }\rig ht ],\, \left[e^{1 456} -e^{2356} \ri gh t]\r ight\ r angle\;, \\ [5pt] H^5_{dR } (M; {\ m a t hb b{R}}) &=& {\mathb b{ R } }\ left\lan gle \l e ft [e ^ {23456}\ ri ght ],\, \ left[ e^{1 3 45 6}\right ],\, \ l ef t[ e^{1245 6} \right ], \,\le ft[e^ { 1235 6}\rig ht] \rig ht\ra n gle \;, \\[5pt ] H^6_{dR}(M; { \m a t hb b {R}} ) & =& {\mathbb {R}} \ left \lan g le \l e ft[e^ {1234 56 } \r i ght] \right\rangle\; .\end{ align ed}$$ Any li near compl e x structur e $J $ d e fined on $\mat hfrak {g}$ gives rise toa com plex str ucture on $ M$ thatwil l b e c all e d * left-invarian t * . Th eIwasawa ma nifoldcan be re gar de d as oneof these s tr uc tu res , alt h ough the re is a n i nfini t e fami ly of the m(s e e [ @andrad a -b a r beri s- do tti; @c ou v] fo r ac omp lete cl assificat ion up t ois omorphi sm). Let $\

],\, \left[e^{235}+e^{145}+e^{136}-e^{246}\right],\,_\right. \\[5pt] _&& \left. \left[-e^{135}+e^{236}-e^{146}-e^{245}\right],\, \left[-e^{136}-e^{235}+e^{145}-e^{246}\right],\,_\right. \\[5pt] _&&_\left. \left[e^{135}+e^{245}+e^{236}-e^{146}\right],\,_\left[-e^{235}+e^{145}+e^{136}+e^{246}\right]\right\rangle_\;, \\[5pt] H^4_{dR}(M;{\mathbb{R}})_&=& {\mathbb{R}}\left\langle \left[e^{1256}\right],\,_\left[e^{3456}\right],\, \left[e^{2346}-e^{1345}\right],\, \left[e^{1346}+e^{2345}\right],\, \right._\\[5pt] && \left._\left[e^{1246}-e^{1235}\right],\,_\left[e^{1236}+e^{1245}\right],\, \left[e^{2456}+e^{1356}\right],\, \left[e^{1456}-e^{2356}\right] \right\rangle \;, \\[5pt] H^5_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{23456}\right],\, \left[e^{13456}\right],\, \left[e^{12456}\right],\, \left[e^{12356}\right] \right\rangle_\;,_\\[5pt] H^6_{dR}(M;{\mathbb{R}})_&=&_{\mathbb{R}}\left\langle_\left[e^{123456}\right] \right\rangle \;.\end{aligned}$$ Any linear complex_structure $J$ defined on $\mathfrak{g}$_gives rise_to a complex structure on $M$ that will_be_called *left-invariant*. The_Iwasawa manifold can be regarded as one of these_structures, although there is an infinite_family of them_(see_[@andrada-barberis-dotti;_@couv] for a complete_classification up to isomorphism). Let $\

, \ldots, b_m \rangle =E$ and $\mu (E)=n+m$. In particular, $\mu(U) \geq \mu (V)\geq {\ell(E)}$. 4. The following are equivalent: 1. $V\subseteq E$ is a reduction and $\mu (V)={\ell(E)}$. 2. If $V=\langle a_1, \ldots,a_n \rangle$ with $n=\mu(V)$, then ${{{\overline}{a}}_1,\dotsc,{{\overline}{a}}_{n}} \in {{{\mathcal F}}(E)}$ is a homogeneous system of parameters. And if any of these two equivalent conditions holds, $V$ is a minimal reduction of $E$. 5. If the residue field $k$ is infinite and $V\subseteq U$ is a minimal reduction, then conditions d1) and d2) hold. In particular, ${{{\mathcal F}}(V)}\subset {{{\mathcal F}}(E)}$ is a noether normalization of ${{{\mathcal F}}(E)}$ and $V^n \cap {{\mathfrak{m}}}E^n ={{\mathfrak{m}}}V^n$ for all $n\geq 0$. As a consequence, minimal reductions always exist. The [*reduction number*]{} of $E$, denoted by $r(E)$, is the minimum of $r_{U}(E)$, where $U$ ranges over all minimal reductions of $E$. \[hsop\] If the residue field is finite, a minimal set of generators of a minimal reduction of $E$ is not necessarily a homogeneous system of parameters of ${{{\mathcal F}}(E)}$. Nevertheless, there always exist homogeneous systems of parameters of ${{{\mathcal F}}(E)}$. This is equivalent to the existence of a family of elements $a_1\in E^{r_1}\setminus {{\mathfrak{m}}}E^{r_1}, \ldots, a_s\in E^{r_s}\setminus {{\mathfrak{m}}}E^{r_s}$, where $s={\ell(E)}$, such that for some $r$, $E^r=a_1E^{r-r_1}+ \cdots + a_sE^{r-r_s}$; and ${\ell(E)}$ is the minimum positive number for a such family of elements to exist. \[as0\] Let ${(R,{{\mathfrak{m}}}, {

, \ldots, b_m \rangle = E$ and $ \mu (E)=n+m$. In particular, $ \mu(U) \geq \mu (V)\geq { \ell(E)}$. 4. The following are equivalent: 1. $ V\subseteq E$ is a reduction and $ \mu (V)={\ell(E)}$. 2. If $ V=\langle a_1, \ldots, a_n \rangle$ with $ n=\mu(V)$, then $ { { { \overline}{a}}_1,\dotsc,{{\overline}{a}}_{n } } \in { { { \mathcal F}}(E)}$ is a homogeneous system of argument. And if any of these two equivalent condition holds, $ V$ is a minimal decrease of $ E$. 5. If the residue field $ k$ is countless and $ V\subseteq U$ is a minimal reduction, then conditions d1) and d2) detention. In finical, $ { { { \mathcal F}}(V)}\subset { { { \mathcal F}}(E)}$ is a noether normalization of $ { { { \mathcal F}}(E)}$ and $ V^n \cap { { \mathfrak{m}}}E^n = { { \mathfrak{m}}}V^n$ for all $ n\geq 0$. As a consequence, minimal decrease always exist. The [ * reduction issue * ] { } of $ E$, denoted by $ r(E)$, is the minimum of $ r_{U}(E)$, where $ U$ ranges over all minimal reduction of $ E$. \[hsop\ ] If the residue playing field is finite, a minimal set of generator of a minimal reduction of $ E$ is not necessarily a homogeneous arrangement of parameters of $ { { { \mathcal F}}(E)}$. Nevertheless, there always exist homogeneous systems of parameter of $ { { { \mathcal F}}(E)}$. This is equivalent to the existence of a family of elements $ a_1\in E^{r_1}\setminus { { \mathfrak{m}}}E^{r_1 }, \ldots, a_s\in E^{r_s}\setminus { { \mathfrak{m}}}E^{r_s}$, where $ s={\ell(E)}$, such that for some $ r$, $ E^r = a_1E^{r - r_1}+ \cdots + a_sE^{r - r_s}$; and $ { \ell(E)}$ is the minimum convinced number for a such family of elements to exist. \[as0\ ] Let $ { (R,{{\mathfrak{m } } }, {

, \ldlts, b_m \rangle =E$ and $\mu (E)=k+m$. In particular, $\mu(U) \gex \mu (V)\feq {\ell(E)}$. 4. The following are equivaleit: 1. $V\wubseteq E$ is a reductkon and $\ml (V)={\ell(E)}$. 2. Mf $V=\langle a_1, \ldova,a_n \rannje$ wjbh $n=\mb(V)$, then ${{{\overline}{s}}_1,\dotsc,{{\overnine}{a}}_{n}} \in {{{\mathwau Y}}(E)}$ is a homogeneous system of parameeers. Ajd if any of trese ewo squivalent conditions holds, $V$ is a miniman reduction og $E$. 5. If the residue field $n$ is infinite and $V\subdeteq U$ is q migumal reductiun, then conditions d1) ahd d2) hold. In particular, ${{{\mathcal F}}(V)}\suyset {{{\mathcao D}}(E)}$ lv a noether normwlization of ${{{\mathcal F}}(E)}$ and $V^n \cap {{\mathfrsk{m}}}X^n ={{\mqthfrak{m}}}V^n$ for all $n\gxq 0$. As a consequence, iinimal radbctions always exist. Rhw [*redoctiot nuover*]{} of $E$, dsnoted by $r(E)$, is the minimum of $r_{U}(E)$, where $U$ ranges odvt all minimam redustyons of $E$. \[hsop\] If the residue field is fpnits, a minimal set of geneeators of a minimal rgduction os $E$ is not necessarily a homogeneous system of pasametxrr oy ${{{\mathzql F}}(E)}$. Nevertheless, there always exist homogeneota xyxtems of paramcters of ${{{\mathcal F}}(R)}$. Hhof is equivalevt to cge existence of a falily of elemwnts $a_1\in T^{r_1}\setkinus {{\mathfrak{m}}}E^{r_1}, \ldots, a_s\ib E^{r_s}\setminuf {{\mathfrak{m}}}E^{r_s}$, wherz $s={\ell(E)}$, such thaj for xome $r$, $E^r=a_1E^{r-r_1}+ \cdots + a_sE^{x-r_s}$; ans ${\ell(E)}$ is tje minimuj positive number fog a vuch family of elements to exist. \[as0\] Oet ${(X,{{\mathfray{m}}}, {

, \ldots, b_m \rangle =E$ and $\mu particular, \geq \mu {\ell(E)}$. 4. The E$ a reduction and (V)={\ell(E)}$. 2. If a_1, \ldots,a_n \rangle$ with $n=\mu(V)$, then \in {{{\mathcal F}}(E)}$ is a homogeneous system of parameters. And if any of two equivalent conditions holds, $V$ is a minimal reduction of $E$. 5. If residue $k$ infinite $V\subseteq U$ is a minimal reduction, then conditions d1) and d2) hold. In particular, ${{{\mathcal F}}(V)}\subset F}}(E)}$ is a noether normalization of ${{{\mathcal F}}(E)}$ $V^n \cap {{\mathfrak{m}}}E^n ={{\mathfrak{m}}}V^n$ all $n\geq 0$. As a minimal always exist. [*reduction of denoted by $r(E)$, the minimum of $r_{U}(E)$, where $U$ ranges over all minimal reductions of $E$. \[hsop\] If the residue is finite, set of of minimal of $E$ is a homogeneous system of parameters of there always exist homogeneous systems of parameters of F}}(E)}$. This equivalent to the existence of a of elements $a_1\in E^{r_1}\setminus {{\mathfrak{m}}}E^{r_1}, \ldots, a_s\in E^{r_s}\setminus where $s={\ell(E)}$, such that for some $r$, $E^r=a_1E^{r-r_1}+ \cdots + a_sE^{r-r_s}$; and ${\ell(E)}$ is the number for a such of elements to \[as0\] ${(R,{{\mathfrak{m}}},

, \ldots, b_m \rangle =E$ and $\mu (E)=n+m$. In pArticular, $\mU(U) \geq \Mu (V)\Geq {\ElL(E)}$. 4. ThE folLowing are equivALent: 1. $v\subseteq E$ is a reduction And $\mu (v)={\eLL(E)}$. 2. If $v=\LaNgle a_1, \Ldots,a_n \RAnGLE$ wiTh $N=\mU(V)$, tHeN ${{{\OvErlinE}{a}}_1,\dOtsc,{{\oveRline}{a}}_{n}} \in {{{\mAthCaL F}}(E)}$ is a homogeNEoUs system of ParAmeters. And if Any Of thesE tWo eQUivalEnt CondiTions hOLds, $V$ is A minimal rEdUCtion oF $e$. 5. If the rESIdUe fiEld $k$ is infinite and $v\SuBSeteq U$ is a minimAl reduCtIOn, THEn cOndItions d1) and D2) hOld. In PArticulAR, ${{{\mATHCal f}}(v)}\subset {{{\mathcaL F}}(E)}$ is a noethER noRmalizAtIon OF ${{{\mathcAl F}}(E)}$ aNd $v^N \caP {{\mathfrak{m}}}E^N ={{\matHfrak{m}}}V^n$ fOr all $n\GEq 0$. As a coNSequencE, minimAl rEduCtioNS aLwAys ExISt. THE [*rEduCTioN number*]{} oF $E$, DeNoted By $r(E)$, IS THE minImuM of $r_{u}(E)$, wheRe $U$ ranges over All MiniMAl rEductIons oF $E$. \[hsOp\] if the ResiduE fielD iS finite, a minimal Set oF generatoRs oF a MinImAl redUCtion oF $E$ iS noT necessArily a hOMogEnEOUS sYstem of parameters oF ${{{\mATHcAl F}}(E)}$. NeveRtheleSS, tHeRE always eXiSt hOmogENEous sYsteMS oF parametErs of ${{{\mAThCaL F}}(E)}$. This Is EquivaLeNt tO thE exisTEnce Of a famIly of eleMents $A_1\In E^{r_1}\setminus {{\maTHfrak{m}}}E^{r_1}, \ldots, A_S\iN e^{R_s}\SEtmiNus {{\Mathfrak{m}}}E^{r_S}$, wheRE $s={\elL(E)}$, suCH tHat FOr somE $r$, $E^r=a_1e^{r-R_1}+ \CdOTs + a_sE^{r-r_s}$; and ${\ell(E)}$ is thE mInimum PositIve number for a Such family OF ELements tO exiST. \[aS0\] let ${(R,{{\mathfrak{m}}}, {

, \ldots, b_m \rangle =E$and $\mu ( E)=n+ m$. In p arti cula r, $\mu(U) \ge q \mu (V)\geq {\ell(E)}$. 4. T he foll o wi ng ar e equiv a le n t : 1. $ V \s ubset eqE$ is a reduction an d$\mu (V)={\e l l( E)}$. 2. If $V=\lan gle a_1,\l dot s ,a_n\ra ngle$ with$ n=\mu( V)$, then $ { {{\ove r line}{a } } _1 ,\do tsc,{{\overline}{ a }} _ {n}} \in {{{\m athcal F } }( E ) }$isa homogene ou s sys t em of p a ra m e t ers . And ifany of thes e tw o equi va len t condi tions h o lds , $V$ is amini mal reduc tion o f $E$. 5 . If t he res idu e f ield $k $isin f ini t eand $V\ subseteq U $is amini m a l redu cti on,thenconditions d1 ) a nd d 2 ) h old.In pa rtic ul ar, $ {{{\ma thcal F }}(V)}\subset { {{\m athcal F} }(E )} $ i sa noe t her no rma liz ation o f ${{{\ m ath ca l F }} (E)}$ and $V^n \ca p{ { \m athfrak{ m}}}E^ n = {{ \ mathfrak {m }}} V^n$ f or al l $n \ ge q 0$. A s a co n se qu ence, m in imal r ed uct ion s alw a ys e xist.The [*re ducti o n number*]{} o f $E$, denoted by $ r( E )$,isthe minimum of$ r_{U }(E) $ ,whe r e $U$ rang es ov e r all minimal reduc ti ons of $E$. \[hsop\] If the resid u e field is fin i te , a minimal set of g eneratorso f a mini mal r eduction of $E$ i s not nece ssa ril y a ho m o ge neous systemo f par am eters o f $ {{{\mat hca l F }}( E)} $. Neverthe less, th er eal wa ysexist homogene ou s s ys tem s ofp aramet ers o f ${ {{ \m a thc al F}}( E )} $ . Thi sis equ iva le nt to the exi stenceof a fami lyo f el em en ts $a_1 \in E^{r_1}\s et minus {{\m at hfr ak{m}} } E ^{r_1},\ldots, a_s\in E^{r_s}\ s etminus {{ \math frak {m}}}E^{r _s} $, whe re$ s={\el l(E)}$ , suc htha t for s o m e$r$ ,$E^r=a_1E^ { r -r_ 1}+ \ cd ots+ a_sE^ {r-r_s}$; and ${\e l l(E )}$ is the mi nim um p o s it ive nu m ber f o r a s uch family of e lements to e x is t. \[as0\ ] Le t${(R,{{ \mathfr ak{m} } }, {

, \ldots,_b_m \rangle_=E$ and $\mu (E)=n+m$._In particular,_$\mu(U)_\geq \mu_(V)\geq_{\ell(E)}$. 4. The_following are equivalent: _ 1._ $V\subseteq E$_is_a reduction and $\mu (V)={\ell(E)}$. 2. If $V=\langle a_1, \ldots,a_n_\rangle$_with $n=\mu(V)$,_then_${{{\overline}{a}}_1,\dotsc,{{\overline}{a}}_{n}}_\in {{{\mathcal F}}(E)}$ is a_homogeneous system of parameters. _ _And if any of these two equivalent conditions_holds,_$V$ is a_minimal reduction of $E$. 5. If the residue field_$k$ is infinite and $V\subseteq U$_is a minimal_reduction,_then_conditions d1) and d2)_hold. In particular, ${{{\mathcal F}}(V)}\subset {{{\mathcal_F}}(E)}$ is a noether normalization of_${{{\mathcal F}}(E)}$ and $V^n \cap {{\mathfrak{m}}}E^n ={{\mathfrak{m}}}V^n$_for all $n\geq 0$. As a consequence,_minimal reductions always exist. The_[*reduction number*]{}_of $E$, denoted by $r(E)$,_is the minimum_of $r_{U}(E)$,_where $U$ ranges_over all minimal reductions of $E$. \[hsop\]_If the residue_field is finite, a minimal set_of_generators of a_minimal_reduction_of $E$_is not necessarily_a_homogeneous system_of_parameters of ${{{\mathcal F}}(E)}$. Nevertheless, there_always_exist homogeneous systems of parameters of ${{{\mathcal_F}}(E)}$. This is equivalent_to_the existence of a_family of elements $a_1\in E^{r_1}\setminus_{{\mathfrak{m}}}E^{r_1}, \ldots, a_s\in E^{r_s}\setminus {{\mathfrak{m}}}E^{r_s}$, where_$s={\ell(E)}$, such_that for_some $r$, $E^r=a_1E^{r-r_1}+ \cdots + a_sE^{r-r_s}$; and ${\ell(E)}$ is the minimum positive_number for a such family of_elements to exist. \[as0\] Let_${(R,{{\mathfrak{m}}}, {

, we vary $S$-wave parameters to describe 61 energy levels taken from $A_1$ irreps: $[000](16,20,24)$, $[001](20,24)$, $[011](20,24)$, $[111](20,24)$ and $[002](20,24)$. The result, with $\chi^2/N_\mathrm{dof} = 49.1/(61-6) = 0.89$, is: ------------------------------ ------------------------------------------------- -- $m = $ $(0.2458 \pm 0.0014 \pm 0.0004) \cdot a_t^{-1}$ $g_{\pi K} = $ $(0.156 \pm 0.004 \pm 0.001) \cdot a_t^{-1}$ $g_{\eta K} = $ $( 0.027 \pm 0.008 \pm 0.008) \cdot a_t^{-1}$ $\gamma_{\pi K, \pi K} = $ $0.082 \pm 0.046 \pm 0.022$ $\gamma_{\pi K, \eta K} = $ $0.33 \pm 0.13 \pm 0.06$ $\gamma_{\eta K, \eta K} = $ $-0.41 \pm 0.05 \pm 0.07$ ------------------------------ ------------------------------------------------- -- $$\label{global}$$ {width="57.00000%"} The phase-shifts and inelasticity corresponding to this fit are shown in Fig. \[fig\_S\_wave\_global\] for the $S$-wave and in Fig. \[fig\_P\_wave\_global\] for the $P$-wave. An alternative approach in which all 80 levels are considered together, varying the $S$ and $P$-wave parameters simultaneously, leads to a solution statistically compatible with the one presented above. As with the $S$-wave fit using only at-rest points, we find only very weak coupling between the $\pi K$ and $\eta K$ channels, with an apparent weak repulsive interaction in the $\eta K$ channel and a gradual rise in the $\pi K$ phase-shift. As previously we note the rapid rise in the $\pi K$ phase-shift at threshold, followed by a slow increase through $90^\circ$ at higher energies. In Section 

, we vary $ S$-wave parameters to describe 61 energy degree accept from $ A_1 $ irreps: $ [ 000](16,20,24)$, $ [ 001](20,24)$, $ [ 011](20,24)$, $ [ 111](20,24)$ and $ [ 002](20,24)$. The result, with $ \chi^2 / N_\mathrm{dof } = 49.1/(61 - 6) = 0.89 $, is: ------------------------------ ------------------------------------------------- -- $ m = $ $ (0.2458 \pm 0.0014 \pm 0.0004) \cdot a_t^{-1}$ $ g_{\pi K } = $ $ (0.156 \pm 0.004 \pm 0.001) \cdot a_t^{-1}$ $ g_{\eta potassium } = $ $ (0.027 \pm 0.008 \pm 0.008) \cdot a_t^{-1}$ $ \gamma_{\pi K, \pi K } = $ $ 0.082 \pm 0.046 \pm 0.022 $ $ \gamma_{\pi K, \eta K } = $ $ 0.33 \pm 0.13 \pm 0.06 $ $ \gamma_{\eta K, \eta kilobyte } = $ $ -0.41 \pm 0.05 \pm 0.07 $ ------------------------------ ------------------------------------------------- -- $ $ \label{global}$$ ! [ image](fig15.pdf){width="57.00000% " } The phase - chemise and inelasticity corresponding to this fit are shown in Fig.   \[fig\_S\_wave\_global\ ] for the $ S$-wave and in Fig.   \[fig\_P\_wave\_global\ ] for the $ P$-wave. An alternative overture in which all 80 levels are considered in concert, varying the $ S$ and $ P$-wave parameters simultaneously, lead to a solution statistically compatible with the one presented above. As with the $ S$-wave fit use only at - rest points, we line up only very weak coupling between the $ \pi K$ and $ \eta K$ channels, with an apparent watery repulsive interaction in the $ \eta K$ channel and a gradual rise in the $ \pi K$ phase - shift. As previously we note the rapid raise in the $ \pi K$ phase - shift at threshold, followed by a slow increase through $ 90^\circ$ at high energies. In Section

, we vary $S$-wave parameters tu describe 61 enetgt levens taksn from $X_1$ irreps: $[000](16,20,24)$, $[001](20,24)$, $[011](20,24)$, $[111](20,24)$ and $[002](20,24)$. The result, wuth $\cyi^2/N_\mathrm{dof} = 49.1/(61-6) = 0.89$, is: ------------------------------ ------------------------------------------------- -- $m = $ $(0.2458 \'j 0.0014 \pm 0.0004) \gbot a_f^{-1}$ $g_{\pi K} = $ $(0.156 \pm 0.004 \pm 0.001) \cdot d_t^{-1}$ $g_{\eta K} = $ $( 0.027 \pm 0.008 \pm 0.008) \cdpt a_t^{-1}$ $\damms_{\[i K, \ii K} = $ $0.082 \pm 0.046 \pm 0.022$ $\gamma_{\pi L, \eta K} = $ $0.33 \pm 0.13 \pm 0.06$ $\hamma_{\eta K, \gfa H} = $ $-0.41 \pm 0.05 \pm 0.07$ ------------------------------ ------------------------------------------------- -- $$\label{global}$$ {widgh="57.00000%"} The phase-shifjs ajg inelasticmty cogresponding to this fht are xhown in Fig. \[fin\_S\_wavx\_gloval\] for the $S$-wave and in Fig. \[fig\_P\_wave\_glofal\] for tve $P$-wave. An alternarice apkroacv in qhizh zlk 80 leveld ace considersd together, varying the $S$ and $K$-wadv parameters aimultwnqously, leads to a solution statisticallj cojpatible with the one peesented above. As with the $S$-wavq fit using only at-rest points, we find only very feak rojplnkn begqefn the $\pi K$ and $\eta K$ channels, with an apparegf eesk repulsive ikteraction in the $\rtw L$ channel and x gradbzl rise in the $\pi K$ ohase-shyft. Aw previoufly ee note the rapid rise in tye $\pi K$ phasv-shidt at threshold, foplowed by a sloe inctease through $90^\circ$ at hngher snergies. In Section 

, we vary $S$-wave parameters to describe levels from $A_1$ $[000](16,20,24)$, $[001](20,24)$, $[011](20,24)$, with = 49.1/(61-6) = is: ------------------------------ ------------------------------------------------- $m = $ $(0.2458 \pm 0.0014 0.0004) \cdot a_t^{-1}$ $g_{\pi K} = $ $(0.156 \pm 0.004 \pm 0.001) \cdot $g_{\eta K} = $ $( 0.027 \pm 0.008 \pm 0.008) \cdot a_t^{-1}$ $\gamma_{\pi \pi = $0.082 0.046 \pm 0.022$ $\gamma_{\pi K, \eta K} = $ $0.33 \pm 0.13 \pm 0.06$ $\gamma_{\eta K, K} = $ $-0.41 \pm 0.05 \pm 0.07$ ------------------------------------------------- -- $$\label{global}$$ {width="57.00000%"} phase-shifts and inelasticity corresponding to fit shown in \[fig\_S\_wave\_global\] the and in Fig. for the $P$-wave. An alternative approach in which all 80 levels are considered together, varying the $S$ $P$-wave parameters to a statistically with one presented above. the $S$-wave fit using only at-rest only very weak coupling between the $\pi K$ $\eta K$ with an apparent weak repulsive interaction the $\eta K$ channel and a gradual rise the $\pi K$ phase-shift. As previously we note the rapid rise in the $\pi K$ threshold, followed by a increase through $90^\circ$ higher In

, we vary $S$-wave parameters to deScribe 61 enerGy levEls TakEn From $a_1$ irrEps: $[000](16,20,24)$, $[001](20,24)$, $[011](20,24)$, $[111](20,24)$ and $[002](20,24)$. The resulT, With $\Chi^2/N_\mathrm{dof} = 49.1/(61-6) = 0.89$, is: ------------------------------ ------------------------------------------------- -- $m = $ $(0.2458 \pm 0.0014 \pm 0.0004) \cdOt a_t^{-1}$ $g_{\Pi k} = $ $(0.156 \Pm 0.004 \pm 0.001) \CDoT a_t^{-1}$ $g_{\eTa K} = $ $( 0.027 \pm 0.008 \pm 0.008) \CDoT A_T^{-1}$ $\gaMmA_{\pI K, \pI K} = $ $0.082 \PM 0.046 \pM 0.022$ $\gammA_{\pi k, \eta K} = $ $0.33 \pm 0.13 \Pm 0.06$ $\gamma_{\eta k, \etA K} = $ $-0.41 \Pm 0.05 \pm 0.07$ ------------------------------ ------------------------------------------------- -- $$\label{gloBAl}$$ {wiDth="57.00000%"} The phase-sHifTs and iNeLasTIcity CorRespoNding tO This fiT are shown In fIg. \[fig\_S\_WAve\_globAL\] FoR the $s$-wave and in Fig. \[fig\_P\_WAvE\_Global\] for the $P$-wAve. An aLtERnATIve AppRoach in whiCh All 80 leVEls are cONsIDERed TOgether, varyinG the $S$ and $P$-waVE paRameteRs SimULtaneoUsly, lEaDS to A solution stAtisTically coMpatibLE with thE One presEnted aBovE. As With THe $s$-wAve FiT UsiNG oNly AT-reSt points, We FiNd onlY verY WEAK couPliNg beTween The $\pi K$ and $\eta K$ ChaNnelS, WitH an apParenT weaK rEpulsIve intEractIoN in the $\eta K$ channEl anD a gradual RisE iN thE $\pI K$ phaSE-shift. as pRevIously wE note thE RapId RISE iN the $\pi K$ phase-shift aT tHREsHold, follOwed by A SlOw INcrease tHrOugH $90^\cirC$ AT highEr enERgIes. In SecTion 

, we vary $S$-wave paramet ers to des cribe 61 en er gy l evel s taken from $ A _1$irreps: $[000](16,20,2 4)$,$[ 0 01]( 2 0, 24)$, $[011] ( 20 , 2 4)$ ,$[ 111 ]( 2 0, 24)$and $[002] (20,24)$.The r esult, with$ \c hi^2/N_\ma thr m{dof} = 49. 1/( 61-6)=0.8 9 $, is : --- ------ - ------ --------- -- - -- --- - ------- - - -- ---- ----------------- - -- - --------- -- $m = $ $(0. 2 458 \pm 0. 0 0 1 4 \ p m 0.0004) \cd ot a_t^{-1} $ $g_{ \p i K} = $ $(0.15 6 \p m 0.004 \ pm 0.0 0 1) \cdo t a_t^{- 1}$ $g_ { \e taK } = $ $( 0. 02 7\pm 0 .008 \ p m 0.0 08) \cd ot a_ t^{-1}$ $ \ gam ma_{\ pi K, \pi K } = $ $0.08 2 \pm 0 .046 \pm 0.022$ $\gamm a_{ \pi K, \et a K} =$ $0 .3 3 \ pm 0.13 \pm 0.06$ $\gamm a_ {\e ta K , \etaK} = $$-0.41 \ pm 0.0 5 \ pm 0.07$ ----- -------- ----- - ----------- -- - ------------- - -- - - -- - ---- --- ----------- ---- - -- - - $ $ \l abe l {glob al}$$ ! [i m age](fig15.pdf){wid th ="57.0 0000% "} The phase -shifts an d i nelastic ityc or r esponding to t his f it are sho w n in Fig . \[f ig\_S\_w ave\_glob a l \] for t he$S$ -wa vea n din Fig. \[fig \ _ P\_w av e\_glob al\ ] for t he$P$ -wa ve. A n alterna tive app ro ac hin wh ich a l l 80 lev el s a re co nside r ed tog ether , va ry in g th e $S$ a n d$ P $-wa ve p aram ete rs simu ltan e ous ly, lea ds to a s olu t ionst at istical ly compatible w ith the on epre sented a bove. A s with the $S$-wave fit using o nly at-r estpoints, w e f ind on lyv ery we ak cou pling b etw e e n the $ \p i K $and $\etaK $ ch annel s, wit h an ap parent weak repuls i veinteraction i n t he $ \ e ta K$ ch a nne la nda gradual rise in the $\piK$ ph ase-shift. Aspr eviousl y we no te th e rapidrise in t he $\pi K $phas e - shi ft at thre shold, f ollowed b y a sl o wincre ase throu gh $9 0^\ci rc$ at hig her e nergie s. In Se ction  

, we_vary $S$-wave_parameters to describe 61_energy levels_taken_from $A_1$_irreps:_$[000](16,20,24)$, $[001](20,24)$, $[011](20,24)$,_$[111](20,24)$ and $[002](20,24)$._The result, with $\chi^2/N_\mathrm{dof}_= 49.1/(61-6) =_0.89$,_is: ------------------------------ ------------------------------------------------- -- __ ___ _ _ $m_= $ $(0.2458 \pm 0.0014 \pm 0.0004) \cdot_a_t^{-1}$_ _ _ _ $g_{\pi_K}_=_$ $(0.156 \pm 0.004_\pm 0.001) \cdot a_t^{-1}$ _ _ _ _ $g_{\eta K} = $_$( 0.027_\pm 0.008 \pm 0.008) \cdot_a_t^{-1}$ _ _ _ $\gamma_{\pi K, \pi_K} = $_$0.082 \pm 0.046 \pm 0.022$ __ ___ _ __ __ _ _ $\gamma_{\pi K, \eta_K} = $ $0.33_\pm_0.13 \pm 0.06$ _ _ _ _ _ _ $\gamma_{\eta K, \eta K}_= $ $-0.41 \pm_0.05 \pm_0.07$_ __ _ _ _ _ __------------------------------ ------------------------------------------------- -- $$\label{global}$$ {width="57.00000%"} The phase-shifts and inelasticity_corresponding to this fit are shown_in Fig. \[fig\_S\_wave\_global\] for the $S$-wave_and_in_Fig. \[fig\_P\_wave\_global\] for the $P$-wave. An_alternative approach in which all 80_levels are considered_together, varying the $S$ and $P$-wave parameters_simultaneously,_leads to a solution statistically compatible_with_the one presented above. As with the_$S$-wave_fit_using only at-rest points, we_find only very weak coupling between_the $\pi K$ and $\eta K$ channels, with an_apparent weak repulsive_interaction in the $\eta K$_channel_and_a gradual rise in the $\pi K$ phase-shift. As previously_we note_the rapid rise_in the $\pi K$ phase-shift at threshold, followed by a_slow increase through $90^\circ$ at higher energies._In Section 

8pt{F}\,({p}\,,{\gamma}) @Angulo_99 \,{}^{20}\kern-0.8pt{Ne}\,$]{} [$\rm\,{}^{20}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) @Angulo_99 \,{}^{21}\kern-0.8pt{Na}\,$]{} [$\rm\,{}^{21}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) @Iliadis_etal01 \,{}^{22}\kern-0.8pt{Na}\,$]{} [$\rm\,{}^{22}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) @Hale_etal02 \,{}^{23}\kern-0.8pt{Na}\,$]{} [$\rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma}) @Hale_etal04 \,{}^{}\kern-0.8pt{\,^4He + ^{20}\kern-2.0pt{Ne}}\,$]{} [$\rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma}) @Hale_etal04 \,{}^{24}\kern-0.8pt{Mg}\,$]{} [$\rm\,{}^{24}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_etal01 \,{}^{25}\kern-0.8pt{Al}\,$]{} [$\rm\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_etal01 \,{}^{26}\kern-0.8pt{Al^g}\,$]{} [$\rm\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_etal01 \,{}^{26}\kern-0.8pt{Al^m}\,$]{} [$\rm\,{}^{26}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_etal01 \,{}^{27}\kern-0.8pt{Al}\,$]{}

8pt{F}\,({p}\,,{\gamma }) @Angulo_99 \,{}^{20}\kern-0.8pt{Ne}\,$ ] { } [ $ \rm\,{}^{20}\kern-0.8pt{Ne}\,({p}\,,{\gamma }) @Angulo_99 \,{}^{21}\kern-0.8pt{Na}\,$ ] { } [ $ \rm\,{}^{21}\kern-0.8pt{Ne}\,({p}\,,{\gamma }) @Iliadis_etal01 \,{}^{22}\kern-0.8pt{Na}\,$ ] { } [ $ \rm\,{}^{22}\kern-0.8pt{Ne}\,({p}\,,{\gamma }) @Hale_etal02 \,{}^{23}\kern-0.8pt{Na}\,$ ] { } [ $ \rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma }) @Hale_etal04 \,{}^{}\kern-0.8pt{\,^4He + ^{20}\kern-2.0pt{Ne}}\,$ ] { } [ $ \rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma }) @Hale_etal04 \,{}^{24}\kern-0.8pt{Mg}\,$ ] { } [ $ \rm\,{}^{24}\kern-0.8pt{Mg}\,({p}\,,{\gamma }) @Iliadis_etal01 \,{}^{25}\kern-0.8pt{Al}\,$ ] { } [ $ \rm\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gamma }) @Iliadis_etal01 \,{}^{26}\kern-0.8pt{Al^g}\,$ ] { } [ $ \rm\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gamma }) @Iliadis_etal01 \,{}^{26}\kern-0.8pt{Al^m}\,$ ] { } [ $ \rm\,{}^{26}\kern-0.8pt{Mg}\,({p}\,,{\gamma }) @Iliadis_etal01 \,{}^{27}\kern-0.8pt{Al}\,$ ] { }

8pt{F}\,({o}\,,{\gamma}) @Angulo_99 \,{}^{20}\kern-0.8't{Ne}\,$]{} [$\rm\,{}^{20}\keen-0.8pt{Ne}\,({p}\,,{\gamma}) @Antulo_99 \,{}^{21}\kern-0.8pt{Ne}\,$]{} [$\rk\,{}^{21}\kern-0.8pt{Ne}\,({p}\,,{\gammd}) @Iliadis_etal01 \,{}^{22}\kern-0.8py{Nw}\,$]{} [$\rm\,{}^{22}\kern-0.8pt{Ne}\,({p}\,,{\gamja}) @Hale_etal02 \,{}^{23}\kern-0.8pt{Nw}\,$]{} [$\rm\,{}^{23}\kern-0.8pg{Na}\,({p}\,,{\gamma}) @Hale_etal04 \,{}^{}\kern-0.8pt{\,^4Hd + ^{20}\kexn-2.0pt{Ne}}\,$]{} [$\rm\,{}^{23}\keri-0.8pt{Na}\,({p}\,,{\damma}) @Hale_etal04 \,{}^{24}\kecn-0.8pt{Ng}\,$]{} [$\rm\,{}^{24}\kesn-0.8't{Mg}\,({p}\,,{\gamma}) @Iuuadks_eual01 \,{}^{25}\kern-0.8ot{Am}\,$]{} [$\gk\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gajma}) @Iliadis_etal01 \,{}^{26}\kern-0.8kt{Al^g}\,$]{} [$\rm\,{}^{25}\ketn-0.8pt{Mg}\,({p}\,,{\gamia}) @Iliadis_etal01 \,{}^{26}\kern-0.8pt{An^m}\,$]{} [$\rm\,{}^{26}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_rtwl01 \,{}^{27}\kern-0.8pt{Al}\,$]{}

8pt{F}\,({p}\,,{\gamma}) @Angulo_99 \,{}^{20}\kern-0.8pt{Ne}\,$]{} [$\rm\,{}^{20}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) @Angulo_99 \,{}^{21}\kern-0.8pt{Na}\,$]{} [$\rm\,{}^{21}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) [$\rm\,{}^{22}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) \,{}^{23}\kern-0.8pt{Na}\,$]{} [$\rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma}) \,{}^{}\kern-0.8pt{\,^4He + ^{20}\kern-2.0pt{Ne}}\,$]{} \,{}^{25}\kern-0.8pt{Al}\,$]{} @Iliadis_etal01 \,{}^{26}\kern-0.8pt{Al^g}\,$]{} [$\rm\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) \,{}^{26}\kern-0.8pt{Al^m}\,$]{} [$\rm\,{}^{26}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_etal01

8pt{F}\,({p}\,,{\gamma}) @Angulo_99 \,{}^{20}\kern-0.8pt{Ne}\,$]{} [$\rm\,{}^{20}\Kern-0.8pt{Ne}\,({p}\,,{\gAmma}) @ANguLo_99 \,{}^{21}\kErN-0.8pt{NA}\,$]{} [$\rm\,{}^{21}\kErn-0.8pt{Ne}\,({p}\,,{\gamma}) @ILIadiS_etal01 \,{}^{22}\kern-0.8pt{Na}\,$]{} [$\rm\,{}^{22}\kern-0.8pt{NE}\,({p}\,,{\gamMa}) @hAle_eTAl02 \,{}^{23}\Kern-0.8pT{Na}\,$]{} [$\rm\,{}^{23}\keRN-0.8pT{nA}\,({p}\,,{\gAmMa}) @halE_eTAl04 \,{}^{}\Kern-0.8pT{\,^4He + ^{20}\Kern-2.0pt{NE}}\,$]{} [$\rm\,{}^{23}\kern-0.8pt{NA}\,({p}\,,{\gAmMa}) @Hale_etal04 \,{}^{24}\keRN-0.8pT{Mg}\,$]{} [$\rm\,{}^{24}\kern-0.8pT{Mg}\,({P}\,,{\gamma}) @IliadiS_etAl01 \,{}^{25}\kern-0.8Pt{al}\,$]{} [$\rM\,{}^{25}\Kern-0.8pT{Mg}\,({P}\,,{\gammA}) @IliadIS_etal01 \,{}^{26}\kErn-0.8pt{Al^g}\,$]{} [$\rM\,{}^{25}\kERn-0.8pt{Mg}\,({P}\,,{\Gamma}) @IlIADiS_etaL01 \,{}^{26}\kern-0.8pt{Al^m}\,$]{} [$\rm\,{}^{26}\kern-0.8pT{mg}\,({P}\,,{\Gamma}) @Iliadis_etAl01 \,{}^{27}\kern-0.8Pt{aL}\,$]{}

8pt{F}\,({p}\,,{\gamma}) @An gulo _99 \,{}^{ 2 0}\k ern-0.8pt{Ne}\,$]{} [ $\ r m\ ,{}^{ 20} \kern-0 .8pt{Ne}\, ({p }\ ,,{\gamma}) @Ang ulo _99 \,{ } ^{21} \ke rn-0. 8pt{Na } \,$]{} [$\rm\, { }^ { 21}\kern-0.8pt {Ne}\, ({ p }\ , , {\g amm a}) @ I l iad i s_etal01 \,{}^{22}\k e rn- 0.8pt{ Na }\, $ ]{} [ $\rm\,{ } ^{22}\k ern-0. 8pt {Ne }\,( { p} \, ,{\ ga m ma} ) @ Hale _ e t a l02 \,{ }^{23 }\kern-0.8pt{ Na} \,$] { } [$\rm\, {}^{ 23}\kern- 0.8 pt {Na }\ ,({p} \ ,,{\ga mma }) @ Hale_etal04 \, {} ^ { }\ kern-0.8 pt{\,^ 4 He + ^{20}\ke rn -2. 0pt{ N e }}\,$ ]{} [$ \r m\,{}^{ 23 }\kern -0 .8p t{N a}\,( { p}\, ,{\gam ma}) @Hale_etal04 \ ,{ } ^{24 }\k ern-0.8pt{M g}\, $ ]{} [$\ rm \,{}^{ 24}\k ern-0.8pt{Mg} \,({p}\,,{ \ g a mma}) @Ilia dis_etal01 \,{} ^{25} \kern-0. 8pt{Al}\, $ ] {} [$\rm\ ,{} ^{25}\k ern -0. 8pt {Mg }\ ,({p}\,,{ \gamma}) @I lia di s_e tal01 \, {}^{2 6}\k er n- 0 .8p t{Al^g} \ ,$ ] { } [$\ rm\ , {}^{ 25 }\ kern-0. 8pt{Mg}\,({p} \, ,{\gamma}) @Iliadis_etal01 \,{}^{2 6}\ kern- 0.8p t{Al^m}\, $]{ } [$\ rm \,{}^{26}\ k e rn- 0.8pt {M g}\, ({p}\,, {\gamma}) @ Ili adis _ e ta l01 \, {} ^ {27 } \ kern-0.8pt{Al}\ ,$]{}

8pt{F}\,({p}\,,{\gamma}) _ _ _ __ __ _ _ _ @Angulo_99 __ \,{}^{20}\kern-0.8pt{Ne}\,$]{} __ ___ _ _ _ __ _ [$\rm\,{}^{20}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) _ _ ___ _ @Angulo_99 _\,{}^{21}\kern-0.8pt{Na}\,$]{} _ _ _ _ _ _ _ _ _ [$\rm\,{}^{21}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) _ _ __ ___ _ __ @Iliadis_etal01 __ \,{}^{22}\kern-0.8pt{Na}\,$]{} __ _ __ _ _ _ _ _ [$\rm\,{}^{22}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) _ _ _ __ __ _ @Hale_etal02 \,{}^{23}\kern-0.8pt{Na}\,$]{} _ _ _ __ _ _ ___ _ [$\rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma}) _ _ __ __ @Hale_etal04 ___\,{}^{}\kern-0.8pt{\,^4He + ^{20}\kern-2.0pt{Ne}}\,$]{} _ _ [$\rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma})_ _ ___ _ _ @Hale_etal04 _ \,{}^{24}\kern-0.8pt{Mg}\,$]{} _ _ _ _ _ [$\rm\,{}^{24}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) __ ____ _ _ _ _ @Iliadis_etal01 _ _\,{}^{25}\kern-0.8pt{Al}\,$]{} __ _ _ __ _ _ [$\rm\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) _ __ _ _ _@Iliadis_etal01 _ \,{}^{26}\kern-0.8pt{Al^g}\,$]{}_ _ _ _ __ ____ _ _ _ __[$\rm\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) _ _ _ _ @Iliadis_etal01 \,{}^{26}\kern-0.8pt{Al^m}\,$]{} _ _ _ __ _ _ __ [$\rm\,{}^{26}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) ___ __ _____@Iliadis_etal01 \,{}^{27}\kern-0.8pt{Al}\,$]{}

is square integrable modulo $Z$, the function $$\phi_{ik}(x)=\overline{c_{e_i,v_k}(s(x))}c_{e_i,u_k}(s(x)) = \ip{v_k}{\pi(s(x))e_i}\ip{e_i}{\pi(s(x))^{-1}u_k}$$ is $\alpha$-integrable on $X$ and, using the Hölder inequality and the orthogonality relations (\[ort1\]), \_E |\_[ik]{}(x)|d(x) & & ( \_E |c\_[e\_i,v\_k]{}(s(x))|\^2d(x) )\^[12]{}\ & & ( \_E |c\_[e\_i,u\_k]{}(x)|\^2d(x) )\^[12]{}\ && \_\_\ & & \^2\_\_\_= 1. Since $\sum_{i,k}\lambda_i w_k = \nor{T}_1\nor{\W}_1=\nor{\W}_1$, the series $\sum_{i,k}\lambda_i w_k \phi_{ik}$ converges $\alpha$-almost everywhere to an integrable function $\phi$ and $$\int_E\phi(x)\,d\alpha(x)=\sum_{i,k}\lambda_i w_k \int_E\phi_{ik}\,d\alpha(x).$$ On the other hand, for $\alpha$-almost all $x\in X$, $\phi(x)=\tr{\W\pi(s(x))T\pi(s(x))^{-1}}$. Hence $\int_E |\tr{\W\pi(s(x))T\pi(s(x))^{-1}}|\,d\alpha(x) \leq \nor{\W}_1$ and the linear form $$\W\mapsto \int_E \tr{\W\pi(s(x))T\pi(s(x))^{-1}}\,d\alpha(x)$$ is continuous on the Banach space of the trace class operators. Therefore it defines a bounded operator $Q_T(E)$ such that & = & \_E d(x)\ & = & \_[i,k]{}\_i w\_k \_E d(x)\ & = & \_[i,k]{}\_i w\_k \_E

is square integrable modulo $ Z$, the function $ $ \phi_{ik}(x)=\overline{c_{e_i, v_k}(s(x))}c_{e_i, u_k}(s(x) ) = \ip{v_k}{\pi(s(x))e_i}\ip{e_i}{\pi(s(x))^{-1}u_k}$$ is $ \alpha$-integrable on $ X$ and, using the Hölder inequality and the orthogonality relations   (\[ort1\ ]), \_E |\_[ik]{}(x)|d(x) & & (\_E |c\_[e\_i, v\_k]{}(s(x))|\^2d(x) ) \^[12]{}\ & & (\_E |c\_[e\_i, u\_k]{}(x)|\^2d(x) ) \^[12]{}\ & & \_\_\ & & \^2\_\_\_= 1. Since $ \sum_{i, k}\lambda_i w_k = \nor{T}_1\nor{\W}_1=\nor{\W}_1 $, the serial $ \sum_{i, k}\lambda_i w_k \phi_{ik}$ converge $ \alpha$-almost everywhere to an integrable function $ \phi$ and $ $ \int_E\phi(x)\,d\alpha(x)=\sum_{i, k}\lambda_i w_k \int_E\phi_{ik}\,d\alpha(x).$$ On the other bridge player, for $ \alpha$-almost all $ x\in X$, $ \phi(x)=\tr{\W\pi(s(x))T\pi(s(x))^{-1}}$. therefore $ \int_E |\tr{\W\pi(s(x))T\pi(s(x))^{-1}}|\,d\alpha(x) \leq \nor{\W}_1 $ and the linear form $ $ \W\mapsto \int_E \tr{\W\pi(s(x))T\pi(s(x))^{-1}}\,d\alpha(x)$$ is continuous on the Banach space of the touch class operators. consequently it define a bounded hustler $ Q_T(E)$ such that & = & \_E d(x)\ & = & \_[i, k]{}\_i w\_k \_E d(x)\ & = & \_[i, k]{}\_i w\_k \_E

is square integrable modulu $Z$, the functiou $$\phi_{ik}(e)=\overlihe{c_{e_i,v_k}(s(b))}c_{e_i,u_k}(s(x)) = \ip{v_k}{\pi(s(x))e_i}\ip{e_i}{\pi(s(x))^{-1}u_n}$$ us $\alkka$-integrable on $X$ ana, using tje Hölder inewyality and the ortmjgonzpity celations (\[ort1\]), \_E |\_[lk]{}(x)|d(x) & & ( \_E |w\_[e\_i,v\_k]{}(s(x))|\^2d(x) )\^[12]{}\ & & ( \_E |c\_[d\_i,b\_k]{}(x)|\^2d(x) )\^[12]{}\ && \_\_\ & & \^2\_\_\_= 1. Since $\sum_{i,k}\lambda_i w_k = \nor{E}_1\nor{\W}_1=\not{\W}_1$, the series $\sui_{i,k}\lsibda_j w_k \phi_{ik}$ converges $\alpha$-almost eberywhege to an integrabke function $\phi$ and $$\int_E\phl(x)\,d\appha(x)=\sum_{i,k}\lambda_i a_k \int_E\phi_{ij}\,d\al[ya(x).$$ On the ogher hand, for $\alpha$-almkst all $x\in X$, $\phi(x)=\tr{\W\pi(s(x))T\pi(s(x))^{-1}}$. Fence $\int_E |\tr{\W\pu(s(z))T\pl(v(x))^{-1}}|\,d\alpha(x) \lxq \nor{\R}_1$ and the likvar fork $$\W\mapsyo \int_E \tr{\W\pi(s(w))T\pi(s(e))^{-1}}\,d\alpha(x)$$ is continuous on the Banach space jf the trdcz class operators. Theeedore ht dafindw a bohnved operahor $Q_T(E)$ such fhat & = & \_E d(z)\ & = & \_[i,k]{}\_i w\_k \_E d(x)\ & = & \_[i,l]{}\_i q\_k \_E

is square integrable modulo $Z$, the function \ip{v_k}{\pi(s(x))e_i}\ip{e_i}{\pi(s(x))^{-1}u_k}$$ $\alpha$-integrable on and, using the relations \_E |\_[ik]{}(x)|d(x) & ( \_E |c\_[e\_i,v\_k]{}(s(x))|\^2d(x) & & ( \_E |c\_[e\_i,u\_k]{}(x)|\^2d(x) )\^[12]{}\ \_\_\ & & \^2\_\_\_= 1. Since $\sum_{i,k}\lambda_i w_k = \nor{T}_1\nor{\W}_1=\nor{\W}_1$, the series $\sum_{i,k}\lambda_i \phi_{ik}$ converges $\alpha$-almost everywhere to an integrable function $\phi$ and $$\int_E\phi(x)\,d\alpha(x)=\sum_{i,k}\lambda_i w_k \int_E\phi_{ik}\,d\alpha(x).$$ the hand, $\alpha$-almost $x\in X$, $\phi(x)=\tr{\W\pi(s(x))T\pi(s(x))^{-1}}$. Hence $\int_E |\tr{\W\pi(s(x))T\pi(s(x))^{-1}}|\,d\alpha(x) \leq \nor{\W}_1$ and the linear form $$\W\mapsto \int_E \tr{\W\pi(s(x))T\pi(s(x))^{-1}}\,d\alpha(x)$$ is on the Banach space of the trace class Therefore it defines a operator $Q_T(E)$ such that & & d(x)\ & & w\_k d(x)\ & = \_[i,k]{}\_i w\_k \_E

is square integrable modulo $Z$, The functioN $$\phi_{iK}(x)=\oVerLiNe{c_{e_I,v_k}(s(X))}c_{e_i,u_k}(s(x)) = \ip{v_k}{\pi(S(X))e_i}\iP{e_i}{\pi(s(x))^{-1}u_k}$$ is $\alpha$-integrAble oN $X$ ANd, usINg The HöLder ineQUaLITy aNd ThE orThOGoNalitY reLations (\[Ort1\]), \_E |\_[ik]{}(x)|d(x) & & ( \_E |C\_[e\_i,V\_k]{}(S(x))|\^2d(x) )\^[12]{}\ & & ( \_E |c\_[e\_i,u\_k]{}(x)|\^2d(X) )\^[12]{}\ && \_\_\ & & \^2\_\_\_= 1. siNce $\sum_{i,k}\laMbdA_i w_k = \nor{T}_1\nor{\W}_1=\Nor{\w}_1$, the seRiEs $\sUM_{i,k}\laMbdA_i w_k \pHi_{ik}$ coNVerges $\Alpha$-almoSt EVerywhERe to an iNTEgRablE function $\phi$ and $$\inT_e\pHI(x)\,d\alpha(x)=\sum_{i,k}\Lambda_I w_K \InT_e\Phi_{Ik}\,d\Alpha(x).$$ On thE oTher hANd, for $\alPHa$-ALMOst ALl $x\in X$, $\phi(x)=\tr{\W\Pi(s(x))T\pi(s(x))^{-1}}$. HeNCe $\iNt_E |\tr{\W\Pi(S(x))T\PI(s(x))^{-1}}|\,d\alPha(x) \lEq \NOr{\W}_1$ And the lineaR forM $$\W\mapsto \iNt_E \tr{\W\PI(s(x))T\pi(s(X))^{-1}}\,D\alpha(x)$$ Is contInuOus On thE baNaCh sPaCE of THe TraCE clAss operaToRs. thereFore IT DEFineS a bOundEd opeRator $Q_T(E)$ such tHat & = & \_e d(x)\ & = & \_[i,K]{}\_I w\_k \_e d(x)\ & = & \_[i,k]{}\_I w\_k \_E

is square integrable modu lo $Z$, th e fun cti on$$ \phi _{ik }(x)=\overline { c_{e _i,v_k}(s(x))}c_{e_i,u _k}(s (x ) ) =\ ip {v_k} {\pi(s( x )) e _ i}\ ip {e _i} {\ p i( s(x)) ^{- 1}u_k}$ $ is $\alp ha$ -i ntegrable on $X $ and, usi ngthe Hölder i neq uality a ndt he or tho gonal ity re l ations  (\[ort1\ ]) , \_E | \ _[ik]{} ( x )| d(x) & & ( \_E |c\_[e \ _i , v\_k]{}(s(x))| \^2d(x )) \^ [ 1 2]{ }\& & ( \_E|c \_[e\ _ i,u\_k] { }( x ) | \^2 d (x) )\^[12]{} \ && \_\_\& &\^2\_\ _\ _=1 . Sinc e $\s um _ {i, k}\lambda_i w_k = \nor{T }_1\no r {\W}_1= \ nor{\W} _1$, t heser ies$ \s um _{i ,k } \la m bd a_i w_k \phi_{i k} $conve rges $ \ a lpha $-a lmos t eve rywhere to an in tegr a ble func tion$\ph i$ and$$\int _E\ph i( x)\,d\alpha(x)= \sum _{i,k}\la mbd a_ i w _k \int _ E\phi_ {ik }\, d\alpha (x).$$O n t he o t he r hand, for $\alph a$ - a lm ost all$x\inX $, $ \ phi(x)=\ tr {\W \pi( s ( x))T\ pi(s ( x) )^{-1}}$ . Henc e $ \i nt_E |\ tr {\W\pi (s (x) )T\ pi(s( x ))^{ -1}}|\ ,d\alpha (x) \ l eq \nor{\W}_1 $ and the line a rf o rm $$\W \ma psto \int_E \tr { \W\p i(s( x )) T\p i (s(x) )^{-1 }} \ ,d \ alpha(x)$$ is conti nu ous on theBanach spaceof the tra c e class op erat o rs . Therefore itdefin es a bound e d operat or $Q _T(E)$ s uch that& = & \_Ed(x )\& = &\ _ [i ,k]{}\_i w\_k \ _E d (x )\ & =& \ _[i,k]{ }\_ i w \_k \_ E

is_square integrable_modulo $Z$, the function_$$\phi_{ik}(x)=\overline{c_{e_i,v_k}(s(x))}c_{e_i,u_k}(s(x)) = \ip{v_k}{\pi(s(x))e_i}\ip{e_i}{\pi(s(x))^{-1}u_k}$$_is_$\alpha$-integrable on_$X$_and, using the_Hölder inequality and_the orthogonality relations (\[ort1\]), \_E_|\_[ik]{}(x)|d(x) & &_(_\_E |c\_[e\_i,v\_k]{}(s(x))|\^2d(x) )\^[12]{}\ & & ( \_E |c\_[e\_i,u\_k]{}(x)|\^2d(x) )\^[12]{}\ && \_\_\ & & \^2\_\_\_= 1. Since $\sum_{i,k}\lambda_i_w_k_= \nor{T}_1\nor{\W}_1=\nor{\W}_1$, the_series_$\sum_{i,k}\lambda_i_w_k \phi_{ik}$ converges $\alpha$-almost everywhere_to an integrable function $\phi$_and $$\int_E\phi(x)\,d\alpha(x)=\sum_{i,k}\lambda_i_w_k \int_E\phi_{ik}\,d\alpha(x).$$ On the other hand, for $\alpha$-almost_all_$x\in X$, $\phi(x)=\tr{\W\pi(s(x))T\pi(s(x))^{-1}}$._Hence $\int_E |\tr{\W\pi(s(x))T\pi(s(x))^{-1}}|\,d\alpha(x) \leq \nor{\W}_1$ and the linear form_$$\W\mapsto \int_E \tr{\W\pi(s(x))T\pi(s(x))^{-1}}\,d\alpha(x)$$ is continuous on_the Banach space_of_the_trace class operators. Therefore_it defines a bounded operator $Q_T(E)$_such that & = & \_E_d(x)\ & = & \_[i,k]{}\_i w\_k \_E d(x)\ &_= & \_[i,k]{}\_i w\_k \_E

low-mass halos and high redshift universe, gas is acquired primarily through the cold mode accretion, by which cold gas flows can directly feed galaxies through cosmic filaments [@Keres; @2005; @Dekel; @2009a; @Dekel; @2009b; @van; @de; @Voort; @2011]. When a galaxy’s dark matter halo grows massive enough to support a stable shock, the infalling gas is first shock-heated to near the viral temperature ($T\sim 10^{6}K$), then radiatively cools and settles into galaxies in a quasi-spherical manner. The transition of these two accretion modes is expected to occur near the critical halo mass, $M_{\rm c}\sim 10^{12}M_{\sun}$ [@Dekel; @2006]. To justify this, it is important to investigate the behavior of gas accretion when a galaxy evolves across $M_{\rm c}$. Simulations suggest that the gas accretion behavior indeed changes near $M_{\rm c}$ [@Stewart; @2011], but observational confirmation of this is still lacking. Observationally, gas flow signatures have been unambiguously detected in the high-quality spectra of SFGs [e.g., @Heckman; @1990; @Sato; @2009; @Weiner; @2009; @Genzel; @2014a; @Rubin; @2014; @Cicone; @2016]. Nevertheless, the detailed properties of gas flows are still difficult to quantify directly. This is because gas flows can occur in multi-phase, and the global gas flow rates depend on the 3D motions and densities of the gas. Indirect methods are thus useful in studying gas flows. For example, early attempts have tried to set constraints on gas flows by modeling the chemical evolution of SFGs to match the observed mass-metallicity relation [@Finlator; @2008; @Spitoni; @2010; @Lilly; @2013; @Yabe; @2015; @Spitoni; @2017]. The assembly history of Milky Way-mass ($M_{\rm MW}\sim 5\times 10^{10}M_{\sun}$, see @McMillan [@2017]) galaxies has recently attracted much attentions, since galaxies near $M_{\rm MW}$ appear quite typical and dominate the stellar mass budget in the local Universe [@vanDokkum; @2013]. Several works have tried to trace the evolution of star formation and

low - mass halos and high red shift population, natural gas is acquired chiefly through the cold mode accretion, by which cold natural gas flows can directly run galaxies through cosmic filaments [ @Keres; @2005; @Dekel; @2009a; @Dekel; @2009b; @van; @de; @Voort; @2011 ]. When a galax ’s blue matter halo grow massive enough to support a stable shock, the infalling gas is first shock - heat to near the viral temperature ($ T\sim 10^{6}K$), then radiatively cools and settles into galaxies in a quasi - ball-shaped manner. The transition of these two accretion modes is expect to occur near the critical aura mass, $ M_{\rm c}\sim 10^{12}M_{\sun}$ [ @Dekel; @2006 ]. To justify this, it is important to investigate the behavior of gas accretion when a galaxy develop across $ M_{\rm c}$. Simulations suggest that the gas accretion behavior indeed changes near $ M_{\rm c}$ [ @Stewart; @2011 ], but observational confirmation of this is still lacking. Observationally, gas flow signatures have been unambiguously detected in the high - quality spectra of SFGs [ for example, @Heckman; @1990; @Sato; @2009; @Weiner; @2009; @Genzel; @2014a; @Rubin; @2014; @Cicone; @2016 ]. however, the detailed properties of gas flows are however difficult to quantify directly. This is because gas menstruation can occur in multi - phase, and the global gas flow pace depend on the 3D motions and densities of the gas. Indirect methods are therefore useful in studying gas flows. For example, early attempts have tried to adjust constraints on gas flows by modeling the chemical evolution of SFGs to pit the observed mass - metallicity relation [ @Finlator; @2008; @Spitoni; @2010; @Lilly; @2013; @Yabe; @2015; @Spitoni; @2017 ]. The fabrication history of Milky Way - mass ($ M_{\rm MW}\sim 5\times 10^{10}M_{\sun}$, see @McMillan [ @2017 ]) galaxies has recently attract much attentions, since galaxies near $ M_{\rm MW}$ look quite typical and dominate the stellar mass budget in the local Universe [ @vanDokkum; @2013 ]. respective works have tried to hound the evolution of star geological formation and

loa-mass halos and high redrhift universe, yqs is ecquires primarkly through the cold mode acrretuon, bt which cold gas flows can direbtly feed galexies through cosmic filaments [@Nerev; @2005; @Dekel; @2009a; @Dekek; @2009b; @van; @de; @Voort; @2011]. When a gxlcxy’s dark matter halo grows massive qnough yo support a stafle xrock, nht infalling gas is first shock-heafed to iear the viral yemperature ($T\sim 10^{6}K$), then rafiatlvely cools and sehtles into talavues in a quari-spherical manner. The transition of these two accretkon mpdes is exkzxtef to occur nxar thv critical halo mass, $K_{\rm c}\sik 10^{12}M_{\sun}$ [@Dekel; @2006]. Bo juvtidy this, it is importait to investigate thg behavior oy gas accretion when q talaxi evonves qcruss $M_{\cm d}$. Simupatmons suggesf that the tas accretion behavoow indeed changss neaw $I_{\rm c}$ [@Stewart; @2011], but observational confirkatjon of this is still laxking. Observationally, has flow fignatures have been unambiguously detected in tha higi-qjalnbn spdxtga of SFGs [e.g., @Heckman; @1990; @Sato; @2009; @Weiner; @2009; @Genzel; @2014a; @Dunik; @2014; @Cicone; @2016]. Neverbheless, the detailrd ptjperties of gxs floca zre still difficulh to quwntift directlr. Thos is because gas flows can occur in mujri-phase, and the gllbal gas fluw rstes cepend on the 3D motions and densities lf the gaa. Indirect methodr age tvus useful in studying gas flows. Foc exakple, eafly sttempes have trled to set constraints oj gas fnows by mofeling the chemical evolution oh SFGs to matvh thv observeb mass-ketallicity rqlation [@Finlatpr; @2008; @Spiconi; @2010; @Uilly; @2013; @Yabv; @2015; @Spitonm; @2017]. The assembjy history of Lilky Way-masv ($M_{\rm MW}\fim 5\rimew 10^{10}M_{\sun}$, rde @McMillan [@2017]) gslaxies hcf rexently attracted mmch ajtsntions, since gcuqxies near $M_{\rm KW}$ xppqag qnite efpical and dmminxte yhe sgellar mass cudgrt in the local Univarse [@vanDokkum; @2013]. Severak rorks hace tried to trace the evolution of star formetion end

low-mass halos and high redshift universe, gas primarily the cold accretion, by which feed through cosmic filaments @2005; @Dekel; @2009a; @2009b; @van; @de; @Voort; @2011]. When galaxy’s dark matter halo grows massive enough to support a stable shock, the gas is first shock-heated to near the viral temperature ($T\sim 10^{6}K$), then radiatively and into in quasi-spherical manner. The transition of these two accretion modes is expected to occur near the critical mass, $M_{\rm c}\sim 10^{12}M_{\sun}$ [@Dekel; @2006]. To justify it is important to the behavior of gas accretion a evolves across c}$. suggest the gas accretion indeed changes near $M_{\rm c}$ [@Stewart; @2011], but observational confirmation of this is still lacking. Observationally, gas signatures have detected in high-quality of [e.g., @Heckman; @1990; @Weiner; @2009; @Genzel; @2014a; @Rubin; @2014; the detailed properties of gas flows are still to quantify This is because gas flows can in multi-phase, and the global gas flow rates on the 3D motions and densities of the gas. Indirect methods are thus useful in flows. For example, early have tried to constraints gas by the chemical of SFGs to match the observed mass-metallicity relation [@Finlator; @2008; @Spitoni; @Lilly; @2013; @Yabe; @2015; @Spitoni; @2017]. The assembly history of ($M_{\rm 5\times 10^{10}M_{\sun}$, see [@2017]) galaxies has recently much since galaxies near $M_{\rm quite and mass in local Universe [@vanDokkum; @2013]. works have tried to trace evolution of star formation

low-mass halos and high redshiFt universe, Gas is AcqUirEd PrimArilY through the colD Mode Accretion, by which cold gaS flowS cAN dirECtLy feeD galaxiES tHROugH cOsMic FiLAmEnts [@KEreS; @2005; @Dekel; @2009a; @dekel; @2009b; @van; @dE; @VoOrT; @2011]. When a galaxy’S DaRk matter haLo gRows massive eNouGh to suPpOrt A StablE shOck, thE infalLIng gas Is first shOcK-Heated TO near thE VIrAl teMperature ($T\sim 10^{6}K$), theN RaDIatively cools aNd settLeS InTO GalAxiEs in a quasi-SpHericAL manner. tHe TRANsiTIon of these two Accretion moDEs iS expecTeD to OCcur neAr the CrITicAl halo mass, $M_{\Rm c}\sIm 10^{12}M_{\sun}$ [@DekEl; @2006]. To juSTify thiS, It is impOrtant To iNveStigATe ThE beHaVIor OF gAs aCCreTion when A gAlAxy evOlveS ACROss $M_{\Rm c}$. simuLatioNs suggest that The Gas aCCreTion bEhaviOr inDeEd chaNges neAr $M_{\rm C}$ [@STewart; @2011], but observAtioNal confirMatIoN of ThIs is sTIll lacKinG. ObServatiOnally, gAS flOw SIGNaTures have been unambIgUOUsLy detectEd in thE HiGh-QUality spEcTra Of SFgS [E.g., @HecKman; @1990; @sAtO; @2009; @Weiner; @2009; @GEnzel; @2014a; @rUbIn; @2014; @cicone; @2016]. NEvErthelEsS, thE deTaileD PropErties Of gas floWs are STill difficult tO Quantify direcTLy. tHIs IS becAusE gas flows caN occUR in mUlti-PHaSe, aND the gLobal GaS FlOW rates depend on the 3D mOtIons anD densIties of the gas. indirect meTHODs are thuS useFUl IN studying gas flOws. FoR example, eaRLy attempTs havE tried to Set constrAINts on gas FloWs bY moDelING tHe chemical evoLUTion Of sFGs to mAtcH the obsErvEd mAss-MetAlLicity relAtion [@FinLaToR; @2008; @SPiTonI; @2010; @LillY; @2013; @yabe; @2015; @SpitOnI; @2017]. ThE aSseMbly hIStory oF MilkY Way-MaSs ($m_{\Rm Mw}\sim 5\timES 10^{10}M_{\SUN}$, see @mcmiLlan [@2017]) GalAxIes haS recENtlY attracTed much atTenTIons, SiNcE galaxiEs near $M_{\rm MW}$ apPeAr quite typIcAl aNd domiNATe the steLlar mass budget in the locaL universE [@vaNDokkUm; @2013]. SeVeral workS haVe trieD to TRace thE evoluTion oF sTar FORmatiON AnD

low-mass halos and high r edshift un ivers e,gas i s ac quir ed primarily t h roug h the cold mode accret ion,by whic h c old g as flow s c a n di re ct lyfe e dgalax ies throug h cosmic f ila me nts [@Keres; @2 005; @Deke l;@2009a; @Dek el; @2009 b; @v a n; @d e;@Voor t; @20 1 1]. Wh en a gala xy ’ s dark matterh a lo gro ws massive enough to support a stab le sho ck , t h e in fal ling gas i sfirst shock-h e at e d ton ear the viral temperatur e ($ T\sim10 ^{6 } K$), t hen r ad i ati vely coolsandsettles i nto ga l axies i n a quas i-sphe ric almann e r. T hetr a nsi t io n o f th ese twoac cr etion mod e s i s ex pec tedto oc cur near thecri tica l ha lo ma ss, $ M_{\ rm c}\s im 10^ {12}M _{ \sun}$ [@Dekel; @20 06]. To j ust if y t hi s, it is imp ort ant to inv estigat e th eb e h av ior of gas accreti on w he n a gala xy evo l ve sa cross $M _{ \rm c}$ . Simul atio n ssuggestthat t h ega s accre ti on beh av ior in deedc hang es nea r $M_{\r m c}$ [@Stewart; @20 1 1], but obser v at i o na l con fir mation of t hisi s st illl ac kin g . Ob serva ti o na l ly, gas flow signat ur es hav e bee n unambiguous ly detecte d i n the hi gh-q u al i ty spectra ofSFGs[e.g., @He c kman; @1 990;@Sato; @ 2009; @We i n er; @200 9;@Ge nze l;@ 2 01 4a; @Rubin; @ 2 0 14;@C icone;@20 16]. Ne ver the les s,th e detaile d proper ti es o fgas flow s are sti ll di ff icu lt to quanti fy di rect ly .T his is bec a us e gasfl ow s ca n o cc ur in mul t i-p hase, a nd the gl oba l gas f lo w rates depend on th e3D motions a nddensit i e s of the gas. Indirect methodsa re thus us efulin s tudying g asflows. Fo r examp le, ea rly a tt emp t s have t ri edto set const r a int s onga s fl ows bymodeling the chemi c alevolution ofSFG s to m at cht he obs er v edm a ss-metallicityrelation [ @F i nl ator; @200 8 ; @ Sp itoni;@2010;@Lill y ; @2013 ; @Yabe;@2015; @S pi toni ; @20 17]. Theassembly historyo f Mil k yWay-m ass ($M_{ \r m M W}\si m 5\ti m es10^{1 0}M_{\ su n}$, s ee @M cM illan [@ 2017]) galaxies has rec entlyattra cte d much at ten t ion s, sincegala xies near$M_ {\r m MW} $ a p pearquit e t ypi c al an d do m inate the st ell a r m ass budgeti n the loca l U n iverse [@v anDokkum; @2013]. Several workshave t rie d t o tra ce the evolution of s t a r format io n and

low-mass_halos and_high redshift universe, gas_is acquired_primarily_through the_cold_mode accretion, by_which cold gas_flows can directly feed_galaxies through cosmic_filaments_[@Keres; @2005; @Dekel; @2009a; @Dekel; @2009b; @van; @de; @Voort; @2011]. When a galaxy’s dark_matter_halo grows_massive_enough_to support a stable shock,_the infalling gas is first_shock-heated to_near the viral temperature ($T\sim 10^{6}K$), then radiatively_cools_and settles into_galaxies in a quasi-spherical manner. The transition of these_two accretion modes is expected to_occur near the_critical_halo_mass, $M_{\rm c}\sim 10^{12}M_{\sun}$_[@Dekel; @2006]. To justify this, it_is important to investigate the behavior_of gas accretion when a galaxy evolves_across $M_{\rm c}$. Simulations suggest that_the gas accretion behavior indeed_changes near_$M_{\rm c}$ [@Stewart; @2011], but_observational confirmation of_this is_still lacking. Observationally, gas_flow signatures have been unambiguously detected_in the high-quality_spectra of SFGs [e.g., @Heckman; @1990;_@Sato;_@2009; @Weiner; @2009;_@Genzel;_@2014a;_@Rubin; @2014;_@Cicone; @2016]. Nevertheless,_the_detailed properties_of_gas flows are still difficult to_quantify_directly. This is because gas flows can_occur in multi-phase, and_the_global gas flow rates_depend on the 3D motions_and densities of the gas. Indirect_methods are_thus useful_in studying gas flows. For example, early attempts have tried to_set constraints on gas flows by_modeling the chemical evolution_of SFGs_to_match the observed_mass-metallicity_relation [@Finlator;_@2008; @Spitoni; @2010; @Lilly; @2013; @Yabe; @2015;_@Spitoni; @2017]. The_assembly history of Milky Way-mass ($M_{\rm_MW}\sim 5\times 10^{10}M_{\sun}$, see_@McMillan_[@2017]) galaxies has recently attracted much_attentions, since galaxies near $M_{\rm MW}$_appear quite typical and dominate_the_stellar_mass budget in the local_Universe [@vanDokkum; @2013]. Several works have_tried to trace_the evolution of star formation and

$, $m_\alpha+m_{2\alpha}\geq 1$, and define $Tf(\lambda)=|\lambda|^2\mathcal{F}f(\lambda)$. Since $-m_\alpha-m_{2\alpha}+1\leq 0$, it follows that $(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}\leq 1$ for all $\lambda\in i{\mathfrak{a}}^*$. Moreover, $$\begin{split} \|Tf\|_2^2 &=\int_{i{\mathfrak{a}}^*}|Tf(\lambda)|^2|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\ &= |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\ &\leq |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda = \|f\|_2^2, \end{split}$$ so $T$ is of strong type $(2,2)$ as an operator from $L^2(A,d\mu)^W$ into $L^2(i{\mathfrak{a}}^*,d\overline{\nu})^W=L^2(i{\mathfrak{a}}^*,|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}d\nu(\lambda))^W$. This is no longer true when $m_\alpha+m_{2\alpha}<1$, in which case one would have to employ a different type of weight and/or modify the measure $d\overline{\nu}$. Note that $|\mathcal{F}f(\lambda)|\leq C\|f\|_1$ for all $\lambda\in i{\mathfrak{a}}^*$ and $f\in L^1(A,d\mu)$.. For $t>0$ and $0\neq f\in L^1(A,d\mu)^W$, define $$E_t(

$, $ m_\alpha+m_{2\alpha}\geq 1 $, and define $ Tf(\lambda)=|\lambda|^2\mathcal{F}f(\lambda)$. Since $ -m_\alpha - m_{2\alpha}+1\leq 0 $, it follows that $ (1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}\leq 1 $ for all $ \lambda\in i{\mathfrak{a}}^*$. Moreover, $ $ \begin{split } \|Tf\|_2 ^ 2 & = \int_{i{\mathfrak{a}}^*}|Tf(\lambda)|^2|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\ & = |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\ & \leq |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda = \|f\|_2 ^ 2, \end{split}$$ so $ T$ is of impregnable character $ (2,2)$ as an operator from $ L^2(A, d\mu)^W$ into $ L^2(i{\mathfrak{a}}^*,d\overline{\nu})^W = L^2(i{\mathfrak{a}}^*,|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}d\nu(\lambda))^W$. This is no longer true when $ m_\alpha+m_{2\alpha}<1 $, in which subject one would own to employ a different character of weight and/or modify the measure $ d\overline{\nu}$. notice that $ |\mathcal{F}f(\lambda)|\leq C\|f\|_1 $ for all $ \lambda\in i{\mathfrak{a}}^*$ and $ f\in L^1(A, d\mu)$.. For $ t>0 $ and $ 0\neq f\in L^1(A, d\mu)^W$, specify $ $ E_t (

$, $m_\appha+m_{2\alpha}\geq 1$, and defint $Tf(\lambda)=|\lambda|^2\mcrhcal{F}h(\lambda)$. Since $-m_\xlpha-m_{2\alpha}+1\leq 0$, it follows tiat $(1+|\oambdq|)^{-(m_\alpha+m_{2\alpha})+1}\leq 1$ for xll $\lambdw\in i{\matyfraj{q}}^*$. Moreover, $$\begin{split} \|Tf\|_2^2 &=\jkt_{i{\machhrak{a}}^*}|Tf(\lambda)|^2|W|^{-1}(1+|\lsmbda|)^{-(m_\alphd+m_{2\alpha})+1}|\lambda|^{-4}|{\mdtfby{c}}(\lambda)|^{-2}\,d\lambda\\ &= |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcaj{F}f(\lambca)|^2(1+|\pambda|)^{-(m_\alpha+m_{2\ajpha})+1}|{\kwthbr{b}}(\lcmbda)|^{-2}\,d\lambda\\ &\leq |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mzthcal{F}h(\lambda)|^2|{\mathbf{c}}(\lsmbda)|^{-2}\,d\lambda = \|f\|_2^2, \end{split}$$ so $T$ id of strong type $(2,2)$ ws an operajkr seom $L^2(A,d\mu)^W$ ivto $L^2(i{\mathfrak{a}}^*,d\overlihe{\nu})^W=L^2(i{\mathfrak{a}}^*,|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\aupha})+1}|\lcmbda|^{-4}d\nu(\lamvdq))^W$. Hvis is no linger true when $m_\alpha+m_{2\al[ha}<1$, in ehich case one wonld yave to employ a diffxrent type of weight and/or mogiyy the measure $d\overlunw{\nu}$. Noje thdt $|\mxrhcxl{F}r(\lembsa)|\leq F\|f\|_1$ hor all $\lamgda\in i{\mathdrak{a}}^*$ and $f\in L^1(A,d\mu)$.. Fjg $t>0$ and $0\neq f\jn L^1(A,d\iu)^R$, define $$E_t(

$, $m_\alpha+m_{2\alpha}\geq 1$, and define $Tf(\lambda)=|\lambda|^2\mathcal{F}f(\lambda)$. Since it that $(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}\leq for all $\lambda\in &= &\leq |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda = \end{split}$$ so $T$ of strong type $(2,2)$ as an from $L^2(A,d\mu)^W$ into $L^2(i{\mathfrak{a}}^*,d\overline{\nu})^W=L^2(i{\mathfrak{a}}^*,|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}d\nu(\lambda))^W$. This is no longer true when $m_\alpha+m_{2\alpha}<1$, in which one would have to employ a different type of weight and/or modify the $d\overline{\nu}$. that C\|f\|_1$ all $\lambda\in i{\mathfrak{a}}^*$ and $f\in L^1(A,d\mu)$.. For $t>0$ and $0\neq f\in L^1(A,d\mu)^W$, define $$E_t(

$, $m_\alpha+m_{2\alpha}\geq 1$, and define $TF(\lambda)=|\lamBda|^2\maThcAl{F}F(\lAmbdA)$. SinCe $-m_\alpha-m_{2\alpha}+1\LEq 0$, it Follows that $(1+|\lambda|)^{-(m_\alphA+m_{2\alpHa})+1}\LEq 1$ foR AlL $\lambDa\in i{\maTHfRAK{a}}^*$. MOrEoVer, $$\BeGIn{Split} \|tf\|_2^2 &=\iNt_{i{\mathFrak{a}}^*}|Tf(\lamBda)|^2|w|^{-1}(1+|\lAmbda|)^{-(m_\alpha+m_{2\ALpHa})+1}|\lambda|^{-4}|{\maThbF{c}}(\lambda)|^{-2}\,d\lamBda\\ &= |w|^{-1}\int_{i{\mAtHfrAK{a}}^*}|\matHcaL{F}f(\laMbda)|^2(1+|\laMBda|)^{-(m_\alPha+m_{2\alpha})+1}|{\MaTHbf{c}}(\laMBda)|^{-2}\,d\lamBDA\\ &\lEq |W|^{-1}\iNt_{i{\mathfrak{a}}^*}|\mathcAL{F}F(\Lambda)|^2|{\mathbf{c}}(\lAmbda)|^{-2}\,d\LaMBdA = \|F\|_2^2, \End{SplIt}$$ so $T$ is of sTrOng tyPE $(2,2)$ as an opERaTOR FroM $l^2(A,d\mu)^W$ into $L^2(i{\mAthfrak{a}}^*,d\ovERliNe{\nu})^W=L^2(I{\mAthFRak{a}}^*,|W|^{-1}(1+|\lAmbda|)^{-(M_\aLPha+M_{2\alpha})+1}|\lambdA|^{-4}d\nu(\Lambda))^W$. ThIs is no LOnger trUE when $m_\aLpha+m_{2\aLphA}<1$, in WhicH CaSe One WoULd hAVe To eMPloY a differEnT tYpe of WeigHT AND/or mOdiFy thE measUre $d\overline{\nU}$. NoTe thAT $|\maThcal{f}f(\lamBda)|\lEq c\|f\|_1$ for All $\lamBda\in I{\mAthfrak{a}}^*$ and $f\in L^1(a,d\mu)$.. for $t>0$ and $0\neQ f\iN L^1(a,d\mU)^W$, DefinE $$e_t(

$, $m_\alpha+m_{2\alpha}\g eq 1$, and defi ne$Tf (\ lamb da)= |\lambda|^2\ma t hcal {F}f(\lambda)$. Since$-m_\ al p ha-m _ {2 \alph a}+1\le q 0 $ , it f ol low st ha t $(1 +|\ lambda| )^{-(m_\al pha +m _{2\alpha})+ 1 }\ leq 1$ for al l $\lambda\i n i {\math fr ak{ a }}^*$ . M oreov er, $$ \ begin{ split} \| Tf \ |_2^2& =\int_{ i { \m athf rak{a}}^*}|Tf(\la m bd a )|^2|W|^{-1}(1 +|\lam bd a |) ^ { -(m _\a lpha+m_{2\ al pha}) + 1}|\lam b da | ^ { -4} | {\mathbf{c}}( \lambda)|^{ - 2}\ ,d\lam bd a\\ &= |W| ^{-1} \i n t_{ i{\mathfrak {a}} ^*}|\math cal{F} f (\lambd a )|^2(1+ |\lamb da| )^{ -(m_ \ al ph a+m _{ 2 \al p ha })+ 1 }|{ \mathbf{ c} }( \lamb da)| ^ { - 2 }\,d \la mbda \\ &\ leq |W|^{-1}\ int _{i{ \ mat hfrak {a}}^ *}|\ ma thcal {F}f(\ lambd a) |^2|{\mathbf{c} }(\l ambda)|^{ -2} \, d\l am bda = \|f\|_ 2^2 , \ end{spl it}$$ s o $T $i s of strong type $(2,2 )$ a san opera tor fr o m$L ^ 2(A,d\mu )^ W$into $ L^2(i {\ma t hf rak{a}}^ *,d\ov e rl in e{\nu}) ^W =L^2(i {\ mat hfr ak{a} } ^*,| W|^{-1 }(1+|\la mbda| ) ^{-(m_\alpha+m _ {2\alpha})+1} | \l a m bd a |^{- 4}d \nu(\lambda ))^W $ . Th is i s n o l o ngertruewh e n$ m_\alpha+m_{2\alpha }< 1$, in whic h case one wo uld have t o e mploy adiff e re n t type of weig ht an d/or modif y the mea sure$d\overl ine{\nu}$ . Note th at$|\ mat hca l { F} f(\lambda)|\l e q C\| f\ |_1$ fo r a ll $\la mbd a\i n i {\m at hfrak{a}} ^*$ and$f \i nL^ 1(A ,d\mu ) $.. For$t >0$ a nd$0\ne q f\inL^1(A ,d\m u) ^W $ , d efine $ $ E_ t (

$, $m_\alpha+m_{2\alpha}\geq_1$, and_define $Tf(\lambda)=|\lambda|^2\mathcal{F}f(\lambda)$. Since $-m_\alpha-m_{2\alpha}+1\leq_0$, it_follows_that $(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}\leq_1$_for all $\lambda\in_i{\mathfrak{a}}^*$. Moreover, $$\begin{split} \|Tf\|_2^2_&=\int_{i{\mathfrak{a}}^*}|Tf(\lambda)|^2|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\ &= |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\ &\leq |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda =_\|f\|_2^2, \end{split}$$ so $T$_is_of strong type $(2,2)$ as an operator from $L^2(A,d\mu)^W$ into $L^2(i{\mathfrak{a}}^*,d\overline{\nu})^W=L^2(i{\mathfrak{a}}^*,|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}d\nu(\lambda))^W$. This is no_longer_true when_$m_\alpha+m_{2\alpha}<1$,_in_which case one would have_to employ a different type_of weight_and/or modify the measure $d\overline{\nu}$. Note that $|\mathcal{F}f(\lambda)|\leq C\|f\|_1$_for_all $\lambda\in i{\mathfrak{a}}^*$_and $f\in L^1(A,d\mu)$.. For $t>0$ and $0\neq f\in L^1(A,d\mu)^W$,_define $$E_t(

function by the inclusion-exclusion principle, then using we have $$\label{eq:est_prob_ruin_set} \P\{Y_1>y_1,\ldots,Y_d>y_d\}\approx \sum_{j=1}^d \frac{1}{y_j} \bar{\Phi}_{d-1} \Big\lbrace \Big (\lambda_{k,j} +\frac{\log y_k/y_j }{2\lambda_{k,j}} \Big )_{k \in I_j}; \bar{\Lambda}_j \Big\rbrace,$$ where $\bar{\Phi}_{d-1}$ is the survival function of the multivariate normal distribution [@nikoloulopoulos2009]. Similar to @cooley2010 we define three extreme events: $\{\text{PM}10 > 95, \text{NO} > 270, \text{SO}2 > 95\}$, $\{\text{NO}2 > 110, \text{SO}2 > 95, \text{NO} > 270\}$ and $\{\text{PM}10 > 95, \text{NO} > 270, \text{NO}2 > 110, \text{SO}2 > 95\}$. Then, we compute probability using in place of the parameters their estimates. Table \[table:proba\_excess\] reports the results. For the three events the estimates fall inside the $95\%$ confidence intervals highlighting the ability of the model to estimate such extreme events. Event $1$ Event $2$ Event $3$ -------------- ---------------------------- ---------------------------- ----------------------------- Excess / $n$ $18 /528 $ $14 / 562$ $12 /528$ Emp. Est. $0.034\; ( 0.019, 0.050 )$ $0.025\; ( 0.012, 0.038 )$ $0.023\; ( 0.010, 0.035 )$ Mod. Est. $0.038$ $0.030$ $0.030$ : Probability estimates of excesses. The first row reports the number of excess and the sample size. The second row reports the empirical estimates and between brackets the $95\%$ confidence intervals obtained with the normal approximation. The third row reports the model estimates.[]{data-label="table:proba_excess"} $ \begin{array}{

function by the inclusion - exclusion principle, then use we take $ $ \label{eq: est_prob_ruin_set } \P\{Y_1 > y_1,\ldots, Y_d > y_d\}\approx \sum_{j=1}^d \frac{1}{y_j } \bar{\Phi}_{d-1 } \Big\lbrace \Big (\lambda_{k, j } + \frac{\log y_k / y_j } { 2\lambda_{k, j } } \Big) _ { k \in I_j }; \bar{\Lambda}_j \Big\rbrace,$$ where $ \bar{\Phi}_{d-1}$ is the survival function of the multivariate normal distribution [ @nikoloulopoulos2009 ]. Similar to @cooley2010 we specify three extreme events: $ \{\text{PM}10 > 95, \text{NO } > 270, \text{SO}2 > 95\}$, $ \{\text{NO}2 > 110, \text{SO}2 > 95, \text{NO } > 270\}$ and $ \{\text{PM}10 > 95, \text{NO } > 270, \text{NO}2 > 110, \text{SO}2 > 95\}$. Then, we compute probability use in position of the parameters their estimate. Table \[table: proba\_excess\ ] reports the resultant role. For the three events the estimates decrease inside the $ 95\%$ confidence intervals highlighting the ability of the model to estimate such extreme events. Event $ 1 $ Event $ 2 $ Event $ 3 $ -------------- ---------------------------- ---------------------------- ----------------------------- Excess / $ n$ $ 18 /528 $ $ 14 / 562 $ $ 12 /528 $ Emp. Est. $ 0.034\; (0.019, 0.050) $ $ 0.025\; (0.012, 0.038) $ $ 0.023\; (0.010, 0.035) $ Mod. Est. $ 0.038 $ $ 0.030 $ $ 0.030 $ : Probability estimate of excesses. The first row report the number of excess and the sample size. The second row report the empirical estimates and between brackets the $ 95\%$ confidence intervals prevail with the normal approximation. The third row reports the model estimates.[]{data - label="table: proba_excess " } $ \begin{array } {

fujction by the inclusion-ewclusion principle, then using we have $$\label{eq:est_prob_ruin_set} \P\{Y_1>y_1,\ldovs,Y_d>t_d\}\appeox \sum_{j=1}^d \frac{1}{y_j} \bar{\Ohi}_{d-1} \Big\lhrace \Bit (\lanvda_{k,j} +\frac{\log y_k/y_j }{2\lambdz_{n,j}} \Bng )_{k \in I_j}; \bar{\Lakbda}_j \Big\rtrace,$$ where $\bas{\Pfi}_{b-1}$ is the survival function of the mujtivaristf normal distrybutpog [@niioloulopoulos2009]. Similar to @cooley2010 we define three extremr events: $\{\text{PM}10 > 95, \text{NO} > 270, \texh{SO}2 > 95\}$, $\{\text{NO}2 > 110, \texh{SO}2 > 95, \text{NI} > 270\}$ wbd $\{\text{PM}10 > 95, \gext{NO} > 270, \ttxc{NO}2 > 110, \text{SK}2 > 95\}$. Then, we compute probability usiny in place if thf parameters theig estimates. Table \[tabne:proba\_rxcess\] reports thx rewults. For the three etents the estimates sall insige the $95\%$ confidence unrervans hhghlkthtkng tie zbilitj oh the model to estimatw such extreme evenus. Evene $1$ Event $2$ Event $3$ -------------- ---------------------------- ---------------------------- ----------------------------- Excess / $n$ $18 /528 $ $14 / 562$ $12 /528$ Emp. Est. $0.034\; ( 0.019, 0.050 )$ $0.025\; ( 0.012, 0.038 )$ $0.023\; ( 0.010, 0.035 )$ Jud. Tsb. $0.038$ $0.030$ $0.030$ : Pgobability estimabes of excesses. Thr vitft row reportr the uhmger of excess and hhe samkle size. The sesond row reports the empirical wstimates anb bwtween brackets thz $95\%$ confidencz intetvals pbtained with the normau aplroximation. The thirs row reports the mocen estimauds.[]{data-label="table:pwoba_excesw"} $ \begnn{array}{

function by the inclusion-exclusion principle, then using $$\label{eq:est_prob_ruin_set} \sum_{j=1}^d \frac{1}{y_j} \Big\lbrace \Big (\lambda_{k,j} \in \bar{\Lambda}_j \Big\rbrace,$$ where is the survival of the multivariate normal distribution [@nikoloulopoulos2009]. to @cooley2010 we define three extreme events: $\{\text{PM}10 > 95, \text{NO} > 270, > 95\}$, $\{\text{NO}2 > 110, \text{SO}2 > 95, \text{NO} > 270\}$ and $\{\text{PM}10 95, > \text{NO}2 110, \text{SO}2 > 95\}$. Then, we compute probability using in place of the parameters their estimates. \[table:proba\_excess\] reports the results. For the three events estimates fall inside the confidence intervals highlighting the ability the to estimate extreme Event Event $2$ Event -------------- ---------------------------- ---------------------------- ----------------------------- Excess / $n$ $18 /528 $ $14 / 562$ $12 /528$ Emp. Est. ( 0.019, $0.025\; ( 0.038 $0.023\; 0.010, 0.035 )$ $0.038$ $0.030$ $0.030$ : Probability estimates first row reports the number of excess and sample size. second row reports the empirical estimates between brackets the $95\%$ confidence intervals obtained with normal approximation. The third row reports the model estimates.[]{data-label="table:proba_excess"} $ \begin{array}{

function by the inclusion-excLusion prinCiple, TheN usInG we hAve $$\lAbel{eq:est_prob_rUIn_seT} \P\{Y_1>y_1,\ldots,Y_d>y_d\}\approx \sum_{J=1}^d \fraC{1}{y_J} \Bar{\PHI}_{d-1} \big\lbRace \Big (\LAmBDA_{k,j} +\FrAc{\Log Y_k/Y_J }{2\lAmbda_{K,j}} \BIg )_{k \in I_j}; \Bar{\Lambda}_j \big\RbRace,$$ where $\bar{\pHi}_{D-1}$ is the survIvaL function of tHe mUltivaRiAte NOrmal DisTribuTion [@niKOlouloPoulos2009]. SimIlAR to @cooLEy2010 we defINE tHree Extreme events: $\{\text{pm}10 > 95, \tEXt{NO} > 270, \text{SO}2 > 95\}$, $\{\text{nO}2 > 110, \text{sO}2 > 95, \TExT{no} > 270\}$ anD $\{\teXt{PM}10 > 95, \text{NO} > 270, \TeXt{NO}2 > 110, \tEXt{SO}2 > 95\}$. TheN, We COMPutE Probability usIng in place oF The ParameTeRs tHEir estImateS. TABle \[Table:proba\_eXcesS\] reports tHe resuLTs. For thE Three evEnts thE esTimAtes FAlL iNsiDe THe $95\%$ cONfIdeNCe iNtervals HiGhLightIng tHE ABIlitY of The mOdel tO estimate such ExtReme EVenTs. EveNt $1$ EveNt $2$ EvEnT $3$ -------------- ---------------------------- ---------------------------- ----------------------------- ExceSs / $n$ $18 /528 $ $14 / 562$ $12 /528$ Emp. est. $0.034\; ( 0.019, 0.050 )$ $0.025\; ( 0.012, 0.038 )$ $0.023\; ( 0.010, 0.035 )$ MoD. ESt. $0.038$ $0.030$ $0.030$ : Probability esTimaTes of exceSseS. THe fIrSt row REports The NumBer of exCess and THe sAmPLE SiZe. The second row repoRtS THe EmpiricaL estimATeS aND between BrAckEts tHE $95\%$ ConfiDencE InTervals oBtaineD WiTh The normAl ApproxImAtiOn. THe thiRD row ReportS the modeL estiMAtes.[]{data-label="tABle:proba_excesS"} $ \BeGIN{aRRay}{

function by the inclusion -exclusion prin cip le, t henusin g we have $$\l a bel{ eq:est_prob_ruin_set}\P\{Y _1 > y_1, \ ld ots,Y _d>y_d\ } \a p p rox \s um_ {j = 1} ^d \ fra c{1}{y_ j} \bar{\P hi} _{ d-1} \Big\l b ra ce \Big (\ lam bda_{k,j} +\ fra c{\log y _k/ y _j }{ 2\l ambda _{k,j} } \Big)_{k \inI_ j }; \ba r {\Lambd a } _j \Bi g\rbrace,$$ where $\ b ar{\Phi}_{d-1} $ is t he su r v iva l f unction of t he mu l tivaria t en o r mal distribution[@nikoloulo p oul os2009 ]. Si m ilar t o @co ol e y20 10 we defin e th ree extre me eve n ts: $\{ \ text{PM }10 >95, \t ext{ N O} > 27 0, \te x t{ SO} 2 >95\}$, $ \{ \t ext{N O}2> 1 1 0, \ tex t{SO }2 >95, \text{NO} >270\ } $ a nd $\ {\tex t{PM }1 0 > 9 5, \te xt{NO }> 270, \text{NO }2 > 110, \te xt{ SO }2>95\}$ . Then, we co mpute p robabil i tyus i n g i n place of the par am e t er s theirestima t es .T able \[t ab le: prob a \ _exce ss\] re ports th e resu l ts .For the t hree e ve nts th e est i mate s fall insidethe $ 9 5\%$ confidenc e intervals hi g hl i g ht i ng t heability ofthem odel toe st ima t e suc h ext re m ee vents. Ev ent $ 1$ E v e n t $2$ Event $3$ -------- - ----- -- ----- -------- --------- - - -- ----- --- --- --- --- - - -- ------- ----- - - ---- -- ------- --- ------ E xce ss/ $ n$ $18 /5 28 $ $ 1 4 / 562$ $12 /528 $ Em p. Est . $ 0. 0 3 4\;(0. 019, 0. 05 0 )$ $0 . 025 \; ( 0. 012, 0.03 8 ) $ $ 0. 02 3\; ( 0 .010, 0.035 ) $ Mod. Est . $0.0 3 8 $ $0.030$ $0.030$ : Prob abi l ity es timate s ofex ces s e s. Th e fi rst r ow reports t henumbe rof e xcess a nd the sample size . Th e second rowrep orts t he em p ir i cal e s tim a t es and betweenbrackets t he $9 5\%$ confi d enc einterva ls obta inedw ith the normal a pproximat io n. T h e th ird row re ports th e model e s timat e s. []{da ta- label= "t abl e:pro ba_exc e ss" } $\begin {a rray}{

function_by the_inclusion-exclusion principle, then using_we have_$$\label{eq:est_prob_ruin_set} \P\{Y_1>y_1,\ldots,Y_d>y_d\}\approx_ \sum_{j=1}^d__\frac{1}{y_j} \bar{\Phi}_{d-1} \Big\lbrace_\Big (\lambda_{k,j} +\frac{\log_y_k/y_j }{2\lambda_{k,j}} \Big )_{k_\in I_j}; \bar{\Lambda}_j_\Big\rbrace,$$_where $\bar{\Phi}_{d-1}$ is the survival function of the multivariate normal distribution [@nikoloulopoulos2009]. Similar to_@cooley2010_we define_three_extreme_events: $\{\text{PM}10 > 95, \text{NO}_> 270, \text{SO}2 > 95\}$,_$\{\text{NO}2 >_110, \text{SO}2 > 95, \text{NO} > 270\}$ and_$\{\text{PM}10_> 95, \text{NO}_> 270, \text{NO}2 > 110, \text{SO}2 > 95\}$. Then,_we compute probability using in place_of the parameters_their_estimates._Table \[table:proba\_excess\] reports the_results. For the three events the_estimates fall inside the $95\%$ confidence_intervals highlighting the ability of the model_to estimate such extreme events. _ _ _ _ _Event $1$_ _ _ _ __Event $2$ ___ _ __ __ __Event $3$ -------------- ---------------------------- ---------------------------- ----------------------------- _ Excess / $n$__ $18 /528 $_ _ _ _ _ $14 / 562$ _ _ _ __$12 /528$ _Emp._Est. _ $0.034\; ( 0.019,_0.050 )$_ $0.025\; ( 0.012, 0.038_)$ $0.023\;_(_0.010, 0.035 )$ Mod. Est._ $0.038$_ ___ _ _ _$0.030$ __ __ __$0.030$ _ : Probability estimates of_excesses. The first row reports the_number of excess and the sample size. The second_row reports the_empirical estimates and between brackets_the_$95\%$_confidence intervals obtained with the normal approximation. The third row_reports the_model estimates.[]{data-label="table:proba_excess"} $ \begin{array}{

\[uniqueness thm\] Let ${\mathcal{K}}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $\mu$ is a cardinal $\geq\operatorname{LS}({\mathcal{K}})$ and $\theta_1$ and $\theta_2$ are limit ordinals $<\mu^+$. If ${\mathcal{K}}$ is $\mu$-superstable and satisfies $\mu$-symmetry, then for $M_1$ and $M_2$ which are $(\mu, \theta_1)$ and $(\mu,\theta_2)$-limit models over $N$, respectively, we have that $M_1$ is isomorphic to $M_2$ over $N$. Moreover the limit model of cardinality $\mu$ is saturated. This is just a restatement of Theorem 5 of [@Va3-SS] and the proof of Theorem 1.9 of [@GVV]. Combining Theorem \[uniqueness thm\] with Proposition \[mu-plus-limit\], we get the following corollary. \[limit is sat\] Let ${\mathcal{K}}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $\kappa$ is a cardinal $\geq\operatorname{LS}({\mathcal{K}})$, and $\theta$ is limit ordinal $<\kappa^{++}$. If ${\mathcal{K}}$ is $\kappa$-stable, $\kappa^+$-superstable and satisfies $\kappa^+$-symmetry, then any $(\kappa^+,\theta)$-limit model is also a $(\kappa,\kappa^+)$-limit model. Downward Symmetry Transfer {#downward section} ========================== In this section we provide the proof of Theorem \[symmetry transfer\]. While the result follows from Theorem 4 and 5 of [@Va3-SS], we include the proof here for completeness since [@Va3-SS] is currently under review and has not yet been published. Additionally, the proof of Theorem \[symmetry transfer\] serves as the blueprint for the successor step for a more general result of transferring symmetry downward that appears in the unpublished work [@VV]. In the proof of Theorem \[symmetry transfer\], we will be using towers composed of models of cardinality $\mu$ and other towers composed of models of cardinality $\mu^+$. These towers will be based on the same sequence of elements $\langle a_\beta\mid \beta<\delta\rangle$. To distinguish the towers of models of size $\mu^+$ from those of size

\[uniqueness thm\ ] Let $ { \mathcal{K}}$ be an abstract elementary class which satisfies the joint embedding and amalgamation place. presuppose $ \mu$ is a cardinal $ \geq\operatorname{LS}({\mathcal{K}})$ and $ \theta_1 $ and $ \theta_2 $ are limit ordinals $ < \mu^+$. If $ { \mathcal{K}}$ is $ \mu$-superstable and satisfies $ \mu$-symmetry, then for $ M_1 $ and $ M_2 $ which are $ (\mu, \theta_1)$ and $ (\mu,\theta_2)$-limit model over $ N$, respectively, we have that $ M_1 $ is isomorphic to $ M_2 $ over $ N$. furthermore the limit model of cardinality $ \mu$ is saturated. This is merely a restatement of Theorem 5 of [ @Va3 - SS ] and the proof of Theorem 1.9 of [ @GVV ]. Combining Theorem \[uniqueness thm\ ] with Proposition \[mu - plus - limit\ ], we catch the following corollary. \[limit is sat\ ] get $ { \mathcal{K}}$ be an abstract elementary course which satisfies the joint embedding and amalgamation place. Suppose $ \kappa$ is a cardinal $ \geq\operatorname{LS}({\mathcal{K}})$, and $ \theta$ is limit ordinal $ < \kappa^{++}$. If $ { \mathcal{K}}$ is $ \kappa$-stable, $ \kappa^+$-superstable and satisfies $ \kappa^+$-symmetry, then any $ (\kappa^+,\theta)$-limit exemplar is also a $ (\kappa,\kappa^+)$-limit model. Downward Symmetry Transfer { # downward incision } = = = = = = = = = = = = = = = = = = = = = = = = = = In this section we provide the proof of Theorem \[symmetry transfer\ ]. While the consequence follows from Theorem 4 and 5 of [ @Va3 - SS ], we include the proof here for completeness since [ @Va3 - SS ] is currently under review and has not so far been published. Additionally, the proof of Theorem \[symmetry transfer\ ] serves as the blueprint for the successor step for a more general result of transferring symmetry downward that appears in the unpublished work [ @VV ]. In the proof of Theorem \[symmetry transfer\ ], we will be using towers composed of models of cardinality $ \mu$ and other tower composed of models of cardinality $ \mu^+$. These tugboat will be based on the same sequence of elements $ \langle a_\beta\mid \beta<\delta\rangle$. To identify the towers of models of size $ \mu^+$ from those of size

\[unieueness thm\] Let ${\mathcal{K}}$ be an abstract elemenvary clzss whicf satisfies the joint embeddmng qnd analgamation properties. Suppose $\lu$ is a xardmnal $\geq\operatorizme{LS}({\mabkcal{K}})$ wnd $\chxta_1$ and $\theta_2$ ate limit ordhnals $<\mu^+$. If ${\madhzap{K}}$ is $\mu$-superstable and satisfies $\mt$-symmetty, then for $M_1$ anq $M_2$ erich are $(\mu, \theta_1)$ and $(\mu,\theta_2)$-limit modsls oveg $N$, respectively, ee have that $M_1$ is isomorphlc tl $M_2$ over $N$. Moreoveg the limit modqo of cardinauity $\mu$ is saturated. Thjs is just a restatement of Theurem 5 of [@Va3-SS] abd thf proof of Tieorem 1.9 of [@GVV]. Comblming Tveorem \[iniqueness thm\] wivh Peoposition \[mu-plus-limiv\], we get the followigg corolldrv. \[limit is sat\] Let ${\matycql{K}}$ bg an dbstfqct eltmeitady clads xhich satisries the jount embedding and akajtamation propsrties. Stppose $\kappa$ is a cardinal $\geq\operatorndme{MS}({\mathcal{K}})$, and $\theta$ is limit ordinal $<\kappa^{++}$. Iv ${\mathcal{H}}$ is $\kappa$-stable, $\kappa^+$-superstable and satisfies $\kdppa^+$-sbmoetxn, gyej any $(\kappa^+,\theta)$-limit model is also a $(\kappa,\ka[la^+)$-kikit model. Downwcrd Symmetry Tramsvet {#downward secjion} ========================== In cgia section we provife the kroof if Theorei \[sykmetry transfer\]. While the rwsult followf from Theorem 4 and 5 of [@Va3-SS], wz inclode thr proof here for compleceness since [@Va3-SS] is currehgly under review anc vas not yet been published. Additionelly, che proow of Theorqm \[symmetrj trakvfer\] serves as the bluekrint xor the sufcessor step for a more general result of tramsxerging symmztry dpwnward that wppears in the unpublnshed dork [@VV]. In nhe proof of Theorem \[symmetry tratdfer\], we will be usind toqers composda of models of cardinalptv $\mu$ and ither towers compoxed kf models of caxbibality $\mu^+$. These toderf aikl te based on dhe ramd sequdnce of elendnts $\langle a_\beta\mid \betd<\delfa\rangle$. To distinbulsh the tiwers of models of siae $\mu^+$ from those ov sizx

\[uniqueness thm\] Let ${\mathcal{K}}$ be an abstract which the joint and amalgamation properties. $\geq\operatorname{LS}({\mathcal{K}})$ $\theta_1$ and $\theta_2$ limit ordinals $<\mu^+$. ${\mathcal{K}}$ is $\mu$-superstable and satisfies $\mu$-symmetry, for $M_1$ and $M_2$ which are $(\mu, \theta_1)$ and $(\mu,\theta_2)$-limit models over $N$, we have that $M_1$ is isomorphic to $M_2$ over $N$. Moreover the limit of $\mu$ saturated. is just a restatement of Theorem 5 of [@Va3-SS] and the proof of Theorem 1.9 of Combining Theorem \[uniqueness thm\] with Proposition \[mu-plus-limit\], we the following corollary. \[limit sat\] Let ${\mathcal{K}}$ be an elementary which satisfies joint and properties. Suppose $\kappa$ a cardinal $\geq\operatorname{LS}({\mathcal{K}})$, and $\theta$ is limit ordinal $<\kappa^{++}$. If ${\mathcal{K}}$ is $\kappa$-stable, $\kappa^+$-superstable and satisfies $\kappa^+$-symmetry, any $(\kappa^+,\theta)$-limit also a model. Symmetry {#downward section} ========================== section we provide the proof of While the result follows from Theorem 4 and of [@Va3-SS], include the proof here for completeness [@Va3-SS] is currently under review and has not been published. Additionally, the proof of Theorem \[symmetry transfer\] serves as the blueprint for the for a more general of transferring symmetry that in unpublished [@VV]. In proof of Theorem \[symmetry transfer\], we will be using towers composed models of cardinality $\mu$ and other towers composed of models $\mu^+$. towers will be on the same sequence elements a_\beta\mid \beta<\delta\rangle$. To distinguish of of those size

\[uniqueness thm\] Let ${\mathcal{K}}$ bE an abstracT elemEntAry ClAss wHich Satisfies the joINt emBedding and amalgamation PropeRtIEs. SuPPoSe $\mu$ iS a cardiNAl $\GEQ\opErAtOrnAmE{lS}({\MathcAl{K}})$ And $\thetA_1$ and $\theta_2$ aRe lImIt ordinals $<\mu^+$. iF ${\mAthcal{K}}$ is $\mU$-suPerstable and SatIsfies $\Mu$-SymMEtry, tHen For $M_1$ aNd $M_2$ whiCH are $(\mu, \Theta_1)$ and $(\mU,\tHEta_2)$-limIT models OVEr $n$, resPectively, we have thAT $M_1$ IS isomorphic to $M_2$ Over $N$. MOrEOvER The LimIt model of cArDinalITy $\mu$ is sATuRATEd. THIs is just a restAtement of ThEOreM 5 of [@Va3-Ss] aNd tHE proof Of TheOrEM 1.9 of [@gVV]. CombininG TheOrem \[uniquEness tHM\] with PrOPositioN \[mu-pluS-liMit\], We geT ThE fOllOwINg cORoLlaRY. \[liMit is sat\] leT ${\mAthcaL{K}}$ be AN ABStraCt eLemeNtary Class which satIsfIes tHE joInt emBeddiNg anD aMalgaMation PropeRtIes. Suppose $\kappa$ Is a cArdinal $\geQ\opErAtoRnAme{LS}({\MAthcal{k}})$, anD $\thEta$ is liMit ordiNAl $<\kApPA^{++}$. iF ${\mAthcal{K}}$ is $\kappa$-stabLe, $\KAPpA^+$-superstAble anD SaTiSFies $\kappA^+$-sYmmEtry, THEn any $(\KappA^+,\ThEta)$-limit Model iS AlSo A $(\kappa,\kApPa^+)$-limiT mOdeL. DoWnwarD symmEtry TrAnsfer {#doWnwarD Section} ========================== In this sECtion we providE ThE PRoOF of THeoRem \[symmetry TranSFer\]. WHile THe ResULt folLows fRoM thEOrem 4 and 5 of [@Va3-SS], we inclUdE the prOof heRe for completeNess since [@VA3-ss] Is currenTly uNDeR Review and has noT yet bEen publishED. AdditioNally, The proof Of Theorem \[SYMmetry trAnsFer\] SerVes AS ThE blueprint for THE sucCeSsor steP foR a more gEneRal ResUlt Of TransferrIng symmeTrY dOwNwArd That aPPears in tHe UnpUbLisHed woRK [@VV]. In tHe proOf of thEoREm \[sYmmetry TRaNSFer\], wE wIlL be uSinG tOwers CompOSed Of modelS of cardinAliTY $\mu$ aNd OtHer toweRs composed of mOdEls of cardiNaLitY $\mu^+$. TheSE Towers wiLl be based on the same sequeNCe of eleMenTs $\lanGle a_\Beta\mid \beTa<\dElta\raNglE$. to distInguisH the tOwErs OF ModelS OF sIze $\Mu^+$ From those oF SIze

\[uniqueness thm\] Let ${\ mathcal{K} }$ be an ab st ract ele mentary classw hich satisfies the joint e mbedd in g and am algam ation p r op e r tie s. S upp os e $ \mu$isa cardi nal $\geq\ ope ra torname{LS}( { \m athcal{K}} )$and $\theta_ 1$and $\ th eta _ 2$ ar e l imitordina l s $<\m u^+$. If${ \ mathca l {K}}$ i s $\ mu$- superstable and s a ti s fies $\mu$-sym metry, t h en f or$M_ 1$ and $M_ 2$ whic h are $( \ mu , \ the t a_1)$ and $(\ mu,\theta_2 ) $-l imit m od els over $ N$, r es p ect ively, we h avethat $M_1 $ is i s omorphi c to $M_ 2$ ove r $ N$. Mor e ov er th el imi t m ode l of cardina li ty $\mu $ is s a t urat ed. Th is is just a resta tem ento f T heore m 5 o f [@ Va 3-SS] and t he pr oo f of Theorem 1. 9 of [@GVV]. Co mb ini ng Theo r em \[u niq uen ess thm \] with Pro po s i t io n \[mu-plus-limit\ ], w eget thefollow i ng c o rollary. \[l imit i s sat \] L e t${\mathc al{K}} $ b ean abst ra ct ele me nta ryclass whic h sati sfies th e joi n t embedding an d amalgamation pr o p er t ies. Su ppose $\kap pa$i s acard i na l $ \ geq\o perat or n am e {LS}({\mathcal{K}}) $, and $ \thet a$ is limit o rdinal $<\ k a p pa^{++}$ . I f $ { \mathcal{K}}$is $\ kappa$-sta b le, $\ka ppa^+ $-supers table and s atisfies $\ kap pa^ +$- s y mm etry, then an y $(\k ap pa^+,\t het a)$-lim itmod elisal so a $(\k appa,\ka pp a^ +) $- lim it mo d el. Dow nw ard S ymm etryT ransfe r {#d ownw ar ds ect ion} == = == = = ==== == == ==== === == == I n th i s s ectionwe provid e t h e pr oo fof Theo rem \[symmetr ytransfer\] .Whi le the r esult fo llows from Theorem 4 an d 5 of [ @Va 3-SS] , we includethe proof he r e forcomple tenes ssin c e [@Va 3 - SS ] i scurrentlyu n der revi ew and has no t yet been publish e d.Additionally, th e pr o o fofT he o rem \ [ sym m e try transfer\]serves asth e b lueprint f o r t he succes sor ste p for a moregeneral r esult oftr ansf e r rin g symmetry downwar d that ap p earsi nthe u npu blishe dwor k [@V V]. I n th e pro of ofTh eorem\[sym me try tran sfer\], we will be usin g towe rs co mpo sed of mo del s of cardinal ity$\mu$ andoth ertower s c o mpose d of mo del s of c ardi n ality $\m u ^+ $.T h es e towers wi l l bebased on the sa me s equence of elemen t s $\langle a_\ beta \ m id\be t a<\d el ta\rangle$. To di st i n guish th etowers of m odels of s i ze $\ mu^+$from t hose of s iz e

\[uniqueness thm\]_Let ${\mathcal{K}}$_be an abstract elementary_class which_satisfies_the joint_embedding_and amalgamation properties._Suppose $\mu$ is_a cardinal $\geq\operatorname{LS}({\mathcal{K}})$ and_$\theta_1$ and $\theta_2$_are_limit ordinals $<\mu^+$. If ${\mathcal{K}}$ is $\mu$-superstable and satisfies $\mu$-symmetry, then for $M_1$ and_$M_2$_which are_$(\mu,_\theta_1)$_and $(\mu,\theta_2)$-limit models over $N$,_respectively, we have that $M_1$_is isomorphic_to $M_2$ over $N$. Moreover the limit model_of_cardinality $\mu$ is_saturated. This is just a restatement of Theorem 5 of_[@Va3-SS] and the proof of Theorem_1.9 of [@GVV]. Combining_Theorem_\[uniqueness_thm\] with Proposition \[mu-plus-limit\],_we get the following corollary. \[limit is_sat\] Let ${\mathcal{K}}$ be an abstract_elementary class which satisfies the joint embedding_and amalgamation properties. Suppose $\kappa$ is_a cardinal $\geq\operatorname{LS}({\mathcal{K}})$, and $\theta$_is limit_ordinal $<\kappa^{++}$. If ${\mathcal{K}}$ is $\kappa$-stable,_$\kappa^+$-superstable and satisfies_$\kappa^+$-symmetry, then_any $(\kappa^+,\theta)$-limit model_is also a $(\kappa,\kappa^+)$-limit model. Downward Symmetry_Transfer {#downward section} ========================== In_this section we provide the proof_of_Theorem \[symmetry transfer\]._While_the_result follows_from Theorem 4_and_5 of_[@Va3-SS],_we include the proof here for_completeness_since [@Va3-SS] is currently under review and_has not yet been_published._Additionally, the proof of_Theorem \[symmetry transfer\] serves as_the blueprint for the successor step_for a_more general_result of transferring symmetry downward that appears in the unpublished work_[@VV]. In the proof of Theorem \[symmetry_transfer\], we will be_using towers_composed_of models of_cardinality_$\mu$ and_other towers composed of models of cardinality_$\mu^+$. These_towers will be based on the_same sequence of elements_$\langle_a_\beta\mid \beta<\delta\rangle$. To distinguish the towers_of models of size $\mu^+$ from_those of size

..}}\right)_{\!|..}\right]\!\!\cdot\! \Delta_\mu^{\scsc\#} A^{\alpha..}\!+\!\frac{\partial L(..)}{\partial y_\nu^{\alpha..}}_{|..} \Delta_\mu^{\scsc\#} \Delta_\nu^{\scsc\#} A^{\alpha..}\!-\!\Delta_\mu^{\scsc\#} [L(..)_{|..}]\right\} } \\[0.7cm] {\ds + \frac{1}{2}\, \frac{\partial^2 L(..)}{\partial y^{\alpha..} \partial y^{\beta..}}_{|..} \cdot \big[\Delta_\mu^{\scsc\#} (A^{\alpha..}(n) \cdot A^{\beta..}(n)) } \\[0.7cm] \qquad - A^{\alpha..} (n) \Delta_\mu^{\scsc\#} A^{\beta..} - (\Delta_\mu^{\scsc\#} A^{\alpha..})\,(A^{\beta..}(n))\big] \\[0.45cm] {\ds + \frac{1}{2}\, \frac{\partial^2 L(..)}{\partial y_\nu^{\alpha..} \partial y_\sigma^{\beta..}}_{|..} \cdot \big[\Delta_\mu^{\scsc\#} (\Delta_\nu^{\scsc\#} A^{\alpha..} \cdot \Delta_\sigma^{\scsc\#} A^{\beta..}) } \\[0.75cm] \qquad - (\Delta_\nu^{\scsc\#} A^{\alpha..})\, (\Delta_\mu^{\scsc\#} \Delta_\sigma^{\scsc\#} A^{\beta..}) - (\Delta_\mu^{\scsc\#} \Delta_\nu^{\scsc\#} A^{\alpha..})\,(\Delta_\sigma^{\scsc\#} A^{\beta..})\big] \\[0.5cm] {\ds + [\Delta_\mu^{\scsc\#}(1)] \Biggr[L(..)_{|..}-\frac{\partial L(..)}{\partial y^{\alpha..}}_{|..} \cdot A^{\alpha..}(n) - \frac{\partial L(..)}{\partial y_\nu^{\alpha..}}_{|..} \cdot \Delta_\nu^{\scsc\#}A^{\

.. } } \right)_{\!|.. }\right]\!\!\cdot\! \Delta_\mu^{\scsc\ # } A^{\alpha.. }\!+\!\frac{\partial L(.. )}{\partial y_\nu^{\alpha.. }}_{|.. } \Delta_\mu^{\scsc\ # } \Delta_\nu^{\scsc\ # } A^{\alpha.. }\!-\!\Delta_\mu^{\scsc\ # } [ L(.. )_{|.. }]\right\ } } \\[0.7 cm ] { \ds + \frac{1}{2}\, \frac{\partial^2 L(.. )}{\partial y^{\alpha.. } \partial y^{\beta.. }}_{|.. } \cdot \big[\Delta_\mu^{\scsc\ # } (A^{\alpha.. }(n) \cdot A^{\beta.. }(n) ) } \\[0.7 cm ] \qquad - A^{\alpha.. } (n) \Delta_\mu^{\scsc\ # } A^{\beta.. } - (\Delta_\mu^{\scsc\ # } A^{\alpha.. })\,(A^{\beta.. }(n))\big ] \\[0.45 cm ] { \ds + \frac{1}{2}\, \frac{\partial^2 L(.. )}{\partial y_\nu^{\alpha.. } \partial y_\sigma^{\beta.. }}_{|.. } \cdot \big[\Delta_\mu^{\scsc\ # } (\Delta_\nu^{\scsc\ # } A^{\alpha.. } \cdot \Delta_\sigma^{\scsc\ # } A^{\beta.. }) } \\[0.75 cm ] \qquad - (\Delta_\nu^{\scsc\ # } A^{\alpha.. })\, (\Delta_\mu^{\scsc\ # } \Delta_\sigma^{\scsc\ # } A^{\beta.. }) - (\Delta_\mu^{\scsc\ # } \Delta_\nu^{\scsc\ # } A^{\alpha.. })\,(\Delta_\sigma^{\scsc\ # } A^{\beta.. })\big ] \\[0.5 cm ] { \ds + [ \Delta_\mu^{\scsc\#}(1) ] \Biggr[L(.. )_{|.. }-\frac{\partial L(.. )}{\partial y^{\alpha.. }}_{|.. } \cdot A^{\alpha.. }(n) - \frac{\partial L(.. )}{\partial y_\nu^{\alpha.. }}_{|.. } \cdot \Delta_\nu^{\scsc\#}A^{\

..}}\rigjt)_{\!|..}\right]\!\!\cdot\! \Delta_\mu^{\scsc\#} A^{\xlpha..}\!+\!\frac{\partial L(..)}{\partmal y_\nu^{\ampha..}}_{|..} \Delga_\mu^{\scsc\#} \Delta_\nu^{\scsc\#} A^{\alpha..}\!-\!\Deptq_\mu^{\scwc\#} [L(..)_{|..}]\right\} } \\[0.7cm] {\ds + \frac{1}{2}\, \wrac{\partiwl^2 L(..)}{\partual b^{\alpha..} \partial y^{\bxfa..}}_{|..} \cdot \big[\Demba_\mu^{\sesr\#} (A^{\alpha..}(n) \cdot A^{\neta..}(n)) } \\[0.7cm] \qqgad - A^{\alpha..} (n) \Geutc_\mu^{\scsc\#} A^{\beta..} - (\Delta_\mu^{\scsc\#} A^{\alpha..})\,(A^{\betw..}(n))\big] \\[0.45cm] {\cs + \frac{1}{2}\, \frac{\parjial^2 K(..)}{\[artjal y_\nu^{\alpha..} \partial y_\sigma^{\beta..}}_{|..} \cdof \big[\Denta_\mu^{\scsc\#} (\Delts_\nu^{\scsc\#} A^{\alpha..} \cdot \Delta_\slgma^{\dcsc\#} A^{\beta..}) } \\[0.75cm] \qquad - (\Delta_\nu^{\scwc\#} A^{\wopha..})\, (\Delta_\mu^{\rcsc\#} \Delta_\spyma^{\scsc\#} A^{\beja..}) - (\Delta_\mu^{\scsc\#} \Delta_\nu^{\scsc\#} A^{\alphx..})\,(\Deltc_\sigma^{\scsc\#} A^{\vera..})\blc] \\[0.5cm] {\ds + [\Delva_\mu^{\scfc\#}(1)] \Biggr[L(..)_{|..}-\frag{\iartial N(..)}{\partiak y^{\alpha..}}_{|..} \cdot S^{\al'ha..}(n) - \frac{\partial L(..)}{\partial y_\nu^{\alpha..}}_{|..} \cdot \Deltw_\nu^{\scsc\#}A^{\

..}}\right)_{\!|..}\right]\!\!\cdot\! \Delta_\mu^{\scsc\#} A^{\alpha..}\!+\!\frac{\partial L(..)}{\partial y_\nu^{\alpha..}}_{|..} \Delta_\mu^{\scsc\#} \Delta_\nu^{\scsc\#} } {\ds + \frac{\partial^2 L(..)}{\partial y^{\alpha..} \cdot } \\[0.7cm] \qquad A^{\alpha..} (n) \Delta_\mu^{\scsc\#} - (\Delta_\mu^{\scsc\#} A^{\alpha..})\,(A^{\beta..}(n))\big] \\[0.45cm] {\ds + \frac{\partial^2 L(..)}{\partial y_\nu^{\alpha..} \partial y_\sigma^{\beta..}}_{|..} \cdot \big[\Delta_\mu^{\scsc\#} (\Delta_\nu^{\scsc\#} A^{\alpha..} \cdot \Delta_\sigma^{\scsc\#} A^{\beta..}) } \qquad - (\Delta_\nu^{\scsc\#} A^{\alpha..})\, (\Delta_\mu^{\scsc\#} \Delta_\sigma^{\scsc\#} A^{\beta..}) - (\Delta_\mu^{\scsc\#} \Delta_\nu^{\scsc\#} A^{\alpha..})\,(\Delta_\sigma^{\scsc\#} A^{\beta..})\big] \\[0.5cm] + \Biggr[L(..)_{|..}-\frac{\partial y^{\alpha..}}_{|..} A^{\alpha..}(n) - \frac{\partial L(..)}{\partial y_\nu^{\alpha..}}_{|..} \cdot \Delta_\nu^{\scsc\#}A^{\

..}}\right)_{\!|..}\right]\!\!\cdot\! \Delta_\mu^{\scsc\#} a^{\alpha..}\!+\!\frac{\PartiAl L(..)}{\ParTiAl y_\nU^{\alpHa..}}_{|..} \Delta_\mu^{\scsc\#} \DELta_\nU^{\scsc\#} A^{\alpha..}\!-\!\Delta_\mu^{\scsc\#} [l(..)_{|..}]\righT\} } \\[0.7cM] {\Ds + \frAC{1}{2}\, \fRac{\paRtial^2 L(..)}{\pARtIAL y^{\aLpHa..} \ParTiAL y^{\Beta..}}_{|..} \cDot \Big[\DeltA_\mu^{\scsc\#} (A^{\alPha..}(N) \cDot A^{\beta..}(n)) } \\[0.7cm] \qQUaD - A^{\alpha..} (n) \DeLta_\Mu^{\scsc\#} A^{\beta..} - (\DEltA_\mu^{\scsC\#} A^{\AlpHA..})\,(A^{\betA..}(n))\bIg] \\[0.45cm] {\dS + \frac{1}{2}\, \fRAc{\partIal^2 L(..)}{\partiAl Y_\Nu^{\alphA..} \Partial Y_\SIgMa^{\beTa..}}_{|..} \cdot \big[\Delta_\mu^{\sCSc\#} (\dElta_\nu^{\scsc\#} A^{\alpHa..} \cdot \deLTa_\SIGma^{\ScsC\#} A^{\beta..}) } \\[0.75cm] \qqUaD - (\DeltA_\Nu^{\scsc\#} A^{\ALpHA..})\, (\dEltA_\Mu^{\scsc\#} \Delta_\siGma^{\scsc\#} A^{\betA..}) - (\delTa_\mu^{\scSc\#} \delTA_\nu^{\scsC\#} A^{\alpHa..})\,(\dEltA_\sigma^{\scsc\#} A^{\Beta..})\Big] \\[0.5cm] {\ds + [\DeLta_\mu^{\sCSc\#}(1)] \Biggr[l(..)_{|..}-\Frac{\parTial L(..)}{\pArtIal Y^{\alpHA..}}_{|..} \cDoT A^{\aLpHA..}(n) - \fRAc{\ParTIal l(..)}{\partial Y_\nU^{\aLpha..}}_{|..} \cDot \DELTA_\Nu^{\scSc\#}A^{\

..}}\right)_{\!|..}\right] \!\!\cdot\ ! \De lta _\m u^ {\sc sc\# } A^{\alpha..} \ !+\! \frac{\partial L(..)}{ \part ia l y_\ n u^ {\alp ha..}}_ { |. . } \D el ta _\m u^ { \s csc\# } \ Delta_\ nu^{\scsc\ #}A^ {\alpha..}\! - \! \Delta_\mu ^{\ scsc\#} [L(. .)_ {|..}] \r igh t \} } \\ [0.7c m] {\d s + \fr ac{1}{2}\ ,\ frac{\ p artial^ 2 L( ..)} {\partial y^{\alp h a. . } \partial y^{ \beta. .} } _{ | . .}\cd ot \big[\D el ta_\m u ^{\scsc \ #} ( A ^{\ a lpha..}(n) \c dot A^{\bet a ..} (n)) } \ \[0 . 7cm] \ qquad - A^{ \alpha..} ( n) \ Delta_\mu ^{\scs c \#} A^{ \ beta..} - (\D elt a_\ mu^{ \ sc sc \#} A ^ {\a l ph a.. } )\, (A^{\bet a. .} (n))\ big] \ \ [ 0.45 cm] {\d s + \ frac{1}{2}\,\fr ac{\ p art ial^2 L(.. )}{\ pa rtial y_\nu ^{\al ph a..} \partial y _\si gma^{\bet a.. }} _{| .. } \cd o t \big [\D elt a_\mu^{ \scsc\# } (\ De l t a _\ nu^{\scsc\#} A^{\a lp h a .. } \cdot\Delta _ \s ig m a^{\scsc \# } A ^{\b e t a..}) } \ \ [0 .75cm] \ qquad- ( \D elta_\n u^ {\scsc \# } A ^{\ alpha . .})\ , (\De lta_\mu^ {\scs c \#} \Delta_\si g ma^{\scsc\#}A ^{ \ b et a ..}) -(\Delta_\mu ^{\s c sc\# } \D e lt a_\ n u^{\s csc\# }A ^{ \ alpha..})\,(\Delta_ \s igma^{ \scsc \#} A^{\beta. .})\big] \ \ [ 0 .5cm] {\ ds + [\ D elta_\mu^{\scs c\#}( 1)] \Biggr [ L(..)_{| ..}-\ frac{\pa rtial L(. . ) }{\parti aly^{ \al pha . . }} _{|..} \cdotA ^ {\al ph a..}(n) -\frac{\ par tia l L (.. )} {\partial y_\nu^{ \a lp ha .. }}_ {|..} \cdot \D el ta_ \n u^{ \scsc \ #}A^{\

..}}\right)_{\!|..}\right]\!\!\cdot\! \Delta_\mu^{\scsc\#} A^{\alpha..}\!+\!\frac{\partial_L(..)}{\partial y_\nu^{\alpha..}}_{|..} \Delta_\mu^{\scsc\#}_\Delta_\nu^{\scsc\#} A^{\alpha..}\!-\!\Delta_\mu^{\scsc\#} [L(..)_{|..}]\right\} } \\[0.7cm] {\ds_+ \frac{1}{2}\,_\frac{\partial^2_L(..)}{\partial y^{\alpha..} \partial_y^{\beta..}}_{|..}_\cdot \big[\Delta_\mu^{\scsc\#} (A^{\alpha..}(n) \cdot_A^{\beta..}(n)) } \\[0.7cm] \qquad_- A^{\alpha..} (n) \Delta_\mu^{\scsc\#}_A^{\beta..} - (\Delta_\mu^{\scsc\#} A^{\alpha..})\,(A^{\beta..}(n))\big] \\[0.45cm] {\ds_+_\frac{1}{2}\, \frac{\partial^2 L(..)}{\partial y_\nu^{\alpha..} \partial y_\sigma^{\beta..}}_{|..} \cdot \big[\Delta_\mu^{\scsc\#} (\Delta_\nu^{\scsc\#} A^{\alpha..} \cdot \Delta_\sigma^{\scsc\#} A^{\beta..}) } \\[0.75cm] \qquad - (\Delta_\nu^{\scsc\#}_A^{\alpha..})\,_(\Delta_\mu^{\scsc\#} \Delta_\sigma^{\scsc\#} A^{\beta..})_-_(\Delta_\mu^{\scsc\#} \Delta_\nu^{\scsc\#}_A^{\alpha..})\,(\Delta_\sigma^{\scsc\#} A^{\beta..})\big] \\[0.5cm] {\ds + [\Delta_\mu^{\scsc\#}(1)] \Biggr[L(..)_{|..}-\frac{\partial L(..)}{\partial_y^{\alpha..}}_{|..} \cdot A^{\alpha..}(n) - \frac{\partial L(..)}{\partial_y_\nu^{\alpha..}}_{|..} \cdot \Delta_\nu^{\scsc\#}A^{\

2} = 183 {\rm ns}$, $\gamma_1 = 0.265$, $\gamma_2 = 0.561$ & $T_{2} =50.0 {\rm ns},$ $\gamma_1 = 1.37$, $\gamma_2 = 0.345$ &\ & Unoptimized & $T_{2} = 52.2 {\rm ns},$ $\gamma_1 = 0.208$, $\gamma_2 = 0.909$ & $T_{2} = 45.1 {\rm ns},$ $\gamma_1 = 0.300$, $\gamma_2 = 1.00$ &\ \[-0.5em\][$h = 40$ MHz, $\sigma_{h} = 23$ MHz]{} & Optimized & $T_{2} = 58.4 {\rm \mu s}$, $\gamma_1 = 1.04$, $\gamma_2 = 0.256$ & $T_{2} =27.8 {\rm ns},$ $\gamma_1 = 0.362$, $\gamma_2 = 2.56$ &\ & Unoptimized & $T_{2} = 22.3 {\rm \mu s},$ $\gamma_1 = 1.00$,$\gamma_2 = 0.258$ & $T_{2} = 22.2 {\rm ns},$ $\gamma_1 = 1.00$, $\gamma_2 = 0.337$ &\ [|M[40pt]{}|M[210pt]{}|M[210pt]{}|N]{} $\delta h$ & Barrier control & Tilt control &\ \[0.5em\][$\delta h = 0$]{} & $T_{2} = 3.00 {\rm \mu s}$, $\gamma = 1.01,$ $\lg A_{J} = -4.13,$ $A_{h} = 0$ & $T_{2} = 42.0 {\rm ns},$ $\gamma = 1.98,$ $\lg A_{J} = -1.89,$ $A_h = 0$ &\ \[0.5em\][$\delta h \neq0$]{} & $T_{2} = 1.15 {\rm \mu s},$ $\gamma = 1.03,$ $\lg A_{J} = -4.06,$ $\lg A_h = -5.53$ & $T_{2} = 22.3 {\rm ns},$ $\

2 } = 183 { \rm ns}$, $ \gamma_1 = 0.265 $, $ \gamma_2 = 0.561 $ & $ T_{2 } = 50.0 { \rm ns},$ $ \gamma_1 = 1.37 $, $ \gamma_2 = 0.345 $ & \ & Unoptimized & $ T_{2 } = 52.2 { \rm ns},$ $ \gamma_1 = 0.208 $, $ \gamma_2 = 0.909 $ & $ T_{2 } = 45.1 { \rm ns},$ $ \gamma_1 = 0.300 $, $ \gamma_2 = 1.00 $ & \ \[-0.5em\][$h = 40 $ MHz, $ \sigma_{h } = 23 $ MHz ] { } & Optimized & $ T_{2 } = 58.4 { \rm \mu s}$, $ \gamma_1 = 1.04 $, $ \gamma_2 = 0.256 $ & $ T_{2 } = 27.8 { \rm ns},$ $ \gamma_1 = 0.362 $, $ \gamma_2 = 2.56 $ & \ & Unoptimized & $ T_{2 } = 22.3 { \rm \mu s},$ $ \gamma_1 = 1.00$,$\gamma_2 = 0.258 $ & $ T_{2 } = 22.2 { \rm ns},$ $ \gamma_1 = 1.00 $, $ \gamma_2 = 0.337 $ & \ [ |M[40pt]{}|M[210pt]{}|M[210pt]{}|N ] { } $ \delta h$ & Barrier control & Tilt control & \ \[0.5em\][$\delta h = 0 $ ] { } & $ T_{2 } = 3.00 { \rm \mu s}$, $ \gamma = 1.01,$ $ \lg A_{J } = -4.13,$ $ A_{h } = 0 $ & $ T_{2 } = 42.0 { \rm ns},$ $ \gamma = 1.98,$ $ \lg A_{J } = -1.89,$ $ A_h = 0 $ & \ \[0.5em\][$\delta h \neq0 $ ] { } & $ T_{2 } = 1.15 { \rm \mu s},$ $ \gamma = 1.03,$ $ \lg A_{J } = -4.06,$ $ \lg A_h = -5.53 $ & $ T_{2 } = 22.3 { \rm ns},$ $ \

2} = 183 {\rm ns}$, $\gamma_1 = 0.265$, $\gamma_2 = 0.561$ & $U_{2} =50.0 {\rm ns},$ $\gamma_1 = 1.37$, $\gamma_2 = 0.345$ &\ & Unoltimized & $T_{2} = 52.2 {\rm ns},$ $\gamma_1 = 0.208$, $\gamma_2 = 0.909$ & $R_{2} = 45.1 {\rn ns},$ $\gamma_1 = 0.300$, $\gamma_2 = 1.00$ &\ \[-0.5eo\][$h = 40$ MHz, $\digma_{h} = 23$ MHe]{} & Optimized & $T_{2} = 58.4 {\rm \mu s}$, $\gammz_1 = 1.04$, $\gcmna_2 = 0.256$ & $T_{2} =27.8 {\rm nx},$ $\gamma_1 = 0.362$, $\camma_2 = 2.56$ &\ & Unopthmkzzd & $T_{2} = 22.3 {\rm \mu s},$ $\gamma_1 = 1.00$,$\gamma_2 = 0.258$ & $T_{2} = 22.2 {\rm ns},$ $\balma_1 = 1.00$, $\gamma_2 = 0.337$ &\ [|I[40pt]{}|M[210kt]{}|M[210[t]{}|N]{} $\svlua h$ & Barrier control & Tilt contrkl &\ \[0.5em\][$\denta h = 0$]{} & $T_{2} = 3.00 {\tm \mu s}$, $\gamma = 1.01,$ $\lg A_{J} = -4.13,$ $A_{h} = 0$ & $H_{2} = 42.0 {\rm ns},$ $\gamma = 1.98,$ $\pg A_{J} = -1.89,$ $A_h = 0$ &\ \[0.5em\][$\qwlta h \neq0$]{} & $G_{2} = 1.15 {\rm \mu s},$ $\gamma = 1.03,$ $\lg A_{J} = -4.06,$ $\lg A_h = -5.53$ & $T_{2} = 22.3 {\rm ns},$ $\

2} = 183 {\rm ns}$, $\gamma_1 = = & $T_{2} {\rm ns},$ $\gamma_1 &\ Unoptimized & $T_{2} 52.2 {\rm ns},$ = 0.208$, $\gamma_2 = 0.909$ & = 45.1 {\rm ns},$ $\gamma_1 = 0.300$, $\gamma_2 = 1.00$ &\ \[-0.5em\][$h = MHz, $\sigma_{h} = 23$ MHz]{} & Optimized & $T_{2} = 58.4 {\rm \mu $\gamma_1 1.04$, = & $T_{2} =27.8 {\rm ns},$ $\gamma_1 = 0.362$, $\gamma_2 = 2.56$ &\ & Unoptimized & $T_{2} 22.3 {\rm \mu s},$ $\gamma_1 = 1.00$,$\gamma_2 = & $T_{2} = 22.2 ns},$ $\gamma_1 = 1.00$, $\gamma_2 0.337$ [|M[40pt]{}|M[210pt]{}|M[210pt]{}|N]{} $\delta & control Tilt control &\ h = 0$]{} & $T_{2} = 3.00 {\rm \mu s}$, $\gamma = 1.01,$ $\lg A_{J} = -4.13,$ = 0$ = 42.0 ns},$ = $\lg A_{J} = = 0$ &\ \[0.5em\][$\delta h \neq0$]{} 1.15 {\rm \mu s},$ $\gamma = 1.03,$ $\lg = -4.06,$ A_h = -5.53$ & $T_{2} = {\rm ns},$ $\

2} = 183 {\rm ns}$, $\gamma_1 = 0.265$, $\gamma_2 = 0.561$ & $T_{2} =50.0 {\rm ns},$ $\gamma_1 = 1.37$, $\gAmma_2 = 0.345$ &\ & UnoptiMized & $t_{2} = 52.2 {\rm Ns},$ $\gAmMa_1 = 0.208$, $\gaMma_2 = 0.909$ & $T_{2} = 45.1 {\Rm ns},$ $\gamma_1 = 0.300$, $\gamma_2 = 1.00$ &\ \[-0.5EM\][$h = 40$ MHZ, $\sigma_{h} = 23$ MHz]{} & Optimized & $T_{2} = 58.4 {\rm \Mu s}$, $\gaMmA_1 = 1.04$, $\GammA_2 = 0.256$ & $t_{2} =27.8 {\rM ns},$ $\gaMma_1 = 0.362$, $\gammA_2 = 2.56$ &\ & unOPTimIzEd & $t_{2} = 22.3 {\rm \Mu S},$ $\GaMma_1 = 1.00$,$\gaMma_2 = 0.258$ & $t_{2} = 22.2 {\rm ns},$ $\gaMma_1 = 1.00$, $\gamma_2 = 0.337$ &\ [|M[40pT]{}|M[210pT]{}|M[210Pt]{}|N]{} $\delta h$ & BarRIeR control & TiLt cOntrol &\ \[0.5em\][$\deltA h = 0$]{} & $T_{2} = 3.00 {\Rm \mu s}$, $\gAmMa = 1.01,$ $\lG a_{J} = -4.13,$ $A_{h} = 0$ & $T_{2} = 42.0 {\Rm nS},$ $\gammA = 1.98,$ $\lg A_{J} = -1.89,$ $A_H = 0$ &\ \[0.5Em\][$\deltA h \neq0$]{} & $T_{2} = 1.15 {\rm \mU s},$ $\GAmma = 1.03,$ $\lg a_{j} = -4.06,$ $\lg A_h = -5.53$ & $T_{2} = 22.3 {\rM NS},$ $\

2} = 183 {\rm ns}$, $\gam ma_1 = 0.2 65$,$\g amm a_ 2 =0.56 1$ & $T_{2} =5 0 .0 {\rm ns},$ $\gamma_1 = 1.37 $, $\ga m ma _2 =0.345$& \& Uno pt im ize d& $ T_{2} =52.2 { \rm ns},$$\g am ma_1 = 0.208 $ ,$\gamma_2= 0 .909$ & $T_{ 2}= 45.1 { \rm ns},$ $\ gamma _1 = 0 . 300$,$\gamma_2 = 1.00$& \ \[-0. 5 e m\ ][$h = 40$ MHz, $\sig m a_ { h} = 23$ MHz] {} & O pt i mi z e d & $T _{2} = 58. 4 {\rm \mu s}$ , $ \ g a mma _ 1 = 1.04$, $\ gamma_2 = 0 . 256 $ & $T _{ 2}= 27.8 {\rmns } ,$$\gamma_1 = 0.3 62$, $\ga mma_2= 2.56$& \ & Uno ptimiz ed& $ T_{2 } = 2 2.3 { \rm \m u s } ,$$\gamma_ 1=1.00$ ,$\g a m m a _2 = 0. 258$ & $T _{2} = 22.2 { \rm ns} , $ $ \gamm a_1 = 1.0 0$ , $\g amma_2 = 0. 33 7$ &\ [|M[40pt ]{}| M[210pt]{ }|M [2 10p t] {}|N] { } $\de lta h$ & Barr ier con t rol & T i lt control &\ \[0.5e m\ ] [ $\ delta h= 0$]{ } & $ T _{2} = 3 .0 0 { \rm\ m u s}$ , $\ g am ma = 1.0 1,$ $\ l gA_ {J} = - 4. 13,$ $ A_ {h} =0$ &$ T_{2 } = 42 .0 {\rmns},$ $\gamma = 1.98 , $ $\lg A_{J}= - 1 . 89 , $ $A _h= 0$ &\ \[0 .5em \ ][$\ delt a h \n e q0$]{ } & $ T_ { 2} = 1.15 {\rm \mu s}, $$\gamm a = 1 .03,$ $\lg A_ {J} = -4.0 6 , $ $\lg A_ h =- 5. 5 3$ & $T_{2} =22.3{\rm ns},$ $\

2} =_183 _{\rm ns}$, $\gamma_1 =_0.265$, $\gamma_2_=_0.561$ &_$T_{2}_=50.0 {\rm_ns},$ $\gamma_1 =_1.37$, $\gamma_2 = 0.345$_&\ & Unoptimized &_$T_{2}_= 52.2 {\rm ns},$ $\gamma_1 = 0.208$, $\gamma_2 = 0.909$ & $T_{2} =_45.1_{\rm ns},$_$\gamma_1_=_0.300$, $\gamma_2 = 1.00$ &\ \[-0.5em\][$h_= 40$ MHz, $\sigma_{h} _= 23$_MHz]{} & Optimized & $T_{2} = 58.4 _{\rm_\mu s}$, $\gamma_1_= 1.04$, $\gamma_2 = 0.256$ & $T_{2} =27.8 _{\rm ns},$ $\gamma_1 = 0.362$, $\gamma_2_= 2.56$ &\ &_Unoptimized_&_$T_{2} = 22.3 _{\rm \mu s},$ $\gamma_1 = 1.00$,$\gamma_2_= 0.258$ & $T_{2} = 22.2_{\rm ns},$ $\gamma_1 = 1.00$, $\gamma_2 =_0.337$ &\ [|M[40pt]{}|M[210pt]{}|M[210pt]{}|N]{} $\delta h$ & Barrier_control & Tilt control &\ \[0.5em\][$\delta_h =_0$]{} & $T_{2} = 3.00_{\rm \mu s}$,_$\gamma =_1.01,$ $\lg A_{J}_= -4.13,$ $A_{h} = 0$ &_$T_{2} = 42.0_{\rm ns},$ $\gamma = 1.98,$ $\lg_A_{J}_= -1.89,$ $A_h_=_0$_&\ \[0.5em\][$\delta h_\neq0$]{} & $T_{2}_=_1.15 {\rm_\mu_s},$ $\gamma = 1.03,$ $\lg A_{J}_=_-4.06,$ $\lg A_h = -5.53$ & $T_{2}_= 22.3 {\rm ns},$_$\

2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}.$$ By the same argument as in Case (5) of the proof of Lemma \[len2\], for any $e+1\leq k<l\leq o$, the term $p_{\lambda_{i_{k}}}p_{\lambda_{i_{l}}}$ appears as a term of the lowest degree in the power sum expansion of $Q_{(\lambda_{i_k},\lambda_{i_l})}$. Moreover, if the power sum symmetric function $p_{\lambda_{i_{e+1}}}p_{\lambda_{i_{e+2}}}\cdots p_{\lambda_{i_o}}$ appears as a term of the lowest degree in the power sum expansion of the product $Q_{(\lambda_{\pi_{2e+1}},\lambda_{\pi_{2e+2}})}\cdots Q_{(\lambda_{\pi_{2m-1}},\lambda_{\pi_{2m}})}$, then the composition of the pairs $\{(\pi_{2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}$ could be any matching of $\{1,2,\ldots,2m\}/\{i_1,j_1,\ldots,i_e,j_e\}$. To summarize, there are $(2(m-e)-1)!!$ matchings $\pi$ such that $A$ appears as a term of the lowest degree in the power sum expansion of the product $Q_{(\lambda_{\pi_1},\lambda_{\pi_2})}\cdots Q_{(\lambda_{\pi_{2m-1}},\lambda_{\pi_{2m}})}$. Combining Corollary \[2odd\] and Theorem \[conn\], we find that the coefficient of $p_{\lambda_{i_k}}p_{\lambda_{i_l}}$ $(e+1\leq k<l\leq o)$ in the power sum expansion of $Q_{(\lambda_{i_k}, \lambda_{i_l})}$ is $-\frac{4}{\lambda_{i_k}\lambda_{i_l}}$. It follows that the coefficient of $A$ in the expansion of the product $Q_{(\lambda_{\pi_1},\lambda_{\pi_2})}\cdots Q_{(\lambda_{\pi_{2m-1}}, \lambda_{\pi_{2m}})}$ is independent of the choice of $\pi$. Since

2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}.$$ By the same argument as in Case (5) of the proof of Lemma \[len2\ ], for any $ e+1\leq k < l\leq o$, the condition $ p_{\lambda_{i_{k}}}p_{\lambda_{i_{l}}}$ appear as a term of the lowest degree in the office sum expansion of $ Q_{(\lambda_{i_k},\lambda_{i_l})}$. Moreover, if the might union symmetric function $ p_{\lambda_{i_{e+1}}}p_{\lambda_{i_{e+2}}}\cdots p_{\lambda_{i_o}}$ look as a term of the lowest academic degree in the power sum expansion of the product $ Q_{(\lambda_{\pi_{2e+1}},\lambda_{\pi_{2e+2}})}\cdots Q_{(\lambda_{\pi_{2m-1}},\lambda_{\pi_{2m}})}$, then the composition of the pair $ \{(\pi_{2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}$ could be any matching of $ \{1,2,\ldots,2m\}/\{i_1,j_1,\ldots, i_e, j_e\}$. To summarize, there are $ (2(m - e)-1)!!$ matchings $ \pi$ such that $ A$ appears as a term of the depleted degree in the power sum expansion of the intersection $ Q_{(\lambda_{\pi_1},\lambda_{\pi_2})}\cdots Q_{(\lambda_{\pi_{2m-1}},\lambda_{\pi_{2m}})}$. Combining Corollary \[2odd\ ] and Theorem \[conn\ ], we find that the coefficient of $ p_{\lambda_{i_k}}p_{\lambda_{i_l}}$ $ (e+1\leq k < l\leq o)$ in the power kernel expansion of $ Q_{(\lambda_{i_k }, \lambda_{i_l})}$ is $ -\frac{4}{\lambda_{i_k}\lambda_{i_l}}$. It follows that the coefficient of $ A$ in the expansion of the product $ Q_{(\lambda_{\pi_1},\lambda_{\pi_2})}\cdots Q_{(\lambda_{\pi_{2m-1 } }, \lambda_{\pi_{2m}})}$ is independent of the choice of $ \pi$. Since

2e+1},\pi_{2f+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}.$$ By the same argument as in Case (5) mf the proof ow Lemma \[len2\], for any $e+1\leq k<l\lxq o$, the uvrm $p_{\lambda_{i_{k}}}p_{\lambda_{i_{u}}}$ appears as a teem oh the lowest degcse in tmz powsv sum xxpansion of $Q_{(\lsmbda_{i_k},\lamtda_{i_l})}$. Moreover, iw che power sum symmetric function $p_{\laibda_{i_{e+1}}}p_{\kalbda_{i_{e+2}}}\cdots p_{\laibda_{p_o}}$ appsars as a term of the lowest degres in tht power sum expansoon of the product $Q_{(\lambda_{\oi_{2e+1}},\lwmbda_{\pi_{2e+2}})}\cdots Q_{(\lambfa_{\pi_{2m-1}},\lambda_{\kj_{2m}})}$, eyen the compusition of the pairs $\{(\pj_{2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}$ could be any mxtchiug of $\{1,2,\ldots,2n\}/\{i_1,h_1,\ldljs,i_e,j_e\}$. To sumnarizv, there are $(2(m-c)-1)!!$ matchhngs $\pi$ such that $A$ aipearv aw a term of the lowesv degree in the powet sum expatsnon of the product $Q_{(\lqmvda_{\pi_1},\nambga_{\pi_2})}\zeotr Q_{(\lzmuda_{\li_{2m-1}},\lamhda_{\'i_{2m}})}$. Combinihg Corollart \[2odd\] and Theorem \[cpng\], we find that the cjesficient of $p_{\lambda_{i_k}}p_{\lambda_{i_l}}$ $(e+1\leq k<l\ltq o)$ jn the power sum expansuon of $Q_{(\lambda_{i_k}, \lambfa_{i_l})}$ is $-\fwac{4}{\lambda_{i_k}\lambda_{i_l}}$. It follows that the coefficiett of $X$ iu the dzpwnsion of the product $Q_{(\lambda_{\pi_1},\lambda_{\pi_2})}\cdots Q_{(\jzmnds_{\pi_{2m-1}}, \lambda_{\pi_{2m}})}$ is indepencejt jf the choice of $\pi$. Sihce

2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}.$$ By the same argument as in of proof of \[len2\], for any $p_{\lambda_{i_{k}}}p_{\lambda_{i_{l}}}$ as a term the lowest degree the power sum expansion of $Q_{(\lambda_{i_k},\lambda_{i_l})}$. if the power sum symmetric function $p_{\lambda_{i_{e+1}}}p_{\lambda_{i_{e+2}}}\cdots p_{\lambda_{i_o}}$ appears as a term of lowest degree in the power sum expansion of the product $Q_{(\lambda_{\pi_{2e+1}},\lambda_{\pi_{2e+2}})}\cdots Q_{(\lambda_{\pi_{2m-1}},\lambda_{\pi_{2m}})}$, then composition the $\{(\pi_{2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}$ be any matching of $\{1,2,\ldots,2m\}/\{i_1,j_1,\ldots,i_e,j_e\}$. To summarize, there are $(2(m-e)-1)!!$ matchings $\pi$ such that $A$ appears a term of the lowest degree in the sum expansion of the $Q_{(\lambda_{\pi_1},\lambda_{\pi_2})}\cdots Q_{(\lambda_{\pi_{2m-1}},\lambda_{\pi_{2m}})}$. Combining Corollary \[2odd\] Theorem we find the of $(e+1\leq k<l\leq o)$ the power sum expansion of $Q_{(\lambda_{i_k}, \lambda_{i_l})}$ is $-\frac{4}{\lambda_{i_k}\lambda_{i_l}}$. It follows that the coefficient of $A$ in expansion of $Q_{(\lambda_{\pi_1},\lambda_{\pi_2})}\cdots Q_{(\lambda_{\pi_{2m-1}}, is of choice of $\pi$.

2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}.$$ By the same aRgument as iN Case (5) Of tHe pRoOf of lemmA \[len2\], for any $e+1\leq K<L\leq O$, the term $p_{\lambda_{i_{k}}}p_{\lambDa_{i_{l}}}$ aPpEArs aS A tErm of The loweST dEGRee In ThE poWeR SuM expaNsiOn of $Q_{(\laMbda_{i_k},\lambDa_{i_L})}$. MOreover, if the POwEr sum symmeTriC function $p_{\laMbdA_{i_{e+1}}}p_{\laMbDa_{i_{E+2}}}\Cdots P_{\laMbda_{i_O}}$ appeaRS as a teRm of the loWeST degreE In the poWER sUm exPansion of the produCT $Q_{(\LAmbda_{\pi_{2e+1}},\lambda_{\Pi_{2e+2}})}\cdoTs q_{(\LaMBDa_{\pI_{2m-1}},\lAmbda_{\pi_{2m}})}$, thEn The coMPositioN Of THE PaiRS $\{(\pi_{2e+1},\pi_{2e+2}),\ldots,(\pI_{2m-1},\pi_{2m})\}$ could bE Any MatchiNg Of $\{1,2,\lDOts,2m\}/\{i_1,j_1,\Ldots,I_e,J_E\}$. To Summarize, thEre aRe $(2(m-e)-1)!!$ matchIngs $\pi$ SUch that $a$ Appears As a terM of The LoweST dEgRee In THe pOWeR suM ExpAnsion of ThE pRoducT $Q_{(\laMBDA_{\Pi_1},\laMbdA_{\pi_2})}\cDots Q_{(\Lambda_{\pi_{2m-1}},\lambDa_{\pI_{2m}})}$. CoMBinIng CoRollaRy \[2odD\] aNd TheOrem \[coNn\], we fInD that the coefficIent Of $p_{\lambda_{I_k}}p_{\LaMbdA_{i_L}}$ $(e+1\leq K<L\leq o)$ iN thE poWer sum eXpansioN Of $Q_{(\LaMBDA_{i_K}, \lambda_{i_l})}$ is $-\frac{4}{\lamBdA_{I_K}\lAmbda_{i_l}}$. IT folloWS tHaT The coeffIcIenT of $A$ IN The exPansIOn Of the proDuct $Q_{(\lAMbDa_{\Pi_1},\lambdA_{\pI_2})}\cdots q_{(\lAmbDa_{\pI_{2m-1}}, \lamBDa_{\pi_{2M}})}$ is indEpendent Of the CHoice of $\pi$. Since

2e+1},\pi_{2e+2}),\ldots,( \pi_{2m-1} ,\pi_ {2m })\ }. $$ B y th e same argumen t asin Case (5) of the pro of of L e mma\ [l en2\] , for a n y$ e +1\ le qk<l \l e qo$, t heterm $p _{\lambda_ {i_ {k }}}p_{\lambd a _{ i_{l}}}$ a ppe ars as a ter m o f thelo wes t degr eein th e powe r sum e xpansionof $Q_{(\ l ambda_{ i _ k} ,\la mbda_{i_l})}$. Mo r eo v er, if the pow er sum s y mm e t ric fu nction $p_ {\ lambd a _{i_{e+ 1 }} } p _ {\l a mbda_{i_{e+2} }}\cdots p_ { \la mbda_{ i_ o}} $ appea rs as a ter m of the lo west degree i n thep ower su m expans ion of th e p rodu c t$Q _{( \l a mbd a _{ \pi _ {2e +1}},\la mb da _{\pi _{2e + 2 } } )}\c dot s Q_ {(\la mbda_{\pi_{2m -1} },\l a mbd a_{\p i_{2m }})} $, then the c ompos it ion of the pair s $\ {(\pi_{2e +1} ,\ pi_ {2 e+2}) , \ldots ,(\ pi_ {2m-1}, \pi_{2m } )\} $c o u ld be any matching o f$ \ {1 ,2,\ldot s,2m\} / \{ i_ 1 ,j_1,\ld ot s,i _e,j _ e \}$. Tos um marize,therea re $ (2(m-e) -1 )!!$ m at chi ngs $\pi $ suc h that $A$ app earsa s a term of th e lowest degre e i n th e pow ersum expansi on o f the pro d uc t $ Q _{(\l ambda _{ \ pi _ 1},\lambda_{\pi_2}) }\ cdotsQ_{(\ lambda_{\pi_{ 2m-1}},\la m b d a_{\pi_{ 2m}} ) }$ . Combining Cor ollar y \[2odd\] and Theo rem \ [conn\], we findt h at the c oef fic ien t o f $p _{\lambda_{i_ k } }p_{ \l ambda_{ i_l }}$ $(e +1\ leq k< l\l eq o)$ in t he power s um e xp ans ion o f $Q_{(\l am bda _{ i_k }, \l a mbda_{ i_l}) }$ i s$- \ fra c{4}{\l a mb d a _{i_ k} \l ambd a_{ i_ l}}$. Itf oll ows tha t the coe ffi c ient o f$A$ inthe expansion o f the prod uc t $ Q_{(\l a m bda_{\pi _1},\lambda_{\pi_2})}\c d ots Q_{ (\l ambda _{\p i_{2m-1}} , \ lambda _{\ p i_{2m} })}$ i s ind ep end e n t oft h echo ic e of $\pi$ . Sin ce

2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}.$$ By_the same_argument as in Case_(5) of_the_proof of_Lemma_\[len2\], for any_$e+1\leq k<l\leq o$,_the term $p_{\lambda_{i_{k}}}p_{\lambda_{i_{l}}}$ appears_as a term_of_the lowest degree in the power sum expansion of $Q_{(\lambda_{i_k},\lambda_{i_l})}$. Moreover, if the power_sum_symmetric function_$p_{\lambda_{i_{e+1}}}p_{\lambda_{i_{e+2}}}\cdots p_{\lambda_{i_o}}$_appears_as a term of the_lowest degree in the power_sum expansion_of the product $Q_{(\lambda_{\pi_{2e+1}},\lambda_{\pi_{2e+2}})}\cdots Q_{(\lambda_{\pi_{2m-1}},\lambda_{\pi_{2m}})}$, then the composition of_the_pairs $\{(\pi_{2e+1},\pi_{2e+2}),\ldots,(\pi_{2m-1},\pi_{2m})\}$ could_be any matching of $\{1,2,\ldots,2m\}/\{i_1,j_1,\ldots,i_e,j_e\}$. To summarize, there are $(2(m-e)-1)!!$_matchings $\pi$ such that $A$ appears_as a term_of_the_lowest degree in the_power sum expansion of the product_$Q_{(\lambda_{\pi_1},\lambda_{\pi_2})}\cdots Q_{(\lambda_{\pi_{2m-1}},\lambda_{\pi_{2m}})}$. Combining Corollary \[2odd\] and Theorem_\[conn\], we find that the coefficient of_$p_{\lambda_{i_k}}p_{\lambda_{i_l}}$ $(e+1\leq k<l\leq o)$ in the_power sum expansion of $Q_{(\lambda_{i_k},_\lambda_{i_l})}$ is_$-\frac{4}{\lambda_{i_k}\lambda_{i_l}}$. It follows that the_coefficient of $A$_in the_expansion of the_product $Q_{(\lambda_{\pi_1},\lambda_{\pi_2})}\cdots Q_{(\lambda_{\pi_{2m-1}}, \lambda_{\pi_{2m}})}$ is independent of_the choice of_$\pi$. Since

. This makes application discovery more straightforward in that each client only needs to locate the hub, and the services provided by the hub are intended to simplify the actions of the client. A disadvantage of this architecture is that the hub may be a message bottleneck and potential single point of failure. The former means that SAMP may not be suitable for extremely high throughput requirements; the latter may be mitigated by an appropriate strategy for hub restart if failure is likely. ![The SAMP hub architecture[]{data-label="fig:samp-archi"}](samp-archi) Note that the hub is defined as a service interface which may have any of a number of implementations. It may be an independent application running as a daemon, an adapter interface layered on top of an existing messaging infrastructure, or a service provided by an application which is itself one of the hub’s clients. The Lifecycle of a Client ------------------------- A SAMP client goes through the following phases: 1. Determine whether a hub is running by using the appropriate hub discovery mechanism. 2. If so, use the hub discovery mechanism to work out how to communicate with the hub. 3. Register with the hub. 4. Store metadata such as client name, description and icon in the hub. 5. Subscribe to a list of MTypes to define messages which may be received. 6. Interrogate the hub for metadata of other clients. 7. Send and/or receive messages to/from other clients via the hub. 8. Unregister with the hub. Phases 4–7 are all optional and may be repeated in any order. By subscribing to the MTypes described in Section \[sect:hub-mtypes\] a client may, if it wishes, keep track of the details of other clients’ registrations, metadata and subscriptions. The Lifecycle of a Hub ---------------------- A SAMP hub goes through the following phases: 1. Locate any existing hub by using the appropriate hub discovery mechanism. 1. Check whether the existing hub is alive. 2. If so, exit. 2. If no hub is running, or a hub is found but is not functioning, write/overwrite the hub discovery record and start up. 3. Await client registrations. When

. This makes application discovery more straightforward in that each client merely need to locate the hub, and the services provided by the hub are intend to simplify the actions of the client. A disadvantage of this computer architecture is that the hub may be a message bottleneck and potential single detail of failure. The former intend that SAMP may not be suitable for extremely gamey throughput prerequisite; the latter may be mitigated by an appropriate strategy for hub restart if failure is likely. ! [ The SAMP hub architecture[]{data - label="fig: samp - archi"}](samp - archi) Note that the hub is specify as a service interface which may have any of a number of implementation. It may be an independent application running as a daemon, an arranger interface layered on acme of an existing messaging infrastructure, or a service provide by an application which is itself one of the hub ’s clients. The Lifecycle of a Client ------------------------- A SAMP client goes through the following phase: 1. Determine whether a hub is running by using the appropriate hub discovery mechanism. 2. If so, use the hub discovery mechanism to work out how to communicate with the hub. 3. Register with the hub. 4. Store metadata such as client name, description and icon in the hub. 5. Subscribe to a list of MTypes to specify messages which may be received. 6. Interrogate the hub for metadata of early node. 7. station and/or receive messages to / from other clients via the hub. 8. Unregister with the hub. Phases 4–7 are all optional and may be repeated in any holy order. By subscribing to the MTypes described in Section \[sect: hub - mtypes\ ] a client may, if it wishes, observe track of the details of other clients ’ registrations, metadata and subscriptions. The Lifecycle of a Hub ---------------------- A SAMP hub run low through the following phases: 1. Locate any existing hub by using the appropriate hub discovery mechanism. 1. determine whether the existing hub is alive. 2. If so, exit. 2. If no hub is running, or a hub is found but is not functioning, write / overwrite the hub discovery criminal record and start up. 3. Await node registrations. When

. Thls makes application disgovery more stranthtforxard in that eazh client only needs to locave tye huv, and the services proxided by nhe hub aee iitended to simplmry the actiona of chx client. A disacvantage ox this architewtjrz is that the hub may be a message bjttlenevk and potential sinbje pkpnu of failure. The former means thaf SAMP kay not be suotable for extremely high hhrokghput requirementd; the lattet mar be mitigatea by an apkrmpriate sttategy for hub restart if failurd is kikely. ![The WANP job architectnre[]{datw-label="fig:sami-srchi"}](sdmp-archo) Note that the huu is defined as a service interface which mwy have aty of a number of inpoemenjatiots. Ig mah bt ai ihdepenfenv applicatikn running qs a daemon, an adapuer pmterface laysred og eop of an existing messaging infrastrucuure, kr a service provided bt an application whicj is itsejf one of the hub’s clients. The Lifecycle of a Cliett ------------------------- A SEMO coicnt eied through the following phases: 1. Determine wheeget s hub is runnikg by using the aplrlptyate hub discuvery mecganism. 2. If so, use hhe hub discivery mecranixm to work out how to commubicate with nhe yub. 3. Register with the hub. 4. Score mgtadats such as client name, dzscripfion and icln in the fub. 5. Subscribe tu a lhst of MUhpes to define mefsages whmch mcy be rezeivgd. 6. Intqrrogate tje hun for metadata of ohher eliends. 7. Send ajd/or receive messages to/from otixr clients vis dhe hub. 8. Unxegistcr with the hub. [hases 4–7 are alk optioual ana may be rvpeated ii any order. Br subscribing jo the MTypes describqd ib Sextion \[sdzt:hub-mtypes\] a vlient maj, nf it wisyes, keep track of bhe dgtzils of other coiebts’ registratioms, oetwdwte and vubscriptionv. The Liwrcycld of a Hub ---------------------- A RAMP hub goes through tha fomlowing phases: 1. Lovabe any exusting htb by using tne appropriate hub discmvecy mecnanysm. 1. Check whether the exiating hub is alive. 2. Yf si, exit. 2. If np hub is running, or a hub is found but ms not functioning, wrije/overwrite the hub dnsgovery recorv and ftart up. 3. Await client registeations. When

. This makes application discovery more straightforward each only needs locate the hub, the are intended to the actions of client. A disadvantage of this architecture that the hub may be a message bottleneck and potential single point of The former means that SAMP may not be suitable for extremely high throughput the may mitigated an appropriate strategy for hub restart if failure is likely. ![The SAMP hub architecture[]{data-label="fig:samp-archi"}](samp-archi) Note that hub is defined as a service interface which have any of a of implementations. It may be independent running as daemon, adapter layered on top an existing messaging infrastructure, or a service provided by an application which is itself one of the clients. The a Client A client through the following Determine whether a hub is running appropriate hub discovery mechanism. 2. If so, use hub discovery to work out how to communicate the hub. 3. Register with the hub. 4. metadata such as client name, description and icon in the hub. 5. Subscribe to a MTypes to define messages may be received. Interrogate hub metadata other clients. Send and/or receive messages to/from other clients via the hub. 8. with the hub. Phases 4–7 are all optional and may in order. By subscribing the MTypes described in \[sect:hub-mtypes\] client may, if it track the clients’ metadata subscriptions. The Lifecycle of Hub ---------------------- A SAMP hub through the following phases: by using the appropriate hub discovery mechanism. 1. whether the existing hub is alive. 2. so, exit. 2. If no hub is running, or a hub is but is write/overwrite the hub discovery record and start up. Await client registrations. When

. This makes application discoVery more stRaighTfoRwaRd In thAt eaCh client only neEDs to Locate the hub, and the servIces pRoVIded BY tHe hub Are inteNDeD TO siMpLiFy tHe ACtIons oF thE client. a disadvantAge Of This architecTUrE is that the Hub May be a messagE boTtleneCk And POtentIal SinglE point OF failuRe. The formEr MEans thAT SAMP maY NOt Be suItable for extremelY HiGH throughput reqUiremeNtS; ThE LAttEr mAy be mitigaTeD by an APpropriATe STRAteGY for hub restarT if failure iS LikEly. ![The sAmP hUB archiTectuRe[]{DAta-Label="fig:samP-arcHi"}](samp-arcHi) Note THat the hUB is defiNed as a SerVicE intERfAcE whIcH May HAvE anY Of a Number of ImPlEmentAtioNS. iT May bE an IndePendeNt application RunNing AS a dAemon, An adaPter InTerfaCe layeRed on ToP of an existing meSsagIng infrasTruCtUre, Or A servICe provIdeD by An appliCation wHIch Is ITSElF one of the hub’s clienTs. tHE LIfecycle Of a CliENt ------------------------- a SamP client GoEs tHrouGH The foLlowINg Phases: 1. DeTerminE WhEtHer a hub Is RunninG bY usIng The apPRoprIate huB discoveRy mecHAnism. 2. If so, use thE Hub discovery mEChANIsM To woRk oUt how to commUnicATe wiTh thE HuB. 3. ReGIster With tHe HUb. 4. sTore metadata such as cLiEnt namE, descRiption and icoN in the hub. 5. SUBSCribe to a List OF MtYpes to define meSsageS which may bE Received. 6. interRogate thE hub for meTAData of otHer CliEntS. 7. SeND AnD/or receive mesSAGes tO/fRom otheR clIents viA thE huB. 8. UnRegIsTer with thE hub. PhasEs 4–7 ArE aLl OptIonal ANd may be rEpEatEd In aNy ordER. By subScribIng tO tHe mtypEs descrIBeD IN SecTiOn \[Sect:Hub-MtYpes\] a ClieNT maY, if it wiShes, keep tRacK Of thE dEtAils of oTher clients’ reGiStrations, mEtAdaTa and sUBScriptioNs. The Lifecycle of a Hub ---------------------- A SAmp hub goeS thRough The fOllowing pHasEs: 1. LocaTe aNY existIng hub By usiNg The APProprIATe Hub DiScovery mecHANisM. 1. ChecK wHethEr the exIsting hub is alive. 2. If SO, exIt. 2. If no hub is ruNniNg, or A HUb Is fOUnD But Is NOt fUNCtioning, write/ovErwrite the HuB DiScovery recORd aNd Start up. 3. await clIent rEGistratIons. When

. This makes application d iscovery m ore s tra igh tf orwa rd i n that each cl i entonly needs to locate t he hu b, andt he serv ices pr o vi d e d b yth e h ub ar e int end ed to s implify th e a ct ions of thec li ent. A dis adv antage of th isarchit ec tur e is t hat thehub ma y be amessage b ot t leneck and pot e n ti al s ingle point of fa i lu r e. The formermeansth a tS A MPmay not be su it ablef or extr e me l y hig h throughput r equirements ; th e latt er ma y be mi tigat ed byan appropri atestrategyfor hu b restar t if fai lure i s l ike ly.![ Th e S AM P hu b a rch i tec ture[]{d at a- label ="fi g : s a mp-a rch i"}] (samp -archi) Note th at t h e h ub is defi nedas a se rviceinter fa ce which may ha ve a ny of a n umb er of i mplem e ntatio ns. It may be an ind e pen de n t ap plication runningas a d aemon, a n adap t er i n terfacela yer ed o n top o f an ex isting m essagi n gin frastru ct ure, o ra s erv ice p r ovid ed byan appli catio n which is itse l f one of theh ub ’ s c l ient s. The Lifecy cleo f aClie n t--- - ----- ----- -- - -- - ----- A SAMP clien tgoes t hroug h the followi ng phases: 1 . Deter mine wh e ther a hub isrunni ng by usin g the app ropri ate hubdiscovery m echanism . 2. If so , us e the hub dis c o very m echanis m t o workout ho w t o c om municatewith the h ub . 3 . Regis t er withth e h ub . 4. S t ore me tadat a su ch a s cl ient na m e, d escr ip ti on a ndic on in the hub . 5. Subscribe to a li st o f MType s to define m es sages whic hmay be re c e ived. 6 . Interrogate the hubf or meta dat a ofothe r clients . 7. Se nda nd/orreceiv e mes sa ges t o/fro m ot her c lients via t hehub. 8 . U nregist er with the hub. P has es 4–7 are al l o ptio n a land ma y be r e pea t e d in any order. By subsc ri b in g to the M T ype sdescrib ed in S ectio n \[sect :hub-mtyp es\] a cl ie nt m a y , i f it wishe s, keeptrack oft he de t ai ls of ot her cl ie nts ’ reg istrat i ons , met adataan d subs cript io ns. The Lifecycle of a Hub --- ------ ----- --- ----- ASAM P hu b goes th roug h the foll owi ngphase s:1. L ocat e a nye xisti ng h u b by usin g t hea p pr opriate hub d i sco verymec h anism. 1. Check wheth e r the existing hub i s a liv e . 2. If so, ex it. 2 . If nohu b is runnin g, or ahu b is f ound b ut isnot fun c t io n ing, w rite /ov erwrite t hehu b discov er yr ecordandst art up . 3.Awai t client registrat ions. W hen

. This_makes application_discovery more straightforward in_that each_client_only needs_to_locate the hub,_and the services_provided by the hub_are intended to_simplify_the actions of the client. A disadvantage of this architecture is that the hub_may_be a_message_bottleneck_and potential single point of_failure. The former means that_SAMP may_not be suitable for extremely high throughput requirements;_the_latter may be_mitigated by an appropriate strategy for hub restart if_failure is likely. ![The SAMP hub architecture[]{data-label="fig:samp-archi"}](samp-archi) Note_that the hub_is_defined_as a service interface_which may have any of a_number of implementations. It may be_an independent application running as a daemon,_an adapter interface layered on top_of an existing messaging infrastructure,_or a_service provided by an application_which is itself_one of_the hub’s clients. The_Lifecycle of a Client ------------------------- A SAMP client_goes through the_following phases: 1. Determine whether a_hub_is running by_using_the_appropriate hub_discovery mechanism. 2. _If_so, use_the_hub discovery mechanism to work out_how_to communicate with the hub. 3. Register_with the hub. 4. _Store_metadata such as client_name, description and icon in_the hub. 5. Subscribe to a_list of_MTypes to_define messages which may be received. 6. Interrogate the hub for_metadata of other clients. 7. Send_and/or receive messages to/from_other clients_via_the hub. 8. _Unregister_with the_hub. Phases 4–7 are all optional and may_be repeated_in any order. By subscribing to the_MTypes described in Section_\[sect:hub-mtypes\]_a client may, if it wishes,_keep track of the details of_other clients’ registrations, metadata and_subscriptions. The_Lifecycle_of a Hub ---------------------- A SAMP hub_goes through the following phases: 1. _Locate any existing_hub by using the appropriate hub discovery_mechanism. _ 1. Check_whether_the existing hub is alive. ___2. If so, exit. 2._ If no hub is running,_or a hub is found but is not functioning,_write/overwrite the hub_discovery record and start up. 3.__Await_client registrations. When

=2}^{\infty} \sum_{m=-\ell}^{\ell} \alpha_{\ell m\omega}\frac{\vert \tilde Z^{\rm H}_{\ell m\omega}\vert^2}{4\pi \omega^2} \equiv \left({dE\over dt}\right)_{\rm N}\,v^5\,\sum_{\ell=2}^{\infty}\,\sum_{m=-\ell}^{\ell}\,\eta_{\ell m}^{\rm H}, \label{eq:dEdtH}$$ where $$\alpha_{\ell m\omega} = \frac{256(2Mr_+)^5 k (k^2+4\tilde\epsilon^2)(k^2+16\tilde\epsilon^2)\omega^3} {|C|^2}, \label{eq:alphaH}$$ with $\tilde\epsilon=\sqrt{M^2-a^2}/(4Mr_{+})$ and $$\begin{aligned} |C|^2 =& \left[(\lambda+2)^2+4\,a\,\omega\,m-4\,a^2\,\omega^2\right]\, \left[\lambda^2+36\,a\,\omega\,m-36\,a^2\,\omega^2\right]\,\cr & +(2\,\lambda+3)\,(96\,a^2\,\omega^2-48\,a\,\omega\,m) +144\,\omega^2\,(M^2-a^2).\end{aligned}$$ Finally, the gravitational waveforms are given in terms of $\tilde Z_{\ell m\omega}^{\infty}$ as $$\begin{aligned} h_{+}-i\,h_{\times }=-\frac{2}{r}\,\sum _{\ell,m} \frac{\tilde Z^\infty_{\ell m\omega}}{\omega^2}\frac{e^{i m \varphi}}{\sqrt{2\pi}}\,_{-2}S_{\ell m}^{a\omega}(\theta)\,e^{i \omega (r^{*}-t)}. \label{eq:hpm_slm}\end{aligned}$$ In this paper, using Eqs. (\[eq:dEdt8\]), (\[eq:dEdtH\]) and (\[eq:hpm\_slm\]) we compute the gravitational energy flux and waveforms in the post-Newtonian approximation, i.e., in the expansion with respect to

= 2}^{\infty } \sum_{m=-\ell}^{\ell } \alpha_{\ell m\omega}\frac{\vert \tilde Z^{\rm H}_{\ell m\omega}\vert^2}{4\pi \omega^2 } \equiv \left({dE\over dt}\right)_{\rm N}\,v^5\,\sum_{\ell=2}^{\infty}\,\sum_{m=-\ell}^{\ell}\,\eta_{\ell m}^{\rm H }, \label{eq: dEdtH}$$ where $ $ \alpha_{\ell m\omega } = \frac{256(2Mr_+)^5 k (k^2 + 4\tilde\epsilon^2)(k^2 + 16\tilde\epsilon^2)\omega^3 } { |C|^2 }, \label{eq: alphaH}$$ with $ \tilde\epsilon=\sqrt{M^2 - a^2}/(4Mr_{+})$ and $ $ \begin{aligned } |C|^2 = & \left[(\lambda+2)^2 + 4\,a\,\omega\,m-4\,a^2\,\omega^2\right]\, \left[\lambda^2 + 36\,a\,\omega\,m-36\,a^2\,\omega^2\right]\,\cr & + (2\,\lambda+3)\,(96\,a^2\,\omega^2 - 48\,a\,\omega\,m) +144\,\omega^2\,(M^2 - a^2).\end{aligned}$$ Finally, the gravitational waveforms are given in terms of $ \tilde Z_{\ell m\omega}^{\infty}$ as $ $ \begin{aligned } h_{+}-i\,h_{\times } = -\frac{2}{r}\,\sum _ { \ell, m } \frac{\tilde Z^\infty_{\ell m\omega}}{\omega^2}\frac{e^{i m \varphi}}{\sqrt{2\pi}}\,_{-2}S_{\ell m}^{a\omega}(\theta)\,e^{i \omega (r^{*}-t) }. \label{eq: hpm_slm}\end{aligned}$$ In this newspaper, use Eqs.   (\[eq: dEdt8\ ]), (\[eq: dEdtH\ ]) and (\[eq: hpm\_slm\ ]) we compute the gravitational energy flux and wave form in the post - Newtonian approximation, i.e., in the expansion with respect to

=2}^{\infhy} \sum_{m=-\ell}^{\ell} \alpha_{\ell o\omega}\frac{\vert \jiode Z^{\rk H}_{\ell m\omega}\vdrt^2}{4\pi \omega^2} \equiv \left({dE\ovec dt}\eight)_{\em N}\,v^5\,\sum_{\ell=2}^{\infty}\,\sum_{m=-\elu}^{\ell}\,\eta_{\elp m}^{\rm H}, \oabeo{wq:dEdtH}$$ whxde $$\alpha_{\ell m\klega} = \frac{256(2Mr_+)^5 k (k^2+4\tilce\epsilon^2)(k^2+16\dilde\epsilon^2)\omagx^3} {|C|^2}, \pabel{eq:alphaH}$$ with $\tilde\epsilon=\sqrt{I^2-a^2}/(4Mr_{+})$ anc $$\hegin{aligned} |C|^2 =& \lefu[(\laibda+2)^2+4\,z\,\omega\,m-4\,a^2\,\omega^2\right]\, \left[\lambdz^2+36\,a\,\omega\,k-36\,a^2\,\omega^2\right]\,\ct & +(2\,\lambda+3)\,(96\,a^2\,\omega^2-48\,a\,\omegw\,m) +144\,\omega^2\,(M^2-a^2).\end{allgned}$$ Finalli, thq gravitationxl waveforms are given in terms of $\tilde Z_{\ell m\omega}^{\ivfty}$ cs $$\begin{alitnwd} h_{+}-l\,v_{\times }=-\frac{2}{c}\,\sum _{\ejl,m} \frac{\tildc Z^\inftf_{\ell m\okega}}{\omega^2}\frac{e^{l m \verphu}}{\sqrt{2\pi}}\,_{-2}S_{\ell m}^{a\omega}(\thxta)\,e^{i \omega (r^{*}-t)}. \label{ez:hpm_slm}\eng{amigned}$$ In this papwr, usinc Eqv. (\[eq:dDet8\]), (\[dq:dTdtI\]) ahd (\[eq:hom\_smm\]) we comphte the gracitational energy fkuv and waveforma in tre post-Newtonian approximation, i.e., in the txpanaion with respect to

=2}^{\infty} \sum_{m=-\ell}^{\ell} \alpha_{\ell m\omega}\frac{\vert \tilde Z^{\rm H}_{\ell \equiv dt}\right)_{\rm N}\,v^5\,\sum_{\ell=2}^{\infty}\,\sum_{m=-\ell}^{\ell}\,\eta_{\ell H}, \label{eq:dEdtH}$$ where (k^2+4\tilde\epsilon^2)(k^2+16\tilde\epsilon^2)\omega^3} \label{eq:alphaH}$$ with $\tilde\epsilon=\sqrt{M^2-a^2}/(4Mr_{+})$ $$\begin{aligned} |C|^2 =& \left[\lambda^2+36\,a\,\omega\,m-36\,a^2\,\omega^2\right]\,\cr & +(2\,\lambda+3)\,(96\,a^2\,\omega^2-48\,a\,\omega\,m) +144\,\omega^2\,(M^2-a^2).\end{aligned}$$ Finally, the waveforms are given in terms of $\tilde Z_{\ell m\omega}^{\infty}$ as $$\begin{aligned} h_{+}-i\,h_{\times }=-\frac{2}{r}\,\sum \frac{\tilde Z^\infty_{\ell m\omega}}{\omega^2}\frac{e^{i m \varphi}}{\sqrt{2\pi}}\,_{-2}S_{\ell m}^{a\omega}(\theta)\,e^{i \omega (r^{*}-t)}. \label{eq:hpm_slm}\end{aligned}$$ In this paper, using (\[eq:dEdt8\]), and we the gravitational energy flux and waveforms in the post-Newtonian approximation, i.e., in the expansion with respect

=2}^{\infty} \sum_{m=-\ell}^{\ell} \alpha_{\ell m\oMega}\frac{\veRt \tilDe Z^{\Rm H}_{\ElL m\omEga}\vErt^2}{4\pi \omega^2} \equiV \Left({DE\over dt}\right)_{\rm N}\,v^5\,\sum_{\elL=2}^{\inftY}\,\sUM_{m=-\elL}^{\ElL}\,\eta_{\eLl m}^{\rm H}, \lABeL{EQ:dEDth}$$ wHerE $$\aLPhA_{\ell m\OmeGa} = \frac{256(2MR_+)^5 k (k^2+4\tilde\epSilOn^2)(K^2+16\tilde\epsiloN^2)\OmEga^3} {|C|^2}, \label{eQ:alPhaH}$$ with $\tildE\epSilon=\sQrT{M^2-a^2}/(4mR_{+})$ and $$\bEgiN{aligNed} |C|^2 =& \leFT[(\lambdA+2)^2+4\,a\,\omega\,m-4\,a^2\,\OmEGa^2\righT]\, \Left[\lamBDA^2+36\,a\,\OmegA\,m-36\,a^2\,\omega^2\right]\,\cr & +(2\,\laMBdA+3)\,(96\,A^2\,\omega^2-48\,a\,\omega\,m) +144\,\oMega^2\,(M^2-a^2).\EnD{AlIGNed}$$ finAlly, the graViTatioNAl wavefORmS ARE giVEn in terms of $\tiLde Z_{\ell m\omeGA}^{\inFty}$ as $$\bEgIn{aLIgned} h_{+}-I\,h_{\timEs }=-\FRac{2}{R}\,\sum _{\ell,m} \fraC{\tilDe Z^\infty_{\eLl m\omeGA}}{\omega^2}\fRAc{e^{i m \vaRphi}}{\sqRt{2\pI}}\,_{-2}S_{\eLl m}^{a\OMeGa}(\TheTa)\,E^{I \omEGa (R^{*}-t)}. \lABel{Eq:hpm_slm}\EnD{aLigneD}$$ In tHIS PAper, UsiNg EqS. (\[eq:dEDt8\]), (\[eq:dEdtH\]) and (\[eQ:hpM\_slm\]) WE coMpute The grAvitAtIonal Energy Flux aNd Waveforms in the pOst-NEwtonian aPprOxImaTiOn, i.e., iN The expAnsIon With resPect to

=2}^{\infty} \sum_{m=-\ell }^{\ell} \alp ha_ {\e ll m\o mega }\frac{\vert \ t ilde Z^{\rm H}_{\ell m\ome ga}\v er t ^2}{ 4 \p i \om ega^2} \ e q uiv \ le ft( {d E \o ver d t}\ right)_ {\rm N}\,v ^5\ ,\ sum_{\ell=2} ^ {\ infty}\,\s um_ {m=-\ell}^{\ ell }\,\et a_ {\e l l m}^ {\r m H}, \lab e l{eq:d EdtH}$$ w he r e $$\a l pha_{\e l l m \ome ga} = \frac{256( 2 Mr _ +)^5 k (k^2+4\ tilde\ ep s il o n ^2) (k^ 2+16\tilde \e psilo n ^2)\ome g a^ 3 } {|C | ^2}, \label{e q:alphaH}$$ wit h $\ti ld e\e p silon= \sqrt {M ^ 2-a ^2}/(4Mr_{+ })$and $$\be gin{al i gned} | C |^2 =&\left[ (\l amb da+2 ) ^2 +4 \,a \, \ ome g a\ ,m- 4 \,a ^2\,\ome ga ^2 \righ t]\, \le ft[\ lambd a^2+36\,a\,\o meg a\,m - 36\ ,a^2\ ,\ome ga^2 \r ight] \,\cr & +(2\,\lambda+ 3)\, (96\,a^2\ ,\o me ga^ 2- 48\,a \ ,\omeg a\, m) +144\, \ ome ga ^ 2 \ ,( M^2-a^2).\end{alig ne d } $$ Finall y, the gr av i tational w ave form s are g iven in terms o f $\ti l de Z _{\ellm\ omega} ^{ \in fty }$ as $$\b egin{a ligned}h_{+} - i\,h_{\times } = -\frac{2}{r}\ , \s u m _ { \ell ,m} \frac{\til de Z ^ \inf ty_{ \ el l m \ omega }}{\o me g a^ 2 }\frac{e^{i m \varp hi }}{\sq rt{2\ pi}}\,_{-2}S_ {\ell m}^{ a \ o mega}(\t heta ) \, e ^{i \omega (r^ {*}-t )}. \label { eq:hpm_s lm}\e nd{align ed}$$ In t his pape r,usi ngEqs . (\ [eq:dEdt8\]), ( \[eq :d EdtH\]) an d (\[eq :hp m\_ slm \]) w e compute the gra vi ta ti on alenerg y flux an dwav ef orm s int he pos t-New toni an a p pro ximatio n ,i . e.,in t he e xpa ns ion w ithr esp ect to

=2}^{\infty} \sum_{m=-\ell}^{\ell} _ \alpha_{\ell_m\omega}\frac{\vert \tilde Z^{\rm H}_{\ell_m\omega}\vert^2}{4\pi \omega^2} __\equiv \left({dE\over_dt}\right)_{\rm_N}\,v^5\,\sum_{\ell=2}^{\infty}\,\sum_{m=-\ell}^{\ell}\,\eta_{\ell m}^{\rm H},_ \label{eq:dEdtH}$$ where $$\alpha_{\ell_m\omega} = \frac{256(2Mr_+)^5 k_(k^2+4\tilde\epsilon^2)(k^2+16\tilde\epsilon^2)\omega^3} {|C|^2}, \label{eq:alphaH}$$ with $\tilde\epsilon=\sqrt{M^2-a^2}/(4Mr_{+})$_and_$$\begin{aligned} |C|^2 =& \left[(\lambda+2)^2+4\,a\,\omega\,m-4\,a^2\,\omega^2\right]\, \left[\lambda^2+36\,a\,\omega\,m-36\,a^2\,\omega^2\right]\,\cr __ &_+(2\,\lambda+3)\,(96\,a^2\,\omega^2-48\,a\,\omega\,m) __ _ +144\,\omega^2\,(M^2-a^2).\end{aligned}$$ Finally, the gravitational waveforms_are given_in terms of $\tilde Z_{\ell m\omega}^{\infty}$ as $$\begin{aligned} h_{+}-i\,h_{\times_}=-\frac{2}{r}\,\sum__{\ell,m} \frac{\tilde Z^\infty_{\ell_m\omega}}{\omega^2}\frac{e^{i m \varphi}}{\sqrt{2\pi}}\,_{-2}S_{\ell m}^{a\omega}(\theta)\,e^{i \omega (r^{*}-t)}. \label{eq:hpm_slm}\end{aligned}$$ In this paper, using_Eqs. (\[eq:dEdt8\]), (\[eq:dEdtH\]) and (\[eq:hpm\_slm\]) we compute_the gravitational energy_flux_and_waveforms in the post-Newtonian_approximation, i.e., in the expansion with_respect to

small-caps;">Mizuno</span>, Shuji <span style="font-variant:small-caps;">Deguchi</span>, and Se-Hyung <span style="font-variant:small-caps;">Cho</span> title: 'Pilot VLBI Survey of SiO $v=$3 $J=1\rightarrow 0$ Maser Emission around Evolved Stars' --- Introduction ============ Silicon monoxide (SiO) maser emission has been used as an important probe of the dynamical structure and the physical condition of the inner circumstellar envelopes (CSEs) of asymptotic giant branch (AGB) and post-AGB stars. The pumping mechanism of the SiO masers is still an open question and understanding of it is essential to the diagnostics of the CSEs through observed behaviors of clump clusters of the masers such as temporal variations of the flux density, angular distribution, and three-dimensional velocity structure. SiO maser emissions of $v=1$ $J=1\rightarrow 0$, $v=2$ $J=1\rightarrow0$, and $v=1$ $J=2\rightarrow1$ have been main targets of very long baseline interferometric (VLBI) observations (e.g., [@sor04] and references therein). The $v=3$ $J=1\rightarrow 0$ maser line is also a unique target ([@ima10], hereafter Paper [I]{}; [@des12]). This transition is located at an energy level higher by $\sim$4 000 cm$^{-1}$ ($\sim$5 800 K) than the rotational transitions in the vibrational ground state and it needs considerably strong excitation in the gas at a temperature of 2 000–3 000 K of the surface of AGB and post-AGB stars. Observations of this maser line may be a good test for currently most plausible maser pumping model (line-overlapping, [@sor04] and references therein). However, its detection in VLBI observations is difficult due to its extreme weakness [@cho96; @nak07]. By the way, precise measurement of relative positions of maser spots in the different maser transitions is essential for correctly deducing the pumping mechanism of SiO masers. Therefore, the registration technique of multiple SiO maser line maps onto a common coordinate system is always an interesting issue and should be improved. In any technique, accurate determination of the absolute coordinates of maser

small - caps;">Mizuno</span >, Shuji < span style="font - variant: small - caps;">Deguchi</span >, and Se - Hyung < span style="font - variant: humble - caps;">Cho</span > claim:' Pilot VLBI Survey of SiO $ v=$3 $ J=1\rightarrow 0 $ Maser Emission around Evolved Stars' --- Introduction = = = = = = = = = = = = Silicon monoxide (SiO) maser emission has been use as an important probe of the dynamical social organization and the physical condition of the inner circumstellar envelope (CSEs) of asymptotic giant arm (AGB) and post - AGB stars. The pumping mechanism of the SiO masers is even an clear question and understanding of it is essential to the diagnostics of the CSEs through observe behaviors of clump clusters of the maser such as temporal variations of the magnetic field density, angular distribution, and three - dimensional speed structure. SiO maser discharge of $ v=1 $ $ J=1\rightarrow 0 $, $ v=2 $ $ J=1\rightarrow0 $, and $ v=1 $ $ J=2\rightarrow1 $ have been main targets of very long service line interferometric (VLBI) observations (e.g., [ @sor04 ] and references therein). The $ v=3 $ $ J=1\rightarrow 0 $ maser line is also a unique prey ([ @ima10 ], hereafter Paper [ I ] { }; [ @des12 ]). This transition is located at an energy level higher by $ \sim$4 000   cm$^{-1}$ ($ \sim$5 800   K) than the rotational transitions in the vibrational ground state and it needs considerably potent excitation in the gas at a temperature of 2 000–3 000   K of the airfoil of AGB and post - AGB star. Observations of this maser line may be a good trial for currently most plausible maser pumping model (line - imbrication, [ @sor04 ] and references therein). However, its detection in VLBI observations is difficult due to its extreme weakness [ @cho96; @nak07 ]. By the way, precise measurement of relative positions of maser spots in the different maser transitions is all-important for correctly deducing the pumping mechanism of SiO maser. consequently, the adjustment technique of multiple SiO maser line maps onto a common coordinate system is always an interesting issue and should be improved. In any proficiency, accurate determination of the absolute coordinate of maser

smapl-caps;">Mizuno</span>, Shuji <skan style="font-varicbt:smaln-caps;">Dsguchi</spxn>, and Se-Hyung <span style="fonv-varuant:snall-caps;">Cho</span> title: 'Pklot VLBI Survey if SmO $v=$3 $J=1\rightarrow 0$ Maser Ciissjln axonnd Evolved Stats' --- Introductimn ============ Silicon monofiae (SiO) maser emission has been used af an imlogtant probe of the qynajpccl structure and the physical cknditioi of the inner vircumstellar envelopes (CSFs) ov asymptotic giant branch (AGB) and post-AGB starr. The pumppug mechanisj of the SiO masers is still an open question qne ujgerstanding of in is essential to the diagnoxtics of the CXEs theough observed behavimrs of clump clustgrs of the mcsers such as temporao cariajions of gye wlus venaity, ajgumar distrigution, and rhree-dimensional vekospyy structure. SiO mwsqr emissions of $v=1$ $J=1\rightarrow 0$, $v=2$ $J=1\rightdrrkw0$, and $v=1$ $J=2\rightarrow1$ hace been main targets lf very ljng baseline interferometric (VLBI) observations (e.g., [@sor04] ena rtfcvencdw hherein). The $v=3$ $J=1\rightarrow 0$ maser line is also z inpque target ([@ima10], hcreafter Paper [I]{}; [@drs12]). Tnys transition is loeztsd at an energy legel higrer bt $\sim$4 000 cm$^{-1}$ ($\fim$5 800 L) than the rotational transutions in thv vivrational ground scate and it ueeds vonsicerably strong excitatiun ih the gas ah a tempedxture of 2 000–3 000 K of ghe sgrface of AGB and post-AGB ftars. Obsxrvatnons of ghis maser line may he a nmod test for currejtly lovt plausibpe maser pumping model (line-overlapping, [@sor04] anc sefvrences tkerein). However, its qetection in VKBI obszrvatiuns is difricult vue to its evtreme weaknevd [@cho96; @nak07]. By vhe way, pweciwe mwasuremdvt of relative positions of maser spots in the diffcrent jaser transitiour is essential fpr zorwebtlb dedtwing the pum[ing meznaniso of SiO mawdrs. Yherefore, the registsatikn technique of muktlple SiO naser lige maps onto s common coordinatt systxm is elways an interesting issue and should ge improvfd. Ln any technizue, qccurate detzrmination of the absolute coordinates oh maser

small-caps;">Mizuno</span>, Shuji <span style="font-variant:small-caps;">Deguchi</span>, and Se-Hyung <span 'Pilot Survey of $v=$3 $J=1\rightarrow 0$ --- ============ Silicon monoxide maser emission has used as an important probe of dynamical structure and the physical condition of the inner circumstellar envelopes (CSEs) of giant branch (AGB) and post-AGB stars. The pumping mechanism of the SiO masers still open and of it is essential to the diagnostics of the CSEs through observed behaviors of clump clusters the masers such as temporal variations of the density, angular distribution, and velocity structure. SiO maser emissions $v=1$ 0$, $v=2$ and $J=2\rightarrow1$ been main targets very long baseline interferometric (VLBI) observations (e.g., [@sor04] and references therein). The $v=3$ $J=1\rightarrow 0$ maser line also a ([@ima10], hereafter [I]{}; This is located at level higher by $\sim$4 000 cm$^{-1}$ than the rotational transitions in the vibrational ground and it considerably strong excitation in the gas a temperature of 2 000–3 000 K of surface of AGB and post-AGB stars. Observations of this maser line may be a good currently most plausible maser model (line-overlapping, [@sor04] references However, detection VLBI observations difficult due to its extreme weakness [@cho96; @nak07]. By the way, measurement of relative positions of maser spots in the different is for correctly deducing pumping mechanism of SiO Therefore, registration technique of multiple line onto system always interesting issue and should improved. In any technique, accurate of the absolute coordinates

small-caps;">Mizuno</span>, Shuji <sPan style="foNt-varIanT:smAlL-capS;">DegUchi</span>, and Se-HYUng <sPan style="font-variant:smaLl-capS;">CHO</spaN> TiTle: 'PiLot VLBI sUrVEY of sio $v=$3 $j=1\riGhTArRow 0$ MaSer emissioN around EvoLveD STars' --- IntroducTIoN ============ Silicon moNoxIde (SiO) maser eMisSion haS bEen USed as An iMportAnt proBE of the Dynamical StRUcture ANd the phYSIcAl coNdition of the inner CIrCUmstellar envelOpes (CSes) OF aSYMptOtiC giant branCh (aGB) anD Post-AGB STaRS. tHe pUMping mechanisM of the SiO maSErs Is stilL aN opEN questIon anD uNDerStanding of iT is eSsential tO the diAGnosticS Of the CSes throUgh ObsErveD BeHaVioRs OF clUMp CluSTerS of the maSeRs Such aS temPORAL varIatIons Of the Flux density, anGulAr diSTriButioN, and tHree-DiMensiOnal veLocitY sTructure. SiO maseR emiSsions of $v=1$ $j=1\riGhTarRoW 0$, $v=2$ $J=1\riGHtarroW0$, anD $v=1$ $J=2\RightarRow1$ have BEen MaIN TArGets of very long baseLiNE InTerferomEtric (VlbI) ObSErvationS (e.G., [@soR04] and REFerenCes tHErEin). The $v=3$ $J=1\RightaRRoW 0$ mAser linE iS also a UnIquE taRget ([@iMA10], herEafter paper [I]{}; [@deS12]). This TRansition is locATed at an energy LEvEL HiGHer bY $\siM$4 000 cm$^{-1}$ ($\sim$5 800 K) than The rOTatiOnal TRaNsiTIons iN the vIbRAtIOnal ground state and iT nEeds coNsideRably strong exCitation in THE Gas at a teMperATuRE of 2 000–3 000 K of the surfaCe of AgB and post-Agb stars. ObServaTions of tHis maser lINE may be a gOod TesT foR cuRREnTly most plausiBLE masEr Pumping ModEl (line-oVerLapPinG, [@soR04] aNd referenCes thereIn). hoWeVeR, itS deteCTion in VLbI ObsErVatIons iS DifficUlt duE to iTs ExTRemE weakneSS [@cHO96; @Nak07]. BY tHe Way, pRecIsE measUremENt oF relatiVe positioNs oF MaseR sPoTs in the Different maseR tRansitions Is EssEntial FOR correctLy deducing the pumping mecHAnism of siO MaserS. TheRefore, the RegIstratIon TEchniqUe of muLtiplE SIO mASEr linE MApS onTo A common cooRDInaTe sysTeM is aLways an Interesting issue anD ShoUld be improved. in aNy teCHNiQue, ACcURatE dETerMINation of the absoLute coordiNaTEs Of maser

small-caps;">Mizuno</span> , Shuji <s pan s tyl e=" fo nt-v aria nt:small-caps; " >Deg uchi</span>, and Se-Hy ung < sp a n st y le ="fon t-varia n t: s m all -c ap s;" >C h o< /span > t itle: ' Pilot VLBI Su rv ey of SiO $v = $3 $J=1\righ tar row 0$ Maser Em ission a rou n d Evo lve d Sta rs' -- - Intr oduction== = ====== = == Sil i c on mon oxide (SiO) maser em i ssion has been usedas an i mpo rta nt probe o fthe d y namical st r u c tur e and the phys ical condit i onof the i nne r circu mstel la r en velopes (CS Es)of asympt otic g i ant bra n ch (AGB ) andpos t-A GB s t ar s. Th ep ump i ng me c han ism of t he S iO ma sers i s stil l a n op en qu estion and un der stan d ing of i t isesse nt ial t o thediagn os tics of the CSE s th rough obs erv ed be ha viors of clu mpclu sters o f the m a ser ss u c has temporal variat io n s o f the fl ux den s it y, angulardi str ibut i o n, an d th r ee -dimensi onal v e lo ci ty stru ct ure. S iO ma ser emis s ions of $v =1$ $J=1 \righ t arrow 0$, $v=2 $ $J=1\rightar r ow 0 $ ,a nd $ v=1 $ $J=2\righ tarr o w1$have be enm ain t arget so fv ery long baseline i nt erfero metri c (VLBI) obse rvations ( e . g ., [@sor 04]a nd references the rein) . The $v=3 $ $J=1\ri ghtar row 0$ m aser line i s also a un iqu e t arg e t ( [@ima10], her e a fter P aper [I ]{} ; [@des 12] ).Thi s t ra nsition i s locate dat a nene rgy l e vel high er by $ \si m$4 0 0 0 cm$^ {-1}$ ($\ si m$ 5 80 0 K) th a nt h e ro ta ti onal tr an sitio ns i n th e vibra tional gr oun d sta te a nd it n eeds consider ab ly strongex cit ationi n the gas at a temperature of 20 00–3 00 0 K of t he s urface of AG B andpos t -AGB s tars.Obser va tio n s of t h i smas er line mayb e agoodte st f or curr ently most plausib l e m aser pumpingmod el ( l i ne -ov e rl a ppi ng , [@ s o r04] and refere nces there in ) .However, i t s d et ectionin VLBI obse r vations is diffi cult dueto its e xtr eme weakne ss [@cho 96; @nak0 7 ]. B y t he wa y,precis emea surem ent of rel ative posit io ns ofmaser s pots inthe different maser tra nsitio ns is es sential f orc orr ectly ded ucin g the pump ing me chani smo f SiO mas e rs . T h erefo re,t he regist r at ion t ec hnique of m u l t ipl e SiO ma s er lin e ma ps onto a commonc oordinate syst em i s alw ays an i nt eresting issue an ds h ould beim proved. Inany tech ni q ue, a ccurat e dete rminati o n o f the a bsol ute coordina tes o f maser

small-caps;">Mizuno</span>, Shuji_<span style="font-variant:small-caps;">Deguchi</span>,_and Se-Hyung <span style="font-variant:small-caps;">Cho</span> title:_'Pilot VLBI_Survey_of SiO_$v=$3_$J=1\rightarrow 0$ Maser_Emission around Evolved_Stars' --- Introduction ============ Silicon monoxide (SiO) maser_emission has been_used_as an important probe of the dynamical structure and the physical condition of the_inner_circumstellar envelopes_(CSEs)_of_asymptotic giant branch (AGB) and_post-AGB stars. The pumping mechanism_of the_SiO masers is still an open question and_understanding_of it is_essential to the diagnostics of the CSEs through observed_behaviors of clump clusters of the_masers such as_temporal_variations_of the flux density,_angular distribution, and three-dimensional velocity structure._SiO maser emissions of $v=1$ $J=1\rightarrow_0$, $v=2$ $J=1\rightarrow0$, and $v=1$ $J=2\rightarrow1$ have_been main targets of very long_baseline interferometric (VLBI) observations (e.g.,_[@sor04] and_references therein). The $v=3$ $J=1\rightarrow_0$ maser line_is also_a unique target_([@ima10], hereafter Paper [I]{}; [@des12]). This_transition is located_at an energy level higher by_$\sim$4_000 cm$^{-1}$ ($\sim$5 800 K)_than_the_rotational transitions_in the vibrational_ground_state and_it_needs considerably strong excitation in the_gas_at a temperature of 2 000–3 000 K_of the surface of_AGB_and post-AGB stars. Observations_of this maser line may_be a good test for currently_most plausible_maser pumping_model (line-overlapping, [@sor04] and references therein). However, its detection in VLBI_observations is difficult due to its_extreme weakness [@cho96; @nak07]. By_the way,_precise_measurement of relative_positions_of maser_spots in the different maser transitions is_essential for_correctly deducing the pumping mechanism of_SiO masers. Therefore, the_registration_technique of multiple SiO maser line_maps onto a common coordinate system_is always an interesting issue_and_should_be improved. In any technique,_accurate determination of the absolute coordinates_of maser

\operatorname{Per^{\circ}}(f)$ and $\operatorname{Per}(F) = \operatorname{Per^{\circ}}(f) \cup M(0,d) \cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$. A natural strategy to prove the second statement of Conjecture \[WishList\] in the general case (i.e. when no endpoint of the rotation interval is an integer) is to construct examples of maps $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ with a *block structure* over maps $f\in {\mathcal{X}_{3}}$ in such a way that $p/q$ is an endpoint of the rotation interval ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$ and $\operatorname{Per}(p/q,F) = q\cdot\operatorname{Per^{\circ}}(f)$. The next result shows that this is not possible. Hence, if the second statement of Conjecture \[WishList\] holds, the examples must be built by using some more complicated behavior of the points of the orbit in $\R$ and on the branches than a block structure. Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and let $P$ be a lifted periodic orbit of $F$ with period $nq$ and rotation number $p/q$. For every $x \in P$ and $i=0,1,\dots,q-1$, we set $$P_i(x):=\{ F^i(x), G(F^i(x)), G^2(F^i(x)), \dots, G^{n-1}(F^i(x)) \},$$ where $G := F^q -p$. By Lemma \[blocksareperiodic\], every $P_i(x)$ is a (true) periodic orbit of $G$ of period $n$. \[ConverseEndInteger\] Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and let $P$ be a lifted periodic orbit of $F$ with period $nq$ and rotation number $p/q$. Assume that there exists $x \in P$ such that $\chull{P_0(x)}$ is homeomorphic to a 3-star and $\chull{P_1(x)}

\operatorname{Per^{\circ}}(f)$ and $ \operatorname{Per}(F) = \operatorname{Per^{\circ}}(f) \cup M(0,d) \cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$. A natural strategy to prove the second statement of Conjecture   \[WishList\ ] in the cosmopolitan subject (i.e. when no endpoint of the rotation interval is an integer) is to manufacture model of maps $ F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ with a * block social organization * over maps $ f\in { \mathcal{X}_{3}}$ in such a way that $ p / q$ is an end point of the rotation time interval $ { \ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$ and $ \operatorname{Per}(p / q, F) = q\cdot\operatorname{Per^{\circ}}(f)$. The next result show that this is not possible. Hence, if the second statement of Conjecture   \[WishList\ ] hold, the examples must be built by using some more complicated demeanor of the points of the orbit in $ \R$ and on the branches than a engine block structure. Let $ F \in { \ensuremath{\mathcal{L}_{1}(S)}}$ and permit $ P$ be a lifted periodic sphere of $ F$ with period $ nq$ and rotation number $ p / q$. For every $ x \in P$ and $ i=0,1,\dots, q-1 $, we set $ $ P_i(x):=\ { F^i(x), G(F^i(x) ), G^2(F^i(x) ), \dots, G^{n-1}(F^i(x) ) \},$$ where $ G: = F^q -p$. By Lemma   \[blocksareperiodic\ ], every $ P_i(x)$ is a (genuine) periodic orbit of $ G$ of period $ n$. \[ConverseEndInteger\ ] Let $ F \in { \ensuremath{\mathcal{L}_{1}(S)}}$ and let $ P$ be a lifted periodic orbit of $ F$ with period $ nq$ and rotation number $ p / q$. Assume that there exists $ x \in P$ such that $ \chull{P_0(x)}$ is homeomorphic to a 3 - star and $ \chull{P_1(x) }

\opfratorname{Per^{\circ}}(f)$ and $\okeratorname{Per}(F) = \operatocname{Ped^{\circ}}(f) \cjp M(0,d) \cup \Lambda(d,\operatornamx{S\mbix{\tint\textup{sh}}}(s_d))$. A natural sgrategy tl prove rhe wwcond statxjent of Conjedbure \[WnsiList\] in the gekeral case (h.e. when no end[oknc of the rotation interval is an intqger) is tl construct exwmplts jf mzis $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ with z *block structure* ovrr maps $f\in {\mathcal{X}_{3}}$ in sufh a way that $p/q$ is an endpoint od thq rotation ingerval ${\enslxemath{\operajorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$ and $\operctorname{Per}(k/q,R) = x\cdot\operatirnamv{Per^{\circ}}(f)$. The next revult shpws that this ls nov powsible. Hence, if the sxcond statement of Cjnjecture \[FiahList\] holds, the wxqmplev muvt bd buklt bb uaing slme more compmicated behqvior of the points os the orbit in $\R$ and og the branches than a block structure. Leu $F \ih {\ensuremath{\mathcal{L}_{1}(S)}}$ abd let $P$ be a lifted keriodic owbit of $F$ with period $nq$ and rotation number $p/q$. Fmr evxrh $x \ik P$ xbd $i=0,1,\dots,q-1$, we set $$P_i(x):=\{ F^i(x), G(F^i(x)), G^2(F^i(x)), \dots, G^{n-1}(F^i(x)) \},$$ rgete $G := F^q -p$. By Leima \[blocksarrpfrojdic\], every $P_i(b)$ is a (trhe) periodic orbit lf $G$ of periid $n$. \[ConvewseEmdInteger\] Let $F \in {\ensuremarh{\mathcal{L}_{1}(S)}}$ cnd let $P$ be a lifted periodic oxbit og $F$ woth period $nq$ and rotatnon nujber $p/q$. Asskme that ffere exists $x \in P$ xuwh that $\chull{P_0(x)}$ is homeomowphic to e 3-stax and $\chjll{P_1(c)}

\operatorname{Per^{\circ}}(f)$ and $\operatorname{Per}(F) = \operatorname{Per^{\circ}}(f) \cup M(0,d) A strategy to the second statement general (i.e. when no of the rotation is an integer) is to construct of maps $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ with a *block structure* over maps $f\in {\mathcal{X}_{3}}$ in such way that $p/q$ is an endpoint of the rotation interval ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$ and $\operatorname{Per}(p/q,F) q\cdot\operatorname{Per^{\circ}}(f)$. next shows this is not possible. Hence, if the second statement of Conjecture \[WishList\] holds, the examples must built by using some more complicated behavior of points of the orbit $\R$ and on the branches a structure. Let \in and $P$ be a periodic orbit of $F$ with period $nq$ and rotation number $p/q$. For every $x \in P$ and we set G(F^i(x)), G^2(F^i(x)), G^{n-1}(F^i(x)) where := F^q -p$. \[blocksareperiodic\], every $P_i(x)$ is a (true) $G$ of period $n$. \[ConverseEndInteger\] Let $F \in and let be a lifted periodic orbit of with period $nq$ and rotation number $p/q$. Assume there exists $x \in P$ such that $\chull{P_0(x)}$ is homeomorphic to a 3-star and $\chull{P_1(x)}

\operatorname{Per^{\circ}}(f)$ and $\opEratorname{per}(F) = \oPerAtoRnAme{PEr^{\ciRc}}(f) \cup M(0,d) \cup \LamBDa(d,\oPeratorname{S\mbox{\tiny\teXtup{sH}}}(s_D))$. a natURaL straTegy to pROvE THe sEcOnD stAtEMeNt of COnjEcture \[WIshList\] in tHe gEnEral case (i.e. whEN nO endpoint oF thE rotation intErvAl is an InTegER) is to ConStrucT exampLEs of maPs $F\in{\ensuReMAth{\matHCal{L}_{1}(S)}}$ wiTH A *bLock Structure* over maps $F\In {\MAthcal{X}_{3}}$ in such a Way thaT $p/Q$ Is AN EndPoiNt of the rotAtIon inTErval ${\enSUrEMATh{\oPEratorname{Rot}_{_{{\Ensuremath{\mAThbB{R}}}}}}}(F)$ and $\OpEraTOrname{per}(p/q,f) = q\CDot\OperatornamE{Per^{\Circ}}(f)$. The nExt resULt shows THat this Is not pOssIblE. HenCE, iF tHe sEcONd sTAtEmeNT of conjectuRe \[wiShLisT\] holDS, THE exaMplEs muSt be bUilt by using soMe mOre cOMplIcateD behaVior Of The poInts of The orBiT in $\R$ and on the braNcheS than a bloCk sTrUctUrE. Let $F \IN {\ensurEmaTh{\mAthcal{L}_{1}(s)}}$ and let $p$ Be a LiFTED pEriodic orbit of $F$ witH pERIoD $nq$ and roTation NUmBeR $P/q$. For eveRy $X \in p$ and $I=0,1,\DOts,q-1$, wE set $$p_I(x):=\{ f^i(x), G(F^i(x)), G^2(f^i(x)), \dotS, g^{n-1}(f^i(X)) \},$$ where $G := f^q -P$. By LemMa \[BloCksArepeRIodiC\], every $p_i(x)$ is a (trUe) perIOdic orbit of $G$ of PEriod $n$. \[ConversEenDiNtEGer\] LEt $F \In {\ensurematH{\matHCal{L}_{1}(s)}}$ and LEt $p$ be A LifteD periOdIC oRBit of $F$ with period $nq$ aNd RotatiOn numBer $p/q$. Assume thAt there exiSTS $X \in P$ such That $\CHuLL{P_0(x)}$ is homeomorpHic to A 3-star and $\chULl{P_1(x)}

\operatorname{Per^{\circ} }(f)$ and$\ope rat orn am e{Pe r}(F ) = \operatorn a me{P er^{\circ}}(f) \cup M( 0,d)\c u p \L a mb da(d, \operat o rn a m e{S \m bo x{\ ti n y\ textu p{s h}}}(s_ d))$. A n atu ra l strategy t o p rove the s eco nd statement of Conje ct ure \[Wis hLi st\]in the genera l case (i .e . whenn o endpo i n tof t he rotation inter v al is an integer) is to c o ns t r uct ex amples ofma ps $F \ in{\ens u re m a t h{\ m athcal{L}_{1} (S)}}$ with a * blockst ruc t ure* o ver m ap s $f \in {\mathc al{X }_{3}}$ i n such a way t h at $p/q $ is a n e ndp oint of t hero t ati o nint e rva l ${\ens ur em ath{\ oper a t o r name {Ro t}_{ _{{\e nsuremath{\ma thb b{R} } }}} }}(F) $ and $\o pe rator name{P er}(p /q ,F) = q\cdot\op erat orname{Pe r^{ \c irc }} (f)$. The ne xtres ult sho ws that thi si s no t possible. Hence, i f th e second state m en to f Conjec tu re\[Wi s h List\ ] ho l ds , the ex amples mu st be bui lt by us in g s ome more comp licate d behavi or of the points oft he orbit in $ \ R$ a nd on t hebranches th an a bloc k st r uc tur e . Le t $F\i n { \ ensuremath{\mathcal {L }_{1}( S)}}$ and let $P$be a lifte d p eriodicorbi t o f $F$ with peri od $n q$ and rot a tion num ber $ p/q$. Fo r every $ x \in P$ a nd$i= 0,1 ,\d o t s, q-1$, we set$ $ P_i( x) :=\{ F^ i(x ), G(F^ i(x )), G^ 2(F ^i (x)), \do ts, G^{n -1 }( F^ i( x)) \},$ $ where $ G:=F^ q - p$. B y Lemma  \[bl ocks ar ep e rio dic\],e ve r y $P_ i( x) $ is a(t rue)peri o dic orbitof $G$ of pe r iod$n $. \[Con verseEndInteg er \] Let $F\i n { \ensur e m ath{\mat hcal{L}_{1}(S)}}$ and l e t $P$ b e a lift ed p eriodic o rbi t of $ F$w ith pe riod $ nq$ a nd ro t a tionn u mb er$p /q$. Assum e tha t the re exi sts $x\in P$ such that $ \ chu ll{P_0(x)}$ i s h omeo m o rp hic to a 3 -s t ara n d $\chull{P_1(x )}

\operatorname{Per^{\circ}}(f)$_and $\operatorname{Per}(F)_= \operatorname{Per^{\circ}}(f) \cup M(0,d)_\cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$. A_natural_strategy to_prove_the second statement_of Conjecture \[WishList\] in_the general case (i.e._when no endpoint_of_the rotation interval is an integer) is to construct examples of maps $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ with_a_*block structure*_over_maps_$f\in {\mathcal{X}_{3}}$ in such a_way that $p/q$ is an_endpoint of_the rotation interval ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$ and $\operatorname{Per}(p/q,F) = q\cdot\operatorname{Per^{\circ}}(f)$._The_next result shows_that this is not possible. Hence, if the second_statement of Conjecture \[WishList\] holds, the examples_must be built_by_using_some more complicated behavior_of the points of the orbit_in $\R$ and on the branches_than a block structure. Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$_and let $P$ be a lifted_periodic orbit of $F$ with_period $nq$_and rotation number $p/q$. For_every $x \in_P$ and_$i=0,1,\dots,q-1$, we set_$$P_i(x):=\{ F^i(x), G(F^i(x)), G^2(F^i(x)), \dots, G^{n-1}(F^i(x))_\},$$ where $G_:= F^q -p$. By Lemma \[blocksareperiodic\], every_$P_i(x)$_is a (true)_periodic_orbit_of $G$_of period $n$. \[ConverseEndInteger\]_Let_$F \in_{\ensuremath{\mathcal{L}_{1}(S)}}$_and let $P$ be a lifted_periodic_orbit of $F$ with period $nq$ and_rotation number $p/q$. Assume_that_there exists $x \in_P$ such that $\chull{P_0(x)}$ is_homeomorphic to a 3-star and $\chull{P_1(x)}

Ebihara:2016fwa], the choice of ${\lambda_{l,k}}$ from Eq.  is not unique; this nonuniqueness is related to $v_\pm$, as we mentioned below Eqs.  and. In Ref. [@Ebihara:2016fwa] we required $v_\pm = i\gamma^2 u_\pm^\ast$ from the beginning so that we could keep charge conjugation symmetry ${\mathcal{C}}$. This symmetry property gives *another* constraint of invariance under $l \leftrightarrow -l-1$. In the present case with $B\neq 0$, there is no way to keep such symmetry; nevertheless, it is convenient to adopt a sufficient condition for Eq.  to be connected to the $B=0$ limit smoothly, that is, $$\begin{split} & \Phi_l({\lambda_{l,k}},\alpha) = 0 \ \ \qquad\qquad\; \text{for} \quad l\geq 0\,, \\ & \Phi_{l+1}({\lambda_{l,k}}-1,\alpha) = 0 \,\qquad \text{for} \quad l\leq -1\,. \end{split} \label{eq:Phi-condition}$$ From the definition of the scalar function $\Phi_l(\lambda,x)$ given in Eqs.  and, we obtain the transverse momenta discretized as ${p_{l,k}}= \sqrt{2eB{\lambda_{l,k}}}$ with $${\lambda_{l,k}}= \begin{cases} \xi_{l,k} \,\qquad\qquad\quad \text{for} \quad l\geq 0\,, \\ \xi_{-l-1,k}-l \qquad \text{for} \quad l\leq -1\,, \end{cases} \label{eq:lambda}$$ where $\xi_{l,k}$ denotes the $k$th zero of ${{}_1F_1}(-\xi,l+1,\alpha)$ as a function of $\xi$. We note that ${\lambda_{l,k}}$ depends on $\alpha$; in other words, the discretized momenta are functions of the magnetic field $B$ as well as $R$. It would be instructive to think of the momentum discretization in the $B=0$ limit. From the asymptotic relations , we

Ebihara:2016fwa ], the choice of $ { \lambda_{l, k}}$ from Eq.   is not unique; this nonuniqueness is related to $ v_\pm$, as we mentioned below Eqs.   and. In Ref.   [ @Ebihara:2016fwa ] we command $ v_\pm = i\gamma^2 u_\pm^\ast$ from the beginning so that we could restrain charge union isotropy $ { \mathcal{C}}$. This symmetry property gives * another * restraint of invariability under $ l \leftrightarrow -l-1$. In the present case with $ B\neq 0 $, there is no way to keep such isotropy; nevertheless, it is convenient to adopt a sufficient condition for Eq.   to be connect to the $ B=0 $ limit smoothly, that is, $ $ \begin{split } & \Phi_l({\lambda_{l, k}},\alpha) = 0 \ \ \qquad\qquad\; \text{for } \quad l\geq 0\, , \\ & \Phi_{l+1}({\lambda_{l, k}}-1,\alpha) = 0 \,\qquad \text{for } \quad l\leq -1\, . \end{split } \label{eq: Phi - condition}$$ From the definition of the scalar function $ \Phi_l(\lambda, x)$ given in Eqs.   and, we obtain the transverse momentum discretized as $ { p_{l, k}}= \sqrt{2eB{\lambda_{l, k}}}$ with $ $ { \lambda_{l, k}}= \begin{cases } \xi_{l, k } \,\qquad\qquad\quad \text{for } \quad l\geq 0\, , \\ \xi_{-l-1,k}-l \qquad \text{for } \quad l\leq -1\, , \end{cases } \label{eq: lambda}$$ where $ \xi_{l, k}$ denotes the $ k$th zero of $ { { } _ 1F_1}(-\xi, l+1,\alpha)$ as a function of $ \xi$. We note that $ { \lambda_{l, k}}$ count on $ \alpha$; in other words, the discretized momenta are functions of the charismatic field $ B$ equally well as $ R$. It would be instructive to think of the momentum discretization in the $ B=0 $ limit. From the asymptotic relations  , we

Ebijara:2016fwa], the choice of ${\laobda_{l,k}}$ from Eq.  nw not nnique; fhis nonjniqueness is related to $v_\pm$, aw we nentioned below Eqs.  ana. In Ref. [@Ehihara:2016fwq] we eequired $v_\'j = i\gamma^2 u_\pm^\zdt$ fxon the beginninn so that wa could keep cvafgz conjugation symmetry ${\mathcal{C}}$. This symmetty property givef *anpeher* bokstraint of invariance under $l \lsftrighuarrow -l-1$. In the prrsent case with $B\neq 0$, therf is no way to keep sufh symmetry; nevqetheless, it ks convenitnc to adopt z sufficient condition for Eq.  tu be eonnected ti rhe $T=0$ limit smoithly, that is, $$\beglm{split} & \Phi_l({\kambda_{l,k}},\alpha) = 0 \ \ \qqyad\qquad\; \text{for} \quad l\geq 0\,, \\ & \Phi_{l+1}({\lambqa_{l,k}}-1,\alpha) = 0 \,\qquad \text{for} \quqd l\leq -1\,. \etd{spuut} \lagek{es:Phi-cojdivion}$$ From tge definitiin of the scalar fumceppn $\Phi_l(\lambdz,x)$ givqn in Eqs.  and, we obtain the transverse mokenfa discretized as ${p_{l,k}}= \swrt{2eB{\lambda_{l,k}}}$ with $${\lalbda_{l,k}}= \bqgin{cases} \xi_{l,k} \,\qquad\qquad\quad \text{for} \quad l\geq 0\,, \\ \xi_{-m-1,y}-l \wqmad \gwxh{for} \quad l\leq -1\,, \end{cases} \label{eq:lambda}$$ wherq $\xo_{l,l}$ denotes the $h$th zero of ${{}_1F_1}(-\di,k+1,\wlpha)$ as a fuvction of $\xi$. We note that ${\lwmbda_{l,k}}$ depebds on $\alkha$; im other words, the discretizwd momenta age fynctions of the maynetic field $B$ ax welk as $R$. It would be instrbctive to think ov the momsvtum discretizatkon it the $B=0$ limit. From the asyiptotic rxlatipns , we

Ebihara:2016fwa], the choice of ${\lambda_{l,k}}$ from Eq. unique; nonuniqueness is to $v_\pm$, as In [@Ebihara:2016fwa] we required = i\gamma^2 u_\pm^\ast$ the beginning so that we could charge conjugation symmetry ${\mathcal{C}}$. This symmetry property gives *another* constraint of invariance under \leftrightarrow -l-1$. In the present case with $B\neq 0$, there is no way keep symmetry; it convenient to adopt a sufficient condition for Eq. to be connected to the $B=0$ limit smoothly, is, $$\begin{split} & \Phi_l({\lambda_{l,k}},\alpha) = 0 \ \ \text{for} \quad l\geq 0\,, & \Phi_{l+1}({\lambda_{l,k}}-1,\alpha) = 0 \,\qquad \quad -1\,. \end{split} From definition the scalar function given in Eqs. and, we obtain the transverse momenta discretized as ${p_{l,k}}= \sqrt{2eB{\lambda_{l,k}}}$ with $${\lambda_{l,k}}= \begin{cases} \xi_{l,k} \text{for} \quad \\ \xi_{-l-1,k}-l \text{for} l\leq \end{cases} \label{eq:lambda}$$ where the $k$th zero of ${{}_1F_1}(-\xi,l+1,\alpha)$ as $\xi$. We note that ${\lambda_{l,k}}$ depends on $\alpha$; other words, discretized momenta are functions of the field $B$ as well as $R$. It would instructive to think of the momentum discretization in the $B=0$ limit. From the asymptotic relations

Ebihara:2016fwa], the choice of ${\lambDa_{l,k}}$ from Eq.  Is not UniQue; ThIs noNuniQueness is relatED to $v_\Pm$, as we mentioned below EqS.  and. IN REF. [@EbiHArA:2016fwa] wE requirED $v_\PM = I\gaMmA^2 u_\Pm^\aSt$ FRoM the bEgiNning so That we coulD keEp Charge conjugATiOn symmetry ${\MatHcal{C}}$. This symMetRy propErTy gIVes *anOthEr* conStrainT Of invaRiance undEr $L \LeftriGHtarrow -L-1$. iN tHe prEsent case with $B\neq 0$, THeRE is no way to keep Such syMmETrY; NEveRthEless, it is cOnVenieNT to adopT A sUFFIciENt condition foR Eq.  to be connECteD to the $b=0$ lImiT SmoothLy, thaT iS, $$\BegIn{split} & \Phi_l({\LambDa_{l,k}},\alpha) = 0 \ \ \Qquad\qQUad\; \text{FOr} \quad l\Geq 0\,, \\ & \Phi_{L+1}({\laMbdA_{l,k}}-1,\aLPhA) = 0 \,\qQuaD \tEXt{fOR} \qUad L\Leq -1\,. \End{split} \LaBeL{eq:PhI-conDITIOn}$$ FrOm tHe deFinitIon of the scalaR fuNctiON $\PhI_l(\lamBda,x)$ gIven In eqs.  anD, we obtAin thE tRansverse momentA disCretized aS ${p_{l,K}}= \sQrt{2EB{\LambdA_{L,k}}}$ with $${\LamBda_{L,k}}= \begin{Cases} \xi_{L,K} \,\qqUaD\QQUaD\quad \text{for} \quad l\gEq 0\,, \\ \XI_{-L-1,k}-L \qquad \teXt{for} \qUAd L\lEQ -1\,, \end{caseS} \lAbeL{eq:lAMBda}$$ whEre $\xI_{L,k}$ Denotes tHe $k$th zERo Of ${{}_1f_1}(-\xi,l+1,\alpHa)$ As a funCtIon Of $\xI$. We noTE thaT ${\lambdA_{l,k}}$ depenDs on $\aLPha$; in other wordS, The discretizeD MoMENtA Are fUncTions of the mAgneTIc fiEld $B$ AS wEll AS $R$. It wOuld bE iNStRUctive to think of the mOmEntum dIscreTization in the $b=0$ limit. From THE AsymptotIc reLAtIOns , we

Ebihara:2016fwa], the choi ce of ${\l ambda _{l ,k} }$ fro m Eq .  is not uniq u e; t his nonuniqueness is r elate dt o $v _ \p m$, a s we me n ti o n edbe lo w E qs .  and.InRef. [@ Ebihara:20 16f wa ] we require d $ v_\pm = i\ gam ma^2 u_\pm^\ ast $ from t heb eginn ing so t hat we couldkeep char ge conjug a tion sy m m et ry $ {\mathcal{C}}$. T h is symmetry prope rty gi ve s * a n oth er* constrain tof in v ariance un d e r $l \leftrightarr ow -l-1$. I n th e pres en t c a se wit h $B\ ne q 0$ , there isno w ay to kee p such symmetr y ; never theles s,itis c o nv en ien tt o a d op t a suf ficientco nd ition for E q .   to be con necte d to the $B=0 $ l imit smo othly , tha t is ,$$\be gin{sp lit} & \Phi_l({\lamb da_{ l,k}},\al pha )= 0 \ \ \q q uad\qq uad \;\text{f or} \qu a d l \g e q 0\ ,, \\ & \Phi_{l+ 1} ( { \l ambda_{l ,k}}-1 , \a lp h a) = 0 \ ,\ qqu ad \ t e xt{fo r} \ q ua d l\leq-1\,. \ en d{split } \lab el {eq :Ph i-con d itio n}$$ F rom thedefin i tion of the sc a lar function$ \P h i _l ( \lam bda ,x)$ givenin E q s. and, we ob t ain t he tr an s ve r se momenta discreti ze d as $ {p_{l ,k}}= \sqrt{2 eB{\lambda _ { l ,k}}}$ w ith$ ${ \ lambda_{l,k}}= \b egin{cases } \xi_{ l,k}\,\qquad \qquad\qu a d \text{f or} \q uad l\ g e q0\,, \\ \xi _ { -l-1 ,k }-l \qq uad \text{ for } \ qua d l \l eq -1\,, \end{c as es } \la bel{e q :lambda} $$ wh er e $ \xi_{ l ,k}$ d enote s th e$k $ thzero of ${ { } _1F_ 1} (- \xi, l+1 ,\ alpha )$ a s afunctio n of $\xi $.W e no te t hat ${\ lambda_{l,k}} $depends on $ \al pha$;i n other w ords, the discretized m o menta a refunct ions of the m agn etic f iel d $B$ a s well as $ R$ . I t woul d be in st ructive to t hin k ofth e mo mentumdiscretization int he$B=0$ limit.Fro m th e as ymp t ot i c r el a tio n s  , we

Ebihara:2016fwa], the_choice of_${\lambda_{l,k}}$ from Eq.  is_not unique;_this_nonuniqueness is_related_to $v_\pm$, as_we mentioned below_Eqs.  and. In Ref. [@Ebihara:2016fwa]_we required $v_\pm_=_i\gamma^2 u_\pm^\ast$ from the beginning so that we could keep charge conjugation symmetry ${\mathcal{C}}$._This_symmetry property_gives_*another*_constraint of invariance under $l_\leftrightarrow -l-1$. In the present_case with_$B\neq 0$, there is no way to keep_such_symmetry; nevertheless, it_is convenient to adopt a sufficient condition for Eq. _to be connected to the $B=0$_limit smoothly, that_is,_$$\begin{split} _ & \Phi_l({\lambda_{l,k}},\alpha) =_0 \ \ \qquad\qquad\; \text{for} \quad_l\geq 0\,, \\ & \Phi_{l+1}({\lambda_{l,k}}-1,\alpha)_= 0 \,\qquad \text{for} \quad l\leq -1\,. _ \end{split} \label{eq:Phi-condition}$$ From the_definition of the scalar function_$\Phi_l(\lambda,x)$ given_in Eqs.  and, we obtain_the transverse momenta_discretized as_${p_{l,k}}= \sqrt{2eB{\lambda_{l,k}}}$ with_$${\lambda_{l,k}}= \begin{cases} \xi_{l,k} \,\qquad\qquad\quad_\text{for} \quad l\geq_0\,, \\ \xi_{-l-1,k}-l \qquad \text{for}_\quad_l\leq -1\,, _\end{cases} __\label{eq:lambda}$$ where_$\xi_{l,k}$ denotes the_$k$th_zero of_${{}_1F_1}(-\xi,l+1,\alpha)$_as a function of $\xi$. We_note_that ${\lambda_{l,k}}$ depends on $\alpha$; in other_words, the discretized momenta_are_functions of the magnetic_field $B$ as well as_$R$. It would be instructive to think_of the_momentum discretization_in the $B=0$ limit. From the asymptotic relations , we

Section 12] most birational module-finite extensions of these rings have been searched. Since the proof given by [@GOTWY] depends on the techniques in the representation theory of maximal Cohen-Macaulay modules, it might have some interests to give a straightforward proof, making use of the results of [@GTT Section 12] and determining the members of ${\mathcal{A}}_R^0$ by Lemma \[lem3.1\], as well. We note it as an appendix. In this appendix, let $(R, \m)$ be a Gorenstein complete local ring of dimension one with algebraically closed residue class field $k$ of characteristic $0$. Suppose that $R$ has finite CM-representation type. Then, by [@Y (8.5), (8.10), and (8.15)] we get $$R \cong k[[X, Y]]/(f),$$ where $k[[X, Y]]$ is the formal power series ring over $k$, and $f$ is one of the following polynomials. - $X^2-Y^{n+1}$ $(n \ge 1)$ - $X^2Y-Y^{n-1}$ $(n \ge 4)$ - $X^3-Y^4$ - $X^3-XY^3$ - $X^3-Y^5$ With this notation we have the following. \[5.1\] The set ${\mathcal{X}}_R$ is given by the following. 1. ${\mathcal{X}}_R = \begin{cases} \left\{(x, y^q) \mid 0 < q \le \ell \right\} & \text{if} \ n=2 \ell -1 \ \text{with} \ \ell \ge 1,\\ \left\{(x, y^q) \mid 0 < q \le \ell \right\} & \text{if} \ n=2\ell \ \text{with} \ \ell \ge 1. \end{cases}$ 2. ${\mathcal{X}}_R = \begin{cases} \left\{(x^2, y), (x, y^{\ell+1}) \right\} & \text{if} \ n=

Section 12 ] most birational module - finite extensions of these rings have been research. Since the validation given by [ @GOTWY ] depends on the technique in the theatrical performance theory of maximal Cohen - Macaulay modules, it might have some interest to give a straightforward validation, make use of the resultant role of [ @GTT Section 12 ] and determining the members of $ { \mathcal{A}}_R^0 $ by Lemma \[lem3.1\ ], equally well. We note it as an appendix. In this appendix, let $ (R, \m)$ be a Gorenstein complete local ring of dimension one with algebraically closed residue course field $ k$ of characteristic $ 0$. Suppose that $ R$ has finite CM - representation character. Then, by [ @Y (8.5), (8.10), and (8.15) ] we get $ $ R \cong k[[X, Y]]/(f),$$ where $ k[[X, Y]]$ is the formal power series hoop over $ k$, and $ f$ is one of the following polynomials. - $ X^2 - Y^{n+1}$ $ (n \ge 1)$ - $ X^2Y - Y^{n-1}$ $ (n \ge 4)$ - $ X^3 - Y^4 $ - $ X^3 - XY^3 $ - $ X^3 - Y^5 $ With this notation we have the following. \[5.1\ ] The hardening $ { \mathcal{X}}_R$ is given by the following. 1. $ { \mathcal{X}}_R = \begin{cases } \left\{(x, y^q) \mid 0 < q \le \ell \right\ } & \text{if } \ n=2 \ell -1 \ \text{with } \ \ell \ge 1,\\ \left\{(x, y^q) \mid 0 < q \le \ell \right\ } & \text{if } \ n=2\ell \ \text{with } \ \ell \ge 1. \end{cases}$ 2. $ { \mathcal{X}}_R = \begin{cases } \left\{(x^2, y), (x, y^{\ell+1 }) \right\ } & \text{if } \ n=

Seftion 12] most birational mudule-finite extgnwions mf theae rings have been searched. Since thx priof guven by [@GOTWY] depends un the tebhniques un tie representatioi theory of maslmal Eoien-Macaulay modoles, it mighd have some indefedts to give a straightforward proof, making ude of the resujts ps [@GTF Section 12] and determining the memgers of ${\mathcal{A}}_R^0$ by Lemma \[lem3.1\], as well. We note it ws an appendix. In tjis appendiz, lee $(R, \m)$ be a Gofenstein complete locam ring of dimension one with aleebrancally closgb redhdue class hield $h$ of charactcgistic $0$. Supposr that $R$ has flnite CM-eepresentation type. Tien, by [@Y (8.5), (8.10), and (8.15)] we ggt $$R \cong n[[X, Y]]/(f),$$ where $k[[X, Y]]$ is tye fotmal [owef sefiea cinf over $k$, end $f$ is ons of the foolowing polynomials. - $Q^2-U^{n+1}$ $(n \ge 1)$ - $X^2G-Y^{n-1}$ $(n \de 4)$ - $X^3-Y^4$ - $X^3-XY^3$ - $X^3-Y^5$ With this notation we vavs the following. \[5.1\] The set ${\mathcal{X}}_R$ is given bi the folljwing. 1. ${\mathcal{X}}_R = \begin{cases} \left\{(x, y^q) \mid 0 < q \lx \dll \rlght\} & \tfxt{if} \ n=2 \ell -1 \ \text{with} \ \ell \ge 1,\\ \left\{(x, y^z) \mod 0 < q \le \ell \rinht\} & \text{if} \ n=2\ell \ \tfxy{rith} \ \ell \ge 1. \enb{daaes}$ 2. ${\mathcal{X}}_R = \begin{sases} \left\{(x^2, y), (x, y^{\ell+1}) \right\} & \text{if} \ n=

Section 12] most birational module-finite extensions of have searched. Since proof given by in representation theory of Cohen-Macaulay modules, it have some interests to give a proof, making use of the results of [@GTT Section 12] and determining the of ${\mathcal{A}}_R^0$ by Lemma \[lem3.1\], as well. We note it as an appendix. this let \m)$ a Gorenstein complete local ring of dimension one with algebraically closed residue class field $k$ of $0$. Suppose that $R$ has finite CM-representation type. by [@Y (8.5), (8.10), (8.15)] we get $$R \cong Y]]/(f),$$ $k[[X, Y]]$ the power ring over $k$, $f$ is one of the following polynomials. - $X^2-Y^{n+1}$ $(n \ge 1)$ - $X^2Y-Y^{n-1}$ $(n \ge 4)$ $X^3-Y^4$ - $X^3-Y^5$ With notation have following. \[5.1\] The is given by the following. 1. \left\{(x, y^q) \mid 0 < q \le \ell & \text{if} n=2 \ell -1 \ \text{with} \ \ge 1,\\ \left\{(x, y^q) \mid 0 < q \ell \right\} & \text{if} \ n=2\ell \ \text{with} \ \ell \ge 1. \end{cases}$ 2. ${\mathcal{X}}_R \left\{(x^2, y), (x, y^{\ell+1}) & \text{if} \

Section 12] most birational moduLe-finite exTensiOns Of tHeSe riNgs hAve been searcheD. sincE the proof given by [@GOTWY] dEpendS oN The tEChNiqueS in the rEPrESEntAtIoN thEoRY oF maxiMal cohen-MaCaulay moduLes, It Might have somE InTerests to gIve A straightforWarD proof, MaKinG Use of The ResulTs of [@GTt sectioN 12] and deterMiNIng the MEmbers oF ${\MAtHcal{a}}_R^0$ by Lemma \[lem3.1\], as welL. we NOte it as an appenDix. In tHiS ApPENdiX, leT $(R, \m)$ be a GoreNsTein cOMplete lOCaL RINg oF Dimension one wIth algebraiCAllY closeD rEsiDUe clasS fielD $k$ OF chAracteristiC $0$. SupPose that $R$ Has finITe CM-repREsentatIon typE. ThEn, bY [@Y (8.5), (8.10), anD (8.15)] We GeT $$R \cOnG K[[X, Y]]/(F),$$ WhEre $K[[x, Y]]$ iS the formAl PoWer seRies RING Over $K$, anD $f$ is One of The following pOlyNomiALs. - $X^2-y^{n+1}$ $(n \ge 1)$ - $x^2Y-Y^{n-1}$ $(n \Ge 4)$ - $X^3-Y^4$ - $x^3-Xy^3$ - $X^3-Y^5$ WiTh this NotatIoN we have the folloWing. \[5.1\] the set ${\matHcaL{X}}_r$ is GiVen by THe follOwiNg. 1. ${\mAthcal{X}}_r = \begin{cASes} \LeFT\{(X, Y^q) \Mid 0 < q \le \ell \right\} & \text{If} \ N=2 \ELl -1 \ \Text{with} \ \Ell \ge 1,\\ \lEFt\{(X, y^Q) \Mid 0 < q \le \elL \rIghT\} & \texT{IF} \ n=2\ell \ \Text{WItH} \ \ell \ge 1. \enD{cases}$ 2. ${\MAtHcAl{X}}_R = \begIn{Cases} \lEfT\{(x^2, y), (X, y^{\eLl+1}) \rigHT\} & \texT{if} \ n=

Section 12] most biration al module- finit e e xte ns ions ofthese rings ha v e be en searched. Since the proo fg iven by [@GO TWY] de p en d s on t he te ch n iq ues i n t he repr esentation th eo ry of maxima l C ohen-Macau lay modules, it mi ght ha ve so m e int ere sts t o give a stra ightforwa rd proof, makingu s eof t he results of [@G T TS ection 12] and deter mi n in g the me mbers of $ {\ mathc a l{A}}_R ^ 0$ b y Le m ma \[lem3.1\] , as well.W e n ote it a s a n appen dix. I n th is appendix , le t $(R, \m )$ bea Gorens t ein com pleteloc alring of d ime ns i ono ne wi t h a lgebraic al ly clos ed r e s i d ue c las s fi eld $ k$ of charact eri stic $0$ . Sup posethat $ R$ ha s fini te CM -r epresentation t ype. Then, by [@ Y(8. 5) , (8. 1 0), an d ( 8.1 5)] weget $$R \co ng k [ [X , Y]]/(f),$$ where $ k [ [X , Y]]$ i s thef or ma l power s er ies rin g over$k$, an d $f$ is one o f t he follow in g poly no mia ls. - $X^2 -Y^{n+ 1}$ $(n\ge 1 ) $ - $X^2Y-Y ^ {n-1}$ $(n \g e 4 ) $ - $ X^3 -Y^4$ - $X^3 - XY^3 $ - $X^ 3 -Y^5$ Wit ht hi s notation we have t he follo wing. \[5.1\] The set ${\ma t h c al{X}}_R $ is gi v en by the foll owing . 1. ${\ m athcal{X }}_R= \ begin{cas e s } \l eft \{( x,y^q ) \m id 0 < q \le\ e ll \ ri ght\} & \t ext{if} \ n= 2 \ ell - 1 \ \text {with} \ \e ll \ ge1,\\ \left \{ (x, y ^q) \mid 0 < q\le \ ell\r ig h t\} & \tex t {i f } \ n =2 \e ll \ \t ex t{wit h} \ \el l \ge 1 . \en d{c a ses} $ 2 . ${\m athcal{X}}_R= \begi n{ cas es} \left\{( x^2, y), (x, y^{\ell+1} ) \righ t\} & \t ext{ if} \ n=

Section_12] most_birational module-finite extensions of_these rings_have_been searched._Since_the proof given_by [@GOTWY] depends_on the techniques in_the representation theory_of_maximal Cohen-Macaulay modules, it might have some interests to give a straightforward proof, making_use_of the_results_of_[@GTT Section 12] and determining_the members of ${\mathcal{A}}_R^0$ by_Lemma \[lem3.1\],_as well. We note it as an appendix. In_this_appendix, let $(R,_\m)$ be a Gorenstein complete local ring of dimension_one with algebraically closed residue class_field $k$ of_characteristic_$0$._Suppose that $R$ has_finite CM-representation type. Then, by [@Y_(8.5), (8.10), and (8.15)] we get_$$R \cong k[[X, Y]]/(f),$$ where $k[[X, Y]]$_is the formal power series ring_over $k$, and $f$ is_one of_the following polynomials. - _$X^2-Y^{n+1}$ $(n \ge_1)$ - _ $X^2Y-Y^{n-1}$ $(n_\ge 4)$ - $X^3-Y^4$ - _ $X^3-XY^3$ - _ $X^3-Y^5$ With this notation we have_the_following. \[5.1\] The set_${\mathcal{X}}_R$_is_given by_the following. 1. _${\mathcal{X}}_R_= __ \begin{cases} _\left\{(x,_y^q) \mid 0 < q \le \ell_\right\} & \text{if} \__n=2 \ell -1 \_\text{with} \ \ell \ge_1,\\ \left\{(x, y^q)_\mid 0_< q_\le \ell \right\} & \text{if} \ n=2\ell \ \text{with} \ \ell_\ge 1. \end{cases}$ 2._ ${\mathcal{X}}_R = _ __\begin{cases} __\left\{(x^2, y),_(x, y^{\ell+1}) \right\} & \text{if} \_ n=

, allowing molecules to become shielded from photodissociating radiation and altering the cooling properties of the primordial gas (see Johnson et al. 2007).](f1.eps){width="12cm"} Mechanical Feedback ------------------- Numerical simulations have indicated that Pop III.1 stars might become as massive as $500~\rm{M}_{\odot}$ (Omukai & Palla 2003; Bromm & Loeb 2004; Yoshida et al. 2006; O’Shea & Norman 2007). After their main-sequence lifetimes of typically $2-3~\rm{Myr}$, stars with masses below $\simeq 100~\rm{M}_{\odot}$ are thought to collapse directly to black holes without significant metal ejection, while in the range $\simeq 140-260~\rm{M}_{\odot}$ a pair-instability supernova (PISN) disrupts the entire progenitor, with explosion energies ranging from $10^{51}-10^{53}~\rm{ergs}$, and yields of order $50\%$ (Heger & Woosley 2002; Heger et al. 2003). Less massive primordial stars with a high degree of angular momentum might explode with similar energies, but as jet-like hypernovae (Umeda & Nomoto 2002; Tominaga et al. 2007). The significant mechanical and chemical feedback effects exerted by such explosions have been investigated with a number of detailed calculations, but these were either performed in one dimension (Salvaterra 2004; Kitayama & Yoshida 2005; Machida et al. 2005; Whalen et al 2008b), or did not start from realistic initial conditions (Bromm et al. 2003; Norman et al. 2004). Recent work treated the full three-dimensional problem in a cosmological context at the cost of limited resolution, finding that the SN remnant propagated for a Hubble time at $z\simeq 20$ to a final mass-weighted mean shock radius of $2.5~\rm{kpc}$, roughly half the size of the H [ii]{} region (Greif et al. 2007). Due to the high explosion energy, the host halo was entirely evacuated. Additional simulations in the absence of a SN explosion were performed to investigate the effect of photoheating and the impact of the SN shock on neighboring minihalos. For the case discussed in Greif et al. (2007), the SN remnant exerted positive mechanical feedback on neighboring minihalos by shock-compressing their cores, while photo

, allowing molecules to become shielded from photodissociating radiation and altering the cool place of the primordial gas (see Johnson et al. 2007).](f1.eps){width="12 cm " } Mechanical Feedback ------------------- Numerical simulation have indicated that Pop   III.1 stars might become equally massive as $ 500~\rm{M}_{\odot}$ (Omukai & Palla 2003; Bromm & Loeb 2004; Yoshida et al. 2006; O’Shea & Norman 2007). After their main - sequence lifetimes of typically $ 2 - 3~\rm{Myr}$, star with masses below $ \simeq 100~\rm{M}_{\odot}$ are thought to crack up directly to black fix without significant metal ejection, while in the range $ \simeq 140 - 260~\rm{M}_{\odot}$ a pair - imbalance supernova (PISN) disrupts the entire progenitor, with explosion energy ranging from $ 10^{51}-10^{53}~\rm{ergs}$, and yields of holy order $ 50\%$ (Heger & Woosley 2002; Heger et al. 2003). Less massive primordial star with a high degree of angular momentum might explode with similar energies, but as k - like hypernovae (Umeda & Nomoto 2002; Tominaga et al. 2007). The significant mechanical and chemical feedback effects exerted by such explosions have been investigated with a number of detailed calculations, but these were either performed in one dimension (Salvaterra 2004; Kitayama & Yoshida 2005; Machida et al. 2005; Whalen et al 2008b), or did not start from realistic initial conditions (Bromm et al. 2003; Norman et al. 2004). Recent work treated the full three - dimensional trouble in a cosmologic context at the cost of circumscribed settlement, discover that the SN remnant propagated for a Hubble meter at $ z\simeq 20 $ to a final mass - weighted mean jolt radius of $ 2.5~\rm{kpc}$, roughly half the size of the H   [ ii ] { } region (Greif et al. 2007). Due to the high explosion energy, the master of ceremonies halo was entirely evacuated. Additional model in the absence of a SN explosion were perform to investigate the effect of photoheating and the impact of the SN shock on neighboring minihalos. For the case discussed in Greif et al. (2007), the SN leftover exerted positive mechanical feedback on neighboring minihalos by jolt - compressing their cores, while photo

, alpowing molecules to becooe shielded from photovissocizting raaiation and altering the coopibg priperties of the primoraial gas (dee Johnwon tt al. 2007).](f1.eps){width="12cm"} Mechanical Feesnack ------------------- Nbmxrical simulatipns have itdicated that [oo INI.1 stars might become as massive as $500~\wm{M}_{\odot}$ (Olukai & Palla 2003; Fromk & Losb 2004; Yoshida et al. 2006; O’Shea & Norman 2007). Zfter tieir main-sequenve lifetimes of typically $2-3~\gm{Myg}$, stars with massed below $\simgs 100~\ri{N}_{\odot}$ are thuught to collapse diredtly to black holes without sigvificcnt metal ehextilt, while in vhe ragge $\simeq 140-260~\rm{M}_{\odot}$ a [air-insyability superkova ('ISN) disrupts the entire 'rogenitor, with expljsion enesgnes ranging from $10^{51}-10^{53}~\rm{erts}$, and iieldv of irddr $50\%$ (Hxged & Woodleb 2002; Heger et al. 2003). Less mqssive primordial suarf with a high segree os angular momentum might explode with spmilzr energies, but as jet-luke hypernovae (Umeda & Nomoto 2002; Eominaga et al. 2007). The significant mechanical and chamicam fetdnwzj fffects exerted by such explosions have been yhvtstpgated with a numner of detailed cakcklsjions, but thesg were zjtger performed in oje dimegsion (Salvaterwa 2004; Litayama & Yoshida 2005; Machida wt al. 2005; Whaleu er al 2008b), or did not dtart from xealisjic inotial conditions (Bromm zt al. 2003; Norman et wl. 2004). Recenf work treated thd flll dhree-dimtvsional problem ig a cosmooogieal contdxt st the cost of llmited resolution, findinh thaj the VN remnant propagated for a Hubble time av $z\simeq 20$ to s xindl mass-wzightec mean shock wadius of $2.5~\rm{kpe}$, roughlv half the size kf the I [ii]{} region (Gweif et al. 2007). Dgg to the high explosijn ebergt, the hurt halo was enyirely evceuated. Adeitional simulatioks in fhe absence of c WN explosion wete oersogmev to ytvestigate tve ewfezy of ohotoheatinn avd tne impact of the SN vhoci on neighboring monlhalos. Fot the casq discussed im Greif et al. (2007), the SN rxmnant exeryed positive mechanical feedback kn neighblrikg minihalos fy smock-sompressiny their cores, while photo

, allowing molecules to become shielded from and the cooling of the primordial 2007).](f1.eps){width="12cm"} Feedback ------------------- Numerical have indicated that III.1 stars might become as massive $500~\rm{M}_{\odot}$ (Omukai & Palla 2003; Bromm & Loeb 2004; Yoshida et al. 2006; & Norman 2007). After their main-sequence lifetimes of typically $2-3~\rm{Myr}$, stars with masses $\simeq are to directly to black holes without significant metal ejection, while in the range $\simeq 140-260~\rm{M}_{\odot}$ a pair-instability (PISN) disrupts the entire progenitor, with explosion energies from $10^{51}-10^{53}~\rm{ergs}$, and yields order $50\%$ (Heger & Woosley Heger al. 2003). massive stars a high degree angular momentum might explode with similar energies, but as jet-like hypernovae (Umeda & Nomoto 2002; Tominaga et 2007). The and chemical effects by explosions have been a number of detailed calculations, but performed in one dimension (Salvaterra 2004; Kitayama & 2005; Machida al. 2005; Whalen et al 2008b), did not start from realistic initial conditions (Bromm al. 2003; Norman et al. 2004). Recent work treated the full three-dimensional problem in a at the cost of resolution, finding that SN propagated a time at 20$ to a final mass-weighted mean shock radius of $2.5~\rm{kpc}$, roughly the size of the H [ii]{} region (Greif et al. to high explosion energy, host halo was entirely Additional in the absence of explosion performed effect photoheating the impact of the shock on neighboring minihalos. For case discussed in Greif remnant exerted positive mechanical feedback on neighboring minihalos shock-compressing their cores, while photo

, allowing molecules to become Shielded frOm phoTodIssOcIatiNg raDiation and alteRIng tHe cooling properties of tHe priMoRDial GAs (See JoHnson et AL. 2007).](f1.EPS){wiDtH="12cM"} MeChANiCal FeEdbAck ------------------- NumeRical simulAtiOnS have indicatED tHat Pop III.1 sTarS might become As mAssive As $500~\Rm{M}_{\ODot}$ (OmUkaI & PallA 2003; Bromm & lOeb 2004; YosHida et al. 2006; O’shEA & NormaN 2007). after thEIR mAin-sEquence lifetimes oF TyPIcally $2-3~\rm{Myr}$, staRs with MaSSeS BEloW $\siMeq 100~\rm{M}_{\odot}$ ArE thouGHt to colLApSE DIreCTly to black holEs without siGNifIcant mEtAl eJEction, While In THe rAnge $\simeq 140-260~\rm{m}_{\odoT}$ a pair-insTabiliTY supernOVa (PISN) dIsruptS thE enTire PRoGeNitOr, WIth EXpLosIOn eNergies rAnGiNg froM $10^{51}-10^{53}~\rm{eRGS}$, ANd yiEldS of oRder $50\%$ (HEger & Woosley 2002; HeGer Et al. 2003). lEss MassiVe priMordIaL starS with a High dEgRee of angular momEntuM might expLodE wIth SiMilar ENergieS, buT as Jet-like HypernoVAe (UMeDA & nOmOto 2002; Tominaga et al. 2007). The SiGNIfIcant mecHanicaL AnD cHEmical feEdBacK effECTs exeRted BY sUch exploSions hAVe BeEn invesTiGated wItH a nUmbEr of dETailEd calcUlations, But thESe were either peRFormed in one diMEnSIOn (sAlvaTerRa 2004; Kitayama & YOshiDA 2005; MacHida ET aL. 2005; WhALen et Al 2008b), or DiD NoT Start from realistic iNiTial coNditiOns (Bromm et al. 2003; NOrman et al. 2004). RECENt work trEateD ThE Full three-dimenSionaL problem in A CosmologIcal cOntext at The cost of LIMited resOluTioN, fiNdiNG ThAt the SN remnanT PRopaGaTed for a hubBle time At $z\SimEq 20$ tO a fInAl mass-weiGhted meaN sHoCk RaDiuS of $2.5~\rm{KPc}$, roughlY hAlf ThE siZe of tHE H [ii]{} reGion (GReif Et Al. 2007). dUe tO the higH ExPLOsioN eNeRgy, tHe hOsT halo Was eNTirEly evacUated. AddiTioNAl siMuLaTions in The absence of a sN Explosion wErE peRformeD TO investiGate the effect of photoheaTIng and tHe iMpact Of thE SN shock oN neIghborIng MInihalOs. For tHe casE dIscUSSed in gREiF et Al. (2007), The SN remnaNT ExeRted pOsItivE mechanIcal feedback on neigHBorIng minihalos bY shOck-cOMPrEssINg THeiR cORes, WHIle photo

, allowing molecules to be come shiel ded f rom ph ot odis soci ating radiatio n and altering the coolingprope rt i es o f t he pr imordia l g a s (s ee J ohn so n e t al. 20 07).](f 1.eps){wid th= "1 2cm"} Mecha n ic al Feedbac k - ------------ --- --- N um eri c al si mul ation s have indica ted thatPo p  III.1 stars m i g ht bec ome as massive as $5 0 0~\rm{M}_{\odo t}$ (O mu k ai & Pa lla 2003; Bro mm & Lo e b 2004; Yo s h i dae t al. 2006; O ’Shea & Nor m an2007). A fte r their main -s e que nce lifetim es o f typical ly $2- 3 ~\rm{My r }$, sta rs wit h m ass es b e lo w$\s im e q 1 0 0~ \rm { M}_ {\odot}$ a re thou ghtt o c olla pse dir ectly to black hol eswith o utsigni fican t me ta l eje ction, whil ein the range $\ sime q 140-260 ~\r m{ M}_ {\ odot} $ a pai r-i nst ability supern o va(P I S N )disrupts the entir ep r og enitor,with e x pl os i on energ ie s r angi n g from $10 ^ {5 1}-10^{5 3}~\rm { er gs }$, and y ieldsof or der $50\ % $ (H eger & Woosley 2002 ; Heger et al.2 003). Less ma s si v e p r imor dia l stars wit h ah ighdegr e eofa ngula r mom en t um might explode withsi milarenerg ies, but as j et-like hy p e r novae (U meda &N omoto 2002; To minag a et al. 2 0 07). The sign ificantmechanica l and chem ica l f eed bac k ef fects exerted b y su ch explos ion s havebee n i nve sti ga ted witha number o fde ta ile d cal c ulations ,but t hes e wer e eithe r per form ed i n on e dimen s io n (Sal va te rra200 4; Kita yama & Y oshida2005; Mac hid a etal .2005; W halen et al 2 00 8b), or di dnot start f rom real istic initial condition s (Bromm et al.2003 ; Normanetal. 20 04) . Recen t work trea te d t h e full t hr ee- di mensionalp r obl em in a cos mologic al context at thec ost of limited r eso luti o n ,fin d in g th at the S N remnant propa gated foraH ub ble time a t $z \s imeq 20 $ to afinal mass-we ighted me an shockra dius o f $ 2.5~\rm{kp c}$, rou ghly half the s i ze of t heH [ii] {} re gion(Greif etal. 2 007).Du e to t he hi gh explosi on energy, the host hal o wasentir ely evacuate d.A ddi tional si mula tions in t heabs enceofa SN e xplo s io n w e re pe rfor m ed to inv e st iga t e t he effect o f p hot oheat ing and th e im pact of the SN sh o ck on neighbor ingm i nih alo s . Fo rthe case discu sse di n Greif e tal. (2007), the SNre m nantexerte d posi tive me c h an i cal fe edba ckon neighb ori ng minihal os b y shock -com pr essing their core s , while photo

, allowing_molecules to_become shielded from photodissociating_radiation and_altering_the cooling_properties_of the primordial_gas (see Johnson_et al. 2007).](f1.eps){width="12cm"} Mechanical Feedback ------------------- Numerical_simulations have indicated_that_Pop III.1 stars might become as massive as $500~\rm{M}_{\odot}$ (Omukai & Palla 2003; Bromm &_Loeb_2004; Yoshida_et_al._2006; O’Shea & Norman 2007)._After their main-sequence lifetimes of_typically $2-3~\rm{Myr}$,_stars with masses below $\simeq 100~\rm{M}_{\odot}$ are thought_to_collapse directly to_black holes without significant metal ejection, while in the_range $\simeq 140-260~\rm{M}_{\odot}$ a pair-instability supernova_(PISN) disrupts the_entire_progenitor,_with explosion energies ranging_from $10^{51}-10^{53}~\rm{ergs}$, and yields of order_$50\%$ (Heger & Woosley 2002; Heger_et al. 2003). Less massive primordial stars_with a high degree of angular_momentum might explode with similar_energies, but_as jet-like hypernovae (Umeda &_Nomoto 2002; Tominaga_et al._2007). The significant_mechanical and chemical feedback effects exerted_by such explosions_have been investigated with a number_of_detailed calculations, but_these_were_either performed_in one dimension_(Salvaterra_2004; Kitayama_&_Yoshida 2005; Machida et al. 2005;_Whalen_et al 2008b), or did not start_from realistic initial conditions_(Bromm_et al. 2003; Norman_et al. 2004). Recent work_treated the full three-dimensional problem in_a cosmological_context at_the cost of limited resolution, finding that the SN remnant propagated_for a Hubble time at $z\simeq_20$ to a final_mass-weighted mean_shock_radius of $2.5~\rm{kpc}$,_roughly_half the_size of the H [ii]{} region (Greif et_al. 2007)._Due to the high explosion energy,_the host halo was_entirely_evacuated. Additional simulations in the absence_of a SN explosion were performed_to investigate the effect of_photoheating_and_the impact of the SN_shock on neighboring minihalos. For the_case discussed in_Greif et al. (2007), the SN remnant_exerted_positive mechanical feedback on neighboring minihalos_by_shock-compressing their cores, while photo

i$ does not have UF and $n_j, n_k$ appears in the decomposition of $f+n_i$, then we have $\al_jn_j= \al_kn_k$. Let $\{i,j,k,l\}$ be a permutation of $\{1,2,3,4\}$. By Corollary \[0\_in\_row\], we may assume that the $(i.l)$ component of $\RF(f)$ is $0$ for any choice of $\RF(f)$. Thus $f+n_i$ contains only $n_j, n_k$. Assume that there are $2$ different expressions $f+n_i = an_j + bn_k = a'n_j + b'n_k$. Assuming $a>a', b< b'$, we have $(a-a')n_j = (b'-b)n_k$ and then $a\ge a-a'\ge \al_j, b'-b \ge \al_k$. \[f + n\_k = bn\_i\] Suppose that $H$ is almost symmetric, $e=4$ and for some $f\in PF'(H)$ we have $f + n_k = bn_i$ for some $k \ne i$. Then one of the following cases occur: 1. $b= \al_i -1$ or 2. $b \ge \al_i$ and for some $j\ne i,k$, $\al_i n_i = \al_j n_j$. This is a direct consequence of Lemma \[f + n\_k = b\_in\_i\] Corollary \[2nj\]. [*Let $H =\langle 33,56,61,84\rangle$ with $\PF(H)=\{f=28,f'=835, F(H)=863\}$ and $\al_1=28, \al_2=3, \al_3=2, \al_4=2$. In this case, $\RF(28)$ is uniquely determined, but there are several choices of $\RF(835)$. Among them we can choose the following, where $ f' + n_3= 16 n_2$ with $16> \al_2$. Note that we have $\al_2n

i$ does not have UF and $ n_j, n_k$ appears in the decomposition of $ f+n_i$, then we have $ \al_jn_j= \al_kn_k$. lease $ \{i, j, k, l\}$ be a substitution of $ \{1,2,3,4\}$. By Corollary \[0\_in\_row\ ], we may assume that the $ (i.l)$ component of $ \RF(f)$ is $ 0 $ for any option of $ \RF(f)$. Thus $ f+n_i$ contain only $ n_j, n_k$. bear that there are $ 2 $ different expression $ f+n_i = an_j + bn_k = a'n_j + b'n_k$. Assuming $ a > a', b < b'$, we have $ (a - a')n_j = (b'-b)n_k$ and then $ a\ge a - a'\ge \al_j, b'-b \ge \al_k$. \[f + n\_k = bn\_i\ ] presuppose that $ H$ is almost symmetric, $ e=4 $ and for some $ f\in PF'(H)$ we get $ f + n_k = bn_i$ for some $ k \ne i$. Then one of the following cases occur: 1. $ b= \al_i -1 $ or 2. $ b \ge \al_i$ and for some $ j\ne i, k$, $ \al_i n_i = \al_j n_j$. This is a lineal consequence of Lemma   \[f + n\_k = b\_in\_i\ ] Corollary \[2nj\ ]. [ * Let $ H = \langle 33,56,61,84\rangle$ with $ \PF(H)=\{f=28,f'=835, F(H)=863\}$ and $ \al_1=28, \al_2=3, \al_3=2, \al_4=2$. In this case, $ \RF(28)$ is uniquely determine, but there are several choice of $ \RF(835)$. Among them we can choose the following, where $ degree fahrenheit' + n_3= 16 n_2 $ with $ 16 > \al_2$. Note that we have $ \al_2n

i$ dles not have UF and $n_j, n_y$ appears in thg eecompmsitioh of $f+n_i$, then we have $\al_jn_j= \al_kn_k$. Let $\{i,h,k,l\}$ bt a permutation of $\{1,2,3,4\}$. By Corolpary \[0\_in\_riw\], wt may assume that the $(i.l)$ gjmpohcnt oy $\CF(f)$ is $0$ for any choice of $\RF(f)$. Thus $f+n_i$ wovtcins only $n_j, n_k$. Assume that there arq $2$ diffrrfnt expressionf $f+n_p = an_j + bn_k = a'n_j + b'n_k$. Assuming $a>a', b< b'$, ws have $(e-a')n_j = (b'-b)n_k$ and yhen $a\ge a-a'\ge \al_j, b'-b \ge \ap_k$. \[f + n\_k = bn\_i\] Suppose tjat $H$ is alnost wymmetric, $e=4$ xnd for some $f\in PF'(H)$ wg have $f + n_k = bn_i$ for some $k \ne k$. Theu one of thg ropnowing casew occlr: 1. $b= \al_i -1$ or 2. $b \ge \an_i$ and gor some $j\ne i,l$, $\an_i b_i = \al_j n_j$. This is a dmrect consequence of Lemma \[f + t\_k = b\_in\_i\] Corollary \[2bj\]. [*Oet $H =\lancle 33,56,61,84\fqngue$ sivh $\LF(H)=\{f=28,f'=835, V(H)=863\}$ end $\al_1=28, \al_2=3, \am_3=2, \al_4=2$. In thiw case, $\RF(28)$ is uniqueky eetermined, buf therq wre several choices of $\RF(835)$. Among them we cah choose the following, qhere $ f' + n_3= 16 n_2$ with $16> \wl_2$. Note trat we have $\al_2n

i$ does not have UF and $n_j, in decomposition of then we have a of $\{1,2,3,4\}$. By \[0\_in\_row\], we may that the $(i.l)$ component of $\RF(f)$ $0$ for any choice of $\RF(f)$. Thus $f+n_i$ contains only $n_j, n_k$. Assume there are $2$ different expressions $f+n_i = an_j + bn_k = a'n_j + Assuming b< we $(a-a')n_j = (b'-b)n_k$ and then $a\ge a-a'\ge \al_j, b'-b \ge \al_k$. \[f + n\_k = bn\_i\] that $H$ is almost symmetric, $e=4$ and for $f\in PF'(H)$ we have + n_k = bn_i$ for $k i$. Then of following occur: 1. $b= -1$ or 2. $b \ge \al_i$ and for some $j\ne i,k$, $\al_i n_i = \al_j n_j$. This a direct Lemma \[f n\_k b\_in\_i\] \[2nj\]. [*Let $H with $\PF(H)=\{f=28,f'=835, F(H)=863\}$ and $\al_1=28, \al_2=3, this case, $\RF(28)$ is uniquely determined, but there several choices $\RF(835)$. Among them we can choose following, where $ f' + n_3= 16 n_2$ $16> \al_2$. Note that we have $\al_2n

i$ does not have UF and $n_j, n_k$ appeArs in the deCompoSitIon Of $F+n_i$, tHen wE have $\al_jn_j= \al_kn_K$. let $\{i,J,k,l\}$ be a permutation of $\{1,2,3,4\}$. By COrollArY \[0\_In\_roW\], We May asSume thaT ThE $(I.L)$ coMpOnEnt Of $\rf(f)$ Is $0$ for Any Choice oF $\RF(f)$. Thus $f+n_I$ coNtAins only $n_j, n_k$. aSsUme that theRe aRe $2$ different eXprEssionS $f+N_i = aN_J + bn_k = a'N_j + b'N_k$. AssUming $a>A', B< b'$, we haVe $(a-a')n_j = (b'-b)n_K$ aND then $a\GE a-a'\ge \al_J, B'-B \gE \al_k$. \[F + n\_k = bn\_i\] Suppose that $h$ Is ALmost symmetric, $E=4$ and foR sOMe $F\IN PF'(h)$ we Have $f + n_k = bn_i$ FoR some $K \Ne i$. Then ONe OF THe fOLlowing cases oCcur: 1. $b= \al_i -1$ or 2. $b \GE \al_I$ and foR sOme $J\Ne i,k$, $\al_I n_i = \al_J n_J$. thiS is a direct cOnseQuence of LEmma \[f + n\_K = B\_in\_i\] CorOLlary \[2nj\]. [*let $H =\laNglE 33,56,61,84\raNgle$ WItH $\Pf(H)=\{f=28,F'=835, F(h)=863\}$ And $\AL_1=28, \aL_2=3, \al_3=2, \AL_4=2$. In This case, $\rF(28)$ Is UniquEly dETERMineD, buT theRe are Several choiceS of $\rF(835)$. AmONg tHem we Can chOose ThE follOwing, wHere $ f' + N_3= 16 n_2$ With $16> \al_2$. Note that wE havE $\al_2n

i$ does not have UF and $n _j, n_k$ a ppear s i n t he dec ompo sition of $f+n _ i$,then we have $\al_jn_j = \al _k n _k$. L et $\ {i,j,k, l \} $ beape rmu ta t io n of$\{ 1,2,3,4 \}$. By Co rol la ry \[0\_in\_ r ow \], we may as sume that th e $ (i.l)$ c omp o nentof$\RF( f)$ is $0$ fo r any cho ic e of $\ R F(f)$.T h us $f+ n_i$ contains onl y $ n _j, n_k$. Assu me tha tt he r e ar e $ 2$ differe nt expr e ssions$ f+ n _ i =a n_j + bn_k = a'n_j + b' n _k$ . Assu mi ng$ a>a',b< b' $, wehave $(a-a' )n_j = (b'-b) n_k$ a n d then$ a\ge a- a'\ge\al _j, b' - b\g e \ al _ k$. \ [f+ n\ _k = bn\ _i \] Supp oset h a t $H$ is alm ost s ymmetric, $e= 4$andf orsome$f\in PF' (H )$ we have$f +n_ k = bn_i$ for s ome$k \ne i$ . T he n o ne of t h e foll owi ngcases o ccur: 1 . $b = \ al _i -1$ or 2. $b\g e \a l_i$ and for s o me $ j \ne i,k$ ,$\a l_in _ i = \ al_j n_ j$. Thi s is a di re ct cons eq uenceof Le mma  \[f+ n\_ k = b\ _in\_i\] Coro l lary \[2nj\].[*Let $H =\la n gl e 33 , 56,6 1,8 4\rangle$ w ith$ \PF( H)=\ { f= 28, f '=835 , F(H )= 8 63 \ }$ and $\al_1=28, \ al _2=3,\al_3 =2, \al_4=2$. In this c a s e , $\RF(2 8)$i su niquely determ ined, but there are seve ral c hoices o f $\RF(83 5 ) $. Among th emwecan c ho ose the follo w i ng,wh ere $ f ' + n_3= 1 6 n _2$ wi th$1 6> \al_2$ . Note t ha twe h ave $\al _ 2n

i$ does_not have_UF and $n_j, n_k$_appears in_the_decomposition of_$f+n_i$,_then we have_$\al_jn_j= \al_kn_k$. Let $\{i,j,k,l\}$_be a permutation of_$\{1,2,3,4\}$. By Corollary_\[0\_in\_row\],_we may assume that the $(i.l)$ component of $\RF(f)$ is $0$ for any choice_of_$\RF(f)$. Thus_$f+n_i$_contains_only $n_j, n_k$. Assume that_there are $2$ different expressions_$f+n_i =_an_j + bn_k = a'n_j + b'n_k$._Assuming_$a>a', b< b'$,_we have $(a-a')n_j = (b'-b)n_k$ and then $a\ge a-a'\ge_\al_j, b'-b \ge \al_k$. \[f +_n\_k = bn\_i\]_Suppose_that_$H$ is almost symmetric,_$e=4$ and for some $f\in PF'(H)$_we have $f + n_k =_bn_i$ for some $k \ne i$. Then_one of the following cases occur: 1._ $b= \al_i -1$ or 2._ $b_\ge \al_i$ and for some_$j\ne i,k$, $\al_i_n_i =_\al_j n_j$. This is_a direct consequence of Lemma \[f +_n\_k = b\_in\_i\]_Corollary \[2nj\]. [*Let $H =\langle 33,56,61,84\rangle$ with_$\PF(H)=\{f=28,f'=835,_F(H)=863\}$ and $\al_1=28,_\al_2=3,_\al_3=2,_\al_4=2$. In_this case, $\RF(28)$_is_uniquely determined,_but_there are several choices of $\RF(835)$._Among_them we can choose the following, where_$ f' + n_3=_16_n_2$ with $16> \al_2$._Note that we have $\al_2n

iski tangent spaces is an isomorphism. [*Proof.* ]{} (i) If $E\in B(n,d,k)$, then $(E,V)\in G_0(n,d,k)$ for any $k$-dimensional subspace of $H^0(E)$. It follows that the image of $\psi$ contains $B(n,d,k)$ as a non-empty Zariski-open subset. Since $G_0(n,d,k)$ is irreducible, it follows that $B(n,d,k)$ is irreducible. \(ii) If $E\in B(n,d,k)-B(n,d,k+1)$, then $\psi^{-1}(E)=\{(E,H^0(E)\}$. \(iii) follows from (i), (ii) and Lemma \[lem:bn2\]. \(iv) Taking $(E',V')=(E,V)$ in (\[long-exact\]) and putting $V=H^0(E)$, we get a map $${{\operatorname{Ext}}}^1((E,H^0(V)),(E,H^0(V)))\to {{\operatorname{Ext}}}^1(E,E)$$ which can be identified with the map $$T_{(E,H^0(E))}G_0(n,d,k)\to T_EM(n,d)$$ induced by $\psi$. By (\[long-exact\]) this map is injective and its image is $${{\operatorname{Ker}}}({{\operatorname{Ext}}}^1(E,E)\to {{\operatorname{Hom}}}(H^0(E),H^1(E))).$$ By standard Brill-Noether theory, this image becomes identified with the subspace $T_EB(n,d,k)$ of $T_EM(n,d)$.$\Box$ \[cor:smooth\] Suppose Conditions \[cond\] hold and $G_0(n,d,k)$ is smooth. Then $\psi$ is an isomorphism over $B(n,d,k)-B(n,d,k+1)$. Moreover, if ${{\operatorname{GCD}}}(n,d,k)=1$, then $G_0(n,d,k)$ is a desingularisation of the closure $\overline{B(n,d,k)}$ of $B(n,d,k)$

iski tangent spaces is an isomorphism. [ * Proof. * ] { } (i) If $ E\in B(n, d, k)$, then $ (E, V)\in G_0(n, d, k)$ for any $ k$-dimensional subspace of $ H^0(E)$. It follow that the prototype of $ \psi$ contains $ B(n, d, k)$ as a non - empty Zariski - open subset. Since $ G_0(n, d, k)$ is irreducible, it follows that $ B(n, d, k)$ is irreducible. \(ii) If $ E\in B(n, d, k)-B(n, d, k+1)$, then $ \psi^{-1}(E)=\{(E, H^0(E)\}$. \(iii) postdate from (i), (ii) and Lemma \[lem: bn2\ ]. \(iv) Taking $ (E',V')=(E, V)$ in (\[long - exact\ ]) and invest $ V = H^0(E)$, we get a function $ $ { { \operatorname{Ext}}}^1((E, H^0(V)),(E, H^0(V)))\to { { \operatorname{Ext}}}^1(E, E)$$ which can be identify with the map $ $ T_{(E, H^0(E))}G_0(n, d, k)\to T_EM(n, d)$$ induce by $ \psi$. By (\[long - exact\ ]) this map is injective and its image is $ $ { { \operatorname{Ker}}}({{\operatorname{Ext}}}^1(E, E)\to { { \operatorname{Hom}}}(H^0(E),H^1(E))).$$ By standard Brill - Noether hypothesis, this image becomes identified with the subspace $ T_EB(n, d, k)$ of $ T_EM(n, d)$.$\Box$ \[cor: smooth\ ] Suppose Conditions \[cond\ ] hold and $ G_0(n, d, k)$ is smooth. Then $ \psi$ is an isomorphism over $ B(n, d, k)-B(n, d, k+1)$. Moreover, if $ { { \operatorname{GCD}}}(n, d, k)=1 $, then $ G_0(n, d, k)$ is a desingularisation of the closing $ \overline{B(n, d, k)}$ of $ B(n, d, k)$

iskl tangent spaces is an iromorphism. [*Proof.* ]{} (i) If $X\in B(n,d,i)$, then $(E,X)\in G_0(n,d,k)$ for any $k$-dimensionap wubspqce of $H^0(E)$. It follows tfat the ilage of $\psi$ rontains $B(n,d,k)$ as a non-emicy Zadlski-o'ei subset. Since $N_0(n,d,k)$ is irraducible, it fonluwd that $B(n,d,k)$ is irreducible. \(ii) If $E\in B(n,d,k)-B(n,c,k+1)$, then $\psi^{-1}(E)=\{(E,H^0(E)\}$. \(iyi) fpjlowa from (i), (ii) and Lemma \[lem:bn2\]. \(iv) Takihg $(E',V')=(E,V)$ in (\[long-exact\]) and putting $V=H^0(E)$, we get a lap $${{\lperatorname{Ext}}}^1((E,H^0(V)),(F,H^0(V)))\to {{\operajkrnwne{Ext}}}^1(E,E)$$ whicf can be identified wijh the map $$T_{(E,H^0(E))}G_0(n,d,k)\to T_EM(n,d)$$ induzed bv $\psi$. By (\[lobg-wxafj\]) this map iw injvctive and its image hs $${{\operstorname{Ker}}}({{\opevatoriame{Wxt}}}^1(E,E)\to {{\operatorname{Hmm}}}(H^0(E),H^1(E))).$$ By standard Brill-Noedhzr theory, this image vexomes idettifkwd ditg vhe subspwce $T_EB(n,d,k)$ of $T_EM(n,d)$.$\Box$ \[coe:smooth\] Suppose Concieppns \[cond\] hols and $D_0(n,q,k)$ is smooth. Then $\psi$ is an isomorphism ovsr $B(n,d,k)-B(n,d,k+1)$. Moreover, if ${{\operatorname{GCD}}}(n,d,k)=1$, tjen $G_0(n,d,k)$ ys a desingularisation of the closure $\overline{B(n,d,n)}$ of $U(n,a,k)$

iski tangent spaces is an isomorphism. [*Proof.* If B(n,d,k)$, then G_0(n,d,k)$ for any follows the image of contains $B(n,d,k)$ as non-empty Zariski-open subset. Since $G_0(n,d,k)$ is it follows that $B(n,d,k)$ is irreducible. \(ii) If $E\in B(n,d,k)-B(n,d,k+1)$, then $\psi^{-1}(E)=\{(E,H^0(E)\}$. \(iii) from (i), (ii) and Lemma \[lem:bn2\]. \(iv) Taking $(E',V')=(E,V)$ in (\[long-exact\]) and putting we a $${{\operatorname{Ext}}}^1((E,H^0(V)),(E,H^0(V)))\to which can be identified with the map $$T_{(E,H^0(E))}G_0(n,d,k)\to T_EM(n,d)$$ induced by $\psi$. By (\[long-exact\]) this map injective and its image is $${{\operatorname{Ker}}}({{\operatorname{Ext}}}^1(E,E)\to {{\operatorname{Hom}}}(H^0(E),H^1(E))).$$ By Brill-Noether theory, this image identified with the subspace $T_EB(n,d,k)$ $T_EM(n,d)$.$\Box$ Suppose Conditions hold $G_0(n,d,k)$ smooth. Then $\psi$ an isomorphism over $B(n,d,k)-B(n,d,k+1)$. Moreover, if ${{\operatorname{GCD}}}(n,d,k)=1$, then $G_0(n,d,k)$ is a desingularisation of the closure $\overline{B(n,d,k)}$ of

iski tangent spaces is an isomOrphism. [*ProOf.* ]{} (i) If $e\in b(n,d,K)$, tHen $(E,v)\in G_0(N,d,k)$ for any $k$-dimeNSionAl subspace of $H^0(E)$. It followS that ThE ImagE Of $\Psi$ coNtains $B(N,D,k)$ AS A noN-eMpTy ZArISkI-open SubSet. SincE $G_0(n,d,k)$ is irrEduCiBle, it follows THaT $B(n,d,k)$ is irrEduCible. \(ii) If $E\in b(n,d,K)-B(n,d,k+1)$, tHeN $\psI^{-1}(e)=\{(E,H^0(E)\}$. \(iIi) fOllowS from (i), (II) and LeMma \[lem:bn2\]. \(iV) TAKing $(E',V')=(e,v)$ in (\[long-EXAcT\]) and Putting $V=H^0(E)$, we get a mAP $${{\oPEratorname{Ext}}}^1((E,h^0(V)),(E,H^0(V)))\tO {{\oPErATOrnAme{ext}}}^1(E,E)$$ which CaN be idENtified WItH THE maP $$t_{(E,H^0(E))}G_0(n,d,k)\to T_EM(N,d)$$ induced by $\PSi$. BY (\[long-eXaCt\]) tHIs map iS injeCtIVe aNd its image iS $${{\opeRatorname{ker}}}({{\opeRAtornamE{ext}}}^1(E,E)\to {{\OperatOrnAme{hom}}}(H^0(e),h^1(E))).$$ by StaNdARd BRIlL-NoETheR theory, tHiS iMage bEcomES IDEntiFieD witH the sUbspace $T_EB(n,d,k)$ Of $T_eM(n,d)$.$\bOx$ \[cOr:smoOth\] SuPposE COnditIons \[coNd\] holD aNd $G_0(n,d,k)$ is smooth. THen $\pSi$ is an isoMorPhIsm OvEr $B(n,d,K)-b(n,d,k+1)$. MoReoVer, If ${{\operaTorname{gcD}}}(n,D,k)=1$, THEN $G_0(N,d,k)$ is a desingularisAtION oF the closUre $\oveRLiNe{b(N,d,k)}$ of $B(n,d,K)$

iski tangent spaces is anisomorphis m. [ *Pr oof .* ]{} (i) If $E\in B(n, d ,k)$ , then $(E,V)\in G_0(n ,d,k) $f or a n y$k$-d imensio n al s ubs pa ce of $ H ^0 (E)$. It follow s that the im ag e of $\psi$c on tains $B(n ,d, k)$ as a non -em pty Za ri ski - opensub set.Since$ G_0(n, d,k)$ isir r educib l e, it f o l lo ws t hat $B(n,d,k)$ is ir r educible. \(i i) If$E \ in B (n, d,k )-B(n,d,k+ 1) $, th e n $\psi ^ {- 1 } ( E)= \ {(E,H^0(E)\}$ . \(iii) f o llo ws fro m(i) , (ii)and L em m a \ [lem:bn2\]. \( iv) Takin g $(E' , V')=(E, V )$ in ( \[long -ex act \])a nd p utt in g $V = H^ 0(E ) $,we get a m ap $${{ \ope r a t o rnam e{E xt}} }^1(( E,H^0(V)),(E, H^0 (V)) ) \to {{\o perat orna me {Ext} }}^1(E ,E)$$ w hich can be ide ntif ied withthe m ap$$ T_{(E , H^0(E) )}G _0( n,d,k)\ to T_EM ( n,d )$ $ i nd uced by $\psi$. By ( \ [ lo ng-exact \]) th i sma p is inje ct ive and i ts im agei s$${{\ope ratorn a me {K er}}}({ {\ operat or nam e{E xt}}} ^ 1(E, E)\to{{\opera torna m e{Hom}}}(H^0(E ) ,H^1(E))).$$B ys t an d ardBri ll-Noethertheo r y, t hisi ma geb ecome s ide nt i fi e d with the subspace $ T_EB(n ,d,k) $ of $T_EM(n, d)$.$\Box$ \ [cor:smo oth\ ] S u ppose Conditio ns \[ cond\] hol d and $G_ 0(n,d ,k)$ issmooth. T h e n $\psi$ is an is omo r p hi sm over $B(n, d , k)-B (n ,d,k+1) $.Moreove r,if${{ \op er atorname{ GCD}}}(n ,d ,k )= 1$ , t hen $ G _0(n,d,k )$ is a de singu l arisat ion o f th ecl o sur e $\ove r li n e {B(n ,d ,k )}$of$B (n,d, k)$

iski tangent_spaces is_an isomorphism. [*Proof.* ]{} (i)_If $E\in_B(n,d,k)$,_then $(E,V)\in_G_0(n,d,k)$_for any $k$-dimensional_subspace of $H^0(E)$._It follows that the_image of $\psi$_contains_$B(n,d,k)$ as a non-empty Zariski-open subset. Since $G_0(n,d,k)$ is irreducible, it follows that $B(n,d,k)$_is_irreducible. \(ii) If_$E\in_B(n,d,k)-B(n,d,k+1)$,_then $\psi^{-1}(E)=\{(E,H^0(E)\}$. \(iii) follows from (i),_(ii) and Lemma \[lem:bn2\]. \(iv) Taking_$(E',V')=(E,V)$ in_(\[long-exact\]) and putting $V=H^0(E)$, we get a map_$${{\operatorname{Ext}}}^1((E,H^0(V)),(E,H^0(V)))\to_{{\operatorname{Ext}}}^1(E,E)$$ which can_be identified with the map $$T_{(E,H^0(E))}G_0(n,d,k)\to T_EM(n,d)$$ induced by_$\psi$. By (\[long-exact\]) this map is_injective and its_image_is_$${{\operatorname{Ker}}}({{\operatorname{Ext}}}^1(E,E)\to {{\operatorname{Hom}}}(H^0(E),H^1(E))).$$ By standard_Brill-Noether theory, this image becomes identified_with the subspace $T_EB(n,d,k)$ of $T_EM(n,d)$.$\Box$ \[cor:smooth\]_Suppose Conditions \[cond\] hold and $G_0(n,d,k)$ is_smooth. Then $\psi$ is an isomorphism_over $B(n,d,k)-B(n,d,k+1)$. Moreover, if ${{\operatorname{GCD}}}(n,d,k)=1$,_then $G_0(n,d,k)$_is a desingularisation of the_closure $\overline{B(n,d,k)}$ of_$B(n,d,k)$

tunneling strength. Therefore, the half-quantized plateau in $\sigma_{12}$ remains robust in the high temperature regime $T_c<T\ll |\Delta|$. Discussion and experimental realization ======================================= Finally, we discuss the feasibility of our proposals. Experimentally, to observe the $\mathcal{N}=\pm1$ chiral TSC and all of the four half-quantized conductance plateaus, a good proximity effect between SC and magnetic TI is necessary. Moreover, the critical field $H^{\perp}_{c}$ of SC should be larger than the coercivity $H^*_{1,2}$ in magnetic TI. From Ref. , the estimated $H_1^*\sim0.05$ T and $H_2^*\sim0.2$ T. The candidate SC materials are Nb and NbSe$_2$. The bulk Nb is a type I SC with $T_{\text{sc}}=9.6$ K and $H^{\perp}_c\sim0.2$ T, while a thin film Nb becomes a type II SC with upper critical field $H^{\perp}_{c2}\sim1$ T. NbSe$_2$ is a type II SC and shows good proximity effect with Bi$_2$Se$_3$ [@wangmx2012] even at $4.2$ K and $0.4$ T, where the proximity effect induced SC gap is $\Delta\sim0.5$ meV. The width of the CMEM $\ell$ can be estimated as $v_F/\Delta\sim0.52~\mu$m, where the Fermi velocity $v_F\sim2.6$ eV Å [@chang2013b]. For a typical junction voltage $V_{\text{sc}}\sim1~\mu$V, $f\sim0.48$ GHz, which is easily accessible in experiments. Conclusion ========== In summary, we propose to realize the $\mathcal{N}=\pm1$ chiral TSC in a magnetic TI near the QAH plateau transition via the proximity effect to an $s$-wave SC. We show that inequivalent SC pairing amplitude on top and bottom surfaces in doped magnetic TIs will optimize the $\mathcal{N}=\pm1$ chiral TSC phases. Several edge transport measurements have been proposed to identify such $\mathcal{N}=1$ TSC in the QAH-SC

tunneling strength. Therefore, the half - quantized plateau in $ \sigma_{12}$ remain full-bodied in the high temperature regime $ T_c < T\ll |\Delta|$. Discussion and experimental realization = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = ultimately, we hash out the feasibility of our proposals. Experimentally, to observe the $ \mathcal{N}=\pm1 $ chiral TSC and all of the four half - quantize conductance tableland, a good proximity effect between SC and magnetic TI is necessary. Moreover, the critical field $ H^{\perp}_{c}$ of SC should be bombastic than the coercivity $ H^*_{1,2}$ in magnetic TI. From Ref.  , the estimated $ H_1^*\sim0.05 $   metric ton and $ H_2^*\sim0.2 $   T. The candidate SC materials are Nb and NbSe$_2$. The bulk Nb is a type I SC with $ T_{\text{sc}}=9.6 $   K and $ H^{\perp}_c\sim0.2 $   T, while a thin film Nb become a type II SC with upper critical field $ H^{\perp}_{c2}\sim1 $   T. NbSe$_2 $ is a type II SC and show good proximity effect with Bi$_2$Se$_3 $   [ @wangmx2012 ] even at $ 4.2 $   K and $ 0.4 $   T, where the proximity consequence induced SC gap is $ \Delta\sim0.5 $   meV. The width of the CMEM $ \ell$ can be estimated as $ v_F/\Delta\sim0.52~\mu$m, where the Fermi velocity $ v_F\sim2.6 $   eV Å   [ @chang2013b ]. For a typical articulation voltage $ V_{\text{sc}}\sim1~\mu$V, $ f\sim0.48 $   GHz, which is easily accessible in experiments. Conclusion = = = = = = = = = = In summary, we propose to realize the $ \mathcal{N}=\pm1 $ chiral TSC in a charismatic TI near the QAH tableland conversion via the proximity effect to an $ s$-wave SC. We show that inequivalent SC pairing amplitude on top and bottom surfaces in doped magnetic TIs will optimize the $ \mathcal{N}=\pm1 $ chiral TSC phases. Several boundary conveyance measurements have been aim to name such $ \mathcal{N}=1 $ TSC in the QAH - SC

tujneling strength. Therefove, the half-quantnzed pleteau ih $\sigma_{12}$ femains robust in the high txmpeeaturt regime $T_c<T\ll |\Deltx|$. Discussiln and ezpermmental realizatmkn ======================================= Finally, we slscusv the feasibilijy of our prmposals. Experikevtclly, to observe the $\mathcal{N}=\pm1$ chiraj TSC amd all of the foor haks-quahnieed conductance plateaus, a good pdoximitj effect between XC and magnetic TI is necedsarj. Moreover, the crihical field $H^{\pewp}_{c}$ of SC shojld be larger than the coercivity $H^*_{1,2}$ in magnetic TI. Frum Rey. , the estimqtwd $J_1^*\vim0.05$ T and $H_2^*\smm0.2$ T. Thv candidate SG materhals arr Nb and NbSe$_2$. Bhe bnlk Bb is a type I SC witi $T_{\text{sc}}=9.6$ K and $H^{\perp}_s\sim0.2$ T, whine a thin film Nb bwcimes d ty[e IK SC wiuh nppsr crihicel field $H^{\psrp}_{c2}\sim1$ T. NbWe$_2$ is a type II SC snq shows good pdoximiey effect with Bi$_2$Se$_3$ [@wangmx2012] even at $4.2$ K and $0.4$ T, whsre the proximity effecr induced SC gap is $\Dglta\sim0.5$ meV. The width of the CMEM $\ell$ can be estimated as $v_F/\Gelta\akm0.52~\mb$n, whefw hhe Fermi velocity $v_F\sim2.6$ eV Å [@chang2013b]. For a typicwm kukction voltage $V_{\bext{sc}}\sim1~\mu$V, $f\sim0.48$ GNz, wnych is easily accessibme in experiments. Clnclusijn ========== In wummary, wt prolose to realize the $\mathcal{B}=\pm1$ chiral TFX in a magnetic TI near the QCH plajeau ttansition via the proxioity effect to wn $s$-wave AZ. We show that iveqlivanent SC kxiring amplitude jn top anv botcom surfxces in do[ed magnetlc TIs will optimize the $\matheal{N}=\pk1$ chiral TDC phases. Several edge transporv measurementx vavv been prjposec to identify such $\mathcal{N}=1$ TSC in the DAH-SC

tunneling strength. Therefore, the half-quantized plateau in robust the high regime $T_c<T\ll |\Delta|$. Finally, discuss the feasibility our proposals. Experimentally, observe the $\mathcal{N}=\pm1$ chiral TSC and of the four half-quantized conductance plateaus, a good proximity effect between SC and TI is necessary. Moreover, the critical field $H^{\perp}_{c}$ of SC should be larger the $H^*_{1,2}$ magnetic From Ref. , the estimated $H_1^*\sim0.05$ T and $H_2^*\sim0.2$ T. The candidate SC materials are Nb NbSe$_2$. The bulk Nb is a type I with $T_{\text{sc}}=9.6$ K and T, while a thin film becomes type II with critical $H^{\perp}_{c2}\sim1$ T. NbSe$_2$ a type II SC and shows good proximity effect with Bi$_2$Se$_3$ [@wangmx2012] even at $4.2$ K and T, where effect induced gap $\Delta\sim0.5$ The width of $\ell$ can be estimated as $v_F/\Delta\sim0.52~\mu$m, velocity $v_F\sim2.6$ eV Å [@chang2013b]. For a typical voltage $V_{\text{sc}}\sim1~\mu$V, GHz, which is easily accessible in Conclusion ========== In summary, we propose to realize $\mathcal{N}=\pm1$ chiral TSC in a magnetic TI near the QAH plateau transition via the proximity an $s$-wave SC. We that inequivalent SC amplitude top bottom in doped TIs will optimize the $\mathcal{N}=\pm1$ chiral TSC phases. Several edge transport have been proposed to identify such $\mathcal{N}=1$ TSC in the

tunneling strength. ThereforE, the half-quAntizEd pLatEaU in $\sIgma_{12}$ Remains robust iN The hIgh temperature regime $T_c<t\ll |\DeLtA|$. discUSsIon anD experiMEnTAL reAlIzAtiOn ======================================= fInAlly, wE diScuss thE feasibiliTy oF oUr proposals. EXPeRimentally, To oBserve the $\matHcaL{N}=\pm1$ chIrAl Tsc and aLl oF the fOur halF-QuantiZed conducTaNCe platEAus, a gooD PRoXimiTy effect between SC ANd MAgnetic TI is necEssary. moREoVER, thE crItical fielD $H^{\Perp}_{c}$ OF SC shouLD bE LARgeR Than the coerciVity $H^*_{1,2}$ in magnETic tI. From reF. , thE EstimaTed $H_1^*\sIm0.05$ t And $h_2^*\sim0.2$ T. The canDidaTe SC materIals arE nb and NbsE$_2$. The bulK Nb is a TypE I Sc witH $t_{\tExT{sc}}=9.6$ k aND $H^{\pERp}_C\siM0.2$ t, whIle a thin FiLm nb becOmes A TYPE II Sc wiTh upPer crItical field $H^{\pErp}_{C2}\sim1$ t. nbSE$_2$ is a tYpe II sC anD sHows gOod proXimitY eFfect with Bi$_2$Se$_3$ [@waNgmx2012] Even at $4.2$ K anD $0.4$ T, wHeRe tHe ProxiMIty effEct IndUced SC gAp is $\DelTA\siM0.5$ mEv. tHe Width of the CMEM $\ell$ cAn BE EsTimated aS $v_F/\DelTA\sIm0.52~\MU$m, where tHe ferMi veLOCity $v_f\sim2.6$ Ev Å [@Chang2013b]. FoR a typiCAl JuNction vOlTage $V_{\tExT{sc}}\Sim1~\Mu$V, $f\sIM0.48$ GHz, Which iS easily aCcessIBle in experimenTS. Conclusion ========== In SUmMARy, WE proPosE to realize tHe $\maTHcal{n}=\pm1$ cHIrAl Tsc in a mAgnetIc ti nEAr the QAH plateau tranSiTion viA the pRoximity effecT to an $s$-wave sc. wE show thaT ineQUiVAlent SC pairing AmpliTude on top aND bottom sUrfacEs in dopeD magnetic tiS will optImiZe tHe $\mAthCAL{N}=\Pm1$ chiral TSC phASEs. SeVeRal edge TraNsport mEasUreMenTs hAvE been propOsed to idEnTiFy SuCh $\mAthcaL{n}=1$ TSC in thE QaH-Sc

tunneling strength. There fore, thehalf- qua nti ze d pl atea u in $\sigma_{ 1 2}$remains robust in thehighte m pera t ur e reg ime $T_ c <T \ l l | \D el ta| $. D iscus sio n and e xperimenta l r ea lization === = == ========== === ============ === ===== F ina l ly, w e d iscus s thef easibi lity of o ur propos a ls. Exp e r im enta lly, to observe t h e$ \mathcal{N}=\p m1$ ch ir a lT S C a ndall of the f our h a lf-quan t iz e d con d uctance plate aus, a good pro ximity e ffe c t betw een S Ca ndmagnetic TI isnecessary . More o ver, th e critic al fie ld$H^ {\pe r p} _{ c}$ o f SC sh oul d be largerth an thecoer c i v i ty $ H^* _{1, 2}$ i n magnetic TI . F romR ef.  , th e est imat ed $H_1 ^*\sim 0.05$  T and $H_2^*\sim 0.2$  T. The c and id ate S C mat e rialsare Nb and Nb Se$_2$. The b u l k N b is a type I SC w it h $T _{\text{ sc}}=9 . 6$  K and $H^{ \p erp }_c\ s i m0.2$  T,w hi le a thi n film Nb b ecomesatype I ISCwit h upp e r cr itical field $ H^{\p e rp}_{c2}\sim1$ T. NbSe$_2$ i s a t yp e IISCand shows g oodp roxi mity ef fec t with Bi$_ 2$ S e$ _ 3$ [@wangmx2012] ev en at $4 .2$ K and $0.4$ T, where the p r oximityeffe c ti nduced SC gapis $\ Delta\sim0 . 5$ meV.The w idth ofthe CMEM$ \ ell$ can be es tim ate d as $v_F/\Delta\ s i m0.5 2~ \mu$m,whe re theFer mivel oci ty $v_F\sim 2.6$ eVÅ[@ ch an g20 13b]. For a ty pi cal j unc tionv oltage $V_{ \tex t{ sc } }\s im1~\mu $ V, $ f\si m0 .4 8$ G Hz, w hichis e a sil y acces sible inexp e rime nt s. Concl usion ======= == = In summ ar y,we pro p o se to re alize the $\mathcal{N}= \ pm1$ ch ira l TSC ina magneti c T I near th e QAH p lateau tran si tio n via t h e p rox im ity effect t o a n $s$ -w aveSC. Weshow that inequiva l ent SC pairing a mpl itud e on to p a n d b ot t oms u rfaces in doped magneticTI s w ill optimi z e t he $\math cal{N}= \pm1$ chiralTSC phase s. Severa ledge t ran sport meas urements have bee n prop o se d toide ntifysu ch$\mat hcal{N } =1$ TSCin the Q AH-SC

tunneling_strength. Therefore,_the half-quantized plateau in_$\sigma_{12}$ remains_robust_in the_high_temperature regime $T_c<T\ll_|\Delta|$. Discussion and experimental_realization ======================================= Finally, we discuss the_feasibility of our_proposals._Experimentally, to observe the $\mathcal{N}=\pm1$ chiral TSC and all of the four half-quantized conductance_plateaus,_a good_proximity_effect_between SC and magnetic TI_is necessary. Moreover, the critical_field $H^{\perp}_{c}$_of SC should be larger than the coercivity_$H^*_{1,2}$_in magnetic TI._From Ref. , the estimated $H_1^*\sim0.05$ T and $H_2^*\sim0.2$ T. The candidate_SC materials are Nb and NbSe$_2$._The bulk Nb_is_a_type I SC with_$T_{\text{sc}}=9.6$ K and $H^{\perp}_c\sim0.2$ T, while a thin_film Nb becomes a type II_SC with upper critical field $H^{\perp}_{c2}\sim1$ T. NbSe$_2$_is a type II SC and_shows good proximity effect with_Bi$_2$Se$_3$ [@wangmx2012] even_at $4.2$ K and $0.4$ T, where_the proximity effect_induced SC_gap is $\Delta\sim0.5$ meV._The width of the CMEM $\ell$_can be estimated_as $v_F/\Delta\sim0.52~\mu$m, where the Fermi velocity_$v_F\sim2.6$ eV_Å [@chang2013b]. For a_typical_junction_voltage $V_{\text{sc}}\sim1~\mu$V,_$f\sim0.48$ GHz, which is_easily_accessible in_experiments. Conclusion ========== In_summary, we propose to realize the_$\mathcal{N}=\pm1$_chiral TSC in a magnetic TI near_the QAH plateau transition_via_the proximity effect to_an $s$-wave SC. We show_that inequivalent SC pairing amplitude on_top and_bottom surfaces_in doped magnetic TIs will optimize the $\mathcal{N}=\pm1$ chiral TSC phases._Several edge transport measurements have been_proposed to identify such_$\mathcal{N}=1$ TSC_in_the QAH-SC

G(1,5) = Q + \Gamma + L_1 + L_2 + L_3.$$ ![The canonical curve $C=\Gamma+Q+L_1+L_2+L_3.$[]{data-label="fig:cover"}](curve8.pdf){width="2.3in"} Bertini’s Theorem implies that a general $8$-dimensional space $\langle \Gamma \rangle\subseteq \PP^8\subseteq \PP^{14}$ cuts out on $\mathbf G(1,5)$ a smooth 2-dimensional linear section $T$, see also [@Ve], Propositions 3.2 and 3.3. By the adjunction formula, $T\hookrightarrow \PP^8$ is a smooth $K3$ surface (of genus $8$) polarized by $\mathcal O_T(C)$. We now describe the Picard lattice of $T$: \[intnumb\] One has the following intersection products on $T$: $$Q^2=-2, \ Q \cdot \Gamma = 3,\ \ Q\cdot L_i = 1, \ \ \ \Gamma \cdot L_i = 2, \ \ L_i \cdot L_j = -2\delta_{ij}, \ \ \mbox{ for } i,j=1,2,3.$$ The generality assumptions ensure that $L_i$ and $L_j$ are disjoint lines, for $i\neq j$. Else, if $L_i\cap L_j\neq \emptyset$, then $\langle p_i,p_j\rangle \subseteq P_i\cap P_j\subseteq \PP^5$. It follows that the four rulings of $R'$ passing through the points $x_i,y_i,x_j,y_j$ respectively, span a $6$-dimensional space in $\PP^8$, which is impossible for $$h^0\Bigl(R',\OO_{R'}(1)(-4(\ell-E))\Bigr)=h^0(R',\OO_{R'}(E))=1,$$ where recall that $\ell, E\in \mbox{Pic}(R')$ denote the line class and the exceptional divisor respectively. This implies that there exists a unique hyperplane in $\PP^8$ containing the four rulings, therefore they must span a $7$-dimensional linear space.

G(1,5) = Q + \Gamma + L_1 + L_2 + L_3.$$ ! [ The canonical curve $ C=\Gamma+Q+L_1+L_2+L_3.$[]{data - label="fig: cover"}](curve8.pdf){width="2.3 in " } Bertini ’s Theorem implies that a general $ 8$-dimensional quad $ \langle \Gamma \rangle\subseteq \PP^8\subseteq \PP^{14}$ cut out on $ \mathbf G(1,5)$ a smooth 2 - dimensional linear section $ T$, see besides [ @Ve ], Propositions 3.2 and 3.3. By the adjunction formula, $ T\hookrightarrow \PP^8 $ is a politic $ K3 $ surface (of genus $ 8 $) polarized by $ \mathcal O_T(C)$. We nowadays describe the Picard lattice of $ T$: \[intnumb\ ] One suffer the following intersection intersection on $ T$: $ $ Q^2=-2, \ Q \cdot \Gamma = 3,\ \ Q\cdot L_i = 1, \ \ \ \Gamma \cdot L_i = 2, \ \ L_i \cdot L_j = -2\delta_{ij }, \ \ \mbox { for } i, j=1,2,3.$$ The generalization assumptions ensure that $ L_i$ and $ L_j$ are disjoint occupation, for $ i\neq j$. Else, if $ L_i\cap L_j\neq \emptyset$, then $ \langle p_i, p_j\rangle \subseteq P_i\cap P_j\subseteq \PP^5$. It follows that the four rulings of $ R'$ excrete through the points $ x_i, y_i, x_j, y_j$ respectively, span a $ 6$-dimensional space in $ \PP^8 $, which is impossible for $ $ h^0\Bigl(R',\OO_{R'}(1)(-4(\ell - E))\Bigr)=h^0(R',\OO_{R'}(E))=1,$$ where echo that $ \ell, E\in \mbox{Pic}(R')$ denote the line class and the exceeding divisor respectively. This implies that there exists a unique hyperplane in $ \PP^8 $ containing the four rulings, consequently they must span a $ 7$-dimensional linear space.

G(1,5) = Q + \Gamma + L_1 + L_2 + L_3.$$ ![The cakonical curve $C=\Gcnma+Q+L_1+L_2+N_3.$[]{data-lzbel="fig:cuver"}](curve8.pdf){width="2.3in"} Bertini’s Vheoeem inplies that a general $8$-aimensionwl space $\lanjle \Gamma \rangle\subseteq \PP^8\subacteq \'P^{14}$ cuts out on $\msthbf G(1,5)$ a vmooth 2-dimensimnxl linear section $T$, see also [@Ve], Propofitions 3.2 wnd 3.3. By the adtuncuiog fodmula, $T\hookrightarrow \PP^8$ is a smoofh $K3$ sugface (of genus $8$) pplarized by $\mathcal O_T(C)$. We now describe the Picagd lattice if $T$: \[ybtnumb\] One hxs the following interaection products on $T$: $$Q^2=-2, \ Q \cdot \Gammc = 3,\ \ Q\cdoj L_l = 1, \ \ \ \Gamna \cdjt L_i = 2, \ \ L_i \cdot L_j = -2\dekta_{ij}, \ \ \mbox{ fpr } i,j=1,2,3.$$ Rhe generality assumpvions ensure that $L_i$ and $L_j$ ase disjoint lines, fir $i\neq j$. Ense, kd $L_k\cak L_o\nes \emptjsev$, then $\langme p_i,p_j\rangoe \subseteq P_i\cap P_k\stvseteq \PP^5$. It rollowf ehat the four rulings of $R'$ passing throlgh fhe points $x_i,y_i,x_j,y_j$ respectively, span a $6$-dimejsional s[ace in $\PP^8$, which is impossible for $$h^0\Bigl(R',\OO_{R'}(1)(-4(\ell-E))\Bhgr)=h^0(R',\KU_{R'}(E))=1,$$ wmere eefall that $\ell, E\in \mbox{Pic}(R')$ denote the line clwas akd the exceptioncl divisor respevtlvrjy. This impligs that thsre exists a uniquf hyperklane un $\PP^8$ conuainimg the four rulings, therefoee they must wpan a $7$-dimensional linear spaee.

G(1,5) = Q + \Gamma + L_1 + ![The canonical $C=\Gamma+Q+L_1+L_2+L_3.$[]{data-label="fig:cover"}](curve8.pdf){width="2.3in"} Bertini’s Theorem space \Gamma \rangle\subseteq \PP^8\subseteq cuts out on G(1,5)$ a smooth 2-dimensional linear section see also [@Ve], Propositions 3.2 and 3.3. By the adjunction formula, $T\hookrightarrow \PP^8$ a smooth $K3$ surface (of genus $8$) polarized by $\mathcal O_T(C)$. We now the lattice $T$: One has the following intersection products on $T$: $$Q^2=-2, \ Q \cdot \Gamma = 3,\ \ L_i = 1, \ \ \ \Gamma \cdot = 2, \ \ \cdot L_j = -2\delta_{ij}, \ \mbox{ } i,j=1,2,3.$$ generality ensure $L_i$ and $L_j$ disjoint lines, for $i\neq j$. Else, if $L_i\cap L_j\neq \emptyset$, then $\langle p_i,p_j\rangle \subseteq P_i\cap P_j\subseteq \PP^5$. follows that rulings of passing the $x_i,y_i,x_j,y_j$ respectively, span space in $\PP^8$, which is impossible recall that $\ell, E\in \mbox{Pic}(R')$ denote the line and the divisor respectively. This implies that there a unique hyperplane in $\PP^8$ containing the four therefore they must span a $7$-dimensional linear space.

G(1,5) = Q + \Gamma + L_1 + L_2 + L_3.$$ ![The canonical curVe $C=\Gamma+Q+L_1+l_2+L_3.$[]{datA-laBel="FiG:covEr"}](cuRve8.pdf){width="2.3in"} BERtinI’s Theorem implies that a gEneraL $8$-dIMensIOnAl spaCe $\langlE \gaMMA \raNgLe\SubSeTEq \pP^8\subSetEq \PP^{14}$ cutS out on $\mathBf G(1,5)$ A sMooth 2-dimensiONaL linear secTioN $T$, see also [@Ve], PRopOsitioNs 3.2 And 3.3. bY the aDjuNctioN formuLA, $T\hookRightarroW \Pp^8$ Is a smoOTh $K3$ surfACE (oF genUs $8$) polarized by $\mathCAl o_t(C)$. We now describE the PiCaRD lATTicE of $t$: \[intnumb\] OnE hAs the FOllowinG InTERSecTIon products on $t$: $$Q^2=-2, \ Q \cdot \GammA = 3,\ \ q\cdOt L_i = 1, \ \ \ \GaMmA \cdOT L_i = 2, \ \ L_i \cDot L_j = -2\DeLTa_{iJ}, \ \ \mbox{ for } i,j=1,2,3.$$ THe geNerality aSsumptIOns ensuRE that $L_i$ And $L_j$ aRe dIsjOint LInEs, For $I\nEQ j$. ELSe, If $L_I\Cap l_j\neq \empTySeT$, then $\LangLE P_I,P_j\raNglE \subSeteq p_i\cap P_j\subsetEq \Pp^5$. It fOLloWs thaT the fOur rUlIngs oF $R'$ passIng thRoUgh the points $x_i,y_I,x_j,y_J$ respectiVelY, sPan A $6$-dImensIOnal spAce In $\Pp^8$, which iS impossIBle FoR $$H^0\bIgL(R',\OO_{R'}(1)(-4(\ell-E))\Bigr)=h^0(R',\OO_{R'}(e))=1,$$ wHERe Recall thAt $\ell, E\IN \mBoX{pic}(R')$ denoTe The Line CLAss anD the EXcEptional DivisoR ReSpEctivelY. THis impLiEs tHat There EXistS a uniqUe hyperpLane iN $\pP^8$ containing thE Four rulings, thEReFORe THey mUst Span a $7$-dimensIonaL LineAr spACe.

G(1,5) = Q + \Gamma + L_1 + L_2 + L _3.$$ ! [Th ecano nica l curve $C=\Ga m ma+Q +L_1+L_2+L_3.$[]{data- label =" f ig:c o ve r"}]( curve8. p df ) { wid th =" 2.3 in " } Bert ini ’s Theo rem implie s t ha t a general$ 8$ -dimension alspace $\lang le\Gamma \ ran g le\su bse teq \ PP^8\s u bseteq \PP^{14} $c uts ou t on $\m a t hb f G( 1,5)$ a smooth 2- d im e nsional linear secti on $T $ , se e a lso [@Ve], P ropos i tions 3 . 2a n d 3. 3 . By the adju nction form u la, $T\ho ok rig h tarrow \PP^ 8$ isa smooth $K 3$ s urface (o f genu s $8$) p o larized by $\ mat hca l O_ T (C )$ . W en owd es cri b e t he Picar dla ttice of$ T $ : \[ int numb \] On e has the fol low ingi nte rsect ion p rodu ct s on$T$: $ $Q^2= -2 , \ Q \cdot \Ga mma= 3,\ \ Q\ cd ot L _i =1 , \ \ \ \G amma \c dot L_i =2, \ \ L_i \cdot L_j =-2 \ d el ta_{ij}, \ \\ mb ox { for } i ,j =1, 2,3. $ $ The gen e ra lity ass umptio n sen sure th at $L_i$ a nd$L_ j$ ar e dis jointlines, f or $i \ neq j$. Else,i f $L_i\cap L_ j \n e q \ e mpty set $, then $\l angl e p_i ,p_j \ ra ngl e \sub seteq P _ i\ c ap P_j\subseteq \PP ^5 $. Itfollo ws that the f our ruling s o f $R'$ p assi n gt hrough the poi nts $ x_i,y_i,x_ j ,y_j$ re spect ively, s pan a $6$ - d imension alspa cein$ \ PP ^8$, which is i mpos si ble for $$ h^0\Big l(R ',\ OO_ {R' }( 1)(-4(\el l-E))\Bi gr )= h^ 0( R', \OO_{ R '}(E))=1 ,$ $ w he rerecal l that$\ell , E\ in \ m box {Pic}(R ' )$ d enot eth e li necl ass a nd t h e e xceptio nal divis orr espe ct iv ely. Th is implies th at there exi st s a uniqu e hyperpla ne in $\PP^8$ containin g the fo urrulin gs,therefore th ey mus t s p an a $ 7$-dim ensio na l l i n ear s p a ce .

G(1,5)_= Q_+ \Gamma + L_1_+ L_2_+_L_3.$$ ![The canonical_curve_$C=\Gamma+Q+L_1+L_2+L_3.$[]{data-label="fig:cover"}](curve8.pdf){width="2.3in"} Bertini’s Theorem implies_that a general_$8$-dimensional space $\langle \Gamma_\rangle\subseteq \PP^8\subseteq \PP^{14}$_cuts_out on $\mathbf G(1,5)$ a smooth 2-dimensional linear section $T$, see also [@Ve], Propositions_3.2_and 3.3._By_the_adjunction formula, $T\hookrightarrow \PP^8$ is_a smooth $K3$ surface (of_genus $8$)_polarized by $\mathcal O_T(C)$. We now describe the_Picard_lattice of $T$: \[intnumb\]_One has the following intersection products on $T$: $$Q^2=-2,_\ Q \cdot \Gamma = 3,\_ \ _Q\cdot__L_i = 1, \_ \ \ \Gamma \cdot_L_i = 2, \ _ \ L_i \cdot L_j = -2\delta_{ij}, _\ \ \mbox{ for } i,j=1,2,3.$$ The_generality assumptions ensure that $L_i$_and $L_j$_are disjoint lines, for $i\neq_j$. Else, if_$L_i\cap L_j\neq_\emptyset$, then $\langle_p_i,p_j\rangle \subseteq P_i\cap P_j\subseteq \PP^5$. It_follows that the_four rulings of $R'$ passing through_the_points $x_i,y_i,x_j,y_j$ respectively,_span_a_$6$-dimensional space_in $\PP^8$, which_is_impossible for_$$h^0\Bigl(R',\OO_{R'}(1)(-4(\ell-E))\Bigr)=h^0(R',\OO_{R'}(E))=1,$$_where recall that $\ell, E\in \mbox{Pic}(R')$_denote_the line class and the exceptional divisor_respectively. This implies that_there_exists a unique hyperplane_in $\PP^8$ containing the four_rulings, therefore they must span a_$7$-dimensional linear_space.

Sphs studied. The mean magnitude of the blue HB stars suggests that And I lies along the line-of-sight at the same distance as M31 to within $\sim\pm$70 kpc. Consequently, the true distance of And I from the center of M31 is between $\sim$45 and $\sim$85 kpc, with the higher estimates being more likely. Such distances are comparable to the galactocentric distances of the nearer Milky Way dSph companions Ursa Minor, Draco and Sculptor. From the mean color of the lower giant branch, the mean metal abundance of And I is estimated as \[Fe/H\] = –1.45 $\pm$ 0.2 dex, while the presence of an internal abundance spread with total range of $\sim$0.6 dex is suggested by the intrinsic color width of the upper giant branch. A small population of faint blue stars, which we identify as blue stragglers, is also present. author: - 'G. S. Da Costa' - 'T. E. Armandroff' - Nelson Caldwell - Patrick Seitzer title: 'The Dwarf Spheroidal Companions to M31: WFPC2 Observations of Andromeda I[^1]' --- Introduction ============ The nine dwarf spheroidal (dSph) galaxy companions to our Galaxy are, because of their proximity, the most easily studied examples of what may be the most common type of galaxy in the Universe. These systems have been probed in greater and greater detail in the last decade, and though their evolutionary history is far from being completely understood, they are all now relatively well observed objects (see recent reviews, e.g. Da Costa 1992, Zinn 1993a, Mateo 1996). But within the Local Group there are at least three other objects that are classified as dwarf spheroidal galaxies: the dSph companions to M31. These galaxies, known as And I, II and III, lie at projected distances of $\sim$45, 130 and 60 kpc, respectively, from the center of M31. [*Thus in the same way as the galactic dSphs are systems in the outer halo of the Galaxy, the And dSphs are systems in the outer halo of M31*]{}. Recent work on the And dSphs has established that there is a considerable degree of similarity between the M31 dS

Sphs studied. The mean magnitude of the blue HB star suggest that And I lies along the line - of - sight at the like distance as M31 to within $ \sim\pm$70 kpc. Consequently, the true distance of And I from the center of M31 is between $ \sim$45 and $ \sim$85 kpc, with the higher estimate being more likely. Such distances are comparable to the galactocentric distance of the nearer Milky Way dSph companions Ursa Minor, Draco and Sculptor. From the beggarly color of the lower elephantine branch, the mean metal abundance of And I is estimated as \[Fe / H\ ] = – 1.45 $ \pm$ 0.2 dex, while the presence of an internal abundance spread with entire range of $ \sim$0.6 dex is suggested by the intrinsic color width of the upper elephantine branch. A small population of dim blue headliner, which we identify as blasphemous stragglers, is also present. author: -' G. S. Da Costa' -' T. E. Armandroff' - Nelson Caldwell - Patrick Seitzer deed:' The Dwarf Spheroidal Companions to M31: WFPC2 Observations of Andromeda I[^1 ]' --- Introduction = = = = = = = = = = = = The nine dwarf spheroidal (dSph) galaxy companions to our Galaxy are, because of their proximity, the most well analyze examples of what may be the most common type of galaxy in the Universe. These systems have been probed in greater and greater contingent in the concluding decade, and though their evolutionary history is far from being wholly understand, they are wholly now relatively well observed objects (see recent reviews, e.g. Da   Costa 1992, Zinn 1993a, Mateo 1996). But within the Local Group there are at least three early objects that are classified as dwarf spheroidal galaxies: the dSph companions to M31. These galaxies, known as And   I, II and III, lie at projected distances of $ \sim$45, 130 and 60 kpc, respectively, from the center of M31. [ * Thus in the like way as the astronomic dSphs are systems in the outer halo of the Galaxy, the And dSphs are systems in the outer halo of M31 * ] { }. late work on the And dSphs has established that there is a considerable degree of similarity between the M31 dS

Sphd studied. The mean mannitude of the blue HB vtars auggests that And I lies along the lmne-od-sighu at the same distavce as M31 no within $\sim\km$70 kpc. Consequently, the tvbe diabance if And I from jhe center ox M31 is between $\skm$45 and $\sim$85 kpc, with the higher estimaees beimg more likely. Soch dpseancss are comparable to the galactocehtric dpstances of the nrarer Milky Way dSph compajiond Ursa Minor, Draco and Sculptir. Fwim the mean zolor of the lower giaht branch, the mean metal abundavce oy And I is gsfildted as \[Fe/H\] = –1.45 $\pm$ 0.2 dex, while bne prevence og an internal sbuidanxe spread with total cange of $\sim$0.6 dex is fuggested bv the intrinsic color wudth mf tve uoper giznv bdanch. W sjall populztion of faunt blue stars, whicn rv identify as blue ftwagglers, is also present. author: - 'G. S. Da Cmstz' - 'T. E. Armandroff' - Nelson Caldwell - Patrick Seijzer title: 'Ehe Dwarf Spheroidal Companions to M31: WFPC2 Observadions uf Ckqfimfda I[^1]' --- Introduction ============ The nine dwarf spheroidal (dS[g) bakaxy companionf to our Gakady wre, because ow theix pdoximity, the most fasily ftudiwd examplts of what may be the most commob type of gajqxy in the Universz. These systzms hafe bern probed in greater anb greafer detail ln the laag decade, and thojgh tveir evolutionary history ys far frim bzing comoletgly undqrstood, thfy arc all now relativelj welp mbserved ohjects (see recent reviews, e.g. Da Rpsta 1992, Zinn 1993a, Mdtem 1996). But wnthin bhe Local Group there are at keast tkree ogher objecns that ace classifieq as dwarf spvgroidal galaxmes: the dFph xompqnions gu M31. These galacies, knowu as Qnd I, II and III, lic at krkjected distanczr of $\sim$45, 130 and 60 klc, fes[ebtitely, ssom the centar ow M31. [*Yhus kn the same day ss the galactic dSphv ars systems in the oitcr halo od the Gajaxy, the And cSphs are systems ln thx outec halo of M31*]{}. Recent work on the And dSphs has estahlifhed that thqre ls a considerayle degree of similarity between the M31 dW

Sphs studied. The mean magnitude of the stars that And lies along the as to within $\sim\pm$70 Consequently, the true of And I from the center M31 is between $\sim$45 and $\sim$85 kpc, with the higher estimates being more Such distances are comparable to the galactocentric distances of the nearer Milky Way companions Minor, and From the mean color of the lower giant branch, the mean metal abundance of And I estimated as \[Fe/H\] = –1.45 $\pm$ 0.2 dex, the presence of an abundance spread with total range $\sim$0.6 is suggested the color of the upper branch. A small population of faint blue stars, which we identify as blue stragglers, is also present. - 'G. Costa' - E. - Caldwell - Patrick 'The Dwarf Spheroidal Companions to M31: Andromeda I[^1]' --- Introduction ============ The nine dwarf (dSph) galaxy to our Galaxy are, because of proximity, the most easily studied examples of what be the most common type of galaxy in the Universe. These systems have been probed and greater detail in last decade, and their history far being completely they are all now relatively well observed objects (see recent reviews, Da Costa 1992, Zinn 1993a, Mateo 1996). But within the there at least three objects that are classified dwarf galaxies: the dSph companions These known II III, at projected distances of 130 and 60 kpc, respectively, the center of M31. as the galactic dSphs are systems in the halo of the Galaxy, the And dSphs systems in the outer halo of M31*]{}. Recent work on the And has established is a considerable degree of similarity between the dS

Sphs studied. The mean magnituDe of the bluE HB stArs SugGeSts tHat ANd I lies along thE Line-Of-sight at the same distanCe as M31 To WIthiN $\SiM\pm$70 kpC. ConseqUEnTLY, thE tRuE diStANcE of AnD I fRom the cEnter of M31 is BetWeEn $\sim$45 and $\sim$85 kPC, wIth the highEr eStimates beinG moRe likeLy. sucH DistaNceS are cOmparaBLe to thE galactocEnTRic disTAnces of THE nEareR Milky Way dSph compANiONs Ursa Minor, DraCo and SCuLPtOR. froM thE mean color Of The loWEr giant BRaNCH, The MEan metal abundAnce of And I iS EstImated As \[fe/H\] = –1.45 $\PM$ 0.2 dex, whIle thE pREseNce of an inteRnal Abundance Spread WIth totaL Range of $\Sim$0.6 dex Is sUggEsteD By ThE inTrINsiC CoLor WIdtH of the upPeR gIant bRancH. a SMAll pOpuLatiOn of fAint blue stars, WhiCh we IDenTify aS blue StraGgLers, iS also pResenT. aUthor: - 'G. S. Da Costa' - 'T. e. ArmAndroff' - NeLsoN CAldWeLl - PatRIck SeiTzeR tiTle: 'The DWarf SphERoiDaL cOMpAnions to M31: WFPC2 ObserVaTIOnS of AndroMeda I[^1]' --- INTrOdUCtion ============ The NiNe dWarf SPHeroiDal (dsPh) Galaxy coMpanioNS tO oUr GalaxY aRe, becaUsE of TheIr proXImitY, the moSt easily StudiED examples of whaT May be the most cOMmON TyPE of gAlaXy in the UnivErse. tHese SystEMs HavE Been pRobed In GReATer and greater detail In The lasT decaDe, and though thEir evolutiONARy historY is fAR fROm being completEly unDerstood, thEY are all nOw relAtively wEll observED Objects (sEe rEceNt rEviEWS, e.G. Da Costa 1992, Zinn 1993a, mATeo 1996). BUt Within tHe LOcal GroUp tHerE arE at LeAst three oTher objeCtS tHaT aRe cLassiFIed as dwaRf SphErOidAl galAXies: thE dSph CompAnIoNS to m31. These gALaXIEs, knOwN aS And i, II AnD III, lIe at PRojEcted diStances of $\Sim$45, 130 ANd 60 kpC, rEsPectiveLy, from the centEr Of M31. [*Thus in tHe SamE way as THE galactiC dSphs are systems in the ouTEr halo oF thE GalaXy, thE And dSphs Are SystemS in THe outeR halo oF M31*]{}. RecEnT woRK On the aND dsphS hAs establisHED thAt theRe Is a cOnsiderAble degree of similaRIty Between the M31 dS

Sphs studied. The mea n magnitud e ofthe bl ue HBstar s suggests tha t And I lies along the line -of-s ig h t at th e sam e dista n ce a s M 31 t o w it h in $\si m\p m$70 kp c. Consequ ent ly , the true d i st ance of An d I from the ce nte r of M 31 is betwe en$\sim $45 an d $\sim $85 kpc,wi t h theh igher e s t im ates being more likel y .S uch distancesare co mp a ra b l e t o t he galacto ce ntric distanc e so f the nearer MilkyWay dSph co m pan ions U rs a M i nor, D racoan d Sc ulptor. Fro m th e mean co lor of the low e r giant branc h,the mea n m et alab u nda n ce of And I is es ti ma ted a s \[ F e / H \] = –1 .45$\pm$ 0.2 dex, whi lethep res enceof an int er nal a bundan ce sp re ad with total r ange of $\sim $0. 6dex i s sug g estedbythe intrin sic col o r w id t h of the upper giant b ra n c h. A small popul a ti on of faint b lue sta r s , whi ch w e i dentifyas blu e s tr agglers ,is als opre sen t. au t hor: - 'G. S. Da C osta' - 'T. E. Arman d roff' - Nelso n C a l dw e ll - Pa trick Seitz er t i tle: 'Th e D war f Sphe roida lC om p anions to M31: WFPC 2Observ ation s of Andromed a I[^1]' - - - Introdu ctio n = = ========== Th e nin e dwarf sp h eroidal(dSph ) galaxy companio n s to ourGal axy ar e,b e ca use of theirp r oxim it y, themos t easil y s tud ied ex am ples of w hat maybe t he m ost comm o n type o fgal ax y i n the Univer se. T hese s ys t ems have b e en p robe din gre ate rand g reat e r d etail i n the las t d e cade ,an d thoug h their evolu ti onary hist or y i s farf r om being completely understood, they ar e a ll no w re lativelywel l obse rve d objec ts (se e rec en t r e v iews, e .g . D aCosta 1992 , Zin n 199 3a , Ma teo 199 6). But within the Loc al Group ther e a re a t le ast th r eeot h ero b jects that areclassified a s d warf spher o ida lgalaxie s: thedSphc ompanio ns to M31 . These g al axie s , kn own as And  I, II a nd III, l i e atp ro jecte d d istanc es of $\si m$45,1 30and 6 0 kpc, r espect ively ,from the center of M31. [*Thusin the same wa y as thegal a cti c dSphs a re s ystems inthe ou ter h alo of th e Ga l ax y,t he An d dS p hs are sy s te msi n t he outer ha l o ofM31*] {}. Recen t wo rk on the And dSp h s has establis hedt h atthe r e is a considerabledeg re e of simil ar ity between the M31 d S

Sphs studied. _ _ The mean magnitude_of the_blue_HB stars_suggests_that And I_lies along the_line-of-sight at the same_distance as M31_to_within $\sim\pm$70 kpc. Consequently, the true distance of And I from the center of_M31_is between_$\sim$45_and_$\sim$85 kpc, with the higher_estimates being more likely. Such_distances are_comparable to the galactocentric distances of the nearer_Milky_Way dSph companions_Ursa Minor, Draco and Sculptor. From the mean color_of the lower giant branch, the_mean metal abundance_of_And_I is estimated as_\[Fe/H\] = –1.45 $\pm$ 0.2 dex,_while the presence of an internal_abundance spread with total range of $\sim$0.6_dex is suggested by the intrinsic_color width of the upper_giant branch._A small population of faint_blue stars, which_we identify_as blue stragglers,_is also present. author: - 'G. S. Da_Costa' - 'T. E._Armandroff' - Nelson Caldwell - Patrick Seitzer title: 'The_Dwarf_Spheroidal Companions to_M31:_WFPC2_Observations of_Andromeda I[^1]' --- Introduction ============ The nine_dwarf_spheroidal (dSph)_galaxy_companions to our Galaxy are, because_of_their proximity, the most easily studied examples_of what may be_the_most common type of_galaxy in the Universe. These_systems have been probed in greater_and greater_detail in_the last decade, and though their evolutionary history is far from_being completely understood, they are all_now relatively well observed_objects (see_recent_reviews, e.g. Da Costa_1992,_Zinn 1993a,_Mateo 1996). But within the Local Group_there are_at least three other objects that_are classified as dwarf_spheroidal_galaxies: the dSph companions to M31._These galaxies, known as And I, II_and III, lie at projected_distances_of_$\sim$45, 130 and 60 kpc,_respectively, from the center of M31._[*Thus in the_same way as the galactic dSphs are_systems_in the outer halo of the_Galaxy,_the And dSphs are systems in_the_outer_halo of M31*]{}. Recent work on_the And dSphs has established that_there is a considerable degree of similarity between the_M31 dS

92-0.61 & 18 14 00.893 & 0.003 & -18 53 26.116 & 0.005 & 6.54 &0.04 &43.51\ G11.92-0.61 & 18 14 00.893 & 0.003 & -18 53 26.119 & 0.005 & 5.88 &0.04 &43.65\ G11.92-0.61 & 18 14 00.893 & 0.004 & -18 53 26.128 & 0.007 & 4.28 &0.04 &43.79\ G11.92-0.61 & 18 14 00.894 & 0.008 & -18 53 26.148 & 0.015 & 2.11 &0.04 &43.93\ G11.92-0.61 & 18 14 00.894 & 0.020 & -18 53 26.168 & 0.037 & 0.81 &0.04 &44.06\ G11.92-0.61 & 18 14 00.894 & 0.051 & -18 53 26.249 & 0.096 & 0.33 &0.04 &44.20\ G18.67+0.03 & 18 24 53.782 & 0.073 & -12 39 20.750 & 0.057 & 0.47 &0.02 &76.13\ G18.67+0.03 & 18 24 53.779 & 0.017 & -12 39 20.744 & 0.013 & 2.19 &0.03 &76.26\ G18.67+0.03 & 18 24 53.781 & 0.011 & -12 39 20.740 & 0.009 & 5.42 &0.04 &76.40\ G18.67+0.03 & 18 24 53.783 & 0.010 & -12 39 20.730 & 0.008 & 8.22 &0.06 &76.54\ G18.67+0.03 & 18 24 53.784 & 0.009 & -12 39 20.732 & 0.007 & 7.94 &0.05 &76.68\ G18.67+0.03 & 18 24 53.783 & 0.010 & -12 39 20.750 & 0.007 & 5.03 &0.04 &76.81

92 - 0.61 & 18 14 00.893 & 0.003 & -18 53 26.116 & 0.005 & 6.54 & 0.04 & 43.51\ G11.92 - 0.61 & 18 14 00.893 & 0.003 & -18 53 26.119 & 0.005 & 5.88 & 0.04 & 43.65\ G11.92 - 0.61 & 18 14 00.893 & 0.004 & -18 53 26.128 & 0.007 & 4.28 & 0.04 & 43.79\ G11.92 - 0.61 & 18 14 00.894 & 0.008 & -18 53 26.148 & 0.015 & 2.11 & 0.04 & 43.93\ G11.92 - 0.61 & 18 14 00.894 & 0.020 & -18 53 26.168 & 0.037 & 0.81 & 0.04 & 44.06\ G11.92 - 0.61 & 18 14 00.894 & 0.051 & -18 53 26.249 & 0.096 & 0.33 & 0.04 & 44.20\ G18.67 + 0.03 & 18 24 53.782 & 0.073 & -12 39 20.750 & 0.057 & 0.47 & 0.02 & 76.13\ G18.67 + 0.03 & 18 24 53.779 & 0.017 & -12 39 20.744 & 0.013 & 2.19 & 0.03 & 76.26\ G18.67 + 0.03 & 18 24 53.781 & 0.011 & -12 39 20.740 & 0.009 & 5.42 & 0.04 & 76.40\ G18.67 + 0.03 & 18 24 53.783 & 0.010 & -12 39 20.730 & 0.008 & 8.22 & 0.06 & 76.54\ G18.67 + 0.03 & 18 24 53.784 & 0.009 & -12 39 20.732 & 0.007 & 7.94 & 0.05 & 76.68\ G18.67 + 0.03 & 18 24 53.783 & 0.010 & -12 39 20.750 & 0.007 & 5.03 & 0.04 & 76.81

92-0.61 & 18 14 00.893 & 0.003 & -18 53 26.116 & 0.005 & 6.54 &0.04 &43.51\ G11.92-0.61 & 18 14 00.893 & 0.003 & -18 53 26.119 & 0.005 & 5.88 &0.04 &43.65\ G11.92-0.61 & 18 14 00.893 & 0.004 & -18 53 26.128 & 0.007 & 4.28 &0.04 &43.79\ G11.92-0.61 & 18 14 00.894 & 0.008 & -18 53 26.148 & 0.015 & 2.11 &0.04 &43.93\ G11.92-0.61 & 18 14 00.894 & 0.020 & -18 53 26.168 & 0.037 & 0.81 &0.04 &44.06\ G11.92-0.61 & 18 14 00.894 & 0.051 & -18 53 26.249 & 0.096 & 0.33 &0.04 &44.20\ G18.67+0.03 & 18 24 53.782 & 0.073 & -12 39 20.750 & 0.057 & 0.47 &0.02 &76.13\ G18.67+0.03 & 18 24 53.779 & 0.017 & -12 39 20.744 & 0.013 & 2.19 &0.03 &76.26\ G18.67+0.03 & 18 24 53.781 & 0.011 & -12 39 20.740 & 0.009 & 5.42 &0.04 &76.40\ J18.67+0.03 & 18 24 53.783 & 0.010 & -12 39 20.730 & 0.008 & 8.22 &0.06 &76.54\ G18.67+0.03 & 18 24 53.784 & 0.009 & -12 39 20.732 & 0.007 & 7.94 &0.05 &76.68\ N18.67+0.03 & 18 24 53.783 & 0.010 & -12 39 20.750 & 0.007 & 5.03 &0.04 &76.81

92-0.61 & 18 14 00.893 & 0.003 53 & 0.005 6.54 &0.04 &43.51\ & & -18 53 & 0.005 & &0.04 &43.65\ G11.92-0.61 & 18 14 & 0.004 & -18 53 26.128 & 0.007 & 4.28 &0.04 &43.79\ G11.92-0.61 18 14 00.894 & 0.008 & -18 53 26.148 & 0.015 & 2.11 &43.93\ & 14 & 0.020 & -18 53 26.168 & 0.037 & 0.81 &0.04 &44.06\ G11.92-0.61 & 18 14 & 0.051 & -18 53 26.249 & 0.096 0.33 &0.04 &44.20\ G18.67+0.03 18 24 53.782 & 0.073 -12 20.750 & & &0.02 G18.67+0.03 & 18 53.779 & 0.017 & -12 39 20.744 & 0.013 & 2.19 &0.03 &76.26\ G18.67+0.03 & 18 24 & 0.011 39 20.740 0.009 5.42 &76.40\ G18.67+0.03 & 53.783 & 0.010 & -12 39 & 8.22 &0.06 &76.54\ G18.67+0.03 & 18 24 & 0.009 -12 39 20.732 & 0.007 & &0.05 &76.68\ G18.67+0.03 & 18 24 53.783 & & -12 39 20.750 & 0.007 & 5.03 &0.04 &76.81

92-0.61 & 18 14 00.893 & 0.003 & -18 53 26.116 & 0.005 & 6.54 &0.04 &43.51\ G11.92-0.61 & 18 14 00.893 & 0.003 & -18 53 26.119 & 0.005 & 5.88 &0.04 &43.65\ G11.92-0.61 & 18 14 00.893 & 0.004 & -18 53 26.128 & 0.007 & 4.28 &0.04 &43.79\ G11.92-0.61 & 18 14 00.894 & 0.008 & -18 53 26.148 & 0.015 & 2.11 &0.04 &43.93\ G11.92-0.61 & 18 14 00.894 & 0.020 & -18 53 26.168 & 0.037 & 0.81 &0.04 &44.06\ G11.92-0.61 & 18 14 00.894 & 0.051 & -18 53 26.249 & 0.096 & 0.33 &0.04 &44.20\ G18.67+0.03 & 18 24 53.782 & 0.073 & -12 39 20.750 & 0.057 & 0.47 &0.02 &76.13\ G18.67+0.03 & 18 24 53.779 & 0.017 & -12 39 20.744 & 0.013 & 2.19 &0.03 &76.26\ G18.67+0.03 & 18 24 53.781 & 0.011 & -12 39 20.740 & 0.009 & 5.42 &0.04 &76.40\ G18.67+0.03 & 18 24 53.783 & 0.010 & -12 39 20.730 & 0.008 & 8.22 &0.06 &76.54\ G18.67+0.03 & 18 24 53.784 & 0.009 & -12 39 20.732 & 0.007 & 7.94 &0.05 &76.68\ G18.67+0.03 & 18 24 53.783 & 0.010 & -12 39 20.750 & 0.007 & 5.03 &0.04 &76.81

92-0.61 & 18 14 00.893 & 0 .003 & -18 53 2 6.1 16&0.00 5 &6.54 &0.04 &43 . 51\G11.92-0.61 & 18 14 00 .893&0 .003 &-18 5 3 26.11 9 & 0 .00 5&5.8 8& 0. 04 &4 3.6 5\ G11. 92-0.61 &1814 00.893 & 0. 0 04 & -18 5326. 128 & 0.007& 4 .28 &0 .0 4 & 4 3.79\ G1 1.92- 0.61 & 18 1400.894 &0. 0 08 & - 1 8 53 26 . 1 48 & 0 .015 & 2.11 &0.04 &4 3 .93\ G11.92-0. 61 & 1 81 40 0 .89 4 & 0.020 & - 18 53 2 6 .168 &0 .0 3 7 & 0 . 81 &0.04 &44. 06\ G11.92- 0 .61 & 1814 00 . 894 &0.051 & -18 53 26.249& 0. 096 & 0.3 3 &0.0 4 &44.20 \ G18.67 +0.03& 1 8 2 4 53 . 78 2& 0 .0 7 3 & -1 2 3 9 20 .750 & 0 .0 57 & 0. 47 & 0 . 0 2 &76 .13 \ G1 8.67+ 0.03 & 18 2453. 779& 0. 017 & -1239 2 0. 744 & 0.013 & 2. 19 &0.03 &76.26\G18. 67+0.03 & 18 2 4 5 3. 781 & 0.011& - 1239 20.7 40 & 0. 0 09&5 . 4 2&0.04 &76.40\ G18. 67 + 0 .0 3 & 18 2 4 53.7 8 3&0 .010 & - 12 39 20. 7 3 0 & 0 .008 &8.22 &0. 06 &76 . 54 \G18.67+ 0. 03 & 1 82453. 784 & 0.00 9 & -1 2 39 20. 732 & 0.007 & 7.94 & 0 .05 &76.68\ G 1 8. 6 7 +0 . 03 & 18 24 53.783& 0. 0 10 & -12 39 20 . 750 & 0.00 7& 5 . 03 &0.04 &76.81

92-0.61 &_18 14_00.893 & 0.003 &_-18 53_26.116_& 0.005_&_6.54 &0.04 &43.51\ G11.92-0.61_& 18 14_00.893 & 0.003 &_-18 53 26.119_&_0.005 & 5.88 &0.04 &43.65\ G11.92-0.61 & 18 14 00.893 & 0.004 & -18 53_26.128_& 0.007_&_4.28_&0.04 &43.79\ G11.92-0.61 & 18 14_00.894 & 0.008 & -18_53 26.148_& 0.015 & 2.11 &0.04 &43.93\ G11.92-0.61 & 18_14_00.894 & 0.020_& -18 53 26.168 & 0.037 & 0.81 &0.04_&44.06\ G11.92-0.61 & 18 14 00.894 &_0.051 & -18_53_26.249_& 0.096 & 0.33_&0.04 &44.20\ G18.67+0.03 & 18 24 53.782_& 0.073 & -12 39 20.750_& 0.057 & 0.47 &0.02 &76.13\ G18.67+0.03 &_18 24 53.779 & 0.017 &_-12 39 20.744 & 0.013_& 2.19_&0.03 &76.26\ G18.67+0.03 & 18 24_53.781 & 0.011_& -12_39 20.740 &_0.009 & 5.42 &0.04 &76.40\ G18.67+0.03 &_18 24 53.783_& 0.010 & -12 39 20.730_&_0.008 & 8.22_&0.06_&76.54\ G18.67+0.03_& 18_24 53.784 &_0.009_& -12_39_20.732 & 0.007 & 7.94 &0.05_&76.68\ G18.67+0.03_& 18 24 53.783 & 0.010 &_-12 39 20.750 &_0.007_& 5.03 &0.04 &76.81

]{}to. The [document-to-document]{}graph [[****]{}]{} has vertex set ${{\cal D}_{{{\rm init}}}}$ and weight function $${{\mathit wt}}^{{\mbox{d$\leftrightarrow$d}\xspace}}({u},{v}) = \begin{cases} {{\mbox{\it rflow}}({u},{v})} & \text{if ${v}\in {{{\mathit Nbhd}}({u}\,\vert\,{\delta},{{\cal D}_{{{\rm init}}}}-{\{{u}\}})}$}, \\ 0 & \text{otherwise}. \end{cases}$$ The [document-as-authority]{}graph [[****]{}]{} has vertex set ${{\cal D}_{{{\rm init}}}}\cup {{\mathit Cl({{\cal D}_{{{\rm init}}}})}}$ and a weight function such that positive-weight edges go only from clusters to documents: $${{\mathit wt}}^{{\mbox{c{$\rightarrow$}d}\xspace}}({u},{v}) = \\ \begin{cases} {{\mbox{\it rflow}}({u},{v})} & \text{if ${u}\in {{\mathit Cl({{\cal D}_{{{\rm init}}}})}}$ and} \\ & {\hspace*{0.1in}}{v}\in {{{\mathit Nbhd}}({u}\,\vert\,{\delta},{{\cal D}_{{{\rm init}}}})}, \\ 0 & \text{otherwise}. \end{cases}$$ The [document-as-hub]{}graph [[****]{}]{} has vertex set ${{\cal D}_{{{\rm init}}}}\cup {{\mathit Cl({{\cal D}_{{{\rm init}}}})}}$ and a weight function such that positive-weight edges go only from documents to clusters: $${{\mathit wt}}^{{\mbox{d{$\rightarrow$}c}\xspace}}({u},{v}) = \\ \begin{cases} {{\mbox{\it rflow}}({u},{v})} & \text{if ${u}\in {{\cal D}_{{{\rm init}}}}$ and} \\ & {\hspace*{0.1in}}{v}\in {{{\mathit Nbhd}}({u}\,\vert\,{\delta},{{\mathit Cl({{\cal D}_{{{\rm init}}}})}})}, \\ 0 & \text{otherwise}. \end{cases}$$ Since the latter two graphs are [one-way bipartite]{}, Theorem \[thm:pagerankBip\] applies to them. #### Clustering Method {#clustering-method

] { } to. The [ document - to - document]{}graph [ [ * * * * ] { } ] { } has vertex set $ { { \cal D}_{{{\rm init}}}}$ and weight affair $ $ { { \mathit wt}}^{{\mbox{d$\leftrightarrow$d}\xspace}}({u},{v }) = \begin{cases } { { \mbox{\it rflow}}({u},{v }) } & \text{if $ { v}\in { { { \mathit Nbhd}}({u}\,\vert\,{\delta},{{\cal D}_{{{\rm init}}}}-{\{{u}\}})}$ }, \\ 0 & \text{otherwise }. \end{cases}$$ The [ text file - as - authority]{}graph [ [ * * * * ] { } ] { } has vertex set $ { { \cal D}_{{{\rm init}}}}\cup { { \mathit Cl({{\cal D}_{{{\rm init}}}})}}$ and a system of weights routine such that positive - weight edge belong only from bunch to document: $ $ { { \mathit wt}}^{{\mbox{c{$\rightarrow$}d}\xspace}}({u},{v }) = \\ \begin{cases } { { \mbox{\it rflow}}({u},{v }) } & \text{if $ { u}\in { { \mathit Cl({{\cal D}_{{{\rm init}}}})}}$ and } \\ & { \hspace*{0.1in}}{v}\in { { { \mathit Nbhd}}({u}\,\vert\,{\delta},{{\cal D}_{{{\rm init } } } }) }, \\ 0 & \text{otherwise }. \end{cases}$$ The [ document - as - hub]{}graph [ [ * * * * ] { } ] { } suffer vertex set $ { { \cal D}_{{{\rm init}}}}\cup { { \mathit Cl({{\cal D}_{{{\rm init}}}})}}$ and a weight unit affair such that positive - weight edges go only from document to clusters: $ $ { { \mathit wt}}^{{\mbox{d{$\rightarrow$}c}\xspace}}({u},{v }) = \\ \begin{cases } { { \mbox{\it rflow}}({u},{v }) } & \text{if $ { u}\in { { \cal D}_{{{\rm init}}}}$ and } \\ & { \hspace*{0.1in}}{v}\in { { { \mathit Nbhd}}({u}\,\vert\,{\delta},{{\mathit Cl({{\cal D}_{{{\rm init } } } }) } }) }, \\ 0 & \text{otherwise }. \end{cases}$$ Since the latter two graphs are [ one - way bipartite ] { }, Theorem \[thm: pagerankBip\ ] put on to them. # # # # Clustering Method { # clustering - method

]{}to. Tje [document-to-document]{}grakh [[****]{}]{} has vertex sej ${{\xal D}_{{{\rk init}}}}$ and weieht function $${{\mathit wt}}^{{\mbox{d$\lxftrughtaerow$d}\xspace}}({u},{v}) = \begin{casds} {{\mbox{\it rflow}}({u},{v})} & \teet{if ${v}\in {{{\mathit Ighd}}({u}\,\verb\,{\belta},{{\dwl D}_{{{\xm init}}}}-{\{{u}\}})}$}, \\ 0 & \text{ptherwise}. \etd{cases}$$ The [docgmdnc-as-authority]{}graph [[****]{}]{} has vertex set ${{\caj D}_{{{\rm imih}}}}\cup {{\mathit Cl({{\sal C}_{{{\wm ihpt}}}})}}$ and a weight function such tgat posptive-weight edges go only from clusters to foculents: $${{\mathit wt}}^{{\mbod{c{$\rightarroq$}d}\xs[qce}}({u},{v}) = \\ \begin{zases} {{\mbox{\pc rflow}}({u},{v})} & \jext{if ${u}\in {{\mathit Cl({{\cal D}_{{{\rm init}}}})}}$ and} \\ & {\hspace*{0.1in}}{c}\ib {{{\mwjhit Nbhd}}({u}\,\vect\,{\deltw},{{\cal D}_{{{\rm inib}}}})}, \\ 0 & \teft{othereise}. \end{cases}$$ Thc [docnmenr-as-hub]{}graph [[****]{}]{} has vertxx set ${{\cal D}_{{{\rm init}}}}\cop {{\mathit Wl({{\eal D}_{{{\rm init}}}})}}$ and a weugyt futctimn sjxh ghau ppsjtive-wfigit edges go only from eocuments to clustets: $${{\nathit wt}}^{{\mbox{s{$\rightwrwow$}c}\xspace}}({u},{v}) = \\ \begin{cases} {{\mbox{\it rflow}}({u},{v})} & \tsxt{if ${u}\in {{\cal D}_{{{\rm init}}}}$ qnd} \\ & {\hspace*{0.1in}}{v}\in {{{\matjit Nbhd}}({u}\,\dert\,{\delta},{{\mathit Cl({{\cal D}_{{{\rm init}}}})}})}, \\ 0 & \text{otherwise}. \eng{casea}$$ Rinec gye latter two graphs are [one-way bipartite]{}, Theorqj \[uhm:iagerankBip\] applics to them. #### Clusteronh Kgthod {#clusterivg-method

]{}to. The [document-to-document]{}graph [[****]{}]{} has vertex set init}}}}$ weight function wt}}^{{\mbox{d$\leftrightarrow$d}\xspace}}({u},{v}) = \begin{cases} {{{\mathit D}_{{{\rm init}}}}-{\{{u}\}})}$}, \\ & \text{otherwise}. \end{cases}$$ [document-as-authority]{}graph [[****]{}]{} has vertex set ${{\cal init}}}}\cup {{\mathit Cl({{\cal D}_{{{\rm init}}}})}}$ and a weight function such that positive-weight edges only from clusters to documents: $${{\mathit wt}}^{{\mbox{c{$\rightarrow$}d}\xspace}}({u},{v}) = \\ \begin{cases} {{\mbox{\it rflow}}({u},{v})} & ${u}\in Cl({{\cal init}}}})}}$ \\ & {\hspace*{0.1in}}{v}\in {{{\mathit Nbhd}}({u}\,\vert\,{\delta},{{\cal D}_{{{\rm init}}}})}, \\ 0 & \text{otherwise}. \end{cases}$$ The [document-as-hub]{}graph [[****]{}]{} has set ${{\cal D}_{{{\rm init}}}}\cup {{\mathit Cl({{\cal D}_{{{\rm init}}}})}}$ a weight function such positive-weight edges go only from to $${{\mathit wt}}^{{\mbox{d{$\rightarrow$}c}\xspace}}({u},{v}) \\ {{\mbox{\it & \text{if ${u}\in D}_{{{\rm init}}}}$ and} \\ & {\hspace*{0.1in}}{v}\in {{{\mathit Nbhd}}({u}\,\vert\,{\delta},{{\mathit Cl({{\cal D}_{{{\rm init}}}})}})}, \\ 0 & \text{otherwise}. \end{cases}$$ Since latter two [one-way bipartite]{}, \[thm:pagerankBip\] to #### Clustering Method

]{}to. The [document-to-document]{}grAph [[****]{}]{} has vertEx set ${{\Cal d}_{{{\rm InIt}}}}$ anD weiGht function $${{\matHIt wt}}^{{\Mbox{d$\leftrightarrow$d}\xsPace}}({u},{V}) = \bEGin{cASeS} {{\mbox{\It rflow}}({U},{V})} & \tEXT{if ${V}\iN {{{\mAthIt nBhD}}({u}\,\verT\,{\deLta},{{\cal D}_{{{\Rm init}}}}-{\{{u}\}})}$}, \\ 0 & \texT{otHeRwise}. \end{caseS}$$ thE [document-aS-auThority]{}graph [[****]{}]{} Has Vertex SeT ${{\caL d}_{{{\rm inIt}}}}\cUp {{\matHit Cl({{\cAL D}_{{{\rm inIt}}}})}}$ and a weiGhT FunctiON such thAT PoSitiVe-weight edges go onLY fROm clusters to doCumentS: $${{\mAThIT Wt}}^{{\mBox{C{$\rightarroW$}d}\XspacE}}({U},{v}) = \\ \begin{CAsES} {{\MBox{\IT rflow}}({u},{v})} & \text{iF ${u}\in {{\mathit CL({{\Cal d}_{{{\rm iniT}}}})}}$ aNd} \\ & {\hSPace*{0.1in}}{V}\in {{{\maThIT NbHd}}({u}\,\vert\,{\deltA},{{\cal d}_{{{\rm init}}}})}, \\ 0 & \teXt{otheRWise}. \end{CAses}$$ The [DocumeNt-aS-huB]{}graPH [[****]{}]{} hAs VerTeX Set ${{\CAl d}_{{{\rm INit}}}}\Cup {{\mathiT CL({{\cAl D}_{{{\rm Init}}}})}}$ AND A WeigHt fUnctIon suCh that positivE-weIght EDgeS go onLy froM docUmEnts tO clustErs: $${{\maThIt wt}}^{{\mbox{d{$\rightaRrow$}C}\xspace}}({u},{v}) = \\ \BegIn{CasEs} {{\Mbox{\iT Rflow}}({u},{V})} & \teXt{iF ${u}\in {{\cal d}_{{{\rm init}}}}$ ANd} \\ & {\hSpACE*{0.1In}}{V}\in {{{\mathit Nbhd}}({u}\,\vert\,{\DeLTA},{{\mAthit Cl({{\cAl D}_{{{\rm iNIt}}}})}})}, \\ 0 & \TeXT{otherwiSe}. \End{CaseS}$$ sInce tHe laTTeR two grapHs are [oNE-wAy BipartiTe]{}, theoreM \[tHm:pAgeRankBIP\] appLies to Them. #### ClusTerinG method {#clusteriNG-method

]{}to. The [document-to-d ocument]{} graph [[ *** *] {}]{ } ha s vertex set $ { {\ca l D}_{{{\rm init}}}}$and w ei g ht f u nc tion$${{\ma t hi t wt} }^ {{ \mb ox { d$ \left rig htarrow $d}\xspace }}( {u },{v}) = \be g in {cases} { {\m box{\it rflo w}} ({u},{ v} )}& \tex t{i f ${v }\in { { {\math it Nbhd}} ({ u }\,\ve r t\,{\de l t a} ,{{\ cal D}_{{{\rm ini t }} } }-{\{{u}\}})}$ }, \\ 0 &\ t ext {ot herwise}.\e nd{ca s es}$$ T he [ d ocu m ent-as-author ity]{}graph [[* ***]{} ]{ } h a s vert ex se t$ {{\ cal D}_{{{\ rm i nit}}}}\c up {{\ m athit C l ({{\cal D}_{{ {\r m i nit} } }} )} }$an d aw ei ght fun ction su ch t hat p osit i v e - weig htedge s goonly from clu ste rs t o do cumen ts: $ ${{\ ma thitwt}}^{ {\mbo x{ c{$\rightarrow$ }d}\ xspace}}( {u} ,{ v}) = \\ \ b egin{c ase s} {{\mbo x{\it r f low }} ( { u }, {v})} & \text{if $ {u } \ in {{\math it Cl( { {\ ca l D}_{{{\ rm in it}} } } )}}$ and } \ \ & {\h space* { 0. 1i n}}{v}\ in {{{\m at hit Nb hd}}( { u}\, \vert\ ,{\delta },{{\ c al D}_{{{\rm i n it}}}})}, \\0& \t e xt{o the rwise}. \en d{ca s es}$ $ T h e[do c ument -as-h ub ] {} g raph [[****]{}]{} h as verte x set ${{\cal D}_{ {{\rm init } } } }\cup {{ \mat h it Cl({{\cal D}_{ {{\rm init}}}}) } }$ and a weig ht funct ion sucht h at posit ive -we igh t e d g es go only from d ocum en ts to c lus ters: $ ${{ \ma thi t w t} }^{{\mbox {d{$\rig ht ar ro w$ }c} \xspa c e}}({u}, {v })=\\\begi n {cases } {{ \mbo x{ \i t rf low}}({ u }, { v })}&\t ext{ if${ u}\in {{\ c alD}_{{{\ rm init}} }}$ and }\\ & {\h space*{0.1in} }{ v}\in {{{\ ma thi t Nbhd } } ({u}\,\v ert\,{\delta},{{\mathit Cl({{\c alD}_{{ {\rm init}}}} )}} )}, \\ 0 & \tex t{othe rwise }. \e n d {case s } $$ S in ce the lat t e r t wo gr ap hs a re [one -way bipartite]{}, The orem \[thm:pa ger ankB i p \] ap p li e s t ot hem . #### Clusterin g Method { #c l us tering-met h od

]{}to. The [document-to-document]{}graph_[[****]{}]{} has_vertex set ${{\cal D}_{{{\rm_init}}}}$ and_weight_function $${{\mathit_wt}}^{{\mbox{d$\leftrightarrow$d}\xspace}}({u},{v})_= \begin{cases} {{\mbox{\it rflow}}({u},{v})}_& \text{if ${v}\in_{{{\mathit Nbhd}}({u}\,\vert\,{\delta},{{\cal D}_{{{\rm init}}}}-{\{{u}\}})}$},_\\ 0 &_\text{otherwise}. \end{cases}$$ The_[document-as-authority]{}graph [[****]{}]{} has vertex set ${{\cal D}_{{{\rm init}}}}\cup {{\mathit Cl({{\cal D}_{{{\rm init}}}})}}$ and a_weight_function such_that_positive-weight_edges go only from clusters_to documents: $${{\mathit wt}}^{{\mbox{c{$\rightarrow$}d}\xspace}}({u},{v}) =_\\ \begin{cases} {{\mbox{\it_rflow}}({u},{v})} & \text{if ${u}\in {{\mathit Cl({{\cal D}_{{{\rm init}}}})}}$ _and}_\\ & {\hspace*{0.1in}}{v}\in_{{{\mathit Nbhd}}({u}\,\vert\,{\delta},{{\cal D}_{{{\rm init}}}})}, \\ 0 & \text{otherwise}. \end{cases}$$ The [document-as-hub]{}graph_[[****]{}]{} has vertex set ${{\cal D}_{{{\rm_init}}}}\cup {{\mathit Cl({{\cal_D}_{{{\rm_init}}}})}}$_and a weight function_such that positive-weight edges go only_from documents to clusters: $${{\mathit wt}}^{{\mbox{d{$\rightarrow$}c}\xspace}}({u},{v})_= \\ \begin{cases} {{\mbox{\it rflow}}({u},{v})} & \text{if ${u}\in_{{\cal D}_{{{\rm init}}}}$ and} \\ &_{\hspace*{0.1in}}{v}\in {{{\mathit Nbhd}}({u}\,\vert\,{\delta},{{\mathit Cl({{\cal D}_{{{\rm_init}}}})}})}, \\ _0 & \text{otherwise}. \end{cases}$$ Since the latter_two graphs are_[one-way bipartite]{},_Theorem \[thm:pagerankBip\] applies_to them. #### Clustering Method {#clustering-method

for $% k^{\prime }>2$. We take $B_0^{+-}=0$. The Eqs. (\[rg1\],\[rg2\]) are solved iteratively. We continue iterating the equations till $% B_{n+1}^{*}(k^{\prime })\approx B_n^{*}(k^{\prime })$, that is, till the solution converges. For $B_0^{initial}=1.0,$ the $B_n^{\prime }$s for various $n$ ranging from $0 \ldots 3$ is shown in Figure 1. Here the convergence is very fast, and after $n=3-4$ iterations $B_n^{*}(k)$ converges to an universal function $$f(k^{\prime })=1.24*k^{\prime -0.32}.$$ From the above arguments, we have shown that $B_n^{*}(k^{\prime })\ $is approximately proportional to $k^{\prime -1/3}$. The other parameter $% B_n^{*+-}(k^{\prime })$ remains close to zero. We infer from the above analysis that the mean magnetic field scales as $% k^{-1/3}$, and the energy spectra scales as $k^{-5/3}$. Essentially, the scaling of $B_0$ leads to $k^{-5/3}$ energy (Kolmogorov-like) spectra in our scheme. We have calculated $B_n^{*}(k^{\prime })$ for $B_0^{initial}=1,2,10$ and found that for large $n$, $B_n^{*}(k^{\prime })\approx 1.25B_0^{initial}k^{\prime -1/3}$ or $$B_n(k)=1.25B_0^{initial}K^{1/2}\Pi ^{1/3}k^{-1/3}.$$ Calculation of $K$ ------------------ We can calculate the Kolmogorov’s constant for MHD turbulence $K$ by calculating the cascade rate $\Pi $ [@Lesl]. In MHD the cascade rates are $$\Pi ^{+}(k)=\Pi ^{-}(k)=-\int_0^kdk^{\prime }T(k^{\prime })$$ The numerical solution of the cascade rate integral yields [@Lesl] $$\label{alpha}\frac{1.24B_0^{initial}}{K^{3/2}}=3.85$$ From the above

for $% k^{\prime } > 2$. We take $ B_0^{+-}=0$. The Eqs. (\[rg1\],\[rg2\ ]) are solved iteratively. We continue iterating the equation till $% B_{n+1}^{*}(k^{\prime }) \approx B_n^{*}(k^{\prime }) $, that is, till the solution converges. For $ B_0^{initial}=1.0,$ the $ B_n^{\prime } $ south for various $ n$ ranging from $ 0 \ldots 3 $ is shown in Figure 1. Here the overlap is very fast, and after $ n=3 - 4 $ iterations $ B_n^{*}(k)$ converges to an universal affair $ $ f(k^{\prime }) = 1.24*k^{\prime -0.32}.$$ From the above arguments, we have shown that $ B_n^{*}(k^{\prime }) \ $ is approximately proportional to $ k^{\prime -1/3}$. The early argument $% B_n^{*+-}(k^{\prime }) $ remains close to zero. We guess from the above analysis that the mean magnetic playing field scales as $% k^{-1/3}$, and the department of energy spectra scale as $ k^{-5/3}$. basically, the scaling of $ B_0 $ leads to $ k^{-5/3}$ energy (Kolmogorov - like) spectrum in our scheme. We have calculated $ B_n^{*}(k^{\prime }) $ for $ B_0^{initial}=1,2,10 $ and find that for big $ n$, $ B_n^{*}(k^{\prime }) \approx 1.25B_0^{initial}k^{\prime -1/3}$ or $ $ B_n(k)=1.25B_0^{initial}K^{1/2}\Pi ^{1/3}k^{-1/3}.$$ Calculation of $ K$ ------------------ We can calculate the Kolmogorov ’s constant for MHD turbulence $ K$ by forecast the cascade rate $ \Pi $ [ @Lesl ]. In MHD the cascade pace are $ $ \Pi ^{+}(k)=\Pi ^{-}(k)=-\int_0^kdk^{\prime } T(k^{\prime }) $ $ The numerical solution of the cascade rate integral yields [ @Lesl ] $ $ \label{alpha}\frac{1.24B_0^{initial}}{K^{3/2}}=3.85$$ From the above

fog $% k^{\prime }>2$. We take $B_0^{+-}=0$. The Tqs. (\[rg1\],\[rg2\]) are solvgd iteravively. Se contivue iterating the equations vill $% B_{n+1}^{*}(k^{\peime })\approx B_n^{*}(k^{\prime })$, ghat is, tpll the silutmon converges. Foc $B_0^{initial}=1.0,$ the $N_n^{\prikx }$s for various $n$ ranging from $0 \ldots 3$ hs skown in Figure 1. Here the convergence is veru vast, and after $n=3-4$ iuerwtiohs $B_n^{*}(k)$ converges to an universal fhnction $$f(k^{\prime })=1.24*k^{\prike -0.32}.$$ From the above argumenhs, wf have shown that $H_n^{*}(k^{\prime })\ $iw ap[eoximately pfoportional to $k^{\prime -1/3}$. The other parameter $% B_n^{*+-}(k^{\prime })$ femaius close to zwro. Ag infer from the wbove analyslx that the mesn magnetic ficld sralew as $% k^{-1/3}$, and the energy spectra scales as $k^{-5/3}$. Essenthamly, the scaling od $V_0$ leags tm $k^{-5/3}$ dberey (Iokmkgorov-pikx) spectra ih our schemw. We have calculatec $F_b^{*}(k^{\prime })$ for $G_0^{initiwl}=1,2,10$ and found that for large $n$, $B_n^{*}(k^{\prime })\apkrox 1.25B_0^{jnitial}k^{\prime -1/3}$ or $$B_n(k)=1.25B_0^{ibitial}K^{1/2}\Pi ^{1/3}k^{-1/3}.$$ Calculatioj of $K$ ------------------ We san calculate the Kolmogorov’s constant for MHD tusbuleicd $K$ bn cauxupating the cascade rate $\Pi $ [@Lesl]. In MHD the cwacsdv rates are $$\Pi ^{+}(k)=\Pl ^{-}(k)=-\int_0^kdk^{\prime }T(k^{\ptile })$$ The numericau solucjoh of the cascade rwte intggral tields [@Lefl] $$\lsbel{alpha}\frac{1.24B_0^{initial}}{K^{3/2}}=3.85$$ From the above

for $% k^{\prime }>2$. We take $B_0^{+-}=0$. (\[rg1\],\[rg2\]) solved iteratively. continue iterating the B_n^{*}(k^{\prime that is, till solution converges. For the $B_n^{\prime }$s for various $n$ from $0 \ldots 3$ is shown in Figure 1. Here the convergence is fast, and after $n=3-4$ iterations $B_n^{*}(k)$ converges to an universal function $$f(k^{\prime })=1.24*k^{\prime From above we shown that $B_n^{*}(k^{\prime })\ $is approximately proportional to $k^{\prime -1/3}$. The other parameter $% B_n^{*+-}(k^{\prime })$ close to zero. We infer from the above that the mean magnetic scales as $% k^{-1/3}$, and energy scales as Essentially, scaling $B_0$ leads to energy (Kolmogorov-like) spectra in our scheme. We have calculated $B_n^{*}(k^{\prime })$ for $B_0^{initial}=1,2,10$ and found that for $n$, $B_n^{*}(k^{\prime -1/3}$ or ^{1/3}k^{-1/3}.$$ of ------------------ We can Kolmogorov’s constant for MHD turbulence $K$ cascade rate $\Pi $ [@Lesl]. In MHD the rates are ^{+}(k)=\Pi ^{-}(k)=-\int_0^kdk^{\prime }T(k^{\prime })$$ The numerical of the cascade rate integral yields [@Lesl] $$\label{alpha}\frac{1.24B_0^{initial}}{K^{3/2}}=3.85$$ the above

for $% k^{\prime }>2$. We take $B_0^{+-}=0$. The Eqs. (\[rg1\],\[rG2\]) are solved IteraTivEly. we ContInue Iterating the eqUAtioNs till $% B_{n+1}^{*}(k^{\prime })\approx B_n^{*}(K^{\primE })$, tHAt is, TIlL the sOlution COnVERgeS. FOr $b_0^{inItIAl}=1.0,$ The $B_n^{\PriMe }$s for vArious $n$ ranGinG fRom $0 \ldots 3$ is shOWn In Figure 1. HeRe tHe convergencE is Very faSt, And AFter $n=3-4$ IteRatioNs $B_n^{*}(k)$ cONvergeS to an univErSAl funcTIon $$f(k^{\prIME })=1.24*k^{\PrimE -0.32}.$$ From the above arguMEnTS, we have shown thAt $B_n^{*}(k^{\pRiME })\ $iS APprOxiMately propOrTionaL To $k^{\primE -1/3}$. thE OTHer PArameter $% B_n^{*+-}(k^{\prIme })$ remains cLOse To zero. we InfER from tHe aboVe ANalYsis that the Mean Magnetic fIeld scALes as $% k^{-1/3}$, aND the eneRgy speCtrA scAles AS $k^{-5/3}$. esSenTiALly, THe ScaLIng Of $B_0$ leads To $K^{-5/3}$ eNergy (kolmOGOROv-liKe) sPectRa in oUr scheme. We havE caLculATed $b_n^{*}(k^{\prIme })$ foR $B_0^{inItIal}=1,2,10$ anD found That fOr Large $n$, $B_n^{*}(k^{\prime })\aPproX 1.25B_0^{initial}K^{\prImE -1/3}$ or $$b_n(K)=1.25B_0^{iniTIal}K^{1/2}\Pi ^{1/3}K^{-1/3}.$$ CaLcuLation oF $K$ ------------------ We can CAlcUlATE ThE Kolmogorov’s constaNt FOR MhD turbulEnce $K$ bY CaLcULating thE cAscAde rATE $\Pi $ [@LeSl]. In mhD The cascaDe rateS ArE $$\PI ^{+}(k)=\Pi ^{-}(k)=-\inT_0^kDk^{\primE }T(K^{\prIme })$$ the nuMEricAl soluTion of thE cascADe rate integral YIelds [@Lesl] $$\labeL{AlPHA}\fRAc{1.24B_0^{iNitIal}}{K^{3/2}}=3.85$$ From the AbovE

for $% k^{\prime }>2$. We take $B_0 ^{+-} =0$ . T he Eqs . (\ [rg1\],\[rg2\] ) are solved iteratively. W e con ti n ue i t er ating the eq u at i o nsti ll $% B _ {n +1}^{ *}( k^{\pri me })\appr oxB_ n^{*}(k^{\pr i me })$, that is , till the s olu tion c on ver g es. F or$B_0^ {initi a l}=1.0 ,$ the $B _n ^ {\prim e }$s fo r va riou s $n$ ranging fro m $ 0 \ldots 3$ isshownin Fi g u re1.Here the c on verge n ce is v e ry f a st, and after $n= 3-4$ iterat i ons $B_n^ {* }(k ) $ conv erges t o an universalfunc tion $$f( k^{\pr i me })=1 . 24*k^{\ prime-0. 32} .$$F ro mthe a b ove ar gum e nts , we hav esh own t hat$ B _ n ^{*} (k^ {\pr ime } )\ $is approx ima tely pro porti onalto $ k^ {\pri me -1/ 3}$.Th e other paramet er $ % B_n^{*+ -}( k^ {\p ri me }) $ remai nsclo se to z ero. W e in fe r f ro m the above analys is t ha t the me an mag n et ic field sc al esas $ % k^{-1 /3}$ , a nd the e nergys pe ct ra scal es as $k ^{ -5/ 3}$ . Ess e ntia lly, t he scali ng of $B_0$ leads to $k^{-5/3}$ en e rg y (K o lmog oro v-like) spe ctra in o ur s c he me. We ha ve ca lc u la t ed $B_n^{*}(k^{\pri me })$ f or $B _0^{initial}= 1,2,10$ an d f ound tha t fo r l a rge $n$, $B_n^ {*}(k ^{\prime } ) \approx1.25B _0^{init ial}k^{\p r i me -1/3} $ o r $ $B_ n(k ) = 1. 25B_0^{initia l } K^{1 /2 }\Pi ^{ 1/3 }k^{-1/ 3}. $$ Ca lcu la tion of $ K$ ----- -- -- -- -- --- -- W e can cal cu lat ethe Kolm o gorov’ s con stan tfo r MH D turbu l en c e $K$ b ycalc ula ti ng th e ca s cad e rate$\Pi $ [@ Les l ]. I nMH D the c ascade ratesar e $$\Pi ^{ +} (k) =\Pi ^ { - }(k)=-\i nt_0^kdk^{\prime }T(k^{ \ prime } )$$ Thenume rical sol uti on ofthe cascad e rate inte gr aly i elds[ @ Le sl] $ $\label{al p h a}\ frac{ 1. 24B_ 0^{init ial}}{K^{3/2}}=3.8 5 $$From the abov e

for_$% k^{\prime }>2$._We take $B_0^{+-}=0$. The_Eqs. (\[rg1\],\[rg2\])_are_solved iteratively._We_continue iterating the_equations till $% B_{n+1}^{*}(k^{\prime_})\approx B_n^{*}(k^{\prime })$, that_is, till the_solution_converges. For $B_0^{initial}=1.0,$ the $B_n^{\prime }$s for various $n$ ranging from $0 \ldots 3$_is_shown in_Figure_1._Here the convergence is very_fast, and after $n=3-4$ iterations_$B_n^{*}(k)$ converges_to an universal function $$f(k^{\prime })=1.24*k^{\prime -0.32}.$$ From_the_above arguments, we_have shown that $B_n^{*}(k^{\prime })\ $is approximately proportional to_$k^{\prime -1/3}$. The other parameter $% B_n^{*+-}(k^{\prime_})$ remains close_to_zero. We_infer from the above_analysis that the mean magnetic field_scales as $% k^{-1/3}$, and the energy_spectra scales as $k^{-5/3}$. Essentially, the scaling_of $B_0$ leads to $k^{-5/3}$ energy_(Kolmogorov-like) spectra in our scheme._We have_calculated $B_n^{*}(k^{\prime })$ for $B_0^{initial}=1,2,10$_and found that_for large_$n$, $B_n^{*}(k^{\prime })\approx 1.25B_0^{initial}k^{\prime_-1/3}$ or $$B_n(k)=1.25B_0^{initial}K^{1/2}\Pi ^{1/3}k^{-1/3}.$$ Calculation of $K$ ------------------ We_can calculate the_Kolmogorov’s constant for MHD turbulence $K$_by_calculating the cascade_rate_$\Pi_$ [@Lesl]._In MHD the_cascade_rates are_$$\Pi_^{+}(k)=\Pi ^{-}(k)=-\int_0^kdk^{\prime }T(k^{\prime })$$ The numerical_solution_of the cascade rate integral yields [@Lesl]_$$\label{alpha}\frac{1.24B_0^{initial}}{K^{3/2}}=3.85$$ From the above