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!:=\!\bar{g}(x^{\pm 1}_1,\ldots,x^{\pm 1})$ has a root $\mu_0\!\in\!({\mathbb{Z}}/p^{2L+1}{\mathbb{Z}})^n\setminus\{{\mathbf{O}}\}$ for some choice of signs, some choice of $i$, and some choice of initial term polynomial $\bar{f}$ of $f$ so that $\bar{g}(x)\!=\!x^{-a_i}\bar{f}(x)$. Writing $\bar{h}(x)\!=\!\gamma_0+\gam... | !: = \!\bar{g}(x^{\pm 1}_1,\ldots, x^{\pm 1})$ has a root $ \mu_0\!\in\!({\mathbb{Z}}/p^{2L+1}{\mathbb{Z}})^n\setminus\{{\mathbf{O}}\}$ for some choice of signs, some option of $ i$, and some option of initial terminus polynomial $ \bar{f}$ of $ f$ so that $ \bar{g}(x)\!=\!x^{-a_i}\bar{f}(x)$. Writing $ \bar{h}(x)\!=\!... | !:=\!\bar{h}(x^{\pm 1}_1,\ldots,x^{\pm 1})$ has a roou $\mu_0\!\in\!({\mathbb{Z}}/p^{2L+1}{\majhvb{Z}})^n\sevminus\{{\mzthbf{O}}\}$ fur some choice of signs, some cyoice of $i$, and some choice uf initiap term pilyninial $\bar{f}$ of $f$ so bkat $\bzv{g}(x)\!=\!x^{-a_n}\ber{f}(x)$. Writing $\bat{h}(x)\!=\!\gamma_0+\gam... | !:=\!\bar{g}(x^{\pm 1}_1,\ldots,x^{\pm 1})$ has a root $\mu_0\!\in\!({\mathbb{Z}}/p^{2L+1}{\mathbb{Z}})^n\setminus\{{\mathbf{O}}\}$ choice signs, some of $i$, and polynomial of $f$ so $\bar{g}(x)\!=\!x^{-a_i}\bar{f}(x)$. Writing $\bar{h}(x)\!=\!\gamma_0+\gamma_{i_1}x^{\alpha_{i_1}}+\cdots+ as before, it is clear that \... | !:=\!\bar{g}(x^{\pm 1}_1,\ldots,x^{\pm 1})$ has a root $\mu_0\!\In\!({\mathbb{Z}}/p^{2l+1}{\mathBb{Z}})^N\seTmInus\{{\MathBf{O}}\}$ for some choiCE of sIgns, some choice of $i$, and soMe choIcE Of inITiAl terM polynoMIaL $\BAr{f}$ Of $F$ sO thAt $\BAr{G}(x)\!=\!x^{-a_i}\Bar{F}(x)$. WritiNg $\bar{h}(x)\!=\!\gamMa_0+\gAm... | !:=\!\bar{g}(x^{\pm 1}_1,\ ldots,x^{\ pm 1} )$has a roo t $\ mu_0\!\in\!({\ m athb b{Z}}/p^{2L+1}{\mathbb {Z}}) ^n \ setm i nu s\{{\ mathbf{ O }} \ } $ f or s ome c h oi ce of si gns, so me choiceof$i $, and somec ho ice of ini tia l term polyn omi al $\b ar {f} $ of $ f$so th at $\b a r{g}(x )\!=\!x^{ -a _ i}\ba... | !:=\!\bar{g}(x^{\pm 1}_1,\ldots,x^{\pm_1})$ has_a root $\mu_0\!\in\!({\mathbb{Z}}/p^{2L+1}{\mathbb{Z}})^n\setminus\{{\mathbf{O}}\}$ for_some choice_of_signs, some_choice_of $i$, and_some choice of_initial term polynomial $\bar{f}$_of $f$ so_that_$\bar{g}(x)\!=\!x^{-a_i}\bar{f}(x)$. Writing $\bar{h}(x)\!=\!\gamma_0+\gam... |
the data where different data sets from the literature have different calibration errors. Also, these discrepancies could be in part attributed to the internal dynamics of the observed galaxies because of the different fiber and/or slit placements. We can improve the fit even further by restricting ourselves to the la... | the data where different data sets from the literature get unlike calibration errors. Also, these discrepancy could be in function attributed to the internal moral force of the observed galaxies because of the different roughage and/or slit placements. We can improve the burst even further by restricting ourselves to t... | thf data where different dxta sets from tkw litecature gave difwerent calibration errors. Aldo, thest discrepancies couud be in iart attrubuttd to the internal dynamigf of bhe oysxrved galaxies necause of dhe different xicex and/or slit placements. We can improde the gih even further by geftridnikg ourselves to the la... | the data where different data sets from have calibration errors. these discrepancies could the dynamics of the galaxies because of different fiber and/or slit placements. We improve the fit even further by restricting ourselves to the largest homogeneous dataset the literature [*i.e.*]{}, the 63 new redshifts measureme... | the data where different data Sets from thE liteRatUre HaVe diFferEnt calibration ERrorS. Also, these discrepancieS coulD bE In paRT aTtribUted to tHE iNTErnAl DyNamIcS Of The obSerVed galaXies becausE of ThE different fiBEr And/or slit pLacEments. We can iMprOve the FiT evEN furtHer By resTrictiNG ourseLves to the La... | the data where differentdata setsfromthe li te ratu re h ave differentc alib ration errors. Also, t hesedi s crep a nc ies c ould be in p art a tt rib ut e dto th e i nternal dynamicsofth e observed g a la xies becau seof the diffe ren t fibe rand / or sl itplace ments. We can improveth e fit e v en furt h e rby ... | the_data where_different data sets from_the literature_have_different calibration_errors._Also, these discrepancies_could be in_part attributed to the_internal dynamics of_the_observed galaxies because of the different fiber and/or slit placements. We can improve the_fit_even further_by_restricting_ourselves to the la... |
1)}
\right)\right\}\right.
\nonumber \\
& & \hspace{-7.5cm}
\times \left. \exp\left\{\sum_{n=0}^{L-1}\mbox{ln}
\left( 1 + A_{n}^{2}\sin^{2}\theta_{n}^{(2)} + A_{n}\sin2\theta_{n}^{(2)}
\right)\right\}\right> \;.
\label{eq:r1r2a}\end{aligned}$$ Using the weak disorder condition we can expand the logarithms and present ... | 1) }
\right)\right\}\right.
\nonumber \\
& & \hspace{-7.5 cm }
\times \left. \exp\left\{\sum_{n=0}^{L-1}\mbox{ln }
\left (1 + A_{n}^{2}\sin^{2}\theta_{n}^{(2) } + A_{n}\sin2\theta_{n}^{(2) }
\right)\right\}\right > \; .
\label{eq: r1r2a}\end{aligned}$$ Using the weak disorder condition we can elaborate ... | 1)}
\rigjt)\right\}\right.
\nonumber \\
& & \hrpace{-7.5cm}
\times \lgfr. \exp\lxft\{\sum_{n=0}^{M-1}\mbox{ln}
\ldft( 1 + A_{n}^{2}\sin^{2}\theta_{n}^{(2)} + A_{n}\sin2\thete_{n}^{(2)}
\ritht)\ritht\}\right> \;.
\label{eq:r1r2a}\end{xligned}$$ Uding the weaj disorder rknditiok we dwn erpend the logaritmms and prevent ... | 1)} \right)\right\}\right. \nonumber \\ & & \hspace{-7.5cm} \exp\left\{\sum_{n=0}^{L-1}\mbox{ln} 1 + + A_{n}\sin2\theta_{n}^{(2)} \right)\right\}\right> disorder we can expand logarithms and present correlator $\left <r_{1, L}^{2}r_{2, L}^{2}\right>$ as $$\begin{aligned} \left<r_{1, L}^{2}r_{2, L}^{2}\right> & = & \lef... | 1)}
\right)\right\}\right.
\nonumber \\
& & \hsPace{-7.5cm}
\timeS \left. \Exp\LefT\{\sUm_{n=0}^{L-1}\Mbox{Ln}
\left( 1 + A_{n}^{2}\sin^{2}\theTA_{n}^{(2)} + A_{n}\Sin2\theta_{n}^{(2)}
\right)\right\}\rigHt> \;.
\labEl{EQ:r1r2a}\ENd{AlignEd}$$ Using THe WEAk dIsOrDer CoNDiTion wE caN expand The logaritHms AnD present ... | 1)}
\right)\right\}\right.
\nonumber \\
& &\hs pa ce{- 7.5c m}
\times \le f t. \ exp\left\{\sum_{n=0}^{ L-1}\ mb o x{ln }
\ left( 1 + A_ { n} ^ { 2}\ si n^ {2} \t h et a_{n} ^{( 2)} + A _{n}\sin2\ the ta _{n}^{(2)}
\ r ig ht)\right\ }\r ight> \;.
\l abe l{eq:r 1r 2a} \ end{a lig ned}$ $ Usin g the w eak disor de... | 1)}
\right)\right\}\right.
\nonumber \\
&_& \hspace{-7.5cm}
\times_ \left. \exp\left\{\sum_{n=0}^{L-1}\mbox{ln}
\left( 1_+ A_{n}^{2}\sin^{2}\theta_{n}^{(2)}_+_A_{n}\sin2\theta_{n}^{(2)}
\right)\right\}\right> \;.
\label{eq:r1r2a}\end{aligned}$$_Using_the weak disorder_condition we can_expand the logarithms and_present ... |
$(X, Y, g, f)$ already existed as a scenario, so we just need to update $g$ and $f$ (with the new and better values) and possibly $Y$ (if we want to fix the observed behavior).
#### Logical formulae and constraints.
Commonly, constraints can be directly expressed in the adaptation domain. Suppose we build a system a... | $ (X, Y, g, f)$ already existed as a scenario, so we just necessitate to update $ g$ and $ f$ (with the newfangled and better values) and possibly $ Y$ (if we want to fasten the observed behavior).
# # # # legitimate formulae and constraints.
normally, constraint can be directly express in the adaptation domain. ... | $(X, J, g, f)$ already existed as a scenario, so cw just need fo updatd $g$ and $f$ (with the new and bxttee valyes) and possibly $Y$ (if de want tl fix thw obwwrved behatjor).
#### Logleal fkvmulaz end constraints.
Gommonly, cotstraints can te dnrectly expressed in the adaptation qomain. Xuopose we build a sjseem z... | $(X, Y, g, f)$ already existed as so just need update $g$ and better and possibly $Y$ we want to the observed behavior). #### Logical formulae constraints. Commonly, constraints can be directly expressed in the adaptation domain. Suppose we a system against an adaptation domain $\mathcal{A} = \{(E_1, \gamma_1, \phi_1),... | $(X, Y, g, f)$ already existed as a scenArio, so we juSt neeD to UpdAtE $g$ anD $f$ (wiTh the new and betTEr vaLues) and possibly $Y$ (if we waNt to fIx THe obSErVed beHavior).
#### LOGiCAL foRmUlAe aNd COnStraiNts.
commonlY, constrainTs cAn Be directly exPReSsed in the aDapTation domain. supPose we BuIld A SysteM a... | $(X, Y, g, f)$ already ex isted as a scen ari o,so wejust need to updat e $g$ and $f$ (with the new andbe t terv al ues)and pos s ib l y $Y $(i f w ew an t tofix the ob served beh avi or ).
#### Log i ca l formulae an d constraint s.
Commo nl y,c onstr ain ts ca n be d i rectly expresse di n thea daptati o ... | $(X,_Y, g,_f)$ already existed as_a scenario,_so_we just_need_to update $g$_and $f$ (with_the new and better_values) and possibly_$Y$_(if we want to fix the observed behavior).
#### Logical formulae and constraints.
Commonly, constraints can_be_directly expressed_in_the_adaptation domain. Suppose we build_a system a... |
ars that have formed by the time that the simulation is stopped have yet commenced hydrogen burning. This justifies neglecting the effects of protostellar feedback in this study. Heating of the dust due to the significant accretion luminosities of the newly-formed protostars will occur (Krumholz 2006), but is unlikely ... | ars that have formed by the time that the simulation is stop have so far commenced hydrogen burning. This justifies neglecting the impression of protostellar feedback in this study. Heating of the debris due to the significant accretion luminosities of the newly - formed protostars will happen (Krumholz 2006), but is i... | ars that have formed by the time that the simulatmon is atopped fave yet commenced hydrogen uurnung. Tyis justifies neglectivg the efvects of prouostellar feedback in this studg. Heaciig of the dust cue to the significant awcfecion luminosities of the newly-formed protosyags will occur (Hrumnjlz 2006), but is unlikely ... | ars that have formed by the time simulation stopped have commenced hydrogen burning. of feedback in this Heating of the due to the significant accretion luminosities the newly-formed protostars will occur (Krumholz 2006), but is unlikely to be important, the temperature of the dust at the onset of dust-induced cooling ... | ars that have formed by the timE that the siMulatIon Is sToPped Have Yet commenced hyDRogeN burning. This justifies nEglecTiNG the EFfEcts oF protosTElLAR feEdBaCk iN tHIs Study. heaTing of tHe dust due tO thE sIgnificant acCReTion luminoSitIes of the newlY-foRmed prOtOstARs wilL ocCur (KrUmholz 2006), BUt is unLikely ... | ars that have formed by th e time tha t the si mul at ionis s topped have ye t com menced hydrogen burnin g. Th is just i fi es ne glectin g t h e ef fe ct s o fp ro toste lla r feedb ack in thi s s tu dy. Heatingo fthe dust d ueto the signi fic ant ac cr eti o n lum ino sitie s of t h e newl y-formedpr o tostar s ... | ars that_have formed_by the time that_the simulation_is_stopped have_yet_commenced hydrogen burning._This justifies neglecting_the effects of protostellar_feedback in this_study._Heating of the dust due to the significant accretion luminosities of the newly-formed protostars_will_occur (Krumholz_2006),_but_is unlikely ... |
\text{ \large 0}& (1-\beta_1)D\\
\end{smallmatrix} \right)$ for suitable $l$ and $T^l$ is a scalar matrix by Lemma \[m7\]. It means that $1-\beta_1^{2}=0$ because if $B=0$ then $T$ is reducible and it can not generate ${\mathrm{Sin}}_{n_1}(q).$ So, $\beta_1=\beta_2$. If $p=2$ then $0=(1-\beta_1^{2})=(1-\beta_1)^{2}$,... | \text { \large 0 } & (1-\beta_1)D\\
\end{smallmatrix } \right)$ for suitable $ l$ and $ T^l$ is a scalar matrix by Lemma \[m7\ ]. It means that $ 1-\beta_1^{2}=0 $ because if $ B=0 $ then $ T$ is reducible and it cannot generate $ { \mathrm{Sin}}_{n_1}(q).$ So, $ \beta_1=\beta_2$. If $ p=2 $ then $ 0=(1-\beta_1^{2})=... |
\tedt{ \large 0}& (1-\beta_1)D\\
\end{smallmxtrix} \right)$ for suitabne $l$ ahd $T^l$ is a scalar matrix by Lemma \[m7\]. Mt mwans ukat $1-\beta_1^{2}=0$ because if $C=0$ then $T$ ps reducivle end it can not gxherate ${\mathrm{Aln}}_{n_1}(q).$ Vi, $\beta_1=\beta_2$. If $k=2$ then $0=(1-\beta_1^{2})=(1-\bata_1)^{2}$,... | \text{ \large 0}& (1-\beta_1)D\\ \end{smallmatrix} \right)$ for and is a matrix by Lemma because $B=0$ then $T$ reducible and it not generate ${\mathrm{Sin}}_{n_1}(q).$ So, $\beta_1=\beta_2$. If then $0=(1-\beta_1^{2})=(1-\beta_1)^{2}$, so $\beta_1=1$ and $\alpha_1=\alpha_2$. Let $p>2$, we have $\beta_1=\beta_2$ and $\... |
\text{ \large 0}& (1-\beta_1)D\\
\end{smallmatRix} \right)$ foR suitAblE $l$ aNd $t^l$ is A scaLar matrix by LemMA \[m7\]. It Means that $1-\beta_1^{2}=0$ because if $b=0$ then $t$ iS ReduCIbLe and It can noT GeNERatE ${\mAtHrm{siN}}_{N_1}(q).$ so, $\betA_1=\beTa_2$. If $p=2$ thEn $0=(1-\beta_1^{2})=(1-\beta_1)^{2}$,... |
\text{ \large 0}& (1-\be ta_1)D\\
\ end{s mal lma tr ix}\rig ht)$ for suita b le $ l$ and $T^l$ is a scal ar ma tr i x by Le mma \ [m7\].I tm e ans t ha t $ 1- \ be ta_1^ {2} =0$ bec ause if $B =0$ t hen $T$ is r e du cible anditcan not gene rat e ${\m at hrm { Sin}} _{n _1}(q ).$ So , $\bet a_1=\beta _2 $ . If $ ... |
\text{_\large 0}&_(1-\beta_1)D\\
\end{smallmatrix} \right)$ for suitable_$l$ and_$T^l$_is a_scalar_matrix by Lemma_\[m7\]. It means_that $1-\beta_1^{2}=0$ because if_$B=0$ then $T$_is_reducible and it can not generate ${\mathrm{Sin}}_{n_1}(q).$ So, $\beta_1=\beta_2$. If $p=2$ then $0=(1-\beta_1^{2})=(1-\beta_1)^{2}$,... |
median}$ Period $\log\left( \frac{M_{BH}}{M_{\odot}} \right)$ $r$ $t_{insp}$ $\Delta t_{GW}$
(days) (pc) (yrs) (ns)
SDSS J121457.39+132024.3 12 14 57... | median}$ Period $ \log\left (\frac{M_{BH}}{M_{\odot } } \right)$ $ r$ $ t_{insp}$ $ \Delta t_{GW}$
(days) (pc) (yrs) (ns)
SDSS J121457.39 +... | medlan}$ Period $\log\left( \nrac{M_{BH}}{M_{\odot}} \rigkr)$ $r$ $t_{inap}$ $\Delta t_{GW}$
(days) (pz) (yrs) (ns)
SDSS J121457.39+132024.3 12 14 57... | median}$ Period $\log\left( \frac{M_{BH}}{M_{\odot}} \right)$ $r$ $t_{insp}$ (days) (yrs) (ns) J121457.39+132024.3 12 14 18.59 9.46 0.011 $1.8 10^3 $ 6.6 J123147.27+101705.3 12 31 47.3 +10 17 1.733 18.83 1851 9.20 0.009 $3.5 \times 10^3 $ 2.3 SDSS J123821.84+030024.2 12 21.8 +03 00 24.6 0.380 18.46 1250 8.92 0.008 $2.2... | median}$ Period $\log\left( \frac{M_{Bh}}{M_{\odot}} \righT)$ $r$ $t_{inSp}$ $\DEltA t_{gW}$
(daYs) (pc) (Yrs) (ns)
SDSS J121457.39+132024.3 12 14 57... | median}$ Period $\log \left( \fr ac{M_ {BH }}{ M_ {\od ot}} \right)$ $r$ $t_{insp}$ $ \Delt at _{GW } $ (days) ( pc) (yrs) (ns) S DSSJ121457.39+132024 . 3 12 14 57.4 +13 20 2 4 .5 1. 494 ... | median}$ _ Period_ $\log\left(_\frac{M_{BH}}{M_{\odot}} \right)$__$r$ __ $t_{insp}$_ _ _ _$\Delta_t_{GW}$
__ ___ _ _ _ __ _ _ _ ___ _ _ _(days) _ _ _ _ _ _ _ _ _ _ __ (pc)___ (yrs)_ __ __ __ (ns)
SDSS J121457.39+132024.3 _12 14 57... |
$. While it is clear that the sequence $\{p_n\}\/$ cannot possibly capture all the information contained in $\phi(z)\/$ as all the off-diagonal matrix elements $\langle m|\hat{\rho}|n\rangle \;,\;m\neq n\/$, are ignored, we can easily show that the PND $\{p_n\}\/$ and the distribution ${\cal P}(I)\/$ determine each oth... | $. While it is clear that the sequence $ \{p_n\}\/$ cannot possibly capture all the information control in $ \phi(z)\/$ as all the away - diagonal matrix elements $ \langle m|\hat{\rho}|n\rangle \;,\;m\neq n\/$, are ignore, we can well show that the PND $ \{p_n\}\/$ and the distribution $ { \cal P}(I)\/$ determine each... | $. Whlle it is clear that the sequence $\{p_n\}\/$ caubot povsibly capture all the information containxd ib $\phi(e)\/$ as all the off-diaeonal matgix elemebts $\oqngle m|\hat{\cgo}|n\rangle \;,\;m\nes n\/$, axe ignored, we cak easily shmw that the PNG $\{o_n\}\/$ and the distribution ${\cal P}(I)\/$ determyne eacn lth... | $. While it is clear that the cannot capture all information contained in matrix $\langle m|\hat{\rho}|n\rangle \;,\;m\neq are ignored, we easily show that the PND $\{p_n\}\/$ the distribution ${\cal P}(I)\/$ determine each other uniquely. To recover ${\cal P}(I)\/$ from we define a generating function $\Lambda(K),\;0\... | $. While it is clear that the sequEnce $\{p_n\}\/$ cannOt posSibLy cApTure All tHe information cONtaiNed in $\phi(z)\/$ as all the off-diAgonaL mATrix ELeMents $\Langle m|\HAt{\RHO}|n\rAnGlE \;,\;m\nEq N\/$, ArE ignoRed, We can eaSily show thAt tHe pND $\{p_n\}\/$ and the dIStRibution ${\caL P}(I)\/$ Determine eacH otH... | $. While it is clear thatthe sequen ce $\ {p_ n\} \/ $ ca nnot possibly capt u re a ll the information con taine di n $\ p hi (z)\/ $ as al l t h e of f- di ago na l m atrix el ements$\langle m |\h at {\rho}|n\ran g le \;,\;m\ne q n \/$, are ign ore d, weca n e a silysho w tha t theP ND $\{ p_n\}\/$an d the d i ... | $. While_it is_clear that the sequence_$\{p_n\}\/$ cannot_possibly_capture all_the_information contained in_$\phi(z)\/$ as all_the off-diagonal matrix elements_$\langle m|\hat{\rho}|n\rangle \;,\;m\neq_n\/$,_are ignored, we can easily show that the PND $\{p_n\}\/$ and the distribution ${\cal_P}(I)\/$_determine each_oth... |
(x + y))
\big( Q_j(\sigma, \sigma) - Q_j(\sigma, \tau) \big)} ~. \end{aligned}$$ It is easy to check that when $\mu = M$, $A_j(s/M^2, \eta)$ coincides with the amplitude $\tilde{A}_j(s, M^2)$ obtained in Ref. [@egt].
Eqs. (\[aleptf\], \[solutionaj\]) describe all invariant amplitudes for $e^+e^-$ -annihilation into ... | (x + y) )
\big (Q_j(\sigma, \sigma) - Q_j(\sigma, \tau) \big) } ~. \end{aligned}$$ It is easy to check that when $ \mu = M$, $ A_j(s / M^2, \eta)$ coincides with the amplitude $ \tilde{A}_j(s, M^2)$ obtained in Ref. [ @egt ].
Eqs. (\[aleptf\ ], \[solutionaj\ ]) identify all changeless amplitudes for $ e^+e^-$ ... | (x + y))
\big( Q_j(\sigma, \sigma) - Q_j(\rigma, \tau) \big)} ~. \gne{alignxd}$$ It ia easy tu check that when $\mu = M$, $A_j(s/M^2, \era)$ councides with the ampligude $\tildv{A}_j(s, M^2)$ obraintd in Ref. [@egt].
Eqs. (\[alxltf\], \[solmcionan\]) desermbe all invariakt amplitudas for $e^+e^-$ -anniviuacion into ... | (x + y)) \big( Q_j(\sigma, \sigma) - \big)} \end{aligned}$$ It easy to check $A_j(s/M^2, coincides with the $\tilde{A}_j(s, M^2)$ obtained Ref. [@egt]. Eqs. (\[aleptf\], \[solutionaj\]) describe invariant amplitudes for $e^+e^-$ -annihilation into a quark or a lepton pair in collinear kinematics (\[tmu\], \[umu\]). Sca... | (x + y))
\big( Q_j(\sigma, \sigma) - Q_j(\sigma, \tAu) \big)} ~. \end{alIgned}$$ it iS eaSy To chEck tHat when $\mu = M$, $A_j(s/M^2, \ETa)$ coIncides with the amplitudE $\tildE{A}_J(S, M^2)$ obTAiNed in ref. [@egt].
EQS. (\[aLEPtf\], \[SoLuTioNaJ\]) DeScribE alL invariAnt amplituDes FoR $e^+e^-$ -annihilatIOn Into ... | (x + y))
\big( Q_j(\sigm a, \sigma) - Q_ j(\ sig ma , \t au)\big)} ~. \end { alig ned}$$ It is easy to c heckth a t wh e n$\mu= M$, $ A _j ( s /M^ 2, \ eta )$ co incid eswith th e amplitud e $ \t ilde{A}_j(s, M^ 2)$ obtain edin Ref. [@eg t].
Eqs. ( \[a l eptf\ ],\[sol utiona j \]) de scribe al li nvaria n t ampl... | (x_+ y))_
\big( Q_j(\sigma, \sigma) -_Q_j(\sigma, \tau)_\big)}_~. \end{aligned}$$_It_is easy to_check that when_$\mu = M$, $A_j(s/M^2,_\eta)$ coincides with_the_amplitude $\tilde{A}_j(s, M^2)$ obtained in Ref. [@egt].
Eqs. (\[aleptf\], \[solutionaj\]) describe all invariant amplitudes for $e^+e^-$ -annihilation_into_... |
lambda}|<\rho \right\}, \quad {\mathbb{D}}_\rho = \left\{ {\lambda}\in{\mathbb{C}}\mid |{\lambda}|<\rho \right\}.$$ Any smooth map $f:{\mathbb{S}}^1\longrightarrow {\mathcal{M}}(2,{\mathbb{C}})$ can be decomposed into its Fourier series $$f({\lambda}) = \sum_{i\in{\mathbb{Z}}}f_i{\lambda}^i.$$ Let $|\cdot|$ denote a no... | lambda}|<\rho \right\ }, \quad { \mathbb{D}}_\rho = \left\ { { \lambda}\in{\mathbb{C}}\mid |{\lambda}|<\rho \right\}.$$ Any smooth map $ f:{\mathbb{S}}^1\longrightarrow { \mathcal{M}}(2,{\mathbb{C}})$ can be decomposed into its Fourier series $ $ f({\lambda }) = \sum_{i\in{\mathbb{Z}}}f_i{\lambda}^i.$$ lease $ |\cdot|$... | lamhda}|<\rho \right\}, \quad {\mathbb{A}}_\rho = \left\{ {\lambbq}\in{\matibb{C}}\mid |{\lambda}|<\rfo \right\}.$$ Any smooth map $f:{\matibb{S}}^1\oongrughtarrow {\mathcal{M}}(2,{\mathcb{C}})$ can bv decompowed mnto its Fourier series $$n({\jambsw}) = \sbm_{m\in{\mathbb{Z}}}f_i{\lamnda}^i.$$ Let $|\cdmt|$ denote a no... | lambda}|<\rho \right\}, \quad {\mathbb{D}}_\rho = \left\{ {\lambda}\in{\mathbb{C}}\mid Any map $f:{\mathbb{S}}^1\longrightarrow can be decomposed = Let $|\cdot|$ denote norm on ${\mathcal{M}}(2,{\mathbb{C}})$. some $\rho>1$ and consider $${\left\Vertf\right\Vert}_\rho := |f_i|\rho^{|i|}.$$ Let $G$ be a Lie group or alg... | lambda}|<\rho \right\}, \quad {\mathbb{D}}_\Rho = \left\{ {\lamBda}\in{\MatHbb{c}}\mId |{\laMbda}|<\Rho \right\}.$$ Any smoOTh maP $f:{\mathbb{S}}^1\longrightarroW {\mathCaL{m}}(2,{\matHBb{c}})$ can bE decompOSeD INto ItS FOurIeR SeRies $$f({\LamBda}) = \sum_{i\In{\mathbb{Z}}}f_I{\laMbDa}^i.$$ Let $|\cdot|$ deNOtE a no... | lambda}|<\rho \right\}, \q uad {\math bb{D} }_\ rho = \le ft\{ {\lambda}\in{ \ math bb{C}}\mid |{\lambda}| <\rho \ r ight \ }. $$ An y smoot h m a p $f :{ \m ath bb { S} }^1\l ong rightar row {\math cal {M }}(2,{\mathb b {C }})$ can b e d ecomposed in toits Fo ur ier serie s $ $f({\ lambda } ) = \s um_{i\in{ \m a... | lambda}|<\rho \right\},_\quad {\mathbb{D}}_\rho_= \left\{ {\lambda}\in{\mathbb{C}}\mid |{\lambda}|<\rho_\right\}.$$ Any_smooth_map $f:{\mathbb{S}}^1\longrightarrow_{\mathcal{M}}(2,{\mathbb{C}})$_can be decomposed_into its Fourier_series $$f({\lambda}) = \sum_{i\in{\mathbb{Z}}}f_i{\lambda}^i.$$_Let $|\cdot|$ denote_a_no... |
starting point is the following classification of homological ring epimorphism from [@BS]:
*([@BS Theorem 5.23])*\[T:BSepi\] Let $R$ be a valuation domain. Then there is a bijection between:
1. non-dense admissible systems $\mathcal{X}$ in $\operatorname*{Spec}(R)$, and
2. epiclasses of homological ring epimorphi... | starting point is the following classification of homologic gang epimorphism from [ @BS ]:
* ([ @BS Theorem 5.23])*\[T: BSepi\ ] Let $ R$ be a valuation knowledge domain. Then there be a bijection between:
1. non - dense admissible systems $ \mathcal{X}$ in $ \operatorname*{Spec}(R)$, and
2. epiclasses of ... | stwrting point is the folluwing classificcrion oh homolkgical rkng epimorphism from [@BS]:
*([@BS Thxoren 5.23])*\[T:BStii\] Let $R$ be a valuatiun domain. Then thwre ms a bijection bxfween:
1. kjn-dehde abmmssible systems $\mathcal{X}$ hn $\operatornama*{Soee}(R)$, and
2. epiclasses of homological rigg epimprohi... | starting point is the following classification of epimorphism [@BS]: *([@BS 5.23])*\[T:BSepi\] Let $R$ there a bijection between: non-dense admissible systems in $\operatorname*{Spec}(R)$, and 2. epiclasses of ring epimorphisms $\lambda: R \rightarrow S$. The bijection consists of two mutually inverse $(i) \rightarrow ... | starting point is the followiNg classifiCatioN of HomOlOgicAl riNg epimorphism fROm [@BS]:
*([@bS Theorem 5.23])*\[T:BSepi\] Let $R$ be a ValuaTiON domAIn. then tHere is a BIjECTioN bEtWeeN:
1. nON-dEnse aDmiSsible sYstems $\mathCal{x}$ iN $\operatornamE*{spEc}(R)$, and
2. epicLasSes of homologIcaL ring ePiMorPHi... | starting point is the fol lowing cla ssifi cat ion o f ho molo gical ring epi m orph ism from [@BS]:
*([@B S The or e m 5. 2 3] )*\[T :BSepi\ ] L e t $R $be ava l ua tiondom ain. Th en there i s a b ijection bet w ee n:
1. no n-d ense admissi ble syste ms $\ m athca l{X }$ in $\ope r atorna me*{Spec} (R ) $... | starting_point is_the following classification of_homological ring_epimorphism_from [@BS]:
*([@BS_Theorem_5.23])*\[T:BSepi\] Let $R$_be a valuation_domain. Then there is_a bijection between:
1.__non-dense admissible systems $\mathcal{X}$ in $\operatorname*{Spec}(R)$, and
2. epiclasses of homological ring epimorphi... |
^{16})-q^2f(q^{4},q^{26})\bigl)\bigr(f(q^{13},q^{17})-qf(q^7,q^{23})\bigl)\notag\\
&\;\;\;+q\bigr(f(q^8,q^{22})-q^2f(q^2,q^{28})\bigl)\bigr(f(q^{11},q^{19})-q^3f(q,q^{29})\bigl).\label{60pre3}\end{aligned}$$ Recall that the Rogers-Ramanujan functions are defined by $$\label{GH}
G(q) :=\sum_{n=0}^{{\infty}}{\dfrac}{q^{n... | ^{16})-q^2f(q^{4},q^{26})\bigl)\bigr(f(q^{13},q^{17})-qf(q^7,q^{23})\bigl)\notag\\
& \;\;\;+q\bigr(f(q^8,q^{22})-q^2f(q^2,q^{28})\bigl)\bigr(f(q^{11},q^{19})-q^3f(q, q^{29})\bigl).\label{60pre3}\end{aligned}$$ Recall that the Rogers - Ramanujan functions are defined by $ $ \label{GH }
G(q): = \sum_{n=0}^{{\infty}}{... | ^{16})-q^2f(q^{4},e^{26})\bigl)\bigr(f(q^{13},q^{17})-qf(q^7,q^{23})\bigl)\notan\\
&\;\;\;+q\bigr(f(q^8,q^{22})-q^2f(q^2,q^{28})\bigl)\bigr(f(q^{11},x^{19})-q^3f(q,q^{29})\bjgl).\label{60ore3}\end{aligned}$$ Recall that thx Roters-Rqmanujan functions are defined hy $$\label{TH}
G(q) :=\wum_{n=0}^{{\infty}}{\dhdac}{q^{n... | ^{16})-q^2f(q^{4},q^{26})\bigl)\bigr(f(q^{13},q^{17})-qf(q^7,q^{23})\bigl)\notag\\ &\;\;\;+q\bigr(f(q^8,q^{22})-q^2f(q^2,q^{28})\bigl)\bigr(f(q^{11},q^{19})-q^3f(q,q^{29})\bigl).\label{60pre3}\end{aligned}$$ Recall that the Rogers-Ramanujan functions by G(q) :=\sum_{n=0}^{{\infty}}{\dfrac}{q^{n^2}}{(q;q)_n} \text{and} ... | ^{16})-q^2f(q^{4},q^{26})\bigl)\bigr(f(q^{13},q^{17})-qf(q^7,q^{23})\bigl)\noTag\\
&\;\;\;+q\bigr(f(q^8,Q^{22})-q^2f(q^2,q^{28})\BigL)\biGr(F(q^{11},q^{19})-q^3F(q,q^{29})\bIgl).\label{60pre3}\end{ALignEd}$$ Recall that the Rogers-RAmanuJaN FuncTIoNs are Defined BY $$\lABEl{Gh}
G(Q) :=\sUm_{n=0}^{{\InFTy}}{\Dfrac}{Q^{n... | ^{16})-q^2f(q^{4},q^{26})\ bigl)\bigr (f(q^ {13 },q ^{ 17}) -qf( q^7,q^{23})\bi g l)\n otag\\
&\;\;\;+q\bigr( f(q^8 ,q ^ {22} ) -q ^2f(q ^2,q^{2 8 }) \ b igl )\ bi gr( f( q ^{ 11},q ^{1 9})-q^3 f(q,q^{29} )\b ig l).\label{60 p re 3}\end{ali gne d}$$ Recalltha t theRo ger s -Rama nuj an fu nction s are d efined by $ $... | ^{16})-q^2f(q^{4},q^{26})\bigl)\bigr(f(q^{13},q^{17})-qf(q^7,q^{23})\bigl)\notag\\
&\;\;\;+q\bigr(f(q^8,q^{22})-q^2f(q^2,q^{28})\bigl)\bigr(f(q^{11},q^{19})-q^3f(q,q^{29})\bigl).\label{60pre3}\end{aligned}$$ Recall_that the_Rogers-Ramanujan functions are defined_by $$\label{GH}
G(q)_:=\sum_{n=0}^{{\infty}}{\dfrac}{q^{n... |
\quad 0\leq t \leq T,$$
and we denote the continuous version of obtained by linear interpolation by $$\label{pir}
\widehat{W}^n_t := \frac{1}{\sqrt{n}} \widehat{Z}^n_{\lfloor nt/T \rfloor}, \quad 0\leq t \leq T.$$ By the central limit theorem; $(W^n, \widehat{W}^n) \Rightarrow (W,W) $ as $n \rightarrow \infty$ on... | \quad 0\leq t \leq T,$$
and we denote the continuous version of obtained by analogue interjection by $ $ \label{pir }
\widehat{W}^n_t: = \frac{1}{\sqrt{n } } \widehat{Z}^n_{\lfloor nt / T \rfloor }, \quad 0\leq thyroxine \leq T.$$ By the central limit theorem; $ (W^n, \widehat{W}^n) \Rightarrow (W, W) $ as $ n... | \quwd 0\leq t \leq T,$$
and we denute the continuous vervion or obtaindd by linear interpolation bb $$\lavel{pie}
\widehat{W}^n_t := \frac{1}{\sdrt{n}} \widejat{Z}^n_{\lflior it/T \rfloor}, \quad 0\leq t \leq T.$$ By bhe cznvral limit theotem; $(W^n, \widehdt{W}^n) \Rightarrof (D,W) $ as $n \rightarrow \infty$ on... | \quad 0\leq t \leq T,$$ and we continuous of obtained linear interpolation by nt/T \quad 0\leq t T.$$ By the limit theorem; $(W^n, \widehat{W}^n) \Rightarrow (W,W) as $n \rightarrow \infty$ on $D([0,T];\mathbb{R}^2)$ ($\Rightarrow$ implies convergence in distribution). i.e., the $(P_n) $ converges to the law $P_0$ on t... | \quad 0\leq t \leq T,$$
and we denote thE continuouS versIon Of oBtAineD by lInear interpolaTIon bY $$\label{pir}
\widehat{W}^n_t := \fraC{1}{\sqrt{N}} \wIDehaT{z}^n_{\LflooR nt/T \rflOOr}, \QUAd 0\lEq T \lEq T.$$ by THe CentrAl lImit theOrem; $(W^n, \wideHat{w}^n) \rightarrow (W,W) $ AS $n \Rightarrow \InfTy$ on... | \quad 0\leq t \leq T,$$
and we den ote t hecon ti nuou s ve rsion of obtai n ed b y linear interpolation by $ $\ l abel { pi r}
\wide h at { W }^n _t : = \ fr a c{ 1}{\s qrt {n}} \w idehat{Z}^ n_{ \l floor nt/T \ r fl oor}, \qua d 0 \leq t \leqT.$ $ By t he ce n trallim it th eorem; $(W^n, \widehat {W } ^n) \R ... | \quad_0\leq t_\leq T,$$
and we denote_the continuous_version_of obtained_by_linear interpolation by_$$\label{pir}
_ \widehat{W}^n_t := \frac{1}{\sqrt{n}}_\widehat{Z}^n_{\lfloor nt/T \rfloor},_\quad_0\leq t \leq T.$$ By the central limit theorem; $(W^n, \widehat{W}^n) \Rightarrow (W,W) $_as_$n \rightarrow_\infty$_on... |
\subseteq \mathbb{Z}_{2}$
------------------------------------------------------------------- ------- ------------------- ---------------------------- ----------------------------------------- ---------------------------------------- --... | \subseteq \mathbb{Z}_{2}$
------------------------------------------------------------------- ------- ------------------- ---------------------------- ----------------------------------------- ---------------------------------------- --------------------------------------------------- ------------------------------... | \subseteq \larhbb{Z}_{2}$
------------------------------------------------------------------- ------- ------------------- ---------------------------- ----------------------------------------- ---------------------------------------- --... | \subseteq \mathbb{Z}_{2}$ ------------------------------------------------------------------- ------- ------------------- ---------------------------- ----------------------------------------- -------------------------------------------------- ------- ---------------------- ------------------------------------------- -... | \subseteq \mathbb{Z}_{2}$
------------------------------------------------------------------- ------- ------------------- ---------------------------- ----------------------------------------- ---------------------------------------- --... | \subseteq \mathb b{Z}_ {2 } $
- -- ----- ------- - -- - - --- -- -- --- -- - -- ----- --- ------- ---------- --- -- -- ------- - - -- ---------- --- -- --------- --- ------ -- --- - ------- ----- ------ - ------ --------- -- - ------ - - ----- - - ... | _ _ _ __ __ _ _ _ __ __ ___ _ _ _ __ _ _ \subseteq \mathbb{Z}_{2}$
-------------------------------------------------------------------_------- ------------------- ----------------------------_-----------------------------------------_----------------------------------------_--... |
consider the corresponding admissible graph $\Gamma$ of type $(n,k,l)$, then an edge $e$ of $\Gamma$ determines either a projection $\pi_e:\mathcal C_{n,k,l}^+\to\mathcal C_{2,0,0}^+$ or $\pi_e:\mathcal C_{n,k,l}^+\to\mathcal C_{1,0,0}^+$.
We now consider the vector space $X=\mathbb K^d$ and two linear (or affine) su... | consider the corresponding admissible graph $ \Gamma$ of type $ (n, k, l)$, then an edge $ e$ of $ \Gamma$ determine either a expulsion $ \pi_e:\mathcal C_{n, k, l}^+\to\mathcal C_{2,0,0}^+$ or $ \pi_e:\mathcal C_{n, k, l}^+\to\mathcal C_{1,0,0}^+$.
We now consider the vector space $ X=\mathbb K^d$ and two analogue ... | cojsider the corresponding admissible grakh $\Gamma$ of tyle $(n,k,l)$, tfen an edge $e$ of $\Gamma$ deterlibes euther a projection $\pi_e:\oathcal C_{j,k,l}^+\to\matycal X_{2,0,0}^+$ or $\pi_e:\mavgcal C_{n,k,l}^+\to\mafmcal E_{1,0,0}^+$.
Wx now consider jhe vector s[ace $X=\mathbb K^g$ xnb two linear (or affine) su... | consider the corresponding admissible graph $\Gamma$ of then edge $e$ $\Gamma$ determines either or C_{n,k,l}^+\to\mathcal C_{1,0,0}^+$. We consider the vector $X=\mathbb K^d$ and two linear (or subspaces $U_i$, $i=1,2$, for which we assume there is a direct sum decomposition X=(U_1\cap U_2)\overset{\perp}\oplus (U_1^\... | consider the corresponding aDmissible gRaph $\GAmmA$ of TyPe $(n,k,L)$, theN an edge $e$ of $\GammA$ DeteRmines either a projectioN $\pi_e:\mAtHCal C_{N,K,l}^+\To\matHcal C_{2,0,0}^+$ or $\PI_e:\MAThcAl c_{n,K,l}^+\tO\mAThCal C_{1,0,0}^+$.
WE noW considEr the vectoR spAcE $X=\mathbb K^d$ anD TwO linear (or aFfiNe) su... | consider the correspondin g admissib le gr aph $\ Ga mma$ oftype $(n,k,l)$ , the n an edge $e$ of $\Gam ma$ d et e rmin e seithe r a pro j ec t i on$\ pi _e: \m a th cal C _{n ,k,l}^+ \to\mathca l C _{ 2,0,0}^+$ or $\ pi_e:\math cal C_{n,k,l}^+ \to \mathc al C_ { 1,0,0 }^+ $.
W e nowc onside r the vec to r spac... | consider_the corresponding_admissible graph $\Gamma$ of_type $(n,k,l)$,_then_an edge_$e$_of $\Gamma$ determines_either a projection_$\pi_e:\mathcal C_{n,k,l}^+\to\mathcal C_{2,0,0}^+$ or_$\pi_e:\mathcal C_{n,k,l}^+\to\mathcal C_{1,0,0}^+$.
We_now_consider the vector space $X=\mathbb K^d$ and two linear (or affine) su... |
term on the right-hand side of Eq. [(\[eq:s\])]{} vanishes. It is furthermore reasonable to suppose that an environmental heat bath in thermal equilibrium leads to a vanishing expectation value of the stochastic force $\hat{\boldsymbol{\xi}}(t)$, Eq. [(\[eq:stochasticforce\])]{}. Consequently, presuming a vanishing sh... | term on the right - hand side of Eq. [ (\[eq: s\ ]) ] { } vanishes. It is furthermore fair to presuppose that an environmental heat bath in thermal chemical equilibrium leave to a vanishing expectation value of the stochastic effect $ \hat{\boldsymbol{\xi}}(t)$, Eq. [ (\[eq: stochasticforce\ ]) ] { }. Consequently,... | tegm on the right-hand side of Eq. [(\[eq:s\])]{} vaniskws. It ms furtgermore feasonable to suppose that ai encironnental heat bath in thdrmal equplibrium oeadw to a vanishing exizctatjln vclne of the stochsstic forca $\hat{\boldsymbon{\xk}}(t)$, Eq. [(\[eq:stochasticforce\])]{}. Consequently, pwesuminb w vanishing sh... | term on the right-hand side of Eq. It furthermore reasonable suppose that an equilibrium to a vanishing value of the force $\hat{\boldsymbol{\xi}}(t)$, Eq. [(\[eq:stochasticforce\])]{}. Consequently, presuming vanishing shift $\mathbf{s}(t)$ seems appropriate for a typical measurement configuration. We will confirm pre... | term on the right-hand side of EQ. [(\[eq:s\])]{} vanishEs. It iS fuRthErMore ReasOnable to supposE That An environmental heat batH in thErMAl eqUIlIbriuM leads tO A vANIshInG eXpeCtATiOn valUe oF the stoChastic forCe $\hAt{\Boldsymbol{\xi}}(T)$, eq. [(\[Eq:stochastIcfOrce\])]{}. ConsequeNtlY, presuMiNg a VAnishIng Sh... | term on the right-hand si de of Eq.[(\[e q:s \]) ]{ } va nish es. It is furt h ermo re reasonable to suppo se th at an e n vi ronme ntal he a tb a thin t her ma l e quili bri um lead s to a van ish in g expectatio n v alue of th e s tochastic fo rce $\hat {\ bol d symbo l{\ xi}}( t)$, E q . [(\[ eq:stocha st i cfo... | term_on the_right-hand side of Eq. [(\[eq:s\])]{}_vanishes. It_is_furthermore reasonable_to_suppose that an_environmental heat bath_in thermal equilibrium leads_to a vanishing_expectation_value of the stochastic force $\hat{\boldsymbol{\xi}}(t)$, Eq. [(\[eq:stochasticforce\])]{}. Consequently, presuming a vanishing sh... |
frac{1}{\sigma^2} \sum_{mn} mn v_{m0}
\sum_{j=1}^\infty u_{jm} \left(\frac{1}{-\lambda_j}\right) v_{nj}.$$
![Autocorrelation time computed (a) numerically using eigenfunction expansion or (b) by simulation using method of batch means. For sufficient cutoff $N$ or trajectory duration $T$, respectively, both methods... | frac{1}{\sigma^2 } \sum_{mn } mn v_{m0 }
\sum_{j=1}^\infty u_{jm } \left(\frac{1}{-\lambda_j}\right) v_{nj}.$$
! [ Autocorrelation time computed (a) numerically using eigenfunction expansion or (boron) by model using method of batch means. For sufficient shortcut $ N$ or trajectory duration $ T$, respectively, ... | fraf{1}{\sigma^2} \sum_{mn} mn v_{m0}
\suo_{j=1}^\infty u_{jm} \lefj(\feac{1}{-\lamuda_j}\riggt) v_{nj}.$$
![Augocorrelation time computed (e) nunericqlly using eigenfunctiun expanspon or (b) vy smmulation using method on batdm meaus. For sufficienj cutoff $N$ os trajectory dgrxtnon $T$, respectively, both methods... | frac{1}{\sigma^2} \sum_{mn} mn v_{m0} \sum_{j=1}^\infty u_{jm} \left(\frac{1}{-\lambda_j}\right) time (a) numerically eigenfunction expansion or of means. For sufficient $N$ or trajectory $T$, respectively, both methods converge to value (dashed line). Parameters: $\theta = h = 0$ and $n_c = 100$. is in units of $1/k_1... | frac{1}{\sigma^2} \sum_{mn} mn v_{m0}
\sum_{j=1}^\infTy u_{jm} \left(\fRac{1}{-\laMbdA_j}\rIgHt) v_{nJ}.$$
![AutOcorrelation tiME comPuted (a) numerically using EigenFuNCtioN ExPansiOn or (b) by SImULAtiOn UsIng MeTHoD of baTch Means. FoR sufficienT cuToFf $N$ or trajectORy Duration $T$, rEspEctively, both MetHods... | frac{1}{\sigma^2} \sum_{mn } mn v_{m0 }
\s um_ {j =1}^ \inf ty u_{jm} \lef t (\fr ac{1}{-\lambda_j}\righ t) v_ {n j }.$$
! [Auto correla t io n tim eco mpu te d ( a) nu mer icallyusing eige nfu nc tion expansi o nor (b) bysim ulation usin g m ethodof ba t ch me ans . For suffi c ient c utoff $N$ o r traje c to... | frac{1}{\sigma^2} \sum_{mn}_mn v_{m0}
_ \sum_{j=1}^\infty_u_{jm} \left(\frac{1}{-\lambda_j}\right)_v_{nj}.$$
![Autocorrelation_time computed_(a)_numerically using eigenfunction_expansion or (b)_by simulation using method_of batch means._For_sufficient cutoff $N$ or trajectory duration $T$, respectively, both methods... |
d_2,\ldots,d_c)$ for $(d_1,d_2,\ldots,d_c)\ne (2,2)$. If $X$ is not a cubic surface or an elliptic curve, then the map $$\mathrm{Aut}(X)\rightarrow \mathrm{Aut}(H^m(X,\mathbb{Q}))$$ is injective where $m = n - c$.
The proposition is well-known for smooth algebraic curves of genus at least two and K3 surfaces. If $X$ i... | d_2,\ldots, d_c)$ for $ (d_1,d_2,\ldots, d_c)\ne (2,2)$. If $ X$ is not a cubic surface or an elliptic curve, then the map $ $ \mathrm{Aut}(X)\rightarrow \mathrm{Aut}(H^m(X,\mathbb{Q}))$$ is injective where $ m = newton - c$.
The suggestion is well - known for smooth algebraic curves of genus at least two and K3 sur... | d_2,\ldlts,d_c)$ for $(d_1,d_2,\ldots,d_c)\ne (2,2)$. In $X$ is not a cubnx surfece or zn ellipgic curve, then the map $$\mathrl{Ayt}(X)\rithtarrow \mathrm{Aut}(H^m(X,\mxthbb{Q}))$$ is injectice wiere $m = n - c$.
The 'dopositljn ia weln-jnown for smoojh algebraic curves of gengs ac least two and K3 surfaces. If $X$ i... | d_2,\ldots,d_c)$ for $(d_1,d_2,\ldots,d_c)\ne (2,2)$. If $X$ is cubic or an curve, then the where = n - The proposition is for smooth algebraic curves of genus least two and K3 surfaces. If $X$ is a surface with nontrivial canonical or the dimension of $X$ is at least $3$, then we have $$\mathrm{Aut}(X)=\mathrm{Aut}_L(... | d_2,\ldots,d_c)$ for $(d_1,d_2,\ldots,d_c)\ne (2,2)$. If $X$ Is not a cubiC surfAce Or aN eLlipTic cUrve, then the map $$\MAthrM{Aut}(X)\rightarrow \mathrm{AUt}(H^m(X,\MaTHbb{Q}))$$ IS iNjectIve wherE $M = n - C$.
tHe pRoPoSitIoN Is Well-kNowN for smoOth algebraIc cUrVes of genus at LEaSt two and K3 sUrfAces. If $X$ i... | d_2,\ldots,d_c)$ for $(d_1 ,d_2,\ldot s,d_c )\n e ( 2, 2)$. If$X$ is not a c u bicsurface or an elliptic curv e, then th e map $$\mat h rm { A ut} (X )\ rig ht a rr ow \m ath rm{Aut} (H^m(X,\ma thb b{ Q}))$$ is in j ec tive where $m = n - c$.
The propo si tio n is w ell -know n fors moothalgebraic c u rves o f ... | d_2,\ldots,d_c)$ for_$(d_1,d_2,\ldots,d_c)\ne (2,2)$._If $X$ is not_a cubic_surface_or an_elliptic_curve, then the_map $$\mathrm{Aut}(X)\rightarrow \mathrm{Aut}(H^m(X,\mathbb{Q}))$$_is injective where $m_= n -_c$.
The_proposition is well-known for smooth algebraic curves of genus at least two and K3_surfaces._If $X$_i... |
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; (left2)_at (-3,-2)
(0,0) node_[$\mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0)_node[circle,fill=black,minimum size=0,inner sep=1]{}_--_(0.3,0) node[circle,fill... |
] considered the correlation functions for the Izergin-Korepin model for massive regime $-1<{q}<0$, within the framework of representation theory of $U_q(A_2^{(2)})$. They gave free field realizations of the vertex operators, and realized integral representation of the correlation functions, as the trace of the vertex ... | ] considered the correlation functions for the Izergin - Korepin model for massive government $ -1<{q}<0 $, within the model of representation theory of $ U_q(A_2^{(2)})$. They give detached field realizations of the vertex hustler, and gain integral theatrical performance of the correlation coefficient functions, as t... | ] cojsidered the correlation functions for jhw Izerjin-Korelin modeu for massive regime $-1<{q}<0$, withii thw franework of representatiun theory of $U_q(A_2^{(2)})$. Rhey tave free hjeld realizatjlns mh the vertex opgrators, and sealized integsau xepresentation of the correlation fugctions, ad the trace of the dertsq ... | ] considered the correlation functions for the for regime $-1<{q}<0$, the framework of gave field realizations of vertex operators, and integral representation of the correlation functions, the trace of the vertex operators. In the limiting case, our integral representation the correlation function for the Izergin-Kore... | ] considered the correlation fUnctions foR the IZerGin-koRepiN modEl for massive reGIme $-1<{q}<0$, Within the framework of rePreseNtATion THeOry of $u_q(A_2^{(2)})$. They GAvE FRee FiElD reAlIZaTions Of tHe verteX operators, And ReAlized integrAL rEpresentatIon Of the correlaTioN functIoNs, aS The trAce Of the Vertex ... | ] considered the correlati on functio ns fo r t heIz ergi n-Ko repin model fo r mas sive regime $-1<{q}<0$ , wit hi n the fr amewo rk of r e pr e s ent at io n t he o ry of $ U_q (A_2^{( 2)})$. The y g av e free field re alizations of the vertexope rators ,and reali zed inte gral r e presen tation of t h e corr ... | ] considered_the correlation_functions for the Izergin-Korepin_model for_massive_regime $-1<{q}<0$,_within_the framework of_representation theory of_$U_q(A_2^{(2)})$. They gave free_field realizations of_the_vertex operators, and realized integral representation of the correlation functions, as the trace of_the_vertex ... |
cases, we initialized with a simple interpolation of each spectral channel by bicubic splines.
Quality Assessment Indices
--------------------------
For simulated data, the spatial and spectral consistency of the fused products with respect to the ground-truth images are numerically evaluated by means of several qua... | cases, we initialized with a simple interpolation of each apparitional distribution channel by bicubic splines.
Quality Assessment Indices
--------------------------
For simulated datum, the spatial and spectral consistency of the fuse products with respect to the background - truth images are numerically measu... | cades, we initialized with x simple interpolation of eadh spectfal channel by bicubic splinxs.
Quqlity Assessment Indices
--------------------------
For simulatef data, tye skatial and spectrem consistency lf tke fused productx with res[ect to the grmuvd-cruth images are numerically evaluatqd by mrajs of several zua... | cases, we initialized with a simple interpolation spectral by bicubic Quality Assessment Indices spatial spectral consistency of fused products with to the ground-truth images are numerically by means of several quality assessment indices. Let $\u^R=\left(u_1^R, \ldots, u_C^R\right)$ be the reference multispectral imag... | cases, we initialized with a siMple interpOlatiOn oF eaCh SpecTral Channel by bicubIC splInes.
Quality Assessment INdiceS
--------------------------
FOR simULaTed daTa, the spATiAL And SpEcTraL cONsIstenCy oF the fusEd products WitH rEspect to the gROuNd-truth imaGes Are numericalLy eValuatEd By mEAns of SevEral qUa... | cases, we initialized wit h a simple inte rpo lat io n of eac h spectral cha n nelby bicubic splines.
Q ualit yA sses s me nt In dices
- - -- - - --- -- -- --- -- - -- -----
F or simu lated data , t he spatial and sp ectral con sis tency of the fu sed pr od uct s with re spect to th e groun d-truth i ma g e... | cases,_we initialized_with a simple interpolation_of each_spectral_channel by_bicubic_splines.
Quality Assessment Indices
--------------------------
For_simulated data, the_spatial and spectral consistency_of the fused_products_with respect to the ground-truth images are numerically evaluated by means of several qua... |
D_2)}} \right \} \\
&\qquad=\frac{1}{2} \log \frac{(1-\rho)^2}{D_1 + D_2 - (1+\rho^2) + 2\rho \sqrt{(1-D_1)(1-D_2)}}
\label{general1}\end{aligned}$$ where the last equation follows from $$\begin{aligned}
{(2-D_1-D_2)}^2 - 4\rho^2 (1-D_1)(1-D_2) \label{2d1d2}
& = (1-\rho^2)(2-D_1-D_2)^2 + \rho^2(D_1-D_2)^2\\
&\geq ... | D_2) } } \right \ } \\
& \qquad=\frac{1}{2 } \log \frac{(1-\rho)^2}{D_1 + D_2 - (1+\rho^2) + 2\rho \sqrt{(1 - D_1)(1 - D_2) } }
\label{general1}\end{aligned}$$ where the last equation follows from $ $ \begin{aligned }
{ (2 - D_1 - D_2)}^2 - 4\rho^2 (1 - D_1)(1 - D_2) \label{2d1d2 }
& = (1-\rho^2)(2 - D_1 ... | D_2)}} \rlght \} \\
&\qquad=\frac{1}{2} \log \frxc{(1-\rho)^2}{D_1 + D_2 - (1+\rho^2) + 2\rho \sxrt{(1-D_1)(1-D_2)}}
\mabel{gendral1}\end{aligned}$$ where the lasv eqyatiob follows from $$\begin{alkgned}
{(2-D_1-D_2)}^2 - 4\gho^2 (1-D_1)(1-D_2) \lavel{2d1v2}
& = (1-\rho^2)(2-D_1-D_2)^2 + \rho^2(D_1-D_2)^2\\
&\jsq ... | D_2)}} \right \} \\ &\qquad=\frac{1}{2} \log \frac{(1-\rho)^2}{D_1 - + 2\rho \label{general1}\end{aligned}$$ where the {(2-D_1-D_2)}^2 4\rho^2 (1-D_1)(1-D_2) \label{2d1d2} = (1-\rho^2)(2-D_1-D_2)^2 + &\geq 0 \label{3d1d2}\end{aligned}$$ and therefore, $$D_1 D_2 - (1+\rho^2)+ 2\rho \sqrt{(1-D_1)(1-D_2)}\leq 1-\rho^2.$$ ... | D_2)}} \right \} \\
&\qquad=\frac{1}{2} \log \frac{(1-\rho)^2}{d_1 + D_2 - (1+\rho^2) + 2\rho \sqRt{(1-D_1)(1-D_2)}}
\lAbeL{geNeRal1}\eNd{alIgned}$$ where the lASt eqUation follows from $$\begin{AlignEd}
{(2-d_1-d_2)}^2 - 4\rho^2 (1-d_1)(1-d_2) \lAbel{2d1D2}
& = (1-\rho^2)(2-D_1-D_2)^2 + \rHO^2(D_1-d_2)^2\\
&\GEq ... | D_2)}} \right \} \\
&\qqua d=\frac{1} {2} \lo g \f rac{ (1-\ rho)^2}{D_1 +D _2 - (1+\rho^2) + 2\rho \ sqrt{ (1 - D_1) ( 1- D_2)} }
\lab e l{ g e ner al 1} \en d{ a li gned} $$where t he last eq uat io n follows fr o m$$\begin{a lig ned}
{(2-D_1 -D_ 2)}^2-4\r h o^2 ( 1-D _1)(1 -D_2)\ label{ 2d1d2}
& = (1-\rh o ^2)(... | D_2)}} \right_\} \\
&\qquad=\frac{1}{2}_ \log \frac{(1-\rho)^2}{D_1_+ D_2_-_(1+\rho^2) +__2\rho \sqrt{(1-D_1)(1-D_2)}}
\label{general1}\end{aligned}$$_where the last_equation follows from $$\begin{aligned}
{(2-D_1-D_2)}^2_- 4\rho^2 (1-D_1)(1-D_2)_\label{2d1d2}_
& = (1-\rho^2)(2-D_1-D_2)^2 + \rho^2(D_1-D_2)^2\\
&\geq ... |
0 \[Esigma\] where $\RR_{\mu\nu}$ and $\Box$ are the Ricci tensor and the D’Alembert operator corresponding to $\gg_{\mu\nu}$.
Vacuum $D$-dimensional equations are thus reduced to scalar-vacuum ones in 4 dimensions. Although such SAS configurations were repeatedly considered [@Rad77; @RSh], it makes sense to return to... | 0 \[Esigma\ ] where $ \RR_{\mu\nu}$ and $ \Box$ are the Ricci tensor and the D’Alembert operator corresponding to $ \gg_{\mu\nu}$.
Vacuum $ D$-dimensional equations are thus dilute to scalar - void ones in 4 dimensions. Although such SAS configuration were repeatedly study [ @Rad77; @RSh ], it makes sense to come ba... | 0 \[Eslgma\] where $\RR_{\mu\nu}$ and $\Bow$ are the Ricci jebsor aid the S’Alemberg operator corresponding to $\jg_{\mu\bu}$.
Vacyum $D$-dimensional equatkons are nhus reduxed uo scalar-vacuum oiss in 4 dimensjlns. Clvhough such SAS configuradions were repaagebly considered [@Rad77; @RSh], it makes sensq to reyugn to... | 0 \[Esigma\] where $\RR_{\mu\nu}$ and $\Box$ are tensor the D’Alembert corresponding to $\gg_{\mu\nu}$. reduced scalar-vacuum ones in dimensions. Although such configurations were repeatedly considered [@Rad77; @RSh], makes sense to return to them to reveal some new features, in particular, connected with higher dimens... | 0 \[Esigma\] where $\RR_{\mu\nu}$ and $\Box$ arE the Ricci tEnsor And The d’ALembErt oPerator correspONdinG to $\gg_{\mu\nu}$.
Vacuum $D$-dimensIonal EqUAtioNS aRe thuS reduceD To SCAlaR-vAcUum OnES iN 4 dimeNsiOns. AlthOugh such SAs coNfIgurations weRE rEpeatedly cOnsIdered [@Rad77; @RSh], It mAkes seNsE to REturn To... | 0 \[Esigma\] where $\RR_{\ mu\nu}$ an d $\B ox$ ar etheRicc i tensor and t h e D’ Alembert operator corr espon di n g to $\ gg_{\ mu\nu}$ .
V a cuu m$D $-d im e ns ional eq uations are thusred uc ed to scalar - va cuum onesin4 dimensions . A lthoug hsuc h SAScon figur ations were r epeatedly c o nsider e d [@Rad ... | 0 \[Esigma\]_where $\RR_{\mu\nu}$_and $\Box$ are the_Ricci tensor_and_the D’Alembert_operator_corresponding to $\gg_{\mu\nu}$.
Vacuum_$D$-dimensional equations are_thus reduced to scalar-vacuum_ones in 4_dimensions._Although such SAS configurations were repeatedly considered [@Rad77; @RSh], it makes sense to return_to... |
X\xrightarrow{1_{X}}X\xrightarrow{}0$ is a left triangle. And for every morphism $z:Y\rightarrow Z$ in $\mathcal{C}$, there exists a left triangle $\Omega
Z\xrightarrow{x}X\xrightarrow{y}Y\xrightarrow{z}Z$.
(LTR2)If $\Omega
Z\xrightarrow{x}X\xrightarrow{y}Y\xrightarrow{z}Z$ is a left triangle, then so is $\Omega Y\xri... | X\xrightarrow{1_{X}}X\xrightarrow{}0 $ is a left triangle. And for every morphism $ z: Y\rightarrow Z$ in $ \mathcal{C}$, there exists a leftover triangulum $ \Omega
Z\xrightarrow{x}X\xrightarrow{y}Y\xrightarrow{z}Z$.
(LTR2)If $ \Omega
Z\xrightarrow{x}X\xrightarrow{y}Y\xrightarrow{z}Z$ is a leftover triangle, th... | X\xrlghtarrow{1_{X}}X\xrightarrow{}0$ ir a left triangle. And hor evedy morphksm $z:Y\rightarrow Z$ in $\mathcap{C}$, thert exists a left trixngle $\Omeha
Z\xrighrarriq{x}X\xrightacdow{y}Y\xrlyhtardlw{z}Z$.
(NVR2)If $\Omega
Z\xrigmtarrow{x}X\xrhghtarrow{y}Y\xrichgaxrow{z}Z$ is a left triangle, then so is $\Omega U\xgi... | X\xrightarrow{1_{X}}X\xrightarrow{}0$ is a left triangle. And for $z:Y\rightarrow in $\mathcal{C}$, exists a left Z\xrightarrow{x}X\xrightarrow{y}Y\xrightarrow{z}Z$ a left triangle, so is $\Omega z}\Omega Z\xrightarrow{x}X\xrightarrow{y}Y$. (LTR3)For any two left $\Omega Z\xrightarrow{x}X\xrightarrow{y}Y\xrightarrow{z}... | X\xrightarrow{1_{X}}X\xrightarrow{}0$ Is a left triAngle. and For EvEry mOrphIsm $z:Y\rightarroW z$ in $\mAthcal{C}$, there exists a lefT triaNgLE $\OmeGA
Z\XrighTarrow{x}x\XrIGHtaRrOw{Y}Y\xRiGHtArrow{Z}Z$.
(LtR2)If $\OmeGa
Z\xrightaRroW{x}x\xrightarrow{Y}y\xRightarrow{Z}Z$ iS a left triangLe, tHen so iS $\OMegA y\xri... | X\xrightarrow{1_{X}}X\xrig htarrow{}0 $ isa l eft t rian gle. And for every morp hism $z:Y\rightarrow Z $ in$\ m athc a l{ C}$,there e x is t s ale ft tr ia n gl e $\O meg a
Z\xri ghtarrow{x }X\ xr ightarrow{y} Y \x rightarrow {z} Z$.
(LTR2)I f $ \Omega
Z \xr i ghtar row {x}X\ xright a rrow{y }Y\xright ar r ow{z}... | X\xrightarrow{1_{X}}X\xrightarrow{}0$ is_a left_triangle. And for every_morphism $z:Y\rightarrow_Z$_in $\mathcal{C}$,_there_exists a left_triangle $\Omega
Z\xrightarrow{x}X\xrightarrow{y}Y\xrightarrow{z}Z$.
(LTR2)If $\Omega
Z\xrightarrow{x}X\xrightarrow{y}Y\xrightarrow{z}Z$_is a left triangle,_then so is_$\Omega_Y\xri... |
Technically, this approach, with $N$ modes being taken into account, means following the dynamics of $7N+3$ real variables. According to our experience, for most of our results, the frequency interval $[0, 30\nu]$ with $N=3000$ together with the realistic assumption of $\Omega_n/\omega_0=0.001\sqrt{\tilde{\omega}_n/\om... | Technically, this approach, with $ N$ modes being taken into account, mean take after the dynamics of $ 7N+3 $ real variables. accord to our experience, for most of our results, the frequency interval $ [ 0, 30\nu]$ with $ N=3000 $ in concert with the realistic assumption of $ \Omega_n/\omega_0=0.001\sqrt{\tilde{\omega... | Tecjnically, this approach, wlth $N$ modes beiny taken into zccount, oeans following the dynamics od $7N+3$ rtcl variables. Accordivg to our experiebce, hor most of our cssults, bke frseueney interval $[0, 30\nu]$ eith $N=3000$ togather with the rdapistic assumption of $\Omega_n/\omega_0=0.001\sqre{\tilde{\okeha}_n/\om... | Technically, this approach, with $N$ modes being account, following the of $7N+3$ real for of our results, frequency interval $[0, with $N=3000$ together with the realistic of $\Omega_n/\omega_0=0.001\sqrt{\tilde{\omega}_n/\omega_0}$ satisfy the requirement mentioned at the end of the previous section: an exciting puls... | Technically, this approach, wiTh $N$ modes beIng taKen IntO aCcouNt, meAns following thE DynaMics of $7N+3$ real variables. AcCordiNg TO our EXpErienCe, for moST oF OUr rEsUlTs, tHe FReQuencY inTerval $[0, 30\nU]$ with $N=3000$ togeTheR wIth the realisTIc Assumption Of $\OMega_n/\omega_0=0.001\sqRt{\tIlde{\omEgA}_n/\oM... | Technically, this approach , with $N$ mode s b ein gtake n in to account, me a ns f ollowing the dynamicsof $7 N+ 3 $ re a lvaria bles. A c co r d ing t oour e x pe rienc e,for mos t of our r esu lt s, the frequ e nc y interval $[ 0, 30\nu]$ w ith $N=30 00 $ t o gethe r w ith t he rea l isticassumptio no f $\Om e ... | Technically, this_approach, with_$N$ modes being taken_into account,_means_following the_dynamics_of $7N+3$ real_variables. According to_our experience, for most_of our results,_the_frequency interval $[0, 30\nu]$ with $N=3000$ together with the realistic assumption of $\Omega_n/\omega_0=0.001\sqrt{\tilde{\omega}_n/\om... |
}}^{(n)}_\text{occ}$ is constructed as before and the transformed density matrix $\tilde{\mathbf{P}}^{(n)}$ is calculated from the transformed eigenvector matrix $\tilde{\mathbf{C}}^{(n)}$, $$\begin{aligned}
\tilde{\mathbf{P}}^{(n)} = 2 \hspace{0.1cm} \tilde{\mathbf{C}}^{(n)}_\text{occ} \left(\tilde{\mathbf{C}}^{(n)}_\... | } } ^{(n)}_\text{occ}$ is constructed as before and the transformed density matrix $ \tilde{\mathbf{P}}^{(n)}$ is account from the transformed eigenvector matrix $ \tilde{\mathbf{C}}^{(n)}$, $ $ \begin{aligned }
\tilde{\mathbf{P}}^{(n) } = 2 \hspace{0.1 cm } \tilde{\mathbf{C}}^{(n)}_\text{occ } \left(\tilde{\mathbf{C... | }}^{(n)}_\tedt{occ}$ is constructed as nefore and the ttabsformxd densjty matrkx $\tilde{\mathbf{P}}^{(n)}$ is calculatxd feom tye transformed eigenveztor matrpx $\tilde{\mqthbh{C}}^{(n)}$, $$\begin{aligned}
\vjlde{\matmyf{P}}^{(n)} = 2 \hspccx{0.1cm} \tilde{\mathbf{G}}^{(n)}_\text{occ} \laft(\tilde{\mathbf{W}}^{(n)}_\... | }}^{(n)}_\text{occ}$ is constructed as before and the matrix is calculated the transformed eigenvector 2 \tilde{\mathbf{C}}^{(n)}_\text{occ} \left(\tilde{\mathbf{C}}^{(n)}_\text{occ}\right)^\mathsf{T} \label{eq:P_T_construction}\end{aligned}$$ density matrix $\mathbf{P}^{(n)}$ the original basis $B$ can be by a back-tr... | }}^{(n)}_\text{occ}$ is constructed as beFore and the TransForMed DeNsitY matRix $\tilde{\mathbf{p}}^{(N)}$ is cAlculated from the transfOrmed EiGEnveCToR matrIx $\tilde{\MAtHBF{C}}^{(n)}$, $$\BeGiN{alIgNEd}
\Tilde{\MatHbf{P}}^{(n)} = 2 \hsPace{0.1cm} \tildE{\maThBf{C}}^{(n)}_\text{occ} \lEFt(\Tilde{\mathbF{C}}^{(n)}_\... | }}^{(n)}_\text{occ}$ is co nstructedas be for e a nd the tra nsformed densi t y ma trix $\tilde{\mathbf{P }}^{( n) } $ is ca lcula ted fro m t h e tr an sf orm ed ei genve cto r matri x $\tilde{ \ma th bf{C}}^{(n)} $ ,$$\begin{a lig ned}
\tilde{ \ma thbf{P }} ^{( n )} =2 \ hspac e{0.1c m } \til de{\mathb f{ C }}^{(... | }}^{(n)}_\text{occ}$ is_constructed as_before and the transformed_density matrix_$\tilde{\mathbf{P}}^{(n)}$_is calculated_from_the transformed eigenvector_matrix $\tilde{\mathbf{C}}^{(n)}$, $$\begin{aligned}
\tilde{\mathbf{P}}^{(n)}_= 2 \hspace{0.1cm} \tilde{\mathbf{C}}^{(n)}_\text{occ}_\left(\tilde{\mathbf{C}}^{(n)}_\... |
bm \psi, \bm \tau, \bm \beta)$ is defined as the corresponding optimization objective.
Denote the estimates of the complex gain, the AoA, and the path delay at the $n$th iteration as $\bm \beta^{(n)} $, $\bm \psi^{(n)} $, and $\bm \tau^{(n)} $, respectively. Utilizing the majorization-minorization (MM) iterative appro... | bm \psi, \bm \tau, \bm \beta)$ is defined as the corresponding optimization objective.
Denote the estimates of the complex amplification, the AoA, and the way delay at the $ n$th iteration as $ \bm \beta^{(n) } $, $ \bm \psi^{(n) } $, and $ \bm \tau^{(n) } $, respectively. Utilizing the majorization - minorization (... | bm \osi, \bm \tau, \bm \beta)$ is denined as the cortewpondiig optijization objective.
Denote the estimatxs od the complex gain, the AoA, xnd the pwth delat at rhe $n$th itxdation as $\bm \gcta^{(n)} $, $\um \psi^{(n)} $, and $\bm \tau^{(n)} $, res[ectively. Utilhzkny the majorization-minorization (MM) itqrative aopro... | bm \psi, \bm \tau, \bm \beta)$ is the optimization objective. the estimates of and path delay at $n$th iteration as \beta^{(n)} $, $\bm \psi^{(n)} $, and \tau^{(n)} $, respectively. Utilizing the majorization-minorization (MM) iterative approach [@log-sum-2; @log-sum-3] and similar [@SFW-BS-GC], the optimization in can... | bm \psi, \bm \tau, \bm \beta)$ is defined As the correSpondIng OptImIzatIon oBjective.
Denote THe esTimates of the complex gaiN, the AOA, ANd thE PaTh delAy at the $N$Th ITEraTiOn As $\bM \bETa^{(N)} $, $\bm \psI^{(n)} $, aNd $\bm \tau^{(N)} $, respectivEly. utIlizing the maJOrIzation-minOriZation (MM) iterAtiVe apprO... | bm \psi, \bm \tau, \bm \be ta)$ is de fined as th ecorr espo nding optimiza t ionobjective.
Denote the esti ma t es o f t he co mplex g a in , the A oA , a nd th e pat h d elay at the $n$th it er ation as $\b m \ beta^{(n)} $, $\bm \psi^{ (n) } $, a nd $\ b m \ta u^{ (n)}$, res p ective ly. Utili zi n g the... | bm \psi,_\bm \tau,_\bm \beta)$ is defined_as the_corresponding_optimization objective.
Denote_the_estimates of the_complex gain, the_AoA, and the path_delay at the_$n$th_iteration as $\bm \beta^{(n)} $, $\bm \psi^{(n)} $, and $\bm \tau^{(n)} $, respectively. Utilizing_the_majorization-minorization (MM)_iterative_appro... |
)$, the commutation relation equal to the corresponding commutator of position and momentum, that is, $[A,C]= i\hbar $. Physically, we are looking for an observable that could act as a *generator of correlations*. One can prove that such an operator $A(X,P)$ can not be expanded in a power series $\sum a_{kr} X^{k}P^{r}... | ) $, the commutation relation equal to the corresponding commutator of placement and momentum, that is, $ [ deoxyadenosine monophosphate, C]= i\hbar $. Physically, we are looking for an observable that could act as a * generator of correlation coefficient *. One can prove that such an operator $ A(X, P)$ cannot be boom... | )$, thf commutation relation edual to the cortewpondiig commhtator ow position and momentum, that iw, $[A,C]= u\hbar $. Physically, we afe lookinh for an obstrvable that coulv act as a *gensvator if correlationx*. One can [rove that sucv xn operator $A(X,P)$ can not be expanded ig a powrr series $\sum a_{kt} X^{k}P^{g}... | )$, the commutation relation equal to the of and momentum, is, $[A,C]= i\hbar for observable that could as a *generator correlations*. One can prove that such operator $A(X,P)$ can not be expanded in a power series $\sum a_{kr} X^{k}P^{r}$ that any power series can be brought to this “normal order” with all of at left ... | )$, the commutation relation equAl to the corRespoNdiNg cOmMutaTor oF position and moMEntuM, that is, $[A,C]= i\hbar $. PhysicalLy, we aRe LOokiNG fOr an oBservabLE tHAT coUlD aCt aS a *GEnEratoR of CorrelaTions*. One caN prOvE that such an oPErAtor $A(X,P)$ can Not Be expanded in A poWer serIeS $\suM A_{kr} X^{k}p^{r}... | )$, the commutation relati on equal t o the co rre sp ondi ng c ommutator of p o siti on and momentum, thatis, $ [A , C]=i \h bar $ . Physi c al l y , w ear e l oo k in g for an observ able thatcou ld act as a *g e ne rator of c orr elations*. O necan pr ov e t h at su chan op erator $A(X,P )$ can no tb e expa n de... | )$, the_commutation relation_equal to the corresponding_commutator of_position_and momentum,_that_is, $[A,C]= i\hbar_$. Physically, we_are looking for an_observable that could_act_as a *generator of correlations*. One can prove that such an operator $A(X,P)$ can_not_be expanded_in_a_power series $\sum a_{kr} X^{k}P^{r}... |
(possible PRG’s; shown as filled circles in Figure 7c). None of these implied star-formation rates are high, compared to normal disk galaxies. For those galaxies with known masses (see Table 2), we use these star-formation rates to compute gas-consumption timescales, assuming a Salpeter initial mass function truncate... | (possible PRG ’s; shown as filled circles in Figure 7c). None of these entail asterisk - formation rates are high, compare to normal disk galaxies. For those galaxy with known masses (attend Table 2), we use these star - geological formation rates to compute flatulence - pulmonary tuberculosis timescales, assumin... | (podsible PRG’s; shown as filued circles in Yugure 7c). None kf these implied star-formation rates aee hith, compared to normal aisk galaqies. For rhost galaxies with kikwn masses (ses Tabnx 2), we use these xtar-formathon rates to cmmouce gas-consumption timescales, assumind a Salleher initial mafs flnstioh truncate... | (possible PRG’s; shown as filled circles in None these implied rates are high, For galaxies with known (see Table 2), use these star-formation rates to compute timescales, assuming a Salpeter initial mass function truncated at $M$$<$0.4[M${_\odot}$]{}(Figure 7d). Only one the galaxies in our sample, PRC C-51, could use... | (possible PRG’s; shown as filled Circles in FIgure 7C). NoNe oF tHese ImplIed star-formatiON ratEs are high, compared to norMal diSk GAlaxIEs. for thOse galaXIeS WIth KnOwN maSsES (sEe TabLe 2), wE use theSe star-formAtiOn Rates to compuTE gAs-consumptIon Timescales, asSumIng a SaLpEteR InitiAl mAss fuNction TRuncatE... | (possible PRG’s; shown as filled ci rcles in Fi gu re 7 c).None of thesei mpli ed star-formation rate s are h i gh,c om pared to nor m al d isk g al axi es . F or th ose galaxi es with kn own masses (seeT ab le 2), weuse these star- for mation r ate s to c omp ute g as-con s umptio n timesca le s , assu m ... | (possible_PRG’s; shown_as filled circles in_Figure 7c). None_of_these implied_star-formation_rates are high,_compared to normal_disk galaxies. For those_galaxies with known_ masses_(see Table 2), we use these star-formation rates to compute gas-consumption timescales, assuming a Salpeter_initial_mass function_truncate... |
}
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![FER comparison of proposed model and the model of [@Chatzigeorgiou2008] for case 0: general MIMO channels with $N_T=1$, $N=N_R=1,2,4$, uncoded.[]{data-label="fig.case.0"}](FER_MIMO_Comparison_Result_Without_Outaget_Uncoded){width="80.00000%"}
![Average FER of the HRS scheme for case 1: $\Omega_... | }
= = = = = = = = = = = = = = = = = =
! [ FER comparison of proposed model and the model of [ @Chatzigeorgiou2008 ] for case 0: cosmopolitan MIMO groove with $ N_T=1 $, $ N = N_R=1,2,4 $, uncoded.[]{data - label="fig.case.0"}](FER_MIMO_Comparison_Result_Without_Outaget_Uncoded){width="80.00000% " }
! [ Average FE... | }
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![FER comparison of proposed oodel and the model of [@Chatzjgeorgioj2008] for case 0: general MIMO chainelw wity $N_T=1$, $N=N_R=1,2,4$, uncoded.[]{data-lacel="fig.casv.0"}](FER_MIMO_Cimpacison_Result_Withonf_Outageb_Bncodsf){widch="80.00000%"}
![Everage FER of jhe HRS scheke for case 1: $\Okeea_... | } ================== ![FER comparison of proposed model model [@Chatzigeorgiou2008] for 0: general MIMO ![Average of the HRS for case 1: uncoded.[]{data-label="fig.case.1"}](UnCoded_L100_case1){width="80.00000%"} ![Theoretical and simulated FER of HRS scheme for case 4 and 5: $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$, coded... | }
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![FER comparison of proposed moDel and the mOdel oF [@ChAtzIgEorgIou2008] fOr case 0: general MimO chAnnels with $N_T=1$, $N=N_R=1,2,4$, uncoded.[]{Data-lAbEL="fig.CAsE.0"}](FER_MiMO_CompARiSON_ReSuLt_witHoUT_OUtageT_UnCoded){wiDth="80.00000%"}
![Average fER Of The HRS scheme FOr Case 1: $\Omega_... | }
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![FE R comparis on of pr opo se d mo deland the modelo f [@ Chatzigeorgiou2008] fo r cas e0 : ge n er al MI MO chan n el s wit h$N _T= 1$ , $ N=N_R =1, 2,4$, u ncoded.[]{ dat a- label="fig.c a se .0"}](FER_ MIM O_Comparison _Re sult_W it hou t _Outa get _Unco ded){w i dth="8 0.00000%" }![Aver a ... | }
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![FER comparison_of proposed_model and the model_of [@Chatzigeorgiou2008]_for_case 0:_general_MIMO channels with_$N_T=1$, $N=N_R=1,2,4$, uncoded.[]{data-label="fig.case.0"}](FER_MIMO_Comparison_Result_Without_Outaget_Uncoded){width="80.00000%"}
![Average_FER of the HRS_scheme for case_1:_$\Omega_... |
and $\ell$ and any subset $S \subset\{1,\dots,N\}$: $$\label{eq:e-commute-uv}
e_k(\mathbf{u}_S)\,e_\ell(\mathbf{v}_S)=e_\ell(\mathbf{v}_S)\,e_k(\mathbf{u}_S);$$
- the commutation relation holds for $|S|\le 3$ and any $k, \ell$;
- the commutation relation holds for $|S|\le 3$ and $k\ell\le 3$.
Explicitly, Th... | and $ \ell$ and any subset $ S \subset\{1,\dots, N\}$: $ $ \label{eq: e - commute - uv }
e_k(\mathbf{u}_S)\,e_\ell(\mathbf{v}_S)=e_\ell(\mathbf{v}_S)\,e_k(\mathbf{u}_S);$$
- the substitution relation back holds for $ |S|\le 3 $ and any $ k, \ell$;
- the commutation relation back hold for $ |S|\le 3 ... | anf $\ell$ and any subset $S \sunset\{1,\dots,N\}$: $$\label{eq:e-commuve-uv}
e_k(\mathbw{u}_S)\,e_\ell(\mathbf{v}_S)=e_\ell(\mathbf{v}_S)\,e_n(\mqthbf{y}_S);$$
- the commutation rdlation hllds for $|S|\le 3$ and any $k, \ell$;
- tmz comjmtatimi relation holdx for $|S|\le 3$ and $k\ell\le 3$.
Ex[lkcntly, Th... | and $\ell$ and any subset $S \subset\{1,\dots,N\}$: - commutation relation for $|S|\le 3$ the relation holds for 3$ and $k\ell\le Explicitly, Theorem \[t es commute uv asserts that the commutation relations hold for all $k$ and $\ell$ and all $S \subset \{1,\dots,N\}$ if and only if the following relations hold: $$\beg... | and $\ell$ and any subset $S \subset\{1,\Dots,N\}$: $$\label{Eq:e-coMmuTe-uV}
e_K(\matHbf{u}_s)\,e_\ell(\mathbf{v}_S)=e_\ELl(\maThbf{v}_S)\,e_k(\mathbf{u}_S);$$
- the comMutatIoN RelaTIoN holdS for $|S|\le 3$ ANd ANY $k, \eLl$;
- ThE coMmUTaTion rElaTion holDs for $|S|\le 3$ anD $k\eLl\Le 3$.
Explicitly, tH... | and $\ell$ and any subset $S \subse t\{1, \do ts, N\ }$:$$\l abel{eq:e-comm u te-u v}
e_k(\mathbf{u}_ S)\,e _\ e ll(\ m at hbf{v }_S)=e_ \ el l ( \ma th bf {v} _S ) \, e_k(\ mat hbf{u}_ S);$$
- th ecommutationr el ation hold s f or $|S|\le 3 $ a nd any $ k,\ ell$;
- th e comm u tation relation h o lds f... | and $\ell$_and any_subset $S \subset\{1,\dots,N\}$: $$\label{eq:e-commute-uv}
_ __e_k(\mathbf{u}_S)\,e_\ell(\mathbf{v}_S)=e_\ell(\mathbf{v}_S)\,e_k(\mathbf{u}_S);$$
- __the commutation relation_holds for $|S|\le_3$ and any $k,_\ell$;
- _the_commutation relation holds for $|S|\le 3$ and $k\ell\le 3$.
Explicitly, Th... |
cumulant and cubic response. The latter could be due to either nonlinear dissipation mechanisms or to dependence on actual current temperature maintained by Joulean heating. But, in any case, the left-hand side can not grow in a more fast way than $\propto t$. On the contrary, in presence of $f^{-\gamma }$ excess nois... | cumulant and cubic response. The latter could be due to either nonlinear dissipation mechanisms or to dependence on actual current temperature sustain by Joulean heating system. But, in any case, the left - bridge player english cannot grow in a more fast room than $ \propto t$. On the reverse, in presence of $ f^{-\ga... | cululant and cubic responst. The latter coulb be dux to eifher nonuinear dissipation mechanismd ir to dependence on actual zurrent tvmperaturw mamntained by Joulxzn heatlug. Buf, in cnb case, the left-mand side cdn not grow in a mlre fast way than $\propto t$. On the cjntrary, ij presence of $s^{-\gamkw }$ esbews nois... | cumulant and cubic response. The latter could to nonlinear dissipation or to dependence by heating. But, in case, the left-hand can not grow in a more way than $\propto t$. On the contrary, in presence of $f^{-\gamma }$ excess the first term on right-hand side should grow as $\propto t^{1+\gamma }$. Hence, sufficiently... | cumulant and cubic response. THe latter coUld be Due To eItHer nOnliNear dissipatioN MechAnisms or to dependence on ActuaL cURrenT TeMperaTure maiNTaINEd bY JOuLeaN hEAtIng. BuT, in Any case, The left-hanD siDe Can not grow in A MoRe fast way tHan $\Propto t$. On the ConTrary, iN pResENce of $F^{-\gaMma }$ exCess noIS... | cumulant and cubic respon se. The la ttercou ldbe due toeither nonline a r di ssipation mechanisms o r tode p ende n ce on a ctual c u rr e n t t em pe rat ur e m ainta ine d by Jo ulean heat ing .But, in anyc as e, the lef t-h and side can no t grow i n a morefas t way than$ \propt o t$. Onth e contr a ry, in... | cumulant_and cubic_response. The latter could_be due_to_either nonlinear_dissipation_mechanisms or to_dependence on actual_current temperature maintained by_Joulean heating. But,_in_any case, the left-hand side can not grow in a more fast way than_$\propto_t$. On_the_contrary,_in presence of $f^{-\gamma }$_excess nois... |
Mitas L, Drobný G, Wagner L K and Schmidt K E 2006 *Phys. Rev. Lett.* **96** 130201
Bajdich M, Mitas L, Wagner L K and Schmidt K E 2008 *Phys. Rev.* B **77** 115112
Lüchow A, Petz R and Scott T C 2007 *J. Chem.Phys.* **126** 144110
F A Reboredo, R Q Hood and P R C Kent 2009 *Phys. Rev.* B **79** 195117
Conroy H 19... | Mitas L, Drobný G, Wagner L K and Schmidt K E 2006 * Phys. Rev. Lett. * * * 96 * * 130201
Bajdich M, Mitas L, Wagner L K and Schmidt K E 2008 * Phys. Rev. * B * * 77 * * 115112
Lüchow A, Petz R and Scott T C 2007 * J. Chem. Phys. * * * 126 * * 144110
F A Reboredo, R Q Hood and P R C Kent 2009 * P... | Mihas L, Drobný G, Wagner L K and Schmidt K G 2006 *Phys. Rxv. Lett.* **96** 130201
Gajdich O, Mitas L, Wagner L K and Schliet K T 2008 *Phys. Rev.* B **77** 115112
Lüchow X, Petz R wnd Scotr T R 2007 *J. Chem.Phys.* **126** 144110
F A Reboredo, R Q Glod cnv P R C Kent 2009 *Pmys. Rev.* B **79** 195117
Cotroy H 19... | Mitas L, Drobný G, Wagner L K K 2006 *Phys. Lett.* **96** 130201 L and Schmidt K 2008 *Phys. Rev.* **77** 115112 Lüchow A, Petz R Scott T C 2007 *J. Chem.Phys.* **126** 144110 F A Reboredo, R Q and P R C Kent 2009 *Phys. Rev.* B **79** 195117 Conroy H *J. Phys.* 1331 C J, Wilson K G and Wilkins J W 1988 *Phys. Rev. Let... | Mitas L, Drobný G, Wagner L K and SChmidt K E 2006 *PhYs. Rev. letT.* **96** 130201
BaJdIch M, mitaS L, Wagner L K and SCHmidT K E 2008 *Phys. Rev.* B **77** 115112
Lüchow A, Petz r and SCoTT T C 2007 *J. cHeM.Phys.* **126** 144110
f A ReborEDo, r q hooD aNd p R C keNT 2009 *PHys. ReV.* B **79** 195117
COnroy H 19... | Mitas L, Drobný G, Wagner L K and S chmid t K E20 06 * Phys . Rev. Lett.** *96* * 130201
Bajdich M, M itasL, Wagn e rL K a nd Schm i dt K E20 08 *P hy s .Rev.* B**77**115112
Lü cho wA, Petz R an d S cott T C 2 007 *J. Chem.Ph ys. * **12 6* * 1 4 4110
FA Reb oredo, R Q Ho od and PRC Kent2 009 *Ph y s .Rev. * ... | Mitas_L, Drobný_G, Wagner L K_and Schmidt_K_E 2006_*Phys. Rev. Lett.* **96**_130201
Bajdich M, Mitas_L, Wagner L_K and Schmidt K_E 2008 *Phys. Rev.* B_**77**_115112
Lüchow A, Petz R and Scott T C 2007 *J. Chem.Phys.* **126** 144110
F A Reboredo, R_Q_Hood and_P_R_C Kent 2009 *Phys. Rev.* B **79**_195117
Conroy H 19... |
M$ will generally have different kinematic properties to the structure involving $T^{(2)}M$.
This difference is difficult to see if we only consider the configuration space itself but not how it is constructed. For example, when $M=\mathbb{R}^n$, both $(T\oplus T)M$ and $T^{(2)}M$ can be considered as (or more properl... | M$ will generally have different kinematic properties to the social organization imply $ T^{(2)}M$.
This difference is difficult to visualize if we entirely consider the configuration outer space itself but not how it is constructed. For example, when $ M=\mathbb{R}^n$, both $ (T\oplus T)M$ and $ T^{(2)}M$ can be co... | M$ wlll generally have diffevent kinematic ptopertiev to tge strucgure involving $T^{(2)}M$.
This differxnce is dufficult to see if we unly conspder the xonfmguration space mfself bmc not mow ic ms constructed. Nor example, when $M=\mathbb{R}^t$, coch $(T\oplus T)M$ and $T^{(2)}M$ can be considereq as (or mlre properl... | M$ will generally have different kinematic properties structure $T^{(2)}M$. This is difficult to the space itself but how it is For example, when $M=\mathbb{R}^n$, both $(T\oplus and $T^{(2)}M$ can be considered as (or more properly, are diffeomorphic to) $\mathbb{R}^{3n}$, there are nontheless significant differences,... | M$ will generally have differeNt kinematiC propErtIes To The sTrucTure involving $T^{(2)}m$.
this Difference is difficult tO see iF wE Only COnSider The confIGuRATioN sPaCe iTsELf But noT hoW it is coNstructed. FOr eXaMple, when $M=\matHBb{r}^n$, both $(T\oplUs T)m$ and $T^{(2)}M$ can be cOnsIdered As (Or mORe proPerL... | M$ will generally have dif ferent kin emati c p rop er ties tothe structurei nvol ving $T^{(2)}M$.
This diff er e ncei sdiffi cult to se e ifwe o nly c o ns iderthe config uration sp ace i tself but no t h ow it is c ons tructed. For ex ample, w hen $M=\m ath bb{R} ^n$, b o th $(T \oplus T) M$ and $T ^ {(2)}M... | M$ will_generally have_different kinematic properties to_the structure_involving_$T^{(2)}M$.
This difference_is_difficult to see_if we only_consider the configuration space_itself but not_how_it is constructed. For example, when $M=\mathbb{R}^n$, both $(T\oplus T)M$ and $T^{(2)}M$ can be_considered_as (or_more_properl... |
, T. Vidick, arXiv:1210.1810 \[quant-ph\].
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C.-E. Bardyn et al., Phys. Rev. A **80**, 062327 (2009).
M. McKague, T.H. Yang, and V. Scarani, arXiv:1203.2976 [\[]{}quant-ph[\]]{}.
J.-D. Bancal et al., Phys. Rev. Lett. 106, 250404 (2011).
F. Magniez et al., in... | , T. Vidick, arXiv:1210.1810 \[quant - ph\ ].
J. Silman et al. , Phys. Rev. Lett. * * 106 * *, 220501 (2011).
C.-E. Bardyn et al. , Phys. Rev. A * * 80 * *, 062327 (2009).
M. McKague, T.H. Yang, and V. Scarani, arXiv:1203.2976 [ \[]{}quant - ph[\ ] ] { }.
J.-D. Bancal et al. , Phys. Rev. Lett. 106, 250404 (... | , T. Gidick, arXiv:1210.1810 \[quant-ph\].
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C.-E. Bardyn dt al., Phys. Rev. A **80**, 062327 (2009).
M. McKague, T.Y. Yant, and V. Scarani, arXiv:1203.2976 [\[]{}duant-ph[\]]{}.
J.-D. Bancal wt ao., Phys. Rev. Lett. 106, 250404 (2011).
F. Magnisd et cl., in... | , T. Vidick, arXiv:1210.1810 \[quant-ph\]. J. Silman Phys. Lett. **106**, (2011). C.-E. Bardyn **80**, (2009). M. McKague, Yang, and V. arXiv:1203.2976 [\[]{}quant-ph[\]]{}. J.-D. Bancal et al., Rev. Lett. 106, 250404 (2011). F. Magniez et al., in Proceedings of the International Colloquium on Automata, Languages and P... | , T. Vidick, arXiv:1210.1810 \[quant-ph\].
J. SilmaN et al., Phys. REv. LetT. **106**, 220501 (2011).
C.-E. barDyN et aL., PhyS. Rev. A **80**, 062327 (2009).
M. McKague, T.h. yang, And V. Scarani, arXiv:1203.2976 [\[]{}quant-pH[\]]{}.
J.-D. BaNcAL et aL., phYs. Rev. lett. 106, 250404 (2011).
F. MaGNiEZ Et aL., iN... | , T. Vidick, arXiv:1210.18 10 \[quant -ph\] .
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J.- D. Ban c al etal., Phys .R ev. Le t t. 1... | , T._Vidick, arXiv:1210.1810_\[quant-ph\].
J. Silman et al.,_Phys. Rev._Lett._**106**, 220501_(2011).
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J.-D. Bancal et al., Phys. Rev. Lett. 106, 250404 (2011).
F. Magniez et_al.,_in... |
d_2$ least significant bits of the binary representation of $x$. We thus decompose the raw keys as $X=({\hat{X}},{\check{X}})$, where ${\hat{X}}$ and ${\check{X}}$ denote the sequence of the $d_2$ most and the $d_1$ least significant bits of each key symbol, respectively. The reconciliation module performs the followin... | d_2 $ least significant bits of the binary representation of $ x$. We thus disintegrate the crude keys as $ X=({\hat{X}},{\check{X}})$, where $ { \hat{X}}$ and $ { \check{X}}$ denote the succession of the $ d_2 $ most and the $ d_1 $ least significant snatch of each key symbol, respectively. The reconciliation module p... | d_2$ lfast significant bits of the binary reptewentatmon of $s$. We thur decompose the raw keys as $E=({\hat{Z}},{\checj{X}})$, where ${\hat{X}}$ and ${\checy{X}}$ denote the seqyenct of the $d_2$ most ais the $d_1$ least dignnfmcant bits of esch key sykbol, respectivalh. Che reconciliation module performs tre follpwln... | d_2$ least significant bits of the binary $x$. thus decompose raw keys as denote sequence of the most and the least significant bits of each key respectively. The reconciliation module performs the following steps: (iii-a) Based on the variance her binned raw key and the samples $X_A^\text{pe}$ and $X_B^\text{pe}$, Ali... | d_2$ least significant bits of thE binary repResenTatIon Of $X$. We tHus dEcompose the raw KEys aS $X=({\hat{X}},{\check{X}})$, where ${\hat{X}}$ aNd ${\cheCk{x}}$ DenoTE tHe seqUence of THe $D_2$ MOst AnD tHe $d_1$ LeASt SigniFicAnt bits Of each key sYmbOl, Respectively. tHe ReconciliaTioN module perfoRms The folLoWin... | d_2$ least significant bit s of the b inary re pre se ntat ionof $x$. We thu s dec ompose the raw keys as $X=( {\ h at{X } }, {\che ck{X}}) $ ,w h ere $ {\ hat {X } }$ and${\ check{X }}$ denote th esequence oft he $d_2$ mos t a nd the $d_1$ le ast si gn ifi c ant b its of e ach ke y symbo l, respec ti v ely. ... | d_2$ least_significant bits_of the binary representation_of $x$._We_thus decompose_the_raw keys as_$X=({\hat{X}},{\check{X}})$, where ${\hat{X}}$_and ${\check{X}}$ denote the_sequence of the_$d_2$_most and the $d_1$ least significant bits of each key symbol, respectively. The reconciliation_module_performs the_followin... |
(u, \omega) = A(u, -\omega)^*$, where $A^*$ is the complex conjugate of $A$. Formally, we consider $\mathbb{R}^N$–valued time series of length $T$, $\left\{ \mb X_t : t =1, \dots, T \right\}$, of the form $$\label{eq:X}
\mb X_t = \int_{-1/2}^{1/2} A(t/T, \omega) \exp{(2 \pi i \omega t)} d \mb Z(\omega),$$ for a time-v... | ( u, \omega) = A(u, -\omega)^*$, where $ A^*$ is the complex conjugate of $ A$. Formally, we consider $ \mathbb{R}^N$–valued time serial of duration $ T$, $ \left\ { \mb X_t: t = 1, \dots, T \right\}$, of the form $ $ \label{eq :X }
\mb X_t = \int_{-1/2}^{1/2 } A(t / T, \omega) \exp{(2 \pi i \omega t) } d \mb Z(\om... | (u, \olega) = A(u, -\omega)^*$, where $A^*$ ir the complex conjugatx of $A$. Rormally, we consider $\mathbb{R}^N$–valued vime seritf of length $T$, $\left\{ \mb X_t : t =1, \dots, T \rigit\}$, of the form $$\legel{eq:X}
\mn X_t = \lnt_{-1/2}^{1/2} A(c/T, \omega) \exp{(2 \pi l \omega t)} g \mb Z(\omega),$$ fos x cime-v... | (u, \omega) = A(u, -\omega)^*$, where $A^*$ complex of $A$. we consider $\mathbb{R}^N$–valued $\left\{ X_t : t \dots, T \right\}$, the form $$\label{eq:X} \mb X_t = A(t/T, \omega) \exp{(2 \pi i \omega t)} d \mb Z(\omega),$$ for a time-varying function $A(u, \omega)$ and an $N$–dimensional mean-zero orthogonal process $... | (u, \omega) = A(u, -\omega)^*$, where $A^*$ is the cOmplex conjUgate Of $A$. forMaLly, wE conSider $\mathbb{R}^N$–vALued Time series of length $T$, $\lefT\{ \mb X_t : T =1, \dOTs, T \rIGhT\}$, of thE form $$\laBEl{EQ:x}
\mb x_t = \InT_{-1/2}^{1/2} A(t/t, \oMEgA) \exp{(2 \pI i \oMega t)} d \mB Z(\omega),$$ for A tiMe-V... | (u, \omega) = A(u, -\omega )^*$, wher e $A^ *$isth e co mple x conjugate of $A$. Formally, we consider $\ma th b b{R} ^ N$ –valu ed time se r i esof l eng th $T $, $\ lef t\{ \mb X_t : t = 1,\d ots, T \righ t \} $, of thefor m $$\label{e q:X }
\mbX_ t = \int_ {-1 /2}^{ 1/2} A ( t/T, \ omega) \e xp { (2 \pi i \omeg ... | (u, \omega)_= A(u,_-\omega)^*$, where $A^*$ is_the complex_conjugate_of $A$._Formally,_we consider $\mathbb{R}^N$–valued_time series of_length $T$, $\left\{ \mb_X_t : t_=1,_\dots, T \right\}$, of the form $$\label{eq:X}
\mb X_t = \int_{-1/2}^{1/2} A(t/T, \omega) \exp{(2 \pi_i_\omega t)}__d_\mb Z(\omega),$$ for a time-v... |
-varying covariate (e.g., presence of emphysema) that is a predictor of both future exposure and of failure. In 1982, the standard analytic approach was to model the conditional probability (i.e., the hazard) of failure time $t$ as a function of past exposure history using a time-dependent Cox proportional hazards mode... | -varying covariate (e.g., presence of emphysema) that is a predictor of both future exposure and of bankruptcy. In 1982, the standard analytic overture was to model the conditional probability (i.e., the hazard) of failure meter $ t$ as a routine of past exposure history using a prison term - subject Cox proportional... | -varjing covariate (e.g., presenct of emphysema) thcr is a predidtor of coth future exposure and of hailyre. Ib 1982, the standard analytkc approabh was to modtl the conditional probabljity (l.e., thz iazard) of failute time $t$ as a function of pxsc exposure history using a time-depenqent Coc oroportional hwzarcf mosv... | -varying covariate (e.g., presence of emphysema) that predictor both future and of failure. approach to model the probability (i.e., the of failure time $t$ as a of past exposure history using a time-dependent Cox proportional hazards model. Robins formally that, even when confounding by unmeasured factors and model sp... | -varying covariate (e.g., presencE of emphyseMa) thaT is A prEdIctoR of bOth future exposURe anD of failure. In 1982, the standarD analYtIC appROaCh was To model THe CONdiTiOnAl pRoBAbIlity (I.e., tHe hazarD) of failure TimE $t$ As a function oF PaSt exposure HisTory using a tiMe-dEpendeNt cox PRoporTioNal haZards mODe... | -varying covariate (e.g.,presence o f emp hys ema )that isa predictor of both future exposure and o f fai lu r e. I n 1 982,the sta n da r d an al yt icap p ro ach w asto mode l the cond iti on al probabili t y(i.e., the ha zard) of fai lur e time $ t$a s a f unc tionof pas t expos ure histo ry usinga time-d e ... | -varying covariate_(e.g., presence of_emphysema) that is a_predictor of_both_future exposure_and_of failure. In_1982, the standard_analytic approach was to_model the conditional_probability_(i.e., the hazard) of failure time $t$ as a function of past exposure history_using_a time-dependent_Cox_proportional_hazards mode... |
is non-zero. This completes the proof of the theorem.
1. The proof of Theorem\[cuplength\] actually shows that if some product $x=x_1\cdots x_t\neq 0$ with $1\leq \mathrm{deg}(x_i)\leq \ell$, then for any vector bundle $\xi$ over $X$ with $\mathrm{charrank}_X(\xi)\geq\ell$ some product of the Stiefel-Whitney classes... | is non - zero. This completes the proof of the theorem.
1. The proof of Theorem\[cuplength\ ] actually shows that if some merchandise $ x = x_1\cdots x_t\neq 0 $ with $ 1\leq \mathrm{deg}(x_i)\leq \ell$, then for any vector package $ \xi$ over $ X$ with $ \mathrm{charrank}_X(\xi)\geq\ell$ some product of the Stief... | is non-zero. This completes uhe proof of the jhworem.
1. The pdoof of Gheorem\[cuplength\] actually shlww thau if some product $x=b_1\cdots x_t\jeq 0$ wity $1\lew \mathrm{deg}(e_j)\leq \ell$, then nor auy vector bundle $\xi$ over $X$ with $\mathrm{chdrfauk}_X(\xi)\geq\ell$ some product of the Stiesel-Whitmej classes... | is non-zero. This completes the proof of 1. proof of actually shows that 0$ $1\leq \mathrm{deg}(x_i)\leq \ell$, for any vector $\xi$ over $X$ with $\mathrm{charrank}_X(\xi)\geq\ell$ some of the Stiefel-Whitney classes of $\xi$ of length greater than or equal to is non-zero. 2. The conclusion of Theorem\[cuplength\] is ... | is non-zero. This completes the Proof of the TheorEm.
1. THe pRoOf of theoRem\[cuplength\] acTUallY shows that if some producT $x=x_1\cdOtS X_t\neQ 0$ WiTh $1\leq \Mathrm{dEG}(x_I)\LEq \eLl$, ThEn fOr ANy VectoR buNdle $\xi$ oVer $X$ with $\maThrM{cHarrank}_X(\xi)\geQ\ElL$ some produCt oF the Stiefel-WHitNey claSsEs... | is non-zero. This complet es the pro of of th e t he orem .
1 . The proof o f The orem\[cuplength\] actu allysh o ws t h at if s ome pro d uc t $x= x_ 1\ cdo ts x_ t\neq 0$ with $ 1\leq \mat hrm {d eg}(x_i)\leq \e ll$, thenfor any vectorbun dle $\ xi $ o v er $X $ w ith $ \mathr m {charr ank}_X(\x i) \ geq\el ... | is_non-zero. This_completes the proof of_the theorem.
1.__The proof_of_Theorem\[cuplength\] actually shows_that if some_product $x=x_1\cdots x_t\neq 0$_with $1\leq \mathrm{deg}(x_i)\leq_\ell$,_then for any vector bundle $\xi$ over $X$ with $\mathrm{charrank}_X(\xi)\geq\ell$ some product of the_Stiefel-Whitney_classes... |
spaced subsets of $S^1_{eucl}$ is somewhat unnatural; we leave open the following question, which would define the ‘upper and lower Rips magnitude’ of the circle intrinsically:
Does this asymptotic behaviour extend to arbitrary finite subsets of $S^1_{eucl}$? For instance, given any $\epsilon>0$, is there a $\delta>0... | spaced subsets of $ S^1_{eucl}$ is somewhat unnatural; we bequeath loose the following question, which would define the ‘ upper and lower Rips order of magnitude ’ of the circle intrinsically:
Does this asymptotic behaviour extend to arbitrary finite subset of $ S^1_{eucl}$? For example, given any $ \epsilon>0 $, is... | spwced subsets of $S^1_{eucl}$ is somewhat unnatorql; we neave kpen the following question, which wonld eefint the ‘upper and lowdr Rips mwgnitude’ of uhe circle intrinsically:
Does thjd asvm'totic behaviout extend to drbitrary finide sbbsets of $S^1_{eucl}$? For instance, given agy $\epsikoj>0$, is there a $\dglta>0... | spaced subsets of $S^1_{eucl}$ is somewhat unnatural; open following question, would define the of circle intrinsically: Does asymptotic behaviour extend arbitrary finite subsets of $S^1_{eucl}$? For given any $\epsilon>0$, is there a $\delta>0$ such that for all finite $A\subseteq with $d_H(A,S^1_{eucl})<\delta$ we ha... | spaced subsets of $S^1_{eucl}$ is somEwhat unnatUral; wE leAve OpEn thE folLowing question, WHich Would define the ‘upper and Lower riPS magNItUde’ of The circLE iNTRinSiCaLly:
doES tHis asYmpTotic beHaviour extEnd To Arbitrary finITe Subsets of $S^1_{EucL}$? For instance, GivEn any $\ePsIloN>0$, Is theRe a $\Delta>0... | spaced subsets of $S^1_{e ucl}$ is s omewh atunn at ural ; we leave open th e fol lowing question, which woul dd efin e t he ‘u pper an d l o w erRi ps ma gn i tu de’ o f t he circ le intrins ica ll y:
Does thi s a symptoticbeh aviour exten d t o arbi tr ary finit e s ubset s of $ S ^1_{eu cl}$? For i n stanc... | spaced_subsets of_$S^1_{eucl}$ is somewhat unnatural;_we leave_open_the following_question,_which would define_the ‘upper and_lower Rips magnitude’ of_the circle intrinsically:
Does_this_asymptotic behaviour extend to arbitrary finite subsets of $S^1_{eucl}$? For instance, given any $\epsilon>0$,_is_there a_$\delta>0... |
zebruch class, with $\theta(fh)=[fh]$ the relative fundamental class.
Moreover, these characteristic classes commute with the corresponding orientations $\theta$ of a smooth morphism (as already explained before). So we only have to show that
- the corresponding Grothendieck transformation $$\ga_{c\ell}=:\La_y^{mot... | zebruch class, with $ \theta(fh)=[fh]$ the relative fundamental class.
furthermore, these characteristic class commute with the corresponding orientation $ \theta$ of a fluent morphism (as already explained ahead). indeed we only suffer to show that
- the corresponding Grothendieck transformation $ $ \ga_{c\el... | zebguch class, with $\theta(fh)=[fm]$ the relative foneamentel clasa.
Moreovef, these characteristic classxs cimmutt with the correspovding orivntations $\theua$ of a smooth moclhism (as alrezfy erpoained before). Xo we only have to show dhxt
- the corresponding Grothendieck twansforkahion $$\ga_{c\ell}=:\La_y^{iot... | zebruch class, with $\theta(fh)=[fh]$ the relative fundamental these classes commute the corresponding orientations (as explained before). So only have to that - the corresponding Grothendieck transformation \bM(\m V/X \xrightarrow{f} Y) \to \bK(X \xrightarrow{f} Y)$$ from Corollary \[twisting\] vanishes on subgroup $\... | zebruch class, with $\theta(fh)=[fh]$ The relativE fundAmeNtaL cLass.
moreOver, these charaCTeriStic classes commute with The coRrESponDInG orieNtationS $\ThETA$ of A sMoOth MoRPhIsm (as AlrEady expLained befoRe). SO wE only have to sHOw That
- the corResPonding GrothEndIeck trAnSfoRMatioN $$\ga_{C\ell}=:\LA_y^{mot... | zebruch class, with $\thet a(fh)=[fh] $ the re lat iv e fu ndam ental class.
M oreo ver, these characteris tic c la s sesc om mutewith th e c o r res po nd ing o r ie ntati ons $\thet a$ of a sm oot hmorphism (as al ready expl ain ed before).Sowe onl yhav e to s how that
- t he cor respondin gG rothen d ieck t... | zebruch class,_with $\theta(fh)=[fh]$_the relative fundamental class.
Moreover,_these characteristic_classes_commute with_the_corresponding orientations $\theta$_of a smooth_morphism (as already explained_before). So we_only_have to show that
- the corresponding Grothendieck transformation $$\ga_{c\ell}=:\La_y^{mot... |
LFPs at each electrode were recorded for 18s while the fly was awake and 18s more after the fly was anaesthetised (isoflurane, 0.6% by volume, through an evaporator). Flies’ unresponsiveness during anaesthesia was confirmed by the absence of behavioural responses to a series of air puffs, and recovery was also confirm... | LFPs at each electrode were recorded for 18s while the fly was awake and eighteen more after the tent-fly was anaesthetised (isoflurane, 0.6% by book, through an evaporator). Flies ’ unresponsiveness during anesthesia was confirmed by the absence of behavioral responses to a series of air travel quilt, and recovery was... | LFOs at each electrode wert recorded for 18s cyile tie fly sas awakd and 18s more after the fly wes abaestyetised (isoflurane, 0.6% by volume, tjrough ab eveporator). Flies’ uidesponslrenesa durnnj anaesthesia wss confirmad by the absetcd lf behavioural responses to a serief of ait ouffs, and recodery ras zlso confirm... | LFPs at each electrode were recorded for the was awake 18s more after 0.6% volume, through an Flies’ unresponsiveness during was confirmed by the absence of responses to a series of air puffs, and recovery was also confirmed after gas was turned off [@CohenEneuro2016]. We used data sampled at 1kHz for the [@CohenEneuro... | LFPs at each electrode were reCorded for 18s While The Fly WaS awaKe anD 18s more after the FLy waS anaesthetised (isofluraNe, 0.6% by vOlUMe, thROuGh an eVaporatOR). FLIEs’ uNrEsPonSiVEnEss duRinG anaestHesia was coNfiRmEd by the absenCE oF behaviourAl rEsponses to a sEriEs of aiR pUffS, And reCovEry waS also cONfirm... | LFPs at each electrode we re recorde d for 18 s w hi le t he f ly was awake a n d 18 s more after the fly w as an ae s thet i se d (is ofluran e ,0 . 6%by v olu me , t hroug h a n evapo rator). Fl ies ’unresponsive n es s during a nae sthesia wascon firmed b y t h e abs enc e ofbehavi o ural r esponsesto a seri e ... | LFPs_at each_electrode were recorded for_18s while_the_fly was_awake_and 18s more_after the fly_was anaesthetised (isoflurane, 0.6%_by volume, through_an_evaporator). Flies’ unresponsiveness during anaesthesia was confirmed by the absence of behavioural responses to_a_series of_air_puffs,_and recovery was also confirm... |
\end{array}\right)V^{\mathrm{T}}.$$ Moreover, amplitudes (\[eq:explicit\_e(p)\]) fulfil the following conditions:
\[seq:amplitudes\] $$\begin{gathered}
p_\mu e^{\mu}_{\phantom{\mu}\sigma}(p) = 0,
\label{eq:amplitudes_eq1} \\
e^{*\mu}_{\phantom{*\mu}\sigma}(p) e_{\mu\sigma^\prime}(p) =
- \delta_{\sig... | \end{array}\right)V^{\mathrm{T}}.$$ Moreover, amplitudes (\[eq: explicit\_e(p)\ ]) fulfil the following condition:
\[seq: amplitudes\ ] $ $ \begin{gathered }
p_\mu e^{\mu}_{\phantom{\mu}\sigma}(p) = 0,
\label{eq: amplitudes_eq1 } \\
e^{*\mu}_{\phantom{*\mu}\sigma}(p) e_{\mu\sigma^\prime}(p) =
... |
\fnd{array}\right)V^{\mathrm{T}}.$$ Moveover, amplitudes (\[eq:expnicit\_e(l)\]) fulfil the following conditions:
\[seq:emplutudew\] $$\begin{gathered}
p_\mu e^{\mu}_{\phantlm{\mu}\sigmq}(p) = 0,
\label{eq:amplitudes_eq1} \\
e^{*\mu}_{\'hentom{*\mu}\sigma}(p) e_{\ku\sigma^\prike}(p) =
- \delta_{\vie... | \end{array}\right)V^{\mathrm{T}}.$$ Moreover, amplitudes (\[eq:explicit\_e(p)\]) fulfil the following $$\begin{gathered} e^{\mu}_{\phantom{\mu}\sigma}(p) = \label{eq:amplitudes_eq1} \\ e^{*\mu}_{\phantom{*\mu}\sigma}(p) \\ e_{\mu\sigma^\prime}(p) = -(VV^{\mathrm{T}})_{\sigma\sigma^\prime}, \\ e^{*\mu}_{\phantom{*\mu}\s... |
\end{array}\right)V^{\mathrm{T}}.$$ MoreOver, amplitUdes (\[eQ:exPliCiT\_e(p)\]) fUlfiL the following cONditIons:
\[seq:amplitudes\] $$\begin{GatheReD}
P_\mu e^{\MU}_{\pHantoM{\mu}\sigmA}(P) = 0,
\lABEl{eQ:aMpLitUdES_eQ1} \\
e^{*\mu}_{\pHanTom{*\mu}\siGma}(p) e_{\mu\sigMa^\pRiMe}(p) =
- \delta_{\sig... |
\end{array}\right)V^{\ mathrm{T}} .$$ M ore ove r, amp litu des (\[eq:expl i cit\ _e(p)\]) fulfil the fo llowi ng cond i ti ons:
\[seq: a mp l i tud es \] $$ \b e gi n{gat her ed}
p_\mu e^{ \mu }_ {\phantom{\m u }\ sigma}(p)= 0 ,
\label{ eq: amplit ud es_ e q1} \ \
e^ {*\mu} _ {\phan tom{*\mu} \s i gma}(... |
_ \end{array}\right)V^{\mathrm{T}}.$$_Moreover, amplitudes (\[eq:explicit\_e(p)\]) fulfil_the following_conditions:
\[seq:amplitudes\]_$$\begin{gathered}
__ p_\mu e^{\mu}_{\phantom{\mu}\sigma}(p)_= 0,
_ \label{eq:amplitudes_eq1} \\
_ e^{*\mu}_{\phantom{*\mu}\sigma}(p)_e_{\mu\sigma^\prime}(p)_=
- \delta_{\sig... |
{aligned}$$
By inverting relation (\[z\_of\_f\]) one obtains an explicit equation for the amplitude of the order parameter as a function of position along the wire (see Fig. \[u\_of\_x\]):
$$\begin{aligned}
f^2(z) &= u_0 + u_1 \sin^2 \big[\text{JacobiAmplitude}\big[z
\sqrt{\frac{u_2}{2}},\frac{u_1}{u_2}\big]\big]\\
&... | { aligned}$$
By inverting relation (\[z\_of\_f\ ]) one obtains an explicit equality for the amplitude of the decree parameter as a function of placement along the telegram (see Fig. \[u\_of\_x\ ] ):
$ $ \begin{aligned }
f^2(z) & = u_0 + u_1 \sin^2 \big[\text{JacobiAmplitude}\big[z
\sqrt{\frac{u_2}{2}},\frac... | {alihned}$$
By inverting relatiok (\[z\_of\_f\]) one obtaiuw an eeplicit equatiov for the amplitude of the ocder paraneter as a function of position along tye wmre (see Fig. \[u\_of\_x\]):
$$\bxfin{aligkzd}
f^2(z) &= m_0 + u_1 \vmn^2 \big[\text{JacoblAmplitude}\bhg[z
\sqrt{\frac{u_2}{2}},\frdc{j_1}{u_2}\yig]\big]\\
&... | {aligned}$$ By inverting relation (\[z\_of\_f\]) one obtains equation the amplitude the order parameter along wire (see Fig. $$\begin{aligned} f^2(z) &= + u_1 \sin^2 \big[\text{JacobiAmplitude}\big[z \sqrt{\frac{u_2}{2}},\frac{u_1}{u_2}\big]\big]\\ &=u_0+u_1 \sqrt{\frac{u_2}{2}},\frac{u_1}{u_2}\big] \label{f_of_z}\end{... | {aligned}$$
By inverting relatioN (\[z\_of\_f\]) one obTains An eXplIcIt eqUatiOn for the amplitUDe of The order parameter as a fuNctioN oF PosiTIoN alonG the wirE (SeE fIg. \[u\_Of\_X\]):
$$\bEgiN{aLIgNed}
f^2(z) &= U_0 + u_1 \sIn^2 \big[\teXt{JacobiAmPliTuDe}\big[z
\sqrt{\frAC{u_2}{2}},\Frac{u_1}{u_2}\big]\bIg]\\
&... | {aligned}$$
By invertingrelation ( \[z\_ of\ _f\ ]) one obt ains an explic i t eq uation for the amplitu de of t h e or d er para meter a s a f unc ti on of p o si tionalo ng thewire (seeFig .\[u\_of\_x\] ) :
$$\begin{ ali gned}
f^2(z) &= u_0 + u _1\ sin^2 \b ig[\t ext{Ja c obiAmp litude}\b ig [ z
\sqr t {\... | {aligned}$$
By inverting_relation (\[z\_of\_f\])_one obtains an explicit_equation for_the_amplitude of_the_order parameter as_a function of_position along the wire_(see Fig. \[u\_of\_x\]):
$$\begin{aligned}
f^2(z) &=_u_0_+ u_1 \sin^2 \big[\text{JacobiAmplitude}\big[z
\sqrt{\frac{u_2}{2}},\frac{u_1}{u_2}\big]\big]\\
&... |
we arrive at the bound $$\label{eqn:gm-am-temp2}
\begin{aligned}
{\operatorname{tr}}\big[ {{\bm{H}}}{{\bm{W}}}^q {{\bm{H}}} {{\bm{Y}}}^{2r-q} \big]
+ {\operatorname{tr}}\big[ {{\bm{H}}}{{\bm{W}}}^{2r-q} {{\bm{H}}} {{\bm{Y}}}^{q} \big]
&\leq \sum_{i,j=1}^d \big( {\left\vert {\lambda_i} \right\vert}^q {\left\vert {\... | we arrive at the bound $ $ \label{eqn: gm - am - temp2 }
\begin{aligned }
{ \operatorname{tr}}\big [ { { \bm{H}}}{{\bm{W}}}^q { { \bm{H } } } { { \bm{Y}}}^{2r - q } \big ]
+ { \operatorname{tr}}\big [ { { \bm{H}}}{{\bm{W}}}^{2r - q } { { \bm{H } } } { { \bm{Y}}}^{q } \big ]
& \leq \sum_{i, j=1}^d \big ({ \l... | we arrive at the bound $$\labtl{eqn:gm-am-temp2}
\begiu{qlignev}
{\operatkrname{tr}}\cig[ {{\bm{H}}}{{\bm{W}}}^q {{\bm{H}}} {{\bm{Y}}}^{2r-q} \big]
+ {\opxratirnamt{nr}}\big[ {{\bm{H}}}{{\bm{W}}}^{2r-q} {{\bm{H}}} {{\bm{H}}}^{q} \big]
&\leq \sum_{u,j=1}^d \uig( {\left\vert {\lamusa_i} \rigmc\vert}^s {\lefc\vxrt {\... | we arrive at the bound $$\label{eqn:gm-am-temp2} \begin{aligned} {{\bm{H}}} \big] + {{\bm{H}}}{{\bm{W}}}^{2r-q} {{\bm{H}}} {{\bm{Y}}}^{q} {\lambda_i} {\left\vert {\smash{\mu_j}} \right\vert}^{2r-q} {\left\vert {\lambda_i} \right\vert}^{2r-q} {\smash{\mu_j}} \right\vert}^{q}\big) \cdot {{{\left\vert { {{\bm{u}}}_i^{*}{{... | we arrive at the bound $$\label{eqN:gm-am-temp2}
\bEgin{aLigNed}
{\OpEratOrnaMe{tr}}\big[ {{\bm{H}}}{{\bm{W}}}^q {{\BM{H}}} {{\bm{y}}}^{2r-q} \big]
+ {\operatorname{tr}}\biG[ {{\bm{H}}}{{\bM{W}}}^{2R-Q} {{\bm{H}}} {{\BM{Y}}}^{Q} \big]
&\lEq \sum_{i,j=1}^D \BiG( {\LEft\VeRt {\LamBdA_I} \rIght\vErt}^Q {\left\veRt {\... | we arrive at the bound $$ \label{eqn :gm-a m-t emp 2}
\be gin{ aligned}
{\ope r ator name{tr}}\big[ {{\bm{H }}}{{ \b m {W}} } ^q {{\b m{H}}}{ {\ b m {Y} }} ^{ 2r- q} \b ig]
+ {\ operato rname{tr}} \bi g[ {{\bm{H}}}{ { \b m{W}}}^{2r -q} {{\bm{H}}}{{\ bm{Y}} }^ {q} \big]
&\l eq \su m _{i,j= 1}^d \big ({ \left\ ... | we_arrive at_the bound $$\label{eqn:gm-am-temp2}
\begin{aligned}
{\operatorname{tr}}\big[ {{\bm{H}}}{{\bm{W}}}^q_{{\bm{H}}} {{\bm{Y}}}^{2r-q}_\big]
+_{\operatorname{tr}}\big[ {{\bm{H}}}{{\bm{W}}}^{2r-q}_{{\bm{H}}}_{{\bm{Y}}}^{q} \big]
_ &\leq_\sum_{i,j=1}^d \big( {\left\vert {\lambda_i}_\right\vert}^q {\left\vert {\... |
bf q} = V/N$ where $V$ is the strength of the impurity potential. In what follows we use standard finite temperature Green’s function formalism [@DS88; @BF04]. Because of the existence of two sub-lattices, the Green’s function can be written as a $2\times 2$ matrix: $$\begin{aligned}
\bm G_{\sigma}(\bm k,{\bf p},\tau) ... | bf q } = V / N$ where $ V$ is the strength of the impurity potential. In what follows we practice standard finite temperature Green ’s affair formalism [ @DS88; @BF04 ]. Because of the existence of two sub - lattice, the Green ’s routine can be written as a $ 2\times 2 $ matrix: $ $ \begin{aligned }
\bm G_{\sigma}(\b... | bf e} = V/N$ where $V$ is the strtngth of the imputiry potxntial. Jn what wollows we use standard finive twmperqture Green’s function wormalism [@DS88; @BF04]. Bwcauww of the eejstence of twk sub-nettices, the Gregn’s function can be writtet xs a $2\times 2$ matrix: $$\begin{aligned}
\bm G_{\sidma}(\bm k,{\nf p},\tau) ... | bf q} = V/N$ where $V$ is of impurity potential. what follows we function [@DS88; @BF04]. Because the existence of sub-lattices, the Green’s function can be as a $2\times 2$ matrix: $$\begin{aligned} \bm G_{\sigma}(\bm k,{\bf p},\tau) = \left(\begin{array}{cc} G_{AA,\sigma} k,{\bf p}, \tau) \hspace{0.5cm} & G_{AB,\sigm... | bf q} = V/N$ where $V$ is the strength oF the impuriTy potEntIal. in What FollOws we use standaRD finIte temperature Green’s fuNctioN fORmalISm [@dS88; @BF04]. BEcause oF ThE EXisTeNcE of TwO SuB-lattIceS, the GreEn’s functioN caN bE written as a $2\tIMeS 2$ matrix: $$\begIn{aLigned}
\bm G_{\sigMa}(\bM k,{\bf p},\tAu) ... | bf q} = V/N$ where $V$ isthe streng th of th e i mp urit y po tential. In wh a t fo llows we use standardfinit et empe r at ure G reen’sf un c t ion f or mal is m [ @DS88 ; @ BF04].Because of th eexistence of tw o sub-latt ice s, the Green ’sfuncti on ca n be w rit ten a s a $2 \ times2$ matrix :$ $\begi n {align... | bf q}_= V/N$_where $V$ is the_strength of_the_impurity potential._In_what follows we_use standard finite_temperature Green’s function formalism_[@DS88; @BF04]. Because_of_the existence of two sub-lattices, the Green’s function can be written as a $2\times_2$_matrix: $$\begin{aligned}
\bm_G_{\sigma}(\bm_k,{\bf_p},\tau) ... |
the data.
A large body of literature has been devoted to studying codes that are minimax optimal with respect to (\[Redundancy\]), exactly [@Topsoe79; @Shtarkov87en2; @Haussler97] or asymptotically [@BarronRissanenYu98; @Grunwald07]. Let us notice that if the minimax expected Shannon redundancy $$\begin{aligned}
\l... | the data.
A large body of literature has been devoted to study code that are minimax optimal with respect to (\[Redundancy\ ]), exactly [ @Topsoe79; @Shtarkov87en2; @Haussler97 ] or asymptotically [ @BarronRissanenYu98; @Grunwald07 ]. get us notice that if the minimax expected Shannon redundancy $ $ \begin{aligned... | thf data.
A large body of littrature has been bwvoted to sthdying cudes that are minimax optimap qith eespect to (\[Redundancy\]), dxactly [@Tlpsoe79; @Shrarkic87en2; @Hausslxd97] or asniptoflcallv [@UarronRissanenYo98; @Grunwald07]. Lat us notice tvag nf the minimax expected Shannon redugdancy $$\nehin{aligned}
\l... | the data. A large body of literature devoted studying codes are minimax optimal [@Topsoe79; @Haussler97] or asymptotically @Grunwald07]. Let us that if the minimax expected Shannon $$\begin{aligned} \label{MinimaxRed} \min_{C} \sup_{P\in\mathcal{M}} {\textbf{E}\, _}{x\sim P} {\left[ |C(x)|+\log P(x) \right]}\end{aligne... | the data.
A large body of literaTure has beeN devoTed To sTuDyinG codEs that are minimAX optImal with respect to (\[RedunDancy\]), ExACtly [@tOpSoe79; @ShTarkov87eN2; @haUSSleR97] oR aSymPtOTiCally [@barRonRissAnenYu98; @GrunWalD07]. LEt us notice thAT iF the minimaX exPected ShannoN reDundanCy $$\BegIN{aligNed}
\L... | the data.
A large body o f literatu re ha s b een d evot ed t o studying cod e s th at are minimax optimal with r e spec t t o (\[ Redunda n cy \ ] ),ex ac tly [ @ To psoe7 9;@Shtark ov87en2; @ Hau ss ler97] or as y mp totically[@B arronRissane nYu 98; @G ru nwa l d07]. Le t usnotice that i f the min im a x exp... | the_data.
A large body_of literature has been_devoted to_studying_codes that_are_minimax optimal with_respect to (\[Redundancy\]),_exactly [@Topsoe79; @Shtarkov87en2; @Haussler97]_or asymptotically [@BarronRissanenYu98;_@Grunwald07]._Let us notice that if the minimax expected Shannon redundancy $$\begin{aligned}
\l... |
mathtt{i})$. If ${{\sc\mbox{C}\hspace{1.0pt}}}$ has more than one object, one can fix an ordering of its objects and define the rank of a finitary $2$-representation of ${{\sc\mbox{C}\hspace{1.0pt}}}$ as a suitable tuple of positive integers. However, in this document we will only consider $2$-categories with a single ... | mathtt{i})$. If $ { { \sc\mbox{C}\hspace{1.0pt}}}$ has more than one object, one can fix an ordering of its object and specify the rank of a finitary $ 2$-representation of $ { { \sc\mbox{C}\hspace{1.0pt}}}$ as a suitable tuple of positive integer. However, in this document we will only consider $ 2$-categories with a ... | matjtt{i})$. If ${{\sc\mbox{C}\hspace{1.0pt}}}$ mas more than ong ibject, one czn fix av ordering of its objects anv dedine uke rank of a finitarh $2$-represejtation if ${{\sr\mbox{C}\hspace{1.0pt}}}$ as a suitable thile oy 'ositive integets. However, it this documend de will only consider $2$-categories with a singke ... | mathtt{i})$. If ${{\sc\mbox{C}\hspace{1.0pt}}}$ has more than one can an ordering its objects and finitary of ${{\sc\mbox{C}\hspace{1.0pt}}}$ as suitable tuple of integers. However, in this document we only consider $2$-categories with a single object. We say that ${{\sc\mbox{C}\hspace{1.0pt}}}$ is [*weakly if it is fi... | mathtt{i})$. If ${{\sc\mbox{C}\hspace{1.0pt}}}$ hAs more than One obJecT, onE cAn fiX an oRdering of its obJEcts And define the rank of a finItary $2$-RePReseNTaTion oF ${{\sc\mbox{c}\HsPACe{1.0pT}}}$ aS a SuiTaBLe Tuple Of pOsitive Integers. HoWevEr, In this documeNT wE will only cOnsIder $2$-categoriEs wIth a siNgLe ... | mathtt{i})$. If ${{\sc\mbo x{C}\hspac e{1.0 pt} }}$ h as m orethan one objec t , on e can fix an orderingof it so bjec t sand d efine t h er a nkof a fi ni t ar y $2$ -re present ation of $ {{\ sc \mbox{C}\hsp a ce {1.0pt}}}$ as a suitabletup le ofpo sit i ve in teg ers.Howeve r , in t his docum en t we wi l l on... | mathtt{i})$. If_${{\sc\mbox{C}\hspace{1.0pt}}}$ has_more than one object,_one can_fix_an ordering_of_its objects and_define the rank_of a finitary $2$-representation_of ${{\sc\mbox{C}\hspace{1.0pt}}}$ as_a_suitable tuple of positive integers. However, in this document we will only consider $2$-categories_with_a single_... |
of the state $s_{i}$. The transition probability matrix is approximated by converting the elements of P by approximating the transition probabilities using $P_{ij}=\nicefrac{N_{ij}}{N_{i}}$. The resulting matrix is often described as a first order Markov Matrix [@Ross2013]. State changes are based on only the observat... | of the state $ s_{i}$. The transition probability matrix is approximated by converting the chemical element of phosphorus by approximating the transition probability use $ P_{ij}=\nicefrac{N_{ij}}{N_{i}}$. The resulting matrix is often identify as a beginning order Markov Matrix [ @Ross2013 ]. state of matter changes a... | of the state $s_{i}$. The transiuion probability matrix ms apprkximated by converting the elements lf P by approximating the travsition pgobabilitues nsing $P_{ij}=\nicefrar{H_{ij}}{N_{i}}$. Tmz reshptiny natrix is oftek described as a first orgef Larkov Matrix [@Ross2013]. State changes arq based oj only the obsgrvat... | of the state $s_{i}$. The transition probability approximated converting the of P by $P_{ij}=\nicefrac{N_{ij}}{N_{i}}$. resulting matrix is described as a order Markov Matrix [@Ross2013]. State changes based on only the observation-to-observation amplitude changes; the matrix is a representation of linearly interpolate... | of the state $s_{i}$. The transition ProbabilitY matrIx iS apPrOximAted By converting thE ElemEnts of P by approximating The trAnSItioN PrObabiLities uSInG $p_{Ij}=\nIcEfRac{n_{iJ}}{n_{i}}$. the reSulTing matRix is often DesCrIbed as a first ORdEr Markov MaTriX [@Ross2013]. State chAngEs are bAsEd oN Only tHe oBservAt... | of the state $s_{i}$. The transitio n pro bab ili ty mat rixis approximate d byconverting the element s ofPb y ap p ro ximat ing the tr a n sit io npro ba b il ities us ing $P_ {ij}=\nice fra c{ N_{ij}}{N_{i } }$ . The resu lti ng matrix is of ten de sc rib e d asa f irstorderM arkovMatrix [@ Ro s s2013] . State... | of_the state_$s_{i}$. The transition probability_matrix is_approximated_by converting_the_elements of P_by approximating the_transition probabilities using $P_{ij}=\nicefrac{N_{ij}}{N_{i}}$._The resulting matrix_is_often described as a first order Markov Matrix [@Ross2013]. State changes are based on_only_the observat... |
the notification (see Supplementary Note 4 for how these load profiles were generated). Subsequently, power flows within the distribution network were calculated. Finally, we make the reasonable assumption that the capacity of each line in the distribution network is limited to 10% over the peak power flow in that lin... | the notification (see Supplementary Note 4 for how these load profiles were generate). Subsequently, exponent flows within the distribution network were calculated. last, we make the reasonable presumption that the capacitance of each line in the distribution net is specify to 10% over the peak might flow in that lin... | thf notification (see Suppltmentary Note 4 for how thxse loas profilds were generated). Subsequentpy, powee flows within the disgribution network wert calculated. Finally, we make ths reavinable assumptlon that tha capacity of aazh line in the distribution network if limitrd to 10% over the keak kowqr fmow in that lin... | the notification (see Supplementary Note 4 for load were generated). power flows within Finally, make the reasonable that the capacity each line in the distribution network limited to 10% over the peak power flow in that line under regular i.e., when no resident receives the notification from the attacker. Finally, we ... | the notification (see SupplemEntary Note 4 For hoW thEse LoAd prOfilEs were generateD). subsEquently, power flows withIn the DiSTribUTiOn netWork werE CaLCUlaTeD. FInaLlY, We Make tHe rEasonabLe assumptiOn tHaT the capacity OF eAch line in tHe dIstribution nEtwOrk is lImIteD To 10% oveR thE peak Power fLOw in thAt lin... | the notification (see Sup plementary Note 4for h ow t hese load profiles were generated). Subsequen tly,po w er f l ow s wit hin the di s t rib ut io n n et w or k wer e c alculat ed. Finall y,we make the re a so nable assu mpt ion that the ca pacity o f e a ch li nein th e dist r ibutio n network i s limit ... | the_notification (see_Supplementary Note 4 for how_these load_profiles_were generated)._Subsequently,_power flows within_the distribution network_were calculated. Finally, we_make the reasonable_assumption_that the capacity of each line in the distribution network is limited to 10%_over_the peak_power_flow_in that lin... |
interesting to note that even though $\delta V$ is added only in region C, the scattering states (WF1, WF2) are affected in all regions L, C, and R reflecting the scattering processes.
To correlate the robustness of transmission $T(E,0)$ against the strength of the disorder potential $\delta V$, we have calculated co... | interesting to note that even though $ \delta V$ is added only in region C, the scatter state (WF1, WF2) are affected in all regions L, C, and R chew over the scattering processes.
To correlate the robustness of infection $ T(E,0)$ against the persuasiveness of the disorder electric potential $ \delta V$, we have ac... | inheresting to note that eyen though $\delta V$ is avded onmy in reeion C, the scattering states (WD1, WF2) qre affected in all reeions L, C, and R rwflerting the scattecjng progzsses.
Fl coxrxlate the robusjness of tratsmission $T(E,0)$ acakndt the strength of the disorder potqntial $\cepta V$, we have salcllwted bo... | interesting to note that even though $\delta added in region the scattering states all L, C, and reflecting the scattering To correlate the robustness of transmission against the strength of the disorder potential $\delta V$, we have calculated configuration (see Table-\[tab1\]) where $\delta V$ is randomly drawn from ... | interesting to note that even Though $\deltA V$ is aDdeD onLy In reGion c, the scattering STateS (WF1, WF2) are affected in all rEgionS L, c, And R REfLectiNg the scATtERIng PrOcEssEs.
tO cOrrelAte The robuStness of trAnsMiSsion $T(E,0)$ againST tHe strength Of tHe disorder poTenTial $\deLtA V$, wE Have cAlcUlateD co... | interesting to note thateven thoug h $\d elt a V $is a dded only in regio n C,the scattering states(WF1, W F 2) a r eaffec ted ina ll r egi on sL,C, an d R r efl ectingthe scatte rin gprocesses.
T ocorrelatethe robustnessoftransm is sio n $T(E ,0) $ aga inst t h e stre ngth of t he disord e r poten t i al $\d el... | interesting_to note_that even though $\delta_V$ is_added_only in_region_C, the scattering_states (WF1, WF2)_are affected in all_regions L, C,_and_R reflecting the scattering processes.
To correlate the robustness of transmission $T(E,0)$ against the strength_of_the disorder_potential_$\delta_V$, we have calculated co... |
s}{d}$. Now recall that ${\mathcal F}^{-1} : L^1 \to L^{\infty}$ and ${\mathcal F}^{-1} : L^2 \to L^2$. Therefore, by real interpolation $$\begin{aligned}
{\mathcal F}^{-1} : (L^1, L^2)_{\theta, 2} \to (L^{\infty}, L^2)_{\theta, 2}\end{aligned}$$ which, by is exactly the statement that $$\begin{aligned}
{\mathcal F}^{-... | s}{d}$. Now recall that $ { \mathcal F}^{-1 }: L^1 \to L^{\infty}$ and $ { \mathcal F}^{-1 }: L^2 \to L^2$. Therefore, by real interjection $ $ \begin{aligned }
{ \mathcal F}^{-1 }: (L^1, L^2)_{\theta, 2 } \to (L^{\infty }, L^2)_{\theta, 2}\end{aligned}$$ which, by is precisely the statement that $ $ \begin{aligned }... | s}{d}$. Jow recall that ${\mathcal N}^{-1} : L^1 \to L^{\infty}$ aue ${\mathral F}^{-1} : M^2 \to L^2$. Tferefore, by real interpolatiln $$\begib{aligned}
{\mathcal F}^{-1} : (L^1, L^2)_{\gheta, 2} \to (L^{\infty}, O^2)_{\theua, 2}\end{aligned}$$ whirg, by is exactmn the wtatement that $$\begin{aligted}
{\mathcal F}^{-... | s}{d}$. Now recall that ${\mathcal F}^{-1} : L^{\infty}$ ${\mathcal F}^{-1} L^2 \to L^2$. {\mathcal : (L^1, L^2)_{\theta, \to (L^{\infty}, L^2)_{\theta, which, by is exactly the statement $$\begin{aligned} {\mathcal F}^{-1} : L^{{\alpha}, 2}({\mathbb R}^d) \to L^{{\beta},2}({\mathbb R}^d)\end{aligned}$$ where $\frac{1}... | s}{d}$. Now recall that ${\mathcal F}^{-1} : L^1 \tO L^{\infty}$ and ${\MathcAl F}^{-1} : l^2 \to l^2$. THereFore, By real interpolATion $$\Begin{aligned}
{\mathcal F}^{-1} : (L^1, L^2)_{\Theta, 2} \To (l^{\InftY}, l^2)_{\tHeta, 2}\eNd{alignED}$$ wHICh, bY iS eXacTlY ThE statEmeNt that $$\bEgin{aligneD}
{\maThCal F}^{-... | s}{d}$. Now recall that ${ \mathcal F }^{-1 } : L^ 1\toL^{\ infty}$ and ${ \ math cal F}^{-1} : L^2 \toL^2$. T h eref o re , byreal in t er p o lat io n$$\ be g in {alig ned }
{\mat hcal F}^{- 1}:(L^1, L^2)_{ \ th eta, 2} \t o ( L^{\infty},L^2 )_{\th et a,2 }\end {al igned }$$ wh i ch, by is exact ly the st a tement... | s}{d}$. Now_recall that_${\mathcal F}^{-1} : L^1_\to L^{\infty}$_and_${\mathcal F}^{-1}_:_L^2 \to L^2$._Therefore, by real_interpolation $$\begin{aligned}
{\mathcal F}^{-1} :_(L^1, L^2)_{\theta, 2}_\to_(L^{\infty}, L^2)_{\theta, 2}\end{aligned}$$ which, by is exactly the statement that $$\begin{aligned}
{\mathcal F}^{-... |
$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ implies that $\chi$ and its inverse $\chi^{-1}$ are elements of $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$. Simple verifications show that $\mathbb{I}=\chi\chi^{-1}=\chi\mathbb{I}\chi^{-1}$. Also, since $\mathfrak{C}$ is a congruence on... | $ \mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ implies that $ \chi$ and its inverse $ \chi^{-1}$ are elements of $ \mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$. Simple verifications picture that $ \mathbb{I}=\chi\chi^{-1}=\chi\mathbb{I}\chi^{-1}$. besides, since $ \mathfrak{C}$ is a co... | $\mahhscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operxtorname{lex}})$ implies thet $\chi$ znd its knverse $\chi^{-1}$ are elements of $\larhscr{U\!O}\!_{\infty}(\mathbb{Z}^n_{\operatofname{lex}})$. Dimple vwrifmcations show thef $\mathbn{N}=\chi\cgl^{-1}=\chi\mctibb{I}\chi^{-1}$. Also, sikce $\mathfran{C}$ is a congruanze on... | $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ implies that $\chi$ and its inverse elements $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$. Simple show that $\mathbb{I}=\chi\chi^{-1}=\chi\mathbb{I}\chi^{-1}$. congruence the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname... | $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\opeRatorname{lEx}})$ impLieS thAt $\Chi$ aNd itS inverse $\chi^{-1}$ are ELemeNts of $\mathscr{I\!O}\!_{\infty}(\matHbb{Z}^n_{\OpERatoRNaMe{lex}})$. simple vERiFICatIoNs ShoW tHAt $\MathbB{I}=\cHi\chi^{-1}=\chI\mathbb{I}\chI^{-1}$. AlSo, Since $\mathfraK{c}$ iS a congruenCe oN... | $\mathscr{I\!O}\!_{\infty }(\mathbb{ Z}^n_ {\o per at orna me{l ex}})$ implies that $\chi$ and its invers e $\c hi ^ {-1} $ a re el ementso f$ \ mat hs cr {I\ !O } \! _{\in fty }(\math bb{Z}^n_{\ ope ra torname{lex} } )$ . Simple v eri fications sh owthat $ \m ath b b{I}= \ch i\chi ^{-1}= \ chi\ma thbb{I}\c hi ^ {-1... | $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$_implies that_$\chi$ and its inverse_$\chi^{-1}$ are_elements_of $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$._Simple_verifications show that_$\mathbb{I}=\chi\chi^{-1}=\chi\mathbb{I}\chi^{-1}$. Also, since_$\mathfrak{C}$ is a congruence_on... |
stray capacitances $C_s$ from the board, components and packaging.
The bandwidth $BW$ of the photoreceiver can be estimated as $$BW=\sqrt{\frac{GBWP}{2\pi R_f\,C_T}},
\label{BW}$$ where $GBWP$ is the gain-bandwidth product of the operational amplifier. The total TIA input current noise $I_{\mathrm{noise}}(f)$ model ... | stray capacitances $ C_s$ from the board, components and promotion.
The bandwidth $ BW$ of the photoreceiver can be estimate as $ $ BW=\sqrt{\frac{GBWP}{2\pi R_f\,C_T } },
\label{BW}$$ where $ GBWP$ is the gain - bandwidth product of the functional amplifier. The total TIA remark current noise $ I_{\mathrm{noise}... | stgay capacitances $C_s$ from the board, components end paciaging.
Thd bandwidth $BW$ of the photorxceicer cqn be estimated as $$BW=\sdrt{\frac{GBAP}{2\pi R_f\,C_R}},
\lauel{BW}$$ where $GBWP$ is the ncin-bahfwidch product of thg operationan amplifier. Tha gocal TIA input current noise $I_{\mathrm{njise}}(f)$ mpdfl ... | stray capacitances $C_s$ from the board, components The $BW$ of photoreceiver can be where is the gain-bandwidth of the operational The total TIA input current noise model can be expressed as $$I_{noise}(f)=\sqrt{i_T^2+i_{TIA}^2(f)}\,\cdot\|\overline{TF}(f)\|, \label{i_noise}$$ where $\|\overline{TF}(f)\|$ is the norma... | stray capacitances $C_s$ from thE board, compOnentS anD paCkAginG.
The Bandwidth $BW$ of tHE phoToreceiver can be estimatEd as $$Bw=\sQRt{\frAC{GbWP}{2\pi r_f\,C_T}},
\labEL{Bw}$$ WHerE $GbWp$ is ThE GaIn-banDwiDth prodUct of the opEraTiOnal amplifieR. thE total TIA iNpuT current noisE $I_{\mAthrm{nOiSe}}(f)$ MOdel ... | stray capacitances $C_s$from the b oard, co mpo ne ntsandpackaging.
Th e ban dwidth $BW$ of the pho torec ei v er c a nbe es timated as $ $BW =\ sq rt{ \f r ac {GBWP }{2 \pi R_f \,C_T}},
\la be l{BW}$$ wher e $ GBWP$ is t hegain-bandwid thproduc toft he op era tiona l ampl i fier.The total T I A inpu t curren t ... | stray_capacitances $C_s$_from the board, components_and packaging.
The_bandwidth_$BW$ of_the_photoreceiver can be_estimated as $$BW=\sqrt{\frac{GBWP}{2\pi_R_f\,C_T}},
\label{BW}$$ where $GBWP$_is the gain-bandwidth_product_of the operational amplifier. The total TIA input current noise $I_{\mathrm{noise}}(f)$ model ... |
{\partial^{2} \vartheta}{\partial x_{k} \partial x_{i}}}
+\frac{\partial}{\partial x_{k}}
\frac{\partial \rho L_0}
{\partial
\frac{\partial \vartheta}{\partial x_{k} }}
-\frac{\partial \rho L_0}{\partial\vartheta}
-\frac{\hbar \rho}{2}
\left( G_1\cos \varphi - G_2\sin \varphi \right)
=0
\label{eq:D6DSV3ARTH}\\
-\frac... | { \partial^{2 } \vartheta}{\partial x_{k } \partial x_{i } } }
+ \frac{\partial}{\partial x_{k } }
\frac{\partial \rho L_0 }
{ \partial
\frac{\partial \vartheta}{\partial x_{k } } }
-\frac{\partial \rho L_0}{\partial\vartheta }
-\frac{\hbar \rho}{2 }
\left (G_1\cos \varphi - G_2\sin \varphi \right)
= ... | {\parhial^{2} \vartheta}{\partial x_{k} \kartial x_{i}}}
+\frac{\parjiql}{\partmal x_{k}}
\fdac{\partixl \rho L_0}
{\partial
\frac{\partial \verthwta}{\paetial x_{k} }}
-\frac{\partial \rfo L_0}{\partiwl\varthera}
-\frec{\hbar \rho}{2}
\left( G_1\rks \varpmn - G_2\sjk \var'hm \right)
=0
\label{ea:D6DSV3ARTH}\\
-\fsac... | {\partial^{2} \vartheta}{\partial x_{k} \partial x_{i}}} +\frac{\partial}{\partial x_{k}} L_0} \frac{\partial \vartheta}{\partial }} -\frac{\partial \rho \varphi G_2\sin \varphi \right) \label{eq:D6DSV3ARTH}\\ -\frac{\partial}{\partial x_{k}} x_{i}} \frac{\partial \rho L_0} {\partial\ \frac{\partial^{2} x_{k} \partial ... | {\partial^{2} \vartheta}{\partial x_{k} \pArtial x_{i}}}
+\frAc{\parTiaL}{\paRtIal x_{K}}
\fraC{\partial \rho L_0}
{\paRTial
\Frac{\partial \vartheta}{\parTial x_{K} }}
-\fRAc{\paRTiAl \rho l_0}{\partiaL\VaRTHetA}
-\fRaC{\hbAr \RHo}{2}
\Left( G_1\Cos \Varphi - G_2\Sin \varphi \rIghT)
=0
\lAbel{eq:D6DSV3ARth}\\
-\fRac... | {\partial^{2} \vartheta}{\ partial x_ {k} \ par tia lx_{i }}}+\frac{\partia l }{\p artial x_{k}}
\frac{\p artia l\ rhoL _0 }
{\p artial\ fr a c {\p ar ti al\v a rt heta} {\p artialx_{k} }}
- \fr ac {\partial \r h oL_0}{\part ial \vartheta}
- \fr ac{\hb ar \r h o}{2}
\l eft(G_1\co s \varp hi - G_2\ si n \varp h i \... | {\partial^{2} \vartheta}{\partial_x_{k} \partial_x_{i}}}
+\frac{\partial}{\partial x_{k}}
\frac{\partial \rho L_0}
{\partial
\frac{\partial_\vartheta}{\partial x_{k}_}}
-\frac{\partial_\rho L_0}{\partial\vartheta}
-\frac{\hbar_\rho}{2}
\left(_G_1\cos \varphi -_G_2\sin \varphi _\right)
=0
\label{eq:D6DSV3ARTH}\\
-\frac... |
u_1,u_2)=\Phi(u_1)+\Phi(u_2)$ and set $\Omega=\Omega_1\times\Omega_2$. Then $$\nabla_G\text{-}\deg(\nabla\Phi, \Omega)=\nabla_G\text{-}\deg(\nabla\Phi_1, \Omega_1)\star\nabla_G\text{-}\deg(\nabla\Phi_2, \Omega_2).$$
\[GLOB\] Fix $\Phi\in C^2_G({\mathcal{H}}\times\Lambda,{\mathbb{R}})$ such that $\nabla_u\Phi(u,\lambda... | u_1,u_2)=\Phi(u_1)+\Phi(u_2)$ and set $ \Omega=\Omega_1\times\Omega_2$. Then $ $ \nabla_G\text{-}\deg(\nabla\Phi, \Omega)=\nabla_G\text{-}\deg(\nabla\Phi_1, \Omega_1)\star\nabla_G\text{-}\deg(\nabla\Phi_2, \Omega_2).$$
\[GLOB\ ] Fix $ \Phi\in C^2_G({\mathcal{H}}\times\Lambda,{\mathbb{R}})$ such that $ \nabla_u\Phi(u... | u_1,u_2)=\Pji(u_1)+\Phi(u_2)$ and set $\Omega=\Omena_1\times\Omega_2$. Theu $$\nabla_J\text{-}\def(\nabla\Phk, \Omega)=\nabla_G\text{-}\deg(\nabla\Phi_1, \Onega_1)\sucr\nabla_G\text{-}\deg(\nabla\Ohi_2, \Omega_2).$$
\[HLOB\] Fix $\Phi\mn C^2_G({\mathcal{H}}\timxa\Lambda,{\mathbb{D}})$ suck vhat $\nabla_u\Phi(u,\kambda... | u_1,u_2)=\Phi(u_1)+\Phi(u_2)$ and set $\Omega=\Omega_1\times\Omega_2$. Then $$\nabla_G\text{-}\deg(\nabla\Phi, \Omega)=\nabla_G\text{-}\deg(\nabla\Phi_1, \[GLOB\] $\Phi\in C^2_G({\mathcal{H}}\times\Lambda,{\mathbb{R}})$ that $\nabla_u\Phi(u,\lambda)=Lu-\nabla_u\eta(u,\lambda)$, where and Suppose that $\nabla_u\Phi(0,\l... | u_1,u_2)=\Phi(u_1)+\Phi(u_2)$ and set $\Omega=\OmegA_1\times\OmegA_2$. Then $$\NabLa_G\TeXt{-}\deG(\nabLa\Phi, \Omega)=\nablA_g\texT{-}\deg(\nabla\Phi_1, \Omega_1)\star\nAbla_G\TeXT{-}\deg(\NAbLa\Phi_2, \omega_2).$$
\[GLob\] FIX $\phi\In c^2_G({\MatHcAL{H}}\Times\lamBda,{\mathBb{R}})$ such thaT $\naBlA_u\Phi(u,\lambda... | u_1,u_2)=\Phi(u_1)+\Phi(u_ 2)$ and se t $\O meg a=\ Om ega_ 1\ti mes\Omega_2$.T hen$$\nabla_G\text{-}\deg (\nab la \ Phi, \O mega) =\nabla _ G\ t e xt{ -} \d eg( \n a bl a\Phi _1, \Omega _1)\star\n abl a_ G\text{-}\de g (\ nabla\Phi_ 2,\Omega_2).$$
\ [GLOB\ ]Fix $\Phi \in C^2_ G({\ma t hcal{H }}\times\ La m bda,{\ ... | u_1,u_2)=\Phi(u_1)+\Phi(u_2)$ and_set $\Omega=\Omega_1\times\Omega_2$._Then $$\nabla_G\text{-}\deg(\nabla\Phi, \Omega)=\nabla_G\text{-}\deg(\nabla\Phi_1, \Omega_1)\star\nabla_G\text{-}\deg(\nabla\Phi_2,_\Omega_2).$$
\[GLOB\] Fix_$\Phi\in_C^2_G({\mathcal{H}}\times\Lambda,{\mathbb{R}})$ such_that_$\nabla_u\Phi(u,\lambda... |
in the opposite direction.
We define the 2-category $2^\prime \text{-}{\operatorname{dgalg}}$ of DG algebras as follows. The objects are DG algebras. For DG algebras ${{\mathcal B}}, {{\mathcal C}}$ the collection of 1-morphisms $1\text{-}\Hom({{\mathcal B}},{{\mathcal C}})$ consists of pairs $(M,\theta)$, where
- ... | in the opposite direction.
We define the 2 - category $ 2^\prime \text{-}{\operatorname{dgalg}}$ of DG algebra as take after. The objects are DG algebras. For DG algebra $ { { \mathcal B } }, { { \mathcal C}}$ the collection of 1 - morphisms $ 1\text{-}\Hom({{\mathcal B}},{{\mathcal C}})$ consist of pairs $ (M,\thet... | in the opposite direction.
Wt define the 2-cateyiry $2^\prmme \texf{-}{\operatofname{dgalg}}$ of DG algebras as filloww. The objects are DG augebras. Flr DG altebres ${{\mathcal B}}, {{\matidal C}}$ tmz colmcctiou if 1-morphisms $1\tgxt{-}\Hom({{\mathcan B}},{{\mathcal C}})$ cmnridts of pairs $(M,\theta)$, where
- ... | in the opposite direction. We define the \text{-}{\operatorname{dgalg}}$ DG algebras follows. The objects algebras B}}, {{\mathcal C}}$ collection of 1-morphisms B}},{{\mathcal C}})$ consists of pairs $(M,\theta)$, - $M\in D({{\mathcal B}}^0 \otimes {{\mathcal C}})$ and there exists an isomorphism (in C}})$) $\theta :{... | in the opposite direction.
We dEfine the 2-caTegorY $2^\prIme \TeXt{-}{\opEratOrname{dgalg}}$ of Dg AlgeBras as follows. The objectS are Dg aLGebrAS. FOr DG aLgebras ${{\MAtHCAl B}}, {{\MaThCal c}}$ tHE cOllecTioN of 1-morpHisms $1\text{-}\HOm({{\mAtHcal B}},{{\mathcal c}})$ CoNsists of paIrs $(m,\theta)$, where
- ... | in the opposite direction .
We defi ne th e 2 -ca te gory $2^ \prime \text{- } {\op eratorname{dgalg}}$ of DG a lg e bras as foll ows. Th e o b j ect sar e D Ga lg ebras . F or DG a lgebras ${ {\m at hcal B}}, {{ \ ma thcal C}}$ th e collection of 1-mor ph ism s $1\t ext {-}\H om({{\ m athcal B}},{{\m at h c... | in_the opposite_direction.
We define the 2-category_$2^\prime \text{-}{\operatorname{dgalg}}$_of_DG algebras_as_follows. The objects_are DG algebras._For DG algebras ${{\mathcal_B}}, {{\mathcal C}}$_the_collection of 1-morphisms $1\text{-}\Hom({{\mathcal B}},{{\mathcal C}})$ consists of pairs $(M,\theta)$, where
- ... |
further out in the stellar atmosphere (see below). We restricted $\log \tau_{\rm c} < -2$ since for larger values of $\tau_{\rm c}$ the outer boundary is placed in a region where the temperature gradient starts to become significant and the periods of all modes become dependent of $\tau_{\rm c}$.
It is clear that the... | further out in the stellar atmosphere (visualize downstairs). We restricted $ \log \tau_{\rm c } < -2 $ since for larger values of $ \tau_{\rm c}$ the extinct boundary is placed in a area where the temperature gradient starts to become significant and the periods of all manner become dependent of $ \tau_{\rm c}$.
It... | fugther out in the stellar atmosphere (see below). Xe restdicted $\lug \tau_{\rm c} < -2$ since for largec vaoues if $\tau_{\rm c}$ the outer buundary id placed in e region where tis tempevcture nradiznv starts to becpme signifhcant and the [efilds of all modes become dependent os $\tau_{\rm c}$.
Lt is clear thwt tnq... | further out in the stellar atmosphere (see restricted \tau_{\rm c} -2$ since for the boundary is placed a region where temperature gradient starts to become significant the periods of all modes become dependent of $\tau_{\rm c}$. It is clear the periods of the normal modes are independent of $\log \tau_{\rm c}$ i.e. ar... | further out in the stellar atmOsphere (see Below). we rEstRiCted $\Log \tAu_{\rm c} < -2$ since for lARger Values of $\tau_{\rm c}$ the outer BoundArY Is plACeD in a rEgion whERe THE teMpErAtuRe GRaDient StaRts to beCome signifIcaNt And the periodS Of All modes beComE dependent of $\Tau_{\Rm c}$.
It iS cLeaR That tHe... | further out in the stella r atmosphe re (s eebel ow ). W e re stricted $\log \tau _{\rm c} < -2$ since f or la rg e r va l ue s of$\tau_{ \ rm c }$th eout er bo undar y i s place d in a reg ion w here the tem p er ature grad ien t starts tobec ome si gn ifi c ant a ndthe p eriods of all modes be co m e depe n dent o... | further_out in_the stellar atmosphere (see_below). We_restricted_$\log \tau_{\rm_c}_< -2$ since_for larger values_of $\tau_{\rm c}$ the_outer boundary is_placed_in a region where the temperature gradient starts to become significant and the periods_of_all modes_become_dependent_of $\tau_{\rm c}$.
It is clear_that the... |
can be written as $$\label{eq: 5.4}
\delta \mbox{\boldmath$r$} \left( t +\delta t \right) \equiv
\mbox{\boldmath$a$}(\mbox{\boldmath$x$}_0
\left( t \right),t) \delta \mbox{\boldmath$r$} \left( t \right) =
\left[ \mbox{\boldmath$I$} + \delta \mbox{\boldmath$a$}
\left(\mbox{\boldmath$x$}_0\... | can be written as $ $ \label{eq: 5.4 }
\delta \mbox{\boldmath$r$ } \left (t + \delta t \right) \equiv
\mbox{\boldmath$a$}(\mbox{\boldmath$x$}_0
\left (t \right),t) \delta \mbox{\boldmath$r$ } \left (t \right) =
\left [ \mbox{\boldmath$I$ } + \delta \mbox{\boldmath$a$ }
\left(\mbo... | caj be written as $$\label{eq: 5.4}
\delta \mbox{\boldmath$c$} \left( f +\delta g \right) \equiv
\mbox{\bolvmaty$a$}(\mboz{\boldmath$x$}_0
\left( t \right),t) \delta \mvox{\biodmath$r$} \lehf( t \rigmc) =
\peft[ \mbox{\boldmath$I$} + \delta \mbmx{\boldmath$a$}
\ueyt(\mbox{\boldmath$x$}_0\... | can be written as $$\label{eq: 5.4} \delta t t \right) \mbox{\boldmath$a$}(\mbox{\boldmath$x$}_0 \left( t \right) \left[ \mbox{\boldmath$I$} + \mbox{\boldmath$a$} \left(\mbox{\boldmath$x$}_0\left( t \right) \right] \delta \mbox{\boldmath$r$} \left( t where $$\label{eq: 5.5} \left[ \delta \mbox{\boldmath$a$} \left(\mbox... | can be written as $$\label{eq: 5.4}
\deltA \mbox{\boldmAth$r$} \lEft( T +\deLtA t \riGht) \eQuiv
\mbox{\boldmaTH$a$}(\mbOx{\boldmath$x$}_0
\left( t \right),t) \Delta \MbOX{\bolDMaTh$r$} \leFt( t \righT) =
\LeFT[ \MboX{\bOlDmaTh$i$} + \DeLta \mbOx{\bOldmath$A$}
\left(\mbox{\bOldMaTh$x$}_0\... | can be written as $$\labe l{eq: 5.4}
\de lta \ mbox {\bo ldmath$r$} \le f t( t +\delta t \right) \eq uiv
\ mb ox{\b oldmath $ a$ } ( \mb ox {\ bol dm a th $x$}_ 0
\left( t \ rig ht ),t) \delta\ mb ox{\boldma th$ r$} \left( t \r ight)= \left [ \mbox {\bold m ath$I$ } + \delt a\ mbox{\ b oldmath $ a $}... | can_be written_as $$\label{eq: 5.4}
_ _\delta_\mbox{\boldmath$r$} \left(_t_+\delta t \right)_\equiv
_ _ \mbox{\boldmath$a$}(\mbox{\boldmath$x$}_0
__ \left( t \right),t) \delta \mbox{\boldmath$r$} \left( t \right) =
__ _\left[__\mbox{\boldmath$I$} + \delta \mbox{\boldmath$a$}
_ \left(\mbox{\boldmath$x$}_0\... |
}=c^{3}a$. The group von Neumann algebra associated with $G$ is a type II$_{1}$ factor, and the preceding corollary easily implies that $\mathfrak{K}_{2}(L(G))=0$ and $\delta
_{0}(L(G))\leq1.$
The next two corollaries follows directly from Corollary 3.
Suppose $\mathcal M$ is a nonprime II$_1$ factor, i.e. $\mathcal
... | } = c^{3}a$. The group von Neumann algebra associated with $ G$ is a type II$_{1}$ factor, and the preceding corollary well entail that $ \mathfrak{K}_{2}(L(G))=0 $ and $ \delta
_ { 0}(L(G))\leq1.$
The next two corollaries follow immediately from Corollary 3.
Suppose $ \mathcal M$ is a nonprime II$_1 $ factor, ... | }=c^{3}a$. Hhe group von Neumann alnebra associated with $G$ is a fype II$_{1}$ wactor, and the preceding corlloary tcsily implies that $\mxthfrak{K}_{2}(L(H))=0$ and $\deota
_{0}(L(J))\leq1.$
The next two corollavnes fkplowv directly from Corollary 3.
Suppose $\mathcdl M$ is a nonprime II$_1$ factor, i.e. $\mathcal
... | }=c^{3}a$. The group von Neumann algebra associated is type II$_{1}$ and the preceding and _{0}(L(G))\leq1.$ The next corollaries follows directly Corollary 3. Suppose $\mathcal M$ is nonprime II$_1$ factor, i.e. $\mathcal M\simeq \mathcal N_1\otimes \mathcal N_2$ for some II$_1$ $ \mathcal N_1, \mathcal N_2$. Then $\m... | }=c^{3}a$. The group von Neumann algebRa associatEd witH $G$ iS a tYpE II$_{1}$ fActoR, and the precediNG corOllary easily implies thaT $\mathFrAK{K}_{2}(L(G))=0$ ANd $\Delta
_{0}(l(G))\leq1.$
ThE NeXT Two CoRoLlaRiES fOllowS diRectly fRom CorollaRy 3.
SUpPose $\mathcal M$ IS a Nonprime II$_1$ FacTor, i.e. $\mathcal
... | }=c^{3}a$. The group von N eumann alg ebraass oci at ed w ith$G$ is a typeI I$_{ 1}$ factor, and the pr ecedi ng coro l la ry ea sily im p li e s th at $ \ma th f ra k{K}_ {2} (L(G))= 0$ and $\d elt a_{0}(L(G))\l e q1 .$
The ne xttwo corollar ies follo ws di r ectly fr om Co rollar y 3.
S uppose $\ ma t hcal M ... | }=c^{3}a$. The_group von_Neumann algebra associated with_$G$ is_a_type II$_{1}$_factor,_and the preceding_corollary easily implies_that $\mathfrak{K}_{2}(L(G))=0$ and $\delta
_{0}(L(G))\leq1.$
The_next two corollaries_follows_directly from Corollary 3.
Suppose $\mathcal M$ is a nonprime II$_1$ factor, i.e. $\mathcal
... |
SN offset {#sec:offset}
@2013Sci...340..170W found evidence for two distinct populations of SNe Ia with respect to their radial distributions in the host galaxies. They found that the high-[$v_{\mathrm{Si}\,\textsc{ii}}$]{} SNe Ia are concentrated in the inner regions of their host galaxies, whereas the normal-[$v_{\... | SN offset { # sec: offset }
@2013Sci... 340.. 170W found evidence for two distinct population of SNe Ia with obedience to their radial distributions in the host galaxies. They establish that the high-[$v_{\mathrm{Si}\,\textsc{ii}}$ ] { } SNe Ia are concentrated in the inner region of their master of ceremonies gal... | SN offset {#sec:offset}
@2013Sci...340..170W fomnd evidence for two divtinct populatkons of SNe Ia with respect vo tyeir eadial distributions iv the hosn galaxiew. Thty found that the high-[$v_{\mabkrm{Si}\,\fcxtsc{ni}}$]{} WNe Ia are congentrated it the inner reciund of their host galaxies, whereas thq normak-[$v_{\... | SN offset {#sec:offset} @2013Sci...340..170W found evidence for populations SNe Ia respect to their galaxies. found that the SNe Ia are in the inner regions of their galaxies, whereas the normal-[$v_{\mathrm{Si}\,\textsc{ii}}$]{} SNe Ia span a wide range of radial distance. next examine this trend in our sample in Fig.... | SN offset {#sec:offset}
@2013Sci...340..170W founD evidence fOr two DisTinCt PopuLatiOns of SNe Ia with REspeCt to their radial distribUtionS iN The hOSt GalaxIes. They FOuND ThaT tHe HigH-[$v_{\MAtHrm{Si}\,\TexTsc{ii}}$]{} SNE Ia are concEntRaTed in the inneR ReGions of theIr hOst galaxies, wHerEas the NoRmaL-[$V_{\... | SN offset {#sec:offset}
@2013Sci.. .340. .17 0Wfo undevid ence for two d i stin ct populations of SNeIa wi th resp e ct to t heir ra d ia l dis tr ib uti on s i n the ho st gala xies. They fo un d that the h i gh -[$v_{\mat hrm {Si}\,\texts c{i i}}$]{ }SNe Ia ar e c oncen trated in the inner re gi o ns oft heir h... | SN_offset {#sec:offset}
@2013Sci...340..170W_found evidence for two_distinct populations_of_SNe Ia_with_respect to their_radial distributions in_the host galaxies. They_found that the_high-[$v_{\mathrm{Si}\,\textsc{ii}}$]{} SNe_Ia are concentrated in the inner regions of their host galaxies, whereas the normal-[$v_{\... |
\sum_{x_1, \ldots, x_{r-1} \in V({\mathcal{H}})} \deg(x_1, x_2, \ldots, x_{r-1}) = \binom{r+1}{r-1} |\mathcal{H}'| = \binom{r+1}{2} |\mathcal{H}'|.$$
On the other hand, again by double counting, $$\begin{aligned}
\sum_{q \in E(\mathcal{H}')} &\sum_{x_1, \ldots, x_{r-1} \in q} \deg(x_1, \ldots, x_{r-1}) \notag\\
& ... | \sum_{x_1, \ldots, x_{r-1 } \in V({\mathcal{H } }) } \deg(x_1, x_2, \ldots, x_{r-1 }) = \binom{r+1}{r-1 } |\mathcal{H}'| = \binom{r+1}{2 } |\mathcal{H}'|.$$
On the other hand, again by double counting, $ $ \begin{aligned }
\sum_{q \in E(\mathcal{H }') } & \sum_{x_1, \ldots, x_{r-1 } \in q } \deg(x_1, \ldots, x_{r-... | \sum_{d_1, \ldots, x_{r-1} \in V({\mathcal{H}})} \aeg(x_1, x_2, \ldots, x_{r-1}) = \binom{c+1}{r-1} |\mathdal{H}'| = \bivom{r+1}{2} |\mathcal{H}'|.$$
On the other haid, atain vy double counting, $$\begkn{aligned}
\dum_{q \in W(\matical{H}')} &\sum_{x_1, \ldots, x_{r-1} \in q} \deg(x_1, \mfots, e_{r-1}) \notag\\
& ... | \sum_{x_1, \ldots, x_{r-1} \in V({\mathcal{H}})} \deg(x_1, x_2, = |\mathcal{H}'| = |\mathcal{H}'|.$$ On the counting, \sum_{q \in E(\mathcal{H}')} \ldots, x_{r-1} \in \deg(x_1, \ldots, x_{r-1}) \notag\\ & = \ldots, x_{r-1} \in V({\mathcal{H}})} \deg(x_1, \ldots, x_{r-1})^2 \notag\\ & \geq \frac{1}{\binom{n}{r-1}} \left... | \sum_{x_1, \ldots, x_{r-1} \in V({\mathcal{H}})} \deg(X_1, x_2, \ldots, x_{r-1}) = \bInom{r+1}{R-1} |\maThcAl{h}'| = \binOm{r+1}{2} |\mAthcal{H}'|.$$
On the otHEr haNd, again by double countinG, $$\begiN{aLIgneD}
\SuM_{q \in E(\Mathcal{h}')} &\SuM_{X_1, \LdoTs, X_{r-1} \In q} \DeG(X_1, \lDots, x_{R-1}) \noTag\\
& ... | \sum_{x_1, \ldots, x_{r-1} \in V({\m athca l{H }}) }\deg (x_1 , x_2, \ldots, x_{r -1}) = \binom{r+1}{r-1 } |\m at h cal{ H }' | = \ binom{r + 1} { 2 } | \m at hca l{ H }' |.$$
On the ot her hand,aga in by double c o un ting, $$\b egi n{aligned}
\ sum _{q \i nE(\ m athca l{H }')}&\sum_ { x_1, \ ldots, x_ {r - 1} \in ... | \sum_{x_1, \ldots,_x_{r-1} \in_V({\mathcal{H}})} \deg(x_1, x_2, \ldots,_x_{r-1}) =_\binom{r+1}{r-1}_|\mathcal{H}'| =_\binom{r+1}{2}_|\mathcal{H}'|.$$
On the other_hand, again by_double counting, $$\begin{aligned}
\sum_{q \in_E(\mathcal{H}')} &\sum_{x_1, \ldots,_x_{r-1}_\in q} \deg(x_1, \ldots, x_{r-1}) \notag\\
& ... |
looks quite similar besides that the factor $\exp
(-Work/T)$ should be replaced with $\exp (-\Delta S)$ where $\Delta
S\{t,Trajectory;Forces\}$ has the sense of increment of entropy (or similar thermodynamic potential) during $Trajectory$.
FLUCTUATIONS OF DISSIPATION
---------------------------
Let $\left\langle J(t... | looks quite similar besides that the factor $ \exp
(-Work / T)$ should be replaced with $ \exp (-\Delta S)$ where $ \Delta
S\{t, Trajectory;Forces\}$ take the common sense of increment of entropy (or similar thermodynamic electric potential) during $ Trajectory$.
FLUCTUATIONS OF waste
-------------------------... | lolks quite similar besider that the factor $\exp
(-Wmrk/T)$ sgould be replaced with $\exp (-\Delta S)$ wiere $\Deltq
S\{t,Trajectory;Forces\}$ har the sende of inxremtnt of entropy (or similar thermkfynakmc potential) duting $Trajectmry$.
FLUCTUATIONV UF DISSIPATION
---------------------------
Let $\left\langle J(t... | looks quite similar besides that the factor should replaced with (-\Delta S)$ where of of entropy (or thermodynamic potential) during FLUCTUATIONS OF DISSIPATION --------------------------- Let $\left\langle _q$ denotes the $n$-th order nonequilibrium cumulant corresponding to the time-cut modification of forces, $x(t)... | looks quite similar besides tHat the factOr $\exp
(-worK/T)$ sHoUld bE repLaced with $\exp (-\DeLTa S)$ wHere $\Delta
S\{t,Trajectory;FOrces\}$ HaS The sENsE of inCrement OF eNTRopY (oR sImiLaR ThErmodYnaMic poteNtial) durinG $TrAjEctory$.
FLUCTUatIoNS OF DISSIpATiON
---------------------------
Let $\left\laNglE J(t... | looks quite similar besid es that th e fac tor $\ ex p
(- Work /T)$ should be repl aced with $\exp (-\Del ta S) $w here $\ Delta
S\{t,T r aj e c tor y; Fo rce s\ } $has t hesense o f incremen t o fentropy (ors im ilar therm ody namic potent ial ) duri ng $T r aject ory $.
F LUCTUA T IONS O F DISSIPA TI O N
---- - ... | looks_quite similar_besides that the factor_$\exp
(-Work/T)$ should_be_replaced with_$\exp_(-\Delta S)$ where_$\Delta
S\{t,Trajectory;Forces\}$ has the_sense of increment of_entropy (or similar_thermodynamic_potential) during $Trajectory$.
FLUCTUATIONS OF DISSIPATION
---------------------------
Let $\left\langle J(t... |
for this trace-U(1) subgroup.
We have demonstrated that, although the trace-U(1) decouples in the limit $k\to 0$, the coupling is not negligibly small at finite momentum scales $k$, as they appear, for example, in scattering experiments. Therefore, observations rule out additional unbroken (massless) trace-U(1) subgr... | for this trace - U(1) subgroup.
We have demonstrated that, although the trace - U(1) decouples in the limit $ k\to 0 $, the yoke is not negligibly humble at finite momentum scales $ k$, as they appear, for case, in disperse experiments. Therefore, observation predominate out additional unplowed (massless) touch - U(... | fog this trace-U(1) subgroup.
We have demonstrajee that, althohgh the grace-U(1) decouples in the limiv $k\ti 0$, tht coupling is not ndgligibly small ar fiiite momentum scemes $k$, as they wppecr, for example, ik scatterinc experiments. Dhdrzfore, observations rule out additionwl unbrpkfn (massless) trwce-U(1) fubgd... | for this trace-U(1) subgroup. We have demonstrated the decouples in limit $k\to 0$, small finite momentum scales as they appear, example, in scattering experiments. Therefore, observations out additional unbroken (massless) trace-U(1) subgroups. An example is the model considered in [@Khoze:2004zc]. In Ref. [@Khoze:200... | for this trace-U(1) subgroup.
We haVe demonstrAted tHat, AltHoUgh tHe trAce-U(1) decouples iN The lImit $k\to 0$, the coupling is noT neglIgIBly sMAlL at fiNite momENtUM ScaLeS $k$, As tHeY ApPear, fOr eXample, iN scatterinG exPeRiments. ThereFOrE, observatiOns Rule out additIonAl unbrOkEn (mASslesS) trAce-U(1) sUbgr... | for this trace-U(1) subgr oup.
We h ave d emo nst ra tedthat , although the trac e-U(1) decouples in th e lim it $k\t o 0 $, th e coupl i ng i s n ot n egl ig i bl y sma llat fini te momentu m s ca les $k$, ast he y appear,for example, in sc atteri ng ex p erime nts . The refore , obser vations r ul e out a d ... | for_this trace-U(1)_subgroup.
We have demonstrated that,_although the_trace-U(1)_decouples in_the_limit $k\to 0$,_the coupling is_not negligibly small at_finite momentum scales_$k$,_as they appear, for example, in scattering experiments. Therefore, observations rule out additional unbroken_(massless)_trace-U(1) subgr... |
10987) of the relations are across-sentence relations. The inter-annotator agreement (IAA) for the final annotation schema is 0.89. It was calculated for the 61 reports annotated by both annotators on a token level using the Cohen’s Kappa coefficient ($\kappa$) [@Cohen:1960].
Discussion and Conclusions {#sec:discussi... | 10987) of the relations are across - sentence relations. The inter - annotator agreement (IAA) for the final note schema is 0.89. It was account for the 61 reports annotated by both annotator on a token horizontal surface using the Cohen ’s Kappa coefficient ($ \kappa$) [ @Cohen:1960 ].
Discussion and Conclusions { ... | 10987) ov the relations are acrors-sentence relajiins. Thx inter-znnotatof agreement (IAA) for the finap qnnotqtion schema is 0.89. It war calculaned for tye 61 ceports annotatev by botm annkbatorv on a token leyel using tve Cohen’s Kappd zozfficient ($\kappa$) [@Cohen:1960].
Discussion and Sonclusoojs {#sec:discussi... | 10987) of the relations are across-sentence relations. agreement for the annotation schema is the reports annotated by annotators on a level using the Cohen’s Kappa coefficient [@Cohen:1960]. Discussion and Conclusions {#sec:discussion} ========================== We presented a manually annotated corpus of and relation... | 10987) of the relations are across-seNtence relaTions. the IntEr-AnnoTatoR agreement (IAA) fOR the Final annotation schema iS 0.89. It waS cALculATeD for tHe 61 reporTS aNNOtaTeD bY boTh ANnOtatoRs oN a token Level using The coHen’s Kappa coeFFiCient ($\kappa$) [@cohEn:1960].
Discussion And concluSiOns {#SEc:disCusSi... | 10987) of the relations a re across- sente nce re la tion s. T he inter-annot a toragreement (IAA) for th e fin al anno t at ion s chema i s 0 . 8 9.It w asca l cu lated fo r the 6 1 reportsann ot ated by both an notators o n a token level us ing th eCoh e n’s K app a coe fficie n t ($\k appa$) [@ Co h en:196 0 ... | 10987)_of the_relations are across-sentence relations._The inter-annotator_agreement_(IAA) for_the_final annotation schema_is 0.89. It_was calculated for the_61 reports annotated_by_both annotators on a token level using the Cohen’s Kappa coefficient ($\kappa$) [@Cohen:1960].
Discussion and_Conclusions_{#sec:discussi... |
rm eje}\sqrt{G M_{\rm rem}R_{\rm eje}}$, respectively, where $R_{\rm torus}$ is typical radius of torus, $R_{\rm eje}$ is the typical distance from merger center to where ejecta occur and $M_{\rm rem}$ is the gravitational mass of remnant core. For dynamical ejecta launched within dynamical timescale ($\leq 10~{\rm ms}... | rm eje}\sqrt{G M_{\rm rem}R_{\rm eje}}$, respectively, where $ R_{\rm torus}$ is typical radius of torus, $ R_{\rm eje}$ is the typical distance from merger plaza to where ejecta happen and $ M_{\rm rem}$ is the gravitational mass of remnant core. For dynamic ejecta launched within dynamic timescale ($ \leq 10~{\rm ms}... | rm fje}\sqrt{G M_{\rm rem}R_{\rm eje}}$, vespectively, whete $R_{\rm tmrus}$ ia typicau radius of torus, $R_{\rm eje}$ is tye tykpcal distance from mefger centvr to wheee eoecta occur and $M_{\rm rem}$ lf ths grarivational mass on remnant cmre. For dynamiwau zjecta launched within dynamical timqscale ($\kee 10~{\rm ms}... | rm eje}\sqrt{G M_{\rm rem}R_{\rm eje}}$, respectively, where is radius of $R_{\rm eje}$ is center where ejecta occur $M_{\rm rem}$ is gravitational mass of remnant core. For ejecta launched within dynamical timescale ($\leq 10~{\rm ms}$), $R_{\rm eje}$ can be estimated the typical radius of the surrounding torus at suc... | rm eje}\sqrt{G M_{\rm rem}R_{\rm eje}}$, resPectively, wHere $R_{\Rm tOruS}$ iS typIcal Radius of torus, $R_{\RM eje}$ Is the typical distance frOm merGeR CentER tO wherE ejecta OCcUR And $m_{\rM rEm}$ iS tHE gRavitAtiOnal masS of remnant CorE. FOr dynamical eJEcTa launched WitHin dynamical TimEscale ($\LeQ 10~{\rm MS}... | rm eje}\sqrt{G M_{\rm rem} R_{\rm eje }}$,res pec ti vely , wh ere $R_{\rm to r us}$ is typical radius oftorus ,$ R_{\ r meje}$ is the ty p i cal d is tan ce fr om me rge r cente r to where ej ec ta occur and $M _{\rm rem} $ i s the gravit ati onal m as s o f remn ant core . Ford ynamic al ejecta l a unched with... | rm eje}\sqrt{G_M_{\rm rem}R_{\rm_eje}}$, respectively, where $R_{\rm_torus}$ is_typical_radius of_torus,_$R_{\rm eje}$ is_the typical distance_from merger center to_where ejecta occur_and_$M_{\rm rem}$ is the gravitational mass of remnant core. For dynamical ejecta launched within_dynamical_timescale ($\leq_10~{\rm_ms}... |
the mean-squared of the spacetime distance tends to a universal constant $\langle\delta X_{\mu}^{2}\rangle/\langle\Delta X_{\mu}\rangle^{2}=2/\pi$ in the extreme IR region of theory. We also argue that this effect is testable by observing a linear dependence between the variance and mean-squared of redshifts from dist... | the mean - squared of the spacetime distance tends to a universal constant $ \langle\delta X_{\mu}^{2}\rangle/\langle\Delta X_{\mu}\rangle^{2}=2/\pi$ in the extreme IR region of hypothesis. We besides argue that this impression is testable by observing a analogue dependence between the variability and mean - squared of... | thf mean-squared of the spagetime distance jebds to a unibersal cunstant $\langle\delta X_{\mu}^{2}\ranglx/\lantle\Deota X_{\mu}\rangle^{2}=2/\pi$ in the extreme PR region of uheory. We also arjhe that this snfect ms testable by pbserving d linear depengevcz between the variance and mean-squarqd of rrddhifts from dift... | the mean-squared of the spacetime distance tends universal $\langle\delta X_{\mu}^{2}\rangle/\langle\Delta in the extreme also that this effect testable by observing linear dependence between the variance and of redshifts from distant spectral lines. The proportionality is $\mathcal{O}(1)$ and expected to identical to ... | the mean-squared of the spacetIme distancE tendS to A unIvErsaL conStant $\langle\delTA X_{\mu}^{2}\Rangle/\langle\Delta X_{\mu}\raNgle^{2}=2/\pI$ iN The eXTrEme IR Region oF ThEORy. WE aLsO arGuE ThAt thiS efFect is tEstable by oBseRvIng a linear dePEnDence betweEn tHe variance anD meAn-squaReD of REdshiFts From dIst... | the mean-squared of the s pacetime d istan ceten ds toa un iversal consta n t $\ langle\delta X_{\mu}^{ 2}\ra ng l e/\l a ng le\De lta X_{ \ mu } \ ran gl e^ {2} =2 / \p i$ in th e extre me IR regi onof theory. Wea ls o argue th atthis effectistestab le by obser vin g a l ineard epende nce betwe en the va r iance ... | the_mean-squared of_the spacetime distance tends_to a_universal_constant $\langle\delta_X_{\mu}^{2}\rangle/\langle\Delta_X_{\mu}\rangle^{2}=2/\pi$ in the_extreme IR region_of theory. We also_argue that this_effect_is testable by observing a linear dependence between the variance and mean-squared of redshifts_from_dist... |
These generate a downward flux in momentum space, but one which is distributed throughout the acceleration region. Combined with the fact that the size of the “box” or region normally increases with energy this also gives an additional loss process because particles can now fall through the back of the “box” as well a... | These generate a downward flux in momentum space, but one which is distribute throughout the acceleration area. Combined with the fact that the size of the “ box ” or area normally increases with department of energy this also gives an extra loss process because particles can nowadays fall through the back of the “ cor... | Thfse generate a downward nlux in momentum space, uut one which ir distributed throughout the axceleeation region. Combined with the fact thqt tie size of the “box” or rennon nkvmallv mncreases with gnergy this dlso gives an ddaicional loss process because particlef can npw fall through jhe bssk or the “box” as well a... | These generate a downward flux in momentum one is distributed the acceleration region. the of the “box” region normally increases energy this also gives an additional process because particles can now fall through the back of the “box” as as being advected out of it (see Fig. 1). Note that particles which through front... | These generate a downward fluX in momentuM spacE, buT onE wHich Is diStributed throuGHout The acceleration region. COmbinEd WIth tHE fAct thAt the siZE oF THe “bOx” Or RegIoN NoRmallY inCreases With energy ThiS aLso gives an adDItIonal loss pRocEss because paRtiCles caN nOw fALl thrOugH the bAck of tHE “box” as Well a... | These generate a downward flux in m oment umspa ce , bu t on e which is dis t ribu ted throughout the acc elera ti o n re g io n. Co mbinedw it h the f ac t t ha t t he si zeof the“box” or r egi on normally in c re ases withene rgy this als o g ives a nadd i tiona l l oss p rocess becaus e particl es can no w fal... | These_generate a_downward flux in momentum_space, but_one_which is_distributed_throughout the acceleration_region. Combined with_the fact that the_size of the_“box”_or region normally increases with energy this also gives an additional loss process because_particles_can now_fall_through_the back of the “box”_as well a... |
{\mathcal{O}}}_0'$-homomorphism $$\label{eq_proof_weierstrass_div_map}
{\ensuremath{\mathcal{O}}}_0^F\ni f\mapsto (f_0',\ldots,f'_{d-1})\in({\ensuremath{\mathcal{O}}}_0^{\prime F})^{\oplus d}$$ descends to an isomorphism $$\label{eq_proof_weierstrass_iso}
{\ensuremath{\mathcal{O}}}_0^F/h{\ensuremath{\mathcal{O}}}_0^F\x... | { \mathcal{O}}}_0'$-homomorphism $ $ \label{eq_proof_weierstrass_div_map }
{ \ensuremath{\mathcal{O}}}_0^F\ni f\mapsto (f_0',\ldots, f'_{d-1})\in({\ensuremath{\mathcal{O}}}_0^{\prime F})^{\oplus d}$$ descends to an isomorphism $ $ \label{eq_proof_weierstrass_iso }
{ \ensuremath{\mathcal{O}}}_0^F / h{\ensuremath{\ma... | {\matjcal{O}}}_0'$-homomorphism $$\label{ed_proof_weierstrass_div_ma'}
{\ensurejath{\mathzal{O}}}_0^F\ni f\mapsto (f_0',\ldots,f'_{d-1})\in({\endueematy{\mathcal{O}}}_0^{\prime F})^{\oplus a}$$ descendd to an usomiephism $$\labxm{eq_proon_ceierabrass_nsi}
{\ensuremath{\matmcal{O}}}_0^F/h{\ensusemath{\mathcal{O}}}_0^X\x... | {\mathcal{O}}}_0'$-homomorphism $$\label{eq_proof_weierstrass_div_map} {\ensuremath{\mathcal{O}}}_0^F\ni f\mapsto (f_0',\ldots,f'_{d-1})\in({\ensuremath{\mathcal{O}}}_0^{\prime F})^{\oplus d}$$ an $$\label{eq_proof_weierstrass_iso} {\ensuremath{\mathcal{O}}}_0^F/h{\ensuremath{\mathcal{O}}}_0^F\xrightarrow{\approx}({\en... | {\mathcal{O}}}_0'$-homomorphism $$\label{Eq_proof_weiErstrAss_Div_MaP}
{\ensUremAth{\mathcal{O}}}_0^F\ni F\MapsTo (f_0',\ldots,f'_{d-1})\in({\ensuremath{\MathcAl{o}}}_0^{\PrimE f})^{\oPlus d}$$ DescendS To AN IsoMoRpHisM $$\lABeL{eq_prOof_WeierstRass_iso}
{\ensUreMaTh{\mathcal{O}}}_0^F/h{\ENsUremath{\matHcaL{O}}}_0^F\x... | {\mathcal{O}}}_0'$-homomor phism $$\l abel{ eq_ pro of _wei erst rass_div_map}{ \ens uremath{\mathcal{O}}}_ 0^F\n if \map s to (f_0 ',\ldot s ,f ' _ {d- 1} )\ in( {\ e ns urema th{ \mathca l{O}}}_0^{ \pr im e F})^{\oplu s d }$$ descen dsto an isomor phi sm $$\ la bel { eq_pr oof _weie rstras s _iso}{\ensurem at h {\m... | {\mathcal{O}}}_0'$-homomorphism $$\label{eq_proof_weierstrass_div_map}
{\ensuremath{\mathcal{O}}}_0^F\ni_f\mapsto (f_0',\ldots,f'_{d-1})\in({\ensuremath{\mathcal{O}}}_0^{\prime_F})^{\oplus d}$$ descends to_an isomorphism_$$\label{eq_proof_weierstrass_iso}
{\ensuremath{\mathcal{O}}}_0^F/h{\ensuremath{\mathcal{O}}}_0^F\x... |
of three coincident counters and two pairs of iron cores $A'$, $A''$ and $B'$, $B''$. The field in the cores is parallel to the axis of the counters and has opposite directions in $A'$, $A''$ (similarly in $B'$, $B''$). The magnetic field is closed by iron bars applied at both ends of $A'$, $A''$ and $B'$, $B''$. Thus... | of three coincident counters and two pairs of iron cores $ A'$, $ A''$ and $ B'$, $ B''$. The field in the core is parallel to the bloc of the counters and has diametric directions in $ A'$, $ A''$ (similarly in $ B'$, $ B''$). The charismatic field is closed by iron cake apply at both ends of $ A'$, $ A''$ and $ B'$, ... | of three coincident countevs and two pairs of iroi cores $A'$, $A''$ and $B'$, $B''$. The field in the cores ms pqralltj to the axis of tfe countegs and haw opkosite directions in $A'$, $A''$ (similadpy iu $U'$, $B''$). The magnetig field is wlosed by iron bxrd applied at both ends of $A'$, $A''$ and $B'$, $B''$. Thus... | of three coincident counters and two pairs cores $A''$ and $B''$. The field to axis of the and has opposite in $A'$, $A''$ (similarly in $B'$, The magnetic field is closed by iron bars applied at both ends of $A''$ and $B'$, $B''$. Thus each pair of iron cores acts like a magnetic (“$c$”) divergent lens for positive ($... | of three coincident counters And two pairS of irOn cOreS $A'$, $a''$ and $b'$, $B''$. ThE field in the corES is pArallel to the axis of the cOunteRs ANd haS OpPositE directIOnS IN $A'$, $A''$ (SiMiLarLy IN $B'$, $b''$). The mAgnEtic fieLd is closed By iRoN bars applied AT bOth ends of $A'$, $a''$ anD $B'$, $B''$. Thus... | of three coincident count ers and tw o pai rsofir on c ores $A'$, $A''$ a n d $B '$, $B''$. The field i n the c o resi sparal lel tot he a xis o fthe c o un tersand has op posite dir ect io ns in $A'$,$ A' '$ (simila rly in $B'$, $B ''$ ). The m agn e tic f iel d isclosed by iro n bars ap pl i ed atb oth end s ... | of_three coincident_counters and two pairs_of iron_cores_$A'$, $A''$_and_$B'$, $B''$. The_field in the_cores is parallel to_the axis of_the_counters and has opposite directions in $A'$, $A''$ (similarly in $B'$, $B''$). The magnetic_field_is closed_by_iron_bars applied at both ends_of $A'$, $A''$ and $B'$,_$B''$. Thus... |
and content.)
Summary tables of spectral analyses for the 1-s peak spectra
------------------------------------------------------------
In this section, we present the tables for the 1-s peak spectra, which are the same set of the tables as those in Section \[sect:T100\_spectra\_tables\] but for the 1-s peak spectra... | and content .)
Summary tables of spectral analyses for the 1 - s acme spectra
------------------------------------------------------------
In this incision, we present the tables for the 1 - s peak spectrum, which are the same set of the mesa as those in Section \[sect: T100\_spectra\_tables\ ] but for the 1 ... | anf content.)
Summary tables uf spectral analyses fmr the 1-s peak rpectra
------------------------------------------------------------
In this section, we prxsenr the tables for the 1-s peak spectra, ahich arw tht same set of the tables as thoac in Vxction \[sect:T100\_specjra\_tables\] bud for the 1-s pedk s'ectra... | and content.) Summary tables of spectral analyses 1-s spectra ------------------------------------------------------------ this section, we 1-s spectra, which are same set of tables as those in Section \[sect:T100\_spectra\_tables\] for the 1-s peak spectral fits. Column Format Unit Description ------------------ -----... | and content.)
Summary tables of Spectral anAlyseS foR thE 1-s Peak SpecTra
------------------------------------------------------------
In this sectiON, we pResent the tables for the 1-s Peak sPeCTra, wHIcH are tHe same sET oF THe tAbLeS as ThOSe In SecTioN \[sect:T100\_sPectra\_tablEs\] bUt For the 1-s peak sPEcTra... | and content.)
Summary ta bles of sp ectra l a nal ys es f or t he 1-s peak sp e ctra
--------------------- ----- -- - ---- - -- ----- ------- - -- - - --- -- --
I nt hi s sec tio n, we p resent the ta bl es for the 1 - speak spect ra, which are t hesame s et of the t abl es as those in Sec tion \[se ct : T100\... | and_content.)
Summary tables_of spectral analyses for_the 1-s_peak_spectra
------------------------------------------------------------
In this_section,_we present the_tables for the_1-s peak spectra, which_are the same_set_of the tables as those in Section \[sect:T100\_spectra\_tables\] but for the 1-s peak spectra... |
.3 28.7 40.9 49.2 43.3 36.8
real-a 98.93 91.34 84.25 82.43 97.31 69.27 75.51 80.70 33.9 **12.5** **12.7** **9.1** **19.8** 31.0 17.1 11.5
(ResNet) Ours **99.37** **94.69** **94.56** ... | .3 28.7 40.9 49.2 43.3 36.8
real - a 98.93 91.34 84.25 82.43 97.31 69.27 75.51 80.70 33.9 * * 12.5 * * * * 12.7 * * * * 9.1 * * * * 19.8 * * 31.0 17.1 11.5
(ResNet) ... | .3 28.7 40.9 49.2 43.3 36.8
ceal-a 98.93 91.34 84.25 82.43 97.31 69.27 75.51 80.70 33.9 **12.5** **12.7** **9.1** **19.8** 31.0 17.1 11.5
(RxaNet) Ours **99.37** **94.69** **94.56** ... | .3 28.7 40.9 49.2 43.3 36.8 real-a 84.25 97.31 69.27 80.70 33.9 **12.5** 11.5 Ours **99.37** **94.69** **93.35** **99.10** **93.04** **91.64** **18.5** **12.6** **13.0** **10.3** 29.7 **11.0** **8.9** ---------- -------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ---... | .3 28.7 40.9 49.2 43.3 36.8
real-a 98.93 91.34 84.25 82.43 97.31 69.27 75.51 80.70 33.9 **12.5** **12.7** **9.1** **19.8** 31.0 17.1 11.5
(ResNet) Ours **99.37** **94.69** **94.56** ... | .3 28.7 40.9 49.2 43 .3 36.8
real -a 98.93 91. 34 84. 2 5 82.43 97 .3 1 69 .27 75.51 80. 70 33.9 ** 12.5** * *12 .7** **9.1 ** **1 9. 8** 3 1.0 17.1 11.5
( Re s Net) Ours ** 99.3 7** **94.69** ** 9 4.56** **93. 35** ... | .3 _ _ 28.7_ __ __40.9 _ _ 49.2 _ __43.3 36.8
__ ___ real-a _ 98.93 _ _ 91.34 __84.25 _ 82.43 _ 97.31 _ _69.27__ _ 75.51 _ 80.70 _ 33.9 _ **12.5** **12.7** _ **9.1** _**19.8** _ 31.0 _ _ _17.1 _ 11.5
_ (ResNet) _ Ours _**99.37**_ **94.69**___**94.56** _ ... |
conductor between the antenna complex and the reaction center. The FMO protein is a trimer where the three identical subunits consist of seven chromophoric sites which are fully characterized [@BioCh2005]. Typical transport times for excitons through FMO are of the order of picoseconds. Recent experiments [@BioEn2007]... | conductor between the antenna complex and the reaction center. The FMO protein is a trimer where the three identical subunits dwell of seven chromophoric web site which are fully characterized [ @BioCh2005 ]. distinctive transport clock time for excitons through FMO are of the order of picoseconds. late experiment ... | cojductor between the antekna complex and jhw reacvion cehter. The FMO protein is a trimer whece tye theee identical subunits consist lf seven chrinophoric smfes whigk are nully rharacterized [@BipCh2005]. Typican transport tiker yor excitons through FMO are of the jrder og oicoseconds. Resent qxpedpmtnts [@BioEn2007]... | conductor between the antenna complex and the The protein is trimer where the seven sites which are characterized [@BioCh2005]. Typical times for excitons through FMO are the order of picoseconds. Recent experiments [@BioEn2007] on the FMO complex show that exciton transport exhibits quantum coherent oscillations up to... | conductor between the antennA complex anD the rEacTioN cEnteR. The fMO protein is a tRImer Where the three identical SubunItS ConsISt Of sevEn chromOPhORIc sItEs WhiCh ARe Fully ChaRacteriZed [@BioCh2005]. TyPicAl Transport timES fOr excitons ThrOugh FMO are of The Order oF pIcoSEcondS. ReCent eXperimENts [@Bioen2007]... | conductor between the ant enna compl ex an d t here acti on c enter. The FMO prot ein is a trimer wherethe t hr e e id e nt icalsubunit s c o n sis tof se ve n c hromo pho ric sit es which a refu lly characte r iz ed [@BioCh 200 5]. Typicaltra nsport t ime s forexc itons throu g h FMOare of th eo rder o f picos... | conductor_between the_antenna complex and the_reaction center._The_FMO protein_is_a trimer where_the three identical_subunits consist of seven_chromophoric sites which_are_fully characterized [@BioCh2005]. Typical transport times for excitons through FMO are of the order of_picoseconds._Recent experiments [@BioEn2007]... |
momentum independent $\Delta_{k}=\Delta$ and $\Delta$ is determined by minimizing the ground state energy. We note that we are considering a band structure with isotropic dispersion and the electron and hole Fermi surfaces are perfectly nested. In this case, the exciton pairing gap $\Delta$ is always non-zero in the m... | momentum independent $ \Delta_{k}=\Delta$ and $ \Delta$ is determined by minimizing the ground department of state department of energy. We note that we are considering a isthmus social organization with isotropic dispersion and the electron and hole Fermi surfaces are perfectly nest. In this case, the exciton pairing ... | molentum independent $\Delta_{y}=\Delta$ and $\Deltc$ is devermines by minkmizing the ground state enecgy. Qe nouv that we are considefing a bajd strucrure qith isotropic disizrsioh and vhe electron anc hole Ferki surfaces ara oexfectly nested. In this case, the excieon paitijg gap $\Delta$ if alewys hon-zero in the m... | momentum independent $\Delta_{k}=\Delta$ and $\Delta$ is determined the state energy. note that we with dispersion and the and hole Fermi are perfectly nested. In this case, exciton pairing gap $\Delta$ is always non-zero in the mean-field theory, although its can be very small. In reality the mean-field gap will be de... | momentum independent $\Delta_{k}=\delta$ and $\DeLta$ is DetErmInEd by MiniMizing the grounD StatE energy. We note that we are ConsiDeRIng a BAnD struCture wiTH iSOTroPiC dIspErSIoN and tHe eLectron And hole FerMi sUrFaces are perfECtLy nested. In ThiS case, the exciTon PairinG gAp $\DELta$ is AlwAys noN-zero iN The m... | momentum independent $\De lta_{k}=\D elta$ an d $ \D elta $ is determined by mini mizing the ground stat e ene rg y . We no te th at we a r ec o nsi de ri ngab an d str uct ure wit h isotropi c d is persion andt he electronand hole Fermisur facesar e p e rfect lyneste d. Int his ca se, the e xc i ton pa i ring ga ... | momentum_independent $\Delta_{k}=\Delta$_and $\Delta$ is determined_by minimizing_the_ground state_energy._We note that_we are considering_a band structure with_isotropic dispersion and_the_electron and hole Fermi surfaces are perfectly nested. In this case, the exciton pairing_gap_$\Delta$ is_always_non-zero_in the m... |
other words, we have the following *girth alternative*:
Let ${\Sigma}$ be a compact orientable surface, and let $\Gamma$ be a finitely generated subgroup of $\Mod({\Sigma})$. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually free-abelian group; moreover, these alternatives are mutually ex... | other words, we have the following * girth option *:
lease $ { \Sigma}$ be a compact orientable surface, and lease $ \Gamma$ be a finitely render subgroup of $ \Mod({\Sigma})$. Then, $ \Gamma$ is either a non - cyclic group with infinite girth or a about free - abelian group; moreover, these alternatives are mutuall... | otjer words, we have the foulowing *girth alternatmve*:
Let ${\Aigma}$ be a compact orientable surfacx, ane let $\Gamma$ be a finitely gdnerated dubgroup of $\Nid({\Sigma})$. Thxh, $\Gamma$ is eifmer a ion-cyclic group with infitite girth or d xixtually free-abelian group; moreover, trese alyegnatives are motualkr ex... | other words, we have the following *girth ${\Sigma}$ a compact surface, and let subgroup $\Mod({\Sigma})$. Then, $\Gamma$ either a non-cyclic with infinite girth or a virtually group; moreover, these alternatives are mutually exclusive. The girth alternative above reduces to case where the interior of ${\Sigma}$ admits... | other words, we have the followIng *girth alTernaTivE*:
LeT ${\SIgma}$ Be a cOmpact orientabLE surFace, and let $\Gamma$ be a finiTely gEnERateD SuBgrouP of $\Mod({\SIGmA})$. tHen, $\gaMmA$ is EiTHeR a non-CycLic grouP with infinIte GiRth or a virtuaLLy Free-abeliaN grOup; moreover, tHesE alterNaTivES are mUtuAlly eX... | other words, we have thefollowing*girt h a lte rn ativ e*:
Let ${\Sigma} $ bea compact orientable s urfac e, andl et $\Ga mma$ be af i nit el ygen er a te d sub gro up of $ \Mod({\Sig ma} )$ . Then, $\Ga m ma $ is eithe r a non-cyclicgro up wit hinf i nitegir th or a vir t uallyfree-abel ia n group ; moreov e r ... | other_words, we_have the following *girth_alternative*:
Let ${\Sigma}$_be_a compact_orientable_surface, and let_$\Gamma$ be a_finitely generated subgroup of_$\Mod({\Sigma})$. Then, $\Gamma$_is_either a non-cyclic group with infinite girth or a virtually free-abelian group; moreover, these_alternatives_are mutually_ex... |
face of ${\mathrm{Newt}}_p(f)$ with inner normal $(v,1)$. [$\blacksquare$]{}
In Example \[ex:newt\] earlier, note that the $3$ lower edges have respective horizontal lengths $2$, $3$, and $1$, and inner normals $(1,1)$, $(0,1)$, and $(-5,1)$. Lemma \[lemma:newt\] then tells us that $f$ has exactly $6$ roots in ${\mat... | face of $ { \mathrm{Newt}}_p(f)$ with inner normal $ (v,1)$. [ $ \blacksquare$ ] { }
In Example \[ex: newt\ ] earlier, note that the $ 3 $ low edge have respective horizontal lengths $ 2 $, $ 3 $, and $ 1 $, and inner convention $ (1,1)$, $ (0,1)$, and $ (-5,1)$. Lemma \[lemma: newt\ ] then tells us that $ f$ has ex... | fafe of ${\mathrm{Newt}}_p(f)$ with lnner normal $(v,1)$. [$\blacksquere$]{}
In Esample \[eb:newt\] earlier, note that the $3$ liwer tbges have respective horizontwl lengtys $2$, $3$, qnd $1$, and iiher normals $(1,1)$, $(0,1)$, wnd $(-5,1)$. Oemma \[lemma:newj\] then tells us that $f$ has ebaetly $6$ roots in ${\mat... | face of ${\mathrm{Newt}}_p(f)$ with inner normal $(v,1)$. Example earlier, note the $3$ lower $2$, and $1$, and normals $(1,1)$, $(0,1)$, $(-5,1)$. Lemma \[lemma:newt\] then tells us $f$ has exactly $6$ roots in ${\mathbb{C}}_3$: $2$ with $3$-adic valuation $1$, $3$ $3$-adic valuation $0$, and $1$ with $3$-adic valuati... | face of ${\mathrm{Newt}}_p(f)$ with innEr normal $(v,1)$. [$\bLacksQuaRe$]{}
IN EXampLe \[ex:Newt\] earlier, notE That The $3$ lower edges have respeCtive HoRIzonTAl LengtHs $2$, $3$, and $1$, anD InNER noRmAlS $(1,1)$, $(0,1)$, anD $(-5,1)$. LEMmA \[lemmA:neWt\] then tElls us that $F$ haS eXactly $6$ roots iN ${\MaT... | face of ${\mathrm{Newt}}_ p(f)$ with inne r n orm al $(v ,1)$ . [$\blacksqua r e$]{ }
In Example \[ex:new t\] e ar l ier, no te th at the$ 3$ l owe red ges h a ve resp ect ive hor izontal le ngt hs $2$, $3$, a n d$1$, and i nne r normals $( 1,1 )$, $( 0, 1)$ , and$(- 5,1)$ . Lemm a \[lem ma:newt\] t h en tel ... | face_of ${\mathrm{Newt}}_p(f)$_with inner normal $(v,1)$._[$\blacksquare$]{}
In Example_\[ex:newt\]_earlier, note_that_the $3$ lower_edges have respective_horizontal lengths $2$, $3$,_and $1$, and_inner_normals $(1,1)$, $(0,1)$, and $(-5,1)$. Lemma \[lemma:newt\] then tells us that $f$ has exactly_$6$_roots in_${\mat... |
\be_{\alpha_j} +\be_{\alpha_j} \be_{\alpha_i}^2 &= (q+q^{-1}) \be_{\alpha_i} \be_{\alpha_j} \be_{\alpha_i},
\ \ \quad\quad &\text{if }& |i-j|=1, \displaybreak[0]\\
\be_{\alpha_i} \be_{\alpha_j} &= \be_{\alpha_j} \be_{\alpha_i}, \ \qquad\qquad\ \ \ \ \qquad &\text{if }& |i-j|>1, \displaybreak[0]\\
\bff_{\alpha_... | \be_{\alpha_j } + \be_{\alpha_j } \be_{\alpha_i}^2 & = (q+q^{-1 }) \be_{\alpha_i } \be_{\alpha_j } \be_{\alpha_i },
\ \ \quad\quad & \text{if } & |i - j|=1, \displaybreak[0]\\
\be_{\alpha_i } \be_{\alpha_j } & = \be_{\alpha_j } \be_{\alpha_i }, \ \qquad\qquad\ \ \ \ \qquad & \text{if } & |i - j|>1, \displaybre... | \be_{\wlpha_j} +\be_{\alpha_j} \be_{\alpha_i}^2 &= (q+q^{-1}) \be_{\alpha_i} \bg_{\aopha_j} \ue_{\alpha_j},
\ \ \qjad\quad &\text{if }& |i-j|=1, \displaybrxak[0]\\
\be_{\alkka_i} \be_{\alpha_j} &= \be_{\alphx_j} \be_{\alphw_i}, \ \qquae\qqued\ \ \ \ \qquad &\text{mr }& |i-j|>1, \dlfplagnreak[0]\\
\bff_{\alpha_... | \be_{\alpha_j} +\be_{\alpha_j} \be_{\alpha_i}^2 &= (q+q^{-1}) \be_{\alpha_i} \be_{\alpha_j} \ &\text{if }& \displaybreak[0]\\ \be_{\alpha_i} \be_{\alpha_j} \ \ \qquad &\text{if |i-j|>1, \displaybreak[0]\\ \bff_{\alpha_i}^2 +\bff_{\alpha_j} \bff_{\alpha_i}^2 &= (q+q^{-1}) \bff_{\alpha_i} \bff_{\alpha_j} \ \ \quad\quad &... | \be_{\alpha_j} +\be_{\alpha_j} \be_{\alpha_i}^2 &= (q+Q^{-1}) \be_{\alpha_i} \bE_{\alphA_j} \bE_{\alPhA_i},
\ \ \quAd\quAd &\text{if }& |i-j|=1, \dispLAybrEak[0]\\
\be_{\alpha_i} \be_{\alpha_j} &= \be_{\aLpha_j} \Be_{\ALpha_I}, \ \QqUad\qqUad\ \ \ \ \qquaD &\TeXT{If }& |i-J|>1, \dIsPlaYbREaK[0]\\
\bff_{\aLphA_... | \be_{\alpha_j} +\be_{\alp ha_j} \be_ {\alp ha_ i}^ 2&= ( q+q^ {-1}) \be_{\al p ha_i } \be_{\alpha_j} \be_{ \alph a_ i }, \ \ \ quad\qu a d& \ tex t{ if }& | i -j |=1,\di splaybr eak[0]\\
\b e_ {\alpha_i} \ b e_ {\alpha_j} &= \be_{\alpha _j} \be_{ \a lph a _i},\ \ qquad \qquad \ \ \ \ \qquad & \t e xt{if} & |i... | \be_{\alpha_j}_+\be_{\alpha_j} \be_{\alpha_i}^2_&= (q+q^{-1}) \be_{\alpha_i} \be_{\alpha_j}_\be_{\alpha_i},
__ \_\_\quad\quad &\text{if }&_|i-j|=1, \displaybreak[0]\\
_\be_{\alpha_i} \be_{\alpha_j} &= \be_{\alpha_j}_\be_{\alpha_i}, \ \qquad\qquad\_\_\ \ \qquad &\text{if }& |i-j|>1, \displaybreak[0]\\
\bff_{\alpha_... |
{{\sf hub}}({u}) = \sum_{{v}\in {V}}
{{{\mathit wt}}{({{u}{\to}{v}})}}\cdot {{\sf auth}}({v}).
\label{eq:hub}$$ Clearly, Equations \[eq:auth\] and \[eq:hub\] are mutually recursive. However, the iterative HITS algorithm[^1] provably converges to (non-identically-zero, non-negative) score functions ${{\sf hub}}^*$ and ... | { { \sf hub}}({u }) = \sum_{{v}\in { V } }
{ { { \mathit wt}}{({{u}{\to}{v}})}}\cdot { { \sf auth}}({v }).
\label{eq: hub}$$ Clearly, Equations \[eq: auth\ ] and \[eq: hub\ ] are mutually recursive. However, the iterative HITS algorithm[^1 ] provably converges to (non - identically - zero, non - damaging) sexual c... | {{\sf jub}}({u}) = \sum_{{v}\in {V}}
{{{\mathit wt}}{({{m}{\to}{v}})}}\cdot {{\sf auth}}({r}).
\oabel{ex:hub}$$ Cmearly, Eduations \[eq:auth\] and \[eq:hub\] arx muruallt recursive. However, thd iteratine HITS aogormthm[^1] provably coiberges bj (noh-ldentncelly-zero, non-negstive) scora functions ${{\sf hjb}}^*$ and ... | {{\sf hub}}({u}) = \sum_{{v}\in {V}} {{{\mathit wt}}{({{u}{\to}{v}})}}\cdot \label{eq:hub}$$ Equations \[eq:auth\] \[eq:hub\] are mutually algorithm[^1] converges to (non-identically-zero, score functions ${{\sf and ${{\sf auth}}^*$ that satisfy the pair of equations. Figure \[fig:hits-icon\] depicts the “iconic” case ... | {{\sf hub}}({u}) = \sum_{{v}\in {V}}
{{{\mathit wt}}{({{u}{\to}{v}})}}\Cdot {{\sf auth}}({V}).
\labeL{eq:Hub}$$ clEarlY, EquAtions \[eq:auth\] anD \[Eq:huB\] are mutually recursive. HOweveR, tHE iteRAtIve HItS algorIThM[^1] PRovAbLy ConVeRGeS to (noN-idEnticalLy-zero, non-nEgaTiVe) score functIOnS ${{\sf hub}}^*$ and ... | {{\sf hub}}({u}) = \sum_{{ v}\in {V}}
{{{ \ma thi twt}} {({{ u}{\to}{v}})}} \ cdot {{\sf auth}}({v}).
\l abel{ eq : hub} $ $Clear ly, Equ a ti o n s \ [e q: aut h\ ] a nd \[ eq: hub\] a re mutuall y r ec ursive. Howe v er , the iter ati ve HITS algo rit hm[^1] p rov a bly c onv erges to (n o n-iden tically-z er o... | {{\sf hub}}({u})_= \sum_{{v}\in_{V}}
{{{\mathit wt}}{({{u}{\to}{v}})}}\cdot {{\sf_auth}}({v}).
\label{eq:hub}$$ Clearly,_Equations_\[eq:auth\] and_\[eq:hub\]_are mutually recursive._However, the iterative_HITS algorithm[^1] provably converges_to (non-identically-zero, non-negative)_score_functions ${{\sf hub}}^*$ and ... |
9}$Fe$_{0.1}$O$_3$.
CONCLUSIONS
===========
The electronic structures of of multiferroic oxides of Ba$_{1-x}$Bi$_x$Ti$_{0.9}$Fe$_{0.1}$O$_3$ ($0 \le x \le 0.12$). have been investigated by employing synchrotron-radiation excited PES and XAS. Via Fe and Ti $2p$ XAS measurements, the valence states of Fe and Ti ions ha... | 9}$Fe$_{0.1}$O$_3$.
CONCLUSIONS
= = = = = = = = = = =
The electronic structures of of multiferroic oxides of Ba$_{1 - x}$Bi$_x$Ti$_{0.9}$Fe$_{0.1}$O$_3 $ ($ 0 \le x \le 0.12 $). have been investigated by use synchrotron - radiation sickness excited PES and XAS. Via Fe and Ti $ 2p$ XAS measurements, the valence ... | 9}$Fe$_{0.1}$O$_3$.
FONCLUSIONS
===========
The electronig structures of of multmferroid oxides of Ba$_{1-x}$Bi$_x$Ti$_{0.9}$Fe$_{0.1}$O$_3$ ($0 \le x \le 0.12$). hate bwen ibvestigated by employivg synchrltron-raduatiib excited 'SS and WCS. Viz Fe cnv Ti $2p$ XAS measorements, the valence statev uf Fe and Ti ions ha... | 9}$Fe$_{0.1}$O$_3$. CONCLUSIONS =========== The electronic structures of oxides Ba$_{1-x}$Bi$_x$Ti$_{0.9}$Fe$_{0.1}$O$_3$ ($0 x \le 0.12$). synchrotron-radiation PES and XAS. Fe and Ti XAS measurements, the valence states of and Ti ions have been determined experimentally. The valence states of Fe ions found to be Fe$^... | 9}$Fe$_{0.1}$O$_3$.
CONCLUSIONS
===========
The electronIc structurEs of oF muLtiFeRroiC oxiDes of Ba$_{1-x}$Bi$_x$Ti$_{0.9}$FE$_{0.1}$o$_3$ ($0 \le x \Le 0.12$). have been investigated By empLoYIng sYNcHrotrOn-radiaTIoN EXciTeD PeS aNd xaS. via Fe And ti $2p$ XAS mEasurementS, thE vAlence states OF FE and Ti ions Ha... | 9}$Fe$_{0.1}$O$_3$.
CONCL USIONS
=== ===== ===
T he ele ctro nic structures of o f multiferroic oxidesof Ba $_ { 1-x} $ Bi $_x$T i$_{0.9 } $F e $ _{0 .1 }$ O$_ 3$ ($ 0 \le x\le 0.1 2$). havebee ninvestigated by employing sy nchrotron-ra dia tion e xc ite d PESand XAS. Via F e and T i $2p$ XA Sm easure m ents... | 9}$Fe$_{0.1}$O$_3$.
CONCLUSIONS
===========
The electronic_structures of_of multiferroic oxides of_Ba$_{1-x}$Bi$_x$Ti$_{0.9}$Fe$_{0.1}$O$_3$ ($0_\le_x \le_0.12$)._have been investigated_by employing synchrotron-radiation_excited PES and XAS._Via Fe and_Ti_$2p$ XAS measurements, the valence states of Fe and Ti ions ha... |
Finally, we summarize in Section \[sec:discussion\].
Models expressed in a programming language\[sec:models\]
========================================================
Probabilistic programming considers models expressed in Turing-complete programming languages. Such models are usually referred to as *probabilistic p... | Finally, we summarize in Section \[sec: discussion\ ].
Models expressed in a programming language\[sec: models\ ]
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Probabilistic scheduling considers mannequin expressed in Turing - accomplished schedu... | Fijally, we summarize in Segtion \[sec:discussnin\].
Modens expdessed iv a programming language\[sec:mldwls\]
========================================================
Pribabilistic programmine considegs models expcessed in Turing-rkmplete progrzlminy oanguages. Such models ara usually refesrdd to as *probabilistic p... | Finally, we summarize in Section \[sec:discussion\]. Models a language\[sec:models\] ======================================================== programming considers models Such are usually referred as *probabilistic programs*—qualified *universal probabilistic programs* when one wishes regard only the broadest class in ... | Finally, we summarize in SectiOn \[sec:discuSsion\].
modEls ExPresSed iN a programming lANguaGe\[sec:models\]
========================================================
ProbabilistIc proGrAMminG CoNsideRs modelS ExPREssEd In turInG-CoMpletE prOgrammiNg languageS. SuCh Models are usuALlY referred tO as *ProbabilistiC p... | Finally, we summarize inSection \[ sec:d isc uss io n\].
Mo dels expressed in a programming language\ [sec: mo d els\ ]
= ===== ======= = == = = === == == === == = == ===== === ======= ========
Pro ba bilistic pro g ra mming cons ide rs models ex pre ssed i nTur i ng-co mpl ete p rogram m ing la nguages.Su c h mod... | Finally,_we summarize_in Section \[sec:discussion\].
Models expressed_in a_programming_language\[sec:models\]
========================================================
Probabilistic programming_considers_models expressed in_Turing-complete programming languages._Such models are usually_referred to as_*probabilistic_p... |
2inrem\]) and using the identity (\[intidentity1\]), we conclude that $$2ig_0\Phi_2^{(2)} = \frac{8g_0}{\pi} \int_{\gamma} k^3\int_{-t}^t I_2(t,s)e^{2ik^4(s - t)}ds dk
= g_0\left(m(t,t) + \frac{i}{2}\bar{g}_0n(t,t)\right).$$ On the other hand, we can write $c^{(3)}$ as the sum of two terms, $c^{(3)} = c_1^{(3)} +c_2^{(... | 2inrem\ ]) and using the identity (\[intidentity1\ ]), we conclude that $ $ 2ig_0\Phi_2^{(2) } = \frac{8g_0}{\pi } \int_{\gamma } k^3\int_{-t}^t I_2(t, s)e^{2ik^4(s - t)}ds dk
= g_0\left(m(t, t) + \frac{i}{2}\bar{g}_0n(t, t)\right).$$ On the other hand, we can spell $ c^{(3)}$ as the kernel of two terms, $ c^{(3) } =... | 2inrfm\]) and using the identitn (\[intidentity1\]), we concluve that $$2ig_0\Phi_2^{(2)} = \wrac{8g_0}{\pi} \int_{\gamma} k^3\int_{-t}^t I_2(t,s)e^{2mk^4(s - t)}ds ek
= g_0\left(m(t,t) + \frac{i}{2}\bar{g}_0v(t,t)\right).$$ Ln the orher yand, we cai write $g^{(3)}$ as fme suk of two terms, $g^{(3)} = c_1^{(3)} +c_2^{(... | 2inrem\]) and using the identity (\[intidentity1\]), we $$2ig_0\Phi_2^{(2)} \frac{8g_0}{\pi} \int_{\gamma} I_2(t,s)e^{2ik^4(s - t)}ds On other hand, we write $c^{(3)}$ as sum of two terms, $c^{(3)} = +c_2^{(3)}$, where $$\label{c13def} c_1^{(3)}= \frac{i}{\pi}\int_{\gamma} k^2\left(\Phi_1(t,k) + i \Phi_1(t, ik) + \frac... | 2inrem\]) and using the identity (\[iNtidentity1\]), We conCluDe tHaT $$2ig_0\PHi_2^{(2)} = \frAc{8g_0}{\pi} \int_{\gamma} k^3\INt_{-t}^t i_2(t,s)e^{2ik^4(s - t)}ds dk
= g_0\left(m(t,t) + \frAc{i}{2}\baR{g}_0N(T,t)\riGHt).$$ on the Other haND, wE CAn wRiTe $C^{(3)}$ as ThE SuM of twO teRms, $c^{(3)} = c_1^{(3)} +c_2^{(... | 2inrem\]) and using the id entity (\[ intid ent ity 1\ ]),we c onclude that $ $ 2ig_ 0\Phi_2^{(2)} = \frac{ 8g_0} {\ p i} \ i nt _{\ga mma} k^ 3 \i n t _{- t} ^t I_ 2( t ,s )e^{2 ik^ 4(s - t )}ds dk
=g_0 \l eft(m(t,t) + \f rac{i}{2}\ bar {g}_0n(t,t)\ rig ht).$$ O n t h e oth erhand, we ca n write $c^{(3)} $a s the... | 2inrem\]) and_using the_identity (\[intidentity1\]), we conclude_that $$2ig_0\Phi_2^{(2)}_=_\frac{8g_0}{\pi} \int_{\gamma}_k^3\int_{-t}^t_I_2(t,s)e^{2ik^4(s - t)}ds_dk
= g_0\left(m(t,t) +_\frac{i}{2}\bar{g}_0n(t,t)\right).$$ On the other_hand, we can_write_$c^{(3)}$ as the sum of two terms, $c^{(3)} = c_1^{(3)} +c_2^{(... |
, collect terms on LHS, and re-state the conjecture as: $$f(\sigma,\lambda_2,\delta)<0,$$ where $$\begin{split}
f(\sigma,\lambda_2,\delta)=&\lambda_2(-44.8\lambda_2-71.68\delta+73.6\sigma+16)\\
&+\delta(12.8+58.88\sigma-37.92\delta)+\\
&\sigma(-20.32\sigma-32.96)+8.48
\end{split}$$
Our goal is to maximiz... | , collect terms on LHS, and re - state the speculation as: $ $ f(\sigma,\lambda_2,\delta)<0,$$ where $ $ \begin{split }
f(\sigma,\lambda_2,\delta)=&\lambda_2(-44.8\lambda_2 - 71.68\delta+73.6\sigma+16)\\
& + \delta(12.8 + 58.88\sigma-37.92\delta)+\\
& \sigma(-20.32\sigma-32.96)+8.48
\end{split}$$
... | , coplect terms on LHS, and rt-state the conjecjuee as: $$h(\sigma,\lzmbda_2,\delga)<0,$$ where $$\begin{split}
f(\sigma,\lalbea_2,\deluc)=&\lambda_2(-44.8\lambda_2-71.68\delta+73.6\siema+16)\\
&+\delta(12.8+58.88\digma-37.92\delra)+\\
&\smgma(-20.32\sigma-32.96)+8.48
\end{sijit}$$
Ohv goan is to maximiz... | , collect terms on LHS, and re-state as: where $$\begin{split} &+\delta(12.8+58.88\sigma-37.92\delta)+\\ &\sigma(-20.32\sigma-32.96)+8.48 \end{split}$$ $f(\sigma,\lambda_2,\delta)$ show that it less than $0$. first identify that $\frac{\partial f(\sigma,\lambda_2,\delta)}{\partial \lambda_2}>0$, $f(\sigma,\lambda_2,\de... | , collect terms on LHS, and re-staTe the conjeCture As: $$f(\SigMa,\LambDa_2,\deLta)<0,$$ where $$\begin{sPLit}
f(\Sigma,\lambda_2,\delta)=&\lambda_2(-44.8\LambdA_2-71.68\dELta+73.6\sIGmA+16)\\
&+\deltA(12.8+58.88\sigma-37.92\dELtA)+\\
&\SIgmA(-20.32\sIgMa-32.96)+8.48
\eNd{SPlIt}$$
Our GoaL is to maXimiz... | , collect terms on LHS, an d re-state thecon jec tu re a s: $ $f(\sigma,\lam b da_2 ,\delta)<0,$$ where $$ \begi n{ s plit }
f(\si gma,\la m bd a _ 2,\ de lt a)= &\ l am bda_2 (-4 4.8\lam bda_2-71.6 8\d el ta+73.6\sigm a +1 6)\\
&+\d elt a(12.8+58.88 \si gma-37 .9 2\d e lta)+ \\ &\si gma(-2 0 .32\si gma-32.96 )+ 8 .... | , collect_terms on_LHS, and re-state the_conjecture as:_$$f(\sigma,\lambda_2,\delta)<0,$$_where $$\begin{split}
_f(\sigma,\lambda_2,\delta)=&\lambda_2(-44.8\lambda_2-71.68\delta+73.6\sigma+16)\\
_&+\delta(12.8+58.88\sigma-37.92\delta)+\\
&\sigma(-20.32\sigma-32.96)+8.48 _ _
_ \end{split}$$
Our_goal_is to maximiz... |
2)$ and space-periodic domains as in our case.
We restrict our considerations to the tridimensional case just for the simplicity, but the technique also works in the two-dimensional case and the results hold for $p\in(1,2)$.
In order to better explain our results we introduce some spaces of functions. For the sequel ... | 2)$ and space - periodic domains as in our case.
We restrict our consideration to the tridimensional subject just for the simplicity, but the technique besides works in the two - dimensional case and the results keep for $ p\in(1,2)$.
In order to better explain our results we introduce some space of functions. Fo... | 2)$ anf space-periodic domains xs in our case.
Wg eestrirt our donsiderxtions to the tridimensional cqse jyst for the simplicity, but the nechnique alsi works in vge two-dliensjlnal rase and the rexults hold for $p\in(1,2)$.
In ordar tl better explain our results we intwoduce xole spaces of fonctipgs. Fkg uhe sequel ... | 2)$ and space-periodic domains as in our restrict considerations to tridimensional case just technique works in the case and the hold for $p\in(1,2)$. In order to explain our results we introduce some spaces of functions. For the sequel it worth to note that vector valued functions are printed in boldface while scalar ... | 2)$ and space-periodic domains as In our case.
WE restRicT ouR cOnsiDeraTions to the tridIMensIonal case just for the simPliciTy, BUt thE TeChniqUe also wORkS IN thE tWo-DimEnSIoNal caSe aNd the reSults hold fOr $p\In(1,2)$.
in order to betTEr Explain our ResUlts we introdUce Some spAcEs oF FunctIonS. For tHe sequEL ... | 2)$ and space-periodic dom ains as in ourcas e.
W e re stri ct our conside r atio ns to the tridimension al ca se just fo r the simpli c it y , bu tth e t ec h ni que a lso worksin the two -di me nsional case an d the resu lts hold for $p \in (1,2)$ .
In order to bett er exp l ain ou r results w e intro d uce ... | 2)$ and_space-periodic domains_as in our case.
We_restrict our_considerations_to the_tridimensional_case just for_the simplicity, but_the technique also works_in the two-dimensional_case_and the results hold for $p\in(1,2)$.
In order to better explain our results we introduce_some_spaces of_functions._For_the sequel ... |
widetilde{\tau}(z)}{\sigma(z)}{\psi}'(z)+
\frac{\widetilde{\sigma}(z)}{\sigma^{2}(z)}\psi(z)=0,$$ where $\sigma(z)$ and $\widetilde{\sigma}(z)$ are polynomials, at most second-degree, and $\widetilde{\tau}(z)$ is a first-degree polynomial.
Using Eq.(2.1) the transformation $$\psi(z)=\Phi(z)\\{y}(z)$$ one reduces it to... | widetilde{\tau}(z)}{\sigma(z)}{\psi}'(z)+
\frac{\widetilde{\sigma}(z)}{\sigma^{2}(z)}\psi(z)=0,$$ where $ \sigma(z)$ and $ \widetilde{\sigma}(z)$ are polynomials, at most second - degree, and $ \widetilde{\tau}(z)$ is a first - degree polynomial.
use Eq.(2.1) the transformation $ $ \psi(z)=\Phi(z)\\{y}(z)$$ one re... | widftilde{\tau}(z)}{\sigma(z)}{\psi}'(z)+
\frac{\didetilde{\sigma}(z)}{\sigma^{2}(z)}\pvi(z)=0,$$ whsre $\sigmx(z)$ and $\widetilde{\sigma}(z)$ are plltnomiqls, at most second-degrde, and $\wifetilde{\tqu}(z)$ ms a first-degree polynomlcl.
Usihn Eq.(2.1) chx transformatiok $$\psi(z)=\Phi(z)\\{y}(s)$$ one reduces ht tl... | widetilde{\tau}(z)}{\sigma(z)}{\psi}'(z)+ \frac{\widetilde{\sigma}(z)}{\sigma^{2}(z)}\psi(z)=0,$$ where $\sigma(z)$ and $\widetilde{\sigma}(z)$ are most and $\widetilde{\tau}(z)$ a first-degree polynomial. one it to the equation $$\sigma(z){y}''+\tau(z){y}'+\lambda{y}=0.$$ The $\Phi(z)$ is defined as the logarithmic \[... | widetilde{\tau}(z)}{\sigma(z)}{\psi}'(z)+
\frAc{\widetildE{\sigmA}(z)}{\sIgmA^{2}(z)}\Psi(z)=0,$$ WherE $\sigma(z)$ and $\wideTIlde{\Sigma}(z)$ are polynomials, at Most sEcONd-deGReE, and $\wIdetildE{\TaU}(Z)$ Is a FiRsT-deGrEE pOlynoMiaL.
Using EQ.(2.1) the transfOrmAtIon $$\psi(z)=\Phi(z)\\{y}(Z)$$ OnE reduces it To... | widetilde{\tau}(z)}{\sigma (z)}{\psi} '(z)+
\f rac {\ wide tild e{\sigma}(z)}{ \ sigm a^{2}(z)}\psi(z)=0,$$where $ \ sigm a (z )$ an d $\wid e ti l d e{\ si gm a}( z) $ a re po lyn omials, at most s eco nd -degree, and $\ widetilde{ \ta u}(z)$ is afir st-deg re e p o lynom ial .
Us ing Eq . (2.1)the trans fo r mat... | widetilde{\tau}(z)}{\sigma(z)}{\psi}'(z)+
\frac{\widetilde{\sigma}(z)}{\sigma^{2}(z)}\psi(z)=0,$$ where_$\sigma(z)$ and_$\widetilde{\sigma}(z)$ are polynomials, at_most second-degree,_and_$\widetilde{\tau}(z)$ is_a_first-degree polynomial.
Using Eq.(2.1)_the transformation $$\psi(z)=\Phi(z)\\{y}(z)$$_one reduces it to... |
ideal as a sub-object of $W$ because of the isomorphism $\mathbf{R}({\mathcal{D}}[s])\simeq\textrm{gr}^V_0(\mathbf{R}({\mathcal{D}}_{x,t})).$
### Condition for the $h$-saturation of $\mathbf{R}(\textrm{gr}^V(N))$
Let $J(f)$ be the ideal generated by $f'_1,\dots,f'_n$. Let us define a morphism of graded $\mathcal{O}$... | ideal as a sub - object of $ W$ because of the isomorphism $ \mathbf{R}({\mathcal{D}}[s])\simeq\textrm{gr}^V_0(\mathbf{R}({\mathcal{D}}_{x, t})).$
# # # Condition for the $ h$-saturation of $ \mathbf{R}(\textrm{gr}^V(N))$
Let $ J(f)$ be the ideal generate by $ f'_1,\dots, f'_n$. lease us define a morphism of grad... | idfal as a sub-object of $W$ necause of the isomorphmsm $\matgbf{R}({\mathzal{D}}[s])\simeq\textrm{gr}^V_0(\mathbf{R}({\mavhcao{D}}_{x,t})).$
### Xondition for the $h$-satjration ov $\mathbf{E}(\texurm{gr}^V(N))$
Let $J(f)$ be vge ideal genedwted uy $f'_1,\dots,f'_n$. Let os define a korphism of grdddd $\mathcal{O}$... | ideal as a sub-object of $W$ because isomorphism ### Condition the $h$-saturation of ideal by $f'_1,\dots,f'_n$. Let define a morphism graded $\mathcal{O}$-algebras $$\varphi_f : \textrm{gr}^F({\mathcal{D}}[s])\simeq\mathcal{O}[s,\xi_1,\dots,\xi_n] \to 0} (\mathcal{O}f+J(f))^dT^d$$ by $\varphi_f (s)=fT$ and for all $i$... | ideal as a sub-object of $W$ becauSe of the isoMorphIsm $\MatHbF{R}({\maThcaL{D}}[s])\simeq\textrm{GR}^V_0(\maThbf{R}({\mathcal{D}}_{x,t})).$
### ConditiOn for ThE $H$-satURaTion oF $\mathbf{r}(\TeXTRm{gR}^V(n))$
LEt $J(F)$ bE ThE ideaL geNerated By $f'_1,\dots,f'_n$. LEt uS dEfine a morphiSM oF graded $\matHcaL{O}$... | ideal as a sub-object of$W$ becaus e ofthe is om orph ism$\mathbf{R}({\ m athc al{D}}[s])\simeq\textr m{gr} ^V _ 0(\m a th bf{R} ({\math c al { D }}_ {x ,t })) .$
# ## Co ndi tion fo r the $h$- sat ur ation of $\m a th bf{R}(\tex trm {gr}^V(N))$
Le t $J(f )$ be the i dea l gen erated by $f' _1,\dots, f' _ n$. Le t ... | ideal_as a_sub-object of $W$ because_of the_isomorphism_$\mathbf{R}({\mathcal{D}}[s])\simeq\textrm{gr}^V_0(\mathbf{R}({\mathcal{D}}_{x,t})).$
### Condition_for_the $h$-saturation of_$\mathbf{R}(\textrm{gr}^V(N))$
Let $J(f)$ be_the ideal generated by_$f'_1,\dots,f'_n$. Let us_define_a morphism of graded $\mathcal{O}$... |
0 & 1\\ 1 & 0
\end{matrix}\right],
\left[\begin{matrix}
0 & 1\\0 & 1\\ 0 & 1
\end{matrix}\right],
\left[\begin{matrix}
1 & 0\\1 & 0\\ 0 & 1
\end{matrix}\right]$$ and can be written as a union of Cartesian products of convex sets, illustrated in Figure \[figure:convexrepresentabl... | 0 & 1\\ 1 & 0
\end{matrix}\right ],
\left[\begin{matrix }
0 & 1\\0 & 1\\ 0 & 1
\end{matrix}\right ],
\left[\begin{matrix }
1 & 0\\1 & 0\\ 0 & 1
\end{matrix}\right]$$ and can be written as a union of Cartesian products of convex sets, illustrate in name \[figure: conve... | 0 & 1\\ 1 & 0
\end{matrix}\right],
\left[\begin{matrix}
0 & 1\\0 & 1\\ 0 & 1
\end{matrix}\right],
\left[\uegib{matrux}
1 & 0\\1 & 0\\ 0 & 1
\end{mxtrix}\righn]$$ and can be xritten as a union of Cavcesiah proburts of convex sgts, illustraded in Figure \[fhgjrz:convexrepresentabl... | 0 & 1\\ 1 & 0 \end{matrix}\right], & & 1\\ & 1 \end{matrix}\right], 0\\ & 1 \end{matrix}\right]$$ can be written a union of Cartesian products of sets, illustrated in Figure \[figure:convexrepresentable\], $$G = \bigcup_{\theta\in\Delta_A} G_\theta,\qquad G_\theta = \{ \theta \times \{ \theta \} \times \Delta_A. $$ As ... | 0 & 1\\ 1 & 0
\end{matrix}\right],
\left[\begin{maTrix}
0 & 1\\0 & 1\\ 0 & 1
\end{matRix}\riGht],
\LefT[\bEgin{MatrIx}
1 & 0\\1 & 0\\ 0 & 1
\end{matrix}\rigHT]$$ and Can be written as a union of carteSiAN proDUcTs of cOnvex seTS, iLLUstRaTeD in fiGUrE \[figuRe:cOnvexrePresentabl... | 0 & 1\\ 1 & 0
\end{ma trix}\righ t], \lef t[\b egin{matrix}
0& 1\\0 & 1\\ 0 & 1
\en d{ m atri x }\ right ],
\le ft [\ beg in { ma trix}
1 & 0 \\1 & 0\\0 & 1
\end{m a tr ix}\right] $$and can be w rit ten as a un i on of Ca rtesi an pro d ucts o f convexse t s, ill u strated i nFigu re \... | 0 &_1\\ 1_& 0
_ _\end{matrix}\right],_
__ _ \left[\begin{matrix}
_ 0_& 1\\0 &_1\\_0 & 1
\end{matrix}\right],
__ \left[\begin{matrix}
___ 1 & 0\\1 &_0\\ 0 & 1
_ _ \end{matrix}\right]$$ and can be written as a_union_of Cartesian products_of convex sets, illustrated in Figure \[figure:convexrepresentabl... |
(m,q)$. The following notation will be used.
- $\omega_{\alpha}:{\mathbb{F}_{q^n}}\to{\mathbb{F}_{q^n}}$, $x\mapsto\alpha x$, for any $\alpha\in{\mathbb{F}_{q^n}}$.
- $\mathcal{F}_n=\{\omega_{\alpha}\colon\alpha\in{\mathbb{F}_{q^n}}\}$, which is a field isomorphic to ${\mathbb{F}_{q^n}}$.
- $\mathcal{F}_{n,q}=... | ( m, q)$. The following notation will be used.
- $ \omega_{\alpha}:{\mathbb{F}_{q^n}}\to{\mathbb{F}_{q^n}}$, $ x\mapsto\alpha x$, for any $ \alpha\in{\mathbb{F}_{q^n}}$.
- $ \mathcal{F}_n=\{\omega_{\alpha}\colon\alpha\in{\mathbb{F}_{q^n}}\}$, which is a field isomorphous to $ { \mathbb{F}_{q^n}}$.
- $... | (m,q)$. Hhe following notation wlll be used.
- $\omggq_{\alpha}:{\kathbb{R}_{q^n}}\to{\matfbb{F}_{q^n}}$, $x\mapsto\alpha x$, for anb $\alpha\in{\nathbb{F}_{q^n}}$.
- $\mathcal{F}_n=\{\ooega_{\alpha}\bolon\alphq\in{\methbb{F}_{q^n}}\}$, which is a field isomkvphic vo ${\mathbb{F}_{q^n}}$.
- $\kathcal{F}_{n,q}=... | (m,q)$. The following notation will be used. $x\mapsto\alpha for any - $\mathcal{F}_n=\{\omega_{\alpha}\colon\alpha\in{\mathbb{F}_{q^n}}\}$, which ${\mathbb{F}_{q^n}}$. $\mathcal{F}_{n,q}=\{\omega_{\alpha}\colon \alpha\in{{\mathbb F}_{q}}\}$, is a subfield $\mathcal{F}_n$ isomorphic to ${{\mathbb F}_{q}}$. - W_1$, $\la... | (m,q)$. The following notation wilL be used.
- $\omeGa_{\alpHa}:{\mAthBb{f}_{q^n}}\tO{\matHbb{F}_{q^n}}$, $x\mapsto\aLPha x$, For any $\alpha\in{\mathbb{F}_{q^n}}$.
- $\MathcAl{f}_N=\{\omeGA_{\aLpha}\cOlon\alpHA\iN{\MAthBb{f}_{q^N}}\}$, whIcH Is A fielD isOmorphiC to ${\mathbb{F}_{Q^n}}$.
- $\mAtHcal{F}_{n,q}=... | (m,q)$. The following nota tion willbe us ed.
- $\o mega _{\alpha}:{\ma t hbb{ F}_{q^n}}\to{\mathbb{F }_{q^ n} } $, $ x \m apsto \alphax $, f oran y$\a lp h a\ in{\m ath bb{F}_{ q^n}}$.
- $\ mathcal{F}_n = \{ \omega_{\a lph a}\colon\alp ha\ in{\ma th bb{ F }_{q^ n}} \}$,whichi s a fi eld isomo rp h ic to$ {\math... | (m,q)$. The_following notation_will be used.
- _ $\omega_{\alpha}:{\mathbb{F}_{q^n}}\to{\mathbb{F}_{q^n}}$,_$x\mapsto\alpha_x$, for_any_$\alpha\in{\mathbb{F}_{q^n}}$.
- _$\mathcal{F}_n=\{\omega_{\alpha}\colon\alpha\in{\mathbb{F}_{q^n}}\}$, which is_a field isomorphic to_${\mathbb{F}_{q^n}}$.
- _$\mathcal{F}_{n,q}=... |
beta
\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right)
\boldsymbol{\mu}_m^{\prime e}-\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2
+\frac{5}{8}|\boldxi|^4\right)
\left(\boldxi\times\boldsymbol{\mu}_p^{\prime\tilde{e}}\right)\right].
\end{split}$$
It is shown that intrinsic dipole moments transform like $\left(1... | beta
\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right)
\boldsymbol{\mu}_m^{\prime e}-\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2
+ \frac{5}{8}|\boldxi|^4\right)
\left(\boldxi\times\boldsymbol{\mu}_p^{\prime\tilde{e}}\right)\right ].
\end{split}$$
It is shown that intrinsic dipole moments transform... | betw
\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxl|^4\right)
\boldsymbol{\mu}_m^{\primx e}-\frac{1}{2}\meft(1-\frac{3}{4}|\coldxi|^2
+\frac{5}{8}|\boldxi|^4\right)
\left(\bolvxi\tumes\bildsymbol{\mu}_p^{\prime\tilde{d}}\right)\rigjt].
\end{splut}$$
It us shown tizt intrlusic slpole noments transfprm like $\laft(1... | beta \left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right) \boldsymbol{\mu}_m^{\prime e}-\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2 +\frac{5}{8}|\boldxi|^4\right) \left(\boldxi\times\boldsymbol{\mu}_p^{\prime\tilde{e}}\right)\right]. \end{split}$$ shown intrinsic dipole transform like $\left(1-\frac{1}{2}|\boldxi... | beta
\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\bolDxi|^4\right)
\boLdsymBol{\Mu}_m^{\PrIme e}-\Frac{1}{2}\Left(1-\frac{3}{4}|\boldxi|^2
+\FRac{5}{8}|\bOldxi|^4\right)
\left(\boldxi\tiMes\boLdSYmboL{\Mu}_P^{\primE\tilde{e}}\RIgHT)\RigHt].
\EnD{spLiT}$$
it Is shoWn tHat intrInsic dipolE moMeNts transform LIkE $\left(1... | beta
\left(1-\frac{1}{2}|\ boldxi|^2+ \frac {3} {8} |\ bold xi|^ 4\right)
\bold s ymbo l{\mu}_m^{\prime e}-\f rac{1 }{ 2 }\le f t( 1-\fr ac{3}{4 } |\ b o ldx i| ^2
+\ fr a c{ 5}{8} |\b oldxi|^ 4\right)
\ lef t( \boldxi\time s \b oldsymbol{ \mu }_p^{\prime\ til de{e}} \r igh t )\rig ht] .
\en d{spli t }$$
I t is show ... | beta
\left(1-\frac{1}{2}|\boldxi|^2+\frac{3}{8}|\boldxi|^4\right)
\boldsymbol{\mu}_m^{\prime e}-\frac{1}{2}\left(1-\frac{3}{4}|\boldxi|^2
+\frac{5}{8}|\boldxi|^4\right)
\left(\boldxi\times\boldsymbol{\mu}_p^{\prime\tilde{e}}\right)\right].
\end{split}$$
It_is shown_that intrinsic dipole moments_transform like_$\left(1... |
\mathrm{d}(x,v)}} \\
& = - {
\lim _{\delta \searrow 0}}{
\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \varphi (x) \chi ^{\;\prime} \left (
\frac{|v|}{\delta} \right )\frac{|v|}{\delta} ( \alpha - \beta
|v|^2) f(x,v)\,{
\mathrm{d}(x,v)}}= 0\end{aligned}$$ by dominated convergence, since $$\left | \chi ^{\;\prime} \le... | \mathrm{d}(x, v) } } \\
& = - {
\lim _ { \delta \searrow 0 } } {
\int _ { { { { \bf}R}}^d \times { { { \bf}R}}^d}\!\!\! \varphi (x) \chi ^{\;\prime } \left (
\frac{|v|}{\delta } \right) \frac{|v|}{\delta } (\alpha - \beta
|v|^2) f(x, v)\, {
\mathrm{d}(x, v)}}= 0\end{aligned}$$ by dominated convergence, since... | \matjrm{d}(x,v)}} \\
& = - {
\lim _{\delta \searvow 0}}{
\int _{{{{\bf}R}}^d \timgs {{{\bf}R}}^d}\!\!\! \tarphi (s) \chi ^{\;\prkme} \left (
\frac{|v|}{\delta} \right )\frec{|v|}{\dwlta} ( \alpha - \beta
|v|^2) f(x,v)\,{
\mathro{d}(x,v)}}= 0\end{apigned}$$ bt donunated contsrgence, since $$\peft | \chi ^{\;\prime} \le... | \mathrm{d}(x,v)}} \\ & = - { \lim 0}}{ _{{{{\bf}R}}^d \times \varphi (x) \chi )\frac{|v|}{\delta} \alpha - \beta f(x,v)\,{ \mathrm{d}(x,v)}}= 0\end{aligned}$$ dominated convergence, since $$\left | \chi \left ( \frac{|v|}{\delta} \right )\frac{|v|}{\delta} ( \alpha - \beta |v|^2) \right |\leq \alpha _{u \geq 0} |\chi ^... | \mathrm{d}(x,v)}} \\
& = - {
\lim _{\delta \searrow 0}}{
\iNt _{{{{\bf}R}}^d \timeS {{{\bf}R}}^d}\!\!\! \VarPhi (X) \cHi ^{\;\prIme} \lEft (
\frac{|v|}{\delta} \rIGht )\fRac{|v|}{\delta} ( \alpha - \beta
|v|^2) f(x,v)\,{
\MathrM{d}(X,V)}}= 0\end{ALiGned}$$ bY dominaTEd CONveRgEnCe, sInCE $$\lEft | \chI ^{\;\prIme} \le... | \mathrm{d}(x,v)}} \\
& = - {
\lim _{ \delt a \ sea rr ow 0 }}{\int _{{{{\bf} R }}^d \times {{{\bf}R}}^d}\ !\!\! \ v arph i ( x) \c hi ^{\; \ pr i m e}\l ef t (
\ f ra c{|v| }{\ delta}\right )\f rac {| v|}{\delta}( \ alpha - \b eta
|v|^2) f(x, v)\ ,{
\ma th rm{ d }(x,v )}} = 0\e nd{ali g ned}$$ by domin at e d c... | \mathrm{d}(x,v)}} \\
&_= -_{
\lim _{\delta \searrow 0}}{
\int__{{{{\bf}R}}^d \times_{{{\bf}R}}^d}\!\!\!_\varphi (x)_\chi_^{\;\prime} \left (
\frac{|v|}{\delta}_\right )\frac{|v|}{\delta} (_\alpha - \beta
|v|^2) f(x,v)\,{
\mathrm{d}(x,v)}}=_0\end{aligned}$$ by dominated_convergence,_since $$\left | \chi ^{\;\prime} \le... |
V + \Psi) \right] - 2c_2 k^2 V =0.\end{aligned}$$
### Synchronous gauge
In the synchronous gauge, where $h_{ij}$ is decomposed into $h$ and $\eta$ as in, we find that $$\begin{aligned}
a^2\delta \rho &= \alpha\left[ 3\mathcal{F}_\mathcal{KK} \delta \mathcal{K} \mathcal{H}^2 + \mathcal{F}_\mathcal{K} \mathcal{H}\left... | V + \Psi) \right ] - 2c_2 k^2 V = 0.\end{aligned}$$
# # # Synchronous gauge
In the synchronous gauge, where $ h_{ij}$ is decomposed into $ h$ and $ \eta$ as in, we find that $ $ \begin{aligned }
a^2\delta \rho & = \alpha\left [ 3\mathcal{F}_\mathcal{KK } \delta \mathcal{K } \mathcal{H}^2 + \mathcal{F}_\mathca... | V + \Osi) \right] - 2c_2 k^2 V =0.\end{aligked}$$
### Synchronous yquge
In the sgnchronojs gauge, where $h_{ij}$ is decomplswd inuj $h$ and $\eta$ as in, de find tjat $$\begib{alijned}
a^2\delta \rho &= \alpha\lenc[ 3\matggal{F}_\mctical{KK} \delta \majhcal{K} \mathcdl{H}^2 + \mathcal{F}_\mdtfccl{K} \mathcal{H}\left... | V + \Psi) \right] - 2c_2 k^2 ### gauge In synchronous gauge, where and as in, we that $$\begin{aligned} a^2\delta &= \alpha\left[ 3\mathcal{F}_\mathcal{KK} \delta \mathcal{K} \mathcal{H}^2 \mathcal{F}_\mathcal{K} \mathcal{H}\left( \frac{1}{2}h' - k^2V \right)\right] +c_{14} \mathcal{F}_\mathcal{K} k^2 (V' + \mathcal{H}... | V + \Psi) \right] - 2c_2 k^2 V =0.\end{aligned}$$
### SynChronous gaUge
In The SynChRonoUs gaUge, where $h_{ij}$ is dECompOsed into $h$ and $\eta$ as in, we fInd thAt $$\BEgin{ALiGned}
a^2\Delta \rhO &= \AlPHA\leFt[ 3\MaThcAl{f}_\MaThcal{kK} \dElta \matHcal{K} \mathcAl{H}^2 + \MaThcal{F}_\mathcaL{k} \mAthcal{H}\lefT... | V + \Psi) \right] - 2c_2 k ^2 V =0.\e nd{al ign ed} $$
## # Sy nchronous gaug e
In the synchronous gauge , whe re $h_{ i j} $ isdecompo s ed i nto $ h$ an d$ \e ta$ a s i n, we f ind that $ $\b eg in{aligned}a ^2 \delta \rh o & = \alpha\le ft[ 3\mat hc al{ F }_\ma thc al{KK } \del t a \mat hcal{K} \ ma t hcal{... | V +_\Psi) \right]_- 2c_2 k^2 V_=0.\end{aligned}$$
### Synchronous_gauge
In_the synchronous_gauge,_where $h_{ij}$ is_decomposed into $h$_and $\eta$ as in,_we find that_$$\begin{aligned}
a^2\delta_\rho &= \alpha\left[ 3\mathcal{F}_\mathcal{KK} \delta \mathcal{K} \mathcal{H}^2 + \mathcal{F}_\mathcal{K} \mathcal{H}\left... |
.25 & 18 44 51.244 & 0.011 & -03 46 03.726 & 0.012 & 0.32 &0.02 &87.34\
G28.83-0.25 & 18 44 50.060 & 0.017 & -03 45 47.709 & 0.019 & 0.19 &0.02 &87.34\
G28.83-0.25 & 18 44 51.507 & 0.020 & -03 45 56.343 & 0.021 & 0.16 &0.01 &87.51\
G28.83-0.25 & 18 44 51.245 & 0.013 & -03 46 03.635 & 0.014 & 0.27 &0.02 &87.51\
G28.83-0... | .25 & 18 44 51.244 & 0.011 & -03 46 03.726 & 0.012 & 0.32 & 0.02 & 87.34\
G28.83 - 0.25 & 18 44 50.060 & 0.017 & -03 45 47.709 & 0.019 & 0.19 & 0.02 & 87.34\
G28.83 - 0.25 & 18 44 51.507 & 0.020 & -03 45 56.343 & 0.021 & 0.16 & 0.01 & 87.51\
G28.83 - 0.25 & 18 44 51.245 & 0.013 & -03 46 03.635 & 0.014 & 0.27 & 0.... | .25 & 18 44 51.244 & 0.011 & -03 46 03.726 & 0.012 & 0.32 &0.02 &87.34\
G28.83-0.25 & 18 44 50.060 & 0.017 & -03 45 47.709 & 0.019 & 0.19 &0.02 &87.34\
G28.83-0.25 & 18 44 51.507 & 0.020 & -03 45 56.343 & 0.021 & 0.16 &0.01 &87.51\
G28.83-0.25 & 18 44 51.245 & 0.013 & -03 46 03.635 & 0.014 & 0.27 &0.02 &87.51\
G28.83-0... | .25 & 18 44 51.244 & 0.011 46 & 0.012 0.32 &0.02 &87.34\ & & -03 45 & 0.019 & &0.02 &87.34\ G28.83-0.25 & 18 44 & 0.020 & -03 45 56.343 & 0.021 & 0.16 &0.01 &87.51\ G28.83-0.25 18 44 51.245 & 0.013 & -03 46 03.635 & 0.014 & 0.27 &87.51\ & 44 & 0.018 & -03 45 47.727 & 0.020 & 0.18 &0.02 &87.51\ G28.83-0.25 & 18 44 & 0.0... | .25 & 18 44 51.244 & 0.011 & -03 46 03.726 & 0.012 & 0.32 &0.02 &87.34\
G28.83-0.25 & 18 44 50.060 & 0.017 & -03 45 47.709 & 0.019 & 0.19 &0.02 &87.34\
G28.83-0.25 & 18 44 51.507 & 0.020 & -03 45 56.343 & 0.021 & 0.16 &0.01 &87.51\
G28.83-0.25 & 18 44 51.245 & 0.013 & -03 46 03.635 & 0.014 & 0.27 &0.02 &87.51\
G28.83-0... | .25 & 18 44 51.244 & 0.011 & -03 4603.72 6 & 0. 01 2 &0.32 &0.02 &87.34\ G28. 83-0.25 & 18 44 50.060 & 0. 01 7 & - 0 345 47 .709 &0 .0 1 9 &0. 19 &0 .0 2 & 87.34 \
G 28.83-0 .25 & 18 4 4 5 1. 507 & 0.020& - 03 45 56.3 43& 0.021 & 0. 16&0.01&8 7.5 1 \
G28 .83 -0.25 & 184 4 51.2 45 & 0.01 3& -03 4 6 03.635 & ... | .25 &_18 44_51.244 & 0.011 &_-03 46_03.726_& 0.012_&_0.32 &0.02 &87.34\
G28.83-0.25_& 18 44_50.060 & 0.017 &_-03 45 47.709_&_0.019 & 0.19 &0.02 &87.34\
G28.83-0.25 & 18 44 51.507 & 0.020 & -03 45_56.343_& 0.021_&_0.16_&0.01 &87.51\
G28.83-0.25 & 18 44_51.245 & 0.013 & -03_46 03.635_& 0.014 & 0.27 &0.02 &87.51\
G28.83-0... |
{e_3}{2}+2 u-2 v} b_1}{32\ 2^{3/4}}+\frac{3 e^{-e_1-\frac{e_3}{2}-\rho} b_1^2}{16 \sqrt{2}}-\frac{3 e^{-e_1+\frac{e_3}{2}+2 u-2 v} b_1^2}{32 \sqrt{2}}\,, \nonumber\\
v'&=& 2 e^{-e_1-\frac{e_3}{2}}-\frac{3}{2} e^{-e_1-\frac{e_3}{2}-\rho}+\frac{1}{2} e^{e_1-\frac{e_3}{2}-\rho}+e^{\frac{e_3}{2}-\rho}+\frac{3 e^{-e_1-\frac... | { e_3}{2}+2 u-2 v } b_1}{32\ 2^{3/4}}+\frac{3 e^{-e_1-\frac{e_3}{2}-\rho } b_1 ^ 2}{16 \sqrt{2}}-\frac{3 e^{-e_1+\frac{e_3}{2}+2 u-2 v } b_1 ^ 2}{32 \sqrt{2}}\, , \nonumber\\
v'&= & 2 e^{-e_1-\frac{e_3}{2}}-\frac{3}{2 } e^{-e_1-\frac{e_3}{2}-\rho}+\frac{1}{2 } e^{e_1-\frac{e_3}{2}-\rho}+e^{\frac{e_3}{2}-\rho}+\frac{3... | {e_3}{2}+2 u-2 v} b_1}{32\ 2^{3/4}}+\frac{3 e^{-e_1-\frac{e_3}{2}-\rho} b_1^2}{16 \sdrt{2}}-\frac{3 e^{-e_1+\frac{e_3}{2}+2 u-2 v} b_1^2}{32 \vqrt{2}}\,, \nknumber\\
v'&=& 2 e^{-e_1-\frac{e_3}{2}}-\frac{3}{2} e^{-e_1-\frac{e_3}{2}-\rho}+\frac{1}{2} e^{w_1-\frac{t_3}{2}-\gho}+e^{\frac{e_3}{2}-\rho}+\frac{3 e^{-e_1-\frxc... | {e_3}{2}+2 u-2 v} b_1}{32\ 2^{3/4}}+\frac{3 e^{-e_1-\frac{e_3}{2}-\rho} b_1^2}{16 u-2 b_1^2}{32 \sqrt{2}}\,, v'&=& 2 e^{-e_1-\frac{e_3}{2}}-\frac{3}{2} e'_1&=& e^{-e_1-\frac{e_3}{2}} -\frac{3}{2} e^{-e_1-\frac{e_3}{2}-\rho} e^{e_1-\frac{e_3}{2}-\rho} +\frac{3}{2} e^{-e_1+\frac{e_3}{2}+2 v} -\frac{1}{4} e^{e_1+\frac{e_3... | {e_3}{2}+2 u-2 v} b_1}{32\ 2^{3/4}}+\frac{3 e^{-e_1-\frac{e_3}{2}-\rho} b_1^2}{16 \sqrt{2}}-\frAc{3 e^{-e_1+\frac{e_3}{2}+2 u-2 V} b_1^2}{32 \sqrT{2}}\,, \noNumBeR\\
v'&=& 2 e^{-e_1-\Frac{E_3}{2}}-\frac{3}{2} e^{-e_1-\frac{e_3}{2}-\rhO}+\Frac{1}{2} E^{e_1-\frac{e_3}{2}-\rho}+e^{\frac{e_3}{2}-\rho}+\fraC{3 e^{-e_1-\frAc... | {e_3}{2}+2 u-2 v} b_1}{32\ 2^{3/4}}+ \frac {3e^{ -e _1-\ frac {e_3}{2}-\rho} b_1^ 2}{16 \sqrt{2}}-\frac{ 3 e^{ -e _ 1+\f r ac {e_3} {2}+2 u - 2v } b_ 1^ 2} {32 \ s qr t{2}} \,, \nonum ber\\
v'&= & 2 e ^{-e_1-\frac { e_ 3}{2}}-\fr ac{ 3}{2} e^{-e_ 1-\ frac{e _3 }{2 } -\rho }+\ frac{ 1}{2}e ^{e_1- \frac{e_3 }{ 2 }-\... | {e_3}{2}+2 u-2_v} b_1}{32\_2^{3/4}}+\frac{3 e^{-e_1-\frac{e_3}{2}-\rho} b_1^2}{16 \sqrt{2}}-\frac{3_e^{-e_1+\frac{e_3}{2}+2 u-2_v}_b_1^2}{32 \sqrt{2}}\,,_\nonumber\\
v'&=&_2 e^{-e_1-\frac{e_3}{2}}-\frac{3}{2} e^{-e_1-\frac{e_3}{2}-\rho}+\frac{1}{2}_e^{e_1-\frac{e_3}{2}-\rho}+e^{\frac{e_3}{2}-\rho}+\frac{3 e^{-e_1-\frac... |
k},\bm{k}')\end{aligned}$$
The linearized gap equation $$\begin{aligned}
-\frac{1}{(2\pi)^2}\sum_{\beta}\oint_{FS}
dk'_{\Vert}\frac{V^{\alpha\beta}(\bm{k,k'})}{v^{\beta}_{F}(\bm{k'})}
\Delta_{\beta}(\bm{k'})=\lambda
\Delta_{\alpha}(\bm{k}).\label{eigenvalue_Tc2}\end{aligned}$$ can be rewritten as $$\begin{aligned}
\fr... | k},\bm{k}')\end{aligned}$$
The linearized gap equation $ $ \begin{aligned }
-\frac{1}{(2\pi)^2}\sum_{\beta}\oint_{FS }
dk'_{\Vert}\frac{V^{\alpha\beta}(\bm{k, k'})}{v^{\beta}_{F}(\bm{k' }) }
\Delta_{\beta}(\bm{k'})=\lambda
\Delta_{\alpha}(\bm{k}).\label{eigenvalue_Tc2}\end{aligned}$$ can be rewritten as $ $ ... | k},\bm{n}')\end{aligned}$$
The linearizea gap equation $$\ywgin{almgned}
-\frzc{1}{(2\pi)^2}\sum_{\bdta}\oint_{FS}
dk'_{\Vert}\frac{V^{\alpha\bete}(\bm{k,j'})}{v^{\betq}_{F}(\bm{k'})}
\Delta_{\beta}(\bm{k'})=\lambdx
\Delta_{\alpja}(\bm{k}).\labwl{eijenvalue_Tc2}\end{alijhed}$$ can be resvitteu es $$\begin{aligned}
\nr... | k},\bm{k}')\end{aligned}$$ The linearized gap equation $$\begin{aligned} -\frac{1}{(2\pi)^2}\sum_{\beta}\oint_{FS} \Delta_{\alpha}(\bm{k}).\label{eigenvalue_Tc2}\end{aligned}$$ be rewritten $$\begin{aligned} \frac{1}{(2\pi)^2}\sum_{\beta}\iint_{\Delta E} where integral $\iint$ is within a narrow window near the FS with... | k},\bm{k}')\end{aligned}$$
The linearizEd gap equatIon $$\beGin{AliGnEd}
-\frAc{1}{(2\pi)^2}\Sum_{\beta}\oint_{FS}
dK'_{\vert}\Frac{V^{\alpha\beta}(\bm{k,k'})}{v^{\betA}_{F}(\bm{k'})}
\deLTa_{\beTA}(\bM{k'})=\lamBda
\DeltA_{\AlPHA}(\bm{K}).\lAbEl{eIgENvAlue_TC2}\enD{aligneD}$$ can be rewrIttEn As $$\begin{alignED}
\fR... | k},\bm{k}')\end{aligned}$$
The line arize d g apeq uati on $ $\begin{aligne d }
-\ frac{1}{(2\pi)^2}\sum_ {\bet a} \ oint _ {F S}
dk '_{\Ver t }\ f r ac{ V^ {\ alp ha \ be ta}(\ bm{ k,k'})} {v^{\beta} _{F }( \bm{k'})}
\D e lt a_{\beta}( \bm {k'})=\lambd a
\ Delta_ {\ alp h a}(\b m{k }).\l abel{e i genval ue_Tc2}\e nd... | k},\bm{k}')\end{aligned}$$
The linearized_gap equation_$$\begin{aligned}
-\frac{1}{(2\pi)^2}\sum_{\beta}\oint_{FS}
dk'_{\Vert}\frac{V^{\alpha\beta}(\bm{k,k'})}{v^{\beta}_{F}(\bm{k'})}
\Delta_{\beta}(\bm{k'})=\lambda
\Delta_{\alpha}(\bm{k}).\label{eigenvalue_Tc2}\end{aligned}$$ can be rewritten_as $$\begin{aligned}
\fr... |
{U}(L_\mathrm{bb}\tau_1)$ [^2] which is composed of disjoint parts. Therefore a combinatorial factor does not appear this time, the saddle point is unique. In spite of this, for all choices of $L_\mathrm{bb}$ the non-compact group $\mathrm{U}(L_\mathrm{bb}\tau_1)$ is unitarily equivalent to the non-compact unitary grou... | { U}(L_\mathrm{bb}\tau_1)$ [ ^2 ] which is composed of disjoint parts. Therefore a combinatorial agent does not look this time, the saddle item is singular. In spite of this, for all choices of $ L_\mathrm{bb}$ the non - compendious group $ \mathrm{U}(L_\mathrm{bb}\tau_1)$ is unitarily equivalent to the non - compact u... | {U}(L_\mwthrm{bb}\tau_1)$ [^2] which is comkosed of disjoint parts. Vherefode a comcinatorial factor does not a'peae thiw time, the saddle poing is uniqle. In spire oh this, for all cikices on $L_\mafmrm{bb}$ vhe non-compact nroup $\mathrk{U}(L_\mathrm{bb}\tau_1)$ ir bnitarily equivalent to the non-compast unitsrj grou... | {U}(L_\mathrm{bb}\tau_1)$ [^2] which is composed of disjoint a factor does appear this time, In of this, for choices of $L_\mathrm{bb}$ non-compact group $\mathrm{U}(L_\mathrm{bb}\tau_1)$ is unitarily equivalent the non-compact unitary group $\mathrm{U}(k_\mathrm{b},k_\mathrm{b})$ because ${\mathrm{Tr\,}}L_\mathrm{bb}\... | {U}(L_\mathrm{bb}\tau_1)$ [^2] which is compoSed of disjoInt paRts. theReFore A comBinatorial factOR doeS not appear this time, the sAddle PoINt is UNiQue. In Spite of THiS, FOr aLl ChOicEs OF $L_\MathrM{bb}$ The non-cOmpact grouP $\maThRm{U}(L_\mathrm{bb}\TAu_1)$ Is unitarilY eqUivalent to thE noN-compaCt UniTAry grOu... | {U}(L_\mathrm{bb}\tau_1)$[^2] which is c omp ose dof d isjo int parts. The r efor e a combinatorial fact or do es nota pp ear t his tim e ,t h e s ad dl e p oi n tis un iqu e. In s pite of th is, f or all choic e sof $L_\mat hrm {bb}$ the no n-c ompact g rou p $\ma thr m{U}( L_\mat h rm{bb} \tau_1)$is unitar i ly e... | {U}(L_\mathrm{bb}\tau_1)$ [^2]_which is_composed of disjoint parts._Therefore a_combinatorial_factor does_not_appear this time,_the saddle point_is unique. In spite_of this, for_all_choices of $L_\mathrm{bb}$ the non-compact group $\mathrm{U}(L_\mathrm{bb}\tau_1)$ is unitarily equivalent to the non-compact unitary_grou... |
sigma^+$ is obtained from $\sigma$ by flipping to $+$ the spin of all $\sigma$-components which interact with the boundary ${\partial}_e(\Delta^\circ)$ and also the spin of sites outside $\Delta^\circ$. Notice that $\sigma^+$ belongs to the support of $\mu^+_{\Delta^\circ}$ and that both $\sigma$ and $\sigma^+$ have th... | sigma^+$ is obtained from $ \sigma$ by flipping to $ + $ the spin of all $ \sigma$-components which interact with the boundary $ { \partial}_e(\Delta^\circ)$ and also the tailspin of web site outside $ \Delta^\circ$. Notice that $ \sigma^+$ belongs to the documentation of $ \mu^+_{\Delta^\circ}$ and that both $ \sigma$... | sigla^+$ is obtained from $\sigmx$ by flipping to $+$ the vpin or all $\siema$-components which interact wuth tye boundary ${\partial}_e(\Deuta^\circ)$ ajd also rhe wpin of sitxa outside $\Delfw^\cire$. Iotice that $\sigka^+$ belongs to the suppord uf $\mu^+_{\Delta^\circ}$ and that both $\sigma$ anq $\sigma^+$ hwve th... | sigma^+$ is obtained from $\sigma$ by flipping the of all which interact with the of sites outside Notice that $\sigma^+$ to the support of $\mu^+_{\Delta^\circ}$ and both $\sigma$ and $\sigma^+$ have the same energy in $\Delta^\circ$, i.e. $$\sum_{B: B \Delta^\circ \neq \emptyset} \Phi_B(\sigma_B) = \sum_{B: B \cap \D... | sigma^+$ is obtained from $\sigma$ bY flipping tO $+$ the sPin Of aLl $\SigmA$-comPonents which inTEracT with the boundary ${\partiaL}_e(\DelTa^\CIrc)$ aND aLso thE spin of SItES OutSiDe $\delTa^\CIrC$. NotiCe tHat $\sigmA^+$ belongs to The SuPport of $\mu^+_{\DelTA^\cIrc}$ and that BotH $\sigma$ and $\sigMa^+$ hAve th... | sigma^+$ is obtained from$\sigma$ b y fli ppi ngto $+$ the spin of all $ \ sigm a$-components which in terac tw itht he boun dary ${ \ pa r t ial }_ e( \De lt a ^\ circ) $ a nd also the spinofsi tes outside$ \D elta^\circ $.Notice that$\s igma^+ $bel o ngs t o t he su pporto f $\mu ^+_{\Delt a^ \ circ}$ and tha t ... | sigma^+$ is_obtained from_$\sigma$ by flipping to_$+$ the_spin_of all_$\sigma$-components_which interact with_the boundary ${\partial}_e(\Delta^\circ)$_and also the spin_of sites outside_$\Delta^\circ$._Notice that $\sigma^+$ belongs to the support of $\mu^+_{\Delta^\circ}$ and that both $\sigma$ and_$\sigma^+$_have th... |
bowls were placed at 7 different positions. At each position, we collect 20 sequences of data with 5 for each bowl-barley combination. In total, we collect 140 sequences of data. 139 of the 140 sequences of data are used to train DCAE and LSTM-RNN. The remaining 1 sequence are used for testing. The parameters of DCAE ... | bowls were placed at 7 different positions. At each placement, we roll up 20 sequences of data with 5 for each bowl - barley combination. In sum, we collect 140 sequences of datum. 139 of the 140 sequences of data are used to discipline DCAE and LSTM - RNN. The remaining 1 sequence are use for testing. The parameters o... | boals were placed at 7 difftrent positions. Aj wach pmsitioh, we coluect 20 sequences of data with 5 dor eqch bowl-barley combinagion. In tltal, we xolltct 140 sequences of data. 139 on the 140 seqbeices of data arg used to trdin DCAE and LVTO-RUN. The remaining 1 sequence are used sor tesyijg. The parametgrs og DCAS ... | bowls were placed at 7 different positions. position, collect 20 of data with In we collect 140 of data. 139 the 140 sequences of data are to train DCAE and LSTM-RNN. The remaining 1 sequence are used for testing. parameters of DCAE and LSTM-RNN are the same as experiment 1. The results DCAE shown Fig.\[dcaeresults\](b... | bowls were placed at 7 differenT positions. at eacH poSitIoN, we cOlleCt 20 sequences of dATa wiTh 5 for each bowl-barley comBinatIoN. in toTAl, We colLect 140 seqUEnCES of DaTa. 139 Of tHe 140 SEqUenceS of Data are Used to traiN DCaE And LSTM-RNN. ThE ReMaining 1 seqUenCe are used for TesTing. ThE pAraMEters Of DcAE ... | bowls were placed at 7 di fferent po sitio ns. At e achposi tion, we colle c t 20 sequences of data wit h 5 f or each bo wl-ba rley co m bi n a tio n. I n t ot a l, we c oll ect 140 sequences of d ata. 139 oft he 140 seque nce s of data ar e u sed to t rai n DCAE an d LST M-RNN. The re maining 1 s e quenc... | bowls_were placed_at 7 different positions._At each_position,_we collect_20_sequences of data_with 5 for_each bowl-barley combination. In_total, we collect_140_sequences of data. 139 of the 140 sequences of data are used to train_DCAE_and LSTM-RNN._The_remaining_1 sequence are used for_testing. The parameters of DCAE_... |
sum_{j=1}^q \log f_{j,\theta}(A_j
Z\mid B_j Z),$$ where $f_{j,\theta}$ is the conditional Gaussian density of $A_j Z$ given $B_j Z$. As proposed by @vecchia and @steinchiwelty, the rank of $B_j$ will generally be larger than that of $A_j$, in which case the main computation in obtaining (\[composite\]) is finding Chole... | sum_{j=1}^q \log f_{j,\theta}(A_j
Z\mid B_j Z),$$ where $ f_{j,\theta}$ is the conditional Gaussian density of $ A_j Z$ given $ B_j Z$. As proposed by @vecchia and @steinchiwelty, the rank of $ B_j$ will by and large be large than that of $ A_j$, in which case the main computation in receive (\[composite\ ]) is findi... | sum_{u=1}^q \log f_{j,\theta}(A_j
Z\mid B_j E),$$ where $f_{j,\theta}$ is the cmnditiknal Gaursian density of $A_j Z$ given $U_j Z$. As peoposed by @vecchia and @steinchiaelty, thw raik of $B_j$ will geisrally nz larfcr thcn that of $A_j$, in which casa the main com[ugacion in obtaining (\[composite\]) is findigg Cholr... | sum_{j=1}^q \log f_{j,\theta}(A_j Z\mid B_j Z),$$ where the Gaussian density $A_j Z$ given @vecchia @steinchiwelty, the rank $B_j$ will generally larger than that of $A_j$, in case the main computation in obtaining (\[composite\]) is finding Cholesky decompositions of the matrices of $B_1Z,\ldots, B_q Z$. For example, ... | sum_{j=1}^q \log f_{j,\theta}(A_j
Z\mid B_j Z),$$ wHere $f_{j,\thetA}$ is thE coNdiTiOnal gausSian density of $A_J z$ givEn $B_j Z$. As proposed by @vecchIa and @StEInchIWeLty, thE rank of $b_J$ wILL geNeRaLly Be LArGer thAn tHat of $A_j$, In which casE thE mAin computatiON iN obtaining (\[ComPosite\]) is findIng chole... | sum_{j=1}^q \log f_{j,\the ta}(A_j
Z\ mid B _jZ), $$ whe re $ f_{j,\theta}$i s th e conditional Gaussian dens it y of$ A_ j Z$given $ B _j Z $.As p rop os e dby @v ecc hia and @steinchi wel ty , the rank o f $ B_j$ willgen erally be la rge r than t hat of $A _j$ , inwhichc ase th e main co mp u tation in obta i ... | sum_{j=1}^q \log_f_{j,\theta}(A_j
Z\mid B_j_Z),$$ where $f_{j,\theta}$ is_the conditional_Gaussian_density of_$A_j_Z$ given $B_j_Z$. As proposed_by @vecchia and @steinchiwelty,_the rank of_$B_j$_will generally be larger than that of $A_j$, in which case the main computation_in_obtaining (\[composite\])_is_finding_Chole... |
, Rev. Sci. Instrum. [**66**]{}, 1394 (1995).
J.A. Carlisle, L.J. Terminello, E.A. Hudson, R.C.C. Perera, J.H. Underwood, T.A. Callcott, J.J. Jia, D.L. Ederer, F.J. Himpsel, and M.G. Samant, Appl. Phys. Lett. [**67**]{}, 34 (1995).
R.C.C. Perera, C.H. Zhang, T.A. Callcott and D.L. Ederer, J. Appl. Phys. [**66**]{}, 3... | , Rev. Sci. Instrum. [ * * 66 * * ] { }, 1394 (1995).
J.A. Carlisle, L.J. Terminello, E.A. Hudson, R.C.C. Perera, J.H. Underwood, T.A. Callcott, J.J. Jia, D.L. Ederer, F.J. Himpsel, and M.G. Samant, Appl. Phys. Lett. [ * * 67 * * ] { }, 34 (1995).
R.C.C. Perera, C.H. Zhang, T.A. Callcott and D.L. Ederer, J. Appl.... | , Reg. Sci. Instrum. [**66**]{}, 1394 (1995).
J.A. Carlirle, L.J. Terminello, E.A. Hndson, R.D.C. Pererx, J.H. Underwood, T.A. Callcott, J.O. Jiq, D.L. Tberer, F.J. Himpsel, and M.G. Samann, Appl. Phts. Lttt. [**67**]{}, 34 (1995).
R.C.C. Perera, R.G. Zhang, T.A. Campcotc end D.L. Ederer, J. Appl. Phys. [**66**]{}, 3... | , Rev. Sci. Instrum. [**66**]{}, 1394 (1995). L.J. E.A. Hudson, Perera, J.H. Underwood, Ederer, Himpsel, and M.G. Appl. Phys. Lett. 34 (1995). R.C.C. Perera, C.H. Zhang, Callcott and D.L. Ederer, J. Appl. Phys. [**66**]{}, 3676 (1989); P.O. Nilsson, J. J.V. Thordson, T.G. Andersson, J. Nordgren, J. Guo and M. Magnuson,... | , Rev. Sci. Instrum. [**66**]{}, 1394 (1995).
J.A. Carlisle, L.J. terminello, e.A. HudSon, r.C.C. peRera, j.H. UnDerwood, T.A. CallcOTt, J.J. jia, D.L. Ederer, F.J. Himpsel, anD M.G. SaMaNT, AppL. phYs. LetT. [**67**]{}, 34 (1995).
R.C.C. PerERa, c.h. zhaNg, t.A. calLcOTt And D.L. edeRer, J. AppL. Phys. [**66**]{}, 3... | , Rev. Sci. Instrum. [**66 **]{}, 139 4 (19 95) .
J. A. C arli sle, L.J. Term i nell o, E.A. Hudson, R.C.C. Pere ra , J.H . U nderw ood, T. A .C a llc ot t, J. J. Ji a, D. L.Ederer, F.J. Himp sel ,and M.G. Sam a nt , Appl. Ph ys. Lett. [**67 **] {}, 34 ( 199 5 ).
R .C. C. Pe rera,C .H. Zh ang, T.A. C a llcott ... | , Rev._Sci. Instrum._[**66**]{}, 1394 (1995).
J.A. Carlisle,_L.J. Terminello,_E.A._Hudson, R.C.C._Perera,_J.H. Underwood, T.A._Callcott, J.J. Jia,_D.L. Ederer, F.J. Himpsel,_and M.G. Samant,_Appl._Phys. Lett. [**67**]{}, 34 (1995).
R.C.C. Perera, C.H. Zhang, T.A. Callcott and D.L. Ederer, J._Appl._Phys. [**66**]{},_3... |
c}},
\widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime })$ is an element of H$^{1}(\mathcal{\mathbf{B}}^c
,\mathbf{Z}_{2}).$ By definition, we have $$\delta _{ij}(x)=\gamma _{ij}(x)\Psi (\gamma _{ij}^{\prime }(x)^{-1}) \:,$$ and $$\rho (\gamma _{ij})=\rho (\gamma _{ij}^{\prime })=g_{ij} \:.$$ This implies that $$\begin{al... | c } },
\widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime }) $ is an element of H$^{1}(\mathcal{\mathbf{B}}^c
, \mathbf{Z}_{2}).$ By definition, we have $ $ \delta _ { ij}(x)=\gamma _ { ij}(x)\Psi (\gamma _ { ij}^{\prime } (x)^{-1 }) \:,$$ and $ $ \rho (\gamma _ { ij})=\rho (\gamma _ { ij}^{\prime }) = g_{ij } \:.$$ Th... | c}},
\wifetilde{\mathcal{\mathbf{P}}^{c}}^{\prlme })$ is an elemeur of H$^{1}(\kathcam{\mathbf{B}}^z
,\mathbf{Z}_{2}).$ By definition, we heve $$\eelta _{ij}(x)=\gamma _{ij}(x)\Psi (\gamma _{ij}^{\prime }(q)^{-1}) \:,$$ and $$\rhi (\ganna _{ij})=\rho (\gejma _{ij}^{\pvnme })=g_{jm} \:.$$ Thns implies that $$\negin{al... | c}}, \widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime })$ is an element of By we have _{ij}(x)=\gamma _{ij}(x)\Psi (\gamma (\gamma (\gamma _{ij}^{\prime })=g_{ij} This implies that \rho (\delta _{ij}) &=&1 \\ &\Longrightarrow _{ij}(x)\in \mathbf{Z}_{2}\end{aligned}$$ i.e., $\delta _{ij}$ is in the center of $\Gamma $ and (... | c}},
\widetilde{\mathcal{\mathbf{P}}^{c}}^{\Prime })$ is an eLemenT of h$^{1}(\maThCal{\mAthbF{B}}^c
,\mathbf{Z}_{2}).$ By deFInitIon, we have $$\delta _{ij}(x)=\gamma _{Ij}(x)\PsI (\gAMma _{iJ}^{\PrIme }(x)^{-1}) \:,$$ aNd $$\rho (\gaMMa _{IJ})=\Rho (\GaMmA _{ij}^{\PrIMe })=G_{ij} \:.$$ ThIs iMplies tHat $$\begin{al... | c}},
\widetilde{\mathcal{\ mathbf{P}} ^{c}} ^{\ pri me })$ isan element ofH $^{1 }(\mathcal{\mathbf{B}} ^c
,\ ma t hbf{ Z }_ {2}). $ Byd ef i n iti on ,weha v e$$\de lta _{ij}( x)=\gamma_{i j} (x)\Psi (\ga m ma _{ij}^{\p rim e }(x)^{-1}) \: ,$$ an d$$\ r ho (\ gam ma _{ ij})=\ r ho (\g amma _{ij }^ { \prime })=g... | c}},
\widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime })$_is an_element of H$^{1}(\mathcal{\mathbf{B}}^c
,\mathbf{Z}_{2}).$ _By definition,_we_have $$\delta__{ij}(x)=\gamma__{ij}(x)\Psi (\gamma _{ij}^{\prime_}(x)^{-1}) \:,$$ and_$$\rho (\gamma _{ij})=\rho (\gamma__{ij}^{\prime })=g_{ij} \:.$$_This_implies that $$\begin{al... |
function transforms under translations by a lattice vector. It will be convenient to define the following $p$ set of wave functions, for $N_{\Phi}\neq 0$: $$\begin{aligned}
& &{\tilde \psi}_{l,k_{1},{\vec c}}^{j} =
e^{-\frac{\pi\omega}{L_{1}L_{2}}\sum y_{\gamma}^{2}}
\prod_{\gamma < \delta} \left(\theta_{1}\left( z_{\... | function transforms under translations by a lattice vector. It will be commodious to specify the following $ p$ set of wave function, for $ N_{\Phi}\neq 0 $: $ $ \begin{aligned }
& & { \tilde \psi}_{l, k_{1},{\vec c}}^{j } =
e^{-\frac{\pi\omega}{L_{1}L_{2}}\sum y_{\gamma}^{2 } }
\prod_{\gamma < \delta } \left(\th... | fujction transforms under uranslations by a latticx vectod. It wilu be convenient to define thx foolowibg $p$ set of wave functkons, for $J_{\Phi}\neq 0$: $$\begmn{aligned}
& &{\tilde \'ai}_{l,k_{1},{\vec c}}^{j} =
e^{-\fdwc{\pi\mnega}{L_{1}L_{2}}\sum y_{\gamka}^{2}}
\prod_{\gammd < \delta} \left(\tvega_{1}\peft( z_{\... | function transforms under translations by a lattice will convenient to the following $p$ $N_{\Phi}\neq $$\begin{aligned} & &{\tilde c}}^{j} = e^{-\frac{\pi\omega}{L_{1}L_{2}}\sum \prod_{\gamma < \delta} \left(\theta_{1}\left( z_{\gamma} - \mid \tau \right)\right)^{\frac{1}{\kappa}} \nonumber \\ & &\sum_{K=0}^{M-1}\sum_... | function transforms under trAnslations By a laTtiCe vEcTor. IT wilL be convenient tO DefiNe the following $p$ set of waVe funCtIOns, fOR $N_{\phi}\neQ 0$: $$\begin{aLIgNED}
& &{\tiLdE \pSi}_{l,K_{1},{\vEC c}}^{J} =
e^{-\fraC{\pi\Omega}{L_{1}L_{2}}\Sum y_{\gamma}^{2}}
\pRod_{\GaMma < \delta} \left(\THeTa_{1}\left( z_{\... | function transforms under translati ons b y a la tt icevect or. It will be conv enient to define the f ollow in g $p$ se t ofwave fu n ct i o ns, f or $N _{ \ Ph i}\ne q 0 $: $$\b egin{align ed}
& &{\tilde \p s i} _{l,k_{1}, {\v ec c}}^{j} =
e^ {-\fra c{ \pi \ omega }{L _{1}L _{2}}\ s um y_{ \gamma}^{ 2} }
\p... | function_transforms under_translations by a lattice_vector. It_will_be convenient_to_define the following_$p$ set of_wave functions, for $N_{\Phi}\neq_0$: $$\begin{aligned}
& &{\tilde_\psi}_{l,k_{1},{\vec_c}}^{j} =
e^{-\frac{\pi\omega}{L_{1}L_{2}}\sum y_{\gamma}^{2}}
\prod_{\gamma < \delta} \left(\theta_{1}\left( z_{\... |
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